id
int64 3
39.4k
| title
stringlengths 1
80
| text
stringlengths 2
313k
| paragraphs
listlengths 1
6.47k
| abstract
stringlengths 1
52k
â | wikitext
stringlengths 10
330k
â | date_created
stringlengths 20
20
â | date_modified
stringlengths 20
20
| templates
sequencelengths 0
20
| url
stringlengths 32
653
|
---|---|---|---|---|---|---|---|---|---|
1,889 | æ§èª²çš(-2012幎床)é«çåŠæ ¡æ°åŠB | 倪åæ
2003ã2012幎ã®æ°åŠBã¯
ã§æ§æãããŠããã
2012ã2021幎ã®æ°åŠ B ã¯ã
ã«ãã£ãŠæ§æãããŠããã
2022幎ãã®æ°åŠBã¯
ã§æ§æãããŠããã
ã»ã³ã¿ãŒè©Šéšã§ã¯æ°åŠ B ã«ãããŠãæ°åãã»ããã¯ãã«ãã»ã確çååžãšçµ±èšçãªæšæž¬ãã® 3 åéã®ãã¡ã®ãããã 2 åéãéžæãã解çããããšã«ãªã£ãŠããã
| [
{
"paragraph_id": 0,
"tag": "p",
"text": "倪åæ",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "2003ã2012幎ã®æ°åŠBã¯",
"title": ""
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ã§æ§æãããŠããã",
"title": ""
},
{
"paragraph_id": 3,
"tag": "p",
"text": "2012ã2021幎ã®æ°åŠ B ã¯ã",
"title": ""
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ã«ãã£ãŠæ§æãããŠããã",
"title": ""
},
{
"paragraph_id": 5,
"tag": "p",
"text": "2022幎ãã®æ°åŠBã¯",
"title": ""
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ã§æ§æãããŠããã",
"title": ""
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ã»ã³ã¿ãŒè©Šéšã§ã¯æ°åŠ B ã«ãããŠãæ°åãã»ããã¯ãã«ãã»ã確çååžãšçµ±èšçãªæšæž¬ãã® 3 åéã®ãã¡ã®ãããã 2 åéãéžæãã解çããããšã«ãªã£ãŠããã",
"title": "åœæã®äœçœ®ä»ã"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "",
"title": "åœæã®äœçœ®ä»ã"
}
] | 倪åæ 2003ã2012幎ã®æ°åŠB㯠æ°å
ãã¯ãã«
çµ±èšãšã³ã³ãã¥ãŒã¿
床æ°ååžã»æ£åžå³
代衚å€ã»åæ£ã»æšæºåå·®ã»çžé¢ä¿æ°
æ°å€è§£æãšã³ã³ãã¥ãŒã¿
ããã°ã©ã ã»ã¢ã«ãŽãªãºã
æŽæ°èšç®ã»è¿äŒŒèšç®ã»æ°å€ç©å ã§æ§æãããŠããã 2012ã2021幎ã®æ°åŠ B ã¯ã æ°å
ãã¯ãã«
確çååžãšçµ±èšçãªæšæž¬ ã«ãã£ãŠæ§æãããŠããã 2022幎ãã®æ°åŠB㯠æ°å
çµ±èšçãªæšæž¬
æ°åŠãšç€ŸäŒç掻 ã§æ§æãããŠããã | {{pathnav|frame=1|é«çåŠæ ¡ã®åŠç¿|é«çåŠæ ¡æ°åŠ}}
2003ã2012幎ã®æ°åŠBã¯
*æ°å
*ãã¯ãã«
*çµ±èšãšã³ã³ãã¥ãŒã¿
**床æ°ååžã»æ£åžå³
**代衚å€ã»åæ£ã»æšæºåå·®ã»çžé¢ä¿æ°
*æ°å€è§£æãšã³ã³ãã¥ãŒã¿
**ããã°ã©ã ã»ã¢ã«ãŽãªãºã
**æŽæ°èšç®ã»è¿äŒŒèšç®ã»æ°å€ç©å
ã§æ§æãããŠããã
2012ã2021幎ã®æ°åŠ B ã¯ã
* [[é«çåŠæ ¡æ°åŠB/æ°å|æ°å]]
* [[é«çåŠæ ¡æ°åŠB/ãã¯ãã«|ãã¯ãã«]]
* [[é«çåŠæ ¡æ°åŠB/確çååžãšçµ±èšçãªæšæž¬|確çååžãšçµ±èšçãªæšæž¬]]
ã«ãã£ãŠæ§æãããŠããã
2022幎ãã®æ°åŠBã¯
*æ°å
*çµ±èšçãªæšæž¬
*[[é«çåŠæ ¡æ°åŠB/æ°åŠãšç€ŸäŒç掻|æ°åŠãšç€ŸäŒç掻]]
ã§æ§æãããŠããã
== åœæã®äœçœ®ä»ã ==
=== æ°åŠ B ãåŠã¶æ矩 ===
*æ°åããã¯ãã«ãçµ±èšã«ã€ããŠç解ãããåºç€çãªç¥èã®ç¿åŸãšæèœã®ç¿çãå³ãã
*äºè±¡ãæ°åŠçã«èå¯ããåŠçããèœåã䌞ã°ããšãšãã«ããããã掻çšããèœåãè²ãŠãã
=== ã»ã³ã¿ãŒè©Šéšã»äºæ¬¡è©Šéšã«ãããŠã®æ°åŠB ===
ã»ã³ã¿ãŒè©Šéšã§ã¯æ°åŠ B ã«ãããŠãæ°åãã»ããã¯ãã«ãã»ã確çååžãšçµ±èšçãªæšæž¬ãã® 3 åéã®ãã¡ã®ãããã 2 åéãéžæãã解çããããšã«ãªã£ãŠããã
{{DEFAULTSORT:æ§1 ãããšããã€ããããããB}}
[[Category:æ°åŠæè²]]
[[Category:é«çåŠæ ¡æ°åŠB|*]] | 2005-05-03T05:50:17Z | 2024-03-19T15:08:30Z | [
"ãã³ãã¬ãŒã:Pathnav"
] | https://ja.wikibooks.org/wiki/%E6%97%A7%E8%AA%B2%E7%A8%8B(-2012%E5%B9%B4%E5%BA%A6)%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E6%95%B0%E5%AD%A6B |
1,892 | é«çåŠæ ¡æ°åŠC/ãã¯ãã« | çç§ã«ãããŠãåã¯å€§ãããšåããæã€éã§ãããšç¿ã£ãã ããã倧ãããšåããæã€éã¯ãåã®ä»ã«ããé床ã颚ã®å¹ãæ¹ãªã©ãããã
äŸãã°ãããå°ç¹ããæå»ã«ããã颚ã®å¹ãæ¹ã¯ã颚éãšé¢šåããæãç«ã€ããã®ããã«ã倧ãããšåããæã€éãå°å
¥ãããšãããããå¹çããæ±ããã
ãã®ããŒãžã§ã¯ã倧ãããšåããæã€éã§ãããã¯ãã«ãæ±ãã
ãŸããå³åœ¢ã®åé¡ã«å¯ŸããŠä»£æ°çãªã¢ãããŒããåããã®ããã¯ãã«ã®å©ç¹ã®äžã€ã§ããã
å¹³é¢äžã®ç¹ S {\displaystyle \mathrm {S} } ããç¹ T {\displaystyle \mathrm {T} } ãžåããç¢å°ãèããããã®ãããªç¢å°ã®ããã«åããæã€ç·åãæåç·åãšããã
ãã®ãšããç¹ S {\displaystyle \mathrm {S} } ãå§ç¹ãç¹ T {\displaystyle \mathrm {T} } ãçµç¹ãšããã
æå¹ç·åã§ã倧ãããšæ¹åãåããã®ã¯ãã¯ãã«ãšããŠåããã®ãšããã
æåç·åã¯äœçœ®ãé·ã(倧ãã)ãåããšããæ
å ±ãæã€ããã¯ãã«ã¯ãæåç·åã®æã€æ
å ±ã®ãã¡ãäœçœ®ã®æ
å ±ãå¿ããŠã倧ãããåãã ãã«çç®ãããã®ãšèããããšãã§ããã
æåç·å S T {\displaystyle \mathrm {ST} } ã§è¡šããããã¯ãã«ã S T â {\displaystyle \mathrm {\vec {ST}} } ãšããããã¯ãã«ã¯äžæå㧠a â {\displaystyle {\vec {a}}} ãªã©ãšè¡šãããããšãããããã¯ãã« a â {\displaystyle {\vec {a}}} ã®å€§ããã | a â | {\displaystyle |{\vec {a}}|} ã§è¡šãã
æåç·å S T {\displaystyle \mathrm {ST} } ãæåç·å S â² T â² {\displaystyle \mathrm {S'T'} } ã«å¯Ÿãã倧ãããçãããåããçãããªããäœçœ®ãéã£ãŠããŠãããã¯ãã«ãšããŠçããã S T â = S â² T â² â {\displaystyle \mathrm {\vec {ST}} =\mathrm {\vec {S'T'}} } ã§ããã
倧ããã 1 ã§ãããã¯ãã«ãåäœãã¯ãã«ãšããã
ãã¯ãã« a â {\displaystyle {\vec {a}}} ã«å¯Ÿãããã¯ãã« a â {\displaystyle {\vec {a}}} ãšæ¹åãéã§ã倧ãããçãããã¯ãã«ãéãã¯ãã«ãšããã â a â {\displaystyle -{\vec {a}}} ãšããã
å§ç¹ãšçµç¹ãçãããã¯ãã«ãé¶ãã¯ãã«ãšããã 0 â {\displaystyle {\vec {0}}} ã§è¡šããä»»æã®ç¹ A {\displaystyle \mathrm {A} } ã«å¯Ÿãã A A â = 0 â {\displaystyle \mathrm {\vec {AA}} ={\vec {0}}} ã§ããããŒããã¯ãã«ã®å€§ãã㯠0 ã§ãåãã¯èããªããã®ãšããã
ãã¯ãã« a â , b â {\displaystyle {\vec {a}},{\vec {b}}} ã«å¯Ÿãã a â = A B â , b â = B C â {\displaystyle {\vec {a}}=\mathrm {\vec {AB}} ,{\vec {b}}=\mathrm {\vec {BC}} } ãšãªãç¹ããšãããã®ãšããã¯ãã«ã®å æ³ã a â + b â = A C â {\displaystyle {\vec {a}}+{\vec {b}}=\mathrm {\vec {AC}} } ã§å®ããã
ãã¯ãã«ã®å æ³ã«ã€ããŠä»¥äžãæãç«ã€ã
ãŸãã a â + 0 â = a â {\displaystyle {\vec {a}}+{\vec {0}}={\vec {a}}} ãšããã
ãã¯ãã« a â , b â {\displaystyle {\vec {a}},{\vec {b}}} ã«å¯Ÿãã a â â b â = a â + ( â b â ) {\displaystyle {\vec {a}}-{\vec {b}}={\vec {a}}+(-{\vec {b}})} ãšããã
ãŒããã¯ãã«ã¯ãªããã¯ãã« a â {\displaystyle {\vec {a}}} ãšå®æ° k {\displaystyle k} ã«å¯Ÿãããã¯ãã«ã®å®æ°å k a â {\displaystyle k{\vec {a}}} ã以äžã®ããã«å®ããã
ãŸããŒããã¯ãã« 0 â {\displaystyle {\vec {0}}} ã«å¯Ÿããå®æ°åã k 0 â = 0 â {\displaystyle k{\vec {0}}={\vec {0}}} ã§å®ããã
以äžã®æ§è³ªããªããã€ã
ãŒããã¯ãã«ã§ã¯ãªããã¯ãã« a â , b â ( â 0 â ) {\displaystyle {\vec {a}},{\vec {b}}\,(\neq {\vec {0}})} ã«å¯Ÿãã a â = A A â² â , b â = B B â² â {\displaystyle {\vec {a}}={\vec {\mathrm {AA'} }},{\vec {b}}={\vec {\mathrm {BB'} }}} ãšãªãç¹ããšãã
ãã®ãšããçŽç· A A â² {\displaystyle \mathrm {AA'} } ãšçŽç· B B â² {\displaystyle \mathrm {BB'} } ãå¹³è¡ã§ãããšãããã¯ãã« a â , b â {\displaystyle {\vec {a}},{\vec {b}}} ã¯å¹³è¡ã§ãããšããã a â ⥠b â {\displaystyle {\vec {a}}\parallel {\vec {b}}} ã§è¡šãã
ãŸããçŽç· A A â² {\displaystyle \mathrm {AA'} } ãšçŽç· B B â² {\displaystyle \mathrm {BB'} } ãåçŽã§ãããšãããã¯ãã« a â , b â {\displaystyle {\vec {a}},{\vec {b}}} ã¯åçŽã§ãããšããã a â ⥠b â {\displaystyle {\vec {a}}\perp {\vec {b}}} ã§è¡šãã
ãã¯ãã« a â , b â {\displaystyle {\vec {a}},{\vec {b}}} ãå¹³è¡ã®ãšããæããã«ãçæ¹ã®ãã¯ãã«ãå®æ°åããã°å€§ãããšåããäžèŽããã®ã§ã
a â ⥠b â ⺠b â = k a â {\displaystyle {\vec {a}}\parallel {\vec {b}}\iff {\vec {b}}=k{\vec {a}}} ãšãªãå®æ° k {\displaystyle k} ãååšãã
ãæãç«ã€ã
ãã¯ãã« a â , b â {\displaystyle {\vec {a}},{\vec {b}}} ããšãã«ãŒããã¯ãã«ã§ãªã( a â , b â â 0 â {\displaystyle {\vec {a}},{\vec {b}}\neq {\vec {0}}} ) ãå¹³è¡ã§ãªããšããä»»æã®ãã¯ãã« p â {\displaystyle {\vec {p}}} ã«å¯ŸããŠã p â = s a â + t b â {\displaystyle {\vec {p}}=s{\vec {a}}+t{\vec {b}}} ãšãªãå®æ° s , t {\displaystyle s,t} ãåãããšãã§ããã
蚌æ
a â = O A â , b â = O B â , p â = O P â {\displaystyle {\vec {a}}={\vec {\mathrm {OA} }},{\vec {b}}={\vec {\mathrm {OB} }},{\vec {p}}={\vec {\mathrm {OP} }}} ãšãªãç¹ããšããç¹ P {\displaystyle \mathrm {P} } ãéããçŽç· O B , O A {\displaystyle \mathrm {OB} ,\mathrm {OA} } ã«å¹³è¡ãªçŽç·ãããããã çŽç· O A , O B {\displaystyle \mathrm {OA} ,\mathrm {OB} } ãšäº€ããç¹ããããã S , T {\displaystyle \mathrm {S,T} } ãšçœ®ãã
ãã®ãšãã O S â = s a â , O T â = t b â {\displaystyle {\vec {\mathrm {OS} }}=s{\vec {a}},{\vec {\mathrm {OT} }}=t{\vec {b}}} ãšãªãå®æ° s , t {\displaystyle s,t} ãåãããšãã§ãããããã§ãåè§åœ¢ O S P T {\displaystyle \mathrm {OSPT} } ã¯å¹³è¡å蟺圢ãªã®ã§ã p â = s a â + t b â {\displaystyle {\vec {p}}=s{\vec {a}}+t{\vec {b}}} ãæãç«ã€ã
ãã¯ãã« a â {\displaystyle {\vec {a}}} ã«å¯ŸããŠã座æšå¹³é¢äžã®åç¹ã O {\displaystyle \mathrm {O} } ãšãããšãã a â = O A â {\displaystyle {\vec {a}}=\mathrm {\vec {OA}} } ãšãªãç¹ A ( a x , a y ) {\displaystyle \mathrm {A} (a_{x},a_{y})} ãåãããšãã§ãããããã§ã ( a x , a y ) {\displaystyle (a_{x},a_{y})} ããã¯ãã« a â {\displaystyle {\vec {a}}} ã®æå衚瀺ãšãã a â = ( a x , a y ) {\displaystyle {\vec {a}}=(a_{x},a_{y})} ããŸãã¯ã瞊ã«äžŠã¹ãŠã a â = ( a x a y ) {\displaystyle {\vec {a}}=\left({\begin{aligned}a_{x}\\a_{y}\end{aligned}}\right)} ãšæžãã
ãã¯ãã« a â , b â {\displaystyle {\vec {a}},{\vec {b}}} ã«å¯ŸããŠã a â = O A â , b â = O B â {\displaystyle {\vec {a}}=\mathrm {\vec {OA}} ,\,{\vec {b}}=\mathrm {\vec {OB}} } ãšãªãç¹ A , B {\displaystyle \mathrm {A} ,\mathrm {B} } ããšãã a â = ( a x , a y ) , b â = ( b x , b y ) {\displaystyle {\vec {a}}=(a_{x},a_{y}),\,{\vec {b}}=(b_{x},b_{y})} ãšãããšã
a â = b â ⺠O A â = O B â ⺠{\displaystyle {\vec {a}}={\vec {b}}\iff {\vec {\mathrm {OA} }}={\vec {\mathrm {OB} }}\iff } ç¹ A , B {\displaystyle \mathrm {A} ,\,\mathrm {B} } ãäžèŽãã ⺠a x = b x {\displaystyle \iff a_{x}=b_{x}} ã〠a y = b y {\displaystyle a_{y}=b_{y}}
ãŸãã a â = ( a x , a y ) {\displaystyle {\vec {a}}=(a_{x},a_{y})} ã«å¯ŸããŠã a â = O A â {\displaystyle {\vec {a}}=\mathrm {\vec {OA}} } ãšãããšãã | a â | {\displaystyle |{\vec {a}}|} ã¯ç·å O A {\displaystyle \mathrm {OA} } ã®é·ããªã®ã§ã
| a â | = a x 2 + a y 2 {\displaystyle |{\vec {a}}|={\sqrt {a_{x}^{2}+a_{y}^{2}}}}
ã§ããã
ãã¯ãã« a â = ( a x , a y ) , b â = ( b x , b y ) {\displaystyle {\vec {a}}=(a_{x},a_{y}),{\vec {b}}=(b_{x},b_{y})} ã«å¯ŸããŠã
a â + b â = ( a x + b x , a y + b y ) {\displaystyle {\vec {a}}+{\vec {b}}=(a_{x}+b_{x},a_{y}+b_{y})}
a â â b â = ( a x â b x , a y â b y ) {\displaystyle {\vec {a}}-{\vec {b}}=(a_{x}-b_{x},a_{y}-b_{y})}
k a â = ( k a x , k a y ) {\displaystyle k{\vec {a}}=(ka_{x},ka_{y})}
ããªããã€ã
ããç¹ãåºæºã«ããŠããã®ç¹ãå§ç¹ãšãããã¯ãã«ã«ã€ããŠèããããšã«ããããã¯ãã«ãçšããŠç¹ã®äœçœ®é¢ä¿ã«ã€ããŠèå¯ããããšãã§ããã
ç¹ã®äœçœ®é¢ä¿åºæºãšãªãç¹ O {\displaystyle {\rm {O}}} ããããããå®ããããã®ãšããç¹ A {\displaystyle {\rm {A}}} ã«å¯ŸããŠããã¯ãã« O A â {\displaystyle {\vec {\rm {OA}}}} ãç¹ A {\displaystyle {\rm {A}}} ã®äœçœ®ãã¯ãã«ãšãããäœçœ®ãã¯ãã« a â {\displaystyle {\vec {a}}} ã§äžããããç¹ A {\displaystyle {\rm {A}}} ã A ( a â ) {\displaystyle \mathrm {A} ({\vec {a}})} ã§è¡šãã
ãŸããç¹ A ( a â ) , B ( b â ) {\displaystyle \mathrm {A} ({\vec {a}}),\,\mathrm {B} ({\vec {b}})} ã®ãšãã A B â = O B â â O A â = b â â a â {\displaystyle {\vec {\rm {AB}}}={\vec {\rm {OB}}}-{\vec {\rm {OA}}}={\vec {b}}-{\vec {a}}} ãæãç«ã€ã
以äžãäœçœ®ãã¯ãã«ã®åºæºç¹ãç¹ O {\displaystyle {\rm {O}}} ãšããã
ç¹ A ( a â ) , B ( b â ) {\displaystyle {\rm {A({\vec {a}}),\,{\rm {B({\vec {b}})}}}}} ãéãç·å A B {\displaystyle \mathrm {AB} } ã m : n {\displaystyle m:n} ã«å
åããç¹ P ( p â ) {\displaystyle \mathrm {P} ({\vec {p}})} ãæ±ããã
A P â = m m + n A B â {\displaystyle {\vec {\mathrm {AP} }}={\frac {m}{m+n}}{\vec {\mathrm {AB} }}} ããã p â â a â = m m + n ( b â â a â ) {\displaystyle {\vec {p}}-{\vec {a}}={\frac {m}{m+n}}({\vec {b}}-{\vec {a}})} ãããã£ãŠã p â = n a â + m b â m + n {\displaystyle {\vec {p}}={\frac {n{\vec {a}}+m{\vec {b}}}{m+n}}} ã§ããã
次ã«ãç¹ A ( a â ) , B ( b â ) {\displaystyle {\rm {A({\vec {a}}),\,{\rm {B({\vec {b}})}}}}} ãéãç·å A B {\displaystyle \mathrm {AB} } ã m : n {\displaystyle m:n} ã«å€åããç¹ Q ( q â ) {\displaystyle \mathrm {Q} ({\vec {q}})} ãæ±ããã
m > n {\displaystyle m>n} ã®å Žåã¯ã A Q â = m m â n A B â {\displaystyle {\vec {\mathrm {AQ} }}={\frac {m}{m-n}}{\vec {\mathrm {AB} }}} ããã q â â a â = m m â n ( b â â a â ) {\displaystyle {\vec {q}}-{\vec {a}}={\frac {m}{m-n}}({\vec {b}}-{\vec {a}})} ãããã£ãŠã q â = â n a â + m b â m â n {\displaystyle {\vec {q}}={\frac {-n{\vec {a}}+m{\vec {b}}}{m-n}}} ã§ããã
m < n {\displaystyle m<n} ã®å Žåã¯ã B Q â = n n â m B A â {\displaystyle {\vec {\mathrm {BQ} }}={\frac {n}{n-m}}{\vec {\mathrm {BA} }}} ã«æ³šæããŠåæ§ã«èšç®ããã°ãåãšåãã q â = â n a â + m b â m â n {\displaystyle {\vec {q}}={\frac {-n{\vec {a}}+m{\vec {b}}}{m-n}}} ãåŸãããã
äžè§åœ¢ A B C {\displaystyle \mathrm {ABC} } ã«å¯Ÿãã A ( a â ) , B ( b â ) , C ( c â ) {\displaystyle \mathrm {A} ({\vec {a}}),\,\mathrm {B} ({\vec {b}}),\,\mathrm {C} ({\vec {c}})} ãšçœ®ãããã®äžè§åœ¢ A B C {\displaystyle \mathrm {ABC} } ã®éå¿ G ( g â ) {\displaystyle \mathrm {G} ({\vec {g}})} ãæ±ããã
ç·å B C {\displaystyle \mathrm {BC} } ã®äžç¹ã M ( m â ) {\displaystyle \mathrm {M} ({\vec {m}})} ãšãããšãç¹ M {\displaystyle \mathrm {M} } ã¯ç·å B C {\displaystyle \mathrm {BC} } ã 1 : 1 {\displaystyle 1:1} ã«å
åããç¹ãªã®ã§ã m â = b â + c â 2 {\displaystyle {\vec {m}}={\frac {{\vec {b}}+{\vec {c}}}{2}}} ã§ããã
ç¹ G {\displaystyle \mathrm {G} } ã¯ç·å A M {\displaystyle \mathrm {AM} } ã 2 : 1 {\displaystyle 2:1} ã«å
åããç¹ãªã®ã§ã g â = a â + b â + c â 3 {\displaystyle {\vec {g}}={\frac {{\vec {a}}+{\vec {b}}+{\vec {c}}}{3}}} ã§ããã
äžè§åœ¢ A B C {\displaystyle \mathrm {ABC} } ã«å¯Ÿãã A ( a â ) , B ( b â ) , C ( c â ) {\displaystyle \mathrm {A} ({\vec {a}}),\,\mathrm {B} ({\vec {b}}),\,\mathrm {C} ({\vec {c}})} ãšçœ®ããããã«ã A B = c , B C = a , C A = b {\displaystyle \mathrm {AB} =c,\,\mathrm {BC} =a,\,\mathrm {CA} =b} ãšçœ®ããäžè§åœ¢ A B C {\displaystyle \mathrm {ABC} } ã®å
å¿ã®äœçœ®ãã¯ãã« I ( i â ) {\displaystyle \mathrm {I} ({\vec {i}})} ãæ±ããã
A {\displaystyle {\rm {A}}} ã®äºçåç·ãšç·å B C {\displaystyle {\rm {BC}}} ã®äº€ç¹ã D ( d â ) {\displaystyle \mathrm {D} ({\vec {d}})} ãšããããã®ãšããäžè§åœ¢ã®äºçåç·ã®æ§è³ªãã B D : D C = c : b {\displaystyle \mathrm {BD} :\mathrm {DC} =c:b} ãããã£ãŠã d â = b b â + c c â b + c {\displaystyle {\vec {d}}={\frac {b{\vec {b}}+c{\vec {c}}}{b+c}}} ã§ããã
ããã§ã A I : I D = B A : B D = c : a c b + c = ( b + c ) : a {\displaystyle \mathrm {AI} :\mathrm {ID} =\mathrm {BA} :\mathrm {BD} =c:{\frac {ac}{b+c}}=(b+c):a} ã§ããã
ãããã£ãŠã i â = a a â + ( b + c ) d â a + b + c = a a â + b b â + c c â a + b + c {\displaystyle {\vec {i}}={\frac {a{\vec {a}}+(b+c){\vec {d}}}{a+b+c}}={\frac {a{\vec {a}}+b{\vec {b}}+c{\vec {c}}}{a+b+c}}} ã§ããã
äžåŠãŸãã¯é«æ ¡ã®çç§ã®ååŠã§ã¯ãååŠçãªä»äºã®å®çŸ©ããªãã£ãããšãããã ããããã®ä»äºã§ã¯ã移åæ¹å以å€ã®åã¯ãä»äºã«å¯äžããªãã£ãããã®ãããªåã®ä»äºã®èšç®ãããã¯ãã«ã®èŠ³ç¹ããã¿ãã°ãå
ç©ãšããæ°ããæŠå¿µãå®çŸ©ã§ããã
ãã¯ãã« a â , b â {\displaystyle {\vec {a}},{\vec {b}}} ã«å¯Ÿãã a â = O A â , b â = O B â {\displaystyle {\vec {a}}={\vec {\mathrm {OA} }},{\vec {b}}={\vec {\mathrm {OB} }}} ãšãªãç¹ O , A , B {\displaystyle \mathrm {O,A,B} } ããšãããã®ãšãã â A O B {\displaystyle \angle \mathrm {AOB} } ããã¯ãã« a â , b â {\displaystyle {\vec {a}},{\vec {b}}} ã®ãªãè§ãšããã
(å³)
ãã¯ãã« a â , b â {\displaystyle {\vec {a}},{\vec {b}}} ã®ãªãè§ã Ξ {\displaystyle \theta } ãšãããšããå
ç© a â â
b â {\displaystyle {\vec {a}}\cdot {\vec {b}}} ã
ã§å®ããã
å®çŸ©ããããã¯ãã«ã®å
ç©ã¯äžæ¹ã®ãã¯ãã«ãããäžæ¹ã®ãã¯ãã«ã«å°åœ±ãããšãã®ã倧ããã®ç©ã§ãããšèšããã
(å³)
ãã¯ãã« a â , b â {\displaystyle {\vec {a}},{\vec {b}}} ã a â = ( a 1 , a 2 ) , b â = ( b 1 , b 2 ) {\displaystyle {\vec {a}}=(a_{1},a_{2}),{\vec {b}}=(b_{1},b_{2})} ãšæå衚瀺ãããšãã®ãå
ç© a â â
b â {\displaystyle {\vec {a}}\cdot {\vec {b}}} ã«ã€ããŠèããŠã¿ããã
ãã¯ãã« a â , b â {\displaystyle {\vec {a}},{\vec {b}}} ã«å¯Ÿãã a â = O A â , b â = O B â {\displaystyle {\vec {a}}={\vec {\mathrm {OA} }},{\vec {b}}={\vec {\mathrm {OB} }}} ãšãªãç¹ O , A , B {\displaystyle \mathrm {O,A,B} } ããšãããã¯ãã« a â , b â {\displaystyle {\vec {a}},{\vec {b}}} ã®ãªãè§ã Ξ {\displaystyle \theta } ãšããããã®ãšã â³ O A B {\displaystyle \triangle \mathrm {OAB} } ã«å¯ŸãäœåŒŠå®çãçšããŠ
A B 2 = O A 2 + O B 2 â 2 â
O A â
O B cos Ξ {\displaystyle \mathrm {\mathrm {AB} } ^{2}=\mathrm {\mathrm {OA} } ^{2}+\mathrm {\mathrm {OB} } ^{2}-2\cdot \mathrm {\mathrm {OA} } \cdot \mathrm {\mathrm {OB} } \cos \theta }
(å³)
ããã§ã A B = | b â â a â | , O A = | a â | , O B = | b â | {\displaystyle \mathrm {\mathrm {AB} } =|{\vec {b}}-{\vec {a}}|,\mathrm {\mathrm {OA} } =|{\vec {a}}|,\mathrm {\mathrm {OB} } =|{\vec {b}}|} ãšã O A â
O B cos Ξ = | a â | | b â | cos Ξ = a â â
b â {\displaystyle \mathrm {\mathrm {OA} } \cdot \mathrm {\mathrm {OB} } \cos \theta =|{\vec {a}}||{\vec {b}}|\cos \theta ={\vec {a}}\cdot {\vec {b}}} ãã
| b â â a â | 2 = | a â | 2 + | b â | 2 â 2 a â â
b â {\displaystyle |{\vec {b}}-{\vec {a}}|^{2}=|{\vec {a}}|^{2}+|{\vec {b}}|^{2}-2{\vec {a}}\cdot {\vec {b}}} ã§ããã®ã§ã a â â
b â = 1 2 ( | a â | 2 + | b â | 2 â | b â â a â | 2 ) {\displaystyle {\vec {a}}\cdot {\vec {b}}={\frac {1}{2}}(|{\vec {a}}|^{2}+|{\vec {b}}|^{2}-|{\vec {b}}-{\vec {a}}|^{2})} ã§ããã
ããã§ã | a â | 2 = a 1 2 + a 2 2 , | b â | 2 = b 1 2 + b 2 2 , | b â â a â | 2 = | ( b 1 â a 1 , b 2 â a 2 ) | 2 = ( b 1 â a 1 ) 2 + ( b 2 â a 2 ) 2 {\displaystyle |{\vec {a}}|^{2}=a_{1}^{2}+a_{2}^{2},|{\vec {b}}|^{2}=b_{1}^{2}+b_{2}^{2},|{\vec {b}}-{\vec {a}}|^{2}=|(b_{1}-a_{1},b_{2}-a_{2})|^{2}=(b_{1}-a_{1})^{2}+(b_{2}-a_{2})^{2}} ãªã®ã§ãããã代å
¥ããã°
a â â
b â = 1 2 ( | a â | 2 + | b â | 2 â | b â â a â | 2 ) {\displaystyle {\vec {a}}\cdot {\vec {b}}={\frac {1}{2}}(|{\vec {a}}|^{2}+|{\vec {b}}|^{2}-|{\vec {b}}-{\vec {a}}|^{2})} = 1 2 [ ( a 1 2 + a 2 2 ) + ( b 1 2 + b 2 2 ) â ( b 1 â a 1 ) 2 + ( b 2 â a 2 ) 2 ] {\displaystyle ={\frac {1}{2}}\left[(a_{1}^{2}+a_{2}^{2})+(b_{1}^{2}+b_{2}^{2})-(b_{1}-a_{1})^{2}+(b_{2}-a_{2})^{2}\right]} = a 1 b 1 + a 2 b 2 {\displaystyle =a_{1}b_{1}+a_{2}b_{2}} ã§ããã
ãããã£ãŠ a â â
b â = a 1 b 1 + a 2 b 2 {\displaystyle {\vec {a}}\cdot {\vec {b}}=a_{1}b_{1}+a_{2}b_{2}} ãåŸãããã
å
ç©ã®æ§è³ª â ãã¯ãã« a â , b â , c â {\displaystyle {\vec {a}},{\vec {b}},{\vec {c}}} ãšå®æ° k {\displaystyle k} ã«å¯Ÿã以äžãæãç«ã€ã
ãããã¯ãã¯ãã«ãæå衚瀺ããŠèšç®ããã°èšŒæã§ããã
蚌æ â a â = ( a 1 , a 2 ) , b â = ( b 1 , b 2 ) , c â = ( c 1 , c 2 ) {\displaystyle {\vec {a}}=(a_{1},a_{2}),{\vec {b}}=(b_{1},b_{2}),{\vec {c}}=(c_{1},c_{2})} ãšããã
æŒç¿åé¡
A ( a â ) , B ( b â ) {\displaystyle \mathrm {A} ({\vec {a}}),\,\mathrm {B} ({\vec {b}})} ãšããã ãã®ãšããç·åOAã1:3ã«åããç¹ãšãç·åOBã5:2ã«åããç¹ããããããA',B'ãšããã
(1) ãã¯ãã« O A â² â , O B â² â {\displaystyle {\vec {OA'}},\,{\vec {OB'}}} ããã¯ãã« a â , b â {\displaystyle {\vec {a}},\,{\vec {b}}} ãçšããŠããããã
(2) ç·åAB'ãšãBA'ã®äº€ç¹ M ã®äœçœ®ãã¯ãã«ããã¯ãã« a â , b â {\displaystyle {\vec {a}},\,{\vec {b}}} ãçšããŠããããã
ãã¯ãã«
ãšã ãã¯ãã«
ã¯äºãã«1次ç¬ç«ãª2æ¬ã®ãã¯ãã«ãªã®ã§ã ããããçšããŠããããå³åœ¢äžã®ç¹ãè¡šãããã¯ãã§ããã
å³åœ¢äžã®ããããã®ç¹ã¯ãç¹Oããã®äœçœ®ãã¯ãã«ã§è¡šãããã äŸãã°ããã¯ãã«
ã¯ãç¹OããèŠãŠ
ãšå¹³è¡ãªæ¹åã®ãã¯ãã«ã§ããããã®å€§ãããã
ã§ããã®ã§ã
ã§è¡šãããã åæ§ã«ããã¯ãã«
ã¯ãç¹OããèŠãŠ
ãšå¹³è¡ãªæ¹åã®ãã¯ãã«ã§ããããã®å€§ãããã
ã§ããã®ã§ã
ã§è¡šãããã
次ã«ãç¹A'ãééããç·åA'Bã«å¹³è¡ãªçŽç·ã ãã¯ãã«
ãš
ãçšããŠèšè¿°ããæ¹æ³ãèããã
ããã§ã¯ã ãã®çŽç·äžã®ç¹ã¯ã ããå®æ° s {\displaystyle s} ãçšããŠã
ã§è¡šããããšã«æ³šç®ããã äŸãã°ã
ã®ãšãããã®åŒãè¡šãç¹ã¯
ã«çããã
ã®ãšãã
ã«çãããããããçŽç· A'Bäžã®ç¹ã§ããã
ãããã«å
ã»ã©æ±ãã
ãšã
ã®å€ãçšãããšã
ãåŸãããã
åæ§ã«ãç·åAB'äžã®ç¹ã¯ããå®æ° t {\displaystyle t} ãçšããŠã
ã§è¡šãããã ããã«å
ã»ã©åŸãå€ã代å
¥ãããšã
ãšãªãã
ãã®ããã«ããããã®çŽç·äžã®ç¹ã s {\displaystyle s} , t {\displaystyle t} ã çšããŠè¡šãããã 次ã«ããããã®åŒãåãç¹ã瀺ãããã« s {\displaystyle s} , t {\displaystyle t} ãå®ããã ãã®ããã«ã¯ã
,
ãçãããšãããŠã s {\displaystyle s} , t {\displaystyle t} ã«é¢ããé£ç«æ¹çšåŒãäœããããã解ãã°ããã äžã®åŒã§
ã®ä¿æ°ãçãããšãããšã
ãåŸããã
ã®ä¿æ°ãçãããšãããšã
ãåŸãããã ãã®åŒãé£ç«ããŠè§£ããšã
,
ãåŸãããã ãã®åŒã
,
ã®ã©ã¡ããã«ä»£å
¥ãããšãæ±ããäœçœ®ãã¯ãã«ãåŸãããã®ã§ããã 代å
¥ãããšãæ±ãããã¯ãã«ã¯ã
ãšãªãã
ç¹ A ( a â ) {\displaystyle \mathrm {A} ({\vec {a}})} ãéãããã¯ãã« d â ( â 0 â ) {\displaystyle {\vec {d}}\,(\neq {\vec {0}})} ã«å¹³è¡ãªçŽç·ã g {\displaystyle g} ãšããã g {\displaystyle g} äžã®ç¹ã P ( p â ) {\displaystyle \mathrm {P} ({\vec {p}})} ãšãããšã A P â = 0 â {\displaystyle {\vec {\mathrm {AP} }}={\vec {0}}} ãŸã㯠A P â ⥠d â {\displaystyle {\vec {\mathrm {AP} }}\parallel {\vec {d}}} ã ãã
ãšãªãå®æ° t {\displaystyle t} ãããã
ããªãã¡ã
ãã£ãŠã
ããããçŽç· g {\displaystyle g} ã®ãã¯ãã«æ¹çšåŒ(vector equation)ãšããã d â {\displaystyle {\vec {d}}} ã g {\displaystyle g} ã®æ¹åãã¯ãã«ãšããããŸãã t {\displaystyle t} ãåªä»å€æ°ãšããã
ç¹Aã®åº§æšã ( x 1 , y 1 ) {\displaystyle (x_{1}\ ,\ y_{1})} ã d â = ( a , b ) {\displaystyle {\vec {d}}=(a\ ,\ b)} ãç¹Pã®åº§æšã ( x , y ) {\displaystyle (x\ ,\ y)} ãšãããšããã¯ãã«æ¹çšåŒ p â = a â + t d â {\displaystyle {\vec {p}}={\vec {a}}+t{\vec {d}}} ã¯
ãšãªãããããã£ãŠ
{ x = x 1 + a t y = y 1 + b t {\displaystyle {\begin{cases}x=x_{1}+at\\y=y_{1}+bt\end{cases}}}
ãããçŽç· g {\displaystyle g} ã®åªä»å€æ°è¡šç€ºãšããã
æŒç¿åé¡
ç¹A ( 1 , 2 ) {\displaystyle (1\ ,\ 2)} ãéãã d â = ( 3 , 5 ) {\displaystyle {\vec {d}}=(3\ ,\ 5)} ã«å¹³è¡ãªçŽç·ã®æ¹çšåŒããåªä»å€æ°tãçšããŠè¡šãã
ãŸããtãæ¶å»ããåŒã§è¡šãã
ãã®çŽç·ã®ãã¯ãã«æ¹çšåŒã¯
ãããã£ãŠ
tãæ¶å»ãããšã次ã®ããã«ãªãã
2ç¹ A ( a â ) , B ( b â ) {\displaystyle \mathrm {A} ({\vec {a}}),\,\mathrm {B} ({\vec {b}})} ãéãçŽç·ã®ãã¯ãã«æ¹çšåŒãèããã
çŽç·ABã¯ãç¹Aãéãã A B â = b â â a â {\displaystyle {\vec {AB}}={\vec {b}}-{\vec {a}}} ãæ¹åãã¯ãã«ãšããçŽç·ãšèããããããããã®ãã¯ãã«æ¹çšåŒã¯
ãšãªããããã¯æ¬¡ã®ããã«æžããã
æŒç¿åé¡
2ç¹A ( 2 , 5 ) {\displaystyle (2\ ,\ 5)} ,B ( â 1 , 3 ) {\displaystyle (-1\ ,\ 3)} ãéãçŽç·ã®æ¹çšåŒããåªä»å€æ°tãçšããŠè¡šãã
ãã®çŽç·ã®ãã¯ãã«æ¹çšåŒã¯
ãããã£ãŠ
ç¹Aãéã£ãŠã 0 â {\displaystyle {\vec {0}}} ã§ãªããã¯ãã«ã n â {\displaystyle {\vec {n}}} ã«åçŽãªçŽç·ãgãšãããgäžã®ç¹ãPãšãããšã A P â = 0 â {\displaystyle {\vec {AP}}={\vec {0}}} ãŸã㯠A P â ⥠n â {\displaystyle {\vec {AP}}\perp {\vec {n}}} ã ãã
ã§ããã
ç¹A,Pã®äœçœ®ãã¯ãã«ãããããã a â , p â {\displaystyle {\vec {a}}\ ,\ {\vec {p}}} ãšãããšã A P â = p â â a â {\displaystyle {\vec {AP}}={\vec {p}}-{\vec {a}}} ã ããã(1)ã¯
ãšãªãã(2)ãç¹Aãéã£ãŠã n â {\displaystyle {\vec {n}}} ã«åçŽãªçŽç·gã®ãã¯ãã«æ¹çšåŒã§ããã n â {\displaystyle {\vec {n}}} ããã®çŽç·ã®æ³ç·ãã¯ãã«(ã»ããããã¯ãã«ãnormal vector)ãšããã
ç¹Aã®åº§æšã ( x 1 , y 1 ) {\displaystyle (x_{1}\ ,\ y_{1})} ã n â = ( a , b ) {\displaystyle {\vec {n}}=(a\ ,\ b)} ãç¹Pã®åº§æšã ( x , y ) {\displaystyle (x\ ,\ y)} ãšãããšã p â â a â = ( x â x 1 , y â y 1 ) {\displaystyle {\vec {p}}-{\vec {a}}=(x-x_{1}\ ,\ y-y_{1})} ã ããã(2)ã¯æ¬¡ã®ããã«ãªãã
ãã®æ¹çšåŒã¯ã â a x 1 â b y 1 = c {\displaystyle -ax_{1}-by_{1}=c} ãšãããšã a x + b y + c = 0 {\displaystyle ax+by+c=0} ãšãªãããã次ã®ããšããããã
çŽç· a x + b y + c = 0 {\displaystyle ax+by+c=0} ã®æ³ç·ãã¯ãã«ã¯ã n â = ( a , b ) {\displaystyle {\vec {n}}=(a\ ,\ b)} ã§ããã
æŒç¿åé¡
ç¹A ( 2 , 5 ) {\displaystyle (2\ ,\ 5)} ãéãã n â = ( 4 , 3 ) {\displaystyle {\vec {n}}=(4\ ,\ 3)} ã«åçŽãªçŽç·ã®æ¹çšåŒãæ±ããã
ã€ãŸã
ãããŸã§ã¯ãå¹³é¢äžã®ãã¯ãã«ã«ã€ããŠèããŠããããããããã¯3次å
空éäžã®ãã¯ãã«ã«ã€ããŠèãããããäžè¬ã«ãã¯ãã«ã¯n次å
(ãŠãŒã¯ãªãã)空éäžã§å®çŸ©ããããšãã§ãããããã®ãããªãã®ã¯é«æ ¡ã§ã¯æ±ããªãã
ä»ãŸã§ã¯ãå¹³é¢äžã®å³åœ¢ããã¯ãã«ãæ°åŒãçšããŠè¡šçŸããæ¹æ³ãåŠãã§æ¥ãã ããã§ãã2次å
ãšã¯ãå¹³é¢ã®ããšã§ãããå¹³é¢äžã®ä»»æã®ç¹ãæå®ããã«ã¯æäœã§ã2以äžã®å®æ°ãå¿
èŠã ãããã®ããã«åŒã°ããŠããã
ãã¡ãã容æã«åããéãã2ã€ä»¥äžã®æ¬¡å
ãæã£ãŠããå³åœ¢ãååšããã äŸãã°ã3次å
ç«äœã®1ã€ã§ããçŽæ¹äœã¯çžŠã暪ãé«ãã®3ã€ã®é·ããæã£ãŠããã®ã§ã3次å
å³åœ¢ãšåŒã°ããã
空éã«1ã€ã®å¹³é¢ããšãããã®äžã«çŽäº€ãã座æšè»ž O x , O y {\displaystyle O_{x}\ ,\ O_{y}} ããšãã次ã«Oãéããã®å¹³é¢ã«åçŽãªçŽç· O z {\displaystyle O_{z}} ãã²ãããã®çŽç·äžã§ãOãåç¹ãšãã座æšãèããã
ãã®3çŽç· O x , O y , O z {\displaystyle O_{x}\ ,\ O_{y}\ ,\ O_{z}} ã¯ãã©ã®2ã€ãäºãã«åçŽã§ããããããã座æšè»žãšãããããããx軞ãy軞ãz軞ãšããã
ãŸããx軞ãšy軞ãšã§å®ãŸãå¹³é¢ãxyå¹³é¢ãšãããy軞ãšz軞ãšã§å®ãŸãå¹³é¢ãyzå¹³é¢ãšãããz軞ãšx軞ãšã§å®ãŸãå¹³é¢ãzxå¹³é¢ãšããããããã座æšå¹³é¢ãšããã
空éå
ã®ç¹Aã«å¯ŸããŠãAãéã£ãŠå座æšå¹³é¢ã«å¹³è¡ãª3ã€ã®å¹³é¢ãã€ãããããããx軞ãy軞ãz軞ãšäº€ããç¹ã A 1 , A 2 , A 3 {\displaystyle A_{1}\ ,\ A_{2}\ ,\ A_{3}} ãšãã A 1 , A 2 , A 3 {\displaystyle A_{1}\ ,\ A_{2}\ ,\ A_{3}} ã®ããããã®è»žäžã§ã®åº§æšã a 1 , a 2 , a 3 {\displaystyle a_{1}\ ,\ a_{2}\ ,\ a_{3}} ãšããã
ãã®ãšãã3ã€ã®æ°ã®çµ
ãç¹Aã®åº§æšãšããã a 1 {\displaystyle a_{1}} ãx座æšãšããã a 2 {\displaystyle a_{2}} ãy座æšãšããã a 3 {\displaystyle a_{3}} ãz座æšãšããã
ãã®ããã«åº§æšã®å®ãããã空éã座æšç©ºéãšåŒã³ãç¹Oã座æšç©ºéã®åç¹ãšããã
ããã§ã¯ãç¹ã«3次å
空éã®å³åœ¢ã«æ³šç®ããã ãŸãã¯ãã¯ãã«ãçšããåã«3次å
空éã®ç©ºéå³åœ¢ããæ°åŒã«ãã£ãŠèšè¿°ããæ¹æ³ãèå¯ããã
2次å
空éã«ãããŠããã£ãšãç°¡åãªå³åœ¢ã¯çŽç·ã§ããããã®åŒã¯äžè¬çã«
ã§è¡šããããã ( a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} ã¯ä»»æã®å®æ°ã) ãã㧠x {\displaystyle x} , y {\displaystyle y} ã¯ã2次å
空éã代衚ãã2ã€ã®ãã©ã¡ãŒã¿ãŒã§ããã3次å
空éãçšãããšãã«ã¯ããããã¯3ã€ã®æåã§è¡šããããããšãæåŸ
ãããã
å®éãã®ãããªåŒã§è¡šããããå³åœ¢ã¯ã3次å
空éã§ãåºæ¬çãªå³åœ¢ã§ãããã€ãŸãã
ããäžã®åŒã®é¡äŒŒç©ãšããŠåŸãããã ( a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} , d {\displaystyle d} ã¯ä»»æã®å®æ°ã)
ãã®ãããªå³åœ¢ã¯ã©ããªå³åœ¢ã«å¯Ÿå¿ããã ããã?
å®éã«ã¯ãã®å³åœ¢ãç¹åŸŽã¥ããã®ã¯ãåŸã«åŠã¶3次å
ãã¯ãã«ãçšããã®ããã£ãšãç°¡åã§ããã®ã§ãããã¯åŸã«ãŸããããšã«ããã
ãããããã 1ã€ãã®åŒããåããããšã¯ã3次å
空éã®åº§æšãè¡šãããã©ã¡ãŒã¿ãŒ
ã®ãã¡ã«1ã€ã®é¢ä¿
ãäžããããšã§ã3次å
空éäžã®å³åœ¢ãæå®ã§ãããšããããšã§ããããã®å Žåã¯ã
ãçšããŠããã
ãã¯ãã«ã䜿ããªããŠãå³åœ¢ç解éãåŸãããåŒãšããŠã
ãæããããã ( a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} , r {\displaystyle r} ã¯ä»»æã®å®æ°ã) ãã®åŒã¯ã2次å
ã§ãããšããã®
ã®åŒã®é¡äŒŒç©ã§ããã2次å
ã®å Žåã¯ãã®åŒã¯ã
äžå¿ ( a , b ) {\displaystyle (a,b)} ååŸ r {\displaystyle r} ã®åã«å¯Ÿå¿ããŠããã 3次å
ã®ãã®åŒã¯ãçµè«ããããšäžå¿ ( a , b , c ) {\displaystyle (a,b,c)} ååŸ r {\displaystyle r} ã®åã«å¯Ÿå¿ããŠããã®ã§ããã
äžã®åŒ
ãæºããããç¹ ( x , y , z ) {\displaystyle (x,y,z)} ãåãããã®ç¹ãšç¹ ( a , b , c ) {\displaystyle (a,b,c)} ãšã®è·é¢ãèããã
空é座æšã«çœ®ãã x {\displaystyle x} 軞ã y {\displaystyle y} 軞ã z {\displaystyle z} 軞ã¯ããããçŽäº€ããŠããã®ã§ã2ç¹ã®è·é¢ã¯3å¹³æ¹ã®å®çãçšããŠ
ã§äžããããã
ããããäžã®åŒããããã§éžãã ç¹ ( x , y , z ) {\displaystyle (x,y,z)} ã¯ãæ¡ä»¶
ãæºãããŠããã®ã§ã2ç¹ã®è·é¢ã¯
ã§ããã ( r > 0 {\displaystyle r>0} ãçšããã)
ãã£ãŠãäžã®åŒãæºããç¹ã¯å
šãŠç¹ ( a , b , c ) {\displaystyle (a,b,c)} ããã®è·é¢ã r {\displaystyle r} ã§ããç¹ã§ãããããã¯äžå¿ ( a , b , c ) {\displaystyle (a,b,c)} ååŸ r {\displaystyle r} ã®åã«ä»ãªããªãã
æŒç¿åé¡
äžå¿
ååŸ
ã®çã®åŒãæ±ããã
ã«ä»£å
¥ããããšã§ã
ãæ±ããããã
æŒç¿åé¡
ãã©ã®ãã㪠çã«å¯Ÿå¿ãããèšç®ããã
ãã®ãããªæ°åŒãçã«å¯Ÿå¿ãããšãã
ã®ä¿æ°ã¯å¿
ãçãããªããŠã¯ãªããªããããã§ãªãå Žåã¯ãã®å³åœ¢ã¯æ¥åäœã«å¯Ÿå¿ããã®ã ããããã¯æå°èŠé ã®ç¯å²å€ã§ããã ããã§ã¯äžã®åŒã¯ãã®æ¡ä»¶ãæºãããŠããã
ããã§ã¯ããã®åŒã
ã®åœ¢ã«æã£ãŠè¡ãããšãéèŠã§ããã
ã®ããããã«ã€ããŠãã®åŒãå¹³æ¹å®æãããšã
ãåŸãããããã£ãŠãäžã®åŒ
ã¯ã äžå¿
ãååŸ
ã®çã«å¯Ÿå¿ããã
次ã«3次å
空éäžã«ããããã¯ãã«ãèå¯ããã 2次å
空éäžã§ã¯ãã¯ãã«ã¯2ã€ã®éã®çµã¿åããã§è¡šããããã ããã¯1ã€ã®ãã¯ãã«ã¯x軞æ¹åã«å¯Ÿå¿ããéãšy軞æ¹åã«å¯Ÿå¿ããéã®2ã€ãæã£ãŠããå¿
èŠããã£ãããã§ããã ãã®ããšããã3次å
空éã®ãã¯ãã«ã¯3ã€ã®éã®çµã¿åããã§æžããããšãäºæ³ãããã ç¹ã« x {\displaystyle x} 軞æ¹åã®æå a {\displaystyle a} , y {\displaystyle y} 軞æ¹åã®æå b {\displaystyle b} , z {\displaystyle z} 軞æ¹åã®æå c {\displaystyle c} ( a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} ã¯ä»»æã®å®æ°ã) ã§è¡šãããããã¯ãã«ãã
ãšæžããŠè¡šããããšã«ããã
2次å
å¹³é¢ã§ã¯ ãããã¯ãã«
ã¯ã ( a {\displaystyle a} , b {\displaystyle b} ã¯ä»»æã®å®æ°ã)
ã®2æ¬ã®ãã¯ãã«ãçšããŠã
ã§è¡šããããã 3次å
空éã§ããã®ãããªèšè¿°æ³ããããäžã§çšãããã¯ãã«
ã¯ã
ãçšããŠ
ãšæžããããã¯ãã«ã«å¯Ÿå¿ããŠããã
3次å
ãã¯ãã«ã«å¯ŸããŠã2次å
ãã¯ãã«ã§å®ããå®çŸ©ãæ§è³ªãã»ãŒãã®ãŸãŸæç«ããã
3次å
ãã¯ãã«ã®å æ³ã¯ãããããã®ãã¯ãã«èŠçŽ ãç¬ç«ã«è¶³ãåãããããšã«ãã£ãŠå®çŸ©ããã
ãŸããããããã®ãã¯ãã«ã®èŠçŽ ãå
šãŠçãããã¯ãã«ã"ãã¯ãã«ãšããŠçãã"ãšè¡šçŸããã
æŒç¿åé¡
ãã¯ãã«ã®å
ãèšç®ããã
ãåŸãããã
ãã¯ãã« a â {\displaystyle {\vec {a}}} , b â {\displaystyle {\vec {b}}} éã®ãã¯ãã«ã®å
ç©ãå¹³é¢ã®å Žåãšåæ§ã«
( Ξ {\displaystyle \theta } ã¯ããã¯ãã« a â {\displaystyle {\vec {a}}} , b â {\displaystyle {\vec {b}}} ã®ãªãè§ã)
åé
æ³åã1次ç¬ç«ã®æ§è³ªããã®ãŸãŸæãç«ã€ã ãã ãã3次å
空éã®å
šãŠã®ãã¯ãã«ã匵ãã«ã¯ã3ã€ã®ç·åœ¢ç¬ç«ãªãã¯ãã«ãæã£ãŠæ¥ãå¿
èŠãããã
ãã®ããšã®èšŒæã¯ããããç·å代æ°åŠãªã©ã«è©³ããã
æŒç¿åé¡
2ã€ã®ãã¯ãã«ã®å
ç©
ãèšç®ããã
2次å
ã®å Žåãšåãããã«ããã§ãããããã®èŠçŽ ã¯äºãã«çŽäº€ããåäœãã¯ãã«
ã«ãã£ãŠåŒµãããŠããããã®ãã以åãšåããèŠçŽ ããšã®èšç®ãå¯èœã§ããã
ãšãªãã
ããããã现ããèšç®ãè¡ãªããšã
ãåŸããããããããã®ãã¯ãã«ã
ã«åŸã£ãŠå±éãã
( i {\displaystyle i} , j {\displaystyle j} ã¯1,2,3ã®ã©ããã) ã代å
¥ããããšã§äžã®åŒãèšç®ã§ããã¯ãã§ããã
ãããã i {\displaystyle i} ãš j {\displaystyle j} ãçãããªããšãã«ã¯
ãæãç«ã€ããšãããäžã®å±éããåŸã®9åã®é
ã®ãã¡ã§ã6ã€ã¯
ã«çããã
ãŸãã i {\displaystyle i} ãš j {\displaystyle j} ãçãããšãã«ã¯
ãæãç«ã€ããšãããäžã®åŒ
ã®å±éã¯
ãšãªã£ãŠç¢ºãã«èŠçŽ ããšã®èšç®ãšäžèŽããã
æŒç¿åé¡
2次å
空éã®ãã¯ãã«ã¯2æ¬ã®1次ç¬ç«ãªãã¯ãã«ãããã°ãå¿
ããããã®ç·åœ¢çµåã«ãã£ãŠèšç®ã§ããã¯ãã§ããã
ããã§ã
ãš
ãçšããŠã
ãã
ã®åœ¢ã«æžããŠã¿ãã ( c {\displaystyle c} , d {\displaystyle d} ã¯ãäœããã®å®æ°ã)
2次å
ã®ãã¯ãã«ã®ä¿æ°ãæ±ããåé¡ã§ããã c {\displaystyle c} , d {\displaystyle d} ã®æåããã®ãŸãŸçšãããšã c {\displaystyle c} , d {\displaystyle d} ã®æºããæ¡ä»¶ã¯
ã€ãŸã
ãšãªãããã㯠c {\displaystyle c} , d {\displaystyle d} ã«é¢ããé£ç«1次æ¹çšåŒã§æžãæããããã
ããã解ããšã
ãåŸãããã
ãã£ãŠã äžã®åŒã¯
ãšæžãã確ãã«2æ¬ã®ç·åœ¢ç¬ç«ãªãã¯ãã«ã«ãã£ãŠä»ã®ãã¯ãã«ãæžãè¡šãããããšãåãã£ãã
ãã®ãããªèšç®ã¯3次å
ãã¯ãã«ã«å¯ŸããŠãå¯èœã§ããããèšç®ææ³ãšããŠ3å
1次é£ç«æ¹çšåŒãæ±ãå¿
èŠããããæå°èŠé ã®ç¯å²å€ã§ãããå®éã®èšç®ææ³ã¯ãç·å代æ°åŠ,ç©çæ°åŠI ç·åœ¢ä»£æ°ãåç
§ã
ãã®è¡šåŒãçšããŠã以åèŠã
ã®å³åœ¢ç解éãè¿°ã¹ãã
ãã®å³åœ¢äžã®ä»»æã®ç¹ã ( x , y , z ) {\displaystyle (x,y,z)} ã§è¡šããã ãã®ç¹ã¯åç¹Oã«å¯Ÿããäœçœ®ãã¯ãã«ãçšãããš ( x , y , z ) {\displaystyle (x,y,z)} ã§äžããããã 䟿å®ã®ããã« ãã®ãã¯ãã«ã x â {\displaystyle {\vec {x}}} ãšæžãããšã«ããã
äžæ¹ããã¯ãã« a â = ( a , b , c ) {\displaystyle {\vec {a}}=(a,b,c)} ãçšãããšãäžã®åŒã¯ãã¯ãã«ã®å
ç©ãçšã㊠a â â
x â = d {\displaystyle {\vec {a}}\cdot {\vec {x}}=d} ã§äžããããã ã€ãŸãããã®åŒã§è¡šããããå³åœ¢ã¯ãããã¯ãã« a â {\displaystyle {\vec {a}}} ãšã®å
ç©ãäžå®ã«ä¿ã€å³åœ¢ã§ããã ãã®å³åœ¢ã¯ãå®éã«ã¯ a â {\displaystyle {\vec {a}}} ã«çŽäº€ããå¹³é¢ã§äžããããã ãªããªããã®ãããªå¹³é¢äžã®ç¹ã¯ãå¿
ãå¹³é¢äžã®ããäžç¹ã®äœçœ®ãã¯ãã«ã«å ããŠã ãã¯ãã« a â {\displaystyle {\vec {a}}} ã«çŽäº€ãããã¯ãã«ãå ãããã®ã§æžãããšãåºæ¥ãã ãããã ãã¯ãã« a â {\displaystyle {\vec {a}}} ã«çŽäº€ãããã¯ãã«ãš ãã¯ãã« a â {\displaystyle {\vec {a}}} ã®å
ç©ã¯å¿
ã0ã§ããã®ã§ã ãã®ãããªç¹ã®éå㯠ãã¯ãã« a â {\displaystyle {\vec {a}}} ãšäžå®ã®å
ç©ãæã€ã®ã§ããã
ãã£ãŠå
ã®åŒ
ã¯ã ãã¯ãã« a â = ( a , b , c ) {\displaystyle {\vec {a}}=(a,b,c)} ã«çŽäº€ããå¹³é¢ã«å¯Ÿå¿ããããšãåãã£ãã 次㫠d {\displaystyle d} ããå³åœ¢ãè¡šããå¹³é¢ãšãåç¹ãšã®è·é¢ã«é¢ä¿ãããããšã瀺ãã
ç¹ã«ããã¯ãã« a â {\displaystyle {\vec {a}}} ã«æ¯äŸããäœçœ®ãã¯ãã«ãæã€ç¹ x â {\displaystyle {\vec {x}}} ãèããããã®ãšããã®ç¹ãšåç¹ãšã®è·é¢ã¯ã å¹³é¢
ãšåç¹ãšã®è·é¢ã«å¯Ÿå¿ããã ãªããªããäœçœ®ãã¯ãã« x â {\displaystyle {\vec {x}}} ã¯ãåç¹ããå¹³é¢
ã«åçŽã«äžãããç·ã«å¯Ÿå¿ããããã§ããã
ãã®ããšãã仮㫠a â {\displaystyle {\vec {a}}} æ¹åã®åäœãã¯ãã«ã n â {\displaystyle {\vec {n}}} ãšæžããå¹³é¢ãšåç¹ãšã®è·é¢ã m {\displaystyle m} ãšæžããšã x â = m n â {\displaystyle {\vec {x}}=m{\vec {n}}} ãåŸãããã ãã®åŒã
ã«ä»£å
¥ãããšã
ãåŸãããããã£ãŠã d {\displaystyle d} ã¯ã å¹³é¢ãšåç¹ã®è·é¢ m {\displaystyle m} ãšãã¯ãã« a â {\displaystyle {\vec {a}}} ã®å€§ãããããããã®ã§ããã
æŒç¿åé¡
ç¹ã«ãã¯ãã«
ãåããšãã©ã®ãããªåŒãåŸãããŠããã®åŒã¯ ã©ã®ãããªå³åœ¢ã«å¯Ÿå¿ãããã
ãã®ãšã
ã¯ã
ã«å¯Ÿå¿ããã
ãã®åŒã¯ z {\displaystyle z} 座æšã d {\displaystyle d} ã«å¯Ÿå¿ãããã以å€ã® x {\displaystyle x} , y {\displaystyle y} 座æšãä»»æã«åããã å¹³é¢ã«å¯Ÿå¿ããŠãããããã㯠x y {\displaystyle xy} å¹³é¢ã«å¹³è¡ã§ããã x y {\displaystyle xy} å¹³é¢ããã®è·é¢ã d {\displaystyle d} ã§ããå¹³é¢ã§ããã ãŸãã x y {\displaystyle xy} å¹³é¢ãšãã¯ãã«
ã¯çŽäº€ããŠããã®ã§ããã®ããšããããã®åŒã¯æ£ããã
å€ç©ã¯é«æ ¡æ°åŠç¯å²å€ã§å
¥è©Šã«ã¯åºãªãããå€ç©ã¯æ°åŠãç©çãªã©ã«å¿çšã§ãã䟿å©ãªã®ã§ããã§æ±ãã
äžæ¬¡å
ãã¯ãã« a â , b â {\displaystyle {\vec {a}},\,{\vec {b}}} ã«å¯Ÿããå€ç© a â à b â {\displaystyle {\vec {a}}\times {\vec {b}}} ã次ãæºãããã®ãšããã
次ã«å€ç©ã®æå衚瀺ãèããŠã¿ããããã®å®çŸ©ããæå衚瀺ãçŽæ¥å°ãã®ã¯é¢åãªã®ã§ã倩äžãçã«æå衚瀺ãäžããŠããããããå€ç©ã®å®çŸ©ãæºããããšã確èªããã
a â = ( a 1 a 2 a 3 ) {\displaystyle {\vec {a}}={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\end{pmatrix}}} ã b â = ( b 1 b 2 b 3 ) {\displaystyle {\vec {b}}={\begin{pmatrix}b_{1}\\b_{2}\\b_{3}\end{pmatrix}}} ãšãããšãã a â à b â = ( a 2 b 3 â a 3 b 2 a 3 b 1 â a 1 b 3 a 1 b 2 â a 2 b 1 ) {\displaystyle {\vec {a}}\times {\vec {b}}={\begin{pmatrix}a_{2}b_{3}-a_{3}b_{2}\\a_{3}b_{1}-a_{1}b_{3}\\a_{1}b_{2}-a_{2}b_{1}\end{pmatrix}}} ã§ããã
ãŸãã¯ã a â à b â {\displaystyle {\vec {a}}\times {\vec {b}}} 㯠a â , b â {\displaystyle {\vec {a}},\,{\vec {b}}} ãããããšåçŽã§ããããšã確èªãããããã¯ã ( a â à b â ) â
a â = 0 {\displaystyle ({\vec {a}}\times {\vec {b}})\cdot {\vec {a}}=0} ãš ( a â Ã b â ) â
b â = 0 {\displaystyle ({\vec {a}}\times {\vec {b}})\cdot {\vec {b}}=0} ã§ããããšãæå衚瀺ã代å
¥ããã°èšŒæã§ããã
次ã«ã | a â à b â | = | a â | | b â | sin Ξ {\displaystyle |{\vec {a}}\times {\vec {b}}|=|{\vec {a}}||{\vec {b}}|\sin \theta } ã蚌æããã | a â à b â | 2 = | a â | 2 | b â | 2 sin 2 Ξ = | â a | 2 | b â | 2 ( 1 â cos 2 Ξ ) {\displaystyle |{\vec {a}}\times {\vec {b}}|^{2}=|{\vec {a}}|^{2}|{\vec {b}}|^{2}\sin ^{2}\theta ={\vec {|}}a|^{2}|{\vec {b}}|^{2}(1-\cos ^{2}\theta )} ãããã§ã cos 2 Ξ = ( a â â
b â ) 2 | a â | 2 | b â | 2 {\displaystyle \cos ^{2}\theta ={\frac {({\vec {a}}\cdot {\vec {b}})^{2}}{|{\vec {a}}|^{2}|{\vec {b}}|^{2}}}} ã代å
¥ãã | a â à b â | 2 = | â a | 2 | b â | 2 â ( a â â
b â ) 2 {\displaystyle |{\vec {a}}\times {\vec {b}}|^{2}={\vec {|}}a|^{2}|{\vec {b}}|^{2}-({\vec {a}}\cdot {\vec {b}})^{2}} ãåŸãããã®åŒã«ãæå衚瀺ã代å
¥ããã°ã䞡蟺ãçããããšã確èªã§ããã
æåŸã«ããã¬ãã³ã°ã®å·Šæã®æ³å㧠a â à b â {\displaystyle {\vec {a}}\times {\vec {b}}} ã¯èŠªæã®æ¹åã§ããããšã確èªããã
a â = ( 1 0 0 ) {\displaystyle {\vec {a}}={\begin{pmatrix}1\\0\\0\end{pmatrix}}} ã b â = ( 0 1 0 ) {\displaystyle {\vec {b}}={\begin{pmatrix}0\\1\\0\end{pmatrix}}} ã®ãšãã a â à b â = ( 0 0 1 ) {\displaystyle {\vec {a}}\times {\vec {b}}={\begin{pmatrix}0\\0\\1\end{pmatrix}}} ã§ããããããããäºçªç®ã®æ§è³ªã確èªã§ããã
å€ç©ã®å¿çš
2ã€ã®ãã¯ãã«ã«åçŽãªãã¯ãã«ãæ±ããããšããªã©ã¯ãå€ç©ã®æå衚瀺ããèšç®ããã°ãé¢åãªèšç®ãããªããŠãæ±ããããã
åé¢äœ O A B C {\displaystyle \mathrm {OABC} } ã®äœç©ã¯ 1 6 | ( O A â à O B â ) â
O C â | {\displaystyle {\frac {1}{6}}|({\vec {\mathrm {OA} }}\times {\vec {\mathrm {OB} }})\cdot {\vec {\mathrm {OC} }}|} ã§ããã å®éã 1 6 | ( O A â à O B â ) â
O C â | = 1 3 | 1 2 O A â à O B â | | h | {\displaystyle {\frac {1}{6}}|({\vec {\mathrm {OA} }}\times {\vec {\mathrm {OB} }})\cdot {\vec {\mathrm {OC} }}|={\frac {1}{3}}\left|{\frac {1}{2}}{\vec {\mathrm {OA} }}\times {\vec {\mathrm {OB} }}\right||h|} ã§ããããã ãã h ã¯ÎABCãåºé¢ãšãããšãã®åé¢äœã®é«ãã§ããã
ãŸããç©çåŠã®ããŒã¬ã³ãåã¯å€ç©ã䜿ããš F â = q v â à B â {\displaystyle {\vec {F}}=q{\vec {v}}\times {\vec {B}}} ãšç°¡æœã«è¡šããã
èŠãæ¹
å³ã®ããã«èŠçŽ ãããåãããã
| [
{
"paragraph_id": 0,
"tag": "p",
"text": "çç§ã«ãããŠãåã¯å€§ãããšåããæã€éã§ãããšç¿ã£ãã ããã倧ãããšåããæã€éã¯ãåã®ä»ã«ããé床ã颚ã®å¹ãæ¹ãªã©ãããã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "äŸãã°ãããå°ç¹ããæå»ã«ããã颚ã®å¹ãæ¹ã¯ã颚éãšé¢šåããæãç«ã€ããã®ããã«ã倧ãããšåããæã€éãå°å
¥ãããšãããããå¹çããæ±ããã",
"title": ""
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ãã®ããŒãžã§ã¯ã倧ãããšåããæã€éã§ãããã¯ãã«ãæ±ãã",
"title": ""
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ãŸããå³åœ¢ã®åé¡ã«å¯ŸããŠä»£æ°çãªã¢ãããŒããåããã®ããã¯ãã«ã®å©ç¹ã®äžã€ã§ããã",
"title": ""
},
{
"paragraph_id": 4,
"tag": "p",
"text": "å¹³é¢äžã®ç¹ S {\\displaystyle \\mathrm {S} } ããç¹ T {\\displaystyle \\mathrm {T} } ãžåããç¢å°ãèããããã®ãããªç¢å°ã®ããã«åããæã€ç·åãæåç·åãšããã",
"title": "å¹³é¢äžã®ãã¯ãã«"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ãã®ãšããç¹ S {\\displaystyle \\mathrm {S} } ãå§ç¹ãç¹ T {\\displaystyle \\mathrm {T} } ãçµç¹ãšããã",
"title": "å¹³é¢äžã®ãã¯ãã«"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "æå¹ç·åã§ã倧ãããšæ¹åãåããã®ã¯ãã¯ãã«ãšããŠåããã®ãšããã",
"title": "å¹³é¢äžã®ãã¯ãã«"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "æåç·åã¯äœçœ®ãé·ã(倧ãã)ãåããšããæ
å ±ãæã€ããã¯ãã«ã¯ãæåç·åã®æã€æ
å ±ã®ãã¡ãäœçœ®ã®æ
å ±ãå¿ããŠã倧ãããåãã ãã«çç®ãããã®ãšèããããšãã§ããã",
"title": "å¹³é¢äžã®ãã¯ãã«"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "æåç·å S T {\\displaystyle \\mathrm {ST} } ã§è¡šããããã¯ãã«ã S T â {\\displaystyle \\mathrm {\\vec {ST}} } ãšããããã¯ãã«ã¯äžæå㧠a â {\\displaystyle {\\vec {a}}} ãªã©ãšè¡šãããããšãããããã¯ãã« a â {\\displaystyle {\\vec {a}}} ã®å€§ããã | a â | {\\displaystyle |{\\vec {a}}|} ã§è¡šãã",
"title": "å¹³é¢äžã®ãã¯ãã«"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "æåç·å S T {\\displaystyle \\mathrm {ST} } ãæåç·å S â² T â² {\\displaystyle \\mathrm {S'T'} } ã«å¯Ÿãã倧ãããçãããåããçãããªããäœçœ®ãéã£ãŠããŠãããã¯ãã«ãšããŠçããã S T â = S â² T â² â {\\displaystyle \\mathrm {\\vec {ST}} =\\mathrm {\\vec {S'T'}} } ã§ããã",
"title": "å¹³é¢äžã®ãã¯ãã«"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "倧ããã 1 ã§ãããã¯ãã«ãåäœãã¯ãã«ãšããã",
"title": "å¹³é¢äžã®ãã¯ãã«"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "ãã¯ãã« a â {\\displaystyle {\\vec {a}}} ã«å¯Ÿãããã¯ãã« a â {\\displaystyle {\\vec {a}}} ãšæ¹åãéã§ã倧ãããçãããã¯ãã«ãéãã¯ãã«ãšããã â a â {\\displaystyle -{\\vec {a}}} ãšããã",
"title": "å¹³é¢äžã®ãã¯ãã«"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "å§ç¹ãšçµç¹ãçãããã¯ãã«ãé¶ãã¯ãã«ãšããã 0 â {\\displaystyle {\\vec {0}}} ã§è¡šããä»»æã®ç¹ A {\\displaystyle \\mathrm {A} } ã«å¯Ÿãã A A â = 0 â {\\displaystyle \\mathrm {\\vec {AA}} ={\\vec {0}}} ã§ããããŒããã¯ãã«ã®å€§ãã㯠0 ã§ãåãã¯èããªããã®ãšããã",
"title": "å¹³é¢äžã®ãã¯ãã«"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ãã¯ãã« a â , b â {\\displaystyle {\\vec {a}},{\\vec {b}}} ã«å¯Ÿãã a â = A B â , b â = B C â {\\displaystyle {\\vec {a}}=\\mathrm {\\vec {AB}} ,{\\vec {b}}=\\mathrm {\\vec {BC}} } ãšãªãç¹ããšãããã®ãšããã¯ãã«ã®å æ³ã a â + b â = A C â {\\displaystyle {\\vec {a}}+{\\vec {b}}=\\mathrm {\\vec {AC}} } ã§å®ããã",
"title": "å¹³é¢äžã®ãã¯ãã«"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "ãã¯ãã«ã®å æ³ã«ã€ããŠä»¥äžãæãç«ã€ã",
"title": "å¹³é¢äžã®ãã¯ãã«"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "ãŸãã a â + 0 â = a â {\\displaystyle {\\vec {a}}+{\\vec {0}}={\\vec {a}}} ãšããã",
"title": "å¹³é¢äžã®ãã¯ãã«"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "ãã¯ãã« a â , b â {\\displaystyle {\\vec {a}},{\\vec {b}}} ã«å¯Ÿãã a â â b â = a â + ( â b â ) {\\displaystyle {\\vec {a}}-{\\vec {b}}={\\vec {a}}+(-{\\vec {b}})} ãšããã",
"title": "å¹³é¢äžã®ãã¯ãã«"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "ãŒããã¯ãã«ã¯ãªããã¯ãã« a â {\\displaystyle {\\vec {a}}} ãšå®æ° k {\\displaystyle k} ã«å¯Ÿãããã¯ãã«ã®å®æ°å k a â {\\displaystyle k{\\vec {a}}} ã以äžã®ããã«å®ããã",
"title": "å¹³é¢äžã®ãã¯ãã«"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "ãŸããŒããã¯ãã« 0 â {\\displaystyle {\\vec {0}}} ã«å¯Ÿããå®æ°åã k 0 â = 0 â {\\displaystyle k{\\vec {0}}={\\vec {0}}} ã§å®ããã",
"title": "å¹³é¢äžã®ãã¯ãã«"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "以äžã®æ§è³ªããªããã€ã",
"title": "å¹³é¢äžã®ãã¯ãã«"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "ãŒããã¯ãã«ã§ã¯ãªããã¯ãã« a â , b â ( â 0 â ) {\\displaystyle {\\vec {a}},{\\vec {b}}\\,(\\neq {\\vec {0}})} ã«å¯Ÿãã a â = A A â² â , b â = B B â² â {\\displaystyle {\\vec {a}}={\\vec {\\mathrm {AA'} }},{\\vec {b}}={\\vec {\\mathrm {BB'} }}} ãšãªãç¹ããšãã",
"title": "ãã¯ãã«ã®å¹³è¡ã»åçŽ"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "ãã®ãšããçŽç· A A â² {\\displaystyle \\mathrm {AA'} } ãšçŽç· B B â² {\\displaystyle \\mathrm {BB'} } ãå¹³è¡ã§ãããšãããã¯ãã« a â , b â {\\displaystyle {\\vec {a}},{\\vec {b}}} ã¯å¹³è¡ã§ãããšããã a â ⥠b â {\\displaystyle {\\vec {a}}\\parallel {\\vec {b}}} ã§è¡šãã",
"title": "ãã¯ãã«ã®å¹³è¡ã»åçŽ"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "ãŸããçŽç· A A â² {\\displaystyle \\mathrm {AA'} } ãšçŽç· B B â² {\\displaystyle \\mathrm {BB'} } ãåçŽã§ãããšãããã¯ãã« a â , b â {\\displaystyle {\\vec {a}},{\\vec {b}}} ã¯åçŽã§ãããšããã a â ⥠b â {\\displaystyle {\\vec {a}}\\perp {\\vec {b}}} ã§è¡šãã",
"title": "ãã¯ãã«ã®å¹³è¡ã»åçŽ"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "ãã¯ãã« a â , b â {\\displaystyle {\\vec {a}},{\\vec {b}}} ãå¹³è¡ã®ãšããæããã«ãçæ¹ã®ãã¯ãã«ãå®æ°åããã°å€§ãããšåããäžèŽããã®ã§ã",
"title": "ãã¯ãã«ã®å¹³è¡ã»åçŽ"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "a â ⥠b â ⺠b â = k a â {\\displaystyle {\\vec {a}}\\parallel {\\vec {b}}\\iff {\\vec {b}}=k{\\vec {a}}} ãšãªãå®æ° k {\\displaystyle k} ãååšãã",
"title": "ãã¯ãã«ã®å¹³è¡ã»åçŽ"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "ãæãç«ã€ã",
"title": "ãã¯ãã«ã®å¹³è¡ã»åçŽ"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "ãã¯ãã« a â , b â {\\displaystyle {\\vec {a}},{\\vec {b}}} ããšãã«ãŒããã¯ãã«ã§ãªã( a â , b â â 0 â {\\displaystyle {\\vec {a}},{\\vec {b}}\\neq {\\vec {0}}} ) ãå¹³è¡ã§ãªããšããä»»æã®ãã¯ãã« p â {\\displaystyle {\\vec {p}}} ã«å¯ŸããŠã p â = s a â + t b â {\\displaystyle {\\vec {p}}=s{\\vec {a}}+t{\\vec {b}}} ãšãªãå®æ° s , t {\\displaystyle s,t} ãåãããšãã§ããã",
"title": "ãã¯ãã«ã®å解"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "蚌æ",
"title": "ãã¯ãã«ã®å解"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "a â = O A â , b â = O B â , p â = O P â {\\displaystyle {\\vec {a}}={\\vec {\\mathrm {OA} }},{\\vec {b}}={\\vec {\\mathrm {OB} }},{\\vec {p}}={\\vec {\\mathrm {OP} }}} ãšãªãç¹ããšããç¹ P {\\displaystyle \\mathrm {P} } ãéããçŽç· O B , O A {\\displaystyle \\mathrm {OB} ,\\mathrm {OA} } ã«å¹³è¡ãªçŽç·ãããããã çŽç· O A , O B {\\displaystyle \\mathrm {OA} ,\\mathrm {OB} } ãšäº€ããç¹ããããã S , T {\\displaystyle \\mathrm {S,T} } ãšçœ®ãã",
"title": "ãã¯ãã«ã®å解"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "ãã®ãšãã O S â = s a â , O T â = t b â {\\displaystyle {\\vec {\\mathrm {OS} }}=s{\\vec {a}},{\\vec {\\mathrm {OT} }}=t{\\vec {b}}} ãšãªãå®æ° s , t {\\displaystyle s,t} ãåãããšãã§ãããããã§ãåè§åœ¢ O S P T {\\displaystyle \\mathrm {OSPT} } ã¯å¹³è¡å蟺圢ãªã®ã§ã p â = s a â + t b â {\\displaystyle {\\vec {p}}=s{\\vec {a}}+t{\\vec {b}}} ãæãç«ã€ã",
"title": "ãã¯ãã«ã®å解"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "ãã¯ãã« a â {\\displaystyle {\\vec {a}}} ã«å¯ŸããŠã座æšå¹³é¢äžã®åç¹ã O {\\displaystyle \\mathrm {O} } ãšãããšãã a â = O A â {\\displaystyle {\\vec {a}}=\\mathrm {\\vec {OA}} } ãšãªãç¹ A ( a x , a y ) {\\displaystyle \\mathrm {A} (a_{x},a_{y})} ãåãããšãã§ãããããã§ã ( a x , a y ) {\\displaystyle (a_{x},a_{y})} ããã¯ãã« a â {\\displaystyle {\\vec {a}}} ã®æå衚瀺ãšãã a â = ( a x , a y ) {\\displaystyle {\\vec {a}}=(a_{x},a_{y})} ããŸãã¯ã瞊ã«äžŠã¹ãŠã a â = ( a x a y ) {\\displaystyle {\\vec {a}}=\\left({\\begin{aligned}a_{x}\\\\a_{y}\\end{aligned}}\\right)} ãšæžãã",
"title": "ãã¯ãã«ã®æå衚瀺"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "ãã¯ãã« a â , b â {\\displaystyle {\\vec {a}},{\\vec {b}}} ã«å¯ŸããŠã a â = O A â , b â = O B â {\\displaystyle {\\vec {a}}=\\mathrm {\\vec {OA}} ,\\,{\\vec {b}}=\\mathrm {\\vec {OB}} } ãšãªãç¹ A , B {\\displaystyle \\mathrm {A} ,\\mathrm {B} } ããšãã a â = ( a x , a y ) , b â = ( b x , b y ) {\\displaystyle {\\vec {a}}=(a_{x},a_{y}),\\,{\\vec {b}}=(b_{x},b_{y})} ãšãããšã",
"title": "ãã¯ãã«ã®æå衚瀺"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "a â = b â ⺠O A â = O B â ⺠{\\displaystyle {\\vec {a}}={\\vec {b}}\\iff {\\vec {\\mathrm {OA} }}={\\vec {\\mathrm {OB} }}\\iff } ç¹ A , B {\\displaystyle \\mathrm {A} ,\\,\\mathrm {B} } ãäžèŽãã ⺠a x = b x {\\displaystyle \\iff a_{x}=b_{x}} ã〠a y = b y {\\displaystyle a_{y}=b_{y}}",
"title": "ãã¯ãã«ã®æå衚瀺"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "ãŸãã a â = ( a x , a y ) {\\displaystyle {\\vec {a}}=(a_{x},a_{y})} ã«å¯ŸããŠã a â = O A â {\\displaystyle {\\vec {a}}=\\mathrm {\\vec {OA}} } ãšãããšãã | a â | {\\displaystyle |{\\vec {a}}|} ã¯ç·å O A {\\displaystyle \\mathrm {OA} } ã®é·ããªã®ã§ã",
"title": "ãã¯ãã«ã®æå衚瀺"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "| a â | = a x 2 + a y 2 {\\displaystyle |{\\vec {a}}|={\\sqrt {a_{x}^{2}+a_{y}^{2}}}}",
"title": "ãã¯ãã«ã®æå衚瀺"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "ã§ããã",
"title": "ãã¯ãã«ã®æå衚瀺"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "",
"title": "ãã¯ãã«ã®æå衚瀺"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "ãã¯ãã« a â = ( a x , a y ) , b â = ( b x , b y ) {\\displaystyle {\\vec {a}}=(a_{x},a_{y}),{\\vec {b}}=(b_{x},b_{y})} ã«å¯ŸããŠã",
"title": "ãã¯ãã«ã®æå衚瀺"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "a â + b â = ( a x + b x , a y + b y ) {\\displaystyle {\\vec {a}}+{\\vec {b}}=(a_{x}+b_{x},a_{y}+b_{y})}",
"title": "ãã¯ãã«ã®æå衚瀺"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "a â â b â = ( a x â b x , a y â b y ) {\\displaystyle {\\vec {a}}-{\\vec {b}}=(a_{x}-b_{x},a_{y}-b_{y})}",
"title": "ãã¯ãã«ã®æå衚瀺"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "k a â = ( k a x , k a y ) {\\displaystyle k{\\vec {a}}=(ka_{x},ka_{y})}",
"title": "ãã¯ãã«ã®æå衚瀺"
},
{
"paragraph_id": 41,
"tag": "p",
"text": "ããªããã€ã",
"title": "ãã¯ãã«ã®æå衚瀺"
},
{
"paragraph_id": 42,
"tag": "p",
"text": "ããç¹ãåºæºã«ããŠããã®ç¹ãå§ç¹ãšãããã¯ãã«ã«ã€ããŠèããããšã«ããããã¯ãã«ãçšããŠç¹ã®äœçœ®é¢ä¿ã«ã€ããŠèå¯ããããšãã§ããã",
"title": "äœçœ®ãã¯ãã«"
},
{
"paragraph_id": 43,
"tag": "p",
"text": "ç¹ã®äœçœ®é¢ä¿åºæºãšãªãç¹ O {\\displaystyle {\\rm {O}}} ããããããå®ããããã®ãšããç¹ A {\\displaystyle {\\rm {A}}} ã«å¯ŸããŠããã¯ãã« O A â {\\displaystyle {\\vec {\\rm {OA}}}} ãç¹ A {\\displaystyle {\\rm {A}}} ã®äœçœ®ãã¯ãã«ãšãããäœçœ®ãã¯ãã« a â {\\displaystyle {\\vec {a}}} ã§äžããããç¹ A {\\displaystyle {\\rm {A}}} ã A ( a â ) {\\displaystyle \\mathrm {A} ({\\vec {a}})} ã§è¡šãã",
"title": "äœçœ®ãã¯ãã«"
},
{
"paragraph_id": 44,
"tag": "p",
"text": "ãŸããç¹ A ( a â ) , B ( b â ) {\\displaystyle \\mathrm {A} ({\\vec {a}}),\\,\\mathrm {B} ({\\vec {b}})} ã®ãšãã A B â = O B â â O A â = b â â a â {\\displaystyle {\\vec {\\rm {AB}}}={\\vec {\\rm {OB}}}-{\\vec {\\rm {OA}}}={\\vec {b}}-{\\vec {a}}} ãæãç«ã€ã",
"title": "äœçœ®ãã¯ãã«"
},
{
"paragraph_id": 45,
"tag": "p",
"text": "以äžãäœçœ®ãã¯ãã«ã®åºæºç¹ãç¹ O {\\displaystyle {\\rm {O}}} ãšããã",
"title": "äœçœ®ãã¯ãã«"
},
{
"paragraph_id": 46,
"tag": "p",
"text": "ç¹ A ( a â ) , B ( b â ) {\\displaystyle {\\rm {A({\\vec {a}}),\\,{\\rm {B({\\vec {b}})}}}}} ãéãç·å A B {\\displaystyle \\mathrm {AB} } ã m : n {\\displaystyle m:n} ã«å
åããç¹ P ( p â ) {\\displaystyle \\mathrm {P} ({\\vec {p}})} ãæ±ããã",
"title": "äœçœ®ãã¯ãã«"
},
{
"paragraph_id": 47,
"tag": "p",
"text": "A P â = m m + n A B â {\\displaystyle {\\vec {\\mathrm {AP} }}={\\frac {m}{m+n}}{\\vec {\\mathrm {AB} }}} ããã p â â a â = m m + n ( b â â a â ) {\\displaystyle {\\vec {p}}-{\\vec {a}}={\\frac {m}{m+n}}({\\vec {b}}-{\\vec {a}})} ãããã£ãŠã p â = n a â + m b â m + n {\\displaystyle {\\vec {p}}={\\frac {n{\\vec {a}}+m{\\vec {b}}}{m+n}}} ã§ããã",
"title": "äœçœ®ãã¯ãã«"
},
{
"paragraph_id": 48,
"tag": "p",
"text": "次ã«ãç¹ A ( a â ) , B ( b â ) {\\displaystyle {\\rm {A({\\vec {a}}),\\,{\\rm {B({\\vec {b}})}}}}} ãéãç·å A B {\\displaystyle \\mathrm {AB} } ã m : n {\\displaystyle m:n} ã«å€åããç¹ Q ( q â ) {\\displaystyle \\mathrm {Q} ({\\vec {q}})} ãæ±ããã",
"title": "äœçœ®ãã¯ãã«"
},
{
"paragraph_id": 49,
"tag": "p",
"text": "m > n {\\displaystyle m>n} ã®å Žåã¯ã A Q â = m m â n A B â {\\displaystyle {\\vec {\\mathrm {AQ} }}={\\frac {m}{m-n}}{\\vec {\\mathrm {AB} }}} ããã q â â a â = m m â n ( b â â a â ) {\\displaystyle {\\vec {q}}-{\\vec {a}}={\\frac {m}{m-n}}({\\vec {b}}-{\\vec {a}})} ãããã£ãŠã q â = â n a â + m b â m â n {\\displaystyle {\\vec {q}}={\\frac {-n{\\vec {a}}+m{\\vec {b}}}{m-n}}} ã§ããã",
"title": "äœçœ®ãã¯ãã«"
},
{
"paragraph_id": 50,
"tag": "p",
"text": "m < n {\\displaystyle m<n} ã®å Žåã¯ã B Q â = n n â m B A â {\\displaystyle {\\vec {\\mathrm {BQ} }}={\\frac {n}{n-m}}{\\vec {\\mathrm {BA} }}} ã«æ³šæããŠåæ§ã«èšç®ããã°ãåãšåãã q â = â n a â + m b â m â n {\\displaystyle {\\vec {q}}={\\frac {-n{\\vec {a}}+m{\\vec {b}}}{m-n}}} ãåŸãããã",
"title": "äœçœ®ãã¯ãã«"
},
{
"paragraph_id": 51,
"tag": "p",
"text": "äžè§åœ¢ A B C {\\displaystyle \\mathrm {ABC} } ã«å¯Ÿãã A ( a â ) , B ( b â ) , C ( c â ) {\\displaystyle \\mathrm {A} ({\\vec {a}}),\\,\\mathrm {B} ({\\vec {b}}),\\,\\mathrm {C} ({\\vec {c}})} ãšçœ®ãããã®äžè§åœ¢ A B C {\\displaystyle \\mathrm {ABC} } ã®éå¿ G ( g â ) {\\displaystyle \\mathrm {G} ({\\vec {g}})} ãæ±ããã",
"title": "äœçœ®ãã¯ãã«"
},
{
"paragraph_id": 52,
"tag": "p",
"text": "ç·å B C {\\displaystyle \\mathrm {BC} } ã®äžç¹ã M ( m â ) {\\displaystyle \\mathrm {M} ({\\vec {m}})} ãšãããšãç¹ M {\\displaystyle \\mathrm {M} } ã¯ç·å B C {\\displaystyle \\mathrm {BC} } ã 1 : 1 {\\displaystyle 1:1} ã«å
åããç¹ãªã®ã§ã m â = b â + c â 2 {\\displaystyle {\\vec {m}}={\\frac {{\\vec {b}}+{\\vec {c}}}{2}}} ã§ããã",
"title": "äœçœ®ãã¯ãã«"
},
{
"paragraph_id": 53,
"tag": "p",
"text": "ç¹ G {\\displaystyle \\mathrm {G} } ã¯ç·å A M {\\displaystyle \\mathrm {AM} } ã 2 : 1 {\\displaystyle 2:1} ã«å
åããç¹ãªã®ã§ã g â = a â + b â + c â 3 {\\displaystyle {\\vec {g}}={\\frac {{\\vec {a}}+{\\vec {b}}+{\\vec {c}}}{3}}} ã§ããã",
"title": "äœçœ®ãã¯ãã«"
},
{
"paragraph_id": 54,
"tag": "p",
"text": "äžè§åœ¢ A B C {\\displaystyle \\mathrm {ABC} } ã«å¯Ÿãã A ( a â ) , B ( b â ) , C ( c â ) {\\displaystyle \\mathrm {A} ({\\vec {a}}),\\,\\mathrm {B} ({\\vec {b}}),\\,\\mathrm {C} ({\\vec {c}})} ãšçœ®ããããã«ã A B = c , B C = a , C A = b {\\displaystyle \\mathrm {AB} =c,\\,\\mathrm {BC} =a,\\,\\mathrm {CA} =b} ãšçœ®ããäžè§åœ¢ A B C {\\displaystyle \\mathrm {ABC} } ã®å
å¿ã®äœçœ®ãã¯ãã« I ( i â ) {\\displaystyle \\mathrm {I} ({\\vec {i}})} ãæ±ããã",
"title": "äœçœ®ãã¯ãã«"
},
{
"paragraph_id": 55,
"tag": "p",
"text": "A {\\displaystyle {\\rm {A}}} ã®äºçåç·ãšç·å B C {\\displaystyle {\\rm {BC}}} ã®äº€ç¹ã D ( d â ) {\\displaystyle \\mathrm {D} ({\\vec {d}})} ãšããããã®ãšããäžè§åœ¢ã®äºçåç·ã®æ§è³ªãã B D : D C = c : b {\\displaystyle \\mathrm {BD} :\\mathrm {DC} =c:b} ãããã£ãŠã d â = b b â + c c â b + c {\\displaystyle {\\vec {d}}={\\frac {b{\\vec {b}}+c{\\vec {c}}}{b+c}}} ã§ããã",
"title": "äœçœ®ãã¯ãã«"
},
{
"paragraph_id": 56,
"tag": "p",
"text": "ããã§ã A I : I D = B A : B D = c : a c b + c = ( b + c ) : a {\\displaystyle \\mathrm {AI} :\\mathrm {ID} =\\mathrm {BA} :\\mathrm {BD} =c:{\\frac {ac}{b+c}}=(b+c):a} ã§ããã",
"title": "äœçœ®ãã¯ãã«"
},
{
"paragraph_id": 57,
"tag": "p",
"text": "ãããã£ãŠã i â = a a â + ( b + c ) d â a + b + c = a a â + b b â + c c â a + b + c {\\displaystyle {\\vec {i}}={\\frac {a{\\vec {a}}+(b+c){\\vec {d}}}{a+b+c}}={\\frac {a{\\vec {a}}+b{\\vec {b}}+c{\\vec {c}}}{a+b+c}}} ã§ããã",
"title": "äœçœ®ãã¯ãã«"
},
{
"paragraph_id": 58,
"tag": "p",
"text": "äžåŠãŸãã¯é«æ ¡ã®çç§ã®ååŠã§ã¯ãååŠçãªä»äºã®å®çŸ©ããªãã£ãããšãããã ããããã®ä»äºã§ã¯ã移åæ¹å以å€ã®åã¯ãä»äºã«å¯äžããªãã£ãããã®ãããªåã®ä»äºã®èšç®ãããã¯ãã«ã®èŠ³ç¹ããã¿ãã°ãå
ç©ãšããæ°ããæŠå¿µãå®çŸ©ã§ããã",
"title": "ãã¯ãã«ã®å
ç©"
},
{
"paragraph_id": 59,
"tag": "p",
"text": "ãã¯ãã« a â , b â {\\displaystyle {\\vec {a}},{\\vec {b}}} ã«å¯Ÿãã a â = O A â , b â = O B â {\\displaystyle {\\vec {a}}={\\vec {\\mathrm {OA} }},{\\vec {b}}={\\vec {\\mathrm {OB} }}} ãšãªãç¹ O , A , B {\\displaystyle \\mathrm {O,A,B} } ããšãããã®ãšãã â A O B {\\displaystyle \\angle \\mathrm {AOB} } ããã¯ãã« a â , b â {\\displaystyle {\\vec {a}},{\\vec {b}}} ã®ãªãè§ãšããã",
"title": "ãã¯ãã«ã®å
ç©"
},
{
"paragraph_id": 60,
"tag": "p",
"text": "(å³)",
"title": "ãã¯ãã«ã®å
ç©"
},
{
"paragraph_id": 61,
"tag": "p",
"text": "ãã¯ãã« a â , b â {\\displaystyle {\\vec {a}},{\\vec {b}}} ã®ãªãè§ã Ξ {\\displaystyle \\theta } ãšãããšããå
ç© a â â
b â {\\displaystyle {\\vec {a}}\\cdot {\\vec {b}}} ã",
"title": "ãã¯ãã«ã®å
ç©"
},
{
"paragraph_id": 62,
"tag": "p",
"text": "ã§å®ããã",
"title": "ãã¯ãã«ã®å
ç©"
},
{
"paragraph_id": 63,
"tag": "p",
"text": "å®çŸ©ããããã¯ãã«ã®å
ç©ã¯äžæ¹ã®ãã¯ãã«ãããäžæ¹ã®ãã¯ãã«ã«å°åœ±ãããšãã®ã倧ããã®ç©ã§ãããšèšããã",
"title": "ãã¯ãã«ã®å
ç©"
},
{
"paragraph_id": 64,
"tag": "p",
"text": "(å³)",
"title": "ãã¯ãã«ã®å
ç©"
},
{
"paragraph_id": 65,
"tag": "p",
"text": "ãã¯ãã« a â , b â {\\displaystyle {\\vec {a}},{\\vec {b}}} ã a â = ( a 1 , a 2 ) , b â = ( b 1 , b 2 ) {\\displaystyle {\\vec {a}}=(a_{1},a_{2}),{\\vec {b}}=(b_{1},b_{2})} ãšæå衚瀺ãããšãã®ãå
ç© a â â
b â {\\displaystyle {\\vec {a}}\\cdot {\\vec {b}}} ã«ã€ããŠèããŠã¿ããã",
"title": "ãã¯ãã«ã®å
ç©"
},
{
"paragraph_id": 66,
"tag": "p",
"text": "ãã¯ãã« a â , b â {\\displaystyle {\\vec {a}},{\\vec {b}}} ã«å¯Ÿãã a â = O A â , b â = O B â {\\displaystyle {\\vec {a}}={\\vec {\\mathrm {OA} }},{\\vec {b}}={\\vec {\\mathrm {OB} }}} ãšãªãç¹ O , A , B {\\displaystyle \\mathrm {O,A,B} } ããšãããã¯ãã« a â , b â {\\displaystyle {\\vec {a}},{\\vec {b}}} ã®ãªãè§ã Ξ {\\displaystyle \\theta } ãšããããã®ãšã â³ O A B {\\displaystyle \\triangle \\mathrm {OAB} } ã«å¯ŸãäœåŒŠå®çãçšããŠ",
"title": "ãã¯ãã«ã®å
ç©"
},
{
"paragraph_id": 67,
"tag": "p",
"text": "A B 2 = O A 2 + O B 2 â 2 â
O A â
O B cos Ξ {\\displaystyle \\mathrm {\\mathrm {AB} } ^{2}=\\mathrm {\\mathrm {OA} } ^{2}+\\mathrm {\\mathrm {OB} } ^{2}-2\\cdot \\mathrm {\\mathrm {OA} } \\cdot \\mathrm {\\mathrm {OB} } \\cos \\theta }",
"title": "ãã¯ãã«ã®å
ç©"
},
{
"paragraph_id": 68,
"tag": "p",
"text": "(å³)",
"title": "ãã¯ãã«ã®å
ç©"
},
{
"paragraph_id": 69,
"tag": "p",
"text": "ããã§ã A B = | b â â a â | , O A = | a â | , O B = | b â | {\\displaystyle \\mathrm {\\mathrm {AB} } =|{\\vec {b}}-{\\vec {a}}|,\\mathrm {\\mathrm {OA} } =|{\\vec {a}}|,\\mathrm {\\mathrm {OB} } =|{\\vec {b}}|} ãšã O A â
O B cos Ξ = | a â | | b â | cos Ξ = a â â
b â {\\displaystyle \\mathrm {\\mathrm {OA} } \\cdot \\mathrm {\\mathrm {OB} } \\cos \\theta =|{\\vec {a}}||{\\vec {b}}|\\cos \\theta ={\\vec {a}}\\cdot {\\vec {b}}} ãã",
"title": "ãã¯ãã«ã®å
ç©"
},
{
"paragraph_id": 70,
"tag": "p",
"text": "| b â â a â | 2 = | a â | 2 + | b â | 2 â 2 a â â
b â {\\displaystyle |{\\vec {b}}-{\\vec {a}}|^{2}=|{\\vec {a}}|^{2}+|{\\vec {b}}|^{2}-2{\\vec {a}}\\cdot {\\vec {b}}} ã§ããã®ã§ã a â â
b â = 1 2 ( | a â | 2 + | b â | 2 â | b â â a â | 2 ) {\\displaystyle {\\vec {a}}\\cdot {\\vec {b}}={\\frac {1}{2}}(|{\\vec {a}}|^{2}+|{\\vec {b}}|^{2}-|{\\vec {b}}-{\\vec {a}}|^{2})} ã§ããã",
"title": "ãã¯ãã«ã®å
ç©"
},
{
"paragraph_id": 71,
"tag": "p",
"text": "ããã§ã | a â | 2 = a 1 2 + a 2 2 , | b â | 2 = b 1 2 + b 2 2 , | b â â a â | 2 = | ( b 1 â a 1 , b 2 â a 2 ) | 2 = ( b 1 â a 1 ) 2 + ( b 2 â a 2 ) 2 {\\displaystyle |{\\vec {a}}|^{2}=a_{1}^{2}+a_{2}^{2},|{\\vec {b}}|^{2}=b_{1}^{2}+b_{2}^{2},|{\\vec {b}}-{\\vec {a}}|^{2}=|(b_{1}-a_{1},b_{2}-a_{2})|^{2}=(b_{1}-a_{1})^{2}+(b_{2}-a_{2})^{2}} ãªã®ã§ãããã代å
¥ããã°",
"title": "ãã¯ãã«ã®å
ç©"
},
{
"paragraph_id": 72,
"tag": "p",
"text": "a â â
b â = 1 2 ( | a â | 2 + | b â | 2 â | b â â a â | 2 ) {\\displaystyle {\\vec {a}}\\cdot {\\vec {b}}={\\frac {1}{2}}(|{\\vec {a}}|^{2}+|{\\vec {b}}|^{2}-|{\\vec {b}}-{\\vec {a}}|^{2})} = 1 2 [ ( a 1 2 + a 2 2 ) + ( b 1 2 + b 2 2 ) â ( b 1 â a 1 ) 2 + ( b 2 â a 2 ) 2 ] {\\displaystyle ={\\frac {1}{2}}\\left[(a_{1}^{2}+a_{2}^{2})+(b_{1}^{2}+b_{2}^{2})-(b_{1}-a_{1})^{2}+(b_{2}-a_{2})^{2}\\right]} = a 1 b 1 + a 2 b 2 {\\displaystyle =a_{1}b_{1}+a_{2}b_{2}} ã§ããã",
"title": "ãã¯ãã«ã®å
ç©"
},
{
"paragraph_id": 73,
"tag": "p",
"text": "ãããã£ãŠ a â â
b â = a 1 b 1 + a 2 b 2 {\\displaystyle {\\vec {a}}\\cdot {\\vec {b}}=a_{1}b_{1}+a_{2}b_{2}} ãåŸãããã",
"title": "ãã¯ãã«ã®å
ç©"
},
{
"paragraph_id": 74,
"tag": "p",
"text": "å
ç©ã®æ§è³ª â ãã¯ãã« a â , b â , c â {\\displaystyle {\\vec {a}},{\\vec {b}},{\\vec {c}}} ãšå®æ° k {\\displaystyle k} ã«å¯Ÿã以äžãæãç«ã€ã",
"title": "ãã¯ãã«ã®å
ç©"
},
{
"paragraph_id": 75,
"tag": "p",
"text": "ãããã¯ãã¯ãã«ãæå衚瀺ããŠèšç®ããã°èšŒæã§ããã",
"title": "ãã¯ãã«ã®å
ç©"
},
{
"paragraph_id": 76,
"tag": "p",
"text": "蚌æ â a â = ( a 1 , a 2 ) , b â = ( b 1 , b 2 ) , c â = ( c 1 , c 2 ) {\\displaystyle {\\vec {a}}=(a_{1},a_{2}),{\\vec {b}}=(b_{1},b_{2}),{\\vec {c}}=(c_{1},c_{2})} ãšããã",
"title": "ãã¯ãã«ã®å
ç©"
},
{
"paragraph_id": 77,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 78,
"tag": "p",
"text": "A ( a â ) , B ( b â ) {\\displaystyle \\mathrm {A} ({\\vec {a}}),\\,\\mathrm {B} ({\\vec {b}})} ãšããã ãã®ãšããç·åOAã1:3ã«åããç¹ãšãç·åOBã5:2ã«åããç¹ããããããA',B'ãšããã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 79,
"tag": "p",
"text": "(1) ãã¯ãã« O A â² â , O B â² â {\\displaystyle {\\vec {OA'}},\\,{\\vec {OB'}}} ããã¯ãã« a â , b â {\\displaystyle {\\vec {a}},\\,{\\vec {b}}} ãçšããŠããããã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 80,
"tag": "p",
"text": "(2) ç·åAB'ãšãBA'ã®äº€ç¹ M ã®äœçœ®ãã¯ãã«ããã¯ãã« a â , b â {\\displaystyle {\\vec {a}},\\,{\\vec {b}}} ãçšããŠããããã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 81,
"tag": "p",
"text": "ãã¯ãã«",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 82,
"tag": "p",
"text": "ãšã ãã¯ãã«",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 83,
"tag": "p",
"text": "ã¯äºãã«1次ç¬ç«ãª2æ¬ã®ãã¯ãã«ãªã®ã§ã ããããçšããŠããããå³åœ¢äžã®ç¹ãè¡šãããã¯ãã§ããã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 84,
"tag": "p",
"text": "å³åœ¢äžã®ããããã®ç¹ã¯ãç¹Oããã®äœçœ®ãã¯ãã«ã§è¡šãããã äŸãã°ããã¯ãã«",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 85,
"tag": "p",
"text": "ã¯ãç¹OããèŠãŠ",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 86,
"tag": "p",
"text": "ãšå¹³è¡ãªæ¹åã®ãã¯ãã«ã§ããããã®å€§ãããã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 87,
"tag": "p",
"text": "ã§ããã®ã§ã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 88,
"tag": "p",
"text": "ã§è¡šãããã åæ§ã«ããã¯ãã«",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 89,
"tag": "p",
"text": "ã¯ãç¹OããèŠãŠ",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 90,
"tag": "p",
"text": "ãšå¹³è¡ãªæ¹åã®ãã¯ãã«ã§ããããã®å€§ãããã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 91,
"tag": "p",
"text": "ã§ããã®ã§ã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 92,
"tag": "p",
"text": "ã§è¡šãããã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 93,
"tag": "p",
"text": "次ã«ãç¹A'ãééããç·åA'Bã«å¹³è¡ãªçŽç·ã ãã¯ãã«",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 94,
"tag": "p",
"text": "ãš",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 95,
"tag": "p",
"text": "ãçšããŠèšè¿°ããæ¹æ³ãèããã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 96,
"tag": "p",
"text": "ããã§ã¯ã ãã®çŽç·äžã®ç¹ã¯ã ããå®æ° s {\\displaystyle s} ãçšããŠã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 97,
"tag": "p",
"text": "ã§è¡šããããšã«æ³šç®ããã äŸãã°ã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 98,
"tag": "p",
"text": "ã®ãšãããã®åŒãè¡šãç¹ã¯",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 99,
"tag": "p",
"text": "ã«çããã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 100,
"tag": "p",
"text": "ã®ãšãã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 101,
"tag": "p",
"text": "ã«çãããããããçŽç· A'Bäžã®ç¹ã§ããã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 102,
"tag": "p",
"text": "ãããã«å
ã»ã©æ±ãã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 103,
"tag": "p",
"text": "ãšã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 104,
"tag": "p",
"text": "ã®å€ãçšãããšã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 105,
"tag": "p",
"text": "ãåŸãããã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 106,
"tag": "p",
"text": "åæ§ã«ãç·åAB'äžã®ç¹ã¯ããå®æ° t {\\displaystyle t} ãçšããŠã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 107,
"tag": "p",
"text": "ã§è¡šãããã ããã«å
ã»ã©åŸãå€ã代å
¥ãããšã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 108,
"tag": "p",
"text": "ãšãªãã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 109,
"tag": "p",
"text": "ãã®ããã«ããããã®çŽç·äžã®ç¹ã s {\\displaystyle s} , t {\\displaystyle t} ã çšããŠè¡šãããã 次ã«ããããã®åŒãåãç¹ã瀺ãããã« s {\\displaystyle s} , t {\\displaystyle t} ãå®ããã ãã®ããã«ã¯ã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 110,
"tag": "p",
"text": ",",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 111,
"tag": "p",
"text": "ãçãããšãããŠã s {\\displaystyle s} , t {\\displaystyle t} ã«é¢ããé£ç«æ¹çšåŒãäœããããã解ãã°ããã äžã®åŒã§",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 112,
"tag": "p",
"text": "ã®ä¿æ°ãçãããšãããšã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 113,
"tag": "p",
"text": "ãåŸããã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 114,
"tag": "p",
"text": "ã®ä¿æ°ãçãããšãããšã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 115,
"tag": "p",
"text": "ãåŸãããã ãã®åŒãé£ç«ããŠè§£ããšã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 116,
"tag": "p",
"text": ",",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 117,
"tag": "p",
"text": "ãåŸãããã ãã®åŒã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 118,
"tag": "p",
"text": ",",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 119,
"tag": "p",
"text": "ã®ã©ã¡ããã«ä»£å
¥ãããšãæ±ããäœçœ®ãã¯ãã«ãåŸãããã®ã§ããã 代å
¥ãããšãæ±ãããã¯ãã«ã¯ã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 120,
"tag": "p",
"text": "ãšãªãã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 121,
"tag": "p",
"text": "ç¹ A ( a â ) {\\displaystyle \\mathrm {A} ({\\vec {a}})} ãéãããã¯ãã« d â ( â 0 â ) {\\displaystyle {\\vec {d}}\\,(\\neq {\\vec {0}})} ã«å¹³è¡ãªçŽç·ã g {\\displaystyle g} ãšããã g {\\displaystyle g} äžã®ç¹ã P ( p â ) {\\displaystyle \\mathrm {P} ({\\vec {p}})} ãšãããšã A P â = 0 â {\\displaystyle {\\vec {\\mathrm {AP} }}={\\vec {0}}} ãŸã㯠A P â ⥠d â {\\displaystyle {\\vec {\\mathrm {AP} }}\\parallel {\\vec {d}}} ã ãã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 122,
"tag": "p",
"text": "ãšãªãå®æ° t {\\displaystyle t} ãããã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 123,
"tag": "p",
"text": "ããªãã¡ã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 124,
"tag": "p",
"text": "ãã£ãŠã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 125,
"tag": "p",
"text": "ããããçŽç· g {\\displaystyle g} ã®ãã¯ãã«æ¹çšåŒ(vector equation)ãšããã d â {\\displaystyle {\\vec {d}}} ã g {\\displaystyle g} ã®æ¹åãã¯ãã«ãšããããŸãã t {\\displaystyle t} ãåªä»å€æ°ãšããã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 126,
"tag": "p",
"text": "ç¹Aã®åº§æšã ( x 1 , y 1 ) {\\displaystyle (x_{1}\\ ,\\ y_{1})} ã d â = ( a , b ) {\\displaystyle {\\vec {d}}=(a\\ ,\\ b)} ãç¹Pã®åº§æšã ( x , y ) {\\displaystyle (x\\ ,\\ y)} ãšãããšããã¯ãã«æ¹çšåŒ p â = a â + t d â {\\displaystyle {\\vec {p}}={\\vec {a}}+t{\\vec {d}}} ã¯",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 127,
"tag": "p",
"text": "ãšãªãããããã£ãŠ",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 128,
"tag": "p",
"text": "{ x = x 1 + a t y = y 1 + b t {\\displaystyle {\\begin{cases}x=x_{1}+at\\\\y=y_{1}+bt\\end{cases}}}",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 129,
"tag": "p",
"text": "ãããçŽç· g {\\displaystyle g} ã®åªä»å€æ°è¡šç€ºãšããã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 130,
"tag": "p",
"text": "",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 131,
"tag": "p",
"text": "",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 132,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 133,
"tag": "p",
"text": "ç¹A ( 1 , 2 ) {\\displaystyle (1\\ ,\\ 2)} ãéãã d â = ( 3 , 5 ) {\\displaystyle {\\vec {d}}=(3\\ ,\\ 5)} ã«å¹³è¡ãªçŽç·ã®æ¹çšåŒããåªä»å€æ°tãçšããŠè¡šãã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 134,
"tag": "p",
"text": "ãŸããtãæ¶å»ããåŒã§è¡šãã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 135,
"tag": "p",
"text": "ãã®çŽç·ã®ãã¯ãã«æ¹çšåŒã¯",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 136,
"tag": "p",
"text": "ãããã£ãŠ",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 137,
"tag": "p",
"text": "tãæ¶å»ãããšã次ã®ããã«ãªãã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 138,
"tag": "p",
"text": "",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 139,
"tag": "p",
"text": "2ç¹ A ( a â ) , B ( b â ) {\\displaystyle \\mathrm {A} ({\\vec {a}}),\\,\\mathrm {B} ({\\vec {b}})} ãéãçŽç·ã®ãã¯ãã«æ¹çšåŒãèããã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 140,
"tag": "p",
"text": "çŽç·ABã¯ãç¹Aãéãã A B â = b â â a â {\\displaystyle {\\vec {AB}}={\\vec {b}}-{\\vec {a}}} ãæ¹åãã¯ãã«ãšããçŽç·ãšèããããããããã®ãã¯ãã«æ¹çšåŒã¯",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 141,
"tag": "p",
"text": "ãšãªããããã¯æ¬¡ã®ããã«æžããã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 142,
"tag": "p",
"text": "",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 143,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 144,
"tag": "p",
"text": "2ç¹A ( 2 , 5 ) {\\displaystyle (2\\ ,\\ 5)} ,B ( â 1 , 3 ) {\\displaystyle (-1\\ ,\\ 3)} ãéãçŽç·ã®æ¹çšåŒããåªä»å€æ°tãçšããŠè¡šãã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 145,
"tag": "p",
"text": "ãã®çŽç·ã®ãã¯ãã«æ¹çšåŒã¯",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 146,
"tag": "p",
"text": "ãããã£ãŠ",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 147,
"tag": "p",
"text": "",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 148,
"tag": "p",
"text": "ç¹Aãéã£ãŠã 0 â {\\displaystyle {\\vec {0}}} ã§ãªããã¯ãã«ã n â {\\displaystyle {\\vec {n}}} ã«åçŽãªçŽç·ãgãšãããgäžã®ç¹ãPãšãããšã A P â = 0 â {\\displaystyle {\\vec {AP}}={\\vec {0}}} ãŸã㯠A P â ⥠n â {\\displaystyle {\\vec {AP}}\\perp {\\vec {n}}} ã ãã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 149,
"tag": "p",
"text": "ã§ããã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 150,
"tag": "p",
"text": "ç¹A,Pã®äœçœ®ãã¯ãã«ãããããã a â , p â {\\displaystyle {\\vec {a}}\\ ,\\ {\\vec {p}}} ãšãããšã A P â = p â â a â {\\displaystyle {\\vec {AP}}={\\vec {p}}-{\\vec {a}}} ã ããã(1)ã¯",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 151,
"tag": "p",
"text": "ãšãªãã(2)ãç¹Aãéã£ãŠã n â {\\displaystyle {\\vec {n}}} ã«åçŽãªçŽç·gã®ãã¯ãã«æ¹çšåŒã§ããã n â {\\displaystyle {\\vec {n}}} ããã®çŽç·ã®æ³ç·ãã¯ãã«(ã»ããããã¯ãã«ãnormal vector)ãšããã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 152,
"tag": "p",
"text": "ç¹Aã®åº§æšã ( x 1 , y 1 ) {\\displaystyle (x_{1}\\ ,\\ y_{1})} ã n â = ( a , b ) {\\displaystyle {\\vec {n}}=(a\\ ,\\ b)} ãç¹Pã®åº§æšã ( x , y ) {\\displaystyle (x\\ ,\\ y)} ãšãããšã p â â a â = ( x â x 1 , y â y 1 ) {\\displaystyle {\\vec {p}}-{\\vec {a}}=(x-x_{1}\\ ,\\ y-y_{1})} ã ããã(2)ã¯æ¬¡ã®ããã«ãªãã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 153,
"tag": "p",
"text": "ãã®æ¹çšåŒã¯ã â a x 1 â b y 1 = c {\\displaystyle -ax_{1}-by_{1}=c} ãšãããšã a x + b y + c = 0 {\\displaystyle ax+by+c=0} ãšãªãããã次ã®ããšããããã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 154,
"tag": "p",
"text": "çŽç· a x + b y + c = 0 {\\displaystyle ax+by+c=0} ã®æ³ç·ãã¯ãã«ã¯ã n â = ( a , b ) {\\displaystyle {\\vec {n}}=(a\\ ,\\ b)} ã§ããã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 155,
"tag": "p",
"text": "",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 156,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 157,
"tag": "p",
"text": "ç¹A ( 2 , 5 ) {\\displaystyle (2\\ ,\\ 5)} ãéãã n â = ( 4 , 3 ) {\\displaystyle {\\vec {n}}=(4\\ ,\\ 3)} ã«åçŽãªçŽç·ã®æ¹çšåŒãæ±ããã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 158,
"tag": "p",
"text": "ã€ãŸã",
"title": "ãã¯ãã«æ¹çšåŒ"
},
{
"paragraph_id": 159,
"tag": "p",
"text": "ãããŸã§ã¯ãå¹³é¢äžã®ãã¯ãã«ã«ã€ããŠèããŠããããããããã¯3次å
空éäžã®ãã¯ãã«ã«ã€ããŠèãããããäžè¬ã«ãã¯ãã«ã¯n次å
(ãŠãŒã¯ãªãã)空éäžã§å®çŸ©ããããšãã§ãããããã®ãããªãã®ã¯é«æ ¡ã§ã¯æ±ããªãã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 160,
"tag": "p",
"text": "ä»ãŸã§ã¯ãå¹³é¢äžã®å³åœ¢ããã¯ãã«ãæ°åŒãçšããŠè¡šçŸããæ¹æ³ãåŠãã§æ¥ãã ããã§ãã2次å
ãšã¯ãå¹³é¢ã®ããšã§ãããå¹³é¢äžã®ä»»æã®ç¹ãæå®ããã«ã¯æäœã§ã2以äžã®å®æ°ãå¿
èŠã ãããã®ããã«åŒã°ããŠããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 161,
"tag": "p",
"text": "ãã¡ãã容æã«åããéãã2ã€ä»¥äžã®æ¬¡å
ãæã£ãŠããå³åœ¢ãååšããã äŸãã°ã3次å
ç«äœã®1ã€ã§ããçŽæ¹äœã¯çžŠã暪ãé«ãã®3ã€ã®é·ããæã£ãŠããã®ã§ã3次å
å³åœ¢ãšåŒã°ããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 162,
"tag": "p",
"text": "空éã«1ã€ã®å¹³é¢ããšãããã®äžã«çŽäº€ãã座æšè»ž O x , O y {\\displaystyle O_{x}\\ ,\\ O_{y}} ããšãã次ã«Oãéããã®å¹³é¢ã«åçŽãªçŽç· O z {\\displaystyle O_{z}} ãã²ãããã®çŽç·äžã§ãOãåç¹ãšãã座æšãèããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 163,
"tag": "p",
"text": "ãã®3çŽç· O x , O y , O z {\\displaystyle O_{x}\\ ,\\ O_{y}\\ ,\\ O_{z}} ã¯ãã©ã®2ã€ãäºãã«åçŽã§ããããããã座æšè»žãšãããããããx軞ãy軞ãz軞ãšããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 164,
"tag": "p",
"text": "ãŸããx軞ãšy軞ãšã§å®ãŸãå¹³é¢ãxyå¹³é¢ãšãããy軞ãšz軞ãšã§å®ãŸãå¹³é¢ãyzå¹³é¢ãšãããz軞ãšx軞ãšã§å®ãŸãå¹³é¢ãzxå¹³é¢ãšããããããã座æšå¹³é¢ãšããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 165,
"tag": "p",
"text": "空éå
ã®ç¹Aã«å¯ŸããŠãAãéã£ãŠå座æšå¹³é¢ã«å¹³è¡ãª3ã€ã®å¹³é¢ãã€ãããããããx軞ãy軞ãz軞ãšäº€ããç¹ã A 1 , A 2 , A 3 {\\displaystyle A_{1}\\ ,\\ A_{2}\\ ,\\ A_{3}} ãšãã A 1 , A 2 , A 3 {\\displaystyle A_{1}\\ ,\\ A_{2}\\ ,\\ A_{3}} ã®ããããã®è»žäžã§ã®åº§æšã a 1 , a 2 , a 3 {\\displaystyle a_{1}\\ ,\\ a_{2}\\ ,\\ a_{3}} ãšããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 166,
"tag": "p",
"text": "ãã®ãšãã3ã€ã®æ°ã®çµ",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 167,
"tag": "p",
"text": "ãç¹Aã®åº§æšãšããã a 1 {\\displaystyle a_{1}} ãx座æšãšããã a 2 {\\displaystyle a_{2}} ãy座æšãšããã a 3 {\\displaystyle a_{3}} ãz座æšãšããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 168,
"tag": "p",
"text": "ãã®ããã«åº§æšã®å®ãããã空éã座æšç©ºéãšåŒã³ãç¹Oã座æšç©ºéã®åç¹ãšããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 169,
"tag": "p",
"text": "ããã§ã¯ãç¹ã«3次å
空éã®å³åœ¢ã«æ³šç®ããã ãŸãã¯ãã¯ãã«ãçšããåã«3次å
空éã®ç©ºéå³åœ¢ããæ°åŒã«ãã£ãŠèšè¿°ããæ¹æ³ãèå¯ããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 170,
"tag": "p",
"text": "2次å
空éã«ãããŠããã£ãšãç°¡åãªå³åœ¢ã¯çŽç·ã§ããããã®åŒã¯äžè¬çã«",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 171,
"tag": "p",
"text": "ã§è¡šããããã ( a {\\displaystyle a} , b {\\displaystyle b} , c {\\displaystyle c} ã¯ä»»æã®å®æ°ã) ãã㧠x {\\displaystyle x} , y {\\displaystyle y} ã¯ã2次å
空éã代衚ãã2ã€ã®ãã©ã¡ãŒã¿ãŒã§ããã3次å
空éãçšãããšãã«ã¯ããããã¯3ã€ã®æåã§è¡šããããããšãæåŸ
ãããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 172,
"tag": "p",
"text": "å®éãã®ãããªåŒã§è¡šããããå³åœ¢ã¯ã3次å
空éã§ãåºæ¬çãªå³åœ¢ã§ãããã€ãŸãã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 173,
"tag": "p",
"text": "ããäžã®åŒã®é¡äŒŒç©ãšããŠåŸãããã ( a {\\displaystyle a} , b {\\displaystyle b} , c {\\displaystyle c} , d {\\displaystyle d} ã¯ä»»æã®å®æ°ã)",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 174,
"tag": "p",
"text": "ãã®ãããªå³åœ¢ã¯ã©ããªå³åœ¢ã«å¯Ÿå¿ããã ããã?",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 175,
"tag": "p",
"text": "å®éã«ã¯ãã®å³åœ¢ãç¹åŸŽã¥ããã®ã¯ãåŸã«åŠã¶3次å
ãã¯ãã«ãçšããã®ããã£ãšãç°¡åã§ããã®ã§ãããã¯åŸã«ãŸããããšã«ããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 176,
"tag": "p",
"text": "ãããããã 1ã€ãã®åŒããåããããšã¯ã3次å
空éã®åº§æšãè¡šãããã©ã¡ãŒã¿ãŒ",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 177,
"tag": "p",
"text": "ã®ãã¡ã«1ã€ã®é¢ä¿",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 178,
"tag": "p",
"text": "ãäžããããšã§ã3次å
空éäžã®å³åœ¢ãæå®ã§ãããšããããšã§ããããã®å Žåã¯ã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 179,
"tag": "p",
"text": "ãçšããŠããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 180,
"tag": "p",
"text": "ãã¯ãã«ã䜿ããªããŠãå³åœ¢ç解éãåŸãããåŒãšããŠã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 181,
"tag": "p",
"text": "ãæããããã ( a {\\displaystyle a} , b {\\displaystyle b} , c {\\displaystyle c} , r {\\displaystyle r} ã¯ä»»æã®å®æ°ã) ãã®åŒã¯ã2次å
ã§ãããšããã®",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 182,
"tag": "p",
"text": "ã®åŒã®é¡äŒŒç©ã§ããã2次å
ã®å Žåã¯ãã®åŒã¯ã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 183,
"tag": "p",
"text": "äžå¿ ( a , b ) {\\displaystyle (a,b)} ååŸ r {\\displaystyle r} ã®åã«å¯Ÿå¿ããŠããã 3次å
ã®ãã®åŒã¯ãçµè«ããããšäžå¿ ( a , b , c ) {\\displaystyle (a,b,c)} ååŸ r {\\displaystyle r} ã®åã«å¯Ÿå¿ããŠããã®ã§ããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 184,
"tag": "p",
"text": "äžã®åŒ",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 185,
"tag": "p",
"text": "ãæºããããç¹ ( x , y , z ) {\\displaystyle (x,y,z)} ãåãããã®ç¹ãšç¹ ( a , b , c ) {\\displaystyle (a,b,c)} ãšã®è·é¢ãèããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 186,
"tag": "p",
"text": "空é座æšã«çœ®ãã x {\\displaystyle x} 軞ã y {\\displaystyle y} 軞ã z {\\displaystyle z} 軞ã¯ããããçŽäº€ããŠããã®ã§ã2ç¹ã®è·é¢ã¯3å¹³æ¹ã®å®çãçšããŠ",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 187,
"tag": "p",
"text": "ã§äžããããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 188,
"tag": "p",
"text": "ããããäžã®åŒããããã§éžãã ç¹ ( x , y , z ) {\\displaystyle (x,y,z)} ã¯ãæ¡ä»¶",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 189,
"tag": "p",
"text": "ãæºãããŠããã®ã§ã2ç¹ã®è·é¢ã¯",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 190,
"tag": "p",
"text": "ã§ããã ( r > 0 {\\displaystyle r>0} ãçšããã)",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 191,
"tag": "p",
"text": "ãã£ãŠãäžã®åŒãæºããç¹ã¯å
šãŠç¹ ( a , b , c ) {\\displaystyle (a,b,c)} ããã®è·é¢ã r {\\displaystyle r} ã§ããç¹ã§ãããããã¯äžå¿ ( a , b , c ) {\\displaystyle (a,b,c)} ååŸ r {\\displaystyle r} ã®åã«ä»ãªããªãã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 192,
"tag": "p",
"text": "",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 193,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 194,
"tag": "p",
"text": "äžå¿",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 195,
"tag": "p",
"text": "ååŸ",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 196,
"tag": "p",
"text": "ã®çã®åŒãæ±ããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 197,
"tag": "p",
"text": "ã«ä»£å
¥ããããšã§ã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 198,
"tag": "p",
"text": "ãæ±ããããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 199,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 200,
"tag": "p",
"text": "ãã©ã®ãã㪠çã«å¯Ÿå¿ãããèšç®ããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 201,
"tag": "p",
"text": "ãã®ãããªæ°åŒãçã«å¯Ÿå¿ãããšãã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 202,
"tag": "p",
"text": "ã®ä¿æ°ã¯å¿
ãçãããªããŠã¯ãªããªããããã§ãªãå Žåã¯ãã®å³åœ¢ã¯æ¥åäœã«å¯Ÿå¿ããã®ã ããããã¯æå°èŠé ã®ç¯å²å€ã§ããã ããã§ã¯äžã®åŒã¯ãã®æ¡ä»¶ãæºãããŠããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 203,
"tag": "p",
"text": "ããã§ã¯ããã®åŒã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 204,
"tag": "p",
"text": "ã®åœ¢ã«æã£ãŠè¡ãããšãéèŠã§ããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 205,
"tag": "p",
"text": "ã®ããããã«ã€ããŠãã®åŒãå¹³æ¹å®æãããšã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 206,
"tag": "p",
"text": "ãåŸãããããã£ãŠãäžã®åŒ",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 207,
"tag": "p",
"text": "ã¯ã äžå¿",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 208,
"tag": "p",
"text": "ãååŸ",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 209,
"tag": "p",
"text": "ã®çã«å¯Ÿå¿ããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 210,
"tag": "p",
"text": "次ã«3次å
空éäžã«ããããã¯ãã«ãèå¯ããã 2次å
空éäžã§ã¯ãã¯ãã«ã¯2ã€ã®éã®çµã¿åããã§è¡šããããã ããã¯1ã€ã®ãã¯ãã«ã¯x軞æ¹åã«å¯Ÿå¿ããéãšy軞æ¹åã«å¯Ÿå¿ããéã®2ã€ãæã£ãŠããå¿
èŠããã£ãããã§ããã ãã®ããšããã3次å
空éã®ãã¯ãã«ã¯3ã€ã®éã®çµã¿åããã§æžããããšãäºæ³ãããã ç¹ã« x {\\displaystyle x} 軞æ¹åã®æå a {\\displaystyle a} , y {\\displaystyle y} 軞æ¹åã®æå b {\\displaystyle b} , z {\\displaystyle z} 軞æ¹åã®æå c {\\displaystyle c} ( a {\\displaystyle a} , b {\\displaystyle b} , c {\\displaystyle c} ã¯ä»»æã®å®æ°ã) ã§è¡šãããããã¯ãã«ãã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 211,
"tag": "p",
"text": "ãšæžããŠè¡šããããšã«ããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 212,
"tag": "p",
"text": "2次å
å¹³é¢ã§ã¯ ãããã¯ãã«",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 213,
"tag": "p",
"text": "ã¯ã ( a {\\displaystyle a} , b {\\displaystyle b} ã¯ä»»æã®å®æ°ã)",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 214,
"tag": "p",
"text": "ã®2æ¬ã®ãã¯ãã«ãçšããŠã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 215,
"tag": "p",
"text": "ã§è¡šããããã 3次å
空éã§ããã®ãããªèšè¿°æ³ããããäžã§çšãããã¯ãã«",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 216,
"tag": "p",
"text": "ã¯ã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 217,
"tag": "p",
"text": "ãçšããŠ",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 218,
"tag": "p",
"text": "ãšæžããããã¯ãã«ã«å¯Ÿå¿ããŠããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 219,
"tag": "p",
"text": "3次å
ãã¯ãã«ã«å¯ŸããŠã2次å
ãã¯ãã«ã§å®ããå®çŸ©ãæ§è³ªãã»ãŒãã®ãŸãŸæç«ããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 220,
"tag": "p",
"text": "3次å
ãã¯ãã«ã®å æ³ã¯ãããããã®ãã¯ãã«èŠçŽ ãç¬ç«ã«è¶³ãåãããããšã«ãã£ãŠå®çŸ©ããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 221,
"tag": "p",
"text": "ãŸããããããã®ãã¯ãã«ã®èŠçŽ ãå
šãŠçãããã¯ãã«ã\"ãã¯ãã«ãšããŠçãã\"ãšè¡šçŸããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 222,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 223,
"tag": "p",
"text": "ãã¯ãã«ã®å",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 224,
"tag": "p",
"text": "ãèšç®ããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 225,
"tag": "p",
"text": "ãåŸãããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 226,
"tag": "p",
"text": "",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 227,
"tag": "p",
"text": "ãã¯ãã« a â {\\displaystyle {\\vec {a}}} , b â {\\displaystyle {\\vec {b}}} éã®ãã¯ãã«ã®å
ç©ãå¹³é¢ã®å Žåãšåæ§ã«",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 228,
"tag": "p",
"text": "( Ξ {\\displaystyle \\theta } ã¯ããã¯ãã« a â {\\displaystyle {\\vec {a}}} , b â {\\displaystyle {\\vec {b}}} ã®ãªãè§ã)",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 229,
"tag": "p",
"text": "åé
æ³åã1次ç¬ç«ã®æ§è³ªããã®ãŸãŸæãç«ã€ã ãã ãã3次å
空éã®å
šãŠã®ãã¯ãã«ã匵ãã«ã¯ã3ã€ã®ç·åœ¢ç¬ç«ãªãã¯ãã«ãæã£ãŠæ¥ãå¿
èŠãããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 230,
"tag": "p",
"text": "",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 231,
"tag": "p",
"text": "ãã®ããšã®èšŒæã¯ããããç·å代æ°åŠãªã©ã«è©³ããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 232,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 233,
"tag": "p",
"text": "2ã€ã®ãã¯ãã«ã®å
ç©",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 234,
"tag": "p",
"text": "ãèšç®ããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 235,
"tag": "p",
"text": "2次å
ã®å Žåãšåãããã«ããã§ãããããã®èŠçŽ ã¯äºãã«çŽäº€ããåäœãã¯ãã«",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 236,
"tag": "p",
"text": "ã«ãã£ãŠåŒµãããŠããããã®ãã以åãšåããèŠçŽ ããšã®èšç®ãå¯èœã§ããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 237,
"tag": "p",
"text": "ãšãªãã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 238,
"tag": "p",
"text": "ããããã现ããèšç®ãè¡ãªããšã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 239,
"tag": "p",
"text": "ãåŸããããããããã®ãã¯ãã«ã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 240,
"tag": "p",
"text": "ã«åŸã£ãŠå±éãã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 241,
"tag": "p",
"text": "( i {\\displaystyle i} , j {\\displaystyle j} ã¯1,2,3ã®ã©ããã) ã代å
¥ããããšã§äžã®åŒãèšç®ã§ããã¯ãã§ããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 242,
"tag": "p",
"text": "ãããã i {\\displaystyle i} ãš j {\\displaystyle j} ãçãããªããšãã«ã¯",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 243,
"tag": "p",
"text": "ãæãç«ã€ããšãããäžã®å±éããåŸã®9åã®é
ã®ãã¡ã§ã6ã€ã¯",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 244,
"tag": "p",
"text": "ã«çããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 245,
"tag": "p",
"text": "ãŸãã i {\\displaystyle i} ãš j {\\displaystyle j} ãçãããšãã«ã¯",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 246,
"tag": "p",
"text": "ãæãç«ã€ããšãããäžã®åŒ",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 247,
"tag": "p",
"text": "ã®å±éã¯",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 248,
"tag": "p",
"text": "ãšãªã£ãŠç¢ºãã«èŠçŽ ããšã®èšç®ãšäžèŽããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 249,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 250,
"tag": "p",
"text": "2次å
空éã®ãã¯ãã«ã¯2æ¬ã®1次ç¬ç«ãªãã¯ãã«ãããã°ãå¿
ããããã®ç·åœ¢çµåã«ãã£ãŠèšç®ã§ããã¯ãã§ããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 251,
"tag": "p",
"text": "ããã§ã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 252,
"tag": "p",
"text": "ãš",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 253,
"tag": "p",
"text": "ãçšããŠã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 254,
"tag": "p",
"text": "ãã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 255,
"tag": "p",
"text": "ã®åœ¢ã«æžããŠã¿ãã ( c {\\displaystyle c} , d {\\displaystyle d} ã¯ãäœããã®å®æ°ã)",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 256,
"tag": "p",
"text": "2次å
ã®ãã¯ãã«ã®ä¿æ°ãæ±ããåé¡ã§ããã c {\\displaystyle c} , d {\\displaystyle d} ã®æåããã®ãŸãŸçšãããšã c {\\displaystyle c} , d {\\displaystyle d} ã®æºããæ¡ä»¶ã¯",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 257,
"tag": "p",
"text": "ã€ãŸã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 258,
"tag": "p",
"text": "ãšãªãããã㯠c {\\displaystyle c} , d {\\displaystyle d} ã«é¢ããé£ç«1次æ¹çšåŒã§æžãæããããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 259,
"tag": "p",
"text": "ããã解ããšã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 260,
"tag": "p",
"text": "ãåŸãããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 261,
"tag": "p",
"text": "ãã£ãŠã äžã®åŒã¯",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 262,
"tag": "p",
"text": "ãšæžãã確ãã«2æ¬ã®ç·åœ¢ç¬ç«ãªãã¯ãã«ã«ãã£ãŠä»ã®ãã¯ãã«ãæžãè¡šãããããšãåãã£ãã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 263,
"tag": "p",
"text": "",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 264,
"tag": "p",
"text": "ãã®ãããªèšç®ã¯3次å
ãã¯ãã«ã«å¯ŸããŠãå¯èœã§ããããèšç®ææ³ãšããŠ3å
1次é£ç«æ¹çšåŒãæ±ãå¿
èŠããããæå°èŠé ã®ç¯å²å€ã§ãããå®éã®èšç®ææ³ã¯ãç·å代æ°åŠ,ç©çæ°åŠI ç·åœ¢ä»£æ°ãåç
§ã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 265,
"tag": "p",
"text": "",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 266,
"tag": "p",
"text": "ãã®è¡šåŒãçšããŠã以åèŠã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 267,
"tag": "p",
"text": "ã®å³åœ¢ç解éãè¿°ã¹ãã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 268,
"tag": "p",
"text": "ãã®å³åœ¢äžã®ä»»æã®ç¹ã ( x , y , z ) {\\displaystyle (x,y,z)} ã§è¡šããã ãã®ç¹ã¯åç¹Oã«å¯Ÿããäœçœ®ãã¯ãã«ãçšãããš ( x , y , z ) {\\displaystyle (x,y,z)} ã§äžããããã 䟿å®ã®ããã« ãã®ãã¯ãã«ã x â {\\displaystyle {\\vec {x}}} ãšæžãããšã«ããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 269,
"tag": "p",
"text": "äžæ¹ããã¯ãã« a â = ( a , b , c ) {\\displaystyle {\\vec {a}}=(a,b,c)} ãçšãããšãäžã®åŒã¯ãã¯ãã«ã®å
ç©ãçšã㊠a â â
x â = d {\\displaystyle {\\vec {a}}\\cdot {\\vec {x}}=d} ã§äžããããã ã€ãŸãããã®åŒã§è¡šããããå³åœ¢ã¯ãããã¯ãã« a â {\\displaystyle {\\vec {a}}} ãšã®å
ç©ãäžå®ã«ä¿ã€å³åœ¢ã§ããã ãã®å³åœ¢ã¯ãå®éã«ã¯ a â {\\displaystyle {\\vec {a}}} ã«çŽäº€ããå¹³é¢ã§äžããããã ãªããªããã®ãããªå¹³é¢äžã®ç¹ã¯ãå¿
ãå¹³é¢äžã®ããäžç¹ã®äœçœ®ãã¯ãã«ã«å ããŠã ãã¯ãã« a â {\\displaystyle {\\vec {a}}} ã«çŽäº€ãããã¯ãã«ãå ãããã®ã§æžãããšãåºæ¥ãã ãããã ãã¯ãã« a â {\\displaystyle {\\vec {a}}} ã«çŽäº€ãããã¯ãã«ãš ãã¯ãã« a â {\\displaystyle {\\vec {a}}} ã®å
ç©ã¯å¿
ã0ã§ããã®ã§ã ãã®ãããªç¹ã®éå㯠ãã¯ãã« a â {\\displaystyle {\\vec {a}}} ãšäžå®ã®å
ç©ãæã€ã®ã§ããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 270,
"tag": "p",
"text": "ãã£ãŠå
ã®åŒ",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 271,
"tag": "p",
"text": "ã¯ã ãã¯ãã« a â = ( a , b , c ) {\\displaystyle {\\vec {a}}=(a,b,c)} ã«çŽäº€ããå¹³é¢ã«å¯Ÿå¿ããããšãåãã£ãã 次㫠d {\\displaystyle d} ããå³åœ¢ãè¡šããå¹³é¢ãšãåç¹ãšã®è·é¢ã«é¢ä¿ãããããšã瀺ãã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 272,
"tag": "p",
"text": "ç¹ã«ããã¯ãã« a â {\\displaystyle {\\vec {a}}} ã«æ¯äŸããäœçœ®ãã¯ãã«ãæã€ç¹ x â {\\displaystyle {\\vec {x}}} ãèããããã®ãšããã®ç¹ãšåç¹ãšã®è·é¢ã¯ã å¹³é¢",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 273,
"tag": "p",
"text": "ãšåç¹ãšã®è·é¢ã«å¯Ÿå¿ããã ãªããªããäœçœ®ãã¯ãã« x â {\\displaystyle {\\vec {x}}} ã¯ãåç¹ããå¹³é¢",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 274,
"tag": "p",
"text": "ã«åçŽã«äžãããç·ã«å¯Ÿå¿ããããã§ããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 275,
"tag": "p",
"text": "ãã®ããšãã仮㫠a â {\\displaystyle {\\vec {a}}} æ¹åã®åäœãã¯ãã«ã n â {\\displaystyle {\\vec {n}}} ãšæžããå¹³é¢ãšåç¹ãšã®è·é¢ã m {\\displaystyle m} ãšæžããšã x â = m n â {\\displaystyle {\\vec {x}}=m{\\vec {n}}} ãåŸãããã ãã®åŒã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 276,
"tag": "p",
"text": "ã«ä»£å
¥ãããšã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 277,
"tag": "p",
"text": "ãåŸãããããã£ãŠã d {\\displaystyle d} ã¯ã å¹³é¢ãšåç¹ã®è·é¢ m {\\displaystyle m} ãšãã¯ãã« a â {\\displaystyle {\\vec {a}}} ã®å€§ãããããããã®ã§ããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 278,
"tag": "p",
"text": "",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 279,
"tag": "p",
"text": "",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 280,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 281,
"tag": "p",
"text": "ç¹ã«ãã¯ãã«",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 282,
"tag": "p",
"text": "ãåããšãã©ã®ãããªåŒãåŸãããŠããã®åŒã¯ ã©ã®ãããªå³åœ¢ã«å¯Ÿå¿ãããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 283,
"tag": "p",
"text": "ãã®ãšã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 284,
"tag": "p",
"text": "ã¯ã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 285,
"tag": "p",
"text": "ã«å¯Ÿå¿ããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 286,
"tag": "p",
"text": "ãã®åŒã¯ z {\\displaystyle z} 座æšã d {\\displaystyle d} ã«å¯Ÿå¿ãããã以å€ã® x {\\displaystyle x} , y {\\displaystyle y} 座æšãä»»æã«åããã å¹³é¢ã«å¯Ÿå¿ããŠãããããã㯠x y {\\displaystyle xy} å¹³é¢ã«å¹³è¡ã§ããã x y {\\displaystyle xy} å¹³é¢ããã®è·é¢ã d {\\displaystyle d} ã§ããå¹³é¢ã§ããã ãŸãã x y {\\displaystyle xy} å¹³é¢ãšãã¯ãã«",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 287,
"tag": "p",
"text": "ã¯çŽäº€ããŠããã®ã§ããã®ããšããããã®åŒã¯æ£ããã",
"title": "空é座æšãšãã¯ãã«"
},
{
"paragraph_id": 288,
"tag": "p",
"text": "å€ç©ã¯é«æ ¡æ°åŠç¯å²å€ã§å
¥è©Šã«ã¯åºãªãããå€ç©ã¯æ°åŠãç©çãªã©ã«å¿çšã§ãã䟿å©ãªã®ã§ããã§æ±ãã",
"title": "çºå±:å€ç©"
},
{
"paragraph_id": 289,
"tag": "p",
"text": "äžæ¬¡å
ãã¯ãã« a â , b â {\\displaystyle {\\vec {a}},\\,{\\vec {b}}} ã«å¯Ÿããå€ç© a â à b â {\\displaystyle {\\vec {a}}\\times {\\vec {b}}} ã次ãæºãããã®ãšããã",
"title": "çºå±:å€ç©"
},
{
"paragraph_id": 290,
"tag": "p",
"text": "次ã«å€ç©ã®æå衚瀺ãèããŠã¿ããããã®å®çŸ©ããæå衚瀺ãçŽæ¥å°ãã®ã¯é¢åãªã®ã§ã倩äžãçã«æå衚瀺ãäžããŠããããããå€ç©ã®å®çŸ©ãæºããããšã確èªããã",
"title": "çºå±:å€ç©"
},
{
"paragraph_id": 291,
"tag": "p",
"text": "a â = ( a 1 a 2 a 3 ) {\\displaystyle {\\vec {a}}={\\begin{pmatrix}a_{1}\\\\a_{2}\\\\a_{3}\\end{pmatrix}}} ã b â = ( b 1 b 2 b 3 ) {\\displaystyle {\\vec {b}}={\\begin{pmatrix}b_{1}\\\\b_{2}\\\\b_{3}\\end{pmatrix}}} ãšãããšãã a â à b â = ( a 2 b 3 â a 3 b 2 a 3 b 1 â a 1 b 3 a 1 b 2 â a 2 b 1 ) {\\displaystyle {\\vec {a}}\\times {\\vec {b}}={\\begin{pmatrix}a_{2}b_{3}-a_{3}b_{2}\\\\a_{3}b_{1}-a_{1}b_{3}\\\\a_{1}b_{2}-a_{2}b_{1}\\end{pmatrix}}} ã§ããã",
"title": "çºå±:å€ç©"
},
{
"paragraph_id": 292,
"tag": "p",
"text": "ãŸãã¯ã a â à b â {\\displaystyle {\\vec {a}}\\times {\\vec {b}}} 㯠a â , b â {\\displaystyle {\\vec {a}},\\,{\\vec {b}}} ãããããšåçŽã§ããããšã確èªãããããã¯ã ( a â à b â ) â
a â = 0 {\\displaystyle ({\\vec {a}}\\times {\\vec {b}})\\cdot {\\vec {a}}=0} ãš ( a â Ã b â ) â
b â = 0 {\\displaystyle ({\\vec {a}}\\times {\\vec {b}})\\cdot {\\vec {b}}=0} ã§ããããšãæå衚瀺ã代å
¥ããã°èšŒæã§ããã",
"title": "çºå±:å€ç©"
},
{
"paragraph_id": 293,
"tag": "p",
"text": "次ã«ã | a â à b â | = | a â | | b â | sin Ξ {\\displaystyle |{\\vec {a}}\\times {\\vec {b}}|=|{\\vec {a}}||{\\vec {b}}|\\sin \\theta } ã蚌æããã | a â à b â | 2 = | a â | 2 | b â | 2 sin 2 Ξ = | â a | 2 | b â | 2 ( 1 â cos 2 Ξ ) {\\displaystyle |{\\vec {a}}\\times {\\vec {b}}|^{2}=|{\\vec {a}}|^{2}|{\\vec {b}}|^{2}\\sin ^{2}\\theta ={\\vec {|}}a|^{2}|{\\vec {b}}|^{2}(1-\\cos ^{2}\\theta )} ãããã§ã cos 2 Ξ = ( a â â
b â ) 2 | a â | 2 | b â | 2 {\\displaystyle \\cos ^{2}\\theta ={\\frac {({\\vec {a}}\\cdot {\\vec {b}})^{2}}{|{\\vec {a}}|^{2}|{\\vec {b}}|^{2}}}} ã代å
¥ãã | a â à b â | 2 = | â a | 2 | b â | 2 â ( a â â
b â ) 2 {\\displaystyle |{\\vec {a}}\\times {\\vec {b}}|^{2}={\\vec {|}}a|^{2}|{\\vec {b}}|^{2}-({\\vec {a}}\\cdot {\\vec {b}})^{2}} ãåŸãããã®åŒã«ãæå衚瀺ã代å
¥ããã°ã䞡蟺ãçããããšã確èªã§ããã",
"title": "çºå±:å€ç©"
},
{
"paragraph_id": 294,
"tag": "p",
"text": "æåŸã«ããã¬ãã³ã°ã®å·Šæã®æ³å㧠a â à b â {\\displaystyle {\\vec {a}}\\times {\\vec {b}}} ã¯èŠªæã®æ¹åã§ããããšã確èªããã",
"title": "çºå±:å€ç©"
},
{
"paragraph_id": 295,
"tag": "p",
"text": "a â = ( 1 0 0 ) {\\displaystyle {\\vec {a}}={\\begin{pmatrix}1\\\\0\\\\0\\end{pmatrix}}} ã b â = ( 0 1 0 ) {\\displaystyle {\\vec {b}}={\\begin{pmatrix}0\\\\1\\\\0\\end{pmatrix}}} ã®ãšãã a â à b â = ( 0 0 1 ) {\\displaystyle {\\vec {a}}\\times {\\vec {b}}={\\begin{pmatrix}0\\\\0\\\\1\\end{pmatrix}}} ã§ããããããããäºçªç®ã®æ§è³ªã確èªã§ããã",
"title": "çºå±:å€ç©"
},
{
"paragraph_id": 296,
"tag": "p",
"text": "å€ç©ã®å¿çš",
"title": "çºå±:å€ç©"
},
{
"paragraph_id": 297,
"tag": "p",
"text": "2ã€ã®ãã¯ãã«ã«åçŽãªãã¯ãã«ãæ±ããããšããªã©ã¯ãå€ç©ã®æå衚瀺ããèšç®ããã°ãé¢åãªèšç®ãããªããŠãæ±ããããã",
"title": "çºå±:å€ç©"
},
{
"paragraph_id": 298,
"tag": "p",
"text": "åé¢äœ O A B C {\\displaystyle \\mathrm {OABC} } ã®äœç©ã¯ 1 6 | ( O A â à O B â ) â
O C â | {\\displaystyle {\\frac {1}{6}}|({\\vec {\\mathrm {OA} }}\\times {\\vec {\\mathrm {OB} }})\\cdot {\\vec {\\mathrm {OC} }}|} ã§ããã å®éã 1 6 | ( O A â à O B â ) â
O C â | = 1 3 | 1 2 O A â à O B â | | h | {\\displaystyle {\\frac {1}{6}}|({\\vec {\\mathrm {OA} }}\\times {\\vec {\\mathrm {OB} }})\\cdot {\\vec {\\mathrm {OC} }}|={\\frac {1}{3}}\\left|{\\frac {1}{2}}{\\vec {\\mathrm {OA} }}\\times {\\vec {\\mathrm {OB} }}\\right||h|} ã§ããããã ãã h ã¯ÎABCãåºé¢ãšãããšãã®åé¢äœã®é«ãã§ããã",
"title": "çºå±:å€ç©"
},
{
"paragraph_id": 299,
"tag": "p",
"text": "ãŸããç©çåŠã®ããŒã¬ã³ãåã¯å€ç©ã䜿ããš F â = q v â à B â {\\displaystyle {\\vec {F}}=q{\\vec {v}}\\times {\\vec {B}}} ãšç°¡æœã«è¡šããã",
"title": "çºå±:å€ç©"
},
{
"paragraph_id": 300,
"tag": "p",
"text": "",
"title": "çºå±:å€ç©"
},
{
"paragraph_id": 301,
"tag": "p",
"text": "èŠãæ¹",
"title": "çºå±:å€ç©"
},
{
"paragraph_id": 302,
"tag": "p",
"text": "å³ã®ããã«èŠçŽ ãããåãããã",
"title": "çºå±:å€ç©"
},
{
"paragraph_id": 303,
"tag": "p",
"text": "",
"title": "çºå±:å€ç©"
},
{
"paragraph_id": 304,
"tag": "p",
"text": "",
"title": "ã³ã©ã ãªã©"
}
] | çç§ã«ãããŠãåã¯å€§ãããšåããæã€éã§ãããšç¿ã£ãã ããã倧ãããšåããæã€éã¯ãåã®ä»ã«ããé床ã颚ã®å¹ãæ¹ãªã©ãããã äŸãã°ãããå°ç¹ããæå»ã«ããã颚ã®å¹ãæ¹ã¯ã颚éãšé¢šåããæãç«ã€ããã®ããã«ã倧ãããšåããæã€éãå°å
¥ãããšãããããå¹çããæ±ããã ãã®ããŒãžã§ã¯ã倧ãããšåããæã€éã§ãããã¯ãã«ãæ±ãã ãŸããå³åœ¢ã®åé¡ã«å¯ŸããŠä»£æ°çãªã¢ãããŒããåããã®ããã¯ãã«ã®å©ç¹ã®äžã€ã§ããã | {{pathnav|frame=1|ã¡ã€ã³ããŒãž|é«çåŠæ ¡ã®åŠç¿|é«çåŠæ ¡æ°åŠ|é«çåŠæ ¡æ°åŠC|pagename=ãã¯ãã«|small=1}}
çç§ã«ãããŠãåã¯å€§ãããšåããæã€éã§ãããšç¿ã£ãã ããã倧ãããšåããæã€éã¯ãåã®ä»ã«ããé床ã颚ã®å¹ãæ¹ãªã©ãããã
äŸãã°ãããå°ç¹ããæå»ã«ããã颚ã®å¹ãæ¹ã¯ã颚éãšé¢šåããæãç«ã€ããã®ããã«ã倧ãããšåããæã€éãå°å
¥ãããšãããããå¹çããæ±ããã
ãã®ããŒãžã§ã¯ã倧ãããšåããæã€éã§ãã'''ãã¯ãã«'''ãæ±ãã
ãŸããå³åœ¢ã®åé¡ã«å¯ŸããŠä»£æ°çãªã¢ãããŒããåããã®ããã¯ãã«ã®å©ç¹ã®äžã€ã§ããã
==å¹³é¢äžã®ãã¯ãã«==
[[ãã¡ã€ã«:SameVectors.png|ãµã ãã€ã«]]
å¹³é¢äžã®ç¹ <math>\mathrm{S}</math> ããç¹ <math>\mathrm{T}</math> ãžåããç¢å°ãèããããã®ãããªç¢å°ã®ããã«åããæã€ç·åã'''æåç·å'''ãšããã
ãã®ãšããç¹ <math>\mathrm{S}</math> ã'''å§ç¹'''ãç¹ <math>\mathrm{T}</math> ã'''çµç¹'''ãšããã
æåç·åã§ã倧ãããšæ¹åãåããã®ã¯ãã¯ãã«ãšããŠåããã®ãšããã
æåç·åã¯äœçœ®ãé·ãïŒå€§ããïŒãåããšããæ
å ±ãæã€ããã¯ãã«ã¯ãæåç·åã®æã€æ
å ±ã®ãã¡ã'''äœçœ®'''ã®æ
å ±ãå¿ããŠã'''倧ãã'''ã'''åã'''ã ãã«çç®ãããã®ãšèããããšãã§ããã
æåç·å <math>\mathrm{ST}</math> ã§è¡šããããã¯ãã«ã <math>\mathrm{\vec{ST}}</math> ãšããããã¯ãã«ã¯äžæå㧠<math>\vec a</math> ãªã©ãšè¡šãããããšããã<ref>ãŸãã¯ã倪æå㧠<math>\bold a</math> ãªã©ãšè¡šèšãããããšããããããããæ¥æ¬ã®é«çåŠæ ¡ã倧åŠå
¥è©Šã§ã¯ <math>\vec \cdot</math> ãã»ãšãã©ã§ããã</ref>ããã¯ãã« <math>\vec a</math> ã®å€§ããã <math>|\vec a|</math> ã§è¡šãã
æåç·å <math>\mathrm{ST}</math>ãæåç·å <math>\mathrm{S'T'}</math> ã«å¯Ÿãã倧ãããçãããåããçãããªããäœçœ®ãéã£ãŠããŠãããã¯ãã«ãšããŠçããã<math>\mathrm{\vec{ST}} = \mathrm{\vec{S'T'}}</math> ã§ããã<ref>ãã¯ãã«ãšããŠçãããŠããæåç·åãšããŠçãããšã¯éããªã</ref>
倧ããã 1 ã§ãããã¯ãã«ã'''åäœãã¯ãã«'''ãšããã
[[ãã¡ã€ã«:Vector-negation.png|ãµã ãã€ã«|ãã¯ãã« <math>\vec A</math> ã®éãã¯ãã«]]
ãã¯ãã« <math>\vec a</math> ã«å¯Ÿãããã¯ãã« <math>\vec a</math> ãšæ¹åã'''é'''ã§ã倧ãããçãããã¯ãã«ã'''éãã¯ãã«'''ãšããã<math>-\vec a</math> ãšããã
å§ç¹ãšçµç¹ãçãããã¯ãã«ã'''é¶ãã¯ãã«'''ãšããã<math>\vec 0 </math> ã§è¡šããä»»æã®ç¹ <math>\mathrm{A}</math> ã«å¯Ÿãã<math>\mathrm{\vec{AA}} = \vec 0</math> ã§ããããŒããã¯ãã«ã®å€§ãã㯠0 ã§ãåãã¯èããªããã®ãšããã
=== ãã¯ãã«ã®å æ³ ===
[[ãã¡ã€ã«:Vector addition explain.svg|ãµã ãã€ã«|ãã¯ãã«ã®å]]
ãã¯ãã« <math>\vec a, \vec b</math> ã«å¯Ÿãã<math>\vec a = \mathrm{\vec{AB}}, \vec b = \mathrm{\vec{BC}}</math> ãšãªãç¹ããšãããã®ãšããã¯ãã«ã®å æ³ã <math>\vec a + \vec b = \mathrm{\vec{AC}}</math> ã§å®ããã
ãã¯ãã«ã®å æ³ã«ã€ããŠä»¥äžãæãç«ã€ã
* <math>\vec a + \vec b = \vec b + \vec a</math>
* <math>(\vec a + \vec b) + \vec c = \vec a +(\vec b + \vec c)</math>
[[ãã¡ã€ã«:Vector commutative.svg|ãµã ãã€ã«|ãã¯ãã«ã®å æ³ã¯å¯æã§ãã]]
ãŸãã<math>\vec a + \vec 0 = \vec a</math> ãšããã
=== ãã¯ãã«ã®æžæ³ ===
ãã¯ãã« <math>\vec a, \vec b</math> ã«å¯Ÿãã <math>\vec a - \vec b = \vec a+ (-\vec b)</math> ãšããã
[[ãã¡ã€ã«:Vector's subtraction.svg|ãµã ãã€ã«|ãã¯ãã«ã®æžæ³]]
=== ãã¯ãã«ã®å®æ°å ===
ãŒããã¯ãã«ã¯ãªããã¯ãã« <math>\vec a</math> ãšå®æ° <math>k</math> ã«å¯Ÿãããã¯ãã«ã®å®æ°å <math>k\vec a</math> ã以äžã®ããã«å®ããã
# <math>k > 0</math> ã®ãšãããã¯ãã« <math>\vec a</math> ãšæ¹åãåãã§ã倧ããã <math>k</math> åããããã¯ãã«
# <math>k = 0</math> ã®ãšãããŒããã¯ãã« <math>\vec 0</math>
# <math>k < 0</math> ã®ãšããéãã¯ãã« <math>-\vec a</math> ãšæ¹åãåãã§ã倧ããã <math>k</math> åããããã¯ãã«
ãŸããŒããã¯ãã« <math>\vec 0</math> ã«å¯Ÿããå®æ°åã <math>k\vec 0 = \vec 0</math> ã§å®ããã
以äžã®æ§è³ªããªããã€ã
* <math>(k+l)\vec a = k\vec a + l\vec a</math>
* <math>k(\vec a + \vec b) = k\vec a + k\vec b</math>
* <math>(kl)\vec a = k(l\vec a)</math>
== ãã¯ãã«ã®å¹³è¡ã»åçŽ ==
ãŒããã¯ãã«ã§ã¯ãªããã¯ãã« <math>\vec a, \vec b \, (\neq \vec 0)</math> ã«å¯Ÿãã<math>\vec a = \vec{\mathrm{AA'}}, \vec b = \vec{\mathrm{BB'}}</math> ãšãªãç¹ããšãã
ãã®ãšããçŽç· <math>\mathrm{AA'}</math> ãšçŽç· <math>\mathrm{BB'}</math> ãå¹³è¡ã§ãããšãããã¯ãã« <math>\vec a, \vec b</math> ã¯å¹³è¡ã§ãããšããã <math>\vec a \parallel \vec b</math> ã§è¡šãã
ãŸããçŽç· <math>\mathrm{AA'}</math> ãšçŽç· <math>\mathrm{BB'}</math> ãåçŽã§ãããšãããã¯ãã« <math>\vec a, \vec b</math> ã¯åçŽã§ãããšããã<math>\vec a \perp \vec b</math> ã§è¡šãã
ãã¯ãã« <math>\vec a, \vec b</math> ãå¹³è¡ã®ãšããæããã«ãçæ¹ã®ãã¯ãã«ãå®æ°åããã°å€§ãããšåããäžèŽããã®ã§ã
<math>\vec a \parallel \vec b \iff \vec b = k\vec a</math> ãšãªãå®æ° <math>k</math> ãååšãã
ãæãç«ã€ã[[ãã¡ã€ã«:Scalar multiplication of vectors.png|ãµã ãã€ã«|337x337ãã¯ã»ã«|ãã¯ãã«ã®å®æ°å]]
== ãã¯ãã«ã®å解 ==
ãã¯ãã« <math>\vec a, \vec b</math> ããšãã«ãŒããã¯ãã«ã§ãªã(<math>\vec a, \vec b \neq \vec 0</math>) ãå¹³è¡ã§ãªããšããä»»æã®ãã¯ãã« <math>\vec p</math> ã«å¯ŸããŠã <math>\vec p = s\vec a + t \vec b</math> ãšãªãå®æ° <math>s,t</math> ãåãããšãã§ããã
'''蚌æ'''<!-- å³ -->
<math>\vec a = \vec{\mathrm{OA}},\vec b = \vec{\mathrm{OB}},\vec p = \vec{\mathrm{OP}}</math> ãšãªãç¹ããšããç¹ <math>\mathrm{P}</math> ãéããçŽç· <math>\mathrm{OB},\mathrm{OA}</math> ã«å¹³è¡ãªçŽç·ãããããã çŽç· <math>\mathrm{OA},\mathrm{OB}</math> ãšäº€ããç¹ããããã <math>\mathrm{S,T}</math> ãšçœ®ãã
ãã®ãšãã <math>\vec \mathrm{OS} = s\vec a,\vec \mathrm{OT} = t\vec b</math> ãšãªãå®æ° <math>s,t</math> ãåãããšãã§ãããããã§ãåè§åœ¢ <math>\mathrm{OSPT}</math> ã¯å¹³è¡å蟺圢ãªã®ã§ã <math>\vec p = s\vec a + t \vec b</math> ãæãç«ã€ã
== ãã¯ãã«ã®æå衚瀺 ==
ãã¯ãã« <math>\vec a</math> ã«å¯ŸããŠã座æšå¹³é¢äžã®åç¹ã <math>\mathrm O</math> ãšãããšãã<math>\vec a = \mathrm{\vec{OA}}</math> ãšãªãç¹ <math>\mathrm A(a_x,a_y)</math> ãåãããšãã§ãããããã§ã <math>(a_x,a_y)</math> ããã¯ãã« <math>\vec a</math> ã®æå衚瀺ãšãã <math>\vec a = (a_x,a_y)</math>ããŸãã¯ã瞊ã«äžŠã¹ãŠã <math>\vec a = \left(\begin{align}a_x\\a_y\end{align}\right)</math> ãšæžãã
ãã¯ãã« <math>\vec a , \vec b</math> ã«å¯ŸããŠã<math>\vec a = \mathrm{\vec{OA}},\, \vec b = \mathrm{\vec{OB}}</math> ãšãªãç¹ <math>\mathrm{A},\mathrm{B}</math> ããšãã<math>\vec a = (a_x,a_y),\, \vec b = (b_x,b_y)</math> ãšãããšã
<math>\vec a = \vec b \iff \vec{\mathrm{OA}} = \vec{\mathrm{OB}} \iff </math>ç¹ <math>\mathrm A ,\, \mathrm B</math> ãäžèŽãã <math>\iff a_x = b_x </math> ã〠<math>a_y = b_y</math>
ãŸãã <math>\vec a = (a_x, a_y)</math> ã«å¯ŸããŠã<math>\vec a = \mathrm{\vec{OA}}</math> ãšãããšãã <math>|\vec a|</math> ã¯ç·å <math>\mathrm{OA}</math> ã®é·ããªã®ã§ã
<math>|\vec a| = \sqrt{a_x^2 + a_y ^2}</math>
ã§ããã
[[ãã¡ã€ã«:Vector in 2D space.png|ãµã ãã€ã«]]
ãã¯ãã« <math>\vec a = (a_x, a_y) ,\vec b = (b_x,b_y)</math> ã«å¯ŸããŠã
<math>\vec a + \vec b = (a_x + b_x, a_y + b_y)</math>
<math>\vec a - \vec b = (a_x-b_x,a_y-b_y)</math>
<math>k\vec a = (ka_x , ka_y)</math>
ããªããã€ã
==äœçœ®ãã¯ãã«==
ããç¹ãåºæºã«ããŠããã®ç¹ãå§ç¹ãšãããã¯ãã«ã«ã€ããŠèããããšã«ããããã¯ãã«ãçšããŠç¹ã®äœçœ®é¢ä¿ã«ã€ããŠèå¯ããããšãã§ããã
ç¹ã®äœçœ®é¢ä¿åºæºãšãªãç¹ <math>\rm O</math> ããããããå®ããããã®ãšããç¹ <math>\rm A</math> ã«å¯ŸããŠããã¯ãã« <math>\vec{\rm {OA }}</math> ãç¹ <math>\rm A</math> ã®äœçœ®ãã¯ãã«ãšãããäœçœ®ãã¯ãã« <math>\vec{a}</math> ã§äžããããç¹ <math>\rm A</math> ã <math>\mathrm{A}(\vec a)</math> ã§è¡šãã
ãŸããç¹ <math>\mathrm A (\vec a),\,\mathrm B(\vec b)</math> ã®ãšãã<math>\vec{\rm{AB}} = \vec{\rm{OB}} - \vec{\rm{OA}} = \vec b- \vec a</math> ãæãç«ã€ã
=== å
åç¹ã»å€åç¹ã®äœçœ®ãã¯ãã« ===
以äžãäœçœ®ãã¯ãã«ã®åºæºç¹ãç¹ <math>\rm O</math> ãšããã
ç¹ <math>\rm A (\vec a),\,\rm B(\vec b)</math> ãéãç·å <math>\mathrm{AB}</math> ã <math>m:n</math> ã«å
åããç¹ <math>\mathrm{P}(\vec p)</math> ãæ±ããã<!-- å³ -->
<math>\vec{\mathrm{AP}} = \frac{m}{m+n}\vec{\mathrm{AB}}</math> ããã<math>\vec p - \vec a = \frac{m}{m+n}(\vec b - \vec a)</math> ãããã£ãŠã<math>\vec p = \frac{n\vec a + m\vec b}{m+n}</math> ã§ããã<ref><math>\vec p =\frac{m}{m+n}(\vec b - \vec a) + \vec a = \left(1-\frac{m}{m+n}\right)\vec a + \frac{m}{m+n}\vec b = \frac{n\vec a + m\vec b}{m+n} </math></ref>
次ã«ãç¹ <math>\rm A (\vec a),\,\rm B(\vec b)</math> ãéãç·å <math>\mathrm{AB}</math> ã <math>m:n</math> ã«å€åããç¹ <math>\mathrm{Q}(\vec q)</math> ãæ±ããã<!-- å³ -->
<math>m > n</math> ã®å Žåã¯ã <math>\vec{\mathrm{AQ}} = \frac{m}{m-n}\vec{\mathrm{AB}}</math> ããã<math>\vec q - \vec a = \frac{m}{m-n}(\vec b - \vec a) </math> ãããã£ãŠã<math>\vec q = \frac{-n\vec a + m\vec b}{m-n}</math> ã§ããã<ref><math>\vec q = \frac{m}{m-n}(\vec b - \vec a) + \vec a = \left(1-\frac{m}{m-n}\right)\vec a + \frac{m}{m-n}\vec b = \frac{-n\vec a + m\vec b}{m-n} </math></ref>
<math>m < n</math> ã®å Žåã¯ã<math>\vec{\mathrm{BQ}} = \frac{n}{n-m}\vec{\mathrm{BA}}</math> ã«æ³šæããŠåæ§ã«èšç®ããã°ãåãšåãã <math>\vec q = \frac{-n\vec a + m\vec b}{m-n}</math> ãåŸãããã<ref><math>m = n</math> ã®å Žåãã€ãŸãç·åã <math>1:1</math> ã«å€åããç¹ã¯ååšããªãããªããªããä»»æã®ç·åABã«å¯ŸããŠAP:BP=1:1ãšãªãç¹Pã¯ç·åABã®çŽè§äºçåç·äžã«ããããç¹Pãç·åABäžã«ããå Žåãããã¯å
åç¹ã§ãããç¹Pãç·åABäžã«ãªãå Žåãããã¯å€åç¹ã§ã¯ããããªãã</ref>
=== äžè§åœ¢ã®éå¿ã®äœçœ®ãã¯ãã« ===
äžè§åœ¢ <math>\mathrm{ABC}</math> ã«å¯Ÿãã <math>\mathrm{A}(\vec a),\, \mathrm{B}(\vec b),\, \mathrm{C}(\vec c)</math> ãšçœ®ãããã®äžè§åœ¢ <math>\mathrm{ABC}</math> ã®éå¿ <math>\mathrm{G}({\vec g})</math> ãæ±ããã<!-- å³ -->
ç·å <math>\mathrm{BC}</math> ã®äžç¹ã <math>\mathrm{M}(\vec m)</math> ãšãããšãç¹ <math>\mathrm M</math> ã¯ç·å <math>\mathrm{BC}</math> ã <math>1:1</math> ã«å
åããç¹ãªã®ã§ã <math>\vec m = \frac{\vec b + \vec c}{2}</math> ã§ããã
ç¹ <math>\mathrm{G}</math> ã¯ç·å <math>\mathrm{AM}</math> ã <math>2:1</math> ã«å
åããç¹ãªã®ã§ã <math>\vec g = \frac{\vec a + \vec b + \vec c}{3}</math> ã§ããã<ref><math>\vec g = \frac{\vec a + 2\vec m}{2+1} = \frac{\vec a + \vec b + \vec c}{3}</math></ref>
=== äžè§åœ¢ã®å
å¿ã®äœçœ®ãã¯ãã« ===
äžè§åœ¢ <math>\mathrm{ABC}</math> ã«å¯Ÿãã <math>\mathrm{A}(\vec a),\, \mathrm{B}(\vec b),\, \mathrm{C}(\vec c)</math> ãšçœ®ããããã«ã<math>\mathrm{AB} = c,\,\mathrm{BC} = a,\, \mathrm{CA} = b</math> ãšçœ®ããäžè§åœ¢ <math>\mathrm{ABC}</math> ã®å
å¿ã®äœçœ®ãã¯ãã« <math>\mathrm{I}(\vec i)</math> ãæ±ããã<ref>ããã§ãç·åã®é·ããšé ç¹ã®äœçœ®ãã¯ãã«ãåãã¢ã«ãã¡ãããã§çœ®ããŠããããèšå· <math>\vec \bullet</math> ã®ã€ããŠãããã®ã¯ããã¯ãã«ãèšå· <math>\vec \bullet</math> ã®ã€ããŠããªããã®ã¯å®æ°ã§ããããšã«æ³šæããã</ref><!-- å³ -->
<math>\rm A</math> ã®äºçåç·ãšç·å <math>\rm{BC}</math> ã®äº€ç¹ã <math>\mathrm{D}(\vec d)</math> ãšããããã®ãšããäžè§åœ¢ã®äºçåç·ã®æ§è³ªãã<math>\mathrm{BD}:\mathrm{DC} = c:b</math> ãããã£ãŠã<math>\vec d = \frac{b\vec b + c\vec c}{b+c}</math> ã§ããã
ããã§ã<math>\mathrm{AI}:\mathrm{ID} = \mathrm{BA}:\mathrm{BD} = c:\frac{ac}{b+c} = (b+c):a</math><ref><math>\mathrm{BD}:\mathrm{DC} = c:b</math> ãã <math>\mathrm{BD} = \frac{c}{b+c}a</math></ref> ã§ããã
ãããã£ãŠã<math>\vec i = \frac{a\vec a + (b+c)\vec d}{a+b+c} = \frac{a\vec a + b\vec b + c\vec c}{a+b+c}</math> ã§ããã
== ãã¯ãã«ã®å
ç© ==
äžåŠãŸãã¯é«æ ¡ã®çç§ã®ååŠã§ã¯ãååŠçãªä»äºã®å®çŸ©ããªãã£ãããšãããã ããããã®ä»äºã§ã¯ã移åæ¹å以å€ã®åã¯ãä»äºã«å¯äžããªãã£ãããã®ãããªåã®ä»äºã®èšç®ãããã¯ãã«ã®èŠ³ç¹ããã¿ãã°ãå
ç©ãšããæ°ããæŠå¿µãå®çŸ©ã§ããã<ref>[[ç©çæ°åŠI]]ãªã©ãåç
§</ref><ref>ããã¯ãå
âç©âãšããååãã€ããŠããããå®æ°ã®âç©âãšã¯æ§åãéããåçŽã«å®æ°ã®ç©ããã¯ãã«ã«æ¡åŒµãããã®ãå
ç©ãšããããã§ã¯ãªããå®æ°ã®ç©ã¯å®æ°ããå®æ°ãžã®æŒç®ã§ãããããã¯ãã«ã®å
ç©ã¯ãã¯ãã«ããå®æ°ãžã®æŒç®ã§ããã</ref>
ãã¯ãã« <math>
\vec a,\vec b
</math> ã«å¯Ÿãã <math>\vec a = \vec{\mathrm{OA}}, \vec b = \vec{\mathrm{OB}}</math> ãšãªãç¹ <math>\mathrm{O,A,B}</math> ããšãããã®ãšãã <math>\angle \mathrm{AOB}</math> ã'''ãã¯ãã« <math>
\vec a,\vec b
</math> ã®ãªãè§'''ãšããã
(å³)
ãã¯ãã« <math>
\vec a,\vec b
</math> ã®ãªãè§ã <math>\theta</math> ãšãããšããå
ç© <math>\vec a \cdot \vec b</math> ã
:<math>
\vec a \cdot \vec b = |\vec a||\vec b| \cos \theta
</math>
ã§å®ããã<ref>å
ç© <math>\vec a \cdot \vec b</math> ã <math>\vec a \vec b</math> ã <math>\vec a \times \vec b</math> ã®ããã«è¡šèšããŠã¯ãããªãã<math>\vec a \times \vec b</math> ã¯ãã¯ãã«ã®å€ç©ïŒç¯å²å€ïŒãè¡šãã</ref>
å®çŸ©ããããã¯ãã«ã®å
ç©ã¯äžæ¹ã®ãã¯ãã«ãããäžæ¹ã®ãã¯ãã«ã«å°åœ±ãããšãã®ã倧ããã®ç©ã§ãããšèšããã
(å³)
=== æå衚瀺ãããå
ç© ===
ãã¯ãã« <math>
\vec a,\vec b
</math> ã <math>\vec a = (a_1,a_2),\vec b = (b_1,b_2)</math> ãšæå衚瀺ãããšãã®ãå
ç© <math>\vec a \cdot \vec b</math> ã«ã€ããŠèããŠã¿ããã
ãã¯ãã« <math>
\vec a,\vec b
</math> ã«å¯Ÿãã <math>\vec a = \vec{\mathrm{OA}}, \vec b = \vec{\mathrm{OB}}</math> ãšãªãç¹ <math>\mathrm{O,A,B}</math> ããšãããã¯ãã« <math>
\vec a,\vec b
</math> ã®ãªãè§ã <math>\theta </math> ãšããããã®ãšã <math>\triangle \mathrm{OAB}</math> ã«å¯ŸãäœåŒŠå®çãçšããŠ
<math>\mathrm{\mathrm{AB}}^2 = \mathrm{\mathrm{OA}}^2 + \mathrm{\mathrm{OB}}^2 - 2 \cdot \mathrm{\mathrm{OA}} \cdot \mathrm{\mathrm{OB}} \cos \theta </math>
(å³)
ããã§ã <math>\mathrm{\mathrm{AB}} = |\vec b - \vec a|,\mathrm{\mathrm{OA}} = |\vec a|,\mathrm{\mathrm{OB}} = |\vec b|</math> ãšã<math>\mathrm{\mathrm{OA}} \cdot \mathrm{\mathrm{OB}} \cos \theta = |\vec a||\vec b|\cos\theta = \vec a \cdot \vec b</math> ãã
<math>|\vec b - \vec a|^2 = |\vec a|^2 + |\vec b|^2 - 2 \vec a \cdot \vec b</math> ã§ããã®ã§ã <math>\vec a \cdot \vec b = \frac{1}{2}(|\vec a|^2 + |\vec b|^2 - |\vec b - \vec a|^2)</math> ã§ããã
ããã§ã <math>|\vec a|^2 = a_1^2 + a_2^2,|\vec b|^2 = b_1^2 + b_2^2,|\vec b - \vec a|^2 = |(b_1 - a_1, b_2 - a_2)|^2 = (b_1 - a_1)^2 + (b_2 - a_2)^2</math> ãªã®ã§ãããã代å
¥ããã°
<math>\vec a \cdot \vec b = \frac{1}{2}(|\vec a|^2 + |\vec b|^2 - |\vec b - \vec a|^2)</math> <math>= \frac{1}{2}\left[(a_1^2 + a_2^2) + (b_1^2 + b_2^2 )- (b_1 - a_1)^2 + (b_2 - a_2)^2\right] </math> <math>= a_1b_1 + a_2b_2 </math> ã§ããã
ãããã£ãŠ <math>\vec a \cdot \vec b = a_1b_1 + a_2b_2</math> ãåŸãããã
=== å
ç©ã®æ§è³ª ===
{{math_theorem|å
ç©ã®æ§è³ª|ãã¯ãã« <math> {\vec {a}},{\vec {b}},{\vec {c}}</math> ãšå®æ° <math> k</math> ã«å¯Ÿã以äžãæãç«ã€ã
#<math> {\vec {a}}\cdot {\vec {b}}={\vec {b}}\cdot {\vec {a}}</math>
#<math> {\vec {a}}\cdot ({\vec {b}}+{\vec {c}})={\vec {a}}\cdot {\vec {b}}+{\vec {a}}\cdot {\vec {c}}</math>
#<math> (k{\vec {a}})\cdot {\vec {b}}=k({\vec {a}}\cdot {\vec {b}})</math>
#<math> 0\leq {\vec {a}}\cdot {\vec {a}}=|{\vec {a}}|^{2}</math>}}
ãããã¯ãã¯ãã«ãæå衚瀺ããŠèšç®ããã°èšŒæã§ããã
{{Math proof|
<math>\vec a = (a_1,a_2),\vec b = (b_1,b_2),\vec c = (c_1,c_2)</math> ãšããã
#<math>\vec a \cdot \vec b = a_1b_1+a_2b_2 = \vec b \cdot \vec a</math>
#<math>{\vec {a}}\cdot ({\vec {b}}+{\vec {c}})=(a_1,a_2)\cdot(b_1+c_1,b_2+c_2) = (a_1b_1+a_1c_1 )+ (a_2b_2+a_2c_2 ) = {\vec {a}}\cdot {\vec {b}}+{\vec {a}}\cdot {\vec {c}}</math>
#<math>(k{\vec {a}})\cdot {\vec {b}}= (ka_1,ka_2)\cdot (b_1,b_2) =k(a_1b_1+a_2b_2) = k({\vec {a}}\cdot {\vec {b}})</math>
#<math>{\vec {a}}\cdot {\vec {a}} = a_1^2 + a_2^2 = |{\vec {a}}|^{2} \ge 0</math>}}
== ãã¯ãã«æ¹çšåŒ ==
{{æŒç¿åé¡|
<math>\mathrm A (\vec a),\, \mathrm B (\vec b)</math>ãšããã
ãã®ãšããç·åOAã1:3ã«åããç¹ãšãç·åOBã5:2ã«åããç¹ããããããA',B'ãšããã
(1) ãã¯ãã« <math>\vec {OA'},\, \vec {OB'}</math> ããã¯ãã«<math>\vec a,\, \vec b</math>ãçšããŠããããã
(2) ç·åAB'ãšãBA'ã®äº€ç¹ M ã®äœçœ®ãã¯ãã«ããã¯ãã«<math>\vec a,\, \vec b</math>ãçšããŠããããã|
ãã¯ãã«
:<math>
\vec a
</math>
ãšã
ãã¯ãã«
:<math>
\vec b
</math>
ã¯äºãã«1次ç¬ç«ãª2æ¬ã®ãã¯ãã«ãªã®ã§ã
ããããçšããŠããããå³åœ¢äžã®ç¹ãè¡šãããã¯ãã§ããã
å³åœ¢äžã®ããããã®ç¹ã¯ãç¹Oããã®äœçœ®ãã¯ãã«ã§è¡šãããã
äŸãã°ããã¯ãã«
:<math>
\vec {OA'}
</math>
ã¯ãç¹OããèŠãŠ
:<math>
\vec a
</math>
ãšå¹³è¡ãªæ¹åã®ãã¯ãã«ã§ããããã®å€§ãããã
:<math>
\frac 1 4
</math>
ã§ããã®ã§ã
:<math>
\vec {OA'}= \frac 1 4 \vec a
</math>
ã§è¡šãããã
åæ§ã«ããã¯ãã«
:<math>
\vec {OB'}
</math>
ã¯ãç¹OããèŠãŠ
:<math>
\vec b
</math>
ãšå¹³è¡ãªæ¹åã®ãã¯ãã«ã§ããããã®å€§ãããã
:<math>
\frac 2 7
</math>
ã§ããã®ã§ã
:<math>
\vec {OB'}= \frac 2 7 \vec b
</math>
ã§è¡šãããã
次ã«ãç¹A'ãééããç·åA'Bã«å¹³è¡ãªçŽç·ã
ãã¯ãã«
:<math>
\vec a
</math>
ãš
:<math>
\vec b
</math>
ãçšããŠèšè¿°ããæ¹æ³ãèããã
ããã§ã¯ã
ãã®çŽç·äžã®ç¹ã¯ã
ããå®æ°<math>s</math>ãçšããŠã
:<math>
\vec{OA'}
+ s(\vec {A'B})
</math>
ã§è¡šããããšã«æ³šç®ããã
äŸãã°ã
:<math>
s=0
</math>
ã®ãšãããã®åŒãè¡šãç¹ã¯
:<math>
\vec{OA'}
</math>
ã«çããã
:<math>
s = 1
</math>
ã®ãšãã
:<math>
\vec {OB}
</math>
ã«çãããããããçŽç·
A'Bäžã®ç¹ã§ããã
ãããã«å
ã»ã©æ±ãã
:<math>
\vec {OA'}
</math>
ãšã
:<math>
\vec{OB}
</math>
ã®å€ãçšãããšã
:<math>
\vec{OA'}
+ s(\vec {A'B})
</math>
:<math>
= \frac 1 4 \vec a + s(\vec b - \frac 1 4 \vec a)
</math>
:<math>
= \frac 1 4 (1 -s ) \vec a + s \vec b
</math>
ãåŸãããã
åæ§ã«ãç·åAB'äžã®ç¹ã¯ããå®æ°
<math>t</math>ãçšããŠã
:<math>
\vec {OB'} + t\vec{B' A}
</math>
ã§è¡šãããã
ããã«å
ã»ã©åŸãå€ã代å
¥ãããšã
:<math>
\vec {OB'} + t\vec{B' A}
</math>
:<math>
= \frac 2 7 \vec b + t(\vec a - \frac 2 7 \vec b)
</math>
:<math>
=(1-t) \frac 2 7 \vec b + t \vec a
</math>
ãšãªãã
ãã®ããã«ããããã®çŽç·äžã®ç¹ã<math>s</math>,<math>t</math>ã
çšããŠè¡šãããã
次ã«ããããã®åŒãåãç¹ã瀺ãããã«
<math>s</math>,<math>t</math>ãå®ããã
ãã®ããã«ã¯ã
:<math>
\vec{OM}= \frac 1 4 (1 -s ) \vec a + s \vec b
</math>
,
:<math>
\vec{OM}=(1-t) \frac 2 7 \vec b + t \vec a
</math>
ãçãããšãããŠã
<math>s</math>,<math>t</math>ã«é¢ããé£ç«æ¹çšåŒãäœããããã解ãã°ããã
äžã®åŒã§
:<math>
\vec a
</math>
ã®ä¿æ°ãçãããšãããšã
:<math>
\frac 1 4 (1-s) = t
</math>
ãåŸããã
:<math>
\vec b
</math>
ã®ä¿æ°ãçãããšãããšã
:<math>
\frac 2 7 (1-t) = s
</math>
ãåŸãããã
ãã®åŒãé£ç«ããŠè§£ããšã
:<math>
s = \frac 3 {13}
</math>
,
:<math>
t = \frac 5 {26}
</math>
ãåŸãããã
ãã®åŒã
:<math>
\vec{OM}= \frac 1 4 (1 -s ) \vec a + s \vec b
</math>
,
:<math>
\vec{OM} =(1-t) \frac 2 7 \vec b + t \vec a
</math>
ã®ã©ã¡ããã«ä»£å
¥ãããšãæ±ããäœçœ®ãã¯ãã«ãåŸãããã®ã§ããã
代å
¥ãããšãæ±ãããã¯ãã«ã¯ã
:<math>
\vec{OM}= \frac 1 4 (1 -\frac 3 {13} ) \vec a + \frac 3 {13} \vec b
</math>
:<math>
= \frac 5 {26} \vec a + \frac 3 {13} \vec b
</math>
ãšãªãã
:çã
:<math>
\vec{OA'} = \frac 1 4 \vec a
</math>
:<math>
\vec {OB'}= \frac 2 7 \vec b
</math>
:<math>
\vec {OM} = \frac 5 {26} \vec a + \frac 3 {13} \vec b
</math>}}
===== åªä»å€æ°ã䜿ã£ãçŽç·ã®ãã¯ãã«æ¹çšåŒ =====
ç¹ <math>\mathrm{A}(\vec a)</math> ãéãããã¯ãã« <math>\vec {d} \, (\neq \vec 0)</math> ã«å¹³è¡ãªçŽç·ã <math>g</math> ãšããã<math>g</math> äžã®ç¹ã <math>\mathrm{P}(\vec p)</math> ãšãããšã<math>\vec \mathrm{AP} = \vec {0}</math>ãŸãã¯<math>\vec \mathrm{AP} \parallel \vec d</math> ã ãã
:<math>\vec \mathrm{AP} = t \vec {d}</math><!-- å³ -->
ãšãªãå®æ° <math>t</math> ãããã
ããªãã¡ã
:<math>\vec {p} - \vec {a} = t \vec {d}</math>
ãã£ãŠã
:<math>\vec {p} = \vec {a} + t \vec {d}</math>
ããããçŽç· <math>g</math> ã®'''ãã¯ãã«æ¹çšåŒ'''ïŒvector equationïŒãšããã <math>\vec{d}</math> ã <math>g</math> ã®'''æ¹åãã¯ãã«'''ãšããããŸãã<math>t</math> ã{{Ruby|'''åªä»å€æ°'''|ã°ããããžããã}}ãšããã
ç¹Aã®åº§æšã<math>(x_1\ ,\ y_1)</math>ã<math>\vec{d} = (a\ ,\ b)</math>ãç¹Pã®åº§æšã<math>(x\ , \ y)</math>ãšãããšããã¯ãã«æ¹çšåŒ <math>\vec {p} = \vec {a} + t \vec {d}</math> ã¯
:<math>(x\ , \ y) = (x_1\ , \ y_1) + t (a\ , \ b) </math>
ãšãªãããããã£ãŠ
<math>\begin{cases} x = x_1 +at \\ y = y_1 +bt\end{cases}</math>
ãããçŽç· <math>g</math> ã®'''åªä»å€æ°è¡šç€º'''ãšããã
{{æŒç¿åé¡|
ç¹A<math>(1\ ,\ 2)</math>ãéãã<math>\vec{d} = (3\ ,\ 5)</math>ã«å¹³è¡ãªçŽç·ã®æ¹çšåŒããåªä»å€æ°tãçšããŠè¡šãã
ãŸããtãæ¶å»ããåŒã§è¡šãã|
ãã®çŽç·ã®ãã¯ãã«æ¹çšåŒã¯
:<math>(x\ , \ y) = (1\ , \ 2) + t (3\ , \ 5) = (1+3t\ , \ 2+5t)</math>
ãããã£ãŠ
:<math>x = 1+3t\ ,\ y = 2+5t</math>
tãæ¶å»ãããšã次ã®ããã«ãªãã
:<math>5x-3y+1=0</math>}}
2ç¹ <math>\mathrm{A}(\vec a),\, \mathrm{B}(\vec b)</math> ãéãçŽç·ã®ãã¯ãã«æ¹çšåŒãèããã
çŽç·ABã¯ãç¹Aãéãã<math>\vec{AB} = \vec{b} - \vec{a}</math>ãæ¹åãã¯ãã«ãšããçŽç·ãšèããããããããã®ãã¯ãã«æ¹çšåŒã¯
:<math>\vec {p} = \vec {a} + t \left(\vec{b} - \vec{a} \right)</math>
ãšãªããããã¯æ¬¡ã®ããã«æžããã
:<math>\vec {p} = (1-t) \vec {a} + t \vec{b}</math>
{{æŒç¿åé¡|
2ç¹A<math>(2\ ,\ 5)</math>ïŒB<math>(-1\ ,\ 3)</math>ãéãçŽç·ã®æ¹çšåŒããåªä»å€æ°tãçšããŠè¡šãã|
ãã®çŽç·ã®ãã¯ãã«æ¹çšåŒã¯
:<math>(x\ , \ y) = (1-t)(2\ , \ 5) + t (-1\ , \ 3) = (2-3t\ , \ 5-2t)</math>
ãããã£ãŠ
:<math>x = 2-3t\ ,\ y = 5-2t</math>}}
===== å
ç©ã䜿ã£ãçŽç·ã®ãã¯ãã«æ¹çšåŒ =====
ç¹Aãéã£ãŠã<math>\vec {0}</math>ã§ãªããã¯ãã«ã<math>\vec {n}</math>ã«åçŽãªçŽç·ãgãšãããgäžã®ç¹ãPãšãããšã<math>\vec {AP} = \vec {0}</math>ãŸãã¯<math>\vec {AP} \perp \vec {n}</math>ã ãã
:<math>\vec {AP} \cdot \vec {n} =0</math>âŠ(1)
ã§ããã
ç¹A,Pã®äœçœ®ãã¯ãã«ãããããã<math>\vec{a}\ ,\ \vec{p}</math>ãšãããšã<math>\vec {AP} = \vec {p} - \vec {a}</math>ã ããã(1)ã¯
:<math>\vec {n} \cdot (\vec {p} - \vec {a}) = 0</math>âŠ(2)
ãšãªãã(2)ãç¹Aãéã£ãŠã<math>\vec {n}</math>ã«åçŽãªçŽç·gã®ãã¯ãã«æ¹çšåŒã§ããã<math>\vec{n}</math>ããã®çŽç·ã®'''æ³ç·ãã¯ãã«'''ïŒã»ããããã¯ãã«ãnormal vectorïŒãšããã
ç¹Aã®åº§æšã<math>(x_1\ ,\ y_1)</math>ã<math>\vec{n} = (a\ ,\ b)</math>ãç¹Pã®åº§æšã<math>(x\ , \ y)</math>ãšãããšã<math>\vec {p} - \vec {a} = (x-x_1\ , \ y-y_1)</math>ã ããã(2)ã¯æ¬¡ã®ããã«ãªãã
<center><math>a(x-x_1)+b(y-y_1)=0</math></center>
ãã®æ¹çšåŒã¯ã<math>-ax_1-by_1=c</math>ãšãããšã<math>ax+by+c=0</math>ãšãªãããã次ã®ããšããããã
'''çŽç·<math>ax+by+c=0</math>ã®æ³ç·ãã¯ãã«ã¯ã<math>\vec{n} = (a\ ,\ b)</math>ã§ããã'''
{{æŒç¿åé¡|
ç¹A<math>(2\ ,\ 5)</math>ãéãã<math>\vec{n} = (4\ ,\ 3)</math>ã«åçŽãªçŽç·ã®æ¹çšåŒãæ±ããã|:<math>4(x-2)+3(y-5)=0</math>
ã€ãŸã
:<math>4x+3y-23=0</math>}}
==空é座æšãšãã¯ãã«==
ãããŸã§ã¯ãå¹³é¢äžã®ãã¯ãã«ã«ã€ããŠèããŠããããããããã¯ïŒæ¬¡å
空éäžã®ãã¯ãã«ã«ã€ããŠèãããããäžè¬ã«ãã¯ãã«ã¯n次å
(ãŠãŒã¯ãªãã)空éäžã§å®çŸ©ããããšãã§ãããããã®ãããªãã®ã¯é«æ ¡ã§ã¯æ±ããªãã
=====空éåº§æš =====
ä»ãŸã§ã¯ãå¹³é¢äžã®å³åœ¢ããã¯ãã«ãæ°åŒãçšããŠè¡šçŸããæ¹æ³ãåŠãã§æ¥ãã
ããã§ãã2次å
ãšã¯ãå¹³é¢ã®ããšã§ãããå¹³é¢äžã®ä»»æã®ç¹ãæå®ããã«ã¯æäœã§ã2以äžã®å®æ°ãå¿
èŠã ãããã®ããã«åŒã°ããŠããã
ãã¡ãã容æã«åããéãã2ã€ä»¥äžã®æ¬¡å
ãæã£ãŠããå³åœ¢ãååšããã
äŸãã°ã3次å
ç«äœã®1ã€ã§ããçŽæ¹äœã¯çžŠã暪ãé«ãã®3ã€ã®é·ããæã£ãŠããã®ã§ã3次å
å³åœ¢ãšåŒã°ããã
空éã«1ã€ã®å¹³é¢ããšãããã®äžã«çŽäº€ãã座æšè»ž<math>O_x\ , \ O_y</math>ããšãã次ã«Oãéããã®å¹³é¢ã«åçŽãªçŽç·<math>O_z</math>ãã²ãããã®çŽç·äžã§ãOãåç¹ãšãã座æšãèããã
ãã®3çŽç·<math>O_x\ , \ O_y\ , \ O_z</math>ã¯ãã©ã®2ã€ãäºãã«åçŽã§ããããããã'''座æšè»ž'''ãšããããããã'''x軞ãy軞ãz軞'''ãšããã
ãŸããx軞ãšy軞ãšã§å®ãŸãå¹³é¢ã'''xyå¹³é¢'''ãšãããy軞ãšz軞ãšã§å®ãŸãå¹³é¢ã'''yzå¹³é¢'''ãšãããz軞ãšx軞ãšã§å®ãŸãå¹³é¢ã'''zxå¹³é¢'''ãšããããããã'''座æšå¹³é¢'''ãšããã
空éå
ã®ç¹Aã«å¯ŸããŠãAãéã£ãŠå座æšå¹³é¢ã«å¹³è¡ãª3ã€ã®å¹³é¢ãã€ãããããããx軞ãy軞ãz軞ãšäº€ããç¹ã<math>A_1\ , \ A_2\ , \ A_3</math>ãšãã<math>A_1\ , \ A_2\ , \ A_3</math>ã®ããããã®è»žäžã§ã®åº§æšã<math>a_1\ , \ a_2\ , \ a_3</math>ãšããã
ãã®ãšãã3ã€ã®æ°ã®çµ
:<math>(a_1\ , \ a_2\ , \ a_3)</math>
ãç¹Aã®'''座æš'''ãšããã<math>a_1</math>ã'''x座æš'''ãšããã<math>a_2</math>ã'''y座æš'''ãšããã<math>a_3</math>ã'''z座æš'''ãšããã
ãã®ããã«åº§æšã®å®ãããã空éã'''座æšç©ºé'''ãšåŒã³ãç¹Oã座æšç©ºéã®'''åç¹'''ãšããã
=====çé¢ã®æ¹çšåŒ =====
ããã§ã¯ãç¹ã«3次å
空éã®å³åœ¢ã«æ³šç®ããã
ãŸãã¯ãã¯ãã«ãçšããåã«3次å
空éã®ç©ºéå³åœ¢ããæ°åŒã«ãã£ãŠèšè¿°ããæ¹æ³ãèå¯ããã
2次å
空éã«ãããŠããã£ãšãç°¡åãªå³åœ¢ã¯çŽç·ã§ããããã®åŒã¯äžè¬çã«
:<math>
a x + by = c
</math>
ã§è¡šããããã
(<math>a</math>,<math>b</math>,<math>c</math>ã¯ä»»æã®å®æ°ã)
ããã§<math>x</math>,<math>y</math>ã¯ã2次å
空éã代衚ãã2ã€ã®ãã©ã¡ãŒã¿ãŒã§ããã3次å
空éãçšãããšãã«ã¯ããããã¯3ã€ã®æåã§è¡šããããããšãæåŸ
ãããã
å®éãã®ãããªåŒã§è¡šããããå³åœ¢ã¯ã3次å
空éã§ãåºæ¬çãªå³åœ¢ã§ãããã€ãŸãã
:<math>
a x + by + cz = d
</math>
ããäžã®åŒã®é¡äŒŒç©ãšããŠåŸãããã
(<math>a</math>,<math>b</math>,<math>c</math>,<math>d</math>ã¯ä»»æã®å®æ°ã)
ãã®ãããªå³åœ¢ã¯ã©ããªå³åœ¢ã«å¯Ÿå¿ããã ããã?
å®éã«ã¯ãã®å³åœ¢ãç¹åŸŽã¥ããã®ã¯ãåŸã«åŠã¶3次å
ãã¯ãã«ãçšããã®ããã£ãšãç°¡åã§ããã®ã§ãããã¯åŸã«ãŸããããšã«ããã
ãããããã 1ã€ãã®åŒããåããããšã¯ã3次å
空éã®åº§æšãè¡šãããã©ã¡ãŒã¿ãŒ
:<math>
x,y,z
</math>
ã®ãã¡ã«1ã€ã®é¢ä¿
:<math>
f(x,y,z)=0
</math>
ãäžããããšã§ã3次å
空éäžã®å³åœ¢ãæå®ã§ãããšããããšã§ããããã®å Žåã¯ã
:<math>
f(x,y,z) =a x + by + cz - d
</math>
ãçšããŠããã
ãã¯ãã«ã䜿ããªããŠãå³åœ¢ç解éãåŸãããåŒãšããŠã
:<math>
(x -a)^2 +
(y -b)^2 +
(z -c)^2
= r^2
</math>
ãæããããã
(<math>a</math>,<math>b</math>,<math>c</math>,<math>r</math>ã¯ä»»æã®å®æ°ã)
ãã®åŒã¯ã2次å
ã§ãããšããã®
:<math>
(x -a)^2 +
(y -b)^2 +
= r^2
</math>
ã®åŒã®é¡äŒŒç©ã§ããã2次å
ã®å Žåã¯ãã®åŒã¯ã
äžå¿<math>
(a,b)
</math>ååŸ<math>
r
</math>ã®åã«å¯Ÿå¿ããŠããã
3次å
ã®ãã®åŒã¯ãçµè«ããããšäžå¿<math>
(a,b,c)
</math>ååŸ<math>
r
</math>ã®åã«å¯Ÿå¿ããŠããã®ã§ããã
*説æ
äžã®åŒ
:<math>
(x -a)^2 +
(y -b)^2 +
(z -c)^2
= r^2
</math>
ãæºããããç¹<math>
(x,y,z)
</math>ãåãããã®ç¹ãšç¹<math>
(a,b,c)
</math>ãšã®è·é¢ãèããã
空é座æšã«çœ®ãã<math>x</math>軞ã
<math>y</math>軞ã
<math>z</math>軞ã¯ããããçŽäº€ããŠããã®ã§ã2ç¹ã®è·é¢ã¯3å¹³æ¹ã®å®çãçšããŠ
:<math>
\sqrt{ (x -a)^2 + (y -b)^2 + (z -c)^2 }
</math>
ã§äžããããã
ããããäžã®åŒããããã§éžãã ç¹<math>
(x,y,z)
</math>ã¯ãæ¡ä»¶
:<math>
(x -a)^2 +
(y -b)^2 +
(z -c)^2
= r^2
</math>
ãæºãããŠããã®ã§ã2ç¹ã®è·é¢ã¯
:<math>
\sqrt{ (x -a)^2 + (y -b)^2 + (z -c)^2 }
</math>
:<math>
= \sqrt{r^2}
</math>
:<math>
= r
</math>
ã§ããã
(<math>r>0</math>ãçšããã)
ãã£ãŠãäžã®åŒãæºããç¹ã¯å
šãŠç¹<math>
(a,b,c)
</math>ããã®è·é¢ã<math>
r
</math>ã§ããç¹ã§ãããããã¯äžå¿<math>
(a,b,c)
</math>ååŸ<math>
r
</math>ã®åã«ä»ãªããªãã
{{æŒç¿åé¡|
äžå¿
:<math>
(3,7,-2)
</math>
ååŸ
:<math>
1
</math>
ã®çã®åŒãæ±ããã|:<math>
(x -a)^2 +
(y -b)^2 +
(z -c)^2
= r^2
</math>
ã«ä»£å
¥ããããšã§ã
:<math>
(x -3)^2 +
(y -7)^2 +
(z +2)^2
= 1
</math>
ãæ±ããããã}}
{{æŒç¿åé¡|
:<math>
x ^ 2 + 2x + y ^ 2 - 8y + z ^ 2 + 6z - 9 = 0
</math>
ãã©ã®ãããª
çã«å¯Ÿå¿ãããèšç®ããã|ãã®ãããªæ°åŒãçã«å¯Ÿå¿ãããšãã
:<math>
x^2,
y^2,
z^2
</math>
ã®ä¿æ°ã¯å¿
ãçãããªããŠã¯ãªããªããããã§ãªãå Žåã¯ãã®å³åœ¢ã¯æ¥åäœã«å¯Ÿå¿ããã®ã ããããã¯æå°èŠé ã®ç¯å²å€ã§ããã
ããã§ã¯äžã®åŒã¯ãã®æ¡ä»¶ãæºãããŠããã
ããã§ã¯ããã®åŒã
:<math>
(x -a)^2 +
(y -b)^2 +
(z -c)^2
= r^2
</math>
ã®åœ¢ã«æã£ãŠè¡ãããšãéèŠã§ããã
:<math>
x,y,z
</math>
ã®ããããã«ã€ããŠãã®åŒãå¹³æ¹å®æãããšã
:<math>
x ^ 2 + 2x + y ^ 2 - 8y + z ^ 2 + 6z - 9 = 0
</math>
:<math>
(x +1 ) ^2 - 1 + (y -4) ^2 -16 +(z +3)^2 -9 -9=0
</math>
:<math>
(x +1 ) ^2 + (y -4) ^2 +(z +3)^2 = 35
</math>
ãåŸãããããã£ãŠãäžã®åŒ
:<math>
x ^ 2 + 2x + y ^ 2 - 8y + z ^ 2 + 6z - 9 = 0
</math>
ã¯ã
äžå¿
:<math>
(-1,4,-3)
</math>
ãååŸ
:<math>
\sqrt{35}
</math>
ã®çã«å¯Ÿå¿ããã}}
=====空éã«ããããã¯ãã«=====
次ã«3次å
空éäžã«ããããã¯ãã«ãèå¯ããã
2次å
空éäžã§ã¯ãã¯ãã«ã¯2ã€ã®éã®çµã¿åããã§è¡šããããã
ããã¯1ã€ã®ãã¯ãã«ã¯x軞æ¹åã«å¯Ÿå¿ããéãšy軞æ¹åã«å¯Ÿå¿ããéã®2ã€ãæã£ãŠããå¿
èŠããã£ãããã§ããã
ãã®ããšããã3次å
空éã®ãã¯ãã«ã¯3ã€ã®éã®çµã¿åããã§æžããããšãäºæ³ãããã
ç¹ã«<math>x</math>軞æ¹åã®æå<math>a</math>,
<math>y</math>軞æ¹åã®æå<math>b</math>,
<math>z</math>軞æ¹åã®æå<math>c</math>
(<math>a</math>,<math>b</math>,<math>c</math>ã¯ä»»æã®å®æ°ã)
ã§è¡šãããããã¯ãã«ãã
:<math>
(a,b,c)
</math>
ãšæžããŠè¡šããããšã«ããã
2次å
å¹³é¢ã§ã¯
ãããã¯ãã«
:<math>
\vec a =(a,b)
</math>
ã¯ã
(<math>a</math>,<math>b</math>ã¯ä»»æã®å®æ°ã)
:<math>
\vec e _1 = (1,0)
</math>
:<math>
\vec e _2 = (0,1)
</math>
ã®2æ¬ã®ãã¯ãã«ãçšããŠã
:<math>
\vec a = a\vec e _1 + b\vec e _2
</math>
ã§è¡šããããã
3次å
空éã§ããã®ãããªèšè¿°æ³ããããäžã§çšãããã¯ãã«
:<math>
\vec a = (a,b,c)
</math>
ã¯ã
:<math>
\vec e _1 = (1,0,0)
</math>
:<math>
\vec e _2 = (0,1,0)
</math>
:<math>
\vec e _3 = (0,0,1)
</math>
ãçšããŠ
:<math>
\vec a = a \vec e _1 + b \vec e _2 + c\vec e _3
</math>
ãšæžããããã¯ãã«ã«å¯Ÿå¿ããŠããã
3次å
ãã¯ãã«ã«å¯ŸããŠã2次å
ãã¯ãã«ã§å®ããå®çŸ©ãæ§è³ªãã»ãŒãã®ãŸãŸæç«ããã
3次å
ãã¯ãã«ã®å æ³ã¯ãããããã®ãã¯ãã«èŠçŽ ãç¬ç«ã«è¶³ãåãããããšã«ãã£ãŠå®çŸ©ããã
:<math>
(x _1,y _1,z _1)+(x _2,y _2,z _2)
</math>
:<math>
=
(x _1+x _2,y _1+y _2,z _1+z _2)
</math>
ãŸããããããã®ãã¯ãã«ã®èŠçŽ ãå
šãŠçãããã¯ãã«ã"ãã¯ãã«ãšããŠçãã"ãšè¡šçŸããã
{{æŒç¿åé¡|
ãã¯ãã«ã®å
:<math>
(1,2,3)+(4,5,6)
</math>
ãèšç®ããã|:<math>
(1,2,3)+(4,5,6)
</math>
:<math>
=(1+4,2+5,3+6)
</math>
:<math>
=(5,7,9)
</math>
ãåŸãããã}}
=====空éãã¯ãã«ã®å
ç©=====
ãã¯ãã«<math>\vec a</math>,<math>\vec b</math>éã®ãã¯ãã«ã®å
ç©ãå¹³é¢ã®å Žåãšåæ§ã«
:<math>
\vec a \cdot\vec b
= |\vec a||\vec b| \cos \theta
</math>
(<math>\theta</math>ã¯ããã¯ãã«<math>\vec a</math>,<math>\vec b</math>ã®ãªãè§ã)
åé
æ³åã1次ç¬ç«ã®æ§è³ªããã®ãŸãŸæãç«ã€ã
ãã ãã3次å
空éã®å
šãŠã®ãã¯ãã«ã匵ãã«ã¯ã3ã€ã®ç·åœ¢ç¬ç«ãªãã¯ãã«ãæã£ãŠæ¥ãå¿
èŠãããã
*泚æ
ãã®ããšã®èšŒæã¯ãããã[[ç·å代æ°åŠ]]ãªã©ã«è©³ããã
{{æŒç¿åé¡|
2ã€ã®ãã¯ãã«ã®å
ç©
:<math>
(1,2,3) \cdot
(4,5,6)
</math>
ãèšç®ããã|
2次å
ã®å Žåãšåãããã«ããã§ãããããã®èŠçŽ ã¯äºãã«çŽäº€ããåäœãã¯ãã«
:<math>
\vec e _1\ ,\ \vec e _2\ ,\ \vec e _3
</math>
ã«ãã£ãŠåŒµãããŠããããã®ãã以åãšåããèŠçŽ ããšã®èšç®ãå¯èœã§ããã
:<math>
(1,2,3) \cdot
(4,5,6)
</math>
:<math>
=1\times 4 + 2 \times 5 + 3 \times 6
</math>
:<math>
= 32
</math>
ãšãªãã
ããããã现ããèšç®ãè¡ãªããšã
:<math>
(1,2,3) \cdot
(4,5,6)
</math>
:<math>
=( \vec e _1
+2\vec e _2
+3\vec e _3)
\cdot
(4\vec e _1
+5\vec e _2
+6\vec e _3)
</math>
ãåŸããããããããã®ãã¯ãã«ã
:<math>
(a+b+c)(x+y+z)
= (ax+ay+az + bx+by+bz+cx+cy+cz)
</math>
ã«åŸã£ãŠå±éãã
:<math>
\vec e _ i \cdot \vec e _j
</math>
(<math>i</math>,<math>j</math>ã¯1,2,3ã®ã©ããã)
ã代å
¥ããããšã§äžã®åŒãèšç®ã§ããã¯ãã§ããã
ãããã
<math>i</math>ãš<math>j</math>ãçãããªããšãã«ã¯
:<math>
\vec e _ i \cdot \vec e _j
</math>
:<math>
=0
</math>
ãæãç«ã€ããšãããäžã®å±éããåŸã®9åã®é
ã®ãã¡ã§ã6ã€ã¯
:<math>
0
</math>
ã«çããã
ãŸãã
<math>i</math>ãš<math>j</math>ãçãããšãã«ã¯
:<math>
\vec e _ i \cdot \vec e _j
</math>
:<math>
=1
</math>
ãæãç«ã€ããšãããäžã®åŒ
:<math>
=( \vec e _1
+2\vec e _2
+3\vec e _3)
\cdot
(4\vec e _1
+5\vec e _2
+6\vec e _3)
</math>
ã®å±éã¯
:<math>
= 4 + 2 \times 5 + 3 \times 6
</math>
:<math>
= 32
</math>
ãšãªã£ãŠç¢ºãã«èŠçŽ ããšã®èšç®ãšäžèŽããã}}
{{æŒç¿åé¡|
2次å
空éã®ãã¯ãã«ã¯2æ¬ã®1次ç¬ç«ãªãã¯ãã«ãããã°ãå¿
ããããã®ç·åœ¢çµåã«ãã£ãŠèšç®ã§ããã¯ãã§ããã
ããã§ã
:<math>
\vec a _1= (1,2)
</math>
ãš
:<math>
\vec a _2= (-5,3)
</math>
ãçšããŠã
:<math>
\vec b = (10,7)
</math>
ãã
:<math>
\vec b = c \vec a _1
+d \vec a _2
</math>
ã®åœ¢ã«æžããŠã¿ãã
(<math>c</math>,<math>d</math>ã¯ãäœããã®å®æ°ã)|2次å
ã®ãã¯ãã«ã®ä¿æ°ãæ±ããåé¡ã§ããã
<math>c</math>,<math>d</math>ã®æåããã®ãŸãŸçšãããšã<math>c</math>,<math>d</math>ã®æºããæ¡ä»¶ã¯
:<math>
c(1,2) + d(-5,3)
= (10,7)
</math>
ã€ãŸã
:<math>
(c-5d , 2c + 3d) =(10,7)
</math>
ãšãªããããã¯
<math>c</math>,<math>d</math>ã«é¢ããé£ç«1次æ¹çšåŒã§æžãæããããã
:<math>\begin{cases}
c -5d = 10\\
2c + 3d = 7
\end{cases}</math>
ããã解ããšã
:<math>
c = 5
</math>
:<math>
d = -1
</math>
ãåŸãããã
ãã£ãŠã
äžã®åŒã¯
:<math>
5(1,2) -(-5,3)
= (10,7)
</math>
ãšæžãã確ãã«2æ¬ã®ç·åœ¢ç¬ç«ãªãã¯ãã«ã«ãã£ãŠä»ã®ãã¯ãã«ãæžãè¡šãããããšãåãã£ãã
*泚æ
ãã®ãããªèšç®ã¯3次å
ãã¯ãã«ã«å¯ŸããŠãå¯èœã§ããããèšç®ææ³ãšããŠ3å
1次é£ç«æ¹çšåŒãæ±ãå¿
èŠããããæå°èŠé ã®ç¯å²å€ã§ãããå®éã®èšç®ææ³ã¯ã[[ç·å代æ°åŠ]],[[ç©çæ°åŠI ç·åœ¢ä»£æ°]]ãåç
§ã}}
ãã®è¡šåŒãçšããŠã以åèŠã
:<math>
a x + by + cz = d
</math>
ã®å³åœ¢ç解éãè¿°ã¹ãã
ãã®å³åœ¢äžã®ä»»æã®ç¹ã<math>
(x,y,z)
</math>ã§è¡šããã
ãã®ç¹ã¯åç¹Oã«å¯Ÿããäœçœ®ãã¯ãã«ãçšãããš<math>
(x,y,z)
</math>ã§äžããããã
䟿å®ã®ããã«
ãã®ãã¯ãã«ã<math>
\vec x
</math>ãšæžãããšã«ããã
äžæ¹ããã¯ãã«<math>
\vec a = (a,b,c)
</math>ãçšãããšãäžã®åŒã¯ãã¯ãã«ã®å
ç©ãçšããŠ<math>
\vec a \cdot \vec x = d
</math>ã§äžããããã
ã€ãŸãããã®åŒã§è¡šããããå³åœ¢ã¯ãããã¯ãã«
<math>
\vec a
</math>
ãšã®å
ç©ãäžå®ã«ä¿ã€å³åœ¢ã§ããã
ãã®å³åœ¢ã¯ãå®éã«ã¯
<math>
\vec a
</math>
ã«çŽäº€ããå¹³é¢ã§äžããããã
ãªããªããã®ãããªå¹³é¢äžã®ç¹ã¯ãå¿
ãå¹³é¢äžã®ããäžç¹ã®äœçœ®ãã¯ãã«ã«å ããŠã
ãã¯ãã«
<math>
\vec a
</math>
ã«çŽäº€ãããã¯ãã«ãå ãããã®ã§æžãããšãåºæ¥ãã
ãããã
ãã¯ãã«
<math>
\vec a
</math>
ã«çŽäº€ãããã¯ãã«ãš
ãã¯ãã«
<math>
\vec a
</math>
ã®å
ç©ã¯å¿
ã0ã§ããã®ã§ã
ãã®ãããªç¹ã®éåã¯
ãã¯ãã«
<math>
\vec a
</math>
ãšäžå®ã®å
ç©ãæã€ã®ã§ããã
ãã£ãŠå
ã®åŒ
:<math>
a x + by + cz = d
</math>
ã¯ã
ãã¯ãã«<math>
\vec a =(a,b,c)
</math>ã«çŽäº€ããå¹³é¢ã«å¯Ÿå¿ããããšãåãã£ãã
次ã«<math>d</math>ããå³åœ¢ãè¡šããå¹³é¢ãšãåç¹ãšã®è·é¢ã«é¢ä¿ãããããšã瀺ãã
ç¹ã«ããã¯ãã«<math>
\vec a
</math>ã«æ¯äŸããäœçœ®ãã¯ãã«ãæã€ç¹<math>
\vec x
</math>ãèããããã®ãšããã®ç¹ãšåç¹ãšã®è·é¢ã¯ã
å¹³é¢
:<math>
a x + by + cz = d
</math>
ãšåç¹ãšã®è·é¢ã«å¯Ÿå¿ããã
ãªããªããäœçœ®ãã¯ãã«<math>
\vec x
</math>ã¯ãåç¹ããå¹³é¢
:<math>
a x + by + cz = d
</math>
ã«åçŽã«äžãããç·ã«å¯Ÿå¿ããããã§ããã
ãã®ããšããä»®ã«<math>
\vec a
</math>æ¹åã®åäœãã¯ãã«ã<math>
\vec n
</math>ãšæžããå¹³é¢ãšåç¹ãšã®è·é¢ã<math>
m
</math>ãšæžããšã<math>
\vec x = m \vec n
</math>ãåŸãããã
ãã®åŒã
:<math>
\vec a \cdot \vec x = d
</math>
ã«ä»£å
¥ãããšã
:<math>
\vec a \cdot m\vec n = d
</math>
:<math>
m|\vec a| = d
</math>
ãåŸãããããã£ãŠã<math>
d
</math>ã¯ã
å¹³é¢ãšåç¹ã®è·é¢<math>
m
</math>ãšãã¯ãã«<math>
\vec a
</math>ã®å€§ãããããããã®ã§ããã
<!-- äžã§ã¯å²åäžè¬çã«3次å
ã®å¹³é¢ãæ±ã£ãããã㯠-->
<!-- å°ãé£ããå
容ã§ãã£ããå®éã®æå°èŠé ã§ã¯ããå°ã -->
<!-- ç°¡åãªå
容ã -->
{{æŒç¿åé¡|
ç¹ã«ãã¯ãã«
:<math>
\vec a = (0,0,1)
</math>
ãåããšãã©ã®ãããªåŒãåŸãããŠããã®åŒã¯
ã©ã®ãããªå³åœ¢ã«å¯Ÿå¿ãããã|ãã®ãšã
:<math>
\vec a \cdot \vec x = d
</math>
ã¯ã
:<math>
(0,0,1)\cdot (x,y,z) = d
</math>
:<math>
z =d
</math>
ã«å¯Ÿå¿ããã
ãã®åŒã¯<math>z</math>座æšã<math>d</math>ã«å¯Ÿå¿ãããã以å€ã®<math>x</math>,<math>y</math>座æšãä»»æã«åããã
å¹³é¢ã«å¯Ÿå¿ããŠããããããã¯
<math>xy</math>å¹³é¢ã«å¹³è¡ã§ããã
<math>xy</math>å¹³é¢ããã®è·é¢ã<math>d</math>ã§ããå¹³é¢ã§ããã
ãŸãã<math>xy</math>å¹³é¢ãšãã¯ãã«
:<math>
\vec a = (0,0,1)
</math>
ã¯çŽäº€ããŠããã®ã§ããã®ããšããããã®åŒã¯æ£ããã}}
:ç
:: <math>xy</math>å¹³é¢ã«å¹³è¡ã§ããã<math>xy</math>å¹³é¢ããã®è·é¢ã<math>d</math>ã§ããå¹³é¢ã
== çºå±:å€ç© ==
å€ç©ã¯é«æ ¡æ°åŠç¯å²å€ã§å
¥è©Šã«ã¯åºãªãããå€ç©ã¯æ°åŠãç©çãªã©ã«å¿çšã§ãã䟿å©ãªã®ã§ããã§æ±ãã
äžæ¬¡å
ãã¯ãã« <math>\vec a ,\, \vec b</math> ã«å¯Ÿããå€ç© <math>\vec a \times \vec b</math> ã次ãæºãããã®ãšããã
# <math>\vec a \times \vec b</math> 㯠<math>\vec a ,\, \vec b</math> ãããããšåçŽ<ref>æ°åŒã§è¡šããš <math>\vec a \times \vec b \perp \vec a </math> ã〠<math>\vec a \times \vec b \perp \vec b </math></ref>
# ãã¬ãã³ã°ã®å·Šæã®æ³åã®æ Œå¥œãããããã®ãšããäžæã <math>\vec a</math> ã人差ãæã <math>\vec b</math> ããšãããšãã<math>\vec a \times \vec b</math> ã¯èŠªæã®æ¹åã§ããã
# ãã¯ãã« <math>\vec a ,\, \vec b</math> ã®ãªãè§ã <math>\theta</math> ãšããã<math>|\vec a \times \vec b| = |\vec a ||\vec b|
\sin\theta</math><ref><math>|\vec a ||\vec b|
\sin\theta</math> ã¯ãã¯ãã« <math>\vec a ,\, \vec b</math> ã®äœãå¹³è¡å蟺圢ã®é¢ç©ã«çããã</ref>
[[ãã¡ã€ã«:Cross product parallelogram.svg|ãµã ãã€ã«|å€ç©ã®æ¹åãè¡šããå³ãäžã®âèšå·ããªãããããã¯ãã¯ãã«ã§ããã]]
次ã«å€ç©ã®æå衚瀺ãèããŠã¿ããããã®å®çŸ©ããæå衚瀺ãçŽæ¥å°ãã®ã¯é¢åãªã®ã§ã倩äžãçã«æå衚瀺ãäžããŠããããããå€ç©ã®å®çŸ©ãæºããããšã確èªããã
<math>\vec a = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}</math> ã<math>\vec b = \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix}</math> ãšãããšãã<math>\vec a \times \vec b = \begin{pmatrix} a_2b_3 - a_3b_2 \\ a_3b_1 - a_1b_3 \\ a_1b_2 - a_2b_1 \end{pmatrix}</math> ã§ããã
ãŸãã¯ã<math>\vec a \times \vec b</math> 㯠<math>\vec a ,\, \vec b</math> ãããããšåçŽã§ããããšã確èªãããããã¯ã<math>(\vec a \times \vec b) \cdot \vec a = 0</math> ãš <math>(\vec a \times \vec b) \cdot \vec b = 0</math> ã§ããããšãæå衚瀺ã代å
¥ããã°èšŒæã§ããã
次ã«ã <math>|\vec a \times \vec b| = |\vec a ||\vec b|
\sin\theta</math> ã蚌æããã<math>|\vec a \times \vec b|^2 = |\vec a |^2|\vec b|^2
\sin^2\theta = \vec | a |^2|\vec b|^2
(1-\cos^2\theta)</math> ãããã§ã <math>\cos^2 \theta = \frac{(\vec a \cdot \vec b)^2}{|\vec a|^2|\vec b|^2}</math> ã代å
¥ãã<math>|\vec a \times \vec b|^2 = \vec |a |^2|\vec b|^2
-(\vec a \cdot \vec b)^2</math> ãåŸãããã®åŒã«ãæå衚瀺ã代å
¥ããã°ã䞡蟺ãçããããšã確èªã§ããã
æåŸã«ããã¬ãã³ã°ã®å·Šæã®æ³å㧠<math>\vec a \times \vec b</math> ã¯èŠªæã®æ¹åã§ããããšã確èªããã
<math>\vec a = \begin{pmatrix} 1 \\ 0 \\ 0\end{pmatrix}</math>ã <math>\vec b = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}</math> ã®ãšãã<math>\vec a \times \vec b = \begin{pmatrix} 0\\ 0 \\ 1 \end{pmatrix}</math> ã§ããããããããäºçªç®ã®æ§è³ªã確èªã§ããã
'''å€ç©ã®å¿çš'''
2ã€ã®ãã¯ãã«ã«åçŽãªãã¯ãã«ãæ±ããããšããªã©ã¯ãå€ç©ã®æå衚瀺ããèšç®ããã°ãé¢åãªèšç®ãããªããŠãæ±ããããã
åé¢äœ <math> \mathrm{OABC}</math> ã®äœç©ã¯ <math> \frac 1 6 |(\vec \mathrm{OA} \times \vec \mathrm{OB})\cdot \vec \mathrm{OC} | </math>
ã§ããã
å®éã <math> \frac 1 6 |(\vec \mathrm{OA} \times \vec \mathrm{OB})\cdot \vec \mathrm{OC} | = \frac 1 3 \left|\frac 1 2 \vec \mathrm{OA} \times \vec \mathrm{OB}\right||h|</math>ã§ããããã ãã h ã¯ÎABCãåºé¢ãšãããšãã®åé¢äœã®é«ãã§ããã
ãŸããç©çåŠã®ããŒã¬ã³ãåã¯å€ç©ã䜿ããš <math>\vec F = q\vec v \times \vec B</math> ãšç°¡æœã«è¡šããã
'''èŠãæ¹'''
å³ã®ããã«èŠçŽ ãããåãããã
[[ãã¡ã€ã«:Cross product mnemonic a b.svg|ãã¬ãŒã ãªã]]
== ã³ã©ã ãªã© ==
{{ã³ã©ã |ãã¯ãã«ã®çè«ã®æŽå²|2=[[File:WilliamRowanHamilton.jpeg|thumb|ããã«ãã³]]
è€çŽ æ°ãšãã¯ãã«ã®çè«ã¯ããããç¬ç«ããçè«ãšããŠæããããŠããããæŽå²çã«ã¯ããã«ãã³ã«ãã£ãŠè€çŽ æ°ãæ¡åŒµããåå
æ°ãçºèŠãããåå
æ°ãå
ã«ã®ãã¹ãªã©ã«ãã£ãŠãã¯ãã«ãçºèŠãããã
[[w:åå
æ°|åå
æ°]]ã¯ã
:a ïŒ bi ïŒ cj ïŒ dk (a,b,c,dã¯å®æ°)
ã®ããã«ãå®æ°ãš3ã€ã®èæ°åäœi,j,kããã¡ããŠè¡šãããæ°ã§ããã
ããã§ãi,j,k 㯠i^2=-1, j^2=-1, k^2=-1 ãæºããæ°ã§ãi,j,k ã¯äºãã«ç°ãªãã
å®æ°ã®åäœ1åã«å ããŠãããã«3ã€ã®åäœ i, ãj,ã k ããã£ãŠããã®ã§ãåèšã§4åã®åäœãããã®ã§åå
æ°ãšããããããã§ããã
ããŠãããã«ãã³ã«ããåå
æ°ã®çºèŠåŸãããã«ç 究ãé²ããšãå³åœ¢ãç©çåŠãªã©ã®åé¡ã解ãããã«ã¯ 2ä¹ããŠ-1ã«ãªãæ§è³ªã¯ã»ãšãã©ã®ç©ºéã»ç«äœïŒ3次å
ã®å³åœ¢ïŒã®åé¡ã解ãå¿çšã®å Žåã«ã¯äžèŠã§ããããšãåãããåŠæ ¡æè²ã®å Žã§ã¯ãã¯ãã«ãšè€çŽ æ°ãå¥ã
ã«æããããã«ãªã£ãããã§ããã
ãããŠãåå
æ°ã®å
¬åŒã®ãã¡ããã¯ãã«ã§ãé¡äŒŒã®å
¬åŒãæãç«ã€å Žåã«ã¯ããã®åå
æ°ã®å
¬åŒããã¯ãã«çšã«æ¹è¯ãããŠãã¯ãã«ã®å
¬åŒãšããŠèŒžå
¥ãããã®ã§ãçµæçã«ããã«ãã³ã¯ãã¯ãã«ã®å
¬åŒã®çºèŠè
ãšããŠã玹ä»ãããããšã«ãªã£ãã
ãŸããåå
æ°ã¯çŸä»£ã§ã¯3DCGãªã©ã®åéã§å¿çšãããŠããã}}
== è泚 ==
<references/>
{{DEFAULTSORT:ãããšããã€ããããããC ãžããšã}}
[[Category:é«çåŠæ ¡æ°åŠC|ãžããšã]] | 2005-05-03T07:08:24Z | 2023-10-31T10:22:04Z | [
"ãã³ãã¬ãŒã:æŒç¿åé¡",
"ãã³ãã¬ãŒã:Ruby",
"ãã³ãã¬ãŒã:ã³ã©ã ",
"ãã³ãã¬ãŒã:Pathnav",
"ãã³ãã¬ãŒã:Math theorem",
"ãã³ãã¬ãŒã:Math proof"
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E6%95%B0%E5%AD%A6C/%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB |
1,893 | æ§èª²çš(2013幎床-2021幎床)é«çåŠæ ¡æ°åŠC | æ°åŠCã¯
ããæ§æãããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "æ°åŠCã¯",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ããæ§æãããã",
"title": ""
}
] | æ°åŠC㯠ãã¯ãã«
å¹³é¢äžã®æ²ç·
è€çŽ æ°å¹³é¢
æ°åŠçãªè¡šçŸã®å·¥å€« ããæ§æãããã | {{pathnav|é«çåŠæ ¡ã®åŠç¿|é«çåŠæ ¡æ°åŠ|frame=1}}
æ°åŠCã¯
*[[é«çåŠæ ¡æ°åŠB/ãã¯ãã«|ãã¯ãã«]]
*[[é«çåŠæ ¡æ°åŠIII/å¹³é¢äžã®æ²ç·|å¹³é¢äžã®æ²ç·]]
*[[é«çåŠæ ¡æ°åŠIII/è€çŽ æ°å¹³é¢|è€çŽ æ°å¹³é¢]]
*[[é«çåŠæ ¡æ°åŠC/æ°åŠçãªè¡šçŸã®å·¥å€«|æ°åŠçãªè¡šçŸã®å·¥å€«]]
ããæ§æãããã
{{DEFAULTSORT:æ§2 ãããšããã€ããããããC}}
[[Category:é«çåŠæ ¡æè²]]
[[Category:é«çåŠæ ¡æ°åŠC|*]]
[[Category:æ°åŠæè²]] | 2005-05-03T07:35:52Z | 2023-12-09T21:37:53Z | [
"ãã³ãã¬ãŒã:Pathnav"
] | https://ja.wikibooks.org/wiki/%E6%97%A7%E8%AA%B2%E7%A8%8B(2013%E5%B9%B4%E5%BA%A6-2021%E5%B9%B4%E5%BA%A6)%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E6%95%B0%E5%AD%A6C |
1,894 | æ§èª²çš(-2012幎床)é«çåŠæ ¡æ°åŠC/è¡å | é«çåŠæ ¡æ°åŠC > è¡å
æ¬é
ã¯é«çåŠæ ¡æ°åŠCã®è¡åã®è§£èª¬ã§ããã
1次æ¹çšåŒ
ãã次ã®ãããªèšæ³ã§è¡šããŠã¿ãã
ããããå匷ããã®ã¯ãé£ç«æ¹çšåŒãšãã¯ãã«ãšã®é¢ä¿ã§ããããããèå¯ããããããããã«ããããã«è¡å(ããããã€)ãšããéãå°å
¥ããã
ãã¯ãã« ( x y ) {\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}} ã«ã æŒç® ( 1 2 2 3 ) {\displaystyle {\begin{pmatrix}1&2\\2&3\end{pmatrix}}} ãæœããŠ(ãã®æŒç®ã®å
容ãããããããã説æãããè¡åãã§ãã)ã çãã®ãã¯ãã« ( 1 2 ) {\displaystyle {\begin{pmatrix}1\\2\end{pmatrix}}} ãåŸãããšããè¡šçŸã«æžãæããã
ãŸãããã®ãããªèšæ³ãããããã次ã«èª¬æããè¡å(ããããã€ãè±:matrix)ãšããéãæ°ãã«å®çŸ©ããã
ãŸããè¡åã©ããã®ç©ã®å®çŸ©ãã
ã¯ã
ãšçããããšå®ããã äœæ
ãã®ããã«å®ããã®ããèãããã 2ã€ã®é£ç«æ¹çšåŒ
ã«ãããŠãäžéçå€æ°p,qãæ¶å»ããŠãå€æ°x,yã«é¢ããäžã€ã®é£ç«æ¹çšåŒãšæžãçŽããš
ãšãªãã å®éãäž2åŒã®p,qã«ãäž2åŒã代å
¥ããŠæŽé ããã°ããã èªè
ã¯ä»£å
¥ããŠç¢ºèªããã ãããè¡åè¡šçŸãããš
ä»æ¹ã2ã€ã®é£ç«æ¹çšåŒãè¡åãçšããŠæžãçŽããš
äžã®åŒãäžã®åŒã«ã圢åŒçã«ä»£å
¥ãããš
2ã€ã®è¡åè¡šçŸåŒãæ¯èŒããã°ã è¡åã®ç©ã®å®ãæ¹ã®åçæ§ãåããã ããã
ç©ã®å®çŸ©åŒã¯ãäžèŠãããšè€éããã«èŠããããããã«è£å©ç·ã
ã®ããã«åŒããŠã¿ãã°åããããã«ãããšãã°åæåŸã®2è¡1åã ( c e + d g ) {\displaystyle {\begin{pmatrix}&\\ce+dg&\qquad \end{pmatrix}}} ã¯ãåæåã®2è¡ãã®ããããã®æå ( c d ) {\displaystyle {\begin{pmatrix}&\\c&d\end{pmatrix}}} ãšãåæåã®1åç® ( e g ) {\displaystyle {\begin{pmatrix}e&\\g&\end{pmatrix}}} ã®æåãšããæããŠè¶³ãããã®ã«ãªã£ãŠããã
äžè¬ã«ãç©ã®åæåŸã®xè¡yåãã¯ãåæåã®xè¡ãã®ããããã®æåãšãåæåã®yåç®ã®ããããã®æåãšããæããŠè¶³ããçµæã«ãªã£ãŠããã
è¡åã©ããã®ç©ã¯ãé åºã«ãã£ãŠçµæãç°ãªãã ããšãã°è¡åA,Bã
ãšãããšãã
ããããã
ãšãªãã
ãã®ããã«ãäžè¬ã®è¡åAãšè¡åBã®ç©ã¯ãäžè¬ã«
ãšãªãã
äžè¿°ã®äŸã¯ã2å
é£ç«äžæ¬¡æ¹çšåŒãåŒ2åã®å Žåã«çžåœããè¡åã ã£ãããäžè¬ã«é£ç«æ¹çšåŒã®å
ã®æ°ã¯2åãšã¯éããªãããæ¹çšåŒã®æ°ã2åãšã¯éããªãã®ã§ãä»ã®å Žåã«ãè¡åãå®çŸ©ã§ããããã«ãè¡åã®å®çŸ©ãæ¡åŒµããã
ã€ãã®ããã«ãæ°å€ã瞊暪ã«äžŠã¹ãŠãããããã®æ®µã®æåã®åæ°ãçãããã®ã è¡å(ããããã€ãè±:matrix) ãšåŒã¶ã
äŸãã°ã
ã¯è¡åã§ããã
ãã£ãœãã
ã¯ãæåã®åæ°ãäžèŽããªãã®ã§ãè¡åã§ã¯ãªãã
è¡åã®äžéšã®ã暪ã«äžŠãã æ°å€ã®ãããŸãã è¡(ããããè±:row) ãšããã瞊ã«äžŠãã æ°å€ã®ãããŸãã å(ãã€ãè±:column) ãšãããããããã®æ°å€ã æå(ããã¶ããè±:element) ãšåŒã¶ã
äŸãã°ã
ã¯2è¡ã3åãããªãè¡åã§ããã
è¡æ°ãmã§ãåæ°ãnã®è¡åã mÃnè¡å ã®ããã«åŒã³ãç¹ã«è¡æ°ãšåæ°ãçããnã§ããè¡åãªãã° n次æ£æ¹è¡å ãšåŒã¶ã
äŸãã°ã
㯠2Ã3è¡å ã§ããã
第 i è¡ç¬¬ j åã®æåã (i, j) æåãšããã
äŸãã°ã
ã® (2, 1) æåã¯4ã§ããã
ã2ã€ã®è¡åãçããããšã¯ãè¡æ°ãšåæ°ãçããããã€å¯Ÿå¿ãã (i, j) æåããã¹ãŠçããããšãšå®ããã
ã€ãŸãã ( a b c d ) = ( e f g h ) {\displaystyle {\begin{pmatrix}a&&b\\c&&d\\\end{pmatrix}}={\begin{pmatrix}e&&f\\g&&h\\\end{pmatrix}}} ãšã¯ã a = e , b = f , c = g , d = h {\displaystyle a=e,b=f,c=g,d=h} ã§ããã
ãã 1è¡ãããªãè¡åãè¡ãã¯ãã«(ããããã¯ãã«ãè±:row vector)ãšããããã 1åãããªãåãã¯ãã«(ãã€ãã¯ãã«ãè±:column vector )ãšããã
ãã®è¡åã®å®çŸ©ã¯ããã¯ãã«ã®å®çŸ©ãæ¡åŒµãããã®ã«ãªã£ãŠããã
ããšãã°ãã¯ãã«(aãb)ãš(cãd)ã®å
ç© ac+bdã¯ãè¡åã®èšæ³ã䜿ããšã
ãšæžããã
å³èŸºã® ( a c + b d ) {\displaystyle {\begin{pmatrix}ac+bd\end{pmatrix}}} ã¯ã1è¡1åã®è¡åã§ããããã®ããã«ãè¡åã§ã¯ã1è¡1åã®è¡åãèªããã
è¡åã®ç©ã® (i, j) æåã®å€ã¯ãå·ŠåŽã®è¡åã® i è¡ã®ãã¯ãã«ãšãå³åŽã®è¡åã®ç¬¬ j åã®ãã¯ãã«ã®å
ç©ã§ããã
ããšãã°ãè¡å A = ( a b c d ) {\displaystyle A={\begin{pmatrix}a&b\\c&d\end{pmatrix}}} ãš B = ( e f g h ) {\displaystyle B={\begin{pmatrix}e&f\\g&h\end{pmatrix}}} ã®ç© A B = ( a e + b g a f + b h c e + d g c f + d h ) {\displaystyle AB={\begin{pmatrix}ae+bg&af+bh\\ce+dg&cf+dh\end{pmatrix}}} ã®(1, 2) æåã§ãã a f + b h {\displaystyle af+bh} ã¯ã
ãã¯ãã« ( a b ) {\displaystyle {\begin{pmatrix}a&b\end{pmatrix}}} ãš ãã¯ãã« ( f h ) {\displaystyle {\begin{pmatrix}f\\h\end{pmatrix}}} ãšã® å
ç©ã«ãªã£ãŠããã
ãã®ããã«èãããšããè¡åããšã¯ããã¯ãã«ã䞊ã¹ããã®ããšãèšããã(ãã ã䞊ã¹ããã¯ãã«ã®æ¬¡å
ã¯åã次å
ã§ãªããã°ãªããªãã)
ããããã°ãé£ç«1次æ¹çšåŒã
ã¯ãè¡åãçšããŠ
ãšè¡šããã
äŸé¡
次ã®w, x, y, zã®å€ãæ±ããã
ããããã
è¡åã®åã»å·®ã»å®æ°åã®å®çŸ©ã¯ã次ã®ããã«ããã¯ãã«ã®åã»å·®ã»å®æ°åãšäŒŒããããªæ§è³ªãæã€ã
è¡åã®åã®å®çŸ©ã¯ãåèŠçŽ ããšã«è¶³ãåãããããšå®çŸ©ãããã
è¡åã®å·®ã®å®çŸ©ã¯ãåèŠçŽ ããšã«åŒããšå®çŸ©ããã
å®æ°åã®å®çŸ©ã¯ãåèŠçŽ ã«å®æ°ãæããããšã«ãã£ãŠå®çŸ©ããã
(-1)A 㯠-A ãšæžãã
äŸé¡
è¡åA,B,Cã
ã§å®çŸ©ãããšãã
ãèšç®ããã
ããããã
ãšãªãã
é¶è¡å
ãã¹ãŠã®æåã0ã§ããè¡åã ãŒãè¡å(ããããããã€ãè±:zero matrix) ãšããã
( 0 0 0 0 0 0 ) {\displaystyle {\begin{pmatrix}0&0&0\\0&0&0\\\end{pmatrix}}} 㯠ãŒãè¡å ã§ããã
Aãè¡åãOãAãšè¡æ°ã»åæ°ãçããé¶è¡åãšãããšã
ãæºããã
äŸé¡
äžã§çšããè¡å A {\displaystyle A} , B {\displaystyle B} , C {\displaystyle C} ã«ã€ããŠã
ãèšç®ããã
ããããã
ã§ããã
ãã®çµæããåããéããäžè¬ã«è¡åã®ç©ã¯
ãšãªãã
ãšãªãå Žåãè¡åAãšè¡åBã¯äº€æå¯èœ(å¯æ)ã§ãããšããã
åäœè¡å
E = ( 1 0 0 1 ) {\displaystyle E={\begin{pmatrix}1&0\\0&1\end{pmatrix}}}
ãã2Ã2ã®åäœè¡å(2次åäœè¡å)ãšåŒã¶ã察è§æåã ãã1ã§ããããã®ä»ã®æåããã¹ãŠ0ã«çããè¡åã§ãããä»»æã®2Ã2è¡åAã«å¯ŸããŠãEã¯
ãæºããã
è¡åAã«å¯ŸããŠãã®è¡åãšã®ç©ãåäœè¡å A A â 1 = A â 1 A = E {\displaystyle AA^{-1}=A^{-1}A=E} ãšãªãè¡å A â 1 {\displaystyle A^{-1}} ãããã®è¡åã®éè¡åãšåŒã¶ããã®ãããªè¡åã¯ããååšããã°åAã«å¯ŸããŠãã ã²ãšã€ã«å®ãŸãããã¡ããäžè¬ã«ã¯Aã«å¯ŸããŠå³åŽããããããå·ŠåŽããããããã«ãã£ãŠç©ã¯ç°ãªãã®ã ãããã®å Žåã¯Aã«å¯ŸããŠå³ãããããŠåäœè¡åã«ãªãã®ãªãã°å·ŠãããããŠãåäœè¡åã«ãªãããéããŸããããã§ããããšã«æ³šæããŠãããéè¡åã®éè¡åã¯ããšã®è¡åã«çããã
2è¡2åã®è¡å A = ( a b c d ) {\displaystyle A={\begin{pmatrix}a&b\\c&d\end{pmatrix}}} ã«ã€ããŠã¯ã a d â b c â 0 {\displaystyle ad-bc\neq 0} ã®ãšã A â 1 = 1 ( a d â b c ) ( d â b â c a ) {\displaystyle A^{-1}={\frac {1}{(ad-bc)}}{\begin{pmatrix}d&-b\\-c&a\end{pmatrix}}} ãšãªãã ad - bc = 0 ã®ãšããéè¡åã¯ååšããªãã
å®éã«è¡åã®ç©ãåãããšã§ããããæ£ããããšã容æã«ãããã
äŸé¡
äžã§å®ããè¡å A {\displaystyle A} , B {\displaystyle B} , C {\displaystyle C} ã®éè¡åãèšç®ããã
è¡åA,B,Cã¯ããããã
ã§ãã£ãã
ããããã
ã§ããã
1次æ¹çšåŒ
ã¯ã
ãšæžããã䞡蟺ã«å·ŠèŸºã®è¡åã®éè¡åãæãããšã
x = 1, y = 0 ãåŸãããå§ãã®é£ç«1次æ¹çšåŒã解ããããšã«ãªãã äžè¬ã«ãé£ç«1次æ¹çšåŒããã äžçµã®è§£ããã€ãšããé£ç«1次æ¹çšåŒã解ãããšã¯éè¡åãæ±ããããšãšåãã§ããã ç¹ã«ã2Ã2è¡åã®éè¡åã¯æ¢ã«å
¬åŒãåŸãããŠããã®ã§ã2å
1次æ¹çšåŒã¯ç°¡åã«è§£ãããšãã§ããã
A = ( a b c d ) , x = ( x y ) , b = ( p q ) {\displaystyle A={\begin{pmatrix}a&&b\\c&&d\end{pmatrix}},\mathbf {x} ={\begin{pmatrix}x\\y\end{pmatrix}},\mathbf {b} ={\begin{pmatrix}p\\q\end{pmatrix}}} ãšãããš
ãšæžãããããã§Aããã®é£ç«1次æ¹çšåŒã®ä¿æ°è¡åãšããããã®æ¹çšåŒã®è§£ã¯ãAãéè¡åãæã€ãšãäžæã«å®ãŸãã x = A â 1 b {\displaystyle \mathbf {x} =A^{-1}\mathbf {b} } ã§ããã
å¹³é¢äžã®ãã¯ãã« a â {\displaystyle {\vec {a}}} ã«å¯ŸããŠå転è¡å R = ( cos c â sin c sin c cos c ) {\displaystyle R={\begin{pmatrix}\cos c&-\sin c\\\sin c&\cos c\end{pmatrix}}} ããããç© R a â {\displaystyle R{\vec {a}}} ã¯ã a â {\displaystyle {\vec {a}}} ãåç¹ãäžå¿ã«ããŠè§åºŠcã ãå転ããããã¯ãã«ã«ãªã£ãŠããã
座æšå€(x,y)ã®ç¹Pãè¡åããããããšã§ç§»åãããã®ãèããã
ã¯ã
x ( a c ) + y ( b d ) {\displaystyle x{\begin{pmatrix}a\\c\end{pmatrix}}+y{\begin{pmatrix}b\\d\end{pmatrix}}} ãšãæžããã
ããã¯ãæ°ããªçŽç·åº§æšãçšæã(æ°åº§æšã®å座æšè»žã®åäœãã¯ãã«ã¯åã®åº§æšãåºæºã«æž¬ããšãããããæ¹åãã¯ãã« ( a c ) {\displaystyle {\begin{pmatrix}a\\c\end{pmatrix}}} ããã³ æ¹åãã¯ãã« ( b d ) {\displaystyle {\begin{pmatrix}b\\d\end{pmatrix}}} ã§ããã)ããã®åº§æšã«åº§æšå€(x,y)ã代å
¥ããããšã§ç¹Pã移åãããã®ããåã®åº§æšç³»ã§æž¬ã£ãå Žåã®åº§æšå€ã«ãªã£ãŠããã
éåžžã®çŽäº€åº§æš(åç¹ã§90°ã§äº€ãã座æš)ã®äžã®ç¹ã®åº§æš(x,y)ã«ã€ããŠãç¹ã®äœçœ®ã¯åããŸãŸãæ°ããªå¥ã®çŽç·åº§æš(çŽäº€ãšã¯éããªã)ã§èŠãå Žåã®åº§æš(z,w)ãèãããæ°ããªå¥åº§æš(çŽç·åº§æš)ã¯ãèšç®ã®éœåäžãåç¹ã ãã¯å
ã®åº§æšãšåããšããããããšã次ã®ããã«ãåã®åº§æšãšæ°ããªåº§æšãšã®é¢ä¿ããè¡åã§è¡šèšã§ããã
ãšãããµããªé¢ä¿åŒã§èšè¿°ã§ããã å®éã«ãããšãã° (x,y)=(0,0) ã®ãšã ( z,w=0,0) ãšãªã£ãŠããã
ããŠã巊蟺㯠z ( a c ) + w ( b d ) {\displaystyle z{\begin{pmatrix}a\\c\end{pmatrix}}+w{\begin{pmatrix}b\\d\end{pmatrix}}} ãšãæžããã
ãã®åŒãã座æšã®å€æã®å¹ŸäœåŠãšããŠèããå Žåã次ã®ãããªçè«ã«ãªãã
ãŸããæ°ããªçŽç·åº§æšã®åº§æšè»žã®åäœãã¯ãã«ã®æ¹åã¯ãããšã®åº§æšç³»ãåºæºã«èŠããšãããããæ¹åãã¯ãã« ( a c ) {\displaystyle {\begin{pmatrix}a\\c\end{pmatrix}}} ããã³ æ¹åãã¯ãã« ( b d ) {\displaystyle {\begin{pmatrix}b\\d\end{pmatrix}}} ã§ããã
ããŠããã®åé¡ã§ã¯ç¹Pã®äœçœ®(xãy)ã¯äœãå€æããŠãããããã£ãŠãåã®åº§æšãåºæºã«ããŠç¹Pã®äœçœ®ãèŠãŠããäœãå€åããªãããã®åé¡ã§å€æŽããã®ã¯åº§æšè»žã®ã»ãã§ãããããæ°ããªåº§æšç³»ã§èŠãç¹Pã®å€(z,w)ã«èå³ãããã®ã§ããã
å¹³é¢å³åœ¢äžã®ç·åã¯ã2è¡2åã®è¡åã§å€æã§ããã
A = ( a b c d ) {\displaystyle A={\begin{pmatrix}a&b\\c&d\end{pmatrix}}} ã§å€æããå Žåã«ã€ããŠã¯ã a d â b c â 0 {\displaystyle ad-bc\neq 0} ã®ãšããç·ã¯ç·ã«å€æãããåè§åœ¢ã¯åè§åœ¢ã«å€æãããäžè§åœ¢ã¯äžè§åœ¢ã«å€æãããã
2è¡2åã®è¡å A = ( a b c d ) {\displaystyle A={\begin{pmatrix}a&b\\c&d\end{pmatrix}}} ã«ã€ããŠã¯ãå³åœ¢ã®é¢ç©ã¯ã a d â b c {\displaystyle ad-bc} åãããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "é«çåŠæ ¡æ°åŠC > è¡å",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "æ¬é
ã¯é«çåŠæ ¡æ°åŠCã®è¡åã®è§£èª¬ã§ããã",
"title": ""
},
{
"paragraph_id": 2,
"tag": "p",
"text": "1次æ¹çšåŒ",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ãã次ã®ãããªèšæ³ã§è¡šããŠã¿ãã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ããããå匷ããã®ã¯ãé£ç«æ¹çšåŒãšãã¯ãã«ãšã®é¢ä¿ã§ããããããèå¯ããããããããã«ããããã«è¡å(ããããã€)ãšããéãå°å
¥ããã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ãã¯ãã« ( x y ) {\\displaystyle {\\begin{pmatrix}x\\\\y\\end{pmatrix}}} ã«ã æŒç® ( 1 2 2 3 ) {\\displaystyle {\\begin{pmatrix}1&2\\\\2&3\\end{pmatrix}}} ãæœããŠ(ãã®æŒç®ã®å
容ãããããããã説æãããè¡åãã§ãã)ã çãã®ãã¯ãã« ( 1 2 ) {\\displaystyle {\\begin{pmatrix}1\\\\2\\end{pmatrix}}} ãåŸãããšããè¡šçŸã«æžãæããã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ãŸãããã®ãããªèšæ³ãããããã次ã«èª¬æããè¡å(ããããã€ãè±:matrix)ãšããéãæ°ãã«å®çŸ©ããã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ãŸããè¡åã©ããã®ç©ã®å®çŸ©ãã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "ã¯ã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "ãšçããããšå®ããã äœæ
ãã®ããã«å®ããã®ããèãããã 2ã€ã®é£ç«æ¹çšåŒ",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "ã«ãããŠãäžéçå€æ°p,qãæ¶å»ããŠãå€æ°x,yã«é¢ããäžã€ã®é£ç«æ¹çšåŒãšæžãçŽããš",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "ãšãªãã å®éãäž2åŒã®p,qã«ãäž2åŒã代å
¥ããŠæŽé ããã°ããã èªè
ã¯ä»£å
¥ããŠç¢ºèªããã ãããè¡åè¡šçŸãããš",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "ä»æ¹ã2ã€ã®é£ç«æ¹çšåŒãè¡åãçšããŠæžãçŽããš",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "äžã®åŒãäžã®åŒã«ã圢åŒçã«ä»£å
¥ãããš",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "2ã€ã®è¡åè¡šçŸåŒãæ¯èŒããã°ã è¡åã®ç©ã®å®ãæ¹ã®åçæ§ãåããã ããã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "ç©ã®å®çŸ©åŒã¯ãäžèŠãããšè€éããã«èŠããããããã«è£å©ç·ã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "ã®ããã«åŒããŠã¿ãã°åããããã«ãããšãã°åæåŸã®2è¡1åã ( c e + d g ) {\\displaystyle {\\begin{pmatrix}&\\\\ce+dg&\\qquad \\end{pmatrix}}} ã¯ãåæåã®2è¡ãã®ããããã®æå ( c d ) {\\displaystyle {\\begin{pmatrix}&\\\\c&d\\end{pmatrix}}} ãšãåæåã®1åç® ( e g ) {\\displaystyle {\\begin{pmatrix}e&\\\\g&\\end{pmatrix}}} ã®æåãšããæããŠè¶³ãããã®ã«ãªã£ãŠããã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "äžè¬ã«ãç©ã®åæåŸã®xè¡yåãã¯ãåæåã®xè¡ãã®ããããã®æåãšãåæåã®yåç®ã®ããããã®æåãšããæããŠè¶³ããçµæã«ãªã£ãŠããã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "è¡åã©ããã®ç©ã¯ãé åºã«ãã£ãŠçµæãç°ãªãã ããšãã°è¡åA,Bã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "ãšãããšãã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "ããããã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "ãšãªãã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "ãã®ããã«ãäžè¬ã®è¡åAãšè¡åBã®ç©ã¯ãäžè¬ã«",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "ãšãªãã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "äžè¿°ã®äŸã¯ã2å
é£ç«äžæ¬¡æ¹çšåŒãåŒ2åã®å Žåã«çžåœããè¡åã ã£ãããäžè¬ã«é£ç«æ¹çšåŒã®å
ã®æ°ã¯2åãšã¯éããªãããæ¹çšåŒã®æ°ã2åãšã¯éããªãã®ã§ãä»ã®å Žåã«ãè¡åãå®çŸ©ã§ããããã«ãè¡åã®å®çŸ©ãæ¡åŒµããã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "ã€ãã®ããã«ãæ°å€ã瞊暪ã«äžŠã¹ãŠãããããã®æ®µã®æåã®åæ°ãçãããã®ã è¡å(ããããã€ãè±:matrix) ãšåŒã¶ã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "äŸãã°ã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "ã¯è¡åã§ããã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "ãã£ãœãã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "ã¯ãæåã®åæ°ãäžèŽããªãã®ã§ãè¡åã§ã¯ãªãã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "è¡åã®äžéšã®ã暪ã«äžŠãã æ°å€ã®ãããŸãã è¡(ããããè±:row) ãšããã瞊ã«äžŠãã æ°å€ã®ãããŸãã å(ãã€ãè±:column) ãšãããããããã®æ°å€ã æå(ããã¶ããè±:element) ãšåŒã¶ã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "äŸãã°ã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "ã¯2è¡ã3åãããªãè¡åã§ããã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "è¡æ°ãmã§ãåæ°ãnã®è¡åã mÃnè¡å ã®ããã«åŒã³ãç¹ã«è¡æ°ãšåæ°ãçããnã§ããè¡åãªãã° n次æ£æ¹è¡å ãšåŒã¶ã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "äŸãã°ã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "㯠2Ã3è¡å ã§ããã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "第 i è¡ç¬¬ j åã®æåã (i, j) æåãšããã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "äŸãã°ã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "ã® (2, 1) æåã¯4ã§ããã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "ã2ã€ã®è¡åãçããããšã¯ãè¡æ°ãšåæ°ãçããããã€å¯Ÿå¿ãã (i, j) æåããã¹ãŠçããããšãšå®ããã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "ã€ãŸãã ( a b c d ) = ( e f g h ) {\\displaystyle {\\begin{pmatrix}a&&b\\\\c&&d\\\\\\end{pmatrix}}={\\begin{pmatrix}e&&f\\\\g&&h\\\\\\end{pmatrix}}} ãšã¯ã a = e , b = f , c = g , d = h {\\displaystyle a=e,b=f,c=g,d=h} ã§ããã",
"title": "é£ç«äžæ¬¡æ¹çšåŒãšè¡å"
},
{
"paragraph_id": 41,
"tag": "p",
"text": "ãã 1è¡ãããªãè¡åãè¡ãã¯ãã«(ããããã¯ãã«ãè±:row vector)ãšããããã 1åãããªãåãã¯ãã«(ãã€ãã¯ãã«ãè±:column vector )ãšããã",
"title": "ãã¯ãã«å
ç©ãšè¡å"
},
{
"paragraph_id": 42,
"tag": "p",
"text": "ãã®è¡åã®å®çŸ©ã¯ããã¯ãã«ã®å®çŸ©ãæ¡åŒµãããã®ã«ãªã£ãŠããã",
"title": "ãã¯ãã«å
ç©ãšè¡å"
},
{
"paragraph_id": 43,
"tag": "p",
"text": "ããšãã°ãã¯ãã«(aãb)ãš(cãd)ã®å
ç© ac+bdã¯ãè¡åã®èšæ³ã䜿ããšã",
"title": "ãã¯ãã«å
ç©ãšè¡å"
},
{
"paragraph_id": 44,
"tag": "p",
"text": "ãšæžããã",
"title": "ãã¯ãã«å
ç©ãšè¡å"
},
{
"paragraph_id": 45,
"tag": "p",
"text": "å³èŸºã® ( a c + b d ) {\\displaystyle {\\begin{pmatrix}ac+bd\\end{pmatrix}}} ã¯ã1è¡1åã®è¡åã§ããããã®ããã«ãè¡åã§ã¯ã1è¡1åã®è¡åãèªããã",
"title": "ãã¯ãã«å
ç©ãšè¡å"
},
{
"paragraph_id": 46,
"tag": "p",
"text": "è¡åã®ç©ã® (i, j) æåã®å€ã¯ãå·ŠåŽã®è¡åã® i è¡ã®ãã¯ãã«ãšãå³åŽã®è¡åã®ç¬¬ j åã®ãã¯ãã«ã®å
ç©ã§ããã",
"title": "ãã¯ãã«å
ç©ãšè¡å"
},
{
"paragraph_id": 47,
"tag": "p",
"text": "ããšãã°ãè¡å A = ( a b c d ) {\\displaystyle A={\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}}} ãš B = ( e f g h ) {\\displaystyle B={\\begin{pmatrix}e&f\\\\g&h\\end{pmatrix}}} ã®ç© A B = ( a e + b g a f + b h c e + d g c f + d h ) {\\displaystyle AB={\\begin{pmatrix}ae+bg&af+bh\\\\ce+dg&cf+dh\\end{pmatrix}}} ã®(1, 2) æåã§ãã a f + b h {\\displaystyle af+bh} ã¯ã",
"title": "ãã¯ãã«å
ç©ãšè¡å"
},
{
"paragraph_id": 48,
"tag": "p",
"text": "ãã¯ãã« ( a b ) {\\displaystyle {\\begin{pmatrix}a&b\\end{pmatrix}}} ãš ãã¯ãã« ( f h ) {\\displaystyle {\\begin{pmatrix}f\\\\h\\end{pmatrix}}} ãšã® å
ç©ã«ãªã£ãŠããã",
"title": "ãã¯ãã«å
ç©ãšè¡å"
},
{
"paragraph_id": 49,
"tag": "p",
"text": "ãã®ããã«èãããšããè¡åããšã¯ããã¯ãã«ã䞊ã¹ããã®ããšãèšããã(ãã ã䞊ã¹ããã¯ãã«ã®æ¬¡å
ã¯åã次å
ã§ãªããã°ãªããªãã)",
"title": "ãã¯ãã«å
ç©ãšè¡å"
},
{
"paragraph_id": 50,
"tag": "p",
"text": "ããããã°ãé£ç«1次æ¹çšåŒã",
"title": "ãã¯ãã«å
ç©ãšè¡å"
},
{
"paragraph_id": 51,
"tag": "p",
"text": "ã¯ãè¡åãçšããŠ",
"title": "ãã¯ãã«å
ç©ãšè¡å"
},
{
"paragraph_id": 52,
"tag": "p",
"text": "ãšè¡šããã",
"title": "ãã¯ãã«å
ç©ãšè¡å"
},
{
"paragraph_id": 53,
"tag": "p",
"text": "äŸé¡",
"title": "ãã¯ãã«å
ç©ãšè¡å"
},
{
"paragraph_id": 54,
"tag": "p",
"text": "次ã®w, x, y, zã®å€ãæ±ããã",
"title": "ãã¯ãã«å
ç©ãšè¡å"
},
{
"paragraph_id": 55,
"tag": "p",
"text": "ããããã",
"title": "ãã¯ãã«å
ç©ãšè¡å"
},
{
"paragraph_id": 56,
"tag": "p",
"text": "è¡åã®åã»å·®ã»å®æ°åã®å®çŸ©ã¯ã次ã®ããã«ããã¯ãã«ã®åã»å·®ã»å®æ°åãšäŒŒããããªæ§è³ªãæã€ã",
"title": "è¡åã®åïŒå·®ïŒå®æ°å"
},
{
"paragraph_id": 57,
"tag": "p",
"text": "è¡åã®åã®å®çŸ©ã¯ãåèŠçŽ ããšã«è¶³ãåãããããšå®çŸ©ãããã",
"title": "è¡åã®åïŒå·®ïŒå®æ°å"
},
{
"paragraph_id": 58,
"tag": "p",
"text": "è¡åã®å·®ã®å®çŸ©ã¯ãåèŠçŽ ããšã«åŒããšå®çŸ©ããã",
"title": "è¡åã®åïŒå·®ïŒå®æ°å"
},
{
"paragraph_id": 59,
"tag": "p",
"text": "å®æ°åã®å®çŸ©ã¯ãåèŠçŽ ã«å®æ°ãæããããšã«ãã£ãŠå®çŸ©ããã",
"title": "è¡åã®åïŒå·®ïŒå®æ°å"
},
{
"paragraph_id": 60,
"tag": "p",
"text": "(-1)A 㯠-A ãšæžãã",
"title": "è¡åã®åïŒå·®ïŒå®æ°å"
},
{
"paragraph_id": 61,
"tag": "p",
"text": "äŸé¡",
"title": "è¡åã®åïŒå·®ïŒå®æ°å"
},
{
"paragraph_id": 62,
"tag": "p",
"text": "è¡åA,B,Cã",
"title": "è¡åã®åïŒå·®ïŒå®æ°å"
},
{
"paragraph_id": 63,
"tag": "p",
"text": "ã§å®çŸ©ãããšãã",
"title": "è¡åã®åïŒå·®ïŒå®æ°å"
},
{
"paragraph_id": 64,
"tag": "p",
"text": "ãèšç®ããã",
"title": "è¡åã®åïŒå·®ïŒå®æ°å"
},
{
"paragraph_id": 65,
"tag": "p",
"text": "ããããã",
"title": "è¡åã®åïŒå·®ïŒå®æ°å"
},
{
"paragraph_id": 66,
"tag": "p",
"text": "ãšãªãã",
"title": "è¡åã®åïŒå·®ïŒå®æ°å"
},
{
"paragraph_id": 67,
"tag": "p",
"text": "é¶è¡å",
"title": "è¡åã®åïŒå·®ïŒå®æ°å"
},
{
"paragraph_id": 68,
"tag": "p",
"text": "ãã¹ãŠã®æåã0ã§ããè¡åã ãŒãè¡å(ããããããã€ãè±:zero matrix) ãšããã",
"title": "è¡åã®åïŒå·®ïŒå®æ°å"
},
{
"paragraph_id": 69,
"tag": "p",
"text": "( 0 0 0 0 0 0 ) {\\displaystyle {\\begin{pmatrix}0&0&0\\\\0&0&0\\\\\\end{pmatrix}}} 㯠ãŒãè¡å ã§ããã",
"title": "è¡åã®åïŒå·®ïŒå®æ°å"
},
{
"paragraph_id": 70,
"tag": "p",
"text": "Aãè¡åãOãAãšè¡æ°ã»åæ°ãçããé¶è¡åãšãããšã",
"title": "è¡åã®åïŒå·®ïŒå®æ°å"
},
{
"paragraph_id": 71,
"tag": "p",
"text": "ãæºããã",
"title": "è¡åã®åïŒå·®ïŒå®æ°å"
},
{
"paragraph_id": 72,
"tag": "p",
"text": "äŸé¡",
"title": "è¡åã®ç©"
},
{
"paragraph_id": 73,
"tag": "p",
"text": "äžã§çšããè¡å A {\\displaystyle A} , B {\\displaystyle B} , C {\\displaystyle C} ã«ã€ããŠã",
"title": "è¡åã®ç©"
},
{
"paragraph_id": 74,
"tag": "p",
"text": "ãèšç®ããã",
"title": "è¡åã®ç©"
},
{
"paragraph_id": 75,
"tag": "p",
"text": "ããããã",
"title": "è¡åã®ç©"
},
{
"paragraph_id": 76,
"tag": "p",
"text": "ã§ããã",
"title": "è¡åã®ç©"
},
{
"paragraph_id": 77,
"tag": "p",
"text": "ãã®çµæããåããéããäžè¬ã«è¡åã®ç©ã¯",
"title": "è¡åã®ç©"
},
{
"paragraph_id": 78,
"tag": "p",
"text": "ãšãªãã",
"title": "è¡åã®ç©"
},
{
"paragraph_id": 79,
"tag": "p",
"text": "ãšãªãå Žåãè¡åAãšè¡åBã¯äº€æå¯èœ(å¯æ)ã§ãããšããã",
"title": "è¡åã®ç©"
},
{
"paragraph_id": 80,
"tag": "p",
"text": "åäœè¡å",
"title": "è¡åã®ç©"
},
{
"paragraph_id": 81,
"tag": "p",
"text": "E = ( 1 0 0 1 ) {\\displaystyle E={\\begin{pmatrix}1&0\\\\0&1\\end{pmatrix}}}",
"title": "è¡åã®ç©"
},
{
"paragraph_id": 82,
"tag": "p",
"text": "ãã2Ã2ã®åäœè¡å(2次åäœè¡å)ãšåŒã¶ã察è§æåã ãã1ã§ããããã®ä»ã®æåããã¹ãŠ0ã«çããè¡åã§ãããä»»æã®2Ã2è¡åAã«å¯ŸããŠãEã¯",
"title": "è¡åã®ç©"
},
{
"paragraph_id": 83,
"tag": "p",
"text": "ãæºããã",
"title": "è¡åã®ç©"
},
{
"paragraph_id": 84,
"tag": "p",
"text": "è¡åAã«å¯ŸããŠãã®è¡åãšã®ç©ãåäœè¡å A A â 1 = A â 1 A = E {\\displaystyle AA^{-1}=A^{-1}A=E} ãšãªãè¡å A â 1 {\\displaystyle A^{-1}} ãããã®è¡åã®éè¡åãšåŒã¶ããã®ãããªè¡åã¯ããååšããã°åAã«å¯ŸããŠãã ã²ãšã€ã«å®ãŸãããã¡ããäžè¬ã«ã¯Aã«å¯ŸããŠå³åŽããããããå·ŠåŽããããããã«ãã£ãŠç©ã¯ç°ãªãã®ã ãããã®å Žåã¯Aã«å¯ŸããŠå³ãããããŠåäœè¡åã«ãªãã®ãªãã°å·ŠãããããŠãåäœè¡åã«ãªãããéããŸããããã§ããããšã«æ³šæããŠãããéè¡åã®éè¡åã¯ããšã®è¡åã«çããã",
"title": "éè¡å"
},
{
"paragraph_id": 85,
"tag": "p",
"text": "2è¡2åã®è¡å A = ( a b c d ) {\\displaystyle A={\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}}} ã«ã€ããŠã¯ã a d â b c â 0 {\\displaystyle ad-bc\\neq 0} ã®ãšã A â 1 = 1 ( a d â b c ) ( d â b â c a ) {\\displaystyle A^{-1}={\\frac {1}{(ad-bc)}}{\\begin{pmatrix}d&-b\\\\-c&a\\end{pmatrix}}} ãšãªãã ad - bc = 0 ã®ãšããéè¡åã¯ååšããªãã",
"title": "éè¡å"
},
{
"paragraph_id": 86,
"tag": "p",
"text": "å®éã«è¡åã®ç©ãåãããšã§ããããæ£ããããšã容æã«ãããã",
"title": "éè¡å"
},
{
"paragraph_id": 87,
"tag": "p",
"text": "äŸé¡",
"title": "éè¡å"
},
{
"paragraph_id": 88,
"tag": "p",
"text": "äžã§å®ããè¡å A {\\displaystyle A} , B {\\displaystyle B} , C {\\displaystyle C} ã®éè¡åãèšç®ããã",
"title": "éè¡å"
},
{
"paragraph_id": 89,
"tag": "p",
"text": "è¡åA,B,Cã¯ããããã",
"title": "éè¡å"
},
{
"paragraph_id": 90,
"tag": "p",
"text": "ã§ãã£ãã",
"title": "éè¡å"
},
{
"paragraph_id": 91,
"tag": "p",
"text": "",
"title": "éè¡å"
},
{
"paragraph_id": 92,
"tag": "p",
"text": "ããããã",
"title": "éè¡å"
},
{
"paragraph_id": 93,
"tag": "p",
"text": "ã§ããã",
"title": "éè¡å"
},
{
"paragraph_id": 94,
"tag": "p",
"text": "1次æ¹çšåŒ",
"title": "éè¡åãçšããé£ç«äžæ¬¡æ¹çšåŒã®è§£æ³"
},
{
"paragraph_id": 95,
"tag": "p",
"text": "ã¯ã",
"title": "éè¡åãçšããé£ç«äžæ¬¡æ¹çšåŒã®è§£æ³"
},
{
"paragraph_id": 96,
"tag": "p",
"text": "ãšæžããã䞡蟺ã«å·ŠèŸºã®è¡åã®éè¡åãæãããšã",
"title": "éè¡åãçšããé£ç«äžæ¬¡æ¹çšåŒã®è§£æ³"
},
{
"paragraph_id": 97,
"tag": "p",
"text": "x = 1, y = 0 ãåŸãããå§ãã®é£ç«1次æ¹çšåŒã解ããããšã«ãªãã äžè¬ã«ãé£ç«1次æ¹çšåŒããã äžçµã®è§£ããã€ãšããé£ç«1次æ¹çšåŒã解ãããšã¯éè¡åãæ±ããããšãšåãã§ããã ç¹ã«ã2Ã2è¡åã®éè¡åã¯æ¢ã«å
¬åŒãåŸãããŠããã®ã§ã2å
1次æ¹çšåŒã¯ç°¡åã«è§£ãããšãã§ããã",
"title": "éè¡åãçšããé£ç«äžæ¬¡æ¹çšåŒã®è§£æ³"
},
{
"paragraph_id": 98,
"tag": "p",
"text": "A = ( a b c d ) , x = ( x y ) , b = ( p q ) {\\displaystyle A={\\begin{pmatrix}a&&b\\\\c&&d\\end{pmatrix}},\\mathbf {x} ={\\begin{pmatrix}x\\\\y\\end{pmatrix}},\\mathbf {b} ={\\begin{pmatrix}p\\\\q\\end{pmatrix}}} ãšãããš",
"title": "éè¡åãçšããé£ç«äžæ¬¡æ¹çšåŒã®è§£æ³"
},
{
"paragraph_id": 99,
"tag": "p",
"text": "ãšæžãããããã§Aããã®é£ç«1次æ¹çšåŒã®ä¿æ°è¡åãšããããã®æ¹çšåŒã®è§£ã¯ãAãéè¡åãæã€ãšãäžæã«å®ãŸãã x = A â 1 b {\\displaystyle \\mathbf {x} =A^{-1}\\mathbf {b} } ã§ããã",
"title": "éè¡åãçšããé£ç«äžæ¬¡æ¹çšåŒã®è§£æ³"
},
{
"paragraph_id": 100,
"tag": "p",
"text": "å¹³é¢äžã®ãã¯ãã« a â {\\displaystyle {\\vec {a}}} ã«å¯ŸããŠå転è¡å R = ( cos c â sin c sin c cos c ) {\\displaystyle R={\\begin{pmatrix}\\cos c&-\\sin c\\\\\\sin c&\\cos c\\end{pmatrix}}} ããããç© R a â {\\displaystyle R{\\vec {a}}} ã¯ã a â {\\displaystyle {\\vec {a}}} ãåç¹ãäžå¿ã«ããŠè§åºŠcã ãå転ããããã¯ãã«ã«ãªã£ãŠããã",
"title": "è¡åã®å¿çš"
},
{
"paragraph_id": 101,
"tag": "p",
"text": "座æšå€(x,y)ã®ç¹Pãè¡åããããããšã§ç§»åãããã®ãèããã",
"title": "è¡åã®å¿çš"
},
{
"paragraph_id": 102,
"tag": "p",
"text": "ã¯ã",
"title": "è¡åã®å¿çš"
},
{
"paragraph_id": 103,
"tag": "p",
"text": "x ( a c ) + y ( b d ) {\\displaystyle x{\\begin{pmatrix}a\\\\c\\end{pmatrix}}+y{\\begin{pmatrix}b\\\\d\\end{pmatrix}}} ãšãæžããã",
"title": "è¡åã®å¿çš"
},
{
"paragraph_id": 104,
"tag": "p",
"text": "ããã¯ãæ°ããªçŽç·åº§æšãçšæã(æ°åº§æšã®å座æšè»žã®åäœãã¯ãã«ã¯åã®åº§æšãåºæºã«æž¬ããšãããããæ¹åãã¯ãã« ( a c ) {\\displaystyle {\\begin{pmatrix}a\\\\c\\end{pmatrix}}} ããã³ æ¹åãã¯ãã« ( b d ) {\\displaystyle {\\begin{pmatrix}b\\\\d\\end{pmatrix}}} ã§ããã)ããã®åº§æšã«åº§æšå€(x,y)ã代å
¥ããããšã§ç¹Pã移åãããã®ããåã®åº§æšç³»ã§æž¬ã£ãå Žåã®åº§æšå€ã«ãªã£ãŠããã",
"title": "è¡åã®å¿çš"
},
{
"paragraph_id": 105,
"tag": "p",
"text": "éåžžã®çŽäº€åº§æš(åç¹ã§90°ã§äº€ãã座æš)ã®äžã®ç¹ã®åº§æš(x,y)ã«ã€ããŠãç¹ã®äœçœ®ã¯åããŸãŸãæ°ããªå¥ã®çŽç·åº§æš(çŽäº€ãšã¯éããªã)ã§èŠãå Žåã®åº§æš(z,w)ãèãããæ°ããªå¥åº§æš(çŽç·åº§æš)ã¯ãèšç®ã®éœåäžãåç¹ã ãã¯å
ã®åº§æšãšåããšããããããšã次ã®ããã«ãåã®åº§æšãšæ°ããªåº§æšãšã®é¢ä¿ããè¡åã§è¡šèšã§ããã",
"title": "è¡åã®å¿çš"
},
{
"paragraph_id": 106,
"tag": "p",
"text": "ãšãããµããªé¢ä¿åŒã§èšè¿°ã§ããã å®éã«ãããšãã° (x,y)=(0,0) ã®ãšã ( z,w=0,0) ãšãªã£ãŠããã",
"title": "è¡åã®å¿çš"
},
{
"paragraph_id": 107,
"tag": "p",
"text": "ããŠã巊蟺㯠z ( a c ) + w ( b d ) {\\displaystyle z{\\begin{pmatrix}a\\\\c\\end{pmatrix}}+w{\\begin{pmatrix}b\\\\d\\end{pmatrix}}} ãšãæžããã",
"title": "è¡åã®å¿çš"
},
{
"paragraph_id": 108,
"tag": "p",
"text": "ãã®åŒãã座æšã®å€æã®å¹ŸäœåŠãšããŠèããå Žåã次ã®ãããªçè«ã«ãªãã",
"title": "è¡åã®å¿çš"
},
{
"paragraph_id": 109,
"tag": "p",
"text": "ãŸããæ°ããªçŽç·åº§æšã®åº§æšè»žã®åäœãã¯ãã«ã®æ¹åã¯ãããšã®åº§æšç³»ãåºæºã«èŠããšãããããæ¹åãã¯ãã« ( a c ) {\\displaystyle {\\begin{pmatrix}a\\\\c\\end{pmatrix}}} ããã³ æ¹åãã¯ãã« ( b d ) {\\displaystyle {\\begin{pmatrix}b\\\\d\\end{pmatrix}}} ã§ããã",
"title": "è¡åã®å¿çš"
},
{
"paragraph_id": 110,
"tag": "p",
"text": "ããŠããã®åé¡ã§ã¯ç¹Pã®äœçœ®(xãy)ã¯äœãå€æããŠãããããã£ãŠãåã®åº§æšãåºæºã«ããŠç¹Pã®äœçœ®ãèŠãŠããäœãå€åããªãããã®åé¡ã§å€æŽããã®ã¯åº§æšè»žã®ã»ãã§ãããããæ°ããªåº§æšç³»ã§èŠãç¹Pã®å€(z,w)ã«èå³ãããã®ã§ããã",
"title": "è¡åã®å¿çš"
},
{
"paragraph_id": 111,
"tag": "p",
"text": "å¹³é¢å³åœ¢äžã®ç·åã¯ã2è¡2åã®è¡åã§å€æã§ããã",
"title": "è¡åã®å¿çš"
},
{
"paragraph_id": 112,
"tag": "p",
"text": "A = ( a b c d ) {\\displaystyle A={\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}}} ã§å€æããå Žåã«ã€ããŠã¯ã a d â b c â 0 {\\displaystyle ad-bc\\neq 0} ã®ãšããç·ã¯ç·ã«å€æãããåè§åœ¢ã¯åè§åœ¢ã«å€æãããäžè§åœ¢ã¯äžè§åœ¢ã«å€æãããã",
"title": "è¡åã®å¿çš"
},
{
"paragraph_id": 113,
"tag": "p",
"text": "2è¡2åã®è¡å A = ( a b c d ) {\\displaystyle A={\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}}} ã«ã€ããŠã¯ãå³åœ¢ã®é¢ç©ã¯ã a d â b c {\\displaystyle ad-bc} åãããã",
"title": "è¡åã®å¿çš"
}
] | é«çåŠæ ¡æ°åŠC > è¡å æ¬é
ã¯é«çåŠæ ¡æ°åŠCã®è¡åã®è§£èª¬ã§ããã | <small>[[é«çåŠæ ¡æ°åŠC]] > è¡å</small>
----
æ¬é
ã¯[[é«çåŠæ ¡æ°åŠC]]ã®è¡åã®è§£èª¬ã§ããã
== é£ç«äžæ¬¡æ¹çšåŒãšè¡å==
1次æ¹çšåŒ
:<math>
\begin{cases}
x + 2y = 1\\
2x + 3y = 2
\end{cases}
</math>
ãã次ã®ãããªèšæ³ã§è¡šããŠã¿ãã
:<math>
\begin{pmatrix}
1 &2\\
2 &3
\end{pmatrix}
\begin{pmatrix}
x\\
y
\end{pmatrix}
=
\begin{pmatrix}
1\\
2
\end{pmatrix}
</math>
ããããå匷ããã®ã¯ãé£ç«æ¹çšåŒãšãã¯ãã«ãšã®é¢ä¿ã§ããããããèå¯ããããããããã«ããããã«'''è¡å'''ïŒããããã€ïŒãšããéãå°å
¥ããã
ãã¯ãã«
<math>
\begin{pmatrix}
x\\
y
\end{pmatrix}
</math> ã«ã
æŒç® <math>
\begin{pmatrix}
1 &2\\
2 &3
\end{pmatrix}
</math> ãæœããŠïŒãã®æŒç®ã®å
容ãããããããã説æãããè¡åãã§ããïŒã
çãã®ãã¯ãã«
<math>
\begin{pmatrix}
1\\
2
\end{pmatrix}
</math> ãåŸãããšããè¡šçŸã«æžãæããã
ãŸãããã®ãããªèšæ³ãããããã次ã«èª¬æãã'''è¡å'''ïŒããããã€ãè±ïŒmatrixïŒãšããéãæ°ãã«å®çŸ©ããã
*è¡åã©ããã®ç©
ãŸããè¡åã©ããã®ç©ã®å®çŸ©ãã
:ç©ã<math>
\begin{pmatrix}
a& b \\
c& d
\end{pmatrix}
\begin{pmatrix}
e& f\\
g& h
\end{pmatrix}
</math>
ã¯ã
:è¡åã<math>
\begin{pmatrix}
ae + bg &af + bh\\
ce + dg &cf + dh
\end{pmatrix}
</math>
ãšçããããšå®ããã<br/>ã
äœæ
ãã®ããã«å®ããã®ããèãããã<br/>
ïŒã€ã®é£ç«æ¹çšåŒ
:<math>
\begin{cases}
ex + fy = p\\
gx + hy = q
\end{cases}
</math>
:<math>
\begin{cases}
ap + bq = u\\
cp + dq = v
\end{cases}
</math>
ã«ãããŠãäžéçå€æ°p,qãæ¶å»ããŠãå€æ°x,yã«é¢ããäžã€ã®é£ç«æ¹çšåŒãšæžãçŽããš
:<math>
\begin{cases}
(ae + bg)x +(af + bh)y = u\\
(ce + dg)x + (cf + dh)y = v
\end{cases}
</math>
ãšãªãã<br/>ã
å®éãäž2åŒã®p,qã«ãäž2åŒã代å
¥ããŠæŽé ããã°ããã
èªè
ã¯ä»£å
¥ããŠç¢ºèªããã<br/>
ãããè¡åè¡šçŸãããš
:<math>
\begin{pmatrix}
ae + bg &af + bh\\
ce + dg &cf + dh
\end{pmatrix}
\begin{pmatrix}
x\\
y
\end{pmatrix}
=
\begin{pmatrix}
u\\
v
\end{pmatrix}
</math>
ä»æ¹ã2ã€ã®é£ç«æ¹çšåŒãè¡åãçšããŠæžãçŽããš
:<math>
\begin{pmatrix}
e &f\\
g &h
\end{pmatrix}
\begin{pmatrix}
x\\
y
\end{pmatrix}
=
\begin{pmatrix}
p\\
q
\end{pmatrix}
</math>
:<math>
\begin{pmatrix}
a &b\\
c &d
\end{pmatrix}
\begin{pmatrix}
p\\
q
\end{pmatrix}
=
\begin{pmatrix}
u\\
v
\end{pmatrix}
</math>
äžã®åŒãäžã®åŒã«ã圢åŒçã«ä»£å
¥ãããš
:<math>
\begin{pmatrix}
a &b\\
c &d
\end{pmatrix}
\begin{pmatrix}
e &f\\
g &h
\end{pmatrix}
\begin{pmatrix}
x\\
y
\end{pmatrix}
=
\begin{pmatrix}
u\\
v
\end{pmatrix}
</math>
ïŒã€ã®è¡åè¡šçŸåŒãæ¯èŒããã°ã
è¡åã®ç©ã®å®ãæ¹ã®åçæ§ãåããã ããã<br/>ã
[[File:è¡åã®å®çŸ©ã®èª¬æå³.svg|thumb|è¡åã®å®çŸ©ã®èª¬æå³]]
ç©ã®å®çŸ©åŒã¯ãäžèŠãããšè€éããã«èŠããããããã«è£å©ç·ã
:[[File:è¡åã®ç©ã®èšç®æ³.svg|500px|è¡åã®ç©ã®èšç®æ³]]
ã®ããã«åŒããŠã¿ãã°åããããã«ãããšãã°åæåŸã®2è¡1åã <math>
\begin{pmatrix}
& \\
ce + dg & \qquad
\end{pmatrix}
</math> ã¯ãåæåã®2è¡ãã®ããããã®æå
<math>
\begin{pmatrix}
& \\
c& d
\end{pmatrix}
</math>
ãšãåæåã®1åç®
<math>
\begin{pmatrix}
e& \\
g&
\end{pmatrix}
</math>ã®æåãšããæããŠè¶³ãããã®ã«ãªã£ãŠããã
äžè¬ã«ãç©ã®åæåŸã®xè¡yåãã¯ãåæåã®xè¡ãã®ããããã®æåãšãåæåã®yåç®ã®ããããã®æåãšããæããŠè¶³ããçµæã«ãªã£ãŠããã
è¡åã©ããã®ç©ã¯ãé åºã«ãã£ãŠçµæãç°ãªãã
ããšãã°è¡åA,Bã
:<math>
A= \begin{pmatrix}2&4\\ 3&3 \end{pmatrix}
</math>
:<math>
B= \begin{pmatrix}7&9\\ 11&5 \end{pmatrix}
</math>
ãšãããšãã
ããããã
:<math>
AB =\begin{pmatrix}58&38\\ 54&42 \end{pmatrix}
</math>
:<math>
BA= \begin{pmatrix}41&55\\ 37&59 \end{pmatrix}
</math>
ãšãªãã
ãã®ããã«ãäžè¬ã®è¡åAãšè¡åBã®ç©ã¯ãäžè¬ã«
:<math>
AB \ne BA
</math>
ãšãªãã
äžè¿°ã®äŸã¯ã2å
é£ç«äžæ¬¡æ¹çšåŒãåŒ2åã®å Žåã«çžåœããè¡åã ã£ãããäžè¬ã«é£ç«æ¹çšåŒã®å
ã®æ°ã¯2åãšã¯éããªãããæ¹çšåŒã®æ°ã2åãšã¯éããªãã®ã§ãä»ã®å Žåã«ãè¡åãå®çŸ©ã§ããããã«ãè¡åã®å®çŸ©ãæ¡åŒµããã
ã€ãã®ããã«ãæ°å€ã瞊暪ã«äžŠã¹ãŠãããããã®æ®µã®æåã®åæ°ãçãããã®ã '''è¡å'''ïŒããããã€ãè±ïŒmatrixïŒ ãšåŒã¶ã
äŸãã°ã
:<math>
\begin{pmatrix}
1&2&3\\
4&5&6\\
\end{pmatrix}
</math>
ã¯è¡åã§ããã
ãã£ãœãã
:<math>
\begin{pmatrix}
1&2&3\\
&5& \\
\end{pmatrix}
</math>
ã¯ãæåã®åæ°ãäžèŽããªãã®ã§ãè¡åã§ã¯ãªãã
[[File:è¡åã®å®çŸ©ã®èª¬æå³.svg|thumb|è¡åã®å®çŸ©ã®èª¬æå³]]
è¡åã®äžéšã®ã暪ã«äžŠãã æ°å€ã®ãããŸãã '''è¡'''ïŒããããè±ïŒrowïŒ ãšããã瞊ã«äžŠãã æ°å€ã®ãããŸãã '''å'''ïŒãã€ãè±ïŒcolumnïŒ ãšãããããããã®æ°å€ã '''æå'''ïŒããã¶ããè±ïŒelementïŒ ãšåŒã¶ã
äŸãã°ã
:<math>
\begin{pmatrix}
1&2&3\\
4&5&6\\
\end{pmatrix}
</math>
ã¯2è¡ã3åãããªãè¡åã§ããã
è¡æ°ã''m''ã§ãåæ°ã''n''ã®è¡åã ''m''Ã''n''è¡å ã®ããã«åŒã³ãç¹ã«è¡æ°ãšåæ°ãçããnã§ããè¡åãªãã° ''n''次æ£æ¹è¡å ãšåŒã¶ã
äŸãã°ã
:<math>
\begin{pmatrix}
1&2&3\\
4&5&6\\
\end{pmatrix}
</math>
㯠2Ã3è¡å ã§ããã
第 ''i'' è¡ç¬¬ ''j'' åã®æåã (''i'', ''j'') æåãšããã
äŸãã°ã
:<math>
\begin{pmatrix}
1&2&3\\
4&5&6\\
\end{pmatrix}
</math>
ã® (2, 1) æåã¯4ã§ããã
*ãè¡åãçããããšã¯
ã2ã€ã®è¡åãçããããšã¯ãè¡æ°ãšåæ°ãçããããã€å¯Ÿå¿ãã (''i'', ''j'') æåããã¹ãŠçããããšãšå®ããã
ã€ãŸãã
<math>
\begin{pmatrix}
a&&b\\
c&&d\\
\end{pmatrix}
=
\begin{pmatrix}
e&&f\\
g&&h\\
\end{pmatrix}
</math>
ããšã¯ãã<math>a = e , b = f , c = g , d = h</math>ãã§ããã
== ãã¯ãã«å
ç©ãšè¡å ==
ãã 1è¡ãããªãè¡åã'''è¡ãã¯ãã«'''ïŒããããã¯ãã«ãè±ïŒrow vectorïŒãšããããã 1åãããªã'''åãã¯ãã«'''ïŒãã€ãã¯ãã«ãè±ïŒcolumn vector ïŒãšããã
ãã®è¡åã®å®çŸ©ã¯ããã¯ãã«ã®å®çŸ©ãæ¡åŒµãããã®ã«ãªã£ãŠããã
ããšãã°ãã¯ãã«ïŒaãbïŒãš(cãd)ã®å
ç© ac+bdã¯ãè¡åã®èšæ³ã䜿ããšã
:<math>
\begin{pmatrix}
a&&b\\
\end{pmatrix}
\begin{pmatrix}
c\\
d
\end{pmatrix}
=
\begin{pmatrix}
ac+bd
\end{pmatrix}
</math>
ãšæžããã
å³èŸºã® <math>
\begin{pmatrix}
ac+bd
\end{pmatrix}
</math>
ã¯ã1è¡1åã®è¡åã§ããããã®ããã«ãè¡åã§ã¯ã1è¡1åã®è¡åãèªããã
è¡åã®ç©ã® (''i'', ''j'') æåã®å€ã¯ãå·ŠåŽã®è¡åã® ''i'' è¡ã®ãã¯ãã«ãšãå³åŽã®è¡åã®ç¬¬ ''j'' åã®ãã¯ãã«ã®å
ç©ã§ããã
ããšãã°ãè¡å<math>A=
\begin{pmatrix}
a& b \\
c& d
\end{pmatrix}
</math>
ãš
<math>B=
\begin{pmatrix}
e& f\\
g& h
\end{pmatrix}
</math>
ã®ç© <math>AB=
\begin{pmatrix}
ae + bg &af + bh\\
ce + dg &cf + dh
\end{pmatrix}
</math> ã®(1, 2) æåã§ãã <math>af+bh</math> ã¯ã
ãã¯ãã« <math>
\begin{pmatrix}
a& b
\end{pmatrix}
</math> ãš ãã¯ãã« <math>
\begin{pmatrix}
f\\
h
\end{pmatrix}
</math> ãšã® å
ç©ã«ãªã£ãŠããã
ãã®ããã«èãããšããè¡åããšã¯ããã¯ãã«ã䞊ã¹ããã®ããšãèšãããïŒãã ã䞊ã¹ããã¯ãã«ã®æ¬¡å
ã¯åã次å
ã§ãªããã°ãªããªããïŒ
----
ããããã°ãé£ç«1次æ¹çšåŒã
:<math>
\begin{cases}
ax + by = p\\
cx + dy = q
\end{cases}
</math>
ã¯ãè¡åãçšããŠ
:<math>
\begin{pmatrix}
a&&b\\
c&&d
\end{pmatrix}
\begin{pmatrix}
x\\
y
\end{pmatrix}
=
\begin{pmatrix}
p\\
q
\end{pmatrix}
</math>
ãšè¡šããã
'''äŸé¡'''
*å
次ã®''w'', ''x'', ''y'', ''z''ã®å€ãæ±ããã
:<math>
\begin{pmatrix} 1&2 \\ 3&4 \end{pmatrix} = \begin{pmatrix} 2w&3x \\ 4y&5z \end{pmatrix}
</math>
*解ç
ããããã
:<math>
w = {1 \over 2}, x = {2 \over 3}, y = {3 \over 4}, z = {4 \over 5}
</math>
== è¡åã®åïŒå·®ïŒå®æ°å ==
è¡åã®åã»å·®ã»å®æ°åã®å®çŸ©ã¯ã次ã®ããã«ããã¯ãã«ã®åã»å·®ã»å®æ°åãšäŒŒããããªæ§è³ªãæã€ã
è¡åã®'''å'''ã®å®çŸ©ã¯ãåèŠçŽ ããšã«è¶³ãåãããããšå®çŸ©ãããã
:<math>
\begin{pmatrix}
a&&b
\\
c&&d
\end{pmatrix}
+
\begin{pmatrix}
e&&f\\
g&&h
\end{pmatrix}
=
\begin{pmatrix}
a+e&&b+f\\
c+g&&d+h
\end{pmatrix}
</math>
è¡åã®'''å·®'''ã®å®çŸ©ã¯ãåèŠçŽ ããšã«åŒããšå®çŸ©ããã
:<math>
\begin{pmatrix}
a&&b
\\
c&&d
\end{pmatrix}
-
\begin{pmatrix}
e&&f\\
g&&h
\end{pmatrix}
=
\begin{pmatrix}
a-e&&b-f\\
c-g&&d-h
\end{pmatrix}
</math>
å®æ°åã®å®çŸ©ã¯ãåèŠçŽ ã«å®æ°ãæããããšã«ãã£ãŠå®çŸ©ããã
:<math>
k
\begin{pmatrix}
a&&b
\\
c&&d
\end{pmatrix}
=
\begin{pmatrix}
ka&&kb\\
kc&&kd
\end{pmatrix}
</math>
(-1)A 㯠-A ãšæžãã
'''äŸé¡'''
*å
è¡åA,B,Cã
:<math>
A= \begin{pmatrix}2&4\\ 3&3 \end{pmatrix}
</math>
:<math>
B= \begin{pmatrix}7&9\\ 11&5 \end{pmatrix}
</math>
:<math>
C= \begin{pmatrix}8&2\\ 13&15 \end{pmatrix}
</math>
ã§å®çŸ©ãããšãã
:<math>
A + B
</math>
:<math>
C + B
</math>
:<math>
C + A
</math>
ãèšç®ããã
*解ç
ããããã
:<math>
A+B =\begin{pmatrix}9&13\\ 14&8 \end{pmatrix}
</math>
:<math>
C+B= \begin{pmatrix}15&11\\ 24&20 \end{pmatrix}
</math>
:<math>
C+A= \begin{pmatrix}10&6\\ 16&18 \end{pmatrix}
</math>
ãšãªãã
'''é¶è¡å'''
ãã¹ãŠã®æåã0ã§ããè¡åã '''ãŒãè¡å'''ïŒããããããã€ãè±ïŒzero matrixïŒ ãšããã
<math>
\begin{pmatrix}
0&0&0\\
0&0&0\\
\end{pmatrix}
</math>
ã㯠ãŒãè¡å ã§ããã
Aãè¡åãOãAãšè¡æ°ã»åæ°ãçããé¶è¡åãšãããšã
:<math>
A + (-A) = (-A) + A = O
</math>
ãæºããã
== è¡åã®ç©==
'''äŸé¡'''
*å
äžã§çšããè¡å<math>A</math>,<math>B</math>,<math>C</math>ã«ã€ããŠã
:<math>
AB
</math>
:<math>
BA
</math>
:<math>
BC
</math>
:<math>
AC
</math>
:<math>
CA
</math>
ãèšç®ããã
*解ç
ããããã
:<math>
AB =\begin{pmatrix}58&38\\ 54&42 \end{pmatrix}
</math>
:<math>
BA= \begin{pmatrix}41&55\\ 37&59 \end{pmatrix}
</math>
:<math>
BC=\begin{pmatrix}173&149\\ 153&97 \end{pmatrix}
</math>
:<math>
AC=\begin{pmatrix}68&64\\ 63&51 \end{pmatrix}
</math>
:<math>
CA=\begin{pmatrix}22&38\\ 71&97 \end{pmatrix}
</math>
ã§ããã
ãã®çµæããåããéããäžè¬ã«è¡åã®ç©ã¯
:<math>
AB \ne BA
</math>
ãšãªãã
:<math>
AB = BA
</math>
ãšãªãå Žåãè¡åAãšè¡åBã¯äº€æå¯èœïŒå¯æïŒã§ãããšããã
'''åäœè¡å'''
<math>
E =
\begin{pmatrix}
1 &0\\
0 &1
\end{pmatrix}
</math>
ãã2Ã2ã®åäœè¡åïŒ2次åäœè¡åïŒãšåŒã¶ã察è§æåã ãã1ã§ããããã®ä»ã®æåããã¹ãŠ0ã«çããè¡åã§ãããä»»æã®2Ã2è¡åAã«å¯ŸããŠãEã¯
:EA = AE = A
ãæºããã
== éè¡å ==
è¡åAã«å¯ŸããŠãã®è¡åãšã®ç©ãåäœè¡å <math>AA^{-1} = A^{-1}A = E</math> ãšãªãè¡å <math>A^{-1}</math> ãããã®è¡åã®'''éè¡å'''ãšåŒã¶ããã®ãããªè¡åã¯ããååšããã°åAã«å¯ŸããŠãã ã²ãšã€ã«å®ãŸãããã¡ããäžè¬ã«ã¯Aã«å¯ŸããŠå³åŽããããããå·ŠåŽããããããã«ãã£ãŠç©ã¯ç°ãªãã®ã ãããã®å Žåã¯Aã«å¯ŸããŠå³ãããããŠåäœè¡åã«ãªãã®ãªãã°å·ŠãããããŠãåäœè¡åã«ãªãããéããŸããããã§ããããšã«æ³šæããŠãããéè¡åã®éè¡åã¯ããšã®è¡åã«çããã
2è¡2åã®è¡å
<math>
A =
\begin{pmatrix}
a &b\\
c &d
\end{pmatrix}
</math>
ã«ã€ããŠã¯ã<math>ad-bc \ne 0</math>ã®ãšã
<math>
A ^{-1} =
\frac 1 {( ad - bc ) }
\begin{pmatrix}
d&-b\\
-c&a
\end{pmatrix}
</math>
ãšãªãã ad - bc = 0 ã®ãšããéè¡åã¯ååšããªãã
å®éã«è¡åã®ç©ãåãããšã§ããããæ£ããããšã容æã«ãããã
'''äŸé¡'''
*åé¡
äžã§å®ããè¡å<math>A</math>,<math>B</math>,<math>C</math>ã®éè¡åãèšç®ããã
è¡åA,B,Cã¯ããããã
:<math>
A= \begin{pmatrix}2&4\\ 3&3 \end{pmatrix}
</math>
:<math>
B= \begin{pmatrix}7&9\\ 11&5 \end{pmatrix}
</math>
:<math>
C= \begin{pmatrix}8&2\\ 13&15 \end{pmatrix}
</math>
ã§ãã£ãã
<br /><br /><br />
*解ç
ããããã
:<math>
A^{-1}=\begin{pmatrix}-{{1}\over{2}}&{{2}\over{3}}\\ {{1}\over{2}}&-{{1}\over{3 }} \end{pmatrix}
</math>
:<math>
B^{-1}=\begin{pmatrix}-{{5}\over{64}}&{{9}\over{64}}\\ {{11}\over{64}}&-{{7 }\over{64}} \end{pmatrix}
</math>
:<math>
C^{-1}=\begin{pmatrix}{{15}\over{94}}&-{{1}\over{47}}\\ -{{13}\over{94}}&{{4 }\over{47}} \end{pmatrix}
</math>
ã§ããã
== éè¡åãçšããé£ç«äžæ¬¡æ¹çšåŒã®è§£æ³ ==
1次æ¹çšåŒ
:<math>
\begin{cases}
x + 2y = 1\\
2x + 3y = 2
\end{cases}
</math>
ã¯ã
:<math>
\begin{pmatrix}
1 &2\\
2 &3
\end{pmatrix}
\begin{pmatrix}
x\\
y
\end{pmatrix}
=
\begin{pmatrix}
1\\
2
\end{pmatrix}
</math>
ãšæžããã䞡蟺ã«å·ŠèŸºã®è¡åã®éè¡åãæãããšã
:<math>
\begin{pmatrix}
1& 0\\
0&1
\end{pmatrix}
\begin{pmatrix}
x\\
y
\end{pmatrix}
=
\begin{pmatrix}
3 &-2\\
-2 &1
\end{pmatrix}
\begin{pmatrix}
1\\
2
\end{pmatrix}
\times (-1)
</math>
<!-- %(0 1)(y) = (-2 1)(2)-->
:<math>
\begin{pmatrix}
x\\
y
\end{pmatrix}
=
\begin{pmatrix}
1\\
0
\end{pmatrix}
</math>
x = 1, y = 0
ãåŸãããå§ãã®é£ç«1次æ¹çšåŒã解ããããšã«ãªãã
äžè¬ã«ãé£ç«1次æ¹çšåŒããã äžçµã®è§£ããã€ãšããé£ç«1次æ¹çšåŒã解ãããšã¯éè¡åãæ±ããããšãšåãã§ããã
ç¹ã«ã2Ã2è¡åã®éè¡åã¯æ¢ã«å
¬åŒãåŸãããŠããã®ã§ã2å
1次æ¹çšåŒã¯ç°¡åã«è§£ãããšãã§ããã
<math>A = \begin{pmatrix}a&&b\\c&&d\end{pmatrix}, \mathbf{x} = \begin{pmatrix}x\\y\end{pmatrix}, \mathbf{b} = \begin{pmatrix}p\\q\end{pmatrix}</math>ãšãããš
:<math>
A\mathbf{x} = \mathbf{b}
</math>
ãšæžãããããã§''A''ããã®é£ç«1次æ¹çšåŒã®ä¿æ°è¡åãšããããã®æ¹çšåŒã®è§£ã¯ã''A''ãéè¡åãæã€ãšãäžæã«å®ãŸãã <math>\mathbf{x} = A^{-1}\mathbf{b}</math> ã§ããã
== è¡åã®å¿çš==
=== å³åœ¢ãžã®å¿çš ===
==== ç¹ã®ç§»å ====
===== å転è¡å =====
å¹³é¢äžã®ãã¯ãã«<math>\vec a</math>ã«å¯ŸããŠå転è¡å
<math>
R =
\begin{pmatrix}
\cos c& -\sin c\\
\sin c & \cos c
\end{pmatrix}
</math>
ããããç©<math>R \vec a </math>ã¯ã<math>\vec a</math>ãåç¹ãäžå¿ã«ããŠè§åºŠcã ãå転ããããã¯ãã«ã«ãªã£ãŠããã
:ïŒèšŒæïŒ
:ãã¯ãã«aã極座æšãçšããŠ<math>a=(r \cos \theta,r \sin \theta)</math>ãšæžãããããšç©<math>R \vec a</math>ã¯
::<math>R \vec a =
\begin{pmatrix}
\cos c& -\sin c\\
\sin c & \cos c
\end{pmatrix}
\begin{pmatrix}
r \cos \theta \\
r \sin \theta
\end{pmatrix}=
\begin{pmatrix}
r (\cos c \cos \theta - \sin c \sin \theta) \\
r (\sin c \cos \theta + \cos c \sin \theta)
\end{pmatrix}=r
\begin{pmatrix}
\cos (c+\theta) \\
\sin (c+\theta)
\end{pmatrix}</math>
:ã§ãããããã¯ç¢ºãã«<math>\vec a</math>ãè§åºŠcã ãå転ããããã¯ãã«ã§ããã
===== äžè¬ã®è¡åã«ããç¹ã®ç§»å =====
座æšå€ïŒx,yïŒã®ç¹Pãè¡åããããããšã§ç§»åãããã®ãèããã
:<math>
\begin{pmatrix}
a&&b\\
c&&d
\end{pmatrix}
\begin{pmatrix}
x\\
y
\end{pmatrix}
=
\begin{pmatrix}
z\\
w
\end{pmatrix}
</math>
ã¯ã
<math>x\begin{pmatrix}a\\c\end{pmatrix} + y\begin{pmatrix}b\\d\end{pmatrix}</math> ãšãæžããã
ããã¯ãæ°ããªçŽç·åº§æšãçšæãïŒæ°åº§æšã®å座æšè»žã®åäœãã¯ãã«ã¯åã®åº§æšãåºæºã«æž¬ããšãããããæ¹åãã¯ãã« <math>\begin{pmatrix}a\\c\end{pmatrix}</math> ããã³ æ¹åãã¯ãã« <math>\begin{pmatrix}b\\d\end{pmatrix}</math> ã§ãããïŒããã®åº§æšã«åº§æšå€ïŒx,yïŒã代å
¥ããããšã§ç¹Pã移åãããã®ããåã®åº§æšç³»ã§æž¬ã£ãå Žåã®åº§æšå€ã«ãªã£ãŠããã
==== 座æšã®å€æ ====
éåžžã®çŽäº€åº§æšïŒåç¹ã§90°ã§äº€ãã座æšïŒã®äžã®ç¹ã®åº§æšïŒx,yïŒã«ã€ããŠãç¹ã®äœçœ®ã¯åããŸãŸãæ°ããªå¥ã®çŽç·åº§æšïŒçŽäº€ãšã¯éããªãïŒã§èŠãå Žåã®åº§æšïŒz,wïŒãèãããæ°ããªå¥åº§æšïŒçŽç·åº§æšïŒã¯ãèšç®ã®éœåäžãåç¹ã ãã¯å
ã®åº§æšãšåããšããããããšã次ã®ããã«ãåã®åº§æšãšæ°ããªåº§æšãšã®é¢ä¿ããè¡åã§è¡šèšã§ããã
:<math>
\begin{pmatrix}
a&&b\\
c&&d
\end{pmatrix}
\begin{pmatrix}
z\\
w
\end{pmatrix}
=
\begin{pmatrix}
x\\
y
\end{pmatrix}
</math>
ãšãããµããªé¢ä¿åŒã§èšè¿°ã§ããã
å®éã«ãããšãã° (x,y)=(0,0) ã®ãšã ïŒ z,w=0,0ïŒ ãšãªã£ãŠããã
ããŠã巊蟺㯠<math>z\begin{pmatrix}a\\c\end{pmatrix} + w\begin{pmatrix}b\\d\end{pmatrix}</math> ãšãæžããã
ãã®åŒãã座æšã®å€æã®å¹ŸäœåŠãšããŠèããå Žåã次ã®ãããªçè«ã«ãªãã
ãŸããæ°ããªçŽç·åº§æšã®åº§æšè»žã®åäœãã¯ãã«ã®æ¹åã¯ãããšã®åº§æšç³»ãåºæºã«èŠããšãããããæ¹åãã¯ãã« <math>\begin{pmatrix}a\\c\end{pmatrix}</math> ããã³ æ¹åãã¯ãã« <math>\begin{pmatrix}b\\d\end{pmatrix}</math> ã§ããã
:ããã§ãããæ°ããªåº§æšç³»ãåºæºã«ããŠãæ°ããªåº§æšè»žã®åäœãã¯ãã«ã®æ°å€ãèŠãŠããçµæã®åäœãã¯ãã«ã®æ°å€ã¯ ïŒ0,1ïŒ ããã³ (1,0) ã«ãªãã ãã§ãããäœãèšç®ããäºã«ãªããªãããªããªãèªå·±ã®åº§æšç³»ã§èªå·±ã®åäœãã¯ãã«ãèŠãŠããïŒ0,1ïŒ ããã³ (1,0) ã§ãããªããããã§ããã
:èšç®ãã¹ãã¯ãæ°ããªåº§æšè»žãåºæºã«ããŠåã®åº§æšè»žãèŠãå Žåã®æ°å€ããããã¯ãåã®åº§æšè»žãåºæºã«ããŠæ°ããªåº§æšè»žãèŠãå Žåã®æ°å€ã§ããã
ããŠããã®åé¡ã§ã¯ç¹ïŒ°ã®äœçœ®ïŒxãyïŒã¯äœãå€æããŠãããããã£ãŠãåã®åº§æšãåºæºã«ããŠç¹Pã®äœçœ®ãèŠãŠããäœãå€åããªãããã®åé¡ã§å€æŽããã®ã¯åº§æšè»žã®ã»ãã§ãããããæ°ããªåº§æšç³»ã§èŠãç¹ïŒ°ã®å€ïŒz,wïŒã«èå³ãããã®ã§ããã
==== ç·ã®ç§»å ====
å¹³é¢å³åœ¢äžã®ç·åã¯ã2è¡2åã®è¡åã§å€æã§ããã
<math>
A =
\begin{pmatrix}
a &b\\
c &d
\end{pmatrix}
</math>
ã§å€æããå Žåã«ã€ããŠã¯ã<math>ad-bc \ne 0</math>ã®ãšããç·ã¯ç·ã«å€æãããåè§åœ¢ã¯åè§åœ¢ã«å€æãããäžè§åœ¢ã¯äžè§åœ¢ã«å€æãããã
==== é¢ã®ç§»å ====
2è¡2åã®è¡å
<math>
A =
\begin{pmatrix}
a &b\\
c &d
\end{pmatrix}
</math>
ã«ã€ããŠã¯ãå³åœ¢ã®é¢ç©ã¯ã<math>ad-bc </math>åãããã
== ç·åœ¢åå ==
== äžåçŽç· ==
== å€éšãªã³ã¯ ==
[https://www.mext.go.jp/content/20210525-mxt_kyoiku01-000009442_1_1.pdf ãé«çåŠæ ¡æ°åŠç§ææïŒè¡åå
¥éïŒãæéšç§åŠç]
[[category:é«çåŠæ ¡æ°åŠ|ããããã€]]
[[ã«ããŽãª:è¡å]] | 2005-05-03T08:02:28Z | 2024-03-05T12:45:31Z | [] | https://ja.wikibooks.org/wiki/%E6%97%A7%E8%AA%B2%E7%A8%8B(-2012%E5%B9%B4%E5%BA%A6)%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E6%95%B0%E5%AD%A6C/%E8%A1%8C%E5%88%97 |
1,895 | é«çåŠæ ¡æ°åŠC/å¹³é¢äžã®æ²ç· | æŸç©ç·(parabola)ãæ¥å(ellipse)ãåæ²ç·(hyperbola)ããŸãšããŠã2次æ²ç·ãåéæ²ç·ãšãããããããã2次æ²ç·ãšåŒã°ããçç±ã¯ãæŸç©ç·ãæ¥åãåæ²ç·ã¯ x , y {\displaystyle x,y} ã®2æ¬¡åŒ F ( x , y ) {\displaystyle F(x,y)} ã«ãã£ãŠ F ( x , y ) = 0 {\displaystyle F(x,y)=0} ã§è¡šãããšãã§ãããŸã x , y {\displaystyle x,y} ã®2æ¬¡åŒ F ( x , y ) {\displaystyle F(x,y)} ã«ãã£ãŠ F ( x , y ) = 0 {\displaystyle F(x,y)=0} ãšè¡šãããæ²ç·ã¯æŸç©ç·ãæ¥åãåæ²ç·ã2çŽç·ã®ããããã«ãªãããã§ããã
åéæ²ç·ãšåŒã°ããçç±ã¯ãåéé¢ããå
šãŠã®æ¯ç·ãšäº€ãããåºé¢ã«å¹³è¡ãªå¹³é¢ã§åæããããšãã®æé¢ãåããå
šãŠã®æ¯ç·ãšäº€ãããåºé¢ã«å¹³è¡ã§ãªãå¹³é¢ã§åæããããšãã®æé¢ãæ¥åããæ¯ç·ã«å¹³è¡ãªé¢ã§åæããããšãã®æé¢ãæŸç©ç·ããæ¯ç·ã«å¹³è¡ã§ãªãå¹³é¢ã§åæããããšãã®æé¢ãåæ²ç·ãšãªãããã§ããã
2次æ²ç·ã¯çŽç·ãåã«ã€ãã§éèŠãªæ²ç·ã§ããã
å¹³é¢äžã«ç¹ F {\displaystyle \mathrm {F} } ãšãç¹ F {\displaystyle \mathrm {F} } ãéããªãçŽç· l {\displaystyle l} ããšãããã®ãšããçŽç· l {\displaystyle l} ããã®è·é¢ãšç¹ F {\displaystyle \mathrm {F} } ããã®è·é¢ãçããç¹ã®è»è·¡ãæŸç©ç·ãšããããã®ãšããç¹ F {\displaystyle \mathrm {F} } ãæŸç©ç·ã®çŠç¹ãçŽç· l {\displaystyle l} ãæŸç©ç·ã®æºç·ãšããã
çŠç¹ã F ( p , 0 ) {\displaystyle \mathrm {F} (p,0)} æºç·ã l : x = â p {\displaystyle l:x=-p} ãšããæŸç©ç·ã®æ¹çšåŒãæ±ããã P ( x , y ) {\displaystyle \mathrm {P} (x,y)} ããã®æŸç©ç·ã®ç¹ãšãããšãç¹ P {\displaystyle \mathrm {P} } ãšçŽç· l {\displaystyle l} ã®è·é¢ã¯ x + p {\displaystyle x+p} ã§ããã P F = ( x â p ) 2 + y 2 {\displaystyle \mathrm {PF} ={\sqrt {(x-p)^{2}+y^{2}}}} ã§ããããªã®ã§ã ( x + p ) 2 = ( x â p ) 2 + y 2 {\displaystyle (x+p)^{2}=(x-p)^{2}+y^{2}} ã§ããããããæŽçããŠã
y 2 = 4 p x {\displaystyle y^{2}=4px}
ãåŸãã
ããã§ãæŸç©ç· y 2 = 4 p x {\displaystyle y^{2}=4px} ã«ãããŠã x {\displaystyle x} ãš y {\displaystyle y} ãå
¥ãæ¿ããã° y = x 2 4 p {\displaystyle y={\frac {x^{2}}{4p}}} ã§ãããããããäžåŠããåŠãã§ããæŸç©ç·ã®å®çŸ©ãšäžèŽããããšããããã
æŒç¿åé¡
æŸç©ç· y = a x 2 ( a â 0 ) {\displaystyle y=ax^{2}\quad (a\neq 0)} ã®çŠç¹ãšæºç·ãæ±ããã
解ç
çŠç¹ ( 0 , 1 4 a ) {\displaystyle \left(0,{\frac {1}{4a}}\right)} æºç· y = â 1 4 a {\displaystyle y=-{\frac {1}{4a}}}
å¹³é¢äžã«ç°ãªã2ç¹ F , F â² {\displaystyle \mathrm {F} ,\mathrm {F'} } ããšãã F {\displaystyle \mathrm {F} } ãšã®è·é¢ãšã F â² {\displaystyle \mathrm {F'} } ãšã®è·é¢ã®åãäžå®ã§ããç¹ã®è»è·¡ãæ¥åãšããããã®ãšããç¹ F , F â² {\displaystyle \mathrm {F} ,\mathrm {F'} } ãæ¥åã®çŠç¹ãšããã
çŠç¹ã F ( c , 0 ) , F â² ( â c , 0 ) {\displaystyle \mathrm {F} (c,0),\mathrm {F'} (-c,0)} ãšãããç¹ P ( x , y ) {\displaystyle \mathrm {P} (x,y)} ãæ¥åäžã®ç¹ã§ãããšãã P F + P F â² = 2 a {\displaystyle \mathrm {PF} +\mathrm {PF'} =2a} ã§ããã P F = 2 a â P F â² {\displaystyle \mathrm {PF} =2a-\mathrm {PF'} } ãã
( x â c ) 2 + y 2 = 2 a â ( x + c ) 2 + y 2 {\displaystyle {\sqrt {(x-c)^{2}+y^{2}}}=2a-{\sqrt {(x+c)^{2}+y^{2}}}}
䞡蟺ã2ä¹ããŠæŽçãããš
a ( x + c ) 2 + y 2 = a 2 + c x {\displaystyle a{\sqrt {(x+c)^{2}+y^{2}}}=a^{2}+cx}
å床ã䞡蟺ã2ä¹ããŠæŽçãããš
( a 2 â c 2 ) x 2 + a 2 y 2 = a 2 ( a 2 â c 2 ) {\displaystyle (a^{2}-c^{2})x^{2}+a^{2}y^{2}=a^{2}(a^{2}-c^{2})}
ãã㧠a 2 â c 2 = b 2 ( b > 0 ) {\displaystyle a^{2}-c^{2}=b^{2}\quad (b>0)} ãšçœ®ãæãããš
b 2 x 2 + a 2 y 2 = a 2 b 2 {\displaystyle b^{2}x^{2}+a^{2}y^{2}=a^{2}b^{2}}
䞡蟺ã a 2 b 2 {\displaystyle a^{2}b^{2}} ã§å²ããš
x 2 a 2 + y 2 b 2 = 1 ( a > b > 0 ) {\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1\quad (a>b>0)}
ãå°ãããã
x軞ãšã®äº€ç¹ã¯ ( a , 0 ) {\displaystyle (a,0)} ã ( â a , 0 ) {\displaystyle (-a,0)} ãy軞ãšã®äº€ç¹ã¯ ( 0 , b ) {\displaystyle (0,b)} ã ( 0 , â b ) {\displaystyle (0,-b)} ãšãªãã
a > b > 0 {\displaystyle a>b>0} ã®ãšãã 2 a {\displaystyle 2a} ã¯é·è»žã®é·ã(é·åŸ)ã 2 b {\displaystyle 2b} ã¯ç軞ã®é·ã(çåŸ)ãšãªããxyå¹³é¢äžã«ã°ã©ããæžããšæšªé·ã®æ¥åã«ãªãããŸãçŠç¹ã¯é·åŸã§ããx軞äžã«ãããã®åº§æšã¯ ( â a 2 â b 2 , 0 ) , ( a 2 â b 2 , 0 ) {\displaystyle (-{\sqrt {a^{2}-b^{2}}},0),({\sqrt {a^{2}-b^{2}}},0)} ãšãªãã
éã«ã b > a > 0 {\displaystyle b>a>0} ã®ãšãã 2 b {\displaystyle 2b} ã¯é·è»žã®é·ã(é·åŸ)ã 2 a {\displaystyle 2a} ã¯ç軞ã®é·ã(çåŸ)ãšãªããxyå¹³é¢äžã«ã°ã©ããæžããšçžŠé·ã®æ¥åã«ãªãããŸãçŠç¹ã¯é·åŸã§ããy軞äžã«ãããã®åº§æšã¯ ( 0 , b 2 â a 2 ) , ( 0 , â b 2 â a 2 ) {\displaystyle (0,{\sqrt {b^{2}-a^{2}}}),(0,-{\sqrt {b^{2}-a^{2}}})} ãšãªãã
2ã€ã®çŠç¹ãè¿ãã»ã©æ¥åã¯åã«è¿ã¥ãã2ã€ã®çŠç¹ãéãªã£ããšã a = b {\displaystyle a=b} ãšãªããæ¥åã¯åã«ãªãã
ã¡ãªã¿ã«ãææã®åšããå
¬è»¢ããææã®è»éã¯ãææãçŠç¹ãšããæ¥åã«ãªãã
å¹³é¢äžã«ç°ãªã2ç¹ F , F â² {\displaystyle \mathrm {F} ,\mathrm {F'} } ããšãã F {\displaystyle \mathrm {F} } ãšã®è·é¢ãšã F â² {\displaystyle \mathrm {F'} } ãšã®è·é¢ã®å·®ãäžå®ã§ããç¹ã®è»è·¡ãåæ²ç·ãšããã2ç¹ F , F â² {\displaystyle \mathrm {F} ,\mathrm {F'} } ãåæ²ç·ã®çŠç¹ãšããã
çŠç¹ã F ( c , 0 ) , F â² ( â c , 0 ) {\displaystyle \mathrm {F} (c,0),\mathrm {F'} (-c,0)} ãšãããç¹ P ( x , y ) {\displaystyle \mathrm {P} (x,y)} ãåæ²ç·äžã®ç¹ã§ãããšãã | P F â P F â² | = 2 a {\displaystyle |\mathrm {PF} -\mathrm {PF'} |=2a} ã§ããã P F = ± 2 a + P F â² {\displaystyle \mathrm {PF} =\pm 2a+\mathrm {PF'} } ãã
( x â c ) 2 + y 2 = ± 2 a + ( x + c ) 2 + y 2 {\displaystyle {\sqrt {(x-c)^{2}+y^{2}}}=\pm 2a+{\sqrt {(x+c)^{2}+y^{2}}}}
䞡蟺ã2ä¹ããŠæŽçãããš
± a ( x + c ) 2 + y 2 = â a 2 â c x {\displaystyle \pm a{\sqrt {(x+c)^{2}+y^{2}}}=-a^{2}-cx}
å床䞡蟺ã2ä¹ããŠæŽçãããš
( c 2 â a 2 ) x 2 â a 2 y 2 = a 2 ( c 2 â a 2 ) {\displaystyle (c^{2}-a^{2})x^{2}-a^{2}y^{2}=a^{2}(c^{2}-a^{2})}
ããã§ã b 2 = c 2 â a 2 ( b > 0 ) {\displaystyle b^{2}=c^{2}-a^{2}\quad (b>0)} ãšããã䞡蟺ã a 2 b 2 {\displaystyle a^{2}b^{2}} ã§å²ãã°
x 2 a 2 â y 2 b 2 = 1 {\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1}
ã§ããã
åæ²ç·ã x 2 a 2 â y 2 b 2 = 1 {\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1} ã§è¡šããããšããçŠç¹ã®åº§æšã¯ ( a 2 + b 2 , 0 ) , ( â a 2 + b 2 , 0 ) {\displaystyle ({\sqrt {a^{2}+b^{2}}},0),(-{\sqrt {a^{2}+b^{2}}},0)} ãšãªãã
éã«ãåæ²ç·ã x 2 a 2 â y 2 b 2 = â 1 {\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=-1} ã§è¡šããããšããçŠç¹ã®åº§æšã¯ ( 0 , a 2 + b 2 ) , ( 0 , â a 2 + b 2 ) {\displaystyle (0,{\sqrt {a^{2}+b^{2}}}),(0,-{\sqrt {a^{2}+b^{2}}})} ãšãªãã
x = f ( t ) , y = g ( t ) {\displaystyle x=f(t),y=g(t)} ã§è¡šãããç¹ P ( x , y ) {\displaystyle \mathrm {P} (x,y)} ã®éåã¯ããæ²ç·ãæãããã®ãããªæ²ç·ã®è¡šç€ºãåªä»å€æ°è¡šç€ºãšããã
åªä»å€æ°è¡šç€ºã§ã¯ F ( x , y ) = 0 {\displaystyle F(x,y)=0} ã®åœ¢ã§ã¯è¡šãã«ããæ²ç·ãç°¡æœã«è¡šãããšãã§ãããäŸãã°ã x = t - sin t, y = 1 - cos t ã§ãããããã¯ãµã€ã¯ãã€ããšåŒã°ããã
x = f ( t ) , y = g ( t ) {\displaystyle x=f(t),y=g(t)} ãšåªä»å€æ°è¡šç€ºãããŠããæ²ç·ã x {\displaystyle x} æ¹åã« p {\displaystyle p} ã y {\displaystyle y} æ¹åã« q {\displaystyle q} ã ãã ãå¹³è¡ç§»åããæ²ç·ã¯ x = f ( t ) + p , y = g ( t ) + q {\displaystyle x=f(t)+p,y=g(t)+q} ãšè¡šããã
x = p t 2 , y = 2 p t p â 0 {\displaystyle x=pt^{2},y=2pt\quad p\neq 0} ã§è¡šãããæ²ç·ã¯ t {\displaystyle t} ãæ¶å»ãããš y 2 = 4 p x {\displaystyle y^{2}=4px} ãšãªãã®ã§æŸç©ç·ã§ããã
å x 2 + y 2 = r 2 {\displaystyle x^{2}+y^{2}=r^{2}} ãåªä»å€æ°è¡šç€ºãããš x = r cos Ξ , y = r sin Ξ {\displaystyle x=r\cos \theta ,y=r\sin \theta } ãšãªãã
ãããŸã§ã®åŠç¿ã§ã¯ã x {\displaystyle x} 軞㚠y {\displaystyle y} 軞ã䜿ã£ã座æšå¹³é¢(çŽäº€åº§æšãšãã) ( x , y ) {\displaystyle (x,y)} 䜿ãããšã§ã座æšå¹³é¢äžã®1ç¹ãå®ããã ããã§åŠã¶æ¥µåº§æšã§ã¯ã ( r , Ξ ) {\displaystyle (r,\theta )} ã®æåã§äžããããåŒã䜿ã£ãŠæ²ç·ãè¡šãããšãèããã
ããäžç¹OãšåçŽç·OXãå®ãããšãå¹³é¢äžã®ç¹Pã¯ãç¹Oããã®è·é¢rãšã â {\displaystyle \angle } XOPã®è§ Ξ ( 0 †Ξ < 2 Ï ) {\displaystyle \theta \,(0\leq \theta <2\pi )} ã®å€§ããã§äžæã«å®ãŸãã
極座æšã®å®çŸ©
åç¹Oãšè»žOXãå®ãããå¹³é¢äžã®ç¹Pã«ã€ããŠãOPéã®è·é¢ãrã â {\displaystyle \angle } XOPã®å€§ãããΞã§è¡šããåº§æš ( r , Ξ ) {\displaystyle (r,\theta )} ã極座æšãšããã ãã®ãšããOã極ãOXãå§ç·ãšããã ãŸãã Ξ {\displaystyle \theta } ãåè§ãšããã
ãŸããçŽäº€åº§æšãšæ¥µåº§æšã®é¢ä¿ã¯æ¬¡ã®ããã«ãªãã
çŽäº€åº§æšãšæ¥µåº§æšã®é¢ä¿
{ r = x 2 + y 2 cos Ξ = x r sin Ξ = y r { x = r cos Ξ y = r sin Ξ {\displaystyle {\begin{cases}r={\sqrt {x^{2}+y^{2}}}\\\cos \theta =\displaystyle {\frac {x}{r}}\\\sin \theta =\displaystyle {\frac {y}{r}}\end{cases}}\,\,{\begin{cases}x=r\cos \theta \\y=r\sin \theta \end{cases}}}
ããã¯çŽæçã«ã¯è€çŽ æ°å¹³é¢äžã®ç¹ã®çµ¶å¯Ÿå€ãšåè§ãå®ãããšãã«äŒŒãŠããã
r = f ( Ξ ) {\displaystyle r=f(\theta )} ã®åœ¢ã§äžããããåŒã極æ¹çšåŒ(ãããã»ããŠããã)ãšããã極æ¹çšåŒã¯rãšÎžã«ã€ããŠã®é¢æ°ã§ãããããããã¯xãšyãžã®å€æãå¯èœã§ããããã£ãŠxyå¹³é¢äžã«æ²ç·ããããŠãããããšã«ãªãã
ããŸããŸãªæ¥µæ¹çšåŒ
(1)äžå¿O,ååŸaã®å r = a {\displaystyle r=a}
(2)äžå¿ ( r 0 , Ξ 0 ) {\displaystyle (r_{0},{\theta }_{0})} ,ååŸaã®å r 2 â 2 r r 0 cos Ξ 0 + r 0 2 = a 2 {\displaystyle r^{2}-2rr_{0}\cos {\theta }_{0}+{r_{0}}^{2}=a^{2}}
(3)極Oãéããå§ç·ãšÎ±ã®è§ããªãçŽç· Ξ = α {\displaystyle \theta =\alpha }
(4)ç¹ ( a , α ) {\displaystyle (a,\alpha )} ãéããOAã«åçŽãªçŽç· r cos ( Ξ â α ) = a {\displaystyle r\cos(\theta -\alpha )=a}
(äŸ)å ( x â 1 ) 2 + y 2 = 1 {\displaystyle (x-1)^{2}+y^{2}=1} ã極æ¹çšåŒã§è¡šã. x = r cos Ξ , y = r sin Ξ {\displaystyle x=r\cos \theta ,y=r\sin \theta } ã代å
¥ããŠæŽçãããš r ( r â 2 cos Ξ ) = 0 {\displaystyle r(r-2\cos \theta )=0}
r = 0 {\displaystyle r=0} ã¯æ¥µãè¡šããã r = 2 cos Ξ {\displaystyle r=2\cos \theta }
ãããŸã§ã«ã2次æ²ç·ãåªä»å€æ°è¡šç€ºã極æ¹çšåŒãªã©ã®æ²ç·ãšãã®æ§è³ªã«ã€ããŠè¿°ã¹ãŠããã以äžã§ã¯ãããããå©çšããŠããŸããŸãªæ²ç·ã®åŒã瀺ããäžè¬ã«æŠåœ¢ãã€ããã®ã¯å°é£ãªãããã³ã³ãã¥ãŒã¿ã䜿çšããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "æŸç©ç·(parabola)ãæ¥å(ellipse)ãåæ²ç·(hyperbola)ããŸãšããŠã2次æ²ç·ãåéæ²ç·ãšãããããããã2次æ²ç·ãšåŒã°ããçç±ã¯ãæŸç©ç·ãæ¥åãåæ²ç·ã¯ x , y {\\displaystyle x,y} ã®2æ¬¡åŒ F ( x , y ) {\\displaystyle F(x,y)} ã«ãã£ãŠ F ( x , y ) = 0 {\\displaystyle F(x,y)=0} ã§è¡šãããšãã§ãããŸã x , y {\\displaystyle x,y} ã®2æ¬¡åŒ F ( x , y ) {\\displaystyle F(x,y)} ã«ãã£ãŠ F ( x , y ) = 0 {\\displaystyle F(x,y)=0} ãšè¡šãããæ²ç·ã¯æŸç©ç·ãæ¥åãåæ²ç·ã2çŽç·ã®ããããã«ãªãããã§ããã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "åéæ²ç·ãšåŒã°ããçç±ã¯ãåéé¢ããå
šãŠã®æ¯ç·ãšäº€ãããåºé¢ã«å¹³è¡ãªå¹³é¢ã§åæããããšãã®æé¢ãåããå
šãŠã®æ¯ç·ãšäº€ãããåºé¢ã«å¹³è¡ã§ãªãå¹³é¢ã§åæããããšãã®æé¢ãæ¥åããæ¯ç·ã«å¹³è¡ãªé¢ã§åæããããšãã®æé¢ãæŸç©ç·ããæ¯ç·ã«å¹³è¡ã§ãªãå¹³é¢ã§åæããããšãã®æé¢ãåæ²ç·ãšãªãããã§ããã",
"title": ""
},
{
"paragraph_id": 2,
"tag": "p",
"text": "2次æ²ç·ã¯çŽç·ãåã«ã€ãã§éèŠãªæ²ç·ã§ããã",
"title": ""
},
{
"paragraph_id": 3,
"tag": "p",
"text": "å¹³é¢äžã«ç¹ F {\\displaystyle \\mathrm {F} } ãšãç¹ F {\\displaystyle \\mathrm {F} } ãéããªãçŽç· l {\\displaystyle l} ããšãããã®ãšããçŽç· l {\\displaystyle l} ããã®è·é¢ãšç¹ F {\\displaystyle \\mathrm {F} } ããã®è·é¢ãçããç¹ã®è»è·¡ãæŸç©ç·ãšããããã®ãšããç¹ F {\\displaystyle \\mathrm {F} } ãæŸç©ç·ã®çŠç¹ãçŽç· l {\\displaystyle l} ãæŸç©ç·ã®æºç·ãšããã",
"title": "æŸç©ç·"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "çŠç¹ã F ( p , 0 ) {\\displaystyle \\mathrm {F} (p,0)} æºç·ã l : x = â p {\\displaystyle l:x=-p} ãšããæŸç©ç·ã®æ¹çšåŒãæ±ããã P ( x , y ) {\\displaystyle \\mathrm {P} (x,y)} ããã®æŸç©ç·ã®ç¹ãšãããšãç¹ P {\\displaystyle \\mathrm {P} } ãšçŽç· l {\\displaystyle l} ã®è·é¢ã¯ x + p {\\displaystyle x+p} ã§ããã P F = ( x â p ) 2 + y 2 {\\displaystyle \\mathrm {PF} ={\\sqrt {(x-p)^{2}+y^{2}}}} ã§ããããªã®ã§ã ( x + p ) 2 = ( x â p ) 2 + y 2 {\\displaystyle (x+p)^{2}=(x-p)^{2}+y^{2}} ã§ããããããæŽçããŠã",
"title": "æŸç©ç·"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "y 2 = 4 p x {\\displaystyle y^{2}=4px}",
"title": "æŸç©ç·"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ãåŸãã",
"title": "æŸç©ç·"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ããã§ãæŸç©ç· y 2 = 4 p x {\\displaystyle y^{2}=4px} ã«ãããŠã x {\\displaystyle x} ãš y {\\displaystyle y} ãå
¥ãæ¿ããã° y = x 2 4 p {\\displaystyle y={\\frac {x^{2}}{4p}}} ã§ãããããããäžåŠããåŠãã§ããæŸç©ç·ã®å®çŸ©ãšäžèŽããããšããããã",
"title": "æŸç©ç·"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "æŸç©ç·"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "æŸç©ç· y = a x 2 ( a â 0 ) {\\displaystyle y=ax^{2}\\quad (a\\neq 0)} ã®çŠç¹ãšæºç·ãæ±ããã",
"title": "æŸç©ç·"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "解ç",
"title": "æŸç©ç·"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "çŠç¹ ( 0 , 1 4 a ) {\\displaystyle \\left(0,{\\frac {1}{4a}}\\right)} æºç· y = â 1 4 a {\\displaystyle y=-{\\frac {1}{4a}}}",
"title": "æŸç©ç·"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "å¹³é¢äžã«ç°ãªã2ç¹ F , F â² {\\displaystyle \\mathrm {F} ,\\mathrm {F'} } ããšãã F {\\displaystyle \\mathrm {F} } ãšã®è·é¢ãšã F â² {\\displaystyle \\mathrm {F'} } ãšã®è·é¢ã®åãäžå®ã§ããç¹ã®è»è·¡ãæ¥åãšããããã®ãšããç¹ F , F â² {\\displaystyle \\mathrm {F} ,\\mathrm {F'} } ãæ¥åã®çŠç¹ãšããã",
"title": "æ¥å"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "",
"title": "æ¥å"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "çŠç¹ã F ( c , 0 ) , F â² ( â c , 0 ) {\\displaystyle \\mathrm {F} (c,0),\\mathrm {F'} (-c,0)} ãšãããç¹ P ( x , y ) {\\displaystyle \\mathrm {P} (x,y)} ãæ¥åäžã®ç¹ã§ãããšãã P F + P F â² = 2 a {\\displaystyle \\mathrm {PF} +\\mathrm {PF'} =2a} ã§ããã P F = 2 a â P F â² {\\displaystyle \\mathrm {PF} =2a-\\mathrm {PF'} } ãã",
"title": "æ¥å"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "( x â c ) 2 + y 2 = 2 a â ( x + c ) 2 + y 2 {\\displaystyle {\\sqrt {(x-c)^{2}+y^{2}}}=2a-{\\sqrt {(x+c)^{2}+y^{2}}}}",
"title": "æ¥å"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "䞡蟺ã2ä¹ããŠæŽçãããš",
"title": "æ¥å"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "a ( x + c ) 2 + y 2 = a 2 + c x {\\displaystyle a{\\sqrt {(x+c)^{2}+y^{2}}}=a^{2}+cx}",
"title": "æ¥å"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "å床ã䞡蟺ã2ä¹ããŠæŽçãããš",
"title": "æ¥å"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "( a 2 â c 2 ) x 2 + a 2 y 2 = a 2 ( a 2 â c 2 ) {\\displaystyle (a^{2}-c^{2})x^{2}+a^{2}y^{2}=a^{2}(a^{2}-c^{2})}",
"title": "æ¥å"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "ãã㧠a 2 â c 2 = b 2 ( b > 0 ) {\\displaystyle a^{2}-c^{2}=b^{2}\\quad (b>0)} ãšçœ®ãæãããš",
"title": "æ¥å"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "b 2 x 2 + a 2 y 2 = a 2 b 2 {\\displaystyle b^{2}x^{2}+a^{2}y^{2}=a^{2}b^{2}}",
"title": "æ¥å"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "䞡蟺ã a 2 b 2 {\\displaystyle a^{2}b^{2}} ã§å²ããš",
"title": "æ¥å"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "x 2 a 2 + y 2 b 2 = 1 ( a > b > 0 ) {\\displaystyle {\\frac {x^{2}}{a^{2}}}+{\\frac {y^{2}}{b^{2}}}=1\\quad (a>b>0)}",
"title": "æ¥å"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "ãå°ãããã",
"title": "æ¥å"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "x軞ãšã®äº€ç¹ã¯ ( a , 0 ) {\\displaystyle (a,0)} ã ( â a , 0 ) {\\displaystyle (-a,0)} ãy軞ãšã®äº€ç¹ã¯ ( 0 , b ) {\\displaystyle (0,b)} ã ( 0 , â b ) {\\displaystyle (0,-b)} ãšãªãã",
"title": "æ¥å"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "a > b > 0 {\\displaystyle a>b>0} ã®ãšãã 2 a {\\displaystyle 2a} ã¯é·è»žã®é·ã(é·åŸ)ã 2 b {\\displaystyle 2b} ã¯ç軞ã®é·ã(çåŸ)ãšãªããxyå¹³é¢äžã«ã°ã©ããæžããšæšªé·ã®æ¥åã«ãªãããŸãçŠç¹ã¯é·åŸã§ããx軞äžã«ãããã®åº§æšã¯ ( â a 2 â b 2 , 0 ) , ( a 2 â b 2 , 0 ) {\\displaystyle (-{\\sqrt {a^{2}-b^{2}}},0),({\\sqrt {a^{2}-b^{2}}},0)} ãšãªãã",
"title": "æ¥å"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "éã«ã b > a > 0 {\\displaystyle b>a>0} ã®ãšãã 2 b {\\displaystyle 2b} ã¯é·è»žã®é·ã(é·åŸ)ã 2 a {\\displaystyle 2a} ã¯ç軞ã®é·ã(çåŸ)ãšãªããxyå¹³é¢äžã«ã°ã©ããæžããšçžŠé·ã®æ¥åã«ãªãããŸãçŠç¹ã¯é·åŸã§ããy軞äžã«ãããã®åº§æšã¯ ( 0 , b 2 â a 2 ) , ( 0 , â b 2 â a 2 ) {\\displaystyle (0,{\\sqrt {b^{2}-a^{2}}}),(0,-{\\sqrt {b^{2}-a^{2}}})} ãšãªãã",
"title": "æ¥å"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "2ã€ã®çŠç¹ãè¿ãã»ã©æ¥åã¯åã«è¿ã¥ãã2ã€ã®çŠç¹ãéãªã£ããšã a = b {\\displaystyle a=b} ãšãªããæ¥åã¯åã«ãªãã",
"title": "æ¥å"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "ã¡ãªã¿ã«ãææã®åšããå
¬è»¢ããææã®è»éã¯ãææãçŠç¹ãšããæ¥åã«ãªãã",
"title": "æ¥å"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "å¹³é¢äžã«ç°ãªã2ç¹ F , F â² {\\displaystyle \\mathrm {F} ,\\mathrm {F'} } ããšãã F {\\displaystyle \\mathrm {F} } ãšã®è·é¢ãšã F â² {\\displaystyle \\mathrm {F'} } ãšã®è·é¢ã®å·®ãäžå®ã§ããç¹ã®è»è·¡ãåæ²ç·ãšããã2ç¹ F , F â² {\\displaystyle \\mathrm {F} ,\\mathrm {F'} } ãåæ²ç·ã®çŠç¹ãšããã",
"title": "åæ²ç·"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "",
"title": "åæ²ç·"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "çŠç¹ã F ( c , 0 ) , F â² ( â c , 0 ) {\\displaystyle \\mathrm {F} (c,0),\\mathrm {F'} (-c,0)} ãšãããç¹ P ( x , y ) {\\displaystyle \\mathrm {P} (x,y)} ãåæ²ç·äžã®ç¹ã§ãããšãã | P F â P F â² | = 2 a {\\displaystyle |\\mathrm {PF} -\\mathrm {PF'} |=2a} ã§ããã P F = ± 2 a + P F â² {\\displaystyle \\mathrm {PF} =\\pm 2a+\\mathrm {PF'} } ãã",
"title": "åæ²ç·"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "( x â c ) 2 + y 2 = ± 2 a + ( x + c ) 2 + y 2 {\\displaystyle {\\sqrt {(x-c)^{2}+y^{2}}}=\\pm 2a+{\\sqrt {(x+c)^{2}+y^{2}}}}",
"title": "åæ²ç·"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "䞡蟺ã2ä¹ããŠæŽçãããš",
"title": "åæ²ç·"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "± a ( x + c ) 2 + y 2 = â a 2 â c x {\\displaystyle \\pm a{\\sqrt {(x+c)^{2}+y^{2}}}=-a^{2}-cx}",
"title": "åæ²ç·"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "å床䞡蟺ã2ä¹ããŠæŽçãããš",
"title": "åæ²ç·"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "( c 2 â a 2 ) x 2 â a 2 y 2 = a 2 ( c 2 â a 2 ) {\\displaystyle (c^{2}-a^{2})x^{2}-a^{2}y^{2}=a^{2}(c^{2}-a^{2})}",
"title": "åæ²ç·"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "ããã§ã b 2 = c 2 â a 2 ( b > 0 ) {\\displaystyle b^{2}=c^{2}-a^{2}\\quad (b>0)} ãšããã䞡蟺ã a 2 b 2 {\\displaystyle a^{2}b^{2}} ã§å²ãã°",
"title": "åæ²ç·"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "x 2 a 2 â y 2 b 2 = 1 {\\displaystyle {\\frac {x^{2}}{a^{2}}}-{\\frac {y^{2}}{b^{2}}}=1}",
"title": "åæ²ç·"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "ã§ããã",
"title": "åæ²ç·"
},
{
"paragraph_id": 41,
"tag": "p",
"text": "åæ²ç·ã x 2 a 2 â y 2 b 2 = 1 {\\displaystyle {\\frac {x^{2}}{a^{2}}}-{\\frac {y^{2}}{b^{2}}}=1} ã§è¡šããããšããçŠç¹ã®åº§æšã¯ ( a 2 + b 2 , 0 ) , ( â a 2 + b 2 , 0 ) {\\displaystyle ({\\sqrt {a^{2}+b^{2}}},0),(-{\\sqrt {a^{2}+b^{2}}},0)} ãšãªãã",
"title": "åæ²ç·"
},
{
"paragraph_id": 42,
"tag": "p",
"text": "éã«ãåæ²ç·ã x 2 a 2 â y 2 b 2 = â 1 {\\displaystyle {\\frac {x^{2}}{a^{2}}}-{\\frac {y^{2}}{b^{2}}}=-1} ã§è¡šããããšããçŠç¹ã®åº§æšã¯ ( 0 , a 2 + b 2 ) , ( 0 , â a 2 + b 2 ) {\\displaystyle (0,{\\sqrt {a^{2}+b^{2}}}),(0,-{\\sqrt {a^{2}+b^{2}}})} ãšãªãã",
"title": "åæ²ç·"
},
{
"paragraph_id": 43,
"tag": "p",
"text": "x = f ( t ) , y = g ( t ) {\\displaystyle x=f(t),y=g(t)} ã§è¡šãããç¹ P ( x , y ) {\\displaystyle \\mathrm {P} (x,y)} ã®éåã¯ããæ²ç·ãæãããã®ãããªæ²ç·ã®è¡šç€ºãåªä»å€æ°è¡šç€ºãšããã",
"title": "åªä»å€æ°è¡šç€º"
},
{
"paragraph_id": 44,
"tag": "p",
"text": "åªä»å€æ°è¡šç€ºã§ã¯ F ( x , y ) = 0 {\\displaystyle F(x,y)=0} ã®åœ¢ã§ã¯è¡šãã«ããæ²ç·ãç°¡æœã«è¡šãããšãã§ãããäŸãã°ã x = t - sin t, y = 1 - cos t ã§ãããããã¯ãµã€ã¯ãã€ããšåŒã°ããã",
"title": "åªä»å€æ°è¡šç€º"
},
{
"paragraph_id": 45,
"tag": "p",
"text": "x = f ( t ) , y = g ( t ) {\\displaystyle x=f(t),y=g(t)} ãšåªä»å€æ°è¡šç€ºãããŠããæ²ç·ã x {\\displaystyle x} æ¹åã« p {\\displaystyle p} ã y {\\displaystyle y} æ¹åã« q {\\displaystyle q} ã ãã ãå¹³è¡ç§»åããæ²ç·ã¯ x = f ( t ) + p , y = g ( t ) + q {\\displaystyle x=f(t)+p,y=g(t)+q} ãšè¡šããã",
"title": "åªä»å€æ°è¡šç€º"
},
{
"paragraph_id": 46,
"tag": "p",
"text": "x = p t 2 , y = 2 p t p â 0 {\\displaystyle x=pt^{2},y=2pt\\quad p\\neq 0} ã§è¡šãããæ²ç·ã¯ t {\\displaystyle t} ãæ¶å»ãããš y 2 = 4 p x {\\displaystyle y^{2}=4px} ãšãªãã®ã§æŸç©ç·ã§ããã",
"title": "åªä»å€æ°è¡šç€º"
},
{
"paragraph_id": 47,
"tag": "p",
"text": "å x 2 + y 2 = r 2 {\\displaystyle x^{2}+y^{2}=r^{2}} ãåªä»å€æ°è¡šç€ºãããš x = r cos Ξ , y = r sin Ξ {\\displaystyle x=r\\cos \\theta ,y=r\\sin \\theta } ãšãªãã",
"title": "åªä»å€æ°è¡šç€º"
},
{
"paragraph_id": 48,
"tag": "p",
"text": "ãããŸã§ã®åŠç¿ã§ã¯ã x {\\displaystyle x} 軞㚠y {\\displaystyle y} 軞ã䜿ã£ã座æšå¹³é¢(çŽäº€åº§æšãšãã) ( x , y ) {\\displaystyle (x,y)} 䜿ãããšã§ã座æšå¹³é¢äžã®1ç¹ãå®ããã ããã§åŠã¶æ¥µåº§æšã§ã¯ã ( r , Ξ ) {\\displaystyle (r,\\theta )} ã®æåã§äžããããåŒã䜿ã£ãŠæ²ç·ãè¡šãããšãèããã",
"title": "極座æšãšæ¥µæ¹çšåŒ"
},
{
"paragraph_id": 49,
"tag": "p",
"text": "ããäžç¹OãšåçŽç·OXãå®ãããšãå¹³é¢äžã®ç¹Pã¯ãç¹Oããã®è·é¢rãšã â {\\displaystyle \\angle } XOPã®è§ Ξ ( 0 †Ξ < 2 Ï ) {\\displaystyle \\theta \\,(0\\leq \\theta <2\\pi )} ã®å€§ããã§äžæã«å®ãŸãã",
"title": "極座æšãšæ¥µæ¹çšåŒ"
},
{
"paragraph_id": 50,
"tag": "p",
"text": "極座æšã®å®çŸ©",
"title": "極座æšãšæ¥µæ¹çšåŒ"
},
{
"paragraph_id": 51,
"tag": "p",
"text": "åç¹Oãšè»žOXãå®ãããå¹³é¢äžã®ç¹Pã«ã€ããŠãOPéã®è·é¢ãrã â {\\displaystyle \\angle } XOPã®å€§ãããΞã§è¡šããåº§æš ( r , Ξ ) {\\displaystyle (r,\\theta )} ã極座æšãšããã ãã®ãšããOã極ãOXãå§ç·ãšããã ãŸãã Ξ {\\displaystyle \\theta } ãåè§ãšããã",
"title": "極座æšãšæ¥µæ¹çšåŒ"
},
{
"paragraph_id": 52,
"tag": "p",
"text": "ãŸããçŽäº€åº§æšãšæ¥µåº§æšã®é¢ä¿ã¯æ¬¡ã®ããã«ãªãã",
"title": "極座æšãšæ¥µæ¹çšåŒ"
},
{
"paragraph_id": 53,
"tag": "p",
"text": "çŽäº€åº§æšãšæ¥µåº§æšã®é¢ä¿",
"title": "極座æšãšæ¥µæ¹çšåŒ"
},
{
"paragraph_id": 54,
"tag": "p",
"text": "{ r = x 2 + y 2 cos Ξ = x r sin Ξ = y r { x = r cos Ξ y = r sin Ξ {\\displaystyle {\\begin{cases}r={\\sqrt {x^{2}+y^{2}}}\\\\\\cos \\theta =\\displaystyle {\\frac {x}{r}}\\\\\\sin \\theta =\\displaystyle {\\frac {y}{r}}\\end{cases}}\\,\\,{\\begin{cases}x=r\\cos \\theta \\\\y=r\\sin \\theta \\end{cases}}}",
"title": "極座æšãšæ¥µæ¹çšåŒ"
},
{
"paragraph_id": 55,
"tag": "p",
"text": "ããã¯çŽæçã«ã¯è€çŽ æ°å¹³é¢äžã®ç¹ã®çµ¶å¯Ÿå€ãšåè§ãå®ãããšãã«äŒŒãŠããã",
"title": "極座æšãšæ¥µæ¹çšåŒ"
},
{
"paragraph_id": 56,
"tag": "p",
"text": "r = f ( Ξ ) {\\displaystyle r=f(\\theta )} ã®åœ¢ã§äžããããåŒã極æ¹çšåŒ(ãããã»ããŠããã)ãšããã極æ¹çšåŒã¯rãšÎžã«ã€ããŠã®é¢æ°ã§ãããããããã¯xãšyãžã®å€æãå¯èœã§ããããã£ãŠxyå¹³é¢äžã«æ²ç·ããããŠãããããšã«ãªãã",
"title": "極座æšãšæ¥µæ¹çšåŒ"
},
{
"paragraph_id": 57,
"tag": "p",
"text": "ããŸããŸãªæ¥µæ¹çšåŒ",
"title": "極座æšãšæ¥µæ¹çšåŒ"
},
{
"paragraph_id": 58,
"tag": "p",
"text": "(1)äžå¿O,ååŸaã®å r = a {\\displaystyle r=a}",
"title": "極座æšãšæ¥µæ¹çšåŒ"
},
{
"paragraph_id": 59,
"tag": "p",
"text": "(2)äžå¿ ( r 0 , Ξ 0 ) {\\displaystyle (r_{0},{\\theta }_{0})} ,ååŸaã®å r 2 â 2 r r 0 cos Ξ 0 + r 0 2 = a 2 {\\displaystyle r^{2}-2rr_{0}\\cos {\\theta }_{0}+{r_{0}}^{2}=a^{2}}",
"title": "極座æšãšæ¥µæ¹çšåŒ"
},
{
"paragraph_id": 60,
"tag": "p",
"text": "(3)極Oãéããå§ç·ãšÎ±ã®è§ããªãçŽç· Ξ = α {\\displaystyle \\theta =\\alpha }",
"title": "極座æšãšæ¥µæ¹çšåŒ"
},
{
"paragraph_id": 61,
"tag": "p",
"text": "(4)ç¹ ( a , α ) {\\displaystyle (a,\\alpha )} ãéããOAã«åçŽãªçŽç· r cos ( Ξ â α ) = a {\\displaystyle r\\cos(\\theta -\\alpha )=a}",
"title": "極座æšãšæ¥µæ¹çšåŒ"
},
{
"paragraph_id": 62,
"tag": "p",
"text": "(äŸ)å ( x â 1 ) 2 + y 2 = 1 {\\displaystyle (x-1)^{2}+y^{2}=1} ã極æ¹çšåŒã§è¡šã. x = r cos Ξ , y = r sin Ξ {\\displaystyle x=r\\cos \\theta ,y=r\\sin \\theta } ã代å
¥ããŠæŽçãããš r ( r â 2 cos Ξ ) = 0 {\\displaystyle r(r-2\\cos \\theta )=0}",
"title": "極座æšãšæ¥µæ¹çšåŒ"
},
{
"paragraph_id": 63,
"tag": "p",
"text": "r = 0 {\\displaystyle r=0} ã¯æ¥µãè¡šããã r = 2 cos Ξ {\\displaystyle r=2\\cos \\theta }",
"title": "極座æšãšæ¥µæ¹çšåŒ"
},
{
"paragraph_id": 64,
"tag": "p",
"text": "ãããŸã§ã«ã2次æ²ç·ãåªä»å€æ°è¡šç€ºã極æ¹çšåŒãªã©ã®æ²ç·ãšãã®æ§è³ªã«ã€ããŠè¿°ã¹ãŠããã以äžã§ã¯ãããããå©çšããŠããŸããŸãªæ²ç·ã®åŒã瀺ããäžè¬ã«æŠåœ¢ãã€ããã®ã¯å°é£ãªãããã³ã³ãã¥ãŒã¿ã䜿çšããã",
"title": "ããŸããŸãªæ²ç·"
}
] | æŸç©ç·(parabola)ãæ¥å(ellipse)ãåæ²ç·(hyperbola)ããŸãšããŠã2次æ²ç·ãåéæ²ç·ãšãããããããã2次æ²ç·ãšåŒã°ããçç±ã¯ãæŸç©ç·ãæ¥åãåæ²ç·ã¯ x , y ã®2æ¬¡åŒ F ã«ãã£ãŠ F = 0 ã§è¡šãããšãã§ãããŸã x , y ã®2æ¬¡åŒ F ã«ãã£ãŠ F = 0 ãšè¡šãããæ²ç·ã¯æŸç©ç·ãæ¥åãåæ²ç·ã2çŽç·ã®ããããã«ãªãããã§ããã åéæ²ç·ãšåŒã°ããçç±ã¯ãåéé¢ããå
šãŠã®æ¯ç·ãšäº€ãããåºé¢ã«å¹³è¡ãªå¹³é¢ã§åæããããšãã®æé¢ãåããå
šãŠã®æ¯ç·ãšäº€ãããåºé¢ã«å¹³è¡ã§ãªãå¹³é¢ã§åæããããšãã®æé¢ãæ¥åããæ¯ç·ã«å¹³è¡ãªé¢ã§åæããããšãã®æé¢ãæŸç©ç·ããæ¯ç·ã«å¹³è¡ã§ãªãå¹³é¢ã§åæããããšãã®æé¢ãåæ²ç·ãšãªãããã§ããã 2次æ²ç·ã¯çŽç·ãåã«ã€ãã§éèŠãªæ²ç·ã§ããã | {{pathnav|é«çåŠæ ¡ã®åŠç¿|é«çåŠæ ¡æ°åŠ|é«çåŠæ ¡æ°åŠC|pagename=å¹³é¢äžã®æ²ç·|frame=1|small=1}}
æŸç©ç·(parabola)ãæ¥å(ellipse)ãåæ²ç·(hyperbola)ããŸãšããŠã2次æ²ç·ãåéæ²ç·ãšãããããããã2次æ²ç·ãšåŒã°ããçç±ã¯ãæŸç©ç·ãæ¥åãåæ²ç·ã¯ <math>x,y</math> ã®2æ¬¡åŒ <math>F(x,y)</math> ã«ãã£ãŠ <math>F(x,y) = 0</math> ã§è¡šãããšãã§ãããŸã <math>x,y</math> ã®2æ¬¡åŒ <math>F(x,y)</math> ã«ãã£ãŠ <math>F(x,y) = 0</math> ãšè¡šãããæ²ç·ã¯æŸç©ç·ãæ¥åãåæ²ç·ã2çŽç·ã®ããããã«ãªãããã§ããã
[[ãã¡ã€ã«:Conic Sections.svg|ãµã ãã€ã«]]
åéæ²ç·ãšåŒã°ããçç±ã¯ãåéé¢ããå
šãŠã®æ¯ç·ãšäº€ãããåºé¢ã«å¹³è¡ãªå¹³é¢ã§åæããããšãã®æé¢ãåããå
šãŠã®æ¯ç·ãšäº€ãããåºé¢ã«å¹³è¡ã§ãªãå¹³é¢ã§åæããããšãã®æé¢ãæ¥åããæ¯ç·ã«å¹³è¡ãªé¢ã§åæããããšãã®æé¢ãæŸç©ç·ããæ¯ç·ã«å¹³è¡ã§ãªãå¹³é¢ã§åæããããšãã®æé¢ãåæ²ç·ãšãªãããã§ããã
2次æ²ç·ã¯çŽç·ãåã«ã€ãã§éèŠãªæ²ç·ã§ããã
==æŸç©ç·==
å¹³é¢äžã«ç¹ <math>\mathrm{F}</math> ãšãç¹ <math>\mathrm{F}</math> ãéããªãçŽç· <math>l</math> ããšãããã®ãšããçŽç· <math>l</math> ããã®è·é¢ãšç¹ <math>\mathrm{F}</math> ããã®è·é¢ãçããç¹ã®è»è·¡ãæŸç©ç·ãšããããã®ãšããç¹ <math>\mathrm{F}</math> ãæŸç©ç·ã®çŠç¹ãçŽç· <math>l</math> ãæŸç©ç·ã®æºç·ãšããã
[[ãã¡ã€ã«:Parabola with focus and directrix.svg|ãµã ãã€ã«]]
çŠç¹ã <math>\mathrm{F}(p,0)</math> æºç·ã <math>l:x=-p</math> ãšããæŸç©ç·ã®æ¹çšåŒãæ±ããã<math>\mathrm{P}(x,y)</math> ããã®æŸç©ç·ã®ç¹ãšãããšãç¹ <math>\mathrm{P}</math> ãšçŽç· <math>l</math> ã®è·é¢ã¯ <math>x+p</math> ã§ããã<math>\mathrm{PF} =\sqrt{ (x-p)^2 + y^2}</math> ã§ããããªã®ã§ã <math>(x+p)^2 = (x-p)^2 + y^2</math> ã§ããããããæŽçããŠã
<math>y^2 = 4px</math>
ãåŸãã
ããã§ãæŸç©ç· <math>y^2 = 4px</math> ã«ãããŠã <math>x</math> ãš <math>y</math> ãå
¥ãæ¿ããã° <math>y = \frac{x^2}{4p}</math> ã§ãããããããäžåŠããåŠãã§ããæŸç©ç·ã®å®çŸ©ãšäžèŽããããšããããã
'''æŒç¿åé¡'''
æŸç©ç· <math>y = ax^2 \quad (a\neq 0)</math> ã®çŠç¹ãšæºç·ãæ±ããã
'''解ç'''
çŠç¹ <math>\left(0,\frac{1}{4a}\right)</math> æºç· <math>y = -\frac{1}{4a}</math>
==æ¥å==
å¹³é¢äžã«ç°ãªã2ç¹ <math>\mathrm{F},\mathrm{F'}</math> ããšãã<math>\mathrm{F}</math> ãšã®è·é¢ãšã <math>\mathrm{F'}</math> ãšã®è·é¢ã®åãäžå®ã§ããç¹ã®è»è·¡ãæ¥åãšããããã®ãšããç¹ <math>\mathrm{F},\mathrm{F'}</math> ãæ¥åã®çŠç¹ãšããã
çŠç¹ã <math>\mathrm{F}(c,0),\mathrm{F'}(-c,0)</math> ãšãããç¹ <math>\mathrm{P}(x,y)</math> ãæ¥åäžã®ç¹ã§ãããšãã <math>\mathrm{PF} + \mathrm{PF'} = 2a</math> ã§ããã<math>\mathrm{PF} = 2a-\mathrm{PF'}</math> ãã
<math>\sqrt{(x-c)^2+y^2}=2a-\sqrt{(x+c)^2+y^2}</math>
䞡蟺ã2ä¹ããŠæŽçãããš
<math>a\sqrt{(x+c)^2+y^2}=a^2+cx</math>
å床ã䞡蟺ã2ä¹ããŠæŽçãããš
<math>(a^2-c^2)x^2+a^2y^2=a^2(a^2-c^2)</math>
ãã㧠<math>a^2-c^2=b^2 \quad(b >0)</math> ãšçœ®ãæãããš
<math>b^2x^2+a^2y^2=a^2b^2</math>
䞡蟺ã <math>a^2b^2</math> ã§å²ããš
<math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a>b>0)</math>
ãå°ãããã
''x''軞ãšã®äº€ç¹ã¯<math>(a,0)</math>ã<math>(-a,0)</math>ã''y''軞ãšã®äº€ç¹ã¯<math>(0,b)</math>ã<math>(0,-b)</math>ãšãªãã
<math>a>b>0</math>ã®ãšãã<math>2a</math>ã¯é·è»žã®é·ãïŒé·åŸïŒã<math>2b</math>ã¯ç軞ã®é·ãïŒçåŸïŒãšãªãã''xy''å¹³é¢äžã«ã°ã©ããæžããšæšªé·ã®æ¥åã«ãªãããŸãçŠç¹ã¯é·åŸã§ãã''x''軞äžã«ãããã®åº§æšã¯<math>(-\sqrt{a^2-b^2},0),(\sqrt{a^2-b^2},0)</math>ãšãªãã
éã«ã<math>b>a>0</math>ã®ãšãã<math>2b</math>ã¯é·è»žã®é·ãïŒé·åŸïŒã<math>2a</math>ã¯ç軞ã®é·ãïŒçåŸïŒãšãªãã''xy''å¹³é¢äžã«ã°ã©ããæžããšçžŠé·ã®æ¥åã«ãªãããŸãçŠç¹ã¯é·åŸã§ãã''y''軞äžã«ãããã®åº§æšã¯<math>(0,\sqrt{b^2-a^2}),(0,-\sqrt{b^2-a^2})</math>ãšãªãã
2ã€ã®çŠç¹ãè¿ãã»ã©æ¥åã¯åã«è¿ã¥ãã2ã€ã®çŠç¹ãéãªã£ããšã <math>a=b</math> ãšãªããæ¥åã¯åã«ãªãã
ã¡ãªã¿ã«ãææã®åšããå
¬è»¢ããææã®è»éã¯ãææãçŠç¹ãšããæ¥åã«ãªãã
==åæ²ç·==
å¹³é¢äžã«ç°ãªã2ç¹ <math>\mathrm{F},\mathrm{F'}</math> ããšãã<math>\mathrm{F}</math> ãšã®è·é¢ãšã <math>\mathrm{F'}</math> ãšã®è·é¢ã®å·®ãäžå®ã§ããç¹ã®è»è·¡ãåæ²ç·ãšããã2ç¹ <math>\mathrm{F},\mathrm{F'}</math> ãåæ²ç·ã®çŠç¹ãšããã
çŠç¹ã <math>\mathrm{F}(c,0),\mathrm{F'}(-c,0)</math> ãšãããç¹ <math>\mathrm{P}(x,y)</math> ãåæ²ç·äžã®ç¹ã§ãããšãã <math>|\mathrm{PF}-\mathrm{PF'}|=2a</math> ã§ããã<math>\mathrm{PF} = \pm 2a + \mathrm{PF'}</math> ãã
<math>\sqrt{(x-c)^2+y^2}=\pm 2a+\sqrt{(x+c)^2+y^2}</math>
䞡蟺ã2ä¹ããŠæŽçãããš
<math>\pm a \sqrt{(x+c)^2+y^2} = -a^2 -cx</math>
å床䞡蟺ã2ä¹ããŠæŽçãããš
<math>(c^2-a^2)x^2 - a^2y^2 = a^2(c^2-a^2)</math>
ããã§ã <math>b^2 = c^2 - a^2 \quad (b > 0)</math> ãšããã䞡蟺ã <math>a^2b^2</math> ã§å²ãã°
<math>\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1</math>
ã§ããã
åæ²ç·ã<math>\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1</math>ã§è¡šããããšããçŠç¹ã®åº§æšã¯<math>(\sqrt{a^2+b^2},0),(-\sqrt{a^2+b^2},0)</math>ãšãªãã
éã«ãåæ²ç·ã<math>\frac{x^2}{a^2} - \frac{y^2}{b^2} = -1</math>ã§è¡šããããšããçŠç¹ã®åº§æšã¯<math>(0,\sqrt{a^2+b^2}),(0,-\sqrt{a^2+b^2})</math>ãšãªãã
== åªä»å€æ°è¡šç€º==
<math>x=f(t),y=g(t)</math> ã§è¡šãããç¹ <math>\mathrm{P}(x,y)</math> ã®éåã¯ããæ²ç·ãæãããã®ãããªæ²ç·ã®è¡šç€ºãåªä»å€æ°è¡šç€ºãšããã
åªä»å€æ°è¡šç€ºã§ã¯ <math>F(x,y)=0</math> ã®åœ¢ã§ã¯è¡šãã«ããæ²ç·ãç°¡æœã«è¡šãããšãã§ãããäŸãã°ã
x = t - sin t,
y = 1 - cos t
ã§ãããããã¯ãµã€ã¯ãã€ããšåŒã°ããã
[[ç»å:Cycloid_function.png|thumb|left|500px|ãµã€ã¯ãã€ã]]
<math>x=f(t),y=g(t)</math> ãšåªä»å€æ°è¡šç€ºãããŠããæ²ç·ã <math>x</math> æ¹åã« <math>p</math>ã <math>y</math> æ¹åã« <math>q</math> ã ãã ãå¹³è¡ç§»åããæ²ç·ã¯ <math>x=f(t)+p,y=g(t)+q</math> ãšè¡šããã
=== äºæ¬¡æ²ç·ã®åªä»å€æ°è¡šç€º ===
<math>x=pt^2,y=2pt \quad p \neq 0</math> ã§è¡šãããæ²ç·ã¯ <math>t</math> ãæ¶å»ãããš <math>y^2=4px</math> ãšãªãã®ã§æŸç©ç·ã§ããã
å <math>x^2+y^2=r^2</math> ãåªä»å€æ°è¡šç€ºãããš <math>x=r\cos \theta,y=r\sin \theta</math> ãšãªãããã®ããšãããäžè§é¢æ°ã®ããšã'''åé¢æ°'''ãšåŒã¶å Žåãããã
æ¥å<Math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1</Math>ãåªä»å€æ°è¡šç€ºãããš<Math>x=b\cos \theta. y=a\sin \theta</Math>ãšãªãã
åæ²ç·<Math>\frac{x^2}{a^2}-\frac{y^2}{b^2}=1</Math>ã®åªä»å€æ°è¡šç€ºã¯<Math>x=\frac{a}{\cos \theta}, y=b\tan \theta</Math>ãšãªãã
'''åæ²ç·é¢æ°ïŒåèïŒ'''
ãã€ãã¢æ°<Math>e</Math>ãçšããŠ<Math>\sinh \theta=\frac{e^\theta - e^{-\theta}}{2}, \cosh \theta=\frac{e^\theta + e^{-\theta}}{2}</Math>ãšå®çŸ©ãããš<Math>\sinh^2 \theta-\cosh^2 \theta=1</Math>ãæãç«ã€ã®ã§ãäžèšã®åæ²ç·ã®åŒã¯<Math>x=a\sinh \theta, y=b\cosh \theta</Math>ãšæžããã
<Math>\tanh \theta=\frac{e^\theta - e^{-\theta}}{e^\theta + e^{-\theta}}</Math>ãšå®ãããšã<Math>\sinh \theta, \cosh \theta, \tanh \theta</Math>ïŒ'''ã·ã£ã€ã³'''ã'''ã³ãã·ã¥'''ã'''ã¿ã³ã'''ãããã¯'''ãã€ãããªãã¯ãµã€ã³'''ã'''ãã€ãããªãã¯ã³ãµã€ã³'''ã'''ãã€ãããªãã¯ã¿ã³ãžã§ã³ã'''ãšèªãïŒã¯äžè§é¢æ°ãšäŒŒãå
¬åŒïŒçžäºé¢ä¿ãå æ³å®çã埮ç©åå
¬åŒãªã©ïŒã幟ã€ãæãç«ã€ãããã§ããã®3ã€ã®é¢æ°ãšãã®éæ°ãçºããŠ'''åæ²ç·é¢æ°'''ãšåŒã¶ããšã«ããã
éäžè§é¢æ°ãšäžŠã³ãåæ²ç·é¢æ°ãšãã®éé¢æ°ã¯å€§åŠå
¥è©Šã«ãããŠããçš®ã®å®ç©åã®åé¡ã解ãéã«åœ¹ç«ã€ããšã§æåã§ããããã®åé¡ã¯ãããã®é¢æ°ã®åŸ®ç©åå
¬åŒãèæ¯ãšããŠããããããããã®é¢æ°ã§çœ®æãããšç°¡åã«è§£ããããã«ãªã£ãŠããã
<Math>y=\cosh x</Math>ã®ã°ã©ãã¯'''æžåç·ïŒã«ãããªãŒïŒ'''ãšåŒã°ããæåãªæ²ç·ãæãããšã§ç¥ãããŠããã
äžè§é¢æ°ïŒåé¢æ°ïŒãšåæ²ç·é¢æ°ã¯éåžžã«äŒŒãæ§è³ªãæã€ããããã¯'''åæ²ç·é¢æ°ã®å®çŸ©åŒãäžè§é¢æ°ã®è€çŽ ææ°é¢æ°è¡šç€ºãå®æ°ç¯å²ã§æžãæãããã®'''ã§ãããæŽã«ã¯'''äž¡è
ãšã第äžçš®äžå®å
šæ¥åç©åã®éé¢æ°ã§å®çŸ©ãããïŒã€ã³ãã®ïŒæ¥åé¢æ°ã®ç¹å¥ãªå Žå'''ãæããŠããããã§ããã
æ²ç·<Math>x=f(t), y=g(t)</Math>ã<Math>x</Math>軞æ¹åã«<Math>p</Math>ã<Math>y</Math>軞æ¹åã«<Math>q</Math>ã ã䞊è¡ç§»åããæ²ç·ã¯<Math>x=f(t) +p, y=g(t) +q</Math>ãšæžãè¡šãããã
ãªããïŒè€çŽ æ°<Math>Z</Math>ã®æ¹çšåŒïŒ<Math>=x+yi</Math>ã®åœ¢ã§è¡šãããåŒã<Math>Z</Math>ã®æ¥µåœ¢åŒãçšããŠè§£ããšäºæ¬¡æ²ç·ã®åªä»å€æ°è¡šç€ºãçŸããå Žåãããã
== 極座æšãšæ¥µæ¹çšåŒ ==
=== æ¥µåº§æš ===
ãããŸã§ã®åŠç¿ã§ã¯ã<math>x</math>軞ãš<math>y</math>軞ã䜿ã£ã座æšå¹³é¢ïŒ'''çŽäº€åº§æš'''ãšããïŒ<math>(x,y)</math>䜿ãããšã§ã座æšå¹³é¢äžã®1ç¹ãå®ããã
ããã§åŠã¶æ¥µåº§æšã§ã¯ã<math>(r, \theta )</math> ã®æåã§äžããããåŒã䜿ã£ãŠæ²ç·ãè¡šãããšãèããã
ããäžç¹OãšåçŽç·OXãå®ãããšãå¹³é¢äžã®ç¹Pã¯ãç¹Oããã®è·é¢rãšã<math>\angle </math>XOPã®è§<math>\theta \,(0 \le \theta < 2 \pi)</math>ã®å€§ããã§äžæã«å®ãŸãã
極座æšã®å®çŸ©
åç¹Oãšè»žOXãå®ãããå¹³é¢äžã®ç¹Pã«ã€ããŠãOPéã®è·é¢ãrã<math>\angle </math>XOPã®å€§ãããΞã§è¡šãã座æš<math>(r, \theta)</math>ã'''極座æš'''ãšããã
ãã®ãšããOã'''極'''ãOXã'''å§ç·'''ãšããã
ãŸãã<math>\theta</math>ã'''åè§'''ãšããã
ãŸããçŽäº€åº§æšãšæ¥µåº§æšã®é¢ä¿ã¯æ¬¡ã®ããã«ãªãã
çŽäº€åº§æšãšæ¥µåº§æšã®é¢ä¿
<math>\begin{cases}r = \sqrt{x^2 + y^2} \\ \cos \theta =\displaystyle{\frac{x}{r}} \\ \sin \theta =\displaystyle{ \frac{y}{r}} \end{cases} \,\,
\begin{cases} x = r\cos \theta \\ y = r\sin \theta \end{cases}</math>
ããã¯çŽæçã«ã¯è€çŽ æ°å¹³é¢äžã®ç¹ã®çµ¶å¯Ÿå€ãšåè§ãå®ãããšãã«äŒŒãŠããã
=== 極æ¹çšåŒ ===
<math>r = f( \theta )</math>ã®åœ¢ã§äžããããåŒã'''極æ¹çšåŒ'''ïŒãããã»ããŠãããïŒãšããã極æ¹çšåŒã¯rãšÎžã«ã€ããŠã®é¢æ°ã§ãããããããã¯xãšyãžã®å€æãå¯èœã§ããããã£ãŠxyå¹³é¢äžã«æ²ç·ããããŠãããããšã«ãªãã
ããŸããŸãªæ¥µæ¹çšåŒ
(1)äžå¿O,ååŸaã®å <math>r=a</math>
(2)äžå¿<math>(r_0,{\theta}_0)</math>,ååŸaã®å <math>r^2-2rr_0\cos {\theta}_0+{r_0}^2=a^2</math>
(3)極Oãéããå§ç·ãšÎ±ã®è§ããªãçŽç·ã<math>\theta=\alpha</math>
(4)ç¹<math>(a,\alpha)</math>ãéããOAã«åçŽãªçŽç·ã<math>r\cos(\theta-\alpha)=a</math>
(äŸïŒå<math>(x-1)^2+y^2=1</math>ã極æ¹çšåŒã§è¡šã.
<math>x = r\cos \theta, y = r\sin \theta</math>ã代å
¥ããŠæŽçãããš
<math>r(r-2\cos\theta)=0</math>
<math>r=0</math>ã¯æ¥µãè¡šãããã<math>r=2\cos\theta</math>
==ããŸããŸãªæ²ç·==
ãããŸã§ã«ã2次æ²ç·ãåªä»å€æ°è¡šç€ºã極æ¹çšåŒãªã©ã®æ²ç·ãšãã®æ§è³ªã«ã€ããŠè¿°ã¹ãŠããã以äžã§ã¯ãããããå©çšããŠããŸããŸãªæ²ç·ã®åŒã瀺ããäžè¬ã«æŠåœ¢ãã€ããã®ã¯å°é£ãªãããã³ã³ãã¥ãŒã¿ã䜿çšããã
*ãµã€ã¯ãã€ã
*ã«ãŒãžãªã€ã
*ã¢ã¹ããã€ã
*ãªããœã³
*ãã©æ²ç·
*ã¬ã ãã¹ã±ãŒã
*ãªãµãŒãžã¥
{{DEFAULTSORT:ãããšããã€ããããããIII ãžããããããã®ããããã}}
[[Category:é«çåŠæ ¡æ°åŠIII|ãžããããããã®ããããã]] | 2005-05-03T08:21:38Z | 2024-02-21T02:30:00Z | [
"ãã³ãã¬ãŒã:Pathnav"
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E6%95%B0%E5%AD%A6C/%E5%B9%B3%E9%9D%A2%E4%B8%8A%E3%81%AE%E6%9B%B2%E7%B7%9A |
1,901 | æ§èª²çš(-2012幎床)é«çåŠæ ¡æ°åŠII | æ°åŠ II ã¯ã
ããæã£ãŠããã
é«çåŠæ ¡æå°èŠç¶±ã®æ°åŠIIã®ç®æšã«ã¯ã
ã åŒãšèšŒæã»é«æ¬¡æ¹çšåŒ,å³åœ¢ãšæ¹çšåŒ,ãããããªé¢æ°åã³åŸ®åã»ç©åã®èãã«ã€ããŠç解ãã,åºç€çãªç¥èã®ç¿åŸãšæèœã®ç¿çãå³ã,äºè±¡ãæ°åŠçã«èå¯ãåŠçããèœåã䌞ã°ããšãšãã«,ãããã掻çšããæ
床ãè²ãŠããã
ãšãããæ°åŠIã§åŠãã èšç®æè¡ãããšã«ãããé«åºŠãªæ°åŠã身ã«ã€ããããšãç®æšãšããŠããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "æ°åŠ II ã¯ã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ããæã£ãŠããã",
"title": ""
},
{
"paragraph_id": 2,
"tag": "p",
"text": "",
"title": ""
},
{
"paragraph_id": 3,
"tag": "p",
"text": "é«çåŠæ ¡æå°èŠç¶±ã®æ°åŠIIã®ç®æšã«ã¯ã",
"title": ""
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ã åŒãšèšŒæã»é«æ¬¡æ¹çšåŒ,å³åœ¢ãšæ¹çšåŒ,ãããããªé¢æ°åã³åŸ®åã»ç©åã®èãã«ã€ããŠç解ãã,åºç€çãªç¥èã®ç¿åŸãšæèœã®ç¿çãå³ã,äºè±¡ãæ°åŠçã«èå¯ãåŠçããèœåã䌞ã°ããšãšãã«,ãããã掻çšããæ
床ãè²ãŠããã",
"title": ""
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ãšãããæ°åŠIã§åŠãã èšç®æè¡ãããšã«ãããé«åºŠãªæ°åŠã身ã«ã€ããããšãç®æšãšããŠããã",
"title": ""
}
] | æ°åŠ II ã¯ã åŒãšèšŒæã»é«æ¬¡æ¹çšåŒ
å³åœ¢ãšæ¹çšåŒ
ææ°é¢æ°ã»å¯Ÿæ°é¢æ°
äžè§é¢æ°
埮åã»ç©åã®èã ããæã£ãŠããã | {{pathnav|é«çåŠæ ¡ã®åŠç¿|é«çåŠæ ¡æ°åŠ|frame=1}}
æ°åŠ II ã¯ã
* [[é«çåŠæ ¡æ°åŠII/åŒãšèšŒæã»é«æ¬¡æ¹çšåŒ|åŒãšèšŒæã»é«æ¬¡æ¹çšåŒ]]
* [[é«çåŠæ ¡æ°åŠII/å³åœ¢ãšæ¹çšåŒ|å³åœ¢ãšæ¹çšåŒ]]
* [[é«çåŠæ ¡æ°åŠII/ææ°é¢æ°ã»å¯Ÿæ°é¢æ°|ææ°é¢æ°ã»å¯Ÿæ°é¢æ°]]
*[[é«çåŠæ ¡æ°åŠII/äžè§é¢æ°|äžè§é¢æ°]]
* [[é«çåŠæ ¡æ°åŠII/埮åã»ç©åã®èã|埮åã»ç©åã®èã]]
ããæã£ãŠããã
=== æ°åŠ II ãåŠã¶æ矩 ===
é«çåŠæ ¡æå°èŠç¶±ã®æ°åŠIIã®ç®æšã«ã¯ã
<blockquote>ããåŒãšèšŒæã»é«æ¬¡æ¹çšåŒïŒå³åœ¢ãšæ¹çšåŒïŒãããããªé¢æ°åã³åŸ®åã»ç©åã®èãã«ã€ããŠç解ããïŒåºç€çãªç¥èã®ç¿åŸãšæèœã®ç¿çãå³ãïŒäºè±¡ãæ°åŠçã«èå¯ãåŠçããèœåã䌞ã°ããšãšãã«ïŒãããã掻çšããæ
床ãè²ãŠããã</blockquote>
ãšãããæ°åŠâ
ã§åŠãã èšç®æè¡ãããšã«ãããé«åºŠãªæ°åŠã身ã«ã€ããããšãç®æšãšããŠããã
[[Category:æ°åŠæè²|æ§1 ãããšããã£ãããããã2]] | 2005-05-04T09:04:50Z | 2023-12-09T21:31:17Z | [
"ãã³ãã¬ãŒã:Pathnav"
] | https://ja.wikibooks.org/wiki/%E6%97%A7%E8%AA%B2%E7%A8%8B(-2012%E5%B9%B4%E5%BA%A6)%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E6%95%B0%E5%AD%A6II |
1,902 | é«çåŠæ ¡æ°åŠII/åŒãšèšŒæã»é«æ¬¡æ¹çšåŒ | æ¬é
ã¯é«çåŠæ ¡æ°åŠIIã®åŒãšèšŒæã»é«æ¬¡æ¹çšåŒã®è§£èª¬ã§ããã
( a + b ) 5 = ( a + b ) ( a + b ) ( a + b ) ( a + b ) ( a + b ) {\displaystyle (a+b)^{5}=(a+b)(a+b)(a+b)(a+b)(a+b)} ã«ã€ããŠèãããããã®åŒãå±éãããšãã a 2 b 3 {\displaystyle a^{2}b^{3}} ã®ä¿æ°ã¯ãå³èŸºã®5åã® ( a + b ) {\displaystyle (a+b)} ãã a {\displaystyle a} ã3ååãçµã¿åããã«çãããã 5 C 2 = 10 {\displaystyle _{5}\mathrm {C} _{2}=10} ã§ããã
ãã®èããæ¡åŒµããŠ
ãå±éããã a r b n â r {\displaystyle a^{r}b^{n-r}} ã®é
ã®ä¿æ°ã¯ãå³èŸºã® n {\displaystyle n} åã® ( a + b ) {\displaystyle (a+b)} ãã a {\displaystyle a} ã r {\displaystyle r} ååãçµã¿åããã«çãããã n C r {\displaystyle _{n}\mathrm {C} _{r}} ã§ããã
ãã£ãŠã次ã®åŒãåŸããã:
æåŸã®åŒã¯æ°Bã®æ°åã§åŠã¶ç·åèšå· Σ {\displaystyle \Sigma } ã§ãããç¥ããªãã®ãªãç¡èŠããŠãè¯ãã ãã®åŒã äºé
å®ç(binomial theorem) ãšããããŸããããããã®é
ã«ãããä¿æ°ãäºé
ä¿æ°(binomial coefficient) ãšåŒã¶ããšãããã
(I)
(II)
(II)
ãããããèšç®ããã
äºé
å®çãçšããŠèšç®ããã°ãããå®éã«èšç®ãè¡ãªããšã
(I)
(II)
(III)
ãšãªãã
ãã¹ãŠã®èªç¶æ°nã«å¯ŸããŠ
(I)
(II)
(III)
ãæãç«ã€ããšã瀺ãã
äºé
å®ç
ã«ã€ããŠa,bã«é©åœãªå€ã代å
¥ããã°ããã
(I) a = 1,b=1ã代å
¥ãããšã
ãšãªã確ãã«äžããããé¢ä¿ãæç«ããããšãåããã
(II) a=2,b=1ã代å
¥ãããšã
ãšãªã確ãã«äžããããé¢ä¿ãæç«ããããšãåããã
(III) a=1,b=-1ã代å
¥ãããšã
ãšãªã確ãã«äžããããé¢ä¿ãæç«ããããšãåããã
äºé
å®çãæ¡åŒµã㊠( a + b + c ) n {\displaystyle (a+b+c)^{n}} ãå±éããããšãèãããã a p b q c r {\displaystyle a^{p}b^{q}c^{r}} ( p + q + r = n ) {\displaystyle (p+q+r=n)} ã®é
ã®ä¿æ°ã¯ n {\displaystyle n} åã® ( a + b + c ) {\displaystyle (a+b+c)} ãã p {\displaystyle p} åã® a {\displaystyle a} ã q {\displaystyle q} åã® b {\displaystyle b} ã r {\displaystyle r} åã® c {\displaystyle c} ãéžã¶çµåãã«çãããã n ! p ! q ! r ! {\displaystyle {\frac {n!}{p!q!r!}}} ã§ããã
ããã§ã¯ãæŽåŒã®é€æ³ãšåæ°åŒã«ã€ããŠæ±ããæŽåŒã®é€æ³ã¯ãæŽåŒãæŽæ°ã®ããã«æ±ãé€æ³ãè¡ãªãèšç®ææ³ã®ããšã§ãããå®éã«æŽæ°ã®é€æ³ãšæŽåŒã®é€æ³ã«ã¯æ·±ãã€ãªããããããæŽåŒã®å æ°å解ãèãããšãã以äžå æ°å解ã§ããªãæŽåŒãååšããããã®æŽåŒãæŽæ°ã§ããçŽ å æ°ã®ããã«æ±ãããšã§æŽåŒã®çŽ å æ°å解ãå¯èœã«ãªãã
äžã§ã¯ãæŽåŒãæŽæ°ã«å¯Ÿå¿ããæ§è³ªãæã€ããšãè¿°ã¹ããæŽæ°ã«ã€ããŠã¯ãããã«çŽ ãª2ã€ã®æŽæ°ãåãããšã§æçæ°ãå®çŸ©ããããšãåºæ¥ããæŽåŒã«å¯ŸããŠãåãäºãæç«ã¡ããã®ãããªåŒãåæ°åŒãšåŒã¶ã
åæ°ãçšããªããšãã«ã¯ãæŽæ°ã®é€æ³ã¯åãšäœããçšããŠå®çŸ©ãããããã®æãå²ãããæ°Bã¯åDãšå²ãæ°AãäœãRãçšããŠ
ã®æ§è³ªãæºããããšãç¥ãããŠãããæŽåŒã«å¯ŸããŠã䌌ãæ§è³ªãæç«ã¡ãå²ãããåŒB(x)ãåD(x)ãšå²ãåŒA(x)ãäœãR(x)ãçšããŠã
ã®å³èŸºã§xã«ã€ããŠ2次ã®é
ãçŸãã巊蟺ãšäžèŽããªããªãããã£ãŠåã¯å®æ°ã§ãããåãaãäœããrãšãããšäžã®åŒã¯ã
ãšãªãããããã¯a=1,r=1ã§æç«ããããã£ãŠå1,äœã1ã§ãããããé«æ¬¡ã®åŒã«å¯ŸããŠãåãæ§ã«çããå®ããŠããã°ãããäŸãšããŠã
ã®ãããªåŒãèããããã®å Žåã
ã§ãB(x)ã3次ãA(x)ã2次ã§ããããšãããD(x)ã¯1次ã§ããããŸããR(x)ã¯2次ããå°ããããšãã1次以äžã®åŒã«ãªããããã§ãD(x)=ax+b,R(x)=cx+dãšãããšã
ãåŸããããå³èŸºãå±éãããšã
ãåŸãããããxã«ã©ããªå€ãå
¥ããŠããã®çåŒãæãç«ããªããã°ãªããªãã®ã§ãa = 1, b = 0, -a +c = 0, -b +d = 0ãåŸãããçµå±a=c=1, b=d=0ãåŸãããã
ãã®æ¹æ³ã¯ã©ã®é€æ³ã«å¯ŸããŠãçšããããšãåºæ¥ããã次æ°ãé«ããªããšèšç®ãé£ãããªããæŽæ°ã®å Žåãšåæ§ãæŽåŒã®é€æ³ã§ãçç®ãçšããããšãåºæ¥ããäžã®äŸãçšããŠçµæã ããæžããšã
ã®ããã«ãªãã)å³ã«æžãããåŒãå²ãããåŒã§ããã)å·Šã«æžãããåŒãå²ãåŒã§ããã--ã®äžçªäžã«æžãããåŒã¯åã§ãããæŽæ°ã®å²ãç®åæ§å·Šã«æžãããæ°ããé ã«å²ã£ãŠãããããã§ã¯æ¬¡æ°ã倧ããé
ãããå
ã«èšç®ãããé
ã§ãããå²ãããåŒã®äžã«ããåŒã¯åã®ç¬¬1é
ãå²ãåŒã«ãããŠåŸãåŒã§ãããããã§ã¯ã x ( x 2 â 1 ) {\displaystyle x(x^{2}-1)} ã§ã x 3 â x {\displaystyle x^{3}-x} ãšãªãããã ããæŽæ°ã®é€æ³ãšåæ§ãäœãããããªããŠã¯ãªããªãããã®åŸãå²ãããåŒãã x 3 â x {\displaystyle x^{3}-x} ãåŒããæ®ã£ãåŒãæ°ããå²ãããåŒãšããŠæ±ããããã§ã¯ãåŸãåŒãå²ãåŒãããäœæ¬¡ã§ããããšãããããã§èšç®ã¯çµäºã§ããã
x 3 + 2 x 2 + 1 {\displaystyle x^{3}+2x^{2}+1} ã x 4 + 4 x 2 + 3 x + 2 {\displaystyle x^{4}+4x^{2}+3x+2} ãã x 2 + 2 x + 6 {\displaystyle x^{2}+2x+6} ã§å²ã£ãåãšäœããæ±ããã
ãã®èšç®ã¯ã¢ãã¡ãŒã·ã§ã³ã䜿ã£ãŠ 詳ãã衚瀺ãããŠãããèšç®ææ³ã¯ã æŽæ°ã®å Žåã®çç®ãšåããããªææ³ã䜿ããã
ãåŸãããã®ã§ãå x {\displaystyle x} ãäœã â 6 x + 1 {\displaystyle -6x+1} ã§ããã
2ã€ç®ã®åŒã«ã€ããŠã¯ã
ãåŸãããã ãã£ãŠãç㯠å x 2 â 2 x + 2 {\displaystyle x^{2}-2x+2} ãäœã 11 x â 10 {\displaystyle 11x-10} ã§ããã
ãããŸã§ã§æŽåŒãæŽæ°ã®ããã«æ±ããæŽåŒã®é€æ³ãè¡ãªãæ¹æ³ã«ã€ããŠè¿°ã¹ããããã§ã¯ãæŽåŒã«å¯ŸããŠåæ°åŒãå®çŸ©ããæ¹æ³ã«ã€ããŠè¿°ã¹ããåæ°åŒãšã¯ãæŽæ°ã«å¯Ÿããåæ°ã®ããã«ãé€æ³ã«ãã£ãŠçããåŒã§ãããããã§ãé€æ³ãè¡ãªãåŒã¯ã©ã®ãããªãã®ã§ãå·®ãæ¯ããªããåæ°åŒã§ã¯ãååã«å²ãããåŒãæžããåæ¯ã«å²ãåŒãæžããäŸãã°ã
ã¯ãååx+1ãåæ¯ x 2 + 4 {\displaystyle x^{2}+4} ã®åæ°åŒã§ãããåæ°åŒã«å¯ŸããŠãçŽåãéåãååšãããçŽåã¯å
±éå æ°ãæã£ãåååæ¯ããã€åæ°åŒã§çšããããããã®æã«ã¯åååæ¯ãå
±éå æ°ã§å²ããåŒãç°¡åã«ããããšãåºæ¥ããéåã¯ãåæ°åŒã®å æ³ã®æã«ããçšããããããåååæ¯ã«åãæŽåŒããããŠãåæ°åŒãå€åããªãæ§è³ªãçšããã
ãç°¡åã«ããããŸãã
ãèšç®ããã
ã«ã€ããŠååãšåæ¯ãå æ°å解ãããšãåæ¹ãšãã«
ãå æ°ãšããŠå«ãã§ããããšãåããããã®ãšããå
±éã®å æ°ã¯çŽåããããšãå¿
èŠã§ãããèšç®ãããå€ã¯ã
ãšãªãã
次ã®åé¡ã§ã¯ã
ãèšç®ããããã®ãšãã䞡蟺ã®åæ¯ãããããå¿
èŠãããããä»åã«ã€ããŠã¯ãåçŽã«ããããã®åæ°åŒã®ååãšåæ¯ã«åã
ã®åæ¯ããããŠåæ¯ãçµ±äžããã°ãããèšç®ãããšã
ãšãªãã åæ°åŒã®ä¹æ³ã¯ãåååæ¯ãå¥ã
ã«ãããã°ããã
次ã®èšç®ãããã
(I)
(II)
(I)
(II)
åæ¯ãç©ã®åœ¢ã§ããåæ°åŒãäºã€ã®åæ°åŒã®åãå·®ã§è¡šãããåŒã«å€åœ¢ããæäœãéšååæ°å解ãšããã
1 x ( x + 1 ) {\displaystyle {\frac {1}{x(x+1)}}} ãš 1 ( x + 1 ) ( x + 3 ) {\displaystyle {\frac {1}{(x+1)(x+3)}}} ãåæ°åŒã®åãŸãã¯å·®ã®åœ¢ã§è¡šãã
ãšå€åœ¢ã§ããã®ã§ã
ãšãªããçŽåãããš
ãšãªãã
次ã®åé¡ã§ã¯ã
ãšå€åœ¢ããããšã«ãã£ãŠã
ãšãªãã
ãšæ±ãŸãã
éšååæ°å解ã®æäœãéã«èŸ¿ããšãåæ°åŒã®éåã®æäœãšäžèŽããã ã€ãŸããéšååæ°å解ã¯éåã®éã®æäœã§ããã ååãå®æ°ã®å Žåã«ã¯ãäžãšåæ§ã®æ¹æ³ã§éšååæ°å解ããããšãã§ããã
1. 3 ( x â 9 ) ( x â 4 ) {\displaystyle {\frac {3}{(x-9)(x-4)}}}
2. 7 ( 3 x â 1 ) ( 5 â 2 x ) {\displaystyle {\frac {7}{(3x-1)(5-2x)}}}
éšååæ°å解ã¯æ°åã®åã®èšç®ãç©åèšç®ã埮åãå©çšããäžçåŒã®èšŒæçã«åœ¹ç«ã€ãéèŠãªå€åœ¢ã§ããã
çåŒ ( a + b ) 2 = a 2 + 2 a b + b 2 {\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}} ã¯ãæå a , b {\displaystyle a,b} ã«ã©ã®ãããªå€ã代å
¥ããŠãæãç«ã€ããã®ãããªçåŒãæçåŒ(ãããšããã)ãšããã çåŒ 1 x â 1 + 1 x + 1 = 2 x x 2 â 1 {\displaystyle {\frac {1}{x-1}}+{\frac {1}{x+1}}={\frac {2x}{x^{2}-1}}} ã¯ã䞡蟺ãšã x = 1 , â 1 {\displaystyle x=1,-1} ã代å
¥ããããšã¯ã§ããªããããã®ä»ã®å€ã§ããã°ä»£å
¥ããããšãã§ãããŸãã©ã®ãããªå€ã代å
¥ããŠãçåŒãæãç«ã£ãŠããããããæçåŒãšåŒã¶ã
ãã£ãœãã x 2 â x â 2 = 0 {\displaystyle x^{2}-x-2=0} ã¯ãx=2 ãŸã㯠x=ãŒ1 ã代å
¥ãããšãã ãæãç«ã€ãããã®ããã«æåã«ç¹å®ã®å€ã代å
¥ãããšãã«ã ãæãç«ã€åŒã®ããšãæ¹çšåŒãšåŒã³ãæçåŒãšã¯åºå¥ããã
çåŒ a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} ã x {\displaystyle x} ã«ã€ããŠã®æçåŒã§ããã®ã¯ã©ã®ãããªå ŽåããèããŠã¿ããã ããåŒãã x {\displaystyle x} ã«ã€ããŠã®æçåŒã§ããããšã¯ããã®åŒã® x {\displaystyle x} ã«ã©ã®ãããªå€ã代å
¥ããŠãããã®çåŒã¯æãç«ã€ãšããæå³ã§ããããªã®ã§ãäŸãã° x {\displaystyle x} ã« â 1 , 0 , 1 {\displaystyle -1\ ,\ 0\ ,\ 1} ã代å
¥ããåŒ
ã¯ãã¹ãŠæãç«ã€å¿
èŠããããããã解ããš
ãªã®ã§ãçåŒ a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} ã x {\displaystyle x} ã«ã€ããŠã®æçåŒã«ãªããªãã°ã a = b = c = 0 {\displaystyle a=b=c=0} ã§ãªããã°ãªããªãããšããããã
äžè¬ã«ãçåŒ a x 2 + b x + c = a â² x 2 + b â² x + c â² {\displaystyle ax^{2}+bx+c=a'x^{2}+b'x+c'} ãæçåŒã§ããããšãšã ( a â a â² ) x 2 + ( b â b â² ) x + ( c â c â² ) = 0 {\displaystyle (a-a')x^{2}+(b-b')x+(c-c')=0} ãæçåŒã§ããããšãšåãã§ããã ãã£ãŠ
ãŸãšãããšæ¬¡ã®ããã«ãªãã
次ã®çåŒã x {\displaystyle x} ã«ã€ããŠã®æçåŒãšãªãããã«ã a , b , c {\displaystyle a\ ,\ b\ ,\ c} ã®å€ãæ±ããã
çåŒã®å³èŸºã x {\displaystyle x} ã«ã€ããŠæŽçãããš
ãã®çåŒã x {\displaystyle x} ã«ã€ããŠã®æçåŒãšãªãã®ã¯ã䞡蟺ã®åã次æ°ã®é
ã®ä¿æ°ãçãããšãã§ããããã£ãŠ
ããã解ããš
æçåŒãå©çšããããšã§ãè€éãªåæ°åŒã®éšååæ°å解ãã§ããã
ãšããã åæ¯ãæã£ãŠ
ããªãã¡
ããã x {\displaystyle x} ã®æçåŒãªã®ã§ãä¿æ°ãæ¯èŒããŠ
ããªãã¡
æåã®çåŒã«ä»£å
¥ããŠã
次ã®åé¡ã¯ã
ãšããããšã«ãããäžã®åé¡ãšåæ§ã«ããŠ
ãšæ±ãŸãã®ã§ã
a~fãå®æ°ãšããã
a x 2 + b y 2 + c x y + d x + e y + f = 0 {\displaystyle ax^{2}+by^{2}+cxy+dx+ey+f=0} ãx, yã«ã€ããŠã®æçåŒã ãšããã
巊蟺ãxã«ã€ããŠæŽçãããšã a x 2 + ( c y + d ) x + ( b y 2 + e y + f ) = 0 {\displaystyle ax^{2}+(cy+d)x+(by^{2}+ey+f)=0} ã§ããã
ãããxã«ã€ããŠã®æçåŒãªã®ã§ã a = 0 , c y + d = 0 , b y 2 + e y + f = 0 {\displaystyle a=0,cy+d=0,by^{2}+ey+f=0} ãæãç«ã€ã
ãããã¯æŽã«yã«ã€ããŠã®æçåŒãªã®ã§ã以äžã®çåŒãåŸãããã
éã«ããããæãç«ãŠã°å
ã®åŒã¯æããã«x, yã«ã€ããŠã®æçåŒã§ãã
x 2 + a x y + 6 y 2 â x + 5 y + b = ( x â 2 y + c ) ( x â 3 y + d ) {\displaystyle x^{2}+axy+6y^{2}-x+5y+b=(x-2y+c)(x-3y+d)} ãx,yã«ã€ããŠã®æçåŒãšãªãããã«a,b,c,dãå®ããã
ããã»ã©çŽ¹ä»ãããæçåŒããšããèšèã䜿ã£ãŠã蚌æãã®æå³ã説æãããªãããçåŒã蚌æããããšã¯ããã®åŒãæçåŒã§ããããšã瀺ãããšã§ããã
äžè¬ã«ãçåŒ A=B ã蚌æããããã«ã¯ã次ã®ãããªæé ã®ãããããå®è¡ããã°ããã
ãã®ãšããå€åœ¢ã¯åå€å€åœ¢ã§ãªããã°ãªããªãããšã«æ³šæã
( a + b ) 2 â ( a â b ) 2 = 4 a b {\displaystyle (a+b)^{2}-(a-b)^{2}=4ab} ãæãç«ã€ããšã蚌æããã
(蚌æ) 巊蟺ãå±éãããšã
ãšãªããããã¯å³èŸºã«çããããã£ãŠãçåŒ ( a + b ) 2 â ( a â b ) 2 = 4 a b {\displaystyle (a+b)^{2}-(a-b)^{2}=4ab} ã¯èšŒæãããã(çµ)
( x + y ) 2 + ( x â y ) 2 = 2 ( x 2 + y 2 ) {\displaystyle (x+y)^{2}+(x-y)^{2}=2(x^{2}+y^{2})} ãæãç«ã€ããšã蚌æããã
巊蟺ãèšç®ãããšã
ããã¯å³èŸºã«çããããã£ãŠçåŒãæãç«ã€ããšã蚌æãããã(çµ)
次ã®çåŒãæãç«ã€ããšã蚌æããã (I)
(I) (巊蟺) = ( 36 a 2 + 84 a b + 49 b 2 ) + ( 49 a 2 â 84 a b + 36 a 2 ) = 85 a 2 + 85 b 2 {\displaystyle =(36a^{2}+84ab+49b^{2})+(49a^{2}-84ab+36a^{2})=85a^{2}+85b^{2}} (å³èŸº) = ( 81 a 2 + 36 a b + 4 b 2 ) + ( 4 a 2 â 36 a b + 81 b 2 ) = 85 a 2 + 85 b 2 {\displaystyle =(81a^{2}+36ab+4b^{2})+(4a^{2}-36ab+81b^{2})=85a^{2}+85b^{2}} 䞡蟺ãšãåãåŒã«ãªããã
æçåŒã§ãªããšãããäžããããæ¡ä»¶ããçåŒã蚌æããããšãã§ããã
ããã
ãã£ãŠã a 3 + b 3 + c 3 = 3 a b c {\displaystyle a^{3}+b^{3}+c^{3}=3abc} ã§ããã
ãŸãã
ãããäžåŒã®å³èŸºãkãšãããšã
ãªã®ã§ã
ãã£ãŠã a + c b + d = a â c b â d {\displaystyle {\frac {a+c}{b+d}}={\frac {a-c}{b-d}}} ã§ããã
ãªããæ¯ a : b {\displaystyle a:b} ã«ã€ã㊠a b {\displaystyle {\frac {a}{b}}} ãæ¯ã®å€ãšããããŸãã a : b = c : d ⺠a b = c d {\displaystyle a:b=c:d\iff {\frac {a}{b}}={\frac {c}{d}}} ãæ¯äŸåŒãšããã
a x = b y = c z {\displaystyle {\frac {a}{x}}={\frac {b}{y}}={\frac {c}{z}}} ãæãç«ã€ãšãã a : b : c = x : y : z {\displaystyle a:b:c=x:y:z} ãšè¡šãããããé£æ¯ãšããã
äžçåŒã®ããŸããŸãªå
¬åŒã«ã€ããŠã¯ã次ã®4ã€ã®åŒãåºæ¬çãªåŒãšããŠå°åºã§ããå Žåãããããã
é«æ ¡æ°åŠã§ã¯ã次ã®4ã€ã®æ§è³ªã äžçåŒã®ãåºæ¬æ§è³ªããªã©ãšããŠçŽ¹ä»ãããŠããã
(3)ãš(4)ã«ã€ããŠã¯ãã²ãšã€ã®æ§è³ªãšã㊠ãŸãšããŠããæ€å®æç§æžããã(â» åæ通ãªã©)ã
æ°åŠIAã§ç¿ã£ãããªãã°ãã®æå³ã®èšå· â¹ {\displaystyle \Longrightarrow } ã䜿ããšã
ãšãæžããã
äžè¿°ã®4ã€ã®åºæ¬æ§è³ªããã
ã蚌æããŠã¿ããã
(蚌æ) ãŸã a>0 ãªã®ã§ãåºæ¬æ§è³ª(2)ãã
ã§ããã
ãã£ãŠã
ãªã®ã§ãåºæ¬æ§è³ª(1)ãã a + b > 0 {\displaystyle a+b>0} ãæãç«ã€ã(çµ)
åæ§ã«ããŠã
ã蚌æã§ããã
ãããŸã§ã«ç€ºããããšãããäžçåŒ A ⧠B {\displaystyle A\geqq B} ã蚌æãããå Žåã«ã¯ã
ã蚌æããã°ããããšãããã£ãããã¡ãã®æ¹ã蚌æããããå Žåãããããã
äžçåŒã蚌æããéã«æ ¹æ ãšããåºæ¬çãªäžçåŒãšããŠã次ã®æ§è³ªãããã
ãã®å®ç(ãå®æ°ã2ä¹ãããšãããªãããŒã以äžã§ããã)ããåºæ¬æ§è³ª(3),(4)ã䜿ã£ãŠèšŒæããŠã¿ããã
(蚌æ)
aãæ£ã®å Žåãšè² ã®å Žåãš0ã®å Žåã®3éãã«å Žåããããã
[aãæ£ã®å Žå] ãã®ãšããåºæ¬æ§è³ª(3)ããã
ã§ãããããªãã¡ã
ã§ããã
[aãè² ã®å Žå] ãã®ãšããåºæ¬æ§è³ª(4)ãã 0 a < a a {\displaystyle 0a<aa} ã§ãããããªãã¡ã
ã§ããã
[aããŒãã®å Žå] ãã®ãšãã a 2 = 0 {\displaystyle a^{2}=0} ã§ããã
ãã£ãŠããã¹ãŠã®å Žåã«ã€ã㊠a 2 ⧠0 {\displaystyle a^{2}\geqq 0} (çµ)
ãã®ããšãšåºæ¬æ§è³ª(1)(2)ããã次ãæãç«ã€ããšããããã
次ã®äžçåŒãæãç«ã€ããšã蚌æããã
(蚌æ)
ã蚌æããã°ããã
巊蟺ãå±éã㊠ãŸãšãããšã
ãšãªãã
äžåŒã®æåŸã®åŒã®é
ã«ã€ããŠã
ã ããã
ã§ããããã£ãŠ
ã§ããã(çµ)
2ã€ã®æ£ã®æ° a, b ã a>b ãŸã㯠aâ§b ãªãã°ã䞡蟺ã2ä¹ããŠã倧å°é¢ä¿ã¯åããŸãŸã§ããã
ã€ãŸãã
ã§ããã
a>bãšãããä»®å®ãããa,b ã¯æ£ã®æ°ãªã®ã§ã ( a + b ) > 0 {\displaystyle (a+b)>0} ã§ãããå¥ã®ä»®å®ããã a > b ãªã®ã§ã ( a â b ) > 0 {\displaystyle (a-b)>0} ã§ãããããã£ãŠã a 2 â b 2 = ( a + b ) ( a â b ) > 0 {\displaystyle a^{2}-b^{2}=(a+b)(a-b)>0}
éã«ã a 2 â b 2 > 0 {\displaystyle a^{2}-b^{2}>0} ã®ãšãã ( a + b ) ( a â b ) > 0 {\displaystyle (a+b)(a-b)>0} ã§ããã a > 0 , b > 0 {\displaystyle a>0,b>0} ãªã®ã§ a + b > 0 {\displaystyle a+b>0} ã§ããããã£ãŠã a â b > 0 {\displaystyle a-b>0} ãªã®ã§ã a > b {\displaystyle a>b} ã§ããã
ãã£ãŠã a > b ⺠a 2 > b 2 {\displaystyle a>b\quad \Longleftrightarrow \quad a^{2}>b^{2}} ã§ããã
aâ§bã®å Žåãåæ§ã«èšŒæã§ããã
ç·Žç¿ãšããŠã次ã®åé¡ãåããŠã¿ããã
a > 0 {\displaystyle a>0} , b > 0 {\displaystyle b>0} ã®ãšãã次ã®äžçåŒã蚌æããã
(蚌æ) äžçåŒã®äž¡èŸºã¯æ£ã§ããã®ã§ã䞡蟺ã®å¹³æ¹ã®å·®ãèããã°ããã䞡蟺ã®å¹³æ¹ã®å·®ã¯
ã§ãããããã§ãa,b ã¯ãšãã«æ£ã®å®æ°ãªã®ã§ã
ã§ããããšãçšããã
ã§ããã®ã§ã
ãšãªãããã£ãŠã
ã§ããã(çµ)
å®æ° a ã®çµ¶å¯Ÿå€ |a| ã«ã€ããŠã
ã§ããããã次ã®ããšãæãç«ã€ã
|a|â§a , |a|⧠ãŒa , |a|=a
ãŸãã2ã€ã®å®æ° a, b ã®çµ¶å¯Ÿå€ |ab| ã«ã€ããŠã¯ã
ãæãç«ã€ã®ã§ãããã«ããã« |ab|â§0 , |a||b|â§0 ãçµã¿åãããŠã
|ab| = |a| |b| ãæãç«ã€ã
(äŸé¡)
次ã®äžçåŒã蚌æããããŸããçå·ãæãç«ã€ã®ã¯ ã©ã®ãããªå Žåãã 調ã¹ãã
䞡蟺ã®å¹³æ¹ã®å·®ãèãããšã
ãããããæ£ãªããäžããããäžçåŒ |a|+|b| ⧠|a+b| ãæ£ããã
ããã§ã |a| |b| ⧠ab ã§ããã®ã§ã
ã§ããã
ãããã£ãŠã |a|+|b| ⧠|a+b| ã§ããã
çå·ãæãç«ã€ã®ã¯ |a| |b| = ab ã®å Žåãããªãã¡ ab ⧠0 ã®å Žåã§ããã(蚌æ ããã)
2ã€ã®æ° a {\displaystyle a} , b {\displaystyle b} ã«å¯Ÿãã a + b 2 {\displaystyle {\frac {a+b}{2}}} ãçžå å¹³å(ããããžããã)ãšèšãã a b {\displaystyle {\sqrt {ab}}} ãçžä¹å¹³å(ããããããžããã)ãšããã
æ¬ããŒãžã§ã¯ã2åã®æ°ã®å¹³åã«ã€ããŠèå¯ããã
çžå å¹³åãšçžä¹å¹³åã«ã€ããŠã次ã®é¢ä¿åŒãæãç«ã€ã
(蚌æ)
a ⧠0 , b ⧠0 {\displaystyle a\geqq 0,b\geqq 0} ã®ãšã
( a â b ) 2 ⧠0 {\displaystyle \left({\sqrt {a}}-{\sqrt {b}}\right)^{2}\geqq 0} ã§ããããã ( a â b ) 2 2 ⧠0 {\displaystyle {\frac {\left({\sqrt {a}}-{\sqrt {b}}\right)^{2}}{2}}\geqq 0} ãããã£ãŠ a + b 2 ⧠a b {\displaystyle {\frac {a+b}{2}}\geqq {\sqrt {ab}}} çå·ãæãç«ã€ã®ã¯ã ( a â b ) 2 = 0 {\displaystyle \left({\sqrt {a}}-{\sqrt {b}}\right)^{2}=0} ã®ãšããããªãã¡ a = b {\displaystyle a=b} ã®ãšãã§ããã(蚌æ ããã)
å
¬åŒã®å©çšã§ã¯ãäžã®åŒ a + b 2 ⧠a b {\displaystyle {\frac {a+b}{2}}\geqq {\sqrt {ab}}} ã®äž¡èŸºã«2ãããã a + b ⧠2 a b {\displaystyle a+b\geqq 2{\sqrt {ab}}} ã®åœ¢ã®åŒã䜿ãå Žåãããã
a > 0 {\displaystyle a>0} , b > 0 {\displaystyle b>0} ã®ãšãã次ã®äžçåŒãæãç«ã€ããšã蚌æããã (I)
(II)
(I) a > 0 {\displaystyle a>0} ã§ããããã 1 a > 0 {\displaystyle {\frac {1}{a}}>0} ãã£ãŠ a + 1 a ⧠2 a à 1 a = 2 {\displaystyle a+{\frac {1}{a}}\geqq 2{\sqrt {a\times {\frac {1}{a}}}}=2} ãããã£ãŠ
(II)
a > 0 {\displaystyle a>0} , b > 0 {\displaystyle b>0} ã§ããããã b a > 0 {\displaystyle {\frac {b}{a}}>0} , a b > 0 {\displaystyle {\frac {a}{b}}>0} ãã£ãŠ b a + a b + 2 ⧠2 b a à a b + 2 = 2 + 2 = 4 {\displaystyle {\frac {b}{a}}+{\frac {a}{b}}+2\geqq 2{\sqrt {{\frac {b}{a}}\times {\frac {a}{b}}}}+2=2+2=4} ãããã£ãŠ
2ä¹ããŠè² ã«ãªãæ°ããšãããã®ãèããããã®ãããªæ°ã¯ãäžåŠã§ç¿ã£ãå®æ°ã®äžã«ã¯ãªãããšããããããªããªãã°ãæ£ã®æ°ã§ãè² ã®æ°ã§ã2ä¹ãããšç¬Šå·ãæã¡æ¶ããŠæ£ã®æ°ã«ãªã£ãŠããŸãããã§ãããããã§é«æ ¡ã§ã¯ã2ä¹ããŠè² ã«ãªããšããæ§è³ªãæã€æ°ã®æŠå¿µãæ°ããå°å
¥ããããšã«ããã
ãšããæ¹çšåŒãèããããã®æ¹çšåŒã®è§£ã¯å®æ°ã«ã¯ãªããããã§ããã®æ¹çšåŒã®è§£ãšãªãæ°ãæ°ããäœãããã®åäœãæå i {\displaystyle i} ã§ããããã
ãã® i {\displaystyle i} ã®ããšãèæ°åäœ(ããããããã)ãšåŒã¶ã(èæ°åäœã®èšå· i ãè±èªã®ã¢ã«ãã¡ãããã®ã¢ã€ã®å°æåã§ã imaginary unit ã«ç±æ¥ãããšèããããŠããã)
1 + i {\displaystyle 1+i} ã 2 + 5 i {\displaystyle 2+5i} ã®ããã«ãèæ°åäœ i {\displaystyle i} ãšå®æ° a , b {\displaystyle a,b} ãçšããŠ
ãšè¡šãããšãã§ããæ°ãè€çŽ æ°(ãµãããã)ãšããããã®ãšããaããã®è€çŽ æ°ã®å®éš(ãã€ã¶)ãšãããbãèéš(ããã¶)ãšããã
äŸãã°ã 1 + i , 2 + 5 i , 9 2 + 7 2 i , 4 i , 3 {\displaystyle 1+i,\quad 2+5i,\quad {\frac {9}{2}}+{\frac {7}{2}}i,\quad 4i,\quad 3} ã¯ãããããè€çŽ æ°ã§ããã
è€çŽ æ° a+bi ã¯(ãã ã aãšbã¯å®æ°)ãbã0ã®å Žåã«ããããå®æ°ãšèŠãããšãã§ããã
èšãæ¹ãããããšãè€çŽ æ°ãåºæºã«èãããšãå®æ°ãšã¯ã a+0i ã®ãããªãèéšã®ä¿æ°ããŒãã«ãªãè€çŽ æ°ã®ããšã§ãããšãèšããã
4iã®ãããªãèéšã0以å€ã§å®éšããŒãã®è€çŽ æ°ãçŽèæ°(ãã
ããããã)ãšåŒã¶ãçŽèæ°ã¯ã2ä¹ãããšè² ã«ãªãæ°ã§ããã å®æ°ãèéšã0ã®è€çŽ æ°ãšèããããã
å®æ°ã§ãªãè€çŽ æ°ã®ããšããèæ°ã(ãããã)ãšããã
2ã€ã®è€çŽ æ° a+bi ãš c+di ãšãçãããšã¯ã
ã§ããããšã§ããã
ã€ãŸãã
ãšãã«ãè€çŽ æ°a+bi ã 0ã§ãããšã¯ãa=0 ã〠b=0 ã§ããããšã§ããã
è€çŽ æ° z = a + b i {\displaystyle z=a+bi} ã«å¯ŸããŠãèéšã®ç¬Šå·ãå転ãããè€çŽ æ° a â b i {\displaystyle a-bi} ã®ããšããå
±åœ¹(ããããã)ãªè€çŽ æ°ããŸãã¯ãè€çŽ æ° z {\displaystyle z} ã®å
±åœ¹ãã®ããã«åŒã³ã z Ì {\displaystyle {\bar {z}}} ã§ããããããªãããå
±åœ¹ãã¯ãå
±è»ãã®åžžçšæŒ¢åã«ããæžãæãã§ããã
å®æ°aãšå
±åœ¹ãªè€çŽ æ°ã¯ããã®å®æ° a èªèº«ã§ããã
è€çŽ æ° z=a+bi ã«ã€ããŠ
è€çŽ æ°ã«ãååæŒç®(å æžä¹é€)ãå®çŸ©ãããã
è€çŽ æ°ã®æŒç®ã§ã¯ãèæ°åäœ i {\displaystyle i} ããéåžžã®æåã®ããã«æ±ã£ãŠèšç®ãããäžè¬ã«è€çŽ æ° z , w {\displaystyle z\ ,\ w} ãã z = a + b i , w = c + d i {\displaystyle z=a+bi\ ,\ w=c+di} ã§äžãããããšã(ãã ã a , b , c , d {\displaystyle a\ ,\ b\ ,\ c\ ,\ d} ã¯å®æ°ãšãã)ã
ãšãããµãã«è€çŽ æ°ã®å æžä¹é€ã®èšç®æ³ãå®ããããŠããã
ä¹æ³ã®å®çŸ©ã¯ãäžèŠãããšé£ãããã«ã¿ããããå®æ°ã®åé
æ³åãšåæ§ã«å±éããŠããæåŸã« iã«ãã€ãã¹1ã代å
¥ããŠãã£ãã ãã§ããã
é€æ³ã®å®çŸ©ã¯ãååãšåæ¯ã«ãåæ¯ãšå
±åœ¹ãªåœ¢ã®åŒã æãç® ããã ãã§ããã
ä¹æ³ãé€æ³ã®å®çŸ©åŒãæèšããå¿
èŠã¯ç¡ããèšç®ã®éã«ã¯ãå¿
èŠã«å¿ããŠåé
æ³åãå
±åœ¹ãªã©ã®ãå¿
èŠãªåŒå€åœ¢ãè¡ãã°ããã
äŸé¡
2ã€ã®è€çŽ æ°
ã«ã€ããŠã a + b {\displaystyle a+b} ãš a b {\displaystyle ab} ãš a b {\displaystyle {\frac {a}{b}}} ããããããèšç®ããã
解ç
ã§ããã
ããããã«ç°¡åã«ã§ããªãã ããããå®ã¯ãã¡ãã£ãšãããã¯ããã¯ãçšããã°ããèŠããã圢ã«ã§ããã
åæ°ã¯åæ¯ãšååã«åãæ°ããããŠããã£ãã®ã§ãåæ¯ãšååã«åæ¯ã®å
±åœ¹ããããŠã¿ãããããšã
ãåŸãããããã®åœ¢ã®ã»ããå
ã®åŒããããã£ãšèŠããã圢ã§ããã
ãã®ãããªæäœãåæ¯ã®å®æ°åãšããããšããããæ°åŠIã§åŠç¿ããå±éã»å æ°å解å
¬åŒ ( a + b ) ( a â b ) = a 2 â b 2 {\displaystyle (a+b)(a-b)=a^{2}-b^{2}} ã®ç°¡åãªå¿çšã§ããã
æ°ã®ç¯å²ãè€çŽ æ°ã«ãŸã§æ¡åŒµãããšãè² ã®æ°ã®å¹³æ¹æ ¹ãèããããšãã§ããã
äŸãšããŠã -5 ã®å¹³æ¹æ ¹ã«ã€ããŠèããŠã¿ããã
ã§ããããã -5 ã®å¹³æ¹æ ¹ã¯ 5 i {\displaystyle {\sqrt {5}}\ i} ãš â 5 i {\displaystyle -{\sqrt {5}}\ i} ã§ããã
â 5 {\displaystyle {\sqrt {-5}}} ãšã¯ã 5 i {\displaystyle {\sqrt {5}}\ i} ã®ããšãšããã â â 5 {\displaystyle -{\sqrt {-5}}} ãšã¯ã â 5 i {\displaystyle -{\sqrt {5}}\ i} ã®ããšã§ããã ãšãã« â 1 = i {\displaystyle {\sqrt {-1}}\ =\ i} ã§ããã
ããŠã-5 ã®å¹³æ¹æ ¹ã¯ãæ¹çšåŒ x 2 = â 5 {\displaystyle x^{2}=-5} ã®è§£ã§ãããã
ãã®æ¹çšåŒã移é
ããããšã«ããã-5 ã®å¹³æ¹æ ¹ã¯ã
ã®è§£ã§ãããšããããã
ããã«å æ°å解ãããããšã«ããã-5 ã®å¹³æ¹æ ¹ã¯æ¹çšåŒ
ã®è§£ã§ããããšããããã
(I) â 2 â 6 {\displaystyle {\sqrt {-2}}\ {\sqrt {-6}}} ãèšç®ããã
(I)
ãã®ããã«ããŸãããã€ãã¹ã®æ°ã®å¹³æ¹æ ¹ãåºãŠãããããŸãèæ°åäœ i ãçšããåŒã«æžãæããã
ãã®ããšãããç®ãããŠããã
(II) 2 â 3 {\displaystyle {\frac {\sqrt {2}}{\sqrt {-3}}}} ãèšç®ããã
(III) 2次æ¹çšåŒ x 2 = â 7 {\displaystyle x^{2}=-7} ã解ãã
(II)
(III)
è€çŽ æ°ã®å¿çšãšããŠãããã§ã¯2次æ¹çšåŒã®æ§è³ªã«ã€ããŠè¿°ã¹ããä»»æã®2次æ¹çšåŒã¯ã解ã®å
¬åŒã«ãã£ãŠè§£ãããããšãé«çåŠæ ¡æ°åŠIã§è¿°ã¹ãããããã解ã®å
¬åŒã«å«ãŸããæ ¹å·ã®äžèº«ãè² ã®æ°ã®å Žåã«ã¯å®æ°è§£ãååšããªãããšã«æ³šæããå¿
èŠãããã2次æ¹çšåŒ
ã®è§£ã®å
¬åŒã¯ã
ã§ããã
å€å¥åŒ D {\displaystyle D} ã¯
ã«ãã£ãŠå®çŸ©ããããå€å¥åŒã¯ã解ã®å
¬åŒã®æ ¹å·(ã«ãŒãèšå·ã®ããš)ã®äžèº«ã«çãããå€å¥åŒã®æ£è² ã«ãã£ãŠ2次æ¹çšåŒãå®æ°è§£ãæã€ãã©ããã決ãŸãã
D {\displaystyle D} ãè² ã®ãšãã«ã¯ãã®2次æ¹çšåŒã¯å®æ°ã®ç¯å²ã«ã¯è§£ãæããªãã
å€å¥åŒ D {\displaystyle D} ãè² ã®æ°ã§ãã£ããšããxã®è§£ã¯ç°ãªã2ã€ã®èæ°ã«ãªãããã®2ã€ã®è§£ã¯ å
±åœ¹ ã®é¢ä¿ã«ãªã£ãŠããã
è€çŽ æ°ãçšããŠã2次æ¹çšåŒ (1)
(2)
(3)
ã解ãã
解ã®å
¬åŒãçšããŠè§£ãã°ããã(1)ã ããèšç®ãããšã
ãšãªãã ä»ãåãããã«æ±ãããšãåºæ¥ãã
以éã®è§£çã¯ã (2)
(3)
ãšãªãã
æ¹çšåŒã®è§£ã§ãå®æ°ã§ãããã®ã å®æ°è§£ ãšããã
æ¹çšåŒã®è§£ã§ãèæ°ã§ãããã®ã èæ°è§£ ãšããã
2次æ¹çšåŒ a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} ã®è§£ã¯ x = â b ± b 2 â 4 a c 2 a {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}} ã§ããã
2次æ¹çšåŒã®è§£ã®çš®é¡ã¯ã解ã®å
¬åŒã®äžã®æ ¹å·ã®äžã®åŒ b 2 â 4 a c {\displaystyle b^{2}-4ac} ã®ç¬Šå·ãèŠãã°å€å¥ããããšãã§ããã
ãã®åŒ b 2 â 4 a c {\displaystyle b^{2}-4ac} ãã2次æ¹çšåŒ a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} ã®å€å¥åŒ(ã¯ãã¹ã€ãã)ãšãããèšå· D {\displaystyle D} ã§è¡šãã
ãŸããé解ãå®æ°è§£ã§ããã®ã§ã
ãšãããã
次ã®2次æ¹çšåŒã®è§£ãå€å¥ããã
(I)
(II)
(III)
(I)
ã ãããç°ãªã2ã€ã®å®æ°è§£ããã€ã
(II)
ã ãããç°ãªã2ã€ã®èæ°è§£ããã€ã
(III)
ã ãããé解ããã€ã
ãŸãã2次æ¹çšåŒ a x 2 + 2 b â² x + c = 0 {\displaystyle ax^{2}+2b'x+c=0} ã®ãšãã D = 4 ( b â² 2 â a c ) {\displaystyle D=4(b'^{2}-ac)} ãšãªãã®ã§ã 2次æ¹çšåŒ a x 2 + 2 b â² x + c = 0 {\displaystyle ax^{2}+2b'x+c=0} ã®å€å¥åŒã«ã¯
ããã¡ããŠãããã
ãããçšããŠãåã®åé¡
ã®è§£ãå€å¥ãããã
a = 4 , b â² = â 10 , c = 25 {\displaystyle a=4\,,\,b'=-10\,,\,c=25} ã§ãããã
ã ãããé解ããã€ã
2次æ¹çšåŒ a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} ã®2ã€ã®è§£ã α {\displaystyle \alpha } , β {\displaystyle \beta } ãšããã ãã®æ¹çšåŒã¯ã
a ( x â α ) ( x â β ) = 0 {\displaystyle a(x-\alpha )(x-\beta )=0}
ãšå€åœ¢ã§ããã
ãããå±éãããšã
a x 2 â a ( α + β ) x + a α β = 0 {\displaystyle ax^{2}-a(\alpha +\beta )x+a\alpha \beta =0}
ä¿æ°ãæ¯èŒããŠã
c = a α β , b = â a ( α + β ) {\displaystyle c=a\alpha \beta ,b=-a(\alpha +\beta )}
ãåŸãã
ãããå€åœ¢ããã°ã α + β = â b a , α β = c a {\displaystyle \alpha +\beta =-{\frac {b}{a}},\alpha \beta ={\frac {c}{a}}} ãšãªãã
2次æ¹çšåŒ 2 x 2 + 4 x + 3 = 0 {\displaystyle 2x^{2}+4x+3=0} ã®2ã€ã®è§£ã α {\displaystyle \alpha } , β {\displaystyle \beta } ãšãããšãã α 2 + β 2 {\displaystyle \alpha ^{2}+\beta ^{2}} ã®å€ãæ±ããã
解ãšä¿æ°ã®é¢ä¿ããã α + β = â 4 2 = â 2 {\displaystyle \alpha +\beta =-{\frac {4}{2}}=-2} , α β = 3 2 {\displaystyle \alpha \beta ={\frac {3}{2}}} α 2 + β 2 = ( α + β ) 2 â 2 α β = ( â 2 ) 2 â 2 à 3 2 = 1 {\displaystyle \alpha ^{2}+\beta ^{2}=(\alpha +\beta )^{2}-2\alpha \beta =(-2)^{2}-2\times {\frac {3}{2}}=1}
2ã€ã®æ° α {\displaystyle \alpha } , β {\displaystyle \beta } ã解ãšãã2次æ¹çšåŒã¯
ãšè¡šãããã巊蟺ãå±éããŠæŽçãããšæ¬¡ã®ããã«ãªãã
次ã®2æ°ã解ãšãã2次æ¹çšåŒãäœãã
(I)
(II)
(I) å ( 3 + 5 ) + ( 3 â 5 ) = 6 {\displaystyle (3+{\sqrt {5}})+(3-{\sqrt {5}})=6} ç© ( 3 + 5 ) ( 3 â 5 ) = 4 {\displaystyle (3+{\sqrt {5}})(3-{\sqrt {5}})=4} ã§ãããã
(II) å ( 2 + 3 i ) + ( 2 â 3 i ) = 4 {\displaystyle (2+3i)+(2-3i)=4} ç© ( 2 + 3 i ) ( 2 â 3 i ) = 13 {\displaystyle (2+3i)(2-3i)=13} ã§ãããã
2次æ¹çšåŒ a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} ã®2ã€ã®è§£ α {\displaystyle \alpha } , β {\displaystyle \beta } ãããããšã2次åŒ
ãå æ°å解ããããšãã§ããã 解ãšä¿æ°ã®é¢ä¿ α + β = â b a {\displaystyle \alpha +\beta =-{\frac {b}{a}}} , α β = c a {\displaystyle \alpha \beta ={\frac {c}{a}}} ããã
2次æ¹çšåŒã¯ãè€çŽ æ°ã®ç¯å²ã§èãããšã€ãã«è§£ããã€ãããè€çŽ æ°ãŸã§äœ¿ã£ãŠãããšãããšã2次åŒã¯å¿
ã1次åŒã®ç©ã«å æ°å解ããããšãã§ããã
è€çŽ æ°ã®ç¯å²ã§èããŠã次ã®2次åŒãå æ°å解ããã
(I)
(II)
(I) 2次æ¹çšåŒ x 2 + 4 x â 1 = 0 {\displaystyle x^{2}+4x-1=0} ã®è§£ã¯
ãã£ãŠ
(II) 2次æ¹çšåŒ 2 x 2 â 3 x + 2 = 0 {\displaystyle 2x^{2}-3x+2=0} ã®è§£ã¯
ãã£ãŠ
3次以äžã®æŽåŒã«ããæ¹çšåŒãèããã äžè¬ã«æ¹çšåŒã P ( x ) = 0 {\displaystyle P(x)=0} ãšãšãã ãã ãã P ( x ) {\displaystyle P(x)} ã¯ãä»»æã®æ¬¡æ°ã®æŽåŒãšããã
P ( x ) {\displaystyle P(x)} ã1æ¬¡åŒ x â a {\displaystyle x-a} ã§å²ã£ããšãã®åã Q ( x ) {\displaystyle Q(x)} ãäœãã R {\displaystyle R} ãšãããšã
ãã®äž¡èŸºã® x {\displaystyle x} ã« a {\displaystyle a} ã代å
¥ãããšã
ã€ãŸãã P ( x ) {\displaystyle P(x)} ã x â a {\displaystyle x-a} ã§å²ã£ããšãã®äœã㯠P ( a ) {\displaystyle P(a)} ã§ããã
æŽåŒ P ( x ) = x 3 â 2 x + 3 {\displaystyle P(x)=x^{3}-2x+3} ã次ã®åŒã§å²ã£ãäœããæ±ããã (I)
(II)
(III)
(I) P ( 2 ) = 2 3 â 2 Ã 2 + 3 = 7 {\displaystyle P(2)=2^{3}-2\times 2+3=7} (II) P ( â 1 ) = ( â 1 ) 3 â 2 Ã ( â 1 ) + 3 = 4 {\displaystyle P(-1)=(-1)^{3}-2\times (-1)+3=4} (III) P ( 1 2 ) = ( 1 2 ) 3 â 2 Ã ( 1 2 ) + 3 = 17 8 {\displaystyle P\left({\frac {1}{2}}\right)=\left({\frac {1}{2}}\right)^{3}-2\times \left({\frac {1}{2}}\right)+3={\frac {17}{8}}}
ããå®æ° a {\displaystyle a} ã«å¯ŸããŠã
ãæãç«ã£ããšããã ãã®ãšããæŽåŒ P ( x ) {\displaystyle P(x)} ã¯ã ( x â a ) {\displaystyle (x-a)} ãå æ°ã«æã€ããšãåãã ãã®ããšãå æ°å®ç(ãããããŠãã)ãšåŒã¶ã
æŽåŒ P ( x ) {\displaystyle P(x)} ã«å¯ŸããŠãå Q ( x ) {\displaystyle Q(x)} ãå²ãåŒ ( x â a ) {\displaystyle (x-a)} ãšãã æŽåŒã®é€æ³ãçšããããã®ãšããå Q ( x ) {\displaystyle Q(x)} ã ( Q ( x ) {\displaystyle Q(x)} ã¯ã P ( x ) {\displaystyle P(x)} ããã1ã ã次æ°ãäœãæŽåŒã§ããã) äœã c {\displaystyle c} ( c {\displaystyle c} ã¯ãå®æ°ã)ãšãããšã æŽåŒ P ( x ) {\displaystyle P(x)} ã¯ã
ãšæžããã ããã§ã c = 0 {\displaystyle c=0} ã§ãªããšã P ( a ) = 0 {\displaystyle P(a)=0} ã¯æºããããªããã ãã®ãšãã P ( x ) {\displaystyle P(x)} ã¯ã ( x â a ) {\displaystyle (x-a)} ã«ãã£ãŠå²ãåããã ãã£ãŠãå æ°å®çã¯æç«ããã
å æ°å®çãçšããããšã§ããã次æ°ã®é«ãæŽåŒãå æ°å解ããããšã åºæ¥ãããã«ãªããäŸãã°ã3次ã®æŽåŒ
ã«ã€ããŠã x = 1 {\displaystyle x=1} ã代å
¥ãããšã
ã¯0ãšãªãããã£ãŠãå æ°å®çãããã®åŒã¯
ãå æ°ãšããŠæã€ã
ããã§ãå®éæŽåŒã®é€æ³ã䜿ã£ãŠèšç®ãããšã
ãåŸãããã
å æ°å®çãçšã㊠(I)
(II)
ãå æ°å解ããã
(I) å æ°å解ã®çµæã(x+æŽæ°)ã®ç©ã®åœ¢ãªããæŽæ°ã¯6ã®å æ°ã§ãªããã°ãªããªãããã®ããã ± 1 , ± 2 , ± 3 , ± 6 {\displaystyle \pm 1,\pm 2,\pm 3,\pm 6} ãåè£ãšãªãããããã«ã€ããŠã¯å®éã«ä»£å
¥ããŠç¢ºããããããªããx=1ã代å
¥ãããšã
ãšãªãã®ã§ã(x-1)ãå æ°ãšãªããå®éã«æŽåŒã®é€æ³ãè¡ãªããšãåãšã㊠x 2 â 5 x + 6 {\displaystyle x^{2}-5x+6} ãåŸããããããã㯠( x â 2 ) ( x â 3 ) {\displaystyle (x-2)(x-3)} ã«å æ°å解ã§ããããã£ãŠçãã¯ã
ãšãªãã (II) ããã§ãå°éã«24ã®å æ°ãåœãŠã¯ããŠãããããªãã24ã®å æ°ã¯æ°ãå€ãã®ã§ããªãã®èšç®ãå¿
èŠãšãªããããã§ã¯ã-2ã代å
¥ãããšã
ãšãªãã(x+2)ãå æ°ã ãšããããé€æ³ãè¡ãªããšã x 2 â x â 12 {\displaystyle x^{2}-x-12} ãåŸããããã(x-4)(x+3)ã«å æ°å解ã§ãããçãã¯ã
ãšãªãã
å æ°å解ãå æ°å®çãå©çšããŠé«æ¬¡æ¹çšåŒã解ããŠã¿ããã
é«æ¬¡æ¹çšåŒ (I)
(II)
(III)
ã解ãã
(I) 巊蟺ã a 3 â b 3 = ( a â b ) ( a 2 + a b + b 2 ) {\displaystyle a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})} ãçšããŠå æ°å解ãããš
ãããã£ãŠ x â 2 = 0 {\displaystyle \ x-2=0} ãŸã㯠x 2 + 2 x + 4 = 0 {\displaystyle \ x^{2}+2x+4=0} ãã£ãŠ
(II) x 2 = X {\displaystyle \ x^{2}=X\ } ãšãããšã
巊蟺ãå æ°å解ãããš
ãã£ãŠ X = 4 , X = â 2 {\displaystyle X=4\ ,\ X=-2} ããã« x 2 = 4 , x 2 = â 2 {\displaystyle x^{2}=4\ ,\ x^{2}=-2} ãããã£ãŠ
(III) P ( x ) = x 3 â 5 x 2 + 7 x â 2 {\displaystyle \ P(x)=x^{3}-5x^{2}+7x-2\ } ãšããã
ãããã£ãŠã x â 2 {\displaystyle \ x-2\ } 㯠P ( x ) {\displaystyle \ P(x)\ } ã®å æ°ã§ããã
ãã£ãŠ ( x â 2 ) ( x 2 â 3 x + 1 ) = 0 {\displaystyle (x-2)(x^{2}-3x+1)=0} x â 2 = 0 {\displaystyle \ x-2=0} ãŸã㯠x 2 â 3 x + 1 = 0 {\displaystyle \ x^{2}-3x+1=0} ãããã£ãŠ
3次æ¹çšåŒ a x 3 + b x 2 + c x + d = 0 {\displaystyle ax^{3}+bx^{2}+cx+d=0} ã®3ã€ã®è§£ã ã α , β , γ {\displaystyle \alpha \ ,\ \beta \ ,\ \gamma } ãšãããš
ãæãç«ã€ã å³èŸºãå±éãããš
ãã£ãŠ
ããã«
ãããã£ãŠã次ã®ããšãæãç«ã€ã
ãã°ãã°èæ°ã¯ãçŸå®ã«ã¯ååšããªãæ°ãã§ãããšèšãããããšããããæŽå²çã«ãèæ°ãæ±ã£ãæ°åŠãèããã¹ãã§ã¯ãªããšèããããæ代ã¯é·ãã£ãããã®æ代ã®å
é²çãªæ°åŠè
ã®äžã«ã¯ãèæ°ãæå¹ã«æŽ»çšããŠç 究ãé²ããäžæ¹ã§ãææãçºè¡šããéã«ã¯èæ°ãè¡šã«åºããã«èšè¿°ããåªåãããããšã§ãç¡çšãªæµæãåããªãããã«å·¥å€«ããè
ããããšèšãããã»ã©ã§ããã
ã ããããèããŠã¿ãã°ãæ°ããçŸå®ã«ååšããããšã¯ã©ãããæå³ãªã®ã ããããçŸå®ã«éçã䜿ã£ãŠçŽã«åãæããªãã°ãååšã®é·ãããæ£ç¢ºã«ååšçãã®ãã®ã«ãããããšã¯äžå¯èœã§ããããã«æããããããã®å²ã«ååšçãšããå®æ°ã¯ãååšããããšæããããã®ã¯ãªãã ããããæ°çŽç·ãå®æ°ã®ãå®åšããä¿¡ãããããªãã°ãè€çŽ æ°ã¯è€çŽ æ°å¹³é¢(æ°åŠCã§ç¿ã)ã®äžã«ååšããã®ã ãããåãã§ã¯ãªãã ãããã
ãã®ããã«èãããšãããããæ°ãšã¯ãã¹ãŠããæå³ã§æ³åäžã®ååšã§ãããããã«å¯ŸããŠãååšããããååšããªãããšããåããç«ãŠãããšããã³ã»ã³ã¹ã§ããããã«æãããããååšããªããããã«æãããã¡ãªèæ°ã§ããããããšãã°ç©çåŠã®äžåéã§ããéåååŠã®ã·ã¥ã¬ãã£ã³ã¬ãŒæ¹çšåŒã«è¡šãããªã©ãå¿çšäžã®ããŸããŸãªå Žé¢ã«ãããŠããèæ°ã䜿ã£ãŠèšè¿°ããããšãèªç¶ãªå¯Ÿè±¡ã¯å€ãã®ã ã
è€çŽ æ°ã©ããã«ã€ããŠããã®å€§å°é¢ä¿ã¯å®çŸ©ããªãããã®çç±ã¯ãã©ã®ããã«å€§å°é¢ä¿ãå®çŸ©ããŠãã䟿å©ãªæ§è³ªãæºããããšãã§ããªãããã§ãããå
·äœçã«èšãã°ãæ¢ã«è¿°ã¹ãå®æ°ã®å€§å°é¢ä¿ã«ã€ããŠã®ãäžçåŒã®åºæ¬æ§è³ª(1)(2)(3)(4)ãã«ãããåŒãæãç«ãããããšãã§ããªãã®ã ã
ããšãã°ã a + b i < a â² + b â² i {\displaystyle a+bi<a'+b'i} ã§ããããšãã a 2 + b 2 < a â² 2 + b â² 2 {\displaystyle a^{2}+b^{2}<a'^{2}+b'^{2}} ã§ããããšãšããŠå®çŸ©ããŠã¿ããããã®ããã«å®çŸ©ãããšãããšãã°1+2i<2-3iã§ããããŸã2+3i<3+4iã§ããããšãããã(1+2i)+(2+3i)=3+5i,(2-3i)+(3+4i)=5+iã§ããã3+5i>5+iãšãªã£ãŠããŸããããã¯åºæ¬æ§è³ª(2)ãæãç«ããªãããšãæå³ããã
ãã¡ããããã¯é©åœã«èããå®çŸ©ãããŸããŸäžé©åã ã£ããšããã ãã®ããšã ããå®ã¯ãä»ã«ã©ã®ããã«å®çŸ©ããŠããã®ãããªå°é£ããã¯éããããªãããšãç¥ãããŠãããããããã«ãè€çŽ æ°ã«ã¯å€§å°é¢ä¿ãå®çŸ©ããªãã®ã§ããã
ä»åºŠã¯ãè€çŽ æ°ã®å¹³æ¹æ ¹ã«ã€ããŠèããŠã¿ããã æ£ã®æ° a {\displaystyle a} ãèãããšãã
ã§ã¯ã
èæ°åäœ i {\displaystyle i} ã®å¹³æ¹æ ¹ãèãããšãããã¯zã«ã€ããŠã®æ¹çšåŒ z 2 = i {\displaystyle z^{2}=i} ã®è§£ z ã®å€ã§ãããããããã解ãã°ãããã©ã®ãããªè€çŽ æ°zãªããã®åŒãæºããããšãã§ããã ãããã
zãè€çŽ æ°ãšãããšã z = x + y i {\displaystyle z=x+yi} (x,yã¯å®æ°)ãšè¡šãããã ( x + y i ) 2 = i â x 2 + 2 x y i â y 2 = i â ( x 2 â y 2 ) + ( 2 x y â 1 ) i = 0 {\displaystyle (x+yi)^{2}=i\Leftrightarrow x^{2}+2xyi-y^{2}=i\Leftrightarrow (x^{2}-y^{2})+(2xy-1)i=0}
x 2 â y 2 , 2 x y â 1 {\displaystyle x^{2}-y^{2},2xy-1} ã¯å®æ°ã§ãããããå®éšãšèéšãå
±ã«0ã«ãªããã°ãªããªãããã { x 2 â y 2 = 0 ( â x = ± y ) 2 x y â 1 = 0 {\displaystyle {\begin{cases}x^{2}-y^{2}=0(\Leftrightarrow x=\pm y)\\2xy-1=0\end{cases}}}
x = y {\displaystyle x=y} ã®ãšãã 2 x 2 = 1 â x = ± 1 2 , y = ± 1 2 {\displaystyle 2x^{2}=1\Leftrightarrow x=\pm {\frac {1}{\sqrt {2}}},y=\pm {\frac {1}{\sqrt {2}}}} (è€å·åé ãx,yã¯å
±ã«å®æ°ã§ãããããæ¡ä»¶ãæºããã)
x = â y {\displaystyle x=-y} ã®ãšãã â 2 y 2 = 1 â y 2 = â 1 2 {\displaystyle -2y^{2}=1\Leftrightarrow y^{2}=-{\frac {1}{2}}} ããã§ããããæºããå®æ°yã¯ååšããªãããäžé©ã
ãã£ãŠã z = ± ( 1 2 + 1 2 i ) {\displaystyle z=\pm \left({\frac {1}{\sqrt {2}}}+{\frac {1}{\sqrt {2}}}i\right)} â
å®éšããŒããèæ
®ã㊠x = 0 {\displaystyle x=0} ã x = ± 3 y {\displaystyle x=\pm {\sqrt {3}}y} ã ããèéšããŒããªã®ã§ãxã®å€ãåè
ã®ãšã y = â 1 {\displaystyle y=-1} ãåŸè
ã®ãšã y = 1 / 2 {\displaystyle y=1/2} ãšãªãããšãããã«ãããã
2次æ¹çšåŒã«ã¯è§£ã®å
¬åŒããããæ¥æ¬ã®äžåŠãé«æ ¡ã§ãç¿ãã2次æ¹çšåŒã®è§£ã®å
¬åŒãçšããã°ãã©ããªä¿æ°ã®2次æ¹çšåŒã§ãã£ãŠã解ãæ±ããããã3次æ¹çšåŒãš4次æ¹çšåŒã«ãã解ã®å
¬åŒã¯ååšããä¿æ°ãã©ããªä¿æ°ã§ãã£ãŠã解ãæ±ããããããããã®è§£ã®å
¬åŒã¯ã代æ°æ¹çšåŒè«ã§è¿°ã¹ãŠããããã«ãä¿æ°ã«æéåã®ååæŒç®ãšæ ¹å·ããšãæäœã®çµã¿åããã§è¡šãããšãã§ããã
5次æ¹çšåŒã§ã¯ã4次以äžã®æ¹çšåŒãšã¯ç¶æ³ãç°ãªããäžè¬ã®5次æ¹çšåŒã®è§£ã¯ã2次æ¹çšåŒã4次æ¹çšåŒã®ããã«ãä¿æ°ã«æéåã®ååæŒç®ãšæ ¹å·ããšãæäœã®çµã¿åããã§è¡šãããšãã§ããªãã®ã§ããããã ãããã§ããªããããšã®èšŒæã¯å®¹æã§ã¯ãªãããã®ããšã蚌æããã«ã¯ãã¬ãã¢çè«ãç解ããå¿
èŠããã(æ¥æ¬ã®å€§åŠã®æšæºçãªã«ãªãã¥ã©ã ã§ã¯ãçåŠéšæ°åŠç§ã®åŠçã®ã¿ã倧åŠ3幎çã§åŠã¶ã®ãäžè¬çãªçšåºŠã®çè«ã§ãã)ã
ãªããããã§èšããè¡šãããšãã§ããªãããšã¯äžè¬ã®æ¹çšåŒã«ã€ããŠã®ããšã§ãããç¹å¥ãª5次æ¹çšåŒã®å Žåã¯ç°¡åã«è§£ãæ±ãããããããšãã°ã x 5 â 32 = 0 {\displaystyle x^{5}-32=0} ã¯è§£ã®ã²ãšã€ãšã㊠x = 2 {\displaystyle x=2} ããã€ããšã¯ãããããããã®æ¹çšåŒã¯ä»ã®è§£ã«ã€ããŠãäžè§é¢æ°ãçšããŠç°¡åã«è¡šããããšãé«çåŠæ ¡æ°åŠC/è€çŽ æ°å¹³é¢ã«ãããŠåŠã¶ã
ãä¿æ°ã«æéåã®ååæŒç®ãšæ ¹å·ããšãæäœã®çµã¿åãããã«æããªããã°ãäžè¬ã®5次æ¹çšåŒã®è§£ãæ±ããæ¹æ³ãååšããããããé«åºŠãªæ°åŠãçšããå¿
èŠããããw:äºæ¬¡æ¹çšåŒã«èšè¿°ãããã®ã§èå³ã®ããèªè
ã¯åç
§ãããšããã
é«çåŠæ ¡ã§è€çŽ æ°ãåºãŠããåéã¯ãã®åéãšæ°åŠCãå¹³é¢äžã®æ²ç·ãšè€çŽ æ°å¹³é¢ãã®ã¿ã§ãããè€çŽ æ°ã®åºæ¬èšç®ãæ¹çšåŒãè€çŽ æ°ç¯å²ã§è§£ãããšãè€çŽ æ°ã®å¹ŸäœåŠçæå³ã«ã€ããŠæ±ã£ãŠããããããã倧åŠæ°åŠã«ãããŠã¯ãé¢æ°ã®å®çŸ©åã»å€åãè€çŽ æ°ç¯å²ã«åºããŠèãããè€çŽ é¢æ°è«ããšãããã®ãæ±ãã
å®æ°ç¯å²ã§ã®é¢æ°ã¯x, yãšãã«äžæ¬¡å
ã®å®æ°è»žãæã€ãããå
¥åå€ãšåºåå€ã®æãã°ã©ããèããã«ã¯äºæ¬¡å
ã®åº§æšå¹³é¢ã§ååã§ãã£ããããããè€çŽ æ°ç¯å²ã§ã®é¢æ°ã¯x, yãšãã«äºæ¬¡å
ã®è€çŽ æ°å¹³é¢ãæã€ãããå
¥åå€ãšåºåå€ã®æãã°ã©ããèããã«ã¯å次å
ã®åº§æšç©ºéãå¿
èŠã§ãããäžæ¬¡å
空éã«äœãæã
ã«ã¯æç»ããããšãã§ããªãããã®ãããè€çŽ é¢æ°è«ã§ã¯é¢æ°ã®ã°ã©ããèããããšã¯åºæ¬çã«ãªãã(ãã ããåºåãããè€çŽ æ°ã®çµ¶å¯Ÿå€ãèããããšã«ãã£ãŠäžæ¬¡å
ã°ã©ãã«èœãšã蟌ãããšã¯å¯èœ)
ã§ã¯äœãèããã®ããšãããšãè€çŽ é¢æ°ã®åŸ®åç©åã§ãããè€çŽ é¢æ°ã®åŸ®åã«é¢é£ããŠãæ£åé¢æ°ããšããçšèªãåºãŠããããè€çŽ é¢æ°è«ã¯ãã®æ£åé¢æ°ãšãããã®ã®æ§è³ªã調ã¹ãåŠåã ãšèšã£ãŠè¯ãã
è€çŽ é¢æ°è«ã¯ç©çåŠã®ç¹ã«æ³¢åã«é¢ããåé(é³ã»é»ç£æ°ãªã©)ã«ãããŠæŽ»èºããŠããããæ³¢åæ¹çšåŒãããã€ã³ããŒãã³ã¹ããšããèšèã¯æåã ããã
ã¡ãªã¿ã«ãè€çŽ æ°ãããã«æ¡åŒµããæ°ãšããŠãw:åå
æ°ããšãããã®ãããããã®åå
æ°ã¯ãã¯ãã«ãè¡åãšæ·±ãé¢ãããååšããŠãããæ·±æããšé¢çœãã®ã ããããããåé·ã«ãªãããå²æããããªããåå
æ°ãããã«æ¡åŒµããå
«å
æ°ãåå
å
æ°ãšããæ°ãç 究ãããŠããã
| [
{
"paragraph_id": 0,
"tag": "p",
"text": "æ¬é
ã¯é«çåŠæ ¡æ°åŠIIã®åŒãšèšŒæã»é«æ¬¡æ¹çšåŒã®è§£èª¬ã§ããã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "( a + b ) 5 = ( a + b ) ( a + b ) ( a + b ) ( a + b ) ( a + b ) {\\displaystyle (a+b)^{5}=(a+b)(a+b)(a+b)(a+b)(a+b)} ã«ã€ããŠèãããããã®åŒãå±éãããšãã a 2 b 3 {\\displaystyle a^{2}b^{3}} ã®ä¿æ°ã¯ãå³èŸºã®5åã® ( a + b ) {\\displaystyle (a+b)} ãã a {\\displaystyle a} ã3ååãçµã¿åããã«çãããã 5 C 2 = 10 {\\displaystyle _{5}\\mathrm {C} _{2}=10} ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ãã®èããæ¡åŒµããŠ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ãå±éããã a r b n â r {\\displaystyle a^{r}b^{n-r}} ã®é
ã®ä¿æ°ã¯ãå³èŸºã® n {\\displaystyle n} åã® ( a + b ) {\\displaystyle (a+b)} ãã a {\\displaystyle a} ã r {\\displaystyle r} ååãçµã¿åããã«çãããã n C r {\\displaystyle _{n}\\mathrm {C} _{r}} ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ãã£ãŠã次ã®åŒãåŸããã:",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "æåŸã®åŒã¯æ°Bã®æ°åã§åŠã¶ç·åèšå· Σ {\\displaystyle \\Sigma } ã§ãããç¥ããªãã®ãªãç¡èŠããŠãè¯ãã ãã®åŒã äºé
å®ç(binomial theorem) ãšããããŸããããããã®é
ã«ãããä¿æ°ãäºé
ä¿æ°(binomial coefficient) ãšåŒã¶ããšãããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "(I)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "(II)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "(II)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "ãããããèšç®ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "äºé
å®çãçšããŠèšç®ããã°ãããå®éã«èšç®ãè¡ãªããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "(I)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "(II)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "(III)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "ãšãªãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "ãã¹ãŠã®èªç¶æ°nã«å¯ŸããŠ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "(I)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "(II)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "(III)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "ãæãç«ã€ããšã瀺ãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "äºé
å®ç",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "ã«ã€ããŠa,bã«é©åœãªå€ã代å
¥ããã°ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "(I) a = 1,b=1ã代å
¥ãããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "ãšãªã確ãã«äžããããé¢ä¿ãæç«ããããšãåããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "(II) a=2,b=1ã代å
¥ãããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "ãšãªã確ãã«äžããããé¢ä¿ãæç«ããããšãåããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "(III) a=1,b=-1ã代å
¥ãããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "ãšãªã確ãã«äžããããé¢ä¿ãæç«ããããšãåããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "äºé
å®çãæ¡åŒµã㊠( a + b + c ) n {\\displaystyle (a+b+c)^{n}} ãå±éããããšãèãããã a p b q c r {\\displaystyle a^{p}b^{q}c^{r}} ( p + q + r = n ) {\\displaystyle (p+q+r=n)} ã®é
ã®ä¿æ°ã¯ n {\\displaystyle n} åã® ( a + b + c ) {\\displaystyle (a+b+c)} ãã p {\\displaystyle p} åã® a {\\displaystyle a} ã q {\\displaystyle q} åã® b {\\displaystyle b} ã r {\\displaystyle r} åã® c {\\displaystyle c} ãéžã¶çµåãã«çãããã n ! p ! q ! r ! {\\displaystyle {\\frac {n!}{p!q!r!}}} ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "ããã§ã¯ãæŽåŒã®é€æ³ãšåæ°åŒã«ã€ããŠæ±ããæŽåŒã®é€æ³ã¯ãæŽåŒãæŽæ°ã®ããã«æ±ãé€æ³ãè¡ãªãèšç®ææ³ã®ããšã§ãããå®éã«æŽæ°ã®é€æ³ãšæŽåŒã®é€æ³ã«ã¯æ·±ãã€ãªããããããæŽåŒã®å æ°å解ãèãããšãã以äžå æ°å解ã§ããªãæŽåŒãååšããããã®æŽåŒãæŽæ°ã§ããçŽ å æ°ã®ããã«æ±ãããšã§æŽåŒã®çŽ å æ°å解ãå¯èœã«ãªãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "äžã§ã¯ãæŽåŒãæŽæ°ã«å¯Ÿå¿ããæ§è³ªãæã€ããšãè¿°ã¹ããæŽæ°ã«ã€ããŠã¯ãããã«çŽ ãª2ã€ã®æŽæ°ãåãããšã§æçæ°ãå®çŸ©ããããšãåºæ¥ããæŽåŒã«å¯ŸããŠãåãäºãæç«ã¡ããã®ãããªåŒãåæ°åŒãšåŒã¶ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "åæ°ãçšããªããšãã«ã¯ãæŽæ°ã®é€æ³ã¯åãšäœããçšããŠå®çŸ©ãããããã®æãå²ãããæ°Bã¯åDãšå²ãæ°AãäœãRãçšããŠ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "ã®æ§è³ªãæºããããšãç¥ãããŠãããæŽåŒã«å¯ŸããŠã䌌ãæ§è³ªãæç«ã¡ãå²ãããåŒB(x)ãåD(x)ãšå²ãåŒA(x)ãäœãR(x)ãçšããŠã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "ã®å³èŸºã§xã«ã€ããŠ2次ã®é
ãçŸãã巊蟺ãšäžèŽããªããªãããã£ãŠåã¯å®æ°ã§ãããåãaãäœããrãšãããšäžã®åŒã¯ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "ãšãªãããããã¯a=1,r=1ã§æç«ããããã£ãŠå1,äœã1ã§ãããããé«æ¬¡ã®åŒã«å¯ŸããŠãåãæ§ã«çããå®ããŠããã°ãããäŸãšããŠã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "ã®ãããªåŒãèããããã®å Žåã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "ã§ãB(x)ã3次ãA(x)ã2次ã§ããããšãããD(x)ã¯1次ã§ããããŸããR(x)ã¯2次ããå°ããããšãã1次以äžã®åŒã«ãªããããã§ãD(x)=ax+b,R(x)=cx+dãšãããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "ãåŸããããå³èŸºãå±éãããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "ãåŸãããããxã«ã©ããªå€ãå
¥ããŠããã®çåŒãæãç«ããªããã°ãªããªãã®ã§ãa = 1, b = 0, -a +c = 0, -b +d = 0ãåŸãããçµå±a=c=1, b=d=0ãåŸãããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "ãã®æ¹æ³ã¯ã©ã®é€æ³ã«å¯ŸããŠãçšããããšãåºæ¥ããã次æ°ãé«ããªããšèšç®ãé£ãããªããæŽæ°ã®å Žåãšåæ§ãæŽåŒã®é€æ³ã§ãçç®ãçšããããšãåºæ¥ããäžã®äŸãçšããŠçµæã ããæžããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 41,
"tag": "p",
"text": "ã®ããã«ãªãã)å³ã«æžãããåŒãå²ãããåŒã§ããã)å·Šã«æžãããåŒãå²ãåŒã§ããã--ã®äžçªäžã«æžãããåŒã¯åã§ãããæŽæ°ã®å²ãç®åæ§å·Šã«æžãããæ°ããé ã«å²ã£ãŠãããããã§ã¯æ¬¡æ°ã倧ããé
ãããå
ã«èšç®ãããé
ã§ãããå²ãããåŒã®äžã«ããåŒã¯åã®ç¬¬1é
ãå²ãåŒã«ãããŠåŸãåŒã§ãããããã§ã¯ã x ( x 2 â 1 ) {\\displaystyle x(x^{2}-1)} ã§ã x 3 â x {\\displaystyle x^{3}-x} ãšãªãããã ããæŽæ°ã®é€æ³ãšåæ§ãäœãããããªããŠã¯ãªããªãããã®åŸãå²ãããåŒãã x 3 â x {\\displaystyle x^{3}-x} ãåŒããæ®ã£ãåŒãæ°ããå²ãããåŒãšããŠæ±ããããã§ã¯ãåŸãåŒãå²ãåŒãããäœæ¬¡ã§ããããšãããããã§èšç®ã¯çµäºã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 42,
"tag": "p",
"text": "x 3 + 2 x 2 + 1 {\\displaystyle x^{3}+2x^{2}+1} ã x 4 + 4 x 2 + 3 x + 2 {\\displaystyle x^{4}+4x^{2}+3x+2} ãã x 2 + 2 x + 6 {\\displaystyle x^{2}+2x+6} ã§å²ã£ãåãšäœããæ±ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 43,
"tag": "p",
"text": "ãã®èšç®ã¯ã¢ãã¡ãŒã·ã§ã³ã䜿ã£ãŠ 詳ãã衚瀺ãããŠãããèšç®ææ³ã¯ã æŽæ°ã®å Žåã®çç®ãšåããããªææ³ã䜿ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 44,
"tag": "p",
"text": "ãåŸãããã®ã§ãå x {\\displaystyle x} ãäœã â 6 x + 1 {\\displaystyle -6x+1} ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 45,
"tag": "p",
"text": "2ã€ç®ã®åŒã«ã€ããŠã¯ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 46,
"tag": "p",
"text": "ãåŸãããã ãã£ãŠãç㯠å x 2 â 2 x + 2 {\\displaystyle x^{2}-2x+2} ãäœã 11 x â 10 {\\displaystyle 11x-10} ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 47,
"tag": "p",
"text": "ãããŸã§ã§æŽåŒãæŽæ°ã®ããã«æ±ããæŽåŒã®é€æ³ãè¡ãªãæ¹æ³ã«ã€ããŠè¿°ã¹ããããã§ã¯ãæŽåŒã«å¯ŸããŠåæ°åŒãå®çŸ©ããæ¹æ³ã«ã€ããŠè¿°ã¹ããåæ°åŒãšã¯ãæŽæ°ã«å¯Ÿããåæ°ã®ããã«ãé€æ³ã«ãã£ãŠçããåŒã§ãããããã§ãé€æ³ãè¡ãªãåŒã¯ã©ã®ãããªãã®ã§ãå·®ãæ¯ããªããåæ°åŒã§ã¯ãååã«å²ãããåŒãæžããåæ¯ã«å²ãåŒãæžããäŸãã°ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 48,
"tag": "p",
"text": "ã¯ãååx+1ãåæ¯ x 2 + 4 {\\displaystyle x^{2}+4} ã®åæ°åŒã§ãããåæ°åŒã«å¯ŸããŠãçŽåãéåãååšãããçŽåã¯å
±éå æ°ãæã£ãåååæ¯ããã€åæ°åŒã§çšããããããã®æã«ã¯åååæ¯ãå
±éå æ°ã§å²ããåŒãç°¡åã«ããããšãåºæ¥ããéåã¯ãåæ°åŒã®å æ³ã®æã«ããçšããããããåååæ¯ã«åãæŽåŒããããŠãåæ°åŒãå€åããªãæ§è³ªãçšããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 49,
"tag": "p",
"text": "ãç°¡åã«ããããŸãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 50,
"tag": "p",
"text": "ãèšç®ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 51,
"tag": "p",
"text": "ã«ã€ããŠååãšåæ¯ãå æ°å解ãããšãåæ¹ãšãã«",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 52,
"tag": "p",
"text": "ãå æ°ãšããŠå«ãã§ããããšãåããããã®ãšããå
±éã®å æ°ã¯çŽåããããšãå¿
èŠã§ãããèšç®ãããå€ã¯ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 53,
"tag": "p",
"text": "ãšãªãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 54,
"tag": "p",
"text": "次ã®åé¡ã§ã¯ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 55,
"tag": "p",
"text": "ãèšç®ããããã®ãšãã䞡蟺ã®åæ¯ãããããå¿
èŠãããããä»åã«ã€ããŠã¯ãåçŽã«ããããã®åæ°åŒã®ååãšåæ¯ã«åã
ã®åæ¯ããããŠåæ¯ãçµ±äžããã°ãããèšç®ãããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 56,
"tag": "p",
"text": "ãšãªãã åæ°åŒã®ä¹æ³ã¯ãåååæ¯ãå¥ã
ã«ãããã°ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 57,
"tag": "p",
"text": "次ã®èšç®ãããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 58,
"tag": "p",
"text": "(I)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 59,
"tag": "p",
"text": "(II)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 60,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 61,
"tag": "p",
"text": "(I)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 62,
"tag": "p",
"text": "(II)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 63,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 64,
"tag": "p",
"text": "åæ¯ãç©ã®åœ¢ã§ããåæ°åŒãäºã€ã®åæ°åŒã®åãå·®ã§è¡šãããåŒã«å€åœ¢ããæäœãéšååæ°å解ãšããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 65,
"tag": "p",
"text": "1 x ( x + 1 ) {\\displaystyle {\\frac {1}{x(x+1)}}} ãš 1 ( x + 1 ) ( x + 3 ) {\\displaystyle {\\frac {1}{(x+1)(x+3)}}} ãåæ°åŒã®åãŸãã¯å·®ã®åœ¢ã§è¡šãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 66,
"tag": "p",
"text": "ãšå€åœ¢ã§ããã®ã§ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 67,
"tag": "p",
"text": "ãšãªããçŽåãããš",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 68,
"tag": "p",
"text": "ãšãªãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 69,
"tag": "p",
"text": "次ã®åé¡ã§ã¯ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 70,
"tag": "p",
"text": "ãšå€åœ¢ããããšã«ãã£ãŠã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 71,
"tag": "p",
"text": "ãšãªãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 72,
"tag": "p",
"text": "ãšæ±ãŸãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 73,
"tag": "p",
"text": "éšååæ°å解ã®æäœãéã«èŸ¿ããšãåæ°åŒã®éåã®æäœãšäžèŽããã ã€ãŸããéšååæ°å解ã¯éåã®éã®æäœã§ããã ååãå®æ°ã®å Žåã«ã¯ãäžãšåæ§ã®æ¹æ³ã§éšååæ°å解ããããšãã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 74,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 75,
"tag": "p",
"text": "1. 3 ( x â 9 ) ( x â 4 ) {\\displaystyle {\\frac {3}{(x-9)(x-4)}}}",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 76,
"tag": "p",
"text": "2. 7 ( 3 x â 1 ) ( 5 â 2 x ) {\\displaystyle {\\frac {7}{(3x-1)(5-2x)}}}",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 77,
"tag": "p",
"text": "éšååæ°å解ã¯æ°åã®åã®èšç®ãç©åèšç®ã埮åãå©çšããäžçåŒã®èšŒæçã«åœ¹ç«ã€ãéèŠãªå€åœ¢ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 78,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 79,
"tag": "p",
"text": "çåŒ ( a + b ) 2 = a 2 + 2 a b + b 2 {\\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}} ã¯ãæå a , b {\\displaystyle a,b} ã«ã©ã®ãããªå€ã代å
¥ããŠãæãç«ã€ããã®ãããªçåŒãæçåŒ(ãããšããã)ãšããã çåŒ 1 x â 1 + 1 x + 1 = 2 x x 2 â 1 {\\displaystyle {\\frac {1}{x-1}}+{\\frac {1}{x+1}}={\\frac {2x}{x^{2}-1}}} ã¯ã䞡蟺ãšã x = 1 , â 1 {\\displaystyle x=1,-1} ã代å
¥ããããšã¯ã§ããªããããã®ä»ã®å€ã§ããã°ä»£å
¥ããããšãã§ãããŸãã©ã®ãããªå€ã代å
¥ããŠãçåŒãæãç«ã£ãŠããããããæçåŒãšåŒã¶ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 80,
"tag": "p",
"text": "ãã£ãœãã x 2 â x â 2 = 0 {\\displaystyle x^{2}-x-2=0} ã¯ãx=2 ãŸã㯠x=ãŒ1 ã代å
¥ãããšãã ãæãç«ã€ãããã®ããã«æåã«ç¹å®ã®å€ã代å
¥ãããšãã«ã ãæãç«ã€åŒã®ããšãæ¹çšåŒãšåŒã³ãæçåŒãšã¯åºå¥ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 81,
"tag": "p",
"text": "çåŒ a x 2 + b x + c = 0 {\\displaystyle ax^{2}+bx+c=0} ã x {\\displaystyle x} ã«ã€ããŠã®æçåŒã§ããã®ã¯ã©ã®ãããªå ŽåããèããŠã¿ããã ããåŒãã x {\\displaystyle x} ã«ã€ããŠã®æçåŒã§ããããšã¯ããã®åŒã® x {\\displaystyle x} ã«ã©ã®ãããªå€ã代å
¥ããŠãããã®çåŒã¯æãç«ã€ãšããæå³ã§ããããªã®ã§ãäŸãã° x {\\displaystyle x} ã« â 1 , 0 , 1 {\\displaystyle -1\\ ,\\ 0\\ ,\\ 1} ã代å
¥ããåŒ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 82,
"tag": "p",
"text": "ã¯ãã¹ãŠæãç«ã€å¿
èŠããããããã解ããš",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 83,
"tag": "p",
"text": "ãªã®ã§ãçåŒ a x 2 + b x + c = 0 {\\displaystyle ax^{2}+bx+c=0} ã x {\\displaystyle x} ã«ã€ããŠã®æçåŒã«ãªããªãã°ã a = b = c = 0 {\\displaystyle a=b=c=0} ã§ãªããã°ãªããªãããšããããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 84,
"tag": "p",
"text": "äžè¬ã«ãçåŒ a x 2 + b x + c = a â² x 2 + b â² x + c â² {\\displaystyle ax^{2}+bx+c=a'x^{2}+b'x+c'} ãæçåŒã§ããããšãšã ( a â a â² ) x 2 + ( b â b â² ) x + ( c â c â² ) = 0 {\\displaystyle (a-a')x^{2}+(b-b')x+(c-c')=0} ãæçåŒã§ããããšãšåãã§ããã ãã£ãŠ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 85,
"tag": "p",
"text": "ãŸãšãããšæ¬¡ã®ããã«ãªãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 86,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 87,
"tag": "p",
"text": "次ã®çåŒã x {\\displaystyle x} ã«ã€ããŠã®æçåŒãšãªãããã«ã a , b , c {\\displaystyle a\\ ,\\ b\\ ,\\ c} ã®å€ãæ±ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 88,
"tag": "p",
"text": "çåŒã®å³èŸºã x {\\displaystyle x} ã«ã€ããŠæŽçãããš",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 89,
"tag": "p",
"text": "ãã®çåŒã x {\\displaystyle x} ã«ã€ããŠã®æçåŒãšãªãã®ã¯ã䞡蟺ã®åã次æ°ã®é
ã®ä¿æ°ãçãããšãã§ããããã£ãŠ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 90,
"tag": "p",
"text": "ããã解ããš",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 91,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 92,
"tag": "p",
"text": "æçåŒãå©çšããããšã§ãè€éãªåæ°åŒã®éšååæ°å解ãã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 93,
"tag": "p",
"text": "ãšããã åæ¯ãæã£ãŠ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 94,
"tag": "p",
"text": "ããªãã¡",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 95,
"tag": "p",
"text": "ããã x {\\displaystyle x} ã®æçåŒãªã®ã§ãä¿æ°ãæ¯èŒããŠ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 96,
"tag": "p",
"text": "ããªãã¡",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 97,
"tag": "p",
"text": "æåã®çåŒã«ä»£å
¥ããŠã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 98,
"tag": "p",
"text": "次ã®åé¡ã¯ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 99,
"tag": "p",
"text": "ãšããããšã«ãããäžã®åé¡ãšåæ§ã«ããŠ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 100,
"tag": "p",
"text": "ãšæ±ãŸãã®ã§ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 101,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 102,
"tag": "p",
"text": "a~fãå®æ°ãšããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 103,
"tag": "p",
"text": "a x 2 + b y 2 + c x y + d x + e y + f = 0 {\\displaystyle ax^{2}+by^{2}+cxy+dx+ey+f=0} ãx, yã«ã€ããŠã®æçåŒã ãšããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 104,
"tag": "p",
"text": "巊蟺ãxã«ã€ããŠæŽçãããšã a x 2 + ( c y + d ) x + ( b y 2 + e y + f ) = 0 {\\displaystyle ax^{2}+(cy+d)x+(by^{2}+ey+f)=0} ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 105,
"tag": "p",
"text": "ãããxã«ã€ããŠã®æçåŒãªã®ã§ã a = 0 , c y + d = 0 , b y 2 + e y + f = 0 {\\displaystyle a=0,cy+d=0,by^{2}+ey+f=0} ãæãç«ã€ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 106,
"tag": "p",
"text": "ãããã¯æŽã«yã«ã€ããŠã®æçåŒãªã®ã§ã以äžã®çåŒãåŸãããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 107,
"tag": "p",
"text": "éã«ããããæãç«ãŠã°å
ã®åŒã¯æããã«x, yã«ã€ããŠã®æçåŒã§ãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 108,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 109,
"tag": "p",
"text": "x 2 + a x y + 6 y 2 â x + 5 y + b = ( x â 2 y + c ) ( x â 3 y + d ) {\\displaystyle x^{2}+axy+6y^{2}-x+5y+b=(x-2y+c)(x-3y+d)} ãx,yã«ã€ããŠã®æçåŒãšãªãããã«a,b,c,dãå®ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 110,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 111,
"tag": "p",
"text": "ããã»ã©çŽ¹ä»ãããæçåŒããšããèšèã䜿ã£ãŠã蚌æãã®æå³ã説æãããªãããçåŒã蚌æããããšã¯ããã®åŒãæçåŒã§ããããšã瀺ãããšã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 112,
"tag": "p",
"text": "äžè¬ã«ãçåŒ A=B ã蚌æããããã«ã¯ã次ã®ãããªæé ã®ãããããå®è¡ããã°ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 113,
"tag": "p",
"text": "ãã®ãšããå€åœ¢ã¯åå€å€åœ¢ã§ãªããã°ãªããªãããšã«æ³šæã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 114,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 115,
"tag": "p",
"text": "( a + b ) 2 â ( a â b ) 2 = 4 a b {\\displaystyle (a+b)^{2}-(a-b)^{2}=4ab} ãæãç«ã€ããšã蚌æããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 116,
"tag": "p",
"text": "(蚌æ) 巊蟺ãå±éãããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 117,
"tag": "p",
"text": "ãšãªããããã¯å³èŸºã«çããããã£ãŠãçåŒ ( a + b ) 2 â ( a â b ) 2 = 4 a b {\\displaystyle (a+b)^{2}-(a-b)^{2}=4ab} ã¯èšŒæãããã(çµ)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 118,
"tag": "p",
"text": "( x + y ) 2 + ( x â y ) 2 = 2 ( x 2 + y 2 ) {\\displaystyle (x+y)^{2}+(x-y)^{2}=2(x^{2}+y^{2})} ãæãç«ã€ããšã蚌æããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 119,
"tag": "p",
"text": "巊蟺ãèšç®ãããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 120,
"tag": "p",
"text": "ããã¯å³èŸºã«çããããã£ãŠçåŒãæãç«ã€ããšã蚌æãããã(çµ)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 121,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 122,
"tag": "p",
"text": "次ã®çåŒãæãç«ã€ããšã蚌æããã (I)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 123,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 124,
"tag": "p",
"text": "(I) (巊蟺) = ( 36 a 2 + 84 a b + 49 b 2 ) + ( 49 a 2 â 84 a b + 36 a 2 ) = 85 a 2 + 85 b 2 {\\displaystyle =(36a^{2}+84ab+49b^{2})+(49a^{2}-84ab+36a^{2})=85a^{2}+85b^{2}} (å³èŸº) = ( 81 a 2 + 36 a b + 4 b 2 ) + ( 4 a 2 â 36 a b + 81 b 2 ) = 85 a 2 + 85 b 2 {\\displaystyle =(81a^{2}+36ab+4b^{2})+(4a^{2}-36ab+81b^{2})=85a^{2}+85b^{2}} 䞡蟺ãšãåãåŒã«ãªããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 125,
"tag": "p",
"text": "æçåŒã§ãªããšãããäžããããæ¡ä»¶ããçåŒã蚌æããããšãã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 126,
"tag": "p",
"text": "ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 127,
"tag": "p",
"text": "ãã£ãŠã a 3 + b 3 + c 3 = 3 a b c {\\displaystyle a^{3}+b^{3}+c^{3}=3abc} ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 128,
"tag": "p",
"text": "ãŸãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 129,
"tag": "p",
"text": "ãããäžåŒã®å³èŸºãkãšãããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 130,
"tag": "p",
"text": "ãªã®ã§ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 131,
"tag": "p",
"text": "ãã£ãŠã a + c b + d = a â c b â d {\\displaystyle {\\frac {a+c}{b+d}}={\\frac {a-c}{b-d}}} ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 132,
"tag": "p",
"text": "ãªããæ¯ a : b {\\displaystyle a:b} ã«ã€ã㊠a b {\\displaystyle {\\frac {a}{b}}} ãæ¯ã®å€ãšããããŸãã a : b = c : d ⺠a b = c d {\\displaystyle a:b=c:d\\iff {\\frac {a}{b}}={\\frac {c}{d}}} ãæ¯äŸåŒãšããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 133,
"tag": "p",
"text": "a x = b y = c z {\\displaystyle {\\frac {a}{x}}={\\frac {b}{y}}={\\frac {c}{z}}} ãæãç«ã€ãšãã a : b : c = x : y : z {\\displaystyle a:b:c=x:y:z} ãšè¡šãããããé£æ¯ãšããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 134,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 135,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 136,
"tag": "p",
"text": "äžçåŒã®ããŸããŸãªå
¬åŒã«ã€ããŠã¯ã次ã®4ã€ã®åŒãåºæ¬çãªåŒãšããŠå°åºã§ããå Žåãããããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 137,
"tag": "p",
"text": "é«æ ¡æ°åŠã§ã¯ã次ã®4ã€ã®æ§è³ªã äžçåŒã®ãåºæ¬æ§è³ªããªã©ãšããŠçŽ¹ä»ãããŠããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 138,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 139,
"tag": "p",
"text": "(3)ãš(4)ã«ã€ããŠã¯ãã²ãšã€ã®æ§è³ªãšã㊠ãŸãšããŠããæ€å®æç§æžããã(â» åæ通ãªã©)ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 140,
"tag": "p",
"text": "æ°åŠIAã§ç¿ã£ãããªãã°ãã®æå³ã®èšå· â¹ {\\displaystyle \\Longrightarrow } ã䜿ããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 141,
"tag": "p",
"text": "ãšãæžããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 142,
"tag": "p",
"text": "äžè¿°ã®4ã€ã®åºæ¬æ§è³ªããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 143,
"tag": "p",
"text": "ã蚌æããŠã¿ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 144,
"tag": "p",
"text": "(蚌æ) ãŸã a>0 ãªã®ã§ãåºæ¬æ§è³ª(2)ãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 145,
"tag": "p",
"text": "ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 146,
"tag": "p",
"text": "ãã£ãŠã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 147,
"tag": "p",
"text": "ãªã®ã§ãåºæ¬æ§è³ª(1)ãã a + b > 0 {\\displaystyle a+b>0} ãæãç«ã€ã(çµ)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 148,
"tag": "p",
"text": "åæ§ã«ããŠã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 149,
"tag": "p",
"text": "ã蚌æã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 150,
"tag": "p",
"text": "ãããŸã§ã«ç€ºããããšãããäžçåŒ A ⧠B {\\displaystyle A\\geqq B} ã蚌æãããå Žåã«ã¯ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 151,
"tag": "p",
"text": "ã蚌æããã°ããããšãããã£ãããã¡ãã®æ¹ã蚌æããããå Žåãããããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 152,
"tag": "p",
"text": "äžçåŒã蚌æããéã«æ ¹æ ãšããåºæ¬çãªäžçåŒãšããŠã次ã®æ§è³ªãããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 153,
"tag": "p",
"text": "ãã®å®ç(ãå®æ°ã2ä¹ãããšãããªãããŒã以äžã§ããã)ããåºæ¬æ§è³ª(3),(4)ã䜿ã£ãŠèšŒæããŠã¿ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 154,
"tag": "p",
"text": "(蚌æ)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 155,
"tag": "p",
"text": "aãæ£ã®å Žåãšè² ã®å Žåãš0ã®å Žåã®3éãã«å Žåããããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 156,
"tag": "p",
"text": "[aãæ£ã®å Žå] ãã®ãšããåºæ¬æ§è³ª(3)ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 157,
"tag": "p",
"text": "ã§ãããããªãã¡ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 158,
"tag": "p",
"text": "ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 159,
"tag": "p",
"text": "[aãè² ã®å Žå] ãã®ãšããåºæ¬æ§è³ª(4)ãã 0 a < a a {\\displaystyle 0a<aa} ã§ãããããªãã¡ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 160,
"tag": "p",
"text": "ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 161,
"tag": "p",
"text": "[aããŒãã®å Žå] ãã®ãšãã a 2 = 0 {\\displaystyle a^{2}=0} ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 162,
"tag": "p",
"text": "ãã£ãŠããã¹ãŠã®å Žåã«ã€ã㊠a 2 ⧠0 {\\displaystyle a^{2}\\geqq 0} (çµ)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 163,
"tag": "p",
"text": "ãã®ããšãšåºæ¬æ§è³ª(1)(2)ããã次ãæãç«ã€ããšããããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 164,
"tag": "p",
"text": "次ã®äžçåŒãæãç«ã€ããšã蚌æããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 165,
"tag": "p",
"text": "(蚌æ)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 166,
"tag": "p",
"text": "ã蚌æããã°ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 167,
"tag": "p",
"text": "巊蟺ãå±éã㊠ãŸãšãããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 168,
"tag": "p",
"text": "ãšãªãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 169,
"tag": "p",
"text": "äžåŒã®æåŸã®åŒã®é
ã«ã€ããŠã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 170,
"tag": "p",
"text": "ã ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 171,
"tag": "p",
"text": "ã§ããããã£ãŠ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 172,
"tag": "p",
"text": "ã§ããã(çµ)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 173,
"tag": "p",
"text": "2ã€ã®æ£ã®æ° a, b ã a>b ãŸã㯠aâ§b ãªãã°ã䞡蟺ã2ä¹ããŠã倧å°é¢ä¿ã¯åããŸãŸã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 174,
"tag": "p",
"text": "ã€ãŸãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 175,
"tag": "p",
"text": "ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 176,
"tag": "p",
"text": "a>bãšãããä»®å®ãããa,b ã¯æ£ã®æ°ãªã®ã§ã ( a + b ) > 0 {\\displaystyle (a+b)>0} ã§ãããå¥ã®ä»®å®ããã a > b ãªã®ã§ã ( a â b ) > 0 {\\displaystyle (a-b)>0} ã§ãããããã£ãŠã a 2 â b 2 = ( a + b ) ( a â b ) > 0 {\\displaystyle a^{2}-b^{2}=(a+b)(a-b)>0}",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 177,
"tag": "p",
"text": "éã«ã a 2 â b 2 > 0 {\\displaystyle a^{2}-b^{2}>0} ã®ãšãã ( a + b ) ( a â b ) > 0 {\\displaystyle (a+b)(a-b)>0} ã§ããã a > 0 , b > 0 {\\displaystyle a>0,b>0} ãªã®ã§ a + b > 0 {\\displaystyle a+b>0} ã§ããããã£ãŠã a â b > 0 {\\displaystyle a-b>0} ãªã®ã§ã a > b {\\displaystyle a>b} ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 178,
"tag": "p",
"text": "ãã£ãŠã a > b ⺠a 2 > b 2 {\\displaystyle a>b\\quad \\Longleftrightarrow \\quad a^{2}>b^{2}} ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 179,
"tag": "p",
"text": "aâ§bã®å Žåãåæ§ã«èšŒæã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 180,
"tag": "p",
"text": "ç·Žç¿ãšããŠã次ã®åé¡ãåããŠã¿ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 181,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 182,
"tag": "p",
"text": "a > 0 {\\displaystyle a>0} , b > 0 {\\displaystyle b>0} ã®ãšãã次ã®äžçåŒã蚌æããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 183,
"tag": "p",
"text": "(蚌æ) äžçåŒã®äž¡èŸºã¯æ£ã§ããã®ã§ã䞡蟺ã®å¹³æ¹ã®å·®ãèããã°ããã䞡蟺ã®å¹³æ¹ã®å·®ã¯",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 184,
"tag": "p",
"text": "ã§ãããããã§ãa,b ã¯ãšãã«æ£ã®å®æ°ãªã®ã§ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 185,
"tag": "p",
"text": "ã§ããããšãçšããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 186,
"tag": "p",
"text": "ã§ããã®ã§ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 187,
"tag": "p",
"text": "ãšãªãããã£ãŠã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 188,
"tag": "p",
"text": "ã§ããã(çµ)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 189,
"tag": "p",
"text": "å®æ° a ã®çµ¶å¯Ÿå€ |a| ã«ã€ããŠã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 190,
"tag": "p",
"text": "ã§ããããã次ã®ããšãæãç«ã€ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 191,
"tag": "p",
"text": "|a|â§a , |a|⧠ãŒa , |a|=a",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 192,
"tag": "p",
"text": "ãŸãã2ã€ã®å®æ° a, b ã®çµ¶å¯Ÿå€ |ab| ã«ã€ããŠã¯ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 193,
"tag": "p",
"text": "ãæãç«ã€ã®ã§ãããã«ããã« |ab|â§0 , |a||b|â§0 ãçµã¿åãããŠã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 194,
"tag": "p",
"text": "|ab| = |a| |b| ãæãç«ã€ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 195,
"tag": "p",
"text": "(äŸé¡)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 196,
"tag": "p",
"text": "次ã®äžçåŒã蚌æããããŸããçå·ãæãç«ã€ã®ã¯ ã©ã®ãããªå Žåãã 調ã¹ãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 197,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 198,
"tag": "p",
"text": "䞡蟺ã®å¹³æ¹ã®å·®ãèãããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 199,
"tag": "p",
"text": "ãããããæ£ãªããäžããããäžçåŒ |a|+|b| ⧠|a+b| ãæ£ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 200,
"tag": "p",
"text": "ããã§ã |a| |b| ⧠ab ã§ããã®ã§ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 201,
"tag": "p",
"text": "ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 202,
"tag": "p",
"text": "ãããã£ãŠã |a|+|b| ⧠|a+b| ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 203,
"tag": "p",
"text": "çå·ãæãç«ã€ã®ã¯ |a| |b| = ab ã®å Žåãããªãã¡ ab ⧠0 ã®å Žåã§ããã(蚌æ ããã)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 204,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 205,
"tag": "p",
"text": "2ã€ã®æ° a {\\displaystyle a} , b {\\displaystyle b} ã«å¯Ÿãã a + b 2 {\\displaystyle {\\frac {a+b}{2}}} ãçžå å¹³å(ããããžããã)ãšèšãã a b {\\displaystyle {\\sqrt {ab}}} ãçžä¹å¹³å(ããããããžããã)ãšããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 206,
"tag": "p",
"text": "æ¬ããŒãžã§ã¯ã2åã®æ°ã®å¹³åã«ã€ããŠèå¯ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 207,
"tag": "p",
"text": "çžå å¹³åãšçžä¹å¹³åã«ã€ããŠã次ã®é¢ä¿åŒãæãç«ã€ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 208,
"tag": "p",
"text": "(蚌æ)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 209,
"tag": "p",
"text": "a ⧠0 , b ⧠0 {\\displaystyle a\\geqq 0,b\\geqq 0} ã®ãšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 210,
"tag": "p",
"text": "( a â b ) 2 ⧠0 {\\displaystyle \\left({\\sqrt {a}}-{\\sqrt {b}}\\right)^{2}\\geqq 0} ã§ããããã ( a â b ) 2 2 ⧠0 {\\displaystyle {\\frac {\\left({\\sqrt {a}}-{\\sqrt {b}}\\right)^{2}}{2}}\\geqq 0} ãããã£ãŠ a + b 2 ⧠a b {\\displaystyle {\\frac {a+b}{2}}\\geqq {\\sqrt {ab}}} çå·ãæãç«ã€ã®ã¯ã ( a â b ) 2 = 0 {\\displaystyle \\left({\\sqrt {a}}-{\\sqrt {b}}\\right)^{2}=0} ã®ãšããããªãã¡ a = b {\\displaystyle a=b} ã®ãšãã§ããã(蚌æ ããã)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 211,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 212,
"tag": "p",
"text": "å
¬åŒã®å©çšã§ã¯ãäžã®åŒ a + b 2 ⧠a b {\\displaystyle {\\frac {a+b}{2}}\\geqq {\\sqrt {ab}}} ã®äž¡èŸºã«2ãããã a + b ⧠2 a b {\\displaystyle a+b\\geqq 2{\\sqrt {ab}}} ã®åœ¢ã®åŒã䜿ãå Žåãããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 213,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 214,
"tag": "p",
"text": "a > 0 {\\displaystyle a>0} , b > 0 {\\displaystyle b>0} ã®ãšãã次ã®äžçåŒãæãç«ã€ããšã蚌æããã (I)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 215,
"tag": "p",
"text": "(II)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 216,
"tag": "p",
"text": "(I) a > 0 {\\displaystyle a>0} ã§ããããã 1 a > 0 {\\displaystyle {\\frac {1}{a}}>0} ãã£ãŠ a + 1 a ⧠2 a à 1 a = 2 {\\displaystyle a+{\\frac {1}{a}}\\geqq 2{\\sqrt {a\\times {\\frac {1}{a}}}}=2} ãããã£ãŠ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 217,
"tag": "p",
"text": "(II)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 218,
"tag": "p",
"text": "a > 0 {\\displaystyle a>0} , b > 0 {\\displaystyle b>0} ã§ããããã b a > 0 {\\displaystyle {\\frac {b}{a}}>0} , a b > 0 {\\displaystyle {\\frac {a}{b}}>0} ãã£ãŠ b a + a b + 2 ⧠2 b a à a b + 2 = 2 + 2 = 4 {\\displaystyle {\\frac {b}{a}}+{\\frac {a}{b}}+2\\geqq 2{\\sqrt {{\\frac {b}{a}}\\times {\\frac {a}{b}}}}+2=2+2=4} ãããã£ãŠ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 219,
"tag": "p",
"text": "2ä¹ããŠè² ã«ãªãæ°ããšãããã®ãèããããã®ãããªæ°ã¯ãäžåŠã§ç¿ã£ãå®æ°ã®äžã«ã¯ãªãããšããããããªããªãã°ãæ£ã®æ°ã§ãè² ã®æ°ã§ã2ä¹ãããšç¬Šå·ãæã¡æ¶ããŠæ£ã®æ°ã«ãªã£ãŠããŸãããã§ãããããã§é«æ ¡ã§ã¯ã2ä¹ããŠè² ã«ãªããšããæ§è³ªãæã€æ°ã®æŠå¿µãæ°ããå°å
¥ããããšã«ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 220,
"tag": "p",
"text": "ãšããæ¹çšåŒãèããããã®æ¹çšåŒã®è§£ã¯å®æ°ã«ã¯ãªããããã§ããã®æ¹çšåŒã®è§£ãšãªãæ°ãæ°ããäœãããã®åäœãæå i {\\displaystyle i} ã§ããããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 221,
"tag": "p",
"text": "ãã® i {\\displaystyle i} ã®ããšãèæ°åäœ(ããããããã)ãšåŒã¶ã(èæ°åäœã®èšå· i ãè±èªã®ã¢ã«ãã¡ãããã®ã¢ã€ã®å°æåã§ã imaginary unit ã«ç±æ¥ãããšèããããŠããã)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 222,
"tag": "p",
"text": "1 + i {\\displaystyle 1+i} ã 2 + 5 i {\\displaystyle 2+5i} ã®ããã«ãèæ°åäœ i {\\displaystyle i} ãšå®æ° a , b {\\displaystyle a,b} ãçšããŠ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 223,
"tag": "p",
"text": "ãšè¡šãããšãã§ããæ°ãè€çŽ æ°(ãµãããã)ãšããããã®ãšããaããã®è€çŽ æ°ã®å®éš(ãã€ã¶)ãšãããbãèéš(ããã¶)ãšããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 224,
"tag": "p",
"text": "äŸãã°ã 1 + i , 2 + 5 i , 9 2 + 7 2 i , 4 i , 3 {\\displaystyle 1+i,\\quad 2+5i,\\quad {\\frac {9}{2}}+{\\frac {7}{2}}i,\\quad 4i,\\quad 3} ã¯ãããããè€çŽ æ°ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 225,
"tag": "p",
"text": "è€çŽ æ° a+bi ã¯(ãã ã aãšbã¯å®æ°)ãbã0ã®å Žåã«ããããå®æ°ãšèŠãããšãã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 226,
"tag": "p",
"text": "èšãæ¹ãããããšãè€çŽ æ°ãåºæºã«èãããšãå®æ°ãšã¯ã a+0i ã®ãããªãèéšã®ä¿æ°ããŒãã«ãªãè€çŽ æ°ã®ããšã§ãããšãèšããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 227,
"tag": "p",
"text": "4iã®ãããªãèéšã0以å€ã§å®éšããŒãã®è€çŽ æ°ãçŽèæ°(ãã
ããããã)ãšåŒã¶ãçŽèæ°ã¯ã2ä¹ãããšè² ã«ãªãæ°ã§ããã å®æ°ãèéšã0ã®è€çŽ æ°ãšèããããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 228,
"tag": "p",
"text": "å®æ°ã§ãªãè€çŽ æ°ã®ããšããèæ°ã(ãããã)ãšããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 229,
"tag": "p",
"text": "2ã€ã®è€çŽ æ° a+bi ãš c+di ãšãçãããšã¯ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 230,
"tag": "p",
"text": "ã§ããããšã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 231,
"tag": "p",
"text": "ã€ãŸãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 232,
"tag": "p",
"text": "ãšãã«ãè€çŽ æ°a+bi ã 0ã§ãããšã¯ãa=0 ã〠b=0 ã§ããããšã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 233,
"tag": "p",
"text": "è€çŽ æ° z = a + b i {\\displaystyle z=a+bi} ã«å¯ŸããŠãèéšã®ç¬Šå·ãå転ãããè€çŽ æ° a â b i {\\displaystyle a-bi} ã®ããšããå
±åœ¹(ããããã)ãªè€çŽ æ°ããŸãã¯ãè€çŽ æ° z {\\displaystyle z} ã®å
±åœ¹ãã®ããã«åŒã³ã z Ì {\\displaystyle {\\bar {z}}} ã§ããããããªãããå
±åœ¹ãã¯ãå
±è»ãã®åžžçšæŒ¢åã«ããæžãæãã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 234,
"tag": "p",
"text": "å®æ°aãšå
±åœ¹ãªè€çŽ æ°ã¯ããã®å®æ° a èªèº«ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 235,
"tag": "p",
"text": "è€çŽ æ° z=a+bi ã«ã€ããŠ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 236,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 237,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 238,
"tag": "p",
"text": "è€çŽ æ°ã«ãååæŒç®(å æžä¹é€)ãå®çŸ©ãããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 239,
"tag": "p",
"text": "è€çŽ æ°ã®æŒç®ã§ã¯ãèæ°åäœ i {\\displaystyle i} ããéåžžã®æåã®ããã«æ±ã£ãŠèšç®ãããäžè¬ã«è€çŽ æ° z , w {\\displaystyle z\\ ,\\ w} ãã z = a + b i , w = c + d i {\\displaystyle z=a+bi\\ ,\\ w=c+di} ã§äžãããããšã(ãã ã a , b , c , d {\\displaystyle a\\ ,\\ b\\ ,\\ c\\ ,\\ d} ã¯å®æ°ãšãã)ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 240,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 241,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 242,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 243,
"tag": "p",
"text": "ãšãããµãã«è€çŽ æ°ã®å æžä¹é€ã®èšç®æ³ãå®ããããŠããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 244,
"tag": "p",
"text": "ä¹æ³ã®å®çŸ©ã¯ãäžèŠãããšé£ãããã«ã¿ããããå®æ°ã®åé
æ³åãšåæ§ã«å±éããŠããæåŸã« iã«ãã€ãã¹1ã代å
¥ããŠãã£ãã ãã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 245,
"tag": "p",
"text": "é€æ³ã®å®çŸ©ã¯ãååãšåæ¯ã«ãåæ¯ãšå
±åœ¹ãªåœ¢ã®åŒã æãç® ããã ãã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 246,
"tag": "p",
"text": "ä¹æ³ãé€æ³ã®å®çŸ©åŒãæèšããå¿
èŠã¯ç¡ããèšç®ã®éã«ã¯ãå¿
èŠã«å¿ããŠåé
æ³åãå
±åœ¹ãªã©ã®ãå¿
èŠãªåŒå€åœ¢ãè¡ãã°ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 247,
"tag": "p",
"text": "äŸé¡",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 248,
"tag": "p",
"text": "2ã€ã®è€çŽ æ°",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 249,
"tag": "p",
"text": "ã«ã€ããŠã a + b {\\displaystyle a+b} ãš a b {\\displaystyle ab} ãš a b {\\displaystyle {\\frac {a}{b}}} ããããããèšç®ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 250,
"tag": "p",
"text": "解ç",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 251,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 252,
"tag": "p",
"text": "ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 253,
"tag": "p",
"text": "ããããã«ç°¡åã«ã§ããªãã ããããå®ã¯ãã¡ãã£ãšãããã¯ããã¯ãçšããã°ããèŠããã圢ã«ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 254,
"tag": "p",
"text": "åæ°ã¯åæ¯ãšååã«åãæ°ããããŠããã£ãã®ã§ãåæ¯ãšååã«åæ¯ã®å
±åœ¹ããããŠã¿ãããããšã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 255,
"tag": "p",
"text": "ãåŸãããããã®åœ¢ã®ã»ããå
ã®åŒããããã£ãšèŠããã圢ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 256,
"tag": "p",
"text": "ãã®ãããªæäœãåæ¯ã®å®æ°åãšããããšããããæ°åŠIã§åŠç¿ããå±éã»å æ°å解å
¬åŒ ( a + b ) ( a â b ) = a 2 â b 2 {\\displaystyle (a+b)(a-b)=a^{2}-b^{2}} ã®ç°¡åãªå¿çšã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 257,
"tag": "p",
"text": "æ°ã®ç¯å²ãè€çŽ æ°ã«ãŸã§æ¡åŒµãããšãè² ã®æ°ã®å¹³æ¹æ ¹ãèããããšãã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 258,
"tag": "p",
"text": "äŸãšããŠã -5 ã®å¹³æ¹æ ¹ã«ã€ããŠèããŠã¿ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 259,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 260,
"tag": "p",
"text": "ã§ããããã -5 ã®å¹³æ¹æ ¹ã¯ 5 i {\\displaystyle {\\sqrt {5}}\\ i} ãš â 5 i {\\displaystyle -{\\sqrt {5}}\\ i} ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 261,
"tag": "p",
"text": "â 5 {\\displaystyle {\\sqrt {-5}}} ãšã¯ã 5 i {\\displaystyle {\\sqrt {5}}\\ i} ã®ããšãšããã â â 5 {\\displaystyle -{\\sqrt {-5}}} ãšã¯ã â 5 i {\\displaystyle -{\\sqrt {5}}\\ i} ã®ããšã§ããã ãšãã« â 1 = i {\\displaystyle {\\sqrt {-1}}\\ =\\ i} ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 262,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 263,
"tag": "p",
"text": "ããŠã-5 ã®å¹³æ¹æ ¹ã¯ãæ¹çšåŒ x 2 = â 5 {\\displaystyle x^{2}=-5} ã®è§£ã§ãããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 264,
"tag": "p",
"text": "ãã®æ¹çšåŒã移é
ããããšã«ããã-5 ã®å¹³æ¹æ ¹ã¯ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 265,
"tag": "p",
"text": "ã®è§£ã§ãããšããããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 266,
"tag": "p",
"text": "ããã«å æ°å解ãããããšã«ããã-5 ã®å¹³æ¹æ ¹ã¯æ¹çšåŒ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 267,
"tag": "p",
"text": "ã®è§£ã§ããããšããããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 268,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 269,
"tag": "p",
"text": "(I) â 2 â 6 {\\displaystyle {\\sqrt {-2}}\\ {\\sqrt {-6}}} ãèšç®ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 270,
"tag": "p",
"text": "(I)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 271,
"tag": "p",
"text": "ãã®ããã«ããŸãããã€ãã¹ã®æ°ã®å¹³æ¹æ ¹ãåºãŠãããããŸãèæ°åäœ i ãçšããåŒã«æžãæããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 272,
"tag": "p",
"text": "ãã®ããšãããç®ãããŠããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 273,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 274,
"tag": "p",
"text": "(II) 2 â 3 {\\displaystyle {\\frac {\\sqrt {2}}{\\sqrt {-3}}}} ãèšç®ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 275,
"tag": "p",
"text": "(III) 2次æ¹çšåŒ x 2 = â 7 {\\displaystyle x^{2}=-7} ã解ãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 276,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 277,
"tag": "p",
"text": "(II)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 278,
"tag": "p",
"text": "(III)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 279,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 280,
"tag": "p",
"text": "è€çŽ æ°ã®å¿çšãšããŠãããã§ã¯2次æ¹çšåŒã®æ§è³ªã«ã€ããŠè¿°ã¹ããä»»æã®2次æ¹çšåŒã¯ã解ã®å
¬åŒã«ãã£ãŠè§£ãããããšãé«çåŠæ ¡æ°åŠIã§è¿°ã¹ãããããã解ã®å
¬åŒã«å«ãŸããæ ¹å·ã®äžèº«ãè² ã®æ°ã®å Žåã«ã¯å®æ°è§£ãååšããªãããšã«æ³šæããå¿
èŠãããã2次æ¹çšåŒ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 281,
"tag": "p",
"text": "ã®è§£ã®å
¬åŒã¯ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 282,
"tag": "p",
"text": "ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 283,
"tag": "p",
"text": "å€å¥åŒ D {\\displaystyle D} ã¯",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 284,
"tag": "p",
"text": "ã«ãã£ãŠå®çŸ©ããããå€å¥åŒã¯ã解ã®å
¬åŒã®æ ¹å·(ã«ãŒãèšå·ã®ããš)ã®äžèº«ã«çãããå€å¥åŒã®æ£è² ã«ãã£ãŠ2次æ¹çšåŒãå®æ°è§£ãæã€ãã©ããã決ãŸãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 285,
"tag": "p",
"text": "D {\\displaystyle D} ãè² ã®ãšãã«ã¯ãã®2次æ¹çšåŒã¯å®æ°ã®ç¯å²ã«ã¯è§£ãæããªãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 286,
"tag": "p",
"text": "å€å¥åŒ D {\\displaystyle D} ãè² ã®æ°ã§ãã£ããšããxã®è§£ã¯ç°ãªã2ã€ã®èæ°ã«ãªãããã®2ã€ã®è§£ã¯ å
±åœ¹ ã®é¢ä¿ã«ãªã£ãŠããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 287,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 288,
"tag": "p",
"text": "è€çŽ æ°ãçšããŠã2次æ¹çšåŒ (1)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 289,
"tag": "p",
"text": "(2)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 290,
"tag": "p",
"text": "(3)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 291,
"tag": "p",
"text": "ã解ãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 292,
"tag": "p",
"text": "解ã®å
¬åŒãçšããŠè§£ãã°ããã(1)ã ããèšç®ãããšã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 293,
"tag": "p",
"text": "ãšãªãã ä»ãåãããã«æ±ãããšãåºæ¥ãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 294,
"tag": "p",
"text": "以éã®è§£çã¯ã (2)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 295,
"tag": "p",
"text": "(3)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 296,
"tag": "p",
"text": "ãšãªãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 297,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 298,
"tag": "p",
"text": "æ¹çšåŒã®è§£ã§ãå®æ°ã§ãããã®ã å®æ°è§£ ãšããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 299,
"tag": "p",
"text": "æ¹çšåŒã®è§£ã§ãèæ°ã§ãããã®ã èæ°è§£ ãšããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 300,
"tag": "p",
"text": "2次æ¹çšåŒ a x 2 + b x + c = 0 {\\displaystyle ax^{2}+bx+c=0} ã®è§£ã¯ x = â b ± b 2 â 4 a c 2 a {\\displaystyle x={\\frac {-b\\pm {\\sqrt {b^{2}-4ac}}}{2a}}} ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 301,
"tag": "p",
"text": "2次æ¹çšåŒã®è§£ã®çš®é¡ã¯ã解ã®å
¬åŒã®äžã®æ ¹å·ã®äžã®åŒ b 2 â 4 a c {\\displaystyle b^{2}-4ac} ã®ç¬Šå·ãèŠãã°å€å¥ããããšãã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 302,
"tag": "p",
"text": "ãã®åŒ b 2 â 4 a c {\\displaystyle b^{2}-4ac} ãã2次æ¹çšåŒ a x 2 + b x + c = 0 {\\displaystyle ax^{2}+bx+c=0} ã®å€å¥åŒ(ã¯ãã¹ã€ãã)ãšãããèšå· D {\\displaystyle D} ã§è¡šãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 303,
"tag": "p",
"text": "ãŸããé解ãå®æ°è§£ã§ããã®ã§ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 304,
"tag": "p",
"text": "ãšãããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 305,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 306,
"tag": "p",
"text": "次ã®2次æ¹çšåŒã®è§£ãå€å¥ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 307,
"tag": "p",
"text": "(I)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 308,
"tag": "p",
"text": "(II)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 309,
"tag": "p",
"text": "(III)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 310,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 311,
"tag": "p",
"text": "(I)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 312,
"tag": "p",
"text": "ã ãããç°ãªã2ã€ã®å®æ°è§£ããã€ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 313,
"tag": "p",
"text": "(II)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 314,
"tag": "p",
"text": "ã ãããç°ãªã2ã€ã®èæ°è§£ããã€ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 315,
"tag": "p",
"text": "(III)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 316,
"tag": "p",
"text": "ã ãããé解ããã€ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 317,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 318,
"tag": "p",
"text": "ãŸãã2次æ¹çšåŒ a x 2 + 2 b â² x + c = 0 {\\displaystyle ax^{2}+2b'x+c=0} ã®ãšãã D = 4 ( b â² 2 â a c ) {\\displaystyle D=4(b'^{2}-ac)} ãšãªãã®ã§ã 2次æ¹çšåŒ a x 2 + 2 b â² x + c = 0 {\\displaystyle ax^{2}+2b'x+c=0} ã®å€å¥åŒã«ã¯",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 319,
"tag": "p",
"text": "ããã¡ããŠãããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 320,
"tag": "p",
"text": "ãããçšããŠãåã®åé¡",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 321,
"tag": "p",
"text": "ã®è§£ãå€å¥ãããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 322,
"tag": "p",
"text": "a = 4 , b â² = â 10 , c = 25 {\\displaystyle a=4\\,,\\,b'=-10\\,,\\,c=25} ã§ãããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 323,
"tag": "p",
"text": "ã ãããé解ããã€ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 324,
"tag": "p",
"text": "2次æ¹çšåŒ a x 2 + b x + c = 0 {\\displaystyle ax^{2}+bx+c=0} ã®2ã€ã®è§£ã α {\\displaystyle \\alpha } , β {\\displaystyle \\beta } ãšããã ãã®æ¹çšåŒã¯ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 325,
"tag": "p",
"text": "a ( x â α ) ( x â β ) = 0 {\\displaystyle a(x-\\alpha )(x-\\beta )=0}",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 326,
"tag": "p",
"text": "ãšå€åœ¢ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 327,
"tag": "p",
"text": "ãããå±éãããšã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 328,
"tag": "p",
"text": "a x 2 â a ( α + β ) x + a α β = 0 {\\displaystyle ax^{2}-a(\\alpha +\\beta )x+a\\alpha \\beta =0}",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 329,
"tag": "p",
"text": "ä¿æ°ãæ¯èŒããŠã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 330,
"tag": "p",
"text": "c = a α β , b = â a ( α + β ) {\\displaystyle c=a\\alpha \\beta ,b=-a(\\alpha +\\beta )}",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 331,
"tag": "p",
"text": "ãåŸãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 332,
"tag": "p",
"text": "ãããå€åœ¢ããã°ã α + β = â b a , α β = c a {\\displaystyle \\alpha +\\beta =-{\\frac {b}{a}},\\alpha \\beta ={\\frac {c}{a}}} ãšãªãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 333,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 334,
"tag": "p",
"text": "2次æ¹çšåŒ 2 x 2 + 4 x + 3 = 0 {\\displaystyle 2x^{2}+4x+3=0} ã®2ã€ã®è§£ã α {\\displaystyle \\alpha } , β {\\displaystyle \\beta } ãšãããšãã α 2 + β 2 {\\displaystyle \\alpha ^{2}+\\beta ^{2}} ã®å€ãæ±ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 335,
"tag": "p",
"text": "解ãšä¿æ°ã®é¢ä¿ããã α + β = â 4 2 = â 2 {\\displaystyle \\alpha +\\beta =-{\\frac {4}{2}}=-2} , α β = 3 2 {\\displaystyle \\alpha \\beta ={\\frac {3}{2}}} α 2 + β 2 = ( α + β ) 2 â 2 α β = ( â 2 ) 2 â 2 à 3 2 = 1 {\\displaystyle \\alpha ^{2}+\\beta ^{2}=(\\alpha +\\beta )^{2}-2\\alpha \\beta =(-2)^{2}-2\\times {\\frac {3}{2}}=1}",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 336,
"tag": "p",
"text": "2ã€ã®æ° α {\\displaystyle \\alpha } , β {\\displaystyle \\beta } ã解ãšãã2次æ¹çšåŒã¯",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 337,
"tag": "p",
"text": "ãšè¡šãããã巊蟺ãå±éããŠæŽçãããšæ¬¡ã®ããã«ãªãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 338,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 339,
"tag": "p",
"text": "次ã®2æ°ã解ãšãã2次æ¹çšåŒãäœãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 340,
"tag": "p",
"text": "(I)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 341,
"tag": "p",
"text": "(II)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 342,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 343,
"tag": "p",
"text": "(I) å ( 3 + 5 ) + ( 3 â 5 ) = 6 {\\displaystyle (3+{\\sqrt {5}})+(3-{\\sqrt {5}})=6} ç© ( 3 + 5 ) ( 3 â 5 ) = 4 {\\displaystyle (3+{\\sqrt {5}})(3-{\\sqrt {5}})=4} ã§ãããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 344,
"tag": "p",
"text": "(II) å ( 2 + 3 i ) + ( 2 â 3 i ) = 4 {\\displaystyle (2+3i)+(2-3i)=4} ç© ( 2 + 3 i ) ( 2 â 3 i ) = 13 {\\displaystyle (2+3i)(2-3i)=13} ã§ãããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 345,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 346,
"tag": "p",
"text": "2次æ¹çšåŒ a x 2 + b x + c = 0 {\\displaystyle ax^{2}+bx+c=0} ã®2ã€ã®è§£ α {\\displaystyle \\alpha } , β {\\displaystyle \\beta } ãããããšã2次åŒ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 347,
"tag": "p",
"text": "ãå æ°å解ããããšãã§ããã 解ãšä¿æ°ã®é¢ä¿ α + β = â b a {\\displaystyle \\alpha +\\beta =-{\\frac {b}{a}}} , α β = c a {\\displaystyle \\alpha \\beta ={\\frac {c}{a}}} ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 348,
"tag": "p",
"text": "2次æ¹çšåŒã¯ãè€çŽ æ°ã®ç¯å²ã§èãããšã€ãã«è§£ããã€ãããè€çŽ æ°ãŸã§äœ¿ã£ãŠãããšãããšã2次åŒã¯å¿
ã1次åŒã®ç©ã«å æ°å解ããããšãã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 349,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 350,
"tag": "p",
"text": "è€çŽ æ°ã®ç¯å²ã§èããŠã次ã®2次åŒãå æ°å解ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 351,
"tag": "p",
"text": "(I)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 352,
"tag": "p",
"text": "(II)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 353,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 354,
"tag": "p",
"text": "(I) 2次æ¹çšåŒ x 2 + 4 x â 1 = 0 {\\displaystyle x^{2}+4x-1=0} ã®è§£ã¯",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 355,
"tag": "p",
"text": "ãã£ãŠ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 356,
"tag": "p",
"text": "(II) 2次æ¹çšåŒ 2 x 2 â 3 x + 2 = 0 {\\displaystyle 2x^{2}-3x+2=0} ã®è§£ã¯",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 357,
"tag": "p",
"text": "ãã£ãŠ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 358,
"tag": "p",
"text": "3次以äžã®æŽåŒã«ããæ¹çšåŒãèããã äžè¬ã«æ¹çšåŒã P ( x ) = 0 {\\displaystyle P(x)=0} ãšãšãã ãã ãã P ( x ) {\\displaystyle P(x)} ã¯ãä»»æã®æ¬¡æ°ã®æŽåŒãšããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 359,
"tag": "p",
"text": "P ( x ) {\\displaystyle P(x)} ã1æ¬¡åŒ x â a {\\displaystyle x-a} ã§å²ã£ããšãã®åã Q ( x ) {\\displaystyle Q(x)} ãäœãã R {\\displaystyle R} ãšãããšã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 360,
"tag": "p",
"text": "ãã®äž¡èŸºã® x {\\displaystyle x} ã« a {\\displaystyle a} ã代å
¥ãããšã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 361,
"tag": "p",
"text": "ã€ãŸãã P ( x ) {\\displaystyle P(x)} ã x â a {\\displaystyle x-a} ã§å²ã£ããšãã®äœã㯠P ( a ) {\\displaystyle P(a)} ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 362,
"tag": "p",
"text": "æŽåŒ P ( x ) = x 3 â 2 x + 3 {\\displaystyle P(x)=x^{3}-2x+3} ã次ã®åŒã§å²ã£ãäœããæ±ããã (I)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 363,
"tag": "p",
"text": "(II)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 364,
"tag": "p",
"text": "(III)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 365,
"tag": "p",
"text": "(I) P ( 2 ) = 2 3 â 2 Ã 2 + 3 = 7 {\\displaystyle P(2)=2^{3}-2\\times 2+3=7} (II) P ( â 1 ) = ( â 1 ) 3 â 2 Ã ( â 1 ) + 3 = 4 {\\displaystyle P(-1)=(-1)^{3}-2\\times (-1)+3=4} (III) P ( 1 2 ) = ( 1 2 ) 3 â 2 Ã ( 1 2 ) + 3 = 17 8 {\\displaystyle P\\left({\\frac {1}{2}}\\right)=\\left({\\frac {1}{2}}\\right)^{3}-2\\times \\left({\\frac {1}{2}}\\right)+3={\\frac {17}{8}}}",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 366,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 367,
"tag": "p",
"text": "ããå®æ° a {\\displaystyle a} ã«å¯ŸããŠã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 368,
"tag": "p",
"text": "ãæãç«ã£ããšããã ãã®ãšããæŽåŒ P ( x ) {\\displaystyle P(x)} ã¯ã ( x â a ) {\\displaystyle (x-a)} ãå æ°ã«æã€ããšãåãã ãã®ããšãå æ°å®ç(ãããããŠãã)ãšåŒã¶ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 369,
"tag": "p",
"text": "æŽåŒ P ( x ) {\\displaystyle P(x)} ã«å¯ŸããŠãå Q ( x ) {\\displaystyle Q(x)} ãå²ãåŒ ( x â a ) {\\displaystyle (x-a)} ãšãã æŽåŒã®é€æ³ãçšããããã®ãšããå Q ( x ) {\\displaystyle Q(x)} ã ( Q ( x ) {\\displaystyle Q(x)} ã¯ã P ( x ) {\\displaystyle P(x)} ããã1ã ã次æ°ãäœãæŽåŒã§ããã) äœã c {\\displaystyle c} ( c {\\displaystyle c} ã¯ãå®æ°ã)ãšãããšã æŽåŒ P ( x ) {\\displaystyle P(x)} ã¯ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 370,
"tag": "p",
"text": "ãšæžããã ããã§ã c = 0 {\\displaystyle c=0} ã§ãªããšã P ( a ) = 0 {\\displaystyle P(a)=0} ã¯æºããããªããã ãã®ãšãã P ( x ) {\\displaystyle P(x)} ã¯ã ( x â a ) {\\displaystyle (x-a)} ã«ãã£ãŠå²ãåããã ãã£ãŠãå æ°å®çã¯æç«ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 371,
"tag": "p",
"text": "å æ°å®çãçšããããšã§ããã次æ°ã®é«ãæŽåŒãå æ°å解ããããšã åºæ¥ãããã«ãªããäŸãã°ã3次ã®æŽåŒ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 372,
"tag": "p",
"text": "ã«ã€ããŠã x = 1 {\\displaystyle x=1} ã代å
¥ãããšã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 373,
"tag": "p",
"text": "ã¯0ãšãªãããã£ãŠãå æ°å®çãããã®åŒã¯",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 374,
"tag": "p",
"text": "ãå æ°ãšããŠæã€ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 375,
"tag": "p",
"text": "ããã§ãå®éæŽåŒã®é€æ³ã䜿ã£ãŠèšç®ãããšã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 376,
"tag": "p",
"text": "ãåŸãããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 377,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 378,
"tag": "p",
"text": "å æ°å®çãçšã㊠(I)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 379,
"tag": "p",
"text": "(II)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 380,
"tag": "p",
"text": "ãå æ°å解ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 381,
"tag": "p",
"text": "(I) å æ°å解ã®çµæã(x+æŽæ°)ã®ç©ã®åœ¢ãªããæŽæ°ã¯6ã®å æ°ã§ãªããã°ãªããªãããã®ããã ± 1 , ± 2 , ± 3 , ± 6 {\\displaystyle \\pm 1,\\pm 2,\\pm 3,\\pm 6} ãåè£ãšãªãããããã«ã€ããŠã¯å®éã«ä»£å
¥ããŠç¢ºããããããªããx=1ã代å
¥ãããšã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 382,
"tag": "p",
"text": "ãšãªãã®ã§ã(x-1)ãå æ°ãšãªããå®éã«æŽåŒã®é€æ³ãè¡ãªããšãåãšã㊠x 2 â 5 x + 6 {\\displaystyle x^{2}-5x+6} ãåŸããããããã㯠( x â 2 ) ( x â 3 ) {\\displaystyle (x-2)(x-3)} ã«å æ°å解ã§ããããã£ãŠçãã¯ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 383,
"tag": "p",
"text": "ãšãªãã (II) ããã§ãå°éã«24ã®å æ°ãåœãŠã¯ããŠãããããªãã24ã®å æ°ã¯æ°ãå€ãã®ã§ããªãã®èšç®ãå¿
èŠãšãªããããã§ã¯ã-2ã代å
¥ãããšã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 384,
"tag": "p",
"text": "ãšãªãã(x+2)ãå æ°ã ãšããããé€æ³ãè¡ãªããšã x 2 â x â 12 {\\displaystyle x^{2}-x-12} ãåŸããããã(x-4)(x+3)ã«å æ°å解ã§ãããçãã¯ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 385,
"tag": "p",
"text": "ãšãªãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 386,
"tag": "p",
"text": "å æ°å解ãå æ°å®çãå©çšããŠé«æ¬¡æ¹çšåŒã解ããŠã¿ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 387,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 388,
"tag": "p",
"text": "é«æ¬¡æ¹çšåŒ (I)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 389,
"tag": "p",
"text": "(II)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 390,
"tag": "p",
"text": "(III)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 391,
"tag": "p",
"text": "ã解ãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 392,
"tag": "p",
"text": "(I) 巊蟺ã a 3 â b 3 = ( a â b ) ( a 2 + a b + b 2 ) {\\displaystyle a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})} ãçšããŠå æ°å解ãããš",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 393,
"tag": "p",
"text": "ãããã£ãŠ x â 2 = 0 {\\displaystyle \\ x-2=0} ãŸã㯠x 2 + 2 x + 4 = 0 {\\displaystyle \\ x^{2}+2x+4=0} ãã£ãŠ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 394,
"tag": "p",
"text": "(II) x 2 = X {\\displaystyle \\ x^{2}=X\\ } ãšãããšã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 395,
"tag": "p",
"text": "巊蟺ãå æ°å解ãããš",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 396,
"tag": "p",
"text": "ãã£ãŠ X = 4 , X = â 2 {\\displaystyle X=4\\ ,\\ X=-2} ããã« x 2 = 4 , x 2 = â 2 {\\displaystyle x^{2}=4\\ ,\\ x^{2}=-2} ãããã£ãŠ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 397,
"tag": "p",
"text": "(III) P ( x ) = x 3 â 5 x 2 + 7 x â 2 {\\displaystyle \\ P(x)=x^{3}-5x^{2}+7x-2\\ } ãšããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 398,
"tag": "p",
"text": "ãããã£ãŠã x â 2 {\\displaystyle \\ x-2\\ } 㯠P ( x ) {\\displaystyle \\ P(x)\\ } ã®å æ°ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 399,
"tag": "p",
"text": "ãã£ãŠ ( x â 2 ) ( x 2 â 3 x + 1 ) = 0 {\\displaystyle (x-2)(x^{2}-3x+1)=0} x â 2 = 0 {\\displaystyle \\ x-2=0} ãŸã㯠x 2 â 3 x + 1 = 0 {\\displaystyle \\ x^{2}-3x+1=0} ãããã£ãŠ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 400,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 401,
"tag": "p",
"text": "3次æ¹çšåŒ a x 3 + b x 2 + c x + d = 0 {\\displaystyle ax^{3}+bx^{2}+cx+d=0} ã®3ã€ã®è§£ã ã α , β , γ {\\displaystyle \\alpha \\ ,\\ \\beta \\ ,\\ \\gamma } ãšãããš",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 402,
"tag": "p",
"text": "ãæãç«ã€ã å³èŸºãå±éãããš",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 403,
"tag": "p",
"text": "ãã£ãŠ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 404,
"tag": "p",
"text": "ããã«",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 405,
"tag": "p",
"text": "ãããã£ãŠã次ã®ããšãæãç«ã€ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 406,
"tag": "p",
"text": "ãã°ãã°èæ°ã¯ãçŸå®ã«ã¯ååšããªãæ°ãã§ãããšèšãããããšããããæŽå²çã«ãèæ°ãæ±ã£ãæ°åŠãèããã¹ãã§ã¯ãªããšèããããæ代ã¯é·ãã£ãããã®æ代ã®å
é²çãªæ°åŠè
ã®äžã«ã¯ãèæ°ãæå¹ã«æŽ»çšããŠç 究ãé²ããäžæ¹ã§ãææãçºè¡šããéã«ã¯èæ°ãè¡šã«åºããã«èšè¿°ããåªåãããããšã§ãç¡çšãªæµæãåããªãããã«å·¥å€«ããè
ããããšèšãããã»ã©ã§ããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 407,
"tag": "p",
"text": "ã ããããèããŠã¿ãã°ãæ°ããçŸå®ã«ååšããããšã¯ã©ãããæå³ãªã®ã ããããçŸå®ã«éçã䜿ã£ãŠçŽã«åãæããªãã°ãååšã®é·ãããæ£ç¢ºã«ååšçãã®ãã®ã«ãããããšã¯äžå¯èœã§ããããã«æããããããã®å²ã«ååšçãšããå®æ°ã¯ãååšããããšæããããã®ã¯ãªãã ããããæ°çŽç·ãå®æ°ã®ãå®åšããä¿¡ãããããªãã°ãè€çŽ æ°ã¯è€çŽ æ°å¹³é¢(æ°åŠCã§ç¿ã)ã®äžã«ååšããã®ã ãããåãã§ã¯ãªãã ãããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 408,
"tag": "p",
"text": "ãã®ããã«èãããšãããããæ°ãšã¯ãã¹ãŠããæå³ã§æ³åäžã®ååšã§ãããããã«å¯ŸããŠãååšããããååšããªãããšããåããç«ãŠãããšããã³ã»ã³ã¹ã§ããããã«æãããããååšããªããããã«æãããã¡ãªèæ°ã§ããããããšãã°ç©çåŠã®äžåéã§ããéåååŠã®ã·ã¥ã¬ãã£ã³ã¬ãŒæ¹çšåŒã«è¡šãããªã©ãå¿çšäžã®ããŸããŸãªå Žé¢ã«ãããŠããèæ°ã䜿ã£ãŠèšè¿°ããããšãèªç¶ãªå¯Ÿè±¡ã¯å€ãã®ã ã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 409,
"tag": "p",
"text": "è€çŽ æ°ã©ããã«ã€ããŠããã®å€§å°é¢ä¿ã¯å®çŸ©ããªãããã®çç±ã¯ãã©ã®ããã«å€§å°é¢ä¿ãå®çŸ©ããŠãã䟿å©ãªæ§è³ªãæºããããšãã§ããªãããã§ãããå
·äœçã«èšãã°ãæ¢ã«è¿°ã¹ãå®æ°ã®å€§å°é¢ä¿ã«ã€ããŠã®ãäžçåŒã®åºæ¬æ§è³ª(1)(2)(3)(4)ãã«ãããåŒãæãç«ãããããšãã§ããªãã®ã ã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 410,
"tag": "p",
"text": "ããšãã°ã a + b i < a â² + b â² i {\\displaystyle a+bi<a'+b'i} ã§ããããšãã a 2 + b 2 < a â² 2 + b â² 2 {\\displaystyle a^{2}+b^{2}<a'^{2}+b'^{2}} ã§ããããšãšããŠå®çŸ©ããŠã¿ããããã®ããã«å®çŸ©ãããšãããšãã°1+2i<2-3iã§ããããŸã2+3i<3+4iã§ããããšãããã(1+2i)+(2+3i)=3+5i,(2-3i)+(3+4i)=5+iã§ããã3+5i>5+iãšãªã£ãŠããŸããããã¯åºæ¬æ§è³ª(2)ãæãç«ããªãããšãæå³ããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 411,
"tag": "p",
"text": "ãã¡ããããã¯é©åœã«èããå®çŸ©ãããŸããŸäžé©åã ã£ããšããã ãã®ããšã ããå®ã¯ãä»ã«ã©ã®ããã«å®çŸ©ããŠããã®ãããªå°é£ããã¯éããããªãããšãç¥ãããŠãããããããã«ãè€çŽ æ°ã«ã¯å€§å°é¢ä¿ãå®çŸ©ããªãã®ã§ããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 412,
"tag": "p",
"text": "ä»åºŠã¯ãè€çŽ æ°ã®å¹³æ¹æ ¹ã«ã€ããŠèããŠã¿ããã æ£ã®æ° a {\\displaystyle a} ãèãããšãã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 413,
"tag": "p",
"text": "ã§ã¯ã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 414,
"tag": "p",
"text": "èæ°åäœ i {\\displaystyle i} ã®å¹³æ¹æ ¹ãèãããšãããã¯zã«ã€ããŠã®æ¹çšåŒ z 2 = i {\\displaystyle z^{2}=i} ã®è§£ z ã®å€ã§ãããããããã解ãã°ãããã©ã®ãããªè€çŽ æ°zãªããã®åŒãæºããããšãã§ããã ãããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 415,
"tag": "p",
"text": "zãè€çŽ æ°ãšãããšã z = x + y i {\\displaystyle z=x+yi} (x,yã¯å®æ°)ãšè¡šãããã ( x + y i ) 2 = i â x 2 + 2 x y i â y 2 = i â ( x 2 â y 2 ) + ( 2 x y â 1 ) i = 0 {\\displaystyle (x+yi)^{2}=i\\Leftrightarrow x^{2}+2xyi-y^{2}=i\\Leftrightarrow (x^{2}-y^{2})+(2xy-1)i=0}",
"title": "ã³ã©ã "
},
{
"paragraph_id": 416,
"tag": "p",
"text": "x 2 â y 2 , 2 x y â 1 {\\displaystyle x^{2}-y^{2},2xy-1} ã¯å®æ°ã§ãããããå®éšãšèéšãå
±ã«0ã«ãªããã°ãªããªãããã { x 2 â y 2 = 0 ( â x = ± y ) 2 x y â 1 = 0 {\\displaystyle {\\begin{cases}x^{2}-y^{2}=0(\\Leftrightarrow x=\\pm y)\\\\2xy-1=0\\end{cases}}}",
"title": "ã³ã©ã "
},
{
"paragraph_id": 417,
"tag": "p",
"text": "x = y {\\displaystyle x=y} ã®ãšãã 2 x 2 = 1 â x = ± 1 2 , y = ± 1 2 {\\displaystyle 2x^{2}=1\\Leftrightarrow x=\\pm {\\frac {1}{\\sqrt {2}}},y=\\pm {\\frac {1}{\\sqrt {2}}}} (è€å·åé ãx,yã¯å
±ã«å®æ°ã§ãããããæ¡ä»¶ãæºããã)",
"title": "ã³ã©ã "
},
{
"paragraph_id": 418,
"tag": "p",
"text": "x = â y {\\displaystyle x=-y} ã®ãšãã â 2 y 2 = 1 â y 2 = â 1 2 {\\displaystyle -2y^{2}=1\\Leftrightarrow y^{2}=-{\\frac {1}{2}}} ããã§ããããæºããå®æ°yã¯ååšããªãããäžé©ã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 419,
"tag": "p",
"text": "ãã£ãŠã z = ± ( 1 2 + 1 2 i ) {\\displaystyle z=\\pm \\left({\\frac {1}{\\sqrt {2}}}+{\\frac {1}{\\sqrt {2}}}i\\right)} â ",
"title": "ã³ã©ã "
},
{
"paragraph_id": 420,
"tag": "p",
"text": "",
"title": "ã³ã©ã "
},
{
"paragraph_id": 421,
"tag": "p",
"text": "å®éšããŒããèæ
®ã㊠x = 0 {\\displaystyle x=0} ã x = ± 3 y {\\displaystyle x=\\pm {\\sqrt {3}}y} ã ããèéšããŒããªã®ã§ãxã®å€ãåè
ã®ãšã y = â 1 {\\displaystyle y=-1} ãåŸè
ã®ãšã y = 1 / 2 {\\displaystyle y=1/2} ãšãªãããšãããã«ãããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 422,
"tag": "p",
"text": "2次æ¹çšåŒã«ã¯è§£ã®å
¬åŒããããæ¥æ¬ã®äžåŠãé«æ ¡ã§ãç¿ãã2次æ¹çšåŒã®è§£ã®å
¬åŒãçšããã°ãã©ããªä¿æ°ã®2次æ¹çšåŒã§ãã£ãŠã解ãæ±ããããã3次æ¹çšåŒãš4次æ¹çšåŒã«ãã解ã®å
¬åŒã¯ååšããä¿æ°ãã©ããªä¿æ°ã§ãã£ãŠã解ãæ±ããããããããã®è§£ã®å
¬åŒã¯ã代æ°æ¹çšåŒè«ã§è¿°ã¹ãŠããããã«ãä¿æ°ã«æéåã®ååæŒç®ãšæ ¹å·ããšãæäœã®çµã¿åããã§è¡šãããšãã§ããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 423,
"tag": "p",
"text": "5次æ¹çšåŒã§ã¯ã4次以äžã®æ¹çšåŒãšã¯ç¶æ³ãç°ãªããäžè¬ã®5次æ¹çšåŒã®è§£ã¯ã2次æ¹çšåŒã4次æ¹çšåŒã®ããã«ãä¿æ°ã«æéåã®ååæŒç®ãšæ ¹å·ããšãæäœã®çµã¿åããã§è¡šãããšãã§ããªãã®ã§ããããã ãããã§ããªããããšã®èšŒæã¯å®¹æã§ã¯ãªãããã®ããšã蚌æããã«ã¯ãã¬ãã¢çè«ãç解ããå¿
èŠããã(æ¥æ¬ã®å€§åŠã®æšæºçãªã«ãªãã¥ã©ã ã§ã¯ãçåŠéšæ°åŠç§ã®åŠçã®ã¿ã倧åŠ3幎çã§åŠã¶ã®ãäžè¬çãªçšåºŠã®çè«ã§ãã)ã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 424,
"tag": "p",
"text": "ãªããããã§èšããè¡šãããšãã§ããªãããšã¯äžè¬ã®æ¹çšåŒã«ã€ããŠã®ããšã§ãããç¹å¥ãª5次æ¹çšåŒã®å Žåã¯ç°¡åã«è§£ãæ±ãããããããšãã°ã x 5 â 32 = 0 {\\displaystyle x^{5}-32=0} ã¯è§£ã®ã²ãšã€ãšã㊠x = 2 {\\displaystyle x=2} ããã€ããšã¯ãããããããã®æ¹çšåŒã¯ä»ã®è§£ã«ã€ããŠãäžè§é¢æ°ãçšããŠç°¡åã«è¡šããããšãé«çåŠæ ¡æ°åŠC/è€çŽ æ°å¹³é¢ã«ãããŠåŠã¶ã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 425,
"tag": "p",
"text": "ãä¿æ°ã«æéåã®ååæŒç®ãšæ ¹å·ããšãæäœã®çµã¿åãããã«æããªããã°ãäžè¬ã®5次æ¹çšåŒã®è§£ãæ±ããæ¹æ³ãååšããããããé«åºŠãªæ°åŠãçšããå¿
èŠããããw:äºæ¬¡æ¹çšåŒã«èšè¿°ãããã®ã§èå³ã®ããèªè
ã¯åç
§ãããšããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 426,
"tag": "p",
"text": "é«çåŠæ ¡ã§è€çŽ æ°ãåºãŠããåéã¯ãã®åéãšæ°åŠCãå¹³é¢äžã®æ²ç·ãšè€çŽ æ°å¹³é¢ãã®ã¿ã§ãããè€çŽ æ°ã®åºæ¬èšç®ãæ¹çšåŒãè€çŽ æ°ç¯å²ã§è§£ãããšãè€çŽ æ°ã®å¹ŸäœåŠçæå³ã«ã€ããŠæ±ã£ãŠããããããã倧åŠæ°åŠã«ãããŠã¯ãé¢æ°ã®å®çŸ©åã»å€åãè€çŽ æ°ç¯å²ã«åºããŠèãããè€çŽ é¢æ°è«ããšãããã®ãæ±ãã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 427,
"tag": "p",
"text": "å®æ°ç¯å²ã§ã®é¢æ°ã¯x, yãšãã«äžæ¬¡å
ã®å®æ°è»žãæã€ãããå
¥åå€ãšåºåå€ã®æãã°ã©ããèããã«ã¯äºæ¬¡å
ã®åº§æšå¹³é¢ã§ååã§ãã£ããããããè€çŽ æ°ç¯å²ã§ã®é¢æ°ã¯x, yãšãã«äºæ¬¡å
ã®è€çŽ æ°å¹³é¢ãæã€ãããå
¥åå€ãšåºåå€ã®æãã°ã©ããèããã«ã¯å次å
ã®åº§æšç©ºéãå¿
èŠã§ãããäžæ¬¡å
空éã«äœãæã
ã«ã¯æç»ããããšãã§ããªãããã®ãããè€çŽ é¢æ°è«ã§ã¯é¢æ°ã®ã°ã©ããèããããšã¯åºæ¬çã«ãªãã(ãã ããåºåãããè€çŽ æ°ã®çµ¶å¯Ÿå€ãèããããšã«ãã£ãŠäžæ¬¡å
ã°ã©ãã«èœãšã蟌ãããšã¯å¯èœ)",
"title": "ã³ã©ã "
},
{
"paragraph_id": 428,
"tag": "p",
"text": "ã§ã¯äœãèããã®ããšãããšãè€çŽ é¢æ°ã®åŸ®åç©åã§ãããè€çŽ é¢æ°ã®åŸ®åã«é¢é£ããŠãæ£åé¢æ°ããšããçšèªãåºãŠããããè€çŽ é¢æ°è«ã¯ãã®æ£åé¢æ°ãšãããã®ã®æ§è³ªã調ã¹ãåŠåã ãšèšã£ãŠè¯ãã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 429,
"tag": "p",
"text": "è€çŽ é¢æ°è«ã¯ç©çåŠã®ç¹ã«æ³¢åã«é¢ããåé(é³ã»é»ç£æ°ãªã©)ã«ãããŠæŽ»èºããŠããããæ³¢åæ¹çšåŒãããã€ã³ããŒãã³ã¹ããšããèšèã¯æåã ããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 430,
"tag": "p",
"text": "ã¡ãªã¿ã«ãè€çŽ æ°ãããã«æ¡åŒµããæ°ãšããŠãw:åå
æ°ããšãããã®ãããããã®åå
æ°ã¯ãã¯ãã«ãè¡åãšæ·±ãé¢ãããååšããŠãããæ·±æããšé¢çœãã®ã ããããããåé·ã«ãªãããå²æããããªããåå
æ°ãããã«æ¡åŒµããå
«å
æ°ãåå
å
æ°ãšããæ°ãç 究ãããŠããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 431,
"tag": "p",
"text": "",
"title": "ã³ã©ã "
}
] | æ¬é
ã¯é«çåŠæ ¡æ°åŠIIã®åŒãšèšŒæã»é«æ¬¡æ¹çšåŒã®è§£èª¬ã§ããã | {{pathnav|é«çåŠæ ¡ã®åŠç¿|é«çåŠæ ¡æ°åŠ|é«çåŠæ ¡æ°åŠII|pagename=åŒãšèšŒæã»é«æ¬¡æ¹çšåŒ|frame=1|small=1}}
æ¬é
ã¯[[é«çåŠæ ¡æ°åŠII]]ã®åŒãšèšŒæã»é«æ¬¡æ¹çšåŒã®è§£èª¬ã§ããã
== åŒãšèšŒæ ==
=== äºé
å®ç ===
<math>(a+b)^5 = (a+b)(a+b)(a+b)(a+b)(a+b)</math> ã«ã€ããŠèãããããã®åŒãå±éãããšãã<math>a^2b^3</math> ã®ä¿æ°ã¯ãå³èŸºã®5åã® <math>(a+b)</math> ãã <math>a</math> ã3ååãçµã¿åããã«çãããã <math>_5\mathrm{C}_2 = 10</math> ã§ããã
ãã®èããæ¡åŒµããŠ
:<math>(a+b)^n = \underbrace{(a+b)(a+b)(a+b)\cdots(a+b)}_n</math>
ãå±éããã<math>a^rb^{n-r}</math>ã®é
ã®ä¿æ°ã¯ãå³èŸºã® <math>n</math> åã® <math>(a+b)</math> ãã <math>a</math> ã <math>r</math> ååãçµã¿åããã«çãããã <math>_n\mathrm{C}_r</math> ã§ããã
ãã£ãŠã次ã®åŒãåŸãããïŒ
:<math>\begin{align}(a+b)^n &= {}_n\mathrm{C}_0 a^n + {}_n\mathrm{C}_1 a^{n-1}b + {}_n\mathrm{C}_2 a^{n-2}b^2 + \cdots \\
&+ {}_n\mathrm{C}_r a^{n-r}b^r + \cdots + {}_n\mathrm{C}_n b^n \\
&= \sum _{r = 0}^n {}_n\operatorname{C}_r a^r b^{n-r}. \\ \end{align}</math>
æåŸã®åŒã¯[[é«çåŠæ ¡æ°åŠB/æ°å|æ°Bã®æ°å]]ã§åŠã¶ç·åèšå· <math>\Sigma</math> ã§ãããç¥ããªãã®ãªãç¡èŠããŠãè¯ãã
ãã®åŒã '''äºé
å®ç'''ïŒbinomial theoremïŒ ãšããããŸããããããã®é
ã«ãããä¿æ°ãäºé
ä¿æ°ïŒbinomial coefficientïŒ ãšåŒã¶ããšãããã
* åé¡äŸ
** åé¡
(I)
:<math>(x+1) ^4</math>
(II)
:<math>(a + 3) ^ 5</math>
(II)
:<math>(a + b) ^ 5</math>
ãããããèšç®ããã
**解ç
äºé
å®çãçšããŠèšç®ããã°ãããå®éã«èšç®ãè¡ãªããšã
(I)
:<math>x^4+4\,x^3+6\,x^2+4\,x+1</math>
(II)
:<math>a^5+15\,a^4+90\,a^3+270\,a^2+405\,a+243</math>
(III)
:<math>b^5+5\,a\,b^4+10\,a^2\,b^3+10\,a^3\,b^2+5\,a^4\,b+a^5</math>
ãšãªãã
** åé¡
ãã¹ãŠã®èªç¶æ°nã«å¯ŸããŠ
(I)
:<math>2^n = \sum _{k=0} ^n n\operatorname{C} _k </math>
(II)
:<math>3^n = \sum _{k=0} ^n 2^k n\operatorname{C} _k </math>
(III)
:<math>0 = \sum _{k=0} ^n (-1)^k n\operatorname{C} _k </math>
ãæãç«ã€ããšã瀺ãã
** 解ç
äºé
å®ç
:<math>(a+b)^n = \sum _{k = 0}^n {} _n\operatorname{C} _k a^k b^{n-k}</math>
ã«ã€ããŠa,bã«é©åœãªå€ã代å
¥ããã°ããã
(I)
a = 1,b=1ã代å
¥ãããšã
:<math>(1+1)^n = \sum _{k = 0}^n {} _n\operatorname{C} _k </math>
:<math>2^n = \sum _{k = 0}^n {} _n\operatorname{C} _k </math>
ãšãªã確ãã«äžããããé¢ä¿ãæç«ããããšãåããã
(II)
a=2,b=1ã代å
¥ãããšã
:<math>(1+2)^n = \sum _{k = 0}^n {} _n\operatorname{C} _k 2^k</math>
:<math>3^n = \sum _{k = 0}^n {} _n\operatorname{C} _k 2^k</math>
ãšãªã確ãã«äžããããé¢ä¿ãæç«ããããšãåããã
(III)
a=1,b=-1ã代å
¥ãããšã
:<math>(1-1)^n = \sum _{k = 0}^n {} _n\operatorname{C} _k (-1)^k</math>
:<math>0 = \sum _{k = 0}^n {} _n\operatorname{C} _k (-1)^k</math>
ãšãªã確ãã«äžããããé¢ä¿ãæç«ããããšãåããã
==== å€é
å®ç ====
äºé
å®çãæ¡åŒµã㊠<math>(a+b+c)^n</math> ãå±éããããšãèãããã<math>a^pb^qc^r</math> <math>(p+q+r = n)</math> ã®é
ã®ä¿æ°ã¯ <math>n</math> åã® <math>(a+b+c)</math> ãã <math>p</math> åã® <math>a</math>ã<math>q</math> åã® <math>b</math> ã <math>r</math> åã® <math>c</math> ãéžã¶[[é«çåŠæ ¡æ°åŠA/å Žåã®æ°ãšç¢ºç#çµã¿åãã|çµåã]]ã«çãããã <math>\frac{n!}{p!q!r!}</math> ã§ããã
=== æŽåŒã®é€æ³ãåæ°åŒ ===
ããã§ã¯ãæŽåŒã®é€æ³ãšåæ°åŒã«ã€ããŠæ±ããæŽåŒã®é€æ³ã¯ãæŽåŒãæŽæ°ã®ããã«æ±ãé€æ³ãè¡ãªãèšç®ææ³ã®ããšã§ãããå®éã«æŽæ°ã®é€æ³ãšæŽåŒã®é€æ³ã«ã¯æ·±ãã€ãªããããããæŽåŒã®å æ°å解ãèãããšãã以äžå æ°å解ã§ããªãæŽåŒãååšããããã®æŽåŒãæŽæ°ã§ããçŽ å æ°ã®ããã«æ±ãããšã§æŽåŒã®çŽ å æ°å解ãå¯èœã«ãªãã
äžã§ã¯ãæŽåŒãæŽæ°ã«å¯Ÿå¿ããæ§è³ªãæã€ããšãè¿°ã¹ããæŽæ°ã«ã€ããŠã¯ãããã«çŽ ãª2ã€ã®æŽæ°ãåãããšã§æçæ°ãå®çŸ©ããããšãåºæ¥ããæŽåŒã«å¯ŸããŠãåãäºãæç«ã¡ããã®ãããªåŒãåæ°åŒãšåŒã¶ã
==== æŽåŒã®é€æ³ ====
åæ°ãçšããªããšãã«ã¯ãæŽæ°ã®é€æ³ã¯åãšäœããçšããŠå®çŸ©ãããããã®æãå²ãããæ°Bã¯åDãšå²ãæ°AãäœãRãçšããŠ
:<math>
B = AD + R
</math>
ã®æ§è³ªãæºããããšãç¥ãããŠãããæŽåŒã«å¯ŸããŠã䌌ãæ§è³ªãæç«ã¡ãå²ãããåŒB(x)ãåD(x)ãšå²ãåŒA(x)ãäœãR(x)ãçšããŠã
:<math>
B(x) = A(x)D(x) + R(x)
</math>ãšæžããããšããB(x)ããA(x)ã«å²ããããšããããã®æãæŽæ°ã®é€æ³ã®æ§è³ªR<Aã«å¯Ÿå¿ããŠãR(x)ã®æ¬¡æ°<A(x)ã®æ¬¡æ°ãæç«ãããå
·äœäŸãšããŠãx +1ããxã§å²ãããšãèãããå²ãåŒã®æ¬¡æ°ã1ã§ããããšããäœãã®æ¬¡æ°ã¯0ãšãªãäœãã¯å®æ°ã§ããå¿
èŠãããããŸããåãxã®é¢æ°ã§ãããš
:<math>
B(x) = A(x)D(x) + R(x)
</math>
ã®å³èŸºã§xã«ã€ããŠ2次ã®é
ãçŸãã巊蟺ãšäžèŽããªããªãããã£ãŠåã¯å®æ°ã§ãããåãaãäœããrãšãããšäžã®åŒã¯ã
:<math>
x+1 = ax + r
</math>
ãšãªãããããã¯a=1,r=1ã§æç«ããããã£ãŠå1,äœã1ã§ãããããé«æ¬¡ã®åŒã«å¯ŸããŠãåãæ§ã«çããå®ããŠããã°ãããäŸãšããŠã
:<math>
x^3 \div (x^2 -1)
</math>
ã®ãããªåŒãèããããã®å Žåã
:<math>
B(x) = A(x)D(x) + R(x)
</math>
ã§ãB(x)ã3次ãA(x)ã2次ã§ããããšãããD(x)ã¯1次ã§ããããŸããR(x)ã¯2次ããå°ããããšãã1次以äžã®åŒã«ãªããããã§ãD(x)=ax+b,R(x)=cx+dãšãããšã
:<math>
x^3 = (x^2-1) (ax+b) + (cx +d)
</math>
ãåŸããããå³èŸºãå±éãããšã
:<math>
x^3 = ax^3 + b x^2 + (-a +c )x + (-b +d)
</math>
ãåŸãããããxã«ã©ããªå€ãå
¥ããŠããã®çåŒãæãç«ããªããã°ãªããªãã®ã§ãa = 1, b = 0, -a +c = 0, -b +d = 0ãåŸãããçµå±a=c=1, b=d=0ãåŸãããã
<!--
<math>
x^3 \div (x^2 -1)
</math>
ã®ãããªåŒãèããã
ãã®åŒã«ã€ããŠã
<math>
x^3 = x(x^2 - 1) +x
</math>
ãšæžãããšãåºæ¥ããããã㯠<math>x^3</math> ã <math>x^2-1</math> ã§å²ã£ãçµæã
å<math>x</math> ,äœã <math>x</math> ãã§ããã®ãšè§£éã§ããã
ãã®ããã«ãæŽåŒã©ããã§å²ãç®ãããããšãåºæ¥ãã
ãã®ãšããå²ãåŒã¯å²ãããåŒããäœæ¬¡ãåã次æ°ã§ãªããŠã¯ãªããªãã
ãŸããäœãã¯å¿
ãå²ãåŒãããäœæ¬¡ã®åŒã«ãªãã
-->
ãã®æ¹æ³ã¯ã©ã®é€æ³ã«å¯ŸããŠãçšããããšãåºæ¥ããã次æ°ãé«ããªããšèšç®ãé£ãããªããæŽæ°ã®å Žåãšåæ§ãæŽåŒã®é€æ³ã§ãçç®ãçšããããšãåºæ¥ããäžã®äŸãçšããŠçµæã ããæžããšã
*å³
ã®ããã«ãªãã)å³ã«æžãããåŒãå²ãããåŒã§ããã)å·Šã«æžãããåŒãå²ãåŒã§ããã--ã®äžçªäžã«æžãããåŒã¯åã§ãããæŽæ°ã®å²ãç®åæ§å·Šã«æžãããæ°ããé ã«å²ã£ãŠãããããã§ã¯æ¬¡æ°ã倧ããé
ãããå
ã«èšç®ãããé
ã§ãããå²ãããåŒã®äžã«ããåŒã¯åã®ç¬¬1é
ãå²ãåŒã«ãããŠåŸãåŒã§ãããããã§ã¯ã<math>x(x^2-1)</math>ã§ã<math>x^3-x</math>ãšãªãããã ããæŽæ°ã®é€æ³ãšåæ§ãäœãããããªããŠã¯ãªããªãããã®åŸãå²ãããåŒãã<math>x^3-x</math>ãåŒããæ®ã£ãåŒãæ°ããå²ãããåŒãšããŠæ±ããããã§ã¯ãåŸãåŒãå²ãåŒãããäœæ¬¡ã§ããããšãããããã§èšç®ã¯çµäºã§ããã
*åé¡äŸ
**åé¡
:
<math>x^3 + 2x ^2 +1</math>ã<math>x ^4 + 4x^2 +3x +2</math>ãã<math>x^2 +2x +6
</math>ã§å²ã£ãåãšäœããæ±ããã
<!--
æŽã«ã
(I)
:<math>
(x ^4 + 2x^3 - 5x^2 +6x -1) \div (x^2 -5x -1 )
</math>
(II)
:<math>
(3 x ^4 - 7x^3 + x^2 +2x -1) \div (x^2 -3x -4 )
</math>
(III)
:<math>
(2x^5 +3 x ^4 - 7x^3 + x^2 +2x -1) \div (x^2 +7x -4 )
</math>
(IV)
:<math>
(2x^5 +3 x ^4 - 7x^3 + x^2 +2x -1) \div (x^3 +4x^2 +7x -4 )
</math>
ãèšç®ããã
åé¡ãå€ãã®ã§ããšããããã³ã¡ã³ãã¢ãŠãã
-->
** 解ç
ãã®èšç®ã¯ã¢ãã¡ãŒã·ã§ã³ã䜿ã£ãŠ
詳ãã衚瀺ãããŠãããèšç®ææ³ã¯ã
æŽæ°ã®å Žåã®çç®ãšåããããªææ³ã䜿ããã
[[ç»å:Fract.gif|frame|right|èšç®ã®ã¢ãã¡ãŒã·ã§ã³]]
:<math>
x^3 + 2x ^2 +1
=
(x^2 +2x +6) x +(1-6x)
</math>
ãåŸãããã®ã§ãå<math>
x</math>ãäœã<math>-6x +1</math>ã§ããã
2ã€ç®ã®åŒã«ã€ããŠã¯ã
:<math>
x ^4 + 4x^2 +3x +2
=
(x^2 - 2x+2)* (x^2 +2x +6)
+ 11x -10
</math>
ãåŸãããã
ãã£ãŠãçã¯
å<math>x^2 - 2x+2</math>ãäœã<math>11x -10</math>ã§ããã
<!--
æŽã«ãæ®ãã®èšç®çµæã¯ã
(I)
:<math>
\left[ x^2+7\,x+31,168\,x+30 \right]
</math>
(II)
:<math>
\left[ 3\,x^2+2\,x+19,67\,x+75 \right]
</math>
(III)
:<math>
\left[ 2\,x^3-11\,x^2+78\,x-589,4437\,x-2357 \right]
</math>
(IV)
:<math>
\left[ 2\,x^2-5\,x-1,48\,x^2-11\,x-5 \right]
</math>
ãåŸãããã
ãã ããå·Šãåãå³ãäœããšãªã£ãŠããã
-->
==== åæ°åŒ ====
ãããŸã§ã§æŽåŒãæŽæ°ã®ããã«æ±ããæŽåŒã®é€æ³ãè¡ãªãæ¹æ³ã«ã€ããŠè¿°ã¹ããããã§ã¯ãæŽåŒã«å¯ŸããŠåæ°åŒãå®çŸ©ããæ¹æ³ã«ã€ããŠè¿°ã¹ããåæ°åŒãšã¯ãæŽæ°ã«å¯Ÿããåæ°ã®ããã«ãé€æ³ã«ãã£ãŠçããåŒã§ãããããã§ãé€æ³ãè¡ãªãåŒã¯ã©ã®ãããªãã®ã§ãå·®ãæ¯ããªããåæ°åŒã§ã¯ãååã«å²ãããåŒãæžããåæ¯ã«å²ãåŒãæžããäŸãã°ã
:<math>
\frac {x+1}{x^2+4}
</math>
ã¯ãååx+1ãåæ¯<math>x^2+4</math>ã®åæ°åŒã§ãããåæ°åŒã«å¯ŸããŠãçŽåãéåãååšãããçŽåã¯å
±éå æ°ãæã£ãåååæ¯ããã€åæ°åŒã§çšããããããã®æã«ã¯åååæ¯ãå
±éå æ°ã§å²ããåŒãç°¡åã«ããããšãåºæ¥ããéåã¯ãåæ°åŒã®å æ³ã®æã«ããçšããããããåååæ¯ã«åãæŽåŒããããŠãåæ°åŒãå€åããªãæ§è³ªãçšããã
* åé¡äŸ
** åé¡
:<math>
\frac {x^2 -1} {x^3 -1}
</math>
ãç°¡åã«ããããŸãã
:<math>
\frac {x+1}{x^2 +2x + 3}
+ \frac {2x + 5} {x^2 +1}
</math>
ãèšç®ããã
** 解ç
:<math>
\frac {x^2 -1} {x^3 -1}
</math>
ã«ã€ããŠååãšåæ¯ãå æ°å解ãããšãåæ¹ãšãã«
:<math>
x-1
</math>
ãå æ°ãšããŠå«ãã§ããããšãåããããã®ãšããå
±éã®å æ°ã¯çŽåããããšãå¿
èŠã§ãããèšç®ãããå€ã¯ã
:<math>
\frac {x^2 -1} {x^3 -1}
</math>
:<math>
= \frac{(x-1)(x+1)}{(x-1)(x^2+x+1)}
</math>
:<math>
= \frac{x+1} { x^2+x+1}
</math>
ãšãªãã
次ã®åé¡ã§ã¯ã
:<math>
\frac {x+1}{x^2 +2x + 3}
+ \frac {2x + 5} {x^2 +1}
</math>
ãèšç®ããããã®ãšãã䞡蟺ã®åæ¯ãããããå¿
èŠãããããä»åã«ã€ããŠã¯ãåçŽã«ããããã®åæ°åŒã®ååãšåæ¯ã«åã
ã®åæ¯ããããŠåæ¯ãçµ±äžããã°ãããèšç®ãããšã
:<math>
\frac {x+1}{x^2 +2x + 3}
+ \frac {2x + 5} {x^2 +1}
</math>
:<math>
= \frac{(x+1)(x^2+1)}{(x^2 +2x + 3)(x^2+1)}
+\frac{(x^2 +2x + 3)(2x + 5)}{(x^2 +2x + 3)(x^2+1)}
</math>
:<math>
= \frac{(x+1)(x^2+1)+(x^2 +2x + 3)(2x + 5)}
{(x^2 +2x + 3)(x^2+1)}
</math>
:<math>
= \frac {3x^3 +10x^2 + 17 x + 16}
{(x^2 +2x + 3)(x^2+1)}
</math>
ãšãªãã
åæ°åŒã®ä¹æ³ã¯ãåååæ¯ãå¥ã
ã«ãããã°ããã
* åé¡äŸ
** åé¡
次ã®èšç®ãããã
(I)
:<math>
\frac {x^2 - y^2} {x^2 - 2xy + y^2} \times \frac {x-y} {x^2 + xy}
</math>
(II)
:<math>
\frac {x^2 + 4x + 3}{x^2 - 6x + 9} \div \frac {x^2 - 3x - 4} {x^2 - x - 6}
</math>
** 解ç
(I)
:<math>
\frac {x^2 - y^2} {x^2 - 2xy + y^2} \times \frac {x-y} {x^2 + xy}
</math>
:<math>
= \frac {(x+y)(x-y)} {(x-y)^2} \times \frac {x-y} {x(x+y)}
</math>
:<math>
= \frac {(x+y)(x-y)(x-y)} {(x-y)^2\ x(x+y)}
</math>
:<math>
= \frac {1} {x}
</math>
(II)
:<math>
\frac {x^2 + 4x + 3}{x^2 - 6x + 9} \div \frac {x^2 - 3x - 4} {x^2 - x - 6}
</math>
:<math>
= \frac {x^2 + 4x + 3}{x^2 - 6x + 9} \times \frac {x^2 - x - 6} {x^2 - 3x - 4}
</math>
:<math>
= \frac {(x+1)(x+3)} {(x-3)^2} \times \frac {(x+2)(x-3)} {(x+1)(x-4)}
</math>
:<math>
= \frac {(x+1)(x+3)(x+2)(x-3)} {(x-3)^2\ (x+1)(x-4)}
</math>
:<math>
= \frac {(x+3)(x+2)} {(x-3)(x-4)}
</math>
===== éšååæ°å解 =====
åæ¯ãç©ã®åœ¢ã§ããåæ°åŒãäºã€ã®åæ°åŒã®åãå·®ã§è¡šãããåŒã«å€åœ¢ããæäœã'''éšååæ°å解'''ãšããã
*åé¡äŸ
<Math> \frac{1}{x (x+1)} </Math>ãš<Math>\frac{1}{(x+1)(x+3)}</Math>ãåæ°åŒã®åãŸãã¯å·®ã®åœ¢ã§è¡šãã
*解ç
:<Math>\frac{1}{x(x+1)} = \frac{(x+1)-x}{x(x+1)}</Math>
ãšå€åœ¢ã§ããã®ã§ã
:<Math>\frac{x+1}{x(x+1)} - \frac{x}{x(x+1)}</Math>
ãšãªããçŽåãããš
:<Math>\frac{1}{x} - \frac{1}{x+1}</Math>
ãšãªãã
次ã®åé¡ã§ã¯ã
:<Math>\frac{1}{(x+1)(x+3)} = \frac{1}{(x+3) - (x+1)} \cdot \frac{(x+3) - (x+1)}{(x+1)(x+3)}</Math>
ãšå€åœ¢ããããšã«ãã£ãŠã
:<Math>\frac{1}{2} \{ \frac{x+3}{(x+1)(x+3)} - \frac{x+1}{(x+1)(x+3)} \}</Math>
ãšãªãã
:<Math>\frac{1}{2} (\frac{1}{x+1} - \frac{1}{x+3}) </Math>
ãšæ±ãŸãã
éšååæ°å解ã®æäœãéã«èŸ¿ããšãåæ°åŒã®éåã®æäœãšäžèŽããã
ã€ãŸãã'''éšååæ°å解ã¯éåã®éã®æäœ'''ã§ããã
ååãå®æ°ã®å Žåã«ã¯ãäžãšåæ§ã®æ¹æ³ã§éšååæ°å解ããããšãã§ããã
*åé¡
**以äžã®åæ°åŒãéšååæ°å解ãã
**#<Math>\frac{3}{(x-9)(x-4)}</Math>
**#<Math>\frac{7}{(3x-1)(5-2x)}</Math>
*解ç
1. <Math>\frac{3}{(x-9) (x-4)} </Math>
:<Math>= \frac{3}{(x-4) - (x-9)} \cdot \frac{(x-4) - (x-9)}{(x-9)(x-4)}</Math>
:<Math>= \frac{3}{5}\{ \frac{x-4}{(x-9)(x-4)} - \frac{x-9}{(x-9)(x-4)} \}</Math>
:<Math>= \frac{3}{5} ( \frac{1}{x-9} - \frac{1}{x-4} )</Math>
2. <Math>\frac{7}{(3x-1)(5-2x)}</Math>
:<Math>= \frac{-7}{(3x-1)(2x-5)} </Math>
:<Math>= \frac{-7}{(3x-1) - (2x-5)} \cdot \frac{(3x-1) - (2x-5)}{(3x-1)(2x-5)} </Math>
:<Math>= \frac{-7}{2(3x-1) - 3(2x-5)} \cdot \frac{2(3x-1) - 3(2x-5)}{(3x-1)(2x-5)} </Math>
:<Math>= \frac{-7}{(6x-2) - (6x-15)} \{ \frac{2(3x-1)}{(3x-1)(2x-5)} - \frac{3(2x-5)}{(3x-1)(2x-5)} \}</Math>
:<Math>= - \frac{7}{13} (\frac{2}{2x-5} - \frac{3}{3x-1})</Math>
:<Math>= \frac{7}{13} (\frac{3}{3x-1} - \frac{2}{2x-5})</Math>
éšååæ°å解ã¯æ°åã®åã®èšç®ãç©åèšç®ã埮åãå©çšããäžçåŒã®èšŒæçã«åœ¹ç«ã€ãéèŠãªå€åœ¢ã§ããã
=== åŒã®èšŒæ ===
==== æçåŒ ====
çåŒ <math>(a+b)^2=a^2+2ab+b^2</math>ã¯ãæå<math>a,b</math>ã«ã©ã®ãããªå€ã代å
¥ããŠãæãç«ã€ããã®ãããªçåŒã'''æçåŒ'''ïŒãããšãããïŒãšããã
çåŒ<math>\frac {1}{x-1} + \frac {1}{x+1} = \frac {2x}{x^2-1}</math>ã¯ã䞡蟺ãšã<math>x=1,-1</math>ã代å
¥ããããšã¯ã§ããªããããã®ä»ã®å€ã§ããã°ä»£å
¥ããããšãã§ãããŸãã©ã®ãããªå€ã代å
¥ããŠãçåŒãæãç«ã£ãŠããããããæçåŒãšåŒã¶ã
ãã£ãœãã<math>x^2 - x - 2 = 0</math> ã¯ãxïŒ2 ãŸã㯠xïŒãŒ1 ã代å
¥ãããšãã ãæãç«ã€ãããã®ããã«æåã«ç¹å®ã®å€ã代å
¥ãããšãã«ã ãæãç«ã€åŒã®ããšãæ¹çšåŒãšåŒã³ãæçåŒãšã¯åºå¥ããã
çåŒ <math>ax^2+bx+c=0</math> ã <math>x</math> ã«ã€ããŠã®æçåŒã§ããã®ã¯ã©ã®ãããªå ŽåããèããŠã¿ããã
ããåŒãã <math>x</math> ã«ã€ããŠã®æçåŒã§ããããšã¯ããã®åŒã®<math>x</math> ã«ã©ã®ãããªå€ã代å
¥ããŠãããã®çåŒã¯æãç«ã€ãšããæå³ã§ããããªã®ã§ãäŸãã° <math>x</math> ã«<math>-1\ ,\ 0\ ,\ 1</math> ã代å
¥ããåŒ
:<math>a-b+c=0</math>
:<math>c=0</math>
:<math>a+b+c=0</math>
ã¯ãã¹ãŠæãç«ã€å¿
èŠããããããã解ããš
:<math>a=b=c=0</math>
ãªã®ã§ãçåŒ <math>ax^2+bx+c=0</math> ã <math>x</math> ã«ã€ããŠã®æçåŒã«ãªããªãã°ã<math>a=b=c=0</math>ã§ãªããã°ãªããªãããšããããã
äžè¬ã«ãçåŒ <math>ax^2+bx+c=a'x^2+b'x+c'</math> ãæçåŒã§ããããšãšã<math>(a-a')x^2+(b-b')x+(c-c')=0</math> ãæçåŒã§ããããšãšåãã§ããã<br>
ãã£ãŠ
:<math>ax^2+bx+c=a'x^2+b'x+c'</math> ã<math>x</math>ã«ã€ããŠã®æçåŒ ã<math>\Leftrightarrow </math>ã <math>a=a'</math> ã〠<math>b=b'</math> ã〠<math>c=c'</math>
ãŸãšãããšæ¬¡ã®ããã«ãªãã
{| style="border:2px solid yellow;width:fit-content" cellspacing=0
|style="background:yellow"|'''æçåŒã®æ§è³ª'''
|-
|style="padding:5px"|
<math>P\ ,\ Q</math> ã <math>x</math> ã«ã€ããŠã®å€é
åŒãŸãã¯åé
åŒãšããã
::<math>P=0</math> ãæçåŒ ã<math>\Leftrightarrow </math> ã <math>P</math>ã®åé
ã®ä¿æ°ã¯ãã¹ãŠ<math>0</math>ã§ããã
::<math>P=Q</math> ãæçåŒ ã<math>\Leftrightarrow </math> ã <math>P</math>ãš <math>Q</math> ã®æ¬¡æ°ã¯çããã䞡蟺ã®åã次æ°ã®é
ã®ä¿æ°ã¯ãããããçããã
|}
* åé¡äŸ
** åé¡
次ã®çåŒã <math>x</math> ã«ã€ããŠã®æçåŒãšãªãããã«ã<math>a\ ,\ b\ ,\ c</math> ã®å€ãæ±ããã
:<math>x^2-3=a(x-1)^2+b(x-1)+c</math>
** 解ç
çåŒã®å³èŸºã <math>x</math> ã«ã€ããŠæŽçãããš
:<math>a(x-1)^2+b(x-1)+c=ax^2-2ax+a+bx-b+c=ax^2+(-2a+b)x+(a-b+c)</math>
:<math>x^2-3=ax^2+(-2a+b)x+(a-b+c)</math>
ãã®çåŒã <math>x</math> ã«ã€ããŠã®æçåŒãšãªãã®ã¯ã䞡蟺ã®åã次æ°ã®é
ã®ä¿æ°ãçãããšãã§ããããã£ãŠ
:<math>a=1</math>
:<math>-2a+b=0</math>
:<math>a-b+c=-3</math>
ããã解ããš
:<math>a=1\ ,\ b=2\ ,\ c=-2</math>
; '''è€éãªéšååæ°å解'''ïŒçºå±ïŒ
æçåŒãå©çšããããšã§ãè€éãªåæ°åŒã®éšååæ°å解ãã§ããã
*åé¡äŸ
**以äžã®åæ°åŒãéšååæ°å解ãã
**#<Math>\frac{3x-5}{(x+2)(2x-1)}</Math>
**#<Math>\frac{1}{(x-1)^2 (x-2)}</Math>
*解ç
:<Math>\frac{3x-5}{(x+3)(2x-1)} = \frac{a}{2x-1} + \frac{b}{x+3}</Math>
ãšããã
åæ¯ãæã£ãŠ
:<Math>3x-5 = a(x+3) + b(2x-1)</Math>
ããªãã¡
:<Math>3x-5 =(a+2b)x + (3a-b) </Math>
ããã<Math>x</Math>ã®æçåŒãªã®ã§ãä¿æ°ãæ¯èŒããŠ
:<Math>a+2b=3</Math>ãã€<Math>3b-a=-5</Math>
ããªãã¡
:<Math>a=-1, b=2</Math>
æåã®çåŒã«ä»£å
¥ããŠã
:<Math>\frac{3x-5}{(x+3)(2x-1)} = \frac{-1}{2x-1} + \frac{2}{x+3}</Math>
:<Math>= \frac{2}{x+3} - \frac{1}{2x-1}</Math>
次ã®åé¡ã¯ã
:<Math>\frac{1}{(x-1)^2 (x-2)} = \frac{a}{x-1} + \frac{b}{(x-1)^2} + \frac{c}{x-2}</Math>
ãšããããšã«ãããäžã®åé¡ãšåæ§ã«ããŠ
:<Math>a=-1, b=-1, c=1</Math>
ãšæ±ãŸãã®ã§ã
:<Math>\frac{1}{(x-1)^2 (x-2)} = - \frac{1}{x-1} - \frac{1}{(x-1)^2} + \frac{1}{x-2}</Math>
:<Math>= \frac{1}{x-2} - \frac{1}{x-1} - \frac{1}{(x-1)^2}</Math>
'''æçåŒãå©çšããéšååæ°å解'''
æ±ãããæ°åã<Math>a,b,c</Math>ãšããã
1. <Math>\frac{px+q}{(x+m)(x+n)} = \frac{a}{x+m} + \frac{b}{x+n}</Math>
2. <Math>\frac{px+q}{(x+m)^2} = \frac{a}{x+m} + \frac{b}{(x+m)^2}</Math>
3. <Math>\frac{px^2 + qx + r}{(x+m)^2 (x+n)} = \frac{a}{x+m} + \frac{b}{(x+m)^2 } + \frac{c}{x+n}</Math>
4. <Math>\frac{px^2 + qx + r}{(x+m)(x^2 + nx + l)} = \frac{a}{x+m} + \frac{bx+c}{x^2 + nx + l}</Math>
ãã®ããã«ãããåŒã<Math>x</Math>ã®æçåŒãšèŠãããšã«ãã£ãŠã<Math>a,b,c</Math>ãæ±ããããéšååæ°å解ãã§ããã
; '''2ã€ã®æåã«ã€ããŠã®æçåŒ'''ïŒçºå±ïŒ
*äŸ
a~fãå®æ°ãšããã
<Math>ax^2+by^2+cxy+dx+ey+f=0</Math>ãx, yã«ã€ããŠã®æçåŒã ãšããã
巊蟺ãxã«ã€ããŠæŽçãããšã<Math>ax^2+(cy+d)x+(by^2+ey+f)=0</Math>ã§ããã
ãããxã«ã€ããŠã®æçåŒãªã®ã§ã<Math>a=0, cy+d=0, by^2+ey+f=0</Math>ãæãç«ã€ã
ãããã¯æŽã«yã«ã€ããŠã®æçåŒãªã®ã§ã以äžã®çåŒãåŸãããã
:<Math>a=b=c=d=e=f=0</Math>
éã«ããããæãç«ãŠã°å
ã®åŒã¯æããã«x, yã«ã€ããŠã®æçåŒã§ãã
**åé¡
<Math>x^2+axy+6y^2-x+5y+b = (x-2y+c)(x-3y+d)</Math>ãx,yã«ã€ããŠã®æçåŒãšãªãããã«a,b,c,dãå®ããã
==== çåŒã®èšŒæ ====
ããã»ã©çŽ¹ä»ãããæçåŒããšããèšèã䜿ã£ãŠã蚌æãã®æå³ã説æãããªãããçåŒã蚌æããããšã¯ããã®åŒãæçåŒã§ããããšã瀺ãããšã§ããã
äžè¬ã«ãçåŒ AïŒB ã蚌æããããã«ã¯ã次ã®ãããªæé ã®ãããããå®è¡ããã°ããã
:(1)ããAãåŒå€åœ¢ããŠBãå°ããããŸã㯠Bãå€åœ¢ããŠAãå°ãã
:(2)ããA,Bãããããå€åœ¢ããŠãåãåŒCãå°ãã
:(3)ããA-BïŒ0 ã瀺ãã
ãã®ãšããå€åœ¢ã¯åå€å€åœ¢ã§ãªããã°ãªããªãããšã«æ³šæã
* äŸé¡ 1
<math>
(a+b)^2-(a-b)^2 = 4ab
</math>
ãæãç«ã€ããšã蚌æããã
ïŒèšŒæïŒ<br>
巊蟺ãå±éãããšã
:(巊蟺)ïŒ<math>
(a^2+2ab+b^2)-(a^2-2ab+b^2) = a^2+2ab+b^2 - a^2+2ab-b^2=4ab
</math>
ãšãªããããã¯å³èŸºã«çããããã£ãŠãçåŒ <math>
(a+b)^2-(a-b)^2 = 4ab
</math> ã¯èšŒæããããïŒçµïŒ
----
* äŸé¡ 2
<math>
(x+y)^2+(x-y)^2 = 2(x^2+y^2)
</math>
ãæãç«ã€ããšã蚌æããã
:ïŒèšŒæïŒ
巊蟺ãèšç®ãããšã
:ïŒå·ŠèŸºïŒ ïŒ <math> (x^2+2xy+y^2)+(x^2-2xy+y^2) = x^2+2xy+y^2 + x^2-2xy+y^2 = 2x^2+2y^2 =2(x^2+y^2) </math>
ããã¯å³èŸºã«çããããã£ãŠçåŒãæãç«ã€ããšã蚌æããããïŒçµïŒ
----
* åé¡äŸ
** åé¡
次ã®çåŒãæãç«ã€ããšã蚌æããã<br>
(I)
:<math>
(6 a + 7 b )^2 + (7 a - 6 b )^2 = (9 a + 2 b )^2 + (2 a - 9 b )^2
</math>
**解ç
(I)<br>
(巊蟺)<math>
= (36 a^2 + 84 a b + 49 b^2) + (49 a^2 - 84 a b + 36 a^2) = 85 a^2 + 85 b^2
</math><br>
(å³èŸº)<math>
= (81 a^2 + 36 a b + 4 b^2) + (4 a^2 - 36 a b + 81 b^2) = 85 a^2 + 85 b^2
</math><br>
䞡蟺ãšãåãåŒã«ãªããã
:<math>
(6 a + 7 b )^2 + (7 a - 6 b )^2 = (9 a + 2 b )^2 + (2 a - 9 b )^2
</math>
æçåŒã§ãªããšãããäžããããæ¡ä»¶ããçåŒã蚌æããããšãã§ããã
*åé¡äŸ
**<Math>a=b=c=0</Math>ã®ãšãã<Math>a^3+b^3+c^3=3abc</Math>ã§ããããšã蚌æããããŸãã<Math>a:b=c:d</Math>ã®ãšãã<Math>\frac{a+c}{b+d} = \frac{a-c}{b-d}</Math>ã蚌æããã
**解ç
:<Math>a+b+c=0 \iff c=-(a+B)</Math>
ããã
:<Math>a^3+b^3+c^3-3abc = a^3+b^3-(a+b)^3+3ab(a+b)</Math>
:<Math>= a^3+b^3-(a^3+3a^2b+3ab^2+b^3)+3a^2b+3ab^2</Math>
:<Math>=0</Math>
ãã£ãŠã<Math>a^3+b^3+c^3=3abc</Math>ã§ããã
ãŸãã
:<Math>a:b=c:d \iff \frac{a}{b} = \frac{c}{d}</Math>
ãããäžåŒã®å³èŸºãkãšãããšã
:<Math>a=bk, c=dk</Math>
ãªã®ã§ã
:<Math>\frac{a+c}{b+d} = \frac{bk+dk}{b+d} = \frac{k(b+d)}{b+d} = k</Math>
:<Math>\frac{a-c}{b-d} = \frac{bk-dk}{b-d} = \frac{k(b-d)}{b-d} = k</Math>
ãã£ãŠã<Math>\frac{a+c}{b+d} = \frac{a-c}{b-d}</Math>ã§ããã
ãªããæ¯<Math>a:b</Math>ã«ã€ããŠ<Math>\frac{a}{b}</Math>ã'''æ¯ã®å€'''ãšããããŸãã<Math>a:b=c:d \iff \frac{a}{b} = \frac{c}{d}</Math>ã'''æ¯äŸåŒ'''ãšããã
<Math>\frac{a}{x} = \frac{b}{y} = \frac{c}{z}</Math>ãæãç«ã€ãšãã<Math>a:b:c=x:y:z</Math>ãšè¡šããããã'''é£æ¯'''ãšããã
*åé¡
**<Math>a:b:c=1:2:3</Math>ã®ãšãã<Math>a+b+c=24</Math>ãæºãã<Math>a,b,c</Math>ãæ±ããã
==== äžçåŒã®èšŒæ ====
äžçåŒã®ããŸããŸãªå
¬åŒã«ã€ããŠã¯ã次ã®4ã€ã®åŒãåºæ¬çãªåŒãšããŠå°åºã§ããå Žåãããããã
é«æ ¡æ°åŠã§ã¯ã次ã®4ã€ã®æ§è³ªã äžçåŒã®ãåºæ¬æ§è³ªããªã©ãšããŠçŽ¹ä»ãããŠããã
{| style="border:2px solid skyblue; width:fit-content" cellspacing=0
|style="background:skyblue"|'''äžçåŒã®åºæ¬æ§è³ª'''
|-
|style="padding:5px"|
:(1)ãã<math> a>b </math> ã〠<math> b>c </math> ãªãã° <math> a>c </math>
:(2)ãã<math> a>b </math> ãªãã° <math> a+c>b+c </math> ã〠<math> a-c>b-c </math>
:(3)ãã<math> a>b </math> ã〠<math> c>0 </math> ãªãã° <math> ac>bc </math> ã§ããã<math> \frac{a}{c} > \frac{b}{c} </math>ã§ããã
:(4)ãã<math> a>b </math> ã〠<math> c<0 </math> ãªãã° <math> ac<bc </math> ã§ããã<math> \frac{a}{c} < \frac{b}{c} </math>ã§ããã
|}
(3)ãš(4)ã«ã€ããŠã¯ãã²ãšã€ã®æ§è³ªãšã㊠ãŸãšããŠããæ€å®æç§æžãããïŒâ» åæ通ãªã©ïŒã
æ°åŠIAã§ç¿ã£ãããªãã°ãã®æå³ã®èšå· <math>\Longrightarrow </math> ã䜿ããšã
{| style="border:2px solid skyblue; width:fit-content" cellspacing=0
|style="background:skyblue"|'''äžçåŒã®åºæ¬æ§è³ª'''
|-
|style="padding:5px"|
:(1)ãã<math> a>b </math> ã〠<math> b>c </math> ã<math>\Longrightarrow </math>ã <math> a>c </math>
:(2)ãã<math> a>b </math> <math>\Longrightarrow </math> <math> a+c>b+c </math> ããã€ã<math> a-c>b-c </math>
:(3)ãã<math> a>b </math> ã〠<math> c>0 </math> ã<math>\Longrightarrow</math>ã <math> ac>bc </math> ã§ããã<math> \frac{a}{c} > \frac{b}{c} </math>ã§ããã
:(4)ãã<math> a>b </math> ã〠<math> c<0 </math> ã<math>\Longrightarrow</math>ã <math> ac<bc </math> ã§ããã<math> \frac{a}{c} < \frac{b}{c} </math>ã§ããã
|}
ãšãæžããã
äžè¿°ã®4ã€ã®åºæ¬æ§è³ªããã
:a>0, ãb>0 ãªãã° aïŒb ïŒ 0
ã蚌æããŠã¿ããã
ïŒèšŒæïŒ
ãŸã a>0 ãªã®ã§ãåºæ¬æ§è³ª(2)ãã
:aïŒb > b
ã§ããã
ãã£ãŠã
:<math> a+b>b </math> ã〠<math> b>0 </math>
ãªã®ã§ãåºæ¬æ§è³ª(1)ãã<math> a+b>0 </math>
ãæãç«ã€ãïŒçµïŒ
åæ§ã«ããŠã
:aïŒ0, ãbïŒ0 ãªãã° aïŒb ïŒ 0
ã蚌æã§ããã
::ïŒâ» èªè
ã¯èªå㧠ããã蚌æããŠã¿ããæ€å®æç§æžã«ãããã®åŒã®èšŒæã¯çç¥ãããŠãããïŒ
ãããŸã§ã«ç€ºããããšãããäžçåŒ <math> A \geqq B </math> ã蚌æãããå Žåã«ã¯ã
: <math> A-B \geqq 0 </math>
ã蚌æããã°ããããšãããã£ãããã¡ãã®æ¹ã蚌æããããå Žåãããããã
äžçåŒã蚌æããéã«æ ¹æ ãšããåºæ¬çãªäžçåŒãšããŠã次ã®æ§è³ªãããã
{| style="border:2px solid skyblue; width:fit-content" cellspacing=0
|style="background:skyblue"|'''å®æ°ã®2ä¹ã®æ§è³ª'''
|-
|style="padding:5px"|
å®æ° a ã«ã€ããŠãããªãã
:<math>a^2 \geqq 0</math>
ãæãç«ã€ã
ãã®åŒã§çå·ãæãç«ã€å Žåãšã¯ã <math>a = 0</math> ã®å Žåã ãã§ããã
|}
ãã®å®çïŒãå®æ°ã2ä¹ãããšãããªãããŒã以äžã§ãããïŒããåºæ¬æ§è³ª(3),(4)ã䜿ã£ãŠèšŒæããŠã¿ããã
'''ïŒèšŒæïŒ'''
aãæ£ã®å Žåãšè² ã®å Žåãš0ã®å Žåã®3éãã«å Žåããããã
'''<nowiki>[aãæ£ã®å Žå]</nowiki>''' <br>
ãã®ãšããåºæ¬æ§è³ª(3)ããã
:<math> aa>0a </math>
ã§ãããããªãã¡ã
:<math> a^2 > 0 </math>
ã§ããã
'''<nowiki>[aãè² ã®å Žå]</nowiki>'''<br>
ãã®ãšããåºæ¬æ§è³ª(4)ãã
<math>0a < aa </math>
ã§ãããããªãã¡ã
: <math> a^2 > 0 </math>
ã§ããã
'''<nowiki>[aããŒãã®å Žå]</nowiki>''' <br>
ãã®ãšãã
<math>a^2=0</math>
ã§ããã
ãã£ãŠããã¹ãŠã®å Žåã«ã€ããŠ<math>a^2 \geqq 0</math>
(çµ)
ãã®ããšãšåºæ¬æ§è³ª(1)(2)ããã次ãæãç«ã€ããšããããã
{| style="border:2px solid skyblue; width:fit-content" cellspacing=0
|style="background:skyblue"|'''å®æ°ã®2ä¹ã©ããã®åã®æ§è³ª'''
|-
|style="padding:5px"|
2ã€ã®å®æ°a,b ã«ã€ã㊠<math>a^2 \geqq 0</math>, ã<math>b^2 \geqq 0</math> ã§ãããããããªãã
:<math>a^2+b^2 \geqq 0</math>
ãæãç«ã€ã
äžåŒã§çå·ãæãç«ã€å Žåãšã¯ã <math>a^2 = 0</math> ã〠<math>b^2 = 0</math> ã®å Žåã ãã§ãããã€ãŸã <math>a = 0</math> ã〠<math>b = 0</math> ã®å Žåã ãã§ããã
|}
** åé¡
次ã®äžçåŒãæãç«ã€ããšã蚌æããã<br>
:<math>
x^2 + 10 y^2 \geqq 6 x y
</math>
(蚌æ)<br>
:<math>
(x^2 + 10 y^2) -(6 x y) \geqq 0
</math>
ã蚌æããã°ããã
巊蟺ãå±éã㊠ãŸãšãããšã
:<math>
(x^2 + 10 y^2) - 6xy = x^2 - 6 x y + 9 y^2 + y^2 = (x - 3 y)^2 + y^2
</math>
ãšãªãã
äžåŒã®æåŸã®åŒã®é
ã«ã€ããŠã
:<math>
(x - 3 y)^2 \geqq 0 , \quad y^2 \geqq 0
</math>
ã ããã
:<math>
(x - 3 y)^2 + y^2 \geqq 0
</math>
ã§ããããã£ãŠ
:<math>
x^2 + 10 y^2 \geqq 6 x y
</math>
ã§ãããïŒçµïŒ
===== æ ¹å·ãå«ãäžçåŒ =====
2ã€ã®æ£ã®æ° a,ãb ã aïŒb ãŸã㯠aâ§b ãªãã°ã䞡蟺ã2ä¹ããŠã倧å°é¢ä¿ã¯åããŸãŸã§ããã
ã€ãŸãã
: <math> a>0 </math>,ã<math> b>0 </math> ã®ãšãã
:
: <math> a > b \quad \Longleftrightarrow \quad a^2 > b^2 </math>
: <math> a > b \quad \Longleftrightarrow \quad a^2 > b^2 </math>
:
: ããã蚌æããã«ã¯ã<math> a^2 - b^2 </math> ã調ã¹ãã°ããã
:<math> a^2 - b^2 = (a+b)(a-b) </math>
ã§ããã
a>bãšãããä»®å®ãããa,b ã¯æ£ã®æ°ãªã®ã§ã<math> (a+b)>0 </math> ã§ãããå¥ã®ä»®å®ããã a > b ãªã®ã§ã<math> (a-b)>0 </math> ã§ãããããã£ãŠã<math> a^2 - b^2 = (a+b)(a-b) >0 </math>
éã«ã<math>a^2-b^2>0</math>ã®ãšãã<math>(a+b)(a-b)>0</math>ã§ããã<math>a>0,b>0</math>ãªã®ã§<math>a+b>0</math>ã§ããããã£ãŠã<math>a-b>0</math>ãªã®ã§ã<math>a>b</math>ã§ããã
ãã£ãŠã<math> a > b \quad \Longleftrightarrow \quad a^2 > b^2 </math> ã§ããã
aâ§bã®å Žåãåæ§ã«èšŒæã§ããã
----
ç·Žç¿ãšããŠã次ã®åé¡ãåããŠã¿ããã
;äŸé¡
<math> a>0 </math>,ã<math> b>0 </math> ã®ãšãã次ã®äžçåŒã蚌æããã
::<math> \sqrt{a} + \sqrt{b} > \sqrt{a+b} </math>
ïŒèšŒæïŒ
äžçåŒã®äž¡èŸºã¯æ£ã§ããã®ã§ã䞡蟺ã®å¹³æ¹ã®å·®ãèããã°ããã䞡蟺ã®å¹³æ¹ã®å·®ã¯
:<math>( \sqrt{a} + \sqrt{b} )^2 - ( \sqrt{a+b} )^2 = a + 2 \sqrt{a} \sqrt{b} + b - (a+b) 2 \sqrt{ab} </math>
ã§ãããããã§ãa,b ã¯ãšãã«æ£ã®å®æ°ãªã®ã§ã
::<math> \sqrt{a} \sqrt{b} = \sqrt{ab} </math>
ã§ããããšãçšããã
:<math> \sqrt{ab} > 0</math>
ã§ããã®ã§ã
:<math>( \sqrt{a} + \sqrt{b} )^2 - ( \sqrt{a+b} )^2 > 0 </math>
ãšãªãããã£ãŠã
:<math> \sqrt{a} + \sqrt{b} > \sqrt{a+b} </math>
ã§ãããïŒçµïŒ
===== 絶察å€ãå«ãäžçåŒ =====
å®æ° a ã®çµ¶å¯Ÿå€ |a| ã«ã€ããŠã
: a ⧠0 ã®ãšã |a|ïŒa , ã
: aïŒ0 ã®ãšã |a|ïŒ ãŒa
ã§ããããã次ã®ããšãæãç«ã€ã
''' |a|â§a , |a|⧠ãŒa ,ã|a|<sup>2</sup>ïŒa<sup>2</sup> '''
ãŸãã2ã€ã®å®æ° a, b ã®çµ¶å¯Ÿå€ |ab| ã«ã€ããŠã¯ã
: |ab| <sup>2</sup> ïŒ (ab)<sup>2</sup> ïŒ a<sup>2</sup> b<sup>2</sup> ïŒ |a|<sup>2</sup> |b|<sup>2</sup> ïŒ (|a| |b|)<sup>2</sup>
ãæãç«ã€ã®ã§ãããã«ããã« |ab|â§0 ,ã|a||b|â§0 ãçµã¿åãããŠã
''' |ab| ïŒ |a| |b| '''
ãæãç«ã€ã
(äŸé¡)
次ã®äžçåŒã蚌æããããŸããçå·ãæãç«ã€ã®ã¯ ã©ã®ãããªå Žåãã 調ã¹ãã
::|a|ïŒ|b| ⧠|aïŒb|
:(蚌æ)
䞡蟺ã®å¹³æ¹ã®å·®ãèãããšã
:: (|a|ïŒ|b|)<sup>2</sup> ㌠|aïŒb|<sup>2</sup> ïŒ |a|<sup>2</sup> ïŒ 2|a| |b| ïŒ |b|<sup>2</sup> ãŒ(a<sup>2</sup> ïŒ 2ab ïŒ b<sup>2</sup> )
:::::::: ïŒ a<sup>2</sup> ïŒ 2|a| |b| ïŒ b<sup>2</sup> ãŒa<sup>2</sup> ㌠2ab ㌠b<sup>2</sup>
:::::::: ïŒ 2|a| |b| ㌠2ab
:::::::: ïŒ 2 ( |a| |b| ㌠ab )
ãããããæ£ãªããäžããããäžçåŒ |a|ïŒ|b| ⧠|aïŒb| ãæ£ããã
ããã§ã |a| |b| ⧠ab ã§ããã®ã§ã
:: ( |a| |b| ㌠ab ) ⧠0
ã§ããã
ãããã£ãŠã |a|ïŒ|b| ⧠|aïŒb| ã§ããã
çå·ãæãç«ã€ã®ã¯ |a| |b| ïŒ ab ã®å Žåãããªãã¡ ab ⧠0 ã®å Žåã§ãããïŒèšŒæ ãããïŒ
{{ã³ã©ã |äžè§äžçåŒ|
ãªã
::<nowiki>|a|ãŒ|b| ⊠|aïŒb| ⊠|a|ïŒ|b| </nowiki>
ã®é¢ä¿åŒã®ããšããäžè§äžçåŒããšããã
}}
==== çžå å¹³åãšçžä¹å¹³å ====
2ã€ã®æ°<math>a</math>,<math>b</math>ã«å¯Ÿãã<math>\frac{a+b}{2}</math>ã'''çžå å¹³å'''ïŒããããžãããïŒãšèšãã<math>\sqrt{ab}</math>ã'''çžä¹å¹³å'''ïŒããããããžãããïŒãšããã
{{ã³ã©ã |çžä¹å¹³åã®äŸãš3ã€ä»¥äžã®ãã®ã®å¹³å|
å¹³åã¯ã3ã€ä»¥äžã®ãã®ã«ãå®çŸ©ãããã3ã€ä»¥äžã®nåã®ãã®ã®çžå å¹³å㯠<math>\frac{a_1 + a_2 + \cdots +a_n }{n}</math> ã§å®çŸ©ãããã
:å¹³åãèããéãã€ãçžå å¹³åã°ãããèããã¡ã ãã以äžã®ãããªç¶æ³ã§ã¯çžä¹å¹³åã®æ¹ãé©åã§ããã
::ãããäŒæ¥ã§ã¯ã2015幎床ã®å£²äžãåºæºã«ãããšã2016幎床ã§ã¯å幎ïŒ2015幎ïŒã®1.5åã®å£²äžã«ãªããŸããã2017幎床ã§ã¯ãå幎ïŒ2016幎ïŒã®2åã®å£²äžã«ãªããŸãããå¹³åãšããŠãäžå¹Žããšã«äœåã®å£²ãäžãã«ãªã£ãŠãã£ãã§ããããïŒ ã
:ïŒçïŒ<math>\sqrt{1.5 \times 2} = \sqrt{3} \fallingdotseq 1.73</math> ãããçŽ 1.73åã
:ãŸãããã®å¿çšäŸã¯ãé
ã3ã€ä»¥äžã®å Žåã®çžä¹å¹³åã®å®çŸ©ã®ä»æ¹ãã瀺åããŠãããããèªè
ã[[é«çåŠæ ¡æ°åŠII/ææ°é¢æ°ã»å¯Ÿæ°é¢æ°|ææ°é¢æ°]]ãç¥ã£ãŠãããªããé
ã3ã€ïŒããã§ã¯ a, b, c ãšããïŒã®å Žåã®çžä¹å¹³åã¯ã
::ïŒ3ã€ã®é
ã®çžä¹å¹³åïŒïŒ<math> (abc)^{ \frac{1}{3} } </math>
:ã«ãªãã
}}
æ¬ããŒãžã§ã¯ã2åã®æ°ã®å¹³åã«ã€ããŠèå¯ããã
çžå å¹³åãšçžä¹å¹³åã«ã€ããŠã次ã®é¢ä¿åŒãæãç«ã€ã
{| style="border:2px solid yellow;width:80%" cellspacing=0
|style="background:yellow"|'''çžå å¹³åãšçžä¹å¹³å'''
|-
|style="padding:5px"|
<math>a \geqq 0</math> ïŒ<math>b \geqq 0</math>ã®ãšãã<br>
<center><math>\frac{a+b}{2} \geqq \sqrt{ab}</math></center><br>
çå·ãæãç«ã€ã®ã¯ã<math>a = b</math>ã®ãšãã§ããã
|}
ïŒèšŒæïŒ
<math>a \geqq 0 , b \geqq 0</math>ã®ãšã
:<math>
\frac{a+b}{2} - \sqrt{ab} = \frac{a+b-2 \sqrt{ab}}{2} = \frac{\left( \sqrt{a} \right) ^2 - 2 \sqrt{a} \sqrt{b} + \left( \sqrt{b} \right) ^2}{2} = \frac{\left( \sqrt{a} - \sqrt{b} \right) ^2 }{2}
</math>
<math> \left( \sqrt{a} - \sqrt{b} \right) ^2 \geqq 0</math>ã§ããããã<math> \frac{\left( \sqrt{a} - \sqrt{b} \right) ^2 }{2} \geqq 0</math><br>
ãããã£ãŠã<math>\frac{a+b}{2} \geqq \sqrt{ab}</math><br>
çå·ãæãç«ã€ã®ã¯ã<math>\left( \sqrt{a} - \sqrt{b} \right) ^2 = 0 </math> ã®ãšããããªãã¡ <math>a = b</math> ã®ãšãã§ããã(蚌æ ããã)
å
¬åŒã®å©çšã§ã¯ãäžã®åŒ <math>\frac{a+b}{2} \geqq \sqrt{ab}</math> ã®äž¡èŸºã«2ãããã <math>a+b \geqq 2 \sqrt{ab}</math> ã®åœ¢ã®åŒã䜿ãå Žåãããã
* åé¡äŸ
** åé¡
<math>a>0</math> ïŒ<math>b>0</math>ã®ãšãã次ã®äžçåŒãæãç«ã€ããšã蚌æããã<br>
(I)
:<math>
a + \frac{1}{a} \geqq 2
</math>
(II)
:<math>
(a+b)\left( \frac{1}{a} + \frac{1}{b} \right) \geqq 4
</math>
**解ç
(I)<math>a>0</math>ã§ããããã<math>\frac{1}{a} >0</math><br>
ãã£ãŠã<math>a + \frac{1}{a} \geqq 2 \sqrt{a \times \frac{1}{a}} = 2</math><br>
ãããã£ãŠ
:<math>
a + \frac{1}{a} \geqq 2
</math>
(II)
:<math>
(a+b)\left( \frac{1}{a} + \frac{1}{b} \right) = 1+ \frac{a}{b} + \frac{b}{a} +1 = \frac{b}{a} + \frac{a}{b} +2
</math>
<math>a>0</math>ïŒ<math>b>0</math>ã§ããããã<math>\frac{b}{a} >0</math>ïŒ<math>\frac{a}{b} >0</math><br>
ãã£ãŠã<math> \frac{b}{a} + \frac{a}{b} +2 \geqq 2 \sqrt{\frac{b}{a} \times \frac{a}{b}} + 2 = 2+2 =4</math><br>
ãããã£ãŠ
:<math>
(a+b)\left( \frac{1}{a} + \frac{1}{b} \right) \geqq 4
</math>
{{ã³ã©ã |3ã€ä»¥äžã®çžä¹å¹³åãšèª¿åå¹³å|
ããèªè
ãææ°é¢æ°ãªã©ãç¥ã£ãŠããã°ã
nåã®ãã®ã®çžä¹å¹³åã¯ã
::<math>\sqrt[n] {a_1 a_2 \cdots a_n }</math>
ãšæžããã
æ°åŠçãªãå¹³åãã«ã¯ãçžå å¹³åãšçžä¹å¹³åã®ã»ãã«ã調åå¹³åãããã
調åå¹³åã¯ãé»æ°åè·¯ã®äžŠåèšç®ã§äœ¿ãããèãæ¹ã§ããã
nåã®ãã®ã®èª¿åå¹³åã¯ã
::<math>\frac{ n}{ \dfrac{1}{a_1} + \dfrac{1}{a_2} + \cdots + \dfrac{1}{a_n} }</math>
ã§å®çŸ©ãããã
äžè¬ã«æ°åŠçã«ã¯ã調åå¹³åãçžä¹å¹³åãçžå å¹³åã®ããã ã«æ¬¡ã®ãããªå€§å°é¢ä¿
:ïŒèª¿åå¹³åïŒ âŠ ïŒçžä¹å¹³åïŒ âŠ ïŒçžå å¹³åïŒ
ãšããé¢ä¿ãæãç«ã€ããšã蚌æãããŠããã
ããªãã¡ãæ°åŒã§æžãã°
::<math>\frac{ n}{ \dfrac{1}{a_1} + \dfrac{1}{a_2} + \cdots + \dfrac{1}{a_n} } \leqq \sqrt[n] {a_1 a_2 \cdots a_n } \leqq \frac{a_1 + a_2 + \cdots +a_n }{n} </math>
ã®é¢ä¿åŒã§ããã
ç°¡æœã«æžããšã
::<Math>\frac{ n}{ \sum_{k=1}^{n} \dfrac{1}{a_k}} \leqq (\prod_{k=1}^{n}a_k)^{\frac{1}{n}} \leqq \frac{\sum_{k=1}^{n} a_n}{n}</Math>
ãšãªãã
}}
== é«æ¬¡æ¹çšåŒ ==
=== è€çŽ æ° ===
2ä¹ããŠè² ã«ãªãæ°ããšãããã®ãèããããã®ãããªæ°ã¯ãäžåŠã§ç¿ã£ãå®æ°ã®äžã«ã¯ãªãããšããããããªããªãã°ãæ£ã®æ°ã§ãè² ã®æ°ã§ã2ä¹ãããšç¬Šå·ãæã¡æ¶ããŠæ£ã®æ°ã«ãªã£ãŠããŸãããã§ãããããã§é«æ ¡ã§ã¯ã2ä¹ããŠè² ã«ãªããšããæ§è³ªãæã€æ°ã®æŠå¿µãæ°ããå°å
¥ããããšã«ããã
:<math>x^2 = -1</math>
ãšããæ¹çšåŒãèããããã®æ¹çšåŒã®è§£ã¯å®æ°ã«ã¯ãªããããã§ããã®æ¹çšåŒã®è§£ãšãªãæ°ãæ°ããäœãããã®åäœãæå <math>i</math> ã§ããããã
ãã® <math>i</math> ã®ããšã'''èæ°åäœ'''ïŒãããããããïŒãšåŒã¶ãïŒèæ°åäœã®èšå· i ãè±èªã®ã¢ã«ãã¡ãããã®ã¢ã€ã®å°æåã§ã imaginary unit ã«ç±æ¥ãããšèããããŠãããïŒ
<math>1+i</math> ã <math>2+5i</math> ã®ããã«ãèæ°åäœ<math>i</math>ãšå®æ°<math>a,b</math>ãçšããŠ
:<math>a+bi</math>
ãšè¡šãããšãã§ããæ°ã'''è€çŽ æ°'''ïŒãµããããïŒãšããããã®ãšãã''a''ããã®è€çŽ æ°ã®'''å®éš'''ïŒãã€ã¶ïŒãšããã''b''ã'''èéš'''ïŒããã¶ïŒãšããã
äŸãã°ã<math>1+i,\quad 2+5i,\quad \frac{9}{2} + \frac{7}{2} i,\quad 4i,\quad 3</math> ã¯ãããããè€çŽ æ°ã§ããã
è€çŽ æ° aïŒbi ã¯ïŒãã ã aãšbã¯å®æ°ïŒãbã0ã®å Žåã«ããããå®æ°ãšèŠãããšãã§ããã
èšãæ¹ãããããšãè€çŽ æ°ãåºæºã«èãããšãå®æ°ãšã¯ã aïŒ0i ã®ãããªãèéšã®ä¿æ°ããŒãã«ãªãè€çŽ æ°ã®ããšã§ãããšãèšããã
4''i''ã®ãããªãèéšã0以å€ã§å®éšããŒãã®è€çŽ æ°ã'''çŽèæ°'''ïŒãã
ãããããïŒãšåŒã¶ãçŽèæ°ã¯ã2ä¹ãããšè² ã«ãªãæ°ã§ããã
å®æ°ãèéšã0ã®è€çŽ æ°ãšèããããã
å®æ°ã§ãªãè€çŽ æ°ã®ããšããèæ°ãïŒããããïŒãšããã
=== è€çŽ æ°ã®æ§è³ª ===
2ã€ã®è€çŽ æ° a+bi ãš c+di ãšãçãããšã¯ã
: aïŒc ã〠bïŒd
ã§ããããšã§ããã
ã€ãŸãã
: a+bi ïŒ c+di ã<math>\Longleftrightarrow</math>ã a=c ã〠bïŒd
ãšãã«ãè€çŽ æ°aïŒbi ã 0ã§ãããšã¯ãaïŒ0 ã〠bïŒ0 ã§ããããšã§ããã
: a+bi ïŒ 0 ã<math>\Longleftrightarrow</math>ã a=0 ã〠bïŒ0
{| style="border:2px solid skyblue;width:80%" cellspacing=0
|style="background:skyblue"|'''è€çŽ æ°ã®çžç'''
|-
|style="padding:5px"|
: a+bi ïŒ c+di ã<math>\Longleftrightarrow</math>ã a=c ã〠bïŒd
: a+bi ïŒ 0 ã<math>\Longleftrightarrow</math>ã a=0 ã〠bïŒ0
|}
;å
±åœ¹
è€çŽ æ°<math>z=a+bi</math>ã«å¯ŸããŠãèéšã®ç¬Šå·ãå転ãããè€çŽ æ°<math>a-bi</math>ã®ããšãã'''å
±åœ¹'''ïŒãããããïŒãªè€çŽ æ°ããŸãã¯ãè€çŽ æ°<math>z</math>ã®å
±åœ¹ãã®ããã«åŒã³ã <math> \bar z </math> ã§ããããããªãããå
±åœ¹ãã¯ãå
±'''è»'''ãã®åžžçšæŒ¢åã«ããæžãæãã§ããã
å®æ°aãšå
±åœ¹ãªè€çŽ æ°ã¯ããã®å®æ° a èªèº«ã§ããã
è€çŽ æ° zïŒa+bi ã«ã€ããŠ
:<math>z+ \bar z =(a+bi)+(a-bi)=2a</math>
:<math>z \bar z =(a+bi)(a-bi)=a^2-abi+abi-b^2 i^2 = a^2-b^2i^2=a^2+b^2</math>
;ååæŒç®
è€çŽ æ°ã«ãååæŒç®ïŒå æžä¹é€ïŒãå®çŸ©ãããã
è€çŽ æ°ã®æŒç®ã§ã¯ãèæ°åäœ<math>i</math>ããéåžžã®æåã®ããã«æ±ã£ãŠèšç®ãããäžè¬ã«è€çŽ æ°<math>z\ ,\ w</math>ãã<math>z=a+bi\ ,\ w=c+di</math>ã§äžãããããšã(ãã ã <math>a\ ,\ b\ ,\ c\ ,\ d</math>ã¯å®æ°ãšãã)ã
:å æ³ãã<math> (a+bi)+(c+di) = (a+c) + (b+d)i </math>
:æžæ³ãã<math> (a+bi)-(c+di) = (a-c) + (b-d)i </math>
:ä¹æ³ãã<math> (a+bi)(c+di) = (ac-bd) + (ad+bc)i </math>
:é€æ³ãã<math> \frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{(c+di)(c-di)} = \frac{ac+bd}{c^2+d^2} + \frac{bc-ad}{c^2+d^2}i </math>ããïŒãã ã <math>c+di \ne 0</math> ãšãããïŒ
ãšãããµãã«è€çŽ æ°ã®å æžä¹é€ã®èšç®æ³ãå®ããããŠããã
ä¹æ³ã®å®çŸ©ã¯ãäžèŠãããšé£ãããã«ã¿ããããå®æ°ã®åé
æ³åãšåæ§ã«å±éããŠããæåŸã« i<sup>2</sup>ã«ãã€ãã¹1ã代å
¥ããŠãã£ãã ãã§ããã
é€æ³ã®å®çŸ©ã¯ãååãšåæ¯ã«ãåæ¯ãšå
±åœ¹ãªåœ¢ã®åŒã æãç® ããã ãã§ããã
ä¹æ³ãé€æ³ã®å®çŸ©åŒãæèšããå¿
èŠã¯ç¡ããèšç®ã®éã«ã¯ãå¿
èŠã«å¿ããŠåé
æ³åãå
±åœ¹ãªã©ã®ãå¿
èŠãªåŒå€åœ¢ãè¡ãã°ããã
'''äŸé¡'''
2ã€ã®è€çŽ æ°
:<math>a=3+i</math>
:<math>b=4 +7i</math>
ã«ã€ããŠã<math>a+b</math> ãš <math>ab</math> ãš <math>\frac a b</math> ããããããèšç®ããã
'''解ç'''
:<math>\begin{align}
a+b&=(3+i)+(4+7i)\\
&=(3+4)+i(1+7)\\
&=7+8i\\
\end{align}</math>
:<math>\begin{align}
ab&=(3+i)(4+7i) \\
&=12+21i+4i+7i^2 \\
&=12+21i+4i+(-7) \\
&=5+25i \\
\end{align}</math>
ã§ããã
:<math>\frac{a}{b}=\frac{3+i}{4+7i}</math>
ããããã«ç°¡åã«ã§ããªãã ããããå®ã¯ãã¡ãã£ãšãããã¯ããã¯ãçšããã°ããèŠããã圢ã«ã§ããã
åæ°ã¯åæ¯ãšååã«åãæ°ããããŠããã£ãã®ã§ãåæ¯ãšååã«åæ¯ã®å
±åœ¹ããããŠã¿ãããããšã
:<math>\begin{align}
\frac{a}{b}&=\frac{3+i}{4+7i} \\
&=\frac{(3+i)(4-7i)}{(4+7i)(4-7i)} \\
&=\frac{12-21i+4i-(-7)}{16-28i+28i-(-49)} \\
&=\frac{19-17i}{65} \\
&=\frac{19}{65}-\frac{17}{65}i \\
\end{align}</math>
ãåŸãããããã®åœ¢ã®ã»ããå
ã®åŒããããã£ãšèŠããã圢ã§ããã
ãã®ãããªæäœãåæ¯ã®å®æ°åãšããããšããããæ°åŠIã§åŠç¿ããå±éã»å æ°å解å
¬åŒ <math>(a+b)(a-b)=a^2-b^2</math>ã®ç°¡åãªå¿çšã§ããã
=== è² ã®æ°ã®å¹³æ¹æ ¹ ===
æ°ã®ç¯å²ãè€çŽ æ°ã«ãŸã§æ¡åŒµãããšãè² ã®æ°ã®å¹³æ¹æ ¹ãèããããšãã§ããã
äŸãšããŠã -5 ã®å¹³æ¹æ ¹ã«ã€ããŠèããŠã¿ããã<br>
:<math>
(\sqrt{5}\ i)^2 = (\sqrt{5})^2\ i^2 = 5 \times (-1) =-5
</math>
:<math>
(- \sqrt{5}\ i)^2 = (-1)^2 \times (\sqrt{5})^2\ i^2 = (+1) \times 5 \times (-1) = -5
</math>
ã§ããããã -5 ã®å¹³æ¹æ ¹ã¯ <math> \sqrt{5}\ i </math> ãš <math> - \sqrt{5}\ i </math> ã§ããã
{| style="border:2px solid skyblue;width:80%" cellspacing=0
|style="background:skyblue"|'''è² ã®æ°ã®å¹³æ¹æ ¹'''
|-
|style="padding:5px"|
<math>a>0</math>ãšãããšããè² ã®æ°<math>-a</math>ã®å¹³æ¹æ ¹ã¯ã<math>\sqrt{a}\ i</math>ãš<math>- \sqrt{a}\ i</math>ã§ããã
|}
<math> \sqrt{-5} </math>ãšã¯ã<math> \sqrt{5}\ i </math> ã®ããšãšããã<math> - \sqrt{-5} </math>ãšã¯ã<math> - \sqrt{5}\ i </math> ã®ããšã§ããã
ãšãã« <math> \sqrt{-1}\ = \ i </math> ã§ããã
ããŠã-5 ã®å¹³æ¹æ ¹ã¯ãæ¹çšåŒ<math>x^2=-5</math> ã®è§£ã§ãããã
ãã®æ¹çšåŒã移é
ããããšã«ããã-5 ã®å¹³æ¹æ ¹ã¯ã
:<math>
x^2+5=0
</math>
ã®è§£ã§ãããšããããã
ããã«å æ°å解ãããããšã«ããã-5 ã®å¹³æ¹æ ¹ã¯æ¹çšåŒ
:<math>
(x + \sqrt{5}\ i)(x - \sqrt{5}\ i) =0
</math>
ã®è§£ã§ããããšããããã
* äŸé¡
(I) ãã<math>\sqrt{-2}\ \sqrt{-6}</math>ããèšç®ããã
* 解ç
(I)
:<math>\sqrt{-2}\ \sqrt{-6} = \sqrt{2}\ i \times \sqrt{6} \ i = \sqrt{12}\ i^2 = -2 \sqrt{3}</math>
ãã®ããã«ããŸãããã€ãã¹ã®æ°ã®å¹³æ¹æ ¹ãåºãŠãããããŸãèæ°åäœ i ãçšããåŒã«æžãæããã
ãã®ããšãããç®ãããŠããã
* åé¡
(II) ãã<math>\frac{\sqrt{2}}{\sqrt{-3}}</math>ããèšç®ããã
(III) ãã2次æ¹çšåŒã<math>x^2=-7</math>ãã解ãã
** 解ç
(II)
:<math>\frac{\sqrt{2}}{\sqrt{-3}} = \frac{\sqrt{2}}{\sqrt{3}\ i} = \frac{\sqrt{2}\ \sqrt{3}\ i}{\sqrt{3}\ i\ \sqrt{3}\ i} = \frac{\sqrt{6}\ i}{3\ i^2} = - \frac{\sqrt{6}}{3} \ i</math>
(III)
:<math>x^2=-7</math>
:<math>x= \pm \sqrt{-7}</math>
:<math>x= \pm \sqrt{7}\ i</math>
=== 2次æ¹çšåŒã®å€å¥åŒ ===
==== 2次æ¹çšåŒã®è§£ãšè€çŽ æ° ====
è€çŽ æ°ã®å¿çšãšããŠãããã§ã¯2次æ¹çšåŒã®æ§è³ªã«ã€ããŠè¿°ã¹ããä»»æã®2次æ¹çšåŒã¯ã解ã®å
¬åŒã«ãã£ãŠè§£ãããããšã[[é«çåŠæ ¡æ°åŠI æ¹çšåŒãšäžçåŒ#äºæ¬¡æ¹çšåŒ|é«çåŠæ ¡æ°åŠI]]ã§è¿°ã¹ãããããã解ã®å
¬åŒã«å«ãŸããæ ¹å·ã®äžèº«ãè² ã®æ°ã®å Žåã«ã¯å®æ°è§£ãååšããªãããšã«æ³šæããå¿
èŠãããã2次æ¹çšåŒ
:<math>
ax^2+bx+c = 0
</math>
ã®è§£ã®å
¬åŒã¯ã
:<math>
x = \frac{-b \pm \sqrt{b^2 - 4ac} }{a}
</math>
ã§ããã
å€å¥åŒ<math>D</math>ã¯
:<math>
D = b^2-4ac
</math>
ã«ãã£ãŠå®çŸ©ããããå€å¥åŒã¯ã解ã®å
¬åŒã®æ ¹å·(ã«ãŒãèšå·ã®ããš)ã®äžèº«ã«çãããå€å¥åŒã®æ£è² ã«ãã£ãŠ2次æ¹çšåŒãå®æ°è§£ãæã€ãã©ããã決ãŸãã
<math>D</math>ãè² ã®ãšãã«ã¯ãã®2次æ¹çšåŒã¯å®æ°ã®ç¯å²ã«ã¯è§£ãæããªãã
å€å¥åŒ<math>D</math>ãè² ã®æ°ã§ãã£ããšããxã®è§£ã¯ç°ãªã2ã€ã®èæ°ã«ãªãããã®2ã€ã®è§£ã¯ å
±åœ¹ ã®é¢ä¿ã«ãªã£ãŠããã
* åé¡äŸ
** åé¡
è€çŽ æ°ãçšããŠã2次æ¹çšåŒ<br>
(1)
:<math>x ^2 + 5x + 9 =0</math>
(2)
:<math>2x ^2 + 5x + 8 =0</math>
(3)
:<math>2x ^2 - 2x + 8 =0</math>
ã解ãã
** 解ç
解ã®å
¬åŒãçšããŠè§£ãã°ããã(1)ã ããèšç®ãããšã
:<math>
x = \frac {- 5 \pm \sqrt{5^2 - 4 \times 1 \times 9}}{2}
</math>
:<math>
= \frac {-5 \pm \sqrt {11} i}{2}
</math>
ãšãªãã
ä»ãåãããã«æ±ãããšãåºæ¥ãã
以éã®è§£çã¯ã<br>
(2)
:<math>
x = \frac {-5 \pm \sqrt {39} i}{4}
</math>
(3)
:<math>
x = \frac {1 \pm \sqrt {15} i}{2}
</math>
ãšãªãã
<!--
(
*å·çè
ã«å¯Ÿãã泚æ
èšç®ã«ã¯[[w:maxima]]ãçšããã
tex(solve(
x ^2 + 5*x + 9 =0,x
));
tex(solve(
2*x ^2 + 5*x + 8 =0,x
));
tex(
solve(
2*x ^2 - 2*x + 8 =0,x
));
)
-->
==== 2次æ¹çšåŒã®å€å¥åŒ ====
æ¹çšåŒã®è§£ã§ãå®æ°ã§ãããã®ã '''å®æ°è§£''' ãšããã
æ¹çšåŒã®è§£ã§ãèæ°ã§ãããã®ã '''èæ°è§£''' ãšããã
2次æ¹çšåŒ <math>ax^2 + bx + c = 0</math> ã®è§£ã¯ <math>x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} </math> ã§ããã
2次æ¹çšåŒã®è§£ã®çš®é¡ã¯ã解ã®å
¬åŒã®äžã®æ ¹å·ã®äžã®åŒ <math>b^2-4ac</math> ã®ç¬Šå·ãèŠãã°å€å¥ããããšãã§ããã
ãã®åŒ <math>b^2-4ac</math> ãã2次æ¹çšåŒ <math>ax^2 + bx + c = 0</math> ã®'''å€å¥åŒ'''ïŒã¯ãã¹ã€ããïŒãšãããèšå· '''<math>D</math>''' ã§è¡šãã
{| style="border:2px solid skyblue;width:80%" cellspacing=0
|style="background:skyblue"|'''å€å¥åŒãšè§£ã®å€å¥'''
|-
|style="padding:5px"|
2次æ¹çšåŒ <math>ax^2 + bx + c = 0</math> ã®å€å¥åŒ <math>D=b^2-4ac</math> ã«ã€ããŠ
::<math>D>0 \quad \Leftrightarrow \quad </math> ç°ãªã2ã€ã®å®æ°è§£ããã€
::<math>D=0 \quad \Leftrightarrow \quad </math> é解ããã€
::<math>D<0 \quad \Leftrightarrow \quad </math> ç°ãªã2ã€ã®èæ°è§£ããã€
|}
ãŸããé解ãå®æ°è§£ã§ããã®ã§ã
::<math>D \geqq 0 \quad \Leftrightarrow \quad </math> å®æ°è§£ããã€
ãšãããã
* åé¡äŸ
** åé¡
次ã®2次æ¹çšåŒã®è§£ãå€å¥ããã
(I)
:<math>
x^2+3\,x-1=0
</math>
(II)
:<math>
2\,x^2-3\,x+2=0
</math>
(III)
:<math>
4\,x^2-20\,x+25=0
</math>
** 解ç
(I)
:<math>
D=3^2-4 \times 1 \times (-1) =13>0
</math>
ã ãããç°ãªã2ã€ã®å®æ°è§£ããã€ã
(II)
:<math>
D=(-3)^2-4 \times 2 \times 2 =-7<0
</math>
ã ãããç°ãªã2ã€ã®èæ°è§£ããã€ã
(III)
:<math>
D=(-20)^2-4 \times 4 \times 25 =0
</math>
ã ãããé解ããã€ã
ãŸãã2次æ¹çšåŒ <math>ax^2 + 2b'x + c = 0</math> ã®ãšãã<math>D=4(b'^2-ac)</math>ãšãªãã®ã§ã
2次æ¹çšåŒ <math>ax^2 + 2b'x + c = 0</math> ã®å€å¥åŒã«ã¯
:<math>
\frac{D}{4} = b'^2-ac
</math>
ããã¡ããŠãããã
ãããçšããŠãåã®åé¡
:(III) ã<math>4\,x^2-20\,x+25=0</math>
ã®è§£ãå€å¥ãããã
<math>a=4 \, , \, b'=-10 \, , \, c=25</math>ãã§ãããã
:<math>
\frac{D}{4} = (-10)^2- 4 \times 25 =0
</math>
ã ãããé解ããã€ã
==== 2次æ¹çšåŒã®è§£ãšä¿æ°ã®é¢ä¿ ====
2次æ¹çšåŒ <math>ax^2 + bx + c = 0</math> ã®2ã€ã®è§£ã <math>\alpha</math> ïŒ<math>\beta</math> ãšããã ãã®æ¹çšåŒã¯ã
<math>a(x-\alpha)(x-\beta) = 0</math>
ãšå€åœ¢ã§ããã
ãããå±éãããšã
<math>ax^2 -a(\alpha + \beta )x+a\alpha \beta = 0</math>
ä¿æ°ãæ¯èŒããŠã
<math>c = a \alpha \beta, b = -a(\alpha + \beta)</math>
ãåŸãã
ãããå€åœ¢ããã°ã<math>\alpha + \beta = -\frac{b}{a}, \alpha \beta = \frac{c}{a}</math>ãšãªãã<br>
{| style="border:2px solid skyblue;width:80%" cellspacing="0"
| style="background:skyblue" |'''解ãšä¿æ°ã®é¢ä¿'''
|-
| style="padding:5px" |
2次æ¹çšåŒ <math>ax^2 + bx + c = 0</math> ã®2ã€ã®è§£ã <math>\alpha</math> ïŒ<math>\beta</math> ãšããã°<br>
<center><math>\alpha + \beta = - \frac{b}{a}</math> ïŒ<math>\alpha \beta = \frac{c}{a}</math><br></center>
|}
* åé¡äŸ
** åé¡
2次æ¹çšåŒ <math>2x^2 + 4x + 3 = 0</math> ã®2ã€ã®è§£ã <math>\alpha</math> ïŒ<math>\beta</math> ãšãããšãã<math>\alpha ^2 + \beta ^2</math> ã®å€ãæ±ããã
** 解ç
解ãšä¿æ°ã®é¢ä¿ããã
<math>\alpha + \beta = - \frac{4}{2} = - 2 </math>ïŒ<math>\alpha \beta = \frac{3}{2}</math><br>
<math>\alpha ^2 + \beta ^2 = (\alpha + \beta )^2 - 2 \alpha \beta = (-2)^2 - 2 \times \frac{3}{2} = 1</math>
==== 2æ°ã解ãšãã2次æ¹çšåŒ ====
2ã€ã®æ° <math>\alpha</math> ïŒ<math>\beta</math> ã解ãšãã2次æ¹çšåŒã¯
:<math>
(x - \alpha) (x - \beta) = 0
</math>
ãšè¡šãããã巊蟺ãå±éããŠæŽçãããšæ¬¡ã®ããã«ãªãã
{| style="border:2px solid skyblue;width:80%" cellspacing=0
|style="background:skyblue"|'''äžãããã2ã€ã®æ°ã解ãšãã2次æ¹çšåŒ'''
|-
|style="padding:5px"|
2æ° <math>\alpha</math> ïŒ<math>\beta</math> ã解ãšãã2次æ¹çšåŒã¯<br>
<center><math>x^2 - (\alpha + \beta ) x + \alpha \beta = 0</math><br></center>
|}
* åé¡äŸ
** åé¡
次ã®2æ°ã解ãšãã2次æ¹çšåŒãäœãã
(I)
:<math>
3 + \sqrt{5} \ , 3 - \sqrt{5}
</math>
(II)
:<math>
2 + 3 i \ , 2 - 3 i
</math>
** 解ç
(I)<br>
åã<math>(3 + \sqrt{5}) + (3 - \sqrt{5}) = 6</math><br>
ç©ã<math>(3 + \sqrt{5}) (3 - \sqrt{5}) = 4</math>ãã§ãããã<br>
:<math>
x^2 - 6 x + 4 =0
</math>
(II)<br>
åã<math>(2 + 3 i) + (2 - 3 i) = 4</math><br>
ç©ã<math>(2 + 3 i) (2 - 3 i) = 13</math>ãã§ãããã<br>
:<math>
x^2 - 4 x + 13 =0
</math>
==== 2次åŒã®å æ°å解 ====
2次æ¹çšåŒ <math>ax^2 + bx + c = 0</math> ã®2ã€ã®è§£ <math>\alpha</math> ïŒ<math>\beta</math> ãããããšã2次åŒ
:<math>ax^2 + bx + c
</math>
ãå æ°å解ããããšãã§ããã<br>
解ãšä¿æ°ã®é¢ä¿ <math>\alpha + \beta = - \frac{b}{a}</math>ïŒ<math>\alpha \beta = \frac{c}{a}</math> ããã
:<math>
ax^2 + bx + c = a \left(x^2 + \frac{b}{a}x + \frac{c}{a} \right) = a \left\{x^2 - (\alpha + \beta )x + \alpha \beta \right\} = a (x - \alpha)(x - \beta)
</math>
{| style="border:2px solid skyblue;width:80%" cellspacing="0"
| style="background:skyblue" |'''解ãšå æ°å解'''
|-
| style="padding:5px" |
2次æ¹çšåŒ <math>ax^2 + bx + c = 0</math> ã®2ã€ã®è§£ã <math>\alpha</math> ïŒ<math>\beta</math> ãšãããš<br>
<center><math>ax^2 + bx + c = a (x - \alpha)(x - \beta)</math><br></center>
|}
2次æ¹çšåŒã¯ãè€çŽ æ°ã®ç¯å²ã§èãããšã€ãã«è§£ããã€ãããè€çŽ æ°ãŸã§äœ¿ã£ãŠãããšãããšã2次åŒã¯å¿
ã1次åŒã®ç©ã«å æ°å解ããããšãã§ããã
* åé¡äŸ
** åé¡
è€çŽ æ°ã®ç¯å²ã§èããŠã次ã®2次åŒãå æ°å解ããã
(I)
:<math>
x^2 + 4 x - 1
</math>
(II)
:<math>
2 x^2 - 3 x + 2
</math>
**解ç
(I)<br>
2次æ¹çšåŒã<math>x^2 + 4 x - 1 = 0</math>ãã®è§£ã¯<br>
:<math>
x = \frac{-4 \pm \sqrt{4^2-4 \times 1 \times (-1)}}{2 \times 1} = \frac{-4 \pm \sqrt{20}}{2} = \frac{-4 \pm 2 \sqrt{5}}{2} = -2 \pm \sqrt{5}
</math>
ãã£ãŠ
:<math>
x^2 + 4 x - 1 = \left\{ x - (-2 + \sqrt{5}) \right\} \left\{ x - (-2 - \sqrt{5}) \right\} = (x + 2 - \sqrt{5}) (x + 2 + \sqrt{5})
</math>
(II)<br>
2次æ¹çšåŒã<math>2 x^2 - 3 x + 2 = 0</math>ãã®è§£ã¯<br>
:<math>
x = \frac{-(-3) \pm \sqrt{(-3)^2-4 \times 2 \times 2}}{2 \times 2} = \frac{3 \pm \sqrt{-7}}{4} = \frac{3 \pm \sqrt{7} i}{4}
</math>
ãã£ãŠ
:<math>
2 x^2 - 3 x + 2 = 2 \left(x- \frac{3 + \sqrt{7}\; i}{4} \right) \left(x- \frac{3 - \sqrt{7}\; i}{4} \right)
</math>
=== é«æ¬¡æ¹çšåŒ ===
3次以äžã®æŽåŒã«ããæ¹çšåŒãèããã
äžè¬ã«æ¹çšåŒã <math>P(x)=0</math>ãšãšãã
ãã ãã<math>P(x)</math>ã¯ãä»»æã®æ¬¡æ°ã®æŽåŒãšããã
==== å°äœã®å®ç ====
<math>P(x)</math>ã1次åŒ<math>x-a</math>ã§å²ã£ããšãã®åã<math>Q(x)</math>ãäœãã<math>R</math>ãšãããšã
:<math>
P(x) = (x-a)Q(x)+R
</math>
ãã®äž¡èŸºã®<math>x</math>ã«<math>a</math>ã代å
¥ãããšã
:<math>
P(a) = (a-a)Q(a)+R = 0 \times Q(a) + R =R
</math>
ã€ãŸãã<math>P(x)</math>ã<math>x-a</math>ã§å²ã£ããšãã®äœãã¯<math>P(a)</math>ã§ããã
{| style="border:2px solid pink;width:80%" cellspacing="0"
| style="background:pink" |'''å°äœã®å®ç'''
|-
| style="padding:5px" |
æŽåŒ<math>P(x)</math>ã<math>x-a</math>ã§å²ã£ããšãã®äœãã¯ã<math>P(a)</math>ã«çããã
|}
* åé¡äŸ
** åé¡
æŽåŒ <math>P(x) = x^3 -2x + 3</math> ã次ã®åŒã§å²ã£ãäœããæ±ããã<br>
(I)
:<math>
x-2
</math>
(II)
:<math>
x+1
</math>
(III)
:<math>
2x-1
</math>
** 解ç
(I)ã<math>P(2) = 2^3 - 2 \times 2 + 3 = 7</math><br>
(II)ã<math>P(-1) = (-1)^3 - 2 \times (-1) + 3 = 4</math><br>
(III)ã<math>P\left( \frac{1}{2} \right) = \left( \frac{1}{2} \right)^3 - 2 \times \left( \frac{1}{2} \right) + 3 = \frac{17}{8}</math>
===== å æ°å®ç =====
ããå®æ°<math>a</math>ã«å¯ŸããŠã
:<math>
P(a) = 0
</math>
ãæãç«ã£ããšããã
ãã®ãšããæŽåŒ<math>P(x)</math> ã¯ã <math>(x-a)</math> ãå æ°ã«æã€ããšãåãã
ãã®ããšãå æ°å®çïŒãããããŠããïŒãšåŒã¶ã
* å°åº
æŽåŒ<math>P(x)</math>ã«å¯ŸããŠãå<math>Q(x)</math>ãå²ãåŒ<math>(x-a)</math>ãšãã
æŽåŒã®é€æ³ãçšããããã®ãšããå<math>Q(x)</math>ã
(<math>Q(x)</math>ã¯ã<math>P(x)</math>ããã1ã ã次æ°ãäœãæŽåŒã§ããã)
äœã<math>c</math>(<math>c</math>ã¯ãå®æ°ã)ãšãããšã
æŽåŒ<math>P(x)</math> ã¯ã
:<math>
P(x) = (x-a)Q(x) + c
</math>
ãšæžããã
ããã§ã <math>c=0</math> ã§ãªããšã <math>P(a)=0</math> ã¯æºããããªããã
ãã®ãšãã<math>P(x)</math>ã¯ã<math>(x-a)</math>ã«ãã£ãŠå²ãåããã
ãã£ãŠãå æ°å®çã¯æç«ããã
{| style="border:2px solid pink;width:80%" cellspacing=0
|style="background:pink"|'''å æ°å®ç'''
|-
|style="padding:5px"|
æŽåŒ<math>P(x)</math>ã«ã€ããŠ<br>
<center><math>P(a)=0 \Leftrightarrow </math> <math>P(x)</math>ã¯<math>x-a</math>ã§å²ããããã</center>
|}
å æ°å®çãçšããããšã§ããã次æ°ã®é«ãæŽåŒãå æ°å解ããããšã
åºæ¥ãããã«ãªããäŸãã°ã3次ã®æŽåŒ
:<math>
x^3 - 1
</math>
ã«ã€ããŠã<math>x=1</math>ã代å
¥ãããšã
:<math>
x^3 - 1
</math>
ã¯0ãšãªãããã£ãŠãå æ°å®çãããã®åŒã¯
:<math>
(x-1)
</math>
ãå æ°ãšããŠæã€ã
ããã§ãå®éæŽåŒã®é€æ³ã䜿ã£ãŠèšç®ãããšã
:<math>
x^3 - 1 = (x-1)(x^2+x+1)
</math>
ãåŸãããã
* åé¡äŸ
** åé¡
å æ°å®çãçšããŠ<br>
(I)
:<math>
x^3-6\,x^2+11\,x-6
</math>
(II)
:<math>
x^3+x^2-14\,x-24
</math>
<!--
(III)
:<math>
x^3+5\,x^2-34\,x-80
</math>
-->
ãå æ°å解ããã
** 解ç
(I)
å æ°å解ã®çµæã(x+æŽæ°)ã®ç©ã®åœ¢ãªããæŽæ°ã¯6ã®å æ°ã§ãªããã°ãªããªãããã®ããã<math>\pm 1, \pm 2,\pm 3,\pm 6</math>ãåè£ãšãªãããããã«ã€ããŠã¯å®éã«ä»£å
¥ããŠç¢ºããããããªããx=1ã代å
¥ãããšã
:<math>
1-6+11-6=0
</math>
ãšãªãã®ã§ã(x-1)ãå æ°ãšãªããå®éã«æŽåŒã®é€æ³ãè¡ãªããšãåãšããŠ<math>x^2-5x+6</math>ãåŸãããããããã¯<math>(x-2)(x-3)</math>ã«å æ°å解ã§ããããã£ãŠçãã¯ã
:<math>
\left(x-3\right)\,\left(x-2\right)\,\left(x-1\right)
</math>
ãšãªãã<br>
(II)
ããã§ãå°éã«24ã®å æ°ãåœãŠã¯ããŠãããããªãã24ã®å æ°ã¯æ°ãå€ãã®ã§ããªãã®èšç®ãå¿
èŠãšãªããããã§ã¯ã-2ã代å
¥ãããšã
:<math>
-8 +4 -14 \cdot (-2) -24 = 0
</math>
ãšãªãã(x+2)ãå æ°ã ãšããããé€æ³ãè¡ãªããšã<math>x^2 -x -12</math>ãåŸããããã(x-4)(x+3)ã«å æ°å解ã§ãããçãã¯ã
:<math>
\left(x-4\right)\,\left(x+2\right)\,\left(x+3\right)
</math>
ãšãªãã
===== é«æ¬¡æ¹çšåŒ =====
å æ°å解ãå æ°å®çãå©çšããŠé«æ¬¡æ¹çšåŒã解ããŠã¿ããã
* åé¡äŸ
** åé¡
é«æ¬¡æ¹çšåŒ<br>
(I)
:<math>
x^3-8=0
</math>
(II)
:<math>
x^4-2x^2-8=0
</math>
(III)
:<math>
x^3-5x^2+7x-2=0
</math>
ã解ãã
**解ç
(I)
巊蟺ã<math>
a^3-b^3=(a-b)(a^2+ab+b^2)
</math>ãçšããŠå æ°å解ãããš
:<math>
(x-2)(x^2+2x+4)=0
</math>
ãããã£ãŠ<math>\ x-2=0</math>ããŸãã¯<math>\ x^2+2x+4=0</math><br>
ãã£ãŠ
:<math>
x=2\ , \ -1 \pm \sqrt{3} i
</math>
(II) ã<math>\ x^2=X\ </math>ãšãããšã
:<math>
X^2-2X-8=0
</math>
巊蟺ãå æ°å解ãããš
:<math>
(X-4)(X+2)=0
</math>
ãã£ãŠã<math>X=4\ ,\ X=-2</math><br>
ããã«ã<math>x^2=4\ ,\ x^2=-2</math><br>
ãããã£ãŠ
:<math>
x= \pm 2\ ,\ \pm \sqrt{2} i
</math>
(III) ã<math>\ P(x)=x^3-5x^2+7x-2\ </math>ãšããã
:<math>
P(2)=2^3-5 \times 2^2+7 \times 2-2=0
</math>
ãããã£ãŠã<math>\ x-2\ </math>ã¯<math>\ P(x)\ </math>ã®å æ°ã§ããã<br>
:<math>
P(x)=(x-2)(x^2-3x+1)
</math>
ãã£ãŠã<math>(x-2)(x^2-3x+1)=0</math><br>
<math>\ x-2=0</math>ããŸãã¯<math>\ x^2-3x+1=0</math><br>
ãããã£ãŠ
:<math>
x= 2\ ,\ \frac{3 \pm \sqrt{5}}{2}
</math>
=====ïŒçºå±ïŒ3次æ¹çšåŒã®è§£ãšä¿æ°ã®é¢ä¿=====
3次æ¹çšåŒ <math>ax^3+ bx^2+ cx+d=0</math> ã®3ã€ã®è§£ã ã<math>\alpha\ ,\ \beta\ ,\ \gamma</math> ãšãããš
:<math>ax^3+ bx^2+ cx+d=a(x- \alpha)(x- \beta)(x- \gamma)</math>
ãæãç«ã€ã<br>
å³èŸºãå±éãããš
:<math>a(x- \alpha)(x- \beta)(x- \gamma)</math>
:<math>=a(x- \alpha)\left\{x^2-(\beta + \gamma)x+ \beta \gamma \right\}</math>
:<math>=a \left\{x^3-(\alpha + \beta + \gamma)x^2+ (\alpha \beta + \beta \gamma + \gamma \alpha)x - \alpha \beta \gamma \right\}</math>
ãã£ãŠ
:<math>ax^3+ bx^2+ cx+d=a \left\{x^3-(\alpha + \beta + \gamma)x^2+ (\alpha \beta + \beta \gamma + \gamma \alpha)x - \alpha \beta \gamma \right\}</math>
ããã«
:<math>b=-a(\alpha + \beta + \gamma)\ ,\ c= a(\alpha \beta + \beta \gamma + \gamma \alpha)\ ,\ d= -a \alpha \beta \gamma</math>
ãããã£ãŠã次ã®ããšãæãç«ã€ã
{| style="border:2px solid pink;width:80%" cellspacing=0
|style="background:pink"|'''3次æ¹çšåŒã®è§£ãšä¿æ°ã®é¢ä¿'''
|-
|style="padding:5px"|
3次æ¹çšåŒ <math>ax^3+ bx^2+ cx+d=0</math> ã®3ã€ã®è§£ã ã<math>\alpha\ ,\ \beta\ ,\ \gamma</math> ãšãããš
<center><math>\alpha + \beta + \gamma =- \frac{b}{a}\ ,\ \alpha \beta + \beta \gamma + \gamma \alpha= \frac{c}{a}\ ,\ \alpha \beta \gamma =- \frac{d}{a}</math></center>
|}
== ã³ã©ã ==
=== è€çŽ æ°ã¯ãååšããããïŒ ===
ãã°ãã°èæ°ã¯ãçŸå®ã«ã¯ååšããªãæ°ãã§ãããšèšãããããšããããæŽå²çã«ãèæ°ãæ±ã£ãæ°åŠãèããã¹ãã§ã¯ãªããšèããããæ代ã¯é·ãã£ãããã®æ代ã®å
é²çãªæ°åŠè
ã®äžã«ã¯ãèæ°ãæå¹ã«æŽ»çšããŠç 究ãé²ããäžæ¹ã§ãææãçºè¡šããéã«ã¯èæ°ãè¡šã«åºããã«èšè¿°ããåªåãããããšã§ãç¡çšãªæµæãåããªãããã«å·¥å€«ããè
ããããšèšãããã»ã©ã§ããã
ã ããããèããŠã¿ãã°ãæ°ããçŸå®ã«ååšããããšã¯ã©ãããæå³ãªã®ã ããããçŸå®ã«éçã䜿ã£ãŠçŽã«åãæããªãã°ãååšã®é·ãããæ£ç¢ºã«ååšçãã®ãã®ã«ãããããšã¯äžå¯èœã§ããããã«æããããããã®å²ã«ååšçãšããå®æ°ã¯ãååšããããšæããããã®ã¯ãªãã ããããæ°çŽç·ãå®æ°ã®ãå®åšããä¿¡ãããããªãã°ãè€çŽ æ°ã¯è€çŽ æ°å¹³é¢ïŒæ°åŠCã§ç¿ãïŒã®äžã«ååšããã®ã ãããåãã§ã¯ãªãã ãããã
ãã®ããã«èãããšãããããæ°ãšã¯ãã¹ãŠããæå³ã§æ³åäžã®ååšã§ãããããã«å¯ŸããŠãååšããããååšããªãããšããåããç«ãŠãããšããã³ã»ã³ã¹ã§ããããã«æãããããååšããªããããã«æãããã¡ãªèæ°ã§ããããããšãã°ç©çåŠã®äžåéã§ããéåååŠã®ã·ã¥ã¬ãã£ã³ã¬ãŒæ¹çšåŒã«è¡šãããªã©ãå¿çšäžã®ããŸããŸãªå Žé¢ã«ãããŠããèæ°ã䜿ã£ãŠèšè¿°ããããšãèªç¶ãªå¯Ÿè±¡ã¯å€ãã®ã ã
=== è€çŽ æ°ã«ã¯ã倧å°é¢ä¿ããªãã ===
è€çŽ æ°ã©ããã«ã€ããŠããã®å€§å°é¢ä¿ã¯å®çŸ©ããªãããã®çç±ã¯ãã©ã®ããã«å€§å°é¢ä¿ãå®çŸ©ããŠãã䟿å©ãªæ§è³ªãæºããããšãã§ããªãããã§ãããå
·äœçã«èšãã°ãæ¢ã«è¿°ã¹ãå®æ°ã®å€§å°é¢ä¿ã«ã€ããŠã®ãäžçåŒã®åºæ¬æ§è³ª(1)(2)(3)(4)ãã«ãããåŒãæãç«ãããããšãã§ããªãã®ã ã
ããšãã°ã<math>a+bi<a'+b'i</math>ã§ããããšãã<math>a^2+b^2<a'^2+b'^2</math>ã§ããããšãšããŠå®çŸ©ããŠã¿ããããã®ããã«å®çŸ©ãããšãããšãã°1+2i<2-3iã§ããããŸã2+3i<3+4iã§ããããšãããã(1+2i)+(2+3i)=3+5i,(2-3i)+(3+4i)=5+iã§ããã3+5i>5+iãšãªã£ãŠããŸããããã¯åºæ¬æ§è³ª(2)ãæãç«ããªãããšãæå³ããã
ãã¡ããããã¯é©åœã«èããå®çŸ©ãããŸããŸäžé©åã ã£ããšããã ãã®ããšã ããå®ã¯ãä»ã«ã©ã®ããã«å®çŸ©ããŠããã®ãããªå°é£ããã¯éããããªãããšãç¥ãããŠãããããããã«ãè€çŽ æ°ã«ã¯å€§å°é¢ä¿ãå®çŸ©ããªãã®ã§ããã
=== è€çŽ æ°ã®å¹³æ¹æ ¹ (â»çºå±) ===
ä»åºŠã¯ãè€çŽ æ°ã®å¹³æ¹æ ¹ã«ã€ããŠèããŠã¿ããã
æ£ã®æ°<math>a</math>ãèãããšãã
:<math>a</math>ã®å¹³æ¹æ ¹ã¯<math>\pm \sqrt{a}</math>
:<math>-a</math>ã®å¹³æ¹æ ¹ã¯<math>\pm \sqrt{a} i</math>
ã§ã¯ã
:<math>\pm a i</math>ã®å¹³æ¹æ ¹ã¯ã©ã®ããã«è¡šããã ãããã
èæ°åäœ<math>i</math>ã®å¹³æ¹æ ¹ãèãããšãããã¯zã«ã€ããŠã®æ¹çšåŒ <math>z^2 = i</math> ã®è§£ z ã®å€ã§ãããããããã解ãã°ãããã©ã®ãããªè€çŽ æ°zãªããã®åŒãæºããããšãã§ããã ãããã
zãè€çŽ æ°ãšãããšã<math>z = x + yi</math>(x,yã¯å®æ°)ãšè¡šãããã
<math>(x + yi)^2 = i \Leftrightarrow x^2 + 2xyi - y^2 = i \Leftrightarrow (x^2-y^2)+(2xy-1)i = 0</math>
<math>x^2-y^2,2xy-1</math>ã¯å®æ°ã§ãããããå®éšãšèéšãå
±ã«ïŒã«ãªããã°ãªããªãããã
<math>\begin{cases}
x^2-y^2=0 (\Leftrightarrow x= \pm y ) \\
2xy-1=0
\end{cases}</math>
<math>x=y</math>ã®ãšãã<math>2x^2=1 \Leftrightarrow x=\pm\frac{1}{\sqrt{2}},y=\pm\frac{1}{\sqrt{2}}</math> (è€å·åé ãx,yã¯å
±ã«å®æ°ã§ãããããæ¡ä»¶ãæºããã)
<math>x=-y</math>ã®ãšãã<math>-2y^2=1 \Leftrightarrow y^2=-\frac{1}{2}</math> ããã§ããããæºããå®æ°yã¯ååšããªãããäžé©ã
ãã£ãŠã<math>z=\pm\left(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}i\right)</math><sub>â </sub>
*åé¡äŸ
** åé¡
:<math>i \,\!</math>ãèæ°åäœãšãããšãã次ã®åãã«çããã
:(I) <math>-i,30i \,\!</math>ã®å¹³æ¹æ ¹ãæ±ããã
:(II) 2次æ¹çšåŒ <math>z^2 - 30i - 16 = 0 \,\!</math> ã解ãã
:(III) 3次æ¹çšåŒ <math>z^3 = i \,\!</math> ã解ãã
** 解ç
:(I)
::<math>\pm\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}i\right) , \pm\left(\sqrt{15}+\sqrt{15}i\right)</math>
:(II)
::<math>z=5+3i , -5-3i \,\!</math>
:(III)
::<math>z=-i,\frac{i\pm\sqrt{3}}{2}</math>
:ä»åæããåé¡ã¯ãå
šãŠ<math>z=x+yi</math>(x,yã¯å®æ°)ãšçœ®ãããšã§æ±ããããã(III)ã¯ã<math>(x+yi)^3-i=x(x^2-3y^2)+(3x^2y-y^3-1)i=0</math>ããã
å®éšããŒããèæ
®ããŠ<math>x=0</math>ã<math>x=\pm\sqrt{3}y</math>ã ããèéšããŒããªã®ã§ãxã®å€ãåè
ã®ãšã<math>y=-1</math>ãåŸè
ã®ãšã<math>y=1/2</math>ãšãªãããšãããã«ãããã
=== é«æ¬¡æ¹çšåŒã®ã解ã®å
Œ΋ ===
2次æ¹çšåŒã«ã¯è§£ã®å
¬åŒããããæ¥æ¬ã®äžåŠãé«æ ¡ã§ãç¿ãã2次æ¹çšåŒã®è§£ã®å
¬åŒãçšããã°ãã©ããªä¿æ°ã®2次æ¹çšåŒã§ãã£ãŠã解ãæ±ããããã3次æ¹çšåŒãš4次æ¹çšåŒã«ãã解ã®å
¬åŒã¯ååšããä¿æ°ãã©ããªä¿æ°ã§ãã£ãŠã解ãæ±ããããããããã®è§£ã®å
¬åŒã¯ã[[代æ°æ¹çšåŒè«]]ã§è¿°ã¹ãŠããããã«ãä¿æ°ã«æéåã®ååæŒç®ãšæ ¹å·ããšãæäœã®çµã¿åããã§è¡šãããšãã§ããã
5次æ¹çšåŒã§ã¯ã4次以äžã®æ¹çšåŒãšã¯ç¶æ³ãç°ãªããäžè¬ã®5次æ¹çšåŒã®è§£ã¯ã2次æ¹çšåŒã4次æ¹çšåŒã®ããã«ãä¿æ°ã«æéåã®ååæŒç®ãšæ ¹å·ããšãæäœã®çµã¿åããã§è¡šãããšãã§ããªãã®ã§ããããã ãããã§ããªããããšã®èšŒæã¯å®¹æã§ã¯ãªãããã®ããšã蚌æããã«ã¯ã[[ã¬ãã¢çè«]]ãç解ããå¿
èŠãããïŒæ¥æ¬ã®å€§åŠã®æšæºçãªã«ãªãã¥ã©ã ã§ã¯ãçåŠéšæ°åŠç§ã®åŠçã®ã¿ã倧åŠ3幎çã§åŠã¶ã®ãäžè¬çãªçšåºŠã®çè«ã§ããïŒã
ãªããããã§èšããè¡šãããšãã§ããªãããšã¯äžè¬ã®æ¹çšåŒã«ã€ããŠã®ããšã§ãããç¹å¥ãª5次æ¹çšåŒã®å Žåã¯ç°¡åã«è§£ãæ±ãããããããšãã°ã<math> x^5 -32 = 0 </math> ã¯è§£ã®ã²ãšã€ãšã㊠<math> x=2 </math> ããã€ããšã¯ãããããããã®æ¹çšåŒã¯ä»ã®è§£ã«ã€ããŠãäžè§é¢æ°ãçšããŠç°¡åã«è¡šããããšã[[é«çåŠæ ¡æ°åŠC/è€çŽ æ°å¹³é¢]]ã«ãããŠåŠã¶ã
ãä¿æ°ã«æéåã®ååæŒç®ãšæ ¹å·ããšãæäœã®çµã¿åãããã«æããªããã°ãäžè¬ã®5次æ¹çšåŒã®è§£ãæ±ããæ¹æ³ãååšããããããé«åºŠãªæ°åŠãçšããå¿
èŠãããã[[w:äºæ¬¡æ¹çšåŒ]]ã«èšè¿°ãããã®ã§èå³ã®ããèªè
ã¯åç
§ãããšããã
=== è€çŽ æ°ãšé¢æ° ===
é«çåŠæ ¡ã§è€çŽ æ°ãåºãŠããåéã¯ãã®åéãšæ°åŠCã[[é«çåŠæ ¡æ°åŠC/å¹³é¢äžã®æ²ç·|å¹³é¢äžã®æ²ç·]]ãšè€çŽ æ°å¹³é¢ãã®ã¿ã§ãããè€çŽ æ°ã®åºæ¬èšç®ãæ¹çšåŒãè€çŽ æ°ç¯å²ã§è§£ãããšãè€çŽ æ°ã®å¹ŸäœåŠçæå³ã«ã€ããŠæ±ã£ãŠããããããã倧åŠæ°åŠã«ãããŠã¯ãé¢æ°ã®å®çŸ©åã»å€åãè€çŽ æ°ç¯å²ã«åºããŠèããã[[è€çŽ 解æåŠ|è€çŽ é¢æ°è«]]ããšãããã®ãæ±ãã
å®æ°ç¯å²ã§ã®é¢æ°ã¯x, yãšãã«äžæ¬¡å
ã®å®æ°è»žãæã€ãããå
¥åå€ãšåºåå€ã®æãã°ã©ããèããã«ã¯äºæ¬¡å
ã®åº§æšå¹³é¢ã§ååã§ãã£ããããããè€çŽ æ°ç¯å²ã§ã®é¢æ°ã¯x, yãšãã«äºæ¬¡å
ã®è€çŽ æ°å¹³é¢ãæã€ãããå
¥åå€ãšåºåå€ã®æãã°ã©ããèããã«ã¯å次å
ã®åº§æšç©ºéãå¿
èŠã§ãããäžæ¬¡å
空éã«äœãæã
ã«ã¯æç»ããããšãã§ããªãããã®ãããè€çŽ é¢æ°è«ã§ã¯é¢æ°ã®ã°ã©ããèããããšã¯åºæ¬çã«ãªããïŒãã ããåºåãããè€çŽ æ°ã®çµ¶å¯Ÿå€ãèããããšã«ãã£ãŠäžæ¬¡å
ã°ã©ãã«èœãšã蟌ãããšã¯å¯èœïŒ
ã§ã¯äœãèããã®ããšãããšãè€çŽ é¢æ°ã®åŸ®åç©åã§ãããè€çŽ é¢æ°ã®åŸ®åã«é¢é£ããŠãæ£åé¢æ°ããšããçšèªãåºãŠããããè€çŽ é¢æ°è«ã¯ãã®æ£åé¢æ°ãšãããã®ã®æ§è³ªã調ã¹ãåŠåã ãšèšã£ãŠè¯ãã
è€çŽ é¢æ°è«ã¯ç©çåŠã®ç¹ã«æ³¢åã«é¢ããåéïŒé³ã»é»ç£æ°ãªã©ïŒã«ãããŠæŽ»èºããŠããããæ³¢åæ¹çšåŒãããã€ã³ããŒãã³ã¹ããšããèšèã¯æåã ããã
ã¡ãªã¿ã«ãè€çŽ æ°ãããã«æ¡åŒµããæ°ãšããŠã[[w:åå
æ°]]ããšãããã®ãããããã®åå
æ°ã¯[[é«çåŠæ ¡æ°åŠC/ãã¯ãã«|ãã¯ãã«]]ã[[é«çåŠæ ¡æ°åŠC/æ°åŠçãªè¡šçŸã®å·¥å€«#è¡åãçšããè¡šçŸãšãã®æŒç®|è¡å]]ãšæ·±ãé¢ãããååšããŠãããæ·±æããšé¢çœãã®ã ããããããåé·ã«ãªãããå²æããããªããåå
æ°ãããã«æ¡åŒµããå
«å
æ°ãåå
å
æ°ãšããæ°ãç 究ãããŠããã
== æŒç¿åé¡ ==
{{DEFAULTSORT:ãããšããã€ããããããII ãããšããããã}}
[[Category:é«çåŠæ ¡æ°åŠII|ãããšããããã]] | 2005-05-04T09:17:55Z | 2024-03-29T05:50:31Z | [
"ãã³ãã¬ãŒã:Pathnav",
"ãã³ãã¬ãŒã:ã³ã©ã "
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E6%95%B0%E5%AD%A6II/%E5%BC%8F%E3%81%A8%E8%A8%BC%E6%98%8E%E3%83%BB%E9%AB%98%E6%AC%A1%E6%96%B9%E7%A8%8B%E5%BC%8F |
1,902 | é«çåŠæ ¡æ°åŠII/åŒãšèšŒæã»é«æ¬¡æ¹çšåŒ | æ¬é
ã¯é«çåŠæ ¡æ°åŠIIã®åŒãšèšŒæã»é«æ¬¡æ¹çšåŒã®è§£èª¬ã§ããã
( a + b ) 5 = ( a + b ) ( a + b ) ( a + b ) ( a + b ) ( a + b ) {\displaystyle (a+b)^{5}=(a+b)(a+b)(a+b)(a+b)(a+b)} ã«ã€ããŠèãããããã®åŒãå±éãããšãã a 2 b 3 {\displaystyle a^{2}b^{3}} ã®ä¿æ°ã¯ãå³èŸºã®5åã® ( a + b ) {\displaystyle (a+b)} ãã a {\displaystyle a} ã3ååãçµã¿åããã«çãããã 5 C 2 = 10 {\displaystyle _{5}\mathrm {C} _{2}=10} ã§ããã
ãã®èããæ¡åŒµããŠ
ãå±éããã a r b n â r {\displaystyle a^{r}b^{n-r}} ã®é
ã®ä¿æ°ã¯ãå³èŸºã® n {\displaystyle n} åã® ( a + b ) {\displaystyle (a+b)} ãã a {\displaystyle a} ã r {\displaystyle r} ååãçµã¿åããã«çãããã n C r {\displaystyle _{n}\mathrm {C} _{r}} ã§ããã
ãã£ãŠã次ã®åŒãåŸããã:
æåŸã®åŒã¯æ°Bã®æ°åã§åŠã¶ç·åèšå· Σ {\displaystyle \Sigma } ã§ãããç¥ããªãã®ãªãç¡èŠããŠãè¯ãã ãã®åŒã äºé
å®ç(binomial theorem) ãšããããŸããããããã®é
ã«ãããä¿æ°ãäºé
ä¿æ°(binomial coefficient) ãšåŒã¶ããšãããã
(I)
(II)
(II)
ãããããèšç®ããã
äºé
å®çãçšããŠèšç®ããã°ãããå®éã«èšç®ãè¡ãªããšã
(I)
(II)
(III)
ãšãªãã
ãã¹ãŠã®èªç¶æ°nã«å¯ŸããŠ
(I)
(II)
(III)
ãæãç«ã€ããšã瀺ãã
äºé
å®ç
ã«ã€ããŠa,bã«é©åœãªå€ã代å
¥ããã°ããã
(I) a = 1,b=1ã代å
¥ãããšã
ãšãªã確ãã«äžããããé¢ä¿ãæç«ããããšãåããã
(II) a=2,b=1ã代å
¥ãããšã
ãšãªã確ãã«äžããããé¢ä¿ãæç«ããããšãåããã
(III) a=1,b=-1ã代å
¥ãããšã
ãšãªã確ãã«äžããããé¢ä¿ãæç«ããããšãåããã
äºé
å®çãæ¡åŒµã㊠( a + b + c ) n {\displaystyle (a+b+c)^{n}} ãå±éããããšãèãããã a p b q c r {\displaystyle a^{p}b^{q}c^{r}} ( p + q + r = n ) {\displaystyle (p+q+r=n)} ã®é
ã®ä¿æ°ã¯ n {\displaystyle n} åã® ( a + b + c ) {\displaystyle (a+b+c)} ãã p {\displaystyle p} åã® a {\displaystyle a} ã q {\displaystyle q} åã® b {\displaystyle b} ã r {\displaystyle r} åã® c {\displaystyle c} ãéžã¶çµåãã«çãããã n ! p ! q ! r ! {\displaystyle {\frac {n!}{p!q!r!}}} ã§ããã
ããã§ã¯ãæŽåŒã®é€æ³ãšåæ°åŒã«ã€ããŠæ±ããæŽåŒã®é€æ³ã¯ãæŽåŒãæŽæ°ã®ããã«æ±ãé€æ³ãè¡ãªãèšç®ææ³ã®ããšã§ãããå®éã«æŽæ°ã®é€æ³ãšæŽåŒã®é€æ³ã«ã¯æ·±ãã€ãªããããããæŽåŒã®å æ°å解ãèãããšãã以äžå æ°å解ã§ããªãæŽåŒãååšããããã®æŽåŒãæŽæ°ã§ããçŽ å æ°ã®ããã«æ±ãããšã§æŽåŒã®çŽ å æ°å解ãå¯èœã«ãªãã
äžã§ã¯ãæŽåŒãæŽæ°ã«å¯Ÿå¿ããæ§è³ªãæã€ããšãè¿°ã¹ããæŽæ°ã«ã€ããŠã¯ãããã«çŽ ãª2ã€ã®æŽæ°ãåãããšã§æçæ°ãå®çŸ©ããããšãåºæ¥ããæŽåŒã«å¯ŸããŠãåãäºãæç«ã¡ããã®ãããªåŒãåæ°åŒãšåŒã¶ã
åæ°ãçšããªããšãã«ã¯ãæŽæ°ã®é€æ³ã¯åãšäœããçšããŠå®çŸ©ãããããã®æãå²ãããæ°Bã¯åDãšå²ãæ°AãäœãRãçšããŠ
ã®æ§è³ªãæºããããšãç¥ãããŠãããæŽåŒã«å¯ŸããŠã䌌ãæ§è³ªãæç«ã¡ãå²ãããåŒB(x)ãåD(x)ãšå²ãåŒA(x)ãäœãR(x)ãçšããŠã
ã®å³èŸºã§xã«ã€ããŠ2次ã®é
ãçŸãã巊蟺ãšäžèŽããªããªãããã£ãŠåã¯å®æ°ã§ãããåãaãäœããrãšãããšäžã®åŒã¯ã
ãšãªãããããã¯a=1,r=1ã§æç«ããããã£ãŠå1,äœã1ã§ãããããé«æ¬¡ã®åŒã«å¯ŸããŠãåãæ§ã«çããå®ããŠããã°ãããäŸãšããŠã
ã®ãããªåŒãèããããã®å Žåã
ã§ãB(x)ã3次ãA(x)ã2次ã§ããããšãããD(x)ã¯1次ã§ããããŸããR(x)ã¯2次ããå°ããããšãã1次以äžã®åŒã«ãªããããã§ãD(x)=ax+b,R(x)=cx+dãšãããšã
ãåŸããããå³èŸºãå±éãããšã
ãåŸãããããxã«ã©ããªå€ãå
¥ããŠããã®çåŒãæãç«ããªããã°ãªããªãã®ã§ãa = 1, b = 0, -a +c = 0, -b +d = 0ãåŸãããçµå±a=c=1, b=d=0ãåŸãããã
ãã®æ¹æ³ã¯ã©ã®é€æ³ã«å¯ŸããŠãçšããããšãåºæ¥ããã次æ°ãé«ããªããšèšç®ãé£ãããªããæŽæ°ã®å Žåãšåæ§ãæŽåŒã®é€æ³ã§ãçç®ãçšããããšãåºæ¥ããäžã®äŸãçšããŠçµæã ããæžããšã
ã®ããã«ãªãã)å³ã«æžãããåŒãå²ãããåŒã§ããã)å·Šã«æžãããåŒãå²ãåŒã§ããã--ã®äžçªäžã«æžãããåŒã¯åã§ãããæŽæ°ã®å²ãç®åæ§å·Šã«æžãããæ°ããé ã«å²ã£ãŠãããããã§ã¯æ¬¡æ°ã倧ããé
ãããå
ã«èšç®ãããé
ã§ãããå²ãããåŒã®äžã«ããåŒã¯åã®ç¬¬1é
ãå²ãåŒã«ãããŠåŸãåŒã§ãããããã§ã¯ã x ( x 2 â 1 ) {\displaystyle x(x^{2}-1)} ã§ã x 3 â x {\displaystyle x^{3}-x} ãšãªãããã ããæŽæ°ã®é€æ³ãšåæ§ãäœãããããªããŠã¯ãªããªãããã®åŸãå²ãããåŒãã x 3 â x {\displaystyle x^{3}-x} ãåŒããæ®ã£ãåŒãæ°ããå²ãããåŒãšããŠæ±ããããã§ã¯ãåŸãåŒãå²ãåŒãããäœæ¬¡ã§ããããšãããããã§èšç®ã¯çµäºã§ããã
x 3 + 2 x 2 + 1 {\displaystyle x^{3}+2x^{2}+1} ã x 4 + 4 x 2 + 3 x + 2 {\displaystyle x^{4}+4x^{2}+3x+2} ãã x 2 + 2 x + 6 {\displaystyle x^{2}+2x+6} ã§å²ã£ãåãšäœããæ±ããã
ãã®èšç®ã¯ã¢ãã¡ãŒã·ã§ã³ã䜿ã£ãŠ 詳ãã衚瀺ãããŠãããèšç®ææ³ã¯ã æŽæ°ã®å Žåã®çç®ãšåããããªææ³ã䜿ããã
ãåŸãããã®ã§ãå x {\displaystyle x} ãäœã â 6 x + 1 {\displaystyle -6x+1} ã§ããã
2ã€ç®ã®åŒã«ã€ããŠã¯ã
ãåŸãããã ãã£ãŠãç㯠å x 2 â 2 x + 2 {\displaystyle x^{2}-2x+2} ãäœã 11 x â 10 {\displaystyle 11x-10} ã§ããã
ãããŸã§ã§æŽåŒãæŽæ°ã®ããã«æ±ããæŽåŒã®é€æ³ãè¡ãªãæ¹æ³ã«ã€ããŠè¿°ã¹ããããã§ã¯ãæŽåŒã«å¯ŸããŠåæ°åŒãå®çŸ©ããæ¹æ³ã«ã€ããŠè¿°ã¹ããåæ°åŒãšã¯ãæŽæ°ã«å¯Ÿããåæ°ã®ããã«ãé€æ³ã«ãã£ãŠçããåŒã§ãããããã§ãé€æ³ãè¡ãªãåŒã¯ã©ã®ãããªãã®ã§ãå·®ãæ¯ããªããåæ°åŒã§ã¯ãååã«å²ãããåŒãæžããåæ¯ã«å²ãåŒãæžããäŸãã°ã
ã¯ãååx+1ãåæ¯ x 2 + 4 {\displaystyle x^{2}+4} ã®åæ°åŒã§ãããåæ°åŒã«å¯ŸããŠãçŽåãéåãååšãããçŽåã¯å
±éå æ°ãæã£ãåååæ¯ããã€åæ°åŒã§çšããããããã®æã«ã¯åååæ¯ãå
±éå æ°ã§å²ããåŒãç°¡åã«ããããšãåºæ¥ããéåã¯ãåæ°åŒã®å æ³ã®æã«ããçšããããããåååæ¯ã«åãæŽåŒããããŠãåæ°åŒãå€åããªãæ§è³ªãçšããã
ãç°¡åã«ããããŸãã
ãèšç®ããã
ã«ã€ããŠååãšåæ¯ãå æ°å解ãããšãåæ¹ãšãã«
ãå æ°ãšããŠå«ãã§ããããšãåããããã®ãšããå
±éã®å æ°ã¯çŽåããããšãå¿
èŠã§ãããèšç®ãããå€ã¯ã
ãšãªãã
次ã®åé¡ã§ã¯ã
ãèšç®ããããã®ãšãã䞡蟺ã®åæ¯ãããããå¿
èŠãããããä»åã«ã€ããŠã¯ãåçŽã«ããããã®åæ°åŒã®ååãšåæ¯ã«åã
ã®åæ¯ããããŠåæ¯ãçµ±äžããã°ãããèšç®ãããšã
ãšãªãã åæ°åŒã®ä¹æ³ã¯ãåååæ¯ãå¥ã
ã«ãããã°ããã
次ã®èšç®ãããã
(I)
(II)
(I)
(II)
åæ¯ãç©ã®åœ¢ã§ããåæ°åŒãäºã€ã®åæ°åŒã®åãå·®ã§è¡šãããåŒã«å€åœ¢ããæäœãéšååæ°å解ãšããã
1 x ( x + 1 ) {\displaystyle {\frac {1}{x(x+1)}}} ãš 1 ( x + 1 ) ( x + 3 ) {\displaystyle {\frac {1}{(x+1)(x+3)}}} ãåæ°åŒã®åãŸãã¯å·®ã®åœ¢ã§è¡šãã
ãšå€åœ¢ã§ããã®ã§ã
ãšãªããçŽåãããš
ãšãªãã
次ã®åé¡ã§ã¯ã
ãšå€åœ¢ããããšã«ãã£ãŠã
ãšãªãã
ãšæ±ãŸãã
éšååæ°å解ã®æäœãéã«èŸ¿ããšãåæ°åŒã®éåã®æäœãšäžèŽããã ã€ãŸããéšååæ°å解ã¯éåã®éã®æäœã§ããã ååãå®æ°ã®å Žåã«ã¯ãäžãšåæ§ã®æ¹æ³ã§éšååæ°å解ããããšãã§ããã
1. 3 ( x â 9 ) ( x â 4 ) {\displaystyle {\frac {3}{(x-9)(x-4)}}}
2. 7 ( 3 x â 1 ) ( 5 â 2 x ) {\displaystyle {\frac {7}{(3x-1)(5-2x)}}}
éšååæ°å解ã¯æ°åã®åã®èšç®ãç©åèšç®ã埮åãå©çšããäžçåŒã®èšŒæçã«åœ¹ç«ã€ãéèŠãªå€åœ¢ã§ããã
çåŒ ( a + b ) 2 = a 2 + 2 a b + b 2 {\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}} ã¯ãæå a , b {\displaystyle a,b} ã«ã©ã®ãããªå€ã代å
¥ããŠãæãç«ã€ããã®ãããªçåŒãæçåŒ(ãããšããã)ãšããã çåŒ 1 x â 1 + 1 x + 1 = 2 x x 2 â 1 {\displaystyle {\frac {1}{x-1}}+{\frac {1}{x+1}}={\frac {2x}{x^{2}-1}}} ã¯ã䞡蟺ãšã x = 1 , â 1 {\displaystyle x=1,-1} ã代å
¥ããããšã¯ã§ããªããããã®ä»ã®å€ã§ããã°ä»£å
¥ããããšãã§ãããŸãã©ã®ãããªå€ã代å
¥ããŠãçåŒãæãç«ã£ãŠããããããæçåŒãšåŒã¶ã
ãã£ãœãã x 2 â x â 2 = 0 {\displaystyle x^{2}-x-2=0} ã¯ãx=2 ãŸã㯠x=ãŒ1 ã代å
¥ãããšãã ãæãç«ã€ãããã®ããã«æåã«ç¹å®ã®å€ã代å
¥ãããšãã«ã ãæãç«ã€åŒã®ããšãæ¹çšåŒãšåŒã³ãæçåŒãšã¯åºå¥ããã
çåŒ a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} ã x {\displaystyle x} ã«ã€ããŠã®æçåŒã§ããã®ã¯ã©ã®ãããªå ŽåããèããŠã¿ããã ããåŒãã x {\displaystyle x} ã«ã€ããŠã®æçåŒã§ããããšã¯ããã®åŒã® x {\displaystyle x} ã«ã©ã®ãããªå€ã代å
¥ããŠãããã®çåŒã¯æãç«ã€ãšããæå³ã§ããããªã®ã§ãäŸãã° x {\displaystyle x} ã« â 1 , 0 , 1 {\displaystyle -1\ ,\ 0\ ,\ 1} ã代å
¥ããåŒ
ã¯ãã¹ãŠæãç«ã€å¿
èŠããããããã解ããš
ãªã®ã§ãçåŒ a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} ã x {\displaystyle x} ã«ã€ããŠã®æçåŒã«ãªããªãã°ã a = b = c = 0 {\displaystyle a=b=c=0} ã§ãªããã°ãªããªãããšããããã
äžè¬ã«ãçåŒ a x 2 + b x + c = a â² x 2 + b â² x + c â² {\displaystyle ax^{2}+bx+c=a'x^{2}+b'x+c'} ãæçåŒã§ããããšãšã ( a â a â² ) x 2 + ( b â b â² ) x + ( c â c â² ) = 0 {\displaystyle (a-a')x^{2}+(b-b')x+(c-c')=0} ãæçåŒã§ããããšãšåãã§ããã ãã£ãŠ
ãŸãšãããšæ¬¡ã®ããã«ãªãã
次ã®çåŒã x {\displaystyle x} ã«ã€ããŠã®æçåŒãšãªãããã«ã a , b , c {\displaystyle a\ ,\ b\ ,\ c} ã®å€ãæ±ããã
çåŒã®å³èŸºã x {\displaystyle x} ã«ã€ããŠæŽçãããš
ãã®çåŒã x {\displaystyle x} ã«ã€ããŠã®æçåŒãšãªãã®ã¯ã䞡蟺ã®åã次æ°ã®é
ã®ä¿æ°ãçãããšãã§ããããã£ãŠ
ããã解ããš
æçåŒãå©çšããããšã§ãè€éãªåæ°åŒã®éšååæ°å解ãã§ããã
ãšããã åæ¯ãæã£ãŠ
ããªãã¡
ããã x {\displaystyle x} ã®æçåŒãªã®ã§ãä¿æ°ãæ¯èŒããŠ
ããªãã¡
æåã®çåŒã«ä»£å
¥ããŠã
次ã®åé¡ã¯ã
ãšããããšã«ãããäžã®åé¡ãšåæ§ã«ããŠ
ãšæ±ãŸãã®ã§ã
a~fãå®æ°ãšããã
a x 2 + b y 2 + c x y + d x + e y + f = 0 {\displaystyle ax^{2}+by^{2}+cxy+dx+ey+f=0} ãx, yã«ã€ããŠã®æçåŒã ãšããã
巊蟺ãxã«ã€ããŠæŽçãããšã a x 2 + ( c y + d ) x + ( b y 2 + e y + f ) = 0 {\displaystyle ax^{2}+(cy+d)x+(by^{2}+ey+f)=0} ã§ããã
ãããxã«ã€ããŠã®æçåŒãªã®ã§ã a = 0 , c y + d = 0 , b y 2 + e y + f = 0 {\displaystyle a=0,cy+d=0,by^{2}+ey+f=0} ãæãç«ã€ã
ãããã¯æŽã«yã«ã€ããŠã®æçåŒãªã®ã§ã以äžã®çåŒãåŸãããã
éã«ããããæãç«ãŠã°å
ã®åŒã¯æããã«x, yã«ã€ããŠã®æçåŒã§ãã
x 2 + a x y + 6 y 2 â x + 5 y + b = ( x â 2 y + c ) ( x â 3 y + d ) {\displaystyle x^{2}+axy+6y^{2}-x+5y+b=(x-2y+c)(x-3y+d)} ãx,yã«ã€ããŠã®æçåŒãšãªãããã«a,b,c,dãå®ããã
ããã»ã©çŽ¹ä»ãããæçåŒããšããèšèã䜿ã£ãŠã蚌æãã®æå³ã説æãããªãããçåŒã蚌æããããšã¯ããã®åŒãæçåŒã§ããããšã瀺ãããšã§ããã
äžè¬ã«ãçåŒ A=B ã蚌æããããã«ã¯ã次ã®ãããªæé ã®ãããããå®è¡ããã°ããã
ãã®ãšããå€åœ¢ã¯åå€å€åœ¢ã§ãªããã°ãªããªãããšã«æ³šæã
( a + b ) 2 â ( a â b ) 2 = 4 a b {\displaystyle (a+b)^{2}-(a-b)^{2}=4ab} ãæãç«ã€ããšã蚌æããã
(蚌æ) 巊蟺ãå±éãããšã
ãšãªããããã¯å³èŸºã«çããããã£ãŠãçåŒ ( a + b ) 2 â ( a â b ) 2 = 4 a b {\displaystyle (a+b)^{2}-(a-b)^{2}=4ab} ã¯èšŒæãããã(çµ)
( x + y ) 2 + ( x â y ) 2 = 2 ( x 2 + y 2 ) {\displaystyle (x+y)^{2}+(x-y)^{2}=2(x^{2}+y^{2})} ãæãç«ã€ããšã蚌æããã
巊蟺ãèšç®ãããšã
ããã¯å³èŸºã«çããããã£ãŠçåŒãæãç«ã€ããšã蚌æãããã(çµ)
次ã®çåŒãæãç«ã€ããšã蚌æããã (I)
(I) (巊蟺) = ( 36 a 2 + 84 a b + 49 b 2 ) + ( 49 a 2 â 84 a b + 36 a 2 ) = 85 a 2 + 85 b 2 {\displaystyle =(36a^{2}+84ab+49b^{2})+(49a^{2}-84ab+36a^{2})=85a^{2}+85b^{2}} (å³èŸº) = ( 81 a 2 + 36 a b + 4 b 2 ) + ( 4 a 2 â 36 a b + 81 b 2 ) = 85 a 2 + 85 b 2 {\displaystyle =(81a^{2}+36ab+4b^{2})+(4a^{2}-36ab+81b^{2})=85a^{2}+85b^{2}} 䞡蟺ãšãåãåŒã«ãªããã
æçåŒã§ãªããšãããäžããããæ¡ä»¶ããçåŒã蚌æããããšãã§ããã
ããã
ãã£ãŠã a 3 + b 3 + c 3 = 3 a b c {\displaystyle a^{3}+b^{3}+c^{3}=3abc} ã§ããã
ãŸãã
ãããäžåŒã®å³èŸºãkãšãããšã
ãªã®ã§ã
ãã£ãŠã a + c b + d = a â c b â d {\displaystyle {\frac {a+c}{b+d}}={\frac {a-c}{b-d}}} ã§ããã
ãªããæ¯ a : b {\displaystyle a:b} ã«ã€ã㊠a b {\displaystyle {\frac {a}{b}}} ãæ¯ã®å€ãšããããŸãã a : b = c : d ⺠a b = c d {\displaystyle a:b=c:d\iff {\frac {a}{b}}={\frac {c}{d}}} ãæ¯äŸåŒãšããã
a x = b y = c z {\displaystyle {\frac {a}{x}}={\frac {b}{y}}={\frac {c}{z}}} ãæãç«ã€ãšãã a : b : c = x : y : z {\displaystyle a:b:c=x:y:z} ãšè¡šãããããé£æ¯ãšããã
äžçåŒã®ããŸããŸãªå
¬åŒã«ã€ããŠã¯ã次ã®4ã€ã®åŒãåºæ¬çãªåŒãšããŠå°åºã§ããå Žåãããããã
é«æ ¡æ°åŠã§ã¯ã次ã®4ã€ã®æ§è³ªã äžçåŒã®ãåºæ¬æ§è³ªããªã©ãšããŠçŽ¹ä»ãããŠããã
(3)ãš(4)ã«ã€ããŠã¯ãã²ãšã€ã®æ§è³ªãšã㊠ãŸãšããŠããæ€å®æç§æžããã(â» åæ通ãªã©)ã
æ°åŠIAã§ç¿ã£ãããªãã°ãã®æå³ã®èšå· â¹ {\displaystyle \Longrightarrow } ã䜿ããšã
ãšãæžããã
äžè¿°ã®4ã€ã®åºæ¬æ§è³ªããã
ã蚌æããŠã¿ããã
(蚌æ) ãŸã a>0 ãªã®ã§ãåºæ¬æ§è³ª(2)ãã
ã§ããã
ãã£ãŠã
ãªã®ã§ãåºæ¬æ§è³ª(1)ãã a + b > 0 {\displaystyle a+b>0} ãæãç«ã€ã(çµ)
åæ§ã«ããŠã
ã蚌æã§ããã
ãããŸã§ã«ç€ºããããšãããäžçåŒ A ⧠B {\displaystyle A\geqq B} ã蚌æãããå Žåã«ã¯ã
ã蚌æããã°ããããšãããã£ãããã¡ãã®æ¹ã蚌æããããå Žåãããããã
äžçåŒã蚌æããéã«æ ¹æ ãšããåºæ¬çãªäžçåŒãšããŠã次ã®æ§è³ªãããã
ãã®å®ç(ãå®æ°ã2ä¹ãããšãããªãããŒã以äžã§ããã)ããåºæ¬æ§è³ª(3),(4)ã䜿ã£ãŠèšŒæããŠã¿ããã
(蚌æ)
aãæ£ã®å Žåãšè² ã®å Žåãš0ã®å Žåã®3éãã«å Žåããããã
[aãæ£ã®å Žå] ãã®ãšããåºæ¬æ§è³ª(3)ããã
ã§ãããããªãã¡ã
ã§ããã
[aãè² ã®å Žå] ãã®ãšããåºæ¬æ§è³ª(4)ãã 0 a < a a {\displaystyle 0a<aa} ã§ãããããªãã¡ã
ã§ããã
[aããŒãã®å Žå] ãã®ãšãã a 2 = 0 {\displaystyle a^{2}=0} ã§ããã
ãã£ãŠããã¹ãŠã®å Žåã«ã€ã㊠a 2 ⧠0 {\displaystyle a^{2}\geqq 0} (çµ)
ãã®ããšãšåºæ¬æ§è³ª(1)(2)ããã次ãæãç«ã€ããšããããã
次ã®äžçåŒãæãç«ã€ããšã蚌æããã
(蚌æ)
ã蚌æããã°ããã
巊蟺ãå±éã㊠ãŸãšãããšã
ãšãªãã
äžåŒã®æåŸã®åŒã®é
ã«ã€ããŠã
ã ããã
ã§ããããã£ãŠ
ã§ããã(çµ)
2ã€ã®æ£ã®æ° a, b ã a>b ãŸã㯠aâ§b ãªãã°ã䞡蟺ã2ä¹ããŠã倧å°é¢ä¿ã¯åããŸãŸã§ããã
ã€ãŸãã
ã§ããã
a>bãšãããä»®å®ãããa,b ã¯æ£ã®æ°ãªã®ã§ã ( a + b ) > 0 {\displaystyle (a+b)>0} ã§ãããå¥ã®ä»®å®ããã a > b ãªã®ã§ã ( a â b ) > 0 {\displaystyle (a-b)>0} ã§ãããããã£ãŠã a 2 â b 2 = ( a + b ) ( a â b ) > 0 {\displaystyle a^{2}-b^{2}=(a+b)(a-b)>0}
éã«ã a 2 â b 2 > 0 {\displaystyle a^{2}-b^{2}>0} ã®ãšãã ( a + b ) ( a â b ) > 0 {\displaystyle (a+b)(a-b)>0} ã§ããã a > 0 , b > 0 {\displaystyle a>0,b>0} ãªã®ã§ a + b > 0 {\displaystyle a+b>0} ã§ããããã£ãŠã a â b > 0 {\displaystyle a-b>0} ãªã®ã§ã a > b {\displaystyle a>b} ã§ããã
ãã£ãŠã a > b ⺠a 2 > b 2 {\displaystyle a>b\quad \Longleftrightarrow \quad a^{2}>b^{2}} ã§ããã
aâ§bã®å Žåãåæ§ã«èšŒæã§ããã
ç·Žç¿ãšããŠã次ã®åé¡ãåããŠã¿ããã
a > 0 {\displaystyle a>0} , b > 0 {\displaystyle b>0} ã®ãšãã次ã®äžçåŒã蚌æããã
(蚌æ) äžçåŒã®äž¡èŸºã¯æ£ã§ããã®ã§ã䞡蟺ã®å¹³æ¹ã®å·®ãèããã°ããã䞡蟺ã®å¹³æ¹ã®å·®ã¯
ã§ãããããã§ãa,b ã¯ãšãã«æ£ã®å®æ°ãªã®ã§ã
ã§ããããšãçšããã
ã§ããã®ã§ã
ãšãªãããã£ãŠã
ã§ããã(çµ)
å®æ° a ã®çµ¶å¯Ÿå€ |a| ã«ã€ããŠã
ã§ããããã次ã®ããšãæãç«ã€ã
|a|â§a , |a|⧠ãŒa , |a|=a
ãŸãã2ã€ã®å®æ° a, b ã®çµ¶å¯Ÿå€ |ab| ã«ã€ããŠã¯ã
ãæãç«ã€ã®ã§ãããã«ããã« |ab|â§0 , |a||b|â§0 ãçµã¿åãããŠã
|ab| = |a| |b| ãæãç«ã€ã
(äŸé¡)
次ã®äžçåŒã蚌æããããŸããçå·ãæãç«ã€ã®ã¯ ã©ã®ãããªå Žåãã 調ã¹ãã
䞡蟺ã®å¹³æ¹ã®å·®ãèãããšã
ãããããæ£ãªããäžããããäžçåŒ |a|+|b| ⧠|a+b| ãæ£ããã
ããã§ã |a| |b| ⧠ab ã§ããã®ã§ã
ã§ããã
ãããã£ãŠã |a|+|b| ⧠|a+b| ã§ããã
çå·ãæãç«ã€ã®ã¯ |a| |b| = ab ã®å Žåãããªãã¡ ab ⧠0 ã®å Žåã§ããã(蚌æ ããã)
2ã€ã®æ° a {\displaystyle a} , b {\displaystyle b} ã«å¯Ÿãã a + b 2 {\displaystyle {\frac {a+b}{2}}} ãçžå å¹³å(ããããžããã)ãšèšãã a b {\displaystyle {\sqrt {ab}}} ãçžä¹å¹³å(ããããããžããã)ãšããã
æ¬ããŒãžã§ã¯ã2åã®æ°ã®å¹³åã«ã€ããŠèå¯ããã
çžå å¹³åãšçžä¹å¹³åã«ã€ããŠã次ã®é¢ä¿åŒãæãç«ã€ã
(蚌æ)
a ⧠0 , b ⧠0 {\displaystyle a\geqq 0,b\geqq 0} ã®ãšã
( a â b ) 2 ⧠0 {\displaystyle \left({\sqrt {a}}-{\sqrt {b}}\right)^{2}\geqq 0} ã§ããããã ( a â b ) 2 2 ⧠0 {\displaystyle {\frac {\left({\sqrt {a}}-{\sqrt {b}}\right)^{2}}{2}}\geqq 0} ãããã£ãŠ a + b 2 ⧠a b {\displaystyle {\frac {a+b}{2}}\geqq {\sqrt {ab}}} çå·ãæãç«ã€ã®ã¯ã ( a â b ) 2 = 0 {\displaystyle \left({\sqrt {a}}-{\sqrt {b}}\right)^{2}=0} ã®ãšããããªãã¡ a = b {\displaystyle a=b} ã®ãšãã§ããã(蚌æ ããã)
å
¬åŒã®å©çšã§ã¯ãäžã®åŒ a + b 2 ⧠a b {\displaystyle {\frac {a+b}{2}}\geqq {\sqrt {ab}}} ã®äž¡èŸºã«2ãããã a + b ⧠2 a b {\displaystyle a+b\geqq 2{\sqrt {ab}}} ã®åœ¢ã®åŒã䜿ãå Žåãããã
a > 0 {\displaystyle a>0} , b > 0 {\displaystyle b>0} ã®ãšãã次ã®äžçåŒãæãç«ã€ããšã蚌æããã (I)
(II)
(I) a > 0 {\displaystyle a>0} ã§ããããã 1 a > 0 {\displaystyle {\frac {1}{a}}>0} ãã£ãŠ a + 1 a ⧠2 a à 1 a = 2 {\displaystyle a+{\frac {1}{a}}\geqq 2{\sqrt {a\times {\frac {1}{a}}}}=2} ãããã£ãŠ
(II)
a > 0 {\displaystyle a>0} , b > 0 {\displaystyle b>0} ã§ããããã b a > 0 {\displaystyle {\frac {b}{a}}>0} , a b > 0 {\displaystyle {\frac {a}{b}}>0} ãã£ãŠ b a + a b + 2 ⧠2 b a à a b + 2 = 2 + 2 = 4 {\displaystyle {\frac {b}{a}}+{\frac {a}{b}}+2\geqq 2{\sqrt {{\frac {b}{a}}\times {\frac {a}{b}}}}+2=2+2=4} ãããã£ãŠ
2ä¹ããŠè² ã«ãªãæ°ããšãããã®ãèããããã®ãããªæ°ã¯ãäžåŠã§ç¿ã£ãå®æ°ã®äžã«ã¯ãªãããšããããããªããªãã°ãæ£ã®æ°ã§ãè² ã®æ°ã§ã2ä¹ãããšç¬Šå·ãæã¡æ¶ããŠæ£ã®æ°ã«ãªã£ãŠããŸãããã§ãããããã§é«æ ¡ã§ã¯ã2ä¹ããŠè² ã«ãªããšããæ§è³ªãæã€æ°ã®æŠå¿µãæ°ããå°å
¥ããããšã«ããã
ãšããæ¹çšåŒãèããããã®æ¹çšåŒã®è§£ã¯å®æ°ã«ã¯ãªããããã§ããã®æ¹çšåŒã®è§£ãšãªãæ°ãæ°ããäœãããã®åäœãæå i {\displaystyle i} ã§ããããã
ãã® i {\displaystyle i} ã®ããšãèæ°åäœ(ããããããã)ãšåŒã¶ã(èæ°åäœã®èšå· i ãè±èªã®ã¢ã«ãã¡ãããã®ã¢ã€ã®å°æåã§ã imaginary unit ã«ç±æ¥ãããšèããããŠããã)
1 + i {\displaystyle 1+i} ã 2 + 5 i {\displaystyle 2+5i} ã®ããã«ãèæ°åäœ i {\displaystyle i} ãšå®æ° a , b {\displaystyle a,b} ãçšããŠ
ãšè¡šãããšãã§ããæ°ãè€çŽ æ°(ãµãããã)ãšããããã®ãšããaããã®è€çŽ æ°ã®å®éš(ãã€ã¶)ãšãããbãèéš(ããã¶)ãšããã
äŸãã°ã 1 + i , 2 + 5 i , 9 2 + 7 2 i , 4 i , 3 {\displaystyle 1+i,\quad 2+5i,\quad {\frac {9}{2}}+{\frac {7}{2}}i,\quad 4i,\quad 3} ã¯ãããããè€çŽ æ°ã§ããã
è€çŽ æ° a+bi ã¯(ãã ã aãšbã¯å®æ°)ãbã0ã®å Žåã«ããããå®æ°ãšèŠãããšãã§ããã
èšãæ¹ãããããšãè€çŽ æ°ãåºæºã«èãããšãå®æ°ãšã¯ã a+0i ã®ãããªãèéšã®ä¿æ°ããŒãã«ãªãè€çŽ æ°ã®ããšã§ãããšãèšããã
4iã®ãããªãèéšã0以å€ã§å®éšããŒãã®è€çŽ æ°ãçŽèæ°(ãã
ããããã)ãšåŒã¶ãçŽèæ°ã¯ã2ä¹ãããšè² ã«ãªãæ°ã§ããã å®æ°ãèéšã0ã®è€çŽ æ°ãšèããããã
å®æ°ã§ãªãè€çŽ æ°ã®ããšããèæ°ã(ãããã)ãšããã
2ã€ã®è€çŽ æ° a+bi ãš c+di ãšãçãããšã¯ã
ã§ããããšã§ããã
ã€ãŸãã
ãšãã«ãè€çŽ æ°a+bi ã 0ã§ãããšã¯ãa=0 ã〠b=0 ã§ããããšã§ããã
è€çŽ æ° z = a + b i {\displaystyle z=a+bi} ã«å¯ŸããŠãèéšã®ç¬Šå·ãå転ãããè€çŽ æ° a â b i {\displaystyle a-bi} ã®ããšããå
±åœ¹(ããããã)ãªè€çŽ æ°ããŸãã¯ãè€çŽ æ° z {\displaystyle z} ã®å
±åœ¹ãã®ããã«åŒã³ã z Ì {\displaystyle {\bar {z}}} ã§ããããããªãããå
±åœ¹ãã¯ãå
±è»ãã®åžžçšæŒ¢åã«ããæžãæãã§ããã
å®æ°aãšå
±åœ¹ãªè€çŽ æ°ã¯ããã®å®æ° a èªèº«ã§ããã
è€çŽ æ° z=a+bi ã«ã€ããŠ
è€çŽ æ°ã«ãååæŒç®(å æžä¹é€)ãå®çŸ©ãããã
è€çŽ æ°ã®æŒç®ã§ã¯ãèæ°åäœ i {\displaystyle i} ããéåžžã®æåã®ããã«æ±ã£ãŠèšç®ãããäžè¬ã«è€çŽ æ° z , w {\displaystyle z\ ,\ w} ãã z = a + b i , w = c + d i {\displaystyle z=a+bi\ ,\ w=c+di} ã§äžãããããšã(ãã ã a , b , c , d {\displaystyle a\ ,\ b\ ,\ c\ ,\ d} ã¯å®æ°ãšãã)ã
ãšãããµãã«è€çŽ æ°ã®å æžä¹é€ã®èšç®æ³ãå®ããããŠããã
ä¹æ³ã®å®çŸ©ã¯ãäžèŠãããšé£ãããã«ã¿ããããå®æ°ã®åé
æ³åãšåæ§ã«å±éããŠããæåŸã« iã«ãã€ãã¹1ã代å
¥ããŠãã£ãã ãã§ããã
é€æ³ã®å®çŸ©ã¯ãååãšåæ¯ã«ãåæ¯ãšå
±åœ¹ãªåœ¢ã®åŒã æãç® ããã ãã§ããã
ä¹æ³ãé€æ³ã®å®çŸ©åŒãæèšããå¿
èŠã¯ç¡ããèšç®ã®éã«ã¯ãå¿
èŠã«å¿ããŠåé
æ³åãå
±åœ¹ãªã©ã®ãå¿
èŠãªåŒå€åœ¢ãè¡ãã°ããã
äŸé¡
2ã€ã®è€çŽ æ°
ã«ã€ããŠã a + b {\displaystyle a+b} ãš a b {\displaystyle ab} ãš a b {\displaystyle {\frac {a}{b}}} ããããããèšç®ããã
解ç
ã§ããã
ããããã«ç°¡åã«ã§ããªãã ããããå®ã¯ãã¡ãã£ãšãããã¯ããã¯ãçšããã°ããèŠããã圢ã«ã§ããã
åæ°ã¯åæ¯ãšååã«åãæ°ããããŠããã£ãã®ã§ãåæ¯ãšååã«åæ¯ã®å
±åœ¹ããããŠã¿ãããããšã
ãåŸãããããã®åœ¢ã®ã»ããå
ã®åŒããããã£ãšèŠããã圢ã§ããã
ãã®ãããªæäœãåæ¯ã®å®æ°åãšããããšããããæ°åŠIã§åŠç¿ããå±éã»å æ°å解å
¬åŒ ( a + b ) ( a â b ) = a 2 â b 2 {\displaystyle (a+b)(a-b)=a^{2}-b^{2}} ã®ç°¡åãªå¿çšã§ããã
æ°ã®ç¯å²ãè€çŽ æ°ã«ãŸã§æ¡åŒµãããšãè² ã®æ°ã®å¹³æ¹æ ¹ãèããããšãã§ããã
äŸãšããŠã -5 ã®å¹³æ¹æ ¹ã«ã€ããŠèããŠã¿ããã
ã§ããããã -5 ã®å¹³æ¹æ ¹ã¯ 5 i {\displaystyle {\sqrt {5}}\ i} ãš â 5 i {\displaystyle -{\sqrt {5}}\ i} ã§ããã
â 5 {\displaystyle {\sqrt {-5}}} ãšã¯ã 5 i {\displaystyle {\sqrt {5}}\ i} ã®ããšãšããã â â 5 {\displaystyle -{\sqrt {-5}}} ãšã¯ã â 5 i {\displaystyle -{\sqrt {5}}\ i} ã®ããšã§ããã ãšãã« â 1 = i {\displaystyle {\sqrt {-1}}\ =\ i} ã§ããã
ããŠã-5 ã®å¹³æ¹æ ¹ã¯ãæ¹çšåŒ x 2 = â 5 {\displaystyle x^{2}=-5} ã®è§£ã§ãããã
ãã®æ¹çšåŒã移é
ããããšã«ããã-5 ã®å¹³æ¹æ ¹ã¯ã
ã®è§£ã§ãããšããããã
ããã«å æ°å解ãããããšã«ããã-5 ã®å¹³æ¹æ ¹ã¯æ¹çšåŒ
ã®è§£ã§ããããšããããã
(I) â 2 â 6 {\displaystyle {\sqrt {-2}}\ {\sqrt {-6}}} ãèšç®ããã
(I)
ãã®ããã«ããŸãããã€ãã¹ã®æ°ã®å¹³æ¹æ ¹ãåºãŠãããããŸãèæ°åäœ i ãçšããåŒã«æžãæããã
ãã®ããšãããç®ãããŠããã
(II) 2 â 3 {\displaystyle {\frac {\sqrt {2}}{\sqrt {-3}}}} ãèšç®ããã
(III) 2次æ¹çšåŒ x 2 = â 7 {\displaystyle x^{2}=-7} ã解ãã
(II)
(III)
è€çŽ æ°ã®å¿çšãšããŠãããã§ã¯2次æ¹çšåŒã®æ§è³ªã«ã€ããŠè¿°ã¹ããä»»æã®2次æ¹çšåŒã¯ã解ã®å
¬åŒã«ãã£ãŠè§£ãããããšãé«çåŠæ ¡æ°åŠIã§è¿°ã¹ãããããã解ã®å
¬åŒã«å«ãŸããæ ¹å·ã®äžèº«ãè² ã®æ°ã®å Žåã«ã¯å®æ°è§£ãååšããªãããšã«æ³šæããå¿
èŠãããã2次æ¹çšåŒ
ã®è§£ã®å
¬åŒã¯ã
ã§ããã
å€å¥åŒ D {\displaystyle D} ã¯
ã«ãã£ãŠå®çŸ©ããããå€å¥åŒã¯ã解ã®å
¬åŒã®æ ¹å·(ã«ãŒãèšå·ã®ããš)ã®äžèº«ã«çãããå€å¥åŒã®æ£è² ã«ãã£ãŠ2次æ¹çšåŒãå®æ°è§£ãæã€ãã©ããã決ãŸãã
D {\displaystyle D} ãè² ã®ãšãã«ã¯ãã®2次æ¹çšåŒã¯å®æ°ã®ç¯å²ã«ã¯è§£ãæããªãã
å€å¥åŒ D {\displaystyle D} ãè² ã®æ°ã§ãã£ããšããxã®è§£ã¯ç°ãªã2ã€ã®èæ°ã«ãªãããã®2ã€ã®è§£ã¯ å
±åœ¹ ã®é¢ä¿ã«ãªã£ãŠããã
è€çŽ æ°ãçšããŠã2次æ¹çšåŒ (1)
(2)
(3)
ã解ãã
解ã®å
¬åŒãçšããŠè§£ãã°ããã(1)ã ããèšç®ãããšã
ãšãªãã ä»ãåãããã«æ±ãããšãåºæ¥ãã
以éã®è§£çã¯ã (2)
(3)
ãšãªãã
æ¹çšåŒã®è§£ã§ãå®æ°ã§ãããã®ã å®æ°è§£ ãšããã
æ¹çšåŒã®è§£ã§ãèæ°ã§ãããã®ã èæ°è§£ ãšããã
2次æ¹çšåŒ a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} ã®è§£ã¯ x = â b ± b 2 â 4 a c 2 a {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}} ã§ããã
2次æ¹çšåŒã®è§£ã®çš®é¡ã¯ã解ã®å
¬åŒã®äžã®æ ¹å·ã®äžã®åŒ b 2 â 4 a c {\displaystyle b^{2}-4ac} ã®ç¬Šå·ãèŠãã°å€å¥ããããšãã§ããã
ãã®åŒ b 2 â 4 a c {\displaystyle b^{2}-4ac} ãã2次æ¹çšåŒ a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} ã®å€å¥åŒ(ã¯ãã¹ã€ãã)ãšãããèšå· D {\displaystyle D} ã§è¡šãã
ãŸããé解ãå®æ°è§£ã§ããã®ã§ã
ãšãããã
次ã®2次æ¹çšåŒã®è§£ãå€å¥ããã
(I)
(II)
(III)
(I)
ã ãããç°ãªã2ã€ã®å®æ°è§£ããã€ã
(II)
ã ãããç°ãªã2ã€ã®èæ°è§£ããã€ã
(III)
ã ãããé解ããã€ã
ãŸãã2次æ¹çšåŒ a x 2 + 2 b â² x + c = 0 {\displaystyle ax^{2}+2b'x+c=0} ã®ãšãã D = 4 ( b â² 2 â a c ) {\displaystyle D=4(b'^{2}-ac)} ãšãªãã®ã§ã 2次æ¹çšåŒ a x 2 + 2 b â² x + c = 0 {\displaystyle ax^{2}+2b'x+c=0} ã®å€å¥åŒã«ã¯
ããã¡ããŠãããã
ãããçšããŠãåã®åé¡
ã®è§£ãå€å¥ãããã
a = 4 , b â² = â 10 , c = 25 {\displaystyle a=4\,,\,b'=-10\,,\,c=25} ã§ãããã
ã ãããé解ããã€ã
2次æ¹çšåŒ a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} ã®2ã€ã®è§£ã α {\displaystyle \alpha } , β {\displaystyle \beta } ãšããã ãã®æ¹çšåŒã¯ã
a ( x â α ) ( x â β ) = 0 {\displaystyle a(x-\alpha )(x-\beta )=0}
ãšå€åœ¢ã§ããã
ãããå±éãããšã
a x 2 â a ( α + β ) x + a α β = 0 {\displaystyle ax^{2}-a(\alpha +\beta )x+a\alpha \beta =0}
ä¿æ°ãæ¯èŒããŠã
c = a α β , b = â a ( α + β ) {\displaystyle c=a\alpha \beta ,b=-a(\alpha +\beta )}
ãåŸãã
ãããå€åœ¢ããã°ã α + β = â b a , α β = c a {\displaystyle \alpha +\beta =-{\frac {b}{a}},\alpha \beta ={\frac {c}{a}}} ãšãªãã
2次æ¹çšåŒ 2 x 2 + 4 x + 3 = 0 {\displaystyle 2x^{2}+4x+3=0} ã®2ã€ã®è§£ã α {\displaystyle \alpha } , β {\displaystyle \beta } ãšãããšãã α 2 + β 2 {\displaystyle \alpha ^{2}+\beta ^{2}} ã®å€ãæ±ããã
解ãšä¿æ°ã®é¢ä¿ããã α + β = â 4 2 = â 2 {\displaystyle \alpha +\beta =-{\frac {4}{2}}=-2} , α β = 3 2 {\displaystyle \alpha \beta ={\frac {3}{2}}} α 2 + β 2 = ( α + β ) 2 â 2 α β = ( â 2 ) 2 â 2 à 3 2 = 1 {\displaystyle \alpha ^{2}+\beta ^{2}=(\alpha +\beta )^{2}-2\alpha \beta =(-2)^{2}-2\times {\frac {3}{2}}=1}
2ã€ã®æ° α {\displaystyle \alpha } , β {\displaystyle \beta } ã解ãšãã2次æ¹çšåŒã¯
ãšè¡šãããã巊蟺ãå±éããŠæŽçãããšæ¬¡ã®ããã«ãªãã
次ã®2æ°ã解ãšãã2次æ¹çšåŒãäœãã
(I)
(II)
(I) å ( 3 + 5 ) + ( 3 â 5 ) = 6 {\displaystyle (3+{\sqrt {5}})+(3-{\sqrt {5}})=6} ç© ( 3 + 5 ) ( 3 â 5 ) = 4 {\displaystyle (3+{\sqrt {5}})(3-{\sqrt {5}})=4} ã§ãããã
(II) å ( 2 + 3 i ) + ( 2 â 3 i ) = 4 {\displaystyle (2+3i)+(2-3i)=4} ç© ( 2 + 3 i ) ( 2 â 3 i ) = 13 {\displaystyle (2+3i)(2-3i)=13} ã§ãããã
2次æ¹çšåŒ a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} ã®2ã€ã®è§£ α {\displaystyle \alpha } , β {\displaystyle \beta } ãããããšã2次åŒ
ãå æ°å解ããããšãã§ããã 解ãšä¿æ°ã®é¢ä¿ α + β = â b a {\displaystyle \alpha +\beta =-{\frac {b}{a}}} , α β = c a {\displaystyle \alpha \beta ={\frac {c}{a}}} ããã
2次æ¹çšåŒã¯ãè€çŽ æ°ã®ç¯å²ã§èãããšã€ãã«è§£ããã€ãããè€çŽ æ°ãŸã§äœ¿ã£ãŠãããšãããšã2次åŒã¯å¿
ã1次åŒã®ç©ã«å æ°å解ããããšãã§ããã
è€çŽ æ°ã®ç¯å²ã§èããŠã次ã®2次åŒãå æ°å解ããã
(I)
(II)
(I) 2次æ¹çšåŒ x 2 + 4 x â 1 = 0 {\displaystyle x^{2}+4x-1=0} ã®è§£ã¯
ãã£ãŠ
(II) 2次æ¹çšåŒ 2 x 2 â 3 x + 2 = 0 {\displaystyle 2x^{2}-3x+2=0} ã®è§£ã¯
ãã£ãŠ
3次以äžã®æŽåŒã«ããæ¹çšåŒãèããã äžè¬ã«æ¹çšåŒã P ( x ) = 0 {\displaystyle P(x)=0} ãšãšãã ãã ãã P ( x ) {\displaystyle P(x)} ã¯ãä»»æã®æ¬¡æ°ã®æŽåŒãšããã
P ( x ) {\displaystyle P(x)} ã1æ¬¡åŒ x â a {\displaystyle x-a} ã§å²ã£ããšãã®åã Q ( x ) {\displaystyle Q(x)} ãäœãã R {\displaystyle R} ãšãããšã
ãã®äž¡èŸºã® x {\displaystyle x} ã« a {\displaystyle a} ã代å
¥ãããšã
ã€ãŸãã P ( x ) {\displaystyle P(x)} ã x â a {\displaystyle x-a} ã§å²ã£ããšãã®äœã㯠P ( a ) {\displaystyle P(a)} ã§ããã
æŽåŒ P ( x ) = x 3 â 2 x + 3 {\displaystyle P(x)=x^{3}-2x+3} ã次ã®åŒã§å²ã£ãäœããæ±ããã (I)
(II)
(III)
(I) P ( 2 ) = 2 3 â 2 Ã 2 + 3 = 7 {\displaystyle P(2)=2^{3}-2\times 2+3=7} (II) P ( â 1 ) = ( â 1 ) 3 â 2 Ã ( â 1 ) + 3 = 4 {\displaystyle P(-1)=(-1)^{3}-2\times (-1)+3=4} (III) P ( 1 2 ) = ( 1 2 ) 3 â 2 Ã ( 1 2 ) + 3 = 17 8 {\displaystyle P\left({\frac {1}{2}}\right)=\left({\frac {1}{2}}\right)^{3}-2\times \left({\frac {1}{2}}\right)+3={\frac {17}{8}}}
ããå®æ° a {\displaystyle a} ã«å¯ŸããŠã
ãæãç«ã£ããšããã ãã®ãšããæŽåŒ P ( x ) {\displaystyle P(x)} ã¯ã ( x â a ) {\displaystyle (x-a)} ãå æ°ã«æã€ããšãåãã ãã®ããšãå æ°å®ç(ãããããŠãã)ãšåŒã¶ã
æŽåŒ P ( x ) {\displaystyle P(x)} ã«å¯ŸããŠãå Q ( x ) {\displaystyle Q(x)} ãå²ãåŒ ( x â a ) {\displaystyle (x-a)} ãšãã æŽåŒã®é€æ³ãçšããããã®ãšããå Q ( x ) {\displaystyle Q(x)} ã ( Q ( x ) {\displaystyle Q(x)} ã¯ã P ( x ) {\displaystyle P(x)} ããã1ã ã次æ°ãäœãæŽåŒã§ããã) äœã c {\displaystyle c} ( c {\displaystyle c} ã¯ãå®æ°ã)ãšãããšã æŽåŒ P ( x ) {\displaystyle P(x)} ã¯ã
ãšæžããã ããã§ã c = 0 {\displaystyle c=0} ã§ãªããšã P ( a ) = 0 {\displaystyle P(a)=0} ã¯æºããããªããã ãã®ãšãã P ( x ) {\displaystyle P(x)} ã¯ã ( x â a ) {\displaystyle (x-a)} ã«ãã£ãŠå²ãåããã ãã£ãŠãå æ°å®çã¯æç«ããã
å æ°å®çãçšããããšã§ããã次æ°ã®é«ãæŽåŒãå æ°å解ããããšã åºæ¥ãããã«ãªããäŸãã°ã3次ã®æŽåŒ
ã«ã€ããŠã x = 1 {\displaystyle x=1} ã代å
¥ãããšã
ã¯0ãšãªãããã£ãŠãå æ°å®çãããã®åŒã¯
ãå æ°ãšããŠæã€ã
ããã§ãå®éæŽåŒã®é€æ³ã䜿ã£ãŠèšç®ãããšã
ãåŸãããã
å æ°å®çãçšã㊠(I)
(II)
ãå æ°å解ããã
(I) å æ°å解ã®çµæã(x+æŽæ°)ã®ç©ã®åœ¢ãªããæŽæ°ã¯6ã®å æ°ã§ãªããã°ãªããªãããã®ããã ± 1 , ± 2 , ± 3 , ± 6 {\displaystyle \pm 1,\pm 2,\pm 3,\pm 6} ãåè£ãšãªãããããã«ã€ããŠã¯å®éã«ä»£å
¥ããŠç¢ºããããããªããx=1ã代å
¥ãããšã
ãšãªãã®ã§ã(x-1)ãå æ°ãšãªããå®éã«æŽåŒã®é€æ³ãè¡ãªããšãåãšã㊠x 2 â 5 x + 6 {\displaystyle x^{2}-5x+6} ãåŸããããããã㯠( x â 2 ) ( x â 3 ) {\displaystyle (x-2)(x-3)} ã«å æ°å解ã§ããããã£ãŠçãã¯ã
ãšãªãã (II) ããã§ãå°éã«24ã®å æ°ãåœãŠã¯ããŠãããããªãã24ã®å æ°ã¯æ°ãå€ãã®ã§ããªãã®èšç®ãå¿
èŠãšãªããããã§ã¯ã-2ã代å
¥ãããšã
ãšãªãã(x+2)ãå æ°ã ãšããããé€æ³ãè¡ãªããšã x 2 â x â 12 {\displaystyle x^{2}-x-12} ãåŸããããã(x-4)(x+3)ã«å æ°å解ã§ãããçãã¯ã
ãšãªãã
å æ°å解ãå æ°å®çãå©çšããŠé«æ¬¡æ¹çšåŒã解ããŠã¿ããã
é«æ¬¡æ¹çšåŒ (I)
(II)
(III)
ã解ãã
(I) 巊蟺ã a 3 â b 3 = ( a â b ) ( a 2 + a b + b 2 ) {\displaystyle a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})} ãçšããŠå æ°å解ãããš
ãããã£ãŠ x â 2 = 0 {\displaystyle \ x-2=0} ãŸã㯠x 2 + 2 x + 4 = 0 {\displaystyle \ x^{2}+2x+4=0} ãã£ãŠ
(II) x 2 = X {\displaystyle \ x^{2}=X\ } ãšãããšã
巊蟺ãå æ°å解ãããš
ãã£ãŠ X = 4 , X = â 2 {\displaystyle X=4\ ,\ X=-2} ããã« x 2 = 4 , x 2 = â 2 {\displaystyle x^{2}=4\ ,\ x^{2}=-2} ãããã£ãŠ
(III) P ( x ) = x 3 â 5 x 2 + 7 x â 2 {\displaystyle \ P(x)=x^{3}-5x^{2}+7x-2\ } ãšããã
ãããã£ãŠã x â 2 {\displaystyle \ x-2\ } 㯠P ( x ) {\displaystyle \ P(x)\ } ã®å æ°ã§ããã
ãã£ãŠ ( x â 2 ) ( x 2 â 3 x + 1 ) = 0 {\displaystyle (x-2)(x^{2}-3x+1)=0} x â 2 = 0 {\displaystyle \ x-2=0} ãŸã㯠x 2 â 3 x + 1 = 0 {\displaystyle \ x^{2}-3x+1=0} ãããã£ãŠ
3次æ¹çšåŒ a x 3 + b x 2 + c x + d = 0 {\displaystyle ax^{3}+bx^{2}+cx+d=0} ã®3ã€ã®è§£ã ã α , β , γ {\displaystyle \alpha \ ,\ \beta \ ,\ \gamma } ãšãããš
ãæãç«ã€ã å³èŸºãå±éãããš
ãã£ãŠ
ããã«
ãããã£ãŠã次ã®ããšãæãç«ã€ã
ãã°ãã°èæ°ã¯ãçŸå®ã«ã¯ååšããªãæ°ãã§ãããšèšãããããšããããæŽå²çã«ãèæ°ãæ±ã£ãæ°åŠãèããã¹ãã§ã¯ãªããšèããããæ代ã¯é·ãã£ãããã®æ代ã®å
é²çãªæ°åŠè
ã®äžã«ã¯ãèæ°ãæå¹ã«æŽ»çšããŠç 究ãé²ããäžæ¹ã§ãææãçºè¡šããéã«ã¯èæ°ãè¡šã«åºããã«èšè¿°ããåªåãããããšã§ãç¡çšãªæµæãåããªãããã«å·¥å€«ããè
ããããšèšãããã»ã©ã§ããã
ã ããããèããŠã¿ãã°ãæ°ããçŸå®ã«ååšããããšã¯ã©ãããæå³ãªã®ã ããããçŸå®ã«éçã䜿ã£ãŠçŽã«åãæããªãã°ãååšã®é·ãããæ£ç¢ºã«ååšçãã®ãã®ã«ãããããšã¯äžå¯èœã§ããããã«æããããããã®å²ã«ååšçãšããå®æ°ã¯ãååšããããšæããããã®ã¯ãªãã ããããæ°çŽç·ãå®æ°ã®ãå®åšããä¿¡ãããããªãã°ãè€çŽ æ°ã¯è€çŽ æ°å¹³é¢(æ°åŠCã§ç¿ã)ã®äžã«ååšããã®ã ãããåãã§ã¯ãªãã ãããã
ãã®ããã«èãããšãããããæ°ãšã¯ãã¹ãŠããæå³ã§æ³åäžã®ååšã§ãããããã«å¯ŸããŠãååšããããååšããªãããšããåããç«ãŠãããšããã³ã»ã³ã¹ã§ããããã«æãããããååšããªããããã«æãããã¡ãªèæ°ã§ããããããšãã°ç©çåŠã®äžåéã§ããéåååŠã®ã·ã¥ã¬ãã£ã³ã¬ãŒæ¹çšåŒã«è¡šãããªã©ãå¿çšäžã®ããŸããŸãªå Žé¢ã«ãããŠããèæ°ã䜿ã£ãŠèšè¿°ããããšãèªç¶ãªå¯Ÿè±¡ã¯å€ãã®ã ã
è€çŽ æ°ã©ããã«ã€ããŠããã®å€§å°é¢ä¿ã¯å®çŸ©ããªãããã®çç±ã¯ãã©ã®ããã«å€§å°é¢ä¿ãå®çŸ©ããŠãã䟿å©ãªæ§è³ªãæºããããšãã§ããªãããã§ãããå
·äœçã«èšãã°ãæ¢ã«è¿°ã¹ãå®æ°ã®å€§å°é¢ä¿ã«ã€ããŠã®ãäžçåŒã®åºæ¬æ§è³ª(1)(2)(3)(4)ãã«ãããåŒãæãç«ãããããšãã§ããªãã®ã ã
ããšãã°ã a + b i < a â² + b â² i {\displaystyle a+bi<a'+b'i} ã§ããããšãã a 2 + b 2 < a â² 2 + b â² 2 {\displaystyle a^{2}+b^{2}<a'^{2}+b'^{2}} ã§ããããšãšããŠå®çŸ©ããŠã¿ããããã®ããã«å®çŸ©ãããšãããšãã°1+2i<2-3iã§ããããŸã2+3i<3+4iã§ããããšãããã(1+2i)+(2+3i)=3+5i,(2-3i)+(3+4i)=5+iã§ããã3+5i>5+iãšãªã£ãŠããŸããããã¯åºæ¬æ§è³ª(2)ãæãç«ããªãããšãæå³ããã
ãã¡ããããã¯é©åœã«èããå®çŸ©ãããŸããŸäžé©åã ã£ããšããã ãã®ããšã ããå®ã¯ãä»ã«ã©ã®ããã«å®çŸ©ããŠããã®ãããªå°é£ããã¯éããããªãããšãç¥ãããŠãããããããã«ãè€çŽ æ°ã«ã¯å€§å°é¢ä¿ãå®çŸ©ããªãã®ã§ããã
ä»åºŠã¯ãè€çŽ æ°ã®å¹³æ¹æ ¹ã«ã€ããŠèããŠã¿ããã æ£ã®æ° a {\displaystyle a} ãèãããšãã
ã§ã¯ã
èæ°åäœ i {\displaystyle i} ã®å¹³æ¹æ ¹ãèãããšãããã¯zã«ã€ããŠã®æ¹çšåŒ z 2 = i {\displaystyle z^{2}=i} ã®è§£ z ã®å€ã§ãããããããã解ãã°ãããã©ã®ãããªè€çŽ æ°zãªããã®åŒãæºããããšãã§ããã ãããã
zãè€çŽ æ°ãšãããšã z = x + y i {\displaystyle z=x+yi} (x,yã¯å®æ°)ãšè¡šãããã ( x + y i ) 2 = i â x 2 + 2 x y i â y 2 = i â ( x 2 â y 2 ) + ( 2 x y â 1 ) i = 0 {\displaystyle (x+yi)^{2}=i\Leftrightarrow x^{2}+2xyi-y^{2}=i\Leftrightarrow (x^{2}-y^{2})+(2xy-1)i=0}
x 2 â y 2 , 2 x y â 1 {\displaystyle x^{2}-y^{2},2xy-1} ã¯å®æ°ã§ãããããå®éšãšèéšãå
±ã«0ã«ãªããã°ãªããªãããã { x 2 â y 2 = 0 ( â x = ± y ) 2 x y â 1 = 0 {\displaystyle {\begin{cases}x^{2}-y^{2}=0(\Leftrightarrow x=\pm y)\\2xy-1=0\end{cases}}}
x = y {\displaystyle x=y} ã®ãšãã 2 x 2 = 1 â x = ± 1 2 , y = ± 1 2 {\displaystyle 2x^{2}=1\Leftrightarrow x=\pm {\frac {1}{\sqrt {2}}},y=\pm {\frac {1}{\sqrt {2}}}} (è€å·åé ãx,yã¯å
±ã«å®æ°ã§ãããããæ¡ä»¶ãæºããã)
x = â y {\displaystyle x=-y} ã®ãšãã â 2 y 2 = 1 â y 2 = â 1 2 {\displaystyle -2y^{2}=1\Leftrightarrow y^{2}=-{\frac {1}{2}}} ããã§ããããæºããå®æ°yã¯ååšããªãããäžé©ã
ãã£ãŠã z = ± ( 1 2 + 1 2 i ) {\displaystyle z=\pm \left({\frac {1}{\sqrt {2}}}+{\frac {1}{\sqrt {2}}}i\right)} â
å®éšããŒããèæ
®ã㊠x = 0 {\displaystyle x=0} ã x = ± 3 y {\displaystyle x=\pm {\sqrt {3}}y} ã ããèéšããŒããªã®ã§ãxã®å€ãåè
ã®ãšã y = â 1 {\displaystyle y=-1} ãåŸè
ã®ãšã y = 1 / 2 {\displaystyle y=1/2} ãšãªãããšãããã«ãããã
2次æ¹çšåŒã«ã¯è§£ã®å
¬åŒããããæ¥æ¬ã®äžåŠãé«æ ¡ã§ãç¿ãã2次æ¹çšåŒã®è§£ã®å
¬åŒãçšããã°ãã©ããªä¿æ°ã®2次æ¹çšåŒã§ãã£ãŠã解ãæ±ããããã3次æ¹çšåŒãš4次æ¹çšåŒã«ãã解ã®å
¬åŒã¯ååšããä¿æ°ãã©ããªä¿æ°ã§ãã£ãŠã解ãæ±ããããããããã®è§£ã®å
¬åŒã¯ã代æ°æ¹çšåŒè«ã§è¿°ã¹ãŠããããã«ãä¿æ°ã«æéåã®ååæŒç®ãšæ ¹å·ããšãæäœã®çµã¿åããã§è¡šãããšãã§ããã
5次æ¹çšåŒã§ã¯ã4次以äžã®æ¹çšåŒãšã¯ç¶æ³ãç°ãªããäžè¬ã®5次æ¹çšåŒã®è§£ã¯ã2次æ¹çšåŒã4次æ¹çšåŒã®ããã«ãä¿æ°ã«æéåã®ååæŒç®ãšæ ¹å·ããšãæäœã®çµã¿åããã§è¡šãããšãã§ããªãã®ã§ããããã ãããã§ããªããããšã®èšŒæã¯å®¹æã§ã¯ãªãããã®ããšã蚌æããã«ã¯ãã¬ãã¢çè«ãç解ããå¿
èŠããã(æ¥æ¬ã®å€§åŠã®æšæºçãªã«ãªãã¥ã©ã ã§ã¯ãçåŠéšæ°åŠç§ã®åŠçã®ã¿ã倧åŠ3幎çã§åŠã¶ã®ãäžè¬çãªçšåºŠã®çè«ã§ãã)ã
ãªããããã§èšããè¡šãããšãã§ããªãããšã¯äžè¬ã®æ¹çšåŒã«ã€ããŠã®ããšã§ãããç¹å¥ãª5次æ¹çšåŒã®å Žåã¯ç°¡åã«è§£ãæ±ãããããããšãã°ã x 5 â 32 = 0 {\displaystyle x^{5}-32=0} ã¯è§£ã®ã²ãšã€ãšã㊠x = 2 {\displaystyle x=2} ããã€ããšã¯ãããããããã®æ¹çšåŒã¯ä»ã®è§£ã«ã€ããŠãäžè§é¢æ°ãçšããŠç°¡åã«è¡šããããšãé«çåŠæ ¡æ°åŠC/è€çŽ æ°å¹³é¢ã«ãããŠåŠã¶ã
ãä¿æ°ã«æéåã®ååæŒç®ãšæ ¹å·ããšãæäœã®çµã¿åãããã«æããªããã°ãäžè¬ã®5次æ¹çšåŒã®è§£ãæ±ããæ¹æ³ãååšããããããé«åºŠãªæ°åŠãçšããå¿
èŠããããw:äºæ¬¡æ¹çšåŒã«èšè¿°ãããã®ã§èå³ã®ããèªè
ã¯åç
§ãããšããã
é«çåŠæ ¡ã§è€çŽ æ°ãåºãŠããåéã¯ãã®åéãšæ°åŠCãå¹³é¢äžã®æ²ç·ãšè€çŽ æ°å¹³é¢ãã®ã¿ã§ãããè€çŽ æ°ã®åºæ¬èšç®ãæ¹çšåŒãè€çŽ æ°ç¯å²ã§è§£ãããšãè€çŽ æ°ã®å¹ŸäœåŠçæå³ã«ã€ããŠæ±ã£ãŠããããããã倧åŠæ°åŠã«ãããŠã¯ãé¢æ°ã®å®çŸ©åã»å€åãè€çŽ æ°ç¯å²ã«åºããŠèãããè€çŽ é¢æ°è«ããšãããã®ãæ±ãã
å®æ°ç¯å²ã§ã®é¢æ°ã¯x, yãšãã«äžæ¬¡å
ã®å®æ°è»žãæã€ãããå
¥åå€ãšåºåå€ã®æãã°ã©ããèããã«ã¯äºæ¬¡å
ã®åº§æšå¹³é¢ã§ååã§ãã£ããããããè€çŽ æ°ç¯å²ã§ã®é¢æ°ã¯x, yãšãã«äºæ¬¡å
ã®è€çŽ æ°å¹³é¢ãæã€ãããå
¥åå€ãšåºåå€ã®æãã°ã©ããèããã«ã¯å次å
ã®åº§æšç©ºéãå¿
èŠã§ãããäžæ¬¡å
空éã«äœãæã
ã«ã¯æç»ããããšãã§ããªãããã®ãããè€çŽ é¢æ°è«ã§ã¯é¢æ°ã®ã°ã©ããèããããšã¯åºæ¬çã«ãªãã(ãã ããåºåãããè€çŽ æ°ã®çµ¶å¯Ÿå€ãèããããšã«ãã£ãŠäžæ¬¡å
ã°ã©ãã«èœãšã蟌ãããšã¯å¯èœ)
ã§ã¯äœãèããã®ããšãããšãè€çŽ é¢æ°ã®åŸ®åç©åã§ãããè€çŽ é¢æ°ã®åŸ®åã«é¢é£ããŠãæ£åé¢æ°ããšããçšèªãåºãŠããããè€çŽ é¢æ°è«ã¯ãã®æ£åé¢æ°ãšãããã®ã®æ§è³ªã調ã¹ãåŠåã ãšèšã£ãŠè¯ãã
è€çŽ é¢æ°è«ã¯ç©çåŠã®ç¹ã«æ³¢åã«é¢ããåé(é³ã»é»ç£æ°ãªã©)ã«ãããŠæŽ»èºããŠããããæ³¢åæ¹çšåŒãããã€ã³ããŒãã³ã¹ããšããèšèã¯æåã ããã
ã¡ãªã¿ã«ãè€çŽ æ°ãããã«æ¡åŒµããæ°ãšããŠãw:åå
æ°ããšãããã®ãããããã®åå
æ°ã¯ãã¯ãã«ãè¡åãšæ·±ãé¢ãããååšããŠãããæ·±æããšé¢çœãã®ã ããããããåé·ã«ãªãããå²æããããªããåå
æ°ãããã«æ¡åŒµããå
«å
æ°ãåå
å
æ°ãšããæ°ãç 究ãããŠããã
| [
{
"paragraph_id": 0,
"tag": "p",
"text": "æ¬é
ã¯é«çåŠæ ¡æ°åŠIIã®åŒãšèšŒæã»é«æ¬¡æ¹çšåŒã®è§£èª¬ã§ããã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "( a + b ) 5 = ( a + b ) ( a + b ) ( a + b ) ( a + b ) ( a + b ) {\\displaystyle (a+b)^{5}=(a+b)(a+b)(a+b)(a+b)(a+b)} ã«ã€ããŠèãããããã®åŒãå±éãããšãã a 2 b 3 {\\displaystyle a^{2}b^{3}} ã®ä¿æ°ã¯ãå³èŸºã®5åã® ( a + b ) {\\displaystyle (a+b)} ãã a {\\displaystyle a} ã3ååãçµã¿åããã«çãããã 5 C 2 = 10 {\\displaystyle _{5}\\mathrm {C} _{2}=10} ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ãã®èããæ¡åŒµããŠ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ãå±éããã a r b n â r {\\displaystyle a^{r}b^{n-r}} ã®é
ã®ä¿æ°ã¯ãå³èŸºã® n {\\displaystyle n} åã® ( a + b ) {\\displaystyle (a+b)} ãã a {\\displaystyle a} ã r {\\displaystyle r} ååãçµã¿åããã«çãããã n C r {\\displaystyle _{n}\\mathrm {C} _{r}} ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ãã£ãŠã次ã®åŒãåŸããã:",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "æåŸã®åŒã¯æ°Bã®æ°åã§åŠã¶ç·åèšå· Σ {\\displaystyle \\Sigma } ã§ãããç¥ããªãã®ãªãç¡èŠããŠãè¯ãã ãã®åŒã äºé
å®ç(binomial theorem) ãšããããŸããããããã®é
ã«ãããä¿æ°ãäºé
ä¿æ°(binomial coefficient) ãšåŒã¶ããšãããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "(I)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "(II)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "(II)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "ãããããèšç®ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "äºé
å®çãçšããŠèšç®ããã°ãããå®éã«èšç®ãè¡ãªããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "(I)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "(II)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "(III)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "ãšãªãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "ãã¹ãŠã®èªç¶æ°nã«å¯ŸããŠ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "(I)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "(II)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "(III)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "ãæãç«ã€ããšã瀺ãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "äºé
å®ç",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "ã«ã€ããŠa,bã«é©åœãªå€ã代å
¥ããã°ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "(I) a = 1,b=1ã代å
¥ãããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "ãšãªã確ãã«äžããããé¢ä¿ãæç«ããããšãåããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "(II) a=2,b=1ã代å
¥ãããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "ãšãªã確ãã«äžããããé¢ä¿ãæç«ããããšãåããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "(III) a=1,b=-1ã代å
¥ãããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "ãšãªã確ãã«äžããããé¢ä¿ãæç«ããããšãåããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "äºé
å®çãæ¡åŒµã㊠( a + b + c ) n {\\displaystyle (a+b+c)^{n}} ãå±éããããšãèãããã a p b q c r {\\displaystyle a^{p}b^{q}c^{r}} ( p + q + r = n ) {\\displaystyle (p+q+r=n)} ã®é
ã®ä¿æ°ã¯ n {\\displaystyle n} åã® ( a + b + c ) {\\displaystyle (a+b+c)} ãã p {\\displaystyle p} åã® a {\\displaystyle a} ã q {\\displaystyle q} åã® b {\\displaystyle b} ã r {\\displaystyle r} åã® c {\\displaystyle c} ãéžã¶çµåãã«çãããã n ! p ! q ! r ! {\\displaystyle {\\frac {n!}{p!q!r!}}} ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "ããã§ã¯ãæŽåŒã®é€æ³ãšåæ°åŒã«ã€ããŠæ±ããæŽåŒã®é€æ³ã¯ãæŽåŒãæŽæ°ã®ããã«æ±ãé€æ³ãè¡ãªãèšç®ææ³ã®ããšã§ãããå®éã«æŽæ°ã®é€æ³ãšæŽåŒã®é€æ³ã«ã¯æ·±ãã€ãªããããããæŽåŒã®å æ°å解ãèãããšãã以äžå æ°å解ã§ããªãæŽåŒãååšããããã®æŽåŒãæŽæ°ã§ããçŽ å æ°ã®ããã«æ±ãããšã§æŽåŒã®çŽ å æ°å解ãå¯èœã«ãªãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "äžã§ã¯ãæŽåŒãæŽæ°ã«å¯Ÿå¿ããæ§è³ªãæã€ããšãè¿°ã¹ããæŽæ°ã«ã€ããŠã¯ãããã«çŽ ãª2ã€ã®æŽæ°ãåãããšã§æçæ°ãå®çŸ©ããããšãåºæ¥ããæŽåŒã«å¯ŸããŠãåãäºãæç«ã¡ããã®ãããªåŒãåæ°åŒãšåŒã¶ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "åæ°ãçšããªããšãã«ã¯ãæŽæ°ã®é€æ³ã¯åãšäœããçšããŠå®çŸ©ãããããã®æãå²ãããæ°Bã¯åDãšå²ãæ°AãäœãRãçšããŠ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "ã®æ§è³ªãæºããããšãç¥ãããŠãããæŽåŒã«å¯ŸããŠã䌌ãæ§è³ªãæç«ã¡ãå²ãããåŒB(x)ãåD(x)ãšå²ãåŒA(x)ãäœãR(x)ãçšããŠã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "ã®å³èŸºã§xã«ã€ããŠ2次ã®é
ãçŸãã巊蟺ãšäžèŽããªããªãããã£ãŠåã¯å®æ°ã§ãããåãaãäœããrãšãããšäžã®åŒã¯ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "ãšãªãããããã¯a=1,r=1ã§æç«ããããã£ãŠå1,äœã1ã§ãããããé«æ¬¡ã®åŒã«å¯ŸããŠãåãæ§ã«çããå®ããŠããã°ãããäŸãšããŠã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "ã®ãããªåŒãèããããã®å Žåã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "ã§ãB(x)ã3次ãA(x)ã2次ã§ããããšãããD(x)ã¯1次ã§ããããŸããR(x)ã¯2次ããå°ããããšãã1次以äžã®åŒã«ãªããããã§ãD(x)=ax+b,R(x)=cx+dãšãããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "ãåŸããããå³èŸºãå±éãããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "ãåŸãããããxã«ã©ããªå€ãå
¥ããŠããã®çåŒãæãç«ããªããã°ãªããªãã®ã§ãa = 1, b = 0, -a +c = 0, -b +d = 0ãåŸãããçµå±a=c=1, b=d=0ãåŸãããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "ãã®æ¹æ³ã¯ã©ã®é€æ³ã«å¯ŸããŠãçšããããšãåºæ¥ããã次æ°ãé«ããªããšèšç®ãé£ãããªããæŽæ°ã®å Žåãšåæ§ãæŽåŒã®é€æ³ã§ãçç®ãçšããããšãåºæ¥ããäžã®äŸãçšããŠçµæã ããæžããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 41,
"tag": "p",
"text": "ã®ããã«ãªãã)å³ã«æžãããåŒãå²ãããåŒã§ããã)å·Šã«æžãããåŒãå²ãåŒã§ããã--ã®äžçªäžã«æžãããåŒã¯åã§ãããæŽæ°ã®å²ãç®åæ§å·Šã«æžãããæ°ããé ã«å²ã£ãŠãããããã§ã¯æ¬¡æ°ã倧ããé
ãããå
ã«èšç®ãããé
ã§ãããå²ãããåŒã®äžã«ããåŒã¯åã®ç¬¬1é
ãå²ãåŒã«ãããŠåŸãåŒã§ãããããã§ã¯ã x ( x 2 â 1 ) {\\displaystyle x(x^{2}-1)} ã§ã x 3 â x {\\displaystyle x^{3}-x} ãšãªãããã ããæŽæ°ã®é€æ³ãšåæ§ãäœãããããªããŠã¯ãªããªãããã®åŸãå²ãããåŒãã x 3 â x {\\displaystyle x^{3}-x} ãåŒããæ®ã£ãåŒãæ°ããå²ãããåŒãšããŠæ±ããããã§ã¯ãåŸãåŒãå²ãåŒãããäœæ¬¡ã§ããããšãããããã§èšç®ã¯çµäºã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 42,
"tag": "p",
"text": "x 3 + 2 x 2 + 1 {\\displaystyle x^{3}+2x^{2}+1} ã x 4 + 4 x 2 + 3 x + 2 {\\displaystyle x^{4}+4x^{2}+3x+2} ãã x 2 + 2 x + 6 {\\displaystyle x^{2}+2x+6} ã§å²ã£ãåãšäœããæ±ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 43,
"tag": "p",
"text": "ãã®èšç®ã¯ã¢ãã¡ãŒã·ã§ã³ã䜿ã£ãŠ 詳ãã衚瀺ãããŠãããèšç®ææ³ã¯ã æŽæ°ã®å Žåã®çç®ãšåããããªææ³ã䜿ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 44,
"tag": "p",
"text": "ãåŸãããã®ã§ãå x {\\displaystyle x} ãäœã â 6 x + 1 {\\displaystyle -6x+1} ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 45,
"tag": "p",
"text": "2ã€ç®ã®åŒã«ã€ããŠã¯ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 46,
"tag": "p",
"text": "ãåŸãããã ãã£ãŠãç㯠å x 2 â 2 x + 2 {\\displaystyle x^{2}-2x+2} ãäœã 11 x â 10 {\\displaystyle 11x-10} ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 47,
"tag": "p",
"text": "ãããŸã§ã§æŽåŒãæŽæ°ã®ããã«æ±ããæŽåŒã®é€æ³ãè¡ãªãæ¹æ³ã«ã€ããŠè¿°ã¹ããããã§ã¯ãæŽåŒã«å¯ŸããŠåæ°åŒãå®çŸ©ããæ¹æ³ã«ã€ããŠè¿°ã¹ããåæ°åŒãšã¯ãæŽæ°ã«å¯Ÿããåæ°ã®ããã«ãé€æ³ã«ãã£ãŠçããåŒã§ãããããã§ãé€æ³ãè¡ãªãåŒã¯ã©ã®ãããªãã®ã§ãå·®ãæ¯ããªããåæ°åŒã§ã¯ãååã«å²ãããåŒãæžããåæ¯ã«å²ãåŒãæžããäŸãã°ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 48,
"tag": "p",
"text": "ã¯ãååx+1ãåæ¯ x 2 + 4 {\\displaystyle x^{2}+4} ã®åæ°åŒã§ãããåæ°åŒã«å¯ŸããŠãçŽåãéåãååšãããçŽåã¯å
±éå æ°ãæã£ãåååæ¯ããã€åæ°åŒã§çšããããããã®æã«ã¯åååæ¯ãå
±éå æ°ã§å²ããåŒãç°¡åã«ããããšãåºæ¥ããéåã¯ãåæ°åŒã®å æ³ã®æã«ããçšããããããåååæ¯ã«åãæŽåŒããããŠãåæ°åŒãå€åããªãæ§è³ªãçšããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 49,
"tag": "p",
"text": "ãç°¡åã«ããããŸãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 50,
"tag": "p",
"text": "ãèšç®ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 51,
"tag": "p",
"text": "ã«ã€ããŠååãšåæ¯ãå æ°å解ãããšãåæ¹ãšãã«",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 52,
"tag": "p",
"text": "ãå æ°ãšããŠå«ãã§ããããšãåããããã®ãšããå
±éã®å æ°ã¯çŽåããããšãå¿
èŠã§ãããèšç®ãããå€ã¯ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 53,
"tag": "p",
"text": "ãšãªãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 54,
"tag": "p",
"text": "次ã®åé¡ã§ã¯ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 55,
"tag": "p",
"text": "ãèšç®ããããã®ãšãã䞡蟺ã®åæ¯ãããããå¿
èŠãããããä»åã«ã€ããŠã¯ãåçŽã«ããããã®åæ°åŒã®ååãšåæ¯ã«åã
ã®åæ¯ããããŠåæ¯ãçµ±äžããã°ãããèšç®ãããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 56,
"tag": "p",
"text": "ãšãªãã åæ°åŒã®ä¹æ³ã¯ãåååæ¯ãå¥ã
ã«ãããã°ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 57,
"tag": "p",
"text": "次ã®èšç®ãããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 58,
"tag": "p",
"text": "(I)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 59,
"tag": "p",
"text": "(II)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 60,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 61,
"tag": "p",
"text": "(I)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 62,
"tag": "p",
"text": "(II)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 63,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 64,
"tag": "p",
"text": "åæ¯ãç©ã®åœ¢ã§ããåæ°åŒãäºã€ã®åæ°åŒã®åãå·®ã§è¡šãããåŒã«å€åœ¢ããæäœãéšååæ°å解ãšããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 65,
"tag": "p",
"text": "1 x ( x + 1 ) {\\displaystyle {\\frac {1}{x(x+1)}}} ãš 1 ( x + 1 ) ( x + 3 ) {\\displaystyle {\\frac {1}{(x+1)(x+3)}}} ãåæ°åŒã®åãŸãã¯å·®ã®åœ¢ã§è¡šãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 66,
"tag": "p",
"text": "ãšå€åœ¢ã§ããã®ã§ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 67,
"tag": "p",
"text": "ãšãªããçŽåãããš",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 68,
"tag": "p",
"text": "ãšãªãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 69,
"tag": "p",
"text": "次ã®åé¡ã§ã¯ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 70,
"tag": "p",
"text": "ãšå€åœ¢ããããšã«ãã£ãŠã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 71,
"tag": "p",
"text": "ãšãªãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 72,
"tag": "p",
"text": "ãšæ±ãŸãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 73,
"tag": "p",
"text": "éšååæ°å解ã®æäœãéã«èŸ¿ããšãåæ°åŒã®éåã®æäœãšäžèŽããã ã€ãŸããéšååæ°å解ã¯éåã®éã®æäœã§ããã ååãå®æ°ã®å Žåã«ã¯ãäžãšåæ§ã®æ¹æ³ã§éšååæ°å解ããããšãã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 74,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 75,
"tag": "p",
"text": "1. 3 ( x â 9 ) ( x â 4 ) {\\displaystyle {\\frac {3}{(x-9)(x-4)}}}",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 76,
"tag": "p",
"text": "2. 7 ( 3 x â 1 ) ( 5 â 2 x ) {\\displaystyle {\\frac {7}{(3x-1)(5-2x)}}}",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 77,
"tag": "p",
"text": "éšååæ°å解ã¯æ°åã®åã®èšç®ãç©åèšç®ã埮åãå©çšããäžçåŒã®èšŒæçã«åœ¹ç«ã€ãéèŠãªå€åœ¢ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 78,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 79,
"tag": "p",
"text": "çåŒ ( a + b ) 2 = a 2 + 2 a b + b 2 {\\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}} ã¯ãæå a , b {\\displaystyle a,b} ã«ã©ã®ãããªå€ã代å
¥ããŠãæãç«ã€ããã®ãããªçåŒãæçåŒ(ãããšããã)ãšããã çåŒ 1 x â 1 + 1 x + 1 = 2 x x 2 â 1 {\\displaystyle {\\frac {1}{x-1}}+{\\frac {1}{x+1}}={\\frac {2x}{x^{2}-1}}} ã¯ã䞡蟺ãšã x = 1 , â 1 {\\displaystyle x=1,-1} ã代å
¥ããããšã¯ã§ããªããããã®ä»ã®å€ã§ããã°ä»£å
¥ããããšãã§ãããŸãã©ã®ãããªå€ã代å
¥ããŠãçåŒãæãç«ã£ãŠããããããæçåŒãšåŒã¶ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 80,
"tag": "p",
"text": "ãã£ãœãã x 2 â x â 2 = 0 {\\displaystyle x^{2}-x-2=0} ã¯ãx=2 ãŸã㯠x=ãŒ1 ã代å
¥ãããšãã ãæãç«ã€ãããã®ããã«æåã«ç¹å®ã®å€ã代å
¥ãããšãã«ã ãæãç«ã€åŒã®ããšãæ¹çšåŒãšåŒã³ãæçåŒãšã¯åºå¥ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 81,
"tag": "p",
"text": "çåŒ a x 2 + b x + c = 0 {\\displaystyle ax^{2}+bx+c=0} ã x {\\displaystyle x} ã«ã€ããŠã®æçåŒã§ããã®ã¯ã©ã®ãããªå ŽåããèããŠã¿ããã ããåŒãã x {\\displaystyle x} ã«ã€ããŠã®æçåŒã§ããããšã¯ããã®åŒã® x {\\displaystyle x} ã«ã©ã®ãããªå€ã代å
¥ããŠãããã®çåŒã¯æãç«ã€ãšããæå³ã§ããããªã®ã§ãäŸãã° x {\\displaystyle x} ã« â 1 , 0 , 1 {\\displaystyle -1\\ ,\\ 0\\ ,\\ 1} ã代å
¥ããåŒ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 82,
"tag": "p",
"text": "ã¯ãã¹ãŠæãç«ã€å¿
èŠããããããã解ããš",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 83,
"tag": "p",
"text": "ãªã®ã§ãçåŒ a x 2 + b x + c = 0 {\\displaystyle ax^{2}+bx+c=0} ã x {\\displaystyle x} ã«ã€ããŠã®æçåŒã«ãªããªãã°ã a = b = c = 0 {\\displaystyle a=b=c=0} ã§ãªããã°ãªããªãããšããããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 84,
"tag": "p",
"text": "äžè¬ã«ãçåŒ a x 2 + b x + c = a â² x 2 + b â² x + c â² {\\displaystyle ax^{2}+bx+c=a'x^{2}+b'x+c'} ãæçåŒã§ããããšãšã ( a â a â² ) x 2 + ( b â b â² ) x + ( c â c â² ) = 0 {\\displaystyle (a-a')x^{2}+(b-b')x+(c-c')=0} ãæçåŒã§ããããšãšåãã§ããã ãã£ãŠ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 85,
"tag": "p",
"text": "ãŸãšãããšæ¬¡ã®ããã«ãªãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 86,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 87,
"tag": "p",
"text": "次ã®çåŒã x {\\displaystyle x} ã«ã€ããŠã®æçåŒãšãªãããã«ã a , b , c {\\displaystyle a\\ ,\\ b\\ ,\\ c} ã®å€ãæ±ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 88,
"tag": "p",
"text": "çåŒã®å³èŸºã x {\\displaystyle x} ã«ã€ããŠæŽçãããš",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 89,
"tag": "p",
"text": "ãã®çåŒã x {\\displaystyle x} ã«ã€ããŠã®æçåŒãšãªãã®ã¯ã䞡蟺ã®åã次æ°ã®é
ã®ä¿æ°ãçãããšãã§ããããã£ãŠ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 90,
"tag": "p",
"text": "ããã解ããš",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 91,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 92,
"tag": "p",
"text": "æçåŒãå©çšããããšã§ãè€éãªåæ°åŒã®éšååæ°å解ãã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 93,
"tag": "p",
"text": "ãšããã åæ¯ãæã£ãŠ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 94,
"tag": "p",
"text": "ããªãã¡",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 95,
"tag": "p",
"text": "ããã x {\\displaystyle x} ã®æçåŒãªã®ã§ãä¿æ°ãæ¯èŒããŠ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 96,
"tag": "p",
"text": "ããªãã¡",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 97,
"tag": "p",
"text": "æåã®çåŒã«ä»£å
¥ããŠã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 98,
"tag": "p",
"text": "次ã®åé¡ã¯ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 99,
"tag": "p",
"text": "ãšããããšã«ãããäžã®åé¡ãšåæ§ã«ããŠ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 100,
"tag": "p",
"text": "ãšæ±ãŸãã®ã§ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 101,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 102,
"tag": "p",
"text": "a~fãå®æ°ãšããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 103,
"tag": "p",
"text": "a x 2 + b y 2 + c x y + d x + e y + f = 0 {\\displaystyle ax^{2}+by^{2}+cxy+dx+ey+f=0} ãx, yã«ã€ããŠã®æçåŒã ãšããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 104,
"tag": "p",
"text": "巊蟺ãxã«ã€ããŠæŽçãããšã a x 2 + ( c y + d ) x + ( b y 2 + e y + f ) = 0 {\\displaystyle ax^{2}+(cy+d)x+(by^{2}+ey+f)=0} ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 105,
"tag": "p",
"text": "ãããxã«ã€ããŠã®æçåŒãªã®ã§ã a = 0 , c y + d = 0 , b y 2 + e y + f = 0 {\\displaystyle a=0,cy+d=0,by^{2}+ey+f=0} ãæãç«ã€ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 106,
"tag": "p",
"text": "ãããã¯æŽã«yã«ã€ããŠã®æçåŒãªã®ã§ã以äžã®çåŒãåŸãããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 107,
"tag": "p",
"text": "éã«ããããæãç«ãŠã°å
ã®åŒã¯æããã«x, yã«ã€ããŠã®æçåŒã§ãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 108,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 109,
"tag": "p",
"text": "x 2 + a x y + 6 y 2 â x + 5 y + b = ( x â 2 y + c ) ( x â 3 y + d ) {\\displaystyle x^{2}+axy+6y^{2}-x+5y+b=(x-2y+c)(x-3y+d)} ãx,yã«ã€ããŠã®æçåŒãšãªãããã«a,b,c,dãå®ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 110,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 111,
"tag": "p",
"text": "ããã»ã©çŽ¹ä»ãããæçåŒããšããèšèã䜿ã£ãŠã蚌æãã®æå³ã説æãããªãããçåŒã蚌æããããšã¯ããã®åŒãæçåŒã§ããããšã瀺ãããšã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 112,
"tag": "p",
"text": "äžè¬ã«ãçåŒ A=B ã蚌æããããã«ã¯ã次ã®ãããªæé ã®ãããããå®è¡ããã°ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 113,
"tag": "p",
"text": "ãã®ãšããå€åœ¢ã¯åå€å€åœ¢ã§ãªããã°ãªããªãããšã«æ³šæã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 114,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 115,
"tag": "p",
"text": "( a + b ) 2 â ( a â b ) 2 = 4 a b {\\displaystyle (a+b)^{2}-(a-b)^{2}=4ab} ãæãç«ã€ããšã蚌æããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 116,
"tag": "p",
"text": "(蚌æ) 巊蟺ãå±éãããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 117,
"tag": "p",
"text": "ãšãªããããã¯å³èŸºã«çããããã£ãŠãçåŒ ( a + b ) 2 â ( a â b ) 2 = 4 a b {\\displaystyle (a+b)^{2}-(a-b)^{2}=4ab} ã¯èšŒæãããã(çµ)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 118,
"tag": "p",
"text": "( x + y ) 2 + ( x â y ) 2 = 2 ( x 2 + y 2 ) {\\displaystyle (x+y)^{2}+(x-y)^{2}=2(x^{2}+y^{2})} ãæãç«ã€ããšã蚌æããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 119,
"tag": "p",
"text": "巊蟺ãèšç®ãããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 120,
"tag": "p",
"text": "ããã¯å³èŸºã«çããããã£ãŠçåŒãæãç«ã€ããšã蚌æãããã(çµ)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 121,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 122,
"tag": "p",
"text": "次ã®çåŒãæãç«ã€ããšã蚌æããã (I)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 123,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 124,
"tag": "p",
"text": "(I) (巊蟺) = ( 36 a 2 + 84 a b + 49 b 2 ) + ( 49 a 2 â 84 a b + 36 a 2 ) = 85 a 2 + 85 b 2 {\\displaystyle =(36a^{2}+84ab+49b^{2})+(49a^{2}-84ab+36a^{2})=85a^{2}+85b^{2}} (å³èŸº) = ( 81 a 2 + 36 a b + 4 b 2 ) + ( 4 a 2 â 36 a b + 81 b 2 ) = 85 a 2 + 85 b 2 {\\displaystyle =(81a^{2}+36ab+4b^{2})+(4a^{2}-36ab+81b^{2})=85a^{2}+85b^{2}} 䞡蟺ãšãåãåŒã«ãªããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 125,
"tag": "p",
"text": "æçåŒã§ãªããšãããäžããããæ¡ä»¶ããçåŒã蚌æããããšãã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 126,
"tag": "p",
"text": "ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 127,
"tag": "p",
"text": "ãã£ãŠã a 3 + b 3 + c 3 = 3 a b c {\\displaystyle a^{3}+b^{3}+c^{3}=3abc} ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 128,
"tag": "p",
"text": "ãŸãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 129,
"tag": "p",
"text": "ãããäžåŒã®å³èŸºãkãšãããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 130,
"tag": "p",
"text": "ãªã®ã§ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 131,
"tag": "p",
"text": "ãã£ãŠã a + c b + d = a â c b â d {\\displaystyle {\\frac {a+c}{b+d}}={\\frac {a-c}{b-d}}} ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 132,
"tag": "p",
"text": "ãªããæ¯ a : b {\\displaystyle a:b} ã«ã€ã㊠a b {\\displaystyle {\\frac {a}{b}}} ãæ¯ã®å€ãšããããŸãã a : b = c : d ⺠a b = c d {\\displaystyle a:b=c:d\\iff {\\frac {a}{b}}={\\frac {c}{d}}} ãæ¯äŸåŒãšããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 133,
"tag": "p",
"text": "a x = b y = c z {\\displaystyle {\\frac {a}{x}}={\\frac {b}{y}}={\\frac {c}{z}}} ãæãç«ã€ãšãã a : b : c = x : y : z {\\displaystyle a:b:c=x:y:z} ãšè¡šãããããé£æ¯ãšããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 134,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 135,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 136,
"tag": "p",
"text": "äžçåŒã®ããŸããŸãªå
¬åŒã«ã€ããŠã¯ã次ã®4ã€ã®åŒãåºæ¬çãªåŒãšããŠå°åºã§ããå Žåãããããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 137,
"tag": "p",
"text": "é«æ ¡æ°åŠã§ã¯ã次ã®4ã€ã®æ§è³ªã äžçåŒã®ãåºæ¬æ§è³ªããªã©ãšããŠçŽ¹ä»ãããŠããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 138,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 139,
"tag": "p",
"text": "(3)ãš(4)ã«ã€ããŠã¯ãã²ãšã€ã®æ§è³ªãšã㊠ãŸãšããŠããæ€å®æç§æžããã(â» åæ通ãªã©)ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 140,
"tag": "p",
"text": "æ°åŠIAã§ç¿ã£ãããªãã°ãã®æå³ã®èšå· â¹ {\\displaystyle \\Longrightarrow } ã䜿ããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 141,
"tag": "p",
"text": "ãšãæžããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 142,
"tag": "p",
"text": "äžè¿°ã®4ã€ã®åºæ¬æ§è³ªããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 143,
"tag": "p",
"text": "ã蚌æããŠã¿ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 144,
"tag": "p",
"text": "(蚌æ) ãŸã a>0 ãªã®ã§ãåºæ¬æ§è³ª(2)ãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 145,
"tag": "p",
"text": "ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 146,
"tag": "p",
"text": "ãã£ãŠã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 147,
"tag": "p",
"text": "ãªã®ã§ãåºæ¬æ§è³ª(1)ãã a + b > 0 {\\displaystyle a+b>0} ãæãç«ã€ã(çµ)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 148,
"tag": "p",
"text": "åæ§ã«ããŠã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 149,
"tag": "p",
"text": "ã蚌æã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 150,
"tag": "p",
"text": "ãããŸã§ã«ç€ºããããšãããäžçåŒ A ⧠B {\\displaystyle A\\geqq B} ã蚌æãããå Žåã«ã¯ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 151,
"tag": "p",
"text": "ã蚌æããã°ããããšãããã£ãããã¡ãã®æ¹ã蚌æããããå Žåãããããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 152,
"tag": "p",
"text": "äžçåŒã蚌æããéã«æ ¹æ ãšããåºæ¬çãªäžçåŒãšããŠã次ã®æ§è³ªãããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 153,
"tag": "p",
"text": "ãã®å®ç(ãå®æ°ã2ä¹ãããšãããªãããŒã以äžã§ããã)ããåºæ¬æ§è³ª(3),(4)ã䜿ã£ãŠèšŒæããŠã¿ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 154,
"tag": "p",
"text": "(蚌æ)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 155,
"tag": "p",
"text": "aãæ£ã®å Žåãšè² ã®å Žåãš0ã®å Žåã®3éãã«å Žåããããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 156,
"tag": "p",
"text": "[aãæ£ã®å Žå] ãã®ãšããåºæ¬æ§è³ª(3)ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 157,
"tag": "p",
"text": "ã§ãããããªãã¡ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 158,
"tag": "p",
"text": "ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 159,
"tag": "p",
"text": "[aãè² ã®å Žå] ãã®ãšããåºæ¬æ§è³ª(4)ãã 0 a < a a {\\displaystyle 0a<aa} ã§ãããããªãã¡ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 160,
"tag": "p",
"text": "ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 161,
"tag": "p",
"text": "[aããŒãã®å Žå] ãã®ãšãã a 2 = 0 {\\displaystyle a^{2}=0} ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 162,
"tag": "p",
"text": "ãã£ãŠããã¹ãŠã®å Žåã«ã€ã㊠a 2 ⧠0 {\\displaystyle a^{2}\\geqq 0} (çµ)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 163,
"tag": "p",
"text": "ãã®ããšãšåºæ¬æ§è³ª(1)(2)ããã次ãæãç«ã€ããšããããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 164,
"tag": "p",
"text": "次ã®äžçåŒãæãç«ã€ããšã蚌æããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 165,
"tag": "p",
"text": "(蚌æ)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 166,
"tag": "p",
"text": "ã蚌æããã°ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 167,
"tag": "p",
"text": "巊蟺ãå±éã㊠ãŸãšãããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 168,
"tag": "p",
"text": "ãšãªãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 169,
"tag": "p",
"text": "äžåŒã®æåŸã®åŒã®é
ã«ã€ããŠã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 170,
"tag": "p",
"text": "ã ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 171,
"tag": "p",
"text": "ã§ããããã£ãŠ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 172,
"tag": "p",
"text": "ã§ããã(çµ)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 173,
"tag": "p",
"text": "2ã€ã®æ£ã®æ° a, b ã a>b ãŸã㯠aâ§b ãªãã°ã䞡蟺ã2ä¹ããŠã倧å°é¢ä¿ã¯åããŸãŸã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 174,
"tag": "p",
"text": "ã€ãŸãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 175,
"tag": "p",
"text": "ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 176,
"tag": "p",
"text": "a>bãšãããä»®å®ãããa,b ã¯æ£ã®æ°ãªã®ã§ã ( a + b ) > 0 {\\displaystyle (a+b)>0} ã§ãããå¥ã®ä»®å®ããã a > b ãªã®ã§ã ( a â b ) > 0 {\\displaystyle (a-b)>0} ã§ãããããã£ãŠã a 2 â b 2 = ( a + b ) ( a â b ) > 0 {\\displaystyle a^{2}-b^{2}=(a+b)(a-b)>0}",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 177,
"tag": "p",
"text": "éã«ã a 2 â b 2 > 0 {\\displaystyle a^{2}-b^{2}>0} ã®ãšãã ( a + b ) ( a â b ) > 0 {\\displaystyle (a+b)(a-b)>0} ã§ããã a > 0 , b > 0 {\\displaystyle a>0,b>0} ãªã®ã§ a + b > 0 {\\displaystyle a+b>0} ã§ããããã£ãŠã a â b > 0 {\\displaystyle a-b>0} ãªã®ã§ã a > b {\\displaystyle a>b} ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 178,
"tag": "p",
"text": "ãã£ãŠã a > b ⺠a 2 > b 2 {\\displaystyle a>b\\quad \\Longleftrightarrow \\quad a^{2}>b^{2}} ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 179,
"tag": "p",
"text": "aâ§bã®å Žåãåæ§ã«èšŒæã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 180,
"tag": "p",
"text": "ç·Žç¿ãšããŠã次ã®åé¡ãåããŠã¿ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 181,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 182,
"tag": "p",
"text": "a > 0 {\\displaystyle a>0} , b > 0 {\\displaystyle b>0} ã®ãšãã次ã®äžçåŒã蚌æããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 183,
"tag": "p",
"text": "(蚌æ) äžçåŒã®äž¡èŸºã¯æ£ã§ããã®ã§ã䞡蟺ã®å¹³æ¹ã®å·®ãèããã°ããã䞡蟺ã®å¹³æ¹ã®å·®ã¯",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 184,
"tag": "p",
"text": "ã§ãããããã§ãa,b ã¯ãšãã«æ£ã®å®æ°ãªã®ã§ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 185,
"tag": "p",
"text": "ã§ããããšãçšããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 186,
"tag": "p",
"text": "ã§ããã®ã§ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 187,
"tag": "p",
"text": "ãšãªãããã£ãŠã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 188,
"tag": "p",
"text": "ã§ããã(çµ)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 189,
"tag": "p",
"text": "å®æ° a ã®çµ¶å¯Ÿå€ |a| ã«ã€ããŠã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 190,
"tag": "p",
"text": "ã§ããããã次ã®ããšãæãç«ã€ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 191,
"tag": "p",
"text": "|a|â§a , |a|⧠ãŒa , |a|=a",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 192,
"tag": "p",
"text": "ãŸãã2ã€ã®å®æ° a, b ã®çµ¶å¯Ÿå€ |ab| ã«ã€ããŠã¯ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 193,
"tag": "p",
"text": "ãæãç«ã€ã®ã§ãããã«ããã« |ab|â§0 , |a||b|â§0 ãçµã¿åãããŠã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 194,
"tag": "p",
"text": "|ab| = |a| |b| ãæãç«ã€ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 195,
"tag": "p",
"text": "(äŸé¡)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 196,
"tag": "p",
"text": "次ã®äžçåŒã蚌æããããŸããçå·ãæãç«ã€ã®ã¯ ã©ã®ãããªå Žåãã 調ã¹ãã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 197,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 198,
"tag": "p",
"text": "䞡蟺ã®å¹³æ¹ã®å·®ãèãããšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 199,
"tag": "p",
"text": "ãããããæ£ãªããäžããããäžçåŒ |a|+|b| ⧠|a+b| ãæ£ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 200,
"tag": "p",
"text": "ããã§ã |a| |b| ⧠ab ã§ããã®ã§ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 201,
"tag": "p",
"text": "ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 202,
"tag": "p",
"text": "ãããã£ãŠã |a|+|b| ⧠|a+b| ã§ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 203,
"tag": "p",
"text": "çå·ãæãç«ã€ã®ã¯ |a| |b| = ab ã®å Žåãããªãã¡ ab ⧠0 ã®å Žåã§ããã(蚌æ ããã)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 204,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 205,
"tag": "p",
"text": "2ã€ã®æ° a {\\displaystyle a} , b {\\displaystyle b} ã«å¯Ÿãã a + b 2 {\\displaystyle {\\frac {a+b}{2}}} ãçžå å¹³å(ããããžããã)ãšèšãã a b {\\displaystyle {\\sqrt {ab}}} ãçžä¹å¹³å(ããããããžããã)ãšããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 206,
"tag": "p",
"text": "æ¬ããŒãžã§ã¯ã2åã®æ°ã®å¹³åã«ã€ããŠèå¯ããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 207,
"tag": "p",
"text": "çžå å¹³åãšçžä¹å¹³åã«ã€ããŠã次ã®é¢ä¿åŒãæãç«ã€ã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 208,
"tag": "p",
"text": "(蚌æ)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 209,
"tag": "p",
"text": "a ⧠0 , b ⧠0 {\\displaystyle a\\geqq 0,b\\geqq 0} ã®ãšã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 210,
"tag": "p",
"text": "( a â b ) 2 ⧠0 {\\displaystyle \\left({\\sqrt {a}}-{\\sqrt {b}}\\right)^{2}\\geqq 0} ã§ããããã ( a â b ) 2 2 ⧠0 {\\displaystyle {\\frac {\\left({\\sqrt {a}}-{\\sqrt {b}}\\right)^{2}}{2}}\\geqq 0} ãããã£ãŠ a + b 2 ⧠a b {\\displaystyle {\\frac {a+b}{2}}\\geqq {\\sqrt {ab}}} çå·ãæãç«ã€ã®ã¯ã ( a â b ) 2 = 0 {\\displaystyle \\left({\\sqrt {a}}-{\\sqrt {b}}\\right)^{2}=0} ã®ãšããããªãã¡ a = b {\\displaystyle a=b} ã®ãšãã§ããã(蚌æ ããã)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 211,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 212,
"tag": "p",
"text": "å
¬åŒã®å©çšã§ã¯ãäžã®åŒ a + b 2 ⧠a b {\\displaystyle {\\frac {a+b}{2}}\\geqq {\\sqrt {ab}}} ã®äž¡èŸºã«2ãããã a + b ⧠2 a b {\\displaystyle a+b\\geqq 2{\\sqrt {ab}}} ã®åœ¢ã®åŒã䜿ãå Žåãããã",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 213,
"tag": "p",
"text": "",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 214,
"tag": "p",
"text": "a > 0 {\\displaystyle a>0} , b > 0 {\\displaystyle b>0} ã®ãšãã次ã®äžçåŒãæãç«ã€ããšã蚌æããã (I)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 215,
"tag": "p",
"text": "(II)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 216,
"tag": "p",
"text": "(I) a > 0 {\\displaystyle a>0} ã§ããããã 1 a > 0 {\\displaystyle {\\frac {1}{a}}>0} ãã£ãŠ a + 1 a ⧠2 a à 1 a = 2 {\\displaystyle a+{\\frac {1}{a}}\\geqq 2{\\sqrt {a\\times {\\frac {1}{a}}}}=2} ãããã£ãŠ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 217,
"tag": "p",
"text": "(II)",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 218,
"tag": "p",
"text": "a > 0 {\\displaystyle a>0} , b > 0 {\\displaystyle b>0} ã§ããããã b a > 0 {\\displaystyle {\\frac {b}{a}}>0} , a b > 0 {\\displaystyle {\\frac {a}{b}}>0} ãã£ãŠ b a + a b + 2 ⧠2 b a à a b + 2 = 2 + 2 = 4 {\\displaystyle {\\frac {b}{a}}+{\\frac {a}{b}}+2\\geqq 2{\\sqrt {{\\frac {b}{a}}\\times {\\frac {a}{b}}}}+2=2+2=4} ãããã£ãŠ",
"title": "åŒãšèšŒæ"
},
{
"paragraph_id": 219,
"tag": "p",
"text": "2ä¹ããŠè² ã«ãªãæ°ããšãããã®ãèããããã®ãããªæ°ã¯ãäžåŠã§ç¿ã£ãå®æ°ã®äžã«ã¯ãªãããšããããããªããªãã°ãæ£ã®æ°ã§ãè² ã®æ°ã§ã2ä¹ãããšç¬Šå·ãæã¡æ¶ããŠæ£ã®æ°ã«ãªã£ãŠããŸãããã§ãããããã§é«æ ¡ã§ã¯ã2ä¹ããŠè² ã«ãªããšããæ§è³ªãæã€æ°ã®æŠå¿µãæ°ããå°å
¥ããããšã«ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 220,
"tag": "p",
"text": "ãšããæ¹çšåŒãèããããã®æ¹çšåŒã®è§£ã¯å®æ°ã«ã¯ãªããããã§ããã®æ¹çšåŒã®è§£ãšãªãæ°ãæ°ããäœãããã®åäœãæå i {\\displaystyle i} ã§ããããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 221,
"tag": "p",
"text": "ãã® i {\\displaystyle i} ã®ããšãèæ°åäœ(ããããããã)ãšåŒã¶ã(èæ°åäœã®èšå· i ãè±èªã®ã¢ã«ãã¡ãããã®ã¢ã€ã®å°æåã§ã imaginary unit ã«ç±æ¥ãããšèããããŠããã)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 222,
"tag": "p",
"text": "1 + i {\\displaystyle 1+i} ã 2 + 5 i {\\displaystyle 2+5i} ã®ããã«ãèæ°åäœ i {\\displaystyle i} ãšå®æ° a , b {\\displaystyle a,b} ãçšããŠ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 223,
"tag": "p",
"text": "ãšè¡šãããšãã§ããæ°ãè€çŽ æ°(ãµãããã)ãšããããã®ãšããaããã®è€çŽ æ°ã®å®éš(ãã€ã¶)ãšãããbãèéš(ããã¶)ãšããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 224,
"tag": "p",
"text": "äŸãã°ã 1 + i , 2 + 5 i , 9 2 + 7 2 i , 4 i , 3 {\\displaystyle 1+i,\\quad 2+5i,\\quad {\\frac {9}{2}}+{\\frac {7}{2}}i,\\quad 4i,\\quad 3} ã¯ãããããè€çŽ æ°ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 225,
"tag": "p",
"text": "è€çŽ æ° a+bi ã¯(ãã ã aãšbã¯å®æ°)ãbã0ã®å Žåã«ããããå®æ°ãšèŠãããšãã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 226,
"tag": "p",
"text": "èšãæ¹ãããããšãè€çŽ æ°ãåºæºã«èãããšãå®æ°ãšã¯ã a+0i ã®ãããªãèéšã®ä¿æ°ããŒãã«ãªãè€çŽ æ°ã®ããšã§ãããšãèšããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 227,
"tag": "p",
"text": "4iã®ãããªãèéšã0以å€ã§å®éšããŒãã®è€çŽ æ°ãçŽèæ°(ãã
ããããã)ãšåŒã¶ãçŽèæ°ã¯ã2ä¹ãããšè² ã«ãªãæ°ã§ããã å®æ°ãèéšã0ã®è€çŽ æ°ãšèããããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 228,
"tag": "p",
"text": "å®æ°ã§ãªãè€çŽ æ°ã®ããšããèæ°ã(ãããã)ãšããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 229,
"tag": "p",
"text": "2ã€ã®è€çŽ æ° a+bi ãš c+di ãšãçãããšã¯ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 230,
"tag": "p",
"text": "ã§ããããšã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 231,
"tag": "p",
"text": "ã€ãŸãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 232,
"tag": "p",
"text": "ãšãã«ãè€çŽ æ°a+bi ã 0ã§ãããšã¯ãa=0 ã〠b=0 ã§ããããšã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 233,
"tag": "p",
"text": "è€çŽ æ° z = a + b i {\\displaystyle z=a+bi} ã«å¯ŸããŠãèéšã®ç¬Šå·ãå転ãããè€çŽ æ° a â b i {\\displaystyle a-bi} ã®ããšããå
±åœ¹(ããããã)ãªè€çŽ æ°ããŸãã¯ãè€çŽ æ° z {\\displaystyle z} ã®å
±åœ¹ãã®ããã«åŒã³ã z Ì {\\displaystyle {\\bar {z}}} ã§ããããããªãããå
±åœ¹ãã¯ãå
±è»ãã®åžžçšæŒ¢åã«ããæžãæãã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 234,
"tag": "p",
"text": "å®æ°aãšå
±åœ¹ãªè€çŽ æ°ã¯ããã®å®æ° a èªèº«ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 235,
"tag": "p",
"text": "è€çŽ æ° z=a+bi ã«ã€ããŠ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 236,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 237,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 238,
"tag": "p",
"text": "è€çŽ æ°ã«ãååæŒç®(å æžä¹é€)ãå®çŸ©ãããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 239,
"tag": "p",
"text": "è€çŽ æ°ã®æŒç®ã§ã¯ãèæ°åäœ i {\\displaystyle i} ããéåžžã®æåã®ããã«æ±ã£ãŠèšç®ãããäžè¬ã«è€çŽ æ° z , w {\\displaystyle z\\ ,\\ w} ãã z = a + b i , w = c + d i {\\displaystyle z=a+bi\\ ,\\ w=c+di} ã§äžãããããšã(ãã ã a , b , c , d {\\displaystyle a\\ ,\\ b\\ ,\\ c\\ ,\\ d} ã¯å®æ°ãšãã)ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 240,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 241,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 242,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 243,
"tag": "p",
"text": "ãšãããµãã«è€çŽ æ°ã®å æžä¹é€ã®èšç®æ³ãå®ããããŠããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 244,
"tag": "p",
"text": "ä¹æ³ã®å®çŸ©ã¯ãäžèŠãããšé£ãããã«ã¿ããããå®æ°ã®åé
æ³åãšåæ§ã«å±éããŠããæåŸã« iã«ãã€ãã¹1ã代å
¥ããŠãã£ãã ãã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 245,
"tag": "p",
"text": "é€æ³ã®å®çŸ©ã¯ãååãšåæ¯ã«ãåæ¯ãšå
±åœ¹ãªåœ¢ã®åŒã æãç® ããã ãã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 246,
"tag": "p",
"text": "ä¹æ³ãé€æ³ã®å®çŸ©åŒãæèšããå¿
èŠã¯ç¡ããèšç®ã®éã«ã¯ãå¿
èŠã«å¿ããŠåé
æ³åãå
±åœ¹ãªã©ã®ãå¿
èŠãªåŒå€åœ¢ãè¡ãã°ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 247,
"tag": "p",
"text": "äŸé¡",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 248,
"tag": "p",
"text": "2ã€ã®è€çŽ æ°",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 249,
"tag": "p",
"text": "ã«ã€ããŠã a + b {\\displaystyle a+b} ãš a b {\\displaystyle ab} ãš a b {\\displaystyle {\\frac {a}{b}}} ããããããèšç®ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 250,
"tag": "p",
"text": "解ç",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 251,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 252,
"tag": "p",
"text": "ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 253,
"tag": "p",
"text": "ããããã«ç°¡åã«ã§ããªãã ããããå®ã¯ãã¡ãã£ãšãããã¯ããã¯ãçšããã°ããèŠããã圢ã«ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 254,
"tag": "p",
"text": "åæ°ã¯åæ¯ãšååã«åãæ°ããããŠããã£ãã®ã§ãåæ¯ãšååã«åæ¯ã®å
±åœ¹ããããŠã¿ãããããšã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 255,
"tag": "p",
"text": "ãåŸãããããã®åœ¢ã®ã»ããå
ã®åŒããããã£ãšèŠããã圢ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 256,
"tag": "p",
"text": "ãã®ãããªæäœãåæ¯ã®å®æ°åãšããããšããããæ°åŠIã§åŠç¿ããå±éã»å æ°å解å
¬åŒ ( a + b ) ( a â b ) = a 2 â b 2 {\\displaystyle (a+b)(a-b)=a^{2}-b^{2}} ã®ç°¡åãªå¿çšã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 257,
"tag": "p",
"text": "æ°ã®ç¯å²ãè€çŽ æ°ã«ãŸã§æ¡åŒµãããšãè² ã®æ°ã®å¹³æ¹æ ¹ãèããããšãã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 258,
"tag": "p",
"text": "äŸãšããŠã -5 ã®å¹³æ¹æ ¹ã«ã€ããŠèããŠã¿ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 259,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 260,
"tag": "p",
"text": "ã§ããããã -5 ã®å¹³æ¹æ ¹ã¯ 5 i {\\displaystyle {\\sqrt {5}}\\ i} ãš â 5 i {\\displaystyle -{\\sqrt {5}}\\ i} ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 261,
"tag": "p",
"text": "â 5 {\\displaystyle {\\sqrt {-5}}} ãšã¯ã 5 i {\\displaystyle {\\sqrt {5}}\\ i} ã®ããšãšããã â â 5 {\\displaystyle -{\\sqrt {-5}}} ãšã¯ã â 5 i {\\displaystyle -{\\sqrt {5}}\\ i} ã®ããšã§ããã ãšãã« â 1 = i {\\displaystyle {\\sqrt {-1}}\\ =\\ i} ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 262,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 263,
"tag": "p",
"text": "ããŠã-5 ã®å¹³æ¹æ ¹ã¯ãæ¹çšåŒ x 2 = â 5 {\\displaystyle x^{2}=-5} ã®è§£ã§ãããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 264,
"tag": "p",
"text": "ãã®æ¹çšåŒã移é
ããããšã«ããã-5 ã®å¹³æ¹æ ¹ã¯ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 265,
"tag": "p",
"text": "ã®è§£ã§ãããšããããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 266,
"tag": "p",
"text": "ããã«å æ°å解ãããããšã«ããã-5 ã®å¹³æ¹æ ¹ã¯æ¹çšåŒ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 267,
"tag": "p",
"text": "ã®è§£ã§ããããšããããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 268,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 269,
"tag": "p",
"text": "(I) â 2 â 6 {\\displaystyle {\\sqrt {-2}}\\ {\\sqrt {-6}}} ãèšç®ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 270,
"tag": "p",
"text": "(I)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 271,
"tag": "p",
"text": "ãã®ããã«ããŸãããã€ãã¹ã®æ°ã®å¹³æ¹æ ¹ãåºãŠãããããŸãèæ°åäœ i ãçšããåŒã«æžãæããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 272,
"tag": "p",
"text": "ãã®ããšãããç®ãããŠããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 273,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 274,
"tag": "p",
"text": "(II) 2 â 3 {\\displaystyle {\\frac {\\sqrt {2}}{\\sqrt {-3}}}} ãèšç®ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 275,
"tag": "p",
"text": "(III) 2次æ¹çšåŒ x 2 = â 7 {\\displaystyle x^{2}=-7} ã解ãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 276,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 277,
"tag": "p",
"text": "(II)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 278,
"tag": "p",
"text": "(III)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 279,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 280,
"tag": "p",
"text": "è€çŽ æ°ã®å¿çšãšããŠãããã§ã¯2次æ¹çšåŒã®æ§è³ªã«ã€ããŠè¿°ã¹ããä»»æã®2次æ¹çšåŒã¯ã解ã®å
¬åŒã«ãã£ãŠè§£ãããããšãé«çåŠæ ¡æ°åŠIã§è¿°ã¹ãããããã解ã®å
¬åŒã«å«ãŸããæ ¹å·ã®äžèº«ãè² ã®æ°ã®å Žåã«ã¯å®æ°è§£ãååšããªãããšã«æ³šæããå¿
èŠãããã2次æ¹çšåŒ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 281,
"tag": "p",
"text": "ã®è§£ã®å
¬åŒã¯ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 282,
"tag": "p",
"text": "ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 283,
"tag": "p",
"text": "å€å¥åŒ D {\\displaystyle D} ã¯",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 284,
"tag": "p",
"text": "ã«ãã£ãŠå®çŸ©ããããå€å¥åŒã¯ã解ã®å
¬åŒã®æ ¹å·(ã«ãŒãèšå·ã®ããš)ã®äžèº«ã«çãããå€å¥åŒã®æ£è² ã«ãã£ãŠ2次æ¹çšåŒãå®æ°è§£ãæã€ãã©ããã決ãŸãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 285,
"tag": "p",
"text": "D {\\displaystyle D} ãè² ã®ãšãã«ã¯ãã®2次æ¹çšåŒã¯å®æ°ã®ç¯å²ã«ã¯è§£ãæããªãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 286,
"tag": "p",
"text": "å€å¥åŒ D {\\displaystyle D} ãè² ã®æ°ã§ãã£ããšããxã®è§£ã¯ç°ãªã2ã€ã®èæ°ã«ãªãããã®2ã€ã®è§£ã¯ å
±åœ¹ ã®é¢ä¿ã«ãªã£ãŠããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 287,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 288,
"tag": "p",
"text": "è€çŽ æ°ãçšããŠã2次æ¹çšåŒ (1)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 289,
"tag": "p",
"text": "(2)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 290,
"tag": "p",
"text": "(3)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 291,
"tag": "p",
"text": "ã解ãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 292,
"tag": "p",
"text": "解ã®å
¬åŒãçšããŠè§£ãã°ããã(1)ã ããèšç®ãããšã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 293,
"tag": "p",
"text": "ãšãªãã ä»ãåãããã«æ±ãããšãåºæ¥ãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 294,
"tag": "p",
"text": "以éã®è§£çã¯ã (2)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 295,
"tag": "p",
"text": "(3)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 296,
"tag": "p",
"text": "ãšãªãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 297,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 298,
"tag": "p",
"text": "æ¹çšåŒã®è§£ã§ãå®æ°ã§ãããã®ã å®æ°è§£ ãšããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 299,
"tag": "p",
"text": "æ¹çšåŒã®è§£ã§ãèæ°ã§ãããã®ã èæ°è§£ ãšããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 300,
"tag": "p",
"text": "2次æ¹çšåŒ a x 2 + b x + c = 0 {\\displaystyle ax^{2}+bx+c=0} ã®è§£ã¯ x = â b ± b 2 â 4 a c 2 a {\\displaystyle x={\\frac {-b\\pm {\\sqrt {b^{2}-4ac}}}{2a}}} ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 301,
"tag": "p",
"text": "2次æ¹çšåŒã®è§£ã®çš®é¡ã¯ã解ã®å
¬åŒã®äžã®æ ¹å·ã®äžã®åŒ b 2 â 4 a c {\\displaystyle b^{2}-4ac} ã®ç¬Šå·ãèŠãã°å€å¥ããããšãã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 302,
"tag": "p",
"text": "ãã®åŒ b 2 â 4 a c {\\displaystyle b^{2}-4ac} ãã2次æ¹çšåŒ a x 2 + b x + c = 0 {\\displaystyle ax^{2}+bx+c=0} ã®å€å¥åŒ(ã¯ãã¹ã€ãã)ãšãããèšå· D {\\displaystyle D} ã§è¡šãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 303,
"tag": "p",
"text": "ãŸããé解ãå®æ°è§£ã§ããã®ã§ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 304,
"tag": "p",
"text": "ãšãããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 305,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 306,
"tag": "p",
"text": "次ã®2次æ¹çšåŒã®è§£ãå€å¥ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 307,
"tag": "p",
"text": "(I)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 308,
"tag": "p",
"text": "(II)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 309,
"tag": "p",
"text": "(III)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 310,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 311,
"tag": "p",
"text": "(I)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 312,
"tag": "p",
"text": "ã ãããç°ãªã2ã€ã®å®æ°è§£ããã€ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 313,
"tag": "p",
"text": "(II)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 314,
"tag": "p",
"text": "ã ãããç°ãªã2ã€ã®èæ°è§£ããã€ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 315,
"tag": "p",
"text": "(III)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 316,
"tag": "p",
"text": "ã ãããé解ããã€ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 317,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 318,
"tag": "p",
"text": "ãŸãã2次æ¹çšåŒ a x 2 + 2 b â² x + c = 0 {\\displaystyle ax^{2}+2b'x+c=0} ã®ãšãã D = 4 ( b â² 2 â a c ) {\\displaystyle D=4(b'^{2}-ac)} ãšãªãã®ã§ã 2次æ¹çšåŒ a x 2 + 2 b â² x + c = 0 {\\displaystyle ax^{2}+2b'x+c=0} ã®å€å¥åŒã«ã¯",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 319,
"tag": "p",
"text": "ããã¡ããŠãããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 320,
"tag": "p",
"text": "ãããçšããŠãåã®åé¡",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 321,
"tag": "p",
"text": "ã®è§£ãå€å¥ãããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 322,
"tag": "p",
"text": "a = 4 , b â² = â 10 , c = 25 {\\displaystyle a=4\\,,\\,b'=-10\\,,\\,c=25} ã§ãããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 323,
"tag": "p",
"text": "ã ãããé解ããã€ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 324,
"tag": "p",
"text": "2次æ¹çšåŒ a x 2 + b x + c = 0 {\\displaystyle ax^{2}+bx+c=0} ã®2ã€ã®è§£ã α {\\displaystyle \\alpha } , β {\\displaystyle \\beta } ãšããã ãã®æ¹çšåŒã¯ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 325,
"tag": "p",
"text": "a ( x â α ) ( x â β ) = 0 {\\displaystyle a(x-\\alpha )(x-\\beta )=0}",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 326,
"tag": "p",
"text": "ãšå€åœ¢ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 327,
"tag": "p",
"text": "ãããå±éãããšã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 328,
"tag": "p",
"text": "a x 2 â a ( α + β ) x + a α β = 0 {\\displaystyle ax^{2}-a(\\alpha +\\beta )x+a\\alpha \\beta =0}",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 329,
"tag": "p",
"text": "ä¿æ°ãæ¯èŒããŠã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 330,
"tag": "p",
"text": "c = a α β , b = â a ( α + β ) {\\displaystyle c=a\\alpha \\beta ,b=-a(\\alpha +\\beta )}",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 331,
"tag": "p",
"text": "ãåŸãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 332,
"tag": "p",
"text": "ãããå€åœ¢ããã°ã α + β = â b a , α β = c a {\\displaystyle \\alpha +\\beta =-{\\frac {b}{a}},\\alpha \\beta ={\\frac {c}{a}}} ãšãªãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 333,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 334,
"tag": "p",
"text": "2次æ¹çšåŒ 2 x 2 + 4 x + 3 = 0 {\\displaystyle 2x^{2}+4x+3=0} ã®2ã€ã®è§£ã α {\\displaystyle \\alpha } , β {\\displaystyle \\beta } ãšãããšãã α 2 + β 2 {\\displaystyle \\alpha ^{2}+\\beta ^{2}} ã®å€ãæ±ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 335,
"tag": "p",
"text": "解ãšä¿æ°ã®é¢ä¿ããã α + β = â 4 2 = â 2 {\\displaystyle \\alpha +\\beta =-{\\frac {4}{2}}=-2} , α β = 3 2 {\\displaystyle \\alpha \\beta ={\\frac {3}{2}}} α 2 + β 2 = ( α + β ) 2 â 2 α β = ( â 2 ) 2 â 2 à 3 2 = 1 {\\displaystyle \\alpha ^{2}+\\beta ^{2}=(\\alpha +\\beta )^{2}-2\\alpha \\beta =(-2)^{2}-2\\times {\\frac {3}{2}}=1}",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 336,
"tag": "p",
"text": "2ã€ã®æ° α {\\displaystyle \\alpha } , β {\\displaystyle \\beta } ã解ãšãã2次æ¹çšåŒã¯",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 337,
"tag": "p",
"text": "ãšè¡šãããã巊蟺ãå±éããŠæŽçãããšæ¬¡ã®ããã«ãªãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 338,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 339,
"tag": "p",
"text": "次ã®2æ°ã解ãšãã2次æ¹çšåŒãäœãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 340,
"tag": "p",
"text": "(I)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 341,
"tag": "p",
"text": "(II)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 342,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 343,
"tag": "p",
"text": "(I) å ( 3 + 5 ) + ( 3 â 5 ) = 6 {\\displaystyle (3+{\\sqrt {5}})+(3-{\\sqrt {5}})=6} ç© ( 3 + 5 ) ( 3 â 5 ) = 4 {\\displaystyle (3+{\\sqrt {5}})(3-{\\sqrt {5}})=4} ã§ãããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 344,
"tag": "p",
"text": "(II) å ( 2 + 3 i ) + ( 2 â 3 i ) = 4 {\\displaystyle (2+3i)+(2-3i)=4} ç© ( 2 + 3 i ) ( 2 â 3 i ) = 13 {\\displaystyle (2+3i)(2-3i)=13} ã§ãããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 345,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 346,
"tag": "p",
"text": "2次æ¹çšåŒ a x 2 + b x + c = 0 {\\displaystyle ax^{2}+bx+c=0} ã®2ã€ã®è§£ α {\\displaystyle \\alpha } , β {\\displaystyle \\beta } ãããããšã2次åŒ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 347,
"tag": "p",
"text": "ãå æ°å解ããããšãã§ããã 解ãšä¿æ°ã®é¢ä¿ α + β = â b a {\\displaystyle \\alpha +\\beta =-{\\frac {b}{a}}} , α β = c a {\\displaystyle \\alpha \\beta ={\\frac {c}{a}}} ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 348,
"tag": "p",
"text": "2次æ¹çšåŒã¯ãè€çŽ æ°ã®ç¯å²ã§èãããšã€ãã«è§£ããã€ãããè€çŽ æ°ãŸã§äœ¿ã£ãŠãããšãããšã2次åŒã¯å¿
ã1次åŒã®ç©ã«å æ°å解ããããšãã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 349,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 350,
"tag": "p",
"text": "è€çŽ æ°ã®ç¯å²ã§èããŠã次ã®2次åŒãå æ°å解ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 351,
"tag": "p",
"text": "(I)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 352,
"tag": "p",
"text": "(II)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 353,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 354,
"tag": "p",
"text": "(I) 2次æ¹çšåŒ x 2 + 4 x â 1 = 0 {\\displaystyle x^{2}+4x-1=0} ã®è§£ã¯",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 355,
"tag": "p",
"text": "ãã£ãŠ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 356,
"tag": "p",
"text": "(II) 2次æ¹çšåŒ 2 x 2 â 3 x + 2 = 0 {\\displaystyle 2x^{2}-3x+2=0} ã®è§£ã¯",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 357,
"tag": "p",
"text": "ãã£ãŠ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 358,
"tag": "p",
"text": "3次以äžã®æŽåŒã«ããæ¹çšåŒãèããã äžè¬ã«æ¹çšåŒã P ( x ) = 0 {\\displaystyle P(x)=0} ãšãšãã ãã ãã P ( x ) {\\displaystyle P(x)} ã¯ãä»»æã®æ¬¡æ°ã®æŽåŒãšããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 359,
"tag": "p",
"text": "P ( x ) {\\displaystyle P(x)} ã1æ¬¡åŒ x â a {\\displaystyle x-a} ã§å²ã£ããšãã®åã Q ( x ) {\\displaystyle Q(x)} ãäœãã R {\\displaystyle R} ãšãããšã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 360,
"tag": "p",
"text": "ãã®äž¡èŸºã® x {\\displaystyle x} ã« a {\\displaystyle a} ã代å
¥ãããšã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 361,
"tag": "p",
"text": "ã€ãŸãã P ( x ) {\\displaystyle P(x)} ã x â a {\\displaystyle x-a} ã§å²ã£ããšãã®äœã㯠P ( a ) {\\displaystyle P(a)} ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 362,
"tag": "p",
"text": "æŽåŒ P ( x ) = x 3 â 2 x + 3 {\\displaystyle P(x)=x^{3}-2x+3} ã次ã®åŒã§å²ã£ãäœããæ±ããã (I)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 363,
"tag": "p",
"text": "(II)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 364,
"tag": "p",
"text": "(III)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 365,
"tag": "p",
"text": "(I) P ( 2 ) = 2 3 â 2 Ã 2 + 3 = 7 {\\displaystyle P(2)=2^{3}-2\\times 2+3=7} (II) P ( â 1 ) = ( â 1 ) 3 â 2 Ã ( â 1 ) + 3 = 4 {\\displaystyle P(-1)=(-1)^{3}-2\\times (-1)+3=4} (III) P ( 1 2 ) = ( 1 2 ) 3 â 2 Ã ( 1 2 ) + 3 = 17 8 {\\displaystyle P\\left({\\frac {1}{2}}\\right)=\\left({\\frac {1}{2}}\\right)^{3}-2\\times \\left({\\frac {1}{2}}\\right)+3={\\frac {17}{8}}}",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 366,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 367,
"tag": "p",
"text": "ããå®æ° a {\\displaystyle a} ã«å¯ŸããŠã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 368,
"tag": "p",
"text": "ãæãç«ã£ããšããã ãã®ãšããæŽåŒ P ( x ) {\\displaystyle P(x)} ã¯ã ( x â a ) {\\displaystyle (x-a)} ãå æ°ã«æã€ããšãåãã ãã®ããšãå æ°å®ç(ãããããŠãã)ãšåŒã¶ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 369,
"tag": "p",
"text": "æŽåŒ P ( x ) {\\displaystyle P(x)} ã«å¯ŸããŠãå Q ( x ) {\\displaystyle Q(x)} ãå²ãåŒ ( x â a ) {\\displaystyle (x-a)} ãšãã æŽåŒã®é€æ³ãçšããããã®ãšããå Q ( x ) {\\displaystyle Q(x)} ã ( Q ( x ) {\\displaystyle Q(x)} ã¯ã P ( x ) {\\displaystyle P(x)} ããã1ã ã次æ°ãäœãæŽåŒã§ããã) äœã c {\\displaystyle c} ( c {\\displaystyle c} ã¯ãå®æ°ã)ãšãããšã æŽåŒ P ( x ) {\\displaystyle P(x)} ã¯ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 370,
"tag": "p",
"text": "ãšæžããã ããã§ã c = 0 {\\displaystyle c=0} ã§ãªããšã P ( a ) = 0 {\\displaystyle P(a)=0} ã¯æºããããªããã ãã®ãšãã P ( x ) {\\displaystyle P(x)} ã¯ã ( x â a ) {\\displaystyle (x-a)} ã«ãã£ãŠå²ãåããã ãã£ãŠãå æ°å®çã¯æç«ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 371,
"tag": "p",
"text": "å æ°å®çãçšããããšã§ããã次æ°ã®é«ãæŽåŒãå æ°å解ããããšã åºæ¥ãããã«ãªããäŸãã°ã3次ã®æŽåŒ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 372,
"tag": "p",
"text": "ã«ã€ããŠã x = 1 {\\displaystyle x=1} ã代å
¥ãããšã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 373,
"tag": "p",
"text": "ã¯0ãšãªãããã£ãŠãå æ°å®çãããã®åŒã¯",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 374,
"tag": "p",
"text": "ãå æ°ãšããŠæã€ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 375,
"tag": "p",
"text": "ããã§ãå®éæŽåŒã®é€æ³ã䜿ã£ãŠèšç®ãããšã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 376,
"tag": "p",
"text": "ãåŸãããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 377,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 378,
"tag": "p",
"text": "å æ°å®çãçšã㊠(I)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 379,
"tag": "p",
"text": "(II)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 380,
"tag": "p",
"text": "ãå æ°å解ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 381,
"tag": "p",
"text": "(I) å æ°å解ã®çµæã(x+æŽæ°)ã®ç©ã®åœ¢ãªããæŽæ°ã¯6ã®å æ°ã§ãªããã°ãªããªãããã®ããã ± 1 , ± 2 , ± 3 , ± 6 {\\displaystyle \\pm 1,\\pm 2,\\pm 3,\\pm 6} ãåè£ãšãªãããããã«ã€ããŠã¯å®éã«ä»£å
¥ããŠç¢ºããããããªããx=1ã代å
¥ãããšã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 382,
"tag": "p",
"text": "ãšãªãã®ã§ã(x-1)ãå æ°ãšãªããå®éã«æŽåŒã®é€æ³ãè¡ãªããšãåãšã㊠x 2 â 5 x + 6 {\\displaystyle x^{2}-5x+6} ãåŸããããããã㯠( x â 2 ) ( x â 3 ) {\\displaystyle (x-2)(x-3)} ã«å æ°å解ã§ããããã£ãŠçãã¯ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 383,
"tag": "p",
"text": "ãšãªãã (II) ããã§ãå°éã«24ã®å æ°ãåœãŠã¯ããŠãããããªãã24ã®å æ°ã¯æ°ãå€ãã®ã§ããªãã®èšç®ãå¿
èŠãšãªããããã§ã¯ã-2ã代å
¥ãããšã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 384,
"tag": "p",
"text": "ãšãªãã(x+2)ãå æ°ã ãšããããé€æ³ãè¡ãªããšã x 2 â x â 12 {\\displaystyle x^{2}-x-12} ãåŸããããã(x-4)(x+3)ã«å æ°å解ã§ãããçãã¯ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 385,
"tag": "p",
"text": "ãšãªãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 386,
"tag": "p",
"text": "å æ°å解ãå æ°å®çãå©çšããŠé«æ¬¡æ¹çšåŒã解ããŠã¿ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 387,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 388,
"tag": "p",
"text": "é«æ¬¡æ¹çšåŒ (I)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 389,
"tag": "p",
"text": "(II)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 390,
"tag": "p",
"text": "(III)",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 391,
"tag": "p",
"text": "ã解ãã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 392,
"tag": "p",
"text": "(I) 巊蟺ã a 3 â b 3 = ( a â b ) ( a 2 + a b + b 2 ) {\\displaystyle a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})} ãçšããŠå æ°å解ãããš",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 393,
"tag": "p",
"text": "ãããã£ãŠ x â 2 = 0 {\\displaystyle \\ x-2=0} ãŸã㯠x 2 + 2 x + 4 = 0 {\\displaystyle \\ x^{2}+2x+4=0} ãã£ãŠ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 394,
"tag": "p",
"text": "(II) x 2 = X {\\displaystyle \\ x^{2}=X\\ } ãšãããšã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 395,
"tag": "p",
"text": "巊蟺ãå æ°å解ãããš",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 396,
"tag": "p",
"text": "ãã£ãŠ X = 4 , X = â 2 {\\displaystyle X=4\\ ,\\ X=-2} ããã« x 2 = 4 , x 2 = â 2 {\\displaystyle x^{2}=4\\ ,\\ x^{2}=-2} ãããã£ãŠ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 397,
"tag": "p",
"text": "(III) P ( x ) = x 3 â 5 x 2 + 7 x â 2 {\\displaystyle \\ P(x)=x^{3}-5x^{2}+7x-2\\ } ãšããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 398,
"tag": "p",
"text": "ãããã£ãŠã x â 2 {\\displaystyle \\ x-2\\ } 㯠P ( x ) {\\displaystyle \\ P(x)\\ } ã®å æ°ã§ããã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 399,
"tag": "p",
"text": "ãã£ãŠ ( x â 2 ) ( x 2 â 3 x + 1 ) = 0 {\\displaystyle (x-2)(x^{2}-3x+1)=0} x â 2 = 0 {\\displaystyle \\ x-2=0} ãŸã㯠x 2 â 3 x + 1 = 0 {\\displaystyle \\ x^{2}-3x+1=0} ãããã£ãŠ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 400,
"tag": "p",
"text": "",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 401,
"tag": "p",
"text": "3次æ¹çšåŒ a x 3 + b x 2 + c x + d = 0 {\\displaystyle ax^{3}+bx^{2}+cx+d=0} ã®3ã€ã®è§£ã ã α , β , γ {\\displaystyle \\alpha \\ ,\\ \\beta \\ ,\\ \\gamma } ãšãããš",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 402,
"tag": "p",
"text": "ãæãç«ã€ã å³èŸºãå±éãããš",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 403,
"tag": "p",
"text": "ãã£ãŠ",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 404,
"tag": "p",
"text": "ããã«",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 405,
"tag": "p",
"text": "ãããã£ãŠã次ã®ããšãæãç«ã€ã",
"title": "é«æ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 406,
"tag": "p",
"text": "ãã°ãã°èæ°ã¯ãçŸå®ã«ã¯ååšããªãæ°ãã§ãããšèšãããããšããããæŽå²çã«ãèæ°ãæ±ã£ãæ°åŠãèããã¹ãã§ã¯ãªããšèããããæ代ã¯é·ãã£ãããã®æ代ã®å
é²çãªæ°åŠè
ã®äžã«ã¯ãèæ°ãæå¹ã«æŽ»çšããŠç 究ãé²ããäžæ¹ã§ãææãçºè¡šããéã«ã¯èæ°ãè¡šã«åºããã«èšè¿°ããåªåãããããšã§ãç¡çšãªæµæãåããªãããã«å·¥å€«ããè
ããããšèšãããã»ã©ã§ããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 407,
"tag": "p",
"text": "ã ããããèããŠã¿ãã°ãæ°ããçŸå®ã«ååšããããšã¯ã©ãããæå³ãªã®ã ããããçŸå®ã«éçã䜿ã£ãŠçŽã«åãæããªãã°ãååšã®é·ãããæ£ç¢ºã«ååšçãã®ãã®ã«ãããããšã¯äžå¯èœã§ããããã«æããããããã®å²ã«ååšçãšããå®æ°ã¯ãååšããããšæããããã®ã¯ãªãã ããããæ°çŽç·ãå®æ°ã®ãå®åšããä¿¡ãããããªãã°ãè€çŽ æ°ã¯è€çŽ æ°å¹³é¢(æ°åŠCã§ç¿ã)ã®äžã«ååšããã®ã ãããåãã§ã¯ãªãã ãããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 408,
"tag": "p",
"text": "ãã®ããã«èãããšãããããæ°ãšã¯ãã¹ãŠããæå³ã§æ³åäžã®ååšã§ãããããã«å¯ŸããŠãååšããããååšããªãããšããåããç«ãŠãããšããã³ã»ã³ã¹ã§ããããã«æãããããååšããªããããã«æãããã¡ãªèæ°ã§ããããããšãã°ç©çåŠã®äžåéã§ããéåååŠã®ã·ã¥ã¬ãã£ã³ã¬ãŒæ¹çšåŒã«è¡šãããªã©ãå¿çšäžã®ããŸããŸãªå Žé¢ã«ãããŠããèæ°ã䜿ã£ãŠèšè¿°ããããšãèªç¶ãªå¯Ÿè±¡ã¯å€ãã®ã ã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 409,
"tag": "p",
"text": "è€çŽ æ°ã©ããã«ã€ããŠããã®å€§å°é¢ä¿ã¯å®çŸ©ããªãããã®çç±ã¯ãã©ã®ããã«å€§å°é¢ä¿ãå®çŸ©ããŠãã䟿å©ãªæ§è³ªãæºããããšãã§ããªãããã§ãããå
·äœçã«èšãã°ãæ¢ã«è¿°ã¹ãå®æ°ã®å€§å°é¢ä¿ã«ã€ããŠã®ãäžçåŒã®åºæ¬æ§è³ª(1)(2)(3)(4)ãã«ãããåŒãæãç«ãããããšãã§ããªãã®ã ã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 410,
"tag": "p",
"text": "ããšãã°ã a + b i < a â² + b â² i {\\displaystyle a+bi<a'+b'i} ã§ããããšãã a 2 + b 2 < a â² 2 + b â² 2 {\\displaystyle a^{2}+b^{2}<a'^{2}+b'^{2}} ã§ããããšãšããŠå®çŸ©ããŠã¿ããããã®ããã«å®çŸ©ãããšãããšãã°1+2i<2-3iã§ããããŸã2+3i<3+4iã§ããããšãããã(1+2i)+(2+3i)=3+5i,(2-3i)+(3+4i)=5+iã§ããã3+5i>5+iãšãªã£ãŠããŸããããã¯åºæ¬æ§è³ª(2)ãæãç«ããªãããšãæå³ããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 411,
"tag": "p",
"text": "ãã¡ããããã¯é©åœã«èããå®çŸ©ãããŸããŸäžé©åã ã£ããšããã ãã®ããšã ããå®ã¯ãä»ã«ã©ã®ããã«å®çŸ©ããŠããã®ãããªå°é£ããã¯éããããªãããšãç¥ãããŠãããããããã«ãè€çŽ æ°ã«ã¯å€§å°é¢ä¿ãå®çŸ©ããªãã®ã§ããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 412,
"tag": "p",
"text": "ä»åºŠã¯ãè€çŽ æ°ã®å¹³æ¹æ ¹ã«ã€ããŠèããŠã¿ããã æ£ã®æ° a {\\displaystyle a} ãèãããšãã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 413,
"tag": "p",
"text": "ã§ã¯ã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 414,
"tag": "p",
"text": "èæ°åäœ i {\\displaystyle i} ã®å¹³æ¹æ ¹ãèãããšãããã¯zã«ã€ããŠã®æ¹çšåŒ z 2 = i {\\displaystyle z^{2}=i} ã®è§£ z ã®å€ã§ãããããããã解ãã°ãããã©ã®ãããªè€çŽ æ°zãªããã®åŒãæºããããšãã§ããã ãããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 415,
"tag": "p",
"text": "zãè€çŽ æ°ãšãããšã z = x + y i {\\displaystyle z=x+yi} (x,yã¯å®æ°)ãšè¡šãããã ( x + y i ) 2 = i â x 2 + 2 x y i â y 2 = i â ( x 2 â y 2 ) + ( 2 x y â 1 ) i = 0 {\\displaystyle (x+yi)^{2}=i\\Leftrightarrow x^{2}+2xyi-y^{2}=i\\Leftrightarrow (x^{2}-y^{2})+(2xy-1)i=0}",
"title": "ã³ã©ã "
},
{
"paragraph_id": 416,
"tag": "p",
"text": "x 2 â y 2 , 2 x y â 1 {\\displaystyle x^{2}-y^{2},2xy-1} ã¯å®æ°ã§ãããããå®éšãšèéšãå
±ã«0ã«ãªããã°ãªããªãããã { x 2 â y 2 = 0 ( â x = ± y ) 2 x y â 1 = 0 {\\displaystyle {\\begin{cases}x^{2}-y^{2}=0(\\Leftrightarrow x=\\pm y)\\\\2xy-1=0\\end{cases}}}",
"title": "ã³ã©ã "
},
{
"paragraph_id": 417,
"tag": "p",
"text": "x = y {\\displaystyle x=y} ã®ãšãã 2 x 2 = 1 â x = ± 1 2 , y = ± 1 2 {\\displaystyle 2x^{2}=1\\Leftrightarrow x=\\pm {\\frac {1}{\\sqrt {2}}},y=\\pm {\\frac {1}{\\sqrt {2}}}} (è€å·åé ãx,yã¯å
±ã«å®æ°ã§ãããããæ¡ä»¶ãæºããã)",
"title": "ã³ã©ã "
},
{
"paragraph_id": 418,
"tag": "p",
"text": "x = â y {\\displaystyle x=-y} ã®ãšãã â 2 y 2 = 1 â y 2 = â 1 2 {\\displaystyle -2y^{2}=1\\Leftrightarrow y^{2}=-{\\frac {1}{2}}} ããã§ããããæºããå®æ°yã¯ååšããªãããäžé©ã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 419,
"tag": "p",
"text": "ãã£ãŠã z = ± ( 1 2 + 1 2 i ) {\\displaystyle z=\\pm \\left({\\frac {1}{\\sqrt {2}}}+{\\frac {1}{\\sqrt {2}}}i\\right)} â ",
"title": "ã³ã©ã "
},
{
"paragraph_id": 420,
"tag": "p",
"text": "",
"title": "ã³ã©ã "
},
{
"paragraph_id": 421,
"tag": "p",
"text": "å®éšããŒããèæ
®ã㊠x = 0 {\\displaystyle x=0} ã x = ± 3 y {\\displaystyle x=\\pm {\\sqrt {3}}y} ã ããèéšããŒããªã®ã§ãxã®å€ãåè
ã®ãšã y = â 1 {\\displaystyle y=-1} ãåŸè
ã®ãšã y = 1 / 2 {\\displaystyle y=1/2} ãšãªãããšãããã«ãããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 422,
"tag": "p",
"text": "2次æ¹çšåŒã«ã¯è§£ã®å
¬åŒããããæ¥æ¬ã®äžåŠãé«æ ¡ã§ãç¿ãã2次æ¹çšåŒã®è§£ã®å
¬åŒãçšããã°ãã©ããªä¿æ°ã®2次æ¹çšåŒã§ãã£ãŠã解ãæ±ããããã3次æ¹çšåŒãš4次æ¹çšåŒã«ãã解ã®å
¬åŒã¯ååšããä¿æ°ãã©ããªä¿æ°ã§ãã£ãŠã解ãæ±ããããããããã®è§£ã®å
¬åŒã¯ã代æ°æ¹çšåŒè«ã§è¿°ã¹ãŠããããã«ãä¿æ°ã«æéåã®ååæŒç®ãšæ ¹å·ããšãæäœã®çµã¿åããã§è¡šãããšãã§ããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 423,
"tag": "p",
"text": "5次æ¹çšåŒã§ã¯ã4次以äžã®æ¹çšåŒãšã¯ç¶æ³ãç°ãªããäžè¬ã®5次æ¹çšåŒã®è§£ã¯ã2次æ¹çšåŒã4次æ¹çšåŒã®ããã«ãä¿æ°ã«æéåã®ååæŒç®ãšæ ¹å·ããšãæäœã®çµã¿åããã§è¡šãããšãã§ããªãã®ã§ããããã ãããã§ããªããããšã®èšŒæã¯å®¹æã§ã¯ãªãããã®ããšã蚌æããã«ã¯ãã¬ãã¢çè«ãç解ããå¿
èŠããã(æ¥æ¬ã®å€§åŠã®æšæºçãªã«ãªãã¥ã©ã ã§ã¯ãçåŠéšæ°åŠç§ã®åŠçã®ã¿ã倧åŠ3幎çã§åŠã¶ã®ãäžè¬çãªçšåºŠã®çè«ã§ãã)ã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 424,
"tag": "p",
"text": "ãªããããã§èšããè¡šãããšãã§ããªãããšã¯äžè¬ã®æ¹çšåŒã«ã€ããŠã®ããšã§ãããç¹å¥ãª5次æ¹çšåŒã®å Žåã¯ç°¡åã«è§£ãæ±ãããããããšãã°ã x 5 â 32 = 0 {\\displaystyle x^{5}-32=0} ã¯è§£ã®ã²ãšã€ãšã㊠x = 2 {\\displaystyle x=2} ããã€ããšã¯ãããããããã®æ¹çšåŒã¯ä»ã®è§£ã«ã€ããŠãäžè§é¢æ°ãçšããŠç°¡åã«è¡šããããšãé«çåŠæ ¡æ°åŠC/è€çŽ æ°å¹³é¢ã«ãããŠåŠã¶ã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 425,
"tag": "p",
"text": "ãä¿æ°ã«æéåã®ååæŒç®ãšæ ¹å·ããšãæäœã®çµã¿åãããã«æããªããã°ãäžè¬ã®5次æ¹çšåŒã®è§£ãæ±ããæ¹æ³ãååšããããããé«åºŠãªæ°åŠãçšããå¿
èŠããããw:äºæ¬¡æ¹çšåŒã«èšè¿°ãããã®ã§èå³ã®ããèªè
ã¯åç
§ãããšããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 426,
"tag": "p",
"text": "é«çåŠæ ¡ã§è€çŽ æ°ãåºãŠããåéã¯ãã®åéãšæ°åŠCãå¹³é¢äžã®æ²ç·ãšè€çŽ æ°å¹³é¢ãã®ã¿ã§ãããè€çŽ æ°ã®åºæ¬èšç®ãæ¹çšåŒãè€çŽ æ°ç¯å²ã§è§£ãããšãè€çŽ æ°ã®å¹ŸäœåŠçæå³ã«ã€ããŠæ±ã£ãŠããããããã倧åŠæ°åŠã«ãããŠã¯ãé¢æ°ã®å®çŸ©åã»å€åãè€çŽ æ°ç¯å²ã«åºããŠèãããè€çŽ é¢æ°è«ããšãããã®ãæ±ãã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 427,
"tag": "p",
"text": "å®æ°ç¯å²ã§ã®é¢æ°ã¯x, yãšãã«äžæ¬¡å
ã®å®æ°è»žãæã€ãããå
¥åå€ãšåºåå€ã®æãã°ã©ããèããã«ã¯äºæ¬¡å
ã®åº§æšå¹³é¢ã§ååã§ãã£ããããããè€çŽ æ°ç¯å²ã§ã®é¢æ°ã¯x, yãšãã«äºæ¬¡å
ã®è€çŽ æ°å¹³é¢ãæã€ãããå
¥åå€ãšåºåå€ã®æãã°ã©ããèããã«ã¯å次å
ã®åº§æšç©ºéãå¿
èŠã§ãããäžæ¬¡å
空éã«äœãæã
ã«ã¯æç»ããããšãã§ããªãããã®ãããè€çŽ é¢æ°è«ã§ã¯é¢æ°ã®ã°ã©ããèããããšã¯åºæ¬çã«ãªãã(ãã ããåºåãããè€çŽ æ°ã®çµ¶å¯Ÿå€ãèããããšã«ãã£ãŠäžæ¬¡å
ã°ã©ãã«èœãšã蟌ãããšã¯å¯èœ)",
"title": "ã³ã©ã "
},
{
"paragraph_id": 428,
"tag": "p",
"text": "ã§ã¯äœãèããã®ããšãããšãè€çŽ é¢æ°ã®åŸ®åç©åã§ãããè€çŽ é¢æ°ã®åŸ®åã«é¢é£ããŠãæ£åé¢æ°ããšããçšèªãåºãŠããããè€çŽ é¢æ°è«ã¯ãã®æ£åé¢æ°ãšãããã®ã®æ§è³ªã調ã¹ãåŠåã ãšèšã£ãŠè¯ãã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 429,
"tag": "p",
"text": "è€çŽ é¢æ°è«ã¯ç©çåŠã®ç¹ã«æ³¢åã«é¢ããåé(é³ã»é»ç£æ°ãªã©)ã«ãããŠæŽ»èºããŠããããæ³¢åæ¹çšåŒãããã€ã³ããŒãã³ã¹ããšããèšèã¯æåã ããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 430,
"tag": "p",
"text": "ã¡ãªã¿ã«ãè€çŽ æ°ãããã«æ¡åŒµããæ°ãšããŠãw:åå
æ°ããšãããã®ãããããã®åå
æ°ã¯ãã¯ãã«ãè¡åãšæ·±ãé¢ãããååšããŠãããæ·±æããšé¢çœãã®ã ããããããåé·ã«ãªãããå²æããããªããåå
æ°ãããã«æ¡åŒµããå
«å
æ°ãåå
å
æ°ãšããæ°ãç 究ãããŠããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 431,
"tag": "p",
"text": "",
"title": "ã³ã©ã "
}
] | æ¬é
ã¯é«çåŠæ ¡æ°åŠIIã®åŒãšèšŒæã»é«æ¬¡æ¹çšåŒã®è§£èª¬ã§ããã | {{pathnav|é«çåŠæ ¡ã®åŠç¿|é«çåŠæ ¡æ°åŠ|é«çåŠæ ¡æ°åŠII|pagename=åŒãšèšŒæã»é«æ¬¡æ¹çšåŒ|frame=1|small=1}}
æ¬é
ã¯[[é«çåŠæ ¡æ°åŠII]]ã®åŒãšèšŒæã»é«æ¬¡æ¹çšåŒã®è§£èª¬ã§ããã
== åŒãšèšŒæ ==
=== äºé
å®ç ===
<math>(a+b)^5 = (a+b)(a+b)(a+b)(a+b)(a+b)</math> ã«ã€ããŠèãããããã®åŒãå±éãããšãã<math>a^2b^3</math> ã®ä¿æ°ã¯ãå³èŸºã®5åã® <math>(a+b)</math> ãã <math>a</math> ã3ååãçµã¿åããã«çãããã <math>_5\mathrm{C}_2 = 10</math> ã§ããã
ãã®èããæ¡åŒµããŠ
:<math>(a+b)^n = \underbrace{(a+b)(a+b)(a+b)\cdots(a+b)}_n</math>
ãå±éããã<math>a^rb^{n-r}</math>ã®é
ã®ä¿æ°ã¯ãå³èŸºã® <math>n</math> åã® <math>(a+b)</math> ãã <math>a</math> ã <math>r</math> ååãçµã¿åããã«çãããã <math>_n\mathrm{C}_r</math> ã§ããã
ãã£ãŠã次ã®åŒãåŸãããïŒ
:<math>\begin{align}(a+b)^n &= {}_n\mathrm{C}_0 a^n + {}_n\mathrm{C}_1 a^{n-1}b + {}_n\mathrm{C}_2 a^{n-2}b^2 + \cdots \\
&+ {}_n\mathrm{C}_r a^{n-r}b^r + \cdots + {}_n\mathrm{C}_n b^n \\
&= \sum _{r = 0}^n {}_n\operatorname{C}_r a^r b^{n-r}. \\ \end{align}</math>
æåŸã®åŒã¯[[é«çåŠæ ¡æ°åŠB/æ°å|æ°Bã®æ°å]]ã§åŠã¶ç·åèšå· <math>\Sigma</math> ã§ãããç¥ããªãã®ãªãç¡èŠããŠãè¯ãã
ãã®åŒã '''äºé
å®ç'''ïŒbinomial theoremïŒ ãšããããŸããããããã®é
ã«ãããä¿æ°ãäºé
ä¿æ°ïŒbinomial coefficientïŒ ãšåŒã¶ããšãããã
* åé¡äŸ
** åé¡
(I)
:<math>(x+1) ^4</math>
(II)
:<math>(a + 3) ^ 5</math>
(II)
:<math>(a + b) ^ 5</math>
ãããããèšç®ããã
**解ç
äºé
å®çãçšããŠèšç®ããã°ãããå®éã«èšç®ãè¡ãªããšã
(I)
:<math>x^4+4\,x^3+6\,x^2+4\,x+1</math>
(II)
:<math>a^5+15\,a^4+90\,a^3+270\,a^2+405\,a+243</math>
(III)
:<math>b^5+5\,a\,b^4+10\,a^2\,b^3+10\,a^3\,b^2+5\,a^4\,b+a^5</math>
ãšãªãã
** åé¡
ãã¹ãŠã®èªç¶æ°nã«å¯ŸããŠ
(I)
:<math>2^n = \sum _{k=0} ^n n\operatorname{C} _k </math>
(II)
:<math>3^n = \sum _{k=0} ^n 2^k n\operatorname{C} _k </math>
(III)
:<math>0 = \sum _{k=0} ^n (-1)^k n\operatorname{C} _k </math>
ãæãç«ã€ããšã瀺ãã
** 解ç
äºé
å®ç
:<math>(a+b)^n = \sum _{k = 0}^n {} _n\operatorname{C} _k a^k b^{n-k}</math>
ã«ã€ããŠa,bã«é©åœãªå€ã代å
¥ããã°ããã
(I)
a = 1,b=1ã代å
¥ãããšã
:<math>(1+1)^n = \sum _{k = 0}^n {} _n\operatorname{C} _k </math>
:<math>2^n = \sum _{k = 0}^n {} _n\operatorname{C} _k </math>
ãšãªã確ãã«äžããããé¢ä¿ãæç«ããããšãåããã
(II)
a=2,b=1ã代å
¥ãããšã
:<math>(1+2)^n = \sum _{k = 0}^n {} _n\operatorname{C} _k 2^k</math>
:<math>3^n = \sum _{k = 0}^n {} _n\operatorname{C} _k 2^k</math>
ãšãªã確ãã«äžããããé¢ä¿ãæç«ããããšãåããã
(III)
a=1,b=-1ã代å
¥ãããšã
:<math>(1-1)^n = \sum _{k = 0}^n {} _n\operatorname{C} _k (-1)^k</math>
:<math>0 = \sum _{k = 0}^n {} _n\operatorname{C} _k (-1)^k</math>
ãšãªã確ãã«äžããããé¢ä¿ãæç«ããããšãåããã
==== å€é
å®ç ====
äºé
å®çãæ¡åŒµã㊠<math>(a+b+c)^n</math> ãå±éããããšãèãããã<math>a^pb^qc^r</math> <math>(p+q+r = n)</math> ã®é
ã®ä¿æ°ã¯ <math>n</math> åã® <math>(a+b+c)</math> ãã <math>p</math> åã® <math>a</math>ã<math>q</math> åã® <math>b</math> ã <math>r</math> åã® <math>c</math> ãéžã¶[[é«çåŠæ ¡æ°åŠA/å Žåã®æ°ãšç¢ºç#çµã¿åãã|çµåã]]ã«çãããã <math>\frac{n!}{p!q!r!}</math> ã§ããã
=== æŽåŒã®é€æ³ãåæ°åŒ ===
ããã§ã¯ãæŽåŒã®é€æ³ãšåæ°åŒã«ã€ããŠæ±ããæŽåŒã®é€æ³ã¯ãæŽåŒãæŽæ°ã®ããã«æ±ãé€æ³ãè¡ãªãèšç®ææ³ã®ããšã§ãããå®éã«æŽæ°ã®é€æ³ãšæŽåŒã®é€æ³ã«ã¯æ·±ãã€ãªããããããæŽåŒã®å æ°å解ãèãããšãã以äžå æ°å解ã§ããªãæŽåŒãååšããããã®æŽåŒãæŽæ°ã§ããçŽ å æ°ã®ããã«æ±ãããšã§æŽåŒã®çŽ å æ°å解ãå¯èœã«ãªãã
äžã§ã¯ãæŽåŒãæŽæ°ã«å¯Ÿå¿ããæ§è³ªãæã€ããšãè¿°ã¹ããæŽæ°ã«ã€ããŠã¯ãããã«çŽ ãª2ã€ã®æŽæ°ãåãããšã§æçæ°ãå®çŸ©ããããšãåºæ¥ããæŽåŒã«å¯ŸããŠãåãäºãæç«ã¡ããã®ãããªåŒãåæ°åŒãšåŒã¶ã
==== æŽåŒã®é€æ³ ====
åæ°ãçšããªããšãã«ã¯ãæŽæ°ã®é€æ³ã¯åãšäœããçšããŠå®çŸ©ãããããã®æãå²ãããæ°Bã¯åDãšå²ãæ°AãäœãRãçšããŠ
:<math>
B = AD + R
</math>
ã®æ§è³ªãæºããããšãç¥ãããŠãããæŽåŒã«å¯ŸããŠã䌌ãæ§è³ªãæç«ã¡ãå²ãããåŒB(x)ãåD(x)ãšå²ãåŒA(x)ãäœãR(x)ãçšããŠã
:<math>
B(x) = A(x)D(x) + R(x)
</math>ãšæžããããšããB(x)ããA(x)ã«å²ããããšããããã®æãæŽæ°ã®é€æ³ã®æ§è³ªR<Aã«å¯Ÿå¿ããŠãR(x)ã®æ¬¡æ°<A(x)ã®æ¬¡æ°ãæç«ãããå
·äœäŸãšããŠãx +1ããxã§å²ãããšãèãããå²ãåŒã®æ¬¡æ°ã1ã§ããããšããäœãã®æ¬¡æ°ã¯0ãšãªãäœãã¯å®æ°ã§ããå¿
èŠãããããŸããåãxã®é¢æ°ã§ãããš
:<math>
B(x) = A(x)D(x) + R(x)
</math>
ã®å³èŸºã§xã«ã€ããŠ2次ã®é
ãçŸãã巊蟺ãšäžèŽããªããªãããã£ãŠåã¯å®æ°ã§ãããåãaãäœããrãšãããšäžã®åŒã¯ã
:<math>
x+1 = ax + r
</math>
ãšãªãããããã¯a=1,r=1ã§æç«ããããã£ãŠå1,äœã1ã§ãããããé«æ¬¡ã®åŒã«å¯ŸããŠãåãæ§ã«çããå®ããŠããã°ãããäŸãšããŠã
:<math>
x^3 \div (x^2 -1)
</math>
ã®ãããªåŒãèããããã®å Žåã
:<math>
B(x) = A(x)D(x) + R(x)
</math>
ã§ãB(x)ã3次ãA(x)ã2次ã§ããããšãããD(x)ã¯1次ã§ããããŸããR(x)ã¯2次ããå°ããããšãã1次以äžã®åŒã«ãªããããã§ãD(x)=ax+b,R(x)=cx+dãšãããšã
:<math>
x^3 = (x^2-1) (ax+b) + (cx +d)
</math>
ãåŸããããå³èŸºãå±éãããšã
:<math>
x^3 = ax^3 + b x^2 + (-a +c )x + (-b +d)
</math>
ãåŸãããããxã«ã©ããªå€ãå
¥ããŠããã®çåŒãæãç«ããªããã°ãªããªãã®ã§ãa = 1, b = 0, -a +c = 0, -b +d = 0ãåŸãããçµå±a=c=1, b=d=0ãåŸãããã
<!--
<math>
x^3 \div (x^2 -1)
</math>
ã®ãããªåŒãèããã
ãã®åŒã«ã€ããŠã
<math>
x^3 = x(x^2 - 1) +x
</math>
ãšæžãããšãåºæ¥ããããã㯠<math>x^3</math> ã <math>x^2-1</math> ã§å²ã£ãçµæã
å<math>x</math> ,äœã <math>x</math> ãã§ããã®ãšè§£éã§ããã
ãã®ããã«ãæŽåŒã©ããã§å²ãç®ãããããšãåºæ¥ãã
ãã®ãšããå²ãåŒã¯å²ãããåŒããäœæ¬¡ãåã次æ°ã§ãªããŠã¯ãªããªãã
ãŸããäœãã¯å¿
ãå²ãåŒãããäœæ¬¡ã®åŒã«ãªãã
-->
ãã®æ¹æ³ã¯ã©ã®é€æ³ã«å¯ŸããŠãçšããããšãåºæ¥ããã次æ°ãé«ããªããšèšç®ãé£ãããªããæŽæ°ã®å Žåãšåæ§ãæŽåŒã®é€æ³ã§ãçç®ãçšããããšãåºæ¥ããäžã®äŸãçšããŠçµæã ããæžããšã
*å³
ã®ããã«ãªãã)å³ã«æžãããåŒãå²ãããåŒã§ããã)å·Šã«æžãããåŒãå²ãåŒã§ããã--ã®äžçªäžã«æžãããåŒã¯åã§ãããæŽæ°ã®å²ãç®åæ§å·Šã«æžãããæ°ããé ã«å²ã£ãŠãããããã§ã¯æ¬¡æ°ã倧ããé
ãããå
ã«èšç®ãããé
ã§ãããå²ãããåŒã®äžã«ããåŒã¯åã®ç¬¬1é
ãå²ãåŒã«ãããŠåŸãåŒã§ãããããã§ã¯ã<math>x(x^2-1)</math>ã§ã<math>x^3-x</math>ãšãªãããã ããæŽæ°ã®é€æ³ãšåæ§ãäœãããããªããŠã¯ãªããªãããã®åŸãå²ãããåŒãã<math>x^3-x</math>ãåŒããæ®ã£ãåŒãæ°ããå²ãããåŒãšããŠæ±ããããã§ã¯ãåŸãåŒãå²ãåŒãããäœæ¬¡ã§ããããšãããããã§èšç®ã¯çµäºã§ããã
*åé¡äŸ
**åé¡
:
<math>x^3 + 2x ^2 +1</math>ã<math>x ^4 + 4x^2 +3x +2</math>ãã<math>x^2 +2x +6
</math>ã§å²ã£ãåãšäœããæ±ããã
<!--
æŽã«ã
(I)
:<math>
(x ^4 + 2x^3 - 5x^2 +6x -1) \div (x^2 -5x -1 )
</math>
(II)
:<math>
(3 x ^4 - 7x^3 + x^2 +2x -1) \div (x^2 -3x -4 )
</math>
(III)
:<math>
(2x^5 +3 x ^4 - 7x^3 + x^2 +2x -1) \div (x^2 +7x -4 )
</math>
(IV)
:<math>
(2x^5 +3 x ^4 - 7x^3 + x^2 +2x -1) \div (x^3 +4x^2 +7x -4 )
</math>
ãèšç®ããã
åé¡ãå€ãã®ã§ããšããããã³ã¡ã³ãã¢ãŠãã
-->
** 解ç
ãã®èšç®ã¯ã¢ãã¡ãŒã·ã§ã³ã䜿ã£ãŠ
詳ãã衚瀺ãããŠãããèšç®ææ³ã¯ã
æŽæ°ã®å Žåã®çç®ãšåããããªææ³ã䜿ããã
[[ç»å:Fract.gif|frame|right|èšç®ã®ã¢ãã¡ãŒã·ã§ã³]]
:<math>
x^3 + 2x ^2 +1
=
(x^2 +2x +6) x +(1-6x)
</math>
ãåŸãããã®ã§ãå<math>
x</math>ãäœã<math>-6x +1</math>ã§ããã
2ã€ç®ã®åŒã«ã€ããŠã¯ã
:<math>
x ^4 + 4x^2 +3x +2
=
(x^2 - 2x+2)* (x^2 +2x +6)
+ 11x -10
</math>
ãåŸãããã
ãã£ãŠãçã¯
å<math>x^2 - 2x+2</math>ãäœã<math>11x -10</math>ã§ããã
<!--
æŽã«ãæ®ãã®èšç®çµæã¯ã
(I)
:<math>
\left[ x^2+7\,x+31,168\,x+30 \right]
</math>
(II)
:<math>
\left[ 3\,x^2+2\,x+19,67\,x+75 \right]
</math>
(III)
:<math>
\left[ 2\,x^3-11\,x^2+78\,x-589,4437\,x-2357 \right]
</math>
(IV)
:<math>
\left[ 2\,x^2-5\,x-1,48\,x^2-11\,x-5 \right]
</math>
ãåŸãããã
ãã ããå·Šãåãå³ãäœããšãªã£ãŠããã
-->
==== åæ°åŒ ====
ãããŸã§ã§æŽåŒãæŽæ°ã®ããã«æ±ããæŽåŒã®é€æ³ãè¡ãªãæ¹æ³ã«ã€ããŠè¿°ã¹ããããã§ã¯ãæŽåŒã«å¯ŸããŠåæ°åŒãå®çŸ©ããæ¹æ³ã«ã€ããŠè¿°ã¹ããåæ°åŒãšã¯ãæŽæ°ã«å¯Ÿããåæ°ã®ããã«ãé€æ³ã«ãã£ãŠçããåŒã§ãããããã§ãé€æ³ãè¡ãªãåŒã¯ã©ã®ãããªãã®ã§ãå·®ãæ¯ããªããåæ°åŒã§ã¯ãååã«å²ãããåŒãæžããåæ¯ã«å²ãåŒãæžããäŸãã°ã
:<math>
\frac {x+1}{x^2+4}
</math>
ã¯ãååx+1ãåæ¯<math>x^2+4</math>ã®åæ°åŒã§ãããåæ°åŒã«å¯ŸããŠãçŽåãéåãååšãããçŽåã¯å
±éå æ°ãæã£ãåååæ¯ããã€åæ°åŒã§çšããããããã®æã«ã¯åååæ¯ãå
±éå æ°ã§å²ããåŒãç°¡åã«ããããšãåºæ¥ããéåã¯ãåæ°åŒã®å æ³ã®æã«ããçšããããããåååæ¯ã«åãæŽåŒããããŠãåæ°åŒãå€åããªãæ§è³ªãçšããã
* åé¡äŸ
** åé¡
:<math>
\frac {x^2 -1} {x^3 -1}
</math>
ãç°¡åã«ããããŸãã
:<math>
\frac {x+1}{x^2 +2x + 3}
+ \frac {2x + 5} {x^2 +1}
</math>
ãèšç®ããã
** 解ç
:<math>
\frac {x^2 -1} {x^3 -1}
</math>
ã«ã€ããŠååãšåæ¯ãå æ°å解ãããšãåæ¹ãšãã«
:<math>
x-1
</math>
ãå æ°ãšããŠå«ãã§ããããšãåããããã®ãšããå
±éã®å æ°ã¯çŽåããããšãå¿
èŠã§ãããèšç®ãããå€ã¯ã
:<math>
\frac {x^2 -1} {x^3 -1}
</math>
:<math>
= \frac{(x-1)(x+1)}{(x-1)(x^2+x+1)}
</math>
:<math>
= \frac{x+1} { x^2+x+1}
</math>
ãšãªãã
次ã®åé¡ã§ã¯ã
:<math>
\frac {x+1}{x^2 +2x + 3}
+ \frac {2x + 5} {x^2 +1}
</math>
ãèšç®ããããã®ãšãã䞡蟺ã®åæ¯ãããããå¿
èŠãããããä»åã«ã€ããŠã¯ãåçŽã«ããããã®åæ°åŒã®ååãšåæ¯ã«åã
ã®åæ¯ããããŠåæ¯ãçµ±äžããã°ãããèšç®ãããšã
:<math>
\frac {x+1}{x^2 +2x + 3}
+ \frac {2x + 5} {x^2 +1}
</math>
:<math>
= \frac{(x+1)(x^2+1)}{(x^2 +2x + 3)(x^2+1)}
+\frac{(x^2 +2x + 3)(2x + 5)}{(x^2 +2x + 3)(x^2+1)}
</math>
:<math>
= \frac{(x+1)(x^2+1)+(x^2 +2x + 3)(2x + 5)}
{(x^2 +2x + 3)(x^2+1)}
</math>
:<math>
= \frac {3x^3 +10x^2 + 17 x + 16}
{(x^2 +2x + 3)(x^2+1)}
</math>
ãšãªãã
åæ°åŒã®ä¹æ³ã¯ãåååæ¯ãå¥ã
ã«ãããã°ããã
* åé¡äŸ
** åé¡
次ã®èšç®ãããã
(I)
:<math>
\frac {x^2 - y^2} {x^2 - 2xy + y^2} \times \frac {x-y} {x^2 + xy}
</math>
(II)
:<math>
\frac {x^2 + 4x + 3}{x^2 - 6x + 9} \div \frac {x^2 - 3x - 4} {x^2 - x - 6}
</math>
** 解ç
(I)
:<math>
\frac {x^2 - y^2} {x^2 - 2xy + y^2} \times \frac {x-y} {x^2 + xy}
</math>
:<math>
= \frac {(x+y)(x-y)} {(x-y)^2} \times \frac {x-y} {x(x+y)}
</math>
:<math>
= \frac {(x+y)(x-y)(x-y)} {(x-y)^2\ x(x+y)}
</math>
:<math>
= \frac {1} {x}
</math>
(II)
:<math>
\frac {x^2 + 4x + 3}{x^2 - 6x + 9} \div \frac {x^2 - 3x - 4} {x^2 - x - 6}
</math>
:<math>
= \frac {x^2 + 4x + 3}{x^2 - 6x + 9} \times \frac {x^2 - x - 6} {x^2 - 3x - 4}
</math>
:<math>
= \frac {(x+1)(x+3)} {(x-3)^2} \times \frac {(x+2)(x-3)} {(x+1)(x-4)}
</math>
:<math>
= \frac {(x+1)(x+3)(x+2)(x-3)} {(x-3)^2\ (x+1)(x-4)}
</math>
:<math>
= \frac {(x+3)(x+2)} {(x-3)(x-4)}
</math>
===== éšååæ°å解 =====
åæ¯ãç©ã®åœ¢ã§ããåæ°åŒãäºã€ã®åæ°åŒã®åãå·®ã§è¡šãããåŒã«å€åœ¢ããæäœã'''éšååæ°å解'''ãšããã
*åé¡äŸ
<Math> \frac{1}{x (x+1)} </Math>ãš<Math>\frac{1}{(x+1)(x+3)}</Math>ãåæ°åŒã®åãŸãã¯å·®ã®åœ¢ã§è¡šãã
*解ç
:<Math>\frac{1}{x(x+1)} = \frac{(x+1)-x}{x(x+1)}</Math>
ãšå€åœ¢ã§ããã®ã§ã
:<Math>\frac{x+1}{x(x+1)} - \frac{x}{x(x+1)}</Math>
ãšãªããçŽåãããš
:<Math>\frac{1}{x} - \frac{1}{x+1}</Math>
ãšãªãã
次ã®åé¡ã§ã¯ã
:<Math>\frac{1}{(x+1)(x+3)} = \frac{1}{(x+3) - (x+1)} \cdot \frac{(x+3) - (x+1)}{(x+1)(x+3)}</Math>
ãšå€åœ¢ããããšã«ãã£ãŠã
:<Math>\frac{1}{2} \{ \frac{x+3}{(x+1)(x+3)} - \frac{x+1}{(x+1)(x+3)} \}</Math>
ãšãªãã
:<Math>\frac{1}{2} (\frac{1}{x+1} - \frac{1}{x+3}) </Math>
ãšæ±ãŸãã
éšååæ°å解ã®æäœãéã«èŸ¿ããšãåæ°åŒã®éåã®æäœãšäžèŽããã
ã€ãŸãã'''éšååæ°å解ã¯éåã®éã®æäœ'''ã§ããã
ååãå®æ°ã®å Žåã«ã¯ãäžãšåæ§ã®æ¹æ³ã§éšååæ°å解ããããšãã§ããã
*åé¡
**以äžã®åæ°åŒãéšååæ°å解ãã
**#<Math>\frac{3}{(x-9)(x-4)}</Math>
**#<Math>\frac{7}{(3x-1)(5-2x)}</Math>
*解ç
1. <Math>\frac{3}{(x-9) (x-4)} </Math>
:<Math>= \frac{3}{(x-4) - (x-9)} \cdot \frac{(x-4) - (x-9)}{(x-9)(x-4)}</Math>
:<Math>= \frac{3}{5}\{ \frac{x-4}{(x-9)(x-4)} - \frac{x-9}{(x-9)(x-4)} \}</Math>
:<Math>= \frac{3}{5} ( \frac{1}{x-9} - \frac{1}{x-4} )</Math>
2. <Math>\frac{7}{(3x-1)(5-2x)}</Math>
:<Math>= \frac{-7}{(3x-1)(2x-5)} </Math>
:<Math>= \frac{-7}{(3x-1) - (2x-5)} \cdot \frac{(3x-1) - (2x-5)}{(3x-1)(2x-5)} </Math>
:<Math>= \frac{-7}{2(3x-1) - 3(2x-5)} \cdot \frac{2(3x-1) - 3(2x-5)}{(3x-1)(2x-5)} </Math>
:<Math>= \frac{-7}{(6x-2) - (6x-15)} \{ \frac{2(3x-1)}{(3x-1)(2x-5)} - \frac{3(2x-5)}{(3x-1)(2x-5)} \}</Math>
:<Math>= - \frac{7}{13} (\frac{2}{2x-5} - \frac{3}{3x-1})</Math>
:<Math>= \frac{7}{13} (\frac{3}{3x-1} - \frac{2}{2x-5})</Math>
éšååæ°å解ã¯æ°åã®åã®èšç®ãç©åèšç®ã埮åãå©çšããäžçåŒã®èšŒæçã«åœ¹ç«ã€ãéèŠãªå€åœ¢ã§ããã
=== åŒã®èšŒæ ===
==== æçåŒ ====
çåŒ <math>(a+b)^2=a^2+2ab+b^2</math>ã¯ãæå<math>a,b</math>ã«ã©ã®ãããªå€ã代å
¥ããŠãæãç«ã€ããã®ãããªçåŒã'''æçåŒ'''ïŒãããšãããïŒãšããã
çåŒ<math>\frac {1}{x-1} + \frac {1}{x+1} = \frac {2x}{x^2-1}</math>ã¯ã䞡蟺ãšã<math>x=1,-1</math>ã代å
¥ããããšã¯ã§ããªããããã®ä»ã®å€ã§ããã°ä»£å
¥ããããšãã§ãããŸãã©ã®ãããªå€ã代å
¥ããŠãçåŒãæãç«ã£ãŠããããããæçåŒãšåŒã¶ã
ãã£ãœãã<math>x^2 - x - 2 = 0</math> ã¯ãxïŒ2 ãŸã㯠xïŒãŒ1 ã代å
¥ãããšãã ãæãç«ã€ãããã®ããã«æåã«ç¹å®ã®å€ã代å
¥ãããšãã«ã ãæãç«ã€åŒã®ããšãæ¹çšåŒãšåŒã³ãæçåŒãšã¯åºå¥ããã
çåŒ <math>ax^2+bx+c=0</math> ã <math>x</math> ã«ã€ããŠã®æçåŒã§ããã®ã¯ã©ã®ãããªå ŽåããèããŠã¿ããã
ããåŒãã <math>x</math> ã«ã€ããŠã®æçåŒã§ããããšã¯ããã®åŒã®<math>x</math> ã«ã©ã®ãããªå€ã代å
¥ããŠãããã®çåŒã¯æãç«ã€ãšããæå³ã§ããããªã®ã§ãäŸãã° <math>x</math> ã«<math>-1\ ,\ 0\ ,\ 1</math> ã代å
¥ããåŒ
:<math>a-b+c=0</math>
:<math>c=0</math>
:<math>a+b+c=0</math>
ã¯ãã¹ãŠæãç«ã€å¿
èŠããããããã解ããš
:<math>a=b=c=0</math>
ãªã®ã§ãçåŒ <math>ax^2+bx+c=0</math> ã <math>x</math> ã«ã€ããŠã®æçåŒã«ãªããªãã°ã<math>a=b=c=0</math>ã§ãªããã°ãªããªãããšããããã
äžè¬ã«ãçåŒ <math>ax^2+bx+c=a'x^2+b'x+c'</math> ãæçåŒã§ããããšãšã<math>(a-a')x^2+(b-b')x+(c-c')=0</math> ãæçåŒã§ããããšãšåãã§ããã<br>
ãã£ãŠ
:<math>ax^2+bx+c=a'x^2+b'x+c'</math> ã<math>x</math>ã«ã€ããŠã®æçåŒ ã<math>\Leftrightarrow </math>ã <math>a=a'</math> ã〠<math>b=b'</math> ã〠<math>c=c'</math>
ãŸãšãããšæ¬¡ã®ããã«ãªãã
{| style="border:2px solid yellow;width:fit-content" cellspacing=0
|style="background:yellow"|'''æçåŒã®æ§è³ª'''
|-
|style="padding:5px"|
<math>P\ ,\ Q</math> ã <math>x</math> ã«ã€ããŠã®å€é
åŒãŸãã¯åé
åŒãšããã
::<math>P=0</math> ãæçåŒ ã<math>\Leftrightarrow </math> ã <math>P</math>ã®åé
ã®ä¿æ°ã¯ãã¹ãŠ<math>0</math>ã§ããã
::<math>P=Q</math> ãæçåŒ ã<math>\Leftrightarrow </math> ã <math>P</math>ãš <math>Q</math> ã®æ¬¡æ°ã¯çããã䞡蟺ã®åã次æ°ã®é
ã®ä¿æ°ã¯ãããããçããã
|}
* åé¡äŸ
** åé¡
次ã®çåŒã <math>x</math> ã«ã€ããŠã®æçåŒãšãªãããã«ã<math>a\ ,\ b\ ,\ c</math> ã®å€ãæ±ããã
:<math>x^2-3=a(x-1)^2+b(x-1)+c</math>
** 解ç
çåŒã®å³èŸºã <math>x</math> ã«ã€ããŠæŽçãããš
:<math>a(x-1)^2+b(x-1)+c=ax^2-2ax+a+bx-b+c=ax^2+(-2a+b)x+(a-b+c)</math>
:<math>x^2-3=ax^2+(-2a+b)x+(a-b+c)</math>
ãã®çåŒã <math>x</math> ã«ã€ããŠã®æçåŒãšãªãã®ã¯ã䞡蟺ã®åã次æ°ã®é
ã®ä¿æ°ãçãããšãã§ããããã£ãŠ
:<math>a=1</math>
:<math>-2a+b=0</math>
:<math>a-b+c=-3</math>
ããã解ããš
:<math>a=1\ ,\ b=2\ ,\ c=-2</math>
; '''è€éãªéšååæ°å解'''ïŒçºå±ïŒ
æçåŒãå©çšããããšã§ãè€éãªåæ°åŒã®éšååæ°å解ãã§ããã
*åé¡äŸ
**以äžã®åæ°åŒãéšååæ°å解ãã
**#<Math>\frac{3x-5}{(x+2)(2x-1)}</Math>
**#<Math>\frac{1}{(x-1)^2 (x-2)}</Math>
*解ç
:<Math>\frac{3x-5}{(x+3)(2x-1)} = \frac{a}{2x-1} + \frac{b}{x+3}</Math>
ãšããã
åæ¯ãæã£ãŠ
:<Math>3x-5 = a(x+3) + b(2x-1)</Math>
ããªãã¡
:<Math>3x-5 =(a+2b)x + (3a-b) </Math>
ããã<Math>x</Math>ã®æçåŒãªã®ã§ãä¿æ°ãæ¯èŒããŠ
:<Math>a+2b=3</Math>ãã€<Math>3b-a=-5</Math>
ããªãã¡
:<Math>a=-1, b=2</Math>
æåã®çåŒã«ä»£å
¥ããŠã
:<Math>\frac{3x-5}{(x+3)(2x-1)} = \frac{-1}{2x-1} + \frac{2}{x+3}</Math>
:<Math>= \frac{2}{x+3} - \frac{1}{2x-1}</Math>
次ã®åé¡ã¯ã
:<Math>\frac{1}{(x-1)^2 (x-2)} = \frac{a}{x-1} + \frac{b}{(x-1)^2} + \frac{c}{x-2}</Math>
ãšããããšã«ãããäžã®åé¡ãšåæ§ã«ããŠ
:<Math>a=-1, b=-1, c=1</Math>
ãšæ±ãŸãã®ã§ã
:<Math>\frac{1}{(x-1)^2 (x-2)} = - \frac{1}{x-1} - \frac{1}{(x-1)^2} + \frac{1}{x-2}</Math>
:<Math>= \frac{1}{x-2} - \frac{1}{x-1} - \frac{1}{(x-1)^2}</Math>
'''æçåŒãå©çšããéšååæ°å解'''
æ±ãããæ°åã<Math>a,b,c</Math>ãšããã
1. <Math>\frac{px+q}{(x+m)(x+n)} = \frac{a}{x+m} + \frac{b}{x+n}</Math>
2. <Math>\frac{px+q}{(x+m)^2} = \frac{a}{x+m} + \frac{b}{(x+m)^2}</Math>
3. <Math>\frac{px^2 + qx + r}{(x+m)^2 (x+n)} = \frac{a}{x+m} + \frac{b}{(x+m)^2 } + \frac{c}{x+n}</Math>
4. <Math>\frac{px^2 + qx + r}{(x+m)(x^2 + nx + l)} = \frac{a}{x+m} + \frac{bx+c}{x^2 + nx + l}</Math>
ãã®ããã«ãããåŒã<Math>x</Math>ã®æçåŒãšèŠãããšã«ãã£ãŠã<Math>a,b,c</Math>ãæ±ããããéšååæ°å解ãã§ããã
; '''2ã€ã®æåã«ã€ããŠã®æçåŒ'''ïŒçºå±ïŒ
*äŸ
a~fãå®æ°ãšããã
<Math>ax^2+by^2+cxy+dx+ey+f=0</Math>ãx, yã«ã€ããŠã®æçåŒã ãšããã
巊蟺ãxã«ã€ããŠæŽçãããšã<Math>ax^2+(cy+d)x+(by^2+ey+f)=0</Math>ã§ããã
ãããxã«ã€ããŠã®æçåŒãªã®ã§ã<Math>a=0, cy+d=0, by^2+ey+f=0</Math>ãæãç«ã€ã
ãããã¯æŽã«yã«ã€ããŠã®æçåŒãªã®ã§ã以äžã®çåŒãåŸãããã
:<Math>a=b=c=d=e=f=0</Math>
éã«ããããæãç«ãŠã°å
ã®åŒã¯æããã«x, yã«ã€ããŠã®æçåŒã§ãã
**åé¡
<Math>x^2+axy+6y^2-x+5y+b = (x-2y+c)(x-3y+d)</Math>ãx,yã«ã€ããŠã®æçåŒãšãªãããã«a,b,c,dãå®ããã
==== çåŒã®èšŒæ ====
ããã»ã©çŽ¹ä»ãããæçåŒããšããèšèã䜿ã£ãŠã蚌æãã®æå³ã説æãããªãããçåŒã蚌æããããšã¯ããã®åŒãæçåŒã§ããããšã瀺ãããšã§ããã
äžè¬ã«ãçåŒ AïŒB ã蚌æããããã«ã¯ã次ã®ãããªæé ã®ãããããå®è¡ããã°ããã
:(1)ããAãåŒå€åœ¢ããŠBãå°ããããŸã㯠Bãå€åœ¢ããŠAãå°ãã
:(2)ããA,Bãããããå€åœ¢ããŠãåãåŒCãå°ãã
:(3)ããA-BïŒ0 ã瀺ãã
ãã®ãšããå€åœ¢ã¯åå€å€åœ¢ã§ãªããã°ãªããªãããšã«æ³šæã
* äŸé¡ 1
<math>
(a+b)^2-(a-b)^2 = 4ab
</math>
ãæãç«ã€ããšã蚌æããã
ïŒèšŒæïŒ<br>
巊蟺ãå±éãããšã
:(巊蟺)ïŒ<math>
(a^2+2ab+b^2)-(a^2-2ab+b^2) = a^2+2ab+b^2 - a^2+2ab-b^2=4ab
</math>
ãšãªããããã¯å³èŸºã«çããããã£ãŠãçåŒ <math>
(a+b)^2-(a-b)^2 = 4ab
</math> ã¯èšŒæããããïŒçµïŒ
----
* äŸé¡ 2
<math>
(x+y)^2+(x-y)^2 = 2(x^2+y^2)
</math>
ãæãç«ã€ããšã蚌æããã
:ïŒèšŒæïŒ
巊蟺ãèšç®ãããšã
:ïŒå·ŠèŸºïŒ ïŒ <math> (x^2+2xy+y^2)+(x^2-2xy+y^2) = x^2+2xy+y^2 + x^2-2xy+y^2 = 2x^2+2y^2 =2(x^2+y^2) </math>
ããã¯å³èŸºã«çããããã£ãŠçåŒãæãç«ã€ããšã蚌æããããïŒçµïŒ
----
* åé¡äŸ
** åé¡
次ã®çåŒãæãç«ã€ããšã蚌æããã<br>
(I)
:<math>
(6 a + 7 b )^2 + (7 a - 6 b )^2 = (9 a + 2 b )^2 + (2 a - 9 b )^2
</math>
**解ç
(I)<br>
(巊蟺)<math>
= (36 a^2 + 84 a b + 49 b^2) + (49 a^2 - 84 a b + 36 a^2) = 85 a^2 + 85 b^2
</math><br>
(å³èŸº)<math>
= (81 a^2 + 36 a b + 4 b^2) + (4 a^2 - 36 a b + 81 b^2) = 85 a^2 + 85 b^2
</math><br>
䞡蟺ãšãåãåŒã«ãªããã
:<math>
(6 a + 7 b )^2 + (7 a - 6 b )^2 = (9 a + 2 b )^2 + (2 a - 9 b )^2
</math>
æçåŒã§ãªããšãããäžããããæ¡ä»¶ããçåŒã蚌æããããšãã§ããã
*åé¡äŸ
**<Math>a+b+c=0</Math>ã®ãšãã<Math>a^3+b^3+c^3=3abc</Math>ã§ããããšã蚌æããããŸãã<Math>a:b=c:d</Math>ã®ãšãã<Math>\frac{a+c}{b+d} = \frac{a-c}{b-d}</Math>ã蚌æããã
**解ç
:<Math>a+b+c=0 \iff c=-(a+b)</Math>
ããã
:<Math>a^3+b^3+c^3-3abc = a^3+b^3-(a+b)^3+3ab(a+b)</Math>
:<Math>= a^3+b^3-(a^3+3a^2b+3ab^2+b^3)+3a^2b+3ab^2</Math>
:<Math>=0</Math>
ãã£ãŠã<Math>a^3+b^3+c^3=3abc</Math>ã§ããã
ãŸãã
:<Math>a:b=c:d \iff \frac{a}{b} = \frac{c}{d}</Math>
ãããäžåŒã®å³èŸºãkãšãããšã
:<Math>a=bk, c=dk</Math>
ãªã®ã§ã
:<Math>\frac{a+c}{b+d} = \frac{bk+dk}{b+d} = \frac{k(b+d)}{b+d} = k</Math>
:<Math>\frac{a-c}{b-d} = \frac{bk-dk}{b-d} = \frac{k(b-d)}{b-d} = k</Math>
ãã£ãŠã<Math>\frac{a+c}{b+d} = \frac{a-c}{b-d}</Math>ã§ããã
ãªããæ¯<Math>a:b</Math>ã«ã€ããŠ<Math>\frac{a}{b}</Math>ã'''æ¯ã®å€'''ãšããããŸãã<Math>a:b=c:d \iff \frac{a}{b} = \frac{c}{d}</Math>ã'''æ¯äŸåŒ'''ãšããã
<Math>\frac{a}{x} = \frac{b}{y} = \frac{c}{z}</Math>ãæãç«ã€ãšãã<Math>a:b:c=x:y:z</Math>ãšè¡šããããã'''é£æ¯'''ãšããã
*åé¡
**<Math>a:b:c=1:2:3</Math>ã®ãšãã<Math>a+b+c=24</Math>ãæºãã<Math>a,b,c</Math>ãæ±ããã
==== äžçåŒã®èšŒæ ====
äžçåŒã®ããŸããŸãªå
¬åŒã«ã€ããŠã¯ã次ã®4ã€ã®åŒãåºæ¬çãªåŒãšããŠå°åºã§ããå Žåãããããã
é«æ ¡æ°åŠã§ã¯ã次ã®4ã€ã®æ§è³ªã äžçåŒã®ãåºæ¬æ§è³ªããªã©ãšããŠçŽ¹ä»ãããŠããã
{| style="border:2px solid skyblue; width:fit-content" cellspacing=0
|style="background:skyblue"|'''äžçåŒã®åºæ¬æ§è³ª'''
|-
|style="padding:5px"|
:(1)ãã<math> a>b </math> ã〠<math> b>c </math> ãªãã° <math> a>c </math>
:(2)ãã<math> a>b </math> ãªãã° <math> a+c>b+c </math> ã〠<math> a-c>b-c </math>
:(3)ãã<math> a>b </math> ã〠<math> c>0 </math> ãªãã° <math> ac>bc </math> ã§ããã<math> \frac{a}{c} > \frac{b}{c} </math>ã§ããã
:(4)ãã<math> a>b </math> ã〠<math> c<0 </math> ãªãã° <math> ac<bc </math> ã§ããã<math> \frac{a}{c} < \frac{b}{c} </math>ã§ããã
|}
(3)ãš(4)ã«ã€ããŠã¯ãã²ãšã€ã®æ§è³ªãšã㊠ãŸãšããŠããæ€å®æç§æžãããïŒâ» åæ通ãªã©ïŒã
æ°åŠIAã§ç¿ã£ãããªãã°ãã®æå³ã®èšå· <math>\Longrightarrow </math> ã䜿ããšã
{| style="border:2px solid skyblue; width:fit-content" cellspacing=0
|style="background:skyblue"|'''äžçåŒã®åºæ¬æ§è³ª'''
|-
|style="padding:5px"|
:(1)ãã<math> a>b </math> ã〠<math> b>c </math> ã<math>\Longrightarrow </math>ã <math> a>c </math>
:(2)ãã<math> a>b </math> <math>\Longrightarrow </math> <math> a+c>b+c </math> ããã€ã<math> a-c>b-c </math>
:(3)ãã<math> a>b </math> ã〠<math> c>0 </math> ã<math>\Longrightarrow</math>ã <math> ac>bc </math> ã§ããã<math> \frac{a}{c} > \frac{b}{c} </math>ã§ããã
:(4)ãã<math> a>b </math> ã〠<math> c<0 </math> ã<math>\Longrightarrow</math>ã <math> ac<bc </math> ã§ããã<math> \frac{a}{c} < \frac{b}{c} </math>ã§ããã
|}
ãšãæžããã
äžè¿°ã®4ã€ã®åºæ¬æ§è³ªããã
:a>0, ãb>0 ãªãã° aïŒb ïŒ 0
ã蚌æããŠã¿ããã
ïŒèšŒæïŒ
ãŸã a>0 ãªã®ã§ãåºæ¬æ§è³ª(2)ãã
:aïŒb > b
ã§ããã
ãã£ãŠã
:<math> a+b>b </math> ã〠<math> b>0 </math>
ãªã®ã§ãåºæ¬æ§è³ª(1)ãã<math> a+b>0 </math>
ãæãç«ã€ãïŒçµïŒ
åæ§ã«ããŠã
:aïŒ0, ãbïŒ0 ãªãã° aïŒb ïŒ 0
ã蚌æã§ããã
::ïŒâ» èªè
ã¯èªå㧠ããã蚌æããŠã¿ããæ€å®æç§æžã«ãããã®åŒã®èšŒæã¯çç¥ãããŠãããïŒ
ãããŸã§ã«ç€ºããããšãããäžçåŒ <math> A \geqq B </math> ã蚌æãããå Žåã«ã¯ã
: <math> A-B \geqq 0 </math>
ã蚌æããã°ããããšãããã£ãããã¡ãã®æ¹ã蚌æããããå Žåãããããã
äžçåŒã蚌æããéã«æ ¹æ ãšããåºæ¬çãªäžçåŒãšããŠã次ã®æ§è³ªãããã
{| style="border:2px solid skyblue; width:fit-content" cellspacing=0
|style="background:skyblue"|'''å®æ°ã®2ä¹ã®æ§è³ª'''
|-
|style="padding:5px"|
å®æ° a ã«ã€ããŠãããªãã
:<math>a^2 \geqq 0</math>
ãæãç«ã€ã
ãã®åŒã§çå·ãæãç«ã€å Žåãšã¯ã <math>a = 0</math> ã®å Žåã ãã§ããã
|}
ãã®å®çïŒãå®æ°ã2ä¹ãããšãããªãããŒã以äžã§ãããïŒããåºæ¬æ§è³ª(3),(4)ã䜿ã£ãŠèšŒæããŠã¿ããã
'''ïŒèšŒæïŒ'''
aãæ£ã®å Žåãšè² ã®å Žåãš0ã®å Žåã®3éãã«å Žåããããã
'''<nowiki>[aãæ£ã®å Žå]</nowiki>''' <br>
ãã®ãšããåºæ¬æ§è³ª(3)ããã
:<math> aa>0a </math>
ã§ãããããªãã¡ã
:<math> a^2 > 0 </math>
ã§ããã
'''<nowiki>[aãè² ã®å Žå]</nowiki>'''<br>
ãã®ãšããåºæ¬æ§è³ª(4)ãã
<math>0a < aa </math>
ã§ãããããªãã¡ã
: <math> a^2 > 0 </math>
ã§ããã
'''<nowiki>[aããŒãã®å Žå]</nowiki>''' <br>
ãã®ãšãã
<math>a^2=0</math>
ã§ããã
ãã£ãŠããã¹ãŠã®å Žåã«ã€ããŠ<math>a^2 \geqq 0</math>
(çµ)
ãã®ããšãšåºæ¬æ§è³ª(1)(2)ããã次ãæãç«ã€ããšããããã
{| style="border:2px solid skyblue; width:fit-content" cellspacing=0
|style="background:skyblue"|'''å®æ°ã®2ä¹ã©ããã®åã®æ§è³ª'''
|-
|style="padding:5px"|
2ã€ã®å®æ°a,b ã«ã€ã㊠<math>a^2 \geqq 0</math>, ã<math>b^2 \geqq 0</math> ã§ãããããããªãã
:<math>a^2+b^2 \geqq 0</math>
ãæãç«ã€ã
äžåŒã§çå·ãæãç«ã€å Žåãšã¯ã <math>a^2 = 0</math> ã〠<math>b^2 = 0</math> ã®å Žåã ãã§ãããã€ãŸã <math>a = 0</math> ã〠<math>b = 0</math> ã®å Žåã ãã§ããã
|}
** åé¡
次ã®äžçåŒãæãç«ã€ããšã蚌æããã<br>
:<math>
x^2 + 10 y^2 \geqq 6 x y
</math>
(蚌æ)<br>
:<math>
(x^2 + 10 y^2) -(6 x y) \geqq 0
</math>
ã蚌æããã°ããã
巊蟺ãå±éã㊠ãŸãšãããšã
:<math>
(x^2 + 10 y^2) - 6xy = x^2 - 6 x y + 9 y^2 + y^2 = (x - 3 y)^2 + y^2
</math>
ãšãªãã
äžåŒã®æåŸã®åŒã®é
ã«ã€ããŠã
:<math>
(x - 3 y)^2 \geqq 0 , \quad y^2 \geqq 0
</math>
ã ããã
:<math>
(x - 3 y)^2 + y^2 \geqq 0
</math>
ã§ããããã£ãŠ
:<math>
x^2 + 10 y^2 \geqq 6 x y
</math>
ã§ãããïŒçµïŒ
===== æ ¹å·ãå«ãäžçåŒ =====
2ã€ã®æ£ã®æ° a,ãb ã aïŒb ãŸã㯠aâ§b ãªãã°ã䞡蟺ã2ä¹ããŠã倧å°é¢ä¿ã¯åããŸãŸã§ããã
ã€ãŸãã
: <math> a>0 </math>,ã<math> b>0 </math> ã®ãšãã
:
: <math> a > b \quad \Longleftrightarrow \quad a^2 > b^2 </math>
: <math> a > b \quad \Longleftrightarrow \quad a^2 > b^2 </math>
:
: ããã蚌æããã«ã¯ã<math> a^2 - b^2 </math> ã調ã¹ãã°ããã
:<math> a^2 - b^2 = (a+b)(a-b) </math>
ã§ããã
a>bãšãããä»®å®ãããa,b ã¯æ£ã®æ°ãªã®ã§ã<math> (a+b)>0 </math> ã§ãããå¥ã®ä»®å®ããã a > b ãªã®ã§ã<math> (a-b)>0 </math> ã§ãããããã£ãŠã<math> a^2 - b^2 = (a+b)(a-b) >0 </math>
éã«ã<math>a^2-b^2>0</math>ã®ãšãã<math>(a+b)(a-b)>0</math>ã§ããã<math>a>0,b>0</math>ãªã®ã§<math>a+b>0</math>ã§ããããã£ãŠã<math>a-b>0</math>ãªã®ã§ã<math>a>b</math>ã§ããã
ãã£ãŠã<math> a > b \quad \Longleftrightarrow \quad a^2 > b^2 </math> ã§ããã
aâ§bã®å Žåãåæ§ã«èšŒæã§ããã
----
ç·Žç¿ãšããŠã次ã®åé¡ãåããŠã¿ããã
;äŸé¡
<math> a>0 </math>,ã<math> b>0 </math> ã®ãšãã次ã®äžçåŒã蚌æããã
::<math> \sqrt{a} + \sqrt{b} > \sqrt{a+b} </math>
ïŒèšŒæïŒ
äžçåŒã®äž¡èŸºã¯æ£ã§ããã®ã§ã䞡蟺ã®å¹³æ¹ã®å·®ãèããã°ããã䞡蟺ã®å¹³æ¹ã®å·®ã¯
:<math>( \sqrt{a} + \sqrt{b} )^2 - ( \sqrt{a+b} )^2 = a + 2 \sqrt{a} \sqrt{b} + b - (a+b) 2 \sqrt{ab} </math>
ã§ãããããã§ãa,b ã¯ãšãã«æ£ã®å®æ°ãªã®ã§ã
::<math> \sqrt{a} \sqrt{b} = \sqrt{ab} </math>
ã§ããããšãçšããã
:<math> \sqrt{ab} > 0</math>
ã§ããã®ã§ã
:<math>( \sqrt{a} + \sqrt{b} )^2 - ( \sqrt{a+b} )^2 > 0 </math>
ãšãªãããã£ãŠã
:<math> \sqrt{a} + \sqrt{b} > \sqrt{a+b} </math>
ã§ãããïŒçµïŒ
===== 絶察å€ãå«ãäžçåŒ =====
å®æ° a ã®çµ¶å¯Ÿå€ |a| ã«ã€ããŠã
: a ⧠0 ã®ãšã |a|ïŒa , ã
: aïŒ0 ã®ãšã |a|ïŒ ãŒa
ã§ããããã次ã®ããšãæãç«ã€ã
''' |a|â§a , |a|⧠ãŒa ,ã|a|<sup>2</sup>ïŒa<sup>2</sup> '''
ãŸãã2ã€ã®å®æ° a, b ã®çµ¶å¯Ÿå€ |ab| ã«ã€ããŠã¯ã
: |ab| <sup>2</sup> ïŒ (ab)<sup>2</sup> ïŒ a<sup>2</sup> b<sup>2</sup> ïŒ |a|<sup>2</sup> |b|<sup>2</sup> ïŒ (|a| |b|)<sup>2</sup>
ãæãç«ã€ã®ã§ãããã«ããã« |ab|â§0 ,ã|a||b|â§0 ãçµã¿åãããŠã
''' |ab| ïŒ |a| |b| '''
ãæãç«ã€ã
(äŸé¡)
次ã®äžçåŒã蚌æããããŸããçå·ãæãç«ã€ã®ã¯ ã©ã®ãããªå Žåãã 調ã¹ãã
::|a|ïŒ|b| ⧠|aïŒb|
:(蚌æ)
䞡蟺ã®å¹³æ¹ã®å·®ãèãããšã
:: (|a|ïŒ|b|)<sup>2</sup> ㌠|aïŒb|<sup>2</sup> ïŒ |a|<sup>2</sup> ïŒ 2|a| |b| ïŒ |b|<sup>2</sup> ãŒ(a<sup>2</sup> ïŒ 2ab ïŒ b<sup>2</sup> )
:::::::: ïŒ a<sup>2</sup> ïŒ 2|a| |b| ïŒ b<sup>2</sup> ãŒa<sup>2</sup> ㌠2ab ㌠b<sup>2</sup>
:::::::: ïŒ 2|a| |b| ㌠2ab
:::::::: ïŒ 2 ( |a| |b| ㌠ab )
ãããããæ£ãªããäžããããäžçåŒ |a|ïŒ|b| ⧠|aïŒb| ãæ£ããã
ããã§ã |a| |b| ⧠ab ã§ããã®ã§ã
:: ( |a| |b| ㌠ab ) ⧠0
ã§ããã
ãããã£ãŠã |a|ïŒ|b| ⧠|aïŒb| ã§ããã
çå·ãæãç«ã€ã®ã¯ |a| |b| ïŒ ab ã®å Žåãããªãã¡ ab ⧠0 ã®å Žåã§ãããïŒèšŒæ ãããïŒ
{{ã³ã©ã |äžè§äžçåŒ|
ãªã
::<nowiki>|a|ãŒ|b| ⊠|aïŒb| ⊠|a|ïŒ|b| </nowiki>
ã®é¢ä¿åŒã®ããšããäžè§äžçåŒããšããã
}}
==== çžå å¹³åãšçžä¹å¹³å ====
2ã€ã®æ°<math>a</math>,<math>b</math>ã«å¯Ÿãã<math>\frac{a+b}{2}</math>ã'''çžå å¹³å'''ïŒããããžãããïŒãšèšãã<math>\sqrt{ab}</math>ã'''çžä¹å¹³å'''ïŒããããããžãããïŒãšããã
{{ã³ã©ã |çžä¹å¹³åã®äŸãš3ã€ä»¥äžã®ãã®ã®å¹³å|
å¹³åã¯ã3ã€ä»¥äžã®ãã®ã«ãå®çŸ©ãããã3ã€ä»¥äžã®nåã®ãã®ã®çžå å¹³å㯠<math>\frac{a_1 + a_2 + \cdots +a_n }{n}</math> ã§å®çŸ©ãããã
:å¹³åãèããéãã€ãçžå å¹³åã°ãããèããã¡ã ãã以äžã®ãããªç¶æ³ã§ã¯çžä¹å¹³åã®æ¹ãé©åã§ããã
::ãããäŒæ¥ã§ã¯ã2015幎床ã®å£²äžãåºæºã«ãããšã2016幎床ã§ã¯å幎ïŒ2015幎ïŒã®1.5åã®å£²äžã«ãªããŸããã2017幎床ã§ã¯ãå幎ïŒ2016幎ïŒã®2åã®å£²äžã«ãªããŸãããå¹³åãšããŠãäžå¹Žããšã«äœåã®å£²ãäžãã«ãªã£ãŠãã£ãã§ããããïŒ ã
:ïŒçïŒ<math>\sqrt{1.5 \times 2} = \sqrt{3} \fallingdotseq 1.73</math> ãããçŽ 1.73åã
:ãŸãããã®å¿çšäŸã¯ãé
ã3ã€ä»¥äžã®å Žåã®çžä¹å¹³åã®å®çŸ©ã®ä»æ¹ãã瀺åããŠãããããèªè
ã[[é«çåŠæ ¡æ°åŠII/ææ°é¢æ°ã»å¯Ÿæ°é¢æ°|ææ°é¢æ°]]ãç¥ã£ãŠãããªããé
ã3ã€ïŒããã§ã¯ a, b, c ãšããïŒã®å Žåã®çžä¹å¹³åã¯ã
::ïŒ3ã€ã®é
ã®çžä¹å¹³åïŒïŒ<math> (abc)^{ \frac{1}{3} } </math>
:ã«ãªãã
}}
æ¬ããŒãžã§ã¯ã2åã®æ°ã®å¹³åã«ã€ããŠèå¯ããã
çžå å¹³åãšçžä¹å¹³åã«ã€ããŠã次ã®é¢ä¿åŒãæãç«ã€ã
{| style="border:2px solid yellow;width:80%" cellspacing=0
|style="background:yellow"|'''çžå å¹³åãšçžä¹å¹³å'''
|-
|style="padding:5px"|
<math>a \geqq 0</math> ïŒ<math>b \geqq 0</math>ã®ãšãã<br>
<center><math>\frac{a+b}{2} \geqq \sqrt{ab}</math></center><br>
çå·ãæãç«ã€ã®ã¯ã<math>a = b</math>ã®ãšãã§ããã
|}
ïŒèšŒæïŒ
<math>a \geqq 0 , b \geqq 0</math>ã®ãšã
:<math>
\frac{a+b}{2} - \sqrt{ab} = \frac{a+b-2 \sqrt{ab}}{2} = \frac{\left( \sqrt{a} \right) ^2 - 2 \sqrt{a} \sqrt{b} + \left( \sqrt{b} \right) ^2}{2} = \frac{\left( \sqrt{a} - \sqrt{b} \right) ^2 }{2}
</math>
<math> \left( \sqrt{a} - \sqrt{b} \right) ^2 \geqq 0</math>ã§ããããã<math> \frac{\left( \sqrt{a} - \sqrt{b} \right) ^2 }{2} \geqq 0</math><br>
ãããã£ãŠã<math>\frac{a+b}{2} \geqq \sqrt{ab}</math><br>
çå·ãæãç«ã€ã®ã¯ã<math>\left( \sqrt{a} - \sqrt{b} \right) ^2 = 0 </math> ã®ãšããããªãã¡ <math>a = b</math> ã®ãšãã§ããã(蚌æ ããã)
å
¬åŒã®å©çšã§ã¯ãäžã®åŒ <math>\frac{a+b}{2} \geqq \sqrt{ab}</math> ã®äž¡èŸºã«2ãããã <math>a+b \geqq 2 \sqrt{ab}</math> ã®åœ¢ã®åŒã䜿ãå Žåãããã
* åé¡äŸ
** åé¡
<math>a>0</math> ïŒ<math>b>0</math>ã®ãšãã次ã®äžçåŒãæãç«ã€ããšã蚌æããã<br>
(I)
:<math>
a + \frac{1}{a} \geqq 2
</math>
(II)
:<math>
(a+b)\left( \frac{1}{a} + \frac{1}{b} \right) \geqq 4
</math>
**解ç
(I)<math>a>0</math>ã§ããããã<math>\frac{1}{a} >0</math><br>
ãã£ãŠã<math>a + \frac{1}{a} \geqq 2 \sqrt{a \times \frac{1}{a}} = 2</math><br>
ãããã£ãŠ
:<math>
a + \frac{1}{a} \geqq 2
</math>
(II)
:<math>
(a+b)\left( \frac{1}{a} + \frac{1}{b} \right) = 1+ \frac{a}{b} + \frac{b}{a} +1 = \frac{b}{a} + \frac{a}{b} +2
</math>
<math>a>0</math>ïŒ<math>b>0</math>ã§ããããã<math>\frac{b}{a} >0</math>ïŒ<math>\frac{a}{b} >0</math><br>
ãã£ãŠã<math> \frac{b}{a} + \frac{a}{b} +2 \geqq 2 \sqrt{\frac{b}{a} \times \frac{a}{b}} + 2 = 2+2 =4</math><br>
ãããã£ãŠ
:<math>
(a+b)\left( \frac{1}{a} + \frac{1}{b} \right) \geqq 4
</math>
{{ã³ã©ã |3ã€ä»¥äžã®çžä¹å¹³åãšèª¿åå¹³å|
ããèªè
ãææ°é¢æ°ãªã©ãç¥ã£ãŠããã°ã
nåã®ãã®ã®çžä¹å¹³åã¯ã
::<math>\sqrt[n] {a_1 a_2 \cdots a_n }</math>
ãšæžããã
æ°åŠçãªãå¹³åãã«ã¯ãçžå å¹³åãšçžä¹å¹³åã®ã»ãã«ã調åå¹³åãããã
調åå¹³åã¯ãé»æ°åè·¯ã®äžŠåèšç®ã§äœ¿ãããèãæ¹ã§ããã
nåã®ãã®ã®èª¿åå¹³åã¯ã
::<math>\frac{ n}{ \dfrac{1}{a_1} + \dfrac{1}{a_2} + \cdots + \dfrac{1}{a_n} }</math>
ã§å®çŸ©ãããã
äžè¬ã«æ°åŠçã«ã¯ã調åå¹³åãçžä¹å¹³åãçžå å¹³åã®ããã ã«æ¬¡ã®ãããªå€§å°é¢ä¿
:ïŒèª¿åå¹³åïŒ âŠ ïŒçžä¹å¹³åïŒ âŠ ïŒçžå å¹³åïŒ
ãšããé¢ä¿ãæãç«ã€ããšã蚌æãããŠããã
ããªãã¡ãæ°åŒã§æžãã°
::<math>\frac{ n}{ \dfrac{1}{a_1} + \dfrac{1}{a_2} + \cdots + \dfrac{1}{a_n} } \leqq \sqrt[n] {a_1 a_2 \cdots a_n } \leqq \frac{a_1 + a_2 + \cdots +a_n }{n} </math>
ã®é¢ä¿åŒã§ããã
ç°¡æœã«æžããšã
::<Math>\frac{ n}{ \sum_{k=1}^{n} \dfrac{1}{a_k}} \leqq (\prod_{k=1}^{n}a_k)^{\frac{1}{n}} \leqq \frac{\sum_{k=1}^{n} a_n}{n}</Math>
ãšãªãã
}}
== é«æ¬¡æ¹çšåŒ ==
=== è€çŽ æ° ===
2ä¹ããŠè² ã«ãªãæ°ããšãããã®ãèããããã®ãããªæ°ã¯ãäžåŠã§ç¿ã£ãå®æ°ã®äžã«ã¯ãªãããšããããããªããªãã°ãæ£ã®æ°ã§ãè² ã®æ°ã§ã2ä¹ãããšç¬Šå·ãæã¡æ¶ããŠæ£ã®æ°ã«ãªã£ãŠããŸãããã§ãããããã§é«æ ¡ã§ã¯ã2ä¹ããŠè² ã«ãªããšããæ§è³ªãæã€æ°ã®æŠå¿µãæ°ããå°å
¥ããããšã«ããã
:<math>x^2 = -1</math>
ãšããæ¹çšåŒãèããããã®æ¹çšåŒã®è§£ã¯å®æ°ã«ã¯ãªããããã§ããã®æ¹çšåŒã®è§£ãšãªãæ°ãæ°ããäœãããã®åäœãæå <math>i</math> ã§ããããã
ãã® <math>i</math> ã®ããšã'''èæ°åäœ'''ïŒãããããããïŒãšåŒã¶ãïŒèæ°åäœã®èšå· i ãè±èªã®ã¢ã«ãã¡ãããã®ã¢ã€ã®å°æåã§ã imaginary unit ã«ç±æ¥ãããšèããããŠãããïŒ
<math>1+i</math> ã <math>2+5i</math> ã®ããã«ãèæ°åäœ<math>i</math>ãšå®æ°<math>a,b</math>ãçšããŠ
:<math>a+bi</math>
ãšè¡šãããšãã§ããæ°ã'''è€çŽ æ°'''ïŒãµããããïŒãšããããã®ãšãã''a''ããã®è€çŽ æ°ã®'''å®éš'''ïŒãã€ã¶ïŒãšããã''b''ã'''èéš'''ïŒããã¶ïŒãšããã
äŸãã°ã<math>1+i,\quad 2+5i,\quad \frac{9}{2} + \frac{7}{2} i,\quad 4i,\quad 3</math> ã¯ãããããè€çŽ æ°ã§ããã
è€çŽ æ° aïŒbi ã¯ïŒãã ã aãšbã¯å®æ°ïŒãbã0ã®å Žåã«ããããå®æ°ãšèŠãããšãã§ããã
èšãæ¹ãããããšãè€çŽ æ°ãåºæºã«èãããšãå®æ°ãšã¯ã aïŒ0i ã®ãããªãèéšã®ä¿æ°ããŒãã«ãªãè€çŽ æ°ã®ããšã§ãããšãèšããã
4''i''ã®ãããªãèéšã0以å€ã§å®éšããŒãã®è€çŽ æ°ã'''çŽèæ°'''ïŒãã
ãããããïŒãšåŒã¶ãçŽèæ°ã¯ã2ä¹ãããšè² ã«ãªãæ°ã§ããã
å®æ°ãèéšã0ã®è€çŽ æ°ãšèããããã
å®æ°ã§ãªãè€çŽ æ°ã®ããšããèæ°ãïŒããããïŒãšããã
=== è€çŽ æ°ã®æ§è³ª ===
2ã€ã®è€çŽ æ° a+bi ãš c+di ãšãçãããšã¯ã
: aïŒc ã〠bïŒd
ã§ããããšã§ããã
ã€ãŸãã
: a+bi ïŒ c+di ã<math>\Longleftrightarrow</math>ã a=c ã〠bïŒd
ãšãã«ãè€çŽ æ°aïŒbi ã 0ã§ãããšã¯ãaïŒ0 ã〠bïŒ0 ã§ããããšã§ããã
: a+bi ïŒ 0 ã<math>\Longleftrightarrow</math>ã a=0 ã〠bïŒ0
{| style="border:2px solid skyblue;width:80%" cellspacing=0
|style="background:skyblue"|'''è€çŽ æ°ã®çžç'''
|-
|style="padding:5px"|
: a+bi ïŒ c+di ã<math>\Longleftrightarrow</math>ã a=c ã〠bïŒd
: a+bi ïŒ 0 ã<math>\Longleftrightarrow</math>ã a=0 ã〠bïŒ0
|}
;å
±åœ¹
è€çŽ æ°<math>z=a+bi</math>ã«å¯ŸããŠãèéšã®ç¬Šå·ãå転ãããè€çŽ æ°<math>a-bi</math>ã®ããšãã'''å
±åœ¹'''ïŒãããããïŒãªè€çŽ æ°ããŸãã¯ãè€çŽ æ°<math>z</math>ã®å
±åœ¹ãã®ããã«åŒã³ã <math> \bar z </math> ã§ããããããªãããå
±åœ¹ãã¯ãå
±'''è»'''ãã®åžžçšæŒ¢åã«ããæžãæãã§ããã
å®æ°aãšå
±åœ¹ãªè€çŽ æ°ã¯ããã®å®æ° a èªèº«ã§ããã
è€çŽ æ° zïŒa+bi ã«ã€ããŠ
:<math>z+ \bar z =(a+bi)+(a-bi)=2a</math>
:<math>z \bar z =(a+bi)(a-bi)=a^2-abi+abi-b^2 i^2 = a^2-b^2i^2=a^2+b^2</math>
;ååæŒç®
è€çŽ æ°ã«ãååæŒç®ïŒå æžä¹é€ïŒãå®çŸ©ãããã
è€çŽ æ°ã®æŒç®ã§ã¯ãèæ°åäœ<math>i</math>ããéåžžã®æåã®ããã«æ±ã£ãŠèšç®ãããäžè¬ã«è€çŽ æ°<math>z\ ,\ w</math>ãã<math>z=a+bi\ ,\ w=c+di</math>ã§äžãããããšã(ãã ã <math>a\ ,\ b\ ,\ c\ ,\ d</math>ã¯å®æ°ãšãã)ã
:å æ³ãã<math> (a+bi)+(c+di) = (a+c) + (b+d)i </math>
:æžæ³ãã<math> (a+bi)-(c+di) = (a-c) + (b-d)i </math>
:ä¹æ³ãã<math> (a+bi)(c+di) = (ac-bd) + (ad+bc)i </math>
:é€æ³ãã<math> \frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{(c+di)(c-di)} = \frac{ac+bd}{c^2+d^2} + \frac{bc-ad}{c^2+d^2}i </math>ããïŒãã ã <math>c+di \ne 0</math> ãšãããïŒ
ãšãããµãã«è€çŽ æ°ã®å æžä¹é€ã®èšç®æ³ãå®ããããŠããã
ä¹æ³ã®å®çŸ©ã¯ãäžèŠãããšé£ãããã«ã¿ããããå®æ°ã®åé
æ³åãšåæ§ã«å±éããŠããæåŸã« i<sup>2</sup>ã«ãã€ãã¹1ã代å
¥ããŠãã£ãã ãã§ããã
é€æ³ã®å®çŸ©ã¯ãååãšåæ¯ã«ãåæ¯ãšå
±åœ¹ãªåœ¢ã®åŒã æãç® ããã ãã§ããã
ä¹æ³ãé€æ³ã®å®çŸ©åŒãæèšããå¿
èŠã¯ç¡ããèšç®ã®éã«ã¯ãå¿
èŠã«å¿ããŠåé
æ³åãå
±åœ¹ãªã©ã®ãå¿
èŠãªåŒå€åœ¢ãè¡ãã°ããã
'''äŸé¡'''
2ã€ã®è€çŽ æ°
:<math>a=3+i</math>
:<math>b=4 +7i</math>
ã«ã€ããŠã<math>a+b</math> ãš <math>ab</math> ãš <math>\frac a b</math> ããããããèšç®ããã
'''解ç'''
:<math>\begin{align}
a+b&=(3+i)+(4+7i)\\
&=(3+4)+i(1+7)\\
&=7+8i\\
\end{align}</math>
:<math>\begin{align}
ab&=(3+i)(4+7i) \\
&=12+21i+4i+7i^2 \\
&=12+21i+4i+(-7) \\
&=5+25i \\
\end{align}</math>
ã§ããã
:<math>\frac{a}{b}=\frac{3+i}{4+7i}</math>
ããããã«ç°¡åã«ã§ããªãã ããããå®ã¯ãã¡ãã£ãšãããã¯ããã¯ãçšããã°ããèŠããã圢ã«ã§ããã
åæ°ã¯åæ¯ãšååã«åãæ°ããããŠããã£ãã®ã§ãåæ¯ãšååã«åæ¯ã®å
±åœ¹ããããŠã¿ãããããšã
:<math>\begin{align}
\frac{a}{b}&=\frac{3+i}{4+7i} \\
&=\frac{(3+i)(4-7i)}{(4+7i)(4-7i)} \\
&=\frac{12-21i+4i-(-7)}{16-28i+28i-(-49)} \\
&=\frac{19-17i}{65} \\
&=\frac{19}{65}-\frac{17}{65}i \\
\end{align}</math>
ãåŸãããããã®åœ¢ã®ã»ããå
ã®åŒããããã£ãšèŠããã圢ã§ããã
ãã®ãããªæäœãåæ¯ã®å®æ°åãšããããšããããæ°åŠIã§åŠç¿ããå±éã»å æ°å解å
¬åŒ <math>(a+b)(a-b)=a^2-b^2</math>ã®ç°¡åãªå¿çšã§ããã
=== è² ã®æ°ã®å¹³æ¹æ ¹ ===
æ°ã®ç¯å²ãè€çŽ æ°ã«ãŸã§æ¡åŒµãããšãè² ã®æ°ã®å¹³æ¹æ ¹ãèããããšãã§ããã
äŸãšããŠã -5 ã®å¹³æ¹æ ¹ã«ã€ããŠèããŠã¿ããã<br>
:<math>
(\sqrt{5}\ i)^2 = (\sqrt{5})^2\ i^2 = 5 \times (-1) =-5
</math>
:<math>
(- \sqrt{5}\ i)^2 = (-1)^2 \times (\sqrt{5})^2\ i^2 = (+1) \times 5 \times (-1) = -5
</math>
ã§ããããã -5 ã®å¹³æ¹æ ¹ã¯ <math> \sqrt{5}\ i </math> ãš <math> - \sqrt{5}\ i </math> ã§ããã
{| style="border:2px solid skyblue;width:80%" cellspacing=0
|style="background:skyblue"|'''è² ã®æ°ã®å¹³æ¹æ ¹'''
|-
|style="padding:5px"|
<math>a>0</math>ãšãããšããè² ã®æ°<math>-a</math>ã®å¹³æ¹æ ¹ã¯ã<math>\sqrt{a}\ i</math>ãš<math>- \sqrt{a}\ i</math>ã§ããã
|}
<math> \sqrt{-5} </math>ãšã¯ã<math> \sqrt{5}\ i </math> ã®ããšãšããã<math> - \sqrt{-5} </math>ãšã¯ã<math> - \sqrt{5}\ i </math> ã®ããšã§ããã
ãšãã« <math> \sqrt{-1}\ = \ i </math> ã§ããã
ããŠã-5 ã®å¹³æ¹æ ¹ã¯ãæ¹çšåŒ<math>x^2=-5</math> ã®è§£ã§ãããã
ãã®æ¹çšåŒã移é
ããããšã«ããã-5 ã®å¹³æ¹æ ¹ã¯ã
:<math>
x^2+5=0
</math>
ã®è§£ã§ãããšããããã
ããã«å æ°å解ãããããšã«ããã-5 ã®å¹³æ¹æ ¹ã¯æ¹çšåŒ
:<math>
(x + \sqrt{5}\ i)(x - \sqrt{5}\ i) =0
</math>
ã®è§£ã§ããããšããããã
* äŸé¡
(I) ãã<math>\sqrt{-2}\ \sqrt{-6}</math>ããèšç®ããã
* 解ç
(I)
:<math>\sqrt{-2}\ \sqrt{-6} = \sqrt{2}\ i \times \sqrt{6} \ i = \sqrt{12}\ i^2 = -2 \sqrt{3}</math>
ãã®ããã«ããŸãããã€ãã¹ã®æ°ã®å¹³æ¹æ ¹ãåºãŠãããããŸãèæ°åäœ i ãçšããåŒã«æžãæããã
ãã®ããšãããç®ãããŠããã
* åé¡
(II) ãã<math>\frac{\sqrt{2}}{\sqrt{-3}}</math>ããèšç®ããã
(III) ãã2次æ¹çšåŒã<math>x^2=-7</math>ãã解ãã
** 解ç
(II)
:<math>\frac{\sqrt{2}}{\sqrt{-3}} = \frac{\sqrt{2}}{\sqrt{3}\ i} = \frac{\sqrt{2}\ \sqrt{3}\ i}{\sqrt{3}\ i\ \sqrt{3}\ i} = \frac{\sqrt{6}\ i}{3\ i^2} = - \frac{\sqrt{6}}{3} \ i</math>
(III)
:<math>x^2=-7</math>
:<math>x= \pm \sqrt{-7}</math>
:<math>x= \pm \sqrt{7}\ i</math>
=== 2次æ¹çšåŒã®å€å¥åŒ ===
==== 2次æ¹çšåŒã®è§£ãšè€çŽ æ° ====
è€çŽ æ°ã®å¿çšãšããŠãããã§ã¯2次æ¹çšåŒã®æ§è³ªã«ã€ããŠè¿°ã¹ããä»»æã®2次æ¹çšåŒã¯ã解ã®å
¬åŒã«ãã£ãŠè§£ãããããšã[[é«çåŠæ ¡æ°åŠI æ¹çšåŒãšäžçåŒ#äºæ¬¡æ¹çšåŒ|é«çåŠæ ¡æ°åŠI]]ã§è¿°ã¹ãããããã解ã®å
¬åŒã«å«ãŸããæ ¹å·ã®äžèº«ãè² ã®æ°ã®å Žåã«ã¯å®æ°è§£ãååšããªãããšã«æ³šæããå¿
èŠãããã2次æ¹çšåŒ
:<math>
ax^2+bx+c = 0
</math>
ã®è§£ã®å
¬åŒã¯ã
:<math>
x = \frac{-b \pm \sqrt{b^2 - 4ac} }{a}
</math>
ã§ããã
å€å¥åŒ<math>D</math>ã¯
:<math>
D = b^2-4ac
</math>
ã«ãã£ãŠå®çŸ©ããããå€å¥åŒã¯ã解ã®å
¬åŒã®æ ¹å·(ã«ãŒãèšå·ã®ããš)ã®äžèº«ã«çãããå€å¥åŒã®æ£è² ã«ãã£ãŠ2次æ¹çšåŒãå®æ°è§£ãæã€ãã©ããã決ãŸãã
<math>D</math>ãè² ã®ãšãã«ã¯ãã®2次æ¹çšåŒã¯å®æ°ã®ç¯å²ã«ã¯è§£ãæããªãã
å€å¥åŒ<math>D</math>ãè² ã®æ°ã§ãã£ããšããxã®è§£ã¯ç°ãªã2ã€ã®èæ°ã«ãªãããã®2ã€ã®è§£ã¯ å
±åœ¹ ã®é¢ä¿ã«ãªã£ãŠããã
* åé¡äŸ
** åé¡
è€çŽ æ°ãçšããŠã2次æ¹çšåŒ<br>
(1)
:<math>x ^2 + 5x + 9 =0</math>
(2)
:<math>2x ^2 + 5x + 8 =0</math>
(3)
:<math>2x ^2 - 2x + 8 =0</math>
ã解ãã
** 解ç
解ã®å
¬åŒãçšããŠè§£ãã°ããã(1)ã ããèšç®ãããšã
:<math>
x = \frac {- 5 \pm \sqrt{5^2 - 4 \times 1 \times 9}}{2}
</math>
:<math>
= \frac {-5 \pm \sqrt {11} i}{2}
</math>
ãšãªãã
ä»ãåãããã«æ±ãããšãåºæ¥ãã
以éã®è§£çã¯ã<br>
(2)
:<math>
x = \frac {-5 \pm \sqrt {39} i}{4}
</math>
(3)
:<math>
x = \frac {1 \pm \sqrt {15} i}{2}
</math>
ãšãªãã
<!--
(
*å·çè
ã«å¯Ÿãã泚æ
èšç®ã«ã¯[[w:maxima]]ãçšããã
tex(solve(
x ^2 + 5*x + 9 =0,x
));
tex(solve(
2*x ^2 + 5*x + 8 =0,x
));
tex(
solve(
2*x ^2 - 2*x + 8 =0,x
));
)
-->
==== 2次æ¹çšåŒã®å€å¥åŒ ====
æ¹çšåŒã®è§£ã§ãå®æ°ã§ãããã®ã '''å®æ°è§£''' ãšããã
æ¹çšåŒã®è§£ã§ãèæ°ã§ãããã®ã '''èæ°è§£''' ãšããã
2次æ¹çšåŒ <math>ax^2 + bx + c = 0</math> ã®è§£ã¯ <math>x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} </math> ã§ããã
2次æ¹çšåŒã®è§£ã®çš®é¡ã¯ã解ã®å
¬åŒã®äžã®æ ¹å·ã®äžã®åŒ <math>b^2-4ac</math> ã®ç¬Šå·ãèŠãã°å€å¥ããããšãã§ããã
ãã®åŒ <math>b^2-4ac</math> ãã2次æ¹çšåŒ <math>ax^2 + bx + c = 0</math> ã®'''å€å¥åŒ'''ïŒã¯ãã¹ã€ããïŒãšãããèšå· '''<math>D</math>''' ã§è¡šãã
{| style="border:2px solid skyblue;width:80%" cellspacing=0
|style="background:skyblue"|'''å€å¥åŒãšè§£ã®å€å¥'''
|-
|style="padding:5px"|
2次æ¹çšåŒ <math>ax^2 + bx + c = 0</math> ã®å€å¥åŒ <math>D=b^2-4ac</math> ã«ã€ããŠ
::<math>D>0 \quad \Leftrightarrow \quad </math> ç°ãªã2ã€ã®å®æ°è§£ããã€
::<math>D=0 \quad \Leftrightarrow \quad </math> é解ããã€
::<math>D<0 \quad \Leftrightarrow \quad </math> ç°ãªã2ã€ã®èæ°è§£ããã€
|}
ãŸããé解ãå®æ°è§£ã§ããã®ã§ã
::<math>D \geqq 0 \quad \Leftrightarrow \quad </math> å®æ°è§£ããã€
ãšãããã
* åé¡äŸ
** åé¡
次ã®2次æ¹çšåŒã®è§£ãå€å¥ããã
(I)
:<math>
x^2+3\,x-1=0
</math>
(II)
:<math>
2\,x^2-3\,x+2=0
</math>
(III)
:<math>
4\,x^2-20\,x+25=0
</math>
** 解ç
(I)
:<math>
D=3^2-4 \times 1 \times (-1) =13>0
</math>
ã ãããç°ãªã2ã€ã®å®æ°è§£ããã€ã
(II)
:<math>
D=(-3)^2-4 \times 2 \times 2 =-7<0
</math>
ã ãããç°ãªã2ã€ã®èæ°è§£ããã€ã
(III)
:<math>
D=(-20)^2-4 \times 4 \times 25 =0
</math>
ã ãããé解ããã€ã
ãŸãã2次æ¹çšåŒ <math>ax^2 + 2b'x + c = 0</math> ã®ãšãã<math>D=4(b'^2-ac)</math>ãšãªãã®ã§ã
2次æ¹çšåŒ <math>ax^2 + 2b'x + c = 0</math> ã®å€å¥åŒã«ã¯
:<math>
\frac{D}{4} = b'^2-ac
</math>
ããã¡ããŠãããã
ãããçšããŠãåã®åé¡
:(III) ã<math>4\,x^2-20\,x+25=0</math>
ã®è§£ãå€å¥ãããã
<math>a=4 \, , \, b'=-10 \, , \, c=25</math>ãã§ãããã
:<math>
\frac{D}{4} = (-10)^2- 4 \times 25 =0
</math>
ã ãããé解ããã€ã
==== 2次æ¹çšåŒã®è§£ãšä¿æ°ã®é¢ä¿ ====
2次æ¹çšåŒ <math>ax^2 + bx + c = 0</math> ã®2ã€ã®è§£ã <math>\alpha</math> ïŒ<math>\beta</math> ãšããã ãã®æ¹çšåŒã¯ã
<math>a(x-\alpha)(x-\beta) = 0</math>
ãšå€åœ¢ã§ããã
ãããå±éãããšã
<math>ax^2 -a(\alpha + \beta )x+a\alpha \beta = 0</math>
ä¿æ°ãæ¯èŒããŠã
<math>c = a \alpha \beta, b = -a(\alpha + \beta)</math>
ãåŸãã
ãããå€åœ¢ããã°ã<math>\alpha + \beta = -\frac{b}{a}, \alpha \beta = \frac{c}{a}</math>ãšãªãã<br>
{| style="border:2px solid skyblue;width:80%" cellspacing="0"
| style="background:skyblue" |'''解ãšä¿æ°ã®é¢ä¿'''
|-
| style="padding:5px" |
2次æ¹çšåŒ <math>ax^2 + bx + c = 0</math> ã®2ã€ã®è§£ã <math>\alpha</math> ïŒ<math>\beta</math> ãšããã°<br>
<center><math>\alpha + \beta = - \frac{b}{a}</math> ïŒ<math>\alpha \beta = \frac{c}{a}</math><br></center>
|}
* åé¡äŸ
** åé¡
2次æ¹çšåŒ <math>2x^2 + 4x + 3 = 0</math> ã®2ã€ã®è§£ã <math>\alpha</math> ïŒ<math>\beta</math> ãšãããšãã<math>\alpha ^2 + \beta ^2</math> ã®å€ãæ±ããã
** 解ç
解ãšä¿æ°ã®é¢ä¿ããã
<math>\alpha + \beta = - \frac{4}{2} = - 2 </math>ïŒ<math>\alpha \beta = \frac{3}{2}</math><br>
<math>\alpha ^2 + \beta ^2 = (\alpha + \beta )^2 - 2 \alpha \beta = (-2)^2 - 2 \times \frac{3}{2} = 1</math>
==== 2æ°ã解ãšãã2次æ¹çšåŒ ====
2ã€ã®æ° <math>\alpha</math> ïŒ<math>\beta</math> ã解ãšãã2次æ¹çšåŒã¯
:<math>
(x - \alpha) (x - \beta) = 0
</math>
ãšè¡šãããã巊蟺ãå±éããŠæŽçãããšæ¬¡ã®ããã«ãªãã
{| style="border:2px solid skyblue;width:80%" cellspacing=0
|style="background:skyblue"|'''äžãããã2ã€ã®æ°ã解ãšãã2次æ¹çšåŒ'''
|-
|style="padding:5px"|
2æ° <math>\alpha</math> ïŒ<math>\beta</math> ã解ãšãã2次æ¹çšåŒã¯<br>
<center><math>x^2 - (\alpha + \beta ) x + \alpha \beta = 0</math><br></center>
|}
* åé¡äŸ
** åé¡
次ã®2æ°ã解ãšãã2次æ¹çšåŒãäœãã
(I)
:<math>
3 + \sqrt{5} \ , 3 - \sqrt{5}
</math>
(II)
:<math>
2 + 3 i \ , 2 - 3 i
</math>
** 解ç
(I)<br>
åã<math>(3 + \sqrt{5}) + (3 - \sqrt{5}) = 6</math><br>
ç©ã<math>(3 + \sqrt{5}) (3 - \sqrt{5}) = 4</math>ãã§ãããã<br>
:<math>
x^2 - 6 x + 4 =0
</math>
(II)<br>
åã<math>(2 + 3 i) + (2 - 3 i) = 4</math><br>
ç©ã<math>(2 + 3 i) (2 - 3 i) = 13</math>ãã§ãããã<br>
:<math>
x^2 - 4 x + 13 =0
</math>
==== 2次åŒã®å æ°å解 ====
2次æ¹çšåŒ <math>ax^2 + bx + c = 0</math> ã®2ã€ã®è§£ <math>\alpha</math> ïŒ<math>\beta</math> ãããããšã2次åŒ
:<math>ax^2 + bx + c
</math>
ãå æ°å解ããããšãã§ããã<br>
解ãšä¿æ°ã®é¢ä¿ <math>\alpha + \beta = - \frac{b}{a}</math>ïŒ<math>\alpha \beta = \frac{c}{a}</math> ããã
:<math>
ax^2 + bx + c = a \left(x^2 + \frac{b}{a}x + \frac{c}{a} \right) = a \left\{x^2 - (\alpha + \beta )x + \alpha \beta \right\} = a (x - \alpha)(x - \beta)
</math>
{| style="border:2px solid skyblue;width:80%" cellspacing="0"
| style="background:skyblue" |'''解ãšå æ°å解'''
|-
| style="padding:5px" |
2次æ¹çšåŒ <math>ax^2 + bx + c = 0</math> ã®2ã€ã®è§£ã <math>\alpha</math> ïŒ<math>\beta</math> ãšãããš<br>
<center><math>ax^2 + bx + c = a (x - \alpha)(x - \beta)</math><br></center>
|}
2次æ¹çšåŒã¯ãè€çŽ æ°ã®ç¯å²ã§èãããšã€ãã«è§£ããã€ãããè€çŽ æ°ãŸã§äœ¿ã£ãŠãããšãããšã2次åŒã¯å¿
ã1次åŒã®ç©ã«å æ°å解ããããšãã§ããã
* åé¡äŸ
** åé¡
è€çŽ æ°ã®ç¯å²ã§èããŠã次ã®2次åŒãå æ°å解ããã
(I)
:<math>
x^2 + 4 x - 1
</math>
(II)
:<math>
2 x^2 - 3 x + 2
</math>
**解ç
(I)<br>
2次æ¹çšåŒã<math>x^2 + 4 x - 1 = 0</math>ãã®è§£ã¯<br>
:<math>
x = \frac{-4 \pm \sqrt{4^2-4 \times 1 \times (-1)}}{2 \times 1} = \frac{-4 \pm \sqrt{20}}{2} = \frac{-4 \pm 2 \sqrt{5}}{2} = -2 \pm \sqrt{5}
</math>
ãã£ãŠ
:<math>
x^2 + 4 x - 1 = \left\{ x - (-2 + \sqrt{5}) \right\} \left\{ x - (-2 - \sqrt{5}) \right\} = (x + 2 - \sqrt{5}) (x + 2 + \sqrt{5})
</math>
(II)<br>
2次æ¹çšåŒã<math>2 x^2 - 3 x + 2 = 0</math>ãã®è§£ã¯<br>
:<math>
x = \frac{-(-3) \pm \sqrt{(-3)^2-4 \times 2 \times 2}}{2 \times 2} = \frac{3 \pm \sqrt{-7}}{4} = \frac{3 \pm \sqrt{7} i}{4}
</math>
ãã£ãŠ
:<math>
2 x^2 - 3 x + 2 = 2 \left(x- \frac{3 + \sqrt{7}\; i}{4} \right) \left(x- \frac{3 - \sqrt{7}\; i}{4} \right)
</math>
=== é«æ¬¡æ¹çšåŒ ===
3次以äžã®æŽåŒã«ããæ¹çšåŒãèããã
äžè¬ã«æ¹çšåŒã <math>P(x)=0</math>ãšãšãã
ãã ãã<math>P(x)</math>ã¯ãä»»æã®æ¬¡æ°ã®æŽåŒãšããã
==== å°äœã®å®ç ====
<math>P(x)</math>ã1次åŒ<math>x-a</math>ã§å²ã£ããšãã®åã<math>Q(x)</math>ãäœãã<math>R</math>ãšãããšã
:<math>
P(x) = (x-a)Q(x)+R
</math>
ãã®äž¡èŸºã®<math>x</math>ã«<math>a</math>ã代å
¥ãããšã
:<math>
P(a) = (a-a)Q(a)+R = 0 \times Q(a) + R =R
</math>
ã€ãŸãã<math>P(x)</math>ã<math>x-a</math>ã§å²ã£ããšãã®äœãã¯<math>P(a)</math>ã§ããã
{| style="border:2px solid pink;width:80%" cellspacing="0"
| style="background:pink" |'''å°äœã®å®ç'''
|-
| style="padding:5px" |
æŽåŒ<math>P(x)</math>ã<math>x-a</math>ã§å²ã£ããšãã®äœãã¯ã<math>P(a)</math>ã«çããã
|}
* åé¡äŸ
** åé¡
æŽåŒ <math>P(x) = x^3 -2x + 3</math> ã次ã®åŒã§å²ã£ãäœããæ±ããã<br>
(I)
:<math>
x-2
</math>
(II)
:<math>
x+1
</math>
(III)
:<math>
2x-1
</math>
** 解ç
(I)ã<math>P(2) = 2^3 - 2 \times 2 + 3 = 7</math><br>
(II)ã<math>P(-1) = (-1)^3 - 2 \times (-1) + 3 = 4</math><br>
(III)ã<math>P\left( \frac{1}{2} \right) = \left( \frac{1}{2} \right)^3 - 2 \times \left( \frac{1}{2} \right) + 3 = \frac{17}{8}</math>
===== å æ°å®ç =====
ããå®æ°<math>a</math>ã«å¯ŸããŠã
:<math>
P(a) = 0
</math>
ãæãç«ã£ããšããã
ãã®ãšããæŽåŒ<math>P(x)</math> ã¯ã <math>(x-a)</math> ãå æ°ã«æã€ããšãåãã
ãã®ããšãå æ°å®çïŒãããããŠããïŒãšåŒã¶ã
* å°åº
æŽåŒ<math>P(x)</math>ã«å¯ŸããŠãå<math>Q(x)</math>ãå²ãåŒ<math>(x-a)</math>ãšãã
æŽåŒã®é€æ³ãçšããããã®ãšããå<math>Q(x)</math>ã
(<math>Q(x)</math>ã¯ã<math>P(x)</math>ããã1ã ã次æ°ãäœãæŽåŒã§ããã)
äœã<math>c</math>(<math>c</math>ã¯ãå®æ°ã)ãšãããšã
æŽåŒ<math>P(x)</math> ã¯ã
:<math>
P(x) = (x-a)Q(x) + c
</math>
ãšæžããã
ããã§ã <math>c=0</math> ã§ãªããšã <math>P(a)=0</math> ã¯æºããããªããã
ãã®ãšãã<math>P(x)</math>ã¯ã<math>(x-a)</math>ã«ãã£ãŠå²ãåããã
ãã£ãŠãå æ°å®çã¯æç«ããã
{| style="border:2px solid pink;width:80%" cellspacing=0
|style="background:pink"|'''å æ°å®ç'''
|-
|style="padding:5px"|
æŽåŒ<math>P(x)</math>ã«ã€ããŠ<br>
<center><math>P(a)=0 \Leftrightarrow </math> <math>P(x)</math>ã¯<math>x-a</math>ã§å²ããããã</center>
|}
å æ°å®çãçšããããšã§ããã次æ°ã®é«ãæŽåŒãå æ°å解ããããšã
åºæ¥ãããã«ãªããäŸãã°ã3次ã®æŽåŒ
:<math>
x^3 - 1
</math>
ã«ã€ããŠã<math>x=1</math>ã代å
¥ãããšã
:<math>
x^3 - 1
</math>
ã¯0ãšãªãããã£ãŠãå æ°å®çãããã®åŒã¯
:<math>
(x-1)
</math>
ãå æ°ãšããŠæã€ã
ããã§ãå®éæŽåŒã®é€æ³ã䜿ã£ãŠèšç®ãããšã
:<math>
x^3 - 1 = (x-1)(x^2+x+1)
</math>
ãåŸãããã
* åé¡äŸ
** åé¡
å æ°å®çãçšããŠ<br>
(I)
:<math>
x^3-6\,x^2+11\,x-6
</math>
(II)
:<math>
x^3+x^2-14\,x-24
</math>
<!--
(III)
:<math>
x^3+5\,x^2-34\,x-80
</math>
-->
ãå æ°å解ããã
** 解ç
(I)
å æ°å解ã®çµæã(x+æŽæ°)ã®ç©ã®åœ¢ãªããæŽæ°ã¯6ã®å æ°ã§ãªããã°ãªããªãããã®ããã<math>\pm 1, \pm 2,\pm 3,\pm 6</math>ãåè£ãšãªãããããã«ã€ããŠã¯å®éã«ä»£å
¥ããŠç¢ºããããããªããx=1ã代å
¥ãããšã
:<math>
1-6+11-6=0
</math>
ãšãªãã®ã§ã(x-1)ãå æ°ãšãªããå®éã«æŽåŒã®é€æ³ãè¡ãªããšãåãšããŠ<math>x^2-5x+6</math>ãåŸãããããããã¯<math>(x-2)(x-3)</math>ã«å æ°å解ã§ããããã£ãŠçãã¯ã
:<math>
\left(x-3\right)\,\left(x-2\right)\,\left(x-1\right)
</math>
ãšãªãã<br>
(II)
ããã§ãå°éã«24ã®å æ°ãåœãŠã¯ããŠãããããªãã24ã®å æ°ã¯æ°ãå€ãã®ã§ããªãã®èšç®ãå¿
èŠãšãªããããã§ã¯ã-2ã代å
¥ãããšã
:<math>
-8 +4 -14 \cdot (-2) -24 = 0
</math>
ãšãªãã(x+2)ãå æ°ã ãšããããé€æ³ãè¡ãªããšã<math>x^2 -x -12</math>ãåŸããããã(x-4)(x+3)ã«å æ°å解ã§ãããçãã¯ã
:<math>
\left(x-4\right)\,\left(x+2\right)\,\left(x+3\right)
</math>
ãšãªãã
===== é«æ¬¡æ¹çšåŒ =====
å æ°å解ãå æ°å®çãå©çšããŠé«æ¬¡æ¹çšåŒã解ããŠã¿ããã
* åé¡äŸ
** åé¡
é«æ¬¡æ¹çšåŒ<br>
(I)
:<math>
x^3-8=0
</math>
(II)
:<math>
x^4-2x^2-8=0
</math>
(III)
:<math>
x^3-5x^2+7x-2=0
</math>
ã解ãã
**解ç
(I)
巊蟺ã<math>
a^3-b^3=(a-b)(a^2+ab+b^2)
</math>ãçšããŠå æ°å解ãããš
:<math>
(x-2)(x^2+2x+4)=0
</math>
ãããã£ãŠ<math>\ x-2=0</math>ããŸãã¯<math>\ x^2+2x+4=0</math><br>
ãã£ãŠ
:<math>
x=2\ , \ -1 \pm \sqrt{3} i
</math>
(II) ã<math>\ x^2=X\ </math>ãšãããšã
:<math>
X^2-2X-8=0
</math>
巊蟺ãå æ°å解ãããš
:<math>
(X-4)(X+2)=0
</math>
ãã£ãŠã<math>X=4\ ,\ X=-2</math><br>
ããã«ã<math>x^2=4\ ,\ x^2=-2</math><br>
ãããã£ãŠ
:<math>
x= \pm 2\ ,\ \pm \sqrt{2} i
</math>
(III) ã<math>\ P(x)=x^3-5x^2+7x-2\ </math>ãšããã
:<math>
P(2)=2^3-5 \times 2^2+7 \times 2-2=0
</math>
ãããã£ãŠã<math>\ x-2\ </math>ã¯<math>\ P(x)\ </math>ã®å æ°ã§ããã<br>
:<math>
P(x)=(x-2)(x^2-3x+1)
</math>
ãã£ãŠã<math>(x-2)(x^2-3x+1)=0</math><br>
<math>\ x-2=0</math>ããŸãã¯<math>\ x^2-3x+1=0</math><br>
ãããã£ãŠ
:<math>
x= 2\ ,\ \frac{3 \pm \sqrt{5}}{2}
</math>
=====ïŒçºå±ïŒ3次æ¹çšåŒã®è§£ãšä¿æ°ã®é¢ä¿=====
3次æ¹çšåŒ <math>ax^3+ bx^2+ cx+d=0</math> ã®3ã€ã®è§£ã ã<math>\alpha\ ,\ \beta\ ,\ \gamma</math> ãšãããš
:<math>ax^3+ bx^2+ cx+d=a(x- \alpha)(x- \beta)(x- \gamma)</math>
ãæãç«ã€ã<br>
å³èŸºãå±éãããš
:<math>a(x- \alpha)(x- \beta)(x- \gamma)</math>
:<math>=a(x- \alpha)\left\{x^2-(\beta + \gamma)x+ \beta \gamma \right\}</math>
:<math>=a \left\{x^3-(\alpha + \beta + \gamma)x^2+ (\alpha \beta + \beta \gamma + \gamma \alpha)x - \alpha \beta \gamma \right\}</math>
ãã£ãŠ
:<math>ax^3+ bx^2+ cx+d=a \left\{x^3-(\alpha + \beta + \gamma)x^2+ (\alpha \beta + \beta \gamma + \gamma \alpha)x - \alpha \beta \gamma \right\}</math>
ããã«
:<math>b=-a(\alpha + \beta + \gamma)\ ,\ c= a(\alpha \beta + \beta \gamma + \gamma \alpha)\ ,\ d= -a \alpha \beta \gamma</math>
ãããã£ãŠã次ã®ããšãæãç«ã€ã
{| style="border:2px solid pink;width:80%" cellspacing=0
|style="background:pink"|'''3次æ¹çšåŒã®è§£ãšä¿æ°ã®é¢ä¿'''
|-
|style="padding:5px"|
3次æ¹çšåŒ <math>ax^3+ bx^2+ cx+d=0</math> ã®3ã€ã®è§£ã ã<math>\alpha\ ,\ \beta\ ,\ \gamma</math> ãšãããš
<center><math>\alpha + \beta + \gamma =- \frac{b}{a}\ ,\ \alpha \beta + \beta \gamma + \gamma \alpha= \frac{c}{a}\ ,\ \alpha \beta \gamma =- \frac{d}{a}</math></center>
|}
== ã³ã©ã ==
=== è€çŽ æ°ã¯ãååšããããïŒ ===
ãã°ãã°èæ°ã¯ãçŸå®ã«ã¯ååšããªãæ°ãã§ãããšèšãããããšããããæŽå²çã«ãèæ°ãæ±ã£ãæ°åŠãèããã¹ãã§ã¯ãªããšèããããæ代ã¯é·ãã£ãããã®æ代ã®å
é²çãªæ°åŠè
ã®äžã«ã¯ãèæ°ãæå¹ã«æŽ»çšããŠç 究ãé²ããäžæ¹ã§ãææãçºè¡šããéã«ã¯èæ°ãè¡šã«åºããã«èšè¿°ããåªåãããããšã§ãç¡çšãªæµæãåããªãããã«å·¥å€«ããè
ããããšèšãããã»ã©ã§ããã
ã ããããèããŠã¿ãã°ãæ°ããçŸå®ã«ååšããããšã¯ã©ãããæå³ãªã®ã ããããçŸå®ã«éçã䜿ã£ãŠçŽã«åãæããªãã°ãååšã®é·ãããæ£ç¢ºã«ååšçãã®ãã®ã«ãããããšã¯äžå¯èœã§ããããã«æããããããã®å²ã«ååšçãšããå®æ°ã¯ãååšããããšæããããã®ã¯ãªãã ããããæ°çŽç·ãå®æ°ã®ãå®åšããä¿¡ãããããªãã°ãè€çŽ æ°ã¯è€çŽ æ°å¹³é¢ïŒæ°åŠCã§ç¿ãïŒã®äžã«ååšããã®ã ãããåãã§ã¯ãªãã ãããã
ãã®ããã«èãããšãããããæ°ãšã¯ãã¹ãŠããæå³ã§æ³åäžã®ååšã§ãããããã«å¯ŸããŠãååšããããååšããªãããšããåããç«ãŠãããšããã³ã»ã³ã¹ã§ããããã«æãããããååšããªããããã«æãããã¡ãªèæ°ã§ããããããšãã°ç©çåŠã®äžåéã§ããéåååŠã®ã·ã¥ã¬ãã£ã³ã¬ãŒæ¹çšåŒã«è¡šãããªã©ãå¿çšäžã®ããŸããŸãªå Žé¢ã«ãããŠããèæ°ã䜿ã£ãŠèšè¿°ããããšãèªç¶ãªå¯Ÿè±¡ã¯å€ãã®ã ã
=== è€çŽ æ°ã«ã¯ã倧å°é¢ä¿ããªãã ===
è€çŽ æ°ã©ããã«ã€ããŠããã®å€§å°é¢ä¿ã¯å®çŸ©ããªãããã®çç±ã¯ãã©ã®ããã«å€§å°é¢ä¿ãå®çŸ©ããŠãã䟿å©ãªæ§è³ªãæºããããšãã§ããªãããã§ãããå
·äœçã«èšãã°ãæ¢ã«è¿°ã¹ãå®æ°ã®å€§å°é¢ä¿ã«ã€ããŠã®ãäžçåŒã®åºæ¬æ§è³ª(1)(2)(3)(4)ãã«ãããåŒãæãç«ãããããšãã§ããªãã®ã ã
ããšãã°ã<math>a+bi<a'+b'i</math>ã§ããããšãã<math>a^2+b^2<a'^2+b'^2</math>ã§ããããšãšããŠå®çŸ©ããŠã¿ããããã®ããã«å®çŸ©ãããšãããšãã°1+2i<2-3iã§ããããŸã2+3i<3+4iã§ããããšãããã(1+2i)+(2+3i)=3+5i,(2-3i)+(3+4i)=5+iã§ããã3+5i>5+iãšãªã£ãŠããŸããããã¯åºæ¬æ§è³ª(2)ãæãç«ããªãããšãæå³ããã
ãã¡ããããã¯é©åœã«èããå®çŸ©ãããŸããŸäžé©åã ã£ããšããã ãã®ããšã ããå®ã¯ãä»ã«ã©ã®ããã«å®çŸ©ããŠããã®ãããªå°é£ããã¯éããããªãããšãç¥ãããŠãããããããã«ãè€çŽ æ°ã«ã¯å€§å°é¢ä¿ãå®çŸ©ããªãã®ã§ããã
=== è€çŽ æ°ã®å¹³æ¹æ ¹ (â»çºå±) ===
ä»åºŠã¯ãè€çŽ æ°ã®å¹³æ¹æ ¹ã«ã€ããŠèããŠã¿ããã
æ£ã®æ°<math>a</math>ãèãããšãã
:<math>a</math>ã®å¹³æ¹æ ¹ã¯<math>\pm \sqrt{a}</math>
:<math>-a</math>ã®å¹³æ¹æ ¹ã¯<math>\pm \sqrt{a} i</math>
ã§ã¯ã
:<math>\pm a i</math>ã®å¹³æ¹æ ¹ã¯ã©ã®ããã«è¡šããã ãããã
èæ°åäœ<math>i</math>ã®å¹³æ¹æ ¹ãèãããšãããã¯zã«ã€ããŠã®æ¹çšåŒ <math>z^2 = i</math> ã®è§£ z ã®å€ã§ãããããããã解ãã°ãããã©ã®ãããªè€çŽ æ°zãªããã®åŒãæºããããšãã§ããã ãããã
zãè€çŽ æ°ãšãããšã<math>z = x + yi</math>(x,yã¯å®æ°)ãšè¡šãããã
<math>(x + yi)^2 = i \Leftrightarrow x^2 + 2xyi - y^2 = i \Leftrightarrow (x^2-y^2)+(2xy-1)i = 0</math>
<math>x^2-y^2,2xy-1</math>ã¯å®æ°ã§ãããããå®éšãšèéšãå
±ã«ïŒã«ãªããã°ãªããªãããã
<math>\begin{cases}
x^2-y^2=0 (\Leftrightarrow x= \pm y ) \\
2xy-1=0
\end{cases}</math>
<math>x=y</math>ã®ãšãã<math>2x^2=1 \Leftrightarrow x=\pm\frac{1}{\sqrt{2}},y=\pm\frac{1}{\sqrt{2}}</math> (è€å·åé ãx,yã¯å
±ã«å®æ°ã§ãããããæ¡ä»¶ãæºããã)
<math>x=-y</math>ã®ãšãã<math>-2y^2=1 \Leftrightarrow y^2=-\frac{1}{2}</math> ããã§ããããæºããå®æ°yã¯ååšããªãããäžé©ã
ãã£ãŠã<math>z=\pm\left(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}i\right)</math><sub>â </sub>
*åé¡äŸ
** åé¡
:<math>i \,\!</math>ãèæ°åäœãšãããšãã次ã®åãã«çããã
:(I) <math>-i,30i \,\!</math>ã®å¹³æ¹æ ¹ãæ±ããã
:(II) 2次æ¹çšåŒ <math>z^2 - 30i - 16 = 0 \,\!</math> ã解ãã
:(III) 3次æ¹çšåŒ <math>z^3 = i \,\!</math> ã解ãã
** 解ç
:(I)
::<math>\pm\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}i\right) , \pm\left(\sqrt{15}+\sqrt{15}i\right)</math>
:(II)
::<math>z=5+3i , -5-3i \,\!</math>
:(III)
::<math>z=-i,\frac{i\pm\sqrt{3}}{2}</math>
:ä»åæããåé¡ã¯ãå
šãŠ<math>z=x+yi</math>(x,yã¯å®æ°)ãšçœ®ãããšã§æ±ããããã(III)ã¯ã<math>(x+yi)^3-i=x(x^2-3y^2)+(3x^2y-y^3-1)i=0</math>ããã
å®éšããŒããèæ
®ããŠ<math>x=0</math>ã<math>x=\pm\sqrt{3}y</math>ã ããèéšããŒããªã®ã§ãxã®å€ãåè
ã®ãšã<math>y=-1</math>ãåŸè
ã®ãšã<math>y=1/2</math>ãšãªãããšãããã«ãããã
=== é«æ¬¡æ¹çšåŒã®ã解ã®å
Œ΋ ===
2次æ¹çšåŒã«ã¯è§£ã®å
¬åŒããããæ¥æ¬ã®äžåŠãé«æ ¡ã§ãç¿ãã2次æ¹çšåŒã®è§£ã®å
¬åŒãçšããã°ãã©ããªä¿æ°ã®2次æ¹çšåŒã§ãã£ãŠã解ãæ±ããããã3次æ¹çšåŒãš4次æ¹çšåŒã«ãã解ã®å
¬åŒã¯ååšããä¿æ°ãã©ããªä¿æ°ã§ãã£ãŠã解ãæ±ããããããããã®è§£ã®å
¬åŒã¯ã[[代æ°æ¹çšåŒè«]]ã§è¿°ã¹ãŠããããã«ãä¿æ°ã«æéåã®ååæŒç®ãšæ ¹å·ããšãæäœã®çµã¿åããã§è¡šãããšãã§ããã
5次æ¹çšåŒã§ã¯ã4次以äžã®æ¹çšåŒãšã¯ç¶æ³ãç°ãªããäžè¬ã®5次æ¹çšåŒã®è§£ã¯ã2次æ¹çšåŒã4次æ¹çšåŒã®ããã«ãä¿æ°ã«æéåã®ååæŒç®ãšæ ¹å·ããšãæäœã®çµã¿åããã§è¡šãããšãã§ããªãã®ã§ããããã ãããã§ããªããããšã®èšŒæã¯å®¹æã§ã¯ãªãããã®ããšã蚌æããã«ã¯ã[[ã¬ãã¢çè«]]ãç解ããå¿
èŠãããïŒæ¥æ¬ã®å€§åŠã®æšæºçãªã«ãªãã¥ã©ã ã§ã¯ãçåŠéšæ°åŠç§ã®åŠçã®ã¿ã倧åŠ3幎çã§åŠã¶ã®ãäžè¬çãªçšåºŠã®çè«ã§ããïŒã
ãªããããã§èšããè¡šãããšãã§ããªãããšã¯äžè¬ã®æ¹çšåŒã«ã€ããŠã®ããšã§ãããç¹å¥ãª5次æ¹çšåŒã®å Žåã¯ç°¡åã«è§£ãæ±ãããããããšãã°ã<math> x^5 -32 = 0 </math> ã¯è§£ã®ã²ãšã€ãšã㊠<math> x=2 </math> ããã€ããšã¯ãããããããã®æ¹çšåŒã¯ä»ã®è§£ã«ã€ããŠãäžè§é¢æ°ãçšããŠç°¡åã«è¡šããããšã[[é«çåŠæ ¡æ°åŠC/è€çŽ æ°å¹³é¢]]ã«ãããŠåŠã¶ã
ãä¿æ°ã«æéåã®ååæŒç®ãšæ ¹å·ããšãæäœã®çµã¿åãããã«æããªããã°ãäžè¬ã®5次æ¹çšåŒã®è§£ãæ±ããæ¹æ³ãååšããããããé«åºŠãªæ°åŠãçšããå¿
èŠãããã[[w:äºæ¬¡æ¹çšåŒ]]ã«èšè¿°ãããã®ã§èå³ã®ããèªè
ã¯åç
§ãããšããã
=== è€çŽ æ°ãšé¢æ° ===
é«çåŠæ ¡ã§è€çŽ æ°ãåºãŠããåéã¯ãã®åéãšæ°åŠCã[[é«çåŠæ ¡æ°åŠC/å¹³é¢äžã®æ²ç·|å¹³é¢äžã®æ²ç·]]ãšè€çŽ æ°å¹³é¢ãã®ã¿ã§ãããè€çŽ æ°ã®åºæ¬èšç®ãæ¹çšåŒãè€çŽ æ°ç¯å²ã§è§£ãããšãè€çŽ æ°ã®å¹ŸäœåŠçæå³ã«ã€ããŠæ±ã£ãŠããããããã倧åŠæ°åŠã«ãããŠã¯ãé¢æ°ã®å®çŸ©åã»å€åãè€çŽ æ°ç¯å²ã«åºããŠèããã[[è€çŽ 解æåŠ|è€çŽ é¢æ°è«]]ããšãããã®ãæ±ãã
å®æ°ç¯å²ã§ã®é¢æ°ã¯x, yãšãã«äžæ¬¡å
ã®å®æ°è»žãæã€ãããå
¥åå€ãšåºåå€ã®æãã°ã©ããèããã«ã¯äºæ¬¡å
ã®åº§æšå¹³é¢ã§ååã§ãã£ããããããè€çŽ æ°ç¯å²ã§ã®é¢æ°ã¯x, yãšãã«äºæ¬¡å
ã®è€çŽ æ°å¹³é¢ãæã€ãããå
¥åå€ãšåºåå€ã®æãã°ã©ããèããã«ã¯å次å
ã®åº§æšç©ºéãå¿
èŠã§ãããäžæ¬¡å
空éã«äœãæã
ã«ã¯æç»ããããšãã§ããªãããã®ãããè€çŽ é¢æ°è«ã§ã¯é¢æ°ã®ã°ã©ããèããããšã¯åºæ¬çã«ãªããïŒãã ããåºåãããè€çŽ æ°ã®çµ¶å¯Ÿå€ãèããããšã«ãã£ãŠäžæ¬¡å
ã°ã©ãã«èœãšã蟌ãããšã¯å¯èœïŒ
ã§ã¯äœãèããã®ããšãããšãè€çŽ é¢æ°ã®åŸ®åç©åã§ãããè€çŽ é¢æ°ã®åŸ®åã«é¢é£ããŠãæ£åé¢æ°ããšããçšèªãåºãŠããããè€çŽ é¢æ°è«ã¯ãã®æ£åé¢æ°ãšãããã®ã®æ§è³ªã調ã¹ãåŠåã ãšèšã£ãŠè¯ãã
è€çŽ é¢æ°è«ã¯ç©çåŠã®ç¹ã«æ³¢åã«é¢ããåéïŒé³ã»é»ç£æ°ãªã©ïŒã«ãããŠæŽ»èºããŠããããæ³¢åæ¹çšåŒãããã€ã³ããŒãã³ã¹ããšããèšèã¯æåã ããã
ã¡ãªã¿ã«ãè€çŽ æ°ãããã«æ¡åŒµããæ°ãšããŠã[[w:åå
æ°]]ããšãããã®ãããããã®åå
æ°ã¯[[é«çåŠæ ¡æ°åŠC/ãã¯ãã«|ãã¯ãã«]]ã[[é«çåŠæ ¡æ°åŠC/æ°åŠçãªè¡šçŸã®å·¥å€«#è¡åãçšããè¡šçŸãšãã®æŒç®|è¡å]]ãšæ·±ãé¢ãããååšããŠãããæ·±æããšé¢çœãã®ã ããããããåé·ã«ãªãããå²æããããªããåå
æ°ãããã«æ¡åŒµããå
«å
æ°ãåå
å
æ°ãšããæ°ãç 究ãããŠããã
== æŒç¿åé¡ ==
{{DEFAULTSORT:ãããšããã€ããããããII ãããšããããã}}
[[Category:é«çåŠæ ¡æ°åŠII|ãããšããããã]] | 2005-05-04T09:17:55Z | 2024-03-30T03:12:47Z | [
"ãã³ãã¬ãŒã:Pathnav",
"ãã³ãã¬ãŒã:ã³ã©ã "
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E6%95%B0%E5%AD%A6II/%E5%BC%8F%E3%81%A8%E8%A8%BC%E6%98%8E%E3%83%BB%E9%AB%98%E6%AC%A1%E6%96%B9%E7%A8%8B%E5%BC%8F |
1,903 | é«çåŠæ ¡æ°åŠII/å³åœ¢ãšæ¹çšåŒ | ããã§ã¯çŽç·ãšåãªã©ã®æ§è³ªã座æšãçšããŠèå¯ããã
座æšå¹³é¢äžã®2ç¹ A ( x 1 , y 1 ) , B ( x 2 , y 2 ) {\displaystyle \mathrm {A} \left(x_{1}\ ,\ y_{1}\right)\ ,\ \mathrm {B} \left(x_{2}\ ,\ y_{2}\right)} éã®è·é¢ A B {\displaystyle \mathrm {A} \mathrm {B} } ãæ±ããŠã¿ãããçŽç· A B {\displaystyle \mathrm {A} \mathrm {B} } ã座æšè»žã«å¹³è¡ã§ãªããšããç¹ C ( x 2 , y 1 ) {\displaystyle \mathrm {C} \left(x_{2}\ ,\ y_{1}\right)} ããšããš
â³ A B C {\displaystyle \triangle \mathrm {A} \mathrm {B} \mathrm {C} } ã¯çŽè§äžè§åœ¢ã§ãããããäžå¹³æ¹ã®å®çãã
ãã®åŒã¯ãçŽç· A B {\displaystyle \mathrm {A} \mathrm {B} } ãx軞ãy軞ã«å¹³è¡ãªãšãã«ãæãç«ã€ã
ç¹ã«ãåç¹ O {\displaystyle \mathrm {O} } ãšç¹ A ( x 1 , y 1 ) {\displaystyle \mathrm {A} \left(x_{1}\ ,\ y_{1}\right)} éã®è·é¢ã¯
ç¹ A ( x 0 , y 0 ) , B ( x 1 , y 1 ) {\displaystyle \mathrm {A} (x_{0},y_{0}),\mathrm {B} (x_{1},y_{1})} ãšå®æ° m , n > 0 {\displaystyle m,n>0} ã«å¯ŸããŠãç·å A B {\displaystyle \mathrm {AB} } äžã®ç¹ P ( x , y ) {\displaystyle \mathrm {P} (x,y)} ãååšããŠã A P : P B = m : n {\displaystyle \mathrm {AP} :\mathrm {PB} =m:n} ãšãªããšããç¹ P {\displaystyle \mathrm {P} } ã A , B {\displaystyle \mathrm {A} ,\mathrm {B} } ã m : n {\displaystyle m:n} ã«å
åããç¹ãšããã
ãŸããç·å A B {\displaystyle \mathrm {AB} } äžã§ãªãç¹ P ( x , y ) {\displaystyle \mathrm {P} (x,y)} ãååšããŠã A P : P B = m : n {\displaystyle \mathrm {AP} :\mathrm {PB} =m:n} ãšãªããšããç¹ P {\displaystyle \mathrm {P} } ã A , B {\displaystyle \mathrm {A} ,\mathrm {B} } ã m : n {\displaystyle m:n} ã«å€åããç¹ãšããã
æ°çŽç·äžã®ç¹ A ( a ) , B ( b ) {\displaystyle \mathrm {A} (a),\mathrm {B} (b)} ã m : n {\displaystyle m:n} ã«å
åããç¹ãšå€åããç¹ãæ±ããã
å
åç¹ã P ( x ) {\displaystyle \mathrm {P} (x)} ãšããã a < b {\displaystyle a<b} ã®ãšãã A P = x â a , P B = b â x {\displaystyle \mathrm {AP} =x-a,\mathrm {PB} =b-x} ãªã®ã§ã m : n = ( x â a ) : ( b â x ) {\displaystyle m:n=(x-a):(b-x)} ãªã®ã§ã n ( x â a ) = m ( b â x ) ⺠x = n a + m b m + n {\displaystyle n(x-a)=m(b-x)\iff x={\frac {na+mb}{m+n}}} ã§ããã a > b {\displaystyle a>b} ã®ãšããåæ§ã
次ã«å€åç¹ãæ±ãããå€åç¹ã P ( x ) {\displaystyle \mathrm {P} (x)} ãšããã a < b {\displaystyle a<b} 㧠m > n {\displaystyle m>n} ã®ãšãã x > b {\displaystyle x>b} ãšãªãã®ã§ã A P = x â a , B P = x â b {\displaystyle \mathrm {AP} =x-a,\mathrm {BP} =x-b} ãªã®ã§ã m : n = ( x â a ) : ( x â b ) {\displaystyle m:n=(x-a):(x-b)} ãªã®ã§ã x = â n a + m b m â n {\displaystyle x={\frac {-na+mb}{m-n}}}
ããã¯ã a > b {\displaystyle a>b} ãŸã㯠m < n {\displaystyle m<n} ã®ãšããåæ§ã
2次å
ã®å Žåã«ã¯ãäžè¬ã«ç¹ãšç¹ãšã®äœçœ®é¢ä¿ã¯ã座æšè»žã«å¹³è¡ã§ãªãããããã®è·é¢ã®å
åã¯è€éã«ãªãããã«æãããããããå®éã«ã¯ãå
åç¹ãå€åç¹ãèšç®ããã«ã¯ãäžã®å
¬åŒãx,y ã®äž¡æ¹åã«å¯ŸããŠçšããã°ãããããã¯ã2ç¹ãã€ãªãç·ãçŽç·ã§ããã®ã§ããã®çŽç·äžã§ããç¹ããã®è·é¢ãäžå®ã®å²åãšãªãç¹ãããã€ãåã£ããšãããã®ç¹ãšå
ã®ç¹ã®x軞æ¹åã®åº§æšã®å€åã®å²åãšy軞æ¹åã®åº§æšã®å€åã®å²åãšçŽç·èªèº«ã®é·ãã®å€åã®å²åã¯ããããçãããªãããã§ããã
ãã£ãŠãäžè¬ã«ç¹ A ( x 0 , y 0 ) , B ( x 1 , y 1 ) {\displaystyle A(x_{0},y_{0}),B(x_{1},y_{1})} ããa:bã«å
åããç¹ãšå€åããç¹ã¯ã
ã§äžããããã
æŒç¿åé¡
ç¹ A ( 1 , 0 ) , B ( â 4 , 7 ) {\displaystyle \mathrm {A} (1,0),\mathrm {B} (-4,7)} ã3:1ã«ããããå
åãå€åããç¹ãæ±ããã
解ç
å
åç¹ã¯ ( â 11 4 , 21 4 ) {\displaystyle \left({\frac {-11}{4}},{\frac {21}{4}}\right)}
å€åç¹ã¯ ( â 13 2 , 21 2 ) {\displaystyle \left({\frac {-13}{2}},{\frac {21}{2}}\right)}
3ç¹ A ( x 1 , y 1 ) , B ( x 2 , y 2 ) , C ( x 3 , y 3 ) {\displaystyle \mathrm {A} \left(x_{1},y_{1}\right),\mathrm {B} \left(x_{2},y_{2}\right),\mathrm {C} \left(x_{3},y_{3}\right)} ãé ç¹ãšããäžè§åœ¢ã®éå¿ G {\displaystyle \mathrm {G} } ã®åº§æšãæ±ããŠã¿ããã ç·å B C {\displaystyle \mathrm {B} \mathrm {C} } ã®äžç¹ M {\displaystyle \mathrm {M} } ã®åº§æšã¯
éå¿ G {\displaystyle \mathrm {G} } ã¯ç·å A M {\displaystyle \mathrm {A} \mathrm {M} } ã2:1ã«å
åããç¹ã§ããããã G {\displaystyle \mathrm {G} } ã®åº§æšã ( x , y ) {\displaystyle (x,y)} ãšãããš
åæ§ã«
ãã£ãŠãéå¿ G {\displaystyle \mathrm {G} } ã®åº§æšã¯
ããç¹ ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} ãéã£ãŠåŸãaã®çŽç·ã®åŒã¯ã y â y 0 = a ( x â x 0 ) {\displaystyle y-y_{0}=a(x-x_{0})} ã§äžãããããããã¯ãåŸããyã®å€åå / {\displaystyle /} xã®å€ååã§è¡šãããã y â y 0 {\displaystyle y-y_{0}} , x â x 0 {\displaystyle x-x_{0}} ã¯ãŸãã«ãy,xããããã®å€ååãã®ãã®ã§ããããšã«ããã
2ç¹ ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} , ( x 1 , y 1 ) {\displaystyle (x_{1},y_{1})} ãéãçŽç·ã¯åŸãã y 0 â y 1 x 0 â x 1 {\displaystyle {\frac {y_{0}-y_{1}}{x_{0}-x_{1}}}} ã§äžããããããšãçšãããšã y â y 0 = y 0 â y 1 x 0 â x 1 ( x â x 0 ) {\displaystyle y-y_{0}={\frac {y_{0}-y_{1}}{x_{0}-x_{1}}}(x-x_{0})} ã§äžããããã
æŒç¿åé¡
ããããã®çŽç·ãè¡šããåŒãèšç®ããã
(i) åŸã-2ã§ãç¹(-3,1)ãéãçŽç·
(ii) 2ç¹(4,3) ,(5,7)ãéãçŽç·
解ç
ãçšããã°ããã
(i)
(ii)
ãŸãçŽç·ã®æ¹çšåŒã¯äžè¬ã« a x + b y + c = 0 {\displaystyle ax+by+c=0} ã§è¡šãããã
ç¹ ( 1 , 4 ) {\displaystyle (1,4)} ãéããçŽç· y = â 2 x + 3 {\displaystyle y=-2x+3} ã«å¹³è¡ãªçŽç·ãåçŽãªçŽç·ã®æ¹çšåŒãæ±ããã
çŽç· y = â 2 x + 3 {\displaystyle y=-2x+3} ã®åŸã㯠â 2 {\displaystyle -2} ã§ããã å¹³è¡ãªçŽç·ã®æ¹çšåŒã¯
åçŽãªçŽç·ã®åŸãã m {\displaystyle m} ãšãããš
ãã£ãŠãåçŽãªçŽç·ã®æ¹çšåŒã¯
ç¹ P {\displaystyle \mathrm {P} } ãšçŽç· l {\displaystyle l} ã«å¯ŸããçŽç· l {\displaystyle l} äžã®ç¹ãšç¹ P {\displaystyle \mathrm {P} } ã®è·é¢ã®æå°å€ãç¹ãšçŽç·ã®è·é¢ãšãããããã¯ç¹ P {\displaystyle \mathrm {P} } ããçŽç· l {\displaystyle l} ã«äžãããåç· P H {\displaystyle \mathrm {PH} } ã®é·ãã«çããã
çŽç· a x + b y + c = 0 {\displaystyle ax+by+c=0} ãšç¹ ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} ã®è·é¢ã¯
ãšè¡šãããã
蚌æ
ç¹ P ( x 0 , y 0 ) {\displaystyle \mathrm {P} (x_{0},y_{0})} ãšçŽç· l : a x + b y + c = 0 a , b â 0 {\displaystyle l:ax+by+c=0\quad a,b\neq 0} ãšããã
ç¹ P {\displaystyle \mathrm {P} } ããçŽç· l {\displaystyle l} ã«åç·ãäžãããåç·ã®è¶³ãç¹ R {\displaystyle R} ãšããã
ãŸããç¹ P {\displaystyle \mathrm {P} } ãã y {\displaystyle y} 軞ã«å¹³è¡ãªçŽç·ãåŒããçŽç· l {\displaystyle l} ãšã®äº€ç¹ãç¹ S {\displaystyle \mathrm {S} } ãšããã
次ã«ãå³ã®ããã«ãçŽç· l {\displaystyle l} äžã®ç¹ T {\displaystyle \mathrm {T} } ã«å¯ŸããŠãçŽç· T V {\displaystyle \mathrm {TV} } ã x {\displaystyle x} 軞ãšå¹³è¡ãšãªãã T V = | b | {\displaystyle \mathrm {TV} =|b|} ãšãªãããã«ç¹ V {\displaystyle \mathrm {V} } ããšããçŽç· V U {\displaystyle \mathrm {VU} } ã y {\displaystyle y} 軞ã«å¹³è¡ã«ãªãç¹ U {\displaystyle \mathrm {U} } ãçŽç· l {\displaystyle l} äžã«åãã
çŽç· l {\displaystyle l} ã®åŸã㯠â a b {\displaystyle -{\frac {a}{b}}} ãšãªãã®ã§ V U = | a | {\displaystyle \mathrm {VU} =|a|} ã§ããã ããã§ã â³ P R S , â³ T V U {\displaystyle \bigtriangleup \mathrm {PRS} ,\bigtriangleup \mathrm {TVU} } ã¯çŽè§äžè§åœ¢ã§ããã â P S R = â T U V {\displaystyle \angle \mathrm {PSR} =\angle \mathrm {TUV} } ãªã®ã§ã â³ P R S âŒâ³ T V U {\displaystyle \bigtriangleup \mathrm {PRS} \sim \bigtriangleup \mathrm {TVU} } ã§ããããããã£ãŠ
ãŸãç¹ S {\displaystyle \mathrm {S} } ã®åº§æšã ( x 0 , m ) {\displaystyle (x_{0},m)} ãšãããšã P S = | y 0 â m | {\displaystyle \mathrm {PS} =|y_{0}-m|} ã§ãç¹ P {\displaystyle \mathrm {P} } ãšçŽç· l {\displaystyle l} ã®è·é¢ P R {\displaystyle \mathrm {PR} } ã¯ã
P R = P S â
T V T U = | y 0 â m | | b | a 2 + b 2 {\displaystyle \mathrm {PR} ={\mathrm {PS} }\cdot {\frac {\mathrm {TV} }{\mathrm {TU} }}={\frac {|y_{0}-m||b|}{\sqrt {a^{2}+b^{2}}}}}
ãšããã§ãç¹ S {\displaystyle \mathrm {S} } ã¯çŽç· l {\displaystyle l} äžã®ç¹ãªã®ã§ã
ã§ãããããã代å
¥ããã°
ãã¯ãã«ã䜿ã£ã蚌æ
ãã§ã«ãã¯ãã«ãç¥ã£ãŠãããªãã°ãã¡ãã®æ¹ãç°¡æœã§ããã
ç¹ P ( x 0 , y 0 ) {\displaystyle \mathrm {P} (x_{0},y_{0})} ãšçŽç· l : a x + b y + c = 0 {\displaystyle l:ax+by+c=0} ãšããç¹ Q ( x 1 , y 1 ) {\displaystyle \mathrm {Q} (x_{1},y_{1})} ãçŽç· l {\displaystyle l} äžã®ç¹ãšãããçŽç· l {\displaystyle l} ã®æ³ç·ã¯ n â := ( a , b ) {\displaystyle {\vec {n}}:=(a,b)} ã§ã Q P â = ( x 0 â x 1 , y 0 â y 1 ) {\displaystyle {\vec {\mathrm {QP} }}=(x_{0}-x_{1},y_{0}-y_{1})} ã§ããã®ã§ãçŽç· l {\displaystyle l} äžã®ç¹ãšç¹ P {\displaystyle \mathrm {P} } ã®è·é¢ d {\displaystyle d} 㯠d = | Q P â â
n â | | n â | | | = | ( x 0 â x 1 , y 0 â y 1 ) â
( a , b ) a 2 + b 2 | = | a x 0 + b y 0 â ( a x 1 + b y 1 ) | a 2 + b 2 = | a x 0 + b y 0 + c | a 2 + b 2 {\displaystyle d=\left|{\vec {\mathrm {QP} }}\cdot {\frac {\vec {n}}{||{\vec {n}}||}}\right|=\left|(x_{0}-x_{1},y_{0}-y_{1})\cdot {\frac {(a,b)}{\sqrt {a^{2}+b^{2}}}}\right|={\frac {|ax_{0}+by_{0}-(ax_{1}+by_{1})|}{\sqrt {a^{2}+b^{2}}}}={\frac {|ax_{0}+by_{0}+c|}{\sqrt {a^{2}+b^{2}}}}} ã§ããã
æŒç¿åé¡
çŽç· x â 2 y â 3 = 0 {\displaystyle x-2y-3=0} ãšç¹ ( 1 , 2 ) {\displaystyle (1,2)} ã®è·é¢ãæ±ãã
解ç
6 5 {\displaystyle {\frac {6}{\sqrt {5}}}}
äžå¿ C ( a , b ) {\displaystyle \mathrm {C} (a,b)} ååŸ r {\displaystyle r} ã®åã¯ã C P = r {\displaystyle \mathrm {CP} =r} ãšãªãç¹ P {\displaystyle \mathrm {P} } ã®éåã§ãããã€ãŸãã r = ( x â a ) 2 + ( y â b ) 2 {\displaystyle r={\sqrt {(x-a)^{2}+(y-b)^{2}}}} ãšãªãç¹ ( x , y ) {\displaystyle (x,y)} ã®éåã§ããããã®æ¹çšåŒã®äž¡èŸºã¯æ£ãªã®ã§2ä¹ããŠ
( x â a ) 2 + ( y â b ) 2 = r 2 {\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}}
ãåŸãããããåã®æ¹çšåŒã§ããã
ç¹ã«åç¹ãäžå¿ã§ååŸ r {\displaystyle r} ã®åã®æ¹çšåŒã¯ x 2 + y 2 = r 2 {\displaystyle x^{2}+y^{2}=r^{2}} ã§äžããããã
æŒç¿åé¡
解ç
æ¹çšåŒ x 2 + y 2 + l x + m y + n = 0 {\displaystyle x^{2}+y^{2}+lx+my+n=0} ã¯ãã€ãåã§ãããšã¯éããªãã
æ¹çšåŒãå€åœ¢ã㊠( x â a ) 2 + ( y â b ) 2 = k {\displaystyle (x-a)^{2}+(y-b)^{2}=k} ãšãªããšã
å x 2 + y 2 = r 2 {\displaystyle x^{2}+y^{2}=r^{2}} äžã®ããç¹ ( x 1 , y 1 ) {\displaystyle (x_{1},y_{1})} ã§æ¥ããæ¥ç·ã®æ¹çšåŒã¯
ã§è¡šãããã
åæ§ã«ãå ( x â a ) 2 + ( y â b ) 2 = r 2 {\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}} äžã®ããç¹ ( x 2 , y 2 ) {\displaystyle (x_{2},y_{2})} ã§æ¥ããæ¥ç·ã®æ¹çšåŒã¯
ã§è¡šãããã
åãšçŽç·ã®äœçœ®é¢ä¿ã«ã€ããŠå€§ãã次ã®3ã€ã«åé¡ããããšãã§ããã
äžè¬ã®åãšçŽç·ã«ã€ããŠãããã®äœçœ®é¢ä¿ãåé¡ããŠã¿ããã
å C : ( x â p ) 2 + ( y â q ) 2 = r 2 {\displaystyle C:(x-p)^{2}+(y-q)^{2}=r^{2}} ãšçŽç· l : a x + b y + c = 0 {\displaystyle l:ax+by+c=0} ã«ã€ããŠãå C {\displaystyle C} ã®äžå¿ ( p , q ) {\displaystyle (p,q)} ãšçŽç· l {\displaystyle l} ã®è·é¢ d := | a q + b q + c | a 2 + b 2 {\displaystyle d:={\frac {|aq+bq+c|}{\sqrt {a^{2}+b^{2}}}}} ãšãããšã
ä»ã«ããåã®æ¹çšåŒãšçŽç·ã®æ¹çšåŒãé£ç«ããŠãã®å®æ°è§£ã®åæ°ã§åé¡ããæ¹æ³ãããããäœçœ®é¢ä¿ãæ±ããã ããªãäžã®æ¹æ³ã®ã»ããèšç®éãå°ãªãã
æŒç¿åé¡
çŽç· 3 x + 4 y = 1 {\displaystyle 3x+4y=1} ãšå ( x â 3 ) 2 + ( y + 2 ) 2 = 14 {\displaystyle (x-3)^{2}+(y+2)^{2}=14} ã®äº€ç¹ã®åº§æšãæ±ãã
解ç
çŽç·ã®æ¹çšåŒã x {\displaystyle x} ã«ã€ããŠè§£ãããããåã®æ¹çšåŒã«ä»£å
¥ããã°ããã
çã㯠( 2 , â 1 ) , ( â 14 5 , 7 5 ) {\displaystyle (2,-1),\left(-{\frac {14}{5}},{\frac {7}{5}}\right)}
ããæ¡ä»¶ãæºããç¹å
šäœãã€ããå³åœ¢ãããã®æ¡ä»¶ãæºããç¹ã®è»è·¡ãšããã
2ç¹ A ( 1 , 0 ) , B ( 3 , 2 ) {\displaystyle \mathrm {A} (1\ ,\ 0)\ ,\ \mathrm {B} (3\ ,\ 2)} ããçè·é¢ã«ããç¹ P {\displaystyle \mathrm {P} } ã®è»è·¡ãæ±ããã
æ¡ä»¶ A P = B P {\displaystyle \mathrm {A} \mathrm {P} =\mathrm {B} \mathrm {P} } ããã A P 2 = B P 2 {\displaystyle \mathrm {A} \mathrm {P} ^{2}=\mathrm {B} \mathrm {P} ^{2}} P {\displaystyle \mathrm {P} } ã®åº§æšã ( x , y ) {\displaystyle (x\ ,\ y)} ãšãããš
ã ãã
æŽçããŠã
ãããã£ãŠãæ±ããè»è·¡ã¯ãçŽç· y = â x + 3 {\displaystyle y=-x+3} ã§ããã
2ç¹ A ( 0 , 0 ) , B ( 3 , 0 ) {\displaystyle \mathrm {A} (0\ ,\ 0)\ ,\ \mathrm {B} (3\ ,\ 0)} ããã®è·é¢ã®æ¯ã 2 : 1 {\displaystyle 2:1} ã§ããç¹ P {\displaystyle \mathrm {P} } ã®è»è·¡ãæ±ããã
P {\displaystyle \mathrm {P} } ã®åº§æšã ( x , y ) {\displaystyle (x\ ,\ y)} ãšããã P {\displaystyle \mathrm {P} } ãæºããæ¡ä»¶ã¯
ããªãã¡
ããã座æšã§è¡šããš
䞡蟺ã2ä¹ããŠãæŽçãããš
ããªãã¡
ãããã£ãŠãæ±ããè»è·¡ã¯ãäžå¿ã ( 4 , 0 ) {\displaystyle (4\ ,\ 0)} ãååŸã 2 {\displaystyle 2} ã®åã§ããã
m , n {\displaystyle m\ ,\ n} ãç°ãªãæ£ã®æ°ãšãããšãã2ç¹ A , B {\displaystyle \mathrm {A} \ ,\ \mathrm {B} } ããã®è·é¢ã®æ¯ã m : n {\displaystyle m:n} ã§ããç¹ã®è»è·¡ã¯ãç·å A B {\displaystyle \mathrm {A} \mathrm {B} } ã m : n {\displaystyle m:n} ã«å
åããç¹ãšãå€åããç¹ãçŽåŸã®äž¡ç«¯ãšããåã§ããããã®åãã¢ããããŠã¹ã®åãšããã
m = n {\displaystyle m=n} ã®ãšãã¯ãç·å A B {\displaystyle \mathrm {A} \mathrm {B} } ã®åçŽäºçåç·ã§ããã
ãã®ããŒãžã®åéã®ããã«ãæ°åŒãã€ãã£ãŠåº§æšã®äœçœ®ããããããŠã幟äœåŠã®åé¡ã解ãææ³ã®ããšã解æ幟äœåŠãšããã
ãªãã幟äœåŠãšããèšèèªäœã¯ãå³åœ¢ã®åŠåãšãããããªæå³ã§ãããå°åŠæ ¡ãäžåŠæ ¡ã§ç¿ã£ãå³åœ¢ã®çè«ã幟äœåŠã§ããã
äžäžãšãŒãããã®æ°åŠè
ãã«ã«ããã解æ幟äœåŠã®ç 究ãé²ããããªãããã«ã«ãã¯ãå²åŠã®æ Œèšãããæããããã«æãããã§ãæåã§ããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ããã§ã¯çŽç·ãšåãªã©ã®æ§è³ªã座æšãçšããŠèå¯ããã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "座æšå¹³é¢äžã®2ç¹ A ( x 1 , y 1 ) , B ( x 2 , y 2 ) {\\displaystyle \\mathrm {A} \\left(x_{1}\\ ,\\ y_{1}\\right)\\ ,\\ \\mathrm {B} \\left(x_{2}\\ ,\\ y_{2}\\right)} éã®è·é¢ A B {\\displaystyle \\mathrm {A} \\mathrm {B} } ãæ±ããŠã¿ãããçŽç· A B {\\displaystyle \\mathrm {A} \\mathrm {B} } ã座æšè»žã«å¹³è¡ã§ãªããšããç¹ C ( x 2 , y 1 ) {\\displaystyle \\mathrm {C} \\left(x_{2}\\ ,\\ y_{1}\\right)} ããšããš",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "â³ A B C {\\displaystyle \\triangle \\mathrm {A} \\mathrm {B} \\mathrm {C} } ã¯çŽè§äžè§åœ¢ã§ãããããäžå¹³æ¹ã®å®çãã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ãã®åŒã¯ãçŽç· A B {\\displaystyle \\mathrm {A} \\mathrm {B} } ãx軞ãy軞ã«å¹³è¡ãªãšãã«ãæãç«ã€ã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ç¹ã«ãåç¹ O {\\displaystyle \\mathrm {O} } ãšç¹ A ( x 1 , y 1 ) {\\displaystyle \\mathrm {A} \\left(x_{1}\\ ,\\ y_{1}\\right)} éã®è·é¢ã¯",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ç¹ A ( x 0 , y 0 ) , B ( x 1 , y 1 ) {\\displaystyle \\mathrm {A} (x_{0},y_{0}),\\mathrm {B} (x_{1},y_{1})} ãšå®æ° m , n > 0 {\\displaystyle m,n>0} ã«å¯ŸããŠãç·å A B {\\displaystyle \\mathrm {AB} } äžã®ç¹ P ( x , y ) {\\displaystyle \\mathrm {P} (x,y)} ãååšããŠã A P : P B = m : n {\\displaystyle \\mathrm {AP} :\\mathrm {PB} =m:n} ãšãªããšããç¹ P {\\displaystyle \\mathrm {P} } ã A , B {\\displaystyle \\mathrm {A} ,\\mathrm {B} } ã m : n {\\displaystyle m:n} ã«å
åããç¹ãšããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ãŸããç·å A B {\\displaystyle \\mathrm {AB} } äžã§ãªãç¹ P ( x , y ) {\\displaystyle \\mathrm {P} (x,y)} ãååšããŠã A P : P B = m : n {\\displaystyle \\mathrm {AP} :\\mathrm {PB} =m:n} ãšãªããšããç¹ P {\\displaystyle \\mathrm {P} } ã A , B {\\displaystyle \\mathrm {A} ,\\mathrm {B} } ã m : n {\\displaystyle m:n} ã«å€åããç¹ãšããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "æ°çŽç·äžã®ç¹ A ( a ) , B ( b ) {\\displaystyle \\mathrm {A} (a),\\mathrm {B} (b)} ã m : n {\\displaystyle m:n} ã«å
åããç¹ãšå€åããç¹ãæ±ããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "å
åç¹ã P ( x ) {\\displaystyle \\mathrm {P} (x)} ãšããã a < b {\\displaystyle a<b} ã®ãšãã A P = x â a , P B = b â x {\\displaystyle \\mathrm {AP} =x-a,\\mathrm {PB} =b-x} ãªã®ã§ã m : n = ( x â a ) : ( b â x ) {\\displaystyle m:n=(x-a):(b-x)} ãªã®ã§ã n ( x â a ) = m ( b â x ) ⺠x = n a + m b m + n {\\displaystyle n(x-a)=m(b-x)\\iff x={\\frac {na+mb}{m+n}}} ã§ããã a > b {\\displaystyle a>b} ã®ãšããåæ§ã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "次ã«å€åç¹ãæ±ãããå€åç¹ã P ( x ) {\\displaystyle \\mathrm {P} (x)} ãšããã a < b {\\displaystyle a<b} 㧠m > n {\\displaystyle m>n} ã®ãšãã x > b {\\displaystyle x>b} ãšãªãã®ã§ã A P = x â a , B P = x â b {\\displaystyle \\mathrm {AP} =x-a,\\mathrm {BP} =x-b} ãªã®ã§ã m : n = ( x â a ) : ( x â b ) {\\displaystyle m:n=(x-a):(x-b)} ãªã®ã§ã x = â n a + m b m â n {\\displaystyle x={\\frac {-na+mb}{m-n}}}",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "ããã¯ã a > b {\\displaystyle a>b} ãŸã㯠m < n {\\displaystyle m<n} ã®ãšããåæ§ã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "2次å
ã®å Žåã«ã¯ãäžè¬ã«ç¹ãšç¹ãšã®äœçœ®é¢ä¿ã¯ã座æšè»žã«å¹³è¡ã§ãªãããããã®è·é¢ã®å
åã¯è€éã«ãªãããã«æãããããããå®éã«ã¯ãå
åç¹ãå€åç¹ãèšç®ããã«ã¯ãäžã®å
¬åŒãx,y ã®äž¡æ¹åã«å¯ŸããŠçšããã°ãããããã¯ã2ç¹ãã€ãªãç·ãçŽç·ã§ããã®ã§ããã®çŽç·äžã§ããç¹ããã®è·é¢ãäžå®ã®å²åãšãªãç¹ãããã€ãåã£ããšãããã®ç¹ãšå
ã®ç¹ã®x軞æ¹åã®åº§æšã®å€åã®å²åãšy軞æ¹åã®åº§æšã®å€åã®å²åãšçŽç·èªèº«ã®é·ãã®å€åã®å²åã¯ããããçãããªãããã§ããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ãã£ãŠãäžè¬ã«ç¹ A ( x 0 , y 0 ) , B ( x 1 , y 1 ) {\\displaystyle A(x_{0},y_{0}),B(x_{1},y_{1})} ããa:bã«å
åããç¹ãšå€åããç¹ã¯ã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "ã§äžããããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "ç¹ A ( 1 , 0 ) , B ( â 4 , 7 ) {\\displaystyle \\mathrm {A} (1,0),\\mathrm {B} (-4,7)} ã3:1ã«ããããå
åãå€åããç¹ãæ±ããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "解ç",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "å
åç¹ã¯ ( â 11 4 , 21 4 ) {\\displaystyle \\left({\\frac {-11}{4}},{\\frac {21}{4}}\\right)}",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "å€åç¹ã¯ ( â 13 2 , 21 2 ) {\\displaystyle \\left({\\frac {-13}{2}},{\\frac {21}{2}}\\right)}",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "3ç¹ A ( x 1 , y 1 ) , B ( x 2 , y 2 ) , C ( x 3 , y 3 ) {\\displaystyle \\mathrm {A} \\left(x_{1},y_{1}\\right),\\mathrm {B} \\left(x_{2},y_{2}\\right),\\mathrm {C} \\left(x_{3},y_{3}\\right)} ãé ç¹ãšããäžè§åœ¢ã®éå¿ G {\\displaystyle \\mathrm {G} } ã®åº§æšãæ±ããŠã¿ããã ç·å B C {\\displaystyle \\mathrm {B} \\mathrm {C} } ã®äžç¹ M {\\displaystyle \\mathrm {M} } ã®åº§æšã¯",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "éå¿ G {\\displaystyle \\mathrm {G} } ã¯ç·å A M {\\displaystyle \\mathrm {A} \\mathrm {M} } ã2:1ã«å
åããç¹ã§ããããã G {\\displaystyle \\mathrm {G} } ã®åº§æšã ( x , y ) {\\displaystyle (x,y)} ãšãããš",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "åæ§ã«",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "ãã£ãŠãéå¿ G {\\displaystyle \\mathrm {G} } ã®åº§æšã¯",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "ããç¹ ( x 0 , y 0 ) {\\displaystyle (x_{0},y_{0})} ãéã£ãŠåŸãaã®çŽç·ã®åŒã¯ã y â y 0 = a ( x â x 0 ) {\\displaystyle y-y_{0}=a(x-x_{0})} ã§äžãããããããã¯ãåŸããyã®å€åå / {\\displaystyle /} xã®å€ååã§è¡šãããã y â y 0 {\\displaystyle y-y_{0}} , x â x 0 {\\displaystyle x-x_{0}} ã¯ãŸãã«ãy,xããããã®å€ååãã®ãã®ã§ããããšã«ããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "2ç¹ ( x 0 , y 0 ) {\\displaystyle (x_{0},y_{0})} , ( x 1 , y 1 ) {\\displaystyle (x_{1},y_{1})} ãéãçŽç·ã¯åŸãã y 0 â y 1 x 0 â x 1 {\\displaystyle {\\frac {y_{0}-y_{1}}{x_{0}-x_{1}}}} ã§äžããããããšãçšãããšã y â y 0 = y 0 â y 1 x 0 â x 1 ( x â x 0 ) {\\displaystyle y-y_{0}={\\frac {y_{0}-y_{1}}{x_{0}-x_{1}}}(x-x_{0})} ã§äžããããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "ããããã®çŽç·ãè¡šããåŒãèšç®ããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "(i) åŸã-2ã§ãç¹(-3,1)ãéãçŽç·",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "(ii) 2ç¹(4,3) ,(5,7)ãéãçŽç·",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "解ç",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "ãçšããã°ããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "(i)",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "(ii)",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "ãŸãçŽç·ã®æ¹çšåŒã¯äžè¬ã« a x + b y + c = 0 {\\displaystyle ax+by+c=0} ã§è¡šãããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "ç¹ ( 1 , 4 ) {\\displaystyle (1,4)} ãéããçŽç· y = â 2 x + 3 {\\displaystyle y=-2x+3} ã«å¹³è¡ãªçŽç·ãåçŽãªçŽç·ã®æ¹çšåŒãæ±ããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "çŽç· y = â 2 x + 3 {\\displaystyle y=-2x+3} ã®åŸã㯠â 2 {\\displaystyle -2} ã§ããã å¹³è¡ãªçŽç·ã®æ¹çšåŒã¯",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "åçŽãªçŽç·ã®åŸãã m {\\displaystyle m} ãšãããš",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "ãã£ãŠãåçŽãªçŽç·ã®æ¹çšåŒã¯",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 41,
"tag": "p",
"text": "ç¹ P {\\displaystyle \\mathrm {P} } ãšçŽç· l {\\displaystyle l} ã«å¯ŸããçŽç· l {\\displaystyle l} äžã®ç¹ãšç¹ P {\\displaystyle \\mathrm {P} } ã®è·é¢ã®æå°å€ãç¹ãšçŽç·ã®è·é¢ãšãããããã¯ç¹ P {\\displaystyle \\mathrm {P} } ããçŽç· l {\\displaystyle l} ã«äžãããåç· P H {\\displaystyle \\mathrm {PH} } ã®é·ãã«çããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 42,
"tag": "p",
"text": "çŽç· a x + b y + c = 0 {\\displaystyle ax+by+c=0} ãšç¹ ( x 0 , y 0 ) {\\displaystyle (x_{0},y_{0})} ã®è·é¢ã¯",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 43,
"tag": "p",
"text": "ãšè¡šãããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 44,
"tag": "p",
"text": "蚌æ",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 45,
"tag": "p",
"text": "ç¹ P ( x 0 , y 0 ) {\\displaystyle \\mathrm {P} (x_{0},y_{0})} ãšçŽç· l : a x + b y + c = 0 a , b â 0 {\\displaystyle l:ax+by+c=0\\quad a,b\\neq 0} ãšããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 46,
"tag": "p",
"text": "ç¹ P {\\displaystyle \\mathrm {P} } ããçŽç· l {\\displaystyle l} ã«åç·ãäžãããåç·ã®è¶³ãç¹ R {\\displaystyle R} ãšããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 47,
"tag": "p",
"text": "ãŸããç¹ P {\\displaystyle \\mathrm {P} } ãã y {\\displaystyle y} 軞ã«å¹³è¡ãªçŽç·ãåŒããçŽç· l {\\displaystyle l} ãšã®äº€ç¹ãç¹ S {\\displaystyle \\mathrm {S} } ãšããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 48,
"tag": "p",
"text": "次ã«ãå³ã®ããã«ãçŽç· l {\\displaystyle l} äžã®ç¹ T {\\displaystyle \\mathrm {T} } ã«å¯ŸããŠãçŽç· T V {\\displaystyle \\mathrm {TV} } ã x {\\displaystyle x} 軞ãšå¹³è¡ãšãªãã T V = | b | {\\displaystyle \\mathrm {TV} =|b|} ãšãªãããã«ç¹ V {\\displaystyle \\mathrm {V} } ããšããçŽç· V U {\\displaystyle \\mathrm {VU} } ã y {\\displaystyle y} 軞ã«å¹³è¡ã«ãªãç¹ U {\\displaystyle \\mathrm {U} } ãçŽç· l {\\displaystyle l} äžã«åãã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 49,
"tag": "p",
"text": "çŽç· l {\\displaystyle l} ã®åŸã㯠â a b {\\displaystyle -{\\frac {a}{b}}} ãšãªãã®ã§ V U = | a | {\\displaystyle \\mathrm {VU} =|a|} ã§ããã ããã§ã â³ P R S , â³ T V U {\\displaystyle \\bigtriangleup \\mathrm {PRS} ,\\bigtriangleup \\mathrm {TVU} } ã¯çŽè§äžè§åœ¢ã§ããã â P S R = â T U V {\\displaystyle \\angle \\mathrm {PSR} =\\angle \\mathrm {TUV} } ãªã®ã§ã â³ P R S âŒâ³ T V U {\\displaystyle \\bigtriangleup \\mathrm {PRS} \\sim \\bigtriangleup \\mathrm {TVU} } ã§ããããããã£ãŠ",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 50,
"tag": "p",
"text": "ãŸãç¹ S {\\displaystyle \\mathrm {S} } ã®åº§æšã ( x 0 , m ) {\\displaystyle (x_{0},m)} ãšãããšã P S = | y 0 â m | {\\displaystyle \\mathrm {PS} =|y_{0}-m|} ã§ãç¹ P {\\displaystyle \\mathrm {P} } ãšçŽç· l {\\displaystyle l} ã®è·é¢ P R {\\displaystyle \\mathrm {PR} } ã¯ã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 51,
"tag": "p",
"text": "P R = P S â
T V T U = | y 0 â m | | b | a 2 + b 2 {\\displaystyle \\mathrm {PR} ={\\mathrm {PS} }\\cdot {\\frac {\\mathrm {TV} }{\\mathrm {TU} }}={\\frac {|y_{0}-m||b|}{\\sqrt {a^{2}+b^{2}}}}}",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 52,
"tag": "p",
"text": "ãšããã§ãç¹ S {\\displaystyle \\mathrm {S} } ã¯çŽç· l {\\displaystyle l} äžã®ç¹ãªã®ã§ã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 53,
"tag": "p",
"text": "ã§ãããããã代å
¥ããã°",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 54,
"tag": "p",
"text": "ãã¯ãã«ã䜿ã£ã蚌æ",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 55,
"tag": "p",
"text": "ãã§ã«ãã¯ãã«ãç¥ã£ãŠãããªãã°ãã¡ãã®æ¹ãç°¡æœã§ããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 56,
"tag": "p",
"text": "ç¹ P ( x 0 , y 0 ) {\\displaystyle \\mathrm {P} (x_{0},y_{0})} ãšçŽç· l : a x + b y + c = 0 {\\displaystyle l:ax+by+c=0} ãšããç¹ Q ( x 1 , y 1 ) {\\displaystyle \\mathrm {Q} (x_{1},y_{1})} ãçŽç· l {\\displaystyle l} äžã®ç¹ãšãããçŽç· l {\\displaystyle l} ã®æ³ç·ã¯ n â := ( a , b ) {\\displaystyle {\\vec {n}}:=(a,b)} ã§ã Q P â = ( x 0 â x 1 , y 0 â y 1 ) {\\displaystyle {\\vec {\\mathrm {QP} }}=(x_{0}-x_{1},y_{0}-y_{1})} ã§ããã®ã§ãçŽç· l {\\displaystyle l} äžã®ç¹ãšç¹ P {\\displaystyle \\mathrm {P} } ã®è·é¢ d {\\displaystyle d} 㯠d = | Q P â â
n â | | n â | | | = | ( x 0 â x 1 , y 0 â y 1 ) â
( a , b ) a 2 + b 2 | = | a x 0 + b y 0 â ( a x 1 + b y 1 ) | a 2 + b 2 = | a x 0 + b y 0 + c | a 2 + b 2 {\\displaystyle d=\\left|{\\vec {\\mathrm {QP} }}\\cdot {\\frac {\\vec {n}}{||{\\vec {n}}||}}\\right|=\\left|(x_{0}-x_{1},y_{0}-y_{1})\\cdot {\\frac {(a,b)}{\\sqrt {a^{2}+b^{2}}}}\\right|={\\frac {|ax_{0}+by_{0}-(ax_{1}+by_{1})|}{\\sqrt {a^{2}+b^{2}}}}={\\frac {|ax_{0}+by_{0}+c|}{\\sqrt {a^{2}+b^{2}}}}} ã§ããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 57,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 58,
"tag": "p",
"text": "çŽç· x â 2 y â 3 = 0 {\\displaystyle x-2y-3=0} ãšç¹ ( 1 , 2 ) {\\displaystyle (1,2)} ã®è·é¢ãæ±ãã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 59,
"tag": "p",
"text": "解ç",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 60,
"tag": "p",
"text": "6 5 {\\displaystyle {\\frac {6}{\\sqrt {5}}}}",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 61,
"tag": "p",
"text": "äžå¿ C ( a , b ) {\\displaystyle \\mathrm {C} (a,b)} ååŸ r {\\displaystyle r} ã®åã¯ã C P = r {\\displaystyle \\mathrm {CP} =r} ãšãªãç¹ P {\\displaystyle \\mathrm {P} } ã®éåã§ãããã€ãŸãã r = ( x â a ) 2 + ( y â b ) 2 {\\displaystyle r={\\sqrt {(x-a)^{2}+(y-b)^{2}}}} ãšãªãç¹ ( x , y ) {\\displaystyle (x,y)} ã®éåã§ããããã®æ¹çšåŒã®äž¡èŸºã¯æ£ãªã®ã§2ä¹ããŠ",
"title": "å"
},
{
"paragraph_id": 62,
"tag": "p",
"text": "( x â a ) 2 + ( y â b ) 2 = r 2 {\\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}}",
"title": "å"
},
{
"paragraph_id": 63,
"tag": "p",
"text": "ãåŸãããããåã®æ¹çšåŒã§ããã",
"title": "å"
},
{
"paragraph_id": 64,
"tag": "p",
"text": "ç¹ã«åç¹ãäžå¿ã§ååŸ r {\\displaystyle r} ã®åã®æ¹çšåŒã¯ x 2 + y 2 = r 2 {\\displaystyle x^{2}+y^{2}=r^{2}} ã§äžããããã",
"title": "å"
},
{
"paragraph_id": 65,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "å"
},
{
"paragraph_id": 66,
"tag": "p",
"text": "解ç",
"title": "å"
},
{
"paragraph_id": 67,
"tag": "p",
"text": "",
"title": "å"
},
{
"paragraph_id": 68,
"tag": "p",
"text": "æ¹çšåŒ x 2 + y 2 + l x + m y + n = 0 {\\displaystyle x^{2}+y^{2}+lx+my+n=0} ã¯ãã€ãåã§ãããšã¯éããªãã",
"title": "å"
},
{
"paragraph_id": 69,
"tag": "p",
"text": "æ¹çšåŒãå€åœ¢ã㊠( x â a ) 2 + ( y â b ) 2 = k {\\displaystyle (x-a)^{2}+(y-b)^{2}=k} ãšãªããšã",
"title": "å"
},
{
"paragraph_id": 70,
"tag": "p",
"text": "å x 2 + y 2 = r 2 {\\displaystyle x^{2}+y^{2}=r^{2}} äžã®ããç¹ ( x 1 , y 1 ) {\\displaystyle (x_{1},y_{1})} ã§æ¥ããæ¥ç·ã®æ¹çšåŒã¯",
"title": "å"
},
{
"paragraph_id": 71,
"tag": "p",
"text": "ã§è¡šãããã",
"title": "å"
},
{
"paragraph_id": 72,
"tag": "p",
"text": "åæ§ã«ãå ( x â a ) 2 + ( y â b ) 2 = r 2 {\\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}} äžã®ããç¹ ( x 2 , y 2 ) {\\displaystyle (x_{2},y_{2})} ã§æ¥ããæ¥ç·ã®æ¹çšåŒã¯",
"title": "å"
},
{
"paragraph_id": 73,
"tag": "p",
"text": "ã§è¡šãããã",
"title": "å"
},
{
"paragraph_id": 74,
"tag": "p",
"text": "åãšçŽç·ã®äœçœ®é¢ä¿ã«ã€ããŠå€§ãã次ã®3ã€ã«åé¡ããããšãã§ããã",
"title": "å"
},
{
"paragraph_id": 75,
"tag": "p",
"text": "äžè¬ã®åãšçŽç·ã«ã€ããŠãããã®äœçœ®é¢ä¿ãåé¡ããŠã¿ããã",
"title": "å"
},
{
"paragraph_id": 76,
"tag": "p",
"text": "å C : ( x â p ) 2 + ( y â q ) 2 = r 2 {\\displaystyle C:(x-p)^{2}+(y-q)^{2}=r^{2}} ãšçŽç· l : a x + b y + c = 0 {\\displaystyle l:ax+by+c=0} ã«ã€ããŠãå C {\\displaystyle C} ã®äžå¿ ( p , q ) {\\displaystyle (p,q)} ãšçŽç· l {\\displaystyle l} ã®è·é¢ d := | a q + b q + c | a 2 + b 2 {\\displaystyle d:={\\frac {|aq+bq+c|}{\\sqrt {a^{2}+b^{2}}}}} ãšãããšã",
"title": "å"
},
{
"paragraph_id": 77,
"tag": "p",
"text": "ä»ã«ããåã®æ¹çšåŒãšçŽç·ã®æ¹çšåŒãé£ç«ããŠãã®å®æ°è§£ã®åæ°ã§åé¡ããæ¹æ³ãããããäœçœ®é¢ä¿ãæ±ããã ããªãäžã®æ¹æ³ã®ã»ããèšç®éãå°ãªãã",
"title": "å"
},
{
"paragraph_id": 78,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "å"
},
{
"paragraph_id": 79,
"tag": "p",
"text": "çŽç· 3 x + 4 y = 1 {\\displaystyle 3x+4y=1} ãšå ( x â 3 ) 2 + ( y + 2 ) 2 = 14 {\\displaystyle (x-3)^{2}+(y+2)^{2}=14} ã®äº€ç¹ã®åº§æšãæ±ãã",
"title": "å"
},
{
"paragraph_id": 80,
"tag": "p",
"text": "解ç",
"title": "å"
},
{
"paragraph_id": 81,
"tag": "p",
"text": "çŽç·ã®æ¹çšåŒã x {\\displaystyle x} ã«ã€ããŠè§£ãããããåã®æ¹çšåŒã«ä»£å
¥ããã°ããã",
"title": "å"
},
{
"paragraph_id": 82,
"tag": "p",
"text": "çã㯠( 2 , â 1 ) , ( â 14 5 , 7 5 ) {\\displaystyle (2,-1),\\left(-{\\frac {14}{5}},{\\frac {7}{5}}\\right)}",
"title": "å"
},
{
"paragraph_id": 83,
"tag": "p",
"text": "ããæ¡ä»¶ãæºããç¹å
šäœãã€ããå³åœ¢ãããã®æ¡ä»¶ãæºããç¹ã®è»è·¡ãšããã",
"title": "å"
},
{
"paragraph_id": 84,
"tag": "p",
"text": "",
"title": "å"
},
{
"paragraph_id": 85,
"tag": "p",
"text": "2ç¹ A ( 1 , 0 ) , B ( 3 , 2 ) {\\displaystyle \\mathrm {A} (1\\ ,\\ 0)\\ ,\\ \\mathrm {B} (3\\ ,\\ 2)} ããçè·é¢ã«ããç¹ P {\\displaystyle \\mathrm {P} } ã®è»è·¡ãæ±ããã",
"title": "å"
},
{
"paragraph_id": 86,
"tag": "p",
"text": "æ¡ä»¶ A P = B P {\\displaystyle \\mathrm {A} \\mathrm {P} =\\mathrm {B} \\mathrm {P} } ããã A P 2 = B P 2 {\\displaystyle \\mathrm {A} \\mathrm {P} ^{2}=\\mathrm {B} \\mathrm {P} ^{2}} P {\\displaystyle \\mathrm {P} } ã®åº§æšã ( x , y ) {\\displaystyle (x\\ ,\\ y)} ãšãããš",
"title": "å"
},
{
"paragraph_id": 87,
"tag": "p",
"text": "ã ãã",
"title": "å"
},
{
"paragraph_id": 88,
"tag": "p",
"text": "æŽçããŠã",
"title": "å"
},
{
"paragraph_id": 89,
"tag": "p",
"text": "ãããã£ãŠãæ±ããè»è·¡ã¯ãçŽç· y = â x + 3 {\\displaystyle y=-x+3} ã§ããã",
"title": "å"
},
{
"paragraph_id": 90,
"tag": "p",
"text": "",
"title": "å"
},
{
"paragraph_id": 91,
"tag": "p",
"text": "",
"title": "å"
},
{
"paragraph_id": 92,
"tag": "p",
"text": "2ç¹ A ( 0 , 0 ) , B ( 3 , 0 ) {\\displaystyle \\mathrm {A} (0\\ ,\\ 0)\\ ,\\ \\mathrm {B} (3\\ ,\\ 0)} ããã®è·é¢ã®æ¯ã 2 : 1 {\\displaystyle 2:1} ã§ããç¹ P {\\displaystyle \\mathrm {P} } ã®è»è·¡ãæ±ããã",
"title": "å"
},
{
"paragraph_id": 93,
"tag": "p",
"text": "P {\\displaystyle \\mathrm {P} } ã®åº§æšã ( x , y ) {\\displaystyle (x\\ ,\\ y)} ãšããã P {\\displaystyle \\mathrm {P} } ãæºããæ¡ä»¶ã¯",
"title": "å"
},
{
"paragraph_id": 94,
"tag": "p",
"text": "ããªãã¡",
"title": "å"
},
{
"paragraph_id": 95,
"tag": "p",
"text": "ããã座æšã§è¡šããš",
"title": "å"
},
{
"paragraph_id": 96,
"tag": "p",
"text": "䞡蟺ã2ä¹ããŠãæŽçãããš",
"title": "å"
},
{
"paragraph_id": 97,
"tag": "p",
"text": "ããªãã¡",
"title": "å"
},
{
"paragraph_id": 98,
"tag": "p",
"text": "ãããã£ãŠãæ±ããè»è·¡ã¯ãäžå¿ã ( 4 , 0 ) {\\displaystyle (4\\ ,\\ 0)} ãååŸã 2 {\\displaystyle 2} ã®åã§ããã",
"title": "å"
},
{
"paragraph_id": 99,
"tag": "p",
"text": "m , n {\\displaystyle m\\ ,\\ n} ãç°ãªãæ£ã®æ°ãšãããšãã2ç¹ A , B {\\displaystyle \\mathrm {A} \\ ,\\ \\mathrm {B} } ããã®è·é¢ã®æ¯ã m : n {\\displaystyle m:n} ã§ããç¹ã®è»è·¡ã¯ãç·å A B {\\displaystyle \\mathrm {A} \\mathrm {B} } ã m : n {\\displaystyle m:n} ã«å
åããç¹ãšãå€åããç¹ãçŽåŸã®äž¡ç«¯ãšããåã§ããããã®åãã¢ããããŠã¹ã®åãšããã",
"title": "å"
},
{
"paragraph_id": 100,
"tag": "p",
"text": "m = n {\\displaystyle m=n} ã®ãšãã¯ãç·å A B {\\displaystyle \\mathrm {A} \\mathrm {B} } ã®åçŽäºçåç·ã§ããã",
"title": "å"
},
{
"paragraph_id": 101,
"tag": "p",
"text": "",
"title": "å"
},
{
"paragraph_id": 102,
"tag": "p",
"text": "ãã®ããŒãžã®åéã®ããã«ãæ°åŒãã€ãã£ãŠåº§æšã®äœçœ®ããããããŠã幟äœåŠã®åé¡ã解ãææ³ã®ããšã解æ幟äœåŠãšããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 103,
"tag": "p",
"text": "ãªãã幟äœåŠãšããèšèèªäœã¯ãå³åœ¢ã®åŠåãšãããããªæå³ã§ãããå°åŠæ ¡ãäžåŠæ ¡ã§ç¿ã£ãå³åœ¢ã®çè«ã幟äœåŠã§ããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 104,
"tag": "p",
"text": "äžäžãšãŒãããã®æ°åŠè
ãã«ã«ããã解æ幟äœåŠã®ç 究ãé²ããããªãããã«ã«ãã¯ãå²åŠã®æ Œèšãããæããããã«æãããã§ãæåã§ããã",
"title": "ã³ã©ã "
}
] | ããã§ã¯çŽç·ãšåãªã©ã®æ§è³ªã座æšãçšããŠèå¯ããã | {{pathnav|é«çåŠæ ¡ã®åŠç¿|é«çåŠæ ¡æ°åŠ|é«çåŠæ ¡æ°åŠII|pagename=å³åœ¢ãšæ¹çšåŒ|frame=1|small=1}}
ããã§ã¯çŽç·ãšåãªã©ã®æ§è³ªã座æšãçšããŠèå¯ããã
==ç¹ãšçŽç·==
===2ç¹éã®è·é¢===
[[ãã¡ã€ã«:Distance_Formula.svg|å³|200x200ãã¯ã»ã«]]
座æšå¹³é¢äžã®2ç¹ <math>\mathrm{A} \left(x _1\ ,\ y _1 \right)\ ,\ \mathrm{B} \left(x _2\ ,\ y _2 \right)</math> éã®è·é¢ <math>\mathrm{A} \mathrm{B}</math> ãæ±ããŠã¿ããã<br>çŽç· <math>\mathrm{A} \mathrm{B}</math> ã座æšè»žã«å¹³è¡ã§ãªããšã<ref>ã€ãŸããçŽç· <math>\mathrm{A} \mathrm{B}</math> ã <math>x</math> 軞ã <math>y</math> 軞 ã®ã©ã¡ããšãå¹³è¡ã§ãªããšã</ref>ãç¹ <math>\mathrm{C} \left(x _2\ ,\ y _1 \right)</math> ããšããš
:<math>
\mathrm{A} \mathrm{C} = |x _2 - x _1|\ ,\ \mathrm{B} \mathrm{C} = |y _2 - y _1|
</math>
<math>\triangle \mathrm{A} \mathrm{B} \mathrm{C}</math> ã¯çŽè§äžè§åœ¢ã§ãããããäžå¹³æ¹ã®å®çãã
:<math>
\mathrm{A} \mathrm{B} = \sqrt{\mathrm{A} \mathrm{C} ^2+ \mathrm{B} \mathrm{C} ^2} = \sqrt{|x _2 - x _1|^2+|y _2 - y _1|^2} = \sqrt{(x _2 - x _1)^2+(y _2 - y _1)^2}
</math>
ãã®åŒã¯ãçŽç· <math>\mathrm{A} \mathrm{B}</math> ãx軞ãy軞ã«å¹³è¡ãªãšãã«ãæãç«ã€<ref>çŽç· <math>\mathrm{A} \mathrm{B}</math> ã <math>x</math> 軞ã«å¹³è¡ãªãšã㯠<math>\mathrm{BC} = 0</math> ã§ããã <math>\mathrm{AC} = \mathrm{AB}</math> ãšãªãããã£ãŠ <math>\mathrm{AB} = \sqrt{\mathrm{AC}^2+\mathrm{BC}^2} </math> ã¯æãç«ã€ãçŽç· <math>\mathrm{A} \mathrm{B}</math> ã <math>y</math> 軞ã«å¹³è¡ãªãšããåæ§</ref>ã
ç¹ã«ãåç¹ <math>\mathrm{O}</math> ãšç¹ <math>\mathrm{A} \left(x _1\ ,\ y _1 \right)</math> éã®è·é¢ã¯
:<math>
\mathrm{O} \mathrm{A} = \sqrt{x _1^2 + y _1^2}
</math>
=== å
åç¹ãšå€åç¹===
ç¹ <math>
\mathrm{A}(x _0,y _0),\mathrm{B}(x _1,y _1)
</math> ãšå®æ° <math>m,n>0</math> ã«å¯ŸããŠãç·å <math>\mathrm{AB}</math> äžã®ç¹ <math>\mathrm{P}(x,y)</math> ãååšããŠã<math>\mathrm{AP}:\mathrm{PB} = m:n</math> ãšãªããšããç¹ <math>\mathrm{P}</math> ã <math>\mathrm{A},\mathrm{B}</math> ã <math>m:n</math> ã«å
åããç¹ãšããã
ãŸããç·å <math>\mathrm{AB}</math> äžã§ãªãç¹ <math>\mathrm{P}(x,y)</math> ãååšããŠã<math>\mathrm{AP}:\mathrm{PB} = m:n</math> ãšãªããšããç¹ <math>\mathrm{P}</math> ã <math>\mathrm{A},\mathrm{B}</math> ã <math>m:n</math> ã«å€åããç¹ãšããã
æ°çŽç·äžã®ç¹ <math>\mathrm{A}(a),\mathrm{B}(b)</math> ã <math>m:n</math> ã«å
åããç¹ãšå€åããç¹ãæ±ããã
å
åç¹ã <math>\mathrm{P}(x)</math> ãšããã<math>a<b</math> ã®ãšãã <math>\mathrm{AP} = x-a,\mathrm{PB}=b-x</math> ãªã®ã§ã <math>m:n=(x-a):(b-x)</math> ãªã®ã§ã <math>n(x-a)=m(b-x) \iff x = \frac{na+mb}{m+n}</math> ã§ããã <math>a>b</math> ã®ãšããåæ§ã
次ã«å€åç¹ãæ±ãããå€åç¹ã <math>\mathrm{P}(x)</math> ãšããã<math>a<b</math> 㧠<math>m>n</math> ã®ãšãã<math>x>b</math> ãšãªãã®ã§ã <math>\mathrm{AP}=x-a,\mathrm{BP}=x-b</math> ãªã®ã§ã<math>m:n=(x-a):(x-b)</math> ãªã®ã§ã<math>x=\frac{-na+mb}{m-n}</math>
ããã¯ã<math>a>b</math> ãŸã㯠<math>m<n</math> ã®ãšããåæ§ã<ref>å€åç¹ã®åº§æšã¯å
åç¹ã®åº§æšã® <math>n</math> ã <math>-n</math> ã«ãããã®ã«çãã</ref>
2次å
ã®å Žåã«ã¯ãäžè¬ã«ç¹ãšç¹ãšã®äœçœ®é¢ä¿ã¯ã座æšè»žã«å¹³è¡ã§ãªãããããã®è·é¢ã®å
åã¯è€éã«ãªãããã«æãããããããå®éã«ã¯ãå
åç¹ãå€åç¹ãèšç®ããã«ã¯ãäžã®å
Œ΋x,y
ã®äž¡æ¹åã«å¯ŸããŠçšããã°ãããããã¯ã2ç¹ãã€ãªãç·ãçŽç·ã§ããã®ã§ããã®çŽç·äžã§ããç¹ããã®è·é¢ãäžå®ã®å²åãšãªãç¹ãããã€ãåã£ããšãããã®ç¹ãšå
ã®ç¹ã®x軞æ¹åã®åº§æšã®å€åã®å²åãšy軞æ¹åã®åº§æšã®å€åã®å²åãšçŽç·èªèº«ã®é·ãã®å€åã®å²åã¯ããããçãããªãããã§ããã
ãã£ãŠãäžè¬ã«ç¹<math>A(x _0,y _0),B(x _1,y _1)</math>ããa:bã«å
åããç¹ãšå€åããç¹ã¯ã
:å
åç¹
:<math>
(\frac {b x _0 + a x _1} {a +b},
\frac {b y _0 + a y _1} {a +b})
</math>
:å€åç¹
:<math>
(\frac {-b x _0 + a x _1} {a -b},
\frac {-b y _0 + a y _1} {a -b})
</math>
:<math>
=
(
\frac {b x _0 - a x _1} {-a +b},
\frac {b y _0 - a y _1} {-a +b}
)
</math>
ã§äžããããã
'''æŒç¿åé¡'''
ç¹ <math>
\mathrm{A}(1,0),\mathrm{B}(-4,7)
</math> ã3:1ã«ããããå
åãå€åããç¹ãæ±ããã
'''解ç'''
å
åç¹ã¯ <math>
\left(\frac {-11}4,\frac{21}4\right)
</math>
å€åç¹ã¯ <math>
\left(\frac {-13}2,\frac{21}2\right)
</math>
===äžè§åœ¢ã®éå¿===
3ç¹<math>\mathrm{A} \left(x _1 , y _1 \right) , \mathrm{B} \left(x _2 , y _2 \right) , \mathrm{C} \left(x _3 , y _3 \right) </math>ãé ç¹ãšããäžè§åœ¢ã®éå¿ <math>\mathrm{G}</math> ã®åº§æšãæ±ããŠã¿ããã<br>
ç·å<math>\mathrm{B} \mathrm{C}</math>ã®äžç¹<math>\mathrm{M}</math>ã®åº§æšã¯
:<math>
\left(\frac {x _2 + x _3}{2} , \frac {y _2 + y _3}{2} \right)
</math>
éå¿<math>\mathrm{G}</math>ã¯ç·å<math>\mathrm{A} \mathrm{M}</math>ã2:1ã«å
åããç¹ã§ããããã<math>\mathrm{G}</math>ã®åº§æšã<math>(x , y)</math>ãšãããš
:<math>
x= \cfrac { 1 \times x _1 + 2 \times \cfrac { x _2 + x _3 } { 2 } } { 2+1 } = \frac { x _1 + x _2 + x _3 } { 3 }</math>
åæ§ã«
:<math>
y = \frac { y _1 + y _2 + y _3 } { 3 }
</math>
ãã£ãŠãéå¿<math>\mathrm{G}</math>ã®åº§æšã¯
:<math>
\left(\frac { x _1 + x _2 + x _3 } { 3 } , \frac { y _1 + y _2 + y _3 } { 3 } \right)
</math>
===çŽç·ã®æ¹çšåŒ===
ããç¹ <math>(x_0,y_0)</math> ãéã£ãŠåŸãaã®çŽç·ã®åŒã¯ã
<math>
y- y_0 = a(x- x_0)
</math>
ã§äžãããããããã¯ãåŸããyã®å€åå<math>/</math>xã®å€ååã§è¡šãããã<math> y-y_0 </math>,<math> x-x_0 </math>ã¯ãŸãã«ãy,xããããã®å€ååãã®ãã®ã§ããããšã«ããã
2ç¹ <math>(x_0,y_0)</math> , <math>(x_1,y_1)</math> ãéãçŽç·ã¯åŸãã <math>\frac{y_0-y_1}{x_0-x_1}</math> ã§äžããããããšãçšãããšã
<math>
y-y_0 = \frac{y_0-y_1}{x_0-x_1}(x-x_0)
</math>
ã§äžããããã
'''æŒç¿åé¡'''
ããããã®çŽç·ãè¡šããåŒãèšç®ããã
(i)
åŸã-2ã§ãç¹(-3,1)ãéãçŽç·
(ii)
2ç¹(4,3) ,(5,7)ãéãçŽç·
'''解ç'''
:<math>
y-y _0 = a(x-x _0)
</math>
:<math>
y-y _0 = \frac{y _0-y _1}{x _0-x _1}(x-x _0)
</math>
ãçšããã°ããã
(i)
:<math>
\left[ y=-2\,x-5 \right]
</math>
(ii)
:<math>
\left[ y=4\,x-13 \right]
</math>
ãŸãçŽç·ã®æ¹çšåŒã¯äžè¬ã« <math>ax+by+c=0</math> ã§è¡šãããã
====2çŽç·ã®å¹³è¡ãšåçŽ====
{| style="border:2px solid orange;width:80%" cellspacing=0
|style="background:orange"|'''2çŽç·ã®å¹³è¡ãåçŽ'''
|-
|style="padding:5px"|
2çŽç·<math>y=m_1 x+n_1\ ,\ y=m_2 x+n_2</math>ã«ã€ããŠ
<center>2çŽç·ãå¹³è¡<math>\Leftrightarrow m_1=m_2</math></center>
<center>2çŽç·ãåçŽ<math>\Leftrightarrow m_1 m_2=-1</math></center>
|}
*åé¡äŸ
**åé¡
ç¹<math>(1,4)</math>ãéããçŽç·<math>y=-2x+3</math>ã«å¹³è¡ãªçŽç·ãåçŽãªçŽç·ã®æ¹çšåŒãæ±ããã
**解ç
çŽç·<math>y=-2x+3</math>ã®åŸãã¯<math>-2</math>ã§ããã<br>
å¹³è¡ãªçŽç·ã®æ¹çšåŒã¯
:<math>y-4=-2(x-1)</math>
:<math>y=-2x+6</math>
åçŽãªçŽç·ã®åŸãã<math>m</math>ãšãããš
:<math>-2m=-1</math>
:<math>m= \frac{1}{2}</math>
ãã£ãŠãåçŽãªçŽç·ã®æ¹çšåŒã¯
:<math>y-4= \frac{1}{2} (x-1)</math>
:<math>y= \frac{1}{2} x+ \frac{7}{2}</math>
===ç¹ãšçŽç·ã®è·é¢===
ç¹ <math>\mathrm{P}</math> ãšçŽç· <math>l</math> ã«å¯ŸããçŽç· <math>l</math> äžã®ç¹ãšç¹ <math>\mathrm{P}</math> ã®è·é¢ã®æå°å€ã'''ç¹ãšçŽç·ã®è·é¢'''ãšãããããã¯ç¹ <math>\mathrm{P}</math> ããçŽç· <math>l</math> ã«äžãããåç· <math>\mathrm{PH}</math> ã®é·ãã«çããã
çŽç· <math>ax+by+c=0</math> ãšç¹ <math>(x_0,y_0)</math> ã®è·é¢ã¯
:<math>\frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}</math>
ãšè¡šãããã
'''蚌æ'''
[[ãã¡ã€ã«:Point-to-line2.svg|ãµã ãã€ã«]]ç¹ <math>\mathrm{P}(x_0,y_0)</math> ãšçŽç· <math>l:ax+by+c=0 \quad a,b\neq 0</math> ãšããã
ç¹ <math>\mathrm{P}</math> ããçŽç· <math>l</math> ã«åç·ãäžãããåç·ã®è¶³ãç¹ <math>R</math> ãšããã
ãŸããç¹ <math>\mathrm{P}</math> ãã <math>y</math> 軞ã«å¹³è¡ãªçŽç·ãåŒããçŽç· <math>l</math> ãšã®äº€ç¹ãç¹ <math>\mathrm S</math> ãšããã
次ã«ãå³ã®ããã«ãçŽç· <math>l</math> äžã®ç¹ <math>\mathrm T</math> ã«å¯ŸããŠãçŽç· <math>\mathrm{TV}</math> ã <math>x</math> 軞ãšå¹³è¡ãšãªãã<math>\mathrm{TV} = |b|</math> ãšãªãããã«ç¹ <math>\mathrm V</math> ããšããçŽç· <math>\mathrm{VU}</math> ã <math>y</math> 軞ã«å¹³è¡ã«ãªãç¹ <math>\mathrm U</math> ãçŽç· <math>l</math> äžã«åãã
çŽç· <math>l</math> ã®åŸã㯠<math>-\frac{a}{b}</math> ãšãªãã®ã§ <math>\mathrm{VU} = |a|</math> ã§ããã
ããã§ã<math>\bigtriangleup \mathrm{PRS},\bigtriangleup \mathrm{TVU}</math> ã¯çŽè§äžè§åœ¢ã§ããã<math>\angle \mathrm{PSR} = \angle \mathrm{TUV}</math><ref>çŽç· <math>\mathrm{PS}</math> ãšçŽç· <math>\mathrm{VU}</math> ã¯å¹³è¡ãªã®ã§</ref> ãªã®ã§ã<math>\bigtriangleup \mathrm{PRS} \sim \bigtriangleup \mathrm{TVU}</math><ref><math>\sim</math> ã¯çžäŒŒãæå³ãã</ref> ã§ããããããã£ãŠ
:<math>\frac{\mathrm{PR}}{\mathrm{PS}} = \frac{\mathrm{TV}}{\mathrm{TU}}</math>
ãŸãç¹ <math>\mathrm S</math> ã®åº§æšã<math>(x_0,m)</math> ãšãããšã<math>\mathrm{PS} = |y_0-m| </math> ã§ãç¹ <math>\mathrm{P}</math> ãšçŽç· <math>l</math> ã®è·é¢ <math> \mathrm{PR}</math> ã¯ã
<math> \mathrm{PR} ={\mathrm{PS}}\cdot \frac{\mathrm{TV}}{\mathrm{TU}} = \frac{|y_0 - m||b|}{\sqrt{a^2 + b^2}} </math>
ãšããã§ãç¹ <math>\mathrm S</math> ã¯çŽç· <math>l</math> äžã®ç¹ãªã®ã§ã
:<math>m = \frac{-ax_0 - c}{b}</math>
ã§ãããããã代å
¥ããã°
:<math> \mathrm{PR} = \frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}}</math>
:ãåŸãã
:
'''ãã¯ãã«ã䜿ã£ã蚌æ'''
ãã§ã«ãã¯ãã«ãç¥ã£ãŠãããªãã°ãã¡ãã®æ¹ãç°¡æœã§ããã
ç¹ <math>\mathrm{P}(x_0,y_0)</math> ãšçŽç· <math>l:ax+by+c=0</math> ãšããç¹ <math>\mathrm{Q}(x_1,y_1)</math> ãçŽç· <math>l</math> äžã®ç¹ãšãããçŽç· <math>l</math> ã®æ³ç·ã¯ <math>\vec n := (a,b)</math> ã§ã<math>\vec{\mathrm{QP}} = (x_0-x_1,y_0-y_1) </math> ã§ããã®ã§ãçŽç· <math>l</math> äžã®ç¹ãšç¹ <math>\mathrm{P}</math> ã®è·é¢ <math>d</math> 㯠<math>d = \left| \vec{ \mathrm{QP} } \cdot \frac{\vec n}{||\vec n||}\right| = \left|(x_0-x_1,y_0-y_1)\cdot \frac{(a,b)}{\sqrt{a^2+b^2}}\right| = \frac{|ax_0 + by_0 - (ax_1 + by_1)|}{\sqrt{a^2+b^2}} = \frac{|ax_0 + by_0 +c|}{\sqrt{a^2+b^2}}</math><ref>ç¹ <math>\mathrm{Q}(x_1,y_1)</math> ã¯çŽç· <math>l</math> äžã®ç¹ãªã®ã§ <math>ax_1+by_1=-c</math> ã§ããã</ref> ã§ããã
'''æŒç¿åé¡'''
çŽç· <math>x-2y-3=0</math> ãšç¹ <math>(1,2)</math> ã®è·é¢ãæ±ãã
'''解ç'''
<math>\frac{6}{\sqrt 5}</math>
==å==
====åã®æ¹çšåŒ====
äžå¿ <math>\mathrm{C}(a,b)</math> ååŸ <math>r</math> ã®åã¯ã<math>\mathrm{CP} =r</math> ãšãªãç¹ <math>\mathrm{P}</math> ã®éåã§ãããã€ãŸãã <math>r = \sqrt{(x-a)^2+(y-b)^2}</math> ãšãªãç¹ <math>(x,y)</math> ã®éåã§ããããã®æ¹çšåŒã®äž¡èŸºã¯æ£ãªã®ã§2ä¹ããŠ
<math>
(x-a)^2+(y-b)^2 = r^2
</math>
ãåŸãããããåã®æ¹çšåŒã§ããã
ç¹ã«åç¹ãäžå¿ã§ååŸ <math>r</math> ã®åã®æ¹çšåŒã¯ <math>
x^2+y^2 = r^2
</math> ã§äžããããã
'''æŒç¿åé¡'''
# äžå¿ <math>(2,4)</math> ååŸ <math>3</math> ã®åã®æ¹çšåŒãæ±ãã
# å <math>
y^2+2\,y+x^2-6\,x+5=0
</math> ã®äžå¿ãšååŸãæ±ãã
'''解ç'''
# <math>
(x-2)^2+(y-4)^2 = 9
</math>
# <math>
y^2+2\,y+x^2-6\,x+5=0 \iff (x-3)^2 + (y +1)^2 = 5
</math> ãªã®ã§äžå¿ <math>
(3,-1)
</math> ååŸ <math>
\sqrt 5
</math>
æ¹çšåŒ <math>x^2+y^2+lx+my+n = 0</math> ã¯ãã€ãåã§ãããšã¯éããªãã
æ¹çšåŒãå€åœ¢ã㊠<math>(x-a)^2+(y-b)^2 = k</math> ãšãªããšã
# <math>k>0</math> ã®ãšãæ¹çšåŒã¯åãè¡šã
# <math>k=0</math> ã®ãšãæ¹çšåŒã¯1ç¹ <math>(a,b)</math> ãè¡šã
# <math>k<0</math> ã®ãšãæ¹çšåŒã®å·ŠèŸºã¯åžžã«æ£ãªã®ã§ãæ¹çšåŒã®è¡šãå³åœ¢ã¯ãªã
==== åã®æ¥ç· ====
å<math>x^2+y^2=r^2</math>äžã®ããç¹<math>(x_1,y_1)</math>ã§æ¥ããæ¥ç·ã®æ¹çšåŒã¯
:<math>x_1x+y_1y=r^2</math>
ã§è¡šãããã
åæ§ã«ãå<math>(x-a)^2+(y-b)^2=r^2</math>äžã®ããç¹<math>(x_2,y_2)</math>ã§æ¥ããæ¥ç·ã®æ¹çšåŒã¯
:<math>(x_2-a)(x-a)+(y_2-b)(y-b)=r^2</math>
ã§è¡šãããã
====åãšçŽç·====
åãšçŽç·ã®äœçœ®é¢ä¿ã«ã€ããŠå€§ãã次ã®3ã€ã«åé¡ããããšãã§ããã
# åãšçŽç·ã2ç¹ã§äº€ãã(çŽç·ãåã®å
éšãéã)
# åãšçŽç·ã1ç¹ã§äº€ãã(çŽç·ãåã®æ¥ç·ãšãªã)
# åãšçŽç·ã¯äº€ãããªã
<!-- ããããã®äœçœ®é¢ä¿ã®å³ -->
äžè¬ã®åãšçŽç·ã«ã€ããŠãããã®äœçœ®é¢ä¿ãåé¡ããŠã¿ããã
å <math>C:(x-p)^2+(y-q)^2 = r^2</math> ãšçŽç· <math>l:ax+by+c=0</math> ã«ã€ããŠãå <math>C</math> ã®äžå¿ <math>(p,q)</math> ãšçŽç· <math>l</math> ã®è·é¢ <math>d := \frac{|aq+bq+c|}{\sqrt{a^2+b^2}}</math> ãšãããšã
# <math>r>d</math> ã®ãšããå <math>C</math> ãšçŽç· <math>l</math> ã¯2ç¹ã§äº€ãã
# <math>r=d</math> ã®ãšããå <math>C</math> ãšçŽç· <math>l</math> ã¯1ç¹ã§äº€ãã
# <math>r<d</math> ã®ãšããå <math>C</math> ãšçŽç· <math>l</math> ã¯äº€ãããªã
ä»ã«ããåã®æ¹çšåŒãšçŽç·ã®æ¹çšåŒãé£ç«ããŠãã®å®æ°è§£ã®åæ°ã§åé¡ããæ¹æ³ãããããäœçœ®é¢ä¿ãæ±ããã ããªãäžã®æ¹æ³ã®ã»ããèšç®éãå°ãªãã
'''æŒç¿åé¡'''
çŽç· <math>
3x + 4y =1
</math> ãšå <math>
(x-3)^2 + (y+2)^2 = 14
</math> ã®äº€ç¹ã®åº§æšãæ±ãã
'''解ç'''
çŽç·ã®æ¹çšåŒã <math>x</math> ã«ã€ããŠè§£ãããããåã®æ¹çšåŒã«ä»£å
¥ããã°ããã
çã㯠<math>(2,-1),\left(-\frac{14}{5},\frac{7}{5}\right)</math>
===è»è·¡ãšé å===
====è»è·¡ãšæ¹çšåŒ====
ããæ¡ä»¶ãæºããç¹å
šäœãã€ããå³åœ¢ãããã®æ¡ä»¶ãæºããç¹ã®'''è»è·¡'''ãšããã
*åé¡äŸ
**åé¡
2ç¹<math>\mathrm{A}(1\ ,\ 0)\ ,\ \mathrm{B}(3\ ,\ 2)</math>ããçè·é¢ã«ããç¹<math>\mathrm{P}</math>ã®è»è·¡ãæ±ããã
**解ç
æ¡ä»¶<math>\mathrm{A} \mathrm{P} = \mathrm{B} \mathrm{P}</math>ããã<math>\mathrm{A} \mathrm{P} ^2 = \mathrm{B} \mathrm{P} ^2</math><br>
<math>\mathrm{P}</math>ã®åº§æšã<math>(x\ ,\ y)</math>ãšãããš
:<math>
\mathrm{A} \mathrm{P} ^2 =(x-1)^2+y^2
</math>
:<math>
\mathrm{B} \mathrm{P} ^2 =(x-3)^2+(y-2)^2
</math>
ã ãã
:<math>
(x-1)^2+y^2=(x-3)^2+(y-2)^2
</math>
æŽçããŠã
:<math>
y=-x+3
</math>
ãããã£ãŠãæ±ããè»è·¡ã¯ãçŽç·<math>y=-x+3</math>ã§ããã
{| style="border:2px solid orange;width:80%" cellspacing=0
|style="background:orange"|'''è»è·¡ãæ±ããæé '''
|-
|style="padding:5px"|
1.æ±ããè»è·¡äžã®ä»»æã®ç¹ã®åº§æšã<math>(x\ ,\ y)</math>ãªã©ã§è¡šããäžããããæ¡ä»¶ã座æšã®éã®é¢ä¿åŒã§è¡šãã
2.è»è·¡ã®æ¹çšåŒãå°ãããã®æ¹çšåŒã®è¡šãå³åœ¢ãæ±ããã
3.ãã®å³åœ¢äžã®ç¹ãæ¡ä»¶ãæºãããŠããããšã確ãããã
|}
*åé¡äŸ
**åé¡
2ç¹<math>\mathrm{A}(0\ ,\ 0)\ ,\ \mathrm{B}(3\ ,\ 0)</math>ããã®è·é¢ã®æ¯ã<math>2:1</math>ã§ããç¹<math>\mathrm{P}</math>ã®è»è·¡ãæ±ããã
**解ç
<math>\mathrm{P}</math>ã®åº§æšã<math>(x\ ,\ y)</math>ãšããã<br>
<math>\mathrm{P}</math>ãæºããæ¡ä»¶ã¯
:<math>
\mathrm{A} \mathrm{P} : \mathrm{B} \mathrm{P} =2:1
</math>
ããªãã¡
:<math>
\mathrm{A} \mathrm{P} =2 \mathrm{B} \mathrm{P}
</math>
ããã座æšã§è¡šããš
:<math>
\sqrt{x^2+y^2} =2 \sqrt{(x-3)^2+y^2}
</math>
䞡蟺ã2ä¹ããŠãæŽçãããš
:<math>
x^2+y^2-8x+12=0
</math>
ããªãã¡
:<math>
(x-4)^2+y^2=2^2
</math>
ãããã£ãŠãæ±ããè»è·¡ã¯ãäžå¿ã<math>(4\ ,\ 0)</math>ãååŸã<math>2</math>ã®åã§ããã
<math>m\ ,\ n</math>ãç°ãªãæ£ã®æ°ãšãããšãã2ç¹<math>\mathrm{A}\ ,\ \mathrm{B}</math>ããã®è·é¢ã®æ¯ã<math>m:n</math>ã§ããç¹ã®è»è·¡ã¯ãç·å<math>\mathrm{A} \mathrm{B}</math>ã<math>m:n</math>ã«å
åããç¹ãšãå€åããç¹ãçŽåŸã®äž¡ç«¯ãšããåã§ããããã®åã'''ã¢ããããŠã¹ã®å'''ãšããã
<math>m=n</math>ã®ãšãã¯ãç·å<math>\mathrm{A} \mathrm{B}</math>ã®åçŽäºçåç·ã§ããã
== ã³ã©ã ==
[[File:Frans Hals - Portret van René Descartes.jpg|thumb|ãã«ã«ã]]
ãã®ããŒãžã®åéã®ããã«ãæ°åŒãã€ãã£ãŠåº§æšã®äœçœ®ããããããŠã幟äœåŠã®åé¡ã解ãææ³ã®ããšã解æ幟äœåŠãšããã
ãªãã幟äœåŠãšããèšèèªäœã¯ãå³åœ¢ã®åŠåãšãããããªæå³ã§ãããå°åŠæ ¡ãäžåŠæ ¡ã§ç¿ã£ãå³åœ¢ã®çè«ã幟äœåŠã§ããã
äžäžãšãŒãããã®æ°åŠè
ãã«ã«ããã解æ幟äœåŠã®ç 究ãé²ããããªãããã«ã«ãã¯ãå²åŠã®æ Œèšãããæããããã«æãããã§ãæåã§ããã
== æŒç¿åé¡ ==
== è泚 ==
<references/>
{{DEFAULTSORT:ãããšããã€ããããããII ããããšã»ããŠããã}}
[[Category:é«çåŠæ ¡æ°åŠII|ããããšã»ããŠããã]]
[[ã«ããŽãª:å³åœ¢]] | 2005-05-04T09:25:38Z | 2023-08-31T10:18:10Z | [
"ãã³ãã¬ãŒã:Pathnav"
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E6%95%B0%E5%AD%A6II/%E5%9B%B3%E5%BD%A2%E3%81%A8%E6%96%B9%E7%A8%8B%E5%BC%8F |
1,903 | é«çåŠæ ¡æ°åŠII/å³åœ¢ãšæ¹çšåŒ | ããã§ã¯çŽç·ãšåãªã©ã®æ§è³ªã座æšãçšããŠèå¯ããã
座æšå¹³é¢äžã®2ç¹ A ( x 1 , y 1 ) , B ( x 2 , y 2 ) {\displaystyle \mathrm {A} \left(x_{1}\ ,\ y_{1}\right)\ ,\ \mathrm {B} \left(x_{2}\ ,\ y_{2}\right)} éã®è·é¢ A B {\displaystyle \mathrm {A} \mathrm {B} } ãæ±ããŠã¿ãããçŽç· A B {\displaystyle \mathrm {A} \mathrm {B} } ã座æšè»žã«å¹³è¡ã§ãªããšããç¹ C ( x 2 , y 1 ) {\displaystyle \mathrm {C} \left(x_{2}\ ,\ y_{1}\right)} ããšããš
â³ A B C {\displaystyle \triangle \mathrm {A} \mathrm {B} \mathrm {C} } ã¯çŽè§äžè§åœ¢ã§ãããããäžå¹³æ¹ã®å®çãã
ãã®åŒã¯ãçŽç· A B {\displaystyle \mathrm {A} \mathrm {B} } ãx軞ãy軞ã«å¹³è¡ãªãšãã«ãæãç«ã€ã
ç¹ã«ãåç¹ O {\displaystyle \mathrm {O} } ãšç¹ A ( x 1 , y 1 ) {\displaystyle \mathrm {A} \left(x_{1}\ ,\ y_{1}\right)} éã®è·é¢ã¯
ç¹ A ( x 0 , y 0 ) , B ( x 1 , y 1 ) {\displaystyle \mathrm {A} (x_{0},y_{0}),\mathrm {B} (x_{1},y_{1})} ãšå®æ° m , n > 0 {\displaystyle m,n>0} ã«å¯ŸããŠãç·å A B {\displaystyle \mathrm {AB} } äžã®ç¹ P ( x , y ) {\displaystyle \mathrm {P} (x,y)} ãååšããŠã A P : P B = m : n {\displaystyle \mathrm {AP} :\mathrm {PB} =m:n} ãšãªããšããç¹ P {\displaystyle \mathrm {P} } ã A , B {\displaystyle \mathrm {A} ,\mathrm {B} } ã m : n {\displaystyle m:n} ã«å
åããç¹ãšããã
ãŸããç·å A B {\displaystyle \mathrm {AB} } äžã§ãªãç¹ P ( x , y ) {\displaystyle \mathrm {P} (x,y)} ãååšããŠã A P : P B = m : n {\displaystyle \mathrm {AP} :\mathrm {PB} =m:n} ãšãªããšããç¹ P {\displaystyle \mathrm {P} } ã A , B {\displaystyle \mathrm {A} ,\mathrm {B} } ã m : n {\displaystyle m:n} ã«å€åããç¹ãšããã
æ°çŽç·äžã®ç¹ A ( a ) , B ( b ) {\displaystyle \mathrm {A} (a),\mathrm {B} (b)} ã m : n {\displaystyle m:n} ã«å
åããç¹ãšå€åããç¹ãæ±ããã
å
åç¹ã P ( x ) {\displaystyle \mathrm {P} (x)} ãšããã a < b {\displaystyle a<b} ã®ãšãã A P = x â a , P B = b â x {\displaystyle \mathrm {AP} =x-a,\mathrm {PB} =b-x} ãªã®ã§ã m : n = ( x â a ) : ( b â x ) {\displaystyle m:n=(x-a):(b-x)} ãªã®ã§ã n ( x â a ) = m ( b â x ) ⺠x = n a + m b m + n {\displaystyle n(x-a)=m(b-x)\iff x={\frac {na+mb}{m+n}}} ã§ããã a > b {\displaystyle a>b} ã®ãšããåæ§ã
次ã«å€åç¹ãæ±ãããå€åç¹ã P ( x ) {\displaystyle \mathrm {P} (x)} ãšããã a < b {\displaystyle a<b} 㧠m > n {\displaystyle m>n} ã®ãšãã x > b {\displaystyle x>b} ãšãªãã®ã§ã A P = x â a , B P = x â b {\displaystyle \mathrm {AP} =x-a,\mathrm {BP} =x-b} ãªã®ã§ã m : n = ( x â a ) : ( x â b ) {\displaystyle m:n=(x-a):(x-b)} ãªã®ã§ã x = â n a + m b m â n {\displaystyle x={\frac {-na+mb}{m-n}}}
ããã¯ã a > b {\displaystyle a>b} ãŸã㯠m < n {\displaystyle m<n} ã®ãšããåæ§ã
2次å
ã®å Žåã«ã¯ãäžè¬ã«ç¹ãšç¹ãšã®äœçœ®é¢ä¿ã¯ã座æšè»žã«å¹³è¡ã§ãªãããããã®è·é¢ã®å
åã¯è€éã«ãªãããã«æãããããããå®éã«ã¯ãå
åç¹ãå€åç¹ãèšç®ããã«ã¯ãäžã®å
¬åŒãx,y ã®äž¡æ¹åã«å¯ŸããŠçšããã°ãããããã¯ã2ç¹ãã€ãªãç·ãçŽç·ã§ããã®ã§ããã®çŽç·äžã§ããç¹ããã®è·é¢ãäžå®ã®å²åãšãªãç¹ãããã€ãåã£ããšãããã®ç¹ãšå
ã®ç¹ã®x軞æ¹åã®åº§æšã®å€åã®å²åãšy軞æ¹åã®åº§æšã®å€åã®å²åãšçŽç·èªèº«ã®é·ãã®å€åã®å²åã¯ããããçãããªãããã§ããã
ãã£ãŠãäžè¬ã«ç¹ A ( x 0 , y 0 ) , B ( x 1 , y 1 ) {\displaystyle A(x_{0},y_{0}),B(x_{1},y_{1})} ããa:bã«å
åããç¹ãšå€åããç¹ã¯ã
ã§äžããããã
æŒç¿åé¡
ç¹ A ( 1 , 0 ) , B ( â 4 , 7 ) {\displaystyle \mathrm {A} (1,0),\mathrm {B} (-4,7)} ã3:1ã«ããããå
åãå€åããç¹ãæ±ããã
解ç
å
åç¹ã¯ ( â 11 4 , 21 4 ) {\displaystyle \left({\frac {-11}{4}},{\frac {21}{4}}\right)}
å€åç¹ã¯ ( â 13 2 , 21 2 ) {\displaystyle \left({\frac {-13}{2}},{\frac {21}{2}}\right)}
3ç¹ A ( x 1 , y 1 ) , B ( x 2 , y 2 ) , C ( x 3 , y 3 ) {\displaystyle \mathrm {A} \left(x_{1},y_{1}\right),\mathrm {B} \left(x_{2},y_{2}\right),\mathrm {C} \left(x_{3},y_{3}\right)} ãé ç¹ãšããäžè§åœ¢ã®éå¿ G {\displaystyle \mathrm {G} } ã®åº§æšãæ±ããŠã¿ããã ç·å B C {\displaystyle \mathrm {B} \mathrm {C} } ã®äžç¹ M {\displaystyle \mathrm {M} } ã®åº§æšã¯
éå¿ G {\displaystyle \mathrm {G} } ã¯ç·å A M {\displaystyle \mathrm {A} \mathrm {M} } ã2:1ã«å
åããç¹ã§ããããã G {\displaystyle \mathrm {G} } ã®åº§æšã ( x , y ) {\displaystyle (x,y)} ãšãããš
åæ§ã«
ãã£ãŠãéå¿ G {\displaystyle \mathrm {G} } ã®åº§æšã¯
ããç¹ ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} ãéã£ãŠåŸãaã®çŽç·ã®åŒã¯ã y â y 0 = a ( x â x 0 ) {\displaystyle y-y_{0}=a(x-x_{0})} ã§äžãããããããã¯ãåŸããyã®å€åå / {\displaystyle /} xã®å€ååã§è¡šãããã y â y 0 {\displaystyle y-y_{0}} , x â x 0 {\displaystyle x-x_{0}} ã¯ãŸãã«ãy,xããããã®å€ååãã®ãã®ã§ããããšã«ããã
2ç¹ ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} , ( x 1 , y 1 ) {\displaystyle (x_{1},y_{1})} ãéãçŽç·ã¯åŸãã y 0 â y 1 x 0 â x 1 {\displaystyle {\frac {y_{0}-y_{1}}{x_{0}-x_{1}}}} ã§äžããããããšãçšãããšã y â y 0 = y 0 â y 1 x 0 â x 1 ( x â x 0 ) {\displaystyle y-y_{0}={\frac {y_{0}-y_{1}}{x_{0}-x_{1}}}(x-x_{0})} ã§äžããããã
æŒç¿åé¡
ããããã®çŽç·ãè¡šããåŒãèšç®ããã
(i) åŸã-2ã§ãç¹(-3,1)ãéãçŽç·
(ii) 2ç¹(4,3) ,(5,7)ãéãçŽç·
解ç
ãçšããã°ããã
(i)
(ii)
ãŸãçŽç·ã®æ¹çšåŒã¯äžè¬ã« a x + b y + c = 0 {\displaystyle ax+by+c=0} ã§è¡šãããã
ç¹ ( 1 , 4 ) {\displaystyle (1,4)} ãéããçŽç· y = â 2 x + 3 {\displaystyle y=-2x+3} ã«å¹³è¡ãªçŽç·ãåçŽãªçŽç·ã®æ¹çšåŒãæ±ããã
çŽç· y = â 2 x + 3 {\displaystyle y=-2x+3} ã®åŸã㯠â 2 {\displaystyle -2} ã§ããã å¹³è¡ãªçŽç·ã®æ¹çšåŒã¯
åçŽãªçŽç·ã®åŸãã m {\displaystyle m} ãšãããš
ãã£ãŠãåçŽãªçŽç·ã®æ¹çšåŒã¯
ç¹ P {\displaystyle \mathrm {P} } ãšçŽç· l {\displaystyle l} ã«å¯ŸããçŽç· l {\displaystyle l} äžã®ç¹ãšç¹ P {\displaystyle \mathrm {P} } ã®è·é¢ã®æå°å€ãç¹ãšçŽç·ã®è·é¢ãšãããããã¯ç¹ P {\displaystyle \mathrm {P} } ããçŽç· l {\displaystyle l} ã«äžãããåç· P H {\displaystyle \mathrm {PH} } ã®é·ãã«çããã
çŽç· a x + b y + c = 0 {\displaystyle ax+by+c=0} ãšç¹ ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} ã®è·é¢ã¯
ãšè¡šãããã
蚌æ
ç¹ P ( x 0 , y 0 ) {\displaystyle \mathrm {P} (x_{0},y_{0})} ãšçŽç· l : a x + b y + c = 0 a , b â 0 {\displaystyle l:ax+by+c=0\quad a,b\neq 0} ãšããã
ç¹ P {\displaystyle \mathrm {P} } ããçŽç· l {\displaystyle l} ã«åç·ãäžãããåç·ã®è¶³ãç¹ R {\displaystyle R} ãšããã
ãŸããç¹ P {\displaystyle \mathrm {P} } ãã y {\displaystyle y} 軞ã«å¹³è¡ãªçŽç·ãåŒããçŽç· l {\displaystyle l} ãšã®äº€ç¹ãç¹ S {\displaystyle \mathrm {S} } ãšããã
次ã«ãå³ã®ããã«ãçŽç· l {\displaystyle l} äžã®ç¹ T {\displaystyle \mathrm {T} } ã«å¯ŸããŠãçŽç· T V {\displaystyle \mathrm {TV} } ã x {\displaystyle x} 軞ãšå¹³è¡ãšãªãã T V = | b | {\displaystyle \mathrm {TV} =|b|} ãšãªãããã«ç¹ V {\displaystyle \mathrm {V} } ããšããçŽç· V U {\displaystyle \mathrm {VU} } ã y {\displaystyle y} 軞ã«å¹³è¡ã«ãªãç¹ U {\displaystyle \mathrm {U} } ãçŽç· l {\displaystyle l} äžã«åãã
çŽç· l {\displaystyle l} ã®åŸã㯠â a b {\displaystyle -{\frac {a}{b}}} ãšãªãã®ã§ V U = | a | {\displaystyle \mathrm {VU} =|a|} ã§ããã ããã§ã â³ P R S , â³ T V U {\displaystyle \bigtriangleup \mathrm {PRS} ,\bigtriangleup \mathrm {TVU} } ã¯çŽè§äžè§åœ¢ã§ããã â P S R = â T U V {\displaystyle \angle \mathrm {PSR} =\angle \mathrm {TUV} } ãªã®ã§ã â³ P R S âŒâ³ T V U {\displaystyle \bigtriangleup \mathrm {PRS} \sim \bigtriangleup \mathrm {TVU} } ã§ããããããã£ãŠ
ãŸãç¹ S {\displaystyle \mathrm {S} } ã®åº§æšã ( x 0 , m ) {\displaystyle (x_{0},m)} ãšãããšã P S = | y 0 â m | {\displaystyle \mathrm {PS} =|y_{0}-m|} ã§ãç¹ P {\displaystyle \mathrm {P} } ãšçŽç· l {\displaystyle l} ã®è·é¢ P R {\displaystyle \mathrm {PR} } ã¯ã
P R = P S â
T V T U = | y 0 â m | | b | a 2 + b 2 {\displaystyle \mathrm {PR} ={\mathrm {PS} }\cdot {\frac {\mathrm {TV} }{\mathrm {TU} }}={\frac {|y_{0}-m||b|}{\sqrt {a^{2}+b^{2}}}}}
ãšããã§ãç¹ S {\displaystyle \mathrm {S} } ã¯çŽç· l {\displaystyle l} äžã®ç¹ãªã®ã§ã
ã§ãããããã代å
¥ããã°
ãã¯ãã«ã䜿ã£ã蚌æ
ãã§ã«ãã¯ãã«ãç¥ã£ãŠãããªãã°ãã¡ãã®æ¹ãç°¡æœã§ããã
ç¹ P ( x 0 , y 0 ) {\displaystyle \mathrm {P} (x_{0},y_{0})} ãšçŽç· l : a x + b y + c = 0 {\displaystyle l:ax+by+c=0} ãšããç¹ Q ( x 1 , y 1 ) {\displaystyle \mathrm {Q} (x_{1},y_{1})} ãçŽç· l {\displaystyle l} äžã®ç¹ãšãããçŽç· l {\displaystyle l} ã®æ³ç·ã¯ n â := ( a , b ) {\displaystyle {\vec {n}}:=(a,b)} ã§ã Q P â = ( x 0 â x 1 , y 0 â y 1 ) {\displaystyle {\vec {\mathrm {QP} }}=(x_{0}-x_{1},y_{0}-y_{1})} ã§ããã®ã§ãçŽç· l {\displaystyle l} äžã®ç¹ãšç¹ P {\displaystyle \mathrm {P} } ã®è·é¢ d {\displaystyle d} 㯠d = | Q P â â
n â | | n â | | | = | ( x 0 â x 1 , y 0 â y 1 ) â
( a , b ) a 2 + b 2 | = | a x 0 + b y 0 â ( a x 1 + b y 1 ) | a 2 + b 2 = | a x 0 + b y 0 + c | a 2 + b 2 {\displaystyle d=\left|{\vec {\mathrm {QP} }}\cdot {\frac {\vec {n}}{||{\vec {n}}||}}\right|=\left|(x_{0}-x_{1},y_{0}-y_{1})\cdot {\frac {(a,b)}{\sqrt {a^{2}+b^{2}}}}\right|={\frac {|ax_{0}+by_{0}-(ax_{1}+by_{1})|}{\sqrt {a^{2}+b^{2}}}}={\frac {|ax_{0}+by_{0}+c|}{\sqrt {a^{2}+b^{2}}}}} ã§ããã
æŒç¿åé¡
çŽç· x â 2 y â 3 = 0 {\displaystyle x-2y-3=0} ãšç¹ ( 1 , 2 ) {\displaystyle (1,2)} ã®è·é¢ãæ±ãã
解ç
6 5 {\displaystyle {\frac {6}{\sqrt {5}}}}
äžå¿ C ( a , b ) {\displaystyle \mathrm {C} (a,b)} ååŸ r {\displaystyle r} ã®åã¯ã C P = r {\displaystyle \mathrm {CP} =r} ãšãªãç¹ P {\displaystyle \mathrm {P} } ã®éåã§ãããã€ãŸãã r = ( x â a ) 2 + ( y â b ) 2 {\displaystyle r={\sqrt {(x-a)^{2}+(y-b)^{2}}}} ãšãªãç¹ ( x , y ) {\displaystyle (x,y)} ã®éåã§ããããã®æ¹çšåŒã®äž¡èŸºã¯æ£ãªã®ã§2ä¹ããŠ
( x â a ) 2 + ( y â b ) 2 = r 2 {\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}}
ãåŸãããããåã®æ¹çšåŒã§ããã
ç¹ã«åç¹ãäžå¿ã§ååŸ r {\displaystyle r} ã®åã®æ¹çšåŒã¯ x 2 + y 2 = r 2 {\displaystyle x^{2}+y^{2}=r^{2}} ã§äžããããã
æŒç¿åé¡
解ç
æ¹çšåŒ x 2 + y 2 + l x + m y + n = 0 {\displaystyle x^{2}+y^{2}+lx+my+n=0} ã¯ãã€ãåã§ãããšã¯éããªãã
æ¹çšåŒãå€åœ¢ã㊠( x â a ) 2 + ( y â b ) 2 = k {\displaystyle (x-a)^{2}+(y-b)^{2}=k} ãšãªããšã
å x 2 + y 2 = r 2 {\displaystyle x^{2}+y^{2}=r^{2}} äžã®ããç¹ ( x 1 , y 1 ) {\displaystyle (x_{1},y_{1})} ã§æ¥ããæ¥ç·ã®æ¹çšåŒã¯
ã§è¡šãããã
åæ§ã«ãå ( x â a ) 2 + ( y â b ) 2 = r 2 {\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}} äžã®ããç¹ ( x 2 , y 2 ) {\displaystyle (x_{2},y_{2})} ã§æ¥ããæ¥ç·ã®æ¹çšåŒã¯
ã§è¡šãããã
åãšçŽç·ã®äœçœ®é¢ä¿ã«ã€ããŠå€§ãã次ã®3ã€ã«åé¡ããããšãã§ããã
äžè¬ã®åãšçŽç·ã«ã€ããŠãããã®äœçœ®é¢ä¿ãåé¡ããŠã¿ããã
å C : ( x â p ) 2 + ( y â q ) 2 = r 2 {\displaystyle C:(x-p)^{2}+(y-q)^{2}=r^{2}} ãšçŽç· l : a x + b y + c = 0 {\displaystyle l:ax+by+c=0} ã«ã€ããŠãå C {\displaystyle C} ã®äžå¿ ( p , q ) {\displaystyle (p,q)} ãšçŽç· l {\displaystyle l} ã®è·é¢ d := | a q + b q + c | a 2 + b 2 {\displaystyle d:={\frac {|aq+bq+c|}{\sqrt {a^{2}+b^{2}}}}} ãšãããšã
ä»ã«ããåã®æ¹çšåŒãšçŽç·ã®æ¹çšåŒãé£ç«ããŠãã®å®æ°è§£ã®åæ°ã§åé¡ããæ¹æ³ãããããäœçœ®é¢ä¿ãæ±ããã ããªãäžã®æ¹æ³ã®ã»ããèšç®éãå°ãªãã
æŒç¿åé¡
çŽç· 3 x + 4 y = 1 {\displaystyle 3x+4y=1} ãšå ( x â 3 ) 2 + ( y + 2 ) 2 = 14 {\displaystyle (x-3)^{2}+(y+2)^{2}=14} ã®äº€ç¹ã®åº§æšãæ±ãã
解ç
çŽç·ã®æ¹çšåŒã x {\displaystyle x} ã«ã€ããŠè§£ãããããåã®æ¹çšåŒã«ä»£å
¥ããã°ããã
çã㯠( 2 , â 1 ) , ( â 14 5 , 7 5 ) {\displaystyle (2,-1),\left(-{\frac {14}{5}},{\frac {7}{5}}\right)}
ããæ¡ä»¶ãæºããç¹å
šäœãã€ããå³åœ¢ãããã®æ¡ä»¶ãæºããç¹ã®è»è·¡ãšããã
2ç¹ A ( 1 , 0 ) , B ( 3 , 2 ) {\displaystyle \mathrm {A} (1\ ,\ 0)\ ,\ \mathrm {B} (3\ ,\ 2)} ããçè·é¢ã«ããç¹ P {\displaystyle \mathrm {P} } ã®è»è·¡ãæ±ããã
æ¡ä»¶ A P = B P {\displaystyle \mathrm {A} \mathrm {P} =\mathrm {B} \mathrm {P} } ããã A P 2 = B P 2 {\displaystyle \mathrm {A} \mathrm {P} ^{2}=\mathrm {B} \mathrm {P} ^{2}} P {\displaystyle \mathrm {P} } ã®åº§æšã ( x , y ) {\displaystyle (x\ ,\ y)} ãšãããš
ã ãã
æŽçããŠã
ãããã£ãŠãæ±ããè»è·¡ã¯ãçŽç· y = â x + 3 {\displaystyle y=-x+3} ã§ããã
2ç¹ A ( 0 , 0 ) , B ( 3 , 0 ) {\displaystyle \mathrm {A} (0\ ,\ 0)\ ,\ \mathrm {B} (3\ ,\ 0)} ããã®è·é¢ã®æ¯ã 2 : 1 {\displaystyle 2:1} ã§ããç¹ P {\displaystyle \mathrm {P} } ã®è»è·¡ãæ±ããã
P {\displaystyle \mathrm {P} } ã®åº§æšã ( x , y ) {\displaystyle (x\ ,\ y)} ãšããã P {\displaystyle \mathrm {P} } ãæºããæ¡ä»¶ã¯
ããªãã¡
ããã座æšã§è¡šããš
䞡蟺ã2ä¹ããŠãæŽçãããš
ããªãã¡
ãããã£ãŠãæ±ããè»è·¡ã¯ãäžå¿ã ( 4 , 0 ) {\displaystyle (4\ ,\ 0)} ãååŸã 2 {\displaystyle 2} ã®åã§ããã
m , n {\displaystyle m\ ,\ n} ãç°ãªãæ£ã®æ°ãšãããšãã2ç¹ A , B {\displaystyle \mathrm {A} \ ,\ \mathrm {B} } ããã®è·é¢ã®æ¯ã m : n {\displaystyle m:n} ã§ããç¹ã®è»è·¡ã¯ãç·å A B {\displaystyle \mathrm {A} \mathrm {B} } ã m : n {\displaystyle m:n} ã«å
åããç¹ãšãå€åããç¹ãçŽåŸã®äž¡ç«¯ãšããåã§ããããã®åãã¢ããããŠã¹ã®åãšããã
m = n {\displaystyle m=n} ã®ãšãã¯ãç·å A B {\displaystyle \mathrm {A} \mathrm {B} } ã®åçŽäºçåç·ã§ããã
ãã®ããŒãžã®åéã®ããã«ãæ°åŒãã€ãã£ãŠåº§æšã®äœçœ®ããããããŠã幟äœåŠã®åé¡ã解ãææ³ã®ããšã解æ幟äœåŠãšããã
ãªãã幟äœåŠãšããèšèèªäœã¯ãå³åœ¢ã®åŠåãšãããããªæå³ã§ãããå°åŠæ ¡ãäžåŠæ ¡ã§ç¿ã£ãå³åœ¢ã®çè«ã幟äœåŠã§ããã
äžäžãšãŒãããã®æ°åŠè
ãã«ã«ããã解æ幟äœåŠã®ç 究ãé²ããããªãããã«ã«ãã¯ãå²åŠã®æ Œèšãããæããããã«æãããã§ãæåã§ããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ããã§ã¯çŽç·ãšåãªã©ã®æ§è³ªã座æšãçšããŠèå¯ããã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "座æšå¹³é¢äžã®2ç¹ A ( x 1 , y 1 ) , B ( x 2 , y 2 ) {\\displaystyle \\mathrm {A} \\left(x_{1}\\ ,\\ y_{1}\\right)\\ ,\\ \\mathrm {B} \\left(x_{2}\\ ,\\ y_{2}\\right)} éã®è·é¢ A B {\\displaystyle \\mathrm {A} \\mathrm {B} } ãæ±ããŠã¿ãããçŽç· A B {\\displaystyle \\mathrm {A} \\mathrm {B} } ã座æšè»žã«å¹³è¡ã§ãªããšããç¹ C ( x 2 , y 1 ) {\\displaystyle \\mathrm {C} \\left(x_{2}\\ ,\\ y_{1}\\right)} ããšããš",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "â³ A B C {\\displaystyle \\triangle \\mathrm {A} \\mathrm {B} \\mathrm {C} } ã¯çŽè§äžè§åœ¢ã§ãããããäžå¹³æ¹ã®å®çãã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ãã®åŒã¯ãçŽç· A B {\\displaystyle \\mathrm {A} \\mathrm {B} } ãx軞ãy軞ã«å¹³è¡ãªãšãã«ãæãç«ã€ã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ç¹ã«ãåç¹ O {\\displaystyle \\mathrm {O} } ãšç¹ A ( x 1 , y 1 ) {\\displaystyle \\mathrm {A} \\left(x_{1}\\ ,\\ y_{1}\\right)} éã®è·é¢ã¯",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ç¹ A ( x 0 , y 0 ) , B ( x 1 , y 1 ) {\\displaystyle \\mathrm {A} (x_{0},y_{0}),\\mathrm {B} (x_{1},y_{1})} ãšå®æ° m , n > 0 {\\displaystyle m,n>0} ã«å¯ŸããŠãç·å A B {\\displaystyle \\mathrm {AB} } äžã®ç¹ P ( x , y ) {\\displaystyle \\mathrm {P} (x,y)} ãååšããŠã A P : P B = m : n {\\displaystyle \\mathrm {AP} :\\mathrm {PB} =m:n} ãšãªããšããç¹ P {\\displaystyle \\mathrm {P} } ã A , B {\\displaystyle \\mathrm {A} ,\\mathrm {B} } ã m : n {\\displaystyle m:n} ã«å
åããç¹ãšããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ãŸããç·å A B {\\displaystyle \\mathrm {AB} } äžã§ãªãç¹ P ( x , y ) {\\displaystyle \\mathrm {P} (x,y)} ãååšããŠã A P : P B = m : n {\\displaystyle \\mathrm {AP} :\\mathrm {PB} =m:n} ãšãªããšããç¹ P {\\displaystyle \\mathrm {P} } ã A , B {\\displaystyle \\mathrm {A} ,\\mathrm {B} } ã m : n {\\displaystyle m:n} ã«å€åããç¹ãšããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "æ°çŽç·äžã®ç¹ A ( a ) , B ( b ) {\\displaystyle \\mathrm {A} (a),\\mathrm {B} (b)} ã m : n {\\displaystyle m:n} ã«å
åããç¹ãšå€åããç¹ãæ±ããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "å
åç¹ã P ( x ) {\\displaystyle \\mathrm {P} (x)} ãšããã a < b {\\displaystyle a<b} ã®ãšãã A P = x â a , P B = b â x {\\displaystyle \\mathrm {AP} =x-a,\\mathrm {PB} =b-x} ãªã®ã§ã m : n = ( x â a ) : ( b â x ) {\\displaystyle m:n=(x-a):(b-x)} ãªã®ã§ã n ( x â a ) = m ( b â x ) ⺠x = n a + m b m + n {\\displaystyle n(x-a)=m(b-x)\\iff x={\\frac {na+mb}{m+n}}} ã§ããã a > b {\\displaystyle a>b} ã®ãšããåæ§ã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "次ã«å€åç¹ãæ±ãããå€åç¹ã P ( x ) {\\displaystyle \\mathrm {P} (x)} ãšããã a < b {\\displaystyle a<b} 㧠m > n {\\displaystyle m>n} ã®ãšãã x > b {\\displaystyle x>b} ãšãªãã®ã§ã A P = x â a , B P = x â b {\\displaystyle \\mathrm {AP} =x-a,\\mathrm {BP} =x-b} ãªã®ã§ã m : n = ( x â a ) : ( x â b ) {\\displaystyle m:n=(x-a):(x-b)} ãªã®ã§ã x = â n a + m b m â n {\\displaystyle x={\\frac {-na+mb}{m-n}}}",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "ããã¯ã a > b {\\displaystyle a>b} ãŸã㯠m < n {\\displaystyle m<n} ã®ãšããåæ§ã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "2次å
ã®å Žåã«ã¯ãäžè¬ã«ç¹ãšç¹ãšã®äœçœ®é¢ä¿ã¯ã座æšè»žã«å¹³è¡ã§ãªãããããã®è·é¢ã®å
åã¯è€éã«ãªãããã«æãããããããå®éã«ã¯ãå
åç¹ãå€åç¹ãèšç®ããã«ã¯ãäžã®å
¬åŒãx,y ã®äž¡æ¹åã«å¯ŸããŠçšããã°ãããããã¯ã2ç¹ãã€ãªãç·ãçŽç·ã§ããã®ã§ããã®çŽç·äžã§ããç¹ããã®è·é¢ãäžå®ã®å²åãšãªãç¹ãããã€ãåã£ããšãããã®ç¹ãšå
ã®ç¹ã®x軞æ¹åã®åº§æšã®å€åã®å²åãšy軞æ¹åã®åº§æšã®å€åã®å²åãšçŽç·èªèº«ã®é·ãã®å€åã®å²åã¯ããããçãããªãããã§ããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ãã£ãŠãäžè¬ã«ç¹ A ( x 0 , y 0 ) , B ( x 1 , y 1 ) {\\displaystyle A(x_{0},y_{0}),B(x_{1},y_{1})} ããa:bã«å
åããç¹ãšå€åããç¹ã¯ã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "ã§äžããããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "ç¹ A ( 1 , 0 ) , B ( â 4 , 7 ) {\\displaystyle \\mathrm {A} (1,0),\\mathrm {B} (-4,7)} ã3:1ã«ããããå
åãå€åããç¹ãæ±ããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "解ç",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "å
åç¹ã¯ ( â 11 4 , 21 4 ) {\\displaystyle \\left({\\frac {-11}{4}},{\\frac {21}{4}}\\right)}",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "å€åç¹ã¯ ( â 13 2 , 21 2 ) {\\displaystyle \\left({\\frac {-13}{2}},{\\frac {21}{2}}\\right)}",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "3ç¹ A ( x 1 , y 1 ) , B ( x 2 , y 2 ) , C ( x 3 , y 3 ) {\\displaystyle \\mathrm {A} \\left(x_{1},y_{1}\\right),\\mathrm {B} \\left(x_{2},y_{2}\\right),\\mathrm {C} \\left(x_{3},y_{3}\\right)} ãé ç¹ãšããäžè§åœ¢ã®éå¿ G {\\displaystyle \\mathrm {G} } ã®åº§æšãæ±ããŠã¿ããã ç·å B C {\\displaystyle \\mathrm {B} \\mathrm {C} } ã®äžç¹ M {\\displaystyle \\mathrm {M} } ã®åº§æšã¯",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "éå¿ G {\\displaystyle \\mathrm {G} } ã¯ç·å A M {\\displaystyle \\mathrm {A} \\mathrm {M} } ã2:1ã«å
åããç¹ã§ããããã G {\\displaystyle \\mathrm {G} } ã®åº§æšã ( x , y ) {\\displaystyle (x,y)} ãšãããš",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "åæ§ã«",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "ãã£ãŠãéå¿ G {\\displaystyle \\mathrm {G} } ã®åº§æšã¯",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "ããç¹ ( x 0 , y 0 ) {\\displaystyle (x_{0},y_{0})} ãéã£ãŠåŸãaã®çŽç·ã®åŒã¯ã y â y 0 = a ( x â x 0 ) {\\displaystyle y-y_{0}=a(x-x_{0})} ã§äžãããããããã¯ãåŸããyã®å€åå / {\\displaystyle /} xã®å€ååã§è¡šãããã y â y 0 {\\displaystyle y-y_{0}} , x â x 0 {\\displaystyle x-x_{0}} ã¯ãŸãã«ãy,xããããã®å€ååãã®ãã®ã§ããããšã«ããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "2ç¹ ( x 0 , y 0 ) {\\displaystyle (x_{0},y_{0})} , ( x 1 , y 1 ) {\\displaystyle (x_{1},y_{1})} ãéãçŽç·ã¯åŸãã y 0 â y 1 x 0 â x 1 {\\displaystyle {\\frac {y_{0}-y_{1}}{x_{0}-x_{1}}}} ã§äžããããããšãçšãããšã y â y 0 = y 0 â y 1 x 0 â x 1 ( x â x 0 ) {\\displaystyle y-y_{0}={\\frac {y_{0}-y_{1}}{x_{0}-x_{1}}}(x-x_{0})} ã§äžããããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "ããããã®çŽç·ãè¡šããåŒãèšç®ããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "(i) åŸã-2ã§ãç¹(-3,1)ãéãçŽç·",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "(ii) 2ç¹(4,3) ,(5,7)ãéãçŽç·",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "解ç",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "ãçšããã°ããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "(i)",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "(ii)",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "ãŸãçŽç·ã®æ¹çšåŒã¯äžè¬ã« a x + b y + c = 0 {\\displaystyle ax+by+c=0} ã§è¡šãããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "ç¹ ( 1 , 4 ) {\\displaystyle (1,4)} ãéããçŽç· y = â 2 x + 3 {\\displaystyle y=-2x+3} ã«å¹³è¡ãªçŽç·ãåçŽãªçŽç·ã®æ¹çšåŒãæ±ããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "çŽç· y = â 2 x + 3 {\\displaystyle y=-2x+3} ã®åŸã㯠â 2 {\\displaystyle -2} ã§ããã å¹³è¡ãªçŽç·ã®æ¹çšåŒã¯",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "åçŽãªçŽç·ã®åŸãã m {\\displaystyle m} ãšãããš",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "ãã£ãŠãåçŽãªçŽç·ã®æ¹çšåŒã¯",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 41,
"tag": "p",
"text": "ç¹ P {\\displaystyle \\mathrm {P} } ãšçŽç· l {\\displaystyle l} ã«å¯ŸããçŽç· l {\\displaystyle l} äžã®ç¹ãšç¹ P {\\displaystyle \\mathrm {P} } ã®è·é¢ã®æå°å€ãç¹ãšçŽç·ã®è·é¢ãšãããããã¯ç¹ P {\\displaystyle \\mathrm {P} } ããçŽç· l {\\displaystyle l} ã«äžãããåç· P H {\\displaystyle \\mathrm {PH} } ã®é·ãã«çããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 42,
"tag": "p",
"text": "çŽç· a x + b y + c = 0 {\\displaystyle ax+by+c=0} ãšç¹ ( x 0 , y 0 ) {\\displaystyle (x_{0},y_{0})} ã®è·é¢ã¯",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 43,
"tag": "p",
"text": "ãšè¡šãããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 44,
"tag": "p",
"text": "蚌æ",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 45,
"tag": "p",
"text": "ç¹ P ( x 0 , y 0 ) {\\displaystyle \\mathrm {P} (x_{0},y_{0})} ãšçŽç· l : a x + b y + c = 0 a , b â 0 {\\displaystyle l:ax+by+c=0\\quad a,b\\neq 0} ãšããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 46,
"tag": "p",
"text": "ç¹ P {\\displaystyle \\mathrm {P} } ããçŽç· l {\\displaystyle l} ã«åç·ãäžãããåç·ã®è¶³ãç¹ R {\\displaystyle R} ãšããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 47,
"tag": "p",
"text": "ãŸããç¹ P {\\displaystyle \\mathrm {P} } ãã y {\\displaystyle y} 軞ã«å¹³è¡ãªçŽç·ãåŒããçŽç· l {\\displaystyle l} ãšã®äº€ç¹ãç¹ S {\\displaystyle \\mathrm {S} } ãšããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 48,
"tag": "p",
"text": "次ã«ãå³ã®ããã«ãçŽç· l {\\displaystyle l} äžã®ç¹ T {\\displaystyle \\mathrm {T} } ã«å¯ŸããŠãçŽç· T V {\\displaystyle \\mathrm {TV} } ã x {\\displaystyle x} 軞ãšå¹³è¡ãšãªãã T V = | b | {\\displaystyle \\mathrm {TV} =|b|} ãšãªãããã«ç¹ V {\\displaystyle \\mathrm {V} } ããšããçŽç· V U {\\displaystyle \\mathrm {VU} } ã y {\\displaystyle y} 軞ã«å¹³è¡ã«ãªãç¹ U {\\displaystyle \\mathrm {U} } ãçŽç· l {\\displaystyle l} äžã«åãã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 49,
"tag": "p",
"text": "çŽç· l {\\displaystyle l} ã®åŸã㯠â a b {\\displaystyle -{\\frac {a}{b}}} ãšãªãã®ã§ V U = | a | {\\displaystyle \\mathrm {VU} =|a|} ã§ããã ããã§ã â³ P R S , â³ T V U {\\displaystyle \\bigtriangleup \\mathrm {PRS} ,\\bigtriangleup \\mathrm {TVU} } ã¯çŽè§äžè§åœ¢ã§ããã â P S R = â T U V {\\displaystyle \\angle \\mathrm {PSR} =\\angle \\mathrm {TUV} } ãªã®ã§ã â³ P R S âŒâ³ T V U {\\displaystyle \\bigtriangleup \\mathrm {PRS} \\sim \\bigtriangleup \\mathrm {TVU} } ã§ããããããã£ãŠ",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 50,
"tag": "p",
"text": "ãŸãç¹ S {\\displaystyle \\mathrm {S} } ã®åº§æšã ( x 0 , m ) {\\displaystyle (x_{0},m)} ãšãããšã P S = | y 0 â m | {\\displaystyle \\mathrm {PS} =|y_{0}-m|} ã§ãç¹ P {\\displaystyle \\mathrm {P} } ãšçŽç· l {\\displaystyle l} ã®è·é¢ P R {\\displaystyle \\mathrm {PR} } ã¯ã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 51,
"tag": "p",
"text": "P R = P S â
T V T U = | y 0 â m | | b | a 2 + b 2 {\\displaystyle \\mathrm {PR} ={\\mathrm {PS} }\\cdot {\\frac {\\mathrm {TV} }{\\mathrm {TU} }}={\\frac {|y_{0}-m||b|}{\\sqrt {a^{2}+b^{2}}}}}",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 52,
"tag": "p",
"text": "ãšããã§ãç¹ S {\\displaystyle \\mathrm {S} } ã¯çŽç· l {\\displaystyle l} äžã®ç¹ãªã®ã§ã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 53,
"tag": "p",
"text": "ã§ãããããã代å
¥ããã°",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 54,
"tag": "p",
"text": "ãã¯ãã«ã䜿ã£ã蚌æ",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 55,
"tag": "p",
"text": "ãã§ã«ãã¯ãã«ãç¥ã£ãŠãããªãã°ãã¡ãã®æ¹ãç°¡æœã§ããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 56,
"tag": "p",
"text": "ç¹ P ( x 0 , y 0 ) {\\displaystyle \\mathrm {P} (x_{0},y_{0})} ãšçŽç· l : a x + b y + c = 0 {\\displaystyle l:ax+by+c=0} ãšããç¹ Q ( x 1 , y 1 ) {\\displaystyle \\mathrm {Q} (x_{1},y_{1})} ãçŽç· l {\\displaystyle l} äžã®ç¹ãšãããçŽç· l {\\displaystyle l} ã®æ³ç·ã¯ n â := ( a , b ) {\\displaystyle {\\vec {n}}:=(a,b)} ã§ã Q P â = ( x 0 â x 1 , y 0 â y 1 ) {\\displaystyle {\\vec {\\mathrm {QP} }}=(x_{0}-x_{1},y_{0}-y_{1})} ã§ããã®ã§ãçŽç· l {\\displaystyle l} äžã®ç¹ãšç¹ P {\\displaystyle \\mathrm {P} } ã®è·é¢ d {\\displaystyle d} 㯠d = | Q P â â
n â | | n â | | | = | ( x 0 â x 1 , y 0 â y 1 ) â
( a , b ) a 2 + b 2 | = | a x 0 + b y 0 â ( a x 1 + b y 1 ) | a 2 + b 2 = | a x 0 + b y 0 + c | a 2 + b 2 {\\displaystyle d=\\left|{\\vec {\\mathrm {QP} }}\\cdot {\\frac {\\vec {n}}{||{\\vec {n}}||}}\\right|=\\left|(x_{0}-x_{1},y_{0}-y_{1})\\cdot {\\frac {(a,b)}{\\sqrt {a^{2}+b^{2}}}}\\right|={\\frac {|ax_{0}+by_{0}-(ax_{1}+by_{1})|}{\\sqrt {a^{2}+b^{2}}}}={\\frac {|ax_{0}+by_{0}+c|}{\\sqrt {a^{2}+b^{2}}}}} ã§ããã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 57,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 58,
"tag": "p",
"text": "çŽç· x â 2 y â 3 = 0 {\\displaystyle x-2y-3=0} ãšç¹ ( 1 , 2 ) {\\displaystyle (1,2)} ã®è·é¢ãæ±ãã",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 59,
"tag": "p",
"text": "解ç",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 60,
"tag": "p",
"text": "6 5 {\\displaystyle {\\frac {6}{\\sqrt {5}}}}",
"title": "ç¹ãšçŽç·"
},
{
"paragraph_id": 61,
"tag": "p",
"text": "äžå¿ C ( a , b ) {\\displaystyle \\mathrm {C} (a,b)} ååŸ r {\\displaystyle r} ã®åã¯ã C P = r {\\displaystyle \\mathrm {CP} =r} ãšãªãç¹ P {\\displaystyle \\mathrm {P} } ã®éåã§ãããã€ãŸãã r = ( x â a ) 2 + ( y â b ) 2 {\\displaystyle r={\\sqrt {(x-a)^{2}+(y-b)^{2}}}} ãšãªãç¹ ( x , y ) {\\displaystyle (x,y)} ã®éåã§ããããã®æ¹çšåŒã®äž¡èŸºã¯æ£ãªã®ã§2ä¹ããŠ",
"title": "å"
},
{
"paragraph_id": 62,
"tag": "p",
"text": "( x â a ) 2 + ( y â b ) 2 = r 2 {\\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}}",
"title": "å"
},
{
"paragraph_id": 63,
"tag": "p",
"text": "ãåŸãããããåã®æ¹çšåŒã§ããã",
"title": "å"
},
{
"paragraph_id": 64,
"tag": "p",
"text": "ç¹ã«åç¹ãäžå¿ã§ååŸ r {\\displaystyle r} ã®åã®æ¹çšåŒã¯ x 2 + y 2 = r 2 {\\displaystyle x^{2}+y^{2}=r^{2}} ã§äžããããã",
"title": "å"
},
{
"paragraph_id": 65,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "å"
},
{
"paragraph_id": 66,
"tag": "p",
"text": "解ç",
"title": "å"
},
{
"paragraph_id": 67,
"tag": "p",
"text": "",
"title": "å"
},
{
"paragraph_id": 68,
"tag": "p",
"text": "æ¹çšåŒ x 2 + y 2 + l x + m y + n = 0 {\\displaystyle x^{2}+y^{2}+lx+my+n=0} ã¯ãã€ãåã§ãããšã¯éããªãã",
"title": "å"
},
{
"paragraph_id": 69,
"tag": "p",
"text": "æ¹çšåŒãå€åœ¢ã㊠( x â a ) 2 + ( y â b ) 2 = k {\\displaystyle (x-a)^{2}+(y-b)^{2}=k} ãšãªããšã",
"title": "å"
},
{
"paragraph_id": 70,
"tag": "p",
"text": "å x 2 + y 2 = r 2 {\\displaystyle x^{2}+y^{2}=r^{2}} äžã®ããç¹ ( x 1 , y 1 ) {\\displaystyle (x_{1},y_{1})} ã§æ¥ããæ¥ç·ã®æ¹çšåŒã¯",
"title": "å"
},
{
"paragraph_id": 71,
"tag": "p",
"text": "ã§è¡šãããã",
"title": "å"
},
{
"paragraph_id": 72,
"tag": "p",
"text": "åæ§ã«ãå ( x â a ) 2 + ( y â b ) 2 = r 2 {\\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}} äžã®ããç¹ ( x 2 , y 2 ) {\\displaystyle (x_{2},y_{2})} ã§æ¥ããæ¥ç·ã®æ¹çšåŒã¯",
"title": "å"
},
{
"paragraph_id": 73,
"tag": "p",
"text": "ã§è¡šãããã",
"title": "å"
},
{
"paragraph_id": 74,
"tag": "p",
"text": "åãšçŽç·ã®äœçœ®é¢ä¿ã«ã€ããŠå€§ãã次ã®3ã€ã«åé¡ããããšãã§ããã",
"title": "å"
},
{
"paragraph_id": 75,
"tag": "p",
"text": "äžè¬ã®åãšçŽç·ã«ã€ããŠãããã®äœçœ®é¢ä¿ãåé¡ããŠã¿ããã",
"title": "å"
},
{
"paragraph_id": 76,
"tag": "p",
"text": "å C : ( x â p ) 2 + ( y â q ) 2 = r 2 {\\displaystyle C:(x-p)^{2}+(y-q)^{2}=r^{2}} ãšçŽç· l : a x + b y + c = 0 {\\displaystyle l:ax+by+c=0} ã«ã€ããŠãå C {\\displaystyle C} ã®äžå¿ ( p , q ) {\\displaystyle (p,q)} ãšçŽç· l {\\displaystyle l} ã®è·é¢ d := | a q + b q + c | a 2 + b 2 {\\displaystyle d:={\\frac {|aq+bq+c|}{\\sqrt {a^{2}+b^{2}}}}} ãšãããšã",
"title": "å"
},
{
"paragraph_id": 77,
"tag": "p",
"text": "ä»ã«ããåã®æ¹çšåŒãšçŽç·ã®æ¹çšåŒãé£ç«ããŠãã®å®æ°è§£ã®åæ°ã§åé¡ããæ¹æ³ãããããäœçœ®é¢ä¿ãæ±ããã ããªãäžã®æ¹æ³ã®ã»ããèšç®éãå°ãªãã",
"title": "å"
},
{
"paragraph_id": 78,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "å"
},
{
"paragraph_id": 79,
"tag": "p",
"text": "çŽç· 3 x + 4 y = 1 {\\displaystyle 3x+4y=1} ãšå ( x â 3 ) 2 + ( y + 2 ) 2 = 14 {\\displaystyle (x-3)^{2}+(y+2)^{2}=14} ã®äº€ç¹ã®åº§æšãæ±ãã",
"title": "å"
},
{
"paragraph_id": 80,
"tag": "p",
"text": "解ç",
"title": "å"
},
{
"paragraph_id": 81,
"tag": "p",
"text": "çŽç·ã®æ¹çšåŒã x {\\displaystyle x} ã«ã€ããŠè§£ãããããåã®æ¹çšåŒã«ä»£å
¥ããã°ããã",
"title": "å"
},
{
"paragraph_id": 82,
"tag": "p",
"text": "çã㯠( 2 , â 1 ) , ( â 14 5 , 7 5 ) {\\displaystyle (2,-1),\\left(-{\\frac {14}{5}},{\\frac {7}{5}}\\right)}",
"title": "å"
},
{
"paragraph_id": 83,
"tag": "p",
"text": "ããæ¡ä»¶ãæºããç¹å
šäœãã€ããå³åœ¢ãããã®æ¡ä»¶ãæºããç¹ã®è»è·¡ãšããã",
"title": "è»è·¡ãšé å"
},
{
"paragraph_id": 84,
"tag": "p",
"text": "",
"title": "è»è·¡ãšé å"
},
{
"paragraph_id": 85,
"tag": "p",
"text": "2ç¹ A ( 1 , 0 ) , B ( 3 , 2 ) {\\displaystyle \\mathrm {A} (1\\ ,\\ 0)\\ ,\\ \\mathrm {B} (3\\ ,\\ 2)} ããçè·é¢ã«ããç¹ P {\\displaystyle \\mathrm {P} } ã®è»è·¡ãæ±ããã",
"title": "è»è·¡ãšé å"
},
{
"paragraph_id": 86,
"tag": "p",
"text": "æ¡ä»¶ A P = B P {\\displaystyle \\mathrm {A} \\mathrm {P} =\\mathrm {B} \\mathrm {P} } ããã A P 2 = B P 2 {\\displaystyle \\mathrm {A} \\mathrm {P} ^{2}=\\mathrm {B} \\mathrm {P} ^{2}} P {\\displaystyle \\mathrm {P} } ã®åº§æšã ( x , y ) {\\displaystyle (x\\ ,\\ y)} ãšãããš",
"title": "è»è·¡ãšé å"
},
{
"paragraph_id": 87,
"tag": "p",
"text": "ã ãã",
"title": "è»è·¡ãšé å"
},
{
"paragraph_id": 88,
"tag": "p",
"text": "æŽçããŠã",
"title": "è»è·¡ãšé å"
},
{
"paragraph_id": 89,
"tag": "p",
"text": "ãããã£ãŠãæ±ããè»è·¡ã¯ãçŽç· y = â x + 3 {\\displaystyle y=-x+3} ã§ããã",
"title": "è»è·¡ãšé å"
},
{
"paragraph_id": 90,
"tag": "p",
"text": "",
"title": "è»è·¡ãšé å"
},
{
"paragraph_id": 91,
"tag": "p",
"text": "",
"title": "è»è·¡ãšé å"
},
{
"paragraph_id": 92,
"tag": "p",
"text": "2ç¹ A ( 0 , 0 ) , B ( 3 , 0 ) {\\displaystyle \\mathrm {A} (0\\ ,\\ 0)\\ ,\\ \\mathrm {B} (3\\ ,\\ 0)} ããã®è·é¢ã®æ¯ã 2 : 1 {\\displaystyle 2:1} ã§ããç¹ P {\\displaystyle \\mathrm {P} } ã®è»è·¡ãæ±ããã",
"title": "è»è·¡ãšé å"
},
{
"paragraph_id": 93,
"tag": "p",
"text": "P {\\displaystyle \\mathrm {P} } ã®åº§æšã ( x , y ) {\\displaystyle (x\\ ,\\ y)} ãšããã P {\\displaystyle \\mathrm {P} } ãæºããæ¡ä»¶ã¯",
"title": "è»è·¡ãšé å"
},
{
"paragraph_id": 94,
"tag": "p",
"text": "ããªãã¡",
"title": "è»è·¡ãšé å"
},
{
"paragraph_id": 95,
"tag": "p",
"text": "ããã座æšã§è¡šããš",
"title": "è»è·¡ãšé å"
},
{
"paragraph_id": 96,
"tag": "p",
"text": "䞡蟺ã2ä¹ããŠãæŽçãããš",
"title": "è»è·¡ãšé å"
},
{
"paragraph_id": 97,
"tag": "p",
"text": "ããªãã¡",
"title": "è»è·¡ãšé å"
},
{
"paragraph_id": 98,
"tag": "p",
"text": "ãããã£ãŠãæ±ããè»è·¡ã¯ãäžå¿ã ( 4 , 0 ) {\\displaystyle (4\\ ,\\ 0)} ãååŸã 2 {\\displaystyle 2} ã®åã§ããã",
"title": "è»è·¡ãšé å"
},
{
"paragraph_id": 99,
"tag": "p",
"text": "m , n {\\displaystyle m\\ ,\\ n} ãç°ãªãæ£ã®æ°ãšãããšãã2ç¹ A , B {\\displaystyle \\mathrm {A} \\ ,\\ \\mathrm {B} } ããã®è·é¢ã®æ¯ã m : n {\\displaystyle m:n} ã§ããç¹ã®è»è·¡ã¯ãç·å A B {\\displaystyle \\mathrm {A} \\mathrm {B} } ã m : n {\\displaystyle m:n} ã«å
åããç¹ãšãå€åããç¹ãçŽåŸã®äž¡ç«¯ãšããåã§ããããã®åãã¢ããããŠã¹ã®åãšããã",
"title": "è»è·¡ãšé å"
},
{
"paragraph_id": 100,
"tag": "p",
"text": "m = n {\\displaystyle m=n} ã®ãšãã¯ãç·å A B {\\displaystyle \\mathrm {A} \\mathrm {B} } ã®åçŽäºçåç·ã§ããã",
"title": "è»è·¡ãšé å"
},
{
"paragraph_id": 101,
"tag": "p",
"text": "",
"title": "è»è·¡ãšé å"
},
{
"paragraph_id": 102,
"tag": "p",
"text": "ãã®ããŒãžã®åéã®ããã«ãæ°åŒãã€ãã£ãŠåº§æšã®äœçœ®ããããããŠã幟äœåŠã®åé¡ã解ãææ³ã®ããšã解æ幟äœåŠãšããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 103,
"tag": "p",
"text": "ãªãã幟äœåŠãšããèšèèªäœã¯ãå³åœ¢ã®åŠåãšãããããªæå³ã§ãããå°åŠæ ¡ãäžåŠæ ¡ã§ç¿ã£ãå³åœ¢ã®çè«ã幟äœåŠã§ããã",
"title": "ã³ã©ã "
},
{
"paragraph_id": 104,
"tag": "p",
"text": "äžäžãšãŒãããã®æ°åŠè
ãã«ã«ããã解æ幟äœåŠã®ç 究ãé²ããããªãããã«ã«ãã¯ãå²åŠã®æ Œèšãããæããããã«æãããã§ãæåã§ããã",
"title": "ã³ã©ã "
}
] | ããã§ã¯çŽç·ãšåãªã©ã®æ§è³ªã座æšãçšããŠèå¯ããã | {{pathnav|é«çåŠæ ¡ã®åŠç¿|é«çåŠæ ¡æ°åŠ|é«çåŠæ ¡æ°åŠII|pagename=å³åœ¢ãšæ¹çšåŒ|frame=1|small=1}}
ããã§ã¯çŽç·ãšåãªã©ã®æ§è³ªã座æšãçšããŠèå¯ããã
==ç¹ãšçŽç·==
===2ç¹éã®è·é¢===
[[ãã¡ã€ã«:Distance_Formula.svg|å³|200x200ãã¯ã»ã«]]
座æšå¹³é¢äžã®2ç¹ <math>\mathrm{A} \left(x _1\ ,\ y _1 \right)\ ,\ \mathrm{B} \left(x _2\ ,\ y _2 \right)</math> éã®è·é¢ <math>\mathrm{A} \mathrm{B}</math> ãæ±ããŠã¿ããã<br>çŽç· <math>\mathrm{A} \mathrm{B}</math> ã座æšè»žã«å¹³è¡ã§ãªããšã<ref>ã€ãŸããçŽç· <math>\mathrm{A} \mathrm{B}</math> ã <math>x</math> 軞ã <math>y</math> 軞 ã®ã©ã¡ããšãå¹³è¡ã§ãªããšã</ref>ãç¹ <math>\mathrm{C} \left(x _2\ ,\ y _1 \right)</math> ããšããš
:<math>
\mathrm{A} \mathrm{C} = |x _2 - x _1|\ ,\ \mathrm{B} \mathrm{C} = |y _2 - y _1|
</math>
<math>\triangle \mathrm{A} \mathrm{B} \mathrm{C}</math> ã¯çŽè§äžè§åœ¢ã§ãããããäžå¹³æ¹ã®å®çãã
:<math>
\mathrm{A} \mathrm{B} = \sqrt{\mathrm{A} \mathrm{C} ^2+ \mathrm{B} \mathrm{C} ^2} = \sqrt{|x _2 - x _1|^2+|y _2 - y _1|^2} = \sqrt{(x _2 - x _1)^2+(y _2 - y _1)^2}
</math>
ãã®åŒã¯ãçŽç· <math>\mathrm{A} \mathrm{B}</math> ãx軞ãy軞ã«å¹³è¡ãªãšãã«ãæãç«ã€<ref>çŽç· <math>\mathrm{A} \mathrm{B}</math> ã <math>x</math> 軞ã«å¹³è¡ãªãšã㯠<math>\mathrm{BC} = 0</math> ã§ããã <math>\mathrm{AC} = \mathrm{AB}</math> ãšãªãããã£ãŠ <math>\mathrm{AB} = \sqrt{\mathrm{AC}^2+\mathrm{BC}^2} </math> ã¯æãç«ã€ãçŽç· <math>\mathrm{A} \mathrm{B}</math> ã <math>y</math> 軞ã«å¹³è¡ãªãšããåæ§</ref>ã
ç¹ã«ãåç¹ <math>\mathrm{O}</math> ãšç¹ <math>\mathrm{A} \left(x _1\ ,\ y _1 \right)</math> éã®è·é¢ã¯
:<math>
\mathrm{O} \mathrm{A} = \sqrt{x _1^2 + y _1^2}
</math>
=== å
åç¹ãšå€åç¹===
ç¹ <math>
\mathrm{A}(x _0,y _0),\mathrm{B}(x _1,y _1)
</math> ãšå®æ° <math>m,n>0</math> ã«å¯ŸããŠãç·å <math>\mathrm{AB}</math> äžã®ç¹ <math>\mathrm{P}(x,y)</math> ãååšããŠã<math>\mathrm{AP}:\mathrm{PB} = m:n</math> ãšãªããšããç¹ <math>\mathrm{P}</math> ã <math>\mathrm{A},\mathrm{B}</math> ã <math>m:n</math> ã«å
åããç¹ãšããã
ãŸããç·å <math>\mathrm{AB}</math> äžã§ãªãç¹ <math>\mathrm{P}(x,y)</math> ãååšããŠã<math>\mathrm{AP}:\mathrm{PB} = m:n</math> ãšãªããšããç¹ <math>\mathrm{P}</math> ã <math>\mathrm{A},\mathrm{B}</math> ã <math>m:n</math> ã«å€åããç¹ãšããã
æ°çŽç·äžã®ç¹ <math>\mathrm{A}(a),\mathrm{B}(b)</math> ã <math>m:n</math> ã«å
åããç¹ãšå€åããç¹ãæ±ããã
å
åç¹ã <math>\mathrm{P}(x)</math> ãšããã<math>a<b</math> ã®ãšãã <math>\mathrm{AP} = x-a,\mathrm{PB}=b-x</math> ãªã®ã§ã <math>m:n=(x-a):(b-x)</math> ãªã®ã§ã <math>n(x-a)=m(b-x) \iff x = \frac{na+mb}{m+n}</math> ã§ããã <math>a>b</math> ã®ãšããåæ§ã
次ã«å€åç¹ãæ±ãããå€åç¹ã <math>\mathrm{P}(x)</math> ãšããã<math>a<b</math> 㧠<math>m>n</math> ã®ãšãã<math>x>b</math> ãšãªãã®ã§ã <math>\mathrm{AP}=x-a,\mathrm{BP}=x-b</math> ãªã®ã§ã<math>m:n=(x-a):(x-b)</math> ãªã®ã§ã<math>x=\frac{-na+mb}{m-n}</math>
ããã¯ã<math>a>b</math> ãŸã㯠<math>m<n</math> ã®ãšããåæ§ã<ref>å€åç¹ã®åº§æšã¯å
åç¹ã®åº§æšã® <math>n</math> ã <math>-n</math> ã«ãããã®ã«çãã</ref>
2次å
ã®å Žåã«ã¯ãäžè¬ã«ç¹ãšç¹ãšã®äœçœ®é¢ä¿ã¯ã座æšè»žã«å¹³è¡ã§ãªãããããã®è·é¢ã®å
åã¯è€éã«ãªãããã«æãããããããå®éã«ã¯ãå
åç¹ãå€åç¹ãèšç®ããã«ã¯ãäžã®å
Œ΋x,y
ã®äž¡æ¹åã«å¯ŸããŠçšããã°ãããããã¯ã2ç¹ãã€ãªãç·ãçŽç·ã§ããã®ã§ããã®çŽç·äžã§ããç¹ããã®è·é¢ãäžå®ã®å²åãšãªãç¹ãããã€ãåã£ããšãããã®ç¹ãšå
ã®ç¹ã®x軞æ¹åã®åº§æšã®å€åã®å²åãšy軞æ¹åã®åº§æšã®å€åã®å²åãšçŽç·èªèº«ã®é·ãã®å€åã®å²åã¯ããããçãããªãããã§ããã
ãã£ãŠãäžè¬ã«ç¹<math>A(x _0,y _0),B(x _1,y _1)</math>ããa:bã«å
åããç¹ãšå€åããç¹ã¯ã
:å
åç¹
:<math>
(\frac {b x _0 + a x _1} {a +b},
\frac {b y _0 + a y _1} {a +b})
</math>
:å€åç¹
:<math>
(\frac {-b x _0 + a x _1} {a -b},
\frac {-b y _0 + a y _1} {a -b})
</math>
:<math>
=
(
\frac {b x _0 - a x _1} {-a +b},
\frac {b y _0 - a y _1} {-a +b}
)
</math>
ã§äžããããã
'''æŒç¿åé¡'''
ç¹ <math>
\mathrm{A}(1,0),\mathrm{B}(-4,7)
</math> ã3:1ã«ããããå
åãå€åããç¹ãæ±ããã
'''解ç'''
å
åç¹ã¯ <math>
\left(\frac {-11}4,\frac{21}4\right)
</math>
å€åç¹ã¯ <math>
\left(\frac {-13}2,\frac{21}2\right)
</math>
===äžè§åœ¢ã®éå¿===
3ç¹<math>\mathrm{A} \left(x _1 , y _1 \right) , \mathrm{B} \left(x _2 , y _2 \right) , \mathrm{C} \left(x _3 , y _3 \right) </math>ãé ç¹ãšããäžè§åœ¢ã®éå¿ <math>\mathrm{G}</math> ã®åº§æšãæ±ããŠã¿ããã<br>
ç·å<math>\mathrm{B} \mathrm{C}</math>ã®äžç¹<math>\mathrm{M}</math>ã®åº§æšã¯
:<math>
\left(\frac {x _2 + x _3}{2} , \frac {y _2 + y _3}{2} \right)
</math>
éå¿<math>\mathrm{G}</math>ã¯ç·å<math>\mathrm{A} \mathrm{M}</math>ã2:1ã«å
åããç¹ã§ããããã<math>\mathrm{G}</math>ã®åº§æšã<math>(x , y)</math>ãšãããš
:<math>
x= \cfrac { 1 \times x _1 + 2 \times \cfrac { x _2 + x _3 } { 2 } } { 2+1 } = \frac { x _1 + x _2 + x _3 } { 3 }</math>
åæ§ã«
:<math>
y = \frac { y _1 + y _2 + y _3 } { 3 }
</math>
ãã£ãŠãéå¿<math>\mathrm{G}</math>ã®åº§æšã¯
:<math>
\left(\frac { x _1 + x _2 + x _3 } { 3 } , \frac { y _1 + y _2 + y _3 } { 3 } \right)
</math>
===çŽç·ã®æ¹çšåŒ===
ããç¹ <math>(x_0,y_0)</math> ãéã£ãŠåŸãaã®çŽç·ã®åŒã¯ã
<math>
y- y_0 = a(x- x_0)
</math>
ã§äžãããããããã¯ãåŸããyã®å€åå<math>/</math>xã®å€ååã§è¡šãããã<math> y-y_0 </math>,<math> x-x_0 </math>ã¯ãŸãã«ãy,xããããã®å€ååãã®ãã®ã§ããããšã«ããã
2ç¹ <math>(x_0,y_0)</math> , <math>(x_1,y_1)</math> ãéãçŽç·ã¯åŸãã <math>\frac{y_0-y_1}{x_0-x_1}</math> ã§äžããããããšãçšãããšã
<math>
y-y_0 = \frac{y_0-y_1}{x_0-x_1}(x-x_0)
</math>
ã§äžããããã
'''æŒç¿åé¡'''
ããããã®çŽç·ãè¡šããåŒãèšç®ããã
(i)
åŸã-2ã§ãç¹(-3,1)ãéãçŽç·
(ii)
2ç¹(4,3) ,(5,7)ãéãçŽç·
'''解ç'''
:<math>
y-y _0 = a(x-x _0)
</math>
:<math>
y-y _0 = \frac{y _0-y _1}{x _0-x _1}(x-x _0)
</math>
ãçšããã°ããã
(i)
:<math>
\left[ y=-2\,x-5 \right]
</math>
(ii)
:<math>
\left[ y=4\,x-13 \right]
</math>
ãŸãçŽç·ã®æ¹çšåŒã¯äžè¬ã« <math>ax+by+c=0</math> ã§è¡šãããã
====2çŽç·ã®å¹³è¡ãšåçŽ====
{| style="border:2px solid orange;width:80%" cellspacing=0
|style="background:orange"|'''2çŽç·ã®å¹³è¡ãåçŽ'''
|-
|style="padding:5px"|
2çŽç·<math>y=m_1 x+n_1\ ,\ y=m_2 x+n_2</math>ã«ã€ããŠ
<center>2çŽç·ãå¹³è¡<math>\Leftrightarrow m_1=m_2</math></center>
<center>2çŽç·ãåçŽ<math>\Leftrightarrow m_1 m_2=-1</math></center>
|}
*åé¡äŸ
**åé¡
ç¹<math>(1,4)</math>ãéããçŽç·<math>y=-2x+3</math>ã«å¹³è¡ãªçŽç·ãåçŽãªçŽç·ã®æ¹çšåŒãæ±ããã
**解ç
çŽç·<math>y=-2x+3</math>ã®åŸãã¯<math>-2</math>ã§ããã<br>
å¹³è¡ãªçŽç·ã®æ¹çšåŒã¯
:<math>y-4=-2(x-1)</math>
:<math>y=-2x+6</math>
åçŽãªçŽç·ã®åŸãã<math>m</math>ãšãããš
:<math>-2m=-1</math>
:<math>m= \frac{1}{2}</math>
ãã£ãŠãåçŽãªçŽç·ã®æ¹çšåŒã¯
:<math>y-4= \frac{1}{2} (x-1)</math>
:<math>y= \frac{1}{2} x+ \frac{7}{2}</math>
===ç¹ãšçŽç·ã®è·é¢===
ç¹ <math>\mathrm{P}</math> ãšçŽç· <math>l</math> ã«å¯ŸããçŽç· <math>l</math> äžã®ç¹ãšç¹ <math>\mathrm{P}</math> ã®è·é¢ã®æå°å€ã'''ç¹ãšçŽç·ã®è·é¢'''ãšãããããã¯ç¹ <math>\mathrm{P}</math> ããçŽç· <math>l</math> ã«äžãããåç· <math>\mathrm{PH}</math> ã®é·ãã«çããã
çŽç· <math>ax+by+c=0</math> ãšç¹ <math>(x_0,y_0)</math> ã®è·é¢ã¯
:<math>\frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}</math>
ãšè¡šãããã
'''蚌æ'''
[[ãã¡ã€ã«:Point-to-line2.svg|ãµã ãã€ã«]]ç¹ <math>\mathrm{P}(x_0,y_0)</math> ãšçŽç· <math>l:ax+by+c=0 \quad a,b\neq 0</math> ãšããã
ç¹ <math>\mathrm{P}</math> ããçŽç· <math>l</math> ã«åç·ãäžãããåç·ã®è¶³ãç¹ <math>R</math> ãšããã
ãŸããç¹ <math>\mathrm{P}</math> ãã <math>y</math> 軞ã«å¹³è¡ãªçŽç·ãåŒããçŽç· <math>l</math> ãšã®äº€ç¹ãç¹ <math>\mathrm S</math> ãšããã
次ã«ãå³ã®ããã«ãçŽç· <math>l</math> äžã®ç¹ <math>\mathrm T</math> ã«å¯ŸããŠãçŽç· <math>\mathrm{TV}</math> ã <math>x</math> 軞ãšå¹³è¡ãšãªãã<math>\mathrm{TV} = |b|</math> ãšãªãããã«ç¹ <math>\mathrm V</math> ããšããçŽç· <math>\mathrm{VU}</math> ã <math>y</math> 軞ã«å¹³è¡ã«ãªãç¹ <math>\mathrm U</math> ãçŽç· <math>l</math> äžã«åãã
çŽç· <math>l</math> ã®åŸã㯠<math>-\frac{a}{b}</math> ãšãªãã®ã§ <math>\mathrm{VU} = |a|</math> ã§ããã
ããã§ã<math>\bigtriangleup \mathrm{PRS},\bigtriangleup \mathrm{TVU}</math> ã¯çŽè§äžè§åœ¢ã§ããã<math>\angle \mathrm{PSR} = \angle \mathrm{TUV}</math><ref>çŽç· <math>\mathrm{PS}</math> ãšçŽç· <math>\mathrm{VU}</math> ã¯å¹³è¡ãªã®ã§</ref> ãªã®ã§ã<math>\bigtriangleup \mathrm{PRS} \sim \bigtriangleup \mathrm{TVU}</math><ref><math>\sim</math> ã¯çžäŒŒãæå³ãã</ref> ã§ããããããã£ãŠ
:<math>\frac{\mathrm{PR}}{\mathrm{PS}} = \frac{\mathrm{TV}}{\mathrm{TU}}</math>
ãŸãç¹ <math>\mathrm S</math> ã®åº§æšã<math>(x_0,m)</math> ãšãããšã<math>\mathrm{PS} = |y_0-m| </math> ã§ãç¹ <math>\mathrm{P}</math> ãšçŽç· <math>l</math> ã®è·é¢ <math> \mathrm{PR}</math> ã¯ã
<math> \mathrm{PR} ={\mathrm{PS}}\cdot \frac{\mathrm{TV}}{\mathrm{TU}} = \frac{|y_0 - m||b|}{\sqrt{a^2 + b^2}} </math>
ãšããã§ãç¹ <math>\mathrm S</math> ã¯çŽç· <math>l</math> äžã®ç¹ãªã®ã§ã
:<math>m = \frac{-ax_0 - c}{b}</math>
ã§ãããããã代å
¥ããã°
:<math> \mathrm{PR} = \frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}}</math>
:ãåŸãã
:
'''ãã¯ãã«ã䜿ã£ã蚌æ'''
ãã§ã«ãã¯ãã«ãç¥ã£ãŠãããªãã°ãã¡ãã®æ¹ãç°¡æœã§ããã
ç¹ <math>\mathrm{P}(x_0,y_0)</math> ãšçŽç· <math>l:ax+by+c=0</math> ãšããç¹ <math>\mathrm{Q}(x_1,y_1)</math> ãçŽç· <math>l</math> äžã®ç¹ãšãããçŽç· <math>l</math> ã®æ³ç·ã¯ <math>\vec n := (a,b)</math> ã§ã<math>\vec{\mathrm{QP}} = (x_0-x_1,y_0-y_1) </math> ã§ããã®ã§ãçŽç· <math>l</math> äžã®ç¹ãšç¹ <math>\mathrm{P}</math> ã®è·é¢ <math>d</math> 㯠<math>d = \left| \vec{ \mathrm{QP} } \cdot \frac{\vec n}{||\vec n||}\right| = \left|(x_0-x_1,y_0-y_1)\cdot \frac{(a,b)}{\sqrt{a^2+b^2}}\right| = \frac{|ax_0 + by_0 - (ax_1 + by_1)|}{\sqrt{a^2+b^2}} = \frac{|ax_0 + by_0 +c|}{\sqrt{a^2+b^2}}</math><ref>ç¹ <math>\mathrm{Q}(x_1,y_1)</math> ã¯çŽç· <math>l</math> äžã®ç¹ãªã®ã§ <math>ax_1+by_1=-c</math> ã§ããã</ref> ã§ããã
'''æŒç¿åé¡'''
çŽç· <math>x-2y-3=0</math> ãšç¹ <math>(1,2)</math> ã®è·é¢ãæ±ãã
'''解ç'''
<math>\frac{6}{\sqrt 5}</math>
==å==
====åã®æ¹çšåŒ====
äžå¿ <math>\mathrm{C}(a,b)</math> ååŸ <math>r</math> ã®åã¯ã<math>\mathrm{CP} =r</math> ãšãªãç¹ <math>\mathrm{P}</math> ã®éåã§ãããã€ãŸãã <math>r = \sqrt{(x-a)^2+(y-b)^2}</math> ãšãªãç¹ <math>(x,y)</math> ã®éåã§ããããã®æ¹çšåŒã®äž¡èŸºã¯æ£ãªã®ã§2ä¹ããŠ
<math>
(x-a)^2+(y-b)^2 = r^2
</math>
ãåŸãããããåã®æ¹çšåŒã§ããã
ç¹ã«åç¹ãäžå¿ã§ååŸ <math>r</math> ã®åã®æ¹çšåŒã¯ <math>
x^2+y^2 = r^2
</math> ã§äžããããã
'''æŒç¿åé¡'''
# äžå¿ <math>(2,4)</math> ååŸ <math>3</math> ã®åã®æ¹çšåŒãæ±ãã
# å <math>
y^2+2\,y+x^2-6\,x+5=0
</math> ã®äžå¿ãšååŸãæ±ãã
'''解ç'''
# <math>
(x-2)^2+(y-4)^2 = 9
</math>
# <math>
y^2+2\,y+x^2-6\,x+5=0 \iff (x-3)^2 + (y +1)^2 = 5
</math> ãªã®ã§äžå¿ <math>
(3,-1)
</math> ååŸ <math>
\sqrt 5
</math>
æ¹çšåŒ <math>x^2+y^2+lx+my+n = 0</math> ã¯ãã€ãåã§ãããšã¯éããªãã
æ¹çšåŒãå€åœ¢ã㊠<math>(x-a)^2+(y-b)^2 = k</math> ãšãªããšã
# <math>k>0</math> ã®ãšãæ¹çšåŒã¯åãè¡šã
# <math>k=0</math> ã®ãšãæ¹çšåŒã¯1ç¹ <math>(a,b)</math> ãè¡šã
# <math>k<0</math> ã®ãšãæ¹çšåŒã®å·ŠèŸºã¯åžžã«æ£ãªã®ã§ãæ¹çšåŒã®è¡šãå³åœ¢ã¯ãªã
==== åã®æ¥ç· ====
å<math>x^2+y^2=r^2</math>äžã®ããç¹<math>(x_1,y_1)</math>ã§æ¥ããæ¥ç·ã®æ¹çšåŒã¯
:<math>x_1x+y_1y=r^2</math>
ã§è¡šãããã
åæ§ã«ãå<math>(x-a)^2+(y-b)^2=r^2</math>äžã®ããç¹<math>(x_2,y_2)</math>ã§æ¥ããæ¥ç·ã®æ¹çšåŒã¯
:<math>(x_2-a)(x-a)+(y_2-b)(y-b)=r^2</math>
ã§è¡šãããã
====åãšçŽç·====
åãšçŽç·ã®äœçœ®é¢ä¿ã«ã€ããŠå€§ãã次ã®3ã€ã«åé¡ããããšãã§ããã
# åãšçŽç·ã2ç¹ã§äº€ãã(çŽç·ãåã®å
éšãéã)
# åãšçŽç·ã1ç¹ã§äº€ãã(çŽç·ãåã®æ¥ç·ãšãªã)
# åãšçŽç·ã¯äº€ãããªã
<!-- ããããã®äœçœ®é¢ä¿ã®å³ -->
äžè¬ã®åãšçŽç·ã«ã€ããŠãããã®äœçœ®é¢ä¿ãåé¡ããŠã¿ããã
å <math>C:(x-p)^2+(y-q)^2 = r^2</math> ãšçŽç· <math>l:ax+by+c=0</math> ã«ã€ããŠãå <math>C</math> ã®äžå¿ <math>(p,q)</math> ãšçŽç· <math>l</math> ã®è·é¢ <math>d := \frac{|aq+bq+c|}{\sqrt{a^2+b^2}}</math> ãšãããšã
# <math>r>d</math> ã®ãšããå <math>C</math> ãšçŽç· <math>l</math> ã¯2ç¹ã§äº€ãã
# <math>r=d</math> ã®ãšããå <math>C</math> ãšçŽç· <math>l</math> ã¯1ç¹ã§äº€ãã
# <math>r<d</math> ã®ãšããå <math>C</math> ãšçŽç· <math>l</math> ã¯äº€ãããªã
ä»ã«ããåã®æ¹çšåŒãšçŽç·ã®æ¹çšåŒãé£ç«ããŠãã®å®æ°è§£ã®åæ°ã§åé¡ããæ¹æ³ãããããäœçœ®é¢ä¿ãæ±ããã ããªãäžã®æ¹æ³ã®ã»ããèšç®éãå°ãªãã
'''æŒç¿åé¡'''
çŽç· <math>
3x + 4y =1
</math> ãšå <math>
(x-3)^2 + (y+2)^2 = 14
</math> ã®äº€ç¹ã®åº§æšãæ±ãã
'''解ç'''
çŽç·ã®æ¹çšåŒã <math>x</math> ã«ã€ããŠè§£ãããããåã®æ¹çšåŒã«ä»£å
¥ããã°ããã
çã㯠<math>(2,-1),\left(-\frac{14}{5},\frac{7}{5}\right)</math>
==è»è·¡ãšé å==
===è»è·¡ãšæ¹çšåŒ===
ããæ¡ä»¶ãæºããç¹å
šäœãã€ããå³åœ¢ãããã®æ¡ä»¶ãæºããç¹ã®'''è»è·¡'''ãšããã
*åé¡äŸ
**åé¡
2ç¹<math>\mathrm{A}(1\ ,\ 0)\ ,\ \mathrm{B}(3\ ,\ 2)</math>ããçè·é¢ã«ããç¹<math>\mathrm{P}</math>ã®è»è·¡ãæ±ããã
**解ç
æ¡ä»¶<math>\mathrm{A} \mathrm{P} = \mathrm{B} \mathrm{P}</math>ããã<math>\mathrm{A} \mathrm{P} ^2 = \mathrm{B} \mathrm{P} ^2</math><br>
<math>\mathrm{P}</math>ã®åº§æšã<math>(x\ ,\ y)</math>ãšãããš
:<math>
\mathrm{A} \mathrm{P} ^2 =(x-1)^2+y^2
</math>
:<math>
\mathrm{B} \mathrm{P} ^2 =(x-3)^2+(y-2)^2
</math>
ã ãã
:<math>
(x-1)^2+y^2=(x-3)^2+(y-2)^2
</math>
æŽçããŠã
:<math>
y=-x+3
</math>
ãããã£ãŠãæ±ããè»è·¡ã¯ãçŽç·<math>y=-x+3</math>ã§ããã
{| style="border:2px solid orange;width:80%" cellspacing=0
|style="background:orange"|'''è»è·¡ãæ±ããæé '''
|-
|style="padding:5px"|
1.æ±ããè»è·¡äžã®ä»»æã®ç¹ã®åº§æšã<math>(x\ ,\ y)</math>ãªã©ã§è¡šããäžããããæ¡ä»¶ã座æšã®éã®é¢ä¿åŒã§è¡šãã
2.è»è·¡ã®æ¹çšåŒãå°ãããã®æ¹çšåŒã®è¡šãå³åœ¢ãæ±ããã
3.ãã®å³åœ¢äžã®ç¹ãæ¡ä»¶ãæºãããŠããããšã確ãããã
|}
*åé¡äŸ
**åé¡
2ç¹<math>\mathrm{A}(0\ ,\ 0)\ ,\ \mathrm{B}(3\ ,\ 0)</math>ããã®è·é¢ã®æ¯ã<math>2:1</math>ã§ããç¹<math>\mathrm{P}</math>ã®è»è·¡ãæ±ããã
**解ç
<math>\mathrm{P}</math>ã®åº§æšã<math>(x\ ,\ y)</math>ãšããã<br>
<math>\mathrm{P}</math>ãæºããæ¡ä»¶ã¯
:<math>
\mathrm{A} \mathrm{P} : \mathrm{B} \mathrm{P} =2:1
</math>
ããªãã¡
:<math>
\mathrm{A} \mathrm{P} =2 \mathrm{B} \mathrm{P}
</math>
ããã座æšã§è¡šããš
:<math>
\sqrt{x^2+y^2} =2 \sqrt{(x-3)^2+y^2}
</math>
䞡蟺ã2ä¹ããŠãæŽçãããš
:<math>
x^2+y^2-8x+12=0
</math>
ããªãã¡
:<math>
(x-4)^2+y^2=2^2
</math>
ãããã£ãŠãæ±ããè»è·¡ã¯ãäžå¿ã<math>(4\ ,\ 0)</math>ãååŸã<math>2</math>ã®åã§ããã
<math>m\ ,\ n</math>ãç°ãªãæ£ã®æ°ãšãããšãã2ç¹<math>\mathrm{A}\ ,\ \mathrm{B}</math>ããã®è·é¢ã®æ¯ã<math>m:n</math>ã§ããç¹ã®è»è·¡ã¯ãç·å<math>\mathrm{A} \mathrm{B}</math>ã<math>m:n</math>ã«å
åããç¹ãšãå€åããç¹ãçŽåŸã®äž¡ç«¯ãšããåã§ããããã®åã'''ã¢ããããŠã¹ã®å'''ãšããã
<math>m=n</math>ã®ãšãã¯ãç·å<math>\mathrm{A} \mathrm{B}</math>ã®åçŽäºçåç·ã§ããã
=== äžçåŒã®è¡šãé å ===
== ã³ã©ã ==
[[File:Frans Hals - Portret van René Descartes.jpg|thumb|ãã«ã«ã]]
ãã®ããŒãžã®åéã®ããã«ãæ°åŒãã€ãã£ãŠåº§æšã®äœçœ®ããããããŠã幟äœåŠã®åé¡ã解ãææ³ã®ããšã解æ幟äœåŠãšããã
ãªãã幟äœåŠãšããèšèèªäœã¯ãå³åœ¢ã®åŠåãšãããããªæå³ã§ãããå°åŠæ ¡ãäžåŠæ ¡ã§ç¿ã£ãå³åœ¢ã®çè«ã幟äœåŠã§ããã
äžäžãšãŒãããã®æ°åŠè
ãã«ã«ããã解æ幟äœåŠã®ç 究ãé²ããããªãããã«ã«ãã¯ãå²åŠã®æ Œèšãããæããããã«æãããã§ãæåã§ããã
== æŒç¿åé¡ ==
== è泚 ==
<references/>
{{DEFAULTSORT:ãããšããã€ããããããII ããããšã»ããŠããã}}
[[Category:é«çåŠæ ¡æ°åŠII|ããããšã»ããŠããã]]
[[ã«ããŽãª:å³åœ¢]] | 2005-05-04T09:25:38Z | 2024-03-29T02:06:20Z | [
"ãã³ãã¬ãŒã:Pathnav"
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E6%95%B0%E5%AD%A6II/%E5%9B%B3%E5%BD%A2%E3%81%A8%E6%96%B9%E7%A8%8B%E5%BC%8F |
1,904 | æçæ¬/æ¥æ¬æç | W:æ¥æ¬æç | [
{
"paragraph_id": 0,
"tag": "p",
"text": "W:æ¥æ¬æç",
"title": ""
}
] | W:æ¥æ¬æç | {{é²æç¶æ³}}
[[W:æ¥æ¬æç]]
==åé¡==
*[[æçæ¬/粟é²æç]]
*[[æçæ¬/æç³æç]]
*[[æçæ¬/äŒåžæç]]
==ãžã£ã³ã«å¥==
===[[æçæ¬/ç±³æç|ç±³æç]]===
*[[æçæ¬/米飯|ã飯]]{{é²æ|50%|2019-08-26}}
*[[æçæ¬/赀飯|赀飯]]
*[[æçæ¬/ç²¥|ç²¥]]
*[[æçæ¬/åµããã飯|åµããã飯]]{{é²æ|75%|2016-01-06}}
*[[æçæ¬/ãã«ãã|ãã«ãã]]{{é²æ|75%|2019-08-26}}
===[[æçæ¬/ãã|ãã]]===
*[[æçæ¬/æ¡ã寿åž|æ¡ã寿åž]]
*[[æçæ¬/倪巻ã寿åž|倪巻ã寿åž]]
*[[æçæ¬/现巻ã寿åž|现巻ã寿åž]]
*[[æçæ¬/æå·»ã寿åž|æå·»ã寿åž]]
*[[æçæ¬/æŒã寿åž|æŒã寿åž]]
*[[æçæ¬/æ±æžåæ£ãã寿åž|æ±æžåæ£ãã寿åž]]
*[[æçæ¬/äºç®ã°ã寿åž|äºç®ã°ã寿åž]]
*[[æçæ¬/çš²è·å¯¿åž|çš²è·å¯¿åž]]
*[[æçæ¬/æãŸã寿åž|æãŸã寿åž]]
*[[æçæ¬/ãªã寿åž|ãªã寿åž]]
===[[æçæ¬/åºèº«|åºèº«]]===
===[[æçæ¬/éæç|éæç]]===
*[[æçæ¬/ãã§ã|ãã§ã]]{{é²æ|50%|2007-10-18}}
*[[æçæ¬/å¯ãé|å¯ãé]]
*[[æçæ¬/ã¡ãããé|ã¡ãããé]]
*[[æçæ¬/ããçŒã|ããçŒã]]
*[[æçæ¬/ããã¶é|ããã¶é]]
*[[æçæ¬/ã¡ãé|ã¡ãé]]
===[[æçæ¬/麺æç|麺æç]]===
*[[æçæ¬/ãã°|ãã°]]
*[[æçæ¬/ãã©ã|ãã©ã]]
*[[æçæ¬/ã©ãŒã¡ã³|ã©ãŒã¡ã³]]
*[[æçæ¬/ãããã|ãããã]]
*[[æçæ¬/å·ã麊|å·ã麊]]
===æ±ç©===
*[[æçæ¬/å³åæ±|å³åæ±]]{{é²æ|75%|2019-08-28}}
*[[æçæ¬/åžãç©|åžãç©]]{{é²æ|75%|2019-08-28}}
===[[æçæ¬/æãç©|æãç©]]===
*[[æçæ¬/倩麩çŸ
|倩麩çŸ
]]{{é²æ|75%|2019-08-29}}
===[[æçæ¬/çŒãç©|çŒãç©]]===
*[[æçæ¬/ç
§ãçŒã|ç
§ãçŒã]]
===[[æçæ¬/ç
®ç©|ç
®ç©]]===
===[[æçæ¬/åãç©ã»ãã²ãã|åãç©ã»ãã²ãã]]===
===ãã®ä»===
*[[æçæ¬/挬ãç©|挬ãç©]]{{é²æ|00%|2019-08-28}}
**[[æ¢
å¹²ã]]{{é²æ|75%|2019-08-28}}
*[[æçæ¬/è±è
|è±è
]]{{é²æ|00%|2019-08-28}}
*[[æçæ¬/é€å|é€å]]
==å°åå¥==
=== åæµ·éã»æ±åå°æ¹ ===
*[[æçæ¬/åæµ·éã®é·åæç|åæµ·éã®é·åæç]]
*[[æçæ¬/é森çã®é·åæç|é森çã®é·åæç]]
*[[æçæ¬/岩æçã®é·åæç|岩æçã®é·åæç]]
*[[æçæ¬/å®®åçã®é·åæç|å®®åçã®é·åæç]]
*[[æçæ¬/ç§ç°çã®é·åæç|ç§ç°çã®é·åæç]]
*[[æçæ¬/山圢çã®é·åæç|山圢çã®é·åæç]]
*[[æçæ¬/çŠå³¶çã®é·åæç|çŠå³¶çã®é·åæç]]
=== é¢æ±å°æ¹===
*[[æçæ¬/èšåçã®é·åæç|èšåçã®é·åæç]]
*[[æçæ¬/æ æšçã®é·åæç|æ æšçã®é·åæç]]
*[[æçæ¬/矀銬çã®é·åæç|矀銬çã®é·åæç]]
*[[æçæ¬/åŒççã®é·åæç|åŒççã®é·åæç]]
*[[æçæ¬/åèçã®é·åæç|åèçã®é·åæç]]
*[[æçæ¬/æ±äº¬éœã®é·åæç|æ±äº¬éœã®é·åæç]]
*[[æçæ¬/ç¥å¥å·çã®é·åæç|ç¥å¥å·çã®é·åæç]]
===äžéšå°æ¹===
*[[æçæ¬/æ°æœçã®é·åæç|æ°æœçã®é·åæç]]
*[[æçæ¬/é·éçã®é·åæç|é·éçã®é·åæç]]
*[[æçæ¬/山梚çã®é·åæç|山梚çã®é·åæç]]
*[[æçæ¬/å¯å±±çã®é·åæç|å¯å±±çã®é·åæç]]
*[[æçæ¬/ç³å·çã®é·åæç|ç³å·çã®é·åæç]]
*[[æçæ¬/çŠäºçã®é·åæç|çŠäºçã®é·åæç]]
*[[æçæ¬/é岡çã®é·åæç|é岡çã®é·åæç]]
*[[æçæ¬/æç¥çã®é·åæç|æç¥çã®é·åæç]]
*[[æçæ¬/å²éçã®é·åæç|å²éçã®é·åæç]]
===è¿ç¿å°æ¹===
*[[æçæ¬/äžéçã®é·åæç|äžéçã®é·åæç]]
*[[æçæ¬/æ»è³çã®é·åæç|æ»è³çã®é·åæç]]
*[[æçæ¬/京éœåºã®é·åæç|京éœåºã®é·åæç]]
*[[æçæ¬/倧éªåºã®é·åæç|倧éªåºã®é·åæç]]
*[[æçæ¬/å
µåº«çã®é·åæç|å
µåº«çã®é·åæç]]
*[[æçæ¬/å¥è¯çã®é·åæç|å¥è¯çã®é·åæç]]
*[[æçæ¬/åæå±±çã®é·åæç|åæå±±çã®é·åæç]]
===äžåœå°æ¹===
*[[æçæ¬/é³¥åçã®é·åæç|é³¥åçã®é·åæç]]
*[[æçæ¬/å³¶æ ¹çã®é·åæç|å³¶æ ¹çã®é·åæç]]
*[[æçæ¬/岡山çã®é·åæç|岡山çã®é·åæç]]
*[[æçæ¬/åºå³¶çã®é·åæç|åºå³¶çã®é·åæç]]
*[[æçæ¬/å±±å£çã®é·åæç|å±±å£çã®é·åæç]]
===ååœå°æ¹===
*[[æçæ¬/埳島çã®é·åæç|埳島çã®é·åæç]]
*[[æçæ¬/éŠå·çã®é·åæç|éŠå·çã®é·åæç]]
*[[æçæ¬/æåªçã®é·åæç|æåªçã®é·åæç]]
*[[æçæ¬/é«ç¥çã®é·åæç|é«ç¥çã®é·åæç]]
===ä¹å·ã»æ²çžå°æ¹===
*[[æçæ¬/çŠå²¡çã®é·åæç|çŠå²¡çã®é·åæç]]
*[[æçæ¬/äœè³çã®é·åæç|äœè³çã®é·åæç]]
*[[æçæ¬/é·åŽçã®é·åæç|é·åŽçã®é·åæç]]
*[[æçæ¬/倧åçã®é·åæç|倧åçã®é·åæç]]
*[[æçæ¬/çæ¬çã®é·åæç|çæ¬çã®é·åæç]]
*[[æçæ¬/å®®åŽçã®é·åæç|å®®åŽçã®é·åæç]]
*[[æçæ¬/鹿å
島çã®é·åæç|鹿å
島çã®é·åæç]]
*[[æçæ¬/æ²çžçã®é·åæç|æ²çžçã®é·åæç]]
[[Category:æ¥æ¬æç|*]]
[[Category:æ¥æ¬|ãããã]]
[[en:Cookbook:Cuisine of Japan]] | 2005-05-05T00:31:48Z | 2023-09-26T12:37:11Z | [
"ãã³ãã¬ãŒã:é²æç¶æ³",
"ãã³ãã¬ãŒã:é²æ"
] | https://ja.wikibooks.org/wiki/%E6%96%99%E7%90%86%E6%9C%AC/%E6%97%A5%E6%9C%AC%E6%96%99%E7%90%86 |
1,913 | é«çåŠæ ¡æ°åŠII/äžè§é¢æ° | ããã§ã¯äžè§é¢æ°ã®å®çŸ©ãããããšãäžè§é¢æ°ã®åºæ¬çãªæ§è³ªãå æ³å®çãäžè§é¢æ°ã®å¿çšã«ã€ããŠåŠã¶ãäžè§é¢æ°ã¯æ³¢ããã¯ãã«ã®å
ç©ãããŒãªãšå€æãªã©ããŸããŸãªåéã§å¿çšãããŠããã
å³å³ã®ããã«ãå®ç¹Oãäžå¿ãšããŠå転ããåçŽç· OP ãèããããã®ãšãã®å転ããåçŽç· OP ã®ããšãååŸãšããã
åçŽç· OX ãè§åºŠã®åºæºãšããããã®åºæºãšãªãåçŽç· OX ã®ããšãå§ç·ãšããã
ååŸãæèšåãã«å転ããå Žåãå転ããè§åºŠã¯è² ã§ãããšããååŸãåæèšåããããå Žåãå転ããè§åºŠã¯æ£ã§ãããšããã
è² ã®è§åºŠã360°以äžå転ããè§åºŠãèãã«å
¥ããè§ã®ããšãäžè¬è§ãšããã
ããŸãŸã§ã¯è§åºŠã®åäœãšããŠäžåšã 360° ãšãã床æ°æ³ã䜿ã£ãŠããããšã ãããããã§ã匧床æ³ã«ããè§åºŠã®è¡šãæ¹ãåŠã¶ã
ååŸ1 ã®æ圢ã«ãããŠåŒ§ã®é·ãã 1 ã®ãšãã®äžå¿è§ã 1 radãåæ§ã«åŒ§ã®é·ããΞã®ãšãã®äžå¿è§ãΞ radãšå®çŸ©ããããã®å®çŸ©ãã 180° =Ï radã360° = 2Ï rad ãããã«
ãšãªãããŸã匧床æ³ã®åäœ(rad)ã¯ãã°ãã°çç¥ãããã
匧床æ³ãçšãããšãäžè§é¢æ°ã®åŸ®ç©åãèããéã«äŸ¿å©ã§ããã(ãã®ããšã¯æ°åŠIIIã§åŠã¶)
æ圢ã®ååŸãr ã匧床æ³ã§å®çŸ©ãããè§åºŠãΞãšãããšãã匧ã®é·ãl ãšé¢ç©S ã¯
ãšè¡šããã
äžè¬è§ã Ξ {\displaystyle \theta } ã®åçŽç·ãšåäœåã亀ããåã P {\displaystyle \mathrm {P} } ãšããããã®ãšãã® P {\displaystyle \mathrm {P} } ã®åº§æšã ( cos Ξ , sin Ξ ) {\displaystyle (\cos \theta ,\sin \theta )} ãšããããšã§ãé¢æ° sin , cos {\displaystyle \sin ,\cos } ãå®ããããŸãã tan Ξ = sin Ξ cos Ξ {\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}} ãšããããšã§é¢æ° tan Ξ {\displaystyle \tan \theta } ãå®ããã tan Ξ {\displaystyle \tan \theta } ã¯äžè¬è§ã Ξ {\displaystyle \theta } ã®ååŸã®åŸãã«çããã
ãŸããäžè§é¢æ°ã®çŽ¯ä¹ã¯ ( sin Ξ ) n = sin n Ξ {\displaystyle (\sin \theta )^{n}=\sin ^{n}\theta } ãšè¡šèšãããã
cos Ξ ã®ã°ã©ã㯠sin Ξ ã®ã°ã©ãã Ξ軞æ¹åã« â Ï 2 {\displaystyle -{\frac {\pi }{2}}} ã ãå¹³è¡ç§»åãããã®ã§ããã
y = sin Ξ {\displaystyle y=\sin \theta } ã y = cos Ξ {\displaystyle y=\cos \theta } ã®åœ¢ãããæ²ç·ã®ããšã æ£åŒŠæ²ç· (ããããããããã)ãšããã
é¢æ° sin , cos {\displaystyle \sin ,\cos } ã®å€åã¯ã©ã¡ããã [ â 1 , 1 ] {\displaystyle [-1,1]} ã§ããã
å³å³ã®ããã« ãè§ Îž ã®ååŸãšåäœåãšã®äº€ç¹ãPãšããŠã çŽç·OPãš çŽç·x=1 ãšã®äº€ç¹ã T ãšãããšã Tã®åº§æšã¯
ã«ãªãã
ãã®ããšãå©çšããŠã y=tan Ξ ã®ã°ã©ããããããšãã§ããã
y=tan Ξ ã®ã°ã©ãã¯ãäžå³ã®ããã«ãªãã
y=tan Ξ ã®ã°ã©ãã§ã¯ãΞã®å€ã Ï 2 {\displaystyle {\frac {\pi }{2}}} ã«è¿ã¥ããŠãããšã çŽç· Ξ = Ï 2 {\displaystyle \theta ={\frac {\pi }{2}}} ã«éããªãè¿ã¥ããŠããã
ãã®ããã«ãæ²ç·ãããçŽç·ã«éãç¡ãè¿ã¥ããŠãããšããè¿ã¥ãããçŽç·ã®ã»ãã 挞è¿ç· (ãããããã)ãšããã
åæ§ã«èãã次ã®çŽç·ã y=tanΞ ã®æŒžè¿ç·ã§ããã
㯠y=tanΞ ã®æŒžè¿ç·ã§ããã
äžè¬ã«ã
ã¯y=tanΞã®ã°ã©ãã®æŒžè¿ç·ã§ããã
äžè¬è§ã Ξ {\displaystyle \theta } ã®ååŸã¯äžå転ããŠãçããã®ã§ãäžè¬è§ã Ξ + 2 Ï {\displaystyle \theta +2\pi } ã®ååŸãšçãããããããäžè§é¢æ°ã®åšææ§
sin ( Ξ + 2 Ï n ) = sin Ξ cos ( Ξ + 2 Ï n ) = cos Ξ tan ( Ξ + 2 Ï n ) = tan Ξ {\displaystyle {\begin{aligned}\sin(\theta +2\pi n)&=\sin \theta \\\cos(\theta +2\pi n)&=\cos \theta \\\tan(\theta +2\pi n)&=\tan \theta \end{aligned}}}
ãåŸãã
ç¹ ( cos Ξ , sin Ξ ) {\displaystyle (\cos \theta ,\sin \theta )} ã Ï {\displaystyle \pi } å転ããç¹ ( cos ( Ξ + Ï ) , sin ( Ξ + Ï ) ) {\displaystyle (\cos(\theta +\pi ),\sin(\theta +\pi ))} ã¯åç¹ãäžå¿ã«ç¹å¯Ÿç§°ç§»åããç¹ ( â cos Ξ , â sin Ξ ) {\displaystyle (-\cos \theta ,-\sin \theta )} ã§ããããšãã
sin ( Ξ + Ï ) = â sin Ξ cos ( Ξ + Ï ) = â cos Ξ tan ( Ξ + Ï ) = tan Ξ {\displaystyle {\begin{aligned}\sin(\theta +\pi )&=-\sin \theta \\\cos(\theta +\pi )&=-\cos \theta \\\tan(\theta +\pi )&=\tan \theta \end{aligned}}}
ãåŸãã
ç¹ ( cos Ξ , sin Ξ ) {\displaystyle (\cos \theta ,\sin \theta )} ã x {\displaystyle x} 軞ã§ç·å¯Ÿç§°ç§»å移åããç¹ã ( cos ( â Ξ ) , sin ( â Ξ ) ) = ( cos Ξ , â sin Ξ ) {\displaystyle (\cos(-\theta ),\sin(-\theta ))=(\cos \theta ,-\sin \theta )} ã§ããããšãã
sin ( â Ξ ) = â sin Ξ cos ( â Ξ ) = cos Ξ tan ( â Ξ ) = â tan Ξ {\displaystyle {\begin{aligned}\sin(-\theta )&=-\sin \theta \\\cos(-\theta )&=\cos \theta \\\tan(-\theta )&=-\tan \theta \end{aligned}}}
ãåŸãã
åäœååšäžã®ç¹ ( cos Ξ , sin Ξ ) {\displaystyle (\cos \theta ,\sin \theta )} ããåç¹ãŸã§ã®è·é¢ã¯ 1 ãªã®ã§ã sin 2 Ξ + cos 2 Ξ = 1 {\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1} ãæãç«ã€ã
ãŸãããã®åŒã«ã tan Ξ = sin Ξ cos Ξ {\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}} ã€ãŸãã sin Ξ = tan Ξ cos Ξ {\displaystyle \sin \theta =\tan \theta \cos \theta } ã代å
¥ããã°ã 1 + tan 2 Ξ = 1 cos 2 Ξ {\displaystyle 1+\tan ^{2}\theta ={\frac {1}{\cos ^{2}\theta }}} ãæãç«ã€ããšããããã
é¢æ° f ( x ) {\displaystyle f(x)} ã«å¯ŸããŠã0 ã§ãªãå®æ° p {\displaystyle p} ãååšããŠã f ( x + p ) = f ( x ) {\displaystyle f(x+p)=f(x)} ãšãªããšãé¢æ° f ( x ) {\displaystyle f(x)} ã¯åšæé¢æ°ãšãããå®æ° p {\displaystyle p} ãäžã®æ§è³ªãæºãããšãã â p , 2 p {\displaystyle -p,2p} ãªã©ãå®æ° p {\displaystyle p} ã0ãé€ãæŽæ°åããæ°ãäžã®æ§è³ªãæºãããããã§ãåšæé¢æ°ãç¹åŸŽã¥ããéãšããŠãäžã®æ§è³ªãæºããå®æ° p {\displaystyle p} ã®å
ãæ£ã§ãã€æå°ã®ãã®ãéžã³ããããåšæãšåŒã¶ã
sin x , cos x {\displaystyle \sin x,\cos x} ã¯åšæã 2 Ï {\displaystyle 2\pi } ãšããåšæé¢æ°ã§ããã tan x {\displaystyle \tan x} ã¯åšæã Ï {\displaystyle \pi } ãšããåšæé¢æ°ã§ããã
æŒç¿åé¡
k {\displaystyle k} ã0ã§ãªãå®æ°ãšãããé¢æ° sin k x {\displaystyle \sin kx} ã®åšæãèšã
解ç
sin k ( x + 2 Ï k ) = sin k x {\displaystyle \sin k\left(x+{\frac {2\pi }{k}}\right)=\sin kx} ãªã®ã§çã㯠2 Ï k {\displaystyle {\frac {2\pi }{k}}} ãããã¯æ£ã§ãããåšæã®æå°æ§ã®æ¡ä»¶ãæºãããŠããã
é¢æ° f ( x ) {\displaystyle f(x)} ã f ( â x ) = f ( x ) {\displaystyle f(-x)=f(x)} ãæºãããšããé¢æ° f ( x ) {\displaystyle f(x)} ã¯å¶é¢æ°ãšãããå¶é¢æ°ã¯ y {\displaystyle y} 軞ã«é¢ããŠå¯Ÿç§°ãªã°ã©ãã«ãªãã
ãŸããé¢æ° f ( x ) {\displaystyle f(x)} ã f ( â x ) = â f ( x ) {\displaystyle f(-x)=-f(x)} ãæºãããšããé¢æ° f ( x ) {\displaystyle f(x)} ã¯å¥é¢æ°ãšãããå¶é¢æ°ã¯åç¹ã«é¢ããŠå¯Ÿè±¡ãªã°ã©ãã«ãªãã
é¢æ° cos Ξ , x 2 n {\displaystyle \cos \theta ,x^{2n}} ( n {\displaystyle n} ã¯æŽæ°)ã¯å¶é¢æ°ãšãªãã
é¢æ° sin x , x 2 n + 1 {\displaystyle \sin x,x^{2n+1}} ( n {\displaystyle n} ã¯æŽæ°)ã¯å¥é¢æ°ãšãªãã
tan Ξ {\displaystyle \tan \theta } ã¯å¶é¢æ°ããããšãå¥é¢æ°ã調ã¹ãã
解ç
ãªã®ã§ã tan Ξ {\displaystyle \tan \theta } ã¯å¥é¢æ°ã§ããã
é¢æ° y = sin ( Ξ â Ï 3 ) {\displaystyle y=\sin \left(\theta -{\frac {\pi }{3}}\right)} ã®ã°ã©ãã¯ã y = sin Ξ {\displaystyle y=\sin \theta } ã®ã°ã©ãã Ξ軞æ¹åã« Ï 3 {\displaystyle {\frac {\pi }{3}}} ã ãå¹³è¡ç§»åããããã®ã«ãªããåšæ㯠2 Ï {\displaystyle 2\pi } ã§ããã(å¹³è¡ç§»åããŠããåšæã¯å€ããããsinΞãšåããåšæ㯠2 Ï {\displaystyle 2\pi } ã®ãŸãŸã§ããã)
é¢æ° y=2sin Ξ ã®ã°ã©ãã®åœ¢ã¯ y=sin Ξ ãy軞æ¹åã«2åã«æ¡å€§ãããã®ã§ãåšæ㯠y=sin Ξ ãšåãã 2Ï ã§ããã
ãŒ1 ⊠sin Ξ ⊠1 ãªã®ã§ã
å€å㯠ãŒ2 ⊠2sin Ξ ⊠2 ã§ããã
é¢æ° y=sin2Ξ ã®ã°ã©ãã¯y軞ãåºæºã«Îžè»žæ¹åã« 1 2 {\displaystyle {\frac {1}{2}}} åã«çž®å°ãããã®ã«ãªã£ãŠããã
ãããã£ãŠãåšæã 1 2 {\displaystyle {\frac {1}{2}}} åã«ãªã£ãŠãããy=sinΞ ã®åšæ㯠2 Ï {\displaystyle 2\pi } ã ãããy=sin2Ξ ã®åšæã¯ Ï {\displaystyle \pi } ã§ããã
äžè§é¢æ°ã®å æ³å®ç
ãæãç«ã€ã
蚌æ
ä»»æã®å®æ° α , β {\displaystyle \alpha ,\beta } ã«å¯Ÿããåäœååšäžã®ç¹ A ( cos α , sin α ) , B ( cos β , sin β ) {\displaystyle \mathrm {A} (\cos \alpha ,\sin \alpha ),\mathrm {B} (\cos \beta ,\sin \beta )} ããšãããã®ãšãã ç·å A B {\displaystyle \mathrm {AB} } ã®é·ãã®2ä¹ A B 2 {\displaystyle \mathrm {AB} ^{2}} ã¯äœåŒŠå®çã䜿ãããšã«ãã
A B 2 = 2 â 2 cos ( α â β ) {\displaystyle \mathrm {AB} ^{2}=2-2\cos(\alpha -\beta )}
ã§ããã次ã«äžå¹³æ¹ã®å®çã䜿ã£ãŠ
A B 2 = ( cos α â cos α ) 2 + ( sin α â sin β ) 2 = 2 â 2 ( cos α cos β + sin α sin β ) {\displaystyle \mathrm {AB} ^{2}=(\cos \alpha -\cos \alpha )^{2}+(\sin \alpha -\sin \beta )^{2}=2-2(\cos \alpha \cos \beta +\sin \alpha \sin \beta )}
ãããæŽçããŠ
cos ( α â β ) = cos α cos β + sin α sin β {\displaystyle \cos(\alpha -\beta )=\cos \alpha \cos \beta +\sin \alpha \sin \beta }
ãåŸãã
cos ( α + β ) = cos ( α â ( â β ) ) = cos α cos ( â β ) + sin α sin ( â β ) = cos α cos β â sin α sin β {\displaystyle \cos(\alpha +\beta )=\cos(\alpha -(-\beta ))=\cos \alpha \cos(-\beta )+\sin \alpha \sin(-\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta }
ã§ããã
以äžããŸãšããŠ
cos ( α ± β ) = cos α cos β â sin α sin β {\displaystyle \cos(\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta }
ãåŸãã
ããã§ã
sin ( α ± β ) = â cos ( α + Ï 2 ± β ) = â { cos ( α + Ï 2 ) cos ( β ) â sin ( α + Ï 2 ) sin β } = sin α cos β ± cos α sin β {\displaystyle \sin(\alpha \pm \beta )=-\cos(\alpha +{\frac {\pi }{2}}\pm \beta )=-\{\cos(\alpha +{\frac {\pi }{2}})\cos(\beta )\mp \sin(\alpha +{\frac {\pi }{2}})\sin \beta \}=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta }
ããã«ã tan x {\displaystyle \tan x} ã«ã€ããŠã
tan ( α ± β ) = sin ( α ± β ) cos ( α ± β ) = sin α cos β ± cos α sin β cos α cos β â sin α sin β = sin α cos β cos α cos β ± cos α sin β cos α cos β cos α cos β cos α cos β â sin α sin β cos α cos β = tan α ± tan β 1 â tan α tan β {\textstyle {\begin{aligned}\tan(\alpha \pm \beta )&={\frac {\sin(\alpha \pm \beta )}{\cos(\alpha \pm \beta )}}\\&={\frac {\sin \alpha \cos \beta \pm \cos \alpha \sin \beta }{\cos \alpha \cos \beta \mp \sin \alpha \sin \beta }}\\&={\cfrac {{\cfrac {\sin \alpha \cos \beta }{\cos \alpha \cos \beta }}\pm {\cfrac {\cos \alpha \sin \beta }{\cos \alpha \cos \beta }}}{{\cfrac {\cos \alpha \cos \beta }{\cos \alpha \cos \beta }}\mp {\cfrac {\sin \alpha \sin \beta }{\cos \alpha \cos \beta }}}}\\&={\frac {\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }}\end{aligned}}}
ãæãç«ã€ã
å æ³å®çãçšããŠä»¥äžã蚌æã§ããã
sin 2 α = sin ( α + α ) = 2 sin α cos α {\displaystyle \sin 2\alpha =\sin(\alpha +\alpha )=2\sin \alpha \cos \alpha }
cos 2 α = cos ( α + α ) = cos 2 α â sin 2 α = 2 cos 2 α â 1 = 1 â 2 sin 2 α {\displaystyle \cos 2\alpha =\cos(\alpha +\alpha )=\cos ^{2}\alpha -\sin ^{2}\alpha =2\cos ^{2}\alpha -1=1-2\sin ^{2}\alpha }
tan 2 α = 2 tan α 1 â tan 2 α {\displaystyle \tan 2\alpha ={\frac {2\tan \alpha }{1-\tan ^{2}\alpha }}}
次ã«ã cos {\displaystyle \cos } ã®åè§ã®å
¬åŒãå€åœ¢ãããš
sin 2 α = 1 â cos 2 α 2 {\displaystyle \sin ^{2}\alpha ={\frac {1-\cos 2\alpha }{2}}}
cos 2 α = 1 + cos 2 α 2 {\displaystyle \cos ^{2}\alpha ={\frac {1+\cos 2\alpha }{2}}}
ã§ããã
æŒç¿åé¡
解ç
sin 15 â = sin ( 45 â â 30 â ) = 6 â 2 4 {\displaystyle \sin 15^{\circ }=\sin(45^{\circ }-30^{\circ })={\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}
cos 15 â = cos ( 45 â â 30 â ) = 6 + 2 4 {\displaystyle \cos 15^{\circ }=\cos(45^{\circ }-30^{\circ })={\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}
tan 2 α = sin 2 α cos 2 α = 1 â cos 2 α 1 + cos 2 α {\displaystyle \tan ^{2}\alpha ={\frac {\sin ^{2}\alpha }{\cos ^{2}\alpha }}={\frac {1-\cos 2\alpha }{1+\cos 2\alpha }}}
ä»ãŸã§ã®å®çããŸãšãããšã次ã®ããã«ãªãã
èŠãæ¹
å æ³å®çã¯ãå²ããã³ã¹ã¢ã¹ã³ã¹ã¢ã¹å²ãããããã³ã¹ã¢ã¹ã³ã¹ã¢ã¹å²ããå²ããããšããèªååãããããŸãã
cos {\displaystyle \cos } ã®åè§ã®å
¬åŒ cos 2 Ξ = 2 cos 2 Ξ â 1 = 1 â 2 sin 2 Ξ {\displaystyle \cos 2\theta =2\cos ^{2}\theta -1=1-2\sin ^{2}\theta } 㯠± 1 â 2 a a a 2 Ξ {\displaystyle \pm 1\mp 2\mathrm {aaa} ^{2}\theta } ãšãã圢ãèŠã㊠sin {\displaystyle \sin } ã¯ç¬Šå·ã â {\displaystyle -} ã1 ã®ç¬Šå·ã¯ãã®éãšèŠããŸãã
2ä¹ã®äžè§é¢æ° sin 2 Ξ = 1 â cos 2 Ξ 2 , cos 2 Ξ = 1 + cos 2 Ξ 2 {\displaystyle \sin ^{2}\theta ={\frac {1-\cos 2\theta }{2}},\cos ^{2}\theta ={\frac {1+\cos 2\theta }{2}}} ã¯ã 1 ± cos 2 Ξ 2 {\displaystyle {\frac {1\pm \cos 2\theta }{2}}} ãšãã圢ãèŠããŠã sin {\displaystyle \sin } ã¯ç¬Šå·ã â {\displaystyle -} ãšèããŸãã
äžè§é¢æ°ã®å
ã«ãããŠã a , b â 0 {\displaystyle a,b\neq 0} ã®ãšã
{ a a 2 + b 2 } 2 + { b a 2 + b 2 } 2 = 1 {\displaystyle \left\{{\dfrac {a}{\sqrt {a^{2}+b^{2}}}}\right\}^{2}+\left\{{\dfrac {b}{\sqrt {a^{2}+b^{2}}}}\right\}^{2}=1} ã§ããã®ã§ãç¹ ( a a 2 + b 2 , b a 2 + b 2 ) {\displaystyle \left({\dfrac {a}{\sqrt {a^{2}+b^{2}}}},{\dfrac {b}{\sqrt {a^{2}+b^{2}}}}\right)} ã¯åäœååšäžã®ç¹ãªã®ã§ã
ãšãªããããªÎ±ããšãããšãã§ãããã®Î±ãçšããŠæ¬¡ã®ãããªå€åœ¢ãã§ããã
æŒç¿åé¡
r , α {\displaystyle r,\alpha } 㯠r > 0 , â Ï â€ Î± < Ï {\displaystyle r>0,-\pi \leq \alpha <\pi } ãæºãããšããã
解ç
sin Ξ â 3 cos Ξ = 2 ( 1 2 sin Ξ â 3 2 cos Ξ ) = 2 ( sin Ξ cos Ï 3 â cos Ξ sin Ï 3 ) = 2 sin ( Ξ â Ï 3 ) {\displaystyle {\begin{aligned}\sin \theta -{\sqrt {3}}\cos \theta &=2\left({\frac {1}{2}}\sin \theta -{\frac {\sqrt {3}}{2}}\cos \theta \right)\\&=2\left(\sin \theta \cos {\frac {\pi }{3}}-\cos \theta \sin {\frac {\pi }{3}}\right)\\&=2\sin \left(\theta -{\frac {\pi }{3}}\right)\\\end{aligned}}}
äžè§é¢æ°ã®å æ³å®çãçšãããšãäžè§é¢æ°ã®åâç©ã®å
¬åŒãããã³ç©âåã®å
¬åŒãåŸãããããããã
ãšãªãã
å æ³å®ç
ããã (1) + (2) ãã
(1) - (2) ãã
(3) + (4) ãã
(3) - (4) ãã
ãåŸãããã
A = α + β , B = α â β {\displaystyle A=\alpha +\beta ,\,B=\alpha -\beta } ãšãããšã α = A + B 2 , β = A â B 2 {\displaystyle \alpha ={\frac {A+B}{2}},\,\beta ={\frac {A-B}{2}}} ã§ããããããç©âåã®å
¬åŒã«ä»£å
¥ããã°ããããã
ãåŸãããã
èŠãæ¹
ç©âåã®å
¬åŒã¯ãäž2ã€ã¯ α {\displaystyle \alpha } 㚠β {\displaystyle \beta } ãå
¥ãæ¿ããã°åãåŒãªã®ã§ãèŠããã®ã¯3åŒã§ããã sin sin {\displaystyle \sin \sin } ã®å
¬åŒã¯ cos cos {\displaystyle \cos \cos } ã®å
¬åŒã®ç¬Šå·ã2〠â {\displaystyle -} ã«ãããã®ã«ãªã£ãŠããã
åâç©ã®å
¬åŒã¯ã a a a â a a a {\displaystyle {\rm {{aaa}-{\rm {aaa}}}}} ã®åŒã¯ a a a + a a a {\displaystyle {\rm {{aaa}+{\rm {aaa}}}}} ã®å
¬åŒã® cos {\displaystyle \cos } ãš sin {\displaystyle \sin } ãéã«ãã圢ã«ãªã£ãŠããã
(1)äžã®åºŠæ°æ³ã§è¡šãããå€ã匧床æ³ãŠè¡šã
1) 150 {\displaystyle 150} 2) 720 {\displaystyle 720}
(2) sin Ï / 2 {\displaystyle \sin \pi /2} ã®å€ãæ±ãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ããã§ã¯äžè§é¢æ°ã®å®çŸ©ãããããšãäžè§é¢æ°ã®åºæ¬çãªæ§è³ªãå æ³å®çãäžè§é¢æ°ã®å¿çšã«ã€ããŠåŠã¶ãäžè§é¢æ°ã¯æ³¢ããã¯ãã«ã®å
ç©ãããŒãªãšå€æãªã©ããŸããŸãªåéã§å¿çšãããŠããã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "å³å³ã®ããã«ãå®ç¹Oãäžå¿ãšããŠå転ããåçŽç· OP ãèããããã®ãšãã®å転ããåçŽç· OP ã®ããšãååŸãšããã",
"title": "äžè¬è§"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "åçŽç· OX ãè§åºŠã®åºæºãšããããã®åºæºãšãªãåçŽç· OX ã®ããšãå§ç·ãšããã",
"title": "äžè¬è§"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ååŸãæèšåãã«å転ããå Žåãå転ããè§åºŠã¯è² ã§ãããšããååŸãåæèšåããããå Žåãå転ããè§åºŠã¯æ£ã§ãããšããã",
"title": "äžè¬è§"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "è² ã®è§åºŠã360°以äžå転ããè§åºŠãèãã«å
¥ããè§ã®ããšãäžè¬è§ãšããã",
"title": "äžè¬è§"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "",
"title": "äžè¬è§"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ããŸãŸã§ã¯è§åºŠã®åäœãšããŠäžåšã 360° ãšãã床æ°æ³ã䜿ã£ãŠããããšã ãããããã§ã匧床æ³ã«ããè§åºŠã®è¡šãæ¹ãåŠã¶ã",
"title": "匧床æ³"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ååŸ1 ã®æ圢ã«ãããŠåŒ§ã®é·ãã 1 ã®ãšãã®äžå¿è§ã 1 radãåæ§ã«åŒ§ã®é·ããΞã®ãšãã®äžå¿è§ãΞ radãšå®çŸ©ããããã®å®çŸ©ãã 180° =Ï radã360° = 2Ï rad ãããã«",
"title": "匧床æ³"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "ãšãªãããŸã匧床æ³ã®åäœ(rad)ã¯ãã°ãã°çç¥ãããã",
"title": "匧床æ³"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "匧床æ³ãçšãããšãäžè§é¢æ°ã®åŸ®ç©åãèããéã«äŸ¿å©ã§ããã(ãã®ããšã¯æ°åŠIIIã§åŠã¶)",
"title": "匧床æ³"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "æ圢ã®ååŸãr ã匧床æ³ã§å®çŸ©ãããè§åºŠãΞãšãããšãã匧ã®é·ãl ãšé¢ç©S ã¯",
"title": "匧床æ³"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "ãšè¡šããã",
"title": "匧床æ³"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "äžè¬è§ã Ξ {\\displaystyle \\theta } ã®åçŽç·ãšåäœåã亀ããåã P {\\displaystyle \\mathrm {P} } ãšããããã®ãšãã® P {\\displaystyle \\mathrm {P} } ã®åº§æšã ( cos Ξ , sin Ξ ) {\\displaystyle (\\cos \\theta ,\\sin \\theta )} ãšããããšã§ãé¢æ° sin , cos {\\displaystyle \\sin ,\\cos } ãå®ããããŸãã tan Ξ = sin Ξ cos Ξ {\\displaystyle \\tan \\theta ={\\frac {\\sin \\theta }{\\cos \\theta }}} ãšããããšã§é¢æ° tan Ξ {\\displaystyle \\tan \\theta } ãå®ããã tan Ξ {\\displaystyle \\tan \\theta } ã¯äžè¬è§ã Ξ {\\displaystyle \\theta } ã®ååŸã®åŸãã«çããã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ãŸããäžè§é¢æ°ã®çŽ¯ä¹ã¯ ( sin Ξ ) n = sin n Ξ {\\displaystyle (\\sin \\theta )^{n}=\\sin ^{n}\\theta } ãšè¡šèšãããã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "cos Ξ ã®ã°ã©ã㯠sin Ξ ã®ã°ã©ãã Ξ軞æ¹åã« â Ï 2 {\\displaystyle -{\\frac {\\pi }{2}}} ã ãå¹³è¡ç§»åãããã®ã§ããã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "y = sin Ξ {\\displaystyle y=\\sin \\theta } ã y = cos Ξ {\\displaystyle y=\\cos \\theta } ã®åœ¢ãããæ²ç·ã®ããšã æ£åŒŠæ²ç· (ããããããããã)ãšããã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "é¢æ° sin , cos {\\displaystyle \\sin ,\\cos } ã®å€åã¯ã©ã¡ããã [ â 1 , 1 ] {\\displaystyle [-1,1]} ã§ããã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "å³å³ã®ããã« ãè§ Îž ã®ååŸãšåäœåãšã®äº€ç¹ãPãšããŠã çŽç·OPãš çŽç·x=1 ãšã®äº€ç¹ã T ãšãããšã Tã®åº§æšã¯",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "ã«ãªãã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "ãã®ããšãå©çšããŠã y=tan Ξ ã®ã°ã©ããããããšãã§ããã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "y=tan Ξ ã®ã°ã©ãã¯ãäžå³ã®ããã«ãªãã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "y=tan Ξ ã®ã°ã©ãã§ã¯ãΞã®å€ã Ï 2 {\\displaystyle {\\frac {\\pi }{2}}} ã«è¿ã¥ããŠãããšã çŽç· Ξ = Ï 2 {\\displaystyle \\theta ={\\frac {\\pi }{2}}} ã«éããªãè¿ã¥ããŠããã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "ãã®ããã«ãæ²ç·ãããçŽç·ã«éãç¡ãè¿ã¥ããŠãããšããè¿ã¥ãããçŽç·ã®ã»ãã 挞è¿ç· (ãããããã)ãšããã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "åæ§ã«èãã次ã®çŽç·ã y=tanΞ ã®æŒžè¿ç·ã§ããã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "㯠y=tanΞ ã®æŒžè¿ç·ã§ããã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "äžè¬ã«ã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "ã¯y=tanΞã®ã°ã©ãã®æŒžè¿ç·ã§ããã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "äžè¬è§ã Ξ {\\displaystyle \\theta } ã®ååŸã¯äžå転ããŠãçããã®ã§ãäžè¬è§ã Ξ + 2 Ï {\\displaystyle \\theta +2\\pi } ã®ååŸãšçãããããããäžè§é¢æ°ã®åšææ§",
"title": "äžè§é¢æ°ã®æ§è³ª"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "sin ( Ξ + 2 Ï n ) = sin Ξ cos ( Ξ + 2 Ï n ) = cos Ξ tan ( Ξ + 2 Ï n ) = tan Ξ {\\displaystyle {\\begin{aligned}\\sin(\\theta +2\\pi n)&=\\sin \\theta \\\\\\cos(\\theta +2\\pi n)&=\\cos \\theta \\\\\\tan(\\theta +2\\pi n)&=\\tan \\theta \\end{aligned}}}",
"title": "äžè§é¢æ°ã®æ§è³ª"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "ãåŸãã",
"title": "äžè§é¢æ°ã®æ§è³ª"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "ç¹ ( cos Ξ , sin Ξ ) {\\displaystyle (\\cos \\theta ,\\sin \\theta )} ã Ï {\\displaystyle \\pi } å転ããç¹ ( cos ( Ξ + Ï ) , sin ( Ξ + Ï ) ) {\\displaystyle (\\cos(\\theta +\\pi ),\\sin(\\theta +\\pi ))} ã¯åç¹ãäžå¿ã«ç¹å¯Ÿç§°ç§»åããç¹ ( â cos Ξ , â sin Ξ ) {\\displaystyle (-\\cos \\theta ,-\\sin \\theta )} ã§ããããšãã",
"title": "äžè§é¢æ°ã®æ§è³ª"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "sin ( Ξ + Ï ) = â sin Ξ cos ( Ξ + Ï ) = â cos Ξ tan ( Ξ + Ï ) = tan Ξ {\\displaystyle {\\begin{aligned}\\sin(\\theta +\\pi )&=-\\sin \\theta \\\\\\cos(\\theta +\\pi )&=-\\cos \\theta \\\\\\tan(\\theta +\\pi )&=\\tan \\theta \\end{aligned}}}",
"title": "äžè§é¢æ°ã®æ§è³ª"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "ãåŸãã",
"title": "äžè§é¢æ°ã®æ§è³ª"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "ç¹ ( cos Ξ , sin Ξ ) {\\displaystyle (\\cos \\theta ,\\sin \\theta )} ã x {\\displaystyle x} 軞ã§ç·å¯Ÿç§°ç§»å移åããç¹ã ( cos ( â Ξ ) , sin ( â Ξ ) ) = ( cos Ξ , â sin Ξ ) {\\displaystyle (\\cos(-\\theta ),\\sin(-\\theta ))=(\\cos \\theta ,-\\sin \\theta )} ã§ããããšãã",
"title": "äžè§é¢æ°ã®æ§è³ª"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "sin ( â Ξ ) = â sin Ξ cos ( â Ξ ) = cos Ξ tan ( â Ξ ) = â tan Ξ {\\displaystyle {\\begin{aligned}\\sin(-\\theta )&=-\\sin \\theta \\\\\\cos(-\\theta )&=\\cos \\theta \\\\\\tan(-\\theta )&=-\\tan \\theta \\end{aligned}}}",
"title": "äžè§é¢æ°ã®æ§è³ª"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "ãåŸãã",
"title": "äžè§é¢æ°ã®æ§è³ª"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "åäœååšäžã®ç¹ ( cos Ξ , sin Ξ ) {\\displaystyle (\\cos \\theta ,\\sin \\theta )} ããåç¹ãŸã§ã®è·é¢ã¯ 1 ãªã®ã§ã sin 2 Ξ + cos 2 Ξ = 1 {\\displaystyle \\sin ^{2}\\theta +\\cos ^{2}\\theta =1} ãæãç«ã€ã",
"title": "äžè§é¢æ°ã®æ§è³ª"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "ãŸãããã®åŒã«ã tan Ξ = sin Ξ cos Ξ {\\displaystyle \\tan \\theta ={\\frac {\\sin \\theta }{\\cos \\theta }}} ã€ãŸãã sin Ξ = tan Ξ cos Ξ {\\displaystyle \\sin \\theta =\\tan \\theta \\cos \\theta } ã代å
¥ããã°ã 1 + tan 2 Ξ = 1 cos 2 Ξ {\\displaystyle 1+\\tan ^{2}\\theta ={\\frac {1}{\\cos ^{2}\\theta }}} ãæãç«ã€ããšããããã",
"title": "äžè§é¢æ°ã®æ§è³ª"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "é¢æ° f ( x ) {\\displaystyle f(x)} ã«å¯ŸããŠã0 ã§ãªãå®æ° p {\\displaystyle p} ãååšããŠã f ( x + p ) = f ( x ) {\\displaystyle f(x+p)=f(x)} ãšãªããšãé¢æ° f ( x ) {\\displaystyle f(x)} ã¯åšæé¢æ°ãšãããå®æ° p {\\displaystyle p} ãäžã®æ§è³ªãæºãããšãã â p , 2 p {\\displaystyle -p,2p} ãªã©ãå®æ° p {\\displaystyle p} ã0ãé€ãæŽæ°åããæ°ãäžã®æ§è³ªãæºãããããã§ãåšæé¢æ°ãç¹åŸŽã¥ããéãšããŠãäžã®æ§è³ªãæºããå®æ° p {\\displaystyle p} ã®å
ãæ£ã§ãã€æå°ã®ãã®ãéžã³ããããåšæãšåŒã¶ã",
"title": "åšæé¢æ°"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "sin x , cos x {\\displaystyle \\sin x,\\cos x} ã¯åšæã 2 Ï {\\displaystyle 2\\pi } ãšããåšæé¢æ°ã§ããã tan x {\\displaystyle \\tan x} ã¯åšæã Ï {\\displaystyle \\pi } ãšããåšæé¢æ°ã§ããã",
"title": "åšæé¢æ°"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "åšæé¢æ°"
},
{
"paragraph_id": 41,
"tag": "p",
"text": "k {\\displaystyle k} ã0ã§ãªãå®æ°ãšãããé¢æ° sin k x {\\displaystyle \\sin kx} ã®åšæãèšã",
"title": "åšæé¢æ°"
},
{
"paragraph_id": 42,
"tag": "p",
"text": "解ç",
"title": "åšæé¢æ°"
},
{
"paragraph_id": 43,
"tag": "p",
"text": "sin k ( x + 2 Ï k ) = sin k x {\\displaystyle \\sin k\\left(x+{\\frac {2\\pi }{k}}\\right)=\\sin kx} ãªã®ã§çã㯠2 Ï k {\\displaystyle {\\frac {2\\pi }{k}}} ãããã¯æ£ã§ãããåšæã®æå°æ§ã®æ¡ä»¶ãæºãããŠããã",
"title": "åšæé¢æ°"
},
{
"paragraph_id": 44,
"tag": "p",
"text": "é¢æ° f ( x ) {\\displaystyle f(x)} ã f ( â x ) = f ( x ) {\\displaystyle f(-x)=f(x)} ãæºãããšããé¢æ° f ( x ) {\\displaystyle f(x)} ã¯å¶é¢æ°ãšãããå¶é¢æ°ã¯ y {\\displaystyle y} 軞ã«é¢ããŠå¯Ÿç§°ãªã°ã©ãã«ãªãã",
"title": "å¶é¢æ°ãšå¥é¢æ°"
},
{
"paragraph_id": 45,
"tag": "p",
"text": "ãŸããé¢æ° f ( x ) {\\displaystyle f(x)} ã f ( â x ) = â f ( x ) {\\displaystyle f(-x)=-f(x)} ãæºãããšããé¢æ° f ( x ) {\\displaystyle f(x)} ã¯å¥é¢æ°ãšãããå¶é¢æ°ã¯åç¹ã«é¢ããŠå¯Ÿè±¡ãªã°ã©ãã«ãªãã",
"title": "å¶é¢æ°ãšå¥é¢æ°"
},
{
"paragraph_id": 46,
"tag": "p",
"text": "é¢æ° cos Ξ , x 2 n {\\displaystyle \\cos \\theta ,x^{2n}} ( n {\\displaystyle n} ã¯æŽæ°)ã¯å¶é¢æ°ãšãªãã",
"title": "å¶é¢æ°ãšå¥é¢æ°"
},
{
"paragraph_id": 47,
"tag": "p",
"text": "é¢æ° sin x , x 2 n + 1 {\\displaystyle \\sin x,x^{2n+1}} ( n {\\displaystyle n} ã¯æŽæ°)ã¯å¥é¢æ°ãšãªãã",
"title": "å¶é¢æ°ãšå¥é¢æ°"
},
{
"paragraph_id": 48,
"tag": "p",
"text": "tan Ξ {\\displaystyle \\tan \\theta } ã¯å¶é¢æ°ããããšãå¥é¢æ°ã調ã¹ãã",
"title": "å¶é¢æ°ãšå¥é¢æ°"
},
{
"paragraph_id": 49,
"tag": "p",
"text": "解ç",
"title": "å¶é¢æ°ãšå¥é¢æ°"
},
{
"paragraph_id": 50,
"tag": "p",
"text": "ãªã®ã§ã tan Ξ {\\displaystyle \\tan \\theta } ã¯å¥é¢æ°ã§ããã",
"title": "å¶é¢æ°ãšå¥é¢æ°"
},
{
"paragraph_id": 51,
"tag": "p",
"text": "é¢æ° y = sin ( Ξ â Ï 3 ) {\\displaystyle y=\\sin \\left(\\theta -{\\frac {\\pi }{3}}\\right)} ã®ã°ã©ãã¯ã y = sin Ξ {\\displaystyle y=\\sin \\theta } ã®ã°ã©ãã Ξ軞æ¹åã« Ï 3 {\\displaystyle {\\frac {\\pi }{3}}} ã ãå¹³è¡ç§»åããããã®ã«ãªããåšæ㯠2 Ï {\\displaystyle 2\\pi } ã§ããã(å¹³è¡ç§»åããŠããåšæã¯å€ããããsinΞãšåããåšæ㯠2 Ï {\\displaystyle 2\\pi } ã®ãŸãŸã§ããã)",
"title": "ãããããªäžè§é¢æ°"
},
{
"paragraph_id": 52,
"tag": "p",
"text": "",
"title": "ãããããªäžè§é¢æ°"
},
{
"paragraph_id": 53,
"tag": "p",
"text": "é¢æ° y=2sin Ξ ã®ã°ã©ãã®åœ¢ã¯ y=sin Ξ ãy軞æ¹åã«2åã«æ¡å€§ãããã®ã§ãåšæ㯠y=sin Ξ ãšåãã 2Ï ã§ããã",
"title": "ãããããªäžè§é¢æ°"
},
{
"paragraph_id": 54,
"tag": "p",
"text": "ãŒ1 ⊠sin Ξ ⊠1 ãªã®ã§ã",
"title": "ãããããªäžè§é¢æ°"
},
{
"paragraph_id": 55,
"tag": "p",
"text": "å€å㯠ãŒ2 ⊠2sin Ξ ⊠2 ã§ããã",
"title": "ãããããªäžè§é¢æ°"
},
{
"paragraph_id": 56,
"tag": "p",
"text": "é¢æ° y=sin2Ξ ã®ã°ã©ãã¯y軞ãåºæºã«Îžè»žæ¹åã« 1 2 {\\displaystyle {\\frac {1}{2}}} åã«çž®å°ãããã®ã«ãªã£ãŠããã",
"title": "ãããããªäžè§é¢æ°"
},
{
"paragraph_id": 57,
"tag": "p",
"text": "ãããã£ãŠãåšæã 1 2 {\\displaystyle {\\frac {1}{2}}} åã«ãªã£ãŠãããy=sinΞ ã®åšæ㯠2 Ï {\\displaystyle 2\\pi } ã ãããy=sin2Ξ ã®åšæã¯ Ï {\\displaystyle \\pi } ã§ããã",
"title": "ãããããªäžè§é¢æ°"
},
{
"paragraph_id": 58,
"tag": "p",
"text": "äžè§é¢æ°ã®å æ³å®ç",
"title": "å æ³å®ç"
},
{
"paragraph_id": 59,
"tag": "p",
"text": "ãæãç«ã€ã",
"title": "å æ³å®ç"
},
{
"paragraph_id": 60,
"tag": "p",
"text": "蚌æ",
"title": "å æ³å®ç"
},
{
"paragraph_id": 61,
"tag": "p",
"text": "ä»»æã®å®æ° α , β {\\displaystyle \\alpha ,\\beta } ã«å¯Ÿããåäœååšäžã®ç¹ A ( cos α , sin α ) , B ( cos β , sin β ) {\\displaystyle \\mathrm {A} (\\cos \\alpha ,\\sin \\alpha ),\\mathrm {B} (\\cos \\beta ,\\sin \\beta )} ããšãããã®ãšãã ç·å A B {\\displaystyle \\mathrm {AB} } ã®é·ãã®2ä¹ A B 2 {\\displaystyle \\mathrm {AB} ^{2}} ã¯äœåŒŠå®çã䜿ãããšã«ãã",
"title": "å æ³å®ç"
},
{
"paragraph_id": 62,
"tag": "p",
"text": "A B 2 = 2 â 2 cos ( α â β ) {\\displaystyle \\mathrm {AB} ^{2}=2-2\\cos(\\alpha -\\beta )}",
"title": "å æ³å®ç"
},
{
"paragraph_id": 63,
"tag": "p",
"text": "ã§ããã次ã«äžå¹³æ¹ã®å®çã䜿ã£ãŠ",
"title": "å æ³å®ç"
},
{
"paragraph_id": 64,
"tag": "p",
"text": "A B 2 = ( cos α â cos α ) 2 + ( sin α â sin β ) 2 = 2 â 2 ( cos α cos β + sin α sin β ) {\\displaystyle \\mathrm {AB} ^{2}=(\\cos \\alpha -\\cos \\alpha )^{2}+(\\sin \\alpha -\\sin \\beta )^{2}=2-2(\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta )}",
"title": "å æ³å®ç"
},
{
"paragraph_id": 65,
"tag": "p",
"text": "ãããæŽçããŠ",
"title": "å æ³å®ç"
},
{
"paragraph_id": 66,
"tag": "p",
"text": "cos ( α â β ) = cos α cos β + sin α sin β {\\displaystyle \\cos(\\alpha -\\beta )=\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta }",
"title": "å æ³å®ç"
},
{
"paragraph_id": 67,
"tag": "p",
"text": "ãåŸãã",
"title": "å æ³å®ç"
},
{
"paragraph_id": 68,
"tag": "p",
"text": "cos ( α + β ) = cos ( α â ( â β ) ) = cos α cos ( â β ) + sin α sin ( â β ) = cos α cos β â sin α sin β {\\displaystyle \\cos(\\alpha +\\beta )=\\cos(\\alpha -(-\\beta ))=\\cos \\alpha \\cos(-\\beta )+\\sin \\alpha \\sin(-\\beta )=\\cos \\alpha \\cos \\beta -\\sin \\alpha \\sin \\beta }",
"title": "å æ³å®ç"
},
{
"paragraph_id": 69,
"tag": "p",
"text": "ã§ããã",
"title": "å æ³å®ç"
},
{
"paragraph_id": 70,
"tag": "p",
"text": "以äžããŸãšããŠ",
"title": "å æ³å®ç"
},
{
"paragraph_id": 71,
"tag": "p",
"text": "cos ( α ± β ) = cos α cos β â sin α sin β {\\displaystyle \\cos(\\alpha \\pm \\beta )=\\cos \\alpha \\cos \\beta \\mp \\sin \\alpha \\sin \\beta }",
"title": "å æ³å®ç"
},
{
"paragraph_id": 72,
"tag": "p",
"text": "ãåŸãã",
"title": "å æ³å®ç"
},
{
"paragraph_id": 73,
"tag": "p",
"text": "ããã§ã",
"title": "å æ³å®ç"
},
{
"paragraph_id": 74,
"tag": "p",
"text": "sin ( α ± β ) = â cos ( α + Ï 2 ± β ) = â { cos ( α + Ï 2 ) cos ( β ) â sin ( α + Ï 2 ) sin β } = sin α cos β ± cos α sin β {\\displaystyle \\sin(\\alpha \\pm \\beta )=-\\cos(\\alpha +{\\frac {\\pi }{2}}\\pm \\beta )=-\\{\\cos(\\alpha +{\\frac {\\pi }{2}})\\cos(\\beta )\\mp \\sin(\\alpha +{\\frac {\\pi }{2}})\\sin \\beta \\}=\\sin \\alpha \\cos \\beta \\pm \\cos \\alpha \\sin \\beta }",
"title": "å æ³å®ç"
},
{
"paragraph_id": 75,
"tag": "p",
"text": "ããã«ã tan x {\\displaystyle \\tan x} ã«ã€ããŠã",
"title": "å æ³å®ç"
},
{
"paragraph_id": 76,
"tag": "p",
"text": "tan ( α ± β ) = sin ( α ± β ) cos ( α ± β ) = sin α cos β ± cos α sin β cos α cos β â sin α sin β = sin α cos β cos α cos β ± cos α sin β cos α cos β cos α cos β cos α cos β â sin α sin β cos α cos β = tan α ± tan β 1 â tan α tan β {\\textstyle {\\begin{aligned}\\tan(\\alpha \\pm \\beta )&={\\frac {\\sin(\\alpha \\pm \\beta )}{\\cos(\\alpha \\pm \\beta )}}\\\\&={\\frac {\\sin \\alpha \\cos \\beta \\pm \\cos \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta \\mp \\sin \\alpha \\sin \\beta }}\\\\&={\\cfrac {{\\cfrac {\\sin \\alpha \\cos \\beta }{\\cos \\alpha \\cos \\beta }}\\pm {\\cfrac {\\cos \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta }}}{{\\cfrac {\\cos \\alpha \\cos \\beta }{\\cos \\alpha \\cos \\beta }}\\mp {\\cfrac {\\sin \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta }}}}\\\\&={\\frac {\\tan \\alpha \\pm \\tan \\beta }{1\\mp \\tan \\alpha \\tan \\beta }}\\end{aligned}}}",
"title": "å æ³å®ç"
},
{
"paragraph_id": 77,
"tag": "p",
"text": "ãæãç«ã€ã",
"title": "å æ³å®ç"
},
{
"paragraph_id": 78,
"tag": "p",
"text": "å æ³å®çãçšããŠä»¥äžã蚌æã§ããã",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 79,
"tag": "p",
"text": "sin 2 α = sin ( α + α ) = 2 sin α cos α {\\displaystyle \\sin 2\\alpha =\\sin(\\alpha +\\alpha )=2\\sin \\alpha \\cos \\alpha }",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 80,
"tag": "p",
"text": "cos 2 α = cos ( α + α ) = cos 2 α â sin 2 α = 2 cos 2 α â 1 = 1 â 2 sin 2 α {\\displaystyle \\cos 2\\alpha =\\cos(\\alpha +\\alpha )=\\cos ^{2}\\alpha -\\sin ^{2}\\alpha =2\\cos ^{2}\\alpha -1=1-2\\sin ^{2}\\alpha }",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 81,
"tag": "p",
"text": "tan 2 α = 2 tan α 1 â tan 2 α {\\displaystyle \\tan 2\\alpha ={\\frac {2\\tan \\alpha }{1-\\tan ^{2}\\alpha }}}",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 82,
"tag": "p",
"text": "次ã«ã cos {\\displaystyle \\cos } ã®åè§ã®å
¬åŒãå€åœ¢ãããš",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 83,
"tag": "p",
"text": "sin 2 α = 1 â cos 2 α 2 {\\displaystyle \\sin ^{2}\\alpha ={\\frac {1-\\cos 2\\alpha }{2}}}",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 84,
"tag": "p",
"text": "cos 2 α = 1 + cos 2 α 2 {\\displaystyle \\cos ^{2}\\alpha ={\\frac {1+\\cos 2\\alpha }{2}}}",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 85,
"tag": "p",
"text": "ã§ããã",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 86,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 87,
"tag": "p",
"text": "解ç",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 88,
"tag": "p",
"text": "sin 15 â = sin ( 45 â â 30 â ) = 6 â 2 4 {\\displaystyle \\sin 15^{\\circ }=\\sin(45^{\\circ }-30^{\\circ })={\\frac {{\\sqrt {6}}-{\\sqrt {2}}}{4}}}",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 89,
"tag": "p",
"text": "cos 15 â = cos ( 45 â â 30 â ) = 6 + 2 4 {\\displaystyle \\cos 15^{\\circ }=\\cos(45^{\\circ }-30^{\\circ })={\\frac {{\\sqrt {6}}+{\\sqrt {2}}}{4}}}",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 90,
"tag": "p",
"text": "tan 2 α = sin 2 α cos 2 α = 1 â cos 2 α 1 + cos 2 α {\\displaystyle \\tan ^{2}\\alpha ={\\frac {\\sin ^{2}\\alpha }{\\cos ^{2}\\alpha }}={\\frac {1-\\cos 2\\alpha }{1+\\cos 2\\alpha }}}",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 91,
"tag": "p",
"text": "ä»ãŸã§ã®å®çããŸãšãããšã次ã®ããã«ãªãã",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 92,
"tag": "p",
"text": "èŠãæ¹",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 93,
"tag": "p",
"text": "å æ³å®çã¯ãå²ããã³ã¹ã¢ã¹ã³ã¹ã¢ã¹å²ãããããã³ã¹ã¢ã¹ã³ã¹ã¢ã¹å²ããå²ããããšããèªååãããããŸãã",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 94,
"tag": "p",
"text": "cos {\\displaystyle \\cos } ã®åè§ã®å
¬åŒ cos 2 Ξ = 2 cos 2 Ξ â 1 = 1 â 2 sin 2 Ξ {\\displaystyle \\cos 2\\theta =2\\cos ^{2}\\theta -1=1-2\\sin ^{2}\\theta } 㯠± 1 â 2 a a a 2 Ξ {\\displaystyle \\pm 1\\mp 2\\mathrm {aaa} ^{2}\\theta } ãšãã圢ãèŠã㊠sin {\\displaystyle \\sin } ã¯ç¬Šå·ã â {\\displaystyle -} ã1 ã®ç¬Šå·ã¯ãã®éãšèŠããŸãã",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 95,
"tag": "p",
"text": "2ä¹ã®äžè§é¢æ° sin 2 Ξ = 1 â cos 2 Ξ 2 , cos 2 Ξ = 1 + cos 2 Ξ 2 {\\displaystyle \\sin ^{2}\\theta ={\\frac {1-\\cos 2\\theta }{2}},\\cos ^{2}\\theta ={\\frac {1+\\cos 2\\theta }{2}}} ã¯ã 1 ± cos 2 Ξ 2 {\\displaystyle {\\frac {1\\pm \\cos 2\\theta }{2}}} ãšãã圢ãèŠããŠã sin {\\displaystyle \\sin } ã¯ç¬Šå·ã â {\\displaystyle -} ãšèããŸãã",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 96,
"tag": "p",
"text": "äžè§é¢æ°ã®å",
"title": "äžè§é¢æ°ã®åæ"
},
{
"paragraph_id": 97,
"tag": "p",
"text": "ã«ãããŠã a , b â 0 {\\displaystyle a,b\\neq 0} ã®ãšã",
"title": "äžè§é¢æ°ã®åæ"
},
{
"paragraph_id": 98,
"tag": "p",
"text": "{ a a 2 + b 2 } 2 + { b a 2 + b 2 } 2 = 1 {\\displaystyle \\left\\{{\\dfrac {a}{\\sqrt {a^{2}+b^{2}}}}\\right\\}^{2}+\\left\\{{\\dfrac {b}{\\sqrt {a^{2}+b^{2}}}}\\right\\}^{2}=1} ã§ããã®ã§ãç¹ ( a a 2 + b 2 , b a 2 + b 2 ) {\\displaystyle \\left({\\dfrac {a}{\\sqrt {a^{2}+b^{2}}}},{\\dfrac {b}{\\sqrt {a^{2}+b^{2}}}}\\right)} ã¯åäœååšäžã®ç¹ãªã®ã§ã",
"title": "äžè§é¢æ°ã®åæ"
},
{
"paragraph_id": 99,
"tag": "p",
"text": "ãšãªããããªÎ±ããšãããšãã§ãããã®Î±ãçšããŠæ¬¡ã®ãããªå€åœ¢ãã§ããã",
"title": "äžè§é¢æ°ã®åæ"
},
{
"paragraph_id": 100,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "äžè§é¢æ°ã®åæ"
},
{
"paragraph_id": 101,
"tag": "p",
"text": "r , α {\\displaystyle r,\\alpha } 㯠r > 0 , â Ï â€ Î± < Ï {\\displaystyle r>0,-\\pi \\leq \\alpha <\\pi } ãæºãããšããã",
"title": "äžè§é¢æ°ã®åæ"
},
{
"paragraph_id": 102,
"tag": "p",
"text": "解ç",
"title": "äžè§é¢æ°ã®åæ"
},
{
"paragraph_id": 103,
"tag": "p",
"text": "sin Ξ â 3 cos Ξ = 2 ( 1 2 sin Ξ â 3 2 cos Ξ ) = 2 ( sin Ξ cos Ï 3 â cos Ξ sin Ï 3 ) = 2 sin ( Ξ â Ï 3 ) {\\displaystyle {\\begin{aligned}\\sin \\theta -{\\sqrt {3}}\\cos \\theta &=2\\left({\\frac {1}{2}}\\sin \\theta -{\\frac {\\sqrt {3}}{2}}\\cos \\theta \\right)\\\\&=2\\left(\\sin \\theta \\cos {\\frac {\\pi }{3}}-\\cos \\theta \\sin {\\frac {\\pi }{3}}\\right)\\\\&=2\\sin \\left(\\theta -{\\frac {\\pi }{3}}\\right)\\\\\\end{aligned}}}",
"title": "äžè§é¢æ°ã®åæ"
},
{
"paragraph_id": 104,
"tag": "p",
"text": "äžè§é¢æ°ã®å æ³å®çãçšãããšãäžè§é¢æ°ã®åâç©ã®å
¬åŒãããã³ç©âåã®å
¬åŒãåŸãããããããã",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 105,
"tag": "p",
"text": "ãšãªãã",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 106,
"tag": "p",
"text": "å æ³å®ç",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 107,
"tag": "p",
"text": "ããã (1) + (2) ãã",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 108,
"tag": "p",
"text": "(1) - (2) ãã",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 109,
"tag": "p",
"text": "(3) + (4) ãã",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 110,
"tag": "p",
"text": "(3) - (4) ãã",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 111,
"tag": "p",
"text": "ãåŸãããã",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 112,
"tag": "p",
"text": "A = α + β , B = α â β {\\displaystyle A=\\alpha +\\beta ,\\,B=\\alpha -\\beta } ãšãããšã α = A + B 2 , β = A â B 2 {\\displaystyle \\alpha ={\\frac {A+B}{2}},\\,\\beta ={\\frac {A-B}{2}}} ã§ããããããç©âåã®å
¬åŒã«ä»£å
¥ããã°ããããã",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 113,
"tag": "p",
"text": "ãåŸãããã",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 114,
"tag": "p",
"text": "èŠãæ¹",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 115,
"tag": "p",
"text": "ç©âåã®å
¬åŒã¯ãäž2ã€ã¯ α {\\displaystyle \\alpha } 㚠β {\\displaystyle \\beta } ãå
¥ãæ¿ããã°åãåŒãªã®ã§ãèŠããã®ã¯3åŒã§ããã sin sin {\\displaystyle \\sin \\sin } ã®å
¬åŒã¯ cos cos {\\displaystyle \\cos \\cos } ã®å
¬åŒã®ç¬Šå·ã2〠â {\\displaystyle -} ã«ãããã®ã«ãªã£ãŠããã",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 116,
"tag": "p",
"text": "åâç©ã®å
¬åŒã¯ã a a a â a a a {\\displaystyle {\\rm {{aaa}-{\\rm {aaa}}}}} ã®åŒã¯ a a a + a a a {\\displaystyle {\\rm {{aaa}+{\\rm {aaa}}}}} ã®å
¬åŒã® cos {\\displaystyle \\cos } ãš sin {\\displaystyle \\sin } ãéã«ãã圢ã«ãªã£ãŠããã",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 117,
"tag": "p",
"text": "",
"title": "äžè§é¢æ°ã®åºæ¬å
¬åŒ"
},
{
"paragraph_id": 118,
"tag": "p",
"text": "",
"title": "äžè§é¢æ°ã®åºæ¬å
¬åŒ"
},
{
"paragraph_id": 119,
"tag": "p",
"text": "(1)äžã®åºŠæ°æ³ã§è¡šãããå€ã匧床æ³ãŠè¡šã",
"title": "æŒç¿åé¡"
},
{
"paragraph_id": 120,
"tag": "p",
"text": "1) 150 {\\displaystyle 150} 2) 720 {\\displaystyle 720}",
"title": "æŒç¿åé¡"
},
{
"paragraph_id": 121,
"tag": "p",
"text": "(2) sin Ï / 2 {\\displaystyle \\sin \\pi /2} ã®å€ãæ±ãã",
"title": "æŒç¿åé¡"
}
] | ããã§ã¯äžè§é¢æ°ã®å®çŸ©ãããããšãäžè§é¢æ°ã®åºæ¬çãªæ§è³ªãå æ³å®çãäžè§é¢æ°ã®å¿çšã«ã€ããŠåŠã¶ãäžè§é¢æ°ã¯æ³¢ããã¯ãã«ã®å
ç©ãããŒãªãšå€æãªã©ããŸããŸãªåéã§å¿çšãããŠããã | {{Pathnav|é«çåŠæ ¡ã®åŠç¿|é«çåŠæ ¡æ°åŠ|é«çåŠæ ¡æ°åŠII|frame=1}}
ããã§ã¯äžè§é¢æ°ã®å®çŸ©ãããããšãäžè§é¢æ°ã®åºæ¬çãªæ§è³ªãå æ³å®çãäžè§é¢æ°ã®å¿çšã«ã€ããŠåŠã¶ãäžè§é¢æ°ã¯æ³¢ããã¯ãã«ã®å
ç©ãããŒãªãšå€æãªã©ããŸããŸãªåéã§å¿çšãããŠããã
== äžè¬è§ ==
[[File:General angle of trigonometric functions japanese.svg|thumb|300px|]]
[[File:Negative general angle.svg|thumb|300px]]
å³å³ã®ããã«ãå®ç¹Oãäžå¿ãšããŠå転ããåçŽç· OP ãèããããã®ãšãã®å転ããåçŽç· OP ã®ããšã'''ååŸ'''ãšããã
åçŽç· OX ãè§åºŠã®åºæºãšããããã®åºæºãšãªãåçŽç· OX ã®ããšã'''å§ç·'''ãšããã
ååŸãæèšåãã«å転ããå Žåãå転ããè§åºŠã¯è² ã§ãããšããååŸãåæèšåããããå Žåãå転ããè§åºŠã¯æ£ã§ãããšããã
è² ã®è§åºŠã360°以äžå転ããè§åºŠãèãã«å
¥ããè§ã®ããšã'''äžè¬è§'''ãšããã
{{-}}
== åŒ§åºŠæ³ ==
==== ã©ãžã¢ã³ ====
ããŸãŸã§ã¯è§åºŠã®åäœãšããŠäžåšã 360° ãšãã床æ°æ³ã䜿ã£ãŠããããšã ãããããã§ã匧床æ³ã«ããè§åºŠã®è¡šãæ¹ãåŠã¶ã
[[File:1radian japanese.svg|thumb|300px]]
ååŸ1 ã®æ圢ã«ãããŠåŒ§ã®é·ãã 1 ã®ãšãã®äžå¿è§ã 1 radãåæ§ã«åŒ§ã®é·ããθã®ãšãã®äžå¿è§ãθ radãšå®çŸ©ããããã®å®çŸ©ãã 180° =π radã360° = 2π rad ãããã«
:<math>\begin{align}1 ^{\circ} &=\frac{\pi}{180}\, \mathrm{rad} \\
\\
1\, \mathrm{rad} &= \frac {180}{\pi} ^{\circ} \approx 57.3^{\circ}\end{align}</math>
ãšãªãããŸã匧床æ³ã®åäœïŒradïŒã¯ãã°ãã°çç¥ãããã
匧床æ³ãçšãããšãäžè§é¢æ°ã®åŸ®ç©åãèããéã«äŸ¿å©ã§ãããïŒãã®ããšã¯æ°åŠIIIã§åŠã¶ïŒ
==== æ圢ã®åŒ§ã®é·ããšé¢ç© ====
æ圢ã®ååŸã''r'' ã匧床æ³ã§å®çŸ©ãããè§åºŠãθãšãããšãã匧ã®é·ã''l'' ãšé¢ç©''S'' ã¯
:<math>\begin{align}l&=r\theta, \\
\\
S&=\frac{1}{2}r^{2}\theta=\frac{1}{2}rl\end{align}</math>
ãšè¡šããã
== äžè§é¢æ° ==
==== sin ãš cos ã®ã°ã©ã ====
[[File:Sin and cos general angle introduction.svg|thumb|300px|]]
äžè¬è§ã <math>\theta</math> ã®åçŽç·ãšåäœåã亀ããåã <math>\mathrm P</math> ãšããããã®ãšãã® <math>\mathrm P</math> ã®åº§æšã<math>(\cos\theta,\sin\theta)</math> ãšããããšã§ãé¢æ° <math>\sin,\cos</math> ãå®ããããŸãã<math>\tan\theta = \frac{\sin\theta}{\cos\theta}</math> ãšããããšã§é¢æ° <math>\tan\theta</math> ãå®ããã<math>\tan\theta</math> ã¯äžè¬è§ã <math>\theta</math> ã®ååŸã®åŸãã«çããã
* <math>\sin</math> ã¯ãµã€ã³(sine) ãšçºé³ãããæ£åŒŠãšãåŒã°ããã
* <math>\cos</math> ã³ãµã€ã³(cosine) ãšçºé³ãããäœåŒŠãšãåŒã°ããã
* <math>\tan</math> ã¯ã¿ã³ãžã§ã³ã(tangent) ãšçºé³ãããæ£æ¥ãšãåŒã°ããã
ãŸããäžè§é¢æ°ã®çŽ¯ä¹ã¯ <math>(\sin\theta)^n = \sin^n\theta</math> ãšè¡šèšãããã[[ãã¡ã€ã«:Circle cos sin.gif|ãµã ãã€ã«|äžå€®|300x300ãã¯ã»ã«]]
[[File:Y=sin(theta).svg|thumb|500px|left]]
[[File:Y=cos(theta).svg|thumb|500px|left]]
{{-}}
cos Ξ ã®ã°ã©ã㯠sin Ξ ã®ã°ã©ãã Ξ軞æ¹åã« <math> - \frac{ \pi }{2} </math>ã ãå¹³è¡ç§»åãããã®ã§ããã
<math>y = \sin\theta</math> ã <math>y = \cos\theta</math> ã®åœ¢ãããæ²ç·ã®ããšã '''æ£åŒŠæ²ç·''' ïŒãããããããããïŒãšããã
é¢æ° <math>\sin,\cos</math> ã®å€åã¯ã©ã¡ããã<math>[-1,1]</math> ã§ããã
{{-}}
==== tan ã®ã°ã©ã ====
[[File:Tangent function introduction.svg|thumb|300px|]]
å³å³ã®ããã« ãè§ Îž ã®ååŸãšåäœåãšã®äº€ç¹ãPãšããŠã
çŽç·OPãš çŽç·xïŒ1 ãšã®äº€ç¹ã T ãšãããšã
Tã®åº§æšã¯
: T (1, tan Ξ)
ã«ãªãã
ãã®ããšãå©çšããŠã yïŒtan Ξ ã®ã°ã©ããããããšãã§ããã
{{-}}
y=tan Ξ ã®ã°ã©ãã¯ãäžå³ã®ããã«ãªãã<br>
[[File:Y=tan(x).svg|500px|]]
y=tan Ξ ã®ã°ã©ãã§ã¯ãΞã®å€ã <math> \frac{ \pi }{2} </math> ã«è¿ã¥ããŠãããšã
çŽç· <math> \theta = \frac{ \pi }{2} </math> ã«éããªãè¿ã¥ããŠããã
ãã®ããã«ãæ²ç·ãããçŽç·ã«éãç¡ãè¿ã¥ããŠãããšããè¿ã¥ãããçŽç·ã®ã»ãã '''挞è¿ç·''' ïŒããããããïŒãšããã
åæ§ã«èãã次ã®çŽç·ã yïŒtanΞ ã®æŒžè¿ç·ã§ããã
:<math> \cdots , \quad \theta = - \frac{ 3}{2} \pi , \quad \theta = - \frac{ 1}{2} \pi , \quad \theta = \frac{ 1}{2} \pi , \quad \frac{ 3}{2} \pi , \cdots </math>
㯠yïŒtanΞ ã®æŒžè¿ç·ã§ããã
äžè¬ã«ã
:çŽç· <math> \quad \theta = \frac{ \pi }{2} + n \pi </math> ããïŒnã¯æŽæ°ïŒ
ã¯y=tanΞã®ã°ã©ãã®æŒžè¿ç·ã§ããã<ref>é«æ ¡ã»å€§åŠå
¥è©Šã§ã¯äœ¿ãããªããã<math>\sec\theta=\frac{1}{\cos\theta},\csc\theta=\frac{1}{\sin\theta},\cot\theta=\frac{1}{\tan\theta}(=\frac{\cos\theta}{\sin\theta})</math> ãšããŠå®çŸ©ãããäžè§é¢æ°ã䜿ããšãããããããããã®é¢æ°ã¯ãããããã»ã«ã³ããã³ã»ã«ã³ããã³ã¿ã³ãžã§ã³ããšåŒã°ããã</ref>
== äžè§é¢æ°ã®æ§è³ª ==
äžè¬è§ã <math>\theta</math> ã®ååŸã¯äžå転ããŠãçããã®ã§ãäžè¬è§ã <math>\theta+2\pi</math> ã®ååŸãšçãããããããäžè§é¢æ°ã®åšææ§
<math>\begin{align}
\sin(\theta+2\pi n)&=\sin \theta \\
\cos(\theta+2\pi n)&=\cos \theta \\
\tan(\theta+2\pi n)&=\tan \theta
\end{align}</math>
ãåŸãã
ç¹ <math>(\cos\theta,\sin\theta)</math> ã <math>\pi</math> å転ããç¹ <math>(\cos(\theta+\pi),\sin(\theta+\pi))</math> ã¯åç¹ãäžå¿ã«ç¹å¯Ÿç§°ç§»åããç¹ã<math>(-\cos\theta,-\sin\theta)</math> ã§ããããšãã
<math>\begin{align}
\sin(\theta + \pi) &= - \sin \theta \\
\cos(\theta + \pi) &= - \cos \theta \\
\tan(\theta + \pi) &= \tan \theta
\end{align}</math>
ãåŸãã
ç¹ <math>(\cos\theta,\sin\theta)</math> ã <math>x</math> 軞ã§ç·å¯Ÿç§°ç§»å移åããç¹ã <math>(\cos (-\theta),\sin(-\theta)) = (\cos\theta,-\sin\theta)</math> ã§ããããšãã
<math>\begin{align}\sin(-\theta) &= -\sin\theta \\
\cos(-\theta) &= \cos\theta \\
\tan(- \theta) &= -\tan\theta\end{align}</math>
ãåŸãã
* åé¡äŸ
** åé¡
* ::<math>\begin{align}
& \sin(\theta + \frac{\pi}{2}) \\
& \cos(\theta + \frac{\pi}{2}) \\
& \sin(\frac{\pi}{2} -\theta) \\
& \cos(\frac{\pi}{2}- \theta )
\end{align}</math>
*: ãèšç®ããã
** 解ç
*: è§θã«å¯Ÿå¿ããç¹ã P(x, y) ãšããããã®ãšããè§ θ + 90°ã«å¯Ÿå¿ããç¹ã P'(x', y') ãšãããšããã®ç¹ã®åº§æšã¯ãP'(-y, x) ã«å¯Ÿå¿ããããã®ããšãããP'ã«ã€ã㊠sin, cos ãèšç®ãããšã
*:: <math>\begin{align}
x' &= -y \\
&= \cos (\theta + \frac{\pi}{2} )\\
&= -\sin\theta \\
y' &= x \\
&= \sin (\theta + \frac{\pi}{2} ) \\
&= \cos\theta
\end{align}</math>
*: ãåŸãããã
*: åæ§ã«ããŠã90°- θ ã«å¯Ÿå¿ããç¹ã P' '(x' ', y' ') ãšãããšã
*:: <math>\begin{align}
x'' &= y \\
y'' &= x
\end{align}</math>
*: ãšãªãããã£ãŠã
*:: <math>\begin{align}
\sin (\frac{\pi}{2} - \theta) &= \cos\theta \\
\cos (\frac{\pi}{2} - \theta) &= \sin\theta
\end{align}</math>
*: ãåŸãããã
åäœååšäžã®ç¹ <math>(\cos\theta,\sin\theta)</math> ããåç¹ãŸã§ã®è·é¢ã¯ 1 ãªã®ã§ã <math>\sin^2\theta+\cos^2\theta = 1</math> ãæãç«ã€ã
ãŸãããã®åŒã«ã <math>\tan\theta = \frac{\sin\theta}{\cos\theta}</math> ã€ãŸãã <math>\sin\theta = \tan\theta \cos\theta</math> ã代å
¥ããã°ã<math>1+\tan^2\theta = \frac{1}{\cos^2\theta}</math> ãæãç«ã€ããšããããã
== åšæé¢æ° ==
é¢æ° <math>f(x)</math> ã«å¯ŸããŠã0 ã§ãªãå®æ° <math>p</math> ãååšããŠã<math>f(x+p) =f(x)</math> ãšãªããšãé¢æ° <math>f(x)</math> ã¯åšæé¢æ°ãšãããå®æ° <math>p</math> ãäžã®æ§è³ªãæºãããšãã<math>-p,2p</math> ãªã©ãå®æ° <math>p</math> ã0ãé€ãæŽæ°åããæ°ãäžã®æ§è³ªãæºãããããã§ãåšæé¢æ°ãç¹åŸŽã¥ããéãšããŠãäžã®æ§è³ªãæºããå®æ° <math>p</math> ã®å
ãæ£ã§ãã€æå°ã®ãã®ãéžã³ãããã'''åšæ'''ãšåŒã¶ã
<math>\sin x, \cos x</math> ã¯åšæã <math>2\pi</math> ãšããåšæé¢æ°ã§ããã<math>\tan x</math> ã¯åšæã <math>\pi</math> ãšããåšæé¢æ°ã§ããã
'''æŒç¿åé¡'''
<math>k</math> ã0ã§ãªãå®æ°ãšãããé¢æ° <math>\sin kx</math> ã®åšæãèšã
'''解ç'''
<math>\sin k\left(x+\frac{2\pi}{k}\right) = \sin kx</math> ãªã®ã§çã㯠<math>\frac{2\pi}{k}</math> ãããã¯æ£ã§ãããåšæã®æå°æ§ã®æ¡ä»¶ãæºãããŠããã
== å¶é¢æ°ãšå¥é¢æ° ==
é¢æ° <math>f(x)</math> ã <math>f(-x)=f(x)</math> ãæºãããšããé¢æ° <math>f(x)</math> ã¯å¶é¢æ°ãšãããå¶é¢æ°ã¯ <math>y</math> 軞ã«é¢ããŠå¯Ÿç§°ãªã°ã©ãã«ãªãã
ãŸããé¢æ° <math>f(x)</math> ã <math>f(-x)=-f(x)</math> ãæºãããšããé¢æ° <math>f(x)</math> ã¯å¥é¢æ°ãšãããå¶é¢æ°ã¯åç¹ã«é¢ããŠå¯Ÿè±¡ãªã°ã©ãã«ãªãã
é¢æ° <math>\cos\theta,x^{2n}</math> (<math>n</math> ã¯æŽæ°)ã¯å¶é¢æ°ãšãªãã
é¢æ° <math>\sin x , x^{2n+1}</math> (<math>n</math> ã¯æŽæ°)ã¯å¥é¢æ°ãšãªãã
;æŒç¿åé¡
<math>\tan\theta</math> ã¯å¶é¢æ°ããããšãå¥é¢æ°ã調ã¹ãã
'''解ç'''
:<math> \tan( - \theta ) = \frac{\sin(- \theta)} {\cos(-\theta)} = \frac{- \sin(\theta)} {\cos(\theta)} = - \frac{\sin(\theta)} {\cos(\theta)} = - \tan \theta</math>
ãªã®ã§ã <math>\tan\theta</math> ã¯å¥é¢æ°ã§ããã<ref>äžè¬ã«ãé¢æ° <math>f(x) </math> ã«å¯Ÿãã<math>f(x) </math> ãå¶é¢æ°ãå¥é¢æ°ã調ã¹ãã«ã¯ <math>f(-x)</math> ã <math>f(x)</math> ãŸã㯠<math>-f(x)</math> ã®ã©ã¡ãã«çããã調ã¹ãã°ããããŸããã©ã¡ããšãçãããªãå Žåãé¢æ° <math>f(x) </math> ã¯å¶é¢æ°ã§ãå¥é¢æ°ã§ããªãã</ref>
== ãããããªäžè§é¢æ° ==
[[File:Y=sin(theta-pi div 3).svg|thumb|550px|]]
é¢æ° <math> y=\sin \left( \theta - \frac{\pi}{3} \right)</math> ã®ã°ã©ãã¯ã<math> y=\sin \theta </math>ã®ã°ã©ãã Ξ軞æ¹åã« <math> \frac{\pi}{3} </math> ã ãå¹³è¡ç§»åããããã®ã«ãªããåšæ㯠<math> 2 \pi </math> ã§ãããïŒå¹³è¡ç§»åããŠããåšæã¯å€ããããsinΞãšåããåšæ㯠<math> 2 \pi </math> ã®ãŸãŸã§ãããïŒ
{{-}}
[[File:Y=2sin(theta).svg|thumb|550px]]
é¢æ° yïŒ2sin Ξ ã®ã°ã©ãã®åœ¢ã¯ yïŒsin Ξ ãy軞æ¹åã«2åã«æ¡å€§ãããã®ã§ãåšæ㯠yïŒsin Ξ ãšåãã 2Ï ã§ããã
ãŒ1 ⊠sin Ξ ⊠1ãããªã®ã§ã
å€åã¯ãããŒ2 ⊠2sin Ξ ⊠2ããã§ããã
{{-}}
{{-}}
[[File:Y=sin(2 theta) and y=sin(theta).svg|thumb|750px]]
{{-}}
é¢æ° yïŒsin2Ξ ã®ã°ã©ãã¯y軞ãåºæºã«Îžè»žæ¹åã« <math> \frac{1}{2}</math> åã«çž®å°ãããã®ã«ãªã£ãŠããã
ãããã£ãŠãåšæã <math> \frac{1}{2}</math> åã«ãªã£ãŠãããy=sinΞ ã®åšæ㯠<math> 2 \pi </math> ã ãããy=sin2Ξ ã®åšæ㯠<math> \pi </math> ã§ããã
== å æ³å®ç ==
äžè§é¢æ°ã®å æ³å®ç
:<math>\begin{align}
\sin (\alpha \pm \beta) &= \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\
\cos (\alpha \pm \beta) &= \cos \alpha \cos \beta \mp \sin \alpha \sin \beta
\end{align}</math>
ãæãç«ã€ã
'''蚌æ'''
ä»»æã®å®æ° <math>\alpha,\beta</math> ã«å¯Ÿããåäœååšäžã®ç¹ <math>\mathrm{A}(\cos\alpha,\sin\alpha),\mathrm{B}(\cos\beta,\sin\beta)</math> ããšãããã®ãšãã ç·å <math>\mathrm{AB}</math> ã®é·ãã®2ä¹ <math>\mathrm{AB}^2</math> ã¯äœåŒŠå®çã䜿ãããšã«ãã
<math>\mathrm{AB}^2 = 2-2\cos(\alpha-\beta)</math>
ã§ããã次ã«äžå¹³æ¹ã®å®çã䜿ã£ãŠ
<math>\mathrm{AB}^2 = (\cos\alpha -\cos\alpha)^2 + (\sin\alpha-\sin\beta)^2 = 2 - 2(\cos\alpha\cos\beta + \sin\alpha\sin\beta)</math>
ãããæŽçããŠ
<math>\cos(\alpha - \beta)= \cos\alpha\cos\beta + \sin\alpha\sin\beta</math>
ãåŸãã
<math>\cos(\alpha+\beta) = \cos(\alpha-(-\beta)) = \cos\alpha\cos(-\beta) + \sin\alpha\sin(-\beta) =
\cos\alpha\cos\beta - \sin\alpha\sin\beta</math>
ã§ããã
以äžããŸãšããŠ
<math>\cos (\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta</math>
ãåŸãã
ããã§ã
<math>\sin(\alpha \pm \beta) = -\cos(\alpha +\frac{\pi}{2} \pm \beta) =
-\{\cos(\alpha + \frac{\pi}{2})\cos(\beta) \mp \sin(\alpha+\frac{\pi}{2})\sin\beta \} =
\sin\alpha\cos\beta \pm \cos\alpha\sin\beta</math><ref>ãå²ãã(sin)ã³ã¹ã¢ã¹(cos)ã³ã¹ã¢ã¹(cos)å²ãã(sin)ããã³ã¹ã¢ã¹(cos)ã³ã¹ã¢ã¹(cos)å²ãã(sin)å²ãã(sin)ããšããèŠãããããã</ref>
ããã«ã<math>\tan x</math> ã«ã€ããŠã
<math display="inline">\begin{align}
\tan (\alpha\pm\beta) &= \frac {\sin (\alpha\pm\beta) } {\cos (\alpha\pm\beta) } \\
&= \frac { \sin \alpha \cos \beta \pm \cos \alpha \sin \beta } { \cos \alpha \cos \beta \mp \sin \alpha \sin \beta } \\
&= \cfrac { \cfrac { \sin \alpha \cos \beta } { \cos \alpha \cos \beta } \pm \cfrac { \cos \alpha \sin \beta } { \cos \alpha \cos \beta } } { \cfrac { \cos \alpha \cos \beta } { \cos \alpha \cos \beta } \mp \cfrac { \sin \alpha \sin \beta } { \cos \alpha \cos \beta } } \\
&= \frac { \tan \alpha \pm \tan \beta } { 1 \mp \tan \alpha \tan \beta }
\end{align}</math>
ãæãç«ã€ã
== åè§ã®å
¬åŒ ==
å æ³å®çãçšããŠä»¥äžã蚌æã§ããã
<math>\sin 2\alpha = \sin(\alpha + \alpha) = 2\sin\alpha\cos\alpha</math>
<math>\cos 2\alpha = \cos(\alpha+\alpha)=\cos^2\alpha-\sin^2\alpha = 2\cos^2\alpha-1=1-2\sin^2\alpha</math>
<math>\tan 2\alpha = \frac{2\tan \alpha}{1-\tan^2\alpha}</math>
次ã«ã <math>\cos</math> ã®åè§ã®å
¬åŒãå€åœ¢ãããš
<math>\sin^2\alpha = \frac{1-\cos 2\alpha}{2}</math>
<math>\cos^2\alpha = \frac{1+\cos 2\alpha}{2}</math>
ã§ããã
'''æŒç¿åé¡'''
# <math>\sin 15^\circ,\cos 15^\circ</math> ãæ±ãã
# <math>\tan^2 \alpha = \frac{1-\cos 2\alpha}{1+\cos 2\alpha}</math> ã瀺ã
'''解ç'''
<math>\sin 15^\circ = \sin(45^\circ-30^\circ)=\frac{\sqrt 6 - \sqrt 2}{4}</math>
<math>\cos 15^\circ = \cos(45^\circ-30^\circ)=\frac{\sqrt 6 + \sqrt 2}{4}</math>
<math>\tan ^2\alpha = \frac{\sin^2\alpha}{\cos^2\alpha} = \frac{1-\cos 2\alpha}{1+\cos 2\alpha}</math>
ä»ãŸã§ã®å®çããŸãšãããšã次ã®ããã«ãªãã
{| style="border:2px solid yellow;width:80%" cellspacing=0
|style="background:yellow"|'''äžè§é¢æ°ã®å æ³å®ç'''
|-
|style="padding:5px"|
:<math>\begin{align}
\sin (\alpha \pm \beta) &= \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\
\cos (\alpha \pm \beta) &= \cos \alpha \cos \beta \mp \sin \alpha \sin \beta\\
\tan (\alpha \pm \beta) &= \frac { \tan \alpha \pm \tan \beta } { 1 \mp \tan \alpha \tan \beta } \\
\end{align}</math>
|}
{| style="border:2px solidãgreenyellow;width:80%" cellspacing=0
|style="background:greenyellow"|'''2åè§ã®å
¬åŒ'''
|-
|style="padding:5px"|
:<math>\begin{align}
\sin 2 \alpha &= 2 \sin \alpha \cos \alpha \\
\cos 2 \alpha &= \cos ^2 \alpha - \sin ^2 \alpha = 1 - 2 \sin ^2 \alpha = 2 \cos ^2 \alpha - 1 \\
\tan 2 \alpha &= \frac { 2 \tan \alpha } { 1 - \tan ^2 \alpha }
\end{align}</math>
|}
{| style="border:2px solidãskyblue;width:80%" cellspacing=0
|style="background:skyblue"|äžè§é¢æ°ã®2ä¹
|-
|style="padding:5px"|
:<math>\begin{align}
\sin ^2 \alpha &= \frac {1 - \cos 2\alpha }2 \\
\cos ^2 \alpha &= \frac {1 + \cos 2\alpha }2 \\
\tan ^2 \alpha &= \frac {1 - \cos 2\alpha } {1 + \cos 2\alpha }
\end{align}</math>
|}
'''èŠãæ¹'''
å æ³å®çã¯ãå²ããã³ã¹ã¢ã¹ã³ã¹ã¢ã¹å²ãããããã³ã¹ã¢ã¹ã³ã¹ã¢ã¹å²ããå²ããããšããèªååãããããŸãã
<math>\cos</math> ã®åè§ã®å
¬åŒ <math>\cos 2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta</math> 㯠<math>\pm 1 \mp 2\mathrm{aaa}^2\theta</math> ãšãã圢ãèŠã㊠<math>\sin</math> ã¯ç¬Šå·ã <math>-</math>ã1 ã®ç¬Šå·ã¯ãã®éãšèŠããŸãã
2ä¹ã®äžè§é¢æ° <math>\sin^2\theta = \frac{1-\cos 2\theta}{2},\cos^2\theta = \frac{1+\cos 2\theta}{2}</math> ã¯ã<math>\frac{1\pm \cos 2\theta}{2}</math> ãšãã圢ãèŠããŠã <math>\sin</math> ã¯ç¬Šå·ã<math>-</math> ãšèããŸãã
==äžè§é¢æ°ã®åæ==
äžè§é¢æ°ã®å
:<math>
a \sin \theta + b \cos \theta
</math>
ã«ãããŠã<math>a,b\neq 0</math> ã®ãšã
<math>\left\{\dfrac{a}{\sqrt{a^2+b^2}}\right\}^2 + \left\{\dfrac{b}{\sqrt{a^2+b^2}}\right\}^2 = 1</math> ã§ããã®ã§ãç¹ <math>\left(\dfrac{a}{\sqrt{a^2+b^2}},\dfrac{b}{\sqrt{a^2+b^2}}\right)</math> ã¯åäœååšäžã®ç¹ãªã®ã§ã
:<math>
\begin{cases}
\cos \alpha = \dfrac{a}{\sqrt{a^2+b^2}}\\
\sin \alpha = \dfrac{b}{\sqrt{a^2+b^2}}
\end{cases}
</math>
ãšãªããããªαããšãããšãã§ãããã®αãçšããŠæ¬¡ã®ãããªå€åœ¢ãã§ããã
:<math>\begin{align}
a \sin \theta + b \cos \theta & = \sqrt{a^2+b^2}\left( \frac{a}{\sqrt{a^2+b^2}} \sin \theta + \frac{b}{\sqrt{a^2+b^2}} \cos \theta \right) \\
& = \sqrt{a^2+b^2} \left( \sin \theta \cos \alpha + \cos \theta \sin \alpha \right)\\
& = \sqrt{a^2+b^2} \sin \left( \theta + \alpha \right)\\
\end{align}
</math>
'''æŒç¿åé¡'''
<math>r,\alpha</math> 㯠<math>r>0,-\pi\le \alpha< \pi</math> ãæºãããšããã
# <math>\sin \theta - \sqrt{3} \cos \theta</math> ã <math>r \sin \left( \theta + \alpha \right)</math> ã®åœ¢ã«å€åœ¢ããã
# <math>2\cos\theta-2\sin\theta</math> ã <math>r\cos(\theta+\alpha)</math> ã®åœ¢ã«å€åœ¢ããã
'''解ç'''
# <math>r = \sqrt{1^2 + \left( - \sqrt{3} \right)^2} = 2</math> ãã
<math>\begin{align}
\sin \theta - \sqrt{3} \cos \theta & = 2 \left( \frac{1}{2} \sin \theta - \frac{\sqrt{3}}{2} \cos \theta \right) \\
& = 2 \left( \sin \theta \cos \frac{\pi}{3} - \cos \theta \sin \frac{\pi}{3} \right)\\
& = 2 \sin \left( \theta - \frac{\pi}{3} \right)\\
\end{align}
</math>
# <math>2\cos\theta-2\sin\theta=2\sqrt 2\left(\frac{1}{\sqrt 2}\cos\theta-\frac{1}{\sqrt 2}\sin\theta\right)</math> <ref>ããå€åœ¢ããããšã§ãç¹ <math>\left(\frac{1}{\sqrt2},\frac{1}{\sqrt2}\right)</math> ãåäœååšäžã®ç¹ã«ãªã</ref>ããã§ã<math>r\cos(\theta+\alpha) = r(\cos\theta\cos\alpha-\sin\theta\sin\alpha)</math> ã§ããã <math>\cos\alpha=\frac{1}{\sqrt 2},\sin\alpha = \frac{1}{\sqrt 2}</math> ãšãªã <math>\alpha</math> ãšã㊠<math>\alpha = \frac{\pi}{4}</math> ãããã<ref>ããã§ã <math>\alpha</math> ã¯åé¡æã®å¶çŽãæºããããã«éžã¶ã <math>\alpha</math> ã« <math>2\pi</math> ã®æŽæ°åã足ãã <math>\alpha + 2\pi n</math> ãéžãã§ãäžè§é¢æ°ã®åæã¯ã§ããããå®çšçã«ã <math>\alpha</math> ã¯ç°¡åãªãã®ãéžãã æ¹ãããã ããã</ref>ãããã£ãŠã<math>2\cos\theta-2\sin\theta=2\sqrt 2\cos\left(\theta + \frac{\pi}{4}\right)</math>
== åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ ==
äžè§é¢æ°ã®å æ³å®çãçšãããšãäžè§é¢æ°ã®åâç©ã®å
¬åŒãããã³ç©âåã®å
¬åŒãåŸãããããããã
;ç©âåã®å
¬åŒ
:<math>\begin{align}
\sin \alpha \cos \beta &= \frac 1 2 \{\sin (\alpha+\beta) + \sin (\alpha-\beta)\}\\
\cos \alpha \sin \beta &= \frac 1 2 \{\sin (\alpha+\beta) - \sin (\alpha-\beta) \}\\
\cos \alpha \cos \beta &= \frac 1 2 \{\cos (\alpha+\beta) + \cos (\alpha-\beta) \}\\
\sin \alpha \sin \beta &= -\frac 1 2 \{\cos (\alpha+\beta) - \cos (\alpha-\beta) \}
\end{align}</math>
;åâç©ã®å
¬åŒ
:<math>\begin{align}
\sin A + \sin B &= 2 \sin \left(\frac {A+B}2 \right) \cos \left(\frac {A-B}2 \right)\\
\sin A - \sin B &= 2 \cos \left(\frac {A+B}2 \right) \sin \left(\frac {A-B}2 \right)\\
\cos A + \cos B &= 2 \cos \left(\frac {A+B}2 \right) \cos \left(\frac {A-B}2 \right)\\
\cos A - \cos B &= -2 \sin \left(\frac {A+B}2 \right) \sin \left(\frac {A-B}2 \right)
\end{align}</math>
ãšãªãã
;å°åº
å æ³å®ç
:{{åŒçªå·|<math>\sin(\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta</math>|1}}
:{{åŒçªå·|<math>\sin(\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin \beta</math>|2}}
:{{åŒçªå·|<math>\cos(\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta </math>|3}}
:{{åŒçªå·|<math>\cos(\alpha -\beta )=\cos \alpha \cos \beta +\sin \alpha \sin \beta </math>|4}}
ããã [[é«çåŠæ ¡æ°åŠII/äžè§é¢æ°#expr1|(1)]] + [[é«çåŠæ ¡æ°åŠII/äžè§é¢æ°#expr2|(2)]] ãã
:<math>\sin \alpha \cos \beta = \frac 1 2 (\sin (\alpha+\beta) + \sin (\alpha-\beta))</math>
[[é«çåŠæ ¡æ°åŠII/äžè§é¢æ°#expr1|(1)]] - [[é«çåŠæ ¡æ°åŠII/äžè§é¢æ°#expr2|(2)]] ãã
:<math>\cos \alpha \sin \beta = \frac 1 2 (\sin (\alpha+\beta) - \sin (\alpha-\beta) </math>
[[é«çåŠæ ¡æ°åŠII/äžè§é¢æ°#expr3|(3)]] + [[é«çåŠæ ¡æ°åŠII/äžè§é¢æ°#expr4|(4)]] ãã
:<math>\cos \alpha \cos \beta = \frac 1 2 (\cos (\alpha+\beta) + \cos (\alpha-\beta) )</math>
[[é«çåŠæ ¡æ°åŠII/äžè§é¢æ°#expr3|(3)]] - [[é«çåŠæ ¡æ°åŠII/äžè§é¢æ°#expr4|(4)]] ãã
:<math>\sin \alpha \sin \beta = -\frac 1 2 (\cos (\alpha+\beta) - \cos (\alpha-\beta) )</math>
ãåŸãããã
<math>A = \alpha + \beta,\, B = \alpha-\beta</math> ãšãããšã <math>\alpha = \frac{A+B}{2},\, \beta = \frac{A-B}{2}</math> ã§ããããããç©âåã®å
¬åŒã«ä»£å
¥ããã°ããããã
:<math>\begin{align}
\sin A + \sin B &= 2 \sin \left(\frac {A+B}2 \right) \cos \left(\frac {A-B}2 \right)\\
\sin A - \sin B &= 2 \cos \left(\frac {A+B}2 \right) \sin \left(\frac {A-B}2 \right)\\
\cos A + \cos B &= 2 \cos \left(\frac {A+B}2 \right) \cos \left(\frac {A-B}2 \right)\\
\cos A - \cos B &= -2 \sin \left(\frac {A+B}2 \right) \sin \left(\frac {A-B}2 \right)
\end{align}</math>
ãåŸãããã
'''èŠãæ¹'''
ç©âåã®å
¬åŒã¯ãäž2ã€ã¯ <math>\alpha</math> ãš <math>\beta</math> ãå
¥ãæ¿ããã°åãåŒãªã®ã§ãèŠããã®ã¯3åŒã§ããã<math>\sin\sin</math> ã®å
¬åŒã¯ <math>\cos\cos</math> ã®å
¬åŒã®ç¬Šå·ã2〠<math>-</math> ã«ãããã®ã«ãªã£ãŠããã
åâç©ã®å
¬åŒã¯ã<math>\rm{aaa}-\rm{aaa}</math> ã®åŒã¯ <math>\rm{aaa}+\rm{aaa}</math> ã®å
¬åŒã® <math>\cos</math> ãš <math>\sin</math> ãéã«ãã圢ã«ãªã£ãŠããã
== äžè§é¢æ°ã®åºæ¬å
¬åŒ ==
* åšææ§ïŒ''n''ãã¯æŽæ°ïŒ
:<math>\begin{align}
\sin(\theta+2\pi n)&=\sin \theta \\
\cos(\theta+2\pi n)&=\cos \theta \\
\tan(\theta+2\pi n)&=\tan \theta
\end{align}</math>
* å¶é¢æ°ãå¥é¢æ°
:<math>\begin{align}
\sin(-\theta)&=-\sin \theta \\
\cos(-\theta)&=\cos \theta \\
\tan(-\theta)&=-\tan \theta
\end{align}</math>
* <math>\theta+\pi</math>
:<math>\begin{align}
\sin(\theta+\pi)&=-\sin \theta \\
\cos(\theta+\pi)&=-\cos \theta \\
\tan(\theta+\pi)&=\tan \theta
\end{align}</math>
* <math>\pi-\theta</math>
:<math>\begin{align}
\sin(\pi-\theta)&=\sin \theta \\
\cos(\pi-\theta)&=-\cos \theta \\
\tan(\pi-\theta)&=-\tan \theta
\end{align}</math>
* <math>\theta+\frac{1}{2}\pi</math>
:<math>\begin{align}
\sin \left(\theta+\frac{1}{2}\pi \right)&=\cos \theta \\
\cos \left(\theta+\frac{1}{2}\pi \right)&=-\sin \theta \\
\tan \left(\theta+\frac{1}{2}\pi \right)&=-\frac{1}{\tan \theta}
\end{align}</math>
* <math>\frac{\pi}{2}-\theta</math>
:<math>\begin{align}
\sin \left(\frac{\pi}{2}-\theta \right)&=\cos \theta \\
\cos \left(\frac{\pi}{2}-\theta \right)&=\sin \theta \\
\tan \left(\frac{\pi}{2}-\theta \right)&=\frac{1}{\tan \theta}
\end{align}</math>
* åé¡äŸ
** åé¡
*:(i) <math>\sin \frac{10}{3} \pi</math>
*:(ii) <math>\cos \left(- \frac{11}{4} \pi \right)</math>
*:(iii) <math>\tan \frac{31}{6} \pi</math>
*:ã®å€ãæ±ããã
** 解ç
*:(i)
*::<math>
\begin{align}
\sin \frac{10}{3} \pi & = \sin \left(\frac{4}{3}\pi + 2 \pi \right) = \sin \frac{4}{3} \pi \\
& = \sin \left(\frac{\pi}{3} + \pi \right) = - \sin \frac{\pi}{3} \\
& = - \frac{\sqrt{3}}{2}
\end{align}
</math>
*:(ii)
*::<math>
\begin{align}
\cos \left(- \frac{11}{4} \pi \right) & = \cos \frac{11}{4} \pi = \cos \left(\frac{3}{4}\pi + 2 \pi \right)\\
& = \cos \frac{3}{4} \pi = \cos \left(\pi - \frac{\pi}{4}\right)\\
& = - \cos \frac{\pi}{4} = - \frac{1}{\sqrt{2}}
\end{align}
</math>
*:(iii)
*::<math>
\begin{align}
\tan \frac{31}{6} \pi & = \tan \left(\frac{7}{6}\pi + 2 \pi \times 2 \right) = \tan \frac{7}{6} \pi \\
& = \tan \left(\frac{\pi}{6} + \pi \right) = \tan \frac{\pi}{6} \\
& = \frac{1}{\sqrt{3}}
\end{align}
</math>
{{ã³ã©ã |楜åšã®é³ãšäžè§é¢æ°|é³ãæ³¢ã®äžçš®ãªã®ã§ãäžè§é¢æ°ã§è¡šçŸã§ããã
ãªã·ãã¹ã³ãŒã㧠ããã ã®é³ã枬å®ãããšãæ£åŒŠæ³¢ã«è¿ã波圢ã芳枬ãããã
ããããå®éã®æ¥œåšã®é³ã¯ãæ£åŒŠæ³¢ãšã¯éãããªã·ãã¹ã³ãŒãã§ã®ã¿ãŒããã€ãªãªã³ãªã©ã®æ¥œåšã®é³ã枬å®ãããšãæ£åŒŠæ³¢ã§ãªã波圢ãç¹°ãè¿ãããŠããã
ãããå®éã®æ¥œåšã®é³ã®æ³¢åœ¢ã¯ãåšæã®ç°ãªãè€æ°åã®æ£åŒŠæ³¢ãéãåããã波圢ã«ãªã£ãŠããã
:倧åŠãªã©ã§ç¿ãããŒãªãšè§£æã§ããã®ãããªæ£åŒŠæ³¢ã§ãªã波圢ã®è§£æã«ã€ããŠè©³ããç¿ããäžè§é¢æ°ä»¥å€ã®åšæçãªé¢æ°ããäžè§é¢æ°ãä»ããŠè¡šçŸããææ³ãç¥ãããŠããã
}}
{{ã³ã©ã |æ°åŠè
ã¬ãªã³ãã«ãã»ãªã€ã©ãŒ|
[[File:Leonhard_Euler_2.jpg|thumb| ã¬ãªã³ãã«ãã»ãªã€ã©ãŒ(Leonhard Euler 1707幎4æ15æ¥ - 1783幎9æ18æ¥)]]
ããã§ã¯ãææ°é¢æ°ãäžè§é¢æ°ã®å®çŸ©åãå®æ°ãšããŠãããããããã®é¢æ°ã®å®çŸ©åãè€çŽ æ°ãŸã§æ¡åŒµããããšãã§ããã(èå³ã®ããæ欲çãªèªè
ã¯è€çŽ é¢æ°è«ã®æžç±ãèªãã§ã¿ããšãã)
è€çŽ æ°ã«æ¡åŒµããææ°é¢æ°ãäžè§é¢æ°ã§ã¯ <math>e^{i\theta} =\cos\theta+i\sin\theta</math>
ãšããé¢ä¿åŒãæãç«ã€ããã ãã<math>e</math> ã¯ãã€ãã¢æ°ã§ <math>e \approx 2.7</math> ã§ãããããã§ã <math>\theta</math> ã« <math>\pi</math> ã代å
¥ãããš
<math>e^{i\pi}+1=0</math>ãšãªãããã®çåŒã¯ãäžçäžçŸããçåŒããšãåŒã°ããå°èª¬ã«ããªã£ãŠããã®ã§ç¥ã£ãŠãã人ãããã ããã
}}
== æŒç¿åé¡ ==
(1)äžã®åºŠæ°æ³ã§è¡šãããå€ã匧床æ³ãŠè¡šã
1)<math>150</math>
2)<math>720</math>
(2)<math>\sin \pi/2</math>ã®å€ãæ±ãã
== è泚 ==
<references />
{{Wikiversity|Topic:äžè§é¢æ°|äžè§é¢æ°}}
{{DEFAULTSORT:ãããšããã€ããããããII ãããããããã}}
[[Category:é«çåŠæ ¡æ°åŠII|ãããããããã]]
[[ã«ããŽãª:äžè§é¢æ°]] | 2005-05-06T11:30:25Z | 2023-10-01T14:59:30Z | [
"ãã³ãã¬ãŒã:ã³ã©ã ",
"ãã³ãã¬ãŒã:Wikiversity",
"ãã³ãã¬ãŒã:Pathnav",
"ãã³ãã¬ãŒã:-",
"ãã³ãã¬ãŒã:åŒçªå·"
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E6%95%B0%E5%AD%A6II/%E4%B8%89%E8%A7%92%E9%96%A2%E6%95%B0 |
1,913 | é«çåŠæ ¡æ°åŠII/äžè§é¢æ° | ããã§ã¯äžè§é¢æ°ã®å®çŸ©ãããããšãäžè§é¢æ°ã®åºæ¬çãªæ§è³ªãå æ³å®çãäžè§é¢æ°ã®å¿çšã«ã€ããŠåŠã¶ãäžè§é¢æ°ã¯æ³¢ããã¯ãã«ã®å
ç©ãããŒãªãšå€æãªã©ããŸããŸãªåéã§å¿çšãããŠããã
å³å³ã®ããã«ãå®ç¹Oãäžå¿ãšããŠå転ããåçŽç· OP ãèããããã®ãšãã®å転ããåçŽç· OP ã®ããšãååŸãšããã
åçŽç· OX ãè§åºŠã®åºæºãšããããã®åºæºãšãªãåçŽç· OX ã®ããšãå§ç·ãšããã
ååŸãæèšåãã«å転ããå Žåãå転ããè§åºŠã¯è² ã§ãããšããååŸãåæèšåããããå Žåãå転ããè§åºŠã¯æ£ã§ãããšããã
è² ã®è§åºŠã360°以äžå転ããè§åºŠãèãã«å
¥ããè§ã®ããšãäžè¬è§ãšããã
ããŸãŸã§ã¯è§åºŠã®åäœãšããŠäžåšã 360° ãšãã床æ°æ³ã䜿ã£ãŠããããšã ãããããã§ã匧床æ³ã«ããè§åºŠã®è¡šãæ¹ãåŠã¶ã
ååŸ1 ã®æ圢ã«ãããŠåŒ§ã®é·ãã 1 ã®ãšãã®äžå¿è§ã 1 radãåæ§ã«åŒ§ã®é·ããΞã®ãšãã®äžå¿è§ãΞ radãšå®çŸ©ããããã®å®çŸ©ãã 180° =Ï radã360° = 2Ï rad ãããã«
ãšãªãããŸã匧床æ³ã®åäœ(rad)ã¯ãã°ãã°çç¥ãããã
匧床æ³ãçšãããšãäžè§é¢æ°ã®åŸ®ç©åãèããéã«äŸ¿å©ã§ããã(ãã®ããšã¯æ°åŠIIIã§åŠã¶)
æ圢ã®ååŸãr ã匧床æ³ã§å®çŸ©ãããè§åºŠãΞãšãããšãã匧ã®é·ãl ãšé¢ç©S ã¯
ãšè¡šããã
äžè¬è§ã Ξ {\displaystyle \theta } ã®åçŽç·ãšåäœåã亀ããåã P {\displaystyle \mathrm {P} } ãšããããã®ãšãã® P {\displaystyle \mathrm {P} } ã®åº§æšã ( cos Ξ , sin Ξ ) {\displaystyle (\cos \theta ,\sin \theta )} ãšããããšã§ãé¢æ° sin , cos {\displaystyle \sin ,\cos } ãå®ããããŸãã tan Ξ = sin Ξ cos Ξ {\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}} ãšããããšã§é¢æ° tan Ξ {\displaystyle \tan \theta } ãå®ããã tan Ξ {\displaystyle \tan \theta } ã¯äžè¬è§ã Ξ {\displaystyle \theta } ã®ååŸã®åŸãã«çããã
ãŸããäžè§é¢æ°ã®çŽ¯ä¹ã¯ ( sin Ξ ) n = sin n Ξ {\displaystyle (\sin \theta )^{n}=\sin ^{n}\theta } ãšè¡šèšãããã
cos Ξ ã®ã°ã©ã㯠sin Ξ ã®ã°ã©ãã Ξ軞æ¹åã« â Ï 2 {\displaystyle -{\frac {\pi }{2}}} ã ãå¹³è¡ç§»åãããã®ã§ããã
y = sin Ξ {\displaystyle y=\sin \theta } ã y = cos Ξ {\displaystyle y=\cos \theta } ã®åœ¢ãããæ²ç·ã®ããšã æ£åŒŠæ²ç· (ããããããããã)ãšããã
é¢æ° sin , cos {\displaystyle \sin ,\cos } ã®å€åã¯ã©ã¡ããã [ â 1 , 1 ] {\displaystyle [-1,1]} ã§ããã
å³å³ã®ããã« ãè§ Îž ã®ååŸãšåäœåãšã®äº€ç¹ãPãšããŠã çŽç·OPãš çŽç·x=1 ãšã®äº€ç¹ã T ãšãããšã Tã®åº§æšã¯
ã«ãªãã
ãã®ããšãå©çšããŠã y=tan Ξ ã®ã°ã©ããããããšãã§ããã
y=tan Ξ ã®ã°ã©ãã¯ãäžå³ã®ããã«ãªãã
y=tan Ξ ã®ã°ã©ãã§ã¯ãΞã®å€ã Ï 2 {\displaystyle {\frac {\pi }{2}}} ã«è¿ã¥ããŠãããšã çŽç· Ξ = Ï 2 {\displaystyle \theta ={\frac {\pi }{2}}} ã«éããªãè¿ã¥ããŠããã
ãã®ããã«ãæ²ç·ãããçŽç·ã«éãç¡ãè¿ã¥ããŠãããšããè¿ã¥ãããçŽç·ã®ã»ãã 挞è¿ç· (ãããããã)ãšããã
åæ§ã«èãã次ã®çŽç·ã y=tanΞ ã®æŒžè¿ç·ã§ããã
㯠y=tanΞ ã®æŒžè¿ç·ã§ããã
äžè¬ã«ã
ã¯y=tanΞã®ã°ã©ãã®æŒžè¿ç·ã§ããã
äžè¬è§ã Ξ {\displaystyle \theta } ã®ååŸã¯äžå転ããŠãçããã®ã§ãäžè¬è§ã Ξ + 2 Ï {\displaystyle \theta +2\pi } ã®ååŸãšçãããããããäžè§é¢æ°ã®åšææ§
sin ( Ξ + 2 Ï n ) = sin Ξ cos ( Ξ + 2 Ï n ) = cos Ξ tan ( Ξ + 2 Ï n ) = tan Ξ {\displaystyle {\begin{aligned}\sin(\theta +2\pi n)&=\sin \theta \\\cos(\theta +2\pi n)&=\cos \theta \\\tan(\theta +2\pi n)&=\tan \theta \end{aligned}}}
ãåŸãã
ç¹ ( cos Ξ , sin Ξ ) {\displaystyle (\cos \theta ,\sin \theta )} ã Ï {\displaystyle \pi } å転ããç¹ ( cos ( Ξ + Ï ) , sin ( Ξ + Ï ) ) {\displaystyle (\cos(\theta +\pi ),\sin(\theta +\pi ))} ã¯åç¹ãäžå¿ã«ç¹å¯Ÿç§°ç§»åããç¹ ( â cos Ξ , â sin Ξ ) {\displaystyle (-\cos \theta ,-\sin \theta )} ã§ããããšãã
sin ( Ξ + Ï ) = â sin Ξ cos ( Ξ + Ï ) = â cos Ξ tan ( Ξ + Ï ) = tan Ξ {\displaystyle {\begin{aligned}\sin(\theta +\pi )&=-\sin \theta \\\cos(\theta +\pi )&=-\cos \theta \\\tan(\theta +\pi )&=\tan \theta \end{aligned}}}
ãåŸãã
ç¹ ( cos Ξ , sin Ξ ) {\displaystyle (\cos \theta ,\sin \theta )} ã x {\displaystyle x} 軞ã§ç·å¯Ÿç§°ç§»å移åããç¹ã ( cos ( â Ξ ) , sin ( â Ξ ) ) = ( cos Ξ , â sin Ξ ) {\displaystyle (\cos(-\theta ),\sin(-\theta ))=(\cos \theta ,-\sin \theta )} ã§ããããšãã
sin ( â Ξ ) = â sin Ξ cos ( â Ξ ) = cos Ξ tan ( â Ξ ) = â tan Ξ {\displaystyle {\begin{aligned}\sin(-\theta )&=-\sin \theta \\\cos(-\theta )&=\cos \theta \\\tan(-\theta )&=-\tan \theta \end{aligned}}}
ãåŸãã
åäœååšäžã®ç¹ ( cos Ξ , sin Ξ ) {\displaystyle (\cos \theta ,\sin \theta )} ããåç¹ãŸã§ã®è·é¢ã¯ 1 ãªã®ã§ã sin 2 Ξ + cos 2 Ξ = 1 {\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1} ãæãç«ã€ã
ãŸãããã®åŒã«ã tan Ξ = sin Ξ cos Ξ {\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}} ã€ãŸãã sin Ξ = tan Ξ cos Ξ {\displaystyle \sin \theta =\tan \theta \cos \theta } ã代å
¥ããã°ã 1 + tan 2 Ξ = 1 cos 2 Ξ {\displaystyle 1+\tan ^{2}\theta ={\frac {1}{\cos ^{2}\theta }}} ãæãç«ã€ããšããããã
é¢æ° f ( x ) {\displaystyle f(x)} ã«å¯ŸããŠã0 ã§ãªãå®æ° p {\displaystyle p} ãååšããŠã f ( x + p ) = f ( x ) {\displaystyle f(x+p)=f(x)} ãšãªããšãé¢æ° f ( x ) {\displaystyle f(x)} ã¯åšæé¢æ°ãšãããå®æ° p {\displaystyle p} ãäžã®æ§è³ªãæºãããšãã â p , 2 p {\displaystyle -p,2p} ãªã©ãå®æ° p {\displaystyle p} ã0ãé€ãæŽæ°åããæ°ãäžã®æ§è³ªãæºãããããã§ãåšæé¢æ°ãç¹åŸŽã¥ããéãšããŠãäžã®æ§è³ªãæºããå®æ° p {\displaystyle p} ã®å
ãæ£ã§ãã€æå°ã®ãã®ãéžã³ããããåšæãšåŒã¶ã
sin x , cos x {\displaystyle \sin x,\cos x} ã¯åšæã 2 Ï {\displaystyle 2\pi } ãšããåšæé¢æ°ã§ããã tan x {\displaystyle \tan x} ã¯åšæã Ï {\displaystyle \pi } ãšããåšæé¢æ°ã§ããã
æŒç¿åé¡
k {\displaystyle k} ã0ã§ãªãå®æ°ãšãããé¢æ° sin k x {\displaystyle \sin kx} ã®åšæãèšã
解ç
sin k ( x + 2 Ï k ) = sin k x {\displaystyle \sin k\left(x+{\frac {2\pi }{k}}\right)=\sin kx} ãªã®ã§çã㯠2 Ï k {\displaystyle {\frac {2\pi }{k}}} ãããã¯æ£ã§ãããåšæã®æå°æ§ã®æ¡ä»¶ãæºãããŠããã
é¢æ° f ( x ) {\displaystyle f(x)} ã f ( â x ) = f ( x ) {\displaystyle f(-x)=f(x)} ãæºãããšããé¢æ° f ( x ) {\displaystyle f(x)} ã¯å¶é¢æ°ãšãããå¶é¢æ°ã¯ y {\displaystyle y} 軞ã«é¢ããŠå¯Ÿç§°ãªã°ã©ãã«ãªãã
ãŸããé¢æ° f ( x ) {\displaystyle f(x)} ã f ( â x ) = â f ( x ) {\displaystyle f(-x)=-f(x)} ãæºãããšããé¢æ° f ( x ) {\displaystyle f(x)} ã¯å¥é¢æ°ãšãããå¶é¢æ°ã¯åç¹ã«é¢ããŠå¯Ÿè±¡ãªã°ã©ãã«ãªãã
é¢æ° cos Ξ , x 2 n {\displaystyle \cos \theta ,x^{2n}} ( n {\displaystyle n} ã¯æŽæ°)ã¯å¶é¢æ°ãšãªãã
é¢æ° sin x , x 2 n + 1 {\displaystyle \sin x,x^{2n+1}} ( n {\displaystyle n} ã¯æŽæ°)ã¯å¥é¢æ°ãšãªãã
tan Ξ {\displaystyle \tan \theta } ã¯å¶é¢æ°ããããšãå¥é¢æ°ã調ã¹ãã
解ç
ãªã®ã§ã tan Ξ {\displaystyle \tan \theta } ã¯å¥é¢æ°ã§ããã
é¢æ° y = sin ( Ξ â Ï 3 ) {\displaystyle y=\sin \left(\theta -{\frac {\pi }{3}}\right)} ã®ã°ã©ãã¯ã y = sin Ξ {\displaystyle y=\sin \theta } ã®ã°ã©ãã Ξ軞æ¹åã« Ï 3 {\displaystyle {\frac {\pi }{3}}} ã ãå¹³è¡ç§»åããããã®ã«ãªããåšæ㯠2 Ï {\displaystyle 2\pi } ã§ããã(å¹³è¡ç§»åããŠããåšæã¯å€ããããsinΞãšåããåšæ㯠2 Ï {\displaystyle 2\pi } ã®ãŸãŸã§ããã)
é¢æ° y=2sin Ξ ã®ã°ã©ãã®åœ¢ã¯ y=sin Ξ ãy軞æ¹åã«2åã«æ¡å€§ãããã®ã§ãåšæ㯠y=sin Ξ ãšåãã 2Ï ã§ããã
ãŒ1 ⊠sin Ξ ⊠1 ãªã®ã§ã
å€å㯠ãŒ2 ⊠2sin Ξ ⊠2 ã§ããã
é¢æ° y=sin2Ξ ã®ã°ã©ãã¯y軞ãåºæºã«Îžè»žæ¹åã« 1 2 {\displaystyle {\frac {1}{2}}} åã«çž®å°ãããã®ã«ãªã£ãŠããã
ãããã£ãŠãåšæã 1 2 {\displaystyle {\frac {1}{2}}} åã«ãªã£ãŠãããy=sinΞ ã®åšæ㯠2 Ï {\displaystyle 2\pi } ã ãããy=sin2Ξ ã®åšæã¯ Ï {\displaystyle \pi } ã§ããã
äžè§é¢æ°ã®å æ³å®ç
ãæãç«ã€ã
蚌æ
ä»»æã®å®æ° α , β {\displaystyle \alpha ,\beta } ã«å¯Ÿããåäœååšäžã®ç¹ A ( cos α , sin α ) , B ( cos β , sin β ) {\displaystyle \mathrm {A} (\cos \alpha ,\sin \alpha ),\mathrm {B} (\cos \beta ,\sin \beta )} ããšãããã®ãšãã ç·å A B {\displaystyle \mathrm {AB} } ã®é·ãã®2ä¹ A B 2 {\displaystyle \mathrm {AB} ^{2}} ã¯äœåŒŠå®çã䜿ãããšã«ãã
A B 2 = 2 â 2 cos ( α â β ) {\displaystyle \mathrm {AB} ^{2}=2-2\cos(\alpha -\beta )}
ã§ããã次ã«äžå¹³æ¹ã®å®çã䜿ã£ãŠ
A B 2 = ( cos α â cos α ) 2 + ( sin α â sin β ) 2 = 2 â 2 ( cos α cos β + sin α sin β ) {\displaystyle \mathrm {AB} ^{2}=(\cos \alpha -\cos \alpha )^{2}+(\sin \alpha -\sin \beta )^{2}=2-2(\cos \alpha \cos \beta +\sin \alpha \sin \beta )}
ãããæŽçããŠ
cos ( α â β ) = cos α cos β + sin α sin β {\displaystyle \cos(\alpha -\beta )=\cos \alpha \cos \beta +\sin \alpha \sin \beta }
ãåŸãã
cos ( α + β ) = cos ( α â ( â β ) ) = cos α cos ( â β ) + sin α sin ( â β ) = cos α cos β â sin α sin β {\displaystyle \cos(\alpha +\beta )=\cos(\alpha -(-\beta ))=\cos \alpha \cos(-\beta )+\sin \alpha \sin(-\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta }
ã§ããã
以äžããŸãšããŠ
cos ( α ± β ) = cos α cos β â sin α sin β {\displaystyle \cos(\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta }
ãåŸãã
ããã§ã
sin ( α ± β ) = â cos ( α + Ï 2 ± β ) = â { cos ( α + Ï 2 ) cos ( β ) â sin ( α + Ï 2 ) sin β } = sin α cos β ± cos α sin β {\displaystyle \sin(\alpha \pm \beta )=-\cos(\alpha +{\frac {\pi }{2}}\pm \beta )=-\{\cos(\alpha +{\frac {\pi }{2}})\cos(\beta )\mp \sin(\alpha +{\frac {\pi }{2}})\sin \beta \}=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta }
ããã«ã tan x {\displaystyle \tan x} ã«ã€ããŠã
tan ( α ± β ) = sin ( α ± β ) cos ( α ± β ) = sin α cos β ± cos α sin β cos α cos β â sin α sin β = sin α cos β cos α cos β ± cos α sin β cos α cos β cos α cos β cos α cos β â sin α sin β cos α cos β = tan α ± tan β 1 â tan α tan β {\textstyle {\begin{aligned}\tan(\alpha \pm \beta )&={\frac {\sin(\alpha \pm \beta )}{\cos(\alpha \pm \beta )}}\\&={\frac {\sin \alpha \cos \beta \pm \cos \alpha \sin \beta }{\cos \alpha \cos \beta \mp \sin \alpha \sin \beta }}\\&={\cfrac {{\cfrac {\sin \alpha \cos \beta }{\cos \alpha \cos \beta }}\pm {\cfrac {\cos \alpha \sin \beta }{\cos \alpha \cos \beta }}}{{\cfrac {\cos \alpha \cos \beta }{\cos \alpha \cos \beta }}\mp {\cfrac {\sin \alpha \sin \beta }{\cos \alpha \cos \beta }}}}\\&={\frac {\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }}\end{aligned}}}
ãæãç«ã€ã
å æ³å®çãçšããŠä»¥äžã蚌æã§ããã
sin 2 α = sin ( α + α ) = 2 sin α cos α {\displaystyle \sin 2\alpha =\sin(\alpha +\alpha )=2\sin \alpha \cos \alpha }
cos 2 α = cos ( α + α ) = cos 2 α â sin 2 α = 2 cos 2 α â 1 = 1 â 2 sin 2 α {\displaystyle \cos 2\alpha =\cos(\alpha +\alpha )=\cos ^{2}\alpha -\sin ^{2}\alpha =2\cos ^{2}\alpha -1=1-2\sin ^{2}\alpha }
tan 2 α = 2 tan α 1 â tan 2 α {\displaystyle \tan 2\alpha ={\frac {2\tan \alpha }{1-\tan ^{2}\alpha }}}
次ã«ã cos {\displaystyle \cos } ã®åè§ã®å
¬åŒãå€åœ¢ãããš
sin 2 α = 1 â cos 2 α 2 {\displaystyle \sin ^{2}\alpha ={\frac {1-\cos 2\alpha }{2}}}
cos 2 α = 1 + cos 2 α 2 {\displaystyle \cos ^{2}\alpha ={\frac {1+\cos 2\alpha }{2}}}
ã§ããã
æŒç¿åé¡
解ç
sin 15 â = sin ( 45 â â 30 â ) = 6 â 2 4 {\displaystyle \sin 15^{\circ }=\sin(45^{\circ }-30^{\circ })={\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}
cos 15 â = cos ( 45 â â 30 â ) = 6 + 2 4 {\displaystyle \cos 15^{\circ }=\cos(45^{\circ }-30^{\circ })={\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}
tan 2 α = sin 2 α cos 2 α = 1 â cos 2 α 1 + cos 2 α {\displaystyle \tan ^{2}\alpha ={\frac {\sin ^{2}\alpha }{\cos ^{2}\alpha }}={\frac {1-\cos 2\alpha }{1+\cos 2\alpha }}}
ä»ãŸã§ã®å®çããŸãšãããšã次ã®ããã«ãªãã
èŠãæ¹
å æ³å®çã¯ãå²ããã³ã¹ã¢ã¹ã³ã¹ã¢ã¹å²ãããããã³ã¹ã¢ã¹ã³ã¹ã¢ã¹å²ããå²ããããšããèªååãããããŸãã
cos {\displaystyle \cos } ã®åè§ã®å
¬åŒ cos 2 Ξ = 2 cos 2 Ξ â 1 = 1 â 2 sin 2 Ξ {\displaystyle \cos 2\theta =2\cos ^{2}\theta -1=1-2\sin ^{2}\theta } 㯠± 1 â 2 a a a 2 Ξ {\displaystyle \pm 1\mp 2\mathrm {aaa} ^{2}\theta } ãšãã圢ãèŠã㊠sin {\displaystyle \sin } ã¯ç¬Šå·ã â {\displaystyle -} ã1 ã®ç¬Šå·ã¯ãã®éãšèŠããŸãã
2ä¹ã®äžè§é¢æ° sin 2 Ξ = 1 â cos 2 Ξ 2 , cos 2 Ξ = 1 + cos 2 Ξ 2 {\displaystyle \sin ^{2}\theta ={\frac {1-\cos 2\theta }{2}},\cos ^{2}\theta ={\frac {1+\cos 2\theta }{2}}} ã¯ã 1 ± cos 2 Ξ 2 {\displaystyle {\frac {1\pm \cos 2\theta }{2}}} ãšãã圢ãèŠããŠã sin {\displaystyle \sin } ã¯ç¬Šå·ã â {\displaystyle -} ãšèããŸãã
äžè§é¢æ°ã®å
ã«ãããŠã a , b â 0 {\displaystyle a,b\neq 0} ã®ãšã
{ a a 2 + b 2 } 2 + { b a 2 + b 2 } 2 = 1 {\displaystyle \left\{{\dfrac {a}{\sqrt {a^{2}+b^{2}}}}\right\}^{2}+\left\{{\dfrac {b}{\sqrt {a^{2}+b^{2}}}}\right\}^{2}=1} ã§ããã®ã§ãç¹ ( a a 2 + b 2 , b a 2 + b 2 ) {\displaystyle \left({\dfrac {a}{\sqrt {a^{2}+b^{2}}}},{\dfrac {b}{\sqrt {a^{2}+b^{2}}}}\right)} ã¯åäœååšäžã®ç¹ãªã®ã§ã
ãšãªããããªÎ±ããšãããšãã§ãããã®Î±ãçšããŠæ¬¡ã®ãããªå€åœ¢ãã§ããã
æŒç¿åé¡
r , α {\displaystyle r,\alpha } 㯠r > 0 , â Ï â€ Î± < Ï {\displaystyle r>0,-\pi \leq \alpha <\pi } ãæºãããšããã
解ç
sin Ξ â 3 cos Ξ = 2 ( 1 2 sin Ξ â 3 2 cos Ξ ) = 2 ( sin Ξ cos Ï 3 â cos Ξ sin Ï 3 ) = 2 sin ( Ξ â Ï 3 ) {\displaystyle {\begin{aligned}\sin \theta -{\sqrt {3}}\cos \theta &=2\left({\frac {1}{2}}\sin \theta -{\frac {\sqrt {3}}{2}}\cos \theta \right)\\&=2\left(\sin \theta \cos {\frac {\pi }{3}}-\cos \theta \sin {\frac {\pi }{3}}\right)\\&=2\sin \left(\theta -{\frac {\pi }{3}}\right)\\\end{aligned}}}
äžè§é¢æ°ã®å æ³å®çãçšãããšãäžè§é¢æ°ã®åâç©ã®å
¬åŒãããã³ç©âåã®å
¬åŒãåŸãããããããã
ãšãªãã
å æ³å®ç
ããã (1) + (2) ãã
(1) - (2) ãã
(3) + (4) ãã
(3) - (4) ãã
ãåŸãããã
A = α + β , B = α â β {\displaystyle A=\alpha +\beta ,\,B=\alpha -\beta } ãšãããšã α = A + B 2 , β = A â B 2 {\displaystyle \alpha ={\frac {A+B}{2}},\,\beta ={\frac {A-B}{2}}} ã§ããããããç©âåã®å
¬åŒã«ä»£å
¥ããã°ããããã
ãåŸãããã
èŠãæ¹
ç©âåã®å
¬åŒã¯ãäž2ã€ã¯ α {\displaystyle \alpha } 㚠β {\displaystyle \beta } ãå
¥ãæ¿ããã°åãåŒãªã®ã§ãèŠããã®ã¯3åŒã§ããã sin sin {\displaystyle \sin \sin } ã®å
¬åŒã¯ cos cos {\displaystyle \cos \cos } ã®å
¬åŒã®ç¬Šå·ã2〠â {\displaystyle -} ã«ãããã®ã«ãªã£ãŠããã
åâç©ã®å
¬åŒã¯ã a a a â a a a {\displaystyle {\rm {{aaa}-{\rm {aaa}}}}} ã®åŒã¯ a a a + a a a {\displaystyle {\rm {{aaa}+{\rm {aaa}}}}} ã®å
¬åŒã® cos {\displaystyle \cos } ãš sin {\displaystyle \sin } ãéã«ãã圢ã«ãªã£ãŠããã
(1)äžã®åºŠæ°æ³ã§è¡šãããå€ã匧床æ³ãŠè¡šã
1) 150 {\displaystyle 150} 2) 720 {\displaystyle 720}
(2) sin Ï / 2 {\displaystyle \sin \pi /2} ã®å€ãæ±ãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ããã§ã¯äžè§é¢æ°ã®å®çŸ©ãããããšãäžè§é¢æ°ã®åºæ¬çãªæ§è³ªãå æ³å®çãäžè§é¢æ°ã®å¿çšã«ã€ããŠåŠã¶ãäžè§é¢æ°ã¯æ³¢ããã¯ãã«ã®å
ç©ãããŒãªãšå€æãªã©ããŸããŸãªåéã§å¿çšãããŠããã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "å³å³ã®ããã«ãå®ç¹Oãäžå¿ãšããŠå転ããåçŽç· OP ãèããããã®ãšãã®å転ããåçŽç· OP ã®ããšãååŸãšããã",
"title": "äžè¬è§"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "åçŽç· OX ãè§åºŠã®åºæºãšããããã®åºæºãšãªãåçŽç· OX ã®ããšãå§ç·ãšããã",
"title": "äžè¬è§"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ååŸãæèšåãã«å転ããå Žåãå転ããè§åºŠã¯è² ã§ãããšããååŸãåæèšåããããå Žåãå転ããè§åºŠã¯æ£ã§ãããšããã",
"title": "äžè¬è§"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "è² ã®è§åºŠã360°以äžå転ããè§åºŠãèãã«å
¥ããè§ã®ããšãäžè¬è§ãšããã",
"title": "äžè¬è§"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "",
"title": "äžè¬è§"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ããŸãŸã§ã¯è§åºŠã®åäœãšããŠäžåšã 360° ãšãã床æ°æ³ã䜿ã£ãŠããããšã ãããããã§ã匧床æ³ã«ããè§åºŠã®è¡šãæ¹ãåŠã¶ã",
"title": "匧床æ³"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ååŸ1 ã®æ圢ã«ãããŠåŒ§ã®é·ãã 1 ã®ãšãã®äžå¿è§ã 1 radãåæ§ã«åŒ§ã®é·ããΞã®ãšãã®äžå¿è§ãΞ radãšå®çŸ©ããããã®å®çŸ©ãã 180° =Ï radã360° = 2Ï rad ãããã«",
"title": "匧床æ³"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "ãšãªãããŸã匧床æ³ã®åäœ(rad)ã¯ãã°ãã°çç¥ãããã",
"title": "匧床æ³"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "匧床æ³ãçšãããšãäžè§é¢æ°ã®åŸ®ç©åãèããéã«äŸ¿å©ã§ããã(ãã®ããšã¯æ°åŠIIIã§åŠã¶)",
"title": "匧床æ³"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "æ圢ã®ååŸãr ã匧床æ³ã§å®çŸ©ãããè§åºŠãΞãšãããšãã匧ã®é·ãl ãšé¢ç©S ã¯",
"title": "匧床æ³"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "ãšè¡šããã",
"title": "匧床æ³"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "äžè¬è§ã Ξ {\\displaystyle \\theta } ã®åçŽç·ãšåäœåã亀ããåã P {\\displaystyle \\mathrm {P} } ãšããããã®ãšãã® P {\\displaystyle \\mathrm {P} } ã®åº§æšã ( cos Ξ , sin Ξ ) {\\displaystyle (\\cos \\theta ,\\sin \\theta )} ãšããããšã§ãé¢æ° sin , cos {\\displaystyle \\sin ,\\cos } ãå®ããããŸãã tan Ξ = sin Ξ cos Ξ {\\displaystyle \\tan \\theta ={\\frac {\\sin \\theta }{\\cos \\theta }}} ãšããããšã§é¢æ° tan Ξ {\\displaystyle \\tan \\theta } ãå®ããã tan Ξ {\\displaystyle \\tan \\theta } ã¯äžè¬è§ã Ξ {\\displaystyle \\theta } ã®ååŸã®åŸãã«çããã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ãŸããäžè§é¢æ°ã®çŽ¯ä¹ã¯ ( sin Ξ ) n = sin n Ξ {\\displaystyle (\\sin \\theta )^{n}=\\sin ^{n}\\theta } ãšè¡šèšãããã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "cos Ξ ã®ã°ã©ã㯠sin Ξ ã®ã°ã©ãã Ξ軞æ¹åã« â Ï 2 {\\displaystyle -{\\frac {\\pi }{2}}} ã ãå¹³è¡ç§»åãããã®ã§ããã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "y = sin Ξ {\\displaystyle y=\\sin \\theta } ã y = cos Ξ {\\displaystyle y=\\cos \\theta } ã®åœ¢ãããæ²ç·ã®ããšã æ£åŒŠæ²ç· (ããããããããã)ãšããã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "é¢æ° sin , cos {\\displaystyle \\sin ,\\cos } ã®å€åã¯ã©ã¡ããã [ â 1 , 1 ] {\\displaystyle [-1,1]} ã§ããã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "å³å³ã®ããã« ãè§ Îž ã®ååŸãšåäœåãšã®äº€ç¹ãPãšããŠã çŽç·OPãš çŽç·x=1 ãšã®äº€ç¹ã T ãšãããšã Tã®åº§æšã¯",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "ã«ãªãã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "ãã®ããšãå©çšããŠã y=tan Ξ ã®ã°ã©ããããããšãã§ããã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "y=tan Ξ ã®ã°ã©ãã¯ãäžå³ã®ããã«ãªãã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "y=tan Ξ ã®ã°ã©ãã§ã¯ãΞã®å€ã Ï 2 {\\displaystyle {\\frac {\\pi }{2}}} ã«è¿ã¥ããŠãããšã çŽç· Ξ = Ï 2 {\\displaystyle \\theta ={\\frac {\\pi }{2}}} ã«éããªãè¿ã¥ããŠããã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "ãã®ããã«ãæ²ç·ãããçŽç·ã«éãç¡ãè¿ã¥ããŠãããšããè¿ã¥ãããçŽç·ã®ã»ãã 挞è¿ç· (ãããããã)ãšããã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "åæ§ã«èãã次ã®çŽç·ã y=tanΞ ã®æŒžè¿ç·ã§ããã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "㯠y=tanΞ ã®æŒžè¿ç·ã§ããã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "äžè¬ã«ã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "ã¯y=tanΞã®ã°ã©ãã®æŒžè¿ç·ã§ããã",
"title": "äžè§é¢æ°"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "äžè¬è§ã Ξ {\\displaystyle \\theta } ã®ååŸã¯äžå転ããŠãçããã®ã§ãäžè¬è§ã Ξ + 2 Ï {\\displaystyle \\theta +2\\pi } ã®ååŸãšçãããããããäžè§é¢æ°ã®åšææ§",
"title": "äžè§é¢æ°ã®æ§è³ª"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "sin ( Ξ + 2 Ï n ) = sin Ξ cos ( Ξ + 2 Ï n ) = cos Ξ tan ( Ξ + 2 Ï n ) = tan Ξ {\\displaystyle {\\begin{aligned}\\sin(\\theta +2\\pi n)&=\\sin \\theta \\\\\\cos(\\theta +2\\pi n)&=\\cos \\theta \\\\\\tan(\\theta +2\\pi n)&=\\tan \\theta \\end{aligned}}}",
"title": "äžè§é¢æ°ã®æ§è³ª"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "ãåŸãã",
"title": "äžè§é¢æ°ã®æ§è³ª"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "ç¹ ( cos Ξ , sin Ξ ) {\\displaystyle (\\cos \\theta ,\\sin \\theta )} ã Ï {\\displaystyle \\pi } å転ããç¹ ( cos ( Ξ + Ï ) , sin ( Ξ + Ï ) ) {\\displaystyle (\\cos(\\theta +\\pi ),\\sin(\\theta +\\pi ))} ã¯åç¹ãäžå¿ã«ç¹å¯Ÿç§°ç§»åããç¹ ( â cos Ξ , â sin Ξ ) {\\displaystyle (-\\cos \\theta ,-\\sin \\theta )} ã§ããããšãã",
"title": "äžè§é¢æ°ã®æ§è³ª"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "sin ( Ξ + Ï ) = â sin Ξ cos ( Ξ + Ï ) = â cos Ξ tan ( Ξ + Ï ) = tan Ξ {\\displaystyle {\\begin{aligned}\\sin(\\theta +\\pi )&=-\\sin \\theta \\\\\\cos(\\theta +\\pi )&=-\\cos \\theta \\\\\\tan(\\theta +\\pi )&=\\tan \\theta \\end{aligned}}}",
"title": "äžè§é¢æ°ã®æ§è³ª"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "ãåŸãã",
"title": "äžè§é¢æ°ã®æ§è³ª"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "ç¹ ( cos Ξ , sin Ξ ) {\\displaystyle (\\cos \\theta ,\\sin \\theta )} ã x {\\displaystyle x} 軞ã§ç·å¯Ÿç§°ç§»å移åããç¹ã ( cos ( â Ξ ) , sin ( â Ξ ) ) = ( cos Ξ , â sin Ξ ) {\\displaystyle (\\cos(-\\theta ),\\sin(-\\theta ))=(\\cos \\theta ,-\\sin \\theta )} ã§ããããšãã",
"title": "äžè§é¢æ°ã®æ§è³ª"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "sin ( â Ξ ) = â sin Ξ cos ( â Ξ ) = cos Ξ tan ( â Ξ ) = â tan Ξ {\\displaystyle {\\begin{aligned}\\sin(-\\theta )&=-\\sin \\theta \\\\\\cos(-\\theta )&=\\cos \\theta \\\\\\tan(-\\theta )&=-\\tan \\theta \\end{aligned}}}",
"title": "äžè§é¢æ°ã®æ§è³ª"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "ãåŸãã",
"title": "äžè§é¢æ°ã®æ§è³ª"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "åäœååšäžã®ç¹ ( cos Ξ , sin Ξ ) {\\displaystyle (\\cos \\theta ,\\sin \\theta )} ããåç¹ãŸã§ã®è·é¢ã¯ 1 ãªã®ã§ã sin 2 Ξ + cos 2 Ξ = 1 {\\displaystyle \\sin ^{2}\\theta +\\cos ^{2}\\theta =1} ãæãç«ã€ã",
"title": "äžè§é¢æ°ã®æ§è³ª"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "ãŸãããã®åŒã«ã tan Ξ = sin Ξ cos Ξ {\\displaystyle \\tan \\theta ={\\frac {\\sin \\theta }{\\cos \\theta }}} ã€ãŸãã sin Ξ = tan Ξ cos Ξ {\\displaystyle \\sin \\theta =\\tan \\theta \\cos \\theta } ã代å
¥ããã°ã 1 + tan 2 Ξ = 1 cos 2 Ξ {\\displaystyle 1+\\tan ^{2}\\theta ={\\frac {1}{\\cos ^{2}\\theta }}} ãæãç«ã€ããšããããã",
"title": "äžè§é¢æ°ã®æ§è³ª"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "é¢æ° f ( x ) {\\displaystyle f(x)} ã«å¯ŸããŠã0 ã§ãªãå®æ° p {\\displaystyle p} ãååšããŠã f ( x + p ) = f ( x ) {\\displaystyle f(x+p)=f(x)} ãšãªããšãé¢æ° f ( x ) {\\displaystyle f(x)} ã¯åšæé¢æ°ãšãããå®æ° p {\\displaystyle p} ãäžã®æ§è³ªãæºãããšãã â p , 2 p {\\displaystyle -p,2p} ãªã©ãå®æ° p {\\displaystyle p} ã0ãé€ãæŽæ°åããæ°ãäžã®æ§è³ªãæºãããããã§ãåšæé¢æ°ãç¹åŸŽã¥ããéãšããŠãäžã®æ§è³ªãæºããå®æ° p {\\displaystyle p} ã®å
ãæ£ã§ãã€æå°ã®ãã®ãéžã³ããããåšæãšåŒã¶ã",
"title": "åšæé¢æ°"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "sin x , cos x {\\displaystyle \\sin x,\\cos x} ã¯åšæã 2 Ï {\\displaystyle 2\\pi } ãšããåšæé¢æ°ã§ããã tan x {\\displaystyle \\tan x} ã¯åšæã Ï {\\displaystyle \\pi } ãšããåšæé¢æ°ã§ããã",
"title": "åšæé¢æ°"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "åšæé¢æ°"
},
{
"paragraph_id": 41,
"tag": "p",
"text": "k {\\displaystyle k} ã0ã§ãªãå®æ°ãšãããé¢æ° sin k x {\\displaystyle \\sin kx} ã®åšæãèšã",
"title": "åšæé¢æ°"
},
{
"paragraph_id": 42,
"tag": "p",
"text": "解ç",
"title": "åšæé¢æ°"
},
{
"paragraph_id": 43,
"tag": "p",
"text": "sin k ( x + 2 Ï k ) = sin k x {\\displaystyle \\sin k\\left(x+{\\frac {2\\pi }{k}}\\right)=\\sin kx} ãªã®ã§çã㯠2 Ï k {\\displaystyle {\\frac {2\\pi }{k}}} ãããã¯æ£ã§ãããåšæã®æå°æ§ã®æ¡ä»¶ãæºãããŠããã",
"title": "åšæé¢æ°"
},
{
"paragraph_id": 44,
"tag": "p",
"text": "é¢æ° f ( x ) {\\displaystyle f(x)} ã f ( â x ) = f ( x ) {\\displaystyle f(-x)=f(x)} ãæºãããšããé¢æ° f ( x ) {\\displaystyle f(x)} ã¯å¶é¢æ°ãšãããå¶é¢æ°ã¯ y {\\displaystyle y} 軞ã«é¢ããŠå¯Ÿç§°ãªã°ã©ãã«ãªãã",
"title": "å¶é¢æ°ãšå¥é¢æ°"
},
{
"paragraph_id": 45,
"tag": "p",
"text": "ãŸããé¢æ° f ( x ) {\\displaystyle f(x)} ã f ( â x ) = â f ( x ) {\\displaystyle f(-x)=-f(x)} ãæºãããšããé¢æ° f ( x ) {\\displaystyle f(x)} ã¯å¥é¢æ°ãšãããå¶é¢æ°ã¯åç¹ã«é¢ããŠå¯Ÿè±¡ãªã°ã©ãã«ãªãã",
"title": "å¶é¢æ°ãšå¥é¢æ°"
},
{
"paragraph_id": 46,
"tag": "p",
"text": "é¢æ° cos Ξ , x 2 n {\\displaystyle \\cos \\theta ,x^{2n}} ( n {\\displaystyle n} ã¯æŽæ°)ã¯å¶é¢æ°ãšãªãã",
"title": "å¶é¢æ°ãšå¥é¢æ°"
},
{
"paragraph_id": 47,
"tag": "p",
"text": "é¢æ° sin x , x 2 n + 1 {\\displaystyle \\sin x,x^{2n+1}} ( n {\\displaystyle n} ã¯æŽæ°)ã¯å¥é¢æ°ãšãªãã",
"title": "å¶é¢æ°ãšå¥é¢æ°"
},
{
"paragraph_id": 48,
"tag": "p",
"text": "tan Ξ {\\displaystyle \\tan \\theta } ã¯å¶é¢æ°ããããšãå¥é¢æ°ã調ã¹ãã",
"title": "å¶é¢æ°ãšå¥é¢æ°"
},
{
"paragraph_id": 49,
"tag": "p",
"text": "解ç",
"title": "å¶é¢æ°ãšå¥é¢æ°"
},
{
"paragraph_id": 50,
"tag": "p",
"text": "ãªã®ã§ã tan Ξ {\\displaystyle \\tan \\theta } ã¯å¥é¢æ°ã§ããã",
"title": "å¶é¢æ°ãšå¥é¢æ°"
},
{
"paragraph_id": 51,
"tag": "p",
"text": "é¢æ° y = sin ( Ξ â Ï 3 ) {\\displaystyle y=\\sin \\left(\\theta -{\\frac {\\pi }{3}}\\right)} ã®ã°ã©ãã¯ã y = sin Ξ {\\displaystyle y=\\sin \\theta } ã®ã°ã©ãã Ξ軞æ¹åã« Ï 3 {\\displaystyle {\\frac {\\pi }{3}}} ã ãå¹³è¡ç§»åããããã®ã«ãªããåšæ㯠2 Ï {\\displaystyle 2\\pi } ã§ããã(å¹³è¡ç§»åããŠããåšæã¯å€ããããsinΞãšåããåšæ㯠2 Ï {\\displaystyle 2\\pi } ã®ãŸãŸã§ããã)",
"title": "ãããããªäžè§é¢æ°"
},
{
"paragraph_id": 52,
"tag": "p",
"text": "",
"title": "ãããããªäžè§é¢æ°"
},
{
"paragraph_id": 53,
"tag": "p",
"text": "é¢æ° y=2sin Ξ ã®ã°ã©ãã®åœ¢ã¯ y=sin Ξ ãy軞æ¹åã«2åã«æ¡å€§ãããã®ã§ãåšæ㯠y=sin Ξ ãšåãã 2Ï ã§ããã",
"title": "ãããããªäžè§é¢æ°"
},
{
"paragraph_id": 54,
"tag": "p",
"text": "ãŒ1 ⊠sin Ξ ⊠1 ãªã®ã§ã",
"title": "ãããããªäžè§é¢æ°"
},
{
"paragraph_id": 55,
"tag": "p",
"text": "å€å㯠ãŒ2 ⊠2sin Ξ ⊠2 ã§ããã",
"title": "ãããããªäžè§é¢æ°"
},
{
"paragraph_id": 56,
"tag": "p",
"text": "é¢æ° y=sin2Ξ ã®ã°ã©ãã¯y軞ãåºæºã«Îžè»žæ¹åã« 1 2 {\\displaystyle {\\frac {1}{2}}} åã«çž®å°ãããã®ã«ãªã£ãŠããã",
"title": "ãããããªäžè§é¢æ°"
},
{
"paragraph_id": 57,
"tag": "p",
"text": "ãããã£ãŠãåšæã 1 2 {\\displaystyle {\\frac {1}{2}}} åã«ãªã£ãŠãããy=sinΞ ã®åšæ㯠2 Ï {\\displaystyle 2\\pi } ã ãããy=sin2Ξ ã®åšæã¯ Ï {\\displaystyle \\pi } ã§ããã",
"title": "ãããããªäžè§é¢æ°"
},
{
"paragraph_id": 58,
"tag": "p",
"text": "äžè§é¢æ°ã®å æ³å®ç",
"title": "å æ³å®ç"
},
{
"paragraph_id": 59,
"tag": "p",
"text": "ãæãç«ã€ã",
"title": "å æ³å®ç"
},
{
"paragraph_id": 60,
"tag": "p",
"text": "蚌æ",
"title": "å æ³å®ç"
},
{
"paragraph_id": 61,
"tag": "p",
"text": "ä»»æã®å®æ° α , β {\\displaystyle \\alpha ,\\beta } ã«å¯Ÿããåäœååšäžã®ç¹ A ( cos α , sin α ) , B ( cos β , sin β ) {\\displaystyle \\mathrm {A} (\\cos \\alpha ,\\sin \\alpha ),\\mathrm {B} (\\cos \\beta ,\\sin \\beta )} ããšãããã®ãšãã ç·å A B {\\displaystyle \\mathrm {AB} } ã®é·ãã®2ä¹ A B 2 {\\displaystyle \\mathrm {AB} ^{2}} ã¯äœåŒŠå®çã䜿ãããšã«ãã",
"title": "å æ³å®ç"
},
{
"paragraph_id": 62,
"tag": "p",
"text": "A B 2 = 2 â 2 cos ( α â β ) {\\displaystyle \\mathrm {AB} ^{2}=2-2\\cos(\\alpha -\\beta )}",
"title": "å æ³å®ç"
},
{
"paragraph_id": 63,
"tag": "p",
"text": "ã§ããã次ã«äžå¹³æ¹ã®å®çã䜿ã£ãŠ",
"title": "å æ³å®ç"
},
{
"paragraph_id": 64,
"tag": "p",
"text": "A B 2 = ( cos α â cos α ) 2 + ( sin α â sin β ) 2 = 2 â 2 ( cos α cos β + sin α sin β ) {\\displaystyle \\mathrm {AB} ^{2}=(\\cos \\alpha -\\cos \\alpha )^{2}+(\\sin \\alpha -\\sin \\beta )^{2}=2-2(\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta )}",
"title": "å æ³å®ç"
},
{
"paragraph_id": 65,
"tag": "p",
"text": "ãããæŽçããŠ",
"title": "å æ³å®ç"
},
{
"paragraph_id": 66,
"tag": "p",
"text": "cos ( α â β ) = cos α cos β + sin α sin β {\\displaystyle \\cos(\\alpha -\\beta )=\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta }",
"title": "å æ³å®ç"
},
{
"paragraph_id": 67,
"tag": "p",
"text": "ãåŸãã",
"title": "å æ³å®ç"
},
{
"paragraph_id": 68,
"tag": "p",
"text": "cos ( α + β ) = cos ( α â ( â β ) ) = cos α cos ( â β ) + sin α sin ( â β ) = cos α cos β â sin α sin β {\\displaystyle \\cos(\\alpha +\\beta )=\\cos(\\alpha -(-\\beta ))=\\cos \\alpha \\cos(-\\beta )+\\sin \\alpha \\sin(-\\beta )=\\cos \\alpha \\cos \\beta -\\sin \\alpha \\sin \\beta }",
"title": "å æ³å®ç"
},
{
"paragraph_id": 69,
"tag": "p",
"text": "ã§ããã",
"title": "å æ³å®ç"
},
{
"paragraph_id": 70,
"tag": "p",
"text": "以äžããŸãšããŠ",
"title": "å æ³å®ç"
},
{
"paragraph_id": 71,
"tag": "p",
"text": "cos ( α ± β ) = cos α cos β â sin α sin β {\\displaystyle \\cos(\\alpha \\pm \\beta )=\\cos \\alpha \\cos \\beta \\mp \\sin \\alpha \\sin \\beta }",
"title": "å æ³å®ç"
},
{
"paragraph_id": 72,
"tag": "p",
"text": "ãåŸãã",
"title": "å æ³å®ç"
},
{
"paragraph_id": 73,
"tag": "p",
"text": "ããã§ã",
"title": "å æ³å®ç"
},
{
"paragraph_id": 74,
"tag": "p",
"text": "sin ( α ± β ) = â cos ( α + Ï 2 ± β ) = â { cos ( α + Ï 2 ) cos ( β ) â sin ( α + Ï 2 ) sin β } = sin α cos β ± cos α sin β {\\displaystyle \\sin(\\alpha \\pm \\beta )=-\\cos(\\alpha +{\\frac {\\pi }{2}}\\pm \\beta )=-\\{\\cos(\\alpha +{\\frac {\\pi }{2}})\\cos(\\beta )\\mp \\sin(\\alpha +{\\frac {\\pi }{2}})\\sin \\beta \\}=\\sin \\alpha \\cos \\beta \\pm \\cos \\alpha \\sin \\beta }",
"title": "å æ³å®ç"
},
{
"paragraph_id": 75,
"tag": "p",
"text": "ããã«ã tan x {\\displaystyle \\tan x} ã«ã€ããŠã",
"title": "å æ³å®ç"
},
{
"paragraph_id": 76,
"tag": "p",
"text": "tan ( α ± β ) = sin ( α ± β ) cos ( α ± β ) = sin α cos β ± cos α sin β cos α cos β â sin α sin β = sin α cos β cos α cos β ± cos α sin β cos α cos β cos α cos β cos α cos β â sin α sin β cos α cos β = tan α ± tan β 1 â tan α tan β {\\textstyle {\\begin{aligned}\\tan(\\alpha \\pm \\beta )&={\\frac {\\sin(\\alpha \\pm \\beta )}{\\cos(\\alpha \\pm \\beta )}}\\\\&={\\frac {\\sin \\alpha \\cos \\beta \\pm \\cos \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta \\mp \\sin \\alpha \\sin \\beta }}\\\\&={\\cfrac {{\\cfrac {\\sin \\alpha \\cos \\beta }{\\cos \\alpha \\cos \\beta }}\\pm {\\cfrac {\\cos \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta }}}{{\\cfrac {\\cos \\alpha \\cos \\beta }{\\cos \\alpha \\cos \\beta }}\\mp {\\cfrac {\\sin \\alpha \\sin \\beta }{\\cos \\alpha \\cos \\beta }}}}\\\\&={\\frac {\\tan \\alpha \\pm \\tan \\beta }{1\\mp \\tan \\alpha \\tan \\beta }}\\end{aligned}}}",
"title": "å æ³å®ç"
},
{
"paragraph_id": 77,
"tag": "p",
"text": "ãæãç«ã€ã",
"title": "å æ³å®ç"
},
{
"paragraph_id": 78,
"tag": "p",
"text": "å æ³å®çãçšããŠä»¥äžã蚌æã§ããã",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 79,
"tag": "p",
"text": "sin 2 α = sin ( α + α ) = 2 sin α cos α {\\displaystyle \\sin 2\\alpha =\\sin(\\alpha +\\alpha )=2\\sin \\alpha \\cos \\alpha }",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 80,
"tag": "p",
"text": "cos 2 α = cos ( α + α ) = cos 2 α â sin 2 α = 2 cos 2 α â 1 = 1 â 2 sin 2 α {\\displaystyle \\cos 2\\alpha =\\cos(\\alpha +\\alpha )=\\cos ^{2}\\alpha -\\sin ^{2}\\alpha =2\\cos ^{2}\\alpha -1=1-2\\sin ^{2}\\alpha }",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 81,
"tag": "p",
"text": "tan 2 α = 2 tan α 1 â tan 2 α {\\displaystyle \\tan 2\\alpha ={\\frac {2\\tan \\alpha }{1-\\tan ^{2}\\alpha }}}",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 82,
"tag": "p",
"text": "次ã«ã cos {\\displaystyle \\cos } ã®åè§ã®å
¬åŒãå€åœ¢ãããš",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 83,
"tag": "p",
"text": "sin 2 α = 1 â cos 2 α 2 {\\displaystyle \\sin ^{2}\\alpha ={\\frac {1-\\cos 2\\alpha }{2}}}",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 84,
"tag": "p",
"text": "cos 2 α = 1 + cos 2 α 2 {\\displaystyle \\cos ^{2}\\alpha ={\\frac {1+\\cos 2\\alpha }{2}}}",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 85,
"tag": "p",
"text": "ã§ããã",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 86,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 87,
"tag": "p",
"text": "解ç",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 88,
"tag": "p",
"text": "sin 15 â = sin ( 45 â â 30 â ) = 6 â 2 4 {\\displaystyle \\sin 15^{\\circ }=\\sin(45^{\\circ }-30^{\\circ })={\\frac {{\\sqrt {6}}-{\\sqrt {2}}}{4}}}",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 89,
"tag": "p",
"text": "cos 15 â = cos ( 45 â â 30 â ) = 6 + 2 4 {\\displaystyle \\cos 15^{\\circ }=\\cos(45^{\\circ }-30^{\\circ })={\\frac {{\\sqrt {6}}+{\\sqrt {2}}}{4}}}",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 90,
"tag": "p",
"text": "tan 2 α = sin 2 α cos 2 α = 1 â cos 2 α 1 + cos 2 α {\\displaystyle \\tan ^{2}\\alpha ={\\frac {\\sin ^{2}\\alpha }{\\cos ^{2}\\alpha }}={\\frac {1-\\cos 2\\alpha }{1+\\cos 2\\alpha }}}",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 91,
"tag": "p",
"text": "ä»ãŸã§ã®å®çããŸãšãããšã次ã®ããã«ãªãã",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 92,
"tag": "p",
"text": "èŠãæ¹",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 93,
"tag": "p",
"text": "å æ³å®çã¯ãå²ããã³ã¹ã¢ã¹ã³ã¹ã¢ã¹å²ãããããã³ã¹ã¢ã¹ã³ã¹ã¢ã¹å²ããå²ããããšããèªååãããããŸãã",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 94,
"tag": "p",
"text": "cos {\\displaystyle \\cos } ã®åè§ã®å
¬åŒ cos 2 Ξ = 2 cos 2 Ξ â 1 = 1 â 2 sin 2 Ξ {\\displaystyle \\cos 2\\theta =2\\cos ^{2}\\theta -1=1-2\\sin ^{2}\\theta } 㯠± 1 â 2 a a a 2 Ξ {\\displaystyle \\pm 1\\mp 2\\mathrm {aaa} ^{2}\\theta } ãšãã圢ãèŠã㊠sin {\\displaystyle \\sin } ã¯ç¬Šå·ã â {\\displaystyle -} ã1 ã®ç¬Šå·ã¯ãã®éãšèŠããŸãã",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 95,
"tag": "p",
"text": "2ä¹ã®äžè§é¢æ° sin 2 Ξ = 1 â cos 2 Ξ 2 , cos 2 Ξ = 1 + cos 2 Ξ 2 {\\displaystyle \\sin ^{2}\\theta ={\\frac {1-\\cos 2\\theta }{2}},\\cos ^{2}\\theta ={\\frac {1+\\cos 2\\theta }{2}}} ã¯ã 1 ± cos 2 Ξ 2 {\\displaystyle {\\frac {1\\pm \\cos 2\\theta }{2}}} ãšãã圢ãèŠããŠã sin {\\displaystyle \\sin } ã¯ç¬Šå·ã â {\\displaystyle -} ãšèããŸãã",
"title": "åè§ã®å
¬åŒ"
},
{
"paragraph_id": 96,
"tag": "p",
"text": "äžè§é¢æ°ã®å",
"title": "äžè§é¢æ°ã®åæ"
},
{
"paragraph_id": 97,
"tag": "p",
"text": "ã«ãããŠã a , b â 0 {\\displaystyle a,b\\neq 0} ã®ãšã",
"title": "äžè§é¢æ°ã®åæ"
},
{
"paragraph_id": 98,
"tag": "p",
"text": "{ a a 2 + b 2 } 2 + { b a 2 + b 2 } 2 = 1 {\\displaystyle \\left\\{{\\dfrac {a}{\\sqrt {a^{2}+b^{2}}}}\\right\\}^{2}+\\left\\{{\\dfrac {b}{\\sqrt {a^{2}+b^{2}}}}\\right\\}^{2}=1} ã§ããã®ã§ãç¹ ( a a 2 + b 2 , b a 2 + b 2 ) {\\displaystyle \\left({\\dfrac {a}{\\sqrt {a^{2}+b^{2}}}},{\\dfrac {b}{\\sqrt {a^{2}+b^{2}}}}\\right)} ã¯åäœååšäžã®ç¹ãªã®ã§ã",
"title": "äžè§é¢æ°ã®åæ"
},
{
"paragraph_id": 99,
"tag": "p",
"text": "ãšãªããããªÎ±ããšãããšãã§ãããã®Î±ãçšããŠæ¬¡ã®ãããªå€åœ¢ãã§ããã",
"title": "äžè§é¢æ°ã®åæ"
},
{
"paragraph_id": 100,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "äžè§é¢æ°ã®åæ"
},
{
"paragraph_id": 101,
"tag": "p",
"text": "r , α {\\displaystyle r,\\alpha } 㯠r > 0 , â Ï â€ Î± < Ï {\\displaystyle r>0,-\\pi \\leq \\alpha <\\pi } ãæºãããšããã",
"title": "äžè§é¢æ°ã®åæ"
},
{
"paragraph_id": 102,
"tag": "p",
"text": "解ç",
"title": "äžè§é¢æ°ã®åæ"
},
{
"paragraph_id": 103,
"tag": "p",
"text": "sin Ξ â 3 cos Ξ = 2 ( 1 2 sin Ξ â 3 2 cos Ξ ) = 2 ( sin Ξ cos Ï 3 â cos Ξ sin Ï 3 ) = 2 sin ( Ξ â Ï 3 ) {\\displaystyle {\\begin{aligned}\\sin \\theta -{\\sqrt {3}}\\cos \\theta &=2\\left({\\frac {1}{2}}\\sin \\theta -{\\frac {\\sqrt {3}}{2}}\\cos \\theta \\right)\\\\&=2\\left(\\sin \\theta \\cos {\\frac {\\pi }{3}}-\\cos \\theta \\sin {\\frac {\\pi }{3}}\\right)\\\\&=2\\sin \\left(\\theta -{\\frac {\\pi }{3}}\\right)\\\\\\end{aligned}}}",
"title": "äžè§é¢æ°ã®åæ"
},
{
"paragraph_id": 104,
"tag": "p",
"text": "äžè§é¢æ°ã®å æ³å®çãçšãããšãäžè§é¢æ°ã®åâç©ã®å
¬åŒãããã³ç©âåã®å
¬åŒãåŸãããããããã",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 105,
"tag": "p",
"text": "ãšãªãã",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 106,
"tag": "p",
"text": "å æ³å®ç",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 107,
"tag": "p",
"text": "ããã (1) + (2) ãã",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 108,
"tag": "p",
"text": "(1) - (2) ãã",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 109,
"tag": "p",
"text": "(3) + (4) ãã",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 110,
"tag": "p",
"text": "(3) - (4) ãã",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 111,
"tag": "p",
"text": "ãåŸãããã",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 112,
"tag": "p",
"text": "A = α + β , B = α â β {\\displaystyle A=\\alpha +\\beta ,\\,B=\\alpha -\\beta } ãšãããšã α = A + B 2 , β = A â B 2 {\\displaystyle \\alpha ={\\frac {A+B}{2}},\\,\\beta ={\\frac {A-B}{2}}} ã§ããããããç©âåã®å
¬åŒã«ä»£å
¥ããã°ããããã",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 113,
"tag": "p",
"text": "ãåŸãããã",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 114,
"tag": "p",
"text": "èŠãæ¹",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 115,
"tag": "p",
"text": "ç©âåã®å
¬åŒã¯ãäž2ã€ã¯ α {\\displaystyle \\alpha } 㚠β {\\displaystyle \\beta } ãå
¥ãæ¿ããã°åãåŒãªã®ã§ãèŠããã®ã¯3åŒã§ããã sin sin {\\displaystyle \\sin \\sin } ã®å
¬åŒã¯ cos cos {\\displaystyle \\cos \\cos } ã®å
¬åŒã®ç¬Šå·ã2〠â {\\displaystyle -} ã«ãããã®ã«ãªã£ãŠããã",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 116,
"tag": "p",
"text": "åâç©ã®å
¬åŒã¯ã a a a â a a a {\\displaystyle {\\rm {{aaa}-{\\rm {aaa}}}}} ã®åŒã¯ a a a + a a a {\\displaystyle {\\rm {{aaa}+{\\rm {aaa}}}}} ã®å
¬åŒã® cos {\\displaystyle \\cos } ãš sin {\\displaystyle \\sin } ãéã«ãã圢ã«ãªã£ãŠããã",
"title": "åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ"
},
{
"paragraph_id": 117,
"tag": "p",
"text": "",
"title": "äžè§é¢æ°ã®åºæ¬å
¬åŒ"
},
{
"paragraph_id": 118,
"tag": "p",
"text": "",
"title": "äžè§é¢æ°ã®åºæ¬å
¬åŒ"
},
{
"paragraph_id": 119,
"tag": "p",
"text": "(1)äžã®åºŠæ°æ³ã§è¡šãããå€ã匧床æ³ãŠè¡šã",
"title": "æŒç¿åé¡"
},
{
"paragraph_id": 120,
"tag": "p",
"text": "1) 150 {\\displaystyle 150} 2) 720 {\\displaystyle 720}",
"title": "æŒç¿åé¡"
},
{
"paragraph_id": 121,
"tag": "p",
"text": "(2) sin Ï / 2 {\\displaystyle \\sin \\pi /2} ã®å€ãæ±ãã",
"title": "æŒç¿åé¡"
}
] | ããã§ã¯äžè§é¢æ°ã®å®çŸ©ãããããšãäžè§é¢æ°ã®åºæ¬çãªæ§è³ªãå æ³å®çãäžè§é¢æ°ã®å¿çšã«ã€ããŠåŠã¶ãäžè§é¢æ°ã¯æ³¢ããã¯ãã«ã®å
ç©ãããŒãªãšå€æãªã©ããŸããŸãªåéã§å¿çšãããŠããã | {{Pathnav|é«çåŠæ ¡ã®åŠç¿|é«çåŠæ ¡æ°åŠ|é«çåŠæ ¡æ°åŠII|frame=1}}
ããã§ã¯äžè§é¢æ°ã®å®çŸ©ãããããšãäžè§é¢æ°ã®åºæ¬çãªæ§è³ªãå æ³å®çãäžè§é¢æ°ã®å¿çšã«ã€ããŠåŠã¶ãäžè§é¢æ°ã¯æ³¢ããã¯ãã«ã®å
ç©ãããŒãªãšå€æãªã©ããŸããŸãªåéã§å¿çšãããŠããã
== äžè¬è§ ==
[[File:General angle of trigonometric functions japanese.svg|thumb|300px|]]
[[File:Negative general angle.svg|thumb|300px]]
å³å³ã®ããã«ãå®ç¹Oãäžå¿ãšããŠå転ããåçŽç· OP ãèããããã®ãšãã®å転ããåçŽç· OP ã®ããšã'''ååŸ'''ãšããã
åçŽç· OX ãè§åºŠã®åºæºãšããããã®åºæºãšãªãåçŽç· OX ã®ããšã'''å§ç·'''ãšããã
ååŸãæèšåãã«å転ããå Žåãå転ããè§åºŠã¯è² ã§ãããšããååŸãåæèšåããããå Žåãå転ããè§åºŠã¯æ£ã§ãããšããã
è² ã®è§åºŠã360°以äžå転ããè§åºŠãèãã«å
¥ããè§ã®ããšã'''äžè¬è§'''ãšããã
{{-}}
== åŒ§åºŠæ³ ==
==== ã©ãžã¢ã³ ====
ããŸãŸã§ã¯è§åºŠã®åäœãšããŠäžåšã 360° ãšãã床æ°æ³ã䜿ã£ãŠããããšã ãããããã§ã匧床æ³ã«ããè§åºŠã®è¡šãæ¹ãåŠã¶ã
[[File:1radian japanese.svg|thumb|300px]]
ååŸ1 ã®æ圢ã«ãããŠåŒ§ã®é·ãã 1 ã®ãšãã®äžå¿è§ã 1 radãåæ§ã«åŒ§ã®é·ããθã®ãšãã®äžå¿è§ãθ radãšå®çŸ©ããããã®å®çŸ©ãã 180° =π radã360° = 2π rad ãããã«
:<math>\begin{align}1 ^{\circ} &=\frac{\pi}{180}\, \mathrm{rad} \\
\\
1\, \mathrm{rad} &= \frac {180}{\pi} ^{\circ} \approx 57.3^{\circ}\end{align}</math>
ãšãªãããŸã匧床æ³ã®åäœïŒradïŒã¯ãã°ãã°çç¥ãããã
匧床æ³ãçšãããšãäžè§é¢æ°ã®åŸ®ç©åãèããéã«äŸ¿å©ã§ãããïŒãã®ããšã¯æ°åŠIIIã§åŠã¶ïŒ
==== æ圢ã®åŒ§ã®é·ããšé¢ç© ====
æ圢ã®ååŸã''r'' ã匧床æ³ã§å®çŸ©ãããè§åºŠãθãšãããšãã匧ã®é·ã''l'' ãšé¢ç©''S'' ã¯
:<math>\begin{align}l&=r\theta, \\
\\
S&=\frac{1}{2}r^{2}\theta=\frac{1}{2}rl\end{align}</math>
ãšè¡šããã
== äžè§é¢æ° ==
==== sin ãš cos ã®ã°ã©ã ====
[[File:Sin and cos general angle introduction.svg|thumb|300px|]]
äžè¬è§ã <math>\theta</math> ã®åçŽç·ãšåäœåã亀ããåã <math>\mathrm P</math> ãšããããã®ãšãã® <math>\mathrm P</math> ã®åº§æšã<math>(\cos\theta,\sin\theta)</math> ãšããããšã§ãé¢æ° <math>\sin,\cos</math> ãå®ããããŸãã<math>\tan\theta = \frac{\sin\theta}{\cos\theta}</math> ãšããããšã§é¢æ° <math>\tan\theta</math> ãå®ããã<math>\tan\theta</math> ã¯äžè¬è§ã <math>\theta</math> ã®ååŸã®åŸãã«çããã
* <math>\sin</math> ã¯ãµã€ã³(sine) ãšçºé³ãããæ£åŒŠãšãåŒã°ããã
* <math>\cos</math> ã³ãµã€ã³(cosine) ãšçºé³ãããäœåŒŠãšãåŒã°ããã
* <math>\tan</math> ã¯ã¿ã³ãžã§ã³ã(tangent) ãšçºé³ãããæ£æ¥ãšãåŒã°ããã
ãŸããäžè§é¢æ°ã®çŽ¯ä¹ã¯ <math>(\sin\theta)^n = \sin^n\theta</math> ãšè¡šèšãããã[[ãã¡ã€ã«:Circle cos sin.gif|ãµã ãã€ã«|äžå€®|300x300ãã¯ã»ã«]]
[[File:Y=sin(theta).svg|thumb|500px|left]]
[[File:Y=cos(theta).svg|thumb|500px|left]]
{{-}}
cos Ξ ã®ã°ã©ã㯠sin Ξ ã®ã°ã©ãã Ξ軞æ¹åã« <math> - \frac{ \pi }{2} </math>ã ãå¹³è¡ç§»åãããã®ã§ããã
<math>y = \sin\theta</math> ã <math>y = \cos\theta</math> ã®åœ¢ãããæ²ç·ã®ããšã '''æ£åŒŠæ²ç·''' ïŒãããããããããïŒãšããã
é¢æ° <math>\sin,\cos</math> ã®å€åã¯ã©ã¡ããã<math>[-1,1]</math> ã§ããã
{{-}}
==== tan ã®ã°ã©ã ====
[[File:Tangent function introduction.svg|thumb|300px|]]
å³å³ã®ããã« ãè§ Îž ã®ååŸãšåäœåãšã®äº€ç¹ãPãšããŠã
çŽç·OPãš çŽç·xïŒ1 ãšã®äº€ç¹ã T ãšãããšã
Tã®åº§æšã¯
: T (1, tan Ξ)
ã«ãªãã
ãã®ããšãå©çšããŠã yïŒtan Ξ ã®ã°ã©ããããããšãã§ããã
{{-}}
y=tan Ξ ã®ã°ã©ãã¯ãäžå³ã®ããã«ãªãã<br>
[[File:Y=tan(x).svg|500px|]]
y=tan Ξ ã®ã°ã©ãã§ã¯ãΞã®å€ã <math> \frac{ \pi }{2} </math> ã«è¿ã¥ããŠãããšã
çŽç· <math> \theta = \frac{ \pi }{2} </math> ã«éããªãè¿ã¥ããŠããã
ãã®ããã«ãæ²ç·ãããçŽç·ã«éãç¡ãè¿ã¥ããŠãããšããè¿ã¥ãããçŽç·ã®ã»ãã '''挞è¿ç·''' ïŒããããããïŒãšããã
åæ§ã«èãã次ã®çŽç·ã yïŒtanΞ ã®æŒžè¿ç·ã§ããã
:<math> \cdots , \quad \theta = - \frac{ 3}{2} \pi , \quad \theta = - \frac{ 1}{2} \pi , \quad \theta = \frac{ 1}{2} \pi , \quad \frac{ 3}{2} \pi , \cdots </math>
㯠yïŒtanΞ ã®æŒžè¿ç·ã§ããã
äžè¬ã«ã
:çŽç· <math> \quad \theta = \frac{ \pi }{2} + n \pi </math> ããïŒnã¯æŽæ°ïŒ
ã¯y=tanΞã®ã°ã©ãã®æŒžè¿ç·ã§ããã<ref>é«æ ¡ã»å€§åŠå
¥è©Šã§ã¯äœ¿ãããªããã<math>\sec\theta=\frac{1}{\cos\theta},\csc\theta=\frac{1}{\sin\theta},\cot\theta=\frac{1}{\tan\theta}(=\frac{\cos\theta}{\sin\theta})</math> ãšããŠå®çŸ©ãããäžè§é¢æ°ã䜿ããšãããããããããã®é¢æ°ã¯ãããããã»ã«ã³ããã³ã»ã«ã³ããã³ã¿ã³ãžã§ã³ããšåŒã°ããã</ref>
== äžè§é¢æ°ã®æ§è³ª ==
äžè¬è§ã <math>\theta</math> ã®ååŸã¯äžå転ããŠãçããã®ã§ãäžè¬è§ã <math>\theta+2\pi</math> ã®ååŸãšçãããããããäžè§é¢æ°ã®åšææ§
<math>\begin{align}
\sin(\theta+2\pi n)&=\sin \theta \\
\cos(\theta+2\pi n)&=\cos \theta \\
\tan(\theta+2\pi n)&=\tan \theta
\end{align}</math>
ãåŸãã
ç¹ <math>(\cos\theta,\sin\theta)</math> ã <math>\pi</math> å転ããç¹ <math>(\cos(\theta+\pi),\sin(\theta+\pi))</math> ã¯åç¹ãäžå¿ã«ç¹å¯Ÿç§°ç§»åããç¹ã<math>(-\cos\theta,-\sin\theta)</math> ã§ããããšãã
<math>\begin{align}
\sin(\theta + \pi) &= - \sin \theta \\
\cos(\theta + \pi) &= - \cos \theta \\
\tan(\theta + \pi) &= \tan \theta
\end{align}</math>
ãåŸãã
ç¹ <math>(\cos\theta,\sin\theta)</math> ã <math>x</math> 軞ã§ç·å¯Ÿç§°ç§»å移åããç¹ã <math>(\cos (-\theta),\sin(-\theta)) = (\cos\theta,-\sin\theta)</math> ã§ããããšãã
<math>\begin{align}\sin(-\theta) &= -\sin\theta \\
\cos(-\theta) &= \cos\theta \\
\tan(- \theta) &= -\tan\theta\end{align}</math>
ãåŸãã
* åé¡äŸ
** åé¡
* ::<math>\begin{align}
& \sin(\theta + \frac{\pi}{2}) \\
& \cos(\theta + \frac{\pi}{2}) \\
& \sin(\frac{\pi}{2} -\theta) \\
& \cos(\frac{\pi}{2}- \theta )
\end{align}</math>
*: ãèšç®ããã
** 解ç
*: è§θã«å¯Ÿå¿ããç¹ã P(x, y) ãšããããã®ãšããè§ θ + 90°ã«å¯Ÿå¿ããç¹ã P'(x', y') ãšãããšããã®ç¹ã®åº§æšã¯ãP'(-y, x) ã«å¯Ÿå¿ããããã®ããšãããP'ã«ã€ã㊠sin, cos ãèšç®ãããšã
*:: <math>\begin{align}
x' &= -y \\
&= \cos (\theta + \frac{\pi}{2} )\\
&= -\sin\theta \\
y' &= x \\
&= \sin (\theta + \frac{\pi}{2} ) \\
&= \cos\theta
\end{align}</math>
*: ãåŸãããã
*: åæ§ã«ããŠã90°- θ ã«å¯Ÿå¿ããç¹ã P' '(x' ', y' ') ãšãããšã
*:: <math>\begin{align}
x'' &= y \\
y'' &= x
\end{align}</math>
*: ãšãªãããã£ãŠã
*:: <math>\begin{align}
\sin (\frac{\pi}{2} - \theta) &= \cos\theta \\
\cos (\frac{\pi}{2} - \theta) &= \sin\theta
\end{align}</math>
*: ãåŸãããã
åäœååšäžã®ç¹ <math>(\cos\theta,\sin\theta)</math> ããåç¹ãŸã§ã®è·é¢ã¯ 1 ãªã®ã§ã <math>\sin^2\theta+\cos^2\theta = 1</math> ãæãç«ã€ã
ãŸãããã®åŒã«ã <math>\tan\theta = \frac{\sin\theta}{\cos\theta}</math> ã€ãŸãã <math>\sin\theta = \tan\theta \cos\theta</math> ã代å
¥ããã°ã<math>1+\tan^2\theta = \frac{1}{\cos^2\theta}</math> ãæãç«ã€ããšããããã
== åšæé¢æ° ==
é¢æ° <math>f(x)</math> ã«å¯ŸããŠã0 ã§ãªãå®æ° <math>p</math> ãååšããŠã<math>f(x+p) =f(x)</math> ãšãªããšãé¢æ° <math>f(x)</math> ã¯åšæé¢æ°ãšãããå®æ° <math>p</math> ãäžã®æ§è³ªãæºãããšãã<math>-p,2p</math> ãªã©ãå®æ° <math>p</math> ã0ãé€ãæŽæ°åããæ°ãäžã®æ§è³ªãæºãããããã§ãåšæé¢æ°ãç¹åŸŽã¥ããéãšããŠãäžã®æ§è³ªãæºããå®æ° <math>p</math> ã®å
ãæ£ã§ãã€æå°ã®ãã®ãéžã³ãããã'''åšæ'''ãšåŒã¶ã
<math>\sin x, \cos x</math> ã¯åšæã <math>2\pi</math> ãšããåšæé¢æ°ã§ããã<math>\tan x</math> ã¯åšæã <math>\pi</math> ãšããåšæé¢æ°ã§ããã
'''æŒç¿åé¡'''
<math>k</math> ã0ã§ãªãå®æ°ãšãããé¢æ° <math>\sin kx</math> ã®åšæãèšã
'''解ç'''
<math>\sin k\left(x+\frac{2\pi}{k}\right) = \sin kx</math> ãªã®ã§çã㯠<math>\frac{2\pi}{k}</math> ãããã¯æ£ã§ãããåšæã®æå°æ§ã®æ¡ä»¶ãæºãããŠããã
== å¶é¢æ°ãšå¥é¢æ° ==
é¢æ° <math>f(x)</math> ã <math>f(-x)=f(x)</math> ãæºãããšããé¢æ° <math>f(x)</math> ã¯å¶é¢æ°ãšãããå¶é¢æ°ã¯ <math>y</math> 軞ã«é¢ããŠå¯Ÿç§°ãªã°ã©ãã«ãªãã
ãŸããé¢æ° <math>f(x)</math> ã <math>f(-x)=-f(x)</math> ãæºãããšããé¢æ° <math>f(x)</math> ã¯å¥é¢æ°ãšãããå¶é¢æ°ã¯åç¹ã«é¢ããŠå¯Ÿè±¡ãªã°ã©ãã«ãªãã
é¢æ° <math>\cos\theta,x^{2n}</math> (<math>n</math> ã¯æŽæ°)ã¯å¶é¢æ°ãšãªãã
é¢æ° <math>\sin x , x^{2n+1}</math> (<math>n</math> ã¯æŽæ°)ã¯å¥é¢æ°ãšãªãã
;æŒç¿åé¡
<math>\tan\theta</math> ã¯å¶é¢æ°ããããšãå¥é¢æ°ã調ã¹ãã
'''解ç'''
:<math> \tan( - \theta ) = \frac{\sin(- \theta)} {\cos(-\theta)} = \frac{- \sin(\theta)} {\cos(\theta)} = - \frac{\sin(\theta)} {\cos(\theta)} = - \tan \theta</math>
ãªã®ã§ã <math>\tan\theta</math> ã¯å¥é¢æ°ã§ããã<ref>äžè¬ã«ãé¢æ° <math>f(x) </math> ã«å¯Ÿãã<math>f(x) </math> ãå¶é¢æ°ãå¥é¢æ°ã調ã¹ãã«ã¯ <math>f(-x)</math> ã <math>f(x)</math> ãŸã㯠<math>-f(x)</math> ã®ã©ã¡ãã«çããã調ã¹ãã°ããããŸããã©ã¡ããšãçãããªãå Žåãé¢æ° <math>f(x) </math> ã¯å¶é¢æ°ã§ãå¥é¢æ°ã§ããªãã</ref>
== ãããããªäžè§é¢æ° ==
[[File:Y=sin(theta-pi div 3).svg|thumb|550px|]]
é¢æ° <math> y=\sin \left( \theta - \frac{\pi}{3} \right)</math> ã®ã°ã©ãã¯ã<math> y=\sin \theta </math>ã®ã°ã©ãã Ξ軞æ¹åã« <math> \frac{\pi}{3} </math> ã ãå¹³è¡ç§»åããããã®ã«ãªããåšæ㯠<math> 2 \pi </math> ã§ãããïŒå¹³è¡ç§»åããŠããåšæã¯å€ããããsinΞãšåããåšæ㯠<math> 2 \pi </math> ã®ãŸãŸã§ãããïŒ
{{-}}
[[File:Y=2sin(theta).svg|thumb|550px]]
é¢æ° yïŒ2sin Ξ ã®ã°ã©ãã®åœ¢ã¯ yïŒsin Ξ ãy軞æ¹åã«2åã«æ¡å€§ãããã®ã§ãåšæ㯠yïŒsin Ξ ãšåãã 2Ï ã§ããã
ãŒ1 ⊠sin Ξ ⊠1ãããªã®ã§ã
å€åã¯ãããŒ2 ⊠2sin Ξ ⊠2ããã§ããã
{{-}}
{{-}}
[[File:Y=sin(2 theta) and y=sin(theta).svg|thumb|750px]]
{{-}}
é¢æ° yïŒsin2Ξ ã®ã°ã©ãã¯y軞ãåºæºã«Îžè»žæ¹åã« <math> \frac{1}{2}</math> åã«çž®å°ãããã®ã«ãªã£ãŠããã
ãããã£ãŠãåšæã <math> \frac{1}{2}</math> åã«ãªã£ãŠãããy=sinΞ ã®åšæ㯠<math> 2 \pi </math> ã ãããy=sin2Ξ ã®åšæ㯠<math> \pi </math> ã§ããã
== å æ³å®ç ==
äžè§é¢æ°ã®å æ³å®ç
:<math>\begin{align}
\sin (\alpha \pm \beta) &= \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\
\cos (\alpha \pm \beta) &= \cos \alpha \cos \beta \mp \sin \alpha \sin \beta
\end{align}</math>
ãæãç«ã€ã
'''蚌æ'''
ä»»æã®å®æ° <math>\alpha,\beta</math> ã«å¯Ÿããåäœååšäžã®ç¹ <math>\mathrm{A}(\cos\alpha,\sin\alpha),\mathrm{B}(\cos\beta,\sin\beta)</math> ããšãããã®ãšãã ç·å <math>\mathrm{AB}</math> ã®é·ãã®2ä¹ <math>\mathrm{AB}^2</math> ã¯äœåŒŠå®çã䜿ãããšã«ãã
<math>\mathrm{AB}^2 = 2-2\cos(\alpha-\beta)</math>
ã§ããã次ã«äžå¹³æ¹ã®å®çã䜿ã£ãŠ
<math>\mathrm{AB}^2 = (\cos\alpha -\cos\alpha)^2 + (\sin\alpha-\sin\beta)^2 = 2 - 2(\cos\alpha\cos\beta + \sin\alpha\sin\beta)</math>
ãããæŽçããŠ
<math>\cos(\alpha - \beta)= \cos\alpha\cos\beta + \sin\alpha\sin\beta</math>
ãåŸãã
<math>\cos(\alpha+\beta) = \cos(\alpha-(-\beta)) = \cos\alpha\cos(-\beta) + \sin\alpha\sin(-\beta) =
\cos\alpha\cos\beta - \sin\alpha\sin\beta</math>
ã§ããã
以äžããŸãšããŠ
<math>\cos (\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta</math>
ãåŸãã
ããã§ã
<math>\sin(\alpha \pm \beta) = -\cos(\alpha +\frac{\pi}{2} \pm \beta) =
-\{\cos(\alpha + \frac{\pi}{2})\cos(\beta) \mp \sin(\alpha+\frac{\pi}{2})\sin\beta \} =
\sin\alpha\cos\beta \pm \cos\alpha\sin\beta</math><ref>ãå²ãã(sin)ã³ã¹ã¢ã¹(cos)ã³ã¹ã¢ã¹(cos)å²ãã(sin)ããã³ã¹ã¢ã¹(cos)ã³ã¹ã¢ã¹(cos)å²ãã(sin)å²ãã(sin)ããšããèŠãããããã</ref>
ããã«ã<math>\tan x</math> ã«ã€ããŠã
<math display="inline">\begin{align}
\tan (\alpha\pm\beta) &= \frac {\sin (\alpha\pm\beta) } {\cos (\alpha\pm\beta) } \\
&= \frac { \sin \alpha \cos \beta \pm \cos \alpha \sin \beta } { \cos \alpha \cos \beta \mp \sin \alpha \sin \beta } \\
&= \cfrac { \cfrac { \sin \alpha \cos \beta } { \cos \alpha \cos \beta } \pm \cfrac { \cos \alpha \sin \beta } { \cos \alpha \cos \beta } } { \cfrac { \cos \alpha \cos \beta } { \cos \alpha \cos \beta } \mp \cfrac { \sin \alpha \sin \beta } { \cos \alpha \cos \beta } } \\
&= \frac { \tan \alpha \pm \tan \beta } { 1 \mp \tan \alpha \tan \beta }
\end{align}</math>
ãæãç«ã€ã
== åè§ã®å
¬åŒ ==
å æ³å®çãçšããŠä»¥äžã蚌æã§ããã
<math>\sin 2\alpha = \sin(\alpha + \alpha) = 2\sin\alpha\cos\alpha</math>
<math>\cos 2\alpha = \cos(\alpha+\alpha)=\cos^2\alpha-\sin^2\alpha = 2\cos^2\alpha-1=1-2\sin^2\alpha</math>
<math>\tan 2\alpha = \frac{2\tan \alpha}{1-\tan^2\alpha}</math>
次ã«ã <math>\cos</math> ã®åè§ã®å
¬åŒãå€åœ¢ãããš
<math>\sin^2\alpha = \frac{1-\cos 2\alpha}{2}</math>
<math>\cos^2\alpha = \frac{1+\cos 2\alpha}{2}</math>
ã§ããã
'''æŒç¿åé¡'''
# <math>\sin 15^\circ,\cos 15^\circ</math> ãæ±ãã
# <math>\tan^2 \alpha = \frac{1-\cos 2\alpha}{1+\cos 2\alpha}</math> ã瀺ã
'''解ç'''
<math>\sin 15^\circ = \sin(45^\circ-30^\circ)=\frac{\sqrt 6 - \sqrt 2}{4}</math>
<math>\cos 15^\circ = \cos(45^\circ-30^\circ)=\frac{\sqrt 6 + \sqrt 2}{4}</math>
<math>\tan ^2\alpha = \frac{\sin^2\alpha}{\cos^2\alpha} = \frac{1-\cos 2\alpha}{1+\cos 2\alpha}</math>
ä»ãŸã§ã®å®çããŸãšãããšã次ã®ããã«ãªãã
{| style="border:2px solid yellow;width:80%" cellspacing=0
|style="background:yellow"|'''äžè§é¢æ°ã®å æ³å®ç'''
|-
|style="padding:5px"|
:<math>\begin{align}
\sin (\alpha \pm \beta) &= \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\
\cos (\alpha \pm \beta) &= \cos \alpha \cos \beta \mp \sin \alpha \sin \beta\\
\tan (\alpha \pm \beta) &= \frac { \tan \alpha \pm \tan \beta } { 1 \mp \tan \alpha \tan \beta } \\
\end{align}</math>
|}
{| style="border:2px solidãgreenyellow;width:80%" cellspacing=0
|style="background:greenyellow"|'''2åè§ã®å
¬åŒ'''
|-
|style="padding:5px"|
:<math>\begin{align}
\sin 2 \alpha &= 2 \sin \alpha \cos \alpha \\
\cos 2 \alpha &= \cos ^2 \alpha - \sin ^2 \alpha = 1 - 2 \sin ^2 \alpha = 2 \cos ^2 \alpha - 1 \\
\tan 2 \alpha &= \frac { 2 \tan \alpha } { 1 - \tan ^2 \alpha }
\end{align}</math>
|}
{| style="border:2px solidãskyblue;width:80%" cellspacing=0
|style="background:skyblue"|äžè§é¢æ°ã®2ä¹
|-
|style="padding:5px"|
:<math>\begin{align}
\sin ^2 \alpha &= \frac {1 - \cos 2\alpha }2 \\
\cos ^2 \alpha &= \frac {1 + \cos 2\alpha }2 \\
\tan ^2 \alpha &= \frac {1 - \cos 2\alpha } {1 + \cos 2\alpha }
\end{align}</math>
|}
'''èŠãæ¹'''
å æ³å®çã¯ãå²ããã³ã¹ã¢ã¹ã³ã¹ã¢ã¹å²ãããããã³ã¹ã¢ã¹ã³ã¹ã¢ã¹å²ããå²ããããšããèªååãããããŸãã
<math>\cos</math> ã®åè§ã®å
¬åŒ <math>\cos 2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta</math> 㯠<math>\pm 1 \mp 2\mathrm{aaa}^2\theta</math> ãšãã圢ãèŠã㊠<math>\sin</math> ã¯ç¬Šå·ã <math>-</math>ã1 ã®ç¬Šå·ã¯ãã®éãšèŠããŸãã
2ä¹ã®äžè§é¢æ° <math>\sin^2\theta = \frac{1-\cos 2\theta}{2},\cos^2\theta = \frac{1+\cos 2\theta}{2}</math> ã¯ã<math>\frac{1\pm \cos 2\theta}{2}</math> ãšãã圢ãèŠããŠã <math>\sin</math> ã¯ç¬Šå·ã<math>-</math> ãšèããŸãã
==äžè§é¢æ°ã®åæ==
äžè§é¢æ°ã®å
:<math>
a \sin \theta + b \cos \theta
</math>
ã«ãããŠã<math>a,b\neq 0</math> ã®ãšã
<math>\left\{\dfrac{a}{\sqrt{a^2+b^2}}\right\}^2 + \left\{\dfrac{b}{\sqrt{a^2+b^2}}\right\}^2 = 1</math> ã§ããã®ã§ãç¹ <math>\left(\dfrac{a}{\sqrt{a^2+b^2}},\dfrac{b}{\sqrt{a^2+b^2}}\right)</math> ã¯åäœååšäžã®ç¹ãªã®ã§ã
:<math>
\begin{cases}
\cos \alpha = \dfrac{a}{\sqrt{a^2+b^2}}\\
\sin \alpha = \dfrac{b}{\sqrt{a^2+b^2}}
\end{cases}
</math>
ãšãªããããªαããšãããšãã§ãããã®αãçšããŠæ¬¡ã®ãããªå€åœ¢ãã§ããã
:<math>\begin{align}
a \sin \theta + b \cos \theta & = \sqrt{a^2+b^2}\left( \frac{a}{\sqrt{a^2+b^2}} \sin \theta + \frac{b}{\sqrt{a^2+b^2}} \cos \theta \right) \\
& = \sqrt{a^2+b^2} \left( \sin \theta \cos \alpha + \cos \theta \sin \alpha \right)\\
& = \sqrt{a^2+b^2} \sin \left( \theta + \alpha \right)\\
\end{align}
</math>
'''æŒç¿åé¡'''
<math>r,\alpha</math> 㯠<math>r>0,-\pi\le \alpha< \pi</math> ãæºãããšããã
# <math>\sin \theta - \sqrt{3} \cos \theta</math> ã <math>r \sin \left( \theta + \alpha \right)</math> ã®åœ¢ã«å€åœ¢ããã
# <math>2\cos\theta-2\sin\theta</math> ã <math>r\cos(\theta+\alpha)</math> ã®åœ¢ã«å€åœ¢ããã
'''解ç'''
# <math>r = \sqrt{1^2 + \left( - \sqrt{3} \right)^2} = 2</math> ãã
<math>\begin{align}
\sin \theta - \sqrt{3} \cos \theta & = 2 \left( \frac{1}{2} \sin \theta - \frac{\sqrt{3}}{2} \cos \theta \right) \\
& = 2 \left( \sin \theta \cos \frac{\pi}{3} - \cos \theta \sin \frac{\pi}{3} \right)\\
& = 2 \sin \left( \theta - \frac{\pi}{3} \right)\\
\end{align}
</math>
# <math>2\cos\theta-2\sin\theta=2\sqrt 2\left(\frac{1}{\sqrt 2}\cos\theta-\frac{1}{\sqrt 2}\sin\theta\right)</math> <ref>ããå€åœ¢ããããšã§ãç¹ <math>\left(\frac{1}{\sqrt2},\frac{1}{\sqrt2}\right)</math> ãåäœååšäžã®ç¹ã«ãªã</ref>ããã§ã<math>r\cos(\theta+\alpha) = r(\cos\theta\cos\alpha-\sin\theta\sin\alpha)</math> ã§ããã <math>\cos\alpha=\frac{1}{\sqrt 2},\sin\alpha = \frac{1}{\sqrt 2}</math> ãšãªã <math>\alpha</math> ãšã㊠<math>\alpha = \frac{\pi}{4}</math> ãããã<ref>ããã§ã <math>\alpha</math> ã¯åé¡æã®å¶çŽãæºããããã«éžã¶ã <math>\alpha</math> ã« <math>2\pi</math> ã®æŽæ°åã足ãã <math>\alpha + 2\pi n</math> ãéžãã§ãäžè§é¢æ°ã®åæã¯ã§ããããå®çšçã«ã <math>\alpha</math> ã¯ç°¡åãªãã®ãéžãã æ¹ãããã ããã</ref>ãããã£ãŠã<math>2\cos\theta-2\sin\theta=2\sqrt 2\cos\left(\theta + \frac{\pi}{4}\right)</math>
== åããç©ãžã®å
¬åŒãšç©ããåãžã®å
¬åŒ ==
äžè§é¢æ°ã®å æ³å®çãçšãããšãäžè§é¢æ°ã®åâç©ã®å
¬åŒãããã³ç©âåã®å
¬åŒãåŸãããããããã
;ç©âåã®å
¬åŒ
:<math>\begin{align}
\sin \alpha \cos \beta &= \frac 1 2 \{\sin (\alpha+\beta) + \sin (\alpha-\beta)\}\\
\cos \alpha \sin \beta &= \frac 1 2 \{\sin (\alpha+\beta) - \sin (\alpha-\beta) \}\\
\cos \alpha \cos \beta &= \frac 1 2 \{\cos (\alpha+\beta) + \cos (\alpha-\beta) \}\\
\sin \alpha \sin \beta &= -\frac 1 2 \{\cos (\alpha+\beta) - \cos (\alpha-\beta) \}
\end{align}</math>
;åâç©ã®å
¬åŒ
:<math>\begin{align}
\sin A + \sin B &= 2 \sin \left(\frac {A+B}2 \right) \cos \left(\frac {A-B}2 \right)\\
\sin A - \sin B &= 2 \cos \left(\frac {A+B}2 \right) \sin \left(\frac {A-B}2 \right)\\
\cos A + \cos B &= 2 \cos \left(\frac {A+B}2 \right) \cos \left(\frac {A-B}2 \right)\\
\cos A - \cos B &= -2 \sin \left(\frac {A+B}2 \right) \sin \left(\frac {A-B}2 \right)
\end{align}</math>
ãšãªãã
;å°åº
å æ³å®ç
:{{åŒçªå·|<math>\sin(\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta</math>|1}}
:{{åŒçªå·|<math>\sin(\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin \beta</math>|2}}
:{{åŒçªå·|<math>\cos(\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta </math>|3}}
:{{åŒçªå·|<math>\cos(\alpha -\beta )=\cos \alpha \cos \beta +\sin \alpha \sin \beta </math>|4}}
ããã [[é«çåŠæ ¡æ°åŠII/äžè§é¢æ°#expr1|(1)]] + [[é«çåŠæ ¡æ°åŠII/äžè§é¢æ°#expr2|(2)]] ãã
:<math>\sin \alpha \cos \beta = \frac 1 2 (\sin (\alpha+\beta) + \sin (\alpha-\beta))</math>
[[é«çåŠæ ¡æ°åŠII/äžè§é¢æ°#expr1|(1)]] - [[é«çåŠæ ¡æ°åŠII/äžè§é¢æ°#expr2|(2)]] ãã
:<math>\cos \alpha \sin \beta = \frac 1 2 (\sin (\alpha+\beta) - \sin (\alpha-\beta) </math>
[[é«çåŠæ ¡æ°åŠII/äžè§é¢æ°#expr3|(3)]] + [[é«çåŠæ ¡æ°åŠII/äžè§é¢æ°#expr4|(4)]] ãã
:<math>\cos \alpha \cos \beta = \frac 1 2 (\cos (\alpha+\beta) + \cos (\alpha-\beta) )</math>
[[é«çåŠæ ¡æ°åŠII/äžè§é¢æ°#expr3|(3)]] - [[é«çåŠæ ¡æ°åŠII/äžè§é¢æ°#expr4|(4)]] ãã
:<math>\sin \alpha \sin \beta = -\frac 1 2 (\cos (\alpha+\beta) - \cos (\alpha-\beta) )</math>
ãåŸãããã
<math>A = \alpha + \beta,\, B = \alpha-\beta</math> ãšãããšã <math>\alpha = \frac{A+B}{2},\, \beta = \frac{A-B}{2}</math> ã§ããããããç©âåã®å
¬åŒã«ä»£å
¥ããã°ããããã
:<math>\begin{align}
\sin A + \sin B &= 2 \sin \left(\frac {A+B}2 \right) \cos \left(\frac {A-B}2 \right)\\
\sin A - \sin B &= 2 \cos \left(\frac {A+B}2 \right) \sin \left(\frac {A-B}2 \right)\\
\cos A + \cos B &= 2 \cos \left(\frac {A+B}2 \right) \cos \left(\frac {A-B}2 \right)\\
\cos A - \cos B &= -2 \sin \left(\frac {A+B}2 \right) \sin \left(\frac {A-B}2 \right)
\end{align}</math>
ãåŸãããã
'''èŠãæ¹'''
ç©âåã®å
¬åŒã¯ãäž2ã€ã¯ <math>\alpha</math> ãš <math>\beta</math> ãå
¥ãæ¿ããã°åãåŒãªã®ã§ãèŠããã®ã¯3åŒã§ããã<math>\sin\sin</math> ã®å
¬åŒã¯ <math>\cos\cos</math> ã®å
¬åŒã®ç¬Šå·ã2〠<math>-</math> ã«ãããã®ã«ãªã£ãŠããã
åâç©ã®å
¬åŒã¯ã<math>\rm{aaa}-\rm{aaa}</math> ã®åŒã¯ <math>\rm{aaa}+\rm{aaa}</math> ã®å
¬åŒã® <math>\cos</math> ãš <math>\sin</math> ãéã«ãã圢ã«ãªã£ãŠããã
== äžè§é¢æ°ã®åºæ¬å
¬åŒ ==
* åšææ§ïŒ''n''ãã¯æŽæ°ïŒ
:<math>\begin{align}
\sin(\theta+2\pi n)&=\sin \theta \\
\cos(\theta+2\pi n)&=\cos \theta \\
\tan(\theta+2\pi n)&=\tan \theta
\end{align}</math>
* å¶é¢æ°ãå¥é¢æ°
:<math>\begin{align}
\sin(-\theta)&=-\sin \theta \\
\cos(-\theta)&=\cos \theta \\
\tan(-\theta)&=-\tan \theta
\end{align}</math>
* <math>\theta+\pi</math>
:<math>\begin{align}
\sin(\theta+\pi)&=-\sin \theta \\
\cos(\theta+\pi)&=-\cos \theta \\
\tan(\theta+\pi)&=\tan \theta
\end{align}</math>
* <math>\pi-\theta</math>
:<math>\begin{align}
\sin(\pi-\theta)&=\sin \theta \\
\cos(\pi-\theta)&=-\cos \theta \\
\tan(\pi-\theta)&=-\tan \theta
\end{align}</math>
* <math>\theta+\frac{1}{2}\pi</math>
:<math>\begin{align}
\sin \left(\theta+\frac{1}{2}\pi \right)&=\cos \theta \\
\cos \left(\theta+\frac{1}{2}\pi \right)&=-\sin \theta \\
\tan \left(\theta+\frac{1}{2}\pi \right)&=-\frac{1}{\tan \theta}
\end{align}</math>
* <math>\frac{\pi}{2}-\theta</math>
:<math>\begin{align}
\sin \left(\frac{\pi}{2}-\theta \right)&=\cos \theta \\
\cos \left(\frac{\pi}{2}-\theta \right)&=\sin \theta \\
\tan \left(\frac{\pi}{2}-\theta \right)&=\frac{1}{\tan \theta}
\end{align}</math>
* åé¡äŸ
** åé¡
*:(i) <math>\sin \frac{10}{3} \pi</math>
*:(ii) <math>\cos \left(- \frac{11}{4} \pi \right)</math>
*:(iii) <math>\tan \frac{31}{6} \pi</math>
*:ã®å€ãæ±ããã
** 解ç
*:(i)
*::<math>
\begin{align}
\sin \frac{10}{3} \pi & = \sin \left(\frac{4}{3}\pi + 2 \pi \right) = \sin \frac{4}{3} \pi \\
& = \sin \left(\frac{\pi}{3} + \pi \right) = - \sin \frac{\pi}{3} \\
& = - \frac{\sqrt{3}}{2}
\end{align}
</math>
*:(ii)
*::<math>
\begin{align}
\cos \left(- \frac{11}{4} \pi \right) & = \cos \frac{11}{4} \pi = \cos \left(\frac{3}{4}\pi + 2 \pi \right)\\
& = \cos \frac{3}{4} \pi = \cos \left(\pi - \frac{\pi}{4}\right)\\
& = - \cos \frac{\pi}{4} = - \frac{1}{\sqrt{2}}
\end{align}
</math>
*:(iii)
*::<math>
\begin{align}
\tan \frac{31}{6} \pi & = \tan \left(\frac{7}{6}\pi + 2 \pi \times 2 \right) = \tan \frac{7}{6} \pi \\
& = \tan \left(\frac{\pi}{6} + \pi \right) = \tan \frac{\pi}{6} \\
& = \frac{1}{\sqrt{3}}
\end{align}
</math>
{{ã³ã©ã |楜åšã®é³ãšäžè§é¢æ°|é³ãæ³¢ã®äžçš®ãªã®ã§ãäžè§é¢æ°ã§è¡šçŸã§ããã
ãªã·ãã¹ã³ãŒã㧠é³å ã®é³ã枬å®ãããšãæ£åŒŠæ³¢ã«è¿ã波圢ã芳枬ãããã
ããããå®éã®æ¥œåšã®é³ã¯ãæ£åŒŠæ³¢ãšã¯éãããªã·ãã¹ã³ãŒãã§ã®ã¿ãŒããã€ãªãªã³ãªã©ã®æ¥œåšã®é³ã枬å®ãããšãæ£åŒŠæ³¢ã§ãªã波圢ãç¹°ãè¿ãããŠããã
ãããå®éã®æ¥œåšã®é³ã®æ³¢åœ¢ã¯ãåšæã®ç°ãªãè€æ°åã®æ£åŒŠæ³¢ãéãåããã波圢ã«ãªã£ãŠããã
:倧åŠãªã©ã§ç¿ãããŒãªãšè§£æã§ããã®ãããªæ£åŒŠæ³¢ã§ãªã波圢ã®è§£æã«ã€ããŠè©³ããç¿ããäžè§é¢æ°ä»¥å€ã®åšæçãªé¢æ°ããäžè§é¢æ°ãä»ããŠè¡šçŸããææ³ãç¥ãããŠããã
}}
{{ã³ã©ã |æ°åŠè
ã¬ãªã³ãã«ãã»ãªã€ã©ãŒ|
[[File:Leonhard_Euler_2.jpg|thumb| ã¬ãªã³ãã«ãã»ãªã€ã©ãŒ(Leonhard Euler 1707幎4æ15æ¥ - 1783幎9æ18æ¥)]]
ããã§ã¯ãææ°é¢æ°ãäžè§é¢æ°ã®å®çŸ©åãå®æ°ãšããŠãããããããã®é¢æ°ã®å®çŸ©åãè€çŽ æ°ãŸã§æ¡åŒµããããšãã§ããã(èå³ã®ããæ欲çãªèªè
ã¯è€çŽ é¢æ°è«ã®æžç±ãèªãã§ã¿ããšãã)
è€çŽ æ°ã«æ¡åŒµããææ°é¢æ°ãäžè§é¢æ°ã§ã¯ <math>e^{i\theta} =\cos\theta+i\sin\theta</math>
ãšããé¢ä¿åŒãæãç«ã€ããã ãã<math>e</math> ã¯ãã€ãã¢æ°ã§ <math>e \approx 2.7</math> ã§ãããããã§ã <math>\theta</math> ã« <math>\pi</math> ã代å
¥ãããš
<math>e^{i\pi}+1=0</math>ãšãªãããã®çåŒã¯ãäžçäžçŸããçåŒããšãåŒã°ããå°èª¬ã«ããªã£ãŠããã®ã§ç¥ã£ãŠãã人ãããã ããã
}}
{{ã³ã©ã |ååšçã®å€|
ååšçã¯<Math>\pi \fallingdotseq 3.14</Math>ãåºãçšããããŠããããå®ã¯<Math>\tau \fallingdotseq 6.28</Math>ãçšããæ¹ãè¯ãããšããæèŠãããã
åã®å®çŸ©ããä»»æã®ç¹ããã®è·é¢ïŒååŸïŒãçããç¹ã®éåãã§ããããšãããæ°åŠã§åãè°è«ããéã¯ååŸãåºæ¬ã«ããããšãå€ããçŽåŸã¯ïŒå·¥åŠãªã©ãé€ãã°ïŒååšçã決ãããšããããããåºãŠããªãããã®ãããåã絡ãæ°åŠå
¬åŒã«ã¯å€ãã®å Žåä¿æ°2ãã€ããŠããŸããããã§ã<Math>\tau = 2 \pi</Math>ãçšããã°ãå€ãã®å
¬åŒãç°¡åã«(ãããŠäž»åŒµè
ã«ããã°ãæ¬è³ªçã«ã)æžããã以äžã«äŸãæããã
匧床æ³ã§ã¯ãäžåšã<Math>\tau</Math>ã©ãžã¢ã³ãšãªããåãšæ圢ã®æ¯ã<Math>\tau</Math>ã®ä¿æ°ã«äžèŽããããã®ãããäžåšä»¥äžã®ç¯å²ã«ãããŠ<Math>\tau</Math>ã®ä¿æ°ãä»®åæ°ã«ãªãäºããªãã
æ圢ã«ã€ããŠã匧é·ã®å
¬åŒã¯<Math>\theta r</Math>ãé¢ç©ã®å
¬åŒã¯<Math>\frac{1}{2} \theta r^2</Math>ãåã«ã€ããŠãååšã®å
¬åŒã¯<Math>\tau r</Math>ãé¢ç©ã®å
¬åŒã¯<Math>\frac{1}{2} \tau r^2</Math>ããã®ããšãããããããã«ãååšã»åã®é¢ç©ãæ±ãããæãæ圢ã®å
¬åŒã«<Math>\theta = \tau</Math>ã代å
¥ããã ãã§ãããå
¬åŒã®çµ±äžåãå³ããããªããé¢ç©å
¬åŒã«ä¿æ°<Math>\frac{1}{2}</Math>ãã€ããŠããŸã£ãŠããããããã¯ãä¿æ°ã1ã®äžæ¬¡åŒïŒåŒ§é·ã»ååšã®å
¬åŒïŒã[[é«çåŠæ ¡æ°åŠII/埮åã»ç©åã®èã#äžå®ç©å|ç©å]]ããŠããããšè§£éããã°èªç¶ãªããšã§ããã
äžè§é¢æ°ã«ã€ããŠãsinãšcosã®åšæã¯<Math>\tau</Math>ãtanã®åšæã¯<Math>\frac{\tau}{2}</Math>ãšãªããsinãšcosã¯åã1åšãtanã¯åãååšãããšå
ã«æ»ãããšã端çã«ç€ºãããã
ãªã€ã©ãŒã®å
¬åŒ<math>e^{i\theta} =\cos\theta+i\sin\theta</math>
ã«ã€ããŠã<Math>\theta = \tau</Math>ãšãããš<Math>e^{i\tau} = 1</Math>ã§ããããã®åŒã¯åº§æšå¹³é¢äžã§ç¹ïŒ1, 0ïŒããåäœåãäžåšãããšå
ã®ç¹ã«æ»ãããšã瀺ãããŸãã<Math>\theta</Math>ã«äžåšïŒ<Math>\tau</Math>ã©ãžã¢ã³ïŒã代å
¥ããŠããã®ã§sinã0ãcosã1ã§ããããšãçŽæçã«ç解ã§ããã
ä»ã«ãã[[é«çåŠæ ¡æ°åŠB/確çååžãšçµ±èšçãªæšæž¬#æ£èŠååž|æ£èŠååž]]ã®ç¢ºçå¯åºŠé¢æ°ããã£ã©ãã¯å®æ°ãªã©ã<Math>2 \pi</Math>ãç»å ŽããåŒã¯éåžžã«å€ãããããã<Math>\tau</Math>ã«çœ®ãæããããšã«ãã£ãŠãåŒãç°¡æœã«æžãããšãã§ããããã ãã<Math>2 \pi</Math>ã«ããã«ä¿æ°ãããã£ãŠãããã®ã«ã€ããŠã¯ããŸãå€ãããªãã
ãããŸã§ååšç<Math>\tau</Math>ã«ã€ããŠçŽ¹ä»ããŠããããçŸåšã®æ°åŠçã§<Math>\tau</Math>ã䜿ãããããšã¯ããŸãå€ããªããããã¯ãå·¥åŠã«å¿çšããéã«ã¯<Math>\pi</Math>ã®æ¹ãéœåãè¯ãããšãéå»ã«<Math>\pi</Math>ãåºã䜿ãããŠããããä»æŽå€ããã®ã¯å°é£ã§ããããšãçç±ã§ããã䜿ããããšãã¯æåã«ã<Math>2 \pi = \tau</Math>ãšããããšæã£ãŠãã䜿ãã°è¯ãã ããã
}}
== æŒç¿åé¡ ==
(1)äžã®åºŠæ°æ³ã§è¡šãããå€ã匧床æ³ãŠè¡šã
1)<math>150</math>
2)<math>720</math>
(2)<math>\sin \pi/2</math>ã®å€ãæ±ãã
== è泚 ==
<references />
{{Wikiversity|Topic:äžè§é¢æ°|äžè§é¢æ°}}
{{DEFAULTSORT:ãããšããã€ããããããII ãããããããã}}
[[Category:é«çåŠæ ¡æ°åŠII|ãããããããã]]
[[ã«ããŽãª:äžè§é¢æ°]] | 2005-05-06T11:30:25Z | 2024-03-29T02:57:06Z | [
"ãã³ãã¬ãŒã:Pathnav",
"ãã³ãã¬ãŒã:-",
"ãã³ãã¬ãŒã:åŒçªå·",
"ãã³ãã¬ãŒã:ã³ã©ã ",
"ãã³ãã¬ãŒã:Wikiversity"
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E6%95%B0%E5%AD%A6II/%E4%B8%89%E8%A7%92%E9%96%A2%E6%95%B0 |
1,914 | é«çåŠæ ¡æ°åŠII/埮åã»ç©åã®èã | ããã§ã¯åŸ®åç©åã®æŠå¿µã«ã€ããŠç解ããå€é
åŒé¢æ°ã®åŸ®åç©åãåŠã¶ããŸãã埮åã®å¿çšãå¿çšããŠæ¥ç·ã®æ¹çšåŒãã°ã©ãã®æŠåœ¢ãªã©ãæ±ããããç©åãå¿çšããŠã°ã©ãã®é¢ç©ãæ±ããã埮åç©åã¯ç©çåŠãå·¥åŠãªã©ããŸããŸãªåéã§å¿çšãããŠããã
äžåŠæ ¡ã§ã¯ãäžæ¬¡é¢æ°ãš y = a x 2 {\displaystyle y=ax^{2}} ã®å€åã®å²åãæ±ããã ãããããã§ã¯ãåããã®ãå¹³åå€åçãšåŒã¶ããšã«ãããäžè¬ã®é¢æ° y = f ( x ) {\displaystyle y=f(x)} ã®å¹³åå€åçãèããŠã¿ãããäžåŠæ ¡ã§åŠç¿ããããšãšåæ§ã«èãããšã y = f ( x ) {\displaystyle y=f(x)} ã«ãããŠã x {\displaystyle x} ã a {\displaystyle a} ãã b {\displaystyle b} ãŸã§å€åãããšãã®å¹³åå€åçã¯ãã y {\displaystyle y} ã®å€åé/ x {\displaystyle x} ã®å€åéãã§æ±ãããããã€ãŸãã æ§æ解æ倱æ (SVG(ãã©ãŠã¶ã®ãã©ã°ã€ã³ã§ MathML ãæå¹ã«ããããšãã§ããŸã): ãµãŒããŒãhttp://localhost:6011/ja.wikibooks.org/v1/ãããç¡å¹ãªå¿ç ("Math extension cannot connect to Restbase."):): {\displaystyle \frac{f(b)-f(a)}{b-a}} ã§ããã
äŸ
y = x 2 + 2 x + 1 {\displaystyle y=x^{2}+2x+1} ã«ãããŠã x {\displaystyle x} ã-1ãã3ãŸã§å€åãããšãã®å¹³åå€åçãæ±ããã
( 3 2 + 2 â
3 + 1 ) â ( ( â 1 ) 2 + 2 â
( â 1 ) + 1 ) 3 â ( â 1 ) {\displaystyle {\frac {(3^{2}+2\cdot 3+1)-((-1)^{2}+2\cdot (-1)+1)}{3-(-1)}}} = 4 {\displaystyle =4}
é¢æ° f ( x ) {\displaystyle f(x)} ã«ãããŠã x {\displaystyle x} ã a {\displaystyle a} ãšã¯ç°ãªãå€ããšããªããéããªã a {\displaystyle a} ã«è¿ã¥ããšãã f ( x ) {\displaystyle f(x)} ãéããªã A {\displaystyle A} ã«è¿ã¥ãããšãã lim x â a f ( x ) = A {\displaystyle \lim _{x\rightarrow a}f(x)=A} ãšããã
lim x â 0 3 x {\displaystyle \lim _{x\rightarrow 0}3x} ãæ±ããã
x {\displaystyle x} ãã 1 , 0.1 , 0.01 , 0.001 , ⯠{\displaystyle 1,0.1,0.01,0.001,\cdots } ãšéããªã0ã«è¿ã¥ããŠã¿ãããããšã 3 x {\displaystyle 3x} ã¯ã 3 , 0.3 , 0.03 , 0.003 , ⯠{\displaystyle 3,0.3,0.03,0.003,\cdots } ãšãéããªã0ã«è¿ã¥ãããšããããã
ãã£ãŠã x {\displaystyle x} ãéããªã0ã«è¿ã¥ãããšã 3 x {\displaystyle 3x} ã¯éããªã0ã«è¿ã¥ãã®ã§ã lim x â 0 3 x = 0 {\displaystyle \lim _{x\rightarrow 0}3x=0} ã§ããã
次ã«ã lim x â 1 x 2 â 1 x â 1 {\displaystyle \lim _{x\rightarrow 1}{\frac {x^{2}-1}{x-1}}} ãæ±ããã
x {\displaystyle x} ãã 1.1 , 1.01 , 1.001 , 0.0001 , 1.00001 , ⯠{\displaystyle 1.1,1.01,1.001,0.0001,1.00001,\cdots } ãšãéããªã1ã«è¿ã¥ããŠã¿ããšã x 2 â 1 x â 1 {\displaystyle {\frac {x^{2}-1}{x-1}}} ã¯ã 2.1 , 2.01 , 2.001 , 2.0001 , 2.00001 , ⯠{\displaystyle 2.1,2.01,2.001,2.0001,2.00001,\cdots } ãšãéããªã2ã«è¿ã¥ãã
ãªã®ã§ã lim x â 1 x 2 â 1 x â 1 = 2 {\displaystyle \lim _{x\rightarrow 1}{\frac {x^{2}-1}{x-1}}=2} ã§ããã
ããã¯ãåŒã«å€ã代å
¥ããåã«ãåŒèªäœãçŽåããŠããŸã£ãæ¹ãç°¡åã«èšç®ã§ãããããªãã¡ã x 2 â 1 x â 1 = ( x + 1 ) ( x â 1 ) x â 1 {\displaystyle {\frac {x^{2}-1}{x-1}}={\frac {(x+1)(x-1)}{x-1}}} ã§ããã x {\displaystyle x} ã1ãšã¯ç°ãªãå€ãåããªããéããªã1ã«è¿ã¥ãããšã x â 1 {\displaystyle x\neq 1} ãªã®ã§ãããã¯çŽåã§ãã x 2 â 1 x â 1 = ( x + 1 ) ( x â 1 ) x â 1 = x + 1 {\displaystyle {\frac {x^{2}-1}{x-1}}={\frac {(x+1)(x-1)}{x-1}}=x+1} ã§ããã
ãªã®ã§ã lim x â 1 x 2 â 1 x â 1 {\displaystyle \lim _{x\rightarrow 1}{\frac {x^{2}-1}{x-1}}} ãæ±ããã«ã¯ã lim x â 1 ( x + 1 ) {\displaystyle \lim _{x\rightarrow 1}(x+1)} ãæ±ããã°è¯ãã
lim x â 1 ( x + 1 ) = 2 {\displaystyle \lim _{x\rightarrow 1}(x+1)=2} ã§ããã®ã§ã lim x â 1 x 2 â 1 x â 1 = 2 {\displaystyle \lim _{x\rightarrow 1}{\frac {x^{2}-1}{x-1}}=2} ãšæ±ããããšãã§ããã
â»çºå± æåã®äŸã§ã¯ã x {\displaystyle x} ãã 1 , 0.1 , 0.01 , 0.001 , ⯠{\displaystyle 1,0.1,0.01,0.001,\cdots } ãšãéããªã0ã«è¿ã¥ãããã 2 , 0.2 , 0.02 , 0.002 , ⯠{\displaystyle 2,0.2,0.02,0.002,\cdots } ãã â 1 , â 0.1 , â 0.01 , â 0.001 , ⯠{\displaystyle -1,-0.1,-0.01,-0.001,\cdots } ã®ããã«è¿ã¥ããŠã¿ãŠã x {\displaystyle x} ã¯éããªã0ã«è¿ã¥ããä»ã«ãã 1 , â 0.1 , 0.01 , â 0.001 , ⯠{\displaystyle 1,-0.1,0.01,-0.001,\cdots } ã 0.1 , 0.5 , 0.01 , 0.05 , ⯠{\displaystyle 0.1,0.5,0.01,0.05,\cdots } ãªã© x {\displaystyle x} ã0ã«è¿ã¥ãããæ¹æ³ã¯ãããã§ãèããããã
ãã¡ããããã®äŸã§ã¯ã x {\displaystyle x} ãã©ã®ããã«è¿ã¥ãããšããŠã極éã®å€ã¯å€ãããªãã
ãããã x {\displaystyle x} ãã 1 , 0.1 , 0.01 , 0.001 , ⯠{\displaystyle 1,0.1,0.01,0.001,\cdots } ãšè¿ã¥ãããšãã f ( x ) {\displaystyle f(x)} 㯠α {\displaystyle \alpha } ã«è¿ã¥ããã x {\displaystyle x} ãã 2 , 0.2 , 0.02 , 0.002 , ⯠{\displaystyle 2,0.2,0.02,0.002,\cdots } ãšè¿ã¥ãããã f ( x ) {\displaystyle f(x)} 㯠α {\displaystyle \alpha } ã«è¿ã¥ããªãããããªé¢æ° f ( x ) {\displaystyle f(x)} ã ã£ãŠããã ããã
ãªã x {\displaystyle x} ã 1 , 0.1 , 0.01 , 0.001 , ⯠{\displaystyle 1,0.1,0.01,0.001,\cdots } ãšãè¿ã¥ããã ãã§ã極éã®å€ãæ±ããããšãåºæ¥ãã®ã?ãšçåã«æã人ãããããç¥ããªãã
極éãå³å¯ã«å®çŸ©ããã«ã¯ãã€ãã·ãã³ãã«ã¿è«æ³ã䜿ãå¿
èŠããããããããé«æ ¡çã«ã¯å°ãé£ãããšèãã人ãå€ãã®ã§é«æ ¡ã§ã¯ããŸãæããããŠããªãã
ãªã®ã§ããã®æ¬ã§ã¯ãã€ãã·ãã³ãã«ã¿è«æ³ã䜿ãããææ§ãªæ¹æ³ã§æ¥µéãå®çŸ©ããããªã®ã§ãäžã®ãããªçåãæã£ã人ã¯ããã®çåã«ã€ããŠæ·±ãèããã«å
ã«é²ãããã€ãã·ãã³ãã«ã¿è«æ³ãåŠã¶ãããŠã»ããã
é¢æ° y = f ( x ) {\displaystyle y=f(x)} ã®åŸãã«ã€ããŠèããŠã¿ããã
x {\displaystyle x} ã a {\displaystyle a} ãã a + h {\displaystyle a+h} ãŸã§å€åãããšãã®å¹³åå€åçã¯
f ( a + h ) â f ( a ) h {\displaystyle {\frac {f(a+h)-f(a)}{h}}}
ã§ããããã®ãšãã h {\displaystyle h} ãéããªã0ã«è¿ã¥ããã° a {\displaystyle a} ã§ã®åŸããæ±ããããšãã§ãããã€ãŸããé¢æ° y = f ( x ) {\displaystyle y=f(x)} ã® a {\displaystyle a} ã§ã®åŸãã¯
lim h â 0 f ( a + h ) â f ( a ) h {\displaystyle \lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}}
ã§äžãããããããã x = a {\displaystyle x=a} ã«ããã埮åä¿æ°ãšããã
ãŸã
f â² ( x ) = lim h â 0 f ( x + h ) â f ( x ) h {\displaystyle f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}}
ã§äžããããé¢æ° f â² ( x ) {\displaystyle f'(x)} ãé¢æ° f ( x ) {\displaystyle f(x)} ã®å°é¢æ°ãšããã
é¢æ° f ( x ) {\displaystyle f(x)} ã®å°é¢æ°ã¯ d f d x {\displaystyle {\frac {df}{dx}}} ãšè¡šãããããšãããã
ããã§ãããã€ãã®é¢æ°ã®å°é¢æ°ãæ±ããŠã¿ããã
ã§ããã
n {\displaystyle n} ãèªç¶æ°ãšãããé¢æ° f ( x ) = x n {\displaystyle f(x)=x^{n}} ã®å°é¢æ°ã¯äºé
å®çãå¿çšã
ãšæ±ãããã
é¢æ° f ( x ) , g ( x ) {\displaystyle f(x),g(x)} ã«å¯Ÿã次ãæãç«ã€ã
蚌æ
æŒç¿åé¡
次ã®é¢æ°ã埮åãã
1. f ( x ) = 2 x 3 + 4 x 2 â 5 x â 1 {\displaystyle f(x)=2x^{3}+4x^{2}-5x-1} 2. f ( x ) = ( 2 x + 3 ) ( 3 x â 5 ) {\displaystyle f(x)=(2x+3)(3x-5)}
解ç
1.
2. f ( x ) = 6 x 2 â x â 15 {\displaystyle f(x)=6x^{2}-x-15} ã§ãããã
æ²ç· y = f ( x ) {\displaystyle y=f(x)} äžã®ç¹ ( t , f ( t ) ) {\displaystyle (t,f(t))} ã«ãããæ¥ç·ã®æ¹çšåŒãæ±ããããã®æ¥ç·ã®åŸã㯠f â² ( t ) {\displaystyle f'(t)} ã§ãããç¹ ( t , f ( t ) ) {\displaystyle (t,f(t))} ãéãã®ã§ãæ¹çšåŒã¯ y = f â² ( t ) ( x â t ) + f ( t ) {\displaystyle y=f'(t)(x-t)+f(t)} ã§äžãããããå®éã x = t {\displaystyle x=t} ãšãããš y = f ( t ) {\displaystyle y=f(t)} ãšãªãã®ã§ãã®æ¹çšåŒã¯ç¹ ( t , f ( t ) ) {\displaystyle (t,f(t))} ãéãããšããããã x {\displaystyle x} ã®ä¿æ°ã¯ f â² ( t ) {\displaystyle f'(t)} ãªã®ã§åŸã㯠f â² ( t ) {\displaystyle f'(t)} ã§ããã
æ²ç· y = f ( x ) {\displaystyle y=f(x)} äžã®ç¹ ( t , f ( t ) ) {\displaystyle (t,f(t))} ã«ãããæ³ç·ã®æ¹çšåŒã¯ã y = â 1 f â² ( t ) ( x â t ) + f ( t ) {\displaystyle y=-{\frac {1}{f'(t)}}(x-t)+f(t)} ã§äžããããã
f'(x)ã¯ãfã®åŸããè¡šããã®ã§ã f â² ( x ) > 0 {\displaystyle f'(x)>0} ã®ç¹ã§ã¯ãfã¯å¢å€§ãã f â² ( x ) < 0 {\displaystyle f'(x)<0} ã®ç¹ã§ã¯ãfã¯æžå°ããããšããããã
ãããããšã«é¢æ°ã®æŠåœ¢ãæãããšãã§ããã
äŸ
y = x 3 {\displaystyle y=x^{3}} ã®å¢æžã調ã¹ã
䞡蟺ãxã§åŸ®åãããš
f ( x ) = x 3 â 3 x {\displaystyle f(x)=x^{3}-3x} ã埮åãããš
å¢æžè¡šã¯æ¬¡ã®ããã«ãªãã
ãã®é¢æ°ã®ã°ã©ãã¯ã x = â 1 {\displaystyle x=-1} ãå¢ã«ããŠå¢å ããæžå°ã®ç¶æ
ã«å€ããã x = 1 {\displaystyle x=1} ãå¢ã«ããŠæžå°ããå¢å ã®ç¶æ
ã«å€ããã ãã®ãšãã f ( x ) {\displaystyle f(x)} 㯠x = â 1 {\displaystyle x=-1} ã«ãããŠæ¥µå€§(ãããã ã)ã«ãªããšããããã®ãšãã® f ( x ) {\displaystyle f(x)} ã®å€ f ( â 1 ) = 2 {\displaystyle f(-1)=2} ã極倧å€(ãããã ãã¡)ãšããããŸãã x = 1 {\displaystyle x=1} ã«ãããŠæ¥µå°(ãããããã)ã«ãªããšããããã®ãšãã® f ( x ) {\displaystyle f(x)} ã®å€ f ( 1 ) = â 2 {\displaystyle f(1)=-2} ã極å°å€(ããããããã¡)ãšããã極倧å€ãšæ¥µå°å€ãåãããŠæ¥µå€(ãããã¡)ãšããã
äžå®ç©å(indefinite integral)ãšã¯ã埮åããããã®é¢æ°ã«ãªãé¢æ°ãæ±ããæäœã§ããã
ã€ãŸããé¢æ° f ( x ) {\displaystyle f(x)} ã«å¯ŸããŠã F â² ( x ) = f ( x ) {\displaystyle F'(x)=f(x)} ãšãªããé¢æ° F ( x ) {\displaystyle F(x)} ãæ±ããæäœã§ããã
ãã®ãšã F ( x ) {\displaystyle F(x)} ãã f ( x ) {\displaystyle f(x)} ã®åå§é¢æ°(primitive function)ãšåŒã¶ã
äŸãã°ã 1 2 x 2 {\displaystyle {\frac {1}{2}}x^{2}} ã¯åŸ®åãããšã x {\displaystyle x} ã«ãªãã®ã§ã 1 2 x 2 {\displaystyle {\frac {1}{2}}x^{2}} 㯠x {\displaystyle x} ã®åå§é¢æ°ã§ããã
ãããã 1 2 x 2 + 1 {\displaystyle {\frac {1}{2}}x^{2}+1} ãã 1 2 x 2 + 3 {\displaystyle {\frac {1}{2}}x^{2}+3} ãªã©ã埮åãããš x {\displaystyle x} ã«ãªãã®ã§ã 1 2 x 2 + 1 {\displaystyle {\frac {1}{2}}x^{2}+1} ãã 1 2 x 2 + 3 {\displaystyle {\frac {1}{2}}x^{2}+3} ã x {\displaystyle x} ã®åå§é¢æ°ã§ããã
äžè¬ã«ã 1 2 x 2 + C {\displaystyle {\frac {1}{2}}x^{2}+C} (Cã¯ä»»æã®å®æ°)ã§è¡šãããé¢æ°ã¯ã x {\displaystyle x} ã®åå§é¢æ°ã§ããã
x {\displaystyle x} ã®åå§é¢æ°ã¯äžã€ã ãã§ã¯ãªããç¡æ°ã«ããã®ã ã
äžè¬ã«ãé¢æ° f ( x ) {\displaystyle f(x)} ã®åå§é¢æ°ã®äžã€ã F ( x ) {\displaystyle F(x)} ãšãããšããåå§é¢æ°ã«ä»»æã®å®æ°ã足ããé¢æ° F ( x ) + C {\displaystyle F(x)+C} ã f ( x ) {\displaystyle f(x)} ã®åå§é¢æ°ã«ãªãã
ãªããªãã F ( x ) {\displaystyle F(x)} ã f ( x ) {\displaystyle f(x)} ã®åå§é¢æ°ã§ãããã€ãŸãã F â² ( x ) = f ( x ) {\displaystyle F'(x)=f(x)} ã®ãšãã ( F ( x ) + C ) â² = F â² ( x ) + ( C ) â² = F â² ( x ) = f ( x ) {\displaystyle {(F(x)+C)}'=F'(x)+{(C)}'=F'(x)=f(x)} ãšãªãããã ã
ãŸããé¢æ° f ( x ) {\displaystyle f(x)} ã®åå§é¢æ°ã®äžã€ã F ( x ) {\displaystyle F(x)} ã§ãããšãããã¹ãŠã®é¢æ° f ( x ) {\displaystyle f(x)} ã®åå§é¢æ°ã¯ F ( x ) + C {\displaystyle F(x)+C} ã®åœ¢ã«æžããã
F ( x ) + C {\displaystyle F(x)+C} ã®åœ¢ã«æžããªãé¢æ° G ( x ) {\displaystyle G(x)} ãé¢æ° f ( x ) {\displaystyle f(x)} ã®åå§é¢æ°ã§ãããšä»®å®ããããã®ãšãã h ( x ) = F ( x ) â G ( x ) {\displaystyle h(x)=F(x)-G(x)} ãšãããšãé¢æ° h ( x ) {\displaystyle h(x)} ã¯å®æ°ã§ã¯ãªãã
ãã®ãšãã h â² ( x ) = { F ( x ) â G ( x ) } â² = F â² ( x ) â G â² ( x ) = f ( x ) â f ( x ) = 0 {\displaystyle h'(x)=\{F(x)-G(x)\}'=F'(x)-G'(x)=f(x)-f(x)=0} ã§ããã¯ãã ããé¢æ° h ( x ) {\displaystyle h(x)} ã¯å®æ°ã§ã¯ãªãã®ã§ h â² ( x ) = 0 {\displaystyle h'(x)=0} ãšãªããªããããã¯ççŸãªã®ã§ããã¹ãŠã®é¢æ° f ( x ) {\displaystyle f(x)} ã®åå§é¢æ°ã¯ F ( x ) + C {\displaystyle F(x)+C} ã®åœ¢ã«æžããããšã蚌æã§ããã
é¢æ° f ( x ) {\displaystyle f(x)} ã®åå§é¢æ°ã®å
šäœãã â« f ( x ) d x {\displaystyle \int f(x)dx} ãšè¡šãããã®è¡šèšæ³ã¯æåã¯å¥åŠã«æãã ãããããã®ããã«è¡šèšããçç±ã¯åŸã«èª¬æããã®ã§ãä»ã¯ããã®ãŸãŸèŠããŠæ¬²ããã
ãŸãšãããšãé¢æ° f ( x ) {\displaystyle f(x)} ã®åå§é¢æ°ã®å
šäœ â« f ( x ) d x {\displaystyle \int f(x)dx} ã¯ã f ( x ) {\displaystyle f(x)} ã®åå§é¢æ°ã®äžã€ã F ( x ) {\displaystyle F(x)} ãšããŠããã®é¢æ°ã«ä»»æã®å®æ°ã足ããé¢æ° F ( x ) + C {\displaystyle F(x)+C} ã§è¡šããããã€ãŸãã
C {\displaystyle C} ã¯ä»»æã®å®æ°ãšãããããã®ä»»æã®å®æ° C {\displaystyle C} ãç©åå®æ°(constant of integration)ãšåŒã¶ã
â»æ³šæ â« f ( x ) d x {\displaystyle \int f(x)dx} ã¯å®çŸ©ã«ãããããã«ã f ( x ) {\displaystyle f(x)} ã®åå§é¢æ°ã®å
šäœãè¡šããŠãããã€ãŸãã f ( x ) {\displaystyle f(x)} ã®åå§é¢æ°ã®äžã€ã F ( x ) {\displaystyle F(x)} ãšãããšãã â« f ( x ) d x = F ( x ) + C {\displaystyle \int f(x)dx=F(x)+C} ã®å³èŸº F ( x ) + C {\displaystyle F(x)+C} ã¯ã F ( x ) {\displaystyle F(x)} ã«å®æ°ã足ããé¢æ°ã®å
šäœãè¡šããŠãããã€ãŸãã F ( x ) + C {\displaystyle F(x)+C} ã¯ã F ( x ) + 1 {\displaystyle F(x)+1} ãã F ( x ) â 23 {\displaystyle F(x)-23} ãã F ( x ) â 5 Ï {\displaystyle F(x)-5\pi } ãªã©ã®ã F ( x ) {\displaystyle F(x)} ã«å®æ°ã足ããé¢æ°ãã¹ãŠããŸãšã㊠F ( x ) + C {\displaystyle F(x)+C} ãšè¡šããŠããããã®ããšããããµãã«ãªã£ãŠãããšãé倧ãªééããèµ·ããå¯èœæ§ãããã®ã§ã泚æãå¿
èŠã§ããã
é¢æ° f ( x ) = x n {\displaystyle f(x)=x^{n}} (ãã ã n {\displaystyle n} ã¯èªç¶æ°)ã®äžå®ç©åãæ±ããŠã¿ãããã倩äžãçã ãã F ( x ) = 1 n + 1 x n + 1 + C {\displaystyle F(x)={\frac {1}{n+1}}x^{n+1}+C} ( C {\displaystyle C} ã¯ä»»æã®å®æ°)ãšãããšã F â² ( x ) = x n {\displaystyle F'(x)=x^{n}} ãšãªãã®ã§ã 1 n + 1 x n + 1 + C {\displaystyle {\frac {1}{n+1}}x^{n+1}+C} ã¯åå§é¢æ°ã§ããããšããããã
ãããã£ãŠ â« x n d x = 1 n + 1 x n + 1 + C {\displaystyle \int x^{n}dx={\frac {1}{n+1}}x^{n+1}+C}
é¢æ° f ( x ) , g ( x ) {\displaystyle f(x),g(x)} ã®åå§é¢æ°ãããããã F ( x ) , G ( x ) {\displaystyle F(x),G(x)} ãšããã a {\displaystyle a} ãä»»æã®å®æ°å®æ°ãšãããš
{ F ( x ) + G ( x ) } â² = F â² ( x ) + G â² ( x ) = f ( x ) + g ( x ) {\displaystyle \{F(x)+G(x)\}'=F'(x)+G'(x)=f(x)+g(x)}
{ a F ( x ) } â² = a F â² ( x ) = a f ( x ) {\displaystyle \{aF(x)\}'=aF'(x)=af(x)}
ãšãªãã®ã§ã
â« { f ( x ) + g ( x ) } d x = â« f ( x ) d x + â« g ( x ) d x {\displaystyle \int \{f(x)+g(x)\}dx=\int f(x)dx+\int g(x)dx}
â« a f ( x ) d x = a â« f ( x ) d x {\displaystyle \int af(x)dx=a\int f(x)dx}
ãæãç«ã€ããšãåããã
æŒç¿åé¡
äžå®ç©å â« ( x 8 + 2 x 2 â 6 x + 9 ) d x {\displaystyle \int (x^{8}+2x^{2}-6x+9)dx} ãæ±ãã
解ç
â« ( x 8 + 2 x 2 â 6 x + 9 ) d x = â« x 8 d x + 2 â« x 2 d x â 6 â« x d x + 9 â« d x = x 9 9 + 2 x 3 3 â 3 x 2 + 9 x + C {\displaystyle \int (x^{8}+2x^{2}-6x+9)dx=\int x^{8}\,dx+2\int x^{2}\,dx-6\int x\,dx+9\int dx={\frac {x^{9}}{9}}+{\frac {2x^{3}}{3}}-3x^{2}+9x+C} ( C {\displaystyle C} ã¯ç©åå®æ°)
é¢æ° f ( x ) {\displaystyle f(x)} ã®åå§é¢æ°ã®äžã€ã F ( x ) {\displaystyle F(x)} ãšããããã®åå§é¢æ°ã«å€ã代å
¥ããŠããã®å€ã®å·®ãæ±ããæäœããå®ç©åãšåŒã³ã â« a b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)dx} ãšæžããã€ãŸãã
ã§ããã
[ f ( x ) ] a b = f ( b ) â f ( a ) {\displaystyle [f(x)]_{a}^{b}=f(b)-f(a)} ãšããã
ãã®ããã«ãããšã â« a b f ( x ) d x = [ F ( x ) ] a b = F ( b ) â F ( a ) {\displaystyle \int _{a}^{b}f(x)dx=[F(x)]_{a}^{b}=F(b)-F(a)} ãšèšç®ã§ããã
å®ç©åã®å€ã¯åå§é¢æ°ã®éžæã«ãããªããå®éãåå§é¢æ°ãšããŠã F ( x ) + C {\displaystyle F(x)+C} ãéžã³ãå®ç©åãèšç®ãããšã â« a b f ( x ) d x = ( F ( b ) + C ) â ( F ( a ) + C ) = F ( b ) â F ( a ) {\displaystyle \int _{a}^{b}f(x)dx=(F(b)+C)-(F(a)+C)=F(b)-F(a)}
ãšãªããåå§é¢æ°ãšããŠã©ããéžãã§ãå®ç©åã®å€ã¯äžå®ã§ããããšããããã
é¢æ° f ( x ) , g ( x ) {\displaystyle f(x),g(x)} ã«å¯ŸããŠãåå§é¢æ°ããããã F ( x ) , G ( x ) {\displaystyle F(x),G(x)} ãšããã k {\displaystyle k} ãå®æ°ãšããŠã
â« a b k f ( x ) d x = k F ( b ) â k F ( a ) = k ( F ( b ) â F ( a ) ) = k â« a b f ( x ) d x {\displaystyle \int _{a}^{b}kf(x)\,dx=kF(b)-kF(a)=k(F(b)-F(a))=k\int _{a}^{b}f(x)\,dx}
â« a b { f ( x ) + g ( x ) } d x = [ F ( x ) + G ( x ) ] a b = F ( b ) + G ( b ) â ( F ( a ) + G ( a ) ) = F ( b ) â F ( a ) + G ( b ) â G ( a ) = â« a b f ( x ) d x + â« a b g ( x ) d x {\displaystyle \int _{a}^{b}\{f(x)+g(x)\}dx=[F(x)+G(x)]_{a}^{b}=F(b)+G(b)-(F(a)+G(a))=F(b)-F(a)+G(b)-G(a)=\int _{a}^{b}f(x)\,dx+\int _{a}^{b}g(x)\,dx}
â« a a f ( x ) d x = F ( a ) â F ( a ) = 0 {\displaystyle \int _{a}^{a}f(x)\,dx=F(a)-F(a)=0}
â« b a f ( x ) d x = F ( a ) â F ( b ) = â ( F ( b ) â F ( a ) ) = â â« a b f ( x ) d x {\displaystyle \int _{b}^{a}f(x)\,dx=F(a)-F(b)=-(F(b)-F(a))=-\int _{a}^{b}f(x)\,dx}
â« a b f ( x ) d x = F ( b ) â F ( a ) = ( F ( b ) â F ( c ) ) + ( F ( c ) â F ( a ) ) = â« a c f ( x ) d x + â« c b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)\,dx=F(b)-F(a)=(F(b)-F(c))+(F(c)-F(a))=\int _{a}^{c}f(x)\,dx+\int _{c}^{b}f(x)\,dx}
ãæãç«ã€ã
â« 2 5 x 3 d x {\displaystyle \int _{2}^{5}x^{3}dx} ãæ±ããã
1 4 x 4 {\displaystyle {\frac {1}{4}}x^{4}} ã¯ã埮åãããšã x 3 {\displaystyle x^{3}} ãªã®ã§ã 1 4 x 4 {\displaystyle {\frac {1}{4}}x^{4}} 㯠x 3 {\displaystyle x^{3}} ã®åå§é¢æ°ã®äžã€ã§ããããã£ãŠ â« 2 5 x 3 d x = [ 1 4 x 4 ] 2 5 = 1 4 5 4 â 1 4 2 4 = 609 4 {\displaystyle \int _{2}^{5}x^{3}dx=\left[{\frac {1}{4}}x^{4}\right]_{2}^{5}={\frac {1}{4}}5^{4}-{\frac {1}{4}}2^{4}={\frac {609}{4}}} ã§ããã
1 4 x 4 + 1 {\displaystyle {\frac {1}{4}}x^{4}+1} ãã埮åãããšã x 3 {\displaystyle x^{3}} ãªã®ã§ã 1 4 x 4 + 1 {\displaystyle {\frac {1}{4}}x^{4}+1} 㯠x 3 {\displaystyle x^{3}} ã®åå§é¢æ°ã®äžã€ã§ããããã£ãŠã â« 2 5 x 3 d x = [ 1 4 x 4 + 1 ] 2 5 = ( 1 4 5 4 + 1 ) â ( 1 4 2 4 + 1 ) = 609 4 {\displaystyle \int _{2}^{5}x^{3}dx=\left[{\frac {1}{4}}x^{4}+1\right]_{2}^{5}=\left({\frac {1}{4}}5^{4}+1\right)-\left({\frac {1}{4}}2^{4}+1\right)={\frac {609}{4}}} ãšæ±ããããšãã§ããã
aãå®æ°ãšãããšããå®ç©å â« a x f ( t ) d t {\displaystyle \int _{a}^{x}f(t)\,dt} ã¯xã®é¢æ°ã«ãªãã é¢æ° f ( t ) {\displaystyle f(t)} ã®åå§é¢æ°ã®äžã€ã F ( t ) {\displaystyle F(t)} ãšãããš
ãã®äž¡èŸºãxã§åŸ®åãããšã F ( a ) {\displaystyle F(a)} ã¯å®æ°ã§ãããã
é¢æ° f ( x ) {\displaystyle f(x)} ã a ⊠x ⊠b {\displaystyle a\leqq x\leqq b} ã®ç¯å²ã§åžžã«æ£ã§ãããšããããã®ãšããå®ç©å â« a b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)dx} ã«ãã£ãŠãé¢æ° f ( x ) {\displaystyle f(x)} ã®ã°ã©ããšãçŽç· x = a {\displaystyle x=a} ãçŽç· x = b {\displaystyle x=b} ã x {\displaystyle x} 軞ã§å²ãŸããéšåã®é¢ç©ãæ±ããããšãã§ããã
é¢æ° f ( x ) {\displaystyle f(x)} ã®ã°ã©ããšãçŽç· x = a {\displaystyle x=a} ãçŽç· x = c {\displaystyle x=c} ãšã x {\displaystyle x} 軞ã§å²ãŸããéšåã®é¢ç©ã S ( c ) {\displaystyle S(c)} ãšããããšã«ãã£ãŠãé¢æ° S ( x ) {\displaystyle S(x)} ãå®ããã( a ⊠x ⊠b {\displaystyle a\leqq x\leqq b} ãšãã)
é¢æ° f ( x ) {\displaystyle f(x)} ã®ã°ã©ããšãçŽç· x = c {\displaystyle x=c} ãçŽç· x = c + h {\displaystyle x=c+h} ãšã x {\displaystyle x} 軞ã§å²ãŸããéšåã®é¢ç©ãèãã( a ⊠c + h ⊠b {\displaystyle a\leqq c+h\leqq b} ãšãã)ãããã¯ã S ( c + h ) â S ( c ) {\displaystyle S(c+h)-S(c)} ã§ãããããã§ã c < t < c + h {\displaystyle c<t<c+h} ãªã t {\displaystyle t} ããšã£ãŠããŠããã®ç¹ã«ããã f ( x ) {\displaystyle f(x)} ã®å€ f ( t ) {\displaystyle f(t)} ãé«ããšããé·æ¹åœ¢ã®é¢ç©ãèããããšã§ã t {\displaystyle t} ãäžæã«ãšãã°ã S ( c + h ) â S ( c ) = h â
f ( t ) {\displaystyle S(c+h)-S(c)=h\cdot f(t)} ãšã§ããã䞡蟺ã h {\displaystyle h} ã§å²ãã h â 0 {\displaystyle h\to 0} ã®æ¥µéãèãããšã
ã§ãããã巊蟺ã¯åŸ®åã®å®çŸ©ãã S â² ( c ) {\displaystyle S'(c)} ã§ããã lim h â 0 t = c {\displaystyle \lim _{h\to 0}t=c} ã§ããããšã«æ³šæãããšå³èŸºã¯ f ( c ) {\displaystyle f(c)} ã§ãããæåã c {\displaystyle c} ãã x {\displaystyle x} ã«åãæãããšãçµå±
ãåŸããããã€ãŸãã S ( x ) {\displaystyle S(x)} 㯠f ( x ) {\displaystyle f(x)} ã®åå§é¢æ°ã®äžã€ã§ããããšãåããã
ãã£ãŠã â« a b f ( x ) d x = S ( b ) â S ( a ) {\displaystyle \int _{a}^{b}f(x)dx=S(b)-S(a)} ã§ãããããã®åŒã®å³èŸºã¯ãé¢æ° f ( x ) {\displaystyle f(x)} ã®ã°ã©ããšãçŽç· x = a {\displaystyle x=a} ãçŽç· x = b {\displaystyle x=b} ãšã x {\displaystyle x} 軞ã§å²ãŸããé¢ç©ã§ããããã£ãŠã巊蟺 â« a b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)dx} ã¯ãé¢æ° f ( x ) {\displaystyle f(x)} ã®ã°ã©ããšãçŽç· x = a {\displaystyle x=a} ãçŽç· x = b {\displaystyle x=b} ãšã x {\displaystyle x} 軞ã§å²ãŸããé¢ç©ãè¡šããŠããã
æŽå²çã«ã¯ãç©åã¯ãé¢æ°ã®ã°ã©ãã§å²ãŸããéšåã®é¢ç©ãæ±ããããã«èãåºãããããã®ç¯ã§è¿°ã¹ããããªåŸ®åãšã®é¢é£ã¯ç©åèªäœã®çºæãããã£ãšåŸã«ãªã£ãŠçºèŠãããããšã§ããã
äŸãšããŠã 0 ⊠x ⊠1 {\displaystyle 0\leqq x\leqq 1} ã®ç¯å²ã§ãy = xã®ã°ã©ããšx軞ã§ã¯ããŸããéšåã®é¢ç©ããç©åãçšããŠèšç®ããã ( å®éã«ã¯ããã¯äžè§åœ¢ãªã®ã§ãç©åãçšããªããŠãé¢ç©ãèšç®ããããšãåºæ¥ãã ç㯠1 2 {\displaystyle {\frac {1}{2}}} ãšãªãã ) å®ç©åãè¡ãªããšã â« 0 1 x d x {\displaystyle \int _{0}^{1}xdx} = 1 2 [ x 2 ] 0 1 {\displaystyle ={\frac {1}{2}}[x^{2}]_{0}^{1}} = 1 2 [ 1 2 â 0 2 ] {\displaystyle ={\frac {1}{2}}[1^{2}-0^{2}]} = 1 2 [ 1 â 0 ] {\displaystyle ={\frac {1}{2}}[1-0]}
= 1 2 {\displaystyle ={\frac {1}{2}}} ãšãªã確ãã«äžèŽããã
æŒç¿åé¡
æŸç©ç· y = 5 â x 2 {\displaystyle y=5-x^{2}} ãšx軞ããã³2çŽç· x = â 1 , x = 2 {\displaystyle x=-1\ ,\ x=2} ã§å²ãŸããéšåã®é¢ç©Sãæ±ããã
解ç
ãã®æŸç©ç·ã¯ â 1 †x †2 {\displaystyle -1\leq x\leq 2} ã§x軞ã®äžåŽã«ããããã
a †x †b {\displaystyle a\leq x\leq b} ã«ãããŠãåžžã« f ( x ) ⥠g ( x ) {\displaystyle f(x)\geq g(x)} ã§ãããšãã2ã€ã®æ²ç· y = f ( x ) , y = g ( x ) {\displaystyle y=f(x)\ ,\ y=g(x)} ã«æãŸããéšåã®é¢ç©Sã¯ã次ã®åŒã§è¡šãããã
æŸç©ç· y = x 2 â 1 {\displaystyle y=x^{2}-1} ãšçŽç· y = x + 1 {\displaystyle y=x+1} ã«ãã£ãŠå²ãŸããéšåã®é¢ç©Sãæ±ããã
æŸç©ç·ãšçŽç·ã®äº€ç¹ã®x座æšã¯
â 1 †x †2 {\displaystyle -1\leq x\leq 2} ã®ç¯å²ã§ x 2 â 1 †x + 1 {\displaystyle x^{2}-1\leq x+1} ãã
a †x †b {\displaystyle a\leq x\leq b} ã§ã f ( x ) †0 {\displaystyle f(x)\leq 0} ã®ãšããx軞 y = 0 {\displaystyle y=0} ãšæ²ç· y = f ( x ) {\displaystyle y=f(x)} ã«ãã£ãŠæãŸããŠãããšèããããã®ã§ã
ãšãªãã
æŸç©ç· y = x 2 â 2 x {\displaystyle y=x^{2}-2x} ãšx軞ã§å²ãŸããéšåã®é¢ç©Sãæ±ããã
æŸç©ç·ãšx軞ã®äº€ç¹ã®x座æšã¯
ãã®æŸç©ç·ã¯ 0 †x †2 {\displaystyle 0\leq x\leq 2} ã§x軞ã®äžåŽã«ããããã
é«æ ¡æ°åŠãããŠãããšãå°æ¥åŸ®åãšãç©åãšãäœã«äœ¿ã?ããšæã人ã®æ¹ãå€ããšæãã確ãã«æ¥åžžç掻ã§ã¯ãç©åãªã©ã®é«åºŠãª æ°åŠã¯äœ¿ããªããã ããã®äžæ¹è£ã§ã¯ç©åã 埮åãé«æ ¡æ°åŠã§ã¯åãŸããªããããªæ°åŠã䜿ãããŠãããäŸãã°å°é¢šã®é²è·¯äºæ³ã ããã¯ç©åã䜿ãå°é¢šã®é²è·¯ãäºæž¬ããŠãã ä»ã«ãã»ãã¥ãªãã£ã®åŒ·åãªã©ã«ãæ°åŠã¯äœ¿ãããŠãããæ¥åžžç掻ã§ã¯æ°åŠã¯äœ¿ããªãã æ°åŠã«èŠªãã¿ãæã£ãŠã¿ãŠã¯ã©ãã ãããã
(1) F ( x ) = 2 x 2 {\displaystyle F(x)=2x^{2}} ã®ãšã f ( x ) {\displaystyle f(x)} ãæ±ããããã ã F â² ( x ) {\displaystyle F'(x)}
(3)åå§é¢æ°ãå®ç©åãæ±ãã
3) lim x â 0 â« x 5 2 x d x {\displaystyle \lim _{x\rightarrow 0}\int _{x}^{5}2xdx}
4) â« â 60 60 sin x + cos 2 x d x {\displaystyle \int _{-60}^{60}\sin x+\cos ^{2}xdx}
(1) f ( x ) = x 3 {\displaystyle f(x)=x^{3}} åªä¹ã®åŸ®å㯠y â² = n x n â 1 {\displaystyle y'=nx^{n}-1} ã§ããããäžå®ç©åã®å®çŸ©ãã f ( x ) = x 3 {\displaystyle f(x)=x^{3}} ã§ããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ããã§ã¯åŸ®åç©åã®æŠå¿µã«ã€ããŠç解ããå€é
åŒé¢æ°ã®åŸ®åç©åãåŠã¶ããŸãã埮åã®å¿çšãå¿çšããŠæ¥ç·ã®æ¹çšåŒãã°ã©ãã®æŠåœ¢ãªã©ãæ±ããããç©åãå¿çšããŠã°ã©ãã®é¢ç©ãæ±ããã埮åç©åã¯ç©çåŠãå·¥åŠãªã©ããŸããŸãªåéã§å¿çšãããŠããã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "äžåŠæ ¡ã§ã¯ãäžæ¬¡é¢æ°ãš y = a x 2 {\\displaystyle y=ax^{2}} ã®å€åã®å²åãæ±ããã ãããããã§ã¯ãåããã®ãå¹³åå€åçãšåŒã¶ããšã«ãããäžè¬ã®é¢æ° y = f ( x ) {\\displaystyle y=f(x)} ã®å¹³åå€åçãèããŠã¿ãããäžåŠæ ¡ã§åŠç¿ããããšãšåæ§ã«èãããšã y = f ( x ) {\\displaystyle y=f(x)} ã«ãããŠã x {\\displaystyle x} ã a {\\displaystyle a} ãã b {\\displaystyle b} ãŸã§å€åãããšãã®å¹³åå€åçã¯ãã y {\\displaystyle y} ã®å€åé/ x {\\displaystyle x} ã®å€åéãã§æ±ãããããã€ãŸãã æ§æ解æ倱æ (SVG(ãã©ãŠã¶ã®ãã©ã°ã€ã³ã§ MathML ãæå¹ã«ããããšãã§ããŸã): ãµãŒããŒãhttp://localhost:6011/ja.wikibooks.org/v1/ãããç¡å¹ãªå¿ç (\"Math extension cannot connect to Restbase.\"):): {\\displaystyle \\frac{f(b)-f(a)}{b-a}} ã§ããã",
"title": "å¹³åå€åç"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "äŸ",
"title": "å¹³åå€åç"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "y = x 2 + 2 x + 1 {\\displaystyle y=x^{2}+2x+1} ã«ãããŠã x {\\displaystyle x} ã-1ãã3ãŸã§å€åãããšãã®å¹³åå€åçãæ±ããã",
"title": "å¹³åå€åç"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "( 3 2 + 2 â
3 + 1 ) â ( ( â 1 ) 2 + 2 â
( â 1 ) + 1 ) 3 â ( â 1 ) {\\displaystyle {\\frac {(3^{2}+2\\cdot 3+1)-((-1)^{2}+2\\cdot (-1)+1)}{3-(-1)}}} = 4 {\\displaystyle =4}",
"title": "å¹³åå€åç"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "é¢æ° f ( x ) {\\displaystyle f(x)} ã«ãããŠã x {\\displaystyle x} ã a {\\displaystyle a} ãšã¯ç°ãªãå€ããšããªããéããªã a {\\displaystyle a} ã«è¿ã¥ããšãã f ( x ) {\\displaystyle f(x)} ãéããªã A {\\displaystyle A} ã«è¿ã¥ãããšãã lim x â a f ( x ) = A {\\displaystyle \\lim _{x\\rightarrow a}f(x)=A} ãšããã",
"title": "極é"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "lim x â 0 3 x {\\displaystyle \\lim _{x\\rightarrow 0}3x} ãæ±ããã",
"title": "極é"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "x {\\displaystyle x} ãã 1 , 0.1 , 0.01 , 0.001 , ⯠{\\displaystyle 1,0.1,0.01,0.001,\\cdots } ãšéããªã0ã«è¿ã¥ããŠã¿ãããããšã 3 x {\\displaystyle 3x} ã¯ã 3 , 0.3 , 0.03 , 0.003 , ⯠{\\displaystyle 3,0.3,0.03,0.003,\\cdots } ãšãéããªã0ã«è¿ã¥ãããšããããã",
"title": "極é"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "ãã£ãŠã x {\\displaystyle x} ãéããªã0ã«è¿ã¥ãããšã 3 x {\\displaystyle 3x} ã¯éããªã0ã«è¿ã¥ãã®ã§ã lim x â 0 3 x = 0 {\\displaystyle \\lim _{x\\rightarrow 0}3x=0} ã§ããã",
"title": "極é"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "次ã«ã lim x â 1 x 2 â 1 x â 1 {\\displaystyle \\lim _{x\\rightarrow 1}{\\frac {x^{2}-1}{x-1}}} ãæ±ããã",
"title": "極é"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "x {\\displaystyle x} ãã 1.1 , 1.01 , 1.001 , 0.0001 , 1.00001 , ⯠{\\displaystyle 1.1,1.01,1.001,0.0001,1.00001,\\cdots } ãšãéããªã1ã«è¿ã¥ããŠã¿ããšã x 2 â 1 x â 1 {\\displaystyle {\\frac {x^{2}-1}{x-1}}} ã¯ã 2.1 , 2.01 , 2.001 , 2.0001 , 2.00001 , ⯠{\\displaystyle 2.1,2.01,2.001,2.0001,2.00001,\\cdots } ãšãéããªã2ã«è¿ã¥ãã",
"title": "極é"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "ãªã®ã§ã lim x â 1 x 2 â 1 x â 1 = 2 {\\displaystyle \\lim _{x\\rightarrow 1}{\\frac {x^{2}-1}{x-1}}=2} ã§ããã",
"title": "極é"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "ããã¯ãåŒã«å€ã代å
¥ããåã«ãåŒèªäœãçŽåããŠããŸã£ãæ¹ãç°¡åã«èšç®ã§ãããããªãã¡ã x 2 â 1 x â 1 = ( x + 1 ) ( x â 1 ) x â 1 {\\displaystyle {\\frac {x^{2}-1}{x-1}}={\\frac {(x+1)(x-1)}{x-1}}} ã§ããã x {\\displaystyle x} ã1ãšã¯ç°ãªãå€ãåããªããéããªã1ã«è¿ã¥ãããšã x â 1 {\\displaystyle x\\neq 1} ãªã®ã§ãããã¯çŽåã§ãã x 2 â 1 x â 1 = ( x + 1 ) ( x â 1 ) x â 1 = x + 1 {\\displaystyle {\\frac {x^{2}-1}{x-1}}={\\frac {(x+1)(x-1)}{x-1}}=x+1} ã§ããã",
"title": "極é"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ãªã®ã§ã lim x â 1 x 2 â 1 x â 1 {\\displaystyle \\lim _{x\\rightarrow 1}{\\frac {x^{2}-1}{x-1}}} ãæ±ããã«ã¯ã lim x â 1 ( x + 1 ) {\\displaystyle \\lim _{x\\rightarrow 1}(x+1)} ãæ±ããã°è¯ãã",
"title": "極é"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "lim x â 1 ( x + 1 ) = 2 {\\displaystyle \\lim _{x\\rightarrow 1}(x+1)=2} ã§ããã®ã§ã lim x â 1 x 2 â 1 x â 1 = 2 {\\displaystyle \\lim _{x\\rightarrow 1}{\\frac {x^{2}-1}{x-1}}=2} ãšæ±ããããšãã§ããã",
"title": "極é"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "â»çºå± æåã®äŸã§ã¯ã x {\\displaystyle x} ãã 1 , 0.1 , 0.01 , 0.001 , ⯠{\\displaystyle 1,0.1,0.01,0.001,\\cdots } ãšãéããªã0ã«è¿ã¥ãããã 2 , 0.2 , 0.02 , 0.002 , ⯠{\\displaystyle 2,0.2,0.02,0.002,\\cdots } ãã â 1 , â 0.1 , â 0.01 , â 0.001 , ⯠{\\displaystyle -1,-0.1,-0.01,-0.001,\\cdots } ã®ããã«è¿ã¥ããŠã¿ãŠã x {\\displaystyle x} ã¯éããªã0ã«è¿ã¥ããä»ã«ãã 1 , â 0.1 , 0.01 , â 0.001 , ⯠{\\displaystyle 1,-0.1,0.01,-0.001,\\cdots } ã 0.1 , 0.5 , 0.01 , 0.05 , ⯠{\\displaystyle 0.1,0.5,0.01,0.05,\\cdots } ãªã© x {\\displaystyle x} ã0ã«è¿ã¥ãããæ¹æ³ã¯ãããã§ãèããããã",
"title": "極é"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "ãã¡ããããã®äŸã§ã¯ã x {\\displaystyle x} ãã©ã®ããã«è¿ã¥ãããšããŠã極éã®å€ã¯å€ãããªãã",
"title": "極é"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "ãããã x {\\displaystyle x} ãã 1 , 0.1 , 0.01 , 0.001 , ⯠{\\displaystyle 1,0.1,0.01,0.001,\\cdots } ãšè¿ã¥ãããšãã f ( x ) {\\displaystyle f(x)} 㯠α {\\displaystyle \\alpha } ã«è¿ã¥ããã x {\\displaystyle x} ãã 2 , 0.2 , 0.02 , 0.002 , ⯠{\\displaystyle 2,0.2,0.02,0.002,\\cdots } ãšè¿ã¥ãããã f ( x ) {\\displaystyle f(x)} 㯠α {\\displaystyle \\alpha } ã«è¿ã¥ããªãããããªé¢æ° f ( x ) {\\displaystyle f(x)} ã ã£ãŠããã ããã",
"title": "極é"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "ãªã x {\\displaystyle x} ã 1 , 0.1 , 0.01 , 0.001 , ⯠{\\displaystyle 1,0.1,0.01,0.001,\\cdots } ãšãè¿ã¥ããã ãã§ã極éã®å€ãæ±ããããšãåºæ¥ãã®ã?ãšçåã«æã人ãããããç¥ããªãã",
"title": "極é"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "極éãå³å¯ã«å®çŸ©ããã«ã¯ãã€ãã·ãã³ãã«ã¿è«æ³ã䜿ãå¿
èŠããããããããé«æ ¡çã«ã¯å°ãé£ãããšèãã人ãå€ãã®ã§é«æ ¡ã§ã¯ããŸãæããããŠããªãã",
"title": "極é"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "ãªã®ã§ããã®æ¬ã§ã¯ãã€ãã·ãã³ãã«ã¿è«æ³ã䜿ãããææ§ãªæ¹æ³ã§æ¥µéãå®çŸ©ããããªã®ã§ãäžã®ãããªçåãæã£ã人ã¯ããã®çåã«ã€ããŠæ·±ãèããã«å
ã«é²ãããã€ãã·ãã³ãã«ã¿è«æ³ãåŠã¶ãããŠã»ããã",
"title": "極é"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "é¢æ° y = f ( x ) {\\displaystyle y=f(x)} ã®åŸãã«ã€ããŠèããŠã¿ããã",
"title": "埮åä¿æ°ãšå°é¢æ°"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "x {\\displaystyle x} ã a {\\displaystyle a} ãã a + h {\\displaystyle a+h} ãŸã§å€åãããšãã®å¹³åå€åçã¯",
"title": "埮åä¿æ°ãšå°é¢æ°"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "f ( a + h ) â f ( a ) h {\\displaystyle {\\frac {f(a+h)-f(a)}{h}}}",
"title": "埮åä¿æ°ãšå°é¢æ°"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "ã§ããããã®ãšãã h {\\displaystyle h} ãéããªã0ã«è¿ã¥ããã° a {\\displaystyle a} ã§ã®åŸããæ±ããããšãã§ãããã€ãŸããé¢æ° y = f ( x ) {\\displaystyle y=f(x)} ã® a {\\displaystyle a} ã§ã®åŸãã¯",
"title": "埮åä¿æ°ãšå°é¢æ°"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "lim h â 0 f ( a + h ) â f ( a ) h {\\displaystyle \\lim _{h\\to 0}{\\frac {f(a+h)-f(a)}{h}}}",
"title": "埮åä¿æ°ãšå°é¢æ°"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "ã§äžãããããããã x = a {\\displaystyle x=a} ã«ããã埮åä¿æ°ãšããã",
"title": "埮åä¿æ°ãšå°é¢æ°"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "ãŸã",
"title": "埮åä¿æ°ãšå°é¢æ°"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "f â² ( x ) = lim h â 0 f ( x + h ) â f ( x ) h {\\displaystyle f'(x)=\\lim _{h\\to 0}{\\frac {f(x+h)-f(x)}{h}}}",
"title": "埮åä¿æ°ãšå°é¢æ°"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "ã§äžããããé¢æ° f â² ( x ) {\\displaystyle f'(x)} ãé¢æ° f ( x ) {\\displaystyle f(x)} ã®å°é¢æ°ãšããã",
"title": "埮åä¿æ°ãšå°é¢æ°"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "é¢æ° f ( x ) {\\displaystyle f(x)} ã®å°é¢æ°ã¯ d f d x {\\displaystyle {\\frac {df}{dx}}} ãšè¡šãããããšãããã",
"title": "埮åä¿æ°ãšå°é¢æ°"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "ããã§ãããã€ãã®é¢æ°ã®å°é¢æ°ãæ±ããŠã¿ããã",
"title": "埮åä¿æ°ãšå°é¢æ°"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "ã§ããã",
"title": "埮åä¿æ°ãšå°é¢æ°"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "n {\\displaystyle n} ãèªç¶æ°ãšãããé¢æ° f ( x ) = x n {\\displaystyle f(x)=x^{n}} ã®å°é¢æ°ã¯äºé
å®çãå¿çšã",
"title": "埮åä¿æ°ãšå°é¢æ°"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "ãšæ±ãããã",
"title": "埮åä¿æ°ãšå°é¢æ°"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "é¢æ° f ( x ) , g ( x ) {\\displaystyle f(x),g(x)} ã«å¯Ÿã次ãæãç«ã€ã",
"title": "åã»å·®åã³å®æ°åã®å°é¢æ°"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "蚌æ",
"title": "åã»å·®åã³å®æ°åã®å°é¢æ°"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "åã»å·®åã³å®æ°åã®å°é¢æ°"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "次ã®é¢æ°ã埮åãã",
"title": "åã»å·®åã³å®æ°åã®å°é¢æ°"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "1. f ( x ) = 2 x 3 + 4 x 2 â 5 x â 1 {\\displaystyle f(x)=2x^{3}+4x^{2}-5x-1} 2. f ( x ) = ( 2 x + 3 ) ( 3 x â 5 ) {\\displaystyle f(x)=(2x+3)(3x-5)}",
"title": "åã»å·®åã³å®æ°åã®å°é¢æ°"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "解ç",
"title": "åã»å·®åã³å®æ°åã®å°é¢æ°"
},
{
"paragraph_id": 41,
"tag": "p",
"text": "1.",
"title": "åã»å·®åã³å®æ°åã®å°é¢æ°"
},
{
"paragraph_id": 42,
"tag": "p",
"text": "2. f ( x ) = 6 x 2 â x â 15 {\\displaystyle f(x)=6x^{2}-x-15} ã§ãããã",
"title": "åã»å·®åã³å®æ°åã®å°é¢æ°"
},
{
"paragraph_id": 43,
"tag": "p",
"text": "æ²ç· y = f ( x ) {\\displaystyle y=f(x)} äžã®ç¹ ( t , f ( t ) ) {\\displaystyle (t,f(t))} ã«ãããæ¥ç·ã®æ¹çšåŒãæ±ããããã®æ¥ç·ã®åŸã㯠f â² ( t ) {\\displaystyle f'(t)} ã§ãããç¹ ( t , f ( t ) ) {\\displaystyle (t,f(t))} ãéãã®ã§ãæ¹çšåŒã¯ y = f â² ( t ) ( x â t ) + f ( t ) {\\displaystyle y=f'(t)(x-t)+f(t)} ã§äžãããããå®éã x = t {\\displaystyle x=t} ãšãããš y = f ( t ) {\\displaystyle y=f(t)} ãšãªãã®ã§ãã®æ¹çšåŒã¯ç¹ ( t , f ( t ) ) {\\displaystyle (t,f(t))} ãéãããšããããã x {\\displaystyle x} ã®ä¿æ°ã¯ f â² ( t ) {\\displaystyle f'(t)} ãªã®ã§åŸã㯠f â² ( t ) {\\displaystyle f'(t)} ã§ããã",
"title": "å°é¢æ°ã®å¿çš"
},
{
"paragraph_id": 44,
"tag": "p",
"text": "æ²ç· y = f ( x ) {\\displaystyle y=f(x)} äžã®ç¹ ( t , f ( t ) ) {\\displaystyle (t,f(t))} ã«ãããæ³ç·ã®æ¹çšåŒã¯ã y = â 1 f â² ( t ) ( x â t ) + f ( t ) {\\displaystyle y=-{\\frac {1}{f'(t)}}(x-t)+f(t)} ã§äžããããã",
"title": "å°é¢æ°ã®å¿çš"
},
{
"paragraph_id": 45,
"tag": "p",
"text": "f'(x)ã¯ãfã®åŸããè¡šããã®ã§ã f â² ( x ) > 0 {\\displaystyle f'(x)>0} ã®ç¹ã§ã¯ãfã¯å¢å€§ãã f â² ( x ) < 0 {\\displaystyle f'(x)<0} ã®ç¹ã§ã¯ãfã¯æžå°ããããšããããã",
"title": "å°é¢æ°ã®å¿çš"
},
{
"paragraph_id": 46,
"tag": "p",
"text": "ãããããšã«é¢æ°ã®æŠåœ¢ãæãããšãã§ããã",
"title": "å°é¢æ°ã®å¿çš"
},
{
"paragraph_id": 47,
"tag": "p",
"text": "",
"title": "å°é¢æ°ã®å¿çš"
},
{
"paragraph_id": 48,
"tag": "p",
"text": "äŸ",
"title": "å°é¢æ°ã®å¿çš"
},
{
"paragraph_id": 49,
"tag": "p",
"text": "y = x 3 {\\displaystyle y=x^{3}} ã®å¢æžã調ã¹ã",
"title": "å°é¢æ°ã®å¿çš"
},
{
"paragraph_id": 50,
"tag": "p",
"text": "䞡蟺ãxã§åŸ®åãããš",
"title": "å°é¢æ°ã®å¿çš"
},
{
"paragraph_id": 51,
"tag": "p",
"text": "f ( x ) = x 3 â 3 x {\\displaystyle f(x)=x^{3}-3x} ã埮åãããš",
"title": "å°é¢æ°ã®å¿çš"
},
{
"paragraph_id": 52,
"tag": "p",
"text": "å¢æžè¡šã¯æ¬¡ã®ããã«ãªãã",
"title": "å°é¢æ°ã®å¿çš"
},
{
"paragraph_id": 53,
"tag": "p",
"text": "ãã®é¢æ°ã®ã°ã©ãã¯ã x = â 1 {\\displaystyle x=-1} ãå¢ã«ããŠå¢å ããæžå°ã®ç¶æ
ã«å€ããã x = 1 {\\displaystyle x=1} ãå¢ã«ããŠæžå°ããå¢å ã®ç¶æ
ã«å€ããã ãã®ãšãã f ( x ) {\\displaystyle f(x)} 㯠x = â 1 {\\displaystyle x=-1} ã«ãããŠæ¥µå€§(ãããã ã)ã«ãªããšããããã®ãšãã® f ( x ) {\\displaystyle f(x)} ã®å€ f ( â 1 ) = 2 {\\displaystyle f(-1)=2} ã極倧å€(ãããã ãã¡)ãšããããŸãã x = 1 {\\displaystyle x=1} ã«ãããŠæ¥µå°(ãããããã)ã«ãªããšããããã®ãšãã® f ( x ) {\\displaystyle f(x)} ã®å€ f ( 1 ) = â 2 {\\displaystyle f(1)=-2} ã極å°å€(ããããããã¡)ãšããã極倧å€ãšæ¥µå°å€ãåãããŠæ¥µå€(ãããã¡)ãšããã",
"title": "å°é¢æ°ã®å¿çš"
},
{
"paragraph_id": 54,
"tag": "p",
"text": "äžå®ç©å(indefinite integral)ãšã¯ã埮åããããã®é¢æ°ã«ãªãé¢æ°ãæ±ããæäœã§ããã",
"title": "äžå®ç©å"
},
{
"paragraph_id": 55,
"tag": "p",
"text": "ã€ãŸããé¢æ° f ( x ) {\\displaystyle f(x)} ã«å¯ŸããŠã F â² ( x ) = f ( x ) {\\displaystyle F'(x)=f(x)} ãšãªããé¢æ° F ( x ) {\\displaystyle F(x)} ãæ±ããæäœã§ããã",
"title": "äžå®ç©å"
},
{
"paragraph_id": 56,
"tag": "p",
"text": "ãã®ãšã F ( x ) {\\displaystyle F(x)} ãã f ( x ) {\\displaystyle f(x)} ã®åå§é¢æ°(primitive function)ãšåŒã¶ã",
"title": "äžå®ç©å"
},
{
"paragraph_id": 57,
"tag": "p",
"text": "äŸãã°ã 1 2 x 2 {\\displaystyle {\\frac {1}{2}}x^{2}} ã¯åŸ®åãããšã x {\\displaystyle x} ã«ãªãã®ã§ã 1 2 x 2 {\\displaystyle {\\frac {1}{2}}x^{2}} 㯠x {\\displaystyle x} ã®åå§é¢æ°ã§ããã",
"title": "äžå®ç©å"
},
{
"paragraph_id": 58,
"tag": "p",
"text": "ãããã 1 2 x 2 + 1 {\\displaystyle {\\frac {1}{2}}x^{2}+1} ãã 1 2 x 2 + 3 {\\displaystyle {\\frac {1}{2}}x^{2}+3} ãªã©ã埮åãããš x {\\displaystyle x} ã«ãªãã®ã§ã 1 2 x 2 + 1 {\\displaystyle {\\frac {1}{2}}x^{2}+1} ãã 1 2 x 2 + 3 {\\displaystyle {\\frac {1}{2}}x^{2}+3} ã x {\\displaystyle x} ã®åå§é¢æ°ã§ããã",
"title": "äžå®ç©å"
},
{
"paragraph_id": 59,
"tag": "p",
"text": "äžè¬ã«ã 1 2 x 2 + C {\\displaystyle {\\frac {1}{2}}x^{2}+C} (Cã¯ä»»æã®å®æ°)ã§è¡šãããé¢æ°ã¯ã x {\\displaystyle x} ã®åå§é¢æ°ã§ããã",
"title": "äžå®ç©å"
},
{
"paragraph_id": 60,
"tag": "p",
"text": "x {\\displaystyle x} ã®åå§é¢æ°ã¯äžã€ã ãã§ã¯ãªããç¡æ°ã«ããã®ã ã",
"title": "äžå®ç©å"
},
{
"paragraph_id": 61,
"tag": "p",
"text": "",
"title": "äžå®ç©å"
},
{
"paragraph_id": 62,
"tag": "p",
"text": "äžè¬ã«ãé¢æ° f ( x ) {\\displaystyle f(x)} ã®åå§é¢æ°ã®äžã€ã F ( x ) {\\displaystyle F(x)} ãšãããšããåå§é¢æ°ã«ä»»æã®å®æ°ã足ããé¢æ° F ( x ) + C {\\displaystyle F(x)+C} ã f ( x ) {\\displaystyle f(x)} ã®åå§é¢æ°ã«ãªãã",
"title": "äžå®ç©å"
},
{
"paragraph_id": 63,
"tag": "p",
"text": "ãªããªãã F ( x ) {\\displaystyle F(x)} ã f ( x ) {\\displaystyle f(x)} ã®åå§é¢æ°ã§ãããã€ãŸãã F â² ( x ) = f ( x ) {\\displaystyle F'(x)=f(x)} ã®ãšãã ( F ( x ) + C ) â² = F â² ( x ) + ( C ) â² = F â² ( x ) = f ( x ) {\\displaystyle {(F(x)+C)}'=F'(x)+{(C)}'=F'(x)=f(x)} ãšãªãããã ã",
"title": "äžå®ç©å"
},
{
"paragraph_id": 64,
"tag": "p",
"text": "ãŸããé¢æ° f ( x ) {\\displaystyle f(x)} ã®åå§é¢æ°ã®äžã€ã F ( x ) {\\displaystyle F(x)} ã§ãããšãããã¹ãŠã®é¢æ° f ( x ) {\\displaystyle f(x)} ã®åå§é¢æ°ã¯ F ( x ) + C {\\displaystyle F(x)+C} ã®åœ¢ã«æžããã",
"title": "äžå®ç©å"
},
{
"paragraph_id": 65,
"tag": "p",
"text": "F ( x ) + C {\\displaystyle F(x)+C} ã®åœ¢ã«æžããªãé¢æ° G ( x ) {\\displaystyle G(x)} ãé¢æ° f ( x ) {\\displaystyle f(x)} ã®åå§é¢æ°ã§ãããšä»®å®ããããã®ãšãã h ( x ) = F ( x ) â G ( x ) {\\displaystyle h(x)=F(x)-G(x)} ãšãããšãé¢æ° h ( x ) {\\displaystyle h(x)} ã¯å®æ°ã§ã¯ãªãã",
"title": "äžå®ç©å"
},
{
"paragraph_id": 66,
"tag": "p",
"text": "ãã®ãšãã h â² ( x ) = { F ( x ) â G ( x ) } â² = F â² ( x ) â G â² ( x ) = f ( x ) â f ( x ) = 0 {\\displaystyle h'(x)=\\{F(x)-G(x)\\}'=F'(x)-G'(x)=f(x)-f(x)=0} ã§ããã¯ãã ããé¢æ° h ( x ) {\\displaystyle h(x)} ã¯å®æ°ã§ã¯ãªãã®ã§ h â² ( x ) = 0 {\\displaystyle h'(x)=0} ãšãªããªããããã¯ççŸãªã®ã§ããã¹ãŠã®é¢æ° f ( x ) {\\displaystyle f(x)} ã®åå§é¢æ°ã¯ F ( x ) + C {\\displaystyle F(x)+C} ã®åœ¢ã«æžããããšã蚌æã§ããã",
"title": "äžå®ç©å"
},
{
"paragraph_id": 67,
"tag": "p",
"text": "é¢æ° f ( x ) {\\displaystyle f(x)} ã®åå§é¢æ°ã®å
šäœãã â« f ( x ) d x {\\displaystyle \\int f(x)dx} ãšè¡šãããã®è¡šèšæ³ã¯æåã¯å¥åŠã«æãã ãããããã®ããã«è¡šèšããçç±ã¯åŸã«èª¬æããã®ã§ãä»ã¯ããã®ãŸãŸèŠããŠæ¬²ããã",
"title": "äžå®ç©å"
},
{
"paragraph_id": 68,
"tag": "p",
"text": "ãŸãšãããšãé¢æ° f ( x ) {\\displaystyle f(x)} ã®åå§é¢æ°ã®å
šäœ â« f ( x ) d x {\\displaystyle \\int f(x)dx} ã¯ã f ( x ) {\\displaystyle f(x)} ã®åå§é¢æ°ã®äžã€ã F ( x ) {\\displaystyle F(x)} ãšããŠããã®é¢æ°ã«ä»»æã®å®æ°ã足ããé¢æ° F ( x ) + C {\\displaystyle F(x)+C} ã§è¡šããããã€ãŸãã",
"title": "äžå®ç©å"
},
{
"paragraph_id": 69,
"tag": "p",
"text": "C {\\displaystyle C} ã¯ä»»æã®å®æ°ãšãããããã®ä»»æã®å®æ° C {\\displaystyle C} ãç©åå®æ°(constant of integration)ãšåŒã¶ã",
"title": "äžå®ç©å"
},
{
"paragraph_id": 70,
"tag": "p",
"text": "â»æ³šæ â« f ( x ) d x {\\displaystyle \\int f(x)dx} ã¯å®çŸ©ã«ãããããã«ã f ( x ) {\\displaystyle f(x)} ã®åå§é¢æ°ã®å
šäœãè¡šããŠãããã€ãŸãã f ( x ) {\\displaystyle f(x)} ã®åå§é¢æ°ã®äžã€ã F ( x ) {\\displaystyle F(x)} ãšãããšãã â« f ( x ) d x = F ( x ) + C {\\displaystyle \\int f(x)dx=F(x)+C} ã®å³èŸº F ( x ) + C {\\displaystyle F(x)+C} ã¯ã F ( x ) {\\displaystyle F(x)} ã«å®æ°ã足ããé¢æ°ã®å
šäœãè¡šããŠãããã€ãŸãã F ( x ) + C {\\displaystyle F(x)+C} ã¯ã F ( x ) + 1 {\\displaystyle F(x)+1} ãã F ( x ) â 23 {\\displaystyle F(x)-23} ãã F ( x ) â 5 Ï {\\displaystyle F(x)-5\\pi } ãªã©ã®ã F ( x ) {\\displaystyle F(x)} ã«å®æ°ã足ããé¢æ°ãã¹ãŠããŸãšã㊠F ( x ) + C {\\displaystyle F(x)+C} ãšè¡šããŠããããã®ããšããããµãã«ãªã£ãŠãããšãé倧ãªééããèµ·ããå¯èœæ§ãããã®ã§ã泚æãå¿
èŠã§ããã",
"title": "äžå®ç©å"
},
{
"paragraph_id": 71,
"tag": "p",
"text": "é¢æ° f ( x ) = x n {\\displaystyle f(x)=x^{n}} (ãã ã n {\\displaystyle n} ã¯èªç¶æ°)ã®äžå®ç©åãæ±ããŠã¿ãããã倩äžãçã ãã F ( x ) = 1 n + 1 x n + 1 + C {\\displaystyle F(x)={\\frac {1}{n+1}}x^{n+1}+C} ( C {\\displaystyle C} ã¯ä»»æã®å®æ°)ãšãããšã F â² ( x ) = x n {\\displaystyle F'(x)=x^{n}} ãšãªãã®ã§ã 1 n + 1 x n + 1 + C {\\displaystyle {\\frac {1}{n+1}}x^{n+1}+C} ã¯åå§é¢æ°ã§ããããšããããã",
"title": "äžå®ç©å"
},
{
"paragraph_id": 72,
"tag": "p",
"text": "ãããã£ãŠ â« x n d x = 1 n + 1 x n + 1 + C {\\displaystyle \\int x^{n}dx={\\frac {1}{n+1}}x^{n+1}+C}",
"title": "äžå®ç©å"
},
{
"paragraph_id": 73,
"tag": "p",
"text": "é¢æ° f ( x ) , g ( x ) {\\displaystyle f(x),g(x)} ã®åå§é¢æ°ãããããã F ( x ) , G ( x ) {\\displaystyle F(x),G(x)} ãšããã a {\\displaystyle a} ãä»»æã®å®æ°å®æ°ãšãããš",
"title": "äžå®ç©å"
},
{
"paragraph_id": 74,
"tag": "p",
"text": "{ F ( x ) + G ( x ) } â² = F â² ( x ) + G â² ( x ) = f ( x ) + g ( x ) {\\displaystyle \\{F(x)+G(x)\\}'=F'(x)+G'(x)=f(x)+g(x)}",
"title": "äžå®ç©å"
},
{
"paragraph_id": 75,
"tag": "p",
"text": "{ a F ( x ) } â² = a F â² ( x ) = a f ( x ) {\\displaystyle \\{aF(x)\\}'=aF'(x)=af(x)}",
"title": "äžå®ç©å"
},
{
"paragraph_id": 76,
"tag": "p",
"text": "ãšãªãã®ã§ã",
"title": "äžå®ç©å"
},
{
"paragraph_id": 77,
"tag": "p",
"text": "â« { f ( x ) + g ( x ) } d x = â« f ( x ) d x + â« g ( x ) d x {\\displaystyle \\int \\{f(x)+g(x)\\}dx=\\int f(x)dx+\\int g(x)dx}",
"title": "äžå®ç©å"
},
{
"paragraph_id": 78,
"tag": "p",
"text": "â« a f ( x ) d x = a â« f ( x ) d x {\\displaystyle \\int af(x)dx=a\\int f(x)dx}",
"title": "äžå®ç©å"
},
{
"paragraph_id": 79,
"tag": "p",
"text": "ãæãç«ã€ããšãåããã",
"title": "äžå®ç©å"
},
{
"paragraph_id": 80,
"tag": "p",
"text": "",
"title": "äžå®ç©å"
},
{
"paragraph_id": 81,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "äžå®ç©å"
},
{
"paragraph_id": 82,
"tag": "p",
"text": "äžå®ç©å â« ( x 8 + 2 x 2 â 6 x + 9 ) d x {\\displaystyle \\int (x^{8}+2x^{2}-6x+9)dx} ãæ±ãã",
"title": "äžå®ç©å"
},
{
"paragraph_id": 83,
"tag": "p",
"text": "解ç",
"title": "äžå®ç©å"
},
{
"paragraph_id": 84,
"tag": "p",
"text": "â« ( x 8 + 2 x 2 â 6 x + 9 ) d x = â« x 8 d x + 2 â« x 2 d x â 6 â« x d x + 9 â« d x = x 9 9 + 2 x 3 3 â 3 x 2 + 9 x + C {\\displaystyle \\int (x^{8}+2x^{2}-6x+9)dx=\\int x^{8}\\,dx+2\\int x^{2}\\,dx-6\\int x\\,dx+9\\int dx={\\frac {x^{9}}{9}}+{\\frac {2x^{3}}{3}}-3x^{2}+9x+C} ( C {\\displaystyle C} ã¯ç©åå®æ°)",
"title": "äžå®ç©å"
},
{
"paragraph_id": 85,
"tag": "p",
"text": "é¢æ° f ( x ) {\\displaystyle f(x)} ã®åå§é¢æ°ã®äžã€ã F ( x ) {\\displaystyle F(x)} ãšããããã®åå§é¢æ°ã«å€ã代å
¥ããŠããã®å€ã®å·®ãæ±ããæäœããå®ç©åãšåŒã³ã â« a b f ( x ) d x {\\displaystyle \\int _{a}^{b}f(x)dx} ãšæžããã€ãŸãã",
"title": "å®ç©å"
},
{
"paragraph_id": 86,
"tag": "p",
"text": "ã§ããã",
"title": "å®ç©å"
},
{
"paragraph_id": 87,
"tag": "p",
"text": "[ f ( x ) ] a b = f ( b ) â f ( a ) {\\displaystyle [f(x)]_{a}^{b}=f(b)-f(a)} ãšããã",
"title": "å®ç©å"
},
{
"paragraph_id": 88,
"tag": "p",
"text": "ãã®ããã«ãããšã â« a b f ( x ) d x = [ F ( x ) ] a b = F ( b ) â F ( a ) {\\displaystyle \\int _{a}^{b}f(x)dx=[F(x)]_{a}^{b}=F(b)-F(a)} ãšèšç®ã§ããã",
"title": "å®ç©å"
},
{
"paragraph_id": 89,
"tag": "p",
"text": "å®ç©åã®å€ã¯åå§é¢æ°ã®éžæã«ãããªããå®éãåå§é¢æ°ãšããŠã F ( x ) + C {\\displaystyle F(x)+C} ãéžã³ãå®ç©åãèšç®ãããšã â« a b f ( x ) d x = ( F ( b ) + C ) â ( F ( a ) + C ) = F ( b ) â F ( a ) {\\displaystyle \\int _{a}^{b}f(x)dx=(F(b)+C)-(F(a)+C)=F(b)-F(a)}",
"title": "å®ç©å"
},
{
"paragraph_id": 90,
"tag": "p",
"text": "ãšãªããåå§é¢æ°ãšããŠã©ããéžãã§ãå®ç©åã®å€ã¯äžå®ã§ããããšããããã",
"title": "å®ç©å"
},
{
"paragraph_id": 91,
"tag": "p",
"text": "é¢æ° f ( x ) , g ( x ) {\\displaystyle f(x),g(x)} ã«å¯ŸããŠãåå§é¢æ°ããããã F ( x ) , G ( x ) {\\displaystyle F(x),G(x)} ãšããã k {\\displaystyle k} ãå®æ°ãšããŠã",
"title": "å®ç©å"
},
{
"paragraph_id": 92,
"tag": "p",
"text": "â« a b k f ( x ) d x = k F ( b ) â k F ( a ) = k ( F ( b ) â F ( a ) ) = k â« a b f ( x ) d x {\\displaystyle \\int _{a}^{b}kf(x)\\,dx=kF(b)-kF(a)=k(F(b)-F(a))=k\\int _{a}^{b}f(x)\\,dx}",
"title": "å®ç©å"
},
{
"paragraph_id": 93,
"tag": "p",
"text": "â« a b { f ( x ) + g ( x ) } d x = [ F ( x ) + G ( x ) ] a b = F ( b ) + G ( b ) â ( F ( a ) + G ( a ) ) = F ( b ) â F ( a ) + G ( b ) â G ( a ) = â« a b f ( x ) d x + â« a b g ( x ) d x {\\displaystyle \\int _{a}^{b}\\{f(x)+g(x)\\}dx=[F(x)+G(x)]_{a}^{b}=F(b)+G(b)-(F(a)+G(a))=F(b)-F(a)+G(b)-G(a)=\\int _{a}^{b}f(x)\\,dx+\\int _{a}^{b}g(x)\\,dx}",
"title": "å®ç©å"
},
{
"paragraph_id": 94,
"tag": "p",
"text": "â« a a f ( x ) d x = F ( a ) â F ( a ) = 0 {\\displaystyle \\int _{a}^{a}f(x)\\,dx=F(a)-F(a)=0}",
"title": "å®ç©å"
},
{
"paragraph_id": 95,
"tag": "p",
"text": "â« b a f ( x ) d x = F ( a ) â F ( b ) = â ( F ( b ) â F ( a ) ) = â â« a b f ( x ) d x {\\displaystyle \\int _{b}^{a}f(x)\\,dx=F(a)-F(b)=-(F(b)-F(a))=-\\int _{a}^{b}f(x)\\,dx}",
"title": "å®ç©å"
},
{
"paragraph_id": 96,
"tag": "p",
"text": "â« a b f ( x ) d x = F ( b ) â F ( a ) = ( F ( b ) â F ( c ) ) + ( F ( c ) â F ( a ) ) = â« a c f ( x ) d x + â« c b f ( x ) d x {\\displaystyle \\int _{a}^{b}f(x)\\,dx=F(b)-F(a)=(F(b)-F(c))+(F(c)-F(a))=\\int _{a}^{c}f(x)\\,dx+\\int _{c}^{b}f(x)\\,dx}",
"title": "å®ç©å"
},
{
"paragraph_id": 97,
"tag": "p",
"text": "ãæãç«ã€ã",
"title": "å®ç©å"
},
{
"paragraph_id": 98,
"tag": "p",
"text": "â« 2 5 x 3 d x {\\displaystyle \\int _{2}^{5}x^{3}dx} ãæ±ããã",
"title": "å®ç©å"
},
{
"paragraph_id": 99,
"tag": "p",
"text": "1 4 x 4 {\\displaystyle {\\frac {1}{4}}x^{4}} ã¯ã埮åãããšã x 3 {\\displaystyle x^{3}} ãªã®ã§ã 1 4 x 4 {\\displaystyle {\\frac {1}{4}}x^{4}} 㯠x 3 {\\displaystyle x^{3}} ã®åå§é¢æ°ã®äžã€ã§ããããã£ãŠ â« 2 5 x 3 d x = [ 1 4 x 4 ] 2 5 = 1 4 5 4 â 1 4 2 4 = 609 4 {\\displaystyle \\int _{2}^{5}x^{3}dx=\\left[{\\frac {1}{4}}x^{4}\\right]_{2}^{5}={\\frac {1}{4}}5^{4}-{\\frac {1}{4}}2^{4}={\\frac {609}{4}}} ã§ããã",
"title": "å®ç©å"
},
{
"paragraph_id": 100,
"tag": "p",
"text": "1 4 x 4 + 1 {\\displaystyle {\\frac {1}{4}}x^{4}+1} ãã埮åãããšã x 3 {\\displaystyle x^{3}} ãªã®ã§ã 1 4 x 4 + 1 {\\displaystyle {\\frac {1}{4}}x^{4}+1} 㯠x 3 {\\displaystyle x^{3}} ã®åå§é¢æ°ã®äžã€ã§ããããã£ãŠã â« 2 5 x 3 d x = [ 1 4 x 4 + 1 ] 2 5 = ( 1 4 5 4 + 1 ) â ( 1 4 2 4 + 1 ) = 609 4 {\\displaystyle \\int _{2}^{5}x^{3}dx=\\left[{\\frac {1}{4}}x^{4}+1\\right]_{2}^{5}=\\left({\\frac {1}{4}}5^{4}+1\\right)-\\left({\\frac {1}{4}}2^{4}+1\\right)={\\frac {609}{4}}} ãšæ±ããããšãã§ããã",
"title": "å®ç©å"
},
{
"paragraph_id": 101,
"tag": "p",
"text": "aãå®æ°ãšãããšããå®ç©å â« a x f ( t ) d t {\\displaystyle \\int _{a}^{x}f(t)\\,dt} ã¯xã®é¢æ°ã«ãªãã é¢æ° f ( t ) {\\displaystyle f(t)} ã®åå§é¢æ°ã®äžã€ã F ( t ) {\\displaystyle F(t)} ãšãããš",
"title": "埮åç©ååŠã®åºæ¬å®ç"
},
{
"paragraph_id": 102,
"tag": "p",
"text": "ãã®äž¡èŸºãxã§åŸ®åãããšã F ( a ) {\\displaystyle F(a)} ã¯å®æ°ã§ãããã",
"title": "埮åç©ååŠã®åºæ¬å®ç"
},
{
"paragraph_id": 103,
"tag": "p",
"text": "é¢æ° f ( x ) {\\displaystyle f(x)} ã a ⊠x ⊠b {\\displaystyle a\\leqq x\\leqq b} ã®ç¯å²ã§åžžã«æ£ã§ãããšããããã®ãšããå®ç©å â« a b f ( x ) d x {\\displaystyle \\int _{a}^{b}f(x)dx} ã«ãã£ãŠãé¢æ° f ( x ) {\\displaystyle f(x)} ã®ã°ã©ããšãçŽç· x = a {\\displaystyle x=a} ãçŽç· x = b {\\displaystyle x=b} ã x {\\displaystyle x} 軞ã§å²ãŸããéšåã®é¢ç©ãæ±ããããšãã§ããã",
"title": "å®ç©åãšé¢ç©"
},
{
"paragraph_id": 104,
"tag": "p",
"text": "é¢æ° f ( x ) {\\displaystyle f(x)} ã®ã°ã©ããšãçŽç· x = a {\\displaystyle x=a} ãçŽç· x = c {\\displaystyle x=c} ãšã x {\\displaystyle x} 軞ã§å²ãŸããéšåã®é¢ç©ã S ( c ) {\\displaystyle S(c)} ãšããããšã«ãã£ãŠãé¢æ° S ( x ) {\\displaystyle S(x)} ãå®ããã( a ⊠x ⊠b {\\displaystyle a\\leqq x\\leqq b} ãšãã)",
"title": "å®ç©åãšé¢ç©"
},
{
"paragraph_id": 105,
"tag": "p",
"text": "é¢æ° f ( x ) {\\displaystyle f(x)} ã®ã°ã©ããšãçŽç· x = c {\\displaystyle x=c} ãçŽç· x = c + h {\\displaystyle x=c+h} ãšã x {\\displaystyle x} 軞ã§å²ãŸããéšåã®é¢ç©ãèãã( a ⊠c + h ⊠b {\\displaystyle a\\leqq c+h\\leqq b} ãšãã)ãããã¯ã S ( c + h ) â S ( c ) {\\displaystyle S(c+h)-S(c)} ã§ãããããã§ã c < t < c + h {\\displaystyle c<t<c+h} ãªã t {\\displaystyle t} ããšã£ãŠããŠããã®ç¹ã«ããã f ( x ) {\\displaystyle f(x)} ã®å€ f ( t ) {\\displaystyle f(t)} ãé«ããšããé·æ¹åœ¢ã®é¢ç©ãèããããšã§ã t {\\displaystyle t} ãäžæã«ãšãã°ã S ( c + h ) â S ( c ) = h â
f ( t ) {\\displaystyle S(c+h)-S(c)=h\\cdot f(t)} ãšã§ããã䞡蟺ã h {\\displaystyle h} ã§å²ãã h â 0 {\\displaystyle h\\to 0} ã®æ¥µéãèãããšã",
"title": "å®ç©åãšé¢ç©"
},
{
"paragraph_id": 106,
"tag": "p",
"text": "ã§ãããã巊蟺ã¯åŸ®åã®å®çŸ©ãã S â² ( c ) {\\displaystyle S'(c)} ã§ããã lim h â 0 t = c {\\displaystyle \\lim _{h\\to 0}t=c} ã§ããããšã«æ³šæãããšå³èŸºã¯ f ( c ) {\\displaystyle f(c)} ã§ãããæåã c {\\displaystyle c} ãã x {\\displaystyle x} ã«åãæãããšãçµå±",
"title": "å®ç©åãšé¢ç©"
},
{
"paragraph_id": 107,
"tag": "p",
"text": "ãåŸããããã€ãŸãã S ( x ) {\\displaystyle S(x)} 㯠f ( x ) {\\displaystyle f(x)} ã®åå§é¢æ°ã®äžã€ã§ããããšãåããã",
"title": "å®ç©åãšé¢ç©"
},
{
"paragraph_id": 108,
"tag": "p",
"text": "ãã£ãŠã â« a b f ( x ) d x = S ( b ) â S ( a ) {\\displaystyle \\int _{a}^{b}f(x)dx=S(b)-S(a)} ã§ãããããã®åŒã®å³èŸºã¯ãé¢æ° f ( x ) {\\displaystyle f(x)} ã®ã°ã©ããšãçŽç· x = a {\\displaystyle x=a} ãçŽç· x = b {\\displaystyle x=b} ãšã x {\\displaystyle x} 軞ã§å²ãŸããé¢ç©ã§ããããã£ãŠã巊蟺 â« a b f ( x ) d x {\\displaystyle \\int _{a}^{b}f(x)dx} ã¯ãé¢æ° f ( x ) {\\displaystyle f(x)} ã®ã°ã©ããšãçŽç· x = a {\\displaystyle x=a} ãçŽç· x = b {\\displaystyle x=b} ãšã x {\\displaystyle x} 軞ã§å²ãŸããé¢ç©ãè¡šããŠããã",
"title": "å®ç©åãšé¢ç©"
},
{
"paragraph_id": 109,
"tag": "p",
"text": "æŽå²çã«ã¯ãç©åã¯ãé¢æ°ã®ã°ã©ãã§å²ãŸããéšåã®é¢ç©ãæ±ããããã«èãåºãããããã®ç¯ã§è¿°ã¹ããããªåŸ®åãšã®é¢é£ã¯ç©åèªäœã®çºæãããã£ãšåŸã«ãªã£ãŠçºèŠãããããšã§ããã",
"title": "å®ç©åãšé¢ç©"
},
{
"paragraph_id": 110,
"tag": "p",
"text": "äŸãšããŠã 0 ⊠x ⊠1 {\\displaystyle 0\\leqq x\\leqq 1} ã®ç¯å²ã§ãy = xã®ã°ã©ããšx軞ã§ã¯ããŸããéšåã®é¢ç©ããç©åãçšããŠèšç®ããã ( å®éã«ã¯ããã¯äžè§åœ¢ãªã®ã§ãç©åãçšããªããŠãé¢ç©ãèšç®ããããšãåºæ¥ãã ç㯠1 2 {\\displaystyle {\\frac {1}{2}}} ãšãªãã ) å®ç©åãè¡ãªããšã â« 0 1 x d x {\\displaystyle \\int _{0}^{1}xdx} = 1 2 [ x 2 ] 0 1 {\\displaystyle ={\\frac {1}{2}}[x^{2}]_{0}^{1}} = 1 2 [ 1 2 â 0 2 ] {\\displaystyle ={\\frac {1}{2}}[1^{2}-0^{2}]} = 1 2 [ 1 â 0 ] {\\displaystyle ={\\frac {1}{2}}[1-0]}",
"title": "å®ç©åãšé¢ç©"
},
{
"paragraph_id": 111,
"tag": "p",
"text": "= 1 2 {\\displaystyle ={\\frac {1}{2}}} ãšãªã確ãã«äžèŽããã",
"title": "å®ç©åãšé¢ç©"
},
{
"paragraph_id": 112,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "å®ç©åãšé¢ç©"
},
{
"paragraph_id": 113,
"tag": "p",
"text": "æŸç©ç· y = 5 â x 2 {\\displaystyle y=5-x^{2}} ãšx軞ããã³2çŽç· x = â 1 , x = 2 {\\displaystyle x=-1\\ ,\\ x=2} ã§å²ãŸããéšåã®é¢ç©Sãæ±ããã",
"title": "å®ç©åãšé¢ç©"
},
{
"paragraph_id": 114,
"tag": "p",
"text": "解ç",
"title": "å®ç©åãšé¢ç©"
},
{
"paragraph_id": 115,
"tag": "p",
"text": "ãã®æŸç©ç·ã¯ â 1 †x †2 {\\displaystyle -1\\leq x\\leq 2} ã§x軞ã®äžåŽã«ããããã",
"title": "å®ç©åãšé¢ç©"
},
{
"paragraph_id": 116,
"tag": "p",
"text": "a †x †b {\\displaystyle a\\leq x\\leq b} ã«ãããŠãåžžã« f ( x ) ⥠g ( x ) {\\displaystyle f(x)\\geq g(x)} ã§ãããšãã2ã€ã®æ²ç· y = f ( x ) , y = g ( x ) {\\displaystyle y=f(x)\\ ,\\ y=g(x)} ã«æãŸããéšåã®é¢ç©Sã¯ã次ã®åŒã§è¡šãããã",
"title": "å®ç©åãšé¢ç©"
},
{
"paragraph_id": 117,
"tag": "p",
"text": "æŸç©ç· y = x 2 â 1 {\\displaystyle y=x^{2}-1} ãšçŽç· y = x + 1 {\\displaystyle y=x+1} ã«ãã£ãŠå²ãŸããéšåã®é¢ç©Sãæ±ããã",
"title": "å®ç©åãšé¢ç©"
},
{
"paragraph_id": 118,
"tag": "p",
"text": "æŸç©ç·ãšçŽç·ã®äº€ç¹ã®x座æšã¯",
"title": "å®ç©åãšé¢ç©"
},
{
"paragraph_id": 119,
"tag": "p",
"text": "â 1 †x †2 {\\displaystyle -1\\leq x\\leq 2} ã®ç¯å²ã§ x 2 â 1 †x + 1 {\\displaystyle x^{2}-1\\leq x+1} ãã",
"title": "å®ç©åãšé¢ç©"
},
{
"paragraph_id": 120,
"tag": "p",
"text": "a †x †b {\\displaystyle a\\leq x\\leq b} ã§ã f ( x ) †0 {\\displaystyle f(x)\\leq 0} ã®ãšããx軞 y = 0 {\\displaystyle y=0} ãšæ²ç· y = f ( x ) {\\displaystyle y=f(x)} ã«ãã£ãŠæãŸããŠãããšèããããã®ã§ã",
"title": "å®ç©åãšé¢ç©"
},
{
"paragraph_id": 121,
"tag": "p",
"text": "ãšãªãã",
"title": "å®ç©åãšé¢ç©"
},
{
"paragraph_id": 122,
"tag": "p",
"text": "æŸç©ç· y = x 2 â 2 x {\\displaystyle y=x^{2}-2x} ãšx軞ã§å²ãŸããéšåã®é¢ç©Sãæ±ããã",
"title": "å®ç©åãšé¢ç©"
},
{
"paragraph_id": 123,
"tag": "p",
"text": "æŸç©ç·ãšx軞ã®äº€ç¹ã®x座æšã¯",
"title": "å®ç©åãšé¢ç©"
},
{
"paragraph_id": 124,
"tag": "p",
"text": "ãã®æŸç©ç·ã¯ 0 †x †2 {\\displaystyle 0\\leq x\\leq 2} ã§x軞ã®äžåŽã«ããããã",
"title": "å®ç©åãšé¢ç©"
},
{
"paragraph_id": 125,
"tag": "p",
"text": "é«æ ¡æ°åŠãããŠãããšãå°æ¥åŸ®åãšãç©åãšãäœã«äœ¿ã?ããšæã人ã®æ¹ãå€ããšæãã確ãã«æ¥åžžç掻ã§ã¯ãç©åãªã©ã®é«åºŠãª æ°åŠã¯äœ¿ããªããã ããã®äžæ¹è£ã§ã¯ç©åã 埮åãé«æ ¡æ°åŠã§ã¯åãŸããªããããªæ°åŠã䜿ãããŠãããäŸãã°å°é¢šã®é²è·¯äºæ³ã ããã¯ç©åã䜿ãå°é¢šã®é²è·¯ãäºæž¬ããŠãã ä»ã«ãã»ãã¥ãªãã£ã®åŒ·åãªã©ã«ãæ°åŠã¯äœ¿ãããŠãããæ¥åžžç掻ã§ã¯æ°åŠã¯äœ¿ããªãã æ°åŠã«èŠªãã¿ãæã£ãŠã¿ãŠã¯ã©ãã ãããã",
"title": "æ¬åœã«ã¡ãã£ãšããäœè«"
},
{
"paragraph_id": 126,
"tag": "p",
"text": "(1) F ( x ) = 2 x 2 {\\displaystyle F(x)=2x^{2}} ã®ãšã f ( x ) {\\displaystyle f(x)} ãæ±ããããã ã F â² ( x ) {\\displaystyle F'(x)}",
"title": "æŒç¿åé¡"
},
{
"paragraph_id": 127,
"tag": "p",
"text": "(3)åå§é¢æ°ãå®ç©åãæ±ãã",
"title": "æŒç¿åé¡"
},
{
"paragraph_id": 128,
"tag": "p",
"text": "3) lim x â 0 â« x 5 2 x d x {\\displaystyle \\lim _{x\\rightarrow 0}\\int _{x}^{5}2xdx}",
"title": "æŒç¿åé¡"
},
{
"paragraph_id": 129,
"tag": "p",
"text": "4) â« â 60 60 sin x + cos 2 x d x {\\displaystyle \\int _{-60}^{60}\\sin x+\\cos ^{2}xdx}",
"title": "æŒç¿åé¡"
},
{
"paragraph_id": 130,
"tag": "p",
"text": "(1) f ( x ) = x 3 {\\displaystyle f(x)=x^{3}} åªä¹ã®åŸ®å㯠y â² = n x n â 1 {\\displaystyle y'=nx^{n}-1} ã§ããããäžå®ç©åã®å®çŸ©ãã f ( x ) = x 3 {\\displaystyle f(x)=x^{3}} ã§ããã",
"title": "æŒç¿åé¡ã®è§£çãšãã®æåŒã"
}
] | ããã§ã¯åŸ®åç©åã®æŠå¿µã«ã€ããŠç解ããå€é
åŒé¢æ°ã®åŸ®åç©åãåŠã¶ããŸãã埮åã®å¿çšãå¿çšããŠæ¥ç·ã®æ¹çšåŒãã°ã©ãã®æŠåœ¢ãªã©ãæ±ããããç©åãå¿çšããŠã°ã©ãã®é¢ç©ãæ±ããã埮åç©åã¯ç©çåŠãå·¥åŠãªã©ããŸããŸãªåéã§å¿çšãããŠããã | {{Pathnav|é«çåŠæ ¡ã®åŠç¿|é«çåŠæ ¡æ°åŠ|é«çåŠæ ¡æ°åŠII|pagename=埮åã»ç©åã®èã|frame=1|small=1}}
ããã§ã¯åŸ®åç©åã®æŠå¿µã«ã€ããŠç解ããå€é
åŒé¢æ°ã®åŸ®åç©åãåŠã¶ããŸãã埮åã®å¿çšãå¿çšããŠæ¥ç·ã®æ¹çšåŒãã°ã©ãã®æŠåœ¢ãªã©ãæ±ããããç©åãå¿çšããŠã°ã©ãã®é¢ç©ãæ±ããã埮åç©åã¯ç©çåŠãå·¥åŠãªã©ããŸããŸãªåéã§å¿çšãããŠããã
== å¹³åå€åç ==
[[ãã¡ã€ã«:埮å.svg|ãµã ãã€ã«|å¹³åå€åçã®å³]]
äžåŠæ ¡ã§ã¯ãäžæ¬¡é¢æ°ãš<math>y=ax^2</math>ã®'''å€åã®å²å'''ãæ±ããã ãããããã§ã¯ãåããã®ã'''å¹³åå€åç'''ãšåŒã¶ããšã«ãããäžè¬ã®é¢æ° <math>y=f(x)</math> ã®å¹³åå€åçãèããŠã¿ãããäžåŠæ ¡ã§åŠç¿ããããšãšåæ§ã«èãããšã <math>y=f(x)</math> ã«ãããŠã <math>x</math> ã <math>a</math> ãã <math>b</math> ãŸã§å€åãããšãã®å¹³åå€åçã¯ãã <math>y</math> ã®å€åé/ <math>x</math> ã®å€åéãã§æ±ãããããã€ãŸãã <math>\frac{f(b)-f(a)}{b-a}</math> ã§ããã
'''äŸ'''
<math>y=x^2 + 2x + 1</math> ã«ãããŠã <math>x</math> ã-1ãã3ãŸã§å€åãããšãã®å¹³åå€åçãæ±ããã
<math>\frac{(3^2 + 2\cdot 3+1)-((-1)^2 + 2 \cdot (-1) + 1)}{3-(-1)} </math><math>=4</math>
== 極é ==
é¢æ° <math>f(x)</math> ã«ãããŠã <math>x</math> ã <math>a</math> ãšã¯ç°ãªãå€ããšããªããéããªã <math>a</math> ã«è¿ã¥ããšãã <math>f(x)</math> ãéããªã <math>A</math> ã«è¿ã¥ãããšãã <math>
\lim_{x\rightarrow a} f(x) = A
</math> ãšããã
==== äŸ ====
<math>
\lim_{x\rightarrow 0} 3x
</math>ãæ±ããã
<math>x</math>ãã<math>1,0.1,0.01,0.001,\cdots</math>ãšéããªã0ã«è¿ã¥ããŠã¿ãããããšã<math>3x</math>ã¯ã<math>3,0.3,0.03,0.003,\cdots</math>ãšãéããªã0ã«è¿ã¥ãããšããããã
ãã£ãŠã<math>x</math>ãéããªã0ã«è¿ã¥ãããšã<math>3x</math>ã¯éããªã0ã«è¿ã¥ãã®ã§ã<math>
\lim_{x\rightarrow 0} 3x = 0
</math>ã§ããã
次ã«ã
<math>
\lim_{x\rightarrow 1} \frac{x^2 -1 }{x-1}
</math>ãæ±ããã
<math>x</math>ãã<math>1.1,1.01,1.001,0.0001,1.00001,\cdots</math>ãšãéããªã1ã«è¿ã¥ããŠã¿ããšã<math>\frac{x^2 -1 }{x-1} </math>ã¯ã<math>2.1,2.01,2.001,2.0001,2.00001,\cdots</math>ãšãéããªã2ã«è¿ã¥ãã
ãªã®ã§ã<math>
\lim_{x\rightarrow 1} \frac{x^2 -1 }{x-1} = 2
</math>ã§ããã
ããã¯ãåŒã«å€ã代å
¥ããåã«ãåŒèªäœãçŽåããŠããŸã£ãæ¹ãç°¡åã«èšç®ã§ãããããªãã¡ã
<math>\frac{x^2 -1 }{x-1} = \frac{(x+1)(x-1)}{x-1}</math>ã§ããã<math>x</math>ã1ãšã¯ç°ãªãå€ãåããªããéããªã1ã«è¿ã¥ãããšã<math>x \neq 1</math>ãªã®ã§ãããã¯çŽåã§ãã<math>\frac{x^2 -1 }{x-1} = \frac{(x+1)(x-1)}{x-1} = x+1</math>ã§ããã
ãªã®ã§ã<math>
\lim_{x\rightarrow 1} \frac{x^2 -1 }{x-1}
</math>ãæ±ããã«ã¯ã<math>
\lim_{x\rightarrow 1} (x+1)
</math>ãæ±ããã°è¯ãã
<math>
\lim_{x\rightarrow 1} (x+1) = 2
</math>ã§ããã®ã§ã<math>
\lim_{x\rightarrow 1} \frac{x^2 -1 }{x-1} = 2
</math>ãšæ±ããããšãã§ããã
â»çºå±ãæåã®äŸã§ã¯ã<math>x</math>ãã<math>1,0.1,0.01,0.001,\cdots</math>ãšãéããªã0ã«è¿ã¥ãããã<math>2,0.2,0.02,0.002,\cdots</math>ãã<math>-1,-0.1,-0.01,-0.001,\cdots</math>ã®ããã«è¿ã¥ããŠã¿ãŠã<math>x</math>ã¯éããªã0ã«è¿ã¥ããä»ã«ãã<math>1,-0.1,0.01,-0.001,\cdots</math>ã<math>0.1,0.5,0.01,0.05,\cdots</math>ãªã©<math>x</math>ã0ã«è¿ã¥ãããæ¹æ³ã¯ãããã§ãèããããã
ãã¡ããããã®äŸã§ã¯ã<math>x</math>ãã©ã®ããã«è¿ã¥ãããšããŠã極éã®å€ã¯å€ãããªãã
ãããã<math>x</math>ãã<math>1,0.1,0.01,0.001,\cdots</math>ãšè¿ã¥ãããšãã<math>f(x)</math>ã¯<math>\alpha</math>ã«è¿ã¥ããã<math>x</math>ãã<math>2,0.2,0.02,0.002,\cdots</math>ãšè¿ã¥ãããã<math>f(x)</math>ã¯<math>\alpha</math>ã«è¿ã¥ããªãããããªé¢æ°<math>f(x)</math>ã ã£ãŠããã ããã
ãªã<math>x</math>ã<math>1,0.1,0.01,0.001,\cdots</math>ãšãè¿ã¥ããã ãã§ã極éã®å€ãæ±ããããšãåºæ¥ãã®ã?ãšçåã«æã人ãããããç¥ããªãã
極éãå³å¯ã«å®çŸ©ããã«ã¯ã[[解æåŠåºç€/極é#極éã®åœ¢åŒçãªå®çŸ©|ã€ãã·ãã³ãã«ã¿è«æ³]]ã䜿ãå¿
èŠããããããããé«æ ¡çã«ã¯å°ãé£ãããšèãã人ãå€ãã®ã§é«æ ¡ã§ã¯ããŸãæããããŠããªãã
ãªã®ã§ããã®æ¬ã§ã¯ãã€ãã·ãã³ãã«ã¿è«æ³ã䜿ãããææ§ãªæ¹æ³ã§æ¥µéãå®çŸ©ããããªã®ã§ãäžã®ãããªçåãæã£ã人ã¯ããã®çåã«ã€ããŠæ·±ãèããã«å
ã«é²ããã[[解æåŠåºç€/極é#極éã®åœ¢åŒçãªå®çŸ©|ã€ãã·ãã³ãã«ã¿è«æ³]]ãåŠã¶ãããŠã»ããã
[[ãã¡ã€ã«:å¹³åå€åç.svg|ãµã ãã€ã«|å¹³åå€åç]]
== 埮åä¿æ°ãšå°é¢æ° ==
[[ãã¡ã€ã«:Derivative GIF.gif|220x220px|hã0ã«è¿ã¥ãããšãã®ã¢ãã¡ãŒã·ã§ã³|ãµã ãã€ã«]]
é¢æ° <math>y = f(x)</math> ã®åŸãã«ã€ããŠèããŠã¿ããã
<math>x</math> ã <math>a</math> ãã <math>a + h</math> ãŸã§å€åãããšãã®å¹³åå€åçã¯
<math>\frac{f(a+h)-f(a)}{h}</math>
ã§ããããã®ãšãã <math>h</math> ãéããªã0ã«è¿ã¥ããã° <math>a</math> ã§ã®åŸããæ±ããããšãã§ãããã€ãŸããé¢æ° <math>y = f(x)</math> ã® <math>a</math> ã§ã®åŸãã¯
<math>\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}</math>
ã§äžãããããããã <math>x = a</math> ã«ããã'''埮åä¿æ°'''ãšããã
ãŸã
<math>f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}</math>
ã§äžããããé¢æ° <math>f'(x)</math> ãé¢æ° <math>f(x)</math> ã®'''å°é¢æ°'''ãšããã
é¢æ° <math>f(x)</math> ã®å°é¢æ°ã¯<math>\frac{df}{dx}</math>ãšè¡šãããããšãããã
ããã§ãããã€ãã®é¢æ°ã®å°é¢æ°ãæ±ããŠã¿ããã
*<math>f(x) = 1</math>
{|
|-
|<math>f'(x)</math>
|<math>= \lim _{h\rightarrow 0} \frac {f(x+h) - f(x)} h</math>
|-
|
|<math>= \lim _{h\rightarrow 0} \frac {1 - 1} h</math>
|-
|
|<math>= \lim _{h\rightarrow 0} 0</math>
|-
|
|<math>= 0</math>
|}
*<math>f(x) = x</math>
{|
|-
|<math>f'(x)</math>
|<math>= \lim _{h\rightarrow 0} \frac {f(x+h) - f(x)} h</math>
|-
|
|<math>= \lim _{h\rightarrow 0} \frac {x+h - x} h</math>
|-
|
|<math>= \lim _{h\rightarrow 0} 1</math>
|-
|
|<math>= 1</math>
|}
*<math>f(x) = x^2</math>
{|
|-
|<math>f'(x)</math>
|<math>= \lim _{h\rightarrow 0} \frac {f(x+h) - f(x)} h</math>
|-
|
|<math>= \lim _{h\rightarrow 0} \frac {(x+h)^2 - x^2} h</math>
|-
|
|<math>= \lim _{h\rightarrow 0} \frac{2hx + h^2} h</math>
|-
|
|<math>= \lim _{h\rightarrow 0} (2x + h)</math>
|-
|
|<math>= 2x </math>
|}
ã§ããã
<math>n</math> ãèªç¶æ°ãšãããé¢æ° <math>f(x) = x^n</math> ã®å°é¢æ°ã¯äºé
å®çãå¿çšã
{|
|-
|<math>f'(x)</math>
|<math>= \lim _{h\rightarrow 0} \frac {f(x+h) - f(x)} h</math>
|-
|
|<math>= \lim _{h\rightarrow 0} \frac {(x+h)^n - x^n} h</math>
|-
|
|<math>= \lim _{h\rightarrow 0} \frac{(x^n + _nC_1x^{n-1}h + _nC_2x^{n-2}h^2\cdots + h^n) - x^n} h</math>
|-
|
|<math>= \lim _{h\rightarrow 0} (_nC_1x^{n-1} + _nC_2x^{n-2}h + \cdots + h^{n-1})</math>
|-
|
|<math>= nx^{n-1} </math>
|}
ãšæ±ãããã
== åã»å·®åã³å®æ°åã®å°é¢æ° ==
é¢æ° <math>f(x), g(x)</math> ã«å¯Ÿã次ãæãç«ã€ã
# <math>\{f(x) \pm g(x)\}' = f'(x) \pm g'(x)</math> (è€å·åé )
# <math>\{ kf(x) \}' = kf'(x)</math>
'''蚌æ'''
# <math>\{f(x) \pm g(x)\}' = \lim_{h\to 0}\frac{f(x+h) \pm g(x+h)-\{f(x) \pm g(x)\}}{h} = \lim_{h\to 0}\{\frac{f(x+h) - f(x)}{h} \pm \frac{g(x+h) - g(x)}{h}\} = f'(x) \pm g'(x)</math>
# <math>\{ kf(x) \}' = \lim_{h\to 0}\frac{kf(x+h) - kf(x)}{h} = \lim_{h\to 0}k\frac{f(x+h) - f(x)}{h} = kf'(x)</math>
'''æŒç¿åé¡'''
次ã®é¢æ°ã埮åãã
1. <math>f(x)=2x^3+4x^2-5x-1</math><br>2. <math>f(x)=(2x+3)(3x-5)</math>
'''解ç'''
1.
:<math>\begin{align}
f'(x) & = (2x^3+4x^2-5x-1)' \\
& = 2(x^3)'+4(x^2)'-5(x)'-(1)' \\
& = 2 \times 3x^2 + 4 \times 2x -5 \times 1 - 0 \\
& = 6x^2+8x-5
\end{align}
</math>
2. <math>f(x)=6x^2-x-15</math> ã§ãããã
:<math>\begin{align}
f'(x) & = (6x^2-x-15)' \\
& = 6(x^2)'-(x)'-(15)' \\
& = 6 \times 2x - 1 - 0 \\
& = 12x-1
\end{align}
</math>
== å°é¢æ°ã®å¿çš ==
=== æ¥ç·ã®æ¹çšåŒ ===
æ²ç· <math>y = f(x)</math> äžã®ç¹ <math>(t, f(t))</math> ã«ãããæ¥ç·ã®æ¹çšåŒãæ±ããããã®æ¥ç·ã®åŸã㯠<math>f'(t)</math>ã§ãããç¹ <math>(t, f(t))</math> ãéãã®ã§ãæ¹çšåŒã¯ <math>y = f'(t)(x-t) + f(t)</math> ã§äžãããããå®éã<math>x = t </math> ãšãããš <math>y = f(t)</math> ãšãªãã®ã§ãã®æ¹çšåŒã¯ç¹ <math>(t, f(t))</math> ãéãããšããããã <math>x</math> ã®ä¿æ°ã¯ <math>f'(t)</math> ãªã®ã§åŸã㯠<math>f'(t)</math> ã§ããã
=== æ³ç·ã®æ¹çšåŒ ===
æ²ç· <math>y = f(x)</math> äžã®ç¹ <math>(t, f(t))</math> ã«ãããæ³ç·ã®æ¹çšåŒã¯ã<math> y = -\frac{1}{f'(t)}(x-t)+f(t) </math> ã§äžããããã
=== é¢æ°å€ã®å¢æž ===
f'(x)ã¯ãfã®åŸããè¡šããã®ã§ã <math>f'(x)>0</math> ã®ç¹ã§ã¯ãfã¯å¢å€§ãã <math>f'(x)<0</math> ã®ç¹ã§ã¯ãfã¯æžå°ããããšããããã
ãããããšã«é¢æ°ã®æŠåœ¢ãæãããšãã§ããã
'''äŸ'''
<math>y=x^3</math> ã®å¢æžã調ã¹ã
䞡蟺ã''x''ã§åŸ®åãããš
:<math>y'=3x^2</math>
:ãšãªããããã¯0ãé€ãåžžã«æ£ãªã®ã§ã <math>y=x^3</math> ã¯åžžã«å¢å ããããšããããã
=== é¢æ°ã®æ¥µå€§ã»æ¥µå° ===
<math>f(x)=x^3 - 3x</math>ã埮åãããš
:<math>f'(x)=3x^2 -3 =3(x+1)(x-1)</math>
å¢æžè¡šã¯æ¬¡ã®ããã«ãªãã
<table border="1" cellpadding="2">
<tr><th><center><math>x</math></center></th><th><center><math>\cdots</math></center> </th><th><center> <math>-1</math></center> </th><th><center><math>\cdots</math></center></th><th><center><math>1</math></center> </th><th><center><math>\cdots</math></center></th></tr>
<tr><th><center><math>f'(x)</math></center></th><td><center><math>+</math></center></td><td><center> <math>0</math> </center></td><th><center><math>-</math></center></th><td><center><math>0</math></center> </td><td><center>+</center></td></tr>
<tr><th><center><math>f(x)</math></center></th><td><center><math>\nearrow</math></center></td><td><center> <math>2</math> </center></td><th><center><math>\searrow </math></center></th><td><center><math>-2</math></center> </td><td><center><math>\nearrow</math></center></td></tr>
</table>
ãã®é¢æ°ã®ã°ã©ãã¯ã<math>x=-1</math>ãå¢ã«ããŠå¢å ããæžå°ã®ç¶æ
ã«å€ããã<math>x=1</math>ãå¢ã«ããŠæžå°ããå¢å ã®ç¶æ
ã«å€ããã<br>
ãã®ãšãã<math>f(x)</math>ã¯<math>x=-1</math>ã«ãããŠ'''極倧'''ïŒãããã ãïŒã«ãªããšããããã®ãšãã®<math>f(x)</math>ã®å€<math>f(-1)=2</math>ã'''極倧å€'''ïŒãããã ãã¡ïŒãšããããŸãã<math>x=1</math>ã«ãããŠ'''極å°'''ïŒããããããïŒã«ãªããšããããã®ãšãã®<math>f(x)</math>ã®å€<math>f(1)=-2</math>ã'''極å°å€'''ïŒããããããã¡ïŒãšããã極倧å€ãšæ¥µå°å€ãåãããŠ'''極å€'''ïŒãããã¡ïŒãšããã
== äžå®ç©å ==
'''äžå®ç©å'''(indefinite integral)ãšã¯ã埮åããããã®é¢æ°ã«ãªãé¢æ°ãæ±ããæäœã§ããã
ã€ãŸããé¢æ°<math>f(x)</math>ã«å¯ŸããŠã<math>F'(x)=f(x)</math>ãšãªããé¢æ°<math>F(x)</math>ãæ±ããæäœã§ããã
ãã®ãšã<math>F(x)</math>ãã<math>f(x)</math>ã®'''åå§é¢æ°'''(primitive function)ãšåŒã¶ã
äŸãã°ã<math>\frac{1}{2}x^2</math>ã¯åŸ®åãããšã<math>x</math>ã«ãªãã®ã§ã<math>\frac{1}{2}x^2</math>ã¯<math>x</math>ã®åå§é¢æ°ã§ããã
ãããã<math>\frac{1}{2}x^2+1</math>ãã<math>\frac{1}{2}x^2+3</math>ãªã©ã埮åãããš<math>x</math>ã«ãªãã®ã§ã<math>\frac{1}{2}x^2+1</math>ãã<math>\frac{1}{2}x^2+3</math>ã<math>x</math>ã®åå§é¢æ°ã§ããã
äžè¬ã«ã<math>\frac{1}{2}x^2 + C</math>(Cã¯ä»»æã®å®æ°)ã§è¡šãããé¢æ°ã¯ã<math>x</math>ã®åå§é¢æ°ã§ããã
<math>x</math>ã®åå§é¢æ°ã¯äžã€ã ãã§ã¯ãªããç¡æ°ã«ããã®ã ã
äžè¬ã«ãé¢æ° <math>f(x)</math> ã®åå§é¢æ°ã®'''äžã€'''ã <math>F(x)</math> ãšãããšããåå§é¢æ°ã«ä»»æã®å®æ°ã足ããé¢æ° <math>F(x) + C</math> ã <math>f(x)</math> ã®åå§é¢æ°ã«ãªãã
ãªããªãã<math>F(x)</math>ã<math>f(x)</math>ã®åå§é¢æ°ã§ãããã€ãŸãã<math>F'(x)=f(x)</math>ã®ãšãã<math>{(F(x) + C)}' = F'(x) + {(C)}' = F'(x) = f(x)</math>ãšãªãããã ã
ãŸããé¢æ° <math>f(x)</math> ã®åå§é¢æ°ã®äžã€ã <math>F(x)</math> ã§ãããšãããã¹ãŠã®é¢æ° <math>f(x)</math> ã®åå§é¢æ°ã¯ <math>F(x) + C</math> ã®åœ¢ã«æžããã
<math>F(x) + C</math> ã®åœ¢ã«æžããªãé¢æ° <math>G(x)</math>ãé¢æ° <math>f(x)</math> ã®åå§é¢æ°ã§ãããšä»®å®ããããã®ãšãã<math>h(x)=F(x)-G(x)</math>ãšãããšãé¢æ° <math>h(x)</math> ã¯å®æ°ã§ã¯ãªãã
ãã®ãšãã <math>h'(x)=\{F(x)-G(x)\}'=F'(x)-G'(x)=f(x)-f(x)=0</math> ã§ããã¯ãã ããé¢æ° <math>h(x)</math> ã¯å®æ°ã§ã¯ãªãã®ã§ <math>h'(x) = 0</math> ãšãªããªããããã¯ççŸãªã®ã§ããã¹ãŠã®é¢æ° <math>f(x)</math> ã®åå§é¢æ°ã¯ <math>F(x) + C</math>ã®åœ¢ã«æžããããšã蚌æã§ããã
é¢æ°<math>f(x)</math>ã®åå§é¢æ°ã®'''å
šäœ'''ãã<math>\int f(x)dx </math> ãšè¡šãããã®è¡šèšæ³ã¯æåã¯å¥åŠã«æãã ãããããã®ããã«è¡šèšããçç±ã¯åŸã«èª¬æããã®ã§ãä»ã¯ããã®ãŸãŸèŠããŠæ¬²ããã
ãŸãšãããšãé¢æ° <math>f(x)</math> ã®åå§é¢æ°ã®å
šäœ<math>\int f(x)dx </math>ã¯ã<math>f(x)</math>ã®åå§é¢æ°ã®äžã€ã <math>F(x)</math> ãšããŠããã®é¢æ°ã«ä»»æã®å®æ°ã足ããé¢æ°<math>F(x) + C</math>ã§è¡šããããã€ãŸãã
:<math>
\int f(x)dx = F(x)+ C
</math>
<math>C</math>ã¯ä»»æã®å®æ°ãšãããããã®ä»»æã®å®æ° <math>C</math> ã'''ç©åå®æ°'''(constant of integration)ãšåŒã¶ã
â»æ³šæã<math>\int f(x)dx </math>ã¯å®çŸ©ã«ãããããã«ã<math>f(x)</math>ã®åå§é¢æ°ã®'''å
šäœ'''ãè¡šããŠãããã€ãŸãã<math>f(x)</math>ã®åå§é¢æ°ã®äžã€ã<math>F(x)</math>ãšãããšãã<math>
\int f(x)dx = F(x)+ C
</math>ã®å³èŸº<math>F(x) + C</math>ã¯ã<math>F(x)</math>ã«å®æ°ã足ããé¢æ°ã®å
šäœãè¡šããŠãããã€ãŸãã<math>F(x) + C</math>ã¯ã<math>F(x)+1</math>ãã<math>F(x)-23</math>ãã<math>F(x)-5\pi</math>ãªã©ã®ã<math>F(x)</math>ã«å®æ°ã足ããé¢æ°ãã¹ãŠããŸãšããŠ<math>F(x)+C</math>ãšè¡šããŠããããã®ããšããããµãã«ãªã£ãŠãããšã'''é倧ãªééã'''ãèµ·ããå¯èœæ§ãããã®ã§ã泚æãå¿
èŠã§ããã
é¢æ° <math>f(x)=x^n</math> (ãã ã <math>n</math> ã¯èªç¶æ°)ã®äžå®ç©åãæ±ããŠã¿ãããã倩äžãçã ãã<math>F(x) = \frac{1}{n+1}x^{n+1}+C</math> (<math>C</math> ã¯ä»»æã®å®æ°)ãšãããšã <math>F'(x) = x^n</math> ãšãªãã®ã§ã <math>\frac{1}{n+1}x^{n+1}+C</math> ã¯åå§é¢æ°ã§ããããšããããã
ãããã£ãŠ <math>\int x^n dx =\frac{1}{n+1}x^{n+1}+C </math>
é¢æ° <math>f(x),g(x)</math> ã®åå§é¢æ°ãããããã <math>F(x),G(x)</math> ãšããã<math>a</math> ãä»»æã®å®æ°å®æ°ãšãããš
<math>\{F(x)+G(x)\}'=F'(x)+G'(x)=f(x)+g(x)</math>
<math>\{aF(x)\}' = aF'(x)=af(x)</math>
ãšãªãã®ã§ã
<math>\int \{ f(x) + g(x) \} dx = \int f(x) dx + \int g(x) dx</math>
<math>\int af(x) dx = a \int f(x) dx</math>
ãæãç«ã€ããšãåããã
'''æŒç¿åé¡'''
äžå®ç©å <math>\int (x^8+2x^2-6x+9)dx</math> ãæ±ãã
'''解ç'''
<math>\int (x^8+2x^2-6x+9)dx = \int x^8 \,dx + 2\int x^2\,dx -6\int x \,dx +9\int dx = \frac{x^9}{9}+\frac{2x^3}{3}-3x^2 + 9x + C</math> (<math>C</math> ã¯ç©åå®æ°)
== å®ç©å ==
é¢æ°<math>f(x)</math>ã®åå§é¢æ°ã®äžã€ã<math>F(x)</math>ãšããããã®åå§é¢æ°ã«å€ã代å
¥ããŠããã®å€ã®å·®ãæ±ããæäœãã'''å®ç©å'''ãšåŒã³ã<math>\int ^b_a f(x) dx</math>ãšæžããã€ãŸãã
:<math>
\int ^b_a f(x) dx = F(b) - F(a)
</math>
ã§ããã
<math>[f(x)]_a^b = f(b)-f(a)</math><ref><math>f(x)|_a^b</math> ã§è¡šãããæããã</ref>ãšããã
ãã®ããã«ãããšã<math>\int ^b_a f(x) dx =[F(x)]_a^b = F(b) - F(a)</math>ãšèšç®ã§ããã
å®ç©åã®å€ã¯åå§é¢æ°ã®éžæã«ãããªããå®éãåå§é¢æ°ãšããŠã <math>F(x)+C</math> ãéžã³ãå®ç©åãèšç®ãããšã<math>
\int ^b_a f(x) dx = (F(b)+C) - (F(a)+C) = F(b)-F(a)
</math>
ãšãªããåå§é¢æ°ãšããŠã©ããéžãã§ãå®ç©åã®å€ã¯äžå®ã§ããããšããããã<ref>ãªã®ã§ãå®éã«å®ç©åã®èšç®ãããå Žåãåå§é¢æ°ãšããŠå®æ°é
ã0ãšãªãé¢æ°ãéžãã æ¹ãèšç®ããããããªãã</ref>
é¢æ° <math>f(x),g(x)</math> ã«å¯ŸããŠãåå§é¢æ°ããããã <math>F(x),G(x)</math> ãšããã <math>k</math> ãå®æ°ãšããŠã
<math>\int_a^b kf(x)\,dx = kF(b)-kF(a)=k(F(b)-F(a)) = k\int_a^b f(x)\,dx </math>
<math>\int_a^b \{f(x)+g(x)\}dx=[F(x)+G(x)]_a^b = F(b)+G(b)-(F(a)+G(a))=F(b)-F(a)+G(b)-G(a) = \int_a^bf(x)\,dx+\int_a^bg(x)\,dx</math>
<math>\int_a^af(x)\,dx = F(a)-F(a)=0 </math>
<math>\int_b^a f(x)\,dx=F(a)-F(b)=-(F(b)-F(a))=-\int_a^bf(x)\,dx</math>
<math>\int_a^b f(x)\,dx =F(b)-F(a)=(F(b)-F(c))+(F(c)-F(a)) = \int_a^c f(x)\,dx + \int_c^b f(x) \, dx </math>
ãæãç«ã€ã
===== äŸ =====
<math>\int_2^5x^3dx</math>ãæ±ããã
<math>\frac{1}{4}x^4</math>ã¯ã埮åãããšã<math>x^3</math>ãªã®ã§ã<math>\frac{1}{4}x^4</math>ã¯<math>x^3</math>ã®åå§é¢æ°ã®äžã€ã§ããããã£ãŠ<math>\int_2^5x^3dx = \left[\frac{1}{4}x^4\right]_2^5 = \frac{1}{4}5^4 - \frac{1}{4}2^4 = \frac{609}{4}</math>ã§ããã
<math>\frac{1}{4}x^4+1</math>ãã埮åãããšã<math>x^3</math>ãªã®ã§ã<math>\frac{1}{4}x^4+1
</math>ã¯<math>x^3</math>ã®åå§é¢æ°ã®äžã€ã§ããããã£ãŠã<math>\int_2^5x^3dx = \left[\frac{1}{4}x^4+1\right]_2^5 = \left(\frac{1}{4}5^4 + 1\right) - \left(\frac{1}{4}2^4 + 1\right) = \frac{609}{4}</math>ãšæ±ããããšãã§ããã
== 埮åç©ååŠã®åºæ¬å®ç ==
aãå®æ°ãšãããšããå®ç©å<math> \int_a^x f(t)\,dt</math>ã¯xã®é¢æ°ã«ãªãã<br>
é¢æ°<math>f(t)</math>ã®åå§é¢æ°ã®äžã€ã<math>F(t)</math>ãšãããš
:<math>\int_a^x f(t)\,dt=F(x)-F(a)</math>
ãã®äž¡èŸºãxã§åŸ®åãããšã<math>F(a)</math>ã¯å®æ°ã§ãããã
:<math>\frac{d}{dx} \int_a^x f(t)\,dt=\frac{d}{dx} F(x) = f(x)</math><!-- ãã®åŸ®åç©ååŠã®åºæ¬å®çã¯ãç©åããé¢æ°ã埮åãããšå
ã®é¢æ°ã«æ»ãããšããããšã䞻匵ããŠãããã€ãŸãã埮åãšç©åã¯éã®æŒç®ã§ãããšããããšã§ãããããããæã
ã¯äžå®ç©åãã埮åãããå
ã®é¢æ°ã«ãªãé¢æ°ããšå®çŸ©ããŠããã®ã§ãã£ããå®çŸ©ãããã®å®çãæãç«ã€ã®ã¯åœç¶ã®ããã«æããåºæ¬å®çãªããŠä»°ã
ãããååã€ããããããšã«çåãæãã人ãããããç¥ããªããåŸè¿°ããããç©åã¯é¢ç©ãæ±ããããšãšå¯æ¥ãª -->
{| style="border:2px solid pink;width:80%" cellspacing="0"
| style="background:pink" |'''<math>\int_a^x f(t)\,dt</math>ã®å°é¢æ°'''
|-
| style="padding:5px" |
<center><math>\frac{d}{dx} \int_a^x f(t)\,dt= f(x)</math></center>
|}
== å®ç©åãšé¢ç© ==
é¢æ°<math>f(x)</math>ã<math>a \leqq x \leqq b</math>ã®ç¯å²ã§åžžã«æ£ã§ãããšããããã®ãšããå®ç©å<math>\int _a^b f(x) dx</math>ã«ãã£ãŠãé¢æ°<math>f(x)</math>ã®ã°ã©ããšãçŽç·<math>x=a</math>ãçŽç·<math>x=b</math>ã<math>x</math>軞ã§å²ãŸããéšåã®é¢ç©ãæ±ããããšãã§ããã<!-- å³ -->
é¢æ°<math>f(x)</math>ã®ã°ã©ããšãçŽç·<math>x=a</math>ãçŽç·<math>x=c</math>ãšã<math>x</math>軞ã§å²ãŸããéšåã®é¢ç©ã<math>S(c)</math>ãšããããšã«ãã£ãŠãé¢æ°<math>S(x)</math>ãå®ããã(<math>a \leqq x \leqq b</math>ãšãã)
é¢æ°<math>f(x)</math>ã®ã°ã©ããšãçŽç·<math>x=c</math>ãçŽç·<math>x=c+h</math>ãšã<math>x</math>軞ã§å²ãŸããéšåã®é¢ç©ãèãã(<math>a \leqq c+h \leqq b</math>ãšãã)ãããã¯ã<math>S(c+h)-S(c)</math>ã§ãããããã§ã<math>c<t<c+h</math>ãªã<math>t</math>ããšã£ãŠããŠããã®ç¹ã«ããã<math>f(x)</math>ã®å€<math>f(t)</math>ãé«ããšããé·æ¹åœ¢ã®é¢ç©ãèããããšã§ã<math>t</math>ãäžæã«ãšãã°ã<math> S(c+h) - S(c)=h \cdot f(t) </math>ãšã§ããã䞡蟺ã<math>h</math>ã§å²ãã<math>h \to 0</math>ã®æ¥µéãèãããšã
:<math>\lim_{h \to 0} \frac{S(c+h) - S(c)}{h} =\lim_{h \to 0} f(t)</math>
ã§ãããã巊蟺ã¯åŸ®åã®å®çŸ©ãã<math>S'(c)</math>ã§ããã<math>\lim_{h \to 0} t=c</math>ã§ããããšã«æ³šæãããšå³èŸºã¯<math>f(c)</math>ã§ãããæåã<math>c</math>ãã<math>x</math>ã«åãæãããšãçµå±
:<math>S'(x)=f(x)</math>
ãåŸããããã€ãŸãã<math>S(x)</math>ã¯<math>f(x)</math>ã®åå§é¢æ°ã®äžã€ã§ããããšãåããã
ãã£ãŠã<math>\int _a^b f(x) dx = S(b) - S(a)</math>ã§ãããããã®åŒã®å³èŸºã¯ãé¢æ°<math>f(x)</math>ã®ã°ã©ããšãçŽç·<math>x=a</math>ãçŽç·<math>x=b</math>ãšã<math>x</math>軞ã§å²ãŸããé¢ç©ã§ããããã£ãŠã巊蟺<math>\int _a^b f(x) dx</math>ã¯ãé¢æ°<math>f(x)</math>ã®ã°ã©ããšãçŽç·<math>x=a</math>ãçŽç·<math>x=b</math>ãšã<math>x</math>軞ã§å²ãŸããé¢ç©ãè¡šããŠããã
{| style="border:2px solid pink;width:80%" cellspacing="0"
| style="background:pink" |'''å®ç©åãšé¢ç©ã®é¢ä¿'''
|-
| style="padding:5px" |
<math>a \le x \le b</math>ãã§ãã<math>f(x) \ge 0</math>ãã®ãšããçŽç· <math>x=a,x=b</math> ãš <math>x</math> 軞ã <math>f(x)</math> ã§å²ãŸããé¢ç© <math>S</math> ã¯
<center><math>S= \int_a^b f(x)\,dx</math></center>
|}
æŽå²çã«ã¯ãç©åã¯ãé¢æ°ã®ã°ã©ãã§å²ãŸããéšåã®é¢ç©ãæ±ããããã«èãåºãããããã®ç¯ã§è¿°ã¹ããããªåŸ®åãšã®é¢é£ã¯ç©åèªäœã®çºæãããã£ãšåŸã«ãªã£ãŠçºèŠãããããšã§ããã
äŸãšããŠã
<math>0 \leqq x \leqq 1</math>ã®ç¯å²ã§ãy = xã®ã°ã©ããšx軞ã§ã¯ããŸããéšåã®é¢ç©ããç©åãçšããŠèšç®ããã
(
å®éã«ã¯ããã¯äžè§åœ¢ãªã®ã§ãç©åãçšããªããŠãé¢ç©ãèšç®ããããšãåºæ¥ãã
çã¯<math> \frac 1 2</math> ãšãªãã
)
å®ç©åãè¡ãªããšã
<math>
\int_0^1 x dx
</math>
<math>
= \frac 1 2 [x^2]^1_0
</math>
<math>
= \frac 1 2 [1^2 - 0^2]
</math>
<math>
= \frac 1 2 [1 - 0]
</math>
<math>
= \frac 1 2
</math>
ãšãªã確ãã«äžèŽããã
'''æŒç¿åé¡'''
æŸç©ç·<math>y=5-x^2</math>ãšx軞ããã³2çŽç·<math>x=-1\ ,\ x=2</math>ã§å²ãŸããéšåã®é¢ç©Sãæ±ããã
'''解ç'''
ãã®æŸç©ç·ã¯<math>-1 \le x \le 2</math>ã§x軞ã®äžåŽã«ããããã
:<math>S= \int_{-1}^{2} (5-x^2)\,dx=\left[5x - \frac{x^3}{3} \right]^{2}_{-1} =12</math><br>
<br>
<math>a \le x \le b</math>ãã«ãããŠãåžžã«ã<math>f(x) \ge g(x)</math>ãã§ãããšãã2ã€ã®æ²ç·ã<math>y=f(x)\ ,\ y=g(x)</math>ãã«æãŸããéšåã®é¢ç©Sã¯ã次ã®åŒã§è¡šãããã
{| style="border:2px solid pink;width:80%" cellspacing="0"
| style="background:pink" |æ²ç· <math>y=f(x),y=g(x)</math> ã®éã®é¢ç©
|-
| style="padding:5px" |
<math>a \le x \le b</math>ãã§ãã<math>f(x) \ge g(x)</math>ãã®ãšãã
<center><math>S= \int_a^b \left\{ f(x)-g(x) \right\}\,dx</math></center>
|}
* åé¡äŸ
** åé¡
æŸç©ç·<math>y=x^2 -1</math>ãšçŽç·<math>y=x+1</math>ã«ãã£ãŠå²ãŸããéšåã®é¢ç©Sãæ±ããã
** 解ç
æŸç©ç·ãšçŽç·ã®äº€ç¹ã®x座æšã¯
:<math>x^2 -1=x+1</math>
:<math>x^2 -x-2=0</math>
:<math>x=-1\ ,\ x=2</math>
<math>-1 \le x \le 2</math>ã®ç¯å²ã§<math>x^2 -1 \le x+1</math>ãã
:<math>S= \int_{-1}^{2} \left\{ (x+1)-(x^2 -1) \right\}\,dx= \int_{-1}^{2} (-x^2+x+2)\,dx=\left[- \frac{x^3}{3} + \frac{x^2}{2} +2x \right]^{2}_{-1} = \frac{9}{2}</math><br>
<br>
<br>
<math>a \le x \le b</math>ãã§ãã<math>f(x) \le 0</math>ãã®ãšããx軞<math>y=0</math>ãšæ²ç·<math>y=f(x)</math>ã«ãã£ãŠæãŸããŠãããšèããããã®ã§ã
:<math>S= \int_a^b \left\{ 0-f(x) \right\}\,dx = - \int_a^b f(x)\,dx</math>
ãšãªãã
{| style="border:2px solid pink;width:80%" cellspacing=0
|style="background:pink"|'''é¢ç©(3)'''
|-
|style="padding:5px"|
<math>a \le x \le b</math>ãã§ãã<math>f(x) \le 0</math>ãã®ãšãã
<center><math>S=- \int_a^b f(x)\,dx</math></center>
|}
* åé¡äŸ
** åé¡
æŸç©ç·<math>y=x^2 -2x</math>ãšx軞ã§å²ãŸããéšåã®é¢ç©Sãæ±ããã
** 解ç
æŸç©ç·ãšx軞ã®äº€ç¹ã®x座æšã¯
:<math>x^2 -2x=0</math>
:<math>x=0\ ,\ x=2</math>
ãã®æŸç©ç·ã¯<math>0 \le x \le 2</math>ã§x軞ã®äžåŽã«ããããã
:<math>S=- \int_0^2 (x^2 -2x)\,dx=- \left[\frac{x^3}{3} -x^2 \right]^{2}_{0} = \frac{4}{3}</math>
{{ã³ã©ã |ç©çåŠãšåŸ®åç©å|
[[File:GodfreyKneller-IsaacNewton-1689.jpg|thumb|ãã¥ãŒãã³]]
埮åç©åã¯ãç©çåŠã§ãéåæ¹çšåŒã®èšç®ãªã©ã«å¿çšãããŠããã
1600幎代ããã¥ãŒãã³ãªã©ã®ç 究ã«ãããéåã®æ³åã埮åç©åã䜿ã£ãåŒã§è¡šçŸã§ããããšã解æãããã
ãªãããã¥ãŒãã³ã¯èæžãšããŠãããªã³ããã¢ããããããããã®èæžã§ãã¥ãŒãã³ã¯éåã®æ³åã埮åç©åã§è¡šãããããšãè¿°ã¹ãååŠïŒããããïŒã®çè«ãé²æ©ãããã
ãªãã埮åç©åãç 究ããåæ代ã®æ°åŠè
ã«ã¯ããã¥ãŒãã³ã®ä»ã«ãã©ã€ãããããããã
}}
==æ¬åœã«ã¡ãã£ãšããäœè«==
é«æ ¡æ°åŠãããŠãããšãå°æ¥åŸ®åãšãç©åãšãäœã«äœ¿ãïŒããšæã人ã®æ¹ãå€ããšæãã確ãã«æ¥åžžç掻ã§ã¯ãç©åãªã©ã®é«åºŠãª
æ°åŠã¯äœ¿ããªããã ããã®äžæ¹è£ã§ã¯ç©åã
埮åãé«æ ¡æ°åŠã§ã¯åãŸããªããããªæ°åŠã䜿ãããŠãããäŸãã°å°é¢šã®é²è·¯äºæ³ã
ããã¯ç©åã䜿ãå°é¢šã®é²è·¯ãäºæž¬ããŠãããä»ã«ãã»ãã¥ãªãã£ã®åŒ·åãªã©ã«ãæ°åŠã¯äœ¿ãããŠãããæ¥åžžç掻ã§ã¯æ°åŠã¯äœ¿ããªãããæ°åŠã«èŠªãã¿ãæã£ãŠã¿ãŠã¯ã©ãã ãããã
== æŒç¿åé¡ ==
(1)<math>F(x)=2x^2</math>ã®ãšã
<math>f(x)</math>ãæ±ããããã ã<math>F'(x)</math>
(2)<math>\lim_{x\rightarrow c}x^2+x=11</math>ãšãªã<math>c</math>ãæ±ãã
(3)åå§é¢æ°ãå®ç©åãæ±ãã
1)<math>\int ^5_3 2x^9+(6x-2x^3)dx</math>
2)<math>\int \sin x+\tan xdx</math>
3)<math>\lim_{x\rightarrow0}\int ^5_x 2xdx</math>
4)<math>\int ^{60}_{-60} \sin x+\cos^2xdx</math>
==æŒç¿åé¡ã®è§£çãšãã®æåŒã==
(1)<math>f(x)=x^3</math>
åªä¹ã®åŸ®åã¯<math>y'=nx^n-1</math>
ã§ããããäžå®ç©åã®å®çŸ©ãã<math>f(x)=x^3</math>ã§ããã
== è泚 ==
<references/>
{{DEFAULTSORT:ãããšããã€ããããããII ã²ãµããããµã}}
[[Category:é«çåŠæ ¡æ°åŠII|ã²ãµããããµã]]
[[ã«ããŽãª:埮åç©ååŠ]] | 2005-05-06T11:56:28Z | 2023-11-09T05:59:48Z | [
"ãã³ãã¬ãŒã:Pathnav",
"ãã³ãã¬ãŒã:ã³ã©ã "
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E6%95%B0%E5%AD%A6II/%E5%BE%AE%E5%88%86%E3%83%BB%E7%A9%8D%E5%88%86%E3%81%AE%E8%80%83%E3%81%88 |
1,925 | èçœè³ª | èçœè³ª
çç©ã¯çŽ°èå
ã现èéã®æ§ã
ãªæ©èœã«æ¯ããããŠãã,ãã®æ©èœã¯èçœè³ªãªãããŠã¯ãªãããªã. ããã§ã¯,èçœè³ªã®æ§é ãšãã®æ©èœã«ã€ããŠåºæ¬çãªäºé
ã解説ãã.
èçœè³ªãšã¯,ã¢ããé
žãã¢ãããŒãšããããªããŒ(ããªãããã)ã®ãã¡,æ©èœãæã€ãã®ã®ããšã§ãã. | [
{
"paragraph_id": 0,
"tag": "p",
"text": "èçœè³ª",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "çç©ã¯çŽ°èå
ã现èéã®æ§ã
ãªæ©èœã«æ¯ããããŠãã,ãã®æ©èœã¯èçœè³ªãªãããŠã¯ãªãããªã. ããã§ã¯,èçœè³ªã®æ§é ãšãã®æ©èœã«ã€ããŠåºæ¬çãªäºé
ã解説ãã.",
"title": "åºè«"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "èçœè³ªãšã¯,ã¢ããé
žãã¢ãããŒãšããããªããŒ(ããªãããã)ã®ãã¡,æ©èœãæã€ãã®ã®ããšã§ãã.",
"title": "èçœè³ªãšã¯"
}
] | èçœè³ª | '''èçœè³ª'''
==åºè«==
çç©ã¯çŽ°èå
ã现èéã®æ§ã
ãªæ©èœã«æ¯ããããŠããïŒãã®æ©èœã¯èçœè³ªãªãããŠã¯ãªãããªãïŒ
ããã§ã¯ïŒèçœè³ªã®æ§é ãšãã®æ©èœã«ã€ããŠåºæ¬çãªäºé
ã解説ããïŒ
==èçœè³ªãšã¯==
èçœè³ªãšã¯ïŒã¢ããé
žãã¢ãããŒãšããããªããŒïŒããªããããïŒã®ãã¡ïŒæ©èœãæã€ãã®ã®ããšã§ããïŒ
[[Category:é«çåŠæ ¡æè²|çããã¯ããã€]]
[[Category:çç©åŠ|é«ããã¯ããã€]] | null | 2007-01-20T16:23:15Z | [] | https://ja.wikibooks.org/wiki/%E8%9B%8B%E7%99%BD%E8%B3%AA |
1,927 | HTML/ãªã¹ã | ç®æ¡æžããæžãããå Žåãªã©ãäžèšã®ç¯ã®ããã«ãªã¹ã(List)ãå®çŸ©ããŸãã
ãªã¹ãã«ã¯ãé äžåãªã¹ãããåºåãªã¹ãããšãå®çŸ©ãªã¹ããããããŸãã ãã®ãã¡ãé äžåãªã¹ãããšãåºåãªã¹ããã®ããŒã¯ã¢ããã¯å
±éããŠããŸãããå®çŸ©ãªã¹ããã®ããŒã¯ã¢ããã¯2ã€ãšã¯ç°ãªããŸãã
ãé äžåãªã¹ãããšãåºåãªã¹ããã§ã¯
ã®ããã«ãå
éšãLIèŠçŽ ãçšããŠåèŠçŽ ãæå®ãå€åŽã®ãªã¹ãèŠçŽ (ULèŠçŽ ãªã©)ã§è¡šç€ºæ¹æ³ãæå®ããŸãã ãªããå
éšã®LIèŠçŽ ã®åé ã®ãlãã¯ãå°æåã®Lã§ãã®ã§æ³šæããŠãã ããã å
éšã®LIèŠçŽ ã«ãã£ãŠèšè¿°ãããéšåã¯ããŠã§ããã©ãŠã¶ã®ãŠãŒã¶ãŒãšãŒãžã§ã³ãã»ã¹ã¿ã€ã«ã·ãŒãã§ã¯è¡é ã«ã€ã³ãã³ãããšããåé
ç®ã¯åŒ·å¶çã«æ¹è¡ãããŸãã ãŸããè¡é ã«ã¯é»äžžãæ°åã衚瀺ããããã®ããããŸãããŠã§ããã©ãŠã¶ã®çš®é¡ããŠã§ãããŒãžåŽã®èšå®ã«ãã£ãŠé»äžžãæ°å以å€ã®ãã®ã衚瀺ãããå ŽåããããŸãã
ULèŠçŽ ã¯ã¢ã€ãã ã®ãªã¹ããè¡šããŸãããã¢ã€ãã ã®é åºã¯éèŠã§ã¯ãããŸããã ULèŠçŽ ã®å
容ã«ã¯LIèŠçŽ ããèš±ãããŸããã
ãªã¹ãã®äžã«ãªã¹ããå
¥ããããšãã§ããŸãã
ãã®éãå
åŽã®ãªã¹ãã¯ã€ã³ãã³ããããäºã«æ³šæããŠãã ããã
衚瀺çµæã®äŸ
æ®éã®å®å¹ç°å¢ã§ã¯ãã€ã³ãã³ããé©çšãããã®ã¯ãªã¹ãé
ç®ã ãã§ãªããå
åŽã®ulã¿ã°ã§å²ãŸããéšåå
šäœã«ãªããŸãã ããšãã°ã
ãœãŒã¹äŸ
ãšãããšã
ãªã¹ãäžã§ãªã¹ãé
ç®å€ã«ããã¹ããæžããŠããç¹ã«è¿œå ã®ã€ã³ãã³ããªã©ã¯ç¡ãã®ã§ãé
ç®ã®åèãªã©ãæžãããå Žåã«ã¯ããã®ãŸãŸã§ã¯äžäŸ¿ã§ãã
ããã§ãäžèšã®ããã« div ã¿ã°ã«ããã¹ã¿ã€ã«æå®ãªã©ãçšããŠãã€ã³ãã³ããåºæ¥ãŸãã(ãªã¹ãã«éãããäžè¬çã«HTMLã§ã€ã³ãã³ããããå Žåã®ææ³ã§ãã)
ãœãŒã¹äŸ
CSSã®list-style-typeããããã£ãULèŠçŽ ã«é©çšãããšããªã¹ãå
šäœã®ãªã¹ãããŒã¯ã®çš®é¡ãå€æŽããããšãåºæ¥ãŸãã CSSã®list-style-typeããããã£ãULèŠçŽ ã®åèŠçŽ ã®LIèŠçŽ ã«é©çšãããšãåã
ã®ãªã¹ãèŠçŽ ã®ãªã¹ãããŒã¯ã®çš®é¡ãå€æŽããããšãåºæ¥ãŸãã ãªã¹ãããŒã¯ã®ãã¶ã€ã³ã䜿ãããªã¹ãããŒã¯ã®çš®é¡ã¯ãŠã§ããã©ãŠã¶ã«ãã£ãŠç°ãªãå ŽåããããŸãã
OLèŠçŽ ã¯ãã¢ã€ãã ã®ãªã¹ããè¡šããã¢ã€ãã ãæå³çã«é åºä»ããããŠããé åºãå€æŽãããšããã¥ã¡ã³ãã®æå³ãå€ãããããªã±ãŒã¹ã«çšããããŸãã äžè¬çãªãŠã§ããã©ãŠã¶ã§ã¯1, 2, 3, ... ã A, B, C, ... ãšã¬ã³ããªã³ã°ãããäºãå€ããåã
ã®é
ç®ã¯ULèŠçŽ ã®æãšåæ§LIèŠçŽ ãçšããŸãã
CSSã®list-style-typeããããã£ãOLèŠçŽ ã«é©çšãããšãªã¹ãå
šäœã®åæè¡šçŸãå€æŽããããšãåºæ¥ãŸãã CSSã®list-style-typeããããã£ãOLèŠçŽ ã®åèŠçŽ ã®LIèŠçŽ ã«é©çšãããšãåã
ã®ãªã¹ãèŠçŽ ã®åæè¡šçŸãå€æŽããããšãåºæ¥ãŸãã åæè¡šçŸã®ãã¶ã€ã³ã䜿ããåæè¡šçŸã¯ãŠã§ããã©ãŠã¶ã«ãã£ãŠç°ãªãå ŽåããããŸãã
OLèŠçŽ ã«valueå±æ§ãæå®ãããšéå§çªå·ãå€æŽå¯èœã§ãããäŸãã°valueå±æ§ã«5ãæå®ãããšLIèŠçŽ ã«ã¯äžããé çªã«5, 6, 7, ...ãšããçªå·ãæ¯ãããããŸããåå¥ã®liå±æ§ã«startå±æ§ãå€æŽããããšã§ãªã¹ãã®éäžããéå§çªå·ãå€æŽããããšãåºæ¥ããäŸãã°ãªã¹ãäžã®3çªç®ã«ãããªã¹ãã«9ãšããvalueå±æ§å€ãä»äžããå Žåããã®ãªã¹ãã¯äžçªç®ã®é
ç®ãã9, 10, 11, ...ãšããçªå·ãæ¯ãããããã«ãªãã
çšèªã®å®çŸ©ã®ãããªååãšèª¬æã察ã«ãªã£ããªã¹ãã«ã¯DLèŠçŽ ãçšããŸãã DTèŠçŽ ã¯ãULèŠçŽ ãOLèŠçŽ ãšéããLIèŠçŽ ã§ã¯ãªãDTèŠçŽ ãšDDèŠçŽ ãšDIVèŠçŽ ããæ§æãããŸãã DTèŠçŽ ã¯å®çŸ©ãããçšèª(åå)ã瀺ããDDèŠçŽ ã¯çšèªã®èª¬æã瀺ããŸãã DTèŠçŽ ã«ã¯DTèŠçŽ ãšDDèŠçŽ ãšDIVèŠçŽ ã®ã¿ãå«ãããšãåºæ¥ãŸãã DTèŠçŽ ã¯ã€ã³ã©ã€ã³èŠçŽ ãå«ãããšãã§ããŸãã DDèŠçŽ ã¯ãããã¯ã¬ãã«èŠçŽ ãå«ãããšãåºæ¥ãŸãã
次ã®äŸã¯ããŠã£ãããã¯ã¹ã®å§åŠ¹ãããžã§ã¯ãã説æããŠããŸãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ç®æ¡æžããæžãããå Žåãªã©ãäžèšã®ç¯ã®ããã«ãªã¹ã(List)ãå®çŸ©ããŸãã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ãªã¹ãã«ã¯ãé äžåãªã¹ãããåºåãªã¹ãããšãå®çŸ©ãªã¹ããããããŸãã ãã®ãã¡ãé äžåãªã¹ãããšãåºåãªã¹ããã®ããŒã¯ã¢ããã¯å
±éããŠããŸãããå®çŸ©ãªã¹ããã®ããŒã¯ã¢ããã¯2ã€ãšã¯ç°ãªããŸãã",
"title": "æŠèŠ"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ãé äžåãªã¹ãããšãåºåãªã¹ããã§ã¯",
"title": "æŠèŠ"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ã®ããã«ãå
éšãLIèŠçŽ ãçšããŠåèŠçŽ ãæå®ãå€åŽã®ãªã¹ãèŠçŽ (ULèŠçŽ ãªã©)ã§è¡šç€ºæ¹æ³ãæå®ããŸãã ãªããå
éšã®LIèŠçŽ ã®åé ã®ãlãã¯ãå°æåã®Lã§ãã®ã§æ³šæããŠãã ããã å
éšã®LIèŠçŽ ã«ãã£ãŠèšè¿°ãããéšåã¯ããŠã§ããã©ãŠã¶ã®ãŠãŒã¶ãŒãšãŒãžã§ã³ãã»ã¹ã¿ã€ã«ã·ãŒãã§ã¯è¡é ã«ã€ã³ãã³ãããšããåé
ç®ã¯åŒ·å¶çã«æ¹è¡ãããŸãã ãŸããè¡é ã«ã¯é»äžžãæ°åã衚瀺ããããã®ããããŸãããŠã§ããã©ãŠã¶ã®çš®é¡ããŠã§ãããŒãžåŽã®èšå®ã«ãã£ãŠé»äžžãæ°å以å€ã®ãã®ã衚瀺ãããå ŽåããããŸãã",
"title": "æŠèŠ"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ULèŠçŽ ã¯ã¢ã€ãã ã®ãªã¹ããè¡šããŸãããã¢ã€ãã ã®é åºã¯éèŠã§ã¯ãããŸããã ULèŠçŽ ã®å
容ã«ã¯LIèŠçŽ ããèš±ãããŸããã",
"title": "é äžåãªã¹ã"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ãªã¹ãã®äžã«ãªã¹ããå
¥ããããšãã§ããŸãã",
"title": "é äžåãªã¹ã"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ãã®éãå
åŽã®ãªã¹ãã¯ã€ã³ãã³ããããäºã«æ³šæããŠãã ããã",
"title": "é äžåãªã¹ã"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "衚瀺çµæã®äŸ",
"title": "é äžåãªã¹ã"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "æ®éã®å®å¹ç°å¢ã§ã¯ãã€ã³ãã³ããé©çšãããã®ã¯ãªã¹ãé
ç®ã ãã§ãªããå
åŽã®ulã¿ã°ã§å²ãŸããéšåå
šäœã«ãªããŸãã ããšãã°ã",
"title": "é äžåãªã¹ã"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "ãœãŒã¹äŸ",
"title": "é äžåãªã¹ã"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "ãšãããšã",
"title": "é äžåãªã¹ã"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "ãªã¹ãäžã§ãªã¹ãé
ç®å€ã«ããã¹ããæžããŠããç¹ã«è¿œå ã®ã€ã³ãã³ããªã©ã¯ç¡ãã®ã§ãé
ç®ã®åèãªã©ãæžãããå Žåã«ã¯ããã®ãŸãŸã§ã¯äžäŸ¿ã§ãã",
"title": "é äžåãªã¹ã"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "ããã§ãäžèšã®ããã« div ã¿ã°ã«ããã¹ã¿ã€ã«æå®ãªã©ãçšããŠãã€ã³ãã³ããåºæ¥ãŸãã(ãªã¹ãã«éãããäžè¬çã«HTMLã§ã€ã³ãã³ããããå Žåã®ææ³ã§ãã)",
"title": "é äžåãªã¹ã"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ãœãŒã¹äŸ",
"title": "é äžåãªã¹ã"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "",
"title": "é äžåãªã¹ã"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "CSSã®list-style-typeããããã£ãULèŠçŽ ã«é©çšãããšããªã¹ãå
šäœã®ãªã¹ãããŒã¯ã®çš®é¡ãå€æŽããããšãåºæ¥ãŸãã CSSã®list-style-typeããããã£ãULèŠçŽ ã®åèŠçŽ ã®LIèŠçŽ ã«é©çšãããšãåã
ã®ãªã¹ãèŠçŽ ã®ãªã¹ãããŒã¯ã®çš®é¡ãå€æŽããããšãåºæ¥ãŸãã ãªã¹ãããŒã¯ã®ãã¶ã€ã³ã䜿ãããªã¹ãããŒã¯ã®çš®é¡ã¯ãŠã§ããã©ãŠã¶ã«ãã£ãŠç°ãªãå ŽåããããŸãã",
"title": "é äžåãªã¹ã"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "OLèŠçŽ ã¯ãã¢ã€ãã ã®ãªã¹ããè¡šããã¢ã€ãã ãæå³çã«é åºä»ããããŠããé åºãå€æŽãããšããã¥ã¡ã³ãã®æå³ãå€ãããããªã±ãŒã¹ã«çšããããŸãã äžè¬çãªãŠã§ããã©ãŠã¶ã§ã¯1, 2, 3, ... ã A, B, C, ... ãšã¬ã³ããªã³ã°ãããäºãå€ããåã
ã®é
ç®ã¯ULèŠçŽ ã®æãšåæ§LIèŠçŽ ãçšããŸãã",
"title": "åºåãªã¹ã"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "CSSã®list-style-typeããããã£ãOLèŠçŽ ã«é©çšãããšãªã¹ãå
šäœã®åæè¡šçŸãå€æŽããããšãåºæ¥ãŸãã CSSã®list-style-typeããããã£ãOLèŠçŽ ã®åèŠçŽ ã®LIèŠçŽ ã«é©çšãããšãåã
ã®ãªã¹ãèŠçŽ ã®åæè¡šçŸãå€æŽããããšãåºæ¥ãŸãã åæè¡šçŸã®ãã¶ã€ã³ã䜿ããåæè¡šçŸã¯ãŠã§ããã©ãŠã¶ã«ãã£ãŠç°ãªãå ŽåããããŸãã",
"title": "åºåãªã¹ã"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "OLèŠçŽ ã«valueå±æ§ãæå®ãããšéå§çªå·ãå€æŽå¯èœã§ãããäŸãã°valueå±æ§ã«5ãæå®ãããšLIèŠçŽ ã«ã¯äžããé çªã«5, 6, 7, ...ãšããçªå·ãæ¯ãããããŸããåå¥ã®liå±æ§ã«startå±æ§ãå€æŽããããšã§ãªã¹ãã®éäžããéå§çªå·ãå€æŽããããšãåºæ¥ããäŸãã°ãªã¹ãäžã®3çªç®ã«ãããªã¹ãã«9ãšããvalueå±æ§å€ãä»äžããå Žåããã®ãªã¹ãã¯äžçªç®ã®é
ç®ãã9, 10, 11, ...ãšããçªå·ãæ¯ãããããã«ãªãã",
"title": "åºåãªã¹ã"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "çšèªã®å®çŸ©ã®ãããªååãšèª¬æã察ã«ãªã£ããªã¹ãã«ã¯DLèŠçŽ ãçšããŸãã DTèŠçŽ ã¯ãULèŠçŽ ãOLèŠçŽ ãšéããLIèŠçŽ ã§ã¯ãªãDTèŠçŽ ãšDDèŠçŽ ãšDIVèŠçŽ ããæ§æãããŸãã DTèŠçŽ ã¯å®çŸ©ãããçšèª(åå)ã瀺ããDDèŠçŽ ã¯çšèªã®èª¬æã瀺ããŸãã DTèŠçŽ ã«ã¯DTèŠçŽ ãšDDèŠçŽ ãšDIVèŠçŽ ã®ã¿ãå«ãããšãåºæ¥ãŸãã DTèŠçŽ ã¯ã€ã³ã©ã€ã³èŠçŽ ãå«ãããšãã§ããŸãã DDèŠçŽ ã¯ãããã¯ã¬ãã«èŠçŽ ãå«ãããšãåºæ¥ãŸãã",
"title": "å®çŸ©ãªã¹ã"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "次ã®äŸã¯ããŠã£ãããã¯ã¹ã®å§åŠ¹ãããžã§ã¯ãã説æããŠããŸãã",
"title": "å®çŸ©ãªã¹ã"
}
] | ç®æ¡æžããæžãããå Žåãªã©ãäžèšã®ç¯ã®ããã«ãªã¹ã(List)ãå®çŸ©ããŸãã | {{Pathnav|HTML|frame=1|small=1}}
ç®æ¡æžããæžãããå Žåãªã©ãäžèšã®ç¯ã®ããã«ãªã¹ã(List)ãå®çŸ©ããŸã<ref>HTML5ã«ã¯ List ãšããåé¡ã¯ãªãHTML4ãŸã§ List ãšãããèŠçŽ ã¯ãPèŠçŽ ãMAINèŠçŽ ãªã©ãšãšãã« [https://html.spec.whatwg.org/multipage/grouping-content.html#grouping-content §4.4 Grouping content]ã«åé¡ãããŸããã</ref>ã
== æŠèŠ ==
ãªã¹ãã«ã¯ãé äžåãªã¹ãããåºåãªã¹ãããšãå®çŸ©ãªã¹ããããããŸãã
ãã®ãã¡ãé äžåãªã¹ãããšãåºåãªã¹ããã®ããŒã¯ã¢ããã¯å
±éããŠããŸãããå®çŸ©ãªã¹ããã®ããŒã¯ã¢ããã¯2ã€ãšã¯ç°ãªããŸãã
ãé äžåãªã¹ãããšãåºåãªã¹ããã§ã¯
<pre>
<ãªã¹ãèŠçŽ >
<li>ãã¯ã</li>
<li>å³åæ±</li>
<li>çŒãé</li>
</ãªã¹ãèŠçŽ >
</pre>
ã®ããã«ãå
éšãLIèŠçŽ ãçšããŠåèŠçŽ ãæå®ãå€åŽã®ãªã¹ãèŠçŽ (ULèŠçŽ ãªã©)ã§è¡šç€ºæ¹æ³ãæå®ããŸãã
ãªããå
éšã®LIèŠçŽ ã®åé ã®ãlãã¯ãå°æåã®Lã§ãã®ã§æ³šæããŠãã ãã<ref>å
é ãæ°åã§å§ãŸãèŠçŽ ã¯ãããŸããã</ref>ã
å
éšã®LIèŠçŽ ã«ãã£ãŠèšè¿°ãããéšåã¯ããŠã§ããã©ãŠã¶ã®ãŠãŒã¶ãŒãšãŒãžã§ã³ãã»ã¹ã¿ã€ã«ã·ãŒãã§ã¯è¡é ã«ã€ã³ãã³ãããšããåé
ç®ã¯åŒ·å¶çã«æ¹è¡ãããŸãã
ãŸããè¡é ã«ã¯é»äžžãæ°åã衚瀺ããããã®ããããŸãããŠã§ããã©ãŠã¶ã®çš®é¡ããŠã§ãããŒãžåŽã®èšå®ã«ãã£ãŠé»äžžãæ°å以å€ã®ãã®ã衚瀺ãããå ŽåããããŸãã
== é äžåãªã¹ã ==
ULèŠçŽ ã¯ã¢ã€ãã ã®ãªã¹ããè¡šããŸãããã¢ã€ãã ã®é åºã¯éèŠã§ã¯ãããŸããã
ULèŠçŽ ã®å
容ã«ã¯LIèŠçŽ ããèš±ãããŸããã
=== å
¥åäŸ ===
<syntaxhighlight lang="html5">
<ul>
<li>ãã¯ã</li>
<li>å³åæ±</li>
<li>çŒãé</li>
</ul>
</syntaxhighlight>
=== è¡šç€ºäŸ ===
<ul>
<li>ãã¯ã</li>
<li>å³åæ±</li>
<li>çŒãé</li>
</ul>
=== ãªã¹ãã®å
¥ãå ===
ãªã¹ãã®äžã«ãªã¹ããå
¥ããããšãã§ããŸãã
ãã®éãå
åŽã®ãªã¹ãã¯ã€ã³ãã³ããããäºã«æ³šæããŠãã ããã
<syntaxhighlight lang="html5">
<ul>
<li>ãã¯ã</li>
<li>å³åæ±</li>
<ul>
<li>èµ€å³å</li>
<li>çœå³å</li>
</ul>
<li>çŒãé</li>
</ul>
</syntaxhighlight>
衚瀺çµæã®äŸ
<ul>
<li>ãã¯ã</li>
<li>å³åæ±</li>
<ul>
<li>èµ€å³å</li>
<li>çœå³å</li>
</ul>
<li>çŒãé</li>
</ul>
æ®éã®å®å¹ç°å¢ã§ã¯ãã€ã³ãã³ããé©çšãããã®ã¯ãªã¹ãé
ç®ã ãã§ãªããå
åŽã®ulã¿ã°ã§å²ãŸããéšåå
šäœã«ãªããŸãã
ããšãã°ã
ãœãŒã¹äŸ
<syntaxhighlight lang="html5">
<ul>
<li>ãã¯ã</li>
<li>å³åæ±</li>
<ul>
<li>èµ€å³å</li>ããã<br>ããããã<br>ãã<br>ãããã
<li>çœå³å</li>
</ul>
<li>çŒãé</li>
</ul>
</syntaxhighlight>
ãšãããšã
;å®è¡çµæ
<ul>
<li>ãã¯ã</li>
<li>å³åæ±</li>
<ul>
<li>èµ€å³å</li>ããã<br>ããããã<br>ãã<br>ãããã
<li>çœå³å</li>
</ul>
<li>çŒãé</li>
</ul>
=== ãªã¹ãäžã§ã®è¿œèšãªã©ã®ã€ã³ãã³ã ===
ãªã¹ãäžã§ãªã¹ãé
ç®å€ã«ããã¹ããæžããŠããç¹ã«è¿œå ã®ã€ã³ãã³ããªã©ã¯ç¡ãã®ã§ãé
ç®ã®åèãªã©ãæžãããå Žåã«ã¯ããã®ãŸãŸã§ã¯äžäŸ¿ã§ãã
ããã§ãäžèšã®ããã« div ã¿ã°ã«ããã¹ã¿ã€ã«æå®ãªã©ãçšããŠãã€ã³ãã³ããåºæ¥ãŸããïŒãªã¹ãã«éãããäžè¬çã«HTMLã§ã€ã³ãã³ããããå Žåã®ææ³ã§ããïŒ
ãœãŒã¹äŸ
<syntaxhighlight lang="html5">
<ul>
<li>ãã¯ã</li>
<li>å³åæ±</li>
<ul>
<li>èµ€å³å</li><div style="margin-left: 1em;">倧è±ãå€ã</div>
<li>çœå³å</li><div style="margin-left: 1em;">ç±³ãå€ã</div>
</ul>
<li>çŒãé</li>
</ul>
</syntaxhighlight>
;å®è¡çµæ
<ul>
<li>ãã¯ã</li>
<li>å³åæ±</li>
<ul>
<li>èµ€å³å</li><div style="margin-left: 1em;">倧è±ãå€ã</div>
<li>çœå³å</li><div style="margin-left: 1em;">ç±³ãå€ã</div>
</ul>
<li>çŒãé</li>
</ul>
=== 詳现èšå® ===
==== ãªã¹ãããŒã¯ã®çš®é¡ãå€ãã ====
[[CSS]]ã®list-style-typeããããã£ãULèŠçŽ ã«é©çšãããšããªã¹ãå
šäœã®ãªã¹ãããŒã¯ã®çš®é¡ãå€æŽããããšãåºæ¥ãŸãã
CSSã®list-style-typeããããã£ãULèŠçŽ ã®åèŠçŽ ã®LIèŠçŽ ã«é©çšãããšãåã
ã®ãªã¹ãèŠçŽ ã®ãªã¹ãããŒã¯ã®çš®é¡ãå€æŽããããšãåºæ¥ãŸãã
ãªã¹ãããŒã¯ã®ãã¶ã€ã³ã䜿ãããªã¹ãããŒã¯ã®çš®é¡ã¯ãŠã§ããã©ãŠã¶ã«ãã£ãŠç°ãªãå ŽåããããŸãã
; list-style-type: disc
: é»äžž
; list-style-type: circle
: çœäžž
; list-style-type: square
: åè§å
== åºåãªã¹ã ==
OLèŠçŽ ã¯ãã¢ã€ãã ã®ãªã¹ããè¡šããã¢ã€ãã ãæå³çã«é åºä»ããããŠããé åºãå€æŽãããšããã¥ã¡ã³ãã®æå³ãå€ãããããªã±ãŒã¹ã«çšããããŸãã
äžè¬çãªãŠã§ããã©ãŠã¶ã§ã¯1, 2, 3, ... ã A, B, C, ... ãšã¬ã³ããªã³ã°ãããäºãå€ããåã
ã®é
ç®ã¯ULèŠçŽ ã®æãšåæ§LIèŠçŽ ãçšããŸãã
=== å
¥åäŸ ===
<syntaxhighlight lang="html5">
<ol>
<li>ââé§
ã§é»è»ã«ä¹ã</li>
<li>ÃÃé§
ã§ä¹ãæãã</li>
<li>â³â³é§
ã§éãã</li>
</ol>
</syntaxhighlight>
=== è¡šç€ºäŸ ===
<ol>
<li>ââé§
ã§é»è»ã«ä¹ã</li>
<li>ÃÃé§
ã§ä¹ãæãã</li>
<li>â³â³é§
ã§éãã</li>
</ol>
=== 詳现èšå® ===
==== çªå·ã®çš®é¡ãå€ãã ====
[[CSS]]ã®list-style-typeããããã£ãOLèŠçŽ ã«é©çšãããšãªã¹ãå
šäœã®åæè¡šçŸãå€æŽããããšãåºæ¥ãŸãã
CSSã®list-style-typeããããã£ãOLèŠçŽ ã®åèŠçŽ ã®LIèŠçŽ ã«é©çšãããšãåã
ã®ãªã¹ãèŠçŽ ã®åæè¡šçŸãå€æŽããããšãåºæ¥ãŸãã
åæè¡šçŸã®ãã¶ã€ã³ã䜿ããåæè¡šçŸã¯ãŠã§ããã©ãŠã¶ã«ãã£ãŠç°ãªãå ŽåããããŸãã
; list-style-type<nowiki>:</nowiki> decimal
: <ol style="list-style-type:decimal"><li>ç®çšæ°å<li>ç®çšæ°å<li>ç®çšæ°å</ol>
; list-style-type<nowiki>:</nowiki> lower-latin
: <ol style="list-style-type:lower-latin"><li>ã¢ã«ãã¡ãããå°æå<li>ã¢ã«ãã¡ãããå°æå<li>ã¢ã«ãã¡ãããå°æå</ol>
; list-style-type<nowiki>:</nowiki> upper-latin
: <ol style="list-style-type:upper-latin"><li>ã¢ã«ãã¡ããã倧æå<li>ã¢ã«ãã¡ããã倧æå<li>ã¢ã«ãã¡ããã倧æå</ol>
; list-style-type<nowiki>:</nowiki> lower-roman
: <ol style="list-style-type:lower-roman"><li>ããŒãæ°åå°æå<li>ããŒãæ°åå°æå<li>ããŒãæ°åå°æå</ol>
; list-style-type<nowiki>:</nowiki> upper-roman
: <ol style="list-style-type:upper-roman"><li>ããŒãæ°å倧æå<li>ããŒãæ°å倧æå<li>ããŒãæ°å倧æå</ol>
; list-style-type<nowiki>:</nowiki> lower-greek
: <ol style="list-style-type:lower-greek"><li>ã®ãªã·ã£æåå°æå<li>ã®ãªã·ã£æåå°æå<li>ã®ãªã·ã£æåå°æå</ol>
; list-style-type<nowiki>:</nowiki> upper-greek
: <ol style="list-style-type:upper-greek"><li>ã®ãªã·ã£æå倧æå<li>ã®ãªã·ã£æå倧æå<li>ã®ãªã·ã£æå倧æå</ol>
; list-style-type<nowiki>:</nowiki> cjk-decimal
: <ol style="list-style-type:cjk-decimal"><li>挢æ°å<li>挢æ°å<li>挢æ°å</ol>
; list-style-type<nowiki>:</nowiki> katakana-iroha
: <ol style="list-style-type:katakana-iroha"><li>çä»®åã€ãã<li>çä»®åã€ãã<li>çä»®åã€ãã</ol>
; list-style-type<nowiki>:</nowiki> cjk-earthly-branch
: <ol style="list-style-type:cjk-earthly-branch"><li>åäºæ¯<li>åäºæ¯<li>åäºæ¯</ol>
; list-style-type<nowiki>:</nowiki> cjk-heavenly-stem
: <ol style="list-style-type:cjk-heavenly-stem"><li>åå¹²<li>åå¹²<li>åå¹²</ol>
; list-style-type<nowiki>:</nowiki> thai
: <ol style="list-style-type:thai"><li>ã¿ã€æå<li>ã¿ã€æå<li>ã¿ã€æå</ol>
==== æ°åã®é çªãå€ãã ====
OLèŠçŽ ã«valueå±æ§ãæå®ãããšéå§çªå·ãå€æŽå¯èœã§ãããäŸãã°valueå±æ§ã«5ãæå®ãããšLIèŠçŽ ã«ã¯äžããé çªã«5, 6, 7, ...ãšããçªå·ãæ¯ãããããŸããåå¥ã®liå±æ§ã«startå±æ§ãå€æŽããããšã§ãªã¹ãã®éäžããéå§çªå·ãå€æŽããããšãåºæ¥ããäŸãã°ãªã¹ãäžã®3çªç®ã«ãããªã¹ãã«9ãšããvalueå±æ§å€ãä»äžããå Žåããã®ãªã¹ãã¯äžçªç®ã®é
ç®ãã9, 10, 11, ...ãšããçªå·ãæ¯ãããããã«ãªãã
=== å
¥åäŸ ===
<syntaxhighlight lang="html5">
<ol>
<li>ââé§
ã§é»è»ã«ä¹ã</li>
<li value="5">ÃÃé§
ã§ä¹ãæãã</li>
<li>â³â³é§
ã§éãã</li>
</ol>
</syntaxhighlight>
=== è¡šç€ºäŸ ===
<ol>
<li>ââé§
ã§é»è»ã«ä¹ã</li>
<li value="5">ÃÃé§
ã§ä¹ãæãã</li>
<li>â³â³é§
ã§éãã</li>
</ol>
== å®çŸ©ãªã¹ã ==
çšèªã®å®çŸ©ã®ãããªååãšèª¬æã察ã«ãªã£ããªã¹ãã«ã¯DLèŠçŽ ãçšããŸãã
DTèŠçŽ ã¯ãULèŠçŽ ãOLèŠçŽ ãšéããLIèŠçŽ ã§ã¯ãªãDTèŠçŽ ãšDDèŠçŽ ãš<ins>DIVèŠçŽ </ins><ref name="dl_w_div" />ããæ§æãããŸãã
DTèŠçŽ ã¯å®çŸ©ãããçšèªïŒååïŒã瀺ããDDèŠçŽ ã¯çšèªã®èª¬æã瀺ããŸãã
DTèŠçŽ ã«ã¯DTèŠçŽ ãšDDèŠçŽ <ins>ãšDIVèŠçŽ </ins><ref name="dl_w_div">HTML5ã§ã¯DLèŠçŽ ã®çŽäžã®åèŠçŽ ã«DIVèŠçŽ ãèš±ãããããã«ãªããŸããã</ref>ã®ã¿ãå«ãããšãåºæ¥ãŸãã
DTèŠçŽ ã¯ã€ã³ã©ã€ã³èŠçŽ ãå«ãããšãã§ããŸãã
DDèŠçŽ ã¯ãããã¯ã¬ãã«èŠçŽ ãå«ãããšãåºæ¥ãŸãã
次ã®äŸã¯ããŠã£ãããã¯ã¹ã®å§åŠ¹ãããžã§ã¯ãã説æããŠããŸãã
=== èšè¿°äŸ ===
<syntaxhighlight lang="html5">
<dl>
<dt>ã¡ã¿ãŠã£ã</dt>
<dd>å
šãããžã§ã¯ãã®è£å©çãããžã§ã¯ãã§ãã</dd>
<dt>ãŠã£ãããã£ã¢</dt>
<dd>çŸç§äºå
žãäœæãããããžã§ã¯ãã§ãã</dd>
<dt>ãŠã£ã¯ã·ã§ããªãŒ</dt>
<dd>èŸæžã»ã·ãœãŒã©ã¹äœæãããžã§ã¯ãã§ãã</dd>
</dl>
</syntaxhighlight>
=== è¡šç€ºäŸ ===
<dl>
<dt>ã¡ã¿ãŠã£ã</dt>
<dd>å
šãããžã§ã¯ãã®è£å©çãããžã§ã¯ãã§ãã</dd>
<dt>ãŠã£ãããã£ã¢</dt>
<dd>çŸç§äºå
žãäœæãããããžã§ã¯ãã§ãã</dd>
<dt>ãŠã£ã¯ã·ã§ããªãŒ</dt>
<dd>èŸæžã»ã·ãœãŒã©ã¹äœæãããžã§ã¯ãã§ãã</dd>
</dl>
== è泚 ==
<references />
== å€éšãªã³ã¯ ==
* [https://html.spec.whatwg.org/multipage/grouping-content.html#the-ol-element HTML Living Standard::§4.4.5 The ol element]
* [https://html.spec.whatwg.org/multipage/grouping-content.html#the-ul-element HTML Living Standard::§4.4.6 The ul element]
* [https://html.spec.whatwg.org/multipage/grouping-content.html#the-li-element HTML Living Standard::§4.4.8 The ul element]
* [https://html.spec.whatwg.org/multipage/grouping-content.html#the-dl-element HTML Living Standard::§4.4.9 The dl element]
* [https://html.spec.whatwg.org/multipage/grouping-content.html#the-dt-element HTML Living Standard::§4.4.10 The dt element]
* [https://html.spec.whatwg.org/multipage/grouping-content.html#the-dd-element HTML Living Standard::§4.4.11 The dd element]
* [https://drafts.csswg.org/css-lists-3/#text-markers CSS Lists and Counters Module Level 3::§3.4. Text-based Markers: the list-style-type property]
[[Category:HTML|HTML ãããš]] | 2005-05-07T10:39:22Z | 2023-07-25T11:40:40Z | [
"ãã³ãã¬ãŒã:Pathnav"
] | https://ja.wikibooks.org/wiki/HTML/%E3%83%AA%E3%82%B9%E3%83%88 |
1,928 | æ§èª²çš(-2012幎床)é«çåŠæ ¡æ°åŠA | å¹³æ15幎(2003幎)ããå¹³æ23幎(2011幎)ãŸã§ã®éã«é«çåŠæ ¡ã«å
¥åŠãã人ã®å±¥ä¿®ããç§ç®ãæ°åŠAãã¯ä»¥äžã®åå
ãããªã£ãŠããŸãã
å¹³æ24幎(2012幎)ãã什å3幎(2021幎)ãŸã§ã®éã«é«çåŠæ ¡ã«å
¥åŠãã人ã®å±¥ä¿®ããç§ç®ãæ°åŠAãã¯ã以äžã®åå
ãããªã£ãŠããŸãã
什å4幎(2022幎)以éã«é«çåŠæ ¡ã«å
¥åŠãã人ã¯ã以äžã®åå
ãããªããæ°åŠAãã履修ããŸãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "å¹³æ15幎(2003幎)ããå¹³æ23幎(2011幎)ãŸã§ã®éã«é«çåŠæ ¡ã«å
¥åŠãã人ã®å±¥ä¿®ããç§ç®ãæ°åŠAãã¯ä»¥äžã®åå
ãããªã£ãŠããŸãã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "å¹³æ24幎(2012幎)ãã什å3幎(2021幎)ãŸã§ã®éã«é«çåŠæ ¡ã«å
¥åŠãã人ã®å±¥ä¿®ããç§ç®ãæ°åŠAãã¯ã以äžã®åå
ãããªã£ãŠããŸãã",
"title": ""
},
{
"paragraph_id": 2,
"tag": "p",
"text": "什å4幎(2022幎)以éã«é«çåŠæ ¡ã«å
¥åŠãã人ã¯ã以äžã®åå
ãããªããæ°åŠAãã履修ããŸãã",
"title": ""
}
] | å¹³æ15幎(2003幎)ããå¹³æ23幎(2011幎)ãŸã§ã®éã«é«çåŠæ ¡ã«å
¥åŠãã人ã®å±¥ä¿®ããç§ç®ãæ°åŠAãã¯ä»¥äžã®åå
ãããªã£ãŠããŸãã åæ°ã®åŠç
確ç
åœé¡ãšèšŒæ
å¹³é¢å³åœ¢ å¹³æ24幎(2012幎)ãã什å3幎(2021幎ïŒãŸã§ã®éã«é«çåŠæ ¡ã«å
¥åŠãã人ã®å±¥ä¿®ããç§ç®ãæ°åŠAãã¯ã以äžã®åå
ãããªã£ãŠããŸãã å Žåã®æ°ãšç¢ºç
æŽæ°ã®æ§è³ª
å³åœ¢ã®æ§è³ª 什å4幎(2022幎)以éã«é«çåŠæ ¡ã«å
¥åŠãã人ã¯ã以äžã®åå
ãããªããæ°åŠAãã履修ããŸãã å³åœ¢ã®æ§è³ª
å Žåã®æ°ãšç¢ºç
æ°åŠãšäººéã®æŽ»å | {{pathnav|é«çåŠæ ¡ã®åŠç¿|é«çåŠæ ¡æ°åŠ|frame=1}}
å¹³æ15幎(2003幎)ããå¹³æ23幎(2011幎)ãŸã§ã®éã«é«çåŠæ ¡ã«å
¥åŠãã人ã®å±¥ä¿®ããç§ç®ãæ°åŠAãã¯ä»¥äžã®åå
ãããªã£ãŠããŸãã
*åæ°ã®åŠç
*確ç
*[[é«çåŠæ ¡æ°åŠA/éåãšè«ç|åœé¡ãšèšŒæ]]
*å¹³é¢å³åœ¢
å¹³æ24幎(2012幎)ãã什å3幎(2021幎ïŒãŸã§ã®éã«é«çåŠæ ¡ã«å
¥åŠãã人ã®å±¥ä¿®ããç§ç®ãæ°åŠAãã¯ã以äžã®åå
ãããªã£ãŠããŸãã
* [[é«çåŠæ ¡æ°åŠA/å Žåã®æ°ãšç¢ºç|å Žåã®æ°ãšç¢ºç]]
* [[é«çåŠæ ¡æ°åŠA/æŽæ°ã®æ§è³ª|æŽæ°ã®æ§è³ª]]
* [[é«çåŠæ ¡æ°åŠA/å³åœ¢ã®æ§è³ª|å³åœ¢ã®æ§è³ª]]
什å4幎(2022幎)以éã«é«çåŠæ ¡ã«å
¥åŠãã人ã¯ã以äžã®åå
ãããªããæ°åŠAãã履修ããŸãã
* [[é«çåŠæ ¡æ°åŠA/å³åœ¢ã®æ§è³ª|å³åœ¢ã®æ§è³ª]]
* [[é«çåŠæ ¡æ°åŠA/å Žåã®æ°ãšç¢ºç|å Žåã®æ°ãšç¢ºç]]
* [[é«çåŠæ ¡æ°åŠA/æ°åŠãšäººéã®æŽ»å|æ°åŠãšäººéã®æŽ»å]]
{{DEFAULTSORT:æ§1 ãããšããã€ããããããA}}
[[Category:æ°åŠ]]
[[Category:æ°åŠæè²]]
[[Category:åŠæ ¡æè²]]
[[Category:æ®éæè²]]
[[Category:åŸæäžçæè²]]
[[Category:é«çåŠæ ¡æè²]]
[[Category:é«çåŠæ ¡æ°åŠA|*]] | 2005-05-08T02:33:21Z | 2024-03-19T14:00:46Z | [
"ãã³ãã¬ãŒã:Pathnav"
] | https://ja.wikibooks.org/wiki/%E6%97%A7%E8%AA%B2%E7%A8%8B(-2012%E5%B9%B4%E5%BA%A6)%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E6%95%B0%E5%AD%A6A |
1,929 | æ§èª²çš(-2012幎床)é«çåŠæ ¡æ°åŠA/éåãšè«ç | äžåŠã§ã¯ãããšãã°ãèªç¶æ°ã®ãã€ãŸãããšãã9以äžã®èªç¶æ°ã®ãã€ãŸãããšããè² ã®æŽæ°ã®ãã€ãŸããã®ãããªãã®ããéå(ãã
ããã)ãšèªãã§ããã
ã§ã¯ãæ°åŠã§ãããéåããšã¯äœããããããèããŠãããã
æ°åŠã§ã¯ãããéãŸãã®ãã¡ãããã«ãããã«å±ããŠãããå±ããŠãªãããæ確ã«åºå¥ã§ããæ¡ä»¶ã®ããç©ã®ãã€ãŸããéå(ãã
ããããè±:set)ãšãããäŸãã°ããèªç¶æ°ãã¯ãn > 0ãšãªãæŽæ°n ã®å
šäœããšããåºå¥å¯èœãªæ¡ä»¶ãããã®ã§éåãšãããã
ãããã倧ããªæ°ããšãããã€ãŸãã¯ãã©ããããã倧ããªãæ°ãšãããã®ããã¯ã£ããããªããããæ°åŠã®ãéåãã§ã¯ãªãã
ãã ããã倧ããªæ°ããäŸãã°ã1å以äžã®æŽæ°ããšåºå¥ã§ããããã«å®çŸ©ããã°éåã«ãªãããã
ããŠãæ°åŠçãªãéåããæ§æãããã®äžã€äžã€ã®ããšãããã®éåã® èŠçŽ ( ããããè±:element)ãšããã
ããšãã°ããèªç¶æ°ã®éåãã®èŠçŽ ãªããèªç¶æ°1ãèªç¶æ°2ãèªç¶æ°3ãã»ã»ã»ãªã©ã®ã²ãšã€ã²ãšã€ã®èªç¶æ°ãããããèŠçŽ ã§ããã
ã 1 ã¯èªç¶æ°ã®éåã®èŠçŽ ã§ããããšãããã
ã 27 ã¯èªç¶æ°ã®éåã®èŠçŽ ã§ããããšãããã
(â» ç¯å²å€? )ãªããæ°åŠçã«ã¯ãåºå¥ãã¯ã£ãããããããã°ãäŸãã°ãâ³â³é«æ ¡ã®ä»ã®3幎Bçµã®çåŸå
šå¡ãçãéåãšããŠèããããšãã§ãããããªãããããéåããšã¯ãèªç¶æ°ãããæŽæ°ããªã©ã®æ°ã§ãªããŠãããã
(ç¯å²å€)æ°ãšæ°ãšã®å¯Ÿå¿é¢ä¿ã§ãããé¢æ°ãããéåã®åèŠçŽ ãšéåã®åèŠçŽ ãšã®å¯Ÿå¿é¢ä¿ãžãšæ¡åŒµããããšãã§ããã(ãã®éåã¯æ°ã®éåã§ãªããŠãè¯ãã)ãã®ãããªå¯Ÿå¿é¢ä¿ãååãšåŒã¶ã詳ããã¯å€§åŠã®ãéåè«ãã§æ±ããããå
šå°ãããåå°ããªã©ãç¥ã£ãŠãããšèšŒæã«äŸ¿å©ãªç¥èãããã (ç¯å²å€ãããŸã§)
aãéåAã®èŠçŽ ã§ãããšããããã®ãšããaã¯éåAã«å±ãã(ãããã)ãšãããèšå·ã§ã
ãšè¡šãã
bãAã®èŠçŽ ã§ãªããšãã¯ã
ãšè¡šãã
éåããããããšããäž»ã«2çš®é¡ã®æ¹æ³ãããã(äŸã¯ã10以äžã®èªç¶æ°ã®ãã¡å¶æ°ã§ãããã®ãã®éåãè¡šãã)
ã§ãã
ããšãã°ãã10以äžã®èªç¶æ°ã®ãã¡å¶æ°ã§ãããã®ãã®éåãè¡šãå Žåã(1) ã®æ¹æ³(èŠçŽ ãæžã䞊ã¹ãæ¹æ³)ã§ã¯ã
ãšãªãã
äžæ¹ã(2)ã®æ¹æ³(èŠçŽ ã®æºããæ¡ä»¶ãè¿°ã¹ãæ¹æ³)ã§ã¯ã
ãªã©ã®ããã«ãªã(äœéãããã)ã
100以äžã®èªç¶æ°ã®éå A ããããã»ã©ã®(1)ãèŠçŽ ãæžã䞊ã¹ãæ¹æ³ãã®æ¹æ³ã§æžãå Žåã
ãšãªãããŸãããã®èšæ³ã®ãã»ã»ã»ãã®ããã«ãèŠçŽ ã®åæ°ããšãŠãå€ãå Žåãç¡æ°ã«ããå Žåã«ã¯ã{ }èšå·å
ã®èŠçŽ ã®éäžããã»ã»ã»ããŸãã¯ã......ããã...ããªã©ã®ç¹ã
ã§çç¥ããŠããã
100以äžã®å¶æ°ã®éå B ã¯ããã®èšæ³(èŠçŽ ãæžã䞊ã¹ãæ¹æ³)ã§ã¯ã
ã®ããã«ãªãã
æ£ã®å¶æ°å
šäœã®éåã®èŠçŽ ã¯(1)ãèŠçŽ ãæžã䞊ã¹ãæ¹æ³ãã®æ¹æ³ã§æžãå Žåã
ã®ããã«ãæžããã
2ã€ã®éåA,BããããxâA ãªãã° xâBãæãç«ã€ãšããAã¯Bã® éšåéå (ã¶ã¶ããã
ããããè±:subset)ã§ãããšããããBã¯Aãå«ããããAã¯Bã«å«ãŸããããšããããã®ç¶æ
ãèšå·ã§
ãŸãã¯
ã§è¡šãã
è£è¶³
Aã®éšåéåã«ã¯Aèªèº«ãããã(ã€ãŸã A â A ã§ãã)ã
ãŸããA,B ã®éåã®èŠçŽ ãåããšãã
ã§è¡šãã
éå A = {1, 2, 3} ãš éå B = {1, 2 , 3 , 4, 5} ããããšããA 㯠Bã®éšåéåã§ããã
2ã€ã®éåA,Bããããšã ãããã®äž¡æ¹ã®èŠçŽ ã§ãããã®ã®éåã AãšBã® å
±ééšå(ãããã€ãã¶ã¶ã)ãšåŒã³ã
ãšæžãã
ãŸããéåA,Bã®å°ãªããšãã©ã¡ããäžæ¹ã«ã¯å±ããŠããèŠçŽ ãããªãéåã®ããšããAãšBã®åéå(ããã
ããããè±:union)ãšåŒã³ã
ãšæžãã
3ã€ã®éå A, B, C ã«ã€ããŠã¯ã3ã€ã®ã©ãã«ãå±ããèŠçŽ å
šäœã®éåã A,B,C ã®å
±ééšåãšåŒã³ã A â© B â© C ã§è¡šãã
ãŸããéå A, B, C ã®å°ãªããšã1ã€ã«å±ããèŠçŽ ã®éåã A,B,C ã®åéåãšåŒã³ã A ⪠B ⪠C ã§è¡šãã
ããšãã°ãã10以äžã®èªç¶æ°ã®ãã¡ã®å¶æ°ãã®éåAãšãã10以äžã®èªç¶æ°ã®ãã¡ã®å¥æ°ãã®éåBã«ã€ããŠãéåAãšéåBã®å
±ééšåã«ã¯ãäœãèŠçŽ ãç¡ãã
ãã®äŸã®ããã«ããèŠçŽ ããªã«ããªãããšããå Žåãããã®ã§ãæ°åŠã§ã¯ãèŠçŽ ããªã«ããªããå Žåãã²ãšã€ã®éåãšããŠèããã
èŠçŽ ããããªãéåã®ããšã 空éå(ãããã
ããããè±:empty set ããã㯠null set)ãšãããèšå·ã¯
ã§ããããã
ã®ãªã·ã£æåã®ãã¡ã€(Ï, Ï {\displaystyle \phi \ } )ã§è¡šãããããšãå€ãããããå³å¯ã«ã¯ããã¯èª€ãã§ãããäžã®èšå·ã®ä»ã« â
{\displaystyle \emptyset } çãçšãããããããã®æç§æžã§ã¯ã â
{\displaystyle \varnothing } ãçšããã
ã©ã®ãããªéåAã«ãã空éåã¯éšåéåãšããŠå«ãŸããã
ã€ãŸãã空éåã§ãªãããéåãAãšãããšã
ã§ããã
éå { 1, 2 } ã®éšåéåããã¹ãŠåæãããšã次ã®4ã€ã®éåã«ãªãã
éå U ã1ã€èšå®ãããã®éåã®èŠçŽ ãéšåéåã®ã¿ãèããå Žåãèããããã®ãããªãšããéåUã å
šäœéå(ãããããã
ããããè±:universal set) ãšããã
å
šäœéåUã®èŠçŽ ã®ãã¡ãéåAã«å±ããªããã®å
šäœãããªãéåã®ããšãAã® è£éå (ã»ãã
ããããè±:complement)ãšãããèšå·ã§è£éå㯠A Ì {\displaystyle {\overline {A}}} ãšè¡šãã
ããªãã¡
ã§ããã
è£éåã«ã€ããŠã次ã®ããšãæãç«ã€ã
äžã®å³ãçšããŠäžã®æ³åãæ£ããäºã確ããããã
A={x|xã¯1以äž20以äžã®2ã®åæ°}ã»B={y|yã¯1以äž20以äžã®3ã®åæ°}ãšããæã以äžã«é©ããéåã®èŠçŽ ãåæããããã ããå
šäœéåU={z|zã¯1以äž20以äžã®æŽæ°}ãšããã
éåã®èŠçŽ ã®ããšãããµããåŒã³æ¹ã§ãããå
ã(ãã)ãšãèšããŸãã
念ã®ããäŸã瀺ããšãããšãã°ã1以äžãã€12以äžã®å¶æ°ã®éåããšèšãã°
ããšãã°ãã10以äžã®èªç¶æ°ã®ãã¡å¶æ°ã§ãããã®ãã®éå
ãéåAãšããå Žåã
ãã4ãã¯éåAã®èŠçŽ ã§ããããšèšããŸãããåæ§ã«ã4ãã¯éåAãŒå
ã§ãããšãèšããŸãã
äŸãšããŠã4ãããããŸããããå¥ã«ã6ãã§ãã10ãã§ãæ§ããŸãããã2ããã6ããã8ããã10ãããããããäžè¿°ããéåAã®èŠçŽ (å
)ã§ãã
(æ°åŠçã«)æ£ãããã©ãããæ確ã«å€æã§ãã䞻匵ãåœé¡(ããã ããè±: proposition)ãšåŒã¶ã äŸãã°ãã7ã¯çŽ æ°ã§ãããã¯åœé¡ã®äŸã§ããã (äžæ¹ãã5000ã¯å€§ããæ°ã§ããããªã©ã¯åœé¡ãšã¯ãªããªãããªããªãã倧ããããšããèšèã®å€æã䞻芳çãªãã®ã§ãããå€æã«æ確ãªåºæºãèšå®ã§ããªãããã§ããã)
ããåœé¡ãæ確ã«æ£ãã(ãšèšŒæããã)ãšãããã®åœé¡ã¯ç(ãããè±:truth)ã§ãããšåŒã¶ã(ããšãã°ãåœé¡ã7ã¯çŽ æ°ã§ãããã¯çã§ããã) åœé¡ãçã§ãªããšããåœé¡ã¯åœ(ããè±:false)ã§ãããšèšããããšãã°ãåœé¡ ã ãã x 2 = 4 {\displaystyle x^{2}=4} ã§ããã° x = 2 {\displaystyle x=2} ã§ããã ã ã¯ãåœã®åœé¡ã§ããã ãã®æ¹çšåŒã¯ x = â 2 {\displaystyle x=-2} ã解ã«æã€ã
äžã®åœé¡ã" x 2 = 4 {\displaystyle x^{2}=4} ãªãã° x = 2 {\displaystyle x=2} ã§ãã"ã㯠x = â 2 {\displaystyle x=-2} ãããŠã¯ãŸãã®ã§åœã«ãªã£ãã åœé¡ p â q {\displaystyle {\rm {p\Rightarrow q}}} ãåœã§ãããšãã¯ã p {\displaystyle p} ã¯æºããã q {\displaystyle q} ãæºãããªãäŸãååšããããã®ãããªäŸãåäŸ(ã¯ããã)ãšãããåœé¡ãåœã§ããããšã瀺ãã«ã¯ãåäŸã1ã€ãããã°ããã
åœé¡ã¯ããpãªãã°qã§ãããã®åœ¢åŒã§æžãããå Žåãå€ãã
ã pãªãã°qã§ããããšããåœé¡ããèšå·ã â {\displaystyle \Rightarrow } ããçšããŠ
ãšæžãã
ãŸãããã®æ¡ä»¶pããã®åœé¡ã®ä»®å®(ããŠããè±:assumption)ãšãããæ¡ä»¶qããã®åœé¡ã®çµè«(ãã€ãã)ãšåŒã¶ã
次ã®åœé¡ã®çåœãå€å®ããåœã®å Žåã¯åäŸãæããã
æ¡ä»¶ãæ¡ä»¶ãå«ãåœé¡ãèããããšã¯ãéåãèããããšãšåãã§ããã
ããšãã°ãå®æ° x ã«ã€ããŠãx>3 ãªãã° x>1 ã§ããããšããåœé¡ã¯çã§ããã
ããã§ãx>3 ã§ããããšããæ¡ä»¶ã p ãšãããŸããx>3 ã§ããæ°ã®éåã P ãšããããã€ãŸã P={x| x>3 }ã§ããã
åæ§ã«ããx>1ã§ããããšããæ¡ä»¶ã q ãšããx>1ã§ããæ°ã®éåã Q ãšããããã€ãŸã Q={x| x>1 }ã§ããã
ãã®ãšããåœé¡ p â¹ q {\displaystyle {\rm {p\Longrightarrow q}}} ã¯çã§ããããããã¯éåã®å
å«é¢ä¿ PâQ ãæãç«ã€ããšã«å¯Ÿå¿ããŠããã
2ã€ã®æ¡ä»¶ p,q ã«ã€ããŠãåœé¡ãpâqããçã§ãããšãã
ãšããã
2ã€ã®æ¡ä»¶ p.q ã«ã€ããŠã
åœé¡ãpâqããšåœé¡ãqâpãã®äž¡æ¹ãšãçã§ãããšããããã
ãšæžãã
ãšããã
ãã®ãšããpãšqãå
¥ãæ¿ããããšã§ã
ãšããããããšããããã
p ⺠q {\displaystyle {\rm {p\Longleftrightarrow q}}} ã§ãããšããpãšqã¯ãåå€(ã©ãã¡)ã§ããããšããã
æ¡ä»¶ p,q ãæºãããã®ã®éåããããã P,Q ãšããã
ãã®ãšããæ¡ä»¶ãpãã€qãããã³ãpãŸãã¯qããããããå³ã¯ãããããå³å³ã®ããã«ãªãã
æ¡ä»¶pã«å¯Ÿããpã§ãªããã®åœ¢ã®æ¡ä»¶ã pã® åŠå® (ã²ãŠããè±:negation)ãšãããèšå·ã¯ p Ì {\displaystyle {\overline {p}}} ã§è¡šãã
(â» é«æ ¡ã§ã¯ç¿ããªãããåŠå®ã®æå³ãšããŠã ¬ p {\displaystyle \lnot {p}} ãšããèšå·ã¬ããããã)
æ¡ä»¶ãèããããšã¯éåãèããããšãšåããªã®ã§ãéåã«ããããã»ã¢ã«ã¬ã³ã®æ³åãšåæ§ã«ãæ¡ä»¶ã«ãããŠãããã»ã¢ã«ã¬ã³ã®æ³åããªãç«ã€ã
åœé¡ã p â¹ q {\displaystyle {\rm {p\Longrightarrow q}}} ãã«å¯ŸããŠ
ãšåŒã¶ã
ãããã¯ããããã«å³å³ã®ãããªé¢ä¿ã«ããã
ããšãã°ã ããšã®åœé¡ã
ã ãšãããšã
ãã®åœé¡ã®å Žåãããšã®åœé¡ãšå¯Ÿå¶ã¯ããšãã«çã§ããã
ãã£ãœãéã«ã€ããŠã¯ x = -3 ãšããåäŸãããã®ã§ããã®åœé¡ã®å Žåãéã¯æ£ãããªãããŸããè£ãåæ§ã«ãæ£ãããªãã
ãã®ãããªäŸããã次ã®ããšãåããã
ã§ã¯ãããšã®åœé¡ãšå¯Ÿå¶ãšã®é¢ä¿ã¯ãã©ããªãã ãããã
ãã®èå¯ããããããæ¡ä»¶pãæºãããã®ãéåPã«å¯Ÿå¿ãããåæ§ã«æ¡ä»¶qãæºãããã®ãéåQã«å¯Ÿå¿ãããŠã¿ããã
å³ã®éåã®å³ã¯ãpâqãçã§ããããšãè¡šãå³ã§ããããã®å³ã§ã¯ãPã«å±ããŠããèŠçŽ ã¯ãQã«ãå±ããŠããã(ã€ãŸã P â Q {\displaystyle {\rm {P\subset Q}}} ã§ããã)äžæ¹ãQã«å±ããŠãããªãèŠçŽ ã¯ãPã«ãå±ããŠããªãã(ã€ãŸã Q Ì â P Ì {\displaystyle {\rm {{\overline {Q}}\subset {\overline {P}}}}} ã§ããã) ãã®ããšããããåããããã«ã
ã€ãŸããäžè¬ã®åœé¡ã«ãããŠãããšã®åœé¡ãšå¯Ÿå¶ãšã®çåœã¯äžèŽããã
ããåœé¡ã®çµè«ãåŠå®ããŠããã®åŠå®ã®ããšã§ççŸãèµ·ããããšãè¿°ã¹ãããšã§ã ãã®åœé¡ãçã§ããããšãå°åºããä»æ¹ãèçæ³(ã¯ããã»ããè±: proof by contradiction ãªã©)ãšåŒã¶ã
ããšãã°ããAã§ã¯ãªãããšã蚌æããããšããåé¡ã解ãæã¯ãAã§ãããšä»®å®ããããšæžãåºããŠãä»®å®ããããšãšççŸããéšåãäœã£ãŠãççŸããã®ã§Aã§ã¯ãªããããšèšŒæãçµããã
çŽ æ°ã¯ç¡éã«ååšããã
çŽ æ°ãæéåã§ãã£ããšä»®å®ããããã¹ãŠã®çŽ æ°ã®ç©ã a {\displaystyle a} ãšãããšã a + 1 {\displaystyle a+1} ã¯ã©ã®çŽ æ°ã§å²ã£ãŠã1äœãããšã«ãªãã1以å€ã®èªç¶æ°ã§ãã£ãŠãçŽ æ°ã®ç©ã«å解ã§ããªããã®ãååšããããšã«ãªãã a + 1 {\displaystyle a+1} ã®çŽæ°ã®ãã¡1以å€ã§æãå°ãããã®ã b {\displaystyle b} ãšãããšã b {\displaystyle b} ã¯1ãš b {\displaystyle b} 以å€ã®çŽæ°ãæããªãããããã£ãŠ b {\displaystyle b} ãçŽ æ°ã§ããããšã«ãªããã a + 1 {\displaystyle a+1} ãã©ã®çŽ æ°ã§ãå²ãåããªãããšãšççŸããããããã£ãŠãçŽ æ°ã¯æéåã§ã¯ãªããâ | [
{
"paragraph_id": 0,
"tag": "p",
"text": "äžåŠã§ã¯ãããšãã°ãèªç¶æ°ã®ãã€ãŸãããšãã9以äžã®èªç¶æ°ã®ãã€ãŸãããšããè² ã®æŽæ°ã®ãã€ãŸããã®ãããªãã®ããéå(ãã
ããã)ãšèªãã§ããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ã§ã¯ãæ°åŠã§ãããéåããšã¯äœããããããèããŠãããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "æ°åŠã§ã¯ãããéãŸãã®ãã¡ãããã«ãããã«å±ããŠãããå±ããŠãªãããæ確ã«åºå¥ã§ããæ¡ä»¶ã®ããç©ã®ãã€ãŸããéå(ãã
ããããè±:set)ãšãããäŸãã°ããèªç¶æ°ãã¯ãn > 0ãšãªãæŽæ°n ã®å
šäœããšããåºå¥å¯èœãªæ¡ä»¶ãããã®ã§éåãšãããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ãããã倧ããªæ°ããšãããã€ãŸãã¯ãã©ããããã倧ããªãæ°ãšãããã®ããã¯ã£ããããªããããæ°åŠã®ãéåãã§ã¯ãªãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ãã ããã倧ããªæ°ããäŸãã°ã1å以äžã®æŽæ°ããšåºå¥ã§ããããã«å®çŸ©ããã°éåã«ãªãããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ããŠãæ°åŠçãªãéåããæ§æãããã®äžã€äžã€ã®ããšãããã®éåã® èŠçŽ ( ããããè±:element)ãšããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ããšãã°ããèªç¶æ°ã®éåãã®èŠçŽ ãªããèªç¶æ°1ãèªç¶æ°2ãèªç¶æ°3ãã»ã»ã»ãªã©ã®ã²ãšã€ã²ãšã€ã®èªç¶æ°ãããããèŠçŽ ã§ããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ã 1 ã¯èªç¶æ°ã®éåã®èŠçŽ ã§ããããšãããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "ã 27 ã¯èªç¶æ°ã®éåã®èŠçŽ ã§ããããšãããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "",
"title": "éåãšè«ç"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "(â» ç¯å²å€? )ãªããæ°åŠçã«ã¯ãåºå¥ãã¯ã£ãããããããã°ãäŸãã°ãâ³â³é«æ ¡ã®ä»ã®3幎Bçµã®çåŸå
šå¡ãçãéåãšããŠèããããšãã§ãããããªãããããéåããšã¯ãèªç¶æ°ãããæŽæ°ããªã©ã®æ°ã§ãªããŠãããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "(ç¯å²å€)æ°ãšæ°ãšã®å¯Ÿå¿é¢ä¿ã§ãããé¢æ°ãããéåã®åèŠçŽ ãšéåã®åèŠçŽ ãšã®å¯Ÿå¿é¢ä¿ãžãšæ¡åŒµããããšãã§ããã(ãã®éåã¯æ°ã®éåã§ãªããŠãè¯ãã)ãã®ãããªå¯Ÿå¿é¢ä¿ãååãšåŒã¶ã詳ããã¯å€§åŠã®ãéåè«ãã§æ±ããããå
šå°ãããåå°ããªã©ãç¥ã£ãŠãããšèšŒæã«äŸ¿å©ãªç¥èãããã (ç¯å²å€ãããŸã§)",
"title": "éåãšè«ç"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "aãéåAã®èŠçŽ ã§ãããšããããã®ãšããaã¯éåAã«å±ãã(ãããã)ãšãããèšå·ã§ã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ãšè¡šãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "bãAã®èŠçŽ ã§ãªããšãã¯ã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "ãšè¡šãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "éåããããããšããäž»ã«2çš®é¡ã®æ¹æ³ãããã(äŸã¯ã10以äžã®èªç¶æ°ã®ãã¡å¶æ°ã§ãããã®ãã®éåãè¡šãã)",
"title": "éåãšè«ç"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "ã§ãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "ããšãã°ãã10以äžã®èªç¶æ°ã®ãã¡å¶æ°ã§ãããã®ãã®éåãè¡šãå Žåã(1) ã®æ¹æ³(èŠçŽ ãæžã䞊ã¹ãæ¹æ³)ã§ã¯ã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "ãšãªãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "äžæ¹ã(2)ã®æ¹æ³(èŠçŽ ã®æºããæ¡ä»¶ãè¿°ã¹ãæ¹æ³)ã§ã¯ã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "ãªã©ã®ããã«ãªã(äœéãããã)ã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "",
"title": "éåãšè«ç"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "100以äžã®èªç¶æ°ã®éå A ããããã»ã©ã®(1)ãèŠçŽ ãæžã䞊ã¹ãæ¹æ³ãã®æ¹æ³ã§æžãå Žåã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "ãšãªãããŸãããã®èšæ³ã®ãã»ã»ã»ãã®ããã«ãèŠçŽ ã®åæ°ããšãŠãå€ãå Žåãç¡æ°ã«ããå Žåã«ã¯ã{ }èšå·å
ã®èŠçŽ ã®éäžããã»ã»ã»ããŸãã¯ã......ããã...ããªã©ã®ç¹ã
ã§çç¥ããŠããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "100以äžã®å¶æ°ã®éå B ã¯ããã®èšæ³(èŠçŽ ãæžã䞊ã¹ãæ¹æ³)ã§ã¯ã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "ã®ããã«ãªãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "æ£ã®å¶æ°å
šäœã®éåã®èŠçŽ ã¯(1)ãèŠçŽ ãæžã䞊ã¹ãæ¹æ³ãã®æ¹æ³ã§æžãå Žåã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "ã®ããã«ãæžããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "2ã€ã®éåA,BããããxâA ãªãã° xâBãæãç«ã€ãšããAã¯Bã® éšåéå (ã¶ã¶ããã
ããããè±:subset)ã§ãããšããããBã¯Aãå«ããããAã¯Bã«å«ãŸããããšããããã®ç¶æ
ãèšå·ã§",
"title": "éåãšè«ç"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "ãŸãã¯",
"title": "éåãšè«ç"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "ã§è¡šãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "è£è¶³",
"title": "éåãšè«ç"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "Aã®éšåéåã«ã¯Aèªèº«ãããã(ã€ãŸã A â A ã§ãã)ã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "ãŸããA,B ã®éåã®èŠçŽ ãåããšãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "ã§è¡šãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "",
"title": "éåãšè«ç"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "éå A = {1, 2, 3} ãš éå B = {1, 2 , 3 , 4, 5} ããããšããA 㯠Bã®éšåéåã§ããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "",
"title": "éåãšè«ç"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "2ã€ã®éåA,Bããããšã ãããã®äž¡æ¹ã®èŠçŽ ã§ãããã®ã®éåã AãšBã® å
±ééšå(ãããã€ãã¶ã¶ã)ãšåŒã³ã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "ãšæžãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 41,
"tag": "p",
"text": "",
"title": "éåãšè«ç"
},
{
"paragraph_id": 42,
"tag": "p",
"text": "ãŸããéåA,Bã®å°ãªããšãã©ã¡ããäžæ¹ã«ã¯å±ããŠããèŠçŽ ãããªãéåã®ããšããAãšBã®åéå(ããã
ããããè±:union)ãšåŒã³ã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 43,
"tag": "p",
"text": "ãšæžãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 44,
"tag": "p",
"text": "",
"title": "éåãšè«ç"
},
{
"paragraph_id": 45,
"tag": "p",
"text": "3ã€ã®éå A, B, C ã«ã€ããŠã¯ã3ã€ã®ã©ãã«ãå±ããèŠçŽ å
šäœã®éåã A,B,C ã®å
±ééšåãšåŒã³ã A â© B â© C ã§è¡šãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 46,
"tag": "p",
"text": "",
"title": "éåãšè«ç"
},
{
"paragraph_id": 47,
"tag": "p",
"text": "ãŸããéå A, B, C ã®å°ãªããšã1ã€ã«å±ããèŠçŽ ã®éåã A,B,C ã®åéåãšåŒã³ã A ⪠B ⪠C ã§è¡šãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 48,
"tag": "p",
"text": "ããšãã°ãã10以äžã®èªç¶æ°ã®ãã¡ã®å¶æ°ãã®éåAãšãã10以äžã®èªç¶æ°ã®ãã¡ã®å¥æ°ãã®éåBã«ã€ããŠãéåAãšéåBã®å
±ééšåã«ã¯ãäœãèŠçŽ ãç¡ãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 49,
"tag": "p",
"text": "ãã®äŸã®ããã«ããèŠçŽ ããªã«ããªãããšããå Žåãããã®ã§ãæ°åŠã§ã¯ãèŠçŽ ããªã«ããªããå Žåãã²ãšã€ã®éåãšããŠèããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 50,
"tag": "p",
"text": "èŠçŽ ããããªãéåã®ããšã 空éå(ãããã
ããããè±:empty set ããã㯠null set)ãšãããèšå·ã¯",
"title": "éåãšè«ç"
},
{
"paragraph_id": 51,
"tag": "p",
"text": "ã§ããããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 52,
"tag": "p",
"text": "ã®ãªã·ã£æåã®ãã¡ã€(Ï, Ï {\\displaystyle \\phi \\ } )ã§è¡šãããããšãå€ãããããå³å¯ã«ã¯ããã¯èª€ãã§ãããäžã®èšå·ã®ä»ã« â
{\\displaystyle \\emptyset } çãçšãããããããã®æç§æžã§ã¯ã â
{\\displaystyle \\varnothing } ãçšããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 53,
"tag": "p",
"text": "",
"title": "éåãšè«ç"
},
{
"paragraph_id": 54,
"tag": "p",
"text": "ã©ã®ãããªéåAã«ãã空éåã¯éšåéåãšããŠå«ãŸããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 55,
"tag": "p",
"text": "ã€ãŸãã空éåã§ãªãããéåãAãšãããšã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 56,
"tag": "p",
"text": "ã§ããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 57,
"tag": "p",
"text": "",
"title": "éåãšè«ç"
},
{
"paragraph_id": 58,
"tag": "p",
"text": "éå { 1, 2 } ã®éšåéåããã¹ãŠåæãããšã次ã®4ã€ã®éåã«ãªãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 59,
"tag": "p",
"text": "",
"title": "éåãšè«ç"
},
{
"paragraph_id": 60,
"tag": "p",
"text": "",
"title": "éåãšè«ç"
},
{
"paragraph_id": 61,
"tag": "p",
"text": "éå U ã1ã€èšå®ãããã®éåã®èŠçŽ ãéšåéåã®ã¿ãèããå Žåãèããããã®ãããªãšããéåUã å
šäœéå(ãããããã
ããããè±:universal set) ãšããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 62,
"tag": "p",
"text": "å
šäœéåUã®èŠçŽ ã®ãã¡ãéåAã«å±ããªããã®å
šäœãããªãéåã®ããšãAã® è£éå (ã»ãã
ããããè±:complement)ãšãããèšå·ã§è£éå㯠A Ì {\\displaystyle {\\overline {A}}} ãšè¡šãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 63,
"tag": "p",
"text": "ããªãã¡",
"title": "éåãšè«ç"
},
{
"paragraph_id": 64,
"tag": "p",
"text": "ã§ããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 65,
"tag": "p",
"text": "è£éåã«ã€ããŠã次ã®ããšãæãç«ã€ã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 66,
"tag": "p",
"text": "",
"title": "éåãšè«ç"
},
{
"paragraph_id": 67,
"tag": "p",
"text": "äžã®å³ãçšããŠäžã®æ³åãæ£ããäºã確ããããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 68,
"tag": "p",
"text": "A={x|xã¯1以äž20以äžã®2ã®åæ°}ã»B={y|yã¯1以äž20以äžã®3ã®åæ°}ãšããæã以äžã«é©ããéåã®èŠçŽ ãåæããããã ããå
šäœéåU={z|zã¯1以äž20以äžã®æŽæ°}ãšããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 69,
"tag": "p",
"text": "",
"title": "éåãšè«ç"
},
{
"paragraph_id": 70,
"tag": "p",
"text": "",
"title": "éåãšè«ç"
},
{
"paragraph_id": 71,
"tag": "p",
"text": "éåã®èŠçŽ ã®ããšãããµããåŒã³æ¹ã§ãããå
ã(ãã)ãšãèšããŸãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 72,
"tag": "p",
"text": "念ã®ããäŸã瀺ããšãããšãã°ã1以äžãã€12以äžã®å¶æ°ã®éåããšèšãã°",
"title": "éåãšè«ç"
},
{
"paragraph_id": 73,
"tag": "p",
"text": "ããšãã°ãã10以äžã®èªç¶æ°ã®ãã¡å¶æ°ã§ãããã®ãã®éå",
"title": "éåãšè«ç"
},
{
"paragraph_id": 74,
"tag": "p",
"text": "ãéåAãšããå Žåã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 75,
"tag": "p",
"text": "ãã4ãã¯éåAã®èŠçŽ ã§ããããšèšããŸãããåæ§ã«ã4ãã¯éåAãŒå
ã§ãããšãèšããŸãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 76,
"tag": "p",
"text": "äŸãšããŠã4ãããããŸããããå¥ã«ã6ãã§ãã10ãã§ãæ§ããŸãããã2ããã6ããã8ããã10ãããããããäžè¿°ããéåAã®èŠçŽ (å
)ã§ãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 77,
"tag": "p",
"text": "(æ°åŠçã«)æ£ãããã©ãããæ確ã«å€æã§ãã䞻匵ãåœé¡(ããã ããè±: proposition)ãšåŒã¶ã äŸãã°ãã7ã¯çŽ æ°ã§ãããã¯åœé¡ã®äŸã§ããã (äžæ¹ãã5000ã¯å€§ããæ°ã§ããããªã©ã¯åœé¡ãšã¯ãªããªãããªããªãã倧ããããšããèšèã®å€æã䞻芳çãªãã®ã§ãããå€æã«æ確ãªåºæºãèšå®ã§ããªãããã§ããã)",
"title": "éåãšè«ç"
},
{
"paragraph_id": 78,
"tag": "p",
"text": "ããåœé¡ãæ確ã«æ£ãã(ãšèšŒæããã)ãšãããã®åœé¡ã¯ç(ãããè±:truth)ã§ãããšåŒã¶ã(ããšãã°ãåœé¡ã7ã¯çŽ æ°ã§ãããã¯çã§ããã) åœé¡ãçã§ãªããšããåœé¡ã¯åœ(ããè±:false)ã§ãããšèšããããšãã°ãåœé¡ ã ãã x 2 = 4 {\\displaystyle x^{2}=4} ã§ããã° x = 2 {\\displaystyle x=2} ã§ããã ã ã¯ãåœã®åœé¡ã§ããã ãã®æ¹çšåŒã¯ x = â 2 {\\displaystyle x=-2} ã解ã«æã€ã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 79,
"tag": "p",
"text": "äžã®åœé¡ã\" x 2 = 4 {\\displaystyle x^{2}=4} ãªãã° x = 2 {\\displaystyle x=2} ã§ãã\"ã㯠x = â 2 {\\displaystyle x=-2} ãããŠã¯ãŸãã®ã§åœã«ãªã£ãã åœé¡ p â q {\\displaystyle {\\rm {p\\Rightarrow q}}} ãåœã§ãããšãã¯ã p {\\displaystyle p} ã¯æºããã q {\\displaystyle q} ãæºãããªãäŸãååšããããã®ãããªäŸãåäŸ(ã¯ããã)ãšãããåœé¡ãåœã§ããããšã瀺ãã«ã¯ãåäŸã1ã€ãããã°ããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 80,
"tag": "p",
"text": "åœé¡ã¯ããpãªãã°qã§ãããã®åœ¢åŒã§æžãããå Žåãå€ãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 81,
"tag": "p",
"text": "ã pãªãã°qã§ããããšããåœé¡ããèšå·ã â {\\displaystyle \\Rightarrow } ããçšããŠ",
"title": "éåãšè«ç"
},
{
"paragraph_id": 82,
"tag": "p",
"text": "ãšæžãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 83,
"tag": "p",
"text": "ãŸãããã®æ¡ä»¶pããã®åœé¡ã®ä»®å®(ããŠããè±:assumption)ãšãããæ¡ä»¶qããã®åœé¡ã®çµè«(ãã€ãã)ãšåŒã¶ã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 84,
"tag": "p",
"text": "",
"title": "éåãšè«ç"
},
{
"paragraph_id": 85,
"tag": "p",
"text": "次ã®åœé¡ã®çåœãå€å®ããåœã®å Žåã¯åäŸãæããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 86,
"tag": "p",
"text": "",
"title": "éåãšè«ç"
},
{
"paragraph_id": 87,
"tag": "p",
"text": "æ¡ä»¶ãæ¡ä»¶ãå«ãåœé¡ãèããããšã¯ãéåãèããããšãšåãã§ããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 88,
"tag": "p",
"text": "ããšãã°ãå®æ° x ã«ã€ããŠãx>3 ãªãã° x>1 ã§ããããšããåœé¡ã¯çã§ããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 89,
"tag": "p",
"text": "ããã§ãx>3 ã§ããããšããæ¡ä»¶ã p ãšãããŸããx>3 ã§ããæ°ã®éåã P ãšããããã€ãŸã P={x| x>3 }ã§ããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 90,
"tag": "p",
"text": "åæ§ã«ããx>1ã§ããããšããæ¡ä»¶ã q ãšããx>1ã§ããæ°ã®éåã Q ãšããããã€ãŸã Q={x| x>1 }ã§ããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 91,
"tag": "p",
"text": "ãã®ãšããåœé¡ p â¹ q {\\displaystyle {\\rm {p\\Longrightarrow q}}} ã¯çã§ããããããã¯éåã®å
å«é¢ä¿ PâQ ãæãç«ã€ããšã«å¯Ÿå¿ããŠããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 92,
"tag": "p",
"text": "2ã€ã®æ¡ä»¶ p,q ã«ã€ããŠãåœé¡ãpâqããçã§ãããšãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 93,
"tag": "p",
"text": "ãšããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 94,
"tag": "p",
"text": "2ã€ã®æ¡ä»¶ p.q ã«ã€ããŠã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 95,
"tag": "p",
"text": "åœé¡ãpâqããšåœé¡ãqâpãã®äž¡æ¹ãšãçã§ãããšããããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 96,
"tag": "p",
"text": "ãšæžãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 97,
"tag": "p",
"text": "ãšããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 98,
"tag": "p",
"text": "ãã®ãšããpãšqãå
¥ãæ¿ããããšã§ã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 99,
"tag": "p",
"text": "ãšããããããšããããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 100,
"tag": "p",
"text": "p ⺠q {\\displaystyle {\\rm {p\\Longleftrightarrow q}}} ã§ãããšããpãšqã¯ãåå€(ã©ãã¡)ã§ããããšããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 101,
"tag": "p",
"text": "æ¡ä»¶ p,q ãæºãããã®ã®éåããããã P,Q ãšããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 102,
"tag": "p",
"text": "ãã®ãšããæ¡ä»¶ãpãã€qãããã³ãpãŸãã¯qããããããå³ã¯ãããããå³å³ã®ããã«ãªãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 103,
"tag": "p",
"text": "æ¡ä»¶pã«å¯Ÿããpã§ãªããã®åœ¢ã®æ¡ä»¶ã pã® åŠå® (ã²ãŠããè±:negation)ãšãããèšå·ã¯ p Ì {\\displaystyle {\\overline {p}}} ã§è¡šãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 104,
"tag": "p",
"text": "(â» é«æ ¡ã§ã¯ç¿ããªãããåŠå®ã®æå³ãšããŠã ¬ p {\\displaystyle \\lnot {p}} ãšããèšå·ã¬ããããã)",
"title": "éåãšè«ç"
},
{
"paragraph_id": 105,
"tag": "p",
"text": "æ¡ä»¶ãèããããšã¯éåãèããããšãšåããªã®ã§ãéåã«ããããã»ã¢ã«ã¬ã³ã®æ³åãšåæ§ã«ãæ¡ä»¶ã«ãããŠãããã»ã¢ã«ã¬ã³ã®æ³åããªãç«ã€ã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 106,
"tag": "p",
"text": "",
"title": "éåãšè«ç"
},
{
"paragraph_id": 107,
"tag": "p",
"text": "åœé¡ã p â¹ q {\\displaystyle {\\rm {p\\Longrightarrow q}}} ãã«å¯ŸããŠ",
"title": "éåãšè«ç"
},
{
"paragraph_id": 108,
"tag": "p",
"text": "ãšåŒã¶ã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 109,
"tag": "p",
"text": "ãããã¯ããããã«å³å³ã®ãããªé¢ä¿ã«ããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 110,
"tag": "p",
"text": "ããšãã°ã ããšã®åœé¡ã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 111,
"tag": "p",
"text": "ã ãšãããšã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 112,
"tag": "p",
"text": "ãã®åœé¡ã®å Žåãããšã®åœé¡ãšå¯Ÿå¶ã¯ããšãã«çã§ããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 113,
"tag": "p",
"text": "ãã£ãœãéã«ã€ããŠã¯ x = -3 ãšããåäŸãããã®ã§ããã®åœé¡ã®å Žåãéã¯æ£ãããªãããŸããè£ãåæ§ã«ãæ£ãããªãã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 114,
"tag": "p",
"text": "ãã®ãããªäŸããã次ã®ããšãåããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 115,
"tag": "p",
"text": "ã§ã¯ãããšã®åœé¡ãšå¯Ÿå¶ãšã®é¢ä¿ã¯ãã©ããªãã ãããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 116,
"tag": "p",
"text": "ãã®èå¯ããããããæ¡ä»¶pãæºãããã®ãéåPã«å¯Ÿå¿ãããåæ§ã«æ¡ä»¶qãæºãããã®ãéåQã«å¯Ÿå¿ãããŠã¿ããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 117,
"tag": "p",
"text": "å³ã®éåã®å³ã¯ãpâqãçã§ããããšãè¡šãå³ã§ããããã®å³ã§ã¯ãPã«å±ããŠããèŠçŽ ã¯ãQã«ãå±ããŠããã(ã€ãŸã P â Q {\\displaystyle {\\rm {P\\subset Q}}} ã§ããã)äžæ¹ãQã«å±ããŠãããªãèŠçŽ ã¯ãPã«ãå±ããŠããªãã(ã€ãŸã Q Ì â P Ì {\\displaystyle {\\rm {{\\overline {Q}}\\subset {\\overline {P}}}}} ã§ããã) ãã®ããšããããåããããã«ã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 118,
"tag": "p",
"text": "ã€ãŸããäžè¬ã®åœé¡ã«ãããŠãããšã®åœé¡ãšå¯Ÿå¶ãšã®çåœã¯äžèŽããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 119,
"tag": "p",
"text": "ããåœé¡ã®çµè«ãåŠå®ããŠããã®åŠå®ã®ããšã§ççŸãèµ·ããããšãè¿°ã¹ãããšã§ã ãã®åœé¡ãçã§ããããšãå°åºããä»æ¹ãèçæ³(ã¯ããã»ããè±: proof by contradiction ãªã©)ãšåŒã¶ã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 120,
"tag": "p",
"text": "ããšãã°ããAã§ã¯ãªãããšã蚌æããããšããåé¡ã解ãæã¯ãAã§ãããšä»®å®ããããšæžãåºããŠãä»®å®ããããšãšççŸããéšåãäœã£ãŠãççŸããã®ã§Aã§ã¯ãªããããšèšŒæãçµããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 121,
"tag": "p",
"text": "çŽ æ°ã¯ç¡éã«ååšããã",
"title": "éåãšè«ç"
},
{
"paragraph_id": 122,
"tag": "p",
"text": "çŽ æ°ãæéåã§ãã£ããšä»®å®ããããã¹ãŠã®çŽ æ°ã®ç©ã a {\\displaystyle a} ãšãããšã a + 1 {\\displaystyle a+1} ã¯ã©ã®çŽ æ°ã§å²ã£ãŠã1äœãããšã«ãªãã1以å€ã®èªç¶æ°ã§ãã£ãŠãçŽ æ°ã®ç©ã«å解ã§ããªããã®ãååšããããšã«ãªãã a + 1 {\\displaystyle a+1} ã®çŽæ°ã®ãã¡1以å€ã§æãå°ãããã®ã b {\\displaystyle b} ãšãããšã b {\\displaystyle b} ã¯1ãš b {\\displaystyle b} 以å€ã®çŽæ°ãæããªãããããã£ãŠ b {\\displaystyle b} ãçŽ æ°ã§ããããšã«ãªããã a + 1 {\\displaystyle a+1} ãã©ã®çŽ æ°ã§ãå²ãåããªãããšãšççŸããããããã£ãŠãçŽ æ°ã¯æéåã§ã¯ãªããâ ",
"title": "éåãšè«ç"
}
] | null | == éåãšè«ç ==
=== éåãšã¯ ===
äžåŠã§ã¯ãããšãã°ãèªç¶æ°ã®ãã€ãŸãããšãã9以äžã®èªç¶æ°ã®ãã€ãŸãããšããè² ã®æŽæ°ã®ãã€ãŸããã®ãããªãã®ããéåïŒãã
ãããïŒãšèªãã§ããã
ã§ã¯ãæ°åŠã§ãããéåããšã¯äœããããããèããŠãããã
æ°åŠã§ã¯ãããéãŸãã®ãã¡ãããã«ãããã«å±ããŠãããå±ããŠãªãããæ確ã«åºå¥ã§ããæ¡ä»¶ã®ããç©ã®ãã€ãŸãã'''éå'''ïŒãã
ããããè±ïŒsetïŒãšãããäŸãã°ããèªç¶æ°ãã¯ã''n'' > 0ãšãªãæŽæ°''n'' ã®å
šäœããšããåºå¥å¯èœãªæ¡ä»¶ãããã®ã§éåãšãããã
ãããã倧ããªæ°ããšãããã€ãŸãã¯ãã©ããããã倧ããªãæ°ãšãããã®ããã¯ã£ããããªããããæ°åŠã®ãéåãã§ã¯ãªãã
ãã ããã倧ããªæ°ããäŸãã°ã1å以äžã®æŽæ°ããšåºå¥ã§ããããã«å®çŸ©ããã°éåã«ãªãããã
ããŠãæ°åŠçãªãéåããæ§æãããã®äžã€äžã€ã®ããšãããã®éåã® '''èŠçŽ '''ïŒ ããããè±ïŒelementïŒãšããã
äŸ
7以äžã®èªç¶æ°ã®éåã®èŠçŽ ã¯ã1ãš2ãš3ãš4ãš5ãš6ãš7 ã§ããã
ãªãããèŠçŽ ããšããèšèã䜿ãå Žåãã¹ã€ã«éåã®ãªãã¿ãå
šéšã䞊ã¹ãå¿
èŠã¯ç¡ãã
ããšãã°ãã2ã¯ã7以äžã®èªç¶æ°ã®éåã®ãã¡ã®èŠçŽ ã§ããããšèšã£ãŠã倧äžå€«ã§ããåã®æã®ã2ãã1以äžãã7以äžã®å¥ã®æŽæ°ã«çœ®ãæããŠã倧äžå€«ã§ãã
äžè¿°ããããã«ãèŠçŽ ãšã¯éåãæ§æãããã®ã®äžã€äžã€ã®ããšã§ãã
äŸ
ãèªç¶æ°ã®éåãã®èŠçŽ ãªããèªç¶æ°1ãèªç¶æ°2ãèªç¶æ°3ãã»ã»ã»ãªã©ã®ã²ãšã€ã²ãšã€ã®èªç¶æ°ãããããèŠçŽ ã§ããã
ã 1 ã¯èªç¶æ°ã®éåã®èŠçŽ ã§ããããšãããã
ã 27 ã¯èªç¶æ°ã®éåã®èŠçŽ ã§ããããšãããã
èªç¶æ°ã®éåã®ããã«ãéåã¯å¿
ãããèŠçŽ ãæéã§ãªããŠãæ§ããŸããïŒéåã®èŠçŽ ã¯ç¡éã§ãè¯ãïŒã
ïŒâ» ç¯å²å€ïŒ ïŒãªããæ°åŠçã«ã¯ãåºå¥ãã¯ã£ãããããããã°ãäŸãã°ãâ³â³é«æ ¡ã®ä»ã®3幎Bçµã®çåŸå
šå¡ãçãéåãšããŠèããããšãã§ãããããªãããããéåããšã¯ãèªç¶æ°ãããæŽæ°ããªã©ã®æ°ã§ãªããŠãããã
ïŒç¯å²å€ïŒæ°ãšæ°ãšã®å¯Ÿå¿é¢ä¿ã§ãããé¢æ°ãããéåã®åèŠçŽ ãšéåã®åèŠçŽ ãšã®å¯Ÿå¿é¢ä¿ãžãšæ¡åŒµããããšãã§ãããïŒãã®éåã¯æ°ã®éåã§ãªããŠãè¯ããïŒãã®ãããªå¯Ÿå¿é¢ä¿ã'''åå'''ãšåŒã¶ã詳ããã¯å€§åŠã®ãéåè«ãã§æ±ããããå
šå°ãããåå°ããªã©ãç¥ã£ãŠãããšèšŒæã«äŸ¿å©ãªç¥èãããã
ïŒç¯å²å€ãããŸã§ïŒ
=== éåãèŠçŽ ã®é¢ä¿ã®è¡šãæ¹ ===
==== éåãšèŠçŽ ====
[[File:Mathematical set A with element a with no element b.svg|thumb|]]
aãéåAã®èŠçŽ ã§ãããšããããã®ãšããaã¯éåAã«'''å±ãã'''(ãããã)ãšãããèšå·ã§ã
:a ∈ A
::ãŸã㯠éåãã«
:A <math> \ni </math> a
ãšè¡šãã
bãAã®èŠçŽ ã§ãªããšãã¯ã
:b <math>\notin</math> A
::ãŸã㯠éåãã«
:A [[File:Set symbol of not-element.svg|20px]] b
ãšè¡šãã
éåããããããšããäž»ã«2çš®é¡ã®æ¹æ³ããããïŒäŸã¯ã10以äžã®èªç¶æ°ã®ãã¡å¶æ°ã§ãããã®ãã®éåãè¡šããïŒ
:ïŒ1ïŒãèŠçŽ ãæžã䞊ã¹ãæ¹æ³
:ïŒ2ïŒãèŠçŽ ã®æºããæ¡ä»¶ãè¿°ã¹ãæ¹æ³
ã§ãã
[[File:Set of natural numbers less than 10.svg|thumb|]]
ããšãã°ãã10以äžã®èªç¶æ°ã®ãã¡å¶æ°ã§ãããã®ãã®éåãè¡šãå ŽåãïŒ1ïŒãã®æ¹æ³ïŒèŠçŽ ãæžã䞊ã¹ãæ¹æ³ïŒã§ã¯ã
: {2, ã4, ã6, ã8, ã10}
ãšãªãã
äžæ¹ãïŒ2ïŒã®æ¹æ³ïŒèŠçŽ ã®æºããæ¡ä»¶ãè¿°ã¹ãæ¹æ³ïŒã§ã¯ã
:{ x | x=2n (nã¯èªç¶æ°), 2 ≤ x≤ 10 }
:{ x | xã¯2以äž10以äžã®å¶æ° }
:{ 2n | 1 ≤ n≤ 5 (nã¯èªç¶æ°) }
:{ 2n | nã¯1以äž5以äžã®èªç¶æ° }
ãªã©ã®ããã«ãªãïŒäœéããããïŒã
;åè
100以äžã®èªç¶æ°ã®éå A ããããã»ã©ã®ïŒ1ïŒãèŠçŽ ãæžã䞊ã¹ãæ¹æ³ãã®æ¹æ³ã§æžãå Žåã
:A ïŒ {1, ã2, ã3, ã4, ã»ã»ã» , 99, ã100}
ãšãªãããŸãããã®èšæ³ã®ãã»ã»ã»ãã®ããã«ãèŠçŽ ã®åæ°ããšãŠãå€ãå Žåãç¡æ°ã«ããå Žåã«ã¯ãïœ ïœèšå·å
ã®èŠçŽ ã®éäžããã»ã»ã»ããŸãã¯ãâŠâŠãããâŠããªã©ã®ç¹ã
ã§çç¥ããŠããã
:ïŒâ» ãªããã¯ãŒããã§ãâŠâŠããªã©ã®çãç¹ã
ãåºãããå Žåããäžç¹ãªãŒããŒãã§å€æãããšåºããç¹ã6ã€ãã£ãŠããäžç¹ãªãŒããŒãã§åºãã
:æ±äº¬æžç±ã®æ€å®æç§æžã§ãçãã»ãã®äžç¹ãªãŒããŒã䜿ã£ãŠãããåæ通ãªã©ã¯ã6ã€ã®ç¹ã®é·ãäžç¹ãªãŒããŒã䜿ã£ãŠãããïŒ
100以äžã®å¶æ°ã®éå B ã¯ããã®èšæ³ïŒèŠçŽ ãæžã䞊ã¹ãæ¹æ³ïŒã§ã¯ã
:B ïŒ {2,ã 4, ã6, ã»ã»ã» , 98, ã100}
ã®ããã«ãªãã
æ£ã®å¶æ°å
šäœã®éåã®èŠçŽ ã¯ïŒ1ïŒãèŠçŽ ãæžã䞊ã¹ãæ¹æ³ãã®æ¹æ³ã§æžãå Žåã
:{2, ã4, ã6, ãã»ã»ã»}
ã®ããã«ãæžããã
==== éåã©ãã ====
===== éšåéå =====
[[File:Venn A subset B 2.svg|thumb|éåAãéåBã®éšåéåã§ããå Žå]]
2ã€ã®éåA,Bããããx∈A ãªãã° x∈Bãæãç«ã€ãšããAã¯Bã® '''éšåéå''' (ã¶ã¶ããã
ããããè±ïŒsubset)ã§ãããšããããBã¯Aãå«ããããAã¯Bã«å«ãŸããããšããããã®ç¶æ
ãèšå·ã§
:A ⊂ B
ãŸãã¯
:B ⊃ A
ã§è¡šãã
'''è£è¶³'''ãã
Aã®éšåéåã«ã¯Aèªèº«ããããïŒã€ãŸããA ⊂ Aãã§ããïŒã
ãŸããA,B ã®éåã®èŠçŽ ãåããšãã
:A ïŒ B
ã§è¡šãã
{{-}}
;äŸ
éå A ïŒ {1, 2, 3} ãš éå B ïŒ {1, 2 , 3 , 4, 5} ããããšããA 㯠Bã®éšåéåã§ããã
----
===== å
±ééšåãšåéå =====
[[File:Subset intersection A and B.svg|thumb|è²ã®éšåã¯<br>å
±ééšå A ∩ B ]]
2ã€ã®éåA,Bããããšã
ãããã®äž¡æ¹ã®èŠçŽ ã§ãããã®ã®éåã
AãšBã® '''å
±ééšå'''ïŒãããã€ãã¶ã¶ãïŒãšåŒã³ã
:A ∩ B
ãšæžãã
{{-}}
[[File:Subset union A or B.svg|thumb|è²ã®éšåã¯<br>åéå A ∪ B]]
ãŸããéåA,Bã®å°ãªããšãã©ã¡ããäžæ¹ã«ã¯å±ããŠããèŠçŽ ãããªãéåã®ããšããAãšBã®'''åéå'''ïŒããã
ããããè±ïŒunionïŒãšåŒã³ã
:A ∪ B
ãšæžãã
{{-}}
===== 3ã€ã®éåã®å
±ééšåãšåéå =====
[[File:Intersection A and B and C.svg|thumb|A ∩ B ∩ C ]]
3ã€ã®éå A, B, C ã«ã€ããŠã¯ã3ã€ã®ã©ãã«ãå±ããèŠçŽ å
šäœã®éåã A,B,C ã®å
±ééšåãšåŒã³ã<br>
'''A ∩ B ∩ C''' ã§è¡šãã
{{-}}
[[File:Venn union A or B or C.svg|thumb|A ∪ B ∪ C]]
ãŸããéå A, B, C ã®å°ãªããšã1ã€ã«å±ããèŠçŽ ã®éåã A,B,C ã®åéåãšåŒã³ã<br>
'''A ∪ B ∪ C''' ã§è¡šãã
{{-}}
----
===== 空éå =====
ããšãã°ãã10以äžã®èªç¶æ°ã®ãã¡ã®å¶æ°ãã®éåAãšãã10以äžã®èªç¶æ°ã®ãã¡ã®å¥æ°ãã®éåBã«ã€ããŠãéåAãšéåBã®å
±ééšåã«ã¯ãäœãèŠçŽ ãç¡ãã
ãã®äŸã®ããã«ããèŠçŽ ããªã«ããªãããšããå Žåãããã®ã§ãæ°åŠã§ã¯ãèŠçŽ ããªã«ããªããå Žåãã²ãšã€ã®éåãšããŠèããã
èŠçŽ ããããªãéåã®ããšã '''空éå'''(ãããã
ããããè±ïŒempty set ããã㯠null set)ãšãããèšå·ã¯
:<math>\varnothing</math>
ã§ããããã
ã®ãªã·ã£æåã®ãã¡ã€ïŒφ,<math>\phi\ </math>ïŒã§è¡šãããããšãå€ãããããå³å¯ã«ã¯ããã¯èª€ãã§ãããäžã®èšå·ã®ä»ã«<math>\empty</math>çãçšãããããããã®æç§æžã§ã¯ã<math>\varnothing</math>ãçšããã
;è£è¶³
ã©ã®ãããªéåAã«ãã空éåã¯éšåéåãšããŠå«ãŸããã
ã€ãŸãã空éåã§ãªãããéåãAãšãããšã
:<math>\varnothing</math> ⊂ A
ã§ããã
;äŸ
éå { 1, 2 } ã®éšåéåããã¹ãŠåæãããšã次ã®4ã€ã®éåã«ãªãã
:â
,ã{1} ,ã{2} ,ã{1,2}
----
{{-}}
===== å
šäœéåã»è£éå =====
[[File:Universal set and complement.svg|thumb|è²ã€ãã®éšåãè£éå <math>\overline{A}</math> ]]
éå U ã1ã€èšå®ãããã®éåã®èŠçŽ ãéšåéåã®ã¿ãèããå Žåãèããããã®ãããªãšããéåUã '''å
šäœéå'''ïŒãããããã
ããããè±ïŒuniversal setïŒ ãšããã
å
šäœéåUã®èŠçŽ ã®ãã¡ãéåAã«å±ããªããã®å
šäœãããªãéåã®ããšãAã® '''è£éå''' ïŒã»ãã
ããããè±ïŒcomplementïŒãšãããèšå·ã§è£éå㯠<math>\overline{A}</math>ãšè¡šãã
ããªãã¡
:<math>\overline{A}</math> ïŒ {x | xâU ã〠x <math>\notin</math> A}
ã§ããã
::<math>\overline{ (\overline{ A }) }</math> 㯠<math>\overline{A}</math> ã®è£éåãè¡šãããã<math>\overline{ (\overline{ A }) }</math> </span> 㯠<math>\overline{ \overline{ A } }</math> ãšæžãå Žåãããã
è£éåã«ã€ããŠã次ã®ããšãæãç«ã€ã
Aâ©<span style="text-decoration: overline">A</span>ïŒâ
, ããAâª<span style="text-decoration: overline">A</span>ïŒU , ãã<math>\overline{ (\overline{ A }) }</math> ïŒAã
;ãã»ã¢ã«ã¬ã³ã®æ³å<ref>
[[File:AugustusDeMorgan.png|thumb|ãã»ã¢ã«ã¬ã³]]
[[w:ãªãŒã¬ã¹ã¿ã¹ã»ãã»ã¢ã«ã¬ã³]]ã¯19äžçŽã€ã®ãªã¹ã®æ°åŠè
ã</ref>
:<math>\overline{A \cap B}=\overline{A} \cup \overline{B}</math>
: ã
:<math>\overline{A \cup B}=\overline{A} \cap \overline{B}</math>
äžã®å³ãçšããŠäžã®æ³åãæ£ããäºã確ããããã
<gallery>
Image:Venn-diagram-AB.png|2ã€ã®éåã®äžéšã«éãªã£ãŠããéšåãããå Žå
Image:Venn-diagram-ABC.png|çæ¹ãããçæ¹ã®éšåéåã§ããå ŽåïŒAãšBïŒãšãéåå士ãç¬ç«ããŠããå ŽåïŒAãšCïŒ
</gallery>
<gallery widths=300px heights=300px>
Image:Venn_diagram_ABC_RGB.png|3ã€ããå ŽåïŒAãšBãBãšCãCãšAïŒ
Image:Venn diagram ABCD RGB.png|4ã€ããå ŽåïŒAãšBãAãšCãAãšDãBãšCãBãšDãCãšDïŒ
</gallery>
* åé¡
A={x|xã¯1以äž20以äžã®2ã®åæ°}ã»B={y|yã¯1以äž20以äžã®3ã®åæ°}ãšããæã以äžã«é©ããéåã®èŠçŽ ãåæããããã ããå
šäœéåU={z|zã¯1以äž20以äžã®æŽæ°}ãšããã
# <math>\overline{A}</math>
# <math>\overline{B}</math>
# <math>A \cap B</math>
# <math>A \cup B</math>
# <math>A \cup \overline{B}</math>
# <math>\overline{A \cap B}</math>
# <math>\overline{A} \cup \overline{B}</math>
# <math>A \cap \overline{A}</math>
# <math>B \cup \overline{B}</math>
# <math>A \cap \varnothing</math>
# <math>B \cup \varnothing</math>
# <math>A \setminus B</math>
# <math>B \setminus A</math>
# <math>(A \setminus B) \cap (B \setminus A)</math>
* 解ç
# <math>\overline{A}</math>={1,3,5,7,9,11,13,15,17,19}
# <math>\overline{B}</math>={1,2,4,5,7,8,10,11,13,14,16,17,19,20}
# <math>A \cap B</math>={6,12,18}
# <math>A \cup B</math>={2,3,4,6,8,9,10,12,14,15,16,18,20}
# <math>A \cup \overline{B}</math>={1,2,4,5,6,7,8,10,11,12,13,14,16,17,18,19,20}
# <math>\overline{A \cap B}</math>={1,2,3,4,5,7,8,9,10,11,13,14,15,16,17,19,20}
# <math>\overline{A} \cup \overline{B}</math>={1,2,3,4,5,7,8,9,10,11,13,14,15,16,17,19,20}
# <math>A \cap \overline{A} = \varnothing</math>
# <math>B \cup \overline{B} = U</math>={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}
# <math>A \cap \varnothing = \varnothing</math>
# <math>B \cup \varnothing</math>= {3,6,9,12,15,18}
# <math>A \setminus B</math>={2,4,8,10,14,16,20}
# <math>B \setminus A</math>={3,9,15}
# <math>(A \setminus B) \cap (B \setminus A) = \varnothing</math>
{{-}}
===== ïŒâ» ç¯å²å€ïŒéåã®ãèŠçŽ ãã¯ãå
ããšããã =====
:ïŒâ» ç·šéè
ãžïŒã³ã©ã åããããšæããŸããããèšå·ãå¹²æžããŠè¡šç€ºãäžæãåºæ¥ãªãã®ã§ãç¬ç«ããç¯ãšããŸãã
:â» å³æžé€šãªã©ã§æ°åŠæžã調ã¹ãããã«ãå
ããšããè¡šèšãåºãŠãããšæããŸãã®ã§ã玹ä»ããŠãããŸãã
<!-- çŸåœ¹é«æ ¡çã«ã¯åœé¢ã¯äžèŠãªæ
å ±ãªã®ã§ãç« æ«ã«çœ®ãããšæããŸãã -->
éåã®èŠçŽ ã®ããšãããµããåŒã³æ¹ã§ããã'''å
'''ãïŒããïŒãšãèšããŸãã
念ã®ããäŸã瀺ããšãããšãã°ã1以äžãã€12以äžã®å¶æ°ã®éåããšèšãã°
ããšãã°ãã10以äžã®èªç¶æ°ã®ãã¡å¶æ°ã§ãããã®ãã®éå
: {2, ã4, ã6, ã8, ã10}
ãéåAãšããå Žåã
ãã4ãã¯éåAã®èŠçŽ ã§ããããšèšããŸãããåæ§ã«ãã4ãã¯éåAã®å
ã§ããããšãèšããŸãã
äŸãšããŠã4ãããããŸããããå¥ã«ã6ãã§ãã10ãã§ãæ§ããŸãããã2ããã6ããã8ããã10ãããããããäžè¿°ããéåAã®èŠçŽ ïŒå
ïŒã§ãã
=== åœé¡ãšèšŒæ ===
==== åœé¡ãšæ¡ä»¶ ====
ïŒæ°åŠçã«ïŒæ£ãããã©ãããæ確ã«å€æã§ãã䞻匵ã'''åœé¡'''ïŒããã ããè±: propositionïŒãšåŒã¶ã
äŸãã°ãã7ã¯çŽ æ°ã§ãããã¯åœé¡ã®äŸã§ããã
ïŒäžæ¹ãã5000ã¯å€§ããæ°ã§ããããªã©ã¯åœé¡ãšã¯ãªããªãããªããªãã倧ããããšããèšèã®å€æã䞻芳çãªãã®ã§ãããå€æã«æ確ãªåºæºãèšå®ã§ããªãããã§ãããïŒ
ããåœé¡ãæ確ã«æ£ããïŒãšèšŒæãããïŒãšãããã®åœé¡ã¯'''ç'''ïŒãããè±ïŒtruthïŒã§ãããšåŒã¶ã(ããšãã°ãåœé¡ã7ã¯çŽ æ°ã§ãããã¯çã§ããã)
åœé¡ãçã§ãªããšããåœé¡ã¯'''åœ'''ïŒããè±ïŒfalseïŒã§ãããšèšããããšãã°ãåœé¡ ã ãã <math>x^2 = 4</math> ã§ããã° <math>x = 2</math> ã§ããã ã ã¯ãåœã®åœé¡ã§ããã
ãã®æ¹çšåŒã¯<math>x = -2</math>ã解ã«æã€ã
äžã®åœé¡ã"<math>x^2 = 4</math>ãªãã°<math>x = 2</math>ã§ãã"ãã¯<math>x = -2</math>ãããŠã¯ãŸãã®ã§åœã«ãªã£ãã
åœé¡<math>\rm p \Rightarrow q</math>ãåœã§ãããšãã¯ã<math>p</math>ã¯æºããã<math>q</math>ãæºãããªãäŸãååšããããã®ãããªäŸã'''åäŸ'''ïŒã¯ãããïŒãšããã'''åœé¡ãåœã§ããããšã瀺ãã«ã¯ãåäŸã1ã€ãããã°ããã'''
åœé¡ã¯ããpãªãã°qã§ãããã®åœ¢åŒã§æžãããå Žåãå€ãã
ã pãªãã°qã§ããããšããåœé¡ããèšå·ã<math>\Rightarrow</math>ããçšããŠ
:<math> \rm p \Rightarrow q </math>
ãšæžãã
ãŸãããã®æ¡ä»¶pããã®åœé¡ã®'''ä»®å®'''ïŒããŠããè±ïŒassumptionïŒãšãããæ¡ä»¶qããã®åœé¡ã®'''çµè«'''ïŒãã€ããïŒãšåŒã¶ã
* åé¡
次ã®åœé¡ã®çåœãå€å®ããåœã®å Žåã¯åäŸãæããã
# ã<math>ab>0</math>ãªãã°<math>a>0</math>ãã€<math>b>0</math>ã§ããã
# ã<math>a \ge b</math>ãã€<math>b \ge a</math>ãªãã°<math>a=b</math>ã§ããã
# ãæ£äžè§åœ¢ã2ã€çšæããã°ãããã¯çžäŒŒã§ããã
# ãçŽ æ°ãªãã°å¥æ°ã§ããã
* 解ç
# ãåœ(åäŸïŒ<math>a=-1 , b=-2</math>ãªã©)
# ãç
# ãç
# ãåœ(åäŸïŒ2)
==== åœé¡ãšéå ====
[[File:Sets contraposition diagram.svg|thumb|]]
æ¡ä»¶ãæ¡ä»¶ãå«ãåœé¡ãèããããšã¯ãéåãèããããšãšåãã§ããã
ããšãã°ãå®æ° x ã«ã€ããŠãxïŒ3 ãªãã° xïŒ1 ã§ããããšããåœé¡ã¯çã§ããã
ããã§ãxïŒ3 ã§ããããšããæ¡ä»¶ã p ãšãããŸããxïŒ3 ã§ããæ°ã®éåã P ãšããããã€ãŸã PïŒïœx| x>3 ïœã§ããã
åæ§ã«ããxïŒ1ã§ããããšããæ¡ä»¶ã q ãšããxïŒ1ã§ããæ°ã®éåã Q ãšããããã€ãŸã QïŒïœx| x>1 ïœã§ããã
ãã®ãšããåœé¡ <math>\rm p \Longrightarrow q</math> ã¯çã§ããããããã¯éåã®å
å«é¢ä¿ PâQ ãæãç«ã€ããšã«å¯Ÿå¿ããŠããã
==== å¿
èŠæ¡ä»¶ãšååæ¡ä»¶ ====
[[File:å¿
èŠæ¡ä»¶ãšååæ¡ä»¶.svg|thumb|]]
2ã€ã®æ¡ä»¶ p,q ã«ã€ããŠãåœé¡ãpâqããçã§ãããšãã
:pã¯qã§ããããã® '''ååæ¡ä»¶''' ïŒãã
ãã¶ã ãããããïŒã§ãã
:qã¯pã§ããããã® '''å¿
èŠæ¡ä»¶''' ïŒã²ã€ãã ãããããïŒã§ãã
ãšããã
2ã€ã®æ¡ä»¶ p.q ã«ã€ããŠã
åœé¡ãpâqããšåœé¡ãqâpãã®äž¡æ¹ãšãçã§ãããšããããã
:<math>\rm p \Longleftrightarrow q</math>
ãšæžãã
:pã¯qã§ããããã®'''å¿
èŠååæ¡ä»¶'''ã§ãã
ãšããã
ãã®ãšããpãšqãå
¥ãæ¿ããããšã§ã
:qã¯pã§ããããã®å¿
èŠååæ¡ä»¶ã§ãã
ãšããããããšããããã
<math>\rm p \Longleftrightarrow q</math> ã§ãããšããpãšqã¯ã'''åå€'''ïŒã©ãã¡ïŒã§ããããšããã
==== ããã€ãããŸãã¯ããšåŠå® ====
[[File:Logic intersection P and Q.svg|thumb|ãpãã€qãã«å¯Ÿå¿ãã<br> Pâ©Q]]
[[File:Logic union P or Q.svg|thumb|ãpãŸãã¯qãã«å¯Ÿå¿ãã<br> PâªQ]]
æ¡ä»¶ p,q ãæºãããã®ã®éåããããã P,Q ãšããã
ãã®ãšããæ¡ä»¶ãpãã€qãããã³ãpãŸãã¯qããããããå³ã¯ãããããå³å³ã®ããã«ãªãã
:â» æ°åŠã«ãããããŸãã¯ãã®äœ¿ãæ¹ã§ã¯ããpãŸãã¯qãã¯ãæ¡ä»¶pãšæ¡ä»¶qã®å°ãªããšãã©ã¡ããäžæ¹ã§ãæãç«ã£ãŠããã°ããã
æ¡ä»¶pã«å¯Ÿããpã§ãªããã®åœ¢ã®æ¡ä»¶ã pã® '''åŠå®''' ïŒã²ãŠããè±ïŒnegationïŒãšãããèšå·ã¯ <math>\overline{p}</math>ã§è¡šãã
ïŒâ» é«æ ¡ã§ã¯ç¿ããªãããåŠå®ã®æå³ãšããŠã <math>\lnot{p}</math>ãšããèšå·ã¬ãããããïŒ
æ¡ä»¶ãèããããšã¯éåãèããããšãšåããªã®ã§ãéåã«ããããã»ã¢ã«ã¬ã³ã®æ³åãšåæ§ã«ãæ¡ä»¶ã«ãããŠãããã»ã¢ã«ã¬ã³ã®æ³åããªãç«ã€ã
'''ãã»ã¢ã«ã¬ã³ã®æ³å'''
<span style="text-decoration: overline">p ã〠q</span> <math> \Longleftrightarrow </math> <span style="text-decoration: overline"> p </span> ãŸã㯠<span style="text-decoration: overline"> q </span>
<span style="text-decoration: overline">p ãŸã㯠q</span> <math> \Longleftrightarrow </math> <span style="text-decoration: overline"> p </span> ã〠<span style="text-decoration: overline"> q </span>
==== éã»è£ã»å¯Ÿå¶ ====
[[File:Contraposition etc japanese.svg|thumb|400px]]
åœé¡ã <math>\rm p \Longrightarrow q</math> ãã«å¯ŸããŠ
:åœé¡ã<math>\rm q \Longrightarrow p</math>ãã åœé¡ã <math>\rm p \Longrightarrow q</math> ãã® '''é''' ïŒããããè±ïŒconverseïŒ
:åœé¡ã<math>\rm \overline{p} \Longrightarrow \overline{q}</math>ãã åœé¡ã <math>\rm p \Longrightarrow q</math> ãã® '''è£''' ïŒãããè±ïŒinverseïŒ
:åœé¡ã<math>\rm \overline{q} \Longrightarrow \overline{p}</math>ãã åœé¡ã <math>\rm p \Longrightarrow q</math> ãã® '''察å¶''' ïŒãããããè±ïŒcontrapositionïŒ
ãšåŒã¶ã
ãããã¯ããããã«å³å³ã®ãããªé¢ä¿ã«ããã
{{-}}
----
ããšãã°ã
ããšã®åœé¡ã
:ã<math>x=3 \Longrightarrow x^2=9 </math>ã
ã ãšãããšã
:è£ã¯ã<math>x\ne 3 \Longrightarrow x^2 \ne 9 </math>ã ã§ãã
:éã¯ã<math>x^2=9 \Longrightarrow x=3 </math>ã ã§ãã
:察å¶ã¯ã<math>x^2 \ne 9 \Longrightarrow x\ne 3 </math>ãã§ããã
ãã®åœé¡ã®å Žåãããšã®åœé¡ãšå¯Ÿå¶ã¯ããšãã«çã§ããã
ãã£ãœãéã«ã€ããŠã¯ x ïŒ -3 ãšããåäŸãããã®ã§ããã®åœé¡ã®å Žåãéã¯æ£ãããªãããŸããè£ãåæ§ã«ãæ£ãããªãã
ãã®ãããªäŸããã次ã®ããšãåããã
ããåœé¡ãçã§ãã£ãŠãããã®åœé¡ã®éã¯ãããªããããçãšã¯éããªãã
ãŸããããåœé¡ãçã§ãã£ãŠãããã®åœé¡ã®è£ã¯ãããªããããçãšã¯éããªãã
ã§ã¯ãããšã®åœé¡ãšå¯Ÿå¶ãšã®é¢ä¿ã¯ãã©ããªãã ãããã
[[File:Sets contraposition diagram.svg|thumb|]]
ãã®èå¯ããããããæ¡ä»¶pãæºãããã®ãéåPã«å¯Ÿå¿ãããåæ§ã«æ¡ä»¶qãæºãããã®ãéåQã«å¯Ÿå¿ãããŠã¿ããã
å³ã®éåã®å³ã¯ãpâqãçã§ããããšãè¡šãå³ã§ããããã®å³ã§ã¯ãPã«å±ããŠããèŠçŽ ã¯ãQã«ãå±ããŠãããïŒã€ãŸã <math>\rm P \subset Q</math>ã§ãããïŒäžæ¹ãQã«å±ããŠãããªãèŠçŽ ã¯ãPã«ãå±ããŠããªããïŒã€ãŸã <math>\rm \overline{Q} \subset \overline{P}</math> ã§ãããïŒ
ãã®ããšããããåããããã«ã
ããåœé¡ãçã§ãããšãããã®åœé¡ã®å¯Ÿå¶ãçãšãªãã
ããåœé¡ãåœã§ãããšãããã®å¯Ÿå¶ãåœã§ããã
ã€ãŸããäžè¬ã®åœé¡ã«ãããŠãããšã®åœé¡ãšå¯Ÿå¶ãšã®çåœã¯äžèŽããã
==== èçæ³ ====
ããåœé¡ã®çµè«ãåŠå®ããŠããã®åŠå®ã®ããšã§ççŸãèµ·ããããšãè¿°ã¹ãããšã§ã
ãã®åœé¡ãçã§ããããšãå°åºããä»æ¹ã'''èçæ³'''ïŒã¯ããã»ããè±: proof by contradiction ãªã©ïŒãšåŒã¶ã
ããšãã°ããAã§ã¯ãªãããšã蚌æããããšããåé¡ã解ãæã¯ãAã§ãããšä»®å®ããããšæžãåºããŠãä»®å®ããããšãšççŸããéšåãäœã£ãŠãççŸããã®ã§Aã§ã¯ãªããããšèšŒæãçµããã
* äŸé¡
çŽ æ°ã¯ç¡éã«ååšããã
* 蚌æ
çŽ æ°ãæéåã§ãã£ããšä»®å®ããããã¹ãŠã®çŽ æ°ã®ç©ã<math>a</math>ãšãããšã<math>a+1</math>ã¯ã©ã®çŽ æ°ã§å²ã£ãŠã1äœãããšã«ãªãã1以å€ã®èªç¶æ°ã§ãã£ãŠãçŽ æ°ã®ç©ã«å解ã§ããªããã®ãååšããããšã«ãªãã<math>a+1</math>ã®çŽæ°ã®ãã¡1以å€ã§æãå°ãããã®ã<math>b</math>ãšãããšã<math>b</math>ã¯1ãš<math>b</math>以å€ã®çŽæ°ãæããªãããããã£ãŠ<math>b</math>ãçŽ æ°ã§ããããšã«ãªããã<math>a+1</math>ãã©ã®çŽ æ°ã§ãå²ãåããªãããšãšççŸããããããã£ãŠãçŽ æ°ã¯æéåã§ã¯ãªããâ
{{ã³ã©ã |èçæ³ãšæ¥åžžçãªæè|
èçæ³ã¯å€ãã®é«æ ¡çãèŠæãšããŠããŸãããAã§ãããããšã蚌æããããã«ããããããAã§ãªãããšä»®å®ããŠççŸãå°ããšããè«çã®å±éãäžèªç¶ã«æãããããããèŠææèã«ã€ãªãã£ãŠããããã§ããããããèçæ³ã®çºæ³ã¯ç§ãã¡ã®æ¥åžžçãªæèã§ããã䜿ãããŠããŸããããã§ã¯ããã®äŸãããã€ã玹ä»ããŸãããã
äžã€ç®ã¯ã¢ãªãã€èšŒæã§ããããããªããäºä»¶ã®ç¯äººã§ãããšçããããšããŸãããã®ãšãããèªåãç¯äººã§ã¯ãªãããšããããšãã©ã®ããã«èšŒæããŸããããã ãèªåã¯ç¯äººã§ã¯ãªãããšèšãã ãã§ã¯èª¬åŸåããããŸããããã®å Žåãç¯è¡çŸå ŽãAé§
ã§ãã£ããããèªåã¯äºä»¶ãèµ·ãããšãã«ã¯Bé§
ã«ãããããšã蚌æã§ãããã€ãŸãã¢ãªãã€ãæãç«ã€ãªãã°èªåãç¯äººã§ã¯ãªããšããæåãªèšŒæ ãšãªããŸãããã®ä»çµã¿ãç°¡åãªæã«ãããšä»¥äžã®ããã«ãªããŸãã
:ç§ãç¯äººã ãšä»®å®ãããšãAé§
ã«ããããšã«ãªãã
:ãããç§ã¯Bé§
ã«ããïŒïŒAé§
ã«ããªãã£ãïŒã
:ç§ãåãæéã«Aé§
ãšBé§
ã®äž¡æ¹ã«ããããšã¯ã§ããªãã®ã§ãåœåã®ä»®å®ãšççŸããã
:ããã«ç§ãç¯äººã ãšããä»®å®ããŸã¡ãã£ãŠããã®ã§ãç§ã¯ç¯äººã§ã¯ãªãã
ã¢ãªãã€ã瀺ãããšã§èªåã®ç¡å®ã蚌æãããšããã®ã¯ãå®ã¯ããããä»çµã¿ã«ãªã£ãŠããã®ã§ãããªããã¢ãªãã€ã瀺ããŠç¡å®ã蚌æããæ¹æ³ã«ã¯ããã¯ãé«æ ¡çã®å€ããèŠæãšãã察å¶èšŒææ³ã䜿ãæ¹æ³ããããŸãããã¡ãã¯ã¿ãªããã§èããŠã¿ãŠãã ããã
äºã€ç®ã¯æ¶å»æ³ã§ã®éžæè¢ã®éžã³æ¹ã§ããããšãã°ãå¹³æ30幎床ãå«çãã®ã»ã³ã¿ãŒè©Šéšã®å€§å1å7ïŒããŒã¯ã·ãŒãçªå·7ïŒã®åé¡ãèŠãŠã¿ãŸããããïŒãã°ãšæãããçŸè±¡ãèµ·ãããããªã³ã¯ã¯è²ŒããŸããããææ°ã§ãããåèªç¢ºèªããŠäžãããïŒããã®æ£è§£ãå°ãåºãã®ã«èçæ³ã®æç« ãå©çšããŠã¿ãŸãããã
:1ãæ£è§£ã ãšä»®å®ãããšã1ã®éžæè¢ã¯ã°ã©ããæ£ãã説æããŠããã
:ãããã2015幎ã®ããã3ïŒæ¥äŒç¬ïŒãš2050幎ã®ããã3ïŒæ¥äŒéïŒã¯ç°ãªãã®ã§ãæ£ãã説æã«ãªã£ãŠããããä»®å®ãšççŸããã
:ããã«1ãæ£è§£ãšããä»®å®ã¯èª€ã£ãŠããã
:ã ããã1ã®éžæè¢ã¯æ£ãããªãã®ã§æ¶ãã
ãããç¹°ãè¿ããšã誀ã£ãéžæè¢ãæ¶å»ããŠæ£è§£ãå°ãããšãã§ããŸããïŒãå«çãã®ç¥èã¯å
šããããªãã®ã§ãçãããææŠããŠã¿ãŠãã ããïŒã
ãã¡ãããã¢ãªãã€èšŒæã«ããŠãæ¶å»æ³ã«ããŠãããã€ãããããæäœã§è§£ããŠããããã§ã¯ãããŸãããããããèçæ³ã®ããšã¯æèããªãã§è§£ãã®ãåœããåã§ããããå®ã¯èçæ³ã®èãæ¹ã¯æ¬æ¥ãäœæ°ãªãå®è¡ã§ãããããèªç¶ãªçºæ³ãªã®ã§ããã»ãã«ããçããã®æ¥åžžçãªèãæ¹ã®äžã«èçæ³ã®åœ¢ã«ãããã®ãããã¯ãã§ããéã«äžèŠæ£ããèçæ³ã«èŠããŠããå®ã¯ã€ã³ãããªè«çå±éã®ãã®ãããã§ããããããããããšãæ¢ããŠããã®ãè«ççã«èããããã®ãã¬ãŒãã³ã°ã«ãªããŸãããã²ãææŠããŠã¿ãŠãã ããã
ãåèæç®ã
*ãè«ççã«èããããšãïŒå±±äžæ£ç·è, 岩波æžåºïŒå²©æ³¢ãžã¥ãã¢æ°æžïŒïŒ
*ãè«ççã«èããæžãåãïŒåæ²¢å
éè, å
æ瀟ïŒå
æ瀟æ°æžïŒïŒ
}}
== è泚 ==
<references/>
[[category:é«çåŠæ ¡æ°åŠA|ããããããšããã]] | 2005-05-08T02:54:42Z | 2024-03-04T17:41:53Z | [
"ãã³ãã¬ãŒã:-",
"ãã³ãã¬ãŒã:ã³ã©ã "
] | https://ja.wikibooks.org/wiki/%E6%97%A7%E8%AA%B2%E7%A8%8B(-2012%E5%B9%B4%E5%BA%A6)%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E6%95%B0%E5%AD%A6A/%E9%9B%86%E5%90%88%E3%81%A8%E8%AB%96%E7%90%86 |
1,931 | é«çåŠæ ¡æ°åŠA/å Žåã®æ°ãšç¢ºç | ããšãã°ããã¯ãããäžåã«äžŠã¹ãå Žåã䞊ã¹æ¹ã®æ°ã«ã¯ãããã€ãã®æ¹æ³ãããããã£ããã«å
šãŠã®äžŠã³æ¹ãè©Šãããšããæéããããã°å®éšå¯èœã§ããã
ãã®ããã«ããå
šéšã§äœéãããããããšããããã®ãäœéããã®ãäœãã«ãããæ°åããå Žåã®æ°(ã°ããã®ãã) ãšåŒã¶ã
ãã®ããã«äºæã«ã¯ããããã®ããæ¹ãå
šéšã§äœéãããããæ°ããããšãåºæ¥ãäºæãããã
ããäºæã«ã€ããŠ(ãã®ããšãèµ·ãããã)å Žåã®æ°ãæ£ç¢ºã«æ°ããããšãç解ã®åºç€ã§ããããã®äºæã«ã€ããŠãã©ã®ããšãèµ·ãããããã©ã®ããšãèµ·ããã¥ããããèŠåããããã®åºç€ãšãªãã ã€ãŸããå Žåã®æ°ã¯äºæãèµ·ãããã確çãšå¯æ¥ãªé¢ä¿ã«ããã
äŸãã°ãããŒã«ãŒãªã©ã®ã«ãŒãã²ãŒã ã§ã¯éããããšãé£ãã圹ã¯é«ãã©ã³ã¯ãäžããããŠãããã ããã¯èµ·ããã«ãã圹ãåºæ¥ããã©ã³ãã®çµã¿åããã®çŸããã確çãå°ããããšã«ããã ãã®ããšã¯ã52æã®ã«ãŒããã5æãåŒããŠæ¥ããšãã«å
šãŠã®ã«ãŒããåŒã確çãåãã§ãããšãããšãããã圹ã«å¯Ÿå¿ããã«ãŒãã®çµã¿åãããåŒãå Žåã®æ°ãããå°ãªãããšã«å¯Ÿå¿ããã
ãã®ããã«ãå Žåã®æ°ã¯äºæãèµ·ãããã確çãšå¯æ¥ãªé¢ä¿ã«ããã
ã«ãŒãã²ãŒã ã®ããã«ç¢ºçãå
·äœçã«èšç®ã§ããå Žåã®ä»ã«ãã確çã®èãæ¹ãçšããŠèšç®ãããäºæã¯å€ãããã
ããšãã°ãä¿éº(ã»ãã)ãšåŒã°ãããã®ã¯ããäºæã«å€æ®µãã€ãããã®ã§ãããã ä¿éºãäžãããªããŠã¯ãªããªãäºæãèµ·ããã«ãããšå®¢èŠ³çã«æããããã®ã»ã©ããã®ãã®ã®å€æ®µãäžãããšããç¹åŸŽãããã äŸãã°ãèªåè»ä¿éºã«å å
¥ããã®ã«å¿
èŠãªä»£éã¯è¥è
ã§ã¯é«ãã幎什ãéããããšã«äœããªã£ãŠããã ããã¯ãè¥è
ã¯èªåè»ã®å
èš±ãååŸããŠæéãçãå Žåãå€ããä¿éºéã®æ¯æãå¿
èŠãšããèªåè»äºæ
ããããå¯èœæ§ãé«ãããšã«ããã ãã£ãœãã幎什ãéãããã®ã«ã€ããŠã¯é転ã®æéãæãšãšãã«äžéãããšäžè¬ã«èããããã®ã§ä¿éºããããããã®ä»£éã¯å°ãªããªãã®ã§ããã ãŸããåãè¥è
ã§ãæ¢ã«äœåºŠãäºæ
ãéãããã®ã¯åã幎代ã®ä»ã®è¥è
ãããä¿éºæãé«ããªãåŸåãããã ããã¯ãäœåºŠãäºæ
ãéãããã®ã¯é転ã®ä»æ¹ã«äœããã®åé¡ãããåŸåããããããã«ãã£ãŠãµããã³äºæ
ããããå¯èœæ§ãéåžžã®ãã®ãšæ¯ã¹ãŠããé«ããšèããããããšã«ããã
éè¡ã®èè³(ããã)ã§ããã¯ã確çã®èããçšããŠé«ãå©çãåºãããšãå®è·µãããŠããã èè³ã§ããã¯ãä¿éºæ¥ãšããªããããã貞åãã«ãªãå¯èœæ§ãé«ãçžæã«å¯ŸããŠã¯é«ãéå©ã§è³éã貞ãä»ãã ããå®å®ããè³éãæã£ãŠããçžæã«å¯ŸããŠã¯ããäœãéå©ã§è³éã貞ãä»ããããšãå®è¡ããŠæ¥ãã
å©çãå®å®çã«çšŒãæ¹æ³ãšããŠãããã€ãã®äŒç€Ÿãçºè¡ããäºãã«æ§è³ªã®ç°ãªã£ãæ ªãªã©ãåãããŠè³Œå
¥å
ãåæ£ããããšã§æ ªã®å€æ®µãäžãã£ããšãã§ãå€æ®µãããŸãæžãããšãç¡ãããã«ããæ¹æ³ãèæ¡ãããŠããã (ãã ããå€æ®µãæžãã¥ããã®ãšåæ§ã«ãå€æ®µã¯äžããã¥ããã) ããã¯ãæ§è³ªã®ç°ãªã£ãååãåãããŠæ±ãããšã§ãå€æ®µãæ¥å€ãã確çãäžããããšãåºæ¥ãããšãè¡šãããŠããã
ãããã確çã§ã¯ãå¿
ãããäºæž¬ããéãã«äºãé²ãããã§ã¯ç¡ãããšã«æ³šæããå¿
èŠãããã
ãã®ç« ã§ã¯å Žåã®æ°ãšç¢ºçã®èšç®æ³ã玹ä»ããããŸãå
ã«æ§ã
ãªäºæã®å Žåã®æ°ã®èšç®æ³ãæ±ãããã®çµæãçšããŠããäºæãèµ·ãã確çãèšç®ããæ¹æ³ã玹ä»ããã
ããã§ã¯ãæééå A ã®èŠçŽ ã®åæ°ã n(A) ã§è¡šãã
ããšãã°ã10以äžã®èªç¶æ°ã®éåã U ãšããŠããã®ãã¡ å¶æ°ã®éåã A ãšããå Žåã
ãªã®ã§ãAã®èŠçŽ ã®åæ°ã¯5åãªã®ã§
ã§ããã
ãªãã U={1, 2, 3, 4, 5, 6 , 7, 8, 9, 10} ã§èŠçŽ ã®åæ°ã¯10åãªã®ã§
ã§ããã
次ã®ãããªåé¡ãèããŠã¿ããã 100ãŸã§ã®èªç¶æ°ã®ãã¡ã2ãŸãã¯3ã®åæ°ã¯äœåããã?
ãã®ãããªåé¡ã®è§£æ³ãèãããããæºåã®åé¡ãšããŠããŸã10ãŸã§ã®èªç¶æ°ã§èããŠã¿ããã
å
çšã®äŸé¡ã§2ã®åæ°ã«ã€ããŠã¯èããã®ã§ã次ã®åé¡ãšããŠ10ãŸã§ã®3ã®åæ°ã®åæ°ã«ã€ããŠèãããã
10以äžã®èªç¶æ°ã®éåã U ãšããŠããã®ãã¡ 3ã®åæ°ã®éåã B ãšããå Žåã
B={3, 6 , 9} ãªã®ã§ãBã®èŠçŽ ã®åæ°ã¯3åãªã®ã§
ã§ããã
ããŠã
ã«ã¯å
±éã㊠6 ãšããèŠçŽ ãå«ãŸããŠããã
èªç¶æ°10ãŸã§ã«ãã2ãŸãã¯3ã®åæ°ã«ãããèŠçŽ ã¯ã
ã§ãããèŠçŽ ã®åæ°ããããããš 7åã§ããã
äžæ¹ã
ã§ããã1åå€ãã
ãã®ããã«1åå€ããªã£ãŠããŸã£ãåå ã¯ã éåAãšéåBã«å
±éããŠå«ãŸããŠããèŠçŽ 6 ãäºéã«æ°ããŠããŸã£ãŠããããã§ããã
äžè¬ã«ã2ã€ã®éåA,Bã®èŠçŽ ã®åæ° n(A) ãš n(B) ãçšããŠãAãŸãã¯Bã®æ¡ä»¶ãæºããèŠçŽ ã®åæ°ããããããå Žåã«ã¯ãAãšBã«å
±éããŠå«ãŸããŠããèŠçŽ ã®åæ°ãå·®ãåŒããªããã°ãªããªãã
ãã®ããšãåŒã§è¡šããš
ã«ãªãã
ãã ãããâªããšã¯åéåã®èšå·ã§ã AâªB ãšã¯ éåAãšéåBã®åéåã®ããšã§ããã
ãâ©ããšã¯å
±ééšåã®èšå·ã§ã ãAâ©Bããšã¯ éåAãšéåBã®å
±ééšåã®ããšã§ããã
ã§ã¯ããã®å
¬åŒãåèã«ã㊠100ãŸã§ã®èªç¶æ°ã®ãã¡ã2ãŸãã¯3ã®åæ°ã¯äœåããã? ã®çããæ±ãããã
100ãŸã§ã®èªç¶æ°ã®ãã¡ã®ã2ã®åæ°ã®éåãAãšããŠã3ã®åæ°ã®éåãBãšãããš
ããã«ã2ã®åæ°ã§ããã3ã®åæ°ã§ãããæ°ã®éå Aâ©B ãšã¯ãã€ãŸã6ã®åæ°ã®éåã®ããšã§ãã(ãªããªã 2 ãš 3 ã®æå°å
¬åæ°ã 6 ãªã®ã§)ã 96÷6=16 ãªã®ã§ãAâ©B ã®èŠçŽ ã®åæ°ã¯ 16 åãã€ãŸã n(Aâ©B)= 16 ã§ããã
ãããŠãå
¬åŒ
ãé©çšãããšã
ã§ããã
ãã£ãŠã100ãŸã§ã®èªç¶æ°ã®ãã¡ã®2ãŸãã¯3ã®åæ°ã®åæ°ã¯ 67å ã§ããã
3ã€ã®æééåã®åéåã®èŠçŽ ã®åæ°ã«ã€ããŠã¯ã次ã®å
¬åŒãæãç«ã€
n(AâªBâªC) = n(A) + n(B) + n(C) ân(Aâ©B) ân(Bâ©C) ân(Câ©A) + n(Aâ©Bâ©C)
å³ã®å³ãåèã«ãäžã®å
¬åŒã蚌æããã
100以äžã®èªç¶æ°ã®ãã¡ã2ã®åæ°ãŸãã¯3ã®åæ°ãŸãã¯5ã®åæ°ã§ãããã®ã®åæ°ãæ±ããã
(解æ³)
ãŸãã100以äžã®èªç¶æ°ã®ãã¡ã
ãšããã
100÷2=50ãªã®ã§ã100ã¯50çªç®ã®2ã®åæ°ã§ããããã£ãŠ100以äžã®2ã®åæ°ã¯50åã§ãããåæ§ã«èããŠèŠçŽ ã®åæ°ãæ±ãããšã
ã§ããã
äžæ¹ã100以äžã®èªç¶æ°ã®ãã¡
ãšãªãã
ãã£ãŠãå
ã»ã©ãšåæ§ã«èãããš
ãŸãã100以äžã®èªç¶æ°ã®ãã¡ã
Aâ©Bâ©C ã®èŠçŽ ã®åæ°ã¯
ã§ããã
ãã£ãŠã
ãªã®ã§ã100以äžã®èªç¶æ°ã®ãã¡ã®2ã®åæ°ãŸãã¯3ã®åæ°ãŸãã¯5ã®åæ°ã§ãããã®ã®åæ°ã¯ 74åã§ããã
ããšãã°å€§äžå°3åã®ãµã€ã³ãããµã£ãŠãç®ã®åã5ã«ãªãç®ã®çµã¯ãäœéãããã ãããã
ãã®ãããªåé¡ã解ãæ¹æ³ã®ã²ãšã€ãšããŠãå³ã®ããã«ãçµã¿åãããç·åœããã§æžãæ¹æ³ãããã
倧äžå°ã®åèš3åã®ãµã€ã³ãããããã A,B,C ãšããŠè¡šãããããã®æåã«ãã©ã®ç®ãåºãã°åèš5ã«ãªãããèãããšãçµæã¯å³ã®ããã«ãªãã
ãã®ãããªå³ã 暹圢å³(ãã
ããã) ãšããã
3åã®ãµã€ã³ãããµããšããç®ã®åã6ã«ãªãå Žåã¯äœéããããã
æåã«ãnåã®ç°ãªã£ããã®ã䞊ã¹æããå Žåã®æ°ãæ°ããã ãŸãæåã«äžŠã¹ããã®ã¯nåã次ã«äžŠã¹ããã®ã¯(n-1)åããã®æ¬¡ã«äžŠã¹ããã®ã¯(n-2)å ... ãšã ãã ããšéžã¹ããã®ã®æ°ãæžã£ãŠè¡ããæåŸã«ã¯1åããæ®ããªããªãããšã«æ³šç®ãããšããã®äºæã«é¢ããå Žåã®æ°ã¯
ãšãªãã1ããnãŸã§ã®èªç¶æ°ã®ç©ã«ãªãã ãã®æ°ã éä¹ (ããããããfactorial)ãšåŒã³ãéä¹nã®èšå·ã¯ n ! {\displaystyle n!} ã§è¡šãã
ããªãã¡ãéä¹ã¯
n ! = n ( n â 1 ) ( n â 2 ) ⯠3 â
2 â
1 {\displaystyle n!=n(n-1)(n-2)\cdots 3\cdot 2\cdot 1}
ãšå®çŸ©ãããããã®éä¹ã®èšå·ã䜿ãã°ããã®åé¡ã®ãšãã®å Žåã®æ°ã¯ n!ã§ãããšèšãããšãåºæ¥ãã
ãããããèšç®ããã
ãçšããŠèšç®ããã°ããã çãã¯ã
ãšãªãã
ããããã«1ãã5ãŸã§ã®æ°åãæžããã5æã®ã«ãŒãã眮ããŠããã ãã®ã«ãŒãã䞊ã¹æãããšãã (I)ã«ãŒãã®äžŠã¹æ¹ã®æ°ã (II)å¶æ°ãåŸãããã«ãŒãã®äžŠã¹æ¹ã®æ°ã (III)å¥æ°ãåºãã«ãŒãã®äžŠã¹æ¹ã®æ°ããããããèšç®ããã
(I) ã«ãŒãã®æ°ã5æã§ãããããåºå¥ã§ããããšãããã«ãŒãã®äžŠã¹æ¹ã®æ°ã¯
ãšãªãã120ãšãªãã
(II) å¶æ°ãåŸãããã«ã¯äžã®äœã§ããæãå³ã«åºãã«ãŒãããå¶æ°ãšãªãã°ããã ãã®ãããªã«ãŒãã¯2ãš4ã§ãããããããã«å¯ŸããŠåŸã®4æã¯èªç±ã«éžãã§ããã ãã®ããããã®ãããªã«ãŒãã®äžŠã¹æ¹ã¯ã
ãšãªãã
(III) å¥æ°ãåŸãããã«ã¯äžã®äœã§ããæãå³ã«åºãã«ãŒãããå¥æ°ãšãªãã°ããã ãã®ãããªã«ãŒãã¯1,3,5ã§ãããããããã«å¯ŸããŠåŸã®4æã¯èªç±ã«éžãã§ããã ãã®ããããã®ãããªã«ãŒãã®äžŠã¹æ¹ã¯ã
ãšãªããäžæ¹ã5æã®ã«ãŒãã䞊ã¹æããŠåŸãããæ°ã¯å¿
ãå¶æ°ãå¥æ°ã® ã©ã¡ããã§ããã®ã§ã(I)ã®çµæãã(II)ã®çµæãåŒãããšã«ãã£ãŠã (III)ã®çµæã¯åŸãããã¯ãã ããå®éã«ãããèšç®ãããš
ãšãªãã確ãã«ãã®ããã«ãªã£ãŠããã
0,1,2,3,5ãæžããã5æã®ã«ãŒãããããããã䞊ã³æãããšãã
ãããããæ±ããã
(I) å
é ã0ã«ãªã£ããšãã«ã¯5æ¡ã®æ°ã«ãªããªãããšã«æ³šæããã°ãããæ±ããå Žåã®æ°ã¯
ãšãªãã
(II) æåã0ã§ãªãæåŸã0ã2ã§ããæ°ãæ°ããã°ããããŸããæåŸã0ã§ãããšãã«ã¯ãæ®ãã®4æã¯ä»»æã§ããã®ã§
éãã®çµã¿åãããããã
次ã«ãæåŸã2ã§ãããšãã«ã¯æåã¯0ã§ãã£ãŠã¯ãããªãã®ã§ã
éãããã 2ã€ãåãããæ°ã5æ¡ã®å¶æ°ãåŸãããå Žåã®æ°ã§ãããçãã¯ã
ãšãªãã
(III) (I)ã®çµæãã(II)ã®çµæãåŒãã°ããããããã§ã¯ãã®çµæãæ£ãããã©ãã 確ãããããã«ã5æ¡ã®å¥æ°ãåŸãããçµã¿åãããæ°ãäžããŠã¿ãã 5æ¡ã®å¥æ°ãåŸãããã«ã¯æåŸã®æ°ã¯1,3,5ã®ããããã§ãªããŠã¯ãªããªãã ãã®ãã¡ã®ã©ã®å Žåã«ã€ããŠã5æ¡ã®æ°ãåŸãããã«ã¯æåã®æ°ã0㧠åã£ãŠã¯ãªããªãã®ã§ããããã®å Žåã®æ°ã¯ã
ãšãªãããã5æ¡ã®å¥æ°ãåŸãå Žåã®æ°ã§ããã (II)ã®çµæãšè¶³ãåããããšç¢ºãã«(I)ã®çµæãšçãã96ãåŸãã
(IV) 5ã®åæ°ãåŸãããã«ã¯æåŸã®æ°ã0ã5ã§ããã°ããã ãã®ãšãæåŸã0ã«ãªãå Žåã®æ°ã¯ä»ã®4ã€ãä»»æã§ãããã
ååšããã次ã«ãæåŸã5ã«ãªãå Žåã®æ°ã¯æåã®æ°ã0ã§ãã£ãŠã¯ãªããªããã
ã ãååšããã ãã£ãŠçãã¯
ãšãªãã
nåã®ç°ãªã£ããã®ããråãéžãã§ãé çªãã€ããŠäžŠã¹ãä»æ¹ã®æ°ãã n P r {\displaystyle {}_{n}\mathrm {P} _{r}} ãšæžãã ãŸãããã®ãããªèšç®ã®ä»æ¹ã é å (ãã
ããã€ãè±:permutation) ãšããã
nåã®ç°ãªã£ããã®ããråãéžãã§é çªãã€ããŠäžŠã¹ãä»æ¹ã®æ°ã®ããš ãã
ã®ããã«èšãã
æåã«äžŠã¹ããã®ã¯néãã次ã«äžŠã¹ããã®ã¯ (nâ1)éã ããã®æ¬¡ã«äžŠã¹ããã®ã¯ (nâ2)éã ,... æåŸã«ã¯ (nâ(râ1))éã ãšããããã«ãã ãã ãéžã¹ããã®ã®æ°ãæžã£ãŠè¡ãããšã«æ³šç®ãããšãé åã®ç·æ°ãšããŠ
ãåŸãããã
äžè¬ã« n P r {\displaystyle {}_{n}\mathrm {P} _{r}} ã§ã¯ n ⧠r ã§ããã
(I)
(II)
(III)
(IV)
(V)
(VI)
ãããããèšç®ããã
ãããã
ãçšããŠèšç®ããã°ããã
çµæã¯ã (I)
(II)
(III)
(IV)
(V)
(VI)
ãšãªãã
(V)ãš(VI)ã«ã€ããŠã¯äžè¬çã«æŽæ°nã«å¯ŸããŠ
ãåŸãããããã®ãšã
ã¯å
ã
ã®é åã®å®çŸ©ãããããš"nåã®ãã®ã®äžãã1ã€ãéžã°ãªãå Žåã®æ°"ã«å¯Ÿå¿ããŠãããå°ã
äžèªç¶ãªããã«æãããããã®ããã«å€ã眮ããŠãããšäŸ¿å©ã§ããããéåžžãã®ããã«çœ®ãã®ã§ãããããŸããå®éã®å Žåã®æ°ã®èšç®ã§ãã®ãããªå€ãæ±ãããšã¯å€ãã¯ãªããšãããã
A, B, C, D, E ã®5人ãå圢ã«æãã€ãªãã§èŒªãã€ãããšãããã®äžŠã³æ¹ã¯äœéããããã
ãã®ãããªåé¡ã®å Žåãå³ã®ããã«ãå転ãããšéãªã䞊ã³ã¯åã䞊ã³ã§ãããšèããã
解ãæ¹ã®èãæ¹ã¯æ°çš®é¡ããã
ã©ã¡ãã«ãããçµæã¯
ã§ããã
äžè¬ã« ç°ãªã nå ã®ãã®ãå圢ã«äžŠã¹ããã®ãåé åãšããã
åé åã®ç·æ°ãšããŠã次ã®ããšãæãç«ã€ã
ç°ãªã nå ã®åé åã®ç·æ°ã¯ ( n â 1 ) ! {\displaystyle (n-1)!} ã§ããã
nåã®ç°ãªã£ããã®ããråãéžãã§ãé çªãã€ããã«äžŠã¹ãä»æ¹ã®æ°ãã n C r {\displaystyle {}_{n}\mathrm {C} _{r}} ãšæžãããã®ãããªèšç®ã çµã¿åãã(combination) ãšããã äŸãã°ãããã€ãããããŒã«ã«çªå·ããµã£ãŠãããªã©ã®æ¹æ³ã§ãããããã®ããŒã«ãåºå¥ã§ããnåã®ããŒã«ãå
¥ã£ãç®±ã®äžããråã®ããŒã«ãåãã ãæãåãã ããããŒã«ãåãã ããé ã«äžŠã¹ããšãããšããã®å Žåã®æ°ã¯é å n P r {\displaystyle {}_{n}\mathrm {P} _{r}} ã«å¯Ÿå¿ããã
äžæ¹ãåãã ããããŒã«ã®çš®é¡ãéèŠã§ããåãã ããé çªãç¹ã«å¿
èŠã§ãªããšãã«ã¯ããã®å Žåã®æ°ã¯çµã¿åãã n C r {\displaystyle {}_{n}\mathrm {C} _{r}} ã«å¯Ÿå¿ããããããã®æ°ã¯ãäºãã«ç°ãªã£ãå Žåã®æ°ã§ãããäºãã«ç°ãªã£ãèšç®æ³ãå¿
èŠãšãªãã
n C r {\displaystyle {}_{n}\mathrm {C} _{r}} ã¯ã n P r {\displaystyle {}_{n}\mathrm {P} _{r}} éãã®äžŠã¹æ¹ãäœã£ãåŸã«ãããã®äžŠã³ãç¡èŠãããã®ã«çãããããã§ãråãåãã ããŠäœã£ã䞊ã³ã«ã€ããŠã䞊ã¹æ¹ãç¡èŠãããšr!åã®äžŠã³ãåäžèŠãããããšããããã
ãªããªããråã®ãäºãã«åºå¥ã§ããæ°ãèªç±ã«äžŠã³æããå Žåã®æ°ã¯r!ã§ãããããããå
šãŠåäžèŠããããšããã°å
šäœã®å Žåã®æ°ã¯ r!ã®åã ãæžãããšã«ãªãããã§ããããã£ãŠã
ãåŸãããã
æŒç¿åé¡
次ã®å€ãèšç®ãã
(I)
(II)
(III)
(VI)
ããããã«ã€ããŠ
ãçšããŠèšç®ããã°ããã
(I)
(II)
(III)
(VI)
ãšãªãã(IV)ã«ã€ããŠã¯äžè¬ã«æŽæ°nã«å¯ŸããŠ
ãå®çŸ©ããã
ããã¯ããšããšã®çµã¿åããã®èšç®ãšããŠã¯nåã®ç©äœã®ãªããã0åã®ç©äœãéžã¶å Žåã®æ°ã«å¯Ÿå¿ããŠããã å®éã«ã¯ãã®ãããªå Žåã®æ°ãèšç®ããããšèããããšã¯ããŸãç¡ããšæãããããèšç®ã®äŸ¿å®äžã®ããå®çŸ©ãäžã®ããã«ããã ãŸããäžã®èšç®ã§ã¯
ã®åŒããã®ãŸãŸçšãããšã
ã€ãŸãã
ãšãªã£ãŠããã
å®éã«ã¯éä¹ã®èšç®ã¯æŽæ°nã«ã€ããŠã¯nãã1ãŸã§ãäžãããªããããç®ããŠãããšããä»æ¹ã§èšç®ãããŠããã®ã§ãäžã®çµæã¯åŠã«æããã ãããå®éã«ã¯ãããé²ãã çè«ã«ãã£ãŠãã®çµæã¯æ£åœåãããã®ã§ããã ãã®å Žåã䟿å®äž
ã0ã®éä¹ã®å®çŸ©ãšããŠåããããã®ã§ããã
æŒç¿åé¡
5åã®ããŒã«ãå
¥ã£ãããŒã«å
¥ããã2ã€ã®ããŒã«ãåãã ããšã(ããŒã«ã¯ãããã åºå¥ã§ãããã®ãšããã)2ã€ã®ããŒã«ã®éžã³æ¹ã¯ã äœéããããèšç®ããã
ããŒã«ã®åãã ãæ¹ã¯çµã¿åããã®æ°ãçšããŠèšç®ã§ããã 5ã€ã®ããŒã«ã®äžãã2ã€ãåãã ãã®ã§ãããããã®å Žåã®æ°ã¯ã
ãšãªãããã£ãŠãããŒã«ã®åãã ãæ¹ã¯10éãã§ããããšããããã
æŒç¿åé¡
6åã®äºãã«åºå¥ã§ããããŒã«ãå
¥ã£ãç®±ãããã ãã®äžãã (I)3ã€ã®ããŒã«ãš2ã€ã®ããŒã«ãåãã ãæ¹æ³ã®å Žåã®æ°ã(II)2ã€ã®ããŒã«ãåãåºãããšã2åããè¿ãããããããå¥ã®äºãã«åºå¥ã§ããè¢ã«ãããå Žåã®æ°ã(III)2ã€ã®ããŒã«ãåãåºãããšã2åããè¿ãããããããå¥ã®äºãã«åºå¥ã§ããªãè¢ã«ãããå Žåã®æ°ããããããèšç®ããã
(I) æåã«ããŒã«ãåãã ããšãã«ã¯ã6ã€ã®ããŒã«ã®äžãã3ã€ã®ããŒã«ãåãã ãããšãããã®å Žåã®æ°ã¯
ã ãããããŸãã次ã«ãããåãé€ããäžãã2ã€ã®ããŒã«ãåãé€ããšãã«ã¯ ãã®åãã ãæ¹ã¯ã
ã ãããã ãã£ãŠããã®ãšãã®å Žåã®æ°ã¯
ã ãã«ãªããå®éãã®å€ãèšç®ãããšã
ãšãªãã60éãã§ããããšãåããã
(II)
(I)ã®å Žåãšåæ§ã«6ã€ã®ããŒã«ã®äžãã2ã€ã®ããŒã«ã åãã ãããšãããã®å Žåã®æ°ã¯
ã ãããããŸãã次ã«ãããåãé€ããäžãã2ã€ã®ããŒã«ãåãé€ããšãã«ã¯ ãã®åãã ãæ¹ã¯ã
ã ãããã ãã£ãŠããã®ãšãã®å Žåã®æ°ã¯
ã ãã«ãªããå®éãã®å€ãèšç®ãããšã
ãšãªãã90éãã§ããããšãåããã
(III) (II)ãšåãèšç®ã§å€ãæ±ããããšãåºæ¥ãããä»åã¯ããŒã«ããããè¢ã äºãã«åºå¥ã§ããªãããšã«æ³šæããªããŠã¯ãªããªãã ãã®ããšã«ãã£ãŠãèµ·ããããå Žåã®æ°ã¯(II)ã®å Žåã®ååã«ãªãã®ã§ æ±ããå Žåã®æ°ã¯45éããšãªãã
n C r {\displaystyle {}_{n}\mathrm {C} _{r}} ã«ã€ããŠä»¥äžã®åŒãæãç«ã€ã
å°åº
ãçšãããšã
ãåŸããã瀺ãããã
åæ§ã«
ãçšãããšã
ãšãªã瀺ãããã
æåã®åŒã¯ãç°ãªãnåã®ãã®ã®ãã¡råã«Xãšããã©ãã«ãã€ããæ®ãã®n-råã«Yãšããã©ãã«ãã€ããå Žåã®æ°ããæ±ããããšãã§ãããç°ãªãnåã®ãã®ã®ãã¡ããråãéžã³ã©ãã«Xãã€ããæ®ãã«ã©ãã«Yãã€ããå Žåã®æ°ã¯ n C r {\displaystyle _{n}\mathrm {C} _{r}} ã§ãããç°ãªãnåã®ãã®ã®ãã¡ããn-råãéžã³ãã©ãã«Yãã€ããæ®ãã«ã©ãã«Xãã€ããå Žåã®æ°ã¯ n C n â r {\displaystyle _{n}\mathrm {C} _{n-r}} ã§ãããåœç¶ãåè
ãšåŸè
ã®å Žåã®æ°ã¯çããã®ã§ãããããã n C r = n C n â r {\displaystyle _{n}\mathrm {C} _{r}=_{n}\mathrm {C} _{n-r}} ãæ±ããããã
2ã€ç®ã®åŒã¯ã "nåã®ãã®ããråãéžã¶ä»æ¹ã®æ°ã¯ã次ã®æ°ã®åã§ããã æåã®1ã€ãéžã°ãã«ä»ã®n-1åããråãéžã¶ä»æ¹ã®æ°ãšãæåã®1ã€ãéžãã§ä»ã®n-1åããr-1åãéžã¶ä»æ¹ã®æ°ãšã® åã§ããã" ãšããããšãè¡šãããŠããã
ãçšã㊠(I)
(II)
(III)
(VI)
ãããããèšç®ããã
äžã®åŒãçšããŠèšç®ããããšãåºæ¥ãããã¡ããçŽæ¥ã«èšç®ããŠã çããåŸãããšãåºæ¥ãããéåžžã¯ç°¡ååããŠããèšç®ããæ¹ã楜ã§ããã (I)
(II)
(III)
(VI)
ãšãªãã
å³ã®ãããªã«ãŒããå·Šäžã®ç¹ããå³äžã®ç¹ãŸã§æ©ããŠè¡ã人ãããã ãã ãããã®äººã¯å³ãäžã«ããé²ããªããšããããã®ãšãã
ãèšç®ããããã ãaç¹ã¯*ãšæžãããŠããç¹ã®ããäžã®éè·¯ã®ããšããããŠããã ããããã®ã«ãŒãã¯éåããŠããªã瞊4ã€ã暪5ã€ã®ç¢ç€ç®äžã®ã«ãŒãã« ãªã£ãŠããããšã«æ³šæããã
___________
|_|_|_|_|_|
|_|_|*|_|_|
|_|_|_|_|_|
|_|_|_|_|_|
(I) å·Šäžã«ãã人ã¯9åé²ãããšã§å³äžã®ç¹ã«èŸ¿ãçããããã®ãããå·Šäžã«ãã人ãéžã³ããã«ãŒãã®æ°ã¯9åã®ãã¡ã®ã©ã®åã§å³ã§ã¯ãªãäžã éžã¶ãã®å Žåã®æ°ã«çããããã®ãããªå Žåã®æ°ã¯ã9åã®ãã¡ããèªç±ã«4ã€ã®å Žæãéžã¶æ¹æ³ã«çãããçµã¿åãããçšããŠæžãããšãåºæ¥ããå®éã«9åã®ãã¡ããèªç±ã«4ã€ã®å Žæãéžã¶æ¹æ³ã¯ã
ã§æžãããããã®éãèšç®ãããšã
ãåŸãããã
(II) aç¹ãééããŠé²ãã«ãŒãã®æ°ã¯aç¹ã®å·Šã®ç¹ãŸã§ãã£ãŠããaç¹ãééããaç¹ã®å³ã®ç¹ãéã£ãŠå³äžã®ç¹ãŸã§ããä»æ¹ã®æ°ã«çããã ããããã®ã«ãŒãã®æ°ã¯(I)ã®æ¹æ³ãçšããŠèšç®ããããšãã§ããããã®æ°ãå®éã«èšç®ãããšã
ãšãªãã36éãã§ããããšãåããã
æŒç¿åé¡ r n C r = n n â 1 C r â 1 {\displaystyle r_{n}\mathrm {C} _{r}=n_{n-1}\mathrm {C} _{r-1}} ã瀺ã
r n C r = r n ! r ! ( n â r ) ! = n ( n â 1 ) ! ( r â 1 ) ( ( n â 1 ) â ( r â 1 ) ) ! = n n â 1 C r â 1 {\displaystyle r_{n}\mathrm {C} _{r}=r{\frac {n!}{r!(n-r)!}}=n{\frac {(n-1)!}{(r-1)((n-1)-(r-1))!}}=n_{n-1}\mathrm {C} _{r-1}}
ç°ãªãnåã®ç©ºç®±ã«råã®ãã®ãå
¥ããå Žåã®æ°ãéè€çµã¿åãããšããã n H r {\displaystyle _{n}\mathrm {H} _{r}} ã§è¡šãã
éè€çµåãã«ã€ããŠæ¬¡ã®ããã«èå¯ããã
x 1 , x 2 , ⯠, x n , r {\displaystyle x_{1},x_{2},\cdots ,x_{n},r} ãéè² æŽæ°ãšããæ¹çšåŒ x 1 + x 2 + ⯠+ x n = r {\displaystyle x_{1}+x_{2}+\cdots +x_{n}=r} ã®è§£ã®åæ°ã«ã€ããŠèããããã®è§£ã®åæ°ã¯ x 1 , x 2 , ⯠, x n {\displaystyle x_{1},x_{2},\cdots ,x_{n}} ã« r {\displaystyle r} åã®1ãåé
ããå Žåã®æ°ãšèããããšãã§ããã®ã§ãéè€çµã¿åããã®å®çŸ©ããã n H r {\displaystyle _{n}\mathrm {H} _{r}} ã§ããã
ãŸãããã®æ¹çšåŒã®éè² æŽæ°è§£ã®åæ°ã¯ãråã®âã«n-1åã®åºåãã眮ãå Žåã®æ°ãšãèãããããã€ãŸããâââ...ââ(rå)ã«n-1åã®åºåã|ã䞊ã¹ããšâ|ââ|...â|âã®ããã«ãªããããã§ãå·Šããé ã«åºåãã§åºåãããâã®åæ°ãããããã x 1 , x 2 , ⯠, x n {\displaystyle x_{1},x_{2},\cdots ,x_{n}} ãšãããšãããã¯æ¹çšåŒã®è§£ãšãªãã
ãã®å Žåã®æ°ã¯ãråã®âãšn-1åã®åºåã|ã䞊ã¹ããå Žåã®æ°ãªã®ã§ã n + r â 1 C r {\displaystyle _{n+r-1}\mathrm {C} _{r}} ã§ãããæ¹çšåŒã®éè² æŽæ°è§£ã®åæ°ã«ã€ããŠ2éãã®æ¹æ³ã§æ±ãŸã£ãã®ã§ãããã¯çããã n H r = n + r â 1 C r {\displaystyle _{n}\mathrm {H} _{r}=_{n+r-1}\mathrm {C} _{r}} ãæãç«ã€ã
ããå Žåã®æ°ããå®éã«çŸãããå²åã®ããšã確ç(ãããã€ãè±:probability)ãšåŒã¶ã
ããå Žåã®æ°ãå®éã«çŸãããå²åã¯ããã®å Žåã®æ°ãå²ãç®ã§ããã®äºæã«ãããŠèµ·ããåŸãå
šãŠã®äºæã®å Žåã®æ°ã§å²ã£ããã®ã«çããã
ããšãã°ãå
šãçããå²åã§å
šãŠã®é¢ãåºãããããããµã£ããšãã«1ãåºã確ç㯠1 6 {\displaystyle {\frac {1}{6}}} ã§ããã ããã¯1ãåºãå Žåã®æ°1ãã1,2,3,4,5,6ã®ãããããåºãå Žåã®æ°6ã§å²ã£ããã®ã«çããã
èµ€ç2åãšçœç3åãå
¥ã£ãè¢ãããçã2ååæã«åãåºãããã®ãšãã2åãšãçœçãåºã確çãæ±ããã
èµ€çœããããŠ5åã®çãã2åãåãåºãæ¹æ³ã¯
ãã®ãã¡ã2åãšãçœçã«ãªãå Žåã¯
ãã£ãŠæ±ãã確ç㯠3 10 {\displaystyle {\frac {3}{10}}}
確çã®å®çŸ©ããã次ã®æ§è³ªãåŸãããã
2ã€ã®äºè±¡A,Bãåæã«èµ·ãããªããšããäºè±¡AãšBã¯äºãã«æå(ã¯ãã¯ããè±:exclusive)ã§ããããŸãã¯AãšBã¯æåäºè±¡ã§ãããšããã
ç·å7人ã女å5人ã®äžãããããåŒãã§3人ã®å§å¡ãéžã¶ãšãã3人ãšãåæ§ã§ãã確çãæ±ããã
12人ã®äžãã3人ã®å§å¡ãéžã¶å Žåã®æ°ã¯
ããã§ãã3人ãšãç·åã§ãããäºè±¡ãAãã3人ãšã女åã§ãããäºè±¡ãBãšãããšãã3人ãšãåæ§ã§ãããäºè±¡ã¯ãåäºè±¡A ⪠Bã§ãããããããAãšBã¯æåäºè±¡ã§ããã
ãã£ãŠæ±ãã確ç㯠P ( A ⪠B ) = P ( A ) + P ( B ) = 35 220 + 10 220 = 45 220 = 9 44 {\displaystyle P(A\cup B)=P(A)+P(B)={\frac {35}{220}}+{\frac {10}{220}}={\frac {45}{220}}={\frac {9}{44}}}
äºè±¡Aã«å¯ŸããŠããAã§ãªããäºè±¡ã A Ì {\displaystyle {\overline {A}}} ã§è¡šããAã®äœäºè±¡(ããããã)ãšããã
èµ€ç5åãçœç3åã®èš8åå
¥ã£ãŠããè¢ãã3åã®çãåãåºããšããå°ãªããšã1åã¯çœçã§ãã確çãæ±ããã
8åã®çãã3åã®çãåãåºãå Žåã®æ°ã¯
ããŸããå°ãªããšã1åã¯çœçã§ãããäºè±¡ãAãšãããšã A Ì {\displaystyle {\overline {A}}} ã¯ã3åãšãèµ€çã§ããããšããäºè±¡ã ãã
ãã£ãŠæ±ãã確çã¯
ãããã«ä»ã®çµæã«å¯ŸããŠåœ±é¿ããããŒããªãæäœãç¹°ãããããšããããããã®è©Šè¡ã¯ç¬ç«(ã©ããã€ãè±:independent)ã§ãããšèšããç¬ç«ãªè©Šè¡ã«ã€ããŠã¯ãããè©Šè¡ã®èµ·ãã確çãå®ããããŠããŠããããnåç¹°ããããããšããããããèµ·ãã確çã¯ãããããã®è©Šè¡ãèµ·ãã確çã®ç©ãšãªãã
èµ€ç3åãçœç2åã®èš5åå
¥ã£ãŠããè¢ãããããã®äžãã1åã®çãåãåºããŠè²ã確ãããŠããè¢ã«æ»ããåã³1åãåãåºããšãã1åç®ã¯èµ€çã2åç®ã¯çœçãåãåºã確çãæ±ããã
1åç®ã«åãåºããçãè¢ã«æ»ãã®ã§ãã1åç®ã«åãåºããè©Šè¡ãšã2åç®ã«åãåºããè©Šè¡ãšã¯äºãã«ç¬ç«ã§ããã 1åç®ã«åãåºãã1åãèµ€çã§ãã確ç㯠3 5 {\displaystyle {\frac {3}{5}}}
2åç®ã«åãåºãã1åãçœçã§ãã確ç㯠2 5 {\displaystyle {\frac {2}{5}}} ãããã£ãŠæ±ãã確çã¯
åãè©Šè¡ãäœåãç¹°ãè¿ããŠè¡ããšããååã®è©Šè¡ã¯ç¬ç«ã§ããããã®äžé£ã®ç¬ç«ãªè©Šè¡ããŸãšããŠèãããšãããããå埩詊è¡(ã¯ãã·ã ããã)ãšããã
1åã®ããããã5åæãããšãã3ã®åæ°ã®ç®ã4ååºã確çãæ±ããã
1åã®ããããã1åæãããšãã3ã®åæ°ã®ç®ãåºã確çã¯
ãã£ãŠã1åã®ããããã5åæãããšãã3ã®åæ°ã®ç®ã4ååºã確çã¯
èšå·ãΣãã«ã€ããŠã¯ãã¡ããåç
§ã
ããè©Šè¡ããã£ããšãã ãã®è©Šè¡ã§åŸããããšæåŸ
ãããå€ã®ããšãæåŸ
å€(ãããã¡ãè±:expected value)ãšãããæåŸ
å€ã¯ãnåã®äºè±¡ r k ( k = 1 , 2 , ⯠, n ) {\displaystyle r_{k}\ (k=1,2,\cdots ,n)} ã«å¯ŸããŠãåã
v k {\displaystyle v_{k}} ãšããå€ãåŸãããäºè±¡ r k {\displaystyle r_{k}} ãèµ·ãã確çã p k {\displaystyle p_{k}} ã§äžããããŠãããšãã
ã«ãã£ãŠäžãããããäŸãã°ãããããããµã£ããšãåºãç®ã®æåŸ
å€ã¯ã
ãšãªãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ããšãã°ããã¯ãããäžåã«äžŠã¹ãå Žåã䞊ã¹æ¹ã®æ°ã«ã¯ãããã€ãã®æ¹æ³ãããããã£ããã«å
šãŠã®äžŠã³æ¹ãè©Šãããšããæéããããã°å®éšå¯èœã§ããã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ãã®ããã«ããå
šéšã§äœéãããããããšããããã®ãäœéããã®ãäœãã«ãããæ°åããå Žåã®æ°(ã°ããã®ãã) ãšåŒã¶ã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ãã®ããã«äºæã«ã¯ããããã®ããæ¹ãå
šéšã§äœéãããããæ°ããããšãåºæ¥ãäºæãããã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ããäºæã«ã€ããŠ(ãã®ããšãèµ·ãããã)å Žåã®æ°ãæ£ç¢ºã«æ°ããããšãç解ã®åºç€ã§ããããã®äºæã«ã€ããŠãã©ã®ããšãèµ·ãããããã©ã®ããšãèµ·ããã¥ããããèŠåããããã®åºç€ãšãªãã ã€ãŸããå Žåã®æ°ã¯äºæãèµ·ãããã確çãšå¯æ¥ãªé¢ä¿ã«ããã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "äŸãã°ãããŒã«ãŒãªã©ã®ã«ãŒãã²ãŒã ã§ã¯éããããšãé£ãã圹ã¯é«ãã©ã³ã¯ãäžããããŠãããã ããã¯èµ·ããã«ãã圹ãåºæ¥ããã©ã³ãã®çµã¿åããã®çŸããã確çãå°ããããšã«ããã ãã®ããšã¯ã52æã®ã«ãŒããã5æãåŒããŠæ¥ããšãã«å
šãŠã®ã«ãŒããåŒã確çãåãã§ãããšãããšãããã圹ã«å¯Ÿå¿ããã«ãŒãã®çµã¿åãããåŒãå Žåã®æ°ãããå°ãªãããšã«å¯Ÿå¿ããã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ãã®ããã«ãå Žåã®æ°ã¯äºæãèµ·ãããã確çãšå¯æ¥ãªé¢ä¿ã«ããã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ã«ãŒãã²ãŒã ã®ããã«ç¢ºçãå
·äœçã«èšç®ã§ããå Žåã®ä»ã«ãã確çã®èãæ¹ãçšããŠèšç®ãããäºæã¯å€ãããã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ããšãã°ãä¿éº(ã»ãã)ãšåŒã°ãããã®ã¯ããäºæã«å€æ®µãã€ãããã®ã§ãããã ä¿éºãäžãããªããŠã¯ãªããªãäºæãèµ·ããã«ãããšå®¢èŠ³çã«æããããã®ã»ã©ããã®ãã®ã®å€æ®µãäžãããšããç¹åŸŽãããã äŸãã°ãèªåè»ä¿éºã«å å
¥ããã®ã«å¿
èŠãªä»£éã¯è¥è
ã§ã¯é«ãã幎什ãéããããšã«äœããªã£ãŠããã ããã¯ãè¥è
ã¯èªåè»ã®å
èš±ãååŸããŠæéãçãå Žåãå€ããä¿éºéã®æ¯æãå¿
èŠãšããèªåè»äºæ
ããããå¯èœæ§ãé«ãããšã«ããã ãã£ãœãã幎什ãéãããã®ã«ã€ããŠã¯é転ã®æéãæãšãšãã«äžéãããšäžè¬ã«èããããã®ã§ä¿éºããããããã®ä»£éã¯å°ãªããªãã®ã§ããã ãŸããåãè¥è
ã§ãæ¢ã«äœåºŠãäºæ
ãéãããã®ã¯åã幎代ã®ä»ã®è¥è
ãããä¿éºæãé«ããªãåŸåãããã ããã¯ãäœåºŠãäºæ
ãéãããã®ã¯é転ã®ä»æ¹ã«äœããã®åé¡ãããåŸåããããããã«ãã£ãŠãµããã³äºæ
ããããå¯èœæ§ãéåžžã®ãã®ãšæ¯ã¹ãŠããé«ããšèããããããšã«ããã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "éè¡ã®èè³(ããã)ã§ããã¯ã確çã®èããçšããŠé«ãå©çãåºãããšãå®è·µãããŠããã èè³ã§ããã¯ãä¿éºæ¥ãšããªããããã貞åãã«ãªãå¯èœæ§ãé«ãçžæã«å¯ŸããŠã¯é«ãéå©ã§è³éã貞ãä»ãã ããå®å®ããè³éãæã£ãŠããçžæã«å¯ŸããŠã¯ããäœãéå©ã§è³éã貞ãä»ããããšãå®è¡ããŠæ¥ãã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "å©çãå®å®çã«çšŒãæ¹æ³ãšããŠãããã€ãã®äŒç€Ÿãçºè¡ããäºãã«æ§è³ªã®ç°ãªã£ãæ ªãªã©ãåãããŠè³Œå
¥å
ãåæ£ããããšã§æ ªã®å€æ®µãäžãã£ããšãã§ãå€æ®µãããŸãæžãããšãç¡ãããã«ããæ¹æ³ãèæ¡ãããŠããã (ãã ããå€æ®µãæžãã¥ããã®ãšåæ§ã«ãå€æ®µã¯äžããã¥ããã) ããã¯ãæ§è³ªã®ç°ãªã£ãååãåãããŠæ±ãããšã§ãå€æ®µãæ¥å€ãã確çãäžããããšãåºæ¥ãããšãè¡šãããŠããã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "ãããã確çã§ã¯ãå¿
ãããäºæž¬ããéãã«äºãé²ãããã§ã¯ç¡ãããšã«æ³šæããå¿
èŠãããã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "ãã®ç« ã§ã¯å Žåã®æ°ãšç¢ºçã®èšç®æ³ã玹ä»ããããŸãå
ã«æ§ã
ãªäºæã®å Žåã®æ°ã®èšç®æ³ãæ±ãããã®çµæãçšããŠããäºæãèµ·ãã確çãèšç®ããæ¹æ³ã玹ä»ããã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "ããã§ã¯ãæééå A ã®èŠçŽ ã®åæ°ã n(A) ã§è¡šãã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ããšãã°ã10以äžã®èªç¶æ°ã®éåã U ãšããŠããã®ãã¡ å¶æ°ã®éåã A ãšããå Žåã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "ãªã®ã§ãAã®èŠçŽ ã®åæ°ã¯5åãªã®ã§",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "ã§ããã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "ãªãã U={1, 2, 3, 4, 5, 6 , 7, 8, 9, 10} ã§èŠçŽ ã®åæ°ã¯10åãªã®ã§",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "ã§ããã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "次ã®ãããªåé¡ãèããŠã¿ããã 100ãŸã§ã®èªç¶æ°ã®ãã¡ã2ãŸãã¯3ã®åæ°ã¯äœåããã?",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "ãã®ãããªåé¡ã®è§£æ³ãèãããããæºåã®åé¡ãšããŠããŸã10ãŸã§ã®èªç¶æ°ã§èããŠã¿ããã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "å
çšã®äŸé¡ã§2ã®åæ°ã«ã€ããŠã¯èããã®ã§ã次ã®åé¡ãšããŠ10ãŸã§ã®3ã®åæ°ã®åæ°ã«ã€ããŠèãããã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "10以äžã®èªç¶æ°ã®éåã U ãšããŠããã®ãã¡ 3ã®åæ°ã®éåã B ãšããå Žåã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "B={3, 6 , 9} ãªã®ã§ãBã®èŠçŽ ã®åæ°ã¯3åãªã®ã§",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "ã§ããã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "ããŠã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "ã«ã¯å
±éã㊠6 ãšããèŠçŽ ãå«ãŸããŠããã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "èªç¶æ°10ãŸã§ã«ãã2ãŸãã¯3ã®åæ°ã«ãããèŠçŽ ã¯ã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "ã§ãããèŠçŽ ã®åæ°ããããããš 7åã§ããã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "äžæ¹ã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "ã§ããã1åå€ãã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "ãã®ããã«1åå€ããªã£ãŠããŸã£ãåå ã¯ã éåAãšéåBã«å
±éããŠå«ãŸããŠããèŠçŽ 6 ãäºéã«æ°ããŠããŸã£ãŠããããã§ããã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "äžè¬ã«ã2ã€ã®éåA,Bã®èŠçŽ ã®åæ° n(A) ãš n(B) ãçšããŠãAãŸãã¯Bã®æ¡ä»¶ãæºããèŠçŽ ã®åæ°ããããããå Žåã«ã¯ãAãšBã«å
±éããŠå«ãŸããŠããèŠçŽ ã®åæ°ãå·®ãåŒããªããã°ãªããªãã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "ãã®ããšãåŒã§è¡šããš",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "ã«ãªãã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "ãã ãããâªããšã¯åéåã®èšå·ã§ã AâªB ãšã¯ éåAãšéåBã®åéåã®ããšã§ããã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "ãâ©ããšã¯å
±ééšåã®èšå·ã§ã ãAâ©Bããšã¯ éåAãšéåBã®å
±ééšåã®ããšã§ããã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "ã§ã¯ããã®å
¬åŒãåèã«ã㊠100ãŸã§ã®èªç¶æ°ã®ãã¡ã2ãŸãã¯3ã®åæ°ã¯äœåããã? ã®çããæ±ãããã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "100ãŸã§ã®èªç¶æ°ã®ãã¡ã®ã2ã®åæ°ã®éåãAãšããŠã3ã®åæ°ã®éåãBãšãããš",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "ããã«ã2ã®åæ°ã§ããã3ã®åæ°ã§ãããæ°ã®éå Aâ©B ãšã¯ãã€ãŸã6ã®åæ°ã®éåã®ããšã§ãã(ãªããªã 2 ãš 3 ã®æå°å
¬åæ°ã 6 ãªã®ã§)ã 96÷6=16 ãªã®ã§ãAâ©B ã®èŠçŽ ã®åæ°ã¯ 16 åãã€ãŸã n(Aâ©B)= 16 ã§ããã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "ãããŠãå
¬åŒ",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 41,
"tag": "p",
"text": "ãé©çšãããšã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 42,
"tag": "p",
"text": "ã§ããã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 43,
"tag": "p",
"text": "ãã£ãŠã100ãŸã§ã®èªç¶æ°ã®ãã¡ã®2ãŸãã¯3ã®åæ°ã®åæ°ã¯ 67å ã§ããã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 44,
"tag": "p",
"text": "3ã€ã®æééåã®åéåã®èŠçŽ ã®åæ°ã«ã€ããŠã¯ã次ã®å
¬åŒãæãç«ã€",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 45,
"tag": "p",
"text": "n(AâªBâªC) = n(A) + n(B) + n(C) ân(Aâ©B) ân(Bâ©C) ân(Câ©A) + n(Aâ©Bâ©C)",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 46,
"tag": "p",
"text": "å³ã®å³ãåèã«ãäžã®å
¬åŒã蚌æããã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 47,
"tag": "p",
"text": "",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 48,
"tag": "p",
"text": "100以äžã®èªç¶æ°ã®ãã¡ã2ã®åæ°ãŸãã¯3ã®åæ°ãŸãã¯5ã®åæ°ã§ãããã®ã®åæ°ãæ±ããã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 49,
"tag": "p",
"text": "(解æ³)",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 50,
"tag": "p",
"text": "ãŸãã100以äžã®èªç¶æ°ã®ãã¡ã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 51,
"tag": "p",
"text": "ãšããã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 52,
"tag": "p",
"text": "100÷2=50ãªã®ã§ã100ã¯50çªç®ã®2ã®åæ°ã§ããããã£ãŠ100以äžã®2ã®åæ°ã¯50åã§ãããåæ§ã«èããŠèŠçŽ ã®åæ°ãæ±ãããšã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 53,
"tag": "p",
"text": "ã§ããã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 54,
"tag": "p",
"text": "äžæ¹ã100以äžã®èªç¶æ°ã®ãã¡",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 55,
"tag": "p",
"text": "ãšãªãã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 56,
"tag": "p",
"text": "ãã£ãŠãå
ã»ã©ãšåæ§ã«èãããš",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 57,
"tag": "p",
"text": "ãŸãã100以äžã®èªç¶æ°ã®ãã¡ã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 58,
"tag": "p",
"text": "Aâ©Bâ©C ã®èŠçŽ ã®åæ°ã¯",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 59,
"tag": "p",
"text": "ã§ããã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 60,
"tag": "p",
"text": "ãã£ãŠã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 61,
"tag": "p",
"text": "ãªã®ã§ã100以äžã®èªç¶æ°ã®ãã¡ã®2ã®åæ°ãŸãã¯3ã®åæ°ãŸãã¯5ã®åæ°ã§ãããã®ã®åæ°ã¯ 74åã§ããã",
"title": "éåã®èŠçŽ ã®åæ°"
},
{
"paragraph_id": 62,
"tag": "p",
"text": "ããšãã°å€§äžå°3åã®ãµã€ã³ãããµã£ãŠãç®ã®åã5ã«ãªãç®ã®çµã¯ãäœéãããã ãããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 63,
"tag": "p",
"text": "ãã®ãããªåé¡ã解ãæ¹æ³ã®ã²ãšã€ãšããŠãå³ã®ããã«ãçµã¿åãããç·åœããã§æžãæ¹æ³ãããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 64,
"tag": "p",
"text": "倧äžå°ã®åèš3åã®ãµã€ã³ãããããã A,B,C ãšããŠè¡šãããããã®æåã«ãã©ã®ç®ãåºãã°åèš5ã«ãªãããèãããšãçµæã¯å³ã®ããã«ãªãã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 65,
"tag": "p",
"text": "ãã®ãããªå³ã 暹圢å³(ãã
ããã) ãšããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 66,
"tag": "p",
"text": "",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 67,
"tag": "p",
"text": "3åã®ãµã€ã³ãããµããšããç®ã®åã6ã«ãªãå Žåã¯äœéããããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 68,
"tag": "p",
"text": "",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 69,
"tag": "p",
"text": "æåã«ãnåã®ç°ãªã£ããã®ã䞊ã¹æããå Žåã®æ°ãæ°ããã ãŸãæåã«äžŠã¹ããã®ã¯nåã次ã«äžŠã¹ããã®ã¯(n-1)åããã®æ¬¡ã«äžŠã¹ããã®ã¯(n-2)å ... ãšã ãã ããšéžã¹ããã®ã®æ°ãæžã£ãŠè¡ããæåŸã«ã¯1åããæ®ããªããªãããšã«æ³šç®ãããšããã®äºæã«é¢ããå Žåã®æ°ã¯",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 70,
"tag": "p",
"text": "ãšãªãã1ããnãŸã§ã®èªç¶æ°ã®ç©ã«ãªãã ãã®æ°ã éä¹ (ããããããfactorial)ãšåŒã³ãéä¹nã®èšå·ã¯ n ! {\\displaystyle n!} ã§è¡šãã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 71,
"tag": "p",
"text": "ããªãã¡ãéä¹ã¯",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 72,
"tag": "p",
"text": "n ! = n ( n â 1 ) ( n â 2 ) ⯠3 â
2 â
1 {\\displaystyle n!=n(n-1)(n-2)\\cdots 3\\cdot 2\\cdot 1}",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 73,
"tag": "p",
"text": "ãšå®çŸ©ãããããã®éä¹ã®èšå·ã䜿ãã°ããã®åé¡ã®ãšãã®å Žåã®æ°ã¯ n!ã§ãããšèšãããšãåºæ¥ãã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 74,
"tag": "p",
"text": "",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 75,
"tag": "p",
"text": "ãããããèšç®ããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 76,
"tag": "p",
"text": "ãçšããŠèšç®ããã°ããã çãã¯ã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 77,
"tag": "p",
"text": "ãšãªãã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 78,
"tag": "p",
"text": "",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 79,
"tag": "p",
"text": "ããããã«1ãã5ãŸã§ã®æ°åãæžããã5æã®ã«ãŒãã眮ããŠããã ãã®ã«ãŒãã䞊ã¹æãããšãã (I)ã«ãŒãã®äžŠã¹æ¹ã®æ°ã (II)å¶æ°ãåŸãããã«ãŒãã®äžŠã¹æ¹ã®æ°ã (III)å¥æ°ãåºãã«ãŒãã®äžŠã¹æ¹ã®æ°ããããããèšç®ããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 80,
"tag": "p",
"text": "(I) ã«ãŒãã®æ°ã5æã§ãããããåºå¥ã§ããããšãããã«ãŒãã®äžŠã¹æ¹ã®æ°ã¯",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 81,
"tag": "p",
"text": "ãšãªãã120ãšãªãã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 82,
"tag": "p",
"text": "(II) å¶æ°ãåŸãããã«ã¯äžã®äœã§ããæãå³ã«åºãã«ãŒãããå¶æ°ãšãªãã°ããã ãã®ãããªã«ãŒãã¯2ãš4ã§ãããããããã«å¯ŸããŠåŸã®4æã¯èªç±ã«éžãã§ããã ãã®ããããã®ãããªã«ãŒãã®äžŠã¹æ¹ã¯ã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 83,
"tag": "p",
"text": "ãšãªãã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 84,
"tag": "p",
"text": "(III) å¥æ°ãåŸãããã«ã¯äžã®äœã§ããæãå³ã«åºãã«ãŒãããå¥æ°ãšãªãã°ããã ãã®ãããªã«ãŒãã¯1,3,5ã§ãããããããã«å¯ŸããŠåŸã®4æã¯èªç±ã«éžãã§ããã ãã®ããããã®ãããªã«ãŒãã®äžŠã¹æ¹ã¯ã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 85,
"tag": "p",
"text": "ãšãªããäžæ¹ã5æã®ã«ãŒãã䞊ã¹æããŠåŸãããæ°ã¯å¿
ãå¶æ°ãå¥æ°ã® ã©ã¡ããã§ããã®ã§ã(I)ã®çµæãã(II)ã®çµæãåŒãããšã«ãã£ãŠã (III)ã®çµæã¯åŸãããã¯ãã ããå®éã«ãããèšç®ãããš",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 86,
"tag": "p",
"text": "ãšãªãã確ãã«ãã®ããã«ãªã£ãŠããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 87,
"tag": "p",
"text": "",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 88,
"tag": "p",
"text": "0,1,2,3,5ãæžããã5æã®ã«ãŒãããããããã䞊ã³æãããšãã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 89,
"tag": "p",
"text": "ãããããæ±ããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 90,
"tag": "p",
"text": "(I) å
é ã0ã«ãªã£ããšãã«ã¯5æ¡ã®æ°ã«ãªããªãããšã«æ³šæããã°ãããæ±ããå Žåã®æ°ã¯",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 91,
"tag": "p",
"text": "ãšãªãã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 92,
"tag": "p",
"text": "(II) æåã0ã§ãªãæåŸã0ã2ã§ããæ°ãæ°ããã°ããããŸããæåŸã0ã§ãããšãã«ã¯ãæ®ãã®4æã¯ä»»æã§ããã®ã§",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 93,
"tag": "p",
"text": "éãã®çµã¿åãããããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 94,
"tag": "p",
"text": "次ã«ãæåŸã2ã§ãããšãã«ã¯æåã¯0ã§ãã£ãŠã¯ãããªãã®ã§ã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 95,
"tag": "p",
"text": "éãããã 2ã€ãåãããæ°ã5æ¡ã®å¶æ°ãåŸãããå Žåã®æ°ã§ãããçãã¯ã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 96,
"tag": "p",
"text": "ãšãªãã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 97,
"tag": "p",
"text": "(III) (I)ã®çµæãã(II)ã®çµæãåŒãã°ããããããã§ã¯ãã®çµæãæ£ãããã©ãã 確ãããããã«ã5æ¡ã®å¥æ°ãåŸãããçµã¿åãããæ°ãäžããŠã¿ãã 5æ¡ã®å¥æ°ãåŸãããã«ã¯æåŸã®æ°ã¯1,3,5ã®ããããã§ãªããŠã¯ãªããªãã ãã®ãã¡ã®ã©ã®å Žåã«ã€ããŠã5æ¡ã®æ°ãåŸãããã«ã¯æåã®æ°ã0㧠åã£ãŠã¯ãªããªãã®ã§ããããã®å Žåã®æ°ã¯ã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 98,
"tag": "p",
"text": "ãšãªãããã5æ¡ã®å¥æ°ãåŸãå Žåã®æ°ã§ããã (II)ã®çµæãšè¶³ãåããããšç¢ºãã«(I)ã®çµæãšçãã96ãåŸãã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 99,
"tag": "p",
"text": "(IV) 5ã®åæ°ãåŸãããã«ã¯æåŸã®æ°ã0ã5ã§ããã°ããã ãã®ãšãæåŸã0ã«ãªãå Žåã®æ°ã¯ä»ã®4ã€ãä»»æã§ãããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 100,
"tag": "p",
"text": "ååšããã次ã«ãæåŸã5ã«ãªãå Žåã®æ°ã¯æåã®æ°ã0ã§ãã£ãŠã¯ãªããªããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 101,
"tag": "p",
"text": "ã ãååšããã ãã£ãŠçãã¯",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 102,
"tag": "p",
"text": "ãšãªãã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 103,
"tag": "p",
"text": "",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 104,
"tag": "p",
"text": "nåã®ç°ãªã£ããã®ããråãéžãã§ãé çªãã€ããŠäžŠã¹ãä»æ¹ã®æ°ãã n P r {\\displaystyle {}_{n}\\mathrm {P} _{r}} ãšæžãã ãŸãããã®ãããªèšç®ã®ä»æ¹ã é å (ãã
ããã€ãè±:permutation) ãšããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 105,
"tag": "p",
"text": "nåã®ç°ãªã£ããã®ããråãéžãã§é çªãã€ããŠäžŠã¹ãä»æ¹ã®æ°ã®ããš ãã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 106,
"tag": "p",
"text": "ã®ããã«èšãã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 107,
"tag": "p",
"text": "æåã«äžŠã¹ããã®ã¯néãã次ã«äžŠã¹ããã®ã¯ (nâ1)éã ããã®æ¬¡ã«äžŠã¹ããã®ã¯ (nâ2)éã ,... æåŸã«ã¯ (nâ(râ1))éã ãšããããã«ãã ãã ãéžã¹ããã®ã®æ°ãæžã£ãŠè¡ãããšã«æ³šç®ãããšãé åã®ç·æ°ãšããŠ",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 108,
"tag": "p",
"text": "ãåŸãããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 109,
"tag": "p",
"text": "äžè¬ã« n P r {\\displaystyle {}_{n}\\mathrm {P} _{r}} ã§ã¯ n ⧠r ã§ããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 110,
"tag": "p",
"text": "",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 111,
"tag": "p",
"text": "(I)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 112,
"tag": "p",
"text": "(II)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 113,
"tag": "p",
"text": "(III)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 114,
"tag": "p",
"text": "(IV)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 115,
"tag": "p",
"text": "(V)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 116,
"tag": "p",
"text": "(VI)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 117,
"tag": "p",
"text": "ãããããèšç®ããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 118,
"tag": "p",
"text": "ãããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 119,
"tag": "p",
"text": "ãçšããŠèšç®ããã°ããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 120,
"tag": "p",
"text": "çµæã¯ã (I)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 121,
"tag": "p",
"text": "(II)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 122,
"tag": "p",
"text": "(III)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 123,
"tag": "p",
"text": "(IV)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 124,
"tag": "p",
"text": "(V)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 125,
"tag": "p",
"text": "(VI)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 126,
"tag": "p",
"text": "ãšãªãã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 127,
"tag": "p",
"text": "(V)ãš(VI)ã«ã€ããŠã¯äžè¬çã«æŽæ°nã«å¯ŸããŠ",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 128,
"tag": "p",
"text": "ãåŸãããããã®ãšã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 129,
"tag": "p",
"text": "ã¯å
ã
ã®é åã®å®çŸ©ãããããš\"nåã®ãã®ã®äžãã1ã€ãéžã°ãªãå Žåã®æ°\"ã«å¯Ÿå¿ããŠãããå°ã
äžèªç¶ãªããã«æãããããã®ããã«å€ã眮ããŠãããšäŸ¿å©ã§ããããéåžžãã®ããã«çœ®ãã®ã§ãããããŸããå®éã®å Žåã®æ°ã®èšç®ã§ãã®ãããªå€ãæ±ãããšã¯å€ãã¯ãªããšãããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 130,
"tag": "p",
"text": "",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 131,
"tag": "p",
"text": "",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 132,
"tag": "p",
"text": "A, B, C, D, E ã®5人ãå圢ã«æãã€ãªãã§èŒªãã€ãããšãããã®äžŠã³æ¹ã¯äœéããããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 133,
"tag": "p",
"text": "ãã®ãããªåé¡ã®å Žåãå³ã®ããã«ãå転ãããšéãªã䞊ã³ã¯åã䞊ã³ã§ãããšèããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 134,
"tag": "p",
"text": "解ãæ¹ã®èãæ¹ã¯æ°çš®é¡ããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 135,
"tag": "p",
"text": "",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 136,
"tag": "p",
"text": "ã©ã¡ãã«ãããçµæã¯",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 137,
"tag": "p",
"text": "ã§ããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 138,
"tag": "p",
"text": "äžè¬ã« ç°ãªã nå ã®ãã®ãå圢ã«äžŠã¹ããã®ãåé åãšããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 139,
"tag": "p",
"text": "åé åã®ç·æ°ãšããŠã次ã®ããšãæãç«ã€ã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 140,
"tag": "p",
"text": "ç°ãªã nå ã®åé åã®ç·æ°ã¯ ( n â 1 ) ! {\\displaystyle (n-1)!} ã§ããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 141,
"tag": "p",
"text": "nåã®ç°ãªã£ããã®ããråãéžãã§ãé çªãã€ããã«äžŠã¹ãä»æ¹ã®æ°ãã n C r {\\displaystyle {}_{n}\\mathrm {C} _{r}} ãšæžãããã®ãããªèšç®ã çµã¿åãã(combination) ãšããã äŸãã°ãããã€ãããããŒã«ã«çªå·ããµã£ãŠãããªã©ã®æ¹æ³ã§ãããããã®ããŒã«ãåºå¥ã§ããnåã®ããŒã«ãå
¥ã£ãç®±ã®äžããråã®ããŒã«ãåãã ãæãåãã ããããŒã«ãåãã ããé ã«äžŠã¹ããšãããšããã®å Žåã®æ°ã¯é å n P r {\\displaystyle {}_{n}\\mathrm {P} _{r}} ã«å¯Ÿå¿ããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 142,
"tag": "p",
"text": "äžæ¹ãåãã ããããŒã«ã®çš®é¡ãéèŠã§ããåãã ããé çªãç¹ã«å¿
èŠã§ãªããšãã«ã¯ããã®å Žåã®æ°ã¯çµã¿åãã n C r {\\displaystyle {}_{n}\\mathrm {C} _{r}} ã«å¯Ÿå¿ããããããã®æ°ã¯ãäºãã«ç°ãªã£ãå Žåã®æ°ã§ãããäºãã«ç°ãªã£ãèšç®æ³ãå¿
èŠãšãªãã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 143,
"tag": "p",
"text": "n C r {\\displaystyle {}_{n}\\mathrm {C} _{r}} ã¯ã n P r {\\displaystyle {}_{n}\\mathrm {P} _{r}} éãã®äžŠã¹æ¹ãäœã£ãåŸã«ãããã®äžŠã³ãç¡èŠãããã®ã«çãããããã§ãråãåãã ããŠäœã£ã䞊ã³ã«ã€ããŠã䞊ã¹æ¹ãç¡èŠãããšr!åã®äžŠã³ãåäžèŠãããããšããããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 144,
"tag": "p",
"text": "ãªããªããråã®ãäºãã«åºå¥ã§ããæ°ãèªç±ã«äžŠã³æããå Žåã®æ°ã¯r!ã§ãããããããå
šãŠåäžèŠããããšããã°å
šäœã®å Žåã®æ°ã¯ r!ã®åã ãæžãããšã«ãªãããã§ããããã£ãŠã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 145,
"tag": "p",
"text": "ãåŸãããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 146,
"tag": "p",
"text": "",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 147,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 148,
"tag": "p",
"text": "次ã®å€ãèšç®ãã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 149,
"tag": "p",
"text": "(I)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 150,
"tag": "p",
"text": "(II)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 151,
"tag": "p",
"text": "(III)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 152,
"tag": "p",
"text": "(VI)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 153,
"tag": "p",
"text": "",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 154,
"tag": "p",
"text": "ããããã«ã€ããŠ",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 155,
"tag": "p",
"text": "ãçšããŠèšç®ããã°ããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 156,
"tag": "p",
"text": "(I)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 157,
"tag": "p",
"text": "(II)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 158,
"tag": "p",
"text": "(III)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 159,
"tag": "p",
"text": "(VI)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 160,
"tag": "p",
"text": "ãšãªãã(IV)ã«ã€ããŠã¯äžè¬ã«æŽæ°nã«å¯ŸããŠ",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 161,
"tag": "p",
"text": "ãå®çŸ©ããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 162,
"tag": "p",
"text": "ããã¯ããšããšã®çµã¿åããã®èšç®ãšããŠã¯nåã®ç©äœã®ãªããã0åã®ç©äœãéžã¶å Žåã®æ°ã«å¯Ÿå¿ããŠããã å®éã«ã¯ãã®ãããªå Žåã®æ°ãèšç®ããããšèããããšã¯ããŸãç¡ããšæãããããèšç®ã®äŸ¿å®äžã®ããå®çŸ©ãäžã®ããã«ããã ãŸããäžã®èšç®ã§ã¯",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 163,
"tag": "p",
"text": "ã®åŒããã®ãŸãŸçšãããšã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 164,
"tag": "p",
"text": "ã€ãŸãã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 165,
"tag": "p",
"text": "ãšãªã£ãŠããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 166,
"tag": "p",
"text": "å®éã«ã¯éä¹ã®èšç®ã¯æŽæ°nã«ã€ããŠã¯nãã1ãŸã§ãäžãããªããããç®ããŠãããšããä»æ¹ã§èšç®ãããŠããã®ã§ãäžã®çµæã¯åŠã«æããã ãããå®éã«ã¯ãããé²ãã çè«ã«ãã£ãŠãã®çµæã¯æ£åœåãããã®ã§ããã ãã®å Žåã䟿å®äž",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 167,
"tag": "p",
"text": "ã0ã®éä¹ã®å®çŸ©ãšããŠåããããã®ã§ããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 168,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 169,
"tag": "p",
"text": "5åã®ããŒã«ãå
¥ã£ãããŒã«å
¥ããã2ã€ã®ããŒã«ãåãã ããšã(ããŒã«ã¯ãããã åºå¥ã§ãããã®ãšããã)2ã€ã®ããŒã«ã®éžã³æ¹ã¯ã äœéããããèšç®ããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 170,
"tag": "p",
"text": "ããŒã«ã®åãã ãæ¹ã¯çµã¿åããã®æ°ãçšããŠèšç®ã§ããã 5ã€ã®ããŒã«ã®äžãã2ã€ãåãã ãã®ã§ãããããã®å Žåã®æ°ã¯ã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 171,
"tag": "p",
"text": "ãšãªãããã£ãŠãããŒã«ã®åãã ãæ¹ã¯10éãã§ããããšããããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 172,
"tag": "p",
"text": "",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 173,
"tag": "p",
"text": "æŒç¿åé¡",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 174,
"tag": "p",
"text": "6åã®äºãã«åºå¥ã§ããããŒã«ãå
¥ã£ãç®±ãããã ãã®äžãã (I)3ã€ã®ããŒã«ãš2ã€ã®ããŒã«ãåãã ãæ¹æ³ã®å Žåã®æ°ã(II)2ã€ã®ããŒã«ãåãåºãããšã2åããè¿ãããããããå¥ã®äºãã«åºå¥ã§ããè¢ã«ãããå Žåã®æ°ã(III)2ã€ã®ããŒã«ãåãåºãããšã2åããè¿ãããããããå¥ã®äºãã«åºå¥ã§ããªãè¢ã«ãããå Žåã®æ°ããããããèšç®ããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 175,
"tag": "p",
"text": "(I) æåã«ããŒã«ãåãã ããšãã«ã¯ã6ã€ã®ããŒã«ã®äžãã3ã€ã®ããŒã«ãåãã ãããšãããã®å Žåã®æ°ã¯",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 176,
"tag": "p",
"text": "ã ãããããŸãã次ã«ãããåãé€ããäžãã2ã€ã®ããŒã«ãåãé€ããšãã«ã¯ ãã®åãã ãæ¹ã¯ã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 177,
"tag": "p",
"text": "ã ãããã ãã£ãŠããã®ãšãã®å Žåã®æ°ã¯",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 178,
"tag": "p",
"text": "ã ãã«ãªããå®éãã®å€ãèšç®ãããšã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 179,
"tag": "p",
"text": "ãšãªãã60éãã§ããããšãåããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 180,
"tag": "p",
"text": "(II)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 181,
"tag": "p",
"text": "(I)ã®å Žåãšåæ§ã«6ã€ã®ããŒã«ã®äžãã2ã€ã®ããŒã«ã åãã ãããšãããã®å Žåã®æ°ã¯",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 182,
"tag": "p",
"text": "ã ãããããŸãã次ã«ãããåãé€ããäžãã2ã€ã®ããŒã«ãåãé€ããšãã«ã¯ ãã®åãã ãæ¹ã¯ã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 183,
"tag": "p",
"text": "ã ãããã ãã£ãŠããã®ãšãã®å Žåã®æ°ã¯",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 184,
"tag": "p",
"text": "ã ãã«ãªããå®éãã®å€ãèšç®ãããšã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 185,
"tag": "p",
"text": "ãšãªãã90éãã§ããããšãåããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 186,
"tag": "p",
"text": "(III) (II)ãšåãèšç®ã§å€ãæ±ããããšãåºæ¥ãããä»åã¯ããŒã«ããããè¢ã äºãã«åºå¥ã§ããªãããšã«æ³šæããªããŠã¯ãªããªãã ãã®ããšã«ãã£ãŠãèµ·ããããå Žåã®æ°ã¯(II)ã®å Žåã®ååã«ãªãã®ã§ æ±ããå Žåã®æ°ã¯45éããšãªãã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 187,
"tag": "p",
"text": "n C r {\\displaystyle {}_{n}\\mathrm {C} _{r}} ã«ã€ããŠä»¥äžã®åŒãæãç«ã€ã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 188,
"tag": "p",
"text": "å°åº",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 189,
"tag": "p",
"text": "ãçšãããšã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 190,
"tag": "p",
"text": "ãåŸããã瀺ãããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 191,
"tag": "p",
"text": "åæ§ã«",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 192,
"tag": "p",
"text": "ãçšãããšã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 193,
"tag": "p",
"text": "ãšãªã瀺ãããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 194,
"tag": "p",
"text": "æåã®åŒã¯ãç°ãªãnåã®ãã®ã®ãã¡råã«Xãšããã©ãã«ãã€ããæ®ãã®n-råã«Yãšããã©ãã«ãã€ããå Žåã®æ°ããæ±ããããšãã§ãããç°ãªãnåã®ãã®ã®ãã¡ããråãéžã³ã©ãã«Xãã€ããæ®ãã«ã©ãã«Yãã€ããå Žåã®æ°ã¯ n C r {\\displaystyle _{n}\\mathrm {C} _{r}} ã§ãããç°ãªãnåã®ãã®ã®ãã¡ããn-råãéžã³ãã©ãã«Yãã€ããæ®ãã«ã©ãã«Xãã€ããå Žåã®æ°ã¯ n C n â r {\\displaystyle _{n}\\mathrm {C} _{n-r}} ã§ãããåœç¶ãåè
ãšåŸè
ã®å Žåã®æ°ã¯çããã®ã§ãããããã n C r = n C n â r {\\displaystyle _{n}\\mathrm {C} _{r}=_{n}\\mathrm {C} _{n-r}} ãæ±ããããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 195,
"tag": "p",
"text": "2ã€ç®ã®åŒã¯ã \"nåã®ãã®ããråãéžã¶ä»æ¹ã®æ°ã¯ã次ã®æ°ã®åã§ããã æåã®1ã€ãéžã°ãã«ä»ã®n-1åããråãéžã¶ä»æ¹ã®æ°ãšãæåã®1ã€ãéžãã§ä»ã®n-1åããr-1åãéžã¶ä»æ¹ã®æ°ãšã® åã§ããã\" ãšããããšãè¡šãããŠããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 196,
"tag": "p",
"text": "",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 197,
"tag": "p",
"text": "ãçšã㊠(I)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 198,
"tag": "p",
"text": "(II)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 199,
"tag": "p",
"text": "(III)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 200,
"tag": "p",
"text": "(VI)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 201,
"tag": "p",
"text": "ãããããèšç®ããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 202,
"tag": "p",
"text": "äžã®åŒãçšããŠèšç®ããããšãåºæ¥ãããã¡ããçŽæ¥ã«èšç®ããŠã çããåŸãããšãåºæ¥ãããéåžžã¯ç°¡ååããŠããèšç®ããæ¹ã楜ã§ããã (I)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 203,
"tag": "p",
"text": "(II)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 204,
"tag": "p",
"text": "(III)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 205,
"tag": "p",
"text": "(VI)",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 206,
"tag": "p",
"text": "ãšãªãã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 207,
"tag": "p",
"text": "",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 208,
"tag": "p",
"text": "å³ã®ãããªã«ãŒããå·Šäžã®ç¹ããå³äžã®ç¹ãŸã§æ©ããŠè¡ã人ãããã ãã ãããã®äººã¯å³ãäžã«ããé²ããªããšããããã®ãšãã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 209,
"tag": "p",
"text": "ãèšç®ããããã ãaç¹ã¯*ãšæžãããŠããç¹ã®ããäžã®éè·¯ã®ããšããããŠããã ããããã®ã«ãŒãã¯éåããŠããªã瞊4ã€ã暪5ã€ã®ç¢ç€ç®äžã®ã«ãŒãã« ãªã£ãŠããããšã«æ³šæããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 210,
"tag": "p",
"text": "___________",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 211,
"tag": "p",
"text": "|_|_|_|_|_|",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 212,
"tag": "p",
"text": "|_|_|*|_|_|",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 213,
"tag": "p",
"text": "|_|_|_|_|_|",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 214,
"tag": "p",
"text": "|_|_|_|_|_|",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 215,
"tag": "p",
"text": "",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 216,
"tag": "p",
"text": "(I) å·Šäžã«ãã人ã¯9åé²ãããšã§å³äžã®ç¹ã«èŸ¿ãçããããã®ãããå·Šäžã«ãã人ãéžã³ããã«ãŒãã®æ°ã¯9åã®ãã¡ã®ã©ã®åã§å³ã§ã¯ãªãäžã éžã¶ãã®å Žåã®æ°ã«çããããã®ãããªå Žåã®æ°ã¯ã9åã®ãã¡ããèªç±ã«4ã€ã®å Žæãéžã¶æ¹æ³ã«çãããçµã¿åãããçšããŠæžãããšãåºæ¥ããå®éã«9åã®ãã¡ããèªç±ã«4ã€ã®å Žæãéžã¶æ¹æ³ã¯ã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 217,
"tag": "p",
"text": "ã§æžãããããã®éãèšç®ãããšã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 218,
"tag": "p",
"text": "ãåŸãããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 219,
"tag": "p",
"text": "(II) aç¹ãééããŠé²ãã«ãŒãã®æ°ã¯aç¹ã®å·Šã®ç¹ãŸã§ãã£ãŠããaç¹ãééããaç¹ã®å³ã®ç¹ãéã£ãŠå³äžã®ç¹ãŸã§ããä»æ¹ã®æ°ã«çããã ããããã®ã«ãŒãã®æ°ã¯(I)ã®æ¹æ³ãçšããŠèšç®ããããšãã§ããããã®æ°ãå®éã«èšç®ãããšã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 220,
"tag": "p",
"text": "ãšãªãã36éãã§ããããšãåããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 221,
"tag": "p",
"text": "æŒç¿åé¡ r n C r = n n â 1 C r â 1 {\\displaystyle r_{n}\\mathrm {C} _{r}=n_{n-1}\\mathrm {C} _{r-1}} ã瀺ã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 222,
"tag": "p",
"text": "r n C r = r n ! r ! ( n â r ) ! = n ( n â 1 ) ! ( r â 1 ) ( ( n â 1 ) â ( r â 1 ) ) ! = n n â 1 C r â 1 {\\displaystyle r_{n}\\mathrm {C} _{r}=r{\\frac {n!}{r!(n-r)!}}=n{\\frac {(n-1)!}{(r-1)((n-1)-(r-1))!}}=n_{n-1}\\mathrm {C} _{r-1}}",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 223,
"tag": "p",
"text": "ç°ãªãnåã®ç©ºç®±ã«råã®ãã®ãå
¥ããå Žåã®æ°ãéè€çµã¿åãããšããã n H r {\\displaystyle _{n}\\mathrm {H} _{r}} ã§è¡šãã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 224,
"tag": "p",
"text": "éè€çµåãã«ã€ããŠæ¬¡ã®ããã«èå¯ããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 225,
"tag": "p",
"text": "x 1 , x 2 , ⯠, x n , r {\\displaystyle x_{1},x_{2},\\cdots ,x_{n},r} ãéè² æŽæ°ãšããæ¹çšåŒ x 1 + x 2 + ⯠+ x n = r {\\displaystyle x_{1}+x_{2}+\\cdots +x_{n}=r} ã®è§£ã®åæ°ã«ã€ããŠèããããã®è§£ã®åæ°ã¯ x 1 , x 2 , ⯠, x n {\\displaystyle x_{1},x_{2},\\cdots ,x_{n}} ã« r {\\displaystyle r} åã®1ãåé
ããå Žåã®æ°ãšèããããšãã§ããã®ã§ãéè€çµã¿åããã®å®çŸ©ããã n H r {\\displaystyle _{n}\\mathrm {H} _{r}} ã§ããã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 226,
"tag": "p",
"text": "ãŸãããã®æ¹çšåŒã®éè² æŽæ°è§£ã®åæ°ã¯ãråã®âã«n-1åã®åºåãã眮ãå Žåã®æ°ãšãèãããããã€ãŸããâââ...ââ(rå)ã«n-1åã®åºåã|ã䞊ã¹ããšâ|ââ|...â|âã®ããã«ãªããããã§ãå·Šããé ã«åºåãã§åºåãããâã®åæ°ãããããã x 1 , x 2 , ⯠, x n {\\displaystyle x_{1},x_{2},\\cdots ,x_{n}} ãšãããšãããã¯æ¹çšåŒã®è§£ãšãªãã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 227,
"tag": "p",
"text": "ãã®å Žåã®æ°ã¯ãråã®âãšn-1åã®åºåã|ã䞊ã¹ããå Žåã®æ°ãªã®ã§ã n + r â 1 C r {\\displaystyle _{n+r-1}\\mathrm {C} _{r}} ã§ãããæ¹çšåŒã®éè² æŽæ°è§£ã®åæ°ã«ã€ããŠ2éãã®æ¹æ³ã§æ±ãŸã£ãã®ã§ãããã¯çããã n H r = n + r â 1 C r {\\displaystyle _{n}\\mathrm {H} _{r}=_{n+r-1}\\mathrm {C} _{r}} ãæãç«ã€ã",
"title": "å Žåã®æ°"
},
{
"paragraph_id": 228,
"tag": "p",
"text": "ããå Žåã®æ°ããå®éã«çŸãããå²åã®ããšã確ç(ãããã€ãè±:probability)ãšåŒã¶ã",
"title": "確ç"
},
{
"paragraph_id": 229,
"tag": "p",
"text": "ããå Žåã®æ°ãå®éã«çŸãããå²åã¯ããã®å Žåã®æ°ãå²ãç®ã§ããã®äºæã«ãããŠèµ·ããåŸãå
šãŠã®äºæã®å Žåã®æ°ã§å²ã£ããã®ã«çããã",
"title": "確ç"
},
{
"paragraph_id": 230,
"tag": "p",
"text": "ããšãã°ãå
šãçããå²åã§å
šãŠã®é¢ãåºãããããããµã£ããšãã«1ãåºã確ç㯠1 6 {\\displaystyle {\\frac {1}{6}}} ã§ããã ããã¯1ãåºãå Žåã®æ°1ãã1,2,3,4,5,6ã®ãããããåºãå Žåã®æ°6ã§å²ã£ããã®ã«çããã",
"title": "確ç"
},
{
"paragraph_id": 231,
"tag": "p",
"text": "èµ€ç2åãšçœç3åãå
¥ã£ãè¢ãããçã2ååæã«åãåºãããã®ãšãã2åãšãçœçãåºã確çãæ±ããã",
"title": "確ç"
},
{
"paragraph_id": 232,
"tag": "p",
"text": "èµ€çœããããŠ5åã®çãã2åãåãåºãæ¹æ³ã¯",
"title": "確ç"
},
{
"paragraph_id": 233,
"tag": "p",
"text": "ãã®ãã¡ã2åãšãçœçã«ãªãå Žåã¯",
"title": "確ç"
},
{
"paragraph_id": 234,
"tag": "p",
"text": "ãã£ãŠæ±ãã確ç㯠3 10 {\\displaystyle {\\frac {3}{10}}}",
"title": "確ç"
},
{
"paragraph_id": 235,
"tag": "p",
"text": "確çã®å®çŸ©ããã次ã®æ§è³ªãåŸãããã",
"title": "確ç"
},
{
"paragraph_id": 236,
"tag": "p",
"text": "2ã€ã®äºè±¡A,Bãåæã«èµ·ãããªããšããäºè±¡AãšBã¯äºãã«æå(ã¯ãã¯ããè±:exclusive)ã§ããããŸãã¯AãšBã¯æåäºè±¡ã§ãããšããã",
"title": "確ç"
},
{
"paragraph_id": 237,
"tag": "p",
"text": "",
"title": "確ç"
},
{
"paragraph_id": 238,
"tag": "p",
"text": "ç·å7人ã女å5人ã®äžãããããåŒãã§3人ã®å§å¡ãéžã¶ãšãã3人ãšãåæ§ã§ãã確çãæ±ããã",
"title": "確ç"
},
{
"paragraph_id": 239,
"tag": "p",
"text": "12人ã®äžãã3人ã®å§å¡ãéžã¶å Žåã®æ°ã¯",
"title": "確ç"
},
{
"paragraph_id": 240,
"tag": "p",
"text": "ããã§ãã3人ãšãç·åã§ãããäºè±¡ãAãã3人ãšã女åã§ãããäºè±¡ãBãšãããšãã3人ãšãåæ§ã§ãããäºè±¡ã¯ãåäºè±¡A ⪠Bã§ãããããããAãšBã¯æåäºè±¡ã§ããã",
"title": "確ç"
},
{
"paragraph_id": 241,
"tag": "p",
"text": "ãã£ãŠæ±ãã確ç㯠P ( A ⪠B ) = P ( A ) + P ( B ) = 35 220 + 10 220 = 45 220 = 9 44 {\\displaystyle P(A\\cup B)=P(A)+P(B)={\\frac {35}{220}}+{\\frac {10}{220}}={\\frac {45}{220}}={\\frac {9}{44}}}",
"title": "確ç"
},
{
"paragraph_id": 242,
"tag": "p",
"text": "äºè±¡Aã«å¯ŸããŠããAã§ãªããäºè±¡ã A Ì {\\displaystyle {\\overline {A}}} ã§è¡šããAã®äœäºè±¡(ããããã)ãšããã",
"title": "確ç"
},
{
"paragraph_id": 243,
"tag": "p",
"text": "èµ€ç5åãçœç3åã®èš8åå
¥ã£ãŠããè¢ãã3åã®çãåãåºããšããå°ãªããšã1åã¯çœçã§ãã確çãæ±ããã",
"title": "確ç"
},
{
"paragraph_id": 244,
"tag": "p",
"text": "8åã®çãã3åã®çãåãåºãå Žåã®æ°ã¯",
"title": "確ç"
},
{
"paragraph_id": 245,
"tag": "p",
"text": "ããŸããå°ãªããšã1åã¯çœçã§ãããäºè±¡ãAãšãããšã A Ì {\\displaystyle {\\overline {A}}} ã¯ã3åãšãèµ€çã§ããããšããäºè±¡ã ãã",
"title": "確ç"
},
{
"paragraph_id": 246,
"tag": "p",
"text": "ãã£ãŠæ±ãã確çã¯",
"title": "確ç"
},
{
"paragraph_id": 247,
"tag": "p",
"text": "ãããã«ä»ã®çµæã«å¯ŸããŠåœ±é¿ããããŒããªãæäœãç¹°ãããããšããããããã®è©Šè¡ã¯ç¬ç«(ã©ããã€ãè±:independent)ã§ãããšèšããç¬ç«ãªè©Šè¡ã«ã€ããŠã¯ãããè©Šè¡ã®èµ·ãã確çãå®ããããŠããŠããããnåç¹°ããããããšããããããèµ·ãã確çã¯ãããããã®è©Šè¡ãèµ·ãã確çã®ç©ãšãªãã",
"title": "確ç"
},
{
"paragraph_id": 248,
"tag": "p",
"text": "",
"title": "確ç"
},
{
"paragraph_id": 249,
"tag": "p",
"text": "èµ€ç3åãçœç2åã®èš5åå
¥ã£ãŠããè¢ãããããã®äžãã1åã®çãåãåºããŠè²ã確ãããŠããè¢ã«æ»ããåã³1åãåãåºããšãã1åç®ã¯èµ€çã2åç®ã¯çœçãåãåºã確çãæ±ããã",
"title": "確ç"
},
{
"paragraph_id": 250,
"tag": "p",
"text": "1åç®ã«åãåºããçãè¢ã«æ»ãã®ã§ãã1åç®ã«åãåºããè©Šè¡ãšã2åç®ã«åãåºããè©Šè¡ãšã¯äºãã«ç¬ç«ã§ããã 1åç®ã«åãåºãã1åãèµ€çã§ãã確ç㯠3 5 {\\displaystyle {\\frac {3}{5}}}",
"title": "確ç"
},
{
"paragraph_id": 251,
"tag": "p",
"text": "2åç®ã«åãåºãã1åãçœçã§ãã確ç㯠2 5 {\\displaystyle {\\frac {2}{5}}} ãããã£ãŠæ±ãã確çã¯",
"title": "確ç"
},
{
"paragraph_id": 252,
"tag": "p",
"text": "åãè©Šè¡ãäœåãç¹°ãè¿ããŠè¡ããšããååã®è©Šè¡ã¯ç¬ç«ã§ããããã®äžé£ã®ç¬ç«ãªè©Šè¡ããŸãšããŠèãããšãããããå埩詊è¡(ã¯ãã·ã ããã)ãšããã",
"title": "確ç"
},
{
"paragraph_id": 253,
"tag": "p",
"text": "1åã®ããããã5åæãããšãã3ã®åæ°ã®ç®ã4ååºã確çãæ±ããã",
"title": "確ç"
},
{
"paragraph_id": 254,
"tag": "p",
"text": "1åã®ããããã1åæãããšãã3ã®åæ°ã®ç®ãåºã確çã¯",
"title": "確ç"
},
{
"paragraph_id": 255,
"tag": "p",
"text": "ãã£ãŠã1åã®ããããã5åæãããšãã3ã®åæ°ã®ç®ã4ååºã確çã¯",
"title": "確ç"
},
{
"paragraph_id": 256,
"tag": "p",
"text": "èšå·ãΣãã«ã€ããŠã¯ãã¡ããåç
§ã",
"title": "確ç"
},
{
"paragraph_id": 257,
"tag": "p",
"text": "ããè©Šè¡ããã£ããšãã ãã®è©Šè¡ã§åŸããããšæåŸ
ãããå€ã®ããšãæåŸ
å€(ãããã¡ãè±:expected value)ãšãããæåŸ
å€ã¯ãnåã®äºè±¡ r k ( k = 1 , 2 , ⯠, n ) {\\displaystyle r_{k}\\ (k=1,2,\\cdots ,n)} ã«å¯ŸããŠãåã
v k {\\displaystyle v_{k}} ãšããå€ãåŸãããäºè±¡ r k {\\displaystyle r_{k}} ãèµ·ãã確çã p k {\\displaystyle p_{k}} ã§äžããããŠãããšãã",
"title": "確ç"
},
{
"paragraph_id": 258,
"tag": "p",
"text": "ã«ãã£ãŠäžãããããäŸãã°ãããããããµã£ããšãåºãç®ã®æåŸ
å€ã¯ã",
"title": "確ç"
},
{
"paragraph_id": 259,
"tag": "p",
"text": "ãšãªãã",
"title": "確ç"
}
] | null | {{pathnav|é«çåŠæ ¡ã®åŠç¿|é«çåŠæ ¡æ°åŠ|é«çåŠæ ¡æ°åŠA|pagename=å Žåã®æ°ãšç¢ºç|frame=1|small=1}}
== ã¯ããã« ==
ããšãã°ããã¯ãããäžåã«äžŠã¹ãå Žåã䞊ã¹æ¹ã®æ°ã«ã¯ãããã€ãã®æ¹æ³ãããããã£ããã«å
šãŠã®äžŠã³æ¹ãè©Šãããšããæéããããã°å®éšå¯èœã§ããã
ãã®ããã«ããå
šéšã§äœéãããããããšããããã®ãäœéããã®ãäœãã«ãããæ°åããå Žåã®æ°ïŒã°ããã®ããïŒ ãšåŒã¶ã
ãã®ããã«äºæã«ã¯ããããã®ããæ¹ãå
šéšã§äœéãããããæ°ããããšãåºæ¥ãäºæãããã
ããäºæã«ã€ããŠïŒãã®ããšãèµ·ããããïŒå Žåã®æ°ãæ£ç¢ºã«æ°ããããšãç解ã®åºç€ã§ããããã®äºæã«ã€ããŠãã©ã®ããšãèµ·ãããããã©ã®ããšãèµ·ããã¥ããããèŠåããããã®åºç€ãšãªãã
ã€ãŸããå Žåã®æ°ã¯äºæãèµ·ãããã確çãšå¯æ¥ãªé¢ä¿ã«ããã
äŸãã°ãããŒã«ãŒãªã©ã®ã«ãŒãã²ãŒã ã§ã¯éããããšãé£ãã圹ã¯é«ãã©ã³ã¯ãäžããããŠãããã
ããã¯èµ·ããã«ãã圹ãåºæ¥ããã©ã³ãã®çµã¿åããã®çŸããã確çãå°ããããšã«ããã
ãã®ããšã¯ã52æã®ã«ãŒããã5æãåŒããŠæ¥ããšãã«å
šãŠã®ã«ãŒããåŒã確çãåãã§ãããšãããšãããã圹ã«å¯Ÿå¿ããã«ãŒãã®çµã¿åãããåŒãå Žåã®æ°ãããå°ãªãããšã«å¯Ÿå¿ããã
ãã®ããã«ãå Žåã®æ°ã¯äºæãèµ·ãããã確çãšå¯æ¥ãªé¢ä¿ã«ããã
ã«ãŒãã²ãŒã ã®ããã«ç¢ºçãå
·äœçã«èšç®ã§ããå Žåã®ä»ã«ãã確çã®èãæ¹ãçšããŠèšç®ãããäºæã¯å€ãããã
* äŸïŒä¿éº
ããšãã°ãä¿éºïŒã»ããïŒãšåŒã°ãããã®ã¯ããäºæã«å€æ®µãã€ãããã®ã§ãããã
ä¿éºãäžãããªããŠã¯ãªããªãäºæãèµ·ããã«ãããšå®¢èŠ³çã«æããããã®ã»ã©ããã®ãã®ã®å€æ®µãäžãããšããç¹åŸŽãããã
äŸãã°ãèªåè»ä¿éºã«å å
¥ããã®ã«å¿
èŠãªä»£éã¯è¥è
ã§ã¯é«ãã幎什ãéããããšã«äœããªã£ãŠããã
ããã¯ãè¥è
ã¯èªåè»ã®å
èš±ãååŸããŠæéãçãå Žåãå€ããä¿éºéã®æ¯æãå¿
èŠãšããèªåè»äºæ
ããããå¯èœæ§ãé«ãããšã«ããã
ãã£ãœãã幎什ãéãããã®ã«ã€ããŠã¯é転ã®æéãæãšãšãã«äžéãããšäžè¬ã«èããããã®ã§ä¿éºããããããã®ä»£éã¯å°ãªããªãã®ã§ããã
ãŸããåãè¥è
ã§ãæ¢ã«äœåºŠãäºæ
ãéãããã®ã¯åã幎代ã®ä»ã®è¥è
ãããä¿éºæãé«ããªãåŸåãããã
ããã¯ãäœåºŠãäºæ
ãéãããã®ã¯é転ã®ä»æ¹ã«äœããã®åé¡ãããåŸåããããããã«ãã£ãŠãµããã³äºæ
ããããå¯èœæ§ãéåžžã®ãã®ãšæ¯ã¹ãŠããé«ããšèããããããšã«ããã
* äŸïŒéè¡ã®èè³ïŒãããïŒ
éè¡ã®èè³ïŒãããïŒã§ããã¯ã確çã®èããçšããŠé«ãå©çãåºãããšãå®è·µãããŠããã
èè³ã§ããã¯ãä¿éºæ¥ãšããªããããã貞åãã«ãªãå¯èœæ§ãé«ãçžæã«å¯ŸããŠã¯é«ãéå©ã§è³éã貞ãä»ãã
ããå®å®ããè³éãæã£ãŠããçžæã«å¯ŸããŠã¯ããäœãéå©ã§è³éã貞ãä»ããããšãå®è¡ããŠæ¥ãã
* äŸïŒæ ªåŒåžå Žã®åæ£æè³
å©çãå®å®çã«çšŒãæ¹æ³ãšããŠãããã€ãã®äŒç€Ÿãçºè¡ããäºãã«æ§è³ªã®ç°ãªã£ãæ ªãªã©ãåãããŠè³Œå
¥å
ãåæ£ããããšã§æ ªã®å€æ®µãäžãã£ããšãã§ãå€æ®µãããŸãæžãããšãç¡ãããã«ããæ¹æ³ãèæ¡ãããŠããã
ïŒãã ããå€æ®µãæžãã¥ããã®ãšåæ§ã«ãå€æ®µã¯äžããã¥ãããïŒ
ããã¯ãæ§è³ªã®ç°ãªã£ãååãåãããŠæ±ãããšã§ãå€æ®µãæ¥å€ãã確çãäžããããšãåºæ¥ãããšãè¡šãããŠããã
ãããã確çã§ã¯ãå¿
ãããäºæž¬ããéãã«äºãé²ãããã§ã¯ç¡ãããšã«æ³šæããå¿
èŠãããã
ãã®ç« ã§ã¯å Žåã®æ°ãšç¢ºçã®èšç®æ³ã玹ä»ããããŸãå
ã«æ§ã
ãªäºæã®å Žåã®æ°ã®èšç®æ³ãæ±ãããã®çµæãçšããŠããäºæãèµ·ãã確çãèšç®ããæ¹æ³ã玹ä»ããã
== éåã®èŠçŽ ã®åæ° ==
==== 2ã€ã®éåã®åéåã®èŠçŽ ã®åæ° ====
:â» ãã®åå
ã§ã¯ãåå
ã[[é«çåŠæ ¡æ°åŠA/éåãšè«ç]]ãã§ç¿ãéåã®èšå·ã䜿ããåãããªããã°ããã¡ãã®ããŒãžãåç
§ããã
ããã§ã¯ãæééå A ã®èŠçŽ ã®åæ°ã n(A) ã§è¡šãã
ããšãã°ã10以äžã®èªç¶æ°ã®éåã U ãšããŠããã®ãã¡ å¶æ°ã®éåã A ãšããå Žåã
:A=ïœ2, 4, 6 , 8, 10ïœ
ãªã®ã§ãAã®èŠçŽ ã®åæ°ã¯5åãªã®ã§
:n(A)ïŒ5
ã§ããã
ãªãã
U=ïœ1, 2, 3, 4, 5, 6 , 7, 8, 9, 10ïœ
ã§èŠçŽ ã®åæ°ã¯10åãªã®ã§
:n(U)ïŒ10
ã§ããã
次ã®ãããªåé¡ãèããŠã¿ããã
100ãŸã§ã®èªç¶æ°ã®ãã¡ã2ãŸãã¯3ã®åæ°ã¯äœåãããïŒ
ãã®ãããªåé¡ã®è§£æ³ãèãããããæºåã®åé¡ãšããŠããŸã10ãŸã§ã®èªç¶æ°ã§èããŠã¿ããã
å
çšã®äŸé¡ã§2ã®åæ°ã«ã€ããŠã¯èããã®ã§ã次ã®åé¡ãšããŠ10ãŸã§ã®3ã®åæ°ã®åæ°ã«ã€ããŠèãããã
10以äžã®èªç¶æ°ã®éåã U ãšããŠããã®ãã¡ 3ã®åæ°ã®éåã B ãšããå Žåã
B=ïœ3, 6 , 9ïœ
ãªã®ã§ãBã®èŠçŽ ã®åæ°ã¯3åãªã®ã§
:n(B)ïŒ3
ã§ããã
ããŠã
:A=ïœ2, 4, 6 , 8, 10ïœ
:B=ïœ3, 6 , 9ïœ
ã«ã¯å
±éã㊠6 ãšããèŠçŽ ãå«ãŸããŠããã
èªç¶æ°10ãŸã§ã«ãã2ãŸãã¯3ã®åæ°ã«ãããèŠçŽ ã¯ã
:{2, 3, 4, 6, 8, 9, 10}
ã§ãããèŠçŽ ã®åæ°ããããããš 7åã§ããã
äžæ¹ã
:n(A)ïŒn(B)ïŒ 5ïŒ3 ïŒ8
ã§ããã1åå€ãã
ãã®ããã«1åå€ããªã£ãŠããŸã£ãåå ã¯ã éåAãšéåBã«å
±éããŠå«ãŸããŠããèŠçŽ 6 ãäºéã«æ°ããŠããŸã£ãŠããããã§ããã
äžè¬ã«ã2ã€ã®éåA,Bã®èŠçŽ ã®åæ° n(A) ãš n(B) ãçšããŠãAãŸãã¯Bã®æ¡ä»¶ãæºããèŠçŽ ã®åæ°ããããããå Žåã«ã¯ãAãšBã«å
±éããŠå«ãŸããŠããèŠçŽ ã®åæ°ãå·®ãåŒããªããã°ãªããªãã
ãã®ããšãåŒã§è¡šããš
:nïŒAâªBïŒ ïŒ n(A)ïŒn(B)ân(Aâ©B)
ã«ãªãã
ãã ãããâªããšã¯åéåã®èšå·ã§ã AâªB ãšã¯ éåAãšéåBã®åéåã®ããšã§ããã
ãâ©ããšã¯å
±ééšåã®èšå·ã§ã ãAâ©Bããšã¯ éåAãšéåBã®å
±ééšåã®ããšã§ããã
ã§ã¯ããã®å
¬åŒãåèã«ããŠ
100ãŸã§ã®èªç¶æ°ã®ãã¡ã2ãŸãã¯3ã®åæ°ã¯äœåãããïŒ
ã®çããæ±ãããã
100ãŸã§ã®èªç¶æ°ã®ãã¡ã®ã2ã®åæ°ã®éåãAãšããŠã3ã®åæ°ã®éåãBãšãããš
:n(A)ïŒ 100/2 ïŒ50 ãªã®ã§ãéåAã®èŠçŽ ã®åæ°ïŒ2ã®åæ°ã®åæ°ïŒã¯ 50åãã€ãŸã n(A)ïŒ 50 ã§ããã
:n(B)ã«ã€ããŠã¯ïŒ»99÷3ïŒ33 ãªã®ã§ éåBã®èŠçŽ ã®åæ°ïŒ3ã®åæ°ã®åæ°ïŒã¯33åãã€ãŸã n(B)ïŒ 33 ã§ããã
ããã«ã2ã®åæ°ã§ããã3ã®åæ°ã§ãããæ°ã®éå Aâ©B ãšã¯ãã€ãŸã6ã®åæ°ã®éåã®ããšã§ããïŒãªããªã 2 ãš 3 ã®æå°å
¬åæ°ã 6 ãªã®ã§ïŒã
96÷6ïŒ16 ãªã®ã§ãAâ©B ã®èŠçŽ ã®åæ°ã¯ 16 åãã€ãŸã n(Aâ©B)ïŒ 16 ã§ããã
ãããŠãå
¬åŒ
:nïŒAâªBïŒ ïŒ n(A)ïŒn(B)ân(Aâ©B)
ãé©çšãããšã
:nïŒAâªBïŒ ïŒ 50 ïŒ 33 â 16 ïŒ 67
ã§ããã
ãã£ãŠã100ãŸã§ã®èªç¶æ°ã®ãã¡ã®2ãŸãã¯3ã®åæ°ã®åæ°ã¯ 67å ã§ããã
==== çºå±: 3ã€ã®éåã®åéåã®èŠçŽ ã®åæ° ====
[[File:Venn diagram of 3 sets.svg|thumb|]]
3ã€ã®æééåã®åéåã®èŠçŽ ã®åæ°ã«ã€ããŠã¯ã次ã®å
¬åŒãæãç«ã€
n(AâªBâªC) ïŒ n(A) ïŒ n(B) ïŒ n(C) ân(Aâ©B) ân(Bâ©C) ân(Câ©A) ïŒ n(Aâ©Bâ©C)
;åé¡
å³ã®å³ãåèã«ãäžã®å
¬åŒã蚌æããã
;äŸé¡
100以äžã®èªç¶æ°ã®ãã¡ã2ã®åæ°ãŸãã¯3ã®åæ°ãŸãã¯5ã®åæ°ã§ãããã®ã®åæ°ãæ±ããã
(解æ³)
ãŸãã100以äžã®èªç¶æ°ã®ãã¡ã
:2ã®åæ°ã®éåãAã
:3ã®åæ°ã®éåãBã
:5ã®åæ°ã®éåãCã
ãšããã
100÷2=50ãªã®ã§ã100ã¯50çªç®ã®2ã®åæ°ã§ããããã£ãŠ100以äžã®2ã®åæ°ã¯50åã§ãããåæ§ã«èããŠèŠçŽ ã®åæ°ãæ±ãããšã
:n(A) ïŒ 50
:n(B) ïŒ 33
:n(C) ïŒ 20
ã§ããã
äžæ¹ã100以äžã®èªç¶æ°ã®ãã¡
:Aâ©B 㯠6ã®åæ°ã®éåã
:Bâ©C 㯠15ã®åæ°ã®éåã
:Câ©A 㯠10ã®åæ°ã®éåã
ãšãªãã
ãã£ãŠãå
ã»ã©ãšåæ§ã«èãããš
:n(Aâ©B) ïŒ 16
:n(Bâ©C) ïŒ 6
:n(Câ©A) ïŒ 10
ãŸãã100以äžã®èªç¶æ°ã®ãã¡ã
:Aâ©Bâ©C 㯠30ã®åæ°ã®éå ãšãªãã
Aâ©Bâ©C ã®èŠçŽ ã®åæ°ã¯
:n(Aâ©Bâ©C) ïŒ 3
ã§ããã
ãã£ãŠã
:n(AâªBâªC) ïŒ n(A) ïŒ n(B) ïŒ n(C) ân(Aâ©B) ân(Bâ©C) ân(Câ©A) ïŒ n(Aâ©Bâ©C) ïŒ 50 ïŒ 33 ïŒ 20 â 16 â 6 â 10 ïŒ 3 ïŒ 74
ãªã®ã§ã100以äžã®èªç¶æ°ã®ãã¡ã®2ã®åæ°ãŸãã¯3ã®åæ°ãŸãã¯5ã®åæ°ã§ãããã®ã®åæ°ã¯ 74åã§ããã
== å Žåã®æ° ==
==== æš¹åœ¢å³ ====
[[File:Tree diagram sum 5 by three numbers.svg|thumb|]]
ããšãã°å€§äžå°3åã®ãµã€ã³ãããµã£ãŠãç®ã®åã5ã«ãªãç®ã®çµã¯ãäœéãããã ãããã
ãã®ãããªåé¡ã解ãæ¹æ³ã®ã²ãšã€ãšããŠãå³ã®ããã«ãçµã¿åãããç·åœããã§æžãæ¹æ³ãããã
倧äžå°ã®åèš3åã®ãµã€ã³ãããããã A,B,C ãšããŠè¡šãããããã®æåã«ãã©ã®ç®ãåºãã°åèš5ã«ãªãããèãããšãçµæã¯å³ã®ããã«ãªãã
ãã®ãããªå³ã '''暹圢å³'''ïŒãã
ãããïŒ ãšããã
;åé¡
3åã®ãµã€ã³ãããµããšããç®ã®åã6ã«ãªãå Žåã¯äœéããããã
==== éä¹ ====
æåã«ãnåã®ç°ãªã£ããã®ã䞊ã¹æããå Žåã®æ°ãæ°ããã
ãŸãæåã«äžŠã¹ããã®ã¯nåã次ã«äžŠã¹ããã®ã¯(n-1)åããã®æ¬¡ã«äžŠã¹ããã®ã¯(n-2)å ... ãšã ãã ããšéžã¹ããã®ã®æ°ãæžã£ãŠè¡ããæåŸã«ã¯1åããæ®ããªããªãããšã«æ³šç®ãããšããã®äºæã«é¢ããå Žåã®æ°ã¯
:<math>
n (n-1) (n-2) \cdots 3 \cdot 2 \cdot 1
</math>
ãšãªãã1ããnãŸã§ã®èªç¶æ°ã®ç©ã«ãªãã
ãã®æ°ã '''éä¹''' ïŒããããããfactorialïŒãšåŒã³ãéä¹nã®èšå·ã¯ <math> n! </math> ã§è¡šãã
ããªãã¡ãéä¹ã¯
{{ããã¹ãããã¯ã¹|<math>n! = n (n-1) (n-2) \cdots 3 \cdot 2 \cdot 1</math>}}
ãšå®çŸ©ãããããã®éä¹ã®èšå·ã䜿ãã°ããã®åé¡ã®ãšãã®å Žåã®æ°ã¯ n!ã§ãããšèšãããšãåºæ¥ãã
* åé¡äŸ
** åé¡
:<math>
3! , \quad 4! , \quad 5! , \quad 6!
</math>
ãããããèšç®ããã
** 解ç
:<math>
n! = 1 \cdot 2 \cdot \cdots n
</math>
ãçšããŠèšç®ããã°ããã
çãã¯ã
:<math>3! = 6</math>
:<math>4! = 24</math>
:<math>5! = 120</math>
:<math>6! = 720</math>
ãšãªãã
** åé¡
ããããã«1ãã5ãŸã§ã®æ°åãæžããã5æã®ã«ãŒãã眮ããŠããã
ãã®ã«ãŒãã䞊ã¹æãããšãã
(I)ã«ãŒãã®äžŠã¹æ¹ã®æ°ã (II)å¶æ°ãåŸãããã«ãŒãã®äžŠã¹æ¹ã®æ°ã (III)å¥æ°ãåºãã«ãŒãã®äžŠã¹æ¹ã®æ°ããããããèšç®ããã
** 解ç
(I)
ã«ãŒãã®æ°ã5æã§ãããããåºå¥ã§ããããšãããã«ãŒãã®äžŠã¹æ¹ã®æ°ã¯
:<math>5!</math>
ãšãªãã120ãšãªãã
(II)
å¶æ°ãåŸãããã«ã¯äžã®äœã§ããæãå³ã«åºãã«ãŒãããå¶æ°ãšãªãã°ããã
ãã®ãããªã«ãŒãã¯2ãš4ã§ãããããããã«å¯ŸããŠåŸã®4æã¯èªç±ã«éžãã§ããã
ãã®ããããã®ãããªã«ãŒãã®äžŠã¹æ¹ã¯ã
:<math>2 \times 4! = 48</math>
ãšãªãã
(III)
å¥æ°ãåŸãããã«ã¯äžã®äœã§ããæãå³ã«åºãã«ãŒãããå¥æ°ãšãªãã°ããã
ãã®ãããªã«ãŒãã¯1,3,5ã§ãããããããã«å¯ŸããŠåŸã®4æã¯èªç±ã«éžãã§ããã
ãã®ããããã®ãããªã«ãŒãã®äžŠã¹æ¹ã¯ã
:<math>3 \times 4! = 72</math>
ãšãªããäžæ¹ã5æã®ã«ãŒãã䞊ã¹æããŠåŸãããæ°ã¯å¿
ãå¶æ°ãå¥æ°ã®
ã©ã¡ããã§ããã®ã§ã(I)ã®çµæãã(II)ã®çµæãåŒãããšã«ãã£ãŠã
(III)ã®çµæã¯åŸãããã¯ãã ããå®éã«ãããèšç®ãããš
:<math>120 - 48 = 72</math>
ãšãªãã確ãã«ãã®ããã«ãªã£ãŠããã
** åé¡
0,1,2,3,5ãæžããã5æã®ã«ãŒãããããããã䞊ã³æãããšãã
:(I)5æ¡ã®æ°ãåŸãããæ°ã (II) 5æ¡ã®å¶æ°ãåŸãããæ°ã(III) 5æ¡ã®å¥æ°ãåŸãããæ°ã(IV) 5æ¡ã®5ã®åæ°ãåŸãããæ°
ãããããæ±ããã
** 解ç
(I)
å
é ã0ã«ãªã£ããšãã«ã¯5æ¡ã®æ°ã«ãªããªãããšã«æ³šæããã°ãããæ±ããå Žåã®æ°ã¯
:<math>4 \times 4! = 96</math>
ãšãªãã
(II)
æåã0ã§ãªãæåŸã0ã2ã§ããæ°ãæ°ããã°ããããŸããæåŸã0ã§ãããšãã«ã¯ãæ®ãã®4æã¯ä»»æã§ããã®ã§
:<math>4! = 24</math>
éãã®çµã¿åãããããã
次ã«ãæåŸã2ã§ãããšãã«ã¯æåã¯0ã§ãã£ãŠã¯ãããªãã®ã§ã
:<math>3 \times 3! = 18</math>
éãããã
2ã€ãåãããæ°ã5æ¡ã®å¶æ°ãåŸãããå Žåã®æ°ã§ãããçãã¯ã
:<math>24 + 18 = 42</math>
ãšãªãã
(III)
(I)ã®çµæãã(II)ã®çµæãåŒãã°ããããããã§ã¯ãã®çµæãæ£ãããã©ãã
確ãããããã«ã5æ¡ã®å¥æ°ãåŸãããçµã¿åãããæ°ãäžããŠã¿ãã
5æ¡ã®å¥æ°ãåŸãããã«ã¯æåŸã®æ°ã¯1,3,5ã®ããããã§ãªããŠã¯ãªããªãã
ãã®ãã¡ã®ã©ã®å Žåã«ã€ããŠã5æ¡ã®æ°ãåŸãããã«ã¯æåã®æ°ã0ã§
åã£ãŠã¯ãªããªãã®ã§ããããã®å Žåã®æ°ã¯ã
:<math>3 \times 3 \times 3! = 54</math>
ãšãªãããã5æ¡ã®å¥æ°ãåŸãå Žåã®æ°ã§ããã
(II)ã®çµæãšè¶³ãåããããšç¢ºãã«(I)ã®çµæãšçãã96ãåŸãã
(IV)
5ã®åæ°ãåŸãããã«ã¯æåŸã®æ°ã0ã5ã§ããã°ããã
ãã®ãšãæåŸã0ã«ãªãå Žåã®æ°ã¯ä»ã®4ã€ãä»»æã§ãããã
:<math>4! = 24</math>
ååšããã次ã«ãæåŸã5ã«ãªãå Žåã®æ°ã¯æåã®æ°ã0ã§ãã£ãŠã¯ãªããªããã
:<math>3 \times 3! = 18</math>
ã ãååšããã
ãã£ãŠçãã¯
:<math>24 + 18=42</math>
ãšãªãã
===== é å =====
nåã®ç°ãªã£ããã®ããråãéžãã§ãé çªãã€ããŠäžŠã¹ãä»æ¹ã®æ°ãã<math> {}_n \mathrm{P}_r </math>ãšæžãã
ãŸãããã®ãããªèšç®ã®ä»æ¹ã '''é å''' ïŒãã
ããã€ãè±ïŒpermutationïŒ ãšããã
nåã®ç°ãªã£ããã®ããråãéžãã§é çªãã€ããŠäžŠã¹ãä»æ¹ã®æ°ã®ããš ãã
:'''nåããråãšãé å'''
ã®ããã«èšãã
æåã«äžŠã¹ããã®ã¯néãã次ã«äžŠã¹ããã®ã¯ (nâ1)éã ããã®æ¬¡ã«äžŠã¹ããã®ã¯ (nâ2)éã ,... æåŸã«ã¯ (nâ(râ1))éã ãšããããã«ãã ãã ãéžã¹ããã®ã®æ°ãæžã£ãŠè¡ãããšã«æ³šç®ãããšãé åã®ç·æ°ãšããŠ
:<math> {}_n \mathrm{P}_r = n (n-1) (n-2) \cdots (n-r+1) = \frac{n!}{(n-r)!}</math>
ãåŸãããã
:â» ãªã <math> {}_n \mathrm{P}_r </math> ã®P ãšã¯ãé åãæå³ããè±èª permutation ã®é æåã§ããã
äžè¬ã« <math> {}_n \mathrm{P}_r </math> ã§ã¯ n ⧠r ã§ããã
* åé¡äŸ
** åé¡
(I)
:<math>{} _5 \mathrm{P} _3</math>
(II)
:<math>{} _4 \mathrm{P} _2</math>
(III)
:<math>{} _7 \mathrm{P} _3</math>
(IV)
:<math>{} _{10} \mathrm{P} _5</math>
(V)
:<math>{} _{10} \mathrm{P} _1</math>
(VI)
:<math>{} _7 \mathrm{P} _0</math>
ãããããèšç®ããã
** 解ç
ãããã
:<math>{} _n \mathrm{P} _r = n (n-1) (n-2) \cdots (n-r+1) = \frac{n!}{(n-r)!}</math>
ãçšããŠèšç®ããã°ããã
çµæã¯ã
(I)
:<math>{} _5 \mathrm{P} _3 = 5 \times 4 \times 3 = 60</math>
(II)
:<math>{} _4 \mathrm{P} _2 = 4 \times 3 = 12</math>
(III)
:<math>{} _7 \mathrm{P} _3 = 7\times 6\times 5 = 210</math>
(IV)
:<math>{} _{10} \mathrm{P} _5 = 10\times 9\times 8\times 7\times 6 = 30240</math>
(V)
:<math>{} _{10} \mathrm{P} _1 = 10 </math>
(VI)
:<math>{} _7 \mathrm{P} _0 = \frac {7!}{7!} = 1</math>
ãšãªãã
(V)ãš(VI)ã«ã€ããŠã¯äžè¬çã«æŽæ°nã«å¯ŸããŠ
:<math>{} _n \mathrm{P} _1 = n</math>
:<math>{} _n \mathrm{P} _0 = 1</math>
ãåŸãããããã®ãšã
:<math>{} _n \mathrm{P} _0 = 1</math>
ã¯å
ã
ã®é åã®å®çŸ©ãããããš"nåã®ãã®ã®äžãã1ã€ãéžã°ãªãå Žåã®æ°"ã«å¯Ÿå¿ããŠãããå°ã
äžèªç¶ãªããã«æãããããã®ããã«å€ã眮ããŠãããšäŸ¿å©ã§ããããéåžžãã®ããã«çœ®ãã®ã§ãããããŸããå®éã®å Žåã®æ°ã®èšç®ã§ãã®ãããªå€ãæ±ãããšã¯å€ãã¯ãªããšãããã
===== åé å =====
[[File:Circular Permutation 5 elements.svg|thumb|800px]]
{{-}}
A, B, C, D, E ã®5人ãå圢ã«æãã€ãªãã§èŒªãã€ãããšãããã®äžŠã³æ¹ã¯äœéããããã
ãã®ãããªåé¡ã®å Žåãå³ã®ããã«ãå転ãããšéãªã䞊ã³ã¯åã䞊ã³ã§ãããšèããã
解ãæ¹ã®èãæ¹ã¯æ°çš®é¡ããã
:1ã€ã®èãæ¹ãšããŠã5人ãå圢ã«äžŠã¶ãšããå³ã®ããã«å転ãããšåãã«ãªã䞊ã³ã¯ã5éããã€ãããšããèãæ¹ã«ããã <math> \frac{ 5! }{ 5 } </math> ãšããèãæ¹ã§ããã
:ããäžã€ã®èãæ¹ãšããŠãAãåºå®ããŠãæ®ãã®4人ã®äžŠã³ãèããã°ãå¥ã
ã®äžŠã³ãäœãããšããèãæ¹ã§ã <math> (5-1)! </math> ãšããèãæ¹ã§ããã
ã©ã¡ãã«ãããçµæã¯
:<math> 4! = 4 \cdot 3 \cdot 2 \cdot = 24 </math> ïŒéãïŒ
ã§ããã
äžè¬ã« ç°ãªã nå ã®ãã®ãå圢ã«äžŠã¹ããã®ãåé åãšããã
åé åã®ç·æ°ãšããŠã次ã®ããšãæãç«ã€ã
ç°ãªã nå ã®åé åã®ç·æ°ã¯ <math> (n-1)! </math> ã§ããã
==== çµã¿åãã ====
nåã®ç°ãªã£ããã®ããråãéžãã§ãé çªãã€ããã«äžŠã¹ãä»æ¹ã®æ°ãã<math> {}_n \mathrm{C}_r </math>ãšæžãããã®ãããªèšç®ã çµã¿åããïŒcombinationïŒ ãšããã
äŸãã°ãããã€ãããããŒã«ã«çªå·ããµã£ãŠãããªã©ã®æ¹æ³ã§ãããããã®ããŒã«ãåºå¥ã§ããnåã®ããŒã«ãå
¥ã£ãç®±ã®äžããråã®ããŒã«ãåãã ãæãåãã ããããŒã«ãåãã ããé ã«äžŠã¹ããšãããšããã®å Žåã®æ°ã¯é å<math>{} _n \mathrm{P} _r</math>ã«å¯Ÿå¿ããã
äžæ¹ãåãã ããããŒã«ã®çš®é¡ãéèŠã§ããåãã ããé çªãç¹ã«å¿
èŠã§ãªããšãã«ã¯ããã®å Žåã®æ°ã¯çµã¿åãã<math>{} _n \mathrm{C} _r</math>ã«å¯Ÿå¿ããããããã®æ°ã¯ãäºãã«ç°ãªã£ãå Žåã®æ°ã§ãããäºãã«ç°ãªã£ãèšç®æ³ãå¿
èŠãšãªãã
<math>{} _n \mathrm{C} _r</math>ã¯ã<math>{} _n \mathrm{P} _r</math>éãã®äžŠã¹æ¹ãäœã£ãåŸã«ãããã®äžŠã³ãç¡èŠãããã®ã«çãããããã§ãråãåãã ããŠäœã£ã䞊ã³ã«ã€ããŠã䞊ã¹æ¹ãç¡èŠãããšr!åã®äžŠã³ãåäžèŠãããããšããããã
ãªããªããråã®ãäºãã«åºå¥ã§ããæ°ãèªç±ã«äžŠã³æããå Žåã®æ°ã¯r!ã§ãããããããå
šãŠåäžèŠããããšããã°å
šäœã®å Žåã®æ°ã¯
r!ã®åã ãæžãããšã«ãªãããã§ããããã£ãŠã
:<math> {}_n \mathrm{C}_r =\frac { {}_n \mathrm{P}_r }{r!} = \frac{n!}{(n-r)!r!}</math>
ãåŸãããã
{{æŒç¿åé¡|
次ã®å€ãèšç®ãã
(I)
:<math>{} _5 \mathrm{C} _2</math>
(II)
:<math>{} _7 \mathrm{C} _3</math>
(III)
:<math>{} _{10} \mathrm{C} _1</math>
(VI)
:<math>{} _8 \mathrm{C} _0</math>
|
ããããã«ã€ããŠ
:<math>{}_n \mathrm{C} _r =\frac { {}_n \mathrm{P} _r }{r!} = \frac{n!}{(n-r)!r!}</math>
ãçšããŠèšç®ããã°ããã
(I)
:<math>{} _5 \mathrm{C} _2 = \frac {5\times 4}{2\times 1} = 10</math>
(II)
:<math>{} _7 \mathrm{C} _3 = \frac { 7\times 6\times 5} { 3\times 2\times 1} = 35</math>
(III)
:<math>{} _{10} \mathrm{C} _1 = \frac {10} {1} = 10</math>
(VI)
:<math>{} _8 \mathrm{C} _0 = 1 </math>
ãšãªãã(IV)ã«ã€ããŠã¯äžè¬ã«æŽæ°nã«å¯ŸããŠ
:<math>{} _n \mathrm{C} _0 = 1</math>
ãå®çŸ©ããã
ããã¯ããšããšã®çµã¿åããã®èšç®ãšããŠã¯nåã®ç©äœã®ãªããã0åã®ç©äœãéžã¶å Žåã®æ°ã«å¯Ÿå¿ããŠããã
å®éã«ã¯ãã®ãããªå Žåã®æ°ãèšç®ããããšèããããšã¯ããŸãç¡ããšæãããããèšç®ã®äŸ¿å®äžã®ããå®çŸ©ãäžã®ããã«ããã
ãŸããäžã®èšç®ã§ã¯
:<math>{} _n \mathrm{C} _r =\frac { {}_n \mathrm{P} _r }{r!}</math>
ã®åŒããã®ãŸãŸçšãããšã
:<math>{} _n \mathrm{C} _0 = \frac {{} _n \mathrm{P} _0} {0!} = \frac 1 {0!} = 1</math>
ã€ãŸãã
:<math>0! = 1</math>
ãšãªã£ãŠããã
å®éã«ã¯éä¹ã®èšç®ã¯æŽæ°nã«ã€ããŠã¯nãã1ãŸã§ãäžãããªããããç®ããŠãããšããä»æ¹ã§èšç®ãããŠããã®ã§ãäžã®çµæã¯åŠã«æããã
ãããå®éã«ã¯ãããé²ãã çè«ã«ãã£ãŠãã®çµæã¯æ£åœåãããã®ã§ããã
ãã®å Žåã䟿å®äž
:<math>0! = 1</math>
ã0ã®éä¹ã®å®çŸ©ãšããŠåããããã®ã§ããã
}}
{{æŒç¿åé¡|
5åã®ããŒã«ãå
¥ã£ãããŒã«å
¥ããã2ã€ã®ããŒã«ãåãã ããšã(ããŒã«ã¯ãããã
åºå¥ã§ãããã®ãšããã)2ã€ã®ããŒã«ã®éžã³æ¹ã¯ã
äœéããããèšç®ããã|ããŒã«ã®åãã ãæ¹ã¯çµã¿åããã®æ°ãçšããŠèšç®ã§ããã
5ã€ã®ããŒã«ã®äžãã2ã€ãåãã ãã®ã§ãããããã®å Žåã®æ°ã¯ã
:<math>{} _5 \mathrm{C} _2 = \frac {5!}{2!3!} = \frac { 5 \cdot 4 \cdot 3 \cdot 2\cdot 1}{(3 \cdot 2 \cdot 1)( \cdot 2 \cdot 1)}</math>
:<math>= 10</math>
ãšãªãããã£ãŠãããŒã«ã®åãã ãæ¹ã¯10éãã§ããããšããããã}}
{{æŒç¿åé¡|
6åã®äºãã«åºå¥ã§ããããŒã«ãå
¥ã£ãç®±ãããã
ãã®äžãã (I)3ã€ã®ããŒã«ãš2ã€ã®ããŒã«ãåãã ãæ¹æ³ã®å Žåã®æ°ã(II)2ã€ã®ããŒã«ãåãåºãããšã2åããè¿ãããããããå¥ã®äºãã«åºå¥ã§ããè¢ã«ãããå Žåã®æ°ã(III)2ã€ã®ããŒã«ãåãåºãããšã2åããè¿ãããããããå¥ã®äºãã«åºå¥ã§ããªãè¢ã«ãããå Žåã®æ°ããããããèšç®ããã|
(I)
æåã«ããŒã«ãåãã ããšãã«ã¯ã6ã€ã®ããŒã«ã®äžãã3ã€ã®ããŒã«ãåãã ãããšãããã®å Žåã®æ°ã¯
:<math>{} _6 \mathrm{C} _3</math>
ã ãããããŸãã次ã«ãããåãé€ããäžãã2ã€ã®ããŒã«ãåãé€ããšãã«ã¯
ãã®åãã ãæ¹ã¯ã
:<math>{} _3 \mathrm{C} _2</math>
ã ãããã
ãã£ãŠããã®ãšãã®å Žåã®æ°ã¯
:<math>{} _6 \mathrm C _3 \times {} _3 \mathrm{C} _2 </math>
ã ãã«ãªããå®éãã®å€ãèšç®ãããšã
:<math>{} _6 \mathrm C _3 \times {} _3 \mathrm{C} _2 = 20 \times 3 = 60</math>
ãšãªãã60éãã§ããããšãåããã
(II)
(I)ã®å Žåãšåæ§ã«6ã€ã®ããŒã«ã®äžãã2ã€ã®ããŒã«ã
åãã ãããšãããã®å Žåã®æ°ã¯
:<math>{} _6 \mathrm{C} _2</math>
ã ãããããŸãã次ã«ãããåãé€ããäžãã2ã€ã®ããŒã«ãåãé€ããšãã«ã¯
ãã®åãã ãæ¹ã¯ã
:<math>{} _4 \mathrm{C} _2</math>
ã ãããã
ãã£ãŠããã®ãšãã®å Žåã®æ°ã¯
:<math>{} _6 \mathrm C _2 \times {} _4 \mathrm{C} _2 </math>
ã ãã«ãªããå®éãã®å€ãèšç®ãããšã
:<math>{} _6 \mathrm C _2 \times {} _4 \mathrm{C} _2 = 15 \times 6 = 90</math>
ãšãªãã90éãã§ããããšãåããã
(III)
(II)ãšåãèšç®ã§å€ãæ±ããããšãåºæ¥ãããä»åã¯ããŒã«ããããè¢ã
äºãã«åºå¥ã§ããªãããšã«æ³šæããªããŠã¯ãªããªãã
ãã®ããšã«ãã£ãŠãèµ·ããããå Žåã®æ°ã¯(II)ã®å Žåã®ååã«ãªãã®ã§
æ±ããå Žåã®æ°ã¯45éããšãªãã}}
<math> {}_n \mathrm{C}_r </math>ã«ã€ããŠä»¥äžã®åŒãæãç«ã€ã
:<math> {}_n \mathrm C_r = _n \mathrm{C} _{n-r}</math>
:<math> {}_n \mathrm C _r = _{n-1} \mathrm C_r + _{n-1} \mathrm{C} _{r-1}</math>
å°åº
:<math> {}_n \mathrm{C}_r = \frac{n!}{(n-r)!r!}</math>
ãçšãããšã
:<math> {}_n \mathrm{C}_{n-r} = \frac{n!}{(n-(n-r))!(n-r)!}</math>
:<math> = \frac{n!}{r!(n-r)!}</math>
:<math> = {}_n \mathrm{C}_r </math>
ãåŸããã瀺ãããã
åæ§ã«
:<math> {}_n \mathrm{C}_r = \frac{n!}{(n-r)!r!}</math>
ãçšãããšã
:<math> {}_{n-1} \mathrm C_r + _{n-1} \mathrm{C} _{r-1}</math>
:<math>= \frac {(n-1)!}{(n-1-r)!r!} +\frac {(n-1)!}{(n-r)!(r-1)!} </math>
:<math>= \frac {(n-r)}n {}_n \mathrm{C}_r +\frac r n {}_n \mathrm{C}_r</math>
:<math>= {}_n \mathrm{C}_r</math>
ãšãªã瀺ãããã
æåã®åŒã¯ãç°ãªãnåã®ãã®ã®ãã¡råã«Xãšããã©ãã«ãã€ããæ®ãã®n-råã«Yãšããã©ãã«ãã€ããå Žåã®æ°ããæ±ããããšãã§ãããç°ãªãnåã®ãã®ã®ãã¡ããråãéžã³ã©ãã«Xãã€ããæ®ãã«ã©ãã«Yãã€ããå Žåã®æ°ã¯<math>_n \mathrm C _r</math> ã§ãããç°ãªãnåã®ãã®ã®ãã¡ããn-råãéžã³ãã©ãã«Yãã€ããæ®ãã«ã©ãã«Xãã€ããå Žåã®æ°ã¯<math>_n \mathrm C _{n-r}</math> ã§ãããåœç¶ãåè
ãšåŸè
ã®å Žåã®æ°ã¯çããã®ã§ãããããã<math>_n \mathrm C _r = _n \mathrm C_{n-r}</math> ãæ±ããããã
2ã€ç®ã®åŒã¯ã
"nåã®ãã®ããråãéžã¶ä»æ¹ã®æ°ã¯ã次ã®æ°ã®åã§ããã
æåã®1ã€ãéžã°ãã«ä»ã®n-1åããråãéžã¶ä»æ¹ã®æ°ãšãæåã®1ã€ãéžãã§ä»ã®n-1åããr-1åãéžã¶ä»æ¹ã®æ°ãšã®
åã§ããã"
ãšããããšãè¡šãããŠããã
* åé¡äŸ
:<math>{} _n \mathrm{C} _r = _n \mathrm{ \mathrm{C}} _{n-r}</math>
ãçšããŠ
(I)
:<math>{} _5 \mathrm{C} _3</math>
(II)
:<math>{} _7 \mathrm{C} _4</math>
(III)
:<math>{} _{10} \mathrm{C} _9</math>
(VI)
:<math>{} _8 \mathrm{C} _5</math>
ãããããèšç®ããã
** 解ç
äžã®åŒãçšããŠèšç®ããããšãåºæ¥ãããã¡ããçŽæ¥ã«èšç®ããŠã
çããåŸãããšãåºæ¥ãããéåžžã¯ç°¡ååããŠããèšç®ããæ¹ã楜ã§ããã
(I)
:<math>{} _5 \mathrm{C} _3 = {} _5 \mathrm{C} _{5-3} = {} _5 \mathrm{C} _2 = 10</math>
(II)
:<math>{} _7 \mathrm{C} _4= {} _7 \mathrm{C} _{7-4}={} _7 \mathrm{C} _3 = 35</math>
(III)
:<math>{} _{10} \mathrm{C} _9= {} _{10} \mathrm{C} _{10-9}= {} _{10} \mathrm{C} _1 = 10</math>
(VI)
:<math>{} _8 \mathrm{C} _5= {} _8 \mathrm{C} _{8-5}= {} _8 \mathrm{C} _3= 56</math>
ãšãªãã
** åé¡
å³ã®ãããªã«ãŒããå·Šäžã®ç¹ããå³äžã®ç¹ãŸã§æ©ããŠè¡ã人ãããã
ãã ãããã®äººã¯å³ãäžã«ããé²ããªããšããããã®ãšãã
:(I) å·Šäžããå³äžãŸã§é²ãä»æ¹ã®æ°
:(II) aç¹ãééããŠå³äžãŸã§é²ãä»æ¹ã®æ°
ãèšç®ããããã ãaç¹ã¯*ãšæžãããŠããç¹ã®ããäžã®éè·¯ã®ããšããããŠããã
ããããã®ã«ãŒãã¯éåããŠããªã瞊4ã€ã暪5ã€ã®ç¢ç€ç®äžã®ã«ãŒãã«
ãªã£ãŠããããšã«æ³šæããã
___________
|_|_|_|_|_|
|_|_|*|_|_|
|_|_|_|_|_|
|_|_|_|_|_|
** 解ç
(I)
å·Šäžã«ãã人ã¯9åé²ãããšã§å³äžã®ç¹ã«èŸ¿ãçããããã®ãããå·Šäžã«ãã人ãéžã³ããã«ãŒãã®æ°ã¯9åã®ãã¡ã®ã©ã®åã§å³ã§ã¯ãªãäžã
éžã¶ãã®å Žåã®æ°ã«çããããã®ãããªå Žåã®æ°ã¯ã9åã®ãã¡ããèªç±ã«4ã€ã®å Žæãéžã¶æ¹æ³ã«çãããçµã¿åãããçšããŠæžãããšãåºæ¥ããå®éã«9åã®ãã¡ããèªç±ã«4ã€ã®å Žæãéžã¶æ¹æ³ã¯ã
:<math>{} _9 \mathrm{C} _4</math>
ã§æžãããããã®éãèšç®ãããšã
:<math>{} _9 \mathrm{C} _4 = 126</math>
ãåŸãããã
(II)
aç¹ãééããŠé²ãã«ãŒãã®æ°ã¯aç¹ã®å·Šã®ç¹ãŸã§ãã£ãŠããaç¹ãééããaç¹ã®å³ã®ç¹ãéã£ãŠå³äžã®ç¹ãŸã§ããä»æ¹ã®æ°ã«çããã
ããããã®ã«ãŒãã®æ°ã¯(I)ã®æ¹æ³ãçšããŠèšç®ããããšãã§ããããã®æ°ãå®éã«èšç®ãããšã
:<math>{} _4 \mathrm{C} _2 \times {} _4 \mathrm{C} _2 = 6 \times 6 = 36 </math>
ãšãªãã36éãã§ããããšãåããã
{{æŒç¿åé¡|<math>r_n \mathrm C _r = n_{n-1} \mathrm C_{r-1}</math>ã瀺ã|<math>r_n \mathrm C _r = r\frac{n!}{r!(n-r)!} = n\frac{(n-1)!}{(r-1)((n-1)-(r-1))!} = n_{n-1} \mathrm C_{r-1}</math>}}
==== éè€çµã¿åãã ====
ç°ãªãnåã®ç©ºç®±ã«råã®ãã®ãå
¥ããå Žåã®æ°ãéè€çµã¿åãããšããã <math>_n \mathrm H_r</math> ã§è¡šãã
éè€çµåãã«ã€ããŠæ¬¡ã®ããã«èå¯ããã
<math>x_1,x_2,\cdots,x_n,r</math> ãéè² æŽæ°ãšããæ¹çšåŒ <math>x_1+x_2 + \cdots +x_n = r</math> ã®è§£ã®åæ°ã«ã€ããŠèããããã®è§£ã®åæ°ã¯ <math>x_1,x_2,\cdots,x_n</math> ã« <math>r</math> åã®1ãåé
ããå Žåã®æ°ãšèããããšãã§ããã®ã§ãéè€çµã¿åããã®å®çŸ©ããã<math>_n \mathrm H_r</math> ã§ããã
ãŸãããã®æ¹çšåŒã®éè² æŽæ°è§£ã®åæ°ã¯ãråã®âã«n-1åã®åºåãã眮ãå Žåã®æ°ãšãèãããããã€ãŸããâââ...ââ(rå)ã«n-1åã®åºåãïœã䞊ã¹ããšâïœââïœ...âïœâã®ããã«ãªããããã§ãå·Šããé ã«åºåãã§åºåãããâã®åæ°ãããããã<math>x_1,x_2,\cdots,x_n</math> ãšãããšãããã¯æ¹çšåŒã®è§£ãšãªãã
ãã®å Žåã®æ°ã¯ãråã®âãšn-1åã®åºåãïœã䞊ã¹ããå Žåã®æ°ãªã®ã§ã<math>_{n+r-1} \mathrm C _r</math> ã§ãããæ¹çšåŒã®éè² æŽæ°è§£ã®åæ°ã«ã€ããŠ2éãã®æ¹æ³ã§æ±ãŸã£ãã®ã§ãããã¯çããã <math>_n \mathrm H_r = _{n+r-1} \mathrm C_r</math> ãæãç«ã€ã
== 確ç ==
==== 確çã®èšç® ====
ããå Žåã®æ°ããå®éã«çŸãããå²åã®ããšã確çïŒãããã€ãè±ïŒprobabilityïŒãšåŒã¶ã
ããå Žåã®æ°ãå®éã«çŸãããå²åã¯ããã®å Žåã®æ°ãå²ãç®ã§ããã®äºæã«ãããŠèµ·ããåŸãå
šãŠã®äºæã®å Žåã®æ°ã§å²ã£ããã®ã«çããã
ããšãã°ãå
šãçããå²åã§å
šãŠã®é¢ãåºãããããããµã£ããšãã«1ãåºã確çã¯<math>\frac 1 6</math>ã§ããã
ããã¯1ãåºãå Žåã®æ°1ãã1,2,3,4,5,6ã®ãããããåºãå Žåã®æ°6ã§å²ã£ããã®ã«çããã
{| style="border:2px solid skyblue;width:80%" cellspacing=0
|style="background:skyblue"|'''äºè±¡Aã®ç¢ºç'''
|-
|style="padding:5px"|
èµ·ãããããã¹ãŠã®å Žåã®æ°ãNãäºè±¡Aã®èµ·ããå Žåã®æ°ãaãšãããšããäºè±¡Aã®èµ·ãã確çP(A)ã¯ä»¥äžã®åŒã§æ±ããããã
:<math>
P(A) = \frac{a}{N}
</math>
|}
* åé¡äŸ
** åé¡
èµ€ç2åãšçœç3åãå
¥ã£ãè¢ãããçã2ååæã«åãåºãããã®ãšãã2åãšãçœçãåºã確çãæ±ããã
** 解ç
èµ€çœããããŠ5åã®çãã2åãåãåºãæ¹æ³ã¯
:<math>{} _5 \mathrm{C} _2 = \frac {5\times 4}{2\times 1} = 10</math>ïŒéãïŒ
ãã®ãã¡ã2åãšãçœçã«ãªãå Žåã¯
:<math>{} _3 \mathrm{C} _2 = \frac {3\times 2}{2\times 1} = 3</math>ïŒéãïŒ
ãã£ãŠæ±ãã確ç㯠<math> \frac {3}{10} </math>
==== 確çã®æ§è³ª ====
確çã®å®çŸ©ããã次ã®æ§è³ªãåŸãããã
{| style="border:2px solid skyblue;width:80%" cellspacing=0
|style="background:skyblue"|'''確çã®æ§è³ª'''
|-
|style="padding:5px"|
ïŒ1ïŒã©ããªäºè±¡Aã«ã€ããŠãã <math>0 \leqq P(A) \leqq 1</math><br>
ïŒ2ïŒæ±ºããŠèµ·ãããªãäºè±¡ã®ç¢ºç㯠0<br>
ïŒ3ïŒå¿
ãèµ·ããäºè±¡ã®ç¢ºç㯠1
|}
==== æåäºè±¡ã®ç¢ºç ====
2ã€ã®äºè±¡A,Bãåæã«èµ·ãããªããšããäºè±¡AãšBã¯äºãã«'''æå'''ïŒã¯ãã¯ããè±ïŒexclusiveïŒã§ããããŸãã¯AãšBã¯'''æåäºè±¡'''ã§ãããšããã
{| style="border:2px solid skyblue;width:80%" cellspacing=0
|style="background:skyblue"|'''æåäºè±¡ã®ç¢ºç'''
|-
|style="padding:5px"|
AãšBãæåäºè±¡ã®ãšããAãŸãã¯Bãèµ·ãã確çã¯
:'''<math>P(A \cup B) = P(A)+P(B)</math>'''
|}
* åé¡äŸ
** åé¡
ç·å7人ã女å5人ã®äžãããããåŒãã§3人ã®å§å¡ãéžã¶ãšãã3人ãšãåæ§ã§ãã確çãæ±ããã
** 解ç
12人ã®äžãã3人ã®å§å¡ãéžã¶å Žåã®æ°ã¯
:<math>{} _{12} \mathrm{C} _3 = \frac {12\times 11\times 10}{3\times 2\times 1} = 220</math>ïŒéãïŒ
ããã§ãã3人ãšãç·åã§ãããäºè±¡ãAãã3人ãšã女åã§ãããäºè±¡ãBãšãããšãã3人ãšãåæ§ã§ãããäºè±¡ã¯ãåäºè±¡A ∪ Bã§ãããããããAãšBã¯æåäºè±¡ã§ããã
:<math>P(A) = \frac {{} _7 \mathrm{C} _3 }{220}= \frac {35}{220}</math>
:<math>P(B) = \frac {{} _5 \mathrm{C} _3 }{220}= \frac {10}{220}</math>
ãã£ãŠæ±ãã確ç㯠<math>P(A \cup B) = P(A)+P(B) = \frac {35}{220} + \frac {10}{220} = \frac {45}{220} = \frac {9}{44}</math>
==== äœäºè±¡ã®ç¢ºç ====
äºè±¡Aã«å¯ŸããŠããAã§ãªããäºè±¡ã<math>\overline{A}</math>ã§è¡šããAã®'''äœäºè±¡'''ïŒãããããïŒãšããã
{| style="border:2px solid skyblue;width:80%" cellspacing=0
|style="background:skyblue"|'''äœäºè±¡ã®ç¢ºç'''
|-
|style="padding:5px"|
Aã®äœäºè±¡ã<math>\overline{A}</math>ãšãããš<br>
:'''<math>P(A) = 1 - P(\overline{A})</math>'''
|}
* åé¡äŸ
** åé¡
èµ€ç5åãçœç3åã®èš8åå
¥ã£ãŠããè¢ãã3åã®çãåãåºããšããå°ãªããšã1åã¯çœçã§ãã確çãæ±ããã
** 解ç
8åã®çãã3åã®çãåãåºãå Žåã®æ°ã¯
:<math>{} _8 \mathrm{C} _3 = \frac {8\times 7\times 6}{3\times 2\times 1} = 56</math>ïŒéãïŒ
ããŸããå°ãªããšã1åã¯çœçã§ãããäºè±¡ãAãšãããšã<math>\overline{A}</math>ã¯ã3åãšãèµ€çã§ããããšããäºè±¡ã ãã
:<math>P(\overline{A}) = \frac {{} _5 \mathrm{C} _3 }{56} = \frac {10}{56} = \frac {5}{28}</math>
ãã£ãŠæ±ãã確çã¯
:<math>P(A) = 1 - P(\overline{A}) = 1 - \frac {5}{28} = \frac {23}{28}</math>
=== ç¬ç«ãªè©Šè¡ãšç¢ºç ===
==== ç¬ç«ãªè©Šè¡ãšç¢ºç ====
ãããã«ä»ã®çµæã«å¯ŸããŠåœ±é¿ããããŒããªãæäœãç¹°ãããããšããããããã®è©Šè¡ã¯'''ç¬ç«'''ïŒã©ããã€ãè±ïŒindependentïŒã§ãããšèšããç¬ç«ãªè©Šè¡ã«ã€ããŠã¯ãããè©Šè¡ã®èµ·ãã確çãå®ããããŠããŠããããnåç¹°ããããããšããããããèµ·ãã確çã¯ãããããã®è©Šè¡ãèµ·ãã確çã®ç©ãšãªãã
{| style="border:2px solid skyblue;width:80%" cellspacing=0
|style="background:skyblue"|'''ç¬ç«ãªè©Šè¡ãšç¢ºç'''
|-
|style="padding:5px"|
2ã€ã®ç¬ç«ãªè©Šè¡S,Tã«ã€ããŠãSã§ã¯äºè±¡AããTã§ã¯äºè±¡Bãèµ·ãã確çã¯<br>
:'''<math>P(A) \times P(B)</math>'''
|}
<br>
* åé¡äŸ
** åé¡
èµ€ç3åãçœç2åã®èš5åå
¥ã£ãŠããè¢ãããããã®äžãã1åã®çãåãåºããŠè²ã確ãããŠããè¢ã«æ»ããåã³1åãåãåºããšãã1åç®ã¯èµ€çã2åç®ã¯çœçãåãåºã確çãæ±ããã
** 解ç
1åç®ã«åãåºããçãè¢ã«æ»ãã®ã§ãã1åç®ã«åãåºããè©Šè¡ãšã2åç®ã«åãåºããè©Šè¡ãšã¯äºãã«ç¬ç«ã§ããã<br>
1åç®ã«åãåºãã1åãèµ€çã§ãã確ç㯠<math>\frac {3}{5}</math><br>
2åç®ã«åãåºãã1åãçœçã§ãã確ç㯠<math>\frac {2}{5}</math><br>
ãããã£ãŠæ±ãã確çã¯
:<math>\frac {3}{5} \times \frac {2}{5} = \frac {6}{25}</math>
==== å埩詊è¡ã®ç¢ºç ====
åãè©Šè¡ãäœåãç¹°ãè¿ããŠè¡ããšããååã®è©Šè¡ã¯ç¬ç«ã§ããããã®äžé£ã®ç¬ç«ãªè©Šè¡ããŸãšããŠèãããšããããã'''å埩詊è¡'''ïŒã¯ãã·ã ãããïŒãšããã
{| style="border:2px solid skyblue;width:80%" cellspacing=0
|style="background:skyblue"|'''å埩詊è¡ã®ç¢ºç'''
|-
|style="padding:5px"|
ããè©Šè¡ã§ãäºè±¡Eã®èµ·ãã確çãpã§ãããšããããã®è©Šè¡ãnåç¹°ãè¿ããšããäºè±¡Eããã®ãã¡råã ãèµ·ãã確çã¯<br>
:'''<math>{} _n \mathrm{C} _r \; p^r \; (1-p)^{n-r}</math>'''
|}
* åé¡äŸ
** åé¡
1åã®ããããã5åæãããšãã3ã®åæ°ã®ç®ã4ååºã確çãæ±ããã
** 解ç
1åã®ããããã1åæãããšãã3ã®åæ°ã®ç®ãåºã確çã¯
:<math>\frac {2}{6} = \frac {1}{3}</math>ã§ããã
ãã£ãŠã1åã®ããããã5åæãããšãã3ã®åæ°ã®ç®ã4ååºã確çã¯
:<math>{} _5 \mathrm{C} _4 \; \left( \frac{1}{3} \right)^4 \; \left(1 - \frac{1}{3} \right)^{5-4} = \frac {10}{243}</math>
==== æåŸ
å€ ====
èšå·ãΣãã«ã€ããŠã¯[[é«çåŠæ ¡æ°åŠB/æ°å#ç·åèšå·Î£|ãã¡ã]]ãåç
§ã
ããè©Šè¡ããã£ããšãã
ãã®è©Šè¡ã§åŸããããšæåŸ
ãããå€ã®ããšãæåŸ
å€ïŒãããã¡ãè±ïŒexpected valueïŒãšãããæåŸ
å€ã¯ã''n''åã®äºè±¡<math>r_k \ (k=1,2,\cdots,n)</math>ã«å¯ŸããŠãåã
<math>v_k</math>ãšããå€ãåŸãããäºè±¡<math>r_k</math>ãèµ·ãã確çã<math>p_k</math>ã§äžããããŠãããšãã
:<math>E = \sum_{k=1}^n v_k p_k</math>
ã«ãã£ãŠäžãããããäŸãã°ãããããããµã£ããšãåºãç®ã®æåŸ
å€ã¯ã
:<math>\frac 1 6 \times 1 +\frac 1 6 \times 2+\frac 1 6 \times 3+\frac 1 6 \times 4+\frac 1 6 \times 5+\frac 1 6 \times 6</math>
:<math>=\frac 1 6 (1 + 2+3+4+5+6)</math>
:<math>= \frac 7 2</math>
ãšãªãã
{{DEFAULTSORT:ãããšããã€ããããããA ã¯ããã®ãããšãããã€}}
[[Category:é«çåŠæ ¡æ°åŠA|ã¯ããã®ãããšãããã€]]
[[ã«ããŽãª:確ç]] | 2005-05-08T03:13:16Z | 2024-02-23T05:38:59Z | [
"ãã³ãã¬ãŒã:-",
"ãã³ãã¬ãŒã:æŒç¿åé¡",
"ãã³ãã¬ãŒã:Pathnav",
"ãã³ãã¬ãŒã:ããã¹ãããã¯ã¹"
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E6%95%B0%E5%AD%A6A/%E5%A0%B4%E5%90%88%E3%81%AE%E6%95%B0%E3%81%A8%E7%A2%BA%E7%8E%87 |
1,933 | é«çåŠæ ¡æ°åŠIII/ç©åæ³ | ããã§ã¯ãæ°åŠIIã®åŸ®åã»ç©åã®èãã§åŠãã ç©åã®æ§è³ªã«ã€ããŠãã詳ããæ±ãããŸããäžè§é¢æ°ãææ°ã»å¯Ÿæ°é¢æ°ãªã©ã®é¢æ°ã®ç©åã«ã€ããŠãåŠç¿ããã
ç©åæ³ã«ã€ããŠ
â« { f ( x ) + g ( x ) } d x = â« f ( x ) d x + â« g ( x ) d x , {\displaystyle \int \{f(x)+g(x)\}dx=\int f(x)dx+\int g(x)dx,} â« a f ( x ) d x = a â« f ( x ) d x {\displaystyle \int af(x)dx=a\int f(x)dx} (aã¯å®æ°)
ãæãç«ã€ã
å°åº
â« { f ( x ) + g ( x ) } d x = â« f ( x ) d x + â« g ( x ) d x {\displaystyle \int \{f(x)+g(x)\}dx=\int f(x)dx+\int g(x)dx}
ã®äž¡èŸºã埮åãããšã
巊蟺 =å³èŸº = f + g {\displaystyle f+g}
ãåŸãã
ãã£ãŠã
â« { f ( x ) + g ( x ) } d x = â« f ( x ) d x + â« g ( x ) d x {\displaystyle \int \{f(x)+g(x)\}dx=\int f(x)dx+\int g(x)dx}
ã®äž¡èŸºã¯äžèŽããã
(å®éã«ã¯2ã€ã®é¢æ°ã®å°é¢æ°ãäžèŽãããšãã ãããã®é¢æ°ã«ã¯å®æ°ã ãã®ã¡ãããããã
ä»®ã«ãF(x)ãšG(x)ãå
±éã®å°é¢æ°h(x)ãæã£ããšããã
ãã®ãšãã
( F ( x ) â G ( x ) ) â² = h ( x ) â h ( x ) = 0 {\displaystyle (F(x)-G(x))'=h(x)-h(x)=0}
ãšãªããã0ã®åå§é¢æ°ã¯å®æ°Cã§ããããšãåããã
ãã£ãŠã䞡蟺ãç©åãããšã
F ( x ) â G ( x ) = C {\displaystyle F(x)-G(x)=C}
ãšãªããF(x)ãšG(x)ã«ã¯å®æ°ã ãã®å·®ãããªãããšã確ãããããã
ãã£ãŠã
â« { f ( x ) + g ( x ) } d x = â« f ( x ) d x + â« g ( x ) d x {\displaystyle \int \{f(x)+g(x)\}dx=\int f(x)dx+\int g(x)dx}
ã¯å®æ°ã ãã®ã¡ãããå«ãã§æãç«ã€åŒã§ããã ããäžè¬ã«ãäžå®ç©åã絡ãçåŒã¯å®æ°åã®å·®ãå«ããŠæãç«ã€ãšããã®ãéäŸã§ããã)
â« a f ( x ) d x = a â« f ( x ) d x {\displaystyle \int af(x)dx=a\int f(x)dx}
ã«ã€ããŠã䞡蟺ã埮åãããšã
巊蟺=å³èŸº= a f(x)
ãåŸãã
ãã£ãŠã
â« a f d x = a â« f d x {\displaystyle \int afdx=a\int fdx}
ãæãç«ã€ããšãåãã
é¢æ° f ( x ) {\displaystyle f(x)} ã®åå§é¢æ°ã F ( x ) {\displaystyle F(x)} ãšãããš
â« a b f ( x ) = F ( b ) â F ( a ) = â ( F ( a ) â F ( b ) ) = â â« b a f ( x ) d x {\displaystyle \int _{a}^{b}f(x)\,=F(b)-F(a)=-(F(a)-F(b))=-\int _{b}^{a}f(x)\,dx} ã§ããã
â« a c f ( x ) d x + â« c b f ( x ) d x = ( F ( c ) â F ( a ) ) + ( F ( b ) â F ( c ) ) = F ( b ) â F ( a ) = â« a b f ( x ) d x {\displaystyle \int _{a}^{c}f(x)\,dx+\int _{c}^{b}f(x)\,dx=(F(c)-F(a))+(F(b)-F(c))=F(b)-F(a)=\int _{a}^{b}f(x)\,dx}
é¢æ°ã®åå§é¢æ°ãæ±ããæ段ãšããŠã ç©åå€æ°ãå¥ã®å€æ°ã§çœ®ãæããŠç©åãè¡ãªãæ段ãç¥ãããŠããã ããã眮æç©åãšåŒã¶ã
â« f ( g ( x ) ) d g ( x ) = â« f ( g ( x ) ) g â² ( x ) d x {\displaystyle \int f(g(x))dg(x)=\int f(g(x))g'(x)dx}
å°åº
â« f ( g ( x ) ) d g ( x ) = F ( g ( x ) ) {\displaystyle \int f(g(x))dg(x)=F(g(x))} ã x {\displaystyle x} ã«ã€ããŠåŸ®åãããšã
F â² ( g ( x ) ) = f ( g ( x ) ) g â² ( x ) {\displaystyle F'(g(x))=f(g(x))g'(x)}
åã³ x {\displaystyle x} ã«ã€ããŠç©åãããšã
â« f ( g ( x ) ) d g ( x ) = â« f ( g ( x ) ) g â² ( x ) d x {\displaystyle \int f(g(x))dg(x)=\int f(g(x))g'(x)dx}
ãŸããç¹ã«
äŸãã°ã â« ( a x + b ) 2 d x {\displaystyle \int (ax+b)^{2}dx} ãèããã
t = a x + b {\displaystyle t=ax+b} ãšçœ®ãã
ãã®äž¡èŸºã埮åãããš d t = a d x {\displaystyle dt=adx} ãæãç«ã€ããšãèæ
®ãããšã
ãšãªãããšããããã
å®éãã®åŒãxã§åŸ®åãããš ( a x + b ) 2 {\displaystyle (ax+b)^{2}} ãšäžèŽããããšãåãã
眮æç©åã䜿ããã«èšç®ããããšãåºæ¥ãã
( C â² = b 3 3 a + C {\displaystyle C'={\frac {b^{3}}{3a}}+C} ãšçœ®ãæããã)
= ( a x + b ) 3 3 a + C {\displaystyle ={\frac {(ax+b)^{3}}{3a}}+C} ãšãªã確ãã«äžèŽããã
é¢æ°ã®ç©ã®ç©åãè¡ãªããšãããé¢æ°ã®åŸ®åã ããåãã ããŠç©åãããšãããŸãç©åã§ããå Žåããããé¢æ° g ( x ) {\displaystyle g(x)} ã®åå§é¢æ°ã G ( x ) {\displaystyle G(x)} ãšãããš
â« f ( x ) g ( x ) d x = f ( x ) G ( x ) â â« f â² ( x ) G ( x ) d x {\displaystyle \int f(x)g(x)\,dx=f(x)G(x)-\int f'(x)G(x)\,dx}
å°åº
ç©ã®åŸ®åæ³ãã { f ( x ) G ( x ) } â² = f â² ( x ) G ( x ) + f ( x ) g ( x ) {\displaystyle \{f(x)G(x)\}'=f'(x)G(x)+f(x)g(x)} ã§ãããããã移é
ããŠ
f ( x ) g ( x ) = { f ( x ) G ( x ) } â² â f â² ( x ) G ( x ) {\displaystyle f(x)g(x)=\{f(x)G(x)\}'-f'(x)G(x)}
ã§ããã䞡蟺ãxã§ç©åããŠ
â« f ( x ) g ( x ) d x = f ( x ) G ( x ) â â« f â² ( x ) G ( x ) d x {\displaystyle \int f(x)g(x)\,dx=f(x)G(x)-\int f'(x)G(x)\,dx}
ãåŸãããã
äŸãã°ã
n â â 1 {\displaystyle n\neq -1} ã®ãšãã ( 1 n + 1 x n + 1 ) â² = x n {\displaystyle \left({\frac {1}{n+1}}x^{n+1}\right)'=x^{n}} ãªã®ã§ã
â« x n d x = 1 n + 1 x n + 1 + C {\displaystyle \int x^{n}dx={\frac {1}{n+1}}x^{n+1}+C}
n = â 1 {\displaystyle n=-1} ã®ãšãã ( log | x | ) â² = 1 x = x â 1 {\displaystyle (\log |x|)'={\frac {1}{x}}=x^{-1}} ãªã®ã§ã
â« x â 1 d x = â« 1 x d x = log | x | + C {\displaystyle \int x^{-1}dx=\int {\frac {1}{x}}dx=\log |x|+C}
ãæãç«ã€ã
ãæãç«ã€ããšãèæ
®ãããšã
ãšãªãããšãåãã
â« tan x d x {\displaystyle \int \tan xdx} ã¯ã眮æç©åæ³ã䜿ã£ãŠ
ããäžè¬ã«æçé¢æ° R ( x , y ) {\displaystyle R(x,y)} ã«å¯ŸããŠã â« R ( sin Ξ , cos Ξ ) d Ξ {\displaystyle \int R(\sin \theta ,\cos \theta )\,d\theta } ã«ã€ããŠèããã t = tan Ξ 2 {\displaystyle t=\tan {\frac {\theta }{2}}} ãšããã tan 2 Ξ 2 + 1 = 1 cos 2 Ξ 2 {\displaystyle \tan ^{2}{\frac {\theta }{2}}+1={\frac {1}{\cos ^{2}{\frac {\theta }{2}}}}} ãã£ãŠ cos 2 Ξ 2 = 1 1 + t 2 {\displaystyle \cos ^{2}{\frac {\theta }{2}}={\frac {1}{1+t^{2}}}} ã§ããã d t d Ξ = d d Ξ tan Ξ 2 = 1 2 cos 2 Ξ 2 = 1 2 ( t 2 + 1 ) {\displaystyle {\frac {dt}{d\theta }}={\frac {d}{d\theta }}\tan {\frac {\theta }{2}}={\frac {1}{2\cos ^{2}{\frac {\theta }{2}}}}={\frac {1}{2}}(t^{2}+1)} ã§ããã cos Ξ = 2 cos 2 Ξ 2 â 1 = 1 â t 2 1 + t 2 {\displaystyle \cos \theta =2\cos ^{2}{\frac {\theta }{2}}-1={\frac {1-t^{2}}{1+t^{2}}}} ã〠sin Ξ = tan Ξ cos Ξ = 2 tan Ξ 2 1 â tan 2 Ξ 2 cos Ξ = 2 t 1 + t 2 {\displaystyle \sin \theta =\tan \theta \cos \theta ={\frac {2\tan {\frac {\theta }{2}}}{1-\tan ^{2}{\frac {\theta }{2}}}}\cos \theta ={\frac {2t}{1+t^{2}}}}
ã§ããããã£ãŠ
â« R ( sin Ξ , cos Ξ ) d Ξ = â« R ( 2 t 1 + t 2 , 1 â t 2 1 + t 2 ) 2 d t 1 + t 2 {\displaystyle \int R(\sin \theta ,\cos \theta )\,d\theta =\int R\left({\frac {2t}{1+t^{2}}},{\frac {1-t^{2}}{1+t^{2}}}\right)\,{\frac {2dt}{1+t^{2}}}}
ãšæçé¢æ°ã®ç©åã«ãã¡èŸŒããã
幟äœåŠçã¯ããã®å€æã¯åäœåäžã®ç¹ P ( cos Ξ , sin Ξ ) {\displaystyle P(\cos \theta ,\sin \theta )} ãšç¹ A ( â 1 , 0 ) {\displaystyle A(-1,0)} ãçµã¶çŽç·ã®åŸé
t {\displaystyle t} ã§å€æãããã®ã§ãããå®éååšè§ã®å®çãã â x A P = 1 2 â x O P = Ξ 2 {\displaystyle \angle xAP={\frac {1}{2}}\angle xOP={\frac {\theta }{2}}} ãã t = tan Ξ 2 . {\displaystyle t=\tan {\frac {\theta }{2}}.}
被ç©åé¢æ°ã®åšæã Ï {\displaystyle \pi } ã®å Žåã¯ã被ç©åé¢æ°ã¯ sin 2 Ξ , cos 2 Ξ {\displaystyle \sin 2\theta ,\cos 2\theta } ã®æçé¢æ°ãªã®ã§ã t = tan Ξ {\displaystyle t=\tan \theta } ãšçœ®æãããšèšç®ã楜ã ã被ç©åé¢æ°ã sin 2 Ξ , cos 2 Ξ , sin Ξ cos Ξ {\displaystyle \sin ^{2}\theta ,\cos ^{2}\theta ,\sin \theta \cos \theta } ã®æçé¢æ°ãšãªããšãããã®ç¯çã«å±ããã t = tan Ξ {\displaystyle t=\tan \theta } ãšçœ®æãããšãã cos 2 Ξ = 1 1 + tan 2 Ξ = 1 1 + t 2 {\displaystyle \cos ^{2}\theta ={\frac {1}{1+\tan ^{2}\theta }}={\frac {1}{1+t^{2}}}} , sin 2 Ξ = tan 2 Ξ cos 2 Ξ = t 2 1 + t 2 {\displaystyle \sin ^{2}\theta =\tan ^{2}\theta \cos ^{2}\theta ={\frac {t^{2}}{1+t^{2}}}} , sin Ξ cos Ξ = ± sin 2 Ξ cos 2 Ξ = t 1 + t 2 {\displaystyle \sin \theta \cos \theta =\pm {\sqrt {\sin ^{2}\theta \cos ^{2}\theta }}={\frac {t}{1+t^{2}}}} ( sin Ξ cos Ξ {\displaystyle \sin \theta \cos \theta } ãš tan Ξ = sin Ξ cos Ξ {\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}} ã®æ£è² ã¯äžèŽãããã), d Ξ = d t 1 + t 2 {\displaystyle d\theta ={\frac {dt}{1+t^{2}}}} ãšãªãã
äŸ â« 1 sin x cos x d x {\displaystyle \int {\frac {1}{\sin x\cos x}}dx} 㯠t = tan x {\displaystyle t=\tan x} ãšçœ®æãããšã â« 1 sin x cos x d x = â« 1 + t 2 t d t 1 + t 2 = ln | tan x | + C . {\displaystyle \int {\frac {1}{\sin x\cos x}}dx=\int {\frac {1+t^{2}}{t}}{\frac {dt}{1+t^{2}}}=\ln |\tan x|+C.} t = tan Ξ 2 {\displaystyle t=\tan {\frac {\theta }{2}}} ãšçœ®æããŠããŸããšã â« 1 sin x cos x d x = â« 1 + t 2 t ( 1 â t 2 ) d t = ln | t 1 â t 2 | + C â² = ln | tan x | + C {\displaystyle \int {\frac {1}{\sin x\cos x}}\,dx=\int {\frac {1+t^{2}}{t(1-t^{2})}}\,dt=\ln \left|{\frac {t}{1-t^{2}}}\right|+C'=\ln |\tan x|+C} ãšèšç®éãå°ãå¢ããã
ææ°é¢æ°ã«ã€ã㊠( e x ) â² = e x {\displaystyle (e^{x})'=e^{x}} ãæãç«ã€ããšãçšãããšã â« e x d x = e x + C {\displaystyle \int e^{x}dx=e^{x}+C} ãåŸãããã
ãŸãã ( a x ln a ) â² = a x {\displaystyle \left({\frac {a^{x}}{\ln a}}\right)'=a^{x}} ãªã®ã§ã â« a x d x = a x ln a {\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\ln a}}} ã§ããã
ãŸãã log | x | {\displaystyle \log |x|} ã® åå§é¢æ°ãæ±ããããšãåºæ¥ãã
ãšãªãã
æçé¢æ° R ( x ) {\displaystyle R(x)} ã«å¯ŸããŠãç©å â« R ( e x ) d x {\displaystyle \int R(e^{x})\,dx} 㯠t = e x {\displaystyle t=e^{x}} ãããš d t d x = e x = t {\displaystyle {\frac {dt}{dx}}=e^{x}=t} ãã
â« R ( e x ) d x = â« R ( t ) d t t . {\displaystyle \int R(e^{x})\,dx=\int R(t){\frac {dt}{t}}.}
æçé¢æ° R ( x , y ) {\displaystyle R(x,y)} ã«å¯ŸããŠãç©å â« R ( x , a x 2 + b x + c ) d x {\displaystyle \int R(x,{\sqrt {ax^{2}+bx+c}})\,dx} ã«ã€ããŠèããããå¹³æ¹æ ¹ã®äžèº«ã¯å¹³æ¹å®æããããšã«ãã£ãŠã p 2 â x 2 , x 2 + p 2 , x 2 â p 2 {\displaystyle {\sqrt {p^{2}-x^{2}}},{\sqrt {x^{2}+p^{2}}},{\sqrt {x^{2}-p^{2}}}} ã®ããããã®åœ¢ã«ãªããããããã®å Žåã«ã€ããŠã x = p sin Ξ , x = p tan Ξ , x = p cos Ξ {\displaystyle x=p\sin \theta ,x=p\tan \theta ,x={\frac {p}{\cos \theta }}} ãšå€æ°å€æãããšäžè§é¢æ°ã®ç©åã«åž°çããã
ãŸãã y 2 = a x 2 + b x + c {\displaystyle y^{2}=ax^{2}+bx+c} ã¯äºæ¬¡æ²ç·ã§ãç¹ã« a > 0 {\displaystyle a>0} ã®ãšãã¯åæ²ç·ãšãªã( y 2 â a ( x + b 2 a ) 2 = â b 2 + 4 a c 4 a {\displaystyle y^{2}-a\left(x+{\frac {b}{2a}}\right)^{2}={\frac {-b^{2}+4ac}{4a}}} ãã)ããã®ãšãã y = ± a x + t {\displaystyle y=\pm {\sqrt {a}}x+t} ããªãã¡ t = â a x + a x 2 + b x + c {\displaystyle t=\mp {\sqrt {a}}x+{\sqrt {ax^{2}+bx+c}}} ãšå€æãããšããŸãèšç®ã§ãã(笊å·ã¯ã©ã¡ããéžæããŠãè¯ã)ã幟äœåŠçã«ã¯ãåæ²ç·ã®æŒžè¿ç·ã«å¹³è¡ã§åçã t {\displaystyle t} ã®çŽç· y = ± a x + t {\displaystyle y=\pm {\sqrt {a}}x+t} ãšåæ²ç·ã®ãã äžã€ã®äº€ç¹ ( x , y ) {\displaystyle (x,y)} ãå€æ° t {\displaystyle t} ã§è¡šãããã®ã§ããã
äŸ â« d x x 2 â 1 {\displaystyle \int {\frac {dx}{\sqrt {x^{2}-1}}}} 㯠t = x + x 2 â 1 {\displaystyle t=x+{\sqrt {x^{2}-1}}} ãšçœ®æãããšã 1 t = x â x 2 â 1 {\displaystyle {\frac {1}{t}}=x-{\sqrt {x^{2}-1}}} ãªã®ã§ã t + 1 t = 2 x {\displaystyle t+{\frac {1}{t}}=2x} ããªãã¡ 2 d x = ( 1 â 1 t 2 ) d t {\displaystyle 2dx=\left(1-{\frac {1}{t^{2}}}\right)dt} ãŸãã t â 1 t = 2 x 2 â 1 {\displaystyle t-{\frac {1}{t}}=2{\sqrt {x^{2}-1}}} .ãªã®ã§ã â« d x x 2 â 1 = â« 1 â 1 t 2 t â 1 t d t = â« d t t = ln | x + x 2 â 1 | + C {\displaystyle \int {\frac {dx}{\sqrt {x^{2}-1}}}=\int {\frac {1-{\frac {1}{t^{2}}}}{t-{\frac {1}{t}}}}dt=\int {\frac {dt}{t}}=\ln |x+{\sqrt {x^{2}-1}}|+C} ã§ããã
ãšããã§ããã®å€æã¯åæ²ç· y 2 = x 2 â 1 {\displaystyle y^{2}=x^{2}-1} ãšçŽç· y = â x + t {\displaystyle y=-x+t} ã®ãã äžã€ã®äº€ç¹ã«ããå€æã§ãã£ãããã®äº€ç¹ãæ¹çšåŒã解ã㊠t {\displaystyle t} ã§è¡šããšã x = 1 2 ( t + 1 t ) , y = 1 2 ( t â 1 t ) {\displaystyle x={\frac {1}{2}}\left(t+{\frac {1}{t}}\right),\,y={\frac {1}{2}}\left(t-{\frac {1}{t}}\right)} ãåŸããããã¯åæ²ç·ã®åªä»å€æ°è¡šç€ºã®äžã€ã§ããããŸãã t â e t {\displaystyle t\rightarrow e^{t}} ãšãããšã x = e t + e â t 2 = cosh t , y = e t â e â t 2 = sinh t . {\displaystyle x={\frac {e^{t}+e^{-t}}{2}}=\cosh t,\,y={\frac {e^{t}-e^{-t}}{2}}=\sinh t.} ãã㯠x > 0 {\displaystyle x>0} ã®éšåã®åæ²ç·ã®åªä»å€æ°è¡šç€ºã§ãããæå³èŸºã¯åæ²ç·é¢æ°ãšåŒã°ããäžè§é¢æ°ãšäŒŒãæ§è³ªãæã€ãé¢æ°åã® h {\displaystyle \mathrm {h} } ã¯hyperbolaã«ç±æ¥ãããäŸãã°ãåæ²ç·ã®æ¹çšåŒããåŸããã cosh 2 t â sinh 2 t = 1 {\displaystyle \cosh ^{2}t-\sinh ^{2}t=1} 㯠sin 2 Ξ + cos 2 Ξ = 1 {\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1} ãšãã䌌ãŠãããäŸç€ºã®äžå®ç©å㯠x = cosh t {\displaystyle x=\cosh t} ãšçœ®æããŠã解ãããšãåºæ¥ãããã»ãšãã©åãããšãªã®ã§çç¥ããã
a < b {\displaystyle a<b} ãšãããç©å â« a b ( x â a ) ( b â x ) d x {\displaystyle \int _{a}^{b}{\sqrt {(x-a)(b-x)}}\,dx} 㯠y = ( x â a ) ( b â x ) {\displaystyle y={\sqrt {(x-a)(b-x)}}} ãšãããšã ( x â a + b 2 ) + y 2 = ( a â b 2 ) 2 {\displaystyle \left(x-{\frac {a+b}{2}}\right)+y^{2}=\left({\frac {a-b}{2}}\right)^{2}} ããã被ç©åé¢æ° y {\displaystyle y} ã¯äžå¿ a + b 2 {\displaystyle {\frac {a+b}{2}}} ã§ååŸ b â a 2 {\displaystyle {\frac {b-a}{2}}} ã®ååšã®äžååã§ãããç©ååºéããã®äž¡ç«¯ãªã®ã§ãç©åã®å€ã¯ååã®é¢ç©ã«çããã â« a b ( x â a ) ( b â x ) d x = Ï 2 ( b â a 2 ) 2 {\displaystyle \int _{a}^{b}{\sqrt {(x-a)(b-x)}}\,dx={\frac {\pi }{2}}\left({\frac {b-a}{2}}\right)^{2}} ã§ããã
äžè¬ã«ãé¢æ° f ( a â x ) {\displaystyle f(a-x)} ã®ã°ã©ãã¯é¢æ° f ( x ) {\displaystyle f(x)} ã®ã°ã©ããçŽç· x = a 2 {\displaystyle x={\frac {a}{2}}} ã§å¯Ÿç§°ç§»åãããã®ã§ããã
åŸã£ãŠãé£ç¶é¢æ° f ( x ) {\displaystyle f(x)} ãåºé [ a + b 2 , b ] {\displaystyle \left[{\frac {a+b}{2}},b\right]} ã§ç©åããå€ â« a + b 2 b f ( x ) d x {\displaystyle \int _{\frac {a+b}{2}}^{b}f(x)\,dx} ãšãé£ç¶é¢æ° f ( a + b â x ) {\displaystyle f(a+b-x)} ãåºé [ a , a + b 2 ] {\displaystyle \left[a,{\frac {a+b}{2}}\right]} ã§ç©åããå€ â« a a + b 2 f ( a + b â x ) d x {\displaystyle \int _{a}^{\frac {a+b}{2}}f(a+b-x)\,dx} ã¯çãã:
ãã®çåŒã¯åã«ã x â a + b â x {\displaystyle x\to a+b-x} ã®å€æ°å€æã«ãã£ãŠãå°åºã§ããã
ãã®çåŒããã â« a b f ( x ) d x = â« a a + b 2 f ( x ) d x + â« a + b 2 b f ( x ) d x = â« a a + b 2 [ f ( x ) + f ( a + b â x ) ] d x {\displaystyle \int _{a}^{b}f(x)\,dx=\int _{a}^{\frac {a+b}{2}}f(x)\,dx+\int _{\frac {a+b}{2}}^{b}f(x)\,dx=\int _{a}^{\frac {a+b}{2}}[f(x)+f(a+b-x)]\,dx} ãå°ãããã
ãã®å
¬åŒã¯ã f ( x ) + f ( a + b â x ) {\displaystyle f(x)+f(a+b-x)} ãç°¡åãªåœ¢ã«ãªãå®ç©åã§åœ¹ã«ç«ã€ã
äŸãã°ã â« 0 Ï 2 sin x sin x + cos x d x = â« 0 Ï 4 [ sin x sin x + cos x + sin ( Ï 2 â x ) sin ( Ï 2 â x ) + cos ( Ï 2 â x ) ] d x = â« 0 Ï 4 [ sin x sin x + cos x + cos x cos x + sin x ] d x = â« 0 Ï 4 d x = Ï 4 . {\displaystyle {\begin{aligned}\int _{0}^{\frac {\pi }{2}}{\frac {\sin x}{\sin x+\cos x}}\,dx&=\int _{0}^{\frac {\pi }{4}}\left[{\frac {\sin x}{\sin x+\cos x}}+{\frac {\sin({\frac {\pi }{2}}-x)}{\sin({\frac {\pi }{2}}-x)+\cos({\frac {\pi }{2}}-x)}}\right]\,dx\\&=\int _{0}^{\frac {\pi }{4}}\left[{\frac {\sin x}{\sin x+\cos x}}+{\frac {\cos x}{\cos x+\sin x}}\right]\,dx\\&=\int _{0}^{\frac {\pi }{4}}dx={\frac {\pi }{4}}.\end{aligned}}}
King Property ã®å¿çšäŸã¯ â« â 1 1 x 2 1 + e x d x = 1 3 {\displaystyle \int _{-1}^{1}{\frac {x^{2}}{1+e^{x}}}\,dx={\frac {1}{3}}} , â« 0 Ï 4 ln ( 1 + tan x ) d x = Ï 8 ln 2 {\displaystyle \int _{0}^{\frac {\pi }{4}}\ln(1+\tan x)\,dx={\frac {\pi }{8}}\ln 2} , â« 0 Ï 2 ln sin x d x = â Ï 2 ln 2 {\displaystyle \int _{0}^{\frac {\pi }{2}}\ln \sin x\,dx=-{\frac {\pi }{2}}\ln 2} ãªã©ããããèšç®ããŠã¿ãã
æŒç¿åé¡1
次ã®äžå®ç©åãæ±ããã
æŒç¿åé¡2
第äžå
第äºå
ããé¢æ°f(x)ã®åå§é¢æ°ãæ±ããæŒç®ã¯ f(x)ãšx軞ã«ã¯ããŸããé åã®é¢ç©ãæ±ããæŒç®ã«çããã ãã®ããšãçšã㊠ããé¢æ°ã«ãã£ãŠäœãããé åã®é¢ç©ãæ±ããããšãåºæ¥ãã
äŸãã°ã â« 0 1 x 2 d x = 1 3 {\displaystyle \int _{0}^{1}x^{2}dx={\frac {1}{3}}} ã¯ãæŸç©ç· y = x 2 {\displaystyle y=x^{2}} ã«ã€ã㊠0 < x < 1 {\displaystyle 0<x<1} ã®ç¯å²ã§ãããŸããé¢ç©ã«çããã
æ¥å x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1} ã®é¢ç© S = Ï a b {\displaystyle S=\pi ab} ã®å°åº
æ¥å x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1} ã y {\displaystyle y} ã«ã€ããŠè§£ããš
ãšãªãããã®ãã¡ y = b a a 2 â x 2 {\displaystyle y={\frac {b}{a}}{\sqrt {a^{2}-x^{2}}}} ã¯åæ¥å(æ¥åã®äžåå)ã瀺ããŠããããã®åæ¥åã®é¢ç©ã2åãããã®ãæ¥åã®é¢ç©Sãšãªãã®ã§
ãšãªãã
ããç«äœ V 0 {\displaystyle V_{0}} ã® x = t {\displaystyle x=t} ã«ãããæé¢ç©ãæéãªå€ã§ããã®å€ã t {\displaystyle t} ã®é¢æ° S ( t ) {\displaystyle S(t)} ãšãªããšãããã®ç«äœãå¹³é¢ x = a {\displaystyle x=a} , x = b {\displaystyle x=b} (ãã ãã a < b {\displaystyle a<b} )ã§åãåã£ãé åã®äœç©ã¯ãåºé¢ç© S ( t ) {\displaystyle S(t)} ã«æ¥µããŠå°ããé«ã d t {\displaystyle dt} ã®ç© S ( t ) d t {\displaystyle S(t)\,dt} ã®åºé [ a , b ] {\displaystyle [a,b]} ã«ããã环ç©ã§ããã®ã§ã以äžã®åŒã§è¡šãããšãã§ããã
(äŸ1)
(äŸ2)
y = f ( x ) ( a †x †b ) {\displaystyle y=f(x)(a\leq x\leq b)} ã§äžããããæ²ç·ãx軞ã®åãã«å転ãããŠäœããã ç«äœã®äœç©Vã¯ã V = â« a b Ï ( f ( x ) ) 2 d x {\displaystyle V=\int _{a}^{b}\pi (f(x))^{2}dx} ã§äžããããã
å°åº
ç«äœãx軞ã«åçŽã§ãããx=cãæºããé¢ãšx=c+hãæºããé¢ã§åããš(hã¯å°ã㪠å®æ°)ããã®åæé¢ã§æãŸããç«äœã¯ååŸ f(c)ã®åãšååŸ f(c+h)ã®å ã§ã¯ããŸããç«äœãšãªãã ããããhã極ããŠå°ãããšãããã®å³åœ¢ã¯ååŸf(c),é«ãhã®åæ±ã§ è¿äŒŒã§ããã ãã£ãŠãã®2ã€ã®é¢ã«é¢ããŠãåŸãããå³åœ¢ã®äœç©ã¯ h Ã Ï ( f ( c ) ) 2 {\displaystyle h\times \pi (f(c))^{2}} ãšãªãã ããã a < c < b {\displaystyle a<c<b} æºããå
šãŠã®cã«ã€ããŠè¶³ãåããããšã S = â« a b Ï ( f ( x ) ) 2 d x {\displaystyle S=\int _{a}^{b}\pi (f(x))^{2}dx} ãåŸãããã
äŸãã°ã y = x 2 ( 0 < x < 1 ) {\displaystyle y=x^{2}~(0<x<1)} ãx軞ã®åãã«å転ãããŠåŸãããå³åœ¢ã®äœç©ã¯ã
S = â« 0 1 Ï ( x 2 ) 2 d x {\displaystyle S=\int _{0}^{1}\pi (x^{2})^{2}dx} = Ï â« 0 1 x 4 d x {\displaystyle =\pi \int _{0}^{1}x^{4}dx} = Ï 5 {\displaystyle ={\frac {\pi }{5}}} ãšãªãã
çã®äœç© V = 4 3 Ï r 3 {\displaystyle V={\frac {4}{3}}\pi r^{3}} ã®å°åº
ååŸrã®çã¯åå y = r 2 â x 2 {\displaystyle y={\sqrt {r^{2}-x^{2}}}} ãx軞ã®åšãã«å転ãããŠã€ããããšãã§ããã
ãŸãäœç©ãrã§åŸ®åãããšçã®è¡šé¢ç© S = 4 Ï r 2 {\displaystyle S=4\pi r^{2}} ãåŸãããã
ãããŸã§ã«åŠãã ããã«ãç©åã¯åŸ®åã®éæŒç®ã§ãããšåæã«ã座æšå¹³é¢äžã§ã®é¢ç©èšç®ã§ãããããã®é
ã§ã¯ã座æšå¹³é¢äžã®é¢ç©èšç®ã®æ¹æ³ã®äžã€ã§ããåºåæ±ç©æ³ãããã³ç©åæ³ãšã®é¢é£ã«ã€ããŠåŠã¶ã
å³å³ã®ãããªããæ²ç· y = f ( x ) {\displaystyle y=f(x)} ããããåçŽã®ãããããã§ã¯ã€ãã« f ( x ) > 0 {\displaystyle f(x)>0} ã§ãããã®ãšããŠèããããã®æ²ç·ãšãx軞ãããã³çŽç· x = a , x = b ( a < b ) {\displaystyle x=a,x=b(a<b)} ã«ãã£ãŠå²ãŸããé åã®é¢ç©Sãæ±ããããã®é¢ç©ã¯#é¢ç©ã®é
ã§åŠãã ããã«ã
ãšç©åæ³ãçšããŠèšç®ããããšãã§ãããã§ã¯ããããããå°ãåå§çãªæ¹æ³ã§è¿äŒŒçã«æ±ããããšãèããŠã¿ããã
æ²ç·ãå«ãå³åœ¢ã®é¢ç©ãæ±ããããšã¯ç°¡åã§ã¯ãªãããäŸãã°äžè§åœ¢ãé·æ¹åœ¢ãå°åœ¢ãªã©ã®çŽç·ã§å²ãŸããå³åœ¢ã®é¢ç©ãæ±ããããšã¯é£ãããªããããã§ãäžå³ã®ããã«y=f(x)ãæ£ã°ã©ãã§è¿äŒŒããé·æ¹åœ¢ã®é¢ç©ã®åãèšç®ããããšã§ãæ±ãããé¢ç©Sã«è¿ãå€ãæ±ããããšãã§ãããå·Šäžã®ããã«æ£ã°ã©ãã®å¹
ã倧ãããšèª€å·®ã倧ããããæ£ã°ã©ãã®å¹
ãçãããã°ããã»ã©ãããªãã¡åå²æ°ãå€ãããã»ã©ãåŸã
ã«æ±ãããé¢ç©ã®å€ã«è¿ã¥ããããšãã§ãããããã§ããã®åºé[a,b]ãnçåãããã®æã®é·æ¹åœ¢ã®é¢ç©ã®ç·åãæ±ãããã®åŸã§ n â â {\displaystyle n\to \infty } ã®æ¥µéãèããããšã«ããããã®ããã«ããŠãåºéã现ããçåå²ããé·æ¹åœ¢ã®é¢ç©ã®ç·åãæ±ããããšã«ããå³åœ¢ã®é¢ç©ãæ±ããæ¹æ³ããåºåæ±ç©æ³ãšåŒã¶ã
y = f ( x ) {\displaystyle y=f(x)} ãæ£ã°ã©ãã§è¿äŒŒãããšããå³å³ã®ããã«ãé·æ¹åœ¢ã®å·Šäžã®é ç¹ãæ²ç·äžã«åãæ¹æ³ãšãå³äžã®é ç¹ãæ²ç·äžã«åãæ¹æ³ããããã©ã¡ãã®æ¹æ³ã§ããåå²æ°ã倧ããããã°ãããæ±ãããé¢ç©ã«è¿ã¥ããããŸãã¯å·Šäžã®é ç¹ãæ²ç·äžã«åãæ¹æ³ã§èããããšã«ããã
ããã§ã¯é¢ç©ãæ±ãããåºéããåçŽã®ãã[0, 1]ãšãããåºé[0, 1]ãnçåãããšããããããã®é·æ¹åœ¢ã®å·Šç«¯ã®x座æšã¯ã
ãšãªããããã§ãäžè¬ã«ç¬¬kçªç®ã®é·æ¹åœ¢ã«ã€ããŠèããããšã«ããããã ãããã¡ã°ãå·ŠåŽã®é·æ¹åœ¢ã第0çªç®ãšãããã¡ã°ãå³åŽã®é·æ¹åœ¢ã第n-1çªç®ãšããã第kçªç®ã®é·æ¹åœ¢ã®å·Šç«¯ã®x座æšã¯ k n {\displaystyle {\frac {k}{n}}} ã§ããããããã®é·æ¹åœ¢ã®é«ã㯠f ( k n ) {\displaystyle f\left({\frac {k}{n}}\right)} ãšãªãããŸãé·æ¹åœ¢ã®å¹
㯠1 n {\displaystyle {\frac {1}{n}}} ã§ããããã®ããããã®é·æ¹åœ¢ã®é¢ç© s k {\displaystyle s_{k}} ã¯ã
ãšãªãããããã£ãŠããããã®é·æ¹åœ¢ã®é¢ç©ã®ç·å S n {\displaystyle S_{n}} ã¯ã
ãã® S n {\displaystyle S_{n}} ã¯ãåºé[0, 1]ãnçåããæã®é·æ¹åœ¢ã®é¢ç©ã®ç·åã§ããããnã倧ããããã°ããã»ã©ã次第ã«ããšã®é¢ç©ã«è¿ã¥ããŠããããããã£ãŠã n â â {\displaystyle n\to \infty } ã®æ¥µéãèãã
ãšãªãããã®ããã«ããŠãæ±ãããé¢ç©ãèšç®ããããšãã§ãããããã«ãããã§ãã®åºéã®é¢ç©ãç©åæ³ã«ããèšç®ã§ããããšããã
ãæãç«ã€ããŸããé·æ¹åœ¢ã®å³äžã®é ç¹ãæ²ç·äžã«åãå Žåã¯ãåæ§ã«ããŠ
ãšãªãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ããã§ã¯ãæ°åŠIIã®åŸ®åã»ç©åã®èãã§åŠãã ç©åã®æ§è³ªã«ã€ããŠãã詳ããæ±ãããŸããäžè§é¢æ°ãææ°ã»å¯Ÿæ°é¢æ°ãªã©ã®é¢æ°ã®ç©åã«ã€ããŠãåŠç¿ããã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ç©åæ³ã«ã€ããŠ",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "â« { f ( x ) + g ( x ) } d x = â« f ( x ) d x + â« g ( x ) d x , {\\displaystyle \\int \\{f(x)+g(x)\\}dx=\\int f(x)dx+\\int g(x)dx,} â« a f ( x ) d x = a â« f ( x ) d x {\\displaystyle \\int af(x)dx=a\\int f(x)dx} (aã¯å®æ°)",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ãæãç«ã€ã",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "å°åº",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "â« { f ( x ) + g ( x ) } d x = â« f ( x ) d x + â« g ( x ) d x {\\displaystyle \\int \\{f(x)+g(x)\\}dx=\\int f(x)dx+\\int g(x)dx}",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ã®äž¡èŸºã埮åãããšã",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "巊蟺 =å³èŸº = f + g {\\displaystyle f+g}",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "ãåŸãã",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "ãã£ãŠã",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "â« { f ( x ) + g ( x ) } d x = â« f ( x ) d x + â« g ( x ) d x {\\displaystyle \\int \\{f(x)+g(x)\\}dx=\\int f(x)dx+\\int g(x)dx}",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "ã®äž¡èŸºã¯äžèŽããã",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "(å®éã«ã¯2ã€ã®é¢æ°ã®å°é¢æ°ãäžèŽãããšãã ãããã®é¢æ°ã«ã¯å®æ°ã ãã®ã¡ãããããã",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ä»®ã«ãF(x)ãšG(x)ãå
±éã®å°é¢æ°h(x)ãæã£ããšããã",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "ãã®ãšãã",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "( F ( x ) â G ( x ) ) â² = h ( x ) â h ( x ) = 0 {\\displaystyle (F(x)-G(x))'=h(x)-h(x)=0}",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "ãšãªããã0ã®åå§é¢æ°ã¯å®æ°Cã§ããããšãåããã",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "ãã£ãŠã䞡蟺ãç©åãããšã",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "F ( x ) â G ( x ) = C {\\displaystyle F(x)-G(x)=C}",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "ãšãªããF(x)ãšG(x)ã«ã¯å®æ°ã ãã®å·®ãããªãããšã確ãããããã",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "ãã£ãŠã",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "â« { f ( x ) + g ( x ) } d x = â« f ( x ) d x + â« g ( x ) d x {\\displaystyle \\int \\{f(x)+g(x)\\}dx=\\int f(x)dx+\\int g(x)dx}",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "ã¯å®æ°ã ãã®ã¡ãããå«ãã§æãç«ã€åŒã§ããã ããäžè¬ã«ãäžå®ç©åã絡ãçåŒã¯å®æ°åã®å·®ãå«ããŠæãç«ã€ãšããã®ãéäŸã§ããã)",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "â« a f ( x ) d x = a â« f ( x ) d x {\\displaystyle \\int af(x)dx=a\\int f(x)dx}",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "ã«ã€ããŠã䞡蟺ã埮åãããšã",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "巊蟺=å³èŸº= a f(x)",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "ãåŸãã",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "ãã£ãŠã",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "â« a f d x = a â« f d x {\\displaystyle \\int afdx=a\\int fdx}",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "ãæãç«ã€ããšãåãã",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "é¢æ° f ( x ) {\\displaystyle f(x)} ã®åå§é¢æ°ã F ( x ) {\\displaystyle F(x)} ãšãããš",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "â« a b f ( x ) = F ( b ) â F ( a ) = â ( F ( a ) â F ( b ) ) = â â« b a f ( x ) d x {\\displaystyle \\int _{a}^{b}f(x)\\,=F(b)-F(a)=-(F(a)-F(b))=-\\int _{b}^{a}f(x)\\,dx} ã§ããã",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "â« a c f ( x ) d x + â« c b f ( x ) d x = ( F ( c ) â F ( a ) ) + ( F ( b ) â F ( c ) ) = F ( b ) â F ( a ) = â« a b f ( x ) d x {\\displaystyle \\int _{a}^{c}f(x)\\,dx+\\int _{c}^{b}f(x)\\,dx=(F(c)-F(a))+(F(b)-F(c))=F(b)-F(a)=\\int _{a}^{b}f(x)\\,dx}",
"title": "ç©åã®åºæ¬çãªæ§è³ª"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "é¢æ°ã®åå§é¢æ°ãæ±ããæ段ãšããŠã ç©åå€æ°ãå¥ã®å€æ°ã§çœ®ãæããŠç©åãè¡ãªãæ段ãç¥ãããŠããã ããã眮æç©åãšåŒã¶ã",
"title": "眮æç©åæ³"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "â« f ( g ( x ) ) d g ( x ) = â« f ( g ( x ) ) g â² ( x ) d x {\\displaystyle \\int f(g(x))dg(x)=\\int f(g(x))g'(x)dx}",
"title": "眮æç©åæ³"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "å°åº",
"title": "眮æç©åæ³"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "â« f ( g ( x ) ) d g ( x ) = F ( g ( x ) ) {\\displaystyle \\int f(g(x))dg(x)=F(g(x))} ã x {\\displaystyle x} ã«ã€ããŠåŸ®åãããšã",
"title": "眮æç©åæ³"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "F â² ( g ( x ) ) = f ( g ( x ) ) g â² ( x ) {\\displaystyle F'(g(x))=f(g(x))g'(x)}",
"title": "眮æç©åæ³"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "åã³ x {\\displaystyle x} ã«ã€ããŠç©åãããšã",
"title": "眮æç©åæ³"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "â« f ( g ( x ) ) d g ( x ) = â« f ( g ( x ) ) g â² ( x ) d x {\\displaystyle \\int f(g(x))dg(x)=\\int f(g(x))g'(x)dx}",
"title": "眮æç©åæ³"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "ãŸããç¹ã«",
"title": "眮æç©åæ³"
},
{
"paragraph_id": 41,
"tag": "p",
"text": "äŸãã°ã â« ( a x + b ) 2 d x {\\displaystyle \\int (ax+b)^{2}dx} ãèããã",
"title": "眮æç©åæ³"
},
{
"paragraph_id": 42,
"tag": "p",
"text": "t = a x + b {\\displaystyle t=ax+b} ãšçœ®ãã",
"title": "眮æç©åæ³"
},
{
"paragraph_id": 43,
"tag": "p",
"text": "ãã®äž¡èŸºã埮åãããš d t = a d x {\\displaystyle dt=adx} ãæãç«ã€ããšãèæ
®ãããšã",
"title": "眮æç©åæ³"
},
{
"paragraph_id": 44,
"tag": "p",
"text": "ãšãªãããšããããã",
"title": "眮æç©åæ³"
},
{
"paragraph_id": 45,
"tag": "p",
"text": "å®éãã®åŒãxã§åŸ®åãããš ( a x + b ) 2 {\\displaystyle (ax+b)^{2}} ãšäžèŽããããšãåãã",
"title": "眮æç©åæ³"
},
{
"paragraph_id": 46,
"tag": "p",
"text": "眮æç©åã䜿ããã«èšç®ããããšãåºæ¥ãã",
"title": "眮æç©åæ³"
},
{
"paragraph_id": 47,
"tag": "p",
"text": "( C â² = b 3 3 a + C {\\displaystyle C'={\\frac {b^{3}}{3a}}+C} ãšçœ®ãæããã)",
"title": "眮æç©åæ³"
},
{
"paragraph_id": 48,
"tag": "p",
"text": "= ( a x + b ) 3 3 a + C {\\displaystyle ={\\frac {(ax+b)^{3}}{3a}}+C} ãšãªã確ãã«äžèŽããã",
"title": "眮æç©åæ³"
},
{
"paragraph_id": 49,
"tag": "p",
"text": "",
"title": "眮æç©åæ³"
},
{
"paragraph_id": 50,
"tag": "p",
"text": "é¢æ°ã®ç©ã®ç©åãè¡ãªããšãããé¢æ°ã®åŸ®åã ããåãã ããŠç©åãããšãããŸãç©åã§ããå Žåããããé¢æ° g ( x ) {\\displaystyle g(x)} ã®åå§é¢æ°ã G ( x ) {\\displaystyle G(x)} ãšãããš",
"title": "éšåç©åæ³"
},
{
"paragraph_id": 51,
"tag": "p",
"text": "â« f ( x ) g ( x ) d x = f ( x ) G ( x ) â â« f â² ( x ) G ( x ) d x {\\displaystyle \\int f(x)g(x)\\,dx=f(x)G(x)-\\int f'(x)G(x)\\,dx}",
"title": "éšåç©åæ³"
},
{
"paragraph_id": 52,
"tag": "p",
"text": "",
"title": "éšåç©åæ³"
},
{
"paragraph_id": 53,
"tag": "p",
"text": "å°åº",
"title": "éšåç©åæ³"
},
{
"paragraph_id": 54,
"tag": "p",
"text": "ç©ã®åŸ®åæ³ãã { f ( x ) G ( x ) } â² = f â² ( x ) G ( x ) + f ( x ) g ( x ) {\\displaystyle \\{f(x)G(x)\\}'=f'(x)G(x)+f(x)g(x)} ã§ãããããã移é
ããŠ",
"title": "éšåç©åæ³"
},
{
"paragraph_id": 55,
"tag": "p",
"text": "f ( x ) g ( x ) = { f ( x ) G ( x ) } â² â f â² ( x ) G ( x ) {\\displaystyle f(x)g(x)=\\{f(x)G(x)\\}'-f'(x)G(x)}",
"title": "éšåç©åæ³"
},
{
"paragraph_id": 56,
"tag": "p",
"text": "ã§ããã䞡蟺ãxã§ç©åããŠ",
"title": "éšåç©åæ³"
},
{
"paragraph_id": 57,
"tag": "p",
"text": "â« f ( x ) g ( x ) d x = f ( x ) G ( x ) â â« f â² ( x ) G ( x ) d x {\\displaystyle \\int f(x)g(x)\\,dx=f(x)G(x)-\\int f'(x)G(x)\\,dx}",
"title": "éšåç©åæ³"
},
{
"paragraph_id": 58,
"tag": "p",
"text": "ãåŸãããã",
"title": "éšåç©åæ³"
},
{
"paragraph_id": 59,
"tag": "p",
"text": "",
"title": "éšåç©åæ³"
},
{
"paragraph_id": 60,
"tag": "p",
"text": "äŸãã°ã",
"title": "éšåç©åæ³"
},
{
"paragraph_id": 61,
"tag": "p",
"text": "",
"title": "éšåç©åæ³"
},
{
"paragraph_id": 62,
"tag": "p",
"text": "n â â 1 {\\displaystyle n\\neq -1} ã®ãšãã ( 1 n + 1 x n + 1 ) â² = x n {\\displaystyle \\left({\\frac {1}{n+1}}x^{n+1}\\right)'=x^{n}} ãªã®ã§ã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 63,
"tag": "p",
"text": "â« x n d x = 1 n + 1 x n + 1 + C {\\displaystyle \\int x^{n}dx={\\frac {1}{n+1}}x^{n+1}+C}",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 64,
"tag": "p",
"text": "n = â 1 {\\displaystyle n=-1} ã®ãšãã ( log | x | ) â² = 1 x = x â 1 {\\displaystyle (\\log |x|)'={\\frac {1}{x}}=x^{-1}} ãªã®ã§ã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 65,
"tag": "p",
"text": "â« x â 1 d x = â« 1 x d x = log | x | + C {\\displaystyle \\int x^{-1}dx=\\int {\\frac {1}{x}}dx=\\log |x|+C}",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 66,
"tag": "p",
"text": "ãæãç«ã€ã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 67,
"tag": "p",
"text": "ãæãç«ã€ããšãèæ
®ãããšã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 68,
"tag": "p",
"text": "ãšãªãããšãåãã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 69,
"tag": "p",
"text": "â« tan x d x {\\displaystyle \\int \\tan xdx} ã¯ã眮æç©åæ³ã䜿ã£ãŠ",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 70,
"tag": "p",
"text": "ããäžè¬ã«æçé¢æ° R ( x , y ) {\\displaystyle R(x,y)} ã«å¯ŸããŠã â« R ( sin Ξ , cos Ξ ) d Ξ {\\displaystyle \\int R(\\sin \\theta ,\\cos \\theta )\\,d\\theta } ã«ã€ããŠèããã t = tan Ξ 2 {\\displaystyle t=\\tan {\\frac {\\theta }{2}}} ãšããã tan 2 Ξ 2 + 1 = 1 cos 2 Ξ 2 {\\displaystyle \\tan ^{2}{\\frac {\\theta }{2}}+1={\\frac {1}{\\cos ^{2}{\\frac {\\theta }{2}}}}} ãã£ãŠ cos 2 Ξ 2 = 1 1 + t 2 {\\displaystyle \\cos ^{2}{\\frac {\\theta }{2}}={\\frac {1}{1+t^{2}}}} ã§ããã d t d Ξ = d d Ξ tan Ξ 2 = 1 2 cos 2 Ξ 2 = 1 2 ( t 2 + 1 ) {\\displaystyle {\\frac {dt}{d\\theta }}={\\frac {d}{d\\theta }}\\tan {\\frac {\\theta }{2}}={\\frac {1}{2\\cos ^{2}{\\frac {\\theta }{2}}}}={\\frac {1}{2}}(t^{2}+1)} ã§ããã cos Ξ = 2 cos 2 Ξ 2 â 1 = 1 â t 2 1 + t 2 {\\displaystyle \\cos \\theta =2\\cos ^{2}{\\frac {\\theta }{2}}-1={\\frac {1-t^{2}}{1+t^{2}}}} ã〠sin Ξ = tan Ξ cos Ξ = 2 tan Ξ 2 1 â tan 2 Ξ 2 cos Ξ = 2 t 1 + t 2 {\\displaystyle \\sin \\theta =\\tan \\theta \\cos \\theta ={\\frac {2\\tan {\\frac {\\theta }{2}}}{1-\\tan ^{2}{\\frac {\\theta }{2}}}}\\cos \\theta ={\\frac {2t}{1+t^{2}}}}",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 71,
"tag": "p",
"text": "ã§ããããã£ãŠ",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 72,
"tag": "p",
"text": "â« R ( sin Ξ , cos Ξ ) d Ξ = â« R ( 2 t 1 + t 2 , 1 â t 2 1 + t 2 ) 2 d t 1 + t 2 {\\displaystyle \\int R(\\sin \\theta ,\\cos \\theta )\\,d\\theta =\\int R\\left({\\frac {2t}{1+t^{2}}},{\\frac {1-t^{2}}{1+t^{2}}}\\right)\\,{\\frac {2dt}{1+t^{2}}}}",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 73,
"tag": "p",
"text": "ãšæçé¢æ°ã®ç©åã«ãã¡èŸŒããã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 74,
"tag": "p",
"text": "幟äœåŠçã¯ããã®å€æã¯åäœåäžã®ç¹ P ( cos Ξ , sin Ξ ) {\\displaystyle P(\\cos \\theta ,\\sin \\theta )} ãšç¹ A ( â 1 , 0 ) {\\displaystyle A(-1,0)} ãçµã¶çŽç·ã®åŸé
t {\\displaystyle t} ã§å€æãããã®ã§ãããå®éååšè§ã®å®çãã â x A P = 1 2 â x O P = Ξ 2 {\\displaystyle \\angle xAP={\\frac {1}{2}}\\angle xOP={\\frac {\\theta }{2}}} ãã t = tan Ξ 2 . {\\displaystyle t=\\tan {\\frac {\\theta }{2}}.}",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 75,
"tag": "p",
"text": "被ç©åé¢æ°ã®åšæã Ï {\\displaystyle \\pi } ã®å Žåã¯ã被ç©åé¢æ°ã¯ sin 2 Ξ , cos 2 Ξ {\\displaystyle \\sin 2\\theta ,\\cos 2\\theta } ã®æçé¢æ°ãªã®ã§ã t = tan Ξ {\\displaystyle t=\\tan \\theta } ãšçœ®æãããšèšç®ã楜ã ã被ç©åé¢æ°ã sin 2 Ξ , cos 2 Ξ , sin Ξ cos Ξ {\\displaystyle \\sin ^{2}\\theta ,\\cos ^{2}\\theta ,\\sin \\theta \\cos \\theta } ã®æçé¢æ°ãšãªããšãããã®ç¯çã«å±ããã t = tan Ξ {\\displaystyle t=\\tan \\theta } ãšçœ®æãããšãã cos 2 Ξ = 1 1 + tan 2 Ξ = 1 1 + t 2 {\\displaystyle \\cos ^{2}\\theta ={\\frac {1}{1+\\tan ^{2}\\theta }}={\\frac {1}{1+t^{2}}}} , sin 2 Ξ = tan 2 Ξ cos 2 Ξ = t 2 1 + t 2 {\\displaystyle \\sin ^{2}\\theta =\\tan ^{2}\\theta \\cos ^{2}\\theta ={\\frac {t^{2}}{1+t^{2}}}} , sin Ξ cos Ξ = ± sin 2 Ξ cos 2 Ξ = t 1 + t 2 {\\displaystyle \\sin \\theta \\cos \\theta =\\pm {\\sqrt {\\sin ^{2}\\theta \\cos ^{2}\\theta }}={\\frac {t}{1+t^{2}}}} ( sin Ξ cos Ξ {\\displaystyle \\sin \\theta \\cos \\theta } ãš tan Ξ = sin Ξ cos Ξ {\\displaystyle \\tan \\theta ={\\frac {\\sin \\theta }{\\cos \\theta }}} ã®æ£è² ã¯äžèŽãããã), d Ξ = d t 1 + t 2 {\\displaystyle d\\theta ={\\frac {dt}{1+t^{2}}}} ãšãªãã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 76,
"tag": "p",
"text": "äŸ â« 1 sin x cos x d x {\\displaystyle \\int {\\frac {1}{\\sin x\\cos x}}dx} 㯠t = tan x {\\displaystyle t=\\tan x} ãšçœ®æãããšã â« 1 sin x cos x d x = â« 1 + t 2 t d t 1 + t 2 = ln | tan x | + C . {\\displaystyle \\int {\\frac {1}{\\sin x\\cos x}}dx=\\int {\\frac {1+t^{2}}{t}}{\\frac {dt}{1+t^{2}}}=\\ln |\\tan x|+C.} t = tan Ξ 2 {\\displaystyle t=\\tan {\\frac {\\theta }{2}}} ãšçœ®æããŠããŸããšã â« 1 sin x cos x d x = â« 1 + t 2 t ( 1 â t 2 ) d t = ln | t 1 â t 2 | + C â² = ln | tan x | + C {\\displaystyle \\int {\\frac {1}{\\sin x\\cos x}}\\,dx=\\int {\\frac {1+t^{2}}{t(1-t^{2})}}\\,dt=\\ln \\left|{\\frac {t}{1-t^{2}}}\\right|+C'=\\ln |\\tan x|+C} ãšèšç®éãå°ãå¢ããã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 77,
"tag": "p",
"text": "ææ°é¢æ°ã«ã€ã㊠( e x ) â² = e x {\\displaystyle (e^{x})'=e^{x}} ãæãç«ã€ããšãçšãããšã â« e x d x = e x + C {\\displaystyle \\int e^{x}dx=e^{x}+C} ãåŸãããã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 78,
"tag": "p",
"text": "ãŸãã ( a x ln a ) â² = a x {\\displaystyle \\left({\\frac {a^{x}}{\\ln a}}\\right)'=a^{x}} ãªã®ã§ã â« a x d x = a x ln a {\\displaystyle \\int a^{x}\\,dx={\\frac {a^{x}}{\\ln a}}} ã§ããã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 79,
"tag": "p",
"text": "ãŸãã log | x | {\\displaystyle \\log |x|} ã® åå§é¢æ°ãæ±ããããšãåºæ¥ãã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 80,
"tag": "p",
"text": "ãšãªãã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 81,
"tag": "p",
"text": "",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 82,
"tag": "p",
"text": "æçé¢æ° R ( x ) {\\displaystyle R(x)} ã«å¯ŸããŠãç©å â« R ( e x ) d x {\\displaystyle \\int R(e^{x})\\,dx} 㯠t = e x {\\displaystyle t=e^{x}} ãããš d t d x = e x = t {\\displaystyle {\\frac {dt}{dx}}=e^{x}=t} ãã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 83,
"tag": "p",
"text": "â« R ( e x ) d x = â« R ( t ) d t t . {\\displaystyle \\int R(e^{x})\\,dx=\\int R(t){\\frac {dt}{t}}.}",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 84,
"tag": "p",
"text": "æçé¢æ° R ( x , y ) {\\displaystyle R(x,y)} ã«å¯ŸããŠãç©å â« R ( x , a x 2 + b x + c ) d x {\\displaystyle \\int R(x,{\\sqrt {ax^{2}+bx+c}})\\,dx} ã«ã€ããŠèããããå¹³æ¹æ ¹ã®äžèº«ã¯å¹³æ¹å®æããããšã«ãã£ãŠã p 2 â x 2 , x 2 + p 2 , x 2 â p 2 {\\displaystyle {\\sqrt {p^{2}-x^{2}}},{\\sqrt {x^{2}+p^{2}}},{\\sqrt {x^{2}-p^{2}}}} ã®ããããã®åœ¢ã«ãªããããããã®å Žåã«ã€ããŠã x = p sin Ξ , x = p tan Ξ , x = p cos Ξ {\\displaystyle x=p\\sin \\theta ,x=p\\tan \\theta ,x={\\frac {p}{\\cos \\theta }}} ãšå€æ°å€æãããšäžè§é¢æ°ã®ç©åã«åž°çããã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 85,
"tag": "p",
"text": "ãŸãã y 2 = a x 2 + b x + c {\\displaystyle y^{2}=ax^{2}+bx+c} ã¯äºæ¬¡æ²ç·ã§ãç¹ã« a > 0 {\\displaystyle a>0} ã®ãšãã¯åæ²ç·ãšãªã( y 2 â a ( x + b 2 a ) 2 = â b 2 + 4 a c 4 a {\\displaystyle y^{2}-a\\left(x+{\\frac {b}{2a}}\\right)^{2}={\\frac {-b^{2}+4ac}{4a}}} ãã)ããã®ãšãã y = ± a x + t {\\displaystyle y=\\pm {\\sqrt {a}}x+t} ããªãã¡ t = â a x + a x 2 + b x + c {\\displaystyle t=\\mp {\\sqrt {a}}x+{\\sqrt {ax^{2}+bx+c}}} ãšå€æãããšããŸãèšç®ã§ãã(笊å·ã¯ã©ã¡ããéžæããŠãè¯ã)ã幟äœåŠçã«ã¯ãåæ²ç·ã®æŒžè¿ç·ã«å¹³è¡ã§åçã t {\\displaystyle t} ã®çŽç· y = ± a x + t {\\displaystyle y=\\pm {\\sqrt {a}}x+t} ãšåæ²ç·ã®ãã äžã€ã®äº€ç¹ ( x , y ) {\\displaystyle (x,y)} ãå€æ° t {\\displaystyle t} ã§è¡šãããã®ã§ããã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 86,
"tag": "p",
"text": "äŸ â« d x x 2 â 1 {\\displaystyle \\int {\\frac {dx}{\\sqrt {x^{2}-1}}}} 㯠t = x + x 2 â 1 {\\displaystyle t=x+{\\sqrt {x^{2}-1}}} ãšçœ®æãããšã 1 t = x â x 2 â 1 {\\displaystyle {\\frac {1}{t}}=x-{\\sqrt {x^{2}-1}}} ãªã®ã§ã t + 1 t = 2 x {\\displaystyle t+{\\frac {1}{t}}=2x} ããªãã¡ 2 d x = ( 1 â 1 t 2 ) d t {\\displaystyle 2dx=\\left(1-{\\frac {1}{t^{2}}}\\right)dt} ãŸãã t â 1 t = 2 x 2 â 1 {\\displaystyle t-{\\frac {1}{t}}=2{\\sqrt {x^{2}-1}}} .ãªã®ã§ã â« d x x 2 â 1 = â« 1 â 1 t 2 t â 1 t d t = â« d t t = ln | x + x 2 â 1 | + C {\\displaystyle \\int {\\frac {dx}{\\sqrt {x^{2}-1}}}=\\int {\\frac {1-{\\frac {1}{t^{2}}}}{t-{\\frac {1}{t}}}}dt=\\int {\\frac {dt}{t}}=\\ln |x+{\\sqrt {x^{2}-1}}|+C} ã§ããã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 87,
"tag": "p",
"text": "ãšããã§ããã®å€æã¯åæ²ç· y 2 = x 2 â 1 {\\displaystyle y^{2}=x^{2}-1} ãšçŽç· y = â x + t {\\displaystyle y=-x+t} ã®ãã äžã€ã®äº€ç¹ã«ããå€æã§ãã£ãããã®äº€ç¹ãæ¹çšåŒã解ã㊠t {\\displaystyle t} ã§è¡šããšã x = 1 2 ( t + 1 t ) , y = 1 2 ( t â 1 t ) {\\displaystyle x={\\frac {1}{2}}\\left(t+{\\frac {1}{t}}\\right),\\,y={\\frac {1}{2}}\\left(t-{\\frac {1}{t}}\\right)} ãåŸããããã¯åæ²ç·ã®åªä»å€æ°è¡šç€ºã®äžã€ã§ããããŸãã t â e t {\\displaystyle t\\rightarrow e^{t}} ãšãããšã x = e t + e â t 2 = cosh t , y = e t â e â t 2 = sinh t . {\\displaystyle x={\\frac {e^{t}+e^{-t}}{2}}=\\cosh t,\\,y={\\frac {e^{t}-e^{-t}}{2}}=\\sinh t.} ãã㯠x > 0 {\\displaystyle x>0} ã®éšåã®åæ²ç·ã®åªä»å€æ°è¡šç€ºã§ãããæå³èŸºã¯åæ²ç·é¢æ°ãšåŒã°ããäžè§é¢æ°ãšäŒŒãæ§è³ªãæã€ãé¢æ°åã® h {\\displaystyle \\mathrm {h} } ã¯hyperbolaã«ç±æ¥ãããäŸãã°ãåæ²ç·ã®æ¹çšåŒããåŸããã cosh 2 t â sinh 2 t = 1 {\\displaystyle \\cosh ^{2}t-\\sinh ^{2}t=1} 㯠sin 2 Ξ + cos 2 Ξ = 1 {\\displaystyle \\sin ^{2}\\theta +\\cos ^{2}\\theta =1} ãšãã䌌ãŠãããäŸç€ºã®äžå®ç©å㯠x = cosh t {\\displaystyle x=\\cosh t} ãšçœ®æããŠã解ãããšãåºæ¥ãããã»ãšãã©åãããšãªã®ã§çç¥ããã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 88,
"tag": "p",
"text": "a < b {\\displaystyle a<b} ãšãããç©å â« a b ( x â a ) ( b â x ) d x {\\displaystyle \\int _{a}^{b}{\\sqrt {(x-a)(b-x)}}\\,dx} 㯠y = ( x â a ) ( b â x ) {\\displaystyle y={\\sqrt {(x-a)(b-x)}}} ãšãããšã ( x â a + b 2 ) + y 2 = ( a â b 2 ) 2 {\\displaystyle \\left(x-{\\frac {a+b}{2}}\\right)+y^{2}=\\left({\\frac {a-b}{2}}\\right)^{2}} ããã被ç©åé¢æ° y {\\displaystyle y} ã¯äžå¿ a + b 2 {\\displaystyle {\\frac {a+b}{2}}} ã§ååŸ b â a 2 {\\displaystyle {\\frac {b-a}{2}}} ã®ååšã®äžååã§ãããç©ååºéããã®äž¡ç«¯ãªã®ã§ãç©åã®å€ã¯ååã®é¢ç©ã«çããã â« a b ( x â a ) ( b â x ) d x = Ï 2 ( b â a 2 ) 2 {\\displaystyle \\int _{a}^{b}{\\sqrt {(x-a)(b-x)}}\\,dx={\\frac {\\pi }{2}}\\left({\\frac {b-a}{2}}\\right)^{2}} ã§ããã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 89,
"tag": "p",
"text": "äžè¬ã«ãé¢æ° f ( a â x ) {\\displaystyle f(a-x)} ã®ã°ã©ãã¯é¢æ° f ( x ) {\\displaystyle f(x)} ã®ã°ã©ããçŽç· x = a 2 {\\displaystyle x={\\frac {a}{2}}} ã§å¯Ÿç§°ç§»åãããã®ã§ããã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 90,
"tag": "p",
"text": "åŸã£ãŠãé£ç¶é¢æ° f ( x ) {\\displaystyle f(x)} ãåºé [ a + b 2 , b ] {\\displaystyle \\left[{\\frac {a+b}{2}},b\\right]} ã§ç©åããå€ â« a + b 2 b f ( x ) d x {\\displaystyle \\int _{\\frac {a+b}{2}}^{b}f(x)\\,dx} ãšãé£ç¶é¢æ° f ( a + b â x ) {\\displaystyle f(a+b-x)} ãåºé [ a , a + b 2 ] {\\displaystyle \\left[a,{\\frac {a+b}{2}}\\right]} ã§ç©åããå€ â« a a + b 2 f ( a + b â x ) d x {\\displaystyle \\int _{a}^{\\frac {a+b}{2}}f(a+b-x)\\,dx} ã¯çãã:",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 91,
"tag": "p",
"text": "ãã®çåŒã¯åã«ã x â a + b â x {\\displaystyle x\\to a+b-x} ã®å€æ°å€æã«ãã£ãŠãå°åºã§ããã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 92,
"tag": "p",
"text": "ãã®çåŒããã â« a b f ( x ) d x = â« a a + b 2 f ( x ) d x + â« a + b 2 b f ( x ) d x = â« a a + b 2 [ f ( x ) + f ( a + b â x ) ] d x {\\displaystyle \\int _{a}^{b}f(x)\\,dx=\\int _{a}^{\\frac {a+b}{2}}f(x)\\,dx+\\int _{\\frac {a+b}{2}}^{b}f(x)\\,dx=\\int _{a}^{\\frac {a+b}{2}}[f(x)+f(a+b-x)]\\,dx} ãå°ãããã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 93,
"tag": "p",
"text": "ãã®å
¬åŒã¯ã f ( x ) + f ( a + b â x ) {\\displaystyle f(x)+f(a+b-x)} ãç°¡åãªåœ¢ã«ãªãå®ç©åã§åœ¹ã«ç«ã€ã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 94,
"tag": "p",
"text": "äŸãã°ã â« 0 Ï 2 sin x sin x + cos x d x = â« 0 Ï 4 [ sin x sin x + cos x + sin ( Ï 2 â x ) sin ( Ï 2 â x ) + cos ( Ï 2 â x ) ] d x = â« 0 Ï 4 [ sin x sin x + cos x + cos x cos x + sin x ] d x = â« 0 Ï 4 d x = Ï 4 . {\\displaystyle {\\begin{aligned}\\int _{0}^{\\frac {\\pi }{2}}{\\frac {\\sin x}{\\sin x+\\cos x}}\\,dx&=\\int _{0}^{\\frac {\\pi }{4}}\\left[{\\frac {\\sin x}{\\sin x+\\cos x}}+{\\frac {\\sin({\\frac {\\pi }{2}}-x)}{\\sin({\\frac {\\pi }{2}}-x)+\\cos({\\frac {\\pi }{2}}-x)}}\\right]\\,dx\\\\&=\\int _{0}^{\\frac {\\pi }{4}}\\left[{\\frac {\\sin x}{\\sin x+\\cos x}}+{\\frac {\\cos x}{\\cos x+\\sin x}}\\right]\\,dx\\\\&=\\int _{0}^{\\frac {\\pi }{4}}dx={\\frac {\\pi }{4}}.\\end{aligned}}}",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 95,
"tag": "p",
"text": "King Property ã®å¿çšäŸã¯ â« â 1 1 x 2 1 + e x d x = 1 3 {\\displaystyle \\int _{-1}^{1}{\\frac {x^{2}}{1+e^{x}}}\\,dx={\\frac {1}{3}}} , â« 0 Ï 4 ln ( 1 + tan x ) d x = Ï 8 ln 2 {\\displaystyle \\int _{0}^{\\frac {\\pi }{4}}\\ln(1+\\tan x)\\,dx={\\frac {\\pi }{8}}\\ln 2} , â« 0 Ï 2 ln sin x d x = â Ï 2 ln 2 {\\displaystyle \\int _{0}^{\\frac {\\pi }{2}}\\ln \\sin x\\,dx=-{\\frac {\\pi }{2}}\\ln 2} ãªã©ããããèšç®ããŠã¿ãã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 96,
"tag": "p",
"text": "",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 97,
"tag": "p",
"text": "æŒç¿åé¡1",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 98,
"tag": "p",
"text": "次ã®äžå®ç©åãæ±ããã",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 99,
"tag": "p",
"text": "",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 100,
"tag": "p",
"text": "æŒç¿åé¡2",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 101,
"tag": "p",
"text": "第äžå",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 102,
"tag": "p",
"text": "第äºå",
"title": "ãããããªé¢æ°ã®ç©å"
},
{
"paragraph_id": 103,
"tag": "p",
"text": "",
"title": "ç©åã®å¿çš"
},
{
"paragraph_id": 104,
"tag": "p",
"text": "ããé¢æ°f(x)ã®åå§é¢æ°ãæ±ããæŒç®ã¯ f(x)ãšx軞ã«ã¯ããŸããé åã®é¢ç©ãæ±ããæŒç®ã«çããã ãã®ããšãçšã㊠ããé¢æ°ã«ãã£ãŠäœãããé åã®é¢ç©ãæ±ããããšãåºæ¥ãã",
"title": "ç©åã®å¿çš"
},
{
"paragraph_id": 105,
"tag": "p",
"text": "äŸãã°ã â« 0 1 x 2 d x = 1 3 {\\displaystyle \\int _{0}^{1}x^{2}dx={\\frac {1}{3}}} ã¯ãæŸç©ç· y = x 2 {\\displaystyle y=x^{2}} ã«ã€ã㊠0 < x < 1 {\\displaystyle 0<x<1} ã®ç¯å²ã§ãããŸããé¢ç©ã«çããã",
"title": "ç©åã®å¿çš"
},
{
"paragraph_id": 106,
"tag": "p",
"text": "æ¥å x 2 a 2 + y 2 b 2 = 1 {\\displaystyle {\\frac {x^{2}}{a^{2}}}+{\\frac {y^{2}}{b^{2}}}=1} ã®é¢ç© S = Ï a b {\\displaystyle S=\\pi ab} ã®å°åº",
"title": "ç©åã®å¿çš"
},
{
"paragraph_id": 107,
"tag": "p",
"text": "æ¥å x 2 a 2 + y 2 b 2 = 1 {\\displaystyle {\\frac {x^{2}}{a^{2}}}+{\\frac {y^{2}}{b^{2}}}=1} ã y {\\displaystyle y} ã«ã€ããŠè§£ããš",
"title": "ç©åã®å¿çš"
},
{
"paragraph_id": 108,
"tag": "p",
"text": "ãšãªãããã®ãã¡ y = b a a 2 â x 2 {\\displaystyle y={\\frac {b}{a}}{\\sqrt {a^{2}-x^{2}}}} ã¯åæ¥å(æ¥åã®äžåå)ã瀺ããŠããããã®åæ¥åã®é¢ç©ã2åãããã®ãæ¥åã®é¢ç©Sãšãªãã®ã§",
"title": "ç©åã®å¿çš"
},
{
"paragraph_id": 109,
"tag": "p",
"text": "ãšãªãã",
"title": "ç©åã®å¿çš"
},
{
"paragraph_id": 110,
"tag": "p",
"text": "ããç«äœ V 0 {\\displaystyle V_{0}} ã® x = t {\\displaystyle x=t} ã«ãããæé¢ç©ãæéãªå€ã§ããã®å€ã t {\\displaystyle t} ã®é¢æ° S ( t ) {\\displaystyle S(t)} ãšãªããšãããã®ç«äœãå¹³é¢ x = a {\\displaystyle x=a} , x = b {\\displaystyle x=b} (ãã ãã a < b {\\displaystyle a<b} )ã§åãåã£ãé åã®äœç©ã¯ãåºé¢ç© S ( t ) {\\displaystyle S(t)} ã«æ¥µããŠå°ããé«ã d t {\\displaystyle dt} ã®ç© S ( t ) d t {\\displaystyle S(t)\\,dt} ã®åºé [ a , b ] {\\displaystyle [a,b]} ã«ããã环ç©ã§ããã®ã§ã以äžã®åŒã§è¡šãããšãã§ããã",
"title": "ç©åã®å¿çš"
},
{
"paragraph_id": 111,
"tag": "p",
"text": "(äŸ1)",
"title": "ç©åã®å¿çš"
},
{
"paragraph_id": 112,
"tag": "p",
"text": "(äŸ2)",
"title": "ç©åã®å¿çš"
},
{
"paragraph_id": 113,
"tag": "p",
"text": "y = f ( x ) ( a †x †b ) {\\displaystyle y=f(x)(a\\leq x\\leq b)} ã§äžããããæ²ç·ãx軞ã®åãã«å転ãããŠäœããã ç«äœã®äœç©Vã¯ã V = â« a b Ï ( f ( x ) ) 2 d x {\\displaystyle V=\\int _{a}^{b}\\pi (f(x))^{2}dx} ã§äžããããã",
"title": "ç©åã®å¿çš"
},
{
"paragraph_id": 114,
"tag": "p",
"text": "å°åº",
"title": "ç©åã®å¿çš"
},
{
"paragraph_id": 115,
"tag": "p",
"text": "ç«äœãx軞ã«åçŽã§ãããx=cãæºããé¢ãšx=c+hãæºããé¢ã§åããš(hã¯å°ã㪠å®æ°)ããã®åæé¢ã§æãŸããç«äœã¯ååŸ f(c)ã®åãšååŸ f(c+h)ã®å ã§ã¯ããŸããç«äœãšãªãã ããããhã極ããŠå°ãããšãããã®å³åœ¢ã¯ååŸf(c),é«ãhã®åæ±ã§ è¿äŒŒã§ããã ãã£ãŠãã®2ã€ã®é¢ã«é¢ããŠãåŸãããå³åœ¢ã®äœç©ã¯ h Ã Ï ( f ( c ) ) 2 {\\displaystyle h\\times \\pi (f(c))^{2}} ãšãªãã ããã a < c < b {\\displaystyle a<c<b} æºããå
šãŠã®cã«ã€ããŠè¶³ãåããããšã S = â« a b Ï ( f ( x ) ) 2 d x {\\displaystyle S=\\int _{a}^{b}\\pi (f(x))^{2}dx} ãåŸãããã",
"title": "ç©åã®å¿çš"
},
{
"paragraph_id": 116,
"tag": "p",
"text": "äŸãã°ã y = x 2 ( 0 < x < 1 ) {\\displaystyle y=x^{2}~(0<x<1)} ãx軞ã®åãã«å転ãããŠåŸãããå³åœ¢ã®äœç©ã¯ã",
"title": "ç©åã®å¿çš"
},
{
"paragraph_id": 117,
"tag": "p",
"text": "S = â« 0 1 Ï ( x 2 ) 2 d x {\\displaystyle S=\\int _{0}^{1}\\pi (x^{2})^{2}dx} = Ï â« 0 1 x 4 d x {\\displaystyle =\\pi \\int _{0}^{1}x^{4}dx} = Ï 5 {\\displaystyle ={\\frac {\\pi }{5}}} ãšãªãã",
"title": "ç©åã®å¿çš"
},
{
"paragraph_id": 118,
"tag": "p",
"text": "çã®äœç© V = 4 3 Ï r 3 {\\displaystyle V={\\frac {4}{3}}\\pi r^{3}} ã®å°åº",
"title": "ç©åã®å¿çš"
},
{
"paragraph_id": 119,
"tag": "p",
"text": "ååŸrã®çã¯åå y = r 2 â x 2 {\\displaystyle y={\\sqrt {r^{2}-x^{2}}}} ãx軞ã®åšãã«å転ãããŠã€ããããšãã§ããã",
"title": "ç©åã®å¿çš"
},
{
"paragraph_id": 120,
"tag": "p",
"text": "ãŸãäœç©ãrã§åŸ®åãããšçã®è¡šé¢ç© S = 4 Ï r 2 {\\displaystyle S=4\\pi r^{2}} ãåŸãããã",
"title": "ç©åã®å¿çš"
},
{
"paragraph_id": 121,
"tag": "p",
"text": "ãããŸã§ã«åŠãã ããã«ãç©åã¯åŸ®åã®éæŒç®ã§ãããšåæã«ã座æšå¹³é¢äžã§ã®é¢ç©èšç®ã§ãããããã®é
ã§ã¯ã座æšå¹³é¢äžã®é¢ç©èšç®ã®æ¹æ³ã®äžã€ã§ããåºåæ±ç©æ³ãããã³ç©åæ³ãšã®é¢é£ã«ã€ããŠåŠã¶ã",
"title": "åºåæ±ç©æ³"
},
{
"paragraph_id": 122,
"tag": "p",
"text": "å³å³ã®ãããªããæ²ç· y = f ( x ) {\\displaystyle y=f(x)} ããããåçŽã®ãããããã§ã¯ã€ãã« f ( x ) > 0 {\\displaystyle f(x)>0} ã§ãããã®ãšããŠèããããã®æ²ç·ãšãx軞ãããã³çŽç· x = a , x = b ( a < b ) {\\displaystyle x=a,x=b(a<b)} ã«ãã£ãŠå²ãŸããé åã®é¢ç©Sãæ±ããããã®é¢ç©ã¯#é¢ç©ã®é
ã§åŠãã ããã«ã",
"title": "åºåæ±ç©æ³"
},
{
"paragraph_id": 123,
"tag": "p",
"text": "ãšç©åæ³ãçšããŠèšç®ããããšãã§ãããã§ã¯ããããããå°ãåå§çãªæ¹æ³ã§è¿äŒŒçã«æ±ããããšãèããŠã¿ããã",
"title": "åºåæ±ç©æ³"
},
{
"paragraph_id": 124,
"tag": "p",
"text": "æ²ç·ãå«ãå³åœ¢ã®é¢ç©ãæ±ããããšã¯ç°¡åã§ã¯ãªãããäŸãã°äžè§åœ¢ãé·æ¹åœ¢ãå°åœ¢ãªã©ã®çŽç·ã§å²ãŸããå³åœ¢ã®é¢ç©ãæ±ããããšã¯é£ãããªããããã§ãäžå³ã®ããã«y=f(x)ãæ£ã°ã©ãã§è¿äŒŒããé·æ¹åœ¢ã®é¢ç©ã®åãèšç®ããããšã§ãæ±ãããé¢ç©Sã«è¿ãå€ãæ±ããããšãã§ãããå·Šäžã®ããã«æ£ã°ã©ãã®å¹
ã倧ãããšèª€å·®ã倧ããããæ£ã°ã©ãã®å¹
ãçãããã°ããã»ã©ãããªãã¡åå²æ°ãå€ãããã»ã©ãåŸã
ã«æ±ãããé¢ç©ã®å€ã«è¿ã¥ããããšãã§ãããããã§ããã®åºé[a,b]ãnçåãããã®æã®é·æ¹åœ¢ã®é¢ç©ã®ç·åãæ±ãããã®åŸã§ n â â {\\displaystyle n\\to \\infty } ã®æ¥µéãèããããšã«ããããã®ããã«ããŠãåºéã现ããçåå²ããé·æ¹åœ¢ã®é¢ç©ã®ç·åãæ±ããããšã«ããå³åœ¢ã®é¢ç©ãæ±ããæ¹æ³ããåºåæ±ç©æ³ãšåŒã¶ã",
"title": "åºåæ±ç©æ³"
},
{
"paragraph_id": 125,
"tag": "p",
"text": "y = f ( x ) {\\displaystyle y=f(x)} ãæ£ã°ã©ãã§è¿äŒŒãããšããå³å³ã®ããã«ãé·æ¹åœ¢ã®å·Šäžã®é ç¹ãæ²ç·äžã«åãæ¹æ³ãšãå³äžã®é ç¹ãæ²ç·äžã«åãæ¹æ³ããããã©ã¡ãã®æ¹æ³ã§ããåå²æ°ã倧ããããã°ãããæ±ãããé¢ç©ã«è¿ã¥ããããŸãã¯å·Šäžã®é ç¹ãæ²ç·äžã«åãæ¹æ³ã§èããããšã«ããã",
"title": "åºåæ±ç©æ³"
},
{
"paragraph_id": 126,
"tag": "p",
"text": "ããã§ã¯é¢ç©ãæ±ãããåºéããåçŽã®ãã[0, 1]ãšãããåºé[0, 1]ãnçåãããšããããããã®é·æ¹åœ¢ã®å·Šç«¯ã®x座æšã¯ã",
"title": "åºåæ±ç©æ³"
},
{
"paragraph_id": 127,
"tag": "p",
"text": "ãšãªããããã§ãäžè¬ã«ç¬¬kçªç®ã®é·æ¹åœ¢ã«ã€ããŠèããããšã«ããããã ãããã¡ã°ãå·ŠåŽã®é·æ¹åœ¢ã第0çªç®ãšãããã¡ã°ãå³åŽã®é·æ¹åœ¢ã第n-1çªç®ãšããã第kçªç®ã®é·æ¹åœ¢ã®å·Šç«¯ã®x座æšã¯ k n {\\displaystyle {\\frac {k}{n}}} ã§ããããããã®é·æ¹åœ¢ã®é«ã㯠f ( k n ) {\\displaystyle f\\left({\\frac {k}{n}}\\right)} ãšãªãããŸãé·æ¹åœ¢ã®å¹
㯠1 n {\\displaystyle {\\frac {1}{n}}} ã§ããããã®ããããã®é·æ¹åœ¢ã®é¢ç© s k {\\displaystyle s_{k}} ã¯ã",
"title": "åºåæ±ç©æ³"
},
{
"paragraph_id": 128,
"tag": "p",
"text": "ãšãªãããããã£ãŠããããã®é·æ¹åœ¢ã®é¢ç©ã®ç·å S n {\\displaystyle S_{n}} ã¯ã",
"title": "åºåæ±ç©æ³"
},
{
"paragraph_id": 129,
"tag": "p",
"text": "ãã® S n {\\displaystyle S_{n}} ã¯ãåºé[0, 1]ãnçåããæã®é·æ¹åœ¢ã®é¢ç©ã®ç·åã§ããããnã倧ããããã°ããã»ã©ã次第ã«ããšã®é¢ç©ã«è¿ã¥ããŠããããããã£ãŠã n â â {\\displaystyle n\\to \\infty } ã®æ¥µéãèãã",
"title": "åºåæ±ç©æ³"
},
{
"paragraph_id": 130,
"tag": "p",
"text": "ãšãªãããã®ããã«ããŠãæ±ãããé¢ç©ãèšç®ããããšãã§ãããããã«ãããã§ãã®åºéã®é¢ç©ãç©åæ³ã«ããèšç®ã§ããããšããã",
"title": "åºåæ±ç©æ³"
},
{
"paragraph_id": 131,
"tag": "p",
"text": "ãæãç«ã€ããŸããé·æ¹åœ¢ã®å³äžã®é ç¹ãæ²ç·äžã«åãå Žåã¯ãåæ§ã«ããŠ",
"title": "åºåæ±ç©æ³"
},
{
"paragraph_id": 132,
"tag": "p",
"text": "ãšãªãã",
"title": "åºåæ±ç©æ³"
}
] | ããã§ã¯ãæ°åŠIIã®åŸ®åã»ç©åã®èãã§åŠãã ç©åã®æ§è³ªã«ã€ããŠãã詳ããæ±ãããŸããäžè§é¢æ°ãææ°ã»å¯Ÿæ°é¢æ°ãªã©ã®é¢æ°ã®ç©åã«ã€ããŠãåŠç¿ããã | {{pathnav|é«çåŠæ ¡ã®åŠç¿|é«çåŠæ ¡æ°åŠ|é«çåŠæ ¡æ°åŠIII|pagename=ç©åæ³|frame=1|small=1}}
ããã§ã¯ãæ°åŠIIã®[[é«çåŠæ ¡æ°åŠII/埮åã»ç©åã®èã|埮åã»ç©åã®èã]]ã§åŠãã ç©åã®æ§è³ªã«ã€ããŠãã詳ããæ±ãããŸããäžè§é¢æ°ãææ°ã»å¯Ÿæ°é¢æ°ãªã©ã®é¢æ°ã®ç©åã«ã€ããŠãåŠç¿ããã
== ç©åã®åºæ¬çãªæ§è³ª ==
ç©åæ³ã«ã€ããŠ
<math>\int \{ f(x) + g(x) \} dx = \int f(x) dx + \int g(x) dx ,</math>
<math>\int af(x) dx = a \int f(x) dx</math>(aã¯å®æ°)
ãæãç«ã€ã
å°åº
<math>\int \{ f(x) + g(x) \} dx = \int f(x) dx + \int g(x) dx</math>
ã®äž¡èŸºã埮åãããšã
巊蟺 =å³èŸº = <math> f + g</math>
ãåŸãã
ãã£ãŠã
<math>\int \{ f(x) + g(x) \} dx = \int f(x) dx + \int g(x) dx</math>
ã®äž¡èŸºã¯äžèŽããã
(å®éã«ã¯2ã€ã®é¢æ°ã®å°é¢æ°ãäžèŽãããšãã
ãããã®é¢æ°ã«ã¯å®æ°ã ãã®ã¡ãããããã
ä»®ã«ãF(x)ãšG(x)ãå
±éã®å°é¢æ°h(x)ãæã£ããšããã
ãã®ãšãã
<math>(F(x)-G(x) )' = h(x)- h(x) = 0</math>
ãšãªããã0ã®åå§é¢æ°ã¯å®æ°Cã§ããããšãåããã
ãã£ãŠã䞡蟺ãç©åãããšã
<math>F(x)-G(x) = C</math>
ãšãªããF(x)ãšG(x)ã«ã¯å®æ°ã ãã®å·®ãããªãããšã確ãããããã
ãã£ãŠã
<math>\int \{ f(x) + g(x) \} dx = \int f(x) dx + \int g(x) dx</math>
ã¯å®æ°ã ãã®ã¡ãããå«ãã§æãç«ã€åŒã§ããã
ããäžè¬ã«ãäžå®ç©åã絡ãçåŒã¯å®æ°åã®å·®ãå«ããŠæãç«ã€ãšããã®ãéäŸã§ããã)
<math>\int af(x) dx = a \int f(x) dx</math>
ã«ã€ããŠã䞡蟺ã埮åãããšã
巊蟺=å³èŸº= a f(x)
ãåŸãã
ãã£ãŠã
<math>\int af dx = a\int f dx</math>
ãæãç«ã€ããšãåãã
é¢æ° <math>f(x)</math> ã®åå§é¢æ°ã <math>F(x)</math> ãšãããš
<math>\int_a^b f(x) \, = F(b)-F(a) = -(F(a)-F(b)) = -\int_b^af(x)\, dx</math> ã§ããã
<math>\int_{a}^{c} f(x) \, dx + \int_{c}^{b} f(x) \, dx = (F(c) - F(a)) + (F(b) - F(c)) = F(b) - F(a) = \int_{a}^{b} f(x) \, dx</math>
==眮æç©åæ³==
é¢æ°ã®åå§é¢æ°ãæ±ããæ段ãšããŠã
ç©åå€æ°ãå¥ã®å€æ°ã§çœ®ãæããŠç©åãè¡ãªãæ段ãç¥ãããŠããã
ããã眮æç©åãšåŒã¶ã
<math>\int f(g(x)) dg(x) = \int f(g(x)) g'(x) dx</math>
å°åº
<math>\int f(g(x)) dg(x) =F(g(x))</math>ã<math>x</math>ã«ã€ããŠåŸ®åãããšã
<math>F'(g(x)) = f(g(x))g'(x)</math>
åã³<math>x</math>ã«ã€ããŠç©åãããšã
<math>\int f(g(x)) dg(x) = \int f(g(x)) g'(x) dx</math>
ãŸããç¹ã«
*<math>\int f(ax+b) dx = \frac{1}{a} \int f(ax+b) d(ax+b)</math>
*<math>\int \{f(x)\}^n f'(x) dx = \frac{1}{n+1} \{f(x)\}^{n+1} + C (n \ne -1)</math>
*<math>\int \frac{f'(x)}{f(x)} dx = \log | f(x) | + C</math>
äŸãã°ã<math>\int (ax+b)^2 dx</math>ãèããã
<math>t = ax+b</math>ãšçœ®ãã
ãã®äž¡èŸºã埮åãããš
<math>dt = adx</math>
ãæãç«ã€ããšãèæ
®ãããšã
{|
|-
|<math>\int t^2 \frac {dt} a</math>
|<math>=\frac{ t^3} {3a} + C</math>
|-
|
|<math>=\frac{ (ax+b)^3} {3a} + C</math>
|}
ãšãªãããšããããã
å®éãã®åŒãxã§åŸ®åãããš
<math>
(ax+b)^2
</math>
ãšäžèŽããããšãåãã
眮æç©åã䜿ããã«èšç®ããããšãåºæ¥ãã
{|
|-
|<math>\int (ax+b)^2 dx</math>
|<math>=\int (a^2x^2+2abx +b^2) dx</math>
|-
|
|<math>= \frac {a^2} 3 x^3 +abx^2 +b^2x + C'</math>
|-
|
|<math>= \frac {a^2} 3 x^3 +abx^2 +b^2x + \frac {b^3} {3a} +C</math>
|}
(<math>C'=\frac {b^3} {3a} +C</math>ãšçœ®ãæããã)
<math>=\frac{ (ax+b)^3} {3a} + C</math>
ãšãªã確ãã«äžèŽããã
==éšåç©åæ³==
é¢æ°ã®ç©ã®ç©åãè¡ãªããšãããé¢æ°ã®åŸ®åã ããåãã ããŠç©åãããšãããŸãç©åã§ããå Žåããããé¢æ° <math>g(x)</math> ã®åå§é¢æ°ã <math>G(x)</math> ãšãããš
<math>\int f(x) g(x) \, dx = f(x) G(x) - \int f'(x) G(x) \, dx</math>
å°åº
ç©ã®åŸ®åæ³ãã <math>\{f(x)G(x)\}' = f'(x)G(x) + f(x)g(x)</math> ã§ãããããã移é
ããŠ
<math>f(x)g(x) = \{f(x)G(x)\}' - f'(x)G(x)</math>
ã§ããã䞡蟺ãxã§ç©åããŠ
<math>\int f(x) g(x) \, dx = f(x) G(x) - \int f'(x) G(x) \, dx</math>
ãåŸãããã
äŸãã°ã
{|
|-
|<math>\int x (ax+b)^3 dx</math>
|<math>=\int x \left(\frac {(ax+b)^4} {4a} \right)' dx</math>
|-
|
|<math>=x \left(\frac {(ax+b)^4} {4a} \right)- \int (x)' \frac {(ax+b)^4} {4a} dx</math>
|-
|
|<math>=x \left(\frac {(ax+b)^4} {4a} \right)- \int (x)' \frac {(ax+b)^4} {4a} dx</math>
|-
|
|<math>=x \left(\frac {(ax+b)^4} {4a} \right)- \int \frac {(ax+b)^4} {4a} dx</math>
|-
|
|<math>=x \left(\frac {(ax+b)^4} {4a} \right)- \frac {(ax+b)^5} {20a^2} </math>
|}
== ãããããªé¢æ°ã®ç©å==
=== å€é
åŒé¢æ°ã®ç©å ===
<math>n \ne -1</math>ã®ãšãã<math>\left(\frac{1}{n+1} x^{n+1}\right)'=x^n</math>ãªã®ã§ã
<math>\int x^n dx = \frac{1}{n+1} x^{n+1} + C</math>
<math>n = -1</math>ã®ãšãã<math>(\log |x| )' = \frac{1}{x} = x^{-1}</math>ãªã®ã§ã
<math>\int x^{-1} dx = \int \frac {1}{x} dx = \log |x| + C</math>
ãæãç«ã€ã
=== äžè§é¢æ°ã®ç©å ===
*<math>(\sin x )' = \cos x</math>
*<math>(\cos x )' = -\sin x</math>
*<math>(\tan x )' = \frac{1}{\cos^2 x}</math>
ãæãç«ã€ããšãèæ
®ãããšã
*<math>\int \cos x dx= \sin x + C</math>
*<math>\int \sin x dx = - \cos x + C</math>
*<math>\int \frac{1}{\cos^2 x } dx = \tan x + C</math>
ãšãªãããšãåãã
<math>\int \tan x dx</math>ã¯ã眮æç©åæ³ã䜿ã£ãŠ
{|
|-
|<math>\int \tan x dx</math>
|<math>=\int \frac{\sin x}{\cos x} dx</math>
|-
|
|<math>=\int \frac{-(\cos x)'}{\cos x} dx</math>
|-
|
|<math>= - \int \frac{(\cos x)'}{\cos x} dx</math>
|-
|
|<math>= - \log | \cos x | + C</math>
|}
:ã
:ãªãåæ§ã«ã<math>\frac{1}{\tan x} = \frac{\cos x}{\sin x}</math>ãã§ããã®ã§ã<math>\int \frac{1}{\tan x} dx = \int \frac{\cos x}{\sin x} dx =\int \frac{(\sin x)'}{\sin x} dx = \log \left|\sin x\right| + C</math>
:ã
ããäžè¬ã«æçé¢æ° <math>R(x,y)</math> ã«å¯ŸããŠã<math>\int R(\sin\theta,\cos\theta) \,d\theta</math> ã«ã€ããŠèããã <math>t = \tan \frac{\theta}{2}</math> ãšããã <math>\tan^2\frac{\theta}{2} + 1 = \frac{1}{\cos^2\frac{\theta}{2}}</math> ãã£ãŠ <math>\cos^2\frac{\theta}{2} = \frac{1}{1+t^2}</math>ã§ããã<math>\frac{dt}{d\theta} = \frac{d}{d\theta}\tan\frac{\theta}{2} = \frac{1}{2\cos^2\frac{\theta}{2}} = \frac{1}{2}(t^2+1)</math> ã§ããã<math>\cos\theta = 2\cos^2\frac{\theta}{2} - 1 = \frac{1-t^2}{1+t^2}</math> ã〠<math>\sin\theta = \tan\theta\cos\theta = \frac{2\tan\frac{\theta}{2}}{1-\tan^2\frac{\theta}{2}}\cos\theta =
\frac{2t}{1+t^2}</math>
ã§ããããã£ãŠ
<math>\int R(\sin\theta,\cos\theta) \,d\theta
= \int R\left(\frac{2t}{1+t^2}, \frac{1-t^2}{1+t^2}\right) \, \frac{2dt}{1+t^2}</math>
ãšæçé¢æ°ã®ç©åã«ãã¡èŸŒããã
幟äœåŠçã¯ããã®å€æã¯åäœåäžã®ç¹ <math>P(\cos \theta, \sin \theta)</math>ãšç¹ <math>A(-1,0)</math> ãçµã¶çŽç·ã®åŸé
<math>t</math> ã§å€æãããã®ã§ãããå®éååšè§ã®å®çãã <math>\angle xAP = \frac 1 2 \angle xOP = \frac \theta 2</math>ãã <math>t = \tan \frac{\theta} 2.</math>
被ç©åé¢æ°ã®åšæã <math>\pi</math> ã®å Žåã¯ã被ç©åé¢æ°ã¯ <math>\sin 2\theta,\cos 2 \theta</math> ã®æçé¢æ°ãªã®ã§ã <math>t = \tan\theta</math> ãšçœ®æãããšèšç®ã楜ã ã被ç©åé¢æ°ã <math>\sin^2\theta,\cos^2\theta,\sin\theta\cos\theta</math> ã®æçé¢æ°ãšãªããšãããã®ç¯çã«å±ããã<math>t = \tan\theta</math> ãšçœ®æãããšãã<math>\cos^2\theta = \frac{1}{1+\tan^2\theta}=\frac{1}{1+t^2}</math>, <math>\sin^2\theta = \tan^2 \theta \cos^2 \theta = \frac{t^2}{1+t^2}</math> , <math>\sin\theta \cos\theta = \pm\sqrt{\sin^2\theta \cos^2\theta} = \frac{t}{1+t^2}</math> (<math>\sin\theta \cos\theta</math> ãš <math>\tan\theta = \frac{\sin\theta}{\cos\theta}</math> ã®æ£è² ã¯äžèŽãããã), <math>d \theta = \frac {dt}{1 + t^2}</math> ãšãªãã
äŸã<math>\int\frac{1}{\sin x \cos x}dx</math> 㯠<math>t = \tan x</math> ãšçœ®æãããšã<math>\int \frac {1}{\sin x \cos x}dx = \int \frac {1+t^2}{t} \frac { dt}{1+t^2} = \ln|\tan x| + C. </math> <math>t = \tan \frac{\theta}{2}</math> ãšçœ®æããŠããŸããšã<math>\int \frac{1}{\sin x \cos x}\,dx = \int \frac {1+t^2}{t(1-t^2)}\,dt = \ln \left|\frac{t}{1-t^2}\right| + C' = \ln|\tan x| + C </math> ãšèšç®éãå°ãå¢ããã
=== ææ°ã»å¯Ÿæ°é¢æ°ã®ç©å ===
ææ°é¢æ°ã«ã€ããŠ
<math>(e^x )' = e^x</math>
ãæãç«ã€ããšãçšãããšã
<math>\int e^x dx = e^x + C</math>
ãåŸãããã
ãŸãã <math>\left(\frac{a^x}{\ln a}\right)' = a^x</math> ãªã®ã§ã <math>\int a^x \, dx=\frac{a^x}{\ln a}</math> ã§ããã
ãŸãã<math>\log |x|</math>ã®
åå§é¢æ°ãæ±ããããšãåºæ¥ãã
{|
|<math>\int \log |x| dx </math>
|<math>=\int (x)' \log |x| dx </math>
|-
|
|<math>=x \log |x| -\int x (\log |x|)' dx </math>
|-
|
|<math>=x \log |x| -\int x \frac 1 x dx </math>
|-
|
|<math>=x \log |x| -\int dx </math>
|-
|
|<math>=x \log |x| -x + C</math>
|}
ãšãªãã
æçé¢æ° <math>R(x)</math> ã«å¯ŸããŠãç©å <math>\int R(e^x) \, dx</math> 㯠<math>t = e^x</math> ãããš <math>\frac{dt}{dx} = e^x = t</math> ãã
<math>\int R(e^x) \, dx = \int R(t) \frac{dt}{t}.</math>
=== äºæ¬¡ç¡çé¢æ°ã®ç©åïŒçºå±ïŒ ===
æçé¢æ° <math>R(x,y)</math> ã«å¯ŸããŠãç©å <math>\int R(x,\sqrt{ax^2 + bx + c}) \, dx</math> ã«ã€ããŠèããããå¹³æ¹æ ¹ã®äžèº«ã¯å¹³æ¹å®æããããšã«ãã£ãŠã<math>\sqrt{p^2-x^2},\sqrt{x^2+p^2},\sqrt{x^2-p^2}</math>ã®ããããã®åœ¢ã«ãªããããããã®å Žåã«ã€ããŠã<math>x = p\sin \theta,x = p\tan\theta,x = \frac{p}{\cos \theta}</math> ãšå€æ°å€æãããšäžè§é¢æ°ã®ç©åã«åž°çããã
ãŸãã<math>y^2 = ax^2 +bx + c</math> ã¯äºæ¬¡æ²ç·ã§ãç¹ã« <math>a>0</math> ã®ãšãã¯åæ²ç·ãšãªãïŒ<math>y^2 -a\left(x+\frac{b}{2a}\right)^2 = \frac{-b^2 + 4ac}{4a}</math>ãã<ref>å³èŸºã0ã®ãšãåæ²ç·ãšã¯ãªããªããããã®ãšãã¯ç°¡åã«å¹³æ¹æ ¹ãå€ãããšãåºæ¥ãã®ã§èããå¿
èŠã¯ãªãã</ref>ïŒããã®ãšãã<math>y=\pm \sqrt a x + t</math> ããªãã¡ <math>t = \mp \sqrt a x + \sqrt{ax^2 + bx + c}</math> ãšå€æãããšããŸãèšç®ã§ããïŒç¬Šå·ã¯ã©ã¡ããéžæããŠãè¯ãïŒã幟äœåŠçã«ã¯ãåæ²ç·ã®æŒžè¿ç·ã«å¹³è¡ã§åçã <math>t</math> ã®çŽç· <math>y=\pm \sqrt a x + t</math> ãšåæ²ç·ã®ãã äžã€ã®äº€ç¹ <math>(x,y)</math> ãå€æ° <math>t</math> ã§è¡šãããã®ã§ããã
äŸ <math>\int \frac{dx}{\sqrt{x^2-1}} </math> 㯠<math>t = x + \sqrt{ x^2-1}</math> ãšçœ®æãããšã<math>\frac 1 t = x - \sqrt{x^2-1}</math> ãªã®ã§ã<math>t + \frac 1 t = 2x</math> ããªãã¡ <math>2dx = \left(1 - \frac 1 {t^2}\right)dt</math> ãŸãã <math>t - \frac 1 t = 2\sqrt{x^2-1}</math>.ãªã®ã§ã<math>\int \frac{dx}{\sqrt{x^2-1}} = \int \frac{1-\frac{1}{t^2}}{t-\frac 1 t}dt = \int \frac{dt}{t} = \ln |x + \sqrt{x^2-1}| + C </math> ã§ããã
ãšããã§ããã®å€æã¯åæ²ç· <math>y^2 = x^2 - 1</math> ãšçŽç· <math>y = -x + t</math> ã®ãã äžã€ã®äº€ç¹ã«ããå€æã§ãã£ãããã®äº€ç¹ãæ¹çšåŒã解ã㊠<math>t</math> ã§è¡šããšã<math>x = \frac 1 2 \left(t + \frac 1 t\right), \, y =\frac 1 2 \left(t - \frac 1 t\right)</math> ãåŸããããã¯åæ²ç·ã®åªä»å€æ°è¡šç€ºã®äžã€ã§ããããŸãã <math>t \rightarrow e^t</math> ãšãããšã<math>x = \frac{e^t + e^{ -t} }{2} = \cosh t, \, y = \frac{e^t - e^{-t}}{2} = \sinh t.</math> ãã㯠<math>x > 0</math> ã®éšåã®åæ²ç·ã®åªä»å€æ°è¡šç€ºã§ãããæå³èŸºã¯åæ²ç·é¢æ°ãšåŒã°ããäžè§é¢æ°ãšäŒŒãæ§è³ªãæã€ãé¢æ°åã® <math>\mathrm{h}</math> ã¯hyperbolaã«ç±æ¥ãããäŸãã°ãåæ²ç·ã®æ¹çšåŒããåŸããã <math>\cosh^2 t - \sinh^2 t = 1</math> 㯠<math>\sin^2\theta + \cos^2\theta = 1</math> ãšãã䌌ãŠãããäŸç€ºã®äžå®ç©å㯠<math>x = \cosh t</math> ãšçœ®æããŠã解ãããšãåºæ¥ãããã»ãšãã©åãããšãªã®ã§çç¥ããã
=== ç¹æ®ãªå®ç©å ===
==== å ====
<math>a < b</math> ãšãããç©å <math>\int_a ^b \sqrt{(x-a)(b-x)}\, dx</math> 㯠<math>y = \sqrt{(x-a)(b-x)}</math> ãšãããšã<math>\left(x-\frac{a+b}{2} \right) + y^2 = \left(\frac{a-b}{2} \right)^2</math> ããã被ç©åé¢æ° <math>y</math> ã¯äžå¿ <math>\frac{a+b}{2}</math> ã§ååŸ <math>\frac{b-a}{2}</math>ã®ååšã®äžååã§ãããç©ååºéããã®äž¡ç«¯ãªã®ã§ãç©åã®å€ã¯ååã®é¢ç©ã«çããã<math>\int_a ^b \sqrt{(x-a)(b-x)} \, dx = \frac{\pi}{2}\left(\frac{b-a}{2}\right)^2</math> ã§ããã
==== King Property ====
äžè¬ã«ãé¢æ° <math>f(a-x)</math> ã®ã°ã©ãã¯é¢æ° <math>f(x)</math> ã®ã°ã©ããçŽç· <math>x = \frac a 2</math> ã§å¯Ÿç§°ç§»åãããã®ã§ããã
åŸã£ãŠãé£ç¶é¢æ° <math>f(x)</math> ãåºé <math>\left[\frac{a+b}{2},b\right]</math> ã§ç©åããå€ <math>\int_{\frac{a+b}{2}}^{b} f(x) \, dx</math> ãšãé£ç¶é¢æ° <math>f(a+b-x)</math> ãåºé <math>\left[a,\frac{a+b}{2}\right]</math> ã§ç©åããå€ <math>\int_{a}^{\frac{a+b}{2}} f(a+b-x)\, dx</math> ã¯çããïŒ
:<math>\int_{\frac{a+b}{2}}^{b} f(x) \, dx = \int_{a}^{\frac{a+b}{2}} f(a+b-x) \, dx.</math>
ãã®çåŒã¯åã«ã <math>x \to a+b-x</math> ã®å€æ°å€æã«ãã£ãŠãå°åºã§ããã
ãã®çåŒããã <math>\int_a^b f(x) \, dx = \int_{a}^{\frac{a+b}{2}} f(x)\, dx +\int_{\frac{a+b}{2}}^{b} f(x) \, dx = \int_{a}^{\frac{a+b}{2}} [f(x) + f(a+b-x)] \, dx </math> ãå°ãããã
ãã®å
¬åŒã¯ã<math>f(x) + f(a+b-x)</math> ãç°¡åãªåœ¢ã«ãªãå®ç©åã§åœ¹ã«ç«ã€ã
äŸãã°ã<math>\begin{align}\int_{0}^{\frac{\pi}{2}} \frac{\sin x}{\sin x + \cos x} \, dx &= \int_{0}^{\frac{\pi}{4}} \left[\frac{\sin x}{\sin x + \cos x} +\frac{\sin (\frac{\pi}{2}-x)}{\sin (\frac{\pi}{2}-x) + \cos (\frac{\pi}{2}-x)}\right]\, dx \\
&= \int_{0}^{\frac{\pi}{4}} \left[\frac{\sin x}{\sin x + \cos x} +\frac{\cos x}{\cos x + \sin x}\right]\, dx \\ &= \int_{0}^{\frac{\pi}{4}}dx = \frac{\pi}{4}.\end{align} </math>
King Property ã®å¿çšäŸã¯ <math>\int_{-1}^{1} \frac{x^2}{1+e^x} \, dx = \frac 1 3</math> , <math>\int_0^{\frac \pi 4} \ln(1+\tan x)\, dx = \frac \pi 8 \ln 2</math> , <math>\int_0^{\frac \pi 2} \ln \sin x \, dx = -\frac{\pi}{2}\ln 2</math> ãªã©ããããèšç®ããŠã¿ãã
'''æŒç¿åé¡1'''
次ã®äžå®ç©åãæ±ããã
:(1)<math>\int \tan xdx</math>
:(2)<math>\int \frac{1}{\cos ^2x}dx</math>
:(3)<math>\int \log xdx</math>
:(4)<math>\int x\log xdx</math>
:(5)<math>\int x^2\log xdx</math>
:(6)<math>\int x^3\log xdx</math>
:(7)<math>\int x\sin xdx</math>
:(8)<math>\int x^2\sin xdx</math>
:(9)<math>\int x^2e^xdx</math>
*解ç
:(1)<math>-\log (\cos x)+C</math>
:(2)<math>\tan x+C</math>
:(3)<math>x\log x-x+C</math>
:(4)<math>\frac{x^2\log x}{2}-\frac{x^2}{4}+C</math>
:(5)<math>\frac{x^3\log x}{3}-\frac{x^3}{9}+C</math>
:(6)<math>\frac{x^4\log x}{4}-\frac{x^4}{16}+C</math>
:(7)<math>\sin x-x\cos x+C</math>
:(8)<math>2x\sin x+(2-x^2)\cos x+C</math>
:(9)<math>(x^2-2x+2)e^x+C</math>
:
'''æŒç¿åé¡2'''
'''第äžå'''
:<math>n</math> ã¯éè² æŽæ°ãšãã<math>I_n = \int_{0}^{\frac \pi 2}\sin^n x \, dx</math> ãšããã
:(1) <math>\int_{0}^{\frac{\pi}{2}}\sin^n x \, dx = \int_{0}^{\frac{\pi}{2}}\cos^n x \, dx</math> ã瀺ãã
:(2) <math>I_n = \frac{n-1}{n}I_{n-2}\quad (n \ge 2)</math> ã瀺ãã
:(3) <math>I_n</math> ãæ±ããã
'''第äºå'''
:<math>m,n</math> ã¯éè² æŽæ°ã<math>\alpha,\beta</math> 㯠<math>\beta > \alpha</math> ãªãå®æ°ãšãã<math>I_{m,n} = \int_\alpha^\beta (x-\alpha)^m(\beta - x)^n \, dx</math> ãšããã
:(1) <math>I_{m,n} = \frac{n}{m+1} I_{m+1,n-1} \quad (n\ge 1) </math> ã瀺ãã
:(2) <math>I_{m,n}</math> ãæ±ããã
==ç©åã®å¿çš==
==== é¢ç©ïŒäœç©====
=====é¢ç©=====
ããé¢æ°f(x)ã®åå§é¢æ°ãæ±ããæŒç®ã¯
f(x)ãšx軞ã«ã¯ããŸããé åã®é¢ç©ãæ±ããæŒç®ã«çããã
ãã®ããšãçšããŠ
ããé¢æ°ã«ãã£ãŠäœãããé åã®é¢ç©ãæ±ããããšãåºæ¥ãã
[[ç»å:Integral_x%5E2_0-1.png|right|x^2ã®0ãã1ãŸã§ã®ç©å]]
äŸãã°ã
<math>
\int _0 ^1 x^2 dx = \frac 1 3
</math>
ã¯ãæŸç©ç·<math> y = x^2</math>ã«ã€ããŠ
<math>0 < x < 1</math>ã®ç¯å²ã§ãããŸããé¢ç©ã«çããã
;æ¥åã®é¢ç©
æ¥å<math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1</math>ã®é¢ç©<math>S=\pi ab</math>ã®å°åº
æ¥å<math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1</math>ã<math>y</math>ã«ã€ããŠè§£ããš
:<math>y=\pm\frac{b}{a}\sqrt{a^2-x^2}</math>
ãšãªãããã®ãã¡<math>y=\frac{b}{a}\sqrt{a^2-x^2}</math>ã¯åæ¥åïŒæ¥åã®äžååïŒã瀺ããŠããããã®åæ¥åã®é¢ç©ã2åãããã®ãæ¥åã®é¢ç©''S''ãšãªãã®ã§
:<math>S=2\int _{-a} ^a \frac{b}{a}\sqrt{a^2-x^2} = \frac{2b}{a}\int _{-a} ^a \sqrt{a^2-x^2} = \frac{2b}{a} \times \frac{\pi a^2}{2} = \pi ab</math>
ãšãªãã
=====äœç©=====
ããç«äœ<math>V_0</math>ã®<math>x = t</math>ã«ãããæé¢ç©ãæéãªå€ã§ããã®å€ã <math>t</math>ã®é¢æ°<math>S(t)</math>ãšãªããšãããã®ç«äœãå¹³é¢<math>x = a</math>ïŒ<math>x = b</math>ïŒãã ãã<math>a < b</math>ïŒã§åãåã£ãé åã®äœç©ã¯ãåºé¢ç©<math>S(t)</math>ã«æ¥µããŠå°ããé«ã<math>dt</math><ref>ãªãããã®æã<math>dt</math>ã<math>S(t)</math>ã«å¯ŸããŠç©ååºéã§åžžã«éçŽæ¹åã®é¢ä¿ã«ããããšãä¿èšŒãããŠããªããã°ãªããªãã</ref>ã®ç©<math>S(t) \, dt</math>ã®åºé<math>[a,b]</math>ã«ããã环ç©ã§ããã®ã§ã以äžã®åŒã§è¡šãããšãã§ããã
:<math> V = \int_a^{b} S(t) \, dt</math>
ïŒäŸ1ïŒ
:<math>O(0,0,0), A(1,0,0), B(1,1,0), C(1,0,2)</math>ã§ããäžè§éãèããã
:ãã®äžè§éãå¹³é¢<math>x=t (0\leqq t \leqq 1)</math>ã§åæãããšãæé¢ã®äžè§åœ¢ã®å座æšã¯<math>A_t(t,0,0), B_t(t,t,0), C_t(t,0,2t)</math>ãšãªãããã®æã<math>\triangle{A_t B_t C_t}</math>ã®é¢ç©<math>S(t)=t^2</math>ãšãªãã
:ããããåºé<math>[0,1]</math>ã§ç©åãããšã
:<math> V = \int_0^{1} S(t) \, dt = \int_0^{1} t^2 \, dt = \left[ \frac{t^3}{3}\right]_{0}^{1} = \frac{1}{3}</math>ãšãªã<ref>äžè§é<math>O-ABC</math>ã¯ã<math>\triangle{ABC}</math>ãåºé¢ïŒ<math>S=1</math>ïŒãšãã<math>OA</math>ãé«ãïŒ<math>1</math>ïŒãšããäžè§éãªã®ã§ãäœç©ã¯ã<math>\frac{1}{3}</math>ãšãªããæ£ããã</ref>ã
ïŒäŸ2ïŒ
:èšå
:#<math>O(0,0,0), A(1,0,0), B(0,1,0), C(1,1,0), D(0,0,1), E(1,0,1), F(0,1,1), G(1,1,1)</math>ã§ããç«æ¹äœãæ³å®ã
:#å¹³é¢<math>x=t (0\leqq t \leqq 1)</math>ã§åæãã<math>\square{O_t A_t B_t C_t}</math>ãåŸãã
:#ç·å<math>O_t A_t , A_t B_t , B_t C_t , C_t O_t </math>ã«ãåã
ç¹<math>O_t, A_t, B_t, C_t</math>ãããé·ã<math>t</math>ã§ããç¹<math>H_t, I_t, J_t, K_t</math>ããšãã<math>\square{H_t I_t J_t K_t}</math>ã<math>S_t</math>ãšããã
:#<math>t</math>ãåºé<math>[0,1]</math>ã§å€åãããæã<math>S_t</math>ãééããéšåã®äœç©<math>V</math>ãæ±ããããªãã<math>S_t</math>ãæ£æ¹åœ¢ã§ãã蚌æã¯çç¥ããŠããã
:解ç
:#<math>S_t</math>ã®1蟺ã®é·ãã<math>l</math>ãšãããšã<math>l^2 = t^2 + (1-t)^2 = 2t^2 - 2t + 1</math>
:#<math>S_t</math>ã®é¢ç©<math>S(t)</math>ã¯<math>l^2</math>ã§ããããã<math>S(t) = 2t^2 - 2t + 1</math>
:#ããããåºé<math>[0,1]</math>ã§ç©åãããšã
:#<math> V = \int_0^{1} S(t) \, dt = \int_0^{1} (2t^2 - 2t + 1) \, dt = \left[ \frac{2t^3}{3} - t^2 +t \right]_{0}^{1} = \frac{2}{3}</math>ãšãªãã
====== å転äœã®äœç© ======
<math>y= f(x) (a \le x \le b )</math>
ã§äžããããæ²ç·ãx軞ã®åãã«å転ãããŠäœããã
ç«äœã®äœç©Vã¯ã
<math>
V = \int _a ^b \pi ( f(x))^2 dx
</math>
ã§äžããããã
å°åº
ç«äœãx軞ã«åçŽã§ãããx=cãæºããé¢ãšx=c+hãæºããé¢ã§åããšïŒhã¯å°ããª
å®æ°ïŒããã®åæé¢ã§æãŸããç«äœã¯ååŸ f(c)ã®åãšååŸ f(c+h)ã®å
ã§ã¯ããŸããç«äœãšãªãã
ããããhã極ããŠå°ãããšãããã®å³åœ¢ã¯ååŸf(c),é«ãhã®åæ±ã§
è¿äŒŒã§ããã
ãã£ãŠãã®2ã€ã®é¢ã«é¢ããŠãåŸãããå³åœ¢ã®äœç©ã¯
<math>
h \times \pi (f(c) )^2
</math>
ãšãªãã
ããã<math>a<c<b</math>æºããå
šãŠã®cã«ã€ããŠè¶³ãåããããšã
<math>
S = \int _a ^b \pi ( f(x))^2 dx
</math>
ãåŸãããã
äŸãã°ã
<math>
y= x^2 ~(0<x<1)
</math>
ãx軞ã®åãã«å転ãããŠåŸãããå³åœ¢ã®äœç©ã¯ã
:å³åœ¢ã®çµµ?
<math>
S = \int_0^1 \pi (x^2)^2 dx
</math>
<math>
=\pi \int_0^1 x^4 dx
</math>
<math>
=\frac {\pi} 5
</math>
ãšãªãã
;çã®äœç©
çã®äœç©<math>V=\frac{4}{3}\pi r^3</math>ã®å°åº
ååŸ''r''ã®çã¯åå<math>y=\sqrt{r^2-x^2}</math>ã''x''軞ã®åšãã«å転ãããŠã€ããããšãã§ããã
:<math>V=\pi \int_{-r}^r \sqrt{r^2-x^2}^2 dx=\pi \int_{-r}^r (r^2-x^2) dx= \frac{4}{3}\pi r^3</math>
ãŸãäœç©ã''r''ã§åŸ®åãããšçã®è¡šé¢ç©<math>S=4\pi r^2</math>ãåŸãããã
== åºåæ±ç©æ³ ==
ãããŸã§ã«åŠãã ããã«ãç©åã¯åŸ®åã®éæŒç®ã§ãããšåæã«ã座æšå¹³é¢äžã§ã®é¢ç©èšç®ã§ãããããã®é
ã§ã¯ã座æšå¹³é¢äžã®é¢ç©èšç®ã®æ¹æ³ã®äžã€ã§ããåºåæ±ç©æ³ãããã³ç©åæ³ãšã®é¢é£ã«ã€ããŠåŠã¶ã
[[File:Riemann Integration 1.png|thumb|300px|é¢ç©èšç®]]
å³å³ã®ãããªããæ²ç·<math>y=f(x)</math>ããããåçŽã®ãããããã§ã¯ã€ãã«<math>f(x)>0</math>ã§ãããã®ãšããŠèããããã®æ²ç·ãšã''x''軞ãããã³çŽç·<math>x = a, x = b (a < b)</math>ã«ãã£ãŠå²ãŸããé åã®é¢ç©''S''ãæ±ããããã®é¢ç©ã¯[[#é¢ç©]]ã®é
ã§åŠãã ããã«ã
: <math>S = \int_a^b f(x)dx</math>
ãšç©åæ³ãçšããŠèšç®ããããšãã§ãããã§ã¯ããããããå°ãåå§çãªæ¹æ³ã§è¿äŒŒçã«æ±ããããšãèããŠã¿ããã
æ²ç·ãå«ãå³åœ¢ã®é¢ç©ãæ±ããããšã¯ç°¡åã§ã¯ãªãããäŸãã°äžè§åœ¢ãé·æ¹åœ¢ãå°åœ¢ãªã©ã®çŽç·ã§å²ãŸããå³åœ¢ã®é¢ç©ãæ±ããããšã¯é£ãããªããããã§ãäžå³ã®ããã«y=f(x)ãæ£ã°ã©ãã§è¿äŒŒããé·æ¹åœ¢ã®é¢ç©ã®åãèšç®ããããšã§ãæ±ãããé¢ç©''S''ã«è¿ãå€ãæ±ããããšãã§ãããå·Šäžã®ããã«æ£ã°ã©ãã®å¹
ã倧ãããšèª€å·®ã倧ããããæ£ã°ã©ãã®å¹
ãçãããã°ããã»ã©ãããªãã¡åå²æ°ãå€ãããã»ã©ãåŸã
ã«æ±ãããé¢ç©ã®å€ã«è¿ã¥ããããšãã§ãããããã§ããã®åºé[''a'',''b'']ã''n''çåãããã®æã®é·æ¹åœ¢ã®é¢ç©ã®ç·åãæ±ãããã®åŸã§<math>n \to \infty</math>ã®æ¥µéãèããããšã«ããããã®ããã«ããŠãåºéã现ããçåå²ããé·æ¹åœ¢ã®é¢ç©ã®ç·åãæ±ããããšã«ããå³åœ¢ã®é¢ç©ãæ±ããæ¹æ³ããåºåæ±ç©æ³ãšåŒã¶ã
:[[File:Riemann Integration 4.png|350px|æ£ã°ã©ãã«ããè¿äŒŒ]][[File:Riemann Integration 5.png|350px|ããã«çŽ°ããªæ£ã°ã©ãã«ããè¿äŒŒ]]
[[File:Integral numericky obd.svg|thumb|å·ŠåŽã§è¿äŒŒ]][[File:Somme-superiori.png|thumb|å³åŽã§è¿äŒŒ]]
<math>y=f(x)</math>ãæ£ã°ã©ãã§è¿äŒŒãããšããå³å³ã®ããã«ãé·æ¹åœ¢ã®å·Šäžã®é ç¹ãæ²ç·äžã«åãæ¹æ³ãšãå³äžã®é ç¹ãæ²ç·äžã«åãæ¹æ³ããããã©ã¡ãã®æ¹æ³ã§ããåå²æ°ã倧ããããã°ãããæ±ãããé¢ç©ã«è¿ã¥ããããŸãã¯å·Šäžã®é ç¹ãæ²ç·äžã«åãæ¹æ³ã§èããããšã«ããã
ããã§ã¯é¢ç©ãæ±ãããåºéããåçŽã®ãã[0, 1]ãšãããåºé[0, 1]ã''n''çåãããšããããããã®é·æ¹åœ¢ã®å·Šç«¯ã®x座æšã¯ã
:<math>0, \frac{1}{n}, \frac{2}{n}, \cdots, \frac{n-1}{n}</math>
ãšãªããããã§ãäžè¬ã«ç¬¬''k''çªç®ã®é·æ¹åœ¢ã«ã€ããŠèããããšã«ããããã ãããã¡ã°ãå·ŠåŽã®é·æ¹åœ¢ã第0çªç®ãšãããã¡ã°ãå³åŽã®é·æ¹åœ¢ã第''n''-1çªç®ãšããã第''k''çªç®ã®é·æ¹åœ¢ã®å·Šç«¯ã®x座æšã¯<math>\frac{k}{n}</math>ã§ããããããã®é·æ¹åœ¢ã®é«ãã¯<math>f\left(\frac{k}{n}\right)</math>ãšãªãããŸãé·æ¹åœ¢ã®å¹
ã¯<math>\frac{1}{n}</math>ã§ããããã®ããããã®é·æ¹åœ¢ã®é¢ç©<math>s_k</math>ã¯ã
:<math>s_k = \frac{1}{n}f\left(\frac{k}{n}\right)</math>
ãšãªãããããã£ãŠããããã®é·æ¹åœ¢ã®é¢ç©ã®ç·å<math>S_n</math>ã¯ã
:<math>S_n = \sum_{k = 0}^{n-1} s_k = \frac{1}{n}\sum_{k = 0}^{n-1} f\left(\frac{k}{n}\right)</math>
ãã®<math>S_n</math>ã¯ãåºé[0, 1]ã''n''çåããæã®é·æ¹åœ¢ã®é¢ç©ã®ç·åã§ãããã''n''ã倧ããããã°ããã»ã©ã次第ã«ããšã®é¢ç©ã«è¿ã¥ããŠããããããã£ãŠã<math>n\to\infty</math>ã®æ¥µéãèãã
:<math>S = \lim_{n\to\infty} S_n = \lim_{n\to\infty} \frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)</math>
ãšãªãããã®ããã«ããŠãæ±ãããé¢ç©ãèšç®ããããšãã§ãããããã«ãããã§ãã®åºéã®é¢ç©ãç©åæ³ã«ããèšç®ã§ããããšããã
:<math>\lim_{n\to\infty} \frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right) = \int_0^1f(x)dx</math>
ãæãç«ã€ããŸããé·æ¹åœ¢ã®å³äžã®é ç¹ãæ²ç·äžã«åãå Žåã¯ãåæ§ã«ããŠ
:<math>S = \lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^{n} f\left(\frac{k}{n}\right) = \int_0^1f(x)dx</math>
ãšãªãã
== æŒç¿åé¡ ==
* [[é«çåŠæ ¡æ°åŠIII ç©åæ³/æŒç¿åé¡|äžå®ç©å44é¡]]
* [[/æŒç¿åé¡]]
== è泚 ==
<references/>
{{DEFAULTSORT:ãããšããã€ããããããIII ãããµãã»ã}}
[[Category:é«çåŠæ ¡æ°åŠIII|ãããµãã»ã]]
[[ã«ããŽãª:ç©åæ³]] | 2005-05-08T05:07:14Z | 2024-03-20T20:55:26Z | [
"ãã³ãã¬ãŒã:Pathnav"
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E6%95%B0%E5%AD%A6III/%E7%A9%8D%E5%88%86%E6%B3%95 |
1,936 | ç©çåŠ | ç©çåŠã«é¢ããææžã»è³æã»æç§æžãåããããæžåº«ã§ããåé²å
容ã¯ä»¥äžãã芧ãã ããã
(note:1åäœã¯35æéã®åŠç¿ã«ãã£ãŠä¿®äºã§ããé
ç®ãè¡šãããŠããŸããäŸãã°ãå€å
žååŠãä¿®åŸããã«ã¯70æéã®åŠç¿ãè¡ãªãããšãæ±ããããŠããŸãã) | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ç©çåŠã«é¢ããææžã»è³æã»æç§æžãåããããæžåº«ã§ããåé²å
容ã¯ä»¥äžãã芧ãã ããã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "",
"title": ""
},
{
"paragraph_id": 2,
"tag": "p",
"text": "(note:1åäœã¯35æéã®åŠç¿ã«ãã£ãŠä¿®äºã§ããé
ç®ãè¡šãããŠããŸããäŸãã°ãå€å
žååŠãä¿®åŸããã«ã¯70æéã®åŠç¿ãè¡ãªãããšãæ±ããããŠããŸãã)",
"title": "åçæè²çšæç§æž"
}
] | ç©çåŠã«é¢ããææžã»è³æã»æç§æžãåããããæžåº«ã§ããåé²å
容ã¯ä»¥äžãã芧ãã ããã | {{Pathnav|ã¡ã€ã³ããŒãž|èªç¶ç§åŠ|frame=1|small=1}}
{| style="float:right"
|-
|{{Wikipedia|ç©çåŠ|ç©çåŠ}}
|-
|{{Wikiversity|School:ç©çåŠ|ç©çåŠ}}
|-
|{{Wikiquote|Category:ç©çåŠè
|ç©çåŠè
}}
|-
|{{Wiktionary|Category:ç©çåŠ|ç©çåŠ}}
|-
|{{Commons|Category:Physics}}
|-
|{{èµæžäžèŠ§}}
|-
|{{é²æç¶æ³}}
|}
[[w:ç©çåŠ|ç©çåŠ]]ã«é¢ããææžã»è³æã»æç§æžãåããããæžåº«ã§ããåé²å
容ã¯ä»¥äžãã芧ãã ããã
== åçæè²çšæç§æž ==
ïŒnote:1åäœã¯35æéã®åŠç¿ã«ãã£ãŠä¿®äºã§ããé
ç®ãè¡šãããŠããŸããäŸãã°ã[[å€å
žååŠ]]ãä¿®åŸããã«ã¯70æéã®åŠç¿ãè¡ãªãããšãæ±ããããŠããŸããïŒ
* [[å°åŠæ ¡çç§]] ?åäœïŒç¹ã«ç©çãååŠçã®åºåããç¡ããããïŒ{{é²æ|25%|2015-03-13}}
* [[äžåŠæ ¡çç§]] ?åäœïŒç¹ã«ç©çãååŠçã®åºåããç¡ããããïŒ{{é²æ|25%|2015-03-13}}
* [[é«çåŠæ ¡ç©ç]] 6åäœ {{é²æ|25%|2015-03-13}}
* [[倧åŠåéšç©ç]] ?åäœ {{é²æ|00%|2015-03-13}}
=== ä»é² ===
* [[åçç©çåŠå
¬åŒé]] {{é²æ|25%|2015-03-13}}
== äžè¬æç§æž ==
*[[ç©çåŠå
¥é]]
*[[ç©çåŠæŠèª¬]]
=== äžè¬æé€èª²ç® ===
* [[ç©çæ°åŠI]] 5åäœ{{é²æ|100%|2023-11-05}}
: ç·åœ¢ä»£æ°ãšè§£æãæ¢ã«åŠãã 人ã«ãšã£ãŠã¯2åäœã§ãã
* [[å€å
žååŠ]] 2åäœ{{é²æ|50%|2023-11-05}}
* [[é»ç£æ°åŠ]] 2åäœ{{é²æ|50%|2023-11-05}}
* [[ç±ååŠ]] 2åäœ{{é²æ|25%|2023-11-05}}
* [[æ¯åãšæ³¢å]] 2åäœ{{é²æ|50%|2023-11-05}}
* [[ç¹æ®çžå¯Ÿè«]] 2åäœ{{é²æ|75%|2023-11-05}}
* [[éåååŠ]] 2åäœ{{é²æ|50%|2023-11-05}}
**[[çžå¯Ÿè«çéåååŠ]]
**[[å Žã®éåè«]]{{é²æ|75%|2023-11-05}}
=== å°éç§ç® ===
* [[解æååŠ]] 2åäœ {{é²æ|25%|2015-03-13}}
* [[é»ç£æ°åŠII]] 2åäœ {{é²æ|25%|2015-03-13}}
* [[éåååŠII]] 2åäœ
* [[ç©çæ°åŠII]] 2åäœ
* [[çµ±èšååŠI]] 2åäœ
* [[çµ±èšååŠII]] 2åäœ
**[[éåçµ±èšååŠ]]
**[[é平衡統èšååŠ]]
* [[é³é¿åŠ]]{{é²æ|25%|2023-11-05}}
* [[äžè¬ååŠ]]
* [[/æµäœååŠ/]]{{é²æ|00%|2023-11-05}}
* [[/é£ç¶äœã®ååŠ/]]{{é²æ|00%|2023-11-05}}
* [[/äžè¬çžå¯Ÿæ§çè« å
¥é/]]
<!--
*[[ååç©çåŠ]]
**[[ååç©çåŠ]]
**[[é«ååç©çåŠ]]
* [[ç©æ§ç©çåŠ]]
**[[åºäœç©çåŠ]]
**[[ç£æ§ç©çåŠ]]
**[[éå±ç©çåŠ]]
**[[åå°äœç©çåŠ]]
**[[äœæž©ç©çåŠ]]
**[[è¡šé¢ç©çåŠ]]
**[[éç·åœ¢ç©çåŠ]]
* [[ãã©ãºãç©çåŠ]]
**[[é»ç£æµäœååŠ]]
-->
=== 倧åŠé¢ç§ç®===
* [[å Žã®éåè«]] 2åäœ
* [[äžè¬çžå¯Ÿæ§çè«]] 2åäœ
* [[è¶
察称æ§]] ?åäœ
<!--:[http://arxiv.org/abs/hep-ph/9709356 ã¬ãã¥ãŒè«æ:Supersymmetry Primerãžã®ãªã³ã¯]-->
* [[匊çè«]] ?åäœ
<!--:[http://arxiv.org/abs/hep-th/9709062 ã¬ãã¥ãŒè«æ:Introduction to Superstring Theoryãžã®ãªã³ã¯]-->
* [[ç©çåŠã®ããã®èšç®æ©ãšãªãŒãã³ãœãŒã¹]] 1åäœ
=== æªåé¡ ===
* [[å
åŠ]]
** [[å
ã®å極]]
* [[ãã³ããŒã¬ã³ã¹ã®æ¬]]
=== é¢é£åé ===
*[[æ°åŠ]]
**[[æ°å€è§£æ]]
*[[倩æåŠ]]
*[[ååŠ]]
*[[çç©åŠ]]
*[[å·¥åŠ]]
*[[å°çç§åŠ]]
*[[å»åŠ]]
*[[å²åŠ]]
*[[å¿çåŠ]]
*[[çµæžåŠ]]
{{NDC|420|*}}
[[Category:èªç¶ç§åŠ|ãµã€ããã]]
[[Category:ç©çåŠ|! ãµã€ããã]]
[[Category:ç©çåŠæè²|! ãµã€ããã]]
[[Category:æžåº«|ãµã€ããã]] | 2005-05-08T05:54:33Z | 2024-03-17T09:11:12Z | [
"ãã³ãã¬ãŒã:NDC",
"ãã³ãã¬ãŒã:Pathnav",
"ãã³ãã¬ãŒã:Wikiversity",
"ãã³ãã¬ãŒã:Wikiquote",
"ãã³ãã¬ãŒã:é²æç¶æ³",
"ãã³ãã¬ãŒã:é²æ",
"ãã³ãã¬ãŒã:Wikipedia",
"ãã³ãã¬ãŒã:Wiktionary",
"ãã³ãã¬ãŒã:Commons",
"ãã³ãã¬ãŒã:èµæžäžèŠ§"
] | https://ja.wikibooks.org/wiki/%E7%89%A9%E7%90%86%E5%AD%A6 |
1,937 | é«çåŠæ ¡ç©ç | äžè¬ã®æç§æžãšã¯å
容ãè¥å¹²éãéšåããããŸãããå匷ã®åèã«ãªãã°å¹žãã§ãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "äžè¬ã®æç§æžãšã¯å
容ãè¥å¹²éãéšåããããŸãããå匷ã®åèã«ãªãã°å¹žãã§ãã",
"title": ""
}
] | äžè¬ã®æç§æžãšã¯å
容ãè¥å¹²éãéšåããããŸãããå匷ã®åèã«ãªãã°å¹žãã§ãã | [[å°åŠæ ¡ã»äžåŠæ ¡ã»é«çåŠæ ¡ã®åŠç¿]]>[[é«çåŠæ ¡ã®åŠç¿]]>[[é«çåŠæ ¡çç§]]>é«çåŠæ ¡ç©ç
{{é²æç¶æ³}}
äžè¬ã®æç§æžãšã¯å
容ãè¥å¹²éãéšåããããŸãããå匷ã®åèã«ãªãã°å¹žãã§ãã
== çŸèª²çšïŒ2012幎床以éå
¥åŠè
çšïŒ ==
*[[é«çåŠæ ¡çç§ ç©çåºç€|ç©çåºç€]]
*[[é«çåŠæ ¡ ç©ç|ç©ç]]
== åè ==
=== ååŒ·æ³ ===
* [[åŠç¿æ¹æ³/é«æ ¡ç©ç]] {{é²æ|50%|2015-12-06}}
=== é¢é£ããã¹ã ===
* [[åçç©çåŠå
¬åŒé]] {{é²æ|50%|2015-12-06}}
* [[倧åŠåéšç©ç]] {{é²æ|00%|2015-12-06}}
== æ§èª²çšïŒ2003幎床ïœ2011幎床å
¥åŠè
çšïŒ ==
* [[é«çåŠæ ¡ç©ç/ç©çI|ç©çI]] {{é²æ|50%|2015-07-24}} 3åäœ
* [[é«çåŠæ ¡ç©ç/ç©çII|ç©çII]] {{é²æ|50%|2017-08-09}} 3åäœ
* [[é«çåŠæ ¡çç§åºç€|çç§åºç€]]{{é²æ|25%|2013-09-16}}
* [[é«çåŠæ ¡çç§ç·åA|çç§ç·åA]]{{é²æ|25%|2013-09-16}}
[[Category:é«çåŠæ ¡æè²|ãµã€ã]]
[[Category:çç§æè²|é«ãµã€ã]]
[[Category:ç©çåŠæè²|é«ãµã€ã]]
[[category:é«æ ¡çç§|ãµã€ã]] | null | 2022-09-17T16:59:31Z | [
"ãã³ãã¬ãŒã:é²æç¶æ³",
"ãã³ãã¬ãŒã:é²æ"
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E7%89%A9%E7%90%86 |
1,938 | é«çåŠæ ¡ç©ç/ç©çI | æ¬é
ã¯é«çåŠæ ¡çç§ã®ç§ç®ã§ãããç©ç Iãã®è§£èª¬ã§ããã
ç©ç I | [
{
"paragraph_id": 0,
"tag": "p",
"text": "æ¬é
ã¯é«çåŠæ ¡çç§ã®ç§ç®ã§ãããç©ç Iãã®è§£èª¬ã§ããã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ç©ç I",
"title": "ç®æ¬¡"
}
] | æ¬é
ã¯é«çåŠæ ¡çç§ã®ç§ç®ã§ãããç©ç Iãã®è§£èª¬ã§ããã | :* [[é«çåŠæ ¡ç©ç]] > ç©çI<br>
:* ç©çIIã®æç§æžãžã®ãªã³ã¯ â [[é«çåŠæ ¡ç©ç/ç©çII|é«çåŠæ ¡çç§ ç©çII]]<br />
----
æ¬é
ã¯é«çåŠæ ¡çç§ã®ç§ç®ã§ãããç©ç Iãã®è§£èª¬ã§ããã
== ç®æ¬¡ ==
{{é²æç¶æ³}}
ç©ç I
* [[é«çåŠæ ¡ç©ç/ç©çI/éåãšãšãã«ã®ãŒ|éåãšãšãã«ã®ãŒ]] {{é²æ|50%|2015-06-27}}
:: [[é«çåŠæ ¡ç©ç/ç©çI/éåãšãšãã«ã®ãŒ/ç©äœã®éå|éåãšãšãã«ã®ãŒ/ç©äœã®éå]] {{é²æ|75%|2015-07-10}}
:: [[é«çåŠæ ¡ç©ç/ç©çI/éåãšãšãã«ã®ãŒ/éåã®æ³å|éåãšãšãã«ã®ãŒ/éåã®æ³å]] {{é²æ|50%|2015-07-10}}
:: [[é«çåŠæ ¡ç©ç/ç©çI/éåãšãšãã«ã®ãŒ/ä»äºãšãšãã«ã®ãŒ|éåãšãšãã«ã®ãŒ/ä»äºãšãšãã«ã®ãŒ]] {{é²æ|50%|2015-07-18}}
* [[é«çåŠæ ¡ç©ç/ç©çI/æ³¢|æ³¢]] {{é²æ|25%|2015-07-24}}
:: [[é«çåŠæ ¡ç©ç/ç©çI/æ³¢/æ³¢ã®æ§è³ª|æ³¢/æ³¢ã®æ§è³ª]] {{é²æ|50%|2015-07-24}}
:: [[é«çåŠæ ¡ç©ç/ç©çI/æ³¢/é³æ³¢ãšæ¯å|æ³¢/é³æ³¢ãšæ¯å]] {{é²æ|50%|2016-01-23}}
:: [[é«çåŠæ ¡ç©ç/ç©çI/æ³¢/å
æ³¢|æ³¢/å
æ³¢]] {{é²æ|00%|2015-07-24}}
* [[é«çåŠæ ¡ç©ç/ç©çI/ç±|ç±]] {{é²æ|25%|2015-07-10}}
::
::
* [[é«çåŠæ ¡ç©ç/ç©çI/é»æ°|é»æ°]] {{é²æ|50%|2016-01-23}}
:ïŒé¢é£ç§ç®ïŒ [[é«çåŠæ ¡ååŠI/é»æ± ãšé»æ°å解]] {{é²æ|75%|2015-12-25}}
== é¢é£ç§ç® ==
* [[é«çåŠæ ¡æ°åŠII/ãããããªé¢æ°]] ïŒäžè§é¢æ°ãææ°é¢æ°ãªã©ïŒ
* [[é«çåŠæ ¡æ°åŠII/埮åã»ç©åã®èã]]
* [[é«çåŠæ ¡æ°åŠII/埮åã»ç©åã®èã]]
[[Category:é«çåŠæ ¡æè²|ç©ãµã€ã1]]
[[Category:ç©çåŠ|é«ãµã€ã1]]
[[Category:ç©çåŠæè²|é«ãµã€ã1]] | null | 2017-06-24T20:07:04Z | [
"ãã³ãã¬ãŒã:é²æç¶æ³",
"ãã³ãã¬ãŒã:é²æ"
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E7%89%A9%E7%90%86/%E7%89%A9%E7%90%86I |
1,939 | é«çåŠæ ¡ç©ç/é»æ° | é«çåŠæ ¡çç§ ç©çI > é»æ°
æ¬é
ã¯é«çåŠæ ¡çç§ ç©çIã®é»æ°ã®è§£èª¬ã§ããã
çŸåšç§ãã¡ã䜿ã£ãŠããå€ãã®è£œåãé»æ°ãçšããŠåããŠããã ããã«ã¯æ§ã
ãªçç±ãèãããããããŸã第äžã«é»æ°ã¯æ§ã
ãªå¥ã®ãšãã«ã®ãŒã«å€æã§ããããšãäŸãã°é»ç±ç·ã䜿ã£ãŠç±ã«ãé»çãçºå
ãã€ãªãŒãã䜿ã£ãŠå
ã«ãã¢ãŒã¿ãŒã䜿ã£ãŠéåã«å€æããããšãåºæ¥ãã次ã«ãé»æ± ãã³ã³ãã³ãµã䜿ã£ãŠãšãã«ã®ãŒãç¶æãããŸãŸæã¡éã¶ããšãåºæ¥ãããšããé»ç·ã䜿ã£ãŠé·è·é¢ãéé»ã§ããããšããŸããé»å補åã®èšç®èœåãä¿¡å·ã®äŒéèœåãåªããŠããããšããŸãæ¯èŒçã«å®å
šã«å°éã®ãšãã«ã®ãŒãåãåºããããšãçãèããããã
é»æ°ãéåã«å€ãããã®ãšã㊠ã¢ãŒã¿ãŒãããã ããã®éã®ãã®ãšããŠã éåãé»æ°ã«å€ããããšãåºæ¥ãã ãããè¡ãªãã®ã¯çºé»æ©ãšåŒã°ããã äŸãã°ãçºé»æã¯äœããã®éåã®ãšãã«ã®ãŒã å©çšããŠé»æ°ãããããŠããã äŸãã°ãæ°Žåçºé»æã§ã¯ã æ°Žã®èœäžããåãå©çšããŠããã 倧éã®æ°Žãèœäžãããšãã«ã¯ 人éãäœå人ãããã£ãŠãããã åºæ¥ãããšããªãããããšãããã ããšãã°ãåãç«ã£ã海岞ç·ãªã©ã¯ äž»ã«æ°Žã®æµãã«ãã£ãŠäœãããŠããã ãã®ããã«ãæ°Žã®åã¯åŒ·å€§ã§ããã®ã§ã ãããäžæãå©çšããæ¹æ³ããããš éœåããããå®éçŸä»£ã§ã¯ é»æ°ãåªä»ãšããŠããã®åã åãã ãããšã«æåããŠããã
é»æ± ã®ããã«é»æ¥µã®+ãš-ãå®ãŸã£ã é»æµãçŽæµé»æµãšåŒã¶ã äžæ¹ãçºé»æããåŸãããé»æµã®ããã« +ãš-ãéãé床ã§å
¥ãæãã é»æµã亀æµé»æµãšåŒã¶ã
å®éã«ã¯ãã€ãªãŒããçšã㊠亀æµãçŽæµã«å€ã㊠䜿ãããšãããè¡ãªãããã
äœããªã空éãå
ãçŽé²ããŠããããã« èŠããããšããããå®éã«ã¯ãã㯠é»æ³¢ãšåããã®ã§ããã é»æ³¢ãšã¯äŸãã°ãæºåž¯é»è©±ã®éä¿¡ã«äœ¿ããããã®ã§ããã é»è·ãæã£ãç©äœãåãããšãå¿
ç¶çã« çãããã®ã§ããã
ãã©ã¹ããã¯ã®äžæ·ããªã©ã§é«ªãããããšåž¯é»ããçŸè±¡ãªã©ã®ããã«ãç©è³ªãé»æ°ã垯ã³ãããšã垯é»(ããã§ã)ãšãããç©äœãããã£ãŠçºçãããéé»æ°ãæ©æŠé»æ°ãšããã ã¬ã©ã¹æ£ãçµ¹ã®åžã§ããããšãã¬ã©ã¹æ£ã¯æ£ã®é»æ°ã«åž¯é»ããçµ¹ã¯è² ã®é»æ°ã«åž¯é»ããã é»æ°ã®éãé»è·(ã§ãããcharge)ãšããããããã¯é»æ°éãšããã
é»è·ã®åäœã¯ã¯ãŒãã³ã§ãããã¯ãŒãã³ã®èšå·ã¯Cã§ããã
éé»æ°ã«ããé»è·ã©ããã«åãåãéé»æ°åãšããã
ãªãã垯é»ããŠããªãç¶æ
ãé»æ°çã«äžæ§ã§ããããšããã
éå±ã®ããã«ãé»æ°ãéããç©äœãå°äœ(ã©ããããconductor)ãšããããã©ã¹ããã¯ãã¬ã©ã¹ããŽã ã®ããã«é»æ°ãéããªãç©è³ªã絶çžäœ(ãã€ãããããinsulator)ãããã¯äžå°äœ(ãµã©ããã)ãšããã
éå±ã¯å°äœã§ããã
é»æ°ã®æ£äœã¯é»å(electron)ãšããç²åã§ããããã®é»åã¯è² é»è·ã垯ã³ãŠããã(é»åã®é»è·ãè² ã«å®çŸ©ãããŠããã®ã¯ã人é¡ãé»åãçºèŠããåã«é»è·ã®æ£è² ã®å®çŸ©ãè¡ãããããšããé»åãèŠã€ãã£ãéã«é»åã®é»è·ã調ã¹ããè² é»è·ã ã£ãããã§ããã)
éå±ãå°äœãªã®ã¯ãéå±äžã®é»åã¯ãããšã®ååãé¢ããŠããã®éå±å
šäœã®äžãèªç±ã«åããããã§ãããéå±äžã®é»åã®ããã«ãç©è³ªäžãèªç±ã«åããç¶æ
ã®é»åããèªç±é»å(ãããã§ãã)ãšããã
é»æµãšã¯ãèªç±é»åã移åããããšã§ããã
ãã£ãœãã絶çžäœã¯ãèªç±é»åããããªãã絶çžäœã®é»åã¯ããã¹ãŠãããšã®ååã«æçž(ããã°ã)ãããŠéã蟌ããããŠããŠãèªç±ã«ã¯åããªãã
æ£é»è·ãšã¯ãç©è³ªã«é»åãæ¬ ä¹ããŠããç¶æ
ã§ããã è² é»è·ãšã¯ãç©è³ªãé»åãå€ãæã£ãŠããç¶æ
ã§ããã
垯é»ããŠããªã絶çžäœã®ç©è³ªããããããããŠãäž¡æ¹ãæ©æŠé»æ°ã«åž¯é»ãããå Žåãçæ¹ã¯æ£é»è·ãçããããçæ¹ã®ç©è³ªã¯è² é»è·ãçããããã®ãšããçºçããæ£é»è·ã®å€§ãããšè² é»è·ã®å€§ããã¯åãã§ããã ããã¯ãé»åã移åããŠãçæ¹ã®ç©è³ªã¯é»åãäžè¶³ããããçæ¹ã¯çéã®é»åãéå°ã«ãªã£ãŠããããã§ããã
ãã®ããã«ãé»åã¯çæãæ¶æ»
ãããªãããããé»è·ä¿åã®æ³åãšèšãããããã¯é»æ°éä¿åã®æ³åãšèšãã
é»æ°çã«äžæ§ã§ãã£ãå°äœã®ç©è³ª(ä»®ã«ç©è³ªAãšãã)ã«åž¯é»ããå¥ã®ç©è³ª(ä»®ã«ç©è³ªBãšãã)ãæ¥è§Šãããã«è¿ã¥ãããšãç©è³ªAã«ã¯ã垯é»ç©è³ªBã®é»è·ã«åŒãå¯ããããŠãç©äœAã®å
éšã§å察笊å·ã®é»è·ã垯é»ç©äœBã«è¿ãåŽã®è¡šé¢ã«çããããŸãã垯é»ç©äœBãšåãé»è·ã¯åçºããã®ã§ãç©äœAå
éšã®åž¯é»ç©äœBãšã¯é ãåŽã®è¡šé¢ã«çããã
ãã®ãããªçŸè±¡ãéé»èªå°(ããã§ãããã©ã;Electrostatic induction)ãšãããéé»èªå°ã§çããé»è·ã®æ£é»è·ã®éãšè² é»è·ã®éã¯çéã§ããã(é»æ°éä¿åã®æ³å)
å°äœã®å
éšã«éé»æ°åã¯ç¡ãããããã£ããšãããšãèªç±é»åãªã©ã®é»è·ãåããé»æµãæµãç¶ããããšã«ãªããããã®ãããªçŸè±¡ã¯å®åšããªãã®ã§äžåçã«ãªãããããã£ãŠãå°äœã®å
éšã«éé»æ°åã¯ç¡ãã
è¡šé¢ã«é»è·ãéãŸãã®ã¯ãå°äœã®å
éšã«éé»æ°åãäœãããªãããã§ããããããã£ãŠéé»èªå°ã§åŒãå¯ããããé»è·ã®å€§ããã¯ãå€éšããå°äœå
éšãžã®éé»æ°åãæã¡æ¶ãã ãã®å€§ããã§ããã
ãã®å°äœå
éšã®é»è·ããŒãã«ãªãæ§è³ªãå¿çšãããšãäžç©ºã®å°äœã§åºæ¥ãç©äœãçšããŠãéé»æ°åãé®èœããããšãã§ããããããéé»é®èœ(ããã§ããããžããelectric shilding)ãšããã
絶çžäœ(ä»®ã«Aãšãã)ã«é»è·ãè¿ã¥ããå Žåã¯ãå°äœãšã¯éããç©äœAã®å
éšã®é»åã¯èªç±ã«è¡šé¢ã«ã¯éãŸããªãããç©äœå
éšã®ååã®æ£è² ã®é»è·ã®æ¥µæ§ãæã£ãéšåããå€éšã®éé»æ°åã«åŒãå¯ããããããã«ãè¿ã¥ããé»è·ã«è¿ãåŽã«ã¯ç°çš®ã®é»è·ãçããé ãåŽã«ã¯ãåçš®ã®é»è·ãçããã ååãååãå€éšã®éé»æ°åã«ãã£ãŠãæ£è² ã®é»è·ã®éšåãçããããšãå極(ã¶ãããã)ãšããããå€éšã®é»åã«ãã£ãŠèµ·ããããã®ãããªå極ã®ããããèªé»å極(ããã§ãã¶ãããããdielectric polarization)ãšããã
絶çžäœã¯ãéé»æ°åã«ããããããšèªé»å極ãè¡ãã®ã§ã絶çžäœã®ããšãèªé»äœ(ããã§ããããdielectric)ãšãããã
å°äœã«éé»èªå°ãããæ£è² ã®é»è·ã¯ãå°äœãåæãªã©ãããã°æ£é»è·ãšè² é»è·ãå¥åã«åãåºãããšãã§ããããããèªé»äœã®æ£è² ã®é»è·ã¯ãååãååãšå¯æ¥ã«çµã³ã€ããŠãããããæ£è² ã®é»è·ãåãããŠåãåºãããšã¯åºæ¥ãªãã
ããç©è³ªãé»æ°ã垯ã³ãŠãã(垯é»ããŠãã)ãšãããã®åž¯é»ã®å€§å°ã®çšåºŠãé»è·(ã§ãããelectric charge)ãšãããããŸããŸãªç©è³ªããããããªæ¹æ³ã§åž¯é»ãããçµæãé»è·ã«ã¯ã垯é»ãã2åã®ãã®ã©ãããè¿ã¥ããæã«åŒã£åŒµãåããã®(åŒåãåã)ãšåçºããããã®(æ¥åãã¯ããã)ã®2çš®é¡ãããããšãåãã£ãã ãã®ãããªã垯é»ããŠããç©äœã«åãåãéé»æ°åãšããã
ã¹ã€ã®åž¯é»ãããã®ããä»ã«ãããã€ãçšæããŠãè¿ã¥ããŠå®éšãã2åã®ç©äœã®çµã¿åãããå€ãããšãçµã¿åããã«ãã£ãŠã2åã®ç©äœã©ããã«åŒåãåãå Žåãããã°ãæ¥åãåãå Žåãããããšãåãã£ãããã®åŒåãšæ¥åã®é¢ä¿ã¯ã垯é»ããé»è·ã®çš®é¡ã«å¿ããããšãããã£ãã
çµè«ãèšããšãé»è·ã«ã¯æ£è² ã®2çš®é¡ããããæ£ã®é»è·ã©ããã®ç©äœãè¿ã¥ãããšãã¯åçºããããè² ã®é»è·ã©ãããè¿ã¥ãããšããåçºããããæ£ãšè² ã®é»è·ãè¿ã¥ããæã«ã¯åŒåãåãã
ã€ãŸããå笊å·ã®é»è·ãè¿ã¥ããå Žåã¯ãåçºåãçãããç°ç¬Šå·ã®é»è·ãè¿ã¥ããå Žåã¯ãåŒåãçããã
éé»æ°ã©ããã®åã®åŒ·ãã¯ãå®éšçã«ã¯ãé»è·ã®éã«åãåã¯ãéåã®å Žåãšåæ§ã«åãåãŒãåã2ç©äœã®éã®è·é¢ã®2ä¹ã«åæ¯äŸããããšãç¥ãããŠãããæŽã«ãé»è·ã®å€§ããã倧ããã»ã©é»è·éã«åãåã倧ããããšãèæ
®ãããšãè·é¢rã ãé¢ããŠãããããé»è· q 1 {\displaystyle q_{1}} ã q 2 {\displaystyle q_{2}} ãæã£ãŠãã2ç©äœã®éã«åãåFã¯ã
ã§äžããããããããã¯ãŒãã³ã®æ³å( Coulomb's law)ãšãããããã§ã k {\displaystyle k} ã¯æ¯äŸå®æ°ã§ãããäž¡é»è·ã®åšå²ã«ããç©äœã®çš®é¡ã«ããå€åããå®æ°ã§ãããç空äžã§ã®é»å Žãèããå Žåã®kã®å€ã¯ã
ã§ããããŸãã ε {\displaystyle \epsilon } ã¯åŸã»ã©ç»å Žããèªé»ç(ããã§ããã€)ãšåŒã°ããç©çå®æ°ã§ãããèªé»çã¯ãäž¡é»è·ã®åšå²ã«ããç©äœã®çš®é¡ã«ããå€åããå®æ°ã§ãããèªé»çã«ã€ããŠã¯ããã®æãåããŠèªãã 段éã§ã¯ããŸã ç¥ããªããŠãè¯ããã®ã¡ã«ç©çIIã§èªé»çã詳ãã解説ããã
èªé»ç ε {\displaystyle \epsilon } ãšã¯ãŒãã³ã®æ¯äŸå®æ°kã«ã¯äžåŒã®é¢ä¿
ãããã
ç©äœã®ãŸããã«èç©ããããã®ãé»è·ãšåŒã¶ãé»æ°åã«ãã£ãŠåçºããã£ãããåŒãã€ããã£ããããç©äœãé»è·ãæã€ç©äœãšåŒã¶ããŸããããã§èŠ³å¯ãããéé»æ°åããã¯ãŒãã³åãšåŒã¶ããšãããã 2åã®é»è·ã©ããããããŒãåã¯åãã§ããããããã£ãŠäœçšã»åäœçšã®æ³åã«åŸã£ãŠããã
ããã§ãé»è·ã®åäœã¯[C]ã§äžãããããèšå·ã®Cã¯ãã¯ãŒãã³ããšèªãã
å³ã®ããã«ã2æ¬ã®ç³žã«ãããããåã質ém[kg]ã§ãåã笊å·ãšå€§ããã®é»è·q[C]ã®çããã¶ãããã£ãŠããããã¯ãã¯ãŒãã³åã§åçºããã®ã§ãå³ã®ããã«ã糞ãè§åºŠÎžããªãã
ãã®ãšãã質émã«ããéåãšãé»è·qã«ããã¯ãŒãã³åãšã®é¢ä¿ã«ã€ããŠãåŒãç«ãŠãããªããå¿
èŠãªãã°ã糞ã®åŒµåã¯T[N]ãšããããšã
解æ³
å³ã®ãããªäœçœ®é¢ä¿ã«ãªãã®ã§ãå³ã®ããã«åŒãç«ãŠãã°ããã
â» äžèšã®2æ¬ã®ç³žã«ã¶ãããã£ãçã®ã¯ãŒãã³åã®äŸé¡ã¯ãé»æ°ç£æ°åŠã®ã©ã®å
¥éæžã«ããããããªå
žåçãªåé¡ã§ããã®ã§ãèªè
ã¯ãã¡ããšç解ããããšã
é»è· q 1 {\displaystyle q_{1}} , q 2 {\displaystyle q_{2}} ã®éã®è·é¢ãrã®å Žåãš2rã®å Žåã§ã¯ãéã«åãåã®å€§ããã¯ã©ã¡ããã©ãã ã倧ãããçããã ãŸããè·é¢ã2rã®æã®2ç¹éã®åã®å€§ãããçããã
ã¯ãŒãã³åã¯ãç©äœéã®è·é¢ã®é2ä¹ã«æ¯äŸããã®ã§ãè·é¢ã2rã®æã¯ãrã®æã®å€§ããã® 1 4 {\displaystyle {\frac {1}{4}}} ãšãªãããŸããåãåã®å€§ããã¯ãã¯ãŒãã³åã®åŒãçšããŠã
ãšãªãã
æ¢ã«ãããé»è·Aã®ãŸããã®å¥ã®é»è·Bã«ã¯ããã®é»è·ããã®è·é¢ã®é2ä¹ã«æ¯äŸããåããããããšãè¿°ã¹ãã
ããã§ãé»è·Bãåããåã¯ããã®é»è·Bã®å€§ããã«æ¯äŸããããšãåãããŠèãããšããã®é»è·Bã®å€§ããã«ããããããé»è·Aã®å€§ããã ãã§æ±ºãŸãéãå°å
¥ããŠãããšéœåããããããã§ããã®ãããªéãšããŠé»å Ž(ã§ãã°)ãå°å
¥ããããã®ãšããé»å Ž E â {\displaystyle {\vec {E}}} ã®äžã«ããé»è· q {\displaystyle q} ã«åãå F â {\displaystyle {\vec {F}}} ã¯ã
ã§äžãããããé»å Žã¯åäœé»è·ã«åãåãšèããããšãã§ããé»å Žã®åäœã¯[N/C]ã§ããããé»å Žãã¯ããé»çã(ã§ããã)ãšãåŒã°ããã
(æ¥æ¬ã®ç©çåŠã§ã¯ãé»å ŽããšåŒã¶ããšãå€ãããŸããæ¥æ¬ã®é»æ°å·¥åŠã§ã¯ãé»çããšåŒã°ããããšãå€ããææ²»æã®ç¿»èš³ã®éã®ãæ¥æ¬åœå
ã®æ¥çããšã®éãã«éããããããªãæ¥æ¬ããŒã«ã«ãªéœåã§ãããåŒã³æ¹ã¯ç©çã®æ¬è³ªãšã¯é¢ä¿ãªãã®ã§ãããã§ã¯ãã©ã¡ãã®è¡šçŸãçšãããã¯ãæ¬æžã§ã¯ç¹ã«ãã ãããªããè±èªã§ã¯ç©çåŠã»é»æ°å·¥åŠãšãâelectric fieldâã§å
±éããŠããã)
äžã®ã¯ãŒãã³åã®çµæãšåããããšãé»è·Aã®ãŸããã«å¥ã®é»è·ãååšããªããšããé»è· q {\displaystyle q} [C]ã®é»è·ããŸãšãé»å Ž E â {\displaystyle {\vec {E}}} ã¯ã
ã§äžããããããã ããrã¯é»è·ããã®è·é¢ã§ããã e â r {\displaystyle {\vec {e}}_{r}} ã¯ãé»è·ãšããç¹ãçµãã çŽç·äžã§ãé»è·ãšå察æ¹åãåããåäœãã¯ãã«ã§ããã
é»è·ã®åãã®é»å Žã¯ãå¹³é¢äžã§æŸå°ç¶ã®ãã¯ãã«ãšãªãããšã«æ³šæã
é»å Žã¯ãã¯ãã«ã§ãããé»è·ã2åãããšãã¯ãããããã®é»è·ãã€ããé»å Žããéãåãããã°ããã
ã§ããã
é»è·ã3å以äžã®ãšãããåæ§ã«éãåãããã°è¯ãã
å³ã®ããã«ãé»è·ããåºãé»å Žã®æ¹åãå³ç€ºãããã®ãé»æ°åç·(ã§ãããããããelectric line of force)ãšããã é»è·ãè€æ°ããå Žåã«ã¯ãå®éã«æ°ãã«çœ®ãããé»è·ãåããåã¯ããããã足ãåããããã®ãšãªãããããã£ãŠãè€æ°ã®é»è·ãããå Žåã®åšå²ã®é»çã¯ãããããã®é»è·ãäœãé»çãã¯ãã«ã®åãšãªã(éãåããã®åç)ã
é»æ°åç·ãå³ç€ºããå Žåã¯ãæ£é»è·ããåç·ãåºãŠãè² é»è·ã§åç·ãåžåãããããã«æžããåç·ã¯ãé»å Žãå³ç€ºãããã®ãªã®ã§ãé»è·ä»¥å€ã®å Žæã§ã¯ãåç·ãåå²ããããšã¯ãªãã åç·ãçæããã®ã¯æ£é»è·ã®å Žæã®ã¿ã§ãããåç·ãæ¶æ»
ããã®ã¯ãè² é»è·ã®å Žæã®ã¿ã§ããã èšãæããã°ãåç·ãé»è·ä»¥å€ã®å Žæã§æ¶æ»
ããããšã¯ãªãããé»è·ä»¥å€ã®å Žæã§åç·ãçæããããšã¯ãªãã
å°äœã®å
éšã®é»å Žã¯ãŒãã§ãã£ããèšãæããã°ãé»æ°åç·ã¯ãå°äœã®å
éšã«ã¯é²å
¥ã§ããªãã
ç¹é»è·ããã¯ãå³ã®ããã«ãæŸå°ç¶ã«é»æ°åç·ãåºããã¯ãŒãã³ã®æ³åã®ä¿æ°ã«ãã
ã®ãã¡ã®ãåæ¯ã® 4 Ï r 2 {\displaystyle 4\pi r^{2}} ã¯ãçã®è¡šé¢ç©ã®å
¬åŒã«çããã®ã§ãé»æ°åç·ã®å¯åºŠã«æ¯äŸããŠãé»å Žã®åŒ·ããããã¯éé»æ°åã®åŒ·ãã決ãŸããšèããããã
éé»èªå°ã§ã¯ãå°äœå
éšã«ã¯éé»æ°åãåããŠããªãã®ã§ãã£ããããã¯ãé»å ŽãšããæŠå¿µãçšããŠèšãæããã°ãå°äœå
éšã®é»å Žã¯ãŒãã§ããããšèšããã
ã¯ãŒãã³åã¯å(ã¡ãã)ã§ãããããããã«éãã£ãŠå¥ã®é»è·ãè¿ã¥ããå Žåã¯ãè¿ã¥ããå¥ã®é»è·ã¯ä»äºãããããšã«ãªãããŸããè¿ã¥ããé»è·ãææŸãã°ãã¯ãŒãã³åã«ãã£ãŠåãåããä»äºãããããšã«ãªããããè¿ã¥ããç¶æ
ã«ããå¥é»è·ã¯äœçœ®ãšãã«ã®ãŒãèããŠããããšã«ãªãã ãããã£ãŠãã¯ãŒãã³åã«å¯ŸããŠãäœçœ®ãšãã«ã®ãŒãå®çŸ©ããããšãã§ããã(ãªããè¡æè»éäžã®ç©äœã®ãããªãå°è¡šãã倧ããé¢ããå Žæã®éåããã¯ãŒãã³åãšåæ§ã«é2ä¹åãªã®ã§ãããã§èããèšç®ææ³ã¯éåã«ããäœçœ®ãšãã«ã®ãŒã«ãå¿çšã§ãããéåå é床gãçšããåmgãšããã®ã¯å°è¡šè¿ãã§ã®è¿äŒŒã«ãããªãã)
ã¯ãŒãã³åã«ããé»å Žã®å®çŸ©ã§ã¯ãåäœé»è·ã«å¯ŸããŠé»å Žãå®çŸ©ããã®ãšåæ§ãäœçœ®ãšãã«ã®ãŒã«å¯ŸããŠããåäœé»è·ã«å¿ããŠå®çŸ©ã§ããéãå°å
¥ãããšéœåãããããã®ãããªéãé»äœ(ã§ãããelectric potential)ãšåŒã¶ãé»äœã®åäœã¯ãã«ããšãããé»äœãäŸãããšãå°è¡šè¿ãã§ã®éåã®äœçœ®ãšãã«ã®ãŒãèããéã®ãghããªã©ã«çžåœããéã§ããã
ã¯ãŒãã³åã®çµæãšã q {\displaystyle q} [C]ã®é»è·ããè·é¢rã ãé¢ããç¹ã®é»äœVã¯ãé»å Žã®ç©åèšç®ã§åŸãããã(ç©åããŸã ç¿ã£ãŠãªãåŠå¹Žã®èªè
ã¯ãåãããªããŠãæ°ã«ããã次ã®çµæãžãšé²ãã§ãã ããã)çµæã®ã¿ãèšããšã
ãšãªãã
é»äœVã®ç¹ã«q[C]ã®é»è·ã眮ãããšãããã®é»è·ã®ã¯ãŒãã³åã«ããäœçœ®ãšãã«ã®ãŒU[J]ã¯ãé»äœVãçšããã°ã
ãšãªãããããã£ãŠãé»äœ V 1 {\displaystyle V_{1}} ãã«ãã®ç¹ããé»äœ V 2 {\displaystyle V_{2}} ãã«ãã®äœçœ®ãžãšé»è·q[C]ãéé»æ°åãåããŠç§»åãããšããéé»æ°åã®ããä»äºW[J]ã¯
ãšãªãã
ãã£ãœããäžæ§ãªé»å Žã«ãããŠã¯ãé»äœã®åŒããé»å ŽãçšããŠç°¡åã«è¡šãããšãã§ãããè·é¢dã ãé¢ããå¹³è¡å¹³æ¿é»æ¥µã®éã«äžæ§ãªé»å Ž E â {\displaystyle {\vec {E}}} ãçããŠãããšãããã®é»çã®äžã«çœ®ããé»è·qã¯éé»æ°å q E â {\displaystyle q{\vec {E}}} ãåããããã®é»è·ãé»çã®åãã«æ²¿ã£ãŠäžæ¹ã®é»æ¥µããä»æ¹ã®é»æ¥µãŸã§ç§»åãããšããé»çã®ããä»äºW㯠W = q E d {\displaystyle W=qEd} ãšãªããããããã2極æ¿ã®é»äœå·®Vã¯ã
ã§è¡šãããšãã§ããããšãããããåŒãå€åœ¢ããŠ
ãšããããšãã§ãããããã§ãåäœãèãããšãå³èŸºã¯é»å§ãè·é¢ã§å²ã£ããã®ã§ãããããé»çã®åäœãšããŠ[N/C]ã®ã»ã[V/m]ãçšããããšãã§ããããšããããã
é»äœã®åäœã¯ãã«ãã§ããããã®éã¯æ¢ã«äžåŠæ ¡çç§ãªã©ã§æ±ã£ãé»å§(ã§ããã€ãvoltage)ã®åäœãšåãåäœã§ãããå®éã«é»æ°åè·¯ã«é»å§ããããããšã¯ãåè·¯äžã®é»åã«é»å ŽããããŠåããããšãšçããã
éé»èªå°ã«ãã£ãŠãå°äœå
éšã®é»å Žã¯ãŒãã§ãã£ãããã®ããšãããå°äœã®è¡šé¢ã¯ãé»äœãçãããå°äœè¡šé¢ã¯äºãã«çé»äœã§ããã
é»äœã®åºæºã¯ãå®çšäžã¯ãå°é¢ã®é»äœããŒãã«çœ®ãããšãå€ããé»æ°åè·¯ã®äžéšã倧å°ã«ã€ãªãããšãæ¥å°(ãã£ã¡)ãŸãã¯ã¢ãŒã¹(earth)ãšãããåè·¯ãã¢ãŒã¹ããŠããã®ã€ãªãã éšåã®é»äœããŒããšèŠãªãããšãå€ãã
çŽç·äžã§è·é¢0, b[m]ã®ç¹ã«ãé»è·q, q'ãæã€ç©äœã眮ããŠããããã®æãäœçœ®a[m](a<b)ã®ç¹ã®é»äœãæ±ããã
é»äœã®åŒãçšããã°ãããé»è·ãè€æ°ãããšãã«ã¯ãé»äœã¯ããããã®é»è·ãã€ããåºãé»è·ã®åã«ãªãããšã«æ³šæãçãã¯ã
ãšãªãã
å°äœè¡šé¢ã¯çé»äœãªã®ã§ããã£ãŠãé»æ°åç·ã¯å°äœè¡šé¢ã«åçŽã§ããã
ãã®ããšãããé»æ°åç·ãšé»å Žã¯åçŽã§ããã
é»å Žãéãåãããããããã«ãé»äœãéãåãããããããªããªãé»äœãšã¯ãé»å ŽãèããŠãçµè·¯ã«ãŠç©åãããã®ã§ããããã
åŠæ ¡ã®ãã¹ããªã©ã§ã¯ãé»äœã®èšç®ã®ãããã¯ãŒãã³åã®æ¹åã®åéããªã©ã«ããèšç®ãã¹ãªã©ããµãããããé»å Žãæ±ããŠããããããç©åããŠãé»äœãæ±ããã®ããèšç®äžã¯å®å
šã§ããã
ã³ã³ãã³ãµãŒ(è±:capacitor ,ããã£ãã·ã¿ããšèªã)ã¯ãå³ã®ããã«ã2æã®é»æ¥µãåãããããåè·¯äžã«é»è·ãèç©ã§ããéšåãäžããçŽ åã§ããã
ã³ã³ãã³ãµãŒã«é»è·ãèããããšãå
é»(ãã
ãã§ã)ãšãããã³ã³ãã³ãµãŒããé»è·ãæŸåºãããããšãæŸé»ãšããã
ã³ã³ãã³ãµã®äž¡ç«¯ã«ããé»äœVãäžãããããšããã³ã³ãã³ãµã«ã¯ãé»äœã«æ¯äŸããé»è·Qãèç©ãããããã®ãšããã³ã³ãã³ãµã®èç©èœåãèšå·ã§ C ãšãããŠã
ãšããŠCãåããCã¯éé»å®¹é(ããã§ãããããããelectric capacitance)ãšåŒã°ããåäœã¯F(ãã¡ã©ããfarad)ã§äžããããã
1ãã¡ã©ãã¯å®çšäžã¯å€§ããããã®ã§ã10ãã¡ã©ããåäœã«ãã1pF(ãã³ãã¡ã©ã)ãã10ãã¡ã©ããåäœã«ãã1ÎŒF(ãã€ã¯ããã¡ã©ã)ã䜿ãããããšãå€ãã
極æ¿ãå¹³è¡ãªã³ã³ãã³ãµãŒãå¹³è¡æ¿ã³ã³ãã³ãµãŒãšããã å¹³è¡æ¿ã³ã³ãã³ãµãŒã®ã極æ¿ã©ããã®é»å Žã¯ãäžæ§ãªé»å Žã§ããã
ãã®å¹³è¡æ¿ã³ã³ãã³ãµãŒã®éé»å®¹éCã®åŒã¯ãåŸè¿°ããçç±ã«ããã
ã§äžãããããããã§ãSã¯å°äœå¹³é¢ã®é¢ç©ã§ãããdã¯å°äœéã®è·é¢ã§ããã
å®éšçã«ãããã®éé»å®¹éã®å
¬åŒã¯ãæ£ããããšã確ãããããŠããã
ããã§äžããéé»å®¹éã¯ãå¹³é¢äžã«é»è·ãäžæ§ã«ååžãããšã®ä»®å®ã§å°ãããããã®ãšããå°äœéã«çããé»çEã¯ãå°äœãæã€é»è·ãQ, -Qãšããæã
ãŸãã極æ¿ã®é»è·å¯åºŠãã極æ¿ã®ã©ãã§ãäžå®ã ãšä»®å®ããŠ(ãã®ããã«ã¯ãã³ã³ãã³ãµãŒã®åºã(ã€ãŸãé¢ç©)ãããã
ãã¶ãã«åºããšä»®å®ããå¿
èŠãããããšãããããã®ãããªä»®å®ã«ãããé»è·å¯åºŠã¯)ã
ã§ããã
é»æ°åç·ã®æ§è³ªãšããŠããã©ã¹ã®é»è·ããçããŠãã€ãã¹ã®é»è·ã§åžåãããã®ã§ããã£ãŠå¹³è¡æ¿ã³ã³ãã³ãµãŒéã®é»æ°åç·ã®ååžã¯ãå³ã®ããã«ãé»æ°åç·ãããã©ã¹æ¥µæ¿ããåçŽã«ããã€ãã¹æ¥µæ¿ãžåãã£ãŠé»æ°åç·ãåºãŠããããŠãã€ãã¹æ¥µæ¿ã«é»æ°åç·ãåžåãããã
é»å Žã¯ãå°äœéã®åç¹ã§ã
ã§äžãããããé»å Žãæ±ããããã®ã§ãããããé»äœãèšç®ã§ãããå°äœéã®åç¹ã§é»å Žã®å€§ãããåäžãªã®ã§ãé»äœã®å€§ããã¯é»å Žã®å€§ããã«2ç¹éã®è·é¢ãããããã®ã«ãªããããã§ãé»äœVã¯ã
ãšãªããããã®åŒãšéé»å®¹éCã®å®çŸ©ãèŠæ¯ã¹ããšã
ãåŸãããã
é»æ± ã®ååŠåå¿ã«ã€ããŠã¯ãå¥ç§ç®ã®ååŠIãªã©ã§è©³ããæ±ãããããã®ç« ã§ã¯ãé»å§ãé»æµã®ç解ã«é¢ããç¹ãéç¹çã«èª¬æãããã
éå±å
çŽ ã®åäœãæ°ŽãŸãã¯æ°Žæº¶æ¶²ã«å
¥ãããšãã®ãéœã€ãªã³ã®ãªãããããã€ãªã³ååŸå(ionization tendency)ãšããã äŸãšããŠãäºéZnãåžå¡©é
žHClã®æ°Žæº¶æ¶²ã«å
¥ãããšãäºéZnã¯æº¶ãããŸãäºéã¯é»åã倱ã£ãŠZnã«ãªãã
äžæ¹ãéAgãåžå¡©é
žã«å
¥ããŠãåå¿ã¯èµ·ãããªãã
ãã®ããã«éå±ã®ã€ãªã³ååŸåã®å€§ããã¯ãç©è³ªããšã«å€§ãããç°ãªãã
äºçš®é¡ã®éå±åäœãé»è§£è³ªæ°Žæº¶æ¶²ã«å
¥ãããšé»æ± ãã§ãããããã¯ã€ãªã³ååŸå(åäœã®éå±ã®ååãæ°ŽãŸãã¯æ°Žæº¶æ¶²äžã§é»åãæŸåºããŠéœã€ãªã³ã«ãªãæ§è³ª)ã倧ããéå±ãé»åãæŸåºããŠéœã€ãªã³ãšãªã£ãŠæº¶ããã€ãªã³ååŸåã®å°ããéå±ãæåºããããã§ããã
ã€ãªã³ååŸåã®å€§ããæ¹ã®éå±ãè² æ¥µ(ãµããã)ãšãããã€ãªã³ååŸåã®å°ããæ¹ã®éå±ãæ£æ¥µ(ããããã)ãšããã ã€ãªã³ååŸåã®å€§ããéå±ã®ã»ãããéœã€ãªã³ã«ãªã£ãŠæº¶ãåºãçµæãéå±æ¿ã«ã¯é»åãå€ãèç©ããã®ã§ãäž¡æ¹ã®éå±æ¿ãé
ç·ã§ã€ãªãã°ãã€ãªã³ååŸåã®å€§ããæ¹ããå°ããæ¹ã«é»åã¯æµããããé»æµãã§ã¯ç¡ãããé»åããšãã£ãŠãããšã«æ³šæãé»åã¯è² é»è·ã§ããã®ã§ãé»æµã®æµããšé»åã®æµãã¯ãéåãã«ãªãã
ããŸããŸãªæº¶æ¶²ãéå±ã®çµã¿åããã§ãã€ãªã³ååŸåã®æ¯èŒã®å®éšãè¡ã£ãçµæãã€ãªã³ååŸåã®å€§ããã決å®ãããã å·Šããé ã«ãã€ãªã³ååŸåã®å€§ããéå±ã䞊ã¹ããšã以äžã®ããã«ãªãã
éå±ããã€ãªã³ååŸåã®å€§ããã®é ã«äžŠã¹ããã®ãéå±ã®ã€ãªã³ååãšããã æ°ŽçŽ ã¯éå±ã§ã¯ç¡ããæ¯èŒã®ãããã€ãªã³ååŸåã«å ããããã éå±ååã¯ãäžèšã®ä»ã«ãããããé«æ ¡ååŠã§ã¯äžèšã®éå±ã®ã¿ã®ã€ãªã³ååãçšããããšãå€ãã ã€ãªã³ååã®èšæ¶ã®ããã®èªååãããšããŠã
ã貞ããããªããŸããããŠã«ããªãã²ã©ãããåéãã
ãªã©ã®ãããªèªååããããããã¡ãªã¿ã«ãã®èªååããã®å Žåã
ãKã ãã ãCa ãªNaããŸMg ãAlããZn ãŠFe ã«Ni ã ãªPbãã²H2 ã©Cu ãHg ãAg ã åéPt,Auãã
ãšå¯Ÿå¿ããŠããã
è² æ¥µ(äºéæ¿)ã§ã®åå¿
æ£æ¥µ(é
æ¿)ã§ã®åå¿
ãã«ã¿ã®é»æ± ã§ã¯ãåŸããã䞡極éã®é»äœå·®(ãé»å§ããšãããã)ã¯ã1.1ãã«ãã§ããã(ãã«ãã®åäœã¯Vãªã®ã§ã1.1Vãšãæžãã)ãã®äž¡æ¥µæ¿ã®é»äœå·®ãèµ·é»åãšãããèµ·é»åã¯ãäž¡é»æ¥µã®éå±ã®çµã¿åããã«ãã£ãŠæ±ºãŸãç©è³ªåºæã§ããã
èµ·é»åã®åäœã®ãã«ãã¯ãéé»æ°åã®é»äœã®åäœã®ãã«ããšåãåäœã§ãããé»æ°åè·¯ã®é»å§ã®ãã«ããšããèµ·é»åã®åäœã®ãã«ãã¯åãåäœã§ããã
ãã«ã¿é»æ± ã®æ§é ã以äžã®ãããªæååã«è¡šããå Žåããã®ãããªè¡šç€ºãé»æ± å³ãããã¯é»æ± åŒãšããã
aqã¯æ°Žã®ããšã§ãããH2SO4aqãšæžããŠãç¡«é
žæ°Žæº¶æ¶²ãè¡šããŠããã
ç©çåŠã®é»æ°åè·¯ã®ç 究ã§ã¯ããã®ãããªé»æ± ãªã©ã®çŸè±¡ã®çºèŠãšçºæã«ãã£ãŠãå®å®ãªçŽæµé»æºãå®éšçã«åŸãããããã«ãªããçŽæµé»æ°åè·¯ã®æ£ç¢ºãªå®éšãå¯èœã«ãªã£ããé»æ± ã®çºæ以åã«ãããã©ã³ã¹äººã®ç©çåŠè
ã¯ãŒãã³ãªã©ã«ããéé»æ°ã«ããé»æ°ååŠã®ç 究ãªã©ã«ãã£ãŠãé»äœå·®ã®æŠå¿µãé»è·ã®æŠå¿µã¯ãã£ããã ãããã®æ代ã®é»æºã¯ãäž»ã«éé»æ°ã«ãããã®ã ã£ãã®ã§ãå®å®é»æºã§ã¯ç¡ãã£ãã
ãããŠãé»æ± ã«ããå®å®ãªé»æºã®çºæã¯ãåæã«å®å®ãªé»æµã®çºæã§ããã£ãããã®ãããªé»æ± ã®çºæãªã©ã«ãããçŽæµé»æ°åè·¯ã®ç 究ãªã©ããããã€ã人ã®ç©çåŠè
ãªãŒã ããããŸããŸãªå°äœã«é»æµãæµãå®éšãšçè«ç 究ãè¡ãããšã«ãããé»æ°åè·¯ã®çè«ã®ãªãŒã ã®æ³å(ãªãŒã ã®ã»ããããOhm's law)ãçºèŠãããã
ãã€ã¯ãªãŒã ã¯é»æ± ã§ã¯ãªãç±é»å¯Ÿ(ãã€ã§ãã€ã)ãšãããã®ã䜿ã£ãŠãé»æ°åè·¯ã«å®å®ããé»æµããªããç 究ããããåœæã®é»æ± ã§ã¯ãèµ·é»åããã ãã«æžã£ãŠããŸãããªãŒã ã¯åœåã¯é»æ± ã§å®éšããããããŸãå®å®é»æµãåŸãããªãã£ãã
ç±é»å¯Ÿãšã¯ããŸãç°ãªãéå±ææã®2æ¬ã®éå±ç·ãæ¥ç¶ããŠ1ã€ã®åè·¯ãã€ããã2ã€ã®æ¥ç¹ã«æž©åºŠå·®ãäžãããšãåè·¯ã«é»å§ãçºçããããé»æµãæµãã(ãã®çŸè±¡ãããŒãŒããã¯å¹æãšãã)ããã®çŸè±¡ãããã¯ã1821幎ã«ãŒãŒããã¯ãçºèŠããããã®ãããªåè·¯ããç±é»å¯Ÿã§ããããªããåã2æ¬ã®éå±ç·ã§ã¯ã枩床差ãäžããŠãé»å§ã¯çºçãããé»æµã¯æµããªãã
ãªãŒã ã¯ããã«ãªã³å€§åŠææããã±ã³ãã«ãã®å©èšã«ãã£ãŠããã®ç±é»å¯Ÿãå®éšã«å©çšããã枩床ãå®å®ãããã®ã¯ãåœæã®æè¡ã§ãæ¯èŒçç°¡åã§ãã£ãã®ã§ãããããŠãªãŒã ã¯å®å®é»æµããã¡ããå®éšãã§ããã®ã§ããã
ãªãŒã ã®æ³å(Ohm's law)ãšã¯ã
ãã»ãšãã©ã®å°äœã§ã¯ãé»æµ I ãæµããŠããå°äœäžã®2ç¹ã®ç¹ P 1 {\displaystyle P_{1}} ãšç¹ P 2 {\displaystyle P_{2}} éã®é»äœå·® E = E 1 â E 2 {\displaystyle E=E_{1}-E_{2}} ã¯ãé»æµ I ã«æ¯äŸãããã
ãšããå®éšæ³åã§ããã 誀解ããããããããªãŒã ã®æ³åã¯ããã®ãããªå®éšæ³åã§ãã£ãŠãã¹ã€ã«æµæã®å®çŸ©åŒã§ã¯ç¡ããåæ§ã«ããªãŒã ã®æ³åã¯ãã¹ã€ã«é»å§ã®å®çŸ©åŒã§ã¯ç¡ãããé»æµã®å®çŸ©åŒã§ãç¡ããäžåŠæ ¡ã®çç§ã§ã®é»æ°åè·¯ã®æè²ã§ã¯ãéå±ã®é»æ°å解ã®èµ·é»åã®æè²ãŸã§ã¯ããªãã®ã§ããšãããã°ãé»å§ã誀解ããŠããé»å§ã¯ãåãªãé»æµã®æ¯äŸéã§ãæµæã¯ãã®æ¯äŸä¿æ°ãã®ãããªèª€è§£ããå Žåãæããããããã®è§£éã¯æããã«èª€è§£ã§ããã ãŸããåå°äœãªã©ã®äžéšã®ææã§ã¯ãé»æµãå¢ãææã®æž©åºŠãäžæãããšæµæãäžããçŸè±¡ãç¥ãããŠããã®ã§ãåå°äœã§ã¯ãªãŒã ã®æ³åãæãç«ããªãå Žåãããããªã®ã§ããªãŒã ã®æ³åãå®çŸ©åŒãšèããã®ã¯äžåçã§ããã
å°ç·ãªã©ã®å°äœå
ã®é»æ°ã®æµããé»æµ(ã§ããã
ããelectric current)ãšãããé»æµã®åŒ·ãã¯ã¢ã³ãã¢ãšããåäœã§è¡šãã1ã¢ã³ãã¢ã®å®çŸ©ã¯æ¬¡ã®éãã§ããã
1ç§éã«1ã¯ãŒãã³(èšå·C)ã®é»æµãééããããšã1ã¢ã³ãã¢ãšããã
ã¢ã³ãã¢ã®èšå·ã¯Aã§ããããŸããé»æµã¯ãåäœæéãããã®é»è·ã®éééã§ãããã®ã§ãé»æµã®åäœã[C/s]ãšæžãå Žåãããã äžè¬çã«ã¯ãé»æµã®åäœã¯ããªãã¹ã[A]ã§è¡šèšããããšãå€ãã
é»æµI[A]ãšæét[S]ã§å°ç·æé¢ãééããé»è·Q[C]ã®é¢ä¿ãåŒã§è¡šããšã
ã§ããã
é»æµã®åãã®åãæ¹ã«ã€ããŠã¯ãèªç±é»åã¯è² é»è·ãæã£ãŠãããããèªç±é»åã®åããšã¯å察åãã«é»æµã®åãããšãããšã«æ³šæããã
次ã«é»æµãšèªç±é»åã®é床ãšã®é¢ä¿ãèããã èªç±é»åã®é»è·ã®çµ¶å¯Ÿå€ãeãšãããšãèªç±é»åã¯è² é»è·ã§ãããããèªç±é»åã®é»è·ã¯ãã€ãã¹ç¬Šå·ãã€ã-eã§ããã
ãã€ã人ã®ç©çåŠè
ãªãŒã ã¯æ¬¡ã®ãããªæ³åãçºèŠããã
ãã»ãšãã©ã®å°äœã§ã¯ãé»æµ I ãæµããŠããå°äœäžã®2ç¹ã®ç¹ P 1 {\displaystyle P_{1}} ãšç¹ P 2 {\displaystyle P_{2}} éã®é»äœå·® E = E 1 â E 2 {\displaystyle E=E_{1}-E_{2}} ã¯ãé»æµ I ã«æ¯äŸãããã
ãã®å®éšæ³åããªãŒã ã®æ³å(Ohm's law)ãšããã åŒã§è¡šããšãé»äœå·®ãVãšããŠãé»æµãIãšããå Žåã«ãæ¯äŸä¿æ°ãRãšããŠã
ã§ããã ããã§ãé»äœãšé»æµã®æ¯äŸä¿æ°Rãé»æ°æµæãããã¯åã«æµæ(resistanceãã¬ãžã¹ã¿ã³ã¹)ãšããã é»æ°æµæã®åäœã¯ãªãŒã ãšèšããèšå·ã¯Î©ã§è¡šãã
æ
£ç¿çã«ãæµæã®èšå·ã¯Rã§ããããå Žåãå€ãã
é»æ°åè·¯ãžãšãã«ã®ãŒãäŸçµŠããé»æºãšããŠå®é»å§ã®çŽæµé»æºãèãããåè·¯ã®2å°ç¹éã«ããäžå®ã®é»å§ãäŸçµŠãç¶ãããã®ã§ãããé»å§æºã®åè·¯å³èšå·ãšããŠã¯ãçšãããããèšå·ã®é·ãåŽãæ£æ¥µã§ããããã©ã¹ã®é»äœã§ãããèšå·ã®çãåŽã¯è² 極ã§ããã
也é»æ± ã¯ãçŽæµé»æºãšããŠåãæ±ã£ãŠè¯ãã
ãªãããããã¯çŽæµé»æºã§ããã亀æµã®å Žåã¯äžè¬åããé»å§æºãšããŠã®èšå·ãçšããããŸãç¹ã«æ£åŒŠæ³¢äº€æµé»å§æºã§ããã°ã®èšå·ãçšããã
æµæåš(resistor)ã¯ãéåžžã¯åã«æµæãšåŒã°ããåè·¯çŽ åã§ãããäžããããé»æ°ãšãã«ã®ãŒãåçŽã«æ¶è²»ããçŽ åã§ãããåè·¯å³èšå·ã¯ãããã¯ã§ããããæ¬æžã§ã¯ãäž¡è
ãšãæµæã®åè·¯å³èšå·ãšããŠçšããããšã«ããã(ç»åçŽ æã®ç¢ºä¿ã®éœåã®ãããäž¡æ¹ã®èšå·ãæ¬æžã§ã¯æ··åšããŸããã容赊ãã ããã)
æ¥æ¬ã§ã¯ãæµæåšã®å³èšå·ã¯ãåŸæ¥ã¯JIS C 0301(1952幎4æå¶å®)ã«åºã¥ããã®ã¶ã®ã¶ã®ç·ç¶ã®å³èšå·ã§å³ç€ºãããŠããããçŸåšã®ãåœéèŠæ Œã®IEC 60617ãå
ã«äœæãããJIS C 0617(1997-1999幎å¶å®)ã§ã¯ã®ã¶ã®ã¶åã®å³èšå·ã¯ç€ºãããªããªããé·æ¹åœ¢ã®ç®±ç¶ã®å³èšå·ã§å³ç€ºããããšã«ãªã£ãŠãããæ§èŠæ Œã§ããJIS C 0301ã¯ãæ°èŠæ ŒJIS C 0617ã®å¶å®ã«äŒŽã£ãŠå»æ¢ããããããæ§èšå·ã§æµæåšãå³ç€ºããå³é¢ã¯ãçŸåšã§ã¯JISéæºæ ãªå³é¢ã«ãªã£ãŠããŸããããããææåã¯ç¡ããããçŸåšãåŸæ¥ã®å³èšå·ãå€çšãããŠããã
è€æ°ã®åè·¯çŽ åã1ã€ã®ç·äžã«é
眮ãããŠãããããªæ¥ç¶ãçŽåæ¥ç¶ãšãããè€æ°ã®åè·¯çŽ åãäºè¡ã«åãããããã«é
眮ãããŠããæ¥ç¶ã䞊åæ¥ç¶ãšããã
çŽåæ¥ç¶ã«ãããŠã¯ãããããã®åè·¯çŽ åã«æµããé»æµã¯å
šãŠçãããäžæ¹ã䞊åæ¥ç¶ã«ãããŠã¯ããããã®åè·¯çŽ åã®äž¡ç«¯ã«ãããé»å§ãå
šãŠçããã
ãŸããçŽåæ¥ç¶ã«ãããŠã¯ããããã®åè·¯çŽ åã«ãããé»å§ã®åãå
šé»å§ãšãªãã䞊åæ¥ç¶ã«ãããŠã¯ããããã®åè·¯çŽ åãæµããé»æµã®åãå
šé»æµãšãªãã
æµæãè€æ°æ¥ç¶ãããŠããå Žåããã®è€æ°ã®æµæããŸãšããŠãããã1ã€ã®æµæãæ¥ç¶ãããŠãããã®ãããªç䟡çãªåè·¯ãèããããšãã§ãããè€æ°ã®æµæãšç䟡ãª1ã€ã®æµæãåææµæãšããã
æµæãnåçŽåã«æ¥ç¶ãããŠããå Žåãèãããæµæ R 1 , R 2 , ⯠, R n {\displaystyle R_{1},R_{2},\cdots ,R_{n}} ãçŽåã«æ¥ç¶ãããŠããå Žåãåæµæãæµããé»æµã¯çããããããiãšãããåæµæ R k ( k = 1 , 2 , ⯠, n ) {\displaystyle R_{k}(k=1,2,\cdots ,n)} ã«ãããé»å§ã v k {\displaystyle v_{k}} ãšãããšããªãŒã ã®æ³åãã
ãæãç«ã€ããã®ãšãçŽåæµæã®äž¡ç«¯ã®é»å§vã¯ã
ã§ããããããšç䟡ãªæµæRã1ã€ã ãæ¥ç¶ãããŠãããããªç䟡åè·¯ãèãããšãã
ãæãç«ã€ããããããã£ãŠãããã®nåã®çŽåæµæã®åææµæRãšããŠ
ãåŸããããªãã¡ãçŽååææµæã¯åæµæã®ç·åãšãªãã
åæ§ã«ãæµæãnå䞊åã«æ¥ç¶ãããŠããå Žåãèãããæµæ R 1 , R 2 , ⯠, R n {\displaystyle R_{1},R_{2},\cdots ,R_{n}} ã䞊åã«æ¥ç¶ãããŠããå Žåãåæµæã®äž¡ç«¯ã®é»å§ã¯çããããããvãšãããåæµæ R k ( k = 1 , 2 , ⯠, n ) {\displaystyle R_{k}(k=1,2,\cdots ,n)} ãæµããé»æµã i k {\displaystyle i_{k}} ãšãããšããªãŒã ã®æ³åãã
ãæãç«ã€ããã®ãšã䞊åæµæãžæµã蟌ãé»æµiã¯ã
ã§ããããããšç䟡ãªæµæRã1ã€ã ãæ¥ç¶ãããŠãããããªç䟡åè·¯ãèãããšãã
ãæãç«ã€ããããããã£ãŠãããã®nåã®äžŠåæµæã®åææµæRãšããŠ
ãåŸããããªãã¡ã䞊ååææµæã®éæ°ã¯åæµæã®éæ°ã®ç·åãšãªãã
æµæRãé»æµIãæµãããšãããã®éšåã®çºç±ã®ãšãã«ã®ãŒã¯ã1ç§ãããã«RI[J/s]ã§ãããããããžã¥ãŒã«ç±ãšãããååã®ç±æ¥ã¯ç©çåŠè
ã®ãžã¥ãŒã«ã調ã¹ãããã§ããããªãŒã ã®æ³åãããV=RIã§ãããã®ã§ããžã¥ãŒã«ç±ã¯VIãšãæžããã
ããã§ãã²ãšãŸããç±ã®èå¯ã«ã¯é¢ããŠã次ã®éãå®çŸ©ãããé»æ°åè·¯ã®ãã2ç¹éãæµããé»æµIãšããã®2ç¹éã®é»å§Vãšã®ç©VIãé»å(power)ãšå®çŸ©ãããé»åã®èšå·ã¯Pã§è¡šããããããšãå€ãã é»åã®åäœã®ãžã¥ãŒã«æ¯ç§[J/s]ã[W]ãšããåäœã§è¡šãããã®åäœWã¯ã¯ãã(Watt)ãšèªãã ã€ãŸãé»åã¯èšå·ã§
ã§ããã
å°ç·ã®å€ªããé·ãã«ãã£ãŠæµæã®å€§ããã¯å€ãããçŽæçã«å€ªãã»ããæµããããã®ã¯åããã ãããããã«äžŠåæ¥ç¶ãšå¯Ÿå¿ãããŠããå°ç·ã倪ãã»ããæµããããã®ã¯åããã ããã å®éã«é»æ°æµæã¯ãå°ç·ã倪ãã«åæ¯äŸããŠå°ãããªãããšãå®éšçã«ç¢ºèªãããŠãããããã§ãã€ãã®ãããªåŒã«ãããŠã¿ããã æµæãR[Ω]ãšããå Žåãå°ç·ã®å€ªããé¢ç©ã§è¡šãA[m]ãšããã°ãæ¯äŸå®æ°ã«kãçšããã°ã
ã§ããã( âã¯ãæ¯äŸé¢ä¿ãè¡šãæ°åŠèšå·ã)
ããã«ãå°ç·ã¯æ質ã倪ããåããªãã°ãå°ç·ãé·ãã»ã©æµæããé·ãã«æ¯äŸããŠæµæã倧ãããªãããšãã確èªãããŠãããããã§ãããã«ãæµæäœã®é·ããèæ
®ããåŒã«è¡šããŠã¿ãã°ã次ã®ããã«ãªããæµæ垯ã®é·ããl[m]ãšããã°
ã§ããã
ããã«ãå°ç·ã®æ質ã«ãã£ãŠãæµæã®å€§ããã¯å€ãããåãé·ãã§åã倪ãã®æµæã§ããæ質ã«ãã£ãŠæµæã®å€§ããã¯ç°ãªããããã§ãæ質ããšã®æ¯äŸå®æ°ãÏãšããã°ãæµæã®åŒã¯ä»¥äžã®åŒã§èšè¿°ãããã
Ïã¯æµæç(ãŠããããã€ãresistivity)ãšåŒã°ãããæµæçã®åäœã¯[Ωm]ã§ããã
ç£ç³ã®ãŸããã«ã¯å¥ã®ç£ç³ãåããåã®ããšãšãªããã®ãçããŠããã ãããç£å Ž(ãã°ãmagnetic field)ãããã¯ç£ç(ããã)ãšåŒã¶ã(æ¥æ¬ã®ç©çåŠã§ã¯ç£å ŽãšåŒã¶ããšãå€ãããŸããæ¥æ¬ã®é»æ°å·¥åŠã§ã¯ç£çãšåŒã°ããããšãå€ããææ²»æã®èš³èªã®éã®ãæ¥æ¬åœå
ã®æ¥çããšã®éãã«éãããå°å瀟äŒçãªäºè±¡ã§ãããåŒã³æ¹ã¯ç©çã®æ¬è³ªãšã¯é¢ä¿ãªãã®ã§ãããã§ã¯ãã©ã¡ãã®è¡šçŸãçšãããã¯ãæ¬æžã§ã¯ç¹ã«ãã ãããªããè±èªã§ã¯ç©çåŠã»é»æ°å·¥åŠãšãâmagnetic fieldâã§å
±éããŠããã)
éãã³ãã«ããããã±ã«ã«ç£ç³ãè¿ã¥ãããšãç£ç³ã«åžãä»ããããã ãŸããéãã³ãã«ããããã±ã«ã«åŒ·ãç£åãäžãããšãéãã³ãã«ããããã±ã«ãã®ãã®ãç£å Žãåšå²ã«åãŒãããã«ãªãã ãã®ãããªãããšããšã¯ç£å Žãæããªãã£ãç©äœãã匷ãç£å Žãåããããšã«ãã£ãŠç£å ŽãåãŒãããã«ãªãçŸè±¡ãç£å(ãããmagnetization)ãšããã ãããã¯é»è·ã®éé»èªå°ãšå¯Ÿå¿ãããŠãç£åã®ããšãç£æ°èªå°(ããããã©ããmagnetic induction)ãšãããã ãããŠãéãã³ãã«ããããã±ã«ã®ããã«ãç£ç³ã«åŒãä»ããããããã«ç£åãããèœåãããç©äœã匷ç£æ§äœ(ãããããããããferromagnet)ãšããã éãšã³ãã«ããšããã±ã«ã¯åŒ·ç£æ§äœã§ããã
é
ã¯ç£åããªãããé
ã¯ç£ç³ã«åŒãã€ããããªãã®ã§ãé
ã¯åŒ·ç£æ§äœã§ã¯ãªãã
éé»èªå°ãå©çšãããéé»é®èœ(ããã§ããããžã)ãšèšããããäžç©ºã®å°äœãã€ãã£ãŠç©è³ªãå²ãããšã§å€éšé»å Žãé®èœããæ¹æ³ããã£ãã®ãšåæ§ã®ãç£æ°ã®é®èœãã匷ç£æ§äœã§ãåºæ¥ããäžç©ºã®åŒ·ç£æ§äœãçšããŠã匷ç£æ§äœã®å
éšã¯ç£å Žãé®èœã§ããããããç£æ°é®èœ(ãããããžããmagnetic shielding)ãšãããç£æ°ã·ãŒã«ããšãããã
ç£å Žã®åããåããããã«å³ç€ºããããç£ç³ã®äœãç£å Žã®æ¹åã¯ãç ã«å«ãŸããç éã®ç²æ«ãç£ç³ã«ãã¡ãã°ããŠããµããããããšã§èŠ³å¯ã§ããã
ãããå³ç€ºãããšãäžå³ã®ããã«ãªãã(ç»åçŽ æã®ç¢ºä¿ã®éœåäžãåçãšå³ç€ºãšã§ã¯ãN極ãšS極ãéã«ãªã£ãŠããŸããã容赊ãã ããã)
ãã®ãããªç£å Žã®å³ãç£åç·(ãããããããmagnetic line of force)ãšãããç£åç·ã®åãã¯ãç£ç³ã®N極ããç£åç·ãåºãŠãS極ã«ç£åç·ãåžåããããšå®çŸ©ããããæ£ç£ç³ã§ã¯ãç£åã®çºçæºãšãªãå Žæããæ£ç£ç³ã®äž¡ç«¯ã®å
端ä»è¿ã«éäžãããããã§ãæ£ç£ç³ã®äž¡ç«¯ã®å
端ä»è¿ãç£æ¥µ(ãããããmagnetic pole)ãšããã
ãã®ãããªç£ç³ã®ã€ããç£åç·ã®åœ¢ã¯ãé»æ°åç·ã§ã®ãç°ç¬Šå·ã®é»è·ã©ãããã€ããé»æ°åç·ã«äŒŒãŠããã
1ã€ã®æ£ç£ç³ã§ã¯N極(north pole)ã®ç£æ°ã®åŒ·ããšãS極(south pole)ã®ç£æ°ã®åŒ·ãã¯çããããŸããç£ç³ã«ã¯ãå¿
ãN極ãšS極ãšãååšãããN極ãšS極ã®ãã©ã¡ããçæ¹ã ããåãåºãããšã¯åºæ¥ãªããããšãæ£ç£ç³ãåæããŠããåæé¢ã«ç£æ¥µãåºçŸããããã®ãããªçŸè±¡ã®ãããçç±ã¯ãããããæ£ç£ç³ãæ§æãã匷ç£æ§äœã®ååã®1åãã€ãå°ããªç£ç³ã§ããããããå°ããªååã®ç£ç³ããããã€ãæŽåããŠã倧ããªæ£ç£ç³ã«ãªã£ãŠããããã§ããã
ä»®æ³çã«ãç£æ¥µãS極ãŸãã¯N極ã®çæ¹ã ãçŸããçŸè±¡ãçè«èšç®ã®ããã«èããããšããããããã®ãããªçåŽã ãã®ç£æ¥µãåç£æ¥µ(ã¢ãããŒã«ãšããã)ãšããããåç£æ¥µã¯å®åšããªãã
æ£ç£ç³ãªã©ããã®ãçåŽã®ç£æ¥µãããã®ãç£æ¥µããã®ç£å Žã®åŒ·ãã®ããšãããã®ãŸãŸãç£æ¥µã®åŒ·ãã(Magnetic charge)ãšåŒã¶ããããã¯ç£è·(ãããmagnetization)ãç£æ°éãšããã
ããããããã®ç£åãšç£å Žã®é¢ä¿ãåŒã§è¡šãããšãèããã ãŸããæ£ç£ç³ã«ã¯ç£æ¥µãäž¡åŽã«2åããã®ã§ãèšç®ãç°¡åã«ããããã«ãæ£ç£ç³ã®äž¡ç«¯ã®è·é¢ã倧ãããå察åŽã®ç£æ¥µã®å€§ãããç¡èŠã§ããç£ç³ãèãããã
ãã®ãããªç£ç³ãçšããŠãå®éšãããšããã次ã®æ³åãåãã£ããç£åã®åŒ·ãã¯2åã®ç©äœã®ç£æ°ém1ããã³m2ã«æ¯äŸãã2åã®ç©äœéã®è·é¢rã®2ä¹ã«åæ¯äŸããã åŒã§è¡šããšã
ã§è¡šãããã(kmã¯æ¯äŸå®æ°) ãããçºèŠè
ã®ã¯ãŒãã³ã®åã«ã¡ãªãã§ãç£æ°ã«é¢ããã¯ãŒãã³ã®æ³åãšãããç£æ°émã®åäœã¯ãŠã§ãŒããšãããèšå·ã¯[Wb]ã§è¡šãã
æ¯äŸå®æ°kmãš1ãŠã§ãŒãã®å€§ãããšã®é¢ä¿ã¯ã1ã¡ãŒãã«é¢ãã1wbã©ããã®ç£æ¥µã«ã¯ãããåãçŽ6.33Ã10ãšããŠã æ¯äŸä¿æ°kmã¯ã
ã§ããã
ã€ãŸãã
ã§ããã
éé»æ°åã«å¯ŸããŠãé»å Žãå®çŸ©ãããããã«ãç£æ°åã«å¯ŸããŠããå Žãå®çŸ©ããããšéœåãè¯ããç£æ°ém1[Wb]ãäœãã次ã®éãç£å Žã®åŒ·ããããã¯ç£å Žã®å€§ãããšèšããèšå·ã¯Hã§è¡šãã
ç£å Žã®åŒ·ãHã®åäœã¯[N/Wb]ã§ãããHãçšãããšãç£æ°ém2[Wb]ã«ã¯ãããç£æ°åf[N]ã¯ã
ãšè¡šããã
ç©çåŠè
ã®ãšã«ã¹ãããã¯ãé»æµã®å®éšãããŠããéã«ãããŸããŸè¿ãã«ãããŠãã£ãæ¹äœç£ç³ãåãã®ã確èªããã圌ã詳ãã調ã¹ãçµæã以äžã®ããšãåãã£ãã
é»æµãæµããŠãããšãã«ã¯ããã®ãŸããã«ã¯ãç£å Žãçãããåãã¯ãé»æµã®æ¹åã«å³ãããé²ãããã«ãå³ãããåãåããšåããªã®ã§ããããå³ããã®æ³åãšããã
ã¢ã³ããŒã«ããç£å Žã®å€§ããã調ã¹ãçµæãç£å Žã®å€§ããHã¯ãé»æµI[A]ãçŽç·çã«æµããŠãããšããçŽç·é»æµã®åšãã®ç£å Žã®å€§ããã¯ãå°ç·ããã®è·é¢ãa[m]ãšãããšãç£å Žã®å€§ããH[N/Wb]ã¯ã
ã§ããããšãç¥ãããŠããã
ãããã¢ã³ããŒã«ã®æ³å(Ampere's law) ãšããã ç£å Žã®å€§ããHã®åäœã¯ã[N/Wb]ã§ãããããã£ãœãã¢ã³ããŒã«ã®æ³åã®åŒãã¿ãã°ã¢ã³ãã¢æ¯ã¡ãŒãã«[A/m]ã§ãããã
å°ç·ãã³ã€ã«ç¶ã«å·»ãã°ãã¢ã³ããŒã«ã®æ³åã§å°ç·ã®åšå²ã«çºçããç£å Žãéãªãããããã®ããã«ããç£å Žã匷ããã³ã€ã«ãé»ç£ç³(ã§ããããããelectromagnet)ãšãããå°ç·ã«é»æµãæµããŠãããšãã«ã®ã¿ãé»ç£ç³ã¯ç£å Žãçºçãããå°ç·ã«é»æµãæµãã®ãæ¢ãããšãé»ç£ç³ã®ç£å Žã¯æ¶ããã
ç£å Žã®å€§ããHã«ã次ã®ç¯ã§æ±ãããŒã¬ã³ãåã®çŸè±¡ã®ãããæ¯äŸä¿æ°ÎŒ(åäœã¯ãã¥ãŒãã³æ¯ã¢ã³ãã¢ã§[N/A])ãæããŠãèšå·Bã§è¡šãã
ãšããããšãããããã®éBãç£æå¯åºŠ(magnetic flux density)ãšãããç£å Žã®å€§ããHã®åããšç£æå¯åºŠBã®åãã¯åãåãã§ããã ãŸããç£å Žã®å€§ããHãšç£æå¯åºŠBã®æ¯äŸä¿æ°ãéç£ç(ãšãããã€ãmagnetic permeability)ãšããã (ããŒã¬ã³ãåã«é¢ããŠã¯ã詳ããã¯ç©çIIã§æ±ããèªè
ãç©çIãåŠã¶åŠå¹Žãªãã°ãèªè
ã¯ãããŒã¬ã³ãåãšããåãããã®ã ãªã»ã»ã»ããšã§ãæã£ãŠããã°ããã)
ãŸããå°ç·ãçšæãããšãããããã®å°ç·ã¯ãéæ¢ããŠãããšããŠãéæ¢ããŠããããåºå®ã¯ããã«ãããå°ç·ã«åãå ããã°ãå°ç·ãåããããã«ããŠããšãããã
ãã®å°ç·ã«é»æµãæµããã ãã§ã¯ãã¹ã€ã«å°ç·ã¯åããªãããããããã®å°ç·ã«ãå€éšã®ç£ç³ã«ããç£å Žãå ãããšãå°ç·ãåãããã®ãããªãç£å Žãšé»æµã®çžäºäœçšã«ãã£ãŠãå°ç·ã«çããåãããŒã¬ã³ãå(ããŒã¬ã³ãããããè±: Lorentz force)ãšããã
ããŒã¬ã³ãåã®åãã¯ãå°ç·ã®é»æµã®åããšç£å Žã®åãã«åçŽã§ãããé»æµIã®åãããç£æå¯åºŠBã®åãã«å³ãããåãåããšåãã§ããã
ãŸããããŒã¬ã³ãåã®å€§ããã¯ãå°ç·ã®é·ãlãšãç£å Žã®å°ç·ãšã®åçŽæ¹åæåã«æ¯äŸããã
ããŒã¬ã³ãåã®å€§ããF[N]ãåŒã§è¡šãã°ãé»æµãšç£å ŽãšãåçŽã ãšããŠãç£å ŽãåããŠããå°ç·ã®åœ¢ç¶ãçŽç·åœ¢ã ãšããŠãé»æµãI[A]ãšããŠãå°ç·ã®é·ããl[m]ãšããŠãå°ç·ã«ããã£ãŠããå€éšç£å Žã®ç£æå¯åºŠãB[N/(Aã»m)]ãšããã°ã
ã§è¡šããã
ããŒã¬ã³ãåã®å
¬åŒã«ãã¯ãŒãã³ã®æ³åãªã©ã§ã¯èŠããããããªæ¯äŸä¿æ°(ä¿æ°Kãªã©ã)ãå«ãŸããªãã®ã¯ãããããããã®ããŒã¬ã³ãåã®çŸè±¡ãå
ã«ãç£æ°éãŠã§ãŒãWbã®åäœããã³ç£æå¯åºŠBã®åäœãã決å®ãããŠããããã§ããã
ãŸãããç£æå¯åºŠãã®å称ãããç£æãã»ãå¯åºŠããšããã®ã¯ãå®ã¯ç£æå¯åºŠã®åäœã®[N/(Aã»m)]ã¯ãåäœãåŒå€åœ¢ãããš[Wb/m]ã§ãããããšãç±æ¥ã§ããããã®åäœ[Wb/m]ããé»æ°å·¥åŠè
ã®ãã¹ã©ã®åã«ã¡ãªã¿ãåäœ[Wb/m] ããã¹ã©ãšèšããèšå·Tã§è¡šãã
ãã®ããŒã¬ã³ãåã®çŸè±¡ããé»æ°æ©åšã®ã¢ãŒã¿(é»åæ©)ã®åçã§ããã
ãªããããã¬ãã³ã°ã®æ³åããšããããŒã¬ã³ãåã«é¢ããæ³åãããããããŒã¬ã³ãåã®èšç®ã«ã¯å®çšçã§ã¯ç¡ããããã¬ãã³ã°ã®åãé¢ããç°ãªãæ³åã幟ã€ããã£ãŠçŽããããééãã®åå ã«ãªããããã®ã§ãæ¬æžã§ã¯æããªãã å®éã«ãå°éçãªç©çèšç®ã§ã¯ããã¬ãã³ã°ã®æ³åã¯ãèšç®ã«ã¯çšããªãã
ãããããã¬ãã³ã°ã®æ³åã«ã¯ããã¬ãã³ã°ã®å³æã®æ³åããšããããšã¯ç°ãªãããã¬ãã³ã°ã®å·Šæã®æ³åãããããã©ã¡ãããã©ã®ç£æ°ã®çŸè±¡ã«çšããæ³åã ã£ãã®ããééãããããã ãããæ¬æžã§ã¯æããªãã
(é»ç£èªå°ã«é¢ããŠã¯ã詳ããã¯ç©çIIã§æ±ãã)
ã¢ã³ããŒã«ã®æ³åã§ã¯ãé»æµã®åšãã«ç£å Žãã§ããã®ã§ãã£ãã
å®ã¯ãç£ç³ãåãããªã©ããŠãç£å Žã䌎ãç©äœãéåãããšããã®ãŸããã«ã¯é»å Žãçããã ä»®ã«ãã³ã€ã«ã®è¿ãã§ãããè¡ãªã£ããšãããšãçããé»å Žã«ãã£ãŠã³ã€ã«ã®äžã«ã¯é»æµãæµããã çããé»å Žã®å€§ããã¯ã
ãšãªãã(ååŸaã®å圢ã®ã³ã€ã«ã®å Žåã) Eã®åäœã¯[V/m]ã§ãããBã®åäœã¯[T]ã§ããã
ãã®çŸè±¡ãé»ç£èªå°(ã§ããããã©ããelectromagnetic induction)ãšãããé»ç£èªå°ã«ãã£ãŠçºçããé»æµãèªå°é»æµãšããã
ãŸããèªå°é»æµã®åãã¯ãç£ç³ã®åãã«ãããã³ã€ã«ã®äžãéãç£æã®å€åã劚ããåãã«ãé»æµãæµããã(èªå°é»æµãã¢ã³ããŒã«ã®æ³åã«åŸããåšå²ã«ç£å Žãäœãã) ãã®èªå°é»æµããã³ã€ã«ã®äžãéãç£æã®å€åã劚ããåãã«èªå°é»æµãæµããçŸè±¡ãã¬ã³ãã®æ³å(Lenz's law)ãšããã
åãé åã«Nåå·»ãããã³ã€ã«ã眮ãããå Žåããã¡ã©ããŒã®é»ç£èªå°ã®æ³åã¯ã次ã®ããã«ãªãã
ããã§ã E {\displaystyle {\mathcal {E}}} ã¯èµ·é»å(ãã«ã ãèšå·ã¯V)ãΊB ã¯ç£æ(ãŠã§ãŒããèšå·ã¯Wb)ãšãããNã¯é»ç·ã®å·»æ°ãšããã
ãã®é»ç£èªå°ã®çŸè±¡ããç«åçºé»ãæ°Žåçºé»ãªã©ã®çºé»æ©ã®åçã§ãããããçã®çºé»ã§ã¯ãæ°žä¹
ç£ç³ãå転ãããããšã§ãçºé»ãããŠãããç«åãæ°Žåãšããã®ã¯ãæ©åšã®å転ãåŸãæ段ã«ãããªãããŸããçºé»æã®çºé»ã«ã¯ãæ°žä¹
ç£ç³ã®å転ãå©çšããŠãããããçºçããé»å§ãé»æµã¯åšæçãªæ³¢åœ¢ã«ãªãã次ã«èª¬æãã亀æµæ³¢åœ¢ã«ãªãã
åè·¯ãžã®å
¥åé»å§ãåšæçã«æéå€åããåè·¯ã®é»å§ããã³é»æµã亀æµ(alternating current)ãšãããããã«å¯Ÿãã也é»æ± ãªã©ã«ãã£ãŠçºçããé»å§ãé»æµã®ããã«ãæéã«ãããäžå®ãªé»å§ãé»æµã¯çŽæµ(direct Current)ãšããã
亀æµæ³¢åœ¢ãäœç§ã§1åšããããšããæéãåšæ(wave period)ãšãããåšæã®èšå·ã¯ T {\displaystyle T} ã§è¡šãåäœã¯ç§[s]ã§ããã
1ç§éã«æ³¢åœ¢ãäœåšããããšããåæ°ãåšæ³¢æ°ãããã¯æ¯åæ°(è±èªã¯ããšãã«frequency)ãšããã é»æ°ã®æ¥çã§ã¯åšæ³¢æ°ãšããçšèªãçšããããšãå€ããç©çã®æ³¢ã®çè«ã§ã¯æ¯åæ°ãšããè¡šçŸãçšããããšãå€ãã
åšæ³¢æ°ã®åäœã¯[1/s]ã§ããããããããã«ã(hertz)ãšããåäœã§è¡šããåäœèšå·HzãçšããŠåšæ³¢æ°fããf[Hz]ãšãããµãã«è¡šãã
亀æµé»æµã亀æµé»å§ãæ£åŒŠæ³¢ã®å Žåã¯ããããã®ãã©ã¡ãŒã¿ãçšããŠ
ãšæžãããšãã§ããã sinãšã¯äžè§é¢æ°ã§ãããç¥ããªããã°æ°åŠIIãªã©ãåèã«ããã ãã®ãšãã®sinã®ä¿æ° I 0 {\displaystyle I_{0}} ã V 0 {\displaystyle V_{0}} ãæ¯å¹
(ããã·ããamplitude)ãšããããŸãæå»t=0ã«ãããé»æµãé»å§ã®å€ã瀺ããæé波圢ã決å®ãã Ξ i {\displaystyle \theta _{i}} ã Ξ v {\displaystyle \theta _{v}} ãåæäœçžãšããã
æ®éç§é«æ ¡ã®é«æ ¡ç©çã§ã¯ã亀æµæ³¢åœ¢ã®èšç®ã«ã¯ãæ£åŒŠæ³¢ã®å Žåãäž»ã«æ±ããæ¹åœ¢æ³¢ãäžè§æ³¢ã®èšç®ã¯ãæ®éã¯æ±ãããªãã ãã ããå·¥æ¥é«æ ¡ã®ææ¥ããå·¥å Žã®å®åã§ã¯æ±ãããšãããã®ã§ãèªè
ã¯æ³¢åœ¢ãåŠãã§ããããšã
çºé»æããäžè¬å®¶åºã«éãããŠããé»å§ã¯äº€æµé»å§ã§ãããæ±æ¥æ¬ã§ã¯50Hzã§ããã西æ¥æ¬ã§ã¯60Hzã§ãããããã¯ææ²»æ代ã®çºé»æ©ã®èŒžå
¥æã«ãæ±æ¥æ¬ã®äºæ¥è
ã¯ãšãŒããããã50Hzçšã®çºé»æ©ã茞å
¥ãã西æ¥æ¬ã®äºæ¥è
ã¯ã¢ã¡ãªã«ãã60Hzã®çºé»æ©ã茞å
¥ããããšã«ããã
çºé»æããäžè¬ã®å®¶åºãªã©ã«éãããé»æµã®åšæ³¢æ°ãåçšåšæ³¢æ°ãšããã
åçšé»æºã®é»å§æ¯å¹
ã¯çŽ140Vã§ããããã㯠100 à 2 {\displaystyle 100\times {\sqrt {2}}} [V]ã§ããã
ãããã«ããšã¯1000Hzã®ããšã§ããããããã«ãã¯kHzãšæžãã
亀æµé»æµã«å¯ŸããŠã¯ãé»æµãšåãæ¯åæ°ã§ãã¢ã³ããŒã«ã®æ³åã§çºçããç£å Žãæ¯åããã
å°ç·ã§ã€ããããã³ã€ã«ã¯ãçŽæµé»æµã§ã¯ããã ã®å°ç·ãšããŠã¯ãããããããã亀æµé»æµã«å¯ŸããŠã¯ãé»ç£èªå°ã«ããèªå·±ã®çºçãããç£å Žã劚ãããããªé»æµããã³èµ·é»åãçºçããããããèªå·±èªå°(self induction)ãšããã
èªå·±èªå°ã«ããèµ·é»åã®å€§ããã¯ãé»æµã®æéå€åçã«æ¯äŸãããèªå·±èªå°ã®èµ·é»åãåŒã§æžãã°ãæ¯äŸä¿æ°ãLãšããŠã
ã§ããã ãã®æ¯äŸä¿æ° L {\displaystyle L} ãèªå·±ã€ã³ãã¯ã¿ã³ã¹(self inductance)ãšãããèªå·±ã€ã³ãã¯ã¿ã³ã¹ã®æ¬¡å
ã¯[Vã»S/m]ã ããããããã³ãªãŒãšããåäœã§è¡šããåäœã«Hãšããèšå·ãçšããã
éå¿ã«äºã€ã®ã³ã€ã«ãå·»ããã³ã€ã«ã®çæ¹ã®é»æµãå€åããããšãã¢ã³ããŒã«ã®æ³åã«ãã£ãŠçããŠããç£æãå€åãããããå察åŽã®ã³ã€ã«ã«ã¯ããã®ç£æå¯åºŠã®å€åãæã¡æ¶ããããªåãã«èµ·é»åãçºçããããã®çŸè±¡ãçžäºèªå°(mutual induction)ãšèšãã
é»å§ãå
¥åãããåŽã®ã³ã€ã«ã1次ã³ã€ã«(primaly coil)ãšèšããèªå°èµ·é»åãçºçãããåŽã®ã³ã€ã«ã2次ã³ã€ã«(secondary coil)ãšããã
çžäºèªå°ã«ããèµ·é»åã®å€§ããã¯ãé»æµã®æéå€åçã«æ¯äŸãããçžäºèªå°ã®èµ·é»åãåŒã§æžãã°ãæ¯äŸä¿æ°ãMãšããŠã(çžäºèªå°ã®æ¯äŸä¿æ°ã¯Lã§ã¯ç¡ãã)åŒã¯ã
ã§ããã ãã®æ¯äŸä¿æ° M {\displaystyle M} ãçžäºã€ã³ãã¯ã¿ã³ã¹(self inductance)ãšãããçžäºã€ã³ãã¯ã¿ã³ã¹ã®æ¬¡å
ã¯ãèªå·±ã€ã³ãã¯ã¿ã³ã¹ã®åäœãšåãã§ãã³ãªãŒ(H)ã§ããã
ãã®çžäºã€ã³ãã¯ã¿ã³ã¹ã®å€§ããã¯ãäž¡æ¹ã®ã³ã€ã«ã®å·»ãæ°ã©ããã®ç©ã«æ¯äŸããã
ç£å Žã®åãã«ãã£ãŠé»å ŽãåŒãèµ·ããããããšãé»ç£èªå°ã®ã»ã¯ã·ã§ã³ã§èŠãã å®éã«ã¯é»å Žã®å€åã«ãã£ãŠç£å ŽãåŒãèµ·ããããããšãç¥ãããŠããã ããã«ãã£ãŠäœããªã空éäžãé»å Žãšç£å ŽãäŒæããŠããããšãäºæ³ãããã
é»ç£æ³¢ã®é床ãç©çåŠè
ã®ãã¯ã¹ãŠã§ã«ãèšç®ã§æ±ãããšãããé»ç£æ³¢ã®é床ã¯ãç空äžã§ã¯åžžã«äžå®ã§ããã€æ³¢ã®é床cãèšç®ã§æ±ãããšããã
ãšãªããæ¢ã«ç¥ãããŠããå
éã«äžèŽããã ãã®ããšãããå
ã¯é»ç£æ³¢ã®äžçš®ã§ããããšãåãã£ããç©çIIã§ãé»ç£æ³¢ã®é床ãæ±ããèšç®ã¯ã詳ããã¯æ±ãã èªè
ãå
éã®æž¬å®å®éšã«ã€ããŠèª¿ã¹ããªããç©çIã®æ³¢åã«é¢ããããŒãžãªã©ã§ãã£ãŸãŒã®å®éšã«ã€ããŠãåç
§ã®ããšã
æ³¢ã¯æ³¢é·Î»ãé·ãã»ã©ãæ¯åæ°fãå°ãããªããæ³¢ã®æ³¢é·Î»ãšæ¯åæ°fã®ç©fλã¯äžå®ã§ãããã¯æ³¢ã®é床vã«çãããã€ãŸã
ã§ããã é»ç£æ³¢ã®å Žåã¯ãé床ãå
éã®cãªã®ã§
ã§ããã
æŸéçšã®ãã¬ããã©ãžãªã®é»æ³¢(ã§ãã±ãradio wave)ã¯ãé»ç£æ³¢(electromagnetic wave)ã®äžçš®ã§ãããæ³¢é·ã0.1mm以äžã®é»ç£æ³¢ãé»æ³¢ã«åé¡ãããããªããé»æ³¢ã®ãã¡ãæ³¢é·ã1mm~1cmã®ããªã¡ãŒãã«ã®é»æ³¢ãããªæ³¢ãšãããåæ§ã«ãæ³¢é·ã1cm~10cmã®é»æ³¢ãã»ã³ãæ³¢ãšãããæ³¢é·10cm~100cm(=1m)ã®é»æ³¢ã¯UHFãšèšããããã¬ãæŸéãªã©ã«äœ¿ãããUHFæŸéã¯ããã®é»æ³¢ã§ãããæ³¢é·1m~10mã®é»æ³¢ã¯VHFãšèšãããããã¬ãæŸéã®VHFæŸéã¯ããã®é»æ³¢ã§ããã
æ³¢é·ã0.1mm以äžã§ãå¯èŠå
ç·(å¯èŠå
ã®æ倧波é·ã¯780ããã¡ãŒãã«çšåºŠ)ãããã¯æ³¢é·ãé·ãé»ç£æ³¢ã¯èµ€å€ç·(ãããããããinfrared raysãã€ã³ãã©ã¬ãŒã ã¬ã€ãº)ãšããããèµ€ãã®ãå€ããšããçç±ã¯ãå¯èŠå
ã®æ倧波é·ã®è²ãèµ€è²ã ããã§ãããèµ€å€ç·ãã®ãã®ã«ã¯è²ã¯ã€ããŠããªããåžè²©ã®èµ€å€ç·ããŒã¿ãŒãªã©ãèµ€è²ã«çºå
ãã補åãããã®ã¯ã䜿çšè
ãåäœç¢ºèªãã§ããããã«ããããã«ã補åã«èµ€è²ã®ã©ã³ãã䜵眮ããŠããããã§ãããèµ€å€ç·ã¯ãç©äœã«åžåããããããåžåã®éãç±ãçºçããã®ã§ãããŒã¿ãŒãªã©ã«å¿çšãããããªãã倪éœå
ã«ãèµ€å€ç·ã¯å«ãŸããã
ããããèµ€å€ç·ãçºèŠãããçµç·¯ã¯ãã€ã®ãªã¹ã®å€©æåŠè
ã®ããŒã·ã§ã«ã倪éœå
ãããªãºã ã§åå
ããéã«ãèµ€è²ã®å
ç·ã®ãšãªãã®ãç®ã«ã¯è²ãèŠããªãéšåã枩床äžæããŠããããšãçºèŠããããšããçµç·¯ãããã
æã
ã人éã®ç®ã«èŠããå¯èŠå
ç·(ãããããããvisible light)ã®æ³¢é·ã¯ãçŽ780ããã¡ãŒãã«ããçŽ380ããã¡ãŒãã«ã®çšåºŠã§ãããå¯èŠå
ã®äžã§æ³¢é·ãæãé·ãé åã®è²ã¯èµ€è²ã§ãããå¯èŠå
ã®äžã§æ³¢é·ãæãçãé åã®è²ã¯çŽ«è²ã§ããã
å
ãã®ãã®ã«ã¯ãè²ã¯ã€ããŠããªããæã
ã人éã®è³ããç®ã«å
¥ã£ãå¯èŠå
ããè²ãšããŠæããã®ã§ããã 倪éœå
ãããªãºã ãªã©ã§åå
(ã¶ããã)ãããšãæ³¢é·ããšã«è»è·¡(ããã)ãããããããã®åå
ããå
ç·ã¯ãä»ã®æ³¢é·ãå«ãŸãããã äžçš®ã®æ³¢é·ãªã®ã§ããã®ãããªå
ç·ããã³å
ãåè²å
(monochromatic light)ãšããã ãŸããçœè²ã¯åè²å
ã§ã¯ãªããçœè²å
(white light)ãšã¯ãå
šãŠã®è²ã®å
ãæ··ãã£ãç¶æ
ã§ããã åæ§ã«ãé»è²ãšããåè²å
ããªããé»è²ãšã¯ãå¯èŠå
ãç¡ãç¶æ
ã§ããã
玫å€ç·(ããããããultraviolet rays)ã¯ååŠåå¿ã«åœ±é¿ãäžããäœçšã匷ãã殺èæ¶æ¯ãªã©ã«å¿çšãããã倪éœå
ã«ã玫å€ç·ã¯å«ãŸããã人éã®èã®æ¥çŒãã®åå ã¯ã玫å€ç·ãã¡ã©ãã³è²çŽ ãé
žåãããããã§ããã
èµ€å€ç·ã¯å€ªéœå
ã®ããªãºã ã«ããåå
ã§çºèŠãããã ãã§ã¯ãåå
ããã玫è²ã®å
ç·ã®ãšãªãã«ãããªã«ãç®ã«ã¯èŠããªãç·ãããã®ã§ã¯?ããšãããµããªããšãåŠè
ãã¡ã«ãã£ãŠèãããã ãã€ãã®ç©çåŠè
ãªãã¿ãŒã«ããååŠçãªå®éšæ¹æ³ãçšããŠã玫å€ç·ã®ååšãå®èšŒãããã
å»ççšã®ã¬ã³ãã²ã³ãªã©ã®ééåçã§çšããããXç·(X-ray)ãé»ç£æ³¢ã®äžçš®ã§ãããçç©ã®çŽ°èãååã¬ãã«ã§å·ã€ããçºããæ§ãæãã ã¬ã³ãç·(gammaârayãγ ray)ãåæ§ã«ãééåçã«ãå¿çšãããããçç©ã®çŽ°èãååã¬ãã«ã§å·ã€ããçºããæ§ãæãã
?? | [
{
"paragraph_id": 0,
"tag": "p",
"text": "é«çåŠæ ¡çç§ ç©çI > é»æ°",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "æ¬é
ã¯é«çåŠæ ¡çç§ ç©çIã®é»æ°ã®è§£èª¬ã§ããã",
"title": ""
},
{
"paragraph_id": 2,
"tag": "p",
"text": "",
"title": ""
},
{
"paragraph_id": 3,
"tag": "p",
"text": "çŸåšç§ãã¡ã䜿ã£ãŠããå€ãã®è£œåãé»æ°ãçšããŠåããŠããã ããã«ã¯æ§ã
ãªçç±ãèãããããããŸã第äžã«é»æ°ã¯æ§ã
ãªå¥ã®ãšãã«ã®ãŒã«å€æã§ããããšãäŸãã°é»ç±ç·ã䜿ã£ãŠç±ã«ãé»çãçºå
ãã€ãªãŒãã䜿ã£ãŠå
ã«ãã¢ãŒã¿ãŒã䜿ã£ãŠéåã«å€æããããšãåºæ¥ãã次ã«ãé»æ± ãã³ã³ãã³ãµã䜿ã£ãŠãšãã«ã®ãŒãç¶æãããŸãŸæã¡éã¶ããšãåºæ¥ãããšããé»ç·ã䜿ã£ãŠé·è·é¢ãéé»ã§ããããšããŸããé»å補åã®èšç®èœåãä¿¡å·ã®äŒéèœåãåªããŠããããšããŸãæ¯èŒçã«å®å
šã«å°éã®ãšãã«ã®ãŒãåãåºããããšãçãèããããã",
"title": "é»æ°"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "",
"title": "é»æ°"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "é»æ°ãéåã«å€ãããã®ãšã㊠ã¢ãŒã¿ãŒãããã ããã®éã®ãã®ãšããŠã éåãé»æ°ã«å€ããããšãåºæ¥ãã ãããè¡ãªãã®ã¯çºé»æ©ãšåŒã°ããã äŸãã°ãçºé»æã¯äœããã®éåã®ãšãã«ã®ãŒã å©çšããŠé»æ°ãããããŠããã äŸãã°ãæ°Žåçºé»æã§ã¯ã æ°Žã®èœäžããåãå©çšããŠããã 倧éã®æ°Žãèœäžãããšãã«ã¯ 人éãäœå人ãããã£ãŠãããã åºæ¥ãããšããªãããããšãããã ããšãã°ãåãç«ã£ã海岞ç·ãªã©ã¯ äž»ã«æ°Žã®æµãã«ãã£ãŠäœãããŠããã ãã®ããã«ãæ°Žã®åã¯åŒ·å€§ã§ããã®ã§ã ãããäžæãå©çšããæ¹æ³ããããš éœåããããå®éçŸä»£ã§ã¯ é»æ°ãåªä»ãšããŠããã®åã åãã ãããšã«æåããŠããã",
"title": "é»æ°"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "é»æ± ã®ããã«é»æ¥µã®+ãš-ãå®ãŸã£ã é»æµãçŽæµé»æµãšåŒã¶ã äžæ¹ãçºé»æããåŸãããé»æµã®ããã« +ãš-ãéãé床ã§å
¥ãæãã é»æµã亀æµé»æµãšåŒã¶ã",
"title": "é»æ°"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "å®éã«ã¯ãã€ãªãŒããçšã㊠亀æµãçŽæµã«å€ã㊠䜿ãããšãããè¡ãªãããã",
"title": "é»æ°"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "äœããªã空éãå
ãçŽé²ããŠããããã« èŠããããšããããå®éã«ã¯ãã㯠é»æ³¢ãšåããã®ã§ããã é»æ³¢ãšã¯äŸãã°ãæºåž¯é»è©±ã®éä¿¡ã«äœ¿ããããã®ã§ããã é»è·ãæã£ãç©äœãåãããšãå¿
ç¶çã« çãããã®ã§ããã",
"title": "é»æ°"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "ãã©ã¹ããã¯ã®äžæ·ããªã©ã§é«ªãããããšåž¯é»ããçŸè±¡ãªã©ã®ããã«ãç©è³ªãé»æ°ã垯ã³ãããšã垯é»(ããã§ã)ãšãããç©äœãããã£ãŠçºçãããéé»æ°ãæ©æŠé»æ°ãšããã ã¬ã©ã¹æ£ãçµ¹ã®åžã§ããããšãã¬ã©ã¹æ£ã¯æ£ã®é»æ°ã«åž¯é»ããçµ¹ã¯è² ã®é»æ°ã«åž¯é»ããã é»æ°ã®éãé»è·(ã§ãããcharge)ãšããããããã¯é»æ°éãšããã",
"title": "éé»æ°"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "é»è·ã®åäœã¯ã¯ãŒãã³ã§ãããã¯ãŒãã³ã®èšå·ã¯Cã§ããã",
"title": "éé»æ°"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "éé»æ°ã«ããé»è·ã©ããã«åãåãéé»æ°åãšããã",
"title": "éé»æ°"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "ãªãã垯é»ããŠããªãç¶æ
ãé»æ°çã«äžæ§ã§ããããšããã",
"title": "éé»æ°"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "éå±ã®ããã«ãé»æ°ãéããç©äœãå°äœ(ã©ããããconductor)ãšããããã©ã¹ããã¯ãã¬ã©ã¹ããŽã ã®ããã«é»æ°ãéããªãç©è³ªã絶çžäœ(ãã€ãããããinsulator)ãããã¯äžå°äœ(ãµã©ããã)ãšããã",
"title": "éé»æ°"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "éå±ã¯å°äœã§ããã",
"title": "éé»æ°"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "é»æ°ã®æ£äœã¯é»å(electron)ãšããç²åã§ããããã®é»åã¯è² é»è·ã垯ã³ãŠããã(é»åã®é»è·ãè² ã«å®çŸ©ãããŠããã®ã¯ã人é¡ãé»åãçºèŠããåã«é»è·ã®æ£è² ã®å®çŸ©ãè¡ãããããšããé»åãèŠã€ãã£ãéã«é»åã®é»è·ã調ã¹ããè² é»è·ã ã£ãããã§ããã)",
"title": "éé»æ°"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "éå±ãå°äœãªã®ã¯ãéå±äžã®é»åã¯ãããšã®ååãé¢ããŠããã®éå±å
šäœã®äžãèªç±ã«åããããã§ãããéå±äžã®é»åã®ããã«ãç©è³ªäžãèªç±ã«åããç¶æ
ã®é»åããèªç±é»å(ãããã§ãã)ãšããã",
"title": "éé»æ°"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "é»æµãšã¯ãèªç±é»åã移åããããšã§ããã",
"title": "éé»æ°"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "ãã£ãœãã絶çžäœã¯ãèªç±é»åããããªãã絶çžäœã®é»åã¯ããã¹ãŠãããšã®ååã«æçž(ããã°ã)ãããŠéã蟌ããããŠããŠãèªç±ã«ã¯åããªãã",
"title": "éé»æ°"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "æ£é»è·ãšã¯ãç©è³ªã«é»åãæ¬ ä¹ããŠããç¶æ
ã§ããã è² é»è·ãšã¯ãç©è³ªãé»åãå€ãæã£ãŠããç¶æ
ã§ããã",
"title": "éé»æ°"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "垯é»ããŠããªã絶çžäœã®ç©è³ªããããããããŠãäž¡æ¹ãæ©æŠé»æ°ã«åž¯é»ãããå Žåãçæ¹ã¯æ£é»è·ãçããããçæ¹ã®ç©è³ªã¯è² é»è·ãçããããã®ãšããçºçããæ£é»è·ã®å€§ãããšè² é»è·ã®å€§ããã¯åãã§ããã ããã¯ãé»åã移åããŠãçæ¹ã®ç©è³ªã¯é»åãäžè¶³ããããçæ¹ã¯çéã®é»åãéå°ã«ãªã£ãŠããããã§ããã",
"title": "éé»æ°"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "ãã®ããã«ãé»åã¯çæãæ¶æ»
ãããªãããããé»è·ä¿åã®æ³åãšèšãããããã¯é»æ°éä¿åã®æ³åãšèšãã",
"title": "éé»æ°"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "é»æ°çã«äžæ§ã§ãã£ãå°äœã®ç©è³ª(ä»®ã«ç©è³ªAãšãã)ã«åž¯é»ããå¥ã®ç©è³ª(ä»®ã«ç©è³ªBãšãã)ãæ¥è§Šãããã«è¿ã¥ãããšãç©è³ªAã«ã¯ã垯é»ç©è³ªBã®é»è·ã«åŒãå¯ããããŠãç©äœAã®å
éšã§å察笊å·ã®é»è·ã垯é»ç©äœBã«è¿ãåŽã®è¡šé¢ã«çããããŸãã垯é»ç©äœBãšåãé»è·ã¯åçºããã®ã§ãç©äœAå
éšã®åž¯é»ç©äœBãšã¯é ãåŽã®è¡šé¢ã«çããã",
"title": "éé»æ°"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "ãã®ãããªçŸè±¡ãéé»èªå°(ããã§ãããã©ã;Electrostatic induction)ãšãããéé»èªå°ã§çããé»è·ã®æ£é»è·ã®éãšè² é»è·ã®éã¯çéã§ããã(é»æ°éä¿åã®æ³å)",
"title": "éé»æ°"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "å°äœã®å
éšã«éé»æ°åã¯ç¡ãããããã£ããšãããšãèªç±é»åãªã©ã®é»è·ãåããé»æµãæµãç¶ããããšã«ãªããããã®ãããªçŸè±¡ã¯å®åšããªãã®ã§äžåçã«ãªãããããã£ãŠãå°äœã®å
éšã«éé»æ°åã¯ç¡ãã",
"title": "éé»æ°"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "è¡šé¢ã«é»è·ãéãŸãã®ã¯ãå°äœã®å
éšã«éé»æ°åãäœãããªãããã§ããããããã£ãŠéé»èªå°ã§åŒãå¯ããããé»è·ã®å€§ããã¯ãå€éšããå°äœå
éšãžã®éé»æ°åãæã¡æ¶ãã ãã®å€§ããã§ããã",
"title": "éé»æ°"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "ãã®å°äœå
éšã®é»è·ããŒãã«ãªãæ§è³ªãå¿çšãããšãäžç©ºã®å°äœã§åºæ¥ãç©äœãçšããŠãéé»æ°åãé®èœããããšãã§ããããããéé»é®èœ(ããã§ããããžããelectric shilding)ãšããã",
"title": "éé»æ°"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "絶çžäœ(ä»®ã«Aãšãã)ã«é»è·ãè¿ã¥ããå Žåã¯ãå°äœãšã¯éããç©äœAã®å
éšã®é»åã¯èªç±ã«è¡šé¢ã«ã¯éãŸããªãããç©äœå
éšã®ååã®æ£è² ã®é»è·ã®æ¥µæ§ãæã£ãéšåããå€éšã®éé»æ°åã«åŒãå¯ããããããã«ãè¿ã¥ããé»è·ã«è¿ãåŽã«ã¯ç°çš®ã®é»è·ãçããé ãåŽã«ã¯ãåçš®ã®é»è·ãçããã ååãååãå€éšã®éé»æ°åã«ãã£ãŠãæ£è² ã®é»è·ã®éšåãçããããšãå極(ã¶ãããã)ãšããããå€éšã®é»åã«ãã£ãŠèµ·ããããã®ãããªå極ã®ããããèªé»å極(ããã§ãã¶ãããããdielectric polarization)ãšããã",
"title": "éé»æ°"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "絶çžäœã¯ãéé»æ°åã«ããããããšèªé»å極ãè¡ãã®ã§ã絶çžäœã®ããšãèªé»äœ(ããã§ããããdielectric)ãšãããã",
"title": "éé»æ°"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "å°äœã«éé»èªå°ãããæ£è² ã®é»è·ã¯ãå°äœãåæãªã©ãããã°æ£é»è·ãšè² é»è·ãå¥åã«åãåºãããšãã§ããããããèªé»äœã®æ£è² ã®é»è·ã¯ãååãååãšå¯æ¥ã«çµã³ã€ããŠãããããæ£è² ã®é»è·ãåãããŠåãåºãããšã¯åºæ¥ãªãã",
"title": "éé»æ°"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "",
"title": "éé»æ°"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "ããç©è³ªãé»æ°ã垯ã³ãŠãã(垯é»ããŠãã)ãšãããã®åž¯é»ã®å€§å°ã®çšåºŠãé»è·(ã§ãããelectric charge)ãšãããããŸããŸãªç©è³ªããããããªæ¹æ³ã§åž¯é»ãããçµæãé»è·ã«ã¯ã垯é»ãã2åã®ãã®ã©ãããè¿ã¥ããæã«åŒã£åŒµãåããã®(åŒåãåã)ãšåçºããããã®(æ¥åãã¯ããã)ã®2çš®é¡ãããããšãåãã£ãã ãã®ãããªã垯é»ããŠããç©äœã«åãåãéé»æ°åãšããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "ã¹ã€ã®åž¯é»ãããã®ããä»ã«ãããã€ãçšæããŠãè¿ã¥ããŠå®éšãã2åã®ç©äœã®çµã¿åãããå€ãããšãçµã¿åããã«ãã£ãŠã2åã®ç©äœã©ããã«åŒåãåãå Žåãããã°ãæ¥åãåãå Žåãããããšãåãã£ãããã®åŒåãšæ¥åã®é¢ä¿ã¯ã垯é»ããé»è·ã®çš®é¡ã«å¿ããããšãããã£ãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "çµè«ãèšããšãé»è·ã«ã¯æ£è² ã®2çš®é¡ããããæ£ã®é»è·ã©ããã®ç©äœãè¿ã¥ãããšãã¯åçºããããè² ã®é»è·ã©ãããè¿ã¥ãããšããåçºããããæ£ãšè² ã®é»è·ãè¿ã¥ããæã«ã¯åŒåãåãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "ã€ãŸããå笊å·ã®é»è·ãè¿ã¥ããå Žåã¯ãåçºåãçãããç°ç¬Šå·ã®é»è·ãè¿ã¥ããå Žåã¯ãåŒåãçããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "éé»æ°ã©ããã®åã®åŒ·ãã¯ãå®éšçã«ã¯ãé»è·ã®éã«åãåã¯ãéåã®å Žåãšåæ§ã«åãåãŒãåã2ç©äœã®éã®è·é¢ã®2ä¹ã«åæ¯äŸããããšãç¥ãããŠãããæŽã«ãé»è·ã®å€§ããã倧ããã»ã©é»è·éã«åãåã倧ããããšãèæ
®ãããšãè·é¢rã ãé¢ããŠãããããé»è· q 1 {\\displaystyle q_{1}} ã q 2 {\\displaystyle q_{2}} ãæã£ãŠãã2ç©äœã®éã«åãåFã¯ã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "ã§äžããããããããã¯ãŒãã³ã®æ³å( Coulomb's law)ãšãããããã§ã k {\\displaystyle k} ã¯æ¯äŸå®æ°ã§ãããäž¡é»è·ã®åšå²ã«ããç©äœã®çš®é¡ã«ããå€åããå®æ°ã§ãããç空äžã§ã®é»å Žãèããå Žåã®kã®å€ã¯ã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "ã§ããããŸãã ε {\\displaystyle \\epsilon } ã¯åŸã»ã©ç»å Žããèªé»ç(ããã§ããã€)ãšåŒã°ããç©çå®æ°ã§ãããèªé»çã¯ãäž¡é»è·ã®åšå²ã«ããç©äœã®çš®é¡ã«ããå€åããå®æ°ã§ãããèªé»çã«ã€ããŠã¯ããã®æãåããŠèªãã 段éã§ã¯ããŸã ç¥ããªããŠãè¯ããã®ã¡ã«ç©çIIã§èªé»çã詳ãã解説ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "èªé»ç ε {\\displaystyle \\epsilon } ãšã¯ãŒãã³ã®æ¯äŸå®æ°kã«ã¯äžåŒã®é¢ä¿",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "ãããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 41,
"tag": "p",
"text": "ç©äœã®ãŸããã«èç©ããããã®ãé»è·ãšåŒã¶ãé»æ°åã«ãã£ãŠåçºããã£ãããåŒãã€ããã£ããããç©äœãé»è·ãæã€ç©äœãšåŒã¶ããŸããããã§èŠ³å¯ãããéé»æ°åããã¯ãŒãã³åãšåŒã¶ããšãããã 2åã®é»è·ã©ããããããŒãåã¯åãã§ããããããã£ãŠäœçšã»åäœçšã®æ³åã«åŸã£ãŠããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 42,
"tag": "p",
"text": "ããã§ãé»è·ã®åäœã¯[C]ã§äžãããããèšå·ã®Cã¯ãã¯ãŒãã³ããšèªãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 43,
"tag": "p",
"text": "",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 44,
"tag": "p",
"text": "å³ã®ããã«ã2æ¬ã®ç³žã«ãããããåã質ém[kg]ã§ãåã笊å·ãšå€§ããã®é»è·q[C]ã®çããã¶ãããã£ãŠããããã¯ãã¯ãŒãã³åã§åçºããã®ã§ãå³ã®ããã«ã糞ãè§åºŠÎžããªãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 45,
"tag": "p",
"text": "ãã®ãšãã質émã«ããéåãšãé»è·qã«ããã¯ãŒãã³åãšã®é¢ä¿ã«ã€ããŠãåŒãç«ãŠãããªããå¿
èŠãªãã°ã糞ã®åŒµåã¯T[N]ãšããããšã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 46,
"tag": "p",
"text": "解æ³",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 47,
"tag": "p",
"text": "å³ã®ãããªäœçœ®é¢ä¿ã«ãªãã®ã§ãå³ã®ããã«åŒãç«ãŠãã°ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 48,
"tag": "p",
"text": "",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 49,
"tag": "p",
"text": "",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 50,
"tag": "p",
"text": "â» äžèšã®2æ¬ã®ç³žã«ã¶ãããã£ãçã®ã¯ãŒãã³åã®äŸé¡ã¯ãé»æ°ç£æ°åŠã®ã©ã®å
¥éæžã«ããããããªå
žåçãªåé¡ã§ããã®ã§ãèªè
ã¯ãã¡ããšç解ããããšã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 51,
"tag": "p",
"text": "",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 52,
"tag": "p",
"text": "é»è· q 1 {\\displaystyle q_{1}} , q 2 {\\displaystyle q_{2}} ã®éã®è·é¢ãrã®å Žåãš2rã®å Žåã§ã¯ãéã«åãåã®å€§ããã¯ã©ã¡ããã©ãã ã倧ãããçããã ãŸããè·é¢ã2rã®æã®2ç¹éã®åã®å€§ãããçããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 53,
"tag": "p",
"text": "ã¯ãŒãã³åã¯ãç©äœéã®è·é¢ã®é2ä¹ã«æ¯äŸããã®ã§ãè·é¢ã2rã®æã¯ãrã®æã®å€§ããã® 1 4 {\\displaystyle {\\frac {1}{4}}} ãšãªãããŸããåãåã®å€§ããã¯ãã¯ãŒãã³åã®åŒãçšããŠã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 54,
"tag": "p",
"text": "ãšãªãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 55,
"tag": "p",
"text": "æ¢ã«ãããé»è·Aã®ãŸããã®å¥ã®é»è·Bã«ã¯ããã®é»è·ããã®è·é¢ã®é2ä¹ã«æ¯äŸããåããããããšãè¿°ã¹ãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 56,
"tag": "p",
"text": "ããã§ãé»è·Bãåããåã¯ããã®é»è·Bã®å€§ããã«æ¯äŸããããšãåãããŠèãããšããã®é»è·Bã®å€§ããã«ããããããé»è·Aã®å€§ããã ãã§æ±ºãŸãéãå°å
¥ããŠãããšéœåããããããã§ããã®ãããªéãšããŠé»å Ž(ã§ãã°)ãå°å
¥ããããã®ãšããé»å Ž E â {\\displaystyle {\\vec {E}}} ã®äžã«ããé»è· q {\\displaystyle q} ã«åãå F â {\\displaystyle {\\vec {F}}} ã¯ã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 57,
"tag": "p",
"text": "ã§äžãããããé»å Žã¯åäœé»è·ã«åãåãšèããããšãã§ããé»å Žã®åäœã¯[N/C]ã§ããããé»å Žãã¯ããé»çã(ã§ããã)ãšãåŒã°ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 58,
"tag": "p",
"text": "(æ¥æ¬ã®ç©çåŠã§ã¯ãé»å ŽããšåŒã¶ããšãå€ãããŸããæ¥æ¬ã®é»æ°å·¥åŠã§ã¯ãé»çããšåŒã°ããããšãå€ããææ²»æã®ç¿»èš³ã®éã®ãæ¥æ¬åœå
ã®æ¥çããšã®éãã«éããããããªãæ¥æ¬ããŒã«ã«ãªéœåã§ãããåŒã³æ¹ã¯ç©çã®æ¬è³ªãšã¯é¢ä¿ãªãã®ã§ãããã§ã¯ãã©ã¡ãã®è¡šçŸãçšãããã¯ãæ¬æžã§ã¯ç¹ã«ãã ãããªããè±èªã§ã¯ç©çåŠã»é»æ°å·¥åŠãšãâelectric fieldâã§å
±éããŠããã)",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 59,
"tag": "p",
"text": "äžã®ã¯ãŒãã³åã®çµæãšåããããšãé»è·Aã®ãŸããã«å¥ã®é»è·ãååšããªããšããé»è· q {\\displaystyle q} [C]ã®é»è·ããŸãšãé»å Ž E â {\\displaystyle {\\vec {E}}} ã¯ã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 60,
"tag": "p",
"text": "ã§äžããããããã ããrã¯é»è·ããã®è·é¢ã§ããã e â r {\\displaystyle {\\vec {e}}_{r}} ã¯ãé»è·ãšããç¹ãçµãã çŽç·äžã§ãé»è·ãšå察æ¹åãåããåäœãã¯ãã«ã§ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 61,
"tag": "p",
"text": "é»è·ã®åãã®é»å Žã¯ãå¹³é¢äžã§æŸå°ç¶ã®ãã¯ãã«ãšãªãããšã«æ³šæã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 62,
"tag": "p",
"text": "é»å Žã¯ãã¯ãã«ã§ãããé»è·ã2åãããšãã¯ãããããã®é»è·ãã€ããé»å Žããéãåãããã°ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 63,
"tag": "p",
"text": "ã§ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 64,
"tag": "p",
"text": "é»è·ã3å以äžã®ãšãããåæ§ã«éãåãããã°è¯ãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 65,
"tag": "p",
"text": "å³ã®ããã«ãé»è·ããåºãé»å Žã®æ¹åãå³ç€ºãããã®ãé»æ°åç·(ã§ãããããããelectric line of force)ãšããã é»è·ãè€æ°ããå Žåã«ã¯ãå®éã«æ°ãã«çœ®ãããé»è·ãåããåã¯ããããã足ãåããããã®ãšãªãããããã£ãŠãè€æ°ã®é»è·ãããå Žåã®åšå²ã®é»çã¯ãããããã®é»è·ãäœãé»çãã¯ãã«ã®åãšãªã(éãåããã®åç)ã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 66,
"tag": "p",
"text": "é»æ°åç·ãå³ç€ºããå Žåã¯ãæ£é»è·ããåç·ãåºãŠãè² é»è·ã§åç·ãåžåãããããã«æžããåç·ã¯ãé»å Žãå³ç€ºãããã®ãªã®ã§ãé»è·ä»¥å€ã®å Žæã§ã¯ãåç·ãåå²ããããšã¯ãªãã åç·ãçæããã®ã¯æ£é»è·ã®å Žæã®ã¿ã§ãããåç·ãæ¶æ»
ããã®ã¯ãè² é»è·ã®å Žæã®ã¿ã§ããã èšãæããã°ãåç·ãé»è·ä»¥å€ã®å Žæã§æ¶æ»
ããããšã¯ãªãããé»è·ä»¥å€ã®å Žæã§åç·ãçæããããšã¯ãªãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 67,
"tag": "p",
"text": "å°äœã®å
éšã®é»å Žã¯ãŒãã§ãã£ããèšãæããã°ãé»æ°åç·ã¯ãå°äœã®å
éšã«ã¯é²å
¥ã§ããªãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 68,
"tag": "p",
"text": "",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 69,
"tag": "p",
"text": "ç¹é»è·ããã¯ãå³ã®ããã«ãæŸå°ç¶ã«é»æ°åç·ãåºããã¯ãŒãã³ã®æ³åã®ä¿æ°ã«ãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 70,
"tag": "p",
"text": "ã®ãã¡ã®ãåæ¯ã® 4 Ï r 2 {\\displaystyle 4\\pi r^{2}} ã¯ãçã®è¡šé¢ç©ã®å
¬åŒã«çããã®ã§ãé»æ°åç·ã®å¯åºŠã«æ¯äŸããŠãé»å Žã®åŒ·ããããã¯éé»æ°åã®åŒ·ãã決ãŸããšèããããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 71,
"tag": "p",
"text": "éé»èªå°ã§ã¯ãå°äœå
éšã«ã¯éé»æ°åãåããŠããªãã®ã§ãã£ããããã¯ãé»å ŽãšããæŠå¿µãçšããŠèšãæããã°ãå°äœå
éšã®é»å Žã¯ãŒãã§ããããšèšããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 72,
"tag": "p",
"text": "ã¯ãŒãã³åã¯å(ã¡ãã)ã§ãããããããã«éãã£ãŠå¥ã®é»è·ãè¿ã¥ããå Žåã¯ãè¿ã¥ããå¥ã®é»è·ã¯ä»äºãããããšã«ãªãããŸããè¿ã¥ããé»è·ãææŸãã°ãã¯ãŒãã³åã«ãã£ãŠåãåããä»äºãããããšã«ãªããããè¿ã¥ããç¶æ
ã«ããå¥é»è·ã¯äœçœ®ãšãã«ã®ãŒãèããŠããããšã«ãªãã ãããã£ãŠãã¯ãŒãã³åã«å¯ŸããŠãäœçœ®ãšãã«ã®ãŒãå®çŸ©ããããšãã§ããã(ãªããè¡æè»éäžã®ç©äœã®ãããªãå°è¡šãã倧ããé¢ããå Žæã®éåããã¯ãŒãã³åãšåæ§ã«é2ä¹åãªã®ã§ãããã§èããèšç®ææ³ã¯éåã«ããäœçœ®ãšãã«ã®ãŒã«ãå¿çšã§ãããéåå é床gãçšããåmgãšããã®ã¯å°è¡šè¿ãã§ã®è¿äŒŒã«ãããªãã)",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 73,
"tag": "p",
"text": "ã¯ãŒãã³åã«ããé»å Žã®å®çŸ©ã§ã¯ãåäœé»è·ã«å¯ŸããŠé»å Žãå®çŸ©ããã®ãšåæ§ãäœçœ®ãšãã«ã®ãŒã«å¯ŸããŠããåäœé»è·ã«å¿ããŠå®çŸ©ã§ããéãå°å
¥ãããšéœåãããããã®ãããªéãé»äœ(ã§ãããelectric potential)ãšåŒã¶ãé»äœã®åäœã¯ãã«ããšãããé»äœãäŸãããšãå°è¡šè¿ãã§ã®éåã®äœçœ®ãšãã«ã®ãŒãèããéã®ãghããªã©ã«çžåœããéã§ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 74,
"tag": "p",
"text": "ã¯ãŒãã³åã®çµæãšã q {\\displaystyle q} [C]ã®é»è·ããè·é¢rã ãé¢ããç¹ã®é»äœVã¯ãé»å Žã®ç©åèšç®ã§åŸãããã(ç©åããŸã ç¿ã£ãŠãªãåŠå¹Žã®èªè
ã¯ãåãããªããŠãæ°ã«ããã次ã®çµæãžãšé²ãã§ãã ããã)çµæã®ã¿ãèšããšã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 75,
"tag": "p",
"text": "ãšãªãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 76,
"tag": "p",
"text": "é»äœVã®ç¹ã«q[C]ã®é»è·ã眮ãããšãããã®é»è·ã®ã¯ãŒãã³åã«ããäœçœ®ãšãã«ã®ãŒU[J]ã¯ãé»äœVãçšããã°ã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 77,
"tag": "p",
"text": "ãšãªãããããã£ãŠãé»äœ V 1 {\\displaystyle V_{1}} ãã«ãã®ç¹ããé»äœ V 2 {\\displaystyle V_{2}} ãã«ãã®äœçœ®ãžãšé»è·q[C]ãéé»æ°åãåããŠç§»åãããšããéé»æ°åã®ããä»äºW[J]ã¯",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 78,
"tag": "p",
"text": "ãšãªãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 79,
"tag": "p",
"text": "",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 80,
"tag": "p",
"text": "ãã£ãœããäžæ§ãªé»å Žã«ãããŠã¯ãé»äœã®åŒããé»å ŽãçšããŠç°¡åã«è¡šãããšãã§ãããè·é¢dã ãé¢ããå¹³è¡å¹³æ¿é»æ¥µã®éã«äžæ§ãªé»å Ž E â {\\displaystyle {\\vec {E}}} ãçããŠãããšãããã®é»çã®äžã«çœ®ããé»è·qã¯éé»æ°å q E â {\\displaystyle q{\\vec {E}}} ãåããããã®é»è·ãé»çã®åãã«æ²¿ã£ãŠäžæ¹ã®é»æ¥µããä»æ¹ã®é»æ¥µãŸã§ç§»åãããšããé»çã®ããä»äºW㯠W = q E d {\\displaystyle W=qEd} ãšãªããããããã2極æ¿ã®é»äœå·®Vã¯ã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 81,
"tag": "p",
"text": "ã§è¡šãããšãã§ããããšãããããåŒãå€åœ¢ããŠ",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 82,
"tag": "p",
"text": "ãšããããšãã§ãããããã§ãåäœãèãããšãå³èŸºã¯é»å§ãè·é¢ã§å²ã£ããã®ã§ãããããé»çã®åäœãšããŠ[N/C]ã®ã»ã[V/m]ãçšããããšãã§ããããšããããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 83,
"tag": "p",
"text": "é»äœã®åäœã¯ãã«ãã§ããããã®éã¯æ¢ã«äžåŠæ ¡çç§ãªã©ã§æ±ã£ãé»å§(ã§ããã€ãvoltage)ã®åäœãšåãåäœã§ãããå®éã«é»æ°åè·¯ã«é»å§ããããããšã¯ãåè·¯äžã®é»åã«é»å ŽããããŠåããããšãšçããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 84,
"tag": "p",
"text": "éé»èªå°ã«ãã£ãŠãå°äœå
éšã®é»å Žã¯ãŒãã§ãã£ãããã®ããšãããå°äœã®è¡šé¢ã¯ãé»äœãçãããå°äœè¡šé¢ã¯äºãã«çé»äœã§ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 85,
"tag": "p",
"text": "é»äœã®åºæºã¯ãå®çšäžã¯ãå°é¢ã®é»äœããŒãã«çœ®ãããšãå€ããé»æ°åè·¯ã®äžéšã倧å°ã«ã€ãªãããšãæ¥å°(ãã£ã¡)ãŸãã¯ã¢ãŒã¹(earth)ãšãããåè·¯ãã¢ãŒã¹ããŠããã®ã€ãªãã éšåã®é»äœããŒããšèŠãªãããšãå€ãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 86,
"tag": "p",
"text": "çŽç·äžã§è·é¢0, b[m]ã®ç¹ã«ãé»è·q, q'ãæã€ç©äœã眮ããŠããããã®æãäœçœ®a[m](a<b)ã®ç¹ã®é»äœãæ±ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 87,
"tag": "p",
"text": "é»äœã®åŒãçšããã°ãããé»è·ãè€æ°ãããšãã«ã¯ãé»äœã¯ããããã®é»è·ãã€ããåºãé»è·ã®åã«ãªãããšã«æ³šæãçãã¯ã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 88,
"tag": "p",
"text": "ãšãªãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 89,
"tag": "p",
"text": "å°äœè¡šé¢ã¯çé»äœãªã®ã§ããã£ãŠãé»æ°åç·ã¯å°äœè¡šé¢ã«åçŽã§ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 90,
"tag": "p",
"text": "ãã®ããšãããé»æ°åç·ãšé»å Žã¯åçŽã§ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 91,
"tag": "p",
"text": "é»å Žãéãåãããããããã«ãé»äœãéãåãããããããªããªãé»äœãšã¯ãé»å ŽãèããŠãçµè·¯ã«ãŠç©åãããã®ã§ããããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 92,
"tag": "p",
"text": "åŠæ ¡ã®ãã¹ããªã©ã§ã¯ãé»äœã®èšç®ã®ãããã¯ãŒãã³åã®æ¹åã®åéããªã©ã«ããèšç®ãã¹ãªã©ããµãããããé»å Žãæ±ããŠããããããç©åããŠãé»äœãæ±ããã®ããèšç®äžã¯å®å
šã§ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 93,
"tag": "p",
"text": "",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 94,
"tag": "p",
"text": "ã³ã³ãã³ãµãŒ(è±:capacitor ,ããã£ãã·ã¿ããšèªã)ã¯ãå³ã®ããã«ã2æã®é»æ¥µãåãããããåè·¯äžã«é»è·ãèç©ã§ããéšåãäžããçŽ åã§ããã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 95,
"tag": "p",
"text": "ã³ã³ãã³ãµãŒã«é»è·ãèããããšãå
é»(ãã
ãã§ã)ãšãããã³ã³ãã³ãµãŒããé»è·ãæŸåºãããããšãæŸé»ãšããã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 96,
"tag": "p",
"text": "ã³ã³ãã³ãµã®äž¡ç«¯ã«ããé»äœVãäžãããããšããã³ã³ãã³ãµã«ã¯ãé»äœã«æ¯äŸããé»è·Qãèç©ãããããã®ãšããã³ã³ãã³ãµã®èç©èœåãèšå·ã§ C ãšãããŠã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 97,
"tag": "p",
"text": "ãšããŠCãåããCã¯éé»å®¹é(ããã§ãããããããelectric capacitance)ãšåŒã°ããåäœã¯F(ãã¡ã©ããfarad)ã§äžããããã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 98,
"tag": "p",
"text": "1ãã¡ã©ãã¯å®çšäžã¯å€§ããããã®ã§ã10ãã¡ã©ããåäœã«ãã1pF(ãã³ãã¡ã©ã)ãã10ãã¡ã©ããåäœã«ãã1ÎŒF(ãã€ã¯ããã¡ã©ã)ã䜿ãããããšãå€ãã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 99,
"tag": "p",
"text": "",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 100,
"tag": "p",
"text": "極æ¿ãå¹³è¡ãªã³ã³ãã³ãµãŒãå¹³è¡æ¿ã³ã³ãã³ãµãŒãšããã å¹³è¡æ¿ã³ã³ãã³ãµãŒã®ã極æ¿ã©ããã®é»å Žã¯ãäžæ§ãªé»å Žã§ããã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 101,
"tag": "p",
"text": "ãã®å¹³è¡æ¿ã³ã³ãã³ãµãŒã®éé»å®¹éCã®åŒã¯ãåŸè¿°ããçç±ã«ããã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 102,
"tag": "p",
"text": "ã§äžãããããããã§ãSã¯å°äœå¹³é¢ã®é¢ç©ã§ãããdã¯å°äœéã®è·é¢ã§ããã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 103,
"tag": "p",
"text": "å®éšçã«ãããã®éé»å®¹éã®å
¬åŒã¯ãæ£ããããšã確ãããããŠããã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 104,
"tag": "p",
"text": "ããã§äžããéé»å®¹éã¯ãå¹³é¢äžã«é»è·ãäžæ§ã«ååžãããšã®ä»®å®ã§å°ãããããã®ãšããå°äœéã«çããé»çEã¯ãå°äœãæã€é»è·ãQ, -Qãšããæã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 105,
"tag": "p",
"text": "ãŸãã極æ¿ã®é»è·å¯åºŠãã極æ¿ã®ã©ãã§ãäžå®ã ãšä»®å®ããŠ(ãã®ããã«ã¯ãã³ã³ãã³ãµãŒã®åºã(ã€ãŸãé¢ç©)ãããã
ãã¶ãã«åºããšä»®å®ããå¿
èŠãããããšãããããã®ãããªä»®å®ã«ãããé»è·å¯åºŠã¯)ã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 106,
"tag": "p",
"text": "ã§ããã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 107,
"tag": "p",
"text": "é»æ°åç·ã®æ§è³ªãšããŠããã©ã¹ã®é»è·ããçããŠãã€ãã¹ã®é»è·ã§åžåãããã®ã§ããã£ãŠå¹³è¡æ¿ã³ã³ãã³ãµãŒéã®é»æ°åç·ã®ååžã¯ãå³ã®ããã«ãé»æ°åç·ãããã©ã¹æ¥µæ¿ããåçŽã«ããã€ãã¹æ¥µæ¿ãžåãã£ãŠé»æ°åç·ãåºãŠããããŠãã€ãã¹æ¥µæ¿ã«é»æ°åç·ãåžåãããã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 108,
"tag": "p",
"text": "é»å Žã¯ãå°äœéã®åç¹ã§ã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 109,
"tag": "p",
"text": "ã§äžãããããé»å Žãæ±ããããã®ã§ãããããé»äœãèšç®ã§ãããå°äœéã®åç¹ã§é»å Žã®å€§ãããåäžãªã®ã§ãé»äœã®å€§ããã¯é»å Žã®å€§ããã«2ç¹éã®è·é¢ãããããã®ã«ãªããããã§ãé»äœVã¯ã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 110,
"tag": "p",
"text": "ãšãªããããã®åŒãšéé»å®¹éCã®å®çŸ©ãèŠæ¯ã¹ããšã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 111,
"tag": "p",
"text": "ãåŸãããã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 112,
"tag": "p",
"text": "",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 113,
"tag": "p",
"text": "é»æ± ã®ååŠåå¿ã«ã€ããŠã¯ãå¥ç§ç®ã®ååŠIãªã©ã§è©³ããæ±ãããããã®ç« ã§ã¯ãé»å§ãé»æµã®ç解ã«é¢ããç¹ãéç¹çã«èª¬æãããã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 114,
"tag": "p",
"text": "éå±å
çŽ ã®åäœãæ°ŽãŸãã¯æ°Žæº¶æ¶²ã«å
¥ãããšãã®ãéœã€ãªã³ã®ãªãããããã€ãªã³ååŸå(ionization tendency)ãšããã äŸãšããŠãäºéZnãåžå¡©é
žHClã®æ°Žæº¶æ¶²ã«å
¥ãããšãäºéZnã¯æº¶ãããŸãäºéã¯é»åã倱ã£ãŠZnã«ãªãã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 115,
"tag": "p",
"text": "äžæ¹ãéAgãåžå¡©é
žã«å
¥ããŠãåå¿ã¯èµ·ãããªãã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 116,
"tag": "p",
"text": "ãã®ããã«éå±ã®ã€ãªã³ååŸåã®å€§ããã¯ãç©è³ªããšã«å€§ãããç°ãªãã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 117,
"tag": "p",
"text": "äºçš®é¡ã®éå±åäœãé»è§£è³ªæ°Žæº¶æ¶²ã«å
¥ãããšé»æ± ãã§ãããããã¯ã€ãªã³ååŸå(åäœã®éå±ã®ååãæ°ŽãŸãã¯æ°Žæº¶æ¶²äžã§é»åãæŸåºããŠéœã€ãªã³ã«ãªãæ§è³ª)ã倧ããéå±ãé»åãæŸåºããŠéœã€ãªã³ãšãªã£ãŠæº¶ããã€ãªã³ååŸåã®å°ããéå±ãæåºããããã§ããã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 118,
"tag": "p",
"text": "ã€ãªã³ååŸåã®å€§ããæ¹ã®éå±ãè² æ¥µ(ãµããã)ãšãããã€ãªã³ååŸåã®å°ããæ¹ã®éå±ãæ£æ¥µ(ããããã)ãšããã ã€ãªã³ååŸåã®å€§ããéå±ã®ã»ãããéœã€ãªã³ã«ãªã£ãŠæº¶ãåºãçµæãéå±æ¿ã«ã¯é»åãå€ãèç©ããã®ã§ãäž¡æ¹ã®éå±æ¿ãé
ç·ã§ã€ãªãã°ãã€ãªã³ååŸåã®å€§ããæ¹ããå°ããæ¹ã«é»åã¯æµããããé»æµãã§ã¯ç¡ãããé»åããšãã£ãŠãããšã«æ³šæãé»åã¯è² é»è·ã§ããã®ã§ãé»æµã®æµããšé»åã®æµãã¯ãéåãã«ãªãã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 119,
"tag": "p",
"text": "ããŸããŸãªæº¶æ¶²ãéå±ã®çµã¿åããã§ãã€ãªã³ååŸåã®æ¯èŒã®å®éšãè¡ã£ãçµæãã€ãªã³ååŸåã®å€§ããã決å®ãããã å·Šããé ã«ãã€ãªã³ååŸåã®å€§ããéå±ã䞊ã¹ããšã以äžã®ããã«ãªãã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 120,
"tag": "p",
"text": "éå±ããã€ãªã³ååŸåã®å€§ããã®é ã«äžŠã¹ããã®ãéå±ã®ã€ãªã³ååãšããã æ°ŽçŽ ã¯éå±ã§ã¯ç¡ããæ¯èŒã®ãããã€ãªã³ååŸåã«å ããããã éå±ååã¯ãäžèšã®ä»ã«ãããããé«æ ¡ååŠã§ã¯äžèšã®éå±ã®ã¿ã®ã€ãªã³ååãçšããããšãå€ãã ã€ãªã³ååã®èšæ¶ã®ããã®èªååãããšããŠã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 121,
"tag": "p",
"text": "ã貞ããããªããŸããããŠã«ããªãã²ã©ãããåéãã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 122,
"tag": "p",
"text": "ãªã©ã®ãããªèªååããããããã¡ãªã¿ã«ãã®èªååããã®å Žåã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 123,
"tag": "p",
"text": "ãKã ãã ãCa ãªNaããŸMg ãAlããZn ãŠFe ã«Ni ã ãªPbãã²H2 ã©Cu ãHg ãAg ã åéPt,Auãã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 124,
"tag": "p",
"text": "ãšå¯Ÿå¿ããŠããã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 125,
"tag": "p",
"text": "è² æ¥µ(äºéæ¿)ã§ã®åå¿",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 126,
"tag": "p",
"text": "æ£æ¥µ(é
æ¿)ã§ã®åå¿",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 127,
"tag": "p",
"text": "ãã«ã¿ã®é»æ± ã§ã¯ãåŸããã䞡極éã®é»äœå·®(ãé»å§ããšãããã)ã¯ã1.1ãã«ãã§ããã(ãã«ãã®åäœã¯Vãªã®ã§ã1.1Vãšãæžãã)ãã®äž¡æ¥µæ¿ã®é»äœå·®ãèµ·é»åãšãããèµ·é»åã¯ãäž¡é»æ¥µã®éå±ã®çµã¿åããã«ãã£ãŠæ±ºãŸãç©è³ªåºæã§ããã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 128,
"tag": "p",
"text": "èµ·é»åã®åäœã®ãã«ãã¯ãéé»æ°åã®é»äœã®åäœã®ãã«ããšåãåäœã§ãããé»æ°åè·¯ã®é»å§ã®ãã«ããšããèµ·é»åã®åäœã®ãã«ãã¯åãåäœã§ããã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 129,
"tag": "p",
"text": "ãã«ã¿é»æ± ã®æ§é ã以äžã®ãããªæååã«è¡šããå Žåããã®ãããªè¡šç€ºãé»æ± å³ãããã¯é»æ± åŒãšããã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 130,
"tag": "p",
"text": "aqã¯æ°Žã®ããšã§ãããH2SO4aqãšæžããŠãç¡«é
žæ°Žæº¶æ¶²ãè¡šããŠããã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 131,
"tag": "p",
"text": "ç©çåŠã®é»æ°åè·¯ã®ç 究ã§ã¯ããã®ãããªé»æ± ãªã©ã®çŸè±¡ã®çºèŠãšçºæã«ãã£ãŠãå®å®ãªçŽæµé»æºãå®éšçã«åŸãããããã«ãªããçŽæµé»æ°åè·¯ã®æ£ç¢ºãªå®éšãå¯èœã«ãªã£ããé»æ± ã®çºæ以åã«ãããã©ã³ã¹äººã®ç©çåŠè
ã¯ãŒãã³ãªã©ã«ããéé»æ°ã«ããé»æ°ååŠã®ç 究ãªã©ã«ãã£ãŠãé»äœå·®ã®æŠå¿µãé»è·ã®æŠå¿µã¯ãã£ããã ãããã®æ代ã®é»æºã¯ãäž»ã«éé»æ°ã«ãããã®ã ã£ãã®ã§ãå®å®é»æºã§ã¯ç¡ãã£ãã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 132,
"tag": "p",
"text": "ãããŠãé»æ± ã«ããå®å®ãªé»æºã®çºæã¯ãåæã«å®å®ãªé»æµã®çºæã§ããã£ãããã®ãããªé»æ± ã®çºæãªã©ã«ãããçŽæµé»æ°åè·¯ã®ç 究ãªã©ããããã€ã人ã®ç©çåŠè
ãªãŒã ããããŸããŸãªå°äœã«é»æµãæµãå®éšãšçè«ç 究ãè¡ãããšã«ãããé»æ°åè·¯ã®çè«ã®ãªãŒã ã®æ³å(ãªãŒã ã®ã»ããããOhm's law)ãçºèŠãããã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 133,
"tag": "p",
"text": "ãã€ã¯ãªãŒã ã¯é»æ± ã§ã¯ãªãç±é»å¯Ÿ(ãã€ã§ãã€ã)ãšãããã®ã䜿ã£ãŠãé»æ°åè·¯ã«å®å®ããé»æµããªããç 究ããããåœæã®é»æ± ã§ã¯ãèµ·é»åããã ãã«æžã£ãŠããŸãããªãŒã ã¯åœåã¯é»æ± ã§å®éšããããããŸãå®å®é»æµãåŸãããªãã£ãã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 134,
"tag": "p",
"text": "ç±é»å¯Ÿãšã¯ããŸãç°ãªãéå±ææã®2æ¬ã®éå±ç·ãæ¥ç¶ããŠ1ã€ã®åè·¯ãã€ããã2ã€ã®æ¥ç¹ã«æž©åºŠå·®ãäžãããšãåè·¯ã«é»å§ãçºçããããé»æµãæµãã(ãã®çŸè±¡ãããŒãŒããã¯å¹æãšãã)ããã®çŸè±¡ãããã¯ã1821幎ã«ãŒãŒããã¯ãçºèŠããããã®ãããªåè·¯ããç±é»å¯Ÿã§ããããªããåã2æ¬ã®éå±ç·ã§ã¯ã枩床差ãäžããŠãé»å§ã¯çºçãããé»æµã¯æµããªãã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 135,
"tag": "p",
"text": "ãªãŒã ã¯ããã«ãªã³å€§åŠææããã±ã³ãã«ãã®å©èšã«ãã£ãŠããã®ç±é»å¯Ÿãå®éšã«å©çšããã枩床ãå®å®ãããã®ã¯ãåœæã®æè¡ã§ãæ¯èŒçç°¡åã§ãã£ãã®ã§ãããããŠãªãŒã ã¯å®å®é»æµããã¡ããå®éšãã§ããã®ã§ããã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 136,
"tag": "p",
"text": "",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 137,
"tag": "p",
"text": "ãªãŒã ã®æ³å(Ohm's law)ãšã¯ã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 138,
"tag": "p",
"text": "ãã»ãšãã©ã®å°äœã§ã¯ãé»æµ I ãæµããŠããå°äœäžã®2ç¹ã®ç¹ P 1 {\\displaystyle P_{1}} ãšç¹ P 2 {\\displaystyle P_{2}} éã®é»äœå·® E = E 1 â E 2 {\\displaystyle E=E_{1}-E_{2}} ã¯ãé»æµ I ã«æ¯äŸãããã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 139,
"tag": "p",
"text": "ãšããå®éšæ³åã§ããã 誀解ããããããããªãŒã ã®æ³åã¯ããã®ãããªå®éšæ³åã§ãã£ãŠãã¹ã€ã«æµæã®å®çŸ©åŒã§ã¯ç¡ããåæ§ã«ããªãŒã ã®æ³åã¯ãã¹ã€ã«é»å§ã®å®çŸ©åŒã§ã¯ç¡ãããé»æµã®å®çŸ©åŒã§ãç¡ããäžåŠæ ¡ã®çç§ã§ã®é»æ°åè·¯ã®æè²ã§ã¯ãéå±ã®é»æ°å解ã®èµ·é»åã®æè²ãŸã§ã¯ããªãã®ã§ããšãããã°ãé»å§ã誀解ããŠããé»å§ã¯ãåãªãé»æµã®æ¯äŸéã§ãæµæã¯ãã®æ¯äŸä¿æ°ãã®ãããªèª€è§£ããå Žåãæããããããã®è§£éã¯æããã«èª€è§£ã§ããã ãŸããåå°äœãªã©ã®äžéšã®ææã§ã¯ãé»æµãå¢ãææã®æž©åºŠãäžæãããšæµæãäžããçŸè±¡ãç¥ãããŠããã®ã§ãåå°äœã§ã¯ãªãŒã ã®æ³åãæãç«ããªãå Žåãããããªã®ã§ããªãŒã ã®æ³åãå®çŸ©åŒãšèããã®ã¯äžåçã§ããã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 140,
"tag": "p",
"text": "å°ç·ãªã©ã®å°äœå
ã®é»æ°ã®æµããé»æµ(ã§ããã
ããelectric current)ãšãããé»æµã®åŒ·ãã¯ã¢ã³ãã¢ãšããåäœã§è¡šãã1ã¢ã³ãã¢ã®å®çŸ©ã¯æ¬¡ã®éãã§ããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 141,
"tag": "p",
"text": "1ç§éã«1ã¯ãŒãã³(èšå·C)ã®é»æµãééããããšã1ã¢ã³ãã¢ãšããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 142,
"tag": "p",
"text": "ã¢ã³ãã¢ã®èšå·ã¯Aã§ããããŸããé»æµã¯ãåäœæéãããã®é»è·ã®éééã§ãããã®ã§ãé»æµã®åäœã[C/s]ãšæžãå Žåãããã äžè¬çã«ã¯ãé»æµã®åäœã¯ããªãã¹ã[A]ã§è¡šèšããããšãå€ãã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 143,
"tag": "p",
"text": "é»æµI[A]ãšæét[S]ã§å°ç·æé¢ãééããé»è·Q[C]ã®é¢ä¿ãåŒã§è¡šããšã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 144,
"tag": "p",
"text": "ã§ããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 145,
"tag": "p",
"text": "é»æµã®åãã®åãæ¹ã«ã€ããŠã¯ãèªç±é»åã¯è² é»è·ãæã£ãŠãããããèªç±é»åã®åããšã¯å察åãã«é»æµã®åãããšãããšã«æ³šæããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 146,
"tag": "p",
"text": "次ã«é»æµãšèªç±é»åã®é床ãšã®é¢ä¿ãèããã èªç±é»åã®é»è·ã®çµ¶å¯Ÿå€ãeãšãããšãèªç±é»åã¯è² é»è·ã§ãããããèªç±é»åã®é»è·ã¯ãã€ãã¹ç¬Šå·ãã€ã-eã§ããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 147,
"tag": "p",
"text": "ãã€ã人ã®ç©çåŠè
ãªãŒã ã¯æ¬¡ã®ãããªæ³åãçºèŠããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 148,
"tag": "p",
"text": "ãã»ãšãã©ã®å°äœã§ã¯ãé»æµ I ãæµããŠããå°äœäžã®2ç¹ã®ç¹ P 1 {\\displaystyle P_{1}} ãšç¹ P 2 {\\displaystyle P_{2}} éã®é»äœå·® E = E 1 â E 2 {\\displaystyle E=E_{1}-E_{2}} ã¯ãé»æµ I ã«æ¯äŸãããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 149,
"tag": "p",
"text": "ãã®å®éšæ³åããªãŒã ã®æ³å(Ohm's law)ãšããã åŒã§è¡šããšãé»äœå·®ãVãšããŠãé»æµãIãšããå Žåã«ãæ¯äŸä¿æ°ãRãšããŠã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 150,
"tag": "p",
"text": "ã§ããã ããã§ãé»äœãšé»æµã®æ¯äŸä¿æ°Rãé»æ°æµæãããã¯åã«æµæ(resistanceãã¬ãžã¹ã¿ã³ã¹)ãšããã é»æ°æµæã®åäœã¯ãªãŒã ãšèšããèšå·ã¯Î©ã§è¡šãã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 151,
"tag": "p",
"text": "æ
£ç¿çã«ãæµæã®èšå·ã¯Rã§ããããå Žåãå€ãã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 152,
"tag": "p",
"text": "é»æ°åè·¯ãžãšãã«ã®ãŒãäŸçµŠããé»æºãšããŠå®é»å§ã®çŽæµé»æºãèãããåè·¯ã®2å°ç¹éã«ããäžå®ã®é»å§ãäŸçµŠãç¶ãããã®ã§ãããé»å§æºã®åè·¯å³èšå·ãšããŠã¯ãçšãããããèšå·ã®é·ãåŽãæ£æ¥µã§ããããã©ã¹ã®é»äœã§ãããèšå·ã®çãåŽã¯è² 極ã§ããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 153,
"tag": "p",
"text": "也é»æ± ã¯ãçŽæµé»æºãšããŠåãæ±ã£ãŠè¯ãã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 154,
"tag": "p",
"text": "ãªãããããã¯çŽæµé»æºã§ããã亀æµã®å Žåã¯äžè¬åããé»å§æºãšããŠã®èšå·ãçšããããŸãç¹ã«æ£åŒŠæ³¢äº€æµé»å§æºã§ããã°ã®èšå·ãçšããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 155,
"tag": "p",
"text": "æµæåš(resistor)ã¯ãéåžžã¯åã«æµæãšåŒã°ããåè·¯çŽ åã§ãããäžããããé»æ°ãšãã«ã®ãŒãåçŽã«æ¶è²»ããçŽ åã§ãããåè·¯å³èšå·ã¯ãããã¯ã§ããããæ¬æžã§ã¯ãäž¡è
ãšãæµæã®åè·¯å³èšå·ãšããŠçšããããšã«ããã(ç»åçŽ æã®ç¢ºä¿ã®éœåã®ãããäž¡æ¹ã®èšå·ãæ¬æžã§ã¯æ··åšããŸããã容赊ãã ããã)",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 156,
"tag": "p",
"text": "æ¥æ¬ã§ã¯ãæµæåšã®å³èšå·ã¯ãåŸæ¥ã¯JIS C 0301(1952幎4æå¶å®)ã«åºã¥ããã®ã¶ã®ã¶ã®ç·ç¶ã®å³èšå·ã§å³ç€ºãããŠããããçŸåšã®ãåœéèŠæ Œã®IEC 60617ãå
ã«äœæãããJIS C 0617(1997-1999幎å¶å®)ã§ã¯ã®ã¶ã®ã¶åã®å³èšå·ã¯ç€ºãããªããªããé·æ¹åœ¢ã®ç®±ç¶ã®å³èšå·ã§å³ç€ºããããšã«ãªã£ãŠãããæ§èŠæ Œã§ããJIS C 0301ã¯ãæ°èŠæ ŒJIS C 0617ã®å¶å®ã«äŒŽã£ãŠå»æ¢ããããããæ§èšå·ã§æµæåšãå³ç€ºããå³é¢ã¯ãçŸåšã§ã¯JISéæºæ ãªå³é¢ã«ãªã£ãŠããŸããããããææåã¯ç¡ããããçŸåšãåŸæ¥ã®å³èšå·ãå€çšãããŠããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 157,
"tag": "p",
"text": "",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 158,
"tag": "p",
"text": "è€æ°ã®åè·¯çŽ åã1ã€ã®ç·äžã«é
眮ãããŠãããããªæ¥ç¶ãçŽåæ¥ç¶ãšãããè€æ°ã®åè·¯çŽ åãäºè¡ã«åãããããã«é
眮ãããŠããæ¥ç¶ã䞊åæ¥ç¶ãšããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 159,
"tag": "p",
"text": "çŽåæ¥ç¶ã«ãããŠã¯ãããããã®åè·¯çŽ åã«æµããé»æµã¯å
šãŠçãããäžæ¹ã䞊åæ¥ç¶ã«ãããŠã¯ããããã®åè·¯çŽ åã®äž¡ç«¯ã«ãããé»å§ãå
šãŠçããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 160,
"tag": "p",
"text": "ãŸããçŽåæ¥ç¶ã«ãããŠã¯ããããã®åè·¯çŽ åã«ãããé»å§ã®åãå
šé»å§ãšãªãã䞊åæ¥ç¶ã«ãããŠã¯ããããã®åè·¯çŽ åãæµããé»æµã®åãå
šé»æµãšãªãã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 161,
"tag": "p",
"text": "æµæãè€æ°æ¥ç¶ãããŠããå Žåããã®è€æ°ã®æµæããŸãšããŠãããã1ã€ã®æµæãæ¥ç¶ãããŠãããã®ãããªç䟡çãªåè·¯ãèããããšãã§ãããè€æ°ã®æµæãšç䟡ãª1ã€ã®æµæãåææµæãšããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 162,
"tag": "p",
"text": "æµæãnåçŽåã«æ¥ç¶ãããŠããå Žåãèãããæµæ R 1 , R 2 , ⯠, R n {\\displaystyle R_{1},R_{2},\\cdots ,R_{n}} ãçŽåã«æ¥ç¶ãããŠããå Žåãåæµæãæµããé»æµã¯çããããããiãšãããåæµæ R k ( k = 1 , 2 , ⯠, n ) {\\displaystyle R_{k}(k=1,2,\\cdots ,n)} ã«ãããé»å§ã v k {\\displaystyle v_{k}} ãšãããšããªãŒã ã®æ³åãã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 163,
"tag": "p",
"text": "ãæãç«ã€ããã®ãšãçŽåæµæã®äž¡ç«¯ã®é»å§vã¯ã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 164,
"tag": "p",
"text": "ã§ããããããšç䟡ãªæµæRã1ã€ã ãæ¥ç¶ãããŠãããããªç䟡åè·¯ãèãããšãã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 165,
"tag": "p",
"text": "ãæãç«ã€ããããããã£ãŠãããã®nåã®çŽåæµæã®åææµæRãšããŠ",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 166,
"tag": "p",
"text": "ãåŸããããªãã¡ãçŽååææµæã¯åæµæã®ç·åãšãªãã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 167,
"tag": "p",
"text": "åæ§ã«ãæµæãnå䞊åã«æ¥ç¶ãããŠããå Žåãèãããæµæ R 1 , R 2 , ⯠, R n {\\displaystyle R_{1},R_{2},\\cdots ,R_{n}} ã䞊åã«æ¥ç¶ãããŠããå Žåãåæµæã®äž¡ç«¯ã®é»å§ã¯çããããããvãšãããåæµæ R k ( k = 1 , 2 , ⯠, n ) {\\displaystyle R_{k}(k=1,2,\\cdots ,n)} ãæµããé»æµã i k {\\displaystyle i_{k}} ãšãããšããªãŒã ã®æ³åãã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 168,
"tag": "p",
"text": "ãæãç«ã€ããã®ãšã䞊åæµæãžæµã蟌ãé»æµiã¯ã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 169,
"tag": "p",
"text": "ã§ããããããšç䟡ãªæµæRã1ã€ã ãæ¥ç¶ãããŠãããããªç䟡åè·¯ãèãããšãã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 170,
"tag": "p",
"text": "ãæãç«ã€ããããããã£ãŠãããã®nåã®äžŠåæµæã®åææµæRãšããŠ",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 171,
"tag": "p",
"text": "ãåŸããããªãã¡ã䞊ååææµæã®éæ°ã¯åæµæã®éæ°ã®ç·åãšãªãã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 172,
"tag": "p",
"text": "æµæRãé»æµIãæµãããšãããã®éšåã®çºç±ã®ãšãã«ã®ãŒã¯ã1ç§ãããã«RI[J/s]ã§ãããããããžã¥ãŒã«ç±ãšãããååã®ç±æ¥ã¯ç©çåŠè
ã®ãžã¥ãŒã«ã調ã¹ãããã§ããããªãŒã ã®æ³åãããV=RIã§ãããã®ã§ããžã¥ãŒã«ç±ã¯VIãšãæžããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 173,
"tag": "p",
"text": "ããã§ãã²ãšãŸããç±ã®èå¯ã«ã¯é¢ããŠã次ã®éãå®çŸ©ãããé»æ°åè·¯ã®ãã2ç¹éãæµããé»æµIãšããã®2ç¹éã®é»å§Vãšã®ç©VIãé»å(power)ãšå®çŸ©ãããé»åã®èšå·ã¯Pã§è¡šããããããšãå€ãã é»åã®åäœã®ãžã¥ãŒã«æ¯ç§[J/s]ã[W]ãšããåäœã§è¡šãããã®åäœWã¯ã¯ãã(Watt)ãšèªãã ã€ãŸãé»åã¯èšå·ã§",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 174,
"tag": "p",
"text": "ã§ããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 175,
"tag": "p",
"text": "å°ç·ã®å€ªããé·ãã«ãã£ãŠæµæã®å€§ããã¯å€ãããçŽæçã«å€ªãã»ããæµããããã®ã¯åããã ãããããã«äžŠåæ¥ç¶ãšå¯Ÿå¿ãããŠããå°ç·ã倪ãã»ããæµããããã®ã¯åããã ããã å®éã«é»æ°æµæã¯ãå°ç·ã倪ãã«åæ¯äŸããŠå°ãããªãããšãå®éšçã«ç¢ºèªãããŠãããããã§ãã€ãã®ãããªåŒã«ãããŠã¿ããã æµæãR[Ω]ãšããå Žåãå°ç·ã®å€ªããé¢ç©ã§è¡šãA[m]ãšããã°ãæ¯äŸå®æ°ã«kãçšããã°ã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 176,
"tag": "p",
"text": "ã§ããã( âã¯ãæ¯äŸé¢ä¿ãè¡šãæ°åŠèšå·ã)",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 177,
"tag": "p",
"text": "ããã«ãå°ç·ã¯æ質ã倪ããåããªãã°ãå°ç·ãé·ãã»ã©æµæããé·ãã«æ¯äŸããŠæµæã倧ãããªãããšãã確èªãããŠãããããã§ãããã«ãæµæäœã®é·ããèæ
®ããåŒã«è¡šããŠã¿ãã°ã次ã®ããã«ãªããæµæ垯ã®é·ããl[m]ãšããã°",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 178,
"tag": "p",
"text": "ã§ããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 179,
"tag": "p",
"text": "ããã«ãå°ç·ã®æ質ã«ãã£ãŠãæµæã®å€§ããã¯å€ãããåãé·ãã§åã倪ãã®æµæã§ããæ質ã«ãã£ãŠæµæã®å€§ããã¯ç°ãªããããã§ãæ質ããšã®æ¯äŸå®æ°ãÏãšããã°ãæµæã®åŒã¯ä»¥äžã®åŒã§èšè¿°ãããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 180,
"tag": "p",
"text": "Ïã¯æµæç(ãŠããããã€ãresistivity)ãšåŒã°ãããæµæçã®åäœã¯[Ωm]ã§ããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 181,
"tag": "p",
"text": "",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 182,
"tag": "p",
"text": "ç£ç³ã®ãŸããã«ã¯å¥ã®ç£ç³ãåããåã®ããšãšãªããã®ãçããŠããã ãããç£å Ž(ãã°ãmagnetic field)ãããã¯ç£ç(ããã)ãšåŒã¶ã(æ¥æ¬ã®ç©çåŠã§ã¯ç£å ŽãšåŒã¶ããšãå€ãããŸããæ¥æ¬ã®é»æ°å·¥åŠã§ã¯ç£çãšåŒã°ããããšãå€ããææ²»æã®èš³èªã®éã®ãæ¥æ¬åœå
ã®æ¥çããšã®éãã«éãããå°å瀟äŒçãªäºè±¡ã§ãããåŒã³æ¹ã¯ç©çã®æ¬è³ªãšã¯é¢ä¿ãªãã®ã§ãããã§ã¯ãã©ã¡ãã®è¡šçŸãçšãããã¯ãæ¬æžã§ã¯ç¹ã«ãã ãããªããè±èªã§ã¯ç©çåŠã»é»æ°å·¥åŠãšãâmagnetic fieldâã§å
±éããŠããã)",
"title": "ç£å"
},
{
"paragraph_id": 183,
"tag": "p",
"text": "éãã³ãã«ããããã±ã«ã«ç£ç³ãè¿ã¥ãããšãç£ç³ã«åžãä»ããããã ãŸããéãã³ãã«ããããã±ã«ã«åŒ·ãç£åãäžãããšãéãã³ãã«ããããã±ã«ãã®ãã®ãç£å Žãåšå²ã«åãŒãããã«ãªãã ãã®ãããªãããšããšã¯ç£å Žãæããªãã£ãç©äœãã匷ãç£å Žãåããããšã«ãã£ãŠç£å ŽãåãŒãããã«ãªãçŸè±¡ãç£å(ãããmagnetization)ãšããã ãããã¯é»è·ã®éé»èªå°ãšå¯Ÿå¿ãããŠãç£åã®ããšãç£æ°èªå°(ããããã©ããmagnetic induction)ãšãããã ãããŠãéãã³ãã«ããããã±ã«ã®ããã«ãç£ç³ã«åŒãä»ããããããã«ç£åãããèœåãããç©äœã匷ç£æ§äœ(ãããããããããferromagnet)ãšããã éãšã³ãã«ããšããã±ã«ã¯åŒ·ç£æ§äœã§ããã",
"title": "ç£å"
},
{
"paragraph_id": 184,
"tag": "p",
"text": "é
ã¯ç£åããªãããé
ã¯ç£ç³ã«åŒãã€ããããªãã®ã§ãé
ã¯åŒ·ç£æ§äœã§ã¯ãªãã",
"title": "ç£å"
},
{
"paragraph_id": 185,
"tag": "p",
"text": "éé»èªå°ãå©çšãããéé»é®èœ(ããã§ããããžã)ãšèšããããäžç©ºã®å°äœãã€ãã£ãŠç©è³ªãå²ãããšã§å€éšé»å Žãé®èœããæ¹æ³ããã£ãã®ãšåæ§ã®ãç£æ°ã®é®èœãã匷ç£æ§äœã§ãåºæ¥ããäžç©ºã®åŒ·ç£æ§äœãçšããŠã匷ç£æ§äœã®å
éšã¯ç£å Žãé®èœã§ããããããç£æ°é®èœ(ãããããžããmagnetic shielding)ãšãããç£æ°ã·ãŒã«ããšãããã",
"title": "ç£å"
},
{
"paragraph_id": 186,
"tag": "p",
"text": "ç£å Žã®åããåããããã«å³ç€ºããããç£ç³ã®äœãç£å Žã®æ¹åã¯ãç ã«å«ãŸããç éã®ç²æ«ãç£ç³ã«ãã¡ãã°ããŠããµããããããšã§èŠ³å¯ã§ããã",
"title": "ç£å"
},
{
"paragraph_id": 187,
"tag": "p",
"text": "ãããå³ç€ºãããšãäžå³ã®ããã«ãªãã(ç»åçŽ æã®ç¢ºä¿ã®éœåäžãåçãšå³ç€ºãšã§ã¯ãN極ãšS極ãéã«ãªã£ãŠããŸããã容赊ãã ããã)",
"title": "ç£å"
},
{
"paragraph_id": 188,
"tag": "p",
"text": "ãã®ãããªç£å Žã®å³ãç£åç·(ãããããããmagnetic line of force)ãšãããç£åç·ã®åãã¯ãç£ç³ã®N極ããç£åç·ãåºãŠãS極ã«ç£åç·ãåžåããããšå®çŸ©ããããæ£ç£ç³ã§ã¯ãç£åã®çºçæºãšãªãå Žæããæ£ç£ç³ã®äž¡ç«¯ã®å
端ä»è¿ã«éäžãããããã§ãæ£ç£ç³ã®äž¡ç«¯ã®å
端ä»è¿ãç£æ¥µ(ãããããmagnetic pole)ãšããã",
"title": "ç£å"
},
{
"paragraph_id": 189,
"tag": "p",
"text": "ãã®ãããªç£ç³ã®ã€ããç£åç·ã®åœ¢ã¯ãé»æ°åç·ã§ã®ãç°ç¬Šå·ã®é»è·ã©ãããã€ããé»æ°åç·ã«äŒŒãŠããã",
"title": "ç£å"
},
{
"paragraph_id": 190,
"tag": "p",
"text": "1ã€ã®æ£ç£ç³ã§ã¯N極(north pole)ã®ç£æ°ã®åŒ·ããšãS極(south pole)ã®ç£æ°ã®åŒ·ãã¯çããããŸããç£ç³ã«ã¯ãå¿
ãN極ãšS極ãšãååšãããN極ãšS極ã®ãã©ã¡ããçæ¹ã ããåãåºãããšã¯åºæ¥ãªããããšãæ£ç£ç³ãåæããŠããåæé¢ã«ç£æ¥µãåºçŸããããã®ãããªçŸè±¡ã®ãããçç±ã¯ãããããæ£ç£ç³ãæ§æãã匷ç£æ§äœã®ååã®1åãã€ãå°ããªç£ç³ã§ããããããå°ããªååã®ç£ç³ããããã€ãæŽåããŠã倧ããªæ£ç£ç³ã«ãªã£ãŠããããã§ããã",
"title": "ç£å"
},
{
"paragraph_id": 191,
"tag": "p",
"text": "ä»®æ³çã«ãç£æ¥µãS極ãŸãã¯N極ã®çæ¹ã ãçŸããçŸè±¡ãçè«èšç®ã®ããã«èããããšããããããã®ãããªçåŽã ãã®ç£æ¥µãåç£æ¥µ(ã¢ãããŒã«ãšããã)ãšããããåç£æ¥µã¯å®åšããªãã",
"title": "ç£å"
},
{
"paragraph_id": 192,
"tag": "p",
"text": "æ£ç£ç³ãªã©ããã®ãçåŽã®ç£æ¥µãããã®ãç£æ¥µããã®ç£å Žã®åŒ·ãã®ããšãããã®ãŸãŸãç£æ¥µã®åŒ·ãã(Magnetic charge)ãšåŒã¶ããããã¯ç£è·(ãããmagnetization)ãç£æ°éãšããã",
"title": "ç£å"
},
{
"paragraph_id": 193,
"tag": "p",
"text": "ããããããã®ç£åãšç£å Žã®é¢ä¿ãåŒã§è¡šãããšãèããã ãŸããæ£ç£ç³ã«ã¯ç£æ¥µãäž¡åŽã«2åããã®ã§ãèšç®ãç°¡åã«ããããã«ãæ£ç£ç³ã®äž¡ç«¯ã®è·é¢ã倧ãããå察åŽã®ç£æ¥µã®å€§ãããç¡èŠã§ããç£ç³ãèãããã",
"title": "ç£å"
},
{
"paragraph_id": 194,
"tag": "p",
"text": "ãã®ãããªç£ç³ãçšããŠãå®éšãããšããã次ã®æ³åãåãã£ããç£åã®åŒ·ãã¯2åã®ç©äœã®ç£æ°ém1ããã³m2ã«æ¯äŸãã2åã®ç©äœéã®è·é¢rã®2ä¹ã«åæ¯äŸããã åŒã§è¡šããšã",
"title": "ç£å"
},
{
"paragraph_id": 195,
"tag": "p",
"text": "ã§è¡šãããã(kmã¯æ¯äŸå®æ°) ãããçºèŠè
ã®ã¯ãŒãã³ã®åã«ã¡ãªãã§ãç£æ°ã«é¢ããã¯ãŒãã³ã®æ³åãšãããç£æ°émã®åäœã¯ãŠã§ãŒããšãããèšå·ã¯[Wb]ã§è¡šãã",
"title": "ç£å"
},
{
"paragraph_id": 196,
"tag": "p",
"text": "æ¯äŸå®æ°kmãš1ãŠã§ãŒãã®å€§ãããšã®é¢ä¿ã¯ã1ã¡ãŒãã«é¢ãã1wbã©ããã®ç£æ¥µã«ã¯ãããåãçŽ6.33Ã10ãšããŠã æ¯äŸä¿æ°kmã¯ã",
"title": "ç£å"
},
{
"paragraph_id": 197,
"tag": "p",
"text": "ã§ããã",
"title": "ç£å"
},
{
"paragraph_id": 198,
"tag": "p",
"text": "ã€ãŸãã",
"title": "ç£å"
},
{
"paragraph_id": 199,
"tag": "p",
"text": "ã§ããã",
"title": "ç£å"
},
{
"paragraph_id": 200,
"tag": "p",
"text": "éé»æ°åã«å¯ŸããŠãé»å Žãå®çŸ©ãããããã«ãç£æ°åã«å¯ŸããŠããå Žãå®çŸ©ããããšéœåãè¯ããç£æ°ém1[Wb]ãäœãã次ã®éãç£å Žã®åŒ·ããããã¯ç£å Žã®å€§ãããšèšããèšå·ã¯Hã§è¡šãã",
"title": "ç£å"
},
{
"paragraph_id": 201,
"tag": "p",
"text": "ç£å Žã®åŒ·ãHã®åäœã¯[N/Wb]ã§ãããHãçšãããšãç£æ°ém2[Wb]ã«ã¯ãããç£æ°åf[N]ã¯ã",
"title": "ç£å"
},
{
"paragraph_id": 202,
"tag": "p",
"text": "ãšè¡šããã",
"title": "ç£å"
},
{
"paragraph_id": 203,
"tag": "p",
"text": "ç©çåŠè
ã®ãšã«ã¹ãããã¯ãé»æµã®å®éšãããŠããéã«ãããŸããŸè¿ãã«ãããŠãã£ãæ¹äœç£ç³ãåãã®ã確èªããã圌ã詳ãã調ã¹ãçµæã以äžã®ããšãåãã£ãã",
"title": "é»æµãã€ããç£å Ž"
},
{
"paragraph_id": 204,
"tag": "p",
"text": "é»æµãæµããŠãããšãã«ã¯ããã®ãŸããã«ã¯ãç£å Žãçãããåãã¯ãé»æµã®æ¹åã«å³ãããé²ãããã«ãå³ãããåãåããšåããªã®ã§ããããå³ããã®æ³åãšããã",
"title": "é»æµãã€ããç£å Ž"
},
{
"paragraph_id": 205,
"tag": "p",
"text": "ã¢ã³ããŒã«ããç£å Žã®å€§ããã調ã¹ãçµæãç£å Žã®å€§ããHã¯ãé»æµI[A]ãçŽç·çã«æµããŠãããšããçŽç·é»æµã®åšãã®ç£å Žã®å€§ããã¯ãå°ç·ããã®è·é¢ãa[m]ãšãããšãç£å Žã®å€§ããH[N/Wb]ã¯ã",
"title": "é»æµãã€ããç£å Ž"
},
{
"paragraph_id": 206,
"tag": "p",
"text": "ã§ããããšãç¥ãããŠããã",
"title": "é»æµãã€ããç£å Ž"
},
{
"paragraph_id": 207,
"tag": "p",
"text": "ãããã¢ã³ããŒã«ã®æ³å(Ampere's law) ãšããã ç£å Žã®å€§ããHã®åäœã¯ã[N/Wb]ã§ãããããã£ãœãã¢ã³ããŒã«ã®æ³åã®åŒãã¿ãã°ã¢ã³ãã¢æ¯ã¡ãŒãã«[A/m]ã§ãããã",
"title": "é»æµãã€ããç£å Ž"
},
{
"paragraph_id": 208,
"tag": "p",
"text": "å°ç·ãã³ã€ã«ç¶ã«å·»ãã°ãã¢ã³ããŒã«ã®æ³åã§å°ç·ã®åšå²ã«çºçããç£å Žãéãªãããããã®ããã«ããç£å Žã匷ããã³ã€ã«ãé»ç£ç³(ã§ããããããelectromagnet)ãšãããå°ç·ã«é»æµãæµããŠãããšãã«ã®ã¿ãé»ç£ç³ã¯ç£å Žãçºçãããå°ç·ã«é»æµãæµãã®ãæ¢ãããšãé»ç£ç³ã®ç£å Žã¯æ¶ããã",
"title": "é»æµãã€ããç£å Ž"
},
{
"paragraph_id": 209,
"tag": "p",
"text": "ç£å Žã®å€§ããHã«ã次ã®ç¯ã§æ±ãããŒã¬ã³ãåã®çŸè±¡ã®ãããæ¯äŸä¿æ°ÎŒ(åäœã¯ãã¥ãŒãã³æ¯ã¢ã³ãã¢ã§[N/A])ãæããŠãèšå·Bã§è¡šãã",
"title": "é»æµãã€ããç£å Ž"
},
{
"paragraph_id": 210,
"tag": "p",
"text": "ãšããããšãããããã®éBãç£æå¯åºŠ(magnetic flux density)ãšãããç£å Žã®å€§ããHã®åããšç£æå¯åºŠBã®åãã¯åãåãã§ããã ãŸããç£å Žã®å€§ããHãšç£æå¯åºŠBã®æ¯äŸä¿æ°ãéç£ç(ãšãããã€ãmagnetic permeability)ãšããã (ããŒã¬ã³ãåã«é¢ããŠã¯ã詳ããã¯ç©çIIã§æ±ããèªè
ãç©çIãåŠã¶åŠå¹Žãªãã°ãèªè
ã¯ãããŒã¬ã³ãåãšããåãããã®ã ãªã»ã»ã»ããšã§ãæã£ãŠããã°ããã)",
"title": "é»æµãã€ããç£å Ž"
},
{
"paragraph_id": 211,
"tag": "p",
"text": "ãŸããå°ç·ãçšæãããšãããããã®å°ç·ã¯ãéæ¢ããŠãããšããŠãéæ¢ããŠããããåºå®ã¯ããã«ãããå°ç·ã«åãå ããã°ãå°ç·ãåããããã«ããŠããšãããã",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 212,
"tag": "p",
"text": "ãã®å°ç·ã«é»æµãæµããã ãã§ã¯ãã¹ã€ã«å°ç·ã¯åããªãããããããã®å°ç·ã«ãå€éšã®ç£ç³ã«ããç£å Žãå ãããšãå°ç·ãåãããã®ãããªãç£å Žãšé»æµã®çžäºäœçšã«ãã£ãŠãå°ç·ã«çããåãããŒã¬ã³ãå(ããŒã¬ã³ãããããè±: Lorentz force)ãšããã",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 213,
"tag": "p",
"text": "ããŒã¬ã³ãåã®åãã¯ãå°ç·ã®é»æµã®åããšç£å Žã®åãã«åçŽã§ãããé»æµIã®åãããç£æå¯åºŠBã®åãã«å³ãããåãåããšåãã§ããã",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 214,
"tag": "p",
"text": "ãŸããããŒã¬ã³ãåã®å€§ããã¯ãå°ç·ã®é·ãlãšãç£å Žã®å°ç·ãšã®åçŽæ¹åæåã«æ¯äŸããã",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 215,
"tag": "p",
"text": "ããŒã¬ã³ãåã®å€§ããF[N]ãåŒã§è¡šãã°ãé»æµãšç£å ŽãšãåçŽã ãšããŠãç£å ŽãåããŠããå°ç·ã®åœ¢ç¶ãçŽç·åœ¢ã ãšããŠãé»æµãI[A]ãšããŠãå°ç·ã®é·ããl[m]ãšããŠãå°ç·ã«ããã£ãŠããå€éšç£å Žã®ç£æå¯åºŠãB[N/(Aã»m)]ãšããã°ã",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 216,
"tag": "p",
"text": "ã§è¡šããã",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 217,
"tag": "p",
"text": "ããŒã¬ã³ãåã®å
¬åŒã«ãã¯ãŒãã³ã®æ³åãªã©ã§ã¯èŠããããããªæ¯äŸä¿æ°(ä¿æ°Kãªã©ã)ãå«ãŸããªãã®ã¯ãããããããã®ããŒã¬ã³ãåã®çŸè±¡ãå
ã«ãç£æ°éãŠã§ãŒãWbã®åäœããã³ç£æå¯åºŠBã®åäœãã決å®ãããŠããããã§ããã",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 218,
"tag": "p",
"text": "ãŸãããç£æå¯åºŠãã®å称ãããç£æãã»ãå¯åºŠããšããã®ã¯ãå®ã¯ç£æå¯åºŠã®åäœã®[N/(Aã»m)]ã¯ãåäœãåŒå€åœ¢ãããš[Wb/m]ã§ãããããšãç±æ¥ã§ããããã®åäœ[Wb/m]ããé»æ°å·¥åŠè
ã®ãã¹ã©ã®åã«ã¡ãªã¿ãåäœ[Wb/m] ããã¹ã©ãšèšããèšå·Tã§è¡šãã",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 219,
"tag": "p",
"text": "ãã®ããŒã¬ã³ãåã®çŸè±¡ããé»æ°æ©åšã®ã¢ãŒã¿(é»åæ©)ã®åçã§ããã",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 220,
"tag": "p",
"text": "",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 221,
"tag": "p",
"text": "ãªããããã¬ãã³ã°ã®æ³åããšããããŒã¬ã³ãåã«é¢ããæ³åãããããããŒã¬ã³ãåã®èšç®ã«ã¯å®çšçã§ã¯ç¡ããããã¬ãã³ã°ã®åãé¢ããç°ãªãæ³åã幟ã€ããã£ãŠçŽããããééãã®åå ã«ãªããããã®ã§ãæ¬æžã§ã¯æããªãã å®éã«ãå°éçãªç©çèšç®ã§ã¯ããã¬ãã³ã°ã®æ³åã¯ãèšç®ã«ã¯çšããªãã",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 222,
"tag": "p",
"text": "ãããããã¬ãã³ã°ã®æ³åã«ã¯ããã¬ãã³ã°ã®å³æã®æ³åããšããããšã¯ç°ãªãããã¬ãã³ã°ã®å·Šæã®æ³åãããããã©ã¡ãããã©ã®ç£æ°ã®çŸè±¡ã«çšããæ³åã ã£ãã®ããééãããããã ãããæ¬æžã§ã¯æããªãã",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 223,
"tag": "p",
"text": "(é»ç£èªå°ã«é¢ããŠã¯ã詳ããã¯ç©çIIã§æ±ãã)",
"title": "é»ç£èªå°"
},
{
"paragraph_id": 224,
"tag": "p",
"text": "ã¢ã³ããŒã«ã®æ³åã§ã¯ãé»æµã®åšãã«ç£å Žãã§ããã®ã§ãã£ãã",
"title": "é»ç£èªå°"
},
{
"paragraph_id": 225,
"tag": "p",
"text": "å®ã¯ãç£ç³ãåãããªã©ããŠãç£å Žã䌎ãç©äœãéåãããšããã®ãŸããã«ã¯é»å Žãçããã ä»®ã«ãã³ã€ã«ã®è¿ãã§ãããè¡ãªã£ããšãããšãçããé»å Žã«ãã£ãŠã³ã€ã«ã®äžã«ã¯é»æµãæµããã çããé»å Žã®å€§ããã¯ã",
"title": "é»ç£èªå°"
},
{
"paragraph_id": 226,
"tag": "p",
"text": "ãšãªãã(ååŸaã®å圢ã®ã³ã€ã«ã®å Žåã) Eã®åäœã¯[V/m]ã§ãããBã®åäœã¯[T]ã§ããã",
"title": "é»ç£èªå°"
},
{
"paragraph_id": 227,
"tag": "p",
"text": "ãã®çŸè±¡ãé»ç£èªå°(ã§ããããã©ããelectromagnetic induction)ãšãããé»ç£èªå°ã«ãã£ãŠçºçããé»æµãèªå°é»æµãšããã",
"title": "é»ç£èªå°"
},
{
"paragraph_id": 228,
"tag": "p",
"text": "ãŸããèªå°é»æµã®åãã¯ãç£ç³ã®åãã«ãããã³ã€ã«ã®äžãéãç£æã®å€åã劚ããåãã«ãé»æµãæµããã(èªå°é»æµãã¢ã³ããŒã«ã®æ³åã«åŸããåšå²ã«ç£å Žãäœãã) ãã®èªå°é»æµããã³ã€ã«ã®äžãéãç£æã®å€åã劚ããåãã«èªå°é»æµãæµããçŸè±¡ãã¬ã³ãã®æ³å(Lenz's law)ãšããã",
"title": "é»ç£èªå°"
},
{
"paragraph_id": 229,
"tag": "p",
"text": "",
"title": "é»ç£èªå°"
},
{
"paragraph_id": 230,
"tag": "p",
"text": "åãé åã«Nåå·»ãããã³ã€ã«ã眮ãããå Žåããã¡ã©ããŒã®é»ç£èªå°ã®æ³åã¯ã次ã®ããã«ãªãã",
"title": "é»ç£èªå°"
},
{
"paragraph_id": 231,
"tag": "p",
"text": "ããã§ã E {\\displaystyle {\\mathcal {E}}} ã¯èµ·é»å(ãã«ã ãèšå·ã¯V)ãΊB ã¯ç£æ(ãŠã§ãŒããèšå·ã¯Wb)ãšãããNã¯é»ç·ã®å·»æ°ãšããã",
"title": "é»ç£èªå°"
},
{
"paragraph_id": 232,
"tag": "p",
"text": "ãã®é»ç£èªå°ã®çŸè±¡ããç«åçºé»ãæ°Žåçºé»ãªã©ã®çºé»æ©ã®åçã§ãããããçã®çºé»ã§ã¯ãæ°žä¹
ç£ç³ãå転ãããããšã§ãçºé»ãããŠãããç«åãæ°Žåãšããã®ã¯ãæ©åšã®å転ãåŸãæ段ã«ãããªãããŸããçºé»æã®çºé»ã«ã¯ãæ°žä¹
ç£ç³ã®å転ãå©çšããŠãããããçºçããé»å§ãé»æµã¯åšæçãªæ³¢åœ¢ã«ãªãã次ã«èª¬æãã亀æµæ³¢åœ¢ã«ãªãã",
"title": "é»ç£èªå°"
},
{
"paragraph_id": 233,
"tag": "p",
"text": "åè·¯ãžã®å
¥åé»å§ãåšæçã«æéå€åããåè·¯ã®é»å§ããã³é»æµã亀æµ(alternating current)ãšãããããã«å¯Ÿãã也é»æ± ãªã©ã«ãã£ãŠçºçããé»å§ãé»æµã®ããã«ãæéã«ãããäžå®ãªé»å§ãé»æµã¯çŽæµ(direct Current)ãšããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 234,
"tag": "p",
"text": "亀æµæ³¢åœ¢ãäœç§ã§1åšããããšããæéãåšæ(wave period)ãšãããåšæã®èšå·ã¯ T {\\displaystyle T} ã§è¡šãåäœã¯ç§[s]ã§ããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 235,
"tag": "p",
"text": "1ç§éã«æ³¢åœ¢ãäœåšããããšããåæ°ãåšæ³¢æ°ãããã¯æ¯åæ°(è±èªã¯ããšãã«frequency)ãšããã é»æ°ã®æ¥çã§ã¯åšæ³¢æ°ãšããçšèªãçšããããšãå€ããç©çã®æ³¢ã®çè«ã§ã¯æ¯åæ°ãšããè¡šçŸãçšããããšãå€ãã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 236,
"tag": "p",
"text": "åšæ³¢æ°ã®åäœã¯[1/s]ã§ããããããããã«ã(hertz)ãšããåäœã§è¡šããåäœèšå·HzãçšããŠåšæ³¢æ°fããf[Hz]ãšãããµãã«è¡šãã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 237,
"tag": "p",
"text": "亀æµé»æµã亀æµé»å§ãæ£åŒŠæ³¢ã®å Žåã¯ããããã®ãã©ã¡ãŒã¿ãçšããŠ",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 238,
"tag": "p",
"text": "ãšæžãããšãã§ããã sinãšã¯äžè§é¢æ°ã§ãããç¥ããªããã°æ°åŠIIãªã©ãåèã«ããã ãã®ãšãã®sinã®ä¿æ° I 0 {\\displaystyle I_{0}} ã V 0 {\\displaystyle V_{0}} ãæ¯å¹
(ããã·ããamplitude)ãšããããŸãæå»t=0ã«ãããé»æµãé»å§ã®å€ã瀺ããæé波圢ã決å®ãã Ξ i {\\displaystyle \\theta _{i}} ã Ξ v {\\displaystyle \\theta _{v}} ãåæäœçžãšããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 239,
"tag": "p",
"text": "æ®éç§é«æ ¡ã®é«æ ¡ç©çã§ã¯ã亀æµæ³¢åœ¢ã®èšç®ã«ã¯ãæ£åŒŠæ³¢ã®å Žåãäž»ã«æ±ããæ¹åœ¢æ³¢ãäžè§æ³¢ã®èšç®ã¯ãæ®éã¯æ±ãããªãã ãã ããå·¥æ¥é«æ ¡ã®ææ¥ããå·¥å Žã®å®åã§ã¯æ±ãããšãããã®ã§ãèªè
ã¯æ³¢åœ¢ãåŠãã§ããããšã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 240,
"tag": "p",
"text": "çºé»æããäžè¬å®¶åºã«éãããŠããé»å§ã¯äº€æµé»å§ã§ãããæ±æ¥æ¬ã§ã¯50Hzã§ããã西æ¥æ¬ã§ã¯60Hzã§ãããããã¯ææ²»æ代ã®çºé»æ©ã®èŒžå
¥æã«ãæ±æ¥æ¬ã®äºæ¥è
ã¯ãšãŒããããã50Hzçšã®çºé»æ©ã茞å
¥ãã西æ¥æ¬ã®äºæ¥è
ã¯ã¢ã¡ãªã«ãã60Hzã®çºé»æ©ã茞å
¥ããããšã«ããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 241,
"tag": "p",
"text": "çºé»æããäžè¬ã®å®¶åºãªã©ã«éãããé»æµã®åšæ³¢æ°ãåçšåšæ³¢æ°ãšããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 242,
"tag": "p",
"text": "åçšé»æºã®é»å§æ¯å¹
ã¯çŽ140Vã§ããããã㯠100 à 2 {\\displaystyle 100\\times {\\sqrt {2}}} [V]ã§ããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 243,
"tag": "p",
"text": "ãããã«ããšã¯1000Hzã®ããšã§ããããããã«ãã¯kHzãšæžãã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 244,
"tag": "p",
"text": "亀æµé»æµã«å¯ŸããŠã¯ãé»æµãšåãæ¯åæ°ã§ãã¢ã³ããŒã«ã®æ³åã§çºçããç£å Žãæ¯åããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 245,
"tag": "p",
"text": "å°ç·ã§ã€ããããã³ã€ã«ã¯ãçŽæµé»æµã§ã¯ããã ã®å°ç·ãšããŠã¯ãããããããã亀æµé»æµã«å¯ŸããŠã¯ãé»ç£èªå°ã«ããèªå·±ã®çºçãããç£å Žã劚ãããããªé»æµããã³èµ·é»åãçºçããããããèªå·±èªå°(self induction)ãšããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 246,
"tag": "p",
"text": "èªå·±èªå°ã«ããèµ·é»åã®å€§ããã¯ãé»æµã®æéå€åçã«æ¯äŸãããèªå·±èªå°ã®èµ·é»åãåŒã§æžãã°ãæ¯äŸä¿æ°ãLãšããŠã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 247,
"tag": "p",
"text": "ã§ããã ãã®æ¯äŸä¿æ° L {\\displaystyle L} ãèªå·±ã€ã³ãã¯ã¿ã³ã¹(self inductance)ãšãããèªå·±ã€ã³ãã¯ã¿ã³ã¹ã®æ¬¡å
ã¯[Vã»S/m]ã ããããããã³ãªãŒãšããåäœã§è¡šããåäœã«Hãšããèšå·ãçšããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 248,
"tag": "p",
"text": "éå¿ã«äºã€ã®ã³ã€ã«ãå·»ããã³ã€ã«ã®çæ¹ã®é»æµãå€åããããšãã¢ã³ããŒã«ã®æ³åã«ãã£ãŠçããŠããç£æãå€åãããããå察åŽã®ã³ã€ã«ã«ã¯ããã®ç£æå¯åºŠã®å€åãæã¡æ¶ããããªåãã«èµ·é»åãçºçããããã®çŸè±¡ãçžäºèªå°(mutual induction)ãšèšãã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 249,
"tag": "p",
"text": "é»å§ãå
¥åãããåŽã®ã³ã€ã«ã1次ã³ã€ã«(primaly coil)ãšèšããèªå°èµ·é»åãçºçãããåŽã®ã³ã€ã«ã2次ã³ã€ã«(secondary coil)ãšããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 250,
"tag": "p",
"text": "çžäºèªå°ã«ããèµ·é»åã®å€§ããã¯ãé»æµã®æéå€åçã«æ¯äŸãããçžäºèªå°ã®èµ·é»åãåŒã§æžãã°ãæ¯äŸä¿æ°ãMãšããŠã(çžäºèªå°ã®æ¯äŸä¿æ°ã¯Lã§ã¯ç¡ãã)åŒã¯ã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 251,
"tag": "p",
"text": "ã§ããã ãã®æ¯äŸä¿æ° M {\\displaystyle M} ãçžäºã€ã³ãã¯ã¿ã³ã¹(self inductance)ãšãããçžäºã€ã³ãã¯ã¿ã³ã¹ã®æ¬¡å
ã¯ãèªå·±ã€ã³ãã¯ã¿ã³ã¹ã®åäœãšåãã§ãã³ãªãŒ(H)ã§ããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 252,
"tag": "p",
"text": "ãã®çžäºã€ã³ãã¯ã¿ã³ã¹ã®å€§ããã¯ãäž¡æ¹ã®ã³ã€ã«ã®å·»ãæ°ã©ããã®ç©ã«æ¯äŸããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 253,
"tag": "p",
"text": "",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 254,
"tag": "p",
"text": "ç£å Žã®åãã«ãã£ãŠé»å ŽãåŒãèµ·ããããããšãé»ç£èªå°ã®ã»ã¯ã·ã§ã³ã§èŠãã å®éã«ã¯é»å Žã®å€åã«ãã£ãŠç£å ŽãåŒãèµ·ããããããšãç¥ãããŠããã ããã«ãã£ãŠäœããªã空éäžãé»å Žãšç£å ŽãäŒæããŠããããšãäºæ³ãããã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 255,
"tag": "p",
"text": "é»ç£æ³¢ã®é床ãç©çåŠè
ã®ãã¯ã¹ãŠã§ã«ãèšç®ã§æ±ãããšãããé»ç£æ³¢ã®é床ã¯ãç空äžã§ã¯åžžã«äžå®ã§ããã€æ³¢ã®é床cãèšç®ã§æ±ãããšããã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 256,
"tag": "p",
"text": "ãšãªããæ¢ã«ç¥ãããŠããå
éã«äžèŽããã ãã®ããšãããå
ã¯é»ç£æ³¢ã®äžçš®ã§ããããšãåãã£ããç©çIIã§ãé»ç£æ³¢ã®é床ãæ±ããèšç®ã¯ã詳ããã¯æ±ãã èªè
ãå
éã®æž¬å®å®éšã«ã€ããŠèª¿ã¹ããªããç©çIã®æ³¢åã«é¢ããããŒãžãªã©ã§ãã£ãŸãŒã®å®éšã«ã€ããŠãåç
§ã®ããšã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 257,
"tag": "p",
"text": "æ³¢ã¯æ³¢é·Î»ãé·ãã»ã©ãæ¯åæ°fãå°ãããªããæ³¢ã®æ³¢é·Î»ãšæ¯åæ°fã®ç©fλã¯äžå®ã§ãããã¯æ³¢ã®é床vã«çãããã€ãŸã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 258,
"tag": "p",
"text": "ã§ããã é»ç£æ³¢ã®å Žåã¯ãé床ãå
éã®cãªã®ã§",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 259,
"tag": "p",
"text": "ã§ããã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 260,
"tag": "p",
"text": "æŸéçšã®ãã¬ããã©ãžãªã®é»æ³¢(ã§ãã±ãradio wave)ã¯ãé»ç£æ³¢(electromagnetic wave)ã®äžçš®ã§ãããæ³¢é·ã0.1mm以äžã®é»ç£æ³¢ãé»æ³¢ã«åé¡ãããããªããé»æ³¢ã®ãã¡ãæ³¢é·ã1mm~1cmã®ããªã¡ãŒãã«ã®é»æ³¢ãããªæ³¢ãšãããåæ§ã«ãæ³¢é·ã1cm~10cmã®é»æ³¢ãã»ã³ãæ³¢ãšãããæ³¢é·10cm~100cm(=1m)ã®é»æ³¢ã¯UHFãšèšããããã¬ãæŸéãªã©ã«äœ¿ãããUHFæŸéã¯ããã®é»æ³¢ã§ãããæ³¢é·1m~10mã®é»æ³¢ã¯VHFãšèšãããããã¬ãæŸéã®VHFæŸéã¯ããã®é»æ³¢ã§ããã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 261,
"tag": "p",
"text": "æ³¢é·ã0.1mm以äžã§ãå¯èŠå
ç·(å¯èŠå
ã®æ倧波é·ã¯780ããã¡ãŒãã«çšåºŠ)ãããã¯æ³¢é·ãé·ãé»ç£æ³¢ã¯èµ€å€ç·(ãããããããinfrared raysãã€ã³ãã©ã¬ãŒã ã¬ã€ãº)ãšããããèµ€ãã®ãå€ããšããçç±ã¯ãå¯èŠå
ã®æ倧波é·ã®è²ãèµ€è²ã ããã§ãããèµ€å€ç·ãã®ãã®ã«ã¯è²ã¯ã€ããŠããªããåžè²©ã®èµ€å€ç·ããŒã¿ãŒãªã©ãèµ€è²ã«çºå
ãã補åãããã®ã¯ã䜿çšè
ãåäœç¢ºèªãã§ããããã«ããããã«ã補åã«èµ€è²ã®ã©ã³ãã䜵眮ããŠããããã§ãããèµ€å€ç·ã¯ãç©äœã«åžåããããããåžåã®éãç±ãçºçããã®ã§ãããŒã¿ãŒãªã©ã«å¿çšãããããªãã倪éœå
ã«ãèµ€å€ç·ã¯å«ãŸããã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 262,
"tag": "p",
"text": "ããããèµ€å€ç·ãçºèŠãããçµç·¯ã¯ãã€ã®ãªã¹ã®å€©æåŠè
ã®ããŒã·ã§ã«ã倪éœå
ãããªãºã ã§åå
ããéã«ãèµ€è²ã®å
ç·ã®ãšãªãã®ãç®ã«ã¯è²ãèŠããªãéšåã枩床äžæããŠããããšãçºèŠããããšããçµç·¯ãããã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 263,
"tag": "p",
"text": "",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 264,
"tag": "p",
"text": "æã
ã人éã®ç®ã«èŠããå¯èŠå
ç·(ãããããããvisible light)ã®æ³¢é·ã¯ãçŽ780ããã¡ãŒãã«ããçŽ380ããã¡ãŒãã«ã®çšåºŠã§ãããå¯èŠå
ã®äžã§æ³¢é·ãæãé·ãé åã®è²ã¯èµ€è²ã§ãããå¯èŠå
ã®äžã§æ³¢é·ãæãçãé åã®è²ã¯çŽ«è²ã§ããã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 265,
"tag": "p",
"text": "å
ãã®ãã®ã«ã¯ãè²ã¯ã€ããŠããªããæã
ã人éã®è³ããç®ã«å
¥ã£ãå¯èŠå
ããè²ãšããŠæããã®ã§ããã 倪éœå
ãããªãºã ãªã©ã§åå
(ã¶ããã)ãããšãæ³¢é·ããšã«è»è·¡(ããã)ãããããããã®åå
ããå
ç·ã¯ãä»ã®æ³¢é·ãå«ãŸãããã äžçš®ã®æ³¢é·ãªã®ã§ããã®ãããªå
ç·ããã³å
ãåè²å
(monochromatic light)ãšããã ãŸããçœè²ã¯åè²å
ã§ã¯ãªããçœè²å
(white light)ãšã¯ãå
šãŠã®è²ã®å
ãæ··ãã£ãç¶æ
ã§ããã åæ§ã«ãé»è²ãšããåè²å
ããªããé»è²ãšã¯ãå¯èŠå
ãç¡ãç¶æ
ã§ããã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 266,
"tag": "p",
"text": "玫å€ç·(ããããããultraviolet rays)ã¯ååŠåå¿ã«åœ±é¿ãäžããäœçšã匷ãã殺èæ¶æ¯ãªã©ã«å¿çšãããã倪éœå
ã«ã玫å€ç·ã¯å«ãŸããã人éã®èã®æ¥çŒãã®åå ã¯ã玫å€ç·ãã¡ã©ãã³è²çŽ ãé
žåãããããã§ããã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 267,
"tag": "p",
"text": "èµ€å€ç·ã¯å€ªéœå
ã®ããªãºã ã«ããåå
ã§çºèŠãããã ãã§ã¯ãåå
ããã玫è²ã®å
ç·ã®ãšãªãã«ãããªã«ãç®ã«ã¯èŠããªãç·ãããã®ã§ã¯?ããšãããµããªããšãåŠè
ãã¡ã«ãã£ãŠèãããã ãã€ãã®ç©çåŠè
ãªãã¿ãŒã«ããååŠçãªå®éšæ¹æ³ãçšããŠã玫å€ç·ã®ååšãå®èšŒãããã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 268,
"tag": "p",
"text": "å»ççšã®ã¬ã³ãã²ã³ãªã©ã®ééåçã§çšããããXç·(X-ray)ãé»ç£æ³¢ã®äžçš®ã§ãããçç©ã®çŽ°èãååã¬ãã«ã§å·ã€ããçºããæ§ãæãã ã¬ã³ãç·(gammaârayãγ ray)ãåæ§ã«ãééåçã«ãå¿çšãããããçç©ã®çŽ°èãååã¬ãã«ã§å·ã€ããçºããæ§ãæãã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 269,
"tag": "p",
"text": "??",
"title": "é»ç£æ³¢"
}
] | é«çåŠæ ¡çç§ ç©çI > é»æ° æ¬é
ã¯é«çåŠæ ¡çç§ ç©çIã®é»æ°ã®è§£èª¬ã§ããã | <small>[[é«çåŠæ ¡ ç©ç]] > é»æ°</small>
----
æ¬é
ã¯[[é«çåŠæ ¡ ç©ç]]ã®é»æ°ã®è§£èª¬ã§ããã
==é»æ°==
===ç掻ã®äžã®é»æ°===
====é»æ°ãšç掻====
çŸåšç§ãã¡ã䜿ã£ãŠããå€ãã®è£œåãé»æ°ãçšããŠåããŠããã
ããã«ã¯æ§ã
ãªçç±ãèãããããããŸã第äžã«é»æ°ã¯æ§ã
ãªå¥ã®ãšãã«ã®ãŒã«å€æã§ããããšãäŸãã°é»ç±ç·ã䜿ã£ãŠç±ã«ãé»çãçºå
ãã€ãªãŒãã䜿ã£ãŠå
ã«ãã¢ãŒã¿ãŒã䜿ã£ãŠéåã«å€æããããšãåºæ¥ãã次ã«ãé»æ± ãã³ã³ãã³ãµã䜿ã£ãŠãšãã«ã®ãŒãç¶æãããŸãŸæã¡éã¶ããšãåºæ¥ãããšããé»ç·ã䜿ã£ãŠé·è·é¢ãéé»ã§ããããšããŸããé»å補åã®èšç®èœåãä¿¡å·ã®äŒéèœåãåªããŠããããšããŸãæ¯èŒçã«å®å
šã«å°éã®ãšãã«ã®ãŒãåãåºããããšãçãèããããã
<!--
%(äŸãã°ãåååã䜿ã£ãŠã湯ã沞ãã人ã¯
% ããªããšæãããã)
-->
====ã¢ãŒã¿ãŒãšçºé»æ©====
é»æ°ãéåã«å€ãããã®ãšããŠ
ã¢ãŒã¿ãŒãããã
ããã®éã®ãã®ãšããŠã
éåãé»æ°ã«å€ããããšãåºæ¥ãã
ãããè¡ãªãã®ã¯çºé»æ©ãšåŒã°ããã
äŸãã°ãçºé»æã¯äœããã®éåã®ãšãã«ã®ãŒã
å©çšããŠé»æ°ãããããŠããã
äŸãã°ãæ°Žåçºé»æã§ã¯ã
æ°Žã®èœäžããåãå©çšããŠããã
倧éã®æ°Žãèœäžãããšãã«ã¯
人éãäœå人ãããã£ãŠãããã
åºæ¥ãããšããªãããããšãããã
ããšãã°ãåãç«ã£ã海岞ç·ãªã©ã¯
äž»ã«æ°Žã®æµãã«ãã£ãŠäœãããŠããã
ãã®ããã«ãæ°Žã®åã¯åŒ·å€§ã§ããã®ã§ã
ãããäžæãå©çšããæ¹æ³ããããš
éœåããããå®éçŸä»£ã§ã¯
é»æ°ãåªä»ãšããŠããã®åã
åãã ãããšã«æåããŠããã
<!--
%ãã£ãšå·¥åŠçãªããšãæžããæ¹ããããã...ã
%
-->
====亀æµãšé»æ³¢====
é»æ± ã®ããã«é»æ¥µã®+ãš-ãå®ãŸã£ã
é»æµãçŽæµé»æµãšåŒã¶ã
äžæ¹ãçºé»æããåŸãããé»æµã®ããã«
+ãš-ãéãé床ã§å
¥ãæãã
é»æµã亀æµé»æµãšåŒã¶ã
å®éã«ã¯ãã€ãªãŒããçšããŠ
亀æµãçŽæµã«å€ããŠ
䜿ãããšãããè¡ãªãããã
äœããªã空éãå
ãçŽé²ããŠããããã«
èŠããããšããããå®éã«ã¯ããã¯
é»æ³¢ãšåããã®ã§ããã
é»æ³¢ãšã¯äŸãã°ãæºåž¯é»è©±ã®éä¿¡ã«äœ¿ããããã®ã§ããã
é»è·ãæã£ãç©äœãåãããšãå¿
ç¶çã«
çãããã®ã§ããã
== éé»æ° ==
ãã©ã¹ããã¯ã®äžæ·ããªã©ã§é«ªãããããšåž¯é»ããçŸè±¡ãªã©ã®ããã«ãç©è³ªãé»æ°ã垯ã³ãããšã'''垯é»'''ïŒããã§ãïŒãšãããç©äœãããã£ãŠçºçãããéé»æ°ã'''æ©æŠé»æ°'''ãšããã
ã¬ã©ã¹æ£ãçµ¹ã®åžã§ããããšãã¬ã©ã¹æ£ã¯æ£ã®é»æ°ã«åž¯é»ããçµ¹ã¯è² ã®é»æ°ã«åž¯é»ããã
é»æ°ã®éã'''é»è·'''ïŒã§ãããchargeïŒãšããããããã¯'''é»æ°é'''ãšããã
é»è·ã®åäœã¯'''ã¯ãŒãã³'''ã§ãããã¯ãŒãã³ã®èšå·ã¯Cã§ããã
éé»æ°ã«ããé»è·ã©ããã«åãåã'''éé»æ°å'''ãšããã
ãªãã垯é»ããŠããªãç¶æ
ãé»æ°çã«äžæ§ã§ããããšããã
éå±ã®ããã«ãé»æ°ãéããç©äœã'''å°äœ'''ïŒã©ããããconductorïŒãšããããã©ã¹ããã¯ãã¬ã©ã¹ããŽã ã®ããã«é»æ°ãéããªãç©è³ªã'''絶çžäœ'''ïŒãã€ãããããinsulatorïŒãããã¯'''äžå°äœ'''ïŒãµã©ãããïŒãšããã
éå±ã¯å°äœã§ããã
é»æ°ã®æ£äœã¯'''é»å'''ïŒelectronïŒãšããç²åã§ããããã®é»åã¯è² é»è·ã垯ã³ãŠãããïŒé»åã®é»è·ãè² ã«å®çŸ©ãããŠããã®ã¯ã人é¡ãé»åãçºèŠããåã«é»è·ã®æ£è² ã®å®çŸ©ãè¡ãããããšããé»åãèŠã€ãã£ãéã«é»åã®é»è·ã調ã¹ããè² é»è·ã ã£ãããã§ãããïŒ
[[File:Metalic bond model.svg|thumb|400px|éå±äžã§ã®èªç±é»åã®æš¡åŒå³]]
éå±ãå°äœãªã®ã¯ãéå±äžã®é»åã¯ãããšã®ååãé¢ããŠããã®éå±å
šäœã®äžãèªç±ã«åããããã§ãããéå±äžã®é»åã®ããã«ãç©è³ªäžãèªç±ã«åããç¶æ
ã®é»åãã'''èªç±é»å'''ïŒãããã§ããïŒãšããã
é»æµãšã¯ãèªç±é»åã移åããããšã§ããã
ãã£ãœãã絶çžäœã¯ãèªç±é»åããããªãã絶çžäœã®é»åã¯ããã¹ãŠãããšã®ååã«æçžïŒããã°ãïŒãããŠéã蟌ããããŠããŠãèªç±ã«ã¯åããªãã
æ£é»è·ãšã¯ãç©è³ªã«é»åãæ¬ ä¹ããŠããç¶æ
ã§ããã
è² é»è·ãšã¯ãç©è³ªãé»åãå€ãæã£ãŠããç¶æ
ã§ããã
垯é»ããŠããªã絶çžäœã®ç©è³ªããããããããŠãäž¡æ¹ãæ©æŠé»æ°ã«åž¯é»ãããå Žåãçæ¹ã¯æ£é»è·ãçããããçæ¹ã®ç©è³ªã¯è² é»è·ãçããããã®ãšããçºçããæ£é»è·ã®å€§ãããšè² é»è·ã®å€§ããã¯åãã§ããã
ããã¯ãé»åã移åããŠãçæ¹ã®ç©è³ªã¯é»åãäžè¶³ããããçæ¹ã¯çéã®é»åãéå°ã«ãªã£ãŠããããã§ããã
ãã®ããã«ãé»åã¯çæãæ¶æ»
ãããªããããã'''é»è·ä¿åå'''ãããã¯'''é»æ°éä¿åå'''ãšèšãã
=== éé»èªå° ===
[[Image:Electrostatic induction.svg|thumb|upright=1.5|å°äœã¯ãè¿ãã®é»è·ã«ãã£ãŠè¡šé¢ã«é»è·ãèªå°ããããç©äœå
éšã®éé»æ°åã®å€§ããã¯ãŒãã§ããã]]
é»æ°çã«äžæ§ã§ãã£ãå°äœã®ç©è³ªïŒä»®ã«ç©è³ªAãšããïŒã«åž¯é»ããå¥ã®ç©è³ªïŒä»®ã«ç©è³ªBãšããïŒãæ¥è§Šãããã«è¿ã¥ãããšãç©è³ªAã«ã¯ã垯é»ç©è³ªBã®é»è·ã«åŒãå¯ããããŠãç©äœAã®å
éšã§å察笊å·ã®é»è·ã垯é»ç©äœBã«è¿ãåŽã®è¡šé¢ã«çããããŸãã垯é»ç©äœBãšåãé»è·ã¯åçºããã®ã§ãç©äœAå
éšã®åž¯é»ç©äœBãšã¯é ãåŽã®è¡šé¢ã«çããã
ãã®ãããªçŸè±¡ã'''éé»èªå°'''ïŒããã§ãããã©ã;Electrostatic inductionïŒãšãããéé»èªå°ã§çããé»è·ã®æ£é»è·ã®éãšè² é»è·ã®éã¯çéã§ãããïŒé»æ°éä¿åã®æ³åïŒ
å°äœã®å
éšã«éé»æ°åã¯ç¡ãããããã£ããšãããšãèªç±é»åãªã©ã®é»è·ãåããé»æµãæµãç¶ããããšã«ãªããããã®ãããªçŸè±¡ã¯å®åšããªãã®ã§äžåçã«ãªãããããã£ãŠãå°äœã®å
éšã«éé»æ°åã¯ç¡ãã
è¡šé¢ã«é»è·ãéãŸãã®ã¯ãå°äœã®å
éšã«éé»æ°åãäœãããªãããã§ããããããã£ãŠéé»èªå°ã§åŒãå¯ããããé»è·ã®å€§ããã¯ãå€éšããå°äœå
éšãžã®éé»æ°åãæã¡æ¶ãã ãã®å€§ããã§ããã
ãã®å°äœå
éšã®é»è·ããŒãã«ãªãæ§è³ªãå¿çšãããšãäžç©ºã®å°äœã§åºæ¥ãç©äœãçšããŠãéé»æ°åãé®èœããããšãã§ããããããéé»é®èœïŒããã§ããããžããelectric shildingïŒãšããã
=== èªé»å極 ===
[[File:Pith ball electroscope operating principle.svg|thumb|300px|èªé»å極ã®æŠå¿µå³]]
絶çžäœïŒä»®ã«AãšããïŒã«é»è·ãè¿ã¥ããå Žåã¯ãå°äœãšã¯éããç©äœAã®å
éšã®é»åã¯èªç±ã«è¡šé¢ã«ã¯éãŸããªãããç©äœå
éšã®ååã®æ£è² ã®é»è·ã®æ¥µæ§ãæã£ãéšåããå€éšã®éé»æ°åã«åŒãå¯ããããããã«ãè¿ã¥ããé»è·ã«è¿ãåŽã«ã¯ç°çš®ã®é»è·ãçããé ãåŽã«ã¯ãåçš®ã®é»è·ãçããã
ååãååãå€éšã®éé»æ°åã«ãã£ãŠãæ£è² ã®é»è·ã®éšåãçããããšã'''å極'''ïŒã¶ããããïŒãšããããå€éšã®é»åã«ãã£ãŠèµ·ããããã®ãããªå極ã®ãããã'''èªé»å極'''ïŒããã§ãã¶ãããããdielectric polarizationïŒãšããã
絶çžäœã¯ãéé»æ°åã«ããããããšèªé»å極ãè¡ãã®ã§ã絶çžäœã®ããšã'''èªé»äœ'''ïŒããã§ããããdielectricïŒãšãããã
å°äœã«éé»èªå°ãããæ£è² ã®é»è·ã¯ãå°äœãåæãªã©ãããã°æ£é»è·ãšè² é»è·ãå¥åã«åãåºãããšãã§ããããããèªé»äœã®æ£è² ã®é»è·ã¯ãååãååãšå¯æ¥ã«çµã³ã€ããŠãããããæ£è² ã®é»è·ãåãããŠåãåºãããšã¯åºæ¥ãªãã
{{clear}}
==é»å Žãšç£å Ž==
===é»è·ãšé»å Ž===
====é»è·====
[[Image:Static repulsion.jpg|thumb|ã»ãããŒãã®åãé»è·ã«ããåçº]]
[[Image:Static attraction.jpg|thumb|ã»ãããŒãã®ç°é»è·ããåŒãå¯ã]]
ããç©è³ªãé»æ°ã垯ã³ãŠããïŒåž¯é»ããŠããïŒãšãããã®åž¯é»ã®å€§å°ã®çšåºŠã'''é»è·'''ïŒã§ãããelectric chargeïŒãšãããããŸããŸãªç©è³ªããããããªæ¹æ³ã§åž¯é»ãããçµæãé»è·ã«ã¯ã垯é»ãã2åã®ãã®ã©ãããè¿ã¥ããæã«åŒã£åŒµãåããã®ïŒåŒåãåãïŒãšåçºããããã®ïŒæ¥åãã¯ãããïŒã®2çš®é¡ãããããšãåãã£ãã
ãã®ãããªã垯é»ããŠããç©äœã«åãåã'''éé»æ°å'''ãšããã
ã¹ã€ã®åž¯é»ãããã®ããä»ã«ãããã€ãçšæããŠãè¿ã¥ããŠå®éšãã2åã®ç©äœã®çµã¿åãããå€ãããšãçµã¿åããã«ãã£ãŠã2åã®ç©äœã©ããã«åŒåãåãå Žåãããã°ãæ¥åãåãå Žåãããããšãåãã£ãããã®åŒåãšæ¥åã®é¢ä¿ã¯ã垯é»ããé»è·ã®çš®é¡ã«å¿ããããšãããã£ãã
==== æ£é»è·ãšè² é»è· ====
çµè«ãèšããšãé»è·ã«ã¯æ£è² ã®2çš®é¡ããããæ£ã®é»è·ã©ããã®ç©äœãè¿ã¥ãããšãã¯åçºããããè² ã®é»è·ã©ãããè¿ã¥ãããšããåçºããããæ£ãšè² ã®é»è·ãè¿ã¥ããæã«ã¯åŒåãåãã
ã€ãŸããå笊å·ã®é»è·ãè¿ã¥ããå Žåã¯ãåçºåãçãããç°ç¬Šå·ã®é»è·ãè¿ã¥ããå Žåã¯ãåŒåãçããã
{{clear}}
==== éé»æ°å ====
[[File:Coulomb.jpg|thumb|150px|ã¯ãŒãã³ã®èåãCharles Augustin de Coulomb]]
[[Image:Bcoulomb.png|thumb|left|300px|ã¯ãŒãã³ãéé»æ°åã®æž¬å®ã«çšããããããèšã]]
[[File:Coulomb torsion.svg|thumb|300px|]]
éé»æ°ã©ããã®åã®åŒ·ãã¯ãå®éšçã«ã¯ãé»è·ã®éã«åãåã¯ãéåã®å Žåãšåæ§ã«åãåãŒãåã2ç©äœã®éã®è·é¢ã®2ä¹ã«åæ¯äŸããããšãç¥ãããŠãããæŽã«ãé»è·ã®å€§ããã倧ããã»ã©é»è·éã«åãåã倧ããããšãèæ
®ãããšãè·é¢''r''ã ãé¢ããŠãããããé»è·<math>q _1</math>ã<math>q _2</math>ãæã£ãŠãã2ç©äœã®éã«åãå''F''ã¯ã
:<math>
F = k\frac{q_1 q_2}{r^2} = \frac 1 {4\pi\epsilon} \frac {q _1 q _2}{r^2}
</math>
ã§äžãããããããã'''ã¯ãŒãã³ã®æ³å'''ïŒ Coulomb's lawïŒãšãããããã§ã<math>k</math>ã¯æ¯äŸå®æ°ã§ãããäž¡é»è·ã®åšå²ã«ããç©äœã®çš®é¡ã«ããå€åããå®æ°ã§ãããç空äžã§ã®é»å Žãèããå Žåã®kã®å€ã¯ã
:<math>k_0 = 9.0 \times 10^9 </math>[Nã»m<sup>2</sup>/C<sup>2</sup>]ïŒã¯ãŒãã³ã®æ¯äŸå®æ°ïŒ
ã§ããããŸãã<math>\epsilon</math>ã¯åŸã»ã©ç»å Žããèªé»çïŒããã§ããã€ïŒãšåŒã°ããç©çå®æ°ã§ãããèªé»çã¯ãäž¡é»è·ã®åšå²ã«ããç©äœã®çš®é¡ã«ããå€åããå®æ°ã§ãããèªé»çã«ã€ããŠã¯ããã®æãåããŠèªãã 段éã§ã¯ããŸã ç¥ããªããŠãè¯ããã®ã¡ã«ç©çIIã§èªé»çã詳ãã解説ããã
èªé»ç<math>\epsilon</math>ãšã¯ãŒãã³ã®æ¯äŸå®æ°kã«ã¯äžåŒã®é¢ä¿
:<math>k= \frac 1 {4\pi\epsilon}</math>
ãããã
ç©äœã®ãŸããã«èç©ããããã®ã'''é»è·'''ãšåŒã¶ãé»æ°åã«ãã£ãŠåçºããã£ãããåŒãã€ããã£ããããç©äœã'''é»è·ãæã€'''ç©äœãšåŒã¶ããŸããããã§èŠ³å¯ãããéé»æ°åãã'''ã¯ãŒãã³å'''ãšåŒã¶ããšãããã
2åã®é»è·ã©ããããããŒãåã¯åãã§ããããããã£ãŠ'''äœçšã»åäœçšã®æ³å'''ã«åŸã£ãŠããã
[[Image:Coulombslawgraph.svg|thumb|center|300px|2åã®ç¹é»è·ã®éã«åãåã®é¢ä¿ã<br>ã¯ãŒãã³ã®æ³åã«ãããšF1=F2ãšãªãã]]
ããã§ãé»è·ã®åäœã¯<nowiki>[C]</nowiki>ã§äžãããããèšå·ã®Cã¯ãã¯ãŒãã³ããšèªãã
{{-}}
----
*äŸé¡
[[File:ã¯ãŒãã³ã®æ³å äŸé¡1.svg|thumb|ã¯ãŒãã³ã®æ³å äŸé¡1]]
å³ã®ããã«ã2æ¬ã®ç³žã«ãããããåã質émkgã§ãåã笊å·ãšå€§ããã®é»è·qCã®çããã¶ãããã£ãŠããããã¯ãã¯ãŒãã³åã§åçºããã®ã§ãå³ã®ããã«ã糞ãè§åºŠÎžããªãã
ãã®ãšãã質émã«ããéåãšãé»è·qã«ããã¯ãŒãã³åãšã®é¢ä¿ã«ã€ããŠãåŒãç«ãŠãããªããå¿
èŠãªãã°ã糞ã®åŒµåã¯TNãšããããšã
{{-}}
解æ³
:[[File:ã¯ãŒãã³ã®æ³å äŸé¡1 解æ³.svg|thumb|left|400px|ã¯ãŒãã³ã®æ³å äŸé¡1 解æ³]]
å³ã®ãããªäœçœ®é¢ä¿ã«ãªãã®ã§ãå³ã®ããã«åŒãç«ãŠãã°ããã
:â» ãã®ããã«ãé»æ°ç£æ°åŠã®åé¡ã§ã¯ãå³ããã¡ããšæžããŠã解æ³ãèããå¿
èŠããããæ°åŒã ãã§èšç®ãããšãç«åŒãã¹ãªã©ã®åå ã«ãªãã
{{-}}
----
â» äžèšã®2æ¬ã®ç³žã«ã¶ãããã£ãçã®ã¯ãŒãã³åã®äŸé¡ã¯ãé»æ°ç£æ°åŠã®ã©ã®å
¥éæžã«ããããããªå
žåçãªåé¡ã§ããã®ã§ãèªè
ã¯ãã¡ããšç解ããããšã
{{-}}
----
*åé¡äŸ
**åé¡
é»è·<math>q _1</math>, <math>q _2</math>ã®éã®è·é¢ãrã®å Žåãš2rã®å Žåã§ã¯ãéã«åãåã®å€§ããã¯ã©ã¡ããã©ãã ã倧ãããçããã
ãŸããè·é¢ã2rã®æã®2ç¹éã®åã®å€§ãããçããã
**解ç
ã¯ãŒãã³åã¯ãç©äœéã®è·é¢ã®é2ä¹ã«æ¯äŸããã®ã§ãè·é¢ã2rã®æã¯ãrã®æã®å€§ããã®<math>\frac 1 4</math>ãšãªãããŸããåãåã®å€§ããã¯ãã¯ãŒãã³åã®åŒãçšããŠã
:<math>
f = \frac 1 {4\pi\epsilon _0} \frac {q _1 q _2}{4r^2}
</math>
ãšãªãã
----
====é»å Ž ====
æ¢ã«ãããé»è·Aã®ãŸããã®å¥ã®é»è·Bã«ã¯ããã®é»è·ããã®è·é¢ã®é2ä¹ã«æ¯äŸããåããããããšãè¿°ã¹ãã
[[File:é»å Žã®éãåãã.svg|thumb|400px|é»å Žã®éãåãã]]
ããã§ãé»è·Bãåããåã¯ããã®é»è·Bã®å€§ããã«æ¯äŸããããšãåãããŠèãããšããã®é»è·Bã®å€§ããã«ããããããé»è·Aã®å€§ããã ãã§æ±ºãŸãéãå°å
¥ããŠãããšéœåããããããã§ããã®ãããªéãšããŠ'''é»å Ž'''ïŒã§ãã°ïŒãå°å
¥ããããã®ãšããé»å Ž<math>\vec E</math>ã®äžã«ããé»è·<math>q</math>ã«åãå<math>\vec F</math>ã¯ã
:<math>\vec F = q \vec E</math>
ã§äžãããããé»å Žã¯åäœé»è·ã«åãåãšèããããšãã§ããé»å Žã®åäœã¯[N/C]ã§ããããé»å Žãã¯ããé»çãïŒã§ãããïŒãšãåŒã°ããã
ïŒæ¥æ¬ã®ç©çåŠã§ã¯ãé»å ŽããšåŒã¶ããšãå€ãããŸããæ¥æ¬ã®é»æ°å·¥åŠã§ã¯ãé»çããšåŒã°ããããšãå€ããææ²»æã®ç¿»èš³ã®éã®ãæ¥æ¬åœå
ã®æ¥çããšã®éãã«éããããããªãæ¥æ¬ããŒã«ã«ãªéœåã§ãããåŒã³æ¹ã¯ç©çã®æ¬è³ªãšã¯é¢ä¿ãªãã®ã§ãããã§ã¯ãã©ã¡ãã®è¡šçŸãçšãããã¯ãæ¬æžã§ã¯ç¹ã«ãã ãããªããè±èªã§ã¯ç©çåŠã»é»æ°å·¥åŠãšãâelectric fieldâã§å
±éããŠãããïŒ
äžã®ã¯ãŒãã³åã®çµæãšåããããšãé»è·Aã®ãŸããã«å¥ã®é»è·ãååšããªããšããé»è·<math>q</math>[C]ã®é»è·ããŸãšãé»å Ž<math>\vec E</math>ã¯ã
:<math>\vec E = \frac 1 {4\pi\epsilon _0} \frac {q}{r^2} \vec e _r</math>
ã§äžããããããã ããrã¯é»è·ããã®è·é¢ã§ããã<math>\vec e _r</math>ã¯ãé»è·ãšããç¹ãçµãã çŽç·äžã§ãé»è·ãšå察æ¹åãåããåäœãã¯ãã«ã§ããã
é»è·ã®åãã®é»å Žã¯ãå¹³é¢äžã§æŸå°ç¶ã®ãã¯ãã«ãšãªãããšã«æ³šæã
{|
| [[File:VFPt minus thumb.svg|150px|thumb|è² é»è·ã®åšãã®é»å Žã®åã]]
| [[File:VFPt plus thumb.svg|150px|thumb|æ£é»è·ã®åšãã®é»è·ã®åã]]
|}
é»å Žã¯ãã¯ãã«ã§ãããé»è·ã2åãããšãã¯ãããããã®é»è·ãã€ããé»å Žããéãåãããã°ããã
:<math> \vec E = \vec {E_1} + \vec {E_2} </math>
ã§ããã
é»è·ã3å以äžã®ãšãããåæ§ã«éãåãããã°è¯ãã
å³ã®ããã«ãé»è·ããåºãé»å Žã®æ¹åãå³ç€ºãããã®ã'''é»æ°åç·'''ïŒã§ãããããããelectric line of forceïŒãšããã
é»è·ãè€æ°ããå Žåã«ã¯ãå®éã«æ°ãã«çœ®ãããé»è·ãåããåã¯ããããã足ãåããããã®ãšãªãããããã£ãŠãè€æ°ã®é»è·ãããå Žåã®åšå²ã®é»çã¯ãããããã®é»è·ãäœãé»çãã¯ãã«ã®åãšãªãïŒéãåããã®åçïŒã
<!-- é»æ°åç· -->
[[File:Camposcargas.PNG|thumb|left|300px|å笊å·ã®é»è·ã©ããïŒå·ŠïŒãè¿ã¥ããå Žåã¯åçºããããç°ãªã笊å·ã®é»è·ã©ããïŒå³ïŒãè¿ã¥ããå Žåã¯åŒãä»ãåãã]]
[[File:VFPt dipole electric manylines.svg|thumb|center|200px|ç°ç¬Šå·ã®é»è·ã©ããã®å Žåã®é»æ°åç·]]
{{clear}}
é»æ°åç·ãå³ç€ºããå Žåã¯ãæ£é»è·ããåç·ãåºãŠãè² é»è·ã§åç·ãåžåãããããã«æžããåç·ã¯ãé»å Žãå³ç€ºãããã®ãªã®ã§ãé»è·ä»¥å€ã®å Žæã§ã¯ãåç·ãåå²ããããšã¯ãªãã
åç·ãçæããã®ã¯æ£é»è·ã®å Žæã®ã¿ã§ãããåç·ãæ¶æ»
ããã®ã¯ãè² é»è·ã®å Žæã®ã¿ã§ããã
èšãæããã°ãåç·ãé»è·ä»¥å€ã®å Žæã§æ¶æ»
ããããšã¯ãªãããé»è·ä»¥å€ã®å Žæã§åç·ãçæããããšã¯ãªãã
å°äœã®å
éšã®é»å Žã¯ãŒãã§ãã£ããèšãæããã°ãé»æ°åç·ã¯ãå°äœã®å
éšã«ã¯é²å
¥ã§ããªãã
[[File:VFPt image charge plane horizontal.svg|200px|thumb|é»æ°åç·ã¯ãå°äœã®å
éšã«ã¯é²å
¥ã§ããªãã]]
<!-- ã¬ãŠã¹ã®æ³å -->
[[File:E FieldOnePointCharge.svg|é»æ°åç·ã®ååž]]
ç¹é»è·ããã¯ãå³ã®ããã«ãæŸå°ç¶ã«é»æ°åç·ãåºããã¯ãŒãã³ã®æ³åã®ä¿æ°ã«ãã
:<math>\frac{1}{4 \pi \epsilon _0} \frac{q_1 q_2}{r^2}</math>
ã®ãã¡ã®ãåæ¯ã®
<math>4 \pi r^2</math>
ã¯ãçã®è¡šé¢ç©ã®å
¬åŒã«çããã®ã§ãé»æ°åç·ã®å¯åºŠã«æ¯äŸããŠãé»å Žã®åŒ·ããããã¯éé»æ°åã®åŒ·ãã決ãŸããšèããããã
éé»èªå°ã§ã¯ãå°äœå
éšã«ã¯éé»æ°åãåããŠããªãã®ã§ãã£ããããã¯ãé»å ŽãšããæŠå¿µãçšããŠèšãæããã°ãå°äœå
éšã®é»å Žã¯ãŒãã§ããããšèšããã
====é»äœ====
ã¯ãŒãã³åã¯åïŒã¡ããïŒã§ãããããããã«éãã£ãŠå¥ã®é»è·ãè¿ã¥ããå Žåã¯ãè¿ã¥ããå¥ã®é»è·ã¯ä»äºãããããšã«ãªãããŸããè¿ã¥ããé»è·ãææŸãã°ãã¯ãŒãã³åã«ãã£ãŠåãåããä»äºãããããšã«ãªããããè¿ã¥ããç¶æ
ã«ããå¥é»è·ã¯äœçœ®ãšãã«ã®ãŒãèããŠããããšã«ãªãã
ãããã£ãŠãã¯ãŒãã³åã«å¯ŸããŠãäœçœ®ãšãã«ã®ãŒãå®çŸ©ããããšãã§ãããïŒãªããè¡æè»éäžã®ç©äœã®ãããªãå°è¡šãã倧ããé¢ããå Žæã®éåããã¯ãŒãã³åãšåæ§ã«é2ä¹åãªã®ã§ãããã§èããèšç®ææ³ã¯éåã«ããäœçœ®ãšãã«ã®ãŒã«ãå¿çšã§ãããéåå é床gãçšããåmgãšããã®ã¯å°è¡šè¿ãã§ã®è¿äŒŒã«ãããªããïŒ
ã¯ãŒãã³åã«ããé»å Žã®å®çŸ©ã§ã¯ãåäœé»è·ã«å¯ŸããŠé»å Žãå®çŸ©ããã®ãšåæ§ãäœçœ®ãšãã«ã®ãŒã«å¯ŸããŠããåäœé»è·ã«å¿ããŠå®çŸ©ã§ããéãå°å
¥ãããšéœåãããããã®ãããªéã'''é»äœ'''ïŒã§ãããelectric potentialïŒãšåŒã¶ãé»äœã®åäœã¯'''ãã«ã'''ãšãããé»äœãäŸãããšãå°è¡šè¿ãã§ã®éåã®äœçœ®ãšãã«ã®ãŒãèããéã®ãghããªã©ã«çžåœããéã§ããã
ã¯ãŒãã³åã®çµæãšã<math>q</math>[C]ã®é»è·ããè·é¢''r''ã ãé¢ããç¹ã®é»äœVã¯ãé»å Žã®ç©åèšç®ã§åŸããããïŒç©åããŸã ç¿ã£ãŠãªãåŠå¹Žã®èªè
ã¯ãåãããªããŠãæ°ã«ããã次ã®çµæãžãšé²ãã§ãã ãããïŒçµæã®ã¿ãèšããšã
:<math>V=\frac{1}{4\pi\epsilon _0} \frac{q}{r}</math>
ãšãªãã
é»äœVã®ç¹ã«''q''[C]ã®é»è·ã眮ãããšãããã®é»è·ã®ã¯ãŒãã³åã«ããäœçœ®ãšãã«ã®ãŒ''U''[J]ã¯ãé»äœVãçšããã°ã
:<math>U = qV</math>
ãšãªãããããã£ãŠãé»äœ<math>V_1</math>ãã«ãã®ç¹ããé»äœ<math>V_2</math>ãã«ãã®äœçœ®ãžãšé»è·''q''[C]ãéé»æ°åãåããŠç§»åãããšããéé»æ°åã®ããä»äº''W''[J]ã¯
:<math>W = q(V_2 - V_1)</math>
ãšãªãã
{{-}}
[[File:äžæ§ãªé»å Ž.svg|thumb|500px|äžæ§ãªé»å Ž]]
ãã£ãœããäžæ§ãªé»å Žã«ãããŠã¯ãé»äœã®åŒããé»å ŽãçšããŠç°¡åã«è¡šãããšãã§ãããè·é¢''d''ã ãé¢ããå¹³è¡å¹³æ¿é»æ¥µã®éã«äžæ§ãªé»å Ž<math>\vec E</math>ãçããŠãããšãããã®é»çã®äžã«çœ®ããé»è·''q''ã¯éé»æ°å<math>q\vec E</math>ãåããããã®é»è·ãé»çã®åãã«æ²¿ã£ãŠäžæ¹ã®é»æ¥µããä»æ¹ã®é»æ¥µãŸã§ç§»åãããšããé»çã®ããä»äº''W''㯠<math>W = qEd</math> ãšãªããããããã2極æ¿ã®é»äœå·®''V''ã¯ã
:<math>V=Ed</math>
ã§è¡šãããšãã§ããããšãããããåŒãå€åœ¢ããŠ
:<math>E= \frac{V}{d}</math>
ãšããããšãã§ãããããã§ãåäœãèãããšãå³èŸºã¯é»å§ãè·é¢ã§å²ã£ããã®ã§ãããããé»çã®åäœãšããŠ[N/C]ã®ã»ã[V/m]ãçšããããšãã§ããããšããããã
é»äœã®åäœã¯'''ãã«ã'''ã§ããããã®éã¯æ¢ã«[[äžåŠæ ¡çç§]]ãªã©ã§æ±ã£ã[[w:é»å§|é»å§]]ïŒã§ããã€ãvoltageïŒã®åäœãš'''åã'''åäœã§ãããå®éã«é»æ°åè·¯ã«é»å§ããããããšã¯ãåè·¯äžã®é»åã«é»å ŽããããŠåããããšãš'''çãã'''ã
éé»èªå°ã«ãã£ãŠãå°äœå
éšã®é»å Žã¯ãŒãã§ãã£ãããã®ããšãããå°äœã®è¡šé¢ã¯ãé»äœãçãããå°äœè¡šé¢ã¯äºãã«çé»äœã§ããã
é»äœã®åºæºã¯ãå®çšäžã¯ãå°é¢ã®é»äœããŒãã«çœ®ãããšãå€ããé»æ°åè·¯ã®äžéšã倧å°ã«ã€ãªãããšãæ¥å°ïŒãã£ã¡ïŒãŸãã¯'''ã¢ãŒã¹'''ïŒearthïŒãšãããåè·¯ãã¢ãŒã¹ããŠããã®ã€ãªãã éšåã®é»äœããŒããšèŠãªãããšãå€ãã
*åé¡äŸ
**åé¡
çŽç·äžã§è·é¢0, b[m]ã®ç¹ã«ãé»è·q, q'ãæã€ç©äœã眮ããŠããããã®æãäœçœ®a[m](a<b)ã®ç¹ã®é»äœãæ±ããã
**解ç
é»äœã®åŒãçšããã°ãããé»è·ãè€æ°ãããšãã«ã¯ãé»äœã¯ããããã®é»è·ãã€ããåºãé»è·ã®åã«ãªãããšã«æ³šæãçãã¯ã
:<math>V = \frac{1}{4\pi\epsilon _0} (\frac{q}{a} + \frac{q'}{b-a})</math>
ãšãªãã
----
å°äœè¡šé¢ã¯çé»äœãªã®ã§ããã£ãŠãé»æ°åç·ã¯å°äœè¡šé¢ã«åçŽã§ããã
ãã®ããšãããé»æ°åç·ãšé»å Žã¯åçŽã§ããã
é»å Žãéãåãããããããã«ãé»äœãéãåãããããããªããªãé»äœãšã¯ãé»å ŽãèããŠãçµè·¯ã«ãŠç©åãããã®ã§ããããã
åŠæ ¡ã®ãã¹ããªã©ã§ã¯ãé»äœã®èšç®ã®ãããã¯ãŒãã³åã®æ¹åã®åéããªã©ã«ããèšç®ãã¹ãªã©ããµãããããé»å Žãæ±ããŠããããããç©åããŠãé»äœãæ±ããã®ããèšç®äžã¯å®å
šã§ããã
== éé»èªå°ãšèªé»å極 ==
=== ã³ã³ãã³ãµãŒ ===
[[File:ã³ã³ãã³ãµãŒ æ§é ãšåç.svg|thumb|400px|ã³ã³ãã³ãµãŒ]]
'''ã³ã³ãã³ãµãŒ'''ïŒè±:capacitor ,ããã£ãã·ã¿ããšèªãïŒã¯ãå³ã®ããã«ã2æã®é»æ¥µãåãããããåè·¯äžã«é»è·ãèç©ã§ããéšåãäžããçŽ åã§ããã
[[File:ã³ã³ãã³ãµãŒ å
é»ã®ä»çµã¿.svg|thumb|500px|ã³ã³ãã³ãµãŒã®å
é»ã®ä»çµã¿]]
ã³ã³ãã³ãµãŒã«é»è·ãèããããšã'''å
é»'''ïŒãã
ãã§ãïŒãšãããã³ã³ãã³ãµãŒããé»è·ãæŸåºãããããšã'''æŸé»'''ãšããã
ã³ã³ãã³ãµã®äž¡ç«¯ã«ããé»äœVãäžãããããšããã³ã³ãã³ãµã«ã¯ãé»äœã«æ¯äŸããé»è·Qãèç©ãããããã®ãšããã³ã³ãã³ãµã®èç©èœåãèšå·ã§ C ãšãããŠã
:<math>Q=CV</math>
ãšããŠCãåããCã¯'''éé»å®¹é'''ïŒããã§ãããããããelectric capacitanceïŒãšåŒã°ããåäœã¯F('''ãã¡ã©ã'''ãfarad)ã§äžããããã
1ãã¡ã©ãã¯å®çšäžã¯å€§ããããã®ã§ã10<sup>-12</sup>ãã¡ã©ããåäœã«ãã1pF(ãã³ãã¡ã©ã)ãã10<sup>-6</sup>ãã¡ã©ããåäœã«ãã1ÎŒF(ãã€ã¯ããã¡ã©ã)ã䜿ãããããšãå€ãã
{{-}}
=== å¹³è¡æ¿ã³ã³ãã³ãµãŒ ===
[[File:å¹³è¡æ¿ã³ã³ãã³ãµãŒ é»å Ž.svg|thumb|400px|å¹³è¡æ¿ã³ã³ãã³ãµãŒã®é»å Ž]]
極æ¿ãå¹³è¡ãªã³ã³ãã³ãµãŒãå¹³è¡æ¿ã³ã³ãã³ãµãŒãšããã
å¹³è¡æ¿ã³ã³ãã³ãµãŒã®ã極æ¿ã©ããã®é»å Žã¯ãäžæ§ãªé»å Žã§ããã
ãã®å¹³è¡æ¿ã³ã³ãã³ãµãŒã®éé»å®¹éCã®åŒã¯ãåŸè¿°ããçç±ã«ããã
:<math>C=\epsilon_0 \frac{S}{d}</math>
ã§äžãããããããã§ãSã¯å°äœå¹³é¢ã®é¢ç©ã§ãããdã¯å°äœéã®è·é¢ã§ããã
å®éšçã«ãããã®éé»å®¹éã®å
¬åŒã¯ãæ£ããããšã確ãããããŠããã
* å¹³è¡æ¿ã³ã³ãã³ãµãŒã®éé»å®¹éã®å
¬åŒã®å°åº
ããã§äžããéé»å®¹éã¯ã'''å¹³é¢äžã«é»è·ãäžæ§ã«ååžãã'''ãšã®ä»®å®ã§å°ãããããã®ãšããå°äœéã«çããé»çEã¯ãå°äœãæã€é»è·ãQ, -Qãšããæã
ãŸãã極æ¿ã®é»è·å¯åºŠãã極æ¿ã®ã©ãã§ãäžå®ã ãšä»®å®ããŠïŒãã®ããã«ã¯ãã³ã³ãã³ãµãŒã®åºãïŒã€ãŸãé¢ç©ïŒãããã
ãã¶ãã«åºããšä»®å®ããå¿
èŠãããããšãããããã®ãããªä»®å®ã«ãããé»è·å¯åºŠã¯ïŒã
:é»è·å¯åºŠïŒ<math>Q/S</math>C/m<sup>2</sup>
ã§ããã
é»æ°åç·ã®æ§è³ªãšããŠããã©ã¹ã®é»è·ããçããŠãã€ãã¹ã®é»è·ã§åžåãããã®ã§ããã£ãŠå¹³è¡æ¿ã³ã³ãã³ãµãŒéã®é»æ°åç·ã®ååžã¯ãå³ã®ããã«ãé»æ°åç·ãããã©ã¹æ¥µæ¿ããåçŽã«ããã€ãã¹æ¥µæ¿ãžåãã£ãŠé»æ°åç·ãåºãŠããããŠãã€ãã¹æ¥µæ¿ã«é»æ°åç·ãåžåãããã
é»å Žã¯ãå°äœéã®åç¹ã§ã
:<math>E = \frac{Q/S}{\epsilon _0} =\frac{Q}{\epsilon _0 S}</math>
ã§äžãããããé»å Žãæ±ããããã®ã§ãããããé»äœãèšç®ã§ãããå°äœéã®åç¹ã§é»å Žã®å€§ãããåäžãªã®ã§ãé»äœã®å€§ããã¯é»å Žã®å€§ããã«2ç¹éã®è·é¢ãããããã®ã«ãªããããã§ãé»äœVã¯ã
:<math>V=Ed=\frac{d}{\epsilon_0S}Q</math>
ãšãªããããã®åŒãšéé»å®¹éCã®å®çŸ©ãèŠæ¯ã¹ããšã
:<math>C=\epsilon_0\frac{S}{d}</math>
ãåŸãããã
== é»æ± ã®ä»çµã¿ ==
é»æ± ã®ååŠåå¿ã«ã€ããŠã¯ãå¥ç§ç®ã®ååŠIãªã©ã§è©³ããæ±ãããããã®ç« ã§ã¯ãé»å§ãé»æµã®ç解ã«é¢ããç¹ãéç¹çã«èª¬æãããã
=== ã€ãªã³ååŸå ===
éå±å
çŽ ã®åäœãæ°ŽãŸãã¯æ°Žæº¶æ¶²ã«å
¥ãããšãã®ãéœã€ãªã³ã®ãªããããã'''ã€ãªã³ååŸå'''ïŒionization tendencyïŒãšããã
äŸãšããŠãäºéZnãåžå¡©é
žHClã®æ°Žæº¶æ¶²ã«å
¥ãããšãäºéZnã¯æº¶ãããŸãäºéã¯é»åã倱ã£ãŠZn<sup>2+</sup>ã«ãªãã
:Zn + 2H<sup>+</sup> â Zn<sup>2+</sup> + H<sub>2</sub>
äžæ¹ãéAgãåžå¡©é
žã«å
¥ããŠãåå¿ã¯èµ·ãããªãã
ãã®ããã«éå±ã®ã€ãªã³ååŸåã®å€§ããã¯ãç©è³ªããšã«å€§ãããç°ãªãã
=== é»æ± ===
äºçš®é¡ã®éå±åäœãé»è§£è³ªæ°Žæº¶æ¶²ã«å
¥ãããšé»æ± ãã§ãããããã¯[[ã€ãªã³ååŸå]]ïŒåäœã®éå±ã®ååãæ°ŽãŸãã¯æ°Žæº¶æ¶²äžã§é»åãæŸåºããŠéœã€ãªã³ã«ãªãæ§è³ªïŒã倧ããéå±ãé»åãæŸåºããŠéœã€ãªã³ãšãªã£ãŠæº¶ããã€ãªã³ååŸåã®å°ããéå±ãæåºããããã§ããã
ã€ãªã³ååŸåã®å€§ããæ¹ã®éå±ã'''è² æ¥µ'''ïŒãµãããïŒãšãããã€ãªã³ååŸåã®å°ããæ¹ã®éå±ã'''æ£æ¥µ'''ïŒãããããïŒãšããã
ã€ãªã³ååŸåã®å€§ããéå±ã®ã»ãããéœã€ãªã³ã«ãªã£ãŠæº¶ãåºãçµæãéå±æ¿ã«ã¯é»åãå€ãèç©ããã®ã§ãäž¡æ¹ã®éå±æ¿ãé
ç·ã§ã€ãªãã°ãã€ãªã³ååŸåã®å€§ããæ¹ããå°ããæ¹ã«é»åã¯æµããããé»æµãã§ã¯ç¡ãããé»åããšãã£ãŠãããšã«æ³šæãé»åã¯è² é»è·ã§ããã®ã§ãé»æµã®æµããšé»åã®æµãã¯ãéåãã«ãªãã
=== ã€ãªã³åå ===
ããŸããŸãªæº¶æ¶²ãéå±ã®çµã¿åããã§ãã€ãªã³ååŸåã®æ¯èŒã®å®éšãè¡ã£ãçµæãã€ãªã³ååŸåã®å€§ããã決å®ãããã
å·Šããé ã«ãã€ãªã³ååŸåã®å€§ããéå±ã䞊ã¹ããšã以äžã®ããã«ãªãã
: K > Ca > Na > Mg > Al > Zn > Fe > Ni > Sn > Pb > (H<sub>2</sub>) > Cu > Hg > Ag > Pt > Au
éå±ããã€ãªã³ååŸåã®å€§ããã®é ã«äžŠã¹ããã®ãéå±ã®'''ã€ãªã³åå'''ãšããã
æ°ŽçŽ ã¯éå±ã§ã¯ç¡ããæ¯èŒã®ãããã€ãªã³ååŸåã«å ããããã
éå±ååã¯ãäžèšã®ä»ã«ãããããé«æ ¡ååŠã§ã¯äžèšã®éå±ã®ã¿ã®ã€ãªã³ååãçšããããšãå€ãã
ã€ãªã³ååã®èšæ¶ã®ããã®èªååãããšããŠã
ã貞ããããªããŸããããŠã«ããªãã²ã©ãããåéãã
ãªã©ã®ãããªèªååããããããã¡ãªã¿ã«ãã®èªååããã®å Žåã
ãKã ãã ãCa ãªNaããŸMg ãAlããZn ãŠFe ã«Ni ã ãªPbãã²H2 ã©Cu ãHg ãAg ã åéPt,Auãã
ãšå¯Ÿå¿ããŠããã
=== ãã«ã¿é»æ± ===
:åžç¡«é
žH<sub>2</sub>SO<sub>4</sub>ã®äžã«äºéæ¿Znãšé
æ¿Cuãå
¥ãããã®ã
è² æ¥µïŒäºéæ¿ïŒã§ã®åå¿
:Zn â Zn<sup>2+</sup> + 2e<sup>-</sup>
æ£æ¥µïŒé
æ¿ïŒã§ã®åå¿
:2H<sup> + </sup> + 2e<sup>-</sup> â H<sub>2</sub>â
==== èµ·é»å ====
ãã«ã¿ã®é»æ± ã§ã¯ãåŸããã䞡極éã®é»äœå·®ïŒãé»å§ããšããããïŒã¯ã1.1ãã«ãã§ããã(ãã«ãã®åäœã¯Vãªã®ã§ã1.1Vãšãæžãã)ãã®äž¡æ¥µæ¿ã®é»äœå·®ã'''èµ·é»å'''ãšãããèµ·é»åã¯ãäž¡é»æ¥µã®éå±ã®çµã¿åããã«ãã£ãŠæ±ºãŸãç©è³ªåºæã§ããã
èµ·é»åã®åäœã®ãã«ãã¯ãéé»æ°åã®é»äœã®åäœã®ãã«ããš'''åã'''åäœã§ãããé»æ°åè·¯ã®é»å§ã®ãã«ããšããèµ·é»åã®åäœã®ãã«ãã¯åãåäœã§ããã
=== é»æ± å³ ===
ãã«ã¿é»æ± ã®æ§é ã以äžã®ãããªæååã«è¡šããå Žåããã®ãããªè¡šç€ºã'''é»æ± å³'''ãããã¯'''é»æ± åŒ'''ãšããã
:(-) Zn | H<sub>2</sub>SO<sub>4</sub>aq |Cu (+)
aqã¯æ°Žã®ããšã§ãããH<sub>2</sub>SO<sub>4</sub>aqãšæžããŠãç¡«é
žæ°Žæº¶æ¶²ãè¡šããŠããã
;é»æ°åè·¯ãšã®é¢é£äºé
ç©çåŠã®é»æ°åè·¯ã®ç 究ã§ã¯ããã®ãããªé»æ± ãªã©ã®çŸè±¡ã®çºèŠãšçºæã«ãã£ãŠãå®å®ãªçŽæµé»æºãå®éšçã«åŸãããããã«ãªããçŽæµé»æ°åè·¯ã®æ£ç¢ºãªå®éšãå¯èœã«ãªã£ããé»æ± ã®çºæ以åã«ãããã©ã³ã¹äººã®ç©çåŠè
ã¯ãŒãã³ãªã©ã«ããéé»æ°ã«ããé»æ°ååŠã®ç 究ãªã©ã«ãã£ãŠãé»äœå·®ã®æŠå¿µãé»è·ã®æŠå¿µã¯ãã£ããã ãããã®æ代ã®é»æºã¯ãäž»ã«éé»æ°ã«ãããã®ã ã£ãã®ã§ãå®å®é»æºã§ã¯ç¡ãã£ãã
ãããŠãé»æ± ã«ããå®å®ãªé»æºã®çºæã¯ãåæã«å®å®ãªé»æµã®çºæã§ããã£ãããã®ãããªé»æ± ã®çºæãªã©ã«ãããçŽæµé»æ°åè·¯ã®ç 究ãªã©ããããã€ã人ã®ç©çåŠè
ãªãŒã ããããŸããŸãªå°äœã«é»æµãæµãå®éšãšçè«ç 究ãè¡ãããšã«ãããé»æ°åè·¯ã®çè«ã®'''ãªãŒã ã®æ³å'''ïŒãªãŒã ã®ã»ããããOhm's lawïŒãçºèŠãããã
[[File:Thermocouples diagram.svg|thumb|ç±é»å¯Ÿã®åçãããçµã¿åããã®éå±AãšBã§ãå³ã®ããã«2ã€ã®æ¥ç¹ã«ç°ãªã枩床ãäžãããšãé»æµãæµããã]]
ãã€ã¯ãªãŒã ã¯é»æ± ã§ã¯ãªãç±é»å¯ŸïŒãã€ã§ãã€ãïŒãšãããã®ã䜿ã£ãŠãé»æ°åè·¯ã«å®å®ããé»æµããªããç 究ããããåœæã®é»æ± ã§ã¯ãèµ·é»åããã ãã«æžã£ãŠããŸãããªãŒã ã¯åœåã¯é»æ± ã§å®éšããããããŸãå®å®é»æµãåŸãããªãã£ãã
ç±é»å¯Ÿãšã¯ããŸãç°ãªãéå±ææã®2æ¬ã®éå±ç·ãæ¥ç¶ããŠïŒã€ã®åè·¯ãã€ããã2ã€ã®æ¥ç¹ã«æž©åºŠå·®ãäžãããšãåè·¯ã«é»å§ãçºçããããé»æµãæµããïŒãã®çŸè±¡ãããŒãŒããã¯å¹æãšããïŒããã®çŸè±¡ãããã¯ã1821幎ã«ãŒãŒããã¯ãçºèŠããããã®ãããªåè·¯ããç±é»å¯Ÿã§ããããªããåã2æ¬ã®éå±ç·ã§ã¯ã枩床差ãäžããŠãé»å§ã¯çºçãããé»æµã¯æµããªãã
ãªãŒã ã¯ããã«ãªã³å€§åŠææããã±ã³ãã«ãã®å©èšã«ãã£ãŠããã®ç±é»å¯Ÿãå®éšã«å©çšããã枩床ãå®å®ãããã®ã¯ãåœæã®æè¡ã§ãæ¯èŒçç°¡åã§ãã£ãã®ã§ãããããŠãªãŒã ã¯å®å®é»æµããã¡ããå®éšãã§ããã®ã§ããã
:â» ç±é»å¯Ÿã«ã€ããŠã¯ãé«æ ¡ã®ç¯å²ãè¶
ãããã倧åŠå
¥è©Šã«ãåºé¡ãããªãã ãããã倧åŠã®ææ¥ã§ãããŸãæ·±å
¥ãããªãã®ã§ãåãããªããã°ãæ°ã«ããªããŠããã
:â» å®ã¯åæ通ã®ãç§åŠãšäººéç掻ãã§ç±é»å¯ŸïŒåæ通ã®æç§æžã§ã¯ãç±é»çŽ åããšèšè¿°ïŒã«ã€ããŠãç±ã®ç©çã®åå
ã§èª¬æããŠããããã ãããããã«ãªãŒã ã®æ³åã®å®éšãšã®é¢é£ãŸã§ã¯èª¬æããŠãªããã»ã»ã»ã
;ãªãŒã ã®æ³åãšã®é¢ä¿
ãªãŒã ã®æ³åïŒOhm's lawïŒãšã¯ã
ãã»ãšãã©ã®å°äœã§ã¯ãé»æµ I ãæµããŠããå°äœäžã®2ç¹ã®ç¹ <math>P_1</math>ãšç¹ <math>P_2</math> éã®é»äœå·® <math>E = E_1 - E_2</math> ã¯ãé»æµ I ã«æ¯äŸãããã
ãšããå®éšæ³åã§ããã
誀解ããããããããªãŒã ã®æ³åã¯ããã®ãããªå®éšæ³åã§ãã£ãŠãã¹ã€ã«æµæã®å®çŸ©åŒã§ã¯ç¡ããåæ§ã«ããªãŒã ã®æ³åã¯ãã¹ã€ã«é»å§ã®å®çŸ©åŒã§ã¯ç¡ãããé»æµã®å®çŸ©åŒã§ãç¡ããäžåŠæ ¡ã®çç§ã§ã®é»æ°åè·¯ã®æè²ã§ã¯ãéå±ã®é»æ°å解ã®èµ·é»åã®æè²ãŸã§ã¯ããªãã®ã§ããšãããã°ãé»å§ã誀解ããŠããé»å§ã¯ãåãªãé»æµã®æ¯äŸéã§ãæµæã¯ãã®æ¯äŸä¿æ°ãã®ãããªèª€è§£ããå Žåãæããããããã®è§£éã¯æããã«èª€è§£ã§ããã
ãŸããåå°äœãªã©ã®äžéšã®ææã§ã¯ãé»æµãå¢ãææã®æž©åºŠãäžæãããšæµæãäžããçŸè±¡ãç¥ãããŠããã®ã§ãåå°äœã§ã¯ãªãŒã ã®æ³åãæãç«ããªãå Žåãããããªã®ã§ããªãŒã ã®æ³åãå®çŸ©åŒãšèããã®ã¯äžåçã§ããã
== é»æµãšé»æ°åè·¯ ==
[[Image:Wheatstonebridge.svg|right|thumb|300px|alt=A Wheatstone bridge has four resistors forming the sides of a diamond shape. A battery is connected across one pair of opposite corners, and a galvanometer across the other pair. |é»æ°åè·¯ã®äŸãèªè
ãããã®å³ã®æå³ãåããããã«ãªãã®ããæ¬ç¯ã®ç®æšã®äžã€ã§ãããã¡ãªã¿ã«ããã€ããã¹ãã³ã»ããªããžãïŒWheatstone bridgeïŒãšããåè·¯ã§ããã<br>R1ãR2ãR3ãRxã¯æµæãV<sub>G</sub>ãäžžã§å²ã£ãŠããèšå·ã¯é»å§èšã<br>AãBãCãDã¯åãªãåè·¯ã®åæµããŠããæ¥ç¹ã]]
å°ç·ãªã©ã®å°äœå
ã®é»æ°ã®æµãã'''é»æµ'''ïŒã§ããã
ããelectric currentïŒãšãããé»æµã®åŒ·ãã¯'''ã¢ã³ãã¢'''ãšããåäœã§è¡šãã1ã¢ã³ãã¢ã®å®çŸ©ã¯æ¬¡ã®éãã§ããã
1ç§éã«1ã¯ãŒãã³ïŒèšå·CïŒã®é»æµãééããããšã1'''ã¢ã³ãã¢'''ãšããã
ã¢ã³ãã¢ã®èšå·ã¯Aã§ããããŸããé»æµã¯ãåäœæéãããã®é»è·ã®éééã§ãããã®ã§ãé»æµã®åäœã[C/s]ãšæžãå Žåãããã
äžè¬çã«ã¯ãé»æµã®åäœã¯ããªãã¹ã[A]ã§è¡šèšããããšãå€ãã
é»æµI[A]ãšæét[S]ã§å°ç·æé¢ãééããé»è·Q[C]ã®é¢ä¿ãåŒã§è¡šããšã
:<math>I=\frac{Q}{t}</math>
ã§ããã
é»æµã®åãã®åãæ¹ã«ã€ããŠã¯ãèªç±é»åã¯è² é»è·ãæã£ãŠãããããèªç±é»åã®åããšã¯å察åãã«é»æµã®åãããšãããšã«æ³šæããã
次ã«é»æµãšèªç±é»åã®é床ãšã®é¢ä¿ãèããã
èªç±é»åã®é»è·ã®çµ¶å¯Ÿå€ãeãšãããšãèªç±é»åã¯è² é»è·ã§ãããããèªç±é»åã®é»è·ã¯ãã€ãã¹ç¬Šå·ãã€ã-eã§ããã
=== ãªãŒã ã®æ³å ===
ãã€ã人ã®ç©çåŠè
ãªãŒã ã¯æ¬¡ã®ãããªæ³åãçºèŠããã
ãã»ãšãã©ã®å°äœã§ã¯ãé»æµ I ãæµããŠããå°äœäžã®2ç¹ã®ç¹ <math>P_1</math>ãšç¹ <math>P_2</math> éã®é»äœå·® <math>E = E_1 - E_2</math> ã¯ãé»æµ I ã«æ¯äŸãããã
ãã®å®éšæ³åã'''ãªãŒã ã®æ³å'''ïŒOhm's lawïŒãšããã
åŒã§è¡šããšãé»äœå·®ãVãšããŠãé»æµãIãšããå Žåã«ãæ¯äŸä¿æ°ãRãšããŠã
:V=RI
ã§ããã
ããã§ãé»äœãšé»æµã®æ¯äŸä¿æ°Rã'''é»æ°æµæ'''ãããã¯åã«'''æµæ'''ïŒresistanceã'''ã¬ãžã¹ã¿ã³ã¹'''ïŒãšããã
é»æ°æµæã®åäœã¯ãªãŒã ãšèšããèšå·ã¯Ωã§è¡šãã
æ
£ç¿çã«ãæµæã®èšå·ã¯Rã§ããããå Žåãå€ãã
=== é»æ°åè·¯ ===
[[File:Ohm's Law with Voltage source TeX.svg|right|thumb|é»æ°åè·¯å³ã®äŸãé»æºã¯äº€æµé»æºãvãé»å§ãRãæµæãiã¯é»æµã]]
é»æ°åè·¯ãžãšãã«ã®ãŒãäŸçµŠããé»æºãšããŠå®é»å§ã®çŽæµé»æºãèãããåè·¯ã®2å°ç¹éã«ããäžå®ã®é»å§ãäŸçµŠãç¶ãããã®ã§ãããé»å§æºã®åè·¯å³èšå·ãšããŠã¯[[File:Cell.svg|30px|é»å§æº]]ãçšãããããèšå·ã®é·ãåŽãæ£æ¥µã§ããããã©ã¹ã®é»äœã§ãããèšå·ã®çãåŽã¯è² 極ã§ããã
也é»æ± ã¯ãçŽæµé»æºãšããŠåãæ±ã£ãŠè¯ãã
ãªãããããã¯çŽæµé»æºã§ããã亀æµã®å Žåã¯äžè¬åããé»å§æºãšããŠ[[File:Voltage Source.svg|30px|亀æµé»å§æº]]ã®èšå·ãçšããããŸãç¹ã«æ£åŒŠæ³¢äº€æµé»å§æºã§ããã°[[File:Voltage Source (AC).svg|30px|æ£åŒŠæ³¢äº€æµé»å§æº]]ã®èšå·ãçšããã
==== æµæåš ====
[[File:3 Resistors.jpg|thumb|æµæ]]
'''æµæåš'''(resistor)ã¯ãéåžžã¯åã«'''æµæ'''ãšåŒã°ããåè·¯çŽ åã§ãããäžããããé»æ°ãšãã«ã®ãŒãåçŽã«æ¶è²»ããçŽ åã§ãããåè·¯å³èšå·ã¯[[File:Resistor symbol America.svg|60px|æµæ]]ãããã¯[[File:Resistor symbol IEC.svg|60px|è² è·]]ã§ããããæ¬æžã§ã¯ãäž¡è
ãšãæµæã®åè·¯å³èšå·ãšããŠçšããããšã«ãããïŒç»åçŽ æã®ç¢ºä¿ã®éœåã®ãããäž¡æ¹ã®èšå·ãæ¬æžã§ã¯æ··åšããŸããã容赊ãã ãããïŒ
{{clear}}
===== æµæåšã®å³èšå· =====
æ¥æ¬ã§ã¯ãæµæåšã®å³èšå·ã¯ãåŸæ¥ã¯JIS C 0301ïŒ1952幎4æå¶å®ïŒã«åºã¥ããã®ã¶ã®ã¶ã®ç·ç¶ã®å³èšå·ã§å³ç€ºãããŠããããçŸåšã®ãåœéèŠæ Œã®IEC 60617ãå
ã«äœæãããJIS C 0617ïŒ1997-1999幎å¶å®ïŒã§ã¯ã®ã¶ã®ã¶åã®å³èšå·ã¯ç€ºãããªããªããé·æ¹åœ¢ã®ç®±ç¶ã®å³èšå·ã§å³ç€ºããããšã«ãªã£ãŠãããæ§èŠæ Œã§ããJIS C 0301ã¯ãæ°èŠæ ŒJIS C 0617ã®å¶å®ã«äŒŽã£ãŠå»æ¢ããããããæ§èšå·ã§æµæåšãå³ç€ºããå³é¢ã¯ãçŸåšã§ã¯JISéæºæ ãªå³é¢ã«ãªã£ãŠããŸããããããææåã¯ç¡ããããçŸåšãåŸæ¥ã®å³èšå·ãå€çšãããŠããã
<gallery>
ãã¡ã€ã«:Resistor_symbol_America.svg|åŸæ¥èŠæ Œã®å³èšå·
ãã¡ã€ã«:Resistor_symbol_IEC.svg|æ°èŠæ Œã®å³èšå·
</gallery>
==== é»æ°åè·¯å³èšå·ã®äŸ ====
<gallery>
ãã¡ã€ã«:åºå®æµæåš.svg|åºå®æµæåš
File:Variable resistor as rheostat symbol GOST.svg|å¯å€æµæåš
ãã¡ã€ã«:é»æ± .svg|é»æ± ãçŽæµé»æºïŒé·ãæ¹ãæ£æ¥µïŒ
File:Voltage Source (AC).svg|亀æµé»æº
ãã¡ã€ã«:SPST-Switch.svg|ã¹ã€ãã
ãã¡ã€ã«:ã³ã³ãã³ãµ.svg|ã³ã³ãã³ãµ
File:Inductor h wikisch.svg|ã³ã€ã«
File:Symbole amperemetre.png|é»æµèš
File:Symbole voltmetre.png|é»å§èš
File:Earth Ground.svg|æ¥å°
ãã¡ã€ã«:Fuse.svg|ãã¥ãŒãº
</gallery>
==== çŽåãšäžŠå ====
è€æ°ã®åè·¯çŽ åã1ã€ã®ç·äžã«é
眮ãããŠãããããªæ¥ç¶ã'''çŽåæ¥ç¶'''ãšãããè€æ°ã®åè·¯çŽ åãäºè¡ã«åãããããã«é
眮ãããŠããæ¥ç¶ã'''䞊åæ¥ç¶'''ãšããã
çŽåæ¥ç¶ã«ãããŠã¯ãããããã®åè·¯çŽ åã«æµããé»æµã¯å
šãŠçãããäžæ¹ã䞊åæ¥ç¶ã«ãããŠã¯ããããã®åè·¯çŽ åã®äž¡ç«¯ã«ãããé»å§ãå
šãŠçããã
ãŸããçŽåæ¥ç¶ã«ãããŠã¯ããããã®åè·¯çŽ åã«ãããé»å§ã®åãå
šé»å§ãšãªãã䞊åæ¥ç¶ã«ãããŠã¯ããããã®åè·¯çŽ åãæµããé»æµã®åãå
šé»æµãšãªãã
==== çŽåã§ã®åææµæ ====
æµæãè€æ°æ¥ç¶ãããŠããå Žåããã®è€æ°ã®æµæããŸãšããŠãããã1ã€ã®æµæãæ¥ç¶ãããŠãããã®ãããªç䟡çãªåè·¯ãèããããšãã§ãããè€æ°ã®æµæãšç䟡ãª1ã€ã®æµæã'''åææµæ'''ãšããã
[[ãã¡ã€ã«:Resistors in series.svg|thumb|çŽåæµæ]]
æµæã''n''åçŽåã«æ¥ç¶ãããŠããå Žåãèãããæµæ<math>R_1, R_2, \cdots, R_n</math>ãçŽåã«æ¥ç¶ãããŠããå Žåãåæµæãæµããé»æµã¯çãããããã''i''ãšãããåæµæ<math>R_k (k = 1, 2, \cdots, n)</math>ã«ãããé»å§ã<math>v_k</math>ãšãããšããªãŒã ã®æ³åãã
:<math>v_k = R_ki (k = 1, 2, \cdots, n)</math>
ãæãç«ã€ããã®ãšãçŽåæµæã®äž¡ç«¯ã®é»å§''v''ã¯ã
:<math>v = \sum_{k=1}^n v_k = \sum_{k=1}^n R_k i = i\sum_{k=1}^n R_k</math>
ã§ããããããšç䟡ãªæµæ''R''ã1ã€ã ãæ¥ç¶ãããŠãããããªç䟡åè·¯ãèãããšãã
:<math>v = Ri</math>
ãæãç«ã€ããããããã£ãŠãããã®''n''åã®çŽåæµæã®åææµæ''R''ãšããŠ
:<math>R = \sum_{k=1}^n R_k</math>
ãåŸããããªãã¡ãçŽååææµæã¯åæµæã®ç·åãšãªãã
==== 䞊åã§ã®åææµæ ====
[[ãã¡ã€ã«:Resistors in parallel.svg|thumb|䞊åæµæ]]
åæ§ã«ãæµæã''n''å䞊åã«æ¥ç¶ãããŠããå Žåãèãããæµæ<math>R_1, R_2, \cdots, R_n</math>ã䞊åã«æ¥ç¶ãããŠããå Žåãåæµæã®äž¡ç«¯ã®é»å§ã¯çãããããã''v''ãšãããåæµæ<math>R_k (k = 1, 2, \cdots, n)</math>ãæµããé»æµã<math>i_k</math>ãšãããšããªãŒã ã®æ³åãã
:<math>v = R_ki_k (k = 1, 2, \cdots, n)</math>
ãæãç«ã€ããã®ãšã䞊åæµæãžæµã蟌ãé»æµ''i''ã¯ã
:<math>i = \sum_{k=1}^n i_k = \sum_{k=1}^n \frac{v}{R_k} = v\sum_{k=1}^n \frac{1}{R_k}</math>
ã§ããããããšç䟡ãªæµæ''R''ã1ã€ã ãæ¥ç¶ãããŠãããããªç䟡åè·¯ãèãããšãã
:<math>v = Ri</math>
ãæãç«ã€ããããããã£ãŠãããã®''n''åã®äžŠåæµæã®åææµæ''R''ãšããŠ
:<math>\frac{1}{R} = \sum_{k=1}^n \frac{1}{R_k}</math>
ãåŸããããªãã¡ã䞊ååææµæã®éæ°ã¯åæµæã®éæ°ã®ç·åãšãªãã
==== é»å ====
æµæRãé»æµIãæµãããšãããã®éšåã®çºç±ã®ãšãã«ã®ãŒã¯ã1ç§ãããã«RI<sup>2</sup>[J/s]ã§ãããããã'''ãžã¥ãŒã«ç±'''ãšãããååã®ç±æ¥ã¯ç©çåŠè
ã®ãžã¥ãŒã«ã調ã¹ãããã§ããããªãŒã ã®æ³åãããV=RIã§ãããã®ã§ããžã¥ãŒã«ç±ã¯VIãšãæžããã
ããã§ãã²ãšãŸããç±ã®èå¯ã«ã¯é¢ããŠã次ã®éãå®çŸ©ãããé»æ°åè·¯ã®ãã2ç¹éãæµããé»æµIãšããã®ïŒç¹éã®é»å§Vãšã®ç©VIã'''é»å'''ïŒpowerïŒãšå®çŸ©ãããé»åã®èšå·ã¯Pã§è¡šããããããšãå€ãã
é»åã®åäœã®ãžã¥ãŒã«æ¯ç§[J/s]ã[W]ãšããåäœã§è¡šãããã®åäœWã¯ã¯ããïŒWattïŒãšèªãã
ã€ãŸãé»åã¯èšå·ã§
:P[W]=VI
ã§ããã
==== æµæç ====
å°ç·ã®å€ªããé·ãã«ãã£ãŠæµæã®å€§ããã¯å€ãããçŽæçã«å€ªãã»ããæµããããã®ã¯åããã ãããããã«äžŠåæ¥ç¶ãšå¯Ÿå¿ãããŠããå°ç·ã倪ãã»ããæµããããã®ã¯åããã ããã
å®éã«é»æ°æµæã¯ãå°ç·ã倪ãã«åæ¯äŸããŠå°ãããªãããšãå®éšçã«ç¢ºèªãããŠãããããã§ãã€ãã®ãããªåŒã«ãããŠã¿ããã
æµæãR[Ω]ãšããå Žåãå°ç·ã®å€ªããé¢ç©ã§è¡šãA[m<sup>2</sup>]ãšããã°ãæ¯äŸå®æ°ã«kãçšããã°ã
:R â 1/A
ã§ãããïŒ âã¯ãæ¯äŸé¢ä¿ãè¡šãæ°åŠèšå·ãïŒ
ããã«ãå°ç·ã¯æ質ã倪ããåããªãã°ãå°ç·ãé·ãã»ã©æµæããé·ãã«æ¯äŸããŠæµæã倧ãããªãããšãã確èªãããŠãããããã§ãããã«ãæµæäœã®é·ããèæ
®ããåŒã«è¡šããŠã¿ãã°ã次ã®ããã«ãªããæµæ垯ã®é·ãã''l''[m]ãšããã°
:R â L/A
ã§ããã
ããã«ãå°ç·ã®æ質ã«ãã£ãŠãæµæã®å€§ããã¯å€ãããåãé·ãã§åã倪ãã®æµæã§ããæ質ã«ãã£ãŠæµæã®å€§ããã¯ç°ãªããããã§ãæ質ããšã®æ¯äŸå®æ°ãρãšããã°ãæµæã®åŒã¯ä»¥äžã®åŒã§èšè¿°ãããã
:<math>R=\rho \frac{l}{A}</math>
ρã¯'''æµæç'''ïŒãŠããããã€ãresistivityïŒãšåŒã°ãããæµæçã®åäœã¯[Ωm]ã§ããã
== ç£å ==
=== ç£å Ž ===
[[File:Magnetic field near pole.svg|thumb|right|200px|æ£ç£ç³ã®åšãã«æ¹äœç£éã眮ããŠç£å Žã®åãã調ã¹ãã]]
ç£ç³ã®ãŸããã«ã¯å¥ã®ç£ç³ãåããåã®ããšãšãªããã®ãçããŠããã
ããã'''ç£å Ž'''ïŒãã°ãmagnetic fieldïŒãããã¯'''ç£ç'''ïŒãããïŒãšåŒã¶ãïŒæ¥æ¬ã®ç©çåŠã§ã¯ç£å ŽãšåŒã¶ããšãå€ãããŸããæ¥æ¬ã®é»æ°å·¥åŠã§ã¯ç£çãšåŒã°ããããšãå€ããææ²»æã®èš³èªã®éã®ãæ¥æ¬åœå
ã®æ¥çããšã®éãã«éãããå°å瀟äŒçãªäºè±¡ã§ãããåŒã³æ¹ã¯ç©çã®æ¬è³ªãšã¯é¢ä¿ãªãã®ã§ãããã§ã¯ãã©ã¡ãã®è¡šçŸãçšãããã¯ãæ¬æžã§ã¯ç¹ã«ãã ãããªããè±èªã§ã¯ç©çåŠã»é»æ°å·¥åŠãšãâmagnetic fieldâã§å
±éããŠãããïŒ
éãã³ãã«ããããã±ã«ã«ç£ç³ãè¿ã¥ãããšãç£ç³ã«åžãä»ããããã
ãŸããéãã³ãã«ããããã±ã«ã«åŒ·ãç£åãäžãããšãéãã³ãã«ããããã±ã«ãã®ãã®ãç£å Žãåšå²ã«åãŒãããã«ãªãã
ãã®ãããªãããšããšã¯ç£å Žãæããªãã£ãç©äœãã匷ãç£å Žãåããããšã«ãã£ãŠç£å ŽãåãŒãããã«ãªãçŸè±¡ã'''ç£å'''ïŒãããmagnetizationïŒãšããã
ãããã¯é»è·ã®éé»èªå°ãšå¯Ÿå¿ãããŠãç£åã®ããšã'''ç£æ°èªå°'''ïŒããããã©ããmagnetic inductionïŒãšãããã
ãããŠãéãã³ãã«ããããã±ã«ã®ããã«ãç£ç³ã«åŒãä»ããããããã«ç£åãããèœåãããç©äœã'''匷ç£æ§äœ'''ïŒãããããããããferromagnetïŒãšããã
éãšã³ãã«ããšããã±ã«ã¯åŒ·ç£æ§äœã§ããã
é
ã¯ç£åããªãããé
ã¯ç£ç³ã«åŒãã€ããããªãã®ã§ãé
ã¯åŒ·ç£æ§äœã§ã¯ãªãã
;ç£æ°é®èœ
éé»èªå°ãå©çšãããéé»é®èœïŒããã§ããããžãïŒãšèšããããäžç©ºã®å°äœãã€ãã£ãŠç©è³ªãå²ãããšã§å€éšé»å Žãé®èœããæ¹æ³ããã£ãã®ãšåæ§ã®ãç£æ°ã®é®èœãã匷ç£æ§äœã§ãåºæ¥ããäžç©ºã®åŒ·ç£æ§äœãçšããŠã匷ç£æ§äœã®å
éšã¯ç£å Žãé®èœã§ãããããã'''ç£æ°é®èœ'''ïŒãããããžããmagnetic shieldingïŒãšãããç£æ°ã·ãŒã«ããšãããã
==== ç£åç· ====
ç£å Žã®åããåããããã«å³ç€ºããããç£ç³ã®äœãç£å Žã®æ¹åã¯ãç ã«å«ãŸããç éã®ç²æ«ãç£ç³ã«ãã¡ãã°ããŠããµããããããšã§èŠ³å¯ã§ããã
[[File:Magnet0873.png|left|300px|ç éã«ããç£åç·ã®èŠ³å¯]]
{{clear}}
ãããå³ç€ºãããšãäžå³ã®ããã«ãªããïŒç»åçŽ æã®ç¢ºä¿ã®éœåäžãåçãšå³ç€ºãšã§ã¯ãN極ãšS極ãéã«ãªã£ãŠããŸããã容赊ãã ãããïŒ
[[File:VFPt cylindrical magnet.svg|thumb|left|300px|ç£åç·ã®å³ç€º]]
ãã®ãããªç£å Žã®å³ã'''ç£åç·'''ïŒãããããããmagnetic line of forceïŒãšãããç£åç·ã®åãã¯ãç£ç³ã®N極ããç£åç·ãåºãŠãS極ã«ç£åç·ãåžåããããšå®çŸ©ããããæ£ç£ç³ã§ã¯ãç£åã®çºçæºãšãªãå Žæããæ£ç£ç³ã®äž¡ç«¯ã®å
端ä»è¿ã«éäžãããããã§ãæ£ç£ç³ã®äž¡ç«¯ã®å
端ä»è¿ã'''ç£æ¥µ'''ïŒãããããmagnetic poleïŒãšããã
ãã®ãããªç£ç³ã®ã€ããç£åç·ã®åœ¢ã¯ãé»æ°åç·ã§ã®ãç°ç¬Šå·ã®é»è·ã©ãããã€ããé»æ°åç·ã«äŒŒãŠããã
[[File:VFPt dipole electric manylines.svg|thumb|center|200px|ç°ç¬Šå·ã®é»è·ã©ããã®å Žåã®é»æ°åç·]]
1ã€ã®æ£ç£ç³ã§ã¯N極ïŒnorth poleïŒã®ç£æ°ã®åŒ·ããšãS極ïŒsouth poleïŒã®ç£æ°ã®åŒ·ãã¯çããããŸããç£ç³ã«ã¯ãå¿
ãN極ãšS極ãšãååšãããN極ãšS極ã®ãã©ã¡ããçæ¹ã ããåãåºãããšã¯åºæ¥ãªããããšãæ£ç£ç³ãåæããŠããåæé¢ã«ç£æ¥µãåºçŸããããã®ãããªçŸè±¡ã®ãããçç±ã¯ãããããæ£ç£ç³ãæ§æãã匷ç£æ§äœã®ååã®1åãã€ãå°ããªç£ç³ã§ããããããå°ããªååã®ç£ç³ããããã€ãæŽåããŠã倧ããªæ£ç£ç³ã«ãªã£ãŠããããã§ããã
ä»®æ³çã«ãç£æ¥µãS極ãŸãã¯N極ã®çæ¹ã ãçŸããçŸè±¡ãçè«èšç®ã®ããã«èããããšããããããã®ãããªçåŽã ãã®ç£æ¥µã'''åç£æ¥µ'''ïŒ'''ã¢ãããŒã«'''ãšãããïŒãšãããã'''åç£æ¥µã¯å®åšããªã'''ã
{{clear}}
æ£ç£ç³ãªã©ããã®ãçåŽã®ç£æ¥µãããã®ãç£æ¥µããã®ç£å Žã®åŒ·ãã®ããšãããã®ãŸãŸãç£æ¥µã®åŒ·ããïŒMagnetic chargeïŒãšåŒã¶ããããã¯'''ç£è·'''ïŒãããmagnetizationïŒã'''ç£æ°é'''ãšããã
ããããããã®ç£åãšç£å Žã®é¢ä¿ãåŒã§è¡šãããšãèããã
ãŸããæ£ç£ç³ã«ã¯ç£æ¥µãäž¡åŽã«2åããã®ã§ãèšç®ãç°¡åã«ããããã«ãæ£ç£ç³ã®äž¡ç«¯ã®è·é¢ã倧ãããå察åŽã®ç£æ¥µã®å€§ãããç¡èŠã§ããç£ç³ãèãããã
ãã®ãããªç£ç³ãçšããŠãå®éšãããšããã次ã®æ³åãåãã£ããç£åã®åŒ·ãã¯2åã®ç©äœã®ç£æ°ém<sub>1</sub>ããã³m<sub>2</sub>ã«æ¯äŸãã2åã®ç©äœéã®è·é¢rã®2ä¹ã«åæ¯äŸããã
åŒã§è¡šããšã
:<math>f = k_m \frac{q_1 q_2}{r^2}</math>
ã§è¡šããããïŒk<sub>m</sub>ã¯æ¯äŸå®æ°ïŒ
ãããçºèŠè
ã®ã¯ãŒãã³ã®åã«ã¡ãªãã§ã'''ç£æ°ã«é¢ããã¯ãŒãã³ã®æ³å'''ãšãããç£æ°émã®åäœã¯'''ãŠã§ãŒã'''ãšãããèšå·ã¯[Wb]ã§è¡šãã
æ¯äŸå®æ°k<sub>m</sub>ãš1ãŠã§ãŒãã®å€§ãããšã®é¢ä¿ã¯ã1ã¡ãŒãã«é¢ãã1wbã©ããã®ç£æ¥µã«ã¯ãããåãçŽ6.33Ã10<sup>4</sup>ãšããŠã
æ¯äŸä¿æ°k<sub>m</sub>ã¯ã
:k<sub>m</sub>â6.33Ã10<sup>4</sup>ã[Nã»m<sup>2</sup>/Wb<sup>2</sup>]
ã§ããã
ã€ãŸãã
:<math>f = k_m \frac{m_1 m_2}{r^2} = 6.33\times10^4 \frac {m_1 m_2}{r^2}</math>
ã§ããã
==== ç£å Žã®åŒ ====
éé»æ°åã«å¯ŸããŠãé»å Žãå®çŸ©ãããããã«ãç£æ°åã«å¯ŸããŠããå Žãå®çŸ©ããããšéœåãè¯ããç£æ°ém<sub>1</sub>[Wb]ãäœãã次ã®éã'''ç£å Žã®åŒ·ã'''ãããã¯'''ç£å Žã®å€§ãã'''ãšèšããèšå·ã¯Hã§è¡šãã
:<math>H = k_m \frac{m_1}{r^2} = 6.33\times10^4 \frac {m_1}{r^2}</math>
ç£å Žã®åŒ·ãHã®åäœã¯[N/Wb]ã§ãããHãçšãããšãç£æ°ém<sub>2</sub>[Wb]ã«ã¯ãããç£æ°åf[N]ã¯ã
:<math>f = m_2H</math>
ãšè¡šããã
== é»æµãã€ããç£å Ž ==
===ã¢ã³ããŒã«ã®æ³å===
[[Image:Electromagnetism.svg|thumb|right|é»æµã®æ¹åãšç£æå¯åºŠã®æ¹åã®é¢ä¿.<br>ç£æã®åãã¯ãå³ããã®æ³åã®åãã§ããã.]]
ç©çåŠè
ã®ãšã«ã¹ãããã¯ãé»æµã®å®éšãããŠããéã«ãããŸããŸè¿ãã«ãããŠãã£ãæ¹äœç£ç³ãåãã®ã確èªããã圌ã詳ãã調ã¹ãçµæã以äžã®ããšãåãã£ãã
é»æµãæµããŠãããšãã«ã¯ããã®ãŸããã«ã¯ãç£å Žãçãããåãã¯ãé»æµã®æ¹åã«å³ãããé²ãããã«ãå³ãããåãåããšåããªã®ã§ãããã'''å³ããã®æ³å'''ãšããã
ã¢ã³ããŒã«ããç£å Žã®å€§ããã調ã¹ãçµæãç£å Žã®å€§ããHã¯ãé»æµI[A]ãçŽç·çã«æµããŠãããšããçŽç·é»æµã®åšãã®ç£å Žã®å€§ããã¯ãå°ç·ããã®è·é¢ãa[m]ãšãããšãç£å Žã®å€§ããH[N/Wb]ã¯ã
:<math>H=\frac{1}{2\pi a}I</math>
ã§ããããšãç¥ãããŠããã
ããã'''ã¢ã³ããŒã«ã®æ³å'''(Ampere's law) ãšããã
ç£å Žã®å€§ããHã®åäœã¯ã[N/Wb]ã§ãããããã£ãœãã¢ã³ããŒã«ã®æ³åã®åŒãã¿ãã°ã¢ã³ãã¢æ¯ã¡ãŒãã«[A/m]ã§ãããã
;é»ç£ç³
[[File:Simple electromagnet2.gif|thumb|é»ç£ç³ã®äŸ.]]
[[ç»å:VFPt Solenoid correct.svg|thumb|right|é»ç£ç³ã³ã€ã«ã«ããçºçããç£çïŒæé¢å³ïŒ]]
å°ç·ãã³ã€ã«ç¶ã«å·»ãã°ãã¢ã³ããŒã«ã®æ³åã§å°ç·ã®åšå²ã«çºçããç£å Žãéãªãããããã®ããã«ããç£å Žã匷ããã³ã€ã«ã'''é»ç£ç³'''ïŒã§ããããããelectromagnetïŒãšãããå°ç·ã«é»æµãæµããŠãããšãã«ã®ã¿ãé»ç£ç³ã¯ç£å Žãçºçãããå°ç·ã«é»æµãæµãã®ãæ¢ãããšãé»ç£ç³ã®ç£å Žã¯æ¶ããã
=== ç£æå¯åºŠ ===
ç£å Žã®å€§ããHã«ã次ã®ç¯ã§æ±ãããŒã¬ã³ãåã®çŸè±¡ã®ãããæ¯äŸä¿æ°μïŒåäœã¯ãã¥ãŒãã³æ¯ã¢ã³ãã¢ã§[N/A<sup>2</sup>]ïŒãæããŠãèšå·Bã§è¡šãã
:B=μH
ãšããããšãããããã®éBã'''ç£æå¯åºŠ'''ïŒmagnetic flux densityïŒãšãããç£å Žã®å€§ããHã®åããšç£æå¯åºŠBã®åãã¯'''åãåã'''ã§ããã
ãŸããç£å Žã®å€§ããHãšç£æå¯åºŠBã®æ¯äŸä¿æ°ã'''éç£ç'''ïŒãšãããã€ãmagnetic permeabilityïŒãšããã
ïŒããŒã¬ã³ãåã«é¢ããŠã¯ã詳ããã¯ç©çIIã§æ±ããèªè
ãç©çIãåŠã¶åŠå¹Žãªãã°ãèªè
ã¯ãããŒã¬ã³ãåãšããåãããã®ã ãªã»ã»ã»ããšã§ãæã£ãŠããã°ãããïŒ
== ããŒã¬ã³ãå ==
[[File:Lorentzkraft-graphic-part1.PNG|thumb|ããŒã¬ã³ãåã®åããé»è·ã§èããå Žåã<br>é床vããç£æå¯åºŠBã«å³ãããåããåããããŒã¬ã³ãåFã®åãã]]
[[File:Lorentzkraft-graphic-part2.PNG|thumb|ããŒã¬ã³ãåã®åãã<br>é»æµIããç£æå¯åºŠBã«å³ãããåããåããããŒã¬ã³ãåFã®åãã]]
ãŸããå°ç·ãçšæãããšãããããã®å°ç·ã¯ãéæ¢ããŠãããšããŠãéæ¢ããŠããããåºå®ã¯ããã«ãããå°ç·ã«åãå ããã°ãå°ç·ãåããããã«ããŠããšãããã
ãã®å°ç·ã«é»æµãæµããã ãã§ã¯ãã¹ã€ã«å°ç·ã¯åããªãããããããã®å°ç·ã«ãå€éšã®ç£ç³ã«ããç£å Žãå ãããšãå°ç·ãåãããã®ãããªãç£å Žãšé»æµã®çžäºäœçšã«ãã£ãŠãå°ç·ã«çããåã'''ããŒã¬ã³ãå'''ïŒããŒã¬ã³ãããããè±: Lorentz forceïŒãšããã
ããŒã¬ã³ãåã®åãã¯ãå°ç·ã®é»æµã®åããšç£å Žã®åãã«åçŽã§ãããé»æµIã®åãããç£æå¯åºŠBã®åãã«å³ãããåãåããšåãã§ããã
ãŸããããŒã¬ã³ãåã®å€§ããã¯ãå°ç·ã®é·ã''l''ãšãç£å Žã®å°ç·ãšã®åçŽæ¹åæåã«æ¯äŸããã
ããŒã¬ã³ãåã®å€§ããF[N]ãåŒã§è¡šãã°ãé»æµãšç£å ŽãšãåçŽã ãšããŠãç£å ŽãåããŠããå°ç·ã®åœ¢ç¶ãçŽç·åœ¢ã ãšããŠãé»æµãI[A]ãšããŠãå°ç·ã®é·ãã''l''[m]ãšããŠãå°ç·ã«ããã£ãŠããå€éšç£å Žã®ç£æå¯åºŠãB[N/(Aã»m)]ãšããã°ã
:<math>F=IBl</math>
ã§è¡šããã
ããŒã¬ã³ãåã®å
¬åŒã«ãã¯ãŒãã³ã®æ³åãªã©ã§ã¯èŠããããããªæ¯äŸä¿æ°ïŒä¿æ°Kãªã©ãïŒãå«ãŸããªãã®ã¯ãããããããã®ããŒã¬ã³ãåã®çŸè±¡ãå
ã«ãç£æ°éãŠã§ãŒãWbã®åäœããã³ç£æå¯åºŠBã®åäœãã決å®ãããŠããããã§ããã
ãŸãããç£æå¯åºŠãã®å称ãããç£æãã»ãå¯åºŠããšããã®ã¯ãå®ã¯ç£æå¯åºŠã®åäœã®[N/(Aã»m)]ã¯ãåäœãåŒå€åœ¢ãããš[Wb/m<sup>2</sup>]ã§ãããããšãç±æ¥ã§ããããã®åäœ[Wb/m<sup>2</sup>]ããé»æ°å·¥åŠè
ã®ãã¹ã©ã®åã«ã¡ãªã¿ãåäœ[Wb/m<sup>2</sup>]
ã'''ãã¹ã©'''ãšèšããèšå·Tã§è¡šãã
:[T]=[Wb/m<sup>2</sup>]
ãã®ããŒã¬ã³ãåã®çŸè±¡ããé»æ°æ©åšã®ã¢ãŒã¿ïŒé»åæ©ïŒã®åçã§ããã
;ãã¬ãã³ã°ã®æ³åã¯é»ç£æ°èšç®ã§ã¯çšããªã
ãªããããã¬ãã³ã°ã®æ³åããšããããŒã¬ã³ãåã«é¢ããæ³åãããããããŒã¬ã³ãåã®èšç®ã«ã¯å®çšçã§ã¯ç¡ããããã¬ãã³ã°ã®åãé¢ããç°ãªãæ³åã幟ã€ããã£ãŠçŽããããééãã®åå ã«ãªããããã®ã§ãæ¬æžã§ã¯æããªãã
å®éã«ãå°éçãªç©çèšç®ã§ã¯ããã¬ãã³ã°ã®æ³åã¯ãèšç®ã«ã¯çšããªãã
ãããããã¬ãã³ã°ã®æ³åã«ã¯ããã¬ãã³ã°ã®å³æã®æ³åããšããããšã¯ç°ãªãããã¬ãã³ã°ã®å·Šæã®æ³åãããããã©ã¡ãããã©ã®ç£æ°ã®çŸè±¡ã«çšããæ³åã ã£ãã®ããééãããããã ãããæ¬æžã§ã¯æããªãã
{{clear}}
==é»ç£èªå°==
ïŒé»ç£èªå°ã«é¢ããŠã¯ã詳ããã¯ç©çIIã§æ±ããïŒ
ã¢ã³ããŒã«ã®æ³åã§ã¯ãé»æµã®åšãã«ç£å Žãã§ããã®ã§ãã£ãã
:ã§ã¯éã«ãç£å ŽãçšããŠé»æµãèµ·ãããããªçŸè±¡ã¯ããã ãããïŒ
å®ã¯ãç£ç³ãåãããªã©ããŠãç£å Žã䌎ãç©äœãéåãããšããã®ãŸããã«ã¯é»å Žãçããã
ä»®ã«ãã³ã€ã«ã®è¿ãã§ãããè¡ãªã£ããšãããšãçããé»å Žã«ãã£ãŠã³ã€ã«ã®äžã«ã¯é»æµãæµããã
çããé»å Žã®å€§ããã¯ã
:<math>\vec E = \frac 1 {2\pi a} \frac {\Delta \vec B}{\Delta t}</math>
ãšãªãã(ååŸaã®å圢ã®ã³ã€ã«ã®å Žåã)
Eã®åäœã¯[V/m]ã§ãããBã®åäœã¯[T]ã§ããã
ãã®çŸè±¡ã'''é»ç£èªå°'''ïŒã§ããããã©ããelectromagnetic inductionïŒãšãããé»ç£èªå°ã«ãã£ãŠçºçããé»æµã'''èªå°é»æµ'''ãšããã
ãŸããèªå°é»æµã®åãã¯ãç£ç³ã®åãã«ãããã³ã€ã«ã®äžãéãç£æã®å€åã劚ããåãã«ãé»æµãæµãããïŒèªå°é»æµãã¢ã³ããŒã«ã®æ³åã«åŸããåšå²ã«ç£å ŽãäœããïŒ
ãã®èªå°é»æµããã³ã€ã«ã®äžãéãç£æã®å€åã劚ããåãã«èªå°é»æµãæµããçŸè±¡ã'''ã¬ã³ãã®æ³å'''ïŒLenz's lawïŒãšããã
åãé åã«''N''åå·»ãããã³ã€ã«ã眮ãããå Žåããã¡ã©ããŒã®é»ç£èªå°ã®æ³åã¯ã次ã®ããã«ãªãã
: <math>\mathcal{E} = - N{{d\Phi_B} \over dt}</math>
ããã§ã<math>\mathcal{E}</math>ã¯èµ·é»åïŒãã«ã ãèšå·ã¯VïŒãΦ<sub>B</sub> ã¯ç£æïŒãŠã§ãŒããèšå·ã¯WbïŒãšããã''N''ã¯é»ç·ã®å·»æ°ãšããã
ãã®é»ç£èªå°ã®çŸè±¡ããç«åçºé»ãæ°Žåçºé»ãªã©ã®çºé»æ©ã®åçã§ãããããçã®çºé»ã§ã¯ãæ°žä¹
ç£ç³ãå転ãããããšã§ãçºé»ãããŠãããç«åãæ°Žåãšããã®ã¯ãæ©åšã®å転ãåŸãæ段ã«ãããªãããŸããçºé»æã®çºé»ã«ã¯ãæ°žä¹
ç£ç³ã®å転ãå©çšããŠãããããçºçããé»å§ãé»æµã¯åšæçãªæ³¢åœ¢ã«ãªãã次ã«èª¬æãã亀æµæ³¢åœ¢ã«ãªãã
== 亀æµåè·¯ ==
[[File:Waveforms.svg|thumb|400px|亀æµæ³¢åœ¢ã®äŸã<br>äžããé ã«ã<br>æ£åŒŠæ³¢ã<br>æ¹åœ¢æ³¢ã<br>äžè§æ³¢ã<br>ã®ãããæ³¢ã]]
åè·¯ãžã®å
¥åé»å§ãåšæçã«æéå€åããåè·¯ã®é»å§ããã³é»æµã'''亀æµ'''ïŒalternating currentïŒãšãããããã«å¯Ÿãã也é»æ± ãªã©ã«ãã£ãŠçºçããé»å§ãé»æµã®ããã«ãæéã«ãããäžå®ãªé»å§ãé»æµã¯'''çŽæµ'''ïŒdirect CurrentïŒãšããã
亀æµæ³¢åœ¢ãäœç§ã§1åšããããšããæéã'''åšæ'''(wave period)ãšãããåšæã®èšå·ã¯<math>T</math>ã§è¡šãåäœã¯ç§[s]ã§ããã
:<math>f = \frac{1}{T}</math>
1ç§éã«æ³¢åœ¢ãäœåšããããšããåæ°ã'''åšæ³¢æ°'''ãããã¯'''æ¯åæ°'''(è±èªã¯ããšãã«frequency)ãšããã
é»æ°ã®æ¥çã§ã¯åšæ³¢æ°ãšããçšèªãçšããããšãå€ããç©çã®æ³¢ã®çè«ã§ã¯æ¯åæ°ãšããè¡šçŸãçšããããšãå€ãã
åšæ³¢æ°ã®åäœã¯[1/s]ã§ããããããã'''ãã«ã'''ïŒhertzïŒãšããåäœã§è¡šããåäœèšå·'''Hz'''ãçšããŠåšæ³¢æ°fããf[Hz]ãšãããµãã«è¡šãã
亀æµé»æµã亀æµé»å§ãæ£åŒŠæ³¢ã®å Žåã¯ããããã®ãã©ã¡ãŒã¿ãçšããŠ
:<math>i(t) = I_0\sin(2\pi ft + \theta_i) = I_0\sin\left(\frac{2\pi}{T}t + \theta_i\right)</math>
:<math>v(t) = V_0\sin(2\pi ft + \theta_v) = V_0\sin\left(\frac{2\pi}{T}t + \theta_v\right)</math>
ãšæžãããšãã§ããã
sinãšã¯äžè§é¢æ°ã§ãããç¥ããªããã°æ°åŠIIãªã©ãåèã«ããã
ãã®ãšãã®sinã®ä¿æ°<math>I_0</math>ã<math>V_0</math>ã'''æ¯å¹
'''(ããã·ããamplitude)ãšããããŸãæå»''t''=0ã«ãããé»æµãé»å§ã®å€ã瀺ããæé波圢ã決å®ãã<math>\theta_i</math>ã<math>\theta_v</math>ã'''åæäœçž'''ãšããã
æ®éç§é«æ ¡ã®é«æ ¡ç©çã§ã¯ã亀æµæ³¢åœ¢ã®èšç®ã«ã¯ãæ£åŒŠæ³¢ã®å Žåãäž»ã«æ±ããæ¹åœ¢æ³¢ãäžè§æ³¢ã®èšç®ã¯ãæ®éã¯æ±ãããªãã
ãã ããå·¥æ¥é«æ ¡ã®ææ¥ããå·¥å Žã®å®åã§ã¯æ±ãããšãããã®ã§ãèªè
ã¯æ³¢åœ¢ãåŠãã§ããããšã
çºé»æããäžè¬å®¶åºã«éãããŠããé»å§ã¯äº€æµé»å§ã§ãããæ±æ¥æ¬ã§ã¯50Hzã§ããã西æ¥æ¬ã§ã¯60Hzã§ãããããã¯ææ²»æ代ã®çºé»æ©ã®èŒžå
¥æã«ãæ±æ¥æ¬ã®äºæ¥è
ã¯ãšãŒããããã50Hzçšã®çºé»æ©ã茞å
¥ãã西æ¥æ¬ã®äºæ¥è
ã¯ã¢ã¡ãªã«ãã60Hzã®çºé»æ©ã茞å
¥ããããšã«ããã
çºé»æããäžè¬ã®å®¶åºãªã©ã«éãããé»æµã®åšæ³¢æ°ã'''åçšåšæ³¢æ°'''ãšããã
åçšé»æºã®é»å§æ¯å¹
ã¯çŽ140Vã§ãããããã¯<math>100\times\sqrt{2}</math>[V]ã§ããã
ãããã«ããšã¯1000Hzã®ããšã§ããããããã«ãã¯kHzãšæžãã
;ã³ã€ã«ã®èªå·±èªå°
亀æµé»æµã«å¯ŸããŠã¯ãé»æµãšåãæ¯åæ°ã§ãã¢ã³ããŒã«ã®æ³åã§çºçããç£å Žãæ¯åããã
å°ç·ã§ã€ããããã³ã€ã«ã¯ãçŽæµé»æµã§ã¯ããã ã®å°ç·ãšããŠã¯ãããããããã亀æµé»æµã«å¯ŸããŠã¯ãé»ç£èªå°ã«ããèªå·±ã®çºçãããç£å Žã劚ãããããªé»æµããã³èµ·é»åãçºçãããããã'''èªå·±èªå°'''ïŒself inductionïŒãšããã
èªå·±èªå°ã«ããèµ·é»åã®å€§ããã¯ãé»æµã®æéå€åçã«æ¯äŸãããèªå·±èªå°ã®èµ·é»åãåŒã§æžãã°ãæ¯äŸä¿æ°ãLãšããŠã
:<math>e=-L\frac{\Delta I}{\Delta t}</math>
ã§ããã
ãã®æ¯äŸä¿æ°<math>L</math>ã'''èªå·±ã€ã³ãã¯ã¿ã³ã¹'''ïŒself inductanceïŒãšãããèªå·±ã€ã³ãã¯ã¿ã³ã¹ã®æ¬¡å
ã¯[Vã»S/m]ã ããããã'''ãã³ãªãŒ'''ãšããåäœã§è¡šããåäœã«Hãšããèšå·ãçšããã
;çžäºèªå°
[[ãã¡ã€ã«:Transformer Flux.svg|thumb|çžäºèªå°ãå©çšããå€å§åšïŒtransformerïŒ]]
éå¿ã«äºã€ã®ã³ã€ã«ãå·»ããã³ã€ã«ã®çæ¹ã®é»æµãå€åããããšãã¢ã³ããŒã«ã®æ³åã«ãã£ãŠçããŠããç£æãå€åãããããå察åŽã®ã³ã€ã«ã«ã¯ããã®ç£æå¯åºŠã®å€åãæã¡æ¶ããããªåãã«èµ·é»åãçºçããããã®çŸè±¡ã'''çžäºèªå°'''ïŒmutual inductionïŒãšèšãã
é»å§ãå
¥åãããåŽã®ã³ã€ã«ã'''1次ã³ã€ã«'''ïŒprimaly coilïŒãšèšããèªå°èµ·é»åãçºçãããåŽã®ã³ã€ã«ã'''2次ã³ã€ã«'''ïŒsecondary coilïŒãšããã
çžäºèªå°ã«ããèµ·é»åã®å€§ããã¯ãé»æµã®æéå€åçã«æ¯äŸãããçžäºèªå°ã®èµ·é»åãåŒã§æžãã°ãæ¯äŸä¿æ°ãMãšããŠãïŒçžäºèªå°ã®æ¯äŸä¿æ°ã¯Lã§ã¯ç¡ããïŒåŒã¯ã
:<math>e=-M\frac{\Delta I}{\Delta t}</math>
ã§ããã
ãã®æ¯äŸä¿æ°<math>M</math>ã'''çžäºã€ã³ãã¯ã¿ã³ã¹'''ïŒself inductanceïŒãšãããçžäºã€ã³ãã¯ã¿ã³ã¹ã®æ¬¡å
ã¯ãèªå·±ã€ã³ãã¯ã¿ã³ã¹ã®åäœãšåãã§'''ãã³ãªãŒ'''ïŒHïŒã§ããã
ãã®çžäºã€ã³ãã¯ã¿ã³ã¹ã®å€§ããã¯ãäž¡æ¹ã®ã³ã€ã«ã®å·»ãæ°ã©ããã®ç©ã«æ¯äŸããã
{{ã³ã©ã |埮åç©åãè€çŽ æ°ã®åè·¯èšç®ã®è©±é¡|
亀æµåè·¯ã®èšç®ããé«æ ¡ã®æ°åŠã«ãã埮åç©åïŒã³ã¶ãããã¶ãïŒãè€çŽ æ°ïŒãµããããïŒã®çè«ãã€ãã£ãŠèšç®ããããšãã§ãããïŒâ» æ°åŠæç§ã§ããæ°åŠ3ã§ç« æ«ã³ã©ã ãªã©ã§èª¬æãããããå ŽåãããïŒåæ通ã®æ°åŠ3æç§æžãªã©ïŒãïŒ
ã ãé«æ ¡çã¯ãç©çã®åŠç¿ã§ã¯ããŸãã¯é»æ°åç·ãç£æç·ãªã©ã®ç¹æ§ãšãã£ãåç·ã®ç¹æ§ã®ã€ã¡ãŒãžãç¿åŸããããèšç®åŒã®ç·Žç¿ã§ãé«æ ¡ç©çã®æç§æžã«ããå·®åïŒãã¶ãïŒèšå·Îãšããã€ãã£ãåççãªèª¬æãç解ããããã«ããã»ããããã
ãã€ã¯å€§äººã®äºæ
ã ãã亀æµåè·¯ã®èšç®æ³ã¯ãåéã«ãã£ãŠç°ãªã£ãŠãããããŸãçµ±äžãããŠãªããïŒâ» é«æ ¡ç©çã®èšæ³ã®ã»ãã«ããããšãã°ããã§ãŒã¶ãŒè¡šç€ºããšããè€çŽ 衚瀺ããšãç°ãªãèšç®æ³ã»èšæ³ããããããã«ãã©ãã©ã¹å€æããšããèšç®æ³ããããïŒ
ãããè€çŽ æ°ã®èšå·ã i ïŒã¢ã€ïŒãšã¯ãããã j ïŒãžã§ã€ïŒã ã£ãããšããåéã«ãã£ãŠéã£ãŠããã
ãªã®ã§ããšããããé«æ ¡çã¯ãé«æ ¡ç©çã®æç§æžã«ãããããªãå®æ°ãå·®åèšæ³Îãªã©ã®èšæ³ã§èšç®ããŠããã°ã倧åŠåéšãªã©ã§ã¯å®å
šã§ããã
ãããèšæ³ã»èšç®æ³ã®éãã¯ãããŸãç©çæ³åçã«ã¯æ¬è³ªçã§ãªãã®ã§ãããŸã埮åç©åã«ããåè·¯ã®èšç®æ³ã«ã¯æ·±å
¥ãããªãã»ããããã
ãŸãã¯åç·ã€ã¡ãŒãžãšããã«ã¿èšå·ãÎãæ¹åŒã®åçç©çã®èšç®ãç¿åŸãããã
:⻠倧åŠãå°éåŠæ ¡ã®é»æ°ç³»ã®åŠæ ¡ã«é²åŠãããšãäžè¿°ã®ããããªåè·¯èšç®æ³ïŒäž»ã«è€çŽ æ°è¡šç€ºãšã©ãã©ã¹å€æïŒãç¿ãã®ã§ãããããã®èšç®æ³ãéèŠããå°éåéã®äººããããããèªåã®å°éåéã®èšç®æ³ã®æ矩ã䞻匵ãããããããé«æ ¡çã«ã¯å€§åŠæå¡ãã¡ã®ã¿ã³ããçãªäºæ
ã¯ã©ãã§ãããã®ã§ãç¡èŠãããã
:ç±³åœã®20äžçŽã®ããŒãã«ç©çåŠè
ãã¡ã€ã³ãã³ããã ãã¶é»æ°å·¥åŠè
ãå«ã£ãŠãïŒåèæç®: ãã¡ã€ã³ãã³ç©çåŠã«ããé»æ°å·¥åŠãžã®ç®èã£ãœãæå¥ãïŒ
}}
== é»ç£æ³¢ ==
[[File:Onde electromagnetique.svg|thumb|400px|é»ç£æ³¢ã®æŠç¥å³ãé»å Žãšç£å Žãšã¯çŽäº€ããŠããã]]
ç£å Žã®åãã«ãã£ãŠé»å ŽãåŒãèµ·ããããããšãé»ç£èªå°ã®ã»ã¯ã·ã§ã³ã§èŠãã
å®éã«ã¯é»å Žã®å€åã«ãã£ãŠç£å ŽãåŒãèµ·ããããããšãç¥ãããŠããã
ããã«ãã£ãŠäœããªã空éäžãé»å Žãšç£å ŽãäŒæããŠããããšãäºæ³ãããã
é»ç£æ³¢ã®é床ãç©çåŠè
ã®ãã¯ã¹ãŠã§ã«ãèšç®ã§æ±ãããšãããé»ç£æ³¢ã®é床ã¯ãç空äžã§ã¯åžžã«äžå®ã§ããã€æ³¢ã®é床cãèšç®ã§æ±ãããšããã
:c=3.0Ã10<sup>8</sup>
ãšãªããæ¢ã«ç¥ãããŠããå
éã«äžèŽããã
ãã®ããšãããå
ã¯é»ç£æ³¢ã®äžçš®ã§ããããšãåãã£ããç©çIIã§ãé»ç£æ³¢ã®é床ãæ±ããèšç®ã¯ã詳ããã¯æ±ãã
èªè
ãå
éã®æž¬å®å®éšã«ã€ããŠèª¿ã¹ããªããç©çIã®æ³¢åã«é¢ããããŒãžãªã©ã§ãã£ãŸãŒã®å®éšã«ã€ããŠãåç
§ã®ããšã
æ³¢ã¯æ³¢é·Î»ãé·ãã»ã©ãæ¯åæ°fãå°ãããªããæ³¢ã®æ³¢é·Î»ãšæ¯åæ°fã®ç©fλã¯äžå®ã§ãããã¯æ³¢ã®é床vã«çãããã€ãŸã
:v=fλ
ã§ããã
é»ç£æ³¢ã®å Žåã¯ãé床ãå
éã®cãªã®ã§
:c=fλ
ã§ããã
=== é»ç£æ³¢ã®åé¡ ===
* é»æ³¢
æŸéçšã®ãã¬ããã©ãžãªã®é»æ³¢ïŒã§ãã±ãradio waveïŒã¯ãé»ç£æ³¢ïŒelectromagnetic waveïŒã®äžçš®ã§ãããæ³¢é·ã0.1mm以äžã®é»ç£æ³¢ãé»æ³¢ã«åé¡ãããããªããé»æ³¢ã®ãã¡ãæ³¢é·ã1mmïœ1cmã®ããªã¡ãŒãã«ã®é»æ³¢ãããªæ³¢ãšãããåæ§ã«ãæ³¢é·ã1cmïœ10cmã®é»æ³¢ãã»ã³ãæ³¢ãšãããæ³¢é·10cmïœ100cm(=1m)ã®é»æ³¢ã¯UHFãšèšããããã¬ãæŸéãªã©ã«äœ¿ãããUHFæŸéã¯ããã®é»æ³¢ã§ãããæ³¢é·1mïœ10mã®é»æ³¢ã¯VHFãšèšãããããã¬ãæŸéã®VHFæŸéã¯ããã®é»æ³¢ã§ããã
* èµ€å€ç·
æ³¢é·ã0.1mm以äžã§ãå¯èŠå
ç·ïŒå¯èŠå
ã®æ倧波é·ã¯780ããã¡ãŒãã«çšåºŠïŒãããã¯æ³¢é·ãé·ãé»ç£æ³¢ã¯èµ€å€ç·ïŒãããããããinfrared raysãã€ã³ãã©ã¬ãŒã ã¬ã€ãºïŒãšããããèµ€ãã®ãå€ããšããçç±ã¯ãå¯èŠå
ã®æ倧波é·ã®è²ãèµ€è²ã ããã§ãããèµ€å€ç·ãã®ãã®ã«ã¯è²ã¯ã€ããŠããªããåžè²©ã®èµ€å€ç·ããŒã¿ãŒãªã©ãèµ€è²ã«çºå
ãã補åãããã®ã¯ã䜿çšè
ãåäœç¢ºèªãã§ããããã«ããããã«ã補åã«èµ€è²ã®ã©ã³ãã䜵眮ããŠããããã§ãããèµ€å€ç·ã¯ãç©äœã«åžåããããããåžåã®éãç±ãçºçããã®ã§ãããŒã¿ãŒãªã©ã«å¿çšãããããªãã倪éœå
ã«ãèµ€å€ç·ã¯å«ãŸããã
:çºèŠã®çµç·¯
ããããèµ€å€ç·ãçºèŠãããçµç·¯ã¯ãã€ã®ãªã¹ã®å€©æåŠè
ã®ããŒã·ã§ã«ã倪éœå
ãããªãºã ã§åå
ããéã«ãèµ€è²ã®å
ç·ã®ãšãªãã®ãç®ã«ã¯è²ãèŠããªãéšåã枩床äžæããŠããããšãçºèŠããããšããçµç·¯ãããã
* å¯èŠå
ç·
[[Image:Linear visible spectrum.svg]]
{| class="wikitable" style="float:right; text-align:right; margin:0px 0px 7px 7px;"
|-
!è²
!æ³¢é·
!ãšãã«ã®ãŒ
|-
| style="background-color:#CEB0F4; text-align:center;" |玫
|380-450 nm
|2.755-3.26 eV
|-
| style="background-color:#B0CCF4; text-align:center;" |é
|450-495 nm
|2.50-2.755 eV
|-
| style="background-color:#B4F4B0; text-align:center;" |ç·
|495-570 nm
|2.175-2.50 eV
|-
| style="background-color:#F4F4B0; text-align:center;" |é»è²
|570-590 nm
|2.10-2.175 eV
|-
| style="background-color:#F4DDB0; text-align:center;" |æ©è²
|590-620 nm
|1.99-2.10 eV
|-
| style="background-color:#F4B0B0; text-align:center;" |èµ€
|620-750 nm
|1.65-1.99 eV
|}
æã
ã人éã®ç®ã«èŠããå¯èŠå
ç·ïŒãããããããvisible lightïŒã®æ³¢é·ã¯ãçŽ780ããã¡ãŒãã«ããçŽ380ããã¡ãŒãã«ã®çšåºŠã§ãããå¯èŠå
ã®äžã§æ³¢é·ãæãé·ãé åã®è²ã¯èµ€è²ã§ãããå¯èŠå
ã®äžã§æ³¢é·ãæãçãé åã®è²ã¯çŽ«è²ã§ããã
å
ãã®ãã®ã«ã¯ãè²ã¯ã€ããŠããªããæã
ã人éã®è³ããç®ã«å
¥ã£ãå¯èŠå
ããè²ãšããŠæããã®ã§ããã
倪éœå
ãããªãºã ãªã©ã§åå
ïŒã¶ãããïŒãããšãæ³¢é·ããšã«è»è·¡ïŒãããïŒãããããããã®åå
ããå
ç·ã¯ãä»ã®æ³¢é·ãå«ãŸãããã äžçš®ã®æ³¢é·ãªã®ã§ããã®ãããªå
ç·ããã³å
ã'''åè²å
'''ïŒmonochromatic lightïŒãšããã
ãŸããçœè²ã¯åè²å
ã§ã¯ãªãã'''çœè²å
'''(white light)ãšã¯ãå
šãŠã®è²ã®å
ãæ··ãã£ãç¶æ
ã§ããã
åæ§ã«ãé»è²ãšããåè²å
ããªããé»è²ãšã¯ãå¯èŠå
ãç¡ãç¶æ
ã§ããã
{{clear}}
* 玫å€ç·
玫å€ç·ïŒããããããultraviolet raysïŒã¯ååŠåå¿ã«åœ±é¿ãäžããäœçšã匷ãã殺èæ¶æ¯ãªã©ã«å¿çšãããã倪éœå
ã«ã玫å€ç·ã¯å«ãŸããã人éã®èã®æ¥çŒãã®åå ã¯ã玫å€ç·ãã¡ã©ãã³è²çŽ ãé
žåãããããã§ããã
:çºèŠã®çµç·¯
èµ€å€ç·ã¯å€ªéœå
ã®ããªãºã ã«ããåå
ã§çºèŠãããã
ãã§ã¯ãåå
ããã玫è²ã®å
ç·ã®ãšãªãã«ãããªã«ãç®ã«ã¯èŠããªãç·ãããã®ã§ã¯ïŒããšãããµããªããšãåŠè
ãã¡ã«ãã£ãŠèãããã
ãã€ãã®ç©çåŠè
ãªãã¿ãŒã«ããååŠçãªå®éšæ¹æ³ãçšããŠã玫å€ç·ã®ååšãå®èšŒãããã
* Xç·ããã³ã¬ã³ãç·
å»ççšã®ã¬ã³ãã²ã³ãªã©ã®ééåçã§çšããããXç·ïŒX-rayïŒãé»ç£æ³¢ã®äžçš®ã§ãããçç©ã®çŽ°èãååã¬ãã«ã§å·ã€ããçºããæ§ãæãã
ã¬ã³ãç·ïŒgammaârayãγ rayïŒãåæ§ã«ãééåçã«ãå¿çšãããããçç©ã®çŽ°èãååã¬ãã«ã§å·ã€ããçºããæ§ãæãã
{{clear}}
----
===é»æ°ã«é¢ããæ¢æ±æŽ»å===
??
[[Category:é«çåŠæ ¡æè²|ç©ãµã€ã1ãŠãã]]
[[ã«ããŽãª:é»æ°|é«ãµã€ã1ãŠãã]]
[[Category:ç©çåŠæè²|é«ãµã€ã1ãŠãã]] | 2005-05-08T07:17:05Z | 2023-07-30T16:15:57Z | [
"ãã³ãã¬ãŒã:Clear",
"ãã³ãã¬ãŒã:-",
"ãã³ãã¬ãŒã:ã³ã©ã "
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E7%89%A9%E7%90%86/%E9%9B%BB%E6%B0%97 |
1,939 | é«çåŠæ ¡ç©çåºç€/é»æ°ãšç£æ° | é«çåŠæ ¡ ç©çåºç€ > é»æ°
æ¬é
ã¯é«çåŠæ ¡ ç©çåºç€ã®é»æ°ãšç£æ°ã®è§£èª¬ã§ããã
çŸåšç§ãã¡ã䜿ã£ãŠããå€ãã®è£œåãé»æ°ãçšããŠåããŠããã ããã«ã¯æ§ã
ãªçç±ãèãããããããŸã第äžã«é»æ°ã¯æ§ã
ãªå¥ã®ãšãã«ã®ãŒã«å€æã§ããããšãäŸãã°é»ç±ç·ã䜿ã£ãŠç±ã«ãé»çãçºå
ãã€ãªãŒãã䜿ã£ãŠå
ã«ãé»åæ©ã䜿ã£ãŠéåã«å€æããããšãåºæ¥ãã次ã«ãé»æ± ãã³ã³ãã³ãµã䜿ã£ãŠãšãã«ã®ãŒãç¶æãããŸãŸæã¡éã¶ããšãåºæ¥ãããšããé»ç·ã䜿ã£ãŠé·è·é¢ãéé»ã§ããããšããŸããé»å補åã®èšç®èœåãä¿¡å·ã®äŒéèœåãåªããŠããããšããŸãæ¯èŒçã«å®å
šã«å°éã®ãšãã«ã®ãŒãåãåºããããšãçãèããããã
é»æ°ãéåã«å€ãããã®ãšããŠé»åæ©(è±: electric motor)ããããããã®éã®ãã®ãšããŠéåãé»æ°ã«å€ããããšãåºæ¥ãããããè¡ãªãã®ã¯çºé»æ©è±: generatorãšåŒã°ãããçºé»æã¯äœããã®éåã®ãšãã«ã®ãŒãå©çšããŠé»æ°ãããããŠãããäŸãã°ãæ°Žåçºé»æã§ã¯ãæ°Žã®èœäžããåãå©çšããŠããã倧éã®æ°Žãèœäžãããšãã«ã¯äººéãäœå人ãããã£ãŠããããåºæ¥ãããšããªãããããšããããäŸãã°ãåãç«ã£ã海岞ç·ãªã©ã¯äž»ã«æ°Žã®æµãã«ãã£ãŠäœãããŠããããã®ããã«ãæ°Žã®åã¯åŒ·å€§ã§ããã®ã§ããããäžæãå©çšããæ¹æ³ããããšéœåããããå®éçŸä»£ã§ã¯é»æ°ãåªä»ãšããŠããã®åãåãã ãããšã«æåããŠããã
é»æ± ã®ããã«é»æ¥µã®+ãš-ãå®ãŸã£ãé»æµãçŽæµé»æµæãã¯çŽæµ(è±: direct current)ãšåŒã¶ãäžæ¹ãçºé»æããåŸãããé»æµã®ããã«+ãš-ãéãé床ã§å
¥æããé»æµã亀æµé»æµæãã¯äº€æµ(è±: alternating current)ãšåŒã¶ã
å®éã«ã¯ãã€ãªãŒããçšã㊠亀æµãçŽæµã«å€ã㊠䜿ãããšãããè¡ãªãããã
äœããªã空éãå
ãçŽé²ããŠããããã« èŠããããšããããå®éã«ã¯ãã㯠é»æ³¢ãšåããã®ã§ããã é»æ³¢ãšã¯äŸãã°ãæºåž¯é»è©±ã®éä¿¡ã«äœ¿ããããã®ã§ããã é»è·ãæã£ãç©äœãåãããšãå¿
ç¶çã« çãããã®ã§ããã
ãã©ã¹ããã¯ã®äžæ·ããªã©ã§é«ªãããããšåž¯é»ããçŸè±¡ãªã©ã®ããã«ãç©è³ªãé»æ°ã垯ã³ãããšã垯é»(ããã§ã)ãšãããç©äœãããã£ãŠçºçãããéé»æ°ãæ©æŠé»æ°ãšããã ã¬ã©ã¹æ£ãçµ¹ã®åžã§ããããšãã¬ã©ã¹æ£ã¯æ£ã®é»æ°ã«åž¯é»ããçµ¹ã¯è² ã®é»æ°ã«åž¯é»ããã é»æ°ã®éãé»è·(ã§ãããcharge)ãšããããããã¯é»æ°éãšããã
é»è·ã®åäœã¯ã¯ãŒãã³ã§ãããã¯ãŒãã³ã®èšå·ã¯Cã§ããã
éé»æ°ã«ããé»è·ã©ããã«åãåãéé»æ°åãšããã
ãªãã垯é»ããŠããªãç¶æ
ãé»æ°çã«äžæ§ã§ããããšããã
éå±ã®ããã«ãé»æ°ãéããç©äœãå°äœ(ã©ããããconductor)ãšããããã©ã¹ããã¯ãã¬ã©ã¹ããŽã ã®ããã«é»æ°ãéããªãç©è³ªã絶çžäœ(ãã€ãããããinsulator)ãããã¯äžå°äœ(ãµã©ããã)ãšããã
éå±ã¯å°äœã§ããã
é»æ°ã®æ£äœã¯é»å(electron)ãšããç²åã§ããããã®é»åã¯è² é»è·ã垯ã³ãŠããã(é»åã®é»è·ãè² ã«å®çŸ©ãããŠããã®ã¯ã人é¡ãé»åãçºèŠããåã«é»è·ã®æ£è² ã®å®çŸ©ãè¡ãããããšããé»åãèŠã€ãã£ãéã«é»åã®é»è·ã調ã¹ããè² é»è·ã ã£ãããã§ããã)
éå±ãå°äœãªã®ã¯ãéå±äžã®é»åã¯ãããšã®ååãé¢ããŠããã®éå±å
šäœã®äžãèªç±ã«åããããã§ãããéå±äžã®é»åã®ããã«ãç©è³ªäžãèªç±ã«åããç¶æ
ã®é»åããèªç±é»å(ãããã§ãã)ãšããã
é»æµãšã¯ãèªç±é»åã移åããããšã§ããã
ãã£ãœãã絶çžäœã¯ãèªç±é»åããããªãã絶çžäœã®é»åã¯ããã¹ãŠãããšã®ååã«æçž(ããã°ã)ãããŠéã蟌ããããŠããŠãèªç±ã«ã¯åããªãã
æ£é»è·ãšã¯ãç©è³ªã«é»åãæ¬ ä¹ããŠããç¶æ
ã§ããã è² é»è·ãšã¯ãç©è³ªãé»åãå€ãæã£ãŠããç¶æ
ã§ããã
垯é»ããŠããªã絶çžäœã®ç©è³ªããããããããŠãäž¡æ¹ãæ©æŠé»æ°ã«åž¯é»ãããå Žåãçæ¹ã¯æ£é»è·ãçããããçæ¹ã®ç©è³ªã¯è² é»è·ãçããããã®ãšããçºçããæ£é»è·ã®å€§ãããšè² é»è·ã®å€§ããã¯åãã§ããã ããã¯ãé»åã移åããŠãçæ¹ã®ç©è³ªã¯é»åãäžè¶³ããããçæ¹ã¯çéã®é»åãéå°ã«ãªã£ãŠããããã§ããã
ãã®ããã«ãé»åã¯çæãæ¶æ»
ãããªãããããé»è·ä¿ååãããã¯é»æ°éä¿ååãšèšãã
é»æ°çã«äžæ§ã§ãã£ãå°äœã®ç©è³ª(ä»®ã«ç©è³ªAãšãã)ã«åž¯é»ããå¥ã®ç©è³ª(ä»®ã«ç©è³ªBãšãã)ãæ¥è§Šãããã«è¿ã¥ãããšãç©è³ªAã«ã¯ã垯é»ç©è³ªBã®é»è·ã«åŒãå¯ããããŠãç©äœAã®å
éšã§å察笊å·ã®é»è·ã垯é»ç©äœBã«è¿ãåŽã®è¡šé¢ã«çããããŸãã垯é»ç©äœBãšåãé»è·ã¯åçºããã®ã§ãç©äœAå
éšã®åž¯é»ç©äœBãšã¯é ãåŽã®è¡šé¢ã«çããã
ãã®ãããªçŸè±¡ãéé»èªå°(ããã§ãããã©ã;Electrostatic induction)ãšãããéé»èªå°ã§çããé»è·ã®æ£é»è·ã®éãšè² é»è·ã®éã¯çéã§ããã(é»æ°éä¿åã®æ³å)
å°äœã®å
éšã«éé»æ°åã¯ç¡ãããããã£ããšãããšãèªç±é»åãªã©ã®é»è·ãåããé»æµãæµãç¶ããããšã«ãªããããã®ãããªçŸè±¡ã¯å®åšããªãã®ã§äžåçã«ãªãããããã£ãŠãå°äœã®å
éšã«éé»æ°åã¯ç¡ãã
è¡šé¢ã«é»è·ãéãŸãã®ã¯ãå°äœã®å
éšã«éé»æ°åãäœãããªãããã§ããããããã£ãŠéé»èªå°ã§åŒãå¯ããããé»è·ã®å€§ããã¯ãå€éšããå°äœå
éšãžã®éé»æ°åãæã¡æ¶ãã ãã®å€§ããã§ããã
ãã®å°äœå
éšã®é»è·ããŒãã«ãªãæ§è³ªãå¿çšãããšãäžç©ºã®å°äœã§åºæ¥ãç©äœãçšããŠãéé»æ°åãé®èœããããšãã§ããããããéé»é®èœ(ããã§ããããžããelectric shilding)ãšããã
絶çžäœ(ä»®ã«Aãšãã)ã«é»è·ãè¿ã¥ããå Žåã¯ãå°äœãšã¯éããç©äœAã®å
éšã®é»åã¯èªç±ã«è¡šé¢ã«ã¯éãŸããªãããç©äœå
éšã®ååã®æ£è² ã®é»è·ã®æ¥µæ§ãæã£ãéšåããå€éšã®éé»æ°åã«åŒãå¯ããããããã«ãè¿ã¥ããé»è·ã«è¿ãåŽã«ã¯ç°çš®ã®é»è·ãçããé ãåŽã«ã¯ãåçš®ã®é»è·ãçããã ååãååãå€éšã®éé»æ°åã«ãã£ãŠãæ£è² ã®é»è·ã®éšåãçããããšãå極(ã¶ãããã)ãšããããå€éšã®é»åã«ãã£ãŠèµ·ããããã®ãããªå極ã®ããããèªé»å極(ããã§ãã¶ãããããdielectric polarization)ãšããã
絶çžäœã¯ãéé»æ°åã«ããããããšèªé»å極ãè¡ãã®ã§ã絶çžäœã®ããšãèªé»äœ(ããã§ããããdielectric)ãšãããã
å°äœã«éé»èªå°ãããæ£è² ã®é»è·ã¯ãå°äœãåæãªã©ãããã°æ£é»è·ãšè² é»è·ãå¥åã«åãåºãããšãã§ããããããèªé»äœã®æ£è² ã®é»è·ã¯ãååãååãšå¯æ¥ã«çµã³ã€ããŠãããããæ£è² ã®é»è·ãåãããŠåãåºãããšã¯åºæ¥ãªãã
ããç©è³ªãé»æ°ã垯ã³ãŠãã(垯é»ããŠãã)ãšãããã®åž¯é»ã®å€§å°ã®çšåºŠãé»è·(ã§ãããelectric charge)ãšãããããŸããŸãªç©è³ªããããããªæ¹æ³ã§åž¯é»ãããçµæãé»è·ã«ã¯ã垯é»ãã2åã®ãã®ã©ãããè¿ã¥ããæã«åŒã£åŒµãåããã®(åŒåãåã)ãšåçºããããã®(æ¥åãã¯ããã)ã®2çš®é¡ãããããšãåãã£ãã ãã®ãããªã垯é»ããŠããç©äœã«åãåãéé»æ°åãšããã
ã¹ã€ã®åž¯é»ãããã®ããä»ã«ãããã€ãçšæããŠãè¿ã¥ããŠå®éšãã2åã®ç©äœã®çµã¿åãããå€ãããšãçµã¿åããã«ãã£ãŠã2åã®ç©äœã©ããã«åŒåãåãå Žåãããã°ãæ¥åãåãå Žåãããããšãåãã£ãããã®åŒåãšæ¥åã®é¢ä¿ã¯ã垯é»ããé»è·ã®çš®é¡ã«å¿ããããšãããã£ãã
çµè«ãèšããšãé»è·ã«ã¯æ£è² ã®2çš®é¡ããããæ£ã®é»è·ã©ããã®ç©äœãè¿ã¥ãããšãã¯åçºããããè² ã®é»è·ã©ãããè¿ã¥ãããšããåçºããããæ£ãšè² ã®é»è·ãè¿ã¥ããæã«ã¯åŒåãåãã
ã€ãŸããå笊å·ã®é»è·ãè¿ã¥ããå Žåã¯ãåçºåãçãããç°ç¬Šå·ã®é»è·ãè¿ã¥ããå Žåã¯ãåŒåãçããã
éé»æ°ã©ããã®åã®åŒ·ãã¯ãå®éšçã«ã¯ãé»è·ã®éã«åãåã¯ãéåã®å Žåãšåæ§ã«åãåãŒãåã2ç©äœã®éã®è·é¢ã®2ä¹ã«åæ¯äŸããããšãç¥ãããŠãããæŽã«ãé»è·ã®å€§ããã倧ããã»ã©é»è·éã«åãåã倧ããããšãèæ
®ãããšãè·é¢rã ãé¢ããŠãããããé»è· q 1 {\displaystyle q_{1}} ã q 2 {\displaystyle q_{2}} ãæã£ãŠãã2ç©äœã®éã«åãåFã¯ã
ã§äžããããããããã¯ãŒãã³ã®æ³å( Coulomb's law)ãšãããããã§ã k {\displaystyle k} ã¯æ¯äŸå®æ°ã§ãããäž¡é»è·ã®åšå²ã«ããç©äœã®çš®é¡ã«ããå€åããå®æ°ã§ãããç空äžã§ã®é»å Žãèããå Žåã®kã®å€ã¯ã
ã§ããããŸãã ε {\displaystyle \epsilon } ã¯åŸã»ã©ç»å Žããèªé»ç(ããã§ããã€)ãšåŒã°ããç©çå®æ°ã§ãããèªé»çã¯ãäž¡é»è·ã®åšå²ã«ããç©äœã®çš®é¡ã«ããå€åããå®æ°ã§ãããèªé»çã«ã€ããŠã¯ããã®æãåããŠèªãã 段éã§ã¯ããŸã ç¥ããªããŠãè¯ããã®ã¡ã«ç©çIIã§èªé»çã詳ãã解説ããã
èªé»ç ε {\displaystyle \epsilon } ãšã¯ãŒãã³ã®æ¯äŸå®æ°kã«ã¯äžåŒã®é¢ä¿
ãããã
ç©äœã®ãŸããã«èç©ããããã®ãé»è·ãšåŒã¶ãé»æ°åã«ãã£ãŠåçºããã£ãããåŒãã€ããã£ããããç©äœãé»è·ãæã€ç©äœãšåŒã¶ããŸããããã§èŠ³å¯ãããéé»æ°åããã¯ãŒãã³åãšåŒã¶ããšãããã 2åã®é»è·ã©ããããããŒãåã¯åãã§ããããããã£ãŠäœçšã»åäœçšã®æ³åã«åŸã£ãŠããã
ããã§ãé»è·ã®åäœã¯[C]ã§äžãããããèšå·ã®Cã¯ãã¯ãŒãã³ããšèªãã
å³ã®ããã«ã2æ¬ã®ç³žã«ãããããåã質ém[kg]ã§ãåã笊å·ãšå€§ããã®é»è·q[C]ã®çããã¶ãããã£ãŠããããã¯ãã¯ãŒãã³åã§åçºããã®ã§ãå³ã®ããã«ã糞ãè§åºŠÎžããªãã
ãã®ãšãã質émã«ããéåãšãé»è·qã«ããã¯ãŒãã³åãšã®é¢ä¿ã«ã€ããŠãåŒãç«ãŠãããªããå¿
èŠãªãã°ã糞ã®åŒµåã¯T[N]ãšããããšã
解æ³
å³ã®ãããªäœçœ®é¢ä¿ã«ãªãã®ã§ãå³ã®ããã«åŒãç«ãŠãã°ããã
â» äžèšã®2æ¬ã®ç³žã«ã¶ãããã£ãçã®ã¯ãŒãã³åã®äŸé¡ã¯ãé»æ°ç£æ°åŠã®ã©ã®å
¥éæžã«ããããããªå
žåçãªåé¡ã§ããã®ã§ãèªè
ã¯ãã¡ããšç解ããããšã
é»è· q 1 {\displaystyle q_{1}} , q 2 {\displaystyle q_{2}} ã®éã®è·é¢ãrã®å Žåãš2rã®å Žåã§ã¯ãéã«åãåã®å€§ããã¯ã©ã¡ããã©ãã ã倧ãããçããã ãŸããè·é¢ã2rã®æã®2ç¹éã®åã®å€§ãããçããã
ã¯ãŒãã³åã¯ãç©äœéã®è·é¢ã®é2ä¹ã«æ¯äŸããã®ã§ãè·é¢ã2rã®æã¯ãrã®æã®å€§ããã® 1 4 {\displaystyle {\frac {1}{4}}} ãšãªãããŸããåãåã®å€§ããã¯ãã¯ãŒãã³åã®åŒãçšããŠã
ãšãªãã
æ¢ã«ãããé»è·Aã®ãŸããã®å¥ã®é»è·Bã«ã¯ããã®é»è·ããã®è·é¢ã®é2ä¹ã«æ¯äŸããåããããããšãè¿°ã¹ãã
ããã§ãé»è·Bãåããåã¯ããã®é»è·Bã®å€§ããã«æ¯äŸããããšãåãããŠèãããšããã®é»è·Bã®å€§ããã«ããããããé»è·Aã®å€§ããã ãã§æ±ºãŸãéãå°å
¥ããŠãããšéœåããããããã§ããã®ãããªéãšããŠé»å Ž(ã§ãã°)ãå°å
¥ããããã®ãšããé»å Ž E â {\displaystyle {\vec {E}}} ã®äžã«ããé»è· q {\displaystyle q} ã«åãå F â {\displaystyle {\vec {F}}} ã¯ã
ã§äžãããããé»å Žã¯åäœé»è·ã«åãåãšèããããšãã§ããé»å Žã®åäœã¯[N/C]ã§ããããé»å Žãã¯ããé»çã(ã§ããã)ãšãåŒã°ããã
(æ¥æ¬ã®ç©çåŠã§ã¯ãé»å ŽããšåŒã¶ããšãå€ãããŸããæ¥æ¬ã®é»æ°å·¥åŠã§ã¯ãé»çããšåŒã°ããããšãå€ããææ²»æã®ç¿»èš³ã®éã®ãæ¥æ¬åœå
ã®æ¥çããšã®éãã«éããããããªãæ¥æ¬ããŒã«ã«ãªéœåã§ãããåŒã³æ¹ã¯ç©çã®æ¬è³ªãšã¯é¢ä¿ãªãã®ã§ãããã§ã¯ãã©ã¡ãã®è¡šçŸãçšãããã¯ãæ¬æžã§ã¯ç¹ã«ãã ãããªããè±èªã§ã¯ç©çåŠã»é»æ°å·¥åŠãšãâelectric fieldâã§å
±éããŠããã)
äžã®ã¯ãŒãã³åã®çµæãšåããããšãé»è·Aã®ãŸããã«å¥ã®é»è·ãååšããªããšããé»è· q {\displaystyle q} [C]ã®é»è·ããŸãšãé»å Ž E â {\displaystyle {\vec {E}}} ã¯ã
ã§äžããããããã ããrã¯é»è·ããã®è·é¢ã§ããã e â r {\displaystyle {\vec {e}}_{r}} ã¯ãé»è·ãšããç¹ãçµãã çŽç·äžã§ãé»è·ãšå察æ¹åãåããåäœãã¯ãã«ã§ããã
é»è·ã®åãã®é»å Žã¯ãå¹³é¢äžã§æŸå°ç¶ã®ãã¯ãã«ãšãªãããšã«æ³šæã
é»å Žã¯ãã¯ãã«ã§ãããé»è·ã2åãããšãã¯ãããããã®é»è·ãã€ããé»å Žããéãåãããã°ããã
ã§ããã
é»è·ã3å以äžã®ãšãããåæ§ã«éãåãããã°è¯ãã
å³ã®ããã«ãé»è·ããåºãé»å Žã®æ¹åãå³ç€ºãããã®ãé»æ°åç·(ã§ãããããããelectric line of force)ãšããã é»è·ãè€æ°ããå Žåã«ã¯ãå®éã«æ°ãã«çœ®ãããé»è·ãåããåã¯ããããã足ãåããããã®ãšãªãããããã£ãŠãè€æ°ã®é»è·ãããå Žåã®åšå²ã®é»çã¯ãããããã®é»è·ãäœãé»çãã¯ãã«ã®åãšãªã(éãåããã®åç)ã
é»æ°åç·ãå³ç€ºããå Žåã¯ãæ£é»è·ããåç·ãåºãŠãè² é»è·ã§åç·ãåžåãããããã«æžããåç·ã¯ãé»å Žãå³ç€ºãããã®ãªã®ã§ãé»è·ä»¥å€ã®å Žæã§ã¯ãåç·ãåå²ããããšã¯ãªãã åç·ãçæããã®ã¯æ£é»è·ã®å Žæã®ã¿ã§ãããåç·ãæ¶æ»
ããã®ã¯ãè² é»è·ã®å Žæã®ã¿ã§ããã èšãæããã°ãåç·ãé»è·ä»¥å€ã®å Žæã§æ¶æ»
ããããšã¯ãªãããé»è·ä»¥å€ã®å Žæã§åç·ãçæããããšã¯ãªãã
å°äœã®å
éšã®é»å Žã¯ãŒãã§ãã£ããèšãæããã°ãé»æ°åç·ã¯ãå°äœã®å
éšã«ã¯é²å
¥ã§ããªãã
ç¹é»è·ããã¯ãå³ã®ããã«ãæŸå°ç¶ã«é»æ°åç·ãåºããã¯ãŒãã³ã®æ³åã®ä¿æ°ã«ãã
ã®ãã¡ã®ãåæ¯ã® 4 Ï r 2 {\displaystyle 4\pi r^{2}} ã¯ãçã®è¡šé¢ç©ã®å
¬åŒã«çããã®ã§ãé»æ°åç·ã®å¯åºŠã«æ¯äŸããŠãé»å Žã®åŒ·ããããã¯éé»æ°åã®åŒ·ãã決ãŸããšèããããã
éé»èªå°ã§ã¯ãå°äœå
éšã«ã¯éé»æ°åãåããŠããªãã®ã§ãã£ããããã¯ãé»å ŽãšããæŠå¿µãçšããŠèšãæããã°ãå°äœå
éšã®é»å Žã¯ãŒãã§ããããšèšããã
ã¯ãŒãã³åã¯å(ã¡ãã)ã§ãããããããã«éãã£ãŠå¥ã®é»è·ãè¿ã¥ããå Žåã¯ãè¿ã¥ããå¥ã®é»è·ã¯ä»äºãããããšã«ãªãããŸããè¿ã¥ããé»è·ãææŸãã°ãã¯ãŒãã³åã«ãã£ãŠåãåããä»äºãããããšã«ãªããããè¿ã¥ããç¶æ
ã«ããå¥é»è·ã¯äœçœ®ãšãã«ã®ãŒãèããŠããããšã«ãªãã ãããã£ãŠãã¯ãŒãã³åã«å¯ŸããŠãäœçœ®ãšãã«ã®ãŒãå®çŸ©ããããšãã§ããã(ãªããè¡æè»éäžã®ç©äœã®ãããªãå°è¡šãã倧ããé¢ããå Žæã®éåããã¯ãŒãã³åãšåæ§ã«é2ä¹åãªã®ã§ãããã§èããèšç®ææ³ã¯éåã«ããäœçœ®ãšãã«ã®ãŒã«ãå¿çšã§ãããéåå é床gãçšããåmgãšããã®ã¯å°è¡šè¿ãã§ã®è¿äŒŒã«ãããªãã)
ã¯ãŒãã³åã«ããé»å Žã®å®çŸ©ã§ã¯ãåäœé»è·ã«å¯ŸããŠé»å Žãå®çŸ©ããã®ãšåæ§ãäœçœ®ãšãã«ã®ãŒã«å¯ŸããŠããåäœé»è·ã«å¿ããŠå®çŸ©ã§ããéãå°å
¥ãããšéœåãããããã®ãããªéãé»äœ(ã§ãããelectric potential)ãšåŒã¶ãé»äœã®åäœã¯ãã«ããšãããé»äœãäŸãããšãå°è¡šè¿ãã§ã®éåã®äœçœ®ãšãã«ã®ãŒãèããéã®ãghããªã©ã«çžåœããéã§ããã
ã¯ãŒãã³åã®çµæãšã q {\displaystyle q} [C]ã®é»è·ããè·é¢rã ãé¢ããç¹ã®é»äœVã¯ãé»å Žã®ç©åèšç®ã§åŸãããã(ç©åããŸã ç¿ã£ãŠãªãåŠå¹Žã®èªè
ã¯ãåãããªããŠãæ°ã«ããã次ã®çµæãžãšé²ãã§ãã ããã)çµæã®ã¿ãèšããšã
ãšãªãã
é»äœVã®ç¹ã«q[C]ã®é»è·ã眮ãããšãããã®é»è·ã®ã¯ãŒãã³åã«ããäœçœ®ãšãã«ã®ãŒU[J]ã¯ãé»äœVãçšããã°ã
ãšãªãããããã£ãŠãé»äœ V 1 {\displaystyle V_{1}} ãã«ãã®ç¹ããé»äœ V 2 {\displaystyle V_{2}} ãã«ãã®äœçœ®ãžãšé»è·q[C]ãéé»æ°åãåããŠç§»åãããšããéé»æ°åã®ããä»äºW[J]ã¯
ãšãªãã
ãã£ãœããäžæ§ãªé»å Žã«ãããŠã¯ãé»äœã®åŒããé»å ŽãçšããŠç°¡åã«è¡šãããšãã§ãããè·é¢dã ãé¢ããå¹³è¡å¹³æ¿é»æ¥µã®éã«äžæ§ãªé»å Ž E â {\displaystyle {\vec {E}}} ãçããŠãããšãããã®é»çã®äžã«çœ®ããé»è·qã¯éé»æ°å q E â {\displaystyle q{\vec {E}}} ãåããããã®é»è·ãé»çã®åãã«æ²¿ã£ãŠäžæ¹ã®é»æ¥µããä»æ¹ã®é»æ¥µãŸã§ç§»åãããšããé»çã®ããä»äºW㯠W = q E d {\displaystyle W=qEd} ãšãªããããããã2極æ¿ã®é»äœå·®Vã¯ã
ã§è¡šãããšãã§ããããšãããããåŒãå€åœ¢ããŠ
ãšããããšãã§ãããããã§ãåäœãèãããšãå³èŸºã¯é»å§ãè·é¢ã§å²ã£ããã®ã§ãããããé»çã®åäœãšããŠ[N/C]ã®ã»ã[V/m]ãçšããããšãã§ããããšããããã
é»äœã®åäœã¯ãã«ãã§ããããã®éã¯æ¢ã«äžåŠæ ¡çç§ãªã©ã§æ±ã£ãé»å§(ã§ããã€ãvoltage)ã®åäœãšåãåäœã§ãããå®éã«é»æ°åè·¯ã«é»å§ããããããšã¯ãåè·¯äžã®é»åã«é»å ŽããããŠåããããšãšçããã
éé»èªå°ã«ãã£ãŠãå°äœå
éšã®é»å Žã¯ãŒãã§ãã£ãããã®ããšãããå°äœã®è¡šé¢ã¯ãé»äœãçãããå°äœè¡šé¢ã¯äºãã«çé»äœã§ããã
é»äœã®åºæºã¯ãå®çšäžã¯ãå°é¢ã®é»äœããŒãã«çœ®ãããšãå€ããé»æ°åè·¯ã®äžéšã倧å°ã«ã€ãªãããšãæ¥å°(ãã£ã¡)ãŸãã¯ã¢ãŒã¹(earth)ãšãããåè·¯ãã¢ãŒã¹ããŠããã®ã€ãªãã éšåã®é»äœããŒããšèŠãªãããšãå€ãã
çŽç·äžã§è·é¢0, b[m]ã®ç¹ã«ãé»è·q, q'ãæã€ç©äœã眮ããŠããããã®æãäœçœ®a[m](a<b)ã®ç¹ã®é»äœãæ±ããã
é»äœã®åŒãçšããã°ãããé»è·ãè€æ°ãããšãã«ã¯ãé»äœã¯ããããã®é»è·ãã€ããåºãé»è·ã®åã«ãªãããšã«æ³šæãçãã¯ã
ãšãªãã
å°äœè¡šé¢ã¯çé»äœãªã®ã§ããã£ãŠãé»æ°åç·ã¯å°äœè¡šé¢ã«åçŽã§ããã
ãã®ããšãããé»æ°åç·ãšé»å Žã¯åçŽã§ããã
é»å Žãéãåãããããããã«ãé»äœãéãåãããããããªããªãé»äœãšã¯ãé»å ŽãèããŠãçµè·¯ã«ãŠç©åãããã®ã§ããããã
åŠæ ¡ã®ãã¹ããªã©ã§ã¯ãé»äœã®èšç®ã®ãããã¯ãŒãã³åã®æ¹åã®åéããªã©ã«ããèšç®ãã¹ãªã©ããµãããããé»å Žãæ±ããŠããããããç©åããŠãé»äœãæ±ããã®ããèšç®äžã¯å®å
šã§ããã
ã³ã³ãã³ãµãŒ(è±:capacitor ,ããã£ãã·ã¿ããšèªã)ã¯ãå³ã®ããã«ã2æã®é»æ¥µãåãããããåè·¯äžã«é»è·ãèç©ã§ããéšåãäžããçŽ åã§ããã
ã³ã³ãã³ãµãŒã«é»è·ãèããããšãå
é»(ãã
ãã§ã)ãšãããã³ã³ãã³ãµãŒããé»è·ãæŸåºãããããšãæŸé»ãšããã
ã³ã³ãã³ãµã®äž¡ç«¯ã«ããé»äœVãäžãããããšããã³ã³ãã³ãµã«ã¯ãé»äœã«æ¯äŸããé»è·Qãèç©ãããããã®ãšããã³ã³ãã³ãµã®èç©èœåãèšå·ã§ C ãšãããŠã
ãšããŠCãåããCã¯éé»å®¹é(ããã§ãããããããelectric capacitance)ãšåŒã°ããåäœã¯F(ãã¡ã©ããfarad)ã§äžããããã
1ãã¡ã©ãã¯å®çšäžã¯å€§ããããã®ã§ã10ãã¡ã©ããåäœã«ãã1pF(ãã³ãã¡ã©ã)ãã10ãã¡ã©ããåäœã«ãã1ÎŒF(ãã€ã¯ããã¡ã©ã)ã䜿ãããããšãå€ãã
極æ¿ãå¹³è¡ãªã³ã³ãã³ãµãŒãå¹³è¡æ¿ã³ã³ãã³ãµãŒãšããã å¹³è¡æ¿ã³ã³ãã³ãµãŒã®ã極æ¿ã©ããã®é»å Žã¯ãäžæ§ãªé»å Žã§ããã
ãã®å¹³è¡æ¿ã³ã³ãã³ãµãŒã®éé»å®¹éCã®åŒã¯ãåŸè¿°ããçç±ã«ããã
ã§äžãããããããã§ãSã¯å°äœå¹³é¢ã®é¢ç©ã§ãããdã¯å°äœéã®è·é¢ã§ããã
å®éšçã«ãããã®éé»å®¹éã®å
¬åŒã¯ãæ£ããããšã確ãããããŠããã
ããã§äžããéé»å®¹éã¯ãå¹³é¢äžã«é»è·ãäžæ§ã«ååžãããšã®ä»®å®ã§å°ãããããã®ãšããå°äœéã«çããé»çEã¯ãå°äœãæã€é»è·ãQ, -Qãšããæã
ãŸãã極æ¿ã®é»è·å¯åºŠãã極æ¿ã®ã©ãã§ãäžå®ã ãšä»®å®ããŠ(ãã®ããã«ã¯ãã³ã³ãã³ãµãŒã®åºã(ã€ãŸãé¢ç©)ãããã
ãã¶ãã«åºããšä»®å®ããå¿
èŠãããããšãããããã®ãããªä»®å®ã«ãããé»è·å¯åºŠã¯)ã
ã§ããã
é»æ°åç·ã®æ§è³ªãšããŠããã©ã¹ã®é»è·ããçããŠãã€ãã¹ã®é»è·ã§åžåãããã®ã§ããã£ãŠå¹³è¡æ¿ã³ã³ãã³ãµãŒéã®é»æ°åç·ã®ååžã¯ãå³ã®ããã«ãé»æ°åç·ãããã©ã¹æ¥µæ¿ããåçŽã«ããã€ãã¹æ¥µæ¿ãžåãã£ãŠé»æ°åç·ãåºãŠããããŠãã€ãã¹æ¥µæ¿ã«é»æ°åç·ãåžåãããã
é»å Žã¯ãå°äœéã®åç¹ã§ã
ã§äžãããããé»å Žãæ±ããããã®ã§ãããããé»äœãèšç®ã§ãããå°äœéã®åç¹ã§é»å Žã®å€§ãããåäžãªã®ã§ãé»äœã®å€§ããã¯é»å Žã®å€§ããã«2ç¹éã®è·é¢ãããããã®ã«ãªããããã§ãé»äœVã¯ã
ãšãªããããã®åŒãšéé»å®¹éCã®å®çŸ©ãèŠæ¯ã¹ããšã
ãåŸãããã
é»æ± ã®ååŠåå¿ã«ã€ããŠã¯ãå¥ç§ç®ã®ååŠIãªã©ã§è©³ããæ±ãããããã®ç« ã§ã¯ãé»å§ãé»æµã®ç解ã«é¢ããç¹ãéç¹çã«èª¬æãããã
éå±å
çŽ ã®åäœãæ°ŽãŸãã¯æ°Žæº¶æ¶²ã«å
¥ãããšãã®ãéœã€ãªã³ã®ãªãããããã€ãªã³ååŸå(ionization tendency)ãšããã äŸãšããŠãäºéZnãåžå¡©é
žHClã®æ°Žæº¶æ¶²ã«å
¥ãããšãäºéZnã¯æº¶ãããŸãäºéã¯é»åã倱ã£ãŠZnã«ãªãã
äžæ¹ãéAgãåžå¡©é
žã«å
¥ããŠãåå¿ã¯èµ·ãããªãã
ãã®ããã«éå±ã®ã€ãªã³ååŸåã®å€§ããã¯ãç©è³ªããšã«å€§ãããç°ãªãã
äºçš®é¡ã®éå±åäœãé»è§£è³ªæ°Žæº¶æ¶²ã«å
¥ãããšé»æ± ãã§ãããããã¯ã€ãªã³ååŸå(åäœã®éå±ã®ååãæ°ŽãŸãã¯æ°Žæº¶æ¶²äžã§é»åãæŸåºããŠéœã€ãªã³ã«ãªãæ§è³ª)ã倧ããéå±ãé»åãæŸåºããŠéœã€ãªã³ãšãªã£ãŠæº¶ããã€ãªã³ååŸåã®å°ããéå±ãæåºããããã§ããã
ã€ãªã³ååŸåã®å€§ããæ¹ã®éå±ãè² æ¥µ(ãµããã)ãšãããã€ãªã³ååŸåã®å°ããæ¹ã®éå±ãæ£æ¥µ(ããããã)ãšããã ã€ãªã³ååŸåã®å€§ããéå±ã®ã»ãããéœã€ãªã³ã«ãªã£ãŠæº¶ãåºãçµæãéå±æ¿ã«ã¯é»åãå€ãèç©ããã®ã§ãäž¡æ¹ã®éå±æ¿ãé
ç·ã§ã€ãªãã°ãã€ãªã³ååŸåã®å€§ããæ¹ããå°ããæ¹ã«é»åã¯æµããããé»æµãã§ã¯ç¡ãããé»åããšãã£ãŠãããšã«æ³šæãé»åã¯è² é»è·ã§ããã®ã§ãé»æµã®æµããšé»åã®æµãã¯ãéåãã«ãªãã
ããŸããŸãªæº¶æ¶²ãéå±ã®çµã¿åããã§ãã€ãªã³ååŸåã®æ¯èŒã®å®éšãè¡ã£ãçµæãã€ãªã³ååŸåã®å€§ããã決å®ãããã å·Šããé ã«ãã€ãªã³ååŸåã®å€§ããéå±ã䞊ã¹ããšã以äžã®ããã«ãªãã
éå±ããã€ãªã³ååŸåã®å€§ããã®é ã«äžŠã¹ããã®ãéå±ã®ã€ãªã³ååãšããã æ°ŽçŽ ã¯éå±ã§ã¯ç¡ããæ¯èŒã®ãããã€ãªã³ååŸåã«å ããããã éå±ååã¯ãäžèšã®ä»ã«ãããããé«æ ¡ååŠã§ã¯äžèšã®éå±ã®ã¿ã®ã€ãªã³ååãçšããããšãå€ãã ã€ãªã³ååã®èšæ¶ã®ããã®èªååãããšããŠã
ã貞ããããªããŸããããŠã«ããªãã²ã©ãããåéãã
ãªã©ã®ãããªèªååããããããã¡ãªã¿ã«ãã®èªååããã®å Žåã
ãKã ãã ãCa ãªNaããŸMg ãAlããZn ãŠFe ã«Ni ã ãªPbãã²H2 ã©Cu ãHg ãAg ã åéPt,Auãã
ãšå¯Ÿå¿ããŠããã
è² æ¥µ(äºéæ¿)ã§ã®åå¿
æ£æ¥µ(é
æ¿)ã§ã®åå¿
ãã«ã¿ã®é»æ± ã§ã¯ãåŸããã䞡極éã®é»äœå·®(ãé»å§ããšãããã)ã¯ã1.1ãã«ãã§ããã(ãã«ãã®åäœã¯Vãªã®ã§ã1.1Vãšãæžãã)ãã®äž¡æ¥µæ¿ã®é»äœå·®ãèµ·é»åãšãããèµ·é»åã¯ãäž¡é»æ¥µã®éå±ã®çµã¿åããã«ãã£ãŠæ±ºãŸãç©è³ªåºæã§ããã
èµ·é»åã®åäœã®ãã«ãã¯ãéé»æ°åã®é»äœã®åäœã®ãã«ããšåãåäœã§ãããé»æ°åè·¯ã®é»å§ã®ãã«ããšããèµ·é»åã®åäœã®ãã«ãã¯åãåäœã§ããã
ãã«ã¿é»æ± ã®æ§é ã以äžã®ãããªæååã«è¡šããå Žåããã®ãããªè¡šç€ºãé»æ± å³ãããã¯é»æ± åŒãšããã
aqã¯æ°Žã®ããšã§ãããH2SO4aqãšæžããŠãç¡«é
žæ°Žæº¶æ¶²ãè¡šããŠããã
ç©çåŠã®é»æ°åè·¯ã®ç 究ã§ã¯ããã®ãããªé»æ± ãªã©ã®çŸè±¡ã®çºèŠãšçºæã«ãã£ãŠãå®å®ãªçŽæµé»æºãå®éšçã«åŸãããããã«ãªããçŽæµé»æ°åè·¯ã®æ£ç¢ºãªå®éšãå¯èœã«ãªã£ããé»æ± ã®çºæ以åã«ãããã©ã³ã¹äººã®ç©çåŠè
ã¯ãŒãã³ãªã©ã«ããéé»æ°ã«ããé»æ°ååŠã®ç 究ãªã©ã«ãã£ãŠãé»äœå·®ã®æŠå¿µãé»è·ã®æŠå¿µã¯ãã£ããã ãããã®æ代ã®é»æºã¯ãäž»ã«éé»æ°ã«ãããã®ã ã£ãã®ã§ãå®å®é»æºã§ã¯ç¡ãã£ãã
ãããŠãé»æ± ã«ããå®å®ãªé»æºã®çºæã¯ãåæã«å®å®ãªé»æµã®çºæã§ããã£ãããã®ãããªé»æ± ã®çºæãªã©ã«ãããçŽæµé»æ°åè·¯ã®ç 究ãªã©ããããã€ã人ã®ç©çåŠè
ãªãŒã ããããŸããŸãªå°äœã«é»æµãæµãå®éšãšçè«ç 究ãè¡ãããšã«ãããé»æ°åè·¯ã®çè«ã®ãªãŒã ã®æ³å(ãªãŒã ã®ã»ããããOhm's law)ãçºèŠãããã
ãã€ã¯ãªãŒã ã¯é»æ± ã§ã¯ãªãç±é»å¯Ÿ(ãã€ã§ãã€ã)ãšãããã®ã䜿ã£ãŠãé»æ°åè·¯ã«å®å®ããé»æµããªããç 究ããããåœæã®é»æ± ã§ã¯ãèµ·é»åããã ãã«æžã£ãŠããŸãããªãŒã ã¯åœåã¯é»æ± ã§å®éšããããããŸãå®å®é»æµãåŸãããªãã£ãã
ç±é»å¯Ÿãšã¯ããŸãç°ãªãéå±ææã®2æ¬ã®éå±ç·ãæ¥ç¶ããŠ1ã€ã®åè·¯ãã€ããã2ã€ã®æ¥ç¹ã«æž©åºŠå·®ãäžãããšãåè·¯ã«é»å§ãçºçããããé»æµãæµãã(ãã®çŸè±¡ãããŒãŒããã¯å¹æãšãã)ããã®çŸè±¡ãããã¯ã1821幎ã«ãŒãŒããã¯ãçºèŠããããã®ãããªåè·¯ããç±é»å¯Ÿã§ããããªããåã2æ¬ã®éå±ç·ã§ã¯ã枩床差ãäžããŠãé»å§ã¯çºçãããé»æµã¯æµããªãã
ãªãŒã ã¯ããã«ãªã³å€§åŠææããã±ã³ãã«ãã®å©èšã«ãã£ãŠããã®ç±é»å¯Ÿãå®éšã«å©çšããã枩床ãå®å®ãããã®ã¯ãåœæã®æè¡ã§ãæ¯èŒçç°¡åã§ãã£ãã®ã§ãããããŠãªãŒã ã¯å®å®é»æµããã¡ããå®éšãã§ããã®ã§ããã
ãªãŒã ã®æ³å(Ohm's law)ãšã¯ã
ãã»ãšãã©ã®å°äœã§ã¯ãé»æµ I ãæµããŠããå°äœäžã®2ç¹ã®ç¹ P 1 {\displaystyle P_{1}} ãšç¹ P 2 {\displaystyle P_{2}} éã®é»äœå·® E = E 1 â E 2 {\displaystyle E=E_{1}-E_{2}} ã¯ãé»æµ I ã«æ¯äŸãããã
ãšããå®éšæ³åã§ããã 誀解ããããããããªãŒã ã®æ³åã¯ããã®ãããªå®éšæ³åã§ãã£ãŠãã¹ã€ã«æµæã®å®çŸ©åŒã§ã¯ç¡ããåæ§ã«ããªãŒã ã®æ³åã¯ãã¹ã€ã«é»å§ã®å®çŸ©åŒã§ã¯ç¡ãããé»æµã®å®çŸ©åŒã§ãç¡ããäžåŠæ ¡ã®çç§ã§ã®é»æ°åè·¯ã®æè²ã§ã¯ãéå±ã®é»æ°å解ã®èµ·é»åã®æè²ãŸã§ã¯ããªãã®ã§ããšãããã°ãé»å§ã誀解ããŠããé»å§ã¯ãåãªãé»æµã®æ¯äŸéã§ãæµæã¯ãã®æ¯äŸä¿æ°ãã®ãããªèª€è§£ããå Žåãæããããããã®è§£éã¯æããã«èª€è§£ã§ããã ãŸããåå°äœãªã©ã®äžéšã®ææã§ã¯ãé»æµãå¢ãææã®æž©åºŠãäžæãããšæµæãäžããçŸè±¡ãç¥ãããŠããã®ã§ãåå°äœã§ã¯ãªãŒã ã®æ³åãæãç«ããªãå Žåãããããªã®ã§ããªãŒã ã®æ³åãå®çŸ©åŒãšèããã®ã¯äžåçã§ããã
å°ç·ãªã©ã®å°äœå
ã®é»æ°ã®æµããé»æµ(ã§ããã
ããelectric current)ãšãããé»æµã®åŒ·ãã¯ã¢ã³ãã¢ãšããåäœã§è¡šãã1ã¢ã³ãã¢ã®å®çŸ©ã¯æ¬¡ã®éãã§ããã
1ç§éã«1ã¯ãŒãã³(èšå·C)ã®é»æµãééããããšã1ã¢ã³ãã¢ãšããã
ã¢ã³ãã¢ã®èšå·ã¯Aã§ããããŸããé»æµã¯ãåäœæéãããã®é»è·ã®éééã§ãããã®ã§ãé»æµã®åäœã[C/s]ãšæžãå Žåãããã äžè¬çã«ã¯ãé»æµã®åäœã¯ããªãã¹ã[A]ã§è¡šèšããããšãå€ãã
é»æµI[A]ãšæét[S]ã§å°ç·æé¢ãééããé»è·Q[C]ã®é¢ä¿ãåŒã§è¡šããšã
ã§ããã
é»æµã®åãã®åãæ¹ã«ã€ããŠã¯ãèªç±é»åã¯è² é»è·ãæã£ãŠãããããèªç±é»åã®åããšã¯å察åãã«é»æµã®åãããšãããšã«æ³šæããã
次ã«é»æµãšèªç±é»åã®é床ãšã®é¢ä¿ãèããã èªç±é»åã®é»è·ã®çµ¶å¯Ÿå€ãeãšãããšãèªç±é»åã¯è² é»è·ã§ãããããèªç±é»åã®é»è·ã¯ãã€ãã¹ç¬Šå·ãã€ã-eã§ããã
ãã€ã人ã®ç©çåŠè
ãªãŒã ã¯æ¬¡ã®ãããªæ³åãçºèŠããã
ãã»ãšãã©ã®å°äœã§ã¯ãé»æµ I ãæµããŠããå°äœäžã®2ç¹ã®ç¹ P 1 {\displaystyle P_{1}} ãšç¹ P 2 {\displaystyle P_{2}} éã®é»äœå·® E = E 1 â E 2 {\displaystyle E=E_{1}-E_{2}} ã¯ãé»æµ I ã«æ¯äŸãããã
ãã®å®éšæ³åããªãŒã ã®æ³å(Ohm's law)ãšããã åŒã§è¡šããšãé»äœå·®ãVãšããŠãé»æµãIãšããå Žåã«ãæ¯äŸä¿æ°ãRãšããŠã
ã§ããã ããã§ãé»äœãšé»æµã®æ¯äŸä¿æ°Rãé»æ°æµæãããã¯åã«æµæ(resistanceãã¬ãžã¹ã¿ã³ã¹)ãšããã é»æ°æµæã®åäœã¯ãªãŒã ãšèšããèšå·ã¯Î©ã§è¡šãã
æ
£ç¿çã«ãæµæã®èšå·ã¯Rã§ããããå Žåãå€ãã
é»æ°åè·¯ãžãšãã«ã®ãŒãäŸçµŠããé»æºãšããŠå®é»å§ã®çŽæµé»æºãèãããåè·¯ã®2å°ç¹éã«ããäžå®ã®é»å§ãäŸçµŠãç¶ãããã®ã§ãããé»å§æºã®åè·¯å³èšå·ãšããŠã¯ãçšãããããèšå·ã®é·ãåŽãæ£æ¥µã§ããããã©ã¹ã®é»äœã§ãããèšå·ã®çãåŽã¯è² 極ã§ããã
也é»æ± ã¯ãçŽæµé»æºãšããŠåãæ±ã£ãŠè¯ãã
ãªãããããã¯çŽæµé»æºã§ããã亀æµã®å Žåã¯äžè¬åããé»å§æºãšããŠã®èšå·ãçšããããŸãç¹ã«æ£åŒŠæ³¢äº€æµé»å§æºã§ããã°ã®èšå·ãçšããã
æµæåš(resistor)ã¯ãéåžžã¯åã«æµæãšåŒã°ããåè·¯çŽ åã§ãããäžããããé»æ°ãšãã«ã®ãŒãåçŽã«æ¶è²»ããçŽ åã§ãããåè·¯å³èšå·ã¯ãããã¯ã§ããããæ¬æžã§ã¯ãäž¡è
ãšãæµæã®åè·¯å³èšå·ãšããŠçšããããšã«ããã(ç»åçŽ æã®ç¢ºä¿ã®éœåã®ãããäž¡æ¹ã®èšå·ãæ¬æžã§ã¯æ··åšããŸããã容赊ãã ããã)
æ¥æ¬ã§ã¯ãæµæåšã®å³èšå·ã¯ãåŸæ¥ã¯JIS C 0301(1952幎4æå¶å®)ã«åºã¥ããã®ã¶ã®ã¶ã®ç·ç¶ã®å³èšå·ã§å³ç€ºãããŠããããçŸåšã®ãåœéèŠæ Œã®IEC 60617ãå
ã«äœæãããJIS C 0617(1997-1999幎å¶å®)ã§ã¯ã®ã¶ã®ã¶åã®å³èšå·ã¯ç€ºãããªããªããé·æ¹åœ¢ã®ç®±ç¶ã®å³èšå·ã§å³ç€ºããããšã«ãªã£ãŠãããæ§èŠæ Œã§ããJIS C 0301ã¯ãæ°èŠæ ŒJIS C 0617ã®å¶å®ã«äŒŽã£ãŠå»æ¢ããããããæ§èšå·ã§æµæåšãå³ç€ºããå³é¢ã¯ãçŸåšã§ã¯JISéæºæ ãªå³é¢ã«ãªã£ãŠããŸããããããææåã¯ç¡ããããçŸåšãåŸæ¥ã®å³èšå·ãå€çšãããŠããã
è€æ°ã®åè·¯çŽ åã1ã€ã®ç·äžã«é
眮ãããŠãããããªæ¥ç¶ãçŽåæ¥ç¶ãšãããè€æ°ã®åè·¯çŽ åãäºè¡ã«åãããããã«é
眮ãããŠããæ¥ç¶ã䞊åæ¥ç¶ãšããã
çŽåæ¥ç¶ã«ãããŠã¯ãããããã®åè·¯çŽ åã«æµããé»æµã¯å
šãŠçãããäžæ¹ã䞊åæ¥ç¶ã«ãããŠã¯ããããã®åè·¯çŽ åã®äž¡ç«¯ã«ãããé»å§ãå
šãŠçããã
ãŸããçŽåæ¥ç¶ã«ãããŠã¯ããããã®åè·¯çŽ åã«ãããé»å§ã®åãå
šé»å§ãšãªãã䞊åæ¥ç¶ã«ãããŠã¯ããããã®åè·¯çŽ åãæµããé»æµã®åãå
šé»æµãšãªãã
æµæãè€æ°æ¥ç¶ãããŠããå Žåããã®è€æ°ã®æµæããŸãšããŠãããã1ã€ã®æµæãæ¥ç¶ãããŠãããã®ãããªç䟡çãªåè·¯ãèããããšãã§ãããè€æ°ã®æµæãšç䟡ãª1ã€ã®æµæãåææµæãšããã
æµæãnåçŽåã«æ¥ç¶ãããŠããå Žåãèãããæµæ R 1 , R 2 , ⯠, R n {\displaystyle R_{1},R_{2},\cdots ,R_{n}} ãçŽåã«æ¥ç¶ãããŠããå Žåãåæµæãæµããé»æµã¯çããããããiãšãããåæµæ R k ( k = 1 , 2 , ⯠, n ) {\displaystyle R_{k}(k=1,2,\cdots ,n)} ã«ãããé»å§ã v k {\displaystyle v_{k}} ãšãããšããªãŒã ã®æ³åãã
ãæãç«ã€ããã®ãšãçŽåæµæã®äž¡ç«¯ã®é»å§vã¯ã
ã§ããããããšç䟡ãªæµæRã1ã€ã ãæ¥ç¶ãããŠãããããªç䟡åè·¯ãèãããšãã
ãæãç«ã€ããããããã£ãŠãããã®nåã®çŽåæµæã®åææµæRãšããŠ
ãåŸããããªãã¡ãçŽååææµæã¯åæµæã®ç·åãšãªãã
åæ§ã«ãæµæãnå䞊åã«æ¥ç¶ãããŠããå Žåãèãããæµæ R 1 , R 2 , ⯠, R n {\displaystyle R_{1},R_{2},\cdots ,R_{n}} ã䞊åã«æ¥ç¶ãããŠããå Žåãåæµæã®äž¡ç«¯ã®é»å§ã¯çããããããvãšãããåæµæ R k ( k = 1 , 2 , ⯠, n ) {\displaystyle R_{k}(k=1,2,\cdots ,n)} ãæµããé»æµã i k {\displaystyle i_{k}} ãšãããšããªãŒã ã®æ³åãã
ãæãç«ã€ããã®ãšã䞊åæµæãžæµã蟌ãé»æµiã¯ã
ã§ããããããšç䟡ãªæµæRã1ã€ã ãæ¥ç¶ãããŠãããããªç䟡åè·¯ãèãããšãã
ãæãç«ã€ããããããã£ãŠãããã®nåã®äžŠåæµæã®åææµæRãšããŠ
ãåŸããããªãã¡ã䞊ååææµæã®éæ°ã¯åæµæã®éæ°ã®ç·åãšãªãã
æµæRãé»æµIãæµãããšãããã®éšåã®çºç±ã®ãšãã«ã®ãŒã¯ã1ç§ãããã«RI[J/s]ã§ãããããããžã¥ãŒã«ç±ãšãããååã®ç±æ¥ã¯ç©çåŠè
ã®ãžã¥ãŒã«ã調ã¹ãããã§ããããªãŒã ã®æ³åãããV=RIã§ãããã®ã§ããžã¥ãŒã«ç±ã¯VIãšãæžããã
ããã§ãã²ãšãŸããç±ã®èå¯ã«ã¯é¢ããŠã次ã®éãå®çŸ©ãããé»æ°åè·¯ã®ãã2ç¹éãæµããé»æµIãšããã®2ç¹éã®é»å§Vãšã®ç©VIãé»å(power)ãšå®çŸ©ãããé»åã®èšå·ã¯Pã§è¡šããããããšãå€ãã é»åã®åäœã®ãžã¥ãŒã«æ¯ç§[J/s]ã[W]ãšããåäœã§è¡šãããã®åäœWã¯ã¯ãã(Watt)ãšèªãã ã€ãŸãé»åã¯èšå·ã§
ã§ããã
å°ç·ã®å€ªããé·ãã«ãã£ãŠæµæã®å€§ããã¯å€ãããçŽæçã«å€ªãã»ããæµããããã®ã¯åããã ãããããã«äžŠåæ¥ç¶ãšå¯Ÿå¿ãããŠããå°ç·ã倪ãã»ããæµããããã®ã¯åããã ããã å®éã«é»æ°æµæã¯ãå°ç·ã倪ãã«åæ¯äŸããŠå°ãããªãããšãå®éšçã«ç¢ºèªãããŠãããããã§ãã€ãã®ãããªåŒã«ãããŠã¿ããã æµæãR[Ω]ãšããå Žåãå°ç·ã®å€ªããé¢ç©ã§è¡šãA[m]ãšããã°ãæ¯äŸå®æ°ã«kãçšããã°ã
ã§ããã( âã¯ãæ¯äŸé¢ä¿ãè¡šãæ°åŠèšå·ã)
ããã«ãå°ç·ã¯æ質ã倪ããåããªãã°ãå°ç·ãé·ãã»ã©æµæããé·ãã«æ¯äŸããŠæµæã倧ãããªãããšãã確èªãããŠãããããã§ãããã«ãæµæäœã®é·ããèæ
®ããåŒã«è¡šããŠã¿ãã°ã次ã®ããã«ãªããæµæ垯ã®é·ããl[m]ãšããã°
ã§ããã
ããã«ãå°ç·ã®æ質ã«ãã£ãŠãæµæã®å€§ããã¯å€ãããåãé·ãã§åã倪ãã®æµæã§ããæ質ã«ãã£ãŠæµæã®å€§ããã¯ç°ãªããããã§ãæ質ããšã®æ¯äŸå®æ°ãÏãšããã°ãæµæã®åŒã¯ä»¥äžã®åŒã§èšè¿°ãããã
Ïã¯æµæç(ãŠããããã€ãresistivity)ãšåŒã°ãããæµæçã®åäœã¯[Ωm]ã§ããã
ç£ç³ã®ãŸããã«ã¯å¥ã®ç£ç³ãåããåã®ããšãšãªããã®ãçããŠããã ãããç£å Ž(ãã°ãmagnetic field)ãããã¯ç£ç(ããã)ãšåŒã¶ã(æ¥æ¬ã®ç©çåŠã§ã¯ç£å ŽãšåŒã¶ããšãå€ãããŸããæ¥æ¬ã®é»æ°å·¥åŠã§ã¯ç£çãšåŒã°ããããšãå€ããææ²»æã®èš³èªã®éã®ãæ¥æ¬åœå
ã®æ¥çããšã®éãã«éãããå°å瀟äŒçãªäºè±¡ã§ãããåŒã³æ¹ã¯ç©çã®æ¬è³ªãšã¯é¢ä¿ãªãã®ã§ãããã§ã¯ãã©ã¡ãã®è¡šçŸãçšãããã¯ãæ¬æžã§ã¯ç¹ã«ãã ãããªããè±èªã§ã¯ç©çåŠã»é»æ°å·¥åŠãšãâmagnetic fieldâã§å
±éããŠããã)
éãã³ãã«ããããã±ã«ã«ç£ç³ãè¿ã¥ãããšãç£ç³ã«åžãä»ããããã ãŸããéãã³ãã«ããããã±ã«ã«åŒ·ãç£åãäžãããšãéãã³ãã«ããããã±ã«ãã®ãã®ãç£å Žãåšå²ã«åãŒãããã«ãªãã ãã®ãããªãããšããšã¯ç£å Žãæããªãã£ãç©äœãã匷ãç£å Žãåããããšã«ãã£ãŠç£å ŽãåãŒãããã«ãªãçŸè±¡ãç£å(ãããmagnetization)ãšããã ãããã¯é»è·ã®éé»èªå°ãšå¯Ÿå¿ãããŠãç£åã®ããšãç£æ°èªå°(ããããã©ããmagnetic induction)ãšãããã ãããŠãéãã³ãã«ããããã±ã«ã®ããã«ãç£ç³ã«åŒãä»ããããããã«ç£åãããèœåãããç©äœã匷ç£æ§äœ(ãããããããããferromagnet)ãšããã éãšã³ãã«ããšããã±ã«ã¯åŒ·ç£æ§äœã§ããã
é
ã¯ç£åããªãããé
ã¯ç£ç³ã«åŒãã€ããããªãã®ã§ãé
ã¯åŒ·ç£æ§äœã§ã¯ãªãã
éé»èªå°ãå©çšãããéé»é®èœ(ããã§ããããžã)ãšèšããããäžç©ºã®å°äœãã€ãã£ãŠç©è³ªãå²ãããšã§å€éšé»å Žãé®èœããæ¹æ³ããã£ãã®ãšåæ§ã®ãç£æ°ã®é®èœãã匷ç£æ§äœã§ãåºæ¥ããäžç©ºã®åŒ·ç£æ§äœãçšããŠã匷ç£æ§äœã®å
éšã¯ç£å Žãé®èœã§ããããããç£æ°é®èœ(ãããããžããmagnetic shielding)ãšãããç£æ°ã·ãŒã«ããšãããã
ç£å Žã®åããåããããã«å³ç€ºããããç£ç³ã®äœãç£å Žã®æ¹åã¯ãç ã«å«ãŸããç éã®ç²æ«ãç£ç³ã«ãã¡ãã°ããŠããµããããããšã§èŠ³å¯ã§ããã
ãããå³ç€ºãããšãäžå³ã®ããã«ãªãã(ç»åçŽ æã®ç¢ºä¿ã®éœåäžãåçãšå³ç€ºãšã§ã¯ãN極ãšS極ãéã«ãªã£ãŠããŸããã容赊ãã ããã)
ãã®ãããªç£å Žã®å³ãç£åç·(ãããããããmagnetic line of force)ãšãããç£åç·ã®åãã¯ãç£ç³ã®N極ããç£åç·ãåºãŠãS極ã«ç£åç·ãåžåããããšå®çŸ©ããããæ£ç£ç³ã§ã¯ãç£åã®çºçæºãšãªãå Žæããæ£ç£ç³ã®äž¡ç«¯ã®å
端ä»è¿ã«éäžãããããã§ãæ£ç£ç³ã®äž¡ç«¯ã®å
端ä»è¿ãç£æ¥µ(ãããããmagnetic pole)ãšããã
ãã®ãããªç£ç³ã®ã€ããç£åç·ã®åœ¢ã¯ãé»æ°åç·ã§ã®ãç°ç¬Šå·ã®é»è·ã©ãããã€ããé»æ°åç·ã«äŒŒãŠããã
1ã€ã®æ£ç£ç³ã§ã¯N極(north pole)ã®ç£æ°ã®åŒ·ããšãS極(south pole)ã®ç£æ°ã®åŒ·ãã¯çããããŸããç£ç³ã«ã¯ãå¿
ãN極ãšS極ãšãååšãããN極ãšS極ã®ãã©ã¡ããçæ¹ã ããåãåºãããšã¯åºæ¥ãªããããšãæ£ç£ç³ãåæããŠããåæé¢ã«ç£æ¥µãåºçŸããããã®ãããªçŸè±¡ã®ãããçç±ã¯ãããããæ£ç£ç³ãæ§æãã匷ç£æ§äœã®ååã®1åãã€ãå°ããªç£ç³ã§ããããããå°ããªååã®ç£ç³ããããã€ãæŽåããŠã倧ããªæ£ç£ç³ã«ãªã£ãŠããããã§ããã
ä»®æ³çã«ãç£æ¥µãS極ãŸãã¯N極ã®çæ¹ã ãçŸããçŸè±¡ãçè«èšç®ã®ããã«èããããšããããããã®ãããªçåŽã ãã®ç£æ¥µãåç£æ¥µ(ã¢ãããŒã«ãšããã)ãšããããåç£æ¥µã¯å®åšããªãã
æ£ç£ç³ãªã©ããã®ãçåŽã®ç£æ¥µãããã®ãç£æ¥µããã®ç£å Žã®åŒ·ãã®ããšãããã®ãŸãŸãç£æ¥µã®åŒ·ãã(Magnetic charge)ãšåŒã¶ããããã¯ç£è·(ãããmagnetization)ãç£æ°éãšããã
ããããããã®ç£åãšç£å Žã®é¢ä¿ãåŒã§è¡šãããšãèããã ãŸããæ£ç£ç³ã«ã¯ç£æ¥µãäž¡åŽã«2åããã®ã§ãèšç®ãç°¡åã«ããããã«ãæ£ç£ç³ã®äž¡ç«¯ã®è·é¢ã倧ãããå察åŽã®ç£æ¥µã®å€§ãããç¡èŠã§ããç£ç³ãèãããã
ãã®ãããªç£ç³ãçšããŠãå®éšãããšããã次ã®æ³åãåãã£ããç£åã®åŒ·ãã¯2åã®ç©äœã®ç£æ°ém1ããã³m2ã«æ¯äŸãã2åã®ç©äœéã®è·é¢rã®2ä¹ã«åæ¯äŸããã åŒã§è¡šããšã
ã§è¡šãããã(kmã¯æ¯äŸå®æ°) ãããçºèŠè
ã®ã¯ãŒãã³ã®åã«ã¡ãªãã§ãç£æ°ã«é¢ããã¯ãŒãã³ã®æ³åãšãããç£æ°émã®åäœã¯ãŠã§ãŒããšãããèšå·ã¯[Wb]ã§è¡šãã
æ¯äŸå®æ°kmãš1ãŠã§ãŒãã®å€§ãããšã®é¢ä¿ã¯ã1ã¡ãŒãã«é¢ãã1wbã©ããã®ç£æ¥µã«ã¯ãããåãçŽ6.33Ã10ãšããŠã æ¯äŸä¿æ°kmã¯ã
ã§ããã
ã€ãŸãã
ã§ããã
éé»æ°åã«å¯ŸããŠãé»å Žãå®çŸ©ãããããã«ãç£æ°åã«å¯ŸããŠããå Žãå®çŸ©ããããšéœåãè¯ããç£æ°ém1[Wb]ãäœãã次ã®éãç£å Žã®åŒ·ããããã¯ç£å Žã®å€§ãããšèšããèšå·ã¯Hã§è¡šãã
ç£å Žã®åŒ·ãHã®åäœã¯[N/Wb]ã§ãããHãçšãããšãç£æ°ém2[Wb]ã«ã¯ãããç£æ°åf[N]ã¯ã
ãšè¡šããã
ç©çåŠè
ã®ãšã«ã¹ãããã¯ãé»æµã®å®éšãããŠããéã«ãããŸããŸè¿ãã«ãããŠãã£ãæ¹äœç£ç³ãåãã®ã確èªããã圌ã詳ãã調ã¹ãçµæã以äžã®ããšãåãã£ãã
é»æµãæµããŠãããšãã«ã¯ããã®ãŸããã«ã¯ãç£å Žãçãããåãã¯ãé»æµã®æ¹åã«å³ãããé²ãããã«ãå³ãããåãåããšåããªã®ã§ããããå³ããã®æ³åãšããã
ã¢ã³ããŒã«ããç£å Žã®å€§ããã調ã¹ãçµæãç£å Žã®å€§ããHã¯ãé»æµI[A]ãçŽç·çã«æµããŠãããšããçŽç·é»æµã®åšãã®ç£å Žã®å€§ããã¯ãå°ç·ããã®è·é¢ãa[m]ãšãããšãç£å Žã®å€§ããH[N/Wb]ã¯ã
ã§ããããšãç¥ãããŠããã
ãããã¢ã³ããŒã«ã®æ³å(Ampere's law) ãšããã ç£å Žã®å€§ããHã®åäœã¯ã[N/Wb]ã§ãããããã£ãœãã¢ã³ããŒã«ã®æ³åã®åŒãã¿ãã°ã¢ã³ãã¢æ¯ã¡ãŒãã«[A/m]ã§ãããã
å°ç·ãã³ã€ã«ç¶ã«å·»ãã°ãã¢ã³ããŒã«ã®æ³åã§å°ç·ã®åšå²ã«çºçããç£å Žãéãªãããããã®ããã«ããç£å Žã匷ããã³ã€ã«ãé»ç£ç³(ã§ããããããelectromagnet)ãšãããå°ç·ã«é»æµãæµããŠãããšãã«ã®ã¿ãé»ç£ç³ã¯ç£å Žãçºçãããå°ç·ã«é»æµãæµãã®ãæ¢ãããšãé»ç£ç³ã®ç£å Žã¯æ¶ããã
ç£å Žã®å€§ããHã«ã次ã®ç¯ã§æ±ãããŒã¬ã³ãåã®çŸè±¡ã®ãããæ¯äŸä¿æ°ÎŒ(åäœã¯ãã¥ãŒãã³æ¯ã¢ã³ãã¢ã§[N/A])ãæããŠãèšå·Bã§è¡šãã
ãšããããšãããããã®éBãç£æå¯åºŠ(magnetic flux density)ãšãããç£å Žã®å€§ããHã®åããšç£æå¯åºŠBã®åãã¯åãåãã§ããã ãŸããç£å Žã®å€§ããHãšç£æå¯åºŠBã®æ¯äŸä¿æ°ãéç£ç(ãšãããã€ãmagnetic permeability)ãšããã (ããŒã¬ã³ãåã«é¢ããŠã¯ã詳ããã¯ç©çIIã§æ±ããèªè
ãç©çIãåŠã¶åŠå¹Žãªãã°ãèªè
ã¯ãããŒã¬ã³ãåãšããåãããã®ã ãªã»ã»ã»ããšã§ãæã£ãŠããã°ããã)
ãŸããå°ç·ãçšæãããšãããããã®å°ç·ã¯ãéæ¢ããŠãããšããŠãéæ¢ããŠããããåºå®ã¯ããã«ãããå°ç·ã«åãå ããã°ãå°ç·ãåããããã«ããŠããšãããã
ãã®å°ç·ã«é»æµãæµããã ãã§ã¯ãã¹ã€ã«å°ç·ã¯åããªãããããããã®å°ç·ã«ãå€éšã®ç£ç³ã«ããç£å Žãå ãããšãå°ç·ãåãããã®ãããªãç£å Žãšé»æµã®çžäºäœçšã«ãã£ãŠãå°ç·ã«çããåãããŒã¬ã³ãå(ããŒã¬ã³ãããããè±: Lorentz force)ãšããã
ããŒã¬ã³ãåã®åãã¯ãå°ç·ã®é»æµã®åããšç£å Žã®åãã«åçŽã§ãããé»æµIã®åãããç£æå¯åºŠBã®åãã«å³ãããåãåããšåãã§ããã
ãŸããããŒã¬ã³ãåã®å€§ããã¯ãå°ç·ã®é·ãlãšãç£å Žã®å°ç·ãšã®åçŽæ¹åæåã«æ¯äŸããã
ããŒã¬ã³ãåã®å€§ããF[N]ãåŒã§è¡šãã°ãé»æµãšç£å ŽãšãåçŽã ãšããŠãç£å ŽãåããŠããå°ç·ã®åœ¢ç¶ãçŽç·åœ¢ã ãšããŠãé»æµãI[A]ãšããŠãå°ç·ã®é·ããl[m]ãšããŠãå°ç·ã«ããã£ãŠããå€éšç£å Žã®ç£æå¯åºŠãB[N/(Aã»m)]ãšããã°ã
ã§è¡šããã
ããŒã¬ã³ãåã®å
¬åŒã«ãã¯ãŒãã³ã®æ³åãªã©ã§ã¯èŠããããããªæ¯äŸä¿æ°(ä¿æ°Kãªã©ã)ãå«ãŸããªãã®ã¯ãããããããã®ããŒã¬ã³ãåã®çŸè±¡ãå
ã«ãç£æ°éãŠã§ãŒãWbã®åäœããã³ç£æå¯åºŠBã®åäœãã決å®ãããŠããããã§ããã
ãŸãããç£æå¯åºŠãã®å称ãããç£æãã»ãå¯åºŠããšããã®ã¯ãå®ã¯ç£æå¯åºŠã®åäœã®[N/(Aã»m)]ã¯ãåäœãåŒå€åœ¢ãããš[Wb/m]ã§ãããããšãç±æ¥ã§ããããã®åäœ[Wb/m]ããé»æ°å·¥åŠè
ã®ãã¹ã©ã®åã«ã¡ãªã¿ãåäœ[Wb/m] ããã¹ã©ãšèšããèšå·Tã§è¡šãã
ãã®ããŒã¬ã³ãåã®çŸè±¡ããé»æ°æ©åšã®ã¢ãŒã¿(é»åæ©)ã®åçã§ããã
ãªããããã¬ãã³ã°ã®æ³åããšããããŒã¬ã³ãåã«é¢ããæ³åãããããããŒã¬ã³ãåã®èšç®ã«ã¯å®çšçã§ã¯ç¡ããããã¬ãã³ã°ã®åãé¢ããç°ãªãæ³åã幟ã€ããã£ãŠçŽããããééãã®åå ã«ãªããããã®ã§ãæ¬æžã§ã¯æããªãã å®éã«ãå°éçãªç©çèšç®ã§ã¯ããã¬ãã³ã°ã®æ³åã¯ãèšç®ã«ã¯çšããªãã
ãããããã¬ãã³ã°ã®æ³åã«ã¯ããã¬ãã³ã°ã®å³æã®æ³åããšããããšã¯ç°ãªãããã¬ãã³ã°ã®å·Šæã®æ³åãããããã©ã¡ãããã©ã®ç£æ°ã®çŸè±¡ã«çšããæ³åã ã£ãã®ããééãããããã ãããæ¬æžã§ã¯æããªãã
(é»ç£èªå°ã«é¢ããŠã¯ã詳ããã¯ç©çIIã§æ±ãã)
ã¢ã³ããŒã«ã®æ³åã§ã¯ãé»æµã®åšãã«ç£å Žãã§ããã®ã§ãã£ãã
å®ã¯ãç£ç³ãåãããªã©ããŠãç£å Žã䌎ãç©äœãéåãããšããã®ãŸããã«ã¯é»å Žãçããã ä»®ã«ãã³ã€ã«ã®è¿ãã§ãããè¡ãªã£ããšãããšãçããé»å Žã«ãã£ãŠã³ã€ã«ã®äžã«ã¯é»æµãæµããã çããé»å Žã®å€§ããã¯ã
ãšãªãã(ååŸaã®å圢ã®ã³ã€ã«ã®å Žåã) Eã®åäœã¯[V/m]ã§ãããBã®åäœã¯[T]ã§ããã
ãã®çŸè±¡ãé»ç£èªå°(ã§ããããã©ããelectromagnetic induction)ãšãããé»ç£èªå°ã«ãã£ãŠçºçããé»æµãèªå°é»æµãšããã
ãŸããèªå°é»æµã®åãã¯ãç£ç³ã®åãã«ãããã³ã€ã«ã®äžãéãç£æã®å€åã劚ããåãã«ãé»æµãæµããã(èªå°é»æµãã¢ã³ããŒã«ã®æ³åã«åŸããåšå²ã«ç£å Žãäœãã) ãã®èªå°é»æµããã³ã€ã«ã®äžãéãç£æã®å€åã劚ããåãã«èªå°é»æµãæµããçŸè±¡ãã¬ã³ãã®æ³å(Lenz's law)ãšããã
åãé åã«Nåå·»ãããã³ã€ã«ã眮ãããå Žåããã¡ã©ããŒã®é»ç£èªå°ã®æ³åã¯ã次ã®ããã«ãªãã
ããã§ã E {\displaystyle {\mathcal {E}}} ã¯èµ·é»å(ãã«ã ãèšå·ã¯V)ãΊB ã¯ç£æ(ãŠã§ãŒããèšå·ã¯Wb)ãšãããNã¯é»ç·ã®å·»æ°ãšããã
ãã®é»ç£èªå°ã®çŸè±¡ããç«åçºé»ãæ°Žåçºé»ãªã©ã®çºé»æ©ã®åçã§ãããããçã®çºé»ã§ã¯ãæ°žä¹
ç£ç³ãå転ãããããšã§ãçºé»ãããŠãããç«åãæ°Žåãšããã®ã¯ãæ©åšã®å転ãåŸãæ段ã«ãããªãããŸããçºé»æã®çºé»ã«ã¯ãæ°žä¹
ç£ç³ã®å転ãå©çšããŠãããããçºçããé»å§ãé»æµã¯åšæçãªæ³¢åœ¢ã«ãªãã次ã«èª¬æãã亀æµæ³¢åœ¢ã«ãªãã
åè·¯ãžã®å
¥åé»å§ãåšæçã«æéå€åããåè·¯ã®é»å§ããã³é»æµã亀æµ(alternating current)ãšãããããã«å¯Ÿãã也é»æ± ãªã©ã«ãã£ãŠçºçããé»å§ãé»æµã®ããã«ãæéã«ãããäžå®ãªé»å§ãé»æµã¯çŽæµ(direct Current)ãšããã
亀æµæ³¢åœ¢ãäœç§ã§1åšããããšããæéãåšæ(wave period)ãšãããåšæã®èšå·ã¯ T {\displaystyle T} ã§è¡šãåäœã¯ç§[s]ã§ããã
1ç§éã«æ³¢åœ¢ãäœåšããããšããåæ°ãåšæ³¢æ°ãããã¯æ¯åæ°(è±èªã¯ããšãã«frequency)ãšããã é»æ°ã®æ¥çã§ã¯åšæ³¢æ°ãšããçšèªãçšããããšãå€ããç©çã®æ³¢ã®çè«ã§ã¯æ¯åæ°ãšããè¡šçŸãçšããããšãå€ãã
åšæ³¢æ°ã®åäœã¯[1/s]ã§ããããããããã«ã(hertz)ãšããåäœã§è¡šããåäœèšå·HzãçšããŠåšæ³¢æ°fããf[Hz]ãšãããµãã«è¡šãã
亀æµé»æµã亀æµé»å§ãæ£åŒŠæ³¢ã®å Žåã¯ããããã®ãã©ã¡ãŒã¿ãçšããŠ
ãšæžãããšãã§ããã sinãšã¯äžè§é¢æ°ã§ãããç¥ããªããã°æ°åŠIIãªã©ãåèã«ããã ãã®ãšãã®sinã®ä¿æ° I 0 {\displaystyle I_{0}} ã V 0 {\displaystyle V_{0}} ãæ¯å¹
(ããã·ããamplitude)ãšããããŸãæå»t=0ã«ãããé»æµãé»å§ã®å€ã瀺ããæé波圢ã決å®ãã Ξ i {\displaystyle \theta _{i}} ã Ξ v {\displaystyle \theta _{v}} ãåæäœçžãšããã
æ®éç§é«æ ¡ã®é«æ ¡ç©çã§ã¯ã亀æµæ³¢åœ¢ã®èšç®ã«ã¯ãæ£åŒŠæ³¢ã®å Žåãäž»ã«æ±ããæ¹åœ¢æ³¢ãäžè§æ³¢ã®èšç®ã¯ãæ®éã¯æ±ãããªãã ãã ããå·¥æ¥é«æ ¡ã®ææ¥ããå·¥å Žã®å®åã§ã¯æ±ãããšãããã®ã§ãèªè
ã¯æ³¢åœ¢ãåŠãã§ããããšã
çºé»æããäžè¬å®¶åºã«éãããŠããé»å§ã¯äº€æµé»å§ã§ãããæ±æ¥æ¬ã§ã¯50Hzã§ããã西æ¥æ¬ã§ã¯60Hzã§ãããããã¯ææ²»æ代ã®çºé»æ©ã®èŒžå
¥æã«ãæ±æ¥æ¬ã®äºæ¥è
ã¯ãšãŒããããã50Hzçšã®çºé»æ©ã茞å
¥ãã西æ¥æ¬ã®äºæ¥è
ã¯ã¢ã¡ãªã«ãã60Hzã®çºé»æ©ã茞å
¥ããããšã«ããã
çºé»æããäžè¬ã®å®¶åºãªã©ã«éãããé»æµã®åšæ³¢æ°ãåçšåšæ³¢æ°ãšããã
åçšé»æºã®é»å§æ¯å¹
ã¯çŽ140Vã§ããããã㯠100 à 2 {\displaystyle 100\times {\sqrt {2}}} [V]ã§ããã
ãããã«ããšã¯1000Hzã®ããšã§ããããããã«ãã¯kHzãšæžãã
亀æµé»æµã«å¯ŸããŠã¯ãé»æµãšåãæ¯åæ°ã§ãã¢ã³ããŒã«ã®æ³åã§çºçããç£å Žãæ¯åããã
å°ç·ã§ã€ããããã³ã€ã«ã¯ãçŽæµé»æµã§ã¯ããã ã®å°ç·ãšããŠã¯ãããããããã亀æµé»æµã«å¯ŸããŠã¯ãé»ç£èªå°ã«ããèªå·±ã®çºçãããç£å Žã劚ãããããªé»æµããã³èµ·é»åãçºçããããããèªå·±èªå°(self induction)ãšããã
èªå·±èªå°ã«ããèµ·é»åã®å€§ããã¯ãé»æµã®æéå€åçã«æ¯äŸãããèªå·±èªå°ã®èµ·é»åãåŒã§æžãã°ãæ¯äŸä¿æ°ãLãšããŠã
ã§ããã ãã®æ¯äŸä¿æ° L {\displaystyle L} ãèªå·±ã€ã³ãã¯ã¿ã³ã¹(self inductance)ãšãããèªå·±ã€ã³ãã¯ã¿ã³ã¹ã®æ¬¡å
ã¯[Vã»S/m]ã ããããããã³ãªãŒãšããåäœã§è¡šããåäœã«Hãšããèšå·ãçšããã
éå¿ã«äºã€ã®ã³ã€ã«ãå·»ããã³ã€ã«ã®çæ¹ã®é»æµãå€åããããšãã¢ã³ããŒã«ã®æ³åã«ãã£ãŠçããŠããç£æãå€åãããããå察åŽã®ã³ã€ã«ã«ã¯ããã®ç£æå¯åºŠã®å€åãæã¡æ¶ããããªåãã«èµ·é»åãçºçããããã®çŸè±¡ãçžäºèªå°(mutual induction)ãšèšãã
é»å§ãå
¥åãããåŽã®ã³ã€ã«ã1次ã³ã€ã«(primaly coil)ãšèšããèªå°èµ·é»åãçºçãããåŽã®ã³ã€ã«ã2次ã³ã€ã«(secondary coil)ãšããã
çžäºèªå°ã«ããèµ·é»åã®å€§ããã¯ãé»æµã®æéå€åçã«æ¯äŸãããçžäºèªå°ã®èµ·é»åãåŒã§æžãã°ãæ¯äŸä¿æ°ãMãšããŠã(çžäºèªå°ã®æ¯äŸä¿æ°ã¯Lã§ã¯ç¡ãã)åŒã¯ã
ã§ããã ãã®æ¯äŸä¿æ° M {\displaystyle M} ãçžäºã€ã³ãã¯ã¿ã³ã¹(self inductance)ãšãããçžäºã€ã³ãã¯ã¿ã³ã¹ã®æ¬¡å
ã¯ãèªå·±ã€ã³ãã¯ã¿ã³ã¹ã®åäœãšåãã§ãã³ãªãŒ(H)ã§ããã
ãã®çžäºã€ã³ãã¯ã¿ã³ã¹ã®å€§ããã¯ãäž¡æ¹ã®ã³ã€ã«ã®å·»ãæ°ã©ããã®ç©ã«æ¯äŸããã
ç£å Žã®åãã«ãã£ãŠé»å ŽãåŒãèµ·ããããããšãé»ç£èªå°ã®ã»ã¯ã·ã§ã³ã§èŠãã å®éã«ã¯é»å Žã®å€åã«ãã£ãŠç£å ŽãåŒãèµ·ããããããšãç¥ãããŠããã ããã«ãã£ãŠäœããªã空éäžãé»å Žãšç£å ŽãäŒæããŠããããšãäºæ³ãããã
é»ç£æ³¢ã®é床ãç©çåŠè
ã®ãã¯ã¹ãŠã§ã«ãèšç®ã§æ±ãããšãããé»ç£æ³¢ã®é床ã¯ãç空äžã§ã¯åžžã«äžå®ã§ããã€æ³¢ã®é床cãèšç®ã§æ±ãããšããã
ãšãªããæ¢ã«ç¥ãããŠããå
éã«äžèŽããã ãã®ããšãããå
ã¯é»ç£æ³¢ã®äžçš®ã§ããããšãåãã£ããç©çIIã§ãé»ç£æ³¢ã®é床ãæ±ããèšç®ã¯ã詳ããã¯æ±ãã èªè
ãå
éã®æž¬å®å®éšã«ã€ããŠèª¿ã¹ããªããç©çIã®æ³¢åã«é¢ããããŒãžãªã©ã§ãã£ãŸãŒã®å®éšã«ã€ããŠãåç
§ã®ããšã
æ³¢ã¯æ³¢é·Î»ãé·ãã»ã©ãæ¯åæ°fãå°ãããªããæ³¢ã®æ³¢é·Î»ãšæ¯åæ°fã®ç©fλã¯äžå®ã§ãããã¯æ³¢ã®é床vã«çãããã€ãŸã
ã§ããã é»ç£æ³¢ã®å Žåã¯ãé床ãå
éã®cãªã®ã§
ã§ããã
æŸéçšã®ãã¬ããã©ãžãªã®é»æ³¢(ã§ãã±ãradio wave)ã¯ãé»ç£æ³¢(electromagnetic wave)ã®äžçš®ã§ãããæ³¢é·ã0.1mm以äžã®é»ç£æ³¢ãé»æ³¢ã«åé¡ãããããªããé»æ³¢ã®ãã¡ãæ³¢é·ã1mm~1cmã®ããªã¡ãŒãã«ã®é»æ³¢ãããªæ³¢ãšãããåæ§ã«ãæ³¢é·ã1cm~10cmã®é»æ³¢ãã»ã³ãæ³¢ãšãããæ³¢é·10cm~100cm(=1m)ã®é»æ³¢ã¯UHFãšèšããããã¬ãæŸéãªã©ã«äœ¿ãããUHFæŸéã¯ããã®é»æ³¢ã§ãããæ³¢é·1m~10mã®é»æ³¢ã¯VHFãšèšãããããã¬ãæŸéã®VHFæŸéã¯ããã®é»æ³¢ã§ããã
æ³¢é·ã0.1mm以äžã§ãå¯èŠå
ç·(å¯èŠå
ã®æ倧波é·ã¯780ããã¡ãŒãã«çšåºŠ)ãããã¯æ³¢é·ãé·ãé»ç£æ³¢ã¯èµ€å€ç·(ãããããããinfrared raysãã€ã³ãã©ã¬ãŒã ã¬ã€ãº)ãšããããèµ€ãã®ãå€ããšããçç±ã¯ãå¯èŠå
ã®æ倧波é·ã®è²ãèµ€è²ã ããã§ãããèµ€å€ç·ãã®ãã®ã«ã¯è²ã¯ã€ããŠããªããåžè²©ã®èµ€å€ç·ããŒã¿ãŒãªã©ãèµ€è²ã«çºå
ãã補åãããã®ã¯ã䜿çšè
ãåäœç¢ºèªãã§ããããã«ããããã«ã補åã«èµ€è²ã®ã©ã³ãã䜵眮ããŠããããã§ãããèµ€å€ç·ã¯ãç©äœã«åžåããããããåžåã®éãç±ãçºçããã®ã§ãããŒã¿ãŒãªã©ã«å¿çšãããããªãã倪éœå
ã«ãèµ€å€ç·ã¯å«ãŸããã
ããããèµ€å€ç·ãçºèŠãããçµç·¯ã¯ãã€ã®ãªã¹ã®å€©æåŠè
ã®ããŒã·ã§ã«ã倪éœå
ãããªãºã ã§åå
ããéã«ãèµ€è²ã®å
ç·ã®ãšãªãã®ãç®ã«ã¯è²ãèŠããªãéšåã枩床äžæããŠããããšãçºèŠããããšããçµç·¯ãããã
æã
ã人éã®ç®ã«èŠããå¯èŠå
ç·(ãããããããvisible light)ã®æ³¢é·ã¯ãçŽ780ããã¡ãŒãã«ããçŽ380ããã¡ãŒãã«ã®çšåºŠã§ãããå¯èŠå
ã®äžã§æ³¢é·ãæãé·ãé åã®è²ã¯èµ€è²ã§ãããå¯èŠå
ã®äžã§æ³¢é·ãæãçãé åã®è²ã¯çŽ«è²ã§ããã
å
ãã®ãã®ã«ã¯ãè²ã¯ã€ããŠããªããæã
ã人éã®è³ããç®ã«å
¥ã£ãå¯èŠå
ããè²ãšããŠæããã®ã§ããã 倪éœå
ãããªãºã ãªã©ã§åå
(ã¶ããã)ãããšãæ³¢é·ããšã«è»è·¡(ããã)ãããããããã®åå
ããå
ç·ã¯ãä»ã®æ³¢é·ãå«ãŸãããã äžçš®ã®æ³¢é·ãªã®ã§ããã®ãããªå
ç·ããã³å
ãåè²å
(monochromatic light)ãšããã ãŸããçœè²ã¯åè²å
ã§ã¯ãªããçœè²å
(white light)ãšã¯ãå
šãŠã®è²ã®å
ãæ··ãã£ãç¶æ
ã§ããã åæ§ã«ãé»è²ãšããåè²å
ããªããé»è²ãšã¯ãå¯èŠå
ãç¡ãç¶æ
ã§ããã
玫å€ç·(ããããããultraviolet rays)ã¯ååŠåå¿ã«åœ±é¿ãäžããäœçšã匷ãã殺èæ¶æ¯ãªã©ã«å¿çšãããã倪éœå
ã«ã玫å€ç·ã¯å«ãŸããã人éã®èã®æ¥çŒãã®åå ã¯ã玫å€ç·ãã¡ã©ãã³è²çŽ ãé
žåãããããã§ããã
èµ€å€ç·ã¯å€ªéœå
ã®ããªãºã ã«ããåå
ã§çºèŠãããã ãã§ã¯ãåå
ããã玫è²ã®å
ç·ã®ãšãªãã«ãããªã«ãç®ã«ã¯èŠããªãç·ãããã®ã§ã¯?ããšãããµããªããšãåŠè
ãã¡ã«ãã£ãŠèãããã ãã€ãã®ç©çåŠè
ãªãã¿ãŒã«ããååŠçãªå®éšæ¹æ³ãçšããŠã玫å€ç·ã®ååšãå®èšŒãããã
å»ççšã®ã¬ã³ãã²ã³ãªã©ã®ééåçã§çšããããXç·(X-ray)ãé»ç£æ³¢ã®äžçš®ã§ãããçç©ã®çŽ°èãååã¬ãã«ã§å·ã€ããçºããæ§ãæãã ã¬ã³ãç·(gammaârayãγ ray)ãåæ§ã«ãééåçã«ãå¿çšãããããçç©ã®çŽ°èãååã¬ãã«ã§å·ã€ããçºããæ§ãæãã
?? | [
{
"paragraph_id": 0,
"tag": "p",
"text": "é«çåŠæ ¡ ç©çåºç€ > é»æ°",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "æ¬é
ã¯é«çåŠæ ¡ ç©çåºç€ã®é»æ°ãšç£æ°ã®è§£èª¬ã§ããã",
"title": ""
},
{
"paragraph_id": 2,
"tag": "p",
"text": "çŸåšç§ãã¡ã䜿ã£ãŠããå€ãã®è£œåãé»æ°ãçšããŠåããŠããã ããã«ã¯æ§ã
ãªçç±ãèãããããããŸã第äžã«é»æ°ã¯æ§ã
ãªå¥ã®ãšãã«ã®ãŒã«å€æã§ããããšãäŸãã°é»ç±ç·ã䜿ã£ãŠç±ã«ãé»çãçºå
ãã€ãªãŒãã䜿ã£ãŠå
ã«ãé»åæ©ã䜿ã£ãŠéåã«å€æããããšãåºæ¥ãã次ã«ãé»æ± ãã³ã³ãã³ãµã䜿ã£ãŠãšãã«ã®ãŒãç¶æãããŸãŸæã¡éã¶ããšãåºæ¥ãããšããé»ç·ã䜿ã£ãŠé·è·é¢ãéé»ã§ããããšããŸããé»å補åã®èšç®èœåãä¿¡å·ã®äŒéèœåãåªããŠããããšããŸãæ¯èŒçã«å®å
šã«å°éã®ãšãã«ã®ãŒãåãåºããããšãçãèããããã",
"title": "é»æ°"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "",
"title": "é»æ°"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "é»æ°ãéåã«å€ãããã®ãšããŠé»åæ©(è±: electric motor)ããããããã®éã®ãã®ãšããŠéåãé»æ°ã«å€ããããšãåºæ¥ãããããè¡ãªãã®ã¯çºé»æ©è±: generatorãšåŒã°ãããçºé»æã¯äœããã®éåã®ãšãã«ã®ãŒãå©çšããŠé»æ°ãããããŠãããäŸãã°ãæ°Žåçºé»æã§ã¯ãæ°Žã®èœäžããåãå©çšããŠããã倧éã®æ°Žãèœäžãããšãã«ã¯äººéãäœå人ãããã£ãŠããããåºæ¥ãããšããªãããããšããããäŸãã°ãåãç«ã£ã海岞ç·ãªã©ã¯äž»ã«æ°Žã®æµãã«ãã£ãŠäœãããŠããããã®ããã«ãæ°Žã®åã¯åŒ·å€§ã§ããã®ã§ããããäžæãå©çšããæ¹æ³ããããšéœåããããå®éçŸä»£ã§ã¯é»æ°ãåªä»ãšããŠããã®åãåãã ãããšã«æåããŠããã",
"title": "é»æ°"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "é»æ± ã®ããã«é»æ¥µã®+ãš-ãå®ãŸã£ãé»æµãçŽæµé»æµæãã¯çŽæµ(è±: direct current)ãšåŒã¶ãäžæ¹ãçºé»æããåŸãããé»æµã®ããã«+ãš-ãéãé床ã§å
¥æããé»æµã亀æµé»æµæãã¯äº€æµ(è±: alternating current)ãšåŒã¶ã",
"title": "é»æ°"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "å®éã«ã¯ãã€ãªãŒããçšã㊠亀æµãçŽæµã«å€ã㊠䜿ãããšãããè¡ãªãããã",
"title": "é»æ°"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "äœããªã空éãå
ãçŽé²ããŠããããã« èŠããããšããããå®éã«ã¯ãã㯠é»æ³¢ãšåããã®ã§ããã é»æ³¢ãšã¯äŸãã°ãæºåž¯é»è©±ã®éä¿¡ã«äœ¿ããããã®ã§ããã é»è·ãæã£ãç©äœãåãããšãå¿
ç¶çã« çãããã®ã§ããã",
"title": "é»æ°"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "ãã©ã¹ããã¯ã®äžæ·ããªã©ã§é«ªãããããšåž¯é»ããçŸè±¡ãªã©ã®ããã«ãç©è³ªãé»æ°ã垯ã³ãããšã垯é»(ããã§ã)ãšãããç©äœãããã£ãŠçºçãããéé»æ°ãæ©æŠé»æ°ãšããã ã¬ã©ã¹æ£ãçµ¹ã®åžã§ããããšãã¬ã©ã¹æ£ã¯æ£ã®é»æ°ã«åž¯é»ããçµ¹ã¯è² ã®é»æ°ã«åž¯é»ããã é»æ°ã®éãé»è·(ã§ãããcharge)ãšããããããã¯é»æ°éãšããã",
"title": "éé»æ°"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "é»è·ã®åäœã¯ã¯ãŒãã³ã§ãããã¯ãŒãã³ã®èšå·ã¯Cã§ããã",
"title": "éé»æ°"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "éé»æ°ã«ããé»è·ã©ããã«åãåãéé»æ°åãšããã",
"title": "éé»æ°"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "ãªãã垯é»ããŠããªãç¶æ
ãé»æ°çã«äžæ§ã§ããããšããã",
"title": "éé»æ°"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "éå±ã®ããã«ãé»æ°ãéããç©äœãå°äœ(ã©ããããconductor)ãšããããã©ã¹ããã¯ãã¬ã©ã¹ããŽã ã®ããã«é»æ°ãéããªãç©è³ªã絶çžäœ(ãã€ãããããinsulator)ãããã¯äžå°äœ(ãµã©ããã)ãšããã",
"title": "éé»æ°"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "éå±ã¯å°äœã§ããã",
"title": "éé»æ°"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "é»æ°ã®æ£äœã¯é»å(electron)ãšããç²åã§ããããã®é»åã¯è² é»è·ã垯ã³ãŠããã(é»åã®é»è·ãè² ã«å®çŸ©ãããŠããã®ã¯ã人é¡ãé»åãçºèŠããåã«é»è·ã®æ£è² ã®å®çŸ©ãè¡ãããããšããé»åãèŠã€ãã£ãéã«é»åã®é»è·ã調ã¹ããè² é»è·ã ã£ãããã§ããã)",
"title": "éé»æ°"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "éå±ãå°äœãªã®ã¯ãéå±äžã®é»åã¯ãããšã®ååãé¢ããŠããã®éå±å
šäœã®äžãèªç±ã«åããããã§ãããéå±äžã®é»åã®ããã«ãç©è³ªäžãèªç±ã«åããç¶æ
ã®é»åããèªç±é»å(ãããã§ãã)ãšããã",
"title": "éé»æ°"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "é»æµãšã¯ãèªç±é»åã移åããããšã§ããã",
"title": "éé»æ°"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "ãã£ãœãã絶çžäœã¯ãèªç±é»åããããªãã絶çžäœã®é»åã¯ããã¹ãŠãããšã®ååã«æçž(ããã°ã)ãããŠéã蟌ããããŠããŠãèªç±ã«ã¯åããªãã",
"title": "éé»æ°"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "æ£é»è·ãšã¯ãç©è³ªã«é»åãæ¬ ä¹ããŠããç¶æ
ã§ããã è² é»è·ãšã¯ãç©è³ªãé»åãå€ãæã£ãŠããç¶æ
ã§ããã",
"title": "éé»æ°"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "垯é»ããŠããªã絶çžäœã®ç©è³ªããããããããŠãäž¡æ¹ãæ©æŠé»æ°ã«åž¯é»ãããå Žåãçæ¹ã¯æ£é»è·ãçããããçæ¹ã®ç©è³ªã¯è² é»è·ãçããããã®ãšããçºçããæ£é»è·ã®å€§ãããšè² é»è·ã®å€§ããã¯åãã§ããã ããã¯ãé»åã移åããŠãçæ¹ã®ç©è³ªã¯é»åãäžè¶³ããããçæ¹ã¯çéã®é»åãéå°ã«ãªã£ãŠããããã§ããã",
"title": "éé»æ°"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "ãã®ããã«ãé»åã¯çæãæ¶æ»
ãããªãããããé»è·ä¿ååãããã¯é»æ°éä¿ååãšèšãã",
"title": "éé»æ°"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "é»æ°çã«äžæ§ã§ãã£ãå°äœã®ç©è³ª(ä»®ã«ç©è³ªAãšãã)ã«åž¯é»ããå¥ã®ç©è³ª(ä»®ã«ç©è³ªBãšãã)ãæ¥è§Šãããã«è¿ã¥ãããšãç©è³ªAã«ã¯ã垯é»ç©è³ªBã®é»è·ã«åŒãå¯ããããŠãç©äœAã®å
éšã§å察笊å·ã®é»è·ã垯é»ç©äœBã«è¿ãåŽã®è¡šé¢ã«çããããŸãã垯é»ç©äœBãšåãé»è·ã¯åçºããã®ã§ãç©äœAå
éšã®åž¯é»ç©äœBãšã¯é ãåŽã®è¡šé¢ã«çããã",
"title": "éé»æ°"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "ãã®ãããªçŸè±¡ãéé»èªå°(ããã§ãããã©ã;Electrostatic induction)ãšãããéé»èªå°ã§çããé»è·ã®æ£é»è·ã®éãšè² é»è·ã®éã¯çéã§ããã(é»æ°éä¿åã®æ³å)",
"title": "éé»æ°"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "å°äœã®å
éšã«éé»æ°åã¯ç¡ãããããã£ããšãããšãèªç±é»åãªã©ã®é»è·ãåããé»æµãæµãç¶ããããšã«ãªããããã®ãããªçŸè±¡ã¯å®åšããªãã®ã§äžåçã«ãªãããããã£ãŠãå°äœã®å
éšã«éé»æ°åã¯ç¡ãã",
"title": "éé»æ°"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "è¡šé¢ã«é»è·ãéãŸãã®ã¯ãå°äœã®å
éšã«éé»æ°åãäœãããªãããã§ããããããã£ãŠéé»èªå°ã§åŒãå¯ããããé»è·ã®å€§ããã¯ãå€éšããå°äœå
éšãžã®éé»æ°åãæã¡æ¶ãã ãã®å€§ããã§ããã",
"title": "éé»æ°"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "ãã®å°äœå
éšã®é»è·ããŒãã«ãªãæ§è³ªãå¿çšãããšãäžç©ºã®å°äœã§åºæ¥ãç©äœãçšããŠãéé»æ°åãé®èœããããšãã§ããããããéé»é®èœ(ããã§ããããžããelectric shilding)ãšããã",
"title": "éé»æ°"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "絶çžäœ(ä»®ã«Aãšãã)ã«é»è·ãè¿ã¥ããå Žåã¯ãå°äœãšã¯éããç©äœAã®å
éšã®é»åã¯èªç±ã«è¡šé¢ã«ã¯éãŸããªãããç©äœå
éšã®ååã®æ£è² ã®é»è·ã®æ¥µæ§ãæã£ãéšåããå€éšã®éé»æ°åã«åŒãå¯ããããããã«ãè¿ã¥ããé»è·ã«è¿ãåŽã«ã¯ç°çš®ã®é»è·ãçããé ãåŽã«ã¯ãåçš®ã®é»è·ãçããã ååãååãå€éšã®éé»æ°åã«ãã£ãŠãæ£è² ã®é»è·ã®éšåãçããããšãå極(ã¶ãããã)ãšããããå€éšã®é»åã«ãã£ãŠèµ·ããããã®ãããªå極ã®ããããèªé»å極(ããã§ãã¶ãããããdielectric polarization)ãšããã",
"title": "éé»æ°"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "絶çžäœã¯ãéé»æ°åã«ããããããšèªé»å極ãè¡ãã®ã§ã絶çžäœã®ããšãèªé»äœ(ããã§ããããdielectric)ãšãããã",
"title": "éé»æ°"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "å°äœã«éé»èªå°ãããæ£è² ã®é»è·ã¯ãå°äœãåæãªã©ãããã°æ£é»è·ãšè² é»è·ãå¥åã«åãåºãããšãã§ããããããèªé»äœã®æ£è² ã®é»è·ã¯ãååãååãšå¯æ¥ã«çµã³ã€ããŠãããããæ£è² ã®é»è·ãåãããŠåãåºãããšã¯åºæ¥ãªãã",
"title": "éé»æ°"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "",
"title": "éé»æ°"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "ããç©è³ªãé»æ°ã垯ã³ãŠãã(垯é»ããŠãã)ãšãããã®åž¯é»ã®å€§å°ã®çšåºŠãé»è·(ã§ãããelectric charge)ãšãããããŸããŸãªç©è³ªããããããªæ¹æ³ã§åž¯é»ãããçµæãé»è·ã«ã¯ã垯é»ãã2åã®ãã®ã©ãããè¿ã¥ããæã«åŒã£åŒµãåããã®(åŒåãåã)ãšåçºããããã®(æ¥åãã¯ããã)ã®2çš®é¡ãããããšãåãã£ãã ãã®ãããªã垯é»ããŠããç©äœã«åãåãéé»æ°åãšããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "ã¹ã€ã®åž¯é»ãããã®ããä»ã«ãããã€ãçšæããŠãè¿ã¥ããŠå®éšãã2åã®ç©äœã®çµã¿åãããå€ãããšãçµã¿åããã«ãã£ãŠã2åã®ç©äœã©ããã«åŒåãåãå Žåãããã°ãæ¥åãåãå Žåãããããšãåãã£ãããã®åŒåãšæ¥åã®é¢ä¿ã¯ã垯é»ããé»è·ã®çš®é¡ã«å¿ããããšãããã£ãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "çµè«ãèšããšãé»è·ã«ã¯æ£è² ã®2çš®é¡ããããæ£ã®é»è·ã©ããã®ç©äœãè¿ã¥ãããšãã¯åçºããããè² ã®é»è·ã©ãããè¿ã¥ãããšããåçºããããæ£ãšè² ã®é»è·ãè¿ã¥ããæã«ã¯åŒåãåãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "ã€ãŸããå笊å·ã®é»è·ãè¿ã¥ããå Žåã¯ãåçºåãçãããç°ç¬Šå·ã®é»è·ãè¿ã¥ããå Žåã¯ãåŒåãçããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "éé»æ°ã©ããã®åã®åŒ·ãã¯ãå®éšçã«ã¯ãé»è·ã®éã«åãåã¯ãéåã®å Žåãšåæ§ã«åãåãŒãåã2ç©äœã®éã®è·é¢ã®2ä¹ã«åæ¯äŸããããšãç¥ãããŠãããæŽã«ãé»è·ã®å€§ããã倧ããã»ã©é»è·éã«åãåã倧ããããšãèæ
®ãããšãè·é¢rã ãé¢ããŠãããããé»è· q 1 {\\displaystyle q_{1}} ã q 2 {\\displaystyle q_{2}} ãæã£ãŠãã2ç©äœã®éã«åãåFã¯ã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "ã§äžããããããããã¯ãŒãã³ã®æ³å( Coulomb's law)ãšãããããã§ã k {\\displaystyle k} ã¯æ¯äŸå®æ°ã§ãããäž¡é»è·ã®åšå²ã«ããç©äœã®çš®é¡ã«ããå€åããå®æ°ã§ãããç空äžã§ã®é»å Žãèããå Žåã®kã®å€ã¯ã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "ã§ããããŸãã ε {\\displaystyle \\epsilon } ã¯åŸã»ã©ç»å Žããèªé»ç(ããã§ããã€)ãšåŒã°ããç©çå®æ°ã§ãããèªé»çã¯ãäž¡é»è·ã®åšå²ã«ããç©äœã®çš®é¡ã«ããå€åããå®æ°ã§ãããèªé»çã«ã€ããŠã¯ããã®æãåããŠèªãã 段éã§ã¯ããŸã ç¥ããªããŠãè¯ããã®ã¡ã«ç©çIIã§èªé»çã詳ãã解説ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "èªé»ç ε {\\displaystyle \\epsilon } ãšã¯ãŒãã³ã®æ¯äŸå®æ°kã«ã¯äžåŒã®é¢ä¿",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "ãããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "ç©äœã®ãŸããã«èç©ããããã®ãé»è·ãšåŒã¶ãé»æ°åã«ãã£ãŠåçºããã£ãããåŒãã€ããã£ããããç©äœãé»è·ãæã€ç©äœãšåŒã¶ããŸããããã§èŠ³å¯ãããéé»æ°åããã¯ãŒãã³åãšåŒã¶ããšãããã 2åã®é»è·ã©ããããããŒãåã¯åãã§ããããããã£ãŠäœçšã»åäœçšã®æ³åã«åŸã£ãŠããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 41,
"tag": "p",
"text": "ããã§ãé»è·ã®åäœã¯[C]ã§äžãããããèšå·ã®Cã¯ãã¯ãŒãã³ããšèªãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 42,
"tag": "p",
"text": "",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 43,
"tag": "p",
"text": "å³ã®ããã«ã2æ¬ã®ç³žã«ãããããåã質ém[kg]ã§ãåã笊å·ãšå€§ããã®é»è·q[C]ã®çããã¶ãããã£ãŠããããã¯ãã¯ãŒãã³åã§åçºããã®ã§ãå³ã®ããã«ã糞ãè§åºŠÎžããªãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 44,
"tag": "p",
"text": "ãã®ãšãã質émã«ããéåãšãé»è·qã«ããã¯ãŒãã³åãšã®é¢ä¿ã«ã€ããŠãåŒãç«ãŠãããªããå¿
èŠãªãã°ã糞ã®åŒµåã¯T[N]ãšããããšã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 45,
"tag": "p",
"text": "解æ³",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 46,
"tag": "p",
"text": "å³ã®ãããªäœçœ®é¢ä¿ã«ãªãã®ã§ãå³ã®ããã«åŒãç«ãŠãã°ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 47,
"tag": "p",
"text": "",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 48,
"tag": "p",
"text": "",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 49,
"tag": "p",
"text": "â» äžèšã®2æ¬ã®ç³žã«ã¶ãããã£ãçã®ã¯ãŒãã³åã®äŸé¡ã¯ãé»æ°ç£æ°åŠã®ã©ã®å
¥éæžã«ããããããªå
žåçãªåé¡ã§ããã®ã§ãèªè
ã¯ãã¡ããšç解ããããšã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 50,
"tag": "p",
"text": "",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 51,
"tag": "p",
"text": "é»è· q 1 {\\displaystyle q_{1}} , q 2 {\\displaystyle q_{2}} ã®éã®è·é¢ãrã®å Žåãš2rã®å Žåã§ã¯ãéã«åãåã®å€§ããã¯ã©ã¡ããã©ãã ã倧ãããçããã ãŸããè·é¢ã2rã®æã®2ç¹éã®åã®å€§ãããçããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 52,
"tag": "p",
"text": "ã¯ãŒãã³åã¯ãç©äœéã®è·é¢ã®é2ä¹ã«æ¯äŸããã®ã§ãè·é¢ã2rã®æã¯ãrã®æã®å€§ããã® 1 4 {\\displaystyle {\\frac {1}{4}}} ãšãªãããŸããåãåã®å€§ããã¯ãã¯ãŒãã³åã®åŒãçšããŠã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 53,
"tag": "p",
"text": "ãšãªãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 54,
"tag": "p",
"text": "æ¢ã«ãããé»è·Aã®ãŸããã®å¥ã®é»è·Bã«ã¯ããã®é»è·ããã®è·é¢ã®é2ä¹ã«æ¯äŸããåããããããšãè¿°ã¹ãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 55,
"tag": "p",
"text": "ããã§ãé»è·Bãåããåã¯ããã®é»è·Bã®å€§ããã«æ¯äŸããããšãåãããŠèãããšããã®é»è·Bã®å€§ããã«ããããããé»è·Aã®å€§ããã ãã§æ±ºãŸãéãå°å
¥ããŠãããšéœåããããããã§ããã®ãããªéãšããŠé»å Ž(ã§ãã°)ãå°å
¥ããããã®ãšããé»å Ž E â {\\displaystyle {\\vec {E}}} ã®äžã«ããé»è· q {\\displaystyle q} ã«åãå F â {\\displaystyle {\\vec {F}}} ã¯ã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 56,
"tag": "p",
"text": "ã§äžãããããé»å Žã¯åäœé»è·ã«åãåãšèããããšãã§ããé»å Žã®åäœã¯[N/C]ã§ããããé»å Žãã¯ããé»çã(ã§ããã)ãšãåŒã°ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 57,
"tag": "p",
"text": "(æ¥æ¬ã®ç©çåŠã§ã¯ãé»å ŽããšåŒã¶ããšãå€ãããŸããæ¥æ¬ã®é»æ°å·¥åŠã§ã¯ãé»çããšåŒã°ããããšãå€ããææ²»æã®ç¿»èš³ã®éã®ãæ¥æ¬åœå
ã®æ¥çããšã®éãã«éããããããªãæ¥æ¬ããŒã«ã«ãªéœåã§ãããåŒã³æ¹ã¯ç©çã®æ¬è³ªãšã¯é¢ä¿ãªãã®ã§ãããã§ã¯ãã©ã¡ãã®è¡šçŸãçšãããã¯ãæ¬æžã§ã¯ç¹ã«ãã ãããªããè±èªã§ã¯ç©çåŠã»é»æ°å·¥åŠãšãâelectric fieldâã§å
±éããŠããã)",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 58,
"tag": "p",
"text": "äžã®ã¯ãŒãã³åã®çµæãšåããããšãé»è·Aã®ãŸããã«å¥ã®é»è·ãååšããªããšããé»è· q {\\displaystyle q} [C]ã®é»è·ããŸãšãé»å Ž E â {\\displaystyle {\\vec {E}}} ã¯ã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 59,
"tag": "p",
"text": "ã§äžããããããã ããrã¯é»è·ããã®è·é¢ã§ããã e â r {\\displaystyle {\\vec {e}}_{r}} ã¯ãé»è·ãšããç¹ãçµãã çŽç·äžã§ãé»è·ãšå察æ¹åãåããåäœãã¯ãã«ã§ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 60,
"tag": "p",
"text": "é»è·ã®åãã®é»å Žã¯ãå¹³é¢äžã§æŸå°ç¶ã®ãã¯ãã«ãšãªãããšã«æ³šæã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 61,
"tag": "p",
"text": "é»å Žã¯ãã¯ãã«ã§ãããé»è·ã2åãããšãã¯ãããããã®é»è·ãã€ããé»å Žããéãåãããã°ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 62,
"tag": "p",
"text": "ã§ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 63,
"tag": "p",
"text": "é»è·ã3å以äžã®ãšãããåæ§ã«éãåãããã°è¯ãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 64,
"tag": "p",
"text": "å³ã®ããã«ãé»è·ããåºãé»å Žã®æ¹åãå³ç€ºãããã®ãé»æ°åç·(ã§ãããããããelectric line of force)ãšããã é»è·ãè€æ°ããå Žåã«ã¯ãå®éã«æ°ãã«çœ®ãããé»è·ãåããåã¯ããããã足ãåããããã®ãšãªãããããã£ãŠãè€æ°ã®é»è·ãããå Žåã®åšå²ã®é»çã¯ãããããã®é»è·ãäœãé»çãã¯ãã«ã®åãšãªã(éãåããã®åç)ã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 65,
"tag": "p",
"text": "é»æ°åç·ãå³ç€ºããå Žåã¯ãæ£é»è·ããåç·ãåºãŠãè² é»è·ã§åç·ãåžåãããããã«æžããåç·ã¯ãé»å Žãå³ç€ºãããã®ãªã®ã§ãé»è·ä»¥å€ã®å Žæã§ã¯ãåç·ãåå²ããããšã¯ãªãã åç·ãçæããã®ã¯æ£é»è·ã®å Žæã®ã¿ã§ãããåç·ãæ¶æ»
ããã®ã¯ãè² é»è·ã®å Žæã®ã¿ã§ããã èšãæããã°ãåç·ãé»è·ä»¥å€ã®å Žæã§æ¶æ»
ããããšã¯ãªãããé»è·ä»¥å€ã®å Žæã§åç·ãçæããããšã¯ãªãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 66,
"tag": "p",
"text": "å°äœã®å
éšã®é»å Žã¯ãŒãã§ãã£ããèšãæããã°ãé»æ°åç·ã¯ãå°äœã®å
éšã«ã¯é²å
¥ã§ããªãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 67,
"tag": "p",
"text": "",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 68,
"tag": "p",
"text": "ç¹é»è·ããã¯ãå³ã®ããã«ãæŸå°ç¶ã«é»æ°åç·ãåºããã¯ãŒãã³ã®æ³åã®ä¿æ°ã«ãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 69,
"tag": "p",
"text": "ã®ãã¡ã®ãåæ¯ã® 4 Ï r 2 {\\displaystyle 4\\pi r^{2}} ã¯ãçã®è¡šé¢ç©ã®å
¬åŒã«çããã®ã§ãé»æ°åç·ã®å¯åºŠã«æ¯äŸããŠãé»å Žã®åŒ·ããããã¯éé»æ°åã®åŒ·ãã決ãŸããšèããããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 70,
"tag": "p",
"text": "éé»èªå°ã§ã¯ãå°äœå
éšã«ã¯éé»æ°åãåããŠããªãã®ã§ãã£ããããã¯ãé»å ŽãšããæŠå¿µãçšããŠèšãæããã°ãå°äœå
éšã®é»å Žã¯ãŒãã§ããããšèšããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 71,
"tag": "p",
"text": "ã¯ãŒãã³åã¯å(ã¡ãã)ã§ãããããããã«éãã£ãŠå¥ã®é»è·ãè¿ã¥ããå Žåã¯ãè¿ã¥ããå¥ã®é»è·ã¯ä»äºãããããšã«ãªãããŸããè¿ã¥ããé»è·ãææŸãã°ãã¯ãŒãã³åã«ãã£ãŠåãåããä»äºãããããšã«ãªããããè¿ã¥ããç¶æ
ã«ããå¥é»è·ã¯äœçœ®ãšãã«ã®ãŒãèããŠããããšã«ãªãã ãããã£ãŠãã¯ãŒãã³åã«å¯ŸããŠãäœçœ®ãšãã«ã®ãŒãå®çŸ©ããããšãã§ããã(ãªããè¡æè»éäžã®ç©äœã®ãããªãå°è¡šãã倧ããé¢ããå Žæã®éåããã¯ãŒãã³åãšåæ§ã«é2ä¹åãªã®ã§ãããã§èããèšç®ææ³ã¯éåã«ããäœçœ®ãšãã«ã®ãŒã«ãå¿çšã§ãããéåå é床gãçšããåmgãšããã®ã¯å°è¡šè¿ãã§ã®è¿äŒŒã«ãããªãã)",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 72,
"tag": "p",
"text": "ã¯ãŒãã³åã«ããé»å Žã®å®çŸ©ã§ã¯ãåäœé»è·ã«å¯ŸããŠé»å Žãå®çŸ©ããã®ãšåæ§ãäœçœ®ãšãã«ã®ãŒã«å¯ŸããŠããåäœé»è·ã«å¿ããŠå®çŸ©ã§ããéãå°å
¥ãããšéœåãããããã®ãããªéãé»äœ(ã§ãããelectric potential)ãšåŒã¶ãé»äœã®åäœã¯ãã«ããšãããé»äœãäŸãããšãå°è¡šè¿ãã§ã®éåã®äœçœ®ãšãã«ã®ãŒãèããéã®ãghããªã©ã«çžåœããéã§ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 73,
"tag": "p",
"text": "ã¯ãŒãã³åã®çµæãšã q {\\displaystyle q} [C]ã®é»è·ããè·é¢rã ãé¢ããç¹ã®é»äœVã¯ãé»å Žã®ç©åèšç®ã§åŸãããã(ç©åããŸã ç¿ã£ãŠãªãåŠå¹Žã®èªè
ã¯ãåãããªããŠãæ°ã«ããã次ã®çµæãžãšé²ãã§ãã ããã)çµæã®ã¿ãèšããšã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 74,
"tag": "p",
"text": "ãšãªãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 75,
"tag": "p",
"text": "é»äœVã®ç¹ã«q[C]ã®é»è·ã眮ãããšãããã®é»è·ã®ã¯ãŒãã³åã«ããäœçœ®ãšãã«ã®ãŒU[J]ã¯ãé»äœVãçšããã°ã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 76,
"tag": "p",
"text": "ãšãªãããããã£ãŠãé»äœ V 1 {\\displaystyle V_{1}} ãã«ãã®ç¹ããé»äœ V 2 {\\displaystyle V_{2}} ãã«ãã®äœçœ®ãžãšé»è·q[C]ãéé»æ°åãåããŠç§»åãããšããéé»æ°åã®ããä»äºW[J]ã¯",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 77,
"tag": "p",
"text": "ãšãªãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 78,
"tag": "p",
"text": "",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 79,
"tag": "p",
"text": "ãã£ãœããäžæ§ãªé»å Žã«ãããŠã¯ãé»äœã®åŒããé»å ŽãçšããŠç°¡åã«è¡šãããšãã§ãããè·é¢dã ãé¢ããå¹³è¡å¹³æ¿é»æ¥µã®éã«äžæ§ãªé»å Ž E â {\\displaystyle {\\vec {E}}} ãçããŠãããšãããã®é»çã®äžã«çœ®ããé»è·qã¯éé»æ°å q E â {\\displaystyle q{\\vec {E}}} ãåããããã®é»è·ãé»çã®åãã«æ²¿ã£ãŠäžæ¹ã®é»æ¥µããä»æ¹ã®é»æ¥µãŸã§ç§»åãããšããé»çã®ããä»äºW㯠W = q E d {\\displaystyle W=qEd} ãšãªããããããã2極æ¿ã®é»äœå·®Vã¯ã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 80,
"tag": "p",
"text": "ã§è¡šãããšãã§ããããšãããããåŒãå€åœ¢ããŠ",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 81,
"tag": "p",
"text": "ãšããããšãã§ãããããã§ãåäœãèãããšãå³èŸºã¯é»å§ãè·é¢ã§å²ã£ããã®ã§ãããããé»çã®åäœãšããŠ[N/C]ã®ã»ã[V/m]ãçšããããšãã§ããããšããããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 82,
"tag": "p",
"text": "é»äœã®åäœã¯ãã«ãã§ããããã®éã¯æ¢ã«äžåŠæ ¡çç§ãªã©ã§æ±ã£ãé»å§(ã§ããã€ãvoltage)ã®åäœãšåãåäœã§ãããå®éã«é»æ°åè·¯ã«é»å§ããããããšã¯ãåè·¯äžã®é»åã«é»å ŽããããŠåããããšãšçããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 83,
"tag": "p",
"text": "éé»èªå°ã«ãã£ãŠãå°äœå
éšã®é»å Žã¯ãŒãã§ãã£ãããã®ããšãããå°äœã®è¡šé¢ã¯ãé»äœãçãããå°äœè¡šé¢ã¯äºãã«çé»äœã§ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 84,
"tag": "p",
"text": "é»äœã®åºæºã¯ãå®çšäžã¯ãå°é¢ã®é»äœããŒãã«çœ®ãããšãå€ããé»æ°åè·¯ã®äžéšã倧å°ã«ã€ãªãããšãæ¥å°(ãã£ã¡)ãŸãã¯ã¢ãŒã¹(earth)ãšãããåè·¯ãã¢ãŒã¹ããŠããã®ã€ãªãã éšåã®é»äœããŒããšèŠãªãããšãå€ãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 85,
"tag": "p",
"text": "çŽç·äžã§è·é¢0, b[m]ã®ç¹ã«ãé»è·q, q'ãæã€ç©äœã眮ããŠããããã®æãäœçœ®a[m](a<b)ã®ç¹ã®é»äœãæ±ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 86,
"tag": "p",
"text": "é»äœã®åŒãçšããã°ãããé»è·ãè€æ°ãããšãã«ã¯ãé»äœã¯ããããã®é»è·ãã€ããåºãé»è·ã®åã«ãªãããšã«æ³šæãçãã¯ã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 87,
"tag": "p",
"text": "ãšãªãã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 88,
"tag": "p",
"text": "å°äœè¡šé¢ã¯çé»äœãªã®ã§ããã£ãŠãé»æ°åç·ã¯å°äœè¡šé¢ã«åçŽã§ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 89,
"tag": "p",
"text": "ãã®ããšãããé»æ°åç·ãšé»å Žã¯åçŽã§ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 90,
"tag": "p",
"text": "é»å Žãéãåãããããããã«ãé»äœãéãåãããããããªããªãé»äœãšã¯ãé»å ŽãèããŠãçµè·¯ã«ãŠç©åãããã®ã§ããããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 91,
"tag": "p",
"text": "åŠæ ¡ã®ãã¹ããªã©ã§ã¯ãé»äœã®èšç®ã®ãããã¯ãŒãã³åã®æ¹åã®åéããªã©ã«ããèšç®ãã¹ãªã©ããµãããããé»å Žãæ±ããŠããããããç©åããŠãé»äœãæ±ããã®ããèšç®äžã¯å®å
šã§ããã",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 92,
"tag": "p",
"text": "",
"title": "é»å Žãšç£å Ž"
},
{
"paragraph_id": 93,
"tag": "p",
"text": "ã³ã³ãã³ãµãŒ(è±:capacitor ,ããã£ãã·ã¿ããšèªã)ã¯ãå³ã®ããã«ã2æã®é»æ¥µãåãããããåè·¯äžã«é»è·ãèç©ã§ããéšåãäžããçŽ åã§ããã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 94,
"tag": "p",
"text": "ã³ã³ãã³ãµãŒã«é»è·ãèããããšãå
é»(ãã
ãã§ã)ãšãããã³ã³ãã³ãµãŒããé»è·ãæŸåºãããããšãæŸé»ãšããã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 95,
"tag": "p",
"text": "ã³ã³ãã³ãµã®äž¡ç«¯ã«ããé»äœVãäžãããããšããã³ã³ãã³ãµã«ã¯ãé»äœã«æ¯äŸããé»è·Qãèç©ãããããã®ãšããã³ã³ãã³ãµã®èç©èœåãèšå·ã§ C ãšãããŠã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 96,
"tag": "p",
"text": "ãšããŠCãåããCã¯éé»å®¹é(ããã§ãããããããelectric capacitance)ãšåŒã°ããåäœã¯F(ãã¡ã©ããfarad)ã§äžããããã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 97,
"tag": "p",
"text": "1ãã¡ã©ãã¯å®çšäžã¯å€§ããããã®ã§ã10ãã¡ã©ããåäœã«ãã1pF(ãã³ãã¡ã©ã)ãã10ãã¡ã©ããåäœã«ãã1ÎŒF(ãã€ã¯ããã¡ã©ã)ã䜿ãããããšãå€ãã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 98,
"tag": "p",
"text": "",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 99,
"tag": "p",
"text": "極æ¿ãå¹³è¡ãªã³ã³ãã³ãµãŒãå¹³è¡æ¿ã³ã³ãã³ãµãŒãšããã å¹³è¡æ¿ã³ã³ãã³ãµãŒã®ã極æ¿ã©ããã®é»å Žã¯ãäžæ§ãªé»å Žã§ããã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 100,
"tag": "p",
"text": "ãã®å¹³è¡æ¿ã³ã³ãã³ãµãŒã®éé»å®¹éCã®åŒã¯ãåŸè¿°ããçç±ã«ããã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 101,
"tag": "p",
"text": "ã§äžãããããããã§ãSã¯å°äœå¹³é¢ã®é¢ç©ã§ãããdã¯å°äœéã®è·é¢ã§ããã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 102,
"tag": "p",
"text": "å®éšçã«ãããã®éé»å®¹éã®å
¬åŒã¯ãæ£ããããšã確ãããããŠããã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 103,
"tag": "p",
"text": "ããã§äžããéé»å®¹éã¯ãå¹³é¢äžã«é»è·ãäžæ§ã«ååžãããšã®ä»®å®ã§å°ãããããã®ãšããå°äœéã«çããé»çEã¯ãå°äœãæã€é»è·ãQ, -Qãšããæã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 104,
"tag": "p",
"text": "ãŸãã極æ¿ã®é»è·å¯åºŠãã極æ¿ã®ã©ãã§ãäžå®ã ãšä»®å®ããŠ(ãã®ããã«ã¯ãã³ã³ãã³ãµãŒã®åºã(ã€ãŸãé¢ç©)ãããã
ãã¶ãã«åºããšä»®å®ããå¿
èŠãããããšãããããã®ãããªä»®å®ã«ãããé»è·å¯åºŠã¯)ã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 105,
"tag": "p",
"text": "ã§ããã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 106,
"tag": "p",
"text": "é»æ°åç·ã®æ§è³ªãšããŠããã©ã¹ã®é»è·ããçããŠãã€ãã¹ã®é»è·ã§åžåãããã®ã§ããã£ãŠå¹³è¡æ¿ã³ã³ãã³ãµãŒéã®é»æ°åç·ã®ååžã¯ãå³ã®ããã«ãé»æ°åç·ãããã©ã¹æ¥µæ¿ããåçŽã«ããã€ãã¹æ¥µæ¿ãžåãã£ãŠé»æ°åç·ãåºãŠããããŠãã€ãã¹æ¥µæ¿ã«é»æ°åç·ãåžåãããã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 107,
"tag": "p",
"text": "é»å Žã¯ãå°äœéã®åç¹ã§ã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 108,
"tag": "p",
"text": "ã§äžãããããé»å Žãæ±ããããã®ã§ãããããé»äœãèšç®ã§ãããå°äœéã®åç¹ã§é»å Žã®å€§ãããåäžãªã®ã§ãé»äœã®å€§ããã¯é»å Žã®å€§ããã«2ç¹éã®è·é¢ãããããã®ã«ãªããããã§ãé»äœVã¯ã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 109,
"tag": "p",
"text": "ãšãªããããã®åŒãšéé»å®¹éCã®å®çŸ©ãèŠæ¯ã¹ããšã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 110,
"tag": "p",
"text": "ãåŸãããã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 111,
"tag": "p",
"text": "",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 112,
"tag": "p",
"text": "é»æ± ã®ååŠåå¿ã«ã€ããŠã¯ãå¥ç§ç®ã®ååŠIãªã©ã§è©³ããæ±ãããããã®ç« ã§ã¯ãé»å§ãé»æµã®ç解ã«é¢ããç¹ãéç¹çã«èª¬æãããã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 113,
"tag": "p",
"text": "éå±å
çŽ ã®åäœãæ°ŽãŸãã¯æ°Žæº¶æ¶²ã«å
¥ãããšãã®ãéœã€ãªã³ã®ãªãããããã€ãªã³ååŸå(ionization tendency)ãšããã äŸãšããŠãäºéZnãåžå¡©é
žHClã®æ°Žæº¶æ¶²ã«å
¥ãããšãäºéZnã¯æº¶ãããŸãäºéã¯é»åã倱ã£ãŠZnã«ãªãã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 114,
"tag": "p",
"text": "äžæ¹ãéAgãåžå¡©é
žã«å
¥ããŠãåå¿ã¯èµ·ãããªãã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 115,
"tag": "p",
"text": "ãã®ããã«éå±ã®ã€ãªã³ååŸåã®å€§ããã¯ãç©è³ªããšã«å€§ãããç°ãªãã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 116,
"tag": "p",
"text": "äºçš®é¡ã®éå±åäœãé»è§£è³ªæ°Žæº¶æ¶²ã«å
¥ãããšé»æ± ãã§ãããããã¯ã€ãªã³ååŸå(åäœã®éå±ã®ååãæ°ŽãŸãã¯æ°Žæº¶æ¶²äžã§é»åãæŸåºããŠéœã€ãªã³ã«ãªãæ§è³ª)ã倧ããéå±ãé»åãæŸåºããŠéœã€ãªã³ãšãªã£ãŠæº¶ããã€ãªã³ååŸåã®å°ããéå±ãæåºããããã§ããã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 117,
"tag": "p",
"text": "ã€ãªã³ååŸåã®å€§ããæ¹ã®éå±ãè² æ¥µ(ãµããã)ãšãããã€ãªã³ååŸåã®å°ããæ¹ã®éå±ãæ£æ¥µ(ããããã)ãšããã ã€ãªã³ååŸåã®å€§ããéå±ã®ã»ãããéœã€ãªã³ã«ãªã£ãŠæº¶ãåºãçµæãéå±æ¿ã«ã¯é»åãå€ãèç©ããã®ã§ãäž¡æ¹ã®éå±æ¿ãé
ç·ã§ã€ãªãã°ãã€ãªã³ååŸåã®å€§ããæ¹ããå°ããæ¹ã«é»åã¯æµããããé»æµãã§ã¯ç¡ãããé»åããšãã£ãŠãããšã«æ³šæãé»åã¯è² é»è·ã§ããã®ã§ãé»æµã®æµããšé»åã®æµãã¯ãéåãã«ãªãã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 118,
"tag": "p",
"text": "ããŸããŸãªæº¶æ¶²ãéå±ã®çµã¿åããã§ãã€ãªã³ååŸåã®æ¯èŒã®å®éšãè¡ã£ãçµæãã€ãªã³ååŸåã®å€§ããã決å®ãããã å·Šããé ã«ãã€ãªã³ååŸåã®å€§ããéå±ã䞊ã¹ããšã以äžã®ããã«ãªãã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 119,
"tag": "p",
"text": "éå±ããã€ãªã³ååŸåã®å€§ããã®é ã«äžŠã¹ããã®ãéå±ã®ã€ãªã³ååãšããã æ°ŽçŽ ã¯éå±ã§ã¯ç¡ããæ¯èŒã®ãããã€ãªã³ååŸåã«å ããããã éå±ååã¯ãäžèšã®ä»ã«ãããããé«æ ¡ååŠã§ã¯äžèšã®éå±ã®ã¿ã®ã€ãªã³ååãçšããããšãå€ãã ã€ãªã³ååã®èšæ¶ã®ããã®èªååãããšããŠã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 120,
"tag": "p",
"text": "ã貞ããããªããŸããããŠã«ããªãã²ã©ãããåéãã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 121,
"tag": "p",
"text": "ãªã©ã®ãããªèªååããããããã¡ãªã¿ã«ãã®èªååããã®å Žåã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 122,
"tag": "p",
"text": "ãKã ãã ãCa ãªNaããŸMg ãAlããZn ãŠFe ã«Ni ã ãªPbãã²H2 ã©Cu ãHg ãAg ã åéPt,Auãã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 123,
"tag": "p",
"text": "ãšå¯Ÿå¿ããŠããã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 124,
"tag": "p",
"text": "è² æ¥µ(äºéæ¿)ã§ã®åå¿",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 125,
"tag": "p",
"text": "æ£æ¥µ(é
æ¿)ã§ã®åå¿",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 126,
"tag": "p",
"text": "ãã«ã¿ã®é»æ± ã§ã¯ãåŸããã䞡極éã®é»äœå·®(ãé»å§ããšãããã)ã¯ã1.1ãã«ãã§ããã(ãã«ãã®åäœã¯Vãªã®ã§ã1.1Vãšãæžãã)ãã®äž¡æ¥µæ¿ã®é»äœå·®ãèµ·é»åãšãããèµ·é»åã¯ãäž¡é»æ¥µã®éå±ã®çµã¿åããã«ãã£ãŠæ±ºãŸãç©è³ªåºæã§ããã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 127,
"tag": "p",
"text": "èµ·é»åã®åäœã®ãã«ãã¯ãéé»æ°åã®é»äœã®åäœã®ãã«ããšåãåäœã§ãããé»æ°åè·¯ã®é»å§ã®ãã«ããšããèµ·é»åã®åäœã®ãã«ãã¯åãåäœã§ããã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 128,
"tag": "p",
"text": "ãã«ã¿é»æ± ã®æ§é ã以äžã®ãããªæååã«è¡šããå Žåããã®ãããªè¡šç€ºãé»æ± å³ãããã¯é»æ± åŒãšããã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 129,
"tag": "p",
"text": "aqã¯æ°Žã®ããšã§ãããH2SO4aqãšæžããŠãç¡«é
žæ°Žæº¶æ¶²ãè¡šããŠããã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 130,
"tag": "p",
"text": "ç©çåŠã®é»æ°åè·¯ã®ç 究ã§ã¯ããã®ãããªé»æ± ãªã©ã®çŸè±¡ã®çºèŠãšçºæã«ãã£ãŠãå®å®ãªçŽæµé»æºãå®éšçã«åŸãããããã«ãªããçŽæµé»æ°åè·¯ã®æ£ç¢ºãªå®éšãå¯èœã«ãªã£ããé»æ± ã®çºæ以åã«ãããã©ã³ã¹äººã®ç©çåŠè
ã¯ãŒãã³ãªã©ã«ããéé»æ°ã«ããé»æ°ååŠã®ç 究ãªã©ã«ãã£ãŠãé»äœå·®ã®æŠå¿µãé»è·ã®æŠå¿µã¯ãã£ããã ãããã®æ代ã®é»æºã¯ãäž»ã«éé»æ°ã«ãããã®ã ã£ãã®ã§ãå®å®é»æºã§ã¯ç¡ãã£ãã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 131,
"tag": "p",
"text": "ãããŠãé»æ± ã«ããå®å®ãªé»æºã®çºæã¯ãåæã«å®å®ãªé»æµã®çºæã§ããã£ãããã®ãããªé»æ± ã®çºæãªã©ã«ãããçŽæµé»æ°åè·¯ã®ç 究ãªã©ããããã€ã人ã®ç©çåŠè
ãªãŒã ããããŸããŸãªå°äœã«é»æµãæµãå®éšãšçè«ç 究ãè¡ãããšã«ãããé»æ°åè·¯ã®çè«ã®ãªãŒã ã®æ³å(ãªãŒã ã®ã»ããããOhm's law)ãçºèŠãããã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 132,
"tag": "p",
"text": "ãã€ã¯ãªãŒã ã¯é»æ± ã§ã¯ãªãç±é»å¯Ÿ(ãã€ã§ãã€ã)ãšãããã®ã䜿ã£ãŠãé»æ°åè·¯ã«å®å®ããé»æµããªããç 究ããããåœæã®é»æ± ã§ã¯ãèµ·é»åããã ãã«æžã£ãŠããŸãããªãŒã ã¯åœåã¯é»æ± ã§å®éšããããããŸãå®å®é»æµãåŸãããªãã£ãã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 133,
"tag": "p",
"text": "ç±é»å¯Ÿãšã¯ããŸãç°ãªãéå±ææã®2æ¬ã®éå±ç·ãæ¥ç¶ããŠ1ã€ã®åè·¯ãã€ããã2ã€ã®æ¥ç¹ã«æž©åºŠå·®ãäžãããšãåè·¯ã«é»å§ãçºçããããé»æµãæµãã(ãã®çŸè±¡ãããŒãŒããã¯å¹æãšãã)ããã®çŸè±¡ãããã¯ã1821幎ã«ãŒãŒããã¯ãçºèŠããããã®ãããªåè·¯ããç±é»å¯Ÿã§ããããªããåã2æ¬ã®éå±ç·ã§ã¯ã枩床差ãäžããŠãé»å§ã¯çºçãããé»æµã¯æµããªãã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 134,
"tag": "p",
"text": "ãªãŒã ã¯ããã«ãªã³å€§åŠææããã±ã³ãã«ãã®å©èšã«ãã£ãŠããã®ç±é»å¯Ÿãå®éšã«å©çšããã枩床ãå®å®ãããã®ã¯ãåœæã®æè¡ã§ãæ¯èŒçç°¡åã§ãã£ãã®ã§ãããããŠãªãŒã ã¯å®å®é»æµããã¡ããå®éšãã§ããã®ã§ããã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 135,
"tag": "p",
"text": "",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 136,
"tag": "p",
"text": "ãªãŒã ã®æ³å(Ohm's law)ãšã¯ã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 137,
"tag": "p",
"text": "ãã»ãšãã©ã®å°äœã§ã¯ãé»æµ I ãæµããŠããå°äœäžã®2ç¹ã®ç¹ P 1 {\\displaystyle P_{1}} ãšç¹ P 2 {\\displaystyle P_{2}} éã®é»äœå·® E = E 1 â E 2 {\\displaystyle E=E_{1}-E_{2}} ã¯ãé»æµ I ã«æ¯äŸãããã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 138,
"tag": "p",
"text": "ãšããå®éšæ³åã§ããã 誀解ããããããããªãŒã ã®æ³åã¯ããã®ãããªå®éšæ³åã§ãã£ãŠãã¹ã€ã«æµæã®å®çŸ©åŒã§ã¯ç¡ããåæ§ã«ããªãŒã ã®æ³åã¯ãã¹ã€ã«é»å§ã®å®çŸ©åŒã§ã¯ç¡ãããé»æµã®å®çŸ©åŒã§ãç¡ããäžåŠæ ¡ã®çç§ã§ã®é»æ°åè·¯ã®æè²ã§ã¯ãéå±ã®é»æ°å解ã®èµ·é»åã®æè²ãŸã§ã¯ããªãã®ã§ããšãããã°ãé»å§ã誀解ããŠããé»å§ã¯ãåãªãé»æµã®æ¯äŸéã§ãæµæã¯ãã®æ¯äŸä¿æ°ãã®ãããªèª€è§£ããå Žåãæããããããã®è§£éã¯æããã«èª€è§£ã§ããã ãŸããåå°äœãªã©ã®äžéšã®ææã§ã¯ãé»æµãå¢ãææã®æž©åºŠãäžæãããšæµæãäžããçŸè±¡ãç¥ãããŠããã®ã§ãåå°äœã§ã¯ãªãŒã ã®æ³åãæãç«ããªãå Žåãããããªã®ã§ããªãŒã ã®æ³åãå®çŸ©åŒãšèããã®ã¯äžåçã§ããã",
"title": "é»æ± ã®ä»çµã¿"
},
{
"paragraph_id": 139,
"tag": "p",
"text": "å°ç·ãªã©ã®å°äœå
ã®é»æ°ã®æµããé»æµ(ã§ããã
ããelectric current)ãšãããé»æµã®åŒ·ãã¯ã¢ã³ãã¢ãšããåäœã§è¡šãã1ã¢ã³ãã¢ã®å®çŸ©ã¯æ¬¡ã®éãã§ããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 140,
"tag": "p",
"text": "1ç§éã«1ã¯ãŒãã³(èšå·C)ã®é»æµãééããããšã1ã¢ã³ãã¢ãšããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 141,
"tag": "p",
"text": "ã¢ã³ãã¢ã®èšå·ã¯Aã§ããããŸããé»æµã¯ãåäœæéãããã®é»è·ã®éééã§ãããã®ã§ãé»æµã®åäœã[C/s]ãšæžãå Žåãããã äžè¬çã«ã¯ãé»æµã®åäœã¯ããªãã¹ã[A]ã§è¡šèšããããšãå€ãã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 142,
"tag": "p",
"text": "é»æµI[A]ãšæét[S]ã§å°ç·æé¢ãééããé»è·Q[C]ã®é¢ä¿ãåŒã§è¡šããšã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 143,
"tag": "p",
"text": "ã§ããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 144,
"tag": "p",
"text": "é»æµã®åãã®åãæ¹ã«ã€ããŠã¯ãèªç±é»åã¯è² é»è·ãæã£ãŠãããããèªç±é»åã®åããšã¯å察åãã«é»æµã®åãããšãããšã«æ³šæããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 145,
"tag": "p",
"text": "次ã«é»æµãšèªç±é»åã®é床ãšã®é¢ä¿ãèããã èªç±é»åã®é»è·ã®çµ¶å¯Ÿå€ãeãšãããšãèªç±é»åã¯è² é»è·ã§ãããããèªç±é»åã®é»è·ã¯ãã€ãã¹ç¬Šå·ãã€ã-eã§ããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 146,
"tag": "p",
"text": "ãã€ã人ã®ç©çåŠè
ãªãŒã ã¯æ¬¡ã®ãããªæ³åãçºèŠããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 147,
"tag": "p",
"text": "ãã»ãšãã©ã®å°äœã§ã¯ãé»æµ I ãæµããŠããå°äœäžã®2ç¹ã®ç¹ P 1 {\\displaystyle P_{1}} ãšç¹ P 2 {\\displaystyle P_{2}} éã®é»äœå·® E = E 1 â E 2 {\\displaystyle E=E_{1}-E_{2}} ã¯ãé»æµ I ã«æ¯äŸãããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 148,
"tag": "p",
"text": "ãã®å®éšæ³åããªãŒã ã®æ³å(Ohm's law)ãšããã åŒã§è¡šããšãé»äœå·®ãVãšããŠãé»æµãIãšããå Žåã«ãæ¯äŸä¿æ°ãRãšããŠã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 149,
"tag": "p",
"text": "ã§ããã ããã§ãé»äœãšé»æµã®æ¯äŸä¿æ°Rãé»æ°æµæãããã¯åã«æµæ(resistanceãã¬ãžã¹ã¿ã³ã¹)ãšããã é»æ°æµæã®åäœã¯ãªãŒã ãšèšããèšå·ã¯Î©ã§è¡šãã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 150,
"tag": "p",
"text": "æ
£ç¿çã«ãæµæã®èšå·ã¯Rã§ããããå Žåãå€ãã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 151,
"tag": "p",
"text": "é»æ°åè·¯ãžãšãã«ã®ãŒãäŸçµŠããé»æºãšããŠå®é»å§ã®çŽæµé»æºãèãããåè·¯ã®2å°ç¹éã«ããäžå®ã®é»å§ãäŸçµŠãç¶ãããã®ã§ãããé»å§æºã®åè·¯å³èšå·ãšããŠã¯ãçšãããããèšå·ã®é·ãåŽãæ£æ¥µã§ããããã©ã¹ã®é»äœã§ãããèšå·ã®çãåŽã¯è² 極ã§ããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 152,
"tag": "p",
"text": "也é»æ± ã¯ãçŽæµé»æºãšããŠåãæ±ã£ãŠè¯ãã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 153,
"tag": "p",
"text": "ãªãããããã¯çŽæµé»æºã§ããã亀æµã®å Žåã¯äžè¬åããé»å§æºãšããŠã®èšå·ãçšããããŸãç¹ã«æ£åŒŠæ³¢äº€æµé»å§æºã§ããã°ã®èšå·ãçšããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 154,
"tag": "p",
"text": "æµæåš(resistor)ã¯ãéåžžã¯åã«æµæãšåŒã°ããåè·¯çŽ åã§ãããäžããããé»æ°ãšãã«ã®ãŒãåçŽã«æ¶è²»ããçŽ åã§ãããåè·¯å³èšå·ã¯ãããã¯ã§ããããæ¬æžã§ã¯ãäž¡è
ãšãæµæã®åè·¯å³èšå·ãšããŠçšããããšã«ããã(ç»åçŽ æã®ç¢ºä¿ã®éœåã®ãããäž¡æ¹ã®èšå·ãæ¬æžã§ã¯æ··åšããŸããã容赊ãã ããã)",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 155,
"tag": "p",
"text": "æ¥æ¬ã§ã¯ãæµæåšã®å³èšå·ã¯ãåŸæ¥ã¯JIS C 0301(1952幎4æå¶å®)ã«åºã¥ããã®ã¶ã®ã¶ã®ç·ç¶ã®å³èšå·ã§å³ç€ºãããŠããããçŸåšã®ãåœéèŠæ Œã®IEC 60617ãå
ã«äœæãããJIS C 0617(1997-1999幎å¶å®)ã§ã¯ã®ã¶ã®ã¶åã®å³èšå·ã¯ç€ºãããªããªããé·æ¹åœ¢ã®ç®±ç¶ã®å³èšå·ã§å³ç€ºããããšã«ãªã£ãŠãããæ§èŠæ Œã§ããJIS C 0301ã¯ãæ°èŠæ ŒJIS C 0617ã®å¶å®ã«äŒŽã£ãŠå»æ¢ããããããæ§èšå·ã§æµæåšãå³ç€ºããå³é¢ã¯ãçŸåšã§ã¯JISéæºæ ãªå³é¢ã«ãªã£ãŠããŸããããããææåã¯ç¡ããããçŸåšãåŸæ¥ã®å³èšå·ãå€çšãããŠããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 156,
"tag": "p",
"text": "",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 157,
"tag": "p",
"text": "è€æ°ã®åè·¯çŽ åã1ã€ã®ç·äžã«é
眮ãããŠãããããªæ¥ç¶ãçŽåæ¥ç¶ãšãããè€æ°ã®åè·¯çŽ åãäºè¡ã«åãããããã«é
眮ãããŠããæ¥ç¶ã䞊åæ¥ç¶ãšããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 158,
"tag": "p",
"text": "çŽåæ¥ç¶ã«ãããŠã¯ãããããã®åè·¯çŽ åã«æµããé»æµã¯å
šãŠçãããäžæ¹ã䞊åæ¥ç¶ã«ãããŠã¯ããããã®åè·¯çŽ åã®äž¡ç«¯ã«ãããé»å§ãå
šãŠçããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 159,
"tag": "p",
"text": "ãŸããçŽåæ¥ç¶ã«ãããŠã¯ããããã®åè·¯çŽ åã«ãããé»å§ã®åãå
šé»å§ãšãªãã䞊åæ¥ç¶ã«ãããŠã¯ããããã®åè·¯çŽ åãæµããé»æµã®åãå
šé»æµãšãªãã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 160,
"tag": "p",
"text": "æµæãè€æ°æ¥ç¶ãããŠããå Žåããã®è€æ°ã®æµæããŸãšããŠãããã1ã€ã®æµæãæ¥ç¶ãããŠãããã®ãããªç䟡çãªåè·¯ãèããããšãã§ãããè€æ°ã®æµæãšç䟡ãª1ã€ã®æµæãåææµæãšããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 161,
"tag": "p",
"text": "æµæãnåçŽåã«æ¥ç¶ãããŠããå Žåãèãããæµæ R 1 , R 2 , ⯠, R n {\\displaystyle R_{1},R_{2},\\cdots ,R_{n}} ãçŽåã«æ¥ç¶ãããŠããå Žåãåæµæãæµããé»æµã¯çããããããiãšãããåæµæ R k ( k = 1 , 2 , ⯠, n ) {\\displaystyle R_{k}(k=1,2,\\cdots ,n)} ã«ãããé»å§ã v k {\\displaystyle v_{k}} ãšãããšããªãŒã ã®æ³åãã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 162,
"tag": "p",
"text": "ãæãç«ã€ããã®ãšãçŽåæµæã®äž¡ç«¯ã®é»å§vã¯ã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 163,
"tag": "p",
"text": "ã§ããããããšç䟡ãªæµæRã1ã€ã ãæ¥ç¶ãããŠãããããªç䟡åè·¯ãèãããšãã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 164,
"tag": "p",
"text": "ãæãç«ã€ããããããã£ãŠãããã®nåã®çŽåæµæã®åææµæRãšããŠ",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 165,
"tag": "p",
"text": "ãåŸããããªãã¡ãçŽååææµæã¯åæµæã®ç·åãšãªãã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 166,
"tag": "p",
"text": "åæ§ã«ãæµæãnå䞊åã«æ¥ç¶ãããŠããå Žåãèãããæµæ R 1 , R 2 , ⯠, R n {\\displaystyle R_{1},R_{2},\\cdots ,R_{n}} ã䞊åã«æ¥ç¶ãããŠããå Žåãåæµæã®äž¡ç«¯ã®é»å§ã¯çããããããvãšãããåæµæ R k ( k = 1 , 2 , ⯠, n ) {\\displaystyle R_{k}(k=1,2,\\cdots ,n)} ãæµããé»æµã i k {\\displaystyle i_{k}} ãšãããšããªãŒã ã®æ³åãã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 167,
"tag": "p",
"text": "ãæãç«ã€ããã®ãšã䞊åæµæãžæµã蟌ãé»æµiã¯ã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 168,
"tag": "p",
"text": "ã§ããããããšç䟡ãªæµæRã1ã€ã ãæ¥ç¶ãããŠãããããªç䟡åè·¯ãèãããšãã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 169,
"tag": "p",
"text": "ãæãç«ã€ããããããã£ãŠãããã®nåã®äžŠåæµæã®åææµæRãšããŠ",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 170,
"tag": "p",
"text": "ãåŸããããªãã¡ã䞊ååææµæã®éæ°ã¯åæµæã®éæ°ã®ç·åãšãªãã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 171,
"tag": "p",
"text": "æµæRãé»æµIãæµãããšãããã®éšåã®çºç±ã®ãšãã«ã®ãŒã¯ã1ç§ãããã«RI[J/s]ã§ãããããããžã¥ãŒã«ç±ãšãããååã®ç±æ¥ã¯ç©çåŠè
ã®ãžã¥ãŒã«ã調ã¹ãããã§ããããªãŒã ã®æ³åãããV=RIã§ãããã®ã§ããžã¥ãŒã«ç±ã¯VIãšãæžããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 172,
"tag": "p",
"text": "ããã§ãã²ãšãŸããç±ã®èå¯ã«ã¯é¢ããŠã次ã®éãå®çŸ©ãããé»æ°åè·¯ã®ãã2ç¹éãæµããé»æµIãšããã®2ç¹éã®é»å§Vãšã®ç©VIãé»å(power)ãšå®çŸ©ãããé»åã®èšå·ã¯Pã§è¡šããããããšãå€ãã é»åã®åäœã®ãžã¥ãŒã«æ¯ç§[J/s]ã[W]ãšããåäœã§è¡šãããã®åäœWã¯ã¯ãã(Watt)ãšèªãã ã€ãŸãé»åã¯èšå·ã§",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 173,
"tag": "p",
"text": "ã§ããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 174,
"tag": "p",
"text": "å°ç·ã®å€ªããé·ãã«ãã£ãŠæµæã®å€§ããã¯å€ãããçŽæçã«å€ªãã»ããæµããããã®ã¯åããã ãããããã«äžŠåæ¥ç¶ãšå¯Ÿå¿ãããŠããå°ç·ã倪ãã»ããæµããããã®ã¯åããã ããã å®éã«é»æ°æµæã¯ãå°ç·ã倪ãã«åæ¯äŸããŠå°ãããªãããšãå®éšçã«ç¢ºèªãããŠãããããã§ãã€ãã®ãããªåŒã«ãããŠã¿ããã æµæãR[Ω]ãšããå Žåãå°ç·ã®å€ªããé¢ç©ã§è¡šãA[m]ãšããã°ãæ¯äŸå®æ°ã«kãçšããã°ã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 175,
"tag": "p",
"text": "ã§ããã( âã¯ãæ¯äŸé¢ä¿ãè¡šãæ°åŠèšå·ã)",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 176,
"tag": "p",
"text": "ããã«ãå°ç·ã¯æ質ã倪ããåããªãã°ãå°ç·ãé·ãã»ã©æµæããé·ãã«æ¯äŸããŠæµæã倧ãããªãããšãã確èªãããŠãããããã§ãããã«ãæµæäœã®é·ããèæ
®ããåŒã«è¡šããŠã¿ãã°ã次ã®ããã«ãªããæµæ垯ã®é·ããl[m]ãšããã°",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 177,
"tag": "p",
"text": "ã§ããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 178,
"tag": "p",
"text": "ããã«ãå°ç·ã®æ質ã«ãã£ãŠãæµæã®å€§ããã¯å€ãããåãé·ãã§åã倪ãã®æµæã§ããæ質ã«ãã£ãŠæµæã®å€§ããã¯ç°ãªããããã§ãæ質ããšã®æ¯äŸå®æ°ãÏãšããã°ãæµæã®åŒã¯ä»¥äžã®åŒã§èšè¿°ãããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 179,
"tag": "p",
"text": "Ïã¯æµæç(ãŠããããã€ãresistivity)ãšåŒã°ãããæµæçã®åäœã¯[Ωm]ã§ããã",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 180,
"tag": "p",
"text": "",
"title": "é»æµãšé»æ°åè·¯"
},
{
"paragraph_id": 181,
"tag": "p",
"text": "ç£ç³ã®ãŸããã«ã¯å¥ã®ç£ç³ãåããåã®ããšãšãªããã®ãçããŠããã ãããç£å Ž(ãã°ãmagnetic field)ãããã¯ç£ç(ããã)ãšåŒã¶ã(æ¥æ¬ã®ç©çåŠã§ã¯ç£å ŽãšåŒã¶ããšãå€ãããŸããæ¥æ¬ã®é»æ°å·¥åŠã§ã¯ç£çãšåŒã°ããããšãå€ããææ²»æã®èš³èªã®éã®ãæ¥æ¬åœå
ã®æ¥çããšã®éãã«éãããå°å瀟äŒçãªäºè±¡ã§ãããåŒã³æ¹ã¯ç©çã®æ¬è³ªãšã¯é¢ä¿ãªãã®ã§ãããã§ã¯ãã©ã¡ãã®è¡šçŸãçšãããã¯ãæ¬æžã§ã¯ç¹ã«ãã ãããªããè±èªã§ã¯ç©çåŠã»é»æ°å·¥åŠãšãâmagnetic fieldâã§å
±éããŠããã)",
"title": "ç£å"
},
{
"paragraph_id": 182,
"tag": "p",
"text": "éãã³ãã«ããããã±ã«ã«ç£ç³ãè¿ã¥ãããšãç£ç³ã«åžãä»ããããã ãŸããéãã³ãã«ããããã±ã«ã«åŒ·ãç£åãäžãããšãéãã³ãã«ããããã±ã«ãã®ãã®ãç£å Žãåšå²ã«åãŒãããã«ãªãã ãã®ãããªãããšããšã¯ç£å Žãæããªãã£ãç©äœãã匷ãç£å Žãåããããšã«ãã£ãŠç£å ŽãåãŒãããã«ãªãçŸè±¡ãç£å(ãããmagnetization)ãšããã ãããã¯é»è·ã®éé»èªå°ãšå¯Ÿå¿ãããŠãç£åã®ããšãç£æ°èªå°(ããããã©ããmagnetic induction)ãšãããã ãããŠãéãã³ãã«ããããã±ã«ã®ããã«ãç£ç³ã«åŒãä»ããããããã«ç£åãããèœåãããç©äœã匷ç£æ§äœ(ãããããããããferromagnet)ãšããã éãšã³ãã«ããšããã±ã«ã¯åŒ·ç£æ§äœã§ããã",
"title": "ç£å"
},
{
"paragraph_id": 183,
"tag": "p",
"text": "é
ã¯ç£åããªãããé
ã¯ç£ç³ã«åŒãã€ããããªãã®ã§ãé
ã¯åŒ·ç£æ§äœã§ã¯ãªãã",
"title": "ç£å"
},
{
"paragraph_id": 184,
"tag": "p",
"text": "éé»èªå°ãå©çšãããéé»é®èœ(ããã§ããããžã)ãšèšããããäžç©ºã®å°äœãã€ãã£ãŠç©è³ªãå²ãããšã§å€éšé»å Žãé®èœããæ¹æ³ããã£ãã®ãšåæ§ã®ãç£æ°ã®é®èœãã匷ç£æ§äœã§ãåºæ¥ããäžç©ºã®åŒ·ç£æ§äœãçšããŠã匷ç£æ§äœã®å
éšã¯ç£å Žãé®èœã§ããããããç£æ°é®èœ(ãããããžããmagnetic shielding)ãšãããç£æ°ã·ãŒã«ããšãããã",
"title": "ç£å"
},
{
"paragraph_id": 185,
"tag": "p",
"text": "ç£å Žã®åããåããããã«å³ç€ºããããç£ç³ã®äœãç£å Žã®æ¹åã¯ãç ã«å«ãŸããç éã®ç²æ«ãç£ç³ã«ãã¡ãã°ããŠããµããããããšã§èŠ³å¯ã§ããã",
"title": "ç£å"
},
{
"paragraph_id": 186,
"tag": "p",
"text": "ãããå³ç€ºãããšãäžå³ã®ããã«ãªãã(ç»åçŽ æã®ç¢ºä¿ã®éœåäžãåçãšå³ç€ºãšã§ã¯ãN極ãšS極ãéã«ãªã£ãŠããŸããã容赊ãã ããã)",
"title": "ç£å"
},
{
"paragraph_id": 187,
"tag": "p",
"text": "ãã®ãããªç£å Žã®å³ãç£åç·(ãããããããmagnetic line of force)ãšãããç£åç·ã®åãã¯ãç£ç³ã®N極ããç£åç·ãåºãŠãS極ã«ç£åç·ãåžåããããšå®çŸ©ããããæ£ç£ç³ã§ã¯ãç£åã®çºçæºãšãªãå Žæããæ£ç£ç³ã®äž¡ç«¯ã®å
端ä»è¿ã«éäžãããããã§ãæ£ç£ç³ã®äž¡ç«¯ã®å
端ä»è¿ãç£æ¥µ(ãããããmagnetic pole)ãšããã",
"title": "ç£å"
},
{
"paragraph_id": 188,
"tag": "p",
"text": "ãã®ãããªç£ç³ã®ã€ããç£åç·ã®åœ¢ã¯ãé»æ°åç·ã§ã®ãç°ç¬Šå·ã®é»è·ã©ãããã€ããé»æ°åç·ã«äŒŒãŠããã",
"title": "ç£å"
},
{
"paragraph_id": 189,
"tag": "p",
"text": "1ã€ã®æ£ç£ç³ã§ã¯N極(north pole)ã®ç£æ°ã®åŒ·ããšãS極(south pole)ã®ç£æ°ã®åŒ·ãã¯çããããŸããç£ç³ã«ã¯ãå¿
ãN極ãšS極ãšãååšãããN極ãšS極ã®ãã©ã¡ããçæ¹ã ããåãåºãããšã¯åºæ¥ãªããããšãæ£ç£ç³ãåæããŠããåæé¢ã«ç£æ¥µãåºçŸããããã®ãããªçŸè±¡ã®ãããçç±ã¯ãããããæ£ç£ç³ãæ§æãã匷ç£æ§äœã®ååã®1åãã€ãå°ããªç£ç³ã§ããããããå°ããªååã®ç£ç³ããããã€ãæŽåããŠã倧ããªæ£ç£ç³ã«ãªã£ãŠããããã§ããã",
"title": "ç£å"
},
{
"paragraph_id": 190,
"tag": "p",
"text": "ä»®æ³çã«ãç£æ¥µãS極ãŸãã¯N極ã®çæ¹ã ãçŸããçŸè±¡ãçè«èšç®ã®ããã«èããããšããããããã®ãããªçåŽã ãã®ç£æ¥µãåç£æ¥µ(ã¢ãããŒã«ãšããã)ãšããããåç£æ¥µã¯å®åšããªãã",
"title": "ç£å"
},
{
"paragraph_id": 191,
"tag": "p",
"text": "æ£ç£ç³ãªã©ããã®ãçåŽã®ç£æ¥µãããã®ãç£æ¥µããã®ç£å Žã®åŒ·ãã®ããšãããã®ãŸãŸãç£æ¥µã®åŒ·ãã(Magnetic charge)ãšåŒã¶ããããã¯ç£è·(ãããmagnetization)ãç£æ°éãšããã",
"title": "ç£å"
},
{
"paragraph_id": 192,
"tag": "p",
"text": "ããããããã®ç£åãšç£å Žã®é¢ä¿ãåŒã§è¡šãããšãèããã ãŸããæ£ç£ç³ã«ã¯ç£æ¥µãäž¡åŽã«2åããã®ã§ãèšç®ãç°¡åã«ããããã«ãæ£ç£ç³ã®äž¡ç«¯ã®è·é¢ã倧ãããå察åŽã®ç£æ¥µã®å€§ãããç¡èŠã§ããç£ç³ãèãããã",
"title": "ç£å"
},
{
"paragraph_id": 193,
"tag": "p",
"text": "ãã®ãããªç£ç³ãçšããŠãå®éšãããšããã次ã®æ³åãåãã£ããç£åã®åŒ·ãã¯2åã®ç©äœã®ç£æ°ém1ããã³m2ã«æ¯äŸãã2åã®ç©äœéã®è·é¢rã®2ä¹ã«åæ¯äŸããã åŒã§è¡šããšã",
"title": "ç£å"
},
{
"paragraph_id": 194,
"tag": "p",
"text": "ã§è¡šãããã(kmã¯æ¯äŸå®æ°) ãããçºèŠè
ã®ã¯ãŒãã³ã®åã«ã¡ãªãã§ãç£æ°ã«é¢ããã¯ãŒãã³ã®æ³åãšãããç£æ°émã®åäœã¯ãŠã§ãŒããšãããèšå·ã¯[Wb]ã§è¡šãã",
"title": "ç£å"
},
{
"paragraph_id": 195,
"tag": "p",
"text": "æ¯äŸå®æ°kmãš1ãŠã§ãŒãã®å€§ãããšã®é¢ä¿ã¯ã1ã¡ãŒãã«é¢ãã1wbã©ããã®ç£æ¥µã«ã¯ãããåãçŽ6.33Ã10ãšããŠã æ¯äŸä¿æ°kmã¯ã",
"title": "ç£å"
},
{
"paragraph_id": 196,
"tag": "p",
"text": "ã§ããã",
"title": "ç£å"
},
{
"paragraph_id": 197,
"tag": "p",
"text": "ã€ãŸãã",
"title": "ç£å"
},
{
"paragraph_id": 198,
"tag": "p",
"text": "ã§ããã",
"title": "ç£å"
},
{
"paragraph_id": 199,
"tag": "p",
"text": "éé»æ°åã«å¯ŸããŠãé»å Žãå®çŸ©ãããããã«ãç£æ°åã«å¯ŸããŠããå Žãå®çŸ©ããããšéœåãè¯ããç£æ°ém1[Wb]ãäœãã次ã®éãç£å Žã®åŒ·ããããã¯ç£å Žã®å€§ãããšèšããèšå·ã¯Hã§è¡šãã",
"title": "ç£å"
},
{
"paragraph_id": 200,
"tag": "p",
"text": "ç£å Žã®åŒ·ãHã®åäœã¯[N/Wb]ã§ãããHãçšãããšãç£æ°ém2[Wb]ã«ã¯ãããç£æ°åf[N]ã¯ã",
"title": "ç£å"
},
{
"paragraph_id": 201,
"tag": "p",
"text": "ãšè¡šããã",
"title": "ç£å"
},
{
"paragraph_id": 202,
"tag": "p",
"text": "ç©çåŠè
ã®ãšã«ã¹ãããã¯ãé»æµã®å®éšãããŠããéã«ãããŸããŸè¿ãã«ãããŠãã£ãæ¹äœç£ç³ãåãã®ã確èªããã圌ã詳ãã調ã¹ãçµæã以äžã®ããšãåãã£ãã",
"title": "é»æµãã€ããç£å Ž"
},
{
"paragraph_id": 203,
"tag": "p",
"text": "é»æµãæµããŠãããšãã«ã¯ããã®ãŸããã«ã¯ãç£å Žãçãããåãã¯ãé»æµã®æ¹åã«å³ãããé²ãããã«ãå³ãããåãåããšåããªã®ã§ããããå³ããã®æ³åãšããã",
"title": "é»æµãã€ããç£å Ž"
},
{
"paragraph_id": 204,
"tag": "p",
"text": "ã¢ã³ããŒã«ããç£å Žã®å€§ããã調ã¹ãçµæãç£å Žã®å€§ããHã¯ãé»æµI[A]ãçŽç·çã«æµããŠãããšããçŽç·é»æµã®åšãã®ç£å Žã®å€§ããã¯ãå°ç·ããã®è·é¢ãa[m]ãšãããšãç£å Žã®å€§ããH[N/Wb]ã¯ã",
"title": "é»æµãã€ããç£å Ž"
},
{
"paragraph_id": 205,
"tag": "p",
"text": "ã§ããããšãç¥ãããŠããã",
"title": "é»æµãã€ããç£å Ž"
},
{
"paragraph_id": 206,
"tag": "p",
"text": "ãããã¢ã³ããŒã«ã®æ³å(Ampere's law) ãšããã ç£å Žã®å€§ããHã®åäœã¯ã[N/Wb]ã§ãããããã£ãœãã¢ã³ããŒã«ã®æ³åã®åŒãã¿ãã°ã¢ã³ãã¢æ¯ã¡ãŒãã«[A/m]ã§ãããã",
"title": "é»æµãã€ããç£å Ž"
},
{
"paragraph_id": 207,
"tag": "p",
"text": "å°ç·ãã³ã€ã«ç¶ã«å·»ãã°ãã¢ã³ããŒã«ã®æ³åã§å°ç·ã®åšå²ã«çºçããç£å Žãéãªãããããã®ããã«ããç£å Žã匷ããã³ã€ã«ãé»ç£ç³(ã§ããããããelectromagnet)ãšãããå°ç·ã«é»æµãæµããŠãããšãã«ã®ã¿ãé»ç£ç³ã¯ç£å Žãçºçãããå°ç·ã«é»æµãæµãã®ãæ¢ãããšãé»ç£ç³ã®ç£å Žã¯æ¶ããã",
"title": "é»æµãã€ããç£å Ž"
},
{
"paragraph_id": 208,
"tag": "p",
"text": "ç£å Žã®å€§ããHã«ã次ã®ç¯ã§æ±ãããŒã¬ã³ãåã®çŸè±¡ã®ãããæ¯äŸä¿æ°ÎŒ(åäœã¯ãã¥ãŒãã³æ¯ã¢ã³ãã¢ã§[N/A])ãæããŠãèšå·Bã§è¡šãã",
"title": "é»æµãã€ããç£å Ž"
},
{
"paragraph_id": 209,
"tag": "p",
"text": "ãšããããšãããããã®éBãç£æå¯åºŠ(magnetic flux density)ãšãããç£å Žã®å€§ããHã®åããšç£æå¯åºŠBã®åãã¯åãåãã§ããã ãŸããç£å Žã®å€§ããHãšç£æå¯åºŠBã®æ¯äŸä¿æ°ãéç£ç(ãšãããã€ãmagnetic permeability)ãšããã (ããŒã¬ã³ãåã«é¢ããŠã¯ã詳ããã¯ç©çIIã§æ±ããèªè
ãç©çIãåŠã¶åŠå¹Žãªãã°ãèªè
ã¯ãããŒã¬ã³ãåãšããåãããã®ã ãªã»ã»ã»ããšã§ãæã£ãŠããã°ããã)",
"title": "é»æµãã€ããç£å Ž"
},
{
"paragraph_id": 210,
"tag": "p",
"text": "ãŸããå°ç·ãçšæãããšãããããã®å°ç·ã¯ãéæ¢ããŠãããšããŠãéæ¢ããŠããããåºå®ã¯ããã«ãããå°ç·ã«åãå ããã°ãå°ç·ãåããããã«ããŠããšãããã",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 211,
"tag": "p",
"text": "ãã®å°ç·ã«é»æµãæµããã ãã§ã¯ãã¹ã€ã«å°ç·ã¯åããªãããããããã®å°ç·ã«ãå€éšã®ç£ç³ã«ããç£å Žãå ãããšãå°ç·ãåãããã®ãããªãç£å Žãšé»æµã®çžäºäœçšã«ãã£ãŠãå°ç·ã«çããåãããŒã¬ã³ãå(ããŒã¬ã³ãããããè±: Lorentz force)ãšããã",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 212,
"tag": "p",
"text": "ããŒã¬ã³ãåã®åãã¯ãå°ç·ã®é»æµã®åããšç£å Žã®åãã«åçŽã§ãããé»æµIã®åãããç£æå¯åºŠBã®åãã«å³ãããåãåããšåãã§ããã",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 213,
"tag": "p",
"text": "ãŸããããŒã¬ã³ãåã®å€§ããã¯ãå°ç·ã®é·ãlãšãç£å Žã®å°ç·ãšã®åçŽæ¹åæåã«æ¯äŸããã",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 214,
"tag": "p",
"text": "ããŒã¬ã³ãåã®å€§ããF[N]ãåŒã§è¡šãã°ãé»æµãšç£å ŽãšãåçŽã ãšããŠãç£å ŽãåããŠããå°ç·ã®åœ¢ç¶ãçŽç·åœ¢ã ãšããŠãé»æµãI[A]ãšããŠãå°ç·ã®é·ããl[m]ãšããŠãå°ç·ã«ããã£ãŠããå€éšç£å Žã®ç£æå¯åºŠãB[N/(Aã»m)]ãšããã°ã",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 215,
"tag": "p",
"text": "ã§è¡šããã",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 216,
"tag": "p",
"text": "ããŒã¬ã³ãåã®å
¬åŒã«ãã¯ãŒãã³ã®æ³åãªã©ã§ã¯èŠããããããªæ¯äŸä¿æ°(ä¿æ°Kãªã©ã)ãå«ãŸããªãã®ã¯ãããããããã®ããŒã¬ã³ãåã®çŸè±¡ãå
ã«ãç£æ°éãŠã§ãŒãWbã®åäœããã³ç£æå¯åºŠBã®åäœãã決å®ãããŠããããã§ããã",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 217,
"tag": "p",
"text": "ãŸãããç£æå¯åºŠãã®å称ãããç£æãã»ãå¯åºŠããšããã®ã¯ãå®ã¯ç£æå¯åºŠã®åäœã®[N/(Aã»m)]ã¯ãåäœãåŒå€åœ¢ãããš[Wb/m]ã§ãããããšãç±æ¥ã§ããããã®åäœ[Wb/m]ããé»æ°å·¥åŠè
ã®ãã¹ã©ã®åã«ã¡ãªã¿ãåäœ[Wb/m] ããã¹ã©ãšèšããèšå·Tã§è¡šãã",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 218,
"tag": "p",
"text": "ãã®ããŒã¬ã³ãåã®çŸè±¡ããé»æ°æ©åšã®ã¢ãŒã¿(é»åæ©)ã®åçã§ããã",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 219,
"tag": "p",
"text": "",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 220,
"tag": "p",
"text": "ãªããããã¬ãã³ã°ã®æ³åããšããããŒã¬ã³ãåã«é¢ããæ³åãããããããŒã¬ã³ãåã®èšç®ã«ã¯å®çšçã§ã¯ç¡ããããã¬ãã³ã°ã®åãé¢ããç°ãªãæ³åã幟ã€ããã£ãŠçŽããããééãã®åå ã«ãªããããã®ã§ãæ¬æžã§ã¯æããªãã å®éã«ãå°éçãªç©çèšç®ã§ã¯ããã¬ãã³ã°ã®æ³åã¯ãèšç®ã«ã¯çšããªãã",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 221,
"tag": "p",
"text": "ãããããã¬ãã³ã°ã®æ³åã«ã¯ããã¬ãã³ã°ã®å³æã®æ³åããšããããšã¯ç°ãªãããã¬ãã³ã°ã®å·Šæã®æ³åãããããã©ã¡ãããã©ã®ç£æ°ã®çŸè±¡ã«çšããæ³åã ã£ãã®ããééãããããã ãããæ¬æžã§ã¯æããªãã",
"title": "ããŒã¬ã³ãå"
},
{
"paragraph_id": 222,
"tag": "p",
"text": "(é»ç£èªå°ã«é¢ããŠã¯ã詳ããã¯ç©çIIã§æ±ãã)",
"title": "é»ç£èªå°"
},
{
"paragraph_id": 223,
"tag": "p",
"text": "ã¢ã³ããŒã«ã®æ³åã§ã¯ãé»æµã®åšãã«ç£å Žãã§ããã®ã§ãã£ãã",
"title": "é»ç£èªå°"
},
{
"paragraph_id": 224,
"tag": "p",
"text": "å®ã¯ãç£ç³ãåãããªã©ããŠãç£å Žã䌎ãç©äœãéåãããšããã®ãŸããã«ã¯é»å Žãçããã ä»®ã«ãã³ã€ã«ã®è¿ãã§ãããè¡ãªã£ããšãããšãçããé»å Žã«ãã£ãŠã³ã€ã«ã®äžã«ã¯é»æµãæµããã çããé»å Žã®å€§ããã¯ã",
"title": "é»ç£èªå°"
},
{
"paragraph_id": 225,
"tag": "p",
"text": "ãšãªãã(ååŸaã®å圢ã®ã³ã€ã«ã®å Žåã) Eã®åäœã¯[V/m]ã§ãããBã®åäœã¯[T]ã§ããã",
"title": "é»ç£èªå°"
},
{
"paragraph_id": 226,
"tag": "p",
"text": "ãã®çŸè±¡ãé»ç£èªå°(ã§ããããã©ããelectromagnetic induction)ãšãããé»ç£èªå°ã«ãã£ãŠçºçããé»æµãèªå°é»æµãšããã",
"title": "é»ç£èªå°"
},
{
"paragraph_id": 227,
"tag": "p",
"text": "ãŸããèªå°é»æµã®åãã¯ãç£ç³ã®åãã«ãããã³ã€ã«ã®äžãéãç£æã®å€åã劚ããåãã«ãé»æµãæµããã(èªå°é»æµãã¢ã³ããŒã«ã®æ³åã«åŸããåšå²ã«ç£å Žãäœãã) ãã®èªå°é»æµããã³ã€ã«ã®äžãéãç£æã®å€åã劚ããåãã«èªå°é»æµãæµããçŸè±¡ãã¬ã³ãã®æ³å(Lenz's law)ãšããã",
"title": "é»ç£èªå°"
},
{
"paragraph_id": 228,
"tag": "p",
"text": "",
"title": "é»ç£èªå°"
},
{
"paragraph_id": 229,
"tag": "p",
"text": "åãé åã«Nåå·»ãããã³ã€ã«ã眮ãããå Žåããã¡ã©ããŒã®é»ç£èªå°ã®æ³åã¯ã次ã®ããã«ãªãã",
"title": "é»ç£èªå°"
},
{
"paragraph_id": 230,
"tag": "p",
"text": "ããã§ã E {\\displaystyle {\\mathcal {E}}} ã¯èµ·é»å(ãã«ã ãèšå·ã¯V)ãΊB ã¯ç£æ(ãŠã§ãŒããèšå·ã¯Wb)ãšãããNã¯é»ç·ã®å·»æ°ãšããã",
"title": "é»ç£èªå°"
},
{
"paragraph_id": 231,
"tag": "p",
"text": "ãã®é»ç£èªå°ã®çŸè±¡ããç«åçºé»ãæ°Žåçºé»ãªã©ã®çºé»æ©ã®åçã§ãããããçã®çºé»ã§ã¯ãæ°žä¹
ç£ç³ãå転ãããããšã§ãçºé»ãããŠãããç«åãæ°Žåãšããã®ã¯ãæ©åšã®å転ãåŸãæ段ã«ãããªãããŸããçºé»æã®çºé»ã«ã¯ãæ°žä¹
ç£ç³ã®å転ãå©çšããŠãããããçºçããé»å§ãé»æµã¯åšæçãªæ³¢åœ¢ã«ãªãã次ã«èª¬æãã亀æµæ³¢åœ¢ã«ãªãã",
"title": "é»ç£èªå°"
},
{
"paragraph_id": 232,
"tag": "p",
"text": "åè·¯ãžã®å
¥åé»å§ãåšæçã«æéå€åããåè·¯ã®é»å§ããã³é»æµã亀æµ(alternating current)ãšãããããã«å¯Ÿãã也é»æ± ãªã©ã«ãã£ãŠçºçããé»å§ãé»æµã®ããã«ãæéã«ãããäžå®ãªé»å§ãé»æµã¯çŽæµ(direct Current)ãšããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 233,
"tag": "p",
"text": "亀æµæ³¢åœ¢ãäœç§ã§1åšããããšããæéãåšæ(wave period)ãšãããåšæã®èšå·ã¯ T {\\displaystyle T} ã§è¡šãåäœã¯ç§[s]ã§ããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 234,
"tag": "p",
"text": "1ç§éã«æ³¢åœ¢ãäœåšããããšããåæ°ãåšæ³¢æ°ãããã¯æ¯åæ°(è±èªã¯ããšãã«frequency)ãšããã é»æ°ã®æ¥çã§ã¯åšæ³¢æ°ãšããçšèªãçšããããšãå€ããç©çã®æ³¢ã®çè«ã§ã¯æ¯åæ°ãšããè¡šçŸãçšããããšãå€ãã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 235,
"tag": "p",
"text": "åšæ³¢æ°ã®åäœã¯[1/s]ã§ããããããããã«ã(hertz)ãšããåäœã§è¡šããåäœèšå·HzãçšããŠåšæ³¢æ°fããf[Hz]ãšãããµãã«è¡šãã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 236,
"tag": "p",
"text": "亀æµé»æµã亀æµé»å§ãæ£åŒŠæ³¢ã®å Žåã¯ããããã®ãã©ã¡ãŒã¿ãçšããŠ",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 237,
"tag": "p",
"text": "ãšæžãããšãã§ããã sinãšã¯äžè§é¢æ°ã§ãããç¥ããªããã°æ°åŠIIãªã©ãåèã«ããã ãã®ãšãã®sinã®ä¿æ° I 0 {\\displaystyle I_{0}} ã V 0 {\\displaystyle V_{0}} ãæ¯å¹
(ããã·ããamplitude)ãšããããŸãæå»t=0ã«ãããé»æµãé»å§ã®å€ã瀺ããæé波圢ã決å®ãã Ξ i {\\displaystyle \\theta _{i}} ã Ξ v {\\displaystyle \\theta _{v}} ãåæäœçžãšããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 238,
"tag": "p",
"text": "æ®éç§é«æ ¡ã®é«æ ¡ç©çã§ã¯ã亀æµæ³¢åœ¢ã®èšç®ã«ã¯ãæ£åŒŠæ³¢ã®å Žåãäž»ã«æ±ããæ¹åœ¢æ³¢ãäžè§æ³¢ã®èšç®ã¯ãæ®éã¯æ±ãããªãã ãã ããå·¥æ¥é«æ ¡ã®ææ¥ããå·¥å Žã®å®åã§ã¯æ±ãããšãããã®ã§ãèªè
ã¯æ³¢åœ¢ãåŠãã§ããããšã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 239,
"tag": "p",
"text": "çºé»æããäžè¬å®¶åºã«éãããŠããé»å§ã¯äº€æµé»å§ã§ãããæ±æ¥æ¬ã§ã¯50Hzã§ããã西æ¥æ¬ã§ã¯60Hzã§ãããããã¯ææ²»æ代ã®çºé»æ©ã®èŒžå
¥æã«ãæ±æ¥æ¬ã®äºæ¥è
ã¯ãšãŒããããã50Hzçšã®çºé»æ©ã茞å
¥ãã西æ¥æ¬ã®äºæ¥è
ã¯ã¢ã¡ãªã«ãã60Hzã®çºé»æ©ã茞å
¥ããããšã«ããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 240,
"tag": "p",
"text": "çºé»æããäžè¬ã®å®¶åºãªã©ã«éãããé»æµã®åšæ³¢æ°ãåçšåšæ³¢æ°ãšããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 241,
"tag": "p",
"text": "åçšé»æºã®é»å§æ¯å¹
ã¯çŽ140Vã§ããããã㯠100 à 2 {\\displaystyle 100\\times {\\sqrt {2}}} [V]ã§ããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 242,
"tag": "p",
"text": "ãããã«ããšã¯1000Hzã®ããšã§ããããããã«ãã¯kHzãšæžãã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 243,
"tag": "p",
"text": "亀æµé»æµã«å¯ŸããŠã¯ãé»æµãšåãæ¯åæ°ã§ãã¢ã³ããŒã«ã®æ³åã§çºçããç£å Žãæ¯åããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 244,
"tag": "p",
"text": "å°ç·ã§ã€ããããã³ã€ã«ã¯ãçŽæµé»æµã§ã¯ããã ã®å°ç·ãšããŠã¯ãããããããã亀æµé»æµã«å¯ŸããŠã¯ãé»ç£èªå°ã«ããèªå·±ã®çºçãããç£å Žã劚ãããããªé»æµããã³èµ·é»åãçºçããããããèªå·±èªå°(self induction)ãšããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 245,
"tag": "p",
"text": "èªå·±èªå°ã«ããèµ·é»åã®å€§ããã¯ãé»æµã®æéå€åçã«æ¯äŸãããèªå·±èªå°ã®èµ·é»åãåŒã§æžãã°ãæ¯äŸä¿æ°ãLãšããŠã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 246,
"tag": "p",
"text": "ã§ããã ãã®æ¯äŸä¿æ° L {\\displaystyle L} ãèªå·±ã€ã³ãã¯ã¿ã³ã¹(self inductance)ãšãããèªå·±ã€ã³ãã¯ã¿ã³ã¹ã®æ¬¡å
ã¯[Vã»S/m]ã ããããããã³ãªãŒãšããåäœã§è¡šããåäœã«Hãšããèšå·ãçšããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 247,
"tag": "p",
"text": "éå¿ã«äºã€ã®ã³ã€ã«ãå·»ããã³ã€ã«ã®çæ¹ã®é»æµãå€åããããšãã¢ã³ããŒã«ã®æ³åã«ãã£ãŠçããŠããç£æãå€åãããããå察åŽã®ã³ã€ã«ã«ã¯ããã®ç£æå¯åºŠã®å€åãæã¡æ¶ããããªåãã«èµ·é»åãçºçããããã®çŸè±¡ãçžäºèªå°(mutual induction)ãšèšãã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 248,
"tag": "p",
"text": "é»å§ãå
¥åãããåŽã®ã³ã€ã«ã1次ã³ã€ã«(primaly coil)ãšèšããèªå°èµ·é»åãçºçãããåŽã®ã³ã€ã«ã2次ã³ã€ã«(secondary coil)ãšããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 249,
"tag": "p",
"text": "çžäºèªå°ã«ããèµ·é»åã®å€§ããã¯ãé»æµã®æéå€åçã«æ¯äŸãããçžäºèªå°ã®èµ·é»åãåŒã§æžãã°ãæ¯äŸä¿æ°ãMãšããŠã(çžäºèªå°ã®æ¯äŸä¿æ°ã¯Lã§ã¯ç¡ãã)åŒã¯ã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 250,
"tag": "p",
"text": "ã§ããã ãã®æ¯äŸä¿æ° M {\\displaystyle M} ãçžäºã€ã³ãã¯ã¿ã³ã¹(self inductance)ãšãããçžäºã€ã³ãã¯ã¿ã³ã¹ã®æ¬¡å
ã¯ãèªå·±ã€ã³ãã¯ã¿ã³ã¹ã®åäœãšåãã§ãã³ãªãŒ(H)ã§ããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 251,
"tag": "p",
"text": "ãã®çžäºã€ã³ãã¯ã¿ã³ã¹ã®å€§ããã¯ãäž¡æ¹ã®ã³ã€ã«ã®å·»ãæ°ã©ããã®ç©ã«æ¯äŸããã",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 252,
"tag": "p",
"text": "",
"title": "亀æµåè·¯"
},
{
"paragraph_id": 253,
"tag": "p",
"text": "ç£å Žã®åãã«ãã£ãŠé»å ŽãåŒãèµ·ããããããšãé»ç£èªå°ã®ã»ã¯ã·ã§ã³ã§èŠãã å®éã«ã¯é»å Žã®å€åã«ãã£ãŠç£å ŽãåŒãèµ·ããããããšãç¥ãããŠããã ããã«ãã£ãŠäœããªã空éäžãé»å Žãšç£å ŽãäŒæããŠããããšãäºæ³ãããã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 254,
"tag": "p",
"text": "é»ç£æ³¢ã®é床ãç©çåŠè
ã®ãã¯ã¹ãŠã§ã«ãèšç®ã§æ±ãããšãããé»ç£æ³¢ã®é床ã¯ãç空äžã§ã¯åžžã«äžå®ã§ããã€æ³¢ã®é床cãèšç®ã§æ±ãããšããã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 255,
"tag": "p",
"text": "ãšãªããæ¢ã«ç¥ãããŠããå
éã«äžèŽããã ãã®ããšãããå
ã¯é»ç£æ³¢ã®äžçš®ã§ããããšãåãã£ããç©çIIã§ãé»ç£æ³¢ã®é床ãæ±ããèšç®ã¯ã詳ããã¯æ±ãã èªè
ãå
éã®æž¬å®å®éšã«ã€ããŠèª¿ã¹ããªããç©çIã®æ³¢åã«é¢ããããŒãžãªã©ã§ãã£ãŸãŒã®å®éšã«ã€ããŠãåç
§ã®ããšã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 256,
"tag": "p",
"text": "æ³¢ã¯æ³¢é·Î»ãé·ãã»ã©ãæ¯åæ°fãå°ãããªããæ³¢ã®æ³¢é·Î»ãšæ¯åæ°fã®ç©fλã¯äžå®ã§ãããã¯æ³¢ã®é床vã«çãããã€ãŸã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 257,
"tag": "p",
"text": "ã§ããã é»ç£æ³¢ã®å Žåã¯ãé床ãå
éã®cãªã®ã§",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 258,
"tag": "p",
"text": "ã§ããã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 259,
"tag": "p",
"text": "æŸéçšã®ãã¬ããã©ãžãªã®é»æ³¢(ã§ãã±ãradio wave)ã¯ãé»ç£æ³¢(electromagnetic wave)ã®äžçš®ã§ãããæ³¢é·ã0.1mm以äžã®é»ç£æ³¢ãé»æ³¢ã«åé¡ãããããªããé»æ³¢ã®ãã¡ãæ³¢é·ã1mm~1cmã®ããªã¡ãŒãã«ã®é»æ³¢ãããªæ³¢ãšãããåæ§ã«ãæ³¢é·ã1cm~10cmã®é»æ³¢ãã»ã³ãæ³¢ãšãããæ³¢é·10cm~100cm(=1m)ã®é»æ³¢ã¯UHFãšèšããããã¬ãæŸéãªã©ã«äœ¿ãããUHFæŸéã¯ããã®é»æ³¢ã§ãããæ³¢é·1m~10mã®é»æ³¢ã¯VHFãšèšãããããã¬ãæŸéã®VHFæŸéã¯ããã®é»æ³¢ã§ããã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 260,
"tag": "p",
"text": "æ³¢é·ã0.1mm以äžã§ãå¯èŠå
ç·(å¯èŠå
ã®æ倧波é·ã¯780ããã¡ãŒãã«çšåºŠ)ãããã¯æ³¢é·ãé·ãé»ç£æ³¢ã¯èµ€å€ç·(ãããããããinfrared raysãã€ã³ãã©ã¬ãŒã ã¬ã€ãº)ãšããããèµ€ãã®ãå€ããšããçç±ã¯ãå¯èŠå
ã®æ倧波é·ã®è²ãèµ€è²ã ããã§ãããèµ€å€ç·ãã®ãã®ã«ã¯è²ã¯ã€ããŠããªããåžè²©ã®èµ€å€ç·ããŒã¿ãŒãªã©ãèµ€è²ã«çºå
ãã補åãããã®ã¯ã䜿çšè
ãåäœç¢ºèªãã§ããããã«ããããã«ã補åã«èµ€è²ã®ã©ã³ãã䜵眮ããŠããããã§ãããèµ€å€ç·ã¯ãç©äœã«åžåããããããåžåã®éãç±ãçºçããã®ã§ãããŒã¿ãŒãªã©ã«å¿çšãããããªãã倪éœå
ã«ãèµ€å€ç·ã¯å«ãŸããã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 261,
"tag": "p",
"text": "ããããèµ€å€ç·ãçºèŠãããçµç·¯ã¯ãã€ã®ãªã¹ã®å€©æåŠè
ã®ããŒã·ã§ã«ã倪éœå
ãããªãºã ã§åå
ããéã«ãèµ€è²ã®å
ç·ã®ãšãªãã®ãç®ã«ã¯è²ãèŠããªãéšåã枩床äžæããŠããããšãçºèŠããããšããçµç·¯ãããã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 262,
"tag": "p",
"text": "",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 263,
"tag": "p",
"text": "æã
ã人éã®ç®ã«èŠããå¯èŠå
ç·(ãããããããvisible light)ã®æ³¢é·ã¯ãçŽ780ããã¡ãŒãã«ããçŽ380ããã¡ãŒãã«ã®çšåºŠã§ãããå¯èŠå
ã®äžã§æ³¢é·ãæãé·ãé åã®è²ã¯èµ€è²ã§ãããå¯èŠå
ã®äžã§æ³¢é·ãæãçãé åã®è²ã¯çŽ«è²ã§ããã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 264,
"tag": "p",
"text": "å
ãã®ãã®ã«ã¯ãè²ã¯ã€ããŠããªããæã
ã人éã®è³ããç®ã«å
¥ã£ãå¯èŠå
ããè²ãšããŠæããã®ã§ããã 倪éœå
ãããªãºã ãªã©ã§åå
(ã¶ããã)ãããšãæ³¢é·ããšã«è»è·¡(ããã)ãããããããã®åå
ããå
ç·ã¯ãä»ã®æ³¢é·ãå«ãŸãããã äžçš®ã®æ³¢é·ãªã®ã§ããã®ãããªå
ç·ããã³å
ãåè²å
(monochromatic light)ãšããã ãŸããçœè²ã¯åè²å
ã§ã¯ãªããçœè²å
(white light)ãšã¯ãå
šãŠã®è²ã®å
ãæ··ãã£ãç¶æ
ã§ããã åæ§ã«ãé»è²ãšããåè²å
ããªããé»è²ãšã¯ãå¯èŠå
ãç¡ãç¶æ
ã§ããã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 265,
"tag": "p",
"text": "玫å€ç·(ããããããultraviolet rays)ã¯ååŠåå¿ã«åœ±é¿ãäžããäœçšã匷ãã殺èæ¶æ¯ãªã©ã«å¿çšãããã倪éœå
ã«ã玫å€ç·ã¯å«ãŸããã人éã®èã®æ¥çŒãã®åå ã¯ã玫å€ç·ãã¡ã©ãã³è²çŽ ãé
žåãããããã§ããã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 266,
"tag": "p",
"text": "èµ€å€ç·ã¯å€ªéœå
ã®ããªãºã ã«ããåå
ã§çºèŠãããã ãã§ã¯ãåå
ããã玫è²ã®å
ç·ã®ãšãªãã«ãããªã«ãç®ã«ã¯èŠããªãç·ãããã®ã§ã¯?ããšãããµããªããšãåŠè
ãã¡ã«ãã£ãŠèãããã ãã€ãã®ç©çåŠè
ãªãã¿ãŒã«ããååŠçãªå®éšæ¹æ³ãçšããŠã玫å€ç·ã®ååšãå®èšŒãããã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 267,
"tag": "p",
"text": "å»ççšã®ã¬ã³ãã²ã³ãªã©ã®ééåçã§çšããããXç·(X-ray)ãé»ç£æ³¢ã®äžçš®ã§ãããçç©ã®çŽ°èãååã¬ãã«ã§å·ã€ããçºããæ§ãæãã ã¬ã³ãç·(gammaârayãγ ray)ãåæ§ã«ãééåçã«ãå¿çšãããããçç©ã®çŽ°èãååã¬ãã«ã§å·ã€ããçºããæ§ãæãã",
"title": "é»ç£æ³¢"
},
{
"paragraph_id": 268,
"tag": "p",
"text": "??",
"title": "é»ç£æ³¢"
}
] | é«çåŠæ ¡ ç©çåºç€ > é»æ° æ¬é
ã¯é«çåŠæ ¡ ç©çåºç€ã®é»æ°ãšç£æ°ã®è§£èª¬ã§ããã | <small>[[é«çåŠæ ¡ ç©çåºç€]] > é»æ°</small>
----
æ¬é
ã¯[[é«çåŠæ ¡ ç©çåºç€]]ã®é»æ°ãšç£æ°ã®è§£èª¬ã§ããã
==é»æ°==
===ç掻ã®äžã®é»æ°===
====é»æ°ãšç掻====
çŸä»£ç€ŸäŒã«ãããŠãç§ãã¡ã®æ¥åžžç掻ã«ã¯é»æ°ãæ¬ ãããªããå€ãã®è£œåãé»æ°ãçšããŠåäœããŠããããã®çç±ã¯ããŸããŸã ããäž»ãªèŠå ã¯ä»¥äžã®éãã ã
ãŸã第äžã«ãé»æ°ã¯æ§ã
ãªå¥ã®ãšãã«ã®ãŒåœ¢æ
ã«å€æã§ããããšãæãããããäŸãã°ãé»ç±ç·ã䜿çšããã°é»æ°ãšãã«ã®ãŒãç±ãšãã«ã®ãŒã«å€æããææ¿ã調çãªã©ã«å©çšããããšãã§ããããŸããé»çãçºå
ãã€ãªãŒãïŒLEDïŒã䜿çšããã°ãé»æ°ãšãã«ã®ãŒãå
ãšãã«ã®ãŒã«å€æããŠç
§æãè¡ãããšãã§ãããããã«ãé»åæ©ã䜿çšããã°ãé»æ°ãšãã«ã®ãŒãæ©æ¢°çãªéåãšãã«ã®ãŒã«å€æããŠæ§ã
ãªæ©åšã茞éæ段ãåããããšãã§ããã
次ã«ãé»æ± ãã³ã³ãã³ãµã䜿çšããŠãšãã«ã®ãŒã貯èµããæã¡éã¶ããšãã§ããç¹ãéèŠã ãããã«ãããã¢ãã€ã«ããã€ã¹ãæºåž¯é»è©±ãªã©ãé»æºã«æ¥ç¶ãããŠããªãå Žæã§ãé»æ°ãå©çšããããšãå¯èœãšãªãããŸããé»ç·ã䜿çšããŠé·è·é¢ãéé»ããããšãã§ãããããçºé»æãã家åºãå·¥å ŽãŸã§é»æ°ãäŸçµŠããããšãå¯èœã ã
ããã«ãé»å補åã¯èšç®èœåãä¿¡å·ã®äŒéèœåãåªããŠãããæ
å ±æè¡ãéä¿¡åéã§åºãå©çšãããŠããããŸããé»æ°ã¯æ¯èŒçã«å®å
šã«åãæ±ãããšãã§ããå°éã®ãšãã«ã®ãŒã§ãå¹ççã«å©çšããããšãã§ããç¹ã倧ããªå©ç¹ã ã
ãã®ããã«ãé»æ°ã¯ç§ãã¡ã®ç掻ã«ãããŠæ¬ ãããªããšãã«ã®ãŒæºãšãªã£ãŠãããæ§ã
ãªé¢ã§äŸ¿å©ããå¹çæ§ãæäŸããŠããã
====é»åæ©ãšçºé»æ©====
é»åæ©ãšçºé»æ©ã¯ãçŸä»£ã®é»æ°å·¥åŠã«ãããŠéèŠãªåœ¹å²ãæãããŠããããããã®è£
眮ã¯ãé»æ°ãšãã«ã®ãŒãšæ©æ¢°çãªéåãšãã«ã®ãŒã®çžäºå€æãå¯èœã«ããããŸããŸãªç£æ¥ãæ¥åžžç掻ã«ãããŠäœ¿çšãããŠããã
;é»åæ©:
:é»åæ©({{Lang-en-short|electric motor}})ã¯é»æ°ãšãã«ã®ãŒãæ©æ¢°çãªéåã«å€æããè£
眮ã ãäžè¬çãªé»åæ©ã¯ãå°ç·ãç£å Žå
ã§åãããšã«ãã£ãŠçºçããåïŒ[[#ããŒã¬ã³ãå|ããŒã¬ã³ãå]]ïŒãå©çšããŠåäœããããã®åçã«åºã¥ããé»åæ©ã¯çŽæµé»åæ©ãšäº€æµé»åæ©ã®äºçš®é¡ã«åãããããçŽæµé»åæ©ã¯çŽæµé»æºããã®é»åã䜿çšãã亀æµé»åæ©ã¯äº€æµé»æºããã®é»åã䜿çšãããé»åæ©ã¯ãå·¥å Žã®æ©æ¢°ãç£æ¥çšæ©åšã家åºçšå®¶é»è£œåãªã©ãããŸããŸãªçšéã«åºã䜿çšãããŠããã
;çºé»æ©:
:çºé»æ©({{Lang-en-short|generator}})ã¯ãéåãšãã«ã®ãŒãé»æ°ãšãã«ã®ãŒã«å€æããè£
眮ã ãäžè¬çãªçºé»æ©ã¯ãå°ç·ãç£å Žå
ã§åãããšã«ãã£ãŠé»æ°ãçºçããããäž»ã«å転éåãå©çšããŠé»æ°ãçæãããããçºé»æ©ã¯æ©æ¢°çãªãšãã«ã®ãŒãé»æ°ãšãã«ã®ãŒã«å€æããçºé»æã§åºã䜿çšãããŠãããçºé»æã§ã¯ãããŸããŸãªãšãã«ã®ãŒæºïŒæ°Žåã颚åãç«åãåååãªã©ïŒãå©çšããããããã®éåãšãã«ã®ãŒãçºé»æ©ã«ãã£ãŠé»æ°ãšãã«ã®ãŒã«å€æãããã
äž¡è
ã¯çžäºã«é¢é£ããŠãããé»åæ©ã¯çºé»æ©ãšåæ§ã®åçã§åäœããŸãããéã®ããã»ã¹ãè¡ãããã€ãŸããé»åæ©ã¯é»æ°ãšãã«ã®ãŒãæ©æ¢°çãªéåã«å€æããçºé»æ©ã¯æ©æ¢°çãªéåãé»æ°ãšãã«ã®ãŒã«å€æããããã®ããã«ãé»åæ©ãšçºé»æ©ã¯çŸä»£ã®ç£æ¥ãç掻ã«ãããŠæ¬ ãããªãè£
眮ã§ããããšãã«ã®ãŒã®å¹ççãªå©çšã«è²¢ç®ããŠããã
====çŽæµã»äº€æµãšé»æ³¢====
çŽæµãšäº€æµã¯é»æ°ã®æµãæ¹ãè¡šãçšèªã§ããããŸãé»æ³¢ã¯é»ç£æ³¢ã®äžçš®ã§ããã以äžã«ããããã®æŠèŠã瀺ãã
; çŽæµ (Direct Current, DC): é»æ± ãªã©ã®é»æºããäŸçµŠãããé»æµã®ãã¡ãé»æ¥µã®æ£æ¥µããè² æ¥µãžäžæ¹åã«æµããé»æµãæããçŽæµã¯äžå®ã®é»å§ãšæ¥µæ§ãæã¡ãäžå®ã®æ¹åã«æµããç¹æ§ãæã€ãçŽæµã®å©ç¹ã¯å®å®æ§ãšå¶åŸ¡ã®å®¹æãã§ãããé»æ± åŒæ©åšãäžéšã®é»åæ©åšã§äœ¿çšãããã
; äº€æµ (Alternating Current, AC): çºé»æãªã©ã®é»æºããäŸçµŠãããé»æµã®ãã¡ãå®æçã«æ£è² ãé転ããé»æµãæãã亀æµã¯å®æçã«æ³¢åœ¢ãå€åããããããé»åã®éé»ãå€å§ãå€æã容æã§ãããé·è·é¢éé»ã«é©ããŠããããŸãã家åºãå·¥æ¥çšé»æ°åè·¯ã§åºã䜿çšãããŠããã
å®éã«ã¯ãçŽæµãšäº€æµã¯æ©åšãåè·¯ã«ãã£ãŠçžäºå€æãããããšãããã
äŸãã°ããã€ãªãŒãã䜿çšããŠäº€æµãçŽæµã«å€æããæŽæµãè¡ãããã
; é»æ³¢ (Electromagnetic Waves)
: é»æ³¢ã¯ãé»ç£æ³¢ã®äžçš®ã§ãããé»å Žãšç£å Žãåšæçã«æ¯åããæ³¢åã ãé»æ³¢ã¯æ§ã
ãªåšæ³¢æ°ãæã¡ãããã«ãã£ãŠç°ãªãç¹æ§ãçšéããããé»æ³¢ã¯äž»ã«æŸéãéä¿¡ãç¡ç·éä¿¡ãã¬ãŒããŒãªã©ã®åéã§åºãå©çšãããŠããã
: é»æ³¢ã®æ¯åã¯ç©ºéãäŒæããé»ç£æ³¢ãšããŠé²è¡ãããé»æ³¢ã¯ç空äžã空æ°äžãäŒæãããããå°äœãå¿
èŠãšããªããããã¯ãé»æ³¢ãé»åã®ç§»åã«äŸåãããé»å Žãšç£å Žã®æ¯åã«ãã£ãŠçããããã ã
: é»æ³¢ã¯åšæ³¢æ°ã«ãã£ãŠåé¡ããããäœåšæ³¢æ°ã®é»æ³¢ã¯ãäž»ã«AMã©ãžãªãå°äžæ³¢ãã¬ããªã©ã®æŸéã«äœ¿çšããããäžæ¹ãé«åšæ³¢æ°ã®é»æ³¢ã¯ãFMã©ãžãªãæºåž¯é»è©±ãWi-Fiãè¡æéä¿¡ãªã©ã®éä¿¡ã«äœ¿çšãããããŸãã極è¶
çæ³¢ã®é»æ³¢ã¯ãã¬ãŒããŒããã¯ãæ³¢ãªãŒãã³ãªã©ã«å©çšãããã
: é»æ³¢ã¯ãã®ç¹æ§ãããæ
å ±ã®éåä¿¡ãç©äœã®æ¢ç¥ã枬å®ãªã©ã«å¹
åºãå¿çšãããŠããããããŠãçŸä»£ã®éä¿¡æè¡ãç¡ç·æè¡ã®çºå±ã«ãããŠéèŠãªåœ¹å²ãæãããŠããã
== éé»æ° ==
ãã©ã¹ããã¯ã®äžæ·ããªã©ã§é«ªãããããšåž¯é»ããçŸè±¡ãªã©ã®ããã«ãç©è³ªãé»æ°ã垯ã³ãããšã'''垯é»'''ïŒããã§ãïŒãšãããç©äœãããã£ãŠçºçãããéé»æ°ã'''æ©æŠé»æ°'''ãšããã
ã¬ã©ã¹æ£ãçµ¹ã®åžã§ããããšãã¬ã©ã¹æ£ã¯æ£ã®é»æ°ã«åž¯é»ããçµ¹ã¯è² ã®é»æ°ã«åž¯é»ããã
é»æ°ã®éã'''é»è·'''ïŒã§ãããchargeïŒãšããããããã¯'''é»æ°é'''ãšããã
é»è·ã®åäœã¯'''ã¯ãŒãã³'''ã§ãããã¯ãŒãã³ã®èšå·ã¯Cã§ããã
éé»æ°ã«ããé»è·ã©ããã«åãåã'''éé»æ°å'''ãšããã
ãªãã垯é»ããŠããªãç¶æ
ãé»æ°çã«äžæ§ã§ããããšããã
éå±ã®ããã«ãé»æ°ãéããç©äœã'''å°äœ'''ïŒã©ããããconductorïŒãšããããã©ã¹ããã¯ãã¬ã©ã¹ããŽã ã®ããã«é»æ°ãéããªãç©è³ªã'''絶çžäœ'''ïŒãã€ãããããinsulatorïŒãããã¯'''äžå°äœ'''ïŒãµã©ãããïŒãšããã
éå±ã¯å°äœã§ããã
é»æ°ã®æ£äœã¯'''é»å'''ïŒelectronïŒãšããç²åã§ããããã®é»åã¯è² é»è·ã垯ã³ãŠãããïŒé»åã®é»è·ãè² ã«å®çŸ©ãããŠããã®ã¯ã人é¡ãé»åãçºèŠããåã«é»è·ã®æ£è² ã®å®çŸ©ãè¡ãããããšããé»åãèŠã€ãã£ãéã«é»åã®é»è·ã調ã¹ããè² é»è·ã ã£ãããã§ãããïŒ
[[File:Metalic bond model.svg|thumb|400px|éå±äžã§ã®èªç±é»åã®æš¡åŒå³]]
éå±ãå°äœãªã®ã¯ãéå±äžã®é»åã¯ãããšã®ååãé¢ããŠããã®éå±å
šäœã®äžãèªç±ã«åããããã§ãããéå±äžã®é»åã®ããã«ãç©è³ªäžãèªç±ã«åããç¶æ
ã®é»åãã'''èªç±é»å'''ïŒãããã§ããïŒãšããã
é»æµãšã¯ãèªç±é»åã移åããããšã§ããã
ãã£ãœãã絶çžäœã¯ãèªç±é»åããããªãã絶çžäœã®é»åã¯ããã¹ãŠãããšã®ååã«æçžïŒããã°ãïŒãããŠéã蟌ããããŠããŠãèªç±ã«ã¯åããªãã
æ£é»è·ãšã¯ãç©è³ªã«é»åãæ¬ ä¹ããŠããç¶æ
ã§ããã
è² é»è·ãšã¯ãç©è³ªãé»åãå€ãæã£ãŠããç¶æ
ã§ããã
垯é»ããŠããªã絶çžäœã®ç©è³ªããããããããŠãäž¡æ¹ãæ©æŠé»æ°ã«åž¯é»ãããå Žåãçæ¹ã¯æ£é»è·ãçããããçæ¹ã®ç©è³ªã¯è² é»è·ãçããããã®ãšããçºçããæ£é»è·ã®å€§ãããšè² é»è·ã®å€§ããã¯åãã§ããã
ããã¯ãé»åã移åããŠãçæ¹ã®ç©è³ªã¯é»åãäžè¶³ããããçæ¹ã¯çéã®é»åãéå°ã«ãªã£ãŠããããã§ããã
ãã®ããã«ãé»åã¯çæãæ¶æ»
ãããªããããã'''é»è·ä¿åå'''ãããã¯'''é»æ°éä¿åå'''ãšèšãã
=== éé»èªå° ===
[[Image:Electrostatic induction.svg|thumb|upright=1.5|å°äœã¯ãè¿ãã®é»è·ã«ãã£ãŠè¡šé¢ã«é»è·ãèªå°ããããç©äœå
éšã®éé»æ°åã®å€§ããã¯ãŒãã§ããã]]
é»æ°çã«äžæ§ã§ãã£ãå°äœã®ç©è³ªïŒä»®ã«ç©è³ªAãšããïŒã«åž¯é»ããå¥ã®ç©è³ªïŒä»®ã«ç©è³ªBãšããïŒãæ¥è§Šãããã«è¿ã¥ãããšãç©è³ªAã«ã¯ã垯é»ç©è³ªBã®é»è·ã«åŒãå¯ããããŠãç©äœAã®å
éšã§å察笊å·ã®é»è·ã垯é»ç©äœBã«è¿ãåŽã®è¡šé¢ã«çããããŸãã垯é»ç©äœBãšåãé»è·ã¯åçºããã®ã§ãç©äœAå
éšã®åž¯é»ç©äœBãšã¯é ãåŽã®è¡šé¢ã«çããã
ãã®ãããªçŸè±¡ã'''éé»èªå°'''ïŒããã§ãããã©ã;Electrostatic inductionïŒãšãããéé»èªå°ã§çããé»è·ã®æ£é»è·ã®éãšè² é»è·ã®éã¯çéã§ãããïŒé»æ°éä¿åã®æ³åïŒ
å°äœã®å
éšã«éé»æ°åã¯ç¡ãããããã£ããšãããšãèªç±é»åãªã©ã®é»è·ãåããé»æµãæµãç¶ããããšã«ãªããããã®ãããªçŸè±¡ã¯å®åšããªãã®ã§äžåçã«ãªãããããã£ãŠãå°äœã®å
éšã«éé»æ°åã¯ç¡ãã
è¡šé¢ã«é»è·ãéãŸãã®ã¯ãå°äœã®å
éšã«éé»æ°åãäœãããªãããã§ããããããã£ãŠéé»èªå°ã§åŒãå¯ããããé»è·ã®å€§ããã¯ãå€éšããå°äœå
éšãžã®éé»æ°åãæã¡æ¶ãã ãã®å€§ããã§ããã
ãã®å°äœå
éšã®é»è·ããŒãã«ãªãæ§è³ªãå¿çšãããšãäžç©ºã®å°äœã§åºæ¥ãç©äœãçšããŠãéé»æ°åãé®èœããããšãã§ããããããéé»é®èœïŒããã§ããããžããelectric shildingïŒãšããã
=== èªé»å極 ===
[[File:Pith ball electroscope operating principle.svg|thumb|300px|èªé»å極ã®æŠå¿µå³]]
絶çžäœïŒä»®ã«AãšããïŒã«é»è·ãè¿ã¥ããå Žåã¯ãå°äœãšã¯éããç©äœAã®å
éšã®é»åã¯èªç±ã«è¡šé¢ã«ã¯éãŸããªãããç©äœå
éšã®ååã®æ£è² ã®é»è·ã®æ¥µæ§ãæã£ãéšåããå€éšã®éé»æ°åã«åŒãå¯ããããããã«ãè¿ã¥ããé»è·ã«è¿ãåŽã«ã¯ç°çš®ã®é»è·ãçããé ãåŽã«ã¯ãåçš®ã®é»è·ãçããã
ååãååãå€éšã®éé»æ°åã«ãã£ãŠãæ£è² ã®é»è·ã®éšåãçããããšã'''å極'''ïŒã¶ããããïŒãšããããå€éšã®é»åã«ãã£ãŠèµ·ããããã®ãããªå極ã®ãããã'''èªé»å極'''ïŒããã§ãã¶ãããããdielectric polarizationïŒãšããã
絶çžäœã¯ãéé»æ°åã«ããããããšèªé»å極ãè¡ãã®ã§ã絶çžäœã®ããšã'''èªé»äœ'''ïŒããã§ããããdielectricïŒãšãããã
å°äœã«éé»èªå°ãããæ£è² ã®é»è·ã¯ãå°äœãåæãªã©ãããã°æ£é»è·ãšè² é»è·ãå¥åã«åãåºãããšãã§ããããããèªé»äœã®æ£è² ã®é»è·ã¯ãååãååãšå¯æ¥ã«çµã³ã€ããŠãããããæ£è² ã®é»è·ãåãããŠåãåºãããšã¯åºæ¥ãªãã
{{clear}}
==é»å Žãšç£å Ž==
===é»è·ãšé»å Ž===
====é»è·====
[[Image:Static repulsion.jpg|thumb|ã»ãããŒãã®åãé»è·ã«ããåçº]]
[[Image:Static attraction.jpg|thumb|ã»ãããŒãã®ç°é»è·ããåŒãå¯ã]]
ããç©è³ªãé»æ°ã垯ã³ãŠããïŒåž¯é»ããŠããïŒãšãããã®åž¯é»ã®å€§å°ã®çšåºŠã'''é»è·'''ïŒã§ãããelectric chargeïŒãšãããããŸããŸãªç©è³ªããããããªæ¹æ³ã§åž¯é»ãããçµæãé»è·ã«ã¯ã垯é»ãã2åã®ãã®ã©ãããè¿ã¥ããæã«åŒã£åŒµãåããã®ïŒåŒåãåãïŒãšåçºããããã®ïŒæ¥åãã¯ãããïŒã®2çš®é¡ãããããšãåãã£ãã
ãã®ãããªã垯é»ããŠããç©äœã«åãåã'''éé»æ°å'''ãšããã
ã¹ã€ã®åž¯é»ãããã®ããä»ã«ãããã€ãçšæããŠãè¿ã¥ããŠå®éšãã2åã®ç©äœã®çµã¿åãããå€ãããšãçµã¿åããã«ãã£ãŠã2åã®ç©äœã©ããã«åŒåãåãå Žåãããã°ãæ¥åãåãå Žåãããããšãåãã£ãããã®åŒåãšæ¥åã®é¢ä¿ã¯ã垯é»ããé»è·ã®çš®é¡ã«å¿ããããšãããã£ãã
==== æ£é»è·ãšè² é»è· ====
çµè«ãèšããšãé»è·ã«ã¯æ£è² ã®2çš®é¡ããããæ£ã®é»è·ã©ããã®ç©äœãè¿ã¥ãããšãã¯åçºããããè² ã®é»è·ã©ãããè¿ã¥ãããšããåçºããããæ£ãšè² ã®é»è·ãè¿ã¥ããæã«ã¯åŒåãåãã
ã€ãŸããå笊å·ã®é»è·ãè¿ã¥ããå Žåã¯ãåçºåãçãããç°ç¬Šå·ã®é»è·ãè¿ã¥ããå Žåã¯ãåŒåãçããã
{{clear}}
==== éé»æ°å ====
[[File:Coulomb.jpg|thumb|150px|ã¯ãŒãã³ã®èåãCharles Augustin de Coulomb]]
[[Image:Bcoulomb.png|thumb|left|300px|ã¯ãŒãã³ãéé»æ°åã®æž¬å®ã«çšããããããèšã]]
[[File:Coulomb torsion.svg|thumb|300px|]]
éé»æ°ã©ããã®åã®åŒ·ãã¯ãå®éšçã«ã¯ãé»è·ã®éã«åãåã¯ãéåã®å Žåãšåæ§ã«åãåãŒãåã2ç©äœã®éã®è·é¢ã®2ä¹ã«åæ¯äŸããããšãç¥ãããŠãããæŽã«ãé»è·ã®å€§ããã倧ããã»ã©é»è·éã«åãåã倧ããããšãèæ
®ãããšãè·é¢''r''ã ãé¢ããŠãããããé»è·<math>q _1</math>ã<math>q _2</math>ãæã£ãŠãã2ç©äœã®éã«åãå''F''ã¯ã
:<math>
F = k\frac{q_1 q_2}{r^2} = \frac 1 {4\pi\epsilon} \frac {q _1 q _2}{r^2}
</math>
ã§äžãããããããã'''ã¯ãŒãã³ã®æ³å'''ïŒ Coulomb's lawïŒãšãããããã§ã<math>k</math>ã¯æ¯äŸå®æ°ã§ãããäž¡é»è·ã®åšå²ã«ããç©äœã®çš®é¡ã«ããå€åããå®æ°ã§ãããç空äžã§ã®é»å Žãèããå Žåã®kã®å€ã¯ã
:<math>k_0 = 9.0 \times 10^9 </math>[Nã»m<sup>2</sup>/C<sup>2</sup>]ïŒã¯ãŒãã³ã®æ¯äŸå®æ°ïŒ
ã§ããããŸãã<math>\epsilon</math>ã¯åŸã»ã©ç»å Žããèªé»çïŒããã§ããã€ïŒãšåŒã°ããç©çå®æ°ã§ãããèªé»çã¯ãäž¡é»è·ã®åšå²ã«ããç©äœã®çš®é¡ã«ããå€åããå®æ°ã§ãããèªé»çã«ã€ããŠã¯ããã®æãåããŠèªãã 段éã§ã¯ããŸã ç¥ããªããŠãè¯ããã®ã¡ã«ç©çIIã§èªé»çã詳ãã解説ããã
èªé»ç<math>\epsilon</math>ãšã¯ãŒãã³ã®æ¯äŸå®æ°kã«ã¯äžåŒã®é¢ä¿
:<math>k= \frac 1 {4\pi\epsilon}</math>
ãããã
ç©äœã®ãŸããã«èç©ããããã®ã'''é»è·'''ãšåŒã¶ãé»æ°åã«ãã£ãŠåçºããã£ãããåŒãã€ããã£ããããç©äœã'''é»è·ãæã€'''ç©äœãšåŒã¶ããŸããããã§èŠ³å¯ãããéé»æ°åãã'''ã¯ãŒãã³å'''ãšåŒã¶ããšãããã
2åã®é»è·ã©ããããããŒãåã¯åãã§ããããããã£ãŠ'''äœçšã»åäœçšã®æ³å'''ã«åŸã£ãŠããã
[[Image:Coulombslawgraph.svg|thumb|center|300px|2åã®ç¹é»è·ã®éã«åãåã®é¢ä¿ã<br>ã¯ãŒãã³ã®æ³åã«ãããšF1=F2ãšãªãã]]
ããã§ãé»è·ã®åäœã¯<nowiki>[C]</nowiki>ã§äžãããããèšå·ã®Cã¯ãã¯ãŒãã³ããšèªãã
{{-}}
----
*äŸé¡
[[File:ã¯ãŒãã³ã®æ³å äŸé¡1.svg|thumb|ã¯ãŒãã³ã®æ³å äŸé¡1]]
å³ã®ããã«ã2æ¬ã®ç³žã«ãããããåã質émkgã§ãåã笊å·ãšå€§ããã®é»è·qCã®çããã¶ãããã£ãŠããããã¯ãã¯ãŒãã³åã§åçºããã®ã§ãå³ã®ããã«ã糞ãè§åºŠÎžããªãã
ãã®ãšãã質émã«ããéåãšãé»è·qã«ããã¯ãŒãã³åãšã®é¢ä¿ã«ã€ããŠãåŒãç«ãŠãããªããå¿
èŠãªãã°ã糞ã®åŒµåã¯TNãšããããšã
{{-}}
解æ³
:[[File:ã¯ãŒãã³ã®æ³å äŸé¡1 解æ³.svg|thumb|left|400px|ã¯ãŒãã³ã®æ³å äŸé¡1 解æ³]]
å³ã®ãããªäœçœ®é¢ä¿ã«ãªãã®ã§ãå³ã®ããã«åŒãç«ãŠãã°ããã
:â» ãã®ããã«ãé»æ°ç£æ°åŠã®åé¡ã§ã¯ãå³ããã¡ããšæžããŠã解æ³ãèããå¿
èŠããããæ°åŒã ãã§èšç®ãããšãç«åŒãã¹ãªã©ã®åå ã«ãªãã
{{-}}
----
â» äžèšã®2æ¬ã®ç³žã«ã¶ãããã£ãçã®ã¯ãŒãã³åã®äŸé¡ã¯ãé»æ°ç£æ°åŠã®ã©ã®å
¥éæžã«ããããããªå
žåçãªåé¡ã§ããã®ã§ãèªè
ã¯ãã¡ããšç解ããããšã
{{-}}
----
*åé¡äŸ
**åé¡
é»è·<math>q _1</math>, <math>q _2</math>ã®éã®è·é¢ãrã®å Žåãš2rã®å Žåã§ã¯ãéã«åãåã®å€§ããã¯ã©ã¡ããã©ãã ã倧ãããçããã
ãŸããè·é¢ã2rã®æã®2ç¹éã®åã®å€§ãããçããã
**解ç
ã¯ãŒãã³åã¯ãç©äœéã®è·é¢ã®é2ä¹ã«æ¯äŸããã®ã§ãè·é¢ã2rã®æã¯ãrã®æã®å€§ããã®<math>\frac 1 4</math>ãšãªãããŸããåãåã®å€§ããã¯ãã¯ãŒãã³åã®åŒãçšããŠã
:<math>
f = \frac 1 {4\pi\epsilon _0} \frac {q _1 q _2}{4r^2}
</math>
ãšãªãã
----
====é»å Ž ====
æ¢ã«ãããé»è·Aã®ãŸããã®å¥ã®é»è·Bã«ã¯ããã®é»è·ããã®è·é¢ã®é2ä¹ã«æ¯äŸããåããããããšãè¿°ã¹ãã
[[File:é»å Žã®éãåãã.svg|thumb|400px|é»å Žã®éãåãã]]
ããã§ãé»è·Bãåããåã¯ããã®é»è·Bã®å€§ããã«æ¯äŸããããšãåãããŠèãããšããã®é»è·Bã®å€§ããã«ããããããé»è·Aã®å€§ããã ãã§æ±ºãŸãéãå°å
¥ããŠãããšéœåããããããã§ããã®ãããªéãšããŠ'''é»å Ž'''ïŒã§ãã°ïŒãå°å
¥ããããã®ãšããé»å Ž<math>\vec E</math>ã®äžã«ããé»è·<math>q</math>ã«åãå<math>\vec F</math>ã¯ã
:<math>\vec F = q \vec E</math>
ã§äžãããããé»å Žã¯åäœé»è·ã«åãåãšèããããšãã§ããé»å Žã®åäœã¯[N/C]ã§ããããé»å Žãã¯ããé»çãïŒã§ãããïŒãšãåŒã°ããã
ïŒæ¥æ¬ã®ç©çåŠã§ã¯ãé»å ŽããšåŒã¶ããšãå€ãããŸããæ¥æ¬ã®é»æ°å·¥åŠã§ã¯ãé»çããšåŒã°ããããšãå€ããææ²»æã®ç¿»èš³ã®éã®ãæ¥æ¬åœå
ã®æ¥çããšã®éãã«éããããããªãæ¥æ¬ããŒã«ã«ãªéœåã§ãããåŒã³æ¹ã¯ç©çã®æ¬è³ªãšã¯é¢ä¿ãªãã®ã§ãããã§ã¯ãã©ã¡ãã®è¡šçŸãçšãããã¯ãæ¬æžã§ã¯ç¹ã«ãã ãããªããè±èªã§ã¯ç©çåŠã»é»æ°å·¥åŠãšãâelectric fieldâã§å
±éããŠãããïŒ
äžã®ã¯ãŒãã³åã®çµæãšåããããšãé»è·Aã®ãŸããã«å¥ã®é»è·ãååšããªããšããé»è·<math>q</math>[C]ã®é»è·ããŸãšãé»å Ž<math>\vec E</math>ã¯ã
:<math>\vec E = \frac 1 {4\pi\epsilon _0} \frac {q}{r^2} \vec e _r</math>
ã§äžããããããã ããrã¯é»è·ããã®è·é¢ã§ããã<math>\vec e _r</math>ã¯ãé»è·ãšããç¹ãçµãã çŽç·äžã§ãé»è·ãšå察æ¹åãåããåäœãã¯ãã«ã§ããã
é»è·ã®åãã®é»å Žã¯ãå¹³é¢äžã§æŸå°ç¶ã®ãã¯ãã«ãšãªãããšã«æ³šæã
{|
| [[File:VFPt minus thumb.svg|150px|thumb|è² é»è·ã®åšãã®é»å Žã®åã]]
| [[File:VFPt plus thumb.svg|150px|thumb|æ£é»è·ã®åšãã®é»è·ã®åã]]
|}
é»å Žã¯ãã¯ãã«ã§ãããé»è·ã2åãããšãã¯ãããããã®é»è·ãã€ããé»å Žããéãåãããã°ããã
:<math> \vec E = \vec {E_1} + \vec {E_2} </math>
ã§ããã
é»è·ã3å以äžã®ãšãããåæ§ã«éãåãããã°è¯ãã
å³ã®ããã«ãé»è·ããåºãé»å Žã®æ¹åãå³ç€ºãããã®ã'''é»æ°åç·'''ïŒã§ãããããããelectric line of forceïŒãšããã
é»è·ãè€æ°ããå Žåã«ã¯ãå®éã«æ°ãã«çœ®ãããé»è·ãåããåã¯ããããã足ãåããããã®ãšãªãããããã£ãŠãè€æ°ã®é»è·ãããå Žåã®åšå²ã®é»çã¯ãããããã®é»è·ãäœãé»çãã¯ãã«ã®åãšãªãïŒéãåããã®åçïŒã
<!-- é»æ°åç· -->
[[File:Camposcargas.PNG|thumb|left|300px|å笊å·ã®é»è·ã©ããïŒå·ŠïŒãè¿ã¥ããå Žåã¯åçºããããç°ãªã笊å·ã®é»è·ã©ããïŒå³ïŒãè¿ã¥ããå Žåã¯åŒãä»ãåãã]]
[[File:VFPt dipole electric manylines.svg|thumb|center|200px|ç°ç¬Šå·ã®é»è·ã©ããã®å Žåã®é»æ°åç·]]
{{clear}}
é»æ°åç·ãå³ç€ºããå Žåã¯ãæ£é»è·ããåç·ãåºãŠãè² é»è·ã§åç·ãåžåãããããã«æžããåç·ã¯ãé»å Žãå³ç€ºãããã®ãªã®ã§ãé»è·ä»¥å€ã®å Žæã§ã¯ãåç·ãåå²ããããšã¯ãªãã
åç·ãçæããã®ã¯æ£é»è·ã®å Žæã®ã¿ã§ãããåç·ãæ¶æ»
ããã®ã¯ãè² é»è·ã®å Žæã®ã¿ã§ããã
èšãæããã°ãåç·ãé»è·ä»¥å€ã®å Žæã§æ¶æ»
ããããšã¯ãªãããé»è·ä»¥å€ã®å Žæã§åç·ãçæããããšã¯ãªãã
å°äœã®å
éšã®é»å Žã¯ãŒãã§ãã£ããèšãæããã°ãé»æ°åç·ã¯ãå°äœã®å
éšã«ã¯é²å
¥ã§ããªãã
[[File:VFPt image charge plane horizontal.svg|200px|thumb|é»æ°åç·ã¯ãå°äœã®å
éšã«ã¯é²å
¥ã§ããªãã]]
<!-- ã¬ãŠã¹ã®æ³å -->
[[File:E FieldOnePointCharge.svg|é»æ°åç·ã®ååž]]
ç¹é»è·ããã¯ãå³ã®ããã«ãæŸå°ç¶ã«é»æ°åç·ãåºããã¯ãŒãã³ã®æ³åã®ä¿æ°ã«ãã
:<math>\frac{1}{4 \pi \epsilon _0} \frac{q_1 q_2}{r^2}</math>
ã®ãã¡ã®ãåæ¯ã®
<math>4 \pi r^2</math>
ã¯ãçã®è¡šé¢ç©ã®å
¬åŒã«çããã®ã§ãé»æ°åç·ã®å¯åºŠã«æ¯äŸããŠãé»å Žã®åŒ·ããããã¯éé»æ°åã®åŒ·ãã決ãŸããšèããããã
éé»èªå°ã§ã¯ãå°äœå
éšã«ã¯éé»æ°åãåããŠããªãã®ã§ãã£ããããã¯ãé»å ŽãšããæŠå¿µãçšããŠèšãæããã°ãå°äœå
éšã®é»å Žã¯ãŒãã§ããããšèšããã
====é»äœ====
ã¯ãŒãã³åã¯åïŒã¡ããïŒã§ãããããããã«éãã£ãŠå¥ã®é»è·ãè¿ã¥ããå Žåã¯ãè¿ã¥ããå¥ã®é»è·ã¯ä»äºãããããšã«ãªãããŸããè¿ã¥ããé»è·ãææŸãã°ãã¯ãŒãã³åã«ãã£ãŠåãåããä»äºãããããšã«ãªããããè¿ã¥ããç¶æ
ã«ããå¥é»è·ã¯äœçœ®ãšãã«ã®ãŒãèããŠããããšã«ãªãã
ãããã£ãŠãã¯ãŒãã³åã«å¯ŸããŠãäœçœ®ãšãã«ã®ãŒãå®çŸ©ããããšãã§ãããïŒãªããè¡æè»éäžã®ç©äœã®ãããªãå°è¡šãã倧ããé¢ããå Žæã®éåããã¯ãŒãã³åãšåæ§ã«é2ä¹åãªã®ã§ãããã§èããèšç®ææ³ã¯éåã«ããäœçœ®ãšãã«ã®ãŒã«ãå¿çšã§ãããéåå é床gãçšããåmgãšããã®ã¯å°è¡šè¿ãã§ã®è¿äŒŒã«ãããªããïŒ
ã¯ãŒãã³åã«ããé»å Žã®å®çŸ©ã§ã¯ãåäœé»è·ã«å¯ŸããŠé»å Žãå®çŸ©ããã®ãšåæ§ãäœçœ®ãšãã«ã®ãŒã«å¯ŸããŠããåäœé»è·ã«å¿ããŠå®çŸ©ã§ããéãå°å
¥ãããšéœåãããããã®ãããªéã'''é»äœ'''ïŒã§ãããelectric potentialïŒãšåŒã¶ãé»äœã®åäœã¯'''ãã«ã'''ãšãããé»äœãäŸãããšãå°è¡šè¿ãã§ã®éåã®äœçœ®ãšãã«ã®ãŒãèããéã®ãghããªã©ã«çžåœããéã§ããã
ã¯ãŒãã³åã®çµæãšã<math>q</math>[C]ã®é»è·ããè·é¢''r''ã ãé¢ããç¹ã®é»äœVã¯ãé»å Žã®ç©åèšç®ã§åŸããããïŒç©åããŸã ç¿ã£ãŠãªãåŠå¹Žã®èªè
ã¯ãåãããªããŠãæ°ã«ããã次ã®çµæãžãšé²ãã§ãã ãããïŒçµæã®ã¿ãèšããšã
:<math>V=\frac{1}{4\pi\epsilon _0} \frac{q}{r}</math>
ãšãªãã
é»äœVã®ç¹ã«''q''[C]ã®é»è·ã眮ãããšãããã®é»è·ã®ã¯ãŒãã³åã«ããäœçœ®ãšãã«ã®ãŒ''U''[J]ã¯ãé»äœVãçšããã°ã
:<math>U = qV</math>
ãšãªãããããã£ãŠãé»äœ<math>V_1</math>ãã«ãã®ç¹ããé»äœ<math>V_2</math>ãã«ãã®äœçœ®ãžãšé»è·''q''[C]ãéé»æ°åãåããŠç§»åãããšããéé»æ°åã®ããä»äº''W''[J]ã¯
:<math>W = q(V_2 - V_1)</math>
ãšãªãã
{{-}}
[[File:äžæ§ãªé»å Ž.svg|thumb|500px|äžæ§ãªé»å Ž]]
ãã£ãœããäžæ§ãªé»å Žã«ãããŠã¯ãé»äœã®åŒããé»å ŽãçšããŠç°¡åã«è¡šãããšãã§ãããè·é¢''d''ã ãé¢ããå¹³è¡å¹³æ¿é»æ¥µã®éã«äžæ§ãªé»å Ž<math>\vec E</math>ãçããŠãããšãããã®é»çã®äžã«çœ®ããé»è·''q''ã¯éé»æ°å<math>q\vec E</math>ãåããããã®é»è·ãé»çã®åãã«æ²¿ã£ãŠäžæ¹ã®é»æ¥µããä»æ¹ã®é»æ¥µãŸã§ç§»åãããšããé»çã®ããä»äº''W''㯠<math>W = qEd</math> ãšãªããããããã2極æ¿ã®é»äœå·®''V''ã¯ã
:<math>V=Ed</math>
ã§è¡šãããšãã§ããããšãããããåŒãå€åœ¢ããŠ
:<math>E= \frac{V}{d}</math>
ãšããããšãã§ãããããã§ãåäœãèãããšãå³èŸºã¯é»å§ãè·é¢ã§å²ã£ããã®ã§ãããããé»çã®åäœãšããŠ[N/C]ã®ã»ã[V/m]ãçšããããšãã§ããããšããããã
é»äœã®åäœã¯'''ãã«ã'''ã§ããããã®éã¯æ¢ã«[[äžåŠæ ¡çç§]]ãªã©ã§æ±ã£ã[[w:é»å§|é»å§]]ïŒã§ããã€ãvoltageïŒã®åäœãš'''åã'''åäœã§ãããå®éã«é»æ°åè·¯ã«é»å§ããããããšã¯ãåè·¯äžã®é»åã«é»å ŽããããŠåããããšãš'''çãã'''ã
éé»èªå°ã«ãã£ãŠãå°äœå
éšã®é»å Žã¯ãŒãã§ãã£ãããã®ããšãããå°äœã®è¡šé¢ã¯ãé»äœãçãããå°äœè¡šé¢ã¯äºãã«çé»äœã§ããã
é»äœã®åºæºã¯ãå®çšäžã¯ãå°é¢ã®é»äœããŒãã«çœ®ãããšãå€ããé»æ°åè·¯ã®äžéšã倧å°ã«ã€ãªãããšãæ¥å°ïŒãã£ã¡ïŒãŸãã¯'''ã¢ãŒã¹'''ïŒearthïŒãšãããåè·¯ãã¢ãŒã¹ããŠããã®ã€ãªãã éšåã®é»äœããŒããšèŠãªãããšãå€ãã
*åé¡äŸ
**åé¡
çŽç·äžã§è·é¢0, b[m]ã®ç¹ã«ãé»è·q, q'ãæã€ç©äœã眮ããŠããããã®æãäœçœ®a[m](a<b)ã®ç¹ã®é»äœãæ±ããã
**解ç
é»äœã®åŒãçšããã°ãããé»è·ãè€æ°ãããšãã«ã¯ãé»äœã¯ããããã®é»è·ãã€ããåºãé»è·ã®åã«ãªãããšã«æ³šæãçãã¯ã
:<math>V = \frac{1}{4\pi\epsilon _0} (\frac{q}{a} + \frac{q'}{b-a})</math>
ãšãªãã
----
å°äœè¡šé¢ã¯çé»äœãªã®ã§ããã£ãŠãé»æ°åç·ã¯å°äœè¡šé¢ã«åçŽã§ããã
ãã®ããšãããé»æ°åç·ãšé»å Žã¯åçŽã§ããã
é»å Žãéãåãããããããã«ãé»äœãéãåãããããããªããªãé»äœãšã¯ãé»å ŽãèããŠãçµè·¯ã«ãŠç©åãããã®ã§ããããã
åŠæ ¡ã®ãã¹ããªã©ã§ã¯ãé»äœã®èšç®ã®ãããã¯ãŒãã³åã®æ¹åã®åéããªã©ã«ããèšç®ãã¹ãªã©ããµãããããé»å Žãæ±ããŠããããããç©åããŠãé»äœãæ±ããã®ããèšç®äžã¯å®å
šã§ããã
== éé»èªå°ãšèªé»å極 ==
=== ã³ã³ãã³ãµãŒ ===
[[File:ã³ã³ãã³ãµãŒ æ§é ãšåç.svg|thumb|400px|ã³ã³ãã³ãµãŒ]]
'''ã³ã³ãã³ãµãŒ'''ïŒè±:capacitor ,ããã£ãã·ã¿ããšèªãïŒã¯ãå³ã®ããã«ã2æã®é»æ¥µãåãããããåè·¯äžã«é»è·ãèç©ã§ããéšåãäžããçŽ åã§ããã
[[File:ã³ã³ãã³ãµãŒ å
é»ã®ä»çµã¿.svg|thumb|500px|ã³ã³ãã³ãµãŒã®å
é»ã®ä»çµã¿]]
ã³ã³ãã³ãµãŒã«é»è·ãèããããšã'''å
é»'''ïŒãã
ãã§ãïŒãšãããã³ã³ãã³ãµãŒããé»è·ãæŸåºãããããšã'''æŸé»'''ãšããã
ã³ã³ãã³ãµã®äž¡ç«¯ã«ããé»äœVãäžãããããšããã³ã³ãã³ãµã«ã¯ãé»äœã«æ¯äŸããé»è·Qãèç©ãããããã®ãšããã³ã³ãã³ãµã®èç©èœåãèšå·ã§ C ãšãããŠã
:<math>Q=CV</math>
ãšããŠCãåããCã¯'''éé»å®¹é'''ïŒããã§ãããããããelectric capacitanceïŒãšåŒã°ããåäœã¯F('''ãã¡ã©ã'''ãfarad)ã§äžããããã
1ãã¡ã©ãã¯å®çšäžã¯å€§ããããã®ã§ã10<sup>-12</sup>ãã¡ã©ããåäœã«ãã1pF(ãã³ãã¡ã©ã)ãã10<sup>-6</sup>ãã¡ã©ããåäœã«ãã1ÎŒF(ãã€ã¯ããã¡ã©ã)ã䜿ãããããšãå€ãã
{{-}}
=== å¹³è¡æ¿ã³ã³ãã³ãµãŒ ===
[[File:å¹³è¡æ¿ã³ã³ãã³ãµãŒ é»å Ž.svg|thumb|400px|å¹³è¡æ¿ã³ã³ãã³ãµãŒã®é»å Ž]]
極æ¿ãå¹³è¡ãªã³ã³ãã³ãµãŒãå¹³è¡æ¿ã³ã³ãã³ãµãŒãšããã
å¹³è¡æ¿ã³ã³ãã³ãµãŒã®ã極æ¿ã©ããã®é»å Žã¯ãäžæ§ãªé»å Žã§ããã
ãã®å¹³è¡æ¿ã³ã³ãã³ãµãŒã®éé»å®¹éCã®åŒã¯ãåŸè¿°ããçç±ã«ããã
:<math>C=\epsilon_0 \frac{S}{d}</math>
ã§äžãããããããã§ãSã¯å°äœå¹³é¢ã®é¢ç©ã§ãããdã¯å°äœéã®è·é¢ã§ããã
å®éšçã«ãããã®éé»å®¹éã®å
¬åŒã¯ãæ£ããããšã確ãããããŠããã
* å¹³è¡æ¿ã³ã³ãã³ãµãŒã®éé»å®¹éã®å
¬åŒã®å°åº
ããã§äžããéé»å®¹éã¯ã'''å¹³é¢äžã«é»è·ãäžæ§ã«ååžãã'''ãšã®ä»®å®ã§å°ãããããã®ãšããå°äœéã«çããé»çEã¯ãå°äœãæã€é»è·ãQ, -Qãšããæã
ãŸãã極æ¿ã®é»è·å¯åºŠãã極æ¿ã®ã©ãã§ãäžå®ã ãšä»®å®ããŠïŒãã®ããã«ã¯ãã³ã³ãã³ãµãŒã®åºãïŒã€ãŸãé¢ç©ïŒãããã
ãã¶ãã«åºããšä»®å®ããå¿
èŠãããããšãããããã®ãããªä»®å®ã«ãããé»è·å¯åºŠã¯ïŒã
:é»è·å¯åºŠïŒ<math>Q/S</math>C/m<sup>2</sup>
ã§ããã
é»æ°åç·ã®æ§è³ªãšããŠããã©ã¹ã®é»è·ããçããŠãã€ãã¹ã®é»è·ã§åžåãããã®ã§ããã£ãŠå¹³è¡æ¿ã³ã³ãã³ãµãŒéã®é»æ°åç·ã®ååžã¯ãå³ã®ããã«ãé»æ°åç·ãããã©ã¹æ¥µæ¿ããåçŽã«ããã€ãã¹æ¥µæ¿ãžåãã£ãŠé»æ°åç·ãåºãŠããããŠãã€ãã¹æ¥µæ¿ã«é»æ°åç·ãåžåãããã
é»å Žã¯ãå°äœéã®åç¹ã§ã
:<math>E = \frac{Q/S}{\epsilon _0} =\frac{Q}{\epsilon _0 S}</math>
ã§äžãããããé»å Žãæ±ããããã®ã§ãããããé»äœãèšç®ã§ãããå°äœéã®åç¹ã§é»å Žã®å€§ãããåäžãªã®ã§ãé»äœã®å€§ããã¯é»å Žã®å€§ããã«2ç¹éã®è·é¢ãããããã®ã«ãªããããã§ãé»äœVã¯ã
:<math>V=Ed=\frac{d}{\epsilon_0S}Q</math>
ãšãªããããã®åŒãšéé»å®¹éCã®å®çŸ©ãèŠæ¯ã¹ããšã
:<math>C=\epsilon_0\frac{S}{d}</math>
ãåŸãããã
== é»æ± ã®ä»çµã¿ ==
é»æ± ã®ååŠåå¿ã«ã€ããŠã¯ãå¥ç§ç®ã®ååŠIãªã©ã§è©³ããæ±ãããããã®ç« ã§ã¯ãé»å§ãé»æµã®ç解ã«é¢ããç¹ãéç¹çã«èª¬æãããã
=== ã€ãªã³ååŸå ===
éå±å
çŽ ã®åäœãæ°ŽãŸãã¯æ°Žæº¶æ¶²ã«å
¥ãããšãã®ãéœã€ãªã³ã®ãªããããã'''ã€ãªã³ååŸå'''ïŒionization tendencyïŒãšããã
äŸãšããŠãäºéZnãåžå¡©é
žHClã®æ°Žæº¶æ¶²ã«å
¥ãããšãäºéZnã¯æº¶ãããŸãäºéã¯é»åã倱ã£ãŠZn<sup>2+</sup>ã«ãªãã
:Zn + 2H<sup>+</sup> â Zn<sup>2+</sup> + H<sub>2</sub>
äžæ¹ãéAgãåžå¡©é
žã«å
¥ããŠãåå¿ã¯èµ·ãããªãã
ãã®ããã«éå±ã®ã€ãªã³ååŸåã®å€§ããã¯ãç©è³ªããšã«å€§ãããç°ãªãã
=== é»æ± ===
äºçš®é¡ã®éå±åäœãé»è§£è³ªæ°Žæº¶æ¶²ã«å
¥ãããšé»æ± ãã§ãããããã¯[[ã€ãªã³ååŸå]]ïŒåäœã®éå±ã®ååãæ°ŽãŸãã¯æ°Žæº¶æ¶²äžã§é»åãæŸåºããŠéœã€ãªã³ã«ãªãæ§è³ªïŒã倧ããéå±ãé»åãæŸåºããŠéœã€ãªã³ãšãªã£ãŠæº¶ããã€ãªã³ååŸåã®å°ããéå±ãæåºããããã§ããã
ã€ãªã³ååŸåã®å€§ããæ¹ã®éå±ã'''è² æ¥µ'''ïŒãµãããïŒãšãããã€ãªã³ååŸåã®å°ããæ¹ã®éå±ã'''æ£æ¥µ'''ïŒãããããïŒãšããã
ã€ãªã³ååŸåã®å€§ããéå±ã®ã»ãããéœã€ãªã³ã«ãªã£ãŠæº¶ãåºãçµæãéå±æ¿ã«ã¯é»åãå€ãèç©ããã®ã§ãäž¡æ¹ã®éå±æ¿ãé
ç·ã§ã€ãªãã°ãã€ãªã³ååŸåã®å€§ããæ¹ããå°ããæ¹ã«é»åã¯æµããããé»æµãã§ã¯ç¡ãããé»åããšãã£ãŠãããšã«æ³šæãé»åã¯è² é»è·ã§ããã®ã§ãé»æµã®æµããšé»åã®æµãã¯ãéåãã«ãªãã
=== ã€ãªã³åå ===
ããŸããŸãªæº¶æ¶²ãéå±ã®çµã¿åããã§ãã€ãªã³ååŸåã®æ¯èŒã®å®éšãè¡ã£ãçµæãã€ãªã³ååŸåã®å€§ããã決å®ãããã
å·Šããé ã«ãã€ãªã³ååŸåã®å€§ããéå±ã䞊ã¹ããšã以äžã®ããã«ãªãã
: K > Ca > Na > Mg > Al > Zn > Fe > Ni > Sn > Pb > (H<sub>2</sub>) > Cu > Hg > Ag > Pt > Au
éå±ããã€ãªã³ååŸåã®å€§ããã®é ã«äžŠã¹ããã®ãéå±ã®'''ã€ãªã³åå'''ãšããã
æ°ŽçŽ ã¯éå±ã§ã¯ç¡ããæ¯èŒã®ãããã€ãªã³ååŸåã«å ããããã
éå±ååã¯ãäžèšã®ä»ã«ãããããé«æ ¡ååŠã§ã¯äžèšã®éå±ã®ã¿ã®ã€ãªã³ååãçšããããšãå€ãã
ã€ãªã³ååã®èšæ¶ã®ããã®èªååãããšããŠã
ã貞ããããªããŸããããŠã«ããªãã²ã©ãããåéãã
ãªã©ã®ãããªèªååããããããã¡ãªã¿ã«ãã®èªååããã®å Žåã
ãKã ãã ãCa ãªNaããŸMg ãAlããZn ãŠFe ã«Ni ã ãªPbãã²H2 ã©Cu ãHg ãAg ã åéPt,Auãã
ãšå¯Ÿå¿ããŠããã
=== ãã«ã¿é»æ± ===
:åžç¡«é
žH<sub>2</sub>SO<sub>4</sub>ã®äžã«äºéæ¿Znãšé
æ¿Cuãå
¥ãããã®ã
è² æ¥µïŒäºéæ¿ïŒã§ã®åå¿
:Zn â Zn<sup>2+</sup> + 2e<sup>-</sup>
æ£æ¥µïŒé
æ¿ïŒã§ã®åå¿
:2H<sup> + </sup> + 2e<sup>-</sup> â H<sub>2</sub>â
==== èµ·é»å ====
ãã«ã¿ã®é»æ± ã§ã¯ãåŸããã䞡極éã®é»äœå·®ïŒãé»å§ããšããããïŒã¯ã1.1ãã«ãã§ããã(ãã«ãã®åäœã¯Vãªã®ã§ã1.1Vãšãæžãã)ãã®äž¡æ¥µæ¿ã®é»äœå·®ã'''èµ·é»å'''ãšãããèµ·é»åã¯ãäž¡é»æ¥µã®éå±ã®çµã¿åããã«ãã£ãŠæ±ºãŸãç©è³ªåºæã§ããã
èµ·é»åã®åäœã®ãã«ãã¯ãéé»æ°åã®é»äœã®åäœã®ãã«ããš'''åã'''åäœã§ãããé»æ°åè·¯ã®é»å§ã®ãã«ããšããèµ·é»åã®åäœã®ãã«ãã¯åãåäœã§ããã
=== é»æ± å³ ===
ãã«ã¿é»æ± ã®æ§é ã以äžã®ãããªæååã«è¡šããå Žåããã®ãããªè¡šç€ºã'''é»æ± å³'''ãããã¯'''é»æ± åŒ'''ãšããã
:(-) Zn | H<sub>2</sub>SO<sub>4</sub>aq |Cu (+)
aqã¯æ°Žã®ããšã§ãããH<sub>2</sub>SO<sub>4</sub>aqãšæžããŠãç¡«é
žæ°Žæº¶æ¶²ãè¡šããŠããã
;é»æ°åè·¯ãšã®é¢é£äºé
ç©çåŠã®é»æ°åè·¯ã®ç 究ã§ã¯ããã®ãããªé»æ± ãªã©ã®çŸè±¡ã®çºèŠãšçºæã«ãã£ãŠãå®å®ãªçŽæµé»æºãå®éšçã«åŸãããããã«ãªããçŽæµé»æ°åè·¯ã®æ£ç¢ºãªå®éšãå¯èœã«ãªã£ããé»æ± ã®çºæ以åã«ãããã©ã³ã¹äººã®ç©çåŠè
ã¯ãŒãã³ãªã©ã«ããéé»æ°ã«ããé»æ°ååŠã®ç 究ãªã©ã«ãã£ãŠãé»äœå·®ã®æŠå¿µãé»è·ã®æŠå¿µã¯ãã£ããã ãããã®æ代ã®é»æºã¯ãäž»ã«éé»æ°ã«ãããã®ã ã£ãã®ã§ãå®å®é»æºã§ã¯ç¡ãã£ãã
ãããŠãé»æ± ã«ããå®å®ãªé»æºã®çºæã¯ãåæã«å®å®ãªé»æµã®çºæã§ããã£ãããã®ãããªé»æ± ã®çºæãªã©ã«ãããçŽæµé»æ°åè·¯ã®ç 究ãªã©ããããã€ã人ã®ç©çåŠè
ãªãŒã ããããŸããŸãªå°äœã«é»æµãæµãå®éšãšçè«ç 究ãè¡ãããšã«ãããé»æ°åè·¯ã®çè«ã®'''ãªãŒã ã®æ³å'''ïŒãªãŒã ã®ã»ããããOhm's lawïŒãçºèŠãããã
[[File:Thermocouples diagram.svg|thumb|ç±é»å¯Ÿã®åçãããçµã¿åããã®éå±AãšBã§ãå³ã®ããã«2ã€ã®æ¥ç¹ã«ç°ãªã枩床ãäžãããšãé»æµãæµããã]]
ãã€ã¯ãªãŒã ã¯é»æ± ã§ã¯ãªãç±é»å¯ŸïŒãã€ã§ãã€ãïŒãšãããã®ã䜿ã£ãŠãé»æ°åè·¯ã«å®å®ããé»æµããªããç 究ããããåœæã®é»æ± ã§ã¯ãèµ·é»åããã ãã«æžã£ãŠããŸãããªãŒã ã¯åœåã¯é»æ± ã§å®éšããããããŸãå®å®é»æµãåŸãããªãã£ãã
ç±é»å¯Ÿãšã¯ããŸãç°ãªãéå±ææã®2æ¬ã®éå±ç·ãæ¥ç¶ããŠïŒã€ã®åè·¯ãã€ããã2ã€ã®æ¥ç¹ã«æž©åºŠå·®ãäžãããšãåè·¯ã«é»å§ãçºçããããé»æµãæµããïŒãã®çŸè±¡ãããŒãŒããã¯å¹æãšããïŒããã®çŸè±¡ãããã¯ã1821幎ã«ãŒãŒããã¯ãçºèŠããããã®ãããªåè·¯ããç±é»å¯Ÿã§ããããªããåã2æ¬ã®éå±ç·ã§ã¯ã枩床差ãäžããŠãé»å§ã¯çºçãããé»æµã¯æµããªãã
ãªãŒã ã¯ããã«ãªã³å€§åŠææããã±ã³ãã«ãã®å©èšã«ãã£ãŠããã®ç±é»å¯Ÿãå®éšã«å©çšããã枩床ãå®å®ãããã®ã¯ãåœæã®æè¡ã§ãæ¯èŒçç°¡åã§ãã£ãã®ã§ãããããŠãªãŒã ã¯å®å®é»æµããã¡ããå®éšãã§ããã®ã§ããã
:â» ç±é»å¯Ÿã«ã€ããŠã¯ãé«æ ¡ã®ç¯å²ãè¶
ãããã倧åŠå
¥è©Šã«ãåºé¡ãããªãã ãããã倧åŠã®ææ¥ã§ãããŸãæ·±å
¥ãããªãã®ã§ãåãããªããã°ãæ°ã«ããªããŠããã
:â» å®ã¯åæ通ã®ãç§åŠãšäººéç掻ãã§ç±é»å¯ŸïŒåæ通ã®æç§æžã§ã¯ãç±é»çŽ åããšèšè¿°ïŒã«ã€ããŠãç±ã®ç©çã®åå
ã§èª¬æããŠããããã ãããããã«ãªãŒã ã®æ³åã®å®éšãšã®é¢é£ãŸã§ã¯èª¬æããŠãªããã»ã»ã»ã
;ãªãŒã ã®æ³åãšã®é¢ä¿
ãªãŒã ã®æ³åïŒOhm's lawïŒãšã¯ã
ãã»ãšãã©ã®å°äœã§ã¯ãé»æµ I ãæµããŠããå°äœäžã®2ç¹ã®ç¹ <math>P_1</math>ãšç¹ <math>P_2</math> éã®é»äœå·® <math>E = E_1 - E_2</math> ã¯ãé»æµ I ã«æ¯äŸãããã
ãšããå®éšæ³åã§ããã
誀解ããããããããªãŒã ã®æ³åã¯ããã®ãããªå®éšæ³åã§ãã£ãŠãã¹ã€ã«æµæã®å®çŸ©åŒã§ã¯ç¡ããåæ§ã«ããªãŒã ã®æ³åã¯ãã¹ã€ã«é»å§ã®å®çŸ©åŒã§ã¯ç¡ãããé»æµã®å®çŸ©åŒã§ãç¡ããäžåŠæ ¡ã®çç§ã§ã®é»æ°åè·¯ã®æè²ã§ã¯ãéå±ã®é»æ°å解ã®èµ·é»åã®æè²ãŸã§ã¯ããªãã®ã§ããšãããã°ãé»å§ã誀解ããŠããé»å§ã¯ãåãªãé»æµã®æ¯äŸéã§ãæµæã¯ãã®æ¯äŸä¿æ°ãã®ãããªèª€è§£ããå Žåãæããããããã®è§£éã¯æããã«èª€è§£ã§ããã
ãŸããåå°äœãªã©ã®äžéšã®ææã§ã¯ãé»æµãå¢ãææã®æž©åºŠãäžæãããšæµæãäžããçŸè±¡ãç¥ãããŠããã®ã§ãåå°äœã§ã¯ãªãŒã ã®æ³åãæãç«ããªãå Žåãããããªã®ã§ããªãŒã ã®æ³åãå®çŸ©åŒãšèããã®ã¯äžåçã§ããã
== é»æµãšé»æ°åè·¯ ==
[[Image:Wheatstonebridge.svg|right|thumb|300px|alt=A Wheatstone bridge has four resistors forming the sides of a diamond shape. A battery is connected across one pair of opposite corners, and a galvanometer across the other pair. |é»æ°åè·¯ã®äŸãèªè
ãããã®å³ã®æå³ãåããããã«ãªãã®ããæ¬ç¯ã®ç®æšã®äžã€ã§ãããã¡ãªã¿ã«ããã€ããã¹ãã³ã»ããªããžãïŒWheatstone bridgeïŒãšããåè·¯ã§ããã<br>R1ãR2ãR3ãRxã¯æµæãV<sub>G</sub>ãäžžã§å²ã£ãŠããèšå·ã¯é»å§èšã<br>AãBãCãDã¯åãªãåè·¯ã®åæµããŠããæ¥ç¹ã]]
å°ç·ãªã©ã®å°äœå
ã®é»æ°ã®æµãã'''é»æµ'''ïŒã§ããã
ããelectric currentïŒãšãããé»æµã®åŒ·ãã¯'''ã¢ã³ãã¢'''ãšããåäœã§è¡šãã1ã¢ã³ãã¢ã®å®çŸ©ã¯æ¬¡ã®éãã§ããã
1ç§éã«1ã¯ãŒãã³ïŒèšå·CïŒã®é»æµãééããããšã1'''ã¢ã³ãã¢'''ãšããã
ã¢ã³ãã¢ã®èšå·ã¯Aã§ããããŸããé»æµã¯ãåäœæéãããã®é»è·ã®éééã§ãããã®ã§ãé»æµã®åäœã[C/s]ãšæžãå Žåãããã
äžè¬çã«ã¯ãé»æµã®åäœã¯ããªãã¹ã[A]ã§è¡šèšããããšãå€ãã
é»æµI[A]ãšæét[S]ã§å°ç·æé¢ãééããé»è·Q[C]ã®é¢ä¿ãåŒã§è¡šããšã
:<math>I=\frac{Q}{t}</math>
ã§ããã
é»æµã®åãã®åãæ¹ã«ã€ããŠã¯ãèªç±é»åã¯è² é»è·ãæã£ãŠãããããèªç±é»åã®åããšã¯å察åãã«é»æµã®åãããšãããšã«æ³šæããã
次ã«é»æµãšèªç±é»åã®é床ãšã®é¢ä¿ãèããã
èªç±é»åã®é»è·ã®çµ¶å¯Ÿå€ãeãšãããšãèªç±é»åã¯è² é»è·ã§ãããããèªç±é»åã®é»è·ã¯ãã€ãã¹ç¬Šå·ãã€ã-eã§ããã
=== ãªãŒã ã®æ³å ===
ãã€ã人ã®ç©çåŠè
ãªãŒã ã¯æ¬¡ã®ãããªæ³åãçºèŠããã
ãã»ãšãã©ã®å°äœã§ã¯ãé»æµ I ãæµããŠããå°äœäžã®2ç¹ã®ç¹ <math>P_1</math>ãšç¹ <math>P_2</math> éã®é»äœå·® <math>E = E_1 - E_2</math> ã¯ãé»æµ I ã«æ¯äŸãããã
ãã®å®éšæ³åã'''ãªãŒã ã®æ³å'''ïŒOhm's lawïŒãšããã
åŒã§è¡šããšãé»äœå·®ãVãšããŠãé»æµãIãšããå Žåã«ãæ¯äŸä¿æ°ãRãšããŠã
:V=RI
ã§ããã
ããã§ãé»äœãšé»æµã®æ¯äŸä¿æ°Rã'''é»æ°æµæ'''ãããã¯åã«'''æµæ'''ïŒresistanceã'''ã¬ãžã¹ã¿ã³ã¹'''ïŒãšããã
é»æ°æµæã®åäœã¯ãªãŒã ãšèšããèšå·ã¯Ωã§è¡šãã
æ
£ç¿çã«ãæµæã®èšå·ã¯Rã§ããããå Žåãå€ãã
=== é»æ°åè·¯ ===
[[File:Ohm's Law with Voltage source TeX.svg|right|thumb|é»æ°åè·¯å³ã®äŸãé»æºã¯äº€æµé»æºãvãé»å§ãRãæµæãiã¯é»æµã]]
é»æ°åè·¯ãžãšãã«ã®ãŒãäŸçµŠããé»æºãšããŠå®é»å§ã®çŽæµé»æºãèãããåè·¯ã®2å°ç¹éã«ããäžå®ã®é»å§ãäŸçµŠãç¶ãããã®ã§ãããé»å§æºã®åè·¯å³èšå·ãšããŠã¯[[File:Cell.svg|30px|é»å§æº]]ãçšãããããèšå·ã®é·ãåŽãæ£æ¥µã§ããããã©ã¹ã®é»äœã§ãããèšå·ã®çãåŽã¯è² 極ã§ããã
也é»æ± ã¯ãçŽæµé»æºãšããŠåãæ±ã£ãŠè¯ãã
ãªãããããã¯çŽæµé»æºã§ããã亀æµã®å Žåã¯äžè¬åããé»å§æºãšããŠ[[File:Voltage Source.svg|30px|亀æµé»å§æº]]ã®èšå·ãçšããããŸãç¹ã«æ£åŒŠæ³¢äº€æµé»å§æºã§ããã°[[File:Voltage Source (AC).svg|30px|æ£åŒŠæ³¢äº€æµé»å§æº]]ã®èšå·ãçšããã
==== æµæåš ====
[[File:3 Resistors.jpg|thumb|æµæ]]
'''æµæåš'''(resistor)ã¯ãéåžžã¯åã«'''æµæ'''ãšåŒã°ããåè·¯çŽ åã§ãããäžããããé»æ°ãšãã«ã®ãŒãåçŽã«æ¶è²»ããçŽ åã§ãããåè·¯å³èšå·ã¯[[File:Resistor symbol America.svg|60px|æµæ]]ãããã¯[[File:Resistor symbol IEC.svg|60px|è² è·]]ã§ããããæ¬æžã§ã¯ãäž¡è
ãšãæµæã®åè·¯å³èšå·ãšããŠçšããããšã«ãããïŒç»åçŽ æã®ç¢ºä¿ã®éœåã®ãããäž¡æ¹ã®èšå·ãæ¬æžã§ã¯æ··åšããŸããã容赊ãã ãããïŒ
{{clear}}
===== æµæåšã®å³èšå· =====
æ¥æ¬ã§ã¯ãæµæåšã®å³èšå·ã¯ãåŸæ¥ã¯JIS C 0301ïŒ1952幎4æå¶å®ïŒã«åºã¥ããã®ã¶ã®ã¶ã®ç·ç¶ã®å³èšå·ã§å³ç€ºãããŠããããçŸåšã®ãåœéèŠæ Œã®IEC 60617ãå
ã«äœæãããJIS C 0617ïŒ1997-1999幎å¶å®ïŒã§ã¯ã®ã¶ã®ã¶åã®å³èšå·ã¯ç€ºãããªããªããé·æ¹åœ¢ã®ç®±ç¶ã®å³èšå·ã§å³ç€ºããããšã«ãªã£ãŠãããæ§èŠæ Œã§ããJIS C 0301ã¯ãæ°èŠæ ŒJIS C 0617ã®å¶å®ã«äŒŽã£ãŠå»æ¢ããããããæ§èšå·ã§æµæåšãå³ç€ºããå³é¢ã¯ãçŸåšã§ã¯JISéæºæ ãªå³é¢ã«ãªã£ãŠããŸããããããææåã¯ç¡ããããçŸåšãåŸæ¥ã®å³èšå·ãå€çšãããŠããã
<gallery>
ãã¡ã€ã«:Resistor_symbol_America.svg|åŸæ¥èŠæ Œã®å³èšå·
ãã¡ã€ã«:Resistor_symbol_IEC.svg|æ°èŠæ Œã®å³èšå·
</gallery>
==== é»æ°åè·¯å³èšå·ã®äŸ ====
<gallery>
ãã¡ã€ã«:åºå®æµæåš.svg|åºå®æµæåš
File:Variable resistor as rheostat symbol GOST.svg|å¯å€æµæåš
ãã¡ã€ã«:é»æ± .svg|é»æ± ãçŽæµé»æºïŒé·ãæ¹ãæ£æ¥µïŒ
File:Voltage Source (AC).svg|亀æµé»æº
ãã¡ã€ã«:SPST-Switch.svg|ã¹ã€ãã
ãã¡ã€ã«:ã³ã³ãã³ãµ.svg|ã³ã³ãã³ãµ
File:Inductor h wikisch.svg|ã³ã€ã«
File:Symbole amperemetre.png|é»æµèš
File:Symbole voltmetre.png|é»å§èš
File:Earth Ground.svg|æ¥å°
ãã¡ã€ã«:Fuse.svg|ãã¥ãŒãº
</gallery>
==== çŽåãšäžŠå ====
è€æ°ã®åè·¯çŽ åã1ã€ã®ç·äžã«é
眮ãããŠãããããªæ¥ç¶ã'''çŽåæ¥ç¶'''ãšãããè€æ°ã®åè·¯çŽ åãäºè¡ã«åãããããã«é
眮ãããŠããæ¥ç¶ã'''䞊åæ¥ç¶'''ãšããã
çŽåæ¥ç¶ã«ãããŠã¯ãããããã®åè·¯çŽ åã«æµããé»æµã¯å
šãŠçãããäžæ¹ã䞊åæ¥ç¶ã«ãããŠã¯ããããã®åè·¯çŽ åã®äž¡ç«¯ã«ãããé»å§ãå
šãŠçããã
ãŸããçŽåæ¥ç¶ã«ãããŠã¯ããããã®åè·¯çŽ åã«ãããé»å§ã®åãå
šé»å§ãšãªãã䞊åæ¥ç¶ã«ãããŠã¯ããããã®åè·¯çŽ åãæµããé»æµã®åãå
šé»æµãšãªãã
==== çŽåã§ã®åææµæ ====
æµæãè€æ°æ¥ç¶ãããŠããå Žåããã®è€æ°ã®æµæããŸãšããŠãããã1ã€ã®æµæãæ¥ç¶ãããŠãããã®ãããªç䟡çãªåè·¯ãèããããšãã§ãããè€æ°ã®æµæãšç䟡ãª1ã€ã®æµæã'''åææµæ'''ãšããã
[[ãã¡ã€ã«:Resistors in series.svg|thumb|çŽåæµæ]]
æµæã''n''åçŽåã«æ¥ç¶ãããŠããå Žåãèãããæµæ<math>R_1, R_2, \cdots, R_n</math>ãçŽåã«æ¥ç¶ãããŠããå Žåãåæµæãæµããé»æµã¯çãããããã''i''ãšãããåæµæ<math>R_k (k = 1, 2, \cdots, n)</math>ã«ãããé»å§ã<math>v_k</math>ãšãããšããªãŒã ã®æ³åãã
:<math>v_k = R_ki (k = 1, 2, \cdots, n)</math>
ãæãç«ã€ããã®ãšãçŽåæµæã®äž¡ç«¯ã®é»å§''v''ã¯ã
:<math>v = \sum_{k=1}^n v_k = \sum_{k=1}^n R_k i = i\sum_{k=1}^n R_k</math>
ã§ããããããšç䟡ãªæµæ''R''ã1ã€ã ãæ¥ç¶ãããŠãããããªç䟡åè·¯ãèãããšãã
:<math>v = Ri</math>
ãæãç«ã€ããããããã£ãŠãããã®''n''åã®çŽåæµæã®åææµæ''R''ãšããŠ
:<math>R = \sum_{k=1}^n R_k</math>
ãåŸããããªãã¡ãçŽååææµæã¯åæµæã®ç·åãšãªãã
==== 䞊åã§ã®åææµæ ====
[[ãã¡ã€ã«:Resistors in parallel.svg|thumb|䞊åæµæ]]
åæ§ã«ãæµæã''n''å䞊åã«æ¥ç¶ãããŠããå Žåãèãããæµæ<math>R_1, R_2, \cdots, R_n</math>ã䞊åã«æ¥ç¶ãããŠããå Žåãåæµæã®äž¡ç«¯ã®é»å§ã¯çãããããã''v''ãšãããåæµæ<math>R_k (k = 1, 2, \cdots, n)</math>ãæµããé»æµã<math>i_k</math>ãšãããšããªãŒã ã®æ³åãã
:<math>v = R_ki_k (k = 1, 2, \cdots, n)</math>
ãæãç«ã€ããã®ãšã䞊åæµæãžæµã蟌ãé»æµ''i''ã¯ã
:<math>i = \sum_{k=1}^n i_k = \sum_{k=1}^n \frac{v}{R_k} = v\sum_{k=1}^n \frac{1}{R_k}</math>
ã§ããããããšç䟡ãªæµæ''R''ã1ã€ã ãæ¥ç¶ãããŠãããããªç䟡åè·¯ãèãããšãã
:<math>v = Ri</math>
ãæãç«ã€ããããããã£ãŠãããã®''n''åã®äžŠåæµæã®åææµæ''R''ãšããŠ
:<math>\frac{1}{R} = \sum_{k=1}^n \frac{1}{R_k}</math>
ãåŸããããªãã¡ã䞊ååææµæã®éæ°ã¯åæµæã®éæ°ã®ç·åãšãªãã
==== é»å ====
æµæRãé»æµIãæµãããšãããã®éšåã®çºç±ã®ãšãã«ã®ãŒã¯ã1ç§ãããã«RI<sup>2</sup>[J/s]ã§ãããããã'''ãžã¥ãŒã«ç±'''ãšãããååã®ç±æ¥ã¯ç©çåŠè
ã®ãžã¥ãŒã«ã調ã¹ãããã§ããããªãŒã ã®æ³åãããV=RIã§ãããã®ã§ããžã¥ãŒã«ç±ã¯VIãšãæžããã
ããã§ãã²ãšãŸããç±ã®èå¯ã«ã¯é¢ããŠã次ã®éãå®çŸ©ãããé»æ°åè·¯ã®ãã2ç¹éãæµããé»æµIãšããã®ïŒç¹éã®é»å§Vãšã®ç©VIã'''é»å'''ïŒpowerïŒãšå®çŸ©ãããé»åã®èšå·ã¯Pã§è¡šããããããšãå€ãã
é»åã®åäœã®ãžã¥ãŒã«æ¯ç§[J/s]ã[W]ãšããåäœã§è¡šãããã®åäœWã¯ã¯ããïŒWattïŒãšèªãã
ã€ãŸãé»åã¯èšå·ã§
:P[W]=VI
ã§ããã
==== æµæç ====
å°ç·ã®å€ªããé·ãã«ãã£ãŠæµæã®å€§ããã¯å€ãããçŽæçã«å€ªãã»ããæµããããã®ã¯åããã ããããŸãã䞊åæ¥ç¶ãšå¯Ÿå¿ãããããšã§ãå°ç·ã倪ãã»ããæµããããããšã¯èšããã
å®éã«é»æ°æµæã¯ãå°ç·ã®å€ªãã«åæ¯äŸããŠå°ãããªãããšãå®éšçã«ç¢ºèªãããŠãããããã§ãã€ãã®ãããªåŒã«ãããŠã¿ããã
æµæãR[Ω]ãšããå Žåãå°ç·ã®å€ªããé¢ç©ã§è¡šãA[m<sup>2</sup>]ãšããã°ãæ¯äŸå®æ°ã«kãçšããã°ã
:R â 1/A
ã§ãããïŒ âã¯ãæ¯äŸé¢ä¿ãè¡šãæ°åŠèšå·ãïŒ
ããã«ãå°ç·ã¯æ質ã倪ããåããªãã°ãå°ç·ãé·ãã»ã©æµæããé·ãã«æ¯äŸããŠæµæã倧ãããªãããšãã確èªãããŠãããããã§ãããã«ãæµæäœã®é·ããèæ
®ããåŒã«è¡šããŠã¿ãã°ã次ã®ããã«ãªããæµæ垯ã®é·ãã''l''[m]ãšããã°
:R â L/A
ã§ããã
ããã«ãå°ç·ã®æ質ã«ãã£ãŠãæµæã®å€§ããã¯å€ãããåãé·ãã§åã倪ãã®æµæã§ããæ質ã«ãã£ãŠæµæã®å€§ããã¯ç°ãªããããã§ãæ質ããšã®æ¯äŸå®æ°ãρãšããã°ãæµæã®åŒã¯ä»¥äžã®åŒã§èšè¿°ãããã
:<math>R=\rho \frac{l}{A}</math>
ρã¯'''æµæç'''ïŒãŠããããã€ãresistivityïŒãšåŒã°ãããæµæçã®åäœã¯[Ωm]ã§ããã
== ç£å ==
=== ç£å Ž ===
[[File:Magnetic field near pole.svg|thumb|right|200px|æ£ç£ç³ã®åšãã«æ¹äœç£éã眮ããŠç£å Žã®åãã調ã¹ãã]]
ç£ç³ã®ãŸããã«ã¯å¥ã®ç£ç³ãåããåã®ããšãšãªããã®ãçããŠããã
ããã'''ç£å Ž'''ïŒãã°ãmagnetic fieldïŒãããã¯'''ç£ç'''ïŒãããïŒãšåŒã¶ãïŒæ¥æ¬ã®ç©çåŠã§ã¯ç£å ŽãšåŒã¶ããšãå€ãããŸããæ¥æ¬ã®é»æ°å·¥åŠã§ã¯ç£çãšåŒã°ããããšãå€ããææ²»æã®èš³èªã®éã®ãæ¥æ¬åœå
ã®æ¥çããšã®éãã«éãããå°å瀟äŒçãªäºè±¡ã§ãããåŒã³æ¹ã¯ç©çã®æ¬è³ªãšã¯é¢ä¿ãªãã®ã§ãããã§ã¯ãã©ã¡ãã®è¡šçŸãçšãããã¯ãæ¬æžã§ã¯ç¹ã«ãã ãããªããè±èªã§ã¯ç©çåŠã»é»æ°å·¥åŠãšãâmagnetic fieldâã§å
±éããŠãããïŒ
éãã³ãã«ããããã±ã«ã«ç£ç³ãè¿ã¥ãããšãç£ç³ã«åžãä»ããããã
ãŸããéãã³ãã«ããããã±ã«ã«åŒ·ãç£åãäžãããšãéãã³ãã«ããããã±ã«ãã®ãã®ãç£å Žãåšå²ã«åãŒãããã«ãªãã
ãã®ãããªãããšããšã¯ç£å Žãæããªãã£ãç©äœãã匷ãç£å Žãåããããšã«ãã£ãŠç£å ŽãåãŒãããã«ãªãçŸè±¡ã'''ç£å'''ïŒãããmagnetizationïŒãšããã
ãããã¯é»è·ã®éé»èªå°ãšå¯Ÿå¿ãããŠãç£åã®ããšã'''ç£æ°èªå°'''ïŒããããã©ããmagnetic inductionïŒãšãããã
ãããŠãéãã³ãã«ããããã±ã«ã®ããã«ãç£ç³ã«åŒãä»ããããããã«ç£åãããèœåãããç©äœã'''匷ç£æ§äœ'''ïŒãããããããããferromagnetïŒãšããã
éãšã³ãã«ããšããã±ã«ã¯åŒ·ç£æ§äœã§ããã
é
ã¯ç£åããªãããé
ã¯ç£ç³ã«åŒãã€ããããªãã®ã§ãé
ã¯åŒ·ç£æ§äœã§ã¯ãªãã
;ç£æ°é®èœ
éé»èªå°ãå©çšãããéé»é®èœïŒããã§ããããžãïŒãšèšããããäžç©ºã®å°äœãã€ãã£ãŠç©è³ªãå²ãããšã§å€éšé»å Žãé®èœããæ¹æ³ããã£ãã®ãšåæ§ã®ãç£æ°ã®é®èœãã匷ç£æ§äœã§ãåºæ¥ããäžç©ºã®åŒ·ç£æ§äœãçšããŠã匷ç£æ§äœã®å
éšã¯ç£å Žãé®èœã§ãããããã'''ç£æ°é®èœ'''ïŒãããããžããmagnetic shieldingïŒãšãããç£æ°ã·ãŒã«ããšãããã
==== ç£åç· ====
ç£å Žã®åããåããããã«å³ç€ºããããç£ç³ã®äœãç£å Žã®æ¹åã¯ãç ã«å«ãŸããç éã®ç²æ«ãç£ç³ã«ãã¡ãã°ããŠããµããããããšã§èŠ³å¯ã§ããã
[[File:Magnet0873.png|left|300px|ç éã«ããç£åç·ã®èŠ³å¯]]
{{clear}}
ãããå³ç€ºãããšãäžå³ã®ããã«ãªããïŒç»åçŽ æã®ç¢ºä¿ã®éœåäžãåçãšå³ç€ºãšã§ã¯ãN極ãšS極ãéã«ãªã£ãŠããŸããã容赊ãã ãããïŒ
[[File:VFPt cylindrical magnet.svg|thumb|left|300px|ç£åç·ã®å³ç€º]]
ãã®ãããªç£å Žã®å³ã'''ç£åç·'''ïŒãããããããmagnetic line of forceïŒãšãããç£åç·ã®åãã¯ãç£ç³ã®N極ããç£åç·ãåºãŠãS極ã«ç£åç·ãåžåããããšå®çŸ©ããããæ£ç£ç³ã§ã¯ãç£åã®çºçæºãšãªãå Žæããæ£ç£ç³ã®äž¡ç«¯ã®å
端ä»è¿ã«éäžãããããã§ãæ£ç£ç³ã®äž¡ç«¯ã®å
端ä»è¿ã'''ç£æ¥µ'''ïŒãããããmagnetic poleïŒãšããã
ãã®ãããªç£ç³ã®ã€ããç£åç·ã®åœ¢ã¯ãé»æ°åç·ã§ã®ãç°ç¬Šå·ã®é»è·ã©ãããã€ããé»æ°åç·ã«äŒŒãŠããã
[[File:VFPt dipole electric manylines.svg|thumb|center|200px|ç°ç¬Šå·ã®é»è·ã©ããã®å Žåã®é»æ°åç·]]
1ã€ã®æ£ç£ç³ã§ã¯N極ïŒnorth poleïŒã®ç£æ°ã®åŒ·ããšãS極ïŒsouth poleïŒã®ç£æ°ã®åŒ·ãã¯çããããŸããç£ç³ã«ã¯ãå¿
ãN極ãšS極ãšãååšãããN極ãšS極ã®ãã©ã¡ããçæ¹ã ããåãåºãããšã¯åºæ¥ãªããããšãæ£ç£ç³ãåæããŠããåæé¢ã«ç£æ¥µãåºçŸããããã®ãããªçŸè±¡ã®ãããçç±ã¯ãããããæ£ç£ç³ãæ§æãã匷ç£æ§äœã®ååã®1åãã€ãå°ããªç£ç³ã§ããããããå°ããªååã®ç£ç³ããããã€ãæŽåããŠã倧ããªæ£ç£ç³ã«ãªã£ãŠããããã§ããã
ä»®æ³çã«ãç£æ¥µãS極ãŸãã¯N極ã®çæ¹ã ãçŸããçŸè±¡ãçè«èšç®ã®ããã«èããããšããããããã®ãããªçåŽã ãã®ç£æ¥µã'''åç£æ¥µ'''ïŒ'''ã¢ãããŒã«'''ãšãããïŒãšãããã'''åç£æ¥µã¯å®åšããªã'''ã
{{clear}}
æ£ç£ç³ãªã©ããã®ãçåŽã®ç£æ¥µãããã®ãç£æ¥µããã®ç£å Žã®åŒ·ãã®ããšãããã®ãŸãŸãç£æ¥µã®åŒ·ããïŒMagnetic chargeïŒãšåŒã¶ããããã¯'''ç£è·'''ïŒãããmagnetizationïŒã'''ç£æ°é'''ãšããã
ããããããã®ç£åãšç£å Žã®é¢ä¿ãåŒã§è¡šãããšãèããã
ãŸããæ£ç£ç³ã«ã¯ç£æ¥µãäž¡åŽã«2åããã®ã§ãèšç®ãç°¡åã«ããããã«ãæ£ç£ç³ã®äž¡ç«¯ã®è·é¢ã倧ãããå察åŽã®ç£æ¥µã®å€§ãããç¡èŠã§ããç£ç³ãèãããã
ãã®ãããªç£ç³ãçšããŠãå®éšãããšããã次ã®æ³åãåãã£ããç£åã®åŒ·ãã¯2åã®ç©äœã®ç£æ°ém<sub>1</sub>ããã³m<sub>2</sub>ã«æ¯äŸãã2åã®ç©äœéã®è·é¢rã®2ä¹ã«åæ¯äŸããã
åŒã§è¡šããšã
:<math>f = k_m \frac{q_1 q_2}{r^2}</math>
ã§è¡šããããïŒk<sub>m</sub>ã¯æ¯äŸå®æ°ïŒ
ãããçºèŠè
ã®ã¯ãŒãã³ã®åã«ã¡ãªãã§ã'''ç£æ°ã«é¢ããã¯ãŒãã³ã®æ³å'''ãšãããç£æ°émã®åäœã¯'''ãŠã§ãŒã'''ãšãããèšå·ã¯[Wb]ã§è¡šãã
æ¯äŸå®æ°k<sub>m</sub>ãš1ãŠã§ãŒãã®å€§ãããšã®é¢ä¿ã¯ã1ã¡ãŒãã«é¢ãã1wbã©ããã®ç£æ¥µã«ã¯ãããåãçŽ6.33Ã10<sup>4</sup>ãšããŠã
æ¯äŸä¿æ°k<sub>m</sub>ã¯ã
:k<sub>m</sub>â6.33Ã10<sup>4</sup>ã[Nã»m<sup>2</sup>/Wb<sup>2</sup>]
ã§ããã
ã€ãŸãã
:<math>f = k_m \frac{m_1 m_2}{r^2} = 6.33\times10^4 \frac {m_1 m_2}{r^2}</math>
ã§ããã
==== ç£å Žã®åŒ ====
éé»æ°åã«å¯ŸããŠãé»å Žãå®çŸ©ãããããã«ãç£æ°åã«å¯ŸããŠããå Žãå®çŸ©ããããšéœåãè¯ããç£æ°ém<sub>1</sub>[Wb]ãäœãã次ã®éã'''ç£å Žã®åŒ·ã'''ãããã¯'''ç£å Žã®å€§ãã'''ãšèšããèšå·ã¯Hã§è¡šãã
:<math>H = k_m \frac{m_1}{r^2} = 6.33\times10^4 \frac {m_1}{r^2}</math>
ç£å Žã®åŒ·ãHã®åäœã¯[N/Wb]ã§ãããHãçšãããšãç£æ°ém<sub>2</sub>[Wb]ã«ã¯ãããç£æ°åf[N]ã¯ã
:<math>f = m_2H</math>
ãšè¡šããã
== é»æµãã€ããç£å Ž ==
===ã¢ã³ããŒã«ã®æ³å===
[[Image:Electromagnetism.svg|thumb|right|é»æµã®æ¹åãšç£æå¯åºŠã®æ¹åã®é¢ä¿.<br>ç£æã®åãã¯ãå³ããã®æ³åã®åãã§ããã.]]
ç©çåŠè
ã®ãšã«ã¹ãããã¯ãé»æµã®å®éšãããŠããéã«ãããŸããŸè¿ãã«ãããŠãã£ãæ¹äœç£ç³ãåãã®ã確èªããã圌ã詳ãã調ã¹ãçµæã以äžã®ããšãåãã£ãã
é»æµãæµããŠãããšãã«ã¯ããã®ãŸããã«ã¯ãç£å Žãçãããåãã¯ãé»æµã®æ¹åã«å³ãããé²ãããã«ãå³ãããåãåããšåããªã®ã§ãããã'''å³ããã®æ³å'''ãšããã
ã¢ã³ããŒã«ããç£å Žã®å€§ããã調ã¹ãçµæãç£å Žã®å€§ããHã¯ãé»æµI[A]ãçŽç·çã«æµããŠãããšããçŽç·é»æµã®åšãã®ç£å Žã®å€§ããã¯ãå°ç·ããã®è·é¢ãa[m]ãšãããšãç£å Žã®å€§ããH[N/Wb]ã¯ã
:<math>H=\frac{1}{2\pi a}I</math>
ã§ããããšãç¥ãããŠããã
ããã'''ã¢ã³ããŒã«ã®æ³å'''(Ampere's law) ãšããã
ç£å Žã®å€§ããHã®åäœã¯ã[N/Wb]ã§ãããããã£ãœãã¢ã³ããŒã«ã®æ³åã®åŒãã¿ãã°ã¢ã³ãã¢æ¯ã¡ãŒãã«[A/m]ã§ãããã
;é»ç£ç³
[[File:Simple electromagnet2.gif|thumb|é»ç£ç³ã®äŸ.]]
[[ç»å:VFPt Solenoid correct.svg|thumb|right|é»ç£ç³ã³ã€ã«ã«ããçºçããç£çïŒæé¢å³ïŒ]]
å°ç·ãã³ã€ã«ç¶ã«å·»ãã°ãã¢ã³ããŒã«ã®æ³åã§å°ç·ã®åšå²ã«çºçããç£å Žãéãªãããããã®ããã«ããç£å Žã匷ããã³ã€ã«ã'''é»ç£ç³'''ïŒã§ããããããelectromagnetïŒãšãããå°ç·ã«é»æµãæµããŠãããšãã«ã®ã¿ãé»ç£ç³ã¯ç£å Žãçºçãããå°ç·ã«é»æµãæµãã®ãæ¢ãããšãé»ç£ç³ã®ç£å Žã¯æ¶ããã
=== ç£æå¯åºŠ ===
ç£å Žã®å€§ããHã«ã次ã®ç¯ã§æ±ãããŒã¬ã³ãåã®çŸè±¡ã®ãããæ¯äŸä¿æ°μïŒåäœã¯ãã¥ãŒãã³æ¯ã¢ã³ãã¢ã§[N/A<sup>2</sup>]ïŒãæããŠãèšå·Bã§è¡šãã
:B=μH
ãšããããšãããããã®éBã'''ç£æå¯åºŠ'''ïŒmagnetic flux densityïŒãšãããç£å Žã®å€§ããHã®åããšç£æå¯åºŠBã®åãã¯'''åãåã'''ã§ããã
ãŸããç£å Žã®å€§ããHãšç£æå¯åºŠBã®æ¯äŸä¿æ°ã'''éç£ç'''ïŒãšãããã€ãmagnetic permeabilityïŒãšããã
ïŒããŒã¬ã³ãåã«é¢ããŠã¯ã詳ããã¯ç©çIIã§æ±ããèªè
ãç©çIãåŠã¶åŠå¹Žãªãã°ãèªè
ã¯ãããŒã¬ã³ãåãšããåãããã®ã ãªã»ã»ã»ããšã§ãæã£ãŠããã°ãããïŒ
== ããŒã¬ã³ãå ==
[[File:Lorentzkraft-graphic-part1.PNG|thumb|ããŒã¬ã³ãåã®åããé»è·ã§èããå Žåã<br>é床vããç£æå¯åºŠBã«å³ãããåããåããããŒã¬ã³ãåFã®åãã]]
[[File:Lorentzkraft-graphic-part2.PNG|thumb|ããŒã¬ã³ãåã®åãã<br>é»æµIããç£æå¯åºŠBã«å³ãããåããåããããŒã¬ã³ãåFã®åãã]]
ãŸããå°ç·ãçšæãããšãããããã®å°ç·ã¯åºå®ãããã«éæ¢ããŠãããšããŠãããå°ç·ã«åãå ããã°ãå°ç·ãåããããã«ããŠããšãããã
ãã®å°ç·ã«é»æµãæµããã ãã§ã¯ãã¹ã€ã«å°ç·ã¯åããªãããããããã®å°ç·ã«ãå€éšã®ç£ç³ã«ããç£å Žãå ãããšãå°ç·ãåãããã®ãããªãç£å Žãšé»æµã®çžäºäœçšã«ãã£ãŠãå°ç·ã«çããåã'''ããŒã¬ã³ãå'''ïŒããŒã¬ã³ãããããè±: Lorentz forceïŒãšããã
ããŒã¬ã³ãåã®åãã¯ãå°ç·ã®é»æµã®åããšç£å Žã®åãã«åçŽã§ãããé»æµIã®åãããç£æå¯åºŠBã®åãã«å³ãããåãåããšåãã§ããã
ãŸããããŒã¬ã³ãåã®å€§ããã¯ãå°ç·ã®é·ã''l''ãšãç£å Žã®å°ç·ãšã®åçŽæ¹åæåã«æ¯äŸããã
ããŒã¬ã³ãåã®å€§ããF[N]ãåŒã§è¡šãã°ãé»æµãšç£å ŽãšãåçŽã ãšããŠãç£å ŽãåããŠããå°ç·ã®åœ¢ç¶ãçŽç·åœ¢ã ãšããŠãé»æµãI[A]ãšããŠãå°ç·ã®é·ãã''l''[m]ãšããŠãå°ç·ã«ããã£ãŠããå€éšç£å Žã®ç£æå¯åºŠãB[N/(Aã»m)]ãšããã°ã
:<math>F=IBl</math>
ã§è¡šããã
ããŒã¬ã³ãåã®å
¬åŒã«ãã¯ãŒãã³ã®æ³åãªã©ã§ã¯èŠããããããªæ¯äŸä¿æ°ïŒä¿æ°Kãªã©ãïŒãå«ãŸããªãã®ã¯ãããããããã®ããŒã¬ã³ãåã®çŸè±¡ãå
ã«ãç£æ°éãŠã§ãŒãWbã®åäœããã³ç£æå¯åºŠBã®åäœãã決å®ãããŠããããã§ããã
ãŸãããç£æå¯åºŠãã®å称ãããç£æãã»ãå¯åºŠããšããã®ã¯ãå®ã¯ç£æå¯åºŠã®åäœã®[N/(Aã»m)]ã¯ãåäœãåŒå€åœ¢ãããš[Wb/m<sup>2</sup>]ã§ãããããšãç±æ¥ã§ããããã®åäœ[Wb/m<sup>2</sup>]ããé»æ°å·¥åŠè
ã®ãã¹ã©ã®åã«ã¡ãªã¿ãåäœ[Wb/m<sup>2</sup>]
ã'''ãã¹ã©'''ãšèšããèšå·Tã§è¡šãã
:[T]=[Wb/m<sup>2</sup>]
ãã®ããŒã¬ã³ãåã®çŸè±¡ããé»æ°æ©åšã®ã¢ãŒã¿ïŒé»åæ©ïŒã®åçã§ããã
;ãã¬ãã³ã°ã®æ³åã¯é»ç£æ°èšç®ã§ã¯çšããªã
ãªããããã¬ãã³ã°ã®æ³åããšããããŒã¬ã³ãåã«é¢ããæ³åãããããããŒã¬ã³ãåã®èšç®ã«ã¯å®çšçã§ã¯ç¡ããããã¬ãã³ã°ã®åãé¢ããç°ãªãæ³åã幟ã€ããã£ãŠçŽããããééãã®åå ã«ãªããããã®ã§ãæ¬æžã§ã¯æããªãã
å®éã«ãå°éçãªç©çèšç®ã§ã¯ããã¬ãã³ã°ã®æ³åã¯ãèšç®ã«ã¯çšããªãã
ãããããã¬ãã³ã°ã®æ³åã«ã¯ããã¬ãã³ã°ã®å³æã®æ³åããšããããšã¯ç°ãªãããã¬ãã³ã°ã®å·Šæã®æ³åãããããã©ã¡ãããã©ã®ç£æ°ã®çŸè±¡ã«çšããæ³åã ã£ãã®ããééãããããã ãããæ¬æžã§ã¯æããªãã
{{clear}}
==é»ç£èªå°==
ïŒé»ç£èªå°ã«é¢ããŠã¯ã詳ããã¯ç©çIIã§æ±ããïŒ
ã¢ã³ããŒã«ã®æ³åã§ã¯ãé»æµã®åšãã«ç£å Žãã§ããã®ã§ãã£ãã
:ã§ã¯éã«ãç£å ŽãçšããŠé»æµãèµ·ãããããªçŸè±¡ã¯ããã ãããïŒ
å®ã¯ãç£ç³ãåãããªã©ããŠãç£å Žã䌎ãç©äœãéåãããšããã®ãŸããã«ã¯é»å Žãçããã
ä»®ã«ãã³ã€ã«ã®è¿ãã§ãããè¡ãªã£ããšãããšãçããé»å Žã«ãã£ãŠã³ã€ã«ã®äžã«ã¯é»æµãæµããã
çããé»å Žã®å€§ããã¯ã
:<math>\vec E = \frac 1 {2\pi a} \frac {\Delta \vec B}{\Delta t}</math>
ãšãªãã(ååŸaã®å圢ã®ã³ã€ã«ã®å Žåã)
Eã®åäœã¯[V/m]ã§ãããBã®åäœã¯[T]ã§ããã
ãã®çŸè±¡ã'''é»ç£èªå°'''ïŒã§ããããã©ããelectromagnetic inductionïŒãšãããé»ç£èªå°ã«ãã£ãŠçºçããé»æµã'''èªå°é»æµ'''ãšããã
ãŸããèªå°é»æµã®åãã¯ãç£ç³ã®åãã«ãããã³ã€ã«ã®äžãéãç£æã®å€åã劚ããåãã«ãé»æµãæµãããïŒèªå°é»æµãã¢ã³ããŒã«ã®æ³åã«åŸããåšå²ã«ç£å ŽãäœããïŒ
ãã®èªå°é»æµããã³ã€ã«ã®äžãéãç£æã®å€åã劚ããåãã«èªå°é»æµãæµããçŸè±¡ã'''ã¬ã³ãã®æ³å'''ïŒLenz's lawïŒãšããã
åãé åã«''N''åå·»ãããã³ã€ã«ã眮ãããå Žåããã¡ã©ããŒã®é»ç£èªå°ã®æ³åã¯ã次ã®ããã«ãªãã
: <math>\mathcal{E} = - N{{d\Phi_B} \over dt}</math>
ããã§ã<math>\mathcal{E}</math>ã¯èµ·é»åïŒãã«ã ãèšå·ã¯VïŒãΦ<sub>B</sub> ã¯ç£æïŒãŠã§ãŒããèšå·ã¯WbïŒãšããã''N''ã¯é»ç·ã®å·»æ°ãšããã
ãã®é»ç£èªå°ã®çŸè±¡ããç«åçºé»ãæ°Žåçºé»ãªã©ã®çºé»æ©ã®åçã§ãããããçã®çºé»ã§ã¯ãæ°žä¹
ç£ç³ãå転ãããããšã§ãçºé»ãããŠãããç«åãæ°Žåãšããã®ã¯ãæ©åšã®å転ãåŸãæ段ã«ãããªãããŸããçºé»æã®çºé»ã«ã¯ãæ°žä¹
ç£ç³ã®å転ãå©çšããŠãããããçºçããé»å§ãé»æµã¯åšæçãªæ³¢åœ¢ã«ãªãã次ã«èª¬æãã亀æµæ³¢åœ¢ã«ãªãã
== 亀æµåè·¯ ==
[[File:Waveforms.svg|thumb|400px|亀æµæ³¢åœ¢ã®äŸã<br>äžããé ã«ã<br>æ£åŒŠæ³¢ã<br>æ¹åœ¢æ³¢ã<br>äžè§æ³¢ã<br>ã®ãããæ³¢ã]]
åè·¯ãžã®å
¥åé»å§ãåšæçã«æéå€åããåè·¯ã®é»å§ããã³é»æµã'''亀æµ'''ïŒalternating currentïŒãšãããããã«å¯Ÿãã也é»æ± ãªã©ã«ãã£ãŠçºçããé»å§ãé»æµã®ããã«ãæéã«ãããäžå®ãªé»å§ãé»æµã¯'''çŽæµ'''ïŒdirect CurrentïŒãšããã
亀æµæ³¢åœ¢ãäœç§ã§1åšããããšããæéã'''åšæ'''(wave period)ãšãããåšæã®èšå·ã¯<math>T</math>ã§è¡šãåäœã¯ç§[s]ã§ããã
:<math>f = \frac{1}{T}</math>
1ç§éã«æ³¢åœ¢ãäœåšããããšããåæ°ã'''åšæ³¢æ°'''ãããã¯'''æ¯åæ°'''(è±èªã¯ããšãã«frequency)ãšããã
é»æ°ã®æ¥çã§ã¯åšæ³¢æ°ãšããçšèªãçšããããšãå€ããç©çã®æ³¢ã®çè«ã§ã¯æ¯åæ°ãšããè¡šçŸãçšããããšãå€ãã
åšæ³¢æ°ã®åäœã¯[1/s]ã§ããããããã'''ãã«ã'''ïŒhertzïŒãšããåäœã§è¡šããåäœèšå·'''Hz'''ãçšããŠåšæ³¢æ°fããf[Hz]ãšãããµãã«è¡šãã
亀æµé»æµã亀æµé»å§ãæ£åŒŠæ³¢ã®å Žåã¯ããããã®ãã©ã¡ãŒã¿ãçšããŠ
:<math>i(t) = I_0\sin(2\pi ft + \theta_i) = I_0\sin\left(\frac{2\pi}{T}t + \theta_i\right)</math>
:<math>v(t) = V_0\sin(2\pi ft + \theta_v) = V_0\sin\left(\frac{2\pi}{T}t + \theta_v\right)</math>
ãšæžãããšãã§ããã
sinãšã¯äžè§é¢æ°ã§ãããç¥ããªããã°æ°åŠIIãªã©ãåèã«ããã
ãã®ãšãã®sinã®ä¿æ°<math>I_0</math>ã<math>V_0</math>ã'''æ¯å¹
'''(ããã·ããamplitude)ãšããããŸãæå»''t''=0ã«ãããé»æµãé»å§ã®å€ã瀺ããæé波圢ã決å®ãã<math>\theta_i</math>ã<math>\theta_v</math>ã'''åæäœçž'''ãšããã
æ®éç§é«æ ¡ã®é«æ ¡ç©çã§ã¯ã亀æµæ³¢åœ¢ã®èšç®ã«ã¯ãæ£åŒŠæ³¢ã®å Žåãäž»ã«æ±ããæ¹åœ¢æ³¢ãäžè§æ³¢ã®èšç®ã¯ãæ®éã¯æ±ãããªãã
ãã ããå·¥æ¥é«æ ¡ã®ææ¥ããå·¥å Žã®å®åã§ã¯æ±ãããšãããã®ã§ãèªè
ã¯æ³¢åœ¢ãåŠãã§ããããšã
çºé»æããäžè¬å®¶åºã«éãããŠããé»å§ã¯äº€æµé»å§ã§ãããæ±æ¥æ¬ã§ã¯50Hzã§ããã西æ¥æ¬ã§ã¯60Hzã§ãããããã¯ææ²»æ代ã®çºé»æ©ã®èŒžå
¥æã«ãæ±æ¥æ¬ã®äºæ¥è
ã¯ãšãŒããããã50Hzçšã®çºé»æ©ã茞å
¥ãã西æ¥æ¬ã®äºæ¥è
ã¯ã¢ã¡ãªã«ãã60Hzã®çºé»æ©ã茞å
¥ããããšã«ããã
çºé»æããäžè¬ã®å®¶åºãªã©ã«éãããé»æµã®åšæ³¢æ°ã'''åçšåšæ³¢æ°'''ãšããã
åçšé»æºã®é»å§æ¯å¹
ã¯çŽ140Vã§ãããããã¯<math>100\times\sqrt{2}</math>[V]ã§ããã
ãããã«ããšã¯1000Hzã®ããšã§ããããããã«ãã¯kHzãšæžãã
;ã³ã€ã«ã®èªå·±èªå°
亀æµé»æµã«å¯ŸããŠã¯ãé»æµãšåãæ¯åæ°ã§ãã¢ã³ããŒã«ã®æ³åã§çºçããç£å Žãæ¯åããã
å°ç·ã§ã€ããããã³ã€ã«ã¯ãçŽæµé»æµã§ã¯ããã ã®å°ç·ãšããŠã¯ãããããããã亀æµé»æµã«å¯ŸããŠã¯ãé»ç£èªå°ã«ããèªå·±ã®çºçãããç£å Žã劚ãããããªé»æµããã³èµ·é»åãçºçãããããã'''èªå·±èªå°'''ïŒself inductionïŒãšããã
èªå·±èªå°ã«ããèµ·é»åã®å€§ããã¯ãé»æµã®æéå€åçã«æ¯äŸãããèªå·±èªå°ã®èµ·é»åãåŒã§æžãã°ãæ¯äŸä¿æ°ãLãšããŠã
:<math>e=-L\frac{\Delta I}{\Delta t}</math>
ã§ããã
ãã®æ¯äŸä¿æ°<math>L</math>ã'''èªå·±ã€ã³ãã¯ã¿ã³ã¹'''ïŒself inductanceïŒãšãããèªå·±ã€ã³ãã¯ã¿ã³ã¹ã®æ¬¡å
ã¯[Vã»S/m]ã ããããã'''ãã³ãªãŒ'''ãšããåäœã§è¡šããåäœã«Hãšããèšå·ãçšããã
;çžäºèªå°
[[ãã¡ã€ã«:Transformer Flux.svg|thumb|çžäºèªå°ãå©çšããå€å§åšïŒtransformerïŒ]]
éå¿ã«äºã€ã®ã³ã€ã«ãå·»ããã³ã€ã«ã®çæ¹ã®é»æµãå€åããããšãã¢ã³ããŒã«ã®æ³åã«ãã£ãŠçããŠããç£æãå€åãããããå察åŽã®ã³ã€ã«ã«ã¯ããã®ç£æå¯åºŠã®å€åãæã¡æ¶ããããªåãã«èµ·é»åãçºçããããã®çŸè±¡ã'''çžäºèªå°'''ïŒmutual inductionïŒãšèšãã
é»å§ãå
¥åãããåŽã®ã³ã€ã«ã'''1次ã³ã€ã«'''ïŒprimaly coilïŒãšèšããèªå°èµ·é»åãçºçãããåŽã®ã³ã€ã«ã'''2次ã³ã€ã«'''ïŒsecondary coilïŒãšããã
çžäºèªå°ã«ããèµ·é»åã®å€§ããã¯ãé»æµã®æéå€åçã«æ¯äŸãããçžäºèªå°ã®èµ·é»åãåŒã§æžãã°ãæ¯äŸä¿æ°ãMãšããŠãïŒçžäºèªå°ã®æ¯äŸä¿æ°ã¯Lã§ã¯ç¡ããïŒåŒã¯ã
:<math>e=-M\frac{\Delta I}{\Delta t}</math>
ã§ããã
ãã®æ¯äŸä¿æ°<math>M</math>ã'''çžäºã€ã³ãã¯ã¿ã³ã¹'''ïŒself inductanceïŒãšãããçžäºã€ã³ãã¯ã¿ã³ã¹ã®æ¬¡å
ã¯ãèªå·±ã€ã³ãã¯ã¿ã³ã¹ã®åäœãšåãã§'''ãã³ãªãŒ'''ïŒHïŒã§ããã
ãã®çžäºã€ã³ãã¯ã¿ã³ã¹ã®å€§ããã¯ãäž¡æ¹ã®ã³ã€ã«ã®å·»ãæ°ã©ããã®ç©ã«æ¯äŸããã
{{ã³ã©ã |埮åç©åãè€çŽ æ°ã®åè·¯èšç®ã®è©±é¡|
亀æµåè·¯ã®èšç®ããé«æ ¡ã®æ°åŠã«ãã埮åç©åïŒã³ã¶ãããã¶ãïŒãè€çŽ æ°ïŒãµããããïŒã®çè«ãã€ãã£ãŠèšç®ããããšãã§ãããïŒâ» æ°åŠæç§ã§ããæ°åŠ3ã§ç« æ«ã³ã©ã ãªã©ã§èª¬æãããããå ŽåãããïŒåæ通ã®æ°åŠ3æç§æžãªã©ïŒãïŒ
ã ãé«æ ¡çã¯ãç©çã®åŠç¿ã§ã¯ããŸãã¯é»æ°åç·ãç£æç·ãªã©ã®ç¹æ§ãšãã£ãåç·ã®ç¹æ§ã®ã€ã¡ãŒãžãç¿åŸããããèšç®åŒã®ç·Žç¿ã§ãé«æ ¡ç©çã®æç§æžã«ããå·®åïŒãã¶ãïŒèšå·Îãšããã€ãã£ãåççãªèª¬æãç解ããããã«ããã»ããããã
ãã€ã¯å€§äººã®äºæ
ã ãã亀æµåè·¯ã®èšç®æ³ã¯ãåéã«ãã£ãŠç°ãªã£ãŠãããããŸãçµ±äžãããŠãªããïŒâ» é«æ ¡ç©çã®èšæ³ã®ã»ãã«ããããšãã°ããã§ãŒã¶ãŒè¡šç€ºããšããè€çŽ 衚瀺ããšãç°ãªãèšç®æ³ã»èšæ³ããããããã«ãã©ãã©ã¹å€æããšããèšç®æ³ããããïŒ
ãããè€çŽ æ°ã®èšå·ã i ïŒã¢ã€ïŒãšã¯ãããã j ïŒãžã§ã€ïŒã ã£ãããšããåéã«ãã£ãŠéã£ãŠããã
ãªã®ã§ããšããããé«æ ¡çã¯ãé«æ ¡ç©çã®æç§æžã«ãããããªãå®æ°ãå·®åèšæ³Îãªã©ã®èšæ³ã§èšç®ããŠããã°ã倧åŠåéšãªã©ã§ã¯å®å
šã§ããã
ãããèšæ³ã»èšç®æ³ã®éãã¯ãããŸãç©çæ³åçã«ã¯æ¬è³ªçã§ãªãã®ã§ãããŸã埮åç©åã«ããåè·¯ã®èšç®æ³ã«ã¯æ·±å
¥ãããªãã»ããããã
ãŸãã¯åç·ã€ã¡ãŒãžãšããã«ã¿èšå·ãÎãæ¹åŒã®åçç©çã®èšç®ãç¿åŸãããã
:⻠倧åŠãå°éåŠæ ¡ã®é»æ°ç³»ã®åŠæ ¡ã«é²åŠãããšãäžè¿°ã®ããããªåè·¯èšç®æ³ïŒäž»ã«è€çŽ æ°è¡šç€ºãšã©ãã©ã¹å€æïŒãç¿ãã®ã§ãããããã®èšç®æ³ãéèŠããå°éåéã®äººããããããèªåã®å°éåéã®èšç®æ³ã®æ矩ã䞻匵ãããããããé«æ ¡çã«ã¯å€§åŠæå¡ãã¡ã®ã¿ã³ããçãªäºæ
ã¯ã©ãã§ãããã®ã§ãç¡èŠãããã
:ç±³åœã®20äžçŽã®ããŒãã«ç©çåŠè
ãã¡ã€ã³ãã³ããã ãã¶é»æ°å·¥åŠè
ãå«ã£ãŠãïŒåèæç®: ãã¡ã€ã³ãã³ç©çåŠã«ããé»æ°å·¥åŠãžã®ç®èã£ãœãæå¥ãïŒ
}}
== é»ç£æ³¢ ==
[[File:Onde electromagnetique.svg|thumb|400px|é»ç£æ³¢ã®æŠç¥å³ãé»å Žãšç£å Žãšã¯çŽäº€ããŠããã]]
ç£å Žã®åãã«ãã£ãŠé»å ŽãåŒãèµ·ããããããšãé»ç£èªå°ã®ã»ã¯ã·ã§ã³ã§èŠãã
å®éã«ã¯é»å Žã®å€åã«ãã£ãŠç£å ŽãåŒãèµ·ããããããšãç¥ãããŠããã
ããã«ãã£ãŠäœããªã空éäžãé»å Žãšç£å ŽãäŒæããŠããããšãäºæ³ãããã
é»ç£æ³¢ã®é床ãç©çåŠè
ã®ãã¯ã¹ãŠã§ã«ãèšç®ã§æ±ãããšãããé»ç£æ³¢ã®é床ã¯ãç空äžã§ã¯åžžã«äžå®ã§ããã€æ³¢ã®é床cãèšç®ã§æ±ãããšããã
:c=3.0Ã10<sup>8</sup>
ãšãªããæ¢ã«ç¥ãããŠããå
éã«äžèŽããã
ãã®ããšãããå
ã¯é»ç£æ³¢ã®äžçš®ã§ããããšãåãã£ããç©çIIã§ãé»ç£æ³¢ã®é床ãæ±ããèšç®ã¯ã詳ããã¯æ±ãã
èªè
ãå
éã®æž¬å®å®éšã«ã€ããŠèª¿ã¹ããªããç©çIã®æ³¢åã«é¢ããããŒãžãªã©ã§ãã£ãŸãŒã®å®éšã«ã€ããŠãåç
§ã®ããšã
æ³¢ã¯æ³¢é·Î»ãé·ãã»ã©ãæ¯åæ°fãå°ãããªããæ³¢ã®æ³¢é·Î»ãšæ¯åæ°fã®ç©fλã¯äžå®ã§ãããã¯æ³¢ã®é床vã«çãããã€ãŸã
:v=fλ
ã§ããã
é»ç£æ³¢ã®å Žåã¯ãé床ãå
éã®cãªã®ã§
:c=fλ
ã§ããã
=== é»ç£æ³¢ã®åé¡ ===
* é»æ³¢
æŸéçšã®ãã¬ããã©ãžãªã®é»æ³¢ïŒã§ãã±ãradio waveïŒã¯ãé»ç£æ³¢ïŒelectromagnetic waveïŒã®äžçš®ã§ãããæ³¢é·ã0.1mm以äžã®é»ç£æ³¢ãé»æ³¢ã«åé¡ãããããªããé»æ³¢ã®ãã¡ãæ³¢é·ã1mmïœ1cmã®ããªã¡ãŒãã«ã®é»æ³¢ãããªæ³¢ãšãããåæ§ã«ãæ³¢é·ã1cmïœ10cmã®é»æ³¢ãã»ã³ãæ³¢ãšãããæ³¢é·10cmïœ100cm(=1m)ã®é»æ³¢ã¯UHFãšèšããããã¬ãæŸéãªã©ã«äœ¿ãããUHFæŸéã¯ããã®é»æ³¢ã§ãããæ³¢é·1mïœ10mã®é»æ³¢ã¯VHFãšèšãããããã¬ãæŸéã®VHFæŸéã¯ããã®é»æ³¢ã§ããã
* èµ€å€ç·
æ³¢é·ã0.1mm以äžã§ãå¯èŠå
ç·ïŒå¯èŠå
ã®æ倧波é·ã¯780ããã¡ãŒãã«çšåºŠïŒãããã¯æ³¢é·ãé·ãé»ç£æ³¢ã¯èµ€å€ç·ïŒãããããããinfrared raysãã€ã³ãã©ã¬ãŒã ã¬ã€ãºïŒãšããããèµ€ãã®ãå€ããšããçç±ã¯ãå¯èŠå
ã®æ倧波é·ã®è²ãèµ€è²ã ããã§ãããèµ€å€ç·ãã®ãã®ã«ã¯è²ã¯ã€ããŠããªããåžè²©ã®èµ€å€ç·ããŒã¿ãŒãªã©ãèµ€è²ã«çºå
ãã補åãããã®ã¯ã䜿çšè
ãåäœç¢ºèªãã§ããããã«ããããã«ã補åã«èµ€è²ã®ã©ã³ãã䜵眮ããŠããããã§ãããèµ€å€ç·ã¯ãç©äœã«åžåããããããåžåã®éãç±ãçºçããã®ã§ãããŒã¿ãŒãªã©ã«å¿çšãããããªãã倪éœå
ã«ãèµ€å€ç·ã¯å«ãŸããã
:çºèŠã®çµç·¯
ããããèµ€å€ç·ãçºèŠãããçµç·¯ã¯ãã€ã®ãªã¹ã®å€©æåŠè
ã®ããŒã·ã§ã«ã倪éœå
ãããªãºã ã§åå
ããéã«ãèµ€è²ã®å
ç·ã®ãšãªãã®ãç®ã«ã¯è²ãèŠããªãéšåã枩床äžæããŠããããšãçºèŠããããšããçµç·¯ãããã
* å¯èŠå
ç·
[[Image:Linear visible spectrum.svg]]
{| class="wikitable" style="float:right; text-align:right; margin:0px 0px 7px 7px;"
|-
!è²
!æ³¢é·
!ãšãã«ã®ãŒ
|-
| style="background-color:#CEB0F4; text-align:center;" |玫
|380-450 nm
|2.755-3.26 eV
|-
| style="background-color:#B0CCF4; text-align:center;" |é
|450-495 nm
|2.50-2.755 eV
|-
| style="background-color:#B4F4B0; text-align:center;" |ç·
|495-570 nm
|2.175-2.50 eV
|-
| style="background-color:#F4F4B0; text-align:center;" |é»è²
|570-590 nm
|2.10-2.175 eV
|-
| style="background-color:#F4DDB0; text-align:center;" |æ©è²
|590-620 nm
|1.99-2.10 eV
|-
| style="background-color:#F4B0B0; text-align:center;" |èµ€
|620-750 nm
|1.65-1.99 eV
|}
æã
ã人éã®ç®ã«èŠããå¯èŠå
ç·ïŒãããããããvisible lightïŒã®æ³¢é·ã¯ãçŽ780ããã¡ãŒãã«ããçŽ380ããã¡ãŒãã«ã®çšåºŠã§ãããå¯èŠå
ã®äžã§æ³¢é·ãæãé·ãé åã®è²ã¯èµ€è²ã§ãããå¯èŠå
ã®äžã§æ³¢é·ãæãçãé åã®è²ã¯çŽ«è²ã§ããã
å
ãã®ãã®ã«ã¯ãè²ã¯ã€ããŠããªããæã
ã人éã®è³ããç®ã«å
¥ã£ãå¯èŠå
ããè²ãšããŠæããã®ã§ããã
倪éœå
ãããªãºã ãªã©ã§åå
ïŒã¶ãããïŒãããšãæ³¢é·ããšã«è»è·¡ïŒãããïŒãããããããã®åå
ããå
ç·ã¯ãä»ã®æ³¢é·ãå«ãŸãããã äžçš®ã®æ³¢é·ãªã®ã§ããã®ãããªå
ç·ããã³å
ã'''åè²å
'''ïŒmonochromatic lightïŒãšããã
ãŸããçœè²ã¯åè²å
ã§ã¯ãªãã'''çœè²å
'''(white light)ãšã¯ãå
šãŠã®è²ã®å
ãæ··ãã£ãç¶æ
ã§ããã
åæ§ã«ãé»è²ãšããåè²å
ããªããé»è²ãšã¯ãå¯èŠå
ãç¡ãç¶æ
ã§ããã
{{clear}}
* 玫å€ç·
玫å€ç·ïŒããããããultraviolet raysïŒã¯ååŠåå¿ã«åœ±é¿ãäžããäœçšã匷ãã殺èæ¶æ¯ãªã©ã«å¿çšãããã倪éœå
ã«ã玫å€ç·ã¯å«ãŸããã人éã®èã®æ¥çŒãã®åå ã¯ã玫å€ç·ãã¡ã©ãã³è²çŽ ãé
žåãããããã§ããã
:çºèŠã®çµç·¯
èµ€å€ç·ã¯å€ªéœå
ã®ããªãºã ã«ããåå
ã§çºèŠãããã
ãã§ã¯ãåå
ããã玫è²ã®å
ç·ã®ãšãªãã«ãããªã«ãç®ã«ã¯èŠããªãç·ãããã®ã§ã¯ïŒããšãããµããªããšãåŠè
ãã¡ã«ãã£ãŠèãããã
ãã€ãã®ç©çåŠè
ãªãã¿ãŒã«ããååŠçãªå®éšæ¹æ³ãçšããŠã玫å€ç·ã®ååšãå®èšŒãããã
* Xç·ããã³ã¬ã³ãç·
å»ççšã®ã¬ã³ãã²ã³ãªã©ã®ééåçã§çšããããXç·ïŒX-rayïŒãé»ç£æ³¢ã®äžçš®ã§ãããçç©ã®çŽ°èãååã¬ãã«ã§å·ã€ããçºããæ§ãæãã
ã¬ã³ãç·ïŒgammaârayãγ rayïŒãåæ§ã«ãééåçã«ãå¿çšãããããçç©ã®çŽ°èãååã¬ãã«ã§å·ã€ããçºããæ§ãæãã
{{clear}}
----
===é»æ°ã«é¢ããæ¢æ±æŽ»å===
??
[[Category:é«çåŠæ ¡æè²|ç©ãµã€ã1ãŠãã]]
[[ã«ããŽãª:é»æ°|é«ãµã€ã1ãŠãã]]
[[Category:ç©çåŠæè²|é«ãµã€ã1ãŠãã]] | 2005-05-08T07:17:05Z | 2024-02-05T02:40:56Z | [
"ãã³ãã¬ãŒã:Clear",
"ãã³ãã¬ãŒã:-",
"ãã³ãã¬ãŒã:ã³ã©ã ",
"ãã³ãã¬ãŒã:Lang-en-short"
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E7%89%A9%E7%90%86%E5%9F%BA%E7%A4%8E/%E9%9B%BB%E6%B0%97%E3%81%A8%E7%A3%81%E6%B0%97 |
1,940 | é«çåŠæ ¡ç©ç/ç©çI/æ³¢ | é«çåŠæ ¡çç§ ç©çI > æ³¢ | [
{
"paragraph_id": 0,
"tag": "p",
"text": "é«çåŠæ ¡çç§ ç©çI > æ³¢",
"title": ""
}
] | é«çåŠæ ¡çç§ ç©çI > æ³¢ | <small>[[é«çåŠæ ¡çç§ ç©çI]] > æ³¢</small>
----
== æ³¢ã®æ§è³ª ==
:[[é«çåŠæ ¡çç§ ç©çI æ³¢/æ³¢ã®æ§è³ª]] {{é²æ|50%|2015-07-24}}
== é³ ==
:[[é«çåŠæ ¡çç§ ç©çI æ³¢/é³æ³¢ãšæ¯å|æ³¢/é³æ³¢ãšæ¯å]] {{é²æ|50%|2016-01-23}}
== å
==
:[[é«çåŠæ ¡çç§ ç©çI æ³¢/å
æ³¢|æ³¢/å
æ³¢]] {{é²æ|00%|2015-07-24}}
[[Category:é«çåŠæ ¡æè²|ç©ãµã€ã1ãªã¿]]
[[ã«ããŽãª:æ¯åãšæ³¢å|é«ãµã€ã1ãªã¿]]
[[Category:ç©çåŠæè²|é«ãµã€ã1ãªã¿]] | null | 2023-02-01T08:42:34Z | [
"ãã³ãã¬ãŒã:é²æ"
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E7%89%A9%E7%90%86/%E7%89%A9%E7%90%86I/%E6%B3%A2 |
1,941 | é«çåŠæ ¡ç©ç/ç©çI/éåãšãšãã«ã®ãŒ | é«çåŠæ ¡çç§ ç©çI > éåãšãšãã«ã®ãŒ
æ¬é
ã¯é«çåŠæ ¡çç§ ç©çIã®éåãšãšãã«ã®ãŒã®è§£èª¬ã§ããã
(2015-07-10)
åãå ããŠã䌞ã³çž®ã¿ãããã倧ãããç©äœãåäœ(ãããããrigid body)ãšãããããã«å¯ŸããŠããããªã©ã®äŒžã³çž®ã¿ãããç©äœã¯åŒŸæ§äœ(elastic body)ãšããã 以äžã®èšè¿°ã§ã¯ãããã«ãåäœã«ã€ããŠèããã
åäœã«åãæãã£ãŠããç®æããäœçšç¹(ããããŠããpoint of action)ãšèšããäœçšç·ããåã®æ¹åãžå»¶é·ããçŽç·ãäœçšç·(line of action)ãšããã åäœã¯åãå ããäœçœ®ã«ãã£ãŠãåãæ¹ãç°ãªããåã®å ãæ¹ã«ãã£ãŠã䞊é²éåã®ä»ã«å転éåãããå Žåãããã ãŸãããŠãã®åçãèããã°ãåã倧ããã®åãå ããŠããäœçšç¹ã®äœçœ®ã«ãã£ãŠãåäœã«äžãã圱é¿ã¯ç°ãªãããã®ããšãããŠãã®æ¯ç¹ãšäœçšç¹ãšã®è·é¢LãšãåFã®åçŽæ¹åæåF sinΞãšã®ç©ãèãããšå¥œéœåã§ããããã®ç©FL sinΞããåã®ã¢ãŒã¡ã³ã(moment of force)ãšèšãããããã¯åã«ã¢ãŒã¡ã³ã(moment)ãšããã
ãŠã以å€ã®åäœã«å¯ŸããŠããä»»æã®ç¹Oããã®è·é¢ãèãããããæ¯ç¹ãšããŠããã®ç¹Oããã®è·é¢Lãšåã®åçŽæ¹åæåF sinΞéœã®ç©ã§ã¢ãŒã¡ã³ããå®çŸ©ãããã¢ãŒã¡ã³ãã®åäœã¯[Nã»m]ã§ããã ã¢ãŒã¡ã³ããMãšè¡šããå Žåã
ã§ããã åäœã«æããåãè€æ°åãæãå Žåã«ã€ããŠã¯ããã®åã«ããå転æ¹åãåºæºã«ããå転æ¹åãšéã®å Žåã¯ããã€ãã¹ç¬Šå·ã«åãã åã®ã¢ãŒã¡ã³ããé£ãåã£ãŠããå Žåã¯ãã¢ãŒã¡ã³ãã®åèšããŒãã«ãªãããã®å Žåã¯ãåäœã¯å転ããªãã
åäœã«åã倧ããã®åãå察æ¹åã«æãã£ãŠããå Žåããã®åã®å¯Ÿããå¶å(ããããããcouple of force)ãšããã
åäœã«ã¯å€§ããããã£ããããã®å€§ãããç¡èŠããŠãç©äœã質éãæã£ãç¹ãšããŠæ±ãå Žåã¯ãããã質ç¹ãšããã 質ç¹ã¯ãåã®ã¢ãŒã¡ã³ããæããªãã
(Center of Gravity)
(2015-07-10)
éåããŠããç©äœAãéæ¢ããŠããç©äœBã«è¡çªããŠããã®éæ¢ç©äœBãåããããšãããã ãã®ãšããéæ¢ããŠããç©äœãåãåºãé床ã®å€§ããã¯ãç©äœAã®è³ªémAã倧ããã»ã©ãè¡çªãããç©äœBã®é床ã倧ããªé床ã§åãåºãã ããããŸããç©äœAã®é床vAã倧ããã»ã©ãè¡çªãããç©äœBã®é床ã倧ãããªãã ããã
ãã®ããšãããé床vã§éåããŠãã質émã®ç©äœã«é¢ããŠãç©äœã®é床vãšè³ªémã®ç©ã§å®ããããémvãå®çŸ©ãããšéœåãããããã§ããã
ç©äœãåããŠãããšããç©äœã®é床ãšè³ªéã®ç©mvãç©äœã®éåé(ããã©ãããããmomentum)ãšåŒã³ãèšå·ã¯äžè¬ã«pã§è¡šã
ãšå®çŸ©ããã
ç©äœã«å¯ŸããŠåfã Î t {\displaystyle \Delta t} ã®éã ã åããããšãã
ãšããŠãPãåç©(ãããããimpulse)ãšåŒã¶ã ããã§ãåç©ãéåéã®å€åçã§ããããšã瀺ãã å®éããç©äœã«çãæé Î t {\displaystyle \Delta t} ã®éå
ãããã£ããšãããšã
ãšãªãããããã¯éåéã®æéå€åç
ã«æé Î t {\displaystyle \Delta t} ãããããã®ã§ãéåéã®æéå€åã«çããããšãåããã ãã£ãŠãç©äœã«ãããåç©ã¯ãç©äœã®éåéã®å€åéã«çããããšãåãã£ãã
ããã§ã¯ãçæéã®éåéã®å€åçãšããŠã Î p Î t {\displaystyle {\frac {\Delta p}{\Delta t}}} ãšããèšè¿°ãçšããŠããããæ¬æ¥ãã®éã¯w:埮åãçšããŠå®çŸ©ãããããã ããæå°èŠé ã®éœåã®ãããããã§ã¯ãã®ãããªèšè¿°ã¯ããŠããªãã埮åãçšããå°åºã«ã€ããŠã¯ãå€å
žååŠãåç
§ã
éæ¢ããŠããç©äœã«æé Î t {\displaystyle \Delta t} ã®éããæ¹åã«äžæ§ãªåfãããããç©äœãåŸã éåéã¯ã©ãã ãããæŽã«ãç©äœã®è³ªéãmãšãããšãç©äœããã®æ¹åã« åŸãé床ã¯ã©ãã ããã
éåéã®å€ååã¯ç©äœãåããåç©ã«çããã®ã§ãç©äœãåããåç©ãèšç®ããã° ãããç©äœãåããåç©ã¯
ã«çããã®ã§ãç©äœãåŸãéåéã
ã«çãããæŽã«ãéåéã
ãæºããããšãèãããšãç©äœã®é床ã¯
ãšãªãã
éåéã¯ãç©äœãå
šãåãåããªããšãä¿åããã ããã¯ç©äœã«åãåããªããšãã«ã¯ãç©äœã®åããåç©ã¯0ã§ããç©äœã®éåé å€åã0ã§ããããšããåœç¶ã§ããã
ããã«ãè€æ°ã®ç©äœã®éåéã«ã€ããŠã¯ãå¥ã®éèŠãªæ§è³ªãèŠããããããã¯ã è€æ°ã®ç©äœã®ãã€éåéã®ç·åã¯ãããã®ç©äœã®éã®è¡çªã«éã㊠ä¿åãããšããããšã§ããã ããã¯ã€ãŸããäŸãã°ãã2ã€ã®ç©äœãè¡çªãããšããå§ãã«2ç©äœãããããæã£ãŠãã éåéã®åã¯è¡çªãçµãã£ãåŸã«2ç©äœãæã£ãŠããéåéã®åã«çãããšããããšã§ ããã ããã§ãããã€ãã®ç©äœããããšããããã®æã€éåéã®ç·åãã察å¿ããç©äœç³»ã® å
šéåéãšããã
ç©äœã®è¡çªã«ã€ããŠãéåéã¯åžžã«ä¿åãããããããç©äœç³»ã®å
šãšãã«ã®ãŒã¯ åžžã«ä¿åãããšã¯éããªããäžè¬ã«ç©äœã®è¡çªã«ã€ããŠãšãã«ã®ãŒã¯åžžã«å€±ãããŠããã ãã£ãšãç©äœç³»ã«éããªãå
šãšãã«ã®ãŒã¯åžžã«äžå®ã§ããã®ã§ãç©äœãæã£ãŠãã ãšãã«ã®ãŒã¯é³ãç±ã®åœ¢ã§ç©äœç³»ã®å€ã«éããŠè¡ãã®ã§ãããç©äœãè¡çªã«ã€ã㊠倱ããšãã«ã®ãŒã¯è¡çªã«é¢ããç©äœãæã£ãŠããç©æ§å®æ°ã«ãã£ãŠæ±ºãŸãã ãã®ä¿æ°ãw:åçºä¿æ°eãšåŒã¶ãåçºä¿æ°ã¯ãç©äœãè¡çªããããååŸã® ç©äœéã®çžå¯Ÿé床ã®æ¯ã«ãã£ãŠå®ããããã ç¹ã«ç©äœ1ãšç©äœ2ãè¡çªåã«é床 v 1 {\displaystyle v_{1}} , v 2 {\displaystyle v_{2}} ãæã£ãŠãããè¡çªåŸã« é床 v 1 â² {\displaystyle v_{1}'} , v 2 â² {\displaystyle v_{2}'} ãæã£ããšãããšãåçºä¿æ°eã¯ã
ã§å®ãããããããã§ãå³èŸºã®å§ãã® â {\displaystyle -} 笊åã¯ãè¡çªã®ååŸã§ç©äœã®é床ã ãã倧ããç©äœã¯ãè¡çªåã«ããå°ããé床ãæã£ãŠããç©äœããã è¡çªåŸã«ã¯ããå°ããé床ãæã€ããšã«ãªãããã§ããã ãã®ãããåçºä¿æ°ã¯äžè¬ã«æ£ã®æ°ã§ããã ãŸãåçºä¿æ°ã¯1ããå°ããæ°ã§ãããç©äœéã®çžå¯Ÿé床ã¯è¡çªåãã è¡çªåŸã®æ¹ãå°ãããªããç¹ã«e=1ã®ãšããå®å
šåŒŸæ§è¡çªãšåŒã³ 0 < e < 1 {\displaystyle 0<e<1} ã®ãšããé匟æ§è¡çªãšåŒã¶ãå®å
šåŒŸæ§è¡çªã®ãšãã¯ã ãšãã«ã®ãŒã¯å€±ãããªãããšãç¥ãããŠãããäžæ¹ãé匟æ§è¡çªã® ãšãã¯ç©äœç³»ã®å
šãšãã«ã®ãŒã¯å€±ãããã
ããéæ¢ããŠããç©äœ2ã«éåépã§éåããŠããç©äœãè¡çªããããã®ãšãã è¡çªããåŸã®ç©äœ2ãéåé p 2 {\displaystyle p_{2}} ãåŸããšãããšãè¡çªåŸã®ç©äœ1ã®éåé㯠ã©ãã ããšãªã£ããã
éåéã®ä¿ååãèãããšãè¡çªã®ååŸã§ç©äœ1ãšç©äœ2ã§æ§æãããç©äœç³»ã® å
šéåéã¯ä¿åãããããã§ãè¡çªåã®ç©äœç³»ã®å
šéåéã¯pã§ããã®ã§ã è¡çªåŸã®ç©äœç³»ã®å
šéåéãpãšãªããæŽã«ãç©äœ2ã®è¡çªåŸã®éåéã p 2 {\displaystyle p_{2}} ãªã®ã§ãç©äœ1ã®éåéã¯
ãšãªãã
ããã§ãç©äœç³»ã®å
šéåéãä¿åãããããšã¯ãéåã«é¢ããw:äœçšåäœçšã®æ³åããåŸãã äœçšåäœçšã®æ³åãçšãããšãç©äœç³»ã®éã®è¡çªã«éããŠãè¡çªã«é¢ãã ããããã®ç©äœãåããåã¯ã倧ãããçããåãã¯å察ãšãªãã ãã®ãšããããããã®åã«å¯ŸããŠãè¡çªã®æé Î t {\displaystyle \Delta t} ãããããã®ã¯ è¡çªã«éããŠããããã®ç©äœãåãåãåç©ã«çãããããã§ã è¡çªã«é¢ããŠåãåã®åç©ãå
šãŠã®ç©äœã«ã€ããŠè¶³ãåããããšããããã® åã¯äžã®ããšãã0ãšãªããããããå
šéåéã®èšç®ã§ã¯ãŸãã«ãã®ãã㪠å
šç©äœã«ã€ããŠã®éåéã®ç·åãèšç®ããŠããã®ã§ãè¡çªã«ãã£ãŠåŸããããã㪠åç©ã®ç·åã¯0ã«çããããã£ãŠãè¡çªã«éããŠç©äœç³»ã®æã€å
šéåéã¯ä¿åãããã
質émã®2ã€ã®ç©äœãé床 v 1 {\displaystyle v_{1}} , v 2 {\displaystyle v_{2}} ã§ç§»åããŠããããããã®ç©äœãè¡çªãããšãã è¡çªåŸã®ããããã®ç©äœã®é床ãããšãã«ã®ãŒä¿ååãšéåéä¿ååãçšã㊠èšç®ããããã ããç©äœã®è¡çªã«é¢ããŠãšãã«ã®ãŒã¯ä¿åãããšããã
ãã®åé¡ã¯2ã€ã®åã倧ããã®ç©äœãç°ãªã£ãé床ã§ã¶ã€ãããšã ãã®çµæãã©ããªãããèšç®ããåé¡ã§ããã å®éšã®çµæã«ãããšãäžæ¹ãéæ¢ããŠããäžæ¹ãåããŠãããšãã åããŠããç©äœã¯éæ¢ããéæ¢ããŠããç©äœã¯åããŠããç©äœãæã£ãŠãã é床ãšåãé床ã§åãã ãããšãç¥ãããŠãããããã§ã¯ããããã® çµæãèšç®ã«ãã£ãŠç¢ºãããããããšãèŠãããšãåºæ¥ãã è¡çªåŸã®ç©äœã®é床ãããããç©äœ1ã«ã€ããŠã¯ v 1 â² {\displaystyle v_{1}'} ,ç©äœ2ã«ã€ããŠã¯ v 2 â² {\displaystyle v_{2}'} ãšããããã®ãšããç©äœã®è¡çªã«ã€ããŠå
šãšãã«ã®ãŒãä¿åãããããšã çšãããšã
ãåŸããããæŽã«ãç©äœã®è¡çªã«ã€ããŠç©äœç³»ã®å
šéåéãä¿åãããããšãçšãããšã
ãããã¯ã v 1 â² {\displaystyle v'_{1}} , v 2 â² {\displaystyle v'_{2}} ã«ã€ããŠã®2次æ¹çšåŒã§ããã解ãããšãåºæ¥ãã å®éèšç®ãããšã解ãšããŠ
ãåŸããããåè
ã®è§£ã¯è¡çªã«éããŠç©äœã®é床ãå€åããªãããšã 瀺ããŠããããããã¯å®éã®æ
åµãšããŠèãé£ãã®ã§ãåŸè
ã®è§£ãçŸå®ã®è§£ãšãªãã ãã®çµæãèŠããšãç©äœãæã€é床ãå
¥ãæ¿ããããšãåããã
ãã®ããšã¯å®éã«åã倧ããã®çãçšããŠå®éšãè¡ããšã確ãããããšãã§ããã
çŸå®ã®ç©äœã®éåã«ãããŠã¯ããã 1ã€ã®åã ãã§è¡šãããããããªéåã¯æ°å°ãªããããã€ãã®ç©äœããåããåãããã¿åã£ãŠç©äœã®éåã®ãããã決ãŸã£ãŠããããšãå€ãã
äŸãã°ã空æ°äžã«ååšããç©äœã«åããããŠéåãããããšãèããŠã¿ããããã§ã¯ãç©äœã¯ããã«åããããŠãã人éãéå
·ããåãåãããããããäžæ¹ã§ç©äœã¯ç©ºæ°ãšè¡çªããããšã§ç©ºæ°ã®ååããåãåããããšã«ãªãããã®ãããäžè¬ã«ç©ºæ°äžã§ç©äœãè¡ãªãéåã¯ãåããããŠãã人éãæå³ãããã®ãšãããåŸåããããå®éã«ãã®ãããªå¯Ÿå¿ããåã«ãã£ãŠç©äœã®éåã®æ§åã倧ãã圱é¿ãåãããã©ããã¯ãæ±ãçŸè±¡ã®æ§åã«ãã£ãŠå€§ããå€ãã£ãŠãããåé
çšåºŠã®å€§ããã®ç©äœãçšããçæéã®æž¬å®ãªãã空æ°æµæã®åœ±é¿ã¯ç¡èŠããŠãå·®ãæ¯ããªããšæããããããããäŸãã°ãã±ããã倧æ°åã«çªå
¥ãããšãã®ãã±ããã®éåã¯ã空æ°æµæã«ãã£ãŠå€§ãã圱é¿ãã空æ°æµæã®åœ±é¿ãç¡èŠããŠéåã®æ§åã解æããããšã¯é©åã§ã¯ãªãã
ãã®ããã«ã察象ãšããç©äœã®éåã®æ§åã«äŒŽã£ãŠãã©ã®åãéèŠã«ãªãããæ£ããèŠæãããšãå¿
èŠãšãªãã
?? | [
{
"paragraph_id": 0,
"tag": "p",
"text": "é«çåŠæ ¡çç§ ç©çI > éåãšãšãã«ã®ãŒ",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "æ¬é
ã¯é«çåŠæ ¡çç§ ç©çIã®éåãšãšãã«ã®ãŒã®è§£èª¬ã§ããã",
"title": ""
},
{
"paragraph_id": 2,
"tag": "p",
"text": "(2015-07-10)",
"title": "åäœã«åãåã®é£ãåã"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "åãå ããŠã䌞ã³çž®ã¿ãããã倧ãããç©äœãåäœ(ãããããrigid body)ãšãããããã«å¯ŸããŠããããªã©ã®äŒžã³çž®ã¿ãããç©äœã¯åŒŸæ§äœ(elastic body)ãšããã 以äžã®èšè¿°ã§ã¯ãããã«ãåäœã«ã€ããŠèããã",
"title": "åäœã«åãåã®é£ãåã"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "åäœã«åãæãã£ãŠããç®æããäœçšç¹(ããããŠããpoint of action)ãšèšããäœçšç·ããåã®æ¹åãžå»¶é·ããçŽç·ãäœçšç·(line of action)ãšããã åäœã¯åãå ããäœçœ®ã«ãã£ãŠãåãæ¹ãç°ãªããåã®å ãæ¹ã«ãã£ãŠã䞊é²éåã®ä»ã«å転éåãããå Žåãããã ãŸãããŠãã®åçãèããã°ãåã倧ããã®åãå ããŠããäœçšç¹ã®äœçœ®ã«ãã£ãŠãåäœã«äžãã圱é¿ã¯ç°ãªãããã®ããšãããŠãã®æ¯ç¹ãšäœçšç¹ãšã®è·é¢LãšãåFã®åçŽæ¹åæåF sinΞãšã®ç©ãèãããšå¥œéœåã§ããããã®ç©FL sinΞããåã®ã¢ãŒã¡ã³ã(moment of force)ãšèšãããããã¯åã«ã¢ãŒã¡ã³ã(moment)ãšããã",
"title": "åäœã«åãåã®é£ãåã"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ãŠã以å€ã®åäœã«å¯ŸããŠããä»»æã®ç¹Oããã®è·é¢ãèãããããæ¯ç¹ãšããŠããã®ç¹Oããã®è·é¢Lãšåã®åçŽæ¹åæåF sinΞéœã®ç©ã§ã¢ãŒã¡ã³ããå®çŸ©ãããã¢ãŒã¡ã³ãã®åäœã¯[Nã»m]ã§ããã ã¢ãŒã¡ã³ããMãšè¡šããå Žåã",
"title": "åäœã«åãåã®é£ãåã"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ã§ããã åäœã«æããåãè€æ°åãæãå Žåã«ã€ããŠã¯ããã®åã«ããå転æ¹åãåºæºã«ããå転æ¹åãšéã®å Žåã¯ããã€ãã¹ç¬Šå·ã«åãã åã®ã¢ãŒã¡ã³ããé£ãåã£ãŠããå Žåã¯ãã¢ãŒã¡ã³ãã®åèšããŒãã«ãªãããã®å Žåã¯ãåäœã¯å転ããªãã",
"title": "åäœã«åãåã®é£ãåã"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "",
"title": "åäœã«åãåã®é£ãåã"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "åäœã«åã倧ããã®åãå察æ¹åã«æãã£ãŠããå Žåããã®åã®å¯Ÿããå¶å(ããããããcouple of force)ãšããã",
"title": "åäœã«åãåã®é£ãåã"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "",
"title": "åäœã«åãåã®é£ãåã"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "åäœã«ã¯å€§ããããã£ããããã®å€§ãããç¡èŠããŠãç©äœã質éãæã£ãç¹ãšããŠæ±ãå Žåã¯ãããã質ç¹ãšããã 質ç¹ã¯ãåã®ã¢ãŒã¡ã³ããæããªãã",
"title": "åäœã«åãåã®é£ãåã"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "(Center of Gravity)",
"title": "åäœã«åãåã®é£ãåã"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "(2015-07-10)",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "éåããŠããç©äœAãéæ¢ããŠããç©äœBã«è¡çªããŠããã®éæ¢ç©äœBãåããããšãããã ãã®ãšããéæ¢ããŠããç©äœãåãåºãé床ã®å€§ããã¯ãç©äœAã®è³ªémAã倧ããã»ã©ãè¡çªãããç©äœBã®é床ã倧ããªé床ã§åãåºãã ããããŸããç©äœAã®é床vAã倧ããã»ã©ãè¡çªãããç©äœBã®é床ã倧ãããªãã ããã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "ãã®ããšãããé床vã§éåããŠãã質émã®ç©äœã«é¢ããŠãç©äœã®é床vãšè³ªémã®ç©ã§å®ããããémvãå®çŸ©ãããšéœåãããããã§ããã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "ç©äœãåããŠãããšããç©äœã®é床ãšè³ªéã®ç©mvãç©äœã®éåé(ããã©ãããããmomentum)ãšåŒã³ãèšå·ã¯äžè¬ã«pã§è¡šã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "ãšå®çŸ©ããã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "ç©äœã«å¯ŸããŠåfã Î t {\\displaystyle \\Delta t} ã®éã ã åããããšãã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "ãšããŠãPãåç©(ãããããimpulse)ãšåŒã¶ã ããã§ãåç©ãéåéã®å€åçã§ããããšã瀺ãã å®éããç©äœã«çãæé Î t {\\displaystyle \\Delta t} ã®éå",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "ãããã£ããšãããšã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "ãšãªãããããã¯éåéã®æéå€åç",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "ã«æé Î t {\\displaystyle \\Delta t} ãããããã®ã§ãéåéã®æéå€åã«çããããšãåããã ãã£ãŠãç©äœã«ãããåç©ã¯ãç©äœã®éåéã®å€åéã«çããããšãåãã£ãã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "ããã§ã¯ãçæéã®éåéã®å€åçãšããŠã Î p Î t {\\displaystyle {\\frac {\\Delta p}{\\Delta t}}} ãšããèšè¿°ãçšããŠããããæ¬æ¥ãã®éã¯w:埮åãçšããŠå®çŸ©ãããããã ããæå°èŠé ã®éœåã®ãããããã§ã¯ãã®ãããªèšè¿°ã¯ããŠããªãã埮åãçšããå°åºã«ã€ããŠã¯ãå€å
žååŠãåç
§ã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "éæ¢ããŠããç©äœã«æé Î t {\\displaystyle \\Delta t} ã®éããæ¹åã«äžæ§ãªåfãããããç©äœãåŸã éåéã¯ã©ãã ãããæŽã«ãç©äœã®è³ªéãmãšãããšãç©äœããã®æ¹åã« åŸãé床ã¯ã©ãã ããã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "éåéã®å€ååã¯ç©äœãåããåç©ã«çããã®ã§ãç©äœãåããåç©ãèšç®ããã° ãããç©äœãåããåç©ã¯",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "ã«çããã®ã§ãç©äœãåŸãéåéã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "ã«çãããæŽã«ãéåéã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "ãæºããããšãèãããšãç©äœã®é床ã¯",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "ãšãªãã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "éåéã¯ãç©äœãå
šãåãåããªããšãä¿åããã ããã¯ç©äœã«åãåããªããšãã«ã¯ãç©äœã®åããåç©ã¯0ã§ããç©äœã®éåé å€åã0ã§ããããšããåœç¶ã§ããã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "ããã«ãè€æ°ã®ç©äœã®éåéã«ã€ããŠã¯ãå¥ã®éèŠãªæ§è³ªãèŠããããããã¯ã è€æ°ã®ç©äœã®ãã€éåéã®ç·åã¯ãããã®ç©äœã®éã®è¡çªã«éã㊠ä¿åãããšããããšã§ããã ããã¯ã€ãŸããäŸãã°ãã2ã€ã®ç©äœãè¡çªãããšããå§ãã«2ç©äœãããããæã£ãŠãã éåéã®åã¯è¡çªãçµãã£ãåŸã«2ç©äœãæã£ãŠããéåéã®åã«çãããšããããšã§ ããã ããã§ãããã€ãã®ç©äœããããšããããã®æã€éåéã®ç·åãã察å¿ããç©äœç³»ã® å
šéåéãšããã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "ç©äœã®è¡çªã«ã€ããŠãéåéã¯åžžã«ä¿åãããããããç©äœç³»ã®å
šãšãã«ã®ãŒã¯ åžžã«ä¿åãããšã¯éããªããäžè¬ã«ç©äœã®è¡çªã«ã€ããŠãšãã«ã®ãŒã¯åžžã«å€±ãããŠããã ãã£ãšãç©äœç³»ã«éããªãå
šãšãã«ã®ãŒã¯åžžã«äžå®ã§ããã®ã§ãç©äœãæã£ãŠãã ãšãã«ã®ãŒã¯é³ãç±ã®åœ¢ã§ç©äœç³»ã®å€ã«éããŠè¡ãã®ã§ãããç©äœãè¡çªã«ã€ã㊠倱ããšãã«ã®ãŒã¯è¡çªã«é¢ããç©äœãæã£ãŠããç©æ§å®æ°ã«ãã£ãŠæ±ºãŸãã ãã®ä¿æ°ãw:åçºä¿æ°eãšåŒã¶ãåçºä¿æ°ã¯ãç©äœãè¡çªããããååŸã® ç©äœéã®çžå¯Ÿé床ã®æ¯ã«ãã£ãŠå®ããããã ç¹ã«ç©äœ1ãšç©äœ2ãè¡çªåã«é床 v 1 {\\displaystyle v_{1}} , v 2 {\\displaystyle v_{2}} ãæã£ãŠãããè¡çªåŸã« é床 v 1 â² {\\displaystyle v_{1}'} , v 2 â² {\\displaystyle v_{2}'} ãæã£ããšãããšãåçºä¿æ°eã¯ã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "ã§å®ãããããããã§ãå³èŸºã®å§ãã® â {\\displaystyle -} 笊åã¯ãè¡çªã®ååŸã§ç©äœã®é床ã ãã倧ããç©äœã¯ãè¡çªåã«ããå°ããé床ãæã£ãŠããç©äœããã è¡çªåŸã«ã¯ããå°ããé床ãæã€ããšã«ãªãããã§ããã ãã®ãããåçºä¿æ°ã¯äžè¬ã«æ£ã®æ°ã§ããã ãŸãåçºä¿æ°ã¯1ããå°ããæ°ã§ãããç©äœéã®çžå¯Ÿé床ã¯è¡çªåãã è¡çªåŸã®æ¹ãå°ãããªããç¹ã«e=1ã®ãšããå®å
šåŒŸæ§è¡çªãšåŒã³ 0 < e < 1 {\\displaystyle 0<e<1} ã®ãšããé匟æ§è¡çªãšåŒã¶ãå®å
šåŒŸæ§è¡çªã®ãšãã¯ã ãšãã«ã®ãŒã¯å€±ãããªãããšãç¥ãããŠãããäžæ¹ãé匟æ§è¡çªã® ãšãã¯ç©äœç³»ã®å
šãšãã«ã®ãŒã¯å€±ãããã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "ããéæ¢ããŠããç©äœ2ã«éåépã§éåããŠããç©äœãè¡çªããããã®ãšãã è¡çªããåŸã®ç©äœ2ãéåé p 2 {\\displaystyle p_{2}} ãåŸããšãããšãè¡çªåŸã®ç©äœ1ã®éåé㯠ã©ãã ããšãªã£ããã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "éåéã®ä¿ååãèãããšãè¡çªã®ååŸã§ç©äœ1ãšç©äœ2ã§æ§æãããç©äœç³»ã® å
šéåéã¯ä¿åãããããã§ãè¡çªåã®ç©äœç³»ã®å
šéåéã¯pã§ããã®ã§ã è¡çªåŸã®ç©äœç³»ã®å
šéåéãpãšãªããæŽã«ãç©äœ2ã®è¡çªåŸã®éåéã p 2 {\\displaystyle p_{2}} ãªã®ã§ãç©äœ1ã®éåéã¯",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "ãšãªãã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "ããã§ãç©äœç³»ã®å
šéåéãä¿åãããããšã¯ãéåã«é¢ããw:äœçšåäœçšã®æ³åããåŸãã äœçšåäœçšã®æ³åãçšãããšãç©äœç³»ã®éã®è¡çªã«éããŠãè¡çªã«é¢ãã ããããã®ç©äœãåããåã¯ã倧ãããçããåãã¯å察ãšãªãã ãã®ãšããããããã®åã«å¯ŸããŠãè¡çªã®æé Î t {\\displaystyle \\Delta t} ãããããã®ã¯ è¡çªã«éããŠããããã®ç©äœãåãåãåç©ã«çãããããã§ã è¡çªã«é¢ããŠåãåã®åç©ãå
šãŠã®ç©äœã«ã€ããŠè¶³ãåããããšããããã® åã¯äžã®ããšãã0ãšãªããããããå
šéåéã®èšç®ã§ã¯ãŸãã«ãã®ãã㪠å
šç©äœã«ã€ããŠã®éåéã®ç·åãèšç®ããŠããã®ã§ãè¡çªã«ãã£ãŠåŸããããã㪠åç©ã®ç·åã¯0ã«çããããã£ãŠãè¡çªã«éããŠç©äœç³»ã®æã€å
šéåéã¯ä¿åãããã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "質émã®2ã€ã®ç©äœãé床 v 1 {\\displaystyle v_{1}} , v 2 {\\displaystyle v_{2}} ã§ç§»åããŠããããããã®ç©äœãè¡çªãããšãã è¡çªåŸã®ããããã®ç©äœã®é床ãããšãã«ã®ãŒä¿ååãšéåéä¿ååãçšã㊠èšç®ããããã ããç©äœã®è¡çªã«é¢ããŠãšãã«ã®ãŒã¯ä¿åãããšããã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "ãã®åé¡ã¯2ã€ã®åã倧ããã®ç©äœãç°ãªã£ãé床ã§ã¶ã€ãããšã ãã®çµæãã©ããªãããèšç®ããåé¡ã§ããã å®éšã®çµæã«ãããšãäžæ¹ãéæ¢ããŠããäžæ¹ãåããŠãããšãã åããŠããç©äœã¯éæ¢ããéæ¢ããŠããç©äœã¯åããŠããç©äœãæã£ãŠãã é床ãšåãé床ã§åãã ãããšãç¥ãããŠãããããã§ã¯ããããã® çµæãèšç®ã«ãã£ãŠç¢ºãããããããšãèŠãããšãåºæ¥ãã è¡çªåŸã®ç©äœã®é床ãããããç©äœ1ã«ã€ããŠã¯ v 1 â² {\\displaystyle v_{1}'} ,ç©äœ2ã«ã€ããŠã¯ v 2 â² {\\displaystyle v_{2}'} ãšããããã®ãšããç©äœã®è¡çªã«ã€ããŠå
šãšãã«ã®ãŒãä¿åãããããšã çšãããšã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 41,
"tag": "p",
"text": "ãåŸããããæŽã«ãç©äœã®è¡çªã«ã€ããŠç©äœç³»ã®å
šéåéãä¿åãããããšãçšãããšã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 42,
"tag": "p",
"text": "ãããã¯ã v 1 â² {\\displaystyle v'_{1}} , v 2 â² {\\displaystyle v'_{2}} ã«ã€ããŠã®2次æ¹çšåŒã§ããã解ãããšãåºæ¥ãã å®éèšç®ãããšã解ãšããŠ",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 43,
"tag": "p",
"text": "ãåŸããããåè
ã®è§£ã¯è¡çªã«éããŠç©äœã®é床ãå€åããªãããšã 瀺ããŠããããããã¯å®éã®æ
åµãšããŠèãé£ãã®ã§ãåŸè
ã®è§£ãçŸå®ã®è§£ãšãªãã ãã®çµæãèŠããšãç©äœãæã€é床ãå
¥ãæ¿ããããšãåããã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 44,
"tag": "p",
"text": "ãã®ããšã¯å®éã«åã倧ããã®çãçšããŠå®éšãè¡ããšã確ãããããšãã§ããã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 45,
"tag": "p",
"text": "",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 46,
"tag": "p",
"text": "çŸå®ã®ç©äœã®éåã«ãããŠã¯ããã 1ã€ã®åã ãã§è¡šãããããããªéåã¯æ°å°ãªããããã€ãã®ç©äœããåããåãããã¿åã£ãŠç©äœã®éåã®ãããã決ãŸã£ãŠããããšãå€ãã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 47,
"tag": "p",
"text": "äŸãã°ã空æ°äžã«ååšããç©äœã«åããããŠéåãããããšãèããŠã¿ããããã§ã¯ãç©äœã¯ããã«åããããŠãã人éãéå
·ããåãåãããããããäžæ¹ã§ç©äœã¯ç©ºæ°ãšè¡çªããããšã§ç©ºæ°ã®ååããåãåããããšã«ãªãããã®ãããäžè¬ã«ç©ºæ°äžã§ç©äœãè¡ãªãéåã¯ãåããããŠãã人éãæå³ãããã®ãšãããåŸåããããå®éã«ãã®ãããªå¯Ÿå¿ããåã«ãã£ãŠç©äœã®éåã®æ§åã倧ãã圱é¿ãåãããã©ããã¯ãæ±ãçŸè±¡ã®æ§åã«ãã£ãŠå€§ããå€ãã£ãŠãããåé
çšåºŠã®å€§ããã®ç©äœãçšããçæéã®æž¬å®ãªãã空æ°æµæã®åœ±é¿ã¯ç¡èŠããŠãå·®ãæ¯ããªããšæããããããããäŸãã°ãã±ããã倧æ°åã«çªå
¥ãããšãã®ãã±ããã®éåã¯ã空æ°æµæã«ãã£ãŠå€§ãã圱é¿ãã空æ°æµæã®åœ±é¿ãç¡èŠããŠéåã®æ§åã解æããããšã¯é©åã§ã¯ãªãã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 48,
"tag": "p",
"text": "ãã®ããã«ã察象ãšããç©äœã®éåã®æ§åã«äŒŽã£ãŠãã©ã®åãéèŠã«ãªãããæ£ããèŠæãããšãå¿
èŠãšãªãã",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 49,
"tag": "p",
"text": "",
"title": "çºå±: éåéãšåç©"
},
{
"paragraph_id": 50,
"tag": "p",
"text": "??",
"title": "éåãšãšãã«ã®ãŒã«é¢ããæ¢æ±æŽ»å"
}
] | é«çåŠæ ¡çç§ ç©çI > éåãšãšãã«ã®ãŒ æ¬é
ã¯é«çåŠæ ¡çç§ ç©çIã®éåãšãšãã«ã®ãŒã®è§£èª¬ã§ããã | <small>[[é«çåŠæ ¡çç§ ç©çI]] > éåãšãšãã«ã®ãŒ</small>
----
æ¬é
ã¯[[é«çåŠæ ¡çç§ ç©çI]]ã®éåãšãšãã«ã®ãŒã®è§£èª¬ã§ããã
== ç©äœã®éå ==
:[[é«çåŠæ ¡çç§ ç©çI éåãšãšãã«ã®ãŒ/ç©äœã®éå|éåãšãšãã«ã®ãŒ/ç©äœã®éå]] ã§èšèŒã {{é²æ|75%|2015-07-10}}
== éåã®æ³å ==
:[[é«çåŠæ ¡çç§ ç©çI éåãšãšãã«ã®ãŒ/éåã®æ³å|éåãšãšãã«ã®ãŒ/éåã®æ³å]] ã§èšèŒã {{é²æ|50%|2015-07-10}}
== ä»äºãšãšãã«ã®ãŒ ==
:[[é«çåŠæ ¡çç§ ç©çI éåãšãšãã«ã®ãŒ/ä»äºãšãšãã«ã®ãŒ|éåãšãšãã«ã®ãŒ/ä»äºãšãšãã«ã®ãŒ]] ã§èšèŒã {{é²æ|50%|2015-07-18}}
== åäœã«åãåã®é£ãåã ==
{{é²æ|25%|2015-07-10}}
[[File:basculer.jpg|thumb|left|åäœã«ãåã¯ãã©ãåã ?]]
[[File:Palanca-ejemplo.jpg|thumb|300px|'''ãŠã''' ã䜿ãã°ã100 kg ã®ç©äœã 5kg ã®ç©äœã§æã¡äžããããšãã§ããã]]
[[File:Torque, position, and force.svg|thumb|right|]]
åãå ããŠã䌞ã³çž®ã¿ãããã倧ãããç©äœã'''åäœ'''(ãããããrigid body)ãšãããããã«å¯ŸããŠããããªã©ã®äŒžã³çž®ã¿ãããç©äœã¯åŒŸæ§äœ(elastic body)ãšããã
以äžã®èšè¿°ã§ã¯ãããã«ãåäœã«ã€ããŠèããã
åäœã«åãæãã£ãŠããç®æãã'''äœçšç¹'''(ããããŠããpoint of action)ãšèšããäœçšç·ããåã®æ¹åãžå»¶é·ããçŽç·ã'''äœçšç·'''(line of action)ãšããã
åäœã¯åãå ããäœçœ®ã«ãã£ãŠãåãæ¹ãç°ãªããåã®å ãæ¹ã«ãã£ãŠã䞊é²éåã®ä»ã«å転éåãããå Žåãããã
ãŸãããŠãã®åçãèããã°ãåã倧ããã®åãå ããŠããäœçšç¹ã®äœçœ®ã«ãã£ãŠãåäœã«äžãã圱é¿ã¯ç°ãªãããã®ããšãããŠãã®æ¯ç¹ãšäœçšç¹ãšã®è·é¢LãšãåFã®åçŽæ¹åæåF sinθãšã®ç©ãèãããšå¥œéœåã§ããããã®ç©FL sinθãã'''åã®ã¢ãŒã¡ã³ã'''(moment of force)ãšèšãããããã¯åã«ã¢ãŒã¡ã³ã(moment)ãšããã
ãŠã以å€ã®åäœã«å¯ŸããŠããä»»æã®ç¹Oããã®è·é¢ãèãããããæ¯ç¹ãšããŠããã®ç¹Oããã®è·é¢Lãšåã®åçŽæ¹åæåF sinθéœã®ç©ã§ã¢ãŒã¡ã³ããå®çŸ©ãããã¢ãŒã¡ã³ãã®åäœã¯[Nã»m]ã§ããã
ã¢ãŒã¡ã³ããMãšè¡šããå Žåã
:M=FL sinθ
ã§ããã
åäœã«æããåãè€æ°åãæãå Žåã«ã€ããŠã¯ããã®åã«ããå転æ¹åãåºæºã«ããå転æ¹åãšéã®å Žåã¯ããã€ãã¹ç¬Šå·ã«åãã
åã®ã¢ãŒã¡ã³ããé£ãåã£ãŠããå Žåã¯ãã¢ãŒã¡ã³ãã®åèšããŒãã«ãªãããã®å Žåã¯ãåäœã¯å転ããªãã
{{clear}}
;å¶å
[[File:Koppel van krachten.png|thumb|left|å¶åã®ã€ã¡ãŒãžå³]]
[[Image:couple_phys.jpg|thumb|right|300px|å¶å]]
åäœã«åã倧ããã®åãå察æ¹åã«æãã£ãŠããå Žåããã®åã®å¯Ÿãã'''å¶å'''ïŒããããããcouple of forceïŒãšããã
;質ç¹
åäœã«ã¯å€§ããããã£ããããã®å€§ãããç¡èŠããŠãç©äœã質éãæã£ãç¹ãšããŠæ±ãå Žåã¯ãããã'''質ç¹'''ãšããã
質ç¹ã¯ãåã®ã¢ãŒã¡ã³ããæããªãã
{{clear}}
=== éå¿ ===
[[File:Caisse plan incline basculement.svg|thumb|200px|éå¿ã«ã€ããŠ]]
(Center of Gravity)
{{clear}}
==çºå±: éåéãšåç©==
{{é²æ|25%|2015-07-10}}
[[File:Billard.JPG|300px|right|]]
=== éåé ===
éåããŠããç©äœAãéæ¢ããŠããç©äœBã«è¡çªããŠããã®éæ¢ç©äœBãåããããšãããã
ãã®ãšããéæ¢ããŠããç©äœãåãåºãé床ã®å€§ããã¯ãç©äœAã®è³ªém<sub>A</sub>ã倧ããã»ã©ãè¡çªãããç©äœBã®é床ã倧ããªé床ã§åãåºãã ããããŸããç©äœAã®é床vAã倧ããã»ã©ãè¡çªãããç©äœBã®é床ã倧ãããªãã ããã
ãã®ããšãããé床vã§éåããŠãã質émã®ç©äœã«é¢ããŠãç©äœã®é床vãšè³ªémã®ç©ã§å®ããããémvãå®çŸ©ãããšéœåãããããã§ããã
ç©äœãåããŠãããšããç©äœã®é床ãšè³ªéã®ç©mvãç©äœã®'''éåé'''(ããã©ãããããmomentum)ãšåŒã³ãèšå·ã¯äžè¬ã«pã§è¡šã
:<math>
\vec p = m \vec v
</math>
ãšå®çŸ©ããã
=== éåéä¿åã®æ³å ===
ç©äœã«å¯ŸããŠåfã<math>\Delta t</math>ã®éã ã
åããããšãã
:<math>
P = f \Delta t
</math>
ãšããŠãPã'''åç©'''(ãããããimpulse)ãšåŒã¶ã
ããã§ãåç©ãéåéã®å€åçã§ããããšã瀺ãã
å®éããç©äœã«çãæé<math>\Delta t</math>ã®éå
:<math>
\vec f
</math>
ãããã£ããšãããšã
:<math>
\vec P = \vec f \Delta t
</math>
:<math>
= m\vec a \Delta t
</math>
:<math>
= m \frac {\Delta } {\Delta t}\vec v \Delta t
</math>
:<math>
= \frac {\Delta } {\Delta t} \vec p\Delta t
</math>
ãšãªãããããã¯éåéã®æéå€åç
:<math>
\frac {\Delta } {\Delta t} \vec p
</math>
ã«æé<math>\Delta t</math>ãããããã®ã§ãéåéã®æéå€åã«çããããšãåããã
ãã£ãŠãç©äœã«ãããåç©ã¯ãç©äœã®éåéã®å€åéã«çããããšãåãã£ãã
*çºå± 埮åãšå€åé
ããã§ã¯ãçæéã®éåéã®å€åçãšããŠã<math>\frac {\Delta p}{\Delta t}</math>ãšããèšè¿°ãçšããŠããããæ¬æ¥ãã®éã¯[[w:埮å]]ãçšããŠå®çŸ©ãããããã ããæå°èŠé ã®éœåã®ãããããã§ã¯ãã®ãããªèšè¿°ã¯ããŠããªãã埮åãçšããå°åºã«ã€ããŠã¯ã[[å€å
žååŠ]]ãåç
§ã
*åé¡äŸ
**åé¡
éæ¢ããŠããç©äœã«æé<math>\Delta t</math>ã®éããæ¹åã«äžæ§ãªåfãããããç©äœãåŸã
éåéã¯ã©ãã ãããæŽã«ãç©äœã®è³ªéãmãšãããšãç©äœããã®æ¹åã«
åŸãé床ã¯ã©ãã ããã
**解ç
éåéã®å€ååã¯ç©äœãåããåç©ã«çããã®ã§ãç©äœãåããåç©ãèšç®ããã°
ãããç©äœãåããåç©ã¯
:<math>
f \Delta t
</math>
ã«çããã®ã§ãç©äœãåŸãéåéã
:<math>
f \Delta t
</math>
ã«çãããæŽã«ãéåéã
:<math>
p = m v
</math>
ãæºããããšãèãããšãç©äœã®é床ã¯
:<math>
\frac 1 m f \Delta t
</math>
ãšãªãã
éåéã¯ãç©äœãå
šãåãåããªããšãä¿åããã
ããã¯ç©äœã«åãåããªããšãã«ã¯ãç©äœã®åããåç©ã¯0ã§ããç©äœã®éåé
å€åã0ã§ããããšããåœç¶ã§ããã
ããã«ãè€æ°ã®ç©äœã®éåéã«ã€ããŠã¯ãå¥ã®éèŠãªæ§è³ªãèŠããããããã¯ã
è€æ°ã®ç©äœã®ãã€éåéã®ç·åã¯ãããã®ç©äœã®éã®è¡çªã«éããŠ
ä¿åãããšããããšã§ããã
ããã¯ã€ãŸããäŸãã°ãã2ã€ã®ç©äœãè¡çªãããšããå§ãã«2ç©äœãããããæã£ãŠãã
éåéã®åã¯è¡çªãçµãã£ãåŸã«2ç©äœãæã£ãŠããéåéã®åã«çãããšããããšã§
ããã
ããã§ãããã€ãã®ç©äœããããšããããã®æã€éåéã®ç·åãã察å¿ããç©äœç³»ã®
å
šéåéãšããã
ç©äœã®è¡çªã«ã€ããŠãéåéã¯åžžã«ä¿åãããããããç©äœç³»ã®å
šãšãã«ã®ãŒã¯
åžžã«ä¿åãããšã¯éããªããäžè¬ã«ç©äœã®è¡çªã«ã€ããŠãšãã«ã®ãŒã¯åžžã«å€±ãããŠããã
ãã£ãšãç©äœç³»ã«éããªãå
šãšãã«ã®ãŒã¯åžžã«äžå®ã§ããã®ã§ãç©äœãæã£ãŠãã
ãšãã«ã®ãŒã¯é³ãç±ã®åœ¢ã§ç©äœç³»ã®å€ã«éããŠè¡ãã®ã§ãããç©äœãè¡çªã«ã€ããŠ
倱ããšãã«ã®ãŒã¯è¡çªã«é¢ããç©äœãæã£ãŠããç©æ§å®æ°ã«ãã£ãŠæ±ºãŸãã
ãã®ä¿æ°ã[[w:åçºä¿æ°]]eãšåŒã¶ãåçºä¿æ°ã¯ãç©äœãè¡çªããããååŸã®
ç©äœéã®çžå¯Ÿé床ã®æ¯ã«ãã£ãŠå®ããããã
ç¹ã«ç©äœ1ãšç©äœ2ãè¡çªåã«é床 <math>v _1</math>,<math>v _2</math>ãæã£ãŠãããè¡çªåŸã«
é床<math>v _1'</math>,<math>v _2'</math>ãæã£ããšãããšãåçºä¿æ°eã¯ã
:<math>
e = - \frac {v _1 - v _2} {v _1' - v _2'}
</math>
ã§å®ãããããããã§ãå³èŸºã®å§ãã®<math>-</math>笊åã¯ãè¡çªã®ååŸã§ç©äœã®é床ã
ãã倧ããç©äœã¯ãè¡çªåã«ããå°ããé床ãæã£ãŠããç©äœããã
è¡çªåŸã«ã¯ããå°ããé床ãæã€ããšã«ãªãããã§ããã
ãã®ãããåçºä¿æ°ã¯äžè¬ã«æ£ã®æ°ã§ããã
ãŸãåçºä¿æ°ã¯1ããå°ããæ°ã§ãããç©äœéã®çžå¯Ÿé床ã¯è¡çªåãã
è¡çªåŸã®æ¹ãå°ãããªããç¹ã«e=1ã®ãšããå®å
šåŒŸæ§è¡çªãšåŒã³
<math>0<e<1</math>ã®ãšããé匟æ§è¡çªãšåŒã¶ãå®å
šåŒŸæ§è¡çªã®ãšãã¯ã
ãšãã«ã®ãŒã¯å€±ãããªãããšãç¥ãããŠãããäžæ¹ãé匟æ§è¡çªã®
ãšãã¯ç©äœç³»ã®å
šãšãã«ã®ãŒã¯å€±ãããã
*åé¡äŸ
**åé¡
ããéæ¢ããŠããç©äœ2ã«éåépã§éåããŠããç©äœãè¡çªããããã®ãšãã
è¡çªããåŸã®ç©äœ2ãéåé<math>p _2</math>ãåŸããšãããšãè¡çªåŸã®ç©äœ1ã®éåéã¯
ã©ãã ããšãªã£ããã
**解ç
éåéã®ä¿ååãèãããšãè¡çªã®ååŸã§ç©äœ1ãšç©äœ2ã§æ§æãããç©äœç³»ã®
å
šéåéã¯ä¿åãããããã§ãè¡çªåã®ç©äœç³»ã®å
šéåéã¯pã§ããã®ã§ã
è¡çªåŸã®ç©äœç³»ã®å
šéåéãpãšãªããæŽã«ãç©äœ2ã®è¡çªåŸã®éåéã
<math>p _2</math>ãªã®ã§ãç©äœ1ã®éåéã¯
:<math>
p - p _2
</math>
ãšãªãã
ããã§ãç©äœç³»ã®å
šéåéãä¿åãããããšã¯ãéåã«é¢ãã[[w:äœçšåäœçšã®æ³å]]ããåŸãã
äœçšåäœçšã®æ³åãçšãããšãç©äœç³»ã®éã®è¡çªã«éããŠãè¡çªã«é¢ãã
ããããã®ç©äœãåããåã¯ã倧ãããçããåãã¯å察ãšãªãã
ãã®ãšããããããã®åã«å¯ŸããŠãè¡çªã®æé<math>\Delta t</math>ãããããã®ã¯
è¡çªã«éããŠããããã®ç©äœãåãåãåç©ã«çãããããã§ã
è¡çªã«é¢ããŠåãåã®åç©ãå
šãŠã®ç©äœã«ã€ããŠè¶³ãåããããšããããã®
åã¯äžã®ããšãã0ãšãªããããããå
šéåéã®èšç®ã§ã¯ãŸãã«ãã®ãããª
å
šç©äœã«ã€ããŠã®éåéã®ç·åãèšç®ããŠããã®ã§ãè¡çªã«ãã£ãŠåŸããããããª
åç©ã®ç·åã¯0ã«çããããã£ãŠãè¡çªã«éããŠç©äœç³»ã®æã€å
šéåéã¯ä¿åãããã
*åé¡äŸ
**åé¡
質émã®2ã€ã®ç©äœãé床<math>v _1</math>,<math>v _2</math>
ã§ç§»åããŠããããããã®ç©äœãè¡çªãããšãã
è¡çªåŸã®ããããã®ç©äœã®é床ãããšãã«ã®ãŒä¿ååãšéåéä¿ååãçšããŠ
èšç®ããããã ããç©äœã®è¡çªã«é¢ããŠãšãã«ã®ãŒã¯ä¿åãããšããã
**解ç
ãã®åé¡ã¯2ã€ã®åã倧ããã®ç©äœãç°ãªã£ãé床ã§ã¶ã€ãããšã
ãã®çµæãã©ããªãããèšç®ããåé¡ã§ããã
å®éšã®çµæã«ãããšãäžæ¹ãéæ¢ããŠããäžæ¹ãåããŠãããšãã
åããŠããç©äœã¯éæ¢ããéæ¢ããŠããç©äœã¯åããŠããç©äœãæã£ãŠãã
é床ãšåãé床ã§åãã ãããšãç¥ãããŠãããããã§ã¯ããããã®
çµæãèšç®ã«ãã£ãŠç¢ºãããããããšãèŠãããšãåºæ¥ãã
è¡çªåŸã®ç©äœã®é床ãããããç©äœ1ã«ã€ããŠã¯<math>v _1'</math>,ç©äœ2ã«ã€ããŠã¯
<math>v _2'</math>ãšããããã®ãšããç©äœã®è¡çªã«ã€ããŠå
šãšãã«ã®ãŒãä¿åãããããšã
çšãããšã
:<math>
1/2 m v _1^2 + 1/2 m v _2^2
=
1/2 m v _1'{}^2 + 1/2 m v'{} _2^2
</math>
ãåŸããããæŽã«ãç©äœã®è¡çªã«ã€ããŠç©äœç³»ã®å
šéåéãä¿åãããããšãçšãããšã
:<math>
m v _1
+ m v _2 =
m v _1'
+ m v _2'
</math>
ãããã¯ã<math>v' _1</math>,<math>v '_2</math>ã«ã€ããŠã®2次æ¹çšåŒã§ããã解ãããšãåºæ¥ãã
å®éèšç®ãããšã解ãšããŠ
:<math>
(v '_1 ,v' _2 )=(v _1,v _2),(v _2,v _1)
</math>
ãåŸããããåè
ã®è§£ã¯è¡çªã«éããŠç©äœã®é床ãå€åããªãããšã
瀺ããŠããããããã¯å®éã®æ
åµãšããŠèãé£ãã®ã§ãåŸè
ã®è§£ãçŸå®ã®è§£ãšãªãã
ãã®çµæãèŠããšãç©äœãæã€é床ãå
¥ãæ¿ããããšãåããã
ãã®ããšã¯å®éã«åã倧ããã®çãçšããŠå®éšãè¡ããšã確ãããããšãã§ããã
===æ¥åžžã«èµ·ããç©äœã®éå===
çŸå®ã®ç©äœã®éåã«ãããŠã¯ããã 1ã€ã®åã ãã§è¡šãããããããªéåã¯æ°å°ãªããããã€ãã®ç©äœããåããåãããã¿åã£ãŠç©äœã®éåã®ãããã決ãŸã£ãŠããããšãå€ãã
äŸãã°ã空æ°äžã«ååšããç©äœã«åããããŠéåãããããšãèããŠã¿ããããã§ã¯ãç©äœã¯ããã«åããããŠãã人éãéå
·ããåãåãããããããäžæ¹ã§ç©äœã¯ç©ºæ°ãšè¡çªããããšã§ç©ºæ°ã®ååããåãåããããšã«ãªãããã®ãããäžè¬ã«ç©ºæ°äžã§ç©äœãè¡ãªãéåã¯ãåããããŠãã人éãæå³ãããã®ãšãããåŸåããããå®éã«ãã®ãããªå¯Ÿå¿ããåã«ãã£ãŠç©äœã®éåã®æ§åã倧ãã圱é¿ãåãããã©ããã¯ãæ±ãçŸè±¡ã®æ§åã«ãã£ãŠå€§ããå€ãã£ãŠãããåé
çšåºŠã®å€§ããã®ç©äœãçšããçæéã®æž¬å®ãªãã空æ°æµæã®åœ±é¿ã¯ç¡èŠããŠãå·®ãæ¯ããªããšæããããããããäŸãã°ãã±ããã倧æ°åã«çªå
¥ãããšãã®ãã±ããã®éåã¯ã空æ°æµæã«ãã£ãŠå€§ãã圱é¿ãã空æ°æµæã®åœ±é¿ãç¡èŠããŠéåã®æ§åã解æããããšã¯é©åã§ã¯ãªãã
ãã®ããã«ã察象ãšããç©äœã®éåã®æ§åã«äŒŽã£ãŠãã©ã®åãéèŠã«ãªãããæ£ããèŠæãããšãå¿
èŠãšãªãã
==éåãšãšãã«ã®ãŒã«é¢ããæ¢æ±æŽ»å==
??
[[ã«ããŽãª:ååŠ]]
[[ã«ããŽãª:ãšãã«ã®ãŒ]] | 2005-05-08T07:20:33Z | 2024-02-06T05:19:41Z | [
"ãã³ãã¬ãŒã:é²æ",
"ãã³ãã¬ãŒã:Clear"
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E7%89%A9%E7%90%86/%E7%89%A9%E7%90%86I/%E9%81%8B%E5%8B%95%E3%81%A8%E3%82%A8%E3%83%8D%E3%83%AB%E3%82%AE%E3%83%BC |
1,942 | é«çåŠæ ¡ç©ç/ç©çII | æ¬é
ã¯é«çåŠæ ¡çç§ã®ç§ç®ã§ãããç©ç IIãã®è§£èª¬ã§ããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "æ¬é
ã¯é«çåŠæ ¡çç§ã®ç§ç®ã§ãããç©ç IIãã®è§£èª¬ã§ããã",
"title": ""
}
] | æ¬é
ã¯é«çåŠæ ¡çç§ã®ç§ç®ã§ãããç©ç IIãã®è§£èª¬ã§ããã | :* [[é«çåŠæ ¡ç©ç]] > ç©çII<br />
----
æ¬é
ã¯é«çåŠæ ¡çç§ã®ç§ç®ã§ãããç©ç IIãã®è§£èª¬ã§ããã
{{é²æç¶æ³}}
== æç§æž ==
* [[é«çåŠæ ¡ç©ç/ç©çII/åãšéå|åãšéå]] {{é²æ|100%|2013-09-16}}
* [[é«çåŠæ ¡ç©ç/ç©çII/ç±ååŠ|ç±ååŠ]] {{é²æ|75%|2017-08-09}}
* [[é«çåŠæ ¡ç©ç/ç©çII/é»æ°ãšç£æ°|é»æ°ãšç£æ°]] {{é²æ|50%|2013-09-16}}
:* [[é«çåŠæ ¡ç©ç/ç©çII/ç©è³ªãšåå|ç©è³ªãšåå]] {{é²æ|25%|2013-09-16}}(ãé»æ°ãšç£æ°ãã«å
容ãçµ±åäž)
* [[é«çåŠæ ¡ç©ç/ç©çII/ååãšååæ ž|ååãšååæ ž]] {{é²æ|50%|2017-08-09}}
:* åè [[é«çåŠæ ¡ç©ç/ç©çII/ãã³ãã®ã£ãã|ãã³ãã®ã£ãã]] {{é²æ|25%|2017-08-02}}ïŒ2015幎ã§ã¯ç¯å²å€ïŒïŒ
* [[é«çåŠæ ¡ç©ç/ç©çII/çŽ ç²å|çŽ ç²å]] {{é²æ|75%|2017-08-08}}ïŒå
¥è©Šã«ã¯åºãªãã®ãæ®éïŒ
* [[é«çåŠæ ¡ç©ç/ç©çII/課é¡ç 究|課é¡ç 究]]
== é¢é£ãªã³ã¯ ==
* [[é«çåŠæ ¡æ°åŠII/埮åã»ç©åã®èã]]
* [[é«çåŠæ ¡æ°åŠIII/極é]]
* [[é«çåŠæ ¡æ°åŠIII/埮åæ³]]
* [[é«çåŠæ ¡æ°åŠIII/ç©åæ³]]
* ç©çIã®æç§æžãžã®ãªã³ã¯ â [[é«çåŠæ ¡ç©ç/ç©çI|é«çåŠæ ¡çç§ ç©çI]]<br />
[[Category:é«çåŠæ ¡æè²|ç©ãµã€ã2]]
[[Category:ç©çåŠ|é«ãµã€ã2]]
[[Category:ç©çåŠæè²|é«ãµã€ã2]]
[[Category:é«çåŠæ ¡çç§ ç©çII|*]] | null | 2017-08-09T01:40:12Z | [
"ãã³ãã¬ãŒã:é²æç¶æ³",
"ãã³ãã¬ãŒã:é²æ"
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E7%89%A9%E7%90%86/%E7%89%A9%E7%90%86II |
1,943 | é«çåŠæ ¡ç©ç/ååŠ | é«çåŠæ ¡çç§ ç©çåºç€ã§ã¯ãç©äœã®éåãçŽç·äžã®éåãäžå¿ã«æ±ã£ããç©çã§ã¯ãããè€éãªå¹³é¢äžã®éåãæ±ããå¹³é¢äžã®éåã§ã¯ãçŽç·äžã®éåãšã¯éã£ãŠãç©äœã®äœçœ®ãè¡šããã®ã«å¿
èŠãªéã2ã€ã«ãªãããããã¯éåžž x , y {\displaystyle x,\ y} ãšãããã©ã¡ããæå» t {\displaystyle t} ã®äžæã®é¢æ°ãšãªãã
ãããã®é¢æ°ã¯ã©ããªãã®ã§ãããããããã§ã¯äž»ã«ãå®éã®ç©äœã®éåãšããŠããããããããã®ãæ±ãã
å¹³é¢äž,ããªãã¡2次å
ã«ãããŠ,æå» t {\displaystyle t} ã«ãããäœçœ®ã¯ r â ( t ) = ( x ( t ) , y ( t ) ) {\displaystyle {\overrightarrow {r}}(t)=(x(t),\ y(t))} ,埮å°æé Î t {\displaystyle {\mathit {\Delta }}t} éã®å€äœã¯ Î r â = r â ( t + Î t ) â r â ( t ) = ( Î x , Î y ) {\displaystyle {\mathit {\Delta }}{\overrightarrow {r}}={\overrightarrow {r}}(t+{\mathit {\Delta }}t)-{\overrightarrow {r}}(t)=({\mathit {\Delta }}x,\ {\mathit {\Delta }}y)} ãšå®çŸ©ãããããã®ãšã
ã Î t {\displaystyle {\mathit {\Delta }}t} éã®å¹³åé床, Î t â 0 {\displaystyle {\mathit {\Delta }}t\to 0} ã®æ¥µé
ãæå» t {\displaystyle t} ã§ã®(ç¬é)é床ãšããããªã,æå» t {\displaystyle t} ã§ã®éã(é床ã®å€§ãã)ã¯
ãã®å Žåã,é床ããäœçœ®ãæ±ãŸã,åæåæ¯ã«
ãæãç«ã¡,ãããããã¯ãã«ãçšããŠã²ãšãŸãšãã«ããŠä»»æã®æå» t {\displaystyle t} ã«ãããäœçœ®
ãæ±ããããã
ãŸã,
ã Î t {\displaystyle {\mathit {\Delta }}t} éã®å¹³åå é床, Î t â 0 {\displaystyle {\mathit {\Delta }}t\to 0} ã®æ¥µé
ãæå» t {\displaystyle t} ã§ã®(ç¬é)å é床ãšããã ãã®å Žåã,å é床ããé床ãæ±ãŸã,åæåæ¯ã«
ãæãç«ã¡,ãããããã¯ãã«ãçšããŠã²ãšãŸãšãã«ããŠä»»æã®æå» t {\displaystyle t} ã«ãããé床
ãæ±ããããããªã,ããã r â ( 0 ) , v â ( 0 ) {\displaystyle {\overrightarrow {r}}(0),{\overrightarrow {v}}(0)} ã®å€ãåæå€ãšããã ç¹ã«,å é床äžå®ã®ãšãã®éåã¯çå é床éåãšããã,äžèšã®å
¬åŒ(1.2, 1)ã¯ãããã
ãšãªãã
éåæ¹çšåŒã¯ãåãç©äœãåããå é床ã«æ¯äŸãããšããç¹ã¯ããããªãã ããããä»åã¯åãšå é床ã¯ã©ã¡ãããã¯ãã«éã§ããããã£ãŠãå€å f â = ( f x , f y ) {\displaystyle {\overrightarrow {f}}=(f_{x},\ f_{y})} ãåã,å é床 a â = ( a x , a y ) {\displaystyle {\overrightarrow {a}}=(a_{x},\ a_{y})} ã§éåããç©äœã®éåæ¹çšåŒã¯
ãšããããã éåžžã¯ããã®æ¹çšåŒã解ãå Žåã¯èŠçŽ ããšã«ããã
ãšããããã
æå»t = 0ã«ã
ã
ã§ééããç©äœã®æå»tã§ã®äœçœ®ãæ±ããã
ç©äœã®xæ¹åãšyæ¹åã¯äºãã«ç¬ç«ã«çéçŽç·éåãããã ããã§ã¯xæ¹åãyæ¹åãé床
ãªã®ã§ãçéçŽç·éåã®åŒã®ãã¯ãã«éãšããé
ã«ä»£å
¥ãããšã
ãšãªãã èŠçŽ ããšã«ãããšã
ãšãªãã
æå»t=0ã«åç¹(0,\ 0)ãyæ¹åã«é床 v 0 {\displaystyle v_{0}} ã§çéçŽç·éåããŠãã質émã®ç©äœã«ã xæ¹åã®äžæ§ãªåfããããå§ããããã®å Žåãæå»tã«ãããç©äœã®äœçœ®ãš é床ãæ±ããã
x軞æ¹åã«ã¯çå é床éåãšãªãã ç©äœãåããå é床ã¯ãéåæ¹çšåŒã«ãã
ãšãªãã ããã«xæ¹åã®åé床0,åæäœçœ®0ã§ããããšãçå é床çŽç·éåã®åŒã« 代å
¥ãããšã
ãšãªãã
ããã«ãy軞æ¹åã®éåã¯çééåã§ããããã®åé床ã¯ã v 0 {\displaystyle v_{0}} ,åæäœçœ®ã¯0ã§ããã®ã§ã ãã®å€ãçééåã®åŒã«ä»£å
¥ãããšã
ãåŸãããã
ãã®ç« ã§ã¯éåé(ããã©ãããããmomentum)ãæ±ããéåéã¯ãç©äœã®è¡çªã«çœ®ããŠãšãã«ã®ãŒãšäžŠã³ãä¿åéãšãªãéèŠãªéã§ããããŸãããã®ç« ã§ã¯åç©(ãããããimpulse)ãšããéãå°å
¥ãããåç©ã¯éåéã®æéå€åãè¡šããéã§ããããã®å°åºã¯éåæ¹çšåŒãçšããŠæãããã
ç©äœãåããŠããå Žåãç©äœã®é床ãšè³ªéã®ç©ãç©äœã®éåé
ãšå®çŸ©ãããéåæ¹çšåŒ
ã®äž¡èŸºãæå» t = t 1 {\displaystyle t=t_{1}} ãã t = t 2 {\displaystyle t=t_{2}} ãŸã§ç©åãããš
ãšãªãã v â ( t 1 ) = v 1 â , v â ( t 2 ) = v 2 â {\displaystyle {\overrightarrow {v}}(t_{1})={\vec {v_{1}}},{\overrightarrow {v}}(t_{2})={\vec {v_{2}}}} ãšãããš
ãã®åŒã®å·ŠèŸºã¯éåéå€å,å³èŸºã¯åç©(ãããããimpulse)ã§ããããã£ãŠ,éåéå€åã¯åç©ã«çããããšãåãããéåéå€åã Î p â {\displaystyle {\mathit {\Delta }}{\overrightarrow {p}}} ,åç©ã I â {\displaystyle {\overrightarrow {I}}} ãšãããš
ç¹ã«, f â = {\displaystyle {\overrightarrow {f}}=} äžå®ã®ãšã, t 2 â t 1 = Î t {\displaystyle t_{2}-t_{1}={\mathit {\Delta }}t} ãšãããš
埮åãçšããå°åºã«ã€ããŠã¯ãå€å
žååŠãåç
§ã
éæ¢ããŠããç©äœã«æé Î t {\displaystyle {\mathit {\Delta }}t} ã®éããæ¹åã«äžæ§ãªåfãããããç©äœãåŸã éåéã¯ã©ãã ãããããã«ãç©äœã®è³ªéãmãšãããšãç©äœããã®æ¹åã« åŸãé床ã¯ã©ãã ããã
éåéã®å€ååã¯ç©äœãåããåç©ã«çããã®ã§ãç©äœãåããåç©ãèšç®ããã° ãããç©äœãåããåç©ã¯
ã«çããã®ã§ãç©äœãåŸãéåéã
ã«çãããããã«ãéåéã
ãæºããããšãèãããšãç©äœã®é床ã¯
ãšãªãã
éåéã¯ãç©äœãå
šãåãåããªãå Žåã«ã¯ä¿åããããããã¯ç©äœã«åãåããªãå Žåã«ã¯ãç©äœã®åããåç©ã¯0ã§ããç©äœã®éåéå€åã0ã§ããããšããåœç¶ã§ããã
ããã«ãè€æ°ã®ç©äœã®éåéã«ã€ããŠã¯ãå¥ã®éèŠãªæ§è³ªãèŠããããããã¯ãè€æ°ã®ç©äœã®ãã€éåéã®ç·åã¯ãããã®ç©äœã®éã®è¡çªã«éããŠä¿åãããšããããšã§ãããããã¯ã€ãŸããäŸãã°ãã2ã€ã®ç©äœãè¡çªããå Žåãå§ãã«2ç©äœãããããæã£ãŠããéåéã®åã¯è¡çªãçµãã£ãåŸã«2ç©äœãæã£ãŠããéåéã®åã«çãããšããããšã§ãããããã§ãããã€ãã®ç©äœãããå Žåãããã®æã€éåéã®ç·åãã察å¿ããç©äœç³»ã®å
šéåéãšããã
ç©äœã®è¡çªã«ã€ããŠãéåéã¯åžžã«ä¿åãããããããç©äœç³»ã®å
šãšãã«ã®ãŒã¯åžžã«ä¿åãããšã¯éããªããäžè¬ã«ç©äœã®è¡çªã«ã€ããŠãšãã«ã®ãŒã¯åžžã«å€±ãããŠããããã£ãšãç©äœç³»ã«éããªãå
šãšãã«ã®ãŒã¯åžžã«äžå®ã§ããã®ã§ãç©äœãæã£ãŠãããšãã«ã®ãŒã¯é³ãç±ã®åœ¢ã§ç©äœç³»ã®å€ã«éããŠè¡ãã®ã§ãããç©äœãè¡çªã«ã€ããŠå€±ããšãã«ã®ãŒã¯è¡çªã«é¢ããç©äœãæã£ãŠããç©æ§å®æ°ã«ãã£ãŠæ±ºãŸãããã®ä¿æ°ãåçºä¿æ°(ã¯ãã±ã€ãããããcoefficient of restitution)ãšåŒã³ãeãªã©ã®èšå·ã§æžããåçºä¿æ°ã¯ãç©äœãè¡çªããããååŸã§ã®ç©äœéã®çžå¯Ÿé床ã®æ¯ã«ãã£ãŠå®ããããã ç¹ã«ç©äœ1ãšç©äœ2ãè¡çªåã«é床 v 1 , v 2 {\displaystyle v_{1},\ v_{2}} ãæã£ãŠãããè¡çªåŸã«é床 v 1 â² , v 2 â² {\displaystyle v_{1}',\ v_{2}'} ãæã£ããšãããšãåçºä¿æ°eã¯
ã§å®ãããããããã§ãå³èŸºã®å§ãã® â {\displaystyle -} 笊åã¯ãè¡çªã®ååŸã§ç©äœã®é床ããã倧ããç©äœã¯ãè¡çªåã«ããå°ããé床ãæã£ãŠããç©äœãããè¡çªåŸã«ã¯ããå°ããé床ãæã€ããšã«ãªãããã§ããã ãã®ãããåçºä¿æ°ã¯äžè¬ã«æ£ã®æ°ã§ããã ãŸãåçºä¿æ°ã¯1ããå°ããæ°ã§ãããç©äœéã®çžå¯Ÿé床ã¯è¡çªåããè¡çªåŸã®æ¹ãå°ãããªãã ç¹ã« e = 1 {\displaystyle e=1} ã®å Žåã(å®å
š)匟æ§è¡çª(elastic collision)ãšåŒã³ããã£ãœã 0 < e < 1 {\displaystyle 0<e<1} ã®å Žåãé匟æ§è¡çª(inelastic collision)ãšåŒã¶ã匟æ§è¡çªã®å Žåã¯ãååŠçãšãã«ã®ãŒã¯ä¿åããããšãç¥ãããŠãããäžæ¹ãé匟æ§è¡çªã® å Žåã¯ç©äœç³»ã®å
šãšãã«ã®ãŒã¯å€±ãããã
ããéæ¢ããŠããç©äœ2ã«éåépã§éåããŠããç©äœãè¡çªããããã®å Žåã è¡çªããåŸã®ç©äœ2ãéåé p 2 {\displaystyle p_{2}} ãåŸããšãããšãè¡çªåŸã®ç©äœ1ã®éåé㯠ã©ãã ããšãªã£ããã
éåéä¿ååãèãããšãè¡çªã®ååŸã§ç©äœ1ãšç©äœ2ã§æ§æãããç©äœç³»ã®å
šéåéã¯ä¿åããã ããã§ãè¡çªåã®ç©äœç³»ã®å
šéåéã¯pã§ããã®ã§ãè¡çªåŸã®ç©äœç³»ã®å
šéåéãpãšãªãã ããã«ãç©äœ2ã®è¡çªåŸã®éåéã p 2 {\displaystyle p_{2}} ãªã®ã§ãç©äœ1ã®éåéã¯
ãšãªãã
ããã§ãç©äœç³»ã®å
šéåéãä¿åãããããšã¯ãéåã«é¢ãã äœçšã»åäœçšã®æ³å ããåŸãã äœçšåäœçšã®æ³åãçšãããšãç©äœç³»ã®éã®è¡çªã«éããŠãè¡çªã«é¢ããããããã®ç©äœãåããåã¯ã倧ãããçããåãã¯å察ãšãªãã ãã®å Žåãããããã®åã«å¯ŸããŠãè¡çªã®æé Î t {\displaystyle \Delta t} ãããããã®ã¯ è¡çªã«éããŠããããã®ç©äœãåãåãåç©ã«çããã ããã§ãè¡çªã«é¢ããŠåãåã®åç©ãå
šãŠã®ç©äœã«ã€ããŠè¶³ãåããããšããããã®åã¯ãäžã®ããšãã0ãšãªãã ããããå
šéåéã®èšç®ã§ã¯ãŸãã«ãã®ãããªå
šç©äœã«ã€ããŠã®éåéã®ç·åãèšç®ããŠããã®ã§ã è¡çªã«ãã£ãŠåŸããããããªåç©ã®ç·åã¯ã0ã«çããã ãã£ãŠãè¡çªã«éããŠç©äœç³»ã®æã€å
šéåéã¯ä¿åãããã ãããéåéä¿åå(ããã©ãããã ã»ãããããmomentum conservation law)ãšããã
質émã®2ã€ã®ç©äœãé床 v 1 {\displaystyle v_{1}} , v 2 {\displaystyle v_{2}} ã§ç§»åããŠããããããã®ç©äœãè¡çªããå Žåã è¡çªåŸã®ããããã®ç©äœã®é床ãããšãã«ã®ãŒä¿ååãšéåéä¿ååãçšã㊠èšç®ããããã ããç©äœã®è¡çªã«é¢ããŠãšãã«ã®ãŒã¯ä¿åãããšããã
ãã®åé¡ã¯2ã€ã®åã倧ããã®ç©äœãç°ãªã£ãé床ã§ã¶ã€ããå Žå ãã®çµæãã©ããªãããèšç®ããåé¡ã§ããã å®éšã®çµæã«ãããšãäžæ¹ãéæ¢ããŠããäžæ¹ãåããŠããå Žåã åããŠããç©äœã¯éæ¢ããéæ¢ããŠããç©äœã¯åããŠããç©äœãæã£ãŠãã é床ãšåãé床ã§åãã ãããšãç¥ãããŠãããããã§ã¯ããããã® çµæãèšç®ã«ãã£ãŠç¢ºãããããããšãèŠãããšãåºæ¥ãã è¡çªåŸã®ç©äœã®é床ãããããç©äœ1ã«ã€ããŠã¯ v 1 â² {\displaystyle v_{1}'} ,ç©äœ2ã«ã€ããŠã¯ v 2 â² {\displaystyle v_{2}'} ãšããããã®å Žåãç©äœã®è¡çªã«ã€ããŠå
šãšãã«ã®ãŒãä¿åãããããšã çšãããšã
ãåŸããããããã«ãç©äœã®è¡çªã«ã€ããŠç©äœç³»ã®å
šéåéãä¿åãããããšãçšãããšã
ãããã¯ã v 1 â² {\displaystyle v'_{1}} , v 2 â² {\displaystyle v'_{2}} ã«ã€ããŠã®2次æ¹çšåŒã§ããã解ãããšãåºæ¥ããå®éèšç®ãããšã解ãšããŠ
ãåŸããããåè
ã®è§£ã¯è¡çªã«éããŠç©äœã®é床ãå€åãã¬ããšã瀺ããŠããããããã¯å®éã®æ
åµãšããŠèãé£ãã®ã§ãåŸè
ã®è§£ãçŸå®ã®è§£ãšãªãããã®çµæãèŠããšãç©äœãæã€é床ãå
¥ãæ¿ããããšãåããã
ãã®ããšã¯å®éã«åã倧ããã®çãçšããŠå®éšãè¡ããšã確ãããããšãã§ããã
äœçœ®ã®ã¿ããã¡,倧ããããªãã®ã質ç¹ã§ãããåäœãšã¯,倧ãããããã圢ã倧ãããå€ããã¬ç©äœã®ããšã§ããã
åäœã®éåãèããåã«äžå®å¹³é¢äžã®éåã«ã€ããŠæ¬¡ã®ãããªäžè¬çèå¯ãè¡ãã
æå» t {\displaystyle t} ã«ãã㊠x y {\displaystyle xy} å¹³é¢å
ã®äœçœ® r â = ( x , y ) {\displaystyle {\overrightarrow {r}}=(x,\ y)} ãé床 v â = ( v x , v y ) {\displaystyle {\overrightarrow {v}}=(v_{x},\ v_{y})} ã§éåã,å F â = ( F x , F y ) {\displaystyle {\overrightarrow {F}}=(F_{x},\ F_{y})} ãåããŠãã質é m {\displaystyle m} ã®ç©äœã®éåæ¹çšåŒãæåã«åããŠè¡šãã°
2 Ã x â {\displaystyle \times x-} 1 Ã y {\displaystyle \times y} ãã
ãã®å·ŠèŸºã®
ãåç¹OãŸããã®è§éåéãšããã ãã㧠v â {\displaystyle {\overrightarrow {v}}} ãš r â {\displaystyle {\overrightarrow {r}}} ã®ãªãè§ã Ξ , x {\displaystyle \theta ,\ x} 軞㚠r â {\displaystyle {\overrightarrow {r}}} ã®ãªãè§ã Ï {\displaystyle \phi } ãšãããš
ãããã(3.1)ã«ä»£å
¥ãããš
ãåŸãããã
ç©äœãå転ãããåã®å¹æã®å€§ãããè¡šãéãåã®ã¢ãŒã¡ã³ããšãããæŽã« F â {\displaystyle {\overrightarrow {F}}} ãš r â {\displaystyle {\overrightarrow {r}}} ã®ãªãè§ã Î {\displaystyle {\mathit {\Theta }}} ãšãããš
ãã£ãŠåç¹OãŸããã®åã®ã¢ãŒã¡ã³ãã N {\displaystyle N} ã§è¡šããš
ããã« r sin Î {\displaystyle r\sin {\mathit {\Theta }}} ã¯åç¹ããå F â {\displaystyle {\overrightarrow {F}}} ã®äœçšç·ã«äžããåç·ã®é·ãã§ãã,ãããå F â {\displaystyle {\overrightarrow {F}}} ã®åç¹ã«å¯Ÿããè
ã®é·ããšããããã ãåã®ã¢ãŒã¡ã³ãã¯å F â {\displaystyle {\overrightarrow {F}}} ãäœçœ®ãã¯ãã« r â {\displaystyle {\overrightarrow {r}}} ãåæèšåãã«åãåããæ£ãšããŠãã(æèšåãã®é㯠Π< 0 {\displaystyle {\mathit {\Theta }}<0} 㧠r sin Î < 0 {\displaystyle r\sin {\mathit {\Theta }}<0} ãšèãã)ã 以äžãã,3(è§éåéã®æ¹çšåŒ)ã¯
ããã¯åã®ã¢ãŒã¡ã³ããå ããããçµæãšããŠè§éåéãå€åãããšããå æé¢ä¿ãè¡šããç¹ã« N = 0 {\displaystyle N=0} ãªãã°
ãšãªã,è§éåéãä¿åããã
ç©äœã®åéšåã«åãéåã®äœçšç¹ãéå¿(è±: centre of gravity)æãã¯è³ªéäžå¿(è±: centre of mass)ãšããã n {\displaystyle n} ç©äœ(質é: m 1 , m 2 , ⯠⯠, m n {\displaystyle m_{1},\ m_{2},\ \cdots \cdots ,\ m_{n}} ,äœçœ® r 1 â , r 2 â , ⯠⯠, r n â {\displaystyle {\vec {r_{1}}},\ {\vec {r_{2}}},\ \cdots \cdots ,\ {\vec {r_{n}}}} ( n {\displaystyle n} ã¯èªç¶æ°)ã®éå¿ã®äœçœ® r G â {\displaystyle {\vec {r_{\mathrm {G} }}}} ã¯ä»¥äžã®ããã«å®çŸ©ãããã
ãŸãéå¿é床 v G â {\displaystyle {\vec {v_{\mathrm {G} }}}} 㯠d r k â d t = v k â ( k = 1 , 2 , ⯠⯠, n ) {\displaystyle {\frac {d{\vec {r_{k}}}}{dt}}={\vec {v_{k}}}\ (k=1,\ 2,\ \cdots \cdots ,\ n)} ãšãããš
ããã§ã¯ãåççãªå¹³é¢äžã®éåã®1ã€ãšããŠãåéå(è±: circular motion)ãšåæ¯å(è±: simple harmonic motion)ããã€ãããåéåã¯ãåæ¯ãå(ãããµãããsimple pendlum)ã®éåã®é¡äŒŒç©ãšããŠãéèŠã§ããããããšãšãã«ããã®ããŒãžã§ã¯äžæåŒåã«ããéåãæ±ãã äžæåŒåã¯ããããéåãšåãåã§ããã ç©äœãšç©äœã®éã«å¿
ãçããåã§ãããäžæ¹ãããã®åã¯éåžžã«åŒ±ãããã ææã®ããã«å€§ããªè³ªéãæã£ãç©äœã®éåã«ããé¢ãããªãã ããã§ã¯ã倪éœã®ãŸãããå転ããææã®ãããªå€§ããªã¹ã±ãŒã«ã®éåããã€ããããã®ãããªéåã¯åã«è¿ãè»éãšãªãããšãããããã®ãããææã®éåãç解ããäžã§ãåéåãç解ããããšãéèŠã§ããã
ç©äœãåãæãããã«éåããããšãåéåãšåŒã¶ãåãæããããªéåã¯ãäŸãã°ãå圢ã®ã°ã©ãŠã³ãã®ãŸãããèµ°ã人éã®ããã«äººéãææãæã£ãŠè¡ãªãå Žåãæãããèªç¶çŸè±¡ãšããŠèµ·ããå Žåãå€ããäŸãã°ã倪éœã®ãŸãããåãå°çã®éåããå°çã®åããåãæã®éåã¯ãããããåéåã§èšè¿°ãããããŸããäžå®ã®é·ãããã£ãã²ããšäžå®ã®è³ªéãæã£ãç©äœã§äœãããæ¯ãåã®éåã¯ãã²ããåºå®ããç¹ããäžå®ã®è·é¢ããããŠéåããŠãããããç©äœã¯åè»éäžãéåããŠãããåºãæå³ã§ã®åè»éãšãšãããããšãåºæ¥ããããã§ã¯ããã®ãããªå Žåã®ãã¡ã§ä»£è¡šçãªãã®ãšããŠãå®å
šãªåè»éäžãéåããç©äœã®éåããã€ããã
åè»éäžãéåããç©äœã®åº§æšãäžè¬ã®å Žåãšåæ§
ã§è¡šãããããç¹ã«åè»éãè¡šããé¢æ°ã¯é«çåŠæ ¡æ°åŠII ãããããªé¢æ°ã§æ±ã£ãäžè§é¢æ°ã«å¯Ÿå¿ããŠããã
ããã§ãåéåãäžè§é¢æ°ãçšããŠè¡šãããããšãè¿°ã¹ããããã®ããšã¯é«çåŠæ ¡æ°åŠCã®åªä»å€æ°è¡šç€ºãçšããŠãããåªä»å€æ°è¡šç€ºã«ã€ããŠè©³ããã¯ã察å¿ããé
ãåç
§ããŠã»ããã
ååŸr[m]ã®åäžãçããé床ã§ãåéåããç©äœã®éåãèšè¿°ããããšãèããã ããã«ã座æšãåãå Žååç¹ã®äœçœ®ã¯åéåã®äžå¿ã®äœçœ®ãšããã ãã®å Žåã®ç©äœã®éåã¯ãx, y座æšãçšããŠã
ã«ãã£ãŠæžãããããã ãããã®å Žå Ï {\displaystyle \omega } ã¯è§é床ãšåŒã°ãåäœã¯[rad/s]ã§äžããããããã ããããã§[rad]ã¯w:ã©ãžã¢ã³ã§ãããw:匧床æ³ã«ãã£ãŠè§åºŠãè¡šãããå Žåã®åäœã§ããã匧床æ³ã«ã€ããŠã¯é«çåŠæ ¡æ°åŠII ãããããªé¢æ°ãåç
§ãè§é床ã¯åéåãããŠããç©äœãã©ã®çšåºŠã®æéã§åãäžåšãããã«å¯Ÿå¿ããŠããããªã,é«çåŠæ ¡ã®ç©çã«ãããŠè§é床ã¯ã¹ã«ã©ãŒãšããŠæ±ãããŸãããã®éã¯äžã§åããã®ã ããåéåããŠããç©äœã®é床ã«æ¯äŸããã
ãŸããè§é床ã«å¯Ÿå¿ããŠã
ã§äžããããéãw:åšæãšãããåšæã®åäœã¯[s]ã§ãããåšæã¯ç©äœãäœç§éããšã« åç¶ã1åšããããè¡šããéã§ããããã®å Žåã«ã¯ç©äœã¯T[s]ããšã«åç¶ã1åšãããããã«ã
ãw:æ¯åæ°ãšåŒã¶ãæ¯åæ°ã¯åšæãšã¯éã«ãåäœæéåœããã«ç©äœãåç¶ãäœåšãããã æ°ããéã§ãããæ¯åæ°ã®åäœã«ã¯éåžž[Hz]ãçšãããããã¯ã[1/s]ã«çããåäœã§ããã ãŸããåšæTãšãæ¯åæ°fã¯ãé¢ä¿åŒ
ãæºããããã®åŒã¯ããåéåãããŠããç©äœã«ã€ããŠããã®ç©äœã®åéåã® åšæã«å¯Ÿå¿ããæéã®éã«ã¯ãç©äœã¯åç¶ã1åšã ããããšããããšã«å¯Ÿå¿ããã
ãŸãã
ã®åŒã§ ÎŽ {\displaystyle \delta } ã¯ç©äœã®äœçœ®ã®w:äœçžãšåŒã°ããç©äœãåç¶ã®ã©ã®ç¹ã«ãããã瀺ã å€ã§ããã
ãŸãããã®å Žåã®ç©äœã®é床ã®x, yèŠçŽ ã¯
ã§äžããããããã®åŒãšãåŸã®åéåã®å é床ã®å°åºã«ã€ããŠã¯ãåŸã®çºå±ãåç
§ãããã§ãç©äœã®éããvãšãããšã
ãšãªããç©äœã®é床㯠r Ï {\displaystyle r\omega } ã§äžããããããšãåããã
ããã«ã
ãèšç®ãããšã
ãšãªããåéåãããŠããç©äœã®é床ãšåéåã®äžå¿ãåç¹ãšããå Žåã®åº§æšã¯çŽäº€ããŠããããšãåãããããã«ãåéåãããŠããç©äœã®å é床ã¯ã
ãšãªããããã¯
ã«å¯Ÿå¿ããŠãããåéåããããªãç©äœã®å é床ã¯ãåéåãããç©äœã®åº§æšãš ã¡ããã©å察åãã«ãªãããšãåããã
ããã§ã¯ãåéåã®é床ãšå é床ãäžãããããã®å€ã¯ç©äœã®éåã決ãŸãã°æ±ºãŸãå€ãªã®ã§ãåéåã®åŒããèšç®ã§ããããã ãå®éã«ãããã®åŒãåŸãããã«ã¯ãåéåã®åŒã®åŸ®åãè¡ãå¿
èŠããããããããã§ã¯è©³ããæ±ããªããå°åºã«ã€ããŠã¯ãå€å
žååŠãåç
§ã
ååŸr[m]ã®åäžãè§é床 Ï {\displaystyle \omega } ã§éåããç©äœã®å é床ã®å€§ãããèšç®ããã
ã«æ³šç®ãããšãããå³èŸºã«ã€ããŠåéåãããŠããç©äœã®åº§æšãåžžã«
ãæºããããšã«æ³šç®ãããšã
ãšãªãã
50Hzã§åéåããŠããç©äœã®åéåã®åšæãèšç®ããã
ãçšãããšã
ãšãªãã
以äžãã,åéåã®å é床ã®æåã¯
ãã£ãŠ,åéåããç©äœã®è³ªéã m {\displaystyle m} ,åå¿æ¹åã«åãå,ããªãã¡åå¿å(è±: centripetal force)ã F C {\displaystyle F_{\mathrm {C} }} ,æ¥ç·æ¹åã«åãåã F T {\displaystyle F_{\mathrm {T} }} ãšãããšéåæ¹çšåŒã¯
w:åå¿åãw:é å¿å(centrifugal force)
åéåãšé¢ä¿ã®æ·±ãç©äœã®éåãšããŠãåæ¯å(è±: simple harmonic oscillation)ãããããããåæ¯åã¯ããããæ¯åçŸè±¡ã®åºæ¬ã«ãªã£ãŠãããå¿çšç¯å²ãåºãéåã§ãããåéåãšåæ§ãåæ¯åãäžè§é¢æ°ãçšããŠéåãèšè¿°ãããããŸããåšæãäœçžãããç¹ãåéåãšåãã§ããããŸããåæ¯åã¯æ³¢åã«é¢ããçŸè±¡ãšãé¢ä¿ãæ·±ããäœçžãæ¯å¹
ãªã©ã®éãå
±æããŠããã
ããããã¯ãåæ¯åãããç©äœã®æ§è³ªããã詳ããèŠãŠè¡ãã
åæ¯åã¯æ§ã
ãªæ
åµã§ãããããããåçŽãªäŸãšããŠã¯ããã¯ã®æ³åã§æ¯é
ãããã°ãã«æ¥ç¶ãããç©äœã®éåããããããã§ã¯ãã°ãå®æ° k {\displaystyle k} ã®ã°ãã«è³ªé m {\displaystyle m} ã®ç©äœãæ¥ç¶ãããšãããã°ãã®èªç¶é·ã®äœçœ®ãåç¹ãšããŠæå» t {\displaystyle t} ã«ãããåç¹ããã®ç©äœã®äœçœ®ã x ( t ) {\displaystyle x(t)} ãšããå Žåããã®ç©äœã«é¢ããéåæ¹çšåŒã¯
ã§äžããããããã®æ¹çšåŒã®äž¡èŸºã m {\displaystyle m} ã§å²ããšãå é床㯠d 2 x ( t ) d t 2 = â k m x ( t ) {\displaystyle {\frac {d^{2}x(t)}{dt^{2}}}=-{\frac {k}{m}}x(t)} ã§äžããããããšãåããããã®ããã«ãå é床ãšç©äœã®åº§æšãè² ã®æ¯äŸä¿æ°ãæã£ãŠæ¯äŸé¢ä¿ã«ããåŒããåæ¯åã®éåæ¹çšåŒã§ããããã®å Žåãåæ¯åã®æ¯åäžå¿ã x = x C {\displaystyle x=x_{\mathrm {C} }} (åæ¯åã§ã¯æ¯åäžå¿ã¯å®æ°),æå» t {\displaystyle t} ã«ãããç©äœã®éåãäœçœ® x ( t ) {\displaystyle x(t)} ,é床 v ( t ) {\displaystyle v(t)} ,å é床 a ( t ) {\displaystyle a(t)} ã§è¡šããš
ãšãªãã Ï {\displaystyle \omega } ã¯è§æ¯åæ°, ÎŽ {\displaystyle \delta } ã¯åæäœçžã§ããã
ããã§ãåæ¯åã®éåæ¹çšåŒãšãåæ¯åã®éåã®åŒãäžããããå®éã«ã¯åæ¯åã®éåã®åŒã¯éåæ¹çšåŒããå°åºã§ãããããã«ã€ããŠã¯w:埮åæ¹çšåŒãæ±ãå¿
èŠãããã®ã§è©³ããå°åºã«ã€ããŠã¯ãå€å
žååŠãåç
§ã
sin {\displaystyle \sin } é¢æ°ã¯é¢æ°ã®å€ã®å¢å ã«äŒŽã£ãŠåšæçãªæ¯åãè¡ãªãé¢æ°ãªã®ã§ãç©äœã¯ã x = 0 {\displaystyle x=0} ã®ãŸããã§åšæçãªæ¯åãããããšãåããããã ããäžã®åŒã®äžã§Aã¯w:æ¯å¹
ãšåŒã°ããç©äœã®æ¯åã®ç¯å²ãè¡šãéã§ããã
ãã ãããã®å Žåã«ãããŠã¯ãããã®éã¯ç©äœã®åéåã§ã¯ãªããç©äœã®æ¯åã«ã€ããŠã®éã§ãããããããåäœæéåœããã«äœ[rad]ã ãäœçžãé²ããã®éãšæ¯åã®åšæã®äžã§ãã©ã®äœçœ®ã«ç©äœãããããè¡šãéã«å¯Ÿå¿ããŠããããŸããåšæãšæ¯åæ°ãåéåã®å Žåãšåãå®çŸ©ã§äžããããã
ãŸãããã®å Žåã«ã€ããŠã¯éåæ¹çšåŒããè§æ¯åæ°ã決ãŸã
ã§äžããããã
(4.3)ã
ãšæžçŽã, A cos ÎŽ = a , A sin ÎŽ = b {\displaystyle A\cos \delta =a,\ A\sin \delta =b} ãšãããš
ãšãªã,æ¯å¹
ã¯
質émãæã€ããç©äœã«ã€ããŠãã°ãå®æ° k 1 {\displaystyle k_{1}} ã®ã°ããšã°ãå®æ° k 2 {\displaystyle k_{2}} ã®ã°ãã« ã€ãªãããå Žåã§ã¯ã ã©ã¡ãã®å Žåã®æ¹ãç©äœã®è§é床ã倧ãããªããã ãã ãã k 1 > k 2 {\displaystyle k_{1}>k_{2}} ãæãç«ã€ãšããããŸããåšæãšæ¯åæ°ã«ã€ããŠã¯ã©ããªããã
ãã®å Žåã«ã¯ãã®åæ¯åã®è§æ¯åæ°ã¯ã
ã§äžããããããã®éã¯ã°ãå®æ°kã倧ããã»ã©å€§ããã®ã§ãè§æ¯åæ°ã¯ ã°ãå®æ° k 1 {\displaystyle k_{1}} ãæã€ã°ãã®è§æ¯åæ°ã®æ¹ãã°ãå®æ° k 2 {\displaystyle k_{2}} ãæã€ã°ãã®è§æ¯åæ° ãã倧ãããªãããŸããåæ¯åã®æ¯åæ°ã¯åæ¯åã®è§æ¯åæ°ã«æ¯äŸããã®ã§ã æ¯åæ°ã«ã€ããŠãã ã°ãå®æ° k 1 {\displaystyle k_{1}} ãæã€ã°ãã®æ¯åæ°ã®æ¹ãã°ãå®æ° k 2 {\displaystyle k_{2}} ã æã€ã°ãã®æ¯åæ°ãã倧ãããªããäžæ¹ããã®å Žåã®åšæã«ã€ããŠã¯ã
ãæãç«ã€ãããã°ãå®æ°kãå°ããã»ã©å€§ãããªãããã£ãŠãåšæã«ã€ããŠã¯ ã°ãå®æ° k 2 {\displaystyle k_{2}} ãæã€ã°ãã®åšæã®æ¹ãã°ãå®æ° k 1 {\displaystyle k_{1}} ãæã€ã°ãã®åšæ ãã倧ãããªãã
éåã®ããäžã«é·ãl[m]ã®ã²ãã§ã€ããããç©äœã«ãã£ãŠäœãããç©äœã® éçŽäžåãã«åçŽãªæ¹åã®éåãåæ¯åãšãªãããšãæ±ããã ãã ããæ¯ãåã®åãç¯å²ã¯å°ãããã®ãšããã ãã®ããã«åæ¯åãããæ¯ãåã åæ¯ãå(ãããµãããsimple pendlum) ãšåŒã¶ããšãããã
ã²ã ãåºå®ãããŠããäœçœ®ããéçŽã«äžãããçŽç·ãšãç©äœãã€ãªãããŠãã ã²ã ããªãè§åºŠã Ξ {\displaystyle \theta } ãšããããã®å Žåãå³åœ¢çã«èãããšãã®å Žåã®æ°Žå¹³æ¹åã®éåæ¹çšåŒã¯
ãšãªããããã§ã Ξ {\displaystyle \theta } ãå°ããå Žåã
ãšãªãããšã«æ³šæãããšãéåæ¹çšåŒã¯
ãšãªãå
ã»ã©ã®ã°ãã«ã€ãªãããç©äœã®éåæ¹çšåŒãšçãããªãã
ãã£ãŠããã®ç©äœã®éåãåæ¯åã§èšè¿°ãããããšãåãã£ããããã«ã å
ã»ã©ã®è§æ¯åæ°ãšæ¯èŒãããšããã®å Žåã®è§æ¯åæ° Ï {\displaystyle \omega } ã¯
ãšãªãããšãåããã
ãããã®çµæããå°åŠæ ¡çç§ã®çµæã§ãã
ã®å®éšäºå®ãéåæ¹çšåŒã®çµæãšäžèŽããããšã確ãããããã
ãã®ç« ã§ã¯ãäžæåŒåã«ããéåãæ±ããäžæåŒåã¯å
šãŠã®ç©äœã®éã«ååšããŠãããããã®åãåªä»ããéåãšããŠæåãªãã®ã¯å€ªéœã®åããå転ããå°çã®éåããå°çèªèº«ã®åããå転ããæã®éåã§ãããå®éã«ã¯ãã®ãããªäœãã®åããå転ããæ§é ã¯å®å®å
šäœã«åºãèŠãããã
äŸãã°ã空ã«èŠãããæã¯w:ææãšåŒã°ãããããããã®æã®åãã«ã倪éœã«å¯Ÿããå°çãšåãããã«ãææãåããåã£ãŠãããšèããããå®éã«ãã®ãããªææã確èªãããææãããã(w:ç³»å€ææåç
§ã)
ãã®ããã«å®å®ã®äžã§äžæåŒåã«ããå転éåã¯åºã芳枬ããããããã§ã¯ãã®ãããªéåã¯ç©äœéã«åãã©ã®ãããªåã«ãã£ãŠèšè¿°ãããããèŠãŠããã
æŽå²çã«ã¯ãéã«ãã®ãããªç©äœã®éã®éåã説æãããããªåãèããããšã§ ç©äœéã«åãåãçºèŠããããæŽå²ã«ã€ããŠè©³ããã¯w:ãã¥ãŒãã³ãªã©ãåç
§ã
ãŸãã¯ãç©äœéã«åãäžæåŒå(glavitational constant)ã®æ³åãè¿°ã¹ããçš®ã
ã®èŠ³æž¬ã®çµæã«ãããšã質é m 1 {\displaystyle m_{1}} ãæã€ç©äœãšè³ªé m 2 {\displaystyle m_{2}} ãæã€ç©äœã®éã«ã¯
ã§è¡šããããåãåããããã§Gã¯ç©äœã«ãããªãå®æ°ã§ãäžæåŒåå®æ°ãšããã å€ã¯ G = 6.67 à 10 â 11 [ N â
m 2 / k g 2 ] {\displaystyle G=6.67\times 10^{-11}[{\mathrm {N} \cdot \mathrm {m} ^{2}/\mathrm {kg} ^{2}}]} ã§ããã
äžæåŒåã®æ³å
äžæåŒåã¯ç©äœéã®è·é¢ã®2ä¹ã«éæ¯äŸããåã§ããã
ç©äœã®å°ãªããšãçæ¹ãææã®ããã«å·šå€§ãªå Žåãç©äœéã®è·é¢rã¯ãéå¿éã®è·é¢ã§ããã
å°çã®äžæåŒåãèãããå°çã®è³ªéãMãå°çã®ååŸãRã枬å®ããç©äœã®è³ªéãmãšããå ŽåãéåFã¯
ãšãªãã
ãããå°è¡šè¿ãã§ã¯å€§ããã mg ãšçããã®ã§ã
å€åœ¢ããŠ
ãšãªããèšç®åé¡ã®ããããã®å€åœ¢ãçšããããå Žåãããã
å°çã¯èªè»¢ãããŠãããéåã®èšç®ã§ã¯ãå³å¯ã«ã¯èªè»¢ã«ããé å¿åãèããå¿
èŠãããããããããèªè»¢ã®é å¿åã®å€§ããã¯ãäžæåŒåã® 1 300 {\displaystyle {\frac {1}{300}}} åãŠãã©ãããªãã®ã§ãéåžžã¯èªè»¢ã«ããé å¿åãç¡èŠããå Žåãå€ãã
ãªããå°çã®èªè»¢ã®é å¿åã¯ãèµ€éäžã§ãã£ãšã倧ãããªãã
人工è¡æããå°çã®èªè»¢ãšåãåšæã§ãèªè»¢ãšåãåãã«çéåéåãããã°ããã®äººå·¥è¡æã¯å°äžããèŠãŠãã€ãã«å°é¢ã®äžç©ºã«ããã®ã§ãå°äžã®èŠ³æž¬è
ããã¯éæ¢ããŠèŠããããã®ãããªäººå·¥è¡æã®ããšãéæ¢è¡æãšããã
質émã®ç©äœã質éMã®å€§ããªç©äœã®åãããäžæåŒåã®åãåå¿åãšããŠãååŸrã®åéåãããŠããããã®å Žåã®åéåã®è§é床ãæ±ããã
ååŸrãè§é床 Ï {\displaystyle \omega } ã®åéåãããå Žåã®ç©äœã®åå¿å ã¯
ã§ãããäžæ¹ã質émãšè³ªéMã®ç©äœã®éã®è·é¢ãrã§ããå Žåã2ã€ã®ç©äœéã«åãéåã¯ãéåã®å€æ°ãfãšãããšã
ã§äžããããããã£ãŠããããã®åãçãããªãå Žåã«ã質émã®ç©äœã¯è³ªéMã®ç©äœã®ãŸãããåéåã§å転(å
¬è»¢)ããããšãã§ããããã£ãŠã Ï {\displaystyle \omega } ãæ±ããåŒã¯ã
ãšãªãã
å°çè¡šé¢ã§ã®éåã«ããäœçœ®ãšãã«ã®ãŒãèããããã®ãšåæ§ã«ãäžæåŒåã«ããäœçœ®ãšãã«ã®ãŒãèããããšãã§ããã
äžæåŒåã«ããäœçœ®ãšãã«ã®ãŒãæ±ããã«ã¯ãäžæåŒåãç©åããã°ããã
質éMã®ç©äœããrã®è·é¢ã«è³ªémã®ç©äœãååšãããšããããã ããMã¯mããã¯ãã㫠倧ãããšãããç¡éé ç¹ãåºæºã«ãããš(ã€ãŸãç¡éé ã§ã¯äœçœ®ãšãã«ã®ãŒããŒã)ããã®å Žåã質émã®ç©äœã®äœçœ®ãšãã«ã®ãŒã¯
ã§äžããããã
笊å·ã«ãã€ãã¹ãã€ãããšã®ç©ççãªè§£éã¯ãéåãã€ããã ãç©äœã«è¿ã¥ãã»ã©ããã®ç©äœã®ã€ããã ãéååãè±åºããã«ã¯ããšãã«ã®ãŒãè¿œå çã«å¿
èŠã«ãªãããã§ãããšè§£éã§ããã
ç¡éé ã§ã¯ r=+â ãšããã°ãããçµæã U=0 ã«ãªãã
ãªããäžæåŒåã¯ä¿ååã§ããã®ã§ãäœçœ®ãšãã«ã®ãŒã¯ãç¡éé ç¹ããã®çµè·¯ã«ããããçŸåšã®äœçœ®ã ãã§æ±ºãŸãã
ã®ããã«äžããããããŸãããã®ã°ã©ãã¯çŽèŠ³çãªæå³ãæã£ãŠããã å®ã¯ããã®ã°ã©ãã®åŸãã¯ã°ã©ããè¡šããäœçœ®ãšãã«ã®ãŒãæã€ç¹ã«ç©äœã眮ããå Žåã ãã®ç©äœãåãåããæ¹åãšãã®å€§ãããè¡šãããŠãããããã§ã¯ã äœçœ®ãšãã«ã®ãŒã®åŸããåžžã«r=0ã«èœã¡èŸŒãæ¹åã«çããŠããããç©äœMããè·é¢r (rã¯ä»»æã®å®æ°ã)ã®ç¹ã«éæ¢ããŠããç©äœã¯å¿
ãMã®æ¹åã«åžã蟌ãŸããŠè¡ãããšã è¡šãããŠããã(詳ããã¯å€å
žååŠåç
§ã)
ããææäžã«ããç©äœãå®å®ã®ç¡éé ãŸã§å°éãããããã«å®å®è¹ã«ææäžã§ äžããªããŠã¯ãããªãé床ã¯ã©ã®ããã«è¡šãããããããã ããèšç®ã«ã€ããŠã¯ æåã«å®å®è¹ãåºçºããææ以å€ã®å€©äœããã®åœ±é¿ã¯ç¡èŠãããšããã ãŸããææã®ååŸã¯Rã ææã®è³ªéã¯Mãšããã
ææã®åŒåã«ããäœçœ®ãšãã«ã®ãŒã¯ææè¡šé¢ã§
ã§ãããç¡éåç¹ã§ã¯0ã§ããããã ããmã¯å®å®è¹ã®è³ªéãšããã äžæ¹ãå®å®è¹ãç¡éåç¹ã«éããã«ã¯ãå®å®è¹ã®é床ãç¡éåç¹ã§ã¡ããã©0ã« çãããªãã°ãããããã§ãææäžã§ã®å®å®è¹ã®é床ãvãšãããšã ãšãã«ã®ãŒä¿ååããã
ãšãªãããã£ãŠãã®åŒããvãæ±ããã°ãããçã¯ã
äžèšã®èšç®ããåããããã«ãäžè¬ã«ãäžæåŒåã ããåããŠéåããç©äœã®ååŠçãšãã«ã®ãŒã¯ã
ã§ããã
ä»®ã«é«ãå±±ããç©äœãæ°Žå¹³ã«çºå°ãããšã(空æ°æµæã¯ç¡èŠãã)ãå°çã®ãŸãããåãç¶ããããã«å¿
èŠãªæå°ã®åé床ã®ããšã第äžå®å®é床ãšããã(â» ååã¯æèšããªããŠãããèŠããã¹ãã¯ãèšç®æ¹æ³ã§ããã) 第äžå®å®é床ã¯ãèŠããã«ãé å¿åãšåå¿åãã€ãããããã«å¿
èŠãªåé床ã§ããã
第äžå®å®é床ã¯ãç§éã§ã¯çŽ7.91km/sã§ããã
v1ã«ã€ããŠè§§ãã
ãªããããã R = 6400 à 10 m ã§ããã g = 9.8 m/s ã§ããã
ããã«åé床ã倧ãããªããšãç©äœã¯æ¥åè»éã«ãªãã
åé床ãçŽ11.2km/sã«ãªããšãè»éã¯æŸç©ç·ã«ãªããç©äœã¯ç¡éã®åœŒæ¹ã«é£ãã§ããã ãã®çŽ11.2km/sã®ããšã第äºå®å®é床ãšãããããã¯ãç¡éé ã®ç¹ã§ãé床ã0ãè¶
ããå€ã«ãªãããã«å¿
èŠãªåé床ã§ããã
ãªã®ã§ãèšç®ã§ç¬¬äºå®å®é床ãæ±ããã«ã¯ãšãã«ã®ãŒä¿ååãèšç®ã«ã¯äœ¿ãã
ã®åŒããvãæ±ãã
ã«ããã« G M = g R 2 {\displaystyle GM=gR^{2}} ã代å
¥ããŠã
ããã«é¢ä¿ããå®æ°ã代å
¥ããã°ããã
ãªããããã R = 6400 à 10 m ã§ããã g = 9.8 m/s ã§ããã
åé床 11.2km/s以äžã§ã¯ãè»éã¯åæ²ç·ã«ãªããç©äœã¯ç¡éã®åœŒæ¹ã«é£ãã§ããã
â» æ€å®æç§æžã§ã¯ãè泚ãªã©ã«æžããŠãã£ããããã å°çããå°åºããŠã倪éœç³»ã®å€ã«åºãããã«å¿
èŠãªæå°ã®åé床ã®ããšã第äžå®å®é床(çŽ 16.7 km/s) ã§ããã
ã®ãªã·ã£æ代ããäžäžãŸã§ä¿¡ããããŠãã倩å説(è±: geocentric theory)ã«å¯Ÿã,16äžçŽåã°ã«ã³ãã«ãã¯ã¹ã¯å
šãŠã®ææ(è±: planet)ã倪éœãäžå¿ãšããåéåãããŠããå°å説ãæå±ããããã®åŸãã£ã³ã»ãã©ãŒãšã¯é·å¹Žã«ãããææã®èŠ³æž¬ãè¡ã,ãã®èŠ³æž¬çµæãåŒç¶ãã ã±ãã©ãŒã¯ãããã®çµæãããšã«èšç®ãè¡ã,ææã®éè¡ã«é¢ããæ³å,ã±ãã©ãŒã®æ³å(è±: Kepler's laws)ãçºèŠããããªã,æç§æžã¯å€ªéœãšææã®é¢ä¿ã§è«ããŠããã,ä»ã«ãææãšè¡æ(èªç¶è¡æ,人工è¡æ)ã§ãæãç«ã€ã
ææ(è¡æ)ã¯å€ªéœ(ææ)ã1ã€ã®çŠç¹ãšããæ¥åéåããã(æ¥åè»éã®æ³å)ã
ææ(è¡æ)ãšå€ªéœ(ææ)ãçµã¶ååŸãåäœæéã«æãé¢ç©(é¢ç©é床)ã¯äžå®ã§ãã(é¢ç©é床äžå®)ã
ææ(è¡æ)ã®å
¬è»¢åšæ T {\displaystyle T} ã®2ä¹ã¯æ¥åè»éã®é·ååŸ(åé·è»ž) a {\displaystyle a} ã®3ä¹ã«æ¯äŸããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "é«çåŠæ ¡çç§ ç©çåºç€ã§ã¯ãç©äœã®éåãçŽç·äžã®éåãäžå¿ã«æ±ã£ããç©çã§ã¯ãããè€éãªå¹³é¢äžã®éåãæ±ããå¹³é¢äžã®éåã§ã¯ãçŽç·äžã®éåãšã¯éã£ãŠãç©äœã®äœçœ®ãè¡šããã®ã«å¿
èŠãªéã2ã€ã«ãªãããããã¯éåžž x , y {\\displaystyle x,\\ y} ãšãããã©ã¡ããæå» t {\\displaystyle t} ã®äžæã®é¢æ°ãšãªãã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ãããã®é¢æ°ã¯ã©ããªãã®ã§ãããããããã§ã¯äž»ã«ãå®éã®ç©äœã®éåãšããŠããããããããã®ãæ±ãã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "å¹³é¢äž,ããªãã¡2次å
ã«ãããŠ,æå» t {\\displaystyle t} ã«ãããäœçœ®ã¯ r â ( t ) = ( x ( t ) , y ( t ) ) {\\displaystyle {\\overrightarrow {r}}(t)=(x(t),\\ y(t))} ,埮å°æé Î t {\\displaystyle {\\mathit {\\Delta }}t} éã®å€äœã¯ Î r â = r â ( t + Î t ) â r â ( t ) = ( Î x , Î y ) {\\displaystyle {\\mathit {\\Delta }}{\\overrightarrow {r}}={\\overrightarrow {r}}(t+{\\mathit {\\Delta }}t)-{\\overrightarrow {r}}(t)=({\\mathit {\\Delta }}x,\\ {\\mathit {\\Delta }}y)} ãšå®çŸ©ãããããã®ãšã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ã Î t {\\displaystyle {\\mathit {\\Delta }}t} éã®å¹³åé床, Î t â 0 {\\displaystyle {\\mathit {\\Delta }}t\\to 0} ã®æ¥µé",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ãæå» t {\\displaystyle t} ã§ã®(ç¬é)é床ãšããããªã,æå» t {\\displaystyle t} ã§ã®éã(é床ã®å€§ãã)ã¯",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ãã®å Žåã,é床ããäœçœ®ãæ±ãŸã,åæåæ¯ã«",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ãæãç«ã¡,ãããããã¯ãã«ãçšããŠã²ãšãŸãšãã«ããŠä»»æã®æå» t {\\displaystyle t} ã«ãããäœçœ®",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ãæ±ããããã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "ãŸã,",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "ã Î t {\\displaystyle {\\mathit {\\Delta }}t} éã®å¹³åå é床, Î t â 0 {\\displaystyle {\\mathit {\\Delta }}t\\to 0} ã®æ¥µé",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "ãæå» t {\\displaystyle t} ã§ã®(ç¬é)å é床ãšããã ãã®å Žåã,å é床ããé床ãæ±ãŸã,åæåæ¯ã«",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "ãæãç«ã¡,ãããããã¯ãã«ãçšããŠã²ãšãŸãšãã«ããŠä»»æã®æå» t {\\displaystyle t} ã«ãããé床",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "ãæ±ããããããªã,ããã r â ( 0 ) , v â ( 0 ) {\\displaystyle {\\overrightarrow {r}}(0),{\\overrightarrow {v}}(0)} ã®å€ãåæå€ãšããã ç¹ã«,å é床äžå®ã®ãšãã®éåã¯çå é床éåãšããã,äžèšã®å
¬åŒ(1.2, 1)ã¯ãããã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ãšãªãã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "éåæ¹çšåŒã¯ãåãç©äœãåããå é床ã«æ¯äŸãããšããç¹ã¯ããããªãã ããããä»åã¯åãšå é床ã¯ã©ã¡ãããã¯ãã«éã§ããããã£ãŠãå€å f â = ( f x , f y ) {\\displaystyle {\\overrightarrow {f}}=(f_{x},\\ f_{y})} ãåã,å é床 a â = ( a x , a y ) {\\displaystyle {\\overrightarrow {a}}=(a_{x},\\ a_{y})} ã§éåããç©äœã®éåæ¹çšåŒã¯",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "ãšããããã éåžžã¯ããã®æ¹çšåŒã解ãå Žåã¯èŠçŽ ããšã«ããã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "ãšããããã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "æå»t = 0ã«ã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "ã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "ã§ééããç©äœã®æå»tã§ã®äœçœ®ãæ±ããã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "ç©äœã®xæ¹åãšyæ¹åã¯äºãã«ç¬ç«ã«çéçŽç·éåãããã ããã§ã¯xæ¹åãyæ¹åãé床",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "ãªã®ã§ãçéçŽç·éåã®åŒã®ãã¯ãã«éãšããé",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "ã«ä»£å
¥ãããšã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "ãšãªãã èŠçŽ ããšã«ãããšã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "ãšãªãã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "æå»t=0ã«åç¹(0,\\ 0)ãyæ¹åã«é床 v 0 {\\displaystyle v_{0}} ã§çéçŽç·éåããŠãã質émã®ç©äœã«ã xæ¹åã®äžæ§ãªåfããããå§ããããã®å Žåãæå»tã«ãããç©äœã®äœçœ®ãš é床ãæ±ããã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "x軞æ¹åã«ã¯çå é床éåãšãªãã ç©äœãåããå é床ã¯ãéåæ¹çšåŒã«ãã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "ãšãªãã ããã«xæ¹åã®åé床0,åæäœçœ®0ã§ããããšãçå é床çŽç·éåã®åŒã« 代å
¥ãããšã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "ãšãªãã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "ããã«ãy軞æ¹åã®éåã¯çééåã§ããããã®åé床ã¯ã v 0 {\\displaystyle v_{0}} ,åæäœçœ®ã¯0ã§ããã®ã§ã ãã®å€ãçééåã®åŒã«ä»£å
¥ãããšã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "ãåŸãããã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "ãã®ç« ã§ã¯éåé(ããã©ãããããmomentum)ãæ±ããéåéã¯ãç©äœã®è¡çªã«çœ®ããŠãšãã«ã®ãŒãšäžŠã³ãä¿åéãšãªãéèŠãªéã§ããããŸãããã®ç« ã§ã¯åç©(ãããããimpulse)ãšããéãå°å
¥ãããåç©ã¯éåéã®æéå€åãè¡šããéã§ããããã®å°åºã¯éåæ¹çšåŒãçšããŠæãããã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "ç©äœãåããŠããå Žåãç©äœã®é床ãšè³ªéã®ç©ãç©äœã®éåé",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "ãšå®çŸ©ãããéåæ¹çšåŒ",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "ã®äž¡èŸºãæå» t = t 1 {\\displaystyle t=t_{1}} ãã t = t 2 {\\displaystyle t=t_{2}} ãŸã§ç©åãããš",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "ãšãªãã v â ( t 1 ) = v 1 â , v â ( t 2 ) = v 2 â {\\displaystyle {\\overrightarrow {v}}(t_{1})={\\vec {v_{1}}},{\\overrightarrow {v}}(t_{2})={\\vec {v_{2}}}} ãšãããš",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "ãã®åŒã®å·ŠèŸºã¯éåéå€å,å³èŸºã¯åç©(ãããããimpulse)ã§ããããã£ãŠ,éåéå€åã¯åç©ã«çããããšãåãããéåéå€åã Î p â {\\displaystyle {\\mathit {\\Delta }}{\\overrightarrow {p}}} ,åç©ã I â {\\displaystyle {\\overrightarrow {I}}} ãšãããš",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "ç¹ã«, f â = {\\displaystyle {\\overrightarrow {f}}=} äžå®ã®ãšã, t 2 â t 1 = Î t {\\displaystyle t_{2}-t_{1}={\\mathit {\\Delta }}t} ãšãããš",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "埮åãçšããå°åºã«ã€ããŠã¯ãå€å
žååŠãåç
§ã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "éæ¢ããŠããç©äœã«æé Î t {\\displaystyle {\\mathit {\\Delta }}t} ã®éããæ¹åã«äžæ§ãªåfãããããç©äœãåŸã éåéã¯ã©ãã ãããããã«ãç©äœã®è³ªéãmãšãããšãç©äœããã®æ¹åã« åŸãé床ã¯ã©ãã ããã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "éåéã®å€ååã¯ç©äœãåããåç©ã«çããã®ã§ãç©äœãåããåç©ãèšç®ããã° ãããç©äœãåããåç©ã¯",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 41,
"tag": "p",
"text": "ã«çããã®ã§ãç©äœãåŸãéåéã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 42,
"tag": "p",
"text": "ã«çãããããã«ãéåéã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 43,
"tag": "p",
"text": "ãæºããããšãèãããšãç©äœã®é床ã¯",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 44,
"tag": "p",
"text": "ãšãªãã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 45,
"tag": "p",
"text": "éåéã¯ãç©äœãå
šãåãåããªãå Žåã«ã¯ä¿åããããããã¯ç©äœã«åãåããªãå Žåã«ã¯ãç©äœã®åããåç©ã¯0ã§ããç©äœã®éåéå€åã0ã§ããããšããåœç¶ã§ããã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 46,
"tag": "p",
"text": "ããã«ãè€æ°ã®ç©äœã®éåéã«ã€ããŠã¯ãå¥ã®éèŠãªæ§è³ªãèŠããããããã¯ãè€æ°ã®ç©äœã®ãã€éåéã®ç·åã¯ãããã®ç©äœã®éã®è¡çªã«éããŠä¿åãããšããããšã§ãããããã¯ã€ãŸããäŸãã°ãã2ã€ã®ç©äœãè¡çªããå Žåãå§ãã«2ç©äœãããããæã£ãŠããéåéã®åã¯è¡çªãçµãã£ãåŸã«2ç©äœãæã£ãŠããéåéã®åã«çãããšããããšã§ãããããã§ãããã€ãã®ç©äœãããå Žåãããã®æã€éåéã®ç·åãã察å¿ããç©äœç³»ã®å
šéåéãšããã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 47,
"tag": "p",
"text": "ç©äœã®è¡çªã«ã€ããŠãéåéã¯åžžã«ä¿åãããããããç©äœç³»ã®å
šãšãã«ã®ãŒã¯åžžã«ä¿åãããšã¯éããªããäžè¬ã«ç©äœã®è¡çªã«ã€ããŠãšãã«ã®ãŒã¯åžžã«å€±ãããŠããããã£ãšãç©äœç³»ã«éããªãå
šãšãã«ã®ãŒã¯åžžã«äžå®ã§ããã®ã§ãç©äœãæã£ãŠãããšãã«ã®ãŒã¯é³ãç±ã®åœ¢ã§ç©äœç³»ã®å€ã«éããŠè¡ãã®ã§ãããç©äœãè¡çªã«ã€ããŠå€±ããšãã«ã®ãŒã¯è¡çªã«é¢ããç©äœãæã£ãŠããç©æ§å®æ°ã«ãã£ãŠæ±ºãŸãããã®ä¿æ°ãåçºä¿æ°(ã¯ãã±ã€ãããããcoefficient of restitution)ãšåŒã³ãeãªã©ã®èšå·ã§æžããåçºä¿æ°ã¯ãç©äœãè¡çªããããååŸã§ã®ç©äœéã®çžå¯Ÿé床ã®æ¯ã«ãã£ãŠå®ããããã ç¹ã«ç©äœ1ãšç©äœ2ãè¡çªåã«é床 v 1 , v 2 {\\displaystyle v_{1},\\ v_{2}} ãæã£ãŠãããè¡çªåŸã«é床 v 1 â² , v 2 â² {\\displaystyle v_{1}',\\ v_{2}'} ãæã£ããšãããšãåçºä¿æ°eã¯",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 48,
"tag": "p",
"text": "ã§å®ãããããããã§ãå³èŸºã®å§ãã® â {\\displaystyle -} 笊åã¯ãè¡çªã®ååŸã§ç©äœã®é床ããã倧ããç©äœã¯ãè¡çªåã«ããå°ããé床ãæã£ãŠããç©äœãããè¡çªåŸã«ã¯ããå°ããé床ãæã€ããšã«ãªãããã§ããã ãã®ãããåçºä¿æ°ã¯äžè¬ã«æ£ã®æ°ã§ããã ãŸãåçºä¿æ°ã¯1ããå°ããæ°ã§ãããç©äœéã®çžå¯Ÿé床ã¯è¡çªåããè¡çªåŸã®æ¹ãå°ãããªãã ç¹ã« e = 1 {\\displaystyle e=1} ã®å Žåã(å®å
š)匟æ§è¡çª(elastic collision)ãšåŒã³ããã£ãœã 0 < e < 1 {\\displaystyle 0<e<1} ã®å Žåãé匟æ§è¡çª(inelastic collision)ãšåŒã¶ã匟æ§è¡çªã®å Žåã¯ãååŠçãšãã«ã®ãŒã¯ä¿åããããšãç¥ãããŠãããäžæ¹ãé匟æ§è¡çªã® å Žåã¯ç©äœç³»ã®å
šãšãã«ã®ãŒã¯å€±ãããã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 49,
"tag": "p",
"text": "ããéæ¢ããŠããç©äœ2ã«éåépã§éåããŠããç©äœãè¡çªããããã®å Žåã è¡çªããåŸã®ç©äœ2ãéåé p 2 {\\displaystyle p_{2}} ãåŸããšãããšãè¡çªåŸã®ç©äœ1ã®éåé㯠ã©ãã ããšãªã£ããã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 50,
"tag": "p",
"text": "éåéä¿ååãèãããšãè¡çªã®ååŸã§ç©äœ1ãšç©äœ2ã§æ§æãããç©äœç³»ã®å
šéåéã¯ä¿åããã ããã§ãè¡çªåã®ç©äœç³»ã®å
šéåéã¯pã§ããã®ã§ãè¡çªåŸã®ç©äœç³»ã®å
šéåéãpãšãªãã ããã«ãç©äœ2ã®è¡çªåŸã®éåéã p 2 {\\displaystyle p_{2}} ãªã®ã§ãç©äœ1ã®éåéã¯",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 51,
"tag": "p",
"text": "ãšãªãã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 52,
"tag": "p",
"text": "ããã§ãç©äœç³»ã®å
šéåéãä¿åãããããšã¯ãéåã«é¢ãã äœçšã»åäœçšã®æ³å ããåŸãã äœçšåäœçšã®æ³åãçšãããšãç©äœç³»ã®éã®è¡çªã«éããŠãè¡çªã«é¢ããããããã®ç©äœãåããåã¯ã倧ãããçããåãã¯å察ãšãªãã ãã®å Žåãããããã®åã«å¯ŸããŠãè¡çªã®æé Î t {\\displaystyle \\Delta t} ãããããã®ã¯ è¡çªã«éããŠããããã®ç©äœãåãåãåç©ã«çããã ããã§ãè¡çªã«é¢ããŠåãåã®åç©ãå
šãŠã®ç©äœã«ã€ããŠè¶³ãåããããšããããã®åã¯ãäžã®ããšãã0ãšãªãã ããããå
šéåéã®èšç®ã§ã¯ãŸãã«ãã®ãããªå
šç©äœã«ã€ããŠã®éåéã®ç·åãèšç®ããŠããã®ã§ã è¡çªã«ãã£ãŠåŸããããããªåç©ã®ç·åã¯ã0ã«çããã ãã£ãŠãè¡çªã«éããŠç©äœç³»ã®æã€å
šéåéã¯ä¿åãããã ãããéåéä¿åå(ããã©ãããã ã»ãããããmomentum conservation law)ãšããã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 53,
"tag": "p",
"text": "質émã®2ã€ã®ç©äœãé床 v 1 {\\displaystyle v_{1}} , v 2 {\\displaystyle v_{2}} ã§ç§»åããŠããããããã®ç©äœãè¡çªããå Žåã è¡çªåŸã®ããããã®ç©äœã®é床ãããšãã«ã®ãŒä¿ååãšéåéä¿ååãçšã㊠èšç®ããããã ããç©äœã®è¡çªã«é¢ããŠãšãã«ã®ãŒã¯ä¿åãããšããã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 54,
"tag": "p",
"text": "",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 55,
"tag": "p",
"text": "ãã®åé¡ã¯2ã€ã®åã倧ããã®ç©äœãç°ãªã£ãé床ã§ã¶ã€ããå Žå ãã®çµæãã©ããªãããèšç®ããåé¡ã§ããã å®éšã®çµæã«ãããšãäžæ¹ãéæ¢ããŠããäžæ¹ãåããŠããå Žåã åããŠããç©äœã¯éæ¢ããéæ¢ããŠããç©äœã¯åããŠããç©äœãæã£ãŠãã é床ãšåãé床ã§åãã ãããšãç¥ãããŠãããããã§ã¯ããããã® çµæãèšç®ã«ãã£ãŠç¢ºãããããããšãèŠãããšãåºæ¥ãã è¡çªåŸã®ç©äœã®é床ãããããç©äœ1ã«ã€ããŠã¯ v 1 â² {\\displaystyle v_{1}'} ,ç©äœ2ã«ã€ããŠã¯ v 2 â² {\\displaystyle v_{2}'} ãšããããã®å Žåãç©äœã®è¡çªã«ã€ããŠå
šãšãã«ã®ãŒãä¿åãããããšã çšãããšã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 56,
"tag": "p",
"text": "ãåŸããããããã«ãç©äœã®è¡çªã«ã€ããŠç©äœç³»ã®å
šéåéãä¿åãããããšãçšãããšã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 57,
"tag": "p",
"text": "ãããã¯ã v 1 â² {\\displaystyle v'_{1}} , v 2 â² {\\displaystyle v'_{2}} ã«ã€ããŠã®2次æ¹çšåŒã§ããã解ãããšãåºæ¥ããå®éèšç®ãããšã解ãšããŠ",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 58,
"tag": "p",
"text": "ãåŸããããåè
ã®è§£ã¯è¡çªã«éããŠç©äœã®é床ãå€åãã¬ããšã瀺ããŠããããããã¯å®éã®æ
åµãšããŠèãé£ãã®ã§ãåŸè
ã®è§£ãçŸå®ã®è§£ãšãªãããã®çµæãèŠããšãç©äœãæã€é床ãå
¥ãæ¿ããããšãåããã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 59,
"tag": "p",
"text": "ãã®ããšã¯å®éã«åã倧ããã®çãçšããŠå®éšãè¡ããšã確ãããããšãã§ããã",
"title": "å¹³é¢äžã®éå"
},
{
"paragraph_id": 60,
"tag": "p",
"text": "äœçœ®ã®ã¿ããã¡,倧ããããªãã®ã質ç¹ã§ãããåäœãšã¯,倧ãããããã圢ã倧ãããå€ããã¬ç©äœã®ããšã§ããã",
"title": "è§éåéãšåã®ã¢ãŒã¡ã³ã"
},
{
"paragraph_id": 61,
"tag": "p",
"text": "åäœã®éåãèããåã«äžå®å¹³é¢äžã®éåã«ã€ããŠæ¬¡ã®ãããªäžè¬çèå¯ãè¡ãã",
"title": "è§éåéãšåã®ã¢ãŒã¡ã³ã"
},
{
"paragraph_id": 62,
"tag": "p",
"text": "æå» t {\\displaystyle t} ã«ãã㊠x y {\\displaystyle xy} å¹³é¢å
ã®äœçœ® r â = ( x , y ) {\\displaystyle {\\overrightarrow {r}}=(x,\\ y)} ãé床 v â = ( v x , v y ) {\\displaystyle {\\overrightarrow {v}}=(v_{x},\\ v_{y})} ã§éåã,å F â = ( F x , F y ) {\\displaystyle {\\overrightarrow {F}}=(F_{x},\\ F_{y})} ãåããŠãã質é m {\\displaystyle m} ã®ç©äœã®éåæ¹çšåŒãæåã«åããŠè¡šãã°",
"title": "è§éåéãšåã®ã¢ãŒã¡ã³ã"
},
{
"paragraph_id": 63,
"tag": "p",
"text": "2 Ã x â {\\displaystyle \\times x-} 1 Ã y {\\displaystyle \\times y} ãã",
"title": "è§éåéãšåã®ã¢ãŒã¡ã³ã"
},
{
"paragraph_id": 64,
"tag": "p",
"text": "ãã®å·ŠèŸºã®",
"title": "è§éåéãšåã®ã¢ãŒã¡ã³ã"
},
{
"paragraph_id": 65,
"tag": "p",
"text": "ãåç¹OãŸããã®è§éåéãšããã ãã㧠v â {\\displaystyle {\\overrightarrow {v}}} ãš r â {\\displaystyle {\\overrightarrow {r}}} ã®ãªãè§ã Ξ , x {\\displaystyle \\theta ,\\ x} 軞㚠r â {\\displaystyle {\\overrightarrow {r}}} ã®ãªãè§ã Ï {\\displaystyle \\phi } ãšãããš",
"title": "è§éåéãšåã®ã¢ãŒã¡ã³ã"
},
{
"paragraph_id": 66,
"tag": "p",
"text": "ãããã(3.1)ã«ä»£å
¥ãããš",
"title": "è§éåéãšåã®ã¢ãŒã¡ã³ã"
},
{
"paragraph_id": 67,
"tag": "p",
"text": "ãåŸãããã",
"title": "è§éåéãšåã®ã¢ãŒã¡ã³ã"
},
{
"paragraph_id": 68,
"tag": "p",
"text": "ç©äœãå転ãããåã®å¹æã®å€§ãããè¡šãéãåã®ã¢ãŒã¡ã³ããšãããæŽã« F â {\\displaystyle {\\overrightarrow {F}}} ãš r â {\\displaystyle {\\overrightarrow {r}}} ã®ãªãè§ã Î {\\displaystyle {\\mathit {\\Theta }}} ãšãããš",
"title": "è§éåéãšåã®ã¢ãŒã¡ã³ã"
},
{
"paragraph_id": 69,
"tag": "p",
"text": "ãã£ãŠåç¹OãŸããã®åã®ã¢ãŒã¡ã³ãã N {\\displaystyle N} ã§è¡šããš",
"title": "è§éåéãšåã®ã¢ãŒã¡ã³ã"
},
{
"paragraph_id": 70,
"tag": "p",
"text": "ããã« r sin Î {\\displaystyle r\\sin {\\mathit {\\Theta }}} ã¯åç¹ããå F â {\\displaystyle {\\overrightarrow {F}}} ã®äœçšç·ã«äžããåç·ã®é·ãã§ãã,ãããå F â {\\displaystyle {\\overrightarrow {F}}} ã®åç¹ã«å¯Ÿããè
ã®é·ããšããããã ãåã®ã¢ãŒã¡ã³ãã¯å F â {\\displaystyle {\\overrightarrow {F}}} ãäœçœ®ãã¯ãã« r â {\\displaystyle {\\overrightarrow {r}}} ãåæèšåãã«åãåããæ£ãšããŠãã(æèšåãã®é㯠Π< 0 {\\displaystyle {\\mathit {\\Theta }}<0} 㧠r sin Î < 0 {\\displaystyle r\\sin {\\mathit {\\Theta }}<0} ãšèãã)ã 以äžãã,3(è§éåéã®æ¹çšåŒ)ã¯",
"title": "è§éåéãšåã®ã¢ãŒã¡ã³ã"
},
{
"paragraph_id": 71,
"tag": "p",
"text": "ããã¯åã®ã¢ãŒã¡ã³ããå ããããçµæãšããŠè§éåéãå€åãããšããå æé¢ä¿ãè¡šããç¹ã« N = 0 {\\displaystyle N=0} ãªãã°",
"title": "è§éåéãšåã®ã¢ãŒã¡ã³ã"
},
{
"paragraph_id": 72,
"tag": "p",
"text": "ãšãªã,è§éåéãä¿åããã",
"title": "è§éåéãšåã®ã¢ãŒã¡ã³ã"
},
{
"paragraph_id": 73,
"tag": "p",
"text": "ç©äœã®åéšåã«åãéåã®äœçšç¹ãéå¿(è±: centre of gravity)æãã¯è³ªéäžå¿(è±: centre of mass)ãšããã n {\\displaystyle n} ç©äœ(質é: m 1 , m 2 , ⯠⯠, m n {\\displaystyle m_{1},\\ m_{2},\\ \\cdots \\cdots ,\\ m_{n}} ,äœçœ® r 1 â , r 2 â , ⯠⯠, r n â {\\displaystyle {\\vec {r_{1}}},\\ {\\vec {r_{2}}},\\ \\cdots \\cdots ,\\ {\\vec {r_{n}}}} ( n {\\displaystyle n} ã¯èªç¶æ°)ã®éå¿ã®äœçœ® r G â {\\displaystyle {\\vec {r_{\\mathrm {G} }}}} ã¯ä»¥äžã®ããã«å®çŸ©ãããã",
"title": "è§éåéãšåã®ã¢ãŒã¡ã³ã"
},
{
"paragraph_id": 74,
"tag": "p",
"text": "ãŸãéå¿é床 v G â {\\displaystyle {\\vec {v_{\\mathrm {G} }}}} 㯠d r k â d t = v k â ( k = 1 , 2 , ⯠⯠, n ) {\\displaystyle {\\frac {d{\\vec {r_{k}}}}{dt}}={\\vec {v_{k}}}\\ (k=1,\\ 2,\\ \\cdots \\cdots ,\\ n)} ãšãããš",
"title": "è§éåéãšåã®ã¢ãŒã¡ã³ã"
},
{
"paragraph_id": 75,
"tag": "p",
"text": "ããã§ã¯ãåççãªå¹³é¢äžã®éåã®1ã€ãšããŠãåéå(è±: circular motion)ãšåæ¯å(è±: simple harmonic motion)ããã€ãããåéåã¯ãåæ¯ãå(ãããµãããsimple pendlum)ã®éåã®é¡äŒŒç©ãšããŠãéèŠã§ããããããšãšãã«ããã®ããŒãžã§ã¯äžæåŒåã«ããéåãæ±ãã äžæåŒåã¯ããããéåãšåãåã§ããã ç©äœãšç©äœã®éã«å¿
ãçããåã§ãããäžæ¹ãããã®åã¯éåžžã«åŒ±ãããã ææã®ããã«å€§ããªè³ªéãæã£ãç©äœã®éåã«ããé¢ãããªãã ããã§ã¯ã倪éœã®ãŸãããå転ããææã®ãããªå€§ããªã¹ã±ãŒã«ã®éåããã€ããããã®ãããªéåã¯åã«è¿ãè»éãšãªãããšãããããã®ãããææã®éåãç解ããäžã§ãåéåãç解ããããšãéèŠã§ããã",
"title": "åéå"
},
{
"paragraph_id": 76,
"tag": "p",
"text": "ç©äœãåãæãããã«éåããããšãåéåãšåŒã¶ãåãæããããªéåã¯ãäŸãã°ãå圢ã®ã°ã©ãŠã³ãã®ãŸãããèµ°ã人éã®ããã«äººéãææãæã£ãŠè¡ãªãå Žåãæãããèªç¶çŸè±¡ãšããŠèµ·ããå Žåãå€ããäŸãã°ã倪éœã®ãŸãããåãå°çã®éåããå°çã®åããåãæã®éåã¯ãããããåéåã§èšè¿°ãããããŸããäžå®ã®é·ãããã£ãã²ããšäžå®ã®è³ªéãæã£ãç©äœã§äœãããæ¯ãåã®éåã¯ãã²ããåºå®ããç¹ããäžå®ã®è·é¢ããããŠéåããŠãããããç©äœã¯åè»éäžãéåããŠãããåºãæå³ã§ã®åè»éãšãšãããããšãåºæ¥ããããã§ã¯ããã®ãããªå Žåã®ãã¡ã§ä»£è¡šçãªãã®ãšããŠãå®å
šãªåè»éäžãéåããç©äœã®éåããã€ããã",
"title": "åéå"
},
{
"paragraph_id": 77,
"tag": "p",
"text": "åè»éäžãéåããç©äœã®åº§æšãäžè¬ã®å Žåãšåæ§",
"title": "åéå"
},
{
"paragraph_id": 78,
"tag": "p",
"text": "ã§è¡šãããããç¹ã«åè»éãè¡šããé¢æ°ã¯é«çåŠæ ¡æ°åŠII ãããããªé¢æ°ã§æ±ã£ãäžè§é¢æ°ã«å¯Ÿå¿ããŠããã",
"title": "åéå"
},
{
"paragraph_id": 79,
"tag": "p",
"text": "ããã§ãåéåãäžè§é¢æ°ãçšããŠè¡šãããããšãè¿°ã¹ããããã®ããšã¯é«çåŠæ ¡æ°åŠCã®åªä»å€æ°è¡šç€ºãçšããŠãããåªä»å€æ°è¡šç€ºã«ã€ããŠè©³ããã¯ã察å¿ããé
ãåç
§ããŠã»ããã",
"title": "åéå"
},
{
"paragraph_id": 80,
"tag": "p",
"text": "ååŸr[m]ã®åäžãçããé床ã§ãåéåããç©äœã®éåãèšè¿°ããããšãèããã ããã«ã座æšãåãå Žååç¹ã®äœçœ®ã¯åéåã®äžå¿ã®äœçœ®ãšããã ãã®å Žåã®ç©äœã®éåã¯ãx, y座æšãçšããŠã",
"title": "åéå"
},
{
"paragraph_id": 81,
"tag": "p",
"text": "ã«ãã£ãŠæžãããããã ãããã®å Žå Ï {\\displaystyle \\omega } ã¯è§é床ãšåŒã°ãåäœã¯[rad/s]ã§äžããããããã ããããã§[rad]ã¯w:ã©ãžã¢ã³ã§ãããw:匧床æ³ã«ãã£ãŠè§åºŠãè¡šãããå Žåã®åäœã§ããã匧床æ³ã«ã€ããŠã¯é«çåŠæ ¡æ°åŠII ãããããªé¢æ°ãåç
§ãè§é床ã¯åéåãããŠããç©äœãã©ã®çšåºŠã®æéã§åãäžåšãããã«å¯Ÿå¿ããŠããããªã,é«çåŠæ ¡ã®ç©çã«ãããŠè§é床ã¯ã¹ã«ã©ãŒãšããŠæ±ãããŸãããã®éã¯äžã§åããã®ã ããåéåããŠããç©äœã®é床ã«æ¯äŸããã",
"title": "åéå"
},
{
"paragraph_id": 82,
"tag": "p",
"text": "ãŸããè§é床ã«å¯Ÿå¿ããŠã",
"title": "åéå"
},
{
"paragraph_id": 83,
"tag": "p",
"text": "ã§äžããããéãw:åšæãšãããåšæã®åäœã¯[s]ã§ãããåšæã¯ç©äœãäœç§éããšã« åç¶ã1åšããããè¡šããéã§ããããã®å Žåã«ã¯ç©äœã¯T[s]ããšã«åç¶ã1åšãããããã«ã",
"title": "åéå"
},
{
"paragraph_id": 84,
"tag": "p",
"text": "ãw:æ¯åæ°ãšåŒã¶ãæ¯åæ°ã¯åšæãšã¯éã«ãåäœæéåœããã«ç©äœãåç¶ãäœåšãããã æ°ããéã§ãããæ¯åæ°ã®åäœã«ã¯éåžž[Hz]ãçšãããããã¯ã[1/s]ã«çããåäœã§ããã ãŸããåšæTãšãæ¯åæ°fã¯ãé¢ä¿åŒ",
"title": "åéå"
},
{
"paragraph_id": 85,
"tag": "p",
"text": "ãæºããããã®åŒã¯ããåéåãããŠããç©äœã«ã€ããŠããã®ç©äœã®åéåã® åšæã«å¯Ÿå¿ããæéã®éã«ã¯ãç©äœã¯åç¶ã1åšã ããããšããããšã«å¯Ÿå¿ããã",
"title": "åéå"
},
{
"paragraph_id": 86,
"tag": "p",
"text": "ãŸãã",
"title": "åéå"
},
{
"paragraph_id": 87,
"tag": "p",
"text": "ã®åŒã§ ÎŽ {\\displaystyle \\delta } ã¯ç©äœã®äœçœ®ã®w:äœçžãšåŒã°ããç©äœãåç¶ã®ã©ã®ç¹ã«ãããã瀺ã å€ã§ããã",
"title": "åéå"
},
{
"paragraph_id": 88,
"tag": "p",
"text": "ãŸãããã®å Žåã®ç©äœã®é床ã®x, yèŠçŽ ã¯",
"title": "åéå"
},
{
"paragraph_id": 89,
"tag": "p",
"text": "ã§äžããããããã®åŒãšãåŸã®åéåã®å é床ã®å°åºã«ã€ããŠã¯ãåŸã®çºå±ãåç
§ãããã§ãç©äœã®éããvãšãããšã",
"title": "åéå"
},
{
"paragraph_id": 90,
"tag": "p",
"text": "ãšãªããç©äœã®é床㯠r Ï {\\displaystyle r\\omega } ã§äžããããããšãåããã",
"title": "åéå"
},
{
"paragraph_id": 91,
"tag": "p",
"text": "ããã«ã",
"title": "åéå"
},
{
"paragraph_id": 92,
"tag": "p",
"text": "ãèšç®ãããšã",
"title": "åéå"
},
{
"paragraph_id": 93,
"tag": "p",
"text": "ãšãªããåéåãããŠããç©äœã®é床ãšåéåã®äžå¿ãåç¹ãšããå Žåã®åº§æšã¯çŽäº€ããŠããããšãåãããããã«ãåéåãããŠããç©äœã®å é床ã¯ã",
"title": "åéå"
},
{
"paragraph_id": 94,
"tag": "p",
"text": "ãšãªããããã¯",
"title": "åéå"
},
{
"paragraph_id": 95,
"tag": "p",
"text": "ã«å¯Ÿå¿ããŠãããåéåããããªãç©äœã®å é床ã¯ãåéåãããç©äœã®åº§æšãš ã¡ããã©å察åãã«ãªãããšãåããã",
"title": "åéå"
},
{
"paragraph_id": 96,
"tag": "p",
"text": "ããã§ã¯ãåéåã®é床ãšå é床ãäžãããããã®å€ã¯ç©äœã®éåã決ãŸãã°æ±ºãŸãå€ãªã®ã§ãåéåã®åŒããèšç®ã§ããããã ãå®éã«ãããã®åŒãåŸãããã«ã¯ãåéåã®åŒã®åŸ®åãè¡ãå¿
èŠããããããããã§ã¯è©³ããæ±ããªããå°åºã«ã€ããŠã¯ãå€å
žååŠãåç
§ã",
"title": "åéå"
},
{
"paragraph_id": 97,
"tag": "p",
"text": "ååŸr[m]ã®åäžãè§é床 Ï {\\displaystyle \\omega } ã§éåããç©äœã®å é床ã®å€§ãããèšç®ããã",
"title": "åéå"
},
{
"paragraph_id": 98,
"tag": "p",
"text": "ã«æ³šç®ãããšãããå³èŸºã«ã€ããŠåéåãããŠããç©äœã®åº§æšãåžžã«",
"title": "åéå"
},
{
"paragraph_id": 99,
"tag": "p",
"text": "ãæºããããšã«æ³šç®ãããšã",
"title": "åéå"
},
{
"paragraph_id": 100,
"tag": "p",
"text": "ãšãªãã",
"title": "åéå"
},
{
"paragraph_id": 101,
"tag": "p",
"text": "50Hzã§åéåããŠããç©äœã®åéåã®åšæãèšç®ããã",
"title": "åéå"
},
{
"paragraph_id": 102,
"tag": "p",
"text": "ãçšãããšã",
"title": "åéå"
},
{
"paragraph_id": 103,
"tag": "p",
"text": "ãšãªãã",
"title": "åéå"
},
{
"paragraph_id": 104,
"tag": "p",
"text": "以äžãã,åéåã®å é床ã®æåã¯",
"title": "åéå"
},
{
"paragraph_id": 105,
"tag": "p",
"text": "ãã£ãŠ,åéåããç©äœã®è³ªéã m {\\displaystyle m} ,åå¿æ¹åã«åãå,ããªãã¡åå¿å(è±: centripetal force)ã F C {\\displaystyle F_{\\mathrm {C} }} ,æ¥ç·æ¹åã«åãåã F T {\\displaystyle F_{\\mathrm {T} }} ãšãããšéåæ¹çšåŒã¯",
"title": "åéå"
},
{
"paragraph_id": 106,
"tag": "p",
"text": "w:åå¿åãw:é å¿å(centrifugal force)",
"title": "åéå"
},
{
"paragraph_id": 107,
"tag": "p",
"text": "åéåãšé¢ä¿ã®æ·±ãç©äœã®éåãšããŠãåæ¯å(è±: simple harmonic oscillation)ãããããããåæ¯åã¯ããããæ¯åçŸè±¡ã®åºæ¬ã«ãªã£ãŠãããå¿çšç¯å²ãåºãéåã§ãããåéåãšåæ§ãåæ¯åãäžè§é¢æ°ãçšããŠéåãèšè¿°ãããããŸããåšæãäœçžãããç¹ãåéåãšåãã§ããããŸããåæ¯åã¯æ³¢åã«é¢ããçŸè±¡ãšãé¢ä¿ãæ·±ããäœçžãæ¯å¹
ãªã©ã®éãå
±æããŠããã",
"title": "åéå"
},
{
"paragraph_id": 108,
"tag": "p",
"text": "ããããã¯ãåæ¯åãããç©äœã®æ§è³ªããã詳ããèŠãŠè¡ãã",
"title": "åéå"
},
{
"paragraph_id": 109,
"tag": "p",
"text": "åæ¯åã¯æ§ã
ãªæ
åµã§ãããããããåçŽãªäŸãšããŠã¯ããã¯ã®æ³åã§æ¯é
ãããã°ãã«æ¥ç¶ãããç©äœã®éåããããããã§ã¯ãã°ãå®æ° k {\\displaystyle k} ã®ã°ãã«è³ªé m {\\displaystyle m} ã®ç©äœãæ¥ç¶ãããšãããã°ãã®èªç¶é·ã®äœçœ®ãåç¹ãšããŠæå» t {\\displaystyle t} ã«ãããåç¹ããã®ç©äœã®äœçœ®ã x ( t ) {\\displaystyle x(t)} ãšããå Žåããã®ç©äœã«é¢ããéåæ¹çšåŒã¯",
"title": "åéå"
},
{
"paragraph_id": 110,
"tag": "p",
"text": "ã§äžããããããã®æ¹çšåŒã®äž¡èŸºã m {\\displaystyle m} ã§å²ããšãå é床㯠d 2 x ( t ) d t 2 = â k m x ( t ) {\\displaystyle {\\frac {d^{2}x(t)}{dt^{2}}}=-{\\frac {k}{m}}x(t)} ã§äžããããããšãåããããã®ããã«ãå é床ãšç©äœã®åº§æšãè² ã®æ¯äŸä¿æ°ãæã£ãŠæ¯äŸé¢ä¿ã«ããåŒããåæ¯åã®éåæ¹çšåŒã§ããããã®å Žåãåæ¯åã®æ¯åäžå¿ã x = x C {\\displaystyle x=x_{\\mathrm {C} }} (åæ¯åã§ã¯æ¯åäžå¿ã¯å®æ°),æå» t {\\displaystyle t} ã«ãããç©äœã®éåãäœçœ® x ( t ) {\\displaystyle x(t)} ,é床 v ( t ) {\\displaystyle v(t)} ,å é床 a ( t ) {\\displaystyle a(t)} ã§è¡šããš",
"title": "åéå"
},
{
"paragraph_id": 111,
"tag": "p",
"text": "ãšãªãã Ï {\\displaystyle \\omega } ã¯è§æ¯åæ°, ÎŽ {\\displaystyle \\delta } ã¯åæäœçžã§ããã",
"title": "åéå"
},
{
"paragraph_id": 112,
"tag": "p",
"text": "ããã§ãåæ¯åã®éåæ¹çšåŒãšãåæ¯åã®éåã®åŒãäžããããå®éã«ã¯åæ¯åã®éåã®åŒã¯éåæ¹çšåŒããå°åºã§ãããããã«ã€ããŠã¯w:埮åæ¹çšåŒãæ±ãå¿
èŠãããã®ã§è©³ããå°åºã«ã€ããŠã¯ãå€å
žååŠãåç
§ã",
"title": "åéå"
},
{
"paragraph_id": 113,
"tag": "p",
"text": "sin {\\displaystyle \\sin } é¢æ°ã¯é¢æ°ã®å€ã®å¢å ã«äŒŽã£ãŠåšæçãªæ¯åãè¡ãªãé¢æ°ãªã®ã§ãç©äœã¯ã x = 0 {\\displaystyle x=0} ã®ãŸããã§åšæçãªæ¯åãããããšãåããããã ããäžã®åŒã®äžã§Aã¯w:æ¯å¹
ãšåŒã°ããç©äœã®æ¯åã®ç¯å²ãè¡šãéã§ããã",
"title": "åéå"
},
{
"paragraph_id": 114,
"tag": "p",
"text": "ãã ãããã®å Žåã«ãããŠã¯ãããã®éã¯ç©äœã®åéåã§ã¯ãªããç©äœã®æ¯åã«ã€ããŠã®éã§ãããããããåäœæéåœããã«äœ[rad]ã ãäœçžãé²ããã®éãšæ¯åã®åšæã®äžã§ãã©ã®äœçœ®ã«ç©äœãããããè¡šãéã«å¯Ÿå¿ããŠããããŸããåšæãšæ¯åæ°ãåéåã®å Žåãšåãå®çŸ©ã§äžããããã",
"title": "åéå"
},
{
"paragraph_id": 115,
"tag": "p",
"text": "ãŸãããã®å Žåã«ã€ããŠã¯éåæ¹çšåŒããè§æ¯åæ°ã決ãŸã",
"title": "åéå"
},
{
"paragraph_id": 116,
"tag": "p",
"text": "ã§äžããããã",
"title": "åéå"
},
{
"paragraph_id": 117,
"tag": "p",
"text": "(4.3)ã",
"title": "åéå"
},
{
"paragraph_id": 118,
"tag": "p",
"text": "ãšæžçŽã, A cos ÎŽ = a , A sin ÎŽ = b {\\displaystyle A\\cos \\delta =a,\\ A\\sin \\delta =b} ãšãããš",
"title": "åéå"
},
{
"paragraph_id": 119,
"tag": "p",
"text": "ãšãªã,æ¯å¹
ã¯",
"title": "åéå"
},
{
"paragraph_id": 120,
"tag": "p",
"text": "質émãæã€ããç©äœã«ã€ããŠãã°ãå®æ° k 1 {\\displaystyle k_{1}} ã®ã°ããšã°ãå®æ° k 2 {\\displaystyle k_{2}} ã®ã°ãã« ã€ãªãããå Žåã§ã¯ã ã©ã¡ãã®å Žåã®æ¹ãç©äœã®è§é床ã倧ãããªããã ãã ãã k 1 > k 2 {\\displaystyle k_{1}>k_{2}} ãæãç«ã€ãšããããŸããåšæãšæ¯åæ°ã«ã€ããŠã¯ã©ããªããã",
"title": "åéå"
},
{
"paragraph_id": 121,
"tag": "p",
"text": "ãã®å Žåã«ã¯ãã®åæ¯åã®è§æ¯åæ°ã¯ã",
"title": "åéå"
},
{
"paragraph_id": 122,
"tag": "p",
"text": "ã§äžããããããã®éã¯ã°ãå®æ°kã倧ããã»ã©å€§ããã®ã§ãè§æ¯åæ°ã¯ ã°ãå®æ° k 1 {\\displaystyle k_{1}} ãæã€ã°ãã®è§æ¯åæ°ã®æ¹ãã°ãå®æ° k 2 {\\displaystyle k_{2}} ãæã€ã°ãã®è§æ¯åæ° ãã倧ãããªãããŸããåæ¯åã®æ¯åæ°ã¯åæ¯åã®è§æ¯åæ°ã«æ¯äŸããã®ã§ã æ¯åæ°ã«ã€ããŠãã ã°ãå®æ° k 1 {\\displaystyle k_{1}} ãæã€ã°ãã®æ¯åæ°ã®æ¹ãã°ãå®æ° k 2 {\\displaystyle k_{2}} ã æã€ã°ãã®æ¯åæ°ãã倧ãããªããäžæ¹ããã®å Žåã®åšæã«ã€ããŠã¯ã",
"title": "åéå"
},
{
"paragraph_id": 123,
"tag": "p",
"text": "ãæãç«ã€ãããã°ãå®æ°kãå°ããã»ã©å€§ãããªãããã£ãŠãåšæã«ã€ããŠã¯ ã°ãå®æ° k 2 {\\displaystyle k_{2}} ãæã€ã°ãã®åšæã®æ¹ãã°ãå®æ° k 1 {\\displaystyle k_{1}} ãæã€ã°ãã®åšæ ãã倧ãããªãã",
"title": "åéå"
},
{
"paragraph_id": 124,
"tag": "p",
"text": "éåã®ããäžã«é·ãl[m]ã®ã²ãã§ã€ããããç©äœã«ãã£ãŠäœãããç©äœã® éçŽäžåãã«åçŽãªæ¹åã®éåãåæ¯åãšãªãããšãæ±ããã ãã ããæ¯ãåã®åãç¯å²ã¯å°ãããã®ãšããã ãã®ããã«åæ¯åãããæ¯ãåã åæ¯ãå(ãããµãããsimple pendlum) ãšåŒã¶ããšãããã",
"title": "åéå"
},
{
"paragraph_id": 125,
"tag": "p",
"text": "ã²ã ãåºå®ãããŠããäœçœ®ããéçŽã«äžãããçŽç·ãšãç©äœãã€ãªãããŠãã ã²ã ããªãè§åºŠã Ξ {\\displaystyle \\theta } ãšããããã®å Žåãå³åœ¢çã«èãããšãã®å Žåã®æ°Žå¹³æ¹åã®éåæ¹çšåŒã¯",
"title": "åéå"
},
{
"paragraph_id": 126,
"tag": "p",
"text": "ãšãªããããã§ã Ξ {\\displaystyle \\theta } ãå°ããå Žåã",
"title": "åéå"
},
{
"paragraph_id": 127,
"tag": "p",
"text": "ãšãªãããšã«æ³šæãããšãéåæ¹çšåŒã¯",
"title": "åéå"
},
{
"paragraph_id": 128,
"tag": "p",
"text": "ãšãªãå
ã»ã©ã®ã°ãã«ã€ãªãããç©äœã®éåæ¹çšåŒãšçãããªãã",
"title": "åéå"
},
{
"paragraph_id": 129,
"tag": "p",
"text": "ãã£ãŠããã®ç©äœã®éåãåæ¯åã§èšè¿°ãããããšãåãã£ããããã«ã å
ã»ã©ã®è§æ¯åæ°ãšæ¯èŒãããšããã®å Žåã®è§æ¯åæ° Ï {\\displaystyle \\omega } ã¯",
"title": "åéå"
},
{
"paragraph_id": 130,
"tag": "p",
"text": "ãšãªãããšãåããã",
"title": "åéå"
},
{
"paragraph_id": 131,
"tag": "p",
"text": "ãããã®çµæããå°åŠæ ¡çç§ã®çµæã§ãã",
"title": "åéå"
},
{
"paragraph_id": 132,
"tag": "p",
"text": "ã®å®éšäºå®ãéåæ¹çšåŒã®çµæãšäžèŽããããšã確ãããããã",
"title": "åéå"
},
{
"paragraph_id": 133,
"tag": "p",
"text": "ãã®ç« ã§ã¯ãäžæåŒåã«ããéåãæ±ããäžæåŒåã¯å
šãŠã®ç©äœã®éã«ååšããŠãããããã®åãåªä»ããéåãšããŠæåãªãã®ã¯å€ªéœã®åããå転ããå°çã®éåããå°çèªèº«ã®åããå転ããæã®éåã§ãããå®éã«ã¯ãã®ãããªäœãã®åããå転ããæ§é ã¯å®å®å
šäœã«åºãèŠãããã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 134,
"tag": "p",
"text": "äŸãã°ã空ã«èŠãããæã¯w:ææãšåŒã°ãããããããã®æã®åãã«ã倪éœã«å¯Ÿããå°çãšåãããã«ãææãåããåã£ãŠãããšèããããå®éã«ãã®ãããªææã確èªãããææãããã(w:ç³»å€ææåç
§ã)",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 135,
"tag": "p",
"text": "ãã®ããã«å®å®ã®äžã§äžæåŒåã«ããå転éåã¯åºã芳枬ããããããã§ã¯ãã®ãããªéåã¯ç©äœéã«åãã©ã®ãããªåã«ãã£ãŠèšè¿°ãããããèŠãŠããã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 136,
"tag": "p",
"text": "æŽå²çã«ã¯ãéã«ãã®ãããªç©äœã®éã®éåã説æãããããªåãèããããšã§ ç©äœéã«åãåãçºèŠããããæŽå²ã«ã€ããŠè©³ããã¯w:ãã¥ãŒãã³ãªã©ãåç
§ã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 137,
"tag": "p",
"text": "ãŸãã¯ãç©äœéã«åãäžæåŒå(glavitational constant)ã®æ³åãè¿°ã¹ããçš®ã
ã®èŠ³æž¬ã®çµæã«ãããšã質é m 1 {\\displaystyle m_{1}} ãæã€ç©äœãšè³ªé m 2 {\\displaystyle m_{2}} ãæã€ç©äœã®éã«ã¯",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 138,
"tag": "p",
"text": "ã§è¡šããããåãåããããã§Gã¯ç©äœã«ãããªãå®æ°ã§ãäžæåŒåå®æ°ãšããã å€ã¯ G = 6.67 à 10 â 11 [ N â
m 2 / k g 2 ] {\\displaystyle G=6.67\\times 10^{-11}[{\\mathrm {N} \\cdot \\mathrm {m} ^{2}/\\mathrm {kg} ^{2}}]} ã§ããã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 139,
"tag": "p",
"text": "äžæåŒåã®æ³å",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 140,
"tag": "p",
"text": "äžæåŒåã¯ç©äœéã®è·é¢ã®2ä¹ã«éæ¯äŸããåã§ããã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 141,
"tag": "p",
"text": "ç©äœã®å°ãªããšãçæ¹ãææã®ããã«å·šå€§ãªå Žåãç©äœéã®è·é¢rã¯ãéå¿éã®è·é¢ã§ããã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 142,
"tag": "p",
"text": "å°çã®äžæåŒåãèãããå°çã®è³ªéãMãå°çã®ååŸãRã枬å®ããç©äœã®è³ªéãmãšããå ŽåãéåFã¯",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 143,
"tag": "p",
"text": "ãšãªãã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 144,
"tag": "p",
"text": "ãããå°è¡šè¿ãã§ã¯å€§ããã mg ãšçããã®ã§ã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 145,
"tag": "p",
"text": "å€åœ¢ããŠ",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 146,
"tag": "p",
"text": "ãšãªããèšç®åé¡ã®ããããã®å€åœ¢ãçšããããå Žåãããã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 147,
"tag": "p",
"text": "å°çã¯èªè»¢ãããŠãããéåã®èšç®ã§ã¯ãå³å¯ã«ã¯èªè»¢ã«ããé å¿åãèããå¿
èŠãããããããããèªè»¢ã®é å¿åã®å€§ããã¯ãäžæåŒåã® 1 300 {\\displaystyle {\\frac {1}{300}}} åãŠãã©ãããªãã®ã§ãéåžžã¯èªè»¢ã«ããé å¿åãç¡èŠããå Žåãå€ãã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 148,
"tag": "p",
"text": "ãªããå°çã®èªè»¢ã®é å¿åã¯ãèµ€éäžã§ãã£ãšã倧ãããªãã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 149,
"tag": "p",
"text": "",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 150,
"tag": "p",
"text": "人工è¡æããå°çã®èªè»¢ãšåãåšæã§ãèªè»¢ãšåãåãã«çéåéåãããã°ããã®äººå·¥è¡æã¯å°äžããèŠãŠãã€ãã«å°é¢ã®äžç©ºã«ããã®ã§ãå°äžã®èŠ³æž¬è
ããã¯éæ¢ããŠèŠããããã®ãããªäººå·¥è¡æã®ããšãéæ¢è¡æãšããã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 151,
"tag": "p",
"text": "",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 152,
"tag": "p",
"text": "質émã®ç©äœã質éMã®å€§ããªç©äœã®åãããäžæåŒåã®åãåå¿åãšããŠãååŸrã®åéåãããŠããããã®å Žåã®åéåã®è§é床ãæ±ããã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 153,
"tag": "p",
"text": "ååŸrãè§é床 Ï {\\displaystyle \\omega } ã®åéåãããå Žåã®ç©äœã®åå¿å ã¯",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 154,
"tag": "p",
"text": "ã§ãããäžæ¹ã質émãšè³ªéMã®ç©äœã®éã®è·é¢ãrã§ããå Žåã2ã€ã®ç©äœéã«åãéåã¯ãéåã®å€æ°ãfãšãããšã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 155,
"tag": "p",
"text": "ã§äžããããããã£ãŠããããã®åãçãããªãå Žåã«ã質émã®ç©äœã¯è³ªéMã®ç©äœã®ãŸãããåéåã§å転(å
¬è»¢)ããããšãã§ããããã£ãŠã Ï {\\displaystyle \\omega } ãæ±ããåŒã¯ã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 156,
"tag": "p",
"text": "ãšãªãã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 157,
"tag": "p",
"text": "å°çè¡šé¢ã§ã®éåã«ããäœçœ®ãšãã«ã®ãŒãèããããã®ãšåæ§ã«ãäžæåŒåã«ããäœçœ®ãšãã«ã®ãŒãèããããšãã§ããã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 158,
"tag": "p",
"text": "äžæåŒåã«ããäœçœ®ãšãã«ã®ãŒãæ±ããã«ã¯ãäžæåŒåãç©åããã°ããã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 159,
"tag": "p",
"text": "質éMã®ç©äœããrã®è·é¢ã«è³ªémã®ç©äœãååšãããšããããã ããMã¯mããã¯ãã㫠倧ãããšãããç¡éé ç¹ãåºæºã«ãããš(ã€ãŸãç¡éé ã§ã¯äœçœ®ãšãã«ã®ãŒããŒã)ããã®å Žåã質émã®ç©äœã®äœçœ®ãšãã«ã®ãŒã¯",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 160,
"tag": "p",
"text": "ã§äžããããã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 161,
"tag": "p",
"text": "笊å·ã«ãã€ãã¹ãã€ãããšã®ç©ççãªè§£éã¯ãéåãã€ããã ãç©äœã«è¿ã¥ãã»ã©ããã®ç©äœã®ã€ããã ãéååãè±åºããã«ã¯ããšãã«ã®ãŒãè¿œå çã«å¿
èŠã«ãªãããã§ãããšè§£éã§ããã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 162,
"tag": "p",
"text": "ç¡éé ã§ã¯ r=+â ãšããã°ãããçµæã U=0 ã«ãªãã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 163,
"tag": "p",
"text": "ãªããäžæåŒåã¯ä¿ååã§ããã®ã§ãäœçœ®ãšãã«ã®ãŒã¯ãç¡éé ç¹ããã®çµè·¯ã«ããããçŸåšã®äœçœ®ã ãã§æ±ºãŸãã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 164,
"tag": "p",
"text": "",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 165,
"tag": "p",
"text": "ã®ããã«äžããããããŸãããã®ã°ã©ãã¯çŽèŠ³çãªæå³ãæã£ãŠããã å®ã¯ããã®ã°ã©ãã®åŸãã¯ã°ã©ããè¡šããäœçœ®ãšãã«ã®ãŒãæã€ç¹ã«ç©äœã眮ããå Žåã ãã®ç©äœãåãåããæ¹åãšãã®å€§ãããè¡šãããŠãããããã§ã¯ã äœçœ®ãšãã«ã®ãŒã®åŸããåžžã«r=0ã«èœã¡èŸŒãæ¹åã«çããŠããããç©äœMããè·é¢r (rã¯ä»»æã®å®æ°ã)ã®ç¹ã«éæ¢ããŠããç©äœã¯å¿
ãMã®æ¹åã«åžã蟌ãŸããŠè¡ãããšã è¡šãããŠããã(詳ããã¯å€å
žååŠåç
§ã)",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 166,
"tag": "p",
"text": "",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 167,
"tag": "p",
"text": "ããææäžã«ããç©äœãå®å®ã®ç¡éé ãŸã§å°éãããããã«å®å®è¹ã«ææäžã§ äžããªããŠã¯ãããªãé床ã¯ã©ã®ããã«è¡šãããããããã ããèšç®ã«ã€ããŠã¯ æåã«å®å®è¹ãåºçºããææ以å€ã®å€©äœããã®åœ±é¿ã¯ç¡èŠãããšããã ãŸããææã®ååŸã¯Rã ææã®è³ªéã¯Mãšããã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 168,
"tag": "p",
"text": "ææã®åŒåã«ããäœçœ®ãšãã«ã®ãŒã¯ææè¡šé¢ã§",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 169,
"tag": "p",
"text": "ã§ãããç¡éåç¹ã§ã¯0ã§ããããã ããmã¯å®å®è¹ã®è³ªéãšããã äžæ¹ãå®å®è¹ãç¡éåç¹ã«éããã«ã¯ãå®å®è¹ã®é床ãç¡éåç¹ã§ã¡ããã©0ã« çãããªãã°ãããããã§ãææäžã§ã®å®å®è¹ã®é床ãvãšãããšã ãšãã«ã®ãŒä¿ååããã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 170,
"tag": "p",
"text": "ãšãªãããã£ãŠãã®åŒããvãæ±ããã°ãããçã¯ã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 171,
"tag": "p",
"text": "äžèšã®èšç®ããåããããã«ãäžè¬ã«ãäžæåŒåã ããåããŠéåããç©äœã®ååŠçãšãã«ã®ãŒã¯ã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 172,
"tag": "p",
"text": "ã§ããã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 173,
"tag": "p",
"text": "ä»®ã«é«ãå±±ããç©äœãæ°Žå¹³ã«çºå°ãããšã(空æ°æµæã¯ç¡èŠãã)ãå°çã®ãŸãããåãç¶ããããã«å¿
èŠãªæå°ã®åé床ã®ããšã第äžå®å®é床ãšããã(â» ååã¯æèšããªããŠãããèŠããã¹ãã¯ãèšç®æ¹æ³ã§ããã) 第äžå®å®é床ã¯ãèŠããã«ãé å¿åãšåå¿åãã€ãããããã«å¿
èŠãªåé床ã§ããã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 174,
"tag": "p",
"text": "第äžå®å®é床ã¯ãç§éã§ã¯çŽ7.91km/sã§ããã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 175,
"tag": "p",
"text": "v1ã«ã€ããŠè§§ãã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 176,
"tag": "p",
"text": "ãªããããã R = 6400 à 10 m ã§ããã g = 9.8 m/s ã§ããã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 177,
"tag": "p",
"text": "ããã«åé床ã倧ãããªããšãç©äœã¯æ¥åè»éã«ãªãã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 178,
"tag": "p",
"text": "åé床ãçŽ11.2km/sã«ãªããšãè»éã¯æŸç©ç·ã«ãªããç©äœã¯ç¡éã®åœŒæ¹ã«é£ãã§ããã ãã®çŽ11.2km/sã®ããšã第äºå®å®é床ãšãããããã¯ãç¡éé ã®ç¹ã§ãé床ã0ãè¶
ããå€ã«ãªãããã«å¿
èŠãªåé床ã§ããã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 179,
"tag": "p",
"text": "ãªã®ã§ãèšç®ã§ç¬¬äºå®å®é床ãæ±ããã«ã¯ãšãã«ã®ãŒä¿ååãèšç®ã«ã¯äœ¿ãã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 180,
"tag": "p",
"text": "ã®åŒããvãæ±ãã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 181,
"tag": "p",
"text": "ã«ããã« G M = g R 2 {\\displaystyle GM=gR^{2}} ã代å
¥ããŠã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 182,
"tag": "p",
"text": "ããã«é¢ä¿ããå®æ°ã代å
¥ããã°ããã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 183,
"tag": "p",
"text": "ãªããããã R = 6400 à 10 m ã§ããã g = 9.8 m/s ã§ããã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 184,
"tag": "p",
"text": "åé床 11.2km/s以äžã§ã¯ãè»éã¯åæ²ç·ã«ãªããç©äœã¯ç¡éã®åœŒæ¹ã«é£ãã§ããã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 185,
"tag": "p",
"text": "",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 186,
"tag": "p",
"text": "â» æ€å®æç§æžã§ã¯ãè泚ãªã©ã«æžããŠãã£ããããã å°çããå°åºããŠã倪éœç³»ã®å€ã«åºãããã«å¿
èŠãªæå°ã®åé床ã®ããšã第äžå®å®é床(çŽ 16.7 km/s) ã§ããã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 187,
"tag": "p",
"text": "ã®ãªã·ã£æ代ããäžäžãŸã§ä¿¡ããããŠãã倩å説(è±: geocentric theory)ã«å¯Ÿã,16äžçŽåã°ã«ã³ãã«ãã¯ã¹ã¯å
šãŠã®ææ(è±: planet)ã倪éœãäžå¿ãšããåéåãããŠããå°å説ãæå±ããããã®åŸãã£ã³ã»ãã©ãŒãšã¯é·å¹Žã«ãããææã®èŠ³æž¬ãè¡ã,ãã®èŠ³æž¬çµæãåŒç¶ãã ã±ãã©ãŒã¯ãããã®çµæãããšã«èšç®ãè¡ã,ææã®éè¡ã«é¢ããæ³å,ã±ãã©ãŒã®æ³å(è±: Kepler's laws)ãçºèŠããããªã,æç§æžã¯å€ªéœãšææã®é¢ä¿ã§è«ããŠããã,ä»ã«ãææãšè¡æ(èªç¶è¡æ,人工è¡æ)ã§ãæãç«ã€ã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 188,
"tag": "p",
"text": "ææ(è¡æ)ã¯å€ªéœ(ææ)ã1ã€ã®çŠç¹ãšããæ¥åéåããã(æ¥åè»éã®æ³å)ã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 189,
"tag": "p",
"text": "ææ(è¡æ)ãšå€ªéœ(ææ)ãçµã¶ååŸãåäœæéã«æãé¢ç©(é¢ç©é床)ã¯äžå®ã§ãã(é¢ç©é床äžå®)ã",
"title": "äžæåŒåã®æ³å"
},
{
"paragraph_id": 190,
"tag": "p",
"text": "ææ(è¡æ)ã®å
¬è»¢åšæ T {\\displaystyle T} ã®2ä¹ã¯æ¥åè»éã®é·ååŸ(åé·è»ž) a {\\displaystyle a} ã®3ä¹ã«æ¯äŸããã",
"title": "äžæåŒåã®æ³å"
}
] | null | = ç©äœã®éå =
[[é«çåŠæ ¡çç§ ç©çåºç€]]ã§ã¯ãç©äœã®éåãçŽç·äžã®éåãäžå¿ã«æ±ã£ããç©çã§ã¯ãããè€éãªå¹³é¢äžã®éåãæ±ããå¹³é¢äžã®éåã§ã¯ãçŽç·äžã®éåãšã¯éã£ãŠãç©äœã®äœçœ®ãè¡šããã®ã«å¿
èŠãªéã2ã€ã«ãªãããããã¯éåžž<math>x,\ y</math>ãšãããã©ã¡ããæå»<math>t</math>ã®äžæã®é¢æ°ãšãªãã
ãããã®é¢æ°ã¯ã©ããªãã®ã§ãããããããã§ã¯äž»ã«ãå®éã®ç©äœã®éåãšããŠããããããããã®ãæ±ãã
== å¹³é¢äžã®éå ==
{{See also|[[é«çåŠæ ¡ç©çåºç€/ååŠ#ïŒæ¬¡å
ã»ïŒæ¬¡å
ã«ãããäœçœ®ã»é床ã»å é床|é«çåŠæ ¡ç©çåºç€/ååŠ]]}}
å¹³é¢äžïŒããªãã¡ïŒæ¬¡å
ã«ãããŠïŒæå»<math>t</math>ã«ãããäœçœ®ã¯<math>\overrightarrow r(t)=(x(t),\ y(t))</math>ïŒåŸ®å°æé<math>\mathit{\Delta}t</math>éã®å€äœã¯<math>\mathit{\Delta}\overrightarrow r =\overrightarrow r(t +\mathit{\Delta}t)-\overrightarrow r(t)=(\mathit{\Delta}x,\ \mathit{\Delta}y)</math>ãšå®çŸ©ãããããã®ãšã
:<math>\bar \overrightarrow v =\frac{\overrightarrow r(t +\mathit{\Delta}t)-\overrightarrow r(t)}{\mathit{\Delta}t}=\frac{\mathit{\Delta}\overrightarrow r}{\mathit{\Delta}t}</math>
ã<math>\mathit{\Delta}t</math>éã®å¹³åé床ïŒ<math>\mathit{\Delta}t\to 0</math>ã®æ¥µé
:<math>\overrightarrow v(t)=\lim_{\mathit{\Delta}t\to 0}\frac{\overrightarrow r(t +\mathit{\Delta}t)-\overrightarrow r(t)}{\mathit{\Delta}t}=\frac{d\overrightarrow r(t)}{dt}=\left(\frac{dx(t)}{dt},\ \frac{dy(t)}{dt}\right)=(\dot x(t),\ \dot y(t))=(v_x(t),\ v_y(t))</math>
ãæå»<math>t</math>ã§ã®(ç¬é)é床ãšããããªãïŒæå»<math>t</math>ã§ã®éã(é床ã®å€§ãã)ã¯
:<math>v =|\overrightarrow v|=\sqrt{{v_x}^2 +{v_y}^2}</math>.
ãã®å ŽåãïŒé床ããäœçœ®ãæ±ãŸãïŒåæåæ¯ã«
:<math>x(t)= x(0)+\int _0 ^t v_x(t)dt</math>
:<math>y(t)= y(0)+\int _0 ^t v_y(t)dt</math>
ãæãç«ã¡ïŒãããããã¯ãã«ãçšããŠã²ãšãŸãšãã«ããŠä»»æã®æå»<math>t</math>ã«ãããäœçœ®
:<math>\overrightarrow r(t)=\overrightarrow r(0)+\int _0 ^t\overrightarrow v(t)dt</math> (1.1)
ãæ±ããããã
ãŸãïŒ
:<math>\bar \overrightarrow a =\frac{\overrightarrow v(t +\mathit{\Delta}t)-\overrightarrow v(t)}{\mathit{\Delta}t}=\frac{\mathit{\Delta}\overrightarrow v}{\mathit{\Delta}t}</math> (<math>\mathit{\Delta}\overrightarrow v</math>ã¯åŸ®å°æé<math>\mathit{\Delta}t</math>éã®é床å€å)
ã<math>\mathit{\Delta}t</math>éã®å¹³åå é床ïŒ<math>\mathit{\Delta}t\to 0</math>ã®æ¥µé
:<math>\begin{align}\overrightarrow a(t)=\lim_{\mathit{\Delta}t\to 0}\frac{\overrightarrow v(t +\mathit{\Delta}t)-\overrightarrow v(t)}{\mathit{\Delta}t}& =\frac{d\overrightarrow v(t)}{dt}=\left(\frac{dv_x(t)}{dt},\ \frac{dv_y(t)}{dt}\right)=(\dot v_x(t),\ \dot v_y(t))\\ & =\frac{d^2\overrightarrow r(t)}{dt^2}=\left(\frac{d^2x(t)}{dt^2},\ \frac{d^2y(t)}{dt^2}\right)=(\ddot x(t),\ \ddot y(t))\end{align}</math>
ãæå»<math>t</math>ã§ã®(ç¬é)å é床ãšããã
ãã®å ŽåãïŒå é床ããé床ãæ±ãŸãïŒåæåæ¯ã«
:<math>v_x(t)=v_x(0)+\int _0 ^t\frac{dv_x(t)}{dt}dt</math>
:<math>v_y(t)=v_y(0)+\int _0 ^t\frac{dv_y(t)}{dt}dt</math>
ãæãç«ã¡ïŒãããããã¯ãã«ãçšããŠã²ãšãŸãšãã«ããŠä»»æã®æå»<math>t</math>ã«ãããé床
:<math>\overrightarrow v(t)=\overrightarrow v(0)+\int _0 ^t\overrightarrow a(t)dt</math> (1.2)
ãæ±ããããããªãïŒããã<math>\overrightarrow r(0), \overrightarrow v(0)</math>ã®å€ãåæå€ãšããã
ç¹ã«ïŒå é床äžå®ã®ãšãã®éåã¯'''çå é床éå'''ãšãããïŒäžèšã®å
¬åŒ(1.2, 1)ã¯ãããã
:{|
|-
|<math>\overrightarrow v(t)</math>
|<math>=\overrightarrow v(0)+\int _0 ^t\overrightarrow adt</math> (1.3)
|-
|
|<math>=\overrightarrow v(0)+\overrightarrow at</math>
|}
:<math>\overrightarrow r(t)=\overrightarrow r(0)+\int _0 ^t(\overrightarrow v(0)+\overrightarrow at)dt =\overrightarrow r(0)+\overrightarrow v(0)t +\frac{1}{2}\overrightarrow at^2</math>
ãšãªãã
éåæ¹çšåŒã¯ãåãç©äœãåããå é床ã«æ¯äŸãããšããç¹ã¯ããããªãã
ããããä»åã¯åãšå é床ã¯ã©ã¡ãããã¯ãã«éã§ããããã£ãŠãå€å<math>\overrightarrow f=(f_x,\ f_y)</math>ãåãïŒå é床<math>\overrightarrow a=(a_x,\ a_y)</math>ã§éåããç©äœã®éåæ¹çšåŒã¯
:<math>
m\overrightarrow a =\overrightarrow f
</math>
ãšããããã
éåžžã¯ããã®æ¹çšåŒã解ãå Žåã¯èŠçŽ ããšã«ããã
:<math>
ma_x = f_x
</math>
:<math>
ma_y = f_y
</math>
ãšããããã
*åé¡äŸ
**åé¡
æå»t = 0ã«ã
:<math>
\overrightarrow x = (0,\ 0)
</math>
ã
:<math>
v = \frac 1 {\sqrt 2} (1,\ 1)v _0
</math>
ã§ééããç©äœã®æå»tã§ã®äœçœ®ãæ±ããã
**解ç
ç©äœã®xæ¹åãšyæ¹åã¯äºãã«ç¬ç«ã«çéçŽç·éåãããã
ããã§ã¯xæ¹åãyæ¹åãé床
:<math>
v = \frac 1 {\sqrt 2} v _0
</math>
ãªã®ã§ãçéçŽç·éåã®åŒã®ãã¯ãã«éãšããé
:<math>
\overrightarrow x = \overrightarrow v ( t - t _0) + \overrightarrow x _0
</math>
ã«ä»£å
¥ãããšã
:<math>
\overrightarrow x =
\frac 1 {\sqrt 2} (1,\ 1)v _0 t
</math>
ãšãªãã
èŠçŽ ããšã«ãããšã
:<math>
x = \frac 1 {\sqrt 2} v _0 t
</math>
:<math>
y= \frac 1 {\sqrt 2} v _0 t
</math>
ãšãªãã
** åé¡
æå»t=0ã«åç¹(0,\ 0)ãyæ¹åã«é床<math>v _0</math>ã§çéçŽç·éåããŠãã質émã®ç©äœã«ã
xæ¹åã®äžæ§ãªåfããããå§ããããã®å Žåãæå»tã«ãããç©äœã®äœçœ®ãš
é床ãæ±ããã
** 解ç
x軞æ¹åã«ã¯çå é床éåãšãªãã
ç©äœãåããå é床ã¯ãéåæ¹çšåŒã«ãã
:<math>
a = \frac f m
</math>
ãšãªãã
ããã«xæ¹åã®åé床0ïŒåæäœçœ®0ã§ããããšãçå é床çŽç·éåã®åŒã«
代å
¥ãããšã
:<math>
x = \frac 1 2 a t^2
</math>
:<math>
= \frac 1 2 \frac f m t^2
</math>
:<math>
v = a t
</math>
:<math>
= \frac f m t
</math>
ãšãªãã
ããã«ãy軞æ¹åã®éåã¯çééåã§ããããã®åé床ã¯ã<math>v _0</math>ïŒåæäœçœ®ã¯0ã§ããã®ã§ã
ãã®å€ãçééåã®åŒã«ä»£å
¥ãããšã
:<math>
y = v _0 t
</math>
:<math>
v _y = v _0
</math>
ãåŸãããã
= éåéãšåç© =
ãã®ç« ã§ã¯éåéïŒããã©ãããããmomentumïŒãæ±ããéåéã¯ãç©äœã®è¡çªã«çœ®ããŠãšãã«ã®ãŒãšäžŠã³ãä¿åéãšãªãéèŠãªéã§ããããŸãããã®ç« ã§ã¯åç©ïŒãããããimpulseïŒãšããéãå°å
¥ãããåç©ã¯éåéã®æéå€åãè¡šããéã§ããããã®å°åºã¯éåæ¹çšåŒãçšããŠæãããã
ç©äœãåããŠããå Žåãç©äœã®é床ãšè³ªéã®ç©ãç©äœã®éåé
:<math>\overrightarrow p = m\overrightarrow v</math> (2.1)
ãšå®çŸ©ãããéåæ¹çšåŒ
:<math>m\frac{d\overrightarrow v(t)}{dt}=\overrightarrow f</math> (<math>\overrightarrow v(t)</math>ã¯æå»<math>t</math>ã«ãããé床ïŒ<math>\overrightarrow f</math>ã¯åå)
ã®äž¡èŸºãæå»<math>t = t_1</math>ãã<math>t = t_2</math>ãŸã§ç©åãããš
:<math>\int _{t_1}^{t_2}m\frac{d\overrightarrow v(t)}{dt}dt =\int _{t_1}^{t_2}\overrightarrow fdt</math>
:<math>\therefore\int _{t_1}^{t_2}md\overrightarrow v(t)=\int _{t_1}^{t_2}\overrightarrow fdt</math>
:<math>\therefore[m\overrightarrow v(t)]_{t_1}^{t_2}=\int _{t_1}^{t_2}\overrightarrow fdt</math> (泚ïŒ<math>\overrightarrow f</math>ã¯äžå®ãšã¯éãã¬ã®ã§å³èŸºã¯ç©åå®è¡ã§ããªã)
:<math>\therefore m\overrightarrow v(t_2)- m\overrightarrow v(t_1)=\int _{t_1}^{t_2}\overrightarrow fdt</math>
ãšãªãã<math>\overrightarrow v(t_1)=\vec{v_1}, \overrightarrow v(t_2)=\vec{v_2}</math>ãšãããš
:<math>m\vec{v_2}- m\vec{v_1}=\int _{t_1}^{t_2}\overrightarrow fdt</math>. (2.2)
ãã®åŒã®å·ŠèŸºã¯éåéå€åïŒå³èŸºã¯åç©ïŒãããããimpulseïŒã§ããããã£ãŠïŒ'''éåéå€åã¯åç©ã«çãã'''ããšãåãããéåéå€åã<math>\mathit{\Delta}\overrightarrow p</math>ïŒåç©ã<math>\overrightarrow I</math>ãšãããš
:<math>\mathit{\Delta}\overrightarrow p = m(\vec{v_2}-\vec{v_1}), \overrightarrow I =\int _{t_1}^{t_2}\overrightarrow fdt,\ \mathit{\Delta}\overrightarrow p =\overrightarrow I</math>.
ç¹ã«ïŒ<math>\overrightarrow f =</math>äžå®ã®ãšãïŒ<math>t_2 - t_1 =\mathit{\Delta}t</math>ãšãããš
:<math>\overrightarrow I =\overrightarrow f(t_2 - t_1)=\overrightarrow f\mathit{\Delta}t</math>.
* çºå±: 埮åãšå€åé
埮åãçšããå°åºã«ã€ããŠã¯ã[[å€å
žååŠ]]ãåç
§ã
* åé¡äŸ
** åé¡
éæ¢ããŠããç©äœã«æé<math>\mathit{\Delta}t</math>ã®éããæ¹åã«äžæ§ãªåfãããããç©äœãåŸã
éåéã¯ã©ãã ãããããã«ãç©äœã®è³ªéãmãšãããšãç©äœããã®æ¹åã«
åŸãé床ã¯ã©ãã ããã
** 解ç
éåéã®å€ååã¯ç©äœãåããåç©ã«çããã®ã§ãç©äœãåããåç©ãèšç®ããã°
ãããç©äœãåããåç©ã¯
:<math>
f\mathit{\Delta}t
</math>
ã«çããã®ã§ãç©äœãåŸãéåéã
:<math>
f\mathit{\Delta}t
</math>
ã«çãããããã«ãéåéã
:<math>
p = m v
</math>
ãæºããããšãèãããšãç©äœã®é床ã¯
:<math>
\frac 1 m f\mathit{\Delta}t
</math>
ãšãªãã
éåéã¯ãç©äœãå
šãåãåããªãå Žåã«ã¯ä¿åããããããã¯ç©äœã«åãåããªãå Žåã«ã¯ãç©äœã®åããåç©ã¯0ã§ããç©äœã®éåéå€åã0ã§ããããšããåœç¶ã§ããã
ããã«ãè€æ°ã®ç©äœã®éåéã«ã€ããŠã¯ãå¥ã®éèŠãªæ§è³ªãèŠããããããã¯ãè€æ°ã®ç©äœã®ãã€éåéã®ç·åã¯ãããã®ç©äœã®éã®è¡çªã«éããŠä¿åãããšããããšã§ãããããã¯ã€ãŸããäŸãã°ãã2ã€ã®ç©äœãè¡çªããå Žåãå§ãã«2ç©äœãããããæã£ãŠããéåéã®åã¯è¡çªãçµãã£ãåŸã«2ç©äœãæã£ãŠããéåéã®åã«çãããšããããšã§ãããããã§ãããã€ãã®ç©äœãããå Žåãããã®æã€éåéã®ç·åãã察å¿ããç©äœç³»ã®å
šéåéãšããã
ç©äœã®è¡çªã«ã€ããŠãéåéã¯åžžã«ä¿åãããããããç©äœç³»ã®å
šãšãã«ã®ãŒã¯åžžã«ä¿åãããšã¯éããªããäžè¬ã«ç©äœã®è¡çªã«ã€ããŠãšãã«ã®ãŒã¯åžžã«å€±ãããŠããããã£ãšãç©äœç³»ã«éããªãå
šãšãã«ã®ãŒã¯åžžã«äžå®ã§ããã®ã§ãç©äœãæã£ãŠãããšãã«ã®ãŒã¯é³ãç±ã®åœ¢ã§ç©äœç³»ã®å€ã«éããŠè¡ãã®ã§ãããç©äœãè¡çªã«ã€ããŠå€±ããšãã«ã®ãŒã¯è¡çªã«é¢ããç©äœãæã£ãŠããç©æ§å®æ°ã«ãã£ãŠæ±ºãŸãããã®ä¿æ°ãåçºä¿æ°ïŒã¯ãã±ã€ãããããcoefficient of restitutionïŒãšåŒã³ãeãªã©ã®èšå·ã§æžããåçºä¿æ°ã¯ãç©äœãè¡çªããããååŸã§ã®ç©äœéã®çžå¯Ÿé床ã®æ¯ã«ãã£ãŠå®ããããã
ç¹ã«ç©äœ1ãšç©äœ2ãè¡çªåã«é床 <math>v_1,\ v_2</math>ãæã£ãŠãããè¡çªåŸã«é床<math>v_1',\ v_2'</math>ãæã£ããšãããšãåçºä¿æ°eã¯
:<math>v_1' - v_2' = -e(v_1 - v_2)\quad\therefore e = - \frac {v_1' - v_2'} {v_1 - v_2} </math>
ã§å®ãããããããã§ãå³èŸºã®å§ãã®<math>-</math>笊åã¯ãè¡çªã®ååŸã§ç©äœã®é床ããã倧ããç©äœã¯ãè¡çªåã«ããå°ããé床ãæã£ãŠããç©äœãããè¡çªåŸã«ã¯ããå°ããé床ãæã€ããšã«ãªãããã§ããã
ãã®ãããåçºä¿æ°ã¯äžè¬ã«æ£ã®æ°ã§ããã
ãŸãåçºä¿æ°ã¯1ããå°ããæ°ã§ãããç©äœéã®çžå¯Ÿé床ã¯è¡çªåããè¡çªåŸã®æ¹ãå°ãããªãã
ç¹ã«<math>e = 1</math>ã®å Žåã(å®å
š)匟æ§è¡çªïŒelastic collisionïŒãšåŒã³ããã£ãœã<math>0<e<1</math>ã®å Žåãé匟æ§è¡çªïŒinelastic collisionïŒãšåŒã¶ã匟æ§è¡çªã®å Žåã¯ãååŠçãšãã«ã®ãŒã¯ä¿åããããšãç¥ãããŠãããäžæ¹ãé匟æ§è¡çªã®
å Žåã¯ç©äœç³»ã®å
šãšãã«ã®ãŒã¯å€±ãããã
* åé¡äŸ
** åé¡
ããéæ¢ããŠããç©äœ2ã«éåépã§éåããŠããç©äœãè¡çªããããã®å Žåã
è¡çªããåŸã®ç©äœ2ãéåé<math>p _2</math>ãåŸããšãããšãè¡çªåŸã®ç©äœ1ã®éåéã¯
ã©ãã ããšãªã£ããã
** 解ç
éåéä¿ååãèãããšãè¡çªã®ååŸã§ç©äœ1ãšç©äœ2ã§æ§æãããç©äœç³»ã®å
šéåéã¯ä¿åããã
ããã§ãè¡çªåã®ç©äœç³»ã®å
šéåéã¯pã§ããã®ã§ãè¡çªåŸã®ç©äœç³»ã®å
šéåéãpãšãªãã
ããã«ãç©äœ2ã®è¡çªåŸã®éåéã
<math>p _2</math>ãªã®ã§ãç©äœ1ã®éåéã¯
:<math>
p - p _2
</math>
ãšãªãã
ããã§ãç©äœç³»ã®å
šéåéãä¿åãããããšã¯ãéåã«é¢ãã äœçšã»åäœçšã®æ³å ããåŸãã
äœçšåäœçšã®æ³åãçšãããšãç©äœç³»ã®éã®è¡çªã«éããŠãè¡çªã«é¢ããããããã®ç©äœãåããåã¯ã倧ãããçããåãã¯å察ãšãªãã
ãã®å Žåãããããã®åã«å¯ŸããŠãè¡çªã®æé<math>\Delta t</math>ãããããã®ã¯
è¡çªã«éããŠããããã®ç©äœãåãåãåç©ã«çããã
ããã§ãè¡çªã«é¢ããŠåãåã®åç©ãå
šãŠã®ç©äœã«ã€ããŠè¶³ãåããããšããããã®åã¯ãäžã®ããšãã0ãšãªãã
ããããå
šéåéã®èšç®ã§ã¯ãŸãã«ãã®ãããªå
šç©äœã«ã€ããŠã®éåéã®ç·åãèšç®ããŠããã®ã§ã
è¡çªã«ãã£ãŠåŸããããããªåç©ã®ç·åã¯ã0ã«çããã
ãã£ãŠãè¡çªã«éããŠç©äœç³»ã®æã€å
šéåéã¯ä¿åãããã
ããã'''éåéä¿åå'''ïŒããã©ãããã ã»ãããããmomentum conservation lawïŒãšããã
* åé¡äŸ
** åé¡
質émã®2ã€ã®ç©äœãé床<math>v _1</math>, <math>v _2</math>
ã§ç§»åããŠããããããã®ç©äœãè¡çªããå Žåã
è¡çªåŸã®ããããã®ç©äœã®é床ãããšãã«ã®ãŒä¿ååãšéåéä¿ååãçšããŠ
èšç®ããããã ããç©äœã®è¡çªã«é¢ããŠãšãã«ã®ãŒã¯ä¿åãããšããã
** 解ç
ãã®åé¡ã¯2ã€ã®åã倧ããã®ç©äœãç°ãªã£ãé床ã§ã¶ã€ããå Žå
ãã®çµæãã©ããªãããèšç®ããåé¡ã§ããã
å®éšã®çµæã«ãããšãäžæ¹ãéæ¢ããŠããäžæ¹ãåããŠããå Žåã
åããŠããç©äœã¯éæ¢ããéæ¢ããŠããç©äœã¯åããŠããç©äœãæã£ãŠãã
é床ãšåãé床ã§åãã ãããšãç¥ãããŠãããããã§ã¯ããããã®
çµæãèšç®ã«ãã£ãŠç¢ºãããããããšãèŠãããšãåºæ¥ãã
è¡çªåŸã®ç©äœã®é床ãããããç©äœ1ã«ã€ããŠã¯<math>v _1'</math>ïŒç©äœ2ã«ã€ããŠã¯
<math>v _2'</math>ãšããããã®å Žåãç©äœã®è¡çªã«ã€ããŠå
šãšãã«ã®ãŒãä¿åãããããšã
çšãããšã
:<math>
1/2 m v _1^2 + 1/2 m v _2^2
=
1/2 m v _1'{}^2 + 1/2 m v'{} _2^2
</math>
ãåŸããããããã«ãç©äœã®è¡çªã«ã€ããŠç©äœç³»ã®å
šéåéãä¿åãããããšãçšãããšã
:<math>
m v _1
+ m v _2 =
m v _1'
+ m v _2'
</math>
ãããã¯ã<math>v' _1</math>, <math>v '_2</math>ã«ã€ããŠã®2次æ¹çšåŒã§ããã解ãããšãåºæ¥ããå®éèšç®ãããšã解ãšããŠ
:<math>
(v '_1,\ v' _2 )=(v _1,\ v _2),\ (v _2,\ v _1)
</math>
ãåŸããããåè
ã®è§£ã¯è¡çªã«éããŠç©äœã®é床ãå€åãã¬ããšã瀺ããŠããããããã¯å®éã®æ
åµãšããŠèãé£ãã®ã§ãåŸè
ã®è§£ãçŸå®ã®è§£ãšãªãããã®çµæãèŠããšãç©äœãæã€é床ãå
¥ãæ¿ããããšãåããã
ãã®ããšã¯å®éã«åã倧ããã®çãçšããŠå®éšãè¡ããšã確ãããããšãã§ããã
<!-- ããã¯äŸãã°ã
<math>v _1=v,\ v _2=0</math>ã®æãèãããšãè¡çªåŸã®çµæã¯
<math>v _1=0,\ v _2=v</math>ãšãªããå®éšã®çµæãåçŸããããšã«ãªãã
-->
=åäœã®ã€ãåã=
äœçœ®ã®ã¿ããã¡ïŒå€§ããããªãã®ã質ç¹ã§ããã'''åäœ'''ãšã¯ïŒå€§ãããããã圢ã倧ãããå€ããã¬ç©äœã®ããšã§ããã
==è§éåéãšåã®ã¢ãŒã¡ã³ã==
åäœã®éåãèããåã«äžå®å¹³é¢äžã®éåã«ã€ããŠæ¬¡ã®ãããªäžè¬çèå¯ãè¡ãã
æå»<math>t</math>ã«ãããŠ<math>xy</math>å¹³é¢å
ã®äœçœ®<math>\overrightarrow r=(x,\ y)</math>ãé床<math>\overrightarrow v=(v_x,\ v_y)</math>ã§éåãïŒå<math>\overrightarrow F=(F_x,\ F_y)</math>ãåããŠãã質é<math>m</math>ã®ç©äœã®éåæ¹çšåŒãæåã«åããŠè¡šãã°
:<math>m\frac{dv_x}{dt}=F_x,\qquad\qquad\qquad\qquad\;\cdots\cdots</math>â
:<math>m\frac{dv_y}{dt}=F_y.\qquad\qquad\qquad\qquad\;\cdots\cdots</math>â¡
â¡<math>\times x -</math>â <math>\times y</math>ãã
:<math>m\left(x\frac{dv_y}{dt}-y\frac{dv_x}{dt}\right)=xF_y -yF_x</math>
:<math>\therefore \frac{d}{dt}\{m(xv_y -yv_x)\}=xF_y -yF_x.\cdots</math>â¢
ãã®å·ŠèŸºã®
:<math>l=m(xv_y -yv_x)</math> (3.1)
ãåç¹OãŸããã®è§éåéãšããã
ããã§<math>\overrightarrow v</math>ãš<math>\overrightarrow r</math>ã®ãªãè§ã<math>\theta,\ x</math>軞ãš<math>\overrightarrow r</math>ã®ãªãè§ã<math>\phi</math>ãšãããš
:<math>x=r\cos\phi,\ v_x=v\cos(\theta +\phi),\ y=r\sin\phi,\ v_y=v\sin(\theta +\phi)</math>.
ãããã(3.1)ã«ä»£å
¥ãããš
:<math>l=m(r\cos\phi\cdot v\sin(\theta +\phi)-r\sin\phi\cdot v\cos(\theta +\phi))=mrv\sin\theta</math> (3.1a)
ãåŸãããã
ç©äœãå転ãããåã®å¹æã®å€§ãããè¡šãéã'''åã®ã¢ãŒã¡ã³ã'''ãšãããæŽã«<math>\overrightarrow F</math>ãš<math>\overrightarrow r</math>ã®ãªãè§ã<math>\mathit{\Theta}</math>ãšãããš
:<math>F_x=F\cos(\mathit{\Theta}+\phi),\ F_y=F\sin(\mathit{\Theta}+\phi)</math>.
ãã£ãŠ'''åç¹OãŸããã®åã®ã¢ãŒã¡ã³ã'''ã<math>N</math>ã§è¡šããš
:<math>N=xF_y -yF_x=r\cos\phi\cdot F\sin(\mathit{\Theta}+\phi)-r\sin\phi\cdot F\cos(\mathit{\Theta}+\phi)=Fr\sin\mathit{\Theta}</math>. (3.2)
ããã«<math>r\sin\mathit{\Theta}</math>ã¯åç¹ããå<math>\overrightarrow F</math>ã®äœçšç·ã«äžããåç·ã®é·ãã§ããïŒãããå<math>\overrightarrow F</math>ã®'''åç¹ã«å¯Ÿããè
ã®é·ã'''ãšããããã ãåã®ã¢ãŒã¡ã³ãã¯å<math>\overrightarrow F</math>ãäœçœ®ãã¯ãã«<math>\overrightarrow r</math>ãåæèšåãã«åãåããæ£ãšããŠãã(æèšåãã®éã¯<math>\mathit{\Theta}<0</math>ã§<math>r\sin\mathit{\Theta}<0</math>ãšèãã)ã
以äžããïŒâ¢(è§éåéã®æ¹çšåŒ)ã¯
:<math>\frac{dl}{dt}=N</math>. (3.3)
ããã¯åã®ã¢ãŒã¡ã³ããå ããããçµæãšããŠè§éåéãå€åãããšããå æé¢ä¿ãè¡šããç¹ã«<math>N=0</math>ãªãã°
:<math>\frac{dl}{dt}=0\quad\therefore l=</math>äžå®
ãšãªãïŒè§éåéãä¿åããã
==åäœã«åãåã®ã¢ãŒã¡ã³ã==
==éå¿==
ç©äœã®åéšåã«åãéåã®äœçšç¹ã'''éå¿'''({{Lang-en-short|centre of gravity}})æãã¯è³ªéäžå¿({{Lang-en-short|centre of mass}})ãšããã<math>n</math>ç©äœ(質éïŒ<math>m_1,\ m_2,\ \cdots\cdots,\ m_n</math>ïŒäœçœ®<math>\vec{r_1},\ \vec{r_2},\ \cdots\cdots,\ \vec{r_n}</math> (<math>n</math>ã¯èªç¶æ°)ã®éå¿ã®äœçœ®<math>\vec{r_\mathrm{G}}</math>ã¯ä»¥äžã®ããã«å®çŸ©ãããã
:<math>\vec{r_\mathrm{G}}=\frac{m_1\vec{r_1}+m_2\vec{r_2}+\cdots\cdots +m_n\vec{r_n}}{m_1+m_2+\cdots\cdots +m_n}</math>.
ãŸãéå¿é床<math>\vec{v_\mathrm{G}}</math>ã¯<math>\frac{d\vec{r_k}}{dt}=\vec{v_k}\ (k=1,\ 2,\ \cdots\cdots,\ n)</math>ãšãããš
:<math>\vec{v_\mathrm{G}}=\frac{d\vec{r_\mathrm{G}}}{dt}=\frac{m_1\vec{v_1}+m_2\vec{v_2}+\cdots\cdots +m_n\vec{v_n}}{m_1+m_2+\cdots\cdots +m_n}</math>.
= åéåãšåæ¯å =
ããã§ã¯ãåççãªå¹³é¢äžã®éåã®1ã€ãšããŠãåéå({{Lang-en-short|circular motion}})ãšåæ¯å({{Lang-en-short|simple harmonic motion}})ããã€ãããåéåã¯ãåæ¯ãåïŒãããµãããsimple pendlumïŒã®éåã®é¡äŒŒç©ãšããŠãéèŠã§ããããããšãšãã«ããã®ããŒãžã§ã¯äžæåŒåã«ããéåãæ±ãã
äžæåŒåã¯ããããéåãšåãåã§ããã
ç©äœãšç©äœã®éã«å¿
ãçããåã§ãããäžæ¹ãããã®åã¯éåžžã«åŒ±ãããã
ææã®ããã«å€§ããªè³ªéãæã£ãç©äœã®éåã«ããé¢ãããªãã
ããã§ã¯ã倪éœã®ãŸãããå転ããææã®ãããªå€§ããªã¹ã±ãŒã«ã®éåããã€ããããã®ãããªéåã¯åã«è¿ãè»éãšãªãããšãããããã®ãããææã®éåãç解ããäžã§ãåéåãç解ããããšãéèŠã§ããã
== åéå ==
ç©äœãåãæãããã«éåããããšãåéåãšåŒã¶ãåãæããããªéåã¯ãäŸãã°ãå圢ã®ã°ã©ãŠã³ãã®ãŸãããèµ°ã人éã®ããã«äººéãææãæã£ãŠè¡ãªãå Žåãæãããèªç¶çŸè±¡ãšããŠèµ·ããå Žåãå€ããäŸãã°ã倪éœã®ãŸãããåãå°çã®éåããå°çã®åããåãæã®éåã¯ãããããåéåã§èšè¿°ãããããŸããäžå®ã®é·ãããã£ãã²ããšäžå®ã®è³ªéãæã£ãç©äœã§äœãããæ¯ãåã®éåã¯ãã²ããåºå®ããç¹ããäžå®ã®è·é¢ããããŠéåããŠãããããç©äœã¯åè»éäžãéåããŠãããåºãæå³ã§ã®åè»éãšãšãããããšãåºæ¥ããããã§ã¯ããã®ãããªå Žåã®ãã¡ã§ä»£è¡šçãªãã®ãšããŠãå®å
šãªåè»éäžãéåããç©äœã®éåããã€ããã
åè»éäžãéåããç©äœã®åº§æšãäžè¬ã®å Žåãšåæ§
:<math>\overrightarrow r(t)=(x(t),\ y(t))</math>
ã§è¡šãããããç¹ã«åè»éãè¡šããé¢æ°ã¯[[é«çåŠæ ¡æ°åŠII ãããããªé¢æ°]]ã§æ±ã£ãäžè§é¢æ°ã«å¯Ÿå¿ããŠããã
* çºå±: äžè§é¢æ°ãçšããåã®è¡šç€º
ããã§ãåéåãäžè§é¢æ°ãçšããŠè¡šãããããšãè¿°ã¹ããããã®ããšã¯[[é«çåŠæ ¡æ°åŠC]]ã®'''åªä»å€æ°è¡šç€º'''ãçšããŠãããåªä»å€æ°è¡šç€ºã«ã€ããŠè©³ããã¯ã察å¿ããé
ãåç
§ããŠã»ããã
ååŸr[m]ã®åäžãçããé床ã§ãåéåããç©äœã®éåãèšè¿°ããããšãèããã
ããã«ã座æšãåãå Žååç¹ã®äœçœ®ã¯åéåã®äžå¿ã®äœçœ®ãšããã
ãã®å Žåã®ç©äœã®éåã¯ãx, y座æšãçšããŠã
:<math>
x = r \cos (\omega t +\delta)
</math>
:<math>
y = r \sin (\omega t +\delta)
</math>
ã«ãã£ãŠæžãããããã ãããã®å Žå<math>\omega</math>ã¯è§é床ãšåŒã°ãåäœã¯[rad/s]ã§äžããããããã ããããã§[rad]ã¯[[w:ã©ãžã¢ã³]]ã§ããã[[w:匧床æ³]]ã«ãã£ãŠè§åºŠãè¡šãããå Žåã®åäœã§ããã匧床æ³ã«ã€ããŠã¯[[é«çåŠæ ¡æ°åŠII ãããããªé¢æ°]]ãåç
§ãè§é床ã¯åéåãããŠããç©äœãã©ã®çšåºŠã®æéã§åãäžåšãããã«å¯Ÿå¿ããŠããããªãïŒé«çåŠæ ¡ã®ç©çã«ãããŠè§é床ã¯ã¹ã«ã©ãŒãšããŠæ±ãããŸãããã®éã¯äžã§åããã®ã ããåéåããŠããç©äœã®é床ã«æ¯äŸããã
ãŸããè§é床ã«å¯Ÿå¿ããŠã
:<math>
T = \frac {2\pi} \omega
</math>
ã§äžããããéã[[w:åšæ]]ãšãããåšæã®åäœã¯[s]ã§ãããåšæã¯ç©äœãäœç§éããšã«
åç¶ã1åšããããè¡šããéã§ããããã®å Žåã«ã¯ç©äœã¯T[s]ããšã«åç¶ã1åšãããããã«ã
:<math>
f = \frac \omega {2\pi}
</math>
ã[[w:æ¯åæ°]]ãšåŒã¶ãæ¯åæ°ã¯åšæãšã¯éã«ãåäœæéåœããã«ç©äœãåç¶ãäœåšãããã
æ°ããéã§ãããæ¯åæ°ã®åäœã«ã¯éåžž[Hz]ãçšãããããã¯ã[1/s]ã«çããåäœã§ããã
ãŸããåšæTãšãæ¯åæ°fã¯ãé¢ä¿åŒ
:<math>
Tf = 1
</math>
ãæºããããã®åŒã¯ããåéåãããŠããç©äœã«ã€ããŠããã®ç©äœã®åéåã®
åšæã«å¯Ÿå¿ããæéã®éã«ã¯ãç©äœã¯åç¶ã1åšã ããããšããããšã«å¯Ÿå¿ããã
ãŸãã
:<math>
x = r \cos (\omega t +\delta)
</math>
:<math>
y = r \sin (\omega t +\delta)
</math>
ã®åŒã§<math>\delta</math>ã¯ç©äœã®äœçœ®ã®[[w:äœçž]]ãšåŒã°ããç©äœãåç¶ã®ã©ã®ç¹ã«ãããã瀺ã
å€ã§ããã
ãŸãããã®å Žåã®ç©äœã®é床ã®x, yèŠçŽ ã¯
:<math>v_x =\frac{dx}{dt}= -r \omega \sin \omega t</math>
:<math>v_y =\frac{dy}{dt}= r \omega \cos \omega t</math>
ã§äžããããããã®åŒãšãåŸã®åéåã®å é床ã®å°åºã«ã€ããŠã¯ãåŸã®çºå±ãåç
§ãããã§ãç©äœã®éããvãšãããšã
:<math>
v = \sqrt {v _x ^2 +v _x ^2}
= \sqrt {r^2 \omega^2 (\sin^2 \omega t +\cos^2 \omega t) }
= r \omega
</math>
ãšãªããç©äœã®é床ã¯<math>r\omega</math>ã§äžããããããšãåããã
ããã«ã
:<math>
\overrightarrow r \cdot \overrightarrow v
</math>
ãèšç®ãããšã
:<math>
\overrightarrow r \cdot \overrightarrow v
</math>
:<math>
=( r \cos \omega t,\ r \sin \omega t) \cdot (-r \omega \sin \omega t,\ r \omega \cos \omega t)
</math>
:<math>
= r^2 \omega (\cos \omega t \sin \omega t - \cos \omega t \sin \omega t)
</math>
:<math>
= 0
</math>
ãšãªããåéåãããŠããç©äœã®é床ãšåéåã®äžå¿ãåç¹ãšããå Žåã®åº§æšã¯çŽäº€ããŠããããšãåãããããã«ãåéåãããŠããç©äœã®å é床ã¯ã
:<math>\frac{dv_x}{dt^2}= -r \omega^2 \cos \omega t</math>
:<math>\frac{dv_y}{dt^2}= -r \omega^2 \sin \omega t</math>
ãšãªããããã¯
:<math>\overrightarrow a = -\omega ^2 \overrightarrow r</math>
ã«å¯Ÿå¿ããŠãããåéåããããªãç©äœã®å é床ã¯ãåéåãããç©äœã®åº§æšãš
ã¡ããã©å察åãã«ãªãããšãåããã
* çºå±: åéåã®é床ãšå é床
ããã§ã¯ãåéåã®é床ãšå é床ãäžãããããã®å€ã¯ç©äœã®éåã決ãŸãã°æ±ºãŸãå€ãªã®ã§ãåéåã®åŒããèšç®ã§ããããã ãå®éã«ãããã®åŒãåŸãããã«ã¯ãåéåã®åŒã®'''埮å'''ãè¡ãå¿
èŠããããããããã§ã¯è©³ããæ±ããªããå°åºã«ã€ããŠã¯ã[[å€å
žååŠ]]ãåç
§ã
* åé¡äŸ
** åé¡
ååŸr[m]ã®åäžãè§é床<math>\omega</math>ã§éåããç©äœã®å é床ã®å€§ãããèšç®ããã
** 解ç
:<math>
\overrightarrow a = -\omega^2 \overrightarrow r
</math>
ã«æ³šç®ãããšãããå³èŸºã«ã€ããŠåéåãããŠããç©äœã®åº§æšãåžžã«
:<math>
\overrightarrow r ^2 = r^2
</math>
ãæºããããšã«æ³šç®ãããšã
:<math>
|\overrightarrow a| = \sqrt {\overrightarrow a^2}
</math>
:<math>
= \sqrt {r^2 \omega^4} = r \omega^2
</math>
ãšãªãã
** åé¡
50Hzã§åéåããŠããç©äœã®åéåã®åšæãèšç®ããã
** 解ç
:<math>
T = \frac 1 f
</math>
ãçšãããšã
:<math>
T [\textrm s] = \frac 1 {50}[\textrm s]
</math>
:<math>
= 0.020 [\textrm s]
</math>
ãšãªãã
===åéåã®æ¹çšåŒ===
以äžããïŒåéåã®å é床ã®æåã¯
:åå¿æåïŒ<math>a_\mathrm{C}=r{\omega}^2=\frac{v^2}{r},</math>
:æ¥ç·æåïŒ<math>a_\mathrm{T}=\frac{dv}{dt}</math>.
ãã£ãŠïŒåéåããç©äœã®è³ªéã<math>m</math>ïŒåå¿æ¹åã«åãåïŒããªãã¡'''åå¿å'''({{Lang-en-short|centripetal force}})ã<math>F_\mathrm{C}</math>ïŒæ¥ç·æ¹åã«åãåã<math>F_\mathrm{T}</math>ãšãããšéåæ¹çšåŒã¯
:<math>mr{\omega}^2=F_\mathrm{C}\Longleftrightarrow m\frac{v^2}{r}=F_\mathrm{C},</math> (4.1)
:<math>m\frac{dv}{dt}=F_\mathrm{T}</math>. (4.2)
* â» å·çäžïŒèªè
ã«ååããé¡ãããŸããïŒ
[[w:åå¿å]]ã[[w:é å¿å]]ïŒcentrifugal forceïŒ
== åæ¯å ==
åéåãšé¢ä¿ã®æ·±ãç©äœã®éåãšããŠãåæ¯åïŒ{{Lang-en-short|simple harmonic oscillation}}ïŒãããããããåæ¯åã¯ããããæ¯åçŸè±¡ã®åºæ¬ã«ãªã£ãŠãããå¿çšç¯å²ãåºãéåã§ãããåéåãšåæ§ãåæ¯åãäžè§é¢æ°ãçšããŠéåãèšè¿°ãããããŸããåšæãäœçžãããç¹ãåéåãšåãã§ããããŸããåæ¯åã¯æ³¢åã«é¢ããçŸè±¡ãšãé¢ä¿ãæ·±ããäœçžãæ¯å¹
ãªã©ã®éãå
±æããŠããã
ããããã¯ãåæ¯åãããç©äœã®æ§è³ªããã詳ããèŠãŠè¡ãã
åæ¯åã¯æ§ã
ãªæ
åµã§ãããããããåçŽãªäŸãšããŠã¯'''ããã¯ã®æ³å'''ã§æ¯é
ãããã°ãã«æ¥ç¶ãããç©äœã®éåããããããã§ã¯ãã°ãå®æ°<math>k</math>ã®ã°ãã«è³ªé<math>m</math>ã®ç©äœãæ¥ç¶ãããšãããã°ãã®èªç¶é·ã®äœçœ®ãåç¹ãšããŠæå»<math>t</math>ã«ãããåç¹ããã®ç©äœã®äœçœ®ã<math>x(t)</math>ãšããå Žåããã®ç©äœã«é¢ããéåæ¹çšåŒã¯
:<math>m\frac{d^2x(t)}{dt^2}= - kx(t)</math>
ã§äžããããããã®æ¹çšåŒã®äž¡èŸºã<math>m</math>ã§å²ããšãå é床ã¯<math>\frac{d^2x(t)}{dt^2}= -\frac{k}{m}x(t)</math>ã§äžããããããšãåããããã®ããã«ãå é床ãšç©äœã®åº§æšãè² ã®æ¯äŸä¿æ°ãæã£ãŠæ¯äŸé¢ä¿ã«ããåŒããåæ¯åã®éåæ¹çšåŒã§ããããã®å Žåãåæ¯åã®æ¯åäžå¿ã<math>x = x_\mathrm{C}</math>(åæ¯åã§ã¯æ¯åäžå¿ã¯å®æ°)ïŒæå»<math>t</math>ã«ãããç©äœã®éåãäœçœ®<math>x(t)</math>ïŒé床<math>v(t)</math>ïŒå é床<math>a(t)</math>ã§è¡šããš
:<math>x(t)= x_\mathrm{C}+ A \sin (\omega t +\delta),</math> (4.3)
:<math>v(t)= \frac{dx(t)}{dt} = A\omega\cos (\omega t +\delta),</math> (4.4)
:<math>\begin{align}a(t)=\frac{d^2 x(t)}{dt^2}& = -A\omega ^2 \sin (\omega t +\delta)\\ & =-\omega^2(x(t)- x_\mathrm{C})\end{align}</math> (4.5)
ãšãªãã<math>\omega</math>ã¯è§æ¯åæ°ïŒ<math>\delta</math>ã¯åæäœçžã§ããã
*çºå±: åæ¯åã®éåæ¹çšåŒ
ããã§ãåæ¯åã®éåæ¹çšåŒãšãåæ¯åã®éåã®åŒãäžããããå®éã«ã¯åæ¯åã®éåã®åŒã¯éåæ¹çšåŒããå°åºã§ãããããã«ã€ããŠã¯[[w:埮åæ¹çšåŒ]]ãæ±ãå¿
èŠãããã®ã§è©³ããå°åºã«ã€ããŠã¯ã[[å€å
žååŠ]]ãåç
§ã
<math>\sin</math>é¢æ°ã¯é¢æ°ã®å€ã®å¢å ã«äŒŽã£ãŠåšæçãªæ¯åãè¡ãªãé¢æ°ãªã®ã§ãç©äœã¯ã<math>x=0</math>ã®ãŸããã§åšæçãªæ¯åãããããšãåããããã ããäžã®åŒã®äžã§Aã¯[[w:æ¯å¹
]]ãšåŒã°ããç©äœã®æ¯åã®ç¯å²ãè¡šãéã§ããã
ãã ãããã®å Žåã«ãããŠã¯ãããã®éã¯ç©äœã®åéåã§ã¯ãªããç©äœã®æ¯åã«ã€ããŠã®éã§ãããããããåäœæéåœããã«äœ[rad]ã ãäœçžãé²ããã®éãšæ¯åã®åšæã®äžã§ãã©ã®äœçœ®ã«ç©äœãããããè¡šãéã«å¯Ÿå¿ããŠããããŸããåšæãšæ¯åæ°ãåéåã®å Žåãšåãå®çŸ©ã§äžããããã
:<math>T = \frac {2\pi}\omega</math>
:<math>f =\frac \omega {2\pi}</math>
ãŸãããã®å Žåã«ã€ããŠã¯éåæ¹çšåŒããè§æ¯åæ°ã決ãŸã
:<math>m\frac{d^2 x(t)}{dt^2}=-kx(t)</math>
:<math>\begin{align}\therefore\frac{d^2 x(t)}{dt^2}& =-\frac{k}{m}x(t)\\ & =-\omega^2(x(t)- 0)\end{align}</math>
:<math>\therefore\omega^2=\frac{k}{m}\quad\therefore\omega = \sqrt{\frac{k}{m}}\ (\because\omega >0)</math>
ã§äžããããã
(4.3)ã
:<math>x(t)= x_\mathrm{C}+ A\sin\omega t\cos\delta +A\cos\omega t\sin\delta</math>
ãšæžçŽãïŒ<math>A\cos\delta=a,\ A\sin\delta=b</math>ãšãããš
:<math>x(t)= x_\mathrm{C}+ a\sin\omega t +b\cos\omega t,</math> (4.3a)
:<math>v(t)= \dot x(t)=\omega(a\cos\omega t -b\sin\omega t),</math> (4.4a)
:<math>a(t)= \ddot x(t)=-\omega^2(a\sin\omega t +b\cos\omega t)</math> (4.5a)
ãšãªãïŒæ¯å¹
ã¯
:<math>A=\sqrt{a^2+b^2}</math>. (4.6)
* åé¡äŸ
** åé¡
質émãæã€ããç©äœã«ã€ããŠãã°ãå®æ°<math>k _1</math>ã®ã°ããšã°ãå®æ°<math>k _2</math>ã®ã°ãã«
ã€ãªãããå Žåã§ã¯ã ã©ã¡ãã®å Žåã®æ¹ãç©äœã®è§é床ã倧ãããªããã
ãã ãã<math>k _1>k _2</math>ãæãç«ã€ãšããããŸããåšæãšæ¯åæ°ã«ã€ããŠã¯ã©ããªããã
** 解ç
ãã®å Žåã«ã¯ãã®åæ¯åã®è§æ¯åæ°ã¯ã
:<math>
\omega = \sqrt {\frac k m}
</math>
ã§äžããããããã®éã¯ã°ãå®æ°kã倧ããã»ã©å€§ããã®ã§ãè§æ¯åæ°ã¯
ã°ãå®æ°<math>k _1</math>ãæã€ã°ãã®è§æ¯åæ°ã®æ¹ãã°ãå®æ°<math>k _2</math>ãæã€ã°ãã®è§æ¯åæ°
ãã倧ãããªãããŸããåæ¯åã®æ¯åæ°ã¯åæ¯åã®è§æ¯åæ°ã«æ¯äŸããã®ã§ã
æ¯åæ°ã«ã€ããŠãã ã°ãå®æ°<math>k _1</math>ãæã€ã°ãã®æ¯åæ°ã®æ¹ãã°ãå®æ°<math>k _2</math>ã
æã€ã°ãã®æ¯åæ°ãã倧ãããªããäžæ¹ããã®å Žåã®åšæã«ã€ããŠã¯ã
:<math>
T = \frac {2\pi} \omega = 2\pi \sqrt {\frac m k}
</math>
ãæãç«ã€ãããã°ãå®æ°kãå°ããã»ã©å€§ãããªãããã£ãŠãåšæã«ã€ããŠã¯
ã°ãå®æ°<math>k _2</math>ãæã€ã°ãã®åšæã®æ¹ãã°ãå®æ°<math>k _1</math>ãæã€ã°ãã®åšæ
ãã倧ãããªãã
** åé¡
éåã®ããäžã«é·ãl[m]ã®ã²ãã§ã€ããããç©äœã«ãã£ãŠäœãããç©äœã®
éçŽäžåãã«åçŽãªæ¹åã®éåãåæ¯åãšãªãããšãæ±ããã
ãã ããæ¯ãåã®åãç¯å²ã¯å°ãããã®ãšããã
ãã®ããã«åæ¯åãããæ¯ãåã åæ¯ãåïŒãããµãããsimple pendlumïŒ ãšåŒã¶ããšãããã
** 解ç
ã²ã ãåºå®ãããŠããäœçœ®ããéçŽã«äžãããçŽç·ãšãç©äœãã€ãªãããŠãã ã²ã ããªãè§åºŠã <math>\theta</math> ãšããããã®å Žåãå³åœ¢çã«èãããšãã®å Žåã®æ°Žå¹³æ¹åã®éåæ¹çšåŒã¯
:<math>m a _x =- mg \sin \theta </math>
ãšãªããããã§ã<math>\theta</math> ãå°ããå Žåã
:<math>\theta \sim \frac x l</math>
ãšãªãããšã«æ³šæãããšãéåæ¹çšåŒã¯
:<math>a _x = -g \frac x l</math>
:<math>a _x = - \frac g l x</math>
ãšãªãå
ã»ã©ã®ã°ãã«ã€ãªãããç©äœã®éåæ¹çšåŒãšçãããªãã
ãã£ãŠããã®ç©äœã®éåãåæ¯åã§èšè¿°ãããããšãåãã£ããããã«ã
å
ã»ã©ã®è§æ¯åæ°ãšæ¯èŒãããšããã®å Žåã®è§æ¯åæ°<math>\omega</math>ã¯
:<math>\omega = \sqrt{\frac g l}</math>
ãšãªãããšãåããã
ãããã®çµæãã[[å°åŠæ ¡çç§]]ã®çµæã§ãã
:åæ¯ãåã«ã€ããŠ
::ç©äœã®éãã¯æ¯ãåã®åšæãšé¢ä¿ããªãã
::æ¯ãåã®ã²ãã®é·ããé·ããªãã«ã€ããŠãæ¯ãåã®åšæã¯é·ããªãã
ã®å®éšäºå®ãéåæ¹çšåŒã®çµæãšäžèŽããããšã確ãããããã
= äžæåŒå =
ãã®ç« ã§ã¯ãäžæåŒåã«ããéåãæ±ããäžæåŒåã¯å
šãŠã®ç©äœã®éã«ååšããŠãããããã®åãåªä»ããéåãšããŠæåãªãã®ã¯å€ªéœã®åããå転ããå°çã®éåããå°çèªèº«ã®åããå転ããæã®éåã§ãããå®éã«ã¯ãã®ãããªäœãã®åããå転ããæ§é ã¯å®å®å
šäœã«åºãèŠãããã
äŸãã°ã空ã«èŠãããæã¯[[w:ææ]]ãšåŒã°ãããããããã®æã®åãã«ã倪éœã«å¯Ÿããå°çãšåãããã«ãææãåããåã£ãŠãããšèããããå®éã«ãã®ãããªææã確èªãããææãããã([[w:ç³»å€ææ]]åç
§ã)
ãã®ããã«å®å®ã®äžã§äžæåŒåã«ããå転éåã¯åºã芳枬ããããããã§ã¯ãã®ãããªéåã¯ç©äœéã«åãã©ã®ãããªåã«ãã£ãŠèšè¿°ãããããèŠãŠããã
* çºå±: äžæåŒåçºèŠã®æŽå²
æŽå²çã«ã¯ãéã«ãã®ãããªç©äœã®éã®éåã説æãããããªåãèããããšã§
ç©äœéã«åãåãçºèŠããããæŽå²ã«ã€ããŠè©³ããã¯[[w:ãã¥ãŒãã³]]ãªã©ãåç
§ã
== äžæåŒåã®æ³å ==
ãŸãã¯ãç©äœéã«åãäžæåŒåïŒglavitational constantïŒã®æ³åãè¿°ã¹ããçš®ã
ã®èŠ³æž¬ã®çµæã«ãããšã質é<math>m_1</math>ãæã€ç©äœãšè³ªé<math>m_2</math>ãæã€ç©äœã®éã«ã¯
:<math>F = -G \frac{m _1 m _2}{r^2}</math>
ã§è¡šããããåãåããããã§Gã¯ç©äœã«ãããªãå®æ°ã§ã'''äžæåŒåå®æ°'''ãšããã
å€ã¯<math> G = 6.67 \times 10^{-11} [ {\mathrm{N}\cdot\mathrm{m}^2/\mathrm{kg}^2}] </math> ã§ããã
äžæåŒåã®æ³å
:<math>F = -G \frac{m _1 m _2}{r^2}</math>
::F: äžæåŒå
::G: äžæåŒåå®æ°
::r: ç©äœéã®è·é¢
äžæåŒåã¯ç©äœéã®è·é¢ã®2ä¹ã«éæ¯äŸããåã§ããã
ç©äœã®å°ãªããšãçæ¹ãææã®ããã«å·šå€§ãªå Žåãç©äœéã®è·é¢rã¯ãéå¿éã®è·é¢ã§ããã
å°çã®äžæåŒåãèãããå°çã®è³ªéãMãå°çã®ååŸãRã枬å®ããç©äœã®è³ªéãmãšããå ŽåãéåFã¯
:<math>F = -G \frac{M m}{R^2}</math>
ãšãªãã
ãããå°è¡šè¿ãã§ã¯å€§ããã mg ãšçããã®ã§ã
:<math>G \frac{M m}{R^2} = mg </math>
å€åœ¢ããŠ
:<math>G M = gR^2 </math>
ãšãªããèšç®åé¡ã®ããããã®å€åœ¢ãçšããããå Žåãããã
;å°çã®èªè»¢ã®åœ±é¿
å°çã¯èªè»¢ãããŠãããéåã®èšç®ã§ã¯ãå³å¯ã«ã¯èªè»¢ã«ããé å¿åãèããå¿
èŠãããããããããèªè»¢ã®é å¿åã®å€§ããã¯ãäžæåŒåã®<math>\frac{1}{300}</math>åãŠãã©ãããªãã®ã§ãéåžžã¯èªè»¢ã«ããé å¿åãç¡èŠããå Žåãå€ãã
ãªããå°çã®èªè»¢ã®é å¿åã¯ãèµ€éäžã§ãã£ãšã倧ãããªãã
== éæ¢è¡æ ==
人工è¡æããå°çã®èªè»¢ãšåãåšæã§ãèªè»¢ãšåãåãã«çéåéåãããã°ããã®äººå·¥è¡æã¯å°äžããèŠãŠãã€ãã«å°é¢ã®äžç©ºã«ããã®ã§ãå°äžã®èŠ³æž¬è
ããã¯éæ¢ããŠèŠããããã®ãããªäººå·¥è¡æã®ããšã'''éæ¢è¡æ'''ãšããã
** åé¡
質émã®ç©äœã質éMã®å€§ããªç©äœã®åãããäžæåŒåã®åãåå¿åãšããŠãååŸrã®åéåãããŠããããã®å Žåã®åéåã®è§é床ãæ±ããã
** 解ç
ååŸrãè§é床<math>\omega</math>ã®åéåãããå Žåã®ç©äœã®åå¿å ã¯
:<math>- mr \omega ^2</math>
ã§ãããäžæ¹ã質émãšè³ªéMã®ç©äœã®éã®è·é¢ãrã§ããå Žåã2ã€ã®ç©äœéã«åãéåã¯ãéåã®å€æ°ãfãšãããšã
:<math>f = - G\frac{mM}{r^2}</math>
ã§äžããããããã£ãŠããããã®åãçãããªãå Žåã«ã質émã®ç©äœã¯è³ªéMã®ç©äœã®ãŸãããåéåã§å転ïŒå
¬è»¢ïŒããããšãã§ããããã£ãŠã<math>\omega</math>ãæ±ããåŒã¯ã
:<math>- mr \omega^2 = - G\frac{mM}{r^2}</math>
:<math>\omega = \sqrt { G\frac M{r^3} }</math>
ãšãªãã
== äžæåŒåã«ããäœçœ®ãšãã«ã®ãŒ ==
å°çè¡šé¢ã§ã®éåã«ããäœçœ®ãšãã«ã®ãŒãèããããã®ãšåæ§ã«ãäžæåŒåã«ããäœçœ®ãšãã«ã®ãŒãèããããšãã§ããã
:â» èªè
ãç©åãç¥ã£ãŠãããšãåæã«èª¬æãããæ°åŠ3ã®ç©åããŸãªãã ã»ããç解ã¯æ©ããé²åŠæ ¡ãªã©ã§ã¯ãç©åã§äœçœ®ãšãã«ã®ãŒãæ±ããã®ãå®æ
ã§ããã
äžæåŒåã«ããäœçœ®ãšãã«ã®ãŒãæ±ããã«ã¯ãäžæåŒåãç©åããã°ããã
質éMã®ç©äœããrã®è·é¢ã«è³ªémã®ç©äœãååšãããšããããã ããMã¯mããã¯ããã«
倧ãããšãããç¡éé ç¹ãåºæºã«ãããšïŒã€ãŸãç¡éé ã§ã¯äœçœ®ãšãã«ã®ãŒããŒãïŒããã®å Žåã質émã®ç©äœã®äœçœ®ãšãã«ã®ãŒã¯
:<math>U = -G \frac {mM} r</math>
ã§äžããããã
笊å·ã«ãã€ãã¹ãã€ãããšã®ç©ççãªè§£éã¯ãéåãã€ããã ãç©äœã«è¿ã¥ãã»ã©ããã®ç©äœã®ã€ããã ãéååãè±åºããã«ã¯ããšãã«ã®ãŒãè¿œå çã«å¿
èŠã«ãªãããã§ãããšè§£éã§ããã
ç¡éé ã§ã¯ rïŒïŒâ ãšããã°ãããçµæã UïŒ0 ã«ãªãã
ãªããäžæåŒåã¯ä¿ååã§ããã®ã§ãäœçœ®ãšãã«ã®ãŒã¯ãç¡éé ç¹ããã®çµè·¯ã«ããããçŸåšã®äœçœ®ã ãã§æ±ºãŸãã
* å³åç
§
ã®ããã«äžããããããŸãããã®ã°ã©ãã¯çŽèŠ³çãªæå³ãæã£ãŠããã
å®ã¯ããã®ã°ã©ãã®åŸãã¯ã°ã©ããè¡šããäœçœ®ãšãã«ã®ãŒãæã€ç¹ã«ç©äœã眮ããå Žåã
ãã®ç©äœãåãåããæ¹åãšãã®å€§ãããè¡šãããŠãããããã§ã¯ã
äœçœ®ãšãã«ã®ãŒã®åŸããåžžã«r=0ã«èœã¡èŸŒãæ¹åã«çããŠããããç©äœMããè·é¢r
(rã¯ä»»æã®å®æ°ã)ã®ç¹ã«éæ¢ããŠããç©äœã¯å¿
ãMã®æ¹åã«åžã蟌ãŸããŠè¡ãããšã
è¡šãããŠããã(詳ããã¯[[å€å
žååŠ]]åç
§ã)
* åé¡äŸ
** åé¡
ããææäžã«ããç©äœãå®å®ã®ç¡éé ãŸã§å°éãããããã«å®å®è¹ã«ææäžã§
äžããªããŠã¯ãããªãé床ã¯ã©ã®ããã«è¡šãããããããã ããèšç®ã«ã€ããŠã¯
æåã«å®å®è¹ãåºçºããææ以å€ã®å€©äœããã®åœ±é¿ã¯ç¡èŠãããšããã
ãŸããææã®ååŸã¯Rã ææã®è³ªéã¯Mãšããã
** 解ç
ææã®åŒåã«ããäœçœ®ãšãã«ã®ãŒã¯ææè¡šé¢ã§
:<math>- G\frac {mM} R</math>
ã§ãããç¡éåç¹ã§ã¯0ã§ããããã ããmã¯å®å®è¹ã®è³ªéãšããã
äžæ¹ãå®å®è¹ãç¡éåç¹ã«éããã«ã¯ãå®å®è¹ã®é床ãç¡éåç¹ã§ã¡ããã©0ã«
çãããªãã°ãããããã§ãææäžã§ã®å®å®è¹ã®é床ãvãšãããšã
ãšãã«ã®ãŒä¿ååããã
:<math>\frac 1 2 m v^2 - G\frac {mM} R = 0 - 0</math>
ãšãªãããã£ãŠãã®åŒããvãæ±ããã°ãããçã¯ã
:<math>v = \sqrt {2G\frac {M} R }</math> (ç)
äžèšã®èšç®ããåããããã«ãäžè¬ã«ãäžæåŒåã ããåããŠéåããç©äœã®ååŠçãšãã«ã®ãŒã¯ã
:<math>E = \frac 1 2 m v^2 - G\frac {mM} R = </math> ã'''äžå®'''
ã§ããã
== 人工è¡æã®è»é ==
=== å®å®é床 ===
[[ç»å:Newton Cannon.svg|thumb|300px|Cã第äžå®å®é床ã®è»éã]]
ä»®ã«é«ãå±±ããç©äœãæ°Žå¹³ã«çºå°ãããšãïŒç©ºæ°æµæã¯ç¡èŠããïŒãå°çã®ãŸãããåãç¶ããããã«å¿
èŠãªæå°ã®åé床ã®ããšã'''第äžå®å®é床'''ãšãããïŒâ» ååã¯æèšããªããŠãããèŠããã¹ãã¯ãèšç®æ¹æ³ã§ãããïŒ ç¬¬äžå®å®é床ã¯ãèŠããã«ãé å¿åãšåå¿åãã€ãããããã«å¿
èŠãªåé床ã§ããã
第äžå®å®é床ã¯ãç§éã§ã¯çŽ7.91km/sã§ããã
;第äžå®å®é床ã®èšç®
:<math> m\frac{ {v_1}^2 }{r} = G \frac{mM}{R^2}</math>
v<sub>1</sub>ã«ã€ããŠè§§ãã
:<math> v_1 = \sqrt {gR} </math>
ãªããããã R = 6400 à 10<sup>3</sup> m ã§ããã g = 9.8 m/s<sup>2</sup> ã§ããã
:<math> v_1 = \sqrt {9.8 \times 6400 \times 10^3 } = 7.9 \times 10^3 \textrm {m/s} = 7.9 \textrm {km/s} </math> ïŒçïŒ
----
ããã«åé床ã倧ãããªããšãç©äœã¯æ¥åè»éã«ãªãã
åé床ãçŽ11.2km/sã«ãªããšãè»éã¯æŸç©ç·ã«ãªããç©äœã¯ç¡éã®åœŒæ¹ã«é£ãã§ããã
ãã®çŽ11.2km/sã®ããšã'''第äºå®å®é床'''ãšãããããã¯ãç¡éé ã®ç¹ã§ãé床ã0ãè¶
ããå€ã«ãªãããã«å¿
èŠãªåé床ã§ããã
ãªã®ã§ãèšç®ã§ç¬¬äºå®å®é床ãæ±ããã«ã¯ãšãã«ã®ãŒä¿ååãèšç®ã«ã¯äœ¿ãã
;第äºå®å®é床ã®èšç®
:<math>\frac 1 2 m {v_2}^2 - G\frac {mM} R = 0 - 0</math>
ã®åŒããvãæ±ãã
:<math>v_2 = \sqrt {\frac {2GM} R }</math>
ã«ããã« <math> GM = gR^2 </math> ã代å
¥ããŠã
:<math> v_2 = \sqrt { 2gR }</math>
ããã«é¢ä¿ããå®æ°ã代å
¥ããã°ããã
ãªããããã R = 6400 à 10<sup>3</sup> m ã§ããã g = 9.8 m/s<sup>2</sup> ã§ããã
:<math> v_2 = \sqrt { 2 \times 9.8 \times 6400 \times 10^3 } = 1.1 \times 10^4 \textrm {m/s}</math> ïŒçïŒ
----
åé床 11.2km/s以äžã§ã¯ãè»éã¯åæ²ç·ã«ãªããç©äœã¯ç¡éã®åœŒæ¹ã«é£ãã§ããã
{{ã³ã©ã |ç¡ééç¶æ
|
ïŒäžè¿°ã®åå
ããããããããã«ãïŒå°çã®åšå²ããŸãã£ãŠãã人工è¡æã®ãªãã§ãç©ã®ééããªããªãæµ®ãã¹ãçç±ã¯ãéåãšé å¿åãã€ããã£ãŠããããã§ããããã®ãããªç¶æ
ã®ããšã'''ç¡ééç¶æ
'''ãšããã
äžéã§ã¯åœéå®å®ã¹ããŒã·ã§ã³ã®ãªãã§ç©ãæµ®ãã¶æ åãªã©ãæåã§ãããããããç¡ééç¶æ
ã§ããã
ãã£ããŠããïŒå°è¡šããé¢ããŠïŒéåã匱ãŸã£ããã人工è¡æã®äžãç¡éåã«ãªã£ããã®ã§ã¯ãªãïŒ äžéã«ã¯ãåéãããŠãã人ãå€ãããšãã«å
ç«¥ããã®ç§åŠçªçµãªã©ã§ã¯ã説æãäžååã«ãªããã¡ã§ãèŠèŽè
ã®åäŸã¯ããããåéããããŠããå Žåãå€ããèªè
ã¯ãé«æ ¡çã«ãªã£ãããç解ããªããå¿
èŠããããïŒ
ãããããããåå¿åãšããŠã®éåãç¡ãã®ãªããè¡æã®è»éã¯åè»éã§ã¯ãªãããŸã£ããã«çŽç·è»éã«ãªã£ãŠããŸããå®å®ã®ããªãã«é£ãã§ãã£ãŠãã£ãŠããŸãã ããã
ãã ããæ
£ç¿çã«ã人工è¡æã®ãªãã§ééããªããªãç¶æ
ïŒç¡ééç¶æ
ïŒã®ããšãïŒèª€è§£ã®ããããããåŒã³æ¹ã ãïŒãç¡éåç¶æ
ããšããå Žåãå€ããå³å¯ã«ã¯ãç¡ééç¶æ
ãã§ããã
}}
;ïŒâ» åèïŒ ç¬¬äžå®å®é床
â» æ€å®æç§æžã§ã¯ãè泚ãªã©ã«æžããŠãã£ããããã
å°çããå°åºããŠã倪éœç³»ã®å€ã«åºãããã«å¿
èŠãªæå°ã®åé床ã®ããšã'''第äžå®å®é床'''ïŒçŽ 16.7 km/sïŒ ã§ããã
== ã±ãã©ãŒã®æ³å ==
ã®ãªã·ã£æ代ããäžäžãŸã§ä¿¡ããããŠãã[[w:倩å説|倩å説]]({{Lang-en-short|geocentric theory}})ã«å¯ŸãïŒ16äžçŽåã°ã«[[w:ãã³ã©ãŠã¹ã»ã³ãã«ãã¯ã¹|ã³ãã«ãã¯ã¹]]ã¯å
šãŠã®[[w:ææ|ææ]]({{Lang-en-short|planet}})ã倪éœãäžå¿ãšããåéåãããŠãã[[w:å°å説|å°å説]]ãæå±ããããã®åŸ[[w:ãã£ã³ã»ãã©ãŒãš|ãã£ã³ã»ãã©ãŒãš]]ã¯é·å¹Žã«ãããææã®èŠ³æž¬ãè¡ãïŒãã®èŠ³æž¬çµæãåŒç¶ãã [[w:ãšããã¹ã»ã±ãã©ãŒ|ã±ãã©ãŒ]]ã¯ãããã®çµæãããšã«èšç®ãè¡ãïŒææã®éè¡ã«é¢ããæ³åïŒ'''ã±ãã©ãŒã®æ³å'''({{Lang-en-short|Kepler's laws}})ãçºèŠããããªãïŒæç§æžã¯å€ªéœãšææã®é¢ä¿ã§è«ããŠãããïŒä»ã«ãææãšè¡æ(èªç¶è¡æïŒäººå·¥è¡æ)ã§ãæãç«ã€ã
===ã±ãã©ãŒã®ç¬¬ïŒæ³å===
ææ(è¡æ)ã¯å€ªéœ(ææ)ãïŒã€ã®çŠç¹ãšããæ¥åéåããã(æ¥åè»éã®æ³å)ã
===ã±ãã©ãŒã®ç¬¬ïŒæ³å===
[[File:Elliptical motion of man-made satellight.png|thumb|right|640px|å³ äººå·¥è¡æã®æ¥åéå]]
ææ(è¡æ)ãšå€ªéœ(ææ)ãçµã¶ååŸãåäœæéã«æãé¢ç©('''é¢ç©é床''')ã¯äžå®ã§ãã(é¢ç©é床äžå®)ã
* 蚌æ
:å°çã®åšããéåãã人工è¡æã«ã€ããŠèãããå³å³ã®ããã«å°çã®äžå¿ãåç¹ãšããŠ<math>xy</math>å¹³é¢ããšãïŒå°çã®è³ªéã<math>M</math>ïŒäººå·¥è¡æã®è³ªéã<math>m</math>ïŒäžæåŒåå®æ°ã<math>G</math>ïŒæå»<math>t</math>ã«ããã人工è¡æã®äœçœ®ã<math>\overrightarrow r(t)=(x(t),\ y(t))</math>ãšããã人工è¡æã®è§éåéã<math>l</math>ãšãããš
::<math>l=m\left(x(t)\frac{dy(t)}{dt}-y(t)\frac{dx(t)}{dt}\right)</math>. ((3.1)ãåç
§)
:䞡蟺ãæé埮åããŠ
::<math>\begin{align}\frac{dl}{dt} & =m\left(\frac{dx(t)}{dt}\frac{dy(t)}{dt}+x(t)\frac{d^2y(t)}{dt^2}-\frac{dy(t)}{dt}\frac{dx(t)}{dt}-y(t)\frac{d^2x(t)}{dt^2}\right) \\ & =m\left(x(t)\frac{d^2y(t)}{dt^2}-y(t)\frac{d^2x(t)}{dt^2}\right).\cdots\cdots(*)\end{align}</math>
:ããã§ïŒæå»<math>t</math>ã«ããã人工è¡æã®éåæ¹çšåŒã¯
::<math>m\frac{d^2\overrightarrow r(t)}{dt^2}=-G\frac{Mm}{x(t)^2+y(t)^2}\Longleftrightarrow\begin{cases}m\frac{d^2x(t)}{dt^2}=-G\frac{Mm\cdot x(t)}{(x(t)^2+y(t)^2)^\frac{3}{2}} \\ m\frac{d^2y(t)}{dt^2}=-G\frac{Mm\cdot y(t)}{(x(t)^2+y(t)^2)^\frac{3}{2}}\end{cases}</math>
::<math>\therefore \frac{d^2x(t)}{dt^2}=-G\frac{M\cdot x(t)}{(x(t)^2+y(t)^2)^\frac{3}{2}},\ \frac{d^2y(t)}{dt^2}=-G\frac{M\cdot y(t)}{(x(t)^2+y(t)^2)^\frac{3}{2}}</math>.
:ãããã<math>(*)</math>ã«ä»£å
¥ããŠ
::<math>\frac{dl}{dt}=m\left\{x(t)\cdot\left(-G\frac{M\cdot y(t)}{(x(t)^2+y(t)^2)^\frac{3}{2}}\right)-y(t)\cdot\left(-G\frac{M\cdot x(t)}{(x(t)^2+y(t)^2)^\frac{3}{2}}\right)\right\}=0</math>.
:ããã«è§éåé<math>l</math>ã¯äžå®ã§ãã(è§éåéã¯ä¿åãã)ã
:ããã§ïŒæå»<math>t</math>ã«ããã人工è¡æã®é床<math>\frac{d\overrightarrow r(t)}{dt}=\overrightarrow v(t)</math>ãšãïŒå³ã®ããã«äººå·¥è¡æã®äœçœ®ãã¯ãã«<math>\overrightarrow r(t)</math>ãšé床ãã¯ãã«<math>\overrightarrow v(t)</math>ã®ãªãè§ã<math>\theta</math>ïŒäœçœ®ãã¯ãã«<math>\overrightarrow r(t)</math>ãš<math>x</math>軞ãšã®ãªãè§ã<math>\phi</math>ãšããã以äžãã
::<math>\begin{align}\frac{l}{2m}&=\frac{1}{2}\left(x(t)\frac{dy(t)}{dt}-y(t)\frac{dx(t)}{dt}\right) \\ &=\frac{1}{2}(|\overrightarrow r(t)|\cos\phi\cdot |\overrightarrow v(t)|\sin(\theta+\phi)-|\overrightarrow r(t)|\sin\phi\cdot |\overrightarrow v(t)|\cos(\theta+\phi)) \\ & =\frac{1}{2}(|\overrightarrow r(t)||\overrightarrow v(t)|\{\sin\theta(\cos^2\phi+\sin^2\phi)+\cos\phi\cos\theta\sin\phi-\sin\phi\cos\theta\cos\phi\} \\ & =\frac{1}{2}|\overrightarrow r(t)||\overrightarrow v(t)|\sin\theta=\mathrm{const}.\end{align}</math> (<math>\mathrm{const}.</math>ã¯äžå®ã®æå³)
===ã±ãã©ãŒã®ç¬¬ïŒæ³å===
ææ(è¡æ)ã®å
¬è»¢åšæ<math>T</math>ã®ïŒä¹ã¯æ¥åè»éã®é·ååŸ(åé·è»ž)<math>a</math>ã®ïŒä¹ã«æ¯äŸããã
:<math>\frac{T^2}{a^3}=</math>äžå®ïŒ
[[Category:é«çåŠæ ¡æè²|ç©ãµã€ã2ã¡ãããšãããšã]]
[[Category:ç©çåŠ|é«ãµã€ã2ã¡ãããšãããšã]]
[[Category:ç©çåŠæè²|é«ãµã€ã2ã¡ãããšãããšã]]
[[Category:é«çåŠæ ¡çç§ ç©çII|ã¡ãããšãããšã]] | 2005-05-08T07:30:55Z | 2024-03-02T15:54:32Z | [
"ãã³ãã¬ãŒã:ã³ã©ã ",
"ãã³ãã¬ãŒã:See also",
"ãã³ãã¬ãŒã:Lang-en-short"
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E7%89%A9%E7%90%86/%E5%8A%9B%E5%AD%A6 |
1,944 | é«çåŠæ ¡ç©ç/ç©çII/é»æ°ãšç£æ° | ãŸããé«æ ¡ç©çã§ãããèªé»äœã(ããã§ããã)ãšã¯ãéåžžã®ã»ã©ããã¯ãé²æ¯(ãã€ã«)ãªã©é»æ°ãéããªãç©è³ªã®ãã¡é«ãèªé»çã瀺ããã®ã§ãã
éå±ã¯å°äœãªã®ã§èªé»äœã§ã¯ãããŸããã
ã§ã¯ãèªé»äœã®ç©çã«ã€ããŠã説æããŸãã
ã³ã³ãã³ãµãŒã«èªé»äœãå
¥ãããšãèªé»äœãèªé»å極ãèµ·ãããããã³ã³ãã³ãµã®ãã©ã¹æ¥µæ¿ã§çºçããé»æ°åç·ã®ããã€ããæã¡æ¶ãããŸãã
ãã®çµæãèªé»äœã®å
¥ã£ãã³ã³ãã³ãµãŒã®æ¥µæ¿éã®é»å Žã¯ã極æ¿ã®é»è·å¯åºŠã§çºçããé»è·ãç空äžã§ã€ããé»å Žããã匱ããªããŸãã
ãã®çµæãéé»å®¹éãå€ããŸãã
ããŠãç空äžã®éé»å®¹éã®å
¬åŒã¯ã
ã§ããã
èªé»äœã®ããå Žåã®éé»å®¹éã¯ã
ãšãªããŸãã
ããã§ã ε {\displaystyle \varepsilon } ãèªé»ç(ããã§ããã€)ãšãããŸãã ε 0 {\displaystyle \varepsilon _{0}} ããç空äžã®èªé»çãšãããŸãã
ããã§ãæ¯
ããæ¯èªé»ç(ã² ããã§ããã€)ãšãããŸãã
ã€ãŸãã ε r {\displaystyle \varepsilon _{r}} ã¯æ¯èªé»çã§ãã ãã£ãœãã ε 0 {\displaystyle \varepsilon _{0}} ãã㳠ε {\displaystyle \varepsilon } ã¯ãæ¯èªé»çã§ã¯ãããŸããã
æ¯èªé»ç ε r {\displaystyle \varepsilon _{r}} ããã¡ããã°ãéé»å®¹é C ã®åŒã¯ã
ãšæžããŸãã
U=2â1CV2
ç£ç³ã®ãŸããã«ã¯ç©äœãåããåã®ãããã®ãçããŠããŸãã ãããç£å Ž(ãã°)ãšåŒã¶ãç£ç(ããã)ãšãããã
é»æµãæµããŠãããšãã«ãããã®ãŸããã«ã¯ãå³ããã®æ³å(right-handed screw rule)ã«åŸãåãã«ç£çãçããŸãã é»æµI[A]ãçŽç·çã«æµããŠãããšããç£çã®å€§ãã㯠B = ÎŒ 0 2 Ï a I {\displaystyle B={\frac {\mu _{0}}{2\pi a}}I} ã§ããããšãç¥ãããŠããŸãã
ããã§ãaã¯ç£æå¯åºŠã枬ãç¹ãšãé»ç·ã®è·é¢ã
ãŸãã ÎŒ 0 {\displaystyle \mu _{0}} ã¯ç空ã®éç£ç(ãšãããã€ãpermeability)ãè¡šããå€ã¯ 4 Ï Ã 10 â 7 {\displaystyle 4\pi \times 10^{-7}} [H/m] ã§ãã
ç£å Žã䌎ãç©äœãéåãããšããã®ãŸããã«ã¯é»å Žãçããããšãé»ç£èªå°(ã§ããããã©ããelectromagnetic induction)ãšãããŸãã ä»®ã«ããœã¬ãã€ã(solenoidãã³ã€ã«ã®ããš)ã®è¿ãã§ãããè¡ãªã£ããšãããšãçããé»å Žã«ãã£ãŠãœã¬ãã€ãã®äžã«ã¯é»æµãæµããŸãã çããé»å Žã®å€§ããã¯ã E â = 1 2 Ï a d B â d t {\displaystyle {\vec {E}}={\frac {1}{2\pi a}}{\frac {d{\vec {B}}}{dt}}} ãšãªããŸãã(ååŸaã®å圢ã®ã³ã€ã«ã®å Žåã) Eã®åäœã¯[V/m]ã§ãããBã®åäœã¯[T]ã§ãã
ç£å Žã®åãã«ãã£ãŠé»å ŽãåŒãèµ·ããããããšãé»ç£èªå°ã®ã»ã¯ã·ã§ã³ã§èŠãã
ãŸããå®éã«ã¯é»å Žã®å€åã«ãã£ãŠç£å ŽãåŒãèµ·ããããããšãå®éšã«ãã£ãŠç¥ãããŠããŸãã ããã«ãã£ãŠäœããªã空éäžãé»å Žãšç£å ŽãäŒæããŠããããšãäºæ³ãããŸãã
(:é»ç£æ³¢ã®äŒæã®schematicãªçµµ)
ãŸããç©çå®éšå®¶ãã«ãã¯æŸé»å®éšã«ãããåä¿¡æ©ãåè·¯äžã«ã®ã£ããã®ããåè·¯ãšããŠãéä¿¡åŽã®æŸé»ã«ããé»å Žãé éçã«é¢ããäœçœ®ã«ããåä¿¡åŽã®åè·¯ã«äŒããããšã確èªããã
ãã®å®éšã®éããã«ãã¯åä¿¡åè·¯ã®åãããããããšå€ããŠå®éšããããšã«ãããéä¿¡æ©ã®åãã«å¯ŸããŠã®åä¿¡æ©ã®åãã«ãã£ãŠé»å Žã®äŒããæ¹ãç°ãªãããšãããé»å Žã®é éäœçšã«åå
æ§ãããäºãåãã£ãã
é»å Žã®ãã®äœçšã«ã¯åå
æ§ãããã®ã§ãæ³¢ã§ãããšã¿ãªãããšã¯åŠ¥åœã§ãããã
ãã«ãã®å®éšãããå®éšçã«ãããããšãšããŠ
ãå®éšçã«ããããŸãã
ç©çåŠã§ã¯ããã«ãã®å®éšã®ä»¥åãããçè«ç©çåŠè
ã®ãã¯ã¹ãŠã§ã«ã«ããã
é»ç£æ³¢ãšãããé»å Žãšç£å Žã®çžäºäœçšã«ãã£ãŠç空äžãäŒéããäºæž¬ãããŠããã
ãªã®ã§ããã«ãã®å®éšã¯ããã¯ã¹ãŠã§ã«ã®äºæž¬ããé»ç£æ³¢ã ãšã¿ãªãããã çŸä»£ã§ãç©çåŠè
ã¯ãããã¿ãªããŠããŸãã
ãªãããã¯ã¹ãŠã§ã«ãçè«èšç®ã§æ±ããé»ç£æ³¢ã®é床ãæ±ãããšããããã§ã«ç¥ãããŠããå
éã®å€§ãã(ããã 3Ã10 m/s )ã«ç²ŸåºŠããäžèŽããã
ãã®ããšããããå
ã¯é»ç£æ³¢ã®äžçš®ã§ããããšãåãããŸãã
ãã«ãã®å®éšã§ã¯ãå³å¯ã«ã¯å°ãªããšãæŸé»ã®é»å ŽãäŒããããšãã芳枬ã§ããŠãŸãããããããç£å Žããã®å®éšã§äŒãããšèããŠãæ¯éãçããŠç¡ãããå®éã«äººé¡ã«ã¯æ¯éã¯çããŠãªãã®ã§ãä»ã§ããã«ãã®å®éšããã¯ã¹ãŠã§ã«ã®äºæž¬ããé»ç£æ³¢ã®èšŒæã®å®éšãšããŠäŒããããŠããŸãã
ãªããå
ã«ã¯ãåå°ãå±æãåæããã€ã³ã°ã¹ãªããã®åæãªã©ããããã
ãã«ãã®æŸé»å®éšã®ãããªé»ç£æ³¢ã®ç«è±æŸé»ã®å®éšã§ããå
ã®å®éšãšåæ§ã®é
眮ã§ãéå±æ¿ãé
眮ããŠç¢ºèªããããšã§ãé»ç£æ³¢ãåå°ãå±æãåæããã€ã³ã°ã¹ãªããã®åæãªã©ã®çŸè±¡ãèµ·ããããšããå®éšçã«ã確èªãããŠããŸã(â» åèæç® :å®æåºçã®å°éãç©çãã®æ€å®æç§æž)(â» ã€ã³ã°ã®ã¹ãªããã®é»ç£æ³¢å®éšã«é¢ããŠã¯åæ通ã®æç§æžãç©çãã«ãããŸã)ã
ãããã®ããšããããå
ã¯é»ç£æ³¢ã®äžçš®ã§ãããšã¿ãªãã®ã劥åœã§ããããšãåãããŸãã
ç£ç³ã®ãŸããã«ã¯å¥ã®ç£ç³ãåããåã®ããšãšãªããã®ãçããŠããŸãã ãããç£å Ž(ãã°ãmagnetic field)ãããã¯ç£ç(ããã)ãšåŒã¶ã(æ¥æ¬ã®ç©çåŠã§ã¯ç£å ŽãšåŒã¶ããšãå€ãããŸããæ¥æ¬ã®é»æ°å·¥åŠã§ã¯ç£çãšåŒã°ããããšãå€ããææ²»æã®èš³èªã®éã®ãæ¥æ¬åœå
ã®æ¥çããšã®éãã«éãããå°å瀟äŒçãªäºè±¡ã§ãããåŒã³æ¹ã¯ç©çã®æ¬è³ªãšã¯é¢ä¿ãªãã®ã§ãããã§ã¯ãã©ã¡ãã®è¡šçŸãçšãããã¯ãæ¬æžã§ã¯ç¹ã«ãã ãããŸãããè±èªã§ã¯ç©çåŠã»é»æ°å·¥åŠãšãâmagnetic fieldâã§å
±éããŠããŸãã)
éãã³ãã«ããããã±ã«ã«ç£ç³ãè¿ã¥ãããšãç£ç³ã«åžãä»ããããŸãã ãŸããéãã³ãã«ããããã±ã«ã«åŒ·ãç£åãäžãããšãéãã³ãã«ããããã±ã«ãã®ãã®ãç£å Žãåšå²ã«åãŒãããã«ãªããŸãã ãã®ãããªãããšããšã¯ç£å Žãæããªãã£ãç©äœãã匷ãç£å Žãåããããšã«ãã£ãŠç£å ŽãåãŒãããã«ãªãçŸè±¡ãç£å(ãããmagnetization)ãšãããŸãã
ãããã¯é»è·ã®éé»èªå°ãšå¯Ÿå¿ãããŠãç£åã®ããšãç£æ°èªå°(ããããã©ããmagnetic induction)ãšãããã ãããŠãéãã³ãã«ããããã±ã«ã®ããã«ãç£ç³ã«åŒãä»ããããããã«ç£åãããèœåãããç©äœã匷ç£æ§äœ(ãããããããããferromagnet)ãšãããŸãã éãšã³ãã«ããšããã±ã«ã¯åŒ·ç£æ§äœã§ãã
é
ã¯ç£åããªãããé
ã¯ç£ç³ã«åŒãã€ããããªãã®ã§ãé
ã¯åŒ·ç£æ§äœã§ã¯ãããŸããã
éé»èªå°ãå©çšãããéé»é®èœ(ããã§ããããžã)ãšèšãããŸããäžç©ºã®å°äœãã€ãã£ãŠç©è³ªãå²ãããšã§å€éšé»å Žãé®èœããæ¹æ³ããã£ãã®ãšåæ§ã®ãç£æ°ã®é®èœãã匷ç£æ§äœã§ãåºæ¥ãŸããäžç©ºã®åŒ·ç£æ§äœãçšããŠã匷ç£æ§äœã®å
éšã¯ç£å Žãé®èœã§ããŸãããããç£æ°é®èœ(ãããããžããmagnetic shielding)ãšãããŸããç£æ°ã·ãŒã«ããšãããã
åç£æ§äœãåããã¥ãããããããŸããããåã«ããã®ææã«å ããããç£å Žãæã¡æ¶ãæ¹åã«ãç£åãããã ãã®ææã§ãã
ãããããç£åç·ãšããŸãçžäºäœçšããªãç©è³ªãå€ããããšãã°ãã¬ã©ã¹ãæ°Žã«ããŸããç£æ°ãžã®åœ±é¿ã¯ãç空ã®å Žåãšã»ãšãã©å€ãããŸãããã¬ã©ã¹ãæ°Žã®æ¯éç£ç(ã² ãšãããã€) ÎŒ (ãã¥ãŒ)ã¯ãã»ãŒ1ã§ãã
ãªããéã®æ¯éç£çã¯ãç¶æ
ã«ãã£ãŠéç£çã«æ°çŸãæ°åã®éããããããwikipediaæ¥æ¬èªçã§èª¿ã¹ãå Žåã®éã®éç£çã¯çŽ5000ã§ãã
ã§ã¯ãéç£çãã»ãŒ1ã®ç©è³ªã¯ãç£å Žã®æ¹åã¯ãå€éšç£å ŽãåºæºãšããŠãã©ã¡ãåãã ããã? å€éšç£å Žãæã¡æ¶ãæ¹åã«ç£åããŠããã®ã ããã? ãããšããå€éšç£å Žãšåãæ¹åã«ç£åããŠããã®ã ããã?
ãã®éãããããåžžç£æ§(ãããããã)ãšåç£æ§(ã¯ãããã)ã®ã¡ãããã§ãã
ããç©è³ªããå€éšç£å Žã«ã»ãšãã©åå¿ããŸãããããããå°ãã ãå€éšç£å Žãšåãæ¹åã«ãç£åãããŠããçŸè±¡ã®ããšãåžžç£æ§ãšããã§ãããã®ãããªç©è³ªãåžžç£æ§äœãšãããŸããåžžç£æ§äœãããããç©è³ªãšããŠãã¢ã«ãããŠã ã空æ°ãªã©ãããŸãã
ãã£ãœããããç©è³ªããå€éšç£å Žã«ã»ãšãã©åå¿ããŸãããããããå°ãã ãå€éšç£å Žãæã¡æ¶ãæ¹åã«ãç£åãããŠããçŸè±¡ã®ããšãåç£æ§ãšããã§ãããã®ãããªç©è³ªãåç£æ§äœãšãããŸããåç£æ§äœãããããç©è³ªãšããŠãé
ãæ°Žãæ°ŽçŽ ãªã©ãªã©ãããŸãã
å
çŽ ãååã®çš®é¡ã«ãã£ãŠãç£æ§ã®ã¡ãããããçç±ãšããŠãååŠçµåã§ã®é»åè»éã«åå ããããšèããããŠãŸãã
ååŠã®æç§æžã®çºå±äºé
ã«ããsè»éãããpè»éããªã©ã®çè«ããããããã®çè«ã§ããã®çç±ã説æã§ãããšãããŠããŸãããªããçããå
ã«ãããšããdè»éãã®ç¹åŸŽããç£æ§ã®åå ã§ãã(蚌æã¯çç¥ããŸãã)
ããšããšã(ååŠçµåã§é»åæ®»(ã§ãããã)ã«çºçããããšã®ãããŸã)å€ç«é»åã«ã¯ç£æ§ãããããã®ç£æ§ãé»åã2åããã£ãŠ(å€ç«ã§ãªããªã)é»å察ã«ãªãäºã§ãç£æ§ãæã¡æ¶ããã£ãŠãããšèããããŸãããªããå€ç«é»åãããšããšæã£ãŠããç£æ§ã®ããšãã¹ãã³ãšãããŸããããååŠã®çè«ã§ã¯ãã¹ãã³ãäžç¢å°ãâããšäžç¢å°ãâãã®2çš®é¡ã§ããããäºãå€ãã®ã§ããããã®çç±ã¯ãããšããã©ãã°ãããããç£ç³ã®åãã2çš®é¡(ããšãã°N極ãšS極ãšãã2çš®é¡ã®æ¥µããããŸã)ã ããã§ãã
é»åæ®»ãšã¯ãååŠIã®å§ãã®ã»ãã§ãç¿ãããKæ®»ã¯8åã®é»åãå
¥ãããªã©ã®ãã¢ã¬ã®ããšã§ãã
ãŸãšãããšã
ãç£æ§äœã«ã匷ç£æ§äœããããã®ãªããèªé»äœã«ãã匷èªé»äœããããã®ã?ãã®ãããªçåã¯ããšããããæãã§ãããã
ãã¿ã³é
žé PbTiO 3 {\displaystyle {\ce {PbTiO3}}} ããããªãé
žãªããŠã LiNbO 3 {\displaystyle {\ce {LiNbO3}}} ããã匷èªé»äœãã«åé¡ãããå ŽåããããŸãã
ãããã匷ç£æ§äœãç£æ°ããŒããç£æ°ããŒããã£ã¹ã¯ãªã©ã®èšé²ã¡ãã£ã¢ã«çšããããŠããç¶æ³ãšã¯ç°ãªããã匷èªé»äœãã¯èšé²ã¡ãã£ã¢ã«ã¯çšããããŠããŸãããéå»ã«ã¯ããã®ãããªã匷èªé»äœã¡ã¢ãªãŒããç®æãç 究éçºããã£ãããããã2017幎ã®æç¹ã§ã¯ããŸã ã匷èªé»äœã¡ã¢ãªãŒãã®ãããªããã€ã¹ã¯å®çšåããŠããŸããã
ããããä»ã®çšéã§ããããã®ç©è³ªã¯ç£æ¥ã«å®çšåãããŠããŸãã
ãã¿ã³é
žéãããªãé
žãªããŠã ã¯ããã®ç©è³ªã«å§åããããããšé»å§ãçºçããäºãããå§é»äœ(ãã€ã§ããã)ãšããçŽ åãšããŠæŽ»çšãããŠããŸãã(â» ãé«çåŠæ ¡ååŠI/ã»ã©ããã¯ã¹ãã§ãå§é»æ§ã»ã©ããã¯ã¹ããšããŠå§é»äœã玹ä»ãé«æ ¡ååŠã®ç¯å²å
ã§ãã2017幎ã®çŸåšã§ã¯é«æ ¡3幎ã®éžæååŠ(å°éååŠ)ã®ç¯å²å
ã§ãããã)
ãªãããããã®å§é»äœã«ãé»å§ããããããšãç©è³ªãã²ããã
ãã®ãããå§é»äœã«äº€æµé»å§ãå ããããšã§ãå§é»äœãçæéã§äœåãåšæçã«æ¯åããããšã«ãããå§é»äœã®åšå²ã«ãã空æ°ãæ¯åãããäºãã§ããã®ã§ãè¶
é³æ³¢ãçºçããããã®çŽ åãšããŠããã§ã«å®çšåãããŠããŸãã
ãªããããçš®é¡ã®ç©è³ªããå§åããããããšé»å§ãçºçããçŸè±¡ãèµ·ããç©è³ªã®å Žåããã®ãããªæ§è³ªã®ããšãå§é»æ§(ãã€ã§ããã)ãšãããŸãã
ã±ã€çŽ Si ãã²ã«ãããŠã Ge ã¯ãå°äœãšçµ¶çžäœã®äžéã®æµæçããã€ããšãããã±ã€çŽ (ã·ãªã³ã³)ãã²ã«ãããŠã ãªã©ã¯åå°äœãšèšãããŸãã
ãã®åå°äœã®çµæ¶ã«ããããã«ããªã³Pãªã©ã®äžçŽç©ãå
¥ããããšã§ãæµæçã倧ããäžããããŸãã
ã·ãªã³ã³ååã¯äŸ¡é»åã4åã§ãããã·ãªã³ã³ã®çµæ¶ã¯ã4ã€ã®äŸ¡é»åãå
±æçµåãããŠããŸãã
ããã«ãªã³Pãå ãããšããªã³ã¯äŸ¡é»åã5åãªã®ã§ã1åã®äŸ¡é»åãäœãããã®äœã£ã䟡é»åãèªç±é»åãšããŠãçµæ¶ãåãåããããã«ãªããŸãã
ãã®ãããªä»çµã¿ã§ãã·ãªã³ã³ã«ãªã³ãå ããããšã§ãæµæçã倧ããäœäžããããšããã®ãå®èª¬ã§ãã
ãã®ããã«ãè² ã®é»åãäœãããšã§ãå°é»çãäžãã£ãŠãåå°äœã nååå°äœ ãšãããŸãã(ãnã㯠negative ã®ç¥ã)
ã·ãªã³ã³ã®çµæ¶ã«ãäžçŽç©ãšããŠãããŠçŽ Bãã¢ã«ãããŠã Alãªã©ã䟡é»åã3åã®å
çŽ ãå ãããšãé»åã1åã足ããªããªããŸãã
ãã®ãé»åã®äžè¶³ããã¶ãã®ç©ºåžãããŒã«(postive holeãæ£å)ãšãããŸãã
ããŒã«ã¯æ£é»è·ããã¡ãŸãã
é»å§ãæãããšããã®ããŒã«ãåããããã«è¿ãã®çµåã«ãã£ãé»åã移åããŸãããããšã®é»åããã£ãå Žæã«æ°ããªããŒã«ãã§ããã®ã§ãèŠããäžã¯ããŒã«ãé»åãšéæ¹åã«åããããã«èŠããŸãã
ãã£ãŠãããŒã«ãåãããšã§ãé»æµãæµãããšèŠãªããŸãã
ãŸãããã®ããã«ãæ£ã®é»è·ããã€ç²åã«ãã£ãŠå°é»çãäžãã£ãŠãåå°äœã pååå°äœ ãšãããŸãã(ãpã㯠positive ã®ç¥ã)
nååå°äœã§ã¯ãèªç±é»åãé»æµãéã¶ã
pååå°äœã§ã¯ãããŒã«ãé»æµãéã¶ã
ãã®ããã«ãåå°äœäžã§é»è·é»åã®æ
ãæãããã£ãªã¢(carrier)ãšãããŸãã
ã€ãŸããnååå°äœã®ãã£ãªã¢ã¯é»åã§ãpååå°äœã®ãã£ãªã¢ã¯ããŒã«ã§ãã
pååå°äœãšnååå°äœãæ¥åã(pnæ¥å)ãç©äœããäžæ¹åã®ã¿ã«é»æµãæµãã
ãã®ãããªéšåããã€ãªãŒã(diode)ãšãããŸãã
påŽã«æ£é»å§ãæããnåŽã«è² é»å§ãæããæãé»æµãæµããŸãã
ãã£ãœããpåŽã«è² é»å§ãæããnåŽã«æ£é»å§ãæããŠããé»æµãæµããŸããã
åè·¯ã«ãããŠããã€ãªãŒããé»æµãæµãåããé æ¹å(ãã
ãã»ãããŸããã)ãšãããŸããé æ¹åãšã¯å察åããéæ¹åãšãããŸãããã€ãªãŒãã®éæ¹åã«ã¯ãé»æµã¯æµããŸããã
ãã®ããã«äžæ¹åã«æµããä»çµã¿ã¯ããã€ãªãŒãã§ã¯ãã€ãã®ãããªä»çµã¿ã§ãé»æµãæµããããã§ãã
ãã®ããã«äžæ¹åã«ã ãé»æµãæµãããšãæŽæµ(ãããã
ã)ãšãããŸãããªããåå°äœã䜿ããªããŠããç空管ã§ãæŽæµã ããªãå¯èœã§ãã(ãã ãç空管ã®å Žåãç±ã®çºçãèšå€§ã§ãã£ãããèä¹
æ§ãå£ãã®ã§ãé»åéšåãšããŠã®å®çšæ§ã¯ã空管ã¯äœãã®ã§ãçŸä»£ã¯ç空管ã¯é»åéšåãšããŠã¯äœ¿ãããŠããŸããã)
ããœã³ã³ã§ãããžã¿ã«æ³¢åœ¢ãããžã¿ã«ä¿¡å·ã®ããã«åè§ã®é»æµæ³¢åœ¢ãäœã£ãŠããæ¹æ³ã¯ãããããããã®ãã€ãªãŒããšãåŸè¿°ãããã©ã³ãžã¹ã¿ãšããããŸãçµã¿åãããããšã§ãããžã¿ã«æ³¢åœ¢ãã€ãããšããä»çµã¿ã§ãã(â» æ°ç åºçã®æ€å®æç§æžããããããèŠè§£ã§ãã)
ãã€ãªãŒãã®påŽã«æ£é»å§ãããããã£ãœãnåŽã«è² é»å§ãããããšãpåŽã§ã¯æ£é»æ¥µã®æ£é»å§ããããŒã«ãåçºããŠæ¥åé¢ãžãšåããããã£ãœãnåŽã§ã¯èªç±é»åãè² é»æ¥µããåçºããŠæ¥åé¢ãžãšåããããããŠãæ¥åé¢ã§ãããŒã«ãšèªç±é»åãã§ãããæ¶æ»
ããŸãããã®çµæãèŠæãäžãæ£é»è·ããæ£é»æ¥µããè² é»æ¥µã«ç§»åããã®ãšãåçã®çµæã«ãªããŸãã
ãããŠãæ£é»æ¥µãããã€ãã€ããšããŒã«ãäŸçµŠãããã®ã§ãé»æµãæµãç¶ããŸãã
ãã£ãœããpåŽã«è² é»å§ãæããnåŽã«æ£é»å§ãæããæãpåŽã§ã¯ããŒã«ã¯é»æ¥µ(é»æ¥µã«ã¯è² é»å§ãæãã£ãŠãŸã)ã«åŒãå¯ããããæ¥åé¢ããã¯é ããããŸããåæ§ã«nåŽã§ã¯èªç±é»åãé»æ¥µ(æ£é»å§ãæãã£ãŠãŸã)ã«åŒãå¯ããããæ¥åé¢ããã¯é ããããŸãã
ãã®çµæãæ¥åé¢ã«ã¯ãäœåãªããŒã«ãäœåãªèªç±é»åããªãç¶æ
ãšãªãããã£ãŠæ¥åé¢ã®ä»è¿ã«ã¯ãã£ãªã¢ããªãããã®æ¥åé¢ä»è¿ã®ãã£ãªã¢ã®ç¡ãéšåã¯ç©ºä¹å±€(ãããŒããããdepletion layer)ãšåŒã°ããŸãã
ãããŠããã以éã¯ãããŒã«ãèªç±é»åããããã©ãã«ã移åã®äœå°ããªãã®ã§ããã£ãŠé»æµãæµããŸããã
åå°äœã3ã€npnãŸãã¯pnpã®ããã«çµã¿åããããšãé»æµãå¢å¹
(ãããµã)ããããšãã§ããŸããå¢å¹
äœçš(ãããµãããã)ãšãããŸãã
NPNãšã¯ãç端ããé ã«èŠãŠNåã»Påã»Nåã®é ã«äžŠãã§ããšããäºã§ãã
åæ§ã«ãPNPãšã¯ãç端ããé ã«èŠãŠNåã»Påã»Nåã®é ã«äžŠãã§ããšããäºã§ãã
å¢å¹
ãšãã£ãŠãããã£ããŠç¡ãããšãã«ã®ãŒãçºçããããã§ã¯ãªãã®ã§ãæ··åããªãããã«ã
説æã®ç°¡ç¥åã®ãããå€éšé»æºãçç¥ãããäºãããããå®éã¯å€éšé»æºãå¿
èŠã§ããåå°äœçŽ åã¯å°ããªé»æµããæµããªãã®ã§ãé»æµãæžããããã®æµæçŽ åãšããŠã®ä¿è·æµæ(ã»ããŠãããŸããã)ãå¿
èŠã§ãã
ãªããå³ã®ããã«é·æ¹åœ¢ç¶ã«äžŠãã§ããæ¹åŒã®ãã©ã³ãžã¹ã¿ããã€ããŒã©ãã©ã³ãžã¹ã¿ãšãããŸãã(â» æ€å®æç§æžã®æ°ç åºçã®æç§æžã§ãããã€ããŒã©ãã©ã³ãžã¹ã¿ããã³ã©ã ã§ç¿ãã)
ãã€ããŒã©ãã©ã³ãžã¹ã¿ã«ã¯ã端åãäž»ã«3ã€ãããããšããã¿ãããããŒã¹ãããã³ã¬ã¯ã¿ããšããåèš3ã€ã®ç«¯åããããŸãã
ãã€ããŒã©ãã©ã³ãžã¹ã¿ã§ã®é»æµã®å¢å¹
ãšã¯ãããŒã¹é»æµãå¢å¹
ããŠã³ã¬ã¯ã¿ã«éããã§ã(PNPã®å Žå)ãé»æµã®åãã¯PNPåã®ã°ãããš NPPåã®ã°ãããšã§ã¯ç°ãªãããã©ã¡ãã®å Žåã§ãããŒã¹é»æµãå¢å¹
ããããšããä»çµã¿ã¯å
±éã§ãã
ããŠãæš¡åŒå³ã§ã¯æš¡åŒçã«çãäžã®åå°äœã¯ããããå°ããã«æžãããããå®éã®ãã©ã³ãžã¹ã¿ã¯çãäžã®åå°äœã¯ããã§ã¯ãªãã®ã§ãåèçšåºŠã«ã
æè²ã§ã¯ãåå°äœã®é«æ ¡çãå°éå€(é»åå°æ»ä»¥å€)ã®äººããã«ã¯ããããã€ããŒã©ãã©ã³ãžã¹ã¿ãåçŽãªã®ã§çŽ¹ä»ãããããå®éã«åžè²©ã®ã³ã³ãã¥ãŒã¿éšåãªã©ã§ãã䜿ããããã©ã³ãžã¹ã¿ã®æ¹åŒã¯ããããšã¯åœ¢ç¶ããã£ããç°ãªããŸãã
åžè²©ã®ã³ã³ãã¥ãŒã¿éšåã®ãã©ã³ãžã¹ã¿ã«ã¯ãé»çå¹æãã©ã³ãžã¹ã¿ãšããããæ¹åŒã®ãã®ããããçšããããŸãã(ãã¡ãããé»çå¹æãã©ã³ãžã¹ã¿ã«ãããå¢å¹
ãã®æ©èœããããŸãã)
(⻠詳ããã¯å€§åŠã®é»æ°å·¥åŠãŸãã¯å·¥æ¥é«æ ¡ã®é»ååè·¯ãªã©ã®ç§ç®ã§ç¿ãã)
ãã©ã³ãžã¹ã¿ã¯ãåè·¯å³ã§ã¯ãæš¡åŒçã«äžå³ã®ããã«æžãããŸãã
ãã€ãªãŒãããã©ã³ãžã¹ã¿ã®ä»ã«ãåå°äœãçµã¿åãããé»åéšåã¯ããã®ã§ãã(ä»ã«ãããµã€ãªã¹ã¿ããªã©è²ã
ãšãããŸã)ãé«æ ¡ç©çã®ç¯å²ãè¶
ããã®ã§ã説æã¯çç¥ããŸãã(â» ããä»äºã§å°éçãªæ
å ±ãå¿
èŠã«ãªãã°ãå·¥æ¥é«æ ¡ããã®ãé»ååè·¯ãã®æç§æžã«ãã£ãã詳ããæžããŠããã®ã§ããããèªãã°ããã§ãããªããæžåºã®è³æ Œã³ãŒããŒæ¬ã«ããé»æ°å·¥äºå£«ãé»æ°äž»ä»»æè¡è
è©Šéšãªã©ã®å¯Ÿçåã«ã¯ãã»ãŒé»ååè·¯ãç¯å²å€ãªã®ã§ãããŸãé»ååè·¯ã®èª¬æã¯æžããŠãŸããããªã®ã§ãå·¥æ¥é«æ ¡ãé»ååè·¯ãã®æç§æžããŸãã¯å·¥æ¥é«å°ãªã©ã®åçã®ç§ç®ã®æç§æžãåç
§ã®ããšã)
ããœã³ã³ã®CPUãªã©ã®éšåããäžèº«ã®å€ãã¯åå°äœã§ããããã€ãªãŒãããã©ã³ãžã¹ã¿ãªã©ã®çŽ åãCPUãªã©ã®å
éšã«ãããããããŸãããšèšãããŠããŸãã(â» ä»ã«ããæ°Žæ¶æ¯ååããªã©è²ã
ãšCPUå
ã«ã¯ ããããç©ç2ã®ç¯å²å€ãªã®ã§èª¬æãçç¥ã)
éç©åè·¯ãLSI(Large Scale Integratedã倧èŠæš¡éç©åè·¯)ãªã©ãšèšãããé»åéšåãããªã«ãéç©(ãéç©ããè±èªã§ integrate ã€ã³ãã°ã¬ãŒã ãšãã)ããã®ããšãããšãåå°äœçŽ åãéç©ãããšèšãæå³ã§ãã
ãªãããICã(ã¢ã€ã·ãŒ)ãšã¯ Integrated Circuit ã®ç¥ç§°ã§ããããããåèš³ãããã®ããéç©åè·¯ãã§ãã
ã€ãŸããéç©åè·¯ãLSIã®äžèº«ã¯ãåå°äœã§ããããã©ã³ãžã¹ã¿ãªã©ã®çŽ åãé«å¯åºŠã§ããã®åè·¯äžã«è©°ãŸã£ãŠããŸãã
é»åéšåã®åå°äœã®ææãšããŠã¯ãéåžžã¯ã·ãªã³ã³çµæ¶ã䜿ãããŸãã(â» åæ通ãæ°ç ãªã©ãçµæ¶ã§ããããšãèšåã)
ç 究éçºã§ã¯ã·ãªã³ã³ä»¥å€ã®ææãç 究ãããŠããäžéšã®ç¹æ®çšéã§ã¯GaAsãInGaPãªã©ãå©çšãããŠããã(â» æ°ç ã®æ€å®æç§æžã¯GaAsãInGaPãªã©ã«ã³ã©ã ã§èšå)ããããçŸç¶ã§ã¯ãã·ãªã³ã³ãåžè²©ã®ã³ã³ãã¥ãŒã¿éšåäžã®åå°äœçŽ åã®ææã§ã¯äž»æµã§ãã
ãªããã·ãªã³ã³åå°äœã®ææå
éšã¯ã·ãªã³ã³çµæ¶ã§ããããè¡šé¢ã¯ä¿è·èããã³çµ¶çžã®ããã«é
žåãããããŠãããã·ãªã³ã³åå°äœè¡šé¢ã¯é
žåã·ãªã³ã³ã®ä¿è·èã«ãªã£ãŠããŸããã·ãªã³ã³ãé
žåãããšã絶çžç©ã«ãªãã®ã§ãä¿è·èã«ãªãããã§ã(â» æ°ç åºçã®æç§æžãããèšã£ãŠããŸãã)
åå°äœã®å
éšã«ãæ·»å ç©ãªã©ã§ç¹æ§ãå€ããããšã«ãããæµæãã³ã³ãã³ãµãåå°äœå
éšã«è£œé ã§ããŸãã(â» æ°ç ããæµæãã³ã³ãã³ãµãåå°äœå
éšã§äœã£ãŠããäºã«èšåã)
(â» ç¯å²å€: )ããããã³ã€ã«ã¯åå°äœå
éšã«äœãããšãåºæ¥ç¡ãã§ãã
ç£å ŽBã®äžããé»è·qã®è·é»ç²åãé床vã§éåãããšãããŒã¬ã³ãåã¯ãã¯ãã«å€ç©ãçšã㊠f=qã»vÃB ã®åãç²åã«åãããããã§èŠ³æž¬è
ã®åº§æšç³»ãå€ãããšããŠãåãç²åããç²åãšåãæ¹åã«é床vã§åã座æšåœ¢Kã®äžã®èŠ³æž¬è
ããèŠããã©ããªãã? 座æšç³»Kã§ã¯ãç²åã®é床㯠v(K)=0 ã§ãããç£æã®é床ã Vb ãšãããšãåã®åº§æšç³»ã®ç²åãšã¯å察æ¹åã«åãã®ã§ã
æ°ãã座æšç³»Kãã芳枬ããŠããç²åã f=qã»vÃB ã®å€§ããã®åãåããŠå éãããããšã«ã¯å€ãããŸãããã座æšç³»kã§ã¯ãè·é»ç²åã¯éæ¢ããŠããã®ã«ãããŒã¬ã³ãåãåãããšèããã®ã¯äžåçã§ããç£æã¯ãVb=-v ã§éåããŠããã®ã§ãç£æã®éåã«ãã£ãŠ f=qã»(-Vb)ÃB = -qã»VbÃB ã®åãåãããšèããã¹ãã§ããç²åã質é0ã®è³ªç¹ãšã¿ãªãã°ãéæ¢ããŠããè·é»ç²åã«åãåãŒããã®ã¯ãé»å Žã ãã ãããã€ãŸãé床 Vb ã§éåããç£æãã E=-VbÃB ã®èªå°é»å Žãèªèµ·ããããšã«ãªããŸãããã®ãšããç£å Žãšèªå°ãããé»å Žã¯åçŽã§ãã
ããããéåããé»å Žã¯ç£çãäœãããšããã°ãã¢ã³ããŒã«ã®æ³å ãçŽç·ç¶ã«ç¡éã«é·ãå°ç·ãæµãã é»æµI ã¯è·é¢R ã ãé¢ããå Žæã« Bã»2Ïr=ÎŒI ã®ç£å ŽãäœãŸããããšããçŸè±¡ã¯ããã€ã¯ãå°ç·ã®äžã§è·é»ç²åãéåããããšã«ãã£ãŠãè·é»ç²åãšãã£ããã«ãã®ç²åãäœãé»å Žãåãããã®é»å Žã®éåããç£å Žãèªèµ·ããŠããŸããããšããå¯èœæ§ããããŸãã é»æµãæµããŠããç¡éé·ã®ããŸã£ãããªå°ç·ãèããŸããç·å¯åºŠ q[C/m] ã§ååžããé»è·ã¯ãå³ã®ããã«åç察称ãªé»è·ãäœãŸãã
(â» ããã«å³ãã)
çŽç·ããè·é¢rã®ãšãã®é»æ°åç·ã®å¯åºŠDã¯
ãã£ãŠ
é»æµ I ã¯é»è·ååž q ãé床 Ve ã§éåããŠãããšããŠ
ãšå®çŸ©ããã°ã
é»æµ qVe ãè·é¢ r ã®ãšããã«äœãç£å ŽBã¯ã¢ã³ããŒã«ã®æ³åããã
ãšãªããŸãã
ãã®ãšããç£å Žã®åãã¯ãVe ãã ååŸræ¹å ã«ãããåãåãã§ãã
åããŸã§ãµãããŠãã¯ãã«ç©ã§è¡šãã°ã
ã€ãŸã
ãšãããéèŠãªçµè«ãåŸãããŸãã
ãããã¯ã ÎŒH=B ããã¡ã㊠B=ÎŒH=εΌ Ve ÃE ãã
ã§ãã
ãŸãšã
é床 Vbã§éåããç£æBã¯
ã®èªå°é»å Žãèªèµ·ããŸãã ã»ã»â¡1
é床 Ve ã§éåããé»å Ž E ã¯
ã®èªå°ç£å ŽãäœãŸãã
E,Bã®ãããã«ãD,Hã䜿ã£ãŠè¡šèšããã°ã
ãã€
ããŠãé»ç£æ³¢ãé床Cã§ç空äžãäŒãããšããã°ã Vb = Ve = C ãšããŸãã â¡1åŒãšâ¡2åŒã®å€ç©ããšããšã
ãã£ãŠ
ã§ãã
ãã£ãŠãé»ç£æ³¢ã®é床㯠c = 1 ε ÎŒ {\displaystyle c={\frac {1}{\sqrt {\varepsilon \mu }}}} ãšäºæž¬ã§ããŸãã
ãã®ÎµãšÎŒã«å®æž¬å€ãå
¥ãããšãå
éã®æž¬å®å€ c = 299792458 m / s {\displaystyle c=299792458m/s} ãšãé«ã粟床ã§äžèŽããŸãã
ãã®äºãããå
ã¯ãé»ç£æ³¢ã§ããäºãåãããŸãããŸããé»ç£æ³¢ã¯ãå
é床Cã§ç空äžãäŒããŸãã
ãŸãããããããéåé»å Žã®èªå°ããç£å Žã¯
ãšãå€åœ¢ã§ããŸãã
3åŒããã¬ãŠã¹ã®æ³å(1åŒ) ãšçµã¿åããããšãã¢ã³ããŒã«ã®æ³å(2åŒ)ãåŸãããŸãã ãã£ãŠããé床 Ve ã§éåããé»å Ž E ã¯ã B=εΌ Ve ÃE ã®èªå°ç£å ŽãäœãŸããããšããéçšã劥åœã ã£ãããšãããããŸãã
é»ç£æ³¢ã§ã¯é»å Ž E ãšç£å Ž B ãå
é C ã§éåããŠããã®ã§ ç£æã®éåé床 Vb 㯠Vb = C ã§ãããèªå°é»å Ž E 㯠E =-VbÃB ã§ããã®ã§ãäž¡åŒãã E = -cÃB ã§ãã(é»ç£æ³¢ã®é»å Žãšç£å Žã®é¢ä¿åŒ)ãªã
ã§ããã®ã§ã é»ç£æ³¢ã¯
ã®æ¹åã«é²ãã§ããã¯ãã§ãããšããããšã泚ç®ããŸãããã
ãã® E à H {\displaystyle \mathbb {E} \times \mathbb {H} } ã§å®çŸ©ãããéã ãã€ã³ãã£ã³ã° ãã¯ãã« ãšãã¶ã ããã¯åäœé¢ç©ããšãã£ãŠæµãåºãé»ç£å Žã®ãšãã«ã®ãŒã®æµãã®éãããããã
ããŠãé»ç£å Žã®ãšãã«ã®ãŒå¯åºŠã¯ u = 1 2 ε E 2 + 1 2 ÎŒ H 2 {\displaystyle u={\frac {1}{2}}\varepsilon E^{2}+{\frac {1}{2}}\mu H^{2}} ãªã®ã§ãããã«é»ç£æ³¢ã®é»å Žãšç£å Žã®é¢ä¿åŒ E = â C à B {\displaystyle \mathbb {E} =-\mathbb {C} \times \mathbb {B} } ã代å
¥ããŠã
ã®é¢ä¿ãçšãããšã(ãšãã«ã®ãŒã§ã¯ã2ä¹ã«ãããã€ãã¹ç¬Šå·ããªããªãã®ã§ã絶察å€ãåã£ãŠ|E|=|cÃB| ãšããŠãããšãèšç®ãç°¡åã«ãªãå ŽåããããŸãã)
çµæãšããŠ
ãšãªããŸãã é»ç£æ³¢ããå£ã«ããã£ãŠåžåããããšããåäœæéã«åäœé¢ç©ããã å
éC ã®å€§ããã®äœç©ã®ãªãã®é»ç£æ³¢ãå£ã«è¡çªããã®ã§ã
ã®ãšãã«ã®ãŒããåäœæéã«åäœé¢ç©ã«æµã蟌ãã¯ãã§ãã
s= cã»u ã« u= εã»E^2 ã代å
¥ããŠã ε ÎŒ â
c 2 = 1 {\displaystyle \epsilon \mu \cdot c^{2}=1} ãš |E|=|cÃB|ãå©çšãããšãçµæçã«
ã§ãã
ãã£ãŠãã€ã³ãã£ã³ã° ãã¯ãã« EÃH ã¯åäœé¢ç©ãéã£ãŠæµãåºãé»ç£å Žã®ãšãã«ã®ãŒã®æµããããããã
ãã€ã³ãã£ã³ã° ãã¯ãã« S = EÃH = εΌ(C)EÃH ã¯
ã§ãã
倩äžãçãªèª¬æã§ããããã® G=DÃB ãšããéã¯ãéåéã®å¯åºŠã§ãããã®é G=DÃB ããé»ç£æ³¢ã®ãéåéå¯åºŠã(ããã©ããããã¿ã€ã©)ãšãããŸããå®éã«ãDÃB ã®åäœã¯
ãšãªããŸãã ãããã«ãéåéã®å¯åºŠã®åäœãšçããã
ãšããã§ãã®ã¡ã®åå
ã§ç¿ãããå
é»å¹æã§ã¯ ãšãã«ã®ãŒuãšéåépã®é¢ä¿ã¯ãå
é床Cããã¡ããŠã u=cp ãšæžããŸãã
ãããã
åããŸã§å«ããŠ
ãšãªã£ãŠã確ãã« G = DÃB ã¯éåéå¯åºŠãšãªããŸãã
é·ãLã®ãŸã£ãããªééããé床vã§ç£å ŽBã®äžã暪åããšããŸããç°¡åã®ãããééã®è»žãšé床vã®æ¹åãšç£å ŽBã¯åçŽãšããŸãããã®ãšããééã®äžã®é»è·ã«ãããåããã³é»å Žã¯ããŒã¬ã³ãåã«ããã
é»å ŽEã«ãã£ãŠé·ãLã ããé»è·qãäžãããããããšãã«ã®ãŒã¯ qEL å€åããŸããé»äœã¯ V=EL ã§ãã
ãããããèªå°é»å§ V ã¯ãç£æã®1ç§ãããã®æéå€åã«ãªããŸãã ã§ã¯ãä»®ã«åºå®ãããåè·¯ã®äžã«ãœã¬ãã€ããéããŠããã®ãœã¬ãã€ãã«äº€æµé»æµãæµããå Žåããåè·¯ã«èªå°é»å§ãçºçããã®ã ããããçãã¯ããããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ãŸããé«æ ¡ç©çã§ãããèªé»äœã(ããã§ããã)ãšã¯ãéåžžã®ã»ã©ããã¯ãé²æ¯(ãã€ã«)ãªã©é»æ°ãéããªãç©è³ªã®ãã¡é«ãèªé»çã瀺ããã®ã§ãã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 1,
"tag": "p",
"text": "éå±ã¯å°äœãªã®ã§èªé»äœã§ã¯ãããŸããã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ã§ã¯ãèªé»äœã®ç©çã«ã€ããŠã説æããŸãã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ã³ã³ãã³ãµãŒã«èªé»äœãå
¥ãããšãèªé»äœãèªé»å極ãèµ·ãããããã³ã³ãã³ãµã®ãã©ã¹æ¥µæ¿ã§çºçããé»æ°åç·ã®ããã€ããæã¡æ¶ãããŸãã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ãã®çµæãèªé»äœã®å
¥ã£ãã³ã³ãã³ãµãŒã®æ¥µæ¿éã®é»å Žã¯ã極æ¿ã®é»è·å¯åºŠã§çºçããé»è·ãç空äžã§ã€ããé»å Žããã匱ããªããŸãã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ãã®çµæãéé»å®¹éãå€ããŸãã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ããŠãç空äžã®éé»å®¹éã®å
¬åŒã¯ã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ã§ããã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "èªé»äœã®ããå Žåã®éé»å®¹éã¯ã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "ãšãªããŸãã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "ããã§ã ε {\\displaystyle \\varepsilon } ãèªé»ç(ããã§ããã€)ãšãããŸãã ε 0 {\\displaystyle \\varepsilon _{0}} ããç空äžã®èªé»çãšãããŸãã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "ããã§ãæ¯",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "ããæ¯èªé»ç(ã² ããã§ããã€)ãšãããŸãã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ã€ãŸãã ε r {\\displaystyle \\varepsilon _{r}} ã¯æ¯èªé»çã§ãã ãã£ãœãã ε 0 {\\displaystyle \\varepsilon _{0}} ãã㳠ε {\\displaystyle \\varepsilon } ã¯ãæ¯èªé»çã§ã¯ãããŸããã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "æ¯èªé»ç ε r {\\displaystyle \\varepsilon _{r}} ããã¡ããã°ãéé»å®¹é C ã®åŒã¯ã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "ãšæžããŸãã",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "U=2â1CV2",
"title": "éé»èªå°ãšèªé»å極"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "ç£ç³ã®ãŸããã«ã¯ç©äœãåããåã®ãããã®ãçããŠããŸãã ãããç£å Ž(ãã°)ãšåŒã¶ãç£ç(ããã)ãšãããã",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "é»æµãæµããŠãããšãã«ãããã®ãŸããã«ã¯ãå³ããã®æ³å(right-handed screw rule)ã«åŸãåãã«ç£çãçããŸãã é»æµI[A]ãçŽç·çã«æµããŠãããšããç£çã®å€§ãã㯠B = ÎŒ 0 2 Ï a I {\\displaystyle B={\\frac {\\mu _{0}}{2\\pi a}}I} ã§ããããšãç¥ãããŠããŸãã",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "ããã§ãaã¯ç£æå¯åºŠã枬ãç¹ãšãé»ç·ã®è·é¢ã",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "ãŸãã ÎŒ 0 {\\displaystyle \\mu _{0}} ã¯ç空ã®éç£ç(ãšãããã€ãpermeability)ãè¡šããå€ã¯ 4 Ï Ã 10 â 7 {\\displaystyle 4\\pi \\times 10^{-7}} [H/m] ã§ãã",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "ç£å Žã䌎ãç©äœãéåãããšããã®ãŸããã«ã¯é»å Žãçããããšãé»ç£èªå°(ã§ããããã©ããelectromagnetic induction)ãšãããŸãã ä»®ã«ããœã¬ãã€ã(solenoidãã³ã€ã«ã®ããš)ã®è¿ãã§ãããè¡ãªã£ããšãããšãçããé»å Žã«ãã£ãŠãœã¬ãã€ãã®äžã«ã¯é»æµãæµããŸãã çããé»å Žã®å€§ããã¯ã E â = 1 2 Ï a d B â d t {\\displaystyle {\\vec {E}}={\\frac {1}{2\\pi a}}{\\frac {d{\\vec {B}}}{dt}}} ãšãªããŸãã(ååŸaã®å圢ã®ã³ã€ã«ã®å Žåã) Eã®åäœã¯[V/m]ã§ãããBã®åäœã¯[T]ã§ãã",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "ç£å Žã®åãã«ãã£ãŠé»å ŽãåŒãèµ·ããããããšãé»ç£èªå°ã®ã»ã¯ã·ã§ã³ã§èŠãã",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "ãŸããå®éã«ã¯é»å Žã®å€åã«ãã£ãŠç£å ŽãåŒãèµ·ããããããšãå®éšã«ãã£ãŠç¥ãããŠããŸãã ããã«ãã£ãŠäœããªã空éäžãé»å Žãšç£å ŽãäŒæããŠããããšãäºæ³ãããŸãã",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "(:é»ç£æ³¢ã®äŒæã®schematicãªçµµ)",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "ãŸããç©çå®éšå®¶ãã«ãã¯æŸé»å®éšã«ãããåä¿¡æ©ãåè·¯äžã«ã®ã£ããã®ããåè·¯ãšããŠãéä¿¡åŽã®æŸé»ã«ããé»å Žãé éçã«é¢ããäœçœ®ã«ããåä¿¡åŽã®åè·¯ã«äŒããããšã確èªããã",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "ãã®å®éšã®éããã«ãã¯åä¿¡åè·¯ã®åãããããããšå€ããŠå®éšããããšã«ãããéä¿¡æ©ã®åãã«å¯ŸããŠã®åä¿¡æ©ã®åãã«ãã£ãŠé»å Žã®äŒããæ¹ãç°ãªãããšãããé»å Žã®é éäœçšã«åå
æ§ãããäºãåãã£ãã",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "é»å Žã®ãã®äœçšã«ã¯åå
æ§ãããã®ã§ãæ³¢ã§ãããšã¿ãªãããšã¯åŠ¥åœã§ãããã",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "ãã«ãã®å®éšãããå®éšçã«ãããããšãšããŠ",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "ãå®éšçã«ããããŸãã",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "ç©çåŠã§ã¯ããã«ãã®å®éšã®ä»¥åãããçè«ç©çåŠè
ã®ãã¯ã¹ãŠã§ã«ã«ããã",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "é»ç£æ³¢ãšãããé»å Žãšç£å Žã®çžäºäœçšã«ãã£ãŠç空äžãäŒéããäºæž¬ãããŠããã",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "ãªã®ã§ããã«ãã®å®éšã¯ããã¯ã¹ãŠã§ã«ã®äºæž¬ããé»ç£æ³¢ã ãšã¿ãªãããã çŸä»£ã§ãç©çåŠè
ã¯ãããã¿ãªããŠããŸãã",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "ãªãããã¯ã¹ãŠã§ã«ãçè«èšç®ã§æ±ããé»ç£æ³¢ã®é床ãæ±ãããšããããã§ã«ç¥ãããŠããå
éã®å€§ãã(ããã 3Ã10 m/s )ã«ç²ŸåºŠããäžèŽããã",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "ãã®ããšããããå
ã¯é»ç£æ³¢ã®äžçš®ã§ããããšãåãããŸãã",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "ãã«ãã®å®éšã§ã¯ãå³å¯ã«ã¯å°ãªããšãæŸé»ã®é»å ŽãäŒããããšãã芳枬ã§ããŠãŸãããããããç£å Žããã®å®éšã§äŒãããšèããŠãæ¯éãçããŠç¡ãããå®éã«äººé¡ã«ã¯æ¯éã¯çããŠãªãã®ã§ãä»ã§ããã«ãã®å®éšããã¯ã¹ãŠã§ã«ã®äºæž¬ããé»ç£æ³¢ã®èšŒæã®å®éšãšããŠäŒããããŠããŸãã",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "ãªããå
ã«ã¯ãåå°ãå±æãåæããã€ã³ã°ã¹ãªããã®åæãªã©ããããã",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "ãã«ãã®æŸé»å®éšã®ãããªé»ç£æ³¢ã®ç«è±æŸé»ã®å®éšã§ããå
ã®å®éšãšåæ§ã®é
眮ã§ãéå±æ¿ãé
眮ããŠç¢ºèªããããšã§ãé»ç£æ³¢ãåå°ãå±æãåæããã€ã³ã°ã¹ãªããã®åæãªã©ã®çŸè±¡ãèµ·ããããšããå®éšçã«ã確èªãããŠããŸã(â» åèæç® :å®æåºçã®å°éãç©çãã®æ€å®æç§æž)(â» ã€ã³ã°ã®ã¹ãªããã®é»ç£æ³¢å®éšã«é¢ããŠã¯åæ通ã®æç§æžãç©çãã«ãããŸã)ã",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "ãããã®ããšããããå
ã¯é»ç£æ³¢ã®äžçš®ã§ãããšã¿ãªãã®ã劥åœã§ããããšãåãããŸãã",
"title": "é»æµã«ããç£ç"
},
{
"paragraph_id": 41,
"tag": "p",
"text": "ç£ç³ã®ãŸããã«ã¯å¥ã®ç£ç³ãåããåã®ããšãšãªããã®ãçããŠããŸãã ãããç£å Ž(ãã°ãmagnetic field)ãããã¯ç£ç(ããã)ãšåŒã¶ã(æ¥æ¬ã®ç©çåŠã§ã¯ç£å ŽãšåŒã¶ããšãå€ãããŸããæ¥æ¬ã®é»æ°å·¥åŠã§ã¯ç£çãšåŒã°ããããšãå€ããææ²»æã®èš³èªã®éã®ãæ¥æ¬åœå
ã®æ¥çããšã®éãã«éãããå°å瀟äŒçãªäºè±¡ã§ãããåŒã³æ¹ã¯ç©çã®æ¬è³ªãšã¯é¢ä¿ãªãã®ã§ãããã§ã¯ãã©ã¡ãã®è¡šçŸãçšãããã¯ãæ¬æžã§ã¯ç¹ã«ãã ãããŸãããè±èªã§ã¯ç©çåŠã»é»æ°å·¥åŠãšãâmagnetic fieldâã§å
±éããŠããŸãã)",
"title": "ç£æ§äœ"
},
{
"paragraph_id": 42,
"tag": "p",
"text": "éãã³ãã«ããããã±ã«ã«ç£ç³ãè¿ã¥ãããšãç£ç³ã«åžãä»ããããŸãã ãŸããéãã³ãã«ããããã±ã«ã«åŒ·ãç£åãäžãããšãéãã³ãã«ããããã±ã«ãã®ãã®ãç£å Žãåšå²ã«åãŒãããã«ãªããŸãã ãã®ãããªãããšããšã¯ç£å Žãæããªãã£ãç©äœãã匷ãç£å Žãåããããšã«ãã£ãŠç£å ŽãåãŒãããã«ãªãçŸè±¡ãç£å(ãããmagnetization)ãšãããŸãã",
"title": "ç£æ§äœ"
},
{
"paragraph_id": 43,
"tag": "p",
"text": "ãããã¯é»è·ã®éé»èªå°ãšå¯Ÿå¿ãããŠãç£åã®ããšãç£æ°èªå°(ããããã©ããmagnetic induction)ãšãããã ãããŠãéãã³ãã«ããããã±ã«ã®ããã«ãç£ç³ã«åŒãä»ããããããã«ç£åãããèœåãããç©äœã匷ç£æ§äœ(ãããããããããferromagnet)ãšãããŸãã éãšã³ãã«ããšããã±ã«ã¯åŒ·ç£æ§äœã§ãã",
"title": "ç£æ§äœ"
},
{
"paragraph_id": 44,
"tag": "p",
"text": "é
ã¯ç£åããªãããé
ã¯ç£ç³ã«åŒãã€ããããªãã®ã§ãé
ã¯åŒ·ç£æ§äœã§ã¯ãããŸããã",
"title": "ç£æ§äœ"
},
{
"paragraph_id": 45,
"tag": "p",
"text": "éé»èªå°ãå©çšãããéé»é®èœ(ããã§ããããžã)ãšèšãããŸããäžç©ºã®å°äœãã€ãã£ãŠç©è³ªãå²ãããšã§å€éšé»å Žãé®èœããæ¹æ³ããã£ãã®ãšåæ§ã®ãç£æ°ã®é®èœãã匷ç£æ§äœã§ãåºæ¥ãŸããäžç©ºã®åŒ·ç£æ§äœãçšããŠã匷ç£æ§äœã®å
éšã¯ç£å Žãé®èœã§ããŸãããããç£æ°é®èœ(ãããããžããmagnetic shielding)ãšãããŸããç£æ°ã·ãŒã«ããšãããã",
"title": "ç£æ§äœ"
},
{
"paragraph_id": 46,
"tag": "p",
"text": "åç£æ§äœãåããã¥ãããããããŸããããåã«ããã®ææã«å ããããç£å Žãæã¡æ¶ãæ¹åã«ãç£åãããã ãã®ææã§ãã",
"title": "ç£æ§äœ"
},
{
"paragraph_id": 47,
"tag": "p",
"text": "ãããããç£åç·ãšããŸãçžäºäœçšããªãç©è³ªãå€ããããšãã°ãã¬ã©ã¹ãæ°Žã«ããŸããç£æ°ãžã®åœ±é¿ã¯ãç空ã®å Žåãšã»ãšãã©å€ãããŸãããã¬ã©ã¹ãæ°Žã®æ¯éç£ç(ã² ãšãããã€) ÎŒ (ãã¥ãŒ)ã¯ãã»ãŒ1ã§ãã",
"title": "ç£æ§äœ"
},
{
"paragraph_id": 48,
"tag": "p",
"text": "ãªããéã®æ¯éç£çã¯ãç¶æ
ã«ãã£ãŠéç£çã«æ°çŸãæ°åã®éããããããwikipediaæ¥æ¬èªçã§èª¿ã¹ãå Žåã®éã®éç£çã¯çŽ5000ã§ãã",
"title": "ç£æ§äœ"
},
{
"paragraph_id": 49,
"tag": "p",
"text": "ã§ã¯ãéç£çãã»ãŒ1ã®ç©è³ªã¯ãç£å Žã®æ¹åã¯ãå€éšç£å ŽãåºæºãšããŠãã©ã¡ãåãã ããã? å€éšç£å Žãæã¡æ¶ãæ¹åã«ç£åããŠããã®ã ããã? ãããšããå€éšç£å Žãšåãæ¹åã«ç£åããŠããã®ã ããã?",
"title": "ç£æ§äœ"
},
{
"paragraph_id": 50,
"tag": "p",
"text": "ãã®éãããããåžžç£æ§(ãããããã)ãšåç£æ§(ã¯ãããã)ã®ã¡ãããã§ãã",
"title": "ç£æ§äœ"
},
{
"paragraph_id": 51,
"tag": "p",
"text": "ããç©è³ªããå€éšç£å Žã«ã»ãšãã©åå¿ããŸãããããããå°ãã ãå€éšç£å Žãšåãæ¹åã«ãç£åãããŠããçŸè±¡ã®ããšãåžžç£æ§ãšããã§ãããã®ãããªç©è³ªãåžžç£æ§äœãšãããŸããåžžç£æ§äœãããããç©è³ªãšããŠãã¢ã«ãããŠã ã空æ°ãªã©ãããŸãã",
"title": "ç£æ§äœ"
},
{
"paragraph_id": 52,
"tag": "p",
"text": "ãã£ãœããããç©è³ªããå€éšç£å Žã«ã»ãšãã©åå¿ããŸãããããããå°ãã ãå€éšç£å Žãæã¡æ¶ãæ¹åã«ãç£åãããŠããçŸè±¡ã®ããšãåç£æ§ãšããã§ãããã®ãããªç©è³ªãåç£æ§äœãšãããŸããåç£æ§äœãããããç©è³ªãšããŠãé
ãæ°Žãæ°ŽçŽ ãªã©ãªã©ãããŸãã",
"title": "ç£æ§äœ"
},
{
"paragraph_id": 53,
"tag": "p",
"text": "å
çŽ ãååã®çš®é¡ã«ãã£ãŠãç£æ§ã®ã¡ãããããçç±ãšããŠãååŠçµåã§ã®é»åè»éã«åå ããããšèããããŠãŸãã",
"title": "â» ç¯å²å€: ã¹ãã³ãšç£æ§äœ"
},
{
"paragraph_id": 54,
"tag": "p",
"text": "ååŠã®æç§æžã®çºå±äºé
ã«ããsè»éãããpè»éããªã©ã®çè«ããããããã®çè«ã§ããã®çç±ã説æã§ãããšãããŠããŸãããªããçããå
ã«ãããšããdè»éãã®ç¹åŸŽããç£æ§ã®åå ã§ãã(蚌æã¯çç¥ããŸãã)",
"title": "â» ç¯å²å€: ã¹ãã³ãšç£æ§äœ"
},
{
"paragraph_id": 55,
"tag": "p",
"text": "ããšããšã(ååŠçµåã§é»åæ®»(ã§ãããã)ã«çºçããããšã®ãããŸã)å€ç«é»åã«ã¯ç£æ§ãããããã®ç£æ§ãé»åã2åããã£ãŠ(å€ç«ã§ãªããªã)é»å察ã«ãªãäºã§ãç£æ§ãæã¡æ¶ããã£ãŠãããšèããããŸãããªããå€ç«é»åãããšããšæã£ãŠããç£æ§ã®ããšãã¹ãã³ãšãããŸããããååŠã®çè«ã§ã¯ãã¹ãã³ãäžç¢å°ãâããšäžç¢å°ãâãã®2çš®é¡ã§ããããäºãå€ãã®ã§ããããã®çç±ã¯ãããšããã©ãã°ãããããç£ç³ã®åãã2çš®é¡(ããšãã°N極ãšS極ãšãã2çš®é¡ã®æ¥µããããŸã)ã ããã§ãã",
"title": "â» ç¯å²å€: ã¹ãã³ãšç£æ§äœ"
},
{
"paragraph_id": 56,
"tag": "p",
"text": "é»åæ®»ãšã¯ãååŠIã®å§ãã®ã»ãã§ãç¿ãããKæ®»ã¯8åã®é»åãå
¥ãããªã©ã®ãã¢ã¬ã®ããšã§ãã",
"title": "â» ç¯å²å€: ã¹ãã³ãšç£æ§äœ"
},
{
"paragraph_id": 57,
"tag": "p",
"text": "ãŸãšãããšã",
"title": "â» ç¯å²å€: ã¹ãã³ãšç£æ§äœ"
},
{
"paragraph_id": 58,
"tag": "p",
"text": "ãç£æ§äœã«ã匷ç£æ§äœããããã®ãªããèªé»äœã«ãã匷èªé»äœããããã®ã?ãã®ãããªçåã¯ããšããããæãã§ãããã",
"title": "â» ç¯å²å€: ã匷èªé»äœããšå§é»äœ"
},
{
"paragraph_id": 59,
"tag": "p",
"text": "ãã¿ã³é
žé PbTiO 3 {\\displaystyle {\\ce {PbTiO3}}} ããããªãé
žãªããŠã LiNbO 3 {\\displaystyle {\\ce {LiNbO3}}} ããã匷èªé»äœãã«åé¡ãããå ŽåããããŸãã",
"title": "â» ç¯å²å€: ã匷èªé»äœããšå§é»äœ"
},
{
"paragraph_id": 60,
"tag": "p",
"text": "ãããã匷ç£æ§äœãç£æ°ããŒããç£æ°ããŒããã£ã¹ã¯ãªã©ã®èšé²ã¡ãã£ã¢ã«çšããããŠããç¶æ³ãšã¯ç°ãªããã匷èªé»äœãã¯èšé²ã¡ãã£ã¢ã«ã¯çšããããŠããŸãããéå»ã«ã¯ããã®ãããªã匷èªé»äœã¡ã¢ãªãŒããç®æãç 究éçºããã£ãããããã2017幎ã®æç¹ã§ã¯ããŸã ã匷èªé»äœã¡ã¢ãªãŒãã®ãããªããã€ã¹ã¯å®çšåããŠããŸããã",
"title": "â» ç¯å²å€: ã匷èªé»äœããšå§é»äœ"
},
{
"paragraph_id": 61,
"tag": "p",
"text": "ããããä»ã®çšéã§ããããã®ç©è³ªã¯ç£æ¥ã«å®çšåãããŠããŸãã",
"title": "â» ç¯å²å€: ã匷èªé»äœããšå§é»äœ"
},
{
"paragraph_id": 62,
"tag": "p",
"text": "ãã¿ã³é
žéãããªãé
žãªããŠã ã¯ããã®ç©è³ªã«å§åããããããšé»å§ãçºçããäºãããå§é»äœ(ãã€ã§ããã)ãšããçŽ åãšããŠæŽ»çšãããŠããŸãã(â» ãé«çåŠæ ¡ååŠI/ã»ã©ããã¯ã¹ãã§ãå§é»æ§ã»ã©ããã¯ã¹ããšããŠå§é»äœã玹ä»ãé«æ ¡ååŠã®ç¯å²å
ã§ãã2017幎ã®çŸåšã§ã¯é«æ ¡3幎ã®éžæååŠ(å°éååŠ)ã®ç¯å²å
ã§ãããã)",
"title": "â» ç¯å²å€: ã匷èªé»äœããšå§é»äœ"
},
{
"paragraph_id": 63,
"tag": "p",
"text": "ãªãããããã®å§é»äœã«ãé»å§ããããããšãç©è³ªãã²ããã",
"title": "â» ç¯å²å€: ã匷èªé»äœããšå§é»äœ"
},
{
"paragraph_id": 64,
"tag": "p",
"text": "ãã®ãããå§é»äœã«äº€æµé»å§ãå ããããšã§ãå§é»äœãçæéã§äœåãåšæçã«æ¯åããããšã«ãããå§é»äœã®åšå²ã«ãã空æ°ãæ¯åãããäºãã§ããã®ã§ãè¶
é³æ³¢ãçºçããããã®çŽ åãšããŠããã§ã«å®çšåãããŠããŸãã",
"title": "â» ç¯å²å€: ã匷èªé»äœããšå§é»äœ"
},
{
"paragraph_id": 65,
"tag": "p",
"text": "ãªããããçš®é¡ã®ç©è³ªããå§åããããããšé»å§ãçºçããçŸè±¡ãèµ·ããç©è³ªã®å Žåããã®ãããªæ§è³ªã®ããšãå§é»æ§(ãã€ã§ããã)ãšãããŸãã",
"title": "â» ç¯å²å€: ã匷èªé»äœããšå§é»äœ"
},
{
"paragraph_id": 66,
"tag": "p",
"text": "ã±ã€çŽ Si ãã²ã«ãããŠã Ge ã¯ãå°äœãšçµ¶çžäœã®äžéã®æµæçããã€ããšãããã±ã€çŽ (ã·ãªã³ã³)ãã²ã«ãããŠã ãªã©ã¯åå°äœãšèšãããŸãã",
"title": "åå°äœ"
},
{
"paragraph_id": 67,
"tag": "p",
"text": "ãã®åå°äœã®çµæ¶ã«ããããã«ããªã³Pãªã©ã®äžçŽç©ãå
¥ããããšã§ãæµæçã倧ããäžããããŸãã",
"title": "åå°äœ"
},
{
"paragraph_id": 68,
"tag": "p",
"text": "",
"title": "åå°äœ"
},
{
"paragraph_id": 69,
"tag": "p",
"text": "",
"title": "åå°äœ"
},
{
"paragraph_id": 70,
"tag": "p",
"text": "ã·ãªã³ã³ååã¯äŸ¡é»åã4åã§ãããã·ãªã³ã³ã®çµæ¶ã¯ã4ã€ã®äŸ¡é»åãå
±æçµåãããŠããŸãã",
"title": "åå°äœ"
},
{
"paragraph_id": 71,
"tag": "p",
"text": "ããã«ãªã³Pãå ãããšããªã³ã¯äŸ¡é»åã5åãªã®ã§ã1åã®äŸ¡é»åãäœãããã®äœã£ã䟡é»åãèªç±é»åãšããŠãçµæ¶ãåãåããããã«ãªããŸãã",
"title": "åå°äœ"
},
{
"paragraph_id": 72,
"tag": "p",
"text": "ãã®ãããªä»çµã¿ã§ãã·ãªã³ã³ã«ãªã³ãå ããããšã§ãæµæçã倧ããäœäžããããšããã®ãå®èª¬ã§ãã",
"title": "åå°äœ"
},
{
"paragraph_id": 73,
"tag": "p",
"text": "ãã®ããã«ãè² ã®é»åãäœãããšã§ãå°é»çãäžãã£ãŠãåå°äœã nååå°äœ ãšãããŸãã(ãnã㯠negative ã®ç¥ã)",
"title": "åå°äœ"
},
{
"paragraph_id": 74,
"tag": "p",
"text": "ã·ãªã³ã³ã®çµæ¶ã«ãäžçŽç©ãšããŠãããŠçŽ Bãã¢ã«ãããŠã Alãªã©ã䟡é»åã3åã®å
çŽ ãå ãããšãé»åã1åã足ããªããªããŸãã",
"title": "åå°äœ"
},
{
"paragraph_id": 75,
"tag": "p",
"text": "ãã®ãé»åã®äžè¶³ããã¶ãã®ç©ºåžãããŒã«(postive holeãæ£å)ãšãããŸãã",
"title": "åå°äœ"
},
{
"paragraph_id": 76,
"tag": "p",
"text": "ããŒã«ã¯æ£é»è·ããã¡ãŸãã",
"title": "åå°äœ"
},
{
"paragraph_id": 77,
"tag": "p",
"text": "é»å§ãæãããšããã®ããŒã«ãåããããã«è¿ãã®çµåã«ãã£ãé»åã移åããŸãããããšã®é»åããã£ãå Žæã«æ°ããªããŒã«ãã§ããã®ã§ãèŠããäžã¯ããŒã«ãé»åãšéæ¹åã«åããããã«èŠããŸãã",
"title": "åå°äœ"
},
{
"paragraph_id": 78,
"tag": "p",
"text": "ãã£ãŠãããŒã«ãåãããšã§ãé»æµãæµãããšèŠãªããŸãã",
"title": "åå°äœ"
},
{
"paragraph_id": 79,
"tag": "p",
"text": "ãŸãããã®ããã«ãæ£ã®é»è·ããã€ç²åã«ãã£ãŠå°é»çãäžãã£ãŠãåå°äœã pååå°äœ ãšãããŸãã(ãpã㯠positive ã®ç¥ã)",
"title": "åå°äœ"
},
{
"paragraph_id": 80,
"tag": "p",
"text": "nååå°äœã§ã¯ãèªç±é»åãé»æµãéã¶ã",
"title": "åå°äœ"
},
{
"paragraph_id": 81,
"tag": "p",
"text": "pååå°äœã§ã¯ãããŒã«ãé»æµãéã¶ã",
"title": "åå°äœ"
},
{
"paragraph_id": 82,
"tag": "p",
"text": "ãã®ããã«ãåå°äœäžã§é»è·é»åã®æ
ãæãããã£ãªã¢(carrier)ãšãããŸãã",
"title": "åå°äœ"
},
{
"paragraph_id": 83,
"tag": "p",
"text": "ã€ãŸããnååå°äœã®ãã£ãªã¢ã¯é»åã§ãpååå°äœã®ãã£ãªã¢ã¯ããŒã«ã§ãã",
"title": "åå°äœ"
},
{
"paragraph_id": 84,
"tag": "p",
"text": "pååå°äœãšnååå°äœãæ¥åã(pnæ¥å)ãç©äœããäžæ¹åã®ã¿ã«é»æµãæµãã",
"title": "åå°äœ"
},
{
"paragraph_id": 85,
"tag": "p",
"text": "ãã®ãããªéšåããã€ãªãŒã(diode)ãšãããŸãã",
"title": "åå°äœ"
},
{
"paragraph_id": 86,
"tag": "p",
"text": "påŽã«æ£é»å§ãæããnåŽã«è² é»å§ãæããæãé»æµãæµããŸãã",
"title": "åå°äœ"
},
{
"paragraph_id": 87,
"tag": "p",
"text": "ãã£ãœããpåŽã«è² é»å§ãæããnåŽã«æ£é»å§ãæããŠããé»æµãæµããŸããã",
"title": "åå°äœ"
},
{
"paragraph_id": 88,
"tag": "p",
"text": "åè·¯ã«ãããŠããã€ãªãŒããé»æµãæµãåããé æ¹å(ãã
ãã»ãããŸããã)ãšãããŸããé æ¹åãšã¯å察åããéæ¹åãšãããŸãããã€ãªãŒãã®éæ¹åã«ã¯ãé»æµã¯æµããŸããã",
"title": "åå°äœ"
},
{
"paragraph_id": 89,
"tag": "p",
"text": "ãã®ããã«äžæ¹åã«æµããä»çµã¿ã¯ããã€ãªãŒãã§ã¯ãã€ãã®ãããªä»çµã¿ã§ãé»æµãæµããããã§ãã",
"title": "åå°äœ"
},
{
"paragraph_id": 90,
"tag": "p",
"text": "ãã®ããã«äžæ¹åã«ã ãé»æµãæµãããšãæŽæµ(ãããã
ã)ãšãããŸãããªããåå°äœã䜿ããªããŠããç空管ã§ãæŽæµã ããªãå¯èœã§ãã(ãã ãç空管ã®å Žåãç±ã®çºçãèšå€§ã§ãã£ãããèä¹
æ§ãå£ãã®ã§ãé»åéšåãšããŠã®å®çšæ§ã¯ã空管ã¯äœãã®ã§ãçŸä»£ã¯ç空管ã¯é»åéšåãšããŠã¯äœ¿ãããŠããŸããã)",
"title": "åå°äœ"
},
{
"paragraph_id": 91,
"tag": "p",
"text": "",
"title": "åå°äœ"
},
{
"paragraph_id": 92,
"tag": "p",
"text": "ããœã³ã³ã§ãããžã¿ã«æ³¢åœ¢ãããžã¿ã«ä¿¡å·ã®ããã«åè§ã®é»æµæ³¢åœ¢ãäœã£ãŠããæ¹æ³ã¯ãããããããã®ãã€ãªãŒããšãåŸè¿°ãããã©ã³ãžã¹ã¿ãšããããŸãçµã¿åãããããšã§ãããžã¿ã«æ³¢åœ¢ãã€ãããšããä»çµã¿ã§ãã(â» æ°ç åºçã®æ€å®æç§æžããããããèŠè§£ã§ãã)",
"title": "åå°äœ"
},
{
"paragraph_id": 93,
"tag": "p",
"text": "",
"title": "åå°äœ"
},
{
"paragraph_id": 94,
"tag": "p",
"text": "",
"title": "åå°äœ"
},
{
"paragraph_id": 95,
"tag": "p",
"text": "ãã€ãªãŒãã®påŽã«æ£é»å§ãããããã£ãœãnåŽã«è² é»å§ãããããšãpåŽã§ã¯æ£é»æ¥µã®æ£é»å§ããããŒã«ãåçºããŠæ¥åé¢ãžãšåããããã£ãœãnåŽã§ã¯èªç±é»åãè² é»æ¥µããåçºããŠæ¥åé¢ãžãšåããããããŠãæ¥åé¢ã§ãããŒã«ãšèªç±é»åãã§ãããæ¶æ»
ããŸãããã®çµæãèŠæãäžãæ£é»è·ããæ£é»æ¥µããè² é»æ¥µã«ç§»åããã®ãšãåçã®çµæã«ãªããŸãã",
"title": "åå°äœ"
},
{
"paragraph_id": 96,
"tag": "p",
"text": "ãããŠãæ£é»æ¥µãããã€ãã€ããšããŒã«ãäŸçµŠãããã®ã§ãé»æµãæµãç¶ããŸãã",
"title": "åå°äœ"
},
{
"paragraph_id": 97,
"tag": "p",
"text": "ãã£ãœããpåŽã«è² é»å§ãæããnåŽã«æ£é»å§ãæããæãpåŽã§ã¯ããŒã«ã¯é»æ¥µ(é»æ¥µã«ã¯è² é»å§ãæãã£ãŠãŸã)ã«åŒãå¯ããããæ¥åé¢ããã¯é ããããŸããåæ§ã«nåŽã§ã¯èªç±é»åãé»æ¥µ(æ£é»å§ãæãã£ãŠãŸã)ã«åŒãå¯ããããæ¥åé¢ããã¯é ããããŸãã",
"title": "åå°äœ"
},
{
"paragraph_id": 98,
"tag": "p",
"text": "ãã®çµæãæ¥åé¢ã«ã¯ãäœåãªããŒã«ãäœåãªèªç±é»åããªãç¶æ
ãšãªãããã£ãŠæ¥åé¢ã®ä»è¿ã«ã¯ãã£ãªã¢ããªãããã®æ¥åé¢ä»è¿ã®ãã£ãªã¢ã®ç¡ãéšåã¯ç©ºä¹å±€(ãããŒããããdepletion layer)ãšåŒã°ããŸãã",
"title": "åå°äœ"
},
{
"paragraph_id": 99,
"tag": "p",
"text": "ãããŠããã以éã¯ãããŒã«ãèªç±é»åããããã©ãã«ã移åã®äœå°ããªãã®ã§ããã£ãŠé»æµãæµããŸããã",
"title": "åå°äœ"
},
{
"paragraph_id": 100,
"tag": "p",
"text": "",
"title": "åå°äœ"
},
{
"paragraph_id": 101,
"tag": "p",
"text": "åå°äœã3ã€npnãŸãã¯pnpã®ããã«çµã¿åããããšãé»æµãå¢å¹
(ãããµã)ããããšãã§ããŸããå¢å¹
äœçš(ãããµãããã)ãšãããŸãã",
"title": "åå°äœ"
},
{
"paragraph_id": 102,
"tag": "p",
"text": "NPNãšã¯ãç端ããé ã«èŠãŠNåã»Påã»Nåã®é ã«äžŠãã§ããšããäºã§ãã",
"title": "åå°äœ"
},
{
"paragraph_id": 103,
"tag": "p",
"text": "åæ§ã«ãPNPãšã¯ãç端ããé ã«èŠãŠNåã»Påã»Nåã®é ã«äžŠãã§ããšããäºã§ãã",
"title": "åå°äœ"
},
{
"paragraph_id": 104,
"tag": "p",
"text": "å¢å¹
ãšãã£ãŠãããã£ããŠç¡ãããšãã«ã®ãŒãçºçããããã§ã¯ãªãã®ã§ãæ··åããªãããã«ã",
"title": "åå°äœ"
},
{
"paragraph_id": 105,
"tag": "p",
"text": "説æã®ç°¡ç¥åã®ãããå€éšé»æºãçç¥ãããäºãããããå®éã¯å€éšé»æºãå¿
èŠã§ããåå°äœçŽ åã¯å°ããªé»æµããæµããªãã®ã§ãé»æµãæžããããã®æµæçŽ åãšããŠã®ä¿è·æµæ(ã»ããŠãããŸããã)ãå¿
èŠã§ãã",
"title": "åå°äœ"
},
{
"paragraph_id": 106,
"tag": "p",
"text": "",
"title": "åå°äœ"
},
{
"paragraph_id": 107,
"tag": "p",
"text": "ãªããå³ã®ããã«é·æ¹åœ¢ç¶ã«äžŠãã§ããæ¹åŒã®ãã©ã³ãžã¹ã¿ããã€ããŒã©ãã©ã³ãžã¹ã¿ãšãããŸãã(â» æ€å®æç§æžã®æ°ç åºçã®æç§æžã§ãããã€ããŒã©ãã©ã³ãžã¹ã¿ããã³ã©ã ã§ç¿ãã)",
"title": "åå°äœ"
},
{
"paragraph_id": 108,
"tag": "p",
"text": "ãã€ããŒã©ãã©ã³ãžã¹ã¿ã«ã¯ã端åãäž»ã«3ã€ãããããšããã¿ãããããŒã¹ãããã³ã¬ã¯ã¿ããšããåèš3ã€ã®ç«¯åããããŸãã",
"title": "åå°äœ"
},
{
"paragraph_id": 109,
"tag": "p",
"text": "ãã€ããŒã©ãã©ã³ãžã¹ã¿ã§ã®é»æµã®å¢å¹
ãšã¯ãããŒã¹é»æµãå¢å¹
ããŠã³ã¬ã¯ã¿ã«éããã§ã(PNPã®å Žå)ãé»æµã®åãã¯PNPåã®ã°ãããš NPPåã®ã°ãããšã§ã¯ç°ãªãããã©ã¡ãã®å Žåã§ãããŒã¹é»æµãå¢å¹
ããããšããä»çµã¿ã¯å
±éã§ãã",
"title": "åå°äœ"
},
{
"paragraph_id": 110,
"tag": "p",
"text": "ããŠãæš¡åŒå³ã§ã¯æš¡åŒçã«çãäžã®åå°äœã¯ããããå°ããã«æžãããããå®éã®ãã©ã³ãžã¹ã¿ã¯çãäžã®åå°äœã¯ããã§ã¯ãªãã®ã§ãåèçšåºŠã«ã",
"title": "åå°äœ"
},
{
"paragraph_id": 111,
"tag": "p",
"text": "",
"title": "åå°äœ"
},
{
"paragraph_id": 112,
"tag": "p",
"text": "æè²ã§ã¯ãåå°äœã®é«æ ¡çãå°éå€(é»åå°æ»ä»¥å€)ã®äººããã«ã¯ããããã€ããŒã©ãã©ã³ãžã¹ã¿ãåçŽãªã®ã§çŽ¹ä»ãããããå®éã«åžè²©ã®ã³ã³ãã¥ãŒã¿éšåãªã©ã§ãã䜿ããããã©ã³ãžã¹ã¿ã®æ¹åŒã¯ããããšã¯åœ¢ç¶ããã£ããç°ãªããŸãã",
"title": "åå°äœ"
},
{
"paragraph_id": 113,
"tag": "p",
"text": "åžè²©ã®ã³ã³ãã¥ãŒã¿éšåã®ãã©ã³ãžã¹ã¿ã«ã¯ãé»çå¹æãã©ã³ãžã¹ã¿ãšããããæ¹åŒã®ãã®ããããçšããããŸãã(ãã¡ãããé»çå¹æãã©ã³ãžã¹ã¿ã«ãããå¢å¹
ãã®æ©èœããããŸãã)",
"title": "åå°äœ"
},
{
"paragraph_id": 114,
"tag": "p",
"text": "(⻠詳ããã¯å€§åŠã®é»æ°å·¥åŠãŸãã¯å·¥æ¥é«æ ¡ã®é»ååè·¯ãªã©ã®ç§ç®ã§ç¿ãã)",
"title": "åå°äœ"
},
{
"paragraph_id": 115,
"tag": "p",
"text": "ãã©ã³ãžã¹ã¿ã¯ãåè·¯å³ã§ã¯ãæš¡åŒçã«äžå³ã®ããã«æžãããŸãã",
"title": "åå°äœ"
},
{
"paragraph_id": 116,
"tag": "p",
"text": "",
"title": "åå°äœ"
},
{
"paragraph_id": 117,
"tag": "p",
"text": "ãã€ãªãŒãããã©ã³ãžã¹ã¿ã®ä»ã«ãåå°äœãçµã¿åãããé»åéšåã¯ããã®ã§ãã(ä»ã«ãããµã€ãªã¹ã¿ããªã©è²ã
ãšãããŸã)ãé«æ ¡ç©çã®ç¯å²ãè¶
ããã®ã§ã説æã¯çç¥ããŸãã(â» ããä»äºã§å°éçãªæ
å ±ãå¿
èŠã«ãªãã°ãå·¥æ¥é«æ ¡ããã®ãé»ååè·¯ãã®æç§æžã«ãã£ãã詳ããæžããŠããã®ã§ããããèªãã°ããã§ãããªããæžåºã®è³æ Œã³ãŒããŒæ¬ã«ããé»æ°å·¥äºå£«ãé»æ°äž»ä»»æè¡è
è©Šéšãªã©ã®å¯Ÿçåã«ã¯ãã»ãŒé»ååè·¯ãç¯å²å€ãªã®ã§ãããŸãé»ååè·¯ã®èª¬æã¯æžããŠãŸããããªã®ã§ãå·¥æ¥é«æ ¡ãé»ååè·¯ãã®æç§æžããŸãã¯å·¥æ¥é«å°ãªã©ã®åçã®ç§ç®ã®æç§æžãåç
§ã®ããšã)",
"title": "åå°äœ"
},
{
"paragraph_id": 118,
"tag": "p",
"text": "",
"title": "åå°äœ"
},
{
"paragraph_id": 119,
"tag": "p",
"text": "ããœã³ã³ã®CPUãªã©ã®éšåããäžèº«ã®å€ãã¯åå°äœã§ããããã€ãªãŒãããã©ã³ãžã¹ã¿ãªã©ã®çŽ åãCPUãªã©ã®å
éšã«ãããããããŸãããšèšãããŠããŸãã(â» ä»ã«ããæ°Žæ¶æ¯ååããªã©è²ã
ãšCPUå
ã«ã¯ ããããç©ç2ã®ç¯å²å€ãªã®ã§èª¬æãçç¥ã)",
"title": "åå°äœ"
},
{
"paragraph_id": 120,
"tag": "p",
"text": "éç©åè·¯ãLSI(Large Scale Integratedã倧èŠæš¡éç©åè·¯)ãªã©ãšèšãããé»åéšåãããªã«ãéç©(ãéç©ããè±èªã§ integrate ã€ã³ãã°ã¬ãŒã ãšãã)ããã®ããšãããšãåå°äœçŽ åãéç©ãããšèšãæå³ã§ãã",
"title": "åå°äœ"
},
{
"paragraph_id": 121,
"tag": "p",
"text": "ãªãããICã(ã¢ã€ã·ãŒ)ãšã¯ Integrated Circuit ã®ç¥ç§°ã§ããããããåèš³ãããã®ããéç©åè·¯ãã§ãã",
"title": "åå°äœ"
},
{
"paragraph_id": 122,
"tag": "p",
"text": "ã€ãŸããéç©åè·¯ãLSIã®äžèº«ã¯ãåå°äœã§ããããã©ã³ãžã¹ã¿ãªã©ã®çŽ åãé«å¯åºŠã§ããã®åè·¯äžã«è©°ãŸã£ãŠããŸãã",
"title": "åå°äœ"
},
{
"paragraph_id": 123,
"tag": "p",
"text": "é»åéšåã®åå°äœã®ææãšããŠã¯ãéåžžã¯ã·ãªã³ã³çµæ¶ã䜿ãããŸãã(â» åæ通ãæ°ç ãªã©ãçµæ¶ã§ããããšãèšåã)",
"title": "åå°äœ"
},
{
"paragraph_id": 124,
"tag": "p",
"text": "ç 究éçºã§ã¯ã·ãªã³ã³ä»¥å€ã®ææãç 究ãããŠããäžéšã®ç¹æ®çšéã§ã¯GaAsãInGaPãªã©ãå©çšãããŠããã(â» æ°ç ã®æ€å®æç§æžã¯GaAsãInGaPãªã©ã«ã³ã©ã ã§èšå)ããããçŸç¶ã§ã¯ãã·ãªã³ã³ãåžè²©ã®ã³ã³ãã¥ãŒã¿éšåäžã®åå°äœçŽ åã®ææã§ã¯äž»æµã§ãã",
"title": "åå°äœ"
},
{
"paragraph_id": 125,
"tag": "p",
"text": "ãªããã·ãªã³ã³åå°äœã®ææå
éšã¯ã·ãªã³ã³çµæ¶ã§ããããè¡šé¢ã¯ä¿è·èããã³çµ¶çžã®ããã«é
žåãããããŠãããã·ãªã³ã³åå°äœè¡šé¢ã¯é
žåã·ãªã³ã³ã®ä¿è·èã«ãªã£ãŠããŸããã·ãªã³ã³ãé
žåãããšã絶çžç©ã«ãªãã®ã§ãä¿è·èã«ãªãããã§ã(â» æ°ç åºçã®æç§æžãããèšã£ãŠããŸãã)",
"title": "åå°äœ"
},
{
"paragraph_id": 126,
"tag": "p",
"text": "åå°äœã®å
éšã«ãæ·»å ç©ãªã©ã§ç¹æ§ãå€ããããšã«ãããæµæãã³ã³ãã³ãµãåå°äœå
éšã«è£œé ã§ããŸãã(â» æ°ç ããæµæãã³ã³ãã³ãµãåå°äœå
éšã§äœã£ãŠããäºã«èšåã)",
"title": "åå°äœ"
},
{
"paragraph_id": 127,
"tag": "p",
"text": "(â» ç¯å²å€: )ããããã³ã€ã«ã¯åå°äœå
éšã«äœãããšãåºæ¥ç¡ãã§ãã",
"title": "åå°äœ"
},
{
"paragraph_id": 128,
"tag": "p",
"text": "ç£å ŽBã®äžããé»è·qã®è·é»ç²åãé床vã§éåãããšãããŒã¬ã³ãåã¯ãã¯ãã«å€ç©ãçšã㊠f=qã»vÃB ã®åãç²åã«åãããããã§èŠ³æž¬è
ã®åº§æšç³»ãå€ãããšããŠãåãç²åããç²åãšåãæ¹åã«é床vã§åã座æšåœ¢Kã®äžã®èŠ³æž¬è
ããèŠããã©ããªãã? 座æšç³»Kã§ã¯ãç²åã®é床㯠v(K)=0 ã§ãããç£æã®é床ã Vb ãšãããšãåã®åº§æšç³»ã®ç²åãšã¯å察æ¹åã«åãã®ã§ã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 129,
"tag": "p",
"text": "æ°ãã座æšç³»Kãã芳枬ããŠããç²åã f=qã»vÃB ã®å€§ããã®åãåããŠå éãããããšã«ã¯å€ãããŸãããã座æšç³»kã§ã¯ãè·é»ç²åã¯éæ¢ããŠããã®ã«ãããŒã¬ã³ãåãåãããšèããã®ã¯äžåçã§ããç£æã¯ãVb=-v ã§éåããŠããã®ã§ãç£æã®éåã«ãã£ãŠ f=qã»(-Vb)ÃB = -qã»VbÃB ã®åãåãããšèããã¹ãã§ããç²åã質é0ã®è³ªç¹ãšã¿ãªãã°ãéæ¢ããŠããè·é»ç²åã«åãåãŒããã®ã¯ãé»å Žã ãã ãããã€ãŸãé床 Vb ã§éåããç£æãã E=-VbÃB ã®èªå°é»å Žãèªèµ·ããããšã«ãªããŸãããã®ãšããç£å Žãšèªå°ãããé»å Žã¯åçŽã§ãã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 130,
"tag": "p",
"text": "ããããéåããé»å Žã¯ç£çãäœãããšããã°ãã¢ã³ããŒã«ã®æ³å ãçŽç·ç¶ã«ç¡éã«é·ãå°ç·ãæµãã é»æµI ã¯è·é¢R ã ãé¢ããå Žæã« Bã»2Ïr=ÎŒI ã®ç£å ŽãäœãŸããããšããçŸè±¡ã¯ããã€ã¯ãå°ç·ã®äžã§è·é»ç²åãéåããããšã«ãã£ãŠãè·é»ç²åãšãã£ããã«ãã®ç²åãäœãé»å Žãåãããã®é»å Žã®éåããç£å Žãèªèµ·ããŠããŸããããšããå¯èœæ§ããããŸãã é»æµãæµããŠããç¡éé·ã®ããŸã£ãããªå°ç·ãèããŸããç·å¯åºŠ q[C/m] ã§ååžããé»è·ã¯ãå³ã®ããã«åç察称ãªé»è·ãäœãŸãã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 131,
"tag": "p",
"text": "(â» ããã«å³ãã)",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 132,
"tag": "p",
"text": "çŽç·ããè·é¢rã®ãšãã®é»æ°åç·ã®å¯åºŠDã¯",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 133,
"tag": "p",
"text": "ãã£ãŠ",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 134,
"tag": "p",
"text": "é»æµ I ã¯é»è·ååž q ãé床 Ve ã§éåããŠãããšããŠ",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 135,
"tag": "p",
"text": "ãšå®çŸ©ããã°ã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 136,
"tag": "p",
"text": "é»æµ qVe ãè·é¢ r ã®ãšããã«äœãç£å ŽBã¯ã¢ã³ããŒã«ã®æ³åããã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 137,
"tag": "p",
"text": "ãšãªããŸãã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 138,
"tag": "p",
"text": "ãã®ãšããç£å Žã®åãã¯ãVe ãã ååŸræ¹å ã«ãããåãåãã§ãã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 139,
"tag": "p",
"text": "åããŸã§ãµãããŠãã¯ãã«ç©ã§è¡šãã°ã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 140,
"tag": "p",
"text": "ã€ãŸã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 141,
"tag": "p",
"text": "ãšãããéèŠãªçµè«ãåŸãããŸãã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 142,
"tag": "p",
"text": "ãããã¯ã ÎŒH=B ããã¡ã㊠B=ÎŒH=εΌ Ve ÃE ãã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 143,
"tag": "p",
"text": "ã§ãã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 144,
"tag": "p",
"text": "ãŸãšã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 145,
"tag": "p",
"text": "é床 Vbã§éåããç£æBã¯",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 146,
"tag": "p",
"text": "ã®èªå°é»å Žãèªèµ·ããŸãã ã»ã»â¡1",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 147,
"tag": "p",
"text": "é床 Ve ã§éåããé»å Ž E ã¯",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 148,
"tag": "p",
"text": "ã®èªå°ç£å ŽãäœãŸãã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 149,
"tag": "p",
"text": "E,Bã®ãããã«ãD,Hã䜿ã£ãŠè¡šèšããã°ã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 150,
"tag": "p",
"text": "ãã€",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 151,
"tag": "p",
"text": "ããŠãé»ç£æ³¢ãé床Cã§ç空äžãäŒãããšããã°ã Vb = Ve = C ãšããŸãã â¡1åŒãšâ¡2åŒã®å€ç©ããšããšã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 152,
"tag": "p",
"text": "ãã£ãŠ",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 153,
"tag": "p",
"text": "ã§ãã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 154,
"tag": "p",
"text": "ãã£ãŠãé»ç£æ³¢ã®é床㯠c = 1 ε ÎŒ {\\displaystyle c={\\frac {1}{\\sqrt {\\varepsilon \\mu }}}} ãšäºæž¬ã§ããŸãã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 155,
"tag": "p",
"text": "ãã®ÎµãšÎŒã«å®æž¬å€ãå
¥ãããšãå
éã®æž¬å®å€ c = 299792458 m / s {\\displaystyle c=299792458m/s} ãšãé«ã粟床ã§äžèŽããŸãã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 156,
"tag": "p",
"text": "ãã®äºãããå
ã¯ãé»ç£æ³¢ã§ããäºãåãããŸãããŸããé»ç£æ³¢ã¯ãå
é床Cã§ç空äžãäŒããŸãã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 157,
"tag": "p",
"text": "ãŸãããããããéåé»å Žã®èªå°ããç£å Žã¯",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 158,
"tag": "p",
"text": "ãšãå€åœ¢ã§ããŸãã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 159,
"tag": "p",
"text": "3åŒããã¬ãŠã¹ã®æ³å(1åŒ) ãšçµã¿åããããšãã¢ã³ããŒã«ã®æ³å(2åŒ)ãåŸãããŸãã ãã£ãŠããé床 Ve ã§éåããé»å Ž E ã¯ã B=εΌ Ve ÃE ã®èªå°ç£å ŽãäœãŸããããšããéçšã劥åœã ã£ãããšãããããŸãã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 160,
"tag": "p",
"text": "é»ç£æ³¢ã§ã¯é»å Ž E ãšç£å Ž B ãå
é C ã§éåããŠããã®ã§ ç£æã®éåé床 Vb 㯠Vb = C ã§ãããèªå°é»å Ž E 㯠E =-VbÃB ã§ããã®ã§ãäž¡åŒãã E = -cÃB ã§ãã(é»ç£æ³¢ã®é»å Žãšç£å Žã®é¢ä¿åŒ)ãªã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 161,
"tag": "p",
"text": "ã§ããã®ã§ã é»ç£æ³¢ã¯",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 162,
"tag": "p",
"text": "ã®æ¹åã«é²ãã§ããã¯ãã§ãããšããããšã泚ç®ããŸãããã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 163,
"tag": "p",
"text": "ãã® E à H {\\displaystyle \\mathbb {E} \\times \\mathbb {H} } ã§å®çŸ©ãããéã ãã€ã³ãã£ã³ã° ãã¯ãã« ãšãã¶ã ããã¯åäœé¢ç©ããšãã£ãŠæµãåºãé»ç£å Žã®ãšãã«ã®ãŒã®æµãã®éãããããã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 164,
"tag": "p",
"text": "ããŠãé»ç£å Žã®ãšãã«ã®ãŒå¯åºŠã¯ u = 1 2 ε E 2 + 1 2 ÎŒ H 2 {\\displaystyle u={\\frac {1}{2}}\\varepsilon E^{2}+{\\frac {1}{2}}\\mu H^{2}} ãªã®ã§ãããã«é»ç£æ³¢ã®é»å Žãšç£å Žã®é¢ä¿åŒ E = â C à B {\\displaystyle \\mathbb {E} =-\\mathbb {C} \\times \\mathbb {B} } ã代å
¥ããŠã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 165,
"tag": "p",
"text": "ã®é¢ä¿ãçšãããšã(ãšãã«ã®ãŒã§ã¯ã2ä¹ã«ãããã€ãã¹ç¬Šå·ããªããªãã®ã§ã絶察å€ãåã£ãŠ|E|=|cÃB| ãšããŠãããšãèšç®ãç°¡åã«ãªãå ŽåããããŸãã)",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 166,
"tag": "p",
"text": "çµæãšããŠ",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 167,
"tag": "p",
"text": "ãšãªããŸãã é»ç£æ³¢ããå£ã«ããã£ãŠåžåããããšããåäœæéã«åäœé¢ç©ããã å
éC ã®å€§ããã®äœç©ã®ãªãã®é»ç£æ³¢ãå£ã«è¡çªããã®ã§ã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 168,
"tag": "p",
"text": "ã®ãšãã«ã®ãŒããåäœæéã«åäœé¢ç©ã«æµã蟌ãã¯ãã§ãã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 169,
"tag": "p",
"text": "s= cã»u ã« u= εã»E^2 ã代å
¥ããŠã ε ÎŒ â
c 2 = 1 {\\displaystyle \\epsilon \\mu \\cdot c^{2}=1} ãš |E|=|cÃB|ãå©çšãããšãçµæçã«",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 170,
"tag": "p",
"text": "ã§ãã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 171,
"tag": "p",
"text": "ãã£ãŠãã€ã³ãã£ã³ã° ãã¯ãã« EÃH ã¯åäœé¢ç©ãéã£ãŠæµãåºãé»ç£å Žã®ãšãã«ã®ãŒã®æµããããããã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 172,
"tag": "p",
"text": "ãã€ã³ãã£ã³ã° ãã¯ãã« S = EÃH = εΌ(C)EÃH ã¯",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 173,
"tag": "p",
"text": "ã§ãã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 174,
"tag": "p",
"text": "倩äžãçãªèª¬æã§ããããã® G=DÃB ãšããéã¯ãéåéã®å¯åºŠã§ãããã®é G=DÃB ããé»ç£æ³¢ã®ãéåéå¯åºŠã(ããã©ããããã¿ã€ã©)ãšãããŸããå®éã«ãDÃB ã®åäœã¯",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 175,
"tag": "p",
"text": "ãšãªããŸãã ãããã«ãéåéã®å¯åºŠã®åäœãšçããã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 176,
"tag": "p",
"text": "ãšããã§ãã®ã¡ã®åå
ã§ç¿ãããå
é»å¹æã§ã¯ ãšãã«ã®ãŒuãšéåépã®é¢ä¿ã¯ãå
é床Cããã¡ããŠã u=cp ãšæžããŸãã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 177,
"tag": "p",
"text": "ãããã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 178,
"tag": "p",
"text": "åããŸã§å«ããŠ",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 179,
"tag": "p",
"text": "ãšãªã£ãŠã確ãã« G = DÃB ã¯éåéå¯åºŠãšãªããŸãã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 180,
"tag": "p",
"text": "é·ãLã®ãŸã£ãããªééããé床vã§ç£å ŽBã®äžã暪åããšããŸããç°¡åã®ãããééã®è»žãšé床vã®æ¹åãšç£å ŽBã¯åçŽãšããŸãããã®ãšããééã®äžã®é»è·ã«ãããåããã³é»å Žã¯ããŒã¬ã³ãåã«ããã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 181,
"tag": "p",
"text": "é»å ŽEã«ãã£ãŠé·ãLã ããé»è·qãäžãããããããšãã«ã®ãŒã¯ qEL å€åããŸããé»äœã¯ V=EL ã§ãã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
},
{
"paragraph_id": 182,
"tag": "p",
"text": "ãããããèªå°é»å§ V ã¯ãç£æã®1ç§ãããã®æéå€åã«ãªããŸãã ã§ã¯ãä»®ã«åºå®ãããåè·¯ã®äžã«ãœã¬ãã€ããéããŠããã®ãœã¬ãã€ãã«äº€æµé»æµãæµããå Žåããåè·¯ã«èªå°é»å§ãçºçããã®ã ããããçãã¯ããããã",
"title": "çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ"
}
] | null | {{substub}}
== éé»èªå°ãšèªé»å極 ==
=== ã³ã³ãã³ãµãŒ ===
{{Main|é«çåŠæ ¡ç©ç/ç©çI/é»æ°#ã³ã³ãã³ãµãŒ}}
=== èªé»äœ ===
ãŸããé«æ ¡ç©çã§ãããèªé»äœãïŒããã§ãããïŒãšã¯ãéåžžã®ã»ã©ããã¯ãé²æ¯ïŒãã€ã«ïŒããããã¯éåžžã®ãã©ã¹ããã¯ãªã©ã®ããã«ãé»æ°ãéããªãç©è³ªã§ãããã»ã©ããã¯ããã€ã«ã®ããã«ãç³ã®ãããªæ§è³ªããã€ç©è³ªããèªé»äœã§ããå Žåãå€ãã
ã€ãŸããéå±ã¯ãèªé»äœã§ã¯ãªããéå±ã¯ãèªé»äœã§ã¯ãªããïŒéå±ã¯ïŒå°äœã§ããã
ã§ã¯ãèªé»äœã®ç©çã«ã€ããŠã説æããã
[[File:èªé»äœã³ã³ãã³ãµãŒ.svg|thumb|400px|èªé»äœãå
¥ããã³ã³ãã³ãµãŒ]]
ã³ã³ãã³ãµãŒã«èªé»äœãå
¥ãããšãèªé»äœãèªé»å極ãèµ·ãããããã³ã³ãã³ãµã®ãã©ã¹æ¥µæ¿ã§çºçããé»æ°åç·ã®ããã€ããæã¡æ¶ãããã
ãã®çµæãèªé»äœã®å
¥ã£ãã³ã³ãã³ãµãŒã®æ¥µæ¿éã®é»å Žã¯ã極æ¿ã®é»è·å¯åºŠã§çºçããé»è·ãç空äžã§ã€ããé»å Žããã匱ããªãã
ãã®çµæãéé»å®¹éãå€ããã
ããŠãç空äžã®éé»å®¹éã®å
¬åŒã¯ã
:<math>C=\varepsilon_0 \frac{S}{d}</math>
ã§ãã£ãã
èªé»äœã®ããå Žåã®éé»å®¹éã¯ã
:<math>C=\varepsilon \frac{S}{d}</math>
ãšãªãã
ããã§ã <math>\varepsilon </math>ã'''èªé»ç'''ïŒããã§ããã€ïŒãšããã
<math>\varepsilon_0 </math>ãã'''ç空äžã®èªé»ç'''ãšããã
{| class="wikitable" style="float:right"
|+ ç©è³ªã®æ¯èªé»ç
|- style="background:silver"
! ç©è³ª !! æ¯èªé»ç
|-
| ç©ºæ° (20â) || 1.0005
|-
| ãã©ãã£ã³ (20â) || 2.2
|-
| ããŒã«çŽ (20â) || 3.2
|-
| é²æ¯ || 7.0
|-
| æ°Ž (20â) || çŽ80
|-
| ãã¿ã³é
žããªãŠã || çŽ5000
|-
|}
ããã§ãæ¯
:<math> \varepsilon _r = \frac{\varepsilon}{\varepsilon_0}</math>
ãã'''æ¯èªé»ç'''ïŒã² ããã§ããã€ïŒãšããã
ã€ãŸãã<math> \varepsilon _r </math> ã¯æ¯èªé»çã§ããã
ãã£ãœãã <math> \varepsilon _0 </math> ããã³ <math> \varepsilon </math> ã¯ãæ¯èªé»çã§ã¯ãªãã
æ¯èªé»ç <math> \varepsilon _r </math> ããã¡ããã°ãéé»å®¹é C ã®åŒã¯ã
:<math> C = \varepsilon \frac{S}{d} = \varepsilon _r \varepsilon _0 \frac{S}{d} </math>
ãšæžããã
=== ã³ã³ãã³ãµã®éé»ãšãã«ã®ãŒ ===
:
U=2â»Â¹CV²
=2â»Â¹QV
=(2C)â»Â¹Q²
== é»æµã«ããç£ç ==
ç£ç³ã®ãŸããã«ã¯ç©äœãåããåã®ãããã®ãçããŠããã
ããã'''ç£å Ž'''ïŒãã°ïŒãšåŒã¶ã'''ç£ç'''ïŒãããïŒãšãããã
é»æµãæµããŠãããšãã«ãããã®ãŸããã«ã¯ãå³ããã®æ³åïŒright-handed screw ruleïŒã«åŸãåãã«ç£çãçããã
é»æµI[A]ãçŽç·çã«æµããŠãããšããç£çã®å€§ããã¯
<math>
B = \frac {\mu_0} {2\pi a} I
</math>
ã§ããããšãç¥ãããŠããã
ããã§ãaã¯ç£æå¯åºŠã枬ãç¹ãšãé»ç·ã®è·é¢ã
ãŸãã<math>\mu_0</math>ã¯ç空ã®éç£çïŒãšãããã€ãpermeabilityïŒãè¡šããå€ã¯ <math>4\pi \times 10^{-7}</math>[H/m] ã§ããã
<!-- ã¢ã³ããŒã«ã®æ³å? -->
=== é»ç£èªå°ãšé»ç£æ³¢ ===
==== é»ç£èªå° ====
ç£å Žã䌎ãç©äœãéåãããšããã®ãŸããã«ã¯é»å Žãçããããšã'''é»ç£èªå°'''ïŒã§ããããã©ããelectromagnetic inductionïŒãšããã
ä»®ã«ããœã¬ãã€ãïŒsolenoidãã³ã€ã«ã®ããšïŒã®è¿ãã§ãããè¡ãªã£ããšãããšãçããé»å Žã«ãã£ãŠãœã¬ãã€ãã®äžã«ã¯é»æµãæµããã
çããé»å Žã®å€§ããã¯ã
<math>
\vec E = \frac 1 {2\pi a} \frac {d\vec B}{d t}
</math>
ãšãªãã(ååŸaã®å圢ã®ã³ã€ã«ã®å Žåã)
Eã®åäœã¯[V/m]ã§ãããBã®åäœã¯[T]ã§ããã
==== é»ç£æ³¢ ====
ç£å Žã®åãã«ãã£ãŠé»å ŽãåŒãèµ·ããããããšãé»ç£èªå°ã®ç¯ã§èŠãã
ãŸããå®éã«ã¯é»å Žã®å€åã«ãã£ãŠç£å ŽãåŒãèµ·ããããããšãå®éšã«ãã£ãŠç¥ãããŠããã
ããã«ãã£ãŠäœããªã空éäžãé»å Žãšç£å ŽãäŒæããŠããããšãäºæ³ãããã
(:é»ç£æ³¢ã®äŒæã®schematicãªçµµ)
:â» åžè²©ã®å€§åŠçããæç§æžãèªãã§ãããã¯ã«ãã«ãã®å®éšã説æããŠãªãã®ã§ãé«æ ¡åŽã§èª¬æããã
:â» ãªããé«æ ¡ã§ã¯å°éãç©çãã§ç¿ãå
容ã
ãŸããç©çå®éšå®¶ãã«ãã¯æŸé»å®éšã«ãããåä¿¡æ©ãåè·¯äžã«ã®ã£ããã®ããåè·¯ãšããŠãéä¿¡åŽã®æŸé»ã«ããé»å Žãé éçã«é¢ããäœçœ®ã«ããåä¿¡åŽã®åè·¯ã«äŒããããšã確èªããã
ãã®å®éšã®éããã«ãã¯åä¿¡åè·¯ã®åãããããããšå€ããŠå®éšããããšã«ãããéä¿¡æ©ã®åãã«å¯ŸããŠã®åä¿¡æ©ã®åãã«ãã£ãŠé»å Žã®äŒããæ¹ãç°ãªãããšãããé»å Žã®é éäœçšã«åå
æ§ãããäºãåãã£ãã
:ïŒâ» ç¯å²å€ïŒãªããæ¹è§£ç³ãªã©ã«åå
äœçšã®ããããšã¯ããã§ã«ãã®æ代ã«åãã£ãŠãããšæãããã
é»å Žã®ãã®äœçšã«ã¯åå
æ§ãããã®ã§ãæ³¢ã§ãããšã¿ãªãããšã¯åŠ¥åœã§ãããã
ãã«ãã®å®éšãããå®éšçã«ãããããšãšããŠ
:é»å Žã¯é éäœçšã§äŒããããš
:æŸé»ã¯é»å Žã®é éäœçšãçããããããš
:ãã®é»å Žã®é éäœçšã«ã¯ãåå
äœçšã®ããããš
ãå®éšçã«ãããã
ç©çåŠã§ã¯ããã«ãã®å®éšã®ä»¥åãããçè«ç©çåŠè
ã®ãã¯ã¹ãŠã§ã«ã«ããã
é»ç£æ³¢ãšãããé»å Žãšç£å Žã®çžäºäœçšã«ãã£ãŠç空äžãäŒéããäºæž¬ãããŠããã
ãªã®ã§ããã«ãã®å®éšã¯ããã¯ã¹ãŠã§ã«ã®äºæž¬ããé»ç£æ³¢ã ãšã¿ãªãããã
çŸä»£ã§ãç©çåŠè
ã¯ãããã¿ãªããŠããã
ãªãããã¯ã¹ãŠã§ã«ãçè«èšç®ã§æ±ããé»ç£æ³¢ã®é床ãæ±ãããšããããã§ã«ç¥ãããŠããå
éã®å€§ããïŒããã 3Ã10<sup>8</sup> m/s ïŒã«ç²ŸåºŠããäžèŽããã
ãã®ããšããããå
ã¯é»ç£æ³¢ã®äžçš®ã§ããããšãåããã
:ïŒåæ通ã®æç§æžã«ããäœè«: ïŒäœè«ã§ãããã人é¡åã®ç¡ç·éä¿¡ã«æåãã人ç©ã¯ãã«ã³ãŒãã§ããïŒãã«ãã§ã¯ãªãã®ã§ã誀解ããªãããã«ïŒã
ãã«ãã®å®éšã§ã¯ãå³å¯ã«ã¯å°ãªããšãæŸé»ã®é»å ŽãäŒããããšãã芳枬ã§ããŠãªããããããç£å Žããã®å®éšã§äŒãããšèããŠãæ¯éãçããŠç¡ãããå®éã«äººé¡ã«ã¯æ¯éã¯çããŠãªãã®ã§ãä»ã§ããã«ãã®å®éšããã¯ã¹ãŠã§ã«ã®äºæž¬ããé»ç£æ³¢ã®èšŒæã®å®éšãšããŠäŒããããŠããã
ãªããå
ã«ã¯ãåå°ãå±æãåæããã€ã³ã°ã¹ãªããã®åæãªã©ããããããã«ãã®æŸé»å®éšã®ãããªé»ç£æ³¢ã®ç«è±æŸé»ã®å®éšã§ããå
ã®å®éšãšåæ§ã®é
眮ã§ãéå±æ¿ãé
眮ããŠç¢ºèªããããšã§ãé»ç£æ³¢ãåå°ãå±æãåæããã€ã³ã°ã¹ãªããã®åæãªã©ã®çŸè±¡ãèµ·ããããšããå®éšçã«ã確èªãããŠããïŒâ» åèæç® :å®æåºçã®å°éãç©çãã®æ€å®æç§æžïŒïŒâ» ã€ã³ã°ã®ã¹ãªããã®é»ç£æ³¢å®éšã«é¢ããŠã¯åæ通ã®æç§æžãç©çãã«ããïŒã
ãããã®ããšããããå
ã¯é»ç£æ³¢ã®äžçš®ã§ãããšã¿ãªãã®ã劥åœã§ããããšãåããã
:ïŒâ» ç¯å²å€ :ïŒãŸããé»ç£æ³¢ã®åå°ãå©çšããŠãé»ç£æ³¢ã®æ³¢é·ã枬å®ããããšã«ãã«ãã¯æåãã<ref>西æ¢æçŸã枬ãæ¹ã®ç§åŠå² II ååããçŽ ç²åãžããææ瀟ã2012幎3æ15æ¥ åççºè¡ã45ããŒãž<br>
åæµ·é倧åŠåºçãè¿ä»£ç§åŠã®æºæµïœç©çåŠç·šã1974ïœ1977幎ããåèã«ããããã§ããããåæµ·é倧ã®ãã®æç®ã¯çµ¶ç</ref>ãé»ç£æ³¢ãåå°ãããã°ããã£ãŠããæ³¢ãšå¹²æžããŠå®åžžæ³¢ãã§ããã¯ãã§ããããã«ãã®å®éšäŸã§ã¯åä¿¡æ©ãéä¿¡æ©ããé¢ããš33cmããšã«é¡èãªåå¿ãåºããšããããã®å®éšã§ã¯åæ³¢é·ã33cmã ã£ãã®ã ãšæããããã€ãŸãæ³¢é·66cmã®é»ç£æ³¢ãå®éšã§çãããããšæãããã
:ãã ãããã«ãã®ãããªæ¹æ³ã§æž¬å®ã§ããæ³¢é·ã¯ã人éãèçŒã§ç¢ºèªã§ããŠæã§åããããããªçšåºŠã®æ³¢é·ã®å€§ããã®å Žåã ãã§ããããã€ãŸããã»ã³ãã¡ãŒãã«åäœã1ã¡ãŒãã«ä»¥äžãšãã®ãããªæ³¢é·ã§ããããã£ãœããããæ³¢é·ãããã¡ãŒãã«åäœããã€ã¯ãã¡ãŒãã«åäœãªã©ã®å Žåã¯ãåææ Œåãªã©ã䜿ã£ãŠæ³¢é·ã枬å®ããããšã«ãªãã詳ããã¯ã[[é«çåŠæ ¡ç©ç/ç©çII/ååãšååæ ž]]ãã®ã³ã©ã ãåç
§ããããã©ãŠã³ããŒãã¡ãŒãã©ã¶ãã©ãªãŒããªã©ã®ç©çåŠè
ãã¹ããã¥ã©ã åéãªã©ã®çŽ æãçšããŠåææ ŒåãäœæããŠããã
{{ã³ã©ã |ïŒâ» ç¯å²å€: ïŒå»çMRIã®ç£æ°ã®æ³¢ããç©çåŠçã«ã¯é»ç£æ³¢|
ãã¯ã¹ãŠã§ã«ã®æ¹çšåŒã§ã¯ãäžè¿°ã®ããã«é»å Žã®å€åãçãããšãç£å Žã®å€åãçããŠãããã«ãã®ç£å Žã®å€åã«ãããŸãé»å Žãå€åããŠããã»ã»ã»ãšããçŸè±¡ã埮åæ¹çšåŒã§èšè¿°ããŠããã
ãã¯ã¹ãŠã§ã«æ¹çšåŒã®æ矩ãšããŠç§åŠé¢ã§ã¯ãæŸå°ç·ïŒXç·ïŒããã¬ãé»æ³¢ãã©ãžãªé»æ³¢ãå¯èŠå
ïŒå€ªéœå
ãé»æ°ç
§æãªã©ïŒãããã¹ãŠé»ç£æ³¢ã§ãããšããŠçµ±äžçã«åŒèšç®ãã§ããããã«ãªããšããç§åŠçãªæ矩ããããXç·ãšå¯èŠå
ãšã®éãã¯ãåã«æ³¢é·ïŒããã³ãæ³¢é·ã«ãã£ãŠæ±ºãŸãéåãšãã«ã®ãŒïŒã®å·®ã§ããããšçŸä»£ïŒ21äžçŽïŒã§ã¯èããããŠããã
ããã§ç£æ¥ãžã®å¿çšãšããŠæ°ã«ãªãã®ã¯ã20äžçŽåŸå以éã®å»çã§ã¯ãXç·ã«ããã¬ã³ãã²ã³æ®åœ±ã®ä»£ããã«ç£å Žã䜿ã£ãŠäººäœãªã©ã®å
éšã芳å¯ããMRIãªã©ã®æè¡ãããããšããäºã
MRIã¯ãç£å Žã°ãããåãäžããããŠãXç·ãšéã£ãŠå®å
šæ§ããããšäž»åŒµãããããããããã¯ã¹ãŠã§ã«ã®æ¹çšåŒããã§ã¯ãç£å Žã䜿ã£ã以äžãããšãMRIç£å Žã§ãã£ãŠãé»å Žã掟ççã«çºçããããºã§ããªãããã®é»ç£æ³¢ãçºçããäºã«ãªãïŒæ³¢é·ã¯ãšãããïŒã
ãããã倧åŠã®ç©çåŠã®æç§æžã倧åŠã®é»æ°é»åå·¥åŠã®é»ç£æ³¢å·¥åŠã®æç§æžãèªãã§ããããŸããããã£ãå®çšé¢ã®çåã¯çããŠããªããïŒMRIã®å°éæžã¯ã©ããç¥ããªãããå°ãªããšãããç©çåŠãããé»ç£æ³¢å·¥åŠããªã©ã®ç§ç®ã§ã¯ããŸã£ããæ€èšŒãããŠããªããïŒ
: â» MRI ã¯é«æ ¡ã§ãç¿ãããé»ç£æ³¢ãïŒåŸ®åã䜿ããªãç¯å²ã§ïŒä»çµã¿ã ãæç« ã§é«æ ¡ã§ç¿ããããããMRIã®é»ç£æ³¢ãã©ããªã£ãŠããã倧åŠã§ãããã¢ã«æ±ãããŠããªãã
ãªããMRIã¯ãäœå
ã®æ°ŽçŽ ååãšå
±é³Žããæ³¢é·ã ããéžæçã«äººäœã«ç
§å°ããŠããã®åå¿ã®é»ç£æ³¢ã芳å¯ããããšããä»çµã¿ã§ãããæ žç£æ°å
±é³Žæ³ïŒãããã ãããããã»ãïŒãšããä»çµã¿ã®äžçš®ãïŒãªããé»åã¬ã³ãžãããããšäŒŒããããªä»çµã¿ãïŒ
èªè
ã¯ãäœå
ãé»ç£æ³¢ãéã£ãŠãå¹³æ°ãªã®ãïŒããšããçåããããããããªããããªããšèµ€å€ç·ãäœå
ãééããŠããã®ã§ããã®ç¹ã¯èªè
ã¯å®å¿ããŠãããéè¡ATMãªã©ã«ãããéèèªèšŒãã·ã¹ãã ããèµ€å€ç·ã«ãã芳å¯ã·ã¹ãã ã§ããã
ç
é¢ãéè¡ã§ã¯ãïŒç§åŠãªãã©ã·ãŒã®ãšãŒããïŒå©çšè
ãå®å¿ãããããã«ãããšããã«Xç·ãšMRIãšèµ€å€ç·ãšã®å
±éç¹ïŒãã¹ãŠé»ç£æ³¢ã§ããïŒãæããªãããããç©çåŠã§ã¯ãXç·ãç£å Žã®æ³¢ãèµ€å€ç·ãããã¹ãŠé»ç£æ³¢ã§ããããããã¯é»ç£æ³¢ãçºçãããã¢ãã§ããããšãªã£ãŠããã®ãç©çåŠçãªæ¬åœã®èŠè§£ã§ããã
çŸå®ãšããŠãMRIãéè¡ATMéèèªèšŒã®å©çšã§ããã£ããŠïŒXç·ã®è¢«çã¿ããã«ïŒãMRIã§ïŒãããã¯éè¡ATMã§ïŒã¬ã³æ£è
ãçºçãããã ãšãããé»åã¬ã³ãžã¿ããã«å ç±ããŠç±å·ïŒãã£ãããïŒãããã ãšããããããäºä»¶ã¯ã寡èïŒãã¶ãïŒã«ããŠãç§åŠã®çéã§ã¯èããªãã
ãªããXç·ãšMRIã¯å
ã«ãªãé»ç£ãšãã«ã®ãŒã®çºçã®æ©æ§ãéããããšãã°Xç·ã¯äž»ã«ãæŸé»ã«ãã£ãŠçºçããããXç·ç®¡ããæ¯èŒçã«å€§é»å§ã§ã®æŸé»ç®¡ã®äžçš®ã§ãããïŒäžè¬ã®é»æ°ç
§æãªã©ã§ã¯Xç·ã¯çºçããŠããªãã®ã§ãå®å¿ããŠãããïŒ
MRIã®é»ç£æ³¢çºçè£
眮ã¯ãåºæ¬çã«ã¯é»ç£ç³ã«ããé»ç£ãšãã«ã®ãŒã®çºçã§ããã
éèèªèšŒã·ã¹ãã ãªã©ã®èµ€å€ç·çºçè£
眮ã¯ãåºæ¬çã«èµ€å€ç·LEDãªã©ã®åå°äœïŒLEDã¯åå°äœã®äžçš®ïŒã§ããã
åŠæ ¡æè²ã§ã¯ãåŒã®èšç®ããã¹ãã«åºããããã®ã§ãåŠçã¯ã€ããããããåŒã ãã§äœã§ãèšç®ã§ãããã®ããã«é¯èŠããã¡ã§ããããçŸå®ã«ã¯åŒã«ã¯å«ãŸããŠããªããè£
眮ãªã©ã®æ©æ§ã®æ
å ±ãç§åŠçãªæ€èšŒã«ã¯å¿
èŠã§ããã
}}
== ç£æ§äœ ==
[[File:Magnetic field near pole.svg|thumb|right|200px|æ£ç£ç³ã®åšãã«æ¹äœç£éã眮ããŠç£å Žã®åãã調ã¹ãã]]
ç£ç³ã®ãŸããã«ã¯å¥ã®ç£ç³ãåããåã®ããšãšãªããã®ãçããŠããã
ããã'''ç£å Ž'''ïŒãã°ãmagnetic fieldïŒãããã¯'''ç£ç'''ïŒãããïŒãšåŒã¶ãïŒæ¥æ¬ã®ç©çåŠã§ã¯ç£å ŽãšåŒã¶ããšãå€ãããŸããæ¥æ¬ã®é»æ°å·¥åŠã§ã¯ç£çãšåŒã°ããããšãå€ããææ²»æã®èš³èªã®éã®ãæ¥æ¬åœå
ã®æ¥çããšã®éãã«éãããå°å瀟äŒçãªäºè±¡ã§ãããåŒã³æ¹ã¯ç©çã®æ¬è³ªãšã¯é¢ä¿ãªãã®ã§ãããã§ã¯ãã©ã¡ãã®è¡šçŸãçšãããã¯ãæ¬æžã§ã¯ç¹ã«ãã ãããªããè±èªã§ã¯ç©çåŠã»é»æ°å·¥åŠãšãâmagnetic fieldâã§å
±éããŠãããïŒ
éãã³ãã«ããããã±ã«ã«ç£ç³ãè¿ã¥ãããšãç£ç³ã«åžãä»ããããã
ãŸããéãã³ãã«ããããã±ã«ã«åŒ·ãç£åãäžãããšãéãã³ãã«ããããã±ã«ãã®ãã®ãç£å Žãåšå²ã«åãŒãããã«ãªãã
ãã®ãããªãããšããšã¯ç£å Žãæããªãã£ãç©äœãã匷ãç£å Žãåããããšã«ãã£ãŠç£å ŽãåãŒãããã«ãªãçŸè±¡ã'''ç£å'''ïŒãããmagnetizationïŒãšããã
ãããã¯é»è·ã®éé»èªå°ãšå¯Ÿå¿ãããŠãç£åã®ããšã'''ç£æ°èªå°'''ïŒããããã©ããmagnetic inductionïŒãšãããã
ãããŠãéãã³ãã«ããããã±ã«ã®ããã«ãç£ç³ã«åŒãä»ããããããã«ç£åãããèœåãããç©äœã'''匷ç£æ§äœ'''ïŒãããããããããferromagnetïŒãšããã
éãšã³ãã«ããšããã±ã«ã¯åŒ·ç£æ§äœã§ããã
é
ã¯ç£åããªãããé
ã¯ç£ç³ã«åŒãã€ããããªãã®ã§ãé
ã¯åŒ·ç£æ§äœã§ã¯ãªãã
;ç£æ°é®èœ
éé»èªå°ãå©çšãããéé»é®èœïŒããã§ããããžãïŒãšèšããããäžç©ºã®å°äœãã€ãã£ãŠç©è³ªãå²ãããšã§å€éšé»å Žãé®èœããæ¹æ³ããã£ãã®ãšåæ§ã®ãç£æ°ã®é®èœãã匷ç£æ§äœã§ãåºæ¥ããäžç©ºã®åŒ·ç£æ§äœãçšããŠã匷ç£æ§äœã®å
éšã¯ç£å Žãé®èœã§ãããããã'''ç£æ°é®èœ'''ïŒãããããžããmagnetic shieldingïŒãšãããç£æ°ã·ãŒã«ããšãããã
:ç£æ§äœïŒmagnetic substance
:匷ç£æ§äœïŒferromagnet
:åžžç£æ§äœïŒparamagnetic substance
:åç£æ§äœïŒdiamagnetic snbstance
åç£æ§äœãåããã¥ãããããããªãããåã«ããã®ææã«å ããããç£å Žãæã¡æ¶ãæ¹åã«ãç£åãããã ãã®ææã§ããã
ãããããç£åç·ãšããŸãçžäºäœçšããªãç©è³ªãå€ããããšãã°ãã¬ã©ã¹ãæ°Žã«ãããç£æ°ãžã®åœ±é¿ã¯ãç空ã®å Žåãšã»ãšãã©å€ãããªããã¬ã©ã¹ãæ°Žã®æ¯éç£çïŒã² ãšãããã€ïŒ ÎŒ ïŒãã¥ãŒïŒã¯ãã»ãŒ1ã§ããã
ãªããéã®æ¯éç£çã¯ãç¶æ
ã«ãã£ãŠéç£çã«æ°çŸãæ°åã®éããããããwikipediaæ¥æ¬èªçã§èª¿ã¹ãå Žåã®éã®éç£çã¯çŽ5000ã§ããã
ã§ã¯ãéç£çãã»ãŒ1ã®ç©è³ªã¯ãç£å Žã®æ¹åã¯ãå€éšç£å ŽãåºæºãšããŠãã©ã¡ãåãã ãããïŒ å€éšç£å Žãæã¡æ¶ãæ¹åã«ç£åããŠããã®ã ãããïŒ ãããšããå€éšç£å Žãšåãæ¹åã«ç£åããŠããã®ã ãããïŒ
ãã®éãããããåžžç£æ§ïŒããããããïŒãšåç£æ§ïŒã¯ããããïŒã®ã¡ãããã§ããã
ããç©è³ªããå€éšç£å Žã«ã»ãšãã©åå¿ããªããããããå°ãã ãå€éšç£å Žãšåãæ¹åã«ãç£åãããŠããçŸè±¡ã®ããšãåžžç£æ§ãšããããã®ãããªç©è³ªãåžžç£æ§äœãšãããåžžç£æ§äœãããããç©è³ªãšããŠãã¢ã«ãããŠã ã空æ°ãªã©ããã
äžæ¹ãããç©è³ªããå€éšç£å Žã«ã»ãšãã©åå¿ããªããããããå°ãã ãå€éšç£å Žãæã¡æ¶ãæ¹åã«ãç£åãããŠããçŸè±¡ã®ããšãåç£æ§ãšããããã®ãããªç©è³ªãåç£æ§äœãšãããåç£æ§äœãããããç©è³ªãšããŠãé
ãæ°Žãæ°ŽçŽ ãªã©ãããã
== â» ç¯å²å€: ã¹ãã³ãšç£æ§äœ ==
å
çŽ ãååã®çš®é¡ã«ãã£ãŠãç£æ§ã®ã¡ãããããçç±ãšããŠãååŠçµåã§ã®é»åè»éã«åå ããããšèããããŠããã
ååŠã®æç§æžã®çºå±äºé
ã«ããsè»éãããpè»éããªã©ã®çè«ããããããã®çè«ã§ããã®çç±ã説æã§ãããšãããŠããããªããçãå
ã«ãããšããdè»éãã®ç¹åŸŽããç£æ§ã®åå ã§ãããïŒèšŒæã¯çç¥ïŒ
å
ã
ãïŒååŠçµåã§é»åæ®»ïŒã§ããããïŒã«çºçããããšã®ããïŒå€ç«é»åã«ã¯ç£æ§ãããããã®ç£æ§ãé»åã2åããã£ãŠïŒå€ç«ã§ãªããªãïŒé»å察ã«ãªãäºã§ãç£æ§ãæã¡æ¶ããã£ãŠãããšèããããããªããå€ç«é»åãããšããšæã£ãŠããç£æ§ã®ããšã'''ã¹ãã³'''ãšãããããååŠã®çè«ã§ã¯ãã¹ãã³ãäžç¢å°ãâããšäžç¢å°ãâãã®2çš®é¡ã§ããããäºãå€ãã®ã§ãããããã®çç±ã¯ããšããã©ãã°ãããããç£ç³ã®åãã2çš®é¡ïŒããšãã°N極ãšS極ãšãã2çš®é¡ã®æ¥µãããïŒã§ããããã§ããã
é»åæ®»ãšã¯ãååŠIã®å§ãã®ã»ãã§ãç¿ãããKæ®»ã¯8åã®é»åãå
¥ãããªã©ã®ãã¢ã¬ã®ããšã§ããã
ãŸãšãããšã
:* ããããåç¬ã®1åã®é»åã«ã¯ããã€ã¯ç£æ§ãããããã®ãããå€ç«é»åã«ã¯ç£æ§ãããïŒã¹ãã³ïŒããããŠãã®ç£æ§ãããïŒé»åã®ãã¹ãã³ããšèšãããç£æ§ãããïŒãããããå€ç«é»åãé»å察ã«ãªãããšããçç±ã®ã²ãšã€ã§ãããã€ãŸãããããå
±æçµåãèµ·ããçç±ã®ã²ãšã€ã§ãããã
:* ããããååŠåå¿ã«ãã£ãŠå€ç«é»åã¯ãååŠçµåãšããŠãããã«åšå²ã®ååãååãšçµåããŠããŸãã®ã§ãå€ç«é»åã§ã¯ãªãé»å察ã«ãªã£ãŠããŸãã2åã®å察æ¹åã®ç£æ§ããã£ãé»å察ããç£æ§ãæã¡æ¶ããããããããããã®ãããªçç±ã«ãããå€ãã®ïŒååŠçµåã®çµæã§ããïŒç©è³ªã¯ãå€éšç£å Žãšã®çžäºäœçšã匱ãç©è³ªãå€ãã匷ç£æ§ãšãªãå
çŽ ãååã®ç©è³ªã¯å°ãªããå€ãã®å
çŽ ãååã®ç©è³ªã¯åžžç£æ§ãŸãã¯åç£æ§ã«ãªã£ãŠããŸãã§ãããã
{{ã³ã©ã |â» ç¯å²å€: ããŒããã£ã¹ã¯ã®ãã¹ãã³ãããããšã¯ïŒ|
ãã§ã«ããœã³ã³ãªã©ã®ããŒããã£ã¹ã¯ã®èªã¿ãšããããã®ã»ã³ãµãŒã§ãã¹ãã³ãããããšããæè¡ãå®çšåãããŠãããããããããã¯ããã£ããŠãåé»åã®ã¹ãã³ã«æ
å ±ãèšé²ããŠããããã§ã¯ãªãã
ãããããããŒããã£ã¹ã¯ã®ãã£ã¹ã¯åŽã®æè¡ã§ã¯ãªãããã£ã¹ã¯ã®æ
å ±ãèªã¿åãã»ã³ãµãŒã§ãããããåŽã®æè¡ã§ããã
ãã®ã¹ãã³ãããã¯ãã巚倧ç£æ°æµæå¹æãïŒããã ã ãããŠããã ãããïŒãšèšãããçŸè±¡ãå©çšããŠããããã®ãããªç©ççŸè±¡ã®èµ·ããåçãšããŠä»®èª¬ãšããŠã¹ãã³ãæ³åãããŠããã®ã§ãã¹ãã³ãããããšããã®ã§ããã
ã巚倧ç£æ°æµæå¹æããšã¯ãåãã ãããïŒåã æ°ããã¡ãŒãã«ã»ã©ïŒã®éç£æ§äœã®å°äœéå±ããäžäžã«ç£æ§äœã®å±€ã§æããšããã®äžäžã®ç£æ§äœãåãåãã«ç£åããŠããå Žåãšããã£ãœãå察æ¹åã«ç£åããŠããå Žåãšã§ãæãŸããéç£æ§ã®å°äœéå±ã®é»æ°æµæã®å€ããéã£ãŠããããšããçŸè±¡ã§ããã
ããŒããã£ã¹ã¯ã®å¿çšã®ã»ãã«ããé«ç²ŸåºŠã®ç£æ°ã»ã³ãµãŒãšããŠããã¹ãã³ããããæè¡ã¯å®çšåããŠããã
ãã£ãœãããã®ãã¹ãã³ããããæè¡ãšã¯å¥ã«ãç£æ°æµæå¹æããããœã³ã³ã®ã¡ã¢ãªãŒå
ã«ããåã
ã®ã¡ã¢ãªãŒçŽ åã«å¿çšããäºã§å€§å®¹éãã€é»åæ¶è²»ã®ãããªããç£æ°ã¡ã¢ãªããã€ããããšããç 究éçºããããŠããããšã¬ã¯ãããã¯ã¹ãªãã¬ãã¹ãã³ãããã¯ã¹ããšããŠæåŸ
ãããŠããããããããäžäžã®ç£æ§äœã®ç£åã®åããå€ããããã®é»æ°ã³ã€ã«åè·¯ããã©ããã£ãŠåŸ®å°åããŠãçŽ åãšããŠå€§éã«é
眮ããã°ããã®ãïŒããšããæªè§£æ±ºã®é£é¡ãããããã£ãŠ2017幎ã®æç¹ã§ã¯ããŸã ãé«å®¹éã®ç£æ°ã¡ã¢ãªãŒã¯å®çšåããŠããªãã
}}
== â» ç¯å²å€: ã匷èªé»äœããšå§é»äœ ==
ãç£æ§äœã«ã匷ç£æ§äœããããã®ãªããèªé»äœã«ãã匷èªé»äœããããã®ãïŒãã®ãããªçåã¯ããšããããæãã§ãããã
ãã¿ã³é
žé <chem>PbTiO3</chem> ããããªãé
žãªããŠã <chem>LiNbO3</chem> ããã匷èªé»äœãã«åé¡ãããå Žåãããã
ãããã匷ç£æ§äœãç£æ°ããŒããç£æ°ããŒããã£ã¹ã¯ãªã©ã®èšé²ã¡ãã£ã¢ã«çšããããŠããç¶æ³ãšã¯ç°ãªããã匷èªé»äœãã¯èšé²ã¡ãã£ã¢ã«ã¯çšããããŠããªããéå»ã«ã¯ããã®ãããªã匷èªé»äœã¡ã¢ãªããç®æãç 究éçºããã£ãããããã2017幎ã®æç¹ã§ã¯ããŸã ã匷èªé»äœã¡ã¢ãªãã®ãããªããã€ã¹ã¯å®çšåããŠããªãã
ãã ããä»ã®çšéã§ããããã®ç©è³ªã¯ç£æ¥ã«å®çšåãããŠããã
ãã¿ã³é
žéãããªãé
žãªããŠã ã¯ããã®ç©è³ªã«å§åããããããšé»å§ãçºçããäºãããå§é»äœïŒãã€ã§ãããïŒãšããçŽ åãšããŠæŽ»çšãããŠãããïŒâ» ã[[é«çåŠæ ¡ååŠI/ã»ã©ããã¯ã¹]]ãã§ãå§é»æ§ã»ã©ããã¯ã¹ããšããŠå§é»äœã玹ä»ãé«æ ¡ååŠã®ç¯å²å
ã§ããã2017幎ã®çŸåšã§ã¯é«æ ¡3幎ã®éžæååŠïŒå°éååŠïŒã®ç¯å²å
ã ãããïŒ
ãªãããããã®å§é»äœã«ãé»å§ããããããšãç©è³ªãã²ããã
ãã®ãããå§é»äœã«äº€æµé»å§ãå ããããšã§ãå§é»äœãçæéã§äœåãåšæçã«æ¯åããããšã«ãããå§é»äœã®åšå²ã«ãã空æ°ãæ¯åãããäºãã§ããã®ã§ãè¶
é³æ³¢ãçºçããããã®çŽ åãšããŠããã§ã«å®çšåãããŠããã
ãªããããçš®é¡ã®ç©è³ªããå§åããããããšé»å§ãçºçããçŸè±¡ãèµ·ããç©è³ªã®å Žåããã®ãããªæ§è³ªã®ããšãå§é»æ§ïŒãã€ã§ãããïŒãšããã
== åå°äœ ==
ã±ã€çŽ Si ãã²ã«ãããŠã Ge ã¯ãå°äœãšçµ¶çžäœã®äžéã®æµæçããã€ããšãããã±ã€çŽ ({{Lang-en-short|silicon}})ãã²ã«ãããŠã ({{Lang-en-short|germanium}})ãªã©ã¯åå°äœãšèšãããã
ãã®åå°äœã®çµæ¶ã«ããããã«ããªã³Pãªã©ã®äžçŽç©ãå
¥ããããšã§ãæµæçã倧ããäžããããã
:ïŒâ» ç¯å²å€ã泚é: ïŒæé»ã®åæãããã®ã§ãæ€å®æç§æžã§ã¯ãã¡ãã¡èª¬æãããªããããããªããããããããããœã³ã³ãããã³ã³ãã¥ãŒã¿ããªã©ã®ããŒããŠã§ã¢ã®å
éšã¯ãäž»ã«åå°äœãããªãéšåã§ããã
:ããœã³ã³éšåã®ãã¡ããããããã¡ã¢ãªããããªããšãããããããšãèšãããéšåã®ææã¯ããããŠããäžèšã®ãããªæå³ã§ã®ã·ãªã³ã³åå°äœãããªãéšåã§ããã
=== nååå°äœ ===
ã±ã€çŽ ååã¯äŸ¡é»åã4åã§ãããã±ã€çŽ ã®çµæ¶ã¯ã4ã€ã®äŸ¡é»åãå
±æçµåãããŠããã
ããã«ãªã³Pãå ãããšããªã³ã¯äŸ¡é»åã5åãªã®ã§ã1åã®äŸ¡é»åãäœãããã®äœã£ã䟡é»åãèªç±é»åãšããŠãçµæ¶ãåãåããããã«ãªãã
ãã®ãããªä»çµã¿ã§ãã±ã€çŽ ã«ãªã³ãå ããããšã§ãæµæçã倧ããäžããããšããã®ãå®èª¬ã§ããã
ãã®ããã«ãè² ã®é»åãäœãããšã§ãå°é»çãäžãã£ãŠãåå°äœã '''nååå°äœ''' ãšããã(ãnã㯠negative ã®ç¥ã)
=== pååå°äœ ===
ã·ãªã³ã³ã®çµæ¶ã«ãäžçŽç©ãšããŠãããŠçŽ Bãã¢ã«ãããŠã Alãªã©ã䟡é»åã3åã®å
çŽ ãå ãããšãé»åã1åã足ããªããªãã
ãã®ãé»åã®äžè¶³ããã¶ãã®ç©ºåžã'''æ£å'''(postive holeãããŒã«)ãšããã
æ£åã¯æ£é»è·ããã€ã
é»å§ãæãããšããã®æ£åãåããããã«è¿ãã®çµåã«ãã£ãé»åã移åããããããšã®é»åããã£ãå Žæã«æ°ããªæ£åãã§ããã®ã§ãèŠããäžã¯æ£åãé»åãšéæ¹åã«åããããã«èŠããã
ãã£ãŠãæ£åãåãããšã§ãé»æµãæµããŠããããšèŠãªããã
ãŸãããã®ããã«ãæ£ã®é»è·ããã€ç²åã«ãã£ãŠå°é»çãäžãã£ãŠãåå°äœã '''pååå°äœ''' ãšããã(ãpã㯠positive ã®ç¥ã)
=== ãã£ãªã¢ ===
nååå°äœã§ã¯é»åãé»æµãéã¶ã
pååå°äœã§ã¯æ£åãé»æµãéã¶ã
ãã®ããã«ãåå°äœäžã§ã®é»æµã®æ
ãæãã'''ãã£ãªã¢'''ïŒcarrierïŒãšããã
ã€ãŸããnååå°äœã®ãã£ãªã¢ã¯é»åã§ãpååå°äœã®ãã£ãªã¢ã¯æ£åã§ããã
=== pnæ¥å ===
[[File:ãã€ãªãŒãã®é æ¹å.svg|thumb|300px|ãã€ãªãŒãã®é æ¹åãé»æµã¯æµããã]]
[[File:ãã€ãªãŒãã®éæ¹å.svg|thumb|300px|ãã€ãªãŒãã®éæ¹åãé»æµã¯æµããªãã]]
pååå°äœãšnååå°äœãæ¥åã(pnæ¥å)ãç©äœããäžæ¹åã®ã¿ã«é»æµãæµãã
ãã®ãããªéšåã'''ãã€ãªãŒã'''ïŒdiodeïŒãšããã
påŽã«æ£é»å§ãæããnåŽã«è² é»å§ãæããæãé»æµãæµããã
äžæ¹ãpåŽã«è² é»å§ãæããnåŽã«æ£é»å§ãæããŠããé»æµãæµããªãã
åè·¯ã«ãããŠããã€ãªãŒããé»æµãæµãåãã'''é æ¹å'''ïŒãã
ãã»ãããïŒãšãããé æ¹åãšã¯å察åãã'''éæ¹å'''ãšããããã€ãªãŒãã®éæ¹åã«ã¯ãé»æµã¯æµããªãã
ãã®ããã«äžæ¹åã«æµããä»çµã¿ã¯ããã€ãªãŒãã§ã¯ãã€ãã®ãããªä»çµã¿ã§ãé»æµãæµããããã§ããã
ãã®ããã«äžæ¹åã«ã ãé»æµãæµãããšã'''æŽæµ'''ïŒãããã
ãïŒãšããããªããåå°äœã䜿ããªããŠããç空管ã§ãæŽæµã ããªãå¯èœã§ãããïŒãã ãç空管ã®å Žåãç±ã®çºçãèšå€§ã§ãã£ãããèä¹
æ§ãå£ãã®ã§ãé»åéšåãšããŠã®å®çšæ§ã¯ã空管ã¯äœãã®ã§ãçŸä»£ã¯ç空管ã¯é»åéšåãšããŠã¯äœ¿ãããŠããªããïŒ
ããœã³ã³ã§ãããžã¿ã«æ³¢åœ¢ãããžã¿ã«ä¿¡å·ã®ããã«åè§ã®é»æµæ³¢åœ¢ãäœã£ãŠããæ¹æ³ã¯ãããããããã®ãã€ãªãŒããšãåŸè¿°ãããã©ã³ãžã¹ã¿ãšããããŸãçµåããããšã§ãããžã¿ã«æ³¢åœ¢ãã€ãããšããä»çµã¿ã§ãããïŒâ» æ°ç åºçã®æ€å®æç§æžããããããèŠè§£ã§ãããïŒ
* påŽã«æ£é»å§ãæããnåŽã«è² é»å§ãæããæ
ãã€ãªãŒãã®påŽã«æ£é»å§ããããnåŽã«è² é»å§ãããããšãpåŽã§ã¯æ£é»æ¥µã®æ£é»å§ããæ£åãåçºããŠæ¥åé¢ãžãšåãããnåŽã§ã¯é»åãè² é»æ¥µããåçºããŠæ¥åé¢ãžãšåããããããŠãæ¥åé¢ã§æ£åãšé»åãã§ãããæ¶æ»
ããããã®çµæãèŠæãäžãæ£é»è·ããæ£é»æ¥µããè² é»æ¥µã«ç§»åããã®ãšãåçã®çµæã«ãªãã
ãããŠãæ£é»æ¥µãããã€ãã€ããšæ£åãäŸçµŠãããã®ã§ãé»æµãæµãç¶ããã
* påŽã«è² é»å§ãæããnåŽã«æ£é»å§ãæããæ
ãã£ãœããpåŽã«è² é»å§ãæããnåŽã«æ£é»å§ãæããæãpåŽã§ã¯æ£åã¯é»æ¥µïŒé»æ¥µã«ã¯è² é»å§ãæãã£ãŠããïŒã«åŒãå¯ããããæ¥åé¢ããã¯é ããããåæ§ã«nåŽã§ã¯é»åãé»æ¥µïŒæ£é»å§ãæãã£ãŠãïŒã«åŒãå¯ããããæ¥åé¢ããã¯é ãããã
ãã®çµæãæ¥åé¢ã«ã¯ãäœåãªæ£åãäœåãªé»åããªãç¶æ
ãšãªãããã£ãŠæ¥åé¢ã®ä»è¿ã«ã¯ãã£ãªã¢ããªãããã®æ¥åé¢ä»è¿ã®ãã£ãªã¢ã®ç¡ãéšåã¯'''空ä¹å±€'''ïŒãããŒããããdepletion layerïŒãšåŒã°ããã
ãããŠããã以éã¯ãæ£åãé»åããããã©ãã«ã移åã®äœå°ããªãã®ã§ããã£ãŠé»æµãæµããªãã
{{ã³ã©ã |â» ç¯å²å€: ãåå°äœããšã¯ïŒ|
ç©çåŠãååŠã§ããåå°äœãšã¯ãäžè¿°ã®ããã«ãã·ãªã³ã³ãªã©ã®çµæ¶ããã³ããããã®çµæ¶ã«ãäžçŽç©ãå ããããšã§é»æ°ç¹æ§ã調æŽããç©è³ªã®äºã§ããã
ãã£ãœããç£æ§äœã¯ãåå°äœã§ã¯ãªãã
ããããäžéäžè¬ã§ã¯ã倧äŒæ¥ã®ãåå°äœã¡ãŒã«ãŒããšãããäŒæ¥ãçç£ããé»åéšåãããŸãšããŠãåå°äœããšèšãããããšãããããã®ãããããšãç£æ§äœã掻çšãã補åã§ãããåå°äœãããŸã掻çšããŠããªã補åã§ãã£ãŠããåå°äœãšèšãããããšãå€ãã
ããããäŸãšããŠã¯ãç£æ°ããŒããã£ã¹ã¯ã§ãããåå°äœããšèšãããå Žåãããã
ããããç©çåŠã§ã¯ãç£æ§äœã¯ããã£ããŠåå°äœã§ã¯ãªããååŠã§ãåæ§ã«ããç£æ§äœã¯ããã£ããŠåå°äœã§ã¯ãªãããšããŠæ±ãã
ç£æ§äœã ãã§ãªãã液æ¶ãåæ§ã§ããã åæ§ã«ã液æ¶ãã£ã¹ãã¬ã€ãã液æ¶ã®ã¶ã¶ãã¯ãåå°äœã§ã¯ãªãã
倧åŠã®ç©çãååŠã§ããç£æ§äœã¯ãåå°äœã§ã¯ãªãããšããŠæ±ãã液æ¶ãåæ§ã§ããã倧åŠã§ã¯ã液æ¶ã¯åå°äœã§ã¯ãªãããšããŠæ±ãã
æ¬wikibooksé«æ ¡æç§æžã§ããç£æ§äœã液æ¶ã¯ãåå°äœã§ã¯ãªãããšããŠæ±ãã
ãªããäžåŠé«æ ¡ã®ç€ŸäŒç§ã®å°çç§ç®ã®å·¥æ¥çµ±èšã§ã¯ããã¡ããšãé»åéšåããšããè¡šçŸã§ãåå°äœã液æ¶ãããŒããã£ã¹ã¯ãªã©ãããŸãšããŠè¡šçŸããŠããã
}}
=== ãã©ã³ãžã¹ã¿ ===
[[ãã¡ã€ã«:Transistor description ja.svg|right|frame|NPNåãã©ã³ãžã¹ã¿ã®æš¡åŒå³ïŒãã€ããŒã©ãã©ã³ãžã¹ã¿ïŒ]]
åå°äœã3ã€npnãŸãã¯pnpã®ããã«çµã¿åããããšãé»æµãå¢å¹
ïŒãããµãïŒããããšãã§ããã'''å¢å¹
äœçš'''ïŒãããµããããïŒãšããã
NPNãšã¯ãç端ããé ã«èŠãŠNåã»Påã»Nåã®é ã«äžŠãã§ããšããäºã§ããã
åæ§ã«ãPNPãšã¯ãç端ããé ã«èŠãŠNåã»Påã»Nåã®é ã«äžŠãã§ããšããäºã§ããã
å¢å¹
ãšãã£ãŠãããã£ããŠç¡ãããšãã«ã®ãŒãçºçããããã§ã¯ãªãã®ã§ãæ··åããªãããã«ã
説æã®ç°¡ç¥åã®ãããå€éšé»æºãçç¥ãããäºãããããå®éã¯å€éšé»æºãå¿
èŠã§ãããåå°äœçŽ åã¯å°ããªé»æµããæµãã¬ãããé»æµãæžããããã®æµæçŽ åãšããŠã®ä¿è·æµæïŒã»ããŠãããïŒãå¿
èŠã§ããã
ãªããå³ã®ããã«é·æ¹åœ¢ç¶ã«äžŠãã§ããæ¹åŒã®ãã©ã³ãžã¹ã¿ã'''ãã€ããŒã©ãã©ã³ãžã¹ã¿'''ãšãããïŒâ» æ€å®æç§æžã®æ°ç åºçã®æç§æžã§ãããã€ããŒã©ãã©ã³ãžã¹ã¿ããã³ã©ã ã§ç¿ããïŒ
ãã€ããŒã©ãã©ã³ãžã¹ã¿ã«ã¯ã端åãäž»ã«3ã€ãããããšããã¿ãããããŒã¹ãããã³ã¬ã¯ã¿ããšããåèš3ã€ã®ç«¯åãããã
ãã€ããŒã©ãã©ã³ãžã¹ã¿ã§ã®é»æµã®å¢å¹
ãšã¯ãããŒã¹é»æµãå¢å¹
ããŠã³ã¬ã¯ã¿ã«éããã§ããïŒPNPã®å ŽåïŒãé»æµã®åãã¯PNPåã®ã°ãããš NPPåã®ã°ãããšã§ã¯ç°ãªãããã©ã¡ãã®å Žåã§ãããŒã¹é»æµãå¢å¹
ããããšããä»çµã¿ã¯å
±éã§ããã
ããŠãæš¡åŒå³ã§ã¯æš¡åŒçã«çãäžã®åå°äœã¯ããããå°ããã«æžãããããå®éã®ãã©ã³ãžã¹ã¿ã¯çãäžã®åå°äœã¯ããã§ã¯ãªãã®ã§ãåèçšåºŠã«ã
æè²ã§ã¯ãåå°äœã®é«æ ¡çãå°éå€ïŒé»åå°æ»ä»¥å€ïŒã®äººããã«ã¯ããããã€ããŒã©ãã©ã³ãžã¹ã¿ãåçŽãªã®ã§çŽ¹ä»ãããããå®éã«åžè²©ã®ã³ã³ãã¥ãŒã¿éšåãªã©ã§ãã䜿ããããã©ã³ãžã¹ã¿ã®æ¹åŒã¯ããããšã¯åœ¢ç¶ããã£ããç°ãªãã
åžè²©ã®ã³ã³ãã¥ãŒã¿éšåã®ãã©ã³ãžã¹ã¿ã«ã¯ãé»çå¹æãã©ã³ãžã¹ã¿ãšããããæ¹åŒã®ãã®ããããçšãããããïŒãã¡ãããé»çå¹æãã©ã³ãžã¹ã¿ã«ãããå¢å¹
ãã®æ©èœããããïŒ
:ïŒâ» åæ通ã®æ€å®æç§æžã§ããé»çå¹æãã©ã³ãžã¹ã¿ããã³ã©ã æ¬ã§çŽ¹ä»ãããŠãããïŒ
:â» é»çå¹æåã®å Žåã¯ãããœãŒã¹ãããã²ãŒãããããã¬ã€ã³ããªã©ã®ç«¯åããããåçã¯ç°ãªãã®ã§ã察å¿ã¯ããªãã
ïŒâ» 詳ããã¯å€§åŠã®é»æ°å·¥åŠãŸãã¯å·¥æ¥é«æ ¡ã®é»ååè·¯ãªã©ã®ç§ç®ã§ç¿ããïŒ
{{-}}
ãã©ã³ãžã¹ã¿ã¯ãåè·¯å³ã§ã¯ãæš¡åŒçã«äžå³ã®ããã«æžãããã
[[File:NPN transistor symbol jp.svg|thumb|300px|left|NPNãã©ã³ãžã¹ã¿ã®å³èšå·ã]]
[[File:PNP transitor symbol.svg|thumb|center|PNPãã©ã³ãžã¹ã¿ã®å³èšå·ã]]
{{-}}
{{ã³ã©ã |åå°äœã®ç¯å²å€ã®è©±é¡ã®ãããã|
;ãç空管ãã©ã³ãžã¹ã¿ããšã¯å¥ç©
å®ã¯ãé»æµå¢å¹
åè·¯ãã€ããã ããªããç空管ã§ãäœãããçŸä»£ã§ã¯ç空管ã«ã¯çµæžçãªå®çšæ§ãç¡ãã®ã§ãç空管ã®å¢å¹
åè·¯ã¯ãäžè¬ã®è£œåã«ããé»åéšåãšããŠã¯ã䜿ãããŠããªãããªããç空管ã®é»æµå¢å¹
åè·¯ã®ããšãããã©ã³ãžã¹ã¿ããšããã®ã§ãæ··åããªãããã«æ³šæã®ããšã
;é²å
æ©ãªãã§ãæäœæ¥ã§ãã©ã³ãžã¹ã¿ãäœãããšããå ±åãã
åŠè¡æžã®åºå
žã¯ç¡ãã®ã§ãããäžç¢ºãããªæ
å ±ã§ãããã å®ã¯ãã€ãªãŒãããã©ã³ãžã¹ã¿ã¯ãäœãã ããªããææãèããããã€ãŒããªã©ã®é«æž©çšèšåããããã°ãããšã¯ææã®ã·ãªã³ã³ãæ·»å ç©ã®ãªã³ãªã©ã ãã§ãäœããŠããŸããšèšãããŠãããïŒã€ãŸããé²å
æ©ïŒããããïŒãªã©ã®åŸ®çŽ°å å·¥ã®èšåã¯ãç¡ããŠããã€ãªãŒããªã©ãäœããããšãããïŒ
ãããããåå°äœãã©ã³ãžã¹ã¿ã®çºæè
ãè©ŠäœåãšããŠç¹æ¥è§Šãã©ã³ãžã¹ã¿ã補é ããæ代ã«ã¯ããŸã é²å
æ©ãªã©ã®èšåã¯ç¡ãã£ãã®ã ãããèããŠã¿ãã°é²å
æ©ãªãã§ããã©ã³ãžã¹ã¿èªäœãå¯èœãªã®ã¯åœç¶ãšããã°åœç¶ã§ã¯ããã
æŽå²çãªçµç·¯ã§ãçç§æè²ã§ã¯åå°äœå·¥åŠã説æããéã«ããã©ã³ãžã¹ã¿ãªã©ã®çºæåœæã®å
端çè«ã§ãããéåååŠãïŒãããã ããããïŒãšããååã¹ã±ãŒã«ã®äžçã®ç©çæ³åã®çè«ããŸãšããŠèª¬æããã®ã§ãããããåå°äœã®è£œé ã«ãååã¹ã±ãŒã«ã®åŸ®çŽ°å å·¥ã®ããã®èšåãäžå¯æ¬ ã®ããã«æ³åããã¡ã§ããããå®ã¯é²å
æ©ãªã©ã®èšåã¯ãªããŠããã©ã³ãžã¹ã¿ã¯äœããŠããŸããããã
é²å
æ©ãªã©ã䜿ããªãã§ææãšã«ãããªã©ã®æ¯èŒçã«åçŽãªèšåã ãã§æäœæ¥çã«èªäœããåå°äœã¯ãéç©åºŠãäœãã®ã§å®çšã«ã¯ç¡ããªãäºããããå·¥åŠæžãªã©ã§ã¯çŽ¹ä»ã¯ãããªãã®ã§ãããã
ïŒå¥ä»¶ãããããªãããïŒãããããåå°äœã®çºæåœæã¯ã女æ§å·¥å¡ãšãã«çŽ°ããé
ç·äœæ¥ãªã©ããããŠããæ代ããã£ãïŒããã©ã³ãžã¹ã¿ã»ã¬ãŒã«ããšèšãããŠããïŒããšèšããããããã
}}
ãã€ãªãŒãããã©ã³ãžã¹ã¿ã®ä»ã«ãåå°äœãçµã¿åãããé»åéšåã¯ãããïŒä»ã«ãããµã€ãªã¹ã¿ããšãè²ã
ãšããïŒãé«æ ¡ç©çã®ç¯å²ãè¶
ããã®ã§ã説æã¯çç¥ãããïŒâ» ããä»äºã§å°éçãªæ
å ±ãå¿
èŠã«ãªãã°ãå·¥æ¥é«æ ¡ããã®ãé»ååè·¯ãã®æç§æžã«ãã£ãã詳ããæžããŠããã®ã§ããããèªãã°ããããªããæžåºã®è³æ Œã³ãŒããŒæ¬ã«ããé»æ°å·¥äºå£«ãé»æ°äž»ä»»æè¡è
è©Šéšãšãã®å¯Ÿçåã«ã¯ãã»ãŒé»ååè·¯ãç¯å²å€ãªã®ã§ãããŸãé»ååè·¯ã®èª¬æã¯æžããŠãªãããªã®ã§ãå·¥æ¥é«æ ¡ãé»ååè·¯ãã®æç§æžããŸãã¯å·¥æ¥é«å°ãªã©ã®åçã®ç§ç®ã®æç§æžãåç
§ã®ããšãïŒ
:â» éç©åè·¯ã«ã€ããŠã1990幎代ãããã®åèæžã®æ°ç åºçãã£ãŒãåŒã®ç©ç2ã«ãåŸè¿°ã®ãããªéç©åè·¯ãªã©ã®èª¬æããã£ãã
:2010幎以éã®çŸåšããæ
å ±ãæç§ã2000幎代ã«å ãã£ãã®ã§ãCPUãªã©ã®èª¬æã®äžéšããæ
å ±ãæç§ã«ç§»åããŠããã
ããœã³ã³ã®CPUãªã©ã®éšåããäžèº«ã®å€ãã¯åå°äœã§ããããã€ãªãŒãããã©ã³ãžã¹ã¿ãªã©ã®çŽ åãCPUãªã©ã®å
éšã«ããããããããšèšãããŠãããïŒâ» ä»ã«ããæ°Žæ¶æ¯ååããšãè²ã
ãšCPUå
ã«ã¯ ããããç©ç2ã®ç¯å²å€ãªã®ã§èª¬æãçç¥ãïŒ
éç©åè·¯ãLSI(Large Scale Integratedã倧èŠæš¡éç©åè·¯)ãªã©ãšèšãããçµç¹ãããªã«ãéç©ïŒãéç©ããè±èªã§ integrate ã€ã³ãã°ã¬ãŒã ãšããïŒããã®ããšãããšãåå°äœçŽ åãéç©ãããšèšãæå³ã§ããã
ãªãããICãïŒã¢ã€ã·ãŒïŒãšã¯ Integrated Circuit ã®ç¥ç§°ã§ããããããåèš³ãããã®ããéç©åè·¯ãã§ããã
ã€ãŸããéç©åè·¯ãLSIã®äžèº«ã¯ãåå°äœã§ããããã©ã³ãžã¹ã¿ãªã©ã®çŽ åãé«å¯åºŠã§ããã®åè·¯äžã«è©°ãŸã£ãŠããã
é»åéšåã®åå°äœã®ææãšããŠã¯ãéåžžã¯ã·ãªã³ã³çµæ¶ã䜿ããããïŒâ» åæ通ãæ°ç ãªã©ãçµæ¶ã§ããããšãèšåãïŒ
ç 究éçºã§ã¯ã·ãªã³ã³ä»¥å€ã®ææãç 究ãããŠããäžéšã®ç¹æ®çšéã§ã¯GaAsãInGaPãªã©ãå©çšãããŠãããïŒâ» æ°ç ã®æ€å®æç§æžã¯GaAsãInGaPãªã©ã«ã³ã©ã ã§èšåïŒããããçŸç¶ã§ã¯ãã·ãªã³ã³ãåžè²©ã®ã³ã³ãã¥ãŒã¿éšåäžã®åå°äœçŽ åã®ææã§ã¯äž»æµã§ããã
ãªããã·ãªã³ã³åå°äœã®ææå
éšã¯ã·ãªã³ã³çµæ¶ã§ããããè¡šé¢ã¯ä¿è·èããã³çµ¶çžã®ããã«é
žåãããããŠãããã·ãªã³ã³åå°äœè¡šé¢ã¯é
žåã·ãªã³ã³ã®ä¿è·èã«ãªã£ãŠãããã·ãªã³ã³ãé
žåãããšã絶çžç©ã«ãªãã®ã§ãä¿è·èã«ãªãããã§ããïŒâ» æ°ç åºçã®æç§æžãããèšã£ãŠãããïŒ
åå°äœã®å
éšã«ãæ·»å ç©ãªã©ã§ç¹æ§ãå€ããããšã«ãããæµæãã³ã³ãã³ãµãåå°äœå
éšã«è£œé ã§ãããïŒâ» æ°ç ããæµæãã³ã³ãã³ãµãåå°äœå
éšã§äœã£ãŠããäºã«èšåãïŒ
ïŒâ» ç¯å²å€: ïŒããããã³ã€ã«ã¯åå°äœå
éšã«äœãããšãåºæ¥ç¡ãã
== çºå±ïŒ çžå¯Ÿè«ã®äžæ¬¡è¿äŒŒ ==
=== éåããç£æã¯é»å Žãèªèµ·ãã ===
ç£å ŽBã®äžããé»è·qã®è·é»ç²åãé床vã§éåãããšãããŒã¬ã³ãåã¯ãã¯ãã«å€ç©ãçšããŠãfïŒqã»vÃBãã®åãç²åã«åãããããã§èŠ³æž¬è
ã®åº§æšç³»ãå€ãããšããŠãåãç²åããç²åãšåãæ¹åã«é床ïœã§åã座æšåœ¢ïŒ«ã®äžã®èŠ³æž¬è
ããèŠããã©ããªããïŒã座æšç³»Kã§ã¯ãç²åã®é床㯠v(K)ïŒ0 ã§ãããç£æã®é床ã V<sub>b</sub> ãšãããšãåã®åº§æšç³»ã®ç²åãšã¯å察æ¹åã«åãã®ã§ã
:V<sub>b</sub> ïŒïŒv ã§ããã
æ°ãã座æšç³»Kãã芳枬ããŠããç²åã fïŒqã»vÃBãã®å€§ããã®åãåããŠå éãããããšã«ã¯å€ãããªããã座æšç³»ïœã§ã¯ãè·é»ç²åã¯éæ¢ããŠããã®ã«ãããŒã¬ã³ãåãåãããšèããã®ã¯äžåçã§ãããç£æã¯ãV<sub>b</sub>ïŒïŒv ã§éåããŠããã®ã§ãç£æã®éåã«ãã£ãŠãfïŒqã»ïŒïŒV<sub>b</sub>ïŒÃB ïŒ ïŒqã»V<sub>b</sub>ÃBãã®åãåãããšèããã¹ãã§ãããç²åã質é0ã®è³ªç¹ãšã¿ãªãã°ãéæ¢ããŠããè·é»ç²åã«åãåãŒããã®ã¯ãé»å Žã ãã ãããã€ãŸãé床 V<sub>b</sub> ã§éåããç£æãã EïŒïŒV<sub>b</sub>ÃB ã®èªå°é»å Žãèªèµ·ããããšã«ãªãããã®ãšããç£å Žãšèªå°ãããé»å Žã¯åçŽã§ããã
=== éåããé»å Žã¯ç£çãäœã ===
ããããéåããé»å Žã¯ç£çãäœãããšããã°ãã¢ã³ããŒã«ã®æ³åããçŽç·ç¶ã«ç¡éã«é·ãå°ç·ãæµãããé»æµïŒ©ãã¯è·é¢ïŒ²ãã ãé¢ããå Žæã«ãBã»2ÏrïŒÎŒIãã®ç£å ŽãäœããããšããçŸè±¡ã¯ããã€ã¯ãå°ç·ã®äžã§è·é»ç²åãéåããããšã«ãã£ãŠãè·é»ç²åãšãã£ããã«ãã®ç²åãäœãé»å Žãåãããã®é»å Žã®éåããç£å Žãèªèµ·ããŠãããããšããå¯èœæ§ãããã
é»æµãæµããŠããç¡éé·ã®ããŸã£ãããªå°ç·ãèãããç·å¯åºŠ q[C/m] ã§ååžããé»è·ã¯ãå³ã®ããã«åç察称ãªé»è·ãäœãã
ïŒâ» ããã«å³ããïŒ
çŽç·ããè·é¢ïœã®ãšãã®é»æ°åç·ã®å¯åºŠDã¯
:DïŒÎµEïŒ <math> \frac{q}{2\pi r}</math>
ãã£ãŠ
:εEã»2Ïr ïŒqãããâ
é»æµ I ã¯é»è·ååž q ãé床 V<sub>e</sub> ã§éåããŠãããšããŠã
:I ïŒ qV<sub>e</sub>
:[A]ïŒ[c/m]ã»[m/s]ïŒ[c/m]
ãšå®çŸ©ããã°ã
é»æµ qV<sub>e</sub> ãè·é¢ r ã®ãšããã«äœãç£å ŽBã¯ã¢ã³ããŒã«ã®æ³åããã
:Bã»2ÏrïŒïŒÎŒIïŒïŒ ÎŒqV<sub>e</sub>ãããâ¡
ãšãªãã
ãã®ãšããç£å Žã®åãã¯ãV<sub>e</sub> ãã ååŸræ¹å ã«ãããåãåãã§ããã
:â¡Ã·â ãã B/εE ïŒ ÎŒ V<sub>e</sub> BïŒÎµÎŒ V<sub>e</sub>ã»E
åããŸã§ãµãããŠãã¯ãã«ç©ã§è¡šãã°ã
:<math>\vec {B} </math>ïŒÎµÎŒ <math>\vec {V_e} \times \vec E</math> ãšãªãã
ã€ãŸã
:é床 V<sub>e</sub> ã§éåããé»å Ž E ã¯ãèªå°ç£å Ž BïŒÎµÎŒV<sub>e</sub>ÃE ãäœãã
ãšãããéèŠãªçµè«ãåŸãããã
ãããã¯ããÎŒHïŒBãããã¡ããŠãBïŒÎŒHïŒÎµÎŒ V<sub>e</sub> ÃEããã
:HïŒÎµÎŒV<sub>e</sub>ÃEããšãªã£ãŠãããã«ãDïŒÎµEãããã
:HïŒÎŒV<sub>e</sub>ÃDã
ã§ããã
ãŸãšã
é床 V<sub>b</sub>ã§éåããç£æBã¯ã
:EïŒïŒV<sub>b</sub>ÃB
ã®èªå°é»å Žãèªèµ·ããããããã»ã»â¡1
é床 V<sub>e</sub> ã§éåããé»å Ž E ã¯
:B ïŒ ÎµÎŒ V<sub>e</sub>ãà Eã
ã®èªå°ç£å Žãäœãã
E,Bã®ãããã«ãD,Hã䜿ã£ãŠè¡šèšããã°ã
:D ïŒ ïŒÎµ V<sub>b</sub> à B
ãã€
:H ïŒ V<sub>e</sub> à DãããïŒã»ã»ã»â¡2ïŒã
ããŠãé»ç£æ³¢ãé床Cã§ç空äžãäŒãããšããã°ã Vb ïŒ Ve ïŒ Cããšããã â¡1åŒãšâ¡2åŒã®å€ç©ããšããšã
: EÃH ïŒ(ïŒV<sub>b</sub>ÃB)à (V<sub>e</sub>ÃD) ïŒ (ïŒCÃÎŒH) à (CÃεE)ã
:ïŒãεΌ ( C<sup>2</sup>) EÃH
ãã£ãŠ
:εΌã»c<sup>2</sup> ïŒ1
ã§ããã
ãã£ãŠãé»ç£æ³¢ã®é床㯠<math> c = \frac{1}{ \sqrt{ \varepsilon \mu} }</math> ãšäºæž¬ã§ããã
ãã®ÎµãšÎŒã«å®æž¬å€ãå
¥ãããšãå
éã®æž¬å®å€ <math> c = 299792458 m/s</math> ãšãé«ã粟床ã§äžèŽããã
ãã®äºãããå
ã¯ãé»ç£æ³¢ã§ããäºãåããããŸããé»ç£æ³¢ã¯ãå
é床Cã§ç空äžãäŒããã
ãŸãããããããéåé»å Žã®èªå°ããç£å Žã¯
:B ïŒ (ïŒ/ C<sup>2</sup> )V<sub>e</sub>ÃEãããâ¢
ãšãå€åœ¢ã§ããã
â¢åŒããã¬ãŠã¹ã®æ³åïŒâ åŒïŒããšçµã¿åããããšãã¢ã³ããŒã«ã®æ³åïŒâ¡åŒïŒãåŸãããã
ãã£ãŠããé床 V<sub>e</sub> ã§éåããé»å Ž E ã¯ããBïŒÎµÎŒ V<sub>e</sub> ÃEãã®èªå°ç£å Žãäœããããšããéçšã劥åœã ã£ãããšããããã
=== ãã€ã³ãã£ã³ã° ãã¯ãã« ===
é»ç£æ³¢ã§ã¯é»å Ž E ãšç£å Ž B ãå
é C ã§éåããŠããã®ã§ãç£æã®éåé床 V<sub>b</sub> 㯠V<sub>b</sub> ïŒ C ã§ãããèªå°é»å Ž E 㯠E ïŒïŒV<sub>b</sub>ÃB ã§ããã®ã§ãäž¡åŒãã E ïŒ ïŒcÃBãã§ãããïŒé»ç£æ³¢ã®é»å Žãšç£å Žã®é¢ä¿åŒïŒãªã
:<math> \mathbb{B} = \mu \mathbb{H} </math>
ã§ããã®ã§ã
é»ç£æ³¢ã¯
:<math> \mathbb{E} \times \mathbb{H} </math>
ã®æ¹åã«é²ãã§ããã¯ãã ããšããããšã泚ç®ãããã
ãã® <math> \mathbb{E} \times \mathbb{H} </math> ã§å®çŸ©ãããéã '''ãã€ã³ãã£ã³ã°ããã¯ãã«''' ãšãã¶ã
ããã¯åäœé¢ç©ããšãã£ãŠæµãåºãé»ç£å Žã®ãšãã«ã®ãŒã®æµãã®éãããããã
ããŠãé»ç£å Žã®ãšãã«ã®ãŒå¯åºŠã¯ <math> u = \frac{1}{2}\varepsilon E^2 + \frac{1}{2}\mu H^2 </math> ãªã®ã§ãããã«é»ç£æ³¢ã®é»å Žãšç£å Žã®é¢ä¿åŒ <math> \mathbb{E} = - \mathbb{C} \times \mathbb{B} </math> ã代å
¥ããŠã
:<math> \varepsilon \mu \cdot c^2 = 1 </math>
ã®é¢ä¿ãçšãããšãïŒãšãã«ã®ãŒã§ã¯ã2ä¹ã«ãããã€ãã¹ç¬Šå·ããªããªãã®ã§ã絶察å€ãåã£ãŠïœïŒ¥ïœïŒïœïœÃïœããšããŠãããšãèšç®ãç°¡åã«ãªãå ŽåããããïŒ
çµæãšããŠã
:<math> u = \varepsilon E^2 </math>ãããïŒé»ç£æ³¢ã®ãšãã«ã®ãŒå¯åºŠïŒ
ãšãªãã
é»ç£æ³¢ããå£ã«ããã£ãŠåžåããããšããåäœæéã«åäœé¢ç©ããã å
éC ã®å€§ããã®äœç©ã®ãªãã®é»ç£æ³¢ãå£ã«è¡çªããã®ã§ãã
:cã»uã
ã®ãšãã«ã®ãŒããåäœæéã«åäœé¢ç©ã«æµã蟌ãã¯ãã§ããã
ïœïŒ cã»uãã« uïŒ Îµã»E^2 ã代å
¥ããŠã <math> \epsilon \mu \cdot c^2 = 1 </math> ãš ïœEïœïŒïœcÃBïœãå©çšãããšãçµæçã«
:ãs ïŒ <math> \frac{1}{ \sqrt{ \varepsilon \mu} } \epsilon E^2 </math> ïŒ<math> \frac{1}{ \sqrt{ \varepsilon \mu} } \epsilon |E||cB| </math> ïŒïœEïœã»ïœHïœ
ã§ããã
ãã£ãŠãã€ã³ãã£ã³ã°ããã¯ãã« EÃH ã¯åäœé¢ç©ãéã£ãŠæµãåºãé»ç£å Žã®ãšãã«ã®ãŒã®æµããããããã
:EÃHã®åäœã¯ã[V/m]ã»[A/m]ïŒ[Vã»AïŒm<sup>2</sup>]ïŒ[WïŒm<sup>2</sup>]
=== ãã€ã³ãã£ã³ã° ãã¯ãã« ãš éåéå¯åºŠ ===
ãã€ã³ãã£ã³ã° ãã¯ãã«ãS ïŒ EÃH ïŒ ÎµÎŒïŒC<sup>2</sup>ïŒEÃH ã¯
:DïŒÎµE ãš BïŒÎŒH ããã¡ã㊠S ïŒ EÃH ïŒïŒC<sup>2</sup>ïŒDÃB ãšãæžããã
:<math> \mathbb{D} \times \mathbb{B} = \frac{1}{c^2} \mathbb{E} \times \mathbb{H} </math>
ã§ããã
倩äžãçãªèª¬æã§ãããããã® GïŒDÃB ãšããéã¯ãéåéã®å¯åºŠã§ããããã®é GïŒDÃB ããé»ç£æ³¢ã®ãéåéå¯åºŠãïŒããã©ããããã¿ã€ã©ïŒãšãããå®éã«ãDÃB ã®åäœã¯
:[DÃB] ïŒ [ïœ1 / (C<sup>2</sup>)ïœ]ã[EÃH] ïŒ [1 / (ïœ/s)<sup>2</sup>] [W/m<sup>2</sup>]
:ïŒ [Nã»sïŒm<sup>3</sup>]
ãšãªãã
ãããã«ãéåéã®å¯åºŠã®åäœãšçããã
* çºå±: å
é»å¹æãšã®é¢ä¿
ãšããã§ãã®ã¡ã®åå
ã§ç¿ãããå
é»å¹æã§ã¯ ãšãã«ã®ãŒuãšéåépã®é¢ä¿ã¯ãå
é床Cããã¡ããŠã uïŒcp ãšæžããã
:sïŒcã»u 㯠sïŒ cu ïŒïœEÃHïœ ã§ããã uïŒcp ãšããããŠã
:sïŒc (cp) ïŒ (c<sup>2</sup>) p ïŒïœEÃHïœ
ãããã
:p ïŒ (1/c<sup>2</sup>) ïœEÃHïœ ïŒ ÎµÎŒ ïœEÃHïœã
: ïŒ ïœÎµEÃÎŒHïœ ïŒ ïœDÃBïœ
åããŸã§å«ããŠ
:p ïŒ DÃB
ãšãªã£ãŠã確ãã« G ïŒ DÃBãã¯éåéå¯åºŠãšãªãã
=== é»ç£èªå°ã®åæ€èš ===
é·ãã®ãŸã£ãããªééããé床ïœã§ç£å ŽïŒ¢ã®äžã暪åããšãããç°¡åã®ãããééã®è»žãšé床ïœã®æ¹åãšç£å ŽïŒ¢ã¯åçŽãšããããã®ãšããééã®äžã®é»è·ã«ãããåããã³é»å Žã¯ããŒã¬ã³ãåã«ããã
:F ïŒ q vÃB
:F/q ïŒ E ïŒ vÃB ã®é»äœããééã®é·ãæ¹åã«æŽŸçããã
é»å ŽïŒ¥ã«ãã£ãŠé·ãã ããé»è·ïœãäžãããããããšãã«ã®ãŒã¯ ïœïŒ¥ïŒ¬ å€åãããé»äœã¯ ïŒïŒ¥ïŒ¬ ã§ããã
:V ïŒ LvB ïŒ â¿ÎŠïŒâ¿tã
ãããããèªå°é»å§ V ã¯ãç£æã®1ç§ãããã®æéå€åã«ãªãã
ã§ã¯ãä»®ã«åºå®ãããåè·¯ã®äžã«ãœã¬ãã€ããéããŠããã®ãœã¬ãã€ãã«äº€æµé»æµãæµããå Žåããåè·¯ã«èªå°é»å§ãçºçããã®ã§ãããããçãã¯ããããã
{{ã³ã©ã |é»æ©èšåãªã©ã®ãåå°äœã|
ã³ã³ãã¥ãŒã¿ä»¥å€ã®çšéã§ããæ¯èŒçã«å€§ãç®ã®é»æµãé»å§ãªã©ãã€ããé»æ©èšåã匷é»ïŒãããã§ãïŒèšåãªã©ã§ãæŽæµãªã©ã®ç®çã§ãé»æ©èšåã®ããã®å°çšã®åå°äœãã€ãªãŒãã䜿ãããšãããã
ããœã³ã³çšã®åå°äœãšãé»æ©èšåçšã®åå°äœãšã¯ãïŒç©çåŠçãªåçã¯ã»ãŒåãã§ãããïŒè£œåãšããŠã¯ããŸã£ããå®æ ŒïŒãŠãããïŒé»æµã»å®æ Œé»å§ãªã©ã®ä»æ§ã®ç°ãªãå¥è£œåãªã®ã§ãæ··åããªãããã«ã䜿çšå¯èœãªé»æµã®å®æ Œå€ããŸã£ããã±ã¿éãã«ïŒããœã³ã³çšãšé»æ°èšåçšãšã§ã¯ïŒéãã®ã§ã絶察ã«æ··åããŠã¯ãããªãã
ããä»®ã«ãæ¬æ¥ãªãé»æ°èšåçšã®åå°äœã§æŽæµãã¹ãå Žæããããœã³ã³çšã®åå°äœã§æŽæµãããšããã£ãšäºæ
ãªã©ã«ã€ãªããå±éºãªã®ã§ã絶察ã«æ··çšãã¬ããšã
}}
[[Category:é«çåŠæ ¡æè²|ç©ãµã€ã2ãŠãããšãã]]
[[Category:é»æ°|é«ãµã€ã2ãŠãããšãã]]
[[Category:ç©çåŠæè²|é«ãµã€ã2ãŠãããšãã]]
[[Category:é«çåŠæ ¡çç§ ç©çII|ãŠãããšãã]] | 2005-05-08T08:12:24Z | 2023-11-23T07:30:47Z | [
"ãã³ãã¬ãŒã:Main",
"ãã³ãã¬ãŒã:ã³ã©ã ",
"ãã³ãã¬ãŒã:-",
"ãã³ãã¬ãŒã:Substub"
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E7%89%A9%E7%90%86/%E7%89%A9%E7%90%86II/%E9%9B%BB%E6%B0%97%E3%81%A8%E7%A3%81%E6%B0%97 |
1,945 | é«çåŠæ ¡ç©ç/ç©çII/ç©è³ªãšåå | é«çåŠæ ¡çç§ ç©çII > ç©è³ªãšåå
æ¬é
ã¯é«çåŠæ ¡çç§ ç©çIIã®ç©è³ªãšååã®è§£èª¬ã§ããã
ç©è³ªã«ã¯åºäœã液äœãæ°äœã®3ã€ã®çžãããã ãããç©è³ªã®äžæ
ãšåŒã¶ã ãããã¯ãããã枩床ã®é«ãé ãã æ°äœã液äœãåºäœãšãªã£ãŠãããã å®éã«ã¯å§åã®å€åã«ãã£ãŠ çžãå€ããããšãããã
ãã¹ãã³ã®äžã«ç©ºæ°ãã€ããŠãããŠãããš äœããæŒãè¿ããŠããããã«æããããããšãåãã ããã¯ããã¹ãã³ã®äžã®ç©ºæ°ãæŒãè¿ããŠããã®ã§ããã 空æ°ã¯å®éã«ã¯æ§ã
ãªçš®é¡ã®æ°äœã«ãã£ãŠ ã§ããŠããããããã®æ°äœã¯ããããã®ååã«ãã£ãŠ ã§ããŠãããããããã®åå㯠ããããã®é床ãæã£ãŠéåããŠããã ãããã®ã©ã³ãã ãªè¡çªãããã¹ãã³ãæŒãè¿ããŠããã®ã§ããã
çæ³æ°äœãèãããšã å§åãšæž©åºŠã®éã«ã¯ P V = n R T {\displaystyle PV=nRT} ã®é¢ä¿ãããããšãç¥ãããŠããã (çæ³æ°äœã®ç¶æ
æ¹çšåŒ) ããã§ãnã¯ã¢ã«æ¿åºŠ(mol/m 3 {\displaystyle {}^{3}} )ã§ããã Tã¯æž©åºŠ[K]ã§ããã
ç©è³ªãäœã圢æ
ã®1ã€ãååãšåŒã¶ã ååã¯ååæ žãšé»åã«ãã£ãŠæ§æãããŠããã ãããã¯å€å
žçã«ã¯å®å®ãªç¶æ
ãšã㊠ååšãåŸãªãããšãç¥ãããŠããã ããããå®å®ã§ããããã®ã¯å®éã«ã¯ 極埮ã®äžçã§ã¯ããããªã¹ã±ãŒã«ã§ã®äžçãš ç©çæ³åãå€ãã£ãŠæ¥ãããšã«ããã ãã®å Žåã¯é»åã¯æ³¢ã§ãããã®ããã«æ¯èãã ãã®æ§è³ªã«ãã£ãŠæ±ºãŸãããé
眮ã«ãããšãã®ã¿ å®å®ã§ããããããšãç¥ãããŠããã
é»åã®ç¶æ
ã«ãã£ãŠ åºäœã®é»æ°çæ§è³ªã決ãŸãã é»åãå±èµ·ããã®ã«ãšãã«ã®ãŒãå
ãããå¿
èŠã§ãªããšãã ãããå°äœãšåŒã¶ã äžæ¹å€ãã®ãšãã«ã®ãŒãå¿
èŠãšãããšãã ããã絶çžäœ(äžå°äœ)ãšåŒã¶ã
ãŸãããã®äžéã«äœçœ®ãããããªãã®ã åå°äœãšåŒã¶ã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "é«çåŠæ ¡çç§ ç©çII > ç©è³ªãšåå",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "æ¬é
ã¯é«çåŠæ ¡çç§ ç©çIIã®ç©è³ªãšååã®è§£èª¬ã§ããã",
"title": ""
},
{
"paragraph_id": 2,
"tag": "p",
"text": "",
"title": ""
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ç©è³ªã«ã¯åºäœã液äœãæ°äœã®3ã€ã®çžãããã ãããç©è³ªã®äžæ
ãšåŒã¶ã ãããã¯ãããã枩床ã®é«ãé ãã æ°äœã液äœãåºäœãšãªã£ãŠãããã å®éã«ã¯å§åã®å€åã«ãã£ãŠ çžãå€ããããšãããã",
"title": "ç©è³ªãšåå"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ãã¹ãã³ã®äžã«ç©ºæ°ãã€ããŠãããŠãããš äœããæŒãè¿ããŠããããã«æããããããšãåãã ããã¯ããã¹ãã³ã®äžã®ç©ºæ°ãæŒãè¿ããŠããã®ã§ããã 空æ°ã¯å®éã«ã¯æ§ã
ãªçš®é¡ã®æ°äœã«ãã£ãŠ ã§ããŠããããããã®æ°äœã¯ããããã®ååã«ãã£ãŠ ã§ããŠãããããããã®åå㯠ããããã®é床ãæã£ãŠéåããŠããã ãããã®ã©ã³ãã ãªè¡çªãããã¹ãã³ãæŒãè¿ããŠããã®ã§ããã",
"title": "ç©è³ªãšåå"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "çæ³æ°äœãèãããšã å§åãšæž©åºŠã®éã«ã¯ P V = n R T {\\displaystyle PV=nRT} ã®é¢ä¿ãããããšãç¥ãããŠããã (çæ³æ°äœã®ç¶æ
æ¹çšåŒ) ããã§ãnã¯ã¢ã«æ¿åºŠ(mol/m 3 {\\displaystyle {}^{3}} )ã§ããã Tã¯æž©åºŠ[K]ã§ããã",
"title": "ç©è³ªãšåå"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "",
"title": "ç©è³ªãšåå"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ç©è³ªãäœã圢æ
ã®1ã€ãååãšåŒã¶ã ååã¯ååæ žãšé»åã«ãã£ãŠæ§æãããŠããã ãããã¯å€å
žçã«ã¯å®å®ãªç¶æ
ãšã㊠ååšãåŸãªãããšãç¥ãããŠããã ããããå®å®ã§ããããã®ã¯å®éã«ã¯ 極埮ã®äžçã§ã¯ããããªã¹ã±ãŒã«ã§ã®äžçãš ç©çæ³åãå€ãã£ãŠæ¥ãããšã«ããã ãã®å Žåã¯é»åã¯æ³¢ã§ãããã®ããã«æ¯èãã ãã®æ§è³ªã«ãã£ãŠæ±ºãŸãããé
眮ã«ãããšãã®ã¿ å®å®ã§ããããããšãç¥ãããŠããã",
"title": "ç©è³ªãšåå"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "",
"title": "ç©è³ªãšåå"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "é»åã®ç¶æ
ã«ãã£ãŠ åºäœã®é»æ°çæ§è³ªã決ãŸãã é»åãå±èµ·ããã®ã«ãšãã«ã®ãŒãå
ãããå¿
èŠã§ãªããšãã ãããå°äœãšåŒã¶ã äžæ¹å€ãã®ãšãã«ã®ãŒãå¿
èŠãšãããšãã ããã絶çžäœ(äžå°äœ)ãšåŒã¶ã",
"title": "ç©è³ªãšåå"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "ãŸãããã®äžéã«äœçœ®ãããããªãã®ã åå°äœãšåŒã¶ã",
"title": "ç©è³ªãšåå"
}
] | é«çåŠæ ¡çç§ ç©çII > ç©è³ªãšåå æ¬é
ã¯é«çåŠæ ¡çç§ ç©çIIã®ç©è³ªãšååã®è§£èª¬ã§ããã | <small>[[é«çåŠæ ¡çç§ ç©çII]] > ç©è³ªãšåå</small>
----
æ¬é
ã¯[[é«çåŠæ ¡çç§ ç©çII]]ã®ç©è³ªãšååã®è§£èª¬ã§ããã
==ç©è³ªãšåå==
===ååãååã®éå===
====ç©è³ªã®äžæ
====
ç©è³ªã«ã¯åºäœã液äœãæ°äœã®3ã€ã®çžãããã
ãããç©è³ªã®äžæ
ãšåŒã¶ã
ãããã¯ãããã枩床ã®é«ãé ãã
æ°äœã液äœãåºäœãšãªã£ãŠãããã
å®éã«ã¯å§åã®å€åã«ãã£ãŠ
çžãå€ããããšãããã
{{See also|é«çåŠæ ¡ç©ç/ç©çI/ç±|é«æ ¡ååŠ ç©è³ªã®äžæ
}}
====ååã®éåãšå§å====
ãã¹ãã³ã®äžã«ç©ºæ°ãã€ããŠãããŠãããš
äœããæŒãè¿ããŠããããã«æããããããšãåãã
ããã¯ããã¹ãã³ã®äžã®ç©ºæ°ãæŒãè¿ããŠããã®ã§ããã
空æ°ã¯å®éã«ã¯æ§ã
ãªçš®é¡ã®æ°äœã«ãã£ãŠ
ã§ããŠããããããã®æ°äœã¯ããããã®ååã«ãã£ãŠ
ã§ããŠãããããããã®ååã¯
ããããã®é床ãæã£ãŠéåããŠããã
ãããã®ã©ã³ãã ãªè¡çªãããã¹ãã³ãæŒãè¿ããŠããã®ã§ããã
<!-- æ°äœéåè«ã䜿ã£ãå§åã®èšç®ãš -->
<!-- å®éšå€ãšã®æ¯èŒ? -->
çæ³æ°äœãèãããšã
å§åãšæž©åºŠã®éã«ã¯
<math>
PV = nRT
</math>
ã®é¢ä¿ãããããšãç¥ãããŠããã
(çæ³æ°äœã®ç¶æ
æ¹çšåŒ)
ããã§ãnã¯ã¢ã«æ¿åºŠ(mol/m<math>{}^3</math>)ã§ããã
Tã¯æž©åºŠ[K]ã§ããã
===ååãé»åãšç©è³ªã®æ§è³ª===
====ååãšé»å====
ç©è³ªãäœã圢æ
ã®1ã€ãååãšåŒã¶ã
ååã¯ååæ žãšé»åã«ãã£ãŠæ§æãããŠããã
ãããã¯å€å
žçã«ã¯å®å®ãªç¶æ
ãšããŠ
ååšãåŸãªãããšãç¥ãããŠããã
ããããå®å®ã§ããããã®ã¯å®éã«ã¯
極埮ã®äžçã§ã¯ããããªã¹ã±ãŒã«ã§ã®äžçãš
ç©çæ³åãå€ãã£ãŠæ¥ãããšã«ããã
ãã®å Žåã¯é»åã¯æ³¢ã§ãããã®ããã«æ¯èãã
ãã®æ§è³ªã«ãã£ãŠæ±ºãŸãããé
眮ã«ãããšãã®ã¿
å®å®ã§ããããããšãç¥ãããŠããã
====åºäœã®æ§è³ªãšé»å====
é»åã®ç¶æ
ã«ãã£ãŠ
åºäœã®é»æ°çæ§è³ªã決ãŸãã
é»åãå±èµ·ããã®ã«ãšãã«ã®ãŒãå
ãããå¿
èŠã§ãªããšãã
ãããå°äœãšåŒã¶ã
äžæ¹å€ãã®ãšãã«ã®ãŒãå¿
èŠãšãããšãã
ããã絶çžäœ(äžå°äœ)ãšåŒã¶ã
ãŸãããã®äžéã«äœçœ®ãããããªãã®ã
åå°äœãšåŒã¶ã
[[Category:é«çåŠæ ¡æè²|ç©ãµã€ã2ãµã€ãã€ãšããã]]
[[Category:ç©çåŠ|é«ãµã€ã2ãµã€ãã€ãšããã]]
[[Category:ç©çåŠæè²|é«ãµã€ã2ãµã€ãã€ãšããã]]
[[Category:é«çåŠæ ¡çç§ ç©çII|ãµã€ãã€ãšããã]] | null | 2022-10-20T01:24:20Z | [
"ãã³ãã¬ãŒã:See also"
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E7%89%A9%E7%90%86/%E7%89%A9%E7%90%86II/%E7%89%A9%E8%B3%AA%E3%81%A8%E5%8E%9F%E5%AD%90 |
1,946 | é«çåŠæ ¡ç©ç/ååç©ç | ããªã«ã³ã®å®éšãšã¯ãé§å¹ããªã©ã§äœæãã油滎ã®åŸ®å°ãªé£æ²«ã«ãXç·ãã©ãžãŠã ãªã©ã§åž¯é»ãããããããŠãå€éšããé»å ŽãåŒç«ããããããšã油滎ã®éå(äžåã)ã®ã»ãã«ãé»å Žã«ããéé»æ°å(äžåãã«ãªãããã«é»æ¥µæ¿ãèšçœ®ãã)ãåãã®ã§ãé£ãåã£ãŠéæ¢ããç¶æ
ã«ãªã£ãæã®é»å Žãããé»è·ã®å€ã確ãããå®éšã§ããã
ãã®å®éšã§ç®åºã»æž¬å®ãããé»è·ã®å€ã 1.6Ã10 [C]ã®æŽæ°åã«ãªã£ãã®ã§ãé»å1åã®é»è·ã 1.6Ã10 [C]ã ãšåãã£ãã
ãªãããã® 1.6Ã10 [C]ã®ããšãé»æ°çŽ é(ã§ãããããã)ãšããã
è² ã®é»è·ã«åž¯é»ãããŠããéå±æ¿ã«ã玫å€ç·ãåœãŠããšãé»åãé£ã³åºããŠããããšãããããŸããæŸé»å®éšçšã®è² 極ã«é»åãåœãŠããšãé»åãé£ã³åºããŠããããšãããããã®çŸè±¡ããå
é»å¹æ(ããã§ã ããããphotoelectric effect)ãšããã1887幎ããã«ãã«ãã£ãŠãå
é»å¹æãçºèŠããããã¬ãŒãã«ãã«ãã£ãŠãå
é»å¹æã®ç¹åŸŽãæããã«ãªã£ãã
åœãŠãå
ã®æ¯åæ°ããäžå®ã®é«ã以äžã ãšãå
é»å¹æãèµ·ããããã®æ¯åæ°ãéçæ¯åæ°(ãããã ããã©ããã)ãšãããéçæ¯åæ°ããäœãå
ã§ã¯ãå
é»å¹æãèµ·ãããªãããŸããéçæ¯åæ°ã®ãšãã®æ³¢é·ããéçæ³¢é·(ãããã ã¯ã¡ãã)ãšããã
ç©è³ªã«ãã£ãŠãéçæ¯åæ°ã¯ç°ãªããäºéçã§ã¯çŽ«å€ç·ã§ãªããšå
é»å¹æãèµ·ããªãããã»ã·ãŠã ã§ã¯å¯èŠå
ã§ãå
é»å¹æãèµ·ããã
å
é»å¹æãšã¯ãç©è³ªäž(äž»ã«éå±)ã®é»åãå
ã®ãšãã«ã®ãŒãåãåã£ãŠå€éšã«é£ã³åºãçŸè±¡ã®ããšã§ããã ãã®é£ã³åºããé»åããå
é»åã(ããã§ãããphotoelectron)ãšããã
å
é»å¹æã«ã¯,次ã®ãããªç¹åŸŽçãªæ§è³ªãããã
ãããã®æ§è³ªã®ãã¡ã1çªããš2çªãã®æ§è³ªã¯ãå€å
žç©çåŠã§ã¯èª¬æã§ããªãã ã€ãŸããå
ããé»ç£æ³¢ãšããæ³¢åã®æ§è³ªã ããæããŠããŠã¯ãã€ãã€ãŸãåããªãã®ã§ããã
ãªããªããä»®ã«ãé»ç£æ³¢ã®é»ç(é»å Ž)ã«ãã£ãŠéå±ããé»åãæŸåºãããšèããå Žåãããå
ã®åŒ·ãã倧ãããªãã°ãæ¯å¹
ã倧ãããªãã®ã§ãé»ç(é»å Ž)ã倧ãããªãã¯ãã§ããã
ããããå®éšçµæã§ã¯ãå
é»åã®éåãšãã«ã®ãŒã¯ãå
ã®åŒ·ãã«ã¯äŸåããªãã
ãã£ãŠãå€å
žååŠã§ã¯èª¬æã§ããªãã
äžè¿°ã®ççŸ(å€å
žçãªé»ç£æ³¢çè«ã§ã¯ãå
é»å¹æã説æã§ããªãããš)ã解決ããããã«ã次ã®ãããªå
éå仮説ãã¢ã€ã³ã·ã¥ã¿ã€ã³ã«ãã£ãŠæå±ãããã
ãã®2ã€ãã®æ¡ä»¶ãå®åŒåãããšã
ãšãªãã
ãã®åŒã«ãããæ¯äŸå®æ°hã¯ãã©ã³ã¯å®æ°ãšãã°ããå®æ°ã§ã
[Jã»s] ãšããå€ããšãã
ä»äºé¢æ°(ãããš ãããããwork function)ãšã¯ãå
é»å¹æãèµ·ããã®ã«å¿
èŠãªæå°ã®ãšãã«ã®ãŒã®ããšã§ãããéå±ã®çš®é¡ããšã«ã決ãŸã£ãå€ã§ããã
ä»äºé¢æ°ã®å€ã W[J] ãšãããšãå
åã®åŸãéåãšãã«ã®ãŒã®æå€§å€ K0 [J] ã«ã€ããŠã次åŒãåŸãããã
ãã®åŒãããå
é»å¹æãèµ·ããæ¡ä»¶ã¯ hÎœâ§W ãšãªãããã㯠K0â§0 ã«çžåœããã
ãããããå
é»å¹æãèµ·ããéçæ¯åæ° Îœ0 ã«ã€ããŠãhÎœ0=W ãæãç«ã€ã
ãã®å
éå仮説ã«ãããå
é»å¹æã®1çªããš2çªãã®æ§è³ªã¯ã容æã«ãççŸãªã説æã§ããããã«ãªã£ããæ³¢åã¯ç²åã®ããã«æ¯èãã®ã§ããã ãªããå
é»å¹æã®3çªãã®æ§è³ªãããããå Žæã®å
ã®åŒ·ãã¯ã ãã®å Žæã®åäœé¢ç©ã«åäœæéãé£æ¥ããå
åã®æ°ã«æ¯äŸããããšãåããã
ãããããå
ã®æ³¢é·ã¯ãã©ããã£ãŠæž¬å®ãããã®ã ãããã
çŸåšã§ã¯ãããšãã°ååã®çºå
ã¹ãã¯ãã«ã®æ³¢é·æž¬å®ãªããåææ Œåãããªãºã ãšããŠäœ¿ãããšã«ãã£ãŠãæ³¢é·ããšã«åããæ³¢é·ã枬å®ãããŠããã(â» åèæç®: å¹é¢šé€š(ã°ããµããã)ãstep-up åºç€ååŠãã梶æ¬èäº ç·šéãç³å·æ¥æš¹ ã»ãèã2015幎åçã25ããŒãž)
ãããŸããªåçãè¿°ã¹ããšãå¯èŠå
ãŠãã©ã®å
ã®æ³¢é·ã®æž¬å®ã¯ãåææ Œåã«ãã£ãŠæž¬å®ããããã ããã§ã¯ãã®åææ Œåã®çŽ°ããæ°çŸããã¡ãŒãã«ãæ°åããã¡ãŒãã«ãŠãã©ã®ééã®æ ŒåããŸãã©ããã£ãŠäœãã®ãããšããåé¡ã«è¡ãçããŠããŸãã
æŽå²çã«ã¯ãäžèšã®ããã«ãå¯èŠå
ã®æ³¢é·ã枬å®ãããŠãã£ãã
ãŸãã1805幎ããã®ãã€ã³ã°ã®å®éšãã§æåãªã€ã³ã°ãã®ç 究ã«ãããå¯èŠå
ã®æ³¢é·ã¯ããããã 100nm(10m) ã 1000nm ã®çšåºŠã§ããããšã¯ããã®é ããããã§ã«äºæ³ãããŠããã
ãã®åŸããã€ãã®ã¬ã³ãºã®ç 磚工ã ã£ããã©ãŠã³ããŒãã¡ãŒããããããåææ Œåãéçºããå¯èŠå
ã®æ³¢é·ã粟å¯ã«æž¬å®ããäºã«æåããããã©ãŠã³ããŒãã¡ãŒã¯åææ Œåãäœãããã«çŽ°ãééãçšããå å·¥è£
眮ã補äœãããã®å å·¥æ©ã§è£œäœãããåææ ŒåãçšããŠãå
ã®æ³¢é·ã®æž¬å®ããå§ããã®ããç 究ã®å§ãŸãã§ããã1821幎ã«ãã©ãŠã³ããŒãã¡ãŒã¯ã1cmãããæ Œåã130æ¬ã䞊ã¹ãåææ Œåã補äœããã
ãŸãã1870幎ã«ã¯ã¢ã¡ãªã«ã®ã©ã¶ãã©ãŒããã¹ããã¥ã©ã ãšããåéãçšããåå°åã®åææ Œåã補äœã(ãã®ã¹ããã¥ã©ã åéã¯å
ã®åå°æ§ãé«ã)ãããã«ãã£ãŠ1mmããã700æ¬ãã®æ Œåã®ããåææ Œåã補äœããã(èŠåºå
ž)
ããã«ãã®ããã®æ代ãéãããã®æœ€æ»ã®ããã«æ°Žéã䜿ãæ°Žéæµ®éæ³ããç 究éçºã§è¡ãããã
ããé«ç²ŸåºŠãªæ³¢é·æž¬å®ããã®ã¡ã®æ代ã®ç©çåŠè
ãã€ã±ã«ãœã³ã«ãã£ãŠãå¹²æžèš(ããããããã)ãšãããã®ãçšããŠ(çžå¯Ÿæ§çè«ã®å
¥éæžã«ããåºãŠããè£
眮ã§ãããé«æ ¡çã¯ããŸã çžå¯Ÿæ§çè«ãç¿ã£ãŠãªãã®ã§ãæ°ã«ããªããŠããã)ãå¹²æžèšã®åå°é¡ã粟å¯ããžã§çŽ°ããåããããšã«ãããé«ç²ŸåºŠãªæ³¢é·æž¬å®åšãã€ããããã®æž¬å®åšã«ãã£ãŠã«ãããŠã ã®èµ€è²ã¹ãã¯ãã«ç·ã枬å®ããçµæã®æ³¢é·ã¯643.84696nmã ã£ãããã€ã±ã«ãœã³ã®æž¬å®æ¹æ³ã¯ãèµ€è²ã¹ãã¯ãã«å
ã®æ³¢é·ããåœæã®ã¡ãŒãã«ååšãšæ¯èŒããããšã§æž¬å®ããã
ãã€ã±ã«ãœã³ã®å¶äœããå¹²æžèšã«ããæ°Žéæµ®éæ³ã®æè¡ãåãå
¥ããããŠããããšããã
ããã«ãããžã®æè¡é©æ°ã§ãããŒãã³ã»ããã(ãã¡ã«ãã³ã»ãããããšãèš³ã)ãšããã匟åæ§ã®ããæ質ã§ããžãã€ããããšã«ãã£ãŠèª€å·®ããªããããŠå¹³ååãããã®ã§ãè¶
絶çã«é«ç²ŸåºŠã®éããããäœãæè¡ããã€ã®ãªã¹ã®ç©çåŠè
ããŒãã¹ã»ã©ã«ãã»ããŒãã³(è±:en:w:Thomas Ralph Merton )ãªã©ã«ãã£ãŠéçºãããã
ãªããçŸä»£ã§ããç 究çšãšããŠå¹²æžèšãçšããæ³¢é·æž¬å®åšãçšããããŠããã(èŠåºå
ž) ã¡ãŒãã«ååšã¯ããã€ã±ã«ãœã³ã®å®éšã®åœæã¯é·ãã®ããããšã®æšæºã ã£ããã1983幎以éã¯ã¡ãŒãã«ååšã¯é·ãã®æšæºã«ã¯çšããããŠããªããçŸåšã®ã¡ãŒãã«å®çŸ©ã¯ä»¥äžã®éãã
倪éœé»æ± ããå
é»å¹æã®ãããªçŸè±¡ã§ããããšèããããŠããã(â» å®æåºçã®æç§æžãªã©ã§ãæ±ã£ãŠãã話é¡ã)
ãªãã倪éœé»æ± ã¯äžè¬çã«åå°äœã§ããããã€ãªãŒãåããPNæ¥åã®éšåã«å
ãåœãŠãå¿
èŠãããã
(PNæ¥åéšå以å€ã®å Žæã«ãå
ãããã£ãŠããçããé»åããé»æµãšããŠåãåºããªããé»æµãšããŠåãåºããããã«ããã«ã¯ãPNæ¥åã®éšåã«ãå
ãåœãŠãå¿
èŠãããããã®ãããPNæ¥åã®çæ¹ã®æ質ããéæããããã«è¿ãå
ééçã®ææã«ããå¿
èŠãããããéæé»æ¥µããšããã)
(â» ç¯å²å€?: ) ãªããçºå
ãã€ãªãŒãåå°äœã¯ããã®éãã¿ãŒã³ãšããŠèããããŠãããå
é»å¹æã§ãããä»äºé¢æ°ãã«ããããšãã«ã®ãŒããã£ãé»æµãæµãããšã«ããããã®åå°äœç©è³ªã®ãä»äºé¢æ°ãã«ããããšãã«ã®ãŒã®å
ããPNæ¥åã®æ¥åé¢ããæŸåºãããããšããä»çµã¿ã§ããã
ãªããCCDã«ã¡ã©ãªã©ã«äœ¿ãããCCDã¯ã倪éœé»æ± ã®ãããªæ©èœããã€åå°äœããé»åæºãšããŠã§ã¯ãªããå
ã®ã»ã³ãµãŒãšããŠæŽ»çšãããšããä»çµã¿ã®åå°äœã§ããã(â» å®æåºçã®æç§æžãªã©ã§ãæ±ã£ãŠãã話é¡ã)
(â» æ®éç§é«æ ¡ã®ãç©çãç³»ç§ç®ã§ã¯ç¿ããªããã)
ç©ççŸè±¡ã®éååãšããŠãå
é»å¹æãç©è³ªæ³¢ã®ã»ãã«ãååã¹ã±ãŒã«ã®ç©ççŸè±¡ã®éååã¯ãããããçš®é¡ã®è¶
äŒå°ç©è³ªã§ã¯ãããã«éããç£æãéååããçŸè±¡ãç¥ãããŠããã(â» å·¥æ¥é«æ ¡ã®ç§ç®ãå·¥æ¥ææãäžå·»(ãŸãã¯ç§ç®ã®åŸå)ã§ç¿ãã)
ç§åŠè
ã¬ã³ãã²ã³ã¯ã1895幎ãæŸé»ç®¡ããã¡ããŠé°æ¥µç·ã®å®éšãããŠãããšããæŸé»ç®¡ã®ã¡ããã«çœ®ããŠãã£ãåç也æ¿ãæå
ããŠããäºã«æ°ä»ããã
圌(ã¬ã³ãã²ã³)ã¯ãé°æ¥µç·ãã¬ã©ã¹ã«åœãã£ããšãããªã«ãæªç¥ã®ãã®ãæŸå°ãããŠããšèããXç·ãšåã¥ããã
ãããŠãããŸããŸãªå®éšã«ãã£ãŠãXç·ã¯æ¬¡ã®æ§è³ªããã€ããšãæããã«ãªã£ãã
ãã®äºãããXç·ã¯ãè·é»ç²åã§ã¯ãªãäºãåããã(çµè«ããããšãXç·ã®æ£äœã¯ãæ³¢é·ã®çãé»ç£æ³¢ã§ããã)
ãŸãã
ãªã©ã®æ§è³ªãããã
ãªãçŸä»£ã§ã¯ãå»ççšã®Xç·ããã¬ã³ãã²ã³ããšãããã
1912幎ãç©çåŠè
ã©ãŠãšã¯ãXç·ãåçµæ¶ã«åœãŠããšãåçãã£ã«ã ã«å³ã®ãããªæç¹ã®æš¡æ§ã«ããããšãçºèŠããããããã©ãŠãšæç¹(ã¯ããŠã)ãšãããçµæ¶äžã®ååãåææ Œåã®åœ¹å²ãããããšã§çºçããå¹²æžçŸè±¡ã§ããã
1912幎ãç©çåŠè
ãã©ãã°ã¯ãåå°ã匷ãããæ¡ä»¶åŒãçºèŠããã
2d sinΞ = n λ
ããããã©ãã°ã®æ¡ä»¶ãšããã
äžåŒã®dã¯æ Œåé¢ã®ééã®å¹
ã§ããã
Xç·ãççŽ å¡ãªã©ã®(éå±ãšã¯éããªã)ç©è³ªã«åœãŠããã®æ£ä¹±ãããããšã®Xç·ã調ã¹ããšãããšã®Xç·ã®æ³¢é·ãããé·ããã®ããæ£ä¹±ããXç·ã«å«ãŸããã ãã®ããã«æ£ä¹±Xç·ã®æ³¢é·ã䌞ã³ãçŸè±¡ã¯ç©çåŠè
ã³ã³ããã³ã«ãã£ãŠè§£æãããã®ã§ãã³ã³ããã³å¹æ(ãŸãã¯ã³ã³ããã³æ£ä¹±)ãšããã
ãã®çŸè±¡ã¯ãXç·ãæ³¢ãšèããã®ã§ã¯èª¬æãã€ããªãã(ããä»®ã«æ³¢ãšèããå Žåãæ£ä¹±å
ã®æ³¢é·ã¯ãå
¥å°Xç·ãšåãæ³¢é·ã«ãªãã¯ãããªããªããæ°Žé¢ã®æ³¢ã«äŸãããªããããæ°Žé¢ãæ£ã§4ç§éã«1åã®ããŒã¹ã§æºãããããæ°Žé¢ã®æ³¢ãã4ç§éã«1åã®ããŒã¹ã§åšæãè¿ããã®ãšãåãçå±ã) ããŠãæ³¢åã®çè«ã§ã³ã³ããã³å¹æã説æã§ããªããªããç²åã®çè«ã§èª¬æãããã°è¯ãã ããã
ãã®åœæãã¢ã€ã³ã·ã¥ã¿ã€ã³ã¯å
éå仮説ã«ããšã¥ããå
åã¯ãšãã«ã®ãŒhÎœããã€ã ãã§ãªããããã«æ¬¡ã®åŒã§è¡šãããéåépããã€ããšãçºèŠããŠããã
p = h Μ c ( = h Μ Μ λ = h λ ) {\displaystyle p={\frac {h\nu }{c}}(={\frac {h\nu }{\nu \lambda }}={\frac {h}{\lambda }})}
ç©çåŠè
ã³ã³ããã³ã¯ããã®çºèŠãå©çšããæ³¢é·Î»ã®Xç·ããéåé h λ {\displaystyle {\frac {h}{\lambda }}} ãšãšãã«ã®ãŒ h c λ {\displaystyle {\frac {hc}{\lambda }}} ãæã€ç²å(å
å)ã®æµããšèãã Xç·ã®æ£ä¹±ãããã®å
åãç©è³ªäžã®ããé»åãšå®å
šåŒŸæ§è¡çªãããçµæãšèããã
解æ³ã¯ãäžèšã®ãšããã
ãšãã«ã®ãŒä¿åã®åŒ
éåéä¿åã®åŒ
äžèšã®3ã€ã®åŒãé£ç«ãããã®é£ç«æ¹çšåŒã解ãããã«vãšÏãé£ç«èšç®ã§æ¶å»ãããŠããã λ â λ â² {\displaystyle \lambda \fallingdotseq \lambda '} ã®ãšã㫠λ â² â λ + h m c ( 1 â cos Ξ ) {\displaystyle \lambda '\fallingdotseq \lambda +{\frac {h}{mc}}(1-\cos \theta )} ãåŸãããã
ãã®åŒãå®éšåŒãšããäžèŽããã®ã§ãã³ã³ããã³ã®èª¬ã®æ£ããã¯å®èšŒãããã
(ç·šéè
ãž: èšè¿°ããŠãã ããã)(Gimyamma ããã解æ³ãæžããŠã¿ãŸããã)
åŒ(1),(2),(3)ããã v {\displaystyle v} ãš Ï {\displaystyle \phi } ãæ¶å»ããŠã λ , λ â² , Ξ {\displaystyle \lambda ,\lambda ',\theta } ã®é¢ä¿åŒãæ±ããã°ããã
( m v sin Ï ) 2 = ( â h λ â² sin Ξ ) 2 {\displaystyle (mv\sin \phi )^{2}=(-{\frac {h}{\lambda '}}\sin \theta )^{2}}
m 2 v 2 = ( h λ â h λ â² cos Ξ ) 2 + ( â h λ â² sin Ξ ) 2 + h 2 λ â² 2 {\displaystyle m^{2}v^{2}=({\frac {h}{\lambda }}-{\frac {h}{\lambda '}}\cos \theta )^{2}+(-{\frac {h}{\lambda '}}\sin \theta )^{2}+{\frac {h^{2}}{\lambda '^{2}}}}
ãåŸãã
åŒ(1)ã®å³èŸºã®ç¬¬2é
ãå€åœ¢ããŠåŒ(4)ã代å
¥ããã
ãããåŒ(1)ã®å³èŸºã«ä»£å
¥ãããš
ãåŸãã
ãã®åŒãåŒ(5)ã®å³èŸºç¬¬2é
ã«ä»£å
¥ãããšã
ãã®åŒã®å³èŸºã®ç¬¬1é
ã移è¡ããåŒãå€åœ¢ãããš
䞡蟺㫠λ λ â² {\displaystyle \lambda \lambda '} ãæãããš
Xç·ã®æ£ä¹±ã§ã¯ã λ â² â λ {\displaystyle \lambda '\fallingdotseq \lambda } ãªã®ã§
æ
ã«åŒ(6)ãã
ããã§ãææã®åŒãå°åºãããã
å
ã®éåé P[kgã»m/s]=hÎœ/c ã«ã€ããŠã
ãŸã cP=hÎœ[J] ãšå€åœ¢ããŠã¿ããšããé床ã«éåéãããããã®ããšãã«ã®ãŒã§ããããšããå
容ã®å
¬åŒã«ãªã£ãŠããã
ãããç解ãããããã²ãšãŸããå
ãç²åã§ãããšåæã«æµäœã§ãããšèããŠããã®é»ç£æ³¢ãåäœäœç©ãããã®éåépãæã£ãŠãããšããŠããã®æµäœã®éåéã®å¯åºŠ(éåéå¯åºŠ)ã p [(kgã»m/s)/m]ãšãããããã®å Žåã®é»ç£æ³¢ã¯æµäœãªã®ã§ãéåéã¯ããã®å¯åºŠã§èããå¿
èŠãããã
é»ç£æ³¢ãç©äœã«ç
§å°ããŠãå
ãç©äœã«åžåããããšããããåå°ã¯ãªããšããŠãå
ã®ãšãã«ã®ãŒã¯ãã¹ãŠç©äœã«åžåããããšãããç°¡åã®ãããç©äœå£ã«åçŽã«å
ãç
§å°ãããšãããç©äœãžã®å
ã®ç
§å°é¢ç©ãA[m]ãšããã
é»ç£æ³¢ã¯å
é c[m/s] ã§é²ãã®ã ãããå£ããcã®è·é¢ã®éã«ãããã¹ãŠã®å
åã¯ããã¹ãŠåäœæéåŸã«åžåãããäºã«ãªããåäœæéã«å£ã«åžåãããå
åã®éã¯ããã®åäœæéã®ããã ã«å£ã«æµã蟌ãã å
åã®éã§ããã®ã§ã
å³ã®ããã«ãä»®ã«åºé¢ãA[m]ãšããŠãé«ãhã c ( hã®å€§ããã¯cã«çãããåäœæét=1ãããããšããã° h=cã»1 ã§ãã)[m]ãšããæ±ã®äœç© AÃc[m]äžã«å«ãŸããå
åã®éã®ç·åã«çããã
ãã£ãœããéåéå¯åºŠã¯ p[(kgã»m/s)/m]ã ã£ãã®ã§ããã®æ± AÃh ã«å«ãŸããéåéã®ç·åã¯ã AÃhÃp[kgã»m/s]ã§ããã
å
ãåžåããç©äœã®éåéã¯ãåäœæéã«Ahpã®éåéãå¢å ããããšã«ãªãããh=cã§ãã£ãã®ã§ãã€ãŸããéåéãåäœæéãããã« Acp[kgã»m/s] ã ãå£ã«æµãããããšã«ãªãã
ãã£ãœããé«æ ¡ç©çã®ååŠã®çè«ã«ããããéåéã®æéãããã®å€åã¯ãåã§ãããã§ãã£ãã®ã§ãã€ãŸãç©äœã¯ãAcp[N]ã®åãåããã
åãåããã®ã¯ç
§å°ãããé¢ã ãããå[N]ãé¢ç©ã§å²ãã°å§åã®æ¬¡å
[N/m]=[Pa]ã«ãªãã
å®éã«é¢ç©ã§å²ãèšç®ãããã°ãå§åãšã㊠cp[N/m]=[Pa]=[J/m] ãåããäºãèšç®çã«åãããããã«ãå§åã®æ¬¡å
ã¯[N/m]=[Pa]=[J/m]ãšå€åœ¢ã§ããã®ã§ããå§åã¯ãåäœäœç©ãããã®ãšãã«ã®ãŒã®å¯åºŠ(ããšãã«ã®ãŒå¯åºŠããšãã)ã§ããããšèãããã
ãšããã° cp ã®æ¬¡å
ã¯ã[å§å]=[ãšãã«ã®ãŒå¯åºŠ] ãšãªãã
ãã®ãšãã«ã®ãŒå¯åºŠã«ãhÎœã察å¿ããŠãããšèããã°ãåççã§ããã
èŠããã«ãå
ã®ãããªãäºå®äžã¯ç¡éã«å§çž®ã§ããæ³¢ã»æµäœã§ã¯ã
å
¬åŒãšããŠãé床ãvãéåéå¯åºŠãpããšãã«ã®ãŒå¯åºŠãεãšããŠèããã°ã
ãšããé¢ä¿ããªããã€ã
(ãªããæ°Žã空æ°ã®ãããªæ®éã®æµäœã§ã¯ãç¡éã«ã¯å§çž®ã§ããªãã®ã§ãäžèšã®å
¬åŒã¯æãç«ããªãã)
ãããããã³ã³ããã³å¹æã®åŠç¿ã§åãã£ãéåéã®å
¬åŒ p = h Îœ c {\displaystyle p={\frac {h\nu }{c}}} ã¯ãéåéå¯åºŠãšãšãã«ã®ãŒå¯åºŠã®é¢ä¿åŒã«ãå
écãšå
é»å¹æã®ãšãã«ã®ãŒhÎœã代å
¥ãããã®ã«ãªã£ãŠããã
äžèšã®èå¯ã¯ãå
ãæµäœãšããŠèããé»ç£æ³¢ã®éåéã ããç²åãšããŠè§£éãããå
åã®éåéã«ãã cP=hÎœ ãšããé¢ä¿ãæãç«ã€ãšèãããã
ããèªè
ããå§åããšãã«ã®ãŒå¯åºŠãšèããã®ãåããã¥ãããã°ãããšãã°ç±ååŠã®ä»äºã®å
¬åŒ W=Pâ¿V ã®é¡æšãããŠã¯ã©ãã? ãªããäžèšã®éåéãšãšãã«ã®ãŒã®é¢ä¿åŒã®å°åºã¯å€§ãŸããªèª¬æã§ãããæ£ç¢ºãªå°åºæ³ã¯ã(倧åŠã§ç¿ã)ãã¯ã¹ãŠã§ã«ã®æ¹çšåŒã«ãããªããã°ãªããªãã
ããããã ãå
ã¯ãé»åã«äœçšãããšãã«ãå
ãç²åãšããŠæ¯èã(ãµããŸã)ã ãšããã®ãæ£ããã ããã
ãã£ãœãããã¿ããã«ãå
ã¯ç²å! å
ã¯æ³¢åã§ã¯ãªã!!ã(Ã)ãšãããã®ã¯ãåãªã銬鹿ã®ã²ãšã€èŠãã§ããã
ãã¯ã¹ãŠã§ã«ã®æ¹çšåŒã§ã¯ãå
(é»ç£æ³¢)ã¯æ³¢åãšããŠããã€ããã®ã§ããã
ããããå
é»å¹æã§èµ·ããçŸè±¡ã§ã¯ãæŸåºé»åã®ãã€éåãšãã«ã®ãŒã¯ãå
ã®åŒ·åºŠãšã¯ç¡é¢ä¿ã§ãããåçŽãªæµäœãšããŠèãããªãã(ããšãã°éå
ããããå
ãéãããããŠã)å
ã®åŒ·åºŠãäžããã°ãéåéå¯åºŠãäžããããºã ãããã®åž°çµã®æŸåºé»åã®ãšãã«ã®ãŒå¯åºŠãäžããããºã ããããšããäºæž¬ãæãç«ã¡ããã ãããããå®éšçµæã¯ãã®äºæž¬ãšã¯ç°ãªããå
é»å¹æã¯å
ã®åŒ·åºŠãšã¯ç¡é¢ä¿ã«å
ã®åšæ³¢æ°ã«ãã£ãŠæŸåºé»åã®ãšãã«ã®ãŒã決ãŸããã»ã»ã»ãšããã®ããå®éšäºå®ã§ããã
ãã®ãããªå®éšçµæããã20äžçŽåé ã®åœæãåèããŠããéåååŠãªã©ãšé¢é£ã¥ããŠããå
ãæ³¢ã§ãããšåæã«ç²åã§ããããšæå®ããã®ãããŒãã«è²¡å£ãªã©ã§ãããå
é»å¹æãå
ã®ç²å説ã®æ ¹æ ã®ã²ãšã€ãšããã®ãã¢ã€ã³ã·ã¥ã¿ã€ã³ã®ä»®èª¬ã§ãããã¢ã€ã³ã·ã¥ã¿ã€ã³ã®ãã®ä»®èª¬ãå®èª¬ãšããŠèªå®ããã®ãããŒãã«è²¡å£ã§ãããçŸåšã®ç©çåŠã§ã¯ãå
é»å¹æãå
å説ã®æ ¹æ ãšããŠé説ã«ãªã£ãŠããã
å
é»å¹æã®å®éšçµæãã®ãã®ã¯ãåã«ãå
é»å¹æã«ããããå
ãããåçŽãªæµäœã»æ³¢åãšããŠã¯èããããªãã ããã»ã»ã»ãšããã ãã®äºã§ããã
çµå±ãç©çåŠã¯å®éšç§åŠã§ãããå®éšçµæã«ããšã¥ãå®éšæ³åãèŠãããããªãããå
åããšããã¢ã€ãã¢ã¯ããå
é»å¹æã®æŸåºé»å1åãããã®ãšãã«ã®ãŒã¯ãå
¥å°å
ã®åŒ·åºŠã«å¯ãããå
ã®æ³¢é·(åšæ³¢æ°)ã«ããããšããäºãèŠããããããããã®æ段ã«ããããã¢ã€ã³ã·ã¥ã¿ã€ã³ãšãã®æ¯æè
ã«ãšã£ãŠã¯ããå
ã®ç²å説ããšããã®ããèŠããããããããã®ã¢ãã«ã ã£ãã ãã§ãã(ç²åãªã®ã«æ³¢é·(åšæ³¢æ°)ãšã¯ãæå³äžæã ã)ããããŠéåéå¯åºŠãšãšãã«ã®ãŒå¯åºŠã®é¢ä¿ vp=ε ãšããç¥èããŸããå
é»å¹æã®å
¬åŒ cP=hÎœ ãèŠããããããããã®æ段ã«ãããªãã
ãã£ããã®å
ã¯ãåçŽãªæ³¢ã§ããªããåçŽãªç²åã§ããªãããã åã«ãå
ã¯å
ã§ãããå
ã§ãããªãã
ãå
ã®ç²å説ããšããã®ã¯ãç空äžã§åªè³ª(ã°ããã€)ããªããŠãå
ãäŒããããšããçšåºŠã®æå³åãã§ãããªãã ãããã¢ã€ã³ã·ã¥ã¿ã€ã³ãç¹æ®çžå¯Ÿæ§çè«ãçºè¡šããåãŸã§ã¯ã(20äžç€ä»¥éããçŸä»£ã§ã¯åŠå®ãããŠãããã)ãã€ãŠããšãŒãã«ããšããå
ãäŒããåªè³ªã®ååšãä¿¡ããããŠããããããã¢ã€ã³ã·ã¥ã¿ã€ã³ã¯çžå¯Ÿæ§çè«ã«ããããšãŒãã«ã®ååšãåŠå®ããã
ãå
ã®ç²å説ããçºè¡šããŠããè
ãåããã¢ã€ã³ã·ã¥ã¿ã€ã³ã ã£ãã®ã§ãããŒãã«è²¡å£ã¯ãæ¬æ¥ãªãç¹çžå¯Ÿæ§çè«ã§ããŒãã«è³ãæãããããã«ãå
å説ã§ããŒãã«è³ãã¢ã€ã³ã·ã¥ã¿ã€ã³ã«æããã ãã ãã
ç©çåŠè
ãã»ããã€ã¯ãæ³¢ãšèããããŠãå
ãç²åã®æ§è³ªããã€ãªãã°ããã£ãšé»åãç²åãšããŠã®æ§è³ªã ãã§ãªããé»åãæ³¢åã®ããã«æ¯èãã ãããšèããã
ãããŠãé»åã ãã§ãªããäžè¬ã®ç²åã«å¯ŸããŠãããã®èããé©çšãã次ã®å
¬åŒãæå±ããã
ããã¯ãã»ããã€ã«ãã仮説ã§ãã£ãããçŸåšã§ã¯æ£ãããšèªããããŠããã
ãã®æ³¢ã¯ãç©è³ªæ³¢(material wave)ãšåŒã°ããããã»ããã€æ³¢(de Broglie wave length)ãšãããã ããªãã¡ãå
åãé»åã«éãããããããç©è³ªã¯ç²åæ§ãšæ³¢åæ§ããããæã€ãšãããã
ãã®ç©è³ªæ³¢ãšãã説ã«ãããšããããé»åç·ãç©è³ªã«åœãŠãã°ãåæãªã©ã®çŸè±¡ãèµ·ããã¯ãã§ããã
1927幎ã1928幎ã«ãããŠãããããœã³ãšã¬ãŒããŒã¯ãããã±ã«ãªã©ã®ç©è³ªã«é»åç·ãåœãŠãå®éšãè¡ããXç·åæãšåæ§ã«é»åç·ã§ãåæãèµ·ããããšãå®èšŒãããæ¥æ¬ã§ã1928幎ã«èæ± æ£å£«(ããã¡ ããã)ãé²æ¯çã«é»åç·ãåœãŠãå®éšã«ããåæãèµ·ããããšã確èªããã
é»åç·ã®æ³¢é·ã¯ãé«é»å§ããããŠé»åãå éããŠé床ãé«ããã°ãç©è³ªæ³¢ã®æ³¢é·ã¯ããªãå°ããã§ããã®ã§ãå¯èŠå
ã®æ³¢é·ãããå°ãããªãã
ãã®ãããå¯èŠå
ã§ã¯èŠ³æž¬ã§ããªããã£ãçµæ¶æ§é ããé»åæ³¢ãXç·ãªã©ã§èŠ³æž¬ã§ããããã«ãªã£ããçç©åŠã§ãŠã€ã«ã¹ãé»åé¡åŸ®é¡ã§èŠ³æž¬ã§ããããã«ãªã£ãã®ããé»åã®ç©è³ªæ³¢ãå¯èŠå
ããã倧å¹
ã«å°ããããã§ããã
äžè¿°ã®ãããªãããŸããŸãªå®éšã®çµæããããã¹ãŠã®ç©è³ªã«ã¯ãååãŠãã©ã®å€§ããã®äžç(以éãåã«ãååã¹ã±ãŒã«ããªã©ãšç¥èšãã)ã§ã¯ãæ³¢åæ§ãšç²åæ§ã®äž¡æ¹ã®æ§è³ªããã€ãšèããããŠããã ãã®ããšãç²åãšæ³¢åã®äºéæ§ãšããã
ãããŠãååã¹ã±ãŒã«ã§ã¯ãããäžã€ã®ç©è³ª(äž»ã«é»åã®ãããªç²å)ã«ã€ããŠããã®äœçœ®ãšéåéã®äž¡æ¹ãåæã«æ±ºå®ããäºã¯ã§ããªãããã®ããšãäžç¢ºå®æ§åç(ãµãããŠããã ããã)ãšããã
ç©çåŠè
ã¬ã€ã¬ãŒãšç©çåŠè
ããŒã¹ãã³ã¯ã(ã©ãžãŠã ããåºãã)αç²åããããéã±ãã«åœãŠãå®éšãè¡ããαç²åã®æ£ä¹±ã®æ§åã調ã¹ãã(ãªããαç²åã®æ£äœã¯ããªãŠã ã®ååæ žã)ãã®çµæãã»ãšãã©ã®Î±ç²åã¯éã±ããçŽ éãããããéã±ãäžã®äžéšã®å Žæã®è¿ããéã£ãαç²åã ãã倧å¹
ã«æ£ä¹±ããçŸè±¡ãçºèŠããã
ãã®å®éšçµæããã©ã¶ãã©ãŒãã¯ãååæ žã®ååšãã€ããšããã
ååã¯ãäžå¿ã«ååæ žãããããã®ãŸãããé»åãéåãããšããã©ã¶ãã©ãŒãã¢ãã«ãšãã°ããã¢ãã«ã«ãã£ãŠèª¬æãããã
åå(atom)ã¯ãå
šäœãšããŠã¯é»æ°çã«äžæ§ã§ãããè² ã®é»è·ãæããé»åãé»åæ®»ã«æã€ã ããã§ãããªã«ã³ã®å®éš ã«ããçµæãªã©ãããé»åã®è³ªéã¯æ°ŽçŽ ã€ãªã³ã®è³ªéã®çŽ1/1840çšåºŠãããªãããšãåãã£ãŠããã ããªãã¡ãååã¯é»åãšéœã€ãªã³ãšãå«ãŸãããã質éã®å€§éšåã¯éœã€ãªã³ããã€ããšãåããã ååæ žã®å€§ããã¯ååå
šäœã®1/10000çšåºŠã§ãããããååã®å€§éšåã¯ç空ã§ããã ååæ žã¯ãæ£ã®é»è·ããã€Zåã®éœå(proton)ãšãé»æ°çã«äžæ§ãª(AâZ)åã®äžæ§å(neutron)ãããªãã éœåãšäžæ§åã®åæ°ã®åèšã質éæ°(mass number)ãšããã éœåãšäžæ§åã®è³ªéã¯ã»ãŒçãããããååæ žã®è³ªéã¯ã質éæ°Aã«ã»ãŒæ¯äŸããã
é«æž©ã®ç©äœããçºå
ãããå
ã«ã¯ãã©ã®(å¯èŠå
ã®)è²ã®æ³¢é·(åšæ³¢æ°)ãããããã®ãããªé£ç¶çãªæ³¢é·ã®å
ãé£ç¶ã¹ãã¯ãã«ãšããã
ãã£ãœãããããªãŠã ãæ°ŽçŽ ãªã©ã®ãç¹å®ã®ç©è³ªã«é»å§ãããããæŸé»ãããšãã«çºå
ããæ³¢é·ã¯ãç¹å®ã®æ°æ¬ã®æ³¢é·ããå«ãŸããŠãããããã®ãããªã¹ãã¯ãã«ãèŒç·(ããã)ãšããã
ãã«ããŒã¯ãæ°ŽçŽ ååã®æ°æ¬ããèŒç·ã®æ³¢é·ãã次ã®å
¬åŒã§è¡šçŸã§ããããšã«æ°ã¥ããã
λ = 3.65 à 10 â 7 m à ( n 2 n 2 â 4 ) {\displaystyle \lambda =3.65\times 10^{-7}\mathrm {m} \times \left({n^{2} \over n^{2}-4}\right)} (ãã ããn=3, 4 , 5 ,6 ,ã»ã»ã»)
äžåŒäžã®ãmãã¯ã¡ãŒãã«åäœãšããæå³ã(äžåŒã®mã¯ä»£æ°ã§ã¯ãªãã®ã§ãééããªãããã«ã)
ãã®åŸãæ°ŽçŽ ä»¥å€ã®ååããå¯èŠå
以å€ã®é åã«ã€ããŠããç©çåŠè
ãã¡ã«ãã£ãŠèª¿ã¹ããã次ã®å
¬åŒãžãšãç©çåŠè
ãªã¥ãŒãããªã«ãã£ãŠããŸãšããããã
äžåŒã®Rã¯ãªã¥ãŒãããªå®æ°ãšããã R = 1.097 à 10 7 / m {\displaystyle R=1.097\times 10^{7}/m} ã§ããã
ã©ã¶ãã©ãŒãã®ååæš¡åã«åŸãã°ãé»åã¯ããŸãã§ææã®å
¬è»¢ã®ããã«ååæ žãäžå¿ãšããåè»éã®äžãäžå®ã®é床ã§éåããã
ååæ žãäžå¿ãšããååŸr[m]ã®åè»éãéãv[m/s]ã§å転ããé»åã®è§éåé r p = r m v {\displaystyle rp=rmv} ã¯ã h 2 Ï {\displaystyle {\frac {h}{2\pi }}} ã®æ£æŽæ°åã«ãªããªããã°ãªããªã(è§éåéã®éåå)ã
ãæºãããã°ãªããªãã
åŸå¹Ž(1924幎)ããã»ããã€ã¯ãç©è³ªç²åã¯æ³¢åæ§ãæã¡ã
ããã«åŸãã°ãããŒã¢ã®éåæ¡ä»¶ã®ä»®å®ã¯ããé»åè»éã®é·ãã¯ãé»åã®ç©è³ªæ³¢ã®æ³¢é·ã®æ£æŽæ°åã§ããããšè¡šçŸã§ããã
é»åã¯ããããŸã£ããšã³ãšã³ã®ãšãã«ã®ãŒããæããªãããã®ãšã³ãšã³ã®ãšãã«ã®ãŒå€ããšãã«ã®ãŒé äœãšããã
æ°ŽçŽ ååã«ãããŠãé»åè»éäžã«ããé»åã®ãšãã«ã®ãŒãæ±ããèšç®ããããããŸãããã®ããã«ã¯ãååã®ååŸãæ±ããå¿
èŠãããã
æ°ŽçŽ ã®é»åãååæ ž H + {\displaystyle H^{+}} ãäžå¿ãšããååŸrã®åè»éäžãäžå®ã®é床vã§éåããŠãããšããã°ãéåæ¹çšåŒã¯
ã§è¡šãããã
äžæ¹ãé»åãå®åžžæ³¢ã®æ¡ä»¶ãæºããå¿
èŠãããã®ã§ãåé
ã®åŒ(1)ããã
ã§ããã
ãã®vãããã»ã©ã®åéåã®åŒã«ä»£å
¥ããŠæŽé ããã°ã
(ãã ããn=1, 2 , 3 ,ã»ã»ã»)
ã«ãªããããããŠãæ°ŽçŽ ååã®é»åã®è»éååŸãæ±ãŸãã
ããã»ã©ã®è»éååŸã®åŒã§n=1ã®ãšãååŸr1ããããŒã¢ååŸããšããã
ååã®äžçã§ããéåãšãã«ã®ãŒKãšäœçœ®ãšãã«ã®ãŒUã®åãããšãã«ã®ãŒã§ããã
äœçœ®ãšãã«ã®ãŒUã¯ããã®æ°ŽçŽ ã®é»åã®å Žåãªããéé»æ°ãšãã«ã®ãŒãæ±ããã°å
åã§ãããé»äœã®åŒã«ãã£ãŠæ±ããããŠã
ãšãªãã
éåãšãã«ã®ãŒKã¯ã K = 1 2 m v 2 {\displaystyle K={\frac {1}{2}}mv^{2}} ãªã®ã§
äžåŒã®å³èŸºç¬¬äžé
ã«ã
m v 2 = k 0 e 2 r {\displaystyle mv^{2}=k_{0}{\frac {e^{2}}{r}}} ã代å
¥ããã°ã
ãšãªãã
ããã«ãããã«é»åã®è»éååŸ r = r n {\displaystyle r=r_{n}} ã®åŒ(3)ã代å
¥ããã°ã
ãšãªãããããæ°ŽçŽ ååã®ãšãã«ã®ãŒæºäœã§ããã
ãšãã«ã®ãŒæºäœã®å
¬åŒãããèŠããšããŸãããšãã«ã®ãŒãããšã³ãšã³ã®å€ã«ãªãããšãåããããŸãããšãã«ã®ãŒãè² ã«ãªãäºããããã
n=1ã®ãšããããã£ãšããšãã«ã®ãŒã®äœãç¶æ
ã§ããããã®ãããn=1ã®ãšããå®å®ãªç¶æ
ã§ããããã£ãŠãé»åã¯éåžžãn=1ã®ç¶æ
ã«ãªãã
ãªãã
æ°ŽçŽ ååã®çºããå
ã®ã¹ãã¯ãã«ã®å®æž¬å€ãè¡šããªã¥ãŒãããªã®çµéšåŒã«ã€ããŠã¯ãæ¢ã«ãæ°ŽçŽ ååã®ã¹ãã¯ãã«ãã®é
ã§ã§èª¬æããã
é»åããšãã«ã®ãŒé äœ E n {\displaystyle E_{n}} ãããäœããšãã«ã®ãŒé äœ E m {\displaystyle E_{m}} ã«é·ç§»ãããšãã«ãæ¯åæ° Îœ = E n â E m h {\displaystyle \nu ={\frac {E_{n}-E_{m}}{h}}} ã®å
åãäžåæŸåºããã
1 λ = E n â E m c h {\displaystyle {\frac {1}{\lambda }}={\frac {E_{n}-E_{m}}{ch}}} ã§äžããããã®ã§ãå³èŸºã®ãšãã«ã®ãŒé äœã«åŒ(4)ã代å
¥ãããš
ãåŸãããã R â 2 Ï 2 k 0 2 m e 4 c h 3 {\displaystyle {\bf {R}}\triangleq {\frac {2\pi ^{2}k_{0}{}^{2}me^{4}}{ch^{3}}}} ã§ããªã¥ãŒãããªå®æ°Rãå®çŸ©ãããšãåŒ(5)ã¯
Rã®å®çŸ©åŒäžã®è«žå®æ°ã«å€ããããŠèšç®ãããš
é©ãã¹ãããšã«ããªã¥ãŒãããªã®çµéšåŒããèŠäºã«å°åºã§ããã®ã§ããã ããã¯ãããŒã¢ã®ä»®èª¬ã®åŠ¥åœæ§ã瀺ããã®ãšèšãããã
(â» æªèšè¿°)
ååæ žã¯ãéœåãšäžæ§åããã§ããŠããã éœåã¯æ£é»è·ããã¡ãäžæ§åã¯é»è·ããããªãã
ã§ã¯ããªããã©ã¹ã®é»è·ããã€éœåã©ãããããªãã¯ãŒãã³åã§åçºããŠããŸããªãã®ã ããã?
ãã®çç±ãšããŠãã€ãŸãéœåã©ãããã¯ãŒãã³åã§åçºããªãããã®çç±ãšããŠã次ã®ãããªçç±ãèããããŠããã
ãŸããéœåã«äžæ§åãè¿ã¥ããŠæ··åãããšããæ žåããšããéåžžã«åŒ·ãçµååãçºçãã ãã®æ žåãéœåå士ã®ã¯ãŒãã³åã«ãã匷ãæ¥åã«æã¡åã€ã®ã§ãéœåãšäžæ§åã¯çµåããŠãããšèããããŠããã(å¿
ããããéœåãšäžæ§åã®åæ°ã¯åäžã§ãªããŠããããå®éã«ãåšæè¡šã«ããããã€ãã®å
çŽ ã§ããéœåãšäžæ§åã®åæ°ã¯ç°ãªãã)
æ¯å©çã«èšãæãã°ãäžæ§åã¯ãéœåãšéœåãçµã³ã€ãããããªã®ãããªåœ¹å²ãããŠãããšãèããããŠããã
ãªããå称ãšããŠãéœåãšäžæ§åããŸãšããŠãæ žåããšåŒã°ããã
ããå
çŽ ã®ååæ žã®éœåã®æ°ã¯ãåšæè¡šã®ååçªå·ãšäžèŽããã
ãŸããéœåãšäžæ§åã®æ°ã®åã¯è³ªéæ°ãšãã°ããã
質éæ°Aã®ååæ žã¯éåžžã«åŒ·ãæ žåã®ããã«ãå°ããªçäœç¶ã®ç©ºéã®äžã«åºãŸã£ãŠããããã®ååŸrã¯ã 1.2 {\displaystyle 1.2} ~ 1.4 à 10 â 15 à A 1 3 {\displaystyle 1.4\times 10^{-15}\times A^{\frac {1}{3}}} ã§ããããšãç¥ãããŠããã
ä»»æã®ååæ žã¯ããããæ§æããæ žåã§ããéœåãšäžæ§åãèªç±ã§ãããšãã®è³ªé(åäœè³ªéãšãã)ã®åãããå°ãã質éããã€ããã®æžã£ã質éãã質éæ¬ æãšåŒã¶ã 質éæ°Aãååçªå·Zã®ååæ žã®è³ªéæ¬ æ Î m {\displaystyle \Delta m} ããåŒã§æžãã°, ååæ žã®è³ªéãmãéœåãšäžæ§åã®åäœè³ªéããããã m p , m n {\displaystyle m_{p},\ m_{n}} ãšãããšãã
枬å®å®éšã®äºå®ãšããŠãéœååç¬ãäžæ§ååç¬ã®è³ªéã®åæ°ãåãããããããã®çµåããååæ žã®ã»ãã質éãäœãã®ã§ãéœåãäžæ§åãçµåãããšè³ªéã®äžéšãæ¬ æãããšããã®ãã枬å®çµæã®äºå®ã§ããã
ãªã®ã§ã質éæ¬ æã®ãšããããã®åå ãšããŠèããããŠããã®ã¯ãéœåãäžæ§åã©ããã®çµåã§ãããšèããããŠããã
ã ããã§ã¯ããªãéœåãäžæ§åãååæ žãšããŠçµåãããšè³ªéãæ¬ æãããã®çç±ãšããŠã¯ããã£ããŠãçµåã ããããšããçç±ã§ã¯èª¬æãã€ããªãã
ãªã®ã§ãç©çåŠè
ãã¡ã¯ã質éæ¬ æã®èµ·ããæ ¹æ¬çãªåå ãšãªãç©çæ³åããŠãã¢ã€ã³ã·ã¥ã¿ã€ã³ã®çžå¯Ÿæ§çè«ãé©çšããŠããã(æ€å®æç§æžã§ããçžå¯Ÿæ§çè«ã®çµæã§ãããšããŠèª¬æããç«å Ž)
(ã¢ã€ã³ã·ã¥ã¿ã€ã³ã®ç¹æ®)çžå¯Ÿæ§çè«ããå°ãããçµæãšããŠ(â» åè: çžå¯Ÿè«ã«ã¯äžè¬çžå¯Ÿè«ãšç¹æ®çžå¯Ÿè«ã®2çš®é¡ããã)ã質émãšãšãã«ã®ãŒEã«ã¯ã
ãšããé¢ä¿åŒããããšãããã
ãªããC ãšã¯å
éã®å€ã§ããã
ãããã¯å¥ã®æžåŒãšããŠãå€åãè¡šããã«ã¿èšå·Îã䜿ãŠã
ãªã©ãšæžãå Žåãããã
ã€ãŸããããäœããã®çç±ã§ãç空ãã質éãçºçãŸãã¯æ¶å€±ããã°ããã®ã¶ãã®è«å€§ãªãšãã«ã®ãŒãçºçãããšããã®ããçžå¯Ÿæ§çè«ã§ã®ã¢ã€ã³ã·ã¥ã¿ã€ã³ãªã©ã®äž»åŒµã§ããã
ããŠãèªç±ãªéœåãšäžæ§åã¯ãæ žåã«ããçµåãããšãããã®çµåãšãã«ã®ãŒã«çžåœããw:ã¬ã³ãç·ãæŸå°ããããšãç¥ãããŠããã
ãããŠãã¬ã³ãç·ã«ããšãã«ã®ãŒãããã
ãªã®ã§ãéœåãšäžæ§åã®çµåãããšãã®ã¬ã³ãç·ã®ãšãã«ã®ãŒã¯ã質éæ¬ æã«ãã£ãŠçãããšèãããšã枬å®çµæãšããžãããåãã(枬å®çµæã¯ããããŸã§è³ªéãæ¬ æããããšãŸã§ã)
æ žåã®çµåã«ãããŠã質éæ¬ æ Î m {\displaystyle \Delta m} ããã¬ã³ãç·ãªã©ã®ãšãã«ã®ãŒã«è»¢åããããšç©çåŠè
ãã¡ã¯èããŠããã
å
çŽ ã®äžã«ã¯ãæŸå°ç·(radiation)ãåºãæ§è³ªããã€ãã®ãããããã®æ§è³ªãæŸå°èœ(radioactivity)ãšããã ãŸããæŸå°èœããã€ç©è³ªã¯æŸå°æ§ç©è³ªãšããããã æŸå°ç·ã«ã¯3çš®é¡ååšããããããαç·ãβç·ãγç·ãšããã
α厩å£ã¯ã芪ååæ žããããªãŠã ååæ žãæŸå°ãããçŸè±¡ã§ããã ãã®ããªãŠã ååæ žã¯Î±ç²åãšãã°ããã α厩å£åŸã芪ååæ žã®è³ªéæ°ã¯4å°ãããªããååçªå·ã¯2å°ãããªãã
β厩å£ã¯ã芪ååæ žã®äžæ§åãéœåãšé»åã«å€åããããšã§ãé»åãæŸå°ãããçŸè±¡ã§ããã (åè: ãã®ãšããåãã¥ãŒããªããšãã°ãã埮å°ãªç²åãåæã«æŸåºããããšãè¿å¹Žã®åŠèª¬ã§ã¯èããããŠããã)
ãªãããã®é»å(ããŒã¿åŽ©å£ãšããŠæŸåºãããé»åã®ããš)ã¯ãβç²åããšããã°ããã
β厩å£åŸã芪ååæ žã®è³ªéæ°ã¯å€åããªãããååçªå·ã¯1å¢å ããã
γç·ã¯ãα厩å£ãŸãã¯Î²åŽ©å£çŽåŸã®é«ãšãã«ã®ãŒã®ååæ žããäœãšãã«ã®ãŒã®å®å®ãªç¶æ
ã«é·ç§»ãããšãã«æŸå°ãããã γç·ã®æ£äœã¯å
åã§ãXç·ããæ³¢é·ã®çãé»ç£æ³¢ã§ããã
α厩å£ãβ厩å£ã«ãã£ãŠããšã®ååæ žã®æ°ã¯åŸã
ã«æžã£ãŠãããããããã®åŽ©å£ã¯ååæ žã®çš®é¡ããšã«æ±ºãŸã£ãäžå®ã®ç¢ºçã§èµ·ããã®ã§ã厩å£ã«ãã£ãŠããšã®ååæ žã®æ°ãæžãé床ã¯ååæ žã®åæ°ã«æ¯äŸããŠå€åããããããã厩å£ã«ãã£ãŠããšã®ååæ žã®æ°ãåæžããã®ã«ãããæéã¯ãååæ žã®çš®é¡ã ãã«ãã£ãŠããŸããããã§ããã®æéã®ããšããã®ååæ žã® åæžæ(ã¯ãããããhalf life ) ãšåŒã¶ã厩å£ã«ãã£ãŠååæ žã®åæ°ãã©ãã ãã«ãªããã¯ããã®åæžæãçšããŠèšè¿°ããããšãã§ãããååæ žã®åæžæãTãæå»tã§ã®ååæ žã®åæ°ãN(t)ãšãããšã
ãæãç«ã€ã
ååæ žã®åŽ©å£é床ã¯ãååæ žã®åæ°ã«æ¯äŸãããšè¿°ã¹ããå®ã¯ãäžã«è¿°ã¹ãå
¬åŒã¯ãã®æ
å ±ã ãããçŽç²ã«æ°åŠçã«å°ãåºãããšãã§ãããã®ã§ãããé«çåŠæ ¡ã§ã¯æ±ããªãæ°åŠãçšããããèå³ã®ããèªè
ã®ããã«ãã®æŠèŠãèšããŠããã
ååæ žã®åæ°ãšåŽ©å£é床ã®éã®æ¯äŸå®æ°ã¯ååæ žã®çš®é¡ã«ãã£ãŠæ±ºãŸãããã®å®æ°ããã®ååæ žã®åŽ©å£å®æ°ãšããã厩å£å®æ°ãλã®ååæ žã®æå»tã§ã®åæ°ãN(t)ãšãããšããã®å€åé床ãããªãã¡N(t)ã®åŸ®åã¯ã
ã§è¡šãããããã®ãããªãããé¢æ°ãšãã®åŸ®åãšã®é¢ä¿ãè¡šããåŒã埮åæ¹çšåŒãšããã埮åæ¹çšåŒãæºãããããªé¢æ°ãæ±ããããšãã埮åæ¹çšåŒã解ããšããã(詳ãã解æ³ã¯è§£æåŠåºç€/垞埮åæ¹çšåŒã§èª¬æãããã)ãã®åŸ®åæ¹çšåŒã解ããš
ãåŸãããã(ãã®åŒã確ãã«å
ã»ã©ã®åŸ®åæ¹çšåŒãæºãããŠããããšã確ãããŠã¿ã)
åæžæTãšã¯ã N ( t ) = 1 2 N ( 0 ) {\displaystyle N(t)={\frac {1}{2}}N(0)} ãšãªãtã®ããšãªã®ã§ãå
ã»ã©ã®åŒãã
ãåŸãããããã£ãŠã
ãåŸãããã
ã©ã¶ãã©ãŒãã¯ãçªçŽ ã¬ã¹ãå¯éããç®±ã«Î±ç·æºããããšãæ£é»è·ããã£ãç²åãçºçããããšãçºèŠããã
ãã®æ£é»è·ã®ç²åããéœåã§ãããã€ãŸããã©ã¶ãã©ãŒãã¯éœåãçºèŠããã
åæã«ãé
žçŽ ãçºçããããšãçºèŠãããã®çç±ã¯çªçŽ ãé
žçŽ ã«å€æãããããã§ãããã€ãŸããååæ žãå€ããåå¿ãçºèŠããã
ãããã®ããšãåŒã«ãŸãšãããšã
ã§ããã
ãã®ããã«ãããå
çŽ ã®ååããå¥ã®å
çŽ ã®ååã«å€ããåå¿ã®ããšã ååæ žåå¿ ãšããããŸãã¯ããæ žåå¿ããšããã
ãŸããå®å®ç·ã®èŠ³æž¬ã«ãããÎŒç²åãšããã®ããçºèŠãããŠããã
ãããããã©ããã£ãŠçŽ ç²åã芳枬ããããšãããšãããã€ãã®æ¹æ³ããããã
ãªã©ã䜿ãããã
(â» é«æ ¡ã§ç¿ãç¯å²å
ãXç·ãååæ žã®åå
ã§ãé§ç®±(ããã°ã)ãç¿ãã)
é§ç®±(ããã°ã)ãšãããèžæ°ã®ã€ãŸã£ãè£
眮ãã€ãããšããªããã®ç²åãééãããšããã®ç²åã®è»è·¡ã§ãæ°äœãã液äœããåçãèµ·ããã®ã§ãè»è·¡ããç®ã«èŠããã®ã§ããã(â» æ€å®æç§æžã§ã¯ãååæ žã®åéã§ãé§ç®±ã«ã€ããŠç¿ãã)(ã€ã¡ãŒãžçã«ã¯ãé£è¡æ©é²ã®ãããªã®ããã€ã¡ãŒãžããŠãã ããã)
ã§ãç£å Žãå ããå Žåã®ãè»è·¡ã®æ²ããããçãªã©ãããæ¯é»è·ãŸã§ãäºæ³ã§ããã
ãã®ããã«ãé§ç®±ãã€ãã£ãå®éšã«ããã20äžçŽååãäžç€ããã«ã¯ããããããªç²åãçºèŠãããã
ÎŒç²å以å€ã«ããéœé»å(ããã§ãã)ããé§ç®±ã«ãã£ãŠçºèŠãããŠããã
(â» ç¯å²å€:)äžçåã§éœé»åãå®éšçã«èŠ³æž¬ããã¢ã³ããŒãœã³ã¯ãé§ç®±ã«éæ¿ãå
¥ããããšã§éœé»åãçºèŠããã
ãšãããã(ÎŒç²åã®çºèŠããã)éœé»åã®ã»ããçºèŠã¯æ©ãã
(â» ç¯å²å€:)ãŸããéœé»åã¯ãéåååŠã®ã·ã¥ã¬ãŒãã£ã³ã¬ãŒæ¹çšåŒã«ãç¹æ®çžå¯Ÿæ§çè«ãšãçµã¿åããããããã£ã©ãã¯ã®æ¹çšåŒããããçè«çã«äºæ³ãããŠããã
ãŸãããéœé»åããšããç©è³ªã1932幎ã«éæ¿ãå
¥ããé§ç®±(ããã°ã)ã®å®éšã§ã¢ã³ããŒãœã³(人å)ã«ãã£ãŠçºèŠãããŠãããéœé»åã¯è³ªéãé»åãšåãã ããé»è·ãé»åã®å察ã§ãã(ã€ãŸãéœé»åã®é»è·ã¯ãã©ã¹eã¯ãŒãã³ã§ãã)ã(â» éæ¿ã«ã€ããŠã¯é«æ ¡ã®ç¯å²å€ã)
ãããŠãé»åãšéœé»åãè¡çªãããšã2mcã®ãšãã«ã®ãŒãæŸåºããŠãæ¶æ»
ããã(ãã®çŸè±¡(é»åãšéœé»åãè¡çªãããš2mcã®ãšãã«ã®ãŒãæŸåºããŠæ¶æ»
ããçŸè±¡)ã®ããšããã察æ¶æ»
ã(ã€ãããããã€)ãšããã)
éœåã«å¯ŸããŠãããåéœåãããããåéœåã¯ãé»è·ãéœåãšå察ã ãã質éãéœåãšåãã§ãããéœåãšè¡çªãããšå¯Ÿæ¶æ»
ãããã
äžæ§åã«å¯ŸããŠãããåäžæ§åãããããåäžæ§åã¯ãé»è·ã¯ãŒãã ã(ãŒãã®é»è·ã®Â±ãå察ã«ããŠããŒãã®ãŸãŸ)ã質éãåãã§ãäžæ§åãšå¯Ÿæ¶æ»
ãããã
éœé»åãåéœåãåäžæ§åã®ãããªç©è³ªããŸãšããŠãåç©è³ªãšããã
(â» ç¯å²å€: )æŸå°æ§åäœäœã®ãªãã«ã¯ã厩å£ã®ãšãã«éœé»åãæŸåºãããã®ããããæå
端ã®ç
é¢ã§äœ¿ãããPET(éœé»åæå±€æ®åæ³)æè¡ã¯ããããå¿çšãããã®ã§ãããããçŽ ããµãããã«ãªãããªãã·ã°ã«ã³ãŒã¹ãšããç©è³ªã¯ã¬ã³çŽ°èã«ããåã蟌ãŸãããPET蚺æã§ã¯ãããã«(ãã«ãªãããªãã·ã°ã«ã³ãŒã¹ã«)æŸå°æ§ã®ããçŽ F ããšãããã æŸå°æ§ãã«ãªãããªãã·ã°ã«ã³ãŒã¹ãçšããŠããã(â» åæ通ã®ãååŠåºç€ãã®æç§æžã«ãçºå±äºé
ãšããŠãã«ãªãããªãã·ã°ã«ã³ãŒã¹ãPET蚺æã§äœ¿ãããŠãããšã玹ä»ãããŠããã)
åç©è³ªãšã¯å¥ã«ãÎŒç²åããå®å®ç·ã®èŠ³æž¬ããã1937幎ã«èŠã€ãã£ãã
ãã®ÎŒç²åã¯ãé»è·ã¯ãé»åãšåãã ãã質éãé»åãšã¯éããÎŒç²åã®è³ªéã¯ããªããšé»åã®çŽ200åã®è³ªéã§ããã
ÎŒç²åã¯ãã¹ã€ã«éœåãé»åã®åç©è³ªã§ã¯ãªãã®ã§ãã¹ã€ã«éœåãšã察æ¶æ»
ãèµ·ãããªãããé»åãšã察æ¶æ»
ãèµ·ãããªãã
ãªããÎŒç²åã«ããåÎŒç²åãšãããåç©è³ªãååšããããšãåãã£ãŠããã
ãã®ãããªç©è³ªããããããã®äœãã§ããå°äžã§èŠã€ãããªãã®ã¯ãåã«å°äžã®å€§æ°ãªã©ãšè¡çªããŠæ¶æ»
ããŠããŸãããã§ããã
ãªã®ã§ãé«å±±ã®é äžä»è¿ãªã©ã§èŠ³æž¬å®éšããããšãÎŒç²åã®çºèŠã®å¯èœæ§ãé«ãŸãã
ãªã21äžçŽã®çŸåšãÎŒç²åã掻çšããæè¡ãšããŠãçŸåšãç«å±±ãªã©ã®å
éšã芳å¯ããã®ã«ã掻çšãããŠãããÎŒç²åã¯ãééåãé«ãããå°äžã®ç©è³ªãšåå¿ããŠããããã«æ¶æ»
ããŠããŸãã®ã§ããã®ãããªæ§è³ªãå©çšããŠãç«å±±å
éšã®ããã«äººéãå
¥ã蟌ããªãå Žæã芳å¯ãããšããæè¡ãããã§ã«ããã
ãã®ãããªèŠ³æž¬ã«äœ¿ãããÎŒç²åãã©ããã£ãŠçºçãããã®ã?
å®å®ç·ããé£ãã§ããÎŒç²åããã®ãŸãŸäœ¿ããšããæ¹æ³ããããããå®è¡ããŠããç 究è
ããããããããšã¯å¥ã®ææ³ãšããŠãå éåšãªã©ã§äººå·¥çã«ÎŒç²åãªã©ãçºçããããšããæ¹æ³ãããã
å éåšã䜿ã£ãæ¹æ³ã¯ãäžèšã®éãã
ãŸããã·ã¯ãããã³ããµã€ã¯ãããã³ã䜿ã£ãŠãé»åãªã©ãè¶
é«éã«å éããããããäžè¬ã®ç©è³ª(ã°ã©ãã¡ã€ããªã©)ã«åœãŠãã
ãããšãåœç¶ãããããªç²åãçºçããã
ãã®ãã¡ãÏäžéåããç£æ°ã«åå¿ãã(ãšèããããŠãã)ã®ã§ã倧ããªé»ç£ç³ã³ã€ã«ã§ãÏäžéåãæç²ããã
ãã®Ïäžéåã厩å£ããŠãÎŒç²åãçºçããã
ããããå®å®ç·ãäœã«ãã£ãŠçºçããŠãããã®çºçåå ã¯ãçŸæç¹ã®äººé¡ã«ã¯äžæã§ããã(â» åèæç®: æ°ç åºçã®è³æéã®ãå³èª¬ç©çã)
è¶
æ°æ(ã¡ãããããã)ççºã«ãã£ãŠå®å®ç·ãçºçããã®ã§ã¯ããšãã説ããããããšã«ããå®å®ç·ã®çºçåå ã«ã€ããŠã¯æªè§£æã§ããã
é»åãéœåãäžæ§åãªã©ã¯ããã¹ãã³ããšããç£ç³ã®ãããªæ§è³ªããã£ãŠãããç£ç³ã«N極ãšS極ãããããã«ãã¹ãã³ã«ãã2çš®é¡ã®åãããããã¹ãã³ã®ãã®2çš®é¡ã®åãã¯ããäžåãããšãäžåããã«ãããäŸãããããç£ç³ã®ç£åã®çºçåå ã¯ãç£ç³äžã®ååã®æå€æ®»é»åã®ã¹ãã³ã®åããåäžæ¹åã«ããã£ãŠãããããã§ãããšèããããŠããã
å
šååã¯ãé»åãéœåãäžæ§åãå«ãã®ã«ããªã®ã«å€ãã®ç©è³ªããããŸãç£æ§ãçºçããªãã®ã¯ãå察笊å·ã®ã¹ãã³ããã€é»åãçµåãããããšã§ãæã¡æ¶ãããããã§ããã
(ãŠã£ãããé»åãšéœåã®ãããªé»è·ããã€ç²åã«ããã¹ãã³ããªããšèª€è§£ããŠãã人ãããããäžæ§åã«ãã¹ãã³ã¯ããã)
äžåŠé«æ ¡ã§èŠ³æž¬ãããããªæ®éã®æ¹æ³ã§ã¯ãã¹ãã³ã芳枬ã§ããªãããååãªã©ã®ç©è³ªã«ç£æ°ãå ãã€ã€é«åšæ³¢ãå ãããªã©ãããšãã¹ãã³ã®åœ±é¿ã«ãã£ãŠããã®ååã®æ¯åããããåšæ³¢æ°ãéããªã©ã®çŸè±¡ããã¡ããŠãéæ¥çã«(é»åãªã©ã®)ã¹ãã³ã芳枬ã§ããã(ãªããæ žç£æ°å
±é³Žæ³(NMRãnuclear magnetic resonance)ã®åçã§ããã â» çè«çãªè§£æã¯ã倧åŠã¬ãã«ã®ååŠã®ç¥èãå¿
èŠã«ãªãã®ã§çç¥ããã) ååäžã®æ°ŽçŽ ååããããçš®ã®æŸå°æ§åäœäœ(äžæ§åããã£ã1åãµããã ãã®åäœäœ)ãªã©ãé«åšæ³¢ã®åœ±é¿ãåããããããã®çç±ã®ã²ãšã€ããã¹ãã³ã«ãããã®ã ãšèããããŠãã(â» ãªããå»çã§çšããããMRI(magnetic resonance imaging)ã¯ããã®æ žç£æ°å
±é³Žæ³(NMR)ãå©çšããŠäººäœå
éšãªã©ã芳枬ããããšããæ©åšã§ããã)
ããŠãå®ã¯çŽ ç²åããã¹ãã³ããã€ã®ãæ®éã§ããã
ÎŒç²åã¯ã¹ãã³ããã€ã
ÎŒç²åã®ãã¹ãã³ããšããæ§è³ªã«ããç£æ°ãšãÎŒç²åã®ééæ§ã®é«ããå©çšããŠãç©è³ªå
éšã®ç£å Žã®èŠ³æž¬æ¹æ³ãšããŠæ¢ã«ç 究ãããŠããããã®ãããªèŠ³æž¬æè¡ããÎŒãªã³ã¹ãã³å転ããšãããè¶
äŒå°äœã®å
éšã®èŠ³æž¬ãªã©ã«ãããã§ã«ãÎŒãªã³ã¹ãã³å転ãã«ãã芳枬ãç 究ãããŠããã
ãŠã£ãããã£ã¢èšäºãw:ãã¥ãªã³ã¹ãã³å転ãã«ãããšãÎŒãªã³ã®åŽ©å£æã«éœé»åãæŸåºããã®ã§ãéœé»åã®èŠ³æž¬æè¡ãå¿
èŠã§ããã(é«æ ¡ã®ç¯å²å€ã§ãããã)ããããã®åŠçã¯ããããããšå匷ããäºãå€ãã
éœåãšäžæ§åã¯ã質éã¯ã»ãšãã©åãã§ãããé»è·ãéãã ãã§ããã
ãããŠãé»åãšæ¯ã¹ããšãéœåãäžæ§åãã質éãããªã倧ããã
ãã®äºããããéœåãäžæ§åã«ããããã«äžèº«ããããå¥ã®ç²åãè©°ãŸã£ãŠããã®ã§ã¯?ããšããçåãçãŸããŠããŠãéœåãäžæ§åã®å
éšã®æ¢çŽ¢ãå§ãŸã£ãã
ããããçŸåšã§ããéœåãäžæ§åã®å
éšã®æ§é ã¯ãå®éšçã«ã¯åãåºããŠã¯ããªãã(â» éœåãäžæ§åã®å
éšæ§é ãšããŠèª¬æãããŠãããã¯ã©ãŒã¯ãã¯ãåç¬ã§ã¯çºèŠãããŠããªããã¯ã©ãŒã¯ã¯åã«ãå
éšã®èª¬æã®ããã®ãæŠå¿µã§ããã)
æŽå²çã«ã¯ããŸããéœåãšäžæ§åã®å
éšæ§é ãšããŠãæ¶ç©ºã®çŽ ç²åãèããããéœåãšäžæ§åã¯ããããã®çŽ ç²åã®ç¶æ
ãéãã ãããšèããããã
ãã£ãœããé»åã«ã¯ãå
éšæ§é ããªãããšèãããŠããã
ããã20äžçŽãªãã°ãéåååŠã§ã¯ããã®ããããã§ã«ãé»åã®ç¶æ
ãšããŠãã¹ãã³ããšããæŠå¿µããã¿ã€ãã£ãŠãããéåååŠã§ã¯ãååŠçµåã§äŸ¡é»åã2åãŸã§çµåããŠé»å察ã«ãªãçç±ã¯ããã®ã¹ãã³ã2çš®é¡ãããªããŠãå察åãã®ã¹ãã³ã®é»å2åã ããçµåããããã§ããããšãããŠããã
ã¹ãã³ã®2çš®é¡ã®ç¶æ
ã¯ããäžåãããäžåãããšãããµãã«ãããäŸããããã(å®éã®æ¹åã§ã¯ãªãã®ã§ãããŸãæ·±å
¥ãããªãããã«ã)
ãã®ãããªéåååŠãåèã«ããŠãéœåãšäžæ§åã§ããã¢ã€ãœã¹ãã³ããšããæŠå¿µãèããããã(â» ãã¢ã€ãœã¹ãã³ãã¯é«æ ¡ç¯å²å€ã)
éœåãšäžæ§åã¯ãã¢ã€ãœã¹ãã³ã®ç¶æ
ãéãã ãããšèããããã
ãã®åŸã20äžçŽåã°é ããããã¢ã€ãœã¹ãã³ããçºå±ããããã¯ã©ãŒã¯ããšããçè«ãæå±ãããã
æ¶ç©ºã®ãã¯ã©ãŒã¯ããšãã3åã®çŽ ç²åãä»®å®ãããšãå®åšã®éœåãäžæ§åã®æãç«ã€ã¢ãã«ããå®éšçµæãããŸã説æã§ããäºãåãã£ãã
é»è·( + 2 3 e {\displaystyle +{\frac {2}{3}}e} )ããã€çŽ ç²åãã¢ããã¯ã©ãŒã¯ããšã±( â 1 3 e {\displaystyle -{\frac {1}{3}}e} )ããã€çŽ ç²åãããŠã³ã¯ã©ãŒã¯ãããã£ãŠã
ãšèãããšããããããªçŽ ç²åå®éšã®çµæãããŸã説æã§ããäºãåãã£ãã
ãªããé»åã«ã¯ããã®ãããªå
éšæ§é ã¯ãªãããšèãããããã
ã¢ããã¯ã©ãŒã¯ã¯ãuããšç¥èšãããããŠã³ã¯ã©ãŒã¯ã¯ãdããšç¥èšãããã
éœåã®ã¯ã©ãŒã¯æ§é ã¯uudãšç¥èšããã(ã¢ãããã¢ãããããŠã³)ã
äžæ§åã®ã¯ã©ãŒã¯æ§é ã¯uddãšç¥èšããã(ã¢ãããããŠã³ãããŠã³)ã
ãªããäžèšã®èª¬æã§ã¯çç¥ãããããããã1950ã60幎代ãããŸã§ã«ãé«å±±ã§ã®å®å®ç·ã®èŠ³æž¬ãããããã¯æŸå°ç·ã®èŠ³æž¬ãããŸããããã¯ãµã€ã¯ãããã³ãªã©ã«ããç²åã®å éåšè¡çªå®éšã«ãããéœåãäžæ§åã®ã»ãã«ããåçšåºŠã®è³ªéã®ããŸããŸãªç²åãçºèŠãããŠããããããæ°çš®ã®ç²åã¯ãäžéåãã«åé¡ãããã
ããããããã¯ã©ãŒã¯ãã®çè«ã¯ããã®ãããª20äžçŽåã°ãããŸã§ã®å®éšã芳枬ããäœçŸåãã®æ°çš®ã®ç²åãçºèŠãããŠããŸãããã®ãããªçµç·¯ããã£ãã®ã§ãã¯ã©ãŒã¯ã®çè«ãæå±ãããã®ã§ããã
ããŠããäžéåã(ã¡ã
ãããããmason ã¡ãœã³)ãšã¯ãããšããšçè«ç©çåŠè
ã®æ¹¯å·ç§æš¹ã1930幎代ã«æå±ãããéœåãšäžæ§åãšãåŒãä»ããŠãããšãããæ¶ç©ºã®ç²åã§ãã£ããã20äžçŽãªãã°ã«æ°çš®ã®ç²åãçºèŠãããéããäžéåãã®ååã䜿ãããããšã«ãªã£ãã
ããŠãå®éšçã«æ¯èŒçæ©ãææããçºèŠããããäžéåãã§ã¯ããÏäžéåãããããããçš®é¡ã®Ïäžéåã¯ãã¢ããã¯ã©ãŒã¯ãšåããŠã³ã¯ã©ãŒã¯ãããªããÏãšç¥èšãããã(ããŠã³ã¯ã©ãŒã¯ã®åç©è³ªããåããŠã³ã¯ã©ãŒã¯ã) Ï= u d Ì {\displaystyle u{\overline {d}}}
å¥ã®ããçš®é¡ã®Ïäžéåã¯ãããŠã³ã¯ã©ãŒã¯ãšåã¢ããã¯ã©ãŒã¯ãããªããÏãšç¥èšããããÏ= u Ì d {\displaystyle {\overline {u}}d}
ãã®ããã«ãããç²åå
ã®ã¯ã©ãŒã¯ã¯åèš2åã®ã§ãã£ãŠãè¯ãå Žåãããã(ããªãããããéœåã®ããã«ã¯ã©ãŒã¯3åã§ãªããŠãããŸããªãå Žåãããã)
(â» ãã®ãããªå®éšäŸãããç²åå
ã«åèš5åã®ã¯ã©ãŒã¯ã7åã®ã¯ã©ãŒã¯ãèããçè«ããããããããé«æ ¡ç©çã®ç¯å²ã倧å¹
ã«è¶
ããã®ã§ã説æãçç¥ã)
ãŸããäžéåã¯ãèªç¶çã§ã¯çæéã®ããã ã ããååšã§ããç²åã ãšããäºãã芳枬å®éšã«ãã£ãŠãåãã£ãŠããã(äžéåã®ååšã§ããæé(ã寿åœã)ã¯çããããã«ãä»ã®å®å®ãªç²åã«å€æããŠããŸãã)
ããããã¢ãããšããŠã³ã ãã§ã¯ã説æããããªãç²åããã©ãã©ããšçºèŠãããŠãããã¯ã©ãŒã¯ã®æå±æã®åœåã¯ãããããã ãã¯ã©ãŒã¯ã®ã¢ãããšããŠã³ã§ããã£ãšãã»ãšãã©ã®äžéåã®æ§é ã説æã§ããã ããã ãšæåŸ
ãããŠããã®ã ããããããããå®å®ç·ãã1940幎代ã«çºèŠããããKäžéåãã®æ§é ã§ãããã¢ãããšããŠã³ã§ã¯èª¬æã§ããªãã£ãã
ãã®ã»ããå éåšã®çºéãªã©ã«ãããã¢ãããšããŠã³ã®çµã¿åããã ãã§èª¬æã§ããæ°ãè¶
ããŠãã©ãã©ããšæ°çš®ã®ãäžéåããçºèŠãããŠããŸãããã¯ãã¢ãããšããŠã³ã ãã§ã¯ãäžéåã®æ§é ã説æãã¥ãããªã£ãŠããäžãÎŒç²åãã説æã§ããªãã
ãŸããå éåšå®éšã«ããã1970幎代ã«ãDäžéåããªã©ãããŸããŸãªäžéåããå®éšçã«å®åšã確èªãããã
ãã®ããã«ãã¢ãããšããŠã³ã ãã§ã¯èª¬æã®ã§ããªãããããããªç²åãååšããããšãåããããã®ãããçŽ ç²åçè«ã§ã¯ããã¢ããã(u)ãšãããŠã³ã(d)ãšãã2çš®é¡ã®ç¶æ
ã®ä»ã«ããããã«ç¶æ
ãèããå¿
èŠã«ãããŸãããããããŠãæ°ããç¶æ
ãšããŠããŸãããã£ãŒã ã(èšå·c)ãšãã¹ãã¬ã³ãžã(èšå·s)ãèãããããå éåšå®éšã®æè¡ãçºå±ããå éåšå®éšã®è¡çªã®ãšãã«ã®ãŒãäžãã£ãŠãããšãããã«ããããã(èšå·t)ãšãããã ã(èšå·b)ãšããã®ãèããããã
ãªããÎŒç²åã«ã¯å
éšæ§é ã¯ãªãããéœåãäžæ§åã«é»åã察å¿ãããã®ãšåæ§ã«(第1äžä»£)ããã£ãŒã ãã¹ãã¬ã³ãžãããªãéœåçã»äžæ§åçãªç²åãšÎŒç²åã察å¿ããã(第2äžä»£)ãåæ§ã«ãããããããã ãããªãç²åã«ÎŒç²åã察å¿ããã(第3äžä»£)ã
é»åãÎŒç²åã¯å
éšæ§é ããããªããšèããããŠããããã¬ããã³ããšãããå
éšæ§é ããããªããšãããã°ã«ãŒãã«åé¡ãããã
ãKäžéåãã¯ã第1äžä»£ã®ã¯ã©ãŒã¯ãšç¬¬2äžä»£ã®ã¯ã©ãŒã¯ããæãç«ã£ãŠããäºããåãã£ãŠããã(â» æ€å®æç§æžã®ç¯å²å
ã)
ãããŠã2017幎ã®çŸåšãŸã§ãã£ãšãã¯ã©ãŒã¯ã®çè«ããçŽ ç²åã®æ£ããçè«ãšãããŠããã
çŽ ç²åã®èŠ³ç¹ããåé¡ããå Žåã®ãéœåãšäžæ§åã®ããã«ãã¯ã©ãŒã¯3åãããªãç²åã®ããšãããŸãšããŠãããªãªã³ã(éç²å)ãšãããÏäžéå(Ï= u d Ì {\displaystyle u{\overline {d}}} )ãªã©ãã¯ã©ãŒã¯ã2åã®ç²åã¯ãããªãªã³ã«å«ãŸãªãã
ããããäžéåã®ãªãã«ããã©ã ãç²å(udsãã¢ããããŠã³ã¹ãã¬ã³ãžã®çµã¿åãã)ã®ããã«ãã¯ã©ãŒã¯3åãããªãç²åããããã©ã ãç²åãªã©ããããªãªã³ã«å«ããã
éœåãšäžæ§åãã©ã ãç²åãªã©ãšãã£ãããªãªã³ã«ãããã«äžéå(äžéåã¯äœçš®é¡ããã)ãå ããã°ã«ãŒãããŸãšããŠãããããã³ããšããã
ãªããæ®éã®ç©è³ªã®ååæ žã§ã¯ãéœåãšäžæ§åãååæ žã«éãŸã£ãŠãããããã®ããã«éœåãšäžæ§åãååæ žã«åŒãåãããåã®ããšãæ žåãšãããæ žåã®æ£äœã¯ããŸã ãããŸã解æãããŠããªã(å°ãªããšãé«æ ¡ã§æããã»ã©ã«ã¯ããŸã å
åã«ã¯è§£æãããŠããªã)ã
ãšããããããªãªã³ã«ã¯ãæ žåãåããé説ã§ã¯ãäžéåã«ããæ žåã¯åããšãããŠãããã€ãŸãããããã³ã«ãæ žåãåãã
ãããã³ã¯ãããããã¯ã©ãŒã¯ããæ§æãããŠããäºããããããããã¯ã©ãŒã¯ã«æ žåãåãã®ã ãããçãªäºããèããããŠããã
çè«ã§ã¯ãã¯ã©ãŒã¯ãšã¯ã©ãŒã¯ã©ãããåŒãä»ãããæ¶ç©ºã®ç²åãšããŠãã°ã«ãŒãªã³ããèããããŠãããç©çåŠè
ããçè«ãæå±ãããŠãããããã®æ£äœã¯ããŸã ãããŸã解æãããŠãªããããããç©çåŠè
ãã¡ã¯ãã°ã«ãŒãªã³ãçºèŠããããšäž»åŒµããŠããã
çŸåšã®ç©çåŠã§ã¯ãã¯ã©ãŒã¯ãåç¬ã§ã¯åãåºããŠããªãã®ãšåæ§ã«ãã°ã«ãŒãªã³ãåç¬ã§ã¯åãåºããŠã¯ããªãã
ããŠãç©çåŠã§ã¯ã20äžçŽãããéåååŠããšããçè«ããã£ãŠããã®çè«ã«ãããç©çæ³åã®æ ¹æºã§ã¯ãæ³¢ãšç²åãåºå¥ããã®ãç¡æå³ã ãšèšãããŠããããã®ããããã€ãŠã¯æ³¢ã ãšèããããŠããé»ç£æ³¢ããå Žåã«ãã£ãŠã¯ãå
åããšããç²åãšããŠæ±ãããããã«ãªã£ãã
ãã®ããã«ãããæ³¢ãåå Ž(ããã°)ãªã©ããçè«é¢ã§ã¯ç²åã«çœ®ãæããŠè§£éããŠæ±ãäœæ¥ã®ããšããç©çåŠã§ã¯äžè¬ã«ãéååããšããã
ã°ã«ãŒãªã³ããã¯ã©ãŒã¯ãšã¯ã©ãŒã¯ãåŒãä»ããåããéååãããã®ã§ããããé»è·ãšã®é¡æšã§ãã¯ã©ãŒã¯ã«ãè²è·(ã«ã©ãŒè·)ãšããã®ãèããŠãããããã®æ§è³ªã¯ãããŸã解æãããŠãªã(å°ãªããšãé«æ ¡ã§æããã»ã©ã«ã¯ããŸã å
åã«ã¯è§£æãããŠããªã)ã
ã°ã«ãŒãªã³ã®ããã«ãåãåªä»ããç²åã®ããšãã²ãŒãžç²åãšããã
éåãåªä»ããæ¶ç©ºã®ç²åã®ããšãéåå(ã°ã©ããã³)ãšãããããŸã çºèŠãããŠããªããç©çåŠè
ãã¡ããã°ã©ããã³ã¯ããŸã æªçºèŠã§ããããšäž»åŒµããŠããã
é»ç£æ°åãåªä»ããç²åã¯å
å(ãã©ãã³)ãšããããããã¯åã«ãé»ç£å Žãä»®æ³çãªç²åãšããŠçœ®ãæããŠæ±ã£ãã ãã§ããããã©ãã³ã¯ãé«æ ¡ç©çã®é»ç£æ°åéã§ç¿ããããªå€å
žçãªé»ç£æ°èšç®ã§ã¯ããŸã£ãã圹ç«ããªãã
ãªããå
åãã²ãŒãžç²åã«å«ããã
ã€ãŸããå
åãã°ã«ãŒãªã³ã¯ãã²ãŒãžç²åã§ããã
ããŒã¿åŽ©å£ãã€ããã©ãåã®ããšãã匱ãåããšããããã®åãåªä»ããç²åãããŠã£ãŒã¯ããœã³ããšããããæ§è³ªã¯ãããåãã£ãŠããªãããããç©çåŠè
ãã¡ã¯ããŠã£ãŒã¯ããœã³ãçºèŠããããšäž»åŒµããŠããã
ãããããããœã³ããšã¯äœã?
éåååŠã®ã»ãã§ã¯ãé»åã®ãããªãäžç®æã«ããã ãæ°åãŸã§ããååšã§ããªãç²åããŸãšããŠãã§ã«ããªã³ãšããããã§ã«ããªã³çã§ãªãå¥çš®ã®ç²åãšããŠããœã³ããããå
åããããœã³ãšããŠæ±ãããã
ããŠã£ãŒã¯ããœã³ããšã¯ãããããã匱ãåãåªä»ããããœã³ã ãããŠã£ãŒã¯ããœã³ãšåŒãã§ããã®ã ããã
ããŠãé»è·ãšã®é¡æšã§ãã匱ãåãã«é¢ããã匱è·ã(ãããã)ãšããã®ãæå±ãããŠãããããããããã®æ§è³ªã¯ãããŸã解æãããŠãªã(å°ãªããšãé«æ ¡ã§æããã»ã©ã«ã¯ããŸã å
åã«ã¯è§£æãããŠããªã)ã
ããŠãã匱ãåãã®ããäžæ¹ãã°ã«ãŒãªã³ã®åªä»ããåã®ããšãã匷ãåããšãããã
1956幎ã«ãé»åã®ã¹ãã³ã®æ¹åãšãããŒã¿åŽ©å£ç²åã®åºãŠæ¥ãæ¹åãšã®é¢ä¿ãèŠãããã®å®éšãšããŠãã³ãã«ãã®æŸå°æ§åäœäœã§ããã³ãã«ã60ããã¡ããŠæ¬¡ã®ãããªå®éšããã¢ã¡ãªã«ã§è¡ãããã
ã³ãã«ãå
çŽ (å
çŽ èšå·: Co )ã®æŸå°æ§åäœäœã§ããã³ãã«ã60ã極äœæž©ã«å·åŽããç£å ŽããããŠå€æ°ã®ã³ãã«ãååã®é»åæ®»ã®å€ç«é»åã¹ãã³ã®æ¹åãããããç¶æ
ã§ãã³ãã«ã60ãããŒã¿åŽ©å£ããŠçºçããããŒã¿ç²åã®åºãæ¹åã調ã¹ãå®éšãã1956幎ã«ã¢ã¡ãªã«ã§è¡ãããã
éãšããã±ã«ãšã³ãã«ãã¯ãããããéå±åäœã§ç£æ§äœã«ãªãå
çŽ ã§ãããå
çŽ åäœã§ç£æ§äœã«ãªãå
çŽ ã¯ããã®3ã€(éãããã±ã«ãã³ãã«ã)ãããªãã(ãªããæŸå°æ§åäœäœã§ãªãéåžžã®ã³ãã«ãã®ååéã¯59ã§ããã)
ãã®3ã€(éãããã±ã«ãã³ãã«ã)ã®ãªãã§ãã³ãã«ããäžçªãç£æ°ã«å¯äžããé»åã®æ°ãå€ãããšãéåååŠã®çè«ã«ããæ¢ã«ç¥ãããããã®ã§(ã³ãã«ãããã£ãšããdè»éã®é»åã®æ°ãå€ã )ãããŒã¿åŽ©å£ãšã¹ãã³ãšã®é¢ä¿ãã¿ãããã®å®éšã«ãã³ãã«ãã®æŸå°æ§åäœäœã§ããã³ãã«ã60ã䜿ãããã®ã§ããã
å®éšã®çµæãã³ãã«ã60ãããŒã¿åŽ©å£ããŠããŒã¿ç²åã®åºãŠããæ¹åã¯ãã³ãã«ã60ã®ã¹ãã³ã®ç£æ°ã®æ¹åãš(åãæ¹åããã)éã®æ¹åã«å€ãæŸåºãããŠããã®ã芳枬ããããããã¯ã2çš®é¡(ã¹ãã³ãšåæ¹åã«ããŒã¿ç²åã®åºãå Žåãšãã¹ãã³ãšå察æ¹åã«ããŒã¿ç²åã®åºãå Žå)ã®åŽ©å£ã®ç¢ºçãç°ãªã£ãŠãããããŒã¿åŽ©å£ã®ç¢ºçã®(ã¹ãã³æ¹åãåºæºãšããå Žåã®)æ¹å察称æ§ãæããŠããããšã«ãªãã
ãã®ãããªå®éšäºå®ã«ãããã匱ãåãã¯é察称ã§ããããšããã®ãå®èª¬ã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ããªã«ã³ã®å®éšãšã¯ãé§å¹ããªã©ã§äœæãã油滎ã®åŸ®å°ãªé£æ²«ã«ãXç·ãã©ãžãŠã ãªã©ã§åž¯é»ãããããããŠãå€éšããé»å ŽãåŒç«ããããããšã油滎ã®éå(äžåã)ã®ã»ãã«ãé»å Žã«ããéé»æ°å(äžåãã«ãªãããã«é»æ¥µæ¿ãèšçœ®ãã)ãåãã®ã§ãé£ãåã£ãŠéæ¢ããç¶æ
ã«ãªã£ãæã®é»å Žãããé»è·ã®å€ã確ãããå®éšã§ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ãã®å®éšã§ç®åºã»æž¬å®ãããé»è·ã®å€ã 1.6Ã10 [C]ã®æŽæ°åã«ãªã£ãã®ã§ãé»å1åã®é»è·ã 1.6Ã10 [C]ã ãšåãã£ãã",
"title": "é»åãšå
"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ãªãããã® 1.6Ã10 [C]ã®ããšãé»æ°çŽ é(ã§ãããããã)ãšããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "",
"title": "é»åãšå
"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "",
"title": "é»åãšå
"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "è² ã®é»è·ã«åž¯é»ãããŠããéå±æ¿ã«ã玫å€ç·ãåœãŠããšãé»åãé£ã³åºããŠããããšãããããŸããæŸé»å®éšçšã®è² 極ã«é»åãåœãŠããšãé»åãé£ã³åºããŠããããšãããããã®çŸè±¡ããå
é»å¹æ(ããã§ã ããããphotoelectric effect)ãšããã1887幎ããã«ãã«ãã£ãŠãå
é»å¹æãçºèŠããããã¬ãŒãã«ãã«ãã£ãŠãå
é»å¹æã®ç¹åŸŽãæããã«ãªã£ãã",
"title": "é»åãšå
"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "åœãŠãå
ã®æ¯åæ°ããäžå®ã®é«ã以äžã ãšãå
é»å¹æãèµ·ããããã®æ¯åæ°ãéçæ¯åæ°(ãããã ããã©ããã)ãšãããéçæ¯åæ°ããäœãå
ã§ã¯ãå
é»å¹æãèµ·ãããªãããŸããéçæ¯åæ°ã®ãšãã®æ³¢é·ããéçæ³¢é·(ãããã ã¯ã¡ãã)ãšããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ç©è³ªã«ãã£ãŠãéçæ¯åæ°ã¯ç°ãªããäºéçã§ã¯çŽ«å€ç·ã§ãªããšå
é»å¹æãèµ·ããªãããã»ã·ãŠã ã§ã¯å¯èŠå
ã§ãå
é»å¹æãèµ·ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "å
é»å¹æãšã¯ãç©è³ªäž(äž»ã«éå±)ã®é»åãå
ã®ãšãã«ã®ãŒãåãåã£ãŠå€éšã«é£ã³åºãçŸè±¡ã®ããšã§ããã ãã®é£ã³åºããé»åããå
é»åã(ããã§ãããphotoelectron)ãšããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "å
é»å¹æã«ã¯,次ã®ãããªç¹åŸŽçãªæ§è³ªãããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "ãããã®æ§è³ªã®ãã¡ã1çªããš2çªãã®æ§è³ªã¯ãå€å
žç©çåŠã§ã¯èª¬æã§ããªãã ã€ãŸããå
ããé»ç£æ³¢ãšããæ³¢åã®æ§è³ªã ããæããŠããŠã¯ãã€ãã€ãŸãåããªãã®ã§ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "ãªããªããä»®ã«ãé»ç£æ³¢ã®é»ç(é»å Ž)ã«ãã£ãŠéå±ããé»åãæŸåºãããšèããå Žåãããå
ã®åŒ·ãã倧ãããªãã°ãæ¯å¹
ã倧ãããªãã®ã§ãé»ç(é»å Ž)ã倧ãããªãã¯ãã§ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "ããããå®éšçµæã§ã¯ãå
é»åã®éåãšãã«ã®ãŒã¯ãå
ã®åŒ·ãã«ã¯äŸåããªãã",
"title": "é»åãšå
"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ãã£ãŠãå€å
žååŠã§ã¯èª¬æã§ããªãã",
"title": "é»åãšå
"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "äžè¿°ã®ççŸ(å€å
žçãªé»ç£æ³¢çè«ã§ã¯ãå
é»å¹æã説æã§ããªãããš)ã解決ããããã«ã次ã®ãããªå
éå仮説ãã¢ã€ã³ã·ã¥ã¿ã€ã³ã«ãã£ãŠæå±ãããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "ãã®2ã€ãã®æ¡ä»¶ãå®åŒåãããšã",
"title": "é»åãšå
"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "ãšãªãã",
"title": "é»åãšå
"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "ãã®åŒã«ãããæ¯äŸå®æ°hã¯ãã©ã³ã¯å®æ°ãšãã°ããå®æ°ã§ã",
"title": "é»åãšå
"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "[Jã»s] ãšããå€ããšãã",
"title": "é»åãšå
"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "ä»äºé¢æ°(ãããš ãããããwork function)ãšã¯ãå
é»å¹æãèµ·ããã®ã«å¿
èŠãªæå°ã®ãšãã«ã®ãŒã®ããšã§ãããéå±ã®çš®é¡ããšã«ã決ãŸã£ãå€ã§ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "ä»äºé¢æ°ã®å€ã W[J] ãšãããšãå
åã®åŸãéåãšãã«ã®ãŒã®æå€§å€ K0 [J] ã«ã€ããŠã次åŒãåŸãããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "ãã®åŒãããå
é»å¹æãèµ·ããæ¡ä»¶ã¯ hÎœâ§W ãšãªãããã㯠K0â§0 ã«çžåœããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "ãããããå
é»å¹æãèµ·ããéçæ¯åæ° Îœ0 ã«ã€ããŠãhÎœ0=W ãæãç«ã€ã",
"title": "é»åãšå
"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "ãã®å
éå仮説ã«ãããå
é»å¹æã®1çªããš2çªãã®æ§è³ªã¯ã容æã«ãççŸãªã説æã§ããããã«ãªã£ããæ³¢åã¯ç²åã®ããã«æ¯èãã®ã§ããã ãªããå
é»å¹æã®3çªãã®æ§è³ªãããããå Žæã®å
ã®åŒ·ãã¯ã ãã®å Žæã®åäœé¢ç©ã«åäœæéãé£æ¥ããå
åã®æ°ã«æ¯äŸããããšãåããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "",
"title": "é»åãšå
"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "ãããããå
ã®æ³¢é·ã¯ãã©ããã£ãŠæž¬å®ãããã®ã ãããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "çŸåšã§ã¯ãããšãã°ååã®çºå
ã¹ãã¯ãã«ã®æ³¢é·æž¬å®ãªããåææ Œåãããªãºã ãšããŠäœ¿ãããšã«ãã£ãŠãæ³¢é·ããšã«åããæ³¢é·ã枬å®ãããŠããã(â» åèæç®: å¹é¢šé€š(ã°ããµããã)ãstep-up åºç€ååŠãã梶æ¬èäº ç·šéãç³å·æ¥æš¹ ã»ãèã2015幎åçã25ããŒãž)",
"title": "é»åãšå
"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "ãããŸããªåçãè¿°ã¹ããšãå¯èŠå
ãŠãã©ã®å
ã®æ³¢é·ã®æž¬å®ã¯ãåææ Œåã«ãã£ãŠæž¬å®ããããã ããã§ã¯ãã®åææ Œåã®çŽ°ããæ°çŸããã¡ãŒãã«ãæ°åããã¡ãŒãã«ãŠãã©ã®ééã®æ ŒåããŸãã©ããã£ãŠäœãã®ãããšããåé¡ã«è¡ãçããŠããŸãã",
"title": "é»åãšå
"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "æŽå²çã«ã¯ãäžèšã®ããã«ãå¯èŠå
ã®æ³¢é·ã枬å®ãããŠãã£ãã",
"title": "é»åãšå
"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "ãŸãã1805幎ããã®ãã€ã³ã°ã®å®éšãã§æåãªã€ã³ã°ãã®ç 究ã«ãããå¯èŠå
ã®æ³¢é·ã¯ããããã 100nm(10m) ã 1000nm ã®çšåºŠã§ããããšã¯ããã®é ããããã§ã«äºæ³ãããŠããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "ãã®åŸããã€ãã®ã¬ã³ãºã®ç 磚工ã ã£ããã©ãŠã³ããŒãã¡ãŒããããããåææ Œåãéçºããå¯èŠå
ã®æ³¢é·ã粟å¯ã«æž¬å®ããäºã«æåããããã©ãŠã³ããŒãã¡ãŒã¯åææ Œåãäœãããã«çŽ°ãééãçšããå å·¥è£
眮ã補äœãããã®å å·¥æ©ã§è£œäœãããåææ ŒåãçšããŠãå
ã®æ³¢é·ã®æž¬å®ããå§ããã®ããç 究ã®å§ãŸãã§ããã1821幎ã«ãã©ãŠã³ããŒãã¡ãŒã¯ã1cmãããæ Œåã130æ¬ã䞊ã¹ãåææ Œåã補äœããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "ãŸãã1870幎ã«ã¯ã¢ã¡ãªã«ã®ã©ã¶ãã©ãŒããã¹ããã¥ã©ã ãšããåéãçšããåå°åã®åææ Œåã補äœã(ãã®ã¹ããã¥ã©ã åéã¯å
ã®åå°æ§ãé«ã)ãããã«ãã£ãŠ1mmããã700æ¬ãã®æ Œåã®ããåææ Œåã補äœããã(èŠåºå
ž)",
"title": "é»åãšå
"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "ããã«ãã®ããã®æ代ãéãããã®æœ€æ»ã®ããã«æ°Žéã䜿ãæ°Žéæµ®éæ³ããç 究éçºã§è¡ãããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "ããé«ç²ŸåºŠãªæ³¢é·æž¬å®ããã®ã¡ã®æ代ã®ç©çåŠè
ãã€ã±ã«ãœã³ã«ãã£ãŠãå¹²æžèš(ããããããã)ãšãããã®ãçšããŠ(çžå¯Ÿæ§çè«ã®å
¥éæžã«ããåºãŠããè£
眮ã§ãããé«æ ¡çã¯ããŸã çžå¯Ÿæ§çè«ãç¿ã£ãŠãªãã®ã§ãæ°ã«ããªããŠããã)ãå¹²æžèšã®åå°é¡ã粟å¯ããžã§çŽ°ããåããããšã«ãããé«ç²ŸåºŠãªæ³¢é·æž¬å®åšãã€ããããã®æž¬å®åšã«ãã£ãŠã«ãããŠã ã®èµ€è²ã¹ãã¯ãã«ç·ã枬å®ããçµæã®æ³¢é·ã¯643.84696nmã ã£ãããã€ã±ã«ãœã³ã®æž¬å®æ¹æ³ã¯ãèµ€è²ã¹ãã¯ãã«å
ã®æ³¢é·ããåœæã®ã¡ãŒãã«ååšãšæ¯èŒããããšã§æž¬å®ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "ãã€ã±ã«ãœã³ã®å¶äœããå¹²æžèšã«ããæ°Žéæµ®éæ³ã®æè¡ãåãå
¥ããããŠããããšããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "ããã«ãããžã®æè¡é©æ°ã§ãããŒãã³ã»ããã(ãã¡ã«ãã³ã»ãããããšãèš³ã)ãšããã匟åæ§ã®ããæ質ã§ããžãã€ããããšã«ãã£ãŠèª€å·®ããªããããŠå¹³ååãããã®ã§ãè¶
絶çã«é«ç²ŸåºŠã®éããããäœãæè¡ããã€ã®ãªã¹ã®ç©çåŠè
ããŒãã¹ã»ã©ã«ãã»ããŒãã³(è±:en:w:Thomas Ralph Merton )ãªã©ã«ãã£ãŠéçºãããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "ãªããçŸä»£ã§ããç 究çšãšããŠå¹²æžèšãçšããæ³¢é·æž¬å®åšãçšããããŠããã(èŠåºå
ž) ã¡ãŒãã«ååšã¯ããã€ã±ã«ãœã³ã®å®éšã®åœæã¯é·ãã®ããããšã®æšæºã ã£ããã1983幎以éã¯ã¡ãŒãã«ååšã¯é·ãã®æšæºã«ã¯çšããããŠããªããçŸåšã®ã¡ãŒãã«å®çŸ©ã¯ä»¥äžã®éãã",
"title": "é»åãšå
"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "",
"title": "é»åãšå
"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "",
"title": "é»åãšå
"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "倪éœé»æ± ããå
é»å¹æã®ãããªçŸè±¡ã§ããããšèããããŠããã(â» å®æåºçã®æç§æžãªã©ã§ãæ±ã£ãŠãã話é¡ã)",
"title": "é»åãšå
"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "ãªãã倪éœé»æ± ã¯äžè¬çã«åå°äœã§ããããã€ãªãŒãåããPNæ¥åã®éšåã«å
ãåœãŠãå¿
èŠãããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 41,
"tag": "p",
"text": "(PNæ¥åéšå以å€ã®å Žæã«ãå
ãããã£ãŠããçããé»åããé»æµãšããŠåãåºããªããé»æµãšããŠåãåºããããã«ããã«ã¯ãPNæ¥åã®éšåã«ãå
ãåœãŠãå¿
èŠãããããã®ãããPNæ¥åã®çæ¹ã®æ質ããéæããããã«è¿ãå
ééçã®ææã«ããå¿
èŠãããããéæé»æ¥µããšããã)",
"title": "é»åãšå
"
},
{
"paragraph_id": 42,
"tag": "p",
"text": "(â» ç¯å²å€?: ) ãªããçºå
ãã€ãªãŒãåå°äœã¯ããã®éãã¿ãŒã³ãšããŠèããããŠãããå
é»å¹æã§ãããä»äºé¢æ°ãã«ããããšãã«ã®ãŒããã£ãé»æµãæµãããšã«ããããã®åå°äœç©è³ªã®ãä»äºé¢æ°ãã«ããããšãã«ã®ãŒã®å
ããPNæ¥åã®æ¥åé¢ããæŸåºãããããšããä»çµã¿ã§ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 43,
"tag": "p",
"text": "ãªããCCDã«ã¡ã©ãªã©ã«äœ¿ãããCCDã¯ã倪éœé»æ± ã®ãããªæ©èœããã€åå°äœããé»åæºãšããŠã§ã¯ãªããå
ã®ã»ã³ãµãŒãšããŠæŽ»çšãããšããä»çµã¿ã®åå°äœã§ããã(â» å®æåºçã®æç§æžãªã©ã§ãæ±ã£ãŠãã話é¡ã)",
"title": "é»åãšå
"
},
{
"paragraph_id": 44,
"tag": "p",
"text": "(â» æ®éç§é«æ ¡ã®ãç©çãç³»ç§ç®ã§ã¯ç¿ããªããã)",
"title": "é»åãšå
"
},
{
"paragraph_id": 45,
"tag": "p",
"text": "ç©ççŸè±¡ã®éååãšããŠãå
é»å¹æãç©è³ªæ³¢ã®ã»ãã«ãååã¹ã±ãŒã«ã®ç©ççŸè±¡ã®éååã¯ãããããçš®é¡ã®è¶
äŒå°ç©è³ªã§ã¯ãããã«éããç£æãéååããçŸè±¡ãç¥ãããŠããã(â» å·¥æ¥é«æ ¡ã®ç§ç®ãå·¥æ¥ææãäžå·»(ãŸãã¯ç§ç®ã®åŸå)ã§ç¿ãã)",
"title": "é»åãšå
"
},
{
"paragraph_id": 46,
"tag": "p",
"text": "ç§åŠè
ã¬ã³ãã²ã³ã¯ã1895幎ãæŸé»ç®¡ããã¡ããŠé°æ¥µç·ã®å®éšãããŠãããšããæŸé»ç®¡ã®ã¡ããã«çœ®ããŠãã£ãåç也æ¿ãæå
ããŠããäºã«æ°ä»ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 47,
"tag": "p",
"text": "圌(ã¬ã³ãã²ã³)ã¯ãé°æ¥µç·ãã¬ã©ã¹ã«åœãã£ããšãããªã«ãæªç¥ã®ãã®ãæŸå°ãããŠããšèããXç·ãšåã¥ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 48,
"tag": "p",
"text": "ãããŠãããŸããŸãªå®éšã«ãã£ãŠãXç·ã¯æ¬¡ã®æ§è³ªããã€ããšãæããã«ãªã£ãã",
"title": "é»åãšå
"
},
{
"paragraph_id": 49,
"tag": "p",
"text": "ãã®äºãããXç·ã¯ãè·é»ç²åã§ã¯ãªãäºãåããã(çµè«ããããšãXç·ã®æ£äœã¯ãæ³¢é·ã®çãé»ç£æ³¢ã§ããã)",
"title": "é»åãšå
"
},
{
"paragraph_id": 50,
"tag": "p",
"text": "ãŸãã",
"title": "é»åãšå
"
},
{
"paragraph_id": 51,
"tag": "p",
"text": "ãªã©ã®æ§è³ªãããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 52,
"tag": "p",
"text": "ãªãçŸä»£ã§ã¯ãå»ççšã®Xç·ããã¬ã³ãã²ã³ããšãããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 53,
"tag": "p",
"text": "",
"title": "é»åãšå
"
},
{
"paragraph_id": 54,
"tag": "p",
"text": "1912幎ãç©çåŠè
ã©ãŠãšã¯ãXç·ãåçµæ¶ã«åœãŠããšãåçãã£ã«ã ã«å³ã®ãããªæç¹ã®æš¡æ§ã«ããããšãçºèŠããããããã©ãŠãšæç¹(ã¯ããŠã)ãšãããçµæ¶äžã®ååãåææ Œåã®åœ¹å²ãããããšã§çºçããå¹²æžçŸè±¡ã§ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 55,
"tag": "p",
"text": "",
"title": "é»åãšå
"
},
{
"paragraph_id": 56,
"tag": "p",
"text": "1912幎ãç©çåŠè
ãã©ãã°ã¯ãåå°ã匷ãããæ¡ä»¶åŒãçºèŠããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 57,
"tag": "p",
"text": "2d sinΞ = n λ",
"title": "é»åãšå
"
},
{
"paragraph_id": 58,
"tag": "p",
"text": "ããããã©ãã°ã®æ¡ä»¶ãšããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 59,
"tag": "p",
"text": "äžåŒã®dã¯æ Œåé¢ã®ééã®å¹
ã§ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 60,
"tag": "p",
"text": "",
"title": "é»åãšå
"
},
{
"paragraph_id": 61,
"tag": "p",
"text": "",
"title": "é»åãšå
"
},
{
"paragraph_id": 62,
"tag": "p",
"text": "Xç·ãççŽ å¡ãªã©ã®(éå±ãšã¯éããªã)ç©è³ªã«åœãŠããã®æ£ä¹±ãããããšã®Xç·ã調ã¹ããšãããšã®Xç·ã®æ³¢é·ãããé·ããã®ããæ£ä¹±ããXç·ã«å«ãŸããã ãã®ããã«æ£ä¹±Xç·ã®æ³¢é·ã䌞ã³ãçŸè±¡ã¯ç©çåŠè
ã³ã³ããã³ã«ãã£ãŠè§£æãããã®ã§ãã³ã³ããã³å¹æ(ãŸãã¯ã³ã³ããã³æ£ä¹±)ãšããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 63,
"tag": "p",
"text": "",
"title": "é»åãšå
"
},
{
"paragraph_id": 64,
"tag": "p",
"text": "ãã®çŸè±¡ã¯ãXç·ãæ³¢ãšèããã®ã§ã¯èª¬æãã€ããªãã(ããä»®ã«æ³¢ãšèããå Žåãæ£ä¹±å
ã®æ³¢é·ã¯ãå
¥å°Xç·ãšåãæ³¢é·ã«ãªãã¯ãããªããªããæ°Žé¢ã®æ³¢ã«äŸãããªããããæ°Žé¢ãæ£ã§4ç§éã«1åã®ããŒã¹ã§æºãããããæ°Žé¢ã®æ³¢ãã4ç§éã«1åã®ããŒã¹ã§åšæãè¿ããã®ãšãåãçå±ã) ããŠãæ³¢åã®çè«ã§ã³ã³ããã³å¹æã説æã§ããªããªããç²åã®çè«ã§èª¬æãããã°è¯ãã ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 65,
"tag": "p",
"text": "ãã®åœæãã¢ã€ã³ã·ã¥ã¿ã€ã³ã¯å
éå仮説ã«ããšã¥ããå
åã¯ãšãã«ã®ãŒhÎœããã€ã ãã§ãªããããã«æ¬¡ã®åŒã§è¡šãããéåépããã€ããšãçºèŠããŠããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 66,
"tag": "p",
"text": "p = h Μ c ( = h Μ Μ λ = h λ ) {\\displaystyle p={\\frac {h\\nu }{c}}(={\\frac {h\\nu }{\\nu \\lambda }}={\\frac {h}{\\lambda }})}",
"title": "é»åãšå
"
},
{
"paragraph_id": 67,
"tag": "p",
"text": "ç©çåŠè
ã³ã³ããã³ã¯ããã®çºèŠãå©çšããæ³¢é·Î»ã®Xç·ããéåé h λ {\\displaystyle {\\frac {h}{\\lambda }}} ãšãšãã«ã®ãŒ h c λ {\\displaystyle {\\frac {hc}{\\lambda }}} ãæã€ç²å(å
å)ã®æµããšèãã Xç·ã®æ£ä¹±ãããã®å
åãç©è³ªäžã®ããé»åãšå®å
šåŒŸæ§è¡çªãããçµæãšèããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 68,
"tag": "p",
"text": "解æ³ã¯ãäžèšã®ãšããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 69,
"tag": "p",
"text": "ãšãã«ã®ãŒä¿åã®åŒ",
"title": "é»åãšå
"
},
{
"paragraph_id": 70,
"tag": "p",
"text": "éåéä¿åã®åŒ",
"title": "é»åãšå
"
},
{
"paragraph_id": 71,
"tag": "p",
"text": "äžèšã®3ã€ã®åŒãé£ç«ãããã®é£ç«æ¹çšåŒã解ãããã«vãšÏãé£ç«èšç®ã§æ¶å»ãããŠããã λ â λ â² {\\displaystyle \\lambda \\fallingdotseq \\lambda '} ã®ãšã㫠λ â² â λ + h m c ( 1 â cos Ξ ) {\\displaystyle \\lambda '\\fallingdotseq \\lambda +{\\frac {h}{mc}}(1-\\cos \\theta )} ãåŸãããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 72,
"tag": "p",
"text": "ãã®åŒãå®éšåŒãšããäžèŽããã®ã§ãã³ã³ããã³ã®èª¬ã®æ£ããã¯å®èšŒãããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 73,
"tag": "p",
"text": "(ç·šéè
ãž: èšè¿°ããŠãã ããã)(Gimyamma ããã解æ³ãæžããŠã¿ãŸããã)",
"title": "é»åãšå
"
},
{
"paragraph_id": 74,
"tag": "p",
"text": "åŒ(1),(2),(3)ããã v {\\displaystyle v} ãš Ï {\\displaystyle \\phi } ãæ¶å»ããŠã λ , λ â² , Ξ {\\displaystyle \\lambda ,\\lambda ',\\theta } ã®é¢ä¿åŒãæ±ããã°ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 75,
"tag": "p",
"text": "( m v sin Ï ) 2 = ( â h λ â² sin Ξ ) 2 {\\displaystyle (mv\\sin \\phi )^{2}=(-{\\frac {h}{\\lambda '}}\\sin \\theta )^{2}}",
"title": "é»åãšå
"
},
{
"paragraph_id": 76,
"tag": "p",
"text": "m 2 v 2 = ( h λ â h λ â² cos Ξ ) 2 + ( â h λ â² sin Ξ ) 2 + h 2 λ â² 2 {\\displaystyle m^{2}v^{2}=({\\frac {h}{\\lambda }}-{\\frac {h}{\\lambda '}}\\cos \\theta )^{2}+(-{\\frac {h}{\\lambda '}}\\sin \\theta )^{2}+{\\frac {h^{2}}{\\lambda '^{2}}}}",
"title": "é»åãšå
"
},
{
"paragraph_id": 77,
"tag": "p",
"text": "ãåŸãã",
"title": "é»åãšå
"
},
{
"paragraph_id": 78,
"tag": "p",
"text": "åŒ(1)ã®å³èŸºã®ç¬¬2é
ãå€åœ¢ããŠåŒ(4)ã代å
¥ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 79,
"tag": "p",
"text": "ãããåŒ(1)ã®å³èŸºã«ä»£å
¥ãããš",
"title": "é»åãšå
"
},
{
"paragraph_id": 80,
"tag": "p",
"text": "ãåŸãã",
"title": "é»åãšå
"
},
{
"paragraph_id": 81,
"tag": "p",
"text": "ãã®åŒãåŒ(5)ã®å³èŸºç¬¬2é
ã«ä»£å
¥ãããšã",
"title": "é»åãšå
"
},
{
"paragraph_id": 82,
"tag": "p",
"text": "ãã®åŒã®å³èŸºã®ç¬¬1é
ã移è¡ããåŒãå€åœ¢ãããš",
"title": "é»åãšå
"
},
{
"paragraph_id": 83,
"tag": "p",
"text": "䞡蟺㫠λ λ â² {\\displaystyle \\lambda \\lambda '} ãæãããš",
"title": "é»åãšå
"
},
{
"paragraph_id": 84,
"tag": "p",
"text": "Xç·ã®æ£ä¹±ã§ã¯ã λ â² â λ {\\displaystyle \\lambda '\\fallingdotseq \\lambda } ãªã®ã§",
"title": "é»åãšå
"
},
{
"paragraph_id": 85,
"tag": "p",
"text": "æ
ã«åŒ(6)ãã",
"title": "é»åãšå
"
},
{
"paragraph_id": 86,
"tag": "p",
"text": "ããã§ãææã®åŒãå°åºãããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 87,
"tag": "p",
"text": "å
ã®éåé P[kgã»m/s]=hÎœ/c ã«ã€ããŠã",
"title": "é»åãšå
"
},
{
"paragraph_id": 88,
"tag": "p",
"text": "ãŸã cP=hÎœ[J] ãšå€åœ¢ããŠã¿ããšããé床ã«éåéãããããã®ããšãã«ã®ãŒã§ããããšããå
容ã®å
¬åŒã«ãªã£ãŠããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 89,
"tag": "p",
"text": "ãããç解ãããããã²ãšãŸããå
ãç²åã§ãããšåæã«æµäœã§ãããšèããŠããã®é»ç£æ³¢ãåäœäœç©ãããã®éåépãæã£ãŠãããšããŠããã®æµäœã®éåéã®å¯åºŠ(éåéå¯åºŠ)ã p [(kgã»m/s)/m]ãšãããããã®å Žåã®é»ç£æ³¢ã¯æµäœãªã®ã§ãéåéã¯ããã®å¯åºŠã§èããå¿
èŠãããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 90,
"tag": "p",
"text": "é»ç£æ³¢ãç©äœã«ç
§å°ããŠãå
ãç©äœã«åžåããããšããããåå°ã¯ãªããšããŠãå
ã®ãšãã«ã®ãŒã¯ãã¹ãŠç©äœã«åžåããããšãããç°¡åã®ãããç©äœå£ã«åçŽã«å
ãç
§å°ãããšãããç©äœãžã®å
ã®ç
§å°é¢ç©ãA[m]ãšããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 91,
"tag": "p",
"text": "é»ç£æ³¢ã¯å
é c[m/s] ã§é²ãã®ã ãããå£ããcã®è·é¢ã®éã«ãããã¹ãŠã®å
åã¯ããã¹ãŠåäœæéåŸã«åžåãããäºã«ãªããåäœæéã«å£ã«åžåãããå
åã®éã¯ããã®åäœæéã®ããã ã«å£ã«æµã蟌ãã å
åã®éã§ããã®ã§ã",
"title": "é»åãšå
"
},
{
"paragraph_id": 92,
"tag": "p",
"text": "",
"title": "é»åãšå
"
},
{
"paragraph_id": 93,
"tag": "p",
"text": "å³ã®ããã«ãä»®ã«åºé¢ãA[m]ãšããŠãé«ãhã c ( hã®å€§ããã¯cã«çãããåäœæét=1ãããããšããã° h=cã»1 ã§ãã)[m]ãšããæ±ã®äœç© AÃc[m]äžã«å«ãŸããå
åã®éã®ç·åã«çããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 94,
"tag": "p",
"text": "ãã£ãœããéåéå¯åºŠã¯ p[(kgã»m/s)/m]ã ã£ãã®ã§ããã®æ± AÃh ã«å«ãŸããéåéã®ç·åã¯ã AÃhÃp[kgã»m/s]ã§ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 95,
"tag": "p",
"text": "å
ãåžåããç©äœã®éåéã¯ãåäœæéã«Ahpã®éåéãå¢å ããããšã«ãªãããh=cã§ãã£ãã®ã§ãã€ãŸããéåéãåäœæéãããã« Acp[kgã»m/s] ã ãå£ã«æµãããããšã«ãªãã",
"title": "é»åãšå
"
},
{
"paragraph_id": 96,
"tag": "p",
"text": "ãã£ãœããé«æ ¡ç©çã®ååŠã®çè«ã«ããããéåéã®æéãããã®å€åã¯ãåã§ãããã§ãã£ãã®ã§ãã€ãŸãç©äœã¯ãAcp[N]ã®åãåããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 97,
"tag": "p",
"text": "åãåããã®ã¯ç
§å°ãããé¢ã ãããå[N]ãé¢ç©ã§å²ãã°å§åã®æ¬¡å
[N/m]=[Pa]ã«ãªãã",
"title": "é»åãšå
"
},
{
"paragraph_id": 98,
"tag": "p",
"text": "å®éã«é¢ç©ã§å²ãèšç®ãããã°ãå§åãšã㊠cp[N/m]=[Pa]=[J/m] ãåããäºãèšç®çã«åãããããã«ãå§åã®æ¬¡å
ã¯[N/m]=[Pa]=[J/m]ãšå€åœ¢ã§ããã®ã§ããå§åã¯ãåäœäœç©ãããã®ãšãã«ã®ãŒã®å¯åºŠ(ããšãã«ã®ãŒå¯åºŠããšãã)ã§ããããšèãããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 99,
"tag": "p",
"text": "ãšããã° cp ã®æ¬¡å
ã¯ã[å§å]=[ãšãã«ã®ãŒå¯åºŠ] ãšãªãã",
"title": "é»åãšå
"
},
{
"paragraph_id": 100,
"tag": "p",
"text": "ãã®ãšãã«ã®ãŒå¯åºŠã«ãhÎœã察å¿ããŠãããšèããã°ãåççã§ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 101,
"tag": "p",
"text": "èŠããã«ãå
ã®ãããªãäºå®äžã¯ç¡éã«å§çž®ã§ããæ³¢ã»æµäœã§ã¯ã",
"title": "é»åãšå
"
},
{
"paragraph_id": 102,
"tag": "p",
"text": "å
¬åŒãšããŠãé床ãvãéåéå¯åºŠãpããšãã«ã®ãŒå¯åºŠãεãšããŠèããã°ã",
"title": "é»åãšå
"
},
{
"paragraph_id": 103,
"tag": "p",
"text": "ãšããé¢ä¿ããªããã€ã",
"title": "é»åãšå
"
},
{
"paragraph_id": 104,
"tag": "p",
"text": "(ãªããæ°Žã空æ°ã®ãããªæ®éã®æµäœã§ã¯ãç¡éã«ã¯å§çž®ã§ããªãã®ã§ãäžèšã®å
¬åŒã¯æãç«ããªãã)",
"title": "é»åãšå
"
},
{
"paragraph_id": 105,
"tag": "p",
"text": "ãããããã³ã³ããã³å¹æã®åŠç¿ã§åãã£ãéåéã®å
¬åŒ p = h Îœ c {\\displaystyle p={\\frac {h\\nu }{c}}} ã¯ãéåéå¯åºŠãšãšãã«ã®ãŒå¯åºŠã®é¢ä¿åŒã«ãå
écãšå
é»å¹æã®ãšãã«ã®ãŒhÎœã代å
¥ãããã®ã«ãªã£ãŠããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 106,
"tag": "p",
"text": "äžèšã®èå¯ã¯ãå
ãæµäœãšããŠèããé»ç£æ³¢ã®éåéã ããç²åãšããŠè§£éãããå
åã®éåéã«ãã cP=hÎœ ãšããé¢ä¿ãæãç«ã€ãšèãããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 107,
"tag": "p",
"text": "ããèªè
ããå§åããšãã«ã®ãŒå¯åºŠãšèããã®ãåããã¥ãããã°ãããšãã°ç±ååŠã®ä»äºã®å
¬åŒ W=Pâ¿V ã®é¡æšãããŠã¯ã©ãã? ãªããäžèšã®éåéãšãšãã«ã®ãŒã®é¢ä¿åŒã®å°åºã¯å€§ãŸããªèª¬æã§ãããæ£ç¢ºãªå°åºæ³ã¯ã(倧åŠã§ç¿ã)ãã¯ã¹ãŠã§ã«ã®æ¹çšåŒã«ãããªããã°ãªããªãã",
"title": "é»åãšå
"
},
{
"paragraph_id": 108,
"tag": "p",
"text": "ããããã ãå
ã¯ãé»åã«äœçšãããšãã«ãå
ãç²åãšããŠæ¯èã(ãµããŸã)ã ãšããã®ãæ£ããã ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 109,
"tag": "p",
"text": "ãã£ãœãããã¿ããã«ãå
ã¯ç²å! å
ã¯æ³¢åã§ã¯ãªã!!ã(Ã)ãšãããã®ã¯ãåãªã銬鹿ã®ã²ãšã€èŠãã§ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 110,
"tag": "p",
"text": "ãã¯ã¹ãŠã§ã«ã®æ¹çšåŒã§ã¯ãå
(é»ç£æ³¢)ã¯æ³¢åãšããŠããã€ããã®ã§ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 111,
"tag": "p",
"text": "ããããå
é»å¹æã§èµ·ããçŸè±¡ã§ã¯ãæŸåºé»åã®ãã€éåãšãã«ã®ãŒã¯ãå
ã®åŒ·åºŠãšã¯ç¡é¢ä¿ã§ãããåçŽãªæµäœãšããŠèãããªãã(ããšãã°éå
ããããå
ãéãããããŠã)å
ã®åŒ·åºŠãäžããã°ãéåéå¯åºŠãäžããããºã ãããã®åž°çµã®æŸåºé»åã®ãšãã«ã®ãŒå¯åºŠãäžããããºã ããããšããäºæž¬ãæãç«ã¡ããã ãããããå®éšçµæã¯ãã®äºæž¬ãšã¯ç°ãªããå
é»å¹æã¯å
ã®åŒ·åºŠãšã¯ç¡é¢ä¿ã«å
ã®åšæ³¢æ°ã«ãã£ãŠæŸåºé»åã®ãšãã«ã®ãŒã決ãŸããã»ã»ã»ãšããã®ããå®éšäºå®ã§ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 112,
"tag": "p",
"text": "ãã®ãããªå®éšçµæããã20äžçŽåé ã®åœæãåèããŠããéåååŠãªã©ãšé¢é£ã¥ããŠããå
ãæ³¢ã§ãããšåæã«ç²åã§ããããšæå®ããã®ãããŒãã«è²¡å£ãªã©ã§ãããå
é»å¹æãå
ã®ç²å説ã®æ ¹æ ã®ã²ãšã€ãšããã®ãã¢ã€ã³ã·ã¥ã¿ã€ã³ã®ä»®èª¬ã§ãããã¢ã€ã³ã·ã¥ã¿ã€ã³ã®ãã®ä»®èª¬ãå®èª¬ãšããŠèªå®ããã®ãããŒãã«è²¡å£ã§ãããçŸåšã®ç©çåŠã§ã¯ãå
é»å¹æãå
å説ã®æ ¹æ ãšããŠé説ã«ãªã£ãŠããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 113,
"tag": "p",
"text": "å
é»å¹æã®å®éšçµæãã®ãã®ã¯ãåã«ãå
é»å¹æã«ããããå
ãããåçŽãªæµäœã»æ³¢åãšããŠã¯èããããªãã ããã»ã»ã»ãšããã ãã®äºã§ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 114,
"tag": "p",
"text": "çµå±ãç©çåŠã¯å®éšç§åŠã§ãããå®éšçµæã«ããšã¥ãå®éšæ³åãèŠãããããªãããå
åããšããã¢ã€ãã¢ã¯ããå
é»å¹æã®æŸåºé»å1åãããã®ãšãã«ã®ãŒã¯ãå
¥å°å
ã®åŒ·åºŠã«å¯ãããå
ã®æ³¢é·(åšæ³¢æ°)ã«ããããšããäºãèŠããããããããã®æ段ã«ããããã¢ã€ã³ã·ã¥ã¿ã€ã³ãšãã®æ¯æè
ã«ãšã£ãŠã¯ããå
ã®ç²å説ããšããã®ããèŠããããããããã®ã¢ãã«ã ã£ãã ãã§ãã(ç²åãªã®ã«æ³¢é·(åšæ³¢æ°)ãšã¯ãæå³äžæã ã)ããããŠéåéå¯åºŠãšãšãã«ã®ãŒå¯åºŠã®é¢ä¿ vp=ε ãšããç¥èããŸããå
é»å¹æã®å
¬åŒ cP=hÎœ ãèŠããããããããã®æ段ã«ãããªãã",
"title": "é»åãšå
"
},
{
"paragraph_id": 115,
"tag": "p",
"text": "ãã£ããã®å
ã¯ãåçŽãªæ³¢ã§ããªããåçŽãªç²åã§ããªãããã åã«ãå
ã¯å
ã§ãããå
ã§ãããªãã",
"title": "é»åãšå
"
},
{
"paragraph_id": 116,
"tag": "p",
"text": "ãå
ã®ç²å説ããšããã®ã¯ãç空äžã§åªè³ª(ã°ããã€)ããªããŠãå
ãäŒããããšããçšåºŠã®æå³åãã§ãããªãã ãããã¢ã€ã³ã·ã¥ã¿ã€ã³ãç¹æ®çžå¯Ÿæ§çè«ãçºè¡šããåãŸã§ã¯ã(20äžç€ä»¥éããçŸä»£ã§ã¯åŠå®ãããŠãããã)ãã€ãŠããšãŒãã«ããšããå
ãäŒããåªè³ªã®ååšãä¿¡ããããŠããããããã¢ã€ã³ã·ã¥ã¿ã€ã³ã¯çžå¯Ÿæ§çè«ã«ããããšãŒãã«ã®ååšãåŠå®ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 117,
"tag": "p",
"text": "ãå
ã®ç²å説ããçºè¡šããŠããè
ãåããã¢ã€ã³ã·ã¥ã¿ã€ã³ã ã£ãã®ã§ãããŒãã«è²¡å£ã¯ãæ¬æ¥ãªãç¹çžå¯Ÿæ§çè«ã§ããŒãã«è³ãæãããããã«ãå
å説ã§ããŒãã«è³ãã¢ã€ã³ã·ã¥ã¿ã€ã³ã«æããã ãã ãã",
"title": "é»åãšå
"
},
{
"paragraph_id": 118,
"tag": "p",
"text": "ç©çåŠè
ãã»ããã€ã¯ãæ³¢ãšèããããŠãå
ãç²åã®æ§è³ªããã€ãªãã°ããã£ãšé»åãç²åãšããŠã®æ§è³ªã ãã§ãªããé»åãæ³¢åã®ããã«æ¯èãã ãããšèããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 119,
"tag": "p",
"text": "ãããŠãé»åã ãã§ãªããäžè¬ã®ç²åã«å¯ŸããŠãããã®èããé©çšãã次ã®å
¬åŒãæå±ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 120,
"tag": "p",
"text": "ããã¯ãã»ããã€ã«ãã仮説ã§ãã£ãããçŸåšã§ã¯æ£ãããšèªããããŠããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 121,
"tag": "p",
"text": "ãã®æ³¢ã¯ãç©è³ªæ³¢(material wave)ãšåŒã°ããããã»ããã€æ³¢(de Broglie wave length)ãšãããã ããªãã¡ãå
åãé»åã«éãããããããç©è³ªã¯ç²åæ§ãšæ³¢åæ§ããããæã€ãšãããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 122,
"tag": "p",
"text": "ãã®ç©è³ªæ³¢ãšãã説ã«ãããšããããé»åç·ãç©è³ªã«åœãŠãã°ãåæãªã©ã®çŸè±¡ãèµ·ããã¯ãã§ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 123,
"tag": "p",
"text": "1927幎ã1928幎ã«ãããŠãããããœã³ãšã¬ãŒããŒã¯ãããã±ã«ãªã©ã®ç©è³ªã«é»åç·ãåœãŠãå®éšãè¡ããXç·åæãšåæ§ã«é»åç·ã§ãåæãèµ·ããããšãå®èšŒãããæ¥æ¬ã§ã1928幎ã«èæ± æ£å£«(ããã¡ ããã)ãé²æ¯çã«é»åç·ãåœãŠãå®éšã«ããåæãèµ·ããããšã確èªããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 124,
"tag": "p",
"text": "é»åç·ã®æ³¢é·ã¯ãé«é»å§ããããŠé»åãå éããŠé床ãé«ããã°ãç©è³ªæ³¢ã®æ³¢é·ã¯ããªãå°ããã§ããã®ã§ãå¯èŠå
ã®æ³¢é·ãããå°ãããªãã",
"title": "é»åãšå
"
},
{
"paragraph_id": 125,
"tag": "p",
"text": "ãã®ãããå¯èŠå
ã§ã¯èŠ³æž¬ã§ããªããã£ãçµæ¶æ§é ããé»åæ³¢ãXç·ãªã©ã§èŠ³æž¬ã§ããããã«ãªã£ããçç©åŠã§ãŠã€ã«ã¹ãé»åé¡åŸ®é¡ã§èŠ³æž¬ã§ããããã«ãªã£ãã®ããé»åã®ç©è³ªæ³¢ãå¯èŠå
ããã倧å¹
ã«å°ããããã§ããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 126,
"tag": "p",
"text": "äžè¿°ã®ãããªãããŸããŸãªå®éšã®çµæããããã¹ãŠã®ç©è³ªã«ã¯ãååãŠãã©ã®å€§ããã®äžç(以éãåã«ãååã¹ã±ãŒã«ããªã©ãšç¥èšãã)ã§ã¯ãæ³¢åæ§ãšç²åæ§ã®äž¡æ¹ã®æ§è³ªããã€ãšèããããŠããã ãã®ããšãç²åãšæ³¢åã®äºéæ§ãšããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 127,
"tag": "p",
"text": "ãããŠãååã¹ã±ãŒã«ã§ã¯ãããäžã€ã®ç©è³ª(äž»ã«é»åã®ãããªç²å)ã«ã€ããŠããã®äœçœ®ãšéåéã®äž¡æ¹ãåæã«æ±ºå®ããäºã¯ã§ããªãããã®ããšãäžç¢ºå®æ§åç(ãµãããŠããã ããã)ãšããã",
"title": "é»åãšå
"
},
{
"paragraph_id": 128,
"tag": "p",
"text": "",
"title": "é»åãšå
"
},
{
"paragraph_id": 129,
"tag": "p",
"text": "ç©çåŠè
ã¬ã€ã¬ãŒãšç©çåŠè
ããŒã¹ãã³ã¯ã(ã©ãžãŠã ããåºãã)αç²åããããéã±ãã«åœãŠãå®éšãè¡ããαç²åã®æ£ä¹±ã®æ§åã調ã¹ãã(ãªããαç²åã®æ£äœã¯ããªãŠã ã®ååæ žã)ãã®çµæãã»ãšãã©ã®Î±ç²åã¯éã±ããçŽ éãããããéã±ãäžã®äžéšã®å Žæã®è¿ããéã£ãαç²åã ãã倧å¹
ã«æ£ä¹±ããçŸè±¡ãçºèŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 130,
"tag": "p",
"text": "ãã®å®éšçµæããã©ã¶ãã©ãŒãã¯ãååæ žã®ååšãã€ããšããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 131,
"tag": "p",
"text": "ååã¯ãäžå¿ã«ååæ žãããããã®ãŸãããé»åãéåãããšããã©ã¶ãã©ãŒãã¢ãã«ãšãã°ããã¢ãã«ã«ãã£ãŠèª¬æãããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 132,
"tag": "p",
"text": "åå(atom)ã¯ãå
šäœãšããŠã¯é»æ°çã«äžæ§ã§ãããè² ã®é»è·ãæããé»åãé»åæ®»ã«æã€ã ããã§ãããªã«ã³ã®å®éš ã«ããçµæãªã©ãããé»åã®è³ªéã¯æ°ŽçŽ ã€ãªã³ã®è³ªéã®çŽ1/1840çšåºŠãããªãããšãåãã£ãŠããã ããªãã¡ãååã¯é»åãšéœã€ãªã³ãšãå«ãŸãããã質éã®å€§éšåã¯éœã€ãªã³ããã€ããšãåããã ååæ žã®å€§ããã¯ååå
šäœã®1/10000çšåºŠã§ãããããååã®å€§éšåã¯ç空ã§ããã ååæ žã¯ãæ£ã®é»è·ããã€Zåã®éœå(proton)ãšãé»æ°çã«äžæ§ãª(AâZ)åã®äžæ§å(neutron)ãããªãã éœåãšäžæ§åã®åæ°ã®åèšã質éæ°(mass number)ãšããã éœåãšäžæ§åã®è³ªéã¯ã»ãŒçãããããååæ žã®è³ªéã¯ã質éæ°Aã«ã»ãŒæ¯äŸããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 133,
"tag": "p",
"text": "é«æž©ã®ç©äœããçºå
ãããå
ã«ã¯ãã©ã®(å¯èŠå
ã®)è²ã®æ³¢é·(åšæ³¢æ°)ãããããã®ãããªé£ç¶çãªæ³¢é·ã®å
ãé£ç¶ã¹ãã¯ãã«ãšããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 134,
"tag": "p",
"text": "ãã£ãœãããããªãŠã ãæ°ŽçŽ ãªã©ã®ãç¹å®ã®ç©è³ªã«é»å§ãããããæŸé»ãããšãã«çºå
ããæ³¢é·ã¯ãç¹å®ã®æ°æ¬ã®æ³¢é·ããå«ãŸããŠãããããã®ãããªã¹ãã¯ãã«ãèŒç·(ããã)ãšããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 135,
"tag": "p",
"text": "ãã«ããŒã¯ãæ°ŽçŽ ååã®æ°æ¬ããèŒç·ã®æ³¢é·ãã次ã®å
¬åŒã§è¡šçŸã§ããããšã«æ°ã¥ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 136,
"tag": "p",
"text": "λ = 3.65 à 10 â 7 m à ( n 2 n 2 â 4 ) {\\displaystyle \\lambda =3.65\\times 10^{-7}\\mathrm {m} \\times \\left({n^{2} \\over n^{2}-4}\\right)} (ãã ããn=3, 4 , 5 ,6 ,ã»ã»ã»)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 137,
"tag": "p",
"text": "äžåŒäžã®ãmãã¯ã¡ãŒãã«åäœãšããæå³ã(äžåŒã®mã¯ä»£æ°ã§ã¯ãªãã®ã§ãééããªãããã«ã)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 138,
"tag": "p",
"text": "ãã®åŸãæ°ŽçŽ ä»¥å€ã®ååããå¯èŠå
以å€ã®é åã«ã€ããŠããç©çåŠè
ãã¡ã«ãã£ãŠèª¿ã¹ããã次ã®å
¬åŒãžãšãç©çåŠè
ãªã¥ãŒãããªã«ãã£ãŠããŸãšããããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 139,
"tag": "p",
"text": "äžåŒã®Rã¯ãªã¥ãŒãããªå®æ°ãšããã R = 1.097 à 10 7 / m {\\displaystyle R=1.097\\times 10^{7}/m} ã§ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 140,
"tag": "p",
"text": "ã©ã¶ãã©ãŒãã®ååæš¡åã«åŸãã°ãé»åã¯ããŸãã§ææã®å
¬è»¢ã®ããã«ååæ žãäžå¿ãšããåè»éã®äžãäžå®ã®é床ã§éåããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 141,
"tag": "p",
"text": "ååæ žãäžå¿ãšããååŸr[m]ã®åè»éãéãv[m/s]ã§å転ããé»åã®è§éåé r p = r m v {\\displaystyle rp=rmv} ã¯ã h 2 Ï {\\displaystyle {\\frac {h}{2\\pi }}} ã®æ£æŽæ°åã«ãªããªããã°ãªããªã(è§éåéã®éåå)ã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 142,
"tag": "p",
"text": "ãæºãããã°ãªããªãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 143,
"tag": "p",
"text": "åŸå¹Ž(1924幎)ããã»ããã€ã¯ãç©è³ªç²åã¯æ³¢åæ§ãæã¡ã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 144,
"tag": "p",
"text": "ããã«åŸãã°ãããŒã¢ã®éåæ¡ä»¶ã®ä»®å®ã¯ããé»åè»éã®é·ãã¯ãé»åã®ç©è³ªæ³¢ã®æ³¢é·ã®æ£æŽæ°åã§ããããšè¡šçŸã§ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 145,
"tag": "p",
"text": "é»åã¯ããããŸã£ããšã³ãšã³ã®ãšãã«ã®ãŒããæããªãããã®ãšã³ãšã³ã®ãšãã«ã®ãŒå€ããšãã«ã®ãŒé äœãšããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 146,
"tag": "p",
"text": "æ°ŽçŽ ååã«ãããŠãé»åè»éäžã«ããé»åã®ãšãã«ã®ãŒãæ±ããèšç®ããããããŸãããã®ããã«ã¯ãååã®ååŸãæ±ããå¿
èŠãããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 147,
"tag": "p",
"text": "æ°ŽçŽ ã®é»åãååæ ž H + {\\displaystyle H^{+}} ãäžå¿ãšããååŸrã®åè»éäžãäžå®ã®é床vã§éåããŠãããšããã°ãéåæ¹çšåŒã¯",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 148,
"tag": "p",
"text": "ã§è¡šãããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 149,
"tag": "p",
"text": "äžæ¹ãé»åãå®åžžæ³¢ã®æ¡ä»¶ãæºããå¿
èŠãããã®ã§ãåé
ã®åŒ(1)ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 150,
"tag": "p",
"text": "ã§ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 151,
"tag": "p",
"text": "ãã®vãããã»ã©ã®åéåã®åŒã«ä»£å
¥ããŠæŽé ããã°ã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 152,
"tag": "p",
"text": "(ãã ããn=1, 2 , 3 ,ã»ã»ã»)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 153,
"tag": "p",
"text": "ã«ãªããããããŠãæ°ŽçŽ ååã®é»åã®è»éååŸãæ±ãŸãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 154,
"tag": "p",
"text": "ããã»ã©ã®è»éååŸã®åŒã§n=1ã®ãšãååŸr1ããããŒã¢ååŸããšããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 155,
"tag": "p",
"text": "",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 156,
"tag": "p",
"text": "ååã®äžçã§ããéåãšãã«ã®ãŒKãšäœçœ®ãšãã«ã®ãŒUã®åãããšãã«ã®ãŒã§ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 157,
"tag": "p",
"text": "äœçœ®ãšãã«ã®ãŒUã¯ããã®æ°ŽçŽ ã®é»åã®å Žåãªããéé»æ°ãšãã«ã®ãŒãæ±ããã°å
åã§ãããé»äœã®åŒã«ãã£ãŠæ±ããããŠã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 158,
"tag": "p",
"text": "ãšãªãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 159,
"tag": "p",
"text": "éåãšãã«ã®ãŒKã¯ã K = 1 2 m v 2 {\\displaystyle K={\\frac {1}{2}}mv^{2}} ãªã®ã§",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 160,
"tag": "p",
"text": "äžåŒã®å³èŸºç¬¬äžé
ã«ã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 161,
"tag": "p",
"text": "m v 2 = k 0 e 2 r {\\displaystyle mv^{2}=k_{0}{\\frac {e^{2}}{r}}} ã代å
¥ããã°ã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 162,
"tag": "p",
"text": "ãšãªãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 163,
"tag": "p",
"text": "ããã«ãããã«é»åã®è»éååŸ r = r n {\\displaystyle r=r_{n}} ã®åŒ(3)ã代å
¥ããã°ã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 164,
"tag": "p",
"text": "ãšãªãããããæ°ŽçŽ ååã®ãšãã«ã®ãŒæºäœã§ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 165,
"tag": "p",
"text": "ãšãã«ã®ãŒæºäœã®å
¬åŒãããèŠããšããŸãããšãã«ã®ãŒãããšã³ãšã³ã®å€ã«ãªãããšãåããããŸãããšãã«ã®ãŒãè² ã«ãªãäºããããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 166,
"tag": "p",
"text": "n=1ã®ãšããããã£ãšããšãã«ã®ãŒã®äœãç¶æ
ã§ããããã®ãããn=1ã®ãšããå®å®ãªç¶æ
ã§ããããã£ãŠãé»åã¯éåžžãn=1ã®ç¶æ
ã«ãªãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 167,
"tag": "p",
"text": "ãªãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 168,
"tag": "p",
"text": "æ°ŽçŽ ååã®çºããå
ã®ã¹ãã¯ãã«ã®å®æž¬å€ãè¡šããªã¥ãŒãããªã®çµéšåŒã«ã€ããŠã¯ãæ¢ã«ãæ°ŽçŽ ååã®ã¹ãã¯ãã«ãã®é
ã§ã§èª¬æããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 169,
"tag": "p",
"text": "é»åããšãã«ã®ãŒé äœ E n {\\displaystyle E_{n}} ãããäœããšãã«ã®ãŒé äœ E m {\\displaystyle E_{m}} ã«é·ç§»ãããšãã«ãæ¯åæ° Îœ = E n â E m h {\\displaystyle \\nu ={\\frac {E_{n}-E_{m}}{h}}} ã®å
åãäžåæŸåºããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 170,
"tag": "p",
"text": "1 λ = E n â E m c h {\\displaystyle {\\frac {1}{\\lambda }}={\\frac {E_{n}-E_{m}}{ch}}} ã§äžããããã®ã§ãå³èŸºã®ãšãã«ã®ãŒé äœã«åŒ(4)ã代å
¥ãããš",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 171,
"tag": "p",
"text": "ãåŸãããã R â 2 Ï 2 k 0 2 m e 4 c h 3 {\\displaystyle {\\bf {R}}\\triangleq {\\frac {2\\pi ^{2}k_{0}{}^{2}me^{4}}{ch^{3}}}} ã§ããªã¥ãŒãããªå®æ°Rãå®çŸ©ãããšãåŒ(5)ã¯",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 172,
"tag": "p",
"text": "Rã®å®çŸ©åŒäžã®è«žå®æ°ã«å€ããããŠèšç®ãããš",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 173,
"tag": "p",
"text": "é©ãã¹ãããšã«ããªã¥ãŒãããªã®çµéšåŒããèŠäºã«å°åºã§ããã®ã§ããã ããã¯ãããŒã¢ã®ä»®èª¬ã®åŠ¥åœæ§ã瀺ããã®ãšèšãããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 174,
"tag": "p",
"text": "",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 175,
"tag": "p",
"text": "(â» æªèšè¿°)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 176,
"tag": "p",
"text": "ååæ žã¯ãéœåãšäžæ§åããã§ããŠããã éœåã¯æ£é»è·ããã¡ãäžæ§åã¯é»è·ããããªãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 177,
"tag": "p",
"text": "ã§ã¯ããªããã©ã¹ã®é»è·ããã€éœåã©ãããããªãã¯ãŒãã³åã§åçºããŠããŸããªãã®ã ããã?",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 178,
"tag": "p",
"text": "ãã®çç±ãšããŠãã€ãŸãéœåã©ãããã¯ãŒãã³åã§åçºããªãããã®çç±ãšããŠã次ã®ãããªçç±ãèããããŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 179,
"tag": "p",
"text": "ãŸããéœåã«äžæ§åãè¿ã¥ããŠæ··åãããšããæ žåããšããéåžžã«åŒ·ãçµååãçºçãã ãã®æ žåãéœåå士ã®ã¯ãŒãã³åã«ãã匷ãæ¥åã«æã¡åã€ã®ã§ãéœåãšäžæ§åã¯çµåããŠãããšèããããŠããã(å¿
ããããéœåãšäžæ§åã®åæ°ã¯åäžã§ãªããŠããããå®éã«ãåšæè¡šã«ããããã€ãã®å
çŽ ã§ããéœåãšäžæ§åã®åæ°ã¯ç°ãªãã)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 180,
"tag": "p",
"text": "æ¯å©çã«èšãæãã°ãäžæ§åã¯ãéœåãšéœåãçµã³ã€ãããããªã®ãããªåœ¹å²ãããŠãããšãèããããŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 181,
"tag": "p",
"text": "ãªããå称ãšããŠãéœåãšäžæ§åããŸãšããŠãæ žåããšåŒã°ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 182,
"tag": "p",
"text": "ããå
çŽ ã®ååæ žã®éœåã®æ°ã¯ãåšæè¡šã®ååçªå·ãšäžèŽããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 183,
"tag": "p",
"text": "ãŸããéœåãšäžæ§åã®æ°ã®åã¯è³ªéæ°ãšãã°ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 184,
"tag": "p",
"text": "質éæ°Aã®ååæ žã¯éåžžã«åŒ·ãæ žåã®ããã«ãå°ããªçäœç¶ã®ç©ºéã®äžã«åºãŸã£ãŠããããã®ååŸrã¯ã 1.2 {\\displaystyle 1.2} ~ 1.4 à 10 â 15 à A 1 3 {\\displaystyle 1.4\\times 10^{-15}\\times A^{\\frac {1}{3}}} ã§ããããšãç¥ãããŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 185,
"tag": "p",
"text": "ä»»æã®ååæ žã¯ããããæ§æããæ žåã§ããéœåãšäžæ§åãèªç±ã§ãããšãã®è³ªé(åäœè³ªéãšãã)ã®åãããå°ãã質éããã€ããã®æžã£ã質éãã質éæ¬ æãšåŒã¶ã 質éæ°Aãååçªå·Zã®ååæ žã®è³ªéæ¬ æ Î m {\\displaystyle \\Delta m} ããåŒã§æžãã°, ååæ žã®è³ªéãmãéœåãšäžæ§åã®åäœè³ªéããããã m p , m n {\\displaystyle m_{p},\\ m_{n}} ãšãããšãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 186,
"tag": "p",
"text": "",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 187,
"tag": "p",
"text": "",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 188,
"tag": "p",
"text": "",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 189,
"tag": "p",
"text": "枬å®å®éšã®äºå®ãšããŠãéœååç¬ãäžæ§ååç¬ã®è³ªéã®åæ°ãåãããããããã®çµåããååæ žã®ã»ãã質éãäœãã®ã§ãéœåãäžæ§åãçµåãããšè³ªéã®äžéšãæ¬ æãããšããã®ãã枬å®çµæã®äºå®ã§ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 190,
"tag": "p",
"text": "ãªã®ã§ã質éæ¬ æã®ãšããããã®åå ãšããŠèããããŠããã®ã¯ãéœåãäžæ§åã©ããã®çµåã§ãããšèããããŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 191,
"tag": "p",
"text": "ã ããã§ã¯ããªãéœåãäžæ§åãååæ žãšããŠçµåãããšè³ªéãæ¬ æãããã®çç±ãšããŠã¯ããã£ããŠãçµåã ããããšããçç±ã§ã¯èª¬æãã€ããªãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 192,
"tag": "p",
"text": "ãªã®ã§ãç©çåŠè
ãã¡ã¯ã質éæ¬ æã®èµ·ããæ ¹æ¬çãªåå ãšãªãç©çæ³åããŠãã¢ã€ã³ã·ã¥ã¿ã€ã³ã®çžå¯Ÿæ§çè«ãé©çšããŠããã(æ€å®æç§æžã§ããçžå¯Ÿæ§çè«ã®çµæã§ãããšããŠèª¬æããç«å Ž)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 193,
"tag": "p",
"text": "(ã¢ã€ã³ã·ã¥ã¿ã€ã³ã®ç¹æ®)çžå¯Ÿæ§çè«ããå°ãããçµæãšããŠ(â» åè: çžå¯Ÿè«ã«ã¯äžè¬çžå¯Ÿè«ãšç¹æ®çžå¯Ÿè«ã®2çš®é¡ããã)ã質émãšãšãã«ã®ãŒEã«ã¯ã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 194,
"tag": "p",
"text": "ãšããé¢ä¿åŒããããšãããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 195,
"tag": "p",
"text": "ãªããC ãšã¯å
éã®å€ã§ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 196,
"tag": "p",
"text": "ãããã¯å¥ã®æžåŒãšããŠãå€åãè¡šããã«ã¿èšå·Îã䜿ãŠã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 197,
"tag": "p",
"text": "ãªã©ãšæžãå Žåãããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 198,
"tag": "p",
"text": "ã€ãŸããããäœããã®çç±ã§ãç空ãã質éãçºçãŸãã¯æ¶å€±ããã°ããã®ã¶ãã®è«å€§ãªãšãã«ã®ãŒãçºçãããšããã®ããçžå¯Ÿæ§çè«ã§ã®ã¢ã€ã³ã·ã¥ã¿ã€ã³ãªã©ã®äž»åŒµã§ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 199,
"tag": "p",
"text": "ããŠãèªç±ãªéœåãšäžæ§åã¯ãæ žåã«ããçµåãããšãããã®çµåãšãã«ã®ãŒã«çžåœããw:ã¬ã³ãç·ãæŸå°ããããšãç¥ãããŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 200,
"tag": "p",
"text": "ãããŠãã¬ã³ãç·ã«ããšãã«ã®ãŒãããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 201,
"tag": "p",
"text": "ãªã®ã§ãéœåãšäžæ§åã®çµåãããšãã®ã¬ã³ãç·ã®ãšãã«ã®ãŒã¯ã質éæ¬ æã«ãã£ãŠçãããšèãããšã枬å®çµæãšããžãããåãã(枬å®çµæã¯ããããŸã§è³ªéãæ¬ æããããšãŸã§ã)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 202,
"tag": "p",
"text": "æ žåã®çµåã«ãããŠã質éæ¬ æ Î m {\\displaystyle \\Delta m} ããã¬ã³ãç·ãªã©ã®ãšãã«ã®ãŒã«è»¢åããããšç©çåŠè
ãã¡ã¯èããŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 203,
"tag": "p",
"text": "å
çŽ ã®äžã«ã¯ãæŸå°ç·(radiation)ãåºãæ§è³ªããã€ãã®ãããããã®æ§è³ªãæŸå°èœ(radioactivity)ãšããã ãŸããæŸå°èœããã€ç©è³ªã¯æŸå°æ§ç©è³ªãšããããã æŸå°ç·ã«ã¯3çš®é¡ååšããããããαç·ãβç·ãγç·ãšããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 204,
"tag": "p",
"text": "α厩å£ã¯ã芪ååæ žããããªãŠã ååæ žãæŸå°ãããçŸè±¡ã§ããã ãã®ããªãŠã ååæ žã¯Î±ç²åãšãã°ããã α厩å£åŸã芪ååæ žã®è³ªéæ°ã¯4å°ãããªããååçªå·ã¯2å°ãããªãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 205,
"tag": "p",
"text": "β厩å£ã¯ã芪ååæ žã®äžæ§åãéœåãšé»åã«å€åããããšã§ãé»åãæŸå°ãããçŸè±¡ã§ããã (åè: ãã®ãšããåãã¥ãŒããªããšãã°ãã埮å°ãªç²åãåæã«æŸåºããããšãè¿å¹Žã®åŠèª¬ã§ã¯èããããŠããã)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 206,
"tag": "p",
"text": "ãªãããã®é»å(ããŒã¿åŽ©å£ãšããŠæŸåºãããé»åã®ããš)ã¯ãβç²åããšããã°ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 207,
"tag": "p",
"text": "β厩å£åŸã芪ååæ žã®è³ªéæ°ã¯å€åããªãããååçªå·ã¯1å¢å ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 208,
"tag": "p",
"text": "γç·ã¯ãα厩å£ãŸãã¯Î²åŽ©å£çŽåŸã®é«ãšãã«ã®ãŒã®ååæ žããäœãšãã«ã®ãŒã®å®å®ãªç¶æ
ã«é·ç§»ãããšãã«æŸå°ãããã γç·ã®æ£äœã¯å
åã§ãXç·ããæ³¢é·ã®çãé»ç£æ³¢ã§ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 209,
"tag": "p",
"text": "α厩å£ãβ厩å£ã«ãã£ãŠããšã®ååæ žã®æ°ã¯åŸã
ã«æžã£ãŠãããããããã®åŽ©å£ã¯ååæ žã®çš®é¡ããšã«æ±ºãŸã£ãäžå®ã®ç¢ºçã§èµ·ããã®ã§ã厩å£ã«ãã£ãŠããšã®ååæ žã®æ°ãæžãé床ã¯ååæ žã®åæ°ã«æ¯äŸããŠå€åããããããã厩å£ã«ãã£ãŠããšã®ååæ žã®æ°ãåæžããã®ã«ãããæéã¯ãååæ žã®çš®é¡ã ãã«ãã£ãŠããŸããããã§ããã®æéã®ããšããã®ååæ žã® åæžæ(ã¯ãããããhalf life ) ãšåŒã¶ã厩å£ã«ãã£ãŠååæ žã®åæ°ãã©ãã ãã«ãªããã¯ããã®åæžæãçšããŠèšè¿°ããããšãã§ãããååæ žã®åæžæãTãæå»tã§ã®ååæ žã®åæ°ãN(t)ãšãããšã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 210,
"tag": "p",
"text": "ãæãç«ã€ã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 211,
"tag": "p",
"text": "ååæ žã®åŽ©å£é床ã¯ãååæ žã®åæ°ã«æ¯äŸãããšè¿°ã¹ããå®ã¯ãäžã«è¿°ã¹ãå
¬åŒã¯ãã®æ
å ±ã ãããçŽç²ã«æ°åŠçã«å°ãåºãããšãã§ãããã®ã§ãããé«çåŠæ ¡ã§ã¯æ±ããªãæ°åŠãçšããããèå³ã®ããèªè
ã®ããã«ãã®æŠèŠãèšããŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 212,
"tag": "p",
"text": "ååæ žã®åæ°ãšåŽ©å£é床ã®éã®æ¯äŸå®æ°ã¯ååæ žã®çš®é¡ã«ãã£ãŠæ±ºãŸãããã®å®æ°ããã®ååæ žã®åŽ©å£å®æ°ãšããã厩å£å®æ°ãλã®ååæ žã®æå»tã§ã®åæ°ãN(t)ãšãããšããã®å€åé床ãããªãã¡N(t)ã®åŸ®åã¯ã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 213,
"tag": "p",
"text": "ã§è¡šãããããã®ãããªãããé¢æ°ãšãã®åŸ®åãšã®é¢ä¿ãè¡šããåŒã埮åæ¹çšåŒãšããã埮åæ¹çšåŒãæºãããããªé¢æ°ãæ±ããããšãã埮åæ¹çšåŒã解ããšããã(詳ãã解æ³ã¯è§£æåŠåºç€/垞埮åæ¹çšåŒã§èª¬æãããã)ãã®åŸ®åæ¹çšåŒã解ããš",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 214,
"tag": "p",
"text": "ãåŸãããã(ãã®åŒã確ãã«å
ã»ã©ã®åŸ®åæ¹çšåŒãæºãããŠããããšã確ãããŠã¿ã)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 215,
"tag": "p",
"text": "åæžæTãšã¯ã N ( t ) = 1 2 N ( 0 ) {\\displaystyle N(t)={\\frac {1}{2}}N(0)} ãšãªãtã®ããšãªã®ã§ãå
ã»ã©ã®åŒãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 216,
"tag": "p",
"text": "ãåŸãããããã£ãŠã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 217,
"tag": "p",
"text": "ãåŸãããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 218,
"tag": "p",
"text": "ã©ã¶ãã©ãŒãã¯ãçªçŽ ã¬ã¹ãå¯éããç®±ã«Î±ç·æºããããšãæ£é»è·ããã£ãç²åãçºçããããšãçºèŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 219,
"tag": "p",
"text": "ãã®æ£é»è·ã®ç²åããéœåã§ãããã€ãŸããã©ã¶ãã©ãŒãã¯éœåãçºèŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 220,
"tag": "p",
"text": "åæã«ãé
žçŽ ãçºçããããšãçºèŠãããã®çç±ã¯çªçŽ ãé
žçŽ ã«å€æãããããã§ãããã€ãŸããååæ žãå€ããåå¿ãçºèŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 221,
"tag": "p",
"text": "ãããã®ããšãåŒã«ãŸãšãããšã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 222,
"tag": "p",
"text": "ã§ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 223,
"tag": "p",
"text": "ãã®ããã«ãããå
çŽ ã®ååããå¥ã®å
çŽ ã®ååã«å€ããåå¿ã®ããšã ååæ žåå¿ ãšããããŸãã¯ããæ žåå¿ããšããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 224,
"tag": "p",
"text": "",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 225,
"tag": "p",
"text": "",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 226,
"tag": "p",
"text": "ãŸããå®å®ç·ã®èŠ³æž¬ã«ãããÎŒç²åãšããã®ããçºèŠãããŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 227,
"tag": "p",
"text": "ãããããã©ããã£ãŠçŽ ç²åã芳枬ããããšãããšãããã€ãã®æ¹æ³ããããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 228,
"tag": "p",
"text": "ãªã©ã䜿ãããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 229,
"tag": "p",
"text": "(â» é«æ ¡ã§ç¿ãç¯å²å
ãXç·ãååæ žã®åå
ã§ãé§ç®±(ããã°ã)ãç¿ãã)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 230,
"tag": "p",
"text": "é§ç®±(ããã°ã)ãšãããèžæ°ã®ã€ãŸã£ãè£
眮ãã€ãããšããªããã®ç²åãééãããšããã®ç²åã®è»è·¡ã§ãæ°äœãã液äœããåçãèµ·ããã®ã§ãè»è·¡ããç®ã«èŠããã®ã§ããã(â» æ€å®æç§æžã§ã¯ãååæ žã®åéã§ãé§ç®±ã«ã€ããŠç¿ãã)(ã€ã¡ãŒãžçã«ã¯ãé£è¡æ©é²ã®ãããªã®ããã€ã¡ãŒãžããŠãã ããã)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 231,
"tag": "p",
"text": "ã§ãç£å Žãå ããå Žåã®ãè»è·¡ã®æ²ããããçãªã©ãããæ¯é»è·ãŸã§ãäºæ³ã§ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 232,
"tag": "p",
"text": "ãã®ããã«ãé§ç®±ãã€ãã£ãå®éšã«ããã20äžçŽååãäžç€ããã«ã¯ããããããªç²åãçºèŠãããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 233,
"tag": "p",
"text": "ÎŒç²å以å€ã«ããéœé»å(ããã§ãã)ããé§ç®±ã«ãã£ãŠçºèŠãããŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 234,
"tag": "p",
"text": "(â» ç¯å²å€:)äžçåã§éœé»åãå®éšçã«èŠ³æž¬ããã¢ã³ããŒãœã³ã¯ãé§ç®±ã«éæ¿ãå
¥ããããšã§éœé»åãçºèŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 235,
"tag": "p",
"text": "ãšãããã(ÎŒç²åã®çºèŠããã)éœé»åã®ã»ããçºèŠã¯æ©ãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 236,
"tag": "p",
"text": "(â» ç¯å²å€:)ãŸããéœé»åã¯ãéåååŠã®ã·ã¥ã¬ãŒãã£ã³ã¬ãŒæ¹çšåŒã«ãç¹æ®çžå¯Ÿæ§çè«ãšãçµã¿åããããããã£ã©ãã¯ã®æ¹çšåŒããããçè«çã«äºæ³ãããŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 237,
"tag": "p",
"text": "ãŸãããéœé»åããšããç©è³ªã1932幎ã«éæ¿ãå
¥ããé§ç®±(ããã°ã)ã®å®éšã§ã¢ã³ããŒãœã³(人å)ã«ãã£ãŠçºèŠãããŠãããéœé»åã¯è³ªéãé»åãšåãã ããé»è·ãé»åã®å察ã§ãã(ã€ãŸãéœé»åã®é»è·ã¯ãã©ã¹eã¯ãŒãã³ã§ãã)ã(â» éæ¿ã«ã€ããŠã¯é«æ ¡ã®ç¯å²å€ã)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 238,
"tag": "p",
"text": "ãããŠãé»åãšéœé»åãè¡çªãããšã2mcã®ãšãã«ã®ãŒãæŸåºããŠãæ¶æ»
ããã(ãã®çŸè±¡(é»åãšéœé»åãè¡çªãããš2mcã®ãšãã«ã®ãŒãæŸåºããŠæ¶æ»
ããçŸè±¡)ã®ããšããã察æ¶æ»
ã(ã€ãããããã€)ãšããã)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 239,
"tag": "p",
"text": "éœåã«å¯ŸããŠãããåéœåãããããåéœåã¯ãé»è·ãéœåãšå察ã ãã質éãéœåãšåãã§ãããéœåãšè¡çªãããšå¯Ÿæ¶æ»
ãããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 240,
"tag": "p",
"text": "äžæ§åã«å¯ŸããŠãããåäžæ§åãããããåäžæ§åã¯ãé»è·ã¯ãŒãã ã(ãŒãã®é»è·ã®Â±ãå察ã«ããŠããŒãã®ãŸãŸ)ã質éãåãã§ãäžæ§åãšå¯Ÿæ¶æ»
ãããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 241,
"tag": "p",
"text": "éœé»åãåéœåãåäžæ§åã®ãããªç©è³ªããŸãšããŠãåç©è³ªãšããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 242,
"tag": "p",
"text": "(â» ç¯å²å€: )æŸå°æ§åäœäœã®ãªãã«ã¯ã厩å£ã®ãšãã«éœé»åãæŸåºãããã®ããããæå
端ã®ç
é¢ã§äœ¿ãããPET(éœé»åæå±€æ®åæ³)æè¡ã¯ããããå¿çšãããã®ã§ãããããçŽ ããµãããã«ãªãããªãã·ã°ã«ã³ãŒã¹ãšããç©è³ªã¯ã¬ã³çŽ°èã«ããåã蟌ãŸãããPET蚺æã§ã¯ãããã«(ãã«ãªãããªãã·ã°ã«ã³ãŒã¹ã«)æŸå°æ§ã®ããçŽ F ããšãããã æŸå°æ§ãã«ãªãããªãã·ã°ã«ã³ãŒã¹ãçšããŠããã(â» åæ通ã®ãååŠåºç€ãã®æç§æžã«ãçºå±äºé
ãšããŠãã«ãªãããªãã·ã°ã«ã³ãŒã¹ãPET蚺æã§äœ¿ãããŠãããšã玹ä»ãããŠããã)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 243,
"tag": "p",
"text": "åç©è³ªãšã¯å¥ã«ãÎŒç²åããå®å®ç·ã®èŠ³æž¬ããã1937幎ã«èŠã€ãã£ãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 244,
"tag": "p",
"text": "ãã®ÎŒç²åã¯ãé»è·ã¯ãé»åãšåãã ãã質éãé»åãšã¯éããÎŒç²åã®è³ªéã¯ããªããšé»åã®çŽ200åã®è³ªéã§ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 245,
"tag": "p",
"text": "ÎŒç²åã¯ãã¹ã€ã«éœåãé»åã®åç©è³ªã§ã¯ãªãã®ã§ãã¹ã€ã«éœåãšã察æ¶æ»
ãèµ·ãããªãããé»åãšã察æ¶æ»
ãèµ·ãããªãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 246,
"tag": "p",
"text": "ãªããÎŒç²åã«ããåÎŒç²åãšãããåç©è³ªãååšããããšãåãã£ãŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 247,
"tag": "p",
"text": "ãã®ãããªç©è³ªããããããã®äœãã§ããå°äžã§èŠã€ãããªãã®ã¯ãåã«å°äžã®å€§æ°ãªã©ãšè¡çªããŠæ¶æ»
ããŠããŸãããã§ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 248,
"tag": "p",
"text": "ãªã®ã§ãé«å±±ã®é äžä»è¿ãªã©ã§èŠ³æž¬å®éšããããšãÎŒç²åã®çºèŠã®å¯èœæ§ãé«ãŸãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 249,
"tag": "p",
"text": "ãªã21äžçŽã®çŸåšãÎŒç²åã掻çšããæè¡ãšããŠãçŸåšãç«å±±ãªã©ã®å
éšã芳å¯ããã®ã«ã掻çšãããŠãããÎŒç²åã¯ãééåãé«ãããå°äžã®ç©è³ªãšåå¿ããŠããããã«æ¶æ»
ããŠããŸãã®ã§ããã®ãããªæ§è³ªãå©çšããŠãç«å±±å
éšã®ããã«äººéãå
¥ã蟌ããªãå Žæã芳å¯ãããšããæè¡ãããã§ã«ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 250,
"tag": "p",
"text": "",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 251,
"tag": "p",
"text": "ãã®ãããªèŠ³æž¬ã«äœ¿ãããÎŒç²åãã©ããã£ãŠçºçãããã®ã?",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 252,
"tag": "p",
"text": "å®å®ç·ããé£ãã§ããÎŒç²åããã®ãŸãŸäœ¿ããšããæ¹æ³ããããããå®è¡ããŠããç 究è
ããããããããšã¯å¥ã®ææ³ãšããŠãå éåšãªã©ã§äººå·¥çã«ÎŒç²åãªã©ãçºçããããšããæ¹æ³ãããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 253,
"tag": "p",
"text": "å éåšã䜿ã£ãæ¹æ³ã¯ãäžèšã®éãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 254,
"tag": "p",
"text": "ãŸããã·ã¯ãããã³ããµã€ã¯ãããã³ã䜿ã£ãŠãé»åãªã©ãè¶
é«éã«å éããããããäžè¬ã®ç©è³ª(ã°ã©ãã¡ã€ããªã©)ã«åœãŠãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 255,
"tag": "p",
"text": "ãããšãåœç¶ãããããªç²åãçºçããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 256,
"tag": "p",
"text": "ãã®ãã¡ãÏäžéåããç£æ°ã«åå¿ãã(ãšèããããŠãã)ã®ã§ã倧ããªé»ç£ç³ã³ã€ã«ã§ãÏäžéåãæç²ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 257,
"tag": "p",
"text": "ãã®Ïäžéåã厩å£ããŠãÎŒç²åãçºçããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 258,
"tag": "p",
"text": "ããããå®å®ç·ãäœã«ãã£ãŠçºçããŠãããã®çºçåå ã¯ãçŸæç¹ã®äººé¡ã«ã¯äžæã§ããã(â» åèæç®: æ°ç åºçã®è³æéã®ãå³èª¬ç©çã)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 259,
"tag": "p",
"text": "è¶
æ°æ(ã¡ãããããã)ççºã«ãã£ãŠå®å®ç·ãçºçããã®ã§ã¯ããšãã説ããããããšã«ããå®å®ç·ã®çºçåå ã«ã€ããŠã¯æªè§£æã§ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 260,
"tag": "p",
"text": "é»åãéœåãäžæ§åãªã©ã¯ããã¹ãã³ããšããç£ç³ã®ãããªæ§è³ªããã£ãŠãããç£ç³ã«N極ãšS極ãããããã«ãã¹ãã³ã«ãã2çš®é¡ã®åãããããã¹ãã³ã®ãã®2çš®é¡ã®åãã¯ããäžåãããšãäžåããã«ãããäŸãããããç£ç³ã®ç£åã®çºçåå ã¯ãç£ç³äžã®ååã®æå€æ®»é»åã®ã¹ãã³ã®åããåäžæ¹åã«ããã£ãŠãããããã§ãããšèããããŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 261,
"tag": "p",
"text": "å
šååã¯ãé»åãéœåãäžæ§åãå«ãã®ã«ããªã®ã«å€ãã®ç©è³ªããããŸãç£æ§ãçºçããªãã®ã¯ãå察笊å·ã®ã¹ãã³ããã€é»åãçµåãããããšã§ãæã¡æ¶ãããããã§ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 262,
"tag": "p",
"text": "(ãŠã£ãããé»åãšéœåã®ãããªé»è·ããã€ç²åã«ããã¹ãã³ããªããšèª€è§£ããŠãã人ãããããäžæ§åã«ãã¹ãã³ã¯ããã)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 263,
"tag": "p",
"text": "äžåŠé«æ ¡ã§èŠ³æž¬ãããããªæ®éã®æ¹æ³ã§ã¯ãã¹ãã³ã芳枬ã§ããªãããååãªã©ã®ç©è³ªã«ç£æ°ãå ãã€ã€é«åšæ³¢ãå ãããªã©ãããšãã¹ãã³ã®åœ±é¿ã«ãã£ãŠããã®ååã®æ¯åããããåšæ³¢æ°ãéããªã©ã®çŸè±¡ããã¡ããŠãéæ¥çã«(é»åãªã©ã®)ã¹ãã³ã芳枬ã§ããã(ãªããæ žç£æ°å
±é³Žæ³(NMRãnuclear magnetic resonance)ã®åçã§ããã â» çè«çãªè§£æã¯ã倧åŠã¬ãã«ã®ååŠã®ç¥èãå¿
èŠã«ãªãã®ã§çç¥ããã) ååäžã®æ°ŽçŽ ååããããçš®ã®æŸå°æ§åäœäœ(äžæ§åããã£ã1åãµããã ãã®åäœäœ)ãªã©ãé«åšæ³¢ã®åœ±é¿ãåããããããã®çç±ã®ã²ãšã€ããã¹ãã³ã«ãããã®ã ãšèããããŠãã(â» ãªããå»çã§çšããããMRI(magnetic resonance imaging)ã¯ããã®æ žç£æ°å
±é³Žæ³(NMR)ãå©çšããŠäººäœå
éšãªã©ã芳枬ããããšããæ©åšã§ããã)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 264,
"tag": "p",
"text": "ããŠãå®ã¯çŽ ç²åããã¹ãã³ããã€ã®ãæ®éã§ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 265,
"tag": "p",
"text": "ÎŒç²åã¯ã¹ãã³ããã€ã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 266,
"tag": "p",
"text": "ÎŒç²åã®ãã¹ãã³ããšããæ§è³ªã«ããç£æ°ãšãÎŒç²åã®ééæ§ã®é«ããå©çšããŠãç©è³ªå
éšã®ç£å Žã®èŠ³æž¬æ¹æ³ãšããŠæ¢ã«ç 究ãããŠããããã®ãããªèŠ³æž¬æè¡ããÎŒãªã³ã¹ãã³å転ããšãããè¶
äŒå°äœã®å
éšã®èŠ³æž¬ãªã©ã«ãããã§ã«ãÎŒãªã³ã¹ãã³å転ãã«ãã芳枬ãç 究ãããŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 267,
"tag": "p",
"text": "ãŠã£ãããã£ã¢èšäºãw:ãã¥ãªã³ã¹ãã³å転ãã«ãããšãÎŒãªã³ã®åŽ©å£æã«éœé»åãæŸåºããã®ã§ãéœé»åã®èŠ³æž¬æè¡ãå¿
èŠã§ããã(é«æ ¡ã®ç¯å²å€ã§ãããã)ããããã®åŠçã¯ããããããšå匷ããäºãå€ãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 268,
"tag": "p",
"text": "éœåãšäžæ§åã¯ã質éã¯ã»ãšãã©åãã§ãããé»è·ãéãã ãã§ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 269,
"tag": "p",
"text": "ãããŠãé»åãšæ¯ã¹ããšãéœåãäžæ§åãã質éãããªã倧ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 270,
"tag": "p",
"text": "ãã®äºããããéœåãäžæ§åã«ããããã«äžèº«ããããå¥ã®ç²åãè©°ãŸã£ãŠããã®ã§ã¯?ããšããçåãçãŸããŠããŠãéœåãäžæ§åã®å
éšã®æ¢çŽ¢ãå§ãŸã£ãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 271,
"tag": "p",
"text": "ããããçŸåšã§ããéœåãäžæ§åã®å
éšã®æ§é ã¯ãå®éšçã«ã¯åãåºããŠã¯ããªãã(â» éœåãäžæ§åã®å
éšæ§é ãšããŠèª¬æãããŠãããã¯ã©ãŒã¯ãã¯ãåç¬ã§ã¯çºèŠãããŠããªããã¯ã©ãŒã¯ã¯åã«ãå
éšã®èª¬æã®ããã®ãæŠå¿µã§ããã)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 272,
"tag": "p",
"text": "æŽå²çã«ã¯ããŸããéœåãšäžæ§åã®å
éšæ§é ãšããŠãæ¶ç©ºã®çŽ ç²åãèããããéœåãšäžæ§åã¯ããããã®çŽ ç²åã®ç¶æ
ãéãã ãããšèããããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 273,
"tag": "p",
"text": "ãã£ãœããé»åã«ã¯ãå
éšæ§é ããªãããšèãããŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 274,
"tag": "p",
"text": "ããã20äžçŽãªãã°ãéåååŠã§ã¯ããã®ããããã§ã«ãé»åã®ç¶æ
ãšããŠãã¹ãã³ããšããæŠå¿µããã¿ã€ãã£ãŠãããéåååŠã§ã¯ãååŠçµåã§äŸ¡é»åã2åãŸã§çµåããŠé»å察ã«ãªãçç±ã¯ããã®ã¹ãã³ã2çš®é¡ãããªããŠãå察åãã®ã¹ãã³ã®é»å2åã ããçµåããããã§ããããšãããŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 275,
"tag": "p",
"text": "ã¹ãã³ã®2çš®é¡ã®ç¶æ
ã¯ããäžåãããäžåãããšãããµãã«ãããäŸããããã(å®éã®æ¹åã§ã¯ãªãã®ã§ãããŸãæ·±å
¥ãããªãããã«ã)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 276,
"tag": "p",
"text": "ãã®ãããªéåååŠãåèã«ããŠãéœåãšäžæ§åã§ããã¢ã€ãœã¹ãã³ããšããæŠå¿µãèããããã(â» ãã¢ã€ãœã¹ãã³ãã¯é«æ ¡ç¯å²å€ã)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 277,
"tag": "p",
"text": "éœåãšäžæ§åã¯ãã¢ã€ãœã¹ãã³ã®ç¶æ
ãéãã ãããšèããããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 278,
"tag": "p",
"text": "ãã®åŸã20äžçŽåã°é ããããã¢ã€ãœã¹ãã³ããçºå±ããããã¯ã©ãŒã¯ããšããçè«ãæå±ãããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 279,
"tag": "p",
"text": "æ¶ç©ºã®ãã¯ã©ãŒã¯ããšãã3åã®çŽ ç²åãä»®å®ãããšãå®åšã®éœåãäžæ§åã®æãç«ã€ã¢ãã«ããå®éšçµæãããŸã説æã§ããäºãåãã£ãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 280,
"tag": "p",
"text": "é»è·( + 2 3 e {\\displaystyle +{\\frac {2}{3}}e} )ããã€çŽ ç²åãã¢ããã¯ã©ãŒã¯ããšã±( â 1 3 e {\\displaystyle -{\\frac {1}{3}}e} )ããã€çŽ ç²åãããŠã³ã¯ã©ãŒã¯ãããã£ãŠã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 281,
"tag": "p",
"text": "ãšèãããšããããããªçŽ ç²åå®éšã®çµæãããŸã説æã§ããäºãåãã£ãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 282,
"tag": "p",
"text": "ãªããé»åã«ã¯ããã®ãããªå
éšæ§é ã¯ãªãããšèãããããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 283,
"tag": "p",
"text": "ã¢ããã¯ã©ãŒã¯ã¯ãuããšç¥èšãããããŠã³ã¯ã©ãŒã¯ã¯ãdããšç¥èšãããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 284,
"tag": "p",
"text": "éœåã®ã¯ã©ãŒã¯æ§é ã¯uudãšç¥èšããã(ã¢ãããã¢ãããããŠã³)ã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 285,
"tag": "p",
"text": "äžæ§åã®ã¯ã©ãŒã¯æ§é ã¯uddãšç¥èšããã(ã¢ãããããŠã³ãããŠã³)ã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 286,
"tag": "p",
"text": "ãªããäžèšã®èª¬æã§ã¯çç¥ãããããããã1950ã60幎代ãããŸã§ã«ãé«å±±ã§ã®å®å®ç·ã®èŠ³æž¬ãããããã¯æŸå°ç·ã®èŠ³æž¬ãããŸããããã¯ãµã€ã¯ãããã³ãªã©ã«ããç²åã®å éåšè¡çªå®éšã«ãããéœåãäžæ§åã®ã»ãã«ããåçšåºŠã®è³ªéã®ããŸããŸãªç²åãçºèŠãããŠããããããæ°çš®ã®ç²åã¯ãäžéåãã«åé¡ãããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 287,
"tag": "p",
"text": "ããããããã¯ã©ãŒã¯ãã®çè«ã¯ããã®ãããª20äžçŽåã°ãããŸã§ã®å®éšã芳枬ããäœçŸåãã®æ°çš®ã®ç²åãçºèŠãããŠããŸãããã®ãããªçµç·¯ããã£ãã®ã§ãã¯ã©ãŒã¯ã®çè«ãæå±ãããã®ã§ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 288,
"tag": "p",
"text": "ããŠããäžéåã(ã¡ã
ãããããmason ã¡ãœã³)ãšã¯ãããšããšçè«ç©çåŠè
ã®æ¹¯å·ç§æš¹ã1930幎代ã«æå±ãããéœåãšäžæ§åãšãåŒãä»ããŠãããšãããæ¶ç©ºã®ç²åã§ãã£ããã20äžçŽãªãã°ã«æ°çš®ã®ç²åãçºèŠãããéããäžéåãã®ååã䜿ãããããšã«ãªã£ãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 289,
"tag": "p",
"text": "ããŠãå®éšçã«æ¯èŒçæ©ãææããçºèŠããããäžéåãã§ã¯ããÏäžéåãããããããçš®é¡ã®Ïäžéåã¯ãã¢ããã¯ã©ãŒã¯ãšåããŠã³ã¯ã©ãŒã¯ãããªããÏãšç¥èšãããã(ããŠã³ã¯ã©ãŒã¯ã®åç©è³ªããåããŠã³ã¯ã©ãŒã¯ã) Ï= u d Ì {\\displaystyle u{\\overline {d}}}",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 290,
"tag": "p",
"text": "å¥ã®ããçš®é¡ã®Ïäžéåã¯ãããŠã³ã¯ã©ãŒã¯ãšåã¢ããã¯ã©ãŒã¯ãããªããÏãšç¥èšããããÏ= u Ì d {\\displaystyle {\\overline {u}}d}",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 291,
"tag": "p",
"text": "ãã®ããã«ãããç²åå
ã®ã¯ã©ãŒã¯ã¯åèš2åã®ã§ãã£ãŠãè¯ãå Žåãããã(ããªãããããéœåã®ããã«ã¯ã©ãŒã¯3åã§ãªããŠãããŸããªãå Žåãããã)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 292,
"tag": "p",
"text": "(â» ãã®ãããªå®éšäŸãããç²åå
ã«åèš5åã®ã¯ã©ãŒã¯ã7åã®ã¯ã©ãŒã¯ãèããçè«ããããããããé«æ ¡ç©çã®ç¯å²ã倧å¹
ã«è¶
ããã®ã§ã説æãçç¥ã)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 293,
"tag": "p",
"text": "ãŸããäžéåã¯ãèªç¶çã§ã¯çæéã®ããã ã ããååšã§ããç²åã ãšããäºãã芳枬å®éšã«ãã£ãŠãåãã£ãŠããã(äžéåã®ååšã§ããæé(ã寿åœã)ã¯çããããã«ãä»ã®å®å®ãªç²åã«å€æããŠããŸãã)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 294,
"tag": "p",
"text": "ããããã¢ãããšããŠã³ã ãã§ã¯ã説æããããªãç²åããã©ãã©ããšçºèŠãããŠãããã¯ã©ãŒã¯ã®æå±æã®åœåã¯ãããããã ãã¯ã©ãŒã¯ã®ã¢ãããšããŠã³ã§ããã£ãšãã»ãšãã©ã®äžéåã®æ§é ã説æã§ããã ããã ãšæåŸ
ãããŠããã®ã ããããããããå®å®ç·ãã1940幎代ã«çºèŠããããKäžéåãã®æ§é ã§ãããã¢ãããšããŠã³ã§ã¯èª¬æã§ããªãã£ãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 295,
"tag": "p",
"text": "ãã®ã»ããå éåšã®çºéãªã©ã«ãããã¢ãããšããŠã³ã®çµã¿åããã ãã§èª¬æã§ããæ°ãè¶
ããŠãã©ãã©ããšæ°çš®ã®ãäžéåããçºèŠãããŠããŸãããã¯ãã¢ãããšããŠã³ã ãã§ã¯ãäžéåã®æ§é ã説æãã¥ãããªã£ãŠããäžãÎŒç²åãã説æã§ããªãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 296,
"tag": "p",
"text": "ãŸããå éåšå®éšã«ããã1970幎代ã«ãDäžéåããªã©ãããŸããŸãªäžéåããå®éšçã«å®åšã確èªãããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 297,
"tag": "p",
"text": "ãã®ããã«ãã¢ãããšããŠã³ã ãã§ã¯èª¬æã®ã§ããªãããããããªç²åãååšããããšãåããããã®ãããçŽ ç²åçè«ã§ã¯ããã¢ããã(u)ãšãããŠã³ã(d)ãšãã2çš®é¡ã®ç¶æ
ã®ä»ã«ããããã«ç¶æ
ãèããå¿
èŠã«ãããŸãããããããŠãæ°ããç¶æ
ãšããŠããŸãããã£ãŒã ã(èšå·c)ãšãã¹ãã¬ã³ãžã(èšå·s)ãèãããããå éåšå®éšã®æè¡ãçºå±ããå éåšå®éšã®è¡çªã®ãšãã«ã®ãŒãäžãã£ãŠãããšãããã«ããããã(èšå·t)ãšãããã ã(èšå·b)ãšããã®ãèããããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 298,
"tag": "p",
"text": "ãªããÎŒç²åã«ã¯å
éšæ§é ã¯ãªãããéœåãäžæ§åã«é»åã察å¿ãããã®ãšåæ§ã«(第1äžä»£)ããã£ãŒã ãã¹ãã¬ã³ãžãããªãéœåçã»äžæ§åçãªç²åãšÎŒç²åã察å¿ããã(第2äžä»£)ãåæ§ã«ãããããããã ãããªãç²åã«ÎŒç²åã察å¿ããã(第3äžä»£)ã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 299,
"tag": "p",
"text": "é»åãÎŒç²åã¯å
éšæ§é ããããªããšèããããŠããããã¬ããã³ããšãããå
éšæ§é ããããªããšãããã°ã«ãŒãã«åé¡ãããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 300,
"tag": "p",
"text": "ãKäžéåãã¯ã第1äžä»£ã®ã¯ã©ãŒã¯ãšç¬¬2äžä»£ã®ã¯ã©ãŒã¯ããæãç«ã£ãŠããäºããåãã£ãŠããã(â» æ€å®æç§æžã®ç¯å²å
ã)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 301,
"tag": "p",
"text": "ãããŠã2017幎ã®çŸåšãŸã§ãã£ãšãã¯ã©ãŒã¯ã®çè«ããçŽ ç²åã®æ£ããçè«ãšãããŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 302,
"tag": "p",
"text": "çŽ ç²åã®èŠ³ç¹ããåé¡ããå Žåã®ãéœåãšäžæ§åã®ããã«ãã¯ã©ãŒã¯3åãããªãç²åã®ããšãããŸãšããŠãããªãªã³ã(éç²å)ãšãããÏäžéå(Ï= u d Ì {\\displaystyle u{\\overline {d}}} )ãªã©ãã¯ã©ãŒã¯ã2åã®ç²åã¯ãããªãªã³ã«å«ãŸãªãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 303,
"tag": "p",
"text": "ããããäžéåã®ãªãã«ããã©ã ãç²å(udsãã¢ããããŠã³ã¹ãã¬ã³ãžã®çµã¿åãã)ã®ããã«ãã¯ã©ãŒã¯3åãããªãç²åããããã©ã ãç²åãªã©ããããªãªã³ã«å«ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 304,
"tag": "p",
"text": "éœåãšäžæ§åãã©ã ãç²åãªã©ãšãã£ãããªãªã³ã«ãããã«äžéå(äžéåã¯äœçš®é¡ããã)ãå ããã°ã«ãŒãããŸãšããŠãããããã³ããšããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 305,
"tag": "p",
"text": "ãªããæ®éã®ç©è³ªã®ååæ žã§ã¯ãéœåãšäžæ§åãååæ žã«éãŸã£ãŠãããããã®ããã«éœåãšäžæ§åãååæ žã«åŒãåãããåã®ããšãæ žåãšãããæ žåã®æ£äœã¯ããŸã ãããŸã解æãããŠããªã(å°ãªããšãé«æ ¡ã§æããã»ã©ã«ã¯ããŸã å
åã«ã¯è§£æãããŠããªã)ã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 306,
"tag": "p",
"text": "ãšããããããªãªã³ã«ã¯ãæ žåãåããé説ã§ã¯ãäžéåã«ããæ žåã¯åããšãããŠãããã€ãŸãããããã³ã«ãæ žåãåãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 307,
"tag": "p",
"text": "ãããã³ã¯ãããããã¯ã©ãŒã¯ããæ§æãããŠããäºããããããããã¯ã©ãŒã¯ã«æ žåãåãã®ã ãããçãªäºããèããããŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 308,
"tag": "p",
"text": "çè«ã§ã¯ãã¯ã©ãŒã¯ãšã¯ã©ãŒã¯ã©ãããåŒãä»ãããæ¶ç©ºã®ç²åãšããŠãã°ã«ãŒãªã³ããèããããŠãããç©çåŠè
ããçè«ãæå±ãããŠãããããã®æ£äœã¯ããŸã ãããŸã解æãããŠãªããããããç©çåŠè
ãã¡ã¯ãã°ã«ãŒãªã³ãçºèŠããããšäž»åŒµããŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 309,
"tag": "p",
"text": "çŸåšã®ç©çåŠã§ã¯ãã¯ã©ãŒã¯ãåç¬ã§ã¯åãåºããŠããªãã®ãšåæ§ã«ãã°ã«ãŒãªã³ãåç¬ã§ã¯åãåºããŠã¯ããªãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 310,
"tag": "p",
"text": "ããŠãç©çåŠã§ã¯ã20äžçŽãããéåååŠããšããçè«ããã£ãŠããã®çè«ã«ãããç©çæ³åã®æ ¹æºã§ã¯ãæ³¢ãšç²åãåºå¥ããã®ãç¡æå³ã ãšèšãããŠããããã®ããããã€ãŠã¯æ³¢ã ãšèããããŠããé»ç£æ³¢ããå Žåã«ãã£ãŠã¯ãå
åããšããç²åãšããŠæ±ãããããã«ãªã£ãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 311,
"tag": "p",
"text": "ãã®ããã«ãããæ³¢ãåå Ž(ããã°)ãªã©ããçè«é¢ã§ã¯ç²åã«çœ®ãæããŠè§£éããŠæ±ãäœæ¥ã®ããšããç©çåŠã§ã¯äžè¬ã«ãéååããšããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 312,
"tag": "p",
"text": "ã°ã«ãŒãªã³ããã¯ã©ãŒã¯ãšã¯ã©ãŒã¯ãåŒãä»ããåããéååãããã®ã§ããããé»è·ãšã®é¡æšã§ãã¯ã©ãŒã¯ã«ãè²è·(ã«ã©ãŒè·)ãšããã®ãèããŠãããããã®æ§è³ªã¯ãããŸã解æãããŠãªã(å°ãªããšãé«æ ¡ã§æããã»ã©ã«ã¯ããŸã å
åã«ã¯è§£æãããŠããªã)ã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 313,
"tag": "p",
"text": "ã°ã«ãŒãªã³ã®ããã«ãåãåªä»ããç²åã®ããšãã²ãŒãžç²åãšããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 314,
"tag": "p",
"text": "éåãåªä»ããæ¶ç©ºã®ç²åã®ããšãéåå(ã°ã©ããã³)ãšãããããŸã çºèŠãããŠããªããç©çåŠè
ãã¡ããã°ã©ããã³ã¯ããŸã æªçºèŠã§ããããšäž»åŒµããŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 315,
"tag": "p",
"text": "é»ç£æ°åãåªä»ããç²åã¯å
å(ãã©ãã³)ãšããããããã¯åã«ãé»ç£å Žãä»®æ³çãªç²åãšããŠçœ®ãæããŠæ±ã£ãã ãã§ããããã©ãã³ã¯ãé«æ ¡ç©çã®é»ç£æ°åéã§ç¿ããããªå€å
žçãªé»ç£æ°èšç®ã§ã¯ããŸã£ãã圹ç«ããªãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 316,
"tag": "p",
"text": "ãªããå
åãã²ãŒãžç²åã«å«ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 317,
"tag": "p",
"text": "ã€ãŸããå
åãã°ã«ãŒãªã³ã¯ãã²ãŒãžç²åã§ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 318,
"tag": "p",
"text": "ããŒã¿åŽ©å£ãã€ããã©ãåã®ããšãã匱ãåããšããããã®åãåªä»ããç²åãããŠã£ãŒã¯ããœã³ããšããããæ§è³ªã¯ãããåãã£ãŠããªãããããç©çåŠè
ãã¡ã¯ããŠã£ãŒã¯ããœã³ãçºèŠããããšäž»åŒµããŠããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 319,
"tag": "p",
"text": "ãããããããœã³ããšã¯äœã?",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 320,
"tag": "p",
"text": "éåååŠã®ã»ãã§ã¯ãé»åã®ãããªãäžç®æã«ããã ãæ°åãŸã§ããååšã§ããªãç²åããŸãšããŠãã§ã«ããªã³ãšããããã§ã«ããªã³çã§ãªãå¥çš®ã®ç²åãšããŠããœã³ããããå
åããããœã³ãšããŠæ±ãããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 321,
"tag": "p",
"text": "ããŠã£ãŒã¯ããœã³ããšã¯ãããããã匱ãåãåªä»ããããœã³ã ãããŠã£ãŒã¯ããœã³ãšåŒãã§ããã®ã ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 322,
"tag": "p",
"text": "ããŠãé»è·ãšã®é¡æšã§ãã匱ãåãã«é¢ããã匱è·ã(ãããã)ãšããã®ãæå±ãããŠãããããããããã®æ§è³ªã¯ãããŸã解æãããŠãªã(å°ãªããšãé«æ ¡ã§æããã»ã©ã«ã¯ããŸã å
åã«ã¯è§£æãããŠããªã)ã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 323,
"tag": "p",
"text": "ããŠãã匱ãåãã®ããäžæ¹ãã°ã«ãŒãªã³ã®åªä»ããåã®ããšãã匷ãåããšãããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 324,
"tag": "p",
"text": "1956幎ã«ãé»åã®ã¹ãã³ã®æ¹åãšãããŒã¿åŽ©å£ç²åã®åºãŠæ¥ãæ¹åãšã®é¢ä¿ãèŠãããã®å®éšãšããŠãã³ãã«ãã®æŸå°æ§åäœäœã§ããã³ãã«ã60ããã¡ããŠæ¬¡ã®ãããªå®éšããã¢ã¡ãªã«ã§è¡ãããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 325,
"tag": "p",
"text": "ã³ãã«ãå
çŽ (å
çŽ èšå·: Co )ã®æŸå°æ§åäœäœã§ããã³ãã«ã60ã極äœæž©ã«å·åŽããç£å ŽããããŠå€æ°ã®ã³ãã«ãååã®é»åæ®»ã®å€ç«é»åã¹ãã³ã®æ¹åãããããç¶æ
ã§ãã³ãã«ã60ãããŒã¿åŽ©å£ããŠçºçããããŒã¿ç²åã®åºãæ¹åã調ã¹ãå®éšãã1956幎ã«ã¢ã¡ãªã«ã§è¡ãããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 326,
"tag": "p",
"text": "éãšããã±ã«ãšã³ãã«ãã¯ãããããéå±åäœã§ç£æ§äœã«ãªãå
çŽ ã§ãããå
çŽ åäœã§ç£æ§äœã«ãªãå
çŽ ã¯ããã®3ã€(éãããã±ã«ãã³ãã«ã)ãããªãã(ãªããæŸå°æ§åäœäœã§ãªãéåžžã®ã³ãã«ãã®ååéã¯59ã§ããã)",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 327,
"tag": "p",
"text": "ãã®3ã€(éãããã±ã«ãã³ãã«ã)ã®ãªãã§ãã³ãã«ããäžçªãç£æ°ã«å¯äžããé»åã®æ°ãå€ãããšãéåååŠã®çè«ã«ããæ¢ã«ç¥ãããããã®ã§(ã³ãã«ãããã£ãšããdè»éã®é»åã®æ°ãå€ã )ãããŒã¿åŽ©å£ãšã¹ãã³ãšã®é¢ä¿ãã¿ãããã®å®éšã«ãã³ãã«ãã®æŸå°æ§åäœäœã§ããã³ãã«ã60ã䜿ãããã®ã§ããã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 328,
"tag": "p",
"text": "å®éšã®çµæãã³ãã«ã60ãããŒã¿åŽ©å£ããŠããŒã¿ç²åã®åºãŠããæ¹åã¯ãã³ãã«ã60ã®ã¹ãã³ã®ç£æ°ã®æ¹åãš(åãæ¹åããã)éã®æ¹åã«å€ãæŸåºãããŠããã®ã芳枬ããããããã¯ã2çš®é¡(ã¹ãã³ãšåæ¹åã«ããŒã¿ç²åã®åºãå Žåãšãã¹ãã³ãšå察æ¹åã«ããŒã¿ç²åã®åºãå Žå)ã®åŽ©å£ã®ç¢ºçãç°ãªã£ãŠãããããŒã¿åŽ©å£ã®ç¢ºçã®(ã¹ãã³æ¹åãåºæºãšããå Žåã®)æ¹å察称æ§ãæããŠããããšã«ãªãã",
"title": "ååã»ååæ žã»çŽ ç²å"
},
{
"paragraph_id": 329,
"tag": "p",
"text": "ãã®ãããªå®éšäºå®ã«ãããã匱ãåãã¯é察称ã§ããããšããã®ãå®èª¬ã",
"title": "ååã»ååæ žã»çŽ ç²å"
}
] | null | == é»åãšå
==
=== ããªã«ã³ã®å®éš ===
ããªã«ã³ã®å®éšãšã¯ãé§å¹ããªã©ã§äœæãã油滎ã®åŸ®å°ãªé£æ²«ã«ãXç·ãã©ãžãŠã ãªã©ã§åž¯é»ãããããããŠãå€éšããé»å ŽãåŒç«ããããããšã油滎ã®éåïŒäžåãïŒã®ã»ãã«ãé»å Žã«ããéé»æ°åïŒäžåãã«ãªãããã«é»æ¥µæ¿ãèšçœ®ããïŒãåãã®ã§ãé£ãåã£ãŠéæ¢ããç¶æ
ã«ãªã£ãæã®é»å Žãããé»è·ã®å€ã確ãããå®éšã§ããã
ãã®å®éšã§ç®åºã»æž¬å®ãããé»è·ã®å€ã 1.6Ã10<sup>-6</sup> Cã®æŽæ°åã«ãªã£ãã®ã§ãé»å1åã®é»è·ã 1.6Ã10<sup>-19</sup> Cã ãšåãã£ãã
ãªãããã® 1.6Ã10<sup>-19</sup> Cã®ããšã'''é»æ°çŽ é'''ïŒã§ããããããïŒãšããã
{{ã³ã©ã |ïŒâ» ç¯å²å€:ïŒããªã«ã³ä»¥åãããé»åã®é»è·ã¯æž¬å®ãããŠãã|
ååŠã®é»æ°å解ã®å®éšã§ãéå±ã®é»æ°å解ã®å®éšã®æã«çºçããæ°äœã垯é»ããŠããããšã¯ãã©ãã¢ãžãšãªã©ã«ãã£ãŠå€ãããç¥ãããŠãããå®éšç©çåŠè
ã¿ãŠã³ãŒã³ãã¯ãçºçããæ°äœã®ã¢ã«æ°ãšãéé»èªå°ãªã©ã«ãã£ãŠçºçããé»è·ã®åèšã枬å®ããããšã«ãããé»å1åãããã®é»è·ãæŠç®ããã
粟床ã¯ãçŸä»£ã®é»åã®é»è·ãšã±ã¿ãåããããã®ç²ŸåºŠã§ãã¿ãŠã³ãŒã³ãã¯é»åã®é»è·ã®æž¬å®å€ãåŸãã
}}
{{ã³ã©ã |ïŒâ» ç¯å²å€:ïŒ ããªã«ã³ã«äžæ£ã®çããã|
äžçååœã®ç©çåŠã®æè²ã§ã¯ã20äžçŽååã®ããªã«ã³ã®å®éšããé»åã®è³ªéãæ±ããå®éšãšããŠãé·ãã玹ä»ãããŠããã
ããã20äžçŽåŸåãããããããªã«ã³ã®å®éšã«å¯Ÿããç念ãç§åŠçããæåºãããŠããããã®çæã®å
容ã¯ãããªã«ã³ã¯ãèªèº«ã®æå±ãã仮説ã«é©åããªã枬å®å€ãã枬å®èª€å·®ã ãšããŠæå®ããŠããŸãã仮説ã«ãããªã枬å®å€ãæé€ããŠããŸã£ãŠããã®ãããããªãããšããçæã§ããã
ãã®çæã«åããåè«ããŸããç§åŠçããæåºãããŠããã
ã©ã¡ããæ£ãããã«ã€ããŠã¯ãé«æ ¡æç§æžã§ã¯èªããããªããšã§ã¯ãªãã®ã§ãããã«ã€ããŠã¯èª¬æãçç¥ããã
ã©ã¡ãã«ãããçŸä»£ã§ã¯ãè«æã®æçš¿ã§ã¯ããã仮説ã«ãããªã枬å®å€ãèªè
ã«ã ãŸã£ãŠæé€ããŠããŸãããªã®ã«ãããšã®å®éšããŒã¿ãã®ãŸãŸã®ããã«è«æçºè¡šããŠããŸã£ãããããŒã¿æ¹ç«ïŒããããïŒã«ããäžæ£è¡çºãšã¿ãªãããã®ãååã§ããã
ããäŸå€çã«ãã©ãããŠãè«æãªã©ã§è€æ°ãã枬å®å€ã®ããã€ããæç²ããããåŸãªããããªäºæ
ã®ããå Žåã«ã¯
:ïŒããšãã°å®éšããŒã¿ã倧éã«ãããããŠããã¹ãŠã玹ä»ããããªãå Žåã
:ãããã¯ã仮説ã®å
容ã説æããããã«ãè€æ°åã®å®éšãããŠããã®ãã¡æã仮説ã«é©åããåã®å®éšããŒã¿ãå
¬è¡šããå Žåããªã©ïŒã
ãã®ãããªå Žåã«ã¯ããŸãè«æã«ãæç²ããéšåçãªããŒã¿ã§ããããšãæèšããªããã°ãªããªãã ããããã©ãããåºæºã§æç²ãè¡ã£ãããæèšããªããã°ãªããªãã ããã
çŸä»£ã®ç§åŠè«æã§ã¯ãå®éšçµæã®ããŒã¿ãæžãéã«ã¯ãååçã«ãå®éã®å®éšã§åŸãããããŒã¿ããã®ãŸãŸèšè¿°ããããã«åªããŠãè«æãæžããªãããã°ãªããªãã
:â» çŸä»£ã§ãããã°ãã°åŠçå®éšãªã©ã§ãæªæ°ããªããŠãã仮説ã«ãããªãå®éšããŒã¿ãããå®éšãã¹ããšæå®ããŠããŸãã枬å®å€ãæžãæããŠããŸã£ããããããã¯ã仮説ã«ãããªã枬å®å€ãé ããŠããŸãäžæ£è¡çºãèµ·ããããšãããããã®ãããªäžæ£è¡çºãããªããããæ°ãã€ããªããã°ãªããªãã
ãã®ããã«ãããªã«ã³ã®å®éšã«ã€ããŠã¯ããããããšåé¡ç¹ãããã®ã§ã倧åŠå
¥è©Šã«ã¯ãããªã«ã³ã®å®éšã«ã€ããŠãããŸãç£æ«ïŒããŸã€ïŒãªããšã¯åºé¡ãããªãã ããããããããªã«ã³ã®å®éšã®çµæãæèšãããããªå
¥è©Šåé¡ãåºé¡ããããšããããåºé¡è
ã®èŠèãçãããã
ãŸããããããããªã«ã³ã®å®éšã®æ¹æ³ã¯ãããŸã粟床ãè¯ããªãã粟床ãæªãå®éšæ¹æ³ã ãããããäžèšã®ãããªçæãæ®ã£ãŠããŸãã®ã§ãããã
}}
=== å
ã®ç²åæ§ ===
==== å
é»å¹æ ====
:ïŒâ» å®éšçµæã°ã©ããè¿œå ããããšãïŒ
[[File:Photoelectric effect diagram no label.svg|thumb|300px|é»åã®éåãšãã«ã®ãŒã®æ倧å€ãšãå
ã®æ¯åæ°ãšã®é¢ä¿]]
è² ã®é»è·ã«åž¯é»ãããŠããéå±æ¿ã«ã玫å€ç·ãåœãŠããšãé»åãé£ã³åºããŠããããšãããããŸããæŸé»å®éšçšã®è² 極ã«é»åãåœãŠããšãé»åãé£ã³åºããŠããããšãããããã®çŸè±¡ãã'''å
é»å¹æ'''ïŒããã§ã ããããphotoelectric effectïŒãšããã1887幎ããã«ãã«ãã£ãŠãå
é»å¹æãçºèŠããããã¬ãŒãã«ãã«ãã£ãŠãå
é»å¹æã®ç¹åŸŽãæããã«ãªã£ãã
åœãŠãå
ã®æ¯åæ°ããäžå®ã®é«ã以äžã ãšãå
é»å¹æãèµ·ããããã®æ¯åæ°ã'''éçæ¯åæ°'''ïŒãããã ããã©ãããïŒãšãããéçæ¯åæ°ããäœãå
ã§ã¯ãå
é»å¹æãèµ·ãããªãããŸããéçæ¯åæ°ã®ãšãã®æ³¢é·ãã'''éçæ³¢é·'''ïŒãããã ã¯ã¡ããïŒãšããã
ç©è³ªã«ãã£ãŠãéçæ¯åæ°ã¯ç°ãªããäºéçã§ã¯çŽ«å€ç·ã§ãªããšå
é»å¹æãèµ·ããªãããã»ã·ãŠã ã§ã¯å¯èŠå
ã§ãå
é»å¹æãèµ·ããã
å
é»å¹æãšã¯ãç©è³ªäžïŒäž»ã«éå±ïŒã®é»åãå
ã®ãšãã«ã®ãŒãåãåã£ãŠå€éšã«é£ã³åºãçŸè±¡ã®ããšã§ããã
ãã®é£ã³åºããé»åããå
é»åãïŒããã§ãããphotoelectronïŒãšããã
å
é»å¹æã«ã¯ïŒæ¬¡ã®ãããªç¹åŸŽçãªæ§è³ªãããã
:* å
é»å¹æã¯ãå
ã®æ¯åæ°ãããæ¯åæ°ïŒéçæ¯åæ°ïŒä»¥äžã§ãªããšèµ·ãããªãã
:* å
é»åã®éåãšãã«ã®ãŒã®æ倧å€ã¯ãåœãŠãå
ã®æ¯åæ°ã®ã¿ã«äŸåããå
ã®åŒ·ãã«ã¯äŸåããªãã
:* åäœæéãããã«é£ã³åºãå
é»åæ°ã¯ãå
ã®åŒ·ãã«æ¯äŸããã
ãããã®æ§è³ªã®ãã¡ã1çªããš2çªãã®æ§è³ªã¯ãå€å
žç©çåŠã§ã¯èª¬æã§ããªãã
ã€ãŸããå
ããé»ç£æ³¢ãšããæ³¢åã®æ§è³ªã ããæããŠããŠã¯ãã€ãã€ãŸãåããªãã®ã§ããã
ãªããªããä»®ã«ãé»ç£æ³¢ã®é»çïŒé»å ŽïŒã«ãã£ãŠéå±ããé»åãæŸåºãããšèããå Žåãããå
ã®åŒ·ãã倧ãããªãã°ãæ¯å¹
ã倧ãããªãã®ã§ãé»çïŒé»å ŽïŒã倧ãããªãã¯ãã§ããã
ããããå®éšçµæã§ã¯ãå
é»åã®éåãšãã«ã®ãŒã¯ãå
ã®åŒ·ãã«ã¯äŸåããªãã
ãã£ãŠãå€å
žååŠã§ã¯èª¬æã§ããªãã
===== ã¢ã€ã³ã·ã¥ã¿ã€ã³ã® å
éå仮説 =====
äžè¿°ã®ççŸïŒå€å
žçãªé»ç£æ³¢çè«ã§ã¯ãå
é»å¹æã説æã§ããªãããšïŒã解決ããããã«ã次ã®ãããª'''å
éå仮説'''ãã¢ã€ã³ã·ã¥ã¿ã€ã³ã«ãã£ãŠæå±ãããã
* å
ã¯ãå
åïŒããããphotonïŒã®æµãã§ãããå
åããå
éåïŒããããããïŒãšãããã
* å
å1åã®ãšãã«ã®ãŒEã¯ãå
ã®æ¯åæ° <math>\nu </math>Hzã«æ¯äŸããã
ãã®2ã€ãã®æ¡ä»¶ãå®åŒåãããšã
:<math>E = h \nu </math>
ãšãªãã
ãã®åŒã«ãããæ¯äŸå®æ°hã¯'''ãã©ã³ã¯å®æ°'''ãšãã°ããå®æ°ã§ã
:<math> h = 6.626 \times 10 ^{-34} </math>
[Jã»s] ãšããå€ããšãã
'''ä»äºé¢æ°'''ïŒãããš ãããããwork functionïŒãšã¯ãå
é»å¹æãèµ·ããã®ã«å¿
èŠãªæå°ã®ãšãã«ã®ãŒã®ããšã§ãããéå±ã®çš®é¡ããšã«ã決ãŸã£ãå€ã§ããã
ä»äºé¢æ°ã®å€ã WJ ãšãããšãå
åã®åŸãéåãšãã«ã®ãŒã®æå€§å€ K<sub>0</sub> J ã«ã€ããŠã次åŒãåŸãããã
:<math> K _0 = h \nu - W </math> (1.1)
ãã®åŒãããå
é»å¹æãèµ·ããæ¡ä»¶ã¯ hÎœâ§W ãšãªãããã㯠K<sub>0</sub>â§0 ã«çžåœããã
ãããããå
é»å¹æãèµ·ããéçæ¯åæ° Îœ<sub>0</sub> ã«ã€ããŠãhÎœ<sub>0</sub>ïŒW ãæãç«ã€ã
ãã®å
éå仮説ã«ãããå
é»å¹æã®1çªããš2çªãã®æ§è³ªã¯ã容æã«ãççŸãªã説æã§ããããã«ãªã£ããæ³¢åã¯ç²åã®ããã«æ¯èãã®ã§ããã
ãªããå
é»å¹æã®3çªãã®æ§è³ªãããããå Žæã®å
ã®åŒ·ãã¯ã
ãã®å Žæã®åäœé¢ç©ã«åäœæéãé£æ¥ããå
åã®æ°ã«æ¯äŸããããšãåããã
{{ã³ã©ã |ïŒâ» ç¯å²å€ïŒãã€ã¯ã説æã®é åºãé|
é«æ ¡ã§å
ã«å
é»å¹æãç¿ãã倧åŠã§ããšããããã©ã³ã¯ã®çè«ãç¿ãã
ããããå®ã¯ãç©çåŠè
ãã©ã³ã¯ãå
ã«ïŒã¢ã€ã³ã·ã¥ã¿ã€ã³ãããæ©ãïŒããšãã«ã®ãŒã®ãããšãã®åäœã hÎœ ã§ããããšãçºèŠãã<ref>å±±æ¬çŸ©éãååã»ååæ žã»åååãã岩波æžåºã2015幎3æ24æ¥ ç¬¬1å·ã94ããŒãž</ref>ããããããã ããããå®æ° h ããã©ã³ã¯å®æ°ãšããã®ã§ãããã€ãŸããããçš®ã®ç©ççŸè±¡ã«ãããŠãšãã«ã®ãŒã®ãããšãã®åäœãhÎœã§ããããšã¯ããã£ããŠã¢ã€ã³ã·ã¥ã¿ã€ã³ãæå±ããã®ã§ã¯ãªãïŒãã©ã³ã¯ã®æå±ã§ããïŒã
ãã©ã³ã¯ã¯ãé«æž©ç©äœã«ãããå
ã®æŸå°ïŒãç±æŸå°ãããç±èŒ»å°ããªã©ãšããïŒã®ç 究ããããã®ãããªçºèŠãããã
ã§ã¯ãã¢ã€ã³ã·ã¥ã¿ã€ã³ãäœãçºèŠããã®ããšãããšã
*ç±èŒ»å°ã ãã§ãªãå
é»å¹æã«ããã©ã³ã¯å®æ°ãé©çšã§ãããšããåŠèª¬ã
*å
ã®ç²å説ã®æå±ã
ã§ããã
ãªããå
é»å¹æã®æ¯äŸä¿æ°ã枬å®ããå®éšã¯ãã¢ã€ã³ã·ã¥ã¿ã€ã³ã®æå±åŸã«ç©çåŠè
ããªã«ã³ã調ã¹ãŠããããããã«ãã©ã³ã¯å®æ°ãšã»ãŒåãæ°å€ã§ããäºã確èªããŠãã<ref>å±±æ¬çŸ©éãååã»ååæ žã»åååãã岩波æžåºã2015幎3æ24æ¥ ç¬¬1å·ã95ããŒãž</ref>ã
ããŠããã©ã³ã¯ã®å®éšãæ¯ãè¿ããšããã©ã³ã¯ã¯ã¢ã€ã³ã·ã¥ã¿ã€ã³ã®ç 究ãšã¯å¥ã«ããããããªããšã調ã¹ãŠããã
ãã€ã¯ã20äžçŽååã®ç©çåŠè
ã®ãŠã£ãŒã³ïŒäººåïŒããã©ã³ã¯ïŒäººåïŒãªã©ãé«æž©ã®ç©äœããåºãŠããå
ã®æ³¢é·ãšåšæ³¢æ°ãåæãããšããã
次ã®ãããªåšæ³¢æ°fãšåšæ³¢æ°Îœã®é¢ä¿åŒãåãã£ãŠããã
:<math>f(\lambda) = \frac{8\pi hc}{\lambda^5}~\frac{1}{e^\frac{h \nu}{kT}-1}</math>
å³èŸºã®ææ°é¢æ°ã®åæ¯ã«ããkããã«ããã³å®æ°ã§ããã
ãããŠãå³èŸºã®ææ°é¢æ°ã®åæ¯ã«ããh ããã©ã³ã¯å®æ°ãšèšãããå®æ°ã§ãããããã¯ãé«æ ¡ãç©çIIãã®ååç©çã®åå
ã§ã®ã¡ã«ç¿ããå
é»å¹æãïŒããã§ããããïŒã«åºãŠãããã©ã³ã¯å®æ° h ãšåãå®æ°ã§ããã
ãã®åŒïŒããã³ããã®åŒã®ã¢ã€ãã¢ã®å
ã«ãªã£ããŠã£ãŒã³ã®å
¬åŒïŒã¯ãå®éšçã«æž¬å®ããŠç¢ºèªã§ããåŒã§ãããïŒããã¡ãŒã¿ãŒãšèšããã枬å®åšããç±é»å¯ŸïŒãã€ã§ãã€ãïŒãšãã°ããåéææãããã€ããã¹ããŒã³ããªããžãšèšãããé»æ°åè·¯ã䜿ããïŒ
ãããŠãå³èŸºã®åæ¯ã«ãã
:<math>~\frac{1}{e^\frac{h \nu}{kT}-1}</math>
ã«æ³šç®ããã
ããã«é«æ ¡æ°åŠã§ç¿ãçæ¯æ°åã®åã®å
¬åŒ
:<math>a+ar+ar^2+\cdots+ar^{n-1}+\cdots = \frac{a}{1-r}</math> ïŒãã ã ïœrïœïŒ1ïŒ
ãæãåºããŠããããåèã«ç¡éçŽæ°
:<math>S= 1+e^{- \frac{ 1 h \nu}{kT} } + e^{- \frac{ 2 h \nu}{kT} } + e^{- \frac{ 3 h \nu}{kT} } \cdots + e^{- \frac{n h \nu}{kT} }+ </math>
ã®åãæ±ããŠã¿ããšïŒææ°éšã«ãã€ãã¹ãã€ããŠããã®ã§ãå¿
ãåæããïŒã
:<math>e^{- \frac{ 1 h \nu}{kT} } S= e^{- \frac{ 1 h \nu}{kT} } + e^{- \frac{ 2 h \nu}{kT} } + e^{- \frac{ 3 h \nu}{kT} } \cdots + e^{- \frac{n h \nu}{kT} }+ </math>
ãšãªãã®ã§ãïŒå³èŸºã©ããã巊蟺ã©ãããïŒèŸºã
ãåŒãç®ããŠ
:<math>(1 - e^{- \frac{ 1 h \nu}{kT} }) S= 1 </math>
巊蟺ã®ä¿æ°ã移é
ããŠ
:<math> S= \frac{1}{1 - e^{- \frac{ h \nu}{kT} } } </math>
ãšããã䌌ãåŒãåºãŠããã
ããã§çµããããŠããŸããšããã©ã³ã¯ã®åŒã®åæ¯ã®ææ°ã®åŒãšã¯ã䌌ãŠéãªãåŒã§çµãã£ãŠããŸããïŒãããããã°ã°ã£ãŠããããã§çµããããŠããŸã£ãŠãããäžå匷ãªäººãå€ããç©çåŠãã¡ã³ãåä¹ããªãããã£ãšå匷ããŠã»ãããïŒ ïŒãªãããã®æ°åSã¯ãéåçµ±èšååŠã«ããããåé
é¢æ°ããšãããïŒ
ãšããã§ãäœãã®ãšãã«ã®ãŒã®å€ãEãšãããšãã<math> e^{-\frac{E}{k T}} </math> ã®ããšãã'''ãã«ããã³å å'''ãšããããããªã倩äžãçã«ååãåºãããããã«ããã³å åã¯ããšãããªã確çã¿ãããªãã®ã§ããã
ãŸããäžèšã®æ°å S ã®ç©ççãªæå³ã¯ã確çèšç®ãããããã®ãå
šç¢ºçã1ãšããããã®èŠæ Œåã®ããã®ä¿æ°ã§ããã
ãŸããèšç®ããããããã«ãã«ããã³å åã次ã®ããã« <math> \beta = \frac{1}{kT} </math>ã䜿ã£ãŠå€åœ¢ãããã
ãããšããã«ããã³å åã¯ã
<math> e^{-\beta E} </math>
ãšãªãã
ãŸããåé
ä¿æ° S ã®åãæ±ããåã®åœ¢ã®åŒããβã䜿ã£ããã«ããã³å åã®åŒã§çœ®ãæãããã
<math>S= 1+e^{- \beta h \nu} + e^{- 2 \beta h \nu} + e^{- 3 \beta h \nu} \cdots + e^{- n \beta h \nu} + </math>
ãšãªãã
ããŠã次ã®æ°å P ãæ±ãããã
:<math> P = (0 h \nu) \cdot 1 + (1 h \nu) \cdot e^{- \frac{ 1 h \nu}{kT} } + (2 h \nu) \cdot e^{- \frac{ 2 h \nu}{kT} } + (3 h \nu) \cdot e^{- \frac{ 3 h \nu}{kT} } \cdots + (n h \nu) \cdot e^{- \frac{n h \nu}{kT} }+ </math>
ãã®æ°åPã®ç©ççãªæå³ã¯ãé£ã³é£ã³ãªãšãã«ã®ãŒ n hÎœ ïŒãã ã nïŒ1,2,3,4ã»ã»ã»ã»ïŒããããšããå Žåã®ç¢ºççãªå¹³åãšãã«ã®ãŒå€ <E> ã«æ¯äŸããæ°ã§ããã
ãªãããšãã«ã®ãŒã®å¹³åå€<E>ã®åŒã¯ãPãSã§å²ã£ãå€ã§ããã
:<math> \left \langle E \right \rangle = \frac{P}{S} </math>
èšç®ãããããã㫠β ã§æžãæãããã
:<math> P = (0 h \nu) \cdot 1 + (1 h \nu) \cdot e^{- \beta h \nu } + (2 h \nu) \cdot e^{- 2 \beta h \nu } + (3 h \nu) \cdot e^{-3 \beta h \nu } \cdots + (n h \nu) \cdot e^{- n \beta h \nu }+ </math>
ããŠãæ°åPãããããSãšæ¯ã¹ãŠã¿ãããæ¯ã¹å®ãããã«åæ²ããŠãããèªè
ã¯äœãæ°ã¥ãããšã¯ãªãããªïŒ ïŒãã³ã: 埮åïŒ
:<math>S= 1+e^{- \beta h \nu} + e^{- 2 \beta h \nu} + e^{- 3 \beta h \nu} \cdots + e^{- n \beta h \nu} + </math>
æ¯ã¹ãŠã¿ããšããªããšæ°åPã®åé
ã¯æ°åSã®åé
ãβã§åŸ®åãããã®ã«ãã€ãã¹ãæããå€ã«ãªã£ãŠããïŒïŒ
ã€ãŸã
:<math> -\frac{\partial}{\partial \beta} S = P </math>
:â» <math> \partial </math> ãšã¯ãå€å€æ°é¢æ°ã®åŸ®åïŒå埮åïŒã®èšå·ã倧åŠã§ç¿ãããã©ãŠã³ã ãã£ãŒããªã©ãšèªãã
ãšããã§ãæ°åSã¯é«æ ¡ã¬ãã«ã®çæ¯çŽæ°ã®åã®å
¬åŒã«ãã
:<math> S= \frac{1}{1 - e^{- \frac{ h \nu}{kT} } } </math>
ãšãæžããã®ã§ãã£ãã
èšç®ããããããã«Î²ã§çœ®æããŠã
:<math> S= \frac{1}{1 - e^{- \frac{ h \nu}{kT} } } = \frac{1}{1 - e^{- \beta h \nu } } </math>
ãšãªãã
ããã«åŸ®åããããããã«
:<math> S= \frac{1}{1 - e^{- \frac{ h \nu}{kT} } } = \frac{1}{1 - e^{- \beta h \nu } } = (1 - e^{- \beta h \nu } ) ^{-1} </math>
ãšæžãæãããã
ãã®æ°åSã®åã®å
¬åŒãšãå
ã»ã©ã®ãã€ãã¹åŸ®åã®åŒ <math> -\frac{\partial}{\partial \beta} S = P </math> ãšãé£ç«ãããŠã¿ããã
ãããšã
<math> P = -\frac{\partial}{\partial \beta} S = -\frac{\partial}{\partial \beta} (1 - e^{- \beta h \nu } ) ^{-1} = (-1) (1 - e^{- \beta h \nu } ) ^{-2} (-1) (-1)(-h \nu) = \frac{h \nu}{ (1 - e^{- \beta h \nu } ) ^{-2}} e^{- \beta h \nu } </math>
ãšPãæ±ããããã
ããããç§ãã¡ãæçµçã«æ±ãããã®ã¯ãPã§ãªããŠããšãã«ã®ãŒã®å¹³åå€ïŒEïŒã§ãã£ãã
ïŒEïŒã®åŒãåæ²ãããšã
:<math> \left \langle E \right \rangle = \frac{P}{S} </math>
ã§ãã£ãã
ãããŠãPãSãçŽæ°åã®åŒãæ±ããããŠããã®ã§ãããã代å
¥ãããšã
:<math> \left \langle E \right \rangle = \frac{P}{S} = \frac{h \nu}{ (1 - e^{- \beta h \nu } ) ^{2}} \frac{1}{ \frac{1}
{ \frac{1}
{
1 - e^{- \beta h \nu }
}
}
}
e^{- \beta h \nu }
=
\frac{h \nu}{ (1 - e^{- \beta h \nu } ) }
e^{- \beta h \nu }
=
\frac{h \nu}{1- e^{- \beta h \nu } }
e^{- \beta h \nu }
</math>
ãšãªããååã®ææ°é¢æ°ãæ¶ãããã«ãåæ¯ãšååã«ãšãã« <math> e^{- \beta h \nu } </math> ãæãç®ããŠçŽåãããã
ãããšã
:
:<math> \left \langle E \right \rangle = \frac{P}{S} =
\frac{h \nu}{1- e^{- \beta h \nu } }
e^{- \beta h \nu }
=
\frac{h \nu}{e^{ \beta h \nu }- 1 }
</math>
ãšãªãã®ã§ãã ãã¶ãã©ã³ã¯ã®åŒã«äŒŒãŠããã
ãã©ã³ã¯ã®åŒãåæ²ãããšã
:<math>f(\lambda) = \frac{8\pi hc}{\lambda^5}~\frac{1}{e^\frac{h \nu}{kT}-1}</math>
ã§ãã£ãã
ãšããã§ãé«æ ¡ç©çã§ç¿ãå
ã®æ³¢é·Î»ãšé床Cãšåšæ³¢æ°Îœã®é¢ä¿åŒ CïŒÎœÎ» ã䜿ãã°ããã©ã³ã¯ã®åŒã¯ã
:<math>f(\lambda) = \frac{8\pi hc}{\lambda^5}~\frac{1}{e^\frac{h \nu}{kT}-1} = \frac{8\pi h \nu \lambda}{\lambda^5}~\frac{1}{e^\frac{h \nu}{kT}-1} = \frac{8\pi }{\lambda^4}~\frac{h \nu}{e^\frac{h \nu}{kT}-1} </math>
ã§ããã
ãã©ã³ã¯ã¯ããã®åŒããåŒå€åœ¢ã§ã
:<math>f(\lambda) = \frac{8\pi hc}{\lambda^5}~\frac{1}{e^\frac{h \nu}{kT}-1} = \frac{8\pi h \nu \lambda}{\lambda^5}~\frac{1}{e^\frac{h \nu}{kT}-1} = \frac{8\pi }{\lambda^4}~\frac{h \nu}{e^\frac{h \nu}{kT}-1} = \frac{8\pi }{\lambda^4}~ \sum_{n=0}^\infty (n h \nu )e^{- \beta n h \nu} </math>
ãšãããµãã«ãçŽæ°ã®åã®åœ¢ã«æžãæããããããšã«æ°ã¥ããã
ãã®ããšã¯ãã€ãŸããé«æž©ç©äœããã®æŸå°ãšãã«ã®ãŒãåºãå
ã®ãšãã«ã®ãŒããæŸå°çŸè±¡ã®ã©ãã㧠<math> h \nu </math> ã®æŽæ°åã ãã«éãããæ©æ§ã®ããããšãæå³ããã
ãã®ãããªæãã®ãã©ã³ã¯ã®ã²ãããã«ãããçŸä»£ã«ãéåååŠããšèšãããåéã19äžçŽã«è±éããã
ã¢ã€ã³ã·ã¥ã¿ã€ã³ã®å
é»å¹æã®å
ç²å説ã¯ãåã«ããã©ã³ã¯ã®ãã®ãããªç 究ãããšã«ãå
é»å¹æã«ãåœãŠã¯ããŠé£æ³ããã ãã§ããã
ãã®åŸãã¢ã¡ãªã«äººã®ç©çåŠè
ã®æž¬å®ãªã©ã«ãããå
é»å¹æã®ä¿æ°ãããã©ã³ã¯ãæŸå°ã®ç 究ã§çšããå®æ° h ãšåãããšã確èªãããã
ãã©ã³ã¯èªèº«ã¯ããããŸã§éååãããŠããã®ã¯ãå®éšçã«ç¢ºèªãããŠããã®ã¯é«æž©ç©äœã®æŸå°ã«ãããçŸè±¡ã ãšæ
éã ã£ãããã§ããã
å
ãæ³¢ãç²åãã®çºèšã«ã¯ãã©ã³ã¯ã¯ããŸãé¢ãããªãã£ãã
ã ãããã®åŸãã¢ã€ã³ã·ã¥ã¿ã€ã³ã®å
ç²å説ãããŒãã«è³ããšã£ãŠããŸã£ãããšãªã©ãããããããäžççã«å
ã®ç²åæ§ã®åŠèª¬ãåºãŸã£ãŠãã£ãã
ãªããã¢ã€ã³ã·ã¥ã¿ã€ã³ã®å
é»å¹æã®ããŒãã«è³ã¯ãæ¬åœã¯çžå¯Ÿæ§çè«ã«ããŒãã«è³ããããããšããŒãã«è³ã®å¯©æ»å¡ã¯èãããããããåœæã®åŠäŒã«ã¯çžå¯Ÿæ§çè«ãžã®å察æèŠãå€ãã£ãã®ã§ãåœããéãã®ãªããããªå
é»å¹æã®ç 究ã§ããŒãã«è³ãã¢ã€ã³ã·ã¥ã¿ã€ã³ã«æäžããã ãã§ããã
ããŠãå
é»å¹æã®åŒ EïŒhÎœïŒW ãšããã©ã³ã¯ã®æŸå°åŒãæ¯ã¹ãã°åããããã©ã¡ããç°¡åãªåŒããšãããšãæããã«å
é»å¹æã®åŒã®ã»ããä¿æ°ã®çš®é¡ãå°ãªãåçŽãªåœ¢ã§ããã
ãªã®ã§ãããã£ãšãå
é»å¹æã®ã»ãããããåºæ¬çãªç©çæ³åã«è¿ãçŸè±¡ãªã®ã ããããšèããã®ã劥åœã§ããããæè²ã§å
ã«å
é»å¹æãæããŠãããããšãããã©ã³ã¯ã®ç 究ææãæããã®ããåççãããããªãã
ãã ãããã®åºæ¬çãªç©ççŸè±¡ãšããããã¯ãããŠå
ã®ç²å説ãã©ããã¯ãå®éšã¯äœãä¿èšŒããŠãããŠãªããåã«ã人éãã¡ããããŒãã«è³ãªã©ã®äººé瀟äŒã®æš©åšã®éœåã«ããšã¥ããŠåæã«ãå
ã®ç²å説ããããå
é»å¹æã®åŒã®åœ¢ãç°¡åã«ãªã£ãŠããçç±ã ããšæã蟌ãã§ããã ãã§ããã
}}
==== åè: å
ã®æ³¢é·ã®æž¬å® ====
:(â» ç¯å²å€)
ãããããå
ã®æ³¢é·ã¯ãã©ããã£ãŠæž¬å®ãããã®ã ãããã
çŸåšã§ã¯ãããšãã°ååã®çºå
ã¹ãã¯ãã«ã®æ³¢é·æž¬å®ãªããåææ Œåãããªãºã ãšããŠäœ¿ãããšã«ãã£ãŠãæ³¢é·ããšã«åããæ³¢é·ã枬å®ãããŠãããïŒâ» åèæç®: å¹é¢šé€šïŒã°ããµãããïŒãstep-up åºç€ååŠãã梶æ¬èäº ç·šéãç³å·æ¥æš¹ ã»ãèã2015幎åçã25ããŒãžïŒ
ãããŸããªåçãè¿°ã¹ããšãå¯èŠå
ãŠãã©ã®å
ã®æ³¢é·ã®æž¬å®ã¯ãåææ Œåã«ãã£ãŠæž¬å®ããããã ããã§ã¯ãã®åææ Œåã®çŽ°ããæ°çŸããã¡ãŒãã«ãæ°åããã¡ãŒãã«ãŠãã©ã®ééã®æ ŒåããŸãã©ããã£ãŠäœãã®ãããšããåé¡ã«è¡ãçããŠããŸãã
æŽå²çã«ã¯ãäžèšã®ããã«ãå¯èŠå
ã®æ³¢é·ã枬å®ãããŠãã£ãã
ãŸãã1805幎ããã®ãã€ã³ã°ã®å®éšãã§æåãªã€ã³ã°ãã®ç 究ã«ãããå¯èŠå
ã®æ³¢é·ã¯ããããã 100nmïŒ10<sup>-7</sup>mïŒ ã 1000nm ã®çšåºŠã§ããããšã¯ããã®é ããããã§ã«äºæ³ãããŠããã
ãã®åŸããã€ãã®ã¬ã³ãºã®ç 磚工ã ã£ããã©ãŠã³ããŒãã¡ãŒããããããåææ Œåãéçºããå¯èŠå
ã®æ³¢é·ã粟å¯ã«æž¬å®ããäºã«æåããããã©ãŠã³ããŒãã¡ãŒã¯åææ Œåãäœãããã«çŽ°ãééãçšããå å·¥è£
眮ã補äœãããã®å å·¥æ©ã§è£œäœãããåææ ŒåãçšããŠãå
ã®æ³¢é·ã®æž¬å®ããå§ããã®ããç 究ã®å§ãŸãã§ããã1821幎ã«ãã©ãŠã³ããŒãã¡ãŒã¯ã1cmãããæ Œåã130æ¬ã䞊ã¹ãåææ Œåã補äœããã<ref>ãçŸä»£ç·åç§åŠæè²å€§ç³»ãSOPHIA21ã第7å·»ãéåãšãšãã«ã®ãŒããè¬è«ç€Ÿãçºè¡ïŒæå59幎4æ21æ¥ç¬¬äžå·çºè¡çºè¡</ref>
ãŸãã1870幎ã«ã¯ã¢ã¡ãªã«ã®ã©ã¶ãã©ãŒããã¹ããã¥ã©ã ãšããåéãçšããåå°åã®åææ Œåã補äœãïŒãã®ã¹ããã¥ã©ã åéã¯å
ã®åå°æ§ãé«ãïŒãããã«ãã£ãŠ1mmããã700æ¬ãã®æ Œåã®ããåææ Œåã補äœãããïŒèŠåºå
žïŒ
ããã«ãã®ããã®æ代ãéãããã®æœ€æ»ã®ããã«æ°Žéã䜿ãæ°Žéæµ®éæ³ããç 究éçºã§è¡ãããã
ããé«ç²ŸåºŠãªæ³¢é·æž¬å®ããã®ã¡ã®æ代ã®ç©çåŠè
ãã€ã±ã«ãœã³ã«ãã£ãŠãå¹²æžèšïŒãããããããïŒãšãããã®ãçšããŠïŒçžå¯Ÿæ§çè«ã®å
¥éæžã«ããåºãŠããè£
眮ã§ãããé«æ ¡çã¯ããŸã çžå¯Ÿæ§çè«ãç¿ã£ãŠãªãã®ã§ãæ°ã«ããªããŠãããïŒãå¹²æžèšã®åå°é¡ã粟å¯ããžã§çŽ°ããåããããšã«ãããé«ç²ŸåºŠãªæ³¢é·æž¬å®åšãã€ããããã®æž¬å®åšã«ãã£ãŠã«ãããŠã ã®èµ€è²ã¹ãã¯ãã«ç·ã枬å®ããçµæã®æ³¢é·ã¯643.84696nmã ã£ãããã€ã±ã«ãœã³ã®æž¬å®æ¹æ³ã¯ãèµ€è²ã¹ãã¯ãã«å
ã®æ³¢é·ããåœæã®ã¡ãŒãã«ååšãšæ¯èŒããããšã§æž¬å®ããã<ref>å·äžèŠªèã»ããæ°å³è©³ãšãªã¢æç§èŸå
žãç©çããåŠç ãçºè¡ïŒ1994幎3æ10æ¥æ°æ¹èšç第äžå·ãP.244 ããã³ P.233</ref>
ãã€ã±ã«ãœã³ã®å¶äœããå¹²æžèšã«ããæ°Žéæµ®éæ³ã®æè¡ãåãå
¥ããããŠããããšãã<ref>ã¯ãªã¹ã»ãšãŽã¡ã³ã¹ èãæ©æ¬æŽã»äžéæ» å
±èš³ã粟å¯ã®æŽå²ãã倧河åºçã2001幎11æ28æ¥ åçã185ããŒãž</ref>ã
ããã«ãããžã®æè¡é©æ°ã§ãããŒãã³ã»ãããïŒãã¡ã«ãã³ã»ãããããšãèš³ãïŒãšããã匟åæ§ã®ããæ質ã§ããžãã€ããããšã«ãã£ãŠèª€å·®ããªããããŠå¹³ååãããã®ã§ãè¶
絶çã«é«ç²ŸåºŠã®éããããäœãæè¡ããã€ã®ãªã¹ã®ç©çåŠè
ããŒãã¹ã»ã©ã«ãã»ããŒãã³ïŒè±:[[:en:w:Thomas Ralph Merton]] ïŒãªã©ã«ãã£ãŠéçºãããã
ãªããçŸä»£ã§ããç 究çšãšããŠå¹²æžèšãçšããæ³¢é·æž¬å®åšãçšããããŠãããïŒèŠåºå
žïŒ ã¡ãŒãã«ååšã¯ããã€ã±ã«ãœã³ã®å®éšã®åœæã¯é·ãã®ããããšã®æšæºã ã£ããã1983幎以éã¯ã¡ãŒãã«ååšã¯é·ãã®æšæºã«ã¯çšããããŠããªããçŸåšã®ã¡ãŒãã«å®çŸ©ã¯ä»¥äžã®éãã
;ã¡ãŒãã«ã®å®çŸ©
:ç空äžã®å
ã®éã ''c'' ãåäœ [[W:ã¡ãŒãã«|m]] [[W:ç§|s]]<sup>â1</sup> ã§è¡šãããšãã«ããã®æ°å€ã {{val|299792458}} ãšå®ããããšã«ãã£ãŠå®çŸ©ãããã
:ããã§ãç§ã¯ã»ã·ãŠã åšæ³¢æ° ''âÎœ''<sub>Cs</sub> ã«ãã£ãŠå®çŸ©ãããã
==== å
é»å¹æã®æž¬å® ====
:ïŒâ» æªèšè¿°ïŒ
:ïŒâ» åè·¯å³ãè¿œå ããããšãïŒ
:ïŒâ» å®éšçµæã°ã©ããè¿œå ããããšãïŒ
[[File:Cellule photoelectriqie.JPG|thumb|300px|å
é»å¹æã®å®éš]]
[[File:Caracteristique courant tension (frequence fixe).JPG|thumb|300px|é»äœãšå
é»æµã®é¢ä¿]]
{{-}}
==== åè ====
倪éœé»æ± ããå
é»å¹æã®ãããªçŸè±¡ã§ããããšèããããŠãããïŒâ» å®æåºçã®æç§æžãªã©ã§ãæ±ã£ãŠãã話é¡ãïŒ
ãªãã倪éœé»æ± ã¯äžè¬çã«åå°äœã§ããããã€ãªãŒãåããPNæ¥åã®éšåã«å
ãåœãŠãå¿
èŠãããã
ïŒPNæ¥åéšå以å€ã®å Žæã«ãå
ãããã£ãŠããçããé»åããé»æµãšããŠåãåºããªããé»æµãšããŠåãåºããããã«ããã«ã¯ãPNæ¥åã®éšåã«ãå
ãåœãŠãå¿
èŠãããããã®ãããPNæ¥åã®çæ¹ã®æ質ããéæããããã«è¿ãå
ééçã®ææã«ããå¿
èŠãããããéæé»æ¥µããšãããïŒ
ïŒâ» ç¯å²å€ïŒ: ïŒ ãªããçºå
ãã€ãªãŒãåå°äœã¯ããã®éãã¿ãŒã³ãšããŠèããããŠãããå
é»å¹æã§ãããä»äºé¢æ°ãã«ããããšãã«ã®ãŒããã£ãé»æµãæµãããšã«ããããã®åå°äœç©è³ªã®ãä»äºé¢æ°ãã«ããããšãã«ã®ãŒã®å
ããPNæ¥åã®æ¥åé¢ããæŸåºãããããšããä»çµã¿ã§ããã
ãªããCCDã«ã¡ã©ãªã©ã«äœ¿ãããCCDã¯ã倪éœé»æ± ã®ãããªæ©èœããã€åå°äœããé»åæºãšããŠã§ã¯ãªããå
ã®ã»ã³ãµãŒãšããŠæŽ»çšãããšããä»çµã¿ã®åå°äœã§ãããïŒâ» å®æåºçã®æç§æžãªã©ã§ãæ±ã£ãŠãã話é¡ãïŒ
=== â» æ®éç§ã®ç¯å²å€: è¶
äŒå°ã®ç£æã®éåå ===
ïŒâ» æ®éç§é«æ ¡ã®ãç©çãç³»ç§ç®ã§ã¯ç¿ããªãããïŒ
ç©ççŸè±¡ã®éååãšããŠãå
é»å¹æãç©è³ªæ³¢ã®ã»ãã«ãååã¹ã±ãŒã«ã®ç©ççŸè±¡ã®éååã¯ãããããçš®é¡ã®è¶
äŒå°ç©è³ªã§ã¯ãããã«éããç£æãéååããçŸè±¡ãç¥ãããŠãããïŒâ» å·¥æ¥é«æ ¡ã®ç§ç®ãå·¥æ¥ææãäžå·»ïŒãŸãã¯ç§ç®ã®åŸåïŒã§ç¿ããïŒ
:â» ãå·¥æ¥ææãã®æç§æžã«ã¯æžãããŠãªã話é¡ã ããç£æã®éååã¯ãžã§ã»ããœã³çŽ åãªã©ãšããŠãé»å§èšæž¬ãªã©ã®åœå®¶æšæºåšãšããŠãæ¥æ¬ããµããäžçã®å·¥æ¥åœã®ååœã§æ¡çšãããŠããã
:â» ã¢ã€ã³ã·ã¥ã¿ã€ã³ã®ãå
éå仮説ããåŒã³åã®ãšããããããŸã§ä»®èª¬ãªäžæ¹ã§ãè¶
äŒå°ã«ãããç£æã®éååã¯ïŒèª¬ã§ã¯ãªãïŒèŠ³æž¬äºå®ã»å®éšäºå®ã§ãããå®éã«è¶
äŒå°ã«éããç£æãèªå°çŸè±¡ã§ã€ããé»å§ã粟å¯ã«æž¬å®ãããšãé»å§ã«ãŒããé段ç¶ã«ã®ã¶ã®ã¶ã«ãªã£ãããããïŒå³å¯ã«èšããšã芳枬ãããã®ã¯ç£æã®ã€ããèªå°é»å§ã®éååã ãã»ã»ã»ïŒ
:â» å·¥æ¥é«æ ¡ã§ã¯ãæè²ã®é åºãšããŠããããããããå
é»å¹æãããããè¶
äŒå°ã§ã®ç£æã®éååããå
ã«æããŠããå¯èœæ§ããããïŒæ®éç§ã®å°éãç©çãã3幎çã§æããäžæ¹ããå·¥æ¥ææãã¯1ïœ2幎çã§æããå Žåãããã®ã§ïŒãã²ãã£ãšãããå°æ¥çã«æ®éç§é«æ ¡ã§ããç£æã®éååããå
ã«æããå¯èœæ§ãããããã
=== Xç· ===
==== Xç·ã®çºèŠ ====
[[File:Rotating anode x-ray tube (labeled).jpg|thumb|250px|Xç·ç®¡<br>é°æ¥µããåºãé°æ¥µç·ãéœæ¥µã«ã¶ã€ãããšãã¶ã€ãã£ãæã«Xç·ãåºãã]]
[[File:Tube RX a fenetre laterale.png|thumb|Xç·ç®¡ã®åç]]
ç§åŠè
ã¬ã³ãã²ã³ã¯ã1895幎ãæŸé»ç®¡ããã¡ããŠé°æ¥µç·ã®å®éšãããŠãããšããæŸé»ç®¡ã®ã¡ããã«çœ®ããŠãã£ãåç也æ¿ãæå
ããŠããäºã«æ°ä»ããã
圌ïŒã¬ã³ãã²ã³ïŒã¯ãé°æ¥µç·ãã¬ã©ã¹ã«åœãã£ããšãããªã«ãæªç¥ã®ãã®ãæŸå°ãããŠããšèããXç·ãšåã¥ããã
ãããŠãããŸããŸãªå®éšã«ãã£ãŠãXç·ã¯æ¬¡ã®æ§è³ªããã€ããšãæããã«ãªã£ãã
:æ§è³ª: ç£å Žãé»å Žã§æ²ãããªãã
ãã®äºãããXç·ã¯ãè·é»ç²åã§ã¯ãªãäºãåãããïŒçµè«ããããšãXç·ã®æ£äœã¯ãæ³¢é·ã®çãé»ç£æ³¢ã§ãããïŒ
ãŸãã
:æ§è³ª: Xç·ãç
§å°ãããç©è³ªã¯ã€ãªã³ã«é»é¢ãããã
:æ§è³ª: å¯èŠå
ç·ãéããªãç©è³ªã§ããXç·ãªãééã§ããå Žåãããã
ãªã©ã®æ§è³ªãããã
ãªãçŸä»£ã§ã¯ãå»ççšã®Xç·ããã¬ã³ãã²ã³ããšãããã
==== Xç·ã®ã¹ãã¯ãã« ====
[[File:TubeSpectrum.jpg|thumb|240px|ç¹æ§Xç·ïŒKç·ïŒ]]
:ç¹æ§Xç·
:é£ç¶Xç·
{{-}}
==== Xç·ã®æ³¢åæ§ ====
1912幎ãç©çåŠè
ã©ãŠãšã¯ãXç·ãåçµæ¶ã«åœãŠããšãåçãã£ã«ã ã«å³ã®ãããªæç¹ã®æš¡æ§ã«ããããšãçºèŠãããããã'''ã©ãŠãšæç¹'''ïŒã¯ããŠãïŒãšãããçµæ¶äžã®ååãåææ Œåã®åœ¹å²ãããããšã§çºçããå¹²æžçŸè±¡ã§ããã
[[File:Bragg diffraction 2.svg|thumb|400px|ãã©ãã°ã®æ¡ä»¶]]
1912幎ãç©çåŠè
ãã©ãã°ã¯ãåå°ã匷ãããæ¡ä»¶åŒãçºèŠããã
:<math>2d\sin\theta = n\lambda</math>
ã'''ãã©ãã°ã®æ¡ä»¶'''ãšããã
äžåŒã®dã¯æ Œåé¢ã®ééã®å¹
ã§ããã
{{-}}
:ïŒâ» ç¯å²å€:ïŒ ãã£ãœããã¬ã©ã¹ãªã©éæ¶è³ªã®ææã®å Žåããã©ãã°åå°ã®ãããªæ確ãªåæã¯èµ·ããªããïŒâ» åèæç®: æ±äº¬ååŠå人ãç¡æ©ååŠ ãã®çŸä»£çã¢ãããŒã 第2çãã平尟äžä¹ ãªã©èã2013幎第2çã2014幎第2å·ïŒ
==== Xç·ã®ç²åæ§ ====
* ã³ã³ããã³å¹æ
Xç·ãççŽ å¡ãªã©ã®ïŒéå±ãšã¯éããªãïŒç©è³ªã«åœãŠããã®æ£ä¹±ãããããšã®Xç·ã調ã¹ããšãããšã®Xç·ã®æ³¢é·ãããé·ããã®ããæ£ä¹±ããXç·ã«å«ãŸããã
ãã®ããã«æ£ä¹±Xç·ã®æ³¢é·ã䌞ã³ãçŸè±¡ã¯ç©çåŠè
ã³ã³ããã³ã«ãã£ãŠè§£æãããã®ã§ã'''ã³ã³ããã³å¹æ'''ïŒãŸãã¯ã³ã³ããã³æ£ä¹±ïŒãšããã
[[File:Compton ex1.jpg||400px|thumb|right|ã³ã³ããã³ã«ããå®éšç¥å³ããªããå³äžã®ãåçµæ¶ãã¯æ³¢é·ã®æž¬å®çšã§ãã <ref>å島鮮ãåçéåååŠãïŒè£³è¯æ¿ã2014幎第40çãåçã¯1972幎ïŒ</ref> ããåçµæ¶ãã®æ質ã¯æ¹è§£ç³ã®çµæ¶ã§ãããæ£ä¹±æ³¢é·ã¯ãã©ãã°åå°ãªã©ã掻çšããŠæž¬å®ãããïŒã³ã³ããã³æ¬äººã®è«æâThe Spectrum of Scattered X-Raysâ(May 9, 1923).ã«ãæ¹è§£ç³ïŒcalciteïŒã䜿ã£ãŠããããšãšããã©ãã°åå°ïŒBragg ?ïŒãããŠããäºãæžãããŠãããïŒ]]
ãã®çŸè±¡ã¯ãXç·ãæ³¢ãšèããã®ã§ã¯èª¬æãã€ããªããïŒããä»®ã«æ³¢ãšèããå Žåãæ£ä¹±å
ã®æ³¢é·ã¯ãå
¥å°Xç·ãšåãæ³¢é·ã«ãªãã¯ãããªããªããæ°Žé¢ã®æ³¢ã«äŸãããªããããæ°Žé¢ãæ£ã§4ç§éã«1åã®ããŒã¹ã§æºãããããæ°Žé¢ã®æ³¢ãã4ç§éã«1åã®ããŒã¹ã§åšæãè¿ããã®ãšãåãçå±ãïŒ
ããŠãæ³¢åã®çè«ã§ã³ã³ããã³å¹æã説æã§ããªããªããç²åã®çè«ã§èª¬æãããã°è¯ãã ããã
ãã®åœæãã¢ã€ã³ã·ã¥ã¿ã€ã³ã¯å
éå仮説ã«ããšã¥ããå
åã¯ãšãã«ã®ãŒhÎœããã€ã ãã§ãªããããã«æ¬¡ã®åŒã§è¡šãããéåépããã€ããšãçºèŠããŠããã
<math>p=\frac{h\nu}{c}(=\frac{h\nu}{\nu \lambda}=\frac{h}{\lambda})</math>
ç©çåŠè
ã³ã³ããã³ã¯ããã®çºèŠãå©çšããæ³¢é·Î»ã®Xç·ããéåé<math>\frac{h}{\lambda}</math>
ãšãšãã«ã®ãŒ<math>\frac{hc}{\lambda}</math>ãæã€ç²åïŒå
åïŒã®æµããšèãã
Xç·ã®æ£ä¹±ãããã®å
åãç©è³ªäžã®ããé»åãšå®å
šåŒŸæ§è¡çªãããçµæãšèããã
:ã³ã³ããã³ã¯ãã®èãã«åºã¥ããå
åãšé»åã®è¡çªåã®éåéåãšãšãã«ã®ãŒåãè¡çªåŸãä¿åããããšä»®å®ããŠèšç®ããŠãå®éšçµæãšè¯ãåãããçµæãåŸãããããšãçºèŠããã
[[File:Compton effect illust.svg|thumb|400px|ã³ã³ããã³å¹æ<br>ãã®å³ãèŠããšãããããç空äžããã ããé»åã«é»ç£æ³¢ãç
§å°ããããã«èŠããããããã§ã¯ãªããã³ã³ããã³å¹æã®çºèŠããã1920幎代ã®åœæã«ã¯ããŸã ã空äžã«é»åããã ããããŠç²ŸåºŠããé»ç£æ³¢ãç
§å°ããæè¡ãªã©ãç¡ããå®éã«ã³ã³ããã³ãè¡ã£ãå®éšã¯ãç³å¢šã®ççŽ ãªã©ã®ç©è³ªã«Xç·ãç
§å°ããå®éšã§ãããå³äžã®é»åã¯ãççŽ ãªã©ã®ååãæäŸããé»åã§ããã<br>ã³ã³ããã³æ¬äººã®è«æã«ããã®ãããªæãã®å³ãæžãããŠãããããã§ãã®ãããªå³ãæ®åãããã®ãšæãããã]]
解æ³ã¯ãäžèšã®ãšããã
:ãšãã«ã®ãŒä¿åã®åŒãç«ãŠãã
:ãããŠãéåéã®ä¿åã®åŒãç«ãŠããå
·äœçã«ã¯ãx軞æ¹åã®éåéã®ä¿åã®åŒãšãy軞æ¹åã®éåéã®ä¿åã®åŒãç«ãŠãã
----
ãšãã«ã®ãŒä¿åã®åŒ
:<math>\frac{hc}{\lambda} = \frac{hc}{\lambda '} + \frac{1}{2}mv^2 \qquad \qquad</math> (1.2a)
éåéä¿åã®åŒ
:x軞: <math> \frac{h}{\lambda} =\frac{h}{\lambda '} \cos \theta + mv \cos \phi \quad</math> (1.2b)
:y軞: <math> 0 =\frac{h}{\lambda '} \sin \theta - mv \sin \phi \qquad</math>(1.2c)
----
äžèšã®3ã€ã®åŒãé£ç«ãããã®é£ç«æ¹çšåŒã解ãããã«vãšÏãé£ç«èšç®ã§æ¶å»ãããŠããã<math>\lambda \fallingdotseq \lambda '</math>ã®ãšãã«
<math>\lambda ' \fallingdotseq \lambda + \frac{h}{mc} (1 -\cos \theta )</math>
ãåŸãããã
ãã®åŒãå®éšåŒãšããäžèŽããã®ã§ãã³ã³ããã³ã®èª¬ã®æ£ããã¯å®èšŒãããã
----
* ã³ã³ããã³å¹æã®é£ç«æ¹çšåŒã®å
·äœçãªè§£æ³
ïŒç·šéè
ãž: èšè¿°ããŠãã ãããïŒïŒGimyamma ããã解æ³ãæžããŠã¿ãŸãããïŒ
----
åŒ(1.2a),(1.2b),(1.2c)ããã<math>v</math>ãš<math>\phi</math>ãæ¶å»ããŠã
<math>\lambda,\lambda ',\theta</math>ã®é¢ä¿åŒãæ±ããã°ããã
:â
°ïŒãŸããåŒ(1.2b),(1.2c)ãã<math>\phi</math>ãæ¶å»ããã
:åŒ(1.2b)ãã
:<math>(mv \cos \phi)^2 = (\frac{h}{\lambda}-\frac{h}{\lambda '} \cos \theta)^2
</math>
:åŒ(1.2c)ãã
:<math>(mv \sin \phi)^2 = (-\frac{h}{\lambda '} \sin \theta)^2</math>
:ãã®äž¡åŒãå ãããš
:<math>m^2 v^2 = (\frac{h}{\lambda}-\frac{h}{\lambda '} \cos \theta)^2+(-\frac{h}{\lambda '} \sin \theta)^2+\frac{h^2}{\lambda '^2}</math>
:ãã®å³èŸºãæŽé ãããšãææã®
:<math>m^2 v^2 =\frac{h^2}{\lambda^2}-2\frac{h^2}{\lambda \lambda '}\cos \theta
+\frac{h^2}{\lambda '^2}\quad</math> (1.2d)
ãåŸãã
:â
±ïŒåŒ(1.2d)ãåŒ(1.2e)ã«ä»£å
¥ããŠvãæ¶å»ãã:
åŒ(1.2a)ã®å³èŸºã®ç¬¬2é
ãå€åœ¢ããŠåŒ(1.2d)ã代å
¥ããã
:<math>\frac{1}{2}mv^2 =\frac{1}{2m}m^2v^2 = \frac{1}{2m}\bigl(\frac{h^2}{\lambda^2}-2\frac{h^2}{\lambda \lambda '}\cos \theta\bigr)+\frac{h^2}{\lambda '^2}</math>
ãããåŒ(1.2a)ã®å³èŸºã«ä»£å
¥ãããš
:<math>\frac{hc}{\lambda} = \frac{hc}{\lambda '} + \frac{1}{2m}\Bigl(\frac{h^2}{\lambda^2}-2\frac{h^2}{\lambda \lambda '}\cos \theta +\frac{h^2}{\lambda '^2}\Bigr)</math>
䞡蟺ã<math>hc</math>ã§å²ããš
:<math>\frac{1}{\lambda} = \frac{1}{\lambda '} + \frac{h}{2mc}\Bigl(\frac{1}{\lambda^2}-2\frac{1}{\lambda \lambda '}\cos \theta +\frac{1}{\lambda '^2}\Bigr)</math> (1.2e)
ãåŸãã
ãã®åŒã®å³èŸºã®ç¬¬2é
ã®æ¬åŒ§å
ã次ã®ããã«å€åœ¢ããã
:<math>\frac{1}{\lambda^2}-2\frac{1}{\lambda \lambda '}\cos \theta +\frac{1}{\lambda '^2}=\bigl(\frac{1}{\lambda}-\frac{1}{\lambda'}\bigr)^2+\frac{2}{\lambda \lambda'}(1-\cos \theta)</math>
ãã®åŒãåŒ(1.2e)ã®å³èŸºç¬¬2é
ã«ä»£å
¥ãããš
:<math>\frac{1}{\lambda} = \frac{1}{\lambda '} +
\frac{h}{2mc}\Bigl(
\bigl(\frac{1}{\lambda}-\frac{1}{\lambda'}\bigr)^2+\frac{2}{\lambda \lambda'}(1-\cos \theta)
\Bigr)</math>
ãã®åŒã®å³èŸºã®ç¬¬ïŒé
ã移è¡ããåŒãå€åœ¢ãããš
:<math>\frac{\lambda'-\lambda}{\lambda\lambda '}=
\frac{h}{2mc}\Bigl(
\bigl(\frac{\lambda'-\lambda}{\lambda \lambda'}\bigr)^2+\frac{2}{\lambda \lambda'}(1-\cos \theta)
\Bigr)</math>
䞡蟺ã«<math>\lambda \lambda'</math>ãæãããš
:<math>\lambda'-\lambda=
\frac{h}{2mc}\Bigl(
\frac{(\lambda'-\lambda)^2}{\lambda \lambda'}+2(1-\cos \theta)
\Bigr)</math> (1.2f)
Xç·ã®æ£ä¹±ã§ã¯ã<math>\lambda'\fallingdotseq \lambda</math>ãªã®ã§
:<math>\frac{(\lambda'-\lambda)^2}{\lambda \lambda'}</math>ã¯ãæ³¢é·ã«æ¯ã¹ãŠéåžžã«å°ããå€ã«ãªãç¡èŠã§ããã
æ
ã«åŒ(1.2f)ãã
:<math>\lambda'-\lambda \fallingdotseq
\frac{h}{mc}
(1-\cos \theta)
\qquad</math> (1.2g)
ããã§ãææã®åŒãå°åºãããã
----
==== ç¯å²å€: å
åã®æµäœååŠç解éãšéåéå¯åºŠ ====
[[File:Photon-fluid-understanding jp.svg|thumb|400px|å
åã®æµäœååŠç解é]]
å
ã®éåé Pkgã»m/s=hÎœ/ïœ ã«ã€ããŠã
ãŸã cPïŒhΜJ ãšå€åœ¢ããŠã¿ããšããé床ã«éåéãããããã®ããšãã«ã®ãŒã§ããããšããå
容ã®å
¬åŒã«ãªã£ãŠããã
ãããç解ãããããã²ãšãŸããå
ãç²åã§ãããšåæã«æµäœã§ãããšèããŠããã®é»ç£æ³¢ãåäœäœç©ãããã®éåépãæã£ãŠãããšããŠããã®æµäœã®éåéã®å¯åºŠïŒéåéå¯åºŠïŒã p ïŒkgã»m/sïŒ/m<sup>3</sup>ãšãããããã®å Žåã®é»ç£æ³¢ã¯æµäœãªã®ã§ãéåéã¯ããã®å¯åºŠã§èããå¿
èŠãããã
é»ç£æ³¢ãç©äœã«ç
§å°ããŠãå
ãç©äœã«åžåããããšããããåå°ã¯ãªããšããŠãå
ã®ãšãã«ã®ãŒã¯ãã¹ãŠç©äœã«åžåããããšãããç°¡åã®ãããç©äœå£ã«åçŽã«å
ãç
§å°ãããšãããç©äœãžã®å
ã®ç
§å°é¢ç©ãAm<sup>2</sup>ãšããã
é»ç£æ³¢ã¯å
é cm/s ã§é²ãã®ã ãããå£ããcã®è·é¢ã®éã«ãããã¹ãŠã®å
åã¯ããã¹ãŠåäœæéåŸã«åžåãããäºã«ãªããåäœæéã«å£ã«åžåãããå
åã®éã¯ããã®åäœæéã®ããã ã«å£ã«æµã蟌ãã å
åã®éã§ããã®ã§ã
å³ã®ããã«ãä»®ã«åºé¢ãAïœ<sup>2</sup>ãšããŠãé«ãhã c ïŒ hã®å€§ããã¯cã«çãããåäœæétïŒ1ãããããšããã° hïŒcã»1 ã§ããïŒïŒ»mãšããæ±ã®äœç©ãAÃcm<sup>3</sup>äžã«å«ãŸããå
åã®éã®ç·åã«çããã
ãã£ãœããéåéå¯åºŠã¯ pïŒkgã»m/sïŒ/m<sup>3</sup>ã ã£ãã®ã§ããã®æ± AÃh ã«å«ãŸããéåéã®ç·åã¯ã
AÃhÃpkgã»m/sã§ããã
å
ãåžåããç©äœã®éåéã¯ãåäœæéã«Ahpã®éåéãå¢å ããããšã«ãªãããhïŒcã§ãã£ãã®ã§ãã€ãŸããéåéãåäœæéãããã« Acpkgã»m/s ã ãå£ã«æµãããããšã«ãªãã
ãã£ãœããé«æ ¡ç©çã®ååŠã®çè«ã«ããããéåéã®æéãããã®å€åã¯ãåã§ãããã§ãã£ãã®ã§ãã€ãŸãç©äœã¯ãAïœpNã®åãåããã
åãåããã®ã¯ç
§å°ãããé¢ã ãããåNãé¢ç©ã§å²ãã°å§åã®æ¬¡å
N/m<sup>2</sup>ïŒïŒ»Paã«ãªãã
å®éã«é¢ç©ã§å²ãèšç®ãããã°ãå§åãšã㊠cpN/ïœ<sup>2</sup>ïŒ[Pa]ïŒ[J/ïœ<sup>3</sup>] ãåããäºãèšç®çã«åãããããã«ãå§åã®æ¬¡å
ã¯ïŒ»N/ïœ<sup>2</sup>ïŒ[Pa]ïŒ[J/ïœ<sup>3</sup>]ãšå€åœ¢ã§ããã®ã§ããå§åã¯ãåäœäœç©ãããã®ãšãã«ã®ãŒã®å¯åºŠ(ããšãã«ã®ãŒå¯åºŠããšãã)ã§ããããšèãããã
ãšããã° cp ã®æ¬¡å
ã¯ãå§åïŒïŒ»ãšãã«ã®ãŒå¯åºŠïŒœ ãšãªãã
ãã®ãšãã«ã®ãŒå¯åºŠã«ãhÎœã察å¿ããŠãããšèããã°ãåççã§ããã
èŠããã«ãå
ã®ãããªãäºå®äžã¯ç¡éã«å§çž®ã§ããæ³¢ã»æµäœã§ã¯ã
å
¬åŒãšããŠãé床ãvãéåéå¯åºŠãpããšãã«ã®ãŒå¯åºŠãεãšããŠèããã°ã
:vpïŒÎµ
ãšããé¢ä¿ããªããã€ã
ïŒãªããæ°Žã空æ°ã®ãããªæ®éã®æµäœã§ã¯ãç¡éã«ã¯å§çž®ã§ããªãã®ã§ãäžèšã®å
¬åŒã¯æãç«ããªããïŒ
ãããããã³ã³ããã³å¹æã®åŠç¿ã§åãã£ãéåéã®å
¬åŒ <math>p=\frac{h\nu}{c}</math> ã¯ãéåéå¯åºŠãšãšãã«ã®ãŒå¯åºŠã®é¢ä¿åŒã«ãå
écãšå
é»å¹æã®ãšãã«ã®ãŒhÎœã代å
¥ãããã®ã«ãªã£ãŠããã
äžèšã®èå¯ã¯ãå
ãæµäœãšããŠèããé»ç£æ³¢ã®éåéã ããç²åãšããŠè§£éãããå
åã®éåéã«ãã
ïœPïŒhÎœ ãšããé¢ä¿ãæãç«ã€ãšèãããã
ããèªè
ããå§åããšãã«ã®ãŒå¯åºŠãšèããã®ãåããã¥ãããã°ãããšãã°ç±ååŠã®ä»äºã®å
¬åŒ W=Pâ¿V ã®é¡æšãããŠã¯ã©ããïŒ ãªããäžèšã®éåéãšãšãã«ã®ãŒã®é¢ä¿åŒã®å°åºã¯å€§ãŸããªèª¬æã§ãããæ£ç¢ºãªå°åºæ³ã¯ãïŒå€§åŠã§ç¿ãïŒãã¯ã¹ãŠã§ã«ã®æ¹çšåŒã«ãããªããã°ãªããªãã
ããããã ãå
ã¯ãé»åã«äœçšãããšãã«ãå
ãç²åãšããŠæ¯èãïŒãµããŸãïŒã ãšããã®ãæ£ããã ããã
ãã£ãœãããã¿ããã«ãå
ã¯ç²åïŒãå
ã¯æ³¢åã§ã¯ãªãïŒïŒãïŒÃïŒãšãããã®ã¯ãåãªã銬鹿ã®ã²ãšã€èŠãã§ããã
ãã¯ã¹ãŠã§ã«ã®æ¹çšåŒã§ã¯ãå
ïŒé»ç£æ³¢ïŒã¯æ³¢åãšããŠããã€ããã®ã§ããã
ããããå
é»å¹æã§èµ·ããçŸè±¡ã§ã¯ãæŸåºé»åã®ãã€éåãšãã«ã®ãŒã¯ãå
ã®åŒ·åºŠãšã¯ç¡é¢ä¿ã§ãããåçŽãªæµäœãšããŠèãããªããïŒããšãã°éå
ããããå
ãéãããããŠãïŒå
ã®åŒ·åºŠãäžããã°ãéåéå¯åºŠãäžããããºã ãããã®åž°çµã®æŸåºé»åã®ãšãã«ã®ãŒå¯åºŠãäžããããºã ããããšããäºæž¬ãæãç«ã¡ããã ãããããå®éšçµæã¯ãã®äºæž¬ãšã¯ç°ãªããå
é»å¹æã¯å
ã®åŒ·åºŠãšã¯ç¡é¢ä¿ã«å
ã®åšæ³¢æ°ã«ãã£ãŠæŸåºé»åã®ãšãã«ã®ãŒã決ãŸããã»ã»ã»ãšããã®ããå®éšäºå®ã§ããã
ãã®ãããªå®éšçµæããã20äžçŽåé ã®åœæãåèããŠããéåååŠãªã©ãšé¢é£ã¥ããŠããå
ãæ³¢ã§ãããšåæã«ç²åã§ããããšæå®ããã®ãããŒãã«è²¡å£ãªã©ã§ãããå
é»å¹æãå
ã®ç²å説ã®æ ¹æ ã®ã²ãšã€ãšããã®ãã¢ã€ã³ã·ã¥ã¿ã€ã³ã®ä»®èª¬ã§ãããã¢ã€ã³ã·ã¥ã¿ã€ã³ã®ãã®ä»®èª¬ãå®èª¬ãšããŠèªå®ããã®ãããŒãã«è²¡å£ã§ãããçŸåšã®ç©çåŠã§ã¯ãå
é»å¹æãå
å説ã®æ ¹æ ãšããŠé説ã«ãªã£ãŠããã
å
é»å¹æã®å®éšçµæãã®ãã®ã¯ãåã«ãå
é»å¹æã«ããããå
ãããåçŽãªæµäœã»æ³¢åãšããŠã¯èããããªãã ããã»ã»ã»ãšããã ãã®äºã§ããã
çµå±ãç©çåŠã¯å®éšç§åŠã§ãããå®éšçµæã«ããšã¥ãå®éšæ³åãèŠãããããªãããå
åããšããã¢ã€ãã¢ã¯ããå
é»å¹æã®æŸåºé»å1åãããã®ãšãã«ã®ãŒã¯ãå
¥å°å
ã®åŒ·åºŠã«å¯ãããå
ã®æ³¢é·ïŒåšæ³¢æ°ïŒã«ããããšããäºãèŠããããããããã®æ段ã«ããããã¢ã€ã³ã·ã¥ã¿ã€ã³ãšãã®æ¯æè
ã«ãšã£ãŠã¯ããå
ã®ç²å説ããšããã®ããèŠããããããããã®ã¢ãã«ã ã£ãã ãã§ããïŒç²åãªã®ã«æ³¢é·ïŒåšæ³¢æ°ïŒãšã¯ãæå³äžæã ãïŒããããŠéåéå¯åºŠãšãšãã«ã®ãŒå¯åºŠã®é¢ä¿ vpïŒÎµ ãšããç¥èããŸããå
é»å¹æã®å
¬åŒ cPïŒhÎœ ãèŠããããããããã®æ段ã«ãããªãã
ãã£ããã®å
ã¯ãåçŽãªæ³¢ã§ããªããåçŽãªç²åã§ããªãããã åã«ãå
ã¯å
ã§ãããå
ã§ãããªãã
ãå
ã®ç²å説ããšããã®ã¯ãç空äžã§åªè³ªïŒã°ããã€ïŒããªããŠãå
ãäŒããããšããçšåºŠã®æå³åãã§ãããªãã ãããã¢ã€ã³ã·ã¥ã¿ã€ã³ãç¹æ®çžå¯Ÿæ§çè«ãçºè¡šããåãŸã§ã¯ãïŒ20äžç€ä»¥éããçŸä»£ã§ã¯åŠå®ãããŠããããïŒãã€ãŠããšãŒãã«ããšããå
ãäŒããåªè³ªã®ååšãä¿¡ããããŠããããããã¢ã€ã³ã·ã¥ã¿ã€ã³ã¯çžå¯Ÿæ§çè«ã«ããããšãŒãã«ã®ååšãåŠå®ããã
ãå
ã®ç²å説ããçºè¡šããŠããè
ãåããã¢ã€ã³ã·ã¥ã¿ã€ã³ã ã£ãã®ã§ãããŒãã«è²¡å£ã¯ãæ¬æ¥ãªãç¹çžå¯Ÿæ§çè«ã§ããŒãã«è³ãæãããããã«ãå
å説ã§ããŒãã«è³ãã¢ã€ã³ã·ã¥ã¿ã€ã³ã«æããã ãã ãã
=== ç²åã®æ³¢åæ§ ===
==== ç©è³ªæ³¢ ====
ç©çåŠè
ãã»ããã€ã¯ãæ³¢ãšèããããŠãå
ãç²åã®æ§è³ªããã€ãªãã°ããã£ãšé»åãç²åãšããŠã®æ§è³ªã ãã§ãªããé»åãæ³¢åã®ããã«æ¯èãã ãããšèããã
ãããŠãé»åã ãã§ãªããäžè¬ã®ç²åã«å¯ŸããŠãããã®èããé©çšãã次ã®å
¬åŒãæå±ããã
:質émãéãvã®ç²åã¯æ³¢åæ§ããã¡ããã®æ³¢é·ã¯æ¬¡åŒã§äžããããã
:<math>\lambda = \frac h {mv} </math>
ããã¯ãã»ããã€ã«ãã仮説ã§ãã£ãããçŸåšã§ã¯æ£ãããšèªããããŠããã
ãã®æ³¢ã¯ã'''ç©è³ªæ³¢'''ïŒmaterial waveïŒãšåŒã°ããã'''ãã»ããã€æ³¢'''ïŒde Broglie wave lengthïŒãšãããã
ããªãã¡ãå
åãé»åã«éãããããããç©è³ªã¯ç²åæ§ãšæ³¢åæ§ããããæã€ãšãããã
ãã®ç©è³ªæ³¢ãšãã説ã«ãããšããããé»åç·ãç©è³ªã«åœãŠãã°ãåæãªã©ã®çŸè±¡ãèµ·ããã¯ãã§ããã
1927幎ã1928幎ã«ãããŠãããããœã³ãšã¬ãŒããŒã¯ãããã±ã«ãªã©ã®ç©è³ªã«é»åç·ãåœãŠãå®éšãè¡ããXç·åæãšåæ§ã«é»åç·ã§ãåæãèµ·ããããšãå®èšŒãããæ¥æ¬ã§ã1928幎ã«èæ± æ£å£«ïŒããã¡ ãããïŒãé²æ¯çã«é»åç·ãåœãŠãå®éšã«ããåæãèµ·ããããšã確èªããã
é»åç·ã®æ³¢é·ã¯ãé«é»å§ããããŠé»åãå éããŠé床ãé«ããã°ãç©è³ªæ³¢ã®æ³¢é·ã¯ããªãå°ããã§ããã®ã§ãå¯èŠå
ã®æ³¢é·ãããå°ãããªãã
ãã®ãããå¯èŠå
ã§ã¯èŠ³æž¬ã§ããªããã£ãçµæ¶æ§é ããé»åæ³¢ãXç·ãªã©ã§èŠ³æž¬ã§ããããã«ãªã£ããçç©åŠã§ãŠã€ã«ã¹ãé»åé¡åŸ®é¡ã§èŠ³æž¬ã§ããããã«ãªã£ãã®ããé»åã®ç©è³ªæ³¢ãå¯èŠå
ããã倧å¹
ã«å°ããããã§ããã
=== ç²åãšæ³¢åã®äºéæ§ ===
{{ã³ã©ã |åè: é»åããŒã ã«ããæ³¢åæ§ã®å¹²æžå®éš|
[[Image:Egun.jpg|thumb|250px|right|ãã©ãŠã³ç®¡ã®é»åé]]
[[ãã¡ã€ã«:double-slit.svg|thumb|right|350px|é»åã®äºéã¹ãªããã®å¹²æžå®éš]]
[[ãã¡ã€ã«:Doubleslitexperiment_results_Tanamura_1.gif|thumb|left|250px|äºéã¹ãªããå®éšã®çµæ]]
:ïŒâ» æ€å®æç§æžã§ç¿ãç¯å²å
ã§ããïŒ
é»åéïŒã§ãããã
ãïŒãšããå®éšè£
眮ããããéãšãã£ãŠããã¹ã€ã«SFã®ãããªå
µåšã§ã¯ãªããé»åéãšã¯åã«é»åãæŸåºããã ãã®è£
眮ã§ããã
ããŠããã®é»åéããã¡ããŠã1åã¥ã€é»åãåœãŠãå®éšããäºéã¹ãªããã䜿ã£ãŠå®éšãããšãå³ã®ããã«ãæ³¢åã®ããã«ãé»åã®å€ãåœãã£ãå Žæãšé»åã®å°ãªãåœããå Žæãšã®çžæš¡æ§ãã§ããã
{{-}}
ãã®ããã«ãé»åã«ãç²åæ§ãšæ³¢åæ§ããããé»åã¯ç²åã§ããã€ã€ãäºéã¹ãªããã«åãã£ãŠé»åãæã¡èŸŒããšå¹²æžãèµ·ãããšããæ³¢åæ§ãæã£ãŠããã
}}
äžè¿°ã®ãããªãããŸããŸãªå®éšã®çµæããããã¹ãŠã®ç©è³ªã«ã¯ãååãŠãã©ã®å€§ããã®äžçïŒä»¥éãåã«ãååã¹ã±ãŒã«ããªã©ãšç¥èšããïŒã§ã¯ãæ³¢åæ§ãšç²åæ§ã®äž¡æ¹ã®æ§è³ªããã€ãšèããããŠããã
ãã®ããšã'''ç²åãšæ³¢åã®äºéæ§'''ãšããã
* åè: äžç¢ºå®æ§åç
[[File:Bundesarchiv Bild183-R57262, Werner Heisenberg.jpg|thumb|ç©çåŠè
ãã€ãŒã³ãã«ã° <br>äžç¢ºå®æ§åçã®äž»èŠãªæå±è
ã§ããã]]
ãããŠãååã¹ã±ãŒã«ã§ã¯ãããäžã€ã®ç©è³ªïŒäž»ã«é»åã®ãããªç²åïŒã«ã€ããŠããã®äœçœ®ãšéåéã®äž¡æ¹ãåæã«æ±ºå®ããäºã¯ã§ããªãããã®ããšã'''äžç¢ºå®æ§åç'''ïŒãµãããŠããã ãããïŒãšããã
{{-}}
== ååã»ååæ žã»çŽ ç²å ==
===åå===
[[File:Geiger-Marsden experiment expectation and result (Japanese).svg|right|400px|thumb|]]
ç©çåŠè
ã¬ã€ã¬ãŒãšç©çåŠè
ããŒã¹ãã³ã¯ãïŒã©ãžãŠã ããåºããïŒÎ±ç²åããããéã±ãã«åœãŠãå®éšãè¡ããαç²åã®æ£ä¹±ã®æ§åã調ã¹ããïŒãªããαç²åã®æ£äœã¯ããªãŠã ã®ååæ žãïŒãã®çµæãã»ãšãã©ã®Î±ç²åã¯éã±ããçŽ éãããããéã±ãäžã®äžéšã®å Žæã®è¿ããéã£ãαç²åã ãã倧å¹
ã«æ£ä¹±ããçŸè±¡ãçºèŠããã
ãã®å®éšçµæããã©ã¶ãã©ãŒãã¯ãååæ žã®ååšãã€ããšããã
ååã¯ãäžå¿ã«ååæ žãããããã®ãŸãããé»åãéåãããšããã©ã¶ãã©ãŒãã¢ãã«ãšãã°ããã¢ãã«ã«ãã£ãŠèª¬æãããã
ååïŒatomïŒã¯ãå
šäœãšããŠã¯é»æ°çã«äžæ§ã§ãããè² ã®é»è·ãæããé»åãé»åæ®»ã«æã€ã
ããã§ãããªã«ã³ã®å®éš ã«ããçµæãªã©ãããé»åã®è³ªéã¯æ°ŽçŽ ã€ãªã³ã®è³ªéã®çŽ1/1840çšåºŠãããªãããšãåãã£ãŠããã
ããªãã¡ãååã¯é»åãšéœã€ãªã³ãšãå«ãŸãããã質éã®å€§éšåã¯éœã€ãªã³ããã€ããšãåããã
ååæ žã®å€§ããã¯ååå
šäœã®1/10000çšåºŠã§ãããããååã®å€§éšåã¯ç空ã§ããã
ååæ žã¯ãæ£ã®é»è·ããã€Zåã®éœåïŒprotonïŒãšãé»æ°çã«äžæ§ãª(AâZ)åã®äžæ§åïŒneutronïŒãããªãã
éœåãšäžæ§åã®åæ°ã®åèšã質éæ°ïŒmass numberïŒãšããã
éœåãšäžæ§åã®è³ªéã¯ã»ãŒçãããããååæ žã®è³ªéã¯ã質éæ°Aã«ã»ãŒæ¯äŸããã
==== æ°ŽçŽ ååã®ã¹ãã¯ãã« ====
é«æž©ã®ç©äœããçºå
ãããå
ã«ã¯ãã©ã®ïŒå¯èŠå
ã®ïŒè²ã®æ³¢é·ïŒåšæ³¢æ°ïŒãããããã®ãããªé£ç¶çãªæ³¢é·ã®å
ãé£ç¶ã¹ãã¯ãã«ãšããã
ãã£ãœãããããªãŠã ãæ°ŽçŽ ãªã©ã®ãç¹å®ã®ç©è³ªã«é»å§ãããããæŸé»ãããšãã«çºå
ããæ³¢é·ã¯ãç¹å®ã®æ°æ¬ã®æ³¢é·ããå«ãŸããŠãããããã®ãããªã¹ãã¯ãã«ãèŒç·ïŒãããïŒãšããã
ãã«ããŒã¯ãæ°ŽçŽ ååã®æ°æ¬ããèŒç·ã®æ³¢é·ãã次ã®å
¬åŒã§è¡šçŸã§ããããšã«æ°ã¥ããã
:<math>\lambda = 3.65 \times 10^{-7} \mathrm{m} \times \left( {n^2 \over n^2 - 4} \right).\quad(n=3,\ 4,\ 5,\ 6,\cdots\cdots)</math> (2.1)
äžåŒäžã®ãmãã¯ã¡ãŒãã«åäœãšããæå³ãïŒäžåŒã®mã¯ä»£æ°ã§ã¯ãªãã®ã§ãééããªãããã«ãïŒ
ãã®åŸãæ°ŽçŽ ä»¥å€ã®ååããå¯èŠå
以å€ã®é åã«ã€ããŠããç©çåŠè
ãã¡ã«ãã£ãŠèª¿ã¹ããã次ã®å
¬åŒãžãšãç©çåŠè
ãªã¥ãŒãããªã«ãã£ãŠããŸãšããããã
:<math>\frac{1}{\lambda} =R \left( \frac{1}{m^2} -\frac{1}{n^2} \right).\ \left(\begin{array}{lcl}m =1,\ 2,\ 3,\cdots\cdots, \\ n = m+1,\ m+2,\ m+3,\cdots\cdots \end{array}\right)</math> (2.2)
äžåŒã®Rã¯ãªã¥ãŒãããªå®æ°ãšããã<math>R=1.097 \times 10^7</math>/mã§ããã
==== éåè«ãšååã®æ§é ====
[[File:Stationary wave Quantum rule in atom.svg|thumb|300px|ååå
ã®å®åžžæ³¢]]
ã©ã¶ãã©ãŒãã®ååæš¡åã«åŸãã°ãé»åã¯ããŸãã§ææã®å
¬è»¢ã®ããã«ååæ žãäžå¿ãšããåè»éã®äžãäžå®ã®é床ã§éåããã
:åéåãã質ç¹ã¯å é床ããã€ã®ã§ããã®ã¢ãã«ã®é»åã¯å é床éåãç¶ããããšã«ãªãã
:ãšãããå€å
žé»ç£æ°åŠã®åéã§ãå é床éåããããªãé»è·ã¯é»ç£æ³¢ãæŸåºããŠããšãã«ã®ãŒã倱ããšããæ³åãæ¢ã«çºèŠãããŠããã
:ãã®æ³åã«ããã°ãååæ žã®åšãããŸããé»åã¯é»ç£æ³¢ãæŸåºãç¶ãããšãã«ã®ãŒã絶ããæžãããŠãããããã«ã€ããŠé»åã¯ååæ žã«åããŠèœäžããŠãããããååæ žãšã®è·é¢ãå°ããããªããååæ žã®åšããå転ãããããŠååæ žã«è¡çªããŠããŸããåè»éã®äžãå®å®çã«éåããããšã¯äžå¯èœãªã®ã§ããã
:ç©çåŠè
ããŒã¢ã¯ã©ã¶ãã©ãŒãã®ååæš¡åã®æ·±å»ãªççŸãå
æããããã«æ°ŽçŽ ååã®æŸåºããç·ã¹ãã¯ãã«ã«ã€ããŠã説æã§ããååæš¡åãäœãããã
:ãã©ã³ã¯ã®æå±ãããšãã«ã®ãŒéååã®èããšã¢ã€ã³ã·ã¥ã¿ã€ã³ã®å
éåè«ãåãå
¥ãã倧èãªä»®èª¬ãç«ãŠã(1913幎ïŒã
*仮説1ïŒéåæ¡ä»¶
ååæ žãäžå¿ãšããååŸïœ[m]ã®åè»éãéãïœ[m/s]ã§å転ããé»åã®è»éè§éåé<math>rp=mrv</math>ã¯<math>\frac{h}{2\pi}</math>ã®æ£æŽæ°åãããšãããªãïŒããªãã¡
:<math>mrv=n\frac{h}{2\pi} \quad (n=1,\ 2,\ 3,\cdots\cdots)</math> (2.3)
ãæºãããã°ãªããªãïŒè§éåéã®éååïŒããã®ç¶æ
ã'''å®åžžç¶æ
'''ïŒãã®æ¡ä»¶ã'''éåæ¡ä»¶'''ãšããã
:ããã§ãm[kg]ã¯é»åã®è³ªéããh㯠[[w:ãã©ã³ã¯å®æ° |ãã©ã³ã¯å®æ°]]ã§ããã
:ãã®ããŒã¢ã®åŒã®æ£æŽæ°nã'''éåæ°'''ïŒããããããïŒãšããã
åŸå¹Ž(1924幎ïŒããã»ããã€ã¯ãç©è³ªç²åã¯æ³¢åæ§ãæã¡ããã®æ³¢ïŒç©è³ªæ³¢ïŒã¯ãæ³¢é·
:<math>\lambda=\frac{h}{p}=\frac{h}{mv}</math>
ããã€ããšæå±ããããŸãïŒ(2.3)ãå€åœ¢ãããš
:<math>2\pi r=n\frac{h}{mv}=n\lambda</math>.
ãããã¯é»åã®è»éäžåšã®é·ããé»åã®ç©è³ªæ³¢ã®æ³¢é·ã®æ£æŽæ°åã®ãšãïŒé»åæ³¢ã¯å®åžžæ³¢ã«ãªãããšã瀺ããŠããã
:ããã¯ãåè»éäžã«å®åžžæ³¢ãã§ããããã®æ¡ä»¶ãšåãã§ããã
:â» æ€å®æç§æžã§ããããŒã¢ã®åŒã®è¡šèšã¯ãé床vãã€ãã£ãŠè¡šãããè¡šèšã§ããã
*仮説2ïŒæ¯åæ°æ¡ä»¶
é»åã¯ããããŸã£ããšã³ãšã³ã®ãšãã«ã®ãŒããæããªãããã®ãšã³ãšã³ã®ãšãã«ã®ãŒå€ããšãã«ã®ãŒé äœãšããã
:é»åããšãã«ã®ãŒé äœã<math>E'</math>ãã<math>E(<E')</math>ã«é·ç§»ããïŒãšãã«ã®ãŒã倱ã)ãšãã«ã¯ã<math>E'-E=h\nu</math>ã§ããŸãæ¯åæ°<math>\nu</math>ã®äžåã®å
åãæŸåºãã
:éã«ãšãã«ã®ãŒé äœEã®é»åãå€éšãããšãã«ã®ãŒ<math>h\nu = E'-E</math>ãåŸããšããšãã«ã®ãŒé äœE'ã«é·ç§»ããã
==== ãšãã«ã®ãŒæºäœ ====
[[File:Circular-motion-electron-in-atom jp.svg|thumb|400px|æ°ŽçŽ ååå
ã§ã®é»åã®åéå]]
æ°ŽçŽ ååã«ãããŠãé»åè»éäžã«ããé»åã®ãšãã«ã®ãŒãæ±ããèšç®ããããããŸãããã®ããã«ã¯ãååã®ååŸãæ±ããå¿
èŠãããã
* ååŸ
æ°ŽçŽ ã®é»åãååæ ž<math>H^+</math>ãäžå¿ãšããååŸïœã®åè»éäžãäžå®ã®é床ïœã§éåããŠãããšããã°ãéåæ¹çšåŒã¯
:<math> m \frac{v^2}{r} = k_0 \frac {e^2}{r^2} </math>
ã§è¡šãããã
äžæ¹ãé»åãå®åžžæ³¢ã®æ¡ä»¶ãæºããå¿
èŠãããã®ã§ãåé
ã®åŒïŒïŒïŒããã
:<math> v (= v_n) = \frac {nh}{2 \pi m r } \qquad \qquad (2)</math>
ã§ããã
ãã®vãããã»ã©ã®åéåã®åŒã«ä»£å
¥ããŠæŽé ããã°ã
:<math> r(=r_n) = \frac {h^2}{4 \pi ^2 k_0 me^2} n^2\qquad \qquad (3)</math>
ïŒãã ããnïŒ1, 2 , 3 ,ã»ã»ã»ïŒ
ã«ãªããããããŠãæ°ŽçŽ ååã®é»åã®è»éååŸãæ±ãŸãã
ããã»ã©ã®è»éååŸã®åŒã§nïŒ1ã®ãšãååŸr<sub>1</sub>ããããŒã¢ååŸããšããã
* ãšãã«ã®ãŒæºäœ
ååã®äžçã§ããéåãšãã«ã®ãŒKãšäœçœ®ãšãã«ã®ãŒUã®åãããšãã«ã®ãŒã§ããã
äœçœ®ãšãã«ã®ãŒUã¯ããã®æ°ŽçŽ ã®é»åã®å Žåãªããéé»æ°ãšãã«ã®ãŒãæ±ããã°å
åã§ãããé»äœã®åŒã«ãã£ãŠæ±ããããŠã
:<math> U = - k_0 \frac {e^2}{r}</math>
ãšãªãã
éåãšãã«ã®ãŒKã¯ã<math> K = \frac{1}{2}mv^2</math>ãªã®ã§
:<math> E = K+U = \frac{1}{2}mv{}^2 - k_0 \frac {e^2}{r}</math>
äžåŒã®å³èŸºç¬¬äžé
ã«ã
:åéåã®æ¹çšåŒ<math> m \frac{v^2}{r} = k_0 \frac {e^2}{r^2} </math>ã®äž¡èŸºã«ïœãæãã
<math> m v^2 = k_0 \frac {e^2}{r} </math>ã代å
¥ããã°ã
:<math>E(= E_n )= K+U = \frac{1}{2} k_0 \frac {e^2}{r}- k_0 \frac {e^2}{r} = - \frac{k_0e^2}{2r} </math>
ãšãªãã
ããã«ãããã«é»åã®è»éååŸ<math>r=r_n</math>ã®åŒ(3)ã代å
¥ããã°ã
:<math>E(=E_n) = -\frac{2\pi ^2 k_0{}^2 me^4} {h^2} \frac{1}{n^2} \quad (n=1,2,3,,,) \qquad \qquad (4)</math>
ãšãªãããããæ°ŽçŽ ååã®ãšãã«ã®ãŒæºäœã§ããã
ãšãã«ã®ãŒæºäœã®å
¬åŒãããèŠããšããŸãããšãã«ã®ãŒãããšã³ãšã³ã®å€ã«ãªãããšãåããããŸãããšãã«ã®ãŒãè² ã«ãªãäºããããã
nïŒ1ã®ãšããããã£ãšããšãã«ã®ãŒã®äœãç¶æ
ã§ããããã®ãããnïŒ1ã®ãšããå®å®ãªç¶æ
ã§ããããã£ãŠãé»åã¯éåžžãnïŒ1ã®ç¶æ
ã«ãªãã
ãªãã
:<math> -\frac{2\pi ^2 k_0{}^2 me^4} {h^2}</math>ã«è«žå®æ°ã®å€ãå
¥ããŠèšç®ãããš
:<math> - \frac{13.6}{n^2} \ \ \mathrm{eV}</math>ãšãªãã®ã§ã
:æ°ŽçŽ ååã®ãšãã«ã®ãŒé äœã¯
:<math>E(=E_n) = - \frac{13.6}{n^2} \ \ \mathrm{eV}</math>ãšæžããã
:<math>E_1</math>ã¯ãçŽ 13.6 eV ã«ãªãããããã¯æ°ŽçŽ ã®ã€ãªã³åãšãã«ã®ãŒã®å€ã§ãããããã¯ãå®éšå€ã«ããããäžèŽããã
=====ãæ°ŽçŽ ååã®ã¹ãã¯ãã«ã®çµéšåŒã®çè«çå°åº =====
æ°ŽçŽ ååã®çºããå
ã®ã¹ãã¯ãã«ã®å®æž¬å€ãè¡šããªã¥ãŒãããªã®çµéšåŒã«ã€ããŠã¯ãæ¢ã«ãæ°ŽçŽ ååã®ã¹ãã¯ãã«ãã®é
ã§ã§èª¬æããã
:ããŒã¢ã®æ°ŽçŽ ååã¢ãã«ã«åºã¥ããŠåŸããããšãã«ã®ãŒé äœãšæ¯åæ°æ¡ä»¶ã®ä»®èª¬ãçšããã°ããã®åŒã以äžã®ããã«çè«çã«å°åºã§ããã
:ä»»æã®æ£æŽæ°ïœãïœïŒïŒïœïŒãèããã
:ãããšãæ¯åæ°æ¡ä»¶ã®ä»®èª¬ã«ãã
é»åããšãã«ã®ãŒé äœ<math>E_n</math>ãããäœããšãã«ã®ãŒé äœ<math>E_m</math>ã«é·ç§»ãããšãã«ãæ¯åæ°
<math>\nu=\frac{E_n-E_m}{h}</math>
ã®å
åãäžåæŸåºããã
:ãã®å
åã®æ³¢é·Î»ã¯
<math>\frac{1}{\lambda} = \frac{E_n-E_m}{ch}</math>
ã§äžããããã®ã§ãå³èŸºã®ãšãã«ã®ãŒé äœã«åŒïŒïŒïŒã代å
¥ãããš
:<math>\frac{1}{\lambda} = \frac{2\pi ^2 k_0{}^2 me^4} {ch^3}(\frac{1}{m^2}-\frac{1}{n^2}) \qquad \qquad (5)</math>
ãåŸãããã
<math>{\bf R}\triangleq \frac{2\pi ^2 k_0{}^2 me^4} {ch^3}</math>
ã§ããªã¥ãŒãããªå®æ°Rãå®çŸ©ãããšãåŒ(5)ã¯
:<math>\frac{1}{\lambda} = {\bf R}(\frac{1}{m^2}-\frac{1}{n^2}) \qquad \qquad (5')</math>
Rã®å®çŸ©åŒäžã®è«žå®æ°ã«å€ããããŠèšç®ãããš
:<math>{\bf R}=1.097\times 10^7 \ [1/m] \qquad \qquad \qquad (6)</math>
é©ãã¹ãããšã«ããªã¥ãŒãããªã®çµéšåŒããèŠäºã«å°åºã§ããã®ã§ããã
ããã¯ãããŒã¢ã®ä»®èª¬ã®åŠ¥åœæ§ã瀺ããã®ãšèšãããã
{{ã³ã©ã |éä¿çãªãå
ã¯ç²åã§ããæ³¢ã§ããããšãã説æ ïŒâ» ç¯å²å€ïŒ|
ç²åãšæ³¢åã®äºéæ§ã«ã€ããŠã®ãå
ã¯ç²åã§ããæ³¢ã§ããããšãã説æã¯ãåããããããæå³ãåéãããããã
ããæ£ããã¯ã
ãååã¹ã±ãŒã«ãŠãã©ã®ç©ççŸè±¡ãæ±ããšãã¯ãå€å
žç©çã®ãããªïŒäººéã®èçŒã§ã芳å¯ã§ããçšåºŠã®å€§ããã®ïŒå·šèŠçãªåéã®ãç²åããšãæ³¢ãã§ã¯ãåºå¥ã§ããªãçŸè±¡ã«ãééãããããšã§ãèšãã»ãããããæ£ç¢ºã§ããã
å®éšäºå®ãšããŠã®ãå
ãã¯ãå·šèŠçãªåéã§ã¯åºæ¬çã«ã¯ãå
ãã¯æ³¢ã®æ§è³ªããã€ãããããå·šèŠçã§ãªãååãªã©ã®åŸ®çŽ°ãªç²åãžã®ãå
ãã®äœçšãªã©ãèããå Žåã«ã¯ãå
é»å¹æã®ããã«äžå®ã®ãšãã«ã®ãŒã®ãããŸãæ¯ã«äžé£ç¶ã«äœçšããçŸè±¡ãããããã§ãããããé£ç¶éã§ããïŒå€å
žçãªïŒæ³¢ãšã¯æ§è³ªãéãã
ããšãã£ãŠãååã«åœããæ³¢ã¯åŸ®èŠçã ãããšãã£ãŠãååã«åœãã£ããæ³¢ãããã£ããŠè³ªéããã€ããã«ãªãããã§ãªãã質éã«ã¯ã€ããŠã¯å€å
žçãªæ³¢ãšåæ§ã«ãååã¹ã±ãŒã«ã®åŸ®èŠçãªæ³¢ã§ãã質éã¯æããªãã
ãã®ããã«ååã¹ã±ãŒã«ã®åŸ®èŠçãªãæ³¢ãã§ãã質éã«ã€ããŠã¯ãå€å
žçãªæ³¢ãšå
±éãããæ³¢ããã®ãã®ã¯è³ªéããããªãã
ãã®ããã«ãååã¹ã±ãŒã«ã®æ³¢ã«ãããŠãã質éãªã© ããã€ãã®æ§è³ªã§ã¯ãå·šèŠã¹ã±ãŒã«ãšã²ãã€ã¥ãåæ§ã®æ§è³ªããã£ãŠããèŠçŽ ãããã
åæ§ã«ãé»ç£æŸå°ã®åé¡ã®ããã«ãé»åãæ³¢ã®æ§è³ªããã€ãçŸè±¡ããããããããé»åã¯è³ªéããã€ããŸããååŠçµåã®ããã«ã¯ïŒååŠ1ã®ææ¥ã§ãç¿ãããã«ïŒã䟡é»åããšããé»åã®ç²å1åãã€ã®åäœãèããã®ã§ããããããã®ããã«ååã¹ã±ãŒã«ã§ãã£ãŠããé»åã¯å€å
žçãªç²åãšå
±éããæ§è³ªã ããã€ãæã£ãŠããã
ããããååããã®é»ç£æŸå°ã®ãªãããšããè«ççã«èšããããšã¯ãåã«ãå·šèŠçãªåéãšãååã®ãããªåŸ®èŠçãªåéã§ã¯ããæ³åãéãããšããäºã ãã§ããããã®äºã ãã§ã¯ãå¿
ããããé»åã¯æ³¢ã§ããããšã¯æå®ã§ããªãããºã§ããã®ã ããããã人é¡ã¯ãç©è³ªæ³¢ãªã©ãã®ä»ã®å®éšçµæãããšã«ã人é¡ã¯ãé»åã¯æ³¢ã§ããããšä»®å®ããŠã20äžçŽåé ããã«ç©çåŠã®æ°çè«ïŒåœæïŒãçè«æ§ç¯ãããããçŸä»£ã§ãç¶ããŠããã
}}
:(â» ç¯å²å€: 倧åŠã®ç¯å²) å®éã®ç¹æ§ã¹ãã¯ãã«ã®æ³¢é·ã¯ãååå
éšã®é»åã®åœ±é¿ã«ãããè¥å¹²ããºã¬ãããããã£ãå
éšé»åã®è£æ£ãèæ
®ããããã粟床ã®é«ãåŒãšããŠãã¢ãŒãºãªãŒã®å
¬åŒããšããã®ãç¥ãããŠããããªãæŽå²çãªé åºã¯ãäžè¿°ã®èª¬æã®é åºãšã¯éã§ããããã€ã¯å
ã«ã¢ãŒãºãªãŒã®åŒãçºèŠãããããšãããã¢ãŒãºãªãŒãšã¯å¥ã«ç¬ç«ã«ç 究ãããŠããäžè¿°ã®ãããªããŒã¢ãã©ã¶ãã©ãŒãã®çè«ãçšãããšãã¢ãŒãºãªãŒã®å
¬åŒãããŸã説æã§ãããšããäºãç©çåŠè
ã³ãã»ã«ã«ãã£ãŠçºèŠããã<ref>å±±æ¬çŸ©éãååã»ååæ žã»åååãã岩波æžåºã2015幎3æ24æ¥ ç¬¬1å·ã140ããŒãž</ref>ããªãã¢ãŒãºãªãŒã®å
¬åŒã«ã€ããŠèª¿ã¹ãããªãã倧åŠç§åŠã®éåååŠãªã©ã®ç§ç®åã®æç§æžã«èšèŒãããã ããã
==== åºåºç¶æ
ãšå±èµ·ç¶æ
====
ïŒâ» æªèšè¿°ïŒ
=== ååæ ž ===
==== ååæ žã®æ§é ====
ååæ žã¯ãéœåãšäžæ§åããã§ããŠããã
éœåã¯æ£é»è·ããã¡ãäžæ§åã¯é»è·ããããªãã
ã§ã¯ããªããã©ã¹ã®é»è·ããã€éœåã©ãããããªãã¯ãŒãã³åã§åçºããŠããŸããªãã®ã ãããïŒ
ãã®çç±ãšããŠãã€ãŸãéœåã©ãããã¯ãŒãã³åã§åçºããªãããã®çç±ãšããŠã次ã®ãããªçç±ãèããããŠããã
ãŸããéœåã«äžæ§åãè¿ã¥ããŠæ··åãããšããæ žåããšããéåžžã«åŒ·ãçµååãçºçãã
ãã®æ žåãéœåå士ã®ã¯ãŒãã³åã«ãã匷ãæ¥åã«æã¡åã€ã®ã§ãéœåãšäžæ§åã¯çµåããŠãããšèããããŠãããïŒå¿
ããããéœåãšäžæ§åã®åæ°ã¯åäžã§ãªããŠããããå®éã«ãåšæè¡šã«ããããã€ãã®å
çŽ ã§ããéœåãšäžæ§åã®åæ°ã¯ç°ãªããïŒ
æ¯å©çã«èšãæãã°ãäžæ§åã¯ãéœåãšéœåãçµã³ã€ãããããªã®ãããªåœ¹å²ãããŠãããšãèããããŠããã
:ïŒâ» ç¯å²å€:ïŒ ååçªå·ã®äœãå
çŽ ã«ãããŠãéœåãšäžæ§åã®åæ°ã¯ãã»ãŒåæ°ã§ããå Žåãå€ããããšãã°ãé
žçŽ ãçªçŽ ã§ã¯ãéœåãšäžæ§åã¯åæ°ã§ãããäžæ¹ãå
çŽ çªå·ã®é«ãå
çŽ ã»ã©ïŒã€ãŸãåšæè¡šã§äžã®ã»ãã®å
çŽ ïŒãéœåãããäžæ§åãå€ããããšãã°ãŠã©ã³235ã¯äžæ§åæ°ãéœåæ°ã®1.5åã§ããããã®ãããªãåšæè¡šã«ãããéœåæ°ãšäžæ§åæ°ã®èŠ³æž¬äºå®ãããïŒåšæè¡šã調ã¹ãã°ãããã«åããïŒãããã«ã¯æ žåã®æ§è³ªãé¢ä¿ããŠãããšèããããŠãã<ref>å±±æ¬çŸ©éãååã»ååæ žã»åååãã岩波æžåºã2015幎3æ24æ¥ ç¬¬1å·ã190ããŒãž</ref>ã
:ïŒçºå±: ïŒãªãã20äžçŽåŸå以éã®çŽ ç²åè«ã§ã¯ãéœåãšäžæ§åã¯ããã«å°ããªç©è³ªããæãç«ã£ãŠãããšããããã ããé«æ ¡çã¯ãã®åå
ã®åŠç¿ã§ã¯ãååæ žã®æ§æèŠçŽ ãšããŠã¯ããšããããéœåãšäžæ§åãŸã§ãèããã°ååã§ããã
ãªããå称ãšããŠãéœåãšäžæ§åããŸãšããŠãæ žåããšåŒã°ããã
ããå
çŽ ã®ååæ žã®éœåã®æ°ã¯ãåšæè¡šã®'''ååçªå·'''ãšäžèŽããã
ãŸããéœåãšäžæ§åã®æ°ã®åã¯'''質éæ°'''ãšãã°ããã
質éæ°Aã®ååæ žã¯éåžžã«åŒ·ãæ žåã®ããã«ãå°ããªçäœç¶ã®ç©ºéã®äžã«åºãŸã£ãŠããããã®ååŸïœã¯ã
<math>1.2</math>ïœ<math>1.4\times 10^{-15}\times A^{\frac{1}{3}}</math>
ã§ããããšãç¥ãããŠããã
==== ååæ žã®çµåãšãã«ã®ãŒ ãšè³ªéæ¬ æ ====
ä»»æã®ååæ žã¯ããããæ§æããæ žåã§ããéœåãšäžæ§åãèªç±ã§ãããšãã®è³ªéïŒåäœè³ªéãšããïŒã®åãããå°ãã質éããã€ããã®æžã£ã質éãã質éæ¬ æãšåŒã¶ã
質éæ°Aãååçªå·Zã®ååæ žã®è³ªéæ¬ æ<math>\Delta m</math>ããåŒã§æžãã°,
ååæ žã®è³ªéãïœãéœåãšäžæ§åã®åäœè³ªéããããã<math>m_p,\ m_n</math>ãšãããšãã
:<math>\Delta m = m_{p}Z+m_{n}(A-Z)- m</math>ã§ããã
:ïŒâ» ç¯å²å€: ïŒååã«ãããããäžè¬ã«è³ªéæ¬ æã®å€§ããã¯ãããæ¬ æã®ãªãç¶æ
ãšããŠä»®å®ããå Žåã®çè«å€ã®1%ãŠãã©<ref>[https://kotobank.jp/word/%E8%B3%AA%E9%87%8F%E6%AC%A0%E6%90%8D-74242 ã³ããã³ã¯ãæ¥æ¬å€§çŸç§å
šæž(ããããã«)ã®è§£èª¬ãïŒåæ±åŒæ²»ãå
å Žä¿éïŒãªã© ]</ref>ã§ããã粟å¯æž¬å®ã§1%ãšããã®ã¯ããã£ãã倧ããå²åã§ããã
:ïŒâ» ç¯å²å€: ïŒè³ªéæ¬ æã®åŸã§ãã質éæ°ïŒæ žåã«ãããéœåãšäžæ§åã®åæ°ã®åèšïŒã¯éåžžã¯ä¿åãããŠãã<ref>å±±æ¬çŸ©éãååã»ååæ žã»åååãã岩波æžåºã2015幎3æ24æ¥ ç¬¬1å·ã222ããŒãž</ref>ãã€ãŸãã質éæ°ãä¿åãããŠããã®ã«ããããããã質éïŒããã°ã©ã ïŒã枬å®ããã°ããã®æ žåã®è³ªéïŒããã°ã©ã ïŒããããã«æ¬ æããŠããã®ã§ããã
{{ã³ã©ã |ååã¬ãã«ã®è³ªéã®æž¬å®æ³|
[[File:Mass spectrometer schematics.png|thumb|right|質éåæåšã®æš¡åŒå³ãè©Šæå°å
¥éšããã³ã€ãªã³æºïŒå·ŠäžïŒãåæéšïŒå·Šäžãç£å ŽåååïŒãã€ãªã³æ€åºéšïŒå³äžïŒãããŒã¿åŠçéšïŒå³äžïŒãããªãã]]
ãããããã©ããã£ãŠååãååã®è³ªéã粟床ãã枬å®ãããïŒ
ããã¯ãã質é枬å®æ³ããšèšãããæè¡åéã«ãªããé«æ ¡ã¬ãã«ã§ã¯èª¬æã§ããªããïŒã¢ã€ã³ã·ã¥ã¿ã€ã³ãæãããŠããããã£ã枬å®æè¡ãç¥ã£ãŠãããã©ãããçããããã§ãããïŒ
åèæç®ãæªå
¥æãªã®ã§ãæèšã¯ã§ããªãããäžè¬ã«ååã¬ãã«ã®è³ªé枬å®æ³ãšããŠç²Ÿå¯ç§åŠã§ããç¥ããããã®ãšããŠãå³å³ã®ãããªãç£å Žã«ãã£ãŠè·é»ç²åãæ²ããæ¹åŒã®ãã®ãããïŒâ» é«æ ¡ç©çã®ããŒã¬ã³ãåã®èšç®ã§ãã䌌ããããªå®éšè£
眮ã§ã®åè»éã®èšç®ãç¿ãïŒã
ãã®ãããªç£å ŽãšããŒã¬ã³ãåãçšããæ¹åŒã«ãã質é枬å®ã¯äžè¬ã«ããç£å Žåååããšããããã
ãã®ãããªè£
眮ã«ãããç£å Žãé»åã®å€§ããã¯å®éšçã«æ±ºå®ã§ããã®ã§ãæ²çã質éã®é¢æ°ã«ãªãã®ã§ãã€ãŸãååŸãã質éãéç®ã§ããã
枬å®å¯Ÿè±¡ã®å
çŽ ææãäžæ§ã®ååã§ãã£ãŠãããã®ååãåºäœãªããããã«é»åããŒã ãåœãŠãŠãé»åã«ãã£ãŠåŒŸãé£ã°ãããææã垯é»ããŠã€ãªã³åããŠããã®ã§ããããããäžèšã®ãããªç£å Žã«ãã質é枬å®ãå¯èœã«ãªãã
:ïŒâ» ããããåèæç®ãæªå
¥æãªã®ã§ãã¯ãããŠãã®æ¹æ³ã§è³ªéæ¬ æã枬å®ã§ãããã©ããã¯æªç¢ºèªãïŒ
æç®ã
:西æ¢æçŸã枬ãæ¹ã®ç§åŠå² II ååããçŽ ç²åãžããææ瀟ã2012幎3æ15æ¥ åççºè¡ã77ããŒãžã
ãã«ãããšã1919幎ã«ç§åŠè
ã¢ã¹ãã³ïŒäººåïŒã«ãã£ãŠè³ªéåæåšãçºæãããã®ã§ã質éæ¬ æããããçšããŠæž¬å®ãããããšãã®æç®ã§ã¯äž»åŒµãããŠããã
éèŠãªããšãšããŠããããååã®è³ªéã¯æž¬å®çã«æ±ºå®ã§ããæ°å€ã§ããããã£ããŠãäœããã®ä»®å®ã«ããšã¥ãçè«èšç®ã§ã¯ãªãããŸããå€å
žç©ç以äžã®ç¥èïŒçžå¯Ÿæ§çè«ãéåååŠãªã©ïŒãå¿
èŠãšããªããå€å
žçãªé»ç£æ°åŠãªã©ã®å€å
žç©çåŠã«ããšã¥ãå®éšè£
眮ã§æž¬å®ã§ããå®éšäºå®ã§ããã
ãªããååŠã®åäœäœã®ååšããã®è³ªéãããã®ããããã®ãããªè£
眮ã§çºèŠãããã
ãªãïŒé«æ ¡ã§ã¯ç¿ããªããïŒãåå質éãããã€ãã®å
çŽ ã§æž¬å®ã§ããã®ã§ã掟ççã«ããŸã
ååŠã®çè«ã§åããååçªå·ZãšååéAãšã枬å®ãããååã®è³ªéã®æž¬å®å€Mãããšã«ã
Z,AããMãæ±ããå
¬åŒãäœæãããïŒã¯ã€ããŒãã«ãŒã®å
¬åŒã1935幎ïŒã
ãŸããååååŸã®äºæ³å€ãªã©ãç®åºãããŠãã£ãïŒã¬ã€ã³ãŠã©ãŒã¿ãŒïŒäººåïŒã1953幎ïŒã
}}
=====ã質éæ¬ æã®åå ã=====
枬å®å®éšã®äºå®ãšããŠãéœååç¬ãäžæ§ååç¬ã®è³ªéã®åæ°ãåãããããããã®çµåããååæ žã®ã»ãã質éãäœãã®ã§ãéœåãäžæ§åãçµåãããšè³ªéã®äžéšãæ¬ æãããšããã®ãã枬å®çµæã®äºå®ã§ããã
ãªã®ã§ã質éæ¬ æã®ãšããããã®åå ãšããŠèããããŠããã®ã¯ãéœåãäžæ§åã©ããã®çµåã§ãã<ref>[https://kotobank.jp/word/%E8%B3%AA%E9%87%8F%E6%AC%A0%E6%90%8D-74242 ã³ããã³ã¯ãäžç倧çŸç§äºå
ž 第ïŒçã®è§£èª¬ããªã© ]</ref>ãšèããããŠããã
ã ããã§ã¯ããªãéœåãäžæ§åãååæ žãšããŠçµåãããšè³ªéãæ¬ æãããã®çç±ãšããŠã¯ããã£ããŠãçµåã ããããšããçç±ã§ã¯èª¬æãã€ããªãã
ãªã®ã§ãç©çåŠè
ãã¡ã¯ã質éæ¬ æã®èµ·ããæ ¹æ¬çãªåå ãšãªãç©çæ³åããŠãã¢ã€ã³ã·ã¥ã¿ã€ã³ã®çžå¯Ÿæ§çè«ãé©çšããŠãããïŒæ€å®æç§æžã§ããçžå¯Ÿæ§çè«ã®çµæã§ãããšããŠèª¬æããç«å ŽïŒ
ïŒã¢ã€ã³ã·ã¥ã¿ã€ã³ã®ç¹æ®ïŒçžå¯Ÿæ§çè«ããå°ãããçµæãšããŠïŒâ» åè: çžå¯Ÿè«ã«ã¯äžè¬çžå¯Ÿè«ãšç¹æ®çžå¯Ÿè«ã®2çš®é¡ãããïŒã質émãšãšãã«ã®ãŒEã«ã¯ã
:<math>E=m c^2</math>
ãšããé¢ä¿åŒããããšãããã
ãªããC ãšã¯å
éã®å€ã§ããã
ãããã¯å¥ã®æžåŒãšããŠãå€åãè¡šããã«ã¿èšå·Îã䜿ãŠã
:<math>E= c^2 \cdot \Delta m </math>
ãªã©ãšæžãå Žåãããã
ã€ãŸããããäœããã®çç±ã§ãç空ãã質éãçºçãŸãã¯æ¶å€±ããã°ããã®ã¶ãã®è«å€§ãªãšãã«ã®ãŒãçºçãããšããã®ããçžå¯Ÿæ§çè«ã§ã®ã¢ã€ã³ã·ã¥ã¿ã€ã³ãªã©ã®äž»åŒµã§ããã
ããŠãèªç±ãªéœåãšäžæ§åã¯ãæ žåã«ããçµåãããšãããã®çµåãšãã«ã®ãŒã«çžåœãã[[w:ã¬ã³ãç·]]ãæŸå°ããããšãç¥ãããŠããã
ãããŠãã¬ã³ãç·ã«ããšãã«ã®ãŒãããã
ãªã®ã§ãéœåãšäžæ§åã®çµåãããšãã®ã¬ã³ãç·ã®ãšãã«ã®ãŒã¯ã質éæ¬ æã«ãã£ãŠçãããšèãããšã枬å®çµæãšããžãããåããïŒæž¬å®çµæã¯ããããŸã§è³ªéãæ¬ æããããšãŸã§ãïŒ
æ žåã®çµåã«ãããŠã質éæ¬ æ<math>\Delta m </math>ããã¬ã³ãç·ãªã©ã®ãšãã«ã®ãŒã«è»¢åããããšç©çåŠè
ãã¡ã¯èããŠããã
==== æŸå°èœãšæŸå°ç· ====
å
çŽ ã®äžã«ã¯ãæŸå°ç·ïŒradiationïŒãåºãæ§è³ªããã€ãã®ãããããã®æ§è³ªãæŸå°èœïŒradioactivityïŒãšããã
ãŸããæŸå°èœããã€ç©è³ªã¯æŸå°æ§ç©è³ªãšããããã
æŸå°ç·ã«ã¯3çš®é¡ååšããããããαç·ãβç·ãγç·ãšããã
α厩å£ã¯ã芪ååæ žããããªãŠã ååæ žãæŸå°ãããçŸè±¡ã§ããã
ãã®ããªãŠã ååæ žã¯Î±ç²åãšãã°ããã
α厩å£åŸã芪ååæ žã®è³ªéæ°ã¯4å°ãããªããååçªå·ã¯2å°ãããªãã
β厩å£ã¯ã芪ååæ žã®äžæ§åãéœåãšé»åã«å€åããããšã§ãé»åãæŸå°ãããçŸè±¡ã§ããã
ïŒåè: ãã®ãšããåãã¥ãŒããªããšãã°ãã埮å°ãªç²åãåæã«æŸåºããããšãè¿å¹Žã®åŠèª¬ã§ã¯èããããŠãããïŒ
ãªãããã®é»å(ããŒã¿åŽ©å£ãšããŠæŸåºãããé»åã®ããš)ã¯ãβç²åããšããã°ããã
β厩å£åŸã芪ååæ žã®è³ªéæ°ã¯å€åããªãããååçªå·ã¯1å¢å ããã
γç·ã¯ãα厩å£ãŸãã¯Î²åŽ©å£çŽåŸã®é«ãšãã«ã®ãŒã®ååæ žããäœãšãã«ã®ãŒã®å®å®ãªç¶æ
ã«é·ç§»ãããšãã«æŸå°ãããã
γç·ã®æ£äœã¯å
åã§ãXç·ããæ³¢é·ã®çãé»ç£æ³¢ã§ããã
α厩å£ãβ厩å£ã«ãã£ãŠããšã®ååæ žã®æ°ã¯åŸã
ã«æžã£ãŠãããããããã®åŽ©å£ã¯ååæ žã®çš®é¡ããšã«æ±ºãŸã£ãäžå®ã®ç¢ºçã§èµ·ããã®ã§ã厩å£ã«ãã£ãŠããšã®ååæ žã®æ°ãæžãé床ã¯ååæ žã®åæ°ã«æ¯äŸããŠå€åããããããã厩å£ã«ãã£ãŠããšã®ååæ žã®æ°ãåæžããã®ã«ãããæéã¯ãååæ žã®çš®é¡ã ãã«ãã£ãŠããŸããããã§ããã®æéã®ããšããã®ååæ žã® '''åæžæ'''ïŒã¯ãããããhalf life ïŒ ãšåŒã¶ã厩å£ã«ãã£ãŠååæ žã®åæ°ãã©ãã ãã«ãªããã¯ããã®åæžæãçšããŠèšè¿°ããããšãã§ãããååæ žã®åæžæãTãæå»tã§ã®ååæ žã®åæ°ãN(t)ãšãããšã
:<math>N(t)=N(0)(\frac{1}{2})^{\frac{t}{T}}</math>
ãæãç«ã€ã
===== çºå±ã»å
¬åŒã®å°åº =====
ååæ žã®åŽ©å£é床ã¯ãååæ žã®åæ°ã«æ¯äŸãããšè¿°ã¹ããå®ã¯ãäžã«è¿°ã¹ãå
¬åŒã¯ãã®æ
å ±ã ãããçŽç²ã«æ°åŠçã«å°ãåºãããšãã§ãããã®ã§ãããé«çåŠæ ¡ã§ã¯æ±ããªãæ°åŠãçšããããèå³ã®ããèªè
ã®ããã«ãã®æŠèŠãèšããŠããã
ååæ žã®åæ°ãšåŽ©å£é床ã®éã®æ¯äŸå®æ°ã¯ååæ žã®çš®é¡ã«ãã£ãŠæ±ºãŸãããã®å®æ°ããã®ååæ žã®åŽ©å£å®æ°ãšããã厩å£å®æ°ãλã®ååæ žã®æå»tã§ã®åæ°ãN(t)ãšãããšããã®å€åé床ãããªãã¡N(t)ã®åŸ®åã¯ã
:<math>\frac{d}{dt} N(t) = -\lambda N(t)</math>
ã§è¡šãããããã®ãããªãããé¢æ°ãšãã®åŸ®åãšã®é¢ä¿ãè¡šããåŒã埮åæ¹çšåŒãšããã埮åæ¹çšåŒãæºãããããªé¢æ°ãæ±ããããšãã埮åæ¹çšåŒã解ããšãããïŒè©³ãã解æ³ã¯[[解æåŠåºç€/垞埮åæ¹çšåŒ]]ã§èª¬æããããïŒãã®åŸ®åæ¹çšåŒã解ããš
:<math>N(t)= N(0) e^{-\lambda t}</math>
ãåŸããããïŒãã®åŒã確ãã«å
ã»ã©ã®åŸ®åæ¹çšåŒãæºãããŠããããšã確ãããŠã¿ãïŒ
åæžæTãšã¯ã<math>N(t)=\frac{1}{2}N(0)</math>ãšãªãtã®ããšãªã®ã§ãå
ã»ã©ã®åŒãã
:<math>T=\frac{\log 2}{\lambda}</math>
ãåŸãããããã£ãŠã
:<math>N(t)=N(0) e^{-\lambda t}=N(0) 2^{\frac{-\lambda t}{\log 2}}=N(0) (\frac{1}{2})^{\frac{t}{T}}</math>
ãåŸãããã
{{ã³ã©ã ||
:ïŒâ» ç¯å²å€: ç§åŠææ³ã«ããã圱é¿.ïŒãäžè¿°ã®ãåæžæã¯ååæ žã®çš®é¡ã«ãã£ãŠæ±ºãŸãããšããäºã¯ãèšãæããã°ãåæžæã¯ååæ žã®çš®é¡ã«ãã£ãŠ'''ãã'''決ãŸããªãããšããäºã§ããããããã®æå³ããäºã¯ããã倩äžãïŒããŸãã ãïŒçãªèª¬æã ããäžè¿°ã®ãããªæŸå°æ§å£å€ãªã©ã®çŸè±¡ã¯ãæŸå°å£å€ã¯ã確çè«çã«çºçããŠããç©ççŸè±¡ã§ããããšããå¯èœæ§ãé«ãããšããæå³ã§ããã
:ããšãã°ããŠã©ã³é±ç³ããïŒçºèŠåœæã®æè¡ã§ã¯ç¡çã ãïŒåå1åã¶ããåãåºããããšãåæžæã®æéãçµã£ãããã£ãŠããã®ãŠã©ã³ååãããã£ããŠã確å®ã«æŸå°å£å€ããŠå¥ååã«å€åããŠããããšã¯'''èšããªã'''ãã®ã§ããã確çè«çã»çµ±èšæ°åŠçã«ãåæžæã®æéãçµéããããã ããããã®ãããã®éã確çã§æŸå°å£å€ããŠããããšããèšããªãããšããæå³ã§ããã
:ïŒãã®ä»ãååæ°ãªã©ã§ã¯æ±ºãŸããªãäºããããã£ããŠå€å
žçãªç±ååŠã®ãããªãè€æ°ã®ååéïŒãããã¯è€æ°ã®ååéïŒã«ãããçžäºäœçšã®çŸè±¡ã§ããªãããšããäºãæå³ããããŸãã枩床ã«ãã£ãŠåæžæãæŸå°å£å€ã®çµæãå€åããªãäºãããååŠåå¿ã§ã¯ãªãããšãèšãã<ref>å±±æ¬çŸ©éãååã»ååæ žã»åååãã岩波æžåºã2015幎3æ24æ¥ ç¬¬1å·ã154ããŒãž</ref>ãïŒ
:ã確çè«ããšããçšèªã«å¯ŸããŠäžæ¹ãã決å®è«ãïŒãã£ãŠãããïŒãšããå¥ã®å²åŠçãªçšèªããããå€å
žååŠããããããã¥ãŒãã³ååŠãªã©ã®ãããªåæå€ãåé床ãã決ãŸãã°åççã«ã¯ãæªæ¥ã®çŸè±¡ã粟床ããèšç®ã§ãããããªçŸè±¡ãããã¯äžç芳ã®ããšããã決å®è«ããšããã
:ãã€ãŠãã¬ãªã¬ãªããã¥ãŒãã³ãªã©ã®å€ãæ代ã®ç©çåŠã¯ã決å®è«çãªäžç芳ãåæãšããŠããïŒã¬ãªã¬ãªãªã©ãã決å®è«ããšããçšèªãç¥ã£ãŠãããã¯ããšãããïŒã
:ããããæŸå°å£å€ãªã©ã®çŸè±¡ãã確çè«çã§ããããšããäºã®æå³ã¯ã€ãŸãããšããã決å®è«ãã«å¯ŸããŠãæŸå°å£å€ãªã©ãåäŸã®ãããªçŸè±¡ã«ãªã£ãŠãããšããæå³ã§ãããããã¯ã€ãŸãããã以åã®æ±ºå®è«çãªäžç芳ã«å¯Ÿããã倧å€é©ãæå³ããã
:èªè
ã®é«æ ¡çããç©ç1ïŒç©çåºç€ïŒãªã©ã®ç§ç®ã§ã確çã®æ¹çšåŒãªã©ãèŠãããšãç¡ãã ãããå°ãªããšãé«æ ¡ç©çã®ïŒç©ç1ãç©çåºç€ã®ãããªïŒååŠã®ç¯å²ã§ã¯ãå°ãªããšãæ³åãèšè¿°ããåŒãšããŠã¯ãææ°ã察æ°ã®åŒããèªè
ã¯è³ªç¹ã®åŠã§ã¯èŠãããšãç¡ãããºã ãïŒç±ååŠã®å
¬åŒã§ã¯ãçºå±çïŒããç¯å²å€ïŒãªåéã®æ³åã«ãææ°ã䜿ãå
¬åŒãè¥å¹²ãããïŒ
:ãªããç±ååŠã§ã¯çµ±èšçãªèãæ¹ãçšããããããã¯æ±ºå®è«çãªäžç芳ã®æ代ã®å€ãç©çåŠè
ã«ãšã£ãŠã¯ããããæ°äœååã1åã ããªãããã®éåã¯æ±ºå®è«çã«èšè¿°ã§ãããããããååã®åæ°ãèšå€§ãããã®ã§ã人éãèšç®ã§ããªããããããããªãçµ±èšçãªèãæ¹ã䜿ã£ãŠããã ãã§ããããšãã颚ã«ã䟿å®çã«çµ±èšãçšããŠããã ãã§ãããšããäžç芳ã§ãããå€å
žç±ååŠã¯æ±ºå®è«ã®ç Žç¶»ãšã¯æãããŠããªãã£ãã
:å®éãé«æ ¡ç©ç2ã®ç±ååŠã§ç¿ããããªæ°äœååéåè«ããçè«ã®åµå§è
ã§ãããã¯ã¹ãŠã§ã«ããã«ããã³ãªã©ã¯ã決å®è«çãªåæã§ãæ°äœååã®éåã解æããŠãã<ref>[http://www.ivis.co.jp/text/201205230606.pdf ç§åŠå²åŠå
¥é2 ãç系人ã«åœ¹ç«ã€ç§åŠå²åŠããèªã]ã25ããŒãž</ref>ã
:ããããååç©çã«ãããæŸå°å£å€ã¯ããã§ã¯ãªããããããæ±ãç©è³ªïŒããšãã°ãŠã©ã³é±ç³ãªã©ïŒãæ°äœã§ããå¿
èŠã¯ç¡ãããïŒãŠã©ã³é±ç³ãªã©ã¯åºäœãªã®ã§éæ¢ããŠããïŒããããéåããŠãªããããŸããå€æ°ã®ååã®éå£ã§ããå¿
èŠããããªããã€ãŸãåççã«ã¯1åã®ååãŸãã¯æ°åçšåºŠãããªãçµæ¶ã§ãã£ãŠãè¯ããã€ãŸãæ°äœç±ååŠã®ãããªå€æ°ã®ç²åãããªãå
éšæ§é ããããªãã«ãé¢ããããæŸå°æ¹å€ã¯ãã®æ³åãè¡šãåºæ¬å
¬åŒã®ãã®ãã®èªäœ4ã«ãçµ±èšçãªåŒãå«ãŸããŠããã
:ãããæŸå°å£å€ã¯ãç©è³ªãæ§æããååãã®ãã®ã®çŸè±¡ã§ããã
:ã ãããç±ååŠã®å Žåãšã¯éããæŸå°å£å€ã¯ãïŒã®ã¡ã®ç©çåŠããã®åŸç¥æµã§ãããïŒæ±ºå®è«çãªäžç芳ã ãã§ã¯èª¬æã§ããªãäºã§ãããã®ã¡ã®éåååŠïŒããããããããïŒãªã©ã«ã€ãªããããããããçŸä»£ç©çåŠããšããããã(æŸå°å£å€ã®åéããåŸç¥æµã ã)ç©çåŠã®æ°ããäžç芳ã«ã€ãªããåéã®å
é§ããšãªã£ããšããæ矩ãããã
::ïŒâ» ã»ã»ã»ãšãããããªæãã®ããšããããç§åŠå²ãªã©ã§èªãããã®ã ããããããããã§ç¢ºèªããç¯å²ã§ã¯è£ã¥ãã«ãªããããªè³æã»è«æãªã©ãåŸãããªãã£ããïŒ
::ïŒãããé«æ ¡ç©çã§ã¯ãäžè¿°ã®ãããªæãã®ããšãç©çç§ã®æåž«ãææ¥äžã«å£é ã§èª¬æãããããå Žåãããããããç§åŠææ³ãªã©ã¯é«æ ¡ç©çã®ç¯å²å€ãªã®ã§ãæ€å®æç§æžã«ã¯èšèŒãããªãããŸããå¿
ç¶çã«å€§åŠå
¥è©Šã«ãåºé¡ãããªãã®ã§ãäžè¿°ã®ç§åŠå²ã®æŽå²èŠ³ã«ã€ããŠã¯äžžæèšã¯äžèŠã
é«æ ¡ç©çã§ã¯ãæŸå°èœã®åéãšã¯å¥ã«ãååã®ãç©è³ªæ³¢ããšããé»åã®ãæ³¢åæ§ããšãã®çŸä»£ç©ççãªæ³¢åã®æŠå¿µããç©ç2ïŒå°éç©çïŒç§ç®ã§ç¿ãããããã£ãæ³¢åæ§ã«é¢ããäºãããã¥ãŒãã³ååŠçãªæ±ºå®è«ã®ç Žç¶»ã«ãªããã ãããã®ãæ³¢åæ§ãããã¬ãã®çè«ãåæãšããªããŠããæŸå°å£å€ãšããå®éšäºå®ã ãã§ããïŒè³ªç¹ãåäœã®éåã®ãããªïŒãã¥ãŒãã³ååŠçãªæ±ºå®è«çãªäžç芳ãããã€ããããããããã®ã§ããã
å®éãã³ã©ã å€ã®äžèšã®æ¬æã®è©±é¡ã§ã¯ãäžåãé»åãååãªã©ã«æ³¢åæ§ããããã©ããã®è©±é¡ã¯ããŠããªãã
ãªããé«æ ¡ç©çã®æè²ã§ã¯ããããæŸå°ç Žå€ã®åå
ã§ãäžè¿°ã®ããã«ãã¥ãŒãã³ååŠã®æ±ºå®è«ãç Žç¶»ããŠããäºãæããããšã«ãããé«æ ¡ã®é ãçŸä»£ç©ççãªäžç芳ã«ãªãããããšã§ãã®ã¡ã®åå
ã®ç©è³ªæ³¢ãé»åã®æ³¢åæ§ãªã©ã®åå
ã«å°å
¥ããããããã«ãããã»ã»ã»ãšãããããªæè²ææ³ãè¡ããããããã
}}
==== ååæ žåå¿ ====
[[File:Cloud chamber ani bionerd.gif|thumb|right|300px|é§ç®±ïŒããã°ãïŒã®å®éšãéœåã¯é»è·ïŒæ£é»è·ïŒããã£ãŠãããããé§ç®±ã§ã芳枬ããããšãã§ããã ïŒâ» ãã®ç»åã¯ãéœåã®èŠ³æž¬å®éšã§ã¯ãªããé§ç®±ã®åç説æã®ããã®ç»åã§ãããïŒ<br>é§ç®±ïŒããã°ãïŒãšãããèžæ°ã®ã€ãŸã£ãè£
眮ãã€ãããšããªããã®ç²åãééãããšããã®ç²åã®è»è·¡ã§ãæ°äœãã液äœããåçãèµ·ããã®ã§ãè»è·¡ããç®ã«èŠããã®ã§ãããïŒã€ã¡ãŒãžçã«ã¯ãé£è¡æ©é²ã®ãããªã®ããã€ã¡ãŒãžããŠãã ãããïŒ
ã§ãç£å Žãå ããå Žåã®ãè»è·¡ã®æ²ããããçãªã©ãããæ¯é»è·ãŸã§ãäºæ³ã§ããã]]
* éœåã®çºèŠ
ã©ã¶ãã©ãŒãã¯ãçªçŽ ã¬ã¹ãå¯éããç®±ã«Î±ç·æºããããšãæ£é»è·ããã£ãç²åãçºçããããšãçºèŠããã
ãã®æ£é»è·ã®ç²åããéœåã§ãããã€ãŸããã©ã¶ãã©ãŒãã¯éœåãçºèŠããã
åæã«ãé
žçŽ ãçºçããããšãçºèŠãããã®çç±ã¯çªçŽ ãé
žçŽ ã«å€æãããããã§ãããã€ãŸããååæ žãå€ããåå¿ãçºèŠããã
ãããã®ããšãåŒã«ãŸãšãããšã
:<math>_{\ 7}^{14} \mathrm{N} + {}_{2}^{4} \mathrm{He} \rightarrow {}_{\ 8}^{17} \mathrm{O} + {}_{1}^{1} \mathrm{H} </math>
ã§ããã
ãã®ããã«ãããå
çŽ ã®ååããå¥ã®å
çŽ ã®ååã«å€ããåå¿ã®ããšã '''ååæ žåå¿''' ãšããããŸãã¯ããæ žåå¿ããšããã
:ïŒâ» ç¯å²å€: ïŒé§ç®±ã¯ãçš®é¡ã«ãããããæ®éããšã¿ããŒã«ãŸãã¯ã¢ã«ãŽã³ã®æ°äœãå°å
¥ããã<ref>å±±æ¬çŸ©éãååã»ååæ žã»åååãã岩波æžåºã2015幎3æ24æ¥ ç¬¬1å·çºè¡ãP80</ref>ã
:é§ç®±ã®ãããªå®éšè£
眮ã®çšéãšããŠãéœåã®å®éšã®çšéã®ã»ããååæ žåå¿ã®åæ°ã芳枬ããç®çã§ã䜿ãããšãåºæ¥ããæŸå°ç·ã®æž¬å®åšã®ãããããã¬ã€ã¬ãŒã«ãŠã³ã¿ãŒãã®åçããé§ç®±ãšé¡äŒŒããŠãããåççãªæŸå°ç·æž¬å®åšã§ããã¬ã€ã¬ãŒã»ãã¥ã©ãŒç®¡ã«ã¯æ°äœïŒã¢ã«ãŽã³ããšãã¬ã³ã¬ã¹ãªã©ã®äžæŽ»æ§ãªæ°äœïŒãå°å
¥ãããŠãããé§ç®±ã®ããã«æ°æ°äœãå°å
¥ãã枬å®ç®¡ã«ãé«é»å§ããããé»æ°æ¥µæ¿ãè¿œå ããããšã§ãæŸå°ç·ããšãããããã«ãããã®ãã¬ã€ã¬ãŒç®¡ã§ãã[http://www.agc.a.u-tokyo.ac.jp/radioecology/pdf/190930_radioecology_supplement2.pdf ]ãç©çåŠè
ã¬ã€ã¬ãŒã¯ããã®ãããªæž¬å®åšãéçºããããã«ååæ žåå¿ã«ãã£ãŠçæãããããªãŠã ååãéããŠæ°äœãšããŠå°å
¥ããïŒâ» wikiè£è¶³: ãã®ããªãŠã ã«æ°äœã®ç¶æ
æ¹çšåŒãªã©ãé©çšããäºã«ãããïŒåœæãšããŠã¯æé«æ°Žæºã®ç²ŸåºŠã§ã¢ãã¬ããå®æ°ã枬å®ããäºã«æåãã<ref>å±±æ¬çŸ©éãååã»ååæ žã»åååãã岩波æžåºã2015幎3æ24æ¥ ç¬¬1å·çºè¡ãP81</ref>ãåœæç¥ãããŠããããã©ã³ã¯ã®ç±èŒ»å°ã®çè«ããç®åºãããã¢ãã¬ããå®æ°ã®å€ãïŒãã«ããã³å®æ° Na k ãšæ°äœå®æ° k ã®æ¯ããã¢ãã¬ããå®æ° Na ãæ±ããããïŒç©çåŠè
ãã©ã³ããã©ãŠã³éåããæ±ããã¢ãã¬ããå®æ°ã«ãã¬ã€ã¬ãŒã®ã¢ãã¬ããå®æ°ã®ç²ŸåºŠã¯å¹æµãã粟床ã§ãã£ã<ref>å±±æ¬çŸ©éãååã»ååæ žã»åååãã岩波æžåºã2015幎3æ24æ¥ ç¬¬1å·çºè¡ãP82</ref>ã
* äžæ§åã®çºèŠ
===çŽ ç²å===
[[File:Cloud chamber ani bionerd.gif|thumb|right|300px|é§ç®±å®éšããµããšçŸããçœãè»è·¡ããè·é»ç²åãæŸå°ç·ãééããè·¡ã]]
[[File:Physicist Studying Alpha Rays GPN-2000-000381.jpg|thumb|right|300px|é§ç®±ãèŠã蟌ãç©çåŠè
ïŒ1957幎ïŒãäžå¿ã«ããããŠã ã眮ãããŠãããããããæŸå°ãããæŸå°ç·ïŒã¢ã«ãã¡ç²åïŒããè±ã³ãã®ãããªåœ¢ã§å¯èŠåãããŠããã]]
ãŸããå®å®ç·ã®èŠ³æž¬ã«ãããÎŒç²åãšããã®ããçºèŠãããŠããã
==== ç¯å²å€: ã©ããã£ãŠçŽ ç²åã芳枬ããã ====
ãããããã©ããã£ãŠçŽ ç²åã芳枬ããããšãããšãããã€ãã®æ¹æ³ããããã
:åç也æ¿ãïŒçŽ ç²å芳枬çšã®ä¹Ÿæ¿ããååæ žä¹Ÿæ¿ããšããïŒ
:é§ç®±
ãªã©ã䜿ãããã
==== é§ç®±ïŒããã°ãïŒ ====
(â» é«æ ¡ã§ç¿ãç¯å²å
ãXç·ãååæ žã®åå
ã§ãé§ç®±ïŒããã°ãïŒãç¿ãã)
é§ç®±ïŒããã°ãïŒãšãããèžæ°ã®ã€ãŸã£ãè£
眮ãã€ãããšããªããã®ç²åãééãããšããã®ç²åã®è»è·¡ã§ãæ°äœãã液äœããåçãèµ·ããã®ã§ãè»è·¡ããç®ã«èŠããã®ã§ãããïŒâ» æ€å®æç§æžã§ã¯ãååæ žã®åéã§ãé§ç®±ã«ã€ããŠç¿ããïŒïŒã€ã¡ãŒãžçã«ã¯ãé£è¡æ©é²ã®ãããªã®ããã€ã¡ãŒãžããŠãã ãããïŒ
ã§ãç£å Žãå ããå Žåã®ãè»è·¡ã®æ²ããããçãªã©ãããæ¯é»è·ãŸã§ãäºæ³ã§ããã
ãã®ããã«ãé§ç®±ãã€ãã£ãå®éšã«ããã20äžçŽååãäžç€ããã«ã¯ããããããªç²åãçºèŠãããã
ÎŒç²å以å€ã«ããéœé»åïŒããã§ããïŒããé§ç®±ã«ãã£ãŠçºèŠãããŠããã
ïŒâ» ç¯å²å€:ïŒäžçåã§éœé»åãå®éšçã«èŠ³æž¬ããã¢ã³ããŒãœã³ã¯ãé§ç®±ã«éæ¿ãå
¥ããããšã§éœé»åãçºèŠããã
ãšããããïŒÎŒç²åã®çºèŠãããïŒéœé»åã®ã»ããçºèŠã¯æ©ãã
ïŒâ» ç¯å²å€:ïŒãŸããéœé»åã¯ãéåååŠã®ã·ã¥ã¬ãŒãã£ã³ã¬ãŒæ¹çšåŒã«ãç¹æ®çžå¯Ÿæ§çè«ãšãçµã¿åããããããã£ã©ãã¯ã®æ¹çšåŒããããçè«çã«äºæ³ãããŠããã
==== åç©è³ª ====
ãŸãããéœé»åããšããç©è³ªã1932幎ã«éæ¿ãå
¥ããé§ç®±ïŒããã°ãïŒã®å®éšã§ã¢ã³ããŒãœã³ïŒäººåïŒã«ãã£ãŠçºèŠãããŠãããéœé»åã¯è³ªéãé»åãšåãã ããé»è·ãé»åã®å察ã§ããïŒã€ãŸãéœé»åã®é»è·ã¯ãã©ã¹eã¯ãŒãã³ã§ããïŒãïŒâ» éæ¿ã«ã€ããŠã¯é«æ ¡ã®ç¯å²å€ãïŒ
ãããŠãé»åãšéœé»åãè¡çªãããšã2mc<sup>2</sup>ã®ãšãã«ã®ãŒãæŸåºããŠãæ¶æ»
ãããïŒãã®çŸè±¡ïŒé»åãšéœé»åãè¡çªãããš2mc<sup>2</sup>ã®ãšãã«ã®ãŒãæŸåºããŠæ¶æ»
ããçŸè±¡ïŒã®ããšããã察æ¶æ»
ãïŒã€ãããããã€ïŒãšãããïŒ
éœåã«å¯ŸããŠãããåéœåãããããåéœåã¯ãé»è·ãéœåãšå察ã ãã質éãéœåãšåãã§ãããéœåãšè¡çªãããšå¯Ÿæ¶æ»
ãããã
äžæ§åã«å¯ŸããŠãããåäžæ§åãããããåäžæ§åã¯ãé»è·ã¯ãŒãã ãïŒãŒãã®é»è·ã®Â±ãå察ã«ããŠããŒãã®ãŸãŸïŒã質éãåãã§ãäžæ§åãšå¯Ÿæ¶æ»
ãããã
éœé»åãåéœåãåäžæ§åã®ãããªç©è³ªããŸãšããŠãåç©è³ªãšããã
ïŒâ» ç¯å²å€: ïŒæŸå°æ§åäœäœã®ãªãã«ã¯ã厩å£ã®ãšãã«éœé»åãæŸåºãããã®ããããæå
端ã®ç
é¢ã§äœ¿ãããPETïŒéœé»åæå±€æ®åæ³ïŒæè¡ã¯ããããå¿çšãããã®ã§ãããããçŽ ããµãããã«ãªãããªãã·ã°ã«ã³ãŒã¹ãšããç©è³ªã¯ã¬ã³çŽ°èã«ããåã蟌ãŸãããPET蚺æã§ã¯ãããã«ïŒãã«ãªãããªãã·ã°ã«ã³ãŒã¹ã«ïŒæŸå°æ§ã®ããçŽ <sup>18</sup>F ããšãããã æŸå°æ§ãã«ãªãããªãã·ã°ã«ã³ãŒã¹ãçšããŠãããïŒâ» åæ通ã®ãååŠåºç€ãã®æç§æžã«ãçºå±äºé
ãšããŠãã«ãªãããªãã·ã°ã«ã³ãŒã¹ãPET蚺æã§äœ¿ãããŠãããšã玹ä»ãããŠãããïŒ
==== ÎŒç²å ====
[[File:Cosmic-radiation-Shower detection--fr.png|thumb|400px|å®å®ç·ã¯ãå³ã®ããã«ãå°çã®å€§æ°åãªã©ã«å«ãŸããååæ žã«è¡çªããããšã«ãããããã€ãã®äºæ¬¡çãªå®å®ç·ãçºçãããå°çã®é«å±±ã§èŠ³æž¬ã§ããå®å®ç·ã¯ãäºæ¬¡çãªå®å®ç·ã®ã»ãã§ããããã£ãœããå®å®ç©ºéãé£ãã§ããå®å®ç·ã¯ãäžæ¬¡å®å®ç·ãšãããïŒâ» é«æ ¡ã®ç¯å²å
ïŒ å®å®ç·ãšããŠèŠ³æž¬ãããÎŒç²åãéœé»åãÏäžéåã¯ããã®ãããªçŸè±¡ã«ãã£ãŠçºçãããšèããããŠããã]]
åç©è³ªãšã¯å¥ã«ãÎŒç²åããå®å®ç·ã®èŠ³æž¬ããã1937幎ã«èŠã€ãã£ãã
ãã®ÎŒç²åã¯ãé»è·ã¯ãé»åãšåãã ãã質éãé»åãšã¯éããÎŒç²åã®è³ªéã¯ããªããšé»åã®çŽ200åã®è³ªéã§ããã
ÎŒç²åã¯ãã¹ã€ã«éœåãé»åã®åç©è³ªã§ã¯ãªãã®ã§ãã¹ã€ã«éœåãšã察æ¶æ»
ãèµ·ãããªãããé»åãšã察æ¶æ»
ãèµ·ãããªãã
ãªããÎŒç²åã«ããåÎŒç²åãšãããåç©è³ªãååšããããšãåãã£ãŠããã
ãã®ãããªç©è³ªããããããã®äœãã§ããå°äžã§èŠã€ãããªãã®ã¯ãåã«å°äžã®å€§æ°ãªã©ãšè¡çªããŠæ¶æ»
ããŠããŸãããã§ããã
ãªã®ã§ãé«å±±ã®é äžä»è¿ãªã©ã§èŠ³æž¬å®éšããããšãÎŒç²åã®çºèŠã®å¯èœæ§ãé«ãŸãã
ãªã21äžçŽã®çŸåšãÎŒç²åã掻çšããæè¡ãšããŠãçŸåšãç«å±±ãªã©ã®å
éšã芳å¯ããã®ã«ã掻çšãããŠãããÎŒç²åã¯ãééåãé«ãããå°äžã®ç©è³ªãšåå¿ããŠããããã«æ¶æ»
ããŠããŸãã®ã§ããã®ãããªæ§è³ªãå©çšããŠãç«å±±å
éšã®ããã«äººéãå
¥ã蟌ããªãå Žæã芳å¯ãããšããæè¡ãããã§ã«ããã
:ÎŒç²åãªã©ã®çŽ ç²åãæ€åºããããã«ãåç也æ¿ã䜿ããéåžžã®åç也æ¿ãšã¯éããç²åç·ã®ãããªçŽ°ãããã®ãæãããããããã«èª¿æŽãããŠããããååæ žä¹Ÿæ¿ããšãããïŒãååæ žä¹Ÿæ¿ãã«ã€ããŠã¯ç¯å²å€ãïŒ
:也æ¿äžã®æåã«ÎŒç²åãåœããããšã§ãé»æ°ååŠçãªåå¿ãèµ·ããã也æ¿ãåå¿ããã
:æ©ã話ãXç·ãšXç·ä¹Ÿæ¿ã®åçãšåããããªåçã§ãÎŒç²åã䜿ã£ãïŒç«å±±ãªã©ã®ïŒå
éšç 究ãè¡ãããŠããè¿å¹Žã¯ãååæ žä¹Ÿæ¿ã®ä»£ããã«ãåå°äœã»ã³ãµãŒã䜿ã£ãŠãæ€åºããŠããïŒèŠããã«ãããžã«ã¡ã®å
ã»ã³ãµãŒãªã©ãšåãåçïŒã
* ÎŒç²åã®çºçæ¹æ³
ãã®ãããªèŠ³æž¬ã«äœ¿ãããÎŒç²åãã©ããã£ãŠçºçãããã®ãïŒ
å®å®ç·ããé£ãã§ããÎŒç²åããã®ãŸãŸäœ¿ããšããæ¹æ³ããããããå®è¡ããŠããç 究è
ããããããããšã¯å¥ã®ææ³ãšããŠãå éåšãªã©ã§äººå·¥çã«ÎŒç²åãªã©ãçºçããããšããæ¹æ³ãããã
å éåšã䜿ã£ãæ¹æ³ã¯ãäžèšã®éãã
ãŸããã·ã¯ãããã³ããµã€ã¯ãããã³ã䜿ã£ãŠãé»åãªã©ãè¶
é«éã«å éããããããäžè¬ã®ç©è³ªïŒã°ã©ãã¡ã€ããªã©ïŒã«åœãŠãã
ãããšãåœç¶ãããããªç²åãçºçããã
ãã®ãã¡ãÏäžéåããç£æ°ã«åå¿ããïŒãšèããããŠããïŒã®ã§ã倧ããªé»ç£ç³ã³ã€ã«ã§ãÏäžéåãæç²ããã
ãã®Ïäžéåã厩å£ããŠãÎŒç²åãçºçããã
==== â» ç¯å²å€: å®å®ç·ã®çºçåå ã¯äžæ ====
ããããå®å®ç·ãäœã«ãã£ãŠçºçããŠãããã®çºçåå ã¯ãçŸæç¹ã®äººé¡ã«ã¯äžæã§ãããïŒâ» åèæç®: æ°ç åºçã®è³æéã®ãå³èª¬ç©çãïŒ
è¶
æ°æïŒã¡ããããããïŒççºã«ãã£ãŠå®å®ç·ãçºçããã®ã§ã¯ããšãã説ããããããšã«ããå®å®ç·ã®çºçåå ã«ã€ããŠã¯æªè§£æã§ããã
==== ç¯å²å€ïŒ: ã¹ãã³ ====
é»åãéœåãäžæ§åãªã©ã¯ããã¹ãã³ããšããç£ç³ã®ãããªæ§è³ªããã£ãŠãããç£ç³ã«N極ãšS極ãããããã«ãã¹ãã³ã«ãã2çš®é¡ã®åãããããã¹ãã³ã®ãã®2çš®é¡ã®åãã¯ããäžåãããšãäžåããã«ãããäŸãããããç£ç³ã®ç£åã®çºçåå ã¯ãç£ç³äžã®ååã®æå€æ®»é»åã®ã¹ãã³ã®åããåäžæ¹åã«ããã£ãŠãããããã§ãããšèããããŠããã
å
šååã¯ãé»åãéœåãäžæ§åãå«ãã®ã«ããªã®ã«å€ãã®ç©è³ªããããŸãç£æ§ãçºçããªãã®ã¯ãå察笊å·ã®ã¹ãã³ããã€é»åãçµåãããããšã§ãæã¡æ¶ãããããã§ããã
ïŒãŠã£ãããé»åãšéœåã®ãããªé»è·ããã€ç²åã«ããã¹ãã³ããªããšèª€è§£ããŠãã人ãããããäžæ§åã«ãã¹ãã³ã¯ãããïŒ
äžåŠé«æ ¡ã§èŠ³æž¬ãããããªæ®éã®æ¹æ³ã§ã¯ãã¹ãã³ã芳枬ã§ããªãããååãªã©ã®ç©è³ªã«ç£æ°ãå ãã€ã€é«åšæ³¢ãå ãããªã©ãããšãã¹ãã³ã®åœ±é¿ã«ãã£ãŠããã®ååã®æ¯åããããåšæ³¢æ°ãéããªã©ã®çŸè±¡ããã¡ããŠãéæ¥çã«ïŒé»åãªã©ã®ïŒã¹ãã³ã芳枬ã§ãããïŒãªããæ žç£æ°å
±é³Žæ³ïŒNMRãnuclear magnetic resonanceïŒã®åçã§ããã â» çè«çãªè§£æã¯ã倧åŠã¬ãã«ã®ååŠã®ç¥èãå¿
èŠã«ãªãã®ã§çç¥ãããïŒ ååäžã®æ°ŽçŽ ååããããçš®ã®æŸå°æ§åäœäœïŒäžæ§åããã£ã1åãµããã ãã®åäœäœïŒãªã©ãé«åšæ³¢ã®åœ±é¿ãåããããããã®çç±ã®ã²ãšã€ããã¹ãã³ã«ãããã®ã ãšèããããŠããïŒâ» ãªããå»çã§çšããããMRIïŒmagnetic resonance imagingïŒã¯ããã®æ žç£æ°å
±é³Žæ³ïŒNMRïŒãå©çšããŠäººäœå
éšãªã©ã芳枬ããããšããæ©åšã§ãããïŒ
ããŠãå®ã¯çŽ ç²åããã¹ãã³ããã€ã®ãæ®éã§ããã
ÎŒç²åã¯ã¹ãã³ããã€ã
ÎŒç²åã®ãã¹ãã³ããšããæ§è³ªã«ããç£æ°ãšãÎŒç²åã®ééæ§ã®é«ããå©çšããŠãç©è³ªå
éšã®ç£å Žã®èŠ³æž¬æ¹æ³ãšããŠæ¢ã«ç 究ãããŠããããã®ãããªèŠ³æž¬æè¡ããÎŒãªã³ã¹ãã³å転ããšãããè¶
äŒå°äœã®å
éšã®èŠ³æž¬ãªã©ã«ãããã§ã«ãÎŒãªã³ã¹ãã³å転ãã«ãã芳枬ãç 究ãããŠããã
ãŠã£ãããã£ã¢èšäºã[[w:ãã¥ãªã³ã¹ãã³å転]]ãã«ãããšãÎŒãªã³ã®åŽ©å£æã«éœé»åãæŸåºããã®ã§ãéœé»åã®èŠ³æž¬æè¡ãå¿
èŠã§ãããïŒé«æ ¡ã®ç¯å²å€ã§ããããïŒããããã®åŠçã¯ããããããšå匷ããäºãå€ãã
==== éœåãšäžæ§åã®ã¢ã€ãœã¹ãã³ ====
éœåãšäžæ§åã¯ã質éã¯ã»ãšãã©åãã§ãããé»è·ãéãã ãã§ããã
ãããŠãé»åãšæ¯ã¹ããšãéœåãäžæ§åãã質éãããªã倧ããã
ãã®äºããããéœåãäžæ§åã«ããããã«äžèº«ããããå¥ã®ç²åãè©°ãŸã£ãŠããã®ã§ã¯ïŒããšããçåãçãŸããŠããŠãéœåãäžæ§åã®å
éšã®æ¢çŽ¢ãå§ãŸã£ãã
ããããçŸåšã§ããéœåãäžæ§åã®å
éšã®æ§é ã¯ãå®éšçã«ã¯åãåºããŠã¯ããªããïŒâ» éœåãäžæ§åã®å
éšæ§é ãšããŠèª¬æãããŠãããã¯ã©ãŒã¯ãã¯ãåç¬ã§ã¯çºèŠãããŠããªããã¯ã©ãŒã¯ã¯åã«ãå
éšã®èª¬æã®ããã®ãæŠå¿µã§ãããïŒ
æŽå²çã«ã¯ããŸããéœåãšäžæ§åã®å
éšæ§é ãšããŠãæ¶ç©ºã®çŽ ç²åãèããããéœåãšäžæ§åã¯ããããã®çŽ ç²åã®ç¶æ
ãéãã ãããšèããããã
ãã£ãœããé»åã«ã¯ãå
éšæ§é ããªãããšèãããŠããã
ããã20äžçŽãªãã°ãéåååŠã§ã¯ããã®ããããã§ã«ãé»åã®ç¶æ
ãšããŠãã¹ãã³ããšããæŠå¿µããã¿ã€ãã£ãŠãããéåååŠã§ã¯ãååŠçµåã§äŸ¡é»åã2åãŸã§çµåããŠé»å察ã«ãªãçç±ã¯ããã®ã¹ãã³ã2çš®é¡ãããªããŠãå察åãã®ã¹ãã³ã®é»å2åã ããçµåããããã§ããããšãããŠããã
ã¹ãã³ã®2çš®é¡ã®ç¶æ
ã¯ããäžåãããäžåãããšãããµãã«ãããäŸãããããïŒå®éã®æ¹åã§ã¯ãªãã®ã§ãããŸãæ·±å
¥ãããªãããã«ãïŒ
ãã®ãããªéåååŠãåèã«ããŠãéœåãšäžæ§åã§ããã¢ã€ãœã¹ãã³ããšããæŠå¿µãèãããããïŒâ» ãã¢ã€ãœã¹ãã³ãã¯é«æ ¡ç¯å²å€ãïŒ
éœåãšäžæ§åã¯ãã¢ã€ãœã¹ãã³ã®ç¶æ
ãéãã ãããšèããããã
==== ã¯ã©ãŒã¯ ====
ãã®åŸã20äžçŽåã°é ããããã¢ã€ãœã¹ãã³ããçºå±ããããã¯ã©ãŒã¯ããšããçè«ãæå±ãããã
æ¶ç©ºã®ãã¯ã©ãŒã¯ããšãã3åã®çŽ ç²åãä»®å®ãããšãå®åšã®éœåãäžæ§åã®æãç«ã€ã¢ãã«ããå®éšçµæãããŸã説æã§ããäºãåãã£ãã
é»è·(<math>+\frac{2}{3}e</math>)ããã€çŽ ç²åãã¢ããã¯ã©ãŒã¯ããšã±(<math>-\frac{1}{3}e</math>)ããã€çŽ ç²åãããŠã³ã¯ã©ãŒã¯ãããã£ãŠã
:<math>\frac{2}{3}e - \frac{1}{3}e - \frac{1}{3}e=0e</math>ã§éœåã
:<math>\frac{2}{3}e + \frac{2}{3}e - \frac{1}{3}e=1e</math>ã§éœåã
ãšèãããšããããããªçŽ ç²åå®éšã®çµæãããŸã説æã§ããäºãåãã£ãã
ãªããé»åã«ã¯ããã®ãããªå
éšæ§é ã¯ãªãããšèãããããã
ã¢ããã¯ã©ãŒã¯ã¯ãuããšç¥èšãããããŠã³ã¯ã©ãŒã¯ã¯ãdããšç¥èšãããã
éœåã®ã¯ã©ãŒã¯æ§é ã¯uudãšç¥èšãããïŒã¢ãããã¢ãããããŠã³ïŒã
äžæ§åã®ã¯ã©ãŒã¯æ§é ã¯uddãšç¥èšãããïŒã¢ãããããŠã³ãããŠã³ïŒã
==== å éåšå®éšãšäžéå ====
ãªããäžèšã®èª¬æã§ã¯çç¥ãããããããã1950ã60幎代ãããŸã§ã«ãé«å±±ã§ã®å®å®ç·ã®èŠ³æž¬ãããããã¯æŸå°ç·ã®èŠ³æž¬ãããŸããããã¯ãµã€ã¯ãããã³ãªã©ã«ããç²åã®å éåšè¡çªå®éšã«ãããéœåãäžæ§åã®ã»ãã«ããåçšåºŠã®è³ªéã®ããŸããŸãªç²åãçºèŠãããŠããããããæ°çš®ã®ç²åã¯ãäžéåãã«åé¡ãããã
ããããããã¯ã©ãŒã¯ãã®çè«ã¯ããã®ãããª20äžçŽåã°ãããŸã§ã®å®éšã芳枬ããäœçŸåãã®æ°çš®ã®ç²åãçºèŠãããŠããŸãããã®ãããªçµç·¯ããã£ãã®ã§ãã¯ã©ãŒã¯ã®çè«ãæå±ãããã®ã§ããã
ããŠããäžéåãïŒã¡ã
ãããããmason ã¡ãœã³ïŒãšã¯ãããšããšçè«ç©çåŠè
ã®æ¹¯å·ç§æš¹ã1930幎代ã«æå±ãããéœåãšäžæ§åãšãåŒãä»ããŠãããšãããæ¶ç©ºã®ç²åã§ãã£ããã20äžçŽãªãã°ã«æ°çš®ã®ç²åãçºèŠãããéããäžéåãã®ååã䜿ãããããšã«ãªã£ãã
ããŠãå®éšçã«æ¯èŒçæ©ãææããçºèŠããããäžéåãã§ã¯ããÏäžéåãããããããçš®é¡ã®Ïäžéåã¯ãã¢ããã¯ã©ãŒã¯ãšåããŠã³ã¯ã©ãŒã¯ãããªããÏ<sup>+</sup>ãšç¥èšããããïŒããŠã³ã¯ã©ãŒã¯ã®åç©è³ªããåããŠã³ã¯ã©ãŒã¯ãïŒ Ï<sup>ïŒ</sup>ïŒ<math>u\overline{d}</math>
å¥ã®ããçš®é¡ã®Ïäžéåã¯ãããŠã³ã¯ã©ãŒã¯ãšåã¢ããã¯ã©ãŒã¯ãããªããÏ<sup>ãŒ</sup>ãšç¥èšããããÏ<sup>-</sup>ïŒ<math>\overline{u}d</math>
ãã®ããã«ãããç²åå
ã®ã¯ã©ãŒã¯ã¯åèš2åã®ã§ãã£ãŠãè¯ãå ŽåããããïŒããªãããããéœåã®ããã«ã¯ã©ãŒã¯3åã§ãªããŠãããŸããªãå ŽåããããïŒ
ïŒâ» ãã®ãããªå®éšäŸãããç²åå
ã«åèš5åã®ã¯ã©ãŒã¯ã7åã®ã¯ã©ãŒã¯ãèããçè«ããããããããé«æ ¡ç©çã®ç¯å²ã倧å¹
ã«è¶
ããã®ã§ã説æãçç¥ãïŒ
ãŸããäžéåã¯ãèªç¶çã§ã¯çæéã®ããã ã ããååšã§ããç²åã ãšããäºãã芳枬å®éšã«ãã£ãŠãåãã£ãŠãããïŒäžéåã®ååšã§ããæéïŒã寿åœãïŒã¯çããããã«ãä»ã®å®å®ãªç²åã«å€æããŠããŸããïŒ
==== 第2äžä»£ä»¥éã®çŽ ç²å ====
ããããã¢ãããšããŠã³ã ãã§ã¯ã説æããããªãç²åããã©ãã©ããšçºèŠãããŠãããã¯ã©ãŒã¯ã®æå±æã®åœåã¯ãããããã ãã¯ã©ãŒã¯ã®ã¢ãããšããŠã³ã§ããã£ãšãã»ãšãã©ã®äžéåã®æ§é ã説æã§ããã ããã ãšæåŸ
ãããŠããã®ã ããããããããå®å®ç·ãã1940幎代ã«çºèŠããããKäžéåãã®æ§é ã§ãããã¢ãããšããŠã³ã§ã¯èª¬æã§ããªãã£ãã
ãã®ã»ããå éåšã®çºéãªã©ã«ãããã¢ãããšããŠã³ã®çµã¿åããã ãã§èª¬æã§ããæ°ãè¶
ããŠãã©ãã©ããšæ°çš®ã®ãäžéåããçºèŠãããŠããŸãããã¯ãã¢ãããšããŠã³ã ãã§ã¯ãäžéåã®æ§é ã説æãã¥ãããªã£ãŠããäžãÎŒç²åãã説æã§ããªãã
ãŸããå éåšå®éšã«ããã1970幎代ã«ãDäžéåããªã©ãããŸããŸãªäžéåããå®éšçã«å®åšã確èªãããã
ãã®ããã«ãã¢ãããšããŠã³ã ãã§ã¯èª¬æã®ã§ããªãããããããªç²åãååšããããšãåããããã®ãããçŽ ç²åçè«ã§ã¯ããã¢ãããïŒuïŒãšãããŠã³ãïŒdïŒãšãã2çš®é¡ã®ç¶æ
ã®ä»ã«ããããã«ç¶æ
ãèããå¿
èŠã«ãããŸãããããããŠãæ°ããç¶æ
ãšããŠããŸãããã£ãŒã ãïŒèšå·cïŒãšãã¹ãã¬ã³ãžãïŒèšå·sïŒãèãããããå éåšå®éšã®æè¡ãçºå±ããå éåšå®éšã®è¡çªã®ãšãã«ã®ãŒãäžãã£ãŠãããšãããã«ãããããïŒèšå·tïŒãšãããã ãïŒèšå·bïŒãšããã®ãèããããã
ãªããÎŒç²åã«ã¯å
éšæ§é ã¯ãªãããéœåãäžæ§åã«é»åã察å¿ãããã®ãšåæ§ã«ïŒç¬¬1äžä»£ïŒããã£ãŒã ãã¹ãã¬ã³ãžãããªãéœåçã»äžæ§åçãªç²åãšÎŒç²åã察å¿ãããïŒç¬¬2äžä»£ïŒãåæ§ã«ãããããããã ãããªãç²åã«ÎŒç²åã察å¿ãããïŒç¬¬3äžä»£ïŒã
{| class="wikitable"
|+ ã¯ã©ãŒã¯ãšã¬ããã³
|-
! çš®é¡ !! é»è· !! 第1äžä»£ !! 第2äžä»£ !! 第3äžä»£
|-
! rowspan="2"| ã¯ã©ãŒã¯
! <math>\frac{2}{3}e</math>
| ã¢ãã (u)
| ãã£ãŒã (c)
| ããã (t)
|-
! <math>-\frac{1}{3}e</math>
| ããŠã³ (d)
| ã¹ãã¬ã³ãž (s)
| ããã (b)
|-
! rowspan="2"| ã¬ããã³
! rowspan="2"| <math>-e</math>
| é»å (e<sup>ãŒ</sup> )
| ÎŒç²å (''ÎŒ''<sup>ãŒ</sup> )
| Ïç²å (''Ï''<sup>ãŒ</sup> )
|-
| é»åãã¥ãŒããªãïŒ''Îœ''<sub>e</sub> ïŒ
| ÎŒãã¥ãŒããªãïŒ''Îœ''<sub>''ÎŒ''</sub> ïŒ
| Ïãã¥ãŒããªãïŒ''Îœ''<sub>''Ï''</sub> ïŒ
|-
|}
é»åãÎŒç²åã¯å
éšæ§é ããããªããšèããããŠããããã¬ããã³ããšãããå
éšæ§é ããããªããšãããã°ã«ãŒãã«åé¡ãããã
ãKäžéåãã¯ã第1äžä»£ã®ã¯ã©ãŒã¯ãšç¬¬2äžä»£ã®ã¯ã©ãŒã¯ããæãç«ã£ãŠããäºããåãã£ãŠãããïŒâ» æ€å®æç§æžã®ç¯å²å
ãïŒ
ãããŠã2017幎ã®çŸåšãŸã§ãã£ãšãã¯ã©ãŒã¯ã®çè«ããçŽ ç²åã®æ£ããçè«ãšãããŠããã
==== çšèª ====
çŽ ç²åã®èŠ³ç¹ããåé¡ããå Žåã®ãéœåãšäžæ§åã®ããã«ãã¯ã©ãŒã¯3åãããªãç²åã®ããšãããŸãšããŠãããªãªã³ãïŒéç²åïŒãšãããÏäžéåïŒÏ<sup>ïŒ</sup>ïŒ<math>u\overline{d}</math>ïŒãªã©ãã¯ã©ãŒã¯ã2åã®ç²åã¯ãããªãªã³ã«å«ãŸãªãã
ããããäžéåã®ãªãã«ããã©ã ãç²åïŒudsãã¢ããããŠã³ã¹ãã¬ã³ãžã®çµã¿åããïŒã®ããã«ãã¯ã©ãŒã¯3åãããªãç²åããããã©ã ãç²åãªã©ããããªãªã³ã«å«ããã
éœåãšäžæ§åãã©ã ãç²åãªã©ãšãã£ãããªãªã³ã«ãããã«äžéåïŒäžéåã¯äœçš®é¡ãããïŒãå ããã°ã«ãŒãããŸãšããŠãããããã³ããšããã
ãªããæ®éã®ç©è³ªã®ååæ žã§ã¯ãéœåãšäžæ§åãååæ žã«éãŸã£ãŠãããããã®ããã«éœåãšäžæ§åãååæ žã«åŒãåãããåã®ããšã'''æ žå'''ãšãããæ žåã®æ£äœã¯ããŸã ãããŸã解æãããŠããªãïŒå°ãªããšãé«æ ¡ã§æããã»ã©ã«ã¯ããŸã å
åã«ã¯è§£æãããŠããªãïŒã
ãšããããããªãªã³ã«ã¯ãæ žåãåããé説ã§ã¯ãäžéåã«ããæ žåã¯åããšãããŠãããã€ãŸãããããã³ã«ãæ žåãåãã
ãããã³ã¯ãããããã¯ã©ãŒã¯ããæ§æãããŠããäºããããããããã¯ã©ãŒã¯ã«æ žåãåãã®ã ãããçãªäºããèããããŠããã
çè«ã§ã¯ãã¯ã©ãŒã¯ãšã¯ã©ãŒã¯ã©ãããåŒãä»ãããæ¶ç©ºã®ç²åãšããŠãã°ã«ãŒãªã³ããèããããŠãããç©çåŠè
ããçè«ãæå±ãããŠãããããã®æ£äœã¯ããŸã ãããŸã解æãããŠãªããããããç©çåŠè
ãã¡ã¯ãã°ã«ãŒãªã³ãçºèŠããããšäž»åŒµããŠããã
çŸåšã®ç©çåŠã§ã¯ãã¯ã©ãŒã¯ãåç¬ã§ã¯åãåºããŠããªãã®ãšåæ§ã«ãã°ã«ãŒãªã³ãåç¬ã§ã¯åãåºããŠã¯ããªãã
ããŠãç©çåŠã§ã¯ã20äžçŽãããéåååŠããšããçè«ããã£ãŠããã®çè«ã«ãããç©çæ³åã®æ ¹æºã§ã¯ãæ³¢ãšç²åãåºå¥ããã®ãç¡æå³ã ãšèšãããŠããããã®ããããã€ãŠã¯æ³¢ã ãšèããããŠããé»ç£æ³¢ããå Žåã«ãã£ãŠã¯ãå
åããšããç²åãšããŠæ±ãããããã«ãªã£ãã
ãã®ããã«ãããæ³¢ãåå ŽïŒããã°ïŒãªã©ããçè«é¢ã§ã¯ç²åã«çœ®ãæããŠè§£éããŠæ±ãäœæ¥ã®ããšããç©çåŠã§ã¯äžè¬ã«ãéååããšããã
ã°ã«ãŒãªã³ããã¯ã©ãŒã¯ãšã¯ã©ãŒã¯ãåŒãä»ããåããéååãããã®ã§ããããé»è·ãšã®é¡æšã§ãã¯ã©ãŒã¯ã«ãè²è·ïŒã«ã©ãŒè·ïŒãšããã®ãèããŠãããããã®æ§è³ªã¯ãããŸã解æãããŠãªãïŒå°ãªããšãé«æ ¡ã§æããã»ã©ã«ã¯ããŸã å
åã«ã¯è§£æãããŠããªãïŒã
ã°ã«ãŒãªã³ã®ããã«ãåãåªä»ããç²åã®ããšãã²ãŒãžç²åãšããã
{| class="wikitable" style="float: right; text-align: center; margin: 2pt;"
|+ 4ã€ã®åãšã²ãŒãžç²å
|-
! åã®çš®é¡
! ã²ãŒãžç²å
|-
! é»ç£æ°å
| ãå
åã<br>ïŒé»ç£å Žãéååãããã®ïŒ
|-
! ã匷ãåã<br>ïŒã¯ã©ãŒã¯ãåŒãä»ãããåã®ããšïŒ
| ã°ã«ãŒãªã³
|-
! ã匱ãåã<br>ïŒÎ²åŽ©å£ãã€ããã©ããåãã®ããšïŒ
| ãŠã£ãŒã¯ããœã³
|-
! äžæåŒåïŒãéåãïŒ<br>
| ã°ã©ããã³<br>ïŒæªçºèŠïŒ
|-
|}
éåãåªä»ããæ¶ç©ºã®ç²åã®ããšãéååïŒã°ã©ããã³ïŒãšãããããŸã çºèŠãããŠããªããç©çåŠè
ãã¡ããã°ã©ããã³ã¯ããŸã æªçºèŠã§ããããšäž»åŒµããŠããã
é»ç£æ°åãåªä»ããç²åã¯å
åïŒãã©ãã³ïŒãšããããããã¯åã«ãé»ç£å Žãä»®æ³çãªç²åãšããŠçœ®ãæããŠæ±ã£ãã ãã§ããããã©ãã³ã¯ãé«æ ¡ç©çã®é»ç£æ°åéã§ç¿ããããªå€å
žçãªé»ç£æ°èšç®ã§ã¯ããŸã£ãã圹ç«ããªãã
ãªããå
åãã²ãŒãžç²åã«å«ããã
ã€ãŸããå
åãã°ã«ãŒãªã³ã¯ãã²ãŒãžç²åã§ããã
ããŒã¿åŽ©å£ãã€ããã©ãåã®ããšãã匱ãåããšããããã®åãåªä»ããç²åãããŠã£ãŒã¯ããœã³ããšããããæ§è³ªã¯ãããåãã£ãŠããªãããããç©çåŠè
ãã¡ã¯ããŠã£ãŒã¯ããœã³ãçºèŠããããšäž»åŒµããŠããã
ãããããããœã³ããšã¯äœãïŒ
éåååŠã®ã»ãã§ã¯ãé»åã®ãããªãäžç®æã«ããã ãæ°åãŸã§ããååšã§ããªãç²åããŸãšããŠãã§ã«ããªã³ãšããããã§ã«ããªã³çã§ãªãå¥çš®ã®ç²åãšããŠããœã³ããããå
åããããœã³ãšããŠæ±ãããã
ããŠã£ãŒã¯ããœã³ããšã¯ãããããã匱ãåãåªä»ããããœã³ã ãããŠã£ãŒã¯ããœã³ãšåŒãã§ããã®ã ããã
ããŠãé»è·ãšã®é¡æšã§ãã匱ãåãã«é¢ããã匱è·ãïŒããããïŒãšããã®ãæå±ãããŠãããããããããã®æ§è³ªã¯ãããŸã解æãããŠãªãïŒå°ãªããšãé«æ ¡ã§æããã»ã©ã«ã¯ããŸã å
åã«ã¯è§£æãããŠããªãïŒã
ããŠãã匱ãåãã®ããäžæ¹ãã°ã«ãŒãªã³ã®åªä»ããåã®ããšãã匷ãåããšãããã
==== â» ç¯å²å€: ã³ãã«ã60ã®ããŒã¿åŽ©å£ãšã匱ãåã ====
1956幎ã«ãé»åã®ã¹ãã³ã®æ¹åãšãããŒã¿åŽ©å£ç²åã®åºãŠæ¥ãæ¹åãšã®é¢ä¿ãèŠãããã®å®éšãšããŠãã³ãã«ãã®æŸå°æ§åäœäœã§ããã³ãã«ã60ããã¡ããŠæ¬¡ã®ãããªå®éšããã¢ã¡ãªã«ã§è¡ãããã
ã³ãã«ãå
çŽ ïŒå
çŽ èšå·: Co ïŒã®æŸå°æ§åäœäœã§ããã³ãã«ã60ã極äœæž©ã«å·åŽããç£å ŽããããŠå€æ°ã®ã³ãã«ãååã®é»åæ®»ã®å€ç«é»åã¹ãã³ã®æ¹åãããããç¶æ
ã§ãã³ãã«ã60ãããŒã¿åŽ©å£ããŠçºçããããŒã¿ç²åã®åºãæ¹åã調ã¹ãå®éšãã1956幎ã«ã¢ã¡ãªã«ã§è¡ãããã
éãšããã±ã«ãšã³ãã«ãã¯ãããããéå±åäœã§ç£æ§äœã«ãªãå
çŽ ã§ãããå
çŽ åäœã§ç£æ§äœã«ãªãå
çŽ ã¯ããã®3ã€ïŒéãããã±ã«ãã³ãã«ãïŒãããªããïŒãªããæŸå°æ§åäœäœã§ãªãéåžžã®ã³ãã«ãã®ååéã¯59ã§ãããïŒ
ãã®3ã€ïŒéãããã±ã«ãã³ãã«ãïŒã®ãªãã§ãã³ãã«ããäžçªãç£æ°ã«å¯äžããé»åã®æ°ãå€ãããšãéåååŠã®çè«ã«ããæ¢ã«ç¥ãããããã®ã§ïŒã³ãã«ãããã£ãšããdè»éã®é»åã®æ°ãå€ã ïŒãããŒã¿åŽ©å£ãšã¹ãã³ãšã®é¢ä¿ãã¿ãããã®å®éšã«ãã³ãã«ãã®æŸå°æ§åäœäœã§ããã³ãã«ã60ã䜿ãããã®ã§ããã
å®éšã®çµæãã³ãã«ã60ãããŒã¿åŽ©å£ããŠããŒã¿ç²åã®åºãŠããæ¹åã¯ãã³ãã«ã60ã®ã¹ãã³ã®ç£æ°ã®æ¹åãšïŒåãæ¹åãããïŒéã®æ¹åã«å€ãæŸåºãããŠããã®ã芳枬ããããããã¯ã2çš®é¡ïŒã¹ãã³ãšåæ¹åã«ããŒã¿ç²åã®åºãå Žåãšãã¹ãã³ãšå察æ¹åã«ããŒã¿ç²åã®åºãå ŽåïŒã®åŽ©å£ã®ç¢ºçãç°ãªã£ãŠãããããŒã¿åŽ©å£ã®ç¢ºçã®ïŒã¹ãã³æ¹åãåºæºãšããå Žåã®ïŒæ¹å察称æ§ãæããŠããããšã«ãªãã
ãã®ãããªå®éšäºå®ã«ãããã匱ãåãã¯é察称ã§ããããšããã®ãå®èª¬ã
{{-}}
== è泚ã»åèæç®ãªã© ==
[[Category:é«çåŠæ ¡æè²|ç©ãµã€ã2ããããšããããã]]
[[Category:ç©çåŠ|é«ãµã€ã2ããããšããããã]]
[[Category:ç©çåŠæè²|é«ãµã€ã2ããããšããããã]]
[[Category:é«çåŠæ ¡çç§ ç©çII|ããããšããããã]] | 2005-05-08T08:19:59Z | 2024-03-04T16:10:52Z | [
"ãã³ãã¬ãŒã:ã³ã©ã ",
"ãã³ãã¬ãŒã:Val",
"ãã³ãã¬ãŒã:-"
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E7%89%A9%E7%90%86/%E5%8E%9F%E5%AD%90%E7%89%A9%E7%90%86 |
1,971 | åšæåŸãšå
çŽ ã®è«žç¹æ§/å
žåå
çŽ /ã¢ã«ã«ãªåé¡éå±å
çŽ | ã¢ã«ã«ãªåé¡éå±ãšã¯ç¬¬2æå
çŽ ã®ããšããã€ãŠã¯ãåšæè¡šã®2æã®ãã¡ãCa(ã«ã«ã·ãŠã )ãSr(ã¹ããã³ããŠã )ãBa(ããªãŠã )ãRa(ã©ãžãŠã )ã®4ã€ã®å
çŽ ãæããŠããããçŸåšã§ã¯ããã«å ããBe(ããªãªãŠã )ãMg(ãã°ãã·ãŠã )ãã¢ã«ã«ãªåé¡éå±ã«å«ããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ã¢ã«ã«ãªåé¡éå±ãšã¯ç¬¬2æå
çŽ ã®ããšããã€ãŠã¯ãåšæè¡šã®2æã®ãã¡ãCa(ã«ã«ã·ãŠã )ãSr(ã¹ããã³ããŠã )ãBa(ããªãŠã )ãRa(ã©ãžãŠã )ã®4ã€ã®å
çŽ ãæããŠããããçŸåšã§ã¯ããã«å ããBe(ããªãªãŠã )ãMg(ãã°ãã·ãŠã )ãã¢ã«ã«ãªåé¡éå±ã«å«ããã",
"title": ""
}
] | ã¢ã«ã«ãªåé¡éå±ãšã¯ç¬¬2æå
çŽ ã®ããšããã€ãŠã¯ãåšæè¡šã®2æã®ãã¡ãCaïŒã«ã«ã·ãŠã ïŒãSrïŒã¹ããã³ããŠã ïŒãBaïŒããªãŠã ïŒãRaïŒã©ãžãŠã ïŒã®4ã€ã®å
çŽ ãæããŠããããçŸåšã§ã¯ããã«å ããBeïŒããªãªãŠã ïŒãMgïŒãã°ãã·ãŠã ïŒãã¢ã«ã«ãªåé¡éå±ã«å«ããã | '''ã¢ã«ã«ãªåé¡éå±'''ãšã¯'''第2æå
çŽ '''ã®ããšããã€ãŠã¯ãåšæè¡šã®2æã®ãã¡ãCaïŒã«ã«ã·ãŠã ïŒãSrïŒã¹ããã³ããŠã ïŒãBaïŒããªãŠã ïŒãRaïŒã©ãžãŠã ïŒã®4ã€ã®å
çŽ ãæããŠããããçŸåšã§ã¯ããã«å ããBeïŒããªãªãŠã ïŒãMgïŒãã°ãã·ãŠã ïŒãã¢ã«ã«ãªåé¡éå±ã«å«ããã
{{stub}}
[[ã«ããŽãª:å
çŽ ]] | 2005-05-09T13:58:57Z | 2023-08-19T09:40:29Z | [
"ãã³ãã¬ãŒã:Stub"
] | https://ja.wikibooks.org/wiki/%E5%91%A8%E6%9C%9F%E5%BE%8B%E3%81%A8%E5%85%83%E7%B4%A0%E3%81%AE%E8%AB%B8%E7%89%B9%E6%80%A7/%E5%85%B8%E5%9E%8B%E5%85%83%E7%B4%A0/%E3%82%A2%E3%83%AB%E3%82%AB%E3%83%AA%E5%9C%9F%E9%A1%9E%E9%87%91%E5%B1%9E%E5%85%83%E7%B4%A0 |
1,972 | åšæåŸãšå
çŽ ã®è«žç¹æ§/å
žåå
çŽ /ã¢ã«ã«ãªéå±å
çŽ | ã¢ã«ã«ãªéå±ãšã¯ãæ°ŽçŽ ãé€ããåšæ衚第1æã®6ã€ã®å
çŽ ãLi(ãªããŠã )ãNa(ãããªãŠã )ãK(ã«ãªãŠã )ãRb(ã«ããžãŠã )ãCs(ã»ã·ãŠã )ãFr(ãã©ã³ã·ãŠã )ãæãã åºåºç¶æ
ã®æå€æ®»é»åé
眮ã¯(ns)(n=2,3,ã»ã»ã»,7)ã§ãããsé»åã1ã€å€±ã£ãŠã1䟡ã®éœã€ãªã³ã«ãªãããããèªç¶çã«ååšããã¢ã«ã«ãªéå±ã¯ãã¹ãŠé
žåæ°+1ã®ç¶æ
ã§ããã
| [
{
"paragraph_id": 0,
"tag": "p",
"text": "ã¢ã«ã«ãªéå±ãšã¯ãæ°ŽçŽ ãé€ããåšæ衚第1æã®6ã€ã®å
çŽ ãLi(ãªããŠã )ãNa(ãããªãŠã )ãK(ã«ãªãŠã )ãRb(ã«ããžãŠã )ãCs(ã»ã·ãŠã )ãFr(ãã©ã³ã·ãŠã )ãæãã åºåºç¶æ
ã®æå€æ®»é»åé
眮ã¯(ns)(n=2,3,ã»ã»ã»,7)ã§ãããsé»åã1ã€å€±ã£ãŠã1䟡ã®éœã€ãªã³ã«ãªãããããèªç¶çã«ååšããã¢ã«ã«ãªéå±ã¯ãã¹ãŠé
žåæ°+1ã®ç¶æ
ã§ããã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "",
"title": ""
}
] | ã¢ã«ã«ãªéå±ãšã¯ãæ°ŽçŽ ãé€ããåšæ衚第1æã®6ã€ã®å
çŽ ãLiïŒãªããŠã ïŒãNaïŒãããªãŠã ïŒãKïŒã«ãªãŠã ïŒãRbïŒã«ããžãŠã ïŒãCsïŒã»ã·ãŠã ïŒãFrïŒãã©ã³ã·ãŠã ïŒãæãã
ãåºåºç¶æ
ã®æå€æ®»é»åé
眮ã¯(ns)1ïŒn=2,3,ã»ã»ã»,7ïŒã§ãããsé»åã1ã€å€±ã£ãŠã1䟡ã®éœã€ãªã³ã«ãªãããããèªç¶çã«ååšããã¢ã«ã«ãªéå±ã¯ãã¹ãŠé
žåæ°ïŒ1ã®ç¶æ
ã§ããã | '''ã¢ã«ã«ãªéå±'''ãšã¯ãæ°ŽçŽ ãé€ããåšæ衚第1æã®6ã€ã®å
çŽ ãLiïŒãªããŠã ïŒãNaïŒãããªãŠã ïŒãKïŒã«ãªãŠã ïŒãRbïŒã«ããžãŠã ïŒãCsïŒã»ã·ãŠã ïŒãFrïŒãã©ã³ã·ãŠã ïŒãæãã
ãåºåºç¶æ
ã®æå€æ®»é»åé
眮ã¯(ns)<sup>1</sup>ïŒn=2,3,ã»ã»ã»,7ïŒã§ãããsé»åã1ã€å€±ã£ãŠã1䟡ã®éœã€ãªã³ã«ãªãããããèªç¶çã«ååšããã¢ã«ã«ãªéå±ã¯ãã¹ãŠé
žåæ°ïŒ1ã®ç¶æ
ã§ããã
== çè²åå¿ ==
{{stub}}
[[ã«ããŽãª:å
çŽ ]] | null | 2023-01-25T13:27:42Z | [
"ãã³ãã¬ãŒã:Stub"
] | https://ja.wikibooks.org/wiki/%E5%91%A8%E6%9C%9F%E5%BE%8B%E3%81%A8%E5%85%83%E7%B4%A0%E3%81%AE%E8%AB%B8%E7%89%B9%E6%80%A7/%E5%85%B8%E5%9E%8B%E5%85%83%E7%B4%A0/%E3%82%A2%E3%83%AB%E3%82%AB%E3%83%AA%E9%87%91%E5%B1%9E%E5%85%83%E7%B4%A0 |
1,975 | æ§èª²çš(-2012幎床)é«çåŠæ ¡æ°åŠB/æ°å€èšç®ãšã³ã³ãã¥ãŒã¿ãŒ | åççãªç®æ³ãæ±ããèšç®æ©ãçšããŠãããèšç®ããæ¹æ³ãåŠã¶ãããã°ã©ã äŸãšããŠã¯ãPythonãšSchemeãšããèšèªãæ±ããèšèªã®è©³çŽ°ã«ç«ã¡å
¥ãããèãæ¹ãåŠã¶ããšãéèŠãšãªãã
ãŠãŒã¯ãªããã®äºé€æ³ã¯2ã€ã®æŽæ°ã®æ倧å
¬çŽæ°ãæ±ããç®æ³ã§ãããããæŽæ°m, n (m > n > 0) ããšãããã®ãšããŠãŒã¯ãªããã®äºé€æ³ã¯
ã§äžããããã
(å°åº)
m,nãäºãã«çŽ ã§ãããšããèãããmãnã§å²ã£ãåãaãäœããrãšãããšãã
ãæãç«ã€ãããã§ãä»®ã«nãrãå
±éå æ°ãæã€ãªããã®å æ°ã¯mã®å æ°ã§ããããããã¯mãnãäºãã«çŽ ã§ããããšã«ççŸããããã£ãŠãnãrã¯äºãã«çŽ ã§ãããããããäžã®1ã2ãè¡ãªããšäºãã«çŽ ã§ããããå°ãã2ã€ã®æŽæ°n,rãåŸãããããããç¹°ãããããšå°ããåŽã®æŽæ°ã¯1ãšãªãã
å®éäœãã2以äžã«ãªããšãã¯2æ°ãäºãã«çŽ ã§ããããšããã次ã®èšç®ã§æŽã«å°ããæ°ãåŸãããäœãã0ã«ãªãããšã¯å°ããæ¹ã®æ°ã1ã§ããå Žåãé€ããŠã2æ°ãäºãã«çŽ ã§ããããšã«åããããã£ãŠã確ãã«å°ããåŽã®æŽæ°ã¯1ãšãªãããã£ãŠãm,nãäºãã«çŽ ã§ãããšããŠãŒã¯ãªããã®äºé€æ³ã¯ç¢ºãããããã次ã«m,nãæ倧å
¬çŽæ°Mãæã€ãšããèããããã®ãšããmãnã§å²ã£ãåãaãäœããrãšãããšãã
ãæãç«ã€ãã
ãèãããšãrãmãnãšåãæ倧å
¬çŽæ°Mãæã€ãr,m,nãMã§å²ã£ããã®ãããããr',m',n'ãšãããšããããã¯äºãã«çŽ ã§ããã(æ倧å
¬çŽæ°ã®å®çŸ©)ããã®ãšãäžã®2æ°ãäºãã«çŽ ã§ãããšãã®ãŠãŒã¯ãªããäºé€æ³ã®å°åºããå°ããæ¹ã®æŽæ°ã¯1ãåŸãããããã£ãŠå
ã®æŽæ°ã«æ»ãããã«Mããããããšã§ããã®æ¹æ³ã2æ°ã®æ倧å
¬çŽæ°Mãäžããããšãåããããã£ãŠãmãnãå
±éå æ°ãæã€å Žåã«ããŠãŒã¯ãªããäºé€æ³ã¯ç€ºãããã
å®éã®èšç®ã«ã¯èšç®æ©ãçšãããš(ç¹ã«2æ°ã倧ãããšãã«ã¯)䟿å©ã§ããã
ããé¢æ°f(x)ãšx軞ãšã®æ¥ç¹ãæ±ããæ¹æ³ã®1ã€ã«ã2åæ³ããããç¹ã«f(x)ãæ±ããç¹ã§æ£ã®åŸããæã£ãŠãããã®ãšããŠèããããã®æ¹æ³ã¯ã
ãã®æ¹æ³ã¯å
ã
ã®ç¯å²[a,b]ã®äžç¹ãåãã解ãäžç¹ããèŠãŠã©ã¡ãã«ããããå€æããç¯å²ãçããŠããæ¹æ³ã§ããã
å°åœ¢å
¬åŒã¯ãããã°ã©ãf(x)ãšx軞ãšx=a,x=bã«å²ãŸããé¢ç©ãè¿äŒŒçã«æ±ããå
¬åŒã§ããããã®å
¬åŒã§ã¯ã[a,b]ã®ç¯å²ãNåã®å°ããç¯å²ã«åããiåç®ã®ç¯å²ãã [ x i , x i + 1 ] {\displaystyle [x_{i},x_{i+1}]} ãšæžãããã®ãšããã®ç¯å²ã«ãããŠã¯æ±ããé¢ç©ãå°åœ¢ã§è¿äŒŒããŠãé¢ç©ã®ããã¯å°ããã
ããã§ãå°åœ¢ã®é¢ç© s i {\displaystyle s_{i}} ã¯
ã§æžãããããšãèæ
®ãããšãæ±ããé¢ç©Sã¯ã
ã§è¿äŒŒã§ããããšãåããã
Pythonã«ããããã°ã©ã äŸã§ã¯ãååŸ1ã®åã®é¢ç©ãè¿äŒŒçã«æ±ããããã«ãã£ãŠ Ï {\displaystyle \pi } ã®å€ãèšç®ããã
å®éã® Ï {\displaystyle \pi } ã®å€ãšè¿ãå€ãåŸãããŠããããšãåããã
Schemeã«ããããã°ã©ã äŸ
ãã¡ããå®éã® Ï {\displaystyle \pi } ã®å€ãšè¿ãå€ãåŸãããŠããããšãåããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "åççãªç®æ³ãæ±ããèšç®æ©ãçšããŠãããèšç®ããæ¹æ³ãåŠã¶ãããã°ã©ã äŸãšããŠã¯ãPythonãšSchemeãšããèšèªãæ±ããèšèªã®è©³çŽ°ã«ç«ã¡å
¥ãããèãæ¹ãåŠã¶ããšãéèŠãšãªãã",
"title": "æ°å€èšç®ãšã³ã³ãã¥ãŒã¿ãŒ"
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ãŠãŒã¯ãªããã®äºé€æ³ã¯2ã€ã®æŽæ°ã®æ倧å
¬çŽæ°ãæ±ããç®æ³ã§ãããããæŽæ°m, n (m > n > 0) ããšãããã®ãšããŠãŒã¯ãªããã®äºé€æ³ã¯",
"title": "æ°å€èšç®ãšã³ã³ãã¥ãŒã¿ãŒ"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ã§äžããããã",
"title": "æ°å€èšç®ãšã³ã³ãã¥ãŒã¿ãŒ"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "(å°åº)",
"title": "æ°å€èšç®ãšã³ã³ãã¥ãŒã¿ãŒ"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "m,nãäºãã«çŽ ã§ãããšããèãããmãnã§å²ã£ãåãaãäœããrãšãããšãã",
"title": "æ°å€èšç®ãšã³ã³ãã¥ãŒã¿ãŒ"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ãæãç«ã€ãããã§ãä»®ã«nãrãå
±éå æ°ãæã€ãªããã®å æ°ã¯mã®å æ°ã§ããããããã¯mãnãäºãã«çŽ ã§ããããšã«ççŸããããã£ãŠãnãrã¯äºãã«çŽ ã§ãããããããäžã®1ã2ãè¡ãªããšäºãã«çŽ ã§ããããå°ãã2ã€ã®æŽæ°n,rãåŸãããããããç¹°ãããããšå°ããåŽã®æŽæ°ã¯1ãšãªãã",
"title": "æ°å€èšç®ãšã³ã³ãã¥ãŒã¿ãŒ"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "å®éäœãã2以äžã«ãªããšãã¯2æ°ãäºãã«çŽ ã§ããããšããã次ã®èšç®ã§æŽã«å°ããæ°ãåŸãããäœãã0ã«ãªãããšã¯å°ããæ¹ã®æ°ã1ã§ããå Žåãé€ããŠã2æ°ãäºãã«çŽ ã§ããããšã«åããããã£ãŠã確ãã«å°ããåŽã®æŽæ°ã¯1ãšãªãããã£ãŠãm,nãäºãã«çŽ ã§ãããšããŠãŒã¯ãªããã®äºé€æ³ã¯ç¢ºãããããã次ã«m,nãæ倧å
¬çŽæ°Mãæã€ãšããèããããã®ãšããmãnã§å²ã£ãåãaãäœããrãšãããšãã",
"title": "æ°å€èšç®ãšã³ã³ãã¥ãŒã¿ãŒ"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ãæãç«ã€ãã",
"title": "æ°å€èšç®ãšã³ã³ãã¥ãŒã¿ãŒ"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "ãèãããšãrãmãnãšåãæ倧å
¬çŽæ°Mãæã€ãr,m,nãMã§å²ã£ããã®ãããããr',m',n'ãšãããšããããã¯äºãã«çŽ ã§ããã(æ倧å
¬çŽæ°ã®å®çŸ©)ããã®ãšãäžã®2æ°ãäºãã«çŽ ã§ãããšãã®ãŠãŒã¯ãªããäºé€æ³ã®å°åºããå°ããæ¹ã®æŽæ°ã¯1ãåŸãããããã£ãŠå
ã®æŽæ°ã«æ»ãããã«Mããããããšã§ããã®æ¹æ³ã2æ°ã®æ倧å
¬çŽæ°Mãäžããããšãåããããã£ãŠãmãnãå
±éå æ°ãæã€å Žåã«ããŠãŒã¯ãªããäºé€æ³ã¯ç€ºãããã",
"title": "æ°å€èšç®ãšã³ã³ãã¥ãŒã¿ãŒ"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "å®éã®èšç®ã«ã¯èšç®æ©ãçšãããš(ç¹ã«2æ°ã倧ãããšãã«ã¯)䟿å©ã§ããã",
"title": "æ°å€èšç®ãšã³ã³ãã¥ãŒã¿ãŒ"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "ããé¢æ°f(x)ãšx軞ãšã®æ¥ç¹ãæ±ããæ¹æ³ã®1ã€ã«ã2åæ³ããããç¹ã«f(x)ãæ±ããç¹ã§æ£ã®åŸããæã£ãŠãããã®ãšããŠèããããã®æ¹æ³ã¯ã",
"title": "æ°å€èšç®ãšã³ã³ãã¥ãŒã¿ãŒ"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "ãã®æ¹æ³ã¯å
ã
ã®ç¯å²[a,b]ã®äžç¹ãåãã解ãäžç¹ããèŠãŠã©ã¡ãã«ããããå€æããç¯å²ãçããŠããæ¹æ³ã§ããã",
"title": "æ°å€èšç®ãšã³ã³ãã¥ãŒã¿ãŒ"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "å°åœ¢å
¬åŒã¯ãããã°ã©ãf(x)ãšx軞ãšx=a,x=bã«å²ãŸããé¢ç©ãè¿äŒŒçã«æ±ããå
¬åŒã§ããããã®å
¬åŒã§ã¯ã[a,b]ã®ç¯å²ãNåã®å°ããç¯å²ã«åããiåç®ã®ç¯å²ãã [ x i , x i + 1 ] {\\displaystyle [x_{i},x_{i+1}]} ãšæžãããã®ãšããã®ç¯å²ã«ãããŠã¯æ±ããé¢ç©ãå°åœ¢ã§è¿äŒŒããŠãé¢ç©ã®ããã¯å°ããã",
"title": "æ°å€èšç®ãšã³ã³ãã¥ãŒã¿ãŒ"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ããã§ãå°åœ¢ã®é¢ç© s i {\\displaystyle s_{i}} ã¯",
"title": "æ°å€èšç®ãšã³ã³ãã¥ãŒã¿ãŒ"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "ã§æžãããããšãèæ
®ãããšãæ±ããé¢ç©Sã¯ã",
"title": "æ°å€èšç®ãšã³ã³ãã¥ãŒã¿ãŒ"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "ã§è¿äŒŒã§ããããšãåããã",
"title": "æ°å€èšç®ãšã³ã³ãã¥ãŒã¿ãŒ"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "Pythonã«ããããã°ã©ã äŸã§ã¯ãååŸ1ã®åã®é¢ç©ãè¿äŒŒçã«æ±ããããã«ãã£ãŠ Ï {\\displaystyle \\pi } ã®å€ãèšç®ããã",
"title": "æ°å€èšç®ãšã³ã³ãã¥ãŒã¿ãŒ"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "å®éã® Ï {\\displaystyle \\pi } ã®å€ãšè¿ãå€ãåŸãããŠããããšãåããã",
"title": "æ°å€èšç®ãšã³ã³ãã¥ãŒã¿ãŒ"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "Schemeã«ããããã°ã©ã äŸ",
"title": "æ°å€èšç®ãšã³ã³ãã¥ãŒã¿ãŒ"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "ãã¡ããå®éã® Ï {\\displaystyle \\pi } ã®å€ãšè¿ãå€ãåŸãããŠããããšãåããã",
"title": "æ°å€èšç®ãšã³ã³ãã¥ãŒã¿ãŒ"
}
] | null | {{pathnav|frame=1|é«çåŠæ ¡æ°åŠ|é«çåŠæ ¡æ°åŠB}}
==æ°å€èšç®ãšã³ã³ãã¥ãŒã¿ãŒ==
åççãªç®æ³ãæ±ããèšç®æ©ãçšããŠãããèšç®ããæ¹æ³ãåŠã¶ãããã°ã©ã äŸãšããŠã¯ã[[Python]]ãš[[Scheme]]ãšããèšèªãæ±ããèšèªã®è©³çŽ°ã«ç«ã¡å
¥ãããèãæ¹ãåŠã¶ããšãéèŠãšãªãã
===æŽæ°ã®ç®æ³===
====ãŠãŒã¯ãªããã®äºé€æ³====
ãŠãŒã¯ãªããã®äºé€æ³ã¯2ã€ã®æŽæ°ã®æ倧å
¬çŽæ°ãæ±ããç®æ³ã§ãããããæŽæ°m, n (m > n > 0) ããšãããã®ãšããŠãŒã¯ãªããã®äºé€æ³ã¯
#mãnã§å²ã£ãäœããèšç®ãããããrãšããããã®ãšãr=0ãªã3ã«é²ã¿ã<math>r \ne 0</math>ãªãã2ã«é²ãã
#mã以åã®nã®å€ã§çœ®ãæããnãrã®å€ã§çœ®ãæãã1ã«æ»ãã
#nã®å€ãæ倧å
¬çŽæ°ãšãªã£ãŠããã
ã§äžããããã
(å°åº)
m,nãäºãã«çŽ ã§ãããšããèãããmãnã§å²ã£ãåãaãäœããrãšãããšãã
:<math>m=na+r</math> ãã ã <math>(r<n<m)</math>
ãæãç«ã€ãããã§ãä»®ã«nãrãå
±éå æ°ãæã€ãªããã®å æ°ã¯mã®å æ°ã§ããããããã¯mãnãäºãã«çŽ ã§ããããšã«ççŸããããã£ãŠãnãrã¯äºãã«çŽ ã§ãããããããäžã®1ã2ãè¡ãªããšäºãã«çŽ ã§ããããå°ãã2ã€ã®æŽæ°n,rãåŸãããããããç¹°ãããããšå°ããåŽã®æŽæ°ã¯1ãšãªãã
å®éäœãã2以äžã«ãªããšãã¯2æ°ãäºãã«çŽ ã§ããããšããã次ã®èšç®ã§æŽã«å°ããæ°ãåŸãããäœãã0ã«ãªãããšã¯å°ããæ¹ã®æ°ã1ã§ããå Žåãé€ããŠã2æ°ãäºãã«çŽ ã§ããããšã«åããããã£ãŠã確ãã«å°ããåŽã®æŽæ°ã¯1ãšãªãããã£ãŠãm,nãäºãã«çŽ ã§ãããšããŠãŒã¯ãªããã®äºé€æ³ã¯ç¢ºãããããã次ã«m,nãæ倧å
¬çŽæ°Mãæã€ãšããèããããã®ãšããmãnã§å²ã£ãåãaãäœããrãšãããšãã
:<math>m=na+r</math> ãã ã <math>(r<n<m)</math>
ãæãç«ã€ãã
:<math>r = m - na</math>
ãèãããšãrãmãnãšåãæ倧å
¬çŽæ°Mãæã€ãr,m,nãMã§å²ã£ããã®ãããããr',m',n'ãšãããšããããã¯äºãã«çŽ ã§ãããïŒæ倧å
¬çŽæ°ã®å®çŸ©ïŒããã®ãšãäžã®2æ°ãäºãã«çŽ ã§ãããšãã®ãŠãŒã¯ãªããäºé€æ³ã®å°åºããå°ããæ¹ã®æŽæ°ã¯1ãåŸãããããã£ãŠå
ã®æŽæ°ã«æ»ãããã«Mããããããšã§ããã®æ¹æ³ã2æ°ã®æ倧å
¬çŽæ°Mãäžããããšãåããããã£ãŠãmãnãå
±éå æ°ãæã€å Žåã«ããŠãŒã¯ãªããäºé€æ³ã¯ç€ºãããã
å®éã®èšç®ã«ã¯èšç®æ©ãçšãããšïŒç¹ã«2æ°ã倧ãããšãã«ã¯ïŒäŸ¿å©ã§ããã
;[https://paiza.io/projects/mArZ05TFJbs_RlcxXzbJPw?language=python3 Pythonã«ããããã°ã©ã äŸ]:<syntaxhighlight lang=python3>
def euclid(m, n):
print(f"euclid({m}, {n})")
if (n == 0):
return m
return euclid(n, m % n)
print(euclid(45,30))
print(euclid(45,28))
print(euclid(30,28))
</syntaxhighlight>
;å®è¡çµæ:<syntaxhighlight lang=text>
euclid(45, 30)
euclid(30, 15)
euclid(15, 0)
15
euclid(45, 28)
euclid(28, 17)
euclid(17, 11)
euclid(11, 6)
euclid(6, 5)
euclid(5, 1)
euclid(1, 0)
1
euclid(30, 28)
euclid(28, 2)
euclid(2, 0)
2
</syntaxhighlight>
;[[Scheme]]ã«ããããã°ã©ã äŸ:<syntaxhighlight lang="Scheme">
(define (euclid m n)
(let ((r (modulo m n)))
(if (zero? r) ;ãããŸã§ãå°åºéçšã®1
n ;ãããå°åºéçšã®3
(euclid n r)))) ;ãããå°åºéçšã®2
;;;å®è¡äŸ
;;> (euclid 45 30)
;;15
;;> (euclid 45 28)
;;1
;;> (euclid 30 28)
;;2
</syntaxhighlight>
===å®æ°ã®ç®æ³===
==== 2åæ³====
ããé¢æ°f(x)ãšx軞ãšã®æ¥ç¹ãæ±ããæ¹æ³ã®1ã€ã«ã2åæ³ããããç¹ã«f(x)ãæ±ããç¹ã§æ£ã®åŸããæã£ãŠãããã®ãšããŠèããããã®æ¹æ³ã¯ã
# ç¯å²[a,b]å
ã«x軞ãšæ±ããé¢æ°f(x)ã®æ¥ç¹ãå«ãŸããããã«ã2æ°a,bãå®ããã
# mid_point = (a+b)/2 ãèšç®ãããããf(mid_point)ãååã«0ã«è¿ããã°4ã«é²ãã
# ããf(mid_point)<math>></math>0ãªããmid_pointã®å€ãbã®å€ã«ä»£å
¥ãã2ã«æ»ãããããf(mid_point)<math><</math>0ãªããmid_pointã®å€ãaã®å€ã«ä»£å
¥ãã2ã«æ»ãã
# mid_pointã®å€ãæ±ããæ¥ç¹ã®x座æšã§ããã
ãã®æ¹æ³ã¯å
ã
ã®ç¯å²[a,b]ã®äžç¹ãåãã解ãäžç¹ããèŠãŠã©ã¡ãã«ããããå€æããç¯å²ãçããŠããæ¹æ³ã§ããã
;[[Python]]ã«ãã[https://paiza.io/projects/mslsT2vksLfwnt8HqWmn-A?language=python3 ã³ãŒãäŸ]:<syntaxhighlight lang=python3>
from math import isfinite
def bisection(func, left: float, right: float) -> float:
# acceptance inspection
assert (callable(func)),"func is not callable."
assert (isfinite(left)),f"The left({left}) is not a finite number."
assert (isfinite(right)),f"The right({right}) side is not a finite number."
assert (left <= right),f"The left({left}) is bigger than the right({right})."
# Implementation of core algorithms
def core(f, low: float, high: float) -> float:
x = (low + high) / 2
fx = f(x)
if (abs(fx) < +1.0e-10):
return x
if fx < 0.0:
low = x
else:
high = x
return core(f, low, high)
return core(func, left, right)
print(bisection(lambda x: x-1, 0.0, 3.0))
print(bisection(lambda x: x*x-1, 0, 3))
</syntaxhighlight>
;å®è¡çµæ:<syntaxhighlight lang=text>
0.9999999999417923
1.0000000000291038
</syntaxhighlight>
:ãã®ã³ãŒãã¯<math>\lambda(x)=x-1</math>ããŸãã¯ã<math>\lambda(x)=x^2-1</math>ã®ãšãã«è©Šããããçµæ㯠0.9999999999417923 ããã³ 1.0000000000291038 ã§ãããå
å1.0ã«è¿ãå€ãè¿ããŠããã
:
;[[Scheme]]ã«ãã[https://paiza.io/projects/4Du9cGTR0Q3-UWWN24JEqw ã³ãŒãäŸ]:<syntaxhighlight lang="Scheme">
(define (bisection f a b) ;æé 1ã
(let ((e (expt 10 -10))
(mid_point (/ (+ a b) 2))) ;æé 2ãäžç¹ã®èšç®ã
(cond ((or (zero? (f mid_point))
(< (- e) (f mid_point) e))
(exact->inexact mid_point)) ;ãããŸã§ãæé 4ã
((> (f mid_point) 0)
(bisection f a mid_point))
(else (bisection f mid_point b))))) ;ãããŸã§ãæé 3
(print (bisection (lambda (x) (- x 1)) 0 3)) ;x-1ã®è§£ã0ã3éã§æ¢ãã
(print (bisection (lambda (x) (- (expt x 2) 1)) 0 3)) ;x^2-1ã®è§£ã0ã3éã§æ¢ãã
</syntaxhighlight>
;å®è¡çµæ:<syntaxhighlight lang=text>
0.9999999999417923
1.0000000000291038
</syntaxhighlight>
:ãã®ã³ãŒãã<math>\lambda(x)=x-1</math>ããŸãã¯ã<math>\lambda(x)=x^2-1</math>ã®ãšãã«è©Šãããã[[Python]]çã®çµæãšäžèŽããŠããã
==== å°åœ¢å
¬åŒ====
å°åœ¢å
¬åŒã¯ãããã°ã©ãf(x)ãšx軞ãšx=a,x=bã«å²ãŸããé¢ç©ãè¿äŒŒçã«æ±ããå
¬åŒã§ããããã®å
¬åŒã§ã¯ã[a,b]ã®ç¯å²ãNåã®å°ããç¯å²ã«åããiåç®ã®ç¯å²ãã<math>[x _i,x _{i+1}]</math>ãšæžãããã®ãšããã®ç¯å²ã«ãããŠã¯æ±ããé¢ç©ãå°åœ¢ã§è¿äŒŒããŠãé¢ç©ã®ããã¯å°ããã
:æ£ç¢ºãªé¢ç©ãšå°åœ¢ã®é¢ç©ã®ããã®çµµ
ããã§ãå°åœ¢ã®é¢ç©<math>s _i</math>ã¯
:<math>
s _i = \frac12 \{ f(x _i)+f(x _{i+1}) \} \cdot (x _{i+1}-x _i )
</math>
ã§æžãããããšãèæ
®ãããšãæ±ããé¢ç©Sã¯ã
:<math>
S=\sum _{i=0} ^N s _i
</math>
ã§è¿äŒŒã§ããããšãåããã
[[Python]]ã«ããããã°ã©ã äŸã§ã¯ãååŸ1ã®åã®ååã®ïŒã®é¢ç©ãè¿äŒŒçã«æ±ããããã«ãã£ãŠ<math>\pi/4</math>ã®å€ãèšç®ããã
;[https://paiza.io/projects/MCQYVUDrCC7EFyh3OFg1bg?language=python3 trapezoid.py]:<syntaxhighlight lang=python3>
from math import sqrt,pi
from numbers import Number
def trapezoid_formula(f, a, b):
assert callable(f), "f must be a callable"
assert isinstance(a, Number), "a must be a number"
assert isinstance(b, Number), "b must be a number"
n = 20
dx = (b - a) / n
sum = 0
for i in range(n):
sum += (f(a + dx * i) + f(a + dx * (i + 1))) * dx / 2
return sum
print(trapezoid_formula(lambda x: sqrt(1 - x ** 2), 0, 1))
print(pi/4)
</syntaxhighlight>
;å®è¡çµæ:<syntaxhighlight lang=text>
0.7821162199387454
0.7853981633974483
</syntaxhighlight>
å®éã®<math>\pi/4</math>ã®å€ãšè¿ãå€ãåŸãããŠããããšãåããã
[[Scheme]]ã«ããããã°ã©ã äŸ
;[https://paiza.io/projects/YAGiEEO9bJtET0tSvA9_2A?language=scheme trapezoid.scm]:<syntaxhighlight lang="Scheme">
(define (trapezoid_formula f a b)
(let ((n 20))
(let ((dx (/ (- b a) n)))
(let loop ((i 0) (sum 0))
(if (= i n)
(exact->inexact sum)
(loop (+ i 1)
(+ sum (* (+ (f (+ a (* dx i)))
(f (+ a (* dx (+ i 1)))))
(/ dx 2)))))))))
(print (trapezoid_formula (lambda (x)
(sqrt (- 1 (expt x 2))))
0 1) )
(print (atan 1.0))
</syntaxhighlight>
;å®è¡çµæ:<syntaxhighlight lang=text>
0.7821162199387455
0.7853981633974483
</syntaxhighlight>
ãã¡ããå®éã®<math>\pi/4</math>ã®å€ãšè¿ãå€ãåŸãããŠããããšãåããã
[[Category:é«çåŠæ ¡æ°åŠB|ããã¡ãããããšããã²ãã]]
[[ã«ããŽãª:ã³ã³ãã¥ãŒã¿]] | 2005-05-11T12:03:16Z | 2024-02-28T22:35:57Z | [
"ãã³ãã¬ãŒã:Pathnav"
] | https://ja.wikibooks.org/wiki/%E6%97%A7%E8%AA%B2%E7%A8%8B(-2012%E5%B9%B4%E5%BA%A6)%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E6%95%B0%E5%AD%A6B/%E6%95%B0%E5%80%A4%E8%A8%88%E7%AE%97%E3%81%A8%E3%82%B3%E3%83%B3%E3%83%94%E3%83%A5%E3%83%BC%E3%82%BF%E3%83%BC |
1,979 | é»ç£æ°åŠ | æ¬é
ã¯ç©çåŠ é»ç£æ°åŠ (Electromagnetism) ã®è§£èª¬ã§ããã
ããã§ã¯é»æ°ã»ç£æ°ãé¢é£ããçŸè±¡ãæ±ããæŽå²çã«ã¯é»å Žãšç£å Žã«ããçžäºäœçšã¯æ©ãããç¥ãããŠãããçŸä»£ã®æè¡ã®å€ãã¯ãããã®åã«ãã£ãŠããããŸããããã ãã§ã¯ãªããäžã®äžã«ååšããåã®ãã¡ã®ã»ãšãã©ã¯é»ç£æ°åã§æžãããããšãç¥ãããŠãããããã¯ãé»ç£æ°åãä»ã®çžäºäœçšãšæ¯ã¹ãŠãå·šèŠçã«èŠãå Žåã«çžå¯Ÿçã«åŒ·ãåã«ãããã®ã§ããããã§ãããäŸå€çã«ã倩äœãšå€©äœã®éã®çžäºäœçšã¯éåã«ãã£ãŠèšè¿°ãããããããã¯æãå
šäœãšããŠé»æ°çã«äžæ§ã§ãããä»ã®å€©äœãšæ¯èŒçå°ããé»ç£çãªçžäºäœçšããæããªãããšã«ããã
ããã§ã¯ãç¹ã«é»ç£æ°ã«ããåã®ãã¡ã®åççãªèšè¿°æ³ãèŠãŠè¡ãããã®éšåã¯ãååŠãçç©ãé»æ°ãªã©ããããåéã«å¿çšããããããçç³»åéã«é²ãå
šãŠã®åŠçãããç¿çããŠãããã°ãªããªãã
ãŸãããã®åéã¯é«çæè²ã®é»æ°ãšç£æ°ã«åœãããååŠè
ã¯è©²åœæç§æžãç解ã®å©ããšãªãããåŠç¿ã«è¡ãè©°ãŸã£ããåç
§ããé¡ããããã
ç©çåŠç§ã«é²ãåŠçã¯ããã®åŸé»ç£æ°åŠII以éã§çžå¯Ÿè«çãªèšè¿°æ³ãšæŽåçãªèšè¿°ã«ããé»ç£æ°åŠãåŠã¶ããšã«ãªããé»ç£æ°åŠã§ã¯ãã®ãããªèŠç¹ã¯çšãããå€å
žçãª3次å
çèšè¿°æ³ã«ãšã©ããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "æ¬é
ã¯ç©çåŠ é»ç£æ°åŠ (Electromagnetism) ã®è§£èª¬ã§ããã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ããã§ã¯é»æ°ã»ç£æ°ãé¢é£ããçŸè±¡ãæ±ããæŽå²çã«ã¯é»å Žãšç£å Žã«ããçžäºäœçšã¯æ©ãããç¥ãããŠãããçŸä»£ã®æè¡ã®å€ãã¯ãããã®åã«ãã£ãŠããããŸããããã ãã§ã¯ãªããäžã®äžã«ååšããåã®ãã¡ã®ã»ãšãã©ã¯é»ç£æ°åã§æžãããããšãç¥ãããŠãããããã¯ãé»ç£æ°åãä»ã®çžäºäœçšãšæ¯ã¹ãŠãå·šèŠçã«èŠãå Žåã«çžå¯Ÿçã«åŒ·ãåã«ãããã®ã§ããããã§ãããäŸå€çã«ã倩äœãšå€©äœã®éã®çžäºäœçšã¯éåã«ãã£ãŠèšè¿°ãããããããã¯æãå
šäœãšããŠé»æ°çã«äžæ§ã§ãããä»ã®å€©äœãšæ¯èŒçå°ããé»ç£çãªçžäºäœçšããæããªãããšã«ããã",
"title": ""
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ããã§ã¯ãç¹ã«é»ç£æ°ã«ããåã®ãã¡ã®åççãªèšè¿°æ³ãèŠãŠè¡ãããã®éšåã¯ãååŠãçç©ãé»æ°ãªã©ããããåéã«å¿çšããããããçç³»åéã«é²ãå
šãŠã®åŠçãããç¿çããŠãããã°ãªããªãã",
"title": ""
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ãŸãããã®åéã¯é«çæè²ã®é»æ°ãšç£æ°ã«åœãããååŠè
ã¯è©²åœæç§æžãç解ã®å©ããšãªãããåŠç¿ã«è¡ãè©°ãŸã£ããåç
§ããé¡ããããã",
"title": ""
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ç©çåŠç§ã«é²ãåŠçã¯ããã®åŸé»ç£æ°åŠII以éã§çžå¯Ÿè«çãªèšè¿°æ³ãšæŽåçãªèšè¿°ã«ããé»ç£æ°åŠãåŠã¶ããšã«ãªããé»ç£æ°åŠã§ã¯ãã®ãããªèŠç¹ã¯çšãããå€å
žçãª3次å
çèšè¿°æ³ã«ãšã©ããã",
"title": ""
}
] | æ¬é
ã¯ç©çåŠ é»ç£æ°åŠ (Electromagnetism) ã®è§£èª¬ã§ããã ããã§ã¯é»æ°ã»ç£æ°ãé¢é£ããçŸè±¡ãæ±ããæŽå²çã«ã¯é»å Žãšç£å Žã«ããçžäºäœçšã¯æ©ãããç¥ãããŠãããçŸä»£ã®æè¡ã®å€ãã¯ãããã®åã«ãã£ãŠããããŸããããã ãã§ã¯ãªããäžã®äžã«ååšããåã®ãã¡ã®ã»ãšãã©ã¯é»ç£æ°åã§æžãããããšãç¥ãããŠãããããã¯ãé»ç£æ°åãä»ã®çžäºäœçšãšæ¯ã¹ãŠãå·šèŠçã«èŠãå Žåã«çžå¯Ÿçã«åŒ·ãåã«ãããã®ã§ããããã§ãããäŸå€çã«ã倩äœãšå€©äœã®éã®çžäºäœçšã¯éåã«ãã£ãŠèšè¿°ãããããããã¯æãå
šäœãšããŠé»æ°çã«äžæ§ã§ãããä»ã®å€©äœãšæ¯èŒçå°ããé»ç£çãªçžäºäœçšããæããªãããšã«ããã ããã§ã¯ãç¹ã«é»ç£æ°ã«ããåã®ãã¡ã®åççãªèšè¿°æ³ãèŠãŠè¡ãããã®éšåã¯ãååŠãçç©ãé»æ°ãªã©ããããåéã«å¿çšããããããçç³»åéã«é²ãå
šãŠã®åŠçãããç¿çããŠãããã°ãªããªãã ãŸãããã®åéã¯é«çæè²ã®é»æ°ãšç£æ°ã«åœãããååŠè
ã¯è©²åœæç§æžãç解ã®å©ããšãªãããåŠç¿ã«è¡ãè©°ãŸã£ããåç
§ããé¡ããããã ç©çåŠç§ã«é²ãåŠçã¯ããã®åŸé»ç£æ°åŠII以éã§çžå¯Ÿè«çãªèšè¿°æ³ãšæŽåçãªèšè¿°ã«ããé»ç£æ°åŠãåŠã¶ããšã«ãªããé»ç£æ°åŠã§ã¯ãã®ãããªèŠç¹ã¯çšãããå€å
žçãª3次å
çèšè¿°æ³ã«ãšã©ããã | {{åä¿è·S}}
{{Pathnav|ã¡ã€ã³ããŒãž|èªç¶ç§åŠ|ç©çåŠ|frame=1|small=1}}
{{Wikiversity|Topic:é»ç£æ°åŠ|é»ç£æ°åŠ|}}
æ¬é
ã¯ç©çåŠ é»ç£æ°åŠ (Electromagnetism) ã®è§£èª¬ã§ããã
ããã§ã¯é»æ°ã»ç£æ°ãé¢é£ããçŸè±¡ãæ±ããæŽå²çã«ã¯é»å Žãšç£å Žã«ããçžäºäœçšã¯æ©ãããç¥ãããŠãããçŸä»£ã®æè¡ã®å€ãã¯ãããã®åã«ãã£ãŠããããŸããããã ãã§ã¯ãªããäžã®äžã«ååšããåã®ãã¡ã®ã»ãšãã©ã¯é»ç£æ°åã§æžãããããšãç¥ãããŠãããããã¯ãé»ç£æ°åãä»ã®çžäºäœçšãšæ¯ã¹ãŠãå·šèŠçã«èŠãå Žåã«çžå¯Ÿçã«åŒ·ãåã«ãããã®ã§ããããã§ãããäŸå€çã«ã倩äœãšå€©äœã®éã®çžäºäœçšã¯éåã«ãã£ãŠèšè¿°ãããããããã¯æãå
šäœãšããŠé»æ°çã«äžæ§ã§ãããä»ã®å€©äœãšæ¯èŒçå°ããé»ç£çãªçžäºäœçšããæããªãããšã«ããã
ããã§ã¯ãç¹ã«é»ç£æ°ã«ããåã®ãã¡ã®åççãªèšè¿°æ³ãèŠãŠè¡ãããã®éšåã¯ãååŠãçç©ãé»æ°ãªã©ããããåéã«å¿çšããããããçç³»åéã«é²ãå
šãŠã®åŠçãããç¿çããŠãããã°ãªããªãã
ãŸãããã®åéã¯é«çæè²ã®[[é«çåŠæ ¡_ç©ç#é»æ°ãšç£æ°|é»æ°ãšç£æ°]]ã«åœãããååŠè
ã¯è©²åœæç§æžãç解ã®å©ããšãªãããåŠç¿ã«è¡ãè©°ãŸã£ããåç
§ããé¡ããããã
ç©çåŠç§ã«é²ãåŠçã¯ããã®åŸ[[é»ç£æ°åŠII]]以éã§çžå¯Ÿè«çãªèšè¿°æ³ãšæŽåçãªèšè¿°ã«ããé»ç£æ°åŠãåŠã¶ããšã«ãªãã[[é»ç£æ°åŠ]]ã§ã¯ãã®ãããªèŠç¹ã¯çšãããå€å
žçãª3次å
çèšè¿°æ³ã«ãšã©ããã
== ç®æ¬¡ ==
# [[é»ç£æ°åŠ/éé»å Ž|éé»å Ž]]{{é²æ|75%|2023-11-05}}
## [[é»ç£æ°åŠ/éé»å Ž#é»è·ã®éã«åãå|é»è·ã®éã«åãå]]
## [[é»ç£æ°åŠ/éé»å Ž#é»ç|é»ç]]
## [[é»ç£æ°åŠ/éé»å Ž#é»äœ|é»äœ]]
## [[é»ç£æ°åŠ/éé»å Ž#èªé»äœ|èªé»äœ]]
# [[é»ç£æ°åŠ/éç£å Ž|éç£å Ž]]{{é²æ|75%|2023-11-05}}
## [[é»ç£æ°åŠ/éç£å Ž#ç£æ°çãªåã®å°å
¥|ç£æ°çãªåã®å°å
¥]]
## [[é»ç£æ°åŠ/éç£å Ž#ç£ç|ç£ç]]
## [[é»ç£æ°åŠ/éç£å Ž#ããª-ãµããŒã«ã®æ³å|ããª-ãµããŒã«ã®æ³å]]
# [[é»ç£æ°åŠ/é»ç£èªå°|é»ç£èªå°]]{{é²æ|25%|2023-11-05}}
# [[é»ç£æ°åŠ/ãã¯ã¹ãŠã§ã«ã®æ¹çšåŒ|ãã¯ã¹ãŠã§ã«ã®æ¹çšåŒ]]
# [[é»ç£æ°åŠ/é»ç£æ³¢ã®åŒã®å°åº|é»ç£æ³¢ã®åŒã®å°åº]]{{é²æ|100%|2023-11-05}}
# [[é»ç£æ°åŠ/é»ç£å Ž|é»ç£å Ž]]{{é²æ|25%|2023-11-05}}
{{DEFAULTSORT:ãŠããããã}}
[[Category:é»ç£æ°åŠ|*]]
{{NDC|427}} | 2005-05-12T11:11:37Z | 2023-11-05T01:48:05Z | [
"ãã³ãã¬ãŒã:Wikiversity",
"ãã³ãã¬ãŒã:é²æ",
"ãã³ãã¬ãŒã:NDC",
"ãã³ãã¬ãŒã:åä¿è·S",
"ãã³ãã¬ãŒã:Pathnav"
] | https://ja.wikibooks.org/wiki/%E9%9B%BB%E7%A3%81%E6%B0%97%E5%AD%A6 |
1,980 | ç¹æ®çžå¯Ÿè« | 倧åŠã®æç§æž èªç¶ç§åŠ: æ°åŠ - ç©çåŠ; å€å
žååŠ éåååŠ - ååŠ; ç¡æ©ååŠ ææ©ååŠ - çç©åŠ; æ€ç©åŠ ç 究æè¡ - å°çç§åŠ - å»åŠ; 解ååŠ èªåŠ: æ¥æ¬èª è±èª ãšã¹ãã©ã³ã æé®®èª ãã³ããŒã¯èª ãã€ãèª ãã©ã³ã¹èª ã©ãã³èª ã«ãŒããã¢èª 人æç§åŠ: æŽå²åŠ; æ¥æ¬å² äžåœå² äžçå² æŽå²èŠ³ - å¿çåŠ - å²åŠ - èžè¡; é³æ¥œ çŸè¡ - æåŠ; å€å
žæåŠ æŒ¢è©© 瀟äŒç§åŠ: æ³åŠ - çµæžåŠ - å°çåŠ - æè²åŠ; åŠæ ¡æè² æè²å² æ
å ±æè¡: æ
å ±å·¥åŠ; MS-DOS/PC DOS UNIX/Linux TeX/LaTeX CGI - ããã°ã©ãã³ã°; BASIC Cèšèª C++ Dèšèª HTML Java JavaScript Lisp Mizar Perl PHP Python Ruby Scheme SVG å°ã»äžã»é«æ ¡ã®æç§æž å°åŠ: åœèª ç€ŸäŒ ç®æ° çç§ è±èª äžåŠ: åœèª ç€ŸäŒ æ°åŠ çç§ è±èª é«æ ¡: åœèª - å°æŽ - å
¬æ° - æ°åŠ; å
¬åŒé - çç§; ç©ç ååŠ å°åŠ çç© - å€åœèª - æ
å ± 解説æžã»å®çšæžã»åèæž è¶£å³: æçæ¬ - ã¹ããŒã - ã²ãŒã è©Šéš: è³æ Œè©Šéš - å
¥åŠè©Šéš ãã®ä»ã®æ¬: é²çœ - ç掻ãšé²è·¯ - ãŠã£ãããã£ã¢ã®æžãæ¹ - ãžã§ãŒã¯é
æ¬é
ã¯ç¹æ®çžå¯Ÿè«ã®è§£èª¬ã§ãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "倧åŠã®æç§æž èªç¶ç§åŠ: æ°åŠ - ç©çåŠ; å€å
žååŠ éåååŠ - ååŠ; ç¡æ©ååŠ ææ©ååŠ - çç©åŠ; æ€ç©åŠ ç 究æè¡ - å°çç§åŠ - å»åŠ; 解ååŠ èªåŠ: æ¥æ¬èª è±èª ãšã¹ãã©ã³ã æé®®èª ãã³ããŒã¯èª ãã€ãèª ãã©ã³ã¹èª ã©ãã³èª ã«ãŒããã¢èª 人æç§åŠ: æŽå²åŠ; æ¥æ¬å² äžåœå² äžçå² æŽå²èŠ³ - å¿çåŠ - å²åŠ - èžè¡; é³æ¥œ çŸè¡ - æåŠ; å€å
žæåŠ æŒ¢è©© 瀟äŒç§åŠ: æ³åŠ - çµæžåŠ - å°çåŠ - æè²åŠ; åŠæ ¡æè² æè²å² æ
å ±æè¡: æ
å ±å·¥åŠ; MS-DOS/PC DOS UNIX/Linux TeX/LaTeX CGI - ããã°ã©ãã³ã°; BASIC Cèšèª C++ Dèšèª HTML Java JavaScript Lisp Mizar Perl PHP Python Ruby Scheme SVG å°ã»äžã»é«æ ¡ã®æç§æž å°åŠ: åœèª ç€ŸäŒ ç®æ° çç§ è±èª äžåŠ: åœèª ç€ŸäŒ æ°åŠ çç§ è±èª é«æ ¡: åœèª - å°æŽ - å
¬æ° - æ°åŠ; å
¬åŒé - çç§; ç©ç ååŠ å°åŠ çç© - å€åœèª - æ
å ± 解説æžã»å®çšæžã»åèæž è¶£å³: æçæ¬ - ã¹ããŒã - ã²ãŒã è©Šéš: è³æ Œè©Šéš - å
¥åŠè©Šéš ãã®ä»ã®æ¬: é²çœ - ç掻ãšé²è·¯ - ãŠã£ãããã£ã¢ã®æžãæ¹ - ãžã§ãŒã¯é",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "æ¬é
ã¯ç¹æ®çžå¯Ÿè«ã®è§£èª¬ã§ãã",
"title": ""
}
] | æ¬é
ã¯ç¹æ®çžå¯Ÿè«ã®è§£èª¬ã§ãã ã¯ããã«
æŽå²çå°å
¥
å
¥é
ãã³ãœã«
èšç®äŸ
æéã®é
ã
ããŒã¬ã³ãåçž®
é床ã®åæå
4å
éåé éåæ¹çšåŒ
é»ç£æ°åŠãžã®å°å
¥ | {{Pathnav|ã¡ã€ã³ããŒãž|èªç¶ç§åŠ|ç©çåŠ|frame=1|small=1}}
{{é²æç¶æ³}}
{{èµæžäžèŠ§}}
æ¬é
ã¯ç¹æ®çžå¯Ÿè«ã®è§£èª¬ã§ãã
* [[ç¹æ®çžå¯Ÿè« ã¯ããã«|ã¯ããã«]]
* [[ç¹æ®çžå¯Ÿè« æŽå²çå°å
¥|æŽå²çå°å
¥]]
* [[ç¹æ®çžå¯Ÿè« å
¥é|å
¥é]]
* [[ç¹æ®çžå¯Ÿè« ãã³ãœã«|ãã³ãœã«]]
* èšç®äŸ
** [[ç¹æ®çžå¯Ÿè« æéã®é
ã|æéã®é
ã]]
** [[ç¹æ®çžå¯Ÿè« ããŒã¬ã³ãåçž®|ããŒã¬ã³ãåçž®]]
** [[ç¹æ®çžå¯Ÿè« é床ã®åæå|é床ã®åæå]]
* [[ç¹æ®çžå¯Ÿè« 4å
éåé|4å
éåé]]
<!-- E = mc^2 !!! -->
* [[ç¹æ®çžå¯Ÿè« éåæ¹çšåŒ|éåæ¹çšåŒ]]
* [[ç¹æ®çžå¯Ÿè« é»ç£æ°åŠãžã®å°å
¥|é»ç£æ°åŠãžã®å°å
¥]]
{{DEFAULTSORT:ãšããããããããã}}
[[Category:ç¹æ®çžå¯Ÿè«|*]]
{{NDC|421.2}} | 2005-05-13T11:01:46Z | 2024-03-16T03:14:04Z | [
"ãã³ãã¬ãŒã:Pathnav",
"ãã³ãã¬ãŒã:é²æç¶æ³",
"ãã³ãã¬ãŒã:èµæžäžèŠ§",
"ãã³ãã¬ãŒã:NDC"
] | https://ja.wikibooks.org/wiki/%E7%89%B9%E6%AE%8A%E7%9B%B8%E5%AF%BE%E8%AB%96 |
1,981 | ç¹æ®çžå¯Ÿè« ã¯ããã« | è·é¢ãšããã®ã¯äŸãã°ã d s 2 = d x 2 + d y 2 + d z 2 {\displaystyle ds^{2}=dx^{2}+dy^{2}+dz^{2}} ãšããããã«ãã®äžã3次å
ã§ããããã3ã€ã®å€æ°x,y,zãçšããŠæžããããããããäžçã«ã¯ããã²ãšã€æéæ¹åã®èªç±åºŠãããããã«æãããã€ãŸãããããã®ãšãã®é£ã®ãã®ãšãããã®ãèããããšãåºæ¥ãããã«ãããæéã®ãããã®ãšãå°ãæéãçµã£ãŠããã®ãããã®ãšãããã®ãèããããšãåºæ¥ãããã®ãšããæéãäžã®åŒã®ãããªè¡šåŒã§è¡šãããããšéœåããããå®éå®éšçã«ã d s 2 = c 2 d t 2 â d x 2 â d y 2 â d z 2 {\displaystyle ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}} ã§ããããšãç¥ãããŠããã
éèŠãªã®ã¯ããã®åŒãã©ããªé床ã§ãã£ãŠããçéçŽç·éåãã芳枬è
ããèŠãå Žåã«ã¯ãåžžã«æãç«ã£ãŠããããšã§ããã ãã®ããã«çéçŽç·éåãã芳枬è
ããèŠãå Žåã«å€åããªãéãããŒã¬ã³ãäžå€éãšãã¶ã
ãã®ããšã¯éã£ãéåãããŠããç©äœããèŠãå Žåã®ãéåã®éããèšç®ããæ¹æ³ãäžããŠããããã®ãããªå Žåã«é¢ããç©äœã®éåãèŠãŠè¡ãããšããã®ææžã®ç®çãšãªãã
æ°åŠçã«ã¯ãã®ãããªå¯Ÿç§°æ§ãæ±ãè¯ãæ¹æ³ãç¥ãããŠããã®ã§ããŸãã¯ãããå°å
¥ããããããçšãããšã d s 2 = c 2 d t 2 â d x 2 â d y 2 â d z 2 {\displaystyle ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}} = η ÎŒ Îœ d x ÎŒ d x Îœ {\displaystyle =\eta _{\mu \nu }dx^{\mu }dx^{\nu }} ãšæžãããšãåºæ¥ãããã®èšæ³ã¯ãã³ãœã«ã®ã»ã¯ã·ã§ã³ã§å°å
¥ããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "è·é¢ãšããã®ã¯äŸãã°ã d s 2 = d x 2 + d y 2 + d z 2 {\\displaystyle ds^{2}=dx^{2}+dy^{2}+dz^{2}} ãšããããã«ãã®äžã3次å
ã§ããããã3ã€ã®å€æ°x,y,zãçšããŠæžããããããããäžçã«ã¯ããã²ãšã€æéæ¹åã®èªç±åºŠãããããã«æãããã€ãŸãããããã®ãšãã®é£ã®ãã®ãšãããã®ãèããããšãåºæ¥ãããã«ãããæéã®ãããã®ãšãå°ãæéãçµã£ãŠããã®ãããã®ãšãããã®ãèããããšãåºæ¥ãããã®ãšããæéãäžã®åŒã®ãããªè¡šåŒã§è¡šãããããšéœåããããå®éå®éšçã«ã d s 2 = c 2 d t 2 â d x 2 â d y 2 â d z 2 {\\displaystyle ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}} ã§ããããšãç¥ãããŠããã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 1,
"tag": "p",
"text": "éèŠãªã®ã¯ããã®åŒãã©ããªé床ã§ãã£ãŠããçéçŽç·éåãã芳枬è
ããèŠãå Žåã«ã¯ãåžžã«æãç«ã£ãŠããããšã§ããã ãã®ããã«çéçŽç·éåãã芳枬è
ããèŠãå Žåã«å€åããªãéãããŒã¬ã³ãäžå€éãšãã¶ã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ãã®ããšã¯éã£ãéåãããŠããç©äœããèŠãå Žåã®ãéåã®éããèšç®ããæ¹æ³ãäžããŠããããã®ãããªå Žåã«é¢ããç©äœã®éåãèŠãŠè¡ãããšããã®ææžã®ç®çãšãªãã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "æ°åŠçã«ã¯ãã®ãããªå¯Ÿç§°æ§ãæ±ãè¯ãæ¹æ³ãç¥ãããŠããã®ã§ããŸãã¯ãããå°å
¥ããããããçšãããšã d s 2 = c 2 d t 2 â d x 2 â d y 2 â d z 2 {\\displaystyle ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}} = η ÎŒ Îœ d x ÎŒ d x Îœ {\\displaystyle =\\eta _{\\mu \\nu }dx^{\\mu }dx^{\\nu }} ãšæžãããšãåºæ¥ãããã®èšæ³ã¯ãã³ãœã«ã®ã»ã¯ã·ã§ã³ã§å°å
¥ããã",
"title": "ã¯ããã«"
}
] | null | ==ã¯ããã«==
è·é¢ãšããã®ã¯äŸãã°ã
<math>
ds^2 = dx^2+dy^2+dz^2
</math>
ãšããããã«ãã®äžã3次å
ã§ããããã3ã€ã®å€æ°x,y,zãçšããŠæžããããããããäžçã«ã¯ããã²ãšã€æéæ¹åã®èªç±åºŠãããããã«æãããã€ãŸãããããã®ãšãã®é£ã®ãã®ãšãããã®ãèããããšãåºæ¥ãããã«ãããæéã®ãããã®ãšãå°ãæéãçµã£ãŠããã®ãããã®ãšãããã®ãèããããšãåºæ¥ãããã®ãšããæéãäžã®åŒã®ãããªè¡šåŒã§è¡šãããããšéœåããããå®éå®éšçã«ã
<math>
ds^2 = c^2 dt^2 -dx^2-dy^2-dz^2
</math>
ã§ããããšãç¥ãããŠããã
éèŠãªã®ã¯ããã®åŒãã©ããªé床ã§ãã£ãŠããçéçŽç·éåãã芳枬è
ããèŠãå Žåã«ã¯ãåžžã«æãç«ã£ãŠããããšã§ããã ãã®ããã«çéçŽç·éåãã芳枬è
ããèŠãå Žåã«å€åããªãéãããŒã¬ã³ãäžå€éãšãã¶ã
ãã®ããšã¯éã£ãéåãããŠããç©äœããèŠãå Žåã®ãéåã®éããèšç®ããæ¹æ³ãäžããŠããããã®ãããªå Žåã«é¢ããç©äœã®éåãèŠãŠè¡ãããšããã®ææžã®ç®çãšãªãã
æ°åŠçã«ã¯ãã®ãããªå¯Ÿç§°æ§ãæ±ãè¯ãæ¹æ³ãç¥ãããŠããã®ã§ããŸãã¯ãããå°å
¥ããããããçšãããšã
<math>
ds^2 = c^2 dt^2 -dx^2-dy^2-dz^2
</math>
<math>
= \eta _{\mu\nu} dx^\mu dx^\nu
</math>
ãšæžãããšãåºæ¥ãããã®èšæ³ã¯ãã³ãœã«ã®ã»ã¯ã·ã§ã³ã§å°å
¥ããã
<!-- 次ã®ã»ã¯ã·ã§ã³ã"ãã³ãœã«"ã§ãããšä»®å®ããŠã¯ãªããªã...ã -->
[[Category:ç¹æ®çžå¯Ÿè«|ã¯ããã«]] | 2005-05-14T04:32:23Z | 2024-03-16T03:15:09Z | [] | https://ja.wikibooks.org/wiki/%E7%89%B9%E6%AE%8A%E7%9B%B8%E5%AF%BE%E8%AB%96_%E3%81%AF%E3%81%98%E3%82%81%E3%81%AB |
1,982 | ç¹æ®çžå¯Ÿè« ãã³ãœã« | ç¹æ®çžå¯Ÿè« > ãã³ãœã«
ããããã¯ãã³ãœã«ãšããéãçšããã æ°åŠçã«ã¯ãéåžžç©çã§æ±ã 3次å
ã®ãã¯ãã«ã¯ã SO(3)矀ãšãã矀ã®è¡šçŸã®1ã€ãšãªã£ãŠããã ããã§ããããŒã¬ã³ãäžå€æ§ã¯ã ããŒã¬ã³ã矀SO(3,1)ã«å¯Ÿå¿ããŠããã ããã矀ã®è¡šçŸãè¯ãç¥ãããŠããã
ãŸãã ããŒã¬ã³ãå€æã§å€åããªãéã ã¹ã«ã©ãŒãšåŒã¶ã 次ã«ãããŒã¬ã³ãå€æã«å¯ŸããŠã A â² ÎŒ = Î Îœ ÎŒ A Îœ {\displaystyle {A'}^{\mu }=\Lambda _{\nu }^{\mu }A^{\nu }} ãšãªãéããã¯ãã«ãšåŒã¶ã
Î Îœ ÎŒ {\displaystyle \Lambda _{\nu }^{\mu }} ã¯ã6ã€ã®4*4ã®è¡åã§äžãããããã¯ãã«ã«å¯ŸããŠã¯ Î Îœ ÎŒ {\displaystyle \Lambda _{\nu }^{\mu }} ã¯ã B 1 = γ ( 1 β 0 0 β 1 0 0 0 0 1 0 0 0 0 1 ) {\displaystyle B_{1}=\gamma {\begin{pmatrix}1&\beta &0&0\\\beta &1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}} , B 2 = γ ( 1 0 0 0 0 1 β 0 0 β 1 0 0 0 0 1 ) {\displaystyle B_{2}=\gamma {\begin{pmatrix}1&0&0&0\\0&1&\beta &0\\0&\beta &1&0\\0&0&0&1\end{pmatrix}}} , B 3 = γ ( 1 0 0 0 0 1 0 0 0 0 1 β 0 0 β 1 ) {\displaystyle B_{3}=\gamma {\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&\beta \\0&0&\beta &1\end{pmatrix}}} , R 1 = ( 1 0 0 0 0 1 0 0 0 0 cos a â sin a 0 0 sin a cos a ) {\displaystyle R_{1}={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&\cos a&-\sin a\\0&0&\sin a&\cos a\end{pmatrix}}} , R 2 = ( 1 0 0 0 0 cos a 0 sin a 0 0 1 0 0 â sin a 0 cos a ) {\displaystyle R_{2}={\begin{pmatrix}1&0&0&0\\0&\cos a&0&\sin a\\0&0&1&0\\0&-\sin a&0&\cos a\end{pmatrix}}} , R 3 = ( 1 0 0 0 0 cos a â sin a 0 0 sin a cos a 0 0 0 0 1 ) {\displaystyle R_{3}={\begin{pmatrix}1&0&0&0\\0&\cos a&-\sin a&0\\0&\sin a&\cos a&0\\0&0&0&1\\\end{pmatrix}}} ã§äžããããã ãã ãããã㧠β = v c {\displaystyle \beta ={\frac {v}{c}}} γ = 1 1 â v 2 / c 2 {\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}} ãçšããã (ããŒã¬ã³ã矀ã®è¡šçŸã®æ£ç¢ºãªå®çŸ©ã¯ãããããç©çæ°åŠãããã㯠æ°åŠã®"ãªãŒçŸ€"ã§äžããããã) ç¹ã«x軞æ¹åã«é床vã§ããã芳枬è
ã®èŠ³å¯ããç©çéã åŸãã«ã¯ p â² ÎŒ = γ ( 1 β 0 0 β 1 0 0 0 0 1 0 0 0 0 1 ) p ÎŒ {\displaystyle p'^{\mu }=\gamma {\begin{pmatrix}1&\beta &0&0\\\beta &1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}p^{\mu }} ãšãªããç¹ã«xæ¹åã ãã«æ³šç®ãããšãã«ã¯ å€åãèµ·ãããªãyãzæ¹åãç¡èŠã㊠å€æè¡åã γ ( 1 β β 1 ) {\displaystyle \gamma {\begin{pmatrix}1&\beta \\\beta &1\end{pmatrix}}} ãšçãæžãããšãããã
ãããããäŸãã°ã A â² ÎŒ A â² Îœ {\displaystyle {A'}^{\mu }{A'}^{\nu }} ãšãããããªéãäœããšã ãã®é㯠A â² ÎŒ A â² Îœ = Î Ï ÎŒ A Ï Î Ï Îœ A Ï {\displaystyle {A'}^{\mu }{A'}^{\nu }=\Lambda _{\rho }^{\mu }A^{\rho }\Lambda _{\sigma }^{\nu }A^{\sigma }} ãšããããã«å€æããããšãåãã ããã§ã T ÎŒ Îœ = Î Ï ÎŒ Î Ï Îœ T Ï Ï {\displaystyle T^{\mu \nu }=\Lambda _{\rho }^{\mu }\Lambda _{\sigma }^{\nu }T^{\rho \sigma }} ãšããããã«æ¯èãéã 2éã®ãã³ãœã«ãšåŒã¶ã ããã¯æ·»åã2ã€ããããšã«ããã ãŸãããã¯ãã«ã¯1éã®ãã³ãœã«ã ã¹ã«ã©ãŒã¯0éã®ãã³ãœã«ãšããããšãã§ããã (ç¹ã«æ·»åãäžã«ãããã®ãåå€ãã³ãœã« ãšåŒã¶ããšãããã)
ããã§ãèšéãã³ãœã«ãšããç¹å¥ãª2éã®ãã³ãœã«ã å®çŸ©ããã η ÎŒ Îœ = η ÎŒ Îœ = ( 1 0 0 0 0 â 1 0 0 0 0 â 1 0 0 0 0 â 1 ) {\displaystyle \eta ^{\mu \nu }=\eta _{\mu \nu }={\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}}} ããã§ããã®éãçšããŠãã¯ãã«ã®2ä¹ ( A ÎŒ ) 2 = η ÎŒ Îœ A ÎŒ A Îœ = ( A 0 ) 2 â ( A 1 ) 2 â ( A 2 ) 2 â ( A 3 ) 2 {\displaystyle {\begin{matrix}(A^{\mu })^{2}=\eta _{\mu \nu }A^{\mu }A^{\nu }\\=(A^{0})^{2}-(A^{1})^{2}-(A^{2})^{2}-(A^{3})^{2}\end{matrix}}} ãåãã
ããããã®æ·»å㯠åãæ·»åãäžäžã«ãããšãã«ã0-3ãŸã§ã®åãåã£ãŠã æã¡æ¶ãããšãåºæ¥ãã äŸãã°ã A ÎŒ A ÎŒ = â m = 0 3 ( A m ) 2 {\displaystyle A^{\mu }A_{\mu }=\sum _{m=0}^{3}(A^{m})^{2}}
äžä»ãæ·»åã®éãå
±å€ãã¯ãã«ãšåŒã³ã察å¿ãã åå€ãã¯ãã«ãšèšéãã³ãœã«ãçšããŠå®çŸ©ããããšãåºæ¥ãã
ãããã®æ·»åã¯ã èšéãã³ãœã« η ÎŒ Îœ = η ÎŒ Îœ = ( 1 0 0 0 0 â 1 0 0 0 0 â 1 0 0 0 0 â 1 ) {\displaystyle \eta ^{\mu \nu }=\eta _{\mu \nu }={\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}}} ã«ãã£ãŠãäžäžã«ç§»åãããããšãåºæ¥ãã äŸãã°ã x ÎŒ = η ÎŒ Îœ x Îœ {\displaystyle x_{\mu }=\eta _{\mu \nu }x^{\nu }} ãšãªããããã«ãã£ãŠäžä»ãæ·»åã®éãå®çŸ©ããããšãåºæ¥ãã ç¹ã«ãäžä»ãæ·»åã ããæã€ãã³ãœã«ãå
±å€ãã³ãœã«ãšåŒã¶ããšãããã ãŸãã äžä»ããšäžä»ãã®æ·»åãäž¡æ¹æã€ãã³ãœã«ãæ··åãã³ãœã«ãš åŒã¶ããšãããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ç¹æ®çžå¯Ÿè« > ãã³ãœã«",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "",
"title": ""
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ããããã¯ãã³ãœã«ãšããéãçšããã æ°åŠçã«ã¯ãéåžžç©çã§æ±ã 3次å
ã®ãã¯ãã«ã¯ã SO(3)矀ãšãã矀ã®è¡šçŸã®1ã€ãšãªã£ãŠããã ããã§ããããŒã¬ã³ãäžå€æ§ã¯ã ããŒã¬ã³ã矀SO(3,1)ã«å¯Ÿå¿ããŠããã ããã矀ã®è¡šçŸãè¯ãç¥ãããŠããã",
"title": "ãã³ãœã«"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ãŸãã ããŒã¬ã³ãå€æã§å€åããªãéã ã¹ã«ã©ãŒãšåŒã¶ã 次ã«ãããŒã¬ã³ãå€æã«å¯ŸããŠã A â² ÎŒ = Î Îœ ÎŒ A Îœ {\\displaystyle {A'}^{\\mu }=\\Lambda _{\\nu }^{\\mu }A^{\\nu }} ãšãªãéããã¯ãã«ãšåŒã¶ã",
"title": "ãã³ãœã«"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "Î Îœ ÎŒ {\\displaystyle \\Lambda _{\\nu }^{\\mu }} ã¯ã6ã€ã®4*4ã®è¡åã§äžãããããã¯ãã«ã«å¯ŸããŠã¯ Î Îœ ÎŒ {\\displaystyle \\Lambda _{\\nu }^{\\mu }} ã¯ã B 1 = γ ( 1 β 0 0 β 1 0 0 0 0 1 0 0 0 0 1 ) {\\displaystyle B_{1}=\\gamma {\\begin{pmatrix}1&\\beta &0&0\\\\\\beta &1&0&0\\\\0&0&1&0\\\\0&0&0&1\\end{pmatrix}}} , B 2 = γ ( 1 0 0 0 0 1 β 0 0 β 1 0 0 0 0 1 ) {\\displaystyle B_{2}=\\gamma {\\begin{pmatrix}1&0&0&0\\\\0&1&\\beta &0\\\\0&\\beta &1&0\\\\0&0&0&1\\end{pmatrix}}} , B 3 = γ ( 1 0 0 0 0 1 0 0 0 0 1 β 0 0 β 1 ) {\\displaystyle B_{3}=\\gamma {\\begin{pmatrix}1&0&0&0\\\\0&1&0&0\\\\0&0&1&\\beta \\\\0&0&\\beta &1\\end{pmatrix}}} , R 1 = ( 1 0 0 0 0 1 0 0 0 0 cos a â sin a 0 0 sin a cos a ) {\\displaystyle R_{1}={\\begin{pmatrix}1&0&0&0\\\\0&1&0&0\\\\0&0&\\cos a&-\\sin a\\\\0&0&\\sin a&\\cos a\\end{pmatrix}}} , R 2 = ( 1 0 0 0 0 cos a 0 sin a 0 0 1 0 0 â sin a 0 cos a ) {\\displaystyle R_{2}={\\begin{pmatrix}1&0&0&0\\\\0&\\cos a&0&\\sin a\\\\0&0&1&0\\\\0&-\\sin a&0&\\cos a\\end{pmatrix}}} , R 3 = ( 1 0 0 0 0 cos a â sin a 0 0 sin a cos a 0 0 0 0 1 ) {\\displaystyle R_{3}={\\begin{pmatrix}1&0&0&0\\\\0&\\cos a&-\\sin a&0\\\\0&\\sin a&\\cos a&0\\\\0&0&0&1\\\\\\end{pmatrix}}} ã§äžããããã ãã ãããã㧠β = v c {\\displaystyle \\beta ={\\frac {v}{c}}} γ = 1 1 â v 2 / c 2 {\\displaystyle \\gamma ={\\frac {1}{\\sqrt {1-v^{2}/c^{2}}}}} ãçšããã (ããŒã¬ã³ã矀ã®è¡šçŸã®æ£ç¢ºãªå®çŸ©ã¯ãããããç©çæ°åŠãããã㯠æ°åŠã®\"ãªãŒçŸ€\"ã§äžããããã) ç¹ã«x軞æ¹åã«é床vã§ããã芳枬è
ã®èŠ³å¯ããç©çéã åŸãã«ã¯ p â² ÎŒ = γ ( 1 β 0 0 β 1 0 0 0 0 1 0 0 0 0 1 ) p ÎŒ {\\displaystyle p'^{\\mu }=\\gamma {\\begin{pmatrix}1&\\beta &0&0\\\\\\beta &1&0&0\\\\0&0&1&0\\\\0&0&0&1\\end{pmatrix}}p^{\\mu }} ãšãªããç¹ã«xæ¹åã ãã«æ³šç®ãããšãã«ã¯ å€åãèµ·ãããªãyãzæ¹åãç¡èŠã㊠å€æè¡åã γ ( 1 β β 1 ) {\\displaystyle \\gamma {\\begin{pmatrix}1&\\beta \\\\\\beta &1\\end{pmatrix}}} ãšçãæžãããšãããã",
"title": "ãã³ãœã«"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ãããããäŸãã°ã A â² ÎŒ A â² Îœ {\\displaystyle {A'}^{\\mu }{A'}^{\\nu }} ãšãããããªéãäœããšã ãã®é㯠A â² ÎŒ A â² Îœ = Î Ï ÎŒ A Ï Î Ï Îœ A Ï {\\displaystyle {A'}^{\\mu }{A'}^{\\nu }=\\Lambda _{\\rho }^{\\mu }A^{\\rho }\\Lambda _{\\sigma }^{\\nu }A^{\\sigma }} ãšããããã«å€æããããšãåãã ããã§ã T ÎŒ Îœ = Î Ï ÎŒ Î Ï Îœ T Ï Ï {\\displaystyle T^{\\mu \\nu }=\\Lambda _{\\rho }^{\\mu }\\Lambda _{\\sigma }^{\\nu }T^{\\rho \\sigma }} ãšããããã«æ¯èãéã 2éã®ãã³ãœã«ãšåŒã¶ã ããã¯æ·»åã2ã€ããããšã«ããã ãŸãããã¯ãã«ã¯1éã®ãã³ãœã«ã ã¹ã«ã©ãŒã¯0éã®ãã³ãœã«ãšããããšãã§ããã (ç¹ã«æ·»åãäžã«ãããã®ãåå€ãã³ãœã« ãšåŒã¶ããšãããã)",
"title": "ãã³ãœã«"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ããã§ãèšéãã³ãœã«ãšããç¹å¥ãª2éã®ãã³ãœã«ã å®çŸ©ããã η ÎŒ Îœ = η ÎŒ Îœ = ( 1 0 0 0 0 â 1 0 0 0 0 â 1 0 0 0 0 â 1 ) {\\displaystyle \\eta ^{\\mu \\nu }=\\eta _{\\mu \\nu }={\\begin{pmatrix}1&0&0&0\\\\0&-1&0&0\\\\0&0&-1&0\\\\0&0&0&-1\\end{pmatrix}}} ããã§ããã®éãçšããŠãã¯ãã«ã®2ä¹ ( A ÎŒ ) 2 = η ÎŒ Îœ A ÎŒ A Îœ = ( A 0 ) 2 â ( A 1 ) 2 â ( A 2 ) 2 â ( A 3 ) 2 {\\displaystyle {\\begin{matrix}(A^{\\mu })^{2}=\\eta _{\\mu \\nu }A^{\\mu }A^{\\nu }\\\\=(A^{0})^{2}-(A^{1})^{2}-(A^{2})^{2}-(A^{3})^{2}\\end{matrix}}} ãåãã",
"title": "ãã³ãœã«"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ããããã®æ·»å㯠åãæ·»åãäžäžã«ãããšãã«ã0-3ãŸã§ã®åãåã£ãŠã æã¡æ¶ãããšãåºæ¥ãã äŸãã°ã A ÎŒ A ÎŒ = â m = 0 3 ( A m ) 2 {\\displaystyle A^{\\mu }A_{\\mu }=\\sum _{m=0}^{3}(A^{m})^{2}}",
"title": "ãã³ãœã«"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "äžä»ãæ·»åã®éãå
±å€ãã¯ãã«ãšåŒã³ã察å¿ãã åå€ãã¯ãã«ãšèšéãã³ãœã«ãçšããŠå®çŸ©ããããšãåºæ¥ãã",
"title": "ãã³ãœã«"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "ãããã®æ·»åã¯ã èšéãã³ãœã« η ÎŒ Îœ = η ÎŒ Îœ = ( 1 0 0 0 0 â 1 0 0 0 0 â 1 0 0 0 0 â 1 ) {\\displaystyle \\eta ^{\\mu \\nu }=\\eta _{\\mu \\nu }={\\begin{pmatrix}1&0&0&0\\\\0&-1&0&0\\\\0&0&-1&0\\\\0&0&0&-1\\end{pmatrix}}} ã«ãã£ãŠãäžäžã«ç§»åãããããšãåºæ¥ãã äŸãã°ã x ÎŒ = η ÎŒ Îœ x Îœ {\\displaystyle x_{\\mu }=\\eta _{\\mu \\nu }x^{\\nu }} ãšãªããããã«ãã£ãŠäžä»ãæ·»åã®éãå®çŸ©ããããšãåºæ¥ãã ç¹ã«ãäžä»ãæ·»åã ããæã€ãã³ãœã«ãå
±å€ãã³ãœã«ãšåŒã¶ããšãããã ãŸãã äžä»ããšäžä»ãã®æ·»åãäž¡æ¹æã€ãã³ãœã«ãæ··åãã³ãœã«ãš åŒã¶ããšãããã",
"title": "ãã³ãœã«"
}
] | ç¹æ®çžå¯Ÿè« > ãã³ãœã« | <small>[[ ç¹æ®çžå¯Ÿè« ]]> ãã³ãœã« </small>
----
==ãã³ãœã«==
ããããã¯ãã³ãœã«ãšããéãçšããã
æ°åŠçã«ã¯ãéåžžç©çã§æ±ã
3次å
ã®ãã¯ãã«ã¯ã
SO(3)矀ãšãã矀ã®è¡šçŸã®1ã€ãšãªã£ãŠããã
ããã§ããããŒã¬ã³ãäžå€æ§ã¯ã
ããŒã¬ã³ã矀SO(3,1)ã«å¯Ÿå¿ããŠããã
ããã矀ã®è¡šçŸãè¯ãç¥ãããŠããã
ãŸãã
ããŒã¬ã³ãå€æã§å€åããªãéã
ã¹ã«ã©ãŒãšåŒã¶ã
次ã«ãããŒã¬ã³ãå€æã«å¯ŸããŠã
<math>
{A'} ^\mu = \Lambda ^\mu _\nu A^\nu
</math>
ãšãªãéããã¯ãã«ãšåŒã¶ã
<math>
\Lambda ^\mu _\nu
</math>
ã¯ã6ã€ã®4*4ã®è¡åã§äžãããããã¯ãã«ã«å¯ŸããŠã¯
<math>
\Lambda ^\mu _\nu
</math>
ã¯ã
<math>
B _1 =\gamma
\begin{pmatrix}
1 &\beta &0&0\\
\beta &1 & 0&0\\
0&0&1&0\\
0&0&0&1
\end{pmatrix}
</math>
,
<math>
B _2 = \gamma
\begin{pmatrix}
1&0&0&0\\
0&1 &\beta &0\\
0&\beta &1 & 0\\
0&0&0&1
\end{pmatrix}
</math>
,
<math>
B _3 =\gamma
\begin{pmatrix}
1&0&0&0\\
0&1&0&0\\
0 &0&1 &\beta \\
0 &0&\beta &1
\end{pmatrix}
</math>
,
<math>
R _1 =
\begin{pmatrix}
1 &0 &0&0\\
0 &1 & 0&0\\
0&0&\cos a & -\sin a\\
0&0&\sin a &\cos a
\end{pmatrix}
</math>
,
<math>
R _2 =
\begin{pmatrix}
1 &0 &0&0\\
0&\cos a &0& \sin a\\
0 &0 & 1&0\\
0&-\sin a &0&\cos a
\end{pmatrix}
</math>
,
<math>
R _3 =
\begin{pmatrix}
1 &0 &0&0\\
0&\cos a & -\sin a&0\\
0&\sin a &\cos a&0\\
0 &0 &0&1\\
\end{pmatrix}
</math>
ã§äžããããã
ãã ããããã§
<math>
\beta = \frac v c
</math>
<math>
\gamma = \frac 1 {\sqrt { 1 - v^2/c^2}}
</math>
ãçšããã
<!-- å®æãªã³ããŒã¢ã³ãããŒã¹ãã¯...ã -->
(ããŒã¬ã³ã矀ã®è¡šçŸã®æ£ç¢ºãªå®çŸ©ã¯ãããããç©çæ°åŠããããã¯
æ°åŠã®"ãªãŒçŸ€"ã§äžããããã)
<!-- (ããŒã¬ã³ã矀ã¯å€å
žãªãŒçŸ€ã«å«ãŸããªãããšã«æ³šæã -->
<!-- ãã®ããæ°åŠã®(å°ãªããšããªãŒçŸ€ã®)æç§æžã«ã¯ãå«ãŸããªãããç¥ããªãã -->
ç¹ã«x軞æ¹åã«é床vã§ããã芳枬è
ã®èŠ³å¯ããç©çéã
åŸãã«ã¯
<math>
p'^\mu = \gamma
\begin{pmatrix}
1&\beta&0&0\\
\beta&1&0&0\\
0&0&1&0\\
0&0&0&1
\end{pmatrix}
p^\mu
</math>
ãšãªããç¹ã«xæ¹åã ãã«æ³šç®ãããšãã«ã¯
å€åãèµ·ãããªãyãzæ¹åãç¡èŠããŠ
å€æè¡åã
<math>
\gamma
\begin{pmatrix}
1&\beta\\
\beta&1
\end{pmatrix}
</math>
ãšçãæžãããšãããã
ãããããäŸãã°ã
<math>
{A'} ^\mu {A'} ^\nu
</math>
ãšãããããªéãäœããšã
ãã®éã¯
<math>
{A'} ^\mu {A'} ^\nu =\Lambda ^\mu _\rho A^\rho \Lambda ^\nu _\sigma A^\sigma
</math>
ãšããããã«å€æããããšãåãã
<!-- ?? -->
ããã§ã
<math>
T^{\mu\nu} = \Lambda ^\mu _\rho \Lambda ^\nu _ \sigma T ^{\rho \sigma}
</math>
ãšããããã«æ¯èãéã
2éã®ãã³ãœã«ãšåŒã¶ã
ããã¯æ·»åã2ã€ããããšã«ããã
ãŸãããã¯ãã«ã¯1éã®ãã³ãœã«ã
ã¹ã«ã©ãŒã¯0éã®ãã³ãœã«ãšããããšãã§ããã
(ç¹ã«æ·»åãäžã«ãããã®ãåå€ãã³ãœã«
ãšåŒã¶ããšãããã)
ããã§ãèšéãã³ãœã«ãšããç¹å¥ãª2éã®ãã³ãœã«ã
å®çŸ©ããã
<math>
\eta^{\mu\nu} =
\eta _{\mu\nu} =
\begin{pmatrix}
1&0&0&0\\
0&-1&0&0\\
0&0&-1&0\\
0&0&0&-1
\end{pmatrix}
</math>
ããã§ããã®éãçšããŠãã¯ãã«ã®2ä¹
<math>
\begin{matrix}
(A^\mu) ^2 = \eta _{\mu\nu} A^\mu A^\nu\\
= (A^0)^2-(A^1)^2 -(A^2)^2 -(A^3)^2
\end{matrix}
</math>
ãåãã
ããããã®æ·»åã¯
åãæ·»åãäžäžã«ãããšãã«ã0-3ãŸã§ã®åãåã£ãŠã
æã¡æ¶ãããšãåºæ¥ãã
äŸãã°ã
<math>
A^\mu A _\mu = \sum _ {m =0} ^3 (A^m )^2
</math>
äžä»ãæ·»åã®éãå
±å€ãã¯ãã«ãšåŒã³ã察å¿ãã
åå€ãã¯ãã«ãšèšéãã³ãœã«ãçšããŠå®çŸ©ããããšãåºæ¥ãã
ãããã®æ·»åã¯ã
èšéãã³ãœã«
<math>
\eta^{\mu\nu} =
\eta _{\mu\nu} =
\begin{pmatrix}
1&0&0&0\\
0&-1&0&0\\
0&0&-1&0\\
0&0&0&-1
\end{pmatrix}
</math>
ã«ãã£ãŠãäžäžã«ç§»åãããããšãåºæ¥ãã
äŸãã°ã
<math>
x _\mu = \eta _{\mu\nu} x^\nu
</math>
ãšãªããããã«ãã£ãŠäžä»ãæ·»åã®éãå®çŸ©ããããšãåºæ¥ãã
ç¹ã«ãäžä»ãæ·»åã ããæã€ãã³ãœã«ãå
±å€ãã³ãœã«ãšåŒã¶ããšãããã
ãŸãã
äžä»ããšäžä»ãã®æ·»åãäž¡æ¹æã€ãã³ãœã«ãæ··åãã³ãœã«ãš
åŒã¶ããšãããã
[[Category:ç¹æ®çžå¯Ÿè«|ãŠããã]] | 2005-05-14T04:40:32Z | 2024-03-16T03:15:52Z | [] | https://ja.wikibooks.org/wiki/%E7%89%B9%E6%AE%8A%E7%9B%B8%E5%AF%BE%E8%AB%96_%E3%83%86%E3%83%B3%E3%82%BD%E3%83%AB |
1,983 | ç¹æ®çžå¯Ÿè« æéã®é
ã | ç¹æ®çžå¯Ÿè« > æéã®é
ã
ããç¹(0,0)ããé床vã§åãã ããç²å㯠éæ¢ããŠãã芳枬è
ããèŠãŠ (ct,vt)ãšãªãæå»ã«ãããŠã èªåèªèº«ããèŠã座æšç³»ã§ã¯ã γ ( 1 â β â β 1 ) ( c t v t ) {\displaystyle \gamma {\begin{pmatrix}1&-\beta \\-\beta &1\end{pmatrix}}{\begin{pmatrix}ct\\vt\end{pmatrix}}} = γ t ( c â β v â c β + v ) {\displaystyle =\gamma t{\begin{pmatrix}c-\beta v\\-c\beta +v\end{pmatrix}}} = γ t ( c â β v 0 ) {\displaystyle =\gamma t{\begin{pmatrix}c-\beta v\\0\end{pmatrix}}} ãšãªããæåŸã®èšç®ã§
ãçšããã ããã§ãç²åãšäžç·ã«åããŠãã芳枬è
ããèŠãŠ ç²åã®äœçœ®åº§æšã0ã§ããããšã¯ã ç²åãšäžç·ã«åã芳枬è
ã«åã£ãŠ ç²åã¯åããŠããªãããã«èŠããããšã«å¯Ÿå¿ããŠããã ç²åãšå
±ã«éåãã芳枬è
ã«åã£ãŠã®æéçµéã¯
ãšãªãããã£ãŠã ç²åãšäžç·ã«åã芳枬è
ã«åã£ãŠåºçºããŠããçµéããæéãã éæ¢ããŠãã芳枬è
ã«åã£ãŠã® æéããããã£ãããšçµéããŠããããšã瀺ããŠããã ããã¯çŽèŠ³çã«ã¯ãç²åãããé床ã§åããŠããåã ããæéã®æ¹åã« éåããŠããé床ãé
ããªã£ããã®ãšã¿ãªãããšãåºæ¥ãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ç¹æ®çžå¯Ÿè« > æéã®é
ã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "",
"title": "æéã®é
ã"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ããç¹(0,0)ããé床vã§åãã ããç²å㯠éæ¢ããŠãã芳枬è
ããèŠãŠ (ct,vt)ãšãªãæå»ã«ãããŠã èªåèªèº«ããèŠã座æšç³»ã§ã¯ã γ ( 1 â β â β 1 ) ( c t v t ) {\\displaystyle \\gamma {\\begin{pmatrix}1&-\\beta \\\\-\\beta &1\\end{pmatrix}}{\\begin{pmatrix}ct\\\\vt\\end{pmatrix}}} = γ t ( c â β v â c β + v ) {\\displaystyle =\\gamma t{\\begin{pmatrix}c-\\beta v\\\\-c\\beta +v\\end{pmatrix}}} = γ t ( c â β v 0 ) {\\displaystyle =\\gamma t{\\begin{pmatrix}c-\\beta v\\\\0\\end{pmatrix}}} ãšãªããæåŸã®èšç®ã§",
"title": "æéã®é
ã"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ãçšããã ããã§ãç²åãšäžç·ã«åããŠãã芳枬è
ããèŠãŠ ç²åã®äœçœ®åº§æšã0ã§ããããšã¯ã ç²åãšäžç·ã«åã芳枬è
ã«åã£ãŠ ç²åã¯åããŠããªãããã«èŠããããšã«å¯Ÿå¿ããŠããã ç²åãšå
±ã«éåãã芳枬è
ã«åã£ãŠã®æéçµéã¯",
"title": "æéã®é
ã"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ãšãªãããã£ãŠã ç²åãšäžç·ã«åã芳枬è
ã«åã£ãŠåºçºããŠããçµéããæéãã éæ¢ããŠãã芳枬è
ã«åã£ãŠã® æéããããã£ãããšçµéããŠããããšã瀺ããŠããã ããã¯çŽèŠ³çã«ã¯ãç²åãããé床ã§åããŠããåã ããæéã®æ¹åã« éåããŠããé床ãé
ããªã£ããã®ãšã¿ãªãããšãåºæ¥ãã",
"title": "æéã®é
ã"
}
] | ç¹æ®çžå¯Ÿè« > æéã®é
ã | <small> [[ç¹æ®çžå¯Ÿè«]] > æéã®é
ã
----
==æéã®é
ã==
ããç¹(0,0)ããé床vã§åãã ããç²åã¯
éæ¢ããŠãã芳枬è
ããèŠãŠ
(ct,vt)ãšãªãæå»ã«ãããŠã
èªåèªèº«ããèŠã座æšç³»ã§ã¯ã
<math>
\gamma
\begin{pmatrix}
1 & -\beta \\
-\beta & 1
\end{pmatrix}
\begin{pmatrix}
ct\\
vt
\end{pmatrix}
</math>
<math>
= \gamma t
\begin{pmatrix}
c -\beta v \\
-c \beta + v
\end{pmatrix}
</math>
<math>
= \gamma t
\begin{pmatrix}
c -\beta v \\
0
\end{pmatrix}
</math>
ãšãªããæåŸã®èšç®ã§
:<math>
\beta = v / c
</math>
ãçšããã
ããã§ãç²åãšäžç·ã«åããŠãã芳枬è
ããèŠãŠ
ç²åã®äœçœ®åº§æšã0ã§ããããšã¯ã
ç²åãšäžç·ã«åã芳枬è
ã«åã£ãŠ
ç²åã¯åããŠããªãããã«èŠããããšã«å¯Ÿå¿ããŠããã
ç²åãšå
±ã«éåãã芳枬è
ã«åã£ãŠã®æéçµéã¯
:<math>
\gamma t (c - \beta v ) = \gamma t(c - v^2 /c)
</math>
:<math>
= \gamma c t(1 - v^2 /c^2)
</math>
:<math>
= ct \sqrt{1-\beta^2}
</math>
:<math>
< ct
</math>
:= (éæ¢ããŠãã芳枬è
ããèŠãå Žåã®ç²åã®æé)
ãšãªãããã£ãŠã
ç²åãšäžç·ã«åã芳枬è
ã«åã£ãŠåºçºããŠããçµéããæéãã
éæ¢ããŠãã芳枬è
ã«åã£ãŠã®
æéããããã£ãããšçµéããŠããããšã瀺ããŠããã
ããã¯çŽèŠ³çã«ã¯ãç²åãããé床ã§åããŠããåã ããæéã®æ¹åã«
éåããŠããé床ãé
ããªã£ããã®ãšã¿ãªãããšãåºæ¥ãã
[[Category:ç¹æ®çžå¯Ÿè«|ãããã®ããã]] | 2005-05-14T04:46:35Z | 2024-03-16T03:16:44Z | [] | https://ja.wikibooks.org/wiki/%E7%89%B9%E6%AE%8A%E7%9B%B8%E5%AF%BE%E8%AB%96_%E6%99%82%E9%96%93%E3%81%AE%E9%81%85%E3%82%8C |
1,992 | ç¹æ®çžå¯Ÿè« ããŒã¬ã³ãåçž® | ç¹æ®çžå¯Ÿè« > ããŒã¬ã³ãåçž®
ãã芳枬è
ã«ãšã£ãŠ æå»0ã§ãx=0ã«å·Šç«¯ãããã x=lã«å³ç«¯ããã æ£ãèããã ãã®ãšãxæ¹åã«é床vã§ç§»åããŠãã 芳枬è
ã«ãšã£ãŠ (0,0)ã¯ãã®ãŸãŸã§ããããã©ã (0,l)ã¯ã γ ( 1 â β â β 1 ) ( 0 l ) {\displaystyle \gamma {\begin{pmatrix}1&-\beta \\-\beta &1\end{pmatrix}}{\begin{pmatrix}0\\l\end{pmatrix}}} = γ ( â β l l ) {\displaystyle =\gamma {\begin{pmatrix}-\beta l\\l\end{pmatrix}}} ãåŸãããå³ç«¯ãšå·Šç«¯ã¯ ç°ãªã£ãæéã«ããããã«èŠããããšãåãã
å³ç«¯ã¯é床vã§åããŠãã芳枬è
ããèŠãŠ é床vã§åããŠããããã«èŠããããšãã å³ç«¯ã®åããŠãã芳枬è
ã«å¯Ÿããéå㯠( x â x 0 = v ( t â t 0 ) {\displaystyle x-x_{0}=v(t-t_{0})} ã«é©åãªå€ã代å
¥ãããšã) x â γ l = v ( t â 1 c γ β l ) {\displaystyle x-\gamma l=v(t-{\frac {1}{c}}\gamma \beta l)} ãšæžãããã t = 0 ãšãããšã x = γ l â 1 c γ β v l {\displaystyle x=\gamma l-{\frac {1}{c}}\gamma \beta vl} , x = γ l ( 1 â β 2 ) {\displaystyle x=\gamma l(1-\beta ^{2})} , x = l 1 â β 2 {\displaystyle x=l{\sqrt {1-\beta ^{2}}}} ãåŸããã x < l {\displaystyle x<l} ã€ãŸããæ£ãçž®ãã§ããããã«èŠããããšãåããã ãã®ããšãããŒã¬ã³ãåçž®ãšåŒã¶ã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ç¹æ®çžå¯Ÿè« > ããŒã¬ã³ãåçž®",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ãã芳枬è
ã«ãšã£ãŠ æå»0ã§ãx=0ã«å·Šç«¯ãããã x=lã«å³ç«¯ããã æ£ãèããã ãã®ãšãxæ¹åã«é床vã§ç§»åããŠãã 芳枬è
ã«ãšã£ãŠ (0,0)ã¯ãã®ãŸãŸã§ããããã©ã (0,l)ã¯ã γ ( 1 â β â β 1 ) ( 0 l ) {\\displaystyle \\gamma {\\begin{pmatrix}1&-\\beta \\\\-\\beta &1\\end{pmatrix}}{\\begin{pmatrix}0\\\\l\\end{pmatrix}}} = γ ( â β l l ) {\\displaystyle =\\gamma {\\begin{pmatrix}-\\beta l\\\\l\\end{pmatrix}}} ãåŸãããå³ç«¯ãšå·Šç«¯ã¯ ç°ãªã£ãæéã«ããããã«èŠããããšãåãã",
"title": "ããŒã¬ã³ãåçž®"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "å³ç«¯ã¯é床vã§åããŠãã芳枬è
ããèŠãŠ é床vã§åããŠããããã«èŠããããšãã å³ç«¯ã®åããŠãã芳枬è
ã«å¯Ÿããéå㯠( x â x 0 = v ( t â t 0 ) {\\displaystyle x-x_{0}=v(t-t_{0})} ã«é©åãªå€ã代å
¥ãããšã) x â γ l = v ( t â 1 c γ β l ) {\\displaystyle x-\\gamma l=v(t-{\\frac {1}{c}}\\gamma \\beta l)} ãšæžãããã t = 0 ãšãããšã x = γ l â 1 c γ β v l {\\displaystyle x=\\gamma l-{\\frac {1}{c}}\\gamma \\beta vl} , x = γ l ( 1 â β 2 ) {\\displaystyle x=\\gamma l(1-\\beta ^{2})} , x = l 1 â β 2 {\\displaystyle x=l{\\sqrt {1-\\beta ^{2}}}} ãåŸããã x < l {\\displaystyle x<l} ã€ãŸããæ£ãçž®ãã§ããããã«èŠããããšãåããã ãã®ããšãããŒã¬ã³ãåçž®ãšåŒã¶ã",
"title": "ããŒã¬ã³ãåçž®"
}
] | ç¹æ®çžå¯Ÿè« > ããŒã¬ã³ãåçž® | <small> [[ç¹æ®çžå¯Ÿè«]] > ããŒã¬ã³ãåçž®
----
==ããŒã¬ã³ãåçž®==
ãã芳枬è
ã«ãšã£ãŠ
æå»0ã§ãx=0ã«å·Šç«¯ãããã
x=lã«å³ç«¯ããã
æ£ãèããã
ãã®ãšãxæ¹åã«é床vã§ç§»åããŠãã
芳枬è
ã«ãšã£ãŠ
(0,0)ã¯ãã®ãŸãŸã§ããããã©ã
(0,l)ã¯ã
<math>
\gamma
\begin{pmatrix}
1 & -\beta \\
-\beta & 1
\end{pmatrix}
\begin{pmatrix}
0 \\
l
\end{pmatrix}
</math>
<math>
=\gamma
\begin{pmatrix}
-\beta l \\
l
\end{pmatrix}
</math>
ãåŸãããå³ç«¯ãšå·Šç«¯ã¯
ç°ãªã£ãæéã«ããããã«èŠããããšãåãã
<!-- æéã®æ¬¡å
ã -->
<!-- æéã§æž¬ããš -->
<!-- \frac 1 c ( -\beta l , l) ã«ãªã...ãããŠãã©ããããã -->
<!-- æéãé·ãã§æž¬ãããšã«ãããããŒã¬ã³ãå€æã«cãã€ããã...ã -->
<!-- åãæéã«çŸãããããã«ãããšã -->
å³ç«¯ã¯é床vã§åããŠãã芳枬è
ããèŠãŠ
é床vã§åããŠããããã«èŠããããšãã
å³ç«¯ã®åããŠãã芳枬è
ã«å¯Ÿããéåã¯
(<math>x-x _0 = v (t - t _0 )</math> ã«é©åãªå€ã代å
¥ãããšã)
<math>
x - \gamma l = v (t - \frac 1 c \gamma \beta l)
</math>
ãšæžãããã
t = 0 ãšãããšã
<math>
x = \gamma l - \frac 1 c \gamma \beta v l
</math>,
<math>
x= \gamma l ( 1 - \beta^2)
</math>,
<math>
x= l \sqrt{ 1 - \beta^2}
</math>
ãåŸããã
<math>
x < l
</math>
ã€ãŸããæ£ãçž®ãã§ããããã«èŠããããšãåããã
ãã®ããšãããŒã¬ã³ãåçž®ãšåŒã¶ã
[[Category:ç¹æ®çžå¯Ÿè«|ããããã€ãããããã]] | 2005-05-14T09:27:48Z | 2024-03-16T03:17:05Z | [] | https://ja.wikibooks.org/wiki/%E7%89%B9%E6%AE%8A%E7%9B%B8%E5%AF%BE%E8%AB%96_%E3%83%AD%E3%83%BC%E3%83%AC%E3%83%B3%E3%83%84%E5%8F%8E%E7%B8%AE |
1,993 | ç¹æ®çžå¯Ÿè« é»ç£æ°åŠãžã®å°å
¥ | ç¹æ®çžå¯Ÿè« > é»ç£æ°åŠãžã®å°å
¥
ããŒã¬ã³ãå€æã«å¯Ÿã㊠ããå€æããããšããèŠè«ã¯ éåžžã«å€å²ã«ããã£ãŠåœãŠã¯ãŸãããšã ç¥ãããŠãããããã®äŸãšã㊠ç¹ã«æåãªãã®ã¯ é»ç£æ°åŠã§ããã 詳现ã¯é»ç£æ°åŠã§è¿°ã¹ããããã é»ç£æ°åŠã®åºç€æ¹çšåŒã¯ â ÎŒ F ÎŒ Îœ = 4 Ï J Îœ {\displaystyle \partial _{\mu }F^{\mu \nu }=4\pi J^{\nu }} , â Ï F ÎŒ Îœ + â Îœ F Ï ÎŒ + â ÎŒ F Îœ Ï = 0 {\displaystyle \partial _{\rho }F_{\mu \nu }+\partial _{\nu }F_{\rho \mu }+\partial _{\mu }F_{\nu \rho }=0} ãšãªãããšãç¥ãããŠããã (Maxwellæ¹çšåŒ) | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ç¹æ®çžå¯Ÿè« > é»ç£æ°åŠãžã®å°å
¥",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "",
"title": ""
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ããŒã¬ã³ãå€æã«å¯Ÿã㊠ããå€æããããšããèŠè«ã¯ éåžžã«å€å²ã«ããã£ãŠåœãŠã¯ãŸãããšã ç¥ãããŠãããããã®äŸãšã㊠ç¹ã«æåãªãã®ã¯ é»ç£æ°åŠã§ããã 詳现ã¯é»ç£æ°åŠã§è¿°ã¹ããããã é»ç£æ°åŠã®åºç€æ¹çšåŒã¯ â ÎŒ F ÎŒ Îœ = 4 Ï J Îœ {\\displaystyle \\partial _{\\mu }F^{\\mu \\nu }=4\\pi J^{\\nu }} , â Ï F ÎŒ Îœ + â Îœ F Ï ÎŒ + â ÎŒ F Îœ Ï = 0 {\\displaystyle \\partial _{\\rho }F_{\\mu \\nu }+\\partial _{\\nu }F_{\\rho \\mu }+\\partial _{\\mu }F_{\\nu \\rho }=0} ãšãªãããšãç¥ãããŠããã (Maxwellæ¹çšåŒ)",
"title": "é»ç£æ°åŠãžã®å°å
¥"
}
] | ç¹æ®çžå¯Ÿè« > é»ç£æ°åŠãžã®å°å
¥ | <small> [[ç¹æ®çžå¯Ÿè«]] > é»ç£æ°åŠãžã®å°å
¥ </small>
----
==[[é»ç£æ°åŠ]]ãžã®å°å
¥==
ããŒã¬ã³ãå€æã«å¯ŸããŠ
ããå€æããããšããèŠè«ã¯
éåžžã«å€å²ã«ããã£ãŠåœãŠã¯ãŸãããšã
ç¥ãããŠãããããã®äŸãšããŠ
ç¹ã«æåãªãã®ã¯
é»ç£æ°åŠã§ããã
詳现ã¯[[é»ç£æ°åŠ]]ã§è¿°ã¹ããããã
é»ç£æ°åŠã®åºç€æ¹çšåŒã¯
<math>
\partial _\mu F^{\mu\nu} = 4\pi J^\nu
</math>,
<math>
\partial_\rho F_{\mu\nu}+ \partial_\nu F_{\rho\mu}+ \partial_\mu F_{\nu\rho} = 0
</math>
ãšãªãããšãç¥ãããŠããã
(Maxwellæ¹çšåŒ)
[[ã«ããŽãª:é»ç£æ°åŠ|ãšããããããããããŠããããããžã®ãšãã«ãã]]
[[Category:ç¹æ®çžå¯Ÿè«|ãŠããããããžã®ãšãã«ãã]] | null | 2022-12-01T04:16:55Z | [] | https://ja.wikibooks.org/wiki/%E7%89%B9%E6%AE%8A%E7%9B%B8%E5%AF%BE%E8%AB%96_%E9%9B%BB%E7%A3%81%E6%B0%97%E5%AD%A6%E3%81%B8%E3%81%AE%E5%B0%8E%E5%85%A5 |
1,994 | ç¹æ®çžå¯Ÿè« éåæ¹çšåŒ | ç¹æ®çžå¯Ÿè« > éåæ¹çšåŒ
SO(3,1)ã®ãã¡ã§ãæåã®3ã¯SO(3)ã®3ãšåäžã§ããã ãã®ããããã3次å
ã®ãã¯ãã«ãåã£ããšã ãããšé©åœãªéãçµã¿åãããŠ4次å
ã®ãã¯ãã«ã äœãããšãåºæ¥ãã d s 2 {\displaystyle ds^{2}} ãã¹ã«ã©ãŒã§ããããšãã x ÎŒ = ( c t x y z ) {\displaystyle x^{\mu }={\begin{pmatrix}ct\\x\\y\\z\end{pmatrix}}} ã®ããã«ãtãšãx,y,zãçµã¿åãããããããã«æããã ããã«ã åºææé d s 2 = d t 2 1 â ( v / c ) 2 {\displaystyle ds^{2}=dt^{2}{\sqrt {1-(v/c)^{2}}}} ãå°å
¥ãããšããã®éã¯ã¹ã«ã©ãŒã«ãªãã
ãã®ãšãã éåæ¹çšåŒã¯ã ããå f ÎŒ {\displaystyle f^{\mu }} ãæ³å®ãããšã (note: å€ãã®å Žåé»ç£æ°åãæ³å®ããŠããã) d p ÎŒ d s = f ÎŒ {\displaystyle {\frac {d{p^{\mu }}}{d{s}}}=f^{\mu }} ãšæžãããã ããã¯ãéåæ¹çšåŒã ããŒã¬ã³ãå€æã«å¯ŸããŠããæ§è³ªã ãã£ãŠããªããŠã¯ãããªããšãã èŠè«ããæ¥ãŠããã ãã¥ãŒãã³ã®æ¹çšåŒ d p â d t = f â {\displaystyle {\frac {d{\vec {p}}}{dt}}={\vec {f}}} ãã䞡蟺ã3次å
ã®ãã¯ãã«ã§ããããšãã SO(3)ã®å€æã«ã€ããŠè¯ãæ§è³ªããã£ãŠããã äžã®åŒã¯ããã®æ¡åŒµãšèããããšãåºæ¥ãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ç¹æ®çžå¯Ÿè« > éåæ¹çšåŒ",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "SO(3,1)ã®ãã¡ã§ãæåã®3ã¯SO(3)ã®3ãšåäžã§ããã ãã®ããããã3次å
ã®ãã¯ãã«ãåã£ããšã ãããšé©åœãªéãçµã¿åãããŠ4次å
ã®ãã¯ãã«ã äœãããšãåºæ¥ãã d s 2 {\\displaystyle ds^{2}} ãã¹ã«ã©ãŒã§ããããšãã x ÎŒ = ( c t x y z ) {\\displaystyle x^{\\mu }={\\begin{pmatrix}ct\\\\x\\\\y\\\\z\\end{pmatrix}}} ã®ããã«ãtãšãx,y,zãçµã¿åãããããããã«æããã ããã«ã åºææé d s 2 = d t 2 1 â ( v / c ) 2 {\\displaystyle ds^{2}=dt^{2}{\\sqrt {1-(v/c)^{2}}}} ãå°å
¥ãããšããã®éã¯ã¹ã«ã©ãŒã«ãªãã",
"title": "éåæ¹çšåŒ"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ãã®ãšãã éåæ¹çšåŒã¯ã ããå f ÎŒ {\\displaystyle f^{\\mu }} ãæ³å®ãããšã (note: å€ãã®å Žåé»ç£æ°åãæ³å®ããŠããã) d p ÎŒ d s = f ÎŒ {\\displaystyle {\\frac {d{p^{\\mu }}}{d{s}}}=f^{\\mu }} ãšæžãããã ããã¯ãéåæ¹çšåŒã ããŒã¬ã³ãå€æã«å¯ŸããŠããæ§è³ªã ãã£ãŠããªããŠã¯ãããªããšãã èŠè«ããæ¥ãŠããã ãã¥ãŒãã³ã®æ¹çšåŒ d p â d t = f â {\\displaystyle {\\frac {d{\\vec {p}}}{dt}}={\\vec {f}}} ãã䞡蟺ã3次å
ã®ãã¯ãã«ã§ããããšãã SO(3)ã®å€æã«ã€ããŠè¯ãæ§è³ªããã£ãŠããã äžã®åŒã¯ããã®æ¡åŒµãšèããããšãåºæ¥ãã",
"title": "éåæ¹çšåŒ"
}
] | ç¹æ®çžå¯Ÿè« > éåæ¹çšåŒ | <small> [[ç¹æ®çžå¯Ÿè«]] > éåæ¹çšåŒ </small>
----
==éåæ¹çšåŒ==
SO(3,1)ã®ãã¡ã§ãæåã®3ã¯SO(3)ã®3ãšåäžã§ããã
ãã®ããããã3次å
ã®ãã¯ãã«ãåã£ããšã
ãããšé©åœãªéãçµã¿åãããŠ4次å
ã®ãã¯ãã«ã
äœãããšãåºæ¥ãã
<math>ds^2</math>ãã¹ã«ã©ãŒã§ããããšãã
<math>
x^\mu =
\begin{pmatrix}
ct \\
x \\
y \\
z
\end{pmatrix}
</math>
ã®ããã«ãtãšãx,y,zãçµã¿åãããããããã«æããã
ããã«ã
åºææé
<math>
ds^2 = dt ^2 \sqrt{1-(v/c)^2}
</math>
ãå°å
¥ãããšããã®éã¯ã¹ã«ã©ãŒã«ãªãã
ãã®ãšãã
éåæ¹çšåŒã¯ã
ããå<math>f^{\mu}</math>ãæ³å®ãããšã
(note:
<!-- %ããã¯æã§æŒããå Žåã®åã§ããããã(?)
-->
å€ãã®å Žåé»ç£æ°åãæ³å®ããŠããã)
<math>
\frac {d {p^\mu }}{d { s} } = f^\mu
</math>
ãšæžãããã
ããã¯ãéåæ¹çšåŒã
ããŒã¬ã³ãå€æã«å¯ŸããŠããæ§è³ªã
ãã£ãŠããªããŠã¯ãããªããšãã
èŠè«ããæ¥ãŠããã
ãã¥ãŒãã³ã®æ¹çšåŒ
<math>
\frac {d {\vec p }}{d t } = \vec f
</math>
ãã䞡蟺ã3次å
ã®ãã¯ãã«ã§ããããšãã
SO(3)ã®å€æã«ã€ããŠè¯ãæ§è³ªããã£ãŠããã
äžã®åŒã¯ããã®æ¡åŒµãšèããããšãåºæ¥ãã
[[Category:ç¹æ®çžå¯Ÿè«|ãããšãã»ããŠããã]] | 2005-05-14T09:33:30Z | 2024-03-16T03:17:50Z | [] | https://ja.wikibooks.org/wiki/%E7%89%B9%E6%AE%8A%E7%9B%B8%E5%AF%BE%E8%AB%96_%E9%81%8B%E5%8B%95%E6%96%B9%E7%A8%8B%E5%BC%8F |
1,996 | çµæžåŠ çŸä»£çµæžã®ä»çµã¿ è²¡æ¿ | çµæžåŠ>çŸä»£çµæžã®ä»çµã¿>財æ¿
æã£ãŠãã ãããããªããæããªããã°ãªããªãçšéã®é¡ã¯ããªãã®åå
¥ã«ãã£ãŠæ±ºãŸã£ãŠããã¯ãã§ãããªããªã,çšéã¯é«æåŸè
ããäœæåŸè
ã«ç§»ããéã ããã§ããé«æåŸã§ããã°ããã»ã©æããªããã°ãªããªãçšéã¯å¢ããããã§ãããã ã,æ¶è²»çšããã°ãçšãªã©ã®éæ¥çšã¯ãã®æ±ºãŸãã«åããŠããŸãã
æ¿åºãããçµæžæŽ»åã®ããšã財æ¿ãšãããŸãããªãã£ãŒãã»ãã¹ã°ã¬ã€ã(Richard Abel Musgrave)ã¯èæžã財æ¿çè«(The Theory of Public Finance 1959)ãã§è²¡æ¿ã®æ©èœãè³æºã®åé
,æåŸååé
,çµæžã®å®å®åãšãã3ã€ã«åé¡ããŸãããã©ããåžå Žã®å€±æãè£ãæ©èœã§ãã19äžçŽãŸã§ã¯è²¡æ¿ã¯å¿
èŠæäœé床ã«ãããŠããŸããã,çŸä»£ã§ã¯è²¡æ¿èŠæš¡ã¯å€§ãããªã,åé¡åãããããªã£ãŠããŸãã
ç§çšãäºç®ãªã©ã®ååã¯,æ¥æ¬åœæ²æ³ 第7ç« è²¡æ¿ã§å®ããããŠããŸãã
è³æ¬äž»çŸ©çµæžã§ã¯åžå ŽãéããŠè³æºãåé
åããŸãã,å
Œ
±è²¡ã¯åžå ŽãéããŠååŒãããã®ã§ã¯ãããŸããããããã£ãŠ,å
Œ
±è²¡ã¯æ¿åºãä»ããŠé
åãããªããã°ãªããŸããã
åžå ŽãæåŸãå
¬å¹³ã«åé
ããããšã¯ã§ããŸãããæ¿åºã,é«æåŸè
ã«ã¯æåŸçšãªã©ã®çŽ¯é²èª²çšã,äœæåŸè
ã«ã¯ç€ŸäŒä¿éãªã©ã®ç§çšã®æ¯æ¿æ¯åºãããããšã§æåŸãåè¡¡ã«ããããšãã財æ¿ã®æ©èœã®ããšã§ãã
åžå Žçµæžã§ã¯æ¯æ°ã®å€åã«æ³¢ããã,éåžžã«äžå®å®ã§ããçµæžã®å®å®åæ©èœãŸãã¯æ¯æ°èª¿ç¯æ©èœãšãããŸãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "çµæžåŠ>çŸä»£çµæžã®ä»çµã¿>財æ¿",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "æã£ãŠãã ãããããªããæããªããã°ãªããªãçšéã®é¡ã¯ããªãã®åå
¥ã«ãã£ãŠæ±ºãŸã£ãŠããã¯ãã§ãããªããªã,çšéã¯é«æåŸè
ããäœæåŸè
ã«ç§»ããéã ããã§ããé«æåŸã§ããã°ããã»ã©æããªããã°ãªããªãçšéã¯å¢ããããã§ãããã ã,æ¶è²»çšããã°ãçšãªã©ã®éæ¥çšã¯ãã®æ±ºãŸãã«åããŠããŸãã",
"title": "çšéïŒæããªããé§ç®ïŒ"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "æ¿åºãããçµæžæŽ»åã®ããšã財æ¿ãšãããŸãããªãã£ãŒãã»ãã¹ã°ã¬ã€ã(Richard Abel Musgrave)ã¯èæžã財æ¿çè«(The Theory of Public Finance 1959)ãã§è²¡æ¿ã®æ©èœãè³æºã®åé
,æåŸååé
,çµæžã®å®å®åãšãã3ã€ã«åé¡ããŸãããã©ããåžå Žã®å€±æãè£ãæ©èœã§ãã19äžçŽãŸã§ã¯è²¡æ¿ã¯å¿
èŠæäœé床ã«ãããŠããŸããã,çŸä»£ã§ã¯è²¡æ¿èŠæš¡ã¯å€§ãããªã,åé¡åãããããªã£ãŠããŸãã",
"title": "財æ¿ãšã¯"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ç§çšãäºç®ãªã©ã®ååã¯,æ¥æ¬åœæ²æ³ 第7ç« è²¡æ¿ã§å®ããããŠããŸãã",
"title": "財æ¿ã®ä»çµã¿"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "è³æ¬äž»çŸ©çµæžã§ã¯åžå ŽãéããŠè³æºãåé
åããŸãã,å
Œ
±è²¡ã¯åžå ŽãéããŠååŒãããã®ã§ã¯ãããŸããããããã£ãŠ,å
Œ
±è²¡ã¯æ¿åºãä»ããŠé
åãããªããã°ãªããŸããã",
"title": "財æ¿æ¿ç"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "åžå ŽãæåŸãå
¬å¹³ã«åé
ããããšã¯ã§ããŸãããæ¿åºã,é«æåŸè
ã«ã¯æåŸçšãªã©ã®çŽ¯é²èª²çšã,äœæåŸè
ã«ã¯ç€ŸäŒä¿éãªã©ã®ç§çšã®æ¯æ¿æ¯åºãããããšã§æåŸãåè¡¡ã«ããããšãã財æ¿ã®æ©èœã®ããšã§ãã",
"title": "財æ¿æ¿ç"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "åžå Žçµæžã§ã¯æ¯æ°ã®å€åã«æ³¢ããã,éåžžã«äžå®å®ã§ããçµæžã®å®å®åæ©èœãŸãã¯æ¯æ°èª¿ç¯æ©èœãšãããŸãã",
"title": "財æ¿æ¿ç"
}
] | çµæžåŠïŒçŸä»£çµæžã®ä»çµã¿ïŒè²¡æ¿ | [[çµæžåŠ]]ïŒ[[çµæžåŠ_çŸä»£çµæžã®ä»çµã¿|çŸä»£çµæžã®ä»çµã¿]]ïŒè²¡æ¿
----
__TOC__
<div style="margin:0px; padding:0px; background-color:#CCFF99; border:solid #00CC00 1px; width:100%;">
== çšéïŒæããªããé§ç®ïŒ ==
æã£ãŠãã ãããããªããæããªããã°ãªããªãçšéã®é¡ã¯ããªãã®åå
¥ã«ãã£ãŠæ±ºãŸã£ãŠããã¯ãã§ãããªããªãïŒçšéã¯é«æåŸè
ããäœæåŸè
ã«ç§»ããéã ããã§ããé«æåŸã§ããã°ããã»ã©æããªããã°ãªããªãçšéã¯å¢ããããã§ãããã ãïŒæ¶è²»çšããã°ãçšãªã©ã®éæ¥çšã¯ãã®æ±ºãŸãã«åããŠããŸãã
</div>
== 財æ¿ãšã¯ ==
[[çµæžåŠ_çŸä»£çµæžã®ä»çµã¿_çµæžäž»äœãšãã®æŽ»å#æ¿åº|æ¿åº]]ãããçµæžæŽ»åã®ããšã'''財æ¿'''ãšãããŸãã[[w:ãªãã£ãŒãã»ãã¹ã°ã¬ã€ã|ãªãã£ãŒãã»ãã¹ã°ã¬ã€ã(Richard Abel Musgrave)]]ã¯èæžã財æ¿çè«(The Theory of Public Finance 1959)ãã§è²¡æ¿ã®æ©èœãè³æºã®åé
ïŒæåŸååé
ïŒçµæžã®å®å®åãšããïŒã€ã«åé¡ããŸãããã©ãã[[çµæžåŠ_çŸä»£çµæžã®å€å®¹_çµæžã®å€å®¹_äžççµæžã®å€å®¹_è³æ¬äž»çŸ©çµæž#åžå Žã®å€±æ|åžå Žã®å€±æ]]ãè£ãæ©èœã§ãã19äžçŽãŸã§ã¯è²¡æ¿ã¯å¿
èŠæäœé床ã«ãããŠããŸãããïŒçŸä»£ã§ã¯è²¡æ¿èŠæš¡ã¯å€§ãããªãïŒåé¡åãããããªã£ãŠããŸãã
== 財æ¿ã®ä»çµã¿ ==
ç§çšãäºç®ãªã©ã®ååã¯ïŒ[[Wikisource:æ¥æ¬åæ²æ³#第äžç« 財æ¿|æ¥æ¬åœæ²æ³ 第ïŒç« 財æ¿]]ã§å®ããããŠããŸãã
=== ç§çš ===
== 財æ¿æ¿ç ==
=== è³æºã®åé
æ©èœ ===
[[çµæžåŠ_çŸä»£çµæžã®å€å®¹_çµæžã®å€å®¹_äžççµæžã®å€å®¹_è³æ¬äž»çŸ©çµæž|è³æ¬äž»çŸ©çµæž]]ã§ã¯åžå ŽãéããŠè³æºãåé
åããŸããïŒ[[çµæžåŠ_çŸä»£çµæžã®å€å®¹_çµæžã®å€å®¹_çµæžãšã¯äœã#財ç©|å
Œ
±è²¡]]ã¯åžå ŽãéããŠååŒãããã®ã§ã¯ãããŸããããããã£ãŠïŒå
Œ
±è²¡ã¯æ¿åºãä»ããŠé
åãããªããã°ãªããŸããã
=== æåŸååé
æ©èœ ===
åžå ŽãæåŸãå
¬å¹³ã«åé
ããããšã¯ã§ããŸãããæ¿åºãïŒé«æåŸè
ã«ã¯æåŸçšãªã©ã®çŽ¯é²èª²çšãïŒäœæåŸè
ã«ã¯[[çµæžåŠ_瀟äŒä¿éå¶åºŠ|瀟äŒä¿é]]ãªã©ã®ç§çšã®æ¯æ¿æ¯åºãããããšã§æåŸãåè¡¡ã«ããããšãã財æ¿ã®æ©èœã®ããšã§ãã
=== çµæžã®å®å®åæ©èœ ===
åžå Žçµæžã§ã¯æ¯æ°ã®å€åã«æ³¢ãããïŒéåžžã«äžå®å®ã§ããçµæžã®å®å®åæ©èœãŸãã¯æ¯æ°èª¿ç¯æ©èœãšãããŸãã
== 財æ¿ã®èª²é¡ ==
{{stub}}
[[Category:çµæžåŠ|*]] | null | 2009-11-26T04:07:28Z | [
"ãã³ãã¬ãŒã:Stub"
] | https://ja.wikibooks.org/wiki/%E7%B5%8C%E6%B8%88%E5%AD%A6_%E7%8F%BE%E4%BB%A3%E7%B5%8C%E6%B8%88%E3%81%AE%E4%BB%95%E7%B5%84%E3%81%BF_%E8%B2%A1%E6%94%BF |
1,999 | å€å
žæåŠ | æåŠ > å€å
žæåŠ > ç®æ¬¡
倧åŠã®æç§æž èªç¶ç§åŠ: æ°åŠ - ç©çåŠ; å€å
žååŠ éåååŠ - ååŠ; ç¡æ©ååŠ ææ©ååŠ - çç©åŠ; æ€ç©åŠ ç 究æè¡ - å°çç§åŠ - å»åŠ; 解ååŠ èªåŠ: æ¥æ¬èª è±èª ãšã¹ãã©ã³ã æé®®èª ãã³ããŒã¯èª ãã€ãèª ãã©ã³ã¹èª ã©ãã³èª ã«ãŒããã¢èª 人æç§åŠ: æŽå²åŠ; æ¥æ¬å² äžåœå² äžçå² æŽå²èŠ³ - å¿çåŠ - å²åŠ - èžè¡; é³æ¥œ çŸè¡ - æåŠ; å€å
žæåŠ æŒ¢è©© 瀟äŒç§åŠ: æ³åŠ - çµæžåŠ - å°çåŠ - æè²åŠ; åŠæ ¡æè² æè²å² æ
å ±æè¡: æ
å ±å·¥åŠ; MS-DOS/PC DOS UNIX/Linux TeX/LaTeX CGI - ããã°ã©ãã³ã°; BASIC Cèšèª C++ Dèšèª HTML Java JavaScript Lisp Mizar Perl PHP Python Ruby Scheme SVG å°ã»äžã»é«æ ¡ã®æç§æž å°åŠ: åœèª ç€ŸäŒ ç®æ° çç§ è±èª äžåŠ: åœèª ç€ŸäŒ æ°åŠ çç§ è±èª é«æ ¡: åœèª - å°æŽ - å
¬æ° - æ°åŠ; å
¬åŒé - çç§; ç©ç ååŠ å°åŠ çç© - å€åœèª - æ
å ± 解説æžã»å®çšæžã»åèæž è¶£å³: æçæ¬ - ã¹ããŒã - ã²ãŒã è©Šéš: è³æ Œè©Šéš - å
¥åŠè©Šéš ãã®ä»ã®æ¬: é²çœ - ç掻ãšé²è·¯ - ãŠã£ãããã£ã¢ã®æžãæ¹ - ãžã§ãŒã¯é | [
{
"paragraph_id": 0,
"tag": "p",
"text": "æåŠ > å€å
žæåŠ > ç®æ¬¡",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "倧åŠã®æç§æž èªç¶ç§åŠ: æ°åŠ - ç©çåŠ; å€å
žååŠ éåååŠ - ååŠ; ç¡æ©ååŠ ææ©ååŠ - çç©åŠ; æ€ç©åŠ ç 究æè¡ - å°çç§åŠ - å»åŠ; 解ååŠ èªåŠ: æ¥æ¬èª è±èª ãšã¹ãã©ã³ã æé®®èª ãã³ããŒã¯èª ãã€ãèª ãã©ã³ã¹èª ã©ãã³èª ã«ãŒããã¢èª 人æç§åŠ: æŽå²åŠ; æ¥æ¬å² äžåœå² äžçå² æŽå²èŠ³ - å¿çåŠ - å²åŠ - èžè¡; é³æ¥œ çŸè¡ - æåŠ; å€å
žæåŠ æŒ¢è©© 瀟äŒç§åŠ: æ³åŠ - çµæžåŠ - å°çåŠ - æè²åŠ; åŠæ ¡æè² æè²å² æ
å ±æè¡: æ
å ±å·¥åŠ; MS-DOS/PC DOS UNIX/Linux TeX/LaTeX CGI - ããã°ã©ãã³ã°; BASIC Cèšèª C++ Dèšèª HTML Java JavaScript Lisp Mizar Perl PHP Python Ruby Scheme SVG å°ã»äžã»é«æ ¡ã®æç§æž å°åŠ: åœèª ç€ŸäŒ ç®æ° çç§ è±èª äžåŠ: åœèª ç€ŸäŒ æ°åŠ çç§ è±èª é«æ ¡: åœèª - å°æŽ - å
¬æ° - æ°åŠ; å
¬åŒé - çç§; ç©ç ååŠ å°åŠ çç© - å€åœèª - æ
å ± 解説æžã»å®çšæžã»åèæž è¶£å³: æçæ¬ - ã¹ããŒã - ã²ãŒã è©Šéš: è³æ Œè©Šéš - å
¥åŠè©Šéš ãã®ä»ã®æ¬: é²çœ - ç掻ãšé²è·¯ - ãŠã£ãããã£ã¢ã®æžãæ¹ - ãžã§ãŒã¯é",
"title": ""
}
] | æåŠ > å€å
žæåŠ > ç®æ¬¡ | [[æåŠ]] > [[å€å
žæåŠ]] > ç®æ¬¡
----
[[ç»å:å€å
žæåŠ_æçµµ.jpg|480px|center]]
{{èµæžäžèŠ§}}
== æ¥æ¬ã®å€å
ž ==
=== 察蚳ã»è§£èª¬ ===
*[[æ¥æ¬ã®å€å
ž]]
* [[/ããã¯æ|ããã¯æ]]{{é²æ|50%|2005-06-11}}
=== å€å
žææ³ ===
* [[å€å
žæåŠ/å€å
žææ³|å€å
žææ³]]
* [[å€å
žæåŠ/å€å
žææ³/æŽå²çä»®åé£ã|æŽå²çä»®åé£ã]]{{é²æ|75%|2005-05-15}}
* [[å€å
žæåŠ/å€å
žææ³/åè©|åè©]]
* [[å€å
žæåŠ/å€å
žææ³/圢容è©|圢容è©]]
* [[å€å
žæåŠ/å€å
žææ³/圢容åè©|圢容åè©]]
* [[å€å
žæåŠ/å€å
žææ³/åè©|åè©]]
* [[å€å
žæåŠ/å€å
žææ³/é£äœè©|é£äœè©]]
* [[å€å
žæåŠ/å€å
žææ³/å¯è©|å¯è©]]
* [[å€å
žæåŠ/å€å
žææ³/æ¥ç¶è©|æ¥ç¶è©]]
* [[å€å
žæåŠ/å€å
žææ³/æåè©|æåè©]]
* [[å€å
žæåŠ/å€å
žææ³/å©åè©|å©åè©]]
* [[å€å
žæåŠ/å€å
žææ³/å©è©|å©è©]]
* [[å€å
žæåŠ/å€å
žææ³/æ¬èª|æ¬èª]]
* [[å€å
žæåŠ/å€å
žææ³/ä¿ãçµã³ã®æ³å|ä¿ãçµã³ã®æ³å]]
* [[å€å
žæåŠ/å€å
žææ³/æã»é£æ|æã»é£æ]]
* [[å€å
žæåŠ/å€å
žææ³/ä¿®èŸ|ä¿®èŸ]]
*[[å€å
žæåŠ/å€å
žææ³/å€å
žçšèªã»åèª|å€å
žçšèªã»åèª]]
{{é²æç¶æ³}}
== äžåœã®å€å
ž ==
===察話ã»è§£èª¬===
*[[äžåœã®å€å
ž]]
===挢æãææ³===
* [[挢詩]]
[[Category:å€å
žæåŠ|*]]
[[Category:æžåº«|ããŠããµããã]] | 2005-05-14T14:28:41Z | 2024-01-22T09:26:08Z | [
"ãã³ãã¬ãŒã:èµæžäžèŠ§",
"ãã³ãã¬ãŒã:é²æ",
"ãã³ãã¬ãŒã:é²æç¶æ³"
] | https://ja.wikibooks.org/wiki/%E5%8F%A4%E5%85%B8%E6%96%87%E5%AD%A6 |
2,003 | HTML/ãªããžã§ã¯ã |
ç»åã®æ¿å
¥ã«ã¯imgèŠçŽ ã䜿çšãããäž»ã«äœ¿çšãããç»å圢åŒã¯JPEGãGIFãPNGã§ããã
ããã¹ããã©ãŠã¶ã®ãããªç»å衚瀺ãåºæ¥ãªããã©ãŠã¶ã§ã¯imgèŠçŽ å
ã®altå±æ§ã§æå®ããæååã衚瀺ãããã
ããã¹ããã©ãŠã¶ãç»å衚瀺ãããªãããã«èšå®ããŠããã¯ã©ã€ã¢ã³ããžã®é
æ
®ãšããŠãaltå±æ§ã¯ä»£æ¿ãšãªãããè¡šçŸãèšèŒããããšãšãªã£ãŠãããåã«è£
食çãªæå³ã§çšããŠããç»åçã®ããã«ä»£æ¿ãã¹ãæåè¡šçŸããªãå Žåã¯ç©ºæååãæå® (alt="") ããããš(çç¥ã¯åºæ¥ãªã)ã
HTMLã®æ¬æå
ã«èšèŒããããããããã衚瀺ãããç»å(äžèšã³ãŒãã§ã¯ sample.png )ãæå
ã®ã³ã³ãã¥ãŒã¿ã«æºåããŠãããŠãHTMLãã¡ã€ã«ãšåããã©ã«ãã«å
¥ããŠããã
ãªã width ã¯ç»åã®æšªå¹
ãheight ã¯ç»åã®çžŠå¹
ã§ããã
imgèŠçŽ ã«ã¯çµäºã¿ã°ããªãããã</img>ã¯èšèŒããªããsrcå±æ§ã«ã¯ã衚瀺ãããç»åãã¡ã€ã«ã®URI(ã¢ãã¬ã¹)ãèšèŒãããçžå¯ŸURI(*) ã§æå®ããããšãå¯èœã§ãããåäžãµã€ãå
ã§ã®ç»åãåç
§ããå Žåã¯çžå¯ŸURIãçšããããšãå€ããwidthå±æ§ãšheightå±æ§ã§ç»åã®æšªå¹
ãšçžŠå¹
ãæå®ã§ããã
(*)ãã®HTMLãã¡ã€ã«ãååšããå°ç¹(äžè¬çã«ã¯ãã£ã¬ã¯ããª)ããã®ç»åãã¡ã€ã«ã®äœçœ®ã«ãã瀺ãããURIãHTML/ãã€ããŒãªã³ã¯#çžå¯Ÿãã¹ãšçµ¶å¯Ÿãã¹ãåç
§ã
äžã®ã³ãŒãã§ã¯ãç»åã®äžéšãç¹å®é åã«ãªã³ã¯ãèšå®ãããã®é åãã¯ãªãã¯ããå Žåæå®ãããªã³ã¯å
ãžé£ã¹ãããã«ããŠããã
決ããŠãç»åã«åãåè§åœ¢ãªã©ãæç»ããããã§ã¯ãªãã®ã§ãæ··åããªãããã«ããªãããªã³ã¯ããé åã®äœçœ®ã¯ãäžèšã³ãŒãã®å Žåãã¢ãã¿ãŒã®èšå®ã倧ãããªã©ã«ãå¯ãããæ®éã®ããŒãããœã³ã³ã§èŠãå Žåãªããªã³ã¯é åã¯æšªã«äžŠãã§ãã(ç»åã«åãåè§åœ¢ã衚瀺ãããããã§ã¯ãªãã®ã§ãå®è¡ããŠãèŠããªã)ã(ãªããHTMLã§åãåè§ãªã©ã®åºæ¬å³åœ¢ãæç»ãããå ŽåãHTML5以éãªããHTML/HTML5#canvasèŠçŽ ãã§è§£èª¬ããæ¹æ³ã§åãªã©ãæç»ã§ããã)
ãªã³ã¯é åã®äœçœ®ã«ããŠã¹ã«ãŒãœã«ãåããã°ãç»é¢å·Šäžãªã©ãããèŠãã°ããªã³ã¯å
ã®å称ã衚瀺ãããŠããã
imgèŠçŽ ã«usemapå±æ§ã§ãããããèšå®ãããã€ã¡ãŒãžãããã®ååãã#ãããåããšæå®ããããšã«ããå©çšå¯èœãšãªããã€ã¡ãŒãžããããèšå®ãããšãã¯mapå±æ§ã§ãããåã®å®çŸ©ãè¡ããareaèŠçŽ ã®spaceå±æ§ã§é åã®åœ¢ç¶ããcoordså±æ§ã§åº§æšãæå®ããã座æšæå®ã®éã座æšã¯ã³ã³ãã§åºåãã
ãªãã€ã¡ãŒãžãããã®å Žåãããã¹ããã©ãŠã¶ãç»åé衚瀺ãšãªã£ãŠãããã©ãŠã¶ã®ããã«ãªã³ã¯å
ãäœã§ãããã瀺ãaltå±æ§ãæå®ãã¹ãã§ããããã®å Žå空æååãæå®ãã¹ãã§ã¯ãªãã
embedèŠçŽ ã¯ãã©ã°ã€ã³ãåã蟌ã¿ããã©ãŠã¶ãçŽæ¥åçã§ããªããã¡ã€ã«ãããŒãžã³ã³ãã³ãã®äžéšãšããŠå©çšãããã®ã§ããããã©ã°ã€ã³ã¯äºããã©ãŠã¶åŽã«ã€ã³ã¹ããŒã«ããŠããå¿
èŠãããã該åœã®ãã©ã°ã€ã³ããªããšã³ã³ãã³ããå©çšã§ããªãå Žåãããããã ãå
šãŠã®ç°å¢ã§ãã©ã°ã€ã³ãå©çšã§ããããã§ã¯ãªãããã<noembed>ã</noembed>å
ã§å©çšã§ããªãç°å¢ã§è¡šç€ºãããå
容ãæžãã¹ãã§ããããã®éããã©ã°ã€ã³ãæå¹ã«ããŠãã ããããããã©ã°ã€ã³ãã€ã³ã¹ããŒã«ãããŠããå¿
èŠããããŸãããªã©ãšããããã±ãŒã¹ããããããã®ãããªèšè¿°ã¯å¥œãŸãããªãã代æ¿çãªå
容(äŸãã°Flashã«ããã¡ãã¥ãŒãåã蟌ãã§ããå Žåã¯HTMLã§ã®ã¡ãã¥ãŒ)ãèšè¿°ããŠããããå¿
èŠããªããã°äœãæžããªãã»ããè¯ãã
objectã¯ãŠã§ãããŒãžã«æ§ã
ãªããŒã¿ãåã蟌ãããã®ãã®ã§ããããã©ãŠã¶ãçŽæ¥åŠçã§ãããã¡ã€ã«ã§ããã°ãã©ãŠã¶ãçŽæ¥åŠçãè¡ããçŽæ¥åŠçãè¡ããªãå Žåã¯ãã©ã°ã€ã³ãå©çšããŠãã¡ã€ã«ãå©çšããããã ãå©çšã§ããªããã©ãŠã¶ãå€ããããäŸãã°ãã©ã°ã€ã³ãåã蟌ã¿ããå Žåã¯embedèŠçŽ ãªã©ä»ã®èŠçŽ ãåã蟌ãå¿
èŠããããobjectèŠçŽ ã«å²ãŸããéšåã¯ããã®ãªããžã§ã¯ããå©çšã§ããå Žåãã©ãŠã¶ã¯paramèŠçŽ ãšmapèŠçŽ ãé€ããå
šãŠã®èŠçŽ ãç¡èŠãããªããžã§ã¯ããå©çšã§ããªãå Žåã¯object, param, mapèŠçŽ ãç¡èŠããŠããã«ããä»ã®èŠçŽ ã®èšè¿°ãé©çšããã
Windows Media Playerã®ãã©ã°ã€ã³ãåã蟌ãã äŸã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ç»åã®æ¿å
¥ã«ã¯imgèŠçŽ ã䜿çšãããäž»ã«äœ¿çšãããç»å圢åŒã¯JPEGãGIFãPNGã§ããã",
"title": "ç»åã®æ¿å
¥"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ããã¹ããã©ãŠã¶ã®ãããªç»å衚瀺ãåºæ¥ãªããã©ãŠã¶ã§ã¯imgèŠçŽ å
ã®altå±æ§ã§æå®ããæååã衚瀺ãããã",
"title": "ç»åã®æ¿å
¥"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ããã¹ããã©ãŠã¶ãç»å衚瀺ãããªãããã«èšå®ããŠããã¯ã©ã€ã¢ã³ããžã®é
æ
®ãšããŠãaltå±æ§ã¯ä»£æ¿ãšãªãããè¡šçŸãèšèŒããããšãšãªã£ãŠãããåã«è£
食çãªæå³ã§çšããŠããç»åçã®ããã«ä»£æ¿ãã¹ãæåè¡šçŸããªãå Žåã¯ç©ºæååãæå® (alt=\"\") ããããš(çç¥ã¯åºæ¥ãªã)ã",
"title": "ç»åã®æ¿å
¥"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "HTMLã®æ¬æå
ã«èšèŒããããããããã衚瀺ãããç»å(äžèšã³ãŒãã§ã¯ sample.png )ãæå
ã®ã³ã³ãã¥ãŒã¿ã«æºåããŠãããŠãHTMLãã¡ã€ã«ãšåããã©ã«ãã«å
¥ããŠããã",
"title": "ç»åã®æ¿å
¥"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ãªã width ã¯ç»åã®æšªå¹
ãheight ã¯ç»åã®çžŠå¹
ã§ããã",
"title": "ç»åã®æ¿å
¥"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "imgèŠçŽ ã«ã¯çµäºã¿ã°ããªãããã</img>ã¯èšèŒããªããsrcå±æ§ã«ã¯ã衚瀺ãããç»åãã¡ã€ã«ã®URI(ã¢ãã¬ã¹)ãèšèŒãããçžå¯ŸURI(*) ã§æå®ããããšãå¯èœã§ãããåäžãµã€ãå
ã§ã®ç»åãåç
§ããå Žåã¯çžå¯ŸURIãçšããããšãå€ããwidthå±æ§ãšheightå±æ§ã§ç»åã®æšªå¹
ãšçžŠå¹
ãæå®ã§ããã",
"title": "ç»åã®æ¿å
¥"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "(*)ãã®HTMLãã¡ã€ã«ãååšããå°ç¹(äžè¬çã«ã¯ãã£ã¬ã¯ããª)ããã®ç»åãã¡ã€ã«ã®äœçœ®ã«ãã瀺ãããURIãHTML/ãã€ããŒãªã³ã¯#çžå¯Ÿãã¹ãšçµ¶å¯Ÿãã¹ãåç
§ã",
"title": "ç»åã®æ¿å
¥"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "äžã®ã³ãŒãã§ã¯ãç»åã®äžéšãç¹å®é åã«ãªã³ã¯ãèšå®ãããã®é åãã¯ãªãã¯ããå Žåæå®ãããªã³ã¯å
ãžé£ã¹ãããã«ããŠããã",
"title": "ã€ã¡ãŒãžããã"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "決ããŠãç»åã«åãåè§åœ¢ãªã©ãæç»ããããã§ã¯ãªãã®ã§ãæ··åããªãããã«ããªãããªã³ã¯ããé åã®äœçœ®ã¯ãäžèšã³ãŒãã®å Žåãã¢ãã¿ãŒã®èšå®ã倧ãããªã©ã«ãå¯ãããæ®éã®ããŒãããœã³ã³ã§èŠãå Žåãªããªã³ã¯é åã¯æšªã«äžŠãã§ãã(ç»åã«åãåè§åœ¢ã衚瀺ãããããã§ã¯ãªãã®ã§ãå®è¡ããŠãèŠããªã)ã(ãªããHTMLã§åãåè§ãªã©ã®åºæ¬å³åœ¢ãæç»ãããå ŽåãHTML5以éãªããHTML/HTML5#canvasèŠçŽ ãã§è§£èª¬ããæ¹æ³ã§åãªã©ãæç»ã§ããã)",
"title": "ã€ã¡ãŒãžããã"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "ãªã³ã¯é åã®äœçœ®ã«ããŠã¹ã«ãŒãœã«ãåããã°ãç»é¢å·Šäžãªã©ãããèŠãã°ããªã³ã¯å
ã®å称ã衚瀺ãããŠããã",
"title": "ã€ã¡ãŒãžããã"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "imgèŠçŽ ã«usemapå±æ§ã§ãããããèšå®ãããã€ã¡ãŒãžãããã®ååãã#ãããåããšæå®ããããšã«ããå©çšå¯èœãšãªããã€ã¡ãŒãžããããèšå®ãããšãã¯mapå±æ§ã§ãããåã®å®çŸ©ãè¡ããareaèŠçŽ ã®spaceå±æ§ã§é åã®åœ¢ç¶ããcoordså±æ§ã§åº§æšãæå®ããã座æšæå®ã®éã座æšã¯ã³ã³ãã§åºåãã",
"title": "ã€ã¡ãŒãžããã"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "ãªãã€ã¡ãŒãžãããã®å Žåãããã¹ããã©ãŠã¶ãç»åé衚瀺ãšãªã£ãŠãããã©ãŠã¶ã®ããã«ãªã³ã¯å
ãäœã§ãããã瀺ãaltå±æ§ãæå®ãã¹ãã§ããããã®å Žå空æååãæå®ãã¹ãã§ã¯ãªãã",
"title": "ã€ã¡ãŒãžããã"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "embedèŠçŽ ã¯ãã©ã°ã€ã³ãåã蟌ã¿ããã©ãŠã¶ãçŽæ¥åçã§ããªããã¡ã€ã«ãããŒãžã³ã³ãã³ãã®äžéšãšããŠå©çšãããã®ã§ããããã©ã°ã€ã³ã¯äºããã©ãŠã¶åŽã«ã€ã³ã¹ããŒã«ããŠããå¿
èŠãããã該åœã®ãã©ã°ã€ã³ããªããšã³ã³ãã³ããå©çšã§ããªãå Žåãããããã ãå
šãŠã®ç°å¢ã§ãã©ã°ã€ã³ãå©çšã§ããããã§ã¯ãªãããã<noembed>ã</noembed>å
ã§å©çšã§ããªãç°å¢ã§è¡šç€ºãããå
容ãæžãã¹ãã§ããããã®éããã©ã°ã€ã³ãæå¹ã«ããŠãã ããããããã©ã°ã€ã³ãã€ã³ã¹ããŒã«ãããŠããå¿
èŠããããŸãããªã©ãšããããã±ãŒã¹ããããããã®ãããªèšè¿°ã¯å¥œãŸãããªãã代æ¿çãªå
容(äŸãã°Flashã«ããã¡ãã¥ãŒãåã蟌ãã§ããå Žåã¯HTMLã§ã®ã¡ãã¥ãŒ)ãèšè¿°ããŠããããå¿
èŠããªããã°äœãæžããªãã»ããè¯ãã",
"title": "ãã«ãã¡ãã£ã¢ã®æ¿å
¥"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "objectã¯ãŠã§ãããŒãžã«æ§ã
ãªããŒã¿ãåã蟌ãããã®ãã®ã§ããããã©ãŠã¶ãçŽæ¥åŠçã§ãããã¡ã€ã«ã§ããã°ãã©ãŠã¶ãçŽæ¥åŠçãè¡ããçŽæ¥åŠçãè¡ããªãå Žåã¯ãã©ã°ã€ã³ãå©çšããŠãã¡ã€ã«ãå©çšããããã ãå©çšã§ããªããã©ãŠã¶ãå€ããããäŸãã°ãã©ã°ã€ã³ãåã蟌ã¿ããå Žåã¯embedèŠçŽ ãªã©ä»ã®èŠçŽ ãåã蟌ãå¿
èŠããããobjectèŠçŽ ã«å²ãŸããéšåã¯ããã®ãªããžã§ã¯ããå©çšã§ããå Žåãã©ãŠã¶ã¯paramèŠçŽ ãšmapèŠçŽ ãé€ããå
šãŠã®èŠçŽ ãç¡èŠãããªããžã§ã¯ããå©çšã§ããªãå Žåã¯object, param, mapèŠçŽ ãç¡èŠããŠããã«ããä»ã®èŠçŽ ã®èšè¿°ãé©çšããã",
"title": "ãã«ãã¡ãã£ã¢ã®æ¿å
¥"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "Windows Media Playerã®ãã©ã°ã€ã³ãåã蟌ãã äŸã",
"title": "ãã«ãã¡ãã£ã¢ã®æ¿å
¥"
}
] | null | {{Pathnav|HTML|frame=1|small=1}}
== ç»åã®æ¿å
¥ ==
ç»åã®æ¿å
¥ã«ã¯imgèŠçŽ ã䜿çšãããäž»ã«äœ¿çšãããç»å圢åŒã¯[[w:JPEG|JPEG]]ã[[w:Graphics Interchange Format|GIF]]ã[[w:Portable Network Graphics|PNG]]ã§ããã
ããã¹ããã©ãŠã¶ã®ãããªç»å衚瀺ãåºæ¥ãªããã©ãŠã¶ã§ã¯imgèŠçŽ å
ã®altå±æ§ã§æå®ããæååã衚瀺ãããã
ããã¹ããã©ãŠã¶ãç»å衚瀺ãããªãããã«èšå®ããŠããã¯ã©ã€ã¢ã³ããžã®é
æ
®ãšããŠãaltå±æ§ã¯ä»£æ¿ãšãªãããè¡šçŸãèšèŒããããšãšãªã£ãŠãããåã«è£
食çãªæå³ã§çšããŠããç»åçã®ããã«ä»£æ¿ãã¹ãæåè¡šçŸããªãå Žåã¯ç©ºæååãæå® (<code>alt=""</code>) ããããšïŒçç¥ã¯åºæ¥ãªãïŒã
=== äœæäŸ ===
HTMLã®æ¬æå
ã«èšèŒããããããããã衚瀺ãããç»åïŒäžèšã³ãŒãã§ã¯ sample.png ïŒãæå
ã®ã³ã³ãã¥ãŒã¿ã«æºåããŠãããŠãHTMLãã¡ã€ã«ãšåããã©ã«ãã«å
¥ããŠããã
<syntaxhighlight lang="html4strict">
<img src="sample.png" width="200" height="100" alt="Wikibooks">
</syntaxhighlight>
ãªã width ã¯ç»åã®æšªå¹
ãheight ã¯ç»åã®çžŠå¹
ã§ããã
imgèŠçŽ ã«ã¯çµäºã¿ã°ããªãããã<code></img></code>ã¯èšèŒããªããsrcå±æ§ã«ã¯ã衚瀺ãããç»åãã¡ã€ã«ã®URIïŒã¢ãã¬ã¹ïŒãèšèŒãããçžå¯ŸURI(*) ã§æå®ããããšãå¯èœã§ãããåäžãµã€ãå
ã§ã®ç»åãåç
§ããå Žåã¯çžå¯ŸURIãçšããããšãå€ããwidthå±æ§ãšheightå±æ§ã§ç»åã®æšªå¹
ãšçžŠå¹
ãæå®ã§ããã
(*)<small>ãã®HTMLãã¡ã€ã«ãååšããå°ç¹ïŒäžè¬çã«ã¯ãã£ã¬ã¯ããªïŒããã®ç»åãã¡ã€ã«ã®äœçœ®ã«ãã瀺ãããURIã[[HTML/ãã€ããŒãªã³ã¯#çžå¯Ÿãã¹ãšçµ¶å¯Ÿãã¹]]ãåç
§ã</small>
== ã€ã¡ãŒãžããã ==
<div class="preoverflow">
<syntaxhighlight lang="html4strict"><img border="0" src="sample.png" usemap="#sample" alt="ãµã³ãã«" width="500" height="200">
<map name="sample">
<area href="http://ja.wikipedia.org/" shape="circle" alt="ãŠã£ãããã£ã¢" coords="100,100,50">
<area href="http://ja.wikinews.org/" shape="square" alt="ãŠã£ã¯ã·ã§ããªãŒ" coords="250,10,300,190">
<area href="http://ja.wikiquote.org/" shape="poly" alt="ãŠã£ãã¯ãªãŒã" coords="350,250,450,190,450,10,350,250">
</map>
</syntaxhighlight>
</div>
äžã®ã³ãŒãã§ã¯ãç»åã®äžéšãç¹å®é åã«ãªã³ã¯ãèšå®ãããã®é åãã¯ãªãã¯ããå Žåæå®ãããªã³ã¯å
ãžé£ã¹ãããã«ããŠããã
決ããŠãç»åã«åãåè§åœ¢ãªã©ãæç»ããããã§ã¯ãªãã®ã§ãæ··åããªãããã«ããªãããªã³ã¯ããé åã®äœçœ®ã¯ãäžèšã³ãŒãã®å Žåãã¢ãã¿ãŒã®èšå®ã倧ãããªã©ã«ãå¯ãããæ®éã®ããŒãããœã³ã³ã§èŠãå Žåãªããªã³ã¯é åã¯æšªã«äžŠãã§ããïŒç»åã«åãåè§åœ¢ã衚瀺ãããããã§ã¯ãªãã®ã§ãå®è¡ããŠãèŠããªãïŒãïŒãªããHTMLã§åãåè§ãªã©ã®åºæ¬å³åœ¢ãæç»ãããå ŽåãHTML5以éãªãã[[HTML/HTML5#canvasèŠçŽ ]]ãã§è§£èª¬ããæ¹æ³ã§åãªã©ãæç»ã§ãããïŒ
ãªã³ã¯é åã®äœçœ®ã«ããŠã¹ã«ãŒãœã«ãåããã°ãç»é¢å·Šäžãªã©ãããèŠãã°ããªã³ã¯å
ã®å称ã衚瀺ãããŠããã
imgèŠçŽ ã«usemapå±æ§ã§ãããããèšå®ãããã€ã¡ãŒãžãããã®ååãã#ãããåããšæå®ããããšã«ããå©çšå¯èœãšãªããã€ã¡ãŒãžããããèšå®ãããšãã¯mapå±æ§ã§ãããåã®å®çŸ©ãè¡ããareaèŠçŽ ã®spaceå±æ§ã§é åã®åœ¢ç¶ããcoordså±æ§ã§åº§æšãæå®ããã座æšæå®ã®éã座æšã¯ã³ã³ãã§åºåãã
ãªãã€ã¡ãŒãžãããã®å Žåãããã¹ããã©ãŠã¶ãç»åé衚瀺ãšãªã£ãŠãããã©ãŠã¶ã®ããã«ãªã³ã¯å
ãäœã§ãããã瀺ãaltå±æ§ãæå®ãã¹ãã§ããããã®å Žå空æååãæå®ãã¹ãã§ã¯ãªãã
{| class="wikitable"
|-
! style="width:3em;" | åœ¢ç¶ !! spaceå±æ§ã®å€ !! coordså±æ§ã®å€
|-
| å圢 || circle || äžå¿ã®åº§æš (x,y) ãšååŸã®å€ãé çªã«æå®ã
|-
| é·æ¹åœ¢ || square || å·Šäžã®åº§æš (x1,y1) ãšå³äžã®åº§æš (x2,y2) ãé çªã«æå®ã
|-
| å€è§åœ¢ || poly || å
šãŠã®è§ã®xãšyã®åº§æšãé çªã«æå®ãæå®æ°ã«éãã¯ãªãããå§ç¹ãšçµç¹ã¯å¿
ãåã座æšãæå®ããªããã°ãªããªãã
|}
== ãã«ãã¡ãã£ã¢ã®æ¿å
¥ ==
=== embedèŠçŽ ===
embedèŠçŽ ã¯ãã©ã°ã€ã³ãåã蟌ã¿ããã©ãŠã¶ãçŽæ¥åçã§ããªããã¡ã€ã«ãããŒãžã³ã³ãã³ãã®äžéšãšããŠå©çšãããã®ã§ããããã©ã°ã€ã³ã¯äºããã©ãŠã¶åŽã«ã€ã³ã¹ããŒã«ããŠããå¿
èŠãããã該åœã®ãã©ã°ã€ã³ããªããšã³ã³ãã³ããå©çšã§ããªãå Žåãããããã ãå
šãŠã®ç°å¢ã§ãã©ã°ã€ã³ãå©çšã§ããããã§ã¯ãªãããã<code><noembed></code>ã<code></noembed></code>å
ã§å©çšã§ããªãç°å¢ã§è¡šç€ºãããå
容ãæžãã¹ãã§ããããã®éããã©ã°ã€ã³ãæå¹ã«ããŠãã ããããããã©ã°ã€ã³ãã€ã³ã¹ããŒã«ãããŠããå¿
èŠããããŸãããªã©ãšããããã±ãŒã¹ããããããã®ãããªèšè¿°ã¯å¥œãŸãããªãã代æ¿çãªå
容ïŒäŸãã°Flashã«ããã¡ãã¥ãŒãåã蟌ãã§ããå Žåã¯HTMLã§ã®ã¡ãã¥ãŒïŒãèšè¿°ããŠããããå¿
èŠããªããã°äœãæžããªãã»ããè¯ãã
;src
:察象ãšãªããã¡ã€ã«ã®æå®ã
;type
:[[w:Multipurpose Internet Mail Extensions|MIME]]ã¿ã€ãã®æå®ããã¡ã€ã«ã¿ã€ããæ£ããæå®ããããšã§ãã©ãŠã¶åŽãé©åãªãã©ã°ã€ã³ãå²ãåœãŠãŠãªããžã§ã¯ããåçãããæå®ããªãã£ãå Žåã¯ãã©ãŠã¶äŸåãšãªãèªåçã«ãã©ã°ã€ã³ãå²ãåœãŠãããããèªåå²ãåœãŠã®ãã©ã°ã€ã³ãåçãããã¡ã€ã«ã«å¯Ÿå¿ããŠããªãå Žåãããã®ã§æå®ããããšãæãŸããã ããã
:ãã ãç¹å®ç°å¢ã§ããå©çšã§ããªããã©ã°ã€ã³ãæå®ãããšãŠãŒã¶ãŒã®å©äŸ¿æ§ãæãªãããšã«ãªãããããã®ãããªå Žåã¯äœããã®é
æ
®ãè¡ãã¹ãã§ããã
;width,height
:ãªããžã§ã¯ãã®å€§ãããpxåäœã§æå®ãããæå®ããªãã£ãå Žåãã©ã°ã€ã³äŸåã«ãªããããã®æ¹æ³ã ãšãã©ã°ã€ã³ãé©åãªãµã€ãºãèªèããã®ã«æéãæããŠããŸããããŒãžã®èªã¿èŸŒã¿éäžã§çªç¶åã蟌ã¿é åã®å€§ãããå€åããŠããŸãå Žåããããåã蟌ã¿é åã®å€§ãããçªç¶å€åããããšã§ãããããäžã«ããã³ã³ãã³ãã®ïŒç»é¢äžã«ïŒè¡šç€ºäœçœ®ããããŠããŸãããå¯èœã§ããã°æå®ããããšãæãŸããã
;autostart/autoplay
:æå®ãããé³æ¥œãåç»ã®åçèšå®ãtrueãŸãã¯1ã§èªååçãå®è¡ããfalseãŸãã¯0ã§èªååçãè¡ããªãããã©ã°ã€ã³ã®çš®é¡ã«ãã£ãŠã©ã¡ãã䜿ãããç°ãªããããé©åã«äœ¿ãåããå¿
èŠãããã
;loop
:ã«ãŒãåçã®èšå®ãloopå±æ§å€ã«å
·äœçãªæ°åãæå®ããå Žåãã®åæ°ã ãã«ãŒãåçãè¡ãããããã«ãªããtrueã-1ãªã©ãæå®ããå Žåã«ã¯ç¡éã«ãŒãåçãšãªãããŸããfalseã0ãæå®ããããšã§èªååçãè¡ããªãããã ãèªååçãç¡éã«ãŒãåçã¯å«ããããããšãå€ãã人ã«ãã£ãŠã¯èªåã§é³æ¥œããªã£ãç¬éã«ããŒãžãéãã人ããããå¿
èŠããªããã°ãªãã¹ãèªåçãªåçãè¡ãã¹ãã§ã¯ãªãã ããã
;hidden
:åã蟌ã¿ãªããžã§ã¯ããé ãèšå®ã1ã§ãªããžã§ã¯ããé ãã0ã§ãªããžã§ã¯ãã衚瀺ããïŒããã©ã«ãïŒããã ããããã¯åçãåæ¢ã®åãæ¿ãããŠãŒã¶ãŒã«è¡ãããªãããšã«ããªãã®ã§loopå±æ§åæ§äœ¿çšã«ã¯æ³šæãå¿
èŠã§ããã
;volume
:åã蟌ã¿ãªããžã§ã¯ãã®åçé³éãWindows Media Playerã®å Žå-1000ãæå°ã§0ãæ倧ãQuickTimeã®å Žå0ãæå°ã§100ãæ倧ãšèšã£ãå
·åã«ãã©ã°ã€ã³ã®çš®é¡ã§å€ã®èšå®æ¹æ³ãç°ãªãããããã©ã°ã€ã³ã®çš®é¡ãæå®ããªãã£ãå Žåãšãã§ããªãããšã«ãªã£ãŠããŸãã
;pluginpage
:ãªããžã§ã¯ããåçãããã©ã°ã€ã³ã®å
¥æå
URLã瀺ããæå®ãããªãã£ãå Žåã®åŠçã¯ãã©ãŠã¶äŸåã
=== objectèŠçŽ ===
objectã¯ãŠã§ãããŒãžã«æ§ã
ãªããŒã¿ãåã蟌ãããã®ãã®ã§ããããã©ãŠã¶ãçŽæ¥åŠçã§ãããã¡ã€ã«ã§ããã°ãã©ãŠã¶ãçŽæ¥åŠçãè¡ããçŽæ¥åŠçãè¡ããªãå Žåã¯ãã©ã°ã€ã³ãå©çšããŠãã¡ã€ã«ãå©çšããããã ãå©çšã§ããªããã©ãŠã¶ãå€ããããäŸãã°ãã©ã°ã€ã³ãåã蟌ã¿ããå Žåã¯embedèŠçŽ ãªã©ä»ã®èŠçŽ ãåã蟌ãå¿
èŠããããobjectèŠçŽ ã«å²ãŸããéšåã¯ããã®ãªããžã§ã¯ããå©çšã§ããå Žåãã©ãŠã¶ã¯paramèŠçŽ ãšmapèŠçŽ ãé€ããå
šãŠã®èŠçŽ ãç¡èŠãããªããžã§ã¯ããå©çšã§ããªãå Žåã¯object, param, mapèŠçŽ ãç¡èŠããŠããã«ããä»ã®èŠçŽ ã®èšè¿°ãé©çšããã
;data
:ãã¡ã€ã«ã®ããå Žæ
;type
:MIMEã¿ã€ãã®èšå®
;width,height
:ãªããžã§ã¯ãã®å€§ãã
;classid
:ãã©ã°ã€ã³ã®çš®é¡ãèå¥ããããã®ã³ãŒãããã©ã°ã€ã³ã®çš®é¡ããšã«æ±ºãŸã£ãå€ãå®ããããŠããã
;codebase
:ãŠãŒã¶ãŒã®ã³ã³ãã¥ãŒã¿ã«ã€ã³ã¹ããŒã«ãããŠããActiveXã®ããŒãžã§ã³ãæ€åºããããã«äœ¿çšããããURLã®æ«å°Ÿã«ãã<code>#Version=</code>ã¯æäœåäœç°å¢ã瀺ããã®ã§ãããŒãžã§ã³ãå€ãå Žåãã©ã°ã€ã³ãšããŠäœ¿çšãããŠããã¢ããªã±ãŒã·ã§ã³ãã¢ããããŒããå®è¡ããå Žåãããã
;paramèŠçŽ
:åã蟌ããªããžã§ã¯ãã«é¢ãã詳现èšå®ãè¡ãèŠçŽ ãEMBEDèŠçŽ ã«ãŠè©³çŽ°èšå®ãè¡ãåå±æ§ã®å€ããparamèŠçŽ ã®nameå±æ§å€ãšvalueå±æ§å€ã«å²ãåœãŠãããããå
šãŠã®èŠçŽ ãparamèŠçŽ ã«å²ãåœãŠãããããã§ã¯ãªããå²ãåœãŠåœ¢åŒãèšå®å¯èœãªãªãã·ã§ã³ã¯ãã©ã°ã€ã³ã®çš®é¡ã«ãã£ãŠç°ãªãã
=== ãã¡ã€ã«ã¿ã€ãã®æå®äŸ ===
{|class="wikitable"
!colspan="2"|[[w:Windows Media Player|Windows Media Player]]
|-
!ã¯ã©ã¹ID
||CLSID:22d6f312-b0f6-11d0-94ab-0080c74c7e95
|-
!Codebase
||<nowiki>http://activex.microsoft.com/activex/controls/mplayer/en/nsmp2inf.cab#Version=6,4,5,715</nowiki>
|-
!MIMEã¿ã€ã
||application/x-mplayer2
|-
!colspan="2"|[[w:Quick Time Player|Quick Time]]
|-
!ã¯ã©ã¹ID
||clsid:02BF25D5-8C17-4B23-BC80-D3488ABDDC6B
|-
!Codebase
||<nowiki>http://www.apple.com/qtactivex/qtplugin.cab</nowiki>
|-
!MIMEã¿ã€ã
||audio/quicktimeïŒãªãŒãã£ãªã®å ŽåïŒ,video/quicktimeïŒåç»ã®å ŽåïŒ
|-
!colspan="2"|[[w:Adobe Flash|Adobe Flash]]
|-
!ã¯ã©ã¹ID
||clsid:D27CDB6E-AE6D-11cf-96B8-444553540000
|-
!Codebase
||<nowiki>http://download.macromedia.com/pub/shockwave/cabs/flash/swflash.cab#version=8,0,22,0</nowiki>
|-
!MIMEã¿ã€ã
||application/x-shockwave-flash
|}
=== èšè¿°äŸ ===
Windows Media Playerã®ãã©ã°ã€ã³ãåã蟌ãã äŸã
<div class="preoverflow">
<syntaxhighlight lang="html4strict">
<object
classid="CLSID:22d6f312-b0f6-11d0-94ab-0080c74c7e95"
codebase="http://activex.microsoft.com/activex/controls/mplayer/en/nsmp2inf.cab#Version=6,4,5,715"
standby="Loading Microsoft Windows Media Player components..."
type="application/x-oleobject"
height="69"
width="300">
<param name="filename" value="http://www.dummyurl.com/file/music/example.mp3">
<param name="autostart" value="true">
<param name="showcontrols" value="true">
<param name="showstatusbar" value="true">
<param name="showpositioncontrols" value="false">
<param name="showtracker" value="true">
<param name="allowchangedisplaysize" value="false">
<param name="autosize" value="False">
<param name="volume" value="-500">
<param name="enablecontextmenu" value="false">
<embed
type="application/x-mplayer2"
src="http://www.dummyurl.com/file/music/example.mp3"
autostart="1"
showcontrols="1"
showpositioncontrols="0"
showtracker="1"
showstatusbar="1"
volume="-500"
enablecontextmenu="0"
nojava="true"
height="69"
width="300">
<noembed>
</object>
</syntaxhighlight>
</div>
[[en:HyperText Markup Language/Images]]
[[it:HTML/Immagini]]
[[Category:HTML|ããµããããš]] | null | 2021-11-01T03:49:56Z | [
"ãã³ãã¬ãŒã:Pathnav"
] | https://ja.wikibooks.org/wiki/HTML/%E3%82%AA%E3%83%96%E3%82%B8%E3%82%A7%E3%82%AF%E3%83%88 |
2,004 | HTML/æ¬æ | 段èœã§ããäºãè¡šãã«ã¯pèŠçŽ (Paragraphã®ç¥)ã䜿ããå€ãã®ãŠã§ããµã€ãã§ã¯æ®µèœã瀺ãããã«brèŠçŽ ãçšããŠãããããã®çšæ³ã¯HTMLã®æ£ããæžãæ¹ã§ãªããé£ç¶ããbrèŠçŽ ã¯äžéšã®ãã©ãŠã¶ã§ã¯ãŸãšããŠäžã€ã®æ¹è¡ãšããŠè¡šç€ºãããŠããŸãã
Strictã§ã¯bodyèŠçŽ çŽäžã«ãããã¯èŠçŽ ã眮ããŠãã®äžã«æ¬æãæžãå¿
èŠããããbodyèŠçŽ çŽäžã«æ¬æããã¹ããæžããŠã¯ãªããªãã
ãšãã«PèŠçŽ ã«ã€ããŠã䜿ãæ¹ãã https://ja.wikiversity.org/wiki/Topic:HTML ã«æžããŠããŸãã䜿ãæ¹ã«ã€ããŠåå ããŠæ¬²ããã§ã
家ã«åž°ããšã楜ãã¿ã«ããŠããããã€ãé£ã¹ãããŠãããä»æ¹ãç¡ãã®ã§PCãèµ·åãããã®ãŠã§ãæ¥èšãæŽæ°ããŠããã
ä»æ¥ã¯å
æ¥è²·ã£ãè³æãåèã«ãã€ã€ããŠã£ãããã£ã¢ã«é
ç®ãäžã€æçš¿ããããšæããããŠäœæéæããã ãããã
èŠåºãã§ããäºãè¡šãã«ã¯h1~h6èŠçŽ (hã¯Headingã®ç¥)ã䜿ããããäžäœã®èŠåºãã»ã©ãhã®åŸã«ç¶ãæ°åã倧ãããªããäžè¬çãªãã©ãŠã¶ã§ã¯æåãµã€ãºãæåã®å€ªããå€åãããããã®èŠçŽ ã倧æåã倪åç®çã§äœ¿çšããŠã¯ãªããªãã
æ¥æ¬åœ(ã«ã»ããããã«ã£ãœããã)ã¯ãã¢ãžã¢(ãŠãŒã©ã·ã¢å€§éž)ã®æ±æ¹ã«ãã島åœã§ããã
åã€ã®å€§ããªå³¶ãåæµ·éãæ¬å·ãååœãä¹å·ãšãå島å島ãå°ç¬ å諞島ãççå島ãªã©åšèŸºã®å°å³¶ãããªãå島(島匧)ãé åã®äžå¿ããªãã
倧åã®å°åã¯æž©åž¯ã«å±ãããåæ¹ã®è«žå³¶ã¯äºç±åž¯ãåæ¹ã¯äºå¯åž¯çæ°åã瀺ããæµ·æŽæ§æ°åã ããã¢ã³ã¹ãŒã³ã®åœ±é¿ãåããå¯æã®å·®ã¯å€§ããã
äžèšäºäŸã®æç« ã¯Wikipediaã«ããæ¥æ¬ã®é
ç®ã®èšè¿°ãå©çšããŠããã
åŒçšã§ããäºãè¡šãã«ã¯ãblockquoteèŠçŽ ãããã¯qèŠçŽ (Quotationã®ç¥)ã䜿ããblockquoteèŠçŽ ã¯ãããã¯ã¬ãã«ã®åŒçšã«äœ¿çšããqèŠçŽ ã¯ã€ã³ã©ã€ã³ã§ã®åŒçšã«äœ¿çšãããäž¡èŠçŽ ã«ã€ããŠãåºå
žã®URIãè¡šããã®ãšããŠciteå±æ§ãããã®ã¿ã€ãã«ãè¡šããã®ãšããŠtitleå±æ§ãå©çšã§ããããŸããåºå
žãããã¯åç
§å
ã瀺ããã®ãšããŠciteèŠçŽ ãããããã®èŠçŽ ã§å²ã£ãéšåãåºå
žã§ããããšã瀺ãããã«äœ¿çšããã
äžè¬çãªãã©ãŠã¶ã§ã¯ãblockquoteèŠçŽ ã¯å·Šå³ã«ã€ã³ãã³ããããç¶æ
ã§ãqèŠçŽ ã¯åŒçšç¬Šã«æ¬ãããç¶æ
(äžéšç°å¢æªå¯Ÿå¿)ã§ãciteèŠçŽ ã¯æäœã§è¡šç€ºãããããªããå·Šå³ã«ç©ºçœãåãããã«blockquoteèŠçŽ ã䜿çšããã±ãŒã¹ãããããããã¯äžé©åã§ããç°å¢ã«ãã£ãŠã¯ãã®å
容ãåŒçšã§ãããšèªèãããããªããã¹ã¿ã€ã«ã·ãŒãã䜿ã£ãŠç©ºçœãåãããšãæãŸããã
ãŠã£ãã¡ãã£ã¢è²¡å£ã«ã€ããŠããŠã£ãããã£ã¢ã§ã¯ã以äžã®æ§ã«èª¬æããŠããã
ãŠã£ãã¡ãã£ã¢è²¡å£ (Wikimedia Foundation Inc.) ã¯ãŠã£ãããã£ã¢ãéå¶ãããã®æ¯äœãšãªãå£äœã§ããã ç±³åœãããªãå·æ³ã«ããéå¶å©çµç¹ã§ããããŠã£ãããã£ã¢ã®åµç«è
ã®äžäººã§ããããžããŒã»ãŠã§ãŒã«ãºã«ãã£ãŠèšç«ãããã 財å£å称ã®ãŠã£ãã¡ãã£ã¢ã¯è±èªçãŠã£ãããã£ã¢ã®åå è
ã·ã§ã«ãã³ã»ã©ã³ããã³ã®åœåã«ããããŠã£ããšãã«ãã¡ãã£ã¢ããé èªãããã
å財å£ã®ç®çã¯ããŠã£ããçšãããªãŒãã³ã³ã³ãã³ãã®ç¥çè³æºãéçºãããããžã§ã¯ãã®ä¿é²ãããã³ãã®è³æºãç¡æãåºåãªãã§åºãå
¬è¡ã«æäŸããããšã«ããã
åé
ç®ã«ããè¥å¹Žå±€åãã®æè²ã³ã³ãã³ãããŠã£ããžã¥ãã¢ãã®äœæã«ã¯èå³ãåŒããšããã§ããã
匷調ãè¡šãã«ã¯emèŠçŽ (EMphasisã®ç¥)ãstrongèŠçŽ ã䜿ããstrongèŠçŽ ã®æ¹ããã匷ã匷調ãè¡šããäžè¬çãªãã©ãŠã¶ã§ã¯emèŠçŽ ã¯æäœåã§ãstrongèŠçŽ ã¯å€ªåã§è¡šç€ºããããäžéšã®é³å£°ãã©ãŠã¶ã¯ãã®èŠçŽ ãèªèãã匷調é³å£°ã§ããã¹ããèªã¿äžããå Žåãããã
åã¯
æãããã寧ãç±ãã
ããã¹ããã¡ã€ã«ã§ã®æ¹è¡ã¯è¡šç€ºã«ã¯ã»ãŒåœ±é¿ããªã(ãã©ãŠã¶ã«ãã£ãŠã¯ã¹ããŒã¹ãéãããšããã)ã匷å¶çã«æ¹è¡ãããããšãã«ã¯<br>ã䜿ãããªããXHTMLã«ãããŠã¯ã<br />(XHTML 1.0ã§ã¯brãš/ã®éã«åè§ã¹ããŒã¹ãå
¥ããããšãæšå¥šãããŠããããå¿
é ã§ã¯ãªã)ãšããããã«å®ããããŠãããHTMLã§ã<br />ã䜿ãããšã¯åºæ¥ãããææ³äžæ£ããæžãæ¹ã§ã¯ãªãã®ã§èŠæ Œã«æ²¿ã£ãHTMLãæžããããšãã¯æ³šæãããã
æ¹è¡ããŸãã ã¯ãããã?
ããããªããšã ã!
XHTMLã®ãšãã¯ããã£ã¡ã§ã ã!
divèŠçŽ ã¯ãããã¯ã¬ãã«èŠçŽ ãspanèŠçŽ ã¯ã€ã³ã©ã€ã³èŠçŽ ã§ãããããã以å€ã®æå³ã¯ãªãåäœã§æå®ããŠãããã©ãŠã¶ãç¹å¥ãªæ±ããè¡ã£ããã衚瀺ãç¹å¥å€åãããããããšã¯ãªã(ãã ãdivèŠçŽ ã¯ãããã¯ã¬ãã«èŠçŽ ã§ãããååŸã«æ¹è¡ãå
¥ã)ãidå±æ§ãclasså±æ§ã䜿ã£ãŠã¹ã¿ã€ã«ã·ãŒããé©çšããããlangå±æ§ãªã©ãæå®ãããäž»ã«ä»ã®èŠçŽ ã§ã¯ä»£çšã§ããªã(ä»ã®èŠçŽ ãçšãããšç¯å²å
ã«äžå¿
èŠãªæ
å ±ãå®çŸ©ããŠããŸã)ããšãè¡ãæ±çšèŠçŽ ãšããŠçšããããã
HTMLã§åè§è±æ°ã® < ã > ãšãã£ãå¶åŸ¡çšã®æåãã®ãã®ã衚瀺ãããå Žåã«ã¯ã
< ãšè¡šç€ºããããªã < ãšå
¥åããã
> ãšè¡šç€ºããããªã > ãšå
¥åããã
ltãgtã®çŽåŸã®èšå·ã¯ã»ãã³ãã³(;)ã§ããã(ã³ãã³(:)ã§ã¯ãªãã)
HTMLã«éãããããã°ã©ãã³ã°ãé¡äŒŒã®ã³ãŒãã£ã³ã°ã«ãããŠããã®ããã«å¶åŸ¡æåãã®ãã®ãå
¥åããããã®å
¥åæ¹æ³ã®ããšãããšã¹ã±ãŒãã·ãŒã±ã³ã¹ããšããã
HTMLã®ãšã¹ã±ãŒãã·ãŒã±ã³ã¹ã«ã€ããŠã¯ãçš®é¡ãå€ãã®ã§ã詳ããã¯ããããªã©ã§æ€çŽ¢ããŠãããããã
ãªãã掟ççãªè©±é¡ã ãã < ãšwebããŒãžã§è¡šç€ºãããå Žåã &lt;ãšHTMLã«å
¥åããã
| [
{
"paragraph_id": 0,
"tag": "p",
"text": "段èœã§ããäºãè¡šãã«ã¯pèŠçŽ (Paragraphã®ç¥)ã䜿ããå€ãã®ãŠã§ããµã€ãã§ã¯æ®µèœã瀺ãããã«brèŠçŽ ãçšããŠãããããã®çšæ³ã¯HTMLã®æ£ããæžãæ¹ã§ãªããé£ç¶ããbrèŠçŽ ã¯äžéšã®ãã©ãŠã¶ã§ã¯ãŸãšããŠäžã€ã®æ¹è¡ãšããŠè¡šç€ºãããŠããŸãã",
"title": "段èœ"
},
{
"paragraph_id": 1,
"tag": "p",
"text": "Strictã§ã¯bodyèŠçŽ çŽäžã«ãããã¯èŠçŽ ã眮ããŠãã®äžã«æ¬æãæžãå¿
èŠããããbodyèŠçŽ çŽäžã«æ¬æããã¹ããæžããŠã¯ãªããªãã",
"title": "段èœ"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ãšãã«PèŠçŽ ã«ã€ããŠã䜿ãæ¹ãã https://ja.wikiversity.org/wiki/Topic:HTML ã«æžããŠããŸãã䜿ãæ¹ã«ã€ããŠåå ããŠæ¬²ããã§ã",
"title": "段èœ"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "家ã«åž°ããšã楜ãã¿ã«ããŠããããã€ãé£ã¹ãããŠãããä»æ¹ãç¡ãã®ã§PCãèµ·åãããã®ãŠã§ãæ¥èšãæŽæ°ããŠããã",
"title": "段èœ"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ä»æ¥ã¯å
æ¥è²·ã£ãè³æãåèã«ãã€ã€ããŠã£ãããã£ã¢ã«é
ç®ãäžã€æçš¿ããããšæããããŠäœæéæããã ãããã",
"title": "段èœ"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "èŠåºãã§ããäºãè¡šãã«ã¯h1~h6èŠçŽ (hã¯Headingã®ç¥)ã䜿ããããäžäœã®èŠåºãã»ã©ãhã®åŸã«ç¶ãæ°åã倧ãããªããäžè¬çãªãã©ãŠã¶ã§ã¯æåãµã€ãºãæåã®å€ªããå€åãããããã®èŠçŽ ã倧æåã倪åç®çã§äœ¿çšããŠã¯ãªããªãã",
"title": "èŠåºã"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "æ¥æ¬åœ(ã«ã»ããããã«ã£ãœããã)ã¯ãã¢ãžã¢(ãŠãŒã©ã·ã¢å€§éž)ã®æ±æ¹ã«ãã島åœã§ããã",
"title": "èŠåºã"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "åã€ã®å€§ããªå³¶ãåæµ·éãæ¬å·ãååœãä¹å·ãšãå島å島ãå°ç¬ å諞島ãççå島ãªã©åšèŸºã®å°å³¶ãããªãå島(島匧)ãé åã®äžå¿ããªãã",
"title": "èŠåºã"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "倧åã®å°åã¯æž©åž¯ã«å±ãããåæ¹ã®è«žå³¶ã¯äºç±åž¯ãåæ¹ã¯äºå¯åž¯çæ°åã瀺ããæµ·æŽæ§æ°åã ããã¢ã³ã¹ãŒã³ã®åœ±é¿ãåããå¯æã®å·®ã¯å€§ããã",
"title": "èŠåºã"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "äžèšäºäŸã®æç« ã¯Wikipediaã«ããæ¥æ¬ã®é
ç®ã®èšè¿°ãå©çšããŠããã",
"title": "èŠåºã"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "åŒçšã§ããäºãè¡šãã«ã¯ãblockquoteèŠçŽ ãããã¯qèŠçŽ (Quotationã®ç¥)ã䜿ããblockquoteèŠçŽ ã¯ãããã¯ã¬ãã«ã®åŒçšã«äœ¿çšããqèŠçŽ ã¯ã€ã³ã©ã€ã³ã§ã®åŒçšã«äœ¿çšãããäž¡èŠçŽ ã«ã€ããŠãåºå
žã®URIãè¡šããã®ãšããŠciteå±æ§ãããã®ã¿ã€ãã«ãè¡šããã®ãšããŠtitleå±æ§ãå©çšã§ããããŸããåºå
žãããã¯åç
§å
ã瀺ããã®ãšããŠciteèŠçŽ ãããããã®èŠçŽ ã§å²ã£ãéšåãåºå
žã§ããããšã瀺ãããã«äœ¿çšããã",
"title": "åŒçšãšåºå
ž"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "äžè¬çãªãã©ãŠã¶ã§ã¯ãblockquoteèŠçŽ ã¯å·Šå³ã«ã€ã³ãã³ããããç¶æ
ã§ãqèŠçŽ ã¯åŒçšç¬Šã«æ¬ãããç¶æ
(äžéšç°å¢æªå¯Ÿå¿)ã§ãciteèŠçŽ ã¯æäœã§è¡šç€ºãããããªããå·Šå³ã«ç©ºçœãåãããã«blockquoteèŠçŽ ã䜿çšããã±ãŒã¹ãããããããã¯äžé©åã§ããç°å¢ã«ãã£ãŠã¯ãã®å
容ãåŒçšã§ãããšèªèãããããªããã¹ã¿ã€ã«ã·ãŒãã䜿ã£ãŠç©ºçœãåãããšãæãŸããã",
"title": "åŒçšãšåºå
ž"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "ãŠã£ãã¡ãã£ã¢è²¡å£ã«ã€ããŠããŠã£ãããã£ã¢ã§ã¯ã以äžã®æ§ã«èª¬æããŠããã",
"title": "匷調"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ãŠã£ãã¡ãã£ã¢è²¡å£ (Wikimedia Foundation Inc.) ã¯ãŠã£ãããã£ã¢ãéå¶ãããã®æ¯äœãšãªãå£äœã§ããã ç±³åœãããªãå·æ³ã«ããéå¶å©çµç¹ã§ããããŠã£ãããã£ã¢ã®åµç«è
ã®äžäººã§ããããžããŒã»ãŠã§ãŒã«ãºã«ãã£ãŠèšç«ãããã 財å£å称ã®ãŠã£ãã¡ãã£ã¢ã¯è±èªçãŠã£ãããã£ã¢ã®åå è
ã·ã§ã«ãã³ã»ã©ã³ããã³ã®åœåã«ããããŠã£ããšãã«ãã¡ãã£ã¢ããé èªãããã",
"title": "匷調"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "å財å£ã®ç®çã¯ããŠã£ããçšãããªãŒãã³ã³ã³ãã³ãã®ç¥çè³æºãéçºãããããžã§ã¯ãã®ä¿é²ãããã³ãã®è³æºãç¡æãåºåãªãã§åºãå
¬è¡ã«æäŸããããšã«ããã",
"title": "匷調"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "åé
ç®ã«ããè¥å¹Žå±€åãã®æè²ã³ã³ãã³ãããŠã£ããžã¥ãã¢ãã®äœæã«ã¯èå³ãåŒããšããã§ããã",
"title": "匷調"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "匷調ãè¡šãã«ã¯emèŠçŽ (EMphasisã®ç¥)ãstrongèŠçŽ ã䜿ããstrongèŠçŽ ã®æ¹ããã匷ã匷調ãè¡šããäžè¬çãªãã©ãŠã¶ã§ã¯emèŠçŽ ã¯æäœåã§ãstrongèŠçŽ ã¯å€ªåã§è¡šç€ºããããäžéšã®é³å£°ãã©ãŠã¶ã¯ãã®èŠçŽ ãèªèãã匷調é³å£°ã§ããã¹ããèªã¿äžããå Žåãããã",
"title": "匷調"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "åã¯",
"title": "匷調"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "æãããã寧ãç±ãã",
"title": "匷調"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "ããã¹ããã¡ã€ã«ã§ã®æ¹è¡ã¯è¡šç€ºã«ã¯ã»ãŒåœ±é¿ããªã(ãã©ãŠã¶ã«ãã£ãŠã¯ã¹ããŒã¹ãéãããšããã)ã匷å¶çã«æ¹è¡ãããããšãã«ã¯<br>ã䜿ãããªããXHTMLã«ãããŠã¯ã<br />(XHTML 1.0ã§ã¯brãš/ã®éã«åè§ã¹ããŒã¹ãå
¥ããããšãæšå¥šãããŠããããå¿
é ã§ã¯ãªã)ãšããããã«å®ããããŠãããHTMLã§ã<br />ã䜿ãããšã¯åºæ¥ãããææ³äžæ£ããæžãæ¹ã§ã¯ãªãã®ã§èŠæ Œã«æ²¿ã£ãHTMLãæžããããšãã¯æ³šæãããã",
"title": "匷調"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "æ¹è¡ããŸãã ã¯ãããã?",
"title": "匷調"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "ããããªããšã ã!",
"title": "匷調"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "XHTMLã®ãšãã¯ããã£ã¡ã§ã ã!",
"title": "匷調"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "divèŠçŽ ã¯ãããã¯ã¬ãã«èŠçŽ ãspanèŠçŽ ã¯ã€ã³ã©ã€ã³èŠçŽ ã§ãããããã以å€ã®æå³ã¯ãªãåäœã§æå®ããŠãããã©ãŠã¶ãç¹å¥ãªæ±ããè¡ã£ããã衚瀺ãç¹å¥å€åãããããããšã¯ãªã(ãã ãdivèŠçŽ ã¯ãããã¯ã¬ãã«èŠçŽ ã§ãããååŸã«æ¹è¡ãå
¥ã)ãidå±æ§ãclasså±æ§ã䜿ã£ãŠã¹ã¿ã€ã«ã·ãŒããé©çšããããlangå±æ§ãªã©ãæå®ãããäž»ã«ä»ã®èŠçŽ ã§ã¯ä»£çšã§ããªã(ä»ã®èŠçŽ ãçšãããšç¯å²å
ã«äžå¿
èŠãªæ
å ±ãå®çŸ©ããŠããŸã)ããšãè¡ãæ±çšèŠçŽ ãšããŠçšããããã",
"title": "匷調"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "",
"title": "匷調"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "HTMLã§åè§è±æ°ã® < ã > ãšãã£ãå¶åŸ¡çšã®æåãã®ãã®ã衚瀺ãããå Žåã«ã¯ã",
"title": "匷調"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "< ãšè¡šç€ºããããªã < ãšå
¥åããã",
"title": "匷調"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "> ãšè¡šç€ºããããªã > ãšå
¥åããã",
"title": "匷調"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "ltãgtã®çŽåŸã®èšå·ã¯ã»ãã³ãã³(;)ã§ããã(ã³ãã³(:)ã§ã¯ãªãã)",
"title": "匷調"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "HTMLã«éãããããã°ã©ãã³ã°ãé¡äŒŒã®ã³ãŒãã£ã³ã°ã«ãããŠããã®ããã«å¶åŸ¡æåãã®ãã®ãå
¥åããããã®å
¥åæ¹æ³ã®ããšãããšã¹ã±ãŒãã·ãŒã±ã³ã¹ããšããã",
"title": "匷調"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "HTMLã®ãšã¹ã±ãŒãã·ãŒã±ã³ã¹ã«ã€ããŠã¯ãçš®é¡ãå€ãã®ã§ã詳ããã¯ããããªã©ã§æ€çŽ¢ããŠãããããã",
"title": "匷調"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "ãªãã掟ççãªè©±é¡ã ãã < ãšwebããŒãžã§è¡šç€ºãããå Žåã &lt;ãšHTMLã«å
¥åããã",
"title": "匷調"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "",
"title": "匷調"
}
] | null | <div class="toclimit-2">__TOC__</div>
== æ®µèœ ==
段èœã§ããäºãè¡šãã«ã¯pèŠçŽ ïŒParagraphã®ç¥ïŒã䜿ããå€ãã®ãŠã§ããµã€ãã§ã¯æ®µèœã瀺ãããã«brèŠçŽ ãçšããŠãããããã®çšæ³ã¯HTMLã®æ£ããæžãæ¹ã§ãªããé£ç¶ããbrèŠçŽ ã¯äžéšã®ãã©ãŠã¶ã§ã¯ãŸãšããŠäžã€ã®æ¹è¡ãšããŠè¡šç€ºãããŠããŸãã
Strictã§ã¯bodyèŠçŽ çŽäžã«ãããã¯èŠçŽ ã眮ããŠãã®äžã«æ¬æãæžãå¿
èŠããããbodyèŠçŽ çŽäžã«æ¬æããã¹ããæžããŠã¯ãªããªãã
=== èšè¿°äŸ ===
<syntaxhighlight lang="html4strict">
<p>家ã«åž°ããšã楜ãã¿ã«ããŠããããã€ãé£ã¹ãããŠãããä»æ¹ãç¡ãã®ã§PCãèµ·åãããã®ãŠã§ãæ¥èšãæŽæ°ããŠããã</p>
<p>ä»æ¥ã¯å
æ¥è²·ã£ãè³æãåèã«ãã€ã€ããŠã£ãããã£ã¢ã«é
ç®ãäžã€æçš¿ããããšæããããŠäœæéæããã ãããã</p>
</syntaxhighlight>
ãšãã«PèŠçŽ ã«ã€ããŠã䜿ãæ¹ãã
https://ja.wikiversity.org/wiki/Topic:HTML
ã«æžããŠããŸãã䜿ãæ¹ã«ã€ããŠåå ããŠæ¬²ããã§ã
=== è¡šç€ºäŸ ===
<div style="border:#000 1px dashed;padding:1em">
<p>家ã«åž°ããšã楜ãã¿ã«ããŠããããã€ãé£ã¹ãããŠãããä»æ¹ãç¡ãã®ã§PCãèµ·åãããã®ãŠã§ãæ¥èšãæŽæ°ããŠããã</p>
<p>ä»æ¥ã¯å
æ¥è²·ã£ãè³æãåèã«ãã€ã€ããŠã£ãããã£ã¢ã«é
ç®ãäžã€æçš¿ããããšæããããŠäœæéæããã ãããã</p>
</div>
== èŠåºã ==
èŠåºãã§ããäºãè¡šãã«ã¯h1ïœh6èŠçŽ ïŒhã¯Headingã®ç¥ïŒã䜿ããããäžäœã®èŠåºãã»ã©ãhã®åŸã«ç¶ãæ°åã倧ãããªããäžè¬çãªãã©ãŠã¶ã§ã¯æåãµã€ãºãæåã®å€ªããå€åãããããã®èŠçŽ ã倧æåã倪åç®çã§äœ¿çšããŠã¯ãªããªãã
=== èšè¿°äŸ ===
<div class="preoverflow">
<syntaxhighlight lang="html4strict">
<h1>æ¥æ¬</h1>
<p>æ¥æ¬åœïŒã«ã»ããããã«ã£ãœãããïŒã¯ãã¢ãžã¢ïŒãŠãŒã©ã·ã¢å€§éžïŒã®æ±æ¹ã«ãã島åœã§ããã</p>
<h2>å°ç</h2>
<p>åã€ã®å€§ããªå³¶ãåæµ·éãæ¬å·ãååœãä¹å·ãšãå島å島ãå°ç¬ å諞島ãççå島ãªã©åšèŸºã®å°å³¶ãããªãå島ïŒå³¶åŒ§ïŒãé åã®äžå¿ããªãã</p>
<h3>æ°å</h3>
<p>倧åã®å°åã¯æž©åž¯ã«å±ãããåæ¹ã®è«žå³¶ã¯äºç±åž¯ãåæ¹ã¯äºå¯åž¯çæ°åã瀺ããæµ·æŽæ§æ°åã ããã¢ã³ã¹ãŒã³ã®åœ±é¿ãåããå¯æã®å·®ã¯å€§ããã</p>
</syntaxhighlight>
</div>
=== è¡šç€ºäŸ ===
<div style="border:#000 1px dashed;padding:1em">
<div style="color: black;background: none;font-weight: normal;margin: 0 0 1em 0;padding-top: .5em;padding-bottom: .17em;font-size:188%;">æ¥æ¬</div>
æ¥æ¬åœïŒã«ã»ããããã«ã£ãœãããïŒã¯ãã¢ãžã¢ïŒãŠãŒã©ã·ã¢å€§éžïŒã®æ±æ¹ã«ãã島åœã§ããã
<div style="color: black; background: none; font-weight: bold; margin: 0 0 1em 0; padding-top: .5em; padding-bottom: .17em; font-size: 150%;">å°ç</div>
åã€ã®å€§ããªå³¶ãåæµ·éãæ¬å·ãååœãä¹å·ãšãå島å島ãå°ç¬ å諞島ãççå島ãªã©åšèŸºã®å°å³¶ãããªãå島ïŒå³¶åŒ§ïŒãé åã®äžå¿ããªãã
<div style="color: black; background: none; font-weight: bold; margin: 0 0 1em 0; padding-top: .5em; padding-bottom: .17em; font-size: 120%;">æ°å</div>
倧åã®å°åã¯æž©åž¯ã«å±ãããåæ¹ã®è«žå³¶ã¯äºç±åž¯ãåæ¹ã¯äºå¯åž¯çæ°åã瀺ããæµ·æŽæ§æ°åã ããã¢ã³ã¹ãŒã³ã®åœ±é¿ãåããå¯æã®å·®ã¯å€§ããã
</div>
äžèšäºäŸã®æç« ã¯Wikipediaã«ãã[[w:æ¥æ¬|æ¥æ¬]]ã®é
ç®ã®èšè¿°ãå©çšããŠããã
== åŒçšãšåºå
ž ==
åŒçšã§ããäºãè¡šãã«ã¯ãblockquoteèŠçŽ ãããã¯qèŠçŽ ïŒQuotationã®ç¥ïŒã䜿ããblockquoteèŠçŽ ã¯ãããã¯ã¬ãã«ã®åŒçšã«äœ¿çšããqèŠçŽ ã¯ã€ã³ã©ã€ã³ã§ã®åŒçšã«äœ¿çšãããäž¡èŠçŽ ã«ã€ããŠãåºå
žã®URIãè¡šããã®ãšããŠciteå±æ§ãããã®ã¿ã€ãã«ãè¡šããã®ãšããŠtitleå±æ§ãå©çšã§ããããŸããåºå
žãããã¯åç
§å
ã瀺ããã®ãšããŠciteèŠçŽ ãããããã®èŠçŽ ã§å²ã£ãéšåãåºå
žã§ããããšã瀺ãããã«äœ¿çšããã
äžè¬çãªãã©ãŠã¶ã§ã¯ãblockquoteèŠçŽ ã¯å·Šå³ã«ã€ã³ãã³ããããç¶æ
ã§ãqèŠçŽ ã¯åŒçšç¬Šã«æ¬ãããç¶æ
ïŒäžéšç°å¢æªå¯Ÿå¿ïŒã§ãciteèŠçŽ ã¯æäœã§è¡šç€ºãããããªããå·Šå³ã«ç©ºçœãåãããã«blockquoteèŠçŽ ã䜿çšããã±ãŒã¹ãããããããã¯äžé©åã§ããç°å¢ã«ãã£ãŠã¯ãã®å
容ãåŒçšã§ãããšèªèãããããªããã¹ã¿ã€ã«ã·ãŒãã䜿ã£ãŠç©ºçœãåãããšãæãŸããã
=== èšè¿°äŸ ===
<div class="preoverflow">
<syntaxhighlight lang="html4strict">
<p>ãŠã£ãã¡ãã£ã¢è²¡å£ã«ã€ããŠã<cite>ãŠã£ãããã£ã¢</cite>ã§ã¯ã以äžã®æ§ã«èª¬æããŠããã</p>
<blockquote cite="http://ja.wikipedia.org/wiki/ãŠã£ãã¡ãã£ã¢" title="ãŠã£ãã¡ãã£ã¢ - Wikipedia">
<p>ãŠã£ãã¡ãã£ã¢è²¡å£ (Wikimedia Foundation Inc.) ã¯ãŠã£ãããã£ã¢ãéå¶ãããã®æ¯äœãšãªãå£äœã§ããã
ç±³åœãããªãå·æ³ã«ããéå¶å©çµç¹ã§ããããŠã£ãããã£ã¢ã®åµç«è
ã®äžäººã§ããããžããŒã»ãŠã§ãŒã«ãºã«ãã£ãŠèšç«ãããã
財å£å称ã®ãŠã£ãã¡ãã£ã¢ã¯è±èªçãŠã£ãããã£ã¢ã®åå è
ã·ã§ã«ãã³ã»ã©ã³ããã³ã®åœåã«ããããŠã£ããšãã«ãã¡ãã£ã¢ããé èªãããã</p>
<p>å財å£ã®ç®çã¯ããŠã£ããçšãããªãŒãã³ã³ã³ãã³ãã®ç¥çè³æºãéçºãããããžã§ã¯ãã®ä¿é²ãããã³ãã®è³æºãç¡æãåºåãªãã§åºãå
¬è¡ã«æäŸããããšã«ããã</p>
</blockquote>
<p>åé
ç®ã«ããè¥å¹Žå±€åãã®æè²ã³ã³ãã³ãããŠã£ããžã¥ãã¢ãã®äœæã«ã¯èå³ãåŒããšããã§ããã</p>
</syntaxhighlight>
</pre>
=== è¡šç€ºäŸ ===
<div style="border:#000 1px dashed;padding:1em">
<p>ãŠã£ãã¡ãã£ã¢è²¡å£ã«ã€ããŠã<cite>ãŠã£ãããã£ã¢</cite>ã§ã¯ã以äžã®æ§ã«èª¬æããŠããã</p>
<blockquote cite="http://ja.wikipedia.org/wiki/ãŠã£ãã¡ãã£ã¢" title="ãŠã£ãã¡ãã£ã¢ - Wikipedia">
<p>ãŠã£ãã¡ãã£ã¢è²¡å£ (Wikimedia Foundation Inc.) ã¯ãŠã£ãããã£ã¢ãéå¶ãããã®æ¯äœãšãªãå£äœã§ããã
ç±³åœãããªãå·æ³ã«ããéå¶å©çµç¹ã§ããããŠã£ãããã£ã¢ã®åµç«è
ã®äžäººã§ããããžããŒã»ãŠã§ãŒã«ãºã«ãã£ãŠèšç«ãããã
財å£å称ã®ãŠã£ãã¡ãã£ã¢ã¯è±èªçãŠã£ãããã£ã¢ã®åå è
ã·ã§ã«ãã³ã»ã©ã³ããã³ã®åœåã«ããããŠã£ããšãã«ãã¡ãã£ã¢ããé èªãããã</p>
<p>å財å£ã®ç®çã¯ããŠã£ããçšãããªãŒãã³ã³ã³ãã³ãã®ç¥çè³æºãéçºãããããžã§ã¯ãã®ä¿é²ãããã³ãã®è³æºãç¡æãåºåãªãã§åºãå
¬è¡ã«æäŸããããšã«ããã</p>
</blockquote>
<p>åé
ç®ã«ããè¥å¹Žå±€åãã®æè²ã³ã³ãã³ãããŠã£ããžã¥ãã¢ãã®äœæã«ã¯èå³ãåŒããšããã§ããã</p>
</div>
== 匷調 ==
匷調ãè¡šãã«ã¯emèŠçŽ ïŒEMphasisã®ç¥ïŒãstrongèŠçŽ ã䜿ããstrongèŠçŽ ã®æ¹ããã匷ã匷調ãè¡šããäžè¬çãªãã©ãŠã¶ã§ã¯emèŠçŽ ã¯æäœåã§ãstrongèŠçŽ ã¯å€ªåã§è¡šç€ºããããäžéšã®é³å£°ãã©ãŠã¶ã¯ãã®èŠçŽ ãèªèãã匷調é³å£°ã§ããã¹ããèªã¿äžããå Žåãããã
=== èšè¿°äŸ ===
<syntaxhighlight lang="html4strict">
<p><em>æã</em>ããã寧ã<strong>ç±ã</strong>ã</p>
</syntaxhighlight>
åã¯
<syntaxhighlight lang="html4strict">
<p><em>æã</em>ããã寧ã<b>ç±ã</b>ã</p>
</syntaxhighlight>
=== è¡šç€ºäŸ ===
<div style="border:#000 1px dashed;padding:1em">
<p><em>æã</em>ããã寧ã<strong>ç±ã</strong>ã</p>
</div>
== æ±çšå±æ§ ==
;lang
:ãã®èŠçŽ å
ã§ã©ã®èšèªã䜿çšãããŠãããã瀺ãã
;id
:ãã®èŠçŽ ã®ååãæå®ããããããæå®ãããèŠçŽ ã«ã¯ãCSSã§èšå®ãããã¹ã¿ã€ã«ãå²ãåœãŠããããããšãå¯èœã§ããã1ã€ã®ããã¥ã¡ã³ãå
ã§1åãã䜿çšããããšãã§ããªãã
;class
:ãã®èŠçŽ ãšCSSãªã©ã§æå®ãããã¯ã©ã¹ãé¢ä¿ã¥ãããidå±æ§ãšåæ§ãCSSã§èšå®ãããã¹ã¿ã€ã«ãå²ãåœãŠãããšãå¯èœã§ãããã1ã€ã®ããã¥ã¡ã³ãå
ã§äœåã䜿çšããããšãã§ããã
;title
:æå®ãããèŠçŽ ã«å¯Ÿããã¿ã€ãã«ã瀺ããç°¡åãªèª¬æãèšè¿°ããããšãå€ããäžéšã®ãã©ãŠã¶ã§ã¯ããã®èŠçŽ ãããŠã¹ãªãŒããŒãããšãã®å
容ãããŒã«ããããšããŠè¡šç€ºããã
;style
:CSSãªã©ã®ã¹ã¿ã€ã«ã·ãŒããèšè¿°ããããããããå²ãåœãŠãªã©ãèšå®ãããŠããªãã¹ã¿ã€ã«ãçŽæ¥ãã®èŠçŽ ã«å¯ŸããŠæå®ã§ããã
;dir
:æåã®æ¹åãæå®ãããå±æ§å€ã«ltr(left to right)ãæå®ãããšæåãå·Šããå³ãžãrtl(right to left)ãæå®ãããšæåãå³ããå·Šãžè¡šç€ºãããã
== ãã®ä»ããã¹ãé¢ä¿ã®èŠçŽ ==
;dfn
:ãã®èªãå®çŸ©å¯Ÿè±¡ã®çšèªã§ããããšã瀺ããæç« äžã§ãã®èªãå§ããŠåºãŠããå Žåãªã©ã«äœ¿çšãããäžè¬çãªãã©ãŠã¶ã§ã¯ãã€ã¿ãªãã¯è¡šç€ºãšãªãã
;abbr,acronym
:abbrã¯ãã®èªãç¥èªã§ããããšããacronymã¯ãã®èªãé åèªã§ããããšã瀺ããtitleå±æ§ãæå®ãããã®èªã®çç¥ããªã圢ãæžãããšãå¿
é ãšãããŠããããã©ãŠã¶ã«ãã£ãŠã¯ããã®èŠçŽ ãæå®ãããèªã¯ç¹ç·è¡šç€ºãããããInternet Explorerã¯å¯Ÿå¿ããŠããªãã
;sup,sub
:supã¯æåãäžä»ãã«ãsubã¯æåãäžä»ãã«ããã
;pre
:æå®ãããããã¹ããçå¹
<!-- ã»ç¡æ¹è¡ -->ã§è¡šç€ºããããœãŒã¹ã³ãŒããªã©ã衚瀺ãããšãã«äœ¿çšãããHTMLã¯æå¹ã ããæ¹è¡ãã¹ããŒã¹ã¯ãã®ãŸãŸè¡šç€ºãããã
;kbd
:æäœæ³èª¬æãªã©ã«ãããŠãããŒããŒãããå
¥åããæåã瀺ããäžè¬çãªãã©ãŠã¶ã§ã¯çå¹
ã§è¡šç€ºãããã
;code
:ãã®éšåããœãŒã¹ã³ãŒãã§ããããšãæ瀺ãããäžè¬çãªãã©ãŠã¶ã§ã¯çå¹
ã§è¡šç€ºãããã
;samp
:ããã«ããå
容ããããã°ã©ã ãªã©ã«ããåºåãããå
容ã®ãµã³ãã«ã§ããããšã瀺ããäžè¬çãªãã©ãŠã¶ã§ã¯çå¹
ã§è¡šç€ºãããã
;var
:å€æ°ãåŒæ°ã瀺ããšãã«äœ¿çšãããäžè¬çãªç°å¢ã§ã¯ã€ã¿ãªãã¯è¡šç€ºãšãªãã
;ins,del
:insèŠçŽ ã¯ãã®éšåãåŸããæ¿å
¥ããå
容ã§ããããšããdelèŠçŽ ã¯ãã®éšåãåŸããåé€ããå
容ã§ããããšã瀺ããdatetimeå±æ§ïŒISO 8601圢åŒïŒãçšããŠæ¿å
¥ã»åé€æ¥æãèšè¿°ããããciteå±æ§ãçšããŠæ
å ±ã®å
žæ ã瀺ããããtitleå±æ§ãçšããŠç°¡åãªèª¬æãèšè¿°ãããããããšãåºæ¥ãã
:datetimeå±æ§ã¯ã幎ïŒåæ¡ïŒ-æïŒäºæ¡ïŒ-æ¥ïŒäºæ¡ïŒTæ:å:ç§+9:00ã®åœ¢åŒïŒæ¥æ¬æéã®å ŽåïŒã§èšè¿°ãããäŸãã°2010幎1æ1æ¥9æã¡ããã©ã§ããã°<code>2010-01-01T09:00:00+9:00</code>ãšããæžåŒã«ãªãã
:æå®äœçœ®ã«å¿ããŠãããã¯èŠçŽ ãšããŠãã€ã³ã©ã€ã³èŠçŽ ãšããŠã䜿çšããããšãåºæ¥ããããããã¯èŠçŽ ãšããŠæ±ã£ãŠãCSSã§ã®ã¹ã¿ã€ã«æå®ã¯ã€ã³ã©ã€ã³èŠçŽ ãšåãæ±ãã«ãªãã
;bdo
:æå®ãããããã¹ãã®è¡šç€ºæ¹åããdirå±æ§ã§èšå®ããã
== 匷å¶æ¹è¡ ==
ããã¹ããã¡ã€ã«ã§ã®æ¹è¡ã¯è¡šç€ºã«ã¯ã»ãŒåœ±é¿ããªãïŒãã©ãŠã¶ã«ãã£ãŠã¯ã¹ããŒã¹ãéãããšãããïŒã匷å¶çã«æ¹è¡ãããããšãã«ã¯<br>ã䜿ãããªãã[[w:Extensible HyperText Markup Language|XHTML]]ã«ãããŠã¯ã<br />ïŒXHTML 1.0ã§ã¯brãš/ã®éã«åè§ã¹ããŒã¹ãå
¥ããããšãæšå¥šãããŠããããå¿
é ã§ã¯ãªãïŒãšããããã«å®ããããŠãããHTMLã§ã<br />ã䜿ãããšã¯åºæ¥ãããææ³äžæ£ããæžãæ¹ã§ã¯ãªãã®ã§èŠæ Œã«æ²¿ã£ãHTMLãæžããããšãã¯æ³šæãããã<!-- å€ãã¢ããªã®ãµããŒãã§XHTMLã®æ
å ±ã䜿ãã®ã§ãHTML5æ代ã§ããXHTMLã®è©±é¡ãæ®ããããšæããŸãã -->
=== èšè¿°äŸ ===
<syntaxhighlight lang="html4strict">
<p>æ¹è¡ããŸãã
ã¯ãããã?</p>
<p>ããããªããšã<br>
ã!</p>
<p>XHTMLã®ãšãã¯ããã£ã¡ã§ã<br />
ã!</p>
</syntaxhighlight>
=== è¡šç€ºäŸ ===
<div style="border:#000 1px dashed;padding:1em">
<p>æ¹è¡ããŸãã
ã¯ãããã?</p>
<p>ããããªããšã<br />
ã!</p>
<p>XHTMLã®ãšãã¯ããã£ã¡ã§ã<br />
ã!</p>
</div>
== div, spanèŠçŽ ==
divèŠçŽ ã¯ãããã¯ã¬ãã«èŠçŽ ãspanèŠçŽ ã¯ã€ã³ã©ã€ã³èŠçŽ ã§ãããããã以å€ã®æå³ã¯ãªãåäœã§æå®ããŠãããã©ãŠã¶ãç¹å¥ãªæ±ããè¡ã£ããã衚瀺ãç¹å¥å€åãããããããšã¯ãªãïŒãã ãdivèŠçŽ ã¯ãããã¯ã¬ãã«èŠçŽ ã§ãããååŸã«æ¹è¡ãå
¥ãïŒãidå±æ§ãclasså±æ§ã䜿ã£ãŠã¹ã¿ã€ã«ã·ãŒããé©çšããããlangå±æ§ãªã©ãæå®ãããäž»ã«ä»ã®èŠçŽ ã§ã¯ä»£çšã§ããªãïŒä»ã®èŠçŽ ãçšãããšç¯å²å
ã«äžå¿
èŠãªæ
å ±ãå®çŸ©ããŠããŸãïŒããšãè¡ãæ±çšèŠçŽ ãšããŠçšããããã
== ãšã¹ã±ãŒãã·ãŒã±ã³ã¹ ==
HTMLã§åè§è±æ°ã® <nowiki> < </nowiki> ã <nowiki> > </nowiki>ãšãã£ãå¶åŸ¡çšã®æåãã®ãã®ã衚瀺ãããå Žåã«ã¯ã
<nowiki> < </nowiki> ãšè¡šç€ºããããªã <code>&lt;</code> ãšå
¥åããã
<nowiki> > </nowiki> ãšè¡šç€ºããããªã <code>&gt;</code> ãšå
¥åããã
ltãgtã®çŽåŸã®èšå·ã¯ã»ãã³ãã³(;)ã§ããã(ã³ãã³(:)ã§ã¯ãªãã)
HTMLã«éãããããã°ã©ãã³ã°ãé¡äŒŒã®ã³ãŒãã£ã³ã°ã«ãããŠããã®ããã«å¶åŸ¡æåãã®ãã®ãå
¥åããããã®å
¥åæ¹æ³ã®ããšãããšã¹ã±ãŒãã·ãŒã±ã³ã¹ããšããã
HTMLã®ãšã¹ã±ãŒãã·ãŒã±ã³ã¹ã«ã€ããŠã¯ãçš®é¡ãå€ãã®ã§ã詳ããã¯ããããªã©ã§æ€çŽ¢ããŠãããããã
ãªãã掟ççãªè©±é¡ã ãã <code>&lt;</code> ãšwebããŒãžã§è¡šç€ºãããå Žåã <code>&amp;lt;</code>ãšHTMLã«å
¥åããã
<!--
以äžã¯[[HTML/è£
食]]åã äžæŠã³ã¡ã³ãã¢ãŠã
== bodyèŠçŽ ==
=== ç»é¢èæ¯ã®èšå® ===
ç»é¢ã®èæ¯ã¯
#èæ¯è²ãèšå®
#èæ¯ã«ç»åãèšå®
ã®2ã€ã«åé¡ã§ããã
äžè¬çã«èŠèŠçãªå¹æã®èª¿æŽã¯[[CSS]]ãçšããŠè¡ãã¹ããšãããŠããããããã§ã¯CSSã䜿ããããã簡䟿ãªHTMLã§èæ¯ãèšå®ããæ¹æ³ã玹ä»ãããèæ¯è²ãèæ¯ç»åãšãèŠçŽ ã«å±æ§ãæå®ããããšã§èšå®ããã
==== èæ¯è² ====
<syntaxhighlight lang="html4strict"><body bgcolor="#FFFFFF"></syntaxhighlight>
bgcolorå±æ§ã«ã¯èšå®ãããè²ãæå®ãããè²ã®æå®ã¯è²ã®å称ïŒwhiteãblackãªã©ïŒããããã¯16é²æ°ã§æå®ããã16é²æ°ã§æå®ããå Žåã¯ãæåã«<nowiki>#</nowiki>ãã€ããRïŒèµ€èŠçŽ ïŒGïŒç·èŠçŽ ïŒBïŒéèŠçŽ ïŒã®é çªã«ãããã00ïœFFã®æ°å€ãæå®ããã
==== èæ¯ç»å ====
<syntaxhighlight lang="html4strict"><body background="wiki.png"></syntaxhighlight>
backgroundå±æ§ã«ã¯èæ¯ç»åã®ãã¡ã€ã«ã®ã¢ãã¬ã¹ãæå®ãããã¢ãã¬ã¹ã®æå®ã¯çµ¶å¯Ÿãã¹ïŒ<nowiki>http://ïœïœ/wiki.png</nowiki>ïŒãçžå¯Ÿãã¹ïŒ./wiki.pngïŒã䜿çšãããç»å圢åŒã¯äžè¬çã«JPEGãGIFãPNGã®ããããã䜿çšããã[[w:Windows bitmap|BMP]]圢åŒã®ç»åã䜿ã人ãããããBMPã¯Windowsçšã®ãã¡ã€ã«åœ¢åŒã§ããããšã«å ããã¡ã€ã«ãµã€ãºãã®ãã®ã倧ããã®ã§äœ¿ããªãããã«ãããã
-->
[[Category:HTML|HTML ã»ããµã]] | null | 2022-10-22T01:50:12Z | [] | https://ja.wikibooks.org/wiki/HTML/%E6%9C%AC%E6%96%87 |
2,007 | ç·å代æ°åŠ |
ãMVã#è¶
絶ãããã/mona(CV:å€å·æ€è)ãHoneyWorksã âæ:mona(CV:å€å·æ€è) âäœè©ã»äœæ²ã»ç·šæ²:HoneyWorks âæè©ç¿»è¯:Fir(@Fir3k0) â»ä»¥AIèŒå©ç¿»è¯,æ以ç®åæ©ç¿»å§w èªå·±æ ¡æ£ææ
ã#è¶
絶ããããã
åã®3åç§ã«ãã ãã åç»ã¯ãã®ãŸãŸã§ è«äœ 絊æ3åéæéå§ åœ±çé暣æŸç就奜
ç§ã®ãããããšã äŒããã°å¬ããã§ã èŠæ¯èœåäœ å³éå°æçå¯æä¹è ææåŸé«èå¢
ç»é¢è¶ãã§è§Šããªããã© ç§ã¯ãããã ã éç¶éçç«é¢æ²èŸŠæ³è§žç¢°å° äœæå°±åšé裡å
ã³ã¡ã³ããæ®ã㊠åãèªç¥ãã㊠ä¹è«çäžçèš è®æèªèå°äœ
ããããåãªã㊠ãããããããã© éç¶å¯æçå©å å€äžåæž
ããããããå㯠ç§ããããªã äœéæŒå¯æçå©å éæè«å±¬
ãããšãããã£ãŠ ãããç§ã ã£ãŠ å°±ç®éåå°è£å¯æ éä¹æ¯æçäžé¢å¢
åã®äžçªã«ãªããŸã§ãããªã çŽå°æçºäœ ç第äžä¹åéœäžæ眷äŒ
åã®å€§å¥œããªäººã¯ã ãŒã? äœ æåæ¡ç人æ¯èª°~å¢?
è¶
絶ãããã mona è¶
çµå¯æç mona
åãæããŠã人ã¯ã ãŒã? äœ æåæç人æ¯èª°~å¢?
è¶
絶ãããã mona è¶
çµå¯æç mona
ç§ã倧奜ãã ãã æ人å士ã ã æä¹æåæ¡äœ äº æ以æ¯å
©æ
çžæ
çæ人å¢
è±å«ä¿®æ¥é 匵ãã æ°åšä¿®è¡ææåªå以赎å¢
ãããã°ãããã®ã¢ããŒã«ã« ãä»ãåããã ãã éå±çŸèªèº«çæ©æ è«åå€å¥éªæäžæå
å§
ç§ã®ãããããšã äŒããã°å¬ããã§ã èŠæ¯èœåäœ å³éå°æçå¯æä¹è ææåŸé«èå¢
身é·ã 足ã®ãµã€ãºã é ããŠããã¯ãã 身é«ä¹å¥œ éè
³å°ºå¯žä¹å¥œ é±èçç£ä¹å¥œ
æšããªãåœç¶ç¥ã£ãŠãã¯ãã ãã? åŠææ¯äž»æšçè©±äœ ç¶ç¶æç¥éå§?
ãã¡ã³ã倧åã£ãŠ åœããåã ãã© éç¶ç²çµ²åŸéèŠ éæ¯çç¶ç¶ç¶ç
ãããªã«å¥œããšã ç§ããããªã äœåæ¡å°éçš®å°æ¥ éæè«å±¬
ã¯ãµãããèšè? ã ã£ãŠæ¬åœã ãã éäºè©±éœèœå°è©äº? å çºéæ¯çå¿è©±åŠ
åã®äžçªã«ãªããŸã§ãããªã çŽå°æçºäœ ç第äžä¹åéœäžæ眷äŒ
åãèã«ããã®ã¯ã ãŒã? è®äœ æçºä¿èçæ¯èª°~å¢?
è¶
絶ãããã mona è¶
çµå¯æç mona
åã幞ãã«ããã®ã ãŒã? è®äœ è®åŸå¹žçŠçæ¯èª°~å¢?
è¶
絶ãããã mona è¶
çµå¯æç mona
ãã£ãšæããŠãããã 責任åãããã å³äŸ¿æåŸæŽæ·±ä¹å¯ä»¥å ææè² èµ·è²¬ä»»å¢
æµ®æ°ãªããŠãããªããã æäžæè®äœ äžå¿å
©æçåŠ
åã®å€§å¥œããªäººã¯ã ãŒã? äœ æåæ¡ç人æ¯èª°~å¢?
è¶
絶ãããã mona è¶
çµå¯æç mona
åãæããŠã人ã¯ã ãŒã? äœ æåæç人æ¯èª°~å¢?
è¶
絶ãããã mona è¶
çµå¯æç mona
ç§ã倧奜ãã ãã æ人å士ã ã æä¹æåæ¡äœ äº æ以æ¯å
©æ
çžæ
çæ人å¢
è±å«ä¿®æ¥é 匵ãã æ°åšä¿®è¡ææåªåä»¥èµŽå¢ | [
{
"paragraph_id": 0,
"tag": "p",
"text": "",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ãMVã#è¶
絶ãããã/mona(CV:å€å·æ€è)ãHoneyWorksã âæ:mona(CV:å€å·æ€è) âäœè©ã»äœæ²ã»ç·šæ²:HoneyWorks âæè©ç¿»è¯:Fir(@Fir3k0) â»ä»¥AIèŒå©ç¿»è¯,æ以ç®åæ©ç¿»å§w èªå·±æ ¡æ£ææ",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ã#è¶
絶ããããã",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "åã®3åç§ã«ãã ãã åç»ã¯ãã®ãŸãŸã§ è«äœ 絊æ3åéæéå§ åœ±çé暣æŸç就奜",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ç§ã®ãããããšã äŒããã°å¬ããã§ã èŠæ¯èœåäœ å³éå°æçå¯æä¹è ææåŸé«èå¢",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ç»é¢è¶ãã§è§Šããªããã© ç§ã¯ãããã ã éç¶éçç«é¢æ²èŸŠæ³è§žç¢°å° äœæå°±åšé裡å",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ã³ã¡ã³ããæ®ã㊠åãèªç¥ãã㊠ä¹è«çäžçèš è®æèªèå°äœ ",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ããããåãªã㊠ãããããããã© éç¶å¯æçå©å å€äžåæž",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "ããããããå㯠ç§ããããªã äœéæŒå¯æçå©å éæè«å±¬",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "ãããšãããã£ãŠ ãããç§ã ã£ãŠ å°±ç®éåå°è£å¯æ éä¹æ¯æçäžé¢å¢",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "åã®äžçªã«ãªããŸã§ãããªã çŽå°æçºäœ ç第äžä¹åéœäžæ眷äŒ",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "åã®å€§å¥œããªäººã¯ã ãŒã? äœ æåæ¡ç人æ¯èª°~å¢?",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "è¶
絶ãããã mona è¶
çµå¯æç mona",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "åãæããŠã人ã¯ã ãŒã? äœ æåæç人æ¯èª°~å¢?",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "è¶
絶ãããã mona è¶
çµå¯æç mona",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "ç§ã倧奜ãã ãã æ人å士ã ã æä¹æåæ¡äœ äº æ以æ¯å
©æ
çžæ
çæ人å¢",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "è±å«ä¿®æ¥é 匵ãã æ°åšä¿®è¡ææåªå以赎å¢",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "ãããã°ãããã®ã¢ããŒã«ã« ãä»ãåããã ãã éå±çŸèªèº«çæ©æ è«åå€å¥éªæäžæå
å§",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "ç§ã®ãããããšã äŒããã°å¬ããã§ã èŠæ¯èœåäœ å³éå°æçå¯æä¹è ææåŸé«èå¢",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "身é·ã 足ã®ãµã€ãºã é ããŠããã¯ãã 身é«ä¹å¥œ éè
³å°ºå¯žä¹å¥œ é±èçç£ä¹å¥œ",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "æšããªãåœç¶ç¥ã£ãŠãã¯ãã ãã? åŠææ¯äž»æšçè©±äœ ç¶ç¶æç¥éå§?",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "ãã¡ã³ã倧åã£ãŠ åœããåã ãã© éç¶ç²çµ²åŸéèŠ éæ¯çç¶ç¶ç¶ç",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "ãããªã«å¥œããšã ç§ããããªã äœåæ¡å°éçš®å°æ¥ éæè«å±¬",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "ã¯ãµãããèšè? ã ã£ãŠæ¬åœã ãã éäºè©±éœèœå°è©äº? å çºéæ¯çå¿è©±åŠ",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "åã®äžçªã«ãªããŸã§ãããªã çŽå°æçºäœ ç第äžä¹åéœäžæ眷äŒ",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "åãèã«ããã®ã¯ã ãŒã? è®äœ æçºä¿èçæ¯èª°~å¢?",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "è¶
絶ãããã mona è¶
çµå¯æç mona",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "åã幞ãã«ããã®ã ãŒã? è®äœ è®åŸå¹žçŠçæ¯èª°~å¢?",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "è¶
絶ãããã mona è¶
çµå¯æç mona",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "ãã£ãšæããŠãããã 責任åãããã å³äŸ¿æåŸæŽæ·±ä¹å¯ä»¥å ææè² èµ·è²¬ä»»å¢",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "æµ®æ°ãªããŠãããªããã æäžæè®äœ äžå¿å
©æçåŠ",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "åã®å€§å¥œããªäººã¯ã ãŒã? äœ æåæ¡ç人æ¯èª°~å¢?",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "è¶
絶ãããã mona è¶
çµå¯æç mona",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "åãæããŠã人ã¯ã ãŒã? äœ æåæç人æ¯èª°~å¢?",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "è¶
絶ãããã mona è¶
çµå¯æç mona",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "ç§ã倧奜ãã ãã æ人å士ã ã æä¹æåæ¡äœ äº æ以æ¯å
©æ
çžæ
çæ人å¢",
"title": "äžç·ã«æããŸãããïŒ"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "è±å«ä¿®æ¥é 匵ãã æ°åšä¿®è¡ææåªå以赎å¢",
"title": "äžç·ã«æããŸãããïŒ"
}
] | null | {{pathnav|frame=1|ã¡ã€ã³ããŒãž|æ°åŠ|代æ°åŠ}}
æ¬é
ã¯ç·åœ¢ä»£æ°åŠã®è§£èª¬ã§ãã
{{é²æç¶æ³}}
== åºè«ã»å°å
¥ ==
* [[/åºè«|åºè«]]
* [[/ãã¯ãã«|ãã¯ãã«]]
** [[é«çåŠæ ¡æ°åŠC/ãã¯ãã«]]ãåç
§ã®ããšã
* [[/è¡åæŠè«|è¡åæŠè«]]
** [[æ§èª²çšé«çåŠæ ¡æ°åŠC/è¡å]]ãåç
§ã®ããšã
== ç·åæ¹çšåŒ ==
* [[/ç·åæ¹çšåŒ|ç·åæ¹çšåŒåºè«]]
* [[/è¡åã®åºæ¬å€åœ¢|è¡åã®åºæ¬å€åœ¢]] {{é²æ|100%|2009-05-31}}
* [[/éè¡å|éè¡å]]ã{{é²æ|100%|2009-06-2}}
* [[/ç·åæ¹çšåŒã®è§£|ç·åæ¹çšåŒã®è§£]]ã{{é²æ|50%|2009-06-28}}
== è¡ååŒ ==
* [[ç·åœ¢ä»£æ°åŠ/è¡ååŒ|è¡ååŒ]] {{é²æ|25%|2021-03-09}}
* [[ç·åœ¢ä»£æ°åŠ/äœå åè¡å|äœå åè¡å]]
* [[/ã¯ã©ã¡ã«ã®å
¬åŒ|ã¯ã©ã¡ã«ã®å
¬åŒ]]
== ç·åœ¢ç©ºé ==
* [[/ç·å空é|ç·å空é]]
* [[/ç·åœ¢åå|ç·åœ¢åå]]
* [[/åºåºãšæ¬¡å
|åºåºãšæ¬¡å
]]
* [[/èšéãã¯ãã«ç©ºé|èšéãã¯ãã«ç©ºé]]
== 察è§åãšåºæå€ ==
* [[/åºæå€ãšåºæãã¯ãã«|åºæå€ãšåºæãã¯ãã«]]
* [[/è¡åã®äžè§å|è¡åã®äžè§å]]
* [[/è¡åã®å¯Ÿè§å|è¡åã®å¯Ÿè§å]] {{é²æ|50%|2018-11-29}}
* [[/äºæ¬¡åœ¢åŒ|äºæ¬¡åœ¢åŒ]]{{é²æ|25%|2020-8-19}}
== ãžã§ã«ãã³æšæºåœ¢ ==
* [[/åå å|åå å]]
* [[/ãžã§ã«ãã³æšæºåœ¢|ãžã§ã«ãã³æšæºåœ¢]]
{{stub}}
[[Category:ç·åœ¢ä»£æ°åŠ|*]]
[[Category:æ°åŠ|ãããããããããã]] | 2005-05-17T01:42:42Z | 2023-11-19T12:19:21Z | [] | https://ja.wikibooks.org/wiki/%E7%B7%9A%E5%9E%8B%E4%BB%A3%E6%95%B0%E5%AD%A6 |
2,008 | ç·å代æ°åŠ/ç·åæ¹çšåŒ | ç·å代æ°åŠ > ç·åæ¹çšåŒ
ç·åæ¹çšåŒ(é£ç«1次æ¹çšåŒ)ãšã¯ã a i , j , b i â K ( 1 †i †m , 1 †j †n ) {\displaystyle a_{i,j},b_{i}\in \mathbf {K} (1\leq i\leq m,1\leq j\leq n)} ãçšããŠ
ã§è¡šããããæ¹çšåŒã§ããã
äžã®é£ç«æ¹çšåŒã¯ã
ãšããã° A x = b {\displaystyle \ Ax=b} ãšè¡åãçšããŠæžããã
ä»®ã«ãAãæ£æ¹è¡åã§éè¡åãæã€ãªãã ãã®åŒã®äžè¬è§£ã¯ã x = A â 1 b {\displaystyle \ x=A^{-1}b} ãšãªãã
ããããããã¯éåžžã«ç¹æ®ãªå Žåã§ãããäžè¬ã«ã¯è§£ãååšããªãããšãããã°ãããã€ãã®è§£ã®éãåãã(æ£ããã¯ç·åœ¢çµå)ãšããŠè¡šããããããšãããã
ãã®ç« ã§ã¯ãéè¡åã®åå®çŸ©ããå§ããè¡åã®åºæ¬å€åœ¢ãéæ°çãå°å
¥ããæçµçã«ã¯äžã®ç·åæ¹çšåŒã®äžè¬è§£ãå°ãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ç·å代æ°åŠ > ç·åæ¹çšåŒ",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ç·åæ¹çšåŒ(é£ç«1次æ¹çšåŒ)ãšã¯ã a i , j , b i â K ( 1 †i †m , 1 †j †n ) {\\displaystyle a_{i,j},b_{i}\\in \\mathbf {K} (1\\leq i\\leq m,1\\leq j\\leq n)} ãçšããŠ",
"title": "ç·åæ¹çšåŒ"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ã§è¡šããããæ¹çšåŒã§ããã",
"title": "ç·åæ¹çšåŒ"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "äžã®é£ç«æ¹çšåŒã¯ã",
"title": "ç·åæ¹çšåŒ"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ãšããã° A x = b {\\displaystyle \\ Ax=b} ãšè¡åãçšããŠæžããã",
"title": "ç·åæ¹çšåŒ"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ä»®ã«ãAãæ£æ¹è¡åã§éè¡åãæã€ãªãã ãã®åŒã®äžè¬è§£ã¯ã x = A â 1 b {\\displaystyle \\ x=A^{-1}b} ãšãªãã",
"title": "ç·åæ¹çšåŒ"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ããããããã¯éåžžã«ç¹æ®ãªå Žåã§ãããäžè¬ã«ã¯è§£ãååšããªãããšãããã°ãããã€ãã®è§£ã®éãåãã(æ£ããã¯ç·åœ¢çµå)ãšããŠè¡šããããããšãããã",
"title": "ç·åæ¹çšåŒ"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ãã®ç« ã§ã¯ãéè¡åã®åå®çŸ©ããå§ããè¡åã®åºæ¬å€åœ¢ãéæ°çãå°å
¥ããæçµçã«ã¯äžã®ç·åæ¹çšåŒã®äžè¬è§£ãå°ãã",
"title": "ç·åæ¹çšåŒ"
}
] | ç·å代æ°åŠ > ç·åæ¹çšåŒ | <small> [[ç·å代æ°åŠ]] > ç·åæ¹çšåŒ </small>
== ç·åæ¹çšåŒ ==
ç·åæ¹çšåŒïŒé£ç«1次æ¹çšåŒïŒãšã¯ã<math> a_{i,j},b_i \in \mathbf K (1 \leq i \leq m,1 \leq j \leq n) </math> ãçšããŠ
:<math>\begin{cases}
a _{1,1}x _1 + \cdots + a _{1,n}x _n = b _1 \\
\vdots \\
a _{m,1}x _1 + \cdots + a _{m,n}x _n = b _m
\end{cases}</math>
ã§è¡šããããæ¹çšåŒã§ããã
äžã®é£ç«æ¹çšåŒã¯ã
:<math>
A = \begin{pmatrix} a_{1,1} & \cdots & a_{1,n}\\
\vdots & \ddots & \vdots\\
a_{m,1} & \cdots & a_{m,n}\\ \end{pmatrix} ,
x = \begin{pmatrix} x_1\\ x_2 \\ \vdots \\ x_n \end{pmatrix} ,
b = \begin{pmatrix} b_1\\ b_2 \\ \vdots \\ b_m \end{pmatrix}</math>
ãšããã°
<math>
\ Ax = b
</math>
ãšè¡åãçšããŠæžããã
ä»®ã«ãAãæ£æ¹è¡åã§éè¡åãæã€ãªãã
ãã®åŒã®äžè¬è§£ã¯ã
<math>
\ x = A^{-1} b
</math>
ãšãªãã
ããããããã¯éåžžã«ç¹æ®ãªå Žåã§ãããäžè¬ã«ã¯è§£ãååšããªãããšãããã°ãããã€ãã®è§£ã®éãåããïŒæ£ããã¯ç·åœ¢çµåïŒãšããŠè¡šããããããšãããã
ãã®ç« ã§ã¯ãéè¡åã®åå®çŸ©ããå§ããè¡åã®åºæ¬å€åœ¢ãéæ°çãå°å
¥ããæçµçã«ã¯äžã®ç·åæ¹çšåŒã®äžè¬è§£ãå°ãã
[[Category:ç·åœ¢ä»£æ°åŠ|ããããã»ããŠããã]] | null | 2022-08-31T07:55:07Z | [] | https://ja.wikibooks.org/wiki/%E7%B7%9A%E5%9E%8B%E4%BB%A3%E6%95%B0%E5%AD%A6/%E7%B7%9A%E5%9E%8B%E6%96%B9%E7%A8%8B%E5%BC%8F |
2,010 | ç·åœ¢ä»£æ°åŠ/è¡ååŒ | 1 , 2 , ⯠, n {\displaystyle {1,2,\cdots ,n}} ãäºãã«éè€ããªãããã«ã 1 , 2 , ⯠, n {\displaystyle {1,2,\cdots ,n}} ã«ãã€ãæäœãn次ã®çœ®æãšããã
眮æ Ï {\displaystyle \sigma } ã«ãã£ãŠiããã€ãããè¡ãå
ã Ï ( i ) {\displaystyle \sigma (i)} ãšè¡šãã
眮æ Ï {\displaystyle \sigma } ã¯ã次ã®ããã«ãäžã«ããšã®å
ããäžã®è¡ãå
ã䞊ã¹ãŠè¡šçŸãããã
ããã¯ãè¡åãšåãè¡šçŸã ããè¡åã§ã¯ãªãããšã«æ³šæããã
äŸãã°ã 1ã2ã«ã2ã3ã«ã3ã1ã«ãã€ã眮æ Ï {\displaystyle \sigma } ã¯ã3次ã®çœ®æã§ããã Ï ( 1 ) = 2 , Ï ( 2 ) = 3 , Ï ( 3 ) = 1 {\displaystyle \sigma (1)=2,\sigma (2)=3,\sigma (3)=1} ãšãªãããã®çœ®æã¯ã Ï = ( 1 2 3 2 3 1 ) {\displaystyle \sigma ={\begin{pmatrix}1&2&3\\2&3&1\end{pmatrix}}} ãšè¡šããã
e = ( 1 2 ⯠n 1 2 ⯠n ) {\displaystyle e={\begin{pmatrix}1&2&\cdots &n\\1&2&\cdots &n\end{pmatrix}}} ã®ããã«ããã¹ãŠã®æŽæ°ãå€åããªã眮æã®ããšãåäœçœ®æãšããã
ãã眮æ Ï {\displaystyle \sigma } ã«å¯Ÿãã Ï â 1 = ( Ï ( 1 ) Ï ( 2 ) â¯ Ï ( n ) 1 2 ⯠n ) {\displaystyle \sigma ^{-1}={\begin{pmatrix}\sigma (1)&\sigma (2)&\cdots &\sigma (n)\\1&2&\cdots &n\end{pmatrix}}} ãé眮æãšããã
n次ã®çœ®æå
šäœã®éåã S n {\displaystyle S_{n}} ãšè¡šãã äŸãã°ã S 3 = { ( 1 2 3 1 2 3 ) , ( 1 2 3 1 3 2 ) , ( 1 2 3 3 2 1 ) , ( 1 2 3 2 1 3 ) , ( 1 2 3 3 1 2 ) , ( 1 2 3 2 3 1 ) } {\displaystyle S_{3}=\left\{{\begin{pmatrix}1&2&3\\1&2&3\end{pmatrix}},{\begin{pmatrix}1&2&3\\1&3&2\end{pmatrix}},{\begin{pmatrix}1&2&3\\3&2&1\end{pmatrix}},{\begin{pmatrix}1&2&3\\2&1&3\end{pmatrix}},{\begin{pmatrix}1&2&3\\3&1&2\end{pmatrix}},{\begin{pmatrix}1&2&3\\2&3&1\end{pmatrix}}\right\}} ã§ããã
n次ã®çœ®æå
šäœã®éåã®åæ°ã n ! {\displaystyle n!} ã§ããããšã¯èªæã§ãããã
眮æ Ï , Ï â S n {\displaystyle \sigma ,\tau \in S_{n}} ã«å¯Ÿãã眮æã®åæã Ï Ï = ( 1 2 ⯠n Ï ( 1 ) Ï ( 2 ) â¯ Ï ( n ) ) ( 1 2 ⯠n Ï ( 1 ) Ï ( 2 ) â¯ Ï ( n ) ) = ( 1 2 ⯠n Ï ( Ï ( 1 ) ) Ï ( Ï ( 2 ) ) â¯ Ï ( Ï ( n ) ) ) {\displaystyle \sigma \tau ={\begin{pmatrix}1&2&\cdots &n\\\sigma (1)&\sigma (2)&\cdots &\sigma (n)\end{pmatrix}}{\begin{pmatrix}1&2&\cdots &n\\\tau (1)&\tau (2)&\cdots &\tau (n)\end{pmatrix}}={\begin{pmatrix}1&2&\cdots &n\\\sigma (\tau (1))&\sigma (\tau (2))&\cdots &\sigma (\tau (n))\end{pmatrix}}} ãšå®ããã ããã¯ã 1 †i †n {\displaystyle 1\leq i\leq n} ã«å¯Ÿãã Ï Ï ( i ) = Ï ( Ï ( i ) ) {\displaystyle \sigma \tau (i)=\sigma (\tau (i))} ãšè¡šèšããããšãã§ããã ãããããšãèšè¿°éãå°ãªããªãã䟿å©ã ããã
眮æã«ã€ããŠã以äžã®æ§è³ªãæãç«ã€ã
Ï = ( 1 2 ⯠i ⯠j ⯠n 1 2 ⯠j ⯠i ⯠n ) {\displaystyle \sigma ={\begin{pmatrix}1&2&\cdots &i&\cdots &j&\cdots n\\1&2&\cdots &j&\cdots &i&\cdots n\end{pmatrix}}} ã®ããã«ãiãšjã ãã亀æãã眮æãäºæãšããã
ä»»æã®çœ®æã¯äºæã®ç©ã§è¡šãããšãã§ããäºæã®åæ°ã®å¶å¥ã¯äºæã®ãšãæ¹ã«ããããåãã§ãããšããæ§è³ªãããã 眮æãäºæã®ç©ã§è¡šãããšããäºæã®åæ°ãå¶æ°åã®çœ®æãå¶çœ®æãå¥æ°åã®çœ®æãå¥çœ®æãšããã
sgn ( Ï ) = { 1 Ï ã å¶ çœ® æ ã® ãš ã â 1 Ï ã å¥ çœ® æ ã® ãš ã {\displaystyle \operatorname {sgn}(\sigma )={\begin{cases}1&\sigma {\text{ã å¶ çœ® æ ã® ãš ã}}\\-1&\sigma {\mbox{ã å¥ çœ® æ ã® ãš ã}}\end{cases}}\ } ã Ï {\displaystyle \sigma } ã®ç¬Šå·ãšããã
è¡å A = ( a 11 ⯠a 1 n â® â± â® a n 1 ⯠a n n ) {\displaystyle A={\begin{pmatrix}a_{11}&\cdots &a_{1n}\\\vdots &\ddots &\vdots \\a_{n1}&\cdots &a_{nn}\end{pmatrix}}} ã«å¯ŸããŠã
| A | = det A = â Ï â S n sgn ( Ï ) a 1 , Ï ( 1 ) ⯠a n , Ï ( n ) {\displaystyle |A|=\det A=\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )a_{1,\sigma (1)}\cdots a_{n,\sigma (n)}} ãAã®è¡ååŒãšããã
â» â Ï â S n {\displaystyle \sum _{\sigma \in S_{n}}} ãšã¯ã Ï {\displaystyle \sigma } ã« S n {\displaystyle S_{n}} ã®å
ããã¹ãŠä»£å
¥ããŠè¶³ãåããããšããæå³ã§ããã ããšãã°ã A = { 1 , 2 , 3 } {\displaystyle A=\{1,2,3\}} ã®ãšãã â i â A {\displaystyle \sum _{i\in A}} ãš â i = 1 3 {\displaystyle \sum _{i=1}^{3}} ã¯åãæå³ã§ããã
2次æ£æ¹è¡å A = ( a b c d ) {\displaystyle A={\begin{pmatrix}a&b\\c&d\end{pmatrix}}} ã®è¡ååŒãæ±ããŠã¿ããã è¡ååŒã®å®çŸ©ã«åœãŠã¯ãããšã | A | = â Ï â S 2 sgn ( Ï ) a 1 , Ï ( 1 ) a n , Ï ( 2 ) {\displaystyle |A|=\sum _{\sigma \in S_{2}}\operatorname {sgn}(\sigma )a_{1,\sigma (1)}a_{n,\sigma (2)}} ã§ããã S 2 = { ( 1 2 1 2 ) , ( 1 2 2 1 ) } , sgn ( 1 2 1 2 ) = 1 , sgn ( 1 2 2 1 ) = â 1 {\displaystyle S_{2}=\left\{{\begin{pmatrix}1&2\\1&2\end{pmatrix}},{\begin{pmatrix}1&2\\2&1\end{pmatrix}}\right\},\ \operatorname {sgn} {\begin{pmatrix}1&2\\1&2\end{pmatrix}}=1,\ \operatorname {sgn} {\begin{pmatrix}1&2\\2&1\end{pmatrix}}=-1} ã§ããããè¡ååŒã¯ a d â b c {\displaystyle ad-bc} ã§ããã
3次ã®è¡ååŒã§ã¯ã
det A = | a b c d e f g h i | = a e i + b f g + c d h â a f h â b d i â c e g {\displaystyle \det A={\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\\\end{vmatrix}}=aei+bfg+cdh-afh-bdi-ceg}
ãšãªãã
ããã¯ããSarrus(ãµã©ã¹)ã®å±éããŸãã¯ãSarrusã®æ¹æ³ããããããããã®æ³ããšåŒã¶ãã®ã§ãå³å³ã®ããã«æãã«æ°ãä¹ãããã®ã®åãšèããããšãã§ããã äŸãã°ã第1é
a e i {\displaystyle aei} ã¯ã1è¡1åã® a {\displaystyle a} ããã3è¡3åã® i {\displaystyle i} ãŸã§ãå³äžã«åãã£ãŠé ã«ä¹ãããã®ã«çããããŸãã次㮠b f g {\displaystyle bfg} ã¯ã1è¡2åã® b {\displaystyle b} ããå§ããŠãå³äžã«åãã£ãŠé ã«ä¹ãããã®ã«çããã2è¡3åã® f {\displaystyle f} ã®æ¬¡ã¯ç«¯ãçªãæããŠã3è¡1åã® g {\displaystyle g} ã«è³ãã第3é
ãåæ§ã§ããã 4ãã6çªç®ã®é
ã¯ãå³äžã«åãã£ãŠã§ã¯ãªãå·Šäž(å³å³ã§ã¯å³äž)ã«åãã£ãŠä¹ããŠã笊å·ãå転ãããã®ã§ããã
4 à 4 {\displaystyle 4\times 4} 以éã®è¡åã§ã¯ãã®ãããªç°¡åãªèšç®æ³ã¯åŸãããªãã é
ã®æ°ã¯ n à n {\displaystyle n\times n} è¡å㧠n ! {\displaystyle n!} åã§ããããã倧ããªè¡åã«ã€ããŠèšç®æ©ã䜿ããã«è¡ååŒãèšç®ããã®ã¯å°é£ã§ããã
è¡ååŒã«ã€ããŠæãç«ã€æ§è³ªã®ãã¡ã以äžã®4ã€ã¯åºæ¬çã§ããã
1. ãš 2. ã®æ§è³ªãåãããŠãåã«ã€ããŠã®å€éç·åæ§ããšããã3. ã®æ§è³ªã¯ãåã«ã€ããŠã®äº€ä»£æ§ããšãããäžè¬ã«ä»»æã®æ£æ¹è¡å A {\displaystyle A} ã«ã€ã㊠| A | = | t A | {\displaystyle |A|=|{}^{t}\!A|} ã§ããããããããã®æ§è³ªã¯è¡ã«ã€ããŠãæãç«ã€ã
ä»»æã®æ£æ¹è¡åã«å¯ŸããŠããå®æ°ã察å¿ä»ããäœçšã®ãã¡ããã®4ã€ã®æ§è³ªãå
šãŠæºããã®ã¯è¡ååŒã ãã§ããããã®æ§è³ªãå®çŸ©ãšããŠè¡ååŒãå°åºã§ããã
| [
{
"paragraph_id": 0,
"tag": "p",
"text": "1 , 2 , ⯠, n {\\displaystyle {1,2,\\cdots ,n}} ãäºãã«éè€ããªãããã«ã 1 , 2 , ⯠, n {\\displaystyle {1,2,\\cdots ,n}} ã«ãã€ãæäœãn次ã®çœ®æãšããã",
"title": "眮æ"
},
{
"paragraph_id": 1,
"tag": "p",
"text": "眮æ Ï {\\displaystyle \\sigma } ã«ãã£ãŠiããã€ãããè¡ãå
ã Ï ( i ) {\\displaystyle \\sigma (i)} ãšè¡šãã",
"title": "眮æ"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "眮æ Ï {\\displaystyle \\sigma } ã¯ã次ã®ããã«ãäžã«ããšã®å
ããäžã®è¡ãå
ã䞊ã¹ãŠè¡šçŸãããã",
"title": "眮æ"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ããã¯ãè¡åãšåãè¡šçŸã ããè¡åã§ã¯ãªãããšã«æ³šæããã",
"title": "眮æ"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "äŸãã°ã 1ã2ã«ã2ã3ã«ã3ã1ã«ãã€ã眮æ Ï {\\displaystyle \\sigma } ã¯ã3次ã®çœ®æã§ããã Ï ( 1 ) = 2 , Ï ( 2 ) = 3 , Ï ( 3 ) = 1 {\\displaystyle \\sigma (1)=2,\\sigma (2)=3,\\sigma (3)=1} ãšãªãããã®çœ®æã¯ã Ï = ( 1 2 3 2 3 1 ) {\\displaystyle \\sigma ={\\begin{pmatrix}1&2&3\\\\2&3&1\\end{pmatrix}}} ãšè¡šããã",
"title": "眮æ"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "e = ( 1 2 ⯠n 1 2 ⯠n ) {\\displaystyle e={\\begin{pmatrix}1&2&\\cdots &n\\\\1&2&\\cdots &n\\end{pmatrix}}} ã®ããã«ããã¹ãŠã®æŽæ°ãå€åããªã眮æã®ããšãåäœçœ®æãšããã",
"title": "眮æ"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ãã眮æ Ï {\\displaystyle \\sigma } ã«å¯Ÿãã Ï â 1 = ( Ï ( 1 ) Ï ( 2 ) â¯ Ï ( n ) 1 2 ⯠n ) {\\displaystyle \\sigma ^{-1}={\\begin{pmatrix}\\sigma (1)&\\sigma (2)&\\cdots &\\sigma (n)\\\\1&2&\\cdots &n\\end{pmatrix}}} ãé眮æãšããã",
"title": "眮æ"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "n次ã®çœ®æå
šäœã®éåã S n {\\displaystyle S_{n}} ãšè¡šãã äŸãã°ã S 3 = { ( 1 2 3 1 2 3 ) , ( 1 2 3 1 3 2 ) , ( 1 2 3 3 2 1 ) , ( 1 2 3 2 1 3 ) , ( 1 2 3 3 1 2 ) , ( 1 2 3 2 3 1 ) } {\\displaystyle S_{3}=\\left\\{{\\begin{pmatrix}1&2&3\\\\1&2&3\\end{pmatrix}},{\\begin{pmatrix}1&2&3\\\\1&3&2\\end{pmatrix}},{\\begin{pmatrix}1&2&3\\\\3&2&1\\end{pmatrix}},{\\begin{pmatrix}1&2&3\\\\2&1&3\\end{pmatrix}},{\\begin{pmatrix}1&2&3\\\\3&1&2\\end{pmatrix}},{\\begin{pmatrix}1&2&3\\\\2&3&1\\end{pmatrix}}\\right\\}} ã§ããã",
"title": "眮æ"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "n次ã®çœ®æå
šäœã®éåã®åæ°ã n ! {\\displaystyle n!} ã§ããããšã¯èªæã§ãããã",
"title": "眮æ"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "眮æ Ï , Ï â S n {\\displaystyle \\sigma ,\\tau \\in S_{n}} ã«å¯Ÿãã眮æã®åæã Ï Ï = ( 1 2 ⯠n Ï ( 1 ) Ï ( 2 ) â¯ Ï ( n ) ) ( 1 2 ⯠n Ï ( 1 ) Ï ( 2 ) â¯ Ï ( n ) ) = ( 1 2 ⯠n Ï ( Ï ( 1 ) ) Ï ( Ï ( 2 ) ) â¯ Ï ( Ï ( n ) ) ) {\\displaystyle \\sigma \\tau ={\\begin{pmatrix}1&2&\\cdots &n\\\\\\sigma (1)&\\sigma (2)&\\cdots &\\sigma (n)\\end{pmatrix}}{\\begin{pmatrix}1&2&\\cdots &n\\\\\\tau (1)&\\tau (2)&\\cdots &\\tau (n)\\end{pmatrix}}={\\begin{pmatrix}1&2&\\cdots &n\\\\\\sigma (\\tau (1))&\\sigma (\\tau (2))&\\cdots &\\sigma (\\tau (n))\\end{pmatrix}}} ãšå®ããã ããã¯ã 1 †i †n {\\displaystyle 1\\leq i\\leq n} ã«å¯Ÿãã Ï Ï ( i ) = Ï ( Ï ( i ) ) {\\displaystyle \\sigma \\tau (i)=\\sigma (\\tau (i))} ãšè¡šèšããããšãã§ããã ãããããšãèšè¿°éãå°ãªããªãã䟿å©ã ããã",
"title": "眮æ"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "眮æã«ã€ããŠã以äžã®æ§è³ªãæãç«ã€ã",
"title": "眮æ"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "Ï = ( 1 2 ⯠i ⯠j ⯠n 1 2 ⯠j ⯠i ⯠n ) {\\displaystyle \\sigma ={\\begin{pmatrix}1&2&\\cdots &i&\\cdots &j&\\cdots n\\\\1&2&\\cdots &j&\\cdots &i&\\cdots n\\end{pmatrix}}} ã®ããã«ãiãšjã ãã亀æãã眮æãäºæãšããã",
"title": "眮æ"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "ä»»æã®çœ®æã¯äºæã®ç©ã§è¡šãããšãã§ããäºæã®åæ°ã®å¶å¥ã¯äºæã®ãšãæ¹ã«ããããåãã§ãããšããæ§è³ªãããã 眮æãäºæã®ç©ã§è¡šãããšããäºæã®åæ°ãå¶æ°åã®çœ®æãå¶çœ®æãå¥æ°åã®çœ®æãå¥çœ®æãšããã",
"title": "眮æ"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "sgn ( Ï ) = { 1 Ï ã å¶ çœ® æ ã® ãš ã â 1 Ï ã å¥ çœ® æ ã® ãš ã {\\displaystyle \\operatorname {sgn}(\\sigma )={\\begin{cases}1&\\sigma {\\text{ã å¶ çœ® æ ã® ãš ã}}\\\\-1&\\sigma {\\mbox{ã å¥ çœ® æ ã® ãš ã}}\\end{cases}}\\ } ã Ï {\\displaystyle \\sigma } ã®ç¬Šå·ãšããã",
"title": "眮æ"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "è¡å A = ( a 11 ⯠a 1 n â® â± â® a n 1 ⯠a n n ) {\\displaystyle A={\\begin{pmatrix}a_{11}&\\cdots &a_{1n}\\\\\\vdots &\\ddots &\\vdots \\\\a_{n1}&\\cdots &a_{nn}\\end{pmatrix}}} ã«å¯ŸããŠã",
"title": "è¡ååŒ"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "| A | = det A = â Ï â S n sgn ( Ï ) a 1 , Ï ( 1 ) ⯠a n , Ï ( n ) {\\displaystyle |A|=\\det A=\\sum _{\\sigma \\in S_{n}}\\operatorname {sgn}(\\sigma )a_{1,\\sigma (1)}\\cdots a_{n,\\sigma (n)}} ãAã®è¡ååŒãšããã",
"title": "è¡ååŒ"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "â» â Ï â S n {\\displaystyle \\sum _{\\sigma \\in S_{n}}} ãšã¯ã Ï {\\displaystyle \\sigma } ã« S n {\\displaystyle S_{n}} ã®å
ããã¹ãŠä»£å
¥ããŠè¶³ãåããããšããæå³ã§ããã ããšãã°ã A = { 1 , 2 , 3 } {\\displaystyle A=\\{1,2,3\\}} ã®ãšãã â i â A {\\displaystyle \\sum _{i\\in A}} ãš â i = 1 3 {\\displaystyle \\sum _{i=1}^{3}} ã¯åãæå³ã§ããã",
"title": "è¡ååŒ"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "2次æ£æ¹è¡å A = ( a b c d ) {\\displaystyle A={\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}}} ã®è¡ååŒãæ±ããŠã¿ããã è¡ååŒã®å®çŸ©ã«åœãŠã¯ãããšã | A | = â Ï â S 2 sgn ( Ï ) a 1 , Ï ( 1 ) a n , Ï ( 2 ) {\\displaystyle |A|=\\sum _{\\sigma \\in S_{2}}\\operatorname {sgn}(\\sigma )a_{1,\\sigma (1)}a_{n,\\sigma (2)}} ã§ããã S 2 = { ( 1 2 1 2 ) , ( 1 2 2 1 ) } , sgn ( 1 2 1 2 ) = 1 , sgn ( 1 2 2 1 ) = â 1 {\\displaystyle S_{2}=\\left\\{{\\begin{pmatrix}1&2\\\\1&2\\end{pmatrix}},{\\begin{pmatrix}1&2\\\\2&1\\end{pmatrix}}\\right\\},\\ \\operatorname {sgn} {\\begin{pmatrix}1&2\\\\1&2\\end{pmatrix}}=1,\\ \\operatorname {sgn} {\\begin{pmatrix}1&2\\\\2&1\\end{pmatrix}}=-1} ã§ããããè¡ååŒã¯ a d â b c {\\displaystyle ad-bc} ã§ããã",
"title": "è¡ååŒ"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "3次ã®è¡ååŒã§ã¯ã",
"title": "è¡ååŒ"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "det A = | a b c d e f g h i | = a e i + b f g + c d h â a f h â b d i â c e g {\\displaystyle \\det A={\\begin{vmatrix}a&b&c\\\\d&e&f\\\\g&h&i\\\\\\end{vmatrix}}=aei+bfg+cdh-afh-bdi-ceg}",
"title": "è¡ååŒ"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "ãšãªãã",
"title": "è¡ååŒ"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "ããã¯ããSarrus(ãµã©ã¹)ã®å±éããŸãã¯ãSarrusã®æ¹æ³ããããããããã®æ³ããšåŒã¶ãã®ã§ãå³å³ã®ããã«æãã«æ°ãä¹ãããã®ã®åãšèããããšãã§ããã äŸãã°ã第1é
a e i {\\displaystyle aei} ã¯ã1è¡1åã® a {\\displaystyle a} ããã3è¡3åã® i {\\displaystyle i} ãŸã§ãå³äžã«åãã£ãŠé ã«ä¹ãããã®ã«çããããŸãã次㮠b f g {\\displaystyle bfg} ã¯ã1è¡2åã® b {\\displaystyle b} ããå§ããŠãå³äžã«åãã£ãŠé ã«ä¹ãããã®ã«çããã2è¡3åã® f {\\displaystyle f} ã®æ¬¡ã¯ç«¯ãçªãæããŠã3è¡1åã® g {\\displaystyle g} ã«è³ãã第3é
ãåæ§ã§ããã 4ãã6çªç®ã®é
ã¯ãå³äžã«åãã£ãŠã§ã¯ãªãå·Šäž(å³å³ã§ã¯å³äž)ã«åãã£ãŠä¹ããŠã笊å·ãå転ãããã®ã§ããã",
"title": "è¡ååŒ"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "4 à 4 {\\displaystyle 4\\times 4} 以éã®è¡åã§ã¯ãã®ãããªç°¡åãªèšç®æ³ã¯åŸãããªãã é
ã®æ°ã¯ n à n {\\displaystyle n\\times n} è¡å㧠n ! {\\displaystyle n!} åã§ããããã倧ããªè¡åã«ã€ããŠèšç®æ©ã䜿ããã«è¡ååŒãèšç®ããã®ã¯å°é£ã§ããã",
"title": "è¡ååŒ"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "è¡ååŒã«ã€ããŠæãç«ã€æ§è³ªã®ãã¡ã以äžã®4ã€ã¯åºæ¬çã§ããã",
"title": "è¡ååŒ"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "1. ãš 2. ã®æ§è³ªãåãããŠãåã«ã€ããŠã®å€éç·åæ§ããšããã3. ã®æ§è³ªã¯ãåã«ã€ããŠã®äº€ä»£æ§ããšãããäžè¬ã«ä»»æã®æ£æ¹è¡å A {\\displaystyle A} ã«ã€ã㊠| A | = | t A | {\\displaystyle |A|=|{}^{t}\\!A|} ã§ããããããããã®æ§è³ªã¯è¡ã«ã€ããŠãæãç«ã€ã",
"title": "è¡ååŒ"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "ä»»æã®æ£æ¹è¡åã«å¯ŸããŠããå®æ°ã察å¿ä»ããäœçšã®ãã¡ããã®4ã€ã®æ§è³ªãå
šãŠæºããã®ã¯è¡ååŒã ãã§ããããã®æ§è³ªãå®çŸ©ãšããŠè¡ååŒãå°åºã§ããã",
"title": "è¡ååŒ"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "",
"title": "è¡ååŒ"
}
] | null | {{ããã²ãŒã·ã§ã³|æ¬=[[ç·å代æ°åŠ]]|åããŒãž=[[ç·å代æ°åŠ/ç·åæ¹çšåŒã®è§£|ç·åæ¹çšåŒã®è§£]]|ããŒãžå=è¡ååŒ|次ããŒãž=[[ç·åœ¢ä»£æ°åŠ/äœå åè¡å|äœå åè¡å]]}}
==眮æ==
===眮æ===
<math>{1,2, \cdots, n}</math>ãäºãã«éè€ããªãããã«ã<math>{1,2, \cdots, n}</math>ã«ãã€ãæäœã'''n次ã®çœ®æ'''ãšããã
眮æ<math>\sigma</math>ã«ãã£ãŠiããã€ãããè¡ãå
ã<math>\sigma (i)</math>ãšè¡šãã
眮æ<math>\sigma</math>ã¯ã次ã®ããã«ãäžã«ããšã®å
ããäžã®è¡ãå
ã䞊ã¹ãŠè¡šçŸãããã
:<math>\sigma = \begin{pmatrix} 1 & 2 & \cdots & n \\ \sigma (1) & \sigma (2) & \cdots & \sigma (n) \end{pmatrix}</math>
ããã¯ãè¡åãšåãè¡šçŸã ããè¡åã§ã¯ãªãããšã«æ³šæããã
äŸãã°ã
1ã2ã«ã2ã3ã«ã3ã1ã«ãã€ã眮æ<math>\sigma</math>ã¯ã3次ã®çœ®æã§ããã<math>\sigma (1) = 2, \sigma (2) = 3, \sigma (3) = 1</math>ãšãªãããã®çœ®æã¯ã
<math>\sigma = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{pmatrix}</math>ãšè¡šããã
===åäœçœ®æ===
<math>e = \begin{pmatrix} 1 & 2 & \cdots & n \\ 1 & 2 & \cdots &n \end{pmatrix}</math>ã®ããã«ããã¹ãŠã®æŽæ°ãå€åããªã眮æã®ããšã'''åäœçœ®æ'''ãšããã
===é眮æ===
ãã眮æ<math>\sigma</math>ã«å¯Ÿãã<math>\sigma ^{-1} = \begin{pmatrix} \sigma (1) & \sigma (2) & \cdots & \sigma (n) \\ 1 & 2 & \cdots & n \end{pmatrix}</math>ã'''é眮æ'''ãšããã
===眮æå
šäœã®éå===
n次ã®çœ®æå
šäœã®éåã<math>S_n</math>ãšè¡šãã
äŸãã°ã<math>S_3 = \left\{ \begin{pmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \end{pmatrix}, \begin{pmatrix} 1 & 2 & 3 \\ 1 & 3 & 2 \end{pmatrix}, \begin{pmatrix} 1 & 2 & 3 \\ 3 & 2 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 2 & 3 \\ 2 & 1 & 3 \end{pmatrix}, \begin{pmatrix} 1 & 2 & 3 \\ 3 & 1 & 2 \end{pmatrix}, \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{pmatrix} \right\}</math>ã§ããã
n次ã®çœ®æå
šäœã®éåã®åæ°ã<math>n!</math>ã§ããããšã¯èªæã§ãããã
===眮æã®åæ===
眮æ<math>\sigma, \tau \in S_n</math>ã«å¯Ÿãã眮æã®åæã<math>\sigma \tau = \begin{pmatrix} 1 & 2 & \cdots & n \\ \sigma (1) & \sigma (2) & \cdots & \sigma (n) \end{pmatrix} \begin{pmatrix} 1 & 2 & \cdots & n \\ \tau (1) & \tau (2) & \cdots & \tau (n) \end{pmatrix} =\begin{pmatrix} 1 & 2 & \cdots & n \\ \sigma (\tau (1)) & \sigma (\tau (2)) & \cdots & \sigma (\tau (n)) \end{pmatrix}</math>ãšå®ããã<br>
ããã¯ã<math>1 \le i \le n</math>ã«å¯Ÿãã<math>\sigma \tau (i) = \sigma (\tau(i))</math>ãšè¡šèšããããšãã§ããã
ãããããšãèšè¿°éãå°ãªããªãã䟿å©ã ããã
===眮æã®æ§è³ª===
眮æã«ã€ããŠã以äžã®æ§è³ªãæãç«ã€ã
#<math>(\sigma \tau) \rho = \sigma (\tau \rho) </math>
#<math> \sigma e = e \sigma = \sigma </math>
#<math>\sigma \sigma^{-1} = \sigma^{-1} \sigma = e</math>
; 蚌æ
#<math>1 \le i \le n</math>ã«å¯Ÿãã<br><math>((\sigma \tau) \rho)(i) = (\sigma \tau) (\rho (i)) = \sigma (\tau (\rho (i)))</math><br><br><math>(\sigma (\tau \rho))(i) = (\sigma)(\tau \rho (i)) = \sigma (\tau (\rho (i)))</math><br><br>ãã£ãŠã<math>(\sigma \tau) \rho = \sigma (\tau \rho) </math>ã§ããã<br><br><br>
#<math>1 \le i \le n</math>ã«å¯Ÿãã<br><math> (\sigma e)(i) = (\sigma (e(i))) = \sigma (i)</math><br><br><math> e \sigma = (e (\sigma(i))) = \sigma (i)</math><br><br>ãã£ãŠ<math> \sigma e = e \sigma = \sigma </math>ã§ããã<br><br><br>
#<math>1 \le i \le n</math>ã«å¯Ÿãã<br><math>(\sigma \sigma^{-1})(i) = (\sigma (\sigma^{-1} (i) )) = i</math><br><br><math>(\sigma^{-1} \sigma)(i) = (\sigma^{-1} (\sigma (i) )) = i</math><br><br>ãã£ãŠ<math>\sigma \sigma^{-1} = \sigma^{-1} \sigma = e</math>ã§ããã
===äºæ===
<math>\sigma = \begin{pmatrix} 1 & 2 & \cdots & i & \cdots & j & \cdots n \\ 1 & 2 & \cdots & j & \cdots & i & \cdots n \end{pmatrix}</math>ã®ããã«ãiãšjã ãã亀æãã眮æã'''äºæ'''ãšããã
ä»»æã®çœ®æã¯äºæã®ç©ã§è¡šãããšãã§ããäºæã®åæ°ã®å¶å¥ã¯äºæã®ãšãæ¹ã«ããããåãã§ãããšããæ§è³ªãããã
眮æãäºæã®ç©ã§è¡šãããšããäºæã®åæ°ãå¶æ°åã®çœ®æã'''å¶çœ®æ'''ãå¥æ°åã®çœ®æã'''å¥çœ®æ'''ãšããã
; 蚌æ
=== 笊å·===
<math>
\sgn(\sigma) = \begin{cases} 1 & \sigma \text{ã å¶ çœ® æ ã® ãš ã} \\ -1 & \sigma \mbox{ã å¥ çœ® æ ã® ãš ã} \end{cases}\
</math> ã <math>\sigma</math> ã®'''笊å·'''ãšããã
==è¡ååŒ==
è¡å
<math>
A =
\begin{pmatrix}
a _{11} & \cdots & a _{1n} \\
\vdots & \ddots & \vdots \\
a _{n1} & \cdots & a _{nn}
\end{pmatrix}
</math>
ã«å¯ŸããŠã
<math>
|A| = \det A = \sum _{\sigma \in S_n} \sgn(\sigma) a _{1, \sigma (1)} \cdots a _{n, \sigma (n)}
</math>
ãAã®è¡ååŒãšããã
â» <math>\sum _{\sigma \in S_n}</math> ãšã¯ã<math>\sigma</math> ã« <math>S_n</math> ã®å
ããã¹ãŠä»£å
¥ããŠè¶³ãåããããšããæå³ã§ããã<br>
ããšãã°ã<math>A=\{1,2,3\}</math> ã®ãšãã<math>\sum_{i \in A}</math> ãš <math>\sum_{i=1}^{3}</math> ã¯åãæå³ã§ããã
2次æ£æ¹è¡å<math>
A =
\begin{pmatrix}
a&b\\
c&d
\end{pmatrix}
</math>ã®è¡ååŒãæ±ããŠã¿ããã<br>
è¡ååŒã®å®çŸ©ã«åœãŠã¯ãããšã<math>|A| = \sum _{\sigma \in S_2} \sgn(\sigma) a _{1, \sigma (1)} a _{n, \sigma (2)}</math> ã§ããã<br>
<math>S_2 = \left\{ \begin{pmatrix} 1 & 2 \\ 1 & 2 \end{pmatrix}, \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} \right\},\ \sgn \begin{pmatrix} 1 & 2 \\ 1 & 2 \end{pmatrix} = 1,\ \sgn \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} = -1</math><br>
ã§ããããè¡ååŒã¯ <math>ad-bc</math> ã§ããã
3次ã®è¡ååŒã§ã¯ã
<math>
\det A =
\begin{vmatrix}
a & b & c \\
d & e & f \\
g & h & i \\
\end{vmatrix}
= aei + bfg + cdh - afh - bdi -ceg
</math>
ãšãªãã
[[File:Schema sarrus-regel.png|alt=|thumb|ãµã©ã¹ã®æ¹æ³: å·Šäžåã®è¡ååŒã¯ãèµ€ç·ã§çµãã æãäžé
ã®ç©ã®åããéç·ã§çµãã éæãäžé
ã®ç©ã®åãåŒãããã®ã«ãªãã]]
ããã¯ããSarrus(ãµã©ã¹)ã®å±éããŸãã¯ãSarrusã®æ¹æ³ããããããããã®æ³ããšåŒã¶ãã®ã§ãå³å³ã®ããã«æãã«æ°ãä¹ãããã®ã®åãšèããããšãã§ããã
äŸãã°ã第1é
<math>aei</math> ã¯ã1è¡1åã® <math>a</math> ããã3è¡3åã® <math>i</math> ãŸã§ãå³äžã«åãã£ãŠé ã«ä¹ãããã®ã«çããããŸãã次㮠<math>bfg</math> ã¯ã1è¡2åã® <math>b</math> ããå§ããŠãå³äžã«åãã£ãŠé ã«ä¹ãããã®ã«çããã2è¡3åã® <math>f</math> ã®æ¬¡ã¯ç«¯ãçªãæããŠã3è¡1åã® <math>g</math> ã«è³ãã第3é
ãåæ§ã§ããã
4ãã6çªç®ã®é
ã¯ãå³äžã«åãã£ãŠã§ã¯ãªãå·ŠäžïŒå³å³ã§ã¯å³äžïŒã«åãã£ãŠä¹ããŠã笊å·ãå転ãããã®ã§ããã
<math>4 \times 4</math> 以éã®è¡åã§ã¯ãã®ãããªç°¡åãªèšç®æ³ã¯åŸãããªãã
é
ã®æ°ã¯ <math>n \times n</math> è¡å㧠<math>n!</math> åã§ããããã倧ããªè¡åã«ã€ããŠèšç®æ©ã䜿ããã«è¡ååŒãèšç®ããã®ã¯å°é£ã§ããã
===è¡ååŒã®åºæ¬æ§è³ª===
è¡ååŒã«ã€ããŠæãç«ã€æ§è³ªã®ãã¡ã以äžã®4ã€ã¯åºæ¬çã§ããã
#<math>\begin{vmatrix}
a_{1,1} & \cdots & a_{1,i} + a_{1,i}' & \cdots & a_{1,n} \\
\vdots & \ddots & \vdots & \ddots & \vdots \\
a_{n,1} & \cdots & a_{n,i} + a_{n,i}' & \cdots & a_{n,n} \\
\end{vmatrix} =
\begin{vmatrix}
a_{1,1} & \cdots & a_{1,i} & \cdots & a_{1,n} \\
\vdots & \ddots & \vdots & \ddots & \vdots \\
a_{n,1} & \cdots & a_{n,i} & \cdots & a_{n,n} \\ \end{vmatrix} +
\begin{vmatrix}
a_{1,1} & \cdots & a_{1,i}' & \cdots & a_{1,n} \\
\vdots & \ddots & \vdots & \ddots & \vdots \\
a_{n,1} & \cdots & a_{n,i}' & \cdots & a_{n,n} \\
\end{vmatrix}
</math>
#<math>\begin{vmatrix}
a_{1,1} & \cdots & c a_{1,i} & \cdots & a_{1,n} \\
\vdots & \ddots & \vdots & \ddots & \vdots \\
a_{n,1} & \cdots & c a_{n,i} & \cdots & a_{n,n} \\ \end{vmatrix} =
c \begin{vmatrix}
a_{1,1} & \cdots & a_{1,i} & \cdots & a_{1,n} \\
\vdots & \ddots & \vdots & \ddots & \vdots \\
a_{n,1} & \cdots & a_{n,i} & \cdots & a_{n,n} \\ \end{vmatrix}
</math>
#<math>\begin{vmatrix}a_{1,1} & \cdots & a_{1,i} & \cdots & a_{1,j} & \cdots & a_{1,n} \\
\vdots & \ddots & \vdots & \ddots& \vdots & \ddots & \vdots \\
a_{n,1} & \cdots & a_{n,i} & \cdots & a_{n,j} & \cdots & a_{n,n} \\ \end{vmatrix}
= - \begin{vmatrix}a_{1,1} & \cdots & a_{1,j} & \cdots & a_{1,i} & \cdots & a_{1,n} \\
\vdots & \ddots & \vdots & \ddots& \vdots & \ddots & \vdots \\
a_{n,1} & \cdots & a_{n,j} & \cdots & a_{n,i} & \cdots & a_{n,n} \\ \end{vmatrix}</math>
#åäœè¡åã®è¡ååŒã¯1ã
1. ãš 2. ã®æ§è³ªãåãããŠãåã«ã€ããŠã®'''å€éç·åæ§'''ããšããã3. ã®æ§è³ªã¯ãåã«ã€ããŠã®'''亀代æ§'''ããšãããäžè¬ã«ä»»æã®æ£æ¹è¡å <math>A</math> ã«ã€ããŠ<math>|A|=|{}^t\!A|</math> ã§ããããããããã®æ§è³ªã¯è¡ã«ã€ããŠãæãç«ã€ã
; 蚌æ
#<math>\sum_{\sigma \in S_n} \sgn(\sigma) a_{1,\sigma(1)} \cdots (a_{i,\sigma(i)} + a_{i,\sigma(i)}') \cdots a_{n,\sigma(n)}
= \sum_{\sigma \in S_n} (\sgn(\sigma) a_{1,\sigma(1)} \cdots a_{i,\sigma(i)} \cdots a_{n,\sigma(n)}
+ \sgn(\sigma) a_{1,\sigma(1)} \cdots a_{i,\sigma(i)}' \cdots a_{n,\sigma(n)})</math><br><math>
= \sum_{\sigma \in S_n} \sgn(\sigma) a_{1,\sigma(1)} \cdots a_{i,\sigma(i)} \cdots a_{n,\sigma(n)}
+ \sum_{\sigma \in S_n} \sgn(\sigma) a_{1,\sigma(1)} \cdots a_{i,\sigma(i)}' \cdots a_{n,\sigma(n)}.</math> ãã£ãŠèšŒæãããã
#<math>\sum_{\sigma \in S_n} \sgn(\sigma) a_{1,\sigma(1)} \cdots c a_{i,\sigma(i)} \cdots a_{n,\sigma(n)}
= c \sum_{\sigma \in S_n} \sgn(\sigma) a_{1,\sigma(1)} \cdots a_{i,\sigma(i)} \cdots a_{n,\sigma(n)}.</math> ãã£ãŠèšŒæãããã
# n次ã®çœ®æ <math>\sigma</math> ã« <math>i,j</math> ã®äºæãåæãã眮æã <math>\tau</math> ãšããããã®ãšã <math>\sigma(i)=\tau(j),\ \sigma(j)=\tau(i),\ \sigma(k)=\tau(k)\ (k\neq i,j)</math> ã§ããããã <math>\sigma</math> ãå¥çœ®æã§ããã° <math>\tau</math> ã¯å¶çœ®æã<math>\sigma</math> ãå¶çœ®æã§ããã° <math>\tau</math> ã¯å¥çœ®æã§ãããã <math>\sgn(\tau) = - \sgn(\sigma)</math> ã§ãããããã«<br><math>
\sum_{\sigma \in S_n} \sgn(\sigma) a_{1,\sigma(1)} \cdots a_{i,\sigma(i)} \cdots a_{j,\sigma(j)} \cdots a_{n,\sigma(n)}
= \sum_{\tau \in S_n} (- \sgn(\tau)) a_{1,\tau(1)} \cdots a_{i,\tau(j)} \cdots a_{j,\tau(i)} \cdots a_{n,\tau(n)}</math><br><math>
= - \sum_{\tau \in S_n} \sgn(\tau) a_{1,\tau(1)} \cdots a_{i,\tau(i)} \cdots a_{j,\tau(j)} \cdots a_{n,\tau(n)}.</math> ãã£ãŠèšŒæãããã
# è¡ååŒãèšç®ãããšã察è§æåã®ç©ã®é
ã1ããã以å€ã®é
ã¯0ã«ãªãããšããçŽã¡ã«åŸãããã
: (転眮ã«ã€ããŠã®äžå€æ§)ãä»»æã®çœ®æãšãã®é眮æã«ã€ããŠç¬Šå·ã¯çããããã<math>\tau = \sigma^{-1}</math> ãšããŠä»¥äžã®ããã«ç€ºãããã
:: <math>|{}^t\!A| = \sum_{\sigma \in S_n} \sgn(\sigma) a_{\sigma(1),1} \cdots a_{\sigma(n),n}
= \sum_{\sigma \in S_n} \sgn(\sigma^{-1}) a_{1,\sigma^{-1}(1)} \cdots a_{n,\sigma^{-1}(n)}
= \sum_{\tau \in S_n} \sgn(\tau) a_{1,\tau(1)} \cdots a_{n,\tau(n)} = |A|.</math>
ä»»æã®æ£æ¹è¡åã«å¯ŸããŠããå®æ°ã察å¿ä»ããäœçšã®ãã¡ããã®4ã€ã®æ§è³ªãå
šãŠæºããã®ã¯è¡ååŒã ãã§ããããã®æ§è³ªãå®çŸ©ãšããŠè¡ååŒãå°åºã§ããã
{{ããã²ãŒã·ã§ã³|æ¬=[[ç·å代æ°åŠ]]|åããŒãž=[[ç·å代æ°åŠ/ç·åæ¹çšåŒã®è§£|ç·åæ¹çšåŒã®è§£]]|ããŒãžå=è¡ååŒ|次ããŒãž=[[ç·åœ¢ä»£æ°åŠ/äœå åè¡å|äœå åè¡å]]}}
[[Category:ç·åœ¢ä»£æ°åŠ|ããããã€ãã]] | null | 2021-03-09T12:38:08Z | [
"ãã³ãã¬ãŒã:ããã²ãŒã·ã§ã³"
] | https://ja.wikibooks.org/wiki/%E7%B7%9A%E5%BD%A2%E4%BB%A3%E6%95%B0%E5%AD%A6/%E8%A1%8C%E5%88%97%E5%BC%8F |
2,013 | ç·åœ¢ä»£æ°åŠ/äœå åè¡å | æ£æ¹è¡å A {\displaystyle A} ã«å¯ŸããŠã è¡åã® i {\displaystyle i} è¡ç®ãš j {\displaystyle j} åç®ãåãé€ããŠåŸãããè¡åã A i j {\displaystyle A_{ij}} ãšè¡šãããã®ãšãã
a ~ i j = ( â 1 ) i + j | A i j | {\displaystyle {\tilde {a}}_{ij}=(-1)^{i+j}|A_{ij}|} ã A {\displaystyle A} ã® ( i , j ) {\displaystyle (i,j)} äœå åãšããã
( 5 0 8 1 9 3 7 5 2 ) {\displaystyle {\begin{pmatrix}5&0&8\\1&9&3\\7&5&2\end{pmatrix}}} ã® ( 2 , 2 ) {\displaystyle (2,2)} äœå åã¯ã ( â 1 ) 2 + 2 | 5 8 7 2 | = â 46 {\displaystyle (-1)^{2+2}{\begin{vmatrix}5&8\\7&2\end{vmatrix}}=-46} ã§ããã
次ã®ããã«ãäœå åãå©çšããããšã§ãè¡ååŒãæ±ããããšãã§ããã
| A | = a j 1 a ~ j 1 + a j 2 a ~ j 2 + ⯠+ a j n a ~ j n ( 1 †j †n ) {\displaystyle |A|=a_{j1}{\tilde {a}}_{j1}+a_{j2}{\tilde {a}}_{j2}+\cdots +a_{jn}{\tilde {a}}_{jn}(1\leq j\leq n)}
| A | = a 1 i a ~ 1 i + a 2 i a ~ 2 i + ⯠+ a n i a ~ n i ( 1 †i †n ) {\displaystyle |A|=a_{1i}{\tilde {a}}_{1i}+a_{2i}{\tilde {a}}_{2i}+\cdots +a_{ni}{\tilde {a}}_{ni}(1\leq i\leq n)}
ãã ãã A {\displaystyle A} 㯠n {\displaystyle n} 次æ£æ¹è¡åã§ããã
ããããäœå åå±éãšããã
蚌æ
A = ( a 11 ⯠a 1 n ⮠Ⱡ⮠a n 1 ⯠a n n ) {\displaystyle A={\begin{pmatrix}a_{11}&\cdots &a_{1n}\\\vdots &\ddots &\vdots \\a_{n1}&\cdots &a_{nn}\end{pmatrix}}}
ãšããããã®ãšãã
ã§ãããããã§ãè¡å A {\displaystyle A} ã® j {\displaystyle j} åç® ( a 1 j a 2 j â® a n j ) {\displaystyle {\begin{pmatrix}a_{1j}\\a_{2j}\\\vdots \\a_{nj}\end{pmatrix}}} ã¯ã a 1 j ( 1 0 â® 0 ) + a 2 j ( 0 1 â® 0 ) + ⯠+ a n j ( 0 0 â® 1 ) {\displaystyle a_{1j}{\begin{pmatrix}1\\0\\\vdots \\0\end{pmatrix}}+a_{2j}{\begin{pmatrix}0\\1\\\vdots \\0\end{pmatrix}}+\cdots +a_{nj}{\begin{pmatrix}0\\0\\\vdots \\1\end{pmatrix}}} ãšè¡šãããšãã§ãã (1)åŒã¯ã | ( a 11 a 21 â® a n 1 ) , ⯠, a 1 j ( 1 0 â® 0 ) + a 2 j ( 0 1 â® 0 ) + ⯠+ a n j ( 0 0 â® 1 ) , ⯠, ( a n 1 a n 2 â® a n n ) | {\displaystyle \left|{\begin{pmatrix}a_{11}\\a_{21}\\\vdots \\a_{n1}\end{pmatrix}},\cdots ,a_{1j}{\begin{pmatrix}1\\0\\\vdots \\0\end{pmatrix}}+a_{2j}{\begin{pmatrix}0\\1\\\vdots \\0\end{pmatrix}}+\cdots +a_{nj}{\begin{pmatrix}0\\0\\\vdots \\1\end{pmatrix}},\cdots ,{\begin{pmatrix}a_{n1}\\a_{n2}\\\vdots \\a_{nn}\end{pmatrix}}\right|} ãšãè¡šãããšãã§ãããããã«ãè¡ååŒã®æ§è³ªã䜿ãã°ã a 1 j | a 11 ⯠1 ⯠a 1 n a 21 ⯠0 ⯠a 2 n â® â± â® â± â® a n 1 ⯠0 ⯠a n n | + a 2 j | a 11 ⯠0 ⯠a 1 n a 21 ⯠1 ⯠a 2 n â® â± â® â± â® a n 1 ⯠0 ⯠a n n | + ⯠+ a n j | a 11 ⯠0 ⯠a 1 n a 21 ⯠0 ⯠a 2 n â® â± â® â± â® a n 1 ⯠1 ⯠a n n | ⯠( 2 ) {\displaystyle a_{1j}{\begin{vmatrix}a_{11}&\cdots &1&\cdots &a_{1n}\\a_{21}&\cdots &0&\cdots &a_{2n}\\\vdots &\ddots &\vdots &\ddots &\vdots \\a_{n1}&\cdots &0&\cdots &a_{nn}\end{vmatrix}}+a_{2j}{\begin{vmatrix}a_{11}&\cdots &0&\cdots &a_{1n}\\a_{21}&\cdots &1&\cdots &a_{2n}\\\vdots &\ddots &\vdots &\ddots &\vdots \\a_{n1}&\cdots &0&\cdots &a_{nn}\end{vmatrix}}+\cdots +a_{nj}{\begin{vmatrix}a_{11}&\cdots &0&\cdots &a_{1n}\\a_{21}&\cdots &0&\cdots &a_{2n}\\\vdots &\ddots &\vdots &\ddots &\vdots \\a_{n1}&\cdots &1&\cdots &a_{nn}\end{vmatrix}}\cdots (2)} ã§ããã
ããã§ã | a 11 ⯠0 ⯠a 1 n a 21 ⯠0 ⯠a 2 n â® â± â® â± â® a i 1 ⯠1 ⯠a i n â® â± â® â± â® a n 1 ⯠0 ⯠a n n | {\displaystyle {\begin{vmatrix}a_{11}&\cdots &0&\cdots &a_{1n}\\a_{21}&\cdots &0&\cdots &a_{2n}\\\vdots &\ddots &\vdots &\ddots &\vdots \\a_{i1}&\cdots &1&\cdots &a_{in}\\\vdots &\ddots &\vdots &\ddots &\vdots \\a_{n1}&\cdots &0&\cdots &a_{nn}\end{vmatrix}}} ã«ã€ããŠèããã
ãã®è¡åã® i {\displaystyle i} è¡ç®ãšã i â 1 {\displaystyle i-1} è¡ç®ãå
¥ãæ¿ãã i â 1 {\displaystyle i-1} è¡ç®ãšã i â 2 {\displaystyle i-2} è¡ç®ãå
¥ãæ¿ãããã»ã»ã» 2 {\displaystyle 2} è¡ç®ãšã 1 {\displaystyle 1} è¡ç®ãå
¥ãæ¿ããããšããæäœããããšã次ã®ãããªè¡åã«ãªãã ( â 1 ) i â 1 | a i 1 ⯠1 ⯠a i n a 11 ⯠0 ⯠a 1 n a 21 ⯠0 ⯠a 2 n â® â± â® â± â® a i â 1 , 1 ⯠0 ⯠a i â 1 , n a i + 1 , 1 ⯠0 ⯠a i + 1 , n â® â± â® â± â® a n 1 ⯠0 ⯠a n n | {\displaystyle (-1)^{i-1}{\begin{vmatrix}a_{i1}&\cdots &1&\cdots &a_{in}\\a_{11}&\cdots &0&\cdots &a_{1n}\\a_{21}&\cdots &0&\cdots &a_{2n}\\\vdots &\ddots &\vdots &\ddots &\vdots \\a_{i-1,1}&\cdots &0&\cdots &a_{i-1,n}\\a_{i+1,1}&\cdots &0&\cdots &a_{i+1,n}\\\vdots &\ddots &\vdots &\ddots &\vdots \\a_{n1}&\cdots &0&\cdots &a_{nn}\end{vmatrix}}}
è¡åã®è¡ãŸãã¯åãå
¥ãæ¿ãããšãè¡ååŒã®å€ã¯ â 1 {\displaystyle -1} åãããã®ã ã£ãããã®æäœã§ã¯ã i â 1 {\displaystyle i-1} åã®å
¥ãæ¿ããè¡ãã®ã§ããã®åŒã¯ã ( â 1 ) i â 1 {\displaystyle (-1)^{i-1}} åãããŠããã
次ã«ãåãããã«ã j {\displaystyle j} åç®ãšã j â 1 {\displaystyle j-1} åç®ãå
¥ãæ¿ããã j â 1 {\displaystyle j-1} åç®ãšã j â 2 {\displaystyle j-2} åç®ãå
¥ãæ¿ãããã»ã»ã» 2 {\displaystyle 2} åç®ãšã 1 {\displaystyle 1} åç®ãå
¥ãæ¿ããããšããæäœãããããããšã次ã®ãããªè¡åã«ãªãã
( â 1 ) i + j | 1 a i 1 ⯠a i , j â 1 a i , j + 1 ⯠a i n 0 a 11 ⯠a 1 , j â 1 a 1 , j + 1 ⯠a 1 n 0 a 12 ⯠a 2 , j â 1 a 2 , j + 1 ⯠a 2 n â® â® â± â® â® â± â® 0 a i â 1 , 1 ⯠a i â 1 , j â 1 a i â 1 , j + 1 ⯠a i â 1 , n 0 a i + 1 , 1 ⯠a i + 1 , j â 1 a i + 1 , j + 1 ⯠a i + 1 , n â® â® â± â® â® â± â® 0 a n 1 ⯠a n , j â 1 a n , j + 1 ⯠a n n | {\displaystyle (-1)^{i+j}{\begin{vmatrix}1&a_{i1}&\cdots &a_{i,j-1}&a_{i,j+1}&\cdots &a_{in}\\0&a_{11}&\cdots &a_{1,j-1}&a_{1,j+1}&\cdots &a_{1n}\\0&a_{12}&\cdots &a_{2,j-1}&a_{2,j+1}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots &\vdots &\ddots &\vdots \\0&a_{i-1,1}&\cdots &a_{i-1,j-1}&a_{i-1,j+1}&\cdots &a_{i-1,n}\\0&a_{i+1,1}&\cdots &a_{i+1,j-1}&a_{i+1,j+1}&\cdots &a_{i+1,n}\\\vdots &\vdots &\ddots &\vdots &\vdots &\ddots &\vdots \\0&a_{n1}&\cdots &a_{n,j-1}&a_{n,j+1}&\cdots &a_{nn}\end{vmatrix}}}
( â 1 ) i + j â 2 = ( â 1 ) i + j {\displaystyle (-1)^{i+j-2}=(-1)^{i+j}} ã§ããããšã«ã€ããŠã®èª¬æã¯äžèŠã§ãããã ããããè¡ååŒã®å®çŸ©ã«åŸã£ãŠå±éããã
äžè¡ç®ã§ã(1,1)èŠçŽ ãéžã°ãªãé
ã¯ãããããäžåç®ã®0ãéžã¶ã®ã§ã0ãšãªãã ãªã®ã§ãäžè¡ç®ã§ã(1,1)èŠçŽ ãéžã¶é
ã ããèããã°è¯ãããããã¯ã | A i , j | {\displaystyle |A_{i,j}|} ãšäžèŽããã ãã£ãŠããã®è¡ååŒã¯ã ( â 1 ) i + j | A i j | = a ~ i j {\displaystyle (-1)^{i+j}|A_{ij}|={\tilde {a}}_{ij}} ã§ããã
ãããã(2)åŒã«ä»£å
¥ããã°ã | A | = a j 1 a ~ j 1 + a j 2 a ~ j 2 + ⯠+ a j n a ~ j n {\displaystyle |A|=a_{j1}{\tilde {a}}_{j1}+a_{j2}{\tilde {a}}_{j2}+\cdots +a_{jn}{\tilde {a}}_{jn}} ãšãªãã蚌æãããã
ãããšåæ§ã®è°è«ãè¡ã«ãè¡ãã°ãããäžæ¹ã®åŒãå°ãããšãã§ããã
A ~ = ( a ~ j , i ) {\displaystyle {\tilde {A}}=({\tilde {a}}_{j,i})} ãAã®äœå åè¡åãšããã
äœå åè¡åã«ã¯ã以äžã®æ§è³ªãããã
蚌æ
A ~ A = ( a ~ 11 ⯠a ~ m 1 â® â± â® a ~ 1 n ⯠a ~ m n ) ( a 11 ⯠a 1 n â® â± â® a n 1 ⯠a m n ) {\displaystyle {\tilde {A}}A={\begin{pmatrix}{\tilde {a}}_{11}&\cdots &{\tilde {a}}_{m1}\\\vdots &\ddots &\vdots \\{\tilde {a}}_{1n}&\cdots &{\tilde {a}}_{mn}\end{pmatrix}}{\begin{pmatrix}a_{11}&\cdots &a_{1n}\\\vdots &\ddots &\vdots \\a_{n1}&\cdots &a_{mn}\end{pmatrix}}} ãªã®ã§ã è¡å A ~ A {\displaystyle {\tilde {A}}A} ã® ( i , j ) {\displaystyle (i,j)} æåã¯ã
a 1 i a ~ 1 j + a 2 i a ~ 2 j + ⯠+ a n i a ~ n j ⯠( 1 ) {\displaystyle a_{1i}{\tilde {a}}_{1j}+a_{2i}{\tilde {a}}_{2j}+\cdots +a_{ni}{\tilde {a}}_{nj}\cdots (1)} ã§ããã
(i) i = j {\displaystyle i=j} ã®ãšã
(ii) i â j {\displaystyle i\neq j} ã®ãšã
ãŸãšãããšã a 1 i a ~ 1 j + a 2 i a ~ 2 j + ⯠+ a n i a ~ n j = { | A | ( i = j ) 0 ( i â j ) {\displaystyle a_{1i}{\tilde {a}}_{1j}+a_{2i}{\tilde {a}}_{2j}+\cdots +a_{ni}{\tilde {a}}_{nj}={\begin{cases}|A|(i=j)\\0(i\neq j)\\\end{cases}}} ã§ããã ãã£ãŠ A ~ A = | A | E {\displaystyle {\tilde {A}}A=|A|E} ã§ãããåæ§ã®è°è«ãè¡ãã°ã A A ~ = | A | E {\displaystyle A{\tilde {A}}=|A|E} ãå°ãããšãã§ããã
| A | â 0 {\displaystyle |A|\neq 0} ã®ãšã A â 1 {\displaystyle A^{-1}} ãååšããã®ã§ã A ~ A = | A | E {\displaystyle {\tilde {A}}A=|A|E} ã« A â 1 {\displaystyle A^{-1}} ãå³ãããã | A | {\displaystyle |A|} ã§å²ãã°ã A â 1 = A ~ | A | {\displaystyle A^{-1}={\frac {\tilde {A}}{|A|}}} ã§ããäºããããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "æ£æ¹è¡å A {\\displaystyle A} ã«å¯ŸããŠã è¡åã® i {\\displaystyle i} è¡ç®ãš j {\\displaystyle j} åç®ãåãé€ããŠåŸãããè¡åã A i j {\\displaystyle A_{ij}} ãšè¡šãããã®ãšãã",
"title": "äœå åè¡å"
},
{
"paragraph_id": 1,
"tag": "p",
"text": "a ~ i j = ( â 1 ) i + j | A i j | {\\displaystyle {\\tilde {a}}_{ij}=(-1)^{i+j}|A_{ij}|} ã A {\\displaystyle A} ã® ( i , j ) {\\displaystyle (i,j)} äœå åãšããã",
"title": "äœå åè¡å"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "( 5 0 8 1 9 3 7 5 2 ) {\\displaystyle {\\begin{pmatrix}5&0&8\\\\1&9&3\\\\7&5&2\\end{pmatrix}}} ã® ( 2 , 2 ) {\\displaystyle (2,2)} äœå åã¯ã ( â 1 ) 2 + 2 | 5 8 7 2 | = â 46 {\\displaystyle (-1)^{2+2}{\\begin{vmatrix}5&8\\\\7&2\\end{vmatrix}}=-46} ã§ããã",
"title": "äœå åè¡å"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "次ã®ããã«ãäœå åãå©çšããããšã§ãè¡ååŒãæ±ããããšãã§ããã",
"title": "äœå åè¡å"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "| A | = a j 1 a ~ j 1 + a j 2 a ~ j 2 + ⯠+ a j n a ~ j n ( 1 †j †n ) {\\displaystyle |A|=a_{j1}{\\tilde {a}}_{j1}+a_{j2}{\\tilde {a}}_{j2}+\\cdots +a_{jn}{\\tilde {a}}_{jn}(1\\leq j\\leq n)}",
"title": "äœå åè¡å"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "| A | = a 1 i a ~ 1 i + a 2 i a ~ 2 i + ⯠+ a n i a ~ n i ( 1 †i †n ) {\\displaystyle |A|=a_{1i}{\\tilde {a}}_{1i}+a_{2i}{\\tilde {a}}_{2i}+\\cdots +a_{ni}{\\tilde {a}}_{ni}(1\\leq i\\leq n)}",
"title": "äœå åè¡å"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ãã ãã A {\\displaystyle A} 㯠n {\\displaystyle n} 次æ£æ¹è¡åã§ããã",
"title": "äœå åè¡å"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ããããäœå åå±éãšããã",
"title": "äœå åè¡å"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "蚌æ",
"title": "äœå åè¡å"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "A = ( a 11 ⯠a 1 n ⮠Ⱡ⮠a n 1 ⯠a n n ) {\\displaystyle A={\\begin{pmatrix}a_{11}&\\cdots &a_{1n}\\\\\\vdots &\\ddots &\\vdots \\\\a_{n1}&\\cdots &a_{nn}\\end{pmatrix}}}",
"title": "äœå åè¡å"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "ãšããããã®ãšãã",
"title": "äœå åè¡å"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "ã§ãããããã§ãè¡å A {\\displaystyle A} ã® j {\\displaystyle j} åç® ( a 1 j a 2 j â® a n j ) {\\displaystyle {\\begin{pmatrix}a_{1j}\\\\a_{2j}\\\\\\vdots \\\\a_{nj}\\end{pmatrix}}} ã¯ã a 1 j ( 1 0 â® 0 ) + a 2 j ( 0 1 â® 0 ) + ⯠+ a n j ( 0 0 â® 1 ) {\\displaystyle a_{1j}{\\begin{pmatrix}1\\\\0\\\\\\vdots \\\\0\\end{pmatrix}}+a_{2j}{\\begin{pmatrix}0\\\\1\\\\\\vdots \\\\0\\end{pmatrix}}+\\cdots +a_{nj}{\\begin{pmatrix}0\\\\0\\\\\\vdots \\\\1\\end{pmatrix}}} ãšè¡šãããšãã§ãã (1)åŒã¯ã | ( a 11 a 21 â® a n 1 ) , ⯠, a 1 j ( 1 0 â® 0 ) + a 2 j ( 0 1 â® 0 ) + ⯠+ a n j ( 0 0 â® 1 ) , ⯠, ( a n 1 a n 2 â® a n n ) | {\\displaystyle \\left|{\\begin{pmatrix}a_{11}\\\\a_{21}\\\\\\vdots \\\\a_{n1}\\end{pmatrix}},\\cdots ,a_{1j}{\\begin{pmatrix}1\\\\0\\\\\\vdots \\\\0\\end{pmatrix}}+a_{2j}{\\begin{pmatrix}0\\\\1\\\\\\vdots \\\\0\\end{pmatrix}}+\\cdots +a_{nj}{\\begin{pmatrix}0\\\\0\\\\\\vdots \\\\1\\end{pmatrix}},\\cdots ,{\\begin{pmatrix}a_{n1}\\\\a_{n2}\\\\\\vdots \\\\a_{nn}\\end{pmatrix}}\\right|} ãšãè¡šãããšãã§ãããããã«ãè¡ååŒã®æ§è³ªã䜿ãã°ã a 1 j | a 11 ⯠1 ⯠a 1 n a 21 ⯠0 ⯠a 2 n â® â± â® â± â® a n 1 ⯠0 ⯠a n n | + a 2 j | a 11 ⯠0 ⯠a 1 n a 21 ⯠1 ⯠a 2 n â® â± â® â± â® a n 1 ⯠0 ⯠a n n | + ⯠+ a n j | a 11 ⯠0 ⯠a 1 n a 21 ⯠0 ⯠a 2 n â® â± â® â± â® a n 1 ⯠1 ⯠a n n | ⯠( 2 ) {\\displaystyle a_{1j}{\\begin{vmatrix}a_{11}&\\cdots &1&\\cdots &a_{1n}\\\\a_{21}&\\cdots &0&\\cdots &a_{2n}\\\\\\vdots &\\ddots &\\vdots &\\ddots &\\vdots \\\\a_{n1}&\\cdots &0&\\cdots &a_{nn}\\end{vmatrix}}+a_{2j}{\\begin{vmatrix}a_{11}&\\cdots &0&\\cdots &a_{1n}\\\\a_{21}&\\cdots &1&\\cdots &a_{2n}\\\\\\vdots &\\ddots &\\vdots &\\ddots &\\vdots \\\\a_{n1}&\\cdots &0&\\cdots &a_{nn}\\end{vmatrix}}+\\cdots +a_{nj}{\\begin{vmatrix}a_{11}&\\cdots &0&\\cdots &a_{1n}\\\\a_{21}&\\cdots &0&\\cdots &a_{2n}\\\\\\vdots &\\ddots &\\vdots &\\ddots &\\vdots \\\\a_{n1}&\\cdots &1&\\cdots &a_{nn}\\end{vmatrix}}\\cdots (2)} ã§ããã",
"title": "äœå åè¡å"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "ããã§ã | a 11 ⯠0 ⯠a 1 n a 21 ⯠0 ⯠a 2 n â® â± â® â± â® a i 1 ⯠1 ⯠a i n â® â± â® â± â® a n 1 ⯠0 ⯠a n n | {\\displaystyle {\\begin{vmatrix}a_{11}&\\cdots &0&\\cdots &a_{1n}\\\\a_{21}&\\cdots &0&\\cdots &a_{2n}\\\\\\vdots &\\ddots &\\vdots &\\ddots &\\vdots \\\\a_{i1}&\\cdots &1&\\cdots &a_{in}\\\\\\vdots &\\ddots &\\vdots &\\ddots &\\vdots \\\\a_{n1}&\\cdots &0&\\cdots &a_{nn}\\end{vmatrix}}} ã«ã€ããŠèããã",
"title": "äœå åè¡å"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ãã®è¡åã® i {\\displaystyle i} è¡ç®ãšã i â 1 {\\displaystyle i-1} è¡ç®ãå
¥ãæ¿ãã i â 1 {\\displaystyle i-1} è¡ç®ãšã i â 2 {\\displaystyle i-2} è¡ç®ãå
¥ãæ¿ãããã»ã»ã» 2 {\\displaystyle 2} è¡ç®ãšã 1 {\\displaystyle 1} è¡ç®ãå
¥ãæ¿ããããšããæäœããããšã次ã®ãããªè¡åã«ãªãã ( â 1 ) i â 1 | a i 1 ⯠1 ⯠a i n a 11 ⯠0 ⯠a 1 n a 21 ⯠0 ⯠a 2 n â® â± â® â± â® a i â 1 , 1 ⯠0 ⯠a i â 1 , n a i + 1 , 1 ⯠0 ⯠a i + 1 , n â® â± â® â± â® a n 1 ⯠0 ⯠a n n | {\\displaystyle (-1)^{i-1}{\\begin{vmatrix}a_{i1}&\\cdots &1&\\cdots &a_{in}\\\\a_{11}&\\cdots &0&\\cdots &a_{1n}\\\\a_{21}&\\cdots &0&\\cdots &a_{2n}\\\\\\vdots &\\ddots &\\vdots &\\ddots &\\vdots \\\\a_{i-1,1}&\\cdots &0&\\cdots &a_{i-1,n}\\\\a_{i+1,1}&\\cdots &0&\\cdots &a_{i+1,n}\\\\\\vdots &\\ddots &\\vdots &\\ddots &\\vdots \\\\a_{n1}&\\cdots &0&\\cdots &a_{nn}\\end{vmatrix}}}",
"title": "äœå åè¡å"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "è¡åã®è¡ãŸãã¯åãå
¥ãæ¿ãããšãè¡ååŒã®å€ã¯ â 1 {\\displaystyle -1} åãããã®ã ã£ãããã®æäœã§ã¯ã i â 1 {\\displaystyle i-1} åã®å
¥ãæ¿ããè¡ãã®ã§ããã®åŒã¯ã ( â 1 ) i â 1 {\\displaystyle (-1)^{i-1}} åãããŠããã",
"title": "äœå åè¡å"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "次ã«ãåãããã«ã j {\\displaystyle j} åç®ãšã j â 1 {\\displaystyle j-1} åç®ãå
¥ãæ¿ããã j â 1 {\\displaystyle j-1} åç®ãšã j â 2 {\\displaystyle j-2} åç®ãå
¥ãæ¿ãããã»ã»ã» 2 {\\displaystyle 2} åç®ãšã 1 {\\displaystyle 1} åç®ãå
¥ãæ¿ããããšããæäœãããããããšã次ã®ãããªè¡åã«ãªãã",
"title": "äœå åè¡å"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "( â 1 ) i + j | 1 a i 1 ⯠a i , j â 1 a i , j + 1 ⯠a i n 0 a 11 ⯠a 1 , j â 1 a 1 , j + 1 ⯠a 1 n 0 a 12 ⯠a 2 , j â 1 a 2 , j + 1 ⯠a 2 n â® â® â± â® â® â± â® 0 a i â 1 , 1 ⯠a i â 1 , j â 1 a i â 1 , j + 1 ⯠a i â 1 , n 0 a i + 1 , 1 ⯠a i + 1 , j â 1 a i + 1 , j + 1 ⯠a i + 1 , n â® â® â± â® â® â± â® 0 a n 1 ⯠a n , j â 1 a n , j + 1 ⯠a n n | {\\displaystyle (-1)^{i+j}{\\begin{vmatrix}1&a_{i1}&\\cdots &a_{i,j-1}&a_{i,j+1}&\\cdots &a_{in}\\\\0&a_{11}&\\cdots &a_{1,j-1}&a_{1,j+1}&\\cdots &a_{1n}\\\\0&a_{12}&\\cdots &a_{2,j-1}&a_{2,j+1}&\\cdots &a_{2n}\\\\\\vdots &\\vdots &\\ddots &\\vdots &\\vdots &\\ddots &\\vdots \\\\0&a_{i-1,1}&\\cdots &a_{i-1,j-1}&a_{i-1,j+1}&\\cdots &a_{i-1,n}\\\\0&a_{i+1,1}&\\cdots &a_{i+1,j-1}&a_{i+1,j+1}&\\cdots &a_{i+1,n}\\\\\\vdots &\\vdots &\\ddots &\\vdots &\\vdots &\\ddots &\\vdots \\\\0&a_{n1}&\\cdots &a_{n,j-1}&a_{n,j+1}&\\cdots &a_{nn}\\end{vmatrix}}}",
"title": "äœå åè¡å"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "( â 1 ) i + j â 2 = ( â 1 ) i + j {\\displaystyle (-1)^{i+j-2}=(-1)^{i+j}} ã§ããããšã«ã€ããŠã®èª¬æã¯äžèŠã§ãããã ããããè¡ååŒã®å®çŸ©ã«åŸã£ãŠå±éããã",
"title": "äœå åè¡å"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "äžè¡ç®ã§ã(1,1)èŠçŽ ãéžã°ãªãé
ã¯ãããããäžåç®ã®0ãéžã¶ã®ã§ã0ãšãªãã ãªã®ã§ãäžè¡ç®ã§ã(1,1)èŠçŽ ãéžã¶é
ã ããèããã°è¯ãããããã¯ã | A i , j | {\\displaystyle |A_{i,j}|} ãšäžèŽããã ãã£ãŠããã®è¡ååŒã¯ã ( â 1 ) i + j | A i j | = a ~ i j {\\displaystyle (-1)^{i+j}|A_{ij}|={\\tilde {a}}_{ij}} ã§ããã",
"title": "äœå åè¡å"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "",
"title": "äœå åè¡å"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "ãããã(2)åŒã«ä»£å
¥ããã°ã | A | = a j 1 a ~ j 1 + a j 2 a ~ j 2 + ⯠+ a j n a ~ j n {\\displaystyle |A|=a_{j1}{\\tilde {a}}_{j1}+a_{j2}{\\tilde {a}}_{j2}+\\cdots +a_{jn}{\\tilde {a}}_{jn}} ãšãªãã蚌æãããã",
"title": "äœå åè¡å"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "ãããšåæ§ã®è°è«ãè¡ã«ãè¡ãã°ãããäžæ¹ã®åŒãå°ãããšãã§ããã",
"title": "äœå åè¡å"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "A ~ = ( a ~ j , i ) {\\displaystyle {\\tilde {A}}=({\\tilde {a}}_{j,i})} ãAã®äœå åè¡åãšããã",
"title": "äœå åè¡å"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "äœå åè¡åã«ã¯ã以äžã®æ§è³ªãããã",
"title": "äœå åè¡å"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "蚌æ",
"title": "äœå åè¡å"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "A ~ A = ( a ~ 11 ⯠a ~ m 1 â® â± â® a ~ 1 n ⯠a ~ m n ) ( a 11 ⯠a 1 n â® â± â® a n 1 ⯠a m n ) {\\displaystyle {\\tilde {A}}A={\\begin{pmatrix}{\\tilde {a}}_{11}&\\cdots &{\\tilde {a}}_{m1}\\\\\\vdots &\\ddots &\\vdots \\\\{\\tilde {a}}_{1n}&\\cdots &{\\tilde {a}}_{mn}\\end{pmatrix}}{\\begin{pmatrix}a_{11}&\\cdots &a_{1n}\\\\\\vdots &\\ddots &\\vdots \\\\a_{n1}&\\cdots &a_{mn}\\end{pmatrix}}} ãªã®ã§ã è¡å A ~ A {\\displaystyle {\\tilde {A}}A} ã® ( i , j ) {\\displaystyle (i,j)} æåã¯ã",
"title": "äœå åè¡å"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "a 1 i a ~ 1 j + a 2 i a ~ 2 j + ⯠+ a n i a ~ n j ⯠( 1 ) {\\displaystyle a_{1i}{\\tilde {a}}_{1j}+a_{2i}{\\tilde {a}}_{2j}+\\cdots +a_{ni}{\\tilde {a}}_{nj}\\cdots (1)} ã§ããã",
"title": "äœå åè¡å"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "(i) i = j {\\displaystyle i=j} ã®ãšã",
"title": "äœå åè¡å"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "(ii) i â j {\\displaystyle i\\neq j} ã®ãšã",
"title": "äœå åè¡å"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "ãŸãšãããšã a 1 i a ~ 1 j + a 2 i a ~ 2 j + ⯠+ a n i a ~ n j = { | A | ( i = j ) 0 ( i â j ) {\\displaystyle a_{1i}{\\tilde {a}}_{1j}+a_{2i}{\\tilde {a}}_{2j}+\\cdots +a_{ni}{\\tilde {a}}_{nj}={\\begin{cases}|A|(i=j)\\\\0(i\\neq j)\\\\\\end{cases}}} ã§ããã ãã£ãŠ A ~ A = | A | E {\\displaystyle {\\tilde {A}}A=|A|E} ã§ãããåæ§ã®è°è«ãè¡ãã°ã A A ~ = | A | E {\\displaystyle A{\\tilde {A}}=|A|E} ãå°ãããšãã§ããã",
"title": "äœå åè¡å"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "| A | â 0 {\\displaystyle |A|\\neq 0} ã®ãšã A â 1 {\\displaystyle A^{-1}} ãååšããã®ã§ã A ~ A = | A | E {\\displaystyle {\\tilde {A}}A=|A|E} ã« A â 1 {\\displaystyle A^{-1}} ãå³ãããã | A | {\\displaystyle |A|} ã§å²ãã°ã A â 1 = A ~ | A | {\\displaystyle A^{-1}={\\frac {\\tilde {A}}{|A|}}} ã§ããäºããããã",
"title": "äœå åè¡å"
}
] | null | {{ããã²ãŒã·ã§ã³|æ¬=[[ç·å代æ°åŠ]]|åããŒãž=[[ç·åœ¢ä»£æ°åŠ/è¡ååŒ|è¡ååŒ]]|ããŒãžå=äœå åè¡å|次ããŒãž=[[ç·å代æ°åŠ/ã¯ã©ã¡ã«ã®å
¬åŒ|ã¯ã©ã¡ã«ã®å
¬åŒ]]}}
==äœå åè¡å==
===äœå å===
æ£æ¹è¡å<math>A</math>ã«å¯ŸããŠã è¡åã®<math>i</math>è¡ç®ãš<math>j</math>åç®ãåãé€ããŠåŸãããè¡åã<math>A_{ij}</math>ãšè¡šãããã®ãšãã
<math>\tilde a_{ij} = (-1)^{i+j} | A_{ij} |</math>
ã<math>A</math>ã®<math>(i,j)</math>'''äœå å'''ãšããã
;äŸ
<math>\begin{pmatrix}
5 & 0 & 8 \\
1 & 9 & 3 \\
7 & 5 & 2
\end{pmatrix}</math>
ã®<math>(2,2)</math>äœå åã¯ã<math>(-1)^{2+2} \begin{vmatrix} 5 & 8 \\ 7 & 2 \end{vmatrix} = -46</math>ã§ããã
===äœå åå±é===
次ã®ããã«ãäœå åãå©çšããããšã§ãè¡ååŒãæ±ããããšãã§ããã
<math>|A| = a_{j1} \tilde a_{j1} + a_{j2} \tilde a_{j2} + \cdots + a_{jn} \tilde a_{jn} (1 \le j \le n)</math>
<math>|A| = a_{1i} \tilde a_{1i} + a_{2i} \tilde a_{2i} + \cdots + a_{ni} \tilde a_{ni} (1\le i \le n)</math>
ãã ãã<math>A</math>ã¯<math>n</math>次æ£æ¹è¡åã§ããã
ãããã'''äœå åå±é'''ãšããã
'''蚌æ'''
<math>A = \begin{pmatrix}
a_{11} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots \\
a_{n1} & \cdots & a_{nn}
\end{pmatrix}</math>
ãšããããã®ãšãã
:<math>|A| = \begin{vmatrix}
a_{11} & \cdots & a_{1j} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots & \ddots & \vdots \\
a_{n1} & \cdots & a_{nj} & \cdots & a_{nn}
\end{vmatrix}</math>
ã§ãããããã§ãè¡å<math>A</math>ã®<math>j</math>åç®<math>\begin{pmatrix} a_{1j} \\ a_{2j} \\ \vdots \\ a_{nj} \end{pmatrix}</math>ã¯ã
<math>a_{1j} \begin{pmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{pmatrix} + a_{2j} \begin{pmatrix} 0 \\ 1 \\ \vdots \\ 0 \end{pmatrix} + \cdots + a_{nj} \begin{pmatrix} 0 \\ 0 \\ \vdots \\ 1 \end{pmatrix} </math>ãšè¡šãããšãã§ãã
(1)åŒã¯ã
<math>
\left| \begin{pmatrix} a_{11} \\ a_{21} \\ \vdots \\ a_{n1} \end{pmatrix}, \cdots, a_{1j} \begin{pmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{pmatrix} + a_{2j} \begin{pmatrix} 0 \\ 1 \\ \vdots \\ 0 \end{pmatrix} + \cdots + a_{nj} \begin{pmatrix} 0 \\ 0 \\ \vdots \\ 1 \end{pmatrix}, \cdots, \begin{pmatrix}a_{n1} \\ a_{n2} \\ \vdots \\ a_{nn} \end{pmatrix} \right|
</math>ãšãè¡šãããšãã§ãããããã«ãè¡ååŒã®æ§è³ªã䜿ãã°ã
<math>
a_{1j} \begin{vmatrix} a_{11} & \cdots & 1 &\cdots& a_{1n} \\ a_{21} & \cdots & 0 & \cdots& a_{2n} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ a_{n1} & \cdots & 0 &\cdots& a_{nn} \end{vmatrix} +
a_{2j} \begin{vmatrix} a_{11} & \cdots & 0 &\cdots& a_{1n} \\ a_{21} & \cdots & 1 & \cdots & a_{2n} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ a_{n1} & \cdots & 0 &\cdots& a_{nn} \end{vmatrix} + \cdots +
a_{nj} \begin{vmatrix} a_{11} & \cdots & 0 &\cdots& a_{1n} \\ a_{21} & \cdots & 0 & \cdots & a_{2n} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ a_{n1} & \cdots & 1 &\cdots& a_{nn} \end{vmatrix} \cdots (2)
</math>
ã§ããã
ããã§ã<math>\begin{vmatrix}
a_{11} & \cdots & 0 &\cdots& a_{1n} \\
a_{21} & \cdots & 0 & \cdots& a_{2n} \\
\vdots & \ddots & \vdots & \ddots & \vdots\\
a_{i1} & \cdots & 1 & \cdots & a_{in} \\
\vdots & \ddots & \vdots & \ddots & \vdots \\
a_{n1} & \cdots & 0 &\cdots& a_{nn}
\end{vmatrix}</math>ã«ã€ããŠèããã
ãã®è¡åã®<math>i</math>è¡ç®ãšã<math>i-1</math>è¡ç®ãå
¥ãæ¿ãã<math>i-1</math>è¡ç®ãšã<math>i-2</math>è¡ç®ãå
¥ãæ¿ãããã»ã»ã»<math>2</math>è¡ç®ãšã<math>1</math>è¡ç®ãå
¥ãæ¿ããããšããæäœããããšã次ã®ãããªè¡åã«ãªãã
<math> (-1)^{i-1} \begin{vmatrix}
a_{i1} & \cdots & 1 & \cdots & a_{in} \\
a_{11} & \cdots & 0 &\cdots& a_{1n} \\
a_{21} & \cdots & 0 & \cdots& a_{2n} \\
\vdots & \ddots & \vdots & \ddots & \vdots\\
a_{i-1,1} & \cdots & 0 & \cdots & a_{i-1,n} \\
a_{i+1,1} & \cdots & 0 & \cdots & a_{i+1,n} \\
\vdots & \ddots & \vdots & \ddots & \vdots\\
a_{n1} & \cdots & 0 &\cdots& a_{nn}
\end{vmatrix}
</math>
è¡åã®è¡ãŸãã¯åãå
¥ãæ¿ãããšãè¡ååŒã®å€ã¯<math>-1</math>åãããã®ã ã£ãããã®æäœã§ã¯ã<math>i-1</math>åã®å
¥ãæ¿ããè¡ãã®ã§ããã®åŒã¯ã<math>(-1)^{i-1}</math>åãããŠããã
次ã«ãåãããã«ã<math>j</math>åç®ãšã<math>j-1</math>åç®ãå
¥ãæ¿ããã<math>j-1</math>åç®ãšã<math>j-2</math>åç®ãå
¥ãæ¿ãããã»ã»ã»<math>2</math>åç®ãšã<math>1</math>åç®ãå
¥ãæ¿ããããšããæäœãããããããšã次ã®ãããªè¡åã«ãªãã<br>
<math> (-1)^{i+j} \begin{vmatrix}
1 & a_{i1} & \cdots & a_{i,j-1}& a_{i,j+1}& \cdots & a_{in} \\
0 & a_{11} & \cdots & a_{1,j-1}&a_{1,j+1}& \cdots & a_{1n} \\
0 & a_{12} & \cdots & a_{2,j-1}&a_{2,j+1}& \cdots & a_{2n} \\
\vdots & \vdots & \ddots& \vdots & \vdots & \ddots & \vdots\\
0 & a_{i-1,1} & \cdots & a_{i-1,j-1} & a_{i-1,j+1}& \cdots & a_{i-1,n} \\
0 & a_{i+1,1} & \cdots & a_{i+1,j-1} & a_{i+1,j+1}& \cdots & a_{i+1,n} \\
\vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\
0 & a_{n1} & \cdots & a_{n,j-1} & a_{n,j+1} & \cdots& a_{nn}
\end{vmatrix}
</math>
<math>(-1)^{i+j-2}=(-1)^{i+j}</math>ã§ããããšã«ã€ããŠã®èª¬æã¯äžèŠã§ãããã
ããããè¡ååŒã®å®çŸ©ã«åŸã£ãŠå±éããã
äžè¡ç®ã§ã(1,1)èŠçŽ ãéžã°ãªãé
ã¯ãããããäžåç®ã®0ãéžã¶ã®ã§ã0ãšãªãã
ãªã®ã§ãäžè¡ç®ã§ã(1,1)èŠçŽ ãéžã¶é
ã ããèããã°è¯ãããããã¯ã<math>|A_{i,j}|</math>ãšäžèŽããã
ãã£ãŠããã®è¡ååŒã¯ã<math>(-1)^{i+j} |A_{ij}| = \tilde a_{ij}</math>ã§ããã
ãããã(2)åŒã«ä»£å
¥ããã°ã<math>|A| = a_{j1} \tilde a_{j1} + a_{j2} \tilde a_{j2} + \cdots + a_{jn} \tilde a_{jn}</math>ãšãªãã蚌æãããã
ãããšåæ§ã®è°è«ãè¡ã«ãè¡ãã°ãããäžæ¹ã®åŒãå°ãããšãã§ããã
===äœå åè¡å===
<math>\tilde A = (\tilde a_{j,i})</math>ãAã®äœå åè¡åãšããã
äœå åè¡åã«ã¯ã以äžã®æ§è³ªãããã
:<math>A \tilde A = \tilde A A = |A|E</math>
'''蚌æ'''
<math>\tilde A A = \begin{pmatrix} \tilde a_{11} & \cdots & \tilde a_{m1} \\ \vdots & \ddots & \vdots \\ \tilde a_{1n} & \cdots & \tilde a_{mn} \end{pmatrix} \begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{n1} & \cdots & a_{mn} \end{pmatrix}</math>ãªã®ã§ã
è¡å<math>\tilde A A</math>ã®<math>(i,j)</math>æåã¯ã
<math>a_{1i} \tilde a_{1j} + a_{2i} \tilde a_{2j} + \cdots + a_{ni} \tilde a_{nj} \cdots (1)</math>ã§ããã
(i)<math>i=j</math>ã®ãšã
:(1)åŒã¯ãè¡å<math>A</math>ã®<math>i</math>åç®ã«é¢ããŠäœå åå±éãããåŒãšäžèŽããã®ã§ã(1)åŒã¯<math>i=j</math>ã®ãšãã<math>|A|</math>ã§ããã<br>
(ii)<math>i\neq j</math>ã®ãšã
:è¡å<math>A</math>ã®<math>i</math>åç®ãè¡å<math>A</math>ã®<math>j</math>åç®ã«ãªã£ãŠããè¡åã®è¡ååŒã«ã€ããŠèããããã®è¡ååŒã¯ä»¥äžã®ããã«ãªãã<br>
:<math>
\begin{vmatrix}
a_{11} & \cdots & a_{1,i-1} & a_{1j} & a_{1,i+1} & \cdots & a_{1j} & \cdots & a_{1n} \\
a_{21} & \cdots & a_{2,i-1} & a_{2j} & a_{2,i+1} & \cdots & a_{2j} & \cdots & a_{2n} \\
\vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\
a_{n1} & \cdots & a_{n,i-1} & a_{nj} & a_{n,i+1} & \cdots & a_{nj} & \cdots & a_{nn} \\
\end{vmatrix}
</math>
:ãã®è¡åã®iåç®ã«ã€ããŠãäœå åå±éãè¡ããšã(1)åŒãšäžèŽããã
:åãåãããè¡åã®è¡ååŒã¯0ã«ãªãã®ã ã£ãããªã®ã§ã(1)åŒã¯ã<math>i\neq j</math>ã®ãšãã0ã§ããã <br>
ãŸãšãããšã<math>a_{1i} \tilde a_{1j} + a_{2i} \tilde a_{2j} + \cdots + a_{ni} \tilde a_{nj} =
\begin{cases}
|A| (i=j) \\
0 (i \neq j) \\
\end{cases}
</math>ã§ããã
ãã£ãŠ<math>\tilde A A = |A|E</math>ã§ãããåæ§ã®è°è«ãè¡ãã°ã<math>A \tilde A = |A|E</math>ãå°ãããšãã§ããã
===éè¡åã®èšç®===
<math>|A| \neq 0</math>ã®ãšã<math>A^{-1}</math>ãååšããã®ã§ã<math>\tilde A A = |A|E</math>ã«<math>A^{-1}</math>ãå³ãããã<math>|A|</math>ã§å²ãã°ã
<math>A^{-1} = \frac{\tilde A}{|A|}</math>ã§ããäºããããã
{{ããã²ãŒã·ã§ã³|æ¬=[[ç·å代æ°åŠ]]|åããŒãž=[[ç·åœ¢ä»£æ°åŠ/è¡ååŒ|è¡ååŒ]]|ããŒãžå=äœå åè¡å|次ããŒãž=[[ç·å代æ°åŠ/ã¯ã©ã¡ã«ã®å
¬åŒ|ã¯ã©ã¡ã«ã®å
¬åŒ]]}}
[[Category:ç·åœ¢ä»£æ°åŠ|ãããããããããã ããããããããã€]] | null | 2021-01-29T11:11:46Z | [
"ãã³ãã¬ãŒã:ããã²ãŒã·ã§ã³"
] | https://ja.wikibooks.org/wiki/%E7%B7%9A%E5%BD%A2%E4%BB%A3%E6%95%B0%E5%AD%A6/%E4%BD%99%E5%9B%A0%E5%AD%90%E8%A1%8C%E5%88%97 |
2,014 | ç·åœ¢ä»£æ°åŠ/éè¡åã®äžè¬å | ç·å代æ°åŠ > éè¡åã®äžè¬å
éè¡åã¯ã
A â 1 = 1 det A C {\displaystyle A^{-1}={\frac {1}{\det A}}C} ã§æžãããã ããã§Cã¯ãAã®äœå åè¡åã§ããã
å°åº
第lè¡ã«ã€ããŠèããã(l = 1 , ... , n) ãã®ãšããlè¡låã«ã€ã㊠ACãèãããšã â m = 1 n a l m c m l {\displaystyle \sum _{m=1}^{n}a_{lm}c_{ml}} = â m = 1 n a l m ( â 1 ) m + l b l m {\displaystyle =\sum _{m=1}^{n}a_{lm}(-1)^{m+l}b_{lm}} , ( b l m {\displaystyle b_{lm}} ã¯ãè¡åAã®è¡lãåmã«é¢ããå°è¡ååŒã) = det A {\displaystyle =\det A} (åŒã®å±éã®é) ãŸããlè¡ã§ãiå(i = 1, ... , n : l 以å€) ã«ã€ã㊠ACãèãããšã â m = 1 n a l m c m i {\displaystyle \sum _{m=1}^{n}a_{lm}c_{mi}} â m = 1 n a l m ( â 1 ) m + i b i m {\displaystyle \sum _{m=1}^{n}a_{lm}(-1)^{m+i}b_{im}} ããã¯ãè¡åAã§ãiè¡ç®ãlè¡ç®ã§çœ®ãæããè¡åã®è¡ååŒã«çããã è¡ååŒã§è¡åã®ãã¡ã®ããè¡ããããåãä»ã®è¡ãä»ã®åãšäžèŽããå Žåã ãã®2ã€ã®è¡ãŸãã¯åããã®å¯äžã¯å¿
ãæã¡æ¶ãããã (å°åº?) ãã£ãŠiåããã®å¯äžã¯0ã«çããã ãã£ãŠæ±ããè¡å ACã¯ã det ( A ) E {\displaystyle \det(A)E} ãšãªãã 1 det A C {\displaystyle {\frac {1}{\det A}}C} ã¯ã(Cã¯Aã®äœå åè¡å) Aã®éè¡åã«çããããšãåãã
å®éã«ã¯ãã®èšç®ã¯å€ãã®èšç®éãå¿
èŠãšããã®ã§ å®çšçãªèšç®ã«ã¯çšããããªãã å®çšçãªèšç®ã«ã¯ã¬ãŠã¹ã®æ¶å»æ³ã çšããããããšãå€ãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ç·å代æ°åŠ > éè¡åã®äžè¬å",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "éè¡åã¯ã",
"title": "éè¡åã®äžè¬å"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "A â 1 = 1 det A C {\\displaystyle A^{-1}={\\frac {1}{\\det A}}C} ã§æžãããã ããã§Cã¯ãAã®äœå åè¡åã§ããã",
"title": "éè¡åã®äžè¬å"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "å°åº",
"title": "éè¡åã®äžè¬å"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "第lè¡ã«ã€ããŠèããã(l = 1 , ... , n) ãã®ãšããlè¡låã«ã€ã㊠ACãèãããšã â m = 1 n a l m c m l {\\displaystyle \\sum _{m=1}^{n}a_{lm}c_{ml}} = â m = 1 n a l m ( â 1 ) m + l b l m {\\displaystyle =\\sum _{m=1}^{n}a_{lm}(-1)^{m+l}b_{lm}} , ( b l m {\\displaystyle b_{lm}} ã¯ãè¡åAã®è¡lãåmã«é¢ããå°è¡ååŒã) = det A {\\displaystyle =\\det A} (åŒã®å±éã®é) ãŸããlè¡ã§ãiå(i = 1, ... , n : l 以å€) ã«ã€ã㊠ACãèãããšã â m = 1 n a l m c m i {\\displaystyle \\sum _{m=1}^{n}a_{lm}c_{mi}} â m = 1 n a l m ( â 1 ) m + i b i m {\\displaystyle \\sum _{m=1}^{n}a_{lm}(-1)^{m+i}b_{im}} ããã¯ãè¡åAã§ãiè¡ç®ãlè¡ç®ã§çœ®ãæããè¡åã®è¡ååŒã«çããã è¡ååŒã§è¡åã®ãã¡ã®ããè¡ããããåãä»ã®è¡ãä»ã®åãšäžèŽããå Žåã ãã®2ã€ã®è¡ãŸãã¯åããã®å¯äžã¯å¿
ãæã¡æ¶ãããã (å°åº?) ãã£ãŠiåããã®å¯äžã¯0ã«çããã ãã£ãŠæ±ããè¡å ACã¯ã det ( A ) E {\\displaystyle \\det(A)E} ãšãªãã 1 det A C {\\displaystyle {\\frac {1}{\\det A}}C} ã¯ã(Cã¯Aã®äœå åè¡å) Aã®éè¡åã«çããããšãåãã",
"title": "éè¡åã®äžè¬å"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "å®éã«ã¯ãã®èšç®ã¯å€ãã®èšç®éãå¿
èŠãšããã®ã§ å®çšçãªèšç®ã«ã¯çšããããªãã å®çšçãªèšç®ã«ã¯ã¬ãŠã¹ã®æ¶å»æ³ã çšããããããšãå€ãã",
"title": "éè¡åã®äžè¬å"
}
] | ç·å代æ°åŠ > éè¡åã®äžè¬å | <small> [[ç·å代æ°åŠ]] > éè¡åã®äžè¬å </small>
----
==éè¡åã®äžè¬å==
éè¡åã¯ã
<math>
A^{-1} = \frac 1 {\det A} C
</math>
ã§æžãããã
ããã§Cã¯ãAã®äœå åè¡åã§ããã
'''å°åº'''
第''l''è¡ã«ã€ããŠèããã(l = 1 , ... , n)
ãã®ãšããlè¡låã«ã€ããŠ
ACãèãããšã
<math>
\sum _{m=1} ^ n a _{lm} c _{ml}
</math>
<math>
=\sum _{m=1} ^ n a _{lm} (-1)^{m+ l} b _{lm}
</math>,
(<math>b _{lm}</math>ã¯ãè¡åAã®è¡lãåmã«é¢ããå°è¡ååŒã)
<math>
=\det A
</math>
(åŒã®å±éã®é)
ãŸããlè¡ã§ãiå(i = 1, ... , n : l 以å€) ã«ã€ããŠ
ACãèãããšã
<math>
\sum _{m=1} ^ n a _{lm} c _{mi}
</math>
<math>
\sum _{m=1} ^ n a _{lm} (-1)^{m+ i} b _{im}
</math>
ããã¯ãè¡åAã§ãiè¡ç®ãlè¡ç®ã§çœ®ãæããè¡åã®è¡ååŒã«çããã
è¡ååŒã§è¡åã®ãã¡ã®ããè¡ããããåãä»ã®è¡ãä»ã®åãšäžèŽããå Žåã
ãã®2ã€ã®è¡ãŸãã¯åããã®å¯äžã¯å¿
ãæã¡æ¶ãããã
(å°åº?)
ãã£ãŠiåããã®å¯äžã¯0ã«çããã
ãã£ãŠæ±ããè¡å
ACã¯ã
<math>
\det (A ) E
</math>
ãšãªãã
<math>
\frac 1 {\det A} C
</math>
ã¯ã(Cã¯Aã®äœå åè¡å)
Aã®éè¡åã«çããããšãåãã
å®éã«ã¯ãã®èšç®ã¯å€ãã®èšç®éãå¿
èŠãšããã®ã§
å®çšçãªèšç®ã«ã¯çšããããªãã
å®çšçãªèšç®ã«ã¯ã¬ãŠã¹ã®æ¶å»æ³ã
çšããããããšãå€ãã
<!-- ã¬ãŠã¹ã®æ¶å»æ³ã¯èšç®æ©ç§åŠãç·åœ¢ä»£æ°ã... -->
<!-- ç·åœ¢ä»£æ°ã ãããªããã£ã±ã...ã -->
[[Category:ç·åœ¢ä»£æ°åŠ|ãããããããã€ã®ãã€ã¯ããã]] | null | 2015-09-13T05:59:54Z | [] | https://ja.wikibooks.org/wiki/%E7%B7%9A%E5%BD%A2%E4%BB%A3%E6%95%B0%E5%AD%A6/%E9%80%86%E8%A1%8C%E5%88%97%E3%81%AE%E4%B8%80%E8%88%AC%E5%9E%8B |
2,019 | ææ©ååŠ/ã¢ã«ã«ã³ | ææ©ååŠ>ã¢ã«ã«ã³
ççŽ éã«åçµåã®ã¿ãå«ãçåæ°ŽçŽ ãã¢ã«ã«ã³ (alkane) ãšããã
ãªã©ã¯ãã¹ãŠã¢ã«ã«ã³ã§ããã
ççŽ ååã1åã®ã¢ã«ã«ã³ã®åååŒã¯ C H 4 {\displaystyle CH_{4}} ã§ããã åãããã«ãççŽ ååã2åã®ã¢ã«ã«ã³ã¯ C 2 H 6 {\displaystyle C_{2}H_{6}} ã3åãªã C 3 H 8 {\displaystyle C_{3}H_{8}} ã4å㧠C 4 H 10 {\displaystyle C_{4}H_{10}} ãã§ããã ãã®ããã«ã¢ã«ã«ã³ã¯äžè¬çã« C n H 2 n + 2 {\displaystyle C_{n}H_{2n+2}} ã§è¡šãããããã®åŒãã¢ã«ã«ã³ã®äžè¬åŒãšããã
ãã®è¡šããåãããšãããã¢ã«ã«ã³ã®ååã¯ãæ°ããè¡šãéšåãšãã¢ã«ã«ã³ããè¡šãã-aneãããæã£ãŠããã
åååŒã¯åãã§ããããæ§é ãæ§è³ªã®ç°ãªãååç©ããäºãã«ç°æ§äœãšåŒã¶ã ç°æ§äœã«ã¯ãæ§é åŒã®éãæ§é ç°æ§äœãšãæ§é åŒã¯åãã ãç«äœæ§é ã®ç°ãªãç«äœç°æ§äœãããã æ§é ç°æ§äœãåã«ç°æ§äœãšåŒã¶ããšãããã
ã¢ã«ã«ã³ã¯ãççŽ ååã4å以äžã®ãšãæ§é ç°æ§äœãæã€ã ãã®ãããåãåååŒãæã€ã¢ã«ã«ã³ã§ãæ§é ç°æ§äœå士ã§åºå¥ããå¿
èŠãããã
äŸãã°ã
ãšããã¢ã«ã«ã³ãèããã
ãŸããã®äžã§äžçªé·ãççŽ ã®éãæ¢ãã äžçªé·ãã®ã¯çãäžã®åã®ççŽ 10åã§ã¯ç¡ãã çãäžã®åã®å·Šãã9åãšã9åç®ããäžã«3åã®ãåãããŠ12åãäžçªé·ãççŽ ã®éã§ããããããäž»éãšããã ãã®ããã«äž»éã¯æ§é åŒã®ã©ãã«æžããŠãããã¯é¢ä¿ãªãã
äž»éã12åãšæ±ºãŸã£ãã®ã§ãã®ã¢ã«ã«ã³ã¯ã~ããã«ã³ãã§çµããã ãã以å€ã®ççŽ ãšæ°ŽçŽ ã®å¡ã¯ããã¹ãŠçœ®æåºãšããŠæ±ãããã ã¢ã«ã«ã³ã®çœ®æåºã¯ãå¥ã®å°ããã¢ã«ã«ã³ããæ°ŽçŽ ååãäžã€åãé€ãããã®ãšããŠè¡šããããããã¢ã«ãã«åº(alkyl group)ãšããã ã¢ã«ãã«åºã®å称ã¯ãã¢ã«ã«ã³ã®aneãylã«çœ®ãæããããšã§äœãã å·Šãã2åç®ã®ççŽ ããåºãŠãã眮æåºã¯ã¡ã¿ã³(methane)ããæ°ŽçŽ ååãäžã€åãé€ãããã®ã«çããã®ã§ãã¡ãã«(methyl)åºãšããããšã«ãªãã åæ§ã«ãå·Šãã3åç®ã®ççŽ ããåºãŠããã®ããšãã«(ethyl)åºãå·Šãã4åç®ã®ççŽ ããåºãŠããã®ããããã«(propyl)åºãå·Šãã9çªç®ã®ççŽ ããå³ã«åºãŠããã®ãã¡ãã«åºã§ããã
ãããããŸãã¢ã«ãã¡ãããé ã«äžŠã¹ãããããšãethylãmethylãpropylã®é ã«ãªãã 次ã«ããšãã«åºããé ã«ãäž»éã®äœçªç®ã®ççŽ ã«ä»ããŠãããã瀺ãã ããã§ãå·Šããæ°ããã®ã§3çªç®ãšããèãæ¹ãšãå³ããæ°ããã®ã§10çªç®ãšããèãæ¹ããããããªãã¹ãçªå·ãå°ãªããªãããã«ã€ããã ãã£ãŠãã3-ãšãã«~ããšãªãã
次ã«ãã¡ãã«åºã¯ãµãã€ä»ããŠããã®ã§ãããž(di)ã¡ãã«ããšãã颚ã«ãããäœçœ®çªå·ã¯ãäžåºŠæ±ºããçªå·ã¯å€ããªãã®ã§ãã2,9-ãžã¡ãã«ããšãªããæåãšæ°åã®éããã€ãã³ã§ã€ãªããšãã3-ãšãã«-2,9-ãžã¡ãã«~ããšãªãã
æåŸã«ãããã«åºã¯ã4-ãããã«ããšãªãã®ã§ããã¹ãŠãã€ãªãã§ãã®ã¢ã«ã«ã³ã®ååã¯ã3-ãšãã«-2,9-ãžã¡ãã«-4-ãããã«ããã«ã³ããšãªãã
åºãäœåãããã¯ã®ãªã·ã£èªã®æ°è©ã䜿ã£ãŠè¡šãã 1ãã10ãŸã§ãé ã«ãã¢ã (mono)ã»ãž (di)ã»ã㪠(tri)ã»ããã© (tetra)ã»ãã³ã¿ (penta)ã»ãããµ (hexa)ã»ããã¿ (hepta)ã»ãªã¯ã¿ (octa)ã»ãã (nona)ã»ãã« (deca)ã§ããã5ãã10ãŸã§ã¯ã¢ã«ã«ã³ã®å称ãšãé¢ä¿ããã
眮æåå¿ãšã¯ãåå(å£)ãä»ã®åå(å£)ãšçœ®ãæããåå¿ã§ããã ã¢ã«ã«ã³ã¯ã玫å€ç·(æ¥å
)ã®ååšäžã§ããã²ã³ãšé£ç¶çã«çœ®æåå¿ãèµ·ããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ææ©ååŠ>ã¢ã«ã«ã³",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ççŽ éã«åçµåã®ã¿ãå«ãçåæ°ŽçŽ ãã¢ã«ã«ã³ (alkane) ãšããã",
"title": "ã¢ã«ã«ã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ãªã©ã¯ãã¹ãŠã¢ã«ã«ã³ã§ããã",
"title": "ã¢ã«ã«ã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ççŽ ååã1åã®ã¢ã«ã«ã³ã®åååŒã¯ C H 4 {\\displaystyle CH_{4}} ã§ããã åãããã«ãççŽ ååã2åã®ã¢ã«ã«ã³ã¯ C 2 H 6 {\\displaystyle C_{2}H_{6}} ã3åãªã C 3 H 8 {\\displaystyle C_{3}H_{8}} ã4å㧠C 4 H 10 {\\displaystyle C_{4}H_{10}} ãã§ããã ãã®ããã«ã¢ã«ã«ã³ã¯äžè¬çã« C n H 2 n + 2 {\\displaystyle C_{n}H_{2n+2}} ã§è¡šãããããã®åŒãã¢ã«ã«ã³ã®äžè¬åŒãšããã",
"title": "ã¢ã«ã«ã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ãã®è¡šããåãããšãããã¢ã«ã«ã³ã®ååã¯ãæ°ããè¡šãéšåãšãã¢ã«ã«ã³ããè¡šãã-aneãããæã£ãŠããã",
"title": "ã¢ã«ã«ã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "åååŒã¯åãã§ããããæ§é ãæ§è³ªã®ç°ãªãååç©ããäºãã«ç°æ§äœãšåŒã¶ã ç°æ§äœã«ã¯ãæ§é åŒã®éãæ§é ç°æ§äœãšãæ§é åŒã¯åãã ãç«äœæ§é ã®ç°ãªãç«äœç°æ§äœãããã æ§é ç°æ§äœãåã«ç°æ§äœãšåŒã¶ããšãããã",
"title": "ã¢ã«ã«ã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ã¢ã«ã«ã³ã¯ãççŽ ååã4å以äžã®ãšãæ§é ç°æ§äœãæã€ã ãã®ãããåãåååŒãæã€ã¢ã«ã«ã³ã§ãæ§é ç°æ§äœå士ã§åºå¥ããå¿
èŠãããã",
"title": "ã¢ã«ã«ã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "äŸãã°ã",
"title": "ã¢ã«ã«ã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "ãšããã¢ã«ã«ã³ãèããã",
"title": "ã¢ã«ã«ã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "ãŸããã®äžã§äžçªé·ãççŽ ã®éãæ¢ãã äžçªé·ãã®ã¯çãäžã®åã®ççŽ 10åã§ã¯ç¡ãã çãäžã®åã®å·Šãã9åãšã9åç®ããäžã«3åã®ãåãããŠ12åãäžçªé·ãççŽ ã®éã§ããããããäž»éãšããã ãã®ããã«äž»éã¯æ§é åŒã®ã©ãã«æžããŠãããã¯é¢ä¿ãªãã",
"title": "ã¢ã«ã«ã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "äž»éã12åãšæ±ºãŸã£ãã®ã§ãã®ã¢ã«ã«ã³ã¯ã~ããã«ã³ãã§çµããã ãã以å€ã®ççŽ ãšæ°ŽçŽ ã®å¡ã¯ããã¹ãŠçœ®æåºãšããŠæ±ãããã ã¢ã«ã«ã³ã®çœ®æåºã¯ãå¥ã®å°ããã¢ã«ã«ã³ããæ°ŽçŽ ååãäžã€åãé€ãããã®ãšããŠè¡šããããããã¢ã«ãã«åº(alkyl group)ãšããã ã¢ã«ãã«åºã®å称ã¯ãã¢ã«ã«ã³ã®aneãylã«çœ®ãæããããšã§äœãã å·Šãã2åç®ã®ççŽ ããåºãŠãã眮æåºã¯ã¡ã¿ã³(methane)ããæ°ŽçŽ ååãäžã€åãé€ãããã®ã«çããã®ã§ãã¡ãã«(methyl)åºãšããããšã«ãªãã åæ§ã«ãå·Šãã3åç®ã®ççŽ ããåºãŠããã®ããšãã«(ethyl)åºãå·Šãã4åç®ã®ççŽ ããåºãŠããã®ããããã«(propyl)åºãå·Šãã9çªç®ã®ççŽ ããå³ã«åºãŠããã®ãã¡ãã«åºã§ããã",
"title": "ã¢ã«ã«ã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "ãããããŸãã¢ã«ãã¡ãããé ã«äžŠã¹ãããããšãethylãmethylãpropylã®é ã«ãªãã 次ã«ããšãã«åºããé ã«ãäž»éã®äœçªç®ã®ççŽ ã«ä»ããŠãããã瀺ãã ããã§ãå·Šããæ°ããã®ã§3çªç®ãšããèãæ¹ãšãå³ããæ°ããã®ã§10çªç®ãšããèãæ¹ããããããªãã¹ãçªå·ãå°ãªããªãããã«ã€ããã ãã£ãŠãã3-ãšãã«~ããšãªãã",
"title": "ã¢ã«ã«ã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "次ã«ãã¡ãã«åºã¯ãµãã€ä»ããŠããã®ã§ãããž(di)ã¡ãã«ããšãã颚ã«ãããäœçœ®çªå·ã¯ãäžåºŠæ±ºããçªå·ã¯å€ããªãã®ã§ãã2,9-ãžã¡ãã«ããšãªããæåãšæ°åã®éããã€ãã³ã§ã€ãªããšãã3-ãšãã«-2,9-ãžã¡ãã«~ããšãªãã",
"title": "ã¢ã«ã«ã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "æåŸã«ãããã«åºã¯ã4-ãããã«ããšãªãã®ã§ããã¹ãŠãã€ãªãã§ãã®ã¢ã«ã«ã³ã®ååã¯ã3-ãšãã«-2,9-ãžã¡ãã«-4-ãããã«ããã«ã³ããšãªãã",
"title": "ã¢ã«ã«ã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "åºãäœåãããã¯ã®ãªã·ã£èªã®æ°è©ã䜿ã£ãŠè¡šãã 1ãã10ãŸã§ãé ã«ãã¢ã (mono)ã»ãž (di)ã»ã㪠(tri)ã»ããã© (tetra)ã»ãã³ã¿ (penta)ã»ãããµ (hexa)ã»ããã¿ (hepta)ã»ãªã¯ã¿ (octa)ã»ãã (nona)ã»ãã« (deca)ã§ããã5ãã10ãŸã§ã¯ã¢ã«ã«ã³ã®å称ãšãé¢ä¿ããã",
"title": "ã¢ã«ã«ã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "眮æåå¿ãšã¯ãåå(å£)ãä»ã®åå(å£)ãšçœ®ãæããåå¿ã§ããã ã¢ã«ã«ã³ã¯ã玫å€ç·(æ¥å
)ã®ååšäžã§ããã²ã³ãšé£ç¶çã«çœ®æåå¿ãèµ·ããã",
"title": "眮æåå¿"
}
] | ææ©ååŠïŒã¢ã«ã«ã³ | [[ææ©ååŠ]]ïŒã¢ã«ã«ã³
== ã¢ã«ã«ã³ã®å®çŸ©ãšåœåæ³ ==
=== ã¢ã«ã«ã³ã®å®çŸ© ===
ççŽ éã«åçµåã®ã¿ãå«ãçåæ°ŽçŽ ãã¢ã«ã«ã³ (alkane) ãšããã
H
|
H-C-H
|
H
H H H
| | |
H-C-C-C-H
| | |
H H H
H H H H
| | | |
H-C-C-C-C-H
| | | |
H | H H
H-C-H
|
H
ãªã©ã¯ãã¹ãŠã¢ã«ã«ã³ã§ããã
=== ã¢ã«ã«ã³ã®äžè¬åŒ ===
ççŽ ååã1åã®ã¢ã«ã«ã³ã®åååŒã¯<math>CH _4</math>ã§ããã<br>
åãããã«ãççŽ ååã2åã®ã¢ã«ã«ã³ã¯<math>C _2 H _6</math>ã3åãªã<math>C _3 H _8</math>ã4åã§<math>C _4 H _{10}</math>ãã§ããã<br>
ãã®ããã«ã¢ã«ã«ã³ã¯äžè¬çã«<math>C _n H _{2n+2}</math>ã§è¡šãããããã®åŒãã¢ã«ã«ã³ã®'''äžè¬åŒ'''ãšããã
=== çŽéã¢ã«ã«ã³ã®åœåæ³ ===
{| class="wikitable"
|+ çŽéã¢ã«ã«ã³ã®åœåæ³
|-
! ççŽ æ° !! åååŒ !! 綎ã !! èªã¿
|-
| 1 || <math>CH _4</math> || methane || ã¡ã¿ã³
|-
|-
| 2 || <math>C _2 H _6</math> || ethane || ãšã¿ã³
|-
| 3 || <math>C _3 H _8</math> || propane || ãããã³
|-
| 4 || <math>C _4 H _{10}</math> || butane || ãã¿ã³
|-
| 5 || <math>C _5 H _{12}</math> || pentane || ãã³ã¿ã³
|-
| 6 || <math>C _6 H _{14}</math> || hexane || ãããµã³
|-
| 7 || <math>C _7 H _{16}</math> || heptane || ããã¿ã³
|-
| 8 || <math>C _8 H _{18}</math> || octane || ãªã¯ã¿ã³
|-
| 9 || <math>C _9 H _{20}</math> || nonane || ããã³
|-
| 10 || <math>C _{10} H _{22}</math> || decane || ãã«ã³
|-
| 11 || <math>C _{11} H _{24}</math> || undecane || ãŠã³ãã«ã³
|-
| 12 || <math>C _{12} H _{26}</math> || dodecane || ããã«ã³
|}
ãã®è¡šããåãããšãããã¢ã«ã«ã³ã®ååã¯ãæ°ããè¡šãéšåãšãã¢ã«ã«ã³ããè¡šãã-aneãããæã£ãŠããã
=== ç°æ§äœ ===
åååŒã¯åãã§ããããæ§é ãæ§è³ªã®ç°ãªãååç©ããäºãã«'''ç°æ§äœ'''ãšåŒã¶ã
ç°æ§äœã«ã¯ã[[w:ååŠåŒ|æ§é åŒ]]ã®éã'''æ§é ç°æ§äœ'''ãšãæ§é åŒã¯åãã ãç«äœæ§é ã®ç°ãªã'''ç«äœç°æ§äœ'''ãããã
æ§é ç°æ§äœãåã«ç°æ§äœãšåŒã¶ããšãããã
=== ã¢ã«ã«ã³ã®ç°æ§äœ ===
ã¢ã«ã«ã³ã¯ãççŽ ååã4å以äžã®ãšãæ§é ç°æ§äœãæã€ã
ãã®ãããåãåååŒãæã€ã¢ã«ã«ã³ã§ãæ§é ç°æ§äœå士ã§åºå¥ããå¿
èŠãããã
=== åå²ã®ããã¢ã«ã«ã³ã®åœåæ³ ===
äŸãã°ã
CH2-CH3 CH3-CH2-CH2
| |
CH3-CH-CH-CH-CH2-CH2-CH2-CH2-CH-CH3
| |
CH3 CH2-CH2-CH3
ãšããã¢ã«ã«ã³ãèããã
ãŸããã®äžã§äžçªé·ãççŽ ã®éãæ¢ãã
äžçªé·ãã®ã¯çãäžã®åã®ççŽ 10å'''ã§ã¯ç¡ã'''ã
çãäžã®åã®å·Šãã9åãšã9åç®ããäžã«3åã®ãåãããŠ12åãäžçªé·ãççŽ ã®éã§ããããããäž»éãšããã
ãã®ããã«'''äž»éã¯æ§é åŒã®ã©ãã«æžããŠãããã¯é¢ä¿ãªã'''ã
äž»éã12åãšæ±ºãŸã£ãã®ã§ãã®ã¢ã«ã«ã³ã¯ãïœããã«ã³ãã§çµããã
ãã以å€ã®ççŽ ãšæ°ŽçŽ ã®å¡ã¯ããã¹ãŠ[[ææ©ååŠ åº|眮æåº]]ãšããŠæ±ãããã
ã¢ã«ã«ã³ã®çœ®æåºã¯ãå¥ã®å°ããã¢ã«ã«ã³ããæ°ŽçŽ ååãäžã€åãé€ãããã®ãšããŠè¡šãããããã'''ã¢ã«ãã«åº'''(alkyl group)ãšããã
ã¢ã«ãã«åºã®å称ã¯ãã¢ã«ã«ã³ã®aneãylã«çœ®ãæããããšã§äœãã
å·Šãã2åç®ã®ççŽ ããåºãŠãã眮æåºã¯ã¡ã¿ã³(methane)ããæ°ŽçŽ ååãäžã€åãé€ãããã®ã«çããã®ã§ãã¡ãã«(methyl)åºãšããããšã«ãªãã
åæ§ã«ãå·Šãã3åç®ã®ççŽ ããåºãŠããã®ããšãã«(ethyl)åºãå·Šãã4åç®ã®ççŽ ããåºãŠããã®ããããã«(propyl)åºãå·Šãã9çªç®ã®ççŽ ããå³ã«åºãŠããã®ãã¡ãã«åºã§ããã
ãããããŸã'''ã¢ã«ãã¡ãããé '''ã«äžŠã¹ãããããšãethylãmethylãpropylã®é ã«ãªãã
次ã«ããšãã«åºããé ã«ãäž»éã®äœçªç®ã®ççŽ ã«ä»ããŠãããã瀺ãã
ããã§ãå·Šããæ°ããã®ã§3çªç®ãšããèãæ¹ãšãå³ããæ°ããã®ã§10çªç®ãšããèãæ¹ããããã'''ãªãã¹ãçªå·ãå°ãªããªãããã«'''ã€ããã
ãã£ãŠãã3-ãšãã«ïœããšãªãã
次ã«ãã¡ãã«åºã¯ãµãã€ä»ããŠããã®ã§ãããž(di)ã¡ãã«ããšãã颚ã«ãããäœçœ®çªå·ã¯ã'''äžåºŠæ±ºããçªå·ã¯å€ããªã'''ã®ã§ãã2,9-ãžã¡ãã«ããšãªãã'''æåãšæ°åã®éããã€ãã³ã§ã€ãªã'''ãšãã3-ãšãã«-2,9-ãžã¡ãã«ïœããšãªãã
æåŸã«ãããã«åºã¯ã4-ãããã«ããšãªãã®ã§ããã¹ãŠãã€ãªãã§ãã®ã¢ã«ã«ã³ã®ååã¯ã3-ãšãã«-2,9-ãžã¡ãã«-4-ãããã«ããã«ã³ããšãªãã
åºãäœåãããã¯ã®ãªã·ã£èªã®æ°è©ã䜿ã£ãŠè¡šãã
1ãã10ãŸã§ãé ã«ãã¢ã (mono)ã»ãž (di)ã»ã㪠(tri)ã»ããã© (tetra)ã»ãã³ã¿ (penta)ã»ãããµ (hexa)ã»ããã¿ (hepta)ã»ãªã¯ã¿ (octa)ã»ãã (nona)ã»ãã« (deca)ã§ããã5ãã10ãŸã§ã¯ã¢ã«ã«ã³ã®å称ãšãé¢ä¿ããã
== ã¢ã«ã«ã³ã®æ§è³ª ==
*æ°Žã«ã¯æº¶ãã«ããããææ©æº¶åªã«ã¯ãã溶ããã
*åžžæž©ã§ã¯åå¿æ§ã«ä¹ãããé
žå¡©åºãšã¯åå¿ãããé
žåæ§ã»éå
æ§ããªãã
*ççŒããããçºç±éã倧ããã
**CH<sub>4</sub>ïŒ2O<sub>2</sub>ïŒCO<sub>2</sub>ïŒ2H<sub>2</sub>OïŒ890kJ
== 眮æåå¿ ==
眮æåå¿ãšã¯ãåå(å£)ãä»ã®åå(å£)ãšçœ®ãæããåå¿ã§ããã
ã¢ã«ã«ã³ã¯ã玫å€ç·ïŒæ¥å
ïŒã®ååšäžã§ããã²ã³ãšé£ç¶çã«çœ®æåå¿ãèµ·ããã
*CH<sub>4</sub>ïŒCl<sub>2</sub>→CH<sub>3</sub>ClïŒHCl
*CH<sub>3</sub>ClïŒCl<sub>2</sub>→CH<sub>2</sub>Cl<sub>2</sub>ïŒHCl
*CH<sub>2</sub>Cl<sub>2</sub>ïŒCl<sub>2</sub>→CHCl<sub>3</sub>ïŒHCl
*CHCl<sub>3</sub>ïŒCl<sub>2</sub>→CCl<sub>4</sub>ïŒHCl
== å€éšãªã³ã¯ ==
{{Wikipedia|ã¢ã«ã«ã³}}
[[ã«ããŽãª:ææ©ååŠ]]
[[en:Organic Chemistry/Alkanes]] | null | 2022-11-23T05:32:44Z | [
"ãã³ãã¬ãŒã:Wikipedia"
] | https://ja.wikibooks.org/wiki/%E6%9C%89%E6%A9%9F%E5%8C%96%E5%AD%A6/%E3%82%A2%E3%83%AB%E3%82%AB%E3%83%B3 |
2,021 | ææ©ååŠ/åº | ææ©ååŠ>åº
åºãšã¯å®èœåºãçåæ°ŽçŽ åºãªã©ã²ãšãŸãšãŸãã®ååå£ãæãã
ã¢ã«ã«ã³ããHååã1åãšãã®ããããã®ãã¢ã«ãã«åºãšãããã¢ã«ã«ã³ã®ååã®aneãylã«å€ããŠåœåããã ãšãã¬ã³ããHååã1åãšãã®ããããã®ãããã«åºãšããã è³éŠæçåæ°ŽçŽ ããHååã1åãšãã®ããããã®ãã¢ãªãŒã«åºãšããã
çåæ°ŽçŽ åºã¯äžè¬ã«ãR-ããšè¡šãããããšãããã
ããã²ã³ååãåºãšããŠåããããã²ã³ååãåºãšããŠåãå Žåãããã²ãåºãšãããåã
åååãšéãååãäžããããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ææ©ååŠ>åº",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "åºãšã¯å®èœåºãçåæ°ŽçŽ åºãªã©ã²ãšãŸãšãŸãã®ååå£ãæãã",
"title": "åºãšã¯äœã"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ã¢ã«ã«ã³ããHååã1åãšãã®ããããã®ãã¢ã«ãã«åºãšãããã¢ã«ã«ã³ã®ååã®aneãylã«å€ããŠåœåããã ãšãã¬ã³ããHååã1åãšãã®ããããã®ãããã«åºãšããã è³éŠæçåæ°ŽçŽ ããHååã1åãšãã®ããããã®ãã¢ãªãŒã«åºãšããã",
"title": "çåæ°ŽçŽ åºã®çš®é¡"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "çåæ°ŽçŽ åºã¯äžè¬ã«ãR-ããšè¡šãããããšãããã",
"title": "çåæ°ŽçŽ åºã®çš®é¡"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ããã²ã³ååãåºãšããŠåããããã²ã³ååãåºãšããŠåãå Žåãããã²ãåºãšãããåã
åååãšéãååãäžããããã",
"title": "ããã²ãåº"
}
] | ææ©ååŠïŒåº | [[ææ©ååŠ]]ïŒåº
==åºãšã¯äœã==
åºãšã¯[[ææ©ååŠ#å®èœåºãšçåæ°ŽçŽ åº|å®èœåºãçåæ°ŽçŽ åº]]ãªã©ã²ãšãŸãšãŸãã®ååå£ãæãã
==å®èœåºã®çš®é¡==
===é
žçŽ ãå«ãååç©===
<table border="1" class="wikitable">
<tr><th colspan="2">å®èœåº</th><th>ååç©ã®äžè¬å</th><th colspan="2">ååç©ã®äŸ</th></tr>
<tr><td rowspan="2">ããããã·åº</td><td rowspan="2">ïŒOH</td><td>[[ææ©ååŠ_ã¢ã«ã³ãŒã«|ã¢ã«ã³ãŒã«]]</td><td>ã¡ã¿ããŒã«</td><td>CH<sub>3</sub>ïŒOH</td></tr>
<tr><td>ãã§ããŒã«é¡</td><td>ãã§ããŒã«</td><td>C<sub>6</sub>H<sub>5</sub>ïŒOH</td></tr>
<tr><td>ã¢ã«ãããåº</td><td>ïŒCHO</td><td>[[ææ©ååŠ_ã¢ã«ããã|ã¢ã«ããã]]</td><td>ã¢ã»ãã¢ã«ããã</td><td>CH<sub>3</sub>ïŒCHO</td></tr>
<tr><td>ã«ã«ããã«åº</td><td>ïŒCO</td><td>[[ææ©ååŠ_ã±ãã³|ã±ãã³]]</td><td>ã¢ã»ãã³</td><td>CH<sub>3</sub>ïŒCOïŒCH<sub>3</sub></td></tr>
<tr><td>ã«ã«ããã·åº</td><td>ïŒCOOH</td><td>[[ææ©ååŠ_ã«ã«ãã³é
ž|ã«ã«ãã³é
ž]]</td><td>é
¢é
ž</<td>CH<sub>3</sub>ïŒCOOH</td></tr>
<tr><td>ãããåº</td><td>ïŒNO<sub>2</sub></td><td>ãããååç©</td><td>ããããã³ãŒã³</td><td>C<sub>6</sub>H<sub>5</sub>ïŒNO<sub>2</sub></td></tr>
<tr><td>ã¢ããåº</td><td>ïŒNH<sub>2</sub></td><td>ã¢ãã³</td><td>ã¢ããªã³</td><td>C<sub>6</sub>H<sub>5</sub>ïŒNH<sub>2</sub></td></tr>
<tr><td>ã¹ã«ãåº</td><td>ïŒSO<sub>3</sub>H</td><td>ã¹ã«ãã³é
ž</td><td>ãã³ãŒã³ã¹ã«ãã³é
ž</td><td>C<sub>6</sub>H<sub>5</sub>ïŒSO<sub>3</sub>H</td></tr>
<tr><td>ãšãŒãã«çµå</td><td>ïŒOïŒ</td><td>[[ææ©ååŠ_ãšãŒãã«|ãšãŒãã«]]</td><td>ãžã¡ãã«ãšãŒãã«</td><td>CH<sub>3</sub>ïŒOïŒCH<sub>3</sub></td></tr>
<tr><td>ãšã¹ãã«çµå</td><td>ïŒCOOïŒ</td><td>[[ææ©ååŠ_ãšã¹ãã«|ãšã¹ãã«]]</td><td>é
¢é
žã¡ãã«</td><td>CH<sub>3</sub>ïŒCOOïŒCH<sub>3</sub></td></tr>
</table>
==çåæ°ŽçŽ åºã®çš®é¡==
<table border="1" class="wikitable">
<tr><th>çåæ°ŽçŽ åºã®ã°ã«ãŒã</th><th colspan="2">çåæ°ŽçŽ åº</th></tr>
<tr><td rowspan="4">ã¢ã«ãã«åº</td><td>ã¡ãã«åº</td><td>CH<sub>3</sub>ïŒ</td></tr>
<tr><td>ãšãã«åº</td><td>C<sub>2</sub>H<sub>5</sub>ïŒ</td></tr>
<tr><td>(ãã«ãã«)ãããã«åº</td><td>CH<sub>3</sub>CH<sub>2</sub>CH<sub>2</sub>ïŒ</td></tr>
<tr><td>ã€ãœãããã«åº</td><td>(CH<sub>3</sub>)<sub>2</sub>CHïŒ</td></tr>
<tr><td colspan="2">ããã«åº</td><td>CH<sub>2</sub>ïŒCHïŒ</td></tr>
<tr><td rowspan="2">ã¢ãªãŒã«åº</td><td>ãã§ãã«åº</td><td>C<sub>6</sub>H<sub>5</sub>ïŒ</td></tr>
<tr><td>ãããã«åº</td><td>C<sub>10</sub>H<sub>7</sub>ïŒ</td></tr>
</table>
[[ææ©ååŠ_ã¢ã«ã«ã³|ã¢ã«ã«ã³]]ããHååã1åãšãã®ããããã®ãã¢ã«ãã«åºãšãããã¢ã«ã«ã³ã®ååã®aneãylã«å€ããŠåœåããã
[[ææ©ååŠ_ã¢ã«ã±ã³|ãšãã¬ã³]]ããHååã1åãšãã®ããããã®ãããã«åºãšããã
è³éŠæçåæ°ŽçŽ ããHååã1åãšãã®ããããã®ãã¢ãªãŒã«åºãšããã
çåæ°ŽçŽ åºã¯äžè¬ã«ãRïŒããšè¡šãããããšãããã
==ããã²ãåº==
ããã²ã³ååãåºãšããŠåããããã²ã³ååãåºãšããŠåãå Žåãããã²ãåºãšãããåã
åååãšéãååãäžããããã
<table border="1" class="wikitable">
<tr><th>ããã²ãåº</th><th>å称</th><th>ååã®è±å</th></tr>
<tr><td>FïŒ</td><td>ãã«ãªãåº(Fluoro)</td><td>ãã«ãªãªã³(Fluorine)</td></tr>
<tr><td>ClïŒ</td><td>ã¯ããåº(Chloro)</td><td>ã¯ããªã³(Chlorine)</td></tr>
<tr><td>BrïŒ</td><td>ããã¢åº(Bromo)</td><td>ãããã³(Bromine)</td></tr>
<tr><td>IïŒ</td><td>ãšãŒãåº(Iodo)</td><td>ãšãŒãã£ã³(Iodine)</td></tr>
<tr><td>AtïŒ</td><td>ã¢ã¹ã¿ãåº(Astato)</td><td>ã¢ã¹ã¿ãã£ã³(Astatine)</td></tr>
</table>
[[ã«ããŽãª:ææ©ååŠ]] | null | 2022-11-23T05:33:30Z | [] | https://ja.wikibooks.org/wiki/%E6%9C%89%E6%A9%9F%E5%8C%96%E5%AD%A6/%E5%9F%BA |
2,022 | ææ©ååŠ/ã¢ã«ã±ã³ | ææ©ååŠ>ã¢ã«ã±ã³
ççŽ éã«Ïçµåã1ã€ã ãå«ãèèªæçåæ°ŽçŽ ãã¢ã«ã±ã³ (alkene) ãšããã ã¢ã«ã±ã³ã¯äžè¬åŒCnH2nã§è¡šãããã å®çŸ©ãããççŽ ååã1ã€ã®ã¢ã«ã±ã³ã¯ååšããªãã
ã¢ã«ã«ã³ã®èªå°Ÿaneãeneã«å€ããã ãšãã³ (ethen)ããããã³ (propene)ãããã³ (buthene)ããã³ãã³ (pentene)ã»ã»ã»
äœãããšãã³ã¯æ£åŒãªå称(äœç³»å)ãããæ
£çšåãšãã¬ã³(ethylene)ã®æ¹ãè¯ã䜿ãããã ãããã³ãæ
£çšåãããã¬ã³(propylene)ãäœç³»åãšåçšåºŠäœ¿ãããã
ããã³ä»¥äžã®é·ãã®ã¢ã«ã±ã³ã«ã¯ãäºéçµåã®äœçœ®ã«ããæ§é ç°æ§äœãååšããã ãã®å Žåãšãäºéçµåãäž»éã®ã©ãã«ããããåºæ¥ãã ãå°ããçªå·ã«ãã£ãŠè¡šãã
CH2=CHCH2CH3ã1-ããã³ã
CH3CH=CHCH3ã2-ããã³ã
ã¡ãªã¿ã«ããã³ã®ç°æ§äœã«ã¯CH2=C(CH3)2(2-ã¡ãã«ãããã³)ãååšããã
äºéçµåãæã€2ã€ã®ççŽ ååãšããã«çµåãã4ã€ã®ååã¯åäžå¹³é¢äžã«ãããäºéçµåã軞ã«ã²ããããã«å転ãããããšã¯ã§ããªãããã®ãããäºéçµåãæã€äž¡æ¹ã®ççŽ ååã«ããããéãåå(å£)ãæ¥ç¶ããŠãããšãã2ã€ã®ç«äœç°æ§äœãååšããããããã·ã¹ã»ãã©ã³ã¹ç°æ§äœãšããã
äŸãã°2-ããã³ã¯CH3>C=C<CH3ãšCH3>C=C<Hã®2ã€ãååšããããã®ãšããäž»é(ççŽ æ°æå€ã®é)ãšãªãççŽ éªšæ Œãäºéçµåã®åãåŽã«ããæ¹ãã·ã¹ (cis) åãå察åŽã«ããæ¹ããã©ã³ã¹ (trans) åãšããã®ã§ãåè
ã¯ãã·ã¹-2-ããã³ããåŸè
ã¯ããã©ã³ã¹-2-ããã³ãã§ããããã£ãŠããã³ã«ã¯æ§é ç°æ§äœã®1-ããã³, 2-ã¡ãã«ãããã³ãå«ãã4çš®ã®ç«äœç°æ§äœãååšããã泚æãã¹ãã¯ãåçš®ã®ååå£ãåãåŽã«ãããå察åŽã«ãããã«ãã£ãŠcis/transãåºå¥ããã®ã§ã¯ãªãããããŸã§äž»éãåãåŽã«ãããå察åŽã«ãããã«ãã£ãŠåºå¥ããç¹ã§ãããäŸãã°ãCH3>C=C<CH3ãšCH3>C=C<C2H5ã§ã¯ãåè
ãtransãåŸè
ãcisã§ããã
äºéçµåã®ãã¡çæ¹ã¯ÏçµåãšåŒã°ããå
ãçµåãããçæ¹ã¯ÏçµåãšåŒã°ãã匱ãçµåã§ãæ°ŽçŽ ãããã²ã³ãªã©ãè¿ã¥ããšÏçµåãåãåå¿ããããããä»å åå¿ãšããã
ã¢ã«ã±ã³å士ãä»å åå¿ãèµ·ãããšãå€æ°ã®ã¢ã«ã±ã³ãã€ãªãã£ã倧ããªååãåºæ¥ãããã®åå¿ãä»å éåãšããã
ä»å éåã¯ããã«åºãæã€ãã®ãèµ·ãããäžè¬çã«æžããš
ãšãªãã
äºéçµåã¯é
žåããããããé
žåå€ãäžãããšäºéçµåãéè£ããã±ãã³ãã¢ã«ããããã«ã«ãã³é
žãªã©ã«ãªãã
éãã³ã¬ã³é
žå¡©ãåé
žåãªã¹ããŠã ã«ããé
žåã§ã¯ã2䟡ã¢ã«ã³ãŒã«(1,2-ãžãªãŒã«)ãçããã
éé
ž-OOHã«ããé
žåã§ã¯ã-C-O-C-ã§æ§æãããäžå¡ç°ååç©ããšããã·ããçããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ææ©ååŠ>ã¢ã«ã±ã³",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ççŽ éã«Ïçµåã1ã€ã ãå«ãèèªæçåæ°ŽçŽ ãã¢ã«ã±ã³ (alkene) ãšããã ã¢ã«ã±ã³ã¯äžè¬åŒCnH2nã§è¡šãããã å®çŸ©ãããççŽ ååã1ã€ã®ã¢ã«ã±ã³ã¯ååšããªãã",
"title": "ã¢ã«ã±ã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ã¢ã«ã«ã³ã®èªå°Ÿaneãeneã«å€ããã ãšãã³ (ethen)ããããã³ (propene)ãããã³ (buthene)ããã³ãã³ (pentene)ã»ã»ã»",
"title": "ã¢ã«ã±ã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "äœãããšãã³ã¯æ£åŒãªå称(äœç³»å)ãããæ
£çšåãšãã¬ã³(ethylene)ã®æ¹ãè¯ã䜿ãããã ãããã³ãæ
£çšåãããã¬ã³(propylene)ãäœç³»åãšåçšåºŠäœ¿ãããã",
"title": "ã¢ã«ã±ã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ããã³ä»¥äžã®é·ãã®ã¢ã«ã±ã³ã«ã¯ãäºéçµåã®äœçœ®ã«ããæ§é ç°æ§äœãååšããã ãã®å Žåãšãäºéçµåãäž»éã®ã©ãã«ããããåºæ¥ãã ãå°ããçªå·ã«ãã£ãŠè¡šãã",
"title": "ã¢ã«ã±ã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "CH2=CHCH2CH3ã1-ããã³ã",
"title": "ã¢ã«ã±ã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "CH3CH=CHCH3ã2-ããã³ã",
"title": "ã¢ã«ã±ã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ã¡ãªã¿ã«ããã³ã®ç°æ§äœã«ã¯CH2=C(CH3)2(2-ã¡ãã«ãããã³)ãååšããã",
"title": "ã¢ã«ã±ã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "äºéçµåãæã€2ã€ã®ççŽ ååãšããã«çµåãã4ã€ã®ååã¯åäžå¹³é¢äžã«ãããäºéçµåã軞ã«ã²ããããã«å転ãããããšã¯ã§ããªãããã®ãããäºéçµåãæã€äž¡æ¹ã®ççŽ ååã«ããããéãåå(å£)ãæ¥ç¶ããŠãããšãã2ã€ã®ç«äœç°æ§äœãååšããããããã·ã¹ã»ãã©ã³ã¹ç°æ§äœãšããã",
"title": "ã¢ã«ã±ã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "äŸãã°2-ããã³ã¯CH3>C=C<CH3ãšCH3>C=C<Hã®2ã€ãååšããããã®ãšããäž»é(ççŽ æ°æå€ã®é)ãšãªãççŽ éªšæ Œãäºéçµåã®åãåŽã«ããæ¹ãã·ã¹ (cis) åãå察åŽã«ããæ¹ããã©ã³ã¹ (trans) åãšããã®ã§ãåè
ã¯ãã·ã¹-2-ããã³ããåŸè
ã¯ããã©ã³ã¹-2-ããã³ãã§ããããã£ãŠããã³ã«ã¯æ§é ç°æ§äœã®1-ããã³, 2-ã¡ãã«ãããã³ãå«ãã4çš®ã®ç«äœç°æ§äœãååšããã泚æãã¹ãã¯ãåçš®ã®ååå£ãåãåŽã«ãããå察åŽã«ãããã«ãã£ãŠcis/transãåºå¥ããã®ã§ã¯ãªãããããŸã§äž»éãåãåŽã«ãããå察åŽã«ãããã«ãã£ãŠåºå¥ããç¹ã§ãããäŸãã°ãCH3>C=C<CH3ãšCH3>C=C<C2H5ã§ã¯ãåè
ãtransãåŸè
ãcisã§ããã",
"title": "ã¢ã«ã±ã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "äºéçµåã®ãã¡çæ¹ã¯ÏçµåãšåŒã°ããå
ãçµåãããçæ¹ã¯ÏçµåãšåŒã°ãã匱ãçµåã§ãæ°ŽçŽ ãããã²ã³ãªã©ãè¿ã¥ããšÏçµåãåãåå¿ããããããä»å åå¿ãšããã",
"title": "ã¢ã«ã±ã³ã®æ§è³ª"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "ã¢ã«ã±ã³å士ãä»å åå¿ãèµ·ãããšãå€æ°ã®ã¢ã«ã±ã³ãã€ãªãã£ã倧ããªååãåºæ¥ãããã®åå¿ãä»å éåãšããã",
"title": "ã¢ã«ã±ã³ã®æ§è³ª"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "ä»å éåã¯ããã«åºãæã€ãã®ãèµ·ãããäžè¬çã«æžããš",
"title": "ã¢ã«ã±ã³ã®æ§è³ª"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ãšãªãã",
"title": "ã¢ã«ã±ã³ã®æ§è³ª"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "äºéçµåã¯é
žåããããããé
žåå€ãäžãããšäºéçµåãéè£ããã±ãã³ãã¢ã«ããããã«ã«ãã³é
žãªã©ã«ãªãã",
"title": "ã¢ã«ã±ã³ã®æ§è³ª"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "éãã³ã¬ã³é
žå¡©ãåé
žåãªã¹ããŠã ã«ããé
žåã§ã¯ã2䟡ã¢ã«ã³ãŒã«(1,2-ãžãªãŒã«)ãçããã",
"title": "ã¢ã«ã±ã³ã®æ§è³ª"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "éé
ž-OOHã«ããé
žåã§ã¯ã-C-O-C-ã§æ§æãããäžå¡ç°ååç©ããšããã·ããçããã",
"title": "ã¢ã«ã±ã³ã®æ§è³ª"
}
] | ææ©ååŠïŒã¢ã«ã±ã³ | [[ææ©ååŠ]]ïŒã¢ã«ã±ã³
== ã¢ã«ã±ã³ã®å®çŸ©ãšåœåæ³ ==
=== ã¢ã«ã±ã³ã®å®çŸ© ===
ççŽ éã«Ïçµåã1ã€ã ãå«ãèèªæçåæ°ŽçŽ ãã¢ã«ã±ã³ (alkene) ãšããã
ã¢ã«ã±ã³ã¯[[ææ©ååŠ_ã¢ã«ã«ã³#ã¢ã«ã«ã³ã®äžè¬åŒ|äžè¬åŒ]]C<sub>n</sub>H<sub>2n</sub>ã§è¡šãããã
å®çŸ©ãããççŽ ååã1ã€ã®ã¢ã«ã±ã³ã¯ååšããªãã
=== åœåæ³ ===
[[ææ©ååŠ_ã¢ã«ã«ã³#åœåæ³|ã¢ã«ã«ã³]]ã®èªå°Ÿaneãeneã«å€ããã
ãšãã³ (ethen)ããããã³ (propene)ãããã³ (buthene)ããã³ãã³ (pentene)ã»ã»ã»
äœãããšãã³ã¯æ£åŒãªå称ïŒäœç³»åïŒãããæ
£çšåãšãã¬ã³(ethylene)ã®æ¹ãè¯ã䜿ãããã
ãããã³ãæ
£çšåãããã¬ã³(propylene)ãäœç³»åãšåçšåºŠäœ¿ãããã
<gallery>
File:Ethene structural.svg|ãšãã³
File:Propene-2D-flat.png|ãããã³
</gallery>
ããã³ä»¥äžã®é·ãã®ã¢ã«ã±ã³ã«ã¯ãäºéçµåã®äœçœ®ã«ãã[[ææ©ååŠ_ã¢ã«ã«ã³#ç°æ§äœ|æ§é ç°æ§äœ]]ãååšããã
ãã®å Žåãšãäºéçµåã[[ææ©ååŠ_ã¢ã«ã«ã³#ã¢ã«ã«ã³ã®è©³ããåœåæ³|äž»é]]ã®ã©ãã«ããããåºæ¥ãã ãå°ããçªå·ã«ãã£ãŠè¡šãã
CH<sub>2</sub>=CHCH<sub>2</sub>CH<sub>3</sub>ã1-ããã³ã
CH<sub>3</sub>CH=CHCH<sub>3</sub>ã2-ããã³ã
ã¡ãªã¿ã«ããã³ã®ç°æ§äœã«ã¯CH<sub>2</sub>=C(CH<sub>3</sub>)<sub>2</sub>(2ïŒã¡ãã«ãããã³)ãååšããã
<gallery>
File:1-butene.svg|1ïŒããã³
File:Cis-2-butene.svg|2ïŒããã³ïŒã·ã¹åïŒ
File:Methylpropene.PNG|2ïŒã¡ãã«ãããã³
</gallery>
=== ã·ã¹ã»ãã©ã³ã¹ç°æ§äœ ===
äºéçµåãæã€2ã€ã®ççŽ ååãšããã«çµåãã4ã€ã®ååã¯åäžå¹³é¢äžã«ãããäºéçµåã軞ã«ã²ããããã«å転ãããããšã¯ã§ããªãããã®ãããäºéçµåãæã€äž¡æ¹ã®ççŽ ååã«ããããéãåå(å£)ãæ¥ç¶ããŠãããšãã2ã€ã®[[ææ©ååŠ_ã¢ã«ã«ã³#ç°æ§äœ|ç«äœç°æ§äœ]]ãååšãããããã'''ã·ã¹ã»ãã©ã³ã¹ç°æ§äœ'''ãšããã
äŸãã°2ïŒããã³ã¯<sub>CH3</sub><sup>H</sup>ïŒCïŒCïŒ<sup>H</sup><sub>CH3</sub>ãš<sub>CH3</sub><sup>H</sup>ïŒCïŒCïŒ<sub>H</sub><sup>CH3</sup>ã®2ã€ãååšããããã®ãšããäž»éïŒççŽ æ°æå€ã®éïŒãšãªãççŽ éªšæ Œãäºéçµåã®åãåŽã«ããæ¹ã'''ã·ã¹''' (cis) åãå察åŽã«ããæ¹ã'''ãã©ã³ã¹''' (trans) åãšããã®ã§ãåè
ã¯ãã·ã¹ïŒ2ïŒããã³ããåŸè
ã¯ããã©ã³ã¹ïŒ2ïŒããã³ãã§ããããã£ãŠããã³ã«ã¯æ§é ç°æ§äœã®1-ããã³, 2-ã¡ãã«ãããã³ãå«ãã4çš®ã®ç«äœç°æ§äœãååšããã泚æãã¹ãã¯ãåçš®ã®ååå£ãåãåŽã«ãããå察åŽã«ãããã«ãã£ãŠcis/transãåºå¥ããã®ã§ã¯ãªãããããŸã§äž»éãåãåŽã«ãããå察åŽã«ãããã«ãã£ãŠåºå¥ããç¹ã§ãããäŸãã°ã<sub>CH3</sub><sup>H</sup>ïŒCïŒCïŒ<sup>C2H5</sup><sub>CH3</sub>ãš<sub>CH3</sub><sup>H</sup>ïŒCïŒCïŒ<sub>C2H5</sub><sup>CH3</sup>ã§ã¯ãåè
ãtransãåŸè
ãcisã§ããã
<gallery>
File:Cis-2-butene.svg|2ïŒããã³ïŒã·ã¹åïŒ
File:Trans-2-butene.svg|2ïŒããã³ïŒãã©ã³ã¹åïŒ
</gallery>
== ã¢ã«ã±ã³ã®æ§è³ª ==
=== ä»å åå¿ ===
äºéçµåã®ãã¡çæ¹ã¯σçµåãšåŒã°ããå
ãçµåãããçæ¹ã¯πçµåãšåŒã°ãã匱ãçµåã§ãæ°ŽçŽ ãããã²ã³ãªã©ãè¿ã¥ããšπçµåãåãåå¿ãããããã'''ä»å åå¿'''ãšããã
*CH<sub>2</sub>ïŒCH<sub>2</sub> ïŒ H<sub>2</sub> → CH<sub>3</sub>ïŒCH<sub>3</sub>
*CH<sub>2</sub>ïŒCH<sub>2</sub> ïŒ Br<sub>2</sub> → CH<sub>2</sub>BrïŒCH<sub>2</sub>Br
**ã¢ã«ã±ã³ã¯1molã«ã€ã1molã®èçŽ æ°Žãè±è²ããã[[ææ©ååŠ_ã¢ã«ã«ã³#ã¢ã«ã«ã³ã®æ§è³ª|ã¢ã«ã«ã³]]ã¯èçŽ æ°Žãè±è²ããªããã[[ææ©ååŠ_ã¢ã«ãã³#ä»å åå¿|ã¢ã«ãã³]]ã¯1molã«ã€ã2molã®èçŽ æ°Žãè±è²ããã®ã§åºå¥ã§ããã
*CH<sub>2</sub>ïŒCH<sub>2</sub> ïŒ H<sub>2</sub>O → CH<sub>3</sub>ïŒCH<sub>2</sub>OH
**äžè¬ã«ã¢ã«ã±ã³ã«æ°Žãä»å ãããšã¢ã«ã³ãŒã«ã«ãªãã
*CH<sub>2</sub>ïŒCHïŒCH<sub>3</sub> ïŒ HCl → CH<sub>3</sub>ïŒCHClïŒCH<sub>3</sub>
**HClãH<sub>2</sub>OçãHXåã®ååç©ãä»å ãããšããHååã¯Cååã«çŽæ¥çµåããHååã®å€ãæ¹ã«çµåãããããããã«ã³ããã³ãåãšããã
**ãã ãããã©ã³é¡(BH<sub>3</sub>, BHR<sub>2</sub>ãªã©)ã®ä»å ã«ãããŠã¯ãHååãCååã«çŽæ¥çµåããHååã®å°ãªãæ¹ã«çµåãããéãã«ã³ããã³ãåãé©çšãããã
=== ä»å éå ===
ã¢ã«ã±ã³å士ãä»å åå¿ãèµ·ãããšãå€æ°ã®ã¢ã«ã±ã³ãã€ãªãã£ã倧ããªååãåºæ¥ãããã®åå¿ã'''ä»å éå'''ãšããã
ä»å éåã¯[[ææ©ååŠ_åº#çåæ°ŽçŽ åºã®çš®é¡|ããã«åº]]ãæã€ãã®ãèµ·ãããäžè¬çã«æžããš
*n CH<sub>2</sub>ïŒCHX → (ïŒCH<sub>2</sub>ïŒCHXïŒ)<sub>n</sub>
ãšãªãã
=== éå
æ§ ===
äºéçµåã¯é
žåããããããé
žåå€ãäžãããšäºéçµåãéè£ããã±ãã³ãã¢ã«ããããã«ã«ãã³é
žãªã©ã«ãªãã
*CH<sub>3</sub>CH<sub>2</sub>CHïŒCHCH<sub>3</sub> ïŒ 4(O) → (CH<sub>3</sub>CH<sub>2</sub>CHO ïŒCH<sub>3</sub>CHO ïŒ 2(O)) → CH<sub>3</sub>CH<sub>2</sub>COOH ïŒ CH<sub>3</sub>COOH
éãã³ã¬ã³é
žå¡©ãåé
žåãªã¹ããŠã ã«ããé
žåã§ã¯ã2䟡ã¢ã«ã³ãŒã«(1,2-ãžãªãŒã«)ãçããã
*CH<sub>2</sub>ïŒCH<sub>2</sub> ïŒ (O) ïŒ H<sub>2</sub>O → CH<sub>2</sub>OHïŒCH<sub>2</sub>OH
éé
ž-OOHã«ããé
žåã§ã¯ã-C-O-C-ã§æ§æãããäžå¡ç°ååç©ããšããã·ããçããã
[[ã«ããŽãª:ææ©ååŠ]]
[[en:Organic Chemistry/Alkenes]] | null | 2022-11-23T05:32:51Z | [] | https://ja.wikibooks.org/wiki/%E6%9C%89%E6%A9%9F%E5%8C%96%E5%AD%A6/%E3%82%A2%E3%83%AB%E3%82%B1%E3%83%B3 |
2,024 | ææ©ååŠ/ã¢ã«ãã³ | ææ©ååŠ>ã¢ã«ãã³
ççŽ éã«äžéçµåã1ã€ã ãå«ãèèªæçåæ°ŽçŽ ãã¢ã«ãã³ (alkyne) ãšããã ã¢ã«ãã³ã¯äžè¬åŒCnH2n-2ã§è¡šãããã å®çŸ©ãããççŽ ååã1ã€ã®ã¢ã«ãã³ã¯ååšããªãã
ã¢ã«ã«ã³ã®èªå°Ÿaneãyneã«å€ããã ãšãã³ (ethyn)ããããã³ (propyne)ãããã³ (buthyne)ããã³ãã³ (pentyne)ã»ã»ã»
äœãããšãã³ããããã³ã¯æ£åŒãªå称(åœéå)ãããæ
£çšåã¢ã»ãã¬ã³ãã¡ãã«ã¢ã»ãã¬ã³ã®æ¹ãè¯ã䜿ãããã
æ§é ç°æ§äœã®åœåã«ã€ããŠã¯ã¢ã«ã±ã³ãšåãã§ããã
Câ¡Cã«çŽæ¥çµåããHååã¯åŒ±ãã€ãªã³æ§ã瀺ãã®ã§ãã¢ã³ã¢ãã¢æ§ç¡é
žé氎溶液([Ag(NH3)2]ãå«ãã 溶液)ã«ã¢ã»ãã¬ã³ãéãããšéã¢ã»ããªã(çœè²æ²æ®¿)ãçãããã¢ã»ããªãã¯äžå®å®ã§ãç¹ã«ä¹Ÿç¥ãããã®ã¯ççºæ§ããããæ«ç«¯äžéçµåã®æ€åºã«çšããããã
H-Câ¡C-H + 2Ag â Ag-Câ¡C-Ag + 2H
H-Câ¡C-CH2-CH3 + Ag â Ag-Câ¡C-CH2-CH3
äžéçµåã®ãã¡1æ¬ã¯Ïçµåãæ®ãã®2æ¬ã¯Ïçµåã§ããããã£ãŠãã¢ã«ãã³ã¯ä»å åå¿ããã
ã¢ã»ãã¬ã³ååã¯å°æ°ã§ä»å éåããã
ã¢ã»ãã¬ã³äžååãéåãããšãã³ãŒã³ãçããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ææ©ååŠ>ã¢ã«ãã³",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ççŽ éã«äžéçµåã1ã€ã ãå«ãèèªæçåæ°ŽçŽ ãã¢ã«ãã³ (alkyne) ãšããã ã¢ã«ãã³ã¯äžè¬åŒCnH2n-2ã§è¡šãããã å®çŸ©ãããççŽ ååã1ã€ã®ã¢ã«ãã³ã¯ååšããªãã",
"title": "ã¢ã«ãã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ã¢ã«ã«ã³ã®èªå°Ÿaneãyneã«å€ããã ãšãã³ (ethyn)ããããã³ (propyne)ãããã³ (buthyne)ããã³ãã³ (pentyne)ã»ã»ã»",
"title": "ã¢ã«ãã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "äœãããšãã³ããããã³ã¯æ£åŒãªå称(åœéå)ãããæ
£çšåã¢ã»ãã¬ã³ãã¡ãã«ã¢ã»ãã¬ã³ã®æ¹ãè¯ã䜿ãããã",
"title": "ã¢ã«ãã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "æ§é ç°æ§äœã®åœåã«ã€ããŠã¯ã¢ã«ã±ã³ãšåãã§ããã",
"title": "ã¢ã«ãã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "Câ¡Cã«çŽæ¥çµåããHååã¯åŒ±ãã€ãªã³æ§ã瀺ãã®ã§ãã¢ã³ã¢ãã¢æ§ç¡é
žé氎溶液([Ag(NH3)2]ãå«ãã 溶液)ã«ã¢ã»ãã¬ã³ãéãããšéã¢ã»ããªã(çœè²æ²æ®¿)ãçãããã¢ã»ããªãã¯äžå®å®ã§ãç¹ã«ä¹Ÿç¥ãããã®ã¯ççºæ§ããããæ«ç«¯äžéçµåã®æ€åºã«çšããããã",
"title": "ã¢ã«ãã³ã®æ§è³ª"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "H-Câ¡C-H + 2Ag â Ag-Câ¡C-Ag + 2H",
"title": "ã¢ã«ãã³ã®æ§è³ª"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "H-Câ¡C-CH2-CH3 + Ag â Ag-Câ¡C-CH2-CH3",
"title": "ã¢ã«ãã³ã®æ§è³ª"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "äžéçµåã®ãã¡1æ¬ã¯Ïçµåãæ®ãã®2æ¬ã¯Ïçµåã§ããããã£ãŠãã¢ã«ãã³ã¯ä»å åå¿ããã",
"title": "ã¢ã«ãã³ã®æ§è³ª"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "ã¢ã»ãã¬ã³ååã¯å°æ°ã§ä»å éåããã",
"title": "ã¢ã«ãã³ã®æ§è³ª"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "ã¢ã»ãã¬ã³äžååãéåãããšãã³ãŒã³ãçããã",
"title": "ã¢ã«ãã³ã®æ§è³ª"
}
] | ææ©ååŠïŒã¢ã«ãã³ | [[ææ©ååŠ]]ïŒã¢ã«ãã³
== ã¢ã«ãã³ã®å®çŸ©ãšåœåæ³ ==
=== ã¢ã«ãã³ã®å®çŸ© ===
ççŽ éã«äžéçµåã1ã€ã ãå«ãèèªæçåæ°ŽçŽ ãã¢ã«ãã³ (alkyne) ãšããã
ã¢ã«ãã³ã¯äžè¬åŒC<sub>n</sub>H<sub>2nïŒ2</sub>ã§è¡šãããã
å®çŸ©ãããççŽ ååã1ã€ã®ã¢ã«ãã³ã¯ååšããªãã
=== åœåæ³ ===
[[ææ©ååŠ_ã¢ã«ã«ã³#åœåæ³|ã¢ã«ã«ã³]]ã®èªå°Ÿaneãyneã«å€ããã
ãšãã³ (ethyn)ããããã³ (propyne)ãããã³ (buthyne)ããã³ãã³ (pentyne)ã»ã»ã»
äœãããšãã³ããããã³ã¯æ£åŒãªå称ïŒåœéåïŒãããæ
£çšåã¢ã»ãã¬ã³ãã¡ãã«ã¢ã»ãã¬ã³ã®æ¹ãè¯ã䜿ãããã
[[ææ©ååŠ_ã¢ã«ã«ã³#ç°æ§äœ|æ§é ç°æ§äœ]]ã®åœåã«ã€ããŠã¯[[ææ©ååŠ_ã¢ã«ã±ã³#åœåæ³|ã¢ã«ã±ã³]]ãšåãã§ããã
== ã¢ã«ãã³ã®æ§è³ª ==
*ã¢ã»ãã¬ã³CH≡CHã¯çŽç·æ§é ããšãã
*[[ææ©ååŠ_ã¢ã«ã«ã³#ã¢ã«ã«ã³ã®æ§è³ª|眮æåå¿]]ã[[ææ©ååŠ_ã¢ã«ã±ã³#ä»å åå¿|ä»å åå¿]]ã®ã©ã¡ããèµ·ããã
=== 眮æåå¿ ===
Câ¡Cã«çŽæ¥çµåããHååã¯åŒ±ãã€ãªã³æ§ã瀺ãã®ã§ãã¢ã³ã¢ãã¢æ§ç¡é
žé氎溶液ïŒïŒ»Ag(NH<sub>3</sub>)<sub>2</sub><sup>ïŒ</sup>ãå«ãã 溶液ïŒã«ã¢ã»ãã¬ã³ãéãããšéã¢ã»ããªãïŒçœè²æ²æ®¿ïŒãçãããã¢ã»ããªãã¯äžå®å®ã§ãç¹ã«ä¹Ÿç¥ãããã®ã¯ççºæ§ããããæ«ç«¯äžéçµåã®æ€åºã«çšããããã
HïŒCâ¡CïŒH ïŒ 2Ag<sup>ïŒ</sup> → AgïŒCâ¡CïŒAg ïŒ 2H<sup>ïŒ</sup>
HïŒCâ¡CïŒCH<sub>2</sub>ïŒCH<sub>3</sub>ã+ Ag<sup>ïŒ</sup> → AgïŒCâ¡CïŒCH<sub>2</sub>ïŒCH<sub>3</sub>
=== ä»å åå¿ ===
äžéçµåã®ãã¡1æ¬ã¯[[ææ©ååŠ_ã¢ã«ã±ã³#ä»å åå¿|σçµå]]ãæ®ãã®2æ¬ã¯[[ææ©ååŠ_ã¢ã«ã±ã³#ä»å åå¿|πçµå]]ã§ããããã£ãŠãã¢ã«ãã³ã¯ä»å åå¿ããã
*HïŒCâ¡CïŒH ïŒ 2H<sub>2</sub> → CH<sub>2</sub>ïŒCH<sub>2</sub> ïŒ H<sub>2</sub> → CH<sub>3</sub>ïŒCH<sub>3</sub>
*HïŒCâ¡CïŒH ïŒ 2Cl<sub>2</sub> → (CHClïŒCHCl ïŒ Cl<sub>2</sub>) → CHCl<sub>2</sub>ïŒCHCl<sub>2</sub>
**æ°ŽçŽ ä»å ã¯ã¢ã«ãã³ããã¢ã«ã±ã³ãã¢ã«ã±ã³ããã¢ã«ã«ã³ãžãšé£ç¶çã«å€åããããããã²ã³ä»å ã¯ã¢ã«ãã³ããäžæ°ã«ã¢ã«ã«ã³ã«ãªãã
*HïŒCâ¡CïŒH ïŒ 2Br<sub>2</sub> → CHBr<sub>2</sub>ïŒCHBr<sub>2</sub>
**1molã«ã€ã2molã®èçŽ ãè±è²ããã®ã§ã[[ææ©ååŠ_ã¢ã«ã«ã³#ã¢ã«ã«ã³ã®æ§è³ª|ã¢ã«ã«ã³]]ã[[ææ©ååŠ_ã¢ã«ã±ã³#ä»å åå¿|ã¢ã«ã±ã³]]ãšåºå¥ã§ããã
*HïŒCâ¡CïŒH ïŒ H<sub>2</sub>O → (CH<sub>2</sub>ïŒCHOH) → CH<sub>3</sub>ïŒCHO
**ããã«ã¢ã«ã³ãŒã«CH<sub>2</sub>ïŒCHOHã¯äžå®å®ãªã®ã§ã[[ææ©ååŠ_åº#å®èœåºã®çš®é¡|ããããã·ã«åº]]ã®Hååãäºéçµåã®å察åŽã«é£ãã§äºå€ç°æ§ãèµ·ãããã¢ã»ãã¢ã«ããããšãªãã
=== ä»å éå ===
ã¢ã»ãã¬ã³ååã¯å°æ°ã§ä»å éåããã
*2 HïŒCâ¡CïŒH → CH<sub>2</sub>ïŒCHïŒCâ¡CH
**çæ¹ã®ååã®äžéçµåã1æ¬éããããçæ¹ã®ååã¯CHéã®çµåãåããŠããããçµåããã
ã¢ã»ãã¬ã³äžååãéåãããšãã³ãŒã³ãçããã
*3 HïŒCâ¡CïŒH → C<sub>6</sub>H<sub>6</sub>
**ãã³ãŒã³C<sub>6</sub>H<sub>6</sub>ã¯è³éŠæååç©ã®æå°åäœã§ããã
CH=CH
/ \
CH CH
\\ //
CH-CH
ãã³ãŒã³C6H6
[[ã«ããŽãª:ææ©ååŠ]]
[[en:Organic Chemistry/Alkynes]] | null | 2022-11-23T05:32:47Z | [] | https://ja.wikibooks.org/wiki/%E6%9C%89%E6%A9%9F%E5%8C%96%E5%AD%A6/%E3%82%A2%E3%83%AB%E3%82%AD%E3%83%B3 |
2,027 | HTML/å€éšãªã³ã¯ | HTMLã®äœæã«åœ¹ç«ã€æ
å ±æºã玹ä»ããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "HTMLã®äœæã«åœ¹ç«ã€æ
å ±æºã玹ä»ããã",
"title": ""
}
] | HTMLã®äœæã«åœ¹ç«ã€æ
å ±æºã玹ä»ããã | {{Pathnav|HTML|frame=1|small=1}}
{{Wikipedia|HTML|HTML}}
HTMLã®äœæã«åœ¹ç«ã€æ
å ±æºã玹ä»ããã
== ãªã³ã¯ ==
* [https://whatwg.org/ Web Hypertext Application Technology Working Group(WHATWG)]ïŒW3Cã«ä»£ãã£ãŠHTML/DOMçã®æšæºçå®ãè¡ãå£äœ
** [https://html.spec.whatwg.org/ HTML Living Standard]ïŒ[https://momdo.github.io/html/ æ¥æ¬èªèš³]ïŒïŒææ°ã®HTMLæšæº
** [https://dom.spec.whatwg.org/ DOM Living Standard]ïŒ[https://triple-underscore.github.io/DOM4-ja.html æ¥æ¬èªèš³]ïŒïŒææ°ã®DOMæšæº
* [https://www.w3.org/ World Wide Web Consortium (W3C)]ïŒHTML4.01ãŸã§ã®æšæºãæå±ããŠããæ©é¢W3Cã®ãµã€ã
** [https://www.w3.org/TR/html5/ https://www.w3.org/TR/html5/]ã¯[https://html.spec.whatwg.org/ https://html.spec.whatwg.org/]ãžã®ãªãã€ã¬ã¯ã
** [https://www.w3.org/TR/ All Standards and Drafts - W3C]ïŒW3Cã®æšæºãšãã®èæ¡ãäžèŠ§ã»æ€çŽ¢ã§ãã
** [https://www.w3.org/MarkUp/ W3C HTML Home Page] -- 2010-12-17: The XHTML2 Working Group is closed.
** [http://validator.w3.org/ The W3C Markup Validation Service]ïŒHTMLææ³ãã§ãã«ãŒïŒå¶äœããHTMLã«èª€ãçãç¡ããããã§ãã¯åºæ¥ã
** [https://www.w3.org/Style/CSS/ W3C Cascading Style Sheets]
** [https://jigsaw.w3.org/css-validator/ W3C CSS æ€èšŒãµãŒãã¹]ïŒCSSææ³ãã§ãã«ãŒ
* [http://www.asahi-net.or.jp/%7Esd5a-ucd/rec-html40j/ HTML 4ä»æ§æžéŠèš³èšç»]ïŒéå
¬åŒã®æ¥æ¬èªèš³
== ãªã³ã¯åã ==
* <del> <!-- http://htmllint.itc.keio.ac.jp/htmllint/htmllint.html --> Another HTML-lint gatewayïŒHTMLææ³ãã§ãã«ãŒïŒæ¥æ¬èªïŒ</del> ⻠以äžããªã³ã¯åã
* <del> <!-- [http://www.mozilla.gr.jp/standards/ Webæšæºæ®åãããžã§ã¯ã]-->Webæšæºæ®åãããžã§ã¯ãïŒ[[w:ãããçµ|ãããçµ]]ã«ããWebæšæºåTips</del>
* <del> <!-- [http://operawiki.info/WebDevToolbar Web Developer Toolbar & Menu for Opera] --> "Web Developer Toolbar & Menu for Opera" - Operaã§äœ¿ããéçºããŒã«ïŒè±èªçïŒ</del>
[[Category:World Wide Web|HTML ãããµããã]]
[[en:HyperText Markup Language/Links]] | null | 2021-06-06T02:35:48Z | [
"ãã³ãã¬ãŒã:Pathnav",
"ãã³ãã¬ãŒã:Wikipedia"
] | https://ja.wikibooks.org/wiki/HTML/%E5%A4%96%E9%83%A8%E3%83%AA%E3%83%B3%E3%82%AF |
2,035 | ç¹æ®çžå¯Ÿè« 4å
éåé | ç¹æ®çžå¯Ÿè« > 4å
éåé
解æååŠãèãããšã空éã®çæ¹æ§ããéåéä¿åã 瀺ãããã®ãšåæ§ã«ãæéã«å¯Ÿããäžæ§æ§ãããšãã«ã®ãŒã® ä¿ååãå°ãåºãããã ãã®ããã x ÎŒ = ( c t x y z ) {\displaystyle x^{\mu }={\begin{pmatrix}ct\\x\\y\\z\end{pmatrix}}} ã®ããã«çµã¿åãããŠ4å
ãã¯ãã«ãäœã£ãããšã«å¯Ÿå¿ããŠã p ÎŒ = ( ε / c p x p y p z ) {\displaystyle p^{\mu }={\begin{pmatrix}\epsilon /c\\p_{x}\\p_{y}\\p_{z}\end{pmatrix}}} ã«ãã£ãŠã4å
ãã¯ãã«ãäœãããšãåºæ¥ãã ããã§ã ε {\displaystyle \epsilon } ã¯ãšãã«ã®ãŒã§ããã ãã®4å
ãã¯ãã«ã4å
éåéãšåŒã¶ã ããéæ¢ããç©äœã«ã€ããŠã¯ p â = 0 {\displaystyle {\vec {p}}=0} ãæãç«ã€ã®ã§ã p ÎŒ = ( ε / c 0 0 0 ) {\displaystyle p^{\mu }={\begin{pmatrix}\epsilon /c\\0\\0\\0\end{pmatrix}}} ãšãªãã ãã®ãšã㮠ε {\displaystyle \epsilon } ã®å€ãããã質émããã€ç©äœã«å¯ŸããŠã mc ãšçœ®ãã ε / c = m c {\displaystyle \epsilon /c=mc} ã€ãŸã, ε = m c 2 {\displaystyle \epsilon =mc^{2}} ã«æ³šæã (ãšãã«ã®ãŒã®å®æ°å€ã¯ã©ã®ããã«ã§ãåããããç¹ã«ãã®ããã« éžã¶ã®ã¯å®éšçã«è³ªéãšãšãã«ã®ãŒã®åå€æ§ãç¥ãããŠããããšã« ãã£ãŠãããã®ãšæãããã) ãã®ãšãã ε {\displaystyle \epsilon } ãš | p â | {\displaystyle |{\vec {p}}|} ã®é¢ä¿ã¯ã4å
éåéã®2ä¹ãããŒã¬ã³ãã¹ã«ã©ãŒã§ããããšãã p ÎŒ p ÎŒ = ε 2 / c 2 â p â 2 = m 2 c 2 {\displaystyle p^{\mu }p_{\mu }=\epsilon ^{2}/c^{2}-{\vec {p}}^{2}=m^{2}c^{2}} ãšãªãã ãã£ãŠã ε = c | p â | 2 + m 2 c 2 {\displaystyle \epsilon =c{\sqrt {|{\vec {p}}|^{2}+m^{2}c^{2}}}} ãåŸãããã p â 2 {\displaystyle {\vec {p}}^{2}} ãå°ãããšããŠãã€ã©ãŒå±éãè¡ãªããšã ε = m c 2 + p â 2 2 m + O ( p â 4 ) {\displaystyle \epsilon =mc^{2}+{\frac {{\vec {p}}^{2}}{2m}}+O({\vec {p}}^{4})} ãåŸãããéåžžã®ãšãã«ã®ãŒãšéåéã®é¢ä¿åŒ ε = p â 2 2 m {\displaystyle \epsilon ={\frac {{\vec {p}}^{2}}{2m}}} ãšãå®æ° m c 2 {\displaystyle mc^{2}} ãé€ããŠäžèŽããã å®æ° m c 2 {\displaystyle mc^{2}} ãéæ¢ãšãã«ã®ãŒãšåŒã¶ããšãããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ç¹æ®çžå¯Ÿè« > 4å
éåé",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "解æååŠãèãããšã空éã®çæ¹æ§ããéåéä¿åã 瀺ãããã®ãšåæ§ã«ãæéã«å¯Ÿããäžæ§æ§ãããšãã«ã®ãŒã® ä¿ååãå°ãåºãããã ãã®ããã x ÎŒ = ( c t x y z ) {\\displaystyle x^{\\mu }={\\begin{pmatrix}ct\\\\x\\\\y\\\\z\\end{pmatrix}}} ã®ããã«çµã¿åãããŠ4å
ãã¯ãã«ãäœã£ãããšã«å¯Ÿå¿ããŠã p ÎŒ = ( ε / c p x p y p z ) {\\displaystyle p^{\\mu }={\\begin{pmatrix}\\epsilon /c\\\\p_{x}\\\\p_{y}\\\\p_{z}\\end{pmatrix}}} ã«ãã£ãŠã4å
ãã¯ãã«ãäœãããšãåºæ¥ãã ããã§ã ε {\\displaystyle \\epsilon } ã¯ãšãã«ã®ãŒã§ããã ãã®4å
ãã¯ãã«ã4å
éåéãšåŒã¶ã ããéæ¢ããç©äœã«ã€ããŠã¯ p â = 0 {\\displaystyle {\\vec {p}}=0} ãæãç«ã€ã®ã§ã p ÎŒ = ( ε / c 0 0 0 ) {\\displaystyle p^{\\mu }={\\begin{pmatrix}\\epsilon /c\\\\0\\\\0\\\\0\\end{pmatrix}}} ãšãªãã ãã®ãšã㮠ε {\\displaystyle \\epsilon } ã®å€ãããã質émããã€ç©äœã«å¯ŸããŠã mc ãšçœ®ãã ε / c = m c {\\displaystyle \\epsilon /c=mc} ã€ãŸã, ε = m c 2 {\\displaystyle \\epsilon =mc^{2}} ã«æ³šæã (ãšãã«ã®ãŒã®å®æ°å€ã¯ã©ã®ããã«ã§ãåããããç¹ã«ãã®ããã« éžã¶ã®ã¯å®éšçã«è³ªéãšãšãã«ã®ãŒã®åå€æ§ãç¥ãããŠããããšã« ãã£ãŠãããã®ãšæãããã) ãã®ãšãã ε {\\displaystyle \\epsilon } ãš | p â | {\\displaystyle |{\\vec {p}}|} ã®é¢ä¿ã¯ã4å
éåéã®2ä¹ãããŒã¬ã³ãã¹ã«ã©ãŒã§ããããšãã p ÎŒ p ÎŒ = ε 2 / c 2 â p â 2 = m 2 c 2 {\\displaystyle p^{\\mu }p_{\\mu }=\\epsilon ^{2}/c^{2}-{\\vec {p}}^{2}=m^{2}c^{2}} ãšãªãã ãã£ãŠã ε = c | p â | 2 + m 2 c 2 {\\displaystyle \\epsilon =c{\\sqrt {|{\\vec {p}}|^{2}+m^{2}c^{2}}}} ãåŸãããã p â 2 {\\displaystyle {\\vec {p}}^{2}} ãå°ãããšããŠãã€ã©ãŒå±éãè¡ãªããšã ε = m c 2 + p â 2 2 m + O ( p â 4 ) {\\displaystyle \\epsilon =mc^{2}+{\\frac {{\\vec {p}}^{2}}{2m}}+O({\\vec {p}}^{4})} ãåŸãããéåžžã®ãšãã«ã®ãŒãšéåéã®é¢ä¿åŒ ε = p â 2 2 m {\\displaystyle \\epsilon ={\\frac {{\\vec {p}}^{2}}{2m}}} ãšãå®æ° m c 2 {\\displaystyle mc^{2}} ãé€ããŠäžèŽããã å®æ° m c 2 {\\displaystyle mc^{2}} ãéæ¢ãšãã«ã®ãŒãšåŒã¶ããšãããã",
"title": "4å
éåé"
}
] | ç¹æ®çžå¯Ÿè« > 4å
éåé | <small> [[ç¹æ®çžå¯Ÿè«]] > 4å
éåé</small>
----
==4å
éåé==
解æååŠãèãããšã空éã®çæ¹æ§ããéåéä¿åã
瀺ãããã®ãšåæ§ã«ãæéã«å¯Ÿããäžæ§æ§ãããšãã«ã®ãŒã®
ä¿ååãå°ãåºãããã
ãã®ããã
<math>
x^\mu =
\begin{pmatrix}
ct \\
x \\
y \\
z
\end{pmatrix}
</math>
ã®ããã«çµã¿åãããŠ4å
ãã¯ãã«ãäœã£ãããšã«å¯Ÿå¿ããŠã
<math>
p^\mu =
\begin{pmatrix}
\epsilon / c \\
p _x \\
p _y \\
p _z
\end{pmatrix}
</math>
ã«ãã£ãŠã4å
ãã¯ãã«ãäœãããšãåºæ¥ãã
ããã§ã
<math>
\epsilon
</math>
ã¯ãšãã«ã®ãŒã§ããã
ãã®4å
ãã¯ãã«ã4å
éåéãšåŒã¶ã
ããéæ¢ããç©äœã«ã€ããŠã¯
<math>
\vec p = 0
</math>
ãæãç«ã€ã®ã§ã
<math>
p^\mu =
\begin{pmatrix}
\epsilon / c \\
0\\
0 \\
0
\end{pmatrix}
</math>
ãšãªãã
ãã®ãšãã® <math>\epsilon</math> ã®å€ãããã質émããã€ç©äœã«å¯ŸããŠã
mc ãšçœ®ãã
<math>
\epsilon / c = mc
</math>
ã€ãŸã,
<math>
\epsilon = mc^2
</math>
ã«æ³šæã
(ãšãã«ã®ãŒã®å®æ°å€ã¯ã©ã®ããã«ã§ãåããããç¹ã«ãã®ããã«
éžã¶ã®ã¯å®éšçã«è³ªéãšãšãã«ã®ãŒã®åå€æ§ãç¥ãããŠããããšã«
ãã£ãŠãããã®ãšæãããã)
<!-- ? -->
ãã®ãšãã
<math>\epsilon</math>ãš
<math>
| \vec p |
</math>
ã®é¢ä¿ã¯ã4å
éåéã®2ä¹ãããŒã¬ã³ãã¹ã«ã©ãŒã§ããããšãã
<math>
p^\mu p _\mu =\epsilon ^2 /c^2 - \vec p^2 = m^2 c^2
</math>
ãšãªãã
ãã£ãŠã
<math>
\epsilon =c \sqrt {|\vec p |^2 + m^2 c^2 }
</math>
ãåŸãããã
<math> \vec p ^2 </math> ãå°ãããšããŠãã€ã©ãŒå±éãè¡ãªããšã
<math>
\epsilon = mc^2 + \frac {\vec p^2} {2m} + O (\vec p^4)
</math>
ãåŸãããéåžžã®ãšãã«ã®ãŒãšéåéã®é¢ä¿åŒ
<math>
\epsilon = \frac {\vec p^2} {2m}
</math>
ãšãå®æ°<math>mc^2</math>ãé€ããŠäžèŽããã
å®æ°<math>mc^2</math>ãéæ¢ãšãã«ã®ãŒãšåŒã¶ããšãããã
[[Category:ç¹æ®çžå¯Ÿè«|ããããããšãããã]] | 2005-05-24T08:24:12Z | 2024-03-16T03:17:34Z | [] | https://ja.wikibooks.org/wiki/%E7%89%B9%E6%AE%8A%E7%9B%B8%E5%AF%BE%E8%AB%96_4%E5%85%83%E9%81%8B%E5%8B%95%E9%87%8F |
2,036 | ç±ååŠ | æ¬é
ã¯ç©çåŠ ç±ååŠã®è§£èª¬ã§ãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "æ¬é
ã¯ç©çåŠ ç±ååŠã®è§£èª¬ã§ãã",
"title": ""
}
] | æ¬é
ã¯ç©çåŠ ç±ååŠã®è§£èª¬ã§ãã ã¯ããã« (2017-05-25)
枩床(ç±ååŠã®ç¬¬0æ³å) (2020-10-21)
ç±ãšä»äº(ç±ååŠã®ç¬¬1æ³å) (2017-05-25)
第äºæ³åããã³å¯ééçšããã³ãšã³ããããŒã«ã€ããŠãé«çåŠæ ¡ç©ç/ç©çII/ç±ååŠã (2017-05-25) ç±ååŠçãªãšãã«ã®ãŒ (2017-05-25) ïŒãã®ãã¹ã®èªç±ãšãã«ã®ãŒãã®å®çŸ©ïŒ | {{Pathnav|ã¡ã€ã³ããŒãž|èªç¶ç§åŠ|ç©çåŠ|frame=1|small=1}}
{{wikiversity|Topic:ç±åŠ|ç±åŠ}}
æ¬é
ã¯ç©çåŠ ç±ååŠã®è§£èª¬ã§ãã
* [[ç±ååŠ/ã¯ããã«|ã¯ããã«]] {{é²æ|25%|2017-05-25}}
* [[ç±ååŠ/枩床|枩床(ç±ååŠã®ç¬¬0æ³å)]] {{é²æ|00%|2020-10-21}}
* [[ç±ååŠ/ç±ãšä»äº|ç±ãšä»äº(ç±ååŠã®ç¬¬1æ³å)]] {{é²æ|00%|2017-05-25}}
* 第äºæ³åããã³å¯ééçšããã³ãšã³ããããŒã«ã€ããŠã[[é«çåŠæ ¡ç©ç/ç©çII/ç±ååŠ]]ã{{é²æ|50%|2017-05-25}} ïŒâ» é«æ ¡ç¯å²å€ã ãã第äºæ³åãå¯ééçšããã³ãšã³ããããŒã«ã€ããŠæžããŠãããïŒ
:* [[ç±ååŠ/ç±ååŠã®ç¬¬2æ³å|ç±ååŠã®ç¬¬2æ³å]] {{é²æ|25%|2017-05-25}}ïŒâ» çŸæç¹ã§ã¯ãã»ãŒé«æ ¡ç©çã®ã³ããŒãå çä¿®æ£ããé¡ãããŸããïŒ
:* [[ç±ååŠ/å¯ééçš|å¯ééçš]] {{é²æ|25%|2017-05-25}} ïŒâ» çŸæç¹ã§ã¯ãã»ãŒé«æ ¡ç©çã®ã³ããŒãå çä¿®æ£ããé¡ãããŸããïŒ
:* [[ç±ååŠ/ãšã³ããããŒ|ãšã³ããããŒ]] {{é²æ|25%|2017-05-25}}ïŒâ» çŸæç¹ã§ã¯ãã»ãŒé«æ ¡ç©çã®ã³ããŒãå çä¿®æ£ããé¡ãããŸããïŒ
* [[ç±ååŠ/ç±ååŠçãªãšãã«ã®ãŒ|ç±ååŠçãªãšãã«ã®ãŒ]] {{é²æ|25%|2017-05-25}} ïŒãã®ãã¹ã®èªç±ãšãã«ã®ãŒãã®å®çŸ©ïŒ
{{stub}}
{{DEFAULTSORT:ãã€ãããã}}
[[Category:ç±ååŠ|*]]
{{NDC|426}} | 2005-05-24T09:01:01Z | 2024-03-17T10:31:29Z | [
"ãã³ãã¬ãŒã:NDC",
"ãã³ãã¬ãŒã:Pathnav",
"ãã³ãã¬ãŒã:Wikiversity",
"ãã³ãã¬ãŒã:é²æ",
"ãã³ãã¬ãŒã:Stub"
] | https://ja.wikibooks.org/wiki/%E7%86%B1%E5%8A%9B%E5%AD%A6 |
2,037 | ç±ååŠ/ã¯ããã« | ç±ååŠ > ã¯ããã«
ãã®åéã¯é«çæè²ã®ç±ååŠã«åœãããŸããååŠè
ã¯è©²åœæç§æžãç解ã®å©ããšãªããŸãã®ã§åŠç¿ã«è¡ãè©°ãŸã£ããåç
§ããŠãã ããã
çŸåšã§ã¯ç±ã100% ã®å¹çã§ãä»äºã«å€ããããšã¯åºæ¥ãªãããšãç¥ãããŠããã ãã®ããšã¯ç±ã埮èŠçãªç©äœã®ä¹±éãªåãããæ§æãããŠããã 確ãã«ãããã¯ãšãã«ã®ãŒãæã£ãŠã¯ããã®ã ãããããã 秩åºã ã£ãä»æ¹ã§åãåºããäœããã®ããšã«åœ¹ç«ãŠãããšã å°é£ã§ããããšã«ãã£ãŠããã
ãã®é
ã§ã¯ã埮èŠçãªç©äœã®åãããåŸãããå·šèŠçãªéã çšããŠãç±ãšä»äºã®é¢ä¿ãèŠãŠããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ç±ååŠ > ã¯ããã«",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ãã®åéã¯é«çæè²ã®ç±ååŠã«åœãããŸããååŠè
ã¯è©²åœæç§æžãç解ã®å©ããšãªããŸãã®ã§åŠç¿ã«è¡ãè©°ãŸã£ããåç
§ããŠãã ããã",
"title": ""
},
{
"paragraph_id": 2,
"tag": "p",
"text": "çŸåšã§ã¯ç±ã100% ã®å¹çã§ãä»äºã«å€ããããšã¯åºæ¥ãªãããšãç¥ãããŠããã ãã®ããšã¯ç±ã埮èŠçãªç©äœã®ä¹±éãªåãããæ§æãããŠããã 確ãã«ãããã¯ãšãã«ã®ãŒãæã£ãŠã¯ããã®ã ãããããã 秩åºã ã£ãä»æ¹ã§åãåºããäœããã®ããšã«åœ¹ç«ãŠãããšã å°é£ã§ããããšã«ãã£ãŠããã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ãã®é
ã§ã¯ã埮èŠçãªç©äœã®åãããåŸãããå·šèŠçãªéã çšããŠãç±ãšä»äºã®é¢ä¿ãèŠãŠããã",
"title": "ã¯ããã«"
}
] | ç±ååŠ > ã¯ããã« ãã®åéã¯é«çæè²ã®ç±ååŠã«åœãããŸããååŠè
ã¯è©²åœæç§æžãç解ã®å©ããšãªããŸãã®ã§åŠç¿ã«è¡ãè©°ãŸã£ããåç
§ããŠãã ããã | <small> [[ç±ååŠ]] > ã¯ããã«</small>
----
ãã®åéã¯é«çæè²ã®[[é«çåŠæ ¡_ç©ç#ç±ååŠ|ç±ååŠ]]ã«åœãããŸããååŠè
ã¯è©²åœæç§æžãç解ã®å©ããšãªããŸãã®ã§åŠç¿ã«è¡ãè©°ãŸã£ããåç
§ããŠãã ããã
==ã¯ããã«==
çŸåšã§ã¯ç±ã100%
ã®å¹çã§ãä»äºã«å€ããããšã¯åºæ¥ãªãããšãç¥ãããŠããã
ãã®ããšã¯ç±ã埮èŠçãªç©äœã®ä¹±éãªåãããæ§æãããŠããã
確ãã«ãããã¯ãšãã«ã®ãŒãæã£ãŠã¯ããã®ã ãããããã
秩åºã ã£ãä»æ¹ã§åãåºããäœããã®ããšã«åœ¹ç«ãŠãããšã
å°é£ã§ããããšã«ãã£ãŠããã
ãã®é
ã§ã¯ã埮èŠçãªç©äœã®åãããåŸãããå·šèŠçãªéã
çšããŠãç±ãšä»äºã®é¢ä¿ãèŠãŠããã
[[Category:ç±ååŠ|ã¯ããã«]] | 2005-05-24T09:02:01Z | 2023-10-24T16:30:30Z | [] | https://ja.wikibooks.org/wiki/%E7%86%B1%E5%8A%9B%E5%AD%A6/%E3%81%AF%E3%81%98%E3%82%81%E3%81%AB |
2,038 | ç±ååŠ/ç±ãšä»äº | ç±ååŠ > ç±ãšä»äº
ããç©äœã«ã€ããŠããšãã«ã®ãŒã®åæ¯ãèããã d Q {\displaystyle dQ} ãç©äœãåãåã£ãç±ã d U {\displaystyle dU} ãç©äœã®å
éšãšãã«ã®ãŒã®å€åã d W {\displaystyle dW} ãç©äœããããä»äº(å€ã«ä»äºããããšãã d W {\displaystyle dW} ã¯è² ã«ãªãã)ãšãããšãã å®éšçã«
ãç¥ãããŠããã ãã®åŒãç±ååŠã®ç¬¬1æ³åãšåŒã¶ã (å®éã«ã¯ç©äœãåãåã£ãç±ã®ãã¡ãããç©äœä»¥å€ã®å€çã«å¯Ÿã㊠è¡ãªãä»äºãåŒãå»ã£ããã®ããç©äœã®æã€å
éšãšãã«ã®ãŒãšåŒãã§ããã ãã®ããããã®åŒã¯å
éšãšãã«ã®ãŒã®å®çŸ©ã®åŒãšããŠã¿ãããšãåºæ¥ãã) ãã®åŒã¯ãç©äœã«ç±ãäžããããšã¯ãŸãã§ç©äœã«ä»äºã ããããšã§ãããã®ããã«æãããããšãããããšããããšã 瀺ããŠããã äŸãã°ãæ°Žã®äžã«é»ç±ç·ããããŠé»æ°ãæµãããšãèããã ãã®ãšããé»æ°ã¯ãã®ãšãã«ã®ãŒãç±ãšããŠæŸåºããã ããã«ãã£ãŠæ°Žã®æž©åºŠãäžããã ããã¯é»æ°ãšãã«ã®ãŒãé»ç±ç·ã«ãã£ãŠç±ãšãã«ã®ãŒã«å€æãã ãããæ°Žã«äžãããããã®ãšè§£éããããšãåºæ¥ãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ç±ååŠ > ç±ãšä»äº",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ããç©äœã«ã€ããŠããšãã«ã®ãŒã®åæ¯ãèããã d Q {\\displaystyle dQ} ãç©äœãåãåã£ãç±ã d U {\\displaystyle dU} ãç©äœã®å
éšãšãã«ã®ãŒã®å€åã d W {\\displaystyle dW} ãç©äœããããä»äº(å€ã«ä»äºããããšãã d W {\\displaystyle dW} ã¯è² ã«ãªãã)ãšãããšãã å®éšçã«",
"title": "ç±ãšä»äº(ç±ååŠã®ç¬¬1æ³å)"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ãç¥ãããŠããã ãã®åŒãç±ååŠã®ç¬¬1æ³åãšåŒã¶ã (å®éã«ã¯ç©äœãåãåã£ãç±ã®ãã¡ãããç©äœä»¥å€ã®å€çã«å¯Ÿã㊠è¡ãªãä»äºãåŒãå»ã£ããã®ããç©äœã®æã€å
éšãšãã«ã®ãŒãšåŒãã§ããã ãã®ããããã®åŒã¯å
éšãšãã«ã®ãŒã®å®çŸ©ã®åŒãšããŠã¿ãããšãåºæ¥ãã) ãã®åŒã¯ãç©äœã«ç±ãäžããããšã¯ãŸãã§ç©äœã«ä»äºã ããããšã§ãããã®ããã«æãããããšãããããšããããšã 瀺ããŠããã äŸãã°ãæ°Žã®äžã«é»ç±ç·ããããŠé»æ°ãæµãããšãèããã ãã®ãšããé»æ°ã¯ãã®ãšãã«ã®ãŒãç±ãšããŠæŸåºããã ããã«ãã£ãŠæ°Žã®æž©åºŠãäžããã ããã¯é»æ°ãšãã«ã®ãŒãé»ç±ç·ã«ãã£ãŠç±ãšãã«ã®ãŒã«å€æãã ãããæ°Žã«äžãããããã®ãšè§£éããããšãåºæ¥ãã",
"title": "ç±ãšä»äº(ç±ååŠã®ç¬¬1æ³å)"
}
] | ç±ååŠ > ç±ãšä»äº | <small> [[ç±ååŠ]] > ç±ãšä»äº</small>
----
==ç±ãšä»äº(ç±ååŠã®ç¬¬1æ³å)==
ããç©äœã«ã€ããŠããšãã«ã®ãŒã®åæ¯ãèããã
<math>dQ</math> ãç©äœãåãåã£ãç±ã<math>dU</math> ãç©äœã®å
éšãšãã«ã®ãŒã®å€åã
<math>dW</math> ãç©äœããããä»äº(å€ã«ä»äºããããšãã<math>dW</math> ã¯è² ã«ãªãã)ãšãããšãã
å®éšçã«
: <math>dQ = dU - dW</math>
ãç¥ãããŠããã
ãã®åŒãç±ååŠã®ç¬¬1æ³åãšåŒã¶ã
(å®éã«ã¯ç©äœãåãåã£ãç±ã®ãã¡ãããç©äœä»¥å€ã®å€çã«å¯ŸããŠ
è¡ãªãä»äºãåŒãå»ã£ããã®ããç©äœã®æã€å
éšãšãã«ã®ãŒãšåŒãã§ããã
ãã®ããããã®åŒã¯å
éšãšãã«ã®ãŒã®å®çŸ©ã®åŒãšããŠã¿ãããšãåºæ¥ãã)
<!-- ? -->
ãã®åŒã¯ãç©äœã«ç±ãäžããããšã¯ãŸãã§ç©äœã«ä»äºã
ããããšã§ãããã®ããã«æãããããšãããããšããããšã
瀺ããŠããã
äŸãã°ãæ°Žã®äžã«é»ç±ç·ããããŠé»æ°ãæµãããšãèããã
ãã®ãšããé»æ°ã¯ãã®ãšãã«ã®ãŒãç±ãšããŠæŸåºããã
ããã«ãã£ãŠæ°Žã®æž©åºŠãäžããã
ããã¯é»æ°ãšãã«ã®ãŒãé»ç±ç·ã«ãã£ãŠç±ãšãã«ã®ãŒã«å€æãã
ãããæ°Žã«äžãããããã®ãšè§£éããããšãåºæ¥ãã
[[Category:ç±ååŠ|ãã€ãšãããš]] | null | 2022-12-01T04:09:27Z | [] | https://ja.wikibooks.org/wiki/%E7%86%B1%E5%8A%9B%E5%AD%A6/%E7%86%B1%E3%81%A8%E4%BB%95%E4%BA%8B |
2,039 | ç±ååŠ/ç±ååŠã®ç¬¬2æ³å | ç±ååŠ > ç±ååŠã®ç¬¬2æ³å
ç±ã®å·šèŠçãªæ§è³ªãšããŠã "枩床ã®äœããã®ãã枩床ã®é«ããã®ã«å¯Ÿã㊠ä»ã®ç©äœã«åœ±é¿ãäžããäºç¡ãã«ç±ãäžããããããšã¯ã§ããªãã" ããšãç¥ãããŠããã ãããç±ååŠã®ç¬¬2æ³åãšããã äŸãã°ãä»®ã«ãã®ããšãå¯èœã ã£ããšãããšã å·ããæ°Žãšç±ã湯ãæ··ãããšã å·ããæ°Žã¯ããå·ããã湯ã¯ããç±ããšããããšã èµ·ããåŸãããšãäºæ³ããããå®éã«ã¯ çµéšçã«ãããã®ããšãèµ·ãããªãããšãç¥ãããŠããã
æ°äœã®å€æ°ã®å€æ°p,V,Tã¯ãçæ³æ°äœã§ããããã¡ã³ãã«ã¯ãŒã«ã¹æ°äœã§ãããç¶æ
æ¹çšåŒ(çæ³æ°äœããã¡ã³ãã«ã¯ãŒã«ã¹æ°äœãã¯ãããã§ã¯åããªã)ããããªãã°ãå€æ°p,V,Tã®ãã¡ã®ãã©ããäºã€ã決ãŸãã°ãæ°äœã®ç¶æ
æ¹çšåŒããæ®ãã®å€æ°ã決ãŸããããããŠ3å€æ°p,V,Tã決ãŸãã
å
éšãšãã«ã®ãŒã¯ãçæ³æ°äœã§ããããã¡ã³ãã«ã¯ãŒã«ã¹æ°äœã§ãããã©ã¡ãã«ããŠããå€æ°p,V,Tã®ãã¡ãã©ããäºã€ã決ãŸãã°ãæ°äœã®æ¹çšåŒããæ®ãã®æ¹çšåŒã決ãŸãã決ãŸã£ã3å€æ°ã®p,V,Tã«ãã£ãŠãå
éšãšãã«ã®ãŒã決ãŸã£ãŠããŸãããã®ãããªãç¶æ
å€æ°ã«ãã£ãŠã®ã¿æ±ºãŸãç©çéãç¶æ
é(ãããããããã)ãšããã 3å€æ°ã®p,V,Tã決ãŸãã°å
éšãšãã«ã®ãŒã決å®ãããã®ã§ãå
éšãšãã«ã€ã®ãŒã¯ç¶æ
éã§ããã å
éšãšãã«ã®ãŒã決ãã3å€æ°ã®ãã¡ãçã«ç¬ç«å€æ°ãªã®ã¯ããã®ãã¡ã®2åã®ã¿ã§ãããå€æ°p,V,Tã®ã©ãã2åãŸã§ç¬ç«å€æ°ã«éžãã§ãããããæ®ãã®1åã¯æ¢ã«éžãã å€æ°ã®åŸå±å€æ°ã«ãªãã
ã©ã®å€æ°ãç¬ç«å€æ°ã«éžã¶ãšãç¥ãããçããæ±ãããããã¯ãåé¡ã«ããã
(å€å€æ°ã®é¢æ°ã®åŸ®åç©åã«ã€ããŠã¯ã倧åŠçç§ç³»ã§æè²ããããå€å€æ°é¢æ°ã®åŸ®åãå埮åãšããã解説ã¯é«æ ¡ã¬ãã«ãè¶
ããã®ã§çç¥ã)
åç¯ã§èšåããã3ã€ã®å€æ°(å§åpãäœç©Vã枩床T)ã®ã»ãããšã³ããããŒSãå
éšãšãã«ã®ãŒUãªã©ãç±ååŠç³»ã®å¹³è¡¡ç¶æ
ãç¹åŸŽä»ããç¶æ
éã§ããã
åç¯ãšåæ§ã5ã€ã®ç¶æ
ép,V,T,U,Sã®ãã¡ä»»æã®2ã€ãç¬ç«å€æ°ã«éžã¶å Žåã«ããæ®ã3ã€ã®å€æ°ã¯ããã2ã€ã®ç¬ç«å€æ°ã§è¡šãããåŸå±å€æ°ãšããŠæ±ããã
ãã®5ã€ã®å€æ°ã®ä»»æã®çµã¿åãããç¬ç«å€æ°ã«ãã€ç¶æ
éã¯ãäžè¬ã«ç±ååŠé¢æ°ãšåŒã°ããã
å
éšãšãã«ã®ãŒU(S,V)ã®ã»ããåŸã®ç« ã«ãŠèšåããããšã³ããããŒS(U,V)ããšã³ã¿ã«ããŒH(S,p)ããã«ã ãã«ãã®èªç±ãšãã«ã®ãŒF(V,T)ãã®ãã¹ã®èªç±ãšãã«ã®ãŒG(T,p)ãªã©ãç±ååŠé¢æ°ã§ããã
(ãã®ç¯ã§ã¯ãé«æ ¡æ°åŠã®æ°åŠIIIçžåœã®åŸ®åç©åãçšãããåãããªããã°æ°åŠIIIãåç
§ã®ããšã)
å§åãpãšæžããšãããäœç©ãVãã¢ã«æ°ãnãæ®éæ°äœå®æ°ãnã枩床ã絶察枩床ã§Tãšããã
ä»äºWã®ãç¬éçãªä»äºã®å€§ããã¯åŸ®åãçšããŠdWãšè¡šãããäœç©Vã®ããã®ç¬éã®äœç©å€åã¯åŸ®åãçšããŠdVãšè¡šãããããããçšããã°ã
d W = p d V {\displaystyle dW=pdV}
ãšåŸ®åæ¹çšåŒã§è¡šããã(å®å§å€åã§ã¯ç¡ãããããã®åŒã®pã¯å€æ°ã§ããã)
äœç©ãV1ããV2ãŸã§å€åãããæã®ä»äºã¯ãç©åãçšããŠä»¥äžã®ããã«æžãè¡šããã
W = â« V 1 V 2 p d V {\displaystyle W=\int _{V_{1}}^{V_{2}}pdV}
ããã«ãç¶æ
æ¹çšåŒã® p V = n R T {\displaystyle pV=nRT} ããçµã¿åãããã
ç©åå€æ°ã®Vã«åãããŠãpãæžãæãããã
p = n R T V {\displaystyle p={\frac {nRT}{V}}}
ã§ããããããããä»äºã®åŒã¯ã
W = â« V 1 V 2 p d V = â« V 1 V 2 n R T V d V = n R T â« V 1 V 2 d V V = n R T log V 2 V 1 {\displaystyle W=\int _{V_{1}}^{V_{2}}pdV=\int _{V_{1}}^{V_{2}}{\frac {nRT}{V}}dV=nRT\int _{V_{1}}^{V_{2}}{\frac {dV}{V}}=nRT\log {\frac {V_{2}}{V_{1}}}}
ãšãªãã(ãªããlogã¯èªç¶å¯Ÿæ°ã§ããã) çµè«ããŸãšãããšã
ã§ããã
å
éšãšãã«ã®ãŒUã¯ãçæ³æ°äœã§ã¯æž©åºŠã®ã¿ã®é¢æ°ã§ãçæž©å€åã§ã¯æž©åºŠãå€åããªãããã
ã§ããã
ãããã£ãŠãçæž©å€åã§ã¯
ã§ããã
ãŸããç±ãšå
éšãšãã«ã®ãŒãšä»äºã®é¢ä¿åŒ
ãã次ã®ããã«åŸ®åæ¹çšåŒã«æžãæãããå
éšãšãã«ã®ãŒã®å€åã埮å°å€åãšããŠdUãšè¡šãããšãããšãç±éQãä»äºWã埮å°å€åã«ãªãã®ã§ã以äžã®æ§ãªåŒã«ãªãã
QãWã®åŸ®åæŒç®èšå·dã®äžã«ç¹ã â² {\displaystyle '} ããä»ããŠããã®ã¯ãå³å¯ã«èšããšãç±éQãä»äºWã¯ç¶æ
éã§ç¡ããããåºå¥ããããã«çšããŠããã
æç±å€åã§ã¯
ãªã®ã§ãã€ãŸãã
ãšãªãã
ä»äºã«é¢ããŠã¯
ã§ããã å
éšãšãã«ã®ãŒã®åŸ®å°å€åã¯ãå®ç©ã¢ã«æ¯ç±ãçšããŠã
ãšæžããã
ãªã®ã§ãããçãåŒ 0 = d U + d â² W {\displaystyle 0=dU+d'W} ã«ä»£å
¥ãã
ãšæžããã 䞡蟺ãpVã§å²ããšã
ã§ããããpV=nRTãå©çšãããšã
ãšãªãã
ãã®åŸ®åæ¹çšåŒã解ãããŸã移é
ããŠã
ãšãªãã ç©åããŠã
ããã§ã C o n s t {\displaystyle Const} ã¯ç©åå®æ°ãšããã(ç©åå®æ°ã C {\displaystyle C} ãšæžããªãã£ãã®ã¯ãæ¯ç±ã®èšå·ãšã®æ··åãé¿ããããã) 察æ°ã®æ§è³ªãããä¿æ°R/Cvã察æ°log()ã®äžã®å€æ°ã®ææ°ã«æã£ãŠããã(æ°åŠIIçžåœ)ã®ã§ãèšç®ãããšã
ããã«ç§»é
ããŠãå€æ°ã巊蟺ã«ãŸãšãããšã
察æ°ã®æ§è³ªããã察æ°å士ã®åã¯ãäžã®å€æ°ã®ç©ã«å€ããããã®ã§ã
ã§ããã 察æ°ã®å®çŸ©ãããèªç¶å¯Ÿæ°ã®åºãeãšããã°
ã§ããã e C o n s t {\displaystyle e^{Const}} ãæ°ãããå¥ã®å®æ°ãšããŠãå®æ°âconstantâãšçœ®ãçŽãã°ã
ã§ããã ããã§æç±å€åã®æž©åºŠãšäœç©ã®é¢ä¿åŒã®å
¬åŒãæ±ãŸã£ãã
ä»äºWãšã®é¢ä¿ãèŠããã®ã§ãå
ã»ã©æ±ããäžã®å
¬åŒãpãšTã®åŒã«æžãæããäºãèãããç¶æ
æ¹çšåŒ p V = n R T {\displaystyle pV=nRT} ãçšããŠTããPãšVãçšããåŒã«æžãæãããšããŸã代å
¥ããããããã«ç¶æ
æ¹çšåŒã
ãšæžãæããŠããããå
¬åŒã«ä»£å
¥ããã°ã
1 n R {\displaystyle {\frac {1}{nR}}} ã¯å®æ°ãªã®ã§ããããå®æ°éšã«ãŸãšããŠããŸãã°ãå¥ã®å®æ°ãConst2ãšã§ã眮ããŠã
ãšæžããã ããã§ãææ°éšã®åŒã¯ããã€ã€ãŒã®åŒ C p = C v + R {\displaystyle Cp=Cv+R} ãããå®å§ã¢ã«æ¯ç±ã§æžãæããå¯èœã§ããã
ã§ããã ããã§ã: C p C V {\displaystyle {\frac {C_{p}}{C_{V}}}} ãæ¯ç±æ¯(heat capacity ratio)ãšèšããæ¯ç±æ¯ã®èšå·ã¯äžè¬ã« γ {\displaystyle \gamma } ã§è¡šãã ãããçšãããšã
ã§ããã
ãŸãã枩床ãšäœç©ã®é¢ä¿åŒ
ã«æ¯ç±æ¯ã代å
¥ãããšã
ã«ãªãã
ãããã®ãå§åãšäœç©ã®å
¬åŒãããã³æž©åºŠãšäœç©ã®å
¬åŒã®äºåŒããã¢ãœã³ã®åŒãšããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ç±ååŠ > ç±ååŠã®ç¬¬2æ³å",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ç±ã®å·šèŠçãªæ§è³ªãšããŠã \"枩床ã®äœããã®ãã枩床ã®é«ããã®ã«å¯Ÿã㊠ä»ã®ç©äœã«åœ±é¿ãäžããäºç¡ãã«ç±ãäžããããããšã¯ã§ããªãã\" ããšãç¥ãããŠããã ãããç±ååŠã®ç¬¬2æ³åãšããã äŸãã°ãä»®ã«ãã®ããšãå¯èœã ã£ããšãããšã å·ããæ°Žãšç±ã湯ãæ··ãããšã å·ããæ°Žã¯ããå·ããã湯ã¯ããç±ããšããããšã èµ·ããåŸãããšãäºæ³ããããå®éã«ã¯ çµéšçã«ãããã®ããšãèµ·ãããªãããšãç¥ãããŠããã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "æ°äœã®å€æ°ã®å€æ°p,V,Tã¯ãçæ³æ°äœã§ããããã¡ã³ãã«ã¯ãŒã«ã¹æ°äœã§ãããç¶æ
æ¹çšåŒ(çæ³æ°äœããã¡ã³ãã«ã¯ãŒã«ã¹æ°äœãã¯ãããã§ã¯åããªã)ããããªãã°ãå€æ°p,V,Tã®ãã¡ã®ãã©ããäºã€ã決ãŸãã°ãæ°äœã®ç¶æ
æ¹çšåŒããæ®ãã®å€æ°ã決ãŸããããããŠ3å€æ°p,V,Tã決ãŸãã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "å
éšãšãã«ã®ãŒã¯ãçæ³æ°äœã§ããããã¡ã³ãã«ã¯ãŒã«ã¹æ°äœã§ãããã©ã¡ãã«ããŠããå€æ°p,V,Tã®ãã¡ãã©ããäºã€ã決ãŸãã°ãæ°äœã®æ¹çšåŒããæ®ãã®æ¹çšåŒã決ãŸãã決ãŸã£ã3å€æ°ã®p,V,Tã«ãã£ãŠãå
éšãšãã«ã®ãŒã決ãŸã£ãŠããŸãããã®ãããªãç¶æ
å€æ°ã«ãã£ãŠã®ã¿æ±ºãŸãç©çéãç¶æ
é(ãããããããã)ãšããã 3å€æ°ã®p,V,Tã決ãŸãã°å
éšãšãã«ã®ãŒã決å®ãããã®ã§ãå
éšãšãã«ã€ã®ãŒã¯ç¶æ
éã§ããã å
éšãšãã«ã®ãŒã決ãã3å€æ°ã®ãã¡ãçã«ç¬ç«å€æ°ãªã®ã¯ããã®ãã¡ã®2åã®ã¿ã§ãããå€æ°p,V,Tã®ã©ãã2åãŸã§ç¬ç«å€æ°ã«éžãã§ãããããæ®ãã®1åã¯æ¢ã«éžãã å€æ°ã®åŸå±å€æ°ã«ãªãã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ã©ã®å€æ°ãç¬ç«å€æ°ã«éžã¶ãšãç¥ãããçããæ±ãããããã¯ãåé¡ã«ããã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "(å€å€æ°ã®é¢æ°ã®åŸ®åç©åã«ã€ããŠã¯ã倧åŠçç§ç³»ã§æè²ããããå€å€æ°é¢æ°ã®åŸ®åãå埮åãšããã解説ã¯é«æ ¡ã¬ãã«ãè¶
ããã®ã§çç¥ã)",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "åç¯ã§èšåããã3ã€ã®å€æ°(å§åpãäœç©Vã枩床T)ã®ã»ãããšã³ããããŒSãå
éšãšãã«ã®ãŒUãªã©ãç±ååŠç³»ã®å¹³è¡¡ç¶æ
ãç¹åŸŽä»ããç¶æ
éã§ããã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "åç¯ãšåæ§ã5ã€ã®ç¶æ
ép,V,T,U,Sã®ãã¡ä»»æã®2ã€ãç¬ç«å€æ°ã«éžã¶å Žåã«ããæ®ã3ã€ã®å€æ°ã¯ããã2ã€ã®ç¬ç«å€æ°ã§è¡šãããåŸå±å€æ°ãšããŠæ±ããã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "ãã®5ã€ã®å€æ°ã®ä»»æã®çµã¿åãããç¬ç«å€æ°ã«ãã€ç¶æ
éã¯ãäžè¬ã«ç±ååŠé¢æ°ãšåŒã°ããã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "å
éšãšãã«ã®ãŒU(S,V)ã®ã»ããåŸã®ç« ã«ãŠèšåããããšã³ããããŒS(U,V)ããšã³ã¿ã«ããŒH(S,p)ããã«ã ãã«ãã®èªç±ãšãã«ã®ãŒF(V,T)ãã®ãã¹ã®èªç±ãšãã«ã®ãŒG(T,p)ãªã©ãç±ååŠé¢æ°ã§ããã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "(ãã®ç¯ã§ã¯ãé«æ ¡æ°åŠã®æ°åŠIIIçžåœã®åŸ®åç©åãçšãããåãããªããã°æ°åŠIIIãåç
§ã®ããšã)",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "å§åãpãšæžããšãããäœç©ãVãã¢ã«æ°ãnãæ®éæ°äœå®æ°ãnã枩床ã絶察枩床ã§Tãšããã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "ä»äºWã®ãç¬éçãªä»äºã®å€§ããã¯åŸ®åãçšããŠdWãšè¡šãããäœç©Vã®ããã®ç¬éã®äœç©å€åã¯åŸ®åãçšããŠdVãšè¡šãããããããçšããã°ã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "d W = p d V {\\displaystyle dW=pdV}",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "ãšåŸ®åæ¹çšåŒã§è¡šããã(å®å§å€åã§ã¯ç¡ãããããã®åŒã®pã¯å€æ°ã§ããã)",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "äœç©ãV1ããV2ãŸã§å€åãããæã®ä»äºã¯ãç©åãçšããŠä»¥äžã®ããã«æžãè¡šããã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "W = â« V 1 V 2 p d V {\\displaystyle W=\\int _{V_{1}}^{V_{2}}pdV}",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "ããã«ãç¶æ
æ¹çšåŒã® p V = n R T {\\displaystyle pV=nRT} ããçµã¿åãããã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "ç©åå€æ°ã®Vã«åãããŠãpãæžãæãããã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "p = n R T V {\\displaystyle p={\\frac {nRT}{V}}}",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "ã§ããããããããä»äºã®åŒã¯ã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "W = â« V 1 V 2 p d V = â« V 1 V 2 n R T V d V = n R T â« V 1 V 2 d V V = n R T log V 2 V 1 {\\displaystyle W=\\int _{V_{1}}^{V_{2}}pdV=\\int _{V_{1}}^{V_{2}}{\\frac {nRT}{V}}dV=nRT\\int _{V_{1}}^{V_{2}}{\\frac {dV}{V}}=nRT\\log {\\frac {V_{2}}{V_{1}}}}",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "ãšãªãã(ãªããlogã¯èªç¶å¯Ÿæ°ã§ããã) çµè«ããŸãšãããšã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "ã§ããã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "å
éšãšãã«ã®ãŒUã¯ãçæ³æ°äœã§ã¯æž©åºŠã®ã¿ã®é¢æ°ã§ãçæž©å€åã§ã¯æž©åºŠãå€åããªãããã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "ã§ããã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "ãããã£ãŠãçæž©å€åã§ã¯",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "ã§ããã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "ãŸããç±ãšå
éšãšãã«ã®ãŒãšä»äºã®é¢ä¿åŒ",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "ãã次ã®ããã«åŸ®åæ¹çšåŒã«æžãæãããå
éšãšãã«ã®ãŒã®å€åã埮å°å€åãšããŠdUãšè¡šãããšãããšãç±éQãä»äºWã埮å°å€åã«ãªãã®ã§ã以äžã®æ§ãªåŒã«ãªãã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "QãWã®åŸ®åæŒç®èšå·dã®äžã«ç¹ã â² {\\displaystyle '} ããä»ããŠããã®ã¯ãå³å¯ã«èšããšãç±éQãä»äºWã¯ç¶æ
éã§ç¡ããããåºå¥ããããã«çšããŠããã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "æç±å€åã§ã¯",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "ãªã®ã§ãã€ãŸãã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "ãšãªãã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "ä»äºã«é¢ããŠã¯",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "ã§ããã å
éšãšãã«ã®ãŒã®åŸ®å°å€åã¯ãå®ç©ã¢ã«æ¯ç±ãçšããŠã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "ãšæžããã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "ãªã®ã§ãããçãåŒ 0 = d U + d â² W {\\displaystyle 0=dU+d'W} ã«ä»£å
¥ãã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "ãšæžããã 䞡蟺ãpVã§å²ããšã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 41,
"tag": "p",
"text": "ã§ããããpV=nRTãå©çšãããšã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 42,
"tag": "p",
"text": "ãšãªãã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 43,
"tag": "p",
"text": "ãã®åŸ®åæ¹çšåŒã解ãããŸã移é
ããŠã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 44,
"tag": "p",
"text": "ãšãªãã ç©åããŠã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 45,
"tag": "p",
"text": "ããã§ã C o n s t {\\displaystyle Const} ã¯ç©åå®æ°ãšããã(ç©åå®æ°ã C {\\displaystyle C} ãšæžããªãã£ãã®ã¯ãæ¯ç±ã®èšå·ãšã®æ··åãé¿ããããã) 察æ°ã®æ§è³ªãããä¿æ°R/Cvã察æ°log()ã®äžã®å€æ°ã®ææ°ã«æã£ãŠããã(æ°åŠIIçžåœ)ã®ã§ãèšç®ãããšã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 46,
"tag": "p",
"text": "ããã«ç§»é
ããŠãå€æ°ã巊蟺ã«ãŸãšãããšã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 47,
"tag": "p",
"text": "察æ°ã®æ§è³ªããã察æ°å士ã®åã¯ãäžã®å€æ°ã®ç©ã«å€ããããã®ã§ã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 48,
"tag": "p",
"text": "ã§ããã 察æ°ã®å®çŸ©ãããèªç¶å¯Ÿæ°ã®åºãeãšããã°",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 49,
"tag": "p",
"text": "ã§ããã e C o n s t {\\displaystyle e^{Const}} ãæ°ãããå¥ã®å®æ°ãšããŠãå®æ°âconstantâãšçœ®ãçŽãã°ã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 50,
"tag": "p",
"text": "ã§ããã ããã§æç±å€åã®æž©åºŠãšäœç©ã®é¢ä¿åŒã®å
¬åŒãæ±ãŸã£ãã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 51,
"tag": "p",
"text": "ä»äºWãšã®é¢ä¿ãèŠããã®ã§ãå
ã»ã©æ±ããäžã®å
¬åŒãpãšTã®åŒã«æžãæããäºãèãããç¶æ
æ¹çšåŒ p V = n R T {\\displaystyle pV=nRT} ãçšããŠTããPãšVãçšããåŒã«æžãæãããšããŸã代å
¥ããããããã«ç¶æ
æ¹çšåŒã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 52,
"tag": "p",
"text": "ãšæžãæããŠããããå
¬åŒã«ä»£å
¥ããã°ã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 53,
"tag": "p",
"text": "1 n R {\\displaystyle {\\frac {1}{nR}}} ã¯å®æ°ãªã®ã§ããããå®æ°éšã«ãŸãšããŠããŸãã°ãå¥ã®å®æ°ãConst2ãšã§ã眮ããŠã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 54,
"tag": "p",
"text": "ãšæžããã ããã§ãææ°éšã®åŒã¯ããã€ã€ãŒã®åŒ C p = C v + R {\\displaystyle Cp=Cv+R} ãããå®å§ã¢ã«æ¯ç±ã§æžãæããå¯èœã§ããã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 55,
"tag": "p",
"text": "ã§ããã ããã§ã: C p C V {\\displaystyle {\\frac {C_{p}}{C_{V}}}} ãæ¯ç±æ¯(heat capacity ratio)ãšèšããæ¯ç±æ¯ã®èšå·ã¯äžè¬ã« γ {\\displaystyle \\gamma } ã§è¡šãã ãããçšãããšã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 56,
"tag": "p",
"text": "ã§ããã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 57,
"tag": "p",
"text": "ãŸãã枩床ãšäœç©ã®é¢ä¿åŒ",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 58,
"tag": "p",
"text": "ã«æ¯ç±æ¯ã代å
¥ãããšã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 59,
"tag": "p",
"text": "ã«ãªãã",
"title": "ç±ååŠã®ç¬¬2æ³å"
},
{
"paragraph_id": 60,
"tag": "p",
"text": "ãããã®ãå§åãšäœç©ã®å
¬åŒãããã³æž©åºŠãšäœç©ã®å
¬åŒã®äºåŒããã¢ãœã³ã®åŒãšããã",
"title": "ç±ååŠã®ç¬¬2æ³å"
}
] | ç±ååŠ > ç±ååŠã®ç¬¬2æ³å | <small> [[ç±ååŠ]] > ç±ååŠã®ç¬¬2æ³å</small>
----
==ç±ååŠã®ç¬¬2æ³å==
ç±ã®å·šèŠçãªæ§è³ªãšããŠã
"枩床ã®äœããã®ãã枩床ã®é«ããã®ã«å¯ŸããŠ
ä»ã®ç©äœã«åœ±é¿ãäžããäºç¡ãã«ç±ãäžããããããšã¯ã§ããªãã"
ããšãç¥ãããŠããã
ãããç±ååŠã®ç¬¬2æ³åãšããã
äŸãã°ãä»®ã«ãã®ããšãå¯èœã ã£ããšãããšã
å·ããæ°Žãšç±ã湯ãæ··ãããšã
å·ããæ°Žã¯ããå·ããã湯ã¯ããç±ããšããããšã
èµ·ããåŸãããšãäºæ³ããããå®éã«ã¯
çµéšçã«ãããã®ããšãèµ·ãããªãããšãç¥ãããŠããã
[[Category:ç±ååŠ|ãã€ããããã®ããã«ã»ããã]]
=== ç¶æ
é ===
æ°äœã®å€æ°ã®å€æ°p,V,Tã¯ãçæ³æ°äœã§ããããã¡ã³ãã«ã¯ãŒã«ã¹æ°äœã§ãããç¶æ
æ¹çšåŒïŒçæ³æ°äœããã¡ã³ãã«ã¯ãŒã«ã¹æ°äœãã¯ãããã§ã¯åããªãïŒããããªãã°ãå€æ°p,V,Tã®ãã¡ã®ãã©ããäºã€ã決ãŸãã°ãæ°äœã®ç¶æ
æ¹çšåŒããæ®ãã®å€æ°ã決ãŸããããããŠ3å€æ°p,V,Tã決ãŸãã
å
éšãšãã«ã®ãŒã¯ãçæ³æ°äœã§ããããã¡ã³ãã«ã¯ãŒã«ã¹æ°äœã§ãããã©ã¡ãã«ããŠããå€æ°p,V,Tã®ãã¡ãã©ããäºã€ã決ãŸãã°ãæ°äœã®æ¹çšåŒããæ®ãã®æ¹çšåŒã決ãŸãã決ãŸã£ã3å€æ°ã®p,V,Tã«ãã£ãŠãå
éšãšãã«ã®ãŒã決ãŸã£ãŠããŸãããã®ãããªãç¶æ
å€æ°ã«ãã£ãŠã®ã¿æ±ºãŸãç©çéã'''ç¶æ
é'''ïŒããããããããïŒãšããã
3å€æ°ã®p,V,Tã決ãŸãã°å
éšãšãã«ã®ãŒã決å®ãããã®ã§ãå
éšãšãã«ã€ã®ãŒã¯ç¶æ
éã§ããã
å
éšãšãã«ã®ãŒã決ãã3å€æ°ã®ãã¡ãçã«ç¬ç«å€æ°ãªã®ã¯ããã®ãã¡ã®2åã®ã¿ã§ãããå€æ°p,V,Tã®ã©ãã2åãŸã§ç¬ç«å€æ°ã«éžãã§ãããããæ®ãã®1åã¯æ¢ã«éžãã å€æ°ã®åŸå±å€æ°ã«ãªãã
ã©ã®å€æ°ãç¬ç«å€æ°ã«éžã¶ãšãç¥ãããçããæ±ãããããã¯ãåé¡ã«ããã
ïŒå€å€æ°ã®é¢æ°ã®åŸ®åç©åã«ã€ããŠã¯ã倧åŠçç§ç³»ã§æè²ããããå€å€æ°é¢æ°ã®åŸ®åãå埮åãšããã解説ã¯é«æ ¡ã¬ãã«ãè¶
ããã®ã§çç¥ãïŒ
=== ç±ååŠé¢æ° ===
åç¯ã§èšåãããïŒã€ã®å€æ°ïŒå§åpãäœç©Vã枩床TïŒã®ã»ãããšã³ããããŒSãå
éšãšãã«ã®ãŒUãªã©ãç±ååŠç³»ã®å¹³è¡¡ç¶æ
ãç¹åŸŽä»ããç¶æ
éã§ããã
åç¯ãšåæ§ã5ã€ã®ç¶æ
ép,V,T,U,Sã®ãã¡ä»»æã®2ã€ãç¬ç«å€æ°ã«éžã¶å Žåã«ããæ®ã3ã€ã®å€æ°ã¯ããã2ã€ã®ç¬ç«å€æ°ã§è¡šãããåŸå±å€æ°ãšããŠæ±ããã
ãã®5ã€ã®å€æ°ã®ä»»æã®çµã¿åãããç¬ç«å€æ°ã«ãã€ç¶æ
éã¯ãäžè¬ã«ç±ååŠé¢æ°ãšåŒã°ããã
å
éšãšãã«ã®ãŒU(S,V)ã®ã»ããåŸã®ç« ã«ãŠèšåããããšã³ããããŒS(U,V)ããšã³ã¿ã«ããŒH(S,p)ããã«ã ãã«ãã®èªç±ãšãã«ã®ãŒF(V,T)ãã®ãã¹ã®èªç±ãšãã«ã®ãŒG(T,p)ãªã©ãç±ååŠé¢æ°ã§ããã
=== çæž©å€å ===
ïŒãã®ç¯ã§ã¯ãé«æ ¡æ°åŠã®æ°åŠIIIçžåœã®åŸ®åç©åãçšãããåãããªããã°æ°åŠIIIãåç
§ã®ããšãïŒ
å§åãpãšæžããšãããäœç©ãVãã¢ã«æ°ãnãæ®éæ°äœå®æ°ãnã枩床ã絶察枩床ã§Tãšããã
ä»äºWã®ãç¬éçãªä»äºã®å€§ããã¯åŸ®åãçšããŠdWãšè¡šãããäœç©Vã®ããã®ç¬éã®äœç©å€åã¯åŸ®åãçšããŠdVãšè¡šãããããããçšããã°ã
<math> dW=pdV </math>
ãšåŸ®åæ¹çšåŒã§è¡šãããïŒå®å§å€åã§ã¯ç¡ãããããã®åŒã®pã¯å€æ°ã§ãããïŒ
äœç©ãV<sub>1</sub>ããV<sub>2</sub>ãŸã§å€åãããæã®ä»äºã¯ãç©åãçšããŠä»¥äžã®ããã«æžãè¡šããã
<math> W=\int_{V_1}^{V_2} p dV </math>
ããã«ãç¶æ
æ¹çšåŒã® <math> pV = nRT </math> ããçµã¿åãããã
ç©åå€æ°ã®Vã«åãããŠãpãæžãæãããã
<math>p=\frac{nRT}{V}</math>
ã§ããããããããä»äºã®åŒã¯ã
<math> W=\int_{V_1}^{V_2} p dV= \int_{V_1}^{V_2} \frac{nRT}{V} dV = nRT\int_{V_1}^{V_2}\frac{dV}{V} = nRT\log \frac{V_2}{V_1 }</math>
ãšãªããïŒãªããlogã¯èªç¶å¯Ÿæ°ã§ãããïŒ
çµè«ããŸãšãããšã
:<math> W = nRT \log{\frac{V_2}{V_1}} </math>
ã§ããã
å
éšãšãã«ã®ãŒUã¯ãçæ³æ°äœã§ã¯æž©åºŠã®ã¿ã®é¢æ°ã§ãçæž©å€åã§ã¯æž©åºŠãå€åããªãããã
:<math>\Delta U=0</math>
ã§ããã
ãããã£ãŠãçæž©å€åã§ã¯
:<math>Q=W</math>
ã§ããã
=== æç±å€å ===
ãŸããç±ãšå
éšãšãã«ã®ãŒãšä»äºã®é¢ä¿åŒ
:<math>Q=U+W</math>
ãã次ã®ããã«åŸ®åæ¹çšåŒã«æžãæãããå
éšãšãã«ã®ãŒã®å€åã埮å°å€åãšããŠdUãšè¡šãããšãããšãç±éQãä»äºWã埮å°å€åã«ãªãã®ã§ã以äžã®æ§ãªåŒã«ãªãã
:<math>d'Q=dU+d'W</math>
QãWã®åŸ®åæŒç®èšå·dã®äžã«ç¹ã<math>'</math>ããä»ããŠããã®ã¯ãå³å¯ã«èšããšãç±éQãä»äºWã¯ç¶æ
éã§ç¡ããããåºå¥ããããã«çšããŠããã
æç±å€åã§ã¯
:<math>d'Q=0</math>
ãªã®ã§ãã€ãŸãã
:<math>0=dU+d'W</math>
ãšãªãã
ä»äºã«é¢ããŠã¯
:<math>d'W=pdV</math>
ã§ããã
å
éšãšãã«ã®ãŒã®åŸ®å°å€åã¯ãå®ç©ã¢ã«æ¯ç±ãçšããŠã
:<math>dU=nC_{V}dT</math>
ãšæžããã
ãªã®ã§ãããçãåŒ <math>0=dU+d'W</math> ã«ä»£å
¥ãã
:<math>0=nC_{V}dT+pdV</math>
ãšæžããã
䞡蟺ãpVã§å²ããšã
:<math>0=\frac{nC_VdT}{pV}+{pdV}{pV}=\frac{nC_VdT}{pV}+\frac{dV}{V}</math>
ã§ããããpV=nRTãå©çšãããšã
:<math>0=\frac{nC_VdT}{nRT}+\frac{dV}{V}=\frac{C_V}{R}\frac{dT}{T}+\frac{dV}{V}</math>
ãšãªãã
ãã®åŸ®åæ¹çšåŒã解ãããŸã移é
ããŠã
:<math>\frac{dT}{T}=-\frac{R}{C_V}\frac{dV}{V}</math>
ãšãªãã
ç©åããŠã
:<math>\log T=- \frac{R}{C_V} \log{V}+Const</math>
ããã§ã<math>Const</math>ã¯ç©åå®æ°ãšãããïŒç©åå®æ°ã <math>C</math> ãšæžããªãã£ãã®ã¯ãæ¯ç±ã®èšå·ãšã®æ··åãé¿ãããããïŒ
察æ°ã®æ§è³ªãããä¿æ°R/Cvã察æ°log()ã®äžã®å€æ°ã®ææ°ã«æã£ãŠãããïŒæ°åŠIIçžåœïŒã®ã§ãèšç®ãããšã
:<math>\log T=-\log{V^{\frac{R}{C_V}}}+Const</math>
ããã«ç§»é
ããŠãå€æ°ã巊蟺ã«ãŸãšãããšã
:<math>\log T+\log V^{\frac{R}{C_V}}=Const</math>
察æ°ã®æ§è³ªããã察æ°å士ã®åã¯ãäžã®å€æ°ã®ç©ã«å€ããããã®ã§ã
:<math>\log TV^{\frac{R}{C_V}}=Const</math>
ã§ããã
察æ°ã®å®çŸ©ãããèªç¶å¯Ÿæ°ã®åºãeãšããã°
:<math>TV^{\frac{R}{C_V}}=e^{Const}</math>
ã§ããã
<math>e^{Const}</math>ãæ°ãããå¥ã®å®æ°ãšããŠãå®æ°âconstantâãšçœ®ãçŽãã°ã
:<math>TV^{\frac{R}{C_V}}=constant</math>
ã§ããã
ããã§æç±å€åã®æž©åºŠãšäœç©ã®é¢ä¿åŒã®å
¬åŒãæ±ãŸã£ãã
;枩床ãšäœç©ã®é¢ä¿åŒ
ä»äºWãšã®é¢ä¿ãèŠããã®ã§ãå
ã»ã©æ±ããäžã®å
¬åŒãpãšTã®åŒã«æžãæããäºãèãããç¶æ
æ¹çšåŒ<math>pV=nRT</math>ãçšããŠTããPãšVãçšããåŒã«æžãæãããšããŸã代å
¥ããããããã«ç¶æ
æ¹çšåŒã
:<math>T=\frac{pV}{nR}</math>
ãšæžãæããŠããããå
¬åŒã«ä»£å
¥ããã°ã
:<math>TV^{\frac{R}{C_V}}=\frac{pV}{nR}V^{\frac{R}{C_V}}=\frac{1}{nR}pVV^{\frac{R}{C_V}}=\frac{1}{nR}pV^{1+\frac{R}{C_V}}=constant</math>
;å§åãšäœç©ã®é¢ä¿åŒ
<math>\frac{1}{nR}</math>ã¯å®æ°ãªã®ã§ããããå®æ°éšã«ãŸãšããŠããŸãã°ãå¥ã®å®æ°ãConst<sub>2</sub>ãšã§ã眮ããŠã
:<math>pV^{1+\frac{R}{C_V}}=Const_2</math>
ãšæžããã
ããã§ãææ°éšã®åŒã¯ããã€ã€ãŒã®åŒ<math>Cp=Cv+R</math>ãããå®å§ã¢ã«æ¯ç±ã§æžãæããå¯èœã§ããã
:<math>pV^{\frac{C_p}{C_V}}=Const_2</math>
ã§ããã
ããã§ã:<math>\frac{C_p}{C_V}</math>ã{{ruby|'''æ¯ç±æ¯'''|ã²ãã€ã²}}ïŒheat capacity ratioïŒãšèšããæ¯ç±æ¯ã®èšå·ã¯äžè¬ã«<math>\gamma</math>ã§è¡šãã
ãããçšãããšã
:<math>pV^{\gamma}=Const_2</math>
ã§ããã
ãŸãã枩床ãšäœç©ã®é¢ä¿åŒ
:<math>TV^{\frac{R}{C_V}}=constant</math>
ã«æ¯ç±æ¯ã代å
¥ãããšã
:<math>TV^{\gamma -1}=constant</math>
ã«ãªãã
ãããã®ãå§åãšäœç©ã®å
¬åŒãããã³æž©åºŠãšäœç©ã®å
¬åŒã®äºåŒã'''ãã¢ãœã³ã®åŒ'''ãšããã | null | 2022-12-01T04:09:28Z | [
"ãã³ãã¬ãŒã:Ruby"
] | https://ja.wikibooks.org/wiki/%E7%86%B1%E5%8A%9B%E5%AD%A6/%E7%86%B1%E5%8A%9B%E5%AD%A6%E3%81%AE%E7%AC%AC2%E6%B3%95%E5%89%87 |
2,040 | ç±ååŠ/ãšã³ããã㌠| ç±ååŠ > ãšã³ããããŒ
ãã枩床Tã®ç©äœã«å¯ŸããŠæºéçã« ç±dQãäžãããããšããã®ç©äœã¯ d S = d Q T {\displaystyle dS={\frac {dQ}{T}}} ã®ãšã³ããããŒãåŸããšããã ãã®å€ãçšããŠç¬¬2æ³åãæžãæããããšãåºæ¥ãã ãã枩床 T 1 {\displaystyle T_{1}} ãš T 2 {\displaystyle T_{2}} ( T 1 > T 2 {\displaystyle T_{1}>T_{2}} )ã®ç©äœã(ç©äœ1,ç©äœ2ãšããã) æ¥è§Šããããšãã第2æ³å㯠ããéã®ç±ã T 1 {\displaystyle T_{1}} ã®ç©äœãã T 2 {\displaystyle T_{2}} ã®ç©äœã«ç§»ãããããšãäºèšããã ãã®ãšããããããã®ç©äœãåŸããšã³ããããŒã®éãèšç®ãããš ç©äœ1ã«ã€ããŠã¯ã d S 1 = â d Q T 1 {\displaystyle dS_{1}=-{\frac {dQ}{T_{1}}}} ãåŸãããç©äœ2ã«ã€ããŠã¯ d S 2 = d Q T 2 {\displaystyle dS_{2}={\frac {dQ}{T_{2}}}} ãåŸãããã2ã€ãåãããå Žåãå
šç³»ãšåŒã³ãå
šç³»ã®ãšã³ããããŒã d S tot {\displaystyle dS_{\textrm {tot}}} ãšæžããšã d S tot = d Q ( 1 T 2 â 1 T 1 ) > 0 {\displaystyle dS_{\textrm {tot}}=dQ({\frac {1}{T_{2}}}-{\frac {1}{T_{1}}})>0} ãåŸãããã ãã®ããšããã第2æ³å㯠"å
šç³»ã®ãšã³ããããŒãå¢å€§ããæ¹åã«ç±ã®ç§»åãèµ·ããã" ãšæžãçŽãããšãåºæ¥ãã
ãŸãã d Q = T d S {\displaystyle dQ=TdS} ã®é¢ä¿ãçšããŠã第1æ³åãæžãæããããšãåºæ¥ãã d Q = d U â d W {\displaystyle dQ=dU-dW} ãæžãæããŠã d U = T d S + d W {\displaystyle dU=TdS+dW} ãåŸãããã ç¹ã«æ°äœã«ã€ã㊠d W = â P d V {\displaystyle dW=-PdV} ãšãªããã®ãšããŠãå§åãå®çŸ©ãããš d U = T d S â P d V {\displaystyle dU=TdS-PdV} ãåŸãããã
(泚æ:ããã¯å¯ééçšãèãããšãã®èšè¿°ã§ããäžå¯ééçšãèãããšã㯠...)
ç±å¹çã®å®çŸ©åŒãšãã«ã«ããŒãµã€ã¯ã«ã®ç±å¹çã®æž©åºŠã®é¢ä¿åŒãé£ç«ãããŠã¿ããã ãŸããé«æž©ç±æºã®æž©åºŠãThãšæžããšããŠãé«æž©ç±æºããç±æ©é¢ã«æž¡ãç±éãQhãšæžããšãããã äœæž©ç±æºã®æž©åºŠã¯TcãšããŠãç±æ©é¢ããäœæž©ç±æºã«æŸç±ãããç±éãQcãšæžããšãããã ç±å¹çeã®å®çŸ©åŒã¯ã
ã§ãã£ãããã£ãœããã«ã«ããŒãµã€ã¯ã«ã®ç±å¹çã¯ã
ã§ããã
ãããããã
ã§ãããããã¯ã
ãšãæžããŠã䞡蟺ã®1ãåŒããŠæ¶å»ããŠã
ãšãªãããã€ãã¹ãããã®ã§ã移é
ããã°ã
ã§ããã æ·»åãåãéã©ããããŸãšããã°ã
ãšãªããããã§ã Q T {\displaystyle {\frac {Q}{T}}} ãæ°ããç©çéãšããŠå®çŸ©ããŠããã®éã¯ãšã³ããããŒ(entropy)ãšåŒã°ããããšã³ããããŒã®èšå·ã¯Sãšçœ®ããšããããŸãããšã³ããããŒã®åäœã¯[J/K]ã§ããã ã€ãŸãã S = Q T {\displaystyle S={\frac {Q}{T}}} ã§ããããããããšãåŒ(1)ã¯
ãšæžããã
ç±æ©é¢ã®åäœã®é åºã¯ããŸãæ©é¢ãé«æž©ç±æºããç±ãè²°ã£ãŠãããäœæž©ç±æºã«ç±ãæž¡ãã®ã§ãã£ãã(éã«å
ã«äœé³ç±æºã«æŸç±ããŠããé«æž©ç±æºã§åžç±ããã®ã¯äžå¯èœã§ãããç±æ©é¢ã¯ãããã£ãŠãªãç±ã¯æž¡ããªããç±ååŠã®ç¬¬äºæ³åããåœç¶ã§ããã)ã ãããæéçã«ã¯ãç±æ©é¢ã®ãšã³ããããŒSã¯ããŸãå
ã«S=Shã«ãªã£ãŠãããæéãçµã£ãŠãããšããS=Scã«ãªã£ãã®ã§ããã ãããŠåŒ(2)ããã S h {\displaystyle S_{h}} ⊠S c {\displaystyle S_{c}} ã§ãããããç±æ©é¢ã®ãšã³ããããŒã¯ãæéãçµã£ãŠãå¢å€§ããããšãåããã
以äžã®è«èšŒãããç±æ©é¢ã®ãšã³ããããŒã¯ãããªããå¢å€§ãããããããšã³ããããŒå¢å€§ã®æ³åãšããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ç±ååŠ > ãšã³ããããŒ",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ãã枩床Tã®ç©äœã«å¯ŸããŠæºéçã« ç±dQãäžãããããšããã®ç©äœã¯ d S = d Q T {\\displaystyle dS={\\frac {dQ}{T}}} ã®ãšã³ããããŒãåŸããšããã ãã®å€ãçšããŠç¬¬2æ³åãæžãæããããšãåºæ¥ãã ãã枩床 T 1 {\\displaystyle T_{1}} ãš T 2 {\\displaystyle T_{2}} ( T 1 > T 2 {\\displaystyle T_{1}>T_{2}} )ã®ç©äœã(ç©äœ1,ç©äœ2ãšããã) æ¥è§Šããããšãã第2æ³å㯠ããéã®ç±ã T 1 {\\displaystyle T_{1}} ã®ç©äœãã T 2 {\\displaystyle T_{2}} ã®ç©äœã«ç§»ãããããšãäºèšããã ãã®ãšããããããã®ç©äœãåŸããšã³ããããŒã®éãèšç®ãããš ç©äœ1ã«ã€ããŠã¯ã d S 1 = â d Q T 1 {\\displaystyle dS_{1}=-{\\frac {dQ}{T_{1}}}} ãåŸãããç©äœ2ã«ã€ããŠã¯ d S 2 = d Q T 2 {\\displaystyle dS_{2}={\\frac {dQ}{T_{2}}}} ãåŸãããã2ã€ãåãããå Žåãå
šç³»ãšåŒã³ãå
šç³»ã®ãšã³ããããŒã d S tot {\\displaystyle dS_{\\textrm {tot}}} ãšæžããšã d S tot = d Q ( 1 T 2 â 1 T 1 ) > 0 {\\displaystyle dS_{\\textrm {tot}}=dQ({\\frac {1}{T_{2}}}-{\\frac {1}{T_{1}}})>0} ãåŸãããã ãã®ããšããã第2æ³å㯠\"å
šç³»ã®ãšã³ããããŒãå¢å€§ããæ¹åã«ç±ã®ç§»åãèµ·ããã\" ãšæžãçŽãããšãåºæ¥ãã",
"title": "ãšã³ããããŒ"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ãŸãã d Q = T d S {\\displaystyle dQ=TdS} ã®é¢ä¿ãçšããŠã第1æ³åãæžãæããããšãåºæ¥ãã d Q = d U â d W {\\displaystyle dQ=dU-dW} ãæžãæããŠã d U = T d S + d W {\\displaystyle dU=TdS+dW} ãåŸãããã ç¹ã«æ°äœã«ã€ã㊠d W = â P d V {\\displaystyle dW=-PdV} ãšãªããã®ãšããŠãå§åãå®çŸ©ãããš d U = T d S â P d V {\\displaystyle dU=TdS-PdV} ãåŸãããã",
"title": "ãšã³ããããŒ"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "(泚æ:ããã¯å¯ééçšãèãããšãã®èšè¿°ã§ããäžå¯ééçšãèãããšã㯠...)",
"title": "ãšã³ããããŒ"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ç±å¹çã®å®çŸ©åŒãšãã«ã«ããŒãµã€ã¯ã«ã®ç±å¹çã®æž©åºŠã®é¢ä¿åŒãé£ç«ãããŠã¿ããã ãŸããé«æž©ç±æºã®æž©åºŠãThãšæžããšããŠãé«æž©ç±æºããç±æ©é¢ã«æž¡ãç±éãQhãšæžããšãããã äœæž©ç±æºã®æž©åºŠã¯TcãšããŠãç±æ©é¢ããäœæž©ç±æºã«æŸç±ãããç±éãQcãšæžããšãããã ç±å¹çeã®å®çŸ©åŒã¯ã",
"title": "ãšã³ããããŒ"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ã§ãã£ãããã£ãœããã«ã«ããŒãµã€ã¯ã«ã®ç±å¹çã¯ã",
"title": "ãšã³ããããŒ"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ã§ããã",
"title": "ãšã³ããããŒ"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ãããããã",
"title": "ãšã³ããããŒ"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "ã§ãããããã¯ã",
"title": "ãšã³ããããŒ"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "ãšãæžããŠã䞡蟺ã®1ãåŒããŠæ¶å»ããŠã",
"title": "ãšã³ããããŒ"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "ãšãªãããã€ãã¹ãããã®ã§ã移é
ããã°ã",
"title": "ãšã³ããããŒ"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "ã§ããã æ·»åãåãéã©ããããŸãšããã°ã",
"title": "ãšã³ããããŒ"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "ãšãªããããã§ã Q T {\\displaystyle {\\frac {Q}{T}}} ãæ°ããç©çéãšããŠå®çŸ©ããŠããã®éã¯ãšã³ããããŒ(entropy)ãšåŒã°ããããšã³ããããŒã®èšå·ã¯Sãšçœ®ããšããããŸãããšã³ããããŒã®åäœã¯[J/K]ã§ããã ã€ãŸãã S = Q T {\\displaystyle S={\\frac {Q}{T}}} ã§ããããããããšãåŒ(1)ã¯",
"title": "ãšã³ããããŒ"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ãšæžããã",
"title": "ãšã³ããããŒ"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "ç±æ©é¢ã®åäœã®é åºã¯ããŸãæ©é¢ãé«æž©ç±æºããç±ãè²°ã£ãŠãããäœæž©ç±æºã«ç±ãæž¡ãã®ã§ãã£ãã(éã«å
ã«äœé³ç±æºã«æŸç±ããŠããé«æž©ç±æºã§åžç±ããã®ã¯äžå¯èœã§ãããç±æ©é¢ã¯ãããã£ãŠãªãç±ã¯æž¡ããªããç±ååŠã®ç¬¬äºæ³åããåœç¶ã§ããã)ã ãããæéçã«ã¯ãç±æ©é¢ã®ãšã³ããããŒSã¯ããŸãå
ã«S=Shã«ãªã£ãŠãããæéãçµã£ãŠãããšããS=Scã«ãªã£ãã®ã§ããã ãããŠåŒ(2)ããã S h {\\displaystyle S_{h}} ⊠S c {\\displaystyle S_{c}} ã§ãããããç±æ©é¢ã®ãšã³ããããŒã¯ãæéãçµã£ãŠãå¢å€§ããããšãåããã",
"title": "ãšã³ããããŒ"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "以äžã®è«èšŒãããç±æ©é¢ã®ãšã³ããããŒã¯ãããªããå¢å€§ãããããããšã³ããããŒå¢å€§ã®æ³åãšããã",
"title": "ãšã³ããããŒ"
}
] | ç±ååŠ > ãšã³ããã㌠| <small> [[ç±ååŠ]] > ãšã³ããããŒ</small>
----
==ãšã³ããããŒ==
ãã枩床Tã®ç©äœã«å¯ŸããŠæºéçã«
ç±dQãäžãããããšããã®ç©äœã¯
<math>
dS = \frac {dQ} T
</math>
ã®ãšã³ããããŒãåŸããšããã
ãã®å€ãçšããŠç¬¬2æ³åãæžãæããããšãåºæ¥ãã
ãã枩床<math>T _1</math>ãš<math>T _2</math>(<math>T _1>T _2</math>)ã®ç©äœã(ç©äœ1,ç©äœ2ãšããã)
æ¥è§Šããããšãã第2æ³åã¯
ããéã®ç±ã<math>T _1</math>ã®ç©äœãã<math>T _2</math>ã®ç©äœã«ç§»ãããããšãäºèšããã
ãã®ãšããããããã®ç©äœãåŸããšã³ããããŒã®éãèšç®ãããš
ç©äœ1ã«ã€ããŠã¯ã
<math>
d S _1 = -\frac {d Q} {T _1}
</math>
ãåŸãããç©äœ2ã«ã€ããŠã¯
<math>
d S _2 = \frac {d Q} {T _2}
</math>
ãåŸãããã2ã€ãåãããå Žåãå
šç³»ãšåŒã³ãå
šç³»ã®ãšã³ããããŒã
<math>
d S _{\textrm{tot}}
</math>
ãšæžããšã
<math>
d S _{\textrm{tot}} = dQ (\frac 1 {T _2} -\frac 1 {T _1}) >0
</math>
ãåŸãããã
ãã®ããšããã第2æ³åã¯
"å
šç³»ã®ãšã³ããããŒãå¢å€§ããæ¹åã«ç±ã®ç§»åãèµ·ããã"
ãšæžãçŽãããšãåºæ¥ãã
ãŸãã
<math>
dQ = TdS
</math>
ã®é¢ä¿ãçšããŠã第1æ³åãæžãæããããšãåºæ¥ãã
<math>
dQ = dU - dW
</math>
ãæžãæããŠã
<math>
dU = TdS + dW
</math>
ãåŸãããã
ç¹ã«æ°äœã«ã€ããŠ
<math>
dW = -P dV
</math>
ãšãªããã®ãšããŠãå§åãå®çŸ©ãããš
<math>
dU = TdS - PdV
</math>
ãåŸãããã
(泚æ:ããã¯å¯ééçšãèãããšãã®èšè¿°ã§ããäžå¯ééçšãèãããšãã¯
...)
{{stub}}
=== ãšã³ããã㌠===
ç±å¹çã®å®çŸ©åŒãšãã«ã«ããŒãµã€ã¯ã«ã®ç±å¹çã®æž©åºŠã®é¢ä¿åŒãé£ç«ãããŠã¿ããã
ãŸããé«æž©ç±æºã®æž©åºŠãT<sub>h</sub>ãšæžããšããŠãé«æž©ç±æºããç±æ©é¢ã«æž¡ãç±éãQ<sub>h</sub>ãšæžããšãããã
äœæž©ç±æºã®æž©åºŠã¯T<sub>c</sub>ãšããŠãç±æ©é¢ããäœæž©ç±æºã«æŸç±ãããç±éãQ<sub>c</sub>ãšæžããšãããã
ç±å¹çeã®å®çŸ©åŒã¯ã
:<math>e=\frac{Q_h-Q_c}{Q_h}</math>
ã§ãã£ãããã£ãœããã«ã«ããŒãµã€ã¯ã«ã®ç±å¹çã¯ã
:<math>e</math>'''âŠ'''<math>\frac{T_h-T_c}{T_h}</math>
ã§ããã
ãããããã
:<math>\frac{Q_h-Q_c}{Q_h}</math>'''âŠ'''<math>\frac{T_h-T_c}{T_h}</math>
ã§ãããããã¯ã
:<math>1-\frac{Q_c}{Q_h}</math>'''âŠ'''<math>1-\frac{T_c}{T_h}</math>
ãšãæžããŠã䞡蟺ã®1ãåŒããŠæ¶å»ããŠã
:<math>-\frac{Q_c}{Q_h}</math>'''âŠ'''<math>-\frac{T_c}{T_h}</math>
ãšãªãããã€ãã¹ãããã®ã§ã移é
ããã°ã
:<math>\frac{T_c}{T_h}</math>'''âŠ'''<math>\frac{Q_c}{Q_h}</math>
ã§ããã
æ·»åãåãéã©ããããŸãšããã°ã
:<math>\frac{Q_h}{T_h}</math>'''âŠ'''<math>\frac{Q_c}{T_c}</math>ããããããïŒ1ïŒ
ãšãªããããã§ã<math>\frac{Q}{T}</math>ãæ°ããç©çéãšããŠå®çŸ©ããŠããã®éã¯'''ãšã³ããããŒ'''ïŒentropyïŒãšåŒã°ããããšã³ããããŒã®èšå·ã¯Sãšçœ®ããšããããŸãããšã³ããããŒã®åäœã¯[J/K]ã§ããã
ã€ãŸãã
<math>S=\frac{Q}{T}</math>
ã§ããããããããšãåŒ(1)ã¯
:<math>S_h</math>'''âŠ'''<math>S_c</math>ããããããïŒ2ïŒ
ãšæžããã
ç±æ©é¢ã®åäœã®é åºã¯ããŸãæ©é¢ãé«æž©ç±æºããç±ãè²°ã£ãŠãããäœæž©ç±æºã«ç±ãæž¡ãã®ã§ãã£ããïŒéã«å
ã«äœé³ç±æºã«æŸç±ããŠããé«æž©ç±æºã§åžç±ããã®ã¯äžå¯èœã§ãããç±æ©é¢ã¯ãããã£ãŠãªãç±ã¯æž¡ããªããç±ååŠã®ç¬¬äºæ³åããåœç¶ã§ãããïŒã ãããæéçã«ã¯ãç±æ©é¢ã®ãšã³ããããŒSã¯ããŸãå
ã«S=S<sub>h</sub>ã«ãªã£ãŠãããæéãçµã£ãŠãããšããS=S<sub>c</sub>ã«ãªã£ãã®ã§ããã
ãããŠåŒ(2)ããã<math>S_h</math>'''âŠ'''<math>S_c</math>ãã§ãããããç±æ©é¢ã®ãšã³ããããŒã¯ãæéãçµã£ãŠãå¢å€§ããããšãåããã
以äžã®è«èšŒãããç±æ©é¢ã®ãšã³ããããŒã¯ãããªããå¢å€§ãããããã'''ãšã³ããããŒå¢å€§ã®æ³å'''ãšããã
[[Category:ç±ååŠ|ãããšãã²ã]] | null | 2022-12-01T04:09:28Z | [
"ãã³ãã¬ãŒã:Stub"
] | https://ja.wikibooks.org/wiki/%E7%86%B1%E5%8A%9B%E5%AD%A6/%E3%82%A8%E3%83%B3%E3%83%88%E3%83%AD%E3%83%94%E3%83%BC |
2,041 | ç±ååŠ/ç±ååŠçãªãšãã«ã®ãŒ | ç±ååŠ > ç±ååŠçãªãšãã«ã®ãŒ
ããç³»ã«ã€ããŠ4ã€ã®ãã©ã¡ãŒã¿ T,S,V,Pãèãããšã ãã®ãã¡ã®2ã€ãå®ãããšãä»ã®2ã€ã¯èªåçã«æ±ºå®ãããã (å°åº?)
ãã㧠d U = T d S â P d V {\displaystyle dU=TdS-PdV} ã®åŒãããå
éšãšãã«ã®ãŒUã«ãšã£ãŠèªç¶ãªå€æ°ã¯ SãšVã§ããããšããããã (T,Pã¯S,Vã®é¢æ°ã§ããã) ãã®ãšããã以å€ã®2ã€ã®éãèªç¶ãªå€æ°ãšã㊠æã€éãå®çŸ©ããã
äŸãã°ãS,Pãèªç¶ãªå€æ°ãæã€éãšã㊠H = U + P V {\displaystyle H=U+PV} ãå®çŸ©ãããHããšã³ã¿ã«ããŒãšåŒã¶ã
(å°åº) d H = d U + d ( P V ) {\displaystyle dH=dU+d(PV)} = T d S â P d V + V d P + P d V {\displaystyle =TdS-PdV+VdP+PdV} = T d S + V d P {\displaystyle =TdS+VdP} ãšãªãã確ãã«SãšPãå€æ°ãšãªã£ãŠããã
åæ§ã«ã㊠F = U â T S {\displaystyle F=U-TS} (ãã«ã ãã«ãã®èªç±ãšãã«ã®ãŒ) ,
G = U â T S + P V = H â T S {\displaystyle G=U-TS+PV=H-TS} (ã®ãã¹ã®èªç±ãšãã«ã®ãŒ) ãå®çŸ©ããã ã®ãã¹ã®èªç±ãšãã«ã®ãŒã¯çæž©çå§ã®æ¡ä»¶ã§è¡ãªããã å®éšã«ãããŠããçšããããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ç±ååŠ > ç±ååŠçãªãšãã«ã®ãŒ",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ããç³»ã«ã€ããŠ4ã€ã®ãã©ã¡ãŒã¿ T,S,V,Pãèãããšã ãã®ãã¡ã®2ã€ãå®ãããšãä»ã®2ã€ã¯èªåçã«æ±ºå®ãããã (å°åº?)",
"title": "ç±ååŠçãªãšãã«ã®ãŒ"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ãã㧠d U = T d S â P d V {\\displaystyle dU=TdS-PdV} ã®åŒãããå
éšãšãã«ã®ãŒUã«ãšã£ãŠèªç¶ãªå€æ°ã¯ SãšVã§ããããšããããã (T,Pã¯S,Vã®é¢æ°ã§ããã) ãã®ãšããã以å€ã®2ã€ã®éãèªç¶ãªå€æ°ãšã㊠æã€éãå®çŸ©ããã",
"title": "ç±ååŠçãªãšãã«ã®ãŒ"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "äŸãã°ãS,Pãèªç¶ãªå€æ°ãæã€éãšã㊠H = U + P V {\\displaystyle H=U+PV} ãå®çŸ©ãããHããšã³ã¿ã«ããŒãšåŒã¶ã",
"title": "ç±ååŠçãªãšãã«ã®ãŒ"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "(å°åº) d H = d U + d ( P V ) {\\displaystyle dH=dU+d(PV)} = T d S â P d V + V d P + P d V {\\displaystyle =TdS-PdV+VdP+PdV} = T d S + V d P {\\displaystyle =TdS+VdP} ãšãªãã確ãã«SãšPãå€æ°ãšãªã£ãŠããã",
"title": "ç±ååŠçãªãšãã«ã®ãŒ"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "åæ§ã«ã㊠F = U â T S {\\displaystyle F=U-TS} (ãã«ã ãã«ãã®èªç±ãšãã«ã®ãŒ) ,",
"title": "ç±ååŠçãªãšãã«ã®ãŒ"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "G = U â T S + P V = H â T S {\\displaystyle G=U-TS+PV=H-TS} (ã®ãã¹ã®èªç±ãšãã«ã®ãŒ) ãå®çŸ©ããã ã®ãã¹ã®èªç±ãšãã«ã®ãŒã¯çæž©çå§ã®æ¡ä»¶ã§è¡ãªããã å®éšã«ãããŠããçšããããã",
"title": "ç±ååŠçãªãšãã«ã®ãŒ"
}
] | ç±ååŠ > ç±ååŠçãªãšãã«ã®ãŒ | <small> [[ç±ååŠ]] > ç±ååŠçãªãšãã«ã®ãŒ</small>
----
==ç±ååŠçãªãšãã«ã®ãŒ==
ããç³»ã«ã€ããŠ4ã€ã®ãã©ã¡ãŒã¿
T,S,V,Pãèãããšã
ãã®ãã¡ã®2ã€ãå®ãããšãä»ã®2ã€ã¯èªåçã«æ±ºå®ãããã
(å°åº?)
<!--
(å°åº? (ç¶æ
æ¹çšåŒã§P,V,Tã¯äºãã«ç§»ãå€ããããšãåºæ¥ãããšã³ããããŒã¯
T = 0ã®ãšãããšã³ããããŒã®0ãšããŠåããäœããã®éçšãã€ãããŠç±ã
äžããŠããã° ... (ããããããããã®ç¶æ
ã«å¯Ÿãããšã³ããããŒã®å€ã
äžæçã«æ±ºãŸããªã???) ))
(çµ±èšååŠãæµçšãããªãç³»ã®åé
é¢æ°ãã
T,Vã®é¢æ°ãšããŠèªç±ãšãã«ã®ãŒFãæ±ããã
<math>
P = -\frac {\partial F}{\partial V } ,S = - \frac {\partial F}{\partial T }
</math>
ãšããŠS,Pãæ±ããããã®ã§ã2ã€ã決ããããšã§
ç³»ã®ç¶æ
ãæå®ãããããšã¯åœç¶ãšãªãã
ãããäžã®è°è«ã®æ¬ ç¹ã¯ã©ãã ããã...?)
(ç±ååŠã§ã¯èªç¶ãªå€æ°ã®ç±ååŠé¢æ°ã埩å
ã§ããæããã¹ãŠã®ç±ååŠçç¶æ
ãèšè¿°ã§ããã
ç±ååŠçç¶æ
ãèšè¿°ããã«ã¯
<math>
dS = \frac {1}{T} dU - \frac {P}{T} dV
</math>
ãçšæãããã®äžæ¬¡åœ¢åŒã«å¯ŸããŠç©åãæœãããšã§å®æ°ã®ä»»ææ§ãé€ããŠ
<math>
S = S(U ,V)
</math>
ãšããŠåçŸã§ãã(ãã¢ã³ã«ã¬ã®è£é¡)
-->
ããã§
<math>
dU = TdS - PdV
</math>
ã®åŒãããå
éšãšãã«ã®ãŒUã«ãšã£ãŠèªç¶ãªå€æ°ã¯
SãšVã§ããããšããããã
(T,Pã¯S,Vã®é¢æ°ã§ããã)
ãã®ãšããã以å€ã®2ã€ã®éãèªç¶ãªå€æ°ãšããŠ
æã€éãå®çŸ©ããã
äŸãã°ãS,Pãèªç¶ãªå€æ°ãæã€éãšããŠ
<math>
H = U + PV
</math>
ãå®çŸ©ãããHããšã³ã¿ã«ããŒãšåŒã¶ã
(å°åº)
<math>
dH = dU + d(PV)
</math>
<math>
= TdS - PdV+ VdP + PdV
</math>
<math>
= TdS + VdP
</math>
ãšãªãã確ãã«SãšPãå€æ°ãšãªã£ãŠããã
åæ§ã«ããŠ
<math>
F = U-TS
</math>
(ãã«ã ãã«ãã®èªç±ãšãã«ã®ãŒ)
,
<math>
G = U-TS + PV = H -TS
</math>
(ã®ãã¹ã®èªç±ãšãã«ã®ãŒ)
ãå®çŸ©ããã
ã®ãã¹ã®èªç±ãšãã«ã®ãŒã¯çæž©çå§ã®æ¡ä»¶ã§è¡ãªããã
å®éšã«ãããŠããçšããããã
[[Category:ç±ååŠ|ãã€ãããããŠããªãããã]]
[[ã«ããŽãª:ãšãã«ã®ãŒ]] | 2005-05-24T09:12:28Z | 2024-02-06T05:19:38Z | [] | https://ja.wikibooks.org/wiki/%E7%86%B1%E5%8A%9B%E5%AD%A6/%E7%86%B1%E5%8A%9B%E5%AD%A6%E7%9A%84%E3%81%AA%E3%82%A8%E3%83%8D%E3%83%AB%E3%82%AE%E3%83%BC |
2,047 | ç¹æ®çžå¯Ÿè« é床ã®åæå | ç¹æ®çžå¯Ÿè« > é床ã®åæå
ããé床 v 1 {\displaystyle v_{1}} ãæã€ç©äœ1ããèŠããšãã« ããé床 v 2 {\displaystyle v_{2}} ãæã€ç©äœ2ã ãéæ¢ããŠãã芳枬è
ããèŠããšãã® é床ãèšç®ããã (NewtonååŠã§ã¯ v 1 + v 2 {\displaystyle v_{1}+v_{2}} ãšãªãããšã«æ³šæã)
ããŒã¬ã³ã矀ã®æ§è³ªãã v 1 {\displaystyle v_{1}} ã䜿ã£ãå€æãš v 1 {\displaystyle v_{1}} ã䜿ã£ãå€æãåãããŠäœ¿ãããšã§ã éæ¢ãã芳枬è
ããèŠãå Žåã®ç©äœ2ã®é床ã æ±ãŸãããšãçšãããšã
γ 1 ( 1 β 1 β 1 1 ) à γ 2 ( 1 β 2 β 2 1 ) = γ 3 ( 1 β 3 β 3 1 ) {\displaystyle \gamma _{1}{\begin{pmatrix}1&\beta _{1}\\\beta _{1}&1\end{pmatrix}}\times \gamma _{2}{\begin{pmatrix}1&\beta _{2}\\\beta _{2}&1\end{pmatrix}}=\gamma _{3}{\begin{pmatrix}1&\beta _{3}\\\beta _{3}&1\end{pmatrix}}} ãšãªãããšãåããã 巊蟺ã®1è¡1åæåãèšç®ãããšã = γ 1 γ 2 ( 1 + β 1 β 2 ) {\displaystyle =\gamma _{1}\gamma _{2}(1+\beta _{1}\beta _{2})} ãšãªãããšããããã å³èŸºã®1è¡1åæåãšèŠããã¹ããšã γ 1 γ 2 ( 1 + β 1 β 2 ) = γ 3 {\displaystyle \gamma _{1}\gamma _{2}(1+\beta _{1}\beta _{2})=\gamma _{3}} ãåŸãããã 䞡蟺ã2ä¹ãããšã 1 1 â v 3 2 / c 2 = 1 1 â v 1 2 / c 2 1 1 â v 2 2 / c 2 ( 1 + v 1 v 2 / c 2 ) 2 {\displaystyle {\frac {1}{1-v_{3}^{2}/c^{2}}}={\frac {1}{1-v_{1}^{2}/c^{2}}}{\frac {1}{1-v_{2}^{2}/c^{2}}}(1+v_{1}v_{2}/c^{2})^{2}} 䞡蟺ã®éæ°ãåããšã 1 â v 3 2 / c 2 = ( 1 â v 1 2 / c 2 ) ( 1 â v 2 2 / c 2 ) 1 ( 1 + v 1 v 2 / c 2 ) 2 {\displaystyle 1-v_{3}^{2}/c^{2}=(1-v_{1}^{2}/c^{2})(1-v_{2}^{2}/c^{2}){\frac {1}{(1+v_{1}v_{2}/c^{2})^{2}}}} ãã£ãŠã ( v 3 / c ) 2 = 1 â 1 ( 1 + v 1 v 2 / c 2 ) 2 ( 1 â v 1 2 / c 2 ) ( 1 â v 2 2 / c 2 ) = 1 ( 1 + v 1 v 2 / c 2 ) 2 ( ( 1 + v 1 v 2 / c 2 ) 2 â ( 1 â v 1 2 / c 2 ) ( 1 â v 2 2 / c 2 ) ) = 1 ( 1 + v 1 v 2 / c 2 ) 2 ( 2 v 1 v 2 / c 2 â ( â v 1 2 / c 2 â v 2 2 / c 2 ) ) = 1 ( 1 + v 1 v 2 / c 2 ) 2 ( v 1 / c + v 2 / c ) 2 {\displaystyle {\begin{matrix}(v_{3}/c)^{2}=1-{\frac {1}{(1+v_{1}v_{2}/c^{2})^{2}}}(1-v_{1}^{2}/c^{2})(1-v_{2}^{2}/c^{2})\\={\frac {1}{(1+v_{1}v_{2}/c^{2})^{2}}}((1+v_{1}v_{2}/c^{2})^{2}-(1-v_{1}^{2}/c^{2})(1-v_{2}^{2}/c^{2}))\\={\frac {1}{(1+v_{1}v_{2}/c^{2})^{2}}}(2v_{1}v_{2}/c^{2}-(-v_{1}^{2}/c^{2}-v_{2}^{2}/c^{2}))\\={\frac {1}{(1+v_{1}v_{2}/c^{2})^{2}}}(v_{1}/c+v_{2}/c)^{2}\end{matrix}}}
ããããã v 3 / c = ( v 1 / c + v 2 / c ) 1 + v 1 v 2 / c 2 {\displaystyle v_{3}/c={\frac {(v_{1}/c+v_{2}/c)}{1+v_{1}v_{2}/c^{2}}}} ãåŸãããã ãã㧠v 2 = c {\displaystyle v_{2}=c} ãšãããšã v 3 / c = ( v 1 / c + 1 ) 1 + v 1 / c {\displaystyle v_{3}/c={\frac {(v_{1}/c+1)}{1+v_{1}/c}}} ã€ãŸãã v 3 = c {\displaystyle v_{3}=c} ãåŸãããã ããã¯ãããéã v 1 {\displaystyle v_{1}} ãæã£ã芳枬è
1ãã芳枬è
1ããèŠãŠ å
éã«è¿ãéãã§åãç©äœ2ãèŠããšããŠããéæ¢ãã芳枬è
ããèŠãç©äœ2ã®éã㯠å
écããéããªãããšã¯ç¡ããšããããšã瀺ããŠããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ç¹æ®çžå¯Ÿè« > é床ã®åæå",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ããé床 v 1 {\\displaystyle v_{1}} ãæã€ç©äœ1ããèŠããšãã« ããé床 v 2 {\\displaystyle v_{2}} ãæã€ç©äœ2ã ãéæ¢ããŠãã芳枬è
ããèŠããšãã® é床ãèšç®ããã (NewtonååŠã§ã¯ v 1 + v 2 {\\displaystyle v_{1}+v_{2}} ãšãªãããšã«æ³šæã)",
"title": "é床ã®åæå"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ããŒã¬ã³ã矀ã®æ§è³ªãã v 1 {\\displaystyle v_{1}} ã䜿ã£ãå€æãš v 1 {\\displaystyle v_{1}} ã䜿ã£ãå€æãåãããŠäœ¿ãããšã§ã éæ¢ãã芳枬è
ããèŠãå Žåã®ç©äœ2ã®é床ã æ±ãŸãããšãçšãããšã",
"title": "é床ã®åæå"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "γ 1 ( 1 β 1 β 1 1 ) à γ 2 ( 1 β 2 β 2 1 ) = γ 3 ( 1 β 3 β 3 1 ) {\\displaystyle \\gamma _{1}{\\begin{pmatrix}1&\\beta _{1}\\\\\\beta _{1}&1\\end{pmatrix}}\\times \\gamma _{2}{\\begin{pmatrix}1&\\beta _{2}\\\\\\beta _{2}&1\\end{pmatrix}}=\\gamma _{3}{\\begin{pmatrix}1&\\beta _{3}\\\\\\beta _{3}&1\\end{pmatrix}}} ãšãªãããšãåããã 巊蟺ã®1è¡1åæåãèšç®ãããšã = γ 1 γ 2 ( 1 + β 1 β 2 ) {\\displaystyle =\\gamma _{1}\\gamma _{2}(1+\\beta _{1}\\beta _{2})} ãšãªãããšããããã å³èŸºã®1è¡1åæåãšèŠããã¹ããšã γ 1 γ 2 ( 1 + β 1 β 2 ) = γ 3 {\\displaystyle \\gamma _{1}\\gamma _{2}(1+\\beta _{1}\\beta _{2})=\\gamma _{3}} ãåŸãããã 䞡蟺ã2ä¹ãããšã 1 1 â v 3 2 / c 2 = 1 1 â v 1 2 / c 2 1 1 â v 2 2 / c 2 ( 1 + v 1 v 2 / c 2 ) 2 {\\displaystyle {\\frac {1}{1-v_{3}^{2}/c^{2}}}={\\frac {1}{1-v_{1}^{2}/c^{2}}}{\\frac {1}{1-v_{2}^{2}/c^{2}}}(1+v_{1}v_{2}/c^{2})^{2}} 䞡蟺ã®éæ°ãåããšã 1 â v 3 2 / c 2 = ( 1 â v 1 2 / c 2 ) ( 1 â v 2 2 / c 2 ) 1 ( 1 + v 1 v 2 / c 2 ) 2 {\\displaystyle 1-v_{3}^{2}/c^{2}=(1-v_{1}^{2}/c^{2})(1-v_{2}^{2}/c^{2}){\\frac {1}{(1+v_{1}v_{2}/c^{2})^{2}}}} ãã£ãŠã ( v 3 / c ) 2 = 1 â 1 ( 1 + v 1 v 2 / c 2 ) 2 ( 1 â v 1 2 / c 2 ) ( 1 â v 2 2 / c 2 ) = 1 ( 1 + v 1 v 2 / c 2 ) 2 ( ( 1 + v 1 v 2 / c 2 ) 2 â ( 1 â v 1 2 / c 2 ) ( 1 â v 2 2 / c 2 ) ) = 1 ( 1 + v 1 v 2 / c 2 ) 2 ( 2 v 1 v 2 / c 2 â ( â v 1 2 / c 2 â v 2 2 / c 2 ) ) = 1 ( 1 + v 1 v 2 / c 2 ) 2 ( v 1 / c + v 2 / c ) 2 {\\displaystyle {\\begin{matrix}(v_{3}/c)^{2}=1-{\\frac {1}{(1+v_{1}v_{2}/c^{2})^{2}}}(1-v_{1}^{2}/c^{2})(1-v_{2}^{2}/c^{2})\\\\={\\frac {1}{(1+v_{1}v_{2}/c^{2})^{2}}}((1+v_{1}v_{2}/c^{2})^{2}-(1-v_{1}^{2}/c^{2})(1-v_{2}^{2}/c^{2}))\\\\={\\frac {1}{(1+v_{1}v_{2}/c^{2})^{2}}}(2v_{1}v_{2}/c^{2}-(-v_{1}^{2}/c^{2}-v_{2}^{2}/c^{2}))\\\\={\\frac {1}{(1+v_{1}v_{2}/c^{2})^{2}}}(v_{1}/c+v_{2}/c)^{2}\\end{matrix}}}",
"title": "é床ã®åæå"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ããããã v 3 / c = ( v 1 / c + v 2 / c ) 1 + v 1 v 2 / c 2 {\\displaystyle v_{3}/c={\\frac {(v_{1}/c+v_{2}/c)}{1+v_{1}v_{2}/c^{2}}}} ãåŸãããã ãã㧠v 2 = c {\\displaystyle v_{2}=c} ãšãããšã v 3 / c = ( v 1 / c + 1 ) 1 + v 1 / c {\\displaystyle v_{3}/c={\\frac {(v_{1}/c+1)}{1+v_{1}/c}}} ã€ãŸãã v 3 = c {\\displaystyle v_{3}=c} ãåŸãããã ããã¯ãããéã v 1 {\\displaystyle v_{1}} ãæã£ã芳枬è
1ãã芳枬è
1ããèŠãŠ å
éã«è¿ãéãã§åãç©äœ2ãèŠããšããŠããéæ¢ãã芳枬è
ããèŠãç©äœ2ã®éã㯠å
écããéããªãããšã¯ç¡ããšããããšã瀺ããŠããã",
"title": "é床ã®åæå"
}
] | ç¹æ®çžå¯Ÿè« > é床ã®åæå | <small> [[ç¹æ®çžå¯Ÿè«]] > é床ã®åæå</small>
----
==é床ã®åæå==
ããé床<math>v _1</math>ãæã€ç©äœ1ããèŠããšãã«
ããé床<math>v _2</math>ãæã€ç©äœ2ã
ãéæ¢ããŠãã芳枬è
ããèŠããšãã®
é床ãèšç®ããã
(NewtonååŠã§ã¯ <math>v _1 +v _2</math>ãšãªãããšã«æ³šæã)
ããŒã¬ã³ã矀ã®æ§è³ªãã
<math>v _1</math>ã䜿ã£ãå€æãš
<math>v _1</math>ã䜿ã£ãå€æãåãããŠäœ¿ãããšã§ã
éæ¢ãã芳枬è
ããèŠãå Žåã®ç©äœ2ã®é床ã
æ±ãŸãããšãçšãããšã
<math>
\gamma _1
\begin{pmatrix}
1&\beta _1\\
\beta _1&1
\end{pmatrix}
\times
\gamma _2
\begin{pmatrix}
1&\beta _2\\
\beta _2&1
\end{pmatrix}
=
\gamma _3
\begin{pmatrix}
1&\beta _3\\
\beta _3&1
\end{pmatrix}
</math>
ãšãªãããšãåããã
巊蟺ã®1è¡1åæåãèšç®ãããšã
<math>
= \gamma _1 \gamma _2 (1+\beta _1\beta _2)
</math>
ãšãªãããšããããã
å³èŸºã®1è¡1åæåãšèŠããã¹ããšã
<math>
\gamma _1 \gamma _2 (1+\beta _1\beta _2) = \gamma _3
</math>
ãåŸãããã
䞡蟺ã2ä¹ãããšã
<math>
\frac 1 {1 - v _3^2/c^2} = \frac 1 {1 - v _1^2/c^2}\frac 1 {1 - v _2^2/c^2}
(1+ v _1 v _2 /c^2)^2
</math>
䞡蟺ã®éæ°ãåããšã
<math>
1 - v _3^2/c^2 = (1 - v _1^2/c^2)(1 - v _2^2/c^2)
\frac 1 {(1+ v _1 v _2 /c^2)^2}
</math>
ãã£ãŠã
<math>
\begin{matrix}
(v _3/c )^2 =
1 - \frac 1 {(1+ v _1 v _2 /c^2)^2} (1 - v _1^2/c^2)(1 - v _2^2/c^2)\\
= \frac 1 {(1+ v _1 v _2 /c^2)^2}
((1+ v _1 v _2 /c^2)^2- (1 - v _1^2/c^2)(1 - v _2^2/c^2))\\
=\frac 1 {(1+ v _1 v _2 /c^2)^2}(2 v _1 v _2 /c^2 -(- v _1^2 /c^2 - v _2^2 /c^2 ))\\
=\frac 1 {(1+ v _1 v _2 /c^2)^2}(v _1/c + v _2/c ) ^2
\end{matrix}
</math>
ããããã
<math>
v _3 /c = \frac {( v _1/c + v _2/c )} {1+ v _1 v _2 /c^2}
</math>
ãåŸãããã
ããã§<math>v _2=c</math>ãšãããšã
<math>
v _3 /c = \frac {( v _1/c + 1 )} {1+ v _1 /c}
</math>
ã€ãŸãã
<math>
v _3 = c
</math>
ãåŸãããã
ããã¯ãããéã<math>v _1</math>ãæã£ã芳枬è
1ãã芳枬è
1ããèŠãŠ
å
éã«è¿ãéãã§åãç©äœ2ãèŠããšããŠããéæ¢ãã芳枬è
ããèŠãç©äœ2ã®éãã¯
å
écããéããªãããšã¯ç¡ããšããããšã瀺ããŠããã
[[Category:ç¹æ®çžå¯Ÿè«|ãããšã®ãããããã]] | 2005-05-24T13:10:27Z | 2024-03-16T03:17:21Z | [] | https://ja.wikibooks.org/wiki/%E7%89%B9%E6%AE%8A%E7%9B%B8%E5%AF%BE%E8%AB%96_%E9%80%9F%E5%BA%A6%E3%81%AE%E5%90%88%E6%88%90%E5%89%87 |
2,048 | ææ©ååŠ/ã·ã¯ãã¢ã«ã«ã³ | ææ©ååŠ>ã·ã¯ãã¢ã«ã«ã³
ã·ã¯ãã¢ã«ã«ã³(cycloalkanes)ã¯äžè¬åŒCnH2nã§è¡šãããç°åŒäžé£œåçåæ°ŽçŽ ã®ç·ç§°ã§ãããã·ã¯ãã¢ã«ã«ã³ã¯C-Cçµåããã¹ãŠåçµåã§ããããšãããäžè¬ã«ã¢ã«ã«ã³ã«äŒŒãæ§è³ªã瀺ããäžè¬åŒãåãã¢ã«ã±ã³ãšã¯ç°ãªããã·ã¯ãã¢ã«ã«ã³ã¯ä»å åå¿ããªããåœåã«ã¯åãççŽ æ°ã®ã¢ã«ã«ã³ã®åã«ãã·ã¯ã(cyclo-)ããã€ãããã·ã¯ãã¢ã«ã«ã³ãæãããããªnåã®ååã§æ§æãããç°ã¯äžè¬ã«nå¡ç°(äžå¡ç°ãåå¡ç°ã...)ãšåŒã°ãããäŸãã°ãã·ã¯ããããã³(n=3)ã¯äžå¡ç°ã§ããã
ååè»éã®æ··æçè«ã«ãããšãççŽ ååã«4åã®åå(矀)ãçµåããææ©ååç©ã«ãããŠãççŽ ååäžã®4åã®çµåã¯spæ··æè»éãšåŒã°ãã4ã€ã®æ··æè»éãšããŠè¡šçŸããããããã«ãããã¡ã¿ã³(CH4)ãªã©ã«ã¿ãããæã察称æ§ãé«ãspæ··æè»éã§ã¯ãççŽ ååãéå¿ãšããŠæ£åé¢äœã®åé ç¹ãžäŒžã³ããæ£åé¢äœåœ¢ãã®åå䟡ç¶æ
ãçŸããããã®ãšãã®çæ³çãªspæ··æè»éã®çµåè§ã¯cos(1/3)â109.5°ãšèšç®ããããã·ã¯ãã¢ã«ã«ã³äžã®ççŽ ååã¯ãã¹ãŠspæ··æã§ããã109.5°ã倧ããé¢ããçµåè§ãæããã·ã¯ãã¢ã«ã«ã³ã¯å€§ããªç°ã²ãã¿ã«ããäžå®å®ã«ãªãã
ã·ã¯ããããã³ä»¥å€ã®ã·ã¯ãã¢ã«ã«ã³ã¯åäžå¹³é¢äžã«å
šãŠã®ççŽ ååãååšããæ§é ããšããªããããç«äœé
座ãèæ
®ãããããšãå€ããã·ã¯ããã³ã¿ã³(n=5)ãšã·ã¯ããããµã³(n=6)ãæ¯èŒãããšãæ£äºè§åœ¢ã®å
è§ã¯108°ãæ£å
è§åœ¢ã®å
è§ã¯120°ã§ãããå¹³é¢ååã§ãããªãã°ã·ã¯ããã³ã¿ã³ãããå®å®ã§ãããšäºæ³ããããããããå®éã«ã¯ã·ã¯ããããµã³ãã»ãŒçæ³çãªåœ¢ç¶ã®spæ··æè»éãæããŠãããã·ã¯ããã³ã¿ã³ãããå®å®ãšãªããn=3â10çšåºŠã®ã·ã¯ãã¢ã«ã«ã³ãæ¯èŒãããšn=6ã«è¿ããã®ã»ã©å®å®ã§ããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ææ©ååŠ>ã·ã¯ãã¢ã«ã«ã³",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ã·ã¯ãã¢ã«ã«ã³(cycloalkanes)ã¯äžè¬åŒCnH2nã§è¡šãããç°åŒäžé£œåçåæ°ŽçŽ ã®ç·ç§°ã§ãããã·ã¯ãã¢ã«ã«ã³ã¯C-Cçµåããã¹ãŠåçµåã§ããããšãããäžè¬ã«ã¢ã«ã«ã³ã«äŒŒãæ§è³ªã瀺ããäžè¬åŒãåãã¢ã«ã±ã³ãšã¯ç°ãªããã·ã¯ãã¢ã«ã«ã³ã¯ä»å åå¿ããªããåœåã«ã¯åãççŽ æ°ã®ã¢ã«ã«ã³ã®åã«ãã·ã¯ã(cyclo-)ããã€ãããã·ã¯ãã¢ã«ã«ã³ãæãããããªnåã®ååã§æ§æãããç°ã¯äžè¬ã«nå¡ç°(äžå¡ç°ãåå¡ç°ã...)ãšåŒã°ãããäŸãã°ãã·ã¯ããããã³(n=3)ã¯äžå¡ç°ã§ããã",
"title": "ã·ã¯ãã¢ã«ã«ã³ã®å®çŸ©"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ååè»éã®æ··æçè«ã«ãããšãççŽ ååã«4åã®åå(矀)ãçµåããææ©ååç©ã«ãããŠãççŽ ååäžã®4åã®çµåã¯spæ··æè»éãšåŒã°ãã4ã€ã®æ··æè»éãšããŠè¡šçŸããããããã«ãããã¡ã¿ã³(CH4)ãªã©ã«ã¿ãããæã察称æ§ãé«ãspæ··æè»éã§ã¯ãççŽ ååãéå¿ãšããŠæ£åé¢äœã®åé ç¹ãžäŒžã³ããæ£åé¢äœåœ¢ãã®åå䟡ç¶æ
ãçŸããããã®ãšãã®çæ³çãªspæ··æè»éã®çµåè§ã¯cos(1/3)â109.5°ãšèšç®ããããã·ã¯ãã¢ã«ã«ã³äžã®ççŽ ååã¯ãã¹ãŠspæ··æã§ããã109.5°ã倧ããé¢ããçµåè§ãæããã·ã¯ãã¢ã«ã«ã³ã¯å€§ããªç°ã²ãã¿ã«ããäžå®å®ã«ãªãã",
"title": "å®å®æ§ãšç«äœé
座"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ã·ã¯ããããã³ä»¥å€ã®ã·ã¯ãã¢ã«ã«ã³ã¯åäžå¹³é¢äžã«å
šãŠã®ççŽ ååãååšããæ§é ããšããªããããç«äœé
座ãèæ
®ãããããšãå€ããã·ã¯ããã³ã¿ã³(n=5)ãšã·ã¯ããããµã³(n=6)ãæ¯èŒãããšãæ£äºè§åœ¢ã®å
è§ã¯108°ãæ£å
è§åœ¢ã®å
è§ã¯120°ã§ãããå¹³é¢ååã§ãããªãã°ã·ã¯ããã³ã¿ã³ãããå®å®ã§ãããšäºæ³ããããããããå®éã«ã¯ã·ã¯ããããµã³ãã»ãŒçæ³çãªåœ¢ç¶ã®spæ··æè»éãæããŠãããã·ã¯ããã³ã¿ã³ãããå®å®ãšãªããn=3â10çšåºŠã®ã·ã¯ãã¢ã«ã«ã³ãæ¯èŒãããšn=6ã«è¿ããã®ã»ã©å®å®ã§ããã",
"title": "å®å®æ§ãšç«äœé
座"
}
] | ææ©ååŠïŒã·ã¯ãã¢ã«ã«ã³ | [[ææ©ååŠ]]ïŒã·ã¯ãã¢ã«ã«ã³
== ã·ã¯ãã¢ã«ã«ã³ã®å®çŸ© ==
ã·ã¯ãã¢ã«ã«ã³(cycloalkanes)ã¯[[ææ©ååŠ_ã¢ã«ã«ã³#ã¢ã«ã«ã³ã®äžè¬åŒ|äžè¬åŒ]]C<sub>n</sub>H<sub>2n</sub>ã§è¡šããã[[ææ©ååŠ#ææ©ååç©ã®åé¡|ç°åŒäžé£œåçåæ°ŽçŽ ]]ã®ç·ç§°ã§ãããã·ã¯ãã¢ã«ã«ã³ã¯C-Cçµåããã¹ãŠåçµåã§ããããšãããäžè¬ã«[[ææ©ååŠ_ã¢ã«ã«ã³#ã¢ã«ã«ã³ã®æ§è³ª|ã¢ã«ã«ã³]]ã«äŒŒãæ§è³ªã瀺ããäžè¬åŒãåã[[ææ©ååŠ_ã¢ã«ã±ã³#ã¢ã«ã±ã³ã®å®çŸ©|ã¢ã«ã±ã³]]ãšã¯ç°ãªããã·ã¯ãã¢ã«ã«ã³ã¯[[ææ©ååŠ_ã¢ã«ã±ã³#ä»å åå¿|ä»å åå¿]]ããªããåœåã«ã¯åãççŽ æ°ã®[[ææ©ååŠ_ã¢ã«ã«ã³#åœåæ³|ã¢ã«ã«ã³]]ã®åã«ãã·ã¯ã(cyclo-)ããã€ãããã·ã¯ãã¢ã«ã«ã³ãæãããããªnåã®ååã§æ§æãããç°ã¯äžè¬ã«nå¡ç°(äžå¡ç°ãåå¡ç°ãâŠ)ãšåŒã°ãããäŸãã°ãã·ã¯ããããã³(n=3)ã¯äžå¡ç°ã§ããã
CH2 CH2-CH2
/ \ | |
CH2-CH2 CH2-CH2
ã·ã¯ããããã³ ã·ã¯ããã¿ã³
CH2 CH2-CH2
/ \ / \
CH2 CH2 CH2 CH2
\ / \ /
CH2-CH2 CH2-CH2
ã·ã¯ããã³ã¿ã³ ã·ã¯ããããµã³
== å®å®æ§ãšç«äœé
座 ==
ååè»éã®æ··æçè«ã«ãããšãççŽ ååã«4åã®ååïŒçŸ€ïŒãçµåããææ©ååç©ã«ãããŠãççŽ ååäžã®4åã®çµåã¯sp<sup>3</sup>æ··æè»éãšåŒã°ãã4ã€ã®æ··æè»éãšããŠè¡šçŸããããããã«ãããã¡ã¿ã³(CH<sub>4</sub>)ãªã©ã«ã¿ãããæã察称æ§ãé«ãsp<sup>3</sup>æ··æè»éã§ã¯ãççŽ ååã[[w:éå¿|éå¿]]ãšããŠ[[w:æ£åé¢äœ|æ£åé¢äœ]]ã®åé ç¹ãžäŒžã³ããæ£åé¢äœåœ¢ãã®åå䟡ç¶æ
ãçŸããããã®ãšãã®çæ³çãªsp<sup>3</sup>æ··æè»éã®çµåè§ã¯cos<sup>â1</sup>(1/3)â109.5°ãšèšç®ããããã·ã¯ãã¢ã«ã«ã³äžã®ççŽ ååã¯ãã¹ãŠsp<sup>3</sup>æ··æã§ããã109.5°ã倧ããé¢ããçµåè§ãæããã·ã¯ãã¢ã«ã«ã³ã¯å€§ããª[[ç°ã²ãã¿|ç°ã²ãã¿]]ã«ããäžå®å®ã«ãªãã
ã·ã¯ããããã³ä»¥å€ã®ã·ã¯ãã¢ã«ã«ã³ã¯åäžå¹³é¢äžã«å
šãŠã®ççŽ ååãååšããæ§é ããšããªãããã[[ç«äœé
座]]ãèæ
®ãããããšãå€ããã·ã¯ããã³ã¿ã³(n=5)ãšã·ã¯ããããµã³(n=6)ãæ¯èŒãããšãæ£äºè§åœ¢ã®å
è§ã¯108°ãæ£å
è§åœ¢ã®å
è§ã¯120°ã§ãããå¹³é¢ååã§ãããªãã°ã·ã¯ããã³ã¿ã³ãããå®å®ã§ãããšäºæ³ããããããããå®éã«ã¯ã·ã¯ããããµã³ãã»ãŒçæ³çãªåœ¢ç¶ã®sp<sup>3</sup>æ··æè»éãæããŠãããã·ã¯ããã³ã¿ã³ãããå®å®ãšãªããn=3â10çšåºŠã®ã·ã¯ãã¢ã«ã«ã³ãæ¯èŒãããšn=6ã«è¿ããã®ã»ã©å®å®ã§ããã
[[ã«ããŽãª:ææ©ååŠ]]
[[en:Organic Chemistry/Cycloalkanes]] | null | 2022-11-23T05:33:17Z | [] | https://ja.wikibooks.org/wiki/%E6%9C%89%E6%A9%9F%E5%8C%96%E5%AD%A6/%E3%82%B7%E3%82%AF%E3%83%AD%E3%82%A2%E3%83%AB%E3%82%AB%E3%83%B3 |
2,049 | ææ©ååŠ/ã·ã¯ãã¢ã«ã±ã³ | ææ©ååŠ>ã·ã¯ãã¢ã«ã±ã³
äºéçµåãã²ãšã€ã ãæã€èç°åŒçåæ°ŽçŽ ã§ããã æ§è³ªã¯ã¢ã«ã±ã³ã«äŒŒãŠããã äžè¬åŒã¯CnH2n-2ã§ãããã¢ã«ãã³ãšåãã§ãããããããã·ã¯ãã¢ã«ã±ã³ã¯çœ®æåå¿ããªãã®ã§åºå¥ã§ããã åœåã¯ã¢ã«ã±ã³ã®åã«ãã·ã¯ã(cyclo)ããã€ããã ã·ã¯ãã¢ã«ã«ã³ãšåãããã·ã¯ããã³ãã³ãšã·ã¯ãããã»ã³ãå®å®ã§ããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ææ©ååŠ>ã·ã¯ãã¢ã«ã±ã³",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "äºéçµåãã²ãšã€ã ãæã€èç°åŒçåæ°ŽçŽ ã§ããã æ§è³ªã¯ã¢ã«ã±ã³ã«äŒŒãŠããã äžè¬åŒã¯CnH2n-2ã§ãããã¢ã«ãã³ãšåãã§ãããããããã·ã¯ãã¢ã«ã±ã³ã¯çœ®æåå¿ããªãã®ã§åºå¥ã§ããã åœåã¯ã¢ã«ã±ã³ã®åã«ãã·ã¯ã(cyclo)ããã€ããã ã·ã¯ãã¢ã«ã«ã³ãšåãããã·ã¯ããã³ãã³ãšã·ã¯ãããã»ã³ãå®å®ã§ããã",
"title": "ã·ã¯ãã¢ã«ã±ã³ã®å®çŸ©ãšæ§è³ª"
}
] | ææ©ååŠïŒã·ã¯ãã¢ã«ã±ã³ | [[ææ©ååŠ]]ïŒã·ã¯ãã¢ã«ã±ã³
==ã·ã¯ãã¢ã«ã±ã³ã®å®çŸ©ãšæ§è³ª==
äºéçµåãã²ãšã€ã ãæã€èç°åŒçåæ°ŽçŽ ã§ããã
æ§è³ªã¯[[ææ©ååŠ_ã¢ã«ã±ã³#ã¢ã«ã±ã³ã®æ§è³ª|ã¢ã«ã±ã³]]ã«äŒŒãŠããã
[[ææ©ååŠ_ã¢ã«ã«ã³#ã¢ã«ã«ã³ã®äžè¬åŒ|äžè¬åŒ]]ã¯C<sub>n</sub>H<sub>2n-2</sub>ã§ããã[[ææ©ååŠ_ã¢ã«ãã³#ã¢ã«ãã³ã®å®çŸ©|ã¢ã«ãã³]]ãšåãã§ãããããããã·ã¯ãã¢ã«ã±ã³ã¯[[ææ©ååŠ_ã¢ã«ã«ã³#眮æåå¿|眮æåå¿]]ããªãã®ã§åºå¥ã§ããã
åœåã¯[[ææ©ååŠ_ã¢ã«ã±ã³#åœåæ³|ã¢ã«ã±ã³]]ã®åã«ãã·ã¯ã(cyclo)ããã€ããã
[[ææ©ååŠ_ã·ã¯ãã¢ã«ã«ã³#å®å®æ§|ã·ã¯ãã¢ã«ã«ã³]]ãšåãããã·ã¯ããã³ãã³ãšã·ã¯ãããã»ã³ãå®å®ã§ããã
[[ã«ããŽãª:ææ©ååŠ]] | null | 2022-11-23T05:33:20Z | [] | https://ja.wikibooks.org/wiki/%E6%9C%89%E6%A9%9F%E5%8C%96%E5%AD%A6/%E3%82%B7%E3%82%AF%E3%83%AD%E3%82%A2%E3%83%AB%E3%82%B1%E3%83%B3 |
2,054 | åçæ°åŠå
¬åŒé/åç幟äœ/äœç© | ãã®ããŒãžã§ã¯äœç©ã®å
¬åŒã®è§£èª¬ãããŸãã
V = abh
V = a
V = Sh
V = 1 3 S h {\displaystyle V={\frac {1}{3}}Sh}
éäœã®é ç¹ããåºé¢ S {\displaystyle S} (å³å³ã§ã¯ A b {\displaystyle A_{b}} )ã«åç·ãäžããŠãé ç¹ãã x ( 0 †x †h ) {\displaystyle x(0\leq x\leq h)} ã®è·é¢ã§åºé¢ãšå¹³è¡ã«éäœãåãåã£ãããšã§åŸãããå³åœ¢ã A x {\displaystyle A_{x}} ãšããã
ãã®æãéäœã®å®çŸ©ããã S {\displaystyle S} ãš A x {\displaystyle A_{x}} ã¯çžäŒŒã§ããã
çžäŒŒãªå³åœ¢ã®é¢ç©æ¯ã¯ãçžäŒŒæ¯ã®2ä¹ã«çããããšããã
S : A x = h 2 : x 2 {\displaystyle S:A_{x}=h^{2}:x^{2}}
åŸã£ãŠã
A x = x 2 S h 2 {\displaystyle A_{x}={\frac {x^{2}S}{h^{2}}}}
éäœã®äœç©ã¯ãå¹³é¢å³åœ¢ A x {\displaystyle A_{x}} ã«é¢ããŠã 0 †x †h {\displaystyle 0\leq x\leq h} ã®åºéã§å€åãã环ç©ãããã®ã§ããããã A x {\displaystyle A_{x}} ãåºé [ 0 , h ] {\displaystyle [0,h]} ã§ç©åããããšã«ããåŸãããã
V = â« 0 h A x d x {\displaystyle V=\int _{0}^{h}A_{x}\,dx} = â« 0 h x 2 S h 2 d x {\displaystyle =\int _{0}^{h}{\frac {x^{2}S}{h^{2}}}\,dx} = S h 2 â« 0 h x 2 d x {\displaystyle ={\frac {S}{h^{2}}}\int _{0}^{h}x^{2}\,dx} = S h 2 [ x 3 3 ] 0 h {\displaystyle ={\frac {S}{h^{2}}}\left[{\frac {x^{3}}{3}}\right]_{0}^{h}} = S h 2 ( h 3 3 ) {\displaystyle ={\frac {S}{h^{2}}}\left({\frac {h^{3}}{3}}\right)} = 1 3 S h {\displaystyle ={\frac {1}{3}}Sh}
äžåºã®é¢ç© s {\displaystyle s} (å³å³ã§ã¯ A 2 {\displaystyle A_{2}} )ãäžåºã®é¢ç© S {\displaystyle S} (å³å³ã§ã¯ A 1 {\displaystyle A_{1}} )ãé«ã h {\displaystyle h} ã®éå°ã®äœç© V {\displaystyle V}
éå°ã¯ãå¥åãåé éäœãã®ãšããã S {\displaystyle S} ãåºãšããéäœ: P 1 {\displaystyle P_{1}} ããã s {\displaystyle s} ãåºãšããçžäŒŒãªéäœ: P 2 {\displaystyle P_{2}} ãé€ãããã®ãšãããã
éäœ: P 1 {\displaystyle P_{1}} ã®é«ãã H {\displaystyle H} ãšãããšãéäœ: P 2 {\displaystyle P_{2}} ã®é«ã㯠H â h {\displaystyle H-h} ãšãªããåã
ã®äœç©ã¯ã
çžäŒŒæ¯ãšé¢ç©æ¯ã®é¢ä¿ããã
åŸã£ãŠã
ããããâ»ã«ä»£å
¥ãããšã以äžã®åŒãåŸãã
ããã³åœ¢ã®äžèŸºããåºé¢ã«åç·ãäžããŠãé ç¹ãã x ( 0 †x †h ) {\displaystyle x(0\leq x\leq h)} ã®è·é¢ã§åºé¢ãšå¹³è¡ã«ããã³åœ¢ãåãåã£ãããšã§åŸãããå³åœ¢(é·æ¹åœ¢)ã S x {\displaystyle S_{x}} ãšããã
ãã®é·æ¹åœ¢ã®çžŠæšªã¯æ¯äŸã®é¢ä¿ãã以äžã®ãšãããšãªãã
ããã³åœ¢ã®äœç©ã¯ãå¹³é¢å³åœ¢ S x {\displaystyle S_{x}} ã«é¢ããŠã 0 †x †h {\displaystyle 0\leq x\leq h} ã®åºéã§å€åãã环ç©ãããã®ã§ããããã S x {\displaystyle S_{x}} ãåºé [ 0 , h ] {\displaystyle [0,h]} ã§ç©åããããšã«ããåŸãããã
V = â« 0 h S x d x {\displaystyle V=\int _{0}^{h}S_{x}\,dx} = â« 0 h ( ( a â c ) b x 2 h 2 + b c x h ) d x {\displaystyle =\int _{0}^{h}\left({\frac {(a-c)bx^{2}}{h^{2}}}+{\frac {bcx}{h}}\right)dx} = b h 2 â« 0 h ( ( a â c ) x 2 + c h x ) d x {\displaystyle ={\frac {b}{h^{2}}}\int _{0}^{h}((a-c)x^{2}+chx)dx} = b h 2 [ ( a â c ) x 3 3 + c h x 2 2 ] 0 h {\displaystyle ={\frac {b}{h^{2}}}\left[{\frac {(a-c)x^{3}}{3}}+{\frac {chx^{2}}{2}}\right]_{0}^{h}} = b h 2 ( ( a â c ) h 3 3 + c h 3 2 ) {\displaystyle ={\frac {b}{h^{2}}}\left({\frac {(a-c)h^{3}}{3}}+{\frac {ch^{3}}{2}}\right)} = b h ( a 3 + c 6 ) {\displaystyle =bh\left({\frac {a}{3}}+{\frac {c}{6}}\right)}
V = 2 12 a 3 {\displaystyle V={\frac {\sqrt {2}}{12}}a^{3}}
æ£åé¢äœã®äœç©ã¯ãç«æ¹äœãšã®é¢ä¿ãããå°åºããããšãã§ããŸãã ç«æ¹äœãšé ç¹ãå
±æããæ£åé¢äœã¯ãå
šãŠã®èŸºãç«æ¹äœã®é¢ã®å¯Ÿè§ç·ã«ãªã£ãŠããŸãã ãã£ãŠãç«æ¹äœããäœã£ãäœç©ãåŒãã°ãæ£åé¢äœã®äœç©ãå°ãåºãããšãã§ããŸãã
æ£åé¢äœã®1蟺ã®é·ããaãšããŸãã äœã£ãéšåã¯å
šéšã§4ã€ãããŸããã蟺ã®é·ãã¯å
šãŠããããçããã®ã§ããããã¯ååã«ãªããŸãã
ç«æ¹äœã®1蟺ã®é·ãã¯ãæ£æ¹åœ¢ã®èŸºãšå¯Ÿè§ç·ã®é·ãã®æ¯ã 1 : 2 {\displaystyle 1:{\sqrt {2}}} ãããã
äœã£ãéšåã¯äžè§éãšã¿ãªãããšãã§ããã®ã§ãè§éã®äœç©ããã
æåŸã«ç«æ¹äœããè§é4ã€ãåŒããŸãã
V = 2 3 a 3 {\displaystyle V={\frac {\sqrt {2}}{3}}a^{3}}
æ£å
«é¢äœã¯ãäœç©ã®çããæ£åè§éã2ã€ãããšèŠãããšãã§ããŸãã ãããã®è§éã®é«ãã¯ãè§éã®åºé¢ã®å¯Ÿè§ç·ã®äº€ç¹ããæ±ããããšãã§ããŸãã åºé¢ã«å¯Ÿããé äžã®é ç¹ãšåºé¢ã®å¯Ÿè§ç·ã®äº€ç¹ãçµã¶çŽç·ã¯åçŽã«ãªãã®ã§ã é«ãã¯ãè§éã®æ¯ç·ãšå¯Ÿè§ç·ãããäžå¹³æ¹ã®å®çã§å°åºã§ããŸãã
察è§ç·ã®é·ãã¯ã
察è§ç·ã¯äºãã®äžç¹ã§äº€ããã®ã§ã
é«ãã¯ãæ¯ç·ãšå¯Ÿè§ç·ã®ååããã
å®ã¯ãæ£å
«é¢äœã¯ã©ãã§æ£åè§é2ã€ã«åé¢ããŠããé«ãã¯åäžã§ããããã察è§ç·ã®ååãæ¢ã«é«ãã«ãªã£ãŠããŸãã æåŸã«ãéäœã®äœç©ã®å
¬åŒããã
V = 15 + 7 5 4 a 3 {\displaystyle V={\frac {15+7{\sqrt {5}}}{4}}a^{3}}
V = 5 ( 3 + 5 ) 12 a 3 {\displaystyle V={\frac {5(3+{\sqrt {5}})}{12}}a^{3}}
V = 4 3 Ï r 3 {\displaystyle V={\frac {4}{3}}\pi r^{3}}
ååŸ r {\displaystyle r} ã®å; C {\displaystyle C} ããåã®äžå¿ããã®è·é¢ R {\displaystyle R} (äœãã r {\displaystyle r} ⊠R {\displaystyle R} ãšãã)ã®çŽç·ã軞ãšããŠå転ãããåç°äœ(ããŒã©ã¹ãããŒããå)
(解æ³) | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ãã®ããŒãžã§ã¯äœç©ã®å
¬åŒã®è§£èª¬ãããŸãã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "V = abh",
"title": "çŽæ¹äœã®äœç©"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "V = a",
"title": "ç«æ¹äœã®äœç©"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "V = Sh",
"title": "æ±äœã®äœç©"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "V = 1 3 S h {\\displaystyle V={\\frac {1}{3}}Sh}",
"title": "éäœã®äœç©"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "éäœã®é ç¹ããåºé¢ S {\\displaystyle S} (å³å³ã§ã¯ A b {\\displaystyle A_{b}} )ã«åç·ãäžããŠãé ç¹ãã x ( 0 †x †h ) {\\displaystyle x(0\\leq x\\leq h)} ã®è·é¢ã§åºé¢ãšå¹³è¡ã«éäœãåãåã£ãããšã§åŸãããå³åœ¢ã A x {\\displaystyle A_{x}} ãšããã",
"title": "éäœã®äœç©"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ãã®æãéäœã®å®çŸ©ããã S {\\displaystyle S} ãš A x {\\displaystyle A_{x}} ã¯çžäŒŒã§ããã",
"title": "éäœã®äœç©"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "çžäŒŒãªå³åœ¢ã®é¢ç©æ¯ã¯ãçžäŒŒæ¯ã®2ä¹ã«çããããšããã",
"title": "éäœã®äœç©"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "S : A x = h 2 : x 2 {\\displaystyle S:A_{x}=h^{2}:x^{2}}",
"title": "éäœã®äœç©"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "åŸã£ãŠã",
"title": "éäœã®äœç©"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "A x = x 2 S h 2 {\\displaystyle A_{x}={\\frac {x^{2}S}{h^{2}}}}",
"title": "éäœã®äœç©"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "éäœã®äœç©ã¯ãå¹³é¢å³åœ¢ A x {\\displaystyle A_{x}} ã«é¢ããŠã 0 †x †h {\\displaystyle 0\\leq x\\leq h} ã®åºéã§å€åãã环ç©ãããã®ã§ããããã A x {\\displaystyle A_{x}} ãåºé [ 0 , h ] {\\displaystyle [0,h]} ã§ç©åããããšã«ããåŸãããã",
"title": "éäœã®äœç©"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "V = â« 0 h A x d x {\\displaystyle V=\\int _{0}^{h}A_{x}\\,dx} = â« 0 h x 2 S h 2 d x {\\displaystyle =\\int _{0}^{h}{\\frac {x^{2}S}{h^{2}}}\\,dx} = S h 2 â« 0 h x 2 d x {\\displaystyle ={\\frac {S}{h^{2}}}\\int _{0}^{h}x^{2}\\,dx} = S h 2 [ x 3 3 ] 0 h {\\displaystyle ={\\frac {S}{h^{2}}}\\left[{\\frac {x^{3}}{3}}\\right]_{0}^{h}} = S h 2 ( h 3 3 ) {\\displaystyle ={\\frac {S}{h^{2}}}\\left({\\frac {h^{3}}{3}}\\right)} = 1 3 S h {\\displaystyle ={\\frac {1}{3}}Sh}",
"title": "éäœã®äœç©"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "äžåºã®é¢ç© s {\\displaystyle s} (å³å³ã§ã¯ A 2 {\\displaystyle A_{2}} )ãäžåºã®é¢ç© S {\\displaystyle S} (å³å³ã§ã¯ A 1 {\\displaystyle A_{1}} )ãé«ã h {\\displaystyle h} ã®éå°ã®äœç© V {\\displaystyle V}",
"title": "éå°ã®äœç©"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "éå°ã¯ãå¥åãåé éäœãã®ãšããã S {\\displaystyle S} ãåºãšããéäœ: P 1 {\\displaystyle P_{1}} ããã s {\\displaystyle s} ãåºãšããçžäŒŒãªéäœ: P 2 {\\displaystyle P_{2}} ãé€ãããã®ãšãããã",
"title": "éå°ã®äœç©"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "éäœ: P 1 {\\displaystyle P_{1}} ã®é«ãã H {\\displaystyle H} ãšãããšãéäœ: P 2 {\\displaystyle P_{2}} ã®é«ã㯠H â h {\\displaystyle H-h} ãšãªããåã
ã®äœç©ã¯ã",
"title": "éå°ã®äœç©"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "çžäŒŒæ¯ãšé¢ç©æ¯ã®é¢ä¿ããã",
"title": "éå°ã®äœç©"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "åŸã£ãŠã",
"title": "éå°ã®äœç©"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "ããããâ»ã«ä»£å
¥ãããšã以äžã®åŒãåŸãã",
"title": "éå°ã®äœç©"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "ããã³åœ¢ã®äžèŸºããåºé¢ã«åç·ãäžããŠãé ç¹ãã x ( 0 †x †h ) {\\displaystyle x(0\\leq x\\leq h)} ã®è·é¢ã§åºé¢ãšå¹³è¡ã«ããã³åœ¢ãåãåã£ãããšã§åŸãããå³åœ¢(é·æ¹åœ¢)ã S x {\\displaystyle S_{x}} ãšããã",
"title": "ããã³åœ¢ã®äœç©"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "ãã®é·æ¹åœ¢ã®çžŠæšªã¯æ¯äŸã®é¢ä¿ãã以äžã®ãšãããšãªãã",
"title": "ããã³åœ¢ã®äœç©"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "ããã³åœ¢ã®äœç©ã¯ãå¹³é¢å³åœ¢ S x {\\displaystyle S_{x}} ã«é¢ããŠã 0 †x †h {\\displaystyle 0\\leq x\\leq h} ã®åºéã§å€åãã环ç©ãããã®ã§ããããã S x {\\displaystyle S_{x}} ãåºé [ 0 , h ] {\\displaystyle [0,h]} ã§ç©åããããšã«ããåŸãããã",
"title": "ããã³åœ¢ã®äœç©"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "V = â« 0 h S x d x {\\displaystyle V=\\int _{0}^{h}S_{x}\\,dx} = â« 0 h ( ( a â c ) b x 2 h 2 + b c x h ) d x {\\displaystyle =\\int _{0}^{h}\\left({\\frac {(a-c)bx^{2}}{h^{2}}}+{\\frac {bcx}{h}}\\right)dx} = b h 2 â« 0 h ( ( a â c ) x 2 + c h x ) d x {\\displaystyle ={\\frac {b}{h^{2}}}\\int _{0}^{h}((a-c)x^{2}+chx)dx} = b h 2 [ ( a â c ) x 3 3 + c h x 2 2 ] 0 h {\\displaystyle ={\\frac {b}{h^{2}}}\\left[{\\frac {(a-c)x^{3}}{3}}+{\\frac {chx^{2}}{2}}\\right]_{0}^{h}} = b h 2 ( ( a â c ) h 3 3 + c h 3 2 ) {\\displaystyle ={\\frac {b}{h^{2}}}\\left({\\frac {(a-c)h^{3}}{3}}+{\\frac {ch^{3}}{2}}\\right)} = b h ( a 3 + c 6 ) {\\displaystyle =bh\\left({\\frac {a}{3}}+{\\frac {c}{6}}\\right)}",
"title": "ããã³åœ¢ã®äœç©"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "V = 2 12 a 3 {\\displaystyle V={\\frac {\\sqrt {2}}{12}}a^{3}}",
"title": "æ£å€é¢äœã®äœç©"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "æ£åé¢äœã®äœç©ã¯ãç«æ¹äœãšã®é¢ä¿ãããå°åºããããšãã§ããŸãã ç«æ¹äœãšé ç¹ãå
±æããæ£åé¢äœã¯ãå
šãŠã®èŸºãç«æ¹äœã®é¢ã®å¯Ÿè§ç·ã«ãªã£ãŠããŸãã ãã£ãŠãç«æ¹äœããäœã£ãäœç©ãåŒãã°ãæ£åé¢äœã®äœç©ãå°ãåºãããšãã§ããŸãã",
"title": "æ£å€é¢äœã®äœç©"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "æ£åé¢äœã®1蟺ã®é·ããaãšããŸãã äœã£ãéšåã¯å
šéšã§4ã€ãããŸããã蟺ã®é·ãã¯å
šãŠããããçããã®ã§ããããã¯ååã«ãªããŸãã",
"title": "æ£å€é¢äœã®äœç©"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "ç«æ¹äœã®1蟺ã®é·ãã¯ãæ£æ¹åœ¢ã®èŸºãšå¯Ÿè§ç·ã®é·ãã®æ¯ã 1 : 2 {\\displaystyle 1:{\\sqrt {2}}} ãããã",
"title": "æ£å€é¢äœã®äœç©"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "äœã£ãéšåã¯äžè§éãšã¿ãªãããšãã§ããã®ã§ãè§éã®äœç©ããã",
"title": "æ£å€é¢äœã®äœç©"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "æåŸã«ç«æ¹äœããè§é4ã€ãåŒããŸãã",
"title": "æ£å€é¢äœã®äœç©"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "",
"title": "æ£å€é¢äœã®äœç©"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "V = 2 3 a 3 {\\displaystyle V={\\frac {\\sqrt {2}}{3}}a^{3}}",
"title": "æ£å€é¢äœã®äœç©"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "æ£å
«é¢äœã¯ãäœç©ã®çããæ£åè§éã2ã€ãããšèŠãããšãã§ããŸãã ãããã®è§éã®é«ãã¯ãè§éã®åºé¢ã®å¯Ÿè§ç·ã®äº€ç¹ããæ±ããããšãã§ããŸãã åºé¢ã«å¯Ÿããé äžã®é ç¹ãšåºé¢ã®å¯Ÿè§ç·ã®äº€ç¹ãçµã¶çŽç·ã¯åçŽã«ãªãã®ã§ã é«ãã¯ãè§éã®æ¯ç·ãšå¯Ÿè§ç·ãããäžå¹³æ¹ã®å®çã§å°åºã§ããŸãã",
"title": "æ£å€é¢äœã®äœç©"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "察è§ç·ã®é·ãã¯ã",
"title": "æ£å€é¢äœã®äœç©"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "察è§ç·ã¯äºãã®äžç¹ã§äº€ããã®ã§ã",
"title": "æ£å€é¢äœã®äœç©"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "é«ãã¯ãæ¯ç·ãšå¯Ÿè§ç·ã®ååããã",
"title": "æ£å€é¢äœã®äœç©"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "å®ã¯ãæ£å
«é¢äœã¯ã©ãã§æ£åè§é2ã€ã«åé¢ããŠããé«ãã¯åäžã§ããããã察è§ç·ã®ååãæ¢ã«é«ãã«ãªã£ãŠããŸãã æåŸã«ãéäœã®äœç©ã®å
¬åŒããã",
"title": "æ£å€é¢äœã®äœç©"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "V = 15 + 7 5 4 a 3 {\\displaystyle V={\\frac {15+7{\\sqrt {5}}}{4}}a^{3}}",
"title": "æ£å€é¢äœã®äœç©"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "V = 5 ( 3 + 5 ) 12 a 3 {\\displaystyle V={\\frac {5(3+{\\sqrt {5}})}{12}}a^{3}}",
"title": "æ£å€é¢äœã®äœç©"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "V = 4 3 Ï r 3 {\\displaystyle V={\\frac {4}{3}}\\pi r^{3}}",
"title": "çã®äœç©"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "ååŸ r {\\displaystyle r} ã®å; C {\\displaystyle C} ããåã®äžå¿ããã®è·é¢ R {\\displaystyle R} (äœãã r {\\displaystyle r} ⊠R {\\displaystyle R} ãšãã)ã®çŽç·ã軞ãšããŠå転ãããåç°äœ(ããŒã©ã¹ãããŒããå)",
"title": "åç°äœïŒããŒã©ã¹ïŒã®äœç©"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "(解æ³)",
"title": "åç°äœïŒããŒã©ã¹ïŒã®äœç©"
}
] | ãã®ããŒãžã§ã¯äœç©ã®å
¬åŒã®è§£èª¬ãããŸãã | ãã®ããŒãžã§ã¯äœç©ã®å
¬åŒã®è§£èª¬ãããŸãã
==çŽæ¹äœã®äœç©==
''V'' = ''abh''
==ç«æ¹äœã®äœç©==
''V'' = ''a''<sup>3</sup>
==æ±äœã®äœç©==
''V'' = ''Sh''
==éäœã®äœç©==
<math>V = \frac{1}{3} Sh</math>
[[File:Right circular cone (parameters r,h,x,Ab,Ax).svg|thumb|200px|right|éäœ]]
éäœã®é ç¹ããåºé¢<math>S</math>ïŒå³å³ã§ã¯<math>A_b</math>ïŒã«åç·ãäžããŠãé ç¹ãã<math>x (0 \leq x \leq h)</math>ã®è·é¢ã§åºé¢ãšå¹³è¡ã«éäœãåãåã£ãããšã§åŸãããå³åœ¢ã<math>A_x</math>ãšããã
ãã®æãéäœã®å®çŸ©ããã<math>S</math>ãš<math>A_x</math>ã¯çžäŒŒã§ããã
çžäŒŒãªå³åœ¢ã®é¢ç©æ¯ã¯ãçžäŒŒæ¯ã®ïŒä¹ã«çããããšããã
<math>S : A_x = h^2 : x^2</math>
åŸã£ãŠã
<math>A_x = \frac{x^2 S}{h^2} </math>
éäœã®äœç©ã¯ãå¹³é¢å³åœ¢<math>A_x</math>ã«é¢ããŠã<math>0 \leq x \leq h</math>ã®åºéã§å€åãã环ç©ãããã®ã§ããããã<math>A_x</math>ãåºé<math>[0,h]</math>ã§ç©åããããšã«ããåŸãããã
<math>V = \int_0^h A_x\,dx </math> <math> = \int_0^h \frac{x^2 S}{h^2}\,dx </math> <math> = \frac{S}{h^2} \int_0^h x^2 \,dx </math> <math> = \frac{S}{h^2} \left[ \frac{x^3}{3}\right]_0^h </math> <math> = \frac{S}{h^2} \left(\frac{h^3}{3}\right) </math><math> = \frac{1}{3} Sh</math>
==éå°ã®äœç©==
[[File:Pyramidenstumpf.svg|thumb|200px|right|éå°]]
äžåºã®é¢ç© <math>s</math>ïŒå³å³ã§ã¯<math>A_2</math>ïŒãäžåºã®é¢ç© <math>S</math>ïŒå³å³ã§ã¯<math>A_1</math>ïŒãé«ã <math>h</math> ã®éå°ã®äœç© <math>V</math>
éå°ã¯ãå¥åãåé éäœãã®ãšããã<math>S</math>ãåºãšããéäœ:<math>P_1</math>ããã<math>s</math>ãåºãšããçžäŒŒãªéäœ:<math>P_2</math>ãé€ãããã®ãšãããã
éäœ:<math>P_1</math>ã®é«ãã <math>H</math>ãšãããšãéäœ:<math>P_2</math>ã®é«ã㯠<math>H-h</math>ãšãªããåã
ã®äœç©ã¯ã
:<math>V_1 = \frac{1}{3} SH</math>, <math>V_2 = \frac{1}{3} s(H-h)</math> ãšãªãã®ã§ãæ±ããäœç©<math>V = \frac{1}{3} ( SH - s(H-h) ) = \frac{1}{3} ( H(S - s) +hs) )</math>(â»)ãšãªãã
çžäŒŒæ¯ãšé¢ç©æ¯ã®é¢ä¿ããã
:<math>S : s = H^2 : (H-h)^2</math>
åŸã£ãŠã
:<math>\sqrt{S} : \sqrt{s} = H : (H-h)</math>
:<math>H\sqrt{s} = (H-h)\sqrt{S}</math>
:<math>H(\sqrt{S}-\sqrt{s}) = h\sqrt{S}</math>
:<math>H = \frac{ h\sqrt{S}}{\sqrt{S}-\sqrt{s}}</math><math>= \frac{ h\sqrt{S}(\sqrt{S}+\sqrt{s})}{S-s}</math><math>= \frac{ h(S+\sqrt{sS})}{S-s}</math>
ããããâ»ã«ä»£å
¥ãããšã以äžã®åŒãåŸãã
:<math>V = \frac h 3 (s + \sqrt{s S} + S) </math>
==ããã³åœ¢ã®äœç©==
[[File:Geometric_wedge.png|right|200px|thumb|ããã³åœ¢]]
* äžåºã 瞊ã®ãªãã ''a''ã暪ã®ãªãã ''b''ã®é·æ¹åœ¢ã瞊ãšå¹³è¡ã§ããäžèŸºã®ãªãã ''c''ãé«ã ''h'' ã®'''ããã³åœ¢'''ã®äœç© ''V''ïŒ
*:<math>V = bh\left(\frac{a}{3}+\frac{c}{6}\right) </math>
ããã³åœ¢ã®äžèŸºããåºé¢ã«åç·ãäžããŠãé ç¹ãã<math>x (0 \leq x \leq h)</math>ã®è·é¢ã§åºé¢ãšå¹³è¡ã«ããã³åœ¢ãåãåã£ãããšã§åŸãããå³åœ¢ïŒé·æ¹åœ¢ïŒã<math>S_x</math>ãšããã
ãã®é·æ¹åœ¢ã®çžŠæšªã¯æ¯äŸã®é¢ä¿ãã以äžã®ãšãããšãªãã
*瞊:<math>\frac{(a-c)x}{h}+c</math>, 暪:<math>\frac{bx}{h}</math>
*<math>S_x = \left(\frac{(a-c)x}{h}+c\right)\left(\frac{bx}{h}\right)</math><math> = \frac{(a-c)bx^2}{h^2}+\frac{bcx}{h}</math>
ããã³åœ¢ã®äœç©ã¯ãå¹³é¢å³åœ¢<math>S_x</math>ã«é¢ããŠã<math>0 \leq x \leq h</math>ã®åºéã§å€åãã环ç©ãããã®ã§ããããã<math>S_x</math>ãåºé<math>[0,h]</math>ã§ç©åããããšã«ããåŸãããã
<math>V = \int_0^h S_x\,dx </math> <math> = \int_0^h \left( \frac{(a-c)bx^2}{h^2}+\frac{bcx}{h}\right)dx </math> <math> = \frac{b}{h^2} \int_0^h ((a-c)x^2+chx)dx </math> <math> = \frac{b}{h^2} \left[ \frac{(a-c)x^3}{3}+\frac{chx^2}{2} \right]_0^h </math> <math> = \frac{b}{h^2} \left(\frac{(a-c)h^3}{3}+\frac{ch^3}{2} \right) </math><math> = bh\left(\frac{a}{3}+\frac{c}{6}\right)</math>
==æ£å€é¢äœã®äœç©==
===æ£åé¢äœã®äœç©===
<math>V = \frac{\sqrt{2}}{12} a^3</math>
[[ç»å:æ£åé¢äœã®äœç©.png|right|]]
:ãŸãåºé¢ããèšç®ããŸãã
:æ£åé¢äœã®é äžã®é ç¹ã¯ãåºé¢ã圢æãã3ç¹ããçããäœçœ®ã«ããã®ã§ã
:ããããçäžãžç·ã䌞ã°ãããšãããã®ç·ãšåºé¢ãšã®äº€ç¹ã¯ã3ç¹ããçããäœçœ®ãå³ã¡äžå¿(å€å¿ãå
å¿ãéå¿ãåå¿)ã«äœçœ®ããããšã«ãªããŸãã
:ããã«åºé¢ã®å³åœ¢ã¯æ£äžè§åœ¢ãªã®ã§ãããããã®ç¹ããäžå¿ããšããã察蟺ã«ç¹ããç·åãåŒããšã3ç·å
šãŠãã察蟺ãåçŽã«2çåããŸãã
:ãã®ãšãããã®ç·åã®é·ã(å³å³äžã®èµ€ç·ã®é·ã)ã¯ãäžå¹³æ¹ã®å®çã«ãã£ãŠã
:<math> \begin{matrix} \sqrt{{\color{Green}a}^2 - \left({1 \over 2}a \right)^2} &=& \sqrt{{\color{Green}a}^2 - {1 \over 4}a^2}
\\ \\ & = & \sqrt{{3 \over 4}a^2}
\\ \\ & = & {\color{Red}{\sqrt{3} \over 2}a} \end{matrix}</math>
:次ã«éç·2æ¬ãšç·ç·1æ¬ã§åœ¢æãããäºç蟺äžè§åœ¢ã«ãç·ç·ã察象ã®è»žãšããç·å¯Ÿç§°ãªäºç蟺äžè§åœ¢ãäœå³ããŸãã
:ãã®äºç蟺äžè§åœ¢ã¯ãåºè§ã30ïŸ(æ£äžè§åœ¢ã®è§ã®2çåç·ã§ãããã)ãªã®ã§ã2ã€ç¹ãããš60ïŸã«ãªããŸãã
:2蟺ãçããããã®éã®è§ã60ïŸã§ããäºç蟺äžè§åœ¢ã¯æ£äžè§åœ¢ãªã®ã§ã
:å³å³äžã®é»ç·å
šäœã®é·ãã¯ãéç·ã®é·ãã«çãããäºç蟺äžè§åœ¢ã®é è§ã®äºçåç·ã¯ãåºèŸºãåçŽã«2çåããããã
:ãã®é»ç·ã®ãã¡æ£äžè§åœ¢ã®å
åŽã«å
¥ãé»ç·ã®é·ãã¯ãéç·ã®é·ãã®ååãã€ãŸããèµ€ç·ã®é·ãã®<math>{1 \over 3}</math>ãšãªããŸãã
:éã«éç·ã®é·ãã¯èµ€ç·ã®é·ãã®<math>{2 \over 3}</math>ãªã®ã§ã
:<math> \begin{matrix} {\color{Red} {\sqrt{3} \over 2}a} \times {2 \over 3} &=& {\sqrt{3} \times 2\!\!\!/ \over 2\!\!\!/ \times 3}a
\\ \\ &=& {\color{Blue}{\sqrt{3} \over 3}a} \end{matrix} </math>
:ç¶ããŠé«ããé«ãã¯ãããŸã§ã«èª¿ã¹ãé·ããšäžå¹³æ¹ã®å®çãå©çšããã°ã
:<math> \begin{matrix} \sqrt{{\color{Green}a}^2 - \left({\color{Blue}{\sqrt{3} \over 3}a} \right)^2}
&=& \sqrt{{\color{Green}a}^2 - {1 \over 3}a^2}
\\ \\ &=& \sqrt{{2 \over 3}a^2}
\\ \\ &=& {\color{Brown} a \sqrt{{2 \over 3}}} \end{matrix} </math>
:åºé¢ç©ãé«ããåºãã®ã§ã
:<math> \begin{matrix}
V &=& {\color{Green}a} \times {\color{Red}{\sqrt{3} \over 2}a}
\times {1 \over 2} \times {\color{Brown}a \sqrt{{2 \over 3}}} \times {1 \over 3}
\\ \\ &=& {{\color{Green}a} \times {\color{Red}a \sqrt{3}\!\!\!/} \times {\color{Brown}a \sqrt{2}}
\over 2 \times {\color{Red} 2} \times 3 \times {\color{Brown}\sqrt{3}\!\!\!/}}
\\ \\ &=& {\sqrt{2} \over 12} a^3
\end{matrix}</math>
====ç«æ¹äœããèãã====
[[ç»å:æ£åé¢äœã®äœç©2.png]]
æ£åé¢äœã®äœç©ã¯ãç«æ¹äœãšã®é¢ä¿ãããå°åºããããšãã§ããŸãã<br>
ç«æ¹äœãšé ç¹ãå
±æããæ£åé¢äœã¯ãå
šãŠã®èŸºãç«æ¹äœã®é¢ã®å¯Ÿè§ç·ã«ãªã£ãŠããŸãã<br>
ãã£ãŠãç«æ¹äœããäœã£ãäœç©ãåŒãã°ãæ£åé¢äœã®äœç©ãå°ãåºãããšãã§ããŸãã
æ£åé¢äœã®1蟺ã®é·ãã''a''ãšããŸãã<br>
äœã£ãéšåã¯å
šéšã§4ã€ãããŸããã蟺ã®é·ãã¯å
šãŠããããçããã®ã§ããããã¯ååã«ãªããŸãã
ç«æ¹äœã®1蟺ã®é·ãã¯ãæ£æ¹åœ¢ã®èŸºãšå¯Ÿè§ç·ã®é·ãã®æ¯ã<math>1 : \sqrt{2}</math>ãããã
:<math> a \times \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}a </math>
äœã£ãéšåã¯äžè§éãšã¿ãªãããšãã§ããã®ã§ãè§éã®äœç©ããã
::<math> \frac{1}{3} \times \frac{1}{2} \times \frac{\sqrt{2}}{2}a \times \frac{\sqrt{2}}{2}a \times \frac{\sqrt{2}}{2}a </math><br>
:<math> = \frac{1}{6} \times \left( \frac{\sqrt{2}}{2}a \right)^3 </math><br>
:<math> = \frac{1}{6} \times \frac{\sqrt{2}}{4}a^3 </math><br>
:<math> = \frac{\sqrt{2}}{24}a^3 </math>
æåŸã«ç«æ¹äœããè§é4ã€ãåŒããŸãã
::<math> \left( \frac{\sqrt{2}}{2}a \right)^3 - 4 \left( \frac{\sqrt{2}}{24}a^3 \right) </math><br>
:<math> = \frac{\sqrt{2}}{4}a^3 - \frac{\sqrt{2}}{6}a^3 </math><br>
:<math> = \frac{3 \sqrt{2}}{12}a^3 - \frac{2 \sqrt{2}}{12}a^3 </math><br>
:<math> = \frac{\sqrt{2}}{12}a^3 </math>
<!--===æ£å
é¢äœã®äœç©===-->
===æ£å
«é¢äœã®äœç©===
<math>V = \frac{\sqrt{2}}{3}a^3</math>
[[ç»å:æ£å
«é¢äœã®äœç©.png|thumb|right|é«ãã¯åºé¢ã®å¯Ÿè§ç·ã®äº€ç¹ããæ±ããããšãã§ããŸãã]]
æ£å
«é¢äœã¯ãäœç©ã®çããæ£åè§éã2ã€ãããšèŠãããšãã§ããŸãã<br>
ãããã®è§éã®é«ãã¯ãè§éã®åºé¢ã®å¯Ÿè§ç·ã®äº€ç¹ããæ±ããããšãã§ããŸãã<br>
åºé¢ã«å¯Ÿããé äžã®é ç¹ãšåºé¢ã®å¯Ÿè§ç·ã®äº€ç¹ãçµã¶çŽç·ã¯åçŽã«ãªãã®ã§ã<br>
é«ãã¯ãè§éã®æ¯ç·ãšå¯Ÿè§ç·ãããäžå¹³æ¹ã®å®çã§å°åºã§ããŸãã
察è§ç·ã®é·ãã¯ã
:<math>\sqrt{a^2 + a^2} = a \sqrt{2}</math>
察è§ç·ã¯äºãã®äžç¹ã§äº€ããã®ã§ã
:<math>\frac{a \sqrt{2}}{2}</math>
é«ãã¯ãæ¯ç·ãšå¯Ÿè§ç·ã®ååããã
::<math>\sqrt{a^2 - \left( \frac{a\sqrt{2}}{2} \right)^2}</math>
:<math>= \sqrt{a^2 - \frac{a^2}{2}}</math>
:<math>= \sqrt{\frac{a^2}{2}}</math>
:<math>= {\color{red}\frac{a \sqrt{2}}{2}}</math>
å®ã¯ãæ£å
«é¢äœã¯ã©ãã§æ£åè§é2ã€ã«åé¢ããŠããé«ãã¯åäžã§ããããã察è§ç·ã®ååãæ¢ã«é«ãã«ãªã£ãŠããŸãã<br>
æåŸã«ãéäœã®äœç©ã®å
¬åŒããã
:<math>V = 2 \times \frac{1}{3} \times a^2 \times {\color{red}\frac{a \sqrt{2}}{2}}</math>
::<math>= \frac{1}{3} \times \sqrt{2}a^3</math>
::<math>= \frac{\sqrt{2}}{3}a^3</math>
===æ£åäºé¢äœã®äœç©===
<math>V = \frac{15+7\sqrt{5}}{4}a^3</math>
===æ£äºåé¢äœã®äœç©===
<math>V = \frac{5(3+\sqrt{5})}{12}a^3</math>
==çã®äœç©==
<math>V = \frac{4}{3}\pi r^3</math>
:<math>x^2 + y^2 + z^2 = r^2</math>ã§ããçãèããã
:<math>x = t</math>ã§ãã®çãåæãããšãååŸ<math>\sqrt{r^2-t^2}</math>ã§ããå;<math>C</math>ãåŸããããã®å;<math>C</math>ã®é¢ç©ã¯<math>\pi (r^2-t^2)</math>ã§ããã
:çã®äœç©ã¯ããã®å;<math>C</math>ã«é¢ããŠã<math>-r \leq t \leq r</math>ã®åºéã§å€åãã环ç©ãããã®ã§ããããã<math>\pi (r^2-t^2)</math>ãåºé<math>[-r,r]</math>ã§ç©åããããšã«ããåŸãããã
:<math>V = \int_{-r}^{r} \pi (r^2-t^2)\,dt </math> = <math>\pi \int_{-r}^{r} (r^2-t^2)\,dt </math> = <math>\pi \int_{-r}^{r} (r^2-t^2)\,dt </math> = <math>\pi \left[ tr^2 - \frac{t^3}{3}\right]_{-r}^{r} </math> = <math>\pi \left\{ \left( r^3 - \frac{r^3}{3}\right) - \left( -r^3 + \frac{r^3}{3}\right) \right\}</math> = <math>\frac{4}{3}\pi r^3</math>
==åç°äœïŒããŒã©ã¹ïŒã®äœç©==
{{wikipedia|ããŒã©ã¹}}
[[File:Torus-rotations-flaeche-r.svg|right|250px|thumb|åç°äœã»ããŒã©ã¹]]
ååŸ<math>r</math>ã®å;<math>C</math>ããåã®äžå¿ããã®è·é¢<math>R</math>ïŒäœãã<math>r</math>ãâŠã<math>R</math>ãšããïŒã®çŽç·ã軞ãšããŠå転ãããåç°äœïŒ[[w:ããŒã©ã¹|ããŒã©ã¹]]ãããŒããåïŒ
:ïŒåèïŒ
:*ãã®æã ååŸ<math>r</math>ããå°ååŸããååŸ<math>R</math>ãã倧ååŸããšåŒã¶ããšãããã
:*åç°äœã®å
çžéšã®åã®ååŸ<math>a</math>ãšå€çžéšã®åã®ååŸ<math>b</math>ãäžããããããšãããããã®æã¯ã以äžã®é¢ä¿ãå©çšãèå¯ã
:*:<math>r = \frac{-a+b}{2}</math>, <math>R = \frac{a+b}{2}</math>
[[File:Superficie tórica.svg|right|250px|thumb|åç°äœã®åæå³åœ¢]]
(解æ³)
:å;<math>C</math>ã®äžå¿ããè·é¢<math>t</math>ïŒ0âŠ<math>t</math>âŠ<math>r</math>ïŒã®äœçœ®ã§ãåç°äœã®å転軞ã«åçŽã«åãåããšãååŸ;<math>R-\sqrt{r^2-t^2}</math>ã®åãå
åŽã®å;<math>C_1</math>ãšããååŸ;<math>R+\sqrt{r^2-t^2}</math>ã®å;<math>C_2</math>ãå€åŽã®åãšããå³åœ¢ãåŸãããã
:ãã®å³åœ¢ã®é¢ç©ã<math>S</math>ãšãããšã
::<math>S = \pi \left( R+\sqrt{r^2-t^2} \right)^2 - \pi \left( R-\sqrt{r^2-t^2} \right)^2 = 4\pi R\sqrt{r^2-t^2}</math>
:ãããã<math>0 \leq t \leq r</math>ã®åºéã§å€åãã环ç©ãããšãåç°äœã®1/2ã®äœç©;<math>V_h</math>ãåŸãããã
:::<math>V_h = \int_{0}^{r} 4\pi R\sqrt{r^2-t^2}dt = 4\pi R \int_{0}^{r} \sqrt{r^2-t^2}dt </math>
:::::<math>\int_{0}^{r} \sqrt{r^2-t^2}dt </math> ã解ããïŒçœ®æç©åæ³ãå©çšïŒ
:::::*<math>t = r\sin{\theta}</math>ãšçœ®ãã
::::::<math>t</math>ã<math>\theta</math>ã§åŸ®åãããšã<math>\frac{dt}{d\theta} = r\cos{\theta}</math>ã<math>\therefore</math>ã<math>dt = r\cos{\theta} d\theta</math>
::::::*<math>t = 0</math>ã®æã<math>\theta = 0</math>
::::::*<math>t = r</math>ã®æã<math>\theta = \frac{\pi}{2}</math>
:::::<math>\int_{0}^{r} \sqrt{r^2-t^2}dt = \int_{0}^{\frac{\pi}{2}} \sqrt{r^2-r^2 \sin ^2 \theta}\cdot(r\cos{\theta}) d\theta = r^2 \int_{0}^{\frac{\pi}{2}} \sqrt{1-\sin ^2 \theta}\cdot (\cos{\theta}) d\theta</math>
:::::<math>= r^2 \int_{0}^{\frac{\pi}{2}} \cos ^2\theta d\theta</math> (<math>\because</math> <math>\sqrt{1-\sin ^2 \theta} = \sqrt{\cos ^2 \theta} = |cos \theta|</math>ã<math>0 \leq \theta \leq \frac{\pi}{2}</math>ã§ããã®ã§ã<math> = cos\theta</math>)
:::::<math>= r^2 \int_{0}^{\frac{\pi}{2}} \frac{1+\cos 2\theta}{2} d\theta</math> (<math>\because</math> <math>\cos ^2 \theta = \frac{1+\cos 2\theta}{2}</math>)
:::::<math>= r^2 \left[ \frac{\theta}{2}+\frac{\sin 2\theta}{4} \right]_{0}^{\frac{\pi}{2}} = \frac{r^2 \pi}{4}</math>
:::<math>V_h = 4\pi R \cdot \frac{r^2 \pi}{4} = \pi^2 r^2 R</math>
:<math>\therefore</math>ã <math>V = 2 \pi^2 r^2 R = (\pi r^2) (2 \pi R)</math>
:::åŸåŒã¯ããå¹³é¢äžã«ããå³åœ¢<math>F</math>ã®é¢ç©ã<math>S</math>ãšãã<math>F</math>ãšåãå¹³é¢äžã«ãã<math>F</math>ãéããªã軞<math>l</math>ã®åšãã§<math>F</math>ãäžå転ãããå転äœã®äœç©ã<math>V</math>ãšãããå転ãããå³åœ¢<math>F</math>ã®éå¿<math>G</math>ããå転軞<math>l</math>ãŸã§ã®è·é¢ã<math>R</math>ãšãããšãã
::::<math>V=2\pi RS</math>
:::ãæãç«ã€ããšãã[[w:ãããã¹ïŒã®ã¥ã«ãã³ã®å®ç|ãããã¹ïŒã®ã¥ã«ãã³ã®å®ç]]第äºå®çãšäžèŽããŠããã
[[Category:æ°åŠæè²|ãããšãããããããããããã]]
[[Category:åçæ°åŠå
¬åŒé|ãããã]] | null | 2021-09-03T22:35:09Z | [
"ãã³ãã¬ãŒã:Wikipedia"
] | https://ja.wikibooks.org/wiki/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%E5%85%AC%E5%BC%8F%E9%9B%86/%E5%88%9D%E7%AD%89%E5%B9%BE%E4%BD%95/%E4%BD%93%E7%A9%8D |
2,056 | åçæ°åŠå
¬åŒé/åç代æ°/å±éå
¬åŒ | ããã§ã¯å±éå
¬åŒã®è§£èª¬ãããŸãã
æŒç¿åé¡ ä»¥äžã®åŒãå±éããã
解ç
蚌æ
æŒç¿åé¡ ä»¥äžã®åŒãå±éããã
解ç
蚌æ
æŒç¿åé¡ ä»¥äžã®åŒãå±éããã
解ç
æŒç¿åé¡ ä»¥äžã®åŒãå±éããã
解ç
蚌æ
æŒç¿åé¡ ä»¥äžã®åŒãå±éããã
解ç | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ããã§ã¯å±éå
¬åŒã®è§£èª¬ãããŸãã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "æŒç¿åé¡ ä»¥äžã®åŒãå±éããã",
"title": "åºæ¬çãªåœ¢"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "解ç",
"title": "åºæ¬çãªåœ¢"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "蚌æ",
"title": "2æ°ã®åã»å·®ã®2ä¹"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "æŒç¿åé¡ ä»¥äžã®åŒãå±éããã",
"title": "2æ°ã®åã»å·®ã®2ä¹"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "解ç",
"title": "2æ°ã®åã»å·®ã®2ä¹"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "蚌æ",
"title": "åãšå·®ã®ç©"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "æŒç¿åé¡ ä»¥äžã®åŒãå±éããã",
"title": "åãšå·®ã®ç©"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "解ç",
"title": "åãšå·®ã®ç©"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "æŒç¿åé¡ ä»¥äžã®åŒãå±éããã",
"title": "äžè¬çãª2次ã®å±éå
¬åŒ"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "解ç",
"title": "äžè¬çãª2次ã®å±éå
¬åŒ"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "",
"title": "äžè¬çãª2次ã®å±éå
¬åŒ"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "蚌æ",
"title": "2æ°ã®åã»å·®ã®3ä¹"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "æŒç¿åé¡ ä»¥äžã®åŒãå±éããã",
"title": "2æ°ã®åã»å·®ã®3ä¹"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "解ç",
"title": "2æ°ã®åã»å·®ã®3ä¹"
}
] | ããã§ã¯å±éå
¬åŒã®è§£èª¬ãããŸãã | {{Pathnav|ã¡ã€ã³ããŒãž|æ°åŠ|frame=1}}
ããã§ã¯å±éå
¬åŒã®è§£èª¬ãããŸãã
== åºæ¬çãªåœ¢ ==
* <math>(a+b)(c+d) = ac + ad + bc + bd</math>
; 蚌æ
: <math>A = (a+b)</math>ãšçœ®ããšããã®åŒã¯ã<math>A(c+d)</math>ãšãªãã
: åé
æ³åãé©çšãããšã<math>Ac+Ad</math>ã
: <math>A</math>ãæ»ããš<math>(a+b)c+(a+b)d</math>ã
: ããããã«åé
æ³åãé©çšãããšã<math>ac + ad + bc + bd</math>ãšãªã蚌æãããã
'''æŒç¿åé¡''' 以äžã®åŒãå±éããã
# <math>(x-2)(2x+5)</math>
# <math>(2a-4b)(5c+d)</math>
'''解ç'''
# <math>2x^2 + x - 10</math>
# <math>10ac + 2ad - 20bc -4bd</math>
== 2æ°ã®åã»å·®ã®2ä¹ ==
* <math>(a+b)^2 = a^2 + 2ab + b^2</math>
* <math>(a-b)^2 = a^2 - 2ab + b^2</math>
'''蚌æ'''
*<math>(a+b)^2 = (a+b)(a+b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2</math>
*<math>(a-b)^2 = (a-b)(a-b) = a^2 -ab -ba + b^2 = a^2 -2ab + b^2</math>
'''æŒç¿åé¡''' 以äžã®åŒãå±éããã
# <math>(x+1)^2</math>
# <math>(2a+4b)^2</math>
# <math>(5a-3b)^2</math>
'''解ç'''
# <math>x^2 + 2x + 1</math>
# <math>4a^2 + 16ab + 16b^2</math>
# <math>25a^2 -30ab + 9b^2</math>
== åãšå·®ã®ç© ==
* <math>(a+b)(a-b) = a^2- b^2</math>
'''蚌æ'''
:<math>(a+b)(a-b) = a^2 -ab + ba - b^2 = a^2 - b^2</math>
'''æŒç¿åé¡'''ã以äžã®åŒãå±éããã
# <math>(5x+1)(5x-1)</math>
# <math>(2a-3b)(2a+3b)</math>
'''解ç'''
# <math>25x^2 - 1</math>
# <math>4a^2 - 9b^2</math>
== äžè¬çãª2次ã®å±éå
¬åŒ ==
* <math>(x+a)(x+b) = x^2 + (a+b)x + ab</math>
* <math>(ax+b)(cx+d) = acx^2 + (ad+bc)x + bd</math>
; 蚌æ
: <math>(x+a)(x+b) = x^2 + bx + ax + ab = x^2 + (a+b)x+ab</math>
: <math>(ax+b)(cx+d) = acx^2 + adx + bcx + bd = acx^2 + (ad + bc)x + bd</math>
'''æŒç¿åé¡'''ã以äžã®åŒãå±éããã
# <math>(x+1)(x+3)</math>
# <math>(2x - 3)( 5x + 5)</math>
# <math>(7ab +9)(-2ab + 10)</math>
'''解ç'''
# <math>x^2 + 4x + 3</math>
# <math>10x^2 -5x -15</math>
# <math>-14a^2b^2 +52ab + 90</math>
== 2æ°ã®åã»å·®ã®3ä¹ ==
* <math>(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3</math>
* <math>(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3</math>
'''蚌æ'''
*<math>(a+b)^3 = (a+b)^2(a+b) = (a^2 + 2ab + b^2)(a+b) = a(a^2 + 2ab + b^2) + b(a^2 + 2ab + b^2) = a^3 + 2a^2b + ab^ 2 + a^2b + 2ab^2 + b^3 = a^3 + 3a^2b + 3ab^2 + b^3</math>
*<math>(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3</math>ã®<math>b</math>ã«<math>-b</math>ã代å
¥ãããšã<math>(a-b)^3 = a^3 + 3a^2(-b) + 3a(-b)^2 + (-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3</math>
'''æŒç¿åé¡'''ã以äžã®åŒãå±éããã
# <math>(2a + b)^3</math>
# <math>(4x^2 - 7)^3</math>
'''解ç'''
# <math>8a^3 + 12a^2b + 6ab^2 + b^3</math>
# <math>64x^6 - 336x^4 + 588x^2 - 343</math>
== 2æ°ã®3ä¹ã®åã»å·® ==
* <math>(a+b)(a^2 - ab + b^2) = a^3 + b^3</math>
* <math>(a-b)(a^2 + ab + b^2) = a^3 - b^3</math>
== 3æ°ã®åã®nä¹ ==
* <math>(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca</math>
* <math>(a+b+c)^3 = a^3 + b^3 + c^3 + 3a^2b + 3ab^2 + 3b^2c + 3bc^2 + 3c^2a + 3a^2c+ 6abc</math>
* <math>(a+b+c)^4 = a^4 + b^4 + c^4 + 4a^3b + 4ab^3 + 4b^3c + 4ca^3 + 4bc^3 + 4c^3a + 6a^2b^2 + 6b^2c^2 + 6c^2a^2 + 12a^2bc + 12ab^2c + 12abc^2</math>
== ãã®ä»ã®å±éå
¬åŒ ==
* <math>(a+b+c)(a^2 + b^2 + c^2 - ab - bc - ca) = a^3 + b^3 + c^3 - 3abc</math>
* <math>(x+a)(x+b)(x+c) = x^3 + (a+b+c)x^2 + (ab + bc + ca)x +abc</math>
* <math>(a-b)(a^{n-1} + a^{n-2}b + a^{n-3}b^2 + \cdots + b^{n-1}) = a^n - b^n</math>
[[Category:æ°åŠæè²|ãããšãããããããããããã ãŠããããããã]]
[[Category:æ°åŠ|ãããšãããããããããããã ãŠããããããã]]
[[Category:åçæ°åŠå
¬åŒé|ãŠããããããã]] | null | 2021-07-09T22:41:26Z | [
"ãã³ãã¬ãŒã:Pathnav"
] | https://ja.wikibooks.org/wiki/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6%E5%85%AC%E5%BC%8F%E9%9B%86/%E5%88%9D%E7%AD%89%E4%BB%A3%E6%95%B0/%E5%B1%95%E9%96%8B%E5%85%AC%E5%BC%8F |
2,058 | ææ©ååŠ/ã¢ã«ã³ãŒã« | ææ©ååŠ>ã¢ã«ã³ãŒã«
èèªæçåæ°ŽçŽ ã®CHéã«Oãå
¥ã£ããã®ãã€ãŸããèèªæçåæ°ŽçŽ åºã«ããããã·åºãçµåãããã®ãäžè¬åŒã¯R-OHã§è¡šããããã¡ãªã¿ã«ãè³éŠæçåæ°ŽçŽ åºã®ãã³ãŒã³ç°ã«ããããã·åºãçŽæ¥çµåãããã®ã¯ãã§ããŒã«é¡ãšåŒã°ããã
åœåã¯ã¢ã«ã«ã³ãã¢ã«ã±ã³ãã¢ã«ãã³ãã·ã¯ãã¢ã«ã«ã³ãããã¯ã·ã¯ãã¢ã«ã±ã³ã®-eã-olã«å€ãããæ
£çšåã¯çåæ°ŽçŽ åºã®ååã®åŸãã«ãã¢ã«ã³ãŒã«ããã€ããã CH3CH2CH2CH2OHã¯1-ãã¿ããŒã«ãCH3CH2CH(OH)CH3ã¯2-ãã¿ããŒã«ã§ããã
ããããã·åºã1ååäžã«nåã€ããŠãããã®ãn䟡ã¢ã«ã³ãŒã«ãšããã2䟡以äžãå€äŸ¡ã¢ã«ã³ãŒã«ãšãããå€äŸ¡ã¢ã«ã³ãŒã«ã®ååã¯-olã®åã«ã®ãªã·ã£èªã®æ°è©ãã€ããã
-OHã®ã€ããŠããCååãnåã®Cååãšçµåã(3-n)åã®HååãšçµåããŠãããšããããã第nçŽã¢ã«ã³ãŒã«ãšããã ãã ãã¡ã¿ããŒã«CH3OHã¯ç¬¬é¶çŽã¢ã«ã³ãŒã«ã§ã¯ãªã第äžçŽã¢ã«ã³ãŒã«ãšããŠæ±ãããã
第äžçŽã¢ã«ã³ãŒã«ã¯é
žåããããšã¢ã«ããããçµãŠã«ã«ãã³é
žã«ãªãã 第äºçŽã¢ã«ã³ãŒã«ã¯é
žåããããšã±ãã³ã«ãªãã 第äžçŽã¢ã«ã³ãŒã«ã¯é
žåããã«ããã
ã¢ã«ã³ãŒã«ã¯å®çŸ©ã«ã瀺ãããŠããããããããã·åºãæããŠããããã®ãããååå
ã«æ¥µæ§ãååšãããåŸã£ãŠã極æ§ãæããç©è³ªãšæ··ãããããåŸåããããããšãã°ãæ°Žãã¢ã³ã¢ãã¢ã§ããããã®æ§è³ªã¯çåæ°ŽçŽ åºã®å€§ãããšããããã·åºã®æ°ãé¢é£ããŠãããçåæ°ŽçŽ åºã®å°ããã¡ã¿ããŒã«ããšã¿ããŒã«ããããããŒã«ã¯å®å
šã«æ°Žãšæ··ãããããããã¿ããŒã«ã¯20°Cã®ç¶æ
ã§ã¯1ãªããã«ã«ã€ãã77gãŸã§ãã溶ããªãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ææ©ååŠ>ã¢ã«ã³ãŒã«",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "èèªæçåæ°ŽçŽ ã®CHéã«Oãå
¥ã£ããã®ãã€ãŸããèèªæçåæ°ŽçŽ åºã«ããããã·åºãçµåãããã®ãäžè¬åŒã¯R-OHã§è¡šããããã¡ãªã¿ã«ãè³éŠæçåæ°ŽçŽ åºã®ãã³ãŒã³ç°ã«ããããã·åºãçŽæ¥çµåãããã®ã¯ãã§ããŒã«é¡ãšåŒã°ããã",
"title": "ã¢ã«ã³ãŒã«ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "åœåã¯ã¢ã«ã«ã³ãã¢ã«ã±ã³ãã¢ã«ãã³ãã·ã¯ãã¢ã«ã«ã³ãããã¯ã·ã¯ãã¢ã«ã±ã³ã®-eã-olã«å€ãããæ
£çšåã¯çåæ°ŽçŽ åºã®ååã®åŸãã«ãã¢ã«ã³ãŒã«ããã€ããã CH3CH2CH2CH2OHã¯1-ãã¿ããŒã«ãCH3CH2CH(OH)CH3ã¯2-ãã¿ããŒã«ã§ããã",
"title": "ã¢ã«ã³ãŒã«ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ããããã·åºã1ååäžã«nåã€ããŠãããã®ãn䟡ã¢ã«ã³ãŒã«ãšããã2䟡以äžãå€äŸ¡ã¢ã«ã³ãŒã«ãšãããå€äŸ¡ã¢ã«ã³ãŒã«ã®ååã¯-olã®åã«ã®ãªã·ã£èªã®æ°è©ãã€ããã",
"title": "ã¢ã«ã³ãŒã«ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "-OHã®ã€ããŠããCååãnåã®Cååãšçµåã(3-n)åã®HååãšçµåããŠãããšããããã第nçŽã¢ã«ã³ãŒã«ãšããã ãã ãã¡ã¿ããŒã«CH3OHã¯ç¬¬é¶çŽã¢ã«ã³ãŒã«ã§ã¯ãªã第äžçŽã¢ã«ã³ãŒã«ãšããŠæ±ãããã",
"title": "ã¢ã«ã³ãŒã«ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "第äžçŽã¢ã«ã³ãŒã«ã¯é
žåããããšã¢ã«ããããçµãŠã«ã«ãã³é
žã«ãªãã 第äºçŽã¢ã«ã³ãŒã«ã¯é
žåããããšã±ãã³ã«ãªãã 第äžçŽã¢ã«ã³ãŒã«ã¯é
žåããã«ããã",
"title": "ã¢ã«ã³ãŒã«ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ã¢ã«ã³ãŒã«ã¯å®çŸ©ã«ã瀺ãããŠããããããããã·åºãæããŠããããã®ãããååå
ã«æ¥µæ§ãååšãããåŸã£ãŠã極æ§ãæããç©è³ªãšæ··ãããããåŸåããããããšãã°ãæ°Žãã¢ã³ã¢ãã¢ã§ããããã®æ§è³ªã¯çåæ°ŽçŽ åºã®å€§ãããšããããã·åºã®æ°ãé¢é£ããŠãããçåæ°ŽçŽ åºã®å°ããã¡ã¿ããŒã«ããšã¿ããŒã«ããããããŒã«ã¯å®å
šã«æ°Žãšæ··ãããããããã¿ããŒã«ã¯20°Cã®ç¶æ
ã§ã¯1ãªããã«ã«ã€ãã77gãŸã§ãã溶ããªãã",
"title": "ã¢ã«ã³ãŒã«ã®æ§è³ª"
}
] | ææ©ååŠïŒã¢ã«ã³ãŒã« | [[ææ©ååŠ]]ïŒã¢ã«ã³ãŒã«
==ã¢ã«ã³ãŒã«ã®å®çŸ©ãšåœåæ³==
èèªæçåæ°ŽçŽ ã®CHéã«Oãå
¥ã£ããã®ãã€ãŸãã[[ææ©ååŠ_åº#çåæ°ŽçŽ åºã®çš®é¡|èèªæçåæ°ŽçŽ åº]]ã«[[ææ©ååŠ_åº#å®èœåºã®çš®é¡|ããããã·åº]]ãçµåãããã®ã[[ææ©ååŠ_ã¢ã«ã«ã³#äžè¬åŒ|äžè¬åŒ]]ã¯RïŒOHã§è¡šããããã¡ãªã¿ã«ãè³éŠæçåæ°ŽçŽ åºã®ãã³ãŒã³ç°ã«ããããã·åºãçŽæ¥çµåãããã®ã¯ãã§ããŒã«é¡ãšåŒã°ããã
åœåã¯[[ææ©ååŠ_ã¢ã«ã«ã³#åœåæ³|ã¢ã«ã«ã³]]ã[[ææ©ååŠ_ã¢ã«ã±ã³#åœåæ³|ã¢ã«ã±ã³]]ã[[ææ©ååŠ_ã¢ã«ãã³#åœåæ³|ã¢ã«ãã³]]ã[[ææ©ååŠ_ã·ã¯ãã¢ã«ã«ã³|ã·ã¯ãã¢ã«ã«ã³]]ãããã¯[[ææ©ååŠ_ã·ã¯ãã¢ã«ã±ã³|ã·ã¯ãã¢ã«ã±ã³]]ã®ïŒeãïŒolã«å€ãããæ
£çšåã¯çåæ°ŽçŽ åºã®ååã®åŸãã«ãã¢ã«ã³ãŒã«ããã€ããã
CH<sub>3</sub>CH<sub>2</sub>CH<sub>2</sub>CH<sub>2</sub>OHã¯1ïŒãã¿ããŒã«ãCH<sub>3</sub>CH<sub>2</sub>CH(OH)CH<sub>3</sub>ã¯2ïŒãã¿ããŒã«ã§ããã
ããããã·åºã1ååäžã«nåã€ããŠãããã®ãn䟡ã¢ã«ã³ãŒã«ãšããã2䟡以äžãå€äŸ¡ã¢ã«ã³ãŒã«ãšãããå€äŸ¡ã¢ã«ã³ãŒã«ã®ååã¯ïŒolã®åã«ã®ãªã·ã£èªã®æ°è©ãã€ããã
ïŒOHã®ã€ããŠããCååãnåã®Cååãšçµåã(3-n)åã®HååãšçµåããŠãããšããããã第nçŽã¢ã«ã³ãŒã«ãšããã
ãã ãã¡ã¿ããŒã«CH<sub>3</sub>OHã¯ç¬¬é¶çŽã¢ã«ã³ãŒã«ã§ã¯ãªã第äžçŽã¢ã«ã³ãŒã«ãšããŠæ±ãããã
C C C
| | |
H-C-OH C-C-OH C-C-OH
| | |
H H C
第äžçŽ 第äºçŽ 第äžçŽ
第äžçŽã¢ã«ã³ãŒã«ã¯é
žåããããš[[ææ©ååŠ_ã¢ã«ããã|ã¢ã«ããã]]ãçµãŠ[[ææ©ååŠ_ã«ã«ãã³é
ž|ã«ã«ãã³é
ž]]ã«ãªãã
第äºçŽã¢ã«ã³ãŒã«ã¯é
žåããããš[[ææ©ååŠ_ã±ãã³|ã±ãã³]]ã«ãªãã
第äžçŽã¢ã«ã³ãŒã«ã¯é
žåããã«ããã
== ã¢ã«ã³ãŒã«ã®æ§è³ª ==
ã¢ã«ã³ãŒã«ã¯å®çŸ©ã«ã瀺ãããŠããããããããã·åºãæããŠããããã®ãããååå
ã«æ¥µæ§ãååšãããåŸã£ãŠã極æ§ãæããç©è³ªãšæ··ãããããåŸåããããããšãã°ãæ°Žãã¢ã³ã¢ãã¢ã§ããããã®æ§è³ªã¯çåæ°ŽçŽ åºã®å€§ãããšããããã·åºã®æ°ãé¢é£ããŠãããçåæ°ŽçŽ åºã®å°ããã¡ã¿ããŒã«ããšã¿ããŒã«ããããããŒã«ã¯å®å
šã«æ°Žãšæ··ãããããããã¿ããŒã«ã¯20âã®ç¶æ
ã§ã¯1ãªããã«ã«ã€ãã77gãŸã§ãã溶ããªãã
[[ã«ããŽãª:ææ©ååŠ]]
[[en:Organic Chemistry/Alcohols]] | null | 2022-11-23T05:32:54Z | [] | https://ja.wikibooks.org/wiki/%E6%9C%89%E6%A9%9F%E5%8C%96%E5%AD%A6/%E3%82%A2%E3%83%AB%E3%82%B3%E3%83%BC%E3%83%AB |
2,060 | ãŠã£ãããã£ã¢ã®æžãæ¹/å
¥éç·š/è³æã®æ¢ãæ¹ | <å
é ã«æ»ã
ã€ã³ã¿ãŒãããäžã§ã®æ
å ±åéã¯ããŠã£ãããã£ã¢ã³ãã¡ã«ãšã£ãŠã¯ãã£ãšã身è¿ãªæ段ã®äžã€ãšãããã§ããããæ€çŽ¢çªã«åèªãå
¥ããã ãã§æ¬²ããæ
å ±ãæã«å
¥ãã€ã³ã¿ãŒãããã¯ããšãŠã䟿å©ãªããŒã«ã§ãã
ãã ã䟿å©ãªåé¢ãæ°ãã€ããªããã°ãããªãããšããããŸãã
æ€çŽ¢ãšã³ãžã³ã«ã¯ããã£ã¬ã¯ããªåãšããããåã®2çš®é¡ã®æ€çŽ¢ãšã³ãžã³ããããŸãããã£ã¬ã¯ããªåã¯äººã®æã§ãµã€ãæ
å ±ãå
¥åããæ€çŽ¢ãšã³ãžã³ã§ãããããåã¯ãã¯ããŒã©ãŒãšããããããã°ã©ã ãèªåçã«ãµã€ãæ
å ±ãåé¡ãããã®ã§ããããããã«é·æã»çæãããã®ã§ããŸã䜿ãåããå¿
èŠããããŸãããçŸåšã§ã¯å€§æã®æ€çŽ¢ãšã³ãžã³ã¯ã©ã¡ãã䜵çšããŠãããã®ãå€ããªã£ãŠããŸãã
ã€ã³ã¿ãŒãããäžã§äœãã質åãããããããã(Googleã§æ€çŽ¢ãã)ãšäžèšã§è¿äºãè¿ã£ãŠãããGoogleã¯ã€ã³ã¿ãŒãããäžã§æ€çŽ¢ããããšãã«éåžžã«åŒ·åãªããŒã«ã«ãªããŸãã
åºæ¬ã®äœ¿ãæ¹ã¯ãhttp://www.google.co.jp/ã«ã¢ã¯ã»ã¹ããŠãããã«ããæ€çŽ¢çªã«èª¿ã¹ããåèªãå
¥ããã ãã§ãããããã§ã¯ããããå°ã䟿å©ãªãã¯ããã¯ãèŠããŠãã£ãŠãã ããã
ãå€ã®æãåºããªã©ãæ¬ã®ã¿ã€ãã«ã決ãŸãæå¥ãåºæåè©ãªã©ãæ€çŽ¢ããããšãã«æŽ»çšã§ãããã¯ããã¯ã§ããæ®éã®æ€çŽ¢ã§ã¯ããå€ããæãåºããšããäºã€ã®åèªãäž¡æ¹åºãŠããããŒãžãæ¢ããŠãããŸããã"å€ã®æãåº"ãšããã°ãå€ã®æãåº ãšç¶ããŠåºãŠæ¥ãå Žåã®ã¿ãæ€çŽ¢ããããšãã§ããŸãã
äžã«æžãããããªæ€çŽ¢ãšã³ãžã³ã¯ãäžã€äžã€ã®æ€çŽ¢ãšã³ãžã³ã§é çªã«èª¿ã¹ãŠè¡ããªããŠã¯ãªããŸãããããããªããããããäžçºã§æ€çŽ¢ããŠããŸããµã€ãããããŸããäŸãã°ãJWordã䜿ã£ãŠæ€çŽ¢ãããŠã¿ãŸããããJWordã¯ããããããªæ€çŽ¢ãµã€ã(Yahooã»Exciteã»BIGLOBEã»Fresheyeã»MSNã»Infoseekã»goo)ãæ¯èŒããªããæ€çŽ¢ããããšãã§ããŸããJWordãã©ã°ã€ã³ã䜿ããšãã€ã³ã¿ãŒããããšã¯ã¹ãããŒã©ãŒãããã¢ãã¬ã¹ããŒã«æ€çŽ¢ãããæååãå
¥åããŠããšã³ã¿ãŒããŒãæŒããšãæ¯èŒæ€çŽ¢ãã§ããŸãããŸããç»é²ãããŠããäŒæ¥ã»ãµã€ãã¯ããµã€ãåãå
¥åãããšãèªåçã«ãã®ãµã€ããžãžã£ã³ãããŸãã
å³æžé€šã«ã¯ãæ¬ã貞ãåºãã®ãšã¯å¥ã«ãè³æãæ
å ±ãåéãããšããéèŠãªåœ¹å²ããããŸããå°åã®æŽå²ãæåãç¹å®ã®äŒæ¥ãåŠæ ¡ã®æŽå²ããããã¯ãªã¯ãšã¹ãã®å°ãªãå€å
žæåŠå
šéãªã©ã¯ãéæ¶æžåº«ã調æ»ç 究宀ãªã©ãäžè¬ã®å©çšè
ããã¯å°ãæã®å±ãã«ãããšããã«ãããŠããã®ãæ®éã§ãã
å³æžé€šã¯ãã¡ãããçã«ãªãŒãã³ã«ããŠããè³æãèªãã ãã§ã沢山ã®æ
å ±ãéããããšãã§ããŸãããéæ¶æžåº«ã調æ»ç 究宀ãå©çšããã°ãçŸç§äºå
žãæžãã®ã«çžå¿ãã沢山ã®æ
å ±ãåŸãããšãã§ããŸããåãã¯å°ããæ·å±
ãé«ããšæãããããããŸãããããã²ããã¯ã³ã©ã³ã¯äžã®å³æžé€šæŽ»çšæ³ãèŠããŸãããã
貎éãªæ¬ã®æå·ãé²ãããã«ãæ¬æ£ã§ã¯ãªã奥ã®æ¬æ£ã«çœ®ããŠããäºããããŸãããããã£ãæ¬ãéæ¶å³æžãšãããŸãã éæ¶å³æžã¯æ¬æ£ãèŠãŠãããããªãã®ã§ãã€ã³ã¿ãŒãããããå³æžé€šã®äžã«ãããŠãå³æžèµæžæ€çŽ¢ãªã©ã®èª¿ã¹ã端æ«çãããå Žåãããã䜿ã£ãŠèª¿ã¹ãŠã¿ãã®ãè¯ãã§ããããããžã£ã³ã«å¥ããåºç幎æ€çŽ¢ããå©çšããŠã¿ããšè¯ãã§ãããã
äžå€®å³æžé€šã倧åŠå³æžé€šãªã©ã倧ããªå³æžé€šã«è¡ãã°ããªãŒãã³ã«ããŠãã空éãšã¯å¥ã«ãçŸç§äºå
žãå°éèŸæžã幎éãªã©ãåºããŠãããŠãã空éããããŸããäžè¬ã«ãããã«ã¯å°ä»»ã®ãåžæžããšåŒã°ããè·å¡ãããŠãè³æãæ¢ãæå©ããããŠãããããšãã§ããŸãã
å
¥ãã«ããé°å²æ°ã®ãšãããå€ããæåã¯ç·åŒµãããããããŸããããåæ°ãåºããŠå
¥ã£ãŠã¿ãŸãããã
å³æžé€šåžæžã¯ãè³æãæ¢ãå°é家ã§ã¯ãããŸããããããšããããåéã«ç²ŸéããŠããããã§ã¯ãããŸãããã§ãã®ã§ã調ã¹ããäºæã«ã€ããŠããã«ããŸããã³ããåºãããéèŠãšãªããŸãã調ã¹ããããšãæ確ã«ããã®ã¯ãã¡ããã§ãã
ãªã©ãæçœã«ããŠããå¿
èŠããããŸããå€åœã®äººåãå°åãªã©ã«ã€ããŠã¯ãã«ã¿ã«ãã§ã®é³ã ãã§ã¯ãªãã¢ã«ãã¡ããããåèªã§ã®ç¶Žãã調ã¹ãŠãããšèª¿æ»ã楜ã«ãªããŸãã
倧æµã®å³æžé€šã§ã¯ãè³æã®ãªã¯ãšã¹ããšããå¶åºŠããããŸããã©ããããµãŒãã¹ããšãããšããã®å³æžé€šã«ã¯ãªãè³æãè¿é£ã®åžã®å³æžé€šãªã©ããåãå¯ããŠãããããæ°ãã賌å
¥ããŠãããããããµãŒãã¹ã§ããå°ãæéã¯ããããŸããã貎éãªè³æãèªãããšãã§ããŸãã®ã§ããã²å©çšããŠã¿ãŸãããã
ããæ¬ããã®äžã«ååšããã®ãã©ããããããã¯ãŸã æµéããŠããæ¬ãªã®ãã©ãããªã©ããããäžã§èª¿ã¹ãæ¹æ³ãããã®ã§ã玹ä»ããŠãããŸãã
ãããäžã«ã¯æžç±ãéèªãåŠè¡è«æãªã©ã®æ
å ±ããŸãšããããŒã¿ããŒã¹ãååšããŸããäžã«ã¯ææã®ãã®ããããŸããã䜿çšããã ããªãç¡æã®ãã®ããããŸãããããäžã§åç
§ã§ããè³æãå€ããããããè³æãçšããã°æ軜ã«èšäºã®ã°ã¬ãŒããäžããäºãã§ããŸããæ§ã
ãªãã®ããããŸãããããã§ã¯ç¥å床ããããç¡æã§å©çšã§ããCiNiiãšåœç«åœäŒå³æžé€šã®ãµãŒãã¹ã玹ä»ããŸãã
åœç«æ
å ±åŠç 究æãéå¶ããåŠè¡æ
å ±ããŒã¿ããŒã¹ã§ãããCiNii ArticlesãããCiNii BooksãããCiNii Dissertationsãã®3ã€ã®ããŒã¿ããŒã¹ããããããããåŠè¡è«æã倧åŠå³æžé€šã®æžç±ãå士è«æãæ€çŽ¢ã§ããŸããäžéšé²èŠ§ãææã®è«æããããŸãããå©çšèªäœã¯ç¡æã§ãã
åœç«åœäŒå³æžé€šã®ãµã€ã(ãã¡ã)ã§ã¯ãåå³æžé€šã«åèµãããŠããæ§ã
ãªè³æãæ€çŽ¢ã§ããŸããå€ãã®ãµãŒãã¹ããããŸããã以äžã«ãã䜿ãããæ©èœãæããŠãããŸãããŸããåœç«å³æžé€šã«ããã°ããã€ãã®ææããŒã¿ããŒã¹ãç¡æã§å©çšå¯èœã§ãããã¡ããèŠããŠãããŸãããã
ã¡ã€ã³ããŒãžããå©çšã§ããæ€çŽ¢æ©èœã§ããããŒã¯ãŒãã«é¢ä¿ããæžç±ãèšäºã»è«æãªã©ãæ€çŽ¢ã§ããŸãã
å€å
žç±ãå€ãæ°èãæ¿åºåè¡ç©ãç§åŠæ åãªã©ã®ããžã¿ã«è³æãæ€çŽ¢ã§ãããµãŒãã¹ã§ãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "<å
é ã«æ»ã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ã€ã³ã¿ãŒãããäžã§ã®æ
å ±åéã¯ããŠã£ãããã£ã¢ã³ãã¡ã«ãšã£ãŠã¯ãã£ãšã身è¿ãªæ段ã®äžã€ãšãããã§ããããæ€çŽ¢çªã«åèªãå
¥ããã ãã§æ¬²ããæ
å ±ãæã«å
¥ãã€ã³ã¿ãŒãããã¯ããšãŠã䟿å©ãªããŒã«ã§ãã",
"title": "æ€çŽ¢ãšã³ãžã³ãå©çšãã"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ãã ã䟿å©ãªåé¢ãæ°ãã€ããªããã°ãããªãããšããããŸãã",
"title": "æ€çŽ¢ãšã³ãžã³ãå©çšãã"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "æ€çŽ¢ãšã³ãžã³ã«ã¯ããã£ã¬ã¯ããªåãšããããåã®2çš®é¡ã®æ€çŽ¢ãšã³ãžã³ããããŸãããã£ã¬ã¯ããªåã¯äººã®æã§ãµã€ãæ
å ±ãå
¥åããæ€çŽ¢ãšã³ãžã³ã§ãããããåã¯ãã¯ããŒã©ãŒãšããããããã°ã©ã ãèªåçã«ãµã€ãæ
å ±ãåé¡ãããã®ã§ããããããã«é·æã»çæãããã®ã§ããŸã䜿ãåããå¿
èŠããããŸãããçŸåšã§ã¯å€§æã®æ€çŽ¢ãšã³ãžã³ã¯ã©ã¡ãã䜵çšããŠãããã®ãå€ããªã£ãŠããŸãã",
"title": "æ€çŽ¢ãšã³ãžã³ãå©çšãã"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ã€ã³ã¿ãŒãããäžã§äœãã質åãããããããã(Googleã§æ€çŽ¢ãã)ãšäžèšã§è¿äºãè¿ã£ãŠãããGoogleã¯ã€ã³ã¿ãŒãããäžã§æ€çŽ¢ããããšãã«éåžžã«åŒ·åãªããŒã«ã«ãªããŸãã",
"title": "æ€çŽ¢ãšã³ãžã³ãå©çšãã"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "åºæ¬ã®äœ¿ãæ¹ã¯ãhttp://www.google.co.jp/ã«ã¢ã¯ã»ã¹ããŠãããã«ããæ€çŽ¢çªã«èª¿ã¹ããåèªãå
¥ããã ãã§ãããããã§ã¯ããããå°ã䟿å©ãªãã¯ããã¯ãèŠããŠãã£ãŠãã ããã",
"title": "æ€çŽ¢ãšã³ãžã³ãå©çšãã"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ãå€ã®æãåºããªã©ãæ¬ã®ã¿ã€ãã«ã決ãŸãæå¥ãåºæåè©ãªã©ãæ€çŽ¢ããããšãã«æŽ»çšã§ãããã¯ããã¯ã§ããæ®éã®æ€çŽ¢ã§ã¯ããå€ããæãåºããšããäºã€ã®åèªãäž¡æ¹åºãŠããããŒãžãæ¢ããŠãããŸããã\"å€ã®æãåº\"ãšããã°ãå€ã®æãåº ãšç¶ããŠåºãŠæ¥ãå Žåã®ã¿ãæ€çŽ¢ããããšãã§ããŸãã",
"title": "æ€çŽ¢ãšã³ãžã³ãå©çšãã"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "äžã«æžãããããªæ€çŽ¢ãšã³ãžã³ã¯ãäžã€äžã€ã®æ€çŽ¢ãšã³ãžã³ã§é çªã«èª¿ã¹ãŠè¡ããªããŠã¯ãªããŸãããããããªããããããäžçºã§æ€çŽ¢ããŠããŸããµã€ãããããŸããäŸãã°ãJWordã䜿ã£ãŠæ€çŽ¢ãããŠã¿ãŸããããJWordã¯ããããããªæ€çŽ¢ãµã€ã(Yahooã»Exciteã»BIGLOBEã»Fresheyeã»MSNã»Infoseekã»goo)ãæ¯èŒããªããæ€çŽ¢ããããšãã§ããŸããJWordãã©ã°ã€ã³ã䜿ããšãã€ã³ã¿ãŒããããšã¯ã¹ãããŒã©ãŒãããã¢ãã¬ã¹ããŒã«æ€çŽ¢ãããæååãå
¥åããŠããšã³ã¿ãŒããŒãæŒããšãæ¯èŒæ€çŽ¢ãã§ããŸãããŸããç»é²ãããŠããäŒæ¥ã»ãµã€ãã¯ããµã€ãåãå
¥åãããšãèªåçã«ãã®ãµã€ããžãžã£ã³ãããŸãã",
"title": "æ€çŽ¢ãšã³ãžã³ãå©çšãã"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "å³æžé€šã«ã¯ãæ¬ã貞ãåºãã®ãšã¯å¥ã«ãè³æãæ
å ±ãåéãããšããéèŠãªåœ¹å²ããããŸããå°åã®æŽå²ãæåãç¹å®ã®äŒæ¥ãåŠæ ¡ã®æŽå²ããããã¯ãªã¯ãšã¹ãã®å°ãªãå€å
žæåŠå
šéãªã©ã¯ãéæ¶æžåº«ã調æ»ç 究宀ãªã©ãäžè¬ã®å©çšè
ããã¯å°ãæã®å±ãã«ãããšããã«ãããŠããã®ãæ®éã§ãã",
"title": "å³æžé€šãå©çšãã"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "å³æžé€šã¯ãã¡ãããçã«ãªãŒãã³ã«ããŠããè³æãèªãã ãã§ã沢山ã®æ
å ±ãéããããšãã§ããŸãããéæ¶æžåº«ã調æ»ç 究宀ãå©çšããã°ãçŸç§äºå
žãæžãã®ã«çžå¿ãã沢山ã®æ
å ±ãåŸãããšãã§ããŸããåãã¯å°ããæ·å±
ãé«ããšæãããããããŸãããããã²ããã¯ã³ã©ã³ã¯äžã®å³æžé€šæŽ»çšæ³ãèŠããŸãããã",
"title": "å³æžé€šãå©çšãã"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "貎éãªæ¬ã®æå·ãé²ãããã«ãæ¬æ£ã§ã¯ãªã奥ã®æ¬æ£ã«çœ®ããŠããäºããããŸãããããã£ãæ¬ãéæ¶å³æžãšãããŸãã éæ¶å³æžã¯æ¬æ£ãèŠãŠãããããªãã®ã§ãã€ã³ã¿ãŒãããããå³æžé€šã®äžã«ãããŠãå³æžèµæžæ€çŽ¢ãªã©ã®èª¿ã¹ã端æ«çãããå Žåãããã䜿ã£ãŠèª¿ã¹ãŠã¿ãã®ãè¯ãã§ããããããžã£ã³ã«å¥ããåºç幎æ€çŽ¢ããå©çšããŠã¿ããšè¯ãã§ãããã",
"title": "å³æžé€šãå©çšãã"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "äžå€®å³æžé€šã倧åŠå³æžé€šãªã©ã倧ããªå³æžé€šã«è¡ãã°ããªãŒãã³ã«ããŠãã空éãšã¯å¥ã«ãçŸç§äºå
žãå°éèŸæžã幎éãªã©ãåºããŠãããŠãã空éããããŸããäžè¬ã«ãããã«ã¯å°ä»»ã®ãåžæžããšåŒã°ããè·å¡ãããŠãè³æãæ¢ãæå©ããããŠãããããšãã§ããŸãã",
"title": "å³æžé€šãå©çšãã"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "å
¥ãã«ããé°å²æ°ã®ãšãããå€ããæåã¯ç·åŒµãããããããŸããããåæ°ãåºããŠå
¥ã£ãŠã¿ãŸãããã",
"title": "å³æžé€šãå©çšãã"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "å³æžé€šåžæžã¯ãè³æãæ¢ãå°é家ã§ã¯ãããŸããããããšããããåéã«ç²ŸéããŠããããã§ã¯ãããŸãããã§ãã®ã§ã調ã¹ããäºæã«ã€ããŠããã«ããŸããã³ããåºãããéèŠãšãªããŸãã調ã¹ããããšãæ確ã«ããã®ã¯ãã¡ããã§ãã",
"title": "å³æžé€šãå©çšãã"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "ãªã©ãæçœã«ããŠããå¿
èŠããããŸããå€åœã®äººåãå°åãªã©ã«ã€ããŠã¯ãã«ã¿ã«ãã§ã®é³ã ãã§ã¯ãªãã¢ã«ãã¡ããããåèªã§ã®ç¶Žãã調ã¹ãŠãããšèª¿æ»ã楜ã«ãªããŸãã",
"title": "å³æžé€šãå©çšãã"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "倧æµã®å³æžé€šã§ã¯ãè³æã®ãªã¯ãšã¹ããšããå¶åºŠããããŸããã©ããããµãŒãã¹ããšãããšããã®å³æžé€šã«ã¯ãªãè³æãè¿é£ã®åžã®å³æžé€šãªã©ããåãå¯ããŠãããããæ°ãã賌å
¥ããŠãããããããµãŒãã¹ã§ããå°ãæéã¯ããããŸããã貎éãªè³æãèªãããšãã§ããŸãã®ã§ããã²å©çšããŠã¿ãŸãããã",
"title": "å³æžé€šãå©çšãã"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "ããæ¬ããã®äžã«ååšããã®ãã©ããããããã¯ãŸã æµéããŠããæ¬ãªã®ãã©ãããªã©ããããäžã§èª¿ã¹ãæ¹æ³ãããã®ã§ã玹ä»ããŠãããŸãã",
"title": "çµã¿åãã"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "ãããäžã«ã¯æžç±ãéèªãåŠè¡è«æãªã©ã®æ
å ±ããŸãšããããŒã¿ããŒã¹ãååšããŸããäžã«ã¯ææã®ãã®ããããŸããã䜿çšããã ããªãç¡æã®ãã®ããããŸãããããäžã§åç
§ã§ããè³æãå€ããããããè³æãçšããã°æ軜ã«èšäºã®ã°ã¬ãŒããäžããäºãã§ããŸããæ§ã
ãªãã®ããããŸãããããã§ã¯ç¥å床ããããç¡æã§å©çšã§ããCiNiiãšåœç«åœäŒå³æžé€šã®ãµãŒãã¹ã玹ä»ããŸãã",
"title": "ããŒã¿ããŒã¹ãå©çšãã"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "åœç«æ
å ±åŠç 究æãéå¶ããåŠè¡æ
å ±ããŒã¿ããŒã¹ã§ãããCiNii ArticlesãããCiNii BooksãããCiNii Dissertationsãã®3ã€ã®ããŒã¿ããŒã¹ããããããããåŠè¡è«æã倧åŠå³æžé€šã®æžç±ãå士è«æãæ€çŽ¢ã§ããŸããäžéšé²èŠ§ãææã®è«æããããŸãããå©çšèªäœã¯ç¡æã§ãã",
"title": "ããŒã¿ããŒã¹ãå©çšãã"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "åœç«åœäŒå³æžé€šã®ãµã€ã(ãã¡ã)ã§ã¯ãåå³æžé€šã«åèµãããŠããæ§ã
ãªè³æãæ€çŽ¢ã§ããŸããå€ãã®ãµãŒãã¹ããããŸããã以äžã«ãã䜿ãããæ©èœãæããŠãããŸãããŸããåœç«å³æžé€šã«ããã°ããã€ãã®ææããŒã¿ããŒã¹ãç¡æã§å©çšå¯èœã§ãããã¡ããèŠããŠãããŸãããã",
"title": "ããŒã¿ããŒã¹ãå©çšãã"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "ã¡ã€ã³ããŒãžããå©çšã§ããæ€çŽ¢æ©èœã§ããããŒã¯ãŒãã«é¢ä¿ããæžç±ãèšäºã»è«æãªã©ãæ€çŽ¢ã§ããŸãã",
"title": "ããŒã¿ããŒã¹ãå©çšãã"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "å€å
žç±ãå€ãæ°èãæ¿åºåè¡ç©ãç§åŠæ åãªã©ã®ããžã¿ã«è³æãæ€çŽ¢ã§ãããµãŒãã¹ã§ãã",
"title": "ããŒã¿ããŒã¹ãå©çšãã"
}
] | ïŒå
é ã«æ»ã
| [[Image:Libri books2.jpg|right|500px]]
{{æžãæ¹å
¥éç·šNav}}
==æ€çŽ¢ãšã³ãžã³ãå©çšãã==
ã€ã³ã¿ãŒãããäžã§ã®æ
å ±åéã¯ããŠã£ãããã£ã¢ã³ãã¡ã«ãšã£ãŠã¯ãã£ãšã身è¿ãªæ段ã®äžã€ãšãããã§ããããæ€çŽ¢çªã«åèªãå
¥ããã ãã§æ¬²ããæ
å ±ãæã«å
¥ãã€ã³ã¿ãŒãããã¯ããšãŠã䟿å©ãªããŒã«ã§ãã
ãã ã䟿å©ãªåé¢ãæ°ãã€ããªããã°ãããªãããšããããŸãã
*ããé¢ïŒãªã³ã©ã€ã³ã«ã¯æ²¢å±±ã®ãµã€ãããããŸããæ¬å±ã§éžãã æžç±ãèªãããããã»ã©é¢çœãç¥èãè©°ãŸã£ããµã€ããå€ãã§ããããããŸãã¡ãžã£ãŒã§ã¯ãªããã®ãææ°æ
å ±ãæ¢ãã®ã«ã¯ãæ Œå¥œã®ããŒã«ã§ãããšããããŸãã
*æªãé¢ïŒãªã³ã©ã€ã³ã§ã¯èª°ããæ
å ±ã®çºä¿¡è
ãšãªããããšãé
åã§ããããã©ããããã¯è£ãè¿ãã°ãæ ¡æ£ãæ»èªãçµãŠããªããããã°æ£ç¢ºæ§ã®æªããæ
å ±ãé£ã³äº€ã£ãŠãããšèšãããšã«ããªããŸãã
===ãã£ã¬ã¯ããªåãšããããåãšã³ãžã³===
[[w:æ€çŽ¢ãšã³ãžã³|æ€çŽ¢ãšã³ãžã³]]ã«ã¯ããã£ã¬ã¯ããªåãšããããåã®2çš®é¡ã®æ€çŽ¢ãšã³ãžã³ããããŸãããã£ã¬ã¯ããªåã¯äººã®æã§ãµã€ãæ
å ±ãå
¥åããæ€çŽ¢ãšã³ãžã³ã§ãããããåã¯ãã¯ããŒã©ãŒãšããããããã°ã©ã ãèªåçã«ãµã€ãæ
å ±ãåé¡ãããã®ã§ããããããã«é·æã»çæãããã®ã§ããŸã䜿ãåããå¿
èŠããããŸãããçŸåšã§ã¯å€§æã®æ€çŽ¢ãšã³ãžã³ã¯ã©ã¡ãã䜵çšããŠãããã®ãå€ããªã£ãŠããŸãã
*ãã£ã¬ã¯ããªå
*:人ã®æã§æŽæ°ããã®ã§ãæ
å ±éã®å°ãªãã¯ãããŸãããå
¬åŒãµã€ããªã©ç¢ºå®ãªæ
å ±ã«ãã©ãçãã確çã¯é«ããªããŸãã倧æã®æ€çŽ¢ãšã³ãžã³ã§ã¯ãªããç¹å®ã®ãžã£ã³ã«ã«ã€ããŠèª¿ã¹ããšããããšãã«äœ¿ãæ©äŒãå€ããããããŸãããããšãã°ãæŽå²ã§ãã£ãããã¡ãã·ã§ã³ã§ãã£ãããšãã£ãç¹å®ã®ãžã£ã³ã«ã«ç¹åããæ€çŽ¢ãšã³ãžã³ããããŸãã
*ããããå
*:æ€çŽ¢æ°ã®å€ããããªããšãã£ãŠãããããåãšã³ãžã³ã®é·æã§ãããã ããã®åé¢ã§ããããããæ
å ±ãåéããããã«ãããŸãæ
å ±ã®ãªããµã€ãããå
šãé¢ä¿ãªããµã€ããå€ãåŒã£ããããšããçæããããŸãã
====Google====
ã€ã³ã¿ãŒãããäžã§äœãã質åããããããããïŒGoogleã§æ€çŽ¢ããïŒãšäžèšã§è¿äºãè¿ã£ãŠãããGoogleã¯ã€ã³ã¿ãŒãããäžã§æ€çŽ¢ããããšãã«éåžžã«åŒ·åãªããŒã«ã«ãªããŸãã
åºæ¬ã®äœ¿ãæ¹ã¯ã[http://www.google.co.jp/ http://www.google.co.jp/]ã«ã¢ã¯ã»ã¹ããŠãããã«ããæ€çŽ¢çªã«èª¿ã¹ããåèªãå
¥ããã ãã§ãããããã§ã¯ããããå°ã䟿å©ãªãã¯ããã¯ãèŠããŠãã£ãŠãã ããã
====åŒçšç¬Š====
ãå€ã®æãåºããªã©ãæ¬ã®ã¿ã€ãã«ã決ãŸãæå¥ãåºæåè©ãªã©ãæ€çŽ¢ããããšãã«æŽ»çšã§ãããã¯ããã¯ã§ããæ®éã®æ€çŽ¢ã§ã¯ããå€ããæãåºããšããäºã€ã®åèªãäž¡æ¹åºãŠããããŒãžãæ¢ããŠãããŸããã'''"å€ã®æãåº"'''ãšããã°ãå€ã®æãåºããšç¶ããŠåºãŠæ¥ãå Žåã®ã¿ãæ€çŽ¢ããããšãã§ããŸãã
===JWORD===
äžã«æžãããããªæ€çŽ¢ãšã³ãžã³ã¯ãäžã€äžã€ã®æ€çŽ¢ãšã³ãžã³ã§é çªã«èª¿ã¹ãŠè¡ããªããŠã¯ãªããŸãããããããªããããããäžçºã§æ€çŽ¢ããŠããŸããµã€ãããããŸããäŸãã°ã[http://www.jword.jp/ JWord]ã䜿ã£ãŠæ€çŽ¢ãããŠã¿ãŸããããJWordã¯ããããããªæ€çŽ¢ãµã€ãïŒYahooã»Exciteã»BIGLOBEã»Fresheyeã»MSNã»Infoseekã»gooïŒãæ¯èŒããªããæ€çŽ¢ããããšãã§ããŸããJWordãã©ã°ã€ã³ã䜿ããšãã€ã³ã¿ãŒããããšã¯ã¹ãããŒã©ãŒãããã¢ãã¬ã¹ããŒã«æ€çŽ¢ãããæååãå
¥åããŠããšã³ã¿ãŒããŒãæŒããšãæ¯èŒæ€çŽ¢ãã§ããŸãããŸããç»é²ãããŠããäŒæ¥ã»ãµã€ãã¯ããµã€ãåãå
¥åãããšãèªåçã«ãã®ãµã€ããžãžã£ã³ãããŸãã
*[http://search.jword.jp/cns.dll?type=sb&fm=11&agent=&partner=AP&lang=euc&name=%A5%A6%A5%A3%A5%AD%A5%E1%A5%C7%A5%A3%A5%A2%BA%E2%C3%C4&sbox11_5=%B8%A1%BA%F7 JWordæ€çŽ¢çµæ ããŠã£ãã¡ãã£ã¢è²¡å£ã]
*[http://search.jword.jp/cns.dll?type=sb&fm=2&agent=&partner=AP&lang=euc&name=%A5%A6%A5%A3%A5%AD%A5%DA%A5%C7%A5%A3%A5%A2&bypass=&selsecategory=&service=jwd&style=1 JWordæ€çŽ¢çµæ ããŠã£ãããã£ã¢ã]
==å³æžé€šãå©çšãã==
å³æžé€šã«ã¯ãæ¬ã貞ãåºãã®ãšã¯å¥ã«ã''è³æãæ
å ±ãåéãã''ãšããéèŠãªåœ¹å²ããããŸããå°åã®æŽå²ãæåãç¹å®ã®äŒæ¥ãåŠæ ¡ã®æŽå²ããããã¯ãªã¯ãšã¹ãã®å°ãªãå€å
žæåŠå
šéãªã©ã¯ãéæ¶æžåº«ã調æ»ç 究宀ãªã©ãäžè¬ã®å©çšè
ããã¯å°ãæã®å±ãã«ãããšããã«ãããŠããã®ãæ®éã§ãã
å³æžé€šã¯ãã¡ãããçã«ãªãŒãã³ã«ããŠããè³æãèªãã ãã§ã沢山ã®æ
å ±ãéããããšãã§ããŸãããéæ¶æžåº«ã調æ»ç 究宀ãå©çšããã°ãçŸç§äºå
žãæžãã®ã«çžå¿ãã沢山ã®æ
å ±ãåŸãããšãã§ããŸããåãã¯å°ããæ·å±
ãé«ããšæãããããããŸãããããã²ããã¯ã³ã©ã³ã¯äžã®å³æžé€šæŽ»çšæ³ãèŠããŸãããã
===éæ¶æžåº«ã«ããæ¬ã®æ¢ãæ¹===
貎éãªæ¬ã®æå·ãé²ãããã«ãæ¬æ£ã§ã¯ãªã奥ã®æ¬æ£ã«çœ®ããŠããäºããããŸãããããã£ãæ¬ãéæ¶å³æžãšãããŸãã
éæ¶å³æžã¯æ¬æ£ãèŠãŠãããããªãã®ã§ãã€ã³ã¿ãŒãããããå³æžé€šã®äžã«ãããŠãå³æžèµæžæ€çŽ¢ãªã©ã®èª¿ã¹ã端æ«çãããå Žåãããã䜿ã£ãŠèª¿ã¹ãŠã¿ãã®ãè¯ãã§ããããããžã£ã³ã«å¥ããåºç幎æ€çŽ¢ããå©çšããŠã¿ããšè¯ãã§ãããã
===調æ»ç 究宀===
äžå€®å³æžé€šã倧åŠå³æžé€šãªã©ã倧ããªå³æžé€šã«è¡ãã°ããªãŒãã³ã«ããŠãã空éãšã¯å¥ã«ãçŸç§äºå
žãå°éèŸæžã幎éãªã©ãåºããŠãããŠãã空éããããŸããäžè¬ã«ãããã«ã¯å°ä»»ã®ãåžæžããšåŒã°ããè·å¡ãããŠãè³æãæ¢ãæå©ããããŠãããããšãã§ããŸãã
å
¥ãã«ããé°å²æ°ã®ãšãããå€ããæåã¯ç·åŒµãããããããŸããããåæ°ãåºããŠå
¥ã£ãŠã¿ãŸãããã
===ãªãã¡ã¬ã³ã¹ãµãŒãã¹ã®äžæãªäœ¿ãæ¹===
å³æžé€šåžæžã¯ãè³æãæ¢ãå°é家ã§ã¯ãããŸããããããšããããåéã«ç²ŸéããŠããããã§ã¯ãããŸãããã§ãã®ã§ã調ã¹ããäºæã«ã€ããŠããã«ããŸããã³ããåºãããéèŠãšãªããŸãã調ã¹ããããšãæ確ã«ããã®ã¯ãã¡ããã§ãã
*äœã®ããã«
*äœã®ãžã£ã³ã«ã«ã€ããŠ
ãªã©ãæçœã«ããŠããå¿
èŠããããŸããå€åœã®äººåãå°åãªã©ã«ã€ããŠã¯ãã«ã¿ã«ãã§ã®é³ã ãã§ã¯ãªãã¢ã«ãã¡ããããåèªã§ã®ç¶Žãã調ã¹ãŠãããšèª¿æ»ã楜ã«ãªããŸãã
=== ãã®å Žã«ãªãæ¬ãèªã ===
倧æµã®å³æžé€šã§ã¯ãè³æã®ãªã¯ãšã¹ããšããå¶åºŠããããŸããã©ããããµãŒãã¹ããšãããšããã®å³æžé€šã«ã¯ãªãè³æãè¿é£ã®åžã®å³æžé€šãªã©ããåãå¯ããŠãããããæ°ãã賌å
¥ããŠãããããããµãŒãã¹ã§ããå°ãæéã¯ããããŸããã貎éãªè³æãèªãããšãã§ããŸãã®ã§ããã²å©çšããŠã¿ãŸãããã
==çµã¿åãã==
ããæ¬ããã®äžã«ååšããã®ãã©ããããããã¯ãŸã æµéããŠããæ¬ãªã®ãã©ãããªã©ããããäžã§èª¿ã¹ãæ¹æ³ãããã®ã§ã玹ä»ããŠãããŸãã
*[http://www.books.or.jp/ Books.or.jp] - æ¥æ¬åœå
ã§åºçããããã€æµéããŠããæ¬ãã©ããã調ã¹ãããã®ãµãŒãã¹ã§ããã€ãŸãã絶çæ¬ãæ¬å±ã§æ¢ããšããæéã®ç¡é§ãçãããšãåºæ¥ãŸãã
*<!--[http://webcat.nii.ac.jp/ NACSIS Webcat] - 倧åŠå³æžé€šã®æèµè³æã®ããŒã¿ããŒã¹ã暪æçã«æ€çŽ¢ã§ããã·ã¹ãã ã--><!--ãªã³ã¯åãã®ãã-->
== ããŒã¿ããŒã¹ãå©çšãã ==
ãããäžã«ã¯æžç±ãéèªãåŠè¡è«æãªã©ã®æ
å ±ããŸãšããããŒã¿ããŒã¹ãååšããŸããäžã«ã¯ææã®ãã®ããããŸããã䜿çšããã ããªãç¡æã®ãã®ããããŸãããããäžã§åç
§ã§ããè³æãå€ããããããè³æãçšããã°æ軜ã«èšäºã®ã°ã¬ãŒããäžããäºãã§ããŸããæ§ã
ãªãã®ããããŸãããããã§ã¯ç¥å床ããããç¡æã§å©çšã§ããCiNiiãšåœç«åœäŒå³æžé€šã®ãµãŒãã¹ã玹ä»ããŸãã
=== CiNii ===
åœç«æ
å ±åŠç 究æãéå¶ããåŠè¡æ
å ±ããŒã¿ããŒã¹ã§ããã'''CiNii Articles'''ããã'''CiNii Books'''ããã'''CiNii Dissertations'''ãã®3ã€ã®ããŒã¿ããŒã¹ããããããããåŠè¡è«æã倧åŠå³æžé€šã®æžç±ãå士è«æãæ€çŽ¢ã§ããŸããäžéšé²èŠ§ãææã®è«æããããŸãããå©çšèªäœã¯ç¡æã§ãã
* [http://ci.nii.ac.jp/ CiNii Articles - æ¥æ¬ã®è«æãããã]
* [http://ci.nii.ac.jp/books/ CiNii Books - 倧åŠå³æžé€šã®æ¬ãããã]
* [http://ci.nii.ac.jp/d/ CiNii Dissertations - æ¥æ¬ã®å士è«æãããã]
=== åœç«åœäŒå³æžé€š ===
åœç«åœäŒå³æžé€šã®ãµã€ãïŒ[http://www.ndl.go.jp/index.html ãã¡ã]ïŒã§ã¯ãåå³æžé€šã«åèµãããŠããæ§ã
ãªè³æãæ€çŽ¢ã§ããŸããå€ãã®ãµãŒãã¹ããããŸããã以äžã«ãã䜿ãããæ©èœãæããŠãããŸãããŸããåœç«å³æžé€šã«ããã°ããã€ãã®ææããŒã¿ããŒã¹ãç¡æã§å©çšå¯èœã§ãããã¡ããèŠããŠãããŸãããã
*åœç«åœäŒå³æžé€šãµãŒã
ã¡ã€ã³ããŒãžããå©çšã§ããæ€çŽ¢æ©èœã§ããããŒã¯ãŒãã«é¢ä¿ããæžç±ãèšäºã»è«æãªã©ãæ€çŽ¢ã§ããŸãã
*åœç«åœäŒå³æžé€šããžã¿ã«ã³ã¬ã¯ã·ã§ã³
å€å
žç±ãå€ãæ°èãæ¿åºåè¡ç©ãç§åŠæ åãªã©ã®ããžã¿ã«è³æãæ€çŽ¢ã§ãããµãŒãã¹ã§ãã<!--ãããã®äžããã€ã³ã¿ãŒãããã§é²èŠ§å¯èœãªè¿ä»£ã®è³æãéãããè¿ä»£ããžã¿ã«ã©ã€ãã©ãªãŒããšãããµãŒãã¹ããããŸãã--><!--çµ±åã®ãã-->
==åèæç®==
*å島éæ¹ ä»ç£ä¿®ãå³æžé€šæŠè«ãïŒæš¹ææ¿ïŒISBN 4-88367-001-5
*é·æŸ€é
ç·ãèãæ
å ±ãšæç®ã®æ¢çŽ¢ç¬¬3çãïŒäžžåïŒISBN 4-621-03943
== å€éšãªã³ã¯ ==
*[http://www.ndl.go.jp/jp/data/theme.html åœç«åœäŒå³æžé€šããŒãå¥èª¿ã¹æ¹æ¡å
]
[[Category:ãŠã£ãããã£ã¢ã®æžãæ¹|ã«ã
ããã04]] | null | 2020-05-06T04:21:02Z | [
"ãã³ãã¬ãŒã:æžãæ¹å
¥éç·šNav"
] | https://ja.wikibooks.org/wiki/%E3%82%A6%E3%82%A3%E3%82%AD%E3%83%9A%E3%83%87%E3%82%A3%E3%82%A2%E3%81%AE%E6%9B%B8%E3%81%8D%E6%96%B9/%E5%85%A5%E9%96%80%E7%B7%A8/%E8%B3%87%E6%96%99%E3%81%AE%E6%8E%A2%E3%81%97%E6%96%B9 |
2,061 | ææ©ååŠ/ãšãŒãã« | ææ©ååŠ>ãšãŒãã«
C-O-Cã®åœ¢ã®çµåããšãŒãã«çµåãšãããããããã€ååç©ããšãŒãã«ãšããã åœåæ³ã¯çåæ°ŽçŽ åºã®å称ã®åŸã«ããšãŒãã«ããã€ããã äŸãã°CH3CH2-O-CH3ã¯ããšãã«ã¡ãã«ãšãŒãã«ãã§ããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ææ©ååŠ>ãšãŒãã«",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "C-O-Cã®åœ¢ã®çµåããšãŒãã«çµåãšãããããããã€ååç©ããšãŒãã«ãšããã åœåæ³ã¯çåæ°ŽçŽ åºã®å称ã®åŸã«ããšãŒãã«ããã€ããã äŸãã°CH3CH2-O-CH3ã¯ããšãã«ã¡ãã«ãšãŒãã«ãã§ããã",
"title": "ãšãŒãã«ã®å®çŸ©ãšåœåæ³"
}
] | ææ©ååŠïŒãšãŒãã« | [[ææ©ååŠ]]ïŒãšãŒãã«
==ãšãŒãã«ã®å®çŸ©ãšåœåæ³==
CïŒOïŒCã®åœ¢ã®çµåããšãŒãã«çµåãšãããããããã€ååç©ããšãŒãã«ãšããã
åœåæ³ã¯[[ææ©ååŠ_åº#çåæ°ŽçŽ åºã®çš®é¡|çåæ°ŽçŽ åº]]ã®å称ã®åŸã«ããšãŒãã«ããã€ããã
äŸãã°CH<sub>3</sub>CH<sub>2</sub>ïŒOïŒCH<sub>3</sub>ã¯ããšãã«ã¡ãã«ãšãŒãã«ãã§ããã
[[ã«ããŽãª:ææ©ååŠ]]
[[en:Organic Chemistry/Ethers]] | null | 2022-11-23T05:33:06Z | [] | https://ja.wikibooks.org/wiki/%E6%9C%89%E6%A9%9F%E5%8C%96%E5%AD%A6/%E3%82%A8%E3%83%BC%E3%83%86%E3%83%AB |
2,063 | ææ©ååŠ/ã±ãã³ | ææ©ååŠ>ã±ãã³
ã±ãã³ã¯ã±ãã³åº-CO-ãæã€ååç©ã§ãããåœéåã¯ã¢ã«ã«ã³ã®èªå°Ÿneãnoneã«å€ããããããŸã䜿ãããªãã
第äºçŽã¢ã«ã³ãŒã«ãé
žåãããšãããããã·ã«åºã2åã®ã¢ã«ã³ãŒã«ãã§ããã
ãã ããã²ãšã€ã®Cå
çŽ ã«ããããã·åºã2åä»ããšãããã«è±æ°Žåå¿ãèµ·ããã ãã£ãŠãã®ç©è³ªã¯äžç¬ååšããã ãã§ããã«å¥ã®ç©è³ªã«å€ããã
ãã®ãšããã®>C=Oã®éšåãã±ãã³åºãšãããç°¡åã«>COãšæžãã
>C=Oã¯å®ã¯äžè¬çã«ã¯ã«ã«ããã«åºãšããããããã ããå³å¯ã«ã¯ã«ã«ããã«åº=ã±ãã³åºã§ã¯ãªãã ãªããªãã>C=Oããã€åºã¯ä»ã«ãã¢ã«ãããåº-CHOãã«ã«ããã·ã«åº-COOHãããããã§ããã ã«ã«ããã«åºã¯ãããã®ç·ç§°ãšããŠååšãããã±ãã³åºã¯ãã«ã«ããã«åºã®ãã¡>COã®äŸ¡æšã®äž¡åŽã«çåæ°ŽçŽ åºãä»ãããã®ã§ãããã«ã«ããã«åºã®äžçš®ã§ããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ææ©ååŠ>ã±ãã³",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ã±ãã³ã¯ã±ãã³åº-CO-ãæã€ååç©ã§ãããåœéåã¯ã¢ã«ã«ã³ã®èªå°Ÿneãnoneã«å€ããããããŸã䜿ãããªãã",
"title": "ã±ãã³ã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "第äºçŽã¢ã«ã³ãŒã«ãé
žåãããšãããããã·ã«åºã2åã®ã¢ã«ã³ãŒã«ãã§ããã",
"title": "ã±ãã³ã®çæ"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ãã ããã²ãšã€ã®Cå
çŽ ã«ããããã·åºã2åä»ããšãããã«è±æ°Žåå¿ãèµ·ããã ãã£ãŠãã®ç©è³ªã¯äžç¬ååšããã ãã§ããã«å¥ã®ç©è³ªã«å€ããã",
"title": "ã±ãã³ã®çæ"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ãã®ãšããã®>C=Oã®éšåãã±ãã³åºãšãããç°¡åã«>COãšæžãã",
"title": "ã±ãã³ã®çæ"
},
{
"paragraph_id": 5,
"tag": "p",
"text": ">C=Oã¯å®ã¯äžè¬çã«ã¯ã«ã«ããã«åºãšããããããã ããå³å¯ã«ã¯ã«ã«ããã«åº=ã±ãã³åºã§ã¯ãªãã ãªããªãã>C=Oããã€åºã¯ä»ã«ãã¢ã«ãããåº-CHOãã«ã«ããã·ã«åº-COOHãããããã§ããã ã«ã«ããã«åºã¯ãããã®ç·ç§°ãšããŠååšãããã±ãã³åºã¯ãã«ã«ããã«åºã®ãã¡>COã®äŸ¡æšã®äž¡åŽã«çåæ°ŽçŽ åºãä»ãããã®ã§ãããã«ã«ããã«åºã®äžçš®ã§ããã",
"title": "ã«ã«ããã«åº"
}
] | ææ©ååŠïŒã±ãã³ | [[ææ©ååŠ]]ïŒã±ãã³
==ã±ãã³ã®å®çŸ©ãšåœåæ³==
ã±ãã³ã¯ã±ãã³åºïŒCOïŒãæã€ååç©ã§ãããåœéåã¯ã¢ã«ã«ã³ã®èªå°Ÿneãnoneã«å€ããããããŸã䜿ãããªãã
*CH<sub>3</sub>COCH<sub>3</sub> 2-ããããã³→'''ã¢ã»ãã³'''
*CH<sub>3</sub>CH<sub>2</sub>COCH<sub>3</sub> 2ïŒãã¿ãã³→'''ãšãã«ã¡ãã«ã±ãã³'''
==ã±ãã³ã®çæ==
[[ææ©ååŠ_ã¢ã«ã³ãŒã«|第äºçŽã¢ã«ã³ãŒã«]]ãé
žåãããšãããããã·ã«åºã2åã®ã¢ã«ã³ãŒã«ãã§ããã
*RïŒCH(OH)ïŒR ïŒ (O) â RïŒC(OH)2ïŒR
ãã ããã²ãšã€ã®Cå
çŽ ã«ããããã·åºã2åä»ããšãããã«è±æ°Žåå¿ãèµ·ããã ãã£ãŠãã®ç©è³ªã¯äžç¬ååšããã ãã§ããã«å¥ã®ç©è³ªã«å€ããã
R R
| |
R-C-OH ---> R-C=O
| -H2O
OH
ãã®ãšããã®ïŒCïŒOã®éšåãã±ãã³åºãšãããç°¡åã«ïŒCOãšæžãã
==ã«ã«ããã«åº==
ïŒCïŒOã¯å®ã¯äžè¬çã«ã¯ã«ã«ããã«åºãšããããããã ããå³å¯ã«ã¯ã«ã«ããã«åºïŒã±ãã³åºã§ã¯ãªãã
ãªããªããïŒCïŒOããã€åºã¯ä»ã«ãã¢ã«ãããåºïŒCHOãã«ã«ããã·ã«åºïŒCOOHãããããã§ããã
ã«ã«ããã«åºã¯ãããã®ç·ç§°ãšããŠååšãããã±ãã³åºã¯ãã«ã«ããã«åºã®ãã¡ïŒCOã®äŸ¡æšã®äž¡åŽã«çåæ°ŽçŽ åºãä»ãããã®ã§ãããã«ã«ããã«åºã®äžçš®ã§ããã
[[ã«ããŽãª:ææ©ååŠ]] | null | 2022-11-23T05:33:13Z | [] | https://ja.wikibooks.org/wiki/%E6%9C%89%E6%A9%9F%E5%8C%96%E5%AD%A6/%E3%82%B1%E3%83%88%E3%83%B3 |
2,064 | ææ©ååŠ/ã«ã«ãã³é
ž | ææ©ååŠ>ã«ã«ãã³é
ž
ã«ã«ããã·ã«åº-COOHããã€ååç©ãã«ã«ãã³é
žãšããã åœéåã¯ã¢ã«ã«ã³ã®ååã®åŸã«ãé
žããã€ãããããŸã䜿ãããªãã HCOOHãè»(ã®)é
žãCH3COOHãé
¢é
žãšããã
ã«ã«ãã³é
žã¯ã«ã«ããã·åºãæã£ãŠããé
žæ§ã§ãããã«ã«ãã³é
žã¯æãå°ããªççŽ äžã€ã®ã®é
žãããççŽ 16åã®ãã«ããã³é
žãªã©å€§ãããæ§ã
ã§ãããã眮æåºãæããªãã«ã«ãã³é
žã¯ååéã倧ãããªãã«ã€ããæ°ŽçŽ ã€ãªã³ã®è§£é¢ãæžã£ãŠãããã€ãŸããé
žãšããŠã®åŒ·ããæžãã®ã§ããããããã©ããã£ãçç±ã«ããã®ããšãããšãã«ã«ããã·åºã®é»åå¯åºŠãé«ããäœããã«ãã£ãŠæ±ºãŸããã¢ã«ãã«åºã¯é»åäŸäžæ§åºã§ããããããçµåããŠããååã眮æåºã¯é»åãã¢ã«ãã«åºããæŒãä»ããããã®ã§(æŒãä»ãããããšããã®ã¯æ¯å©è¡šçŸã§ãã£ãŠãååãåšåããé»åã®åæ°ãå¢ãããšããããã§ã¯ãªãã)ãé»åã®å¯åºŠãé«ããªãããããããšãã«ã«ããã·åºã®é
žçŽ ã®é»åå¯åºŠãé«ããªããããæ°ŽçŽ ãšã®éã®çµåãå
ç¢ã«ãªããçµæãæ°ŽçŽ ã¯ã«ã«ããã·åºããé¢ãã«ãããªããé
žæ§åºŠãäœäžãããã¢ã«ãã«åºãæ§æããççŽ ã®æ°ãå€ãã»ã©ãã®åŸåã¯é¡èã§ããã
éã«ãã«ã«ãã³é
žã®ã«ã«ããã·åºã«é£æ¥ããççŽ ã«é»åæ±åŒæ§åºãçµåããŠããå Žåãé
žæ§åºŠã¯åŒ·ããªããäŸãã°ãã¯ããé
¢é
ž(CH2Cl-COOH)ã¯é
¢é
žããã匷ããé»åæ±åŒæ§åºã®æ°ãå¢ããã°ããã«é
žæ§åºŠã¯åŒ·ããªãã
ã«ã«ãã³é
žã¯ä»¥äžã®åå¿ã«ãã£ãŠçæããã
ã¢ã«ããããé
žåãããšã
ãã®R以å€ã®éšåãã«ã«ããã·åºãšãããç°¡åã«-COOHãšè¡šãã
ã«ã«ãã³é
žã®åå¿ã¯ãã®é
žãšããŠã®åŽé¢ãšã«ã«ããã«ååç©ãšããŠã®åŽé¢ã«ãã£ãŠèµ·ãããã«ã«ããã«ååç©ãšããŠã®åå¿ã¯ã±ãã³ã®ç« ã詳ããããã¢ã«ãããçãšã¢ã«ããŒã«çž®åãªã©ãèµ·ãããããšããããšãä»ãå ããŠãããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ææ©ååŠ>ã«ã«ãã³é
ž",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ã«ã«ããã·ã«åº-COOHããã€ååç©ãã«ã«ãã³é
žãšããã åœéåã¯ã¢ã«ã«ã³ã®ååã®åŸã«ãé
žããã€ãããããŸã䜿ãããªãã HCOOHãè»(ã®)é
žãCH3COOHãé
¢é
žãšããã",
"title": "ã«ã«ãã³é
žã®å®çŸ©ãšåœåæ³"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ã«ã«ãã³é
žã¯ã«ã«ããã·åºãæã£ãŠããé
žæ§ã§ãããã«ã«ãã³é
žã¯æãå°ããªççŽ äžã€ã®ã®é
žãããççŽ 16åã®ãã«ããã³é
žãªã©å€§ãããæ§ã
ã§ãããã眮æåºãæããªãã«ã«ãã³é
žã¯ååéã倧ãããªãã«ã€ããæ°ŽçŽ ã€ãªã³ã®è§£é¢ãæžã£ãŠãããã€ãŸããé
žãšããŠã®åŒ·ããæžãã®ã§ããããããã©ããã£ãçç±ã«ããã®ããšãããšãã«ã«ããã·åºã®é»åå¯åºŠãé«ããäœããã«ãã£ãŠæ±ºãŸããã¢ã«ãã«åºã¯é»åäŸäžæ§åºã§ããããããçµåããŠããååã眮æåºã¯é»åãã¢ã«ãã«åºããæŒãä»ããããã®ã§(æŒãä»ãããããšããã®ã¯æ¯å©è¡šçŸã§ãã£ãŠãååãåšåããé»åã®åæ°ãå¢ãããšããããã§ã¯ãªãã)ãé»åã®å¯åºŠãé«ããªãããããããšãã«ã«ããã·åºã®é
žçŽ ã®é»åå¯åºŠãé«ããªããããæ°ŽçŽ ãšã®éã®çµåãå
ç¢ã«ãªããçµæãæ°ŽçŽ ã¯ã«ã«ããã·åºããé¢ãã«ãããªããé
žæ§åºŠãäœäžãããã¢ã«ãã«åºãæ§æããççŽ ã®æ°ãå€ãã»ã©ãã®åŸåã¯é¡èã§ããã",
"title": "ã«ã«ãã³é
žã®æ§è³ª"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "éã«ãã«ã«ãã³é
žã®ã«ã«ããã·åºã«é£æ¥ããççŽ ã«é»åæ±åŒæ§åºãçµåããŠããå Žåãé
žæ§åºŠã¯åŒ·ããªããäŸãã°ãã¯ããé
¢é
ž(CH2Cl-COOH)ã¯é
¢é
žããã匷ããé»åæ±åŒæ§åºã®æ°ãå¢ããã°ããã«é
žæ§åºŠã¯åŒ·ããªãã",
"title": "ã«ã«ãã³é
žã®æ§è³ª"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ã«ã«ãã³é
žã¯ä»¥äžã®åå¿ã«ãã£ãŠçæããã",
"title": "ã«ã«ãã³é
žã®çæ"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ã¢ã«ããããé
žåãããšã",
"title": "ã«ã«ãã³é
žã®çæ"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ãã®R以å€ã®éšåãã«ã«ããã·åºãšãããç°¡åã«-COOHãšè¡šãã",
"title": "ã«ã«ãã³é
žã®çæ"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ã«ã«ãã³é
žã®åå¿ã¯ãã®é
žãšããŠã®åŽé¢ãšã«ã«ããã«ååç©ãšããŠã®åŽé¢ã«ãã£ãŠèµ·ãããã«ã«ããã«ååç©ãšããŠã®åå¿ã¯ã±ãã³ã®ç« ã詳ããããã¢ã«ãããçãšã¢ã«ããŒã«çž®åãªã©ãèµ·ãããããšããããšãä»ãå ããŠãããã",
"title": "ã«ã«ãã³é
žã®åå¿"
}
] | ææ©ååŠïŒã«ã«ãã³é
ž | [[ææ©ååŠ]]ïŒã«ã«ãã³é
ž
==ã«ã«ãã³é
žã®å®çŸ©ãšåœåæ³==
ã«ã«ããã·ã«åºïŒCOOHããã€ååç©ãã«ã«ãã³é
žãšããã
åœéåã¯ã¢ã«ã«ã³ã®ååã®åŸã«ãé
žããã€ãããããŸã䜿ãããªãã
HCOOHãè»(ã®)é
žãCH<sub>3</sub>COOHãé
¢é
žãšããã
==ã«ã«ãã³é
žã®æ§è³ª==
ã«ã«ãã³é
žã¯ã«ã«ããã·åºãæã£ãŠããé
žæ§ã§ãããã«ã«ãã³é
žã¯æãå°ããªççŽ äžã€ã®ã®é
žãããççŽ 16åã®ãã«ããã³é
žãªã©å€§ãããæ§ã
ã§ãããã眮æåºãæããªãã«ã«ãã³é
žã¯ååéã倧ãããªãã«ã€ããæ°ŽçŽ ã€ãªã³ã®è§£é¢ãæžã£ãŠãããã€ãŸããé
žãšããŠã®åŒ·ããæžãã®ã§ããããããã©ããã£ãçç±ã«ããã®ããšãããšãã«ã«ããã·åºã®é»åå¯åºŠãé«ããäœããã«ãã£ãŠæ±ºãŸããã¢ã«ãã«åºã¯é»åäŸäžæ§åºã§ããããããçµåããŠããååã眮æåºã¯é»åãã¢ã«ãã«åºããæŒãä»ããããã®ã§(æŒãä»ãããããšããã®ã¯æ¯å©è¡šçŸã§ãã£ãŠãååãåšåããé»åã®åæ°ãå¢ãããšããããã§ã¯ãªãã)ãé»åã®å¯åºŠãé«ããªãããããããšãã«ã«ããã·åºã®é
žçŽ ã®é»åå¯åºŠãé«ããªããããæ°ŽçŽ ãšã®éã®çµåãå
ç¢ã«ãªããçµæãæ°ŽçŽ ã¯ã«ã«ããã·åºããé¢ãã«ãããªããé
žæ§åºŠãäœäžãããã¢ã«ãã«åºãæ§æããççŽ ã®æ°ãå€ãã»ã©ãã®åŸåã¯é¡èã§ããã
éã«ãã«ã«ãã³é
žã®ã«ã«ããã·åºã«é£æ¥ããççŽ ã«é»åæ±åŒæ§åºãçµåããŠããå Žåãé
žæ§åºŠã¯åŒ·ããªããäŸãã°ãã¯ããé
¢é
ž(CH<sub>2</sub>Cl-COOH)ã¯é
¢é
žããã匷ããé»åæ±åŒæ§åºã®æ°ãå¢ããã°ããã«é
žæ§åºŠã¯åŒ·ããªãã
==ã«ã«ãã³é
žã®çæ==
ã«ã«ãã³é
žã¯ä»¥äžã®åå¿ã«ãã£ãŠçæããã
*ã¢ã«ã³ãŒã«ã®é
žå(éãã³ã¬ã³é
žã«ãªãŠã KMnO<sub>4</sub>ãªã©ã®åŒ·ãé
žåå€ãçšããã)
*ã¢ã«ãããã®é
žå
*ã¢ããã®å解
*ãšã¹ãã«ã®å解
*ã°ãªãã£ãŒã«è©Šè¬ãšäºé
žåççŽ ã®åå¿
===ã¢ã«ãããã®é
žå===
ã¢ã«ããããé
žåãããšã
H O-H
| ---> |
R-C=O +(O) R-C=O
ãã®R以å€ã®éšåãã«ã«ããã·åºãšãããç°¡åã«ïŒCOOHãšè¡šãã
==ã«ã«ãã³é
žã®åå¿==
*äžããã²ã³åãªã³ãšåå¿ããŠããã²ã³åã¢ã·ã«ãšãªãã
*匷åãªéå
å€(æ°ŽçŽ åãªããŠã ã¢ã«ãããŠã ç)ãšåå¿ããŠã¢ã«ã³ãŒã«ã«ãªãã
*ã¢ã«ã³ãŒã«ãšé
žè§Šåªäžã§åå¿ããŠãšã¹ãã«ãäœãã
*ã¢ã³ã¢ãã¢ãªããã¢ãã³ãšåå¿ããŠã¢ãããšãªãã
*ã¢ã«ã«ãªéå±ãšåå¿ããŠã«ã«ãã³é
žå¡©ãäœããèæ§ãœãŒããšã«ã«ãã³é
žã®å¡©ã¯ãã£ãããšããŠããç¥ãããã
*ç¡«é
žã«ãã£ãŠè±æ°Žããã«ã«ãã³é
žç¡æ°Žç©ãšãªãã
ã«ã«ãã³é
žã®åå¿ã¯ãã®é
žãšããŠã®åŽé¢ãšã«ã«ããã«ååç©ãšããŠã®åŽé¢ã«ãã£ãŠèµ·ãããã«ã«ããã«ååç©ãšããŠã®åå¿ã¯ã±ãã³ã®ç« ã詳ããããã¢ã«ãããçãšã¢ã«ããŒã«çž®åãªã©ãèµ·ãããããšããããšãä»ãå ããŠãããã
[[ã«ããŽãª:ææ©ååŠ]] | null | 2022-11-23T05:33:09Z | [] | https://ja.wikibooks.org/wiki/%E6%9C%89%E6%A9%9F%E5%8C%96%E5%AD%A6/%E3%82%AB%E3%83%AB%E3%83%9C%E3%83%B3%E9%85%B8 |
2,067 | é«çåŠæ ¡æ°åŠI/æ°ãšåŒ | ãã¡ããåç
§
3ã12ãªã©ã®æ°(å®æ°)ãã x {\displaystyle x} ã y {\displaystyle y} ãªã©ã®æå(å€æ°)ãæãããããŠã§ããåŒãé
(ãããterm)ãšããã
次ã®ãããªãã®ãé
ã§ããã
ãã®ããã«äžã€ã®é
ã ãããã§ããŠããåŒãåé
åŒ(ãããããããmonomial)ãšããã
â» ãããããŠãæŽåŒããå®çŸ©ãããšã次ã®ãããªå®çŸ©ã«ãªãã
1ã€ä»¥äžã®åé
åŒã足ãããããŠã§ããåŒãæŽåŒ(ãããã)ãšããã
以äžã¯æŽåŒã®äŸã§ããã
åé
åŒã§ããé
ã1ã€ãããªãæŽåŒã®äžã€ã§ãããšèããããšãã§ããã®ã§ããæŽåŒããšããæŠå¿µã䜿ãããšã«ãããå€é
åŒãšåé
åŒãšã®åºå¥ã®å¿
èŠããªããªãã
x â y {\displaystyle x-y} ã®ããã«æžæ³ãå«ãåŒã¯ã x â y = x + ( â y ) = â y + x {\displaystyle x-y=x+(-y)=-y+x} ãšæžæ³ãå æ³ã«çŽãããšãã§ããã®ã§ã x , â y {\displaystyle x,-y} ãé
ã«ãã€æŽåŒã§ãããšèãããããããªãã¡ãå€é
åŒã®é
ãšã¯ãå€é
åŒã足ãç®ã®åœ¢ã«çŽãããšãã®ãäžã€äžã€ã®è¶³ããããã£ãŠããåŒã®ããšã§ãããããšãã° 5 + a â 13 x 2 y = 5 + a + ( â 13 x 2 y ) {\displaystyle 5+a-13x^{2}y=5+a+(-13x^{2}y)} ã®é
㯠5 , a , â 13 x 2 y {\displaystyle 5,a,-13x^{2}y} ã®3ã€ã§ããã
次ã®åŒã®ãã¡åé
åŒã§ãããã®ãçããã
(1), (2) ãåé
åŒã (3) ã¯é
ã6ã€ããããåé
åŒã§ã¯ãªãã
äžã®å
šãŠã®åŒã¯æŽåŒã§ãããã
3 x 2 {\displaystyle 3x^{2}} + 5 x 2 + 8 x {\displaystyle 5x^{2}+8x} ã® 3 x 2 {\displaystyle 3x^{2}} ãš 5 x 2 {\displaystyle 5x^{2}} ã®ããã«ãå€é
åŒã®æåãšææ°ããŸã£ããåãã§ããé
ãç·ç§°ããŠåé¡é
(ã©ããããããlike terms)ãšããã
åé¡é
ã¯åé
æ³å a b + a c = a ( b + c ) {\displaystyle ab+ac=a(b+c)} ã䜿ã£ãŠãŸãšããããšãã§ãããããšãã° 3 x 2 + 5 x 2 + 8 x = ( 3 + 5 ) x 2 + 8 x = 8 x 2 + 8 x {\displaystyle 3x^{2}+5x^{2}+8x=(3+5)x^{2}+8x=8x^{2}+8x} ã§ããã 8 x 2 {\displaystyle 8x^{2}} ãš 8 x {\displaystyle 8x} ã¯æåã¯åãã§ãããææ°ãç°ãªãã®ã§ãåé¡é
ã§ã¯ãªãã
次ã®å€é
åŒã®åé¡é
ãæŽçããã
3 x {\displaystyle 3x} ãšããåé
åŒã¯ã3ãšããæ°ãš x {\displaystyle x} ãšããæåã«åããŠèããããšãã§ãããæ°ã®éšåãåé
åŒã®ä¿æ°(ãããããcoefficient)ãšããã
ããšãã° â x = ( â 1 ) x {\displaystyle -x=(-1)x} ãšããåé
åŒã®ä¿æ°ã¯ -1 ã§ããã
256 x y 2 {\displaystyle 256xy^{2}} ãšããåé
åŒã¯ã256ãšããæ°ãš x , y , y {\displaystyle x,y,y} ãšããæåã«åããŠèããããšãã§ããã®ã§ããã®åé
åŒã®ä¿æ°ã¯256ã§ãããäžæ¹ãæãããããæåã®æ°ãåé
åŒã®æ¬¡æ°(ããããdegree)ãšããã 256 x y 2 {\displaystyle 256xy^{2}} 㯠x , y , y {\displaystyle x,y,y} ãšãã3ã€ã®æåãæãããããŠã§ããŠããã®ã§ããã®åé
åŒã®æ¬¡æ°ã¯3ã§ããã0ãšããåé
åŒã®æ¬¡æ°ã¯ 0 = 0 x = 0 x 2 = 0 x 3 = ⯠{\displaystyle 0=0x=0x^{2}=0x^{3}=\cdots } ãšäžã€ã«å®ãŸããªãã®ã§ãããã§ã¯èããªãã
åé
åŒã®ä¿æ°ãšæ¬¡æ°ã¯ãåã«æ°ãšæåã«åããŠèããã®ã§ã¯ãªããããæåãå€æ°ãšããŠèŠããšãã«ãæ®ãã®æåãå®æ°ãšããŠæ°ãšåãããã«æ±ãããšãããã
ããšãã° â 5 a b c x 3 {\displaystyle -5abcx^{3}} ãšããåé
åŒãã x 3 {\displaystyle x^{3}} ã ããå€æ°ã§ãæ®ãã®æå a , b , c {\displaystyle a,b,c} ã¯å®æ°ãšèããããšãã§ããã ãã®ãšã ( â 5 a b c ) x 3 {\displaystyle (-5abc)x^{3}} ãšåããããã®ã§ããã®åé
åŒã®ä¿æ°ã¯ â 5 a b c {\displaystyle -5abc} ãå€æ°ã¯ x 3 {\displaystyle x^{3}} ã§ã次æ°ã¯3ã§ãããšãããã
ãã®ããšã â 5 a b c x 3 {\displaystyle -5abcx^{3}} ãšããåé
åŒã¯ãã x {\displaystyle x} ã«çç®ãããšãä¿æ°ã¯ â 5 a b c {\displaystyle -5abc} ã次æ°ã¯3ã§ããããªã©ãšããå Žåãããã
ããã㯠â 5 a b c x 3 {\displaystyle -5abcx^{3}} ã® a {\displaystyle a} ãš b {\displaystyle b} ã«çç®ããã°ã ( â 5 c x 3 ) a b {\displaystyle (-5cx^{3})ab} ãšåãããã a {\displaystyle a} ãš b {\displaystyle b} ã«çç®ãããšãã®ãã®åé
åŒã®ä¿æ°ã¯ â 5 c x 3 {\displaystyle -5cx^{3}} ãå€æ°ã¯ a b {\displaystyle ab} ã§ã次æ°ã¯2ã§ãããšãããã
æ
£ç¿çã«ã¯ a , b , c , ⯠{\displaystyle a,b,c,\cdots } ãªã©ã®ã¢ã«ãã¡ãããã®æåã®æ¹ã®æåãå®æ°ãè¡šãã®ã«äœ¿ãã ⯠, x , y , z {\displaystyle \cdots ,x,y,z} ãªã©ã®ã¢ã«ãã¡ãããã®æåŸã®æ¹ã®æåãå€æ°ãè¡šãã®ã«çšããããäžè¬çã«ã¯ãã®éãã§ãªãã
å€é
åŒã®æ¬¡æ°ãšã¯ãå€é
åŒã®åé¡é
ããŸãšãããšãã«ããã£ãšã次æ°ã®é«ãé
ã®æ¬¡æ°ããããããšãã° x 3 + 3 x 2 y + 2 y {\displaystyle x^{3}+3x^{2}y+2y} ã§ã¯ããã£ãšã次æ°ã®é«ãé
㯠x 3 {\displaystyle x^{3}} ã§ããã®ã§ããã®å€é
åŒã®æ¬¡æ°ã¯3ã§ããããã x 3 + 3 x 2 y + 2 y {\displaystyle x^{3}+3x^{2}y+2y} ( x {\displaystyle x} ã¯å®æ°)ã§ããã°ãããªãã¡å€é
åŒã® y {\displaystyle y} ã«ã€ããŠçç®ãããšããã£ãšã次æ°ã®é«ãé
㯠3 x 2 y {\displaystyle 3x^{2}y} ãš 2 y {\displaystyle 2y} ã§ããã®ã§ããã®å€é
åŒã®æ¬¡æ°ã¯1ã§ããããã®ãšãçç®ããæåãå«ãŸãªãé
x 3 {\displaystyle x^{3}} ã¯å®æ°é
(ãŠããããããconstant term)ãšããŠæ°ãšåãããã«æ±ãããã
次ã®å€é
åŒã® x {\displaystyle x} ãŸã㯠y {\displaystyle y} ã«çç®ãããšãã®æ¬¡æ°ãšå®æ°é
ãããããããã
ããšãã°ã
ã®ããã«ã次æ°ã®é«ãé
ããå
ã«é
ããªãã¹ãããšããéã¹ãã(ããã¹ã)ãšããã
ããŠãåŒã䜿ãç®çã«ãã£ãŠã¯ã次æ°ã®ã²ããé
ããå
ã«æžããã»ãã䟿å©ãªå Žåãããã
ããšãã°ã x {\displaystyle x} ã çŽ0.01 ã®ãããª1æªæºã®å°ããæ°ã®å ŽåãåŒ x 2 + 6 x + 7 {\displaystyle x^{2}+6x+7} ã®å€ãæ±ããããªããæå x {\displaystyle x} ã®æ¬¡æ°ã®å°ããé
ã®ã»ãã圱é¿ãé«ãã
ãªã®ã§ã ç®çã«ãã£ãŠã¯
ã®ããã«ã次æ°ã®ã²ããé
ããå
ã«æžãå Žåãããã
7 + 6 x + x 2 {\displaystyle 7+6x+x^{2}} ã®ããã«ã次æ°ã®äœãé
ããå
ã«é
ããªãã¹ãããšããæã¹ãã(ãããã¹ã)ãšããã
å€é
åŒã«2ã€ä»¥äžã®æåããããšããç¹å®ã®1ã€ã®æåã«æ³šç®ããŠäžŠã³å€ãããšã䜿ãããããªãããšãããã
ããšãã°ã
ã®é
ããxã®æ¬¡æ°ãå€ãé
ããå
ã«äžŠã³ãããåé¡é
ããŸãšãããš
ãšãªãã
ãã®(äŸ2)ã®ããã«ãç¹å®ã®æåã ãã«çç®ããŠããã®æåã®æ¬¡æ°ã®é«ãé ã«äžŠã³ããããšäŸ¿å©ãªããšããã°ãã°ããã
äŸ2ã¯ã x {\displaystyle x} ã«ã€ã㊠éã¹ã ã®é ã«äžŠã³å€ããæŽåŒã§ããã
çç®ããŠãªãæåã«ã€ããŠã¯ã䞊ã³æãã®ãšãã¯å®æ°ã®ããã«æ±ãã
ãã£ãœãã x {\displaystyle x} ã«ã€ããŠã次æ°ã®ã²ããé
ããé ã«äžŠã¹ããšã次ã®ãããªåŒã«ãªãã
ãã®ããã«ãç¹å®ã®æåã®æ¬¡æ°ãäœããã®ããé ã«äžŠã³ããããšäŸ¿å©ãªããšããã°ãã°ããã
äŸ3ã¯ãxã«ã€ã㊠æã¹ã ã®é ã«äžŠã³å€ããæŽåŒã§ããã
ããšãã°ãåŒ
ãšããåŒã®å³èŸº
ã®æ¬¡æ°ã¯ããããã§ããããã
aãšxãçããæåãšããŠæ±ãã®ã§ããã°ã a x {\displaystyle ax} ã®æ¬¡æ°ã¯
ãã 1+1 =2 ãªã®ã§ããã®åŒã®æ¬¡æ°ã¯2ã§ããã(é
bã¯æ¬¡æ°1ãªã®ã§ã a x {\displaystyle ax} ã®æ¬¡æ°2ãããäœãã®ã§ç¡èŠããã)
ãããããããã®åŒããå®æ° a {\displaystyle a} ãä¿æ°ãšããå€æ° x {\displaystyle x} ã«ã€ããŠã®äžæ¬¡é¢æ°ãšã¿ãã®ã§ããã°ãäžæ¬¡åŒãšæãã®ãåççã ããã
ãã®ãããªå Žåãç¹å®ã®æåã ãã«æ³šç®ãããã®åŒã®æ¬¡æ°ãèãããšããã
ããšãã°ãæåxã ãã«æ³šç®ããŠãåŒ a x + b {\displaystyle ax+b} ã®æ¬¡æ°ã決ããŠã¿ããã
ãããšãæåxã«æ³šç®ããå Žåã®åŒ a x + b {\displaystyle ax+b} ã®æ¬¡æ°ã¯1ã«ãªãã
ãªããªã
ãã£ãŠãæå x {\displaystyle x} ã«æ³šç®ããå Žåã®é
a x {\displaystyle ax} ã®æ¬¡æ°ã¯ã 0+1 ãªã®ã§ã1ã§ããã
ãã®ããã«èããå Žåãå¿
èŠã«å¿ããŠã©ã®æåã«æ³šç®ããããæèšããŠãæåâ¯â¯ã«æ³šç®ãã次æ°ãã®ããã«è¿°ã¹ããšããã
å€é
åŒã®ç©ã¯åé
æ³åã䜿ã£ãŠèšç®ããããšãã§ããã
ãã®ããã«å€é
åŒã®ç©ã§è¡šãããåŒãäžã€ã®å€é
åŒã«ç¹°ãåºããããšããå€é
åŒãå±é(ãŠããããexpand)ãããšããã
a {\displaystyle a} ã n {\displaystyle n} åæãããã®ã a n {\displaystyle a^{n}} ãšæžããaã®nä¹(-ããããa to the n-th power)ãšããããã ã a 1 = a {\displaystyle a^{1}=a} ãšå®çŸ©ãããããšãã°ã
ã§ããã a , a 2 , a 3 , a 4 , a 5 , ⯠, a n {\displaystyle a,a^{2},a^{3},a^{4},a^{5},\cdots ,a^{n}} ãç·ç§°ã㊠a {\displaystyle a} ã®çŽ¯ä¹(ããããããexponentiationãåªä¹ãã¹ãããããåªãã¹ã)ãšããã a n {\displaystyle a^{n}} ã® n ãææ°(ããããexponent)ãšãã(a ã¯åº(ãŠããbase)ãšãã)ãããã§ã¯èªç¶æ°ãããªãã¡æ£ã®æŽæ°ã®ææ°ãèããã环ä¹ã¯æ¬¡ã®ããã«èããããšãã§ããã
环ä¹ã©ãããæãããããç©ã¯ã次ã®ããã«èšç®ããããšãã§ããã
环ä¹ã©ãããå²ã£ãåã¯ã次ã®ããã«èšç®ããããšãã§ããã
环ä¹ã®çŽ¯ä¹ã¯ã次ã®ããã«èšç®ããããšãã§ããã
ç©ã®çŽ¯ä¹ã¯ã次ã®ããã«èšç®ããããšãã§ããã
ããããããããŠææ°æ³å(ãããã»ããããexponential law)ãšããã
环ä¹ã®å®çŸ©ããæããã
次ã®åŒãèšç®ããªããã
次ã®åŒãå±éããã
ãŸãšãããšã次ã®ããã«ãªãã
次ã®åŒãå±éããªããã
è€éãªåŒã®å±éã¯ãåŒã®äžéšåãäžã€ã®æåã«ãããŠå
¬åŒã䜿ããšããã
次ã®åŒãå±éããªããã
次ã®åŒãå æ°å解ããªããã
次ã®åŒãå æ°å解ããªããã
次ã®åŒãå æ°å解ããªããã
a=b^2ãæãç«ã€ãšããa=2ãšãªããããªbãããªãã¡ 2 {\displaystyle {\sqrt {2}}} ã®å
·äœçãªå€ãã©ã®ãããªãã®ãã調ã¹ãŠã¿ããã
ãã®ããã«ãbãæ§ã
ã«æ±ºããŠããaã¯ãªããªã2ã«ãªããªãã
å®ã¯ 2 {\displaystyle {\sqrt {2}}} ã¯ãåæ¯ååå
±ã«æŽæ°ã®åæ°ã§è¡šãããšã¯ã§ããªãããã®ããã«æŽæ°ãåæ¯ååã«æã€åæ°ã§è¡šããªããããªæ°ãç¡çæ°ãšãããäŸãã°ãååšçÏã¯ç¡çæ°ã§ãããããã«å¯ŸããŠãæŽæ°ã埪ç°å°æ°ãªã©ãåæ¯ååå
±ã«æŽæ°ã®åæ°ã§è¡šãããšã®ã§ããæ°ãæçæ°ãšããã
æçæ°ãšç¡çæ°ãåãããŠå®æ°ãšãããã©ããªå®æ°ã§ãæ°çŽç·äžã®ç¹ãšããŠè¡šããããŸããã©ããªå®æ°ããæéå°æ°ãããã¯ç¡éå°æ°ãšããŠè¡šããã (äžèšã®ãç¡éå°æ°ãã®ç¯ãåç
§)
2 {\displaystyle {\sqrt {2}}} ãæçæ°ã§ãããšä»®å®ãããšãäºãã«çŽ ãª(1以å€ã«å
¬çŽæ°ããããªã)æŽæ° m, n ãçšããŠã
ãšè¡šããããšãã§ããããã®ãšãã䞡蟺ã2ä¹ããŠåæ¯ãæããšã
ãã£ãŠ m ã¯2ã®åæ°ã§ãããæŽæ° l ãçšã㊠m = 2 l {\displaystyle m=2l} ãšè¡šãããšãã§ãããããã (1) ã®åŒã«ä»£å
¥ããŠæŽçãããšã
ãã£ãŠ n ã2ã®åæ°ã§ãããããã㯠m, n ã2ãå
¬çŽæ°ã«ãã€ããšã«ãªããäºãã«çŽ ãšä»®å®ããããšã«ççŸããããããã£ãŠ 2 {\displaystyle {\sqrt {2}}} ã¯ç¡çæ°ã§ãã(èçæ³)ã
0.1 ã 0.123456789 ã®ããã«ãããäœã§çµããå°æ°ãæéå°æ°ãšããã
äžæ¹ã 0.1234512345 ⯠{\displaystyle 0.1234512345\cdots } ã 3.1415926535 ⯠{\displaystyle 3.1415926535\cdots } ã®ããã«ç¡éã«ç¶ãå°æ°ã ç¡éå°æ°(ããã ããããã)ãšããã
ç¡éå°æ°ã®ãã¡ãããäœããäžãããããé
åã®æ°åã®ç¹°ãè¿ãã«ãªã£ãŠãããã®ã 埪ç°å°æ°(ãã
ããã ããããã)ãšãããäŸãã° 0.3333333333 ⯠{\displaystyle 0.3333333333\cdots } ã 0.1428571428 ⯠{\displaystyle 0.1428571428\cdots } ã 0.1232323232 ⯠{\displaystyle 0.1232323232\cdots } ãªã©ã§ãããç¹°ãè¿ãã®æå°åäœã埪ç°ç¯ãšããã埪ç°å°æ°ã¯åŸªç°ç¯1ã€ãçšã㊠0. 3 Ì {\displaystyle 0.{\dot {3}}} ã 0. 1 Ì 4285 7 Ì {\displaystyle 0.{\dot {1}}4285{\dot {7}}} ã 0.1 2 Ì 3 Ì {\displaystyle 0.1{\dot {2}}{\dot {3}}} ã®ããã«åŸªç°ç¯ã®æåãšæåŸ(埪ç°ç¯ãäžæ¡ã®å Žåã¯ã²ãšã€ã ã)ã®äžã«ç¹ãã€ããŠè¡šãã
å
šãŠã®åŸªç°å°æ°ã¯åæ°ã«çŽããã
ãšçœ®ããšã
ã§ããã(2)ãŒ(1) ãã 9 a = 3 {\displaystyle 9a=3} ããã£ãŠ a = 3 9 = 1 3 {\displaystyle a={\frac {3}{9}}={\frac {1}{3}}} ã§ããã
a = 0. 1 Ì 4285 7 Ì 1000000 a = 142857. 1 Ì 4285 7 Ì 999999 a = 142857 a = 142857 999999 = 1 7 {\displaystyle {\begin{aligned}a&=0.{\dot {1}}4285{\dot {7}}\\1000000a&=142857.{\dot {1}}4285{\dot {7}}\\999999a&=142857\\a&={\frac {142857}{999999}}\ ={\frac {1}{7}}\end{aligned}}}
a = 0.1 2 Ì 3 Ì 100 a = 12.3 2 Ì 3 Ì 99 a = 12.2 a = 12.2 99 = 61 495 {\displaystyle {\begin{aligned}a&=0.1{\dot {2}}{\dot {3}}\\100a&=12.3{\dot {2}}{\dot {3}}\\99a&=12.2\\a&={\frac {12.2}{99}}\ ={\frac {61}{495}}\end{aligned}}}
å®æ° a ã«ã€ããŠãa ã®æ°çŽç·äžã§ã®åç¹ãšã®è·é¢ã a ã®çµ¶å¯Ÿå€ãšããã | a | {\displaystyle |a|} ã§è¡šãã
ããšãã°
ã§ããã
å®çŸ©ãã | a | = | â a | {\displaystyle |a|=|-a|} ããããããŸãã a , b {\displaystyle a,b} ãä»»æã®å®æ°ãšãããšããããããã«å¯Ÿå¿ããæ°çŽç·äžã®ä»»æã®2ç¹ P ( a ) , Q ( b ) {\displaystyle \mathrm {P} (a),\mathrm {Q} (b)} éã®è·é¢ã«ã€ããŠã¯ã次ã®ããšããããã
ä»ã2ä¹ããŠaã«ãªãæ°bãèããã
a = 1 {\displaystyle a=1} ã®ãšãã b = 1 {\displaystyle b=1} ãšããŠçµããã«ããŠã¯ãããªãã確ãã« b = 1 {\displaystyle b=1} ãæ¡ä»¶ãæºããã b = â 1 {\displaystyle b=-1} ãæ¡ä»¶ãæºããããã£ãŠ b = 1 {\displaystyle b=1} ãŸã㯠b = â 1 {\displaystyle b=-1} ã§ããã
äžè¬ã«æ£ã®æ°aã«ã€ããŠa=b^2ãšãªãbã¯äºã€ããããã®äºã€ã¯çµ¶å¯Ÿå€ãçããããã®äºã€ã®bãaã®å¹³æ¹æ ¹ãšãããaã®å¹³æ¹æ ¹ã®ãã¡ãæ£ã§ãããã®ã a {\displaystyle {\sqrt {a}}} ãè² ã§ãããã®ã â a {\displaystyle -{\sqrt {a}}} ãšæžãã a {\displaystyle {\sqrt {a}}} ã¯ãã«ãŒãaããšèªãã
äžæ¹ãè² ã®æ°aã«ã€ããŠèããŠã¿ãŠãäžæãbãèŠã€ããããšã¯ã§ããªããå®éã®ãšãããè² ã®æ°ã®å¹³æ¹æ ¹ã¯å®æ°ã§è¡šãããšã¯ã§ããªãã
2 , 4 , 9 , 12 {\displaystyle 2\ ,\ 4\ ,\ 9\ ,\ 12} ã®å¹³æ¹æ ¹ãæ±ããã
± 2 , ± 2 , ± 3 , ± 2 3 {\displaystyle \pm {\sqrt {2}}\ ,\ \pm 2\ ,\ \pm 3\ ,\ \pm 2{\sqrt {3}}}
ããããã®ã«ãŒããèšç®ãã ± {\displaystyle \pm } ãã€ããã°ããããã ããå¹³æ¹æ ¹ã®ã«ãŒã«ã«åŸã£ãŠãç°¡ååã§ãããã®ã¯ç°¡ååããããšãèŠæ±ãããã
äŸãã°ã 2 {\displaystyle 2} ã«å¯ŸããŠã¯ã ± 2 {\displaystyle \pm {\sqrt {2}}} ãšãªãã
äžè¬ã«ã A 2 = | A | {\displaystyle {\sqrt {A^{2}}}=|A|} ã§ããã
æ ¹å·ã«ã€ããŠã次ã®å
¬åŒãæãç«ã€ã
ãŸãã a b {\displaystyle {\sqrt {ab}}} ãšã¯ãå®çŸ©ã«ããšã¥ããŠèãããšã2ä¹ãããš ab ã«ãªãæ°ã®ãã¡ãæ£ã®ã»ãã®æ°ãšããæå³ã§ããã
ãªã®ã§ãå
¬åŒã a b = a b {\displaystyle {\sqrt {a}}{\sqrt {b}}={\sqrt {ab}}} ãã蚌æããã«ã¯ããã®ããšã蚌æããã°ããã
ãªã®ã§ããŸãã a b {\displaystyle {\sqrt {a}}{\sqrt {b}}} ã2ä¹ãããšã
ãšãªãã
ããã« a b {\displaystyle {\sqrt {a}}{\sqrt {b}}} ã¯ããŸãæ¡ä»¶ã2ä¹ãããšabã«ãªãããæºããã
ãããŠãæ£ã®æ°ã®å¹³æ¹æ ¹ã¯æ£ãªã®ã§ã a b {\displaystyle {\sqrt {a}}{\sqrt {b}}} ãæ£ã§ããããã£ãŠ a b {\displaystyle {\sqrt {a}}{\sqrt {b}}} ã¯ãã2ä¹ãããšabã«ãªããæ°ã®ãã¡ã®æ£ã®ã»ãã§ããã
(蚌æããã)
ããã«ãäžã®å
¬åŒ(1)ã«ããã次ã®å
¬åŒãå°ãããã
èšç®ããã
åæ¯ã«æ ¹å·ãå«ãŸãªãåŒã«ããããšããåæ¯ãæçåãããšãããæçåã¯ãåæ¯ãšååã«åãæ°ããããŠãããããšãå©çšããŠè¡ãã
ããšãã°ã 1 2 {\displaystyle {\frac {1}{\sqrt {2}}}} ãæçåãããšã 1 2 = 1 2 2 2 = 2 2 {\displaystyle {\frac {1}{\sqrt {2}}}\ =\ {\frac {1{\sqrt {2}}}{{\sqrt {2}}{\sqrt {2}}}}\ =\ {\frac {\sqrt {2}}{2}}} ãšãªãã
ãŸãããšãã« a b + c {\displaystyle {\frac {a}{b+c}}} ã«ã€ããŠã b 2 â c 2 = 1 {\displaystyle b^{2}-c^{2}=1} ã®ãšãã a b + c = a ( b â c ) ( b + c ) ( b â c ) = a ( b â c ) b 2 â c 2 = a ( b â c ) 1 = a ( b â c ) {\displaystyle {\frac {a}{b+c}}\ =\ {\frac {a(b-c)}{(b+c)(b-c)}}\ =\ {\frac {a(b-c)}{b^{2}-c^{2}}}\ =\ {\frac {a(b-c)}{1}}\ =\ a(b-c)} ã§ããã
ããšãã°ã a = 1 , b = 2 , c = 1 {\displaystyle a=1,b={\sqrt {2}},c=1} ãšãããšã 1 2 + 1 = 2 â 1 {\displaystyle {\frac {1}{{\sqrt {2}}+1}}={\sqrt {2}}-1} ã§ããã
åæ¯ãæçåããã
äºéæ ¹å·ãšã¯ãæ ¹å·ã2éã«ãªã£ãŠããåŒã®ããšã§ãããäºéæ ¹å·ã¯åžžã«å€ããããã§ã¯ãªããæ ¹å·ã®äžã«å«ãŸããåŒã«ãã£ãŠç°¡åã«ã§ãããã©ããã決ãŸããäžè¬ã«ãæ ¹å·å
ã®åŒãã x 2 {\displaystyle x^{2}} ã®åœ¢ã«å€åœ¢ã§ããå Žåã«ã¯ãå€åŽã®æ ¹å·ãå€ãããšãã§ããã
3 + 2 2 {\displaystyle {\sqrt {3+2{\sqrt {2}}}}} ãç°¡åã«ããã
3 + 2 2 {\displaystyle 3+2{\sqrt {2}}} ã ( ⯠) 2 {\displaystyle (\cdots )^{2}} ã®åœ¢ã«ã§ããããèããã
ä»®ã«ã ( a + b ) 2 {\displaystyle ({\sqrt {a}}+{\sqrt {b}})^{2}} (a,bã¯æ£ã®æŽæ°)ã®åœ¢ã«ã§ãããšãããšã 3 + 2 2 = a + b + 2 a b {\displaystyle 3+2{\sqrt {2}}=a+b+2{\sqrt {ab}}} ãšãªãã
ãæºããæŽæ°a,bãæ¢ãã°ããããã®é¢ä¿ã¯ãa=1,b=2(a,bãå
¥ãæããŠãå¯ã)ã«ãã£ãŠæºããããã®ã§ã 3 + 2 2 = ( 2 + 1 ) 2 {\displaystyle 3+2{\sqrt {2}}\ =\ ({\sqrt {2}}+1)^{2}} ãæãç«ã€ã
ãã£ãŠã 3 + 2 2 = ( 2 + 1 ) 2 = 2 + 1 {\displaystyle {\sqrt {3+2{\sqrt {2}}}}\ =\ {\sqrt {({\sqrt {2}}+1)^{2}}}\ =\ {\sqrt {2}}+1} ãšãªãã
次ã®åŒãèšç®ããã
åã倧ããã®éã=ã§çµãã åŒãæ¹çšåŒãšåŒã¶ããšãæ¢ã«åŠç¿ãããããã§ã¯ãç°ãªã£ãéã®å€§ããã®éããè¡šãèšå·ãå°å
¥ãããã®æ§è³ªã«ã€ããŠãŸãšããã
ããæ°A,BããããšããAãBãã倧ããããšã A > B {\displaystyle A>B} ãšè¡šããAãBããå°ããããšã A < B {\displaystyle A<B} ãšè¡šããããã§ã<ãš>ã®ããšãäžçå·ãšåŒã³ããã®ãããªåŒãäžçåŒãšåŒã¶ããŸãã †, ⥠{\displaystyle \leq ,\geq } ã䌌ãæå³ã®äžçåŒã§ããããããããAãšBãçããå€ã§ããå Žåãå«ããã®ã§ããã
ãªããæ¥æ¬ã®æè²ã«ãããŠã¯ã †, ⥠{\displaystyle \leq ,\geq } ã®ä»£ããã«ãäžçå·ã®äžã«çå·ãèšãã ⊠, ⧠{\displaystyle \leqq ,\geqq } ã䜿ãããšãå€ãã
x > 7 {\displaystyle x>7} ãšããäžçåŒããããšããxã¯7ãã倧ããå®æ°ã§ããããŸãã x ⥠7 {\displaystyle x\geq 7} ã®æã«ã¯ãxã¯7以äžã®å®æ°ã§ããã
äžçåŒã§ã¯çåŒãšåãããã«ã䞡蟺ã«æŒç®ãããŠãäžçå·ã®é¢ä¿ãå€ãããªãããšããããäŸãã°ã䞡蟺ã«åãæ°ã足ããŠãã䞡蟺ã®å€§å°é¢ä¿ã¯å€åããªãããã ãã䞡蟺ã«è² ã®æ°ãããããšãã«ã¯ãäžçå·ã®åããå€åããããšã«æ³šæãå¿
èŠã§ãããããã¯ãè² ã®æ°ãããããšäž¡èŸºã®å€ã¯ã0ãäžå¿ã«æ°çŽç·ãæãè¿ããå°ç¹ã«ç§»ãããããšã«ããã
x > y {\displaystyle x>y} ãæãç«ã€ãšãã«ã¯ã x + 3 > y + 3 {\displaystyle x+3>y+3} ã 4 x > 4 y {\displaystyle 4x>4y} ãæãç«ã€ããŸãã â x < â y {\displaystyle -x<-y} ãæãç«ã€ã
äžçåŒã®æ§è³ªã䜿ã£ãŠ
ã®äž¡èŸºãã3ãåŒããš
ãã£ãŠ
ãšãªãã ãã®ããã«ãäžçåŒã§ã移é
ããããšãã§ããã
ã°ã©ããçšããŠèãããšããäžçåŒã¯ã°ã©ãäžã®é åãè¡šããé åã®å¢çã¯äžçå·ãçå·ã«çœ®ãæããéšåã察å¿ãããããã¯ãäžçå·ãæç«ãããã©ããããã®ç·äžã§å
¥ãæ¿ããããšã«ãã£ãŠããã(詳ããã¯æ°åŠII å³åœ¢ãšæ¹çšåŒã§åŠç¿ããã)
y > x + 1 {\displaystyle y>x+1} , y < 2 x + 1 {\displaystyle y<2x+1} , x < 3 {\displaystyle x<3} ã®ã°ã©ã(æ£ããã¯ãé åã)ãæãã
y > x + 1 {\displaystyle y>x+1} ã®ã°ã©ã(é å)ã¯æ¬¡ã®ããã«ãªãããã ããå¢çã¯å«ãŸãªãã
y < 2 x + 1 {\displaystyle y<2x+1} ã®ã°ã©ã(é å)ã¯æ¬¡ã®ããã«ãªãããã ããå¢çã¯å«ãŸãªãã
x < 3 {\displaystyle x<3} ã®ã°ã©ã(é å)ã¯æ¬¡ã®ããã«ãªãããã ããå¢çã¯å«ãŸãªãã
次ã®äžçåŒã解ãã
ããã€ãã®äžçåŒãçµã¿åããããã®ãé£ç«äžçåŒãšããããããã®äžçåŒãåæã«æºãã x {\displaystyle x} ã®å€ã®ç¯å²ãæ±ããããšããé£ç«äžçåŒã解ããšããã
次ã®é£ç«äžçåŒã解ãã (i)
(ii)
(i) x + 2 < 2 x + 4 {\displaystyle x+2<2x+4} ãã â x < 2 {\displaystyle -x<2}
10 â x ⥠3 x â 6 {\displaystyle 10-x\geq 3x-6} ãã â 4 x ⥠â 16 {\displaystyle -4x\geq -16}
(1),(2)ãåæã«æºãã x {\displaystyle x} ã®å€ã®ç¯å²ã¯
(ii) x ⥠1 â x {\displaystyle x\geq 1-x} ãã 2 x ⥠1 {\displaystyle 2x\geq 1}
2 ( x + 1 ) > x â 2 {\displaystyle 2(x+1)>x-2} ãã 2 x + 2 > x â 2 {\displaystyle 2x+2>x-2}
(1),(2)ãåæã«æºãã x {\displaystyle x} ã®å€ã®ç¯å²ã¯
絶察å€ãå«ãäžçåŒã«ã€ããŠèãããã çµ¶å¯Ÿå€ | x | {\displaystyle |x|} ã¯ãæ°çŽç·äžã§ãåç¹ O {\displaystyle \mathrm {O} } ãšç¹ P ( x ) {\displaystyle \mathrm {P} (x)} ã®éã®è·é¢ãè¡šããŠããã ãããã£ãŠã a > 0 {\displaystyle a>0} ã®ãšã
次ã®äžçåŒã解ãã (i)
(ii)
(iii)
(iv)
(i)
(ii)
(iii)
(iv)
äžè¬ã®äºæ¬¡æ¹çšåŒ a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} ( a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} ã¯å®æ°ã a â 0 {\displaystyle a\neq 0} )ã®è§£ x {\displaystyle x} ãæ±ããå
¬åŒã«ã€ããŠèããã
ããã§æçåŒ x 2 + 2 y x = ( x + y ) 2 â y 2 {\displaystyle x^{2}+2yx=(x+y)^{2}-y^{2}} ãš (1) ã®å·ŠèŸºãä¿æ°æ¯èŒãããšã
ã§ããããã(1) ã®åŒã¯æ¬¡ã®ããã«å€åœ¢ã§ãã(å¹³æ¹å®æ)ã
b 2 â 4 a c ⥠0 {\displaystyle b^{2}-4ac\geq 0} ã®ãšã䞡蟺ã®å¹³æ¹æ ¹ããšããšã
ãããäºæ¬¡æ¹çšåŒã®è§£ã®å
¬åŒ(ã«ãã»ããŠãããã®ããã®ãããããquadratic formula; äºæ¬¡å
¬åŒ)ã§ããã解ã®å
¬åŒãäºæ¬¡æ¹çšåŒã®äžè¬åœ¢ã«ä»£å
¥ãããšãå³èŸºã¯0ã«ãªãã¯ãã§ããã
ã§ããããšãçšãããšã
ãšãªãã確ãã«æ£ããããšããããã
ããããã解ã®å
¬åŒãå æ°å解ãçšããŠè§£ããªããã
çµæã®åŒã«æ ¹å·ãçŸããªãå Žåã«ã¯ãäœããã®ä»æ¹ã§å æ°å解ãã§ããããããããããã®æ¹æ³ã䜿ãã«ãããæ ¹å·ã¯ã§ããéãã®ä»æ¹ã§ç°¡ååããããšãéèŠã§ããã
(i)ã¯ç°¡åã«å æ°å解ã§ããã®ã§ã解ã®å
¬åŒãçšããå¿
èŠã¯ãªãã
ããã
ãçããšãªãã(ii)ã§ã¯ãå æ°å解ãåºæ¥ãªãã®ã§ã解ã®å
¬åŒãçšãããå æ°å解ãã§ãããã©ããã¯å®éã«è©Šè¡é¯èª€ããŠèŠåãããããªãã
ã«ã解ã®å
¬åŒãçšãããšãa=5, b= 2, c=-1ããã
ãšãªãã(iii),(iv)ã§ããå æ°å解ã¯åºæ¥ãªãã®ã§ã解ã®å
¬åŒãçšãããçãã¯ã (iii)
(iv)
(v)
ãšå æ°å解ã§ããã®ã§ãçãã¯
ãšãªãã
å
šåãéããŠãå æ°å解ãå¯èœãªæ¹çšåŒã«å¯ŸããŠãã解ã®å
¬åŒã䜿çšããŠãæ§ããªãã
äºæ¬¡æ¹çšåŒ a x 2 + 2 b â² x + c = 0 ( a â 0 ) {\displaystyle ax^{2}+2b'x+c=0(a\neq 0)} ã«ã€ããŠèããã 解ã®å
¬åŒã« b= 2b' ã代å
¥ãããš
ãã£ãŠãäºæ¬¡æ¹çšåŒ a x 2 + 2 b â² x + c = 0 {\displaystyle ax^{2}+2b'x+c=0} ã®è§£ã¯
ãšãªãã
ãäžã®è§£ã®å
¬åŒãçšããŠè§£ããªããã
äžã®è§£ã®å
¬åŒãçšãããšãa=3, b'= 3, c=-2ããã
ãšãªãã
2次æ¹çšåŒ a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} ã®è§£ã¯ x = â b ± b 2 â 4 a c 2 a {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}} ã§ããã ãã®åŒã®æ ¹å·ã®äžèº«ã ãåãåºãããã®ãå€å¥åŒãšåŒã³ã2次æ¹çšåŒã®è§£ã®åæ°ãç°¡åã«å€å¥ã§ããã
D = b 2 â 4 a c {\displaystyle D=b^{2}-4ac} ã®å€ã«ãã£ãŠæ¬¡ã®ããã«ãªãã
(1) D > 0 {\displaystyle D>0} ã®ãšããç°ãªã2ã€ã®è§£ x = â b + b 2 â 4 a c 2 a {\displaystyle x={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}} ãš x = â b â b 2 â 4 a c 2 a {\displaystyle x={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}} ãæã€ã (2) D = 0 {\displaystyle D=0} ã®ãšãã ± b 2 â 4 a c = ± 0 {\displaystyle \pm {\sqrt {b^{2}-4ac}}=\pm 0} ã§ããã®ã§ã2ã€ã®è§£ã¯äžèŽããŠããã 1ã€ã®è§£ x = â b 2 a {\displaystyle x=-{\frac {b}{2a}}} ãæã€ãããã¯2ã€ã®è§£ãéãªã£ããã®ãšèããŠãé解ãšããã (3) D < 0 {\displaystyle D<0} ã®ãšããå®æ°ã®ç¯å²ã§ã¯è§£ã¯ãªãã
2次æ¹çšåŒ a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} ã®è§£ã®åæ°ã¯ D = b 2 â 4 a c {\displaystyle D=b^{2}-4ac} ã®å€ã§å€å®ã§ããã
次ã®2次æ¹çšåŒã®è§£ã®åæ°ãæ±ããã
(I)
ã ãããå®æ°è§£ã¯ãªãã (II)
ã ãããé解ããã€ã (III)
ã ãããç°ãªã2ã€ã®å®æ°ã®è§£ããã€ã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ãã¡ããåç
§",
"title": "éåãšè«ç"
},
{
"paragraph_id": 1,
"tag": "p",
"text": "3ã12ãªã©ã®æ°(å®æ°)ãã x {\\displaystyle x} ã y {\\displaystyle y} ãªã©ã®æå(å€æ°)ãæãããããŠã§ããåŒãé
(ãããterm)ãšããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "次ã®ãããªãã®ãé
ã§ããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ãã®ããã«äžã€ã®é
ã ãããã§ããŠããåŒãåé
åŒ(ãããããããmonomial)ãšããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "â» ãããããŠãæŽåŒããå®çŸ©ãããšã次ã®ãããªå®çŸ©ã«ãªãã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "1ã€ä»¥äžã®åé
åŒã足ãããããŠã§ããåŒãæŽåŒ(ãããã)ãšããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "以äžã¯æŽåŒã®äŸã§ããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "åé
åŒã§ããé
ã1ã€ãããªãæŽåŒã®äžã€ã§ãããšèããããšãã§ããã®ã§ããæŽåŒããšããæŠå¿µã䜿ãããšã«ãããå€é
åŒãšåé
åŒãšã®åºå¥ã®å¿
èŠããªããªãã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "x â y {\\displaystyle x-y} ã®ããã«æžæ³ãå«ãåŒã¯ã x â y = x + ( â y ) = â y + x {\\displaystyle x-y=x+(-y)=-y+x} ãšæžæ³ãå æ³ã«çŽãããšãã§ããã®ã§ã x , â y {\\displaystyle x,-y} ãé
ã«ãã€æŽåŒã§ãããšèãããããããªãã¡ãå€é
åŒã®é
ãšã¯ãå€é
åŒã足ãç®ã®åœ¢ã«çŽãããšãã®ãäžã€äžã€ã®è¶³ããããã£ãŠããåŒã®ããšã§ãããããšãã° 5 + a â 13 x 2 y = 5 + a + ( â 13 x 2 y ) {\\displaystyle 5+a-13x^{2}y=5+a+(-13x^{2}y)} ã®é
㯠5 , a , â 13 x 2 y {\\displaystyle 5,a,-13x^{2}y} ã®3ã€ã§ããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "次ã®åŒã®ãã¡åé
åŒã§ãããã®ãçããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "(1), (2) ãåé
åŒã (3) ã¯é
ã6ã€ããããåé
åŒã§ã¯ãªãã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "äžã®å
šãŠã®åŒã¯æŽåŒã§ãããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "3 x 2 {\\displaystyle 3x^{2}} + 5 x 2 + 8 x {\\displaystyle 5x^{2}+8x} ã® 3 x 2 {\\displaystyle 3x^{2}} ãš 5 x 2 {\\displaystyle 5x^{2}} ã®ããã«ãå€é
åŒã®æåãšææ°ããŸã£ããåãã§ããé
ãç·ç§°ããŠåé¡é
(ã©ããããããlike terms)ãšããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "åé¡é
ã¯åé
æ³å a b + a c = a ( b + c ) {\\displaystyle ab+ac=a(b+c)} ã䜿ã£ãŠãŸãšããããšãã§ãããããšãã° 3 x 2 + 5 x 2 + 8 x = ( 3 + 5 ) x 2 + 8 x = 8 x 2 + 8 x {\\displaystyle 3x^{2}+5x^{2}+8x=(3+5)x^{2}+8x=8x^{2}+8x} ã§ããã 8 x 2 {\\displaystyle 8x^{2}} ãš 8 x {\\displaystyle 8x} ã¯æåã¯åãã§ãããææ°ãç°ãªãã®ã§ãåé¡é
ã§ã¯ãªãã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "次ã®å€é
åŒã®åé¡é
ãæŽçããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "3 x {\\displaystyle 3x} ãšããåé
åŒã¯ã3ãšããæ°ãš x {\\displaystyle x} ãšããæåã«åããŠèããããšãã§ãããæ°ã®éšåãåé
åŒã®ä¿æ°(ãããããcoefficient)ãšããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "ããšãã° â x = ( â 1 ) x {\\displaystyle -x=(-1)x} ãšããåé
åŒã®ä¿æ°ã¯ -1 ã§ããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "256 x y 2 {\\displaystyle 256xy^{2}} ãšããåé
åŒã¯ã256ãšããæ°ãš x , y , y {\\displaystyle x,y,y} ãšããæåã«åããŠèããããšãã§ããã®ã§ããã®åé
åŒã®ä¿æ°ã¯256ã§ãããäžæ¹ãæãããããæåã®æ°ãåé
åŒã®æ¬¡æ°(ããããdegree)ãšããã 256 x y 2 {\\displaystyle 256xy^{2}} 㯠x , y , y {\\displaystyle x,y,y} ãšãã3ã€ã®æåãæãããããŠã§ããŠããã®ã§ããã®åé
åŒã®æ¬¡æ°ã¯3ã§ããã0ãšããåé
åŒã®æ¬¡æ°ã¯ 0 = 0 x = 0 x 2 = 0 x 3 = ⯠{\\displaystyle 0=0x=0x^{2}=0x^{3}=\\cdots } ãšäžã€ã«å®ãŸããªãã®ã§ãããã§ã¯èããªãã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "åé
åŒã®ä¿æ°ãšæ¬¡æ°ã¯ãåã«æ°ãšæåã«åããŠèããã®ã§ã¯ãªããããæåãå€æ°ãšããŠèŠããšãã«ãæ®ãã®æåãå®æ°ãšããŠæ°ãšåãããã«æ±ãããšãããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "ããšãã° â 5 a b c x 3 {\\displaystyle -5abcx^{3}} ãšããåé
åŒãã x 3 {\\displaystyle x^{3}} ã ããå€æ°ã§ãæ®ãã®æå a , b , c {\\displaystyle a,b,c} ã¯å®æ°ãšèããããšãã§ããã ãã®ãšã ( â 5 a b c ) x 3 {\\displaystyle (-5abc)x^{3}} ãšåããããã®ã§ããã®åé
åŒã®ä¿æ°ã¯ â 5 a b c {\\displaystyle -5abc} ãå€æ°ã¯ x 3 {\\displaystyle x^{3}} ã§ã次æ°ã¯3ã§ãããšãããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "ãã®ããšã â 5 a b c x 3 {\\displaystyle -5abcx^{3}} ãšããåé
åŒã¯ãã x {\\displaystyle x} ã«çç®ãããšãä¿æ°ã¯ â 5 a b c {\\displaystyle -5abc} ã次æ°ã¯3ã§ããããªã©ãšããå Žåãããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "ããã㯠â 5 a b c x 3 {\\displaystyle -5abcx^{3}} ã® a {\\displaystyle a} ãš b {\\displaystyle b} ã«çç®ããã°ã ( â 5 c x 3 ) a b {\\displaystyle (-5cx^{3})ab} ãšåãããã a {\\displaystyle a} ãš b {\\displaystyle b} ã«çç®ãããšãã®ãã®åé
åŒã®ä¿æ°ã¯ â 5 c x 3 {\\displaystyle -5cx^{3}} ãå€æ°ã¯ a b {\\displaystyle ab} ã§ã次æ°ã¯2ã§ãããšãããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "æ
£ç¿çã«ã¯ a , b , c , ⯠{\\displaystyle a,b,c,\\cdots } ãªã©ã®ã¢ã«ãã¡ãããã®æåã®æ¹ã®æåãå®æ°ãè¡šãã®ã«äœ¿ãã ⯠, x , y , z {\\displaystyle \\cdots ,x,y,z} ãªã©ã®ã¢ã«ãã¡ãããã®æåŸã®æ¹ã®æåãå€æ°ãè¡šãã®ã«çšããããäžè¬çã«ã¯ãã®éãã§ãªãã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "å€é
åŒã®æ¬¡æ°ãšã¯ãå€é
åŒã®åé¡é
ããŸãšãããšãã«ããã£ãšã次æ°ã®é«ãé
ã®æ¬¡æ°ããããããšãã° x 3 + 3 x 2 y + 2 y {\\displaystyle x^{3}+3x^{2}y+2y} ã§ã¯ããã£ãšã次æ°ã®é«ãé
㯠x 3 {\\displaystyle x^{3}} ã§ããã®ã§ããã®å€é
åŒã®æ¬¡æ°ã¯3ã§ããããã x 3 + 3 x 2 y + 2 y {\\displaystyle x^{3}+3x^{2}y+2y} ( x {\\displaystyle x} ã¯å®æ°)ã§ããã°ãããªãã¡å€é
åŒã® y {\\displaystyle y} ã«ã€ããŠçç®ãããšããã£ãšã次æ°ã®é«ãé
㯠3 x 2 y {\\displaystyle 3x^{2}y} ãš 2 y {\\displaystyle 2y} ã§ããã®ã§ããã®å€é
åŒã®æ¬¡æ°ã¯1ã§ããããã®ãšãçç®ããæåãå«ãŸãªãé
x 3 {\\displaystyle x^{3}} ã¯å®æ°é
(ãŠããããããconstant term)ãšããŠæ°ãšåãããã«æ±ãããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "次ã®å€é
åŒã® x {\\displaystyle x} ãŸã㯠y {\\displaystyle y} ã«çç®ãããšãã®æ¬¡æ°ãšå®æ°é
ãããããããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "ããšãã°ã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "ã®ããã«ã次æ°ã®é«ãé
ããå
ã«é
ããªãã¹ãããšããéã¹ãã(ããã¹ã)ãšããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "ããŠãåŒã䜿ãç®çã«ãã£ãŠã¯ã次æ°ã®ã²ããé
ããå
ã«æžããã»ãã䟿å©ãªå Žåãããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "ããšãã°ã x {\\displaystyle x} ã çŽ0.01 ã®ãããª1æªæºã®å°ããæ°ã®å ŽåãåŒ x 2 + 6 x + 7 {\\displaystyle x^{2}+6x+7} ã®å€ãæ±ããããªããæå x {\\displaystyle x} ã®æ¬¡æ°ã®å°ããé
ã®ã»ãã圱é¿ãé«ãã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "ãªã®ã§ã ç®çã«ãã£ãŠã¯",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "ã®ããã«ã次æ°ã®ã²ããé
ããå
ã«æžãå Žåãããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "7 + 6 x + x 2 {\\displaystyle 7+6x+x^{2}} ã®ããã«ã次æ°ã®äœãé
ããå
ã«é
ããªãã¹ãããšããæã¹ãã(ãããã¹ã)ãšããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "å€é
åŒã«2ã€ä»¥äžã®æåããããšããç¹å®ã®1ã€ã®æåã«æ³šç®ããŠäžŠã³å€ãããšã䜿ãããããªãããšãããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "ããšãã°ã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "ã®é
ããxã®æ¬¡æ°ãå€ãé
ããå
ã«äžŠã³ãããåé¡é
ããŸãšãããš",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "ãšãªãã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 41,
"tag": "p",
"text": "ãã®(äŸ2)ã®ããã«ãç¹å®ã®æåã ãã«çç®ããŠããã®æåã®æ¬¡æ°ã®é«ãé ã«äžŠã³ããããšäŸ¿å©ãªããšããã°ãã°ããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 42,
"tag": "p",
"text": "äŸ2ã¯ã x {\\displaystyle x} ã«ã€ã㊠éã¹ã ã®é ã«äžŠã³å€ããæŽåŒã§ããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 43,
"tag": "p",
"text": "çç®ããŠãªãæåã«ã€ããŠã¯ã䞊ã³æãã®ãšãã¯å®æ°ã®ããã«æ±ãã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 44,
"tag": "p",
"text": "ãã£ãœãã x {\\displaystyle x} ã«ã€ããŠã次æ°ã®ã²ããé
ããé ã«äžŠã¹ããšã次ã®ãããªåŒã«ãªãã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 45,
"tag": "p",
"text": "ãã®ããã«ãç¹å®ã®æåã®æ¬¡æ°ãäœããã®ããé ã«äžŠã³ããããšäŸ¿å©ãªããšããã°ãã°ããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 46,
"tag": "p",
"text": "äŸ3ã¯ãxã«ã€ã㊠æã¹ã ã®é ã«äžŠã³å€ããæŽåŒã§ããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 47,
"tag": "p",
"text": "",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 48,
"tag": "p",
"text": "ããšãã°ãåŒ",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 49,
"tag": "p",
"text": "ãšããåŒã®å³èŸº",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 50,
"tag": "p",
"text": "ã®æ¬¡æ°ã¯ããããã§ããããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 51,
"tag": "p",
"text": "aãšxãçããæåãšããŠæ±ãã®ã§ããã°ã a x {\\displaystyle ax} ã®æ¬¡æ°ã¯",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 52,
"tag": "p",
"text": "ãã 1+1 =2 ãªã®ã§ããã®åŒã®æ¬¡æ°ã¯2ã§ããã(é
bã¯æ¬¡æ°1ãªã®ã§ã a x {\\displaystyle ax} ã®æ¬¡æ°2ãããäœãã®ã§ç¡èŠããã)",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 53,
"tag": "p",
"text": "ãããããããã®åŒããå®æ° a {\\displaystyle a} ãä¿æ°ãšããå€æ° x {\\displaystyle x} ã«ã€ããŠã®äžæ¬¡é¢æ°ãšã¿ãã®ã§ããã°ãäžæ¬¡åŒãšæãã®ãåççã ããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 54,
"tag": "p",
"text": "ãã®ãããªå Žåãç¹å®ã®æåã ãã«æ³šç®ãããã®åŒã®æ¬¡æ°ãèãããšããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 55,
"tag": "p",
"text": "ããšãã°ãæåxã ãã«æ³šç®ããŠãåŒ a x + b {\\displaystyle ax+b} ã®æ¬¡æ°ã決ããŠã¿ããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 56,
"tag": "p",
"text": "ãããšãæåxã«æ³šç®ããå Žåã®åŒ a x + b {\\displaystyle ax+b} ã®æ¬¡æ°ã¯1ã«ãªãã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 57,
"tag": "p",
"text": "ãªããªã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 58,
"tag": "p",
"text": "ãã£ãŠãæå x {\\displaystyle x} ã«æ³šç®ããå Žåã®é
a x {\\displaystyle ax} ã®æ¬¡æ°ã¯ã 0+1 ãªã®ã§ã1ã§ããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 59,
"tag": "p",
"text": "ãã®ããã«èããå Žåãå¿
èŠã«å¿ããŠã©ã®æåã«æ³šç®ããããæèšããŠãæåâ¯â¯ã«æ³šç®ãã次æ°ãã®ããã«è¿°ã¹ããšããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 60,
"tag": "p",
"text": "å€é
åŒã®ç©ã¯åé
æ³åã䜿ã£ãŠèšç®ããããšãã§ããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 61,
"tag": "p",
"text": "ãã®ããã«å€é
åŒã®ç©ã§è¡šãããåŒãäžã€ã®å€é
åŒã«ç¹°ãåºããããšããå€é
åŒãå±é(ãŠããããexpand)ãããšããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 62,
"tag": "p",
"text": "a {\\displaystyle a} ã n {\\displaystyle n} åæãããã®ã a n {\\displaystyle a^{n}} ãšæžããaã®nä¹(-ããããa to the n-th power)ãšããããã ã a 1 = a {\\displaystyle a^{1}=a} ãšå®çŸ©ãããããšãã°ã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 63,
"tag": "p",
"text": "ã§ããã a , a 2 , a 3 , a 4 , a 5 , ⯠, a n {\\displaystyle a,a^{2},a^{3},a^{4},a^{5},\\cdots ,a^{n}} ãç·ç§°ã㊠a {\\displaystyle a} ã®çŽ¯ä¹(ããããããexponentiationãåªä¹ãã¹ãããããåªãã¹ã)ãšããã a n {\\displaystyle a^{n}} ã® n ãææ°(ããããexponent)ãšãã(a ã¯åº(ãŠããbase)ãšãã)ãããã§ã¯èªç¶æ°ãããªãã¡æ£ã®æŽæ°ã®ææ°ãèããã环ä¹ã¯æ¬¡ã®ããã«èããããšãã§ããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 64,
"tag": "p",
"text": "环ä¹ã©ãããæãããããç©ã¯ã次ã®ããã«èšç®ããããšãã§ããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 65,
"tag": "p",
"text": "环ä¹ã©ãããå²ã£ãåã¯ã次ã®ããã«èšç®ããããšãã§ããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 66,
"tag": "p",
"text": "环ä¹ã®çŽ¯ä¹ã¯ã次ã®ããã«èšç®ããããšãã§ããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 67,
"tag": "p",
"text": "ç©ã®çŽ¯ä¹ã¯ã次ã®ããã«èšç®ããããšãã§ããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 68,
"tag": "p",
"text": "ããããããããŠææ°æ³å(ãããã»ããããexponential law)ãšããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 69,
"tag": "p",
"text": "环ä¹ã®å®çŸ©ããæããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 70,
"tag": "p",
"text": "次ã®åŒãèšç®ããªããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 71,
"tag": "p",
"text": "次ã®åŒãå±éããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 72,
"tag": "p",
"text": "ãŸãšãããšã次ã®ããã«ãªãã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 73,
"tag": "p",
"text": "次ã®åŒãå±éããªããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 74,
"tag": "p",
"text": "è€éãªåŒã®å±éã¯ãåŒã®äžéšåãäžã€ã®æåã«ãããŠå
¬åŒã䜿ããšããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 75,
"tag": "p",
"text": "次ã®åŒãå±éããªããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 76,
"tag": "p",
"text": "次ã®åŒãå æ°å解ããªããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 77,
"tag": "p",
"text": "",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 78,
"tag": "p",
"text": "",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 79,
"tag": "p",
"text": "次ã®åŒãå æ°å解ããªããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 80,
"tag": "p",
"text": "次ã®åŒãå æ°å解ããªããã",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 81,
"tag": "p",
"text": "",
"title": "åŒã®å±éãšå æ°å解"
},
{
"paragraph_id": 82,
"tag": "p",
"text": "a=b^2ãæãç«ã€ãšããa=2ãšãªããããªbãããªãã¡ 2 {\\displaystyle {\\sqrt {2}}} ã®å
·äœçãªå€ãã©ã®ãããªãã®ãã調ã¹ãŠã¿ããã",
"title": "å®æ°"
},
{
"paragraph_id": 83,
"tag": "p",
"text": "ãã®ããã«ãbãæ§ã
ã«æ±ºããŠããaã¯ãªããªã2ã«ãªããªãã",
"title": "å®æ°"
},
{
"paragraph_id": 84,
"tag": "p",
"text": "å®ã¯ 2 {\\displaystyle {\\sqrt {2}}} ã¯ãåæ¯ååå
±ã«æŽæ°ã®åæ°ã§è¡šãããšã¯ã§ããªãããã®ããã«æŽæ°ãåæ¯ååã«æã€åæ°ã§è¡šããªããããªæ°ãç¡çæ°ãšãããäŸãã°ãååšçÏã¯ç¡çæ°ã§ãããããã«å¯ŸããŠãæŽæ°ã埪ç°å°æ°ãªã©ãåæ¯ååå
±ã«æŽæ°ã®åæ°ã§è¡šãããšã®ã§ããæ°ãæçæ°ãšããã",
"title": "å®æ°"
},
{
"paragraph_id": 85,
"tag": "p",
"text": "æçæ°ãšç¡çæ°ãåãããŠå®æ°ãšãããã©ããªå®æ°ã§ãæ°çŽç·äžã®ç¹ãšããŠè¡šããããŸããã©ããªå®æ°ããæéå°æ°ãããã¯ç¡éå°æ°ãšããŠè¡šããã (äžèšã®ãç¡éå°æ°ãã®ç¯ãåç
§)",
"title": "å®æ°"
},
{
"paragraph_id": 86,
"tag": "p",
"text": "2 {\\displaystyle {\\sqrt {2}}} ãæçæ°ã§ãããšä»®å®ãããšãäºãã«çŽ ãª(1以å€ã«å
¬çŽæ°ããããªã)æŽæ° m, n ãçšããŠã",
"title": "å®æ°"
},
{
"paragraph_id": 87,
"tag": "p",
"text": "ãšè¡šããããšãã§ããããã®ãšãã䞡蟺ã2ä¹ããŠåæ¯ãæããšã",
"title": "å®æ°"
},
{
"paragraph_id": 88,
"tag": "p",
"text": "ãã£ãŠ m ã¯2ã®åæ°ã§ãããæŽæ° l ãçšã㊠m = 2 l {\\displaystyle m=2l} ãšè¡šãããšãã§ãããããã (1) ã®åŒã«ä»£å
¥ããŠæŽçãããšã",
"title": "å®æ°"
},
{
"paragraph_id": 89,
"tag": "p",
"text": "ãã£ãŠ n ã2ã®åæ°ã§ãããããã㯠m, n ã2ãå
¬çŽæ°ã«ãã€ããšã«ãªããäºãã«çŽ ãšä»®å®ããããšã«ççŸããããããã£ãŠ 2 {\\displaystyle {\\sqrt {2}}} ã¯ç¡çæ°ã§ãã(èçæ³)ã",
"title": "å®æ°"
},
{
"paragraph_id": 90,
"tag": "p",
"text": "0.1 ã 0.123456789 ã®ããã«ãããäœã§çµããå°æ°ãæéå°æ°ãšããã",
"title": "å®æ°"
},
{
"paragraph_id": 91,
"tag": "p",
"text": "äžæ¹ã 0.1234512345 ⯠{\\displaystyle 0.1234512345\\cdots } ã 3.1415926535 ⯠{\\displaystyle 3.1415926535\\cdots } ã®ããã«ç¡éã«ç¶ãå°æ°ã ç¡éå°æ°(ããã ããããã)ãšããã",
"title": "å®æ°"
},
{
"paragraph_id": 92,
"tag": "p",
"text": "ç¡éå°æ°ã®ãã¡ãããäœããäžãããããé
åã®æ°åã®ç¹°ãè¿ãã«ãªã£ãŠãããã®ã 埪ç°å°æ°(ãã
ããã ããããã)ãšãããäŸãã° 0.3333333333 ⯠{\\displaystyle 0.3333333333\\cdots } ã 0.1428571428 ⯠{\\displaystyle 0.1428571428\\cdots } ã 0.1232323232 ⯠{\\displaystyle 0.1232323232\\cdots } ãªã©ã§ãããç¹°ãè¿ãã®æå°åäœã埪ç°ç¯ãšããã埪ç°å°æ°ã¯åŸªç°ç¯1ã€ãçšã㊠0. 3 Ì {\\displaystyle 0.{\\dot {3}}} ã 0. 1 Ì 4285 7 Ì {\\displaystyle 0.{\\dot {1}}4285{\\dot {7}}} ã 0.1 2 Ì 3 Ì {\\displaystyle 0.1{\\dot {2}}{\\dot {3}}} ã®ããã«åŸªç°ç¯ã®æåãšæåŸ(埪ç°ç¯ãäžæ¡ã®å Žåã¯ã²ãšã€ã ã)ã®äžã«ç¹ãã€ããŠè¡šãã",
"title": "å®æ°"
},
{
"paragraph_id": 93,
"tag": "p",
"text": "å
šãŠã®åŸªç°å°æ°ã¯åæ°ã«çŽããã",
"title": "å®æ°"
},
{
"paragraph_id": 94,
"tag": "p",
"text": "ãšçœ®ããšã",
"title": "å®æ°"
},
{
"paragraph_id": 95,
"tag": "p",
"text": "ã§ããã(2)ãŒ(1) ãã 9 a = 3 {\\displaystyle 9a=3} ããã£ãŠ a = 3 9 = 1 3 {\\displaystyle a={\\frac {3}{9}}={\\frac {1}{3}}} ã§ããã",
"title": "å®æ°"
},
{
"paragraph_id": 96,
"tag": "p",
"text": "a = 0. 1 Ì 4285 7 Ì 1000000 a = 142857. 1 Ì 4285 7 Ì 999999 a = 142857 a = 142857 999999 = 1 7 {\\displaystyle {\\begin{aligned}a&=0.{\\dot {1}}4285{\\dot {7}}\\\\1000000a&=142857.{\\dot {1}}4285{\\dot {7}}\\\\999999a&=142857\\\\a&={\\frac {142857}{999999}}\\ ={\\frac {1}{7}}\\end{aligned}}}",
"title": "å®æ°"
},
{
"paragraph_id": 97,
"tag": "p",
"text": "a = 0.1 2 Ì 3 Ì 100 a = 12.3 2 Ì 3 Ì 99 a = 12.2 a = 12.2 99 = 61 495 {\\displaystyle {\\begin{aligned}a&=0.1{\\dot {2}}{\\dot {3}}\\\\100a&=12.3{\\dot {2}}{\\dot {3}}\\\\99a&=12.2\\\\a&={\\frac {12.2}{99}}\\ ={\\frac {61}{495}}\\end{aligned}}}",
"title": "å®æ°"
},
{
"paragraph_id": 98,
"tag": "p",
"text": "å®æ° a ã«ã€ããŠãa ã®æ°çŽç·äžã§ã®åç¹ãšã®è·é¢ã a ã®çµ¶å¯Ÿå€ãšããã | a | {\\displaystyle |a|} ã§è¡šãã",
"title": "å®æ°"
},
{
"paragraph_id": 99,
"tag": "p",
"text": "ããšãã°",
"title": "å®æ°"
},
{
"paragraph_id": 100,
"tag": "p",
"text": "ã§ããã",
"title": "å®æ°"
},
{
"paragraph_id": 101,
"tag": "p",
"text": "å®çŸ©ãã | a | = | â a | {\\displaystyle |a|=|-a|} ããããããŸãã a , b {\\displaystyle a,b} ãä»»æã®å®æ°ãšãããšããããããã«å¯Ÿå¿ããæ°çŽç·äžã®ä»»æã®2ç¹ P ( a ) , Q ( b ) {\\displaystyle \\mathrm {P} (a),\\mathrm {Q} (b)} éã®è·é¢ã«ã€ããŠã¯ã次ã®ããšããããã",
"title": "å®æ°"
},
{
"paragraph_id": 102,
"tag": "p",
"text": "",
"title": "å®æ°"
},
{
"paragraph_id": 103,
"tag": "p",
"text": "",
"title": "å®æ°"
},
{
"paragraph_id": 104,
"tag": "p",
"text": "ä»ã2ä¹ããŠaã«ãªãæ°bãèããã",
"title": "å®æ°"
},
{
"paragraph_id": 105,
"tag": "p",
"text": "a = 1 {\\displaystyle a=1} ã®ãšãã b = 1 {\\displaystyle b=1} ãšããŠçµããã«ããŠã¯ãããªãã確ãã« b = 1 {\\displaystyle b=1} ãæ¡ä»¶ãæºããã b = â 1 {\\displaystyle b=-1} ãæ¡ä»¶ãæºããããã£ãŠ b = 1 {\\displaystyle b=1} ãŸã㯠b = â 1 {\\displaystyle b=-1} ã§ããã",
"title": "å®æ°"
},
{
"paragraph_id": 106,
"tag": "p",
"text": "äžè¬ã«æ£ã®æ°aã«ã€ããŠa=b^2ãšãªãbã¯äºã€ããããã®äºã€ã¯çµ¶å¯Ÿå€ãçããããã®äºã€ã®bãaã®å¹³æ¹æ ¹ãšãããaã®å¹³æ¹æ ¹ã®ãã¡ãæ£ã§ãããã®ã a {\\displaystyle {\\sqrt {a}}} ãè² ã§ãããã®ã â a {\\displaystyle -{\\sqrt {a}}} ãšæžãã a {\\displaystyle {\\sqrt {a}}} ã¯ãã«ãŒãaããšèªãã",
"title": "å®æ°"
},
{
"paragraph_id": 107,
"tag": "p",
"text": "äžæ¹ãè² ã®æ°aã«ã€ããŠèããŠã¿ãŠãäžæãbãèŠã€ããããšã¯ã§ããªããå®éã®ãšãããè² ã®æ°ã®å¹³æ¹æ ¹ã¯å®æ°ã§è¡šãããšã¯ã§ããªãã",
"title": "å®æ°"
},
{
"paragraph_id": 108,
"tag": "p",
"text": "2 , 4 , 9 , 12 {\\displaystyle 2\\ ,\\ 4\\ ,\\ 9\\ ,\\ 12} ã®å¹³æ¹æ ¹ãæ±ããã",
"title": "å®æ°"
},
{
"paragraph_id": 109,
"tag": "p",
"text": "± 2 , ± 2 , ± 3 , ± 2 3 {\\displaystyle \\pm {\\sqrt {2}}\\ ,\\ \\pm 2\\ ,\\ \\pm 3\\ ,\\ \\pm 2{\\sqrt {3}}}",
"title": "å®æ°"
},
{
"paragraph_id": 110,
"tag": "p",
"text": "ããããã®ã«ãŒããèšç®ãã ± {\\displaystyle \\pm } ãã€ããã°ããããã ããå¹³æ¹æ ¹ã®ã«ãŒã«ã«åŸã£ãŠãç°¡ååã§ãããã®ã¯ç°¡ååããããšãèŠæ±ãããã",
"title": "å®æ°"
},
{
"paragraph_id": 111,
"tag": "p",
"text": "äŸãã°ã 2 {\\displaystyle 2} ã«å¯ŸããŠã¯ã ± 2 {\\displaystyle \\pm {\\sqrt {2}}} ãšãªãã",
"title": "å®æ°"
},
{
"paragraph_id": 112,
"tag": "p",
"text": "äžè¬ã«ã A 2 = | A | {\\displaystyle {\\sqrt {A^{2}}}=|A|} ã§ããã",
"title": "å®æ°"
},
{
"paragraph_id": 113,
"tag": "p",
"text": "æ ¹å·ã«ã€ããŠã次ã®å
¬åŒãæãç«ã€ã",
"title": "å®æ°"
},
{
"paragraph_id": 114,
"tag": "p",
"text": "ãŸãã a b {\\displaystyle {\\sqrt {ab}}} ãšã¯ãå®çŸ©ã«ããšã¥ããŠèãããšã2ä¹ãããš ab ã«ãªãæ°ã®ãã¡ãæ£ã®ã»ãã®æ°ãšããæå³ã§ããã",
"title": "å®æ°"
},
{
"paragraph_id": 115,
"tag": "p",
"text": "ãªã®ã§ãå
¬åŒã a b = a b {\\displaystyle {\\sqrt {a}}{\\sqrt {b}}={\\sqrt {ab}}} ãã蚌æããã«ã¯ããã®ããšã蚌æããã°ããã",
"title": "å®æ°"
},
{
"paragraph_id": 116,
"tag": "p",
"text": "ãªã®ã§ããŸãã a b {\\displaystyle {\\sqrt {a}}{\\sqrt {b}}} ã2ä¹ãããšã",
"title": "å®æ°"
},
{
"paragraph_id": 117,
"tag": "p",
"text": "ãšãªãã",
"title": "å®æ°"
},
{
"paragraph_id": 118,
"tag": "p",
"text": "ããã« a b {\\displaystyle {\\sqrt {a}}{\\sqrt {b}}} ã¯ããŸãæ¡ä»¶ã2ä¹ãããšabã«ãªãããæºããã",
"title": "å®æ°"
},
{
"paragraph_id": 119,
"tag": "p",
"text": "ãããŠãæ£ã®æ°ã®å¹³æ¹æ ¹ã¯æ£ãªã®ã§ã a b {\\displaystyle {\\sqrt {a}}{\\sqrt {b}}} ãæ£ã§ããããã£ãŠ a b {\\displaystyle {\\sqrt {a}}{\\sqrt {b}}} ã¯ãã2ä¹ãããšabã«ãªããæ°ã®ãã¡ã®æ£ã®ã»ãã§ããã",
"title": "å®æ°"
},
{
"paragraph_id": 120,
"tag": "p",
"text": "(蚌æããã)",
"title": "å®æ°"
},
{
"paragraph_id": 121,
"tag": "p",
"text": "ããã«ãäžã®å
¬åŒ(1)ã«ããã次ã®å
¬åŒãå°ãããã",
"title": "å®æ°"
},
{
"paragraph_id": 122,
"tag": "p",
"text": "",
"title": "å®æ°"
},
{
"paragraph_id": 123,
"tag": "p",
"text": "èšç®ããã",
"title": "å®æ°"
},
{
"paragraph_id": 124,
"tag": "p",
"text": "åæ¯ã«æ ¹å·ãå«ãŸãªãåŒã«ããããšããåæ¯ãæçåãããšãããæçåã¯ãåæ¯ãšååã«åãæ°ããããŠãããããšãå©çšããŠè¡ãã",
"title": "å®æ°"
},
{
"paragraph_id": 125,
"tag": "p",
"text": "ããšãã°ã 1 2 {\\displaystyle {\\frac {1}{\\sqrt {2}}}} ãæçåãããšã 1 2 = 1 2 2 2 = 2 2 {\\displaystyle {\\frac {1}{\\sqrt {2}}}\\ =\\ {\\frac {1{\\sqrt {2}}}{{\\sqrt {2}}{\\sqrt {2}}}}\\ =\\ {\\frac {\\sqrt {2}}{2}}} ãšãªãã",
"title": "å®æ°"
},
{
"paragraph_id": 126,
"tag": "p",
"text": "ãŸãããšãã« a b + c {\\displaystyle {\\frac {a}{b+c}}} ã«ã€ããŠã b 2 â c 2 = 1 {\\displaystyle b^{2}-c^{2}=1} ã®ãšãã a b + c = a ( b â c ) ( b + c ) ( b â c ) = a ( b â c ) b 2 â c 2 = a ( b â c ) 1 = a ( b â c ) {\\displaystyle {\\frac {a}{b+c}}\\ =\\ {\\frac {a(b-c)}{(b+c)(b-c)}}\\ =\\ {\\frac {a(b-c)}{b^{2}-c^{2}}}\\ =\\ {\\frac {a(b-c)}{1}}\\ =\\ a(b-c)} ã§ããã",
"title": "å®æ°"
},
{
"paragraph_id": 127,
"tag": "p",
"text": "ããšãã°ã a = 1 , b = 2 , c = 1 {\\displaystyle a=1,b={\\sqrt {2}},c=1} ãšãããšã 1 2 + 1 = 2 â 1 {\\displaystyle {\\frac {1}{{\\sqrt {2}}+1}}={\\sqrt {2}}-1} ã§ããã",
"title": "å®æ°"
},
{
"paragraph_id": 128,
"tag": "p",
"text": "åæ¯ãæçåããã",
"title": "å®æ°"
},
{
"paragraph_id": 129,
"tag": "p",
"text": "äºéæ ¹å·ãšã¯ãæ ¹å·ã2éã«ãªã£ãŠããåŒã®ããšã§ãããäºéæ ¹å·ã¯åžžã«å€ããããã§ã¯ãªããæ ¹å·ã®äžã«å«ãŸããåŒã«ãã£ãŠç°¡åã«ã§ãããã©ããã決ãŸããäžè¬ã«ãæ ¹å·å
ã®åŒãã x 2 {\\displaystyle x^{2}} ã®åœ¢ã«å€åœ¢ã§ããå Žåã«ã¯ãå€åŽã®æ ¹å·ãå€ãããšãã§ããã",
"title": "å®æ°"
},
{
"paragraph_id": 130,
"tag": "p",
"text": "3 + 2 2 {\\displaystyle {\\sqrt {3+2{\\sqrt {2}}}}} ãç°¡åã«ããã",
"title": "å®æ°"
},
{
"paragraph_id": 131,
"tag": "p",
"text": "3 + 2 2 {\\displaystyle 3+2{\\sqrt {2}}} ã ( ⯠) 2 {\\displaystyle (\\cdots )^{2}} ã®åœ¢ã«ã§ããããèããã",
"title": "å®æ°"
},
{
"paragraph_id": 132,
"tag": "p",
"text": "ä»®ã«ã ( a + b ) 2 {\\displaystyle ({\\sqrt {a}}+{\\sqrt {b}})^{2}} (a,bã¯æ£ã®æŽæ°)ã®åœ¢ã«ã§ãããšãããšã 3 + 2 2 = a + b + 2 a b {\\displaystyle 3+2{\\sqrt {2}}=a+b+2{\\sqrt {ab}}} ãšãªãã",
"title": "å®æ°"
},
{
"paragraph_id": 133,
"tag": "p",
"text": "ãæºããæŽæ°a,bãæ¢ãã°ããããã®é¢ä¿ã¯ãa=1,b=2(a,bãå
¥ãæããŠãå¯ã)ã«ãã£ãŠæºããããã®ã§ã 3 + 2 2 = ( 2 + 1 ) 2 {\\displaystyle 3+2{\\sqrt {2}}\\ =\\ ({\\sqrt {2}}+1)^{2}} ãæãç«ã€ã",
"title": "å®æ°"
},
{
"paragraph_id": 134,
"tag": "p",
"text": "ãã£ãŠã 3 + 2 2 = ( 2 + 1 ) 2 = 2 + 1 {\\displaystyle {\\sqrt {3+2{\\sqrt {2}}}}\\ =\\ {\\sqrt {({\\sqrt {2}}+1)^{2}}}\\ =\\ {\\sqrt {2}}+1} ãšãªãã",
"title": "å®æ°"
},
{
"paragraph_id": 135,
"tag": "p",
"text": "次ã®åŒãèšç®ããã",
"title": "å®æ°"
},
{
"paragraph_id": 136,
"tag": "p",
"text": "åã倧ããã®éã=ã§çµãã åŒãæ¹çšåŒãšåŒã¶ããšãæ¢ã«åŠç¿ãããããã§ã¯ãç°ãªã£ãéã®å€§ããã®éããè¡šãèšå·ãå°å
¥ãããã®æ§è³ªã«ã€ããŠãŸãšããã",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 137,
"tag": "p",
"text": "ããæ°A,BããããšããAãBãã倧ããããšã A > B {\\displaystyle A>B} ãšè¡šããAãBããå°ããããšã A < B {\\displaystyle A<B} ãšè¡šããããã§ã<ãš>ã®ããšãäžçå·ãšåŒã³ããã®ãããªåŒãäžçåŒãšåŒã¶ããŸãã †, ⥠{\\displaystyle \\leq ,\\geq } ã䌌ãæå³ã®äžçåŒã§ããããããããAãšBãçããå€ã§ããå Žåãå«ããã®ã§ããã",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 138,
"tag": "p",
"text": "ãªããæ¥æ¬ã®æè²ã«ãããŠã¯ã †, ⥠{\\displaystyle \\leq ,\\geq } ã®ä»£ããã«ãäžçå·ã®äžã«çå·ãèšãã ⊠, ⧠{\\displaystyle \\leqq ,\\geqq } ã䜿ãããšãå€ãã",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 139,
"tag": "p",
"text": "x > 7 {\\displaystyle x>7} ãšããäžçåŒããããšããxã¯7ãã倧ããå®æ°ã§ããããŸãã x ⥠7 {\\displaystyle x\\geq 7} ã®æã«ã¯ãxã¯7以äžã®å®æ°ã§ããã",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 140,
"tag": "p",
"text": "äžçåŒã§ã¯çåŒãšåãããã«ã䞡蟺ã«æŒç®ãããŠãäžçå·ã®é¢ä¿ãå€ãããªãããšããããäŸãã°ã䞡蟺ã«åãæ°ã足ããŠãã䞡蟺ã®å€§å°é¢ä¿ã¯å€åããªãããã ãã䞡蟺ã«è² ã®æ°ãããããšãã«ã¯ãäžçå·ã®åããå€åããããšã«æ³šæãå¿
èŠã§ãããããã¯ãè² ã®æ°ãããããšäž¡èŸºã®å€ã¯ã0ãäžå¿ã«æ°çŽç·ãæãè¿ããå°ç¹ã«ç§»ãããããšã«ããã",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 141,
"tag": "p",
"text": "x > y {\\displaystyle x>y} ãæãç«ã€ãšãã«ã¯ã x + 3 > y + 3 {\\displaystyle x+3>y+3} ã 4 x > 4 y {\\displaystyle 4x>4y} ãæãç«ã€ããŸãã â x < â y {\\displaystyle -x<-y} ãæãç«ã€ã",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 142,
"tag": "p",
"text": "äžçåŒã®æ§è³ªã䜿ã£ãŠ",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 143,
"tag": "p",
"text": "ã®äž¡èŸºãã3ãåŒããš",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 144,
"tag": "p",
"text": "ãã£ãŠ",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 145,
"tag": "p",
"text": "ãšãªãã ãã®ããã«ãäžçåŒã§ã移é
ããããšãã§ããã",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 146,
"tag": "p",
"text": "ã°ã©ããçšããŠèãããšããäžçåŒã¯ã°ã©ãäžã®é åãè¡šããé åã®å¢çã¯äžçå·ãçå·ã«çœ®ãæããéšåã察å¿ãããããã¯ãäžçå·ãæç«ãããã©ããããã®ç·äžã§å
¥ãæ¿ããããšã«ãã£ãŠããã(詳ããã¯æ°åŠII å³åœ¢ãšæ¹çšåŒã§åŠç¿ããã)",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 147,
"tag": "p",
"text": "y > x + 1 {\\displaystyle y>x+1} , y < 2 x + 1 {\\displaystyle y<2x+1} , x < 3 {\\displaystyle x<3} ã®ã°ã©ã(æ£ããã¯ãé åã)ãæãã",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 148,
"tag": "p",
"text": "y > x + 1 {\\displaystyle y>x+1} ã®ã°ã©ã(é å)ã¯æ¬¡ã®ããã«ãªãããã ããå¢çã¯å«ãŸãªãã",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 149,
"tag": "p",
"text": "y < 2 x + 1 {\\displaystyle y<2x+1} ã®ã°ã©ã(é å)ã¯æ¬¡ã®ããã«ãªãããã ããå¢çã¯å«ãŸãªãã",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 150,
"tag": "p",
"text": "x < 3 {\\displaystyle x<3} ã®ã°ã©ã(é å)ã¯æ¬¡ã®ããã«ãªãããã ããå¢çã¯å«ãŸãªãã",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 151,
"tag": "p",
"text": "",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 152,
"tag": "p",
"text": "次ã®äžçåŒã解ãã",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 153,
"tag": "p",
"text": "ããã€ãã®äžçåŒãçµã¿åããããã®ãé£ç«äžçåŒãšããããããã®äžçåŒãåæã«æºãã x {\\displaystyle x} ã®å€ã®ç¯å²ãæ±ããããšããé£ç«äžçåŒã解ããšããã",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 154,
"tag": "p",
"text": "次ã®é£ç«äžçåŒã解ãã (i)",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 155,
"tag": "p",
"text": "(ii)",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 156,
"tag": "p",
"text": "(i) x + 2 < 2 x + 4 {\\displaystyle x+2<2x+4} ãã â x < 2 {\\displaystyle -x<2}",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 157,
"tag": "p",
"text": "10 â x ⥠3 x â 6 {\\displaystyle 10-x\\geq 3x-6} ãã â 4 x ⥠â 16 {\\displaystyle -4x\\geq -16}",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 158,
"tag": "p",
"text": "(1),(2)ãåæã«æºãã x {\\displaystyle x} ã®å€ã®ç¯å²ã¯",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 159,
"tag": "p",
"text": "(ii) x ⥠1 â x {\\displaystyle x\\geq 1-x} ãã 2 x ⥠1 {\\displaystyle 2x\\geq 1}",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 160,
"tag": "p",
"text": "2 ( x + 1 ) > x â 2 {\\displaystyle 2(x+1)>x-2} ãã 2 x + 2 > x â 2 {\\displaystyle 2x+2>x-2}",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 161,
"tag": "p",
"text": "(1),(2)ãåæã«æºãã x {\\displaystyle x} ã®å€ã®ç¯å²ã¯",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 162,
"tag": "p",
"text": "",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 163,
"tag": "p",
"text": "絶察å€ãå«ãäžçåŒã«ã€ããŠèãããã çµ¶å¯Ÿå€ | x | {\\displaystyle |x|} ã¯ãæ°çŽç·äžã§ãåç¹ O {\\displaystyle \\mathrm {O} } ãšç¹ P ( x ) {\\displaystyle \\mathrm {P} (x)} ã®éã®è·é¢ãè¡šããŠããã ãããã£ãŠã a > 0 {\\displaystyle a>0} ã®ãšã",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 164,
"tag": "p",
"text": "",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 165,
"tag": "p",
"text": "次ã®äžçåŒã解ãã (i)",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 166,
"tag": "p",
"text": "(ii)",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 167,
"tag": "p",
"text": "(iii)",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 168,
"tag": "p",
"text": "(iv)",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 169,
"tag": "p",
"text": "(i)",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 170,
"tag": "p",
"text": "(ii)",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 171,
"tag": "p",
"text": "(iii)",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 172,
"tag": "p",
"text": "(iv)",
"title": "äžæ¬¡äžçåŒ"
},
{
"paragraph_id": 173,
"tag": "p",
"text": "äžè¬ã®äºæ¬¡æ¹çšåŒ a x 2 + b x + c = 0 {\\displaystyle ax^{2}+bx+c=0} ( a {\\displaystyle a} , b {\\displaystyle b} , c {\\displaystyle c} ã¯å®æ°ã a â 0 {\\displaystyle a\\neq 0} )ã®è§£ x {\\displaystyle x} ãæ±ããå
¬åŒã«ã€ããŠèããã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 174,
"tag": "p",
"text": "ããã§æçåŒ x 2 + 2 y x = ( x + y ) 2 â y 2 {\\displaystyle x^{2}+2yx=(x+y)^{2}-y^{2}} ãš (1) ã®å·ŠèŸºãä¿æ°æ¯èŒãããšã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 175,
"tag": "p",
"text": "ã§ããããã(1) ã®åŒã¯æ¬¡ã®ããã«å€åœ¢ã§ãã(å¹³æ¹å®æ)ã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 176,
"tag": "p",
"text": "b 2 â 4 a c ⥠0 {\\displaystyle b^{2}-4ac\\geq 0} ã®ãšã䞡蟺ã®å¹³æ¹æ ¹ããšããšã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 177,
"tag": "p",
"text": "ãããäºæ¬¡æ¹çšåŒã®è§£ã®å
¬åŒ(ã«ãã»ããŠãããã®ããã®ãããããquadratic formula; äºæ¬¡å
¬åŒ)ã§ããã解ã®å
¬åŒãäºæ¬¡æ¹çšåŒã®äžè¬åœ¢ã«ä»£å
¥ãããšãå³èŸºã¯0ã«ãªãã¯ãã§ããã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 178,
"tag": "p",
"text": "ã§ããããšãçšãããšã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 179,
"tag": "p",
"text": "ãšãªãã確ãã«æ£ããããšããããã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 180,
"tag": "p",
"text": "ããããã解ã®å
¬åŒãå æ°å解ãçšããŠè§£ããªããã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 181,
"tag": "p",
"text": "çµæã®åŒã«æ ¹å·ãçŸããªãå Žåã«ã¯ãäœããã®ä»æ¹ã§å æ°å解ãã§ããããããããããã®æ¹æ³ã䜿ãã«ãããæ ¹å·ã¯ã§ããéãã®ä»æ¹ã§ç°¡ååããããšãéèŠã§ããã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 182,
"tag": "p",
"text": "(i)ã¯ç°¡åã«å æ°å解ã§ããã®ã§ã解ã®å
¬åŒãçšããå¿
èŠã¯ãªãã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 183,
"tag": "p",
"text": "ããã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 184,
"tag": "p",
"text": "ãçããšãªãã(ii)ã§ã¯ãå æ°å解ãåºæ¥ãªãã®ã§ã解ã®å
¬åŒãçšãããå æ°å解ãã§ãããã©ããã¯å®éã«è©Šè¡é¯èª€ããŠèŠåãããããªãã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 185,
"tag": "p",
"text": "ã«ã解ã®å
¬åŒãçšãããšãa=5, b= 2, c=-1ããã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 186,
"tag": "p",
"text": "ãšãªãã(iii),(iv)ã§ããå æ°å解ã¯åºæ¥ãªãã®ã§ã解ã®å
¬åŒãçšãããçãã¯ã (iii)",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 187,
"tag": "p",
"text": "(iv)",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 188,
"tag": "p",
"text": "(v)",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 189,
"tag": "p",
"text": "ãšå æ°å解ã§ããã®ã§ãçãã¯",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 190,
"tag": "p",
"text": "ãšãªãã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 191,
"tag": "p",
"text": "å
šåãéããŠãå æ°å解ãå¯èœãªæ¹çšåŒã«å¯ŸããŠãã解ã®å
¬åŒã䜿çšããŠãæ§ããªãã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 192,
"tag": "p",
"text": "",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 193,
"tag": "p",
"text": "äºæ¬¡æ¹çšåŒ a x 2 + 2 b â² x + c = 0 ( a â 0 ) {\\displaystyle ax^{2}+2b'x+c=0(a\\neq 0)} ã«ã€ããŠèããã 解ã®å
¬åŒã« b= 2b' ã代å
¥ãããš",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 194,
"tag": "p",
"text": "ãã£ãŠãäºæ¬¡æ¹çšåŒ a x 2 + 2 b â² x + c = 0 {\\displaystyle ax^{2}+2b'x+c=0} ã®è§£ã¯",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 195,
"tag": "p",
"text": "ãšãªãã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 196,
"tag": "p",
"text": "ãäžã®è§£ã®å
¬åŒãçšããŠè§£ããªããã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 197,
"tag": "p",
"text": "äžã®è§£ã®å
¬åŒãçšãããšãa=3, b'= 3, c=-2ããã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 198,
"tag": "p",
"text": "ãšãªãã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 199,
"tag": "p",
"text": "",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 200,
"tag": "p",
"text": "2次æ¹çšåŒ a x 2 + b x + c = 0 {\\displaystyle ax^{2}+bx+c=0} ã®è§£ã¯ x = â b ± b 2 â 4 a c 2 a {\\displaystyle x={\\frac {-b\\pm {\\sqrt {b^{2}-4ac}}}{2a}}} ã§ããã ãã®åŒã®æ ¹å·ã®äžèº«ã ãåãåºãããã®ãå€å¥åŒãšåŒã³ã2次æ¹çšåŒã®è§£ã®åæ°ãç°¡åã«å€å¥ã§ããã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 201,
"tag": "p",
"text": "D = b 2 â 4 a c {\\displaystyle D=b^{2}-4ac} ã®å€ã«ãã£ãŠæ¬¡ã®ããã«ãªãã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 202,
"tag": "p",
"text": "(1) D > 0 {\\displaystyle D>0} ã®ãšããç°ãªã2ã€ã®è§£ x = â b + b 2 â 4 a c 2 a {\\displaystyle x={\\frac {-b+{\\sqrt {b^{2}-4ac}}}{2a}}} ãš x = â b â b 2 â 4 a c 2 a {\\displaystyle x={\\frac {-b-{\\sqrt {b^{2}-4ac}}}{2a}}} ãæã€ã (2) D = 0 {\\displaystyle D=0} ã®ãšãã ± b 2 â 4 a c = ± 0 {\\displaystyle \\pm {\\sqrt {b^{2}-4ac}}=\\pm 0} ã§ããã®ã§ã2ã€ã®è§£ã¯äžèŽããŠããã 1ã€ã®è§£ x = â b 2 a {\\displaystyle x=-{\\frac {b}{2a}}} ãæã€ãããã¯2ã€ã®è§£ãéãªã£ããã®ãšèããŠãé解ãšããã (3) D < 0 {\\displaystyle D<0} ã®ãšããå®æ°ã®ç¯å²ã§ã¯è§£ã¯ãªãã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 203,
"tag": "p",
"text": "2次æ¹çšåŒ a x 2 + b x + c = 0 {\\displaystyle ax^{2}+bx+c=0} ã®è§£ã®åæ°ã¯ D = b 2 â 4 a c {\\displaystyle D=b^{2}-4ac} ã®å€ã§å€å®ã§ããã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 204,
"tag": "p",
"text": "次ã®2次æ¹çšåŒã®è§£ã®åæ°ãæ±ããã",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 205,
"tag": "p",
"text": "",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 206,
"tag": "p",
"text": "(I)",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 207,
"tag": "p",
"text": "ã ãããå®æ°è§£ã¯ãªãã (II)",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 208,
"tag": "p",
"text": "ã ãããé解ããã€ã (III)",
"title": "äºæ¬¡æ¹çšåŒ"
},
{
"paragraph_id": 209,
"tag": "p",
"text": "ã ãããç°ãªã2ã€ã®å®æ°ã®è§£ããã€ã",
"title": "äºæ¬¡æ¹çšåŒ"
}
] | null | == éåãšè«ç ==
[[æ§èª²çš(-2012幎床)é«çåŠæ ¡æ°åŠA/éåãšè«ç|ãã¡ã]]ãåç
§
== åŒã®å±éãšå æ°å解 ==
==== æŽåŒ ====
3ã12ãªã©ã®æ°ïŒå®æ°ïŒãã<math>x</math> ã <math>y</math> ãªã©ã®æåïŒå€æ°ïŒãæãããããŠã§ããåŒã'''é
'''ïŒãããtermïŒãšããã
次ã®ãããªãã®ãé
ã§ããã
* <math>3x</math>
* <math>12y</math>
* <math>0</math>
* <math>-x</math>
* <math>256xy^2</math>
ãã®ããã«äžã€ã®é
ã ãããã§ããŠããåŒã'''åé
åŒ'''ïŒãããããããmonomialïŒãšããã
:ïŒâ» è£è¶³: ãå€é
åŒããšã¯ïŒïŒãããã§ã¯ã'''å€é
åŒ'''ãïŒããããããpolynomialïŒãšã¯ãé
ã2ã€ä»¥äžã®åŒã ãšå®çŸ©ããããããå®ã¯ãé
ã1ã€ã®ãã®ãšè€æ°ã®ãã®ãåºå¥ããããããŸãšããŠæ±ã£ãæ¹ããæ§ã
ãªå®çãèšè¿°ããéã«äŸ¿å©ã«ãªãããã®ãããé«æ ¡æ°åŠä»¥å€ã§ã¯ãé
ã1ã€ã®ãã®ãå«ããŠãå€é
åŒããšå®çŸ©ããå Žåãå€ãããšãããããå€é
åŒããšã¯æåãã¿ãã°ããé
ã®å€ãåŒããšããæå³ãªã®ã§ãé
ã1ã€ã§ããããšå®çŸ©ãããšãå®çŸ©ãšååãäžèŽããŠããããæ··ä¹±ã®åå ã«ããªããããã§æ¥æ¬ã®é«æ ¡æè²ã§ã¯ããé
ã1ã€ä»¥äžã®åŒããšããæŠå¿µã«ã€ããŠã¯æŽåŒïŒããããïŒãšããçšèªã䜿ã£ãŠãããããã§ãããæŽãåŒãšã¯ãæŽçãããåŒãšãããããªæå³ã§ããããã£ããŠãæŽæ°ã®åŒãšããæå³ã§ã¯ãªãããªã®ã§ãä¿æ°ãªã©ã¯å°æ°ãåæ°ã§ãããã
â» ãããããŠãæŽåŒããå®çŸ©ãããšã次ã®ãããªå®çŸ©ã«ãªãã
1ã€ä»¥äžã®åé
åŒã足ãããããŠã§ããåŒã'''æŽåŒ'''ïŒããããïŒãšããã
以äžã¯æŽåŒã®äŸã§ããã
* <math>3x + 12y</math>
* <math>5 + a - 13x^2y</math>
* <math>a^2 + 2ab + b^2</math>
* <math>x - y</math>
* <math>2</math>
åé
åŒã§ããé
ã1ã€ãããªãæŽåŒã®äžã€ã§ãããšèããããšãã§ããã®ã§ããæŽåŒããšããæŠå¿µã䜿ãããšã«ãããå€é
åŒãšåé
åŒãšã®åºå¥ã®å¿
èŠããªããªãã
<math>x - y</math> ã®ããã«æžæ³ãå«ãåŒã¯ã <math>x - y = x + (-y) = -y + x</math> ãšæžæ³ãå æ³ã«çŽãããšãã§ããã®ã§ã<math>x, -y</math> ãé
ã«ãã€æŽåŒã§ãããšèãããããããªãã¡ãå€é
åŒã®é
ãšã¯ãå€é
åŒã足ãç®ã®åœ¢ã«çŽãããšãã®ãäžã€äžã€ã®è¶³ããããã£ãŠããåŒã®ããšã§ãããããšãã° <math>5 + a - 13x^2y = 5 + a + (-13x^2y)</math> ã®é
㯠<math>5, a, -13x^2y</math> ã®3ã€ã§ããã
* åé¡
次ã®åŒã®ãã¡åé
åŒã§ãããã®ãçããã
:(1) ã<math>ax^2 \times bx \times c</math>
:(2)ã<math>-(x^3y^4)(z^5)</math>
:(3) ã<math>a^2 + b^2 + c^2 - ab - bc - ca</math>
* 解ç
(1), (2) ãåé
åŒã (3) ã¯é
ã6ã€ããããåé
åŒã§ã¯ãªãã
* åè
äžã®å
šãŠã®åŒã¯æŽåŒã§ãããã
==== æŽåŒã®æŽç ====
<math>3x^2</math> + <math>5x^2+ 8x</math> ã® <math>3x^2</math> ãš <math>5x^2</math> ã®ããã«ãå€é
åŒã®æåãšææ°ããŸã£ããåãã§ããé
ãç·ç§°ããŠ'''åé¡é
'''ïŒã©ããããããlike termsïŒãšããã
åé¡é
ã¯åé
æ³å <math>ab + ac = a(b + c)</math> ã䜿ã£ãŠãŸãšããããšãã§ãããããšãã° <math>3x^2 + 5x^2 + 8x = (3 + 5)x^2 + 8x = 8x^2 + 8x </math>ã§ããã<math>8x^2</math> ãš <math>8x</math> ã¯æåã¯åãã§ãããææ°ãç°ãªãã®ã§ãåé¡é
ã§ã¯ãªãã
* åé¡
次ã®å€é
åŒã®åé¡é
ãæŽçããã
# ã<math>4x^3 - 3xy - 2 + 1 - 3x^3 + 4xy</math>
# ã<math>2a^2 - 4ab + 2a - 4ab^2 - 4a^2b</math>
# ã<math>9 x^2 y^3 z^4 - 8 z^2 y^3 x^4 + 7zyx - 6xyz + 5 x^2 yz - 4 y^2 x z + 3 z x^2 y - 2 x^4 y^3 z^2</math>
* 解ç
# ã<math>x^3 + xy - 1</math>
# ã<math>2a^2 - 4ab + 2a - 4ab^2 - 4a^2b</math>
# ã<math>-10 x^4 y^3 z^2 + 9 x^2 y^3 z^4 + 8 x^2 yz - 4 x y^2 z + xyz</math>
==== æ¬¡æ° ====
<math>3x</math> ãšããåé
åŒã¯ã3ãšããæ°ãš <math>x</math> ãšããæåã«åããŠèããããšãã§ãããæ°ã®éšåãåé
åŒã®'''ä¿æ°'''ïŒãããããcoefficientïŒãšããã
ããšãã° <math>-x = (-1)x</math> ãšããåé
åŒã®ä¿æ°ã¯ -1 ã§ããã
<math>256xy^2</math> ãšããåé
åŒã¯ã256ãšããæ°ãš <math>x, y, y</math> ãšããæåã«åããŠèããããšãã§ããã®ã§ããã®åé
åŒã®ä¿æ°ã¯256ã§ãããäžæ¹ãæãããããæåã®æ°ãåé
åŒã®'''次æ°'''ïŒããããdegreeïŒãšããã<math>256xy^2</math> 㯠<math>x, y, y</math> ãšãã3ã€ã®æåãæãããããŠã§ããŠããã®ã§ããã®åé
åŒã®æ¬¡æ°ã¯3ã§ããã0ãšããåé
åŒã®æ¬¡æ°ã¯ <math>0 = 0x = 0x^2 = 0x^3 = \cdots </math>ãšäžã€ã«å®ãŸããªãã®ã§ãããã§ã¯èããªãã
åé
åŒã®ä¿æ°ãšæ¬¡æ°ã¯ãåã«æ°ãšæåã«åããŠèããã®ã§ã¯ãªããããæåãå€æ°ãšããŠèŠããšãã«ãæ®ãã®æåãå®æ°ãšããŠæ°ãšåãããã«æ±ãããšãããã
ããšãã° <math>-5abcx^3</math>ãšããåé
åŒãã<math>x^3</math> ã ããå€æ°ã§ãæ®ãã®æå <math>a, b, c</math> ã¯å®æ°ãšèããããšãã§ããã
ãã®ãšã<math>(-5abc)x^3</math> ãšåããããã®ã§ããã®åé
åŒã®ä¿æ°ã¯ <math>-5abc</math>ãå€æ°ã¯ <math>x^3</math> ã§ã次æ°ã¯3ã§ãããšãããã
ãã®ããšã <math>-5abcx^3</math> ãšããåé
åŒã¯ãã<math>x</math> ã«''çç®''ãããšãä¿æ°ã¯ <math>-5abc</math>ã次æ°ã¯3ã§ããããªã©ãšããå Žåãããã
ããã㯠<math>-5abcx^3</math> ã® <math>a</math> ãš <math>b</math>ã«çç®ããã°ã<math>(-5cx^3)ab</math> ãšåãããã<math>a</math> ãš <math>b</math> ã«çç®ãããšãã®ãã®åé
åŒã®ä¿æ°ã¯ <math>-5cx^3</math>ãå€æ°ã¯ <math>ab</math> ã§ã次æ°ã¯2ã§ãããšãããã
æ
£ç¿çã«ã¯ <math>a, b, c, \cdots</math> ãªã©ã®ã¢ã«ãã¡ãããã®æåã®æ¹ã®æåãå®æ°ãè¡šãã®ã«äœ¿ãã<math>\cdots , x, y, z</math> ãªã©ã®ã¢ã«ãã¡ãããã®æåŸã®æ¹ã®æåãå€æ°ãè¡šãã®ã«çšããããäžè¬çã«ã¯ãã®éãã§ãªãã
å€é
åŒã®'''次æ°'''ãšã¯ãå€é
åŒã®åé¡é
ããŸãšãããšãã«ããã£ãšã次æ°ã®é«ãé
ã®æ¬¡æ°ããããããšãã° <math>x^3 + 3 x^2 y + 2y</math> ã§ã¯ããã£ãšã次æ°ã®é«ãé
㯠<math>x^3</math> ã§ããã®ã§ããã®å€é
åŒã®æ¬¡æ°ã¯3ã§ããããã <math>x^3 + 3 x^2 y + 2y</math>ïŒ<math>x</math> ã¯å®æ°ïŒã§ããã°ãããªãã¡å€é
åŒã® <math>y</math> ã«ã€ããŠçç®ãããšããã£ãšã次æ°ã®é«ãé
㯠<math>3 x^2 y</math> ãš <math>2y</math> ã§ããã®ã§ããã®å€é
åŒã®æ¬¡æ°ã¯1ã§ããããã®ãšãçç®ããæåãå«ãŸãªãé
<math>x^3</math> ã¯'''å®æ°é
'''ïŒãŠããããããconstant termïŒãšããŠæ°ãšåãããã«æ±ãããã
* åé¡
次ã®å€é
åŒã® <math>x</math> ãŸã㯠<math>y</math> ã«çç®ãããšãã®æ¬¡æ°ãšå®æ°é
ãããããããã
# <math>x^6 + 10xy^2 + 8x^4y + y^5 - 1</math>
# <math>-ad - bcx^2 - bc + 2 x^3 y^2 + y^{100}</math>
# <math>pxy + q^9 y^2 + pqxy - p^8 q^3 x^2 y + p x^4 y^3 + p q^2 x^3 y^4</math>
* 解ç
# <math>x</math> ã«çç®ãããš6次åŒãå®æ°é
㯠<math>y^5 - 1</math>ã<math>y</math> ã«çç®ãããš5次åŒãå®æ°é
㯠<math>x^6 - 1</math>ã
# <math>x</math> ã«çç®ãããš3次åŒãå®æ°é
㯠<math>-ad - bc + y^{100}</math>ã<math>y</math> ã«çç®ãããš100次åŒãå®æ°é
㯠<math>-ad - bcx^2 - bc</math>ã
# <math>x</math> ã«çç®ãããš4次åŒãå®æ°é
㯠<math>q^9 y^2</math>ã<math>y</math> ã«çç®ãããš4次åŒãå®æ°é
ã¯ååšããªãã
==== éã¹ããšæã¹ã ====
ããšãã°ã
:<math>x^2 + 6x +7 </math>
ã®ããã«ã次æ°ã®é«ãé
ããå
ã«é
ããªãã¹ãããšãã'''éã¹ã'''ãïŒããã¹ãïŒãšããã
:â» ãªãã次æ°ã®å€§å°ã«ã€ããŠã¯ã次æ°ã倧ããããšãã次æ°ãé«ãããšèšã£ããããŠããããã€ãŸãã次æ°ã®å€§å°ã«ã€ããŠã¯ãé«äœã§èšãæããŠãããã
ããŠãåŒã䜿ãç®çã«ãã£ãŠã¯ã次æ°ã®ã²ããé
ããå
ã«æžããã»ãã䟿å©ãªå Žåãããã
ããšãã°ã<math>x</math>ã çŽ0.01 ã®ãããª1æªæºã®å°ããæ°ã®å ŽåãåŒ <math>x^2 + 6x +7 </math> ã®å€ãæ±ããããªããæå<math>x</math>ã®æ¬¡æ°ã®å°ããé
ã®ã»ãã圱é¿ãé«ãã
ãªã®ã§ã ç®çã«ãã£ãŠã¯
:<math>7 + 6x + x^2 </math>
ã®ããã«ã次æ°ã®ã²ããé
ããå
ã«æžãå Žåãããã
<math>7 + 6x + x^2 </math> ã®ããã«ã次æ°ã®äœãé
ããå
ã«é
ããªãã¹ãããšãã'''æã¹ã'''ãïŒãããã¹ãïŒãšããã
==== ç¹å®ã®æåãžã®çç® ====
å€é
åŒã«2ã€ä»¥äžã®æåããããšããç¹å®ã®1ã€ã®æåã«æ³šç®ããŠäžŠã³å€ãããšã䜿ãããããªãããšãããã
ããšãã°ã
:<math>x^3 - 5 + 2xy^3+ 7y^2 + 6x^2y </math>ããïŒäŸ1ïŒ
ã®é
ããxã®æ¬¡æ°ãå€ãé
ããå
ã«äžŠã³ãããåé¡é
ããŸãšãããš
:<math>x^3 + (6y)x^2 + (2 y^3 )x + (7y^2 - 5 ) </math>ããïŒäŸ2ïŒ
ãšãªãã
ãã®ïŒäŸ2ïŒã®ããã«ãç¹å®ã®æåã ãã«çç®ããŠããã®æåã®æ¬¡æ°ã®é«ãé ã«äžŠã³ããããšäŸ¿å©ãªããšããã°ãã°ããã
äŸ2ã¯ã<math>x</math>ã«ã€ã㊠éã¹ã ã®é ã«äžŠã³å€ããæŽåŒã§ããã
çç®ããŠãªãæåã«ã€ããŠã¯ã䞊ã³æãã®ãšãã¯å®æ°ã®ããã«æ±ãã
ãã£ãœãã<math>x</math>ã«ã€ããŠã次æ°ã®ã²ããé
ããé ã«äžŠã¹ããšã次ã®ãããªåŒã«ãªãã
:<math>(7y^2 - 5 ) + (2 y^3 )x + (6y)x^2 + x^3 </math>ããïŒäŸ3ïŒ
ãã®ããã«ãç¹å®ã®æåã®æ¬¡æ°ãäœããã®ããé ã«äžŠã³ããããšäŸ¿å©ãªããšããã°ãã°ããã
äŸ3ã¯ãxã«ã€ã㊠æã¹ã ã®é ã«äžŠã³å€ããæŽåŒã§ããã
==== ç¹å®ã®æåã«æ³šç®ããæ¬¡æ° ====
ããšãã°ãåŒ
:<math>y = ax + b </math>
ãšããåŒã®å³èŸº
:<math>ax+b </math>
ã®æ¬¡æ°ã¯ããããã§ããããã
aãšxãçããæåãšããŠæ±ãã®ã§ããã°ã<math>ax</math>ã®æ¬¡æ°ã¯
:<math>a^1 x^1 </math>
ãã 1ïŒ1 ïŒ2 ãªã®ã§ããã®åŒã®æ¬¡æ°ã¯2ã§ãããïŒé
bã¯æ¬¡æ°1ãªã®ã§ã<math>ax</math>ã®æ¬¡æ°2ãããäœãã®ã§ç¡èŠãããïŒ
ãããããããã®åŒããå®æ°<math>a</math>ãä¿æ°ãšããå€æ°<math>x</math>ã«ã€ããŠã®äžæ¬¡é¢æ°ãšã¿ãã®ã§ããã°ãäžæ¬¡åŒãšæãã®ãåççã ããã
ãã®ãããªå Žåãç¹å®ã®æåã ãã«æ³šç®ãããã®åŒã®æ¬¡æ°ãèãããšããã
ããšãã°ãæåxã ãã«æ³šç®ããŠãåŒ <math>ax + b </math> ã®æ¬¡æ°ã決ããŠã¿ããã
ãããšãæåxã«æ³šç®ããå Žåã®åŒ <math>ax + b </math> ã®æ¬¡æ°ã¯1ã«ãªãã
ãªããªã
:æå<math>x</math>ã«æ³šç®ããå Žåã®åŒ <math>a </math> ã®æ¬¡æ°ã¯0ã§ããã
:æå<math>x</math>ã«æ³šç®ããå Žåã®åŒ <math>b </math> ã®æ¬¡æ°ã¯0ã§ããã
:æå<math>x</math>ã«æ³šç®ããå Žåã®åŒ <math>x </math> ã®æ¬¡æ°ã¯1ã§ããã
ãã£ãŠãæå<math>x</math>ã«æ³šç®ããå Žåã®é
<math>ax</math> ã®æ¬¡æ°ã¯ã 0ïŒ1 ãªã®ã§ã1ã§ããã
ãã®ããã«èããå Žåãå¿
èŠã«å¿ããŠã©ã®æåã«æ³šç®ããããæèšããŠãæåâ¯â¯ã«æ³šç®ãã次æ°ãã®ããã«è¿°ã¹ããšããã
==== å€é
åŒã®èšç® ====
å€é
åŒã®ç©ã¯åé
æ³åã䜿ã£ãŠèšç®ããããšãã§ããã
:<math>
\begin{align}
(a + b)(c + d) &= (a + b)c + (a + b)d \\
&= (ac + bc) + (ad + bd) \\
&= ac + bc + ad + bd
\end{align}
</math>
ãã®ããã«å€é
åŒã®ç©ã§è¡šãããåŒãäžã€ã®å€é
åŒã«ç¹°ãåºããããšããå€é
åŒã'''å±é'''ïŒãŠããããexpandïŒãããšããã
====ææ°æ³å====
<math>a</math> ã <math>n</math> åæãããã®ã <math>a^n</math> ãšæžãã'''aã®nä¹'''ïŒ-ãããã''a'' to the ''n''-th powerïŒãšããããã ã <math>a^1 = a</math> ãšå®çŸ©ãããããšãã°ã
:<math>2^1 = 2</math>
:<math>2^2 = 2 \times 2 = 4</math>
:<math>2^3 = 2 \times 2 \times 2 = 8</math>
:<math>2^4 = 2 \times 2 \times 2 \times 2 = 16</math>
:<math>2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32</math>
:...
ã§ããã<math>a, a^2, a^3, a^4, a^5, \cdots, a^n</math> ãç·ç§°ã㊠<math>a</math> ã®'''环ä¹'''ïŒããããããexponentiationãåªä¹ãã¹ãããããåªãã¹ãïŒãšããã<math>a^n</math> ã® ''n'' ã'''ææ°'''ïŒããããexponentïŒãšããïŒ''a'' ã¯'''åº'''ïŒãŠããbaseïŒãšããïŒãããã§ã¯èªç¶æ°ãããªãã¡æ£ã®æŽæ°ã®ææ°ãèããã环ä¹ã¯æ¬¡ã®ããã«èããããšãã§ããã
:<math>2^1 = 2</math>
:<math>2^2 = 2^1 \times 2 = 2 \times 2 = 4</math>
:<math>2^3 = 2^2 \times 2 = 4 \times 2 = 8</math>
:<math>2^4 = 2^3 \times 2 = 8 \times 2 = 16</math>
:<math>2^5 = 2^4 \times 2 = 16 \times 2 = 32</math>
:<math>\cdots</math>
环ä¹ã©ãããæãããããç©ã¯ã次ã®ããã«èšç®ããããšãã§ããã
:<math>
\begin{align}
a^2 \times a^3 &= (a \times a) \times (a \times a \times a) \\
&= a^{2 + 3} \\
&= a^5
\end{align}
</math>
环ä¹ã©ãããå²ã£ãåã¯ã次ã®ããã«èšç®ããããšãã§ããã
:<math>
\begin{align}
a^3 \div a^2 &= \frac{ a \times a \times a }{ a \times a } \\
&= \frac{a}{1} \\
&= a
\end{align}
</math>
环ä¹ã®çŽ¯ä¹ã¯ã次ã®ããã«èšç®ããããšãã§ããã
:<math>
\begin{align}
(a^2)^3 &= a^2 \times a^2 \times a^2 \\
&= a^{2 + 2 + 2} \\
&= a^{2 \times 3} \\
&= a^6
\end{align}
</math>
ç©ã®çŽ¯ä¹ã¯ã次ã®ããã«èšç®ããããšãã§ããã
:<math>
\begin{align}
(ab)^2 &= a \times b \times a \times b \\
&= a \times a \times b \times b \\
&= a^2 b^2
\end{align}
</math>
ããããããããŠ'''ææ°æ³å'''ïŒãããã»ããããexponential lawïŒãšããã
{| style="border: 2px solid skyblue; width: 80%; " cellspacing=0
| style="background: skyblue;" | '''ææ°æ³å'''
|-
| style="padding: 5px;" |
''m'', ''n'' ãæ£ã®æŽæ°ãšãããšã
*<math>a^m \times a^n = a^{m + n}</math>
*<math>a^m \div a^n = a^{m - n}, m > n</math>
*<math>(a^m)^n = a^{mn}</math>
*<math>(ab)^n = a^n b^n</math>
|}
{{蚌æ|ææ°æ³åã®èšŒæ}}
环ä¹ã®å®çŸ©ããæããã
:<math>
\begin{align}
a^m \times a^n &= \overbrace{ \underbrace{ (a \times a \times \cdots \times a) }_m \times \underbrace{ (a \times a \times \cdots \times a) }_n }^{m + n} \\
&= a^{m + n}
\end{align}
</math>
:<math>
\begin{align}
a^m \div a^n &= \frac{ \overbrace{ a \times a \times \cdots \times a }^m }{ \underbrace{ a \times a \times \cdots \times a }_n } \\
&= \frac{ \overbrace{ a \times a \times \cdots \times a }^n \times \overbrace{ a \times a \times \cdots \times a }^{m - n} }{ \underbrace{ a \times a \times \cdots \times a }_n } \\
&= \frac{ \overbrace{ a \times a \times \cdots \times a }^{m - n} }{1} \\
&= \underbrace{ a \times a \times \cdots \times a }_{m - n} \\
&= a^{m - n}
\end{align}
</math>
:<math>
\begin{align}
(a^m)^n &= \underbrace{ a^m \times a^m \times \cdots \times a^m }_n \\
&= a^{ \overbrace{ m + m + \cdots + m }^n } \\
&= a^{mn}
\end{align}
</math>
:<math>
\begin{align}
(ab)^n &= \underbrace{ (a \times b) \times (a \times b) \times \cdots \times (a \times b) }_n \\
&= \underbrace{ (a \times a \times \cdots \times a) }_n \times \underbrace{ (b \times b \times \cdots \times b) }_n \\
&= a^n b^n
\end{align}
</math>
{{蚌æçµãã}}
*åé¡
次ã®åŒãèšç®ããªããã
# <math>x^4 \times x^3</math>
# <math>(a^3)^4</math>
# <math>(-a^2b)^3</math>
*解ç
# <math>x^4 \times x^3 = x^{4+3} = x^7</math>
# <math>(a^3)^4 = a^{3 \times 4} = a^{12}</math>
# <math>
(-a^2b)^3 = (-1)^3 (a^2)^3 b^3 = -a^{2 \times 3}b^3 = -a^6b^3
</math>
==== ä¹æ³å
¬åŒ ====
* åé¡
次ã®åŒãå±éããã
# <math>(a + b)^2</math>
# <math>(a - b)^2</math>
# <math>(a + b)^3</math>
# <math>(a - b)^3</math>
# <math>(a + b + c)^2</math>
# <math>(a - b - c)^2</math>
* 解ç
# <br> <math style="vertical-align: top;">\begin{align}
(a + b)^2 &= (a + b)(a + b) \\
&= a(a + b) + b(a + b) \\
&= (aa + ab) + (ba + bb) \\
&= aa + ab + ba + bb \\
&= a^2 + 2ab + b^2
\end{align}</math>
# <br> <math style="vertical-align: top;">\begin{align}
(a - b)^2 &= \{ a + (-b) \}^2 \\
&= a^2 + 2a(-b) + (-b)^2 \\
&= a^2 - 2ab + b^2
\end{align}</math>
# <br> <math style="vertical-align: top;">\begin{align}
(a + b)^3 &= (a + b)(a + b)^2 \\
&= (a + b)(a^2 + 2ab + b^2) \\
&= a(a^2 + 2ab + b^2) + b(a^2 + 2ab + b^2) \\
&= (a^3 + 2a^2b + ab^2) + (a^2b + 2ab^2 + b^3) \\
&= a^3 + (2a^2b + a^2b) + (ab^2 + 2ab^2) + b^3 \\
&= a^3 + 3a^2b + 3ab^2 + b^3
\end{align}</math>
# <br> <math style="vertical-align: top;">\begin{align}
(a - b)^3 &= \{ a + (-b) \}^3 \\
&= a^3 + 3a^2(-b) + 3a(-b)^2 + (-b)^3 \\
&= a^3 - 3a^2b + 3ab^2 - b^3
\end{align}</math>
# <br> <math style="vertical-align: top;">\begin{align}
(a + b + c)^2 &= \{ (a + b) + c \}^2 \\
&= (a + b)^2 + 2(a + b)c + c^2 \\
&= (a^2 + 2ab + b^2) + (2ac + 2bc) + c^2 \\
&= a^2 + 2ab + b^2 + 2ac + 2bc + c^2 \\
&= a^2 + b^2 + c^2 + 2ab + 2bc + 2ca
\end{align}</math>
# <br> <math style="vertical-align: top;">\begin{align}
(a - b - c)^2 &= a^2 + (-b)^2 + (-c)^2 + 2a(-b) + 2(-b)(-c) + 2(-c)a \\
&= a^2 + b^2 + c^2 - 2ab + 2bc - 2ca
\end{align}</math>
ãŸãšãããšã次ã®ããã«ãªãã
{| style="border: 2px solid skyblue; width: 80%;" cellspacing=0
| style="background: skyblue;"| '''å±éã®å
¬åŒ'''
|-
| style="padding: 5px;" |
* <math>(a \pm b)^2 = a^2 \pm 2ab + b^2</math>
* <math>(a + b)(a - b) = a^2 - b^2</math>
* <math>(x + a)(x + b) = x^2 + (a + b)x + ab</math>
* <math>(ax + b)(cx + d) = acx^2 + (ad + bc)x + bd</math>
* <math>(a \pm b)^3 = a^3 \pm 3a^2b + 3ab^2 \pm b^3</math>
* <math>(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca</math>
* <math>(a \pm b)(a^2 \mp ab + b^2) = a^3 \pm b^3</math>
* <math>(a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) = a^3 + b^3 + c^3 - 3abc</math>
|}
* åé¡
次ã®åŒãå±éããªããã
# <math>(a + 2b)^2</math>
# <math>(3x - 5y)^2</math>
# <math>(4x - 3y)(4x + 3y)</math>
# <math>(x + 1)(x - 5)</math>
# <math>(3x + 2y)(2x - y)</math>
# <math>(x + 3)(x^2 - 3x + 9)</math>
# <math>(a - 5)(a^2 + 5a + 25)</math>
# <math>(x + 4)^3</math>
# <math>(3a - 2b)^3</math>
*解ç
# <math>a^2+2 \times a \times 2b+(2 b)^2 = a^2+4ab+4 b^2 </math>
# <math> (3x)^2-2 \times 3x \times 5y+(5y)^2 = 9x^2-30xy+25y^2 </math>
# <math> (4x)^2-(3y)^2 = 16x^2-9y ^2 </math>
# <math> x^2+\{ 1+(-5) \}x+1 \times (-5) = x^2-4x-5 </math>
# <math> (3 \times 2)x^2+\{ 3 \times (-y) +2y \times 2 \}x+2y \times (-y) = 6x^2+xy-2y^2 </math>
# <math> \left(x+3\right)\,\left(x^2-x \times 3 +3^2 \right) = x^3+3^3 = x^3+27 </math>
# <math> \left(a-5\right)\,\left(a^2+a \times 5 +5^2 \right) = a^3-5^3 =a ^3-125 </math>
# <math> x^3+3 \times x^2 \times 4 +3 \times x \times 4^2 +4^3 = x^3+12x^2+48x+64 </math>
# <math> (3a)^3-3 \times (3a)^2 \times 2b +3 \times 3a \times (2b)^2 -(2b)^3 = 27a^3-54a^2b+36ab^2-8b^3 </math>
==== ä¹æ³å
¬åŒã®å©çš ====
è€éãªåŒã®å±éã¯ãåŒã®äžéšåãäžã€ã®æåã«ãããŠå
¬åŒã䜿ããšããã
* åé¡
次ã®åŒãå±éããªããã
# <math> (a+3b-2c)^2 </math>
# <math> (x+y+4)(x-3y+4) </math>
# <math> \left(x^2-2x+3\right)\,\left(x^2+2x-3\right) </math>
* 解ç
# <br> <math>a+3b=A</math>ãšãããš<br/><math>\begin{align}
(a+3b-2c)^2 & = (A-2c)^2 \\
& = A^2-4cA+4c^2\\
& = (a+3b)^2-4c(a+3b)+4c^2\\
& = a^2+6ab+9b^2-4ca-12bc+4c^2\\
& = a^2+9b^2+4c^2+6ab-12bc-4ca\\
\end{align}
</math>
# <br> <math>x+4=A</math>ãšãããš<br/><math>\begin{align}
(x+y+4)(x-3y+4) & = (A+y)(A-3y) \\
& = A^2-2yA-3y^2\\
& = (x+4)^2-2y(x+4)-3y^2\\
& = x^2+8x+16-2xy-8y-3y^2\\
& = x^2-3y^2-2xy+8x-8y+16\\
\end{align}
</math>
# <br> <math>2x-3=A</math>ãšãããš<br/><math>\begin{align}
\left(x^2-2x+3\right)\,\left(x^2+2x-3\right) & = \left\{x^2-(2x-3) \right\} \left\{x^2+(2x-3) \right\}\\
& = \left(x^2-A\right)\,\left(x^2+A\right)\\
& = x^4-A^2\\
& = x^4-(2x-3)^2\\
& = x^4-(4x^2-12x+9)\\
& = x^4-4x^2+12x-9\\
\end{align}
</math>
==== å æ°å解 ====
{| style="border:2px solid pink;width:80%" cellspacing=0
|style="background:pink"|'''å æ°å解ã®å
¬åŒ''' 1
|-
|style="padding:5px"|
* <math>a^2+2ab+b^2=(a+b)^2</math>
* <math>a^2-2ab+b^2=(a-b)^2</math>
* <math>a^2-b^2=(a+b)(a-b)</math>
* <math>x^2+(a+b)x+ab=(x+a)(x+b)</math>
* <math>acx^2+(ad+bc)x+bd=(ax+b)(cx+d)</math>
|}
* åé¡
次ã®åŒãå æ°å解ããªããã
# ã<math> 2abc-4ab^2 </math>
# ã<math> x^2+6x+9 </math>
# ã<math> 4a^2-4ab+b^2 </math>
# ã<math> 64x^2-9y^2 </math>
# ã<math> x^2-x-6 </math>
# ã<math> 3x^2+2x-5 </math>
# ã<math> 6x^2+xy-y^2 </math>
* 解ç
# ã<math> {\color{red}2ab} \times c - {\color{red}2ab} \times 2b = {\color{red}2ab}(c-2b) </math>
# ã<math> x^2+2 \times x \times 3+3^2 = (x+3)^2 </math>
# ã<math> (2a)^2-2 \times 2a \times b+b^2 = (2a-b)^2 </math>
# ã<math> (8x)^2-(3y)^2 = (8x+3y)(8x-3y) </math>
# ã<math> x^2+\{ 2+(-3) \}x+2 \times (-3) = (x+2)(x-3) </math>
# ã<math> (1 \times 3)x^2+\{ 1 \times 5 + (-1) \times 3 \}x+(-1) \times 5 = (x-1)(3x+5) </math>
# ã<math> (2 \times 3)x^2+\{ 2 \times (-y) + y \times 3 \}x+y \times (-y) = (2x+y)(3x-y) </math>
==== çºå±ïŒ 3次åŒã®å æ°å解 ====
{| style="border:2px solid pink;width:80%" cellspacing=0
|style="background:pink"|'''å æ°å解ã®å
¬åŒ''' 2
|-
|style="padding:5px"|
* <math>a^3+b^3=(a+b)(a^2-ab+b^2)</math>
* <math>a^3-b^3=(a-b)(a^2+ab+b^2)</math>
* <math>a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)</math>
ïŒåèïŒ
* <math>a^n - b^n = (a - b)(a^{n - 1} + a^{n - 2}b + a^{n - 3}b^2 + \cdots + a^2b^{n - 3} + ab^{n - 2} + b^{n - 1})</math>
|}
* åé¡
次ã®åŒãå æ°å解ããªããã
# ã<math> x^3+8 </math>
# ã<math> 27a^3-64b^3 </math>
* 解ç
# ã<math> x^3+2^3= \left(x+2\right)\,\left(x^2-x \times 2 +2^2 \right) = \left(x+2\right)\,\left(x^2-2x+4 \right) </math>
# ã<math> (3a)^3-(4b)^3= \left(3a-4b\right)\,\{(3a)^2+3a \times 4b +(4b)^2 \} = \left(3a-4b\right)\,\left(9a^2+12ab+16b^2 \right) </math>
==== ãããããªå æ°å解 ====
*åé¡
次ã®åŒãå æ°å解ããªããã
# ãã<math> 3xy^3+81x </math>
# ãã<math> (x-5)^2-9y^2 </math>
# ãã<math> x^2+xy+y-1 </math>
# ãã<math> x^2+xy-2y^2+2x+7y-3 </math>
* 解ç
# <br> <math>\begin{align}
3xy^3+81x & = 3x(y^3+27) \\
& = 3x(y^3+3^3)\\
& = 3x \left(y+3\right)\,\left(y^2-y \times 3 +3^2 \right)\\
& = 3x \left(y+3\right)\,\left(y^2-3y+9 \right)\\
\end{align}
</math>
# ãã<math>x-5=A</math>ãšãããš<br/><math>\begin{align}
(x-5)^2-9y^2 & = A^2-9y^2\\
& = (A+3y)(A-3y)\\
& = \left\{(x-5)+3y \right\} \left\{(x-5)-3y \right\}\\
& = (x+3y-5)(x-3y-5)\\
\end{align}
</math>
# ããæã次æ°ã®äœã <math>y</math> ã«çç®ããŠæŽçãããš<br/><math>\begin{align}
x^2+xy+y-1 & = (x+1)y+ \left(x^2-1\right)\\
& = (x+1)y+(x+1)(x-1)\\
& = (x+1)\left\{y+(x-1) \right\}\\
& = (x+1)(x+y-1)\\
\end{align}
</math>
# ãã<math>x</math> ã«çç®ããŠæŽçãããš<br/><math>\begin{align}
x^2+xy-2y^2+2x+7y-3 & = x^2+(y+2)x-(2y^2-7y+3)\\
& = x^2+(y+2)x-(y-3)(2y-1)\\
& = \left\{x-(y-3) \right\} \left\{x+(2y-1) \right\}\\
& = (x-y+3)(x+2y-1)\\
\end{align}
</math>
{{ã³ã©ã | 察称åŒãšäº€ä»£åŒ |
;察称åŒ
ã<math> a^2 + b^2</math> ã¯ã<math>a</math> ãš <math>b</math> ãå
¥ãæ¿ã㊠<math> b^2 + a^2</math> ã«ããŠããå€ã¯ããšã®åŒãšåããŸãŸã§ããã
ãã®ããã«ãæåãå
¥ãæ¿ããŠãåããŸãŸã«ãªãåŒã®ããšã '''察称åŒ'''ïŒ ãããããããïŒãšããã
<math>a</math>,<math>b</math> ã®å¯Ÿç§°åŒã®ãã¡ãåŒ <math> a + b</math> ãš åŒ <math> ab</math> ã®2ã€ã '''åºæ¬å¯Ÿç§°åŒ''' ãšããã
åºæ¬å¯Ÿç§°åŒãããã®å¯Ÿç§°åŒã¯ãåºæ¬å¯Ÿç§°åŒã®å æžä¹é€ã§è¡šãããšãã§ãããããšãã°ã
:<math> a^2 + b^2 = (a+b)^2 -2ab</math>
ã§ããã
;亀代åŒ
ã<math> a^2 - b^2</math> ã¯ãæåãå
¥ãæ¿ãããšã<math> b^2 - a^2</math> ã«ãªãããããã¯ããšã®åŒã ãŒ1 åãããã®ã§ããããã®ããã«ãæåãå
¥ãæ¿ããããšã§ãããšã®åŒã ãŒ1 åãããã®ã«ãªãåŒã®ããšã '''亀代åŒ''' ïŒããããããïŒãšããã
}}
== å®æ° ==
==== ç¡çæ°ãšæçæ° ====
a=b^2ãæãç«ã€ãšããa=2ãšãªããããªbãããªãã¡<math>\sqrt{2}</math>ã®å
·äœçãªå€ãã©ã®ãããªãã®ãã調ã¹ãŠã¿ããã
{|
|-
|b=1
|a=1
|b=2
|a=4
|-
|b=1.4
|a=1.96
|b=1.5
|a=2.25ã
|-
|b=1.41
|a=1.9881
|b=1.42
|a=2.0164ã
|-
|b=1.414
|a=1.999396
|b=1.415
|a=2.002225ã
|-
|b=1.4142ã
|a=1.99996164ã
|b=1.4143ã
|a=2.00024449ã
|}
ãã®ããã«ãbãæ§ã
ã«æ±ºããŠããaã¯ãªããªã2ã«ãªããªãã
å®ã¯<math>\sqrt{2}</math>ã¯ãåæ¯ååå
±ã«æŽæ°ã®åæ°ã§è¡šãããšã¯ã§ããªãããã®ããã«æŽæ°ãåæ¯ååã«æã€åæ°ã§è¡šããªããããªæ°ã'''ç¡çæ°'''ãšãããäŸãã°ãååšçπã¯ç¡çæ°ã§ãããããã«å¯ŸããŠãæŽæ°ã埪ç°å°æ°ãªã©ãåæ¯ååå
±ã«æŽæ°ã®åæ°ã§è¡šãããšã®ã§ããæ°ã'''æçæ°'''ãšããã
æçæ°ãšç¡çæ°ãåãããŠ'''å®æ°'''ãšãããã©ããªå®æ°ã§ãæ°çŽç·äžã®ç¹ãšããŠè¡šããããŸããã©ããªå®æ°ããæéå°æ°ãããã¯ç¡éå°æ°ãšããŠè¡šããã
(äžèšã®ãç¡éå°æ°ãã®ç¯ãåç
§)
;<math>\sqrt{2}</math>ãç¡çæ°ã§ããããšã®èšŒæïŒçºå±ïŒ
<math>\sqrt{2}</math> ãæçæ°ã§ãããšä»®å®ãããšã[[w:äºãã«çŽ |äºãã«çŽ ]]ãªïŒ1以å€ã«å
¬çŽæ°ããããªãïŒæŽæ° ''m'', ''n'' ãçšããŠã
:<math>\sqrt{2} = \frac{m}{n}</math>
ãšè¡šããããšãã§ããããã®ãšãã䞡蟺ã2ä¹ããŠåæ¯ãæããšã
:<math>2n^2 = m^2</math> ⊠(1)
ãã£ãŠ ''m'' ã¯2ã®åæ°ã§ãããæŽæ° ''l'' ãçšã㊠<math>m = 2l</math> ãšè¡šãããšãã§ãããããã (1) ã®åŒã«ä»£å
¥ããŠæŽçãããšã
:<math>2l^2 = n^2</math>
ãã£ãŠ ''n'' ã2ã®åæ°ã§ãããããã㯠''m'', ''n'' ã2ãå
¬çŽæ°ã«ãã€ããšã«ãªããäºãã«çŽ ãšä»®å®ããããšã«ççŸããããããã£ãŠ <math>\sqrt{2}</math> ã¯ç¡çæ°ã§ããïŒ[[é«çåŠæ ¡æ°åŠA éåãšè«ç#èçæ³|èçæ³]]ïŒã
==== ç¡éå°æ° ====
[[File:Real number category japanese.svg|thumb|400px]]
0.1 ã 0.123456789 ã®ããã«ãããäœã§çµããå°æ°ã'''æéå°æ°'''ãšããã
äžæ¹ã<math>0.1234512345 \cdots</math> ã <math>3.1415926535 \cdots</math> ã®ããã«ç¡éã«ç¶ãå°æ°ã '''ç¡éå°æ°'''ïŒããã ãããããïŒãšããã
ç¡éå°æ°ã®ãã¡ãããäœããäžãããããé
åã®æ°åã®ç¹°ãè¿ãã«ãªã£ãŠãããã®ã '''埪ç°å°æ°'''ïŒãã
ããã ãããããïŒãšãããäŸãã° <math>0.3333333333 \cdots</math> ã <math>0.1428571428 \cdots</math>ã<math>0.1232323232 \cdots</math> ãªã©ã§ãããç¹°ãè¿ãã®æå°åäœã'''埪ç°ç¯'''ãšããã埪ç°å°æ°ã¯åŸªç°ç¯1ã€ãçšããŠ<math>0. \dot{3}</math>ã<math>0. \dot{1} 4285 \dot{7}</math>ã<math>0.1 \dot{2} \dot{3}</math>ã®ããã«åŸªç°ç¯ã®æåãšæåŸ(埪ç°ç¯ãäžæ¡ã®å Žåã¯ã²ãšã€ã ã)ã®äžã«ç¹ãã€ããŠè¡šãã
å
šãŠã®åŸªç°å°æ°ã¯åæ°ã«çŽããã
:<math>a = 0. \dot{3}</math>ãã(1)
ãšçœ®ããšã
:<math>10a = 3. \dot{3}</math>ãã(2)
ã§ããã(2)ãŒ(1) ãã <math>9a = 3</math>ããã£ãŠ <math>a = \frac{3}{9} = \frac{1}{3}</math> ã§ããã
;äŸé¡
* (äŸé¡1) <br/>
<math>\begin{align}
a &= 0. \dot{1} 4285 \dot{7}\\
1000000a &= 142857. \dot{1} 4285 \dot{7}\\
999999a &= 142857\\
a &= \frac{142857}{999999} \ = \frac{1}{7}
\end{align}
</math>
* (äŸé¡2)<br/>
<math>\begin{align}
a &= 0.1 \dot{2} \dot{3}\\
100a &= 12.3 \dot{2} \dot{3}\\
99a &= 12.2\\
a &= \frac{12.2}{99} \ = \frac{61}{495}
\end{align}
</math>
==== çµ¶å¯Ÿå€ ====
å®æ° ''a'' ã«ã€ããŠã''a'' ã®æ°çŽç·äžã§ã®åç¹ãšã®è·é¢ã ''a'' ã®çµ¶å¯Ÿå€ãšããã<math>|a|</math> ã§è¡šãã
{| style="border:2px solid green;width:80%" cellspacing=0
|style="background:lightgreen"| '''絶察å€'''
|-
|style="padding:5px"|
:<math>a \geqq 0</math> ã®ãšããã<math>|a|=a</math><br><br>
:<math>a < 0</math> ã®ãšããã<math>|a|=-a</math>
|}
ããšãã°
:<math>|2|=2</math>
:<math>| -3 | \ = \ -(-3) \ = \ 3</math>
ã§ããã
å®çŸ©ãã <math>|a|=|-a|</math> ããããããŸãã<math>a,b</math>ãä»»æã®å®æ°ãšãããšããããããã«å¯Ÿå¿ããæ°çŽç·äžã®ä»»æã®2ç¹ <math>\mathrm{P} (a) , \mathrm{Q} (b)</math> éã®è·é¢ã«ã€ããŠã¯ã次ã®ããšããããã
{| style="border:2px solid green;width:80%" cellspacing=0
|style="background:lightgreen"| '''2ç¹éã®è·é¢'''
|-
|style="padding:5px"|
æ°çŽç·äžã®2ç¹ <math>\mathrm{P} (a)</math> ãš <math>\mathrm{Q} (b)</math> ã®éã®è·é¢ <math>\mathrm{P} \mathrm{Q}</math> 㯠<math>|b-a|</math> ã§è¡šãããã
|}
* äŸé¡
:2ç¹ <math>\mathrm{P} (5)</math> ãš <math>\mathrm{Q} (-1)</math> ã®éã®è·é¢ãæ±ããã
* 解ç
:<math>\mathrm{P} \mathrm{Q} = |5- (-1) | = 6</math> ãªã®ã§ããã£ãŠPQéã®è·é¢ã¯ 6 ã§ããã
<br>
==== å¹³æ¹æ ¹ ====
ä»ã2ä¹ããŠaã«ãªãæ°bãèããã
<math>a=1</math>ã®ãšãã<math>b=1</math>ãšããŠçµããã«ããŠã¯ãããªãã確ãã«<math>b=1</math>ãæ¡ä»¶ãæºããã<math>b=-1</math>ãæ¡ä»¶ãæºããããã£ãŠ<math>b= 1</math> ãŸã㯠<math>b= -1</math>ã§ããã
:â» ç¥åŒã®èšæ³ã§ã <math>b= 1</math> ãš <math>b= -1</math> ããŸãšã㊠<math>b = \pm 1</math> ãšæžãããšãããã
äžè¬ã«æ£ã®æ°aã«ã€ããŠa=b^2ãšãªãbã¯äºã€ããããã®äºã€ã¯çµ¶å¯Ÿå€ãçããããã®äºã€ã®bãaã®å¹³æ¹æ ¹ãšãããaã®å¹³æ¹æ ¹ã®ãã¡ãæ£ã§ãããã®ã<math>\sqrt{a}</math>ãè² ã§ãããã®ã<math>-\sqrt{a}</math>ãšæžãã<math>\sqrt{a}</math>ã¯ãã«ãŒãaããšèªãã
äžæ¹ãè² ã®æ°aã«ã€ããŠèããŠã¿ãŠãäžæãbãèŠã€ããããšã¯ã§ããªããå®éã®ãšãããè² ã®æ°ã®å¹³æ¹æ ¹ã¯å®æ°ã§è¡šãããšã¯ã§ããªãã
{| style="border:2px solid green;width:80%" cellspacing=0
|style="background:lightgreen"|'''å¹³æ¹æ ¹'''
|-
|style="padding:5px"|
*'''æ£ã®æ°aã®å¹³æ¹æ ¹ã¯ <math> \sqrt{a}</math> ãš<math>- \sqrt{a}</math> ã§ããã'''
*'''è² ã®æ°aã®å¹³æ¹æ ¹ã¯'''å®æ°ã®ç¯å²ã§ã¯'''ååšããªãã'''
|}
:<math> \sqrt{a}</math> ãš<math>- \sqrt{a}</math> ããŸãšã㊠<math>\pm \sqrt{a}</math> ãšæžãããšãããã
* åé¡
<math>2\ ,\ 4\ ,\ 9\ ,\ 12</math>ã®å¹³æ¹æ ¹ãæ±ããã
*解ç
<math>\pm \sqrt 2\ ,\ \pm 2\ ,\ \pm 3\ ,\ \pm 2\sqrt 3</math>
*解説
ããããã®ã«ãŒããèšç®ãã<math>\pm</math>ãã€ããã°ããããã ããå¹³æ¹æ ¹ã®ã«ãŒã«ã«åŸã£ãŠãç°¡ååã§ãããã®ã¯ç°¡ååããããšãèŠæ±ãããã
äŸãã°ã<math>2</math>ã«å¯ŸããŠã¯ã<math>\pm\sqrt 2 </math>ãšãªãã
äžè¬ã«ã<math>\sqrt{A^2} = |A|</math>ã§ããã
====å¹³æ¹æ ¹ãå«ãåŒã®èšç®====
æ ¹å·ã«ã€ããŠã次ã®å
¬åŒãæãç«ã€ã
{| style="border:2px solid green;width:80%" cellspacing=0
|style="background:lightgreen"|'''å¹³æ¹æ ¹ã®å
¬åŒ'''
|-
|style="padding:5px"|
<math> a>0, b>0 </math> ã®ãšã
::<math>\sqrt{a} \sqrt{b}= \sqrt{ab}</math> ãããïŒ1ïŒ
:: ã
::<math>\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}</math> ãããïŒ2ïŒ
|}
;å
¬åŒïŒ1ïŒã®èšŒæ
ãŸãã <math> \sqrt{ab}</math> ãšã¯ãå®çŸ©ã«ããšã¥ããŠèãããšã2ä¹ãããš ab ã«ãªãæ°ã®ãã¡ãæ£ã®ã»ãã®æ°ãšããæå³ã§ããã
ãªã®ã§ãå
¬åŒã <math>\sqrt{a} \sqrt{b}= \sqrt{ab}</math> ã ãã蚌æããã«ã¯ããã®ããšã蚌æããã°ããã
ãªã®ã§ããŸãã<math>\sqrt{a} \sqrt{b} </math> ã2ä¹ãããšã
::<math> (\sqrt{a} \sqrt{b} )^2 = (\sqrt{a})^2 (\sqrt{b})^2 = ab </math>
ãšãªãã
ããã«<math>\sqrt{a} \sqrt{b}</math>ã¯ããŸãæ¡ä»¶ã2ä¹ãããšabã«ãªãããæºããã
ãããŠãæ£ã®æ°ã®å¹³æ¹æ ¹ã¯æ£ãªã®ã§ã<math>\sqrt{a} \sqrt{b} </math> ãæ£ã§ããããã£ãŠ <math>\sqrt{a} \sqrt{b} </math> ã¯ãã2ä¹ãããšabã«ãªããæ°ã®ãã¡ã®æ£ã®ã»ãã§ããã
ïŒèšŒæãããïŒ
ããã«ãäžã®å
¬åŒ(1)ã«ããã次ã®å
¬åŒãå°ãããã
{| style="border:2px solid green;width:80%" cellspacing=0
|style="background:lightgreen"|'''å
¬åŒ'''
|-
|style="padding:5px"|
<math> a>0, k>0 </math> ã®ãšã
::<math>\sqrt{k^2a} = k \sqrt{a}</math>
|}
* åé¡
èšç®ããã
# ã<math>\sqrt{8} \sqrt{14}</math>
# ã<math>2 \sqrt{18} + \sqrt{50}</math>
# ã<math>\left(\sqrt{3} - 2 \sqrt{6}\right)^2</math>
* 解ç
# ã<math>\sqrt{8} \sqrt{14} \ = \ \sqrt{8 \times 14} \ = \ \sqrt{2^4 \times 7} \ = \ 2^2 \sqrt{7} \ = \ 4 \sqrt{7}</math>
# ã<math>2 \sqrt{18} + \sqrt{50} \ = \ 2 \times 3 \sqrt{2} + 5 \sqrt{2} \ = \ (6+5) \sqrt{2} \ = \ 11 \sqrt{2}</math>
# ããŸããä¹æ³å
¬åŒ <math>(a-b)^2 = a^2-2ab+b^2</math>ãå©çšããŠå±éããã詳现ã¯ãä¹æ³å
¬åŒãã®ã»ã¯ã·ã§ã³ãåç
§ã®ããšã<br/><math>\begin{align}
\left(\sqrt{3} - 2 \sqrt{6}\right)^2 \ = \ \left(\sqrt{3}\right)^2 -2 \times \sqrt{3} \times 2 \sqrt{6} + \left(2 \sqrt{6}\right)^2 \ = \ 3-4 \sqrt{18} + 24 \ = \ 27-4 \times 3 \sqrt{2} \ = \ 27-12 \sqrt{2}
\end{align}</math>
åæ¯ã«æ ¹å·ãå«ãŸãªãåŒã«ããããšããåæ¯ã'''æçå'''ãããšãããæçåã¯ãåæ¯ãšååã«åãæ°ããããŠãããããšãå©çšããŠè¡ãã
ããšãã°ã<math>\frac{1}{\sqrt{2}}</math>ãæçåãããšã<math>\frac{1}{\sqrt{2}} \ = \ \frac{1 \sqrt{2}}{\sqrt{2}\sqrt{2}} \ = \ \frac{\sqrt{2}}{2}</math>ãšãªãã
ãŸãããšãã«<math>\frac{a}{b+c}</math>ã«ã€ããŠã<math>b^2-c^2=1</math>ã®ãšãã<br/>
<math>\frac{a}{b+c} \ = \ \frac{a(b-c)}{(b+c)(b-c)} \ = \ \frac{a(b-c)}{b^2-c^2} \ = \ \frac{a(b-c)}{1} \ = \ a(b-c)</math>ã§ããã
ããšãã°ã<math>a=1, b=\sqrt{2}, c=1</math>ãšãããšã<math>\frac{1}{\sqrt{2}+1}=\sqrt{2}-1</math>ã§ããã
* åé¡
åæ¯ãæçåããã
# ã<math>\frac{\sqrt{2}}{\sqrt{12}} </math> <br><br>
# ã<math>\frac{\sqrt{2} + 2 \sqrt{3}}{3 \sqrt{2} - \sqrt{3}} </math>
* 解ç
# ã<math>\frac{\sqrt{2}}{\sqrt{12}} \ = \ \frac{\sqrt{2}}{2 \sqrt{3}} \ = \ \frac{\sqrt{2} \sqrt{3}}{2 \sqrt{3} \sqrt{3}} \ = \ \frac{\sqrt{6}}{6}</math> <br><br>
# ã<math>\frac{\sqrt{2} + 2 \sqrt{3}}{3 \sqrt{2} - \sqrt{3}} \ = \ \frac{(\sqrt{2} + 2 \sqrt{3})(3 \sqrt{2} + \sqrt{3})}{(3 \sqrt{2} - \sqrt{3})(3 \sqrt{2} + \sqrt{3})} \ = \ \frac{6+ \sqrt{6} + 6 \sqrt{6} +6}{(3 \sqrt{2})^2 - (\sqrt{3})^2} \ = \ \frac{12 + 7 \sqrt{6}}{18-3} \ = \ \frac{12 + 7 \sqrt{6}}{15}</math>
====äºéæ ¹å·ïŒçºå±ïŒ====
[[w:äºéæ ¹å·|äºéæ ¹å·]]ãšã¯ãæ ¹å·ã2éã«ãªã£ãŠããåŒã®ããšã§ãããäºéæ ¹å·ã¯åžžã«å€ããããã§ã¯ãªããæ ¹å·ã®äžã«å«ãŸããåŒã«ãã£ãŠç°¡åã«ã§ãããã©ããã決ãŸããäžè¬ã«ãæ ¹å·å
ã®åŒãã<math>x^2</math>ã®åœ¢ã«å€åœ¢ã§ããå Žåã«ã¯ãå€åŽã®æ ¹å·ãå€ãããšãã§ããã
*åé¡
<math>\sqrt{3+2\sqrt 2}</math>ãç°¡åã«ããã
*解ç
<math>3+2\sqrt 2</math>ã<math>( \cdots )^2</math>ã®åœ¢ã«ã§ããããèããã
ä»®ã«ã<math>( \sqrt a + \sqrt b )^2</math>(a,bã¯æ£ã®æŽæ°)ã®åœ¢ã«ã§ãããšãããšã<math>3+2\sqrt 2 = a + b + 2\sqrt{ab}</math>ãšãªãã<br/>
:<math>\begin{cases}
a+b &= 3\\
ab &= 2\\
\end{cases}</math><br/>
ãæºããæŽæ°a,bãæ¢ãã°ããããã®é¢ä¿ã¯ãa=1,b=2(a,bãå
¥ãæããŠãå¯ã)ã«ãã£ãŠæºããããã®ã§ã<math>3+2\sqrt 2 \ = \ (\sqrt 2 + 1)^2</math>ãæãç«ã€ã
ãã£ãŠã<math>\sqrt{3+2\sqrt 2} \ = \ \sqrt{(\sqrt 2 + 1)^2} \ = \ \sqrt 2 + 1</math>ãšãªãã
{| style="border:2px solid green;width:80%" cellspacing=0
|style="background:lightgreen"|'''2éæ ¹å·'''
|-
|style="padding:5px"|
<math>a>0\ ,\ b>0</math> ã®ãšã
:<math>\sqrt{(a+b) +2 \sqrt {ab}}= \sqrt {a} + \sqrt {b}</math>
<math>a>b>0</math> ã®ãšã
:<math>\sqrt{(a+b) -2 \sqrt {ab}}= \sqrt {a} - \sqrt {b}</math>
|}
* åé¡
次ã®åŒãèšç®ããã
# <math>\sqrt{12-6 \sqrt {3}}</math>
# <math>\sqrt{3+ \sqrt {5}}</math>
* 解ç
# <math>\sqrt{12-6 \sqrt {3}} \ = \ \sqrt{12-2 \sqrt {27}} \ = \ \sqrt{(9+3) -2 \sqrt {9 \times 3}} \ = \ \sqrt {9} - \sqrt {3} \ = \ 3- \sqrt {3}</math>
#<math>\sqrt{3+ \sqrt {5}} \ = \ \sqrt{\frac{6+ 2 \sqrt {5}}{2}} \ = \ \frac{\sqrt{(5+1) +2 \sqrt {5 \times 1}}}{\sqrt{2}} \ = \ \frac{\sqrt {5} + \sqrt {1}}{\sqrt {2}} \ = \ \frac{\sqrt {10} + \sqrt {2}}{2}</math>
==äžæ¬¡äžçåŒ==
===äžæ¬¡äžçåŒ===
åã倧ããã®éã=ã§çµãã åŒãæ¹çšåŒãšåŒã¶ããšãæ¢ã«åŠç¿ãããããã§ã¯ãç°ãªã£ãéã®å€§ããã®éããè¡šãèšå·ãå°å
¥ãããã®æ§è³ªã«ã€ããŠãŸãšããã
ããæ°A,BããããšããAãBãã倧ããããšã<math>A > B</math>ãšè¡šããAãBããå°ããããšã<math>A < B</math>ãšè¡šããããã§ã<ãš>ã®ããšã[[w:äžçå·|äžçå·]]ãšåŒã³ããã®ãããªåŒãäžçåŒãšåŒã¶ããŸãã<math>\le,\ge</math>ã䌌ãæå³ã®äžçåŒã§ããããããããAãšBãçããå€ã§ããå Žåãå«ããã®ã§ããã
ãªããæ¥æ¬ã®æè²ã«ãããŠã¯ã<math>\le,\ge</math>ã®ä»£ããã«ãäžçå·ã®äžã«çå·ãèšãã<math>\leqq,\geqq</math>ã䜿ãããšãå€ãã
*äŸ
<math>x>7</math>ãšããäžçåŒããããšããxã¯7ãã倧ããå®æ°ã§ããããŸãã<math>x \ge 7</math>ã®æã«ã¯ãxã¯7以äžã®å®æ°ã§ããã
äžçåŒã§ã¯çåŒãšåãããã«ã䞡蟺ã«æŒç®ãããŠãäžçå·ã®é¢ä¿ãå€ãããªãããšããããäŸãã°ã䞡蟺ã«åãæ°ã足ããŠãã䞡蟺ã®å€§å°é¢ä¿ã¯å€åããªãããã ãã䞡蟺ã«è² ã®æ°ãããããšãã«ã¯ãäžçå·ã®åããå€åããããšã«æ³šæãå¿
èŠã§ãããããã¯ãè² ã®æ°ãããããšäž¡èŸºã®å€ã¯ã0ãäžå¿ã«æ°çŽç·ãæãè¿ããå°ç¹ã«ç§»ãããããšã«ããã
{| style="border:2px solid greenyellow;width:80%" cellspacing=0
|style="background:greenyellow"|'''äžçåŒã®æ§è³ª'''
|-
|style="padding:5px"|1. ãã<math> a<b </math> ãªãã°ã<math> a+c<b+c </math>ïŒ<math> a-c<b-c </math>
|-
|style="padding:5px"|2. ãã<math> a<b </math>ïŒ<math> c>0 </math> ãªãã°ã<math> ac<bc </math>ïŒ<math> \frac {a} {c} < \frac {b} {c}</math>
|-
|style="padding:5px"|3. ãã<math> a<b </math>ïŒ<math> c<0 </math> ãªãã°ã<math> ac>bc</math>ïŒ<math> \frac {a} {c} > \frac {b} {c}</math>
|}
* äŸ
<math>x > y</math>ãæãç«ã€ãšãã«ã¯ã<math>x+3>y+3</math>ã<math>4x > 4y</math>ãæãç«ã€ããŸãã<math> -x < -y</math>ãæãç«ã€ã
äžçåŒã®æ§è³ªã䜿ã£ãŠ
:<math> a {\color{red}+3}<b\; </math>
ã®äž¡èŸºãã3ãåŒããš
:<math> a+3-3<b-3\; </math>
ãã£ãŠ
:<math> a<b {\color{red}-3}\; </math>
ãšãªãã<br>
ãã®ããã«ã'''äžçåŒã§ã移é
ããããšãã§ãã'''ã
ã°ã©ããçšããŠèãããšããäžçåŒã¯ã°ã©ãäžã®é åãè¡šããé åã®å¢çã¯äžçå·ãçå·ã«çœ®ãæããéšåã察å¿ãããããã¯ãäžçå·ãæç«ãããã©ããããã®ç·äžã§å
¥ãæ¿ããããšã«ãã£ãŠãããïŒè©³ããã¯[[é«çåŠæ ¡æ°åŠI å³åœ¢ãšæ¹çšåŒ|æ°åŠII å³åœ¢ãšæ¹çšåŒ]]ã§åŠç¿ãããïŒ
* åé¡
<math>y>x+1</math>,<math>y < 2x+1</math>,<math>x <3</math>ã®ã°ã©ã(æ£ããã¯ãé åã)ãæãã
* 解ç
<math> y>x+1 </math> ã®ã°ã©ã(é å)ã¯æ¬¡ã®ããã«ãªãããã ããå¢çã¯å«ãŸãªãã
[[File:Linear Inequality Y GT Xplus1.png|thumb|none|360px|1次äžçåŒ y>x+1 ãè¡šãã°ã©ãã]]
<math> y<2x+1 </math>ã®ã°ã©ã(é å)ã¯æ¬¡ã®ããã«ãªãããã ããå¢çã¯å«ãŸãªãã
[[File:Linear Inequality Y LT 2Xplus1.png|thumb|none|360px|1次äžçåŒ y<2x+1 ãè¡šãã°ã©ãã]]
<math>x<3</math>ã®ã°ã©ã(é å)ã¯æ¬¡ã®ããã«ãªãããã ããå¢çã¯å«ãŸãªãã
[[File:Linear Inequality X LT 3.png|thumb|none|360px|1次äžçåŒ x<3 ãè¡šãã°ã©ãã]]
* åé¡
次ã®äžçåŒã解ãã
# ãã<math>3x-1 \le 9x-7</math>
# ãã<math>3(x-2)>2(5x-3)</math>
# ãã<math>x+1 < \frac {x-1} {3}</math>
* 解ç
# <br> <math>\begin{align} \quad
3x-1 & \le 9x-7\\
3x-9x & \le -7+1\\
-6x & \le -6\\
x & \ge 1
\end{align}
</math>
# <br> <math>\begin{align} \quad
3(x-2) & > 2(5x-3)\\
3x-6 & > 10x-6\\
3x-10x & > -6+6\\
-7x & > 0\\
x & < 0
\end{align}
</math>
# <br> <math>\begin{align} \quad
x+1 & < \frac {x-1} {3}\\
3x+3 & < x-1\\
3x-x & < -1-3\\
2x & < -4\\
x & < -2
\end{align}
</math>
===é£ç«äžçåŒ===
ããã€ãã®äžçåŒãçµã¿åããããã®ã'''é£ç«äžçåŒ'''ãšããããããã®äžçåŒãåæã«æºãã<math>x</math>ã®å€ã®ç¯å²ãæ±ããããšããé£ç«äžçåŒã'''解ã'''ãšããã
<br>
<br>
*åé¡äŸ
**åé¡
次ã®é£ç«äžçåŒã解ãã<br>
(i)
:<math>\left\{ \begin{matrix} x+2<2x+4 \\ 10-x \ge 3x-6 \end{matrix}\right.</math>
(ii)
:<math>\begin{cases}
x \ge 1-x\\
2(x+1)>x-2
\end{cases}</math>
**解ç
(i)<br>
<math>x+2<2x+4</math>ããã<math>-x<2</math><br>
:<math>x>-2</math>âŠâŠ(1)
<math>10-x \ge 3x-6</math>ããã<math>-4x \ge -16</math><br>
:<math>x \le 4</math>âŠâŠ(2)
(1),(2)ãåæã«æºãã<math>x</math>ã®å€ã®ç¯å²ã¯
:<math>-2<x \le 4</math>
(ii)<br>
<math>x \ge 1-x</math>ããã<math>2x \ge 1</math><br>
:<math>x \ge \frac {1} {2}</math>âŠâŠ(1)
<math>2(x+1)>x-2</math>ããã<math>2x+2>x-2</math><br>
:<math>x>-4</math>âŠâŠ(2)
(1),(2)ãåæã«æºãã<math>x</math>ã®å€ã®ç¯å²ã¯
:<math>x \ge \frac {1} {2}</math>
===絶察å€ãå«ãäžçåŒ===
絶察å€ãå«ãäžçåŒã«ã€ããŠèãããã<br>
絶察å€<math>|x|</math>ã¯ãæ°çŽç·äžã§ãåç¹<math>\mathrm{O}</math>ãšç¹<math>\mathrm{P} (x)</math>ã®éã®è·é¢ãè¡šããŠããã
<br>ãããã£ãŠã<math>a>0</math>ã®ãšã
:<math>|x|<a \Leftrightarrow -a<x<a</math>
:<math>|x|>a \Leftrightarrow x<-a\ ,\ a<x</math>
<br>
<br>
*åé¡äŸ
**åé¡
次ã®äžçåŒã解ãã<br>
(i)
:<math>|x|<5</math>
(ii)
:<math>|x| \ge 4</math>
(iii)
:<math>|x-2| \le 3</math>
(iv)
:<math>|x+3|>1</math>
**解ç
(i)
:<math>|x|<5</math>
:<math>-5<x<5</math>
(ii)
:<math>|x| \ge 4</math>
:<math>x \le -4\ ,\ 4 \le x</math>
(iii)
:<math>|x-2| \le 3</math>
:<math>-3 \le x-2 \le 3</math>
:<math>-1 \le x \le 5</math>
(iv)
:<math>|x+3|>1</math>
:<math>x+3<-1\ ,\ 1<x+3</math>
:<math>x<-4\ ,\ -2<x</math>
==äºæ¬¡æ¹çšåŒ==
===解ã®å
¬åŒ===
äžè¬ã®äºæ¬¡æ¹çšåŒ <math>ax^2 + bx + c = 0</math>ïŒ<math>a</math>, <math>b</math>, <math>c</math> ã¯å®æ°ã<math>a\ne0</math>ïŒã®è§£ <math>x</math> ãæ±ããå
¬åŒã«ã€ããŠèããã
:<math>ax^2 + bx + c = 0</math>
:<math>ax^2 + bx = -c</math>
:<math>x^2 + \frac{b}{a}x = -\frac{c}{a}</math> ⊠(1)
ããã§æçåŒ <math>x^2 + 2yx = (x + y)^2 - y^2</math> ãš (1) ã®å·ŠèŸºãä¿æ°æ¯èŒãããšã
:<math>\begin{cases}
2y &= \frac{b}{a} \\
y &= \frac{b}{2a}
\end{cases}</math>
ã§ããããã(1) ã®åŒã¯æ¬¡ã®ããã«å€åœ¢ã§ããïŒå¹³æ¹å®æïŒã
:<math>\left( x + \frac{b}{2a} \right)^2 - \left( \frac{b}{2a} \right)^2 = -\frac{c}{a}</math>
:<math>\left( x + \frac{b}{2a} \right)^2 = \left( \frac{b}{2a} \right)^2 - \frac{c}{a}</math>
:<math>\left( x + \frac{b}{2a} \right)^2 = \frac{b^2 - 4ac}{4a^2}</math>
<math>b^2 - 4ac \ge 0</math> ã®ãšã䞡蟺ã®å¹³æ¹æ ¹ããšããšã
:<math>\sqrt{ \left( x + \frac{b}{2a} \right)^2 } = \sqrt{ \frac{b^2 - 4ac}{4a^2} }</math>
:<math>\left| x + \frac{b}{2a} \right| = \frac{ \sqrt{b^2 - 4ac} }{2a}</math>
:<math>x + \frac{b}{2a} = \pm \frac{ \sqrt{b^2 - 4ac} }{2a}</math>
:<math>x = \frac{ -b \pm \sqrt{b^2 - 4ac} }{2a}</math>
ããã'''äºæ¬¡æ¹çšåŒã®è§£ã®å
¬åŒ'''ïŒã«ãã»ããŠãããã®ããã®ãããããquadratic formula; äºæ¬¡å
¬åŒïŒã§ããã解ã®å
¬åŒãäºæ¬¡æ¹çšåŒã®äžè¬åœ¢ã«ä»£å
¥ãããšãå³èŸºã¯0ã«ãªãã¯ãã§ããã
:<math>
x^2 = \frac 1 {4a^2} (b^2 \mp 2b\sqrt{b^2-4ac} + b^2 -4ac)
</math>
ã§ããããšãçšãããšã
:<math>
ax^2+bx+c= \frac 1 {4a} (b^2 \mp 2b\sqrt{b^2-4ac} + b^2 -4ac) + \frac b {2a}(-b \pm \sqrt{b^2-4ac}) + c
</math>
:<math>
= \frac 1 {4a} (2b^2 \mp 2b\sqrt{b^2-4ac}) + \frac 1 {2a}(-b^2 \pm b\sqrt{b^2-4ac}) = 0
</math>
ãšãªãã確ãã«æ£ããããšããããã
*åé¡
:(i)<math>
x^2-1=0
</math>
:(ii)<math>
5\,x^2+2\,x-1=0
</math>
:(iii)<math>
x^2+3\,x-1=0
</math>
:(iv)<math>
2\,x^2+3\,x-1=0
</math>
:(v)<math>
2\,x^2+3\,x+1=0
</math>
<!--
:(vi)<math>
7\,x^2+16\,x+4=0
</math>
:(vii)<math>
12\,x^2-29\,x-8=0
</math>
:(viii)<math>
12\,x^2-27\,x-8=0
</math>
-->
ããããã解ã®å
¬åŒãå æ°å解ãçšããŠè§£ããªããã
*解ç
çµæã®åŒã«æ ¹å·ãçŸããªãå Žåã«ã¯ãäœããã®ä»æ¹ã§å æ°å解ãã§ããããããããããã®æ¹æ³ã䜿ãã«ãããæ ¹å·ã¯ã§ããéãã®ä»æ¹ã§ç°¡ååããããšãéèŠã§ããã
(i)ã¯ç°¡åã«å æ°å解ã§ããã®ã§ã解ã®å
¬åŒãçšããå¿
èŠã¯ãªãã
:<math>
x^2-1 = (x+1)(x-1) = 0
</math>
ããã
:<math>
x = \pm 1
</math>
ãçããšãªãã(ii)ã§ã¯ãå æ°å解ãåºæ¥ãªãã®ã§ã解ã®å
¬åŒãçšãããå æ°å解ãã§ãããã©ããã¯å®éã«è©Šè¡é¯èª€ããŠèŠåãããããªãã
:<math>
5\,x^2+2\,x-1=0
</math>
ã«ã解ã®å
¬åŒãçšãããšãa=5, b= 2, c=-1ããã
:<math>
x = \frac 1 {2 \cdot 5} (-2 \pm \sqrt{2^2 - 4 \cdot 5 \cdot (-1)})
</math>
:<math>
= \frac 1 {10} (-2 \pm \sqrt {24} )
</math>
:<math>
= \frac 1 {10} (-2 \pm 2 \sqrt 6 ) = \frac 1 5 (-1 \pm \sqrt 6)
</math>
ãšãªãã(iii),(iv)ã§ããå æ°å解ã¯åºæ¥ãªãã®ã§ã解ã®å
¬åŒãçšãããçãã¯ã
(iii)
:<math>
x = \frac 1 2 (-3 \pm \sqrt{13} )
</math>
(iv)
:<math>
x = \frac 1 4 (-3 \pm \sqrt{17} )
</math>
(v)
:<math>
(2x+1)(x+1) = 2\,x^2+3\,x+1
</math>
ãšå æ°å解ã§ããã®ã§ãçãã¯
:<math>
x=-{{1}\over{2}},x=-1
</math>
ãšãªãã
å
šåãéããŠãå æ°å解ãå¯èœãªæ¹çšåŒã«å¯ŸããŠãã解ã®å
¬åŒã䜿çšããŠãæ§ããªãã
<!--
ããããã®è§£çã¯ã
:(i)<math>
\left[ x=-1,x=1 \right]
</math>
:(ii)<math>
\left[ x=-{{\sqrt{6}+1}\over{5}},x={{\sqrt{6}-1}\over{5}} \right]
</math>
:(iii)<math>
\left[ x=-{{\sqrt{13}+3}\over{2}},x={{\sqrt{13}-3}\over{2}} \right]
</math>
:(iv)<math>
\left[ x=-{{\sqrt{17}+3}\over{4}},x={{\sqrt{17}-3}\over{4}} \right]
</math>
:(v)<math>
\left[ x=-{{1}\over{2}},x=-1 \right]
</math>
:(vi)<math>
\left[ x=-2,x=-{{2}\over{7}} \right]
</math>
:(vii)<math>
\left[ x=-{{1}\over{4}},x={{8}\over{3}} \right]
</math>
:(viii)<math>
\left[ x=-{{\sqrt{1113}-27}\over{24}},x={{\sqrt{1113}+27}\over{24}} \right]
</math>
ãšãªãã
æåŸã®çµæã§
:<math>
1113
</math>
ã¯ã
:<math>
1113 = 3 \times 7 \times 53
</math>
ãšçŽ å æ°å解ãããããããã以äžç°¡åã«ãªããªãã
-->
===<math>ax^2 + 2b'x + c = 0</math> ã®è§£ã®å
¬åŒ===
äºæ¬¡æ¹çšåŒ<math>ax^2 + 2b'x + c = 0(a\ne0)</math>ã«ã€ããŠèããã
解ã®å
¬åŒã« b= 2b' ã代å
¥ãããš
:<math>
x = \frac{-2b' \pm \sqrt{(2b')^2-4ac}}{2a} = \frac{-2b' \pm \sqrt{4(b'^2-ac)}}{2a} = \frac{-2b' \pm 2\sqrt{b'^2-ac}}{2a}
</math>
ãã£ãŠãäºæ¬¡æ¹çšåŒ <math>ax^2 + 2b'x + c = 0</math> ã®è§£ã¯
:<math>
x = \frac{-b' \pm \sqrt{b'^2-ac}}{a}
</math>
ãšãªãã
*åé¡äŸ
**åé¡
:<math>
3\,x^2+6\,x-2=0
</math>
ãäžã®è§£ã®å
¬åŒãçšããŠè§£ããªããã
**解ç
äžã®è§£ã®å
¬åŒãçšãããšãa=3, b'= 3, c=-2ããã
:<math>
x = \frac {-3 \pm \sqrt{3^2 - 3 \cdot (-2)}} {3}
</math>
:<math>
= \frac {-3 \pm \sqrt {15}} {3}
</math>
ãšãªãã
===2次æ¹çšåŒã®è§£ã®åæ°===
2次æ¹çšåŒ <math>ax^2 + bx + c = 0</math> ã®è§£ã¯ <math>x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} </math> ã§ããã<br>
ãã®åŒã®æ ¹å·ã®äžèº«ã ãåãåºãããã®ãå€å¥åŒãšåŒã³ã2次æ¹çšåŒã®è§£ã®åæ°ãç°¡åã«å€å¥ã§ããã
<math>D=b^2-4ac</math>ã®å€ã«ãã£ãŠæ¬¡ã®ããã«ãªãã<br>
(1)ã<math>D>0 </math>ã®ãšããç°ãªã2ã€ã®è§£ã<math>x = \frac{-b + \sqrt{b^2-4ac}}{2a} </math>ãš<math>x = \frac{-b - \sqrt{b^2-4ac}}{2a} </math>ãæã€ã<br>
(2)ã<math>D=0 </math>ã®ãšãã<math> \pm \sqrt{b^2-4ac} = \pm 0 </math> ã§ããã®ã§ã2ã€ã®è§£ã¯äžèŽããŠããã 1ã€ã®è§£<math>x = - \frac{b}{2a} </math>ãæã€ãããã¯2ã€ã®è§£ãéãªã£ããã®ãšèããŠã'''é解'''ãšããã<br>
(3)ã<math>D<0 </math>ã®ãšããå®æ°ã®ç¯å²ã§ã¯è§£ã¯ãªãã<br>
2次æ¹çšåŒ <math>ax^2 + bx + c = 0</math> ã®è§£ã®åæ°ã¯<math>D=b^2-4ac</math>ã®å€ã§å€å®ã§ããã
{| style="border:2px solid red;width:80%" cellspacing=0
|style="background:lightred"|'''2次æ¹çšåŒã®è§£ã®åæ°'''
|-
|style="padding:5px"|
2次æ¹çšåŒ <math>ax^2 + bx + c = 0</math> ã®è§£ã¯ã<math>D=b^2-4ac</math>ãšãããšã
::<math>D>0 \Longleftrightarrow </math> ç°ãªã2ã€ã®å®æ°ã®è§£ããã€
::<math>D=0 \Longleftrightarrow </math> é解ããã€
::<math>D<0 \Longleftrightarrow </math> å®æ°è§£ã¯ãªã
|}
* åé¡
次ã®2次æ¹çšåŒã®è§£ã®åæ°ãæ±ããã
:(I) ã<math> 3\,x^2-4\,x+2=0 </math>
:(II) ã<math> 25\,x^2+20\,x+4=0 </math>
:(III)ã<math> x^2+7\,x+1=0 </math>
* 解ç
(I)
:<math> D=(-4)^2-4 \times 3 \times 2 =-8<0 </math>
ã ãããå®æ°è§£ã¯ãªãã<br>
(II)
:<math> D=20^2-4 \times 25 \times 4 =0 </math>
ã ãããé解ããã€ã<br>
(III)
:<math> D=7^2-4 \times 1 \times 1 =45>0 </math>
ã ãããç°ãªã2ã€ã®å®æ°ã®è§£ããã€ã
== æŒç¿åé¡ ==
{{DEFAULTSORT:ãããšãã}}
[[Category:é«çåŠæ ¡æ°åŠI]] | 2005-05-28T11:17:39Z | 2024-03-04T17:54:39Z | [
"ãã³ãã¬ãŒã:蚌æ",
"ãã³ãã¬ãŒã:蚌æçµãã",
"ãã³ãã¬ãŒã:ã³ã©ã "
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E6%95%B0%E5%AD%A6I/%E6%95%B0%E3%81%A8%E5%BC%8F |
2,072 | äžåŠæ ¡åœèª | äžåŠæ ¡ã®åœèªã¯äž»ã«çŸä»£æãšå€æ(æ¥æ¬å€æã»æŒ¢æ)ãšææ³ã®3ã€ã«åãããŸããäžè¬ã®æç§æžãšã¯å
容ãè¥å¹²éãéšåããããŸãããå匷ã®åèã«ãªãã°å¹žãã§ãã
çŸä»£æ - è¡šçŸ
çŸä»£æ(èäœæš©ã®éœåäžãææ²»æ代~倧æ£æ代ã®äœåã®ã¿ãšãªããŸã)
å€æ
æ
äºæèª
ææ³
çŸä»£æ - è¡šçŸ
çŸä»£æ
ææ³
挢æ 挢æ (2014-10-17)
å€æ
çŸä»£ã®æ¬èª
çŸä»£æ
ãã®ä»
ææ³
挢æ
å€æ
çŸä»£æ :è¿ä»£æåŠãªã©
çŸä»£æ :èªè§£
ææ³
å€å
žåžžè
ãªã³ã¯ | [
{
"paragraph_id": 0,
"tag": "p",
"text": "äžåŠæ ¡ã®åœèªã¯äž»ã«çŸä»£æãšå€æ(æ¥æ¬å€æã»æŒ¢æ)ãšææ³ã®3ã€ã«åãããŸããäžè¬ã®æç§æžãšã¯å
容ãè¥å¹²éãéšåããããŸãããå匷ã®åèã«ãªãã°å¹žãã§ãã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "çŸä»£æ - è¡šçŸ",
"title": "åå
"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "çŸä»£æ(èäœæš©ã®éœåäžãææ²»æ代~倧æ£æ代ã®äœåã®ã¿ãšãªããŸã)",
"title": "åå
"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "å€æ",
"title": "åå
"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "æ
äºæèª",
"title": "åå
"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ææ³",
"title": "åå
"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "çŸä»£æ - è¡šçŸ",
"title": "åå
"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "çŸä»£æ",
"title": "åå
"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "ææ³",
"title": "åå
"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "挢æ 挢æ (2014-10-17)",
"title": "åå
"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "å€æ",
"title": "åå
"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "çŸä»£ã®æ¬èª",
"title": "åå
"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "çŸä»£æ",
"title": "åå
"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ãã®ä»",
"title": "åå
"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "ææ³",
"title": "åå
"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "挢æ",
"title": "åå
"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "å€æ",
"title": "åå
"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "çŸä»£æ :è¿ä»£æåŠãªã©",
"title": "åå
"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "çŸä»£æ :èªè§£",
"title": "åå
"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "ææ³",
"title": "åå
"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "å€å
žåžžè",
"title": "åå
"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "ãªã³ã¯",
"title": "ãã®ä»"
}
] | äžåŠæ ¡ã®åœèªã¯äž»ã«çŸä»£æãšå€æ(æ¥æ¬å€æã»æŒ¢æ)ãšææ³ã®3ã€ã«åãããŸããäžè¬ã®æç§æžãšã¯å
容ãè¥å¹²éãéšåããããŸãããå匷ã®åèã«ãªãã°å¹žãã§ãã | {{Pathnav|ã¡ã€ã³ããŒãž|å°åŠæ ¡ã»äžåŠæ ¡ã»é«çåŠæ ¡ã®åŠç¿|äžåŠæ ¡ã®åŠç¿|äžåŠæ ¡åœèª|frame=1|hide=1}}{{é²æç¶æ³}}äžåŠæ ¡ã®åœèªã¯äž»ã«çŸä»£æãšå€æ(æ¥æ¬å€æã»æŒ¢æ)ãšææ³ã®3ã€ã«åãããŸããäžè¬ã®æç§æžãšã¯å
容ãè¥å¹²éãéšåããããŸãããå匷ã®åèã«ãªãã°å¹žãã§ãã
== åå
==
=== 1幎 ===
* [[äžåŠæ ¡åœèª/çŸä»£æ|çŸä»£æ]]
çŸä»£æ - è¡šçŸ
:[[äžåŠæ ¡åœèª/çŸä»£æ/äœæ]]
:[[äžåŠæ ¡åœèª/çŸä»£æ/ææ³æ]]
:[[äžåŠæ ¡åœèª/çŸä»£æ/説æã®ããã]]
çŸä»£æ(èäœæš©ã®éœåäžãææ²»æ代ïœå€§æ£æ代ã®äœåã®ã¿ãšãªããŸã)
:[[äžåŠæ ¡åœèª/çŸä»£æ/åã£ã¡ãã]]ã (åäœ:{{ruby|å€ç®æŒ±ç³|ãªã€ããããã}}) {{é²æ|25%|2022-12-25}}
:[[äžåŠæ ¡åœèª/çŸä»£æ/èèã®ç³ž]]ã (åäœ:{{ruby|è¥å·éŸä¹ä»|ããããããã
ãã®ãã}}) {{é²æ|100%|2022-1-5}}
* [[äžåŠæ ¡åœèª å€æ|å€æ]] {{é²æ|25%|2014-10-17}}
å€æ
:[[äžåŠæ ¡åœèª å€æ/竹åç©èª]] {{é²æ|50%|2014-10-17}}
æ
äºæèª
:[[äžåŠæ ¡åœèª/æ
äºæèª 1幎|æ
äºæèª 1幎]] {{é²æ|25%|2014-10-17}}
:[[äžåŠæ ¡åœèª 挢æ/ççŸ|{{ruby|ççŸ|ããã
ã}}]] {{é²æ|25%|2014-10-17}}
:[[äžåŠæ ¡åœèª 挢æ/éã¯èããåºã§ãŠèãããéã|éã¯{{ruby|è|ãã}}ããåºã§ãŠèãããéã]] {{é²æ|25%|2014-10-17}}
:[[äžåŠæ ¡åœèª 挢æ/æž©æ
ç¥æ°|æž©æ
ç¥æ°]] {{é²æ|25%|2014-10-17}}
:[[äžåŠæ ¡åœèª 挢æ/äºåæ©çŸæ©|äºåæ©çŸæ©]] {{é²æ|25%|2014-10-17}}
ææ³
:[[äžåŠæ ¡åœèª ææ³|ææ³]]
* è³æç·š(å€æ)
:[[äžåŠæ ¡åœèª å€æ/äŒæŸä¿ç©èª|{{ruby|äŒæŸä¿|ããã»}}ç©èª]]
:[[äžåŠæ ¡åœèª å€æ/åäœæ¥èš|{{ruby|åäœ|ãšã}}æ¥èš]] â»ããéèŠïŒ
:[[äžåŠæ ¡åœèª å€æ/äŒå¢ç©èª|{{ruby|äŒå¢|ãã}}ç©èª]]
:[[äžåŠæ ¡åœèª å€æ/åèšæ|{{ruby|åèšæ|ãã£ããããã}}]] â»ããéèŠïŒã{{ruby|å°åŒéšå
äŸ|ãããã¶ã®ãªãã}}
=== 2幎 ===
çŸä»£æ - è¡šçŸ
:[[äžåŠæ ¡åœèª/çŸä»£æ/è°è«ã®ããã®æèŠãææ¡ã®ããã]]
:[[äžåŠæ ¡åœèª/çŸä»£æ/å ±åæžã®æžãæ¹]]
:[[äžåŠæ ¡åœèª/çŸä»£æ/æçŽã®æžãæ¹]]
çŸä»£æ
:[[äžåŠæ ¡åœèª/çŸä»£æ/èµ°ãã¡ãã¹]]ã (åäœ:{{ruby|倪宰治|ã ããããã}})ã{{é²æ|100%|2014-10-17}}
ææ³
:[[äžåŠæ ¡åœèª ææ³|ææ³]] {{é²æ|50%|2014-10-17}}
挢æ<br />
[[äžåŠæ ¡åœèª 挢æ|挢æ]] {{é²æ|25%|2014-10-17}}
:[[äžåŠæ ¡åœèª 挢æ/æ¥æ|æ¥æ]] ({{ruby|æç«|ãšã»}}) {{é²æ|25%|2014-10-17}}
:[[äžåŠæ ¡åœèª 挢æ/é»é¶Žæ¥Œã«ãŠå浩ç¶ã® åºéµã«ä¹ããéã|{{ruby|é»é¶Žæ¥Œ|ãããããã}}ã«ãŠ{{ruby|å浩ç¶|ãããããã}}ã® {{ruby|åºéµ|ããããã}}ã«{{ruby|ä¹|ã}}ããéã]] ({{ruby|æçœ|ãã¯ã}}) {{é²æ|25%|2014-10-17}}
:[[äžåŠæ ¡åœèª 挢æ/絶å¥|絶å¥]] (æç«) {{é²æ|25%|2014-10-17}}
:[[äžåŠæ ¡åœèª 挢æ/æ¥æ|{{ruby|æ¥æ|ãã
ãããã}}]] ({{ruby|å浩ç¶|ãããããã}}) {{é²æ|25%|2014-10-17}}
:[[äžåŠæ ¡åœèª 挢æ/è«èª|è«èª]] ({{ruby|åå|ããã}})
å€æ
:[[äžåŠæ ¡åœèª å€æ/平家ç©èª|{{ruby|平家|ãžãã}}ç©èª]] {{é²æ|50%|2014-10-17}}
:[[äžåŠæ ¡åœèª å€æ/åŸç¶è|{{ruby|åŸç¶è|ã€ãã¥ããã}}]] {{é²æ|25%|2014-10-17}}
:[[äžåŠæ ¡åœèª å€æ/æèå|{{ruby|æèå|ãŸããã®ããã}}]] {{é²æ|25%|2014-10-17}}
çŸä»£ã®æ¬èª
:[[äžåŠæ ¡åœèª/æ¬èª]]
çŸä»£æ
:[[äžåŠæ ¡åœèª/çŸä»£æ/èµ°ãã¡ãã¹]] (åäœïŒ{{Ruby|[[d:Q317685|倪宰治]]|ã ãã ããã}}) {{é²æ|25%|2014-10-17}}
ãã®ä»
:[[äžåŠæ ¡åœèª/çµµã³ã³ãã®èªã¿æ¹|çµµã³ã³ãã®èªã¿æ¹]]
ææ³
:[[äžåŠæ ¡åœèª ææ³|ææ³]]
----
=== 3幎 ===
挢æ
:[[äžåŠæ ¡åœèª 挢æ/å
äºã®å®è¥¿ã«äœ¿ã²ãããéã|å
äºã®å®è¥¿ã«äœ¿ã²ãããéã]] ({{ruby|çç¶|ããã}}) {{é²æ|00%|2014-10-17}}
:[[äžåŠæ ¡åœèª 挢æ/éå€æ|éå€æ]] ({{ruby|æçœ|ãã¯ã}}) {{é²æ|25%|2014-10-17}}
å€æ
:[[äžåŠæ ¡åœèª å€æ/äžèéã»å€ä»åæéã»æ°å€ä»åæé]]
:[[äžåŠæ ¡åœèª å€æ/ããã®ã»ãé]] {{é²æ|50%|2014-10-17}}
çŸä»£æ :è¿ä»£æåŠãªã©
:[[äžåŠæ ¡åœèª/çŸä»£æ/é¯è¿
|äžåŠæ ¡åœèª/çŸä»£æ/æ
é·]] (åäœïŒ{{Ruby|[[d:Q23114|é¯è¿
]]|ããã}}ã»èš³ïŒ{{ruby|äºäžçŽ
æ¢
|ãã®ããããã°ã}}) (竹å
奜蚳ã¯èäœæš©ä¿è·æéäž) {{é²æ|25%|2014-10-17}}
:[[äžåŠæ ¡åœèª/çŸä»£æ/é«ç¬è]] (åäœïŒ{{Ruby|[[d:Q356960|森éŽå€]]|ãããããã}}) {{é²æ|25%|2014-10-17}}
:[[äžåŠæ ¡åœèª/çŸä»£æ/åæ]] (åäœïŒ{{ruby|島åŽè€æ|ããŸãããšããã}}) {{é²æ|25%|2014-10-17}}
:[[äžåŠæ ¡åœèª/çŸä»£æ/æåŸã®äžå¥]] (åäœïŒæ£®éŽå€)
çŸä»£æ :èªè§£
:[[äžåŠæ ¡åœèª/çŸä»£æ/説ææã»è©è«æ]]
ææ³
:[[äžåŠæ ¡åœèª ææ³|ææ³]]
å€å
žåžžè
:[[äžåŠæ ¡åœèª/å€å
žåžžè]]
== ãã®ä» ==
* [[äžåŠæ ¡åœèª ææ³|ææ³]]
* {{ruby|èªåœ|ãã}}
* æ
äºæèª
* [[äžåŠæ ¡åœèª/è¡šçŸææ³|è¡šçŸææ³]]
* èªåœ
::[[äžåŠæ ¡åœèª/æ
äºæèª 1幎|æ
äºæèª 1幎]]
::[[äžåŠæ ¡åœèª/æ
äºæèª 2幎|æ
äºæèª 2幎]]
::[[äžåŠæ ¡åœèª/æ
äºæèª 3幎|æ
äºæèª 3幎]]
:[[äžåŠæ ¡åœèª/æ
£çšå¥|æ
£çšå¥]]
:[[äžåŠæ ¡åœèª/é¡çŸ©èª|é¡çŸ©èª]]
:[[äžåŠæ ¡åœèª/察矩èª|察矩èª]]
:[[äžåŠæ ¡åœèª/çèª|çèª]]
:[[äžåŠæ ¡åœèª/ããšãã|ããšãã]]
* 挢å
:[[äžåŠæ ¡åœèª/挢å 1幎|挢å 1幎]]
:[[äžåŠæ ¡åœèª/挢å 2幎|挢å 2幎]]
:[[äžåŠæ ¡åœèª/挢å 3幎|挢å 3幎]]
:[[äžåŠæ ¡åœèª/äžåŠæ ¡ã§åŠç¿ãã挢å|äžåŠæ ¡ã§åŠç¿ãã挢å]]
----
* åŠç¿æ¹æ³
:[[åŠç¿æ¹æ³/é«æ ¡åéš/åœèª]]
:[[åŠç¿æ¹æ³/äžåŠæ ¡åœèª]]
----
* [[äžçæè²åæã®åœèª]] {{é²æ|00%|2018-12-08}}
----
* é«æ ¡åéš
ãªã³ã¯
: (2014幎9ææç¹ã§ã®èšäºå
容ã¯ããã«åºé¡åŸå)
:[[é«æ ¡åéšçŸä»£æ]] {{é²æ|00%|2013-12-05}}
:[[é«æ ¡åéšå€æ]] {{é²æ|00%|2010-03-28}}
:[[é«æ ¡åéšæŒ¢æ]] {{é²æ|00%|2014-01-05}}
:[[å
¥è©Šå¯Ÿçåé¡/äžåŠæ ¡åœèª]]
[[Category:äžåŠæ ¡åœèª|*]]
[[Category:äžåŠæ ¡æè²|ããã]]
[[Category:æ¥æ¬ã®åœèªç§æè²|*]]
[[Category:æ¥æ¬èª|ã¡ãããã€ããããã]] | 2005-05-29T11:22:17Z | 2023-07-12T04:49:58Z | [
"ãã³ãã¬ãŒã:Pathnav",
"ãã³ãã¬ãŒã:é²æç¶æ³",
"ãã³ãã¬ãŒã:Ruby",
"ãã³ãã¬ãŒã:é²æ"
] | https://ja.wikibooks.org/wiki/%E4%B8%AD%E5%AD%A6%E6%A0%A1%E5%9B%BD%E8%AA%9E |
2,074 | 解æååŠ | æ¬é
ã¯ç©çåŠ è§£æååŠ ã®è§£èª¬ã§ãã
| [
{
"paragraph_id": 0,
"tag": "p",
"text": "æ¬é
ã¯ç©çåŠ è§£æååŠ ã®è§£èª¬ã§ãã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "",
"title": ""
}
] | æ¬é
ã¯ç©çåŠ è§£æååŠ ã®è§£èª¬ã§ãã ã¯ããã«
éåæ¹çšåŒã®äžè¬å
ã©ã°ã©ã³ãžã¢ã³
æå°äœçšã®åç
éåéãããã«ããã¢ã³ã®å®çŸ©
ä¿ååã®å°åº
ãšãã«ã®ãŒä¿ååã®å°åº
éåéä¿ååã®å°åº | {{Pathnav|ã¡ã€ã³ããŒãž|èªç¶ç§åŠ|ç©çåŠ|frame=1|small=1}}
æ¬é
ã¯ç©çåŠ è§£æååŠ ã®è§£èª¬ã§ãã
* [[解æååŠ ã¯ããã«|ã¯ããã«]]
* [[解æååŠ éåæ¹çšåŒã®äžè¬å|éåæ¹çšåŒã®äžè¬å]]
** [[解æååŠ éåæ¹çšåŒã®äžè¬å#ã©ã°ã©ã³ãžã¢ã³|ã©ã°ã©ã³ãžã¢ã³]]
** [[解æååŠ éåæ¹çšåŒã®äžè¬å#æå°äœçšã®åç|æå°äœçšã®åç]]
** [[解æååŠ éåæ¹çšåŒã®äžè¬å#éåéãããã«ããã¢ã³ã®å®çŸ©|éåéãããã«ããã¢ã³ã®å®çŸ©]]
* [[解æååŠ ä¿ååã®å°åº|ä¿ååã®å°åº]]
** [[解æååŠ ä¿ååã®å°åº#ãšãã«ã®ãŒä¿ååã®å°åº|ãšãã«ã®ãŒä¿ååã®å°åº]]
** [[解æååŠ ä¿ååã®å°åº#éåéä¿ååã®å°åº|éåéä¿ååã®å°åº]]
{{stub}}
{{DEFAULTSORT:ãããããããã}}
[[Category:解æååŠ|*]]
{{NDC|423|ãããããããã}} | 2005-05-30T03:53:31Z | 2023-09-28T17:12:06Z | [
"ãã³ãã¬ãŒã:Pathnav",
"ãã³ãã¬ãŒã:Stub",
"ãã³ãã¬ãŒã:NDC"
] | https://ja.wikibooks.org/wiki/%E8%A7%A3%E6%9E%90%E5%8A%9B%E5%AD%A6 |
2,075 | 解æååŠ ã¯ããã« | 解æååŠã¯ãã¥ãŒãã³ååŠã®å
容ãããæ±çšçã«äœ¿ãã圢ã«å®åŒåãçŽãããã®ã§ããã äŸãã°ãã«ã«ã座æšã§ã®ãã¥ãŒãã³ã®éåæ¹çšåŒã極座æšç³»ãžæžãçŽããšãšãŠãç
©éã«ãªãã ãã®å°é£ããããããã解æååŠã§ã¯ä»»æã®åº§æšç³»ã§ãããäžè¬çãªåœ¢ã§ã®ãã¥ãŒãã³ã®éåæ¹çšåŒãåŸãããšãã§ããã
ç¹ã«ç©çç³»ã®å¯Ÿç§°æ§ãèŠãå Žåã«ãããçšããããããšãå€ãã ãŸããå€ãã®ç©äœãé¢ããåé¡ã«å¯ŸããŠã䜿ãããããšãããã
ãŸãããã®éšåã§å®çŸ©ãããçšèªã¯éåååŠããé»ç£æ°åŠãªã©ä»ã®åéã§ãå€ã䜿ããããããã«ãªãã¥ã©ã ã®äžã§ã¯ãéèŠãªäœçœ®ãå ããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "解æååŠã¯ãã¥ãŒãã³ååŠã®å
容ãããæ±çšçã«äœ¿ãã圢ã«å®åŒåãçŽãããã®ã§ããã äŸãã°ãã«ã«ã座æšã§ã®ãã¥ãŒãã³ã®éåæ¹çšåŒã極座æšç³»ãžæžãçŽããšãšãŠãç
©éã«ãªãã ãã®å°é£ããããããã解æååŠã§ã¯ä»»æã®åº§æšç³»ã§ãããäžè¬çãªåœ¢ã§ã®ãã¥ãŒãã³ã®éåæ¹çšåŒãåŸãããšãã§ããã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ç¹ã«ç©çç³»ã®å¯Ÿç§°æ§ãèŠãå Žåã«ãããçšããããããšãå€ãã ãŸããå€ãã®ç©äœãé¢ããåé¡ã«å¯ŸããŠã䜿ãããããšãããã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ãŸãããã®éšåã§å®çŸ©ãããçšèªã¯éåååŠããé»ç£æ°åŠãªã©ä»ã®åéã§ãå€ã䜿ããããããã«ãªãã¥ã©ã ã®äžã§ã¯ãéèŠãªäœçœ®ãå ããã",
"title": "ã¯ããã«"
}
] | null | {{pathnav|ã¡ã€ã³ããŒãž|èªç¶ç§åŠ|ç©çåŠ|解æååŠ|frame=1}}
==ã¯ããã«==
解æååŠã¯ãã¥ãŒãã³ååŠã®å
容ãããæ±çšçã«äœ¿ãã圢ã«å®åŒåãçŽãããã®ã§ããã
äŸãã°ãã«ã«ã座æšã§ã®ãã¥ãŒãã³ã®éåæ¹çšåŒã極座æšç³»ãžæžãçŽããšãšãŠãç
©éã«ãªãã
ãã®å°é£ããããããã解æååŠã§ã¯ä»»æã®åº§æšç³»ã§ãããäžè¬çãªåœ¢ã§ã®ãã¥ãŒãã³ã®éåæ¹çšåŒãåŸãããšãã§ããã
ç¹ã«ç©çç³»ã®å¯Ÿç§°æ§ãèŠãå Žåã«ãããçšããããããšãå€ãã
ãŸããå€ãã®ç©äœãé¢ããåé¡ã«å¯ŸããŠã䜿ãããããšãããã
ãŸãããã®éšåã§å®çŸ©ãããçšèªã¯éåååŠããé»ç£æ°åŠãªã©ä»ã®åéã§ãå€ã䜿ããããããã«ãªãã¥ã©ã ã®äžã§ã¯ãéèŠãªäœçœ®ãå ããã
{{DEFAULTSORT:ãããããããã ã¯ããã«}}
[[Category:解æååŠ|* ã¯ããã«]] | null | 2015-04-17T15:27:40Z | [
"ãã³ãã¬ãŒã:Pathnav"
] | https://ja.wikibooks.org/wiki/%E8%A7%A3%E6%9E%90%E5%8A%9B%E5%AD%A6_%E3%81%AF%E3%81%98%E3%82%81%E3%81%AB |
2,076 | 解æååŠ éåæ¹çšåŒã®äžè¬å | ããé¢æ° L ( q 1 , q 2 , ⯠, q K , q Ì 1 , q Ì 2 , ⯠, q Ì K ) {\displaystyle L(q_{1},q_{2},\cdots ,q_{K},{\dot {q}}_{1},{\dot {q}}_{2},\cdots ,{\dot {q}}_{K})} ããããšãã«ã
S = â« t 0 t 1 d t L ( q 1 ( t ) , q 2 ( t ) , ⯠, q K ( t ) , q Ì 1 ( t ) , q Ì 2 ( t ) , ⯠, q Ì K ( t ) ) {\displaystyle S=\int _{t_{0}}^{t_{1}}dt\,L(q_{1}(t),q_{2}(t),\cdots ,q_{K}(t),{\dot {q}}_{1}(t),{\dot {q}}_{2}(t),\cdots ,{\dot {q}}_{K}(t))}
ãæå°ã«ãã q i ( t ) {\displaystyle q_{i}(t)} ã¯ã©ã®ãããªãã®ã ãããã
ãŸãã¯ç°¡åãªäŸãšããŠãé¢æ° f ( x ) {\displaystyle f(x)} ãæå°ã«ãã x {\displaystyle x} ã«ã€ããŠèãããã f ( x ) {\displaystyle f(x)} ãæå°å€ãåããšãã f â² ( x ) = 0 {\displaystyle f'(x)=0} ãšãªãã®ã ã£ãã f â² ( x ) = 0 {\displaystyle f'(x)=0} ãšãªãããšã¯ã x {\displaystyle x} ã埮å°é ÎŽ x {\displaystyle \delta x} ã ãå€åããããšãã f ( x ) {\displaystyle f(x)} ã®å€åé ÎŽ f := f ( x + ÎŽ x ) â f ( x ) {\displaystyle \delta f:=f(x+\delta x)-f(x)} 㯠Ύ f = 0 {\displaystyle \delta f=0} ã«ãªããšããããšã§ããã
ããããã®é¡æšã§ã S ( { q i } , { q Ì i } ) {\displaystyle S(\{q_{i}\},\{{\dot {q}}_{i}\})} ãæå°ã«ãã { q i ( t ) } {\displaystyle \{q_{i}(t)\}} ã«ã€ããŠã { q i ( t ) } {\displaystyle \{q_{i}(t)\}} ãå°ãã ãå€åãã㊠{ q i ( t ) + ÎŽ q i ( t ) } {\displaystyle \{q_{i}(t)+\delta q_{i}(t)\}} (ãã ããå¢çæ¡ä»¶ ÎŽ q i ( t 0 ) = ÎŽ q i ( t 1 ) = 0 {\displaystyle \delta q_{i}(t_{0})=\delta q_{i}(t_{1})=0} ã課ã)ãšãããšãã® S {\displaystyle S} ã®å€åé ÎŽ S = S ( { q i ( t ) + ÎŽ q i ( t ) } , { q Ì i ( t ) + ÎŽ q Ì i ( t ) } ) â S ( { q i } , { q Ì i } ) {\displaystyle \delta S=S(\{q_{i}(t)+\delta q_{i}(t)\},\{{\dot {q}}_{i}(t)+\delta {\dot {q}}_{i}(t)\})-S(\{q_{i}\},\{{\dot {q}}_{i}\})} 㯠Ύ S = 0 {\displaystyle \delta S=0} ãšãªããšèããããšãåºæ¥ãã
ÎŽ S = S ( { q i ( t ) + ÎŽ q i ( t ) } , { q Ì i ( t ) + ÎŽ q Ì i ( t ) } ) â S ( { q i } , { q Ì i } ) = â« t 0 t 1 d t L ( { q i ( t ) + ÎŽ q i ( t ) } , { q Ì i ( t ) + ÎŽ q Ì i ( t ) } ) â â« t 0 t 1 d t L ( { q i ( t ) } , { q Ì i ( t ) } ) = â« t 0 t 1 d t [ L ( { q i ( t ) + ÎŽ q i ( t ) } , { q Ì i ( t ) + ÎŽ q Ì i ( t ) } ) â L ( { q i ( t ) } , { q Ì i ( t ) } ) ] = â« t 0 t 1 d t â k = 1 K ( â L â q k ÎŽ q k + â L â q Ì k ÎŽ q Ì k ( t ) ) = â k = 1 K â« t 0 t 1 d t ( â L â q k ÎŽ q k + â L â q Ì k ÎŽ q Ì k ( t ) ) = â k = 1 K [ â L â q Ì k q k ( t ) | t 0 t 1 + â« t 0 t 1 d t ÎŽ q k ( t ) ( â L â q k â d d t â L â q Ì k ) ] = â k = 1 K â« t 0 t 1 d t ÎŽ q k ( t ) ( â L â q k â d d t â L â q Ì k ) = 0 {\displaystyle {\begin{aligned}\delta S&=S(\{q_{i}(t)+\delta q_{i}(t)\},\{{\dot {q}}_{i}(t)+\delta {\dot {q}}_{i}(t)\})-S(\{q_{i}\},\{{\dot {q}}_{i}\})\\&=\int _{t_{0}}^{t_{1}}dt\,L(\{q_{i}(t)+\delta q_{i}(t)\},\{{\dot {q}}_{i}(t)+\delta {\dot {q}}_{i}(t)\})-\int _{t_{0}}^{t_{1}}dt\,L(\{q_{i}(t)\},\{{\dot {q}}_{i}(t)\})\\&=\int _{t_{0}}^{t_{1}}dt\,[L(\{q_{i}(t)+\delta q_{i}(t)\},\{{\dot {q}}_{i}(t)+\delta {\dot {q}}_{i}(t)\})-L(\{q_{i}(t)\},\{{\dot {q}}_{i}(t)\})]\\&=\int _{t_{0}}^{t_{1}}dt\sum _{k=1}^{K}\left({\frac {\partial L}{\partial q_{k}}}\delta q_{k}+{\frac {\partial L}{\partial {\dot {q}}_{k}}}\delta {\dot {q}}_{k}(t)\right)\\&=\sum _{k=1}^{K}\int _{t_{0}}^{t_{1}}dt\left({\frac {\partial L}{\partial q_{k}}}\delta q_{k}+{\frac {\partial L}{\partial {\dot {q}}_{k}}}\delta {\dot {q}}_{k}(t)\right)\\&=\sum _{k=1}^{K}\left[{\frac {\partial L}{\partial {\dot {q}}_{k}}}q_{k}(t)|_{t_{0}}^{t_{1}}+\int _{t_{0}}^{t_{1}}dt\delta q_{k}(t)\left({\frac {\partial L}{\partial q_{k}}}-{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{k}}}\right)\right]\\&=\sum _{k=1}^{K}\int _{t_{0}}^{t_{1}}dt\delta q_{k}(t)\left({\frac {\partial L}{\partial q_{k}}}-{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{k}}}\right)\\&=0\end{aligned}}}
ããã§ã ÎŽ q k ( t ) {\displaystyle \delta q_{k}(t)} ã¯ä»»æã§ããã®ã§ããªã€ã©ãŒ=ã©ã°ã©ã³ãžã¥æ¹çšåŒ
d d t â L â q Ì k â â L â q k = 0 {\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{k}}}-{\frac {\partial L}{\partial q_{k}}}=0}
ãåŸãã
ããŠãå€åæ³ãå©çšããããã€ãã®ç°¡åãªäŸã玹ä»ãããã
æ°Žå¹³ãª2ç¹ãããã®2ç¹éè·é¢ã ãããããé·ãã®ãã¢ã§çµãã å Žåãåœç¶ãããŒãã¯ããããã ãã®ããã«ãããŒããªã©ãåãããæã«ã§ããæ²ç·ã®ããšãæžåç·(ãããããã)ãšããã
èšç®äŸã®ããã«ãå°é¢æ°yâã§å埮åãããšããæäœãå¿
èŠã«ãªãã
ãå€åããšããèããçšããŠãéåæ¹çšåŒã®å®çŸ©ãæ°åŒã§æžãäºãããã®èšäºã§ã¯èããã以äžãååŠã«ãããå€åã®èšç®æ¹æ³ã説æããŠããã
ã§ã¯ãå€åãçšããŠãã¥ãŒãã³æ¹çšåŒãæžãæããããšãèããããŸãå€å
žååŠã§ã®ãã¥ãŒãã³æ¹çšåŒã¯
ã®åœ¢ã§æžãããã
å€åãããããã«ã©ã°ã©ã³ãžã¢ã³ãšããéãå°å
¥ããããŸã ãã©ã°ã©ã³ãžã¢ã³ã®å
·äœçãªåœ¢ã¯åãããªããã©ããã質ç¹ãªã©ã®åº§æšäœçœ®ã q {\displaystyle q} ãšããŠããã®äœçœ®ã®æé埮å(ã€ãŸãé床)ã q Ì {\displaystyle {\dot {q}}} ãšããã°ã
ãšãã圢ã«ãªãäºãåãã£ãŠãããå é床 q Ì {\displaystyle {\ddot {q}}} ã¯èããªããŠè¯ãäºãåãã£ãŠããããã倩äžãçã ãã q Ì {\displaystyle {\dot {q}}} ãéåéãšããã®ä¿æ°åã«çžåœããããã§ãããéåéã¯ãéåããŠãã質ç¹ãªã©ã®ä¿åéã§ããããã£ãœããå é床ã¯ãéåããŠãã質ç¹ã®ä¿åéã§ã¯ãªãããã§ããã(ãªããã©ã°ã©ã³ãžã¢ã³Lã¯ã¹ã«ã©ãŒé(ãã¯ãã«ã§ãªãæ°)ã§ããã)
ã©ã°ã©ã³ãžã¢ã³ãããæéã®ç¯å²ã§ç©åãããã®ãã
ãšæžããäœçšãšåŒã¶ãããã§éåæ¹çšåŒãåŸãããã®åçãšããŠã"éåæ¹çšåŒã¯ãå°ãã ã q , q Ì {\displaystyle q,{\dot {q}}} ãå€åããããšããŠããäœçšãå€åããªããããªå€ãåºã q , q Ì {\displaystyle q,{\dot {q}}} ã®é¢ä¿ã«ãã£ãŠäžããããã"ãšããããšãèŠæ±ããã
ãã®ãšãã q , q Ì {\displaystyle q,{\dot {q}}} ãå€åããããšãã®å®éã®äœçšã®å€å ÎŽS ãèšç®ãããš(ÎŽã¯ãã«ã¿ãšèªã)ã
垞埮åé¢æ° q Ì {\displaystyle {\dot {q}}} ã§å埮åããããšã®æ°åŠçæ£åœæ§ãç解ãã¥ãããããããªãããã²ãšãŸããããèšç®ããŠãããããã詳现ã¯åŸè¿°ããã ããã§ã2è¡ç®ãã3è¡ç®ã§ã¯ãéšåç©åã«ãã£ãŠ
ãšãããå³èŸºã§éšåç©åã§åºãŠããé
ãæ¶ãããã«ã" q , q Ì {\displaystyle q,{\dot {q}}} ã¯ç©åç¯å²ã®äž¡ç«¯ã§ãã t = ti , tf ã§ã¯å€åããªã"ãšããèŠè«ãå ããã
æå°äœçšã®åçã«ãããšããã®ãšãã«ÎŽS = 0 ã§ãªããŠã¯ãªããªããÎŽq ã®å€ã«é¢ãããÎŽS = 0 ãæãç«ã€ããã«ã¯ã
ãæãç«ã€å¿
èŠãããããã£ãŠããã®åŒãéåæ¹çšåŒãšãªãã
ç¹ã«q ãéåžžã®åº§æšx ã§ããæã®ããšãèãããããã§ã
ãšãããšãåŒ(1)ã¯ã
ãšãªããéåžžã®èªç±ãªç²åã®éåæ¹çšåŒã«äžèŽãããããã§ã
ã¯ç²åã®éåãšãã«ã®ãŒã§ããã
ãŸããä¿ååã®äžã§ãç¹ã«ç©äœã®é床ã«ãããªãåãåããŠéåããŠããç²åã«å¯ŸããŠã¯ããã®åã«ãã£ãŠåŸãããäœçœ®ãšãã«ã®ãŒãV (q ) ãç©äœã®éåãšãã«ã®ãŒãT ãšè¡šããšãã
ãšãããšãåŒ(1)ã¯ã
ãšãªãããå³èŸºã¯ä¿ååã«å¯Ÿããåãè¡šããã®ã§ãã®ãšãã®ã©ã°ã©ã³ãžã¢ã³ã¯
ã§äžããããããšãåããã
ãŸããèªç±ãªè§éåéã«å¯Ÿããã©ã°ã©ã³ãžã¢ã³ã¯
ã«ãã£ãŠäžããããããã¯åäœã®è§éåéãæã€(æ
£æ§ã¢ãŒã¡ã³ãã¯åäœä»¥å€æã€ããšãåºæ¥ãªãããšã«æ³šæ)ãšãã«ã®ãŒãè¡šããã
ã©ã°ã©ã³ãžã¢ã³ã¯ãåã«ãé«æ ¡ç©çã§ãç¿ããããªéåæ¹çšåŒã®å®çŸ©ããå€åãšããæ°åŠçææ³ã«ããšã¥ããŠãèšãæãããã®ã§ããã
ã©ã°ã©ã³ãžã¢ã³ã¯ãç©çåŠã«ãããŠå
¬åŒãå°ãããã®ãç©çã®(ã»ãŒå
šãŠã®åéã§ã®)å
±éã®æéã§ããã
ãšããã§ãè§éåéã«é¢ããæ¹çšåŒã¯
ãšæžããã(I ã¯æ
£æ§ã¢ãŒã¡ã³ãã Ï â {\displaystyle {\vec {\omega }}} ã¯è§é床ã N â {\displaystyle {\vec {N}}} ã¯ç©äœã«åãåã®ã¢ãŒã¡ã³ã)ã
è§éåéã®åŒã¯ããã¥ãŒãã³æ¹çšåŒã«äŒŒãŠããã
ãã¥ãŒãã³æ¹çšåŒ
ã©ã°ã©ã³ãžã¢ã³ã¯ããã®ãããªéåæ³åãçµ±äžçã«èšè¿°ã§ããã
çµ±äžçã«èšè¿°ã§ãããšãããå Žåã«ã¯éœåãè¯ãããã®ãããªåº§æšã®èšè¿°æ¹æ³ã®çµ±äžåã®ç®çã§ãããã©ã°ã©ã³ãžã¢ã³ãåŸè¿°ã®ããã«ããã¢ã³ãå©çšãããäºãããã
ã©ã°ã©ã³ãžã¢ã³ãçšãããšããéåép ã¯
ãšå®çŸ©ããããå®éãèªç±ãªç²åã«å¯ŸããŠã¯ã
ãåŸãããæ£ããããšãåãããé床ã«äŸåããåãèããå Žåãp ã¯å¿
ãããäžè¬çãªéåéãšäžèŽããªãã
ãã®ãšããããã§å®çŸ©ããéåéãäžè¬åãããéåéãšåŒãã§éåžžã®éåéãšåºå¥ããã
次ã«ããšãã«ã®ãŒã®èšè¿°ãäžè¬åããããšãèãããããããã説æããããã«ããã¢ã³ H ãããšãã«ã®ãŒãäžè¬åãããã®ã«çžåœããã
L 㯠q , q Ì {\displaystyle q,{\dot {q}}} ãå€æ°ãšããŠçšããéã§ããããããããããããq , p ãå€æ°ãšããŠçšããæ¹ã䟿å©ãªããšãããããã®ãããªéã p , q Ì {\displaystyle p,{\dot {q}}} ã®éã®ã«ãžã£ã³ãã«å€æã«ãã£ãŠäœãããšãåºæ¥ãããããããã«ããã¢ã³H ãšåŒã³ã
ã§å®çŸ©ãããç¹ã« L = T ( q Ì ) â V ( q ) {\displaystyle L=T({\dot {q}})-V(q)} ãæºããå Žåã
ãåŸãããH ã¯ç³»ã®å
šãšãã«ã®ãŒãšäžèŽããããã®çµæã¯ãšãã«ã®ãŒä¿ååã®å°åºã«çšããããã
ããã«ããã¢ã³ H ( { q i } , { p i } ) = T + V {\displaystyle H(\{q_{i}\},\{p_{i}\})\,=T+V} ã«ãããŠ
ãæãç«ã€ããããæ£æºæ¹çšåŒãšããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ããé¢æ° L ( q 1 , q 2 , ⯠, q K , q Ì 1 , q Ì 2 , ⯠, q Ì K ) {\\displaystyle L(q_{1},q_{2},\\cdots ,q_{K},{\\dot {q}}_{1},{\\dot {q}}_{2},\\cdots ,{\\dot {q}}_{K})} ããããšãã«ã",
"title": "å€åæ³"
},
{
"paragraph_id": 1,
"tag": "p",
"text": "S = â« t 0 t 1 d t L ( q 1 ( t ) , q 2 ( t ) , ⯠, q K ( t ) , q Ì 1 ( t ) , q Ì 2 ( t ) , ⯠, q Ì K ( t ) ) {\\displaystyle S=\\int _{t_{0}}^{t_{1}}dt\\,L(q_{1}(t),q_{2}(t),\\cdots ,q_{K}(t),{\\dot {q}}_{1}(t),{\\dot {q}}_{2}(t),\\cdots ,{\\dot {q}}_{K}(t))}",
"title": "å€åæ³"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ãæå°ã«ãã q i ( t ) {\\displaystyle q_{i}(t)} ã¯ã©ã®ãããªãã®ã ãããã",
"title": "å€åæ³"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ãŸãã¯ç°¡åãªäŸãšããŠãé¢æ° f ( x ) {\\displaystyle f(x)} ãæå°ã«ãã x {\\displaystyle x} ã«ã€ããŠèãããã f ( x ) {\\displaystyle f(x)} ãæå°å€ãåããšãã f â² ( x ) = 0 {\\displaystyle f'(x)=0} ãšãªãã®ã ã£ãã f â² ( x ) = 0 {\\displaystyle f'(x)=0} ãšãªãããšã¯ã x {\\displaystyle x} ã埮å°é ÎŽ x {\\displaystyle \\delta x} ã ãå€åããããšãã f ( x ) {\\displaystyle f(x)} ã®å€åé ÎŽ f := f ( x + ÎŽ x ) â f ( x ) {\\displaystyle \\delta f:=f(x+\\delta x)-f(x)} 㯠Ύ f = 0 {\\displaystyle \\delta f=0} ã«ãªããšããããšã§ããã",
"title": "å€åæ³"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ããããã®é¡æšã§ã S ( { q i } , { q Ì i } ) {\\displaystyle S(\\{q_{i}\\},\\{{\\dot {q}}_{i}\\})} ãæå°ã«ãã { q i ( t ) } {\\displaystyle \\{q_{i}(t)\\}} ã«ã€ããŠã { q i ( t ) } {\\displaystyle \\{q_{i}(t)\\}} ãå°ãã ãå€åãã㊠{ q i ( t ) + ÎŽ q i ( t ) } {\\displaystyle \\{q_{i}(t)+\\delta q_{i}(t)\\}} (ãã ããå¢çæ¡ä»¶ ÎŽ q i ( t 0 ) = ÎŽ q i ( t 1 ) = 0 {\\displaystyle \\delta q_{i}(t_{0})=\\delta q_{i}(t_{1})=0} ã課ã)ãšãããšãã® S {\\displaystyle S} ã®å€åé ÎŽ S = S ( { q i ( t ) + ÎŽ q i ( t ) } , { q Ì i ( t ) + ÎŽ q Ì i ( t ) } ) â S ( { q i } , { q Ì i } ) {\\displaystyle \\delta S=S(\\{q_{i}(t)+\\delta q_{i}(t)\\},\\{{\\dot {q}}_{i}(t)+\\delta {\\dot {q}}_{i}(t)\\})-S(\\{q_{i}\\},\\{{\\dot {q}}_{i}\\})} 㯠Ύ S = 0 {\\displaystyle \\delta S=0} ãšãªããšèããããšãåºæ¥ãã",
"title": "å€åæ³"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ÎŽ S = S ( { q i ( t ) + ÎŽ q i ( t ) } , { q Ì i ( t ) + ÎŽ q Ì i ( t ) } ) â S ( { q i } , { q Ì i } ) = â« t 0 t 1 d t L ( { q i ( t ) + ÎŽ q i ( t ) } , { q Ì i ( t ) + ÎŽ q Ì i ( t ) } ) â â« t 0 t 1 d t L ( { q i ( t ) } , { q Ì i ( t ) } ) = â« t 0 t 1 d t [ L ( { q i ( t ) + ÎŽ q i ( t ) } , { q Ì i ( t ) + ÎŽ q Ì i ( t ) } ) â L ( { q i ( t ) } , { q Ì i ( t ) } ) ] = â« t 0 t 1 d t â k = 1 K ( â L â q k ÎŽ q k + â L â q Ì k ÎŽ q Ì k ( t ) ) = â k = 1 K â« t 0 t 1 d t ( â L â q k ÎŽ q k + â L â q Ì k ÎŽ q Ì k ( t ) ) = â k = 1 K [ â L â q Ì k q k ( t ) | t 0 t 1 + â« t 0 t 1 d t ÎŽ q k ( t ) ( â L â q k â d d t â L â q Ì k ) ] = â k = 1 K â« t 0 t 1 d t ÎŽ q k ( t ) ( â L â q k â d d t â L â q Ì k ) = 0 {\\displaystyle {\\begin{aligned}\\delta S&=S(\\{q_{i}(t)+\\delta q_{i}(t)\\},\\{{\\dot {q}}_{i}(t)+\\delta {\\dot {q}}_{i}(t)\\})-S(\\{q_{i}\\},\\{{\\dot {q}}_{i}\\})\\\\&=\\int _{t_{0}}^{t_{1}}dt\\,L(\\{q_{i}(t)+\\delta q_{i}(t)\\},\\{{\\dot {q}}_{i}(t)+\\delta {\\dot {q}}_{i}(t)\\})-\\int _{t_{0}}^{t_{1}}dt\\,L(\\{q_{i}(t)\\},\\{{\\dot {q}}_{i}(t)\\})\\\\&=\\int _{t_{0}}^{t_{1}}dt\\,[L(\\{q_{i}(t)+\\delta q_{i}(t)\\},\\{{\\dot {q}}_{i}(t)+\\delta {\\dot {q}}_{i}(t)\\})-L(\\{q_{i}(t)\\},\\{{\\dot {q}}_{i}(t)\\})]\\\\&=\\int _{t_{0}}^{t_{1}}dt\\sum _{k=1}^{K}\\left({\\frac {\\partial L}{\\partial q_{k}}}\\delta q_{k}+{\\frac {\\partial L}{\\partial {\\dot {q}}_{k}}}\\delta {\\dot {q}}_{k}(t)\\right)\\\\&=\\sum _{k=1}^{K}\\int _{t_{0}}^{t_{1}}dt\\left({\\frac {\\partial L}{\\partial q_{k}}}\\delta q_{k}+{\\frac {\\partial L}{\\partial {\\dot {q}}_{k}}}\\delta {\\dot {q}}_{k}(t)\\right)\\\\&=\\sum _{k=1}^{K}\\left[{\\frac {\\partial L}{\\partial {\\dot {q}}_{k}}}q_{k}(t)|_{t_{0}}^{t_{1}}+\\int _{t_{0}}^{t_{1}}dt\\delta q_{k}(t)\\left({\\frac {\\partial L}{\\partial q_{k}}}-{\\frac {d}{dt}}{\\frac {\\partial L}{\\partial {\\dot {q}}_{k}}}\\right)\\right]\\\\&=\\sum _{k=1}^{K}\\int _{t_{0}}^{t_{1}}dt\\delta q_{k}(t)\\left({\\frac {\\partial L}{\\partial q_{k}}}-{\\frac {d}{dt}}{\\frac {\\partial L}{\\partial {\\dot {q}}_{k}}}\\right)\\\\&=0\\end{aligned}}}",
"title": "å€åæ³"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ããã§ã ÎŽ q k ( t ) {\\displaystyle \\delta q_{k}(t)} ã¯ä»»æã§ããã®ã§ããªã€ã©ãŒ=ã©ã°ã©ã³ãžã¥æ¹çšåŒ",
"title": "å€åæ³"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "d d t â L â q Ì k â â L â q k = 0 {\\displaystyle {\\frac {d}{dt}}{\\frac {\\partial L}{\\partial {\\dot {q}}_{k}}}-{\\frac {\\partial L}{\\partial q_{k}}}=0}",
"title": "å€åæ³"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "ãåŸãã",
"title": "å€åæ³"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "ããŠãå€åæ³ãå©çšããããã€ãã®ç°¡åãªäŸã玹ä»ãããã",
"title": "å€åæ³"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "æ°Žå¹³ãª2ç¹ãããã®2ç¹éè·é¢ã ãããããé·ãã®ãã¢ã§çµãã å Žåãåœç¶ãããŒãã¯ããããã ãã®ããã«ãããŒããªã©ãåãããæã«ã§ããæ²ç·ã®ããšãæžåç·(ãããããã)ãšããã",
"title": "å€åæ³"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "èšç®äŸã®ããã«ãå°é¢æ°yâã§å埮åãããšããæäœãå¿
èŠã«ãªãã",
"title": "å€åæ³"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "ãå€åããšããèããçšããŠãéåæ¹çšåŒã®å®çŸ©ãæ°åŒã§æžãäºãããã®èšäºã§ã¯èããã以äžãååŠã«ãããå€åã®èšç®æ¹æ³ã説æããŠããã",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ã§ã¯ãå€åãçšããŠãã¥ãŒãã³æ¹çšåŒãæžãæããããšãèããããŸãå€å
žååŠã§ã®ãã¥ãŒãã³æ¹çšåŒã¯",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "ã®åœ¢ã§æžãããã",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "å€åãããããã«ã©ã°ã©ã³ãžã¢ã³ãšããéãå°å
¥ããããŸã ãã©ã°ã©ã³ãžã¢ã³ã®å
·äœçãªåœ¢ã¯åãããªããã©ããã質ç¹ãªã©ã®åº§æšäœçœ®ã q {\\displaystyle q} ãšããŠããã®äœçœ®ã®æé埮å(ã€ãŸãé床)ã q Ì {\\displaystyle {\\dot {q}}} ãšããã°ã",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "ãšãã圢ã«ãªãäºãåãã£ãŠãããå é床 q Ì {\\displaystyle {\\ddot {q}}} ã¯èããªããŠè¯ãäºãåãã£ãŠããããã倩äžãçã ãã q Ì {\\displaystyle {\\dot {q}}} ãéåéãšããã®ä¿æ°åã«çžåœããããã§ãããéåéã¯ãéåããŠãã質ç¹ãªã©ã®ä¿åéã§ããããã£ãœããå é床ã¯ãéåããŠãã質ç¹ã®ä¿åéã§ã¯ãªãããã§ããã(ãªããã©ã°ã©ã³ãžã¢ã³Lã¯ã¹ã«ã©ãŒé(ãã¯ãã«ã§ãªãæ°)ã§ããã)",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "ã©ã°ã©ã³ãžã¢ã³ãããæéã®ç¯å²ã§ç©åãããã®ãã",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "ãšæžããäœçšãšåŒã¶ãããã§éåæ¹çšåŒãåŸãããã®åçãšããŠã\"éåæ¹çšåŒã¯ãå°ãã ã q , q Ì {\\displaystyle q,{\\dot {q}}} ãå€åããããšããŠããäœçšãå€åããªããããªå€ãåºã q , q Ì {\\displaystyle q,{\\dot {q}}} ã®é¢ä¿ã«ãã£ãŠäžããããã\"ãšããããšãèŠæ±ããã",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "ãã®ãšãã q , q Ì {\\displaystyle q,{\\dot {q}}} ãå€åããããšãã®å®éã®äœçšã®å€å ÎŽS ãèšç®ãããš(ÎŽã¯ãã«ã¿ãšèªã)ã",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "垞埮åé¢æ° q Ì {\\displaystyle {\\dot {q}}} ã§å埮åããããšã®æ°åŠçæ£åœæ§ãç解ãã¥ãããããããªãããã²ãšãŸããããèšç®ããŠãããããã詳现ã¯åŸè¿°ããã ããã§ã2è¡ç®ãã3è¡ç®ã§ã¯ãéšåç©åã«ãã£ãŠ",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "ãšãããå³èŸºã§éšåç©åã§åºãŠããé
ãæ¶ãããã«ã\" q , q Ì {\\displaystyle q,{\\dot {q}}} ã¯ç©åç¯å²ã®äž¡ç«¯ã§ãã t = ti , tf ã§ã¯å€åããªã\"ãšããèŠè«ãå ããã",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "æå°äœçšã®åçã«ãããšããã®ãšãã«ÎŽS = 0 ã§ãªããŠã¯ãªããªããÎŽq ã®å€ã«é¢ãããÎŽS = 0 ãæãç«ã€ããã«ã¯ã",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "ãæãç«ã€å¿
èŠãããããã£ãŠããã®åŒãéåæ¹çšåŒãšãªãã",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "ç¹ã«q ãéåžžã®åº§æšx ã§ããæã®ããšãèãããããã§ã",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "ãšãããšãåŒ(1)ã¯ã",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "ãšãªããéåžžã®èªç±ãªç²åã®éåæ¹çšåŒã«äžèŽãããããã§ã",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "ã¯ç²åã®éåãšãã«ã®ãŒã§ããã",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "ãŸããä¿ååã®äžã§ãç¹ã«ç©äœã®é床ã«ãããªãåãåããŠéåããŠããç²åã«å¯ŸããŠã¯ããã®åã«ãã£ãŠåŸãããäœçœ®ãšãã«ã®ãŒãV (q ) ãç©äœã®éåãšãã«ã®ãŒãT ãšè¡šããšãã",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "ãšãããšãåŒ(1)ã¯ã",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "ãšãªãããå³èŸºã¯ä¿ååã«å¯Ÿããåãè¡šããã®ã§ãã®ãšãã®ã©ã°ã©ã³ãžã¢ã³ã¯",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "ã§äžããããããšãåããã",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "ãŸããèªç±ãªè§éåéã«å¯Ÿããã©ã°ã©ã³ãžã¢ã³ã¯",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "ã«ãã£ãŠäžããããããã¯åäœã®è§éåéãæã€(æ
£æ§ã¢ãŒã¡ã³ãã¯åäœä»¥å€æã€ããšãåºæ¥ãªãããšã«æ³šæ)ãšãã«ã®ãŒãè¡šããã",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "ã©ã°ã©ã³ãžã¢ã³ã¯ãåã«ãé«æ ¡ç©çã§ãç¿ããããªéåæ¹çšåŒã®å®çŸ©ããå€åãšããæ°åŠçææ³ã«ããšã¥ããŠãèšãæãããã®ã§ããã",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "ã©ã°ã©ã³ãžã¢ã³ã¯ãç©çåŠã«ãããŠå
¬åŒãå°ãããã®ãç©çã®(ã»ãŒå
šãŠã®åéã§ã®)å
±éã®æéã§ããã",
"title": "ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "ãšããã§ãè§éåéã«é¢ããæ¹çšåŒã¯",
"title": "äžè¬å座æš"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "ãšæžããã(I ã¯æ
£æ§ã¢ãŒã¡ã³ãã Ï â {\\displaystyle {\\vec {\\omega }}} ã¯è§é床ã N â {\\displaystyle {\\vec {N}}} ã¯ç©äœã«åãåã®ã¢ãŒã¡ã³ã)ã",
"title": "äžè¬å座æš"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "è§éåéã®åŒã¯ããã¥ãŒãã³æ¹çšåŒã«äŒŒãŠããã",
"title": "äžè¬å座æš"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "ãã¥ãŒãã³æ¹çšåŒ",
"title": "äžè¬å座æš"
},
{
"paragraph_id": 41,
"tag": "p",
"text": "ã©ã°ã©ã³ãžã¢ã³ã¯ããã®ãããªéåæ³åãçµ±äžçã«èšè¿°ã§ããã",
"title": "äžè¬å座æš"
},
{
"paragraph_id": 42,
"tag": "p",
"text": "çµ±äžçã«èšè¿°ã§ãããšãããå Žåã«ã¯éœåãè¯ãããã®ãããªåº§æšã®èšè¿°æ¹æ³ã®çµ±äžåã®ç®çã§ãããã©ã°ã©ã³ãžã¢ã³ãåŸè¿°ã®ããã«ããã¢ã³ãå©çšãããäºãããã",
"title": "äžè¬å座æš"
},
{
"paragraph_id": 43,
"tag": "p",
"text": "ã©ã°ã©ã³ãžã¢ã³ãçšãããšããéåép ã¯",
"title": "éåéãããã«ããã¢ã³ã®å®çŸ©"
},
{
"paragraph_id": 44,
"tag": "p",
"text": "ãšå®çŸ©ããããå®éãèªç±ãªç²åã«å¯ŸããŠã¯ã",
"title": "éåéãããã«ããã¢ã³ã®å®çŸ©"
},
{
"paragraph_id": 45,
"tag": "p",
"text": "ãåŸãããæ£ããããšãåãããé床ã«äŸåããåãèããå Žåãp ã¯å¿
ãããäžè¬çãªéåéãšäžèŽããªãã",
"title": "éåéãããã«ããã¢ã³ã®å®çŸ©"
},
{
"paragraph_id": 46,
"tag": "p",
"text": "ãã®ãšããããã§å®çŸ©ããéåéãäžè¬åãããéåéãšåŒãã§éåžžã®éåéãšåºå¥ããã",
"title": "éåéãããã«ããã¢ã³ã®å®çŸ©"
},
{
"paragraph_id": 47,
"tag": "p",
"text": "次ã«ããšãã«ã®ãŒã®èšè¿°ãäžè¬åããããšãèãããããããã説æããããã«ããã¢ã³ H ãããšãã«ã®ãŒãäžè¬åãããã®ã«çžåœããã",
"title": "éåéãããã«ããã¢ã³ã®å®çŸ©"
},
{
"paragraph_id": 48,
"tag": "p",
"text": "L 㯠q , q Ì {\\displaystyle q,{\\dot {q}}} ãå€æ°ãšããŠçšããéã§ããããããããããããq , p ãå€æ°ãšããŠçšããæ¹ã䟿å©ãªããšãããããã®ãããªéã p , q Ì {\\displaystyle p,{\\dot {q}}} ã®éã®ã«ãžã£ã³ãã«å€æã«ãã£ãŠäœãããšãåºæ¥ãããããããã«ããã¢ã³H ãšåŒã³ã",
"title": "éåéãããã«ããã¢ã³ã®å®çŸ©"
},
{
"paragraph_id": 49,
"tag": "p",
"text": "ã§å®çŸ©ãããç¹ã« L = T ( q Ì ) â V ( q ) {\\displaystyle L=T({\\dot {q}})-V(q)} ãæºããå Žåã",
"title": "éåéãããã«ããã¢ã³ã®å®çŸ©"
},
{
"paragraph_id": 50,
"tag": "p",
"text": "ãåŸãããH ã¯ç³»ã®å
šãšãã«ã®ãŒãšäžèŽããããã®çµæã¯ãšãã«ã®ãŒä¿ååã®å°åºã«çšããããã",
"title": "éåéãããã«ããã¢ã³ã®å®çŸ©"
},
{
"paragraph_id": 51,
"tag": "p",
"text": "ããã«ããã¢ã³ H ( { q i } , { p i } ) = T + V {\\displaystyle H(\\{q_{i}\\},\\{p_{i}\\})\\,=T+V} ã«ãããŠ",
"title": "æ£æºæ¹çšåŒ"
},
{
"paragraph_id": 52,
"tag": "p",
"text": "ãæãç«ã€ããããæ£æºæ¹çšåŒãšããã",
"title": "æ£æºæ¹çšåŒ"
}
] | null | {{Pathnav|ã¡ã€ã³ããŒãž|èªç¶ç§åŠ|ç©çåŠ|解æååŠ|frame=1}}
== å€åæ³ ==
ããé¢æ° <math>L(q_1,q_2,\cdots,q_K,\dot q_1,\dot q_2,\cdots,\dot q_K)</math> ããããšãã«ã
<math>S = \int_{t_0}^{t_1} dt \, L(q_1(t),q_2(t),\cdots,q_K(t),\dot q_1(t),\dot q_2(t),\cdots,\dot q_K(t)) </math>
ãæå°ã«ãã <math>q_i(t) </math> ã¯ã©ã®ãããªãã®ã ãããã
ãŸãã¯ç°¡åãªäŸãšããŠãé¢æ° <math>f(x) </math> ãæå°ã«ãã <math>x </math> ã«ã€ããŠèãããã<math>f(x) </math> ãæå°å€ãåããšãã<math>f'(x) = 0 </math> ãšãªãã®ã ã£ãã<math>f'(x) = 0 </math> ãšãªãããšã¯ã<math>x </math> ã埮å°é <math>\delta x </math> ã ãå€åããããšãã<math>f(x) </math> ã®å€åé <math>\delta f := f(x+\delta x) - f(x) </math> 㯠<math>\delta f = 0 </math> ã«ãªããšããããšã§ããã
ããããã®é¡æšã§ã<math>S(\{q_i\},\{\dot q_i\}) </math> ãæå°ã«ãã <math>\{q_i(t)\} </math> ã«ã€ããŠã<math>\{q_i(t)\} </math> ãå°ãã ãå€åãã㊠<math>\{q_i(t) + \delta q_i(t) \} </math> ïŒãã ããå¢çæ¡ä»¶ <math>\delta q_i(t_0) = \delta q_i(t_1) = 0 </math> ã課ãïŒãšãããšãã® <math>S </math> ã®å€åé <math>\delta S = S(\{q_i(t) + \delta q_i(t) \},\{\dot q_i(t) + \delta \dot q_i(t)\} ) - S(\{q_i\},\{\dot q_i\}) </math> 㯠<math>\delta S = 0 </math> ãšãªããšèããããšãåºæ¥ãã
<math>\begin{align} \delta S &= S(\{q_i(t) + \delta q_i(t) \},\{\dot q_i(t) + \delta \dot q_i(t)\} ) - S(\{q_i\},\{\dot q_i\})\\
&= \int_{t_0}^{t_1} dt \, L(\{q_i(t) + \delta q_i(t) \},\{\dot q_i(t) + \delta \dot q_i(t)\}) - \int_{t_0}^{t_1} dt \, L(\{q_i(t)\},\{\dot q_i(t)\}) \\
&= \int_{t_0}^{t_1} dt \, [L(\{q_i(t) + \delta q_i(t) \},\{\dot q_i(t) + \delta \dot q_i(t)\}) - L(\{q_i(t)\},\{\dot q_i(t)\})] \\
&= \int_{t_0}^{t_1} dt \sum_{k=1}^{K} \left(\frac{\partial L}{\partial q_k}\delta q_k + \frac{\partial L}{\partial \dot q_k}\delta \dot q_k(t)\right) \\
&= \sum_{k=1}^{K} \int_{t_0}^{t_1} dt \left(\frac{\partial L}{\partial q_k}\delta q_k + \frac{\partial L}{\partial \dot q_k}\delta \dot q_k(t)\right) \\
&= \sum_{k=1}^{K}\left[ \frac{\partial L}{\partial \dot q_k} q_k(t)|_{t_0}^{t_1} + \int_{t_0}^{t_1} dt \delta q_k(t)\left(\frac{\partial L}{\partial q_k}- \frac{d}{dt}\frac{\partial L}{\partial \dot q_k}\right)\right] \\
&= \sum_{k=1}^{K}\int_{t_0}^{t_1} dt \delta q_k(t)\left(\frac{\partial L}{\partial q_k}- \frac{d}{dt}\frac{\partial L}{\partial \dot q_k}\right) \\
&= 0
\end{align} </math>
ããã§ã <math>\delta q_k(t) </math> ã¯ä»»æã§ããã®ã§ããªã€ã©ãŒïŒã©ã°ã©ã³ãžã¥æ¹çšåŒ
<math>\frac{d}{dt}\frac{\partial L}{\partial \dot q_k} - \frac{\partial L}{\partial q_k} = 0 </math>
ãåŸãã
ããŠãå€åæ³ãå©çšããããã€ãã®ç°¡åãªäŸã玹ä»ãããã
=== çåšåé¡ ===
=== æžåç· ===
æ°Žå¹³ãª2ç¹ãããã®2ç¹éè·é¢ã ãããããé·ãã®ãã¢ã§çµãã å Žåãåœç¶ãããŒãã¯ããããã
ãã®ããã«ãããŒããªã©ãåãããæã«ã§ããæ²ç·ã®ããšãæžåç·ïŒããããããïŒãšããã
:ïŒâ»ãå³ãè¿œå ããŠãã ãããïŒ
:ïŒâ»ãèšç®äŸãèšè¿°ããŠãã ããïŒ
èšç®äŸã®ããã«ãå°é¢æ°yâã§å埮åãããšããæäœãå¿
èŠã«ãªãã
=== æééäžç· ===
== ã©ã°ã©ã³ãžã¢ã³ãšæå°äœçšã®åç ==
ãå€åããšããèããçšããŠãéåæ¹çšåŒã®å®çŸ©ãæ°åŒã§æžãäºãããã®èšäºã§ã¯èããã以äžãååŠã«ãããå€åã®èšç®æ¹æ³ã説æããŠããã
===ã©ã°ã©ã³ãžã¢ã³===
ã§ã¯ãå€åãçšããŠãã¥ãŒãã³æ¹çšåŒãæžãæããããšãèããããŸãå€å
žååŠã§ã®ãã¥ãŒãã³æ¹çšåŒã¯
:<math>
m \ddot {\vec x} = \vec f
</math>
ã®åœ¢ã§æžãããã
å€åãããããã«'''ã©ã°ã©ã³ãžã¢ã³'''ãšããéãå°å
¥ããããŸã ãã©ã°ã©ã³ãžã¢ã³ã®å
·äœçãªåœ¢ã¯åãããªããã©ããã質ç¹ãªã©ã®åº§æšäœçœ®ã<math>q</math>ãšããŠããã®äœçœ®ã®æé埮åïŒã€ãŸãé床ïŒã<math>\dot q</math>ãšããã°ã
:<math>
L = L (q,\dot q)
</math>
ãšãã圢ã«ãªãäºãåãã£ãŠãããå é床<math>\ddot q</math>ã¯èããªããŠè¯ãäºãåãã£ãŠããããã倩äžãçã ãã<math>\dot q</math>ãéåéãšããã®ä¿æ°åã«çžåœããããã§ãããéåéã¯ãéåããŠãã質ç¹ãªã©ã®ä¿åéã§ããããã£ãœããå é床ã¯ãéåããŠãã質ç¹ã®ä¿åéã§ã¯ãªãããã§ãããïŒãªããã©ã°ã©ã³ãžã¢ã³Lã¯ã¹ã«ã©ãŒéïŒãã¯ãã«ã§ãªãæ°ïŒã§ãããïŒ
===æå°äœçšã®åç===
ã©ã°ã©ã³ãžã¢ã³ãããæéã®ç¯å²ã§ç©åãããã®ãã
:<math>
S= \int dt L
</math>
ãšæžãã'''äœçš'''ãšåŒã¶ãããã§éåæ¹çšåŒãåŸãããã®åçãšããŠã"éåæ¹çšåŒã¯ãå°ãã ã <math>q,\dot q</math> ãå€åããããšããŠããäœçšãå€åããªããããªå€ãåºã <math>q,\dot q</math> ã®é¢ä¿ã«ãã£ãŠäžããããã"ãšããããšãèŠæ±ããã
ãã®ãšãã<math>q, \dot q</math> ãå€åããããšãã®å®éã®äœçšã®å€å ÎŽS ãèšç®ãããšïŒÎŽã¯ãã«ã¿ãšèªãïŒã
:<math>
\begin{align}
\delta S &= \int dt \delta L\\
&= \int dt \frac {\partial L}{\partial q } \delta q+ \frac {\partial L}{\partial {\dot q} } \delta \dot q\\
&= \int dt \frac {\partial L}{\partial q } \delta q- \frac {\partial {}}{\partial t }\frac {\partial L}{\partial {\dot q} } \delta q\\
&= \int dt (\frac {\partial L}{\partial q } - \frac {\partial {}}{\partial t }\frac {\partial L}{\partial {\dot q} }) \delta q\\
\end{align}
</math>
垞埮åé¢æ°<math>\dot q</math>ã§å埮åããããšã®æ°åŠçæ£åœæ§ãç解ãã¥ãããããããªãããã²ãšãŸããããèšç®ããŠãããããã詳现ã¯åŸè¿°ããã
ããã§ã2è¡ç®ãã3è¡ç®ã§ã¯ã<!-- magic variables !! -->éšåç©åã«ãã£ãŠ
:<math>
\begin{align}
\int \delta\dot q f(q) &= [\delta q f(q) ] _{t _i}^{t _f}- \int \delta q \frac {\partial {}}{\partial t } f(q)\\
&= - \int \delta q \frac {\partial {}}{\partial t } f(q)
\end{align}
</math>
ãšãããå³èŸºã§éšåç©åã§åºãŠããé
ãæ¶ãããã«ã"<math>q,\dot q</math> ã¯ç©åç¯å²ã®äž¡ç«¯ã§ãã ''t'' = ''t<sub>i</sub>'' , ''t<sub>f</sub>'' ã§ã¯å€åããªã"ãšããèŠè«ãå ãã<ref group="泚">ãã®èŠè«ãå€ããšå¥ã®å€ãåºãŠæ¥ãŠãããå Žåã«ã¯äŸ¿å©ã«ãªãããã§ããã<!-- 詳ããç¥ã£ãŠãã人ã§wikibooksã«æžãããšãã人ã¯ããã ãããã --><!-- é£ããåé¡ã ...ã --></ref>ã
æå°äœçšã®åçã«ãããšããã®ãšãã«δ''S'' = 0 ã§ãªããŠã¯ãªããªããδ''q'' ã®å€ã«é¢ãããδ''S'' = 0 ãæãç«ã€ããã«ã¯ã
:<math>
\frac {\partial L}{\partial q } - \frac {\partial {}}{\partial t }\frac {\partial L}{\partial {\dot q} }= 0 \qquad (1)
</math>
ãæãç«ã€å¿
èŠãããããã£ãŠããã®åŒãéåæ¹çšåŒãšãªãã
ç¹ã«''q'' ãéåžžã®åº§æš''x'' ã§ããæã®ããšãèãããããã§ã
:<math>
L = \frac 1 2 m \dot x^2
</math>
ãšãããšãåŒ(1)ã¯ã
:<math>
m \ddot x = 0
</math>
ãšãªããéåžžã®èªç±ãªç²åã®éåæ¹çšåŒã«äžèŽãããããã§ã
:<math>
\frac 1 2 m \dot x^2
</math>
ã¯ç²åã®éåãšãã«ã®ãŒã§ããã
ãŸããä¿ååã®äžã§ãç¹ã«ç©äœã®é床ã«ãããªãåãåããŠéåããŠããç²åã«å¯ŸããŠã¯ããã®åã«ãã£ãŠåŸãããäœçœ®ãšãã«ã®ãŒã''V'' (''q'' ) ãç©äœã®éåãšãã«ã®ãŒã''T'' ãšè¡šããšãã
:<math>
L = T(\dot q) - V(q)
</math>
ãšãããšãåŒ(1)ã¯ã
:<math>
m \ddot q = - \frac {\partial V}{\partial q }
</math>
ãšãªãããå³èŸºã¯ä¿ååã«å¯Ÿããåãè¡šããã®ã§ãã®ãšãã®ã©ã°ã©ã³ãžã¢ã³ã¯
:<math>
L = T(\dot q) - V(q)
</math>
ã§äžããããããšãåããã
ãŸããèªç±ãªè§éåéã«å¯Ÿããã©ã°ã©ã³ãžã¢ã³ã¯
:<math>
L = \frac 1 2 I \omega^2
</math>
ã«ãã£ãŠäžããããããã¯åäœã®è§éåéãæã€ïŒæ
£æ§ã¢ãŒã¡ã³ãã¯åäœä»¥å€æã€ããšãåºæ¥ãªãããšã«æ³šæïŒãšãã«ã®ãŒãè¡šããã
<!-- é»ç£æ°å㯠ç©äœã®é床ã«äŸåããåã§ããããã©ãã -->
<!-- ï¿œe -->
<!-- L = T(\dot q) - V(q,\dot q) -->
<!-- \ee -->
<!-- ãšãªãããšã®èª¬æã ã©ãã ã£ãã... -->
ã©ã°ã©ã³ãžã¢ã³ã¯ãåã«ãé«æ ¡ç©çã§ãç¿ããããªéåæ¹çšåŒã®å®çŸ©ããå€åãšããæ°åŠçææ³ã«ããšã¥ããŠãèšãæãããã®ã§ããã
ã©ã°ã©ã³ãžã¢ã³ã¯ãç©çåŠã«ãããŠå
¬åŒãå°ãããã®ãç©çã®ïŒã»ãŒå
šãŠã®åéã§ã®ïŒå
±éã®æéã§ããã
== äžè¬ååº§æš ==
ãšããã§ãè§éåéã«é¢ããæ¹çšåŒã¯
:<math>
I \vec \omega = \vec N
</math>
ãšæžãããïŒ''I'' ã¯æ
£æ§ã¢ãŒã¡ã³ãã<math>\vec \omega </math> ã¯è§é床ã<math>\vec N </math> ã¯ç©äœã«åãåã®ã¢ãŒã¡ã³ãïŒã
è§éåéã®åŒã¯ããã¥ãŒãã³æ¹çšåŒã«äŒŒãŠããã
ãã¥ãŒãã³æ¹çšåŒ
:<math>
m \ddot {\vec x} = \vec f
</math>ãšè¯ã䌌ã圢ã§ããã
ã©ã°ã©ã³ãžã¢ã³ã¯ããã®ãããªéåæ³åãçµ±äžçã«èšè¿°ã§ããã
çµ±äžçã«èšè¿°ã§ãããšãããå Žåã«ã¯éœåãè¯ãããã®ãããªåº§æšã®èšè¿°æ¹æ³ã®çµ±äžåã®ç®çã§ãããã©ã°ã©ã³ãžã¢ã³ãåŸè¿°ã®ããã«ããã¢ã³ãå©çšãããäºãããã
==éåéãããã«ããã¢ã³ã®å®çŸ©==
ã©ã°ã©ã³ãžã¢ã³ãçšãããšããéåé''p'' ã¯
:<math>
p \equiv \frac {\partial L}{\partial {\dot q} }
</math>
ãšå®çŸ©ããããå®éãèªç±ãªç²åã«å¯ŸããŠã¯ã
:<math>
p = m \dot q
</math>
ãåŸãããæ£ããããšãåãããé床ã«äŸåããåãèããå Žåã''p'' ã¯å¿
ãããäžè¬çãªéåéãšäžèŽããªãã
ãã®ãšããããã§å®çŸ©ããéåéãäžè¬åãããéåéãšåŒãã§éåžžã®éåéãšåºå¥ããã
次ã«ããšãã«ã®ãŒã®èšè¿°ãäžè¬åããããšãèãããããããã説æããããã«ããã¢ã³ H ãããšãã«ã®ãŒãäžè¬åãããã®ã«çžåœããã
<!-- ãŸããéåéãçšããŠ<math>\dot q</math>ãæ¶ãå»ã£ãéã -->''L'' ã¯<math>q,\dot q</math> ãå€æ°ãšããŠçšããéã§ãããããããããããã''q'' , ''p'' ãå€æ°ãšããŠçšããæ¹ã䟿å©ãªããšãããããã®ãããªéã<math>p,\dot q</math> ã®éã®ã«ãžã£ã³ãã«å€æã«ãã£ãŠäœãããšãåºæ¥ããããã'''ããã«ããã¢ã³'''''H'' ãšåŒã³ã
:<math>
H \equiv \dot q p -L
</math>
ã§å®çŸ©ãããç¹ã«<math>L=T(\dot q) - V(q)</math>ãæºããå Žåã
:<math>
H = T +V
</math>
ãåŸããã''H'' ã¯ç³»ã®å
šãšãã«ã®ãŒãšäžèŽããããã®çµæã¯ãšãã«ã®ãŒä¿ååã®å°åºã«çšããããã
==æ£æºæ¹çšåŒ==
ããã«ããã¢ã³<math> H(\{q_i\},\{p_i\}) \,= T + V</math>ã«ãããŠ
:<math> \dot{p}_i=-\frac{\partial{}H}{\partial{}q_i} </math>
:<math> \dot{q}_i=\frac{\partial{}H}{\partial{}p_i} </math>
ãæãç«ã€ãããã'''æ£æºæ¹çšåŒ'''ãšããã
==ãã¢ãœã³æ¬åŒ§==
{{stub}}
==è泚==
<references group="泚" />
{{DEFAULTSORT:ãããããããã ãããšãã»ããŠãããã®ãã€ã¯ãã}}
[[Category:解æååŠ|* ãããšãã»ããŠãããã®ãã€ã¯ãã]] | 2005-05-30T04:02:47Z | 2024-03-15T21:42:12Z | [
"ãã³ãã¬ãŒã:Pathnav",
"ãã³ãã¬ãŒã:Stub"
] | https://ja.wikibooks.org/wiki/%E8%A7%A3%E6%9E%90%E5%8A%9B%E5%AD%A6_%E9%81%8B%E5%8B%95%E6%96%B9%E7%A8%8B%E5%BC%8F%E3%81%AE%E4%B8%80%E8%88%AC%E5%8C%96 |
2,077 | 解æååŠ ä¿ååã®å°åº | ã©ã°ã©ã³ãžã¢ã³ã¯ç©çç³»ã®å
šãŠã®æ
å ±ãæ
ã£ãŠããã®ã§ããããçšããŠæ§ã
ãªä¿ååã瀺ãããšãåºæ¥ããäŸãã°ããšãã«ã®ãŒä¿ååãšéåéä¿ååãäŸãšããŠæããããã
ãšãã«ã®ãŒã
ã§å®çŸ©ããããã®è¡šåŒãšããã«ããã¢ã³
ãèŠæ¯ã¹ããšãããã«ããã¢ã³ã¯ç³»ã®å
šãšãã«ã®ãŒã«å¯Ÿå¿ããããšãåãããéåéã®ä¿ååã¯ãã®ãšãã
ãšãªãããšãã«ã®ãŒãæéçã«ä¿åããããšãåãããããã§ã4ãã5è¡ç®ã«ç§»ããšãéåæ¹çšåŒ
ãçšãããå®éã«ã¯ããšãã«ã®ãŒã®ä¿ååã¯æéã®åç¹ãåããããšã«å¯ŸããŠç©çç³»ãå€åããªãããšã«ãã ã
éåéä¿ååã¯ç©çç³»å
šäœãå¹³è¡ç§»åããããšã«ãã£ãŠãç©çç³»ã®éåãå€åããªãããšã«ããããã®ããšã空éçäžæ§æ§ãšåŒã¶ããã®ãšãã©ã°ã©ã³ãžã¢ã³ã«å«ãŸããå
šãŠã®ããq ã«ã€ããŠ
ãšãªãå€æãã»ã©ãããŠãã©ã°ã©ã³ãžã¢ã³ã¯äžå€ã§ãªããŠã¯ãªããªãããã®ãšãã
ãåŸãããããã®ãšãÎŽL = 0 ãšãªãããšãšèŠããã¹ããšã
ãšãªããéåéãæéçã«ä¿åããããšãåããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ã©ã°ã©ã³ãžã¢ã³ã¯ç©çç³»ã®å
šãŠã®æ
å ±ãæ
ã£ãŠããã®ã§ããããçšããŠæ§ã
ãªä¿ååã瀺ãããšãåºæ¥ããäŸãã°ããšãã«ã®ãŒä¿ååãšéåéä¿ååãäŸãšããŠæããããã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ãšãã«ã®ãŒã",
"title": "ãšãã«ã®ãŒä¿ååã®å°åº"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ã§å®çŸ©ããããã®è¡šåŒãšããã«ããã¢ã³",
"title": "ãšãã«ã®ãŒä¿ååã®å°åº"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ãèŠæ¯ã¹ããšãããã«ããã¢ã³ã¯ç³»ã®å
šãšãã«ã®ãŒã«å¯Ÿå¿ããããšãåãããéåéã®ä¿ååã¯ãã®ãšãã",
"title": "ãšãã«ã®ãŒä¿ååã®å°åº"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ãšãªãããšãã«ã®ãŒãæéçã«ä¿åããããšãåãããããã§ã4ãã5è¡ç®ã«ç§»ããšãéåæ¹çšåŒ",
"title": "ãšãã«ã®ãŒä¿ååã®å°åº"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ãçšãããå®éã«ã¯ããšãã«ã®ãŒã®ä¿ååã¯æéã®åç¹ãåããããšã«å¯ŸããŠç©çç³»ãå€åããªãããšã«ãã ã",
"title": "ãšãã«ã®ãŒä¿ååã®å°åº"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "éåéä¿ååã¯ç©çç³»å
šäœãå¹³è¡ç§»åããããšã«ãã£ãŠãç©çç³»ã®éåãå€åããªãããšã«ããããã®ããšã空éçäžæ§æ§ãšåŒã¶ããã®ãšãã©ã°ã©ã³ãžã¢ã³ã«å«ãŸããå
šãŠã®ããq ã«ã€ããŠ",
"title": "éåéä¿ååã®å°åº"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ãšãªãå€æãã»ã©ãããŠãã©ã°ã©ã³ãžã¢ã³ã¯äžå€ã§ãªããŠã¯ãªããªãããã®ãšãã",
"title": "éåéä¿ååã®å°åº"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "ãåŸãããããã®ãšãÎŽL = 0 ãšãªãããšãšèŠããã¹ããšã",
"title": "éåéä¿ååã®å°åº"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "ãšãªããéåéãæéçã«ä¿åããããšãåããã",
"title": "éåéä¿ååã®å°åº"
}
] | ã©ã°ã©ã³ãžã¢ã³ã¯ç©çç³»ã®å
šãŠã®æ
å ±ãæ
ã£ãŠããã®ã§ããããçšããŠæ§ã
ãªä¿ååã瀺ãããšãåºæ¥ããäŸãã°ããšãã«ã®ãŒä¿ååãšéåéä¿ååãäŸãšããŠæããããã | {{pathnav|ã¡ã€ã³ããŒãž|èªç¶ç§åŠ|ç©çåŠ|解æååŠ|frame=1}}
ã©ã°ã©ã³ãžã¢ã³ã¯ç©çç³»ã®å
šãŠã®æ
å ±ãæ
ã£ãŠããã®ã§ããããçšããŠæ§ã
ãªä¿ååã瀺ãããšãåºæ¥ããäŸãã°ããšãã«ã®ãŒä¿ååãšéåéä¿ååãäŸãšããŠæããããã
==ãšãã«ã®ãŒä¿ååã®å°åº==
ãšãã«ã®ãŒã
:<math>
E \equiv p \dot q - L
</math>
ã§å®çŸ©ããããã®è¡šåŒãšããã«ããã¢ã³
:<math>
H = p \dot q - L
</math>
ãèŠæ¯ã¹ããšãããã«ããã¢ã³ã¯ç³»ã®å
šãšãã«ã®ãŒã«å¯Ÿå¿ããããšãåãããéåéã®ä¿ååã¯ãã®ãšãã
:<math>
\begin{align}
\frac {\partial E}{\partial t } &= \frac {\partial {}}{\partial t }(p\dot q - L )\\
&=\frac {\partial {}}{\partial t } \left(\frac {\partial {L}}{\partial {\dot q} } \dot q\right) - \frac {\partial L}{\partial t }\\
&=\frac {\partial p}{\partial t } \dot q + p \frac {\partial {\dot q}}{\partial t } - \frac {\partial L}{\partial t }\\
&=\left(\frac {\partial {}}{\partial t } \frac {\partial {L}}{\partial {\dot q} } \right)\dot q
+\frac {\partial {L}}{\partial {\dot q} } \ddot q
- \frac {\partial L}{\partial t }\\
&= \left(\frac {\partial {L}}{\partial {q} } \dot q\right) +\frac {\partial {L}}{\partial {\dot q} } \ddot q - \frac {\partial L}{\partial t }\\
&= \frac {\partial L}{\partial t }- \frac {\partial L}{\partial t }\\
&= 0
\end{align}
</math>
ãšãªãããšãã«ã®ãŒãæéçã«ä¿åããããšãåãããããã§ã4ãã5è¡ç®ã«ç§»ããšãéåæ¹çšåŒ
:<math>
\frac {\partial L}{\partial q } - \frac {\partial {}}{\partial t }\frac {\partial L}{\partial {\dot q} }= 0
</math>
ãçšãããå®éã«ã¯ããšãã«ã®ãŒã®ä¿ååã¯æéã®åç¹ãåããããšã«å¯ŸããŠç©çç³»ãå€åããªãããšã«ãã<!-- ããšã®å°åº (?) -->
ã
==éåéä¿ååã®å°åº==
éåéä¿ååã¯ç©çç³»å
šäœãå¹³è¡ç§»åããããšã«ãã£ãŠãç©çç³»ã®éåãå€åããªãããšã«ããããã®ããšã空éçäžæ§æ§ãšåŒã¶ããã®ãšãã©ã°ã©ã³ãžã¢ã³ã«å«ãŸããå
šãŠã®ãã''q'' ã«ã€ããŠ
:<math>
q\rightarrow q+a ,\; \dot q \rightarrow \dot q
</math>
ãšãªãå€æãã»ã©ãããŠãã©ã°ã©ã³ãžã¢ã³ã¯äžå€ã§ãªããŠã¯ãªããªãããã®ãšãã
:<math>
\begin{align}
\delta L &= \delta q \frac {\partial L}{\partial q } + \delta \dot q \frac {\partial L}{\partial {\dot q} }\\
&= a \frac {\partial L}{\partial q }\\
&= a \frac {\partial {}}{\partial t } \frac {\partial {L }}{\partial {\dot q} } \\
&= a \frac {\partial p}{\partial t }
\end{align}
</math>
ãåŸãããããã®ãšãδ''L'' = 0 ãšãªãããšãšèŠããã¹ããšã
:<math>
\frac {\partial p}{\partial t } = 0
</math>
ãšãªããéåéãæéçã«ä¿åããããšãåããã
{{DEFAULTSORT:ãããããããã ã»ãããã}}
[[Category:解æååŠ|* ã»ãããã]] | null | 2015-04-17T15:27:49Z | [
"ãã³ãã¬ãŒã:Pathnav"
] | https://ja.wikibooks.org/wiki/%E8%A7%A3%E6%9E%90%E5%8A%9B%E5%AD%A6_%E4%BF%9D%E5%AD%98%E5%89%87%E3%81%AE%E5%B0%8E%E5%87%BA |
2,079 | é«çåŠæ ¡å°åŠ | æ°èª²çš
æ§èª²çš
æã
ã®åšãã«ã¯å®ã«å€ãã®èªç¶ãååšããŠãããäžãèŠãã°å°é¢ãããããäžãèŠãã°ç©ºãå®å®ãããããã®ãããªèªç¶ã¯ãã©ã®ããã«æ§æãããŠããã? ã©ã®ããã«ããŠã§ããã®ã? å°åŠã¯ãã®ãããªèªç¶ãååŠãç©çãçç©ã®åéããç·åçã«ç 究ããåŠåã§ããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "æ°èª²çš",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "æ§èª²çš",
"title": ""
},
{
"paragraph_id": 2,
"tag": "p",
"text": "æã
ã®åšãã«ã¯å®ã«å€ãã®èªç¶ãååšããŠãããäžãèŠãã°å°é¢ãããããäžãèŠãã°ç©ºãå®å®ãããããã®ãããªèªç¶ã¯ãã©ã®ããã«æ§æãããŠããã? ã©ã®ããã«ããŠã§ããã®ã? å°åŠã¯ãã®ãããªèªç¶ãååŠãç©çãçç©ã®åéããç·åçã«ç 究ããåŠåã§ããã",
"title": "å°åŠãšã¯"
}
] | æ°èª²çš æ§èª²çš | {{Pathnav|é«çåŠæ ¡ã®åŠç¿|é«çåŠæ ¡çç§|frame=1}}
{{Pathnav|èªç¶ç§åŠ|å°çç§åŠ|frame=1}}
{{é²æç¶æ³}}
æ°èª²çš
:[[é«çåŠæ ¡ãå°åŠåºç€|å°åŠåºç€]] {{é²æ|00%|2015-06-05}}
:[[é«çåŠæ ¡ å°åŠ|å°åŠ]] {{é²æ|25%|2022-10-26}}
æ§èª²çš
:[[é«çåŠæ ¡çç§åºç€å°åŠåé|çç§åºç€ å°åŠåé]] {{é²æ|25%|2015-06-05}}
:[[çç§ç·åB å°åŠåé]] {{é²æ|25%|2015-06-05}}
:[[å°åŠI|å°åŠI]] 3åäœ {{é²æ|25%|2015-06-05}}
:[[å°åŠII]] 3åäœ {{é²æ|25%|2015-06-05}}
==å°åŠãšã¯==
æã
ã®åšãã«ã¯å®ã«å€ãã®èªç¶ãååšããŠãããäžãèŠãã°å°é¢ãããããäžãèŠãã°ç©ºãå®å®ãããããã®ãããªèªç¶ã¯ãã©ã®ããã«æ§æãããŠãããïŒãã©ã®ããã«ããŠã§ããã®ãïŒãå°åŠã¯ãã®ãããªèªç¶ãååŠãç©çãçç©ã®åéããç·åçã«ç 究ããåŠåã§ããã
== åè ==
*[[åŠç¿æ¹æ³/é«æ ¡å°åŠ]]
*[[å°åŠæ ¡ã»äžåŠæ ¡ã»é«çåŠæ ¡ã®åŠç¿/æ€å®æç§æžã®è³Œå
¥æ¹æ³|æ€å®æç§æžã®è³Œå
¥æ¹æ³]]
[[en:High School Earth Science]]
[[Category:é«çåŠæ ¡æè²|å°*ã¡ãã]]
[[Category:çç§æè²|é«ã¡ãã]]
[[Category:å°çç§åŠ|é«*ã¡ãã]]
[[category:é«æ ¡çç§|ã¡ãã]] | null | 2022-10-29T01:43:51Z | [
"ãã³ãã¬ãŒã:Pathnav",
"ãã³ãã¬ãŒã:é²æç¶æ³",
"ãã³ãã¬ãŒã:é²æ"
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E5%9C%B0%E5%AD%A6 |
2,080 | ææ©ååŠ/ãšã¹ãã« | -CO-O-(ãšã¹ãã«çµå)ãæã€ååç©ããšã¹ãã«ãšãã(ãã ãR2ã¯æ°ŽçŽ ååHãé€ã)ã
ããããã·åºåã³ã«ã«ããã·åºã®éã§ã®è±æ°Žçž®åã
R1 -COOH + R2 -OH â R1-COO-R2 + H2O | [
{
"paragraph_id": 0,
"tag": "p",
"text": "-CO-O-(ãšã¹ãã«çµå)ãæã€ååç©ããšã¹ãã«ãšãã(ãã ãR2ã¯æ°ŽçŽ ååHãé€ã)ã",
"title": "ãšã¹ãã«ã®å®çŸ©"
},
{
"paragraph_id": 1,
"tag": "p",
"text": "",
"title": "ãšã¹ãã«ã®å®çŸ©"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ããããã·åºåã³ã«ã«ããã·åºã®éã§ã®è±æ°Žçž®åã",
"title": "åææ¹æ³"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "R1 -COOH + R2 -OH â R1-COO-R2 + H2O",
"title": "åææ¹æ³"
}
] | null | ==ãšã¹ãã«ã®å®çŸ©==
-CO-O-ïŒãšã¹ãã«çµåïŒãæã€ååç©ããšã¹ãã«ãšããïŒãã ãR2ã¯æ°ŽçŽ ååHãé€ãïŒã
O-R2
/
R1-C
\\
O
==åææ¹æ³==
ããããã·åºåã³ã«ã«ããã·åºã®éã§ã®è±æ°Žçž®åã
R<sub>1</sub> -CO'''OH''' + R<sub>2</sub> -O'''H''' â R<sub>1</sub>-COO-R<sub>2</sub> + H<sub>2</sub>O
[[ã«ããŽãª:ææ©ååŠ]] | null | 2022-11-23T05:33:00Z | [] | https://ja.wikibooks.org/wiki/%E6%9C%89%E6%A9%9F%E5%8C%96%E5%AD%A6/%E3%82%A8%E3%82%B9%E3%83%86%E3%83%AB |
2,081 | é«çåŠæ ¡çç§ | ãã®ããŒãžã¯ãé«æ ¡çç§ã®æç§æžã®æ¬æ£ã§ããæç§ãçç§ãã¯ä»¥äžã®ç§ç®ããæ§æãããŠããŸãã
é«çåŠæ ¡ã§ã¯ãç§åŠãšäººéç掻ããšåºç€ç³»ç§ç®1ç§ç®ãããã¯ãåºç€ç³»ç§ç®3ç§ç®ãå¿
ä¿®ãšãªã£ãŠããŸãã ãã ãããç§åŠãšäººéç掻ãã¯å€§åŠå
¥åŠå
±éãã¹ãã§ã¯åºé¡ãããªããã泚æãå¿
èŠã§ãã
é«çåŠæ ¡ã§ã¯ãçç§åºç€ãããçç§ç·åAãããçç§ç·åBãã®ãã¡1ç§ç®ãå«ã2ç§ç®ãå¿
ä¿®ã«ãªã£ãŠããŸãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ãã®ããŒãžã¯ãé«æ ¡çç§ã®æç§æžã®æ¬æ£ã§ããæç§ãçç§ãã¯ä»¥äžã®ç§ç®ããæ§æãããŠããŸãã",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "é«çåŠæ ¡ã§ã¯ãç§åŠãšäººéç掻ããšåºç€ç³»ç§ç®1ç§ç®ãããã¯ãåºç€ç³»ç§ç®3ç§ç®ãå¿
ä¿®ãšãªã£ãŠããŸãã ãã ãããç§åŠãšäººéç掻ãã¯å€§åŠå
¥åŠå
±éãã¹ãã§ã¯åºé¡ãããªããã泚æãå¿
èŠã§ãã",
"title": ""
},
{
"paragraph_id": 2,
"tag": "p",
"text": "é«çåŠæ ¡ã§ã¯ãçç§åºç€ãããçç§ç·åAãããçç§ç·åBãã®ãã¡1ç§ç®ãå«ã2ç§ç®ãå¿
ä¿®ã«ãªã£ãŠããŸãã",
"title": "æ§èª²çš"
}
] | ãã®ããŒãžã¯ãé«æ ¡çç§ã®æç§æžã®æ¬æ£ã§ããæç§ãçç§ãã¯ä»¥äžã®ç§ç®ããæ§æãããŠããŸãã é«çåŠæ ¡ã§ã¯ãç§åŠãšäººéç掻ããšåºç€ç³»ç§ç®1ç§ç®ãããã¯ãåºç€ç³»ç§ç®3ç§ç®ãå¿
ä¿®ãšãªã£ãŠããŸãã
ãã ãããç§åŠãšäººéç掻ãã¯å€§åŠå
¥åŠå
±éãã¹ãã§ã¯åºé¡ãããªããã泚æãå¿
èŠã§ãã | {{pathnav|é«çåŠæ ¡ã®åŠç¿|frame=1|small=1}}
ãã®ããŒãžã¯ãé«æ ¡çç§ã®æç§æžã®æ¬æ£ã§ããæç§ãçç§ãã¯ä»¥äžã®ç§ç®ããæ§æãããŠããŸãã
é«çåŠæ ¡ã§ã¯ãç§åŠãšäººéç掻ããšåºç€ç³»ç§ç®1ç§ç®ãããã¯ãåºç€ç³»ç§ç®3ç§ç®ãå¿
ä¿®ãšãªã£ãŠããŸãã
ãã ãããç§åŠãšäººéç掻ãã¯å€§åŠå
¥åŠå
±éãã¹ãã§ã¯åºé¡ãããªããã泚æãå¿
èŠã§ãã
== ç©ç ==
*[[é«çåŠæ ¡ç©ç]]
:*[[é«çåŠæ ¡ ç©çåºç€|ç©çåºç€]] 2åäœ
:*[[é«çåŠæ ¡ ç©ç|ç©ç]] 4åäœ
== ååŠ ==
*[[é«çåŠæ ¡ååŠ]]
:*[[é«çåŠæ ¡çç§ ååŠåºç€|ååŠåºç€]] 2åäœ {{é²æ|25%|2015-08-14}}
:*[[é«çåŠæ ¡ ååŠ|ååŠ]] 4åäœ
== çç© ==
*[[é«çåŠæ ¡çç©]]
:*[[é«çåŠæ ¡ çç©åºç€|çç©åºç€]] 2åäœ
:*[[é«çåŠæ ¡ çç©|çç©]] 4åäœ
== å°åŠ ==
*[[é«çåŠæ ¡å°åŠ]]
:*[[é«çåŠæ ¡ å°åŠåºç€|å°åŠåºç€]] 2åäœ
:*[[é«çåŠæ ¡ å°åŠ|å°åŠ]] 4åäœ
== ãã®ä»ã®ç§ç® ==
*[[é«çåŠæ ¡ ç§åŠãšäººéç掻|ç§åŠãšäººéç掻]]
* [[é«çåŠæ ¡ çæ°æ¢ç©¶åºç€|çæ°æ¢ç©¶åºç€]] 1åäœ{{é²æ|00%|2023-02-04}}<br />
== æ§èª²çš ==
é«çåŠæ ¡ã§ã¯ãçç§åºç€ãããçç§ç·åAãããçç§ç·åBãã®ãã¡1ç§ç®ãå«ã2ç§ç®ãå¿
ä¿®ã«ãªã£ãŠããŸãã
* [[é«çåŠæ ¡çç§åºç€|çç§åºç€]] 2åäœ {{é²æ|00%|2015-08-14}}
* [[é«çåŠæ ¡çç§ç·åA|çç§ç·åA]] 2åäœ {{é²æ|00%|2015-08-14}}
* [[é«çåŠæ ¡çç§ç·åB|çç§ç·åB]] 2åäœ {{é²æ|25%|2015-08-14}}
* [[é«çåŠæ ¡ç©ç]] 3åäœ {{é²æ|25%|2015-08-14}}
:* [[é«çåŠæ ¡ç©ç/ç©çI|ç©çI]] 3åäœ {{é²æ|25%|2015-08-14}}
:* [[é«çåŠæ ¡ç©ç/ç©çII|ç©çII]] 3åäœ {{é²æ|25%|2015-08-14}}
* [[é«çåŠæ ¡ååŠ]] {{é²æ|25%|2015-08-14}}
:* [[é«çåŠæ ¡ååŠI|ååŠI]] 3åäœ {{é²æ|25%|2015-08-14}}
:* [[é«çåŠæ ¡ååŠII|ååŠII]] 3åäœ {{é²æ|25%|2015-08-14}}
* [[é«çåŠæ ¡çç©]] {{é²æ|50%|2015-08-14}}
:* [[é«çåŠæ ¡çç©/çç©I|çç©I]] 3åäœ {{é²æ|50%|2015-08-14}}
:* [[é«çåŠæ ¡çç©/çç©II|çç©II]] 3åäœ {{é²æ|50%|2015-08-14}}
* [[é«çåŠæ ¡å°åŠ]] {{é²æ|25%|2015-08-14}}
:* [[å°åŠI]] 3åäœ {{é²æ|25%|2015-08-14}}
:* [[å°åŠII]] 3åäœ {{é²æ|25%|2015-08-14}}
[[Category:é«çåŠæ ¡æè²|ç*]]
[[Category:çç§æè²|é«*]]
[[category:é«æ ¡çç§|*]] | 2005-06-01T07:25:16Z | 2023-10-29T06:12:51Z | [
"ãã³ãã¬ãŒã:Pathnav",
"ãã³ãã¬ãŒã:é²æ"
] | https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E7%90%86%E7%A7%91 |
2,082 | ç ç® | æ°åŠ>ç ç®
倧åŠã®æç§æž èªç¶ç§åŠ: æ°åŠ - ç©çåŠ; å€å
žååŠ éåååŠ - ååŠ; ç¡æ©ååŠ ææ©ååŠ - çç©åŠ; æ€ç©åŠ ç 究æè¡ - å°çç§åŠ - å»åŠ; 解ååŠ èªåŠ: æ¥æ¬èª è±èª ãšã¹ãã©ã³ã æé®®èª ãã³ããŒã¯èª ãã€ãèª ãã©ã³ã¹èª ã©ãã³èª ã«ãŒããã¢èª 人æç§åŠ: æŽå²åŠ; æ¥æ¬å² äžåœå² äžçå² æŽå²èŠ³ - å¿çåŠ - å²åŠ - èžè¡; é³æ¥œ çŸè¡ - æåŠ; å€å
žæåŠ æŒ¢è©© 瀟äŒç§åŠ: æ³åŠ - çµæžåŠ - å°çåŠ - æè²åŠ; åŠæ ¡æè² æè²å² æ
å ±æè¡: æ
å ±å·¥åŠ; MS-DOS/PC DOS UNIX/Linux TeX/LaTeX CGI - ããã°ã©ãã³ã°; BASIC Cèšèª C++ Dèšèª HTML Java JavaScript Lisp Mizar Perl PHP Python Ruby Scheme SVG å°ã»äžã»é«æ ¡ã®æç§æž å°åŠ: åœèª ç€ŸäŒ ç®æ° çç§ è±èª äžåŠ: åœèª ç€ŸäŒ æ°åŠ çç§ è±èª é«æ ¡: åœèª - å°æŽ - å
¬æ° - æ°åŠ; å
¬åŒé - çç§; ç©ç ååŠ å°åŠ çç© - å€åœèª - æ
å ± 解説æžã»å®çšæžã»åèæž è¶£å³: æçæ¬ - ã¹ããŒã - ã²ãŒã è©Šéš: è³æ Œè©Šéš - å
¥åŠè©Šéš ãã®ä»ã®æ¬: é²çœ - ç掻ãšé²è·¯ - ãŠã£ãããã£ã¢ã®æžãæ¹ - ãžã§ãŒã¯é
æãé»åããªãã£ãæ代ãèšç®ã«ã¯ããã°ããçšããããŠããŸããããã®æç§æžã§ã¯ãããã°ããçšããèšç®æ³ãç ç®ãã«ã€ããŠè§£èª¬ããŸãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "æ°åŠ>ç ç®",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "倧åŠã®æç§æž èªç¶ç§åŠ: æ°åŠ - ç©çåŠ; å€å
žååŠ éåååŠ - ååŠ; ç¡æ©ååŠ ææ©ååŠ - çç©åŠ; æ€ç©åŠ ç 究æè¡ - å°çç§åŠ - å»åŠ; 解ååŠ èªåŠ: æ¥æ¬èª è±èª ãšã¹ãã©ã³ã æé®®èª ãã³ããŒã¯èª ãã€ãèª ãã©ã³ã¹èª ã©ãã³èª ã«ãŒããã¢èª 人æç§åŠ: æŽå²åŠ; æ¥æ¬å² äžåœå² äžçå² æŽå²èŠ³ - å¿çåŠ - å²åŠ - èžè¡; é³æ¥œ çŸè¡ - æåŠ; å€å
žæåŠ æŒ¢è©© 瀟äŒç§åŠ: æ³åŠ - çµæžåŠ - å°çåŠ - æè²åŠ; åŠæ ¡æè² æè²å² æ
å ±æè¡: æ
å ±å·¥åŠ; MS-DOS/PC DOS UNIX/Linux TeX/LaTeX CGI - ããã°ã©ãã³ã°; BASIC Cèšèª C++ Dèšèª HTML Java JavaScript Lisp Mizar Perl PHP Python Ruby Scheme SVG å°ã»äžã»é«æ ¡ã®æç§æž å°åŠ: åœèª ç€ŸäŒ ç®æ° çç§ è±èª äžåŠ: åœèª ç€ŸäŒ æ°åŠ çç§ è±èª é«æ ¡: åœèª - å°æŽ - å
¬æ° - æ°åŠ; å
¬åŒé - çç§; ç©ç ååŠ å°åŠ çç© - å€åœèª - æ
å ± 解説æžã»å®çšæžã»åèæž è¶£å³: æçæ¬ - ã¹ããŒã - ã²ãŒã è©Šéš: è³æ Œè©Šéš - å
¥åŠè©Šéš ãã®ä»ã®æ¬: é²çœ - ç掻ãšé²è·¯ - ãŠã£ãããã£ã¢ã®æžãæ¹ - ãžã§ãŒã¯é",
"title": ""
},
{
"paragraph_id": 2,
"tag": "p",
"text": "æãé»åããªãã£ãæ代ãèšç®ã«ã¯ããã°ããçšããããŠããŸããããã®æç§æžã§ã¯ãããã°ããçšããèšç®æ³ãç ç®ãã«ã€ããŠè§£èª¬ããŸãã",
"title": ""
}
] | æ°åŠïŒç ç® æãé»åããªãã£ãæ代ãèšç®ã«ã¯ããã°ããçšããããŠããŸããããã®æç§æžã§ã¯ãããã°ããçšããèšç®æ³ãç ç®ãã«ã€ããŠè§£èª¬ããŸãã ç ç®_åºç€ç¥è (2005-06-01)
ç ç®_å æžç® (2005-06-02)
ç ç®_ä¹ç® (2005-06-05)
ç ç®_é€ç® (2005-06-05)
ç ç®_èŠåç®ã»èªäžç®ã»äŒç¥šç® (2005-06-05)
ç ç®_éå¹³ã»éç«
ç ç®_æŒç¿ | [[æ°åŠ]]ïŒç ç®
{{é²æç¶æ³}}
{{èµæžäžèŠ§}}
æãé»åããªãã£ãæ代ãèšç®ã«ã¯ããã°ããçšããããŠããŸããããã®æç§æžã§ã¯ãããã°ããçšããèšç®æ³ãç ç®ãã«ã€ããŠè§£èª¬ããŸãã
*[[ç ç®_åºç€ç¥è]]{{é²æ|75%|2005-06-01}}
*[[ç ç®_å æžç®]]{{é²æ|75%|2005-06-02}}
*[[ç ç®_ä¹ç®]]{{é²æ|75%|2005-06-05}}
*[[ç ç®_é€ç®]]{{é²æ|75%|2005-06-05}}
*[[ç ç®_èŠåç®ã»èªäžç®ã»äŒç¥šç®]]{{é²æ|25%|2005-06-05}}
*[[ç ç®_éå¹³ã»éç«]]
*[[ç ç®_æŒç¿]]
==é¢ä¿å
ãªã³ã¯==
*[http://www.syuzan.net/ æ¥æ¬ç ç®é£ç]
*[http://www.soroban.or.jp/ å
šåœç ç®æè²é£ç]
*[http://shuzan-gakko.com/ å
šåœç ç®åŠæ ¡é£ç]
[[Category:æžåº«|ãããã]]
[[Category:ç ç®|*]] | null | 2006-12-12T15:30:26Z | [
"ãã³ãã¬ãŒã:é²æç¶æ³",
"ãã³ãã¬ãŒã:èµæžäžèŠ§"
] | https://ja.wikibooks.org/wiki/%E7%8F%A0%E7%AE%97 |
2,083 | ç ç® åºç€ç¥è | æ°åŠ>ç ç®>åºç€ç¥è
ç®ç€(ããã°ã)ã«ã¯ã1ã€ã®è»žã«çã5(=1+4)åãã4çç®ç€ãšã6(=1+5)åãã5çç®ç€ãããã äžè¬çã«ã¯4çç®ç€ã䜿ãããã5çç®ç€ã¯60é²æ³ã®èšç®(äŸãã°æéãªã©)ãããã®ã«äŸ¿å©ã§ããã 以äžå
šãŠã4çç®ç€ã§èª¬æããã
çã1åã®æ¹ãäžã§ããã
ç®ç€ã«ã¯ã3æ¡ããšã«ç¹ãæ¯ã£ãŠãããæ®éãç¹ã®çäžã«äžã®äœããšããããããå·Šã«åã®äœãçŸã®äœ...ãšç¶ãã ç¹ã¯èªåã®äœ¿ãããããšããã§ããŸããªãããããŸãã«å·Šããããšèšç®ã§ããªããªãããšãããã®ã§æ³šæãããã
ç®ç€ã䜿ãåã«ãå
šãŠã®æ¡ã0ã«ãªã»ããããäœæ¥ãå¿
èŠã§ãããé»åã§èšãã°ãCãã§ããã
ããã§äžã®å³ã®ãããªç¶æ
ã«ãªã£ãŠããã¯ãã§ããã
ããŠã1ãã9ãŸã§ã®æ°ãå
¥ããŠã¿ãããå·Šæã§ç®ç€ãæã¡ãå³æã®èŠªæã§äžã®çãã人差ãæã§äžã®å
šãŠã®çãæ±ãã
äžäžã«äœæ¡ã«ãªã£ãŠãåãã§ãããäŸãã°ã1013.25ãªã
ã§ããã
次ã¯è¶³ãç®ãåŒãç®ããã£ãŠã¿ããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "æ°åŠ>ç ç®>åºç€ç¥è",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ç®ç€(ããã°ã)ã«ã¯ã1ã€ã®è»žã«çã5(=1+4)åãã4çç®ç€ãšã6(=1+5)åãã5çç®ç€ãããã äžè¬çã«ã¯4çç®ç€ã䜿ãããã5çç®ç€ã¯60é²æ³ã®èšç®(äŸãã°æéãªã©)ãããã®ã«äŸ¿å©ã§ããã 以äžå
šãŠã4çç®ç€ã§èª¬æããã",
"title": "ç®ç€ã®åºç€ç¥è"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "çã1åã®æ¹ãäžã§ããã",
"title": "æ°ã®å
¥ãæ¹"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ç®ç€ã«ã¯ã3æ¡ããšã«ç¹ãæ¯ã£ãŠãããæ®éãç¹ã®çäžã«äžã®äœããšããããããå·Šã«åã®äœãçŸã®äœ...ãšç¶ãã ç¹ã¯èªåã®äœ¿ãããããšããã§ããŸããªãããããŸãã«å·Šããããšèšç®ã§ããªããªãããšãããã®ã§æ³šæãããã",
"title": "æ°ã®å
¥ãæ¹"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ç®ç€ã䜿ãåã«ãå
šãŠã®æ¡ã0ã«ãªã»ããããäœæ¥ãå¿
èŠã§ãããé»åã§èšãã°ãCãã§ããã",
"title": "æ°ã®å
¥ãæ¹"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ããã§äžã®å³ã®ãããªç¶æ
ã«ãªã£ãŠããã¯ãã§ããã",
"title": "æ°ã®å
¥ãæ¹"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ããŠã1ãã9ãŸã§ã®æ°ãå
¥ããŠã¿ãããå·Šæã§ç®ç€ãæã¡ãå³æã®èŠªæã§äžã®çãã人差ãæã§äžã®å
šãŠã®çãæ±ãã",
"title": "æ°ã®å
¥ãæ¹"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "äžäžã«äœæ¡ã«ãªã£ãŠãåãã§ãããäŸãã°ã1013.25ãªã",
"title": "æ°ã®å
¥ãæ¹"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "ã§ããã",
"title": "æ°ã®å
¥ãæ¹"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "次ã¯è¶³ãç®ãåŒãç®ããã£ãŠã¿ããã",
"title": "æ°ã®å
¥ãæ¹"
}
] | æ°åŠïŒç ç®ïŒåºç€ç¥è | [[æ°åŠ]]ïŒ[[ç ç®]]ïŒåºç€ç¥è
==ç®ç€ã®åºç€ç¥è==
ç®ç€(ããã°ã)ã«ã¯ã1ã€ã®è»žã«çã5(ïŒ1+4)åãã4çç®ç€ãšã6(ïŒ1+5)åãã5çç®ç€ãããã
äžè¬çã«ã¯4çç®ç€ã䜿ãããã5çç®ç€ã¯60é²æ³ã®èšç®(äŸãã°æéãªã©)ãããã®ã«äŸ¿å©ã§ããã
以äžå
šãŠã4çç®ç€ã§èª¬æããã
==æ°ã®å
¥ãæ¹==
çã1åã®æ¹ãäžã§ããã
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
====*========*========*========*========*========*====
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
ç®ç€ã«ã¯ã3æ¡ããšã«ç¹ãæ¯ã£ãŠãããæ®éãç¹ã®çäžã«äžã®äœããšããããããå·Šã«åã®äœãçŸã®äœ...ãšç¶ãã
ç¹ã¯èªåã®äœ¿ãããããšããã§ããŸããªãããããŸãã«å·Šããããšèšç®ã§ããªããªãããšãããã®ã§æ³šæãããã
ç®ç€ã䜿ãåã«ãå
šãŠã®æ¡ã0ã«ãªã»ããããäœæ¥ãå¿
èŠã§ãããé»åã§èšãã°ãCãã§ããã
#å·Šæã§ç®ç€ã®å·Šé
ã®ã»ããæã¡ãäžã®æ¹ãæåã«åŸããã
#æºã®äžã«éãã«æ»ãã
#å³æã®äººå·®ãæãã巊端ããé çªã«ãäžã®çãäžç«¯ã«è¡ãããã«æšªã«ã¹ã©ã€ããããã
ããã§äžã®å³ã®ãããªç¶æ
ã«ãªã£ãŠããã¯ãã§ããã
ããŠã1ãã9ãŸã§ã®æ°ãå
¥ããŠã¿ãããå·Šæã§ç®ç€ãæã¡ãå³æã®èŠªæã§äžã®çãã人差ãæã§äžã®å
šãŠã®çãæ±ãã
< > < > < > < > < >
=== === === === ===
< > < > < > < >
< > < > < > < >
< > < > < > < >
< > < > < > < >
< > < > < > < >
0 1 2 3 4
< > < > < > < > < >
=== === === === ===
< > < > < > < >
< > < > < > < >
< > < > < > < >
< > < > < > < >
< > < > < > < >
5 6 7 8 9
äžäžã«äœæ¡ã«ãªã£ãŠãåãã§ãããäŸãã°ã1013.25ãªã
< >< >< >< >< >
< >
=*========*=======
< > < >< >< >
< > < >< >< >
< >< >< >< > < >
< >< >< > < >< >
< >< >< >< >< >< >
1 0 1 3 .2 5
ã§ããã
次ã¯[[ç ç®_å æžç®|足ãç®ãåŒãç®]]ããã£ãŠã¿ããã
[[Category:ç ç®|ããã¡ãã]] | null | 2006-12-12T15:28:41Z | [] | https://ja.wikibooks.org/wiki/%E7%8F%A0%E7%AE%97_%E5%9F%BA%E7%A4%8E%E7%9F%A5%E8%AD%98 |
2,084 | ç ç® å æžç® | æ°åŠ>ç ç®>å æžç®
1+3ã2+6ãªã©ã®èšç®ã¯å
ã®æ°ãå
¥ããããã«å ããæ°ã®çãå
¥ããã
以äžãå ããæ°ãnãšããã
3+4ã9+2ã¯ãã®ãŸãŸã§ã¯åºæ¥ãªããããã§ãŸã5(10)-nãåŒãããããŠãã®åŸ5(10)ã足ããšããæ¹æ³ããšãã
確ãã«n足ããããšã«ãªãã
6+8ãªã©ã¯ä»ãŸã§ã®æ¹æ³ã§ã¯åºæ¥ãªãããããã®èšç®ãããã«ã¯ãŸãn-5ã足ãããã®åŸã5ãåŒã10ãå ããã
ã§ã確ãã«n足ããããšã«ãªãã
äžæ¡ã®è¶³ãç®ã®ãã¹ãŠã®ããæ¹ããŸãšããŠããã
åŒãç®ã¯è¶³ãç®ã®éãããã°è¯ãã ãã§ããã
å
·äœçã«ã¯ã
ç®ç€ã¯çç®ãšéã£ãŠäžã®äœããèšç®ããã ç¹°ãäžãããæ¡ééãã«ãã泚æããã°ãäœæ¡ã«ãªã£ãŠãåºæ¬ã¯åãã§ããã
次ã®èšç®ãããŠã¿ãã
å æžç®ã®åºæ¬ãåºæ¥ãããèŠåç®ãªã©ã«ãææŠãããã ãããã¯æŽã«æãç®ãåŠãŒãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "æ°åŠ>ç ç®>å æžç®",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "1+3ã2+6ãªã©ã®èšç®ã¯å
ã®æ°ãå
¥ããããã«å ããæ°ã®çãå
¥ããã",
"title": "äžæ¡ã®è¶³ãç®"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "以äžãå ããæ°ãnãšããã",
"title": "äžæ¡ã®è¶³ãç®"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "3+4ã9+2ã¯ãã®ãŸãŸã§ã¯åºæ¥ãªããããã§ãŸã5(10)-nãåŒãããããŠãã®åŸ5(10)ã足ããšããæ¹æ³ããšãã",
"title": "äžæ¡ã®è¶³ãç®"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "確ãã«n足ããããšã«ãªãã",
"title": "äžæ¡ã®è¶³ãç®"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "6+8ãªã©ã¯ä»ãŸã§ã®æ¹æ³ã§ã¯åºæ¥ãªãããããã®èšç®ãããã«ã¯ãŸãn-5ã足ãããã®åŸã5ãåŒã10ãå ããã",
"title": "äžæ¡ã®è¶³ãç®"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ã§ã確ãã«n足ããããšã«ãªãã",
"title": "äžæ¡ã®è¶³ãç®"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "äžæ¡ã®è¶³ãç®ã®ãã¹ãŠã®ããæ¹ããŸãšããŠããã",
"title": "äžæ¡ã®è¶³ãç®"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "åŒãç®ã¯è¶³ãç®ã®éãããã°è¯ãã ãã§ããã",
"title": "äžæ¡ã®åŒãç®"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "å
·äœçã«ã¯ã",
"title": "äžæ¡ã®åŒãç®"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "ç®ç€ã¯çç®ãšéã£ãŠäžã®äœããèšç®ããã ç¹°ãäžãããæ¡ééãã«ãã泚æããã°ãäœæ¡ã«ãªã£ãŠãåºæ¬ã¯åãã§ããã",
"title": "è€æ°æ¡ã®å æžç®"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "次ã®èšç®ãããŠã¿ãã",
"title": "è€æ°æ¡ã®å æžç®"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "å æžç®ã®åºæ¬ãåºæ¥ãããèŠåç®ãªã©ã«ãææŠãããã ãããã¯æŽã«æãç®ãåŠãŒãã",
"title": "è€æ°æ¡ã®å æžç®"
}
] | æ°åŠïŒç ç®ïŒå æžç® | [[æ°åŠ]]ïŒ[[ç ç®]]ïŒå æžç®
==äžæ¡ã®è¶³ãç®==
===ãã®ãŸãŸå ãã===
1+3ã2+6ãªã©ã®èšç®ã¯å
ã®æ°ãå
¥ããããã«å ããæ°ã®çãå
¥ããã
< > < > < > < > |-< >
| < >
=== === === === | === ===
< > < > < > < > 6 < > < >
< > < > | < > < >
< > |-< > < > | < >
< > 3-< > < > < > |-< >
< > |-< > < > < > < >
1 + 3 = 4 2 + 6 = 8
===åŒããŠããå ãã===
以äžãå ããæ°ãnãšããã
3+4ã9+2ã¯ãã®ãŸãŸã§ã¯åºæ¥ãªããããã§ãŸã5(10)-nãåŒãããããŠãã®åŸ5(10)ã足ããšããæ¹æ³ããšãã
*<math>-(5-n)+5 = +n</math>
*<math>-(10-n)+10 = +n</math>
確ãã«n足ããããšã«ãªãã
< > < > 5-< > < > < > < >< > < >< >
< > < > < >-|
=== === === === ====*= ====*= | ====*= ====*=
< > < > < > < > < > < > | < > < >< >
< > < > < > < > < >< > < >-8 1-< >
< > 1-< > < >< > < >< >-| < >< > < >< >
< > < > < >< > < >< >-| < >< > < >< >
< > < > < > < > < > < > < >< > < >< >
3 (- 1 + 5) = 7 0 9 (- 0 8 + 1 0) = 1 1
| | | |
---- +4 ---- ã------- +2 --------
===足ããŠåŒããŠè¶³ã===
6+8ãªã©ã¯ä»ãŸã§ã®æ¹æ³ã§ã¯åºæ¥ãªãããããã®èšç®ãããã«ã¯ãŸãn-5ã足ãããã®åŸã5ãåŒã10ãå ããã
*<math>+(n-5)-5+10 = +n</math>
ã§ã確ãã«n足ããããšã«ãªãã
< > < > < > < >< > < >< >
< > < > < >-5
====*= ====*= ====*= ====*= ====*=
< > < >< > < > < > < >< >
< > < > < >< > 1-< >< > < >
< >< > < >-| < >< > < >< > < >< >
< >< > < >< >-3 < >< > < >< > < >< >
< >< > < >< >-| < > < > < >
0 6 (+ 0 3 - 0 5 + 1 0) = 1 4
| |
----------- +8 ------------
===äžæ¡ã®è¶³ãç®ã®ãŸãšã===
äžæ¡ã®è¶³ãç®ã®ãã¹ãŠã®ããæ¹ããŸãšããŠããã
A:ãã®ãŸãŸè¶³ã
B:5-nåŒããŠ5足ã
C:10-nåŒããŠ10足ã
D:n-5足ããŠ5åŒããŠ10足ã
+| 1 2 3 4 5 6 7 8 9
---------------------
1| A A A B A A A A C
2| A A B B A A A C C
3| A B B B A A C C C
4| B B B B A C C C C
5| A A A A C D D D D
6| A A A C C D D D C
7| A A C C C D D C C
8| A C C C C D C C C
9| C C C C C C C C C
==äžæ¡ã®åŒãç®==
åŒãç®ã¯è¶³ãç®ã®éãããã°è¯ãã ãã§ããã
å
·äœçã«ã¯ã
*ãã®ãŸãŸåŒã
*5åŒããŠ5-n足ã
*10åŒããŠ10-n足ã
*10åŒããŠ5足ããŠn-5åŒã
==è€æ°æ¡ã®å æžç®==
ç®ç€ã¯çç®ãšéã£ãŠäžã®äœããèšç®ããã
ç¹°ãäžãããæ¡ééãã«ãã泚æããã°ãäœæ¡ã«ãªã£ãŠãåºæ¬ã¯åãã§ããã
次ã®èšç®ãããŠã¿ãã
*Q1.13579+8642=
*Q2.31415-27182=
A1.
< >< > < >< > < >< >< >
< >< >< > < >< >< > < >< >
====*========*= ====*========*= ====*========*=
< >< > < >< > < >< > < >< > < >< >< >< >< >
< >< >< >< > < > < >< >< > < >< > < >< >
< >< >< > < > < >< > < > < > < >
< > < >< >< > < >< >< >< >< > < >< >< >< >< >
< >< >< >< > < >< >< >< > < >< >< >< >
1 3 5 7 9 + 0 8 0 0 0 + 0 0 6 0 0
< >< >< >< > < >< >< >< >< > < >< >< >< >< >
< >
====*========*= ====*========*= ====*========*=
< >< >< >< >< > < >< >< >< >< > < >< >< >< >< >
< >< >< > < > < >< >< >< > < >< >< >< >
< >< > < > < >
< >< >< >< >< > < >< >< >< >< > < >< >< >< >< >
< >< >< >< > < >< >< >< >< > < >< >< >< >< >
+0 0 0 4 0 + 0 0 0 0 2 = 2 2 2 2 1
A2.
< >< >< >< > < >< >< >< > < >< >< >< >
< > < > < >
====*========*= ====*========*= ====*========*=
< >< >< >< > < >< >< >< > < >< >< >
< > < > < > < > < > < >< >< > < >
< >< >< >< >< > < >< >< >< > < >< >< >< >< >
< >< >< >< > < >< >< >< >< > < >< >< >< >< >
< >< > < >< > < >< > < >< > < > < >< >
3 1 4 1 5 - 2 0 0 0 0 - 0 7 0 0 0
< >< >< >< > < >< >< >< > < >< >< >< >< >
< > < >
====*========*= ====*========*= ====*========*=
< >< >< > < >< >< > < >< >< >< >
< >< >< > < > < >< >< >< >< > < >< >< >< >< >
< >< >< >< >< > < >< > < >< > < >< > < >< >
< >< > < >< > < >< >< > < > < >< >< >
< > < >< >< > < > < >< >< > < > < >< >< >
-0 0 1 0 0 - 0 0 0 8 0 - 0 0 0 0 2
< >< >< >< >< >
====*========*=
< >< >< >< >
< >< >< >< >< >
< >< > < >< >
< >< >< >
< > < >< >< >
-0 4 2 3 3
å æžç®ã®åºæ¬ãåºæ¥ããã[[ç ç®_èŠåç®ã»èªäžç®ã»äŒç¥šç®|èŠåç®ãªã©]]ã«ãææŠãããã
ãããã¯æŽã«[[ç ç®_ä¹ç®|æãç®]]ãåŠãŒãã
[[Category:ç ç®|ããããã]] | null | 2016-07-10T14:08:48Z | [] | https://ja.wikibooks.org/wiki/%E7%8F%A0%E7%AE%97_%E5%8A%A0%E6%B8%9B%E7%AE%97 |
2,090 | ç¡æ©ååŠã®åºç€/ååã®æ§é | ç¡æ©ååŠ>ç¡æ©ååŠã®åºç€>ååã®æ§é
ååã¯ååæ žãšé»åããæãç«ã£ãŠãããååæ žã¯ããã«éœåãšäžæ§åãåºãŸã£ãŠã§ããŠããã é»åã¯ååæ žã®åšãããã¯ãŒãã³å(éé»æ°å)ãåå¿åãšããŠãè¡æã®ããã«åã£ãŠãããšå€å
žååŠçã«ã¯è§£éãããããªããé»åã¯éœåãäžæ§åã®ããã1800åã®1ã®è³ªéãããããªãããããã®ã¢ãã«ã§ã¯ã倪éœãšå°çã®é¢ä¿ã®ããã«ãååæ žãäžå¿ã«é»åãå
¬è»¢ããŠããŠååæ žã¯äžåã§ãããã®ãšã¿ãªãã
éœå㯠+ e {\displaystyle +e} ãé»å㯠â e {\displaystyle -e} ã®é»è·ã垯ã³ãŠããããã㧠e {\displaystyle e} ã¯é»æ°çŽ é( e = 1.6 à 10 â 19 {\displaystyle e=1.6\times 10^{-19}} C)ã§ããããã®ãããå®å®ã«ãªãããã«(ååå
šäœãšããŠé»æ°çã«äžæ§ãšãªãããã«)ååå
ã®éœåæ°ãšé»åæ°ã¯çãããéœåæ°ãååçªå·ãšãèšããããªãã¡ååã¯éœåã®æ°ã«ãã£ãŠç¹åŸŽã¥ããããåšæè¡šã¯éœåã®æ°ã®é ã«äžŠãã§ãããäŸãã°ãéœåæ°ã1ãªãæ°ŽçŽ ã2ãªãããªãŠã ã3ãªããªããŠã ãšãã£ãå
·åã§ããã äžæ§åã«ã¯é»æ°çã«äžæ§ã§ãããé»è·ã¯ç¡ããéœåãè€æ°ããå Žåãããªãã¡ååçªå·ã2以äžã®å Žåãéœåå士ãé»æ°çã«å€§ããªåçºåãæã€ããäžæ§åãç³ã®åœ¹å²ãæãããŠããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ç¡æ©ååŠ>ç¡æ©ååŠã®åºç€>ååã®æ§é ",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ååã¯ååæ žãšé»åããæãç«ã£ãŠãããååæ žã¯ããã«éœåãšäžæ§åãåºãŸã£ãŠã§ããŠããã é»åã¯ååæ žã®åšãããã¯ãŒãã³å(éé»æ°å)ãåå¿åãšããŠãè¡æã®ããã«åã£ãŠãããšå€å
žååŠçã«ã¯è§£éãããããªããé»åã¯éœåãäžæ§åã®ããã1800åã®1ã®è³ªéãããããªãããããã®ã¢ãã«ã§ã¯ã倪éœãšå°çã®é¢ä¿ã®ããã«ãååæ žãäžå¿ã«é»åãå
¬è»¢ããŠããŠååæ žã¯äžåã§ãããã®ãšã¿ãªãã",
"title": ""
},
{
"paragraph_id": 2,
"tag": "p",
"text": "éœå㯠+ e {\\displaystyle +e} ãé»å㯠â e {\\displaystyle -e} ã®é»è·ã垯ã³ãŠããããã㧠e {\\displaystyle e} ã¯é»æ°çŽ é( e = 1.6 à 10 â 19 {\\displaystyle e=1.6\\times 10^{-19}} C)ã§ããããã®ãããå®å®ã«ãªãããã«(ååå
šäœãšããŠé»æ°çã«äžæ§ãšãªãããã«)ååå
ã®éœåæ°ãšé»åæ°ã¯çãããéœåæ°ãååçªå·ãšãèšããããªãã¡ååã¯éœåã®æ°ã«ãã£ãŠç¹åŸŽã¥ããããåšæè¡šã¯éœåã®æ°ã®é ã«äžŠãã§ãããäŸãã°ãéœåæ°ã1ãªãæ°ŽçŽ ã2ãªãããªãŠã ã3ãªããªããŠã ãšãã£ãå
·åã§ããã äžæ§åã«ã¯é»æ°çã«äžæ§ã§ãããé»è·ã¯ç¡ããéœåãè€æ°ããå Žåãããªãã¡ååçªå·ã2以äžã®å Žåãéœåå士ãé»æ°çã«å€§ããªåçºåãæã€ããäžæ§åãç³ã®åœ¹å²ãæãããŠããã",
"title": ""
}
] | ç¡æ©ååŠïŒç¡æ©ååŠã®åºç€ïŒååã®æ§é ååã¯ååæ žãšé»åããæãç«ã£ãŠãããååæ žã¯ããã«éœåãšäžæ§åãåºãŸã£ãŠã§ããŠããã
é»åã¯ååæ žã®åšãããã¯ãŒãã³åïŒéé»æ°åïŒãåå¿åãšããŠãè¡æã®ããã«åã£ãŠãããšå€å
žååŠçã«ã¯è§£éãããããªããé»åã¯éœåãäžæ§åã®ããã1800åã®1ã®è³ªéãããããªãããããã®ã¢ãã«ã§ã¯ã倪éœãšå°çã®é¢ä¿ã®ããã«ãååæ žãäžå¿ã«é»åãå
¬è»¢ããŠããŠååæ žã¯äžåã§ãããã®ãšã¿ãªãã éœå㯠+ e ãé»å㯠â e ã®é»è·ã垯ã³ãŠããããã㧠e ã¯é»æ°çŽ éã§ããããã®ãããå®å®ã«ãªãããã«ïŒååå
šäœãšããŠé»æ°çã«äžæ§ãšãªãããã«ïŒååå
ã®éœåæ°ãšé»åæ°ã¯çãããéœåæ°ãååçªå·ãšãèšããããªãã¡ååã¯éœåã®æ°ã«ãã£ãŠç¹åŸŽã¥ããããåšæè¡šã¯éœåã®æ°ã®é ã«äžŠãã§ãããäŸãã°ãéœåæ°ã1ãªãæ°ŽçŽ ã2ãªãããªãŠã ã3ãªããªããŠã ãšãã£ãå
·åã§ããã
äžæ§åã«ã¯é»æ°çã«äžæ§ã§ãããé»è·ã¯ç¡ããéœåãè€æ°ããå Žåãããªãã¡ååçªå·ã2以äžã®å Žåãéœåå士ãé»æ°çã«å€§ããªåçºåãæã€ããäžæ§åãç³ã®åœ¹å²ãæãããŠããã | [[ç¡æ©ååŠ]]ïŒ[[ç¡æ©ååŠã®åºç€]]ïŒååã®æ§é
ååã¯'''ååæ ž'''ãš'''é»å'''ããæãç«ã£ãŠãããååæ žã¯ããã«'''éœå'''ãš'''äžæ§å'''ãåºãŸã£ãŠã§ããŠããã
é»åã¯ååæ žã®åšãããã¯ãŒãã³åïŒéé»æ°åïŒãåå¿åãšããŠãè¡æã®ããã«åã£ãŠãããšå€å
žååŠçã«ã¯è§£éãããããªããé»åã¯éœåãäžæ§åã®ããã1800åã®1ã®è³ªéãããããªãããããã®ã¢ãã«ã§ã¯ã倪éœãšå°çã®é¢ä¿ã®ããã«ãååæ žãäžå¿ã«é»åãå
¬è»¢ããŠããŠååæ žã¯äžåã§ãããã®ãšã¿ãªãã
éœåã¯<math>+e</math>ãé»åã¯<math>-e</math>ã®'''é»è·'''ã垯ã³ãŠãããããã§<math>e</math>ã¯é»æ°çŽ éïŒ<math>e=1.6 \times 10^{-19}</math>CïŒã§ããããã®ãããå®å®ã«ãªãããã«ïŒååå
šäœãšããŠé»æ°çã«äžæ§ãšãªãããã«ïŒååå
ã®éœåæ°ãšé»åæ°ã¯çãããéœåæ°ã'''ååçªå·'''ãšãèšããããªãã¡ååã¯éœåã®æ°ã«ãã£ãŠç¹åŸŽã¥ããããåšæè¡šã¯éœåã®æ°ã®é ã«äžŠãã§ãããäŸãã°ãéœåæ°ã1ãªãæ°ŽçŽ ã2ãªãããªãŠã ã3ãªããªããŠã ãšãã£ãå
·åã§ããã
äžæ§åã«ã¯é»æ°çã«äžæ§ã§ãããé»è·ã¯ç¡ããéœåãè€æ°ããå Žåãããªãã¡ååçªå·ã2以äžã®å Žåãéœåå士ãé»æ°çã«å€§ããªåçºåãæã€ããäžæ§åãç³ã®åœ¹å²ãæãããŠããã
[[ã«ããŽãª:ç¡æ©ååŠ|ãããããã®ãããããã®ãããã]] | null | 2022-11-23T12:40:16Z | [] | https://ja.wikibooks.org/wiki/%E7%84%A1%E6%A9%9F%E5%8C%96%E5%AD%A6%E3%81%AE%E5%9F%BA%E7%A4%8E/%E5%8E%9F%E5%AD%90%E3%81%AE%E6%A7%8B%E9%80%A0 |
2,095 | Maxima | æ¬é
ã¯æ°åŒåŠçãœãããŠã§ã¢ãŒMaximaã®å
¥éæžã§ãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "æ¬é
ã¯æ°åŒåŠçãœãããŠã§ã¢ãŒMaximaã®å
¥éæžã§ãã",
"title": ""
}
] | æ¬é
ã¯æ°åŒåŠçãœãããŠã§ã¢ãŒMaximaã®å
¥éæžã§ãã | {{Pathnav|ã¡ã€ã³ããŒãž|æ
å ±æè¡|frame=1}}
æ¬é
ã¯æ°åŒåŠçãœãããŠã§ã¢ãŒMaximaã®å
¥éæžã§ãã
== ç®æ¬¡ ==
{| border="0" align=right width=250px cellpadding="4" cellspacing=0 class="noprint" style="clear: right; border: solid #aaa 1px; margin: 0 0 1em 1em; font-size: 90%; background: #f9f9f9"
|-
|[[ç»å:Wikipedia.png|50px|none|Wikipedia]]
|'''[[w:ã¡ã€ã³ããŒãž|ãŠã£ãããã£ã¢]]'''ã«'''[[w:{{{1|{{PAGENAME}}}}}|{{{2|{{{1|{{PAGENAME}}}}}}}}]]'''ã®èšäºããããŸãã
|}
<noinclude>
* [[Maxima ã¯ããã«|ã¯ããã«]]
** [[Maxima ã¯ããã«#Maximaãšã¯|Maximaãšã¯]]
** [[Maxima ã¯ããã«#Lispãšã¯|Lispãšã¯]]
* Maximaã®ã€ã³ã¹ããŒã«
** [[/Common LispåŠçç³»ã®éžæ|Common LispåŠçç³»ã®éžæ]]
** [[/Maximaã®ããã³ããšã³ã|Maximaã®ããã³ããšã³ã]]
** [[Maxima/ãªããã¯ã¹ã«ãããã€ã³ã¹ããŒã«ã®ä»æ¹| ãªããã¯ã¹ã«ãããã€ã³ã¹ããŒã«ã®ä»æ¹]]
*** [[Maxima/ãªããã¯ã¹ã«ãããã€ã³ã¹ããŒã«ã®ä»æ¹# rpmã®äœ¿ãæ¹| rpmã®äœ¿ãæ¹]]
*** [[Maxima/ãªããã¯ã¹ã«ãããã€ã³ã¹ããŒã«ã®ä»æ¹# å®éã®ã€ã³ã¹ããŒã«| å®éã®ã€ã³ã¹ããŒã«]]
** [[/FreeBSDã«ãããã€ã³ã¹ããŒã«ã®ä»æ¹|FreeBSDã«ãããã€ã³ã¹ããŒã«ã®ä»æ¹]]
** [[/Windowsã«ãããã€ã³ã¹ããŒã«ã®ä»æ¹|Windowsã«ãããã€ã³ã¹ããŒã«ã®ä»æ¹]]
** [[/MacOS Xã«ãããã€ã³ã¹ããŒã«ã®ä»æ¹|MacOS Xã«ãããã€ã³ã¹ããŒã«ã®ä»æ¹]]
* [[Maxima/å
·äœçãªäœ¿ãæ¹| å
·äœçãªäœ¿ãæ¹]]
** [[Maxima/å
·äœçãªäœ¿ãæ¹#ãœããã®äœ¿ãæ¹|ãœããã®äœ¿ãæ¹]]
** [[Maxima/å
·äœçãªäœ¿ãæ¹#åççãªæ°åŠã«å¯Ÿãã䜿çšäŸ|åççãªæ°åŠã«å¯Ÿãã䜿çšäŸ]]
** ææ³ã»èšå·ã»æ°åŠå®æ°
** [[/å€æ°ãšå®æ°|å€æ°ãšå®æ°]]
** çŽæ°ã»æå°å
¬åæ°
** [[/è€çŽ æ°|è€çŽ æ°]]
** ãã¯ãã«æäœ
** [[/è¡åæäœ|è¡åæäœ]]
** æ°å€èšç®æ³
** [[/å€é
åŒã»æçåŒ|å€é
åŒã»æçåŒ]]
** [[/æèãšäºå®|æèãšäºå®]]
** çåŒã»äžçåŒæäœ
** [[/é¢æ°ãå®çŸ©ãã|é¢æ°ãå®çŸ©ãã]]
** ã°ã©ããæžã
** [[/ãã¡ã€ã«æäœã»åºå圢åŒå€æ|ãã¡ã€ã«æäœã»åºå圢åŒå€æ]]
** 極é
** [[/埮åã»ç©å|埮åã»ç©å]]
** [[/ç·åã»ç·ç©ã»ãã€ã©ãŒå±é|ç·åã»ç·ç©ã»ãã€ã©ãŒå±é]]
** 埮åæ¹çšåŒæäœ
** ããŒãªãšå€æã»ã©ãã©ã¹å€æ
** [[/äžè§é¢æ°ã»åæ²ç·é¢æ°|äžè§é¢æ°ã»åæ²ç·é¢æ°]]
** ææ°é¢æ°ã»å¯Ÿæ°é¢æ°
** ããã»ã«é¢æ°
** ã«ãžã£ã³ãã«å€é
åŒã»ã«ãžã£ã³ãã«å¹é¢æ°ã»çé¢èª¿åé¢æ°
** ãã®ä»ã®çŽäº€ç³»å€é
åŒ
** è¶
幟äœé¢æ°
** æ¥åç©åã»æ¥åé¢æ°
** [[亀ç¹èšç® çŽç·ãšçŽç· çŽç·ãšå åãšå|亀ç¹èšç® çŽç·ãšçŽç· çŽç·ãšå åãšå]]
** [[åç·èšç® ç¹ãšçŽç· ç¹ãšå|åç·èšç® ç¹ãšçŽç· ç¹ãšå]]
** [[é¢ç©èšç® 座æšæ³ å暪è·æ³]]
** [[/芳枬æ¹çšåŒ æ£èŠæ¹çšåŒ|芳枬æ¹çšåŒ æ£èŠæ¹çšåŒ]]
** [[/ã·ã³ãã¬ãã¯ã¹æ³|ã·ã³ãã¬ãã¯ã¹æ³]]
** [[/æŒç¿åé¡è§£ç|æŒç¿åé¡è§£ç]]
* å€éšãªã³ã¯
** [http://maxima.sourceforge.net/docs/manual/en/maxima.html#SEC_Top|Maxima Manual] (è±æã®å
¬åŒããã¥ã¢ã«)
* [[/玢åŒ|玢åŒ]]
{{DEFAULTSORT:Maxima}}
[[Category:Maxima|*]]
[[Category:æ°åŒåŠçã·ã¹ãã ]]
[[Category:ãœãããŠã§ã¢ã®ããã¥ã¢ã«]]
{{NDC|007.63}}
{{stub}} | null | 2015-08-08T11:31:31Z | [
"ãã³ãã¬ãŒã:Pathnav",
"ãã³ãã¬ãŒã:NDC",
"ãã³ãã¬ãŒã:Stub"
] | https://ja.wikibooks.org/wiki/Maxima |
2,096 | Maxima ã¯ããã« | Maxima > ã¯ããã«
Maximaã¯ãGnu Public License ã®å
ã§é
åžãããŠããæ°åŒåŠçã·ã¹ãã ã§ããã
ãã®ææžã¯ãMaxima ã䜿ãå§ããããšãã人éã®ããã«ãMaximaã®ã€ã³ã¹ããŒã«ããåºæ¬çãªäœ¿ãæ¹ãŸã§ã説æããããã«æžãããã
Maxima ã¯ãç¡æã§é
åžãããŠããã誰ã§ãå
¥æããããšãã§ãããããããç¡æã§ããããšã¯å¿
ãããå質ãæªãããšãæå³ããããã§ã¯ãªããåžè²©ã®æ°åŒåŠçã·ã¹ãã ãšæ¯èŒããŠæ©èœãå£ãããã§ããªãããŸããäžè¬çãªåžè²©ã®æ°åŒåŠçã·ã¹ãã ã倧å€é«äŸ¡ã§ããããšãèãããšãç¹ã«å人ã§æ°åŒåŠçã·ã¹ãã ã䜿ãããå Žåã«ã¯ãæåãªéžæè¢ãšèšããã ããã
ãªããMaxima ã®ã€ã³ã¹ããŒã«ã¯ãå©çšããç°å¢ã«ãã£ãŠã¯ããé£ãããããããªããããã¯ãMaxima ã LISP ã®äžæ¹èšã§ãã Common Lisp ãšããæ¯èŒçç¥å床ã®äœãèšèªã«ãã£ãŠæžãããŠããããã§ããããã®ãããMaxima ãã€ã³ã¹ããŒã«ããããã«ã¯ããŸã Common Lisp ã®åŠçç³»ãã€ã³ã¹ããŒã«ããããšããå§ããªããã°ãªããªãã
Maxima ã¯1968幎㫠MIT ã«ããã Mac ãããžã§ã¯ãã®äžã€ãšããŠéçºããå§ããŠã1982幎㫠DOE Maxima ãšã㊠MIT ã®ãšãã«ã®ãŒåŠéšã§ãããµã¹å€§åŠã® William F Schelter ææãã¡ã³ããã³ã¹ãããŠããã1998幎㫠Schelter ææã MIT ã®ãšãã«ã®ãŒåŠéšãã Gnu Public License ã®å
ã§é
åžããäºãèš±å¯ããã2000幎ãã sourceforge.net ã«ãŠ Maxima ãšããŠé
åžãšã¡ã³ããã³ã¹ããããŠããã
ãªããSchelter ææã¯2001幎ã«æ»å»ããããMaxima ãèµ·åããçŽåŸã«è¡šç€ºããããDedicated to the memory of William Schelter.ãã®äžæ㯠Schelter ææã®å瞟ã称ãããã®ã§ããã
LISP ãšã¯ã©ã ãèšç®ãå®çŸããé¢æ°åããã°ã©ãã³ã°èšèªã§ãCLOSã®ãããªåã蟌ã¿åã®ãªããžã§ã¯ãæåèšèªãå©çšåºæ¥ãäºããããããããã«ãã«ããã©ãã€ã èšèªãžãšé²åããŠããã1958幎ã«éçºãããäžçã§2çªç®ã«å€ãé«çŽèšèªãšããŠãç¥ãããŠãããçŸåšã¹ã¯ãªããèšèªã§æå㪠Perl ã PythonãRuby ãªã©ã®æºæµã«ãªã£ãŠãããã®ã§ãããæºæµã ãããšãã£ãŠã¹ã¯ãªããèšèªã§ã¯ãªããŠãåŠçç³»ã®äžã«ã³ã³ãã€ã«æ©èœãšã€ã³ã¿ããªã¿æ©èœãæ··åšãããŠãããåŠçé床ã¯ãã€ãã£ãã³ã³ãã€ã©ãæã£ãŠããåŠçç³»ãªãã°äžè¬çã« C++ ããå°ãé
ã Java ããã¯éããšãããããã«ãªã£ãŠããã䜿ãæ¹ã«ãã£ãŠã¯ C ããéããªãå Žåãããäºã¯ç¥ãããŠããããŸããMaximaã§çšããããŠããCommon Lispã¯èšèªä»æ§(ANSI)ã«ã³ã³ãã€ã©ã«é¢ããèšè¿°ãããå¯äžã®èšèªã§ãã(ã€ã³ã¿ããªã¿ã«é¢ããèšè¿°ã¯ãªã)ã
LISP ã®ç¹åŸŽãšããŠã以äžã®ãã®ãæããããã
å眮èšæ³ãšã¯ãäŸãã°ã
ã®æŒç®ã«çžåœããèšæ³ã
ã®ããã«æŒç®å(ããã§ã¯ +)ãåã«ã被æŒç®å(ããã§ã¯ 1 ãš 2)ãåŸã«èšãèšæ³ã§ããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "Maxima > ã¯ããã«",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "Maximaã¯ãGnu Public License ã®å
ã§é
åžãããŠããæ°åŒåŠçã·ã¹ãã ã§ããã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ãã®ææžã¯ãMaxima ã䜿ãå§ããããšãã人éã®ããã«ãMaximaã®ã€ã³ã¹ããŒã«ããåºæ¬çãªäœ¿ãæ¹ãŸã§ã説æããããã«æžãããã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "Maxima ã¯ãç¡æã§é
åžãããŠããã誰ã§ãå
¥æããããšãã§ãããããããç¡æã§ããããšã¯å¿
ãããå質ãæªãããšãæå³ããããã§ã¯ãªããåžè²©ã®æ°åŒåŠçã·ã¹ãã ãšæ¯èŒããŠæ©èœãå£ãããã§ããªãããŸããäžè¬çãªåžè²©ã®æ°åŒåŠçã·ã¹ãã ã倧å€é«äŸ¡ã§ããããšãèãããšãç¹ã«å人ã§æ°åŒåŠçã·ã¹ãã ã䜿ãããå Žåã«ã¯ãæåãªéžæè¢ãšèšããã ããã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ãªããMaxima ã®ã€ã³ã¹ããŒã«ã¯ãå©çšããç°å¢ã«ãã£ãŠã¯ããé£ãããããããªããããã¯ãMaxima ã LISP ã®äžæ¹èšã§ãã Common Lisp ãšããæ¯èŒçç¥å床ã®äœãèšèªã«ãã£ãŠæžãããŠããããã§ããããã®ãããMaxima ãã€ã³ã¹ããŒã«ããããã«ã¯ããŸã Common Lisp ã®åŠçç³»ãã€ã³ã¹ããŒã«ããããšããå§ããªããã°ãªããªãã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "Maxima ã¯1968幎㫠MIT ã«ããã Mac ãããžã§ã¯ãã®äžã€ãšããŠéçºããå§ããŠã1982幎㫠DOE Maxima ãšã㊠MIT ã®ãšãã«ã®ãŒåŠéšã§ãããµã¹å€§åŠã® William F Schelter ææãã¡ã³ããã³ã¹ãããŠããã1998幎㫠Schelter ææã MIT ã®ãšãã«ã®ãŒåŠéšãã Gnu Public License ã®å
ã§é
åžããäºãèš±å¯ããã2000幎ãã sourceforge.net ã«ãŠ Maxima ãšããŠé
åžãšã¡ã³ããã³ã¹ããããŠããã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ãªããSchelter ææã¯2001幎ã«æ»å»ããããMaxima ãèµ·åããçŽåŸã«è¡šç€ºããããDedicated to the memory of William Schelter.ãã®äžæ㯠Schelter ææã®å瞟ã称ãããã®ã§ããã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "LISP ãšã¯ã©ã ãèšç®ãå®çŸããé¢æ°åããã°ã©ãã³ã°èšèªã§ãCLOSã®ãããªåã蟌ã¿åã®ãªããžã§ã¯ãæåèšèªãå©çšåºæ¥ãäºããããããããã«ãã«ããã©ãã€ã èšèªãžãšé²åããŠããã1958幎ã«éçºãããäžçã§2çªç®ã«å€ãé«çŽèšèªãšããŠãç¥ãããŠãããçŸåšã¹ã¯ãªããèšèªã§æå㪠Perl ã PythonãRuby ãªã©ã®æºæµã«ãªã£ãŠãããã®ã§ãããæºæµã ãããšãã£ãŠã¹ã¯ãªããèšèªã§ã¯ãªããŠãåŠçç³»ã®äžã«ã³ã³ãã€ã«æ©èœãšã€ã³ã¿ããªã¿æ©èœãæ··åšãããŠãããåŠçé床ã¯ãã€ãã£ãã³ã³ãã€ã©ãæã£ãŠããåŠçç³»ãªãã°äžè¬çã« C++ ããå°ãé
ã Java ããã¯éããšãããããã«ãªã£ãŠããã䜿ãæ¹ã«ãã£ãŠã¯ C ããéããªãå Žåãããäºã¯ç¥ãããŠããããŸããMaximaã§çšããããŠããCommon Lispã¯èšèªä»æ§(ANSI)ã«ã³ã³ãã€ã©ã«é¢ããèšè¿°ãããå¯äžã®èšèªã§ãã(ã€ã³ã¿ããªã¿ã«é¢ããèšè¿°ã¯ãªã)ã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "LISP ã®ç¹åŸŽãšããŠã以äžã®ãã®ãæããããã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "å眮èšæ³ãšã¯ãäŸãã°ã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "ã®æŒç®ã«çžåœããèšæ³ã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "ã®ããã«æŒç®å(ããã§ã¯ +)ãåã«ã被æŒç®å(ããã§ã¯ 1 ãš 2)ãåŸã«èšãèšæ³ã§ããã",
"title": "ã¯ããã«"
}
] | Maxima > ã¯ããã« | <small> [[Maxima]] > ã¯ããã«</small>
----
== ã¯ããã« ==
=== Maxima ãšã¯ ===
[[w:Maxima|Maxima]]ã¯ãGnu Public License ã®å
ã§é
åžãããŠãã[[w:æ°åŒåŠçã·ã¹ãã |æ°åŒåŠçã·ã¹ãã ]]ã§ããã
ãã®ææžã¯ãMaxima ã䜿ãå§ããããšãã人éã®ããã«ãMaximaã®ã€ã³ã¹ããŒã«ããåºæ¬çãªäœ¿ãæ¹ãŸã§ã説æããããã«æžãããã
Maxima ã¯ãç¡æã§é
åžãããŠããã誰ã§ãå
¥æããããšãã§ãããããããç¡æã§ããããšã¯å¿
ãããå質ãæªãããšãæå³ããããã§ã¯ãªããåžè²©ã®æ°åŒåŠçã·ã¹ãã ãšæ¯èŒããŠæ©èœãå£ãããã§ããªãããŸããäžè¬çãªåžè²©ã®æ°åŒåŠçã·ã¹ãã ã倧å€é«äŸ¡ã§ããããšãèãããšãç¹ã«å人ã§æ°åŒåŠçã·ã¹ãã ã䜿ãããå Žåã«ã¯ãæåãªéžæè¢ãšèšããã ããã
ãªããMaxima ã®ã€ã³ã¹ããŒã«ã¯ãå©çšããç°å¢ã«ãã£ãŠã¯ããé£ãããããããªããããã¯ãMaxima ã [[Lisp|LISP]] ã®äžæ¹èšã§ãã [[w:Common Lisp|Common Lisp]] ãšããæ¯èŒçç¥å床ã®äœãèšèªã«ãã£ãŠæžãããŠããããã§ããããã®ãããMaxima ãã€ã³ã¹ããŒã«ããããã«ã¯ããŸã Common Lisp ã®åŠçç³»ãã€ã³ã¹ããŒã«ããããšããå§ããªããã°ãªããªãã
=== Maxima ã®æŽå² ===
Maxima ã¯1968幎㫠MIT ã«ããã Mac ãããžã§ã¯ã[http://ja.wikipedia.org/wiki/Project_MAC]ã®äžã€ãšããŠéçºããå§ããŠã1982幎㫠DOE Maxima ãšã㊠MIT ã®ãšãã«ã®ãŒåŠéšã§ãããµã¹å€§åŠã® William F Schelter ææãã¡ã³ããã³ã¹ãããŠããã1998幎㫠Schelter ææã MIT ã®ãšãã«ã®ãŒåŠéšãã Gnu Public License ã®å
ã§é
åžããäºãèš±å¯ããã2000幎ãã sourceforge.net ã«ãŠ Maxima ãšããŠé
åžãšã¡ã³ããã³ã¹ããããŠããã
ãªããSchelter ææã¯2001幎ã«æ»å»ããããMaxima ãèµ·åããçŽåŸã«è¡šç€ºããããDedicated to the memory of William Schelter.ãã®äžæ㯠Schelter ææã®å瞟ã称ãããã®ã§ããã
=== LISPãšã¯ ===
[[Lisp|LISP]] ãšã¯[[ã©ã ãèšç®]]ãå®çŸãã[[w:é¢æ°åèšèª|é¢æ°åããã°ã©ãã³ã°èšèª]]ã§ã<abbr title="Common Lisp Object System">CLOS</abbr>ã®ãããªåã蟌ã¿åã®ãªããžã§ã¯ãæåèšèªãå©çšåºæ¥ãäºããããããããã«ãã«ããã©ãã€ã èšèªãžãšé²åããŠããã1958幎ã«éçºãããäžçã§2çªç®ã«å€ãé«çŽèšèªãšããŠãç¥ãããŠãããçŸåšã¹ã¯ãªããèšèªã§æå㪠[[Perl]] ã [[Python]]ã[[Ruby]] ãªã©ã®æºæµã«ãªã£ãŠãããã®ã§ãããæºæµã ãããšãã£ãŠã¹ã¯ãªããèšèªã§ã¯ãªããŠãåŠçç³»ã®äžã«ã³ã³ãã€ã«æ©èœãšã€ã³ã¿ããªã¿æ©èœãæ··åšãããŠãããåŠçé床ã¯ãã€ãã£ãã³ã³ãã€ã©ãæã£ãŠããåŠçç³»ãªãã°äžè¬çã« [[C++]] ããå°ãé
ã [[Java]] ããã¯éããšãããããã«ãªã£ãŠããã䜿ãæ¹ã«ãã£ãŠã¯ [[Cèšèª|C]] ããéããªãå Žåãããäºã¯ç¥ãããŠããããŸããMaximaã§çšããããŠãã[[Common Lisp]]ã¯èšèªä»æ§(ANSI)ã«ã³ã³ãã€ã©ã«é¢ããèšè¿°ãããå¯äžã®èšèªã§ãã(ã€ã³ã¿ããªã¿ã«é¢ããèšè¿°ã¯ãªã)ã
LISP ã®ç¹åŸŽãšããŠã以äžã®ãã®ãæããããã
* [[w:ããŒã©ã³ãèšæ³|å眮èšæ³]]
* SåŒ
* èšèªä»æ§ãèªç±ã«æ¡åŒµåºæ¥ãæè»ã
* åçãªèšèª
* 匷åãªãã¯ãæ©èœ
å眮èšæ³ãšã¯ãäŸãã°ã
1 + 2
ã®æŒç®ã«çžåœããèšæ³ã
(+ 1 2)
ã®ããã«æŒç®åïŒããã§ã¯ +ïŒãåã«ã被æŒç®åïŒããã§ã¯ 1 ãš 2ïŒãåŸã«èšãèšæ³ã§ããã
==å€éšãªã³ã¯==
* [http://maxima.sourceforge.net/ å
¬åŒããŒã ããŒãž] (è±æ)
* [http://maxima.sourceforge.net/docs/manual/en/maxima.html#SEC_Top ããã¥ã¢ã«Maxima 5.9.1] (è±æ)
* [http://www.bekkoame.ne.jp/~ponpoko/Math/maxima/maxima.html æ¥æ¬èªã«ç¿»èš³äžã®ããã¥ã¢ã«]
æ¥æ¬èªã®è§£èª¬
* [http://phe.phyas.aichi-edu.ac.jp/~cyamauch/maxima/ æ°åŒåŠçã·ã¹ãã Maxima] - å
¥éçãªè§£èª¬ã
* [http://www.bekkoame.ne.jp/~ponpoko/Math/maxima/MaximaMAIN.html Maximaã§éãŒã] - ããã¥ã¢ã«ã®æ¥æ¬èªç¿»èš³è
ã«ãã解説ã
[[Category:Maxima|ã¯ããã«]] | null | 2021-10-19T00:19:55Z | [] | https://ja.wikibooks.org/wiki/Maxima_%E3%81%AF%E3%81%98%E3%82%81%E3%81%AB |
2,097 | Maxima/ãªããã¯ã¹ã«ãããã€ã³ã¹ããŒã«ã®ä»æ¹ | Maxima > ãªããã¯ã¹ã«ãããã€ã³ã¹ããŒã«ã®ä»æ¹
Linuxã«Maximaãã€ã³ã¹ããŒã«ããæ¹æ³ã¯ãéåžžãããã±ãŒãžãããŒãžã£ãŒã䜿çšããŠè¡ããŸãã
以äžã¯ãäžè¬çãªLinuxãã£ã¹ããªãã¥ãŒã·ã§ã³ãžã®ã€ã³ã¹ããŒã«æé ã®æŠèŠã§ãã
ãããã®æé ã«åŸããšãLinuxãã£ã¹ããªãã¥ãŒã·ã§ã³ã«Maximaãã€ã³ã¹ããŒã«ã§ããŸãã
Maximaãã€ã³ã¹ããŒã«ããéã®æ³šæç¹ã¯ããã€ããããŸãã以äžã«ããã€ãæããŠã¿ãŸã:
ãããã®æ³šæç¹ãèæ
®ããããšã§ãMaximaã®ã¹ã ãŒãºãªã€ã³ã¹ããŒã«ãšäœ¿çšãå¯èœã«ãªããŸãã
æ°å€æŒç®ãªã©ã®éããæåŸ
ãããªãã°ãæŒç®é床ãè¿œæ±ããCMUCLãCMUCLããæåããããŠæŽ»çºã«ã¡ã³ããã³ã¹ãè¡ãããŠããSBCLã®å©çšã埡å§ãããã CLISPã¯ãäžéã³ãŒããã«ã³ã³ãã€ã«ãããããšã«å¯ŸããŠãCMUCLãSBCLã¯ããã€ãã£ãã³ãŒããã«ã³ã³ãã€ã«ããããããæ°åéããªãã Maximaã®ä»¥åã®ããŒãžã§ã³ã¯GCLãšCLISPãåæãšããŠäœãããŠããã®ã ããMaxima 5.9.*以åŸäž»æµãCMUCLãSBCLã«ç§»ã£ãããã§ããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "Maxima > ãªããã¯ã¹ã«ãããã€ã³ã¹ããŒã«ã®ä»æ¹",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "Linuxã«Maximaãã€ã³ã¹ããŒã«ããæ¹æ³ã¯ãéåžžãããã±ãŒãžãããŒãžã£ãŒã䜿çšããŠè¡ããŸãã",
"title": "Maximaã®Linuxãã£ã¹ããªãã¥ãŒã·ã§ã³ãžã®ã€ã³ã¹ããŒã«"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "以äžã¯ãäžè¬çãªLinuxãã£ã¹ããªãã¥ãŒã·ã§ã³ãžã®ã€ã³ã¹ããŒã«æé ã®æŠèŠã§ãã",
"title": "Maximaã®Linuxãã£ã¹ããªãã¥ãŒã·ã§ã³ãžã®ã€ã³ã¹ããŒã«"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ãããã®æé ã«åŸããšãLinuxãã£ã¹ããªãã¥ãŒã·ã§ã³ã«Maximaãã€ã³ã¹ããŒã«ã§ããŸãã",
"title": "Maximaã®Linuxãã£ã¹ããªãã¥ãŒã·ã§ã³ãžã®ã€ã³ã¹ããŒã«"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "Maximaãã€ã³ã¹ããŒã«ããéã®æ³šæç¹ã¯ããã€ããããŸãã以äžã«ããã€ãæããŠã¿ãŸã:",
"title": "Maximaã®Linuxãã£ã¹ããªãã¥ãŒã·ã§ã³ãžã®ã€ã³ã¹ããŒã«"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ãããã®æ³šæç¹ãèæ
®ããããšã§ãMaximaã®ã¹ã ãŒãºãªã€ã³ã¹ããŒã«ãšäœ¿çšãå¯èœã«ãªããŸãã",
"title": "Maximaã®Linuxãã£ã¹ããªãã¥ãŒã·ã§ã³ãžã®ã€ã³ã¹ããŒã«"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "æ°å€æŒç®ãªã©ã®éããæåŸ
ãããªãã°ãæŒç®é床ãè¿œæ±ããCMUCLãCMUCLããæåããããŠæŽ»çºã«ã¡ã³ããã³ã¹ãè¡ãããŠããSBCLã®å©çšã埡å§ãããã CLISPã¯ãäžéã³ãŒããã«ã³ã³ãã€ã«ãããããšã«å¯ŸããŠãCMUCLãSBCLã¯ããã€ãã£ãã³ãŒããã«ã³ã³ãã€ã«ããããããæ°åéããªãã Maximaã®ä»¥åã®ããŒãžã§ã³ã¯GCLãšCLISPãåæãšããŠäœãããŠããã®ã ããMaxima 5.9.*以åŸäž»æµãCMUCLãSBCLã«ç§»ã£ãããã§ããã",
"title": "Maximaã®Linuxãã£ã¹ããªãã¥ãŒã·ã§ã³ãžã®ã€ã³ã¹ããŒã«"
}
] | Maxima > ãªããã¯ã¹ã«ãããã€ã³ã¹ããŒã«ã®ä»æ¹ | <small> [[Maxima]] > ãªããã¯ã¹ã«ãããã€ã³ã¹ããŒã«ã®ä»æ¹</small>
----
== Maximaã®Linuxãã£ã¹ããªãã¥ãŒã·ã§ã³ãžã®ã€ã³ã¹ããŒã«==
Linuxã«Maximaãã€ã³ã¹ããŒã«ããæ¹æ³ã¯ãéåžžãããã±ãŒãžãããŒãžã£ãŒã䜿çšããŠè¡ããŸãã
以äžã¯ãäžè¬çãªLinuxãã£ã¹ããªãã¥ãŒã·ã§ã³ãžã®ã€ã³ã¹ããŒã«æé ã®æŠèŠã§ãã
;Debian/UbuntuããŒã¹ã®ãã£ã¹ããªãã¥ãŒã·ã§ã³
:ããã±ãŒãžãªã¹ããæŽæ°ããMaximaãã€ã³ã¹ããŒã«ããŸãïŒ
:<syntaxhighlight lang=shell>
sudo apt update
sudo apt install maxima
</syntaxhighlight>
;FedoraããŒã¹ã®ãã£ã¹ããªãã¥ãŒã·ã§ã³
:ããã±ãŒãžãªã¹ããæŽæ°ããMaximaãã€ã³ã¹ããŒã«ããŸãïŒ
:<syntaxhighlight lang=shell>
sudo dnf update
sudo dnf install maxima
</syntaxhighlight>
;Arch Linux
:ããã±ãŒãžãªã¹ããæŽæ°ããMaximaãã€ã³ã¹ããŒã«ããŸãïŒ
:<syntaxhighlight lang=shell>
sudo pacman -Syu
sudo pacman -S maxima
</syntaxhighlight>
;CentOS/RHEL
:CentOSãRHELã§ã¯ãEPELãªããžããªãæå¹ã«ãããã®åŸãMaximaãã€ã³ã¹ããŒã«ããŸãïŒ
:<syntaxhighlight lang=shell>
sudo yum install epel-release
sudo yum install maxima
</syntaxhighlight>
ãããã®æé ã«åŸããšãLinuxãã£ã¹ããªãã¥ãŒã·ã§ã³ã«Maximaãã€ã³ã¹ããŒã«ã§ããŸãã
=== Maximaã€ã³ã¹ããŒã«ã®æ³šæç¹ ===
Maximaãã€ã³ã¹ããŒã«ããéã®æ³šæç¹ã¯ããã€ããããŸãã以äžã«ããã€ãæããŠã¿ãŸãïŒ
;ããã±ãŒãžäŸåé¢ä¿ã®ç¢ºèª: ã€ã³ã¹ããŒã«ããåã«ãMaximaãäŸåããããã±ãŒãžãã·ã¹ãã ã«ã€ã³ã¹ããŒã«ãããŠããããšã確èªããŠãã ãããç¹ã«Linuxã·ã¹ãã ã§ã¯ãå¿
èŠãªã©ã€ãã©ãªãã©ã³ã¿ã€ã ãäžè¶³ããŠããå ŽåããããŸãã
;ããŒãžã§ã³ã®éžæ: ææ°ããŒãžã§ã³ãåžžã«å©çšããããšãæãŸããããã§ã¯ãããŸãããå®å®ããããŒãžã§ã³ããç¹å®ã®æ©èœãäºææ§ãå¿
èŠãªå Žåã¯ãé©åãªããŒãžã§ã³ãéžæããå¿
èŠããããŸãã
;ã·ã¹ãã èŠä»¶: Maximaãå®è¡ããããã«å¿
èŠãªããŒããŠã§ã¢èŠä»¶ãããµããŒããããŠãããªãã¬ãŒãã£ã³ã°ã·ã¹ãã ã確èªããŠãã ãããç¹ã«å€ãããŒããŠã§ã¢ãå€ãããŒãžã§ã³ã®ãªãã¬ãŒãã£ã³ã°ã·ã¹ãã ã§ã¯ãæ£åžžã«åäœããªãå ŽåããããŸãã
;ã»ãã¥ãªãã£: ã€ã³ã¹ããŒã«å
ã®ä¿¡é Œæ§ãéèŠã§ããå
¬åŒã®ãœãŒã¹ãä¿¡é Œã§ããããã±ãŒãžãããŒãžã£ãŒããã®ã€ã³ã¹ããŒã«ãæšå¥šããŸãããŸããäžæ£ãªãœãŒã¹ããã®ããŠã³ããŒãããä¿¡é Œã§ããªããªããžããªããã®ããã±ãŒãžã®ã€ã³ã¹ããŒã«ã¯é¿ããã¹ãã§ãã
;ã¢ã³ã€ã³ã¹ããŒã«æé ã®ç解: ã€ã³ã¹ããŒã«åŸã«Maximaãã¢ã³ã€ã³ã¹ããŒã«ããå Žåãã·ã¹ãã ã«åœ±é¿ãäžããªãããã«ãæ£ããæé ãç解ããŠããããšãéèŠã§ããç¹ã«ãæåã§ã€ã³ã¹ããŒã«ããå Žåã¯ããã¡ã€ã«ãèšå®ãæ®ãå¯èœæ§ããããããããããé©åã«ã¯ãªãŒã³ã¢ããããå¿
èŠããããŸãã
ãããã®æ³šæç¹ãèæ
®ããããšã§ãMaximaã®ã¹ã ãŒãºãªã€ã³ã¹ããŒã«ãšäœ¿çšãå¯èœã«ãªããŸãã
==== Common LispåŠçç³»ã«ã€ããŠã®è£è¶³====
æ°å€æŒç®ãªã©ã®éããæåŸ
ãããªãã°ãæŒç®é床ãè¿œæ±ããCMUCLãCMUCLããæåããããŠæŽ»çºã«ã¡ã³ããã³ã¹ãè¡ãããŠããSBCLã®å©çšã埡å§ãããã
CLISPã¯ãäžéã³ãŒããã«ã³ã³ãã€ã«ãããããšã«å¯ŸããŠãCMUCLãSBCLã¯ããã€ãã£ãã³ãŒããã«ã³ã³ãã€ã«ããããããæ°åéããªãã
Maximaã®ä»¥åã®ããŒãžã§ã³ã¯GCLãšCLISPãåæãšããŠäœãããŠããã®ã ããMaxima 5.9.*以åŸäž»æµãCMUCLãSBCLã«ç§»ã£ãããã§ããã
[[Category:Maxima|ããªã€ããã«ãããããããšããã®ããã]] | 2005-06-02T12:24:00Z | 2024-01-30T05:23:18Z | [] | https://ja.wikibooks.org/wiki/Maxima/%E3%83%AA%E3%83%8A%E3%83%83%E3%82%AF%E3%82%B9%E3%81%AB%E3%81%8A%E3%81%91%E3%82%8B%E3%82%A4%E3%83%B3%E3%82%B9%E3%83%88%E3%83%BC%E3%83%AB%E3%81%AE%E4%BB%95%E6%96%B9 |
2,098 | Maxima/å
·äœçãªäœ¿ãæ¹ | Maxima > å
·äœçãªäœ¿ãæ¹
æ°åŒåŠçã·ã¹ãã ãªã®ã§ã èšç®ãããæ°åŒããªããšäœ¿ãéããªãã äŸãã°ãæ°åŠã®æç§æžãããã°ã èšç®ããããåŒãã¿ã€ããã ããã ãã¡ããå€äŒã¿ã®å®¿é¡ãç°¡åã ã:-) æç« é¡ããèªæžææ³æã¯ç¡çã ã...ã
äŸãã°ãåæ°ã®è¶³ãç® 1/2 + 1/3 ãèšç®ãããããšãããã maximaã¯ãcommand lineã¢ãŒããš batchã¢ãŒãã§äœ¿ãããšãåºæ¥ãã é·ãåŒãæžããšããªã©ã¯åŒã®æžãå€ããã§ããªã ãšæžãã®ã倧å€ã ãã command lineã¢ãŒãã®maximaã§ã¯ã å®éã«åŒã®æžãå€ããã§ããªãã ãã®ããå€ãã®å Žåã«batchã¢ãŒãã 䜿ãããšã«ãªããšæãã ãŸãcommand lineã¢ãŒãã§ã¯ã
ãšæã€ã ãããšãmaximaãèµ·åã
ãšè¡šç€ºãããã®ã§ã(iã¯inputã®ç¥)
ãšããã (æåŸã®;ãå¿ãããšããŸãããããããªãããšã èµ·ããã®ã§æ³šæããããš!) äžæãæžãããã°ã
ãšè¡šç€ºãããã(oã¯outputã®ç¥) ãã£ãŠãçã¯5/6ã ãšåãã
batchã¢ãŒãã§ã¯ããŸãèšç®ããããå
容ã ãã¡ã€ã«ã«æžãåºãã äŸãã°ãaaaãšãããã¡ã€ã«ã䜿ããšããã (Linuxã¯æ¡åŒµåãã€ããªããã¡ã€ã«ã 䜿ãããšããããå®éåé¡ãšã㊠maximaã¯æ¡åŒµåã䜿ã£ãŠãã¡ã€ã«ãå€å®ããã®ã§ã¯ ç¡ããããªã®ã§ãããã§ã¯äœãšã€ããŠãå·®ãæ¯ããªãã) ã€ãŸãã奜ããªãšãã£ã¿ã䜿ã£ãŠaaaãç·šéããã®ã§ããã ãããããã¡ãããã®ããã«æžãã®ã奜ããªäººéãããã ããã
äœã«ããaaaã®çšæãåºæ¥ããªãã
ãšæãŠã°ããã çµæã¯æšæºåºåã«è¡šç€ºãããã ããã çµæãé·ããªã£ãå Žåã«ã¯
ãšãããšãçµæã®åŒã bbbã«æžãããŸããã®ã§ãåŸãããã£ãã èªãããšãåºæ¥ãã
maximaã¯åæ°ã®è¶³ãç®ãè¡ãããšãã§ããã
maximaã¯ç°ãªãåæ¯ã®èšç®ãæ£ããè¡ãªãããšãåºæ¥ãã
maximaã¯åæ°ã®ããç®ãè¡ãªãããšãåºæ¥ãã
maximaã¯åæ°ã®å²ãç®ãè¡ãªãããšãåºæ¥ãã
(ãã£ãã¯çç¥ã§ããªãã)
Maximaã¯æ£è² ã®æ°ãæ··ãã£ãååæŒç®ã è¡ãªãããšãåºæ¥ãã
Maximaã¯æååŒã®ååæŒç®ã è¡ãªãããšãåºæ¥ãã
maximaã¯æ¹çšåŒãæ±ããã
maximaã¯2å
æ¹çšåŒãæ±ããã
maximaã¯å¹³æ¹æ ¹ã®å€ãä»»æã®æ¡ãŸã§æ±ããããšãåºæ¥ãã
Maximaã¯åŒã®å±éãšå æ°å解ãè¡ãªãããšãåºæ¥ãã
maximaã¯äºæ¬¡æ¹çšåŒãæ±ãããšãã§ããã
maximaã¯è€çŽ æ°ããµããŒãããŠããã
Note:maximaã¯ææ°eã®å€ãç¥ã£ãŠããã (å®éã«ã¯éåžžã®ã³ã³ãã¥ãŒã¿ãŒã¯ ãã®å€ãèšç®ã§ããããã«ãªã£ãŠããã¯ãã§ããã )
maximaã¯ã埮ç©åããµããŒãããã
note: maxima ã¯åŸ®åæ³ããµããŒãããã
maximaã¯è¡åã®æŒç®ããµããŒãããã
| [
{
"paragraph_id": 0,
"tag": "p",
"text": "Maxima > å
·äœçãªäœ¿ãæ¹",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "æ°åŒåŠçã·ã¹ãã ãªã®ã§ã èšç®ãããæ°åŒããªããšäœ¿ãéããªãã äŸãã°ãæ°åŠã®æç§æžãããã°ã èšç®ããããåŒãã¿ã€ããã ããã ãã¡ããå€äŒã¿ã®å®¿é¡ãç°¡åã ã:-) æç« é¡ããèªæžææ³æã¯ç¡çã ã...ã",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "äŸãã°ãåæ°ã®è¶³ãç® 1/2 + 1/3 ãèšç®ãããããšãããã maximaã¯ãcommand lineã¢ãŒããš batchã¢ãŒãã§äœ¿ãããšãåºæ¥ãã é·ãåŒãæžããšããªã©ã¯åŒã®æžãå€ããã§ããªã ãšæžãã®ã倧å€ã ãã command lineã¢ãŒãã®maximaã§ã¯ã å®éã«åŒã®æžãå€ããã§ããªãã ãã®ããå€ãã®å Žåã«batchã¢ãŒãã 䜿ãããšã«ãªããšæãã ãŸãcommand lineã¢ãŒãã§ã¯ã",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ãšæã€ã ãããšãmaximaãèµ·åã",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ãšè¡šç€ºãããã®ã§ã(iã¯inputã®ç¥)",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ãšããã (æåŸã®;ãå¿ãããšããŸãããããããªãããšã èµ·ããã®ã§æ³šæããããš!) äžæãæžãããã°ã",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ãšè¡šç€ºãããã(oã¯outputã®ç¥) ãã£ãŠãçã¯5/6ã ãšåãã",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "batchã¢ãŒãã§ã¯ããŸãèšç®ããããå
容ã ãã¡ã€ã«ã«æžãåºãã äŸãã°ãaaaãšãããã¡ã€ã«ã䜿ããšããã (Linuxã¯æ¡åŒµåãã€ããªããã¡ã€ã«ã 䜿ãããšããããå®éåé¡ãšã㊠maximaã¯æ¡åŒµåã䜿ã£ãŠãã¡ã€ã«ãå€å®ããã®ã§ã¯ ç¡ããããªã®ã§ãããã§ã¯äœãšã€ããŠãå·®ãæ¯ããªãã) ã€ãŸãã奜ããªãšãã£ã¿ã䜿ã£ãŠaaaãç·šéããã®ã§ããã ãããããã¡ãããã®ããã«æžãã®ã奜ããªäººéãããã ããã",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "äœã«ããaaaã®çšæãåºæ¥ããªãã",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "ãšæãŠã°ããã çµæã¯æšæºåºåã«è¡šç€ºãããã ããã çµæãé·ããªã£ãå Žåã«ã¯",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "ãšãããšãçµæã®åŒã bbbã«æžãããŸããã®ã§ãåŸãããã£ãã èªãããšãåºæ¥ãã",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "maximaã¯åæ°ã®è¶³ãç®ãè¡ãããšãã§ããã",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "maximaã¯ç°ãªãåæ¯ã®èšç®ãæ£ããè¡ãªãããšãåºæ¥ãã",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "maximaã¯åæ°ã®ããç®ãè¡ãªãããšãåºæ¥ãã",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "maximaã¯åæ°ã®å²ãç®ãè¡ãªãããšãåºæ¥ãã",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "(ãã£ãã¯çç¥ã§ããªãã)",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "Maximaã¯æ£è² ã®æ°ãæ··ãã£ãååæŒç®ã è¡ãªãããšãåºæ¥ãã",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "Maximaã¯æååŒã®ååæŒç®ã è¡ãªãããšãåºæ¥ãã",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "maximaã¯æ¹çšåŒãæ±ããã",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "maximaã¯2å
æ¹çšåŒãæ±ããã",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "maximaã¯å¹³æ¹æ ¹ã®å€ãä»»æã®æ¡ãŸã§æ±ããããšãåºæ¥ãã",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "Maximaã¯åŒã®å±éãšå æ°å解ãè¡ãªãããšãåºæ¥ãã",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "maximaã¯äºæ¬¡æ¹çšåŒãæ±ãããšãã§ããã",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "maximaã¯è€çŽ æ°ããµããŒãããŠããã",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 39,
"tag": "p",
"text": "",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 40,
"tag": "p",
"text": "Note:maximaã¯ææ°eã®å€ãç¥ã£ãŠããã (å®éã«ã¯éåžžã®ã³ã³ãã¥ãŒã¿ãŒã¯ ãã®å€ãèšç®ã§ããããã«ãªã£ãŠããã¯ãã§ããã )",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 41,
"tag": "p",
"text": "maximaã¯ã埮ç©åããµããŒãããã",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 42,
"tag": "p",
"text": "",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 43,
"tag": "p",
"text": "",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 44,
"tag": "p",
"text": "note: maxima ã¯åŸ®åæ³ããµããŒãããã",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 45,
"tag": "p",
"text": "",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 46,
"tag": "p",
"text": "",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 47,
"tag": "p",
"text": "maximaã¯è¡åã®æŒç®ããµããŒãããã",
"title": "å
·äœçãªäœ¿ãæ¹"
},
{
"paragraph_id": 48,
"tag": "p",
"text": "",
"title": "å
·äœçãªäœ¿ãæ¹"
}
] | Maxima > å
·äœçãªäœ¿ãæ¹ | <small> [[Maxima]] > å
·äœçãªäœ¿ãæ¹</small>
----
== å
·äœçãªäœ¿ãæ¹==
===ãœããã®äœ¿ãæ¹===
æ°åŒåŠçã·ã¹ãã ãªã®ã§ã
èšç®ãããæ°åŒããªããšäœ¿ãéããªãã
äŸãã°ãæ°åŠã®æç§æžãããã°ã
èšç®ããããåŒãã¿ã€ããã ããã
<!--
ãã䜿ããã èšç®åŒã«å¯Ÿå¿ãã
maximaã®é¢æ°ã®ãã¡ã®å€ããã
ææžã®äžã«æžããã€ããã ã
ã©ãã ããã?
-->
ãã¡ããå€äŒã¿ã®å®¿é¡ãç°¡åã ã:-)
æç« é¡ããèªæžææ³æã¯ç¡çã ã...ã
äŸãã°ãåæ°ã®è¶³ãç®
1/2 + 1/3
ãèšç®ãããããšãããã
<!--
(ãããåºæ¥ãªã人éã¯
ãããããã£ãšå¢ããŠæ¥ããã ãã...)
-->
maximaã¯ãcommand lineã¢ãŒããš
batchã¢ãŒãã§äœ¿ãããšãåºæ¥ãã
é·ãåŒãæžããšããªã©ã¯åŒã®æžãå€ããã§ããªã
ãšæžãã®ã倧å€ã ãã
command lineã¢ãŒãã®maximaã§ã¯ã
å®éã«åŒã®æžãå€ããã§ããªãã
ãã®ããå€ãã®å Žåã«batchã¢ãŒãã
䜿ãããšã«ãªããšæãã
ãŸãcommand lineã¢ãŒãã§ã¯ã
$maxima
ãšæã€ã
ãããšãmaximaãèµ·åã
(%i1)
ãšè¡šç€ºãããã®ã§ã(iã¯inputã®ç¥)
(%i1) 1/2 + 1/3;
ãšããã
(æåŸã®;ãå¿ãããšããŸãããããããªãããšã
èµ·ããã®ã§æ³šæããããš!)
äžæãæžãããã°ã
(%o1)
5
-
6
ãšè¡šç€ºãããã(oã¯outputã®ç¥)
ãã£ãŠãçã¯5/6ã ãšåãã
batchã¢ãŒãã§ã¯ããŸãèšç®ããããå
容ã
ãã¡ã€ã«ã«æžãåºãã
äŸãã°ãaaaãšãããã¡ã€ã«ã䜿ããšããã
(Linuxã¯æ¡åŒµåãã€ããªããã¡ã€ã«ã
䜿ãããšããããå®éåé¡ãšããŠ
maximaã¯æ¡åŒµåã䜿ã£ãŠãã¡ã€ã«ãå€å®ããã®ã§ã¯
ç¡ããããªã®ã§ãããã§ã¯äœãšã€ããŠãå·®ãæ¯ããªãã)
ã€ãŸãã奜ããªãšãã£ã¿ã䜿ã£ãŠaaaãç·šéããã®ã§ããã
ãããããã¡ãããã®ããã«æžãã®ã奜ããªäººéãããã ããã
$cat >>aaa
1/2 + 1/3;
[Ctrl-d]
äœã«ããaaaã®çšæãåºæ¥ããªãã
$maxima -b aaa
ãšæãŠã°ããã
çµæã¯æšæºåºåã«è¡šç€ºãããã ããã
çµæãé·ããªã£ãå Žåã«ã¯
$maxima -b aaa > bbb
ãšãããšãçµæã®åŒã
bbbã«æžãããŸããã®ã§ãåŸãããã£ãã
èªãããšãåºæ¥ãã
===åççãªæ°åŠã«å¯Ÿãã䜿çšäŸ===
====å°åŠæ ¡====
maximaã¯åæ°ã®è¶³ãç®ãè¡ãããšãã§ããã
command: 3/5 + 1/5;
maximaã¯ç°ãªãåæ¯ã®èšç®ãæ£ããè¡ãªãããšãåºæ¥ãã
command: 1/5 + 1/3;
maximaã¯åæ°ã®ããç®ãè¡ãªãããšãåºæ¥ãã
command: 1/5 * 2/3;
maximaã¯åæ°ã®å²ãç®ãè¡ãªãããšãåºæ¥ãã
command: 1 / (1/2);
command: 1 / (2/3);
(ãã£ãã¯çç¥ã§ããªãã)
====äžåŠæ ¡====
Maximaã¯æ£è² ã®æ°ãæ··ãã£ãååæŒç®ã
è¡ãªãããšãåºæ¥ãã
command: -3 + 4;
command: -3*4;
command: 1*(-1)*(-1);
Maximaã¯æååŒã®ååæŒç®ã
è¡ãªãããšãåºæ¥ãã
command: x+x;
command: x+y;
command: x+3*y+4*y;
command: 2*x * 3*y;
command: x * 2*x;
or command: x*2*x;
maximaã¯æ¹çšåŒãæ±ããã
command:solve([x+3=4],[x]);
command:solve([2*x=1],[x]);
maximaã¯2å
æ¹çšåŒãæ±ããã
command: solve([x+2*y = 1, 2*x+y = 3],[x,y]);
maximaã¯å¹³æ¹æ ¹ã®å€ãä»»æã®æ¡ãŸã§æ±ããããšãåºæ¥ãã
command:bfloat(sqrt(3));
Maximaã¯åŒã®å±éãšå æ°å解ãè¡ãªãããšãåºæ¥ãã
command:expand((a+b)*(c+d));
command:factor(a^2-b^2);
maximaã¯äºæ¬¡æ¹çšåŒãæ±ãããšãã§ããã
command:solve([x^2-1=0],[x]);
==== é«çåŠæ ¡ ====
maximaã¯è€çŽ æ°ããµããŒãããŠããã
command: %iãiã«å¯Ÿå¿ããã
Note:maximaã¯ææ°eã®å€ãç¥ã£ãŠããã
(å®éã«ã¯éåžžã®ã³ã³ãã¥ãŒã¿ãŒã¯
ãã®å€ãèšç®ã§ããããã«ãªã£ãŠããã¯ãã§ããã
)
command:bfloat(%e);
maximaã¯ã埮ç©åããµããŒãããã
command:diff(f(x),x);
command:integrate(f(x),x);
command:integrate(f(x),[x,a,b]);
note: maxima ã¯åŸ®åæ³ããµããŒãããã
command: diff(f(x)+g(x),x);
command: diff(af(x),x);
command: diff(f(x)*g(x),x);
command: diff(1/f(x),x);
maximaã¯è¡åã®æŒç®ããµããŒãããã
command: A:matrix([a,b],[c,d]);
command: B:matrix([e,f],[g,h]);
command: A + B
command: A.B (è¡åã®ç©)
command: A^^-1 (éè¡å)
[[Category:Maxima|ããããŠããªã€ãããã]] | null | 2015-08-08T11:27:56Z | [] | https://ja.wikibooks.org/wiki/Maxima/%E5%85%B7%E4%BD%93%E7%9A%84%E3%81%AA%E4%BD%BF%E3%81%84%E6%96%B9 |
2,102 | ç ç® ä¹ç® | æ°åŠ>ç ç®>ä¹ç®
äžæ¡ã®æãç®ã¯è¶³ãç®ãšä¹ä¹ããåºæ¥ãã°ã§ããã
äŸé¡.123456Ã7=
ãŸã123456ã眮ãã
次ã«1ãæã£ãŠ1Ã7=7ãäžæ¡é¢ããæã«çœ®ãã ããã§æãç®ã®çµæãäºæ¡ã«ãªã£ããšãã¯ãäžã®äœãäžæ¡é¢ãããšããã«çœ®ãã
次ã«2ãæã£ãŠ2Ã7=14ã眮ãããã®ãšãç¹°ãäžãããããããšã«æ³šæã
以äžåãäœæ¥ãç¹°ãè¿ãã
ããããŠåºã864192ãçãã§ãããæåã«äžæ¡é¢ããŠçœ®ããã®ã§äœãäºã€ãããŠããããšã«æ³šæãç¹ã«äžã®äœã0ã«ãªããšãã«ééããããã
ãŸãã¯9Ã123ããã£ãŠã¿ããã
æããããæ°ãäžæ¡ãªã®ã§äžæ¡é¢ããŠ9Ã1=9ã眮ãã
æåã®ãã¡ã¯æããæ°ã¯æ®ããŠããã»ããè¯ãã 次ã«9Ã2=18ã眮ãã
æåŸã«9Ã3=27ã眮ãã
ããã§çã1107ã ãšåããã
次ã«987Ã123ãããŠã¿ããã
ããã»ã©ãã£ãããã«9Ã123ãããã
次ã«ãã®äžãã8Ã123ãããã 次ã®èšç®ã§æ¡ãééããªãããã«ãäžæ¡é¢ããæãäžã®äœãªã®ã§ã1Ã8=8ã¯ä»0ãç«ã£ãŠããäœã§ããã
7Ã123ã¯çç¥ããã®ã§èªåã§ãã£ãŠã¿ããã
çãã¯121401ãšãªã£ãã
æ°åãé·ããªããšèªã¿ã¥ãããããã§äžæ¡ããšã«ã³ã³ããæã€ã
ã³ã³ããæã€ãšãã«ã¯ç®ç€äžã®ç¹ãç®å®ãšãªãã
次ã¯å²ãç®ããã£ãŠã¿ããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "æ°åŠ>ç ç®>ä¹ç®",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "äžæ¡ã®æãç®ã¯è¶³ãç®ãšä¹ä¹ããåºæ¥ãã°ã§ããã",
"title": "äžæ¡ãæãã"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "äŸé¡.123456Ã7=",
"title": "äžæ¡ãæãã"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ãŸã123456ã眮ãã",
"title": "äžæ¡ãæãã"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "次ã«1ãæã£ãŠ1Ã7=7ãäžæ¡é¢ããæã«çœ®ãã ããã§æãç®ã®çµæãäºæ¡ã«ãªã£ããšãã¯ãäžã®äœãäžæ¡é¢ãããšããã«çœ®ãã",
"title": "äžæ¡ãæãã"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "次ã«2ãæã£ãŠ2Ã7=14ã眮ãããã®ãšãç¹°ãäžãããããããšã«æ³šæã",
"title": "äžæ¡ãæãã"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "以äžåãäœæ¥ãç¹°ãè¿ãã",
"title": "äžæ¡ãæãã"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ããããŠåºã864192ãçãã§ãããæåã«äžæ¡é¢ããŠçœ®ããã®ã§äœãäºã€ãããŠããããšã«æ³šæãç¹ã«äžã®äœã0ã«ãªããšãã«ééããããã",
"title": "äžæ¡ãæãã"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "ãŸãã¯9Ã123ããã£ãŠã¿ããã",
"title": "è€æ°æ¡ãæãã"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "æããããæ°ãäžæ¡ãªã®ã§äžæ¡é¢ããŠ9Ã1=9ã眮ãã",
"title": "è€æ°æ¡ãæãã"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "æåã®ãã¡ã¯æããæ°ã¯æ®ããŠããã»ããè¯ãã 次ã«9Ã2=18ã眮ãã",
"title": "è€æ°æ¡ãæãã"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "æåŸã«9Ã3=27ã眮ãã",
"title": "è€æ°æ¡ãæãã"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "ããã§çã1107ã ãšåããã",
"title": "è€æ°æ¡ãæãã"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "次ã«987Ã123ãããŠã¿ããã",
"title": "è€æ°æ¡ãæãã"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "ããã»ã©ãã£ãããã«9Ã123ãããã",
"title": "è€æ°æ¡ãæãã"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "次ã«ãã®äžãã8Ã123ãããã 次ã®èšç®ã§æ¡ãééããªãããã«ãäžæ¡é¢ããæãäžã®äœãªã®ã§ã1Ã8=8ã¯ä»0ãç«ã£ãŠããäœã§ããã",
"title": "è€æ°æ¡ãæãã"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "7Ã123ã¯çç¥ããã®ã§èªåã§ãã£ãŠã¿ããã",
"title": "è€æ°æ¡ãæãã"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "çãã¯121401ãšãªã£ãã",
"title": "è€æ°æ¡ãæãã"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "æ°åãé·ããªããšèªã¿ã¥ãããããã§äžæ¡ããšã«ã³ã³ããæã€ã",
"title": "ã³ã³ããæã€"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "ã³ã³ããæã€ãšãã«ã¯ç®ç€äžã®ç¹ãç®å®ãšãªãã",
"title": "ã³ã³ããæã€"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "次ã¯å²ãç®ããã£ãŠã¿ããã",
"title": "ã³ã³ããæã€"
}
] | æ°åŠïŒç ç®ïŒä¹ç® | [[æ°åŠ]]ïŒ[[ç ç®]]ïŒä¹ç®
==äžæ¡ãæãã==
äžæ¡ã®æãç®ã¯[[ç ç®_å æžç®|足ãç®]]ãšä¹ä¹ããåºæ¥ãã°ã§ããã
äŸé¡.123456×7=
ãŸã123456ã眮ãã
< >< >< >< >< >< >< >< >< >< >< >< > < >< >< >< >
< >< >
====*========*========*========*========*========*====
< >< >< >< > < >
< >< >< >< >< >< >< >< > < >< >< >< > < >< >< >< >
< >< >< >< >< >< >< >< >< > < >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< > < >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< >< > < >< >< >< >< >< >
1 2 3 4 5 6
次ã«1ãæã£ãŠ1×7=7ãäžæ¡é¢ããæã«çœ®ãã
ããã§æãç®ã®çµæãäºæ¡ã«ãªã£ããšãã¯ãäžã®äœãäžæ¡é¢ãããšããã«çœ®ãã
< >< >< >< >< >< > < >< >< >< >< > < >< >< >< >
< > < >< >
====*========*========*========*========*========*====
< > < >< >< > < >
< >< >< >< >< >< >< >< >< >< >< >< >< > < >< >< >< >
< >< >< >< >< >< > < >< > < >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< > < >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< >< > < >< >< >< >< >< >
7 2 3 4 5 6
次ã«2ãæã£ãŠ2×7=14ã眮ãããã®ãšãç¹°ãäžãããããããšã«æ³šæã
< >< >< >< >< >< > < >< >< >< >< > < >< >< >< >
< > < >< >
====*========*========*========*========*========*====
< >< > < >< > < >
< >< >< >< >< >< >< >< >< >< >< >< >< > < >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< > < >< >< > < >< >< >< >< >< >< >
< >< >< >< >< >< >< > < >< >< > < >< >< >< >< >< >
+1 4 3 4 5 6
以äžåãäœæ¥ãç¹°ãè¿ãã
< >< >< >< >< >< > < >< >< >< > < >< >< >< >
< >< > < >< >
====*========*========*========*========*========*====
< >< >< > < > < >
< >< >< >< >< >< >< > < >< >< >< > < >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< > < >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< >< > < >< >< >< >< >< >
+2 1 4 5 6
< >< >< >< >< >< > < > < >< > < >< >< >< >
< >< > < > < >< >
====*========*========*========*========*========*====
< >< >< >< > < >
< >< >< >< >< >< >< > < >< >< >< >< > < >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< > < > < >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
+2 8 5 6
< >< >< >< >< >< > < >< > < >< > < >< >< >< >
< >< > < > < >
====*========*========*========*========*========*====
< >< >< >< > < >
< >< >< >< >< >< >< > < > < >< >< > < >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< > < >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< > < >< >< >< >< >< >< >< >< >
+3 5 6
< >< >< >< >< >< > < >< > < >< >< >< >< >< >< >
< >< > < >
====*========*========*========*========*========*====
< >< >< >< >< >< >
< >< >< >< >< >< >< > < > < >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< >< > < >< >< >< >< >< >
< >< >< >< >< >< > < >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< > < > < >< >< >< >< >< >< >
+4 2
ããããŠåºã864192ãçãã§ãããæåã«äžæ¡é¢ããŠçœ®ããã®ã§äœãäºã€ãããŠããããšã«æ³šæãç¹ã«äžã®äœã0ã«ãªããšãã«ééããããã
==è€æ°æ¡ãæãã==
===äžæ¡Ãè€æ°æ¡===
ãŸãã¯9×123ããã£ãŠã¿ããã
< >< >< >< >< > < >
< >
=======*========*====
< >
< >< >< >< >< >< >< >
< >< >< >< >< >< >< >
< >< >< >< >< >< >< >
< >< >< >< >< > < >
9
æããããæ°ãäžæ¡ãªã®ã§äžæ¡é¢ããŠ9×1=9ã眮ãã
< > < >< >< > < >
< > < >
=======*========*====
< > < >
< >< >< >< >< >< >< >
< >< >< >< >< >< >< >
< >< >< >< >< >< >< >
< > < >< >< > < >
9 9
æåã®ãã¡ã¯æããæ°ã¯æ®ããŠããã»ããè¯ãã
次ã«9×2=18ã眮ãã
< >< > < >< > < >
< > < >
=======*========*====
< > < > < >
< >< >< >< >< >< >
< >< >< >< >< >< >< >
< >< > < >< >< >< >
< >< >< >< >< > < >
+1 8 9
æåŸã«9×3=27ã眮ãã
< >< >< > < >< >< >
< >
=======*========*====
< >< > < >
< >< >< >< >< >
< >< >< > < >< >< >
< >< >< >< >< >< >< >
< >< >< >< >< >< >< >
+2 7
ããã§çã1107ã ãšåããã
===è€æ°æ¡Ãè€æ°æ¡===
次ã«987×123ãããŠã¿ããã
< >< >< >< >< >< >< >< >< >< >< > < >< >< >< >
< >< >< >
====*========*========*========*========*========*====
< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< > < >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< > < >< >< >< >< >
< >< >< >< >< >< >< >< >< >< >< > < >< >< >< >< >< >
9 8 7
ããã»ã©ãã£ãããã«9×123ãããã
< >< >< >< >< >< >< >< >< > < >< > < >< >< >< >
< > < >< >
====*========*========*========*========*========*====
< >< > < > < >< >
< >< >< >< >< >< > < >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< > < >< >< > < >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< > < >< >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
1 1 0 7 8 7
次ã«ãã®äžãã8×123ãããã
次ã®èšç®ã§æ¡ãééããªãããã«ãäžæ¡é¢ããæãäžã®äœãªã®ã§ã1×8=8ã¯ä»0ãç«ã£ãŠããäœã§ããã
< >< >< >< >< >< >< >< > < >< > < >< >< >< >
< >< > < >< >
====*========*========*========*========*========*====
< >< >< >< > < >< >
< >< >< >< >< >< > < >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< > < >< >< > < >< >< >< >
< >< >< >< >< >< >< >< > < >< >< > < >< >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
+8 8 7
< >< >< >< >< >< >< >< >< >< >< >< > < >< >< >< >
< >< >
====*========*========*========*========*========*====
< >< > < > < >< >
< >< >< >< >< >< > < >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< > < >< >< >< >< > < >< >< >< >
< >< >< >< >< >< >< >< >< > < >< > < >< >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
+1 6 8 7
< >< >< >< >< >< >< >< >< > < >< >< > < >< >< >< >
< > < >
====*========*========*========*========*========*====
< >< > < > < >
< >< >< >< >< >< > < >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< > < >< >< >< >< > < >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< > < >< >< >< >< >< >< >
+2 4 7
7×123ã¯çç¥ããã®ã§èªåã§ãã£ãŠã¿ããã
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
====*========*========*========*========*========*====
< >< >< >< > < >
< >< >< >< >< >< > < > < >< > < >< >< >< >< >< >
< >< >< >< >< >< >< > < >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< > < >< >< >< >< >< >< >< >
çãã¯121401ãšãªã£ãã
==ã³ã³ããæã€==
æ°åãé·ããªããšèªã¿ã¥ãããããã§äžæ¡ããšã«ã³ã³ããæã€ã
*123456789â123,456,789
*1099511627776â1,099,511,627,776
ã³ã³ããæã€ãšãã«ã¯ç®ç€äžã®ç¹ãç®å®ãšãªãã
次ã¯[[ç ç®_é€ç®|å²ãç®]]ããã£ãŠã¿ããã
[[Category:ç ç®|ããããã]] | null | 2006-12-12T15:27:40Z | [] | https://ja.wikibooks.org/wiki/%E7%8F%A0%E7%AE%97_%E4%B9%97%E7%AE%97 |
2,103 | ç ç® é€ç® | æ°åŠ>ç ç®>é€ç®
789÷3ããã£ãŠã¿ããã
7÷3=2ããŸã1ã§ããããããŠéã«3Ã2ãããŠ6ãåºãã
å2ãäžæ¡é¢ããå
ã«ç«ãŠã3Ã2=6ãæãã
次ã«18÷3=6ã§ã¡ããã©å²ãåãããå6ãäžæ¡é¢ããå
ã«ç«ãŠ18ãæãã
æåŸã«9÷3=3ã§ã¡ããã©å²ãåããã®ã§å3ãäžæ¡é¢ããå
ã«ç«ãŠ9ãæãã
äœãããªããªã£ãã®ã§ããã§çµããã§ãããäœããåºããããªãå°æ°ç¹ä»¥äžãç¶ããŠããŸããªãã
次ã«5535÷45ããã£ãŠã¿ããã
ãŸã55÷45=1ããŸã10ã§ãããå1ãç«ãŠã45Ã1=45ãæãã
次ã«103÷45=2ããŸã13ã§ãããå2ãç«ãŠã45ÃÃ2=90ãæãã
æåŸã«135÷45=3ã§å²ãåãããå3ãç«ãŠã45Ã3=135ãæãã
ããã§åã¯123ãšåããã
å²ãç®ã¯å°ãæç®ãåãåãããããšãå¿
èŠã«ãªãã®ã§ãæåã®ãã¡ã¯ééããå€ãã
ããã§ãèªä¿¡ã®ç¡ãåãåºããåã³æãçŽãããšããå§ãããã
ãã¡ããa÷b=cãªãcÃb=aã§ããã
å²ãç®ãã§ãããéå¹³ããã£ãŠã¿ããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "æ°åŠ>ç ç®>é€ç®",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "789÷3ããã£ãŠã¿ããã",
"title": "äžæ¡ã®å²ãç®"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "7÷3=2ããŸã1ã§ããããããŠéã«3Ã2ãããŠ6ãåºãã",
"title": "äžæ¡ã®å²ãç®"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "å2ãäžæ¡é¢ããå
ã«ç«ãŠã3Ã2=6ãæãã",
"title": "äžæ¡ã®å²ãç®"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "次ã«18÷3=6ã§ã¡ããã©å²ãåãããå6ãäžæ¡é¢ããå
ã«ç«ãŠ18ãæãã",
"title": "äžæ¡ã®å²ãç®"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "æåŸã«9÷3=3ã§ã¡ããã©å²ãåããã®ã§å3ãäžæ¡é¢ããå
ã«ç«ãŠ9ãæãã",
"title": "äžæ¡ã®å²ãç®"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "äœãããªããªã£ãã®ã§ããã§çµããã§ãããäœããåºããããªãå°æ°ç¹ä»¥äžãç¶ããŠããŸããªãã",
"title": "äžæ¡ã®å²ãç®"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "次ã«5535÷45ããã£ãŠã¿ããã",
"title": "è€æ°æ¡ã§å²ã"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "ãŸã55÷45=1ããŸã10ã§ãããå1ãç«ãŠã45Ã1=45ãæãã",
"title": "è€æ°æ¡ã§å²ã"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "次ã«103÷45=2ããŸã13ã§ãããå2ãç«ãŠã45ÃÃ2=90ãæãã",
"title": "è€æ°æ¡ã§å²ã"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "æåŸã«135÷45=3ã§å²ãåãããå3ãç«ãŠã45Ã3=135ãæãã",
"title": "è€æ°æ¡ã§å²ã"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "ããã§åã¯123ãšåããã",
"title": "è€æ°æ¡ã§å²ã"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "å²ãç®ã¯å°ãæç®ãåãåãããããšãå¿
èŠã«ãªãã®ã§ãæåã®ãã¡ã¯ééããå€ãã",
"title": "æ€ç®"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ããã§ãèªä¿¡ã®ç¡ãåãåºããåã³æãçŽãããšããå§ãããã",
"title": "æ€ç®"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "ãã¡ããa÷b=cãªãcÃb=aã§ããã",
"title": "æ€ç®"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "å²ãç®ãã§ãããéå¹³ããã£ãŠã¿ããã",
"title": "æ€ç®"
}
] | æ°åŠïŒç ç®ïŒé€ç® | [[æ°åŠ]]ïŒ[[ç ç®]]ïŒé€ç®
==äžæ¡ã®å²ãç®==
789÷3ããã£ãŠã¿ããã
< >< >< >< >< >< >< >< > < >< >< >< >< >< >< >
< >< >< >
====*========*========*========*========*========*====
< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< > < >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< > < >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< > < >< >< >< >< >< >< >
7 8 9
7÷3=2ããŸã1ã§ããããããŠéã«3×2ãããŠ6ãåºãã
å2ãäžæ¡é¢ããå
ã«ç«ãŠã3×2=6ãæãã
< >< >< >< >< >< >< >< >< > < >< >< >< >< >< >< >
< >< >
====*========*========*========*========*========*====
< > < >< >< >
< >< >< >< >< >< >< >< > < >< >< >< >< >< >< >< >< >
< >< >< >< >< >< > < >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< > < >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< > < >< >< >< >< >< >< >
2 -6
次ã«18÷3=6ã§ã¡ããã©å²ãåãããå6ãäžæ¡é¢ããå
ã«ç«ãŠ18ãæãã
< >< >< >< >< >< >< > < >< > < >< >< >< >< >< >< >
< > < >
====*========*========*========*========*========*====
< >< > < >
< >< >< >< >< >< >< > < >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< > < >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< > < >< >< >< >< >< >< >
2 6 -1 8
æåŸã«9÷3=3ã§ã¡ããã©å²ãåããã®ã§å3ãäžæ¡é¢ããå
ã«ç«ãŠ9ãæãã
< >< >< >< >< >< >< > < >< >< >< >< >< >< >< >< >< >
< >
====*========*========*========*========*========*====
< >< >< >
< >< >< >< >< >< >< > < >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< > < >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< > < >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
2 6 3
äœãããªããªã£ãã®ã§ããã§çµããã§ãããäœããåºããããªãå°æ°ç¹ä»¥äžãç¶ããŠããŸããªãã
==è€æ°æ¡ã§å²ã==
次ã«5535÷45ããã£ãŠã¿ããã
< >< >< >< >< >< >< > < >< >< >< >< >< >< >
< >< > < >
====*========*========*========*========*========*====
< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< > < >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
5 5 3 5
ãŸã55÷45=1ããŸã10ã§ãããå1ãç«ãŠã45×1=45ãæãã
< >< >< >< >< >< >< >< >< > < >< >< >< >< >< >< >
< >
====*========*========*========*========*========*====
< > < > < >
< >< >< >< >< > < > < >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< > < >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
1 -4 5
次ã«103÷45=2ããŸã13ã§ãããå2ãç«ãŠã45×Ã2=90ãæãã
< >< >< >< >< >< >< >< >< > < >< >< >< >< >< >< >
< >
====*========*========*========*========*========*====
< >< > < >< >
< >< >< >< >< > < >< > < >< >< >< >< >< >< >< >< >
< >< >< >< >< >< > < >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< > < >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
1 2 -9 0
æåŸã«135÷45=3ã§å²ãåãããå3ãç«ãŠã45×3=135ãæãã
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
====*========*========*========*========*========*====
< >< >< >
< >< >< >< >< > < >< >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< > < >< >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< > < >< >< >< >< >< >< >< >< >< >
< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >< >
1 2 3 -1 3 5
ããã§åã¯123ãšåããã
==æ€ç®==
å²ãç®ã¯å°ãæç®ãåãåãããããšãå¿
èŠã«ãªãã®ã§ãæåã®ãã¡ã¯ééããå€ãã
ããã§ãèªä¿¡ã®ç¡ãåãåºããåã³æãçŽãããšããå§ãããã
ãã¡ããa÷b=cãªãc×b=aã§ããã
å²ãç®ãã§ããã[[ç ç®_éå¹³ã»éç«|éå¹³]]ããã£ãŠã¿ããã
[[Category:ç ç®|ãããã]] | null | 2006-12-12T15:27:12Z | [] | https://ja.wikibooks.org/wiki/%E7%8F%A0%E7%AE%97_%E9%99%A4%E7%AE%97 |
2,105 | æ¯åãšæ³¢å | æ¬é
ã¯ç©çåŠ æ¯åãšæ³¢å ã®è§£èª¬ã§ãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "æ¬é
ã¯ç©çåŠ æ¯åãšæ³¢å ã®è§£èª¬ã§ãã",
"title": ""
}
] | æ¬é
ã¯ç©çåŠ æ¯åãšæ³¢å ã®è§£èª¬ã§ãã ã¯ããã«
1ç²åã®æ¯å
åæ¯å
é床ã«æ¯äŸããæµæåãããå Žåã®åæ¯å
匷å¶æ¯å
è€æ°ç²åã®æ¯å
2ç²åã®å Žå
è€æ°ç²åã®å Žå
å€ç²åã®å Žå
é£ç¶æ¥µéãžã®ç§»è¡
æ³¢åæ¹çšåŒã®æ§è³ª
1次å
ã®æ³¢åæ¹çšåŒ
2次å
å¹³é¢äžã®æ³¢
3次å
空éäžã®æ³¢ | {{Pathnav|ã¡ã€ã³ããŒãž|èªç¶ç§åŠ|ç©çåŠ|frame=1|small=1}}
æ¬é
ã¯ç©çåŠ æ¯åãšæ³¢å ã®è§£èª¬ã§ãã
* [[æ¯åãšæ³¢å/ã¯ããã«|ã¯ããã«]]
* [[æ¯åãšæ³¢å/1ç²åã®æ¯å|1ç²åã®æ¯å]]
** [[æ¯åãšæ³¢å/1ç²åã®æ¯å#åæ¯å|åæ¯å]]
** [[æ¯åãšæ³¢å/1ç²åã®æ¯å#é床ã«æ¯äŸããæµæåãããå Žåã®åæ¯å|é床ã«æ¯äŸããæµæåãããå Žåã®åæ¯å]]
** [[æ¯åãšæ³¢å/1ç²åã®æ¯å#匷å¶æ¯å|匷å¶æ¯å]]
* [[æ¯åãšæ³¢å/è€æ°ç²åã®æ¯å|è€æ°ç²åã®æ¯å]]
** [[æ¯åãšæ³¢å/è€æ°ç²åã®æ¯å#2ç²åã®å Žå|2ç²åã®å Žå]]
** [[æ¯åãšæ³¢å/è€æ°ç²åã®æ¯å#è€æ°ç²åã®å Žå|è€æ°ç²åã®å Žå]]
** [[æ¯åãšæ³¢å/è€æ°ç²åã®æ¯å#å€ç²åã®å Žå|å€ç²åã®å Žå]]
** [[æ¯åãšæ³¢å/è€æ°ç²åã®æ¯å#é£ç¶æ¥µéãžã®ç§»è¡|é£ç¶æ¥µéãžã®ç§»è¡]]
* [[æ¯åãšæ³¢å/æ³¢åæ¹çšåŒã®æ§è³ª|æ³¢åæ¹çšåŒã®æ§è³ª]]
** [[æ¯åãšæ³¢å/æ³¢åæ¹çšåŒã®æ§è³ª#1次å
ã®æ³¢åæ¹çšåŒ|1次å
ã®æ³¢åæ¹çšåŒ]]
** [[æ¯åãšæ³¢å/æ³¢åæ¹çšåŒã®æ§è³ª#2次å
å¹³é¢äžã®æ³¢|2次å
å¹³é¢äžã®æ³¢]]
** [[æ¯åãšæ³¢å/æ³¢åæ¹çšåŒã®æ§è³ª#3次å
空éäžã®æ³¢|3次å
空éäžã®æ³¢]]
{{DEFAULTSORT:ãããšããšã¯ãšã}}
[[Category:æ¯åãšæ³¢å|*]]
{{NDC|424}} | 2005-06-04T09:17:34Z | 2024-03-16T02:55:36Z | [
"ãã³ãã¬ãŒã:Pathnav",
"ãã³ãã¬ãŒã:NDC"
] | https://ja.wikibooks.org/wiki/%E6%8C%AF%E5%8B%95%E3%81%A8%E6%B3%A2%E5%8B%95 |
2,106 | æ¯åãšæ³¢å/ã¯ããã« | æ¯åãšæ³¢å > ã¯ããã«
ãã®é
ã§ã¯æ³¢åçŸè±¡ãè¡šããæ¹æ³ãåŠã¶ã
æ³¢åçŸè±¡ã®å°å
¥ã®ããã«ååŠçãªæ¯åãçšããããæ¯åãšããçŸè±¡ã¯ååŠçãªãã®ã«éãããä»åéã§ãçŸããããç¹ã«ãé»æ°åè·¯ã«ãããæ¯å(ã³ã³ãã³ãµãšã³ã€ã«ãªã©ãçšãã)ã¯éèŠãªå¿çšã§ããã
ãŸãããã®åéã¯é«çæè²ã®æ³¢åã«åœãããååŠè
ã¯è©²åœæç§æžãç解ã®å©ããšãªãããåŠç¿ã«è¡ãè©°ãŸã£ããåç
§ããé¡ããããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "æ¯åãšæ³¢å > ã¯ããã«",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ãã®é
ã§ã¯æ³¢åçŸè±¡ãè¡šããæ¹æ³ãåŠã¶ã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "æ³¢åçŸè±¡ã®å°å
¥ã®ããã«ååŠçãªæ¯åãçšããããæ¯åãšããçŸè±¡ã¯ååŠçãªãã®ã«éãããä»åéã§ãçŸããããç¹ã«ãé»æ°åè·¯ã«ãããæ¯å(ã³ã³ãã³ãµãšã³ã€ã«ãªã©ãçšãã)ã¯éèŠãªå¿çšã§ããã",
"title": "ã¯ããã«"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ãŸãããã®åéã¯é«çæè²ã®æ³¢åã«åœãããååŠè
ã¯è©²åœæç§æžãç解ã®å©ããšãªãããåŠç¿ã«è¡ãè©°ãŸã£ããåç
§ããé¡ããããã",
"title": "ã¯ããã«"
}
] | æ¯åãšæ³¢å > ã¯ããã« | <small> [[æ¯åãšæ³¢å]] > ã¯ããã«</small>
----
==ã¯ããã«==
ãã®é
ã§ã¯æ³¢åçŸè±¡ãè¡šããæ¹æ³ãåŠã¶ã
æ³¢åçŸè±¡ã®å°å
¥ã®ããã«ååŠçãªæ¯åãçšããããæ¯åãšããçŸè±¡ã¯ååŠçãªãã®ã«éãããä»åéã§ãçŸããããç¹ã«ãé»æ°åè·¯ã«ãããæ¯å(ã³ã³ãã³ãµãšã³ã€ã«ãªã©ãçšãã)ã¯éèŠãªå¿çšã§ããã
<!-- ãšã¯ããããããã¯ç©çåŠãšãããã㯠-->
<!-- å·¥åŠ _é»æ°åè·¯" çã§æ±ãããã ããã -->
ãŸãããã®åéã¯é«çæè²ã®[[é«çåŠæ ¡_ç©ç#æ³¢å|æ³¢å]]ã«åœãããååŠè
ã¯è©²åœæç§æžãç解ã®å©ããšãªãããåŠç¿ã«è¡ãè©°ãŸã£ããåç
§ããé¡ããããã
{{DEFAULTSORT:ãããšããšã¯ãšãã¯ããã«}}
[[Category:æ¯åãšæ³¢å|0]] | 2005-06-05T02:47:26Z | 2024-03-16T02:56:33Z | [] | https://ja.wikibooks.org/wiki/%E6%8C%AF%E5%8B%95%E3%81%A8%E6%B3%A2%E5%8B%95/%E3%81%AF%E3%81%98%E3%82%81%E3%81%AB |
2,107 | æ¯åãšæ³¢å/1ç²åã®æ¯å | ç©äœãå
ãåãããšãã®éåãèããããã®ãšãéåæ¹çšåŒã¯ã
ã§è¡šããããå€æ°k ããk = m Ï ã®é¢ä¿ã䜿ã£ãŠÏã«çœ®ãæãããšã
ãåŸãããããã®åŒã¯ãå®æ°ä¿æ°ç·åœ¢2é埮åæ¹çšåŒã§ããã®ã§ç°¡åã«è§£ãããšãã§ãã解ã¯ã
ãšãªããããã§A , B ã¯ä»»æå®æ°ã§ããããããã決ããããã«ã¯2ã€ã®åææ¡ä»¶ãå¿
èŠã ããããã§ã¯ç¹ã«ãt = 0 ã§ã
ãšãããš
ã§è¡šãããããã£ãŠããã®åŒã®è§£ã¯
ãšãªãããã®éåãåæ¯åãÏãåºæè§æ¯åæ°ãšåŒã¶ã
次ã«åã
ã§äžããããå Žåãèããã第2é
ã¯ãé床ã«æ¯äŸããåã§ããããã®æ§ãªåã¯ç©ºæ°æµæãªã©ã«èŠãããããã®å Žåã®éåæ¹çšåŒã¯
ãšãªãã ãããå®æ°ä¿æ°ã®2é埮åæ¹çšåŒãªã®ã§è§£ããã®ã ããããã§ã¯å
·äœçã«èšç®ããããŸãã
ãšä»®å®ããããããéåæ¹çšåŒã«ä»£å
¥ãããš
ãåŸãããa ã¯
ã§äžããããããšãåãããããã§ãæµæåã®ä¿æ°ã§ããγãå°ããæ°ã§ãããšãããªããæ ¹å·ã®äžèº«ã¯è² ã«ãªãããã®è§£ã¯è€çŽ æ°ã«ãã£ãŠäžãããããå®éšããšããšäžè¬è§£ã¯
ãšãªã(A , B , α, βã¯ä»»æä¿æ°)ããã®è§£ã§ãä¿æ° exp(-γt /2) ã¯ç²åã®éåãæµæåãåããŠæéçã«æžè¡°ããŠããæ§åã瀺ããŠãããäžæ¹
ã®é
ã¯ããã®ç©äœãÏã«è¿ãè§æ¯åæ°ã§æ¯åããŠããããšã瀺ããŠããããã®æ¯åãæžè¡°æ¯åãšåŒã¶ã
ç©äœãåæ¯åã®åã«å ããŠãåšæçãªå€åãåããŠããå Žåãèããããã®ãšãã
ãšãªã(å€åã®å€§ãããè¡šããã©ã¡ãŒã¿ãšããŠãããã§ã¯ãåŸã®èšç®ãç°¡åã«ããããã«ããã©ã¡ãŒã¿ãml ãšãã圢ã«çœ®ãã)ã
ãã®ãšãéåæ¹çšåŒã¯ã
ãšãªãããã®åŒã¯ãå®æ°ä¿æ°2é垞埮åæ¹çšåŒã«å ããŠãå³èŸºã«é¢æ°é
ãå ãã£ã圢ãããŠããããã®ãšãç¹å¥ãªè§£ã®åœ¢ãäºæž¬ããŠããã®æ¹çšåŒã®ç¹è§£ãæ±ããããšãåºæ¥ããããã§ã¯ã解ã
ãšä»®å®ããããããéåæ¹çšåŒã«ä»£å
¥ãããš
ãåŸãããããã£ãŠã
ã¯ããã®æ¹çšåŒã®ç¹è§£ãšãªãã巊蟺ã®ç·åœ¢æ¹çšåŒã«å¯Ÿããäžè¬è§£ãå ãããšããã®æ¹çšåŒå
šäœã«å¯Ÿãã解ã¯ã
ãšè¡šãããã
ãã®åŒã¯ã2ã€ã®éšåã«åãããŠããããŸãåŸåã®
ã®éšåã¯ãç©äœãå€åãåããŠããŠããåæ¯åãšåãè§æ¯åæ°ã®éåãç¹°ããããããšãè¡šããŠããã次ã«ååã®
ã®éšåã¯ãå€åãšåãè§æ¯åæ° Ï 0 {\displaystyle \omega _{0}} ã®åšæçãªéåãåŒãèµ·ããããããšãè¡šããŠããã
ãã®åŒãèŠããšãÏ = Ï0 ã®ãšããx (t ) ã®å€ãç¡é倧ã«ãªãããã«èŠãããå®éãç³»ã®åºæè§æ¯åæ°Ïãšå€åã®è§æ¯åæ°Ï0 ãè¿ããšããç©äœã«ããããŠå€§ããªæ¯åãåŒãèµ·ããããããšãç¥ãããŠããããã®çŸè±¡ãå
±é³ŽãšåŒã¶ã
å
±é³Žã®å Žåã«x ãç¡é倧ã«ãªããšããçµæãã§ãã®ã¯ããã¡ããç©ççã«ãããããšã§ã¯ãªããäžã®è§£æ³(ç¹ã«ç¹è§£ãšããŠä»®å®ãã圢)ãäžååã ã£ãããšãæå³ãããæ£ãã解ãæ±ããã«ã¯è§£ãšããŠä»®å®ãã圢ãããåºããªããã°ãªããªããããããšã¯å¥ã«ç©ççã«ããããããããæ¹æ³ããããã¢ã€ãã¢ã¯ãäžã®è§£æ³ã§åŸã解ã¯ÏãšÏ0 ãå°ãã§ãéãã°ããŸããããäžæ¹ããã©ã¡ãŒã¿ãé£ç¶çã«å€ãããšçŸè±¡ã(å€å)é£ç¶çã«å€ããã¯ãã ãããÏ = Ï0 ã®å Žåãšããã®ã¯ãÏ0âÏ ã®æ¥µéãšããŠåŸãããã¯ãã§ãããã ããäžã®è§£ã§ãã®æ¥µéãåãã°å
±é³Žã®å Žåã®æ£ãã解ãåŸããããããšããããšã§ãããäœããäžã®è§£ã§åçŽã«ãã®æ¥µéããšã£ãŠããã¡ããããŸããããªããä¿æ°A , B ãæå³ã倱ãããã§ãããããã§A , B ã®ä»£ãã«åžžã«ã¯ã£ããããç©ççæå³ãæã€éãããªãã¡åææ¡ä»¶(t = 0 ã§ã®x , dx /dt ã®å€)ã䜿ã£ãŠäžã®è§£ãæžãçŽãããã®äžã§Ï0âÏ ã®æ¥µéãåããt = 0 ã§x (0) = X , dx (0)/dt = V ãšãããš
ã«ãªãã®ã§è§£ã¯
ãšãªããããããäžã§Ï0âÏ ã®æ¥µéãåããšã極éã¯ç¢ºãã«ååšããŠæ¬¡ã®ããã«ãªã:
ç¹ã«éèŠãªã®ã¯ç¬¬2é
ã§ãcosã«t ãæãã£ãŠããç¹ã§ãããæ¯åã®æ¯å¹
ãæéã«æ¯äŸããŠå¢å€§ããŠããã®ã§ããããã®çµæããå
±é³Žã®æ¬è³ªãåãããã€ãŸãå
±é³Žã§ã¯å€åããäžãããããšãã«ã®ãŒãèç©ãããŠãããšããã®ãæ¬è³ªã§ããã®ããæ¯å¹
ãæéãšãšãã«æãŠããç¡ã倧ãããªã£ãŠããã®ã§ãã(äžäŒããã®æåãªãšããœãŒãã§ãã寺ã®éãæäžæ¬ã§å€§ããåãããšããã®ããããæ£ç¢ºãªåšæã§æã§æŒãããšãç¶ãããšå
±é³Žã«ããå°ãã¥ã€ãšãã«ã®ãŒãèç©ãããŠæåŸã«ã¯å€§ããªæ¯åã«ãªã)ã
ãªããäžã®è§£æ³ã§ã¯çŸè±¡ã®ãã©ã¡ãŒã¿å€åã«å¯Ÿããé£ç¶æ§ãåæãšããŠããããã®åæã¯ãããŠãæãç«ã€ããã©ããªå Žåã§ããšãŸã§ã¯ãããªããä»ã®å Žåã«ã©ããã確èªããã«ã¯åŸããã解ãå
ã®åŸ®åæ¹çšåŒ(ã§Ï0 = Ïãšãããã®)ã«ä»£å
¥ããã°ããã確ãã«è§£ã«ãªã£ãŠããããšãåããããŸããÏ0 ãÏã«è¿ã¥ãã«ã€ãæ¯åã®ã°ã©ããã©ãå€ãã£ãŠè¡ãããèŠãã®ãèå³æ·±ãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "ç©äœãå",
"title": "åæ¯å"
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ãåãããšãã®éåãèããããã®ãšãéåæ¹çšåŒã¯ã",
"title": "åæ¯å"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ã§è¡šããããå€æ°k ããk = m Ï ã®é¢ä¿ã䜿ã£ãŠÏã«çœ®ãæãããšã",
"title": "åæ¯å"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ãåŸãããããã®åŒã¯ãå®æ°ä¿æ°ç·åœ¢2é埮åæ¹çšåŒã§ããã®ã§ç°¡åã«è§£ãããšãã§ãã解ã¯ã",
"title": "åæ¯å"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ãšãªããããã§A , B ã¯ä»»æå®æ°ã§ããããããã決ããããã«ã¯2ã€ã®åææ¡ä»¶ãå¿
èŠã ããããã§ã¯ç¹ã«ãt = 0 ã§ã",
"title": "åæ¯å"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ãšãããš",
"title": "åæ¯å"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ã§è¡šãããããã£ãŠããã®åŒã®è§£ã¯",
"title": "åæ¯å"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ãšãªãããã®éåãåæ¯åãÏãåºæè§æ¯åæ°ãšåŒã¶ã",
"title": "åæ¯å"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "次ã«åã",
"title": "é床ã«æ¯äŸããæµæåãããå Žåã®åæ¯å"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "ã§äžããããå Žåãèããã第2é
ã¯ãé床ã«æ¯äŸããåã§ããããã®æ§ãªåã¯ç©ºæ°æµæãªã©ã«èŠãããããã®å Žåã®éåæ¹çšåŒã¯",
"title": "é床ã«æ¯äŸããæµæåãããå Žåã®åæ¯å"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "ãšãªãã ãããå®æ°ä¿æ°ã®2é埮åæ¹çšåŒãªã®ã§è§£ããã®ã ããããã§ã¯å
·äœçã«èšç®ããããŸãã",
"title": "é床ã«æ¯äŸããæµæåãããå Žåã®åæ¯å"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "ãšä»®å®ããããããéåæ¹çšåŒã«ä»£å
¥ãããš",
"title": "é床ã«æ¯äŸããæµæåãããå Žåã®åæ¯å"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "ãåŸãããa ã¯",
"title": "é床ã«æ¯äŸããæµæåãããå Žåã®åæ¯å"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ã§äžããããããšãåãããããã§ãæµæåã®ä¿æ°ã§ããγãå°ããæ°ã§ãããšãããªããæ ¹å·ã®äžèº«ã¯è² ã«ãªãããã®è§£ã¯è€çŽ æ°ã«ãã£ãŠäžãããããå®éšããšããšäžè¬è§£ã¯",
"title": "é床ã«æ¯äŸããæµæåãããå Žåã®åæ¯å"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "ãšãªã(A , B , α, βã¯ä»»æä¿æ°)ããã®è§£ã§ãä¿æ° exp(-γt /2) ã¯ç²åã®éåãæµæåãåããŠæéçã«æžè¡°ããŠããæ§åã瀺ããŠãããäžæ¹",
"title": "é床ã«æ¯äŸããæµæåãããå Žåã®åæ¯å"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "ã®é
ã¯ããã®ç©äœãÏã«è¿ãè§æ¯åæ°ã§æ¯åããŠããããšã瀺ããŠããããã®æ¯åãæžè¡°æ¯åãšåŒã¶ã",
"title": "é床ã«æ¯äŸããæµæåãããå Žåã®åæ¯å"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "ç©äœãåæ¯åã®åã«å ããŠãåšæçãªå€åãåããŠããå Žåãèããããã®ãšãã",
"title": "匷å¶æ¯å"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "ãšãªã(å€åã®å€§ãããè¡šããã©ã¡ãŒã¿ãšããŠãããã§ã¯ãåŸã®èšç®ãç°¡åã«ããããã«ããã©ã¡ãŒã¿ãml ãšãã圢ã«çœ®ãã)ã",
"title": "匷å¶æ¯å"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "ãã®ãšãéåæ¹çšåŒã¯ã",
"title": "匷å¶æ¯å"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "ãšãªãããã®åŒã¯ãå®æ°ä¿æ°2é垞埮åæ¹çšåŒã«å ããŠãå³èŸºã«é¢æ°é
ãå ãã£ã圢ãããŠããããã®ãšãç¹å¥ãªè§£ã®åœ¢ãäºæž¬ããŠããã®æ¹çšåŒã®ç¹è§£ãæ±ããããšãåºæ¥ããããã§ã¯ã解ã",
"title": "匷å¶æ¯å"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "ãšä»®å®ããããããéåæ¹çšåŒã«ä»£å
¥ãããš",
"title": "匷å¶æ¯å"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "ãåŸãããããã£ãŠã",
"title": "匷å¶æ¯å"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "ã¯ããã®æ¹çšåŒã®ç¹è§£ãšãªãã巊蟺ã®ç·åœ¢æ¹çšåŒã«å¯Ÿããäžè¬è§£ãå ãããšããã®æ¹çšåŒå
šäœã«å¯Ÿãã解ã¯ã",
"title": "匷å¶æ¯å"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "ãšè¡šãããã",
"title": "匷å¶æ¯å"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "ãã®åŒã¯ã2ã€ã®éšåã«åãããŠããããŸãåŸåã®",
"title": "匷å¶æ¯å"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "ã®éšåã¯ãç©äœãå€åãåããŠããŠããåæ¯åãšåãè§æ¯åæ°ã®éåãç¹°ããããããšãè¡šããŠããã次ã«ååã®",
"title": "匷å¶æ¯å"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "ã®éšåã¯ãå€åãšåãè§æ¯åæ° Ï 0 {\\displaystyle \\omega _{0}} ã®åšæçãªéåãåŒãèµ·ããããããšãè¡šããŠããã",
"title": "匷å¶æ¯å"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "ãã®åŒãèŠããšãÏ = Ï0 ã®ãšããx (t ) ã®å€ãç¡é倧ã«ãªãããã«èŠãããå®éãç³»ã®åºæè§æ¯åæ°Ïãšå€åã®è§æ¯åæ°Ï0 ãè¿ããšããç©äœã«ããããŠå€§ããªæ¯åãåŒãèµ·ããããããšãç¥ãããŠããããã®çŸè±¡ãå
±é³ŽãšåŒã¶ã",
"title": "匷å¶æ¯å"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "å
±é³Žã®å Žåã«x ãç¡é倧ã«ãªããšããçµæãã§ãã®ã¯ããã¡ããç©ççã«ãããããšã§ã¯ãªããäžã®è§£æ³(ç¹ã«ç¹è§£ãšããŠä»®å®ãã圢)ãäžååã ã£ãããšãæå³ãããæ£ãã解ãæ±ããã«ã¯è§£ãšããŠä»®å®ãã圢ãããåºããªããã°ãªããªããããããšã¯å¥ã«ç©ççã«ããããããããæ¹æ³ããããã¢ã€ãã¢ã¯ãäžã®è§£æ³ã§åŸã解ã¯ÏãšÏ0 ãå°ãã§ãéãã°ããŸããããäžæ¹ããã©ã¡ãŒã¿ãé£ç¶çã«å€ãããšçŸè±¡ã(å€å)é£ç¶çã«å€ããã¯ãã ãããÏ = Ï0 ã®å Žåãšããã®ã¯ãÏ0âÏ ã®æ¥µéãšããŠåŸãããã¯ãã§ãããã ããäžã®è§£ã§ãã®æ¥µéãåãã°å
±é³Žã®å Žåã®æ£ãã解ãåŸããããããšããããšã§ãããäœããäžã®è§£ã§åçŽã«ãã®æ¥µéããšã£ãŠããã¡ããããŸããããªããä¿æ°A , B ãæå³ã倱ãããã§ãããããã§A , B ã®ä»£ãã«åžžã«ã¯ã£ããããç©ççæå³ãæã€éãããªãã¡åææ¡ä»¶(t = 0 ã§ã®x , dx /dt ã®å€)ã䜿ã£ãŠäžã®è§£ãæžãçŽãããã®äžã§Ï0âÏ ã®æ¥µéãåããt = 0 ã§x (0) = X , dx (0)/dt = V ãšãããš",
"title": "匷å¶æ¯å"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "ã«ãªãã®ã§è§£ã¯",
"title": "匷å¶æ¯å"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "ãšãªããããããäžã§Ï0âÏ ã®æ¥µéãåããšã極éã¯ç¢ºãã«ååšããŠæ¬¡ã®ããã«ãªã:",
"title": "匷å¶æ¯å"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "ç¹ã«éèŠãªã®ã¯ç¬¬2é
ã§ãcosã«t ãæãã£ãŠããç¹ã§ãããæ¯åã®æ¯å¹
ãæéã«æ¯äŸããŠå¢å€§ããŠããã®ã§ããããã®çµæããå
±é³Žã®æ¬è³ªãåãããã€ãŸãå
±é³Žã§ã¯å€åããäžãããããšãã«ã®ãŒãèç©ãããŠãããšããã®ãæ¬è³ªã§ããã®ããæ¯å¹
ãæéãšãšãã«æãŠããç¡ã倧ãããªã£ãŠããã®ã§ãã(äžäŒããã®æåãªãšããœãŒãã§ãã寺ã®éãæäžæ¬ã§å€§ããåãããšããã®ããããæ£ç¢ºãªåšæã§æã§æŒãããšãç¶ãããšå
±é³Žã«ããå°ãã¥ã€ãšãã«ã®ãŒãèç©ãããŠæåŸã«ã¯å€§ããªæ¯åã«ãªã)ã",
"title": "匷å¶æ¯å"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "ãªããäžã®è§£æ³ã§ã¯çŸè±¡ã®ãã©ã¡ãŒã¿å€åã«å¯Ÿããé£ç¶æ§ãåæãšããŠããããã®åæã¯ãããŠãæãç«ã€ããã©ããªå Žåã§ããšãŸã§ã¯ãããªããä»ã®å Žåã«ã©ããã確èªããã«ã¯åŸããã解ãå
ã®åŸ®åæ¹çšåŒ(ã§Ï0 = Ïãšãããã®)ã«ä»£å
¥ããã°ããã確ãã«è§£ã«ãªã£ãŠããããšãåããããŸããÏ0 ãÏã«è¿ã¥ãã«ã€ãæ¯åã®ã°ã©ããã©ãå€ãã£ãŠè¡ãããèŠãã®ãèå³æ·±ãã",
"title": "匷å¶æ¯å"
}
] | null | {{Pathnav|ã¡ã€ã³ããŒãž|èªç¶ç§åŠ|ç©çåŠ|æ¯åãšæ³¢å|frame=1}}
==åæ¯å==
ç©äœãå
:<math>
f(x) = -kx
</math>
ãåãããšãã®éåãèããããã®ãšãéåæ¹çšåŒã¯ã
:<math>
m\ddot x(t) + kx(t) =0
</math>
ã§è¡šããããå€æ°''k'' ãã''k'' = ''m'' ω<sup>2</sup> ã®é¢ä¿ã䜿ã£ãŠωã«çœ®ãæãããšã
:<math>
\ddot x(t) + \omega^2x(t) =0
</math>
ãåŸãããããã®åŒã¯ãå®æ°ä¿æ°ç·åœ¢2é埮åæ¹çšåŒã§ããã®ã§ç°¡åã«è§£ãããšãã§ãã解ã¯ã
:<math>
x(t) = A \cos \omega t + B \sin\omega t
</math>
ãšãªããããã§''A'' , ''B'' ã¯ä»»æå®æ°ã§ããããããã決ããããã«ã¯2ã€ã®åææ¡ä»¶ãå¿
èŠã ããããã§ã¯ç¹ã«ã''t'' = 0 ã§ã
:<math>
x(0)=x _0,\quad v(0)\equiv \dot{x}(t) = v _0
</math>
ãšãããš
:<math>
A = x _0,\quad
B = \frac {v _0} \omega
</math>
ã§è¡šãããããã£ãŠããã®åŒã®è§£ã¯
:<math>\begin{align}
x(t) &= x _0 \cos \omega t + \frac {v _0} \omega \sin\omega t \\
&=\sqrt{x_0^2+\left(\frac{v_0}{\omega}\right)^2}\cos(\omega t-\theta),\quad \tan\theta=\frac{v_0}{x_0\omega}
\end{align}</math>
ãšãªãããã®éåã'''åæ¯å'''ãωã'''åºæè§æ¯åæ°'''ãšåŒã¶ã
==é床ã«æ¯äŸããæµæåãããå Žåã®åæ¯å==
次ã«åã
:<math>
f(x,\dot{x}) = -kx - m\gamma \dot x
</math>
ã§äžããããå Žåãèããã第2é
ã¯ãé床ã«æ¯äŸããåã§ããããã®æ§ãªåã¯ç©ºæ°æµæãªã©ã«èŠãããããã®å Žåã®éåæ¹çšåŒã¯
:<math>
\begin{align}
m\ddot x(t) + m\gamma \dot x(t) + kx(t) &=0\\
\therefore\quad\ddot x(t) + \gamma \dot x(t) + \omega^2 x(t) &=0
\end{align}
</math>
ãšãªãã
ãããå®æ°ä¿æ°ã®2é埮åæ¹çšåŒãªã®ã§è§£ããã®ã ãã<!--çµæãç¬ç¹ãªã®ã§ã-->ããã§ã¯å
·äœçã«èšç®ããããŸãã
:<math>
x(t) = e ^{at}
</math>
ãšä»®å®ããããããéåæ¹çšåŒã«ä»£å
¥ãããš
:<math>
a^2 + \gamma a + \omega^2 = 0
</math>
ãåŸããã''a'' ã¯
:<math>
a _\pm = \frac 1 2 ( -\gamma \pm \sqrt{\gamma ^2 - 4 \omega^2} )
</math>
ã§äžããããããšãåãããããã§ãæµæåã®ä¿æ°ã§ããγãå°ããæ°ã§ãããšãããªã<!-- ããã§ãªããšãã®èšç® -->ãæ ¹å·ã®äžèº«ã¯è² ã«ãªãããã®è§£ã¯è€çŽ æ°ã«ãã£ãŠäžãããããå®éšããšããšäžè¬è§£ã¯
:<math>
\begin{align}
x(t) &= A e^{i a _+ t} + B e^{i a _- t}\\
&= e^{-\gamma t/2 } \{\alpha \sin (t\sqrt { \omega^2 - \gamma^2 / 4} ) +
\beta \cos (t\sqrt {\omega^2 - \gamma^2 / 4} ) \}
\end{align}
</math>
ãšãªãïŒ''A'' , ''B'' , α, βã¯ä»»æä¿æ°ïŒããã®è§£ã§ãä¿æ° exp(-γ''t'' /2) ã¯ç²åã®éåãæµæåãåããŠæéçã«æžè¡°ããŠããæ§åã瀺ããŠãããäžæ¹
:<math>
\alpha \sin (t\sqrt { \omega^2 - \gamma^2 / 4} ) + \beta \cos (t\sqrt {\omega^2 - \gamma^2 / 4} )
</math>
ã®é
ã¯ããã®ç©äœãωã«è¿ãè§æ¯åæ°ã§æ¯åããŠããããšã瀺ããŠããããã®æ¯åã'''æžè¡°æ¯å'''ãšåŒã¶ã
==匷å¶æ¯å==
ç©äœãåæ¯åã®åã«å ããŠãåšæçãªå€åãåããŠããå Žåãèããããã®ãšãã
:<math>
f(x,t) = -kx + m l\sin \omega _0 t
</math>
ãšãªãïŒå€åã®å€§ãããè¡šããã©ã¡ãŒã¿ãšããŠãããã§ã¯ãåŸã®èšç®ãç°¡åã«ããããã«ããã©ã¡ãŒã¿ã''ml'' ãšãã圢ã«çœ®ããïŒã
ãã®ãšãéåæ¹çšåŒã¯ã
:<math>
\begin{align}
m\ddot x(t)+ m \omega ^2 x(t) &= ml \sin \omega _0 t\\
\therefore \quad \ddot x(t)+ \omega ^2 x(t) &= l \sin \omega _0 t
\end{align}
</math>
ãšãªãããã®åŒã¯ãå®æ°ä¿æ°2é垞埮åæ¹çšåŒã«å ããŠãå³èŸºã«é¢æ°é
ãå ãã£ã圢ãããŠããããã®ãšãç¹å¥ãªè§£ã®åœ¢ãäºæž¬ããŠããã®æ¹çšåŒã®ç¹è§£ãæ±ããããšãåºæ¥ããããã§ã¯ã解ã
:<math>
x(t) = C \sin \omega _0 t
</math>
ãšä»®å®ããããããéåæ¹çšåŒã«ä»£å
¥ãããš
:<math>
\begin{align}
& (- C \omega _0 ^2 + \omega^2 C )\sin \omega _0 t = l \sin \omega _0 t\\
& \therefore \quad C = \frac l {\omega^2 - \omega_0^2 }
\end{align}
</math>
ãåŸãããããã£ãŠã
:<math>
x(t) = \frac l {\omega^2 - \omega_0^2)}\sin \omega _0 t
</math>
ã¯ããã®æ¹çšåŒã®ç¹è§£ãšãªãã巊蟺ã®ç·åœ¢æ¹çšåŒã«å¯Ÿããäžè¬è§£ãå ãããšããã®æ¹çšåŒå
šäœã«å¯Ÿãã解ã¯ã
:<math>
x(t) = \frac l {\omega^2 - \omega_0 ^2}\sin \omega _0 t
+ A \sin \omega t + B \cos \omega t
</math>
ãšè¡šãããã
ãã®åŒã¯ã2ã€ã®éšåã«åãããŠããããŸãåŸåã®
:<math>
A \sin \omega t + B \cos \omega t
</math>
ã®éšåã¯ãç©äœãå€åãåããŠããŠããåæ¯åãšåãè§æ¯åæ°ã®éåãç¹°ããããããšãè¡šããŠããã次ã«ååã®
:<math>
\frac l {- \omega _0 ^2 + \omega^2 }\sin \omega _0 t
</math>
ã®éšåã¯ãå€åãšåãè§æ¯åæ°<math>\omega _0</math>ã®åšæçãªéåãåŒãèµ·ããããããšãè¡šããŠããã
ãã®åŒãèŠããšãω = ω<sub>0</sub> ã®ãšãã''x'' (''t'' ) ã®å€ãç¡é倧ã«ãªãããã«èŠãããå®éãç³»ã®åºæè§æ¯åæ°ωãšå€åã®è§æ¯åæ°ω<sub>0</sub> ãè¿ããšããç©äœã«ããããŠå€§ããªæ¯åãåŒãèµ·ããããããšãç¥ãããŠããããã®çŸè±¡ã'''å
±é³Ž'''ãšåŒã¶ã
å
±é³Žã®å Žåã«''x'' ãç¡é倧ã«ãªããšããçµæãã§ãã®ã¯ããã¡ããç©ççã«ãããããšã§ã¯ãªããäžã®è§£æ³ïŒç¹ã«ç¹è§£ãšããŠä»®å®ãã圢ïŒãäžååã ã£ãããšãæå³ãããæ£ãã解ãæ±ããã«ã¯è§£ãšããŠä»®å®ãã圢ãããåºããªããã°ãªããªããããããšã¯å¥ã«ç©ççã«ããããããããæ¹æ³ããããã¢ã€ãã¢ã¯ãäžã®è§£æ³ã§åŸã解ã¯ωãšω<sub>0</sub> ãå°ãã§ãéãã°ããŸããããäžæ¹ããã©ã¡ãŒã¿ãé£ç¶çã«å€ãããšçŸè±¡ãïŒå€åïŒé£ç¶çã«å€ããã¯ãã ãããω = ω<sub>0</sub> ã®å Žåãšããã®ã¯ãω<sub>0</sub>→ω ã®æ¥µéãšããŠåŸãããã¯ãã§ãããã ããäžã®è§£ã§ãã®æ¥µéãåãã°å
±é³Žã®å Žåã®æ£ãã解ãåŸããããããšããããšã§ãããäœããäžã®è§£ã§åçŽã«ãã®æ¥µéããšã£ãŠããã¡ããããŸããããªããä¿æ°''A'' , ''B'' ãæå³ã倱ãããã§ãããããã§''A'' , ''B'' ã®ä»£ãã«åžžã«ã¯ã£ããããç©ççæå³ãæã€éãããªãã¡åææ¡ä»¶ïŒ''t'' = 0 ã§ã®''x'' , ''dx'' /''dt'' ã®å€ïŒã䜿ã£ãŠäžã®è§£ãæžãçŽãããã®äžã§ω<sub>0</sub>→ω ã®æ¥µéãåãã''t'' = 0 ã§''x'' (0) = ''X'' , ''dx'' (0)/''dt'' = ''V'' ãšãããš
:<math>A=\frac{1}{\omega}\left(V+\frac{l\omega_0}{\omega_0^2-\omega^2}\right),\quad B=X</math>
ã«ãªãã®ã§è§£ã¯
:<math>\begin{align}
x(t) &= \frac l {- \omega _0 ^2 + \omega^2 }\sin \omega _0 t + \frac{1}{\omega}\left(V+\frac{l\omega_0}{\omega_0^2-\omega^2}\right)\sin \omega t+X \cos \omega t \\
&= \frac{l}{\omega(\omega_0+\omega)}\frac{\omega\sin \omega_0t-\omega_0\sin\omega t}{-\omega_0+\omega}+ \frac{V}{\omega}\sin \omega t+X \cos \omega t
\end{align}</math>
ãšãªããããããäžã§ω<sub>0</sub>→ω ã®æ¥µéãåããšã極éã¯ç¢ºãã«ååšããŠæ¬¡ã®ããã«ãªãïŒ
:<math>
x(t) = \frac{l}{2\omega^2}(\sin \omega t-\omega t\cos\omega t)+ \frac{V}{\omega}\sin \omega t+X \cos \omega t
</math>
ç¹ã«éèŠãªã®ã¯ç¬¬2é
ã§ãcosã«''t'' ãæãã£ãŠããç¹ã§ãããæ¯åã®æ¯å¹
ãæéã«æ¯äŸããŠå¢å€§ããŠããã®ã§ããããã®çµæããå
±é³Žã®æ¬è³ªãåãããã€ãŸãå
±é³Žã§ã¯å€åããäžãããããšãã«ã®ãŒãèç©ãããŠãããšããã®ãæ¬è³ªã§ããã®ããæ¯å¹
ãæéãšãšãã«æãŠããç¡ã倧ãããªã£ãŠããã®ã§ããïŒäžäŒããã®æåãªãšããœãŒãã§ãã寺ã®éãæäžæ¬ã§å€§ããåãããšããã®ããããæ£ç¢ºãªåšæã§æã§æŒãããšãç¶ãããšå
±é³Žã«ããå°ãã¥ã€ãšãã«ã®ãŒãèç©ãããŠæåŸã«ã¯å€§ããªæ¯åã«ãªãïŒã
ãªããäžã®è§£æ³ã§ã¯çŸè±¡ã®ãã©ã¡ãŒã¿å€åã«å¯Ÿããé£ç¶æ§ãåæãšããŠããããã®åæã¯ãããŠãæãç«ã€ããã©ããªå Žåã§ããšãŸã§ã¯ãããªããä»ã®å Žåã«ã©ããã確èªããã«ã¯åŸããã解ãå
ã®åŸ®åæ¹çšåŒïŒã§ω<sub>0</sub> = ωãšãããã®ïŒã«ä»£å
¥ããã°ããã確ãã«è§£ã«ãªã£ãŠããããšãåããããŸããω<sub>0</sub> ãωã«è¿ã¥ãã«ã€ãæ¯åã®ã°ã©ããã©ãå€ãã£ãŠè¡ãããèŠãã®ãèå³æ·±ãã
<!-- TODO -->
<!--\gamma ã®å€ããããŠèšç®ãçŽããå Žå -->
<!-- ç¹è§£ã®èšç®ãããããé¢åãã...ã -->
<!-- æµæåãå
¥ããå Žåã®èšç® -->
<!-- ä»åã¯ç¹è§£ãæ±ããããã« -->
<!-- A sin \w _0 t + B cos \w _0 t -->
<!-- ãšçœ®ãå¿
èŠãããã -->
{{DEFAULTSORT:ãããšããšã¯ãšããã¡ããããã®ãããšã}}
[[Category:æ¯åãšæ³¢å|ãã¡ããããã®ãããšã]] | 2005-06-05T02:52:08Z | 2024-03-16T02:57:39Z | [
"ãã³ãã¬ãŒã:Pathnav"
] | https://ja.wikibooks.org/wiki/%E6%8C%AF%E5%8B%95%E3%81%A8%E6%B3%A2%E5%8B%95/1%E7%B2%92%E5%AD%90%E3%81%AE%E6%8C%AF%E5%8B%95 |
2,109 | æ¯åãšæ³¢å/è€æ°ç²åã®æ¯å | 質ém1 , m2 ã®2ã€ã®è³ªç¹ããããå®æ°k ã®ããã«ãã£ãŠã€ãªãããŠãã2èªç±åºŠç³»ãèããããã®ãšããããã®æ¹åã«x 軞ãåãããããåããªãç¶æ³ã«ãªã£ãŠãããšãã®è³ªç¹m1 ã®åº§æšãx1 ã質ç¹m2 ã®åº§æšãx2 ãšãããšãéåæ¹çšåŒ
ãåŸãããã座æš
ãå°å
¥ãããåŒ(1.1)ãm2 åãããã®ãããåŒ(1.2)ãm1 åãããã®ãåŒããšã
ãåŸããããããã§ã
ãšçœ®ãããåŒ(2)ã¯åæ¯åã®æ¹çšåŒã§ãããv1 , v2 ããããã質ç¹m1 ã質ç¹m2 ã®é床ãšãããšããã®è§£ã¯v1 = -v2 ã®ããã«åæ¯åãè¡ãªãããã®è§æ¯åæ°ã¯ã
ã§äžããããããšãåããã
ãŸããéåæ¹çšåŒ(1.1), (1.2)ã足ãåããããšã
ãåŸããããããã§ã
ã§ãããããããã2ç©äœã®éåãx , X ã䜿ã£ã座æšã§è¡šãããX ã«ã€ããŠã¯èªç±ãªè³ªç¹ãšåãéåãããããšãåããã
ãã®ãšã2ç©äœã®å Žåã«ãããŠãäžã§å®çŸ©ãããX ãéå¿åº§æšãx ãçžå¯Ÿåº§æšãšåŒã¶ã
åãåé¡ãæŽã«å€ãã®ç²åãæ±ããšãã®ããæ¹ã§æžãããšãåºæ¥ãã åŒ(1.1), (1.2)ã§äžããããéåæ¹çšåŒã¯ãå®æ°ä¿æ°é£ç«2é垞埮åæ¹çšåŒã§ããã®ã§éåžžã®ä»æ¹ã§è§£ãããšãåºæ¥ãããã®æ¹éã«ãããã£ãŠã
(a1 , a2 ã¯å®æ°)ãšããããã®ãšãéåæ¹çšåŒã¯ã
ãããã¯ã
ãšæžãããšãåºæ¥ããããã§a1 = a2 = 0 ã¯ãã®æ¹çšåŒã®è§£ã§ãããããã以å€ã®è§£ããããšã
ãæãç«ã€ããšãå¿
èŠã§ãã(ç·å代æ°ã§ã¯ããã®ãããªæ¹çšåŒãåºææ¹çšåŒãšåŒã¶)ãããã解ããšã
ãã£ãŠã
ãã
ãšãªããããã¯ãäžã§æ±ããå€ãšäžèŽããŠãããçµå±2ç©äœã®å Žåã§ã¯ãç·å代æ°ã®åºææ¹çšåŒã容æã«æ±ãããããšããããšãèšããã
ç²åã®æ°ãããã«å€ãå€èªç±åºŠç³»ã®å Žåããäžã§æ±ããæ¹æ³ãçšããããšãåºæ¥ããç¹ã«éèŠãªã®ã¯ãå
šãŠã®è³ªç¹ãåã質ém ãæã£ãŠãããããå®æ°k ã®ããã§ã€ãªãããŠããå Žåã§ããã
質ç¹ãN åããN èªç±åºŠç³»ãèãããn çªç®ã®è³ªç¹ã®åº§æšãun ãšãããšãéåæ¹çšåŒã¯ã
ãšãªããããã¯ãN å
é£ç«å®æ°ä¿æ°2é垞埮åæ¹çšåŒã§ããã®ã§ããã¯ã解ãããšãåºæ¥ãã
(an ã¯å®æ°)ãšãããšã
ãåŸãããããããè¡åã®åœ¢ã§æžããšã
ãšãªãããã®æ¹çšåŒã解ãã«ã¯äžè¬ã«ã¯ãã®è¡åã®åºææ¹çšåŒã解ããã°ãªããªãã幞ãã«ããã®å Žåã«ã¯åºæãã¯ãã«ã®åœ¢ãç¥ãããŠãããããã¯ã
(dã¯ä»»æã®å®æ°)ãšãªãã å®é
ãèšç®ãããšã第k è¡ç®ã«ã€ããŠ
ãšãªãè¡åããããåŸã®å€ããsin kd Ã(å®æ°) ã®åœ¢ãããŠããããšããããã 確ãã«ãã®ãã¯ãã«ã¯ãäžããããè¡åã®åºæãã¯ãã«ãšãªãã
åç¯ã§N è¡N åã®å€§ããªè¡åã®åºæãã¯ãã«ãç°¡åã«æ±ããããããšãèŠããå®éã«ã¯ãã®ããšã¯äžã§èŠãè¡åã®æ§è³ªã«ãã£ãŠããããã®æ§è³ªãå
·äœçã«èŠãããã«ãç²åã®æ°ãããããŠå€ããç²åãé£ç¶çã«ååžããŠãããšèŠãå Žåãèããã
2é埮å
ãé¢æ£çãªéã«çŽãããšãèãããx ãé¢æ£åããŠxi - 1 , xi , xi + 1 ãªã©ãšãããšããè¿äŒŒçã«
ãšæžããããšã«æ³šç®ãããšã
ãšãªããååã® ui + 1 - 2ui + ui - 1 ã¯éåæ¹çšåŒ(3)ã®å³èŸºã«ãçŸããŠãããããã2é埮åãè¡šããŠããããšãåããã åŒ(3)ã«ä»£å
¥ãããšãv ãããå®æ°ãšããŠ
ãåŸãããããã®æ¹çšåŒãæ³¢åæ¹çšåŒãšåŒã¶ã åŸã«åããããšã ããæ³¢åæ¹çšåŒã¯ç©äœã®éåãéããŠãšãã«ã®ãŒãäŒæ¬ããŠè¡ãæ§åãè¡šãæ¹çšåŒãšãªã£ãŠãããããããå
ã¯ããã®æ¹çšåŒã®æ§è³ªãèŠãŠè¡ãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "質ém1 , m2 ã®2ã€ã®è³ªç¹ããããå®æ°k ã®ããã«ãã£ãŠã€ãªãããŠãã2èªç±åºŠç³»ãèããããã®ãšããããã®æ¹åã«x 軞ãåãããããåããªãç¶æ³ã«ãªã£ãŠãããšãã®è³ªç¹m1 ã®åº§æšãx1 ã質ç¹m2 ã®åº§æšãx2 ãšãããšãéåæ¹çšåŒ",
"title": "2ç²åã®å Žå"
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ãåŸãããã座æš",
"title": "2ç²åã®å Žå"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ãå°å
¥ãããåŒ(1.1)ãm2 åãããã®ãããåŒ(1.2)ãm1 åãããã®ãåŒããšã",
"title": "2ç²åã®å Žå"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ãåŸããããããã§ã",
"title": "2ç²åã®å Žå"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ãšçœ®ãããåŒ(2)ã¯åæ¯åã®æ¹çšåŒã§ãããv1 , v2 ããããã質ç¹m1 ã質ç¹m2 ã®é床ãšãããšããã®è§£ã¯v1 = -v2 ã®ããã«åæ¯åãè¡ãªãããã®è§æ¯åæ°ã¯ã",
"title": "2ç²åã®å Žå"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ã§äžããããããšãåããã",
"title": "2ç²åã®å Žå"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ãŸããéåæ¹çšåŒ(1.1), (1.2)ã足ãåããããšã",
"title": "2ç²åã®å Žå"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ãåŸããããããã§ã",
"title": "2ç²åã®å Žå"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "ã§ãããããããã2ç©äœã®éåãx , X ã䜿ã£ã座æšã§è¡šãããX ã«ã€ããŠã¯èªç±ãªè³ªç¹ãšåãéåãããããšãåããã",
"title": "2ç²åã®å Žå"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "ãã®ãšã2ç©äœã®å Žåã«ãããŠãäžã§å®çŸ©ãããX ãéå¿åº§æšãx ãçžå¯Ÿåº§æšãšåŒã¶ã",
"title": "2ç²åã®å Žå"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "",
"title": "2ç²åã®å Žå"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "åãåé¡ãæŽã«å€ãã®ç²åãæ±ããšãã®ããæ¹ã§æžãããšãåºæ¥ãã åŒ(1.1), (1.2)ã§äžããããéåæ¹çšåŒã¯ãå®æ°ä¿æ°é£ç«2é垞埮åæ¹çšåŒã§ããã®ã§éåžžã®ä»æ¹ã§è§£ãããšãåºæ¥ãããã®æ¹éã«ãããã£ãŠã",
"title": "2ç²åã®å Žå"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "(a1 , a2 ã¯å®æ°)ãšããããã®ãšãéåæ¹çšåŒã¯ã",
"title": "2ç²åã®å Žå"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ãããã¯ã",
"title": "2ç²åã®å Žå"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "ãšæžãããšãåºæ¥ããããã§a1 = a2 = 0 ã¯ãã®æ¹çšåŒã®è§£ã§ãããããã以å€ã®è§£ããããšã",
"title": "2ç²åã®å Žå"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "ãæãç«ã€ããšãå¿
èŠã§ãã(ç·å代æ°ã§ã¯ããã®ãããªæ¹çšåŒãåºææ¹çšåŒãšåŒã¶)ãããã解ããšã",
"title": "2ç²åã®å Žå"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "ãã£ãŠã",
"title": "2ç²åã®å Žå"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "ãã",
"title": "2ç²åã®å Žå"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "ãšãªããããã¯ãäžã§æ±ããå€ãšäžèŽããŠãããçµå±2ç©äœã®å Žåã§ã¯ãç·å代æ°ã®åºææ¹çšåŒã容æã«æ±ãããããšããããšãèšããã",
"title": "2ç²åã®å Žå"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "ç²åã®æ°ãããã«å€ãå€èªç±åºŠç³»ã®å Žåããäžã§æ±ããæ¹æ³ãçšããããšãåºæ¥ããç¹ã«éèŠãªã®ã¯ãå
šãŠã®è³ªç¹ãåã質ém ãæã£ãŠãããããå®æ°k ã®ããã§ã€ãªãããŠããå Žåã§ããã",
"title": "å€ç²åã®å Žå"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "質ç¹ãN åããN èªç±åºŠç³»ãèãããn çªç®ã®è³ªç¹ã®åº§æšãun ãšãããšãéåæ¹çšåŒã¯ã",
"title": "å€ç²åã®å Žå"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "ãšãªããããã¯ãN å
é£ç«å®æ°ä¿æ°2é垞埮åæ¹çšåŒã§ããã®ã§ããã¯ã解ãããšãåºæ¥ãã",
"title": "å€ç²åã®å Žå"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "(an ã¯å®æ°)ãšãããšã",
"title": "å€ç²åã®å Žå"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "ãåŸãããããããè¡åã®åœ¢ã§æžããšã",
"title": "å€ç²åã®å Žå"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "ãšãªãããã®æ¹çšåŒã解ãã«ã¯äžè¬ã«ã¯ãã®è¡åã®åºææ¹çšåŒã解ããã°ãªããªãã幞ãã«ããã®å Žåã«ã¯åºæãã¯ãã«ã®åœ¢ãç¥ãããŠãããããã¯ã",
"title": "å€ç²åã®å Žå"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "(dã¯ä»»æã®å®æ°)ãšãªãã å®é",
"title": "å€ç²åã®å Žå"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "ãèšç®ãããšã第k è¡ç®ã«ã€ããŠ",
"title": "å€ç²åã®å Žå"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "ãšãªãè¡åããããåŸã®å€ããsin kd Ã(å®æ°) ã®åœ¢ãããŠããããšããããã 確ãã«ãã®ãã¯ãã«ã¯ãäžããããè¡åã®åºæãã¯ãã«ãšãªãã",
"title": "å€ç²åã®å Žå"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "åç¯ã§N è¡N åã®å€§ããªè¡åã®åºæãã¯ãã«ãç°¡åã«æ±ããããããšãèŠããå®éã«ã¯ãã®ããšã¯äžã§èŠãè¡åã®æ§è³ªã«ãã£ãŠããããã®æ§è³ªãå
·äœçã«èŠãããã«ãç²åã®æ°ãããããŠå€ããç²åãé£ç¶çã«ååžããŠãããšèŠãå Žåãèããã",
"title": "é£ç¶æ¥µéãžã®ç§»è¡"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "2é埮å",
"title": "é£ç¶æ¥µéãžã®ç§»è¡"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "ãé¢æ£çãªéã«çŽãããšãèãããx ãé¢æ£åããŠxi - 1 , xi , xi + 1 ãªã©ãšãããšããè¿äŒŒçã«",
"title": "é£ç¶æ¥µéãžã®ç§»è¡"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "ãšæžããããšã«æ³šç®ãããšã",
"title": "é£ç¶æ¥µéãžã®ç§»è¡"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "ãšãªããååã® ui + 1 - 2ui + ui - 1 ã¯éåæ¹çšåŒ(3)ã®å³èŸºã«ãçŸããŠãããããã2é埮åãè¡šããŠããããšãåããã åŒ(3)ã«ä»£å
¥ãããšãv ãããå®æ°ãšããŠ",
"title": "é£ç¶æ¥µéãžã®ç§»è¡"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "ãåŸãããããã®æ¹çšåŒãæ³¢åæ¹çšåŒãšåŒã¶ã åŸã«åããããšã ããæ³¢åæ¹çšåŒã¯ç©äœã®éåãéããŠãšãã«ã®ãŒãäŒæ¬ããŠè¡ãæ§åãè¡šãæ¹çšåŒãšãªã£ãŠãããããããå
ã¯ããã®æ¹çšåŒã®æ§è³ªãèŠãŠè¡ãã",
"title": "é£ç¶æ¥µéãžã®ç§»è¡"
}
] | null | {{Pathnav|ã¡ã€ã³ããŒãž|èªç¶ç§åŠ|ç©çåŠ|æ¯åãšæ³¢å|frame=1}}
==2ç²åã®å Žå==
質é''m''<sub>1</sub> , ''m''<sub>2</sub> ã®2ã€ã®è³ªç¹ããããå®æ°''k'' ã®ããã«ãã£ãŠã€ãªãããŠãã2èªç±åºŠç³»ãèããããã®ãšããããã®æ¹åã«''x'' 軞ãåãããããåããªãç¶æ³ã«ãªã£ãŠãããšãã®è³ªç¹''m''<sub>1</sub> ã®åº§æšã''x''<sub>1</sub> ã質ç¹''m''<sub>2</sub> ã®åº§æšã''x''<sub>2</sub> ãšãããšãéåæ¹çšåŒ
:<math>\begin{align}
m _1 \ddot x _1 =& -k (x _1 - x _2)\qquad (1.1)\\
m _2 \ddot x _2 =& k (x _1 - x _2) \qquad (1.2)
\end{align}</math>
ãåŸãããã座æš
:<math>\begin{align}
X &:= \frac{m _1 x _1+m _2 x _2}{m _1 + m _2}\\
x &:= x _1 - x _2
\end{align}</math>
ãå°å
¥ãããåŒ(1.1)ã''m''<sub>2</sub> åãããã®ãããåŒ(1.2)ã''m''<sub>1</sub> åãããã®ãåŒããšã
:<math>
\begin{align}
&m _1 m _2 (\ddot x _1 - \ddot x _2 ) = -k (x _1-x _2) ( m _2 + m _1)\\
&m _1 m _2 \ddot x = -k x ( m _2 + m _1)\\
&\therefore \quad \mu \ddot x = -k x \qquad (2)
\end{align}
</math>
ãåŸããããããã§ã
:<math>
\mu := \frac {m _1 m _2}{m _1 + m _2 }
</math>
ãšçœ®ãããåŒ(2)ã¯åæ¯åã®æ¹çšåŒã§ããã''v''<sub>1</sub> , ''v''<sub>2</sub> ããããã質ç¹''m''<sub>1</sub> ã質ç¹''m''<sub>2</sub> ã®é床ãšãããšããã®è§£ã¯''v''<sub>1</sub> = -''v''<sub>2</sub> ã®ããã«åæ¯åãè¡ãªãããã®è§æ¯åæ°ã¯ã
:<math>\omega_0=
\sqrt {\frac k \mu}
</math>
ã§äžããããããšãåããã
ãŸããéåæ¹çšåŒ(1.1), (1.2)ã足ãåããããšã
:<math>\begin{align}
& m _1 \ddot x _1 +m _2 \ddot x _2 = 0\\
& (m _1 + m _2 )\frac {m _1 \ddot x _1 +m _2 \ddot x _2}{m _1+m _2} = 0\\
&\therefore \quad M \ddot X = 0
\end{align}</math>
ãåŸããããããã§ã
:<math>M := m _1+m _2</math>
ã§ãããããããã2ç©äœã®éåã''x'' , ''X'' ã䜿ã£ã座æšã§è¡šããã''X'' ã«ã€ããŠã¯èªç±ãªè³ªç¹ãšåãéåãããããšãåããã
ãã®ãšã2ç©äœã®å Žåã«ãããŠãäžã§å®çŸ©ããã''X'' ãéå¿åº§æšã''x'' ãçžå¯Ÿåº§æšãšåŒã¶ã
<!-- ãã®éšåã¯"å€å
žååŠ"ã«å
¥ããã¹ãããç¥ããŸããã -->
<!-- Xã¯nåã®ç©äœã«å¯ŸããŠå®çŸ©ã§ããã -->
<!-- å®çŸ©ã¯ã -->
<!-- X = \frac { \sum _ i m _i x _i } {\sum _i m _i} -->
<!-- ãšãªãã -->
<!-- *åè -->
<!-- 2ã€ã®ç©äœã®éåã¯ãã®æ§ã«éå¿åº§æš -->
<!-- ã©ã°ã©ã³ãžã¢ã³ã2座æšã§åé¢ã§ããããšã¯ -->
<!-- å¿
èŠã ããã? -->
åãåé¡ãæŽã«å€ãã®ç²åãæ±ããšãã®ããæ¹ã§æžãããšãåºæ¥ãã åŒ(1.1), (1.2)ã§äžããããéåæ¹çšåŒã¯ãå®æ°ä¿æ°é£ç«2é垞埮åæ¹çšåŒã§ããã®ã§éåžžã®ä»æ¹ã§è§£ãããšãåºæ¥ãããã®æ¹éã«ãããã£ãŠã
:<math>\begin{align}
x _1(t) &= a _1 e^{i\omega t}\\
x _2(t) &= a _2 e^{i\omega t}
\end{align}</math>
ïŒ''a''<sub>1</sub> , ''a''<sub>2</sub> ã¯å®æ°ïŒãšããã<!--(èæ°åäœiãå ããã®ãæ
£çšçã§ããã)-->ãã®ãšãéåæ¹çšåŒã¯ã
:<math>\begin{align}
- m _1\omega^2 a _1 =& -k (a _1 - a _2)\\
- m _2 \omega^2 a _2 =& k (a _1 - a _2)
\end{align}</math>
ãããã¯ã
:<math>\begin{align}
(- m _1 \omega^2 + k) a _1 - k a _2 =& 0\\
-k a _1 + (k - m _2 \omega^2) a _2 =& 0
\end{align}</math>
ãšæžãããšãåºæ¥ããããã§''a''<sub>1</sub> = ''a''<sub>2</sub> = 0 ã¯ãã®æ¹çšåŒã®è§£ã§ãããããã以å€ã®è§£ããããšã
:<math>
\begin{vmatrix}
- m _1 \omega^2 + k& - k \\
-k & k - m _2 \omega^2
\end{vmatrix}
= 0
</math>
ãæãç«ã€ããšãå¿
èŠã§ããïŒç·å代æ°ã§ã¯ããã®ãããªæ¹çšåŒãåºææ¹çšåŒãšåŒã¶ïŒãããã解ããšã
:<math>\begin{align}
m _1 m _2 \omega^4 + k ( -m _1 -m _2) \omega^2 &= 0\\
\omega^2 ( m _1 m _2 \omega^2 - (m _1 + m _2)k ) &=0
\end{align}</math>
ãã£ãŠã
:<math>
\omega^2 = 0, \frac k \mu
</math>
ãã
:<math>
\omega = 0, \pm \sqrt{ \frac k \mu}
</math>
ãšãªããããã¯ãäžã§æ±ããå€ãšäžèŽããŠãããçµå±2ç©äœã®å Žåã§ã¯ãç·å代æ°ã®åºææ¹çšåŒã容æã«æ±ãããããšããããšãèšããã
==è€æ°ç²åã®å Žå==
<!-- 3ã€ã®ç²åã䜿ã£ãå ŽåãäŸã«åã£ãŠåºæºåº§æšã®å°å
¥ã -->
==å€ç²åã®å Žå==
ç²åã®æ°ãããã«å€ãå€èªç±åºŠç³»ã®å Žåããäžã§æ±ããæ¹æ³ãçšããããšãåºæ¥ããç¹ã«éèŠãªã®ã¯ãå
šãŠã®è³ªç¹ãåã質é''m'' ãæã£ãŠãããããå®æ°''k'' ã®ããã§ã€ãªãããŠããå Žåã§ããã
* å³
質ç¹ã''N'' åãã''N'' èªç±åºŠç³»ãèããã''n'' çªç®ã®è³ªç¹ã®åº§æšã''u<sub>n</sub>'' ãšãããšãéåæ¹çšåŒã¯ã
:<math>\begin{align}
m \ddot u _n &= - k( u _n - u _{n-1} ) + k( u _{n+1} -u _n ) \\
&= k( u _{n+1} -2u _n + u _{n-1} ), \\
\ddot u _n &= \omega _0^2( u _{n+1} -2u _n + u _{n-1} ) \qquad (3)
\end{align}</math>
ãšãªããããã¯ã''N'' å
é£ç«å®æ°ä¿æ°2é垞埮åæ¹çšåŒã§ããã®ã§ããã¯ã解ãããšãåºæ¥ãã
:<math>
u _n = a _n e^{i\omega t}
</math>
ïŒ''a<sub>n</sub>'' ã¯å®æ°ïŒãšãããšã
:<math>
-\omega^2 u _n = \omega _0^2( u _{n+1} -2u _n + u _{n-1} )
</math>
ãåŸãããããããè¡åã®åœ¢ã§æžããšã
:<math>
\begin{pmatrix}
-2 + \frac{\omega^2}{ \omega _0^2} &1 & & & 0 \\
1 &-2 + \frac{\omega^2}{ \omega _0^2} &1 \\
&1 &-2 + \frac{\omega^2}{ \omega _0^2} & \ddots\\
&& \ddots & \ddots & 1\\
0&&&1 &-2 + \frac{\omega^2}{ \omega _0^2} \\
\end{pmatrix}
\begin{pmatrix} u_1^2 \\ u_2^2 \\ u_3^2 \\ \vdots \\ u_N^2 \end{pmatrix}
= \boldsymbol{0}
</math>
ãšãªãããã®æ¹çšåŒã解ãã«ã¯äžè¬ã«ã¯ãã®è¡åã®åºææ¹çšåŒã解ããã°ãªããªãã幞ãã«ããã®å Žåã«ã¯åºæãã¯ãã«ã®åœ¢ãç¥ãããŠãããããã¯ã
:<math>
\begin{pmatrix} u_1^2 \\ u_2^2 \\ u_3^2 \\ \vdots \\ u_N^2 \end{pmatrix}
=
\begin{pmatrix}
\sin d \\
\sin 2 d \\
\sin 3 d \\
\vdots \\
\sin N d \\
\end{pmatrix}
</math>
ïŒ''d''ã¯ä»»æã®å®æ°ïŒãšãªãã
<!-- TODO -->
<!-- åºå®ç«¯ãšèªç±ç«¯ãšçœ®ãããšãã® dã®å€ã -->
<!-- (åŸã«ãé£ç¶æ¥µéãåã£ããšã -->
<!-- (ï¿œrac 1 {v^2}\frac {\partial^2 {}}{\partial^2 t } - \frac {\partial^2 {}}{\partial^2 x } )u(x,t) = 0 -->
<!-- ã®å®åžžè§£ãã\sin ï¿œrac x l ãªã©ã§äžããããããšã«ããã -->
å®é
:<math>
\begin{pmatrix}
-2 + \frac{\omega^2}{ \omega _0^2} &1 & & & 0 \\
1 &-2 + \frac{\omega^2}{ \omega _0^2} &1 \\
&1 &-2 + \frac{\omega^2}{ \omega _0^2} & \ddots\\
&& \ddots & \ddots & 1\\
0&&&1 &-2 + \frac{\omega^2}{ \omega _0^2} \\
\end{pmatrix}\begin{pmatrix}
\sin d \\
\sin 2 d \\
\sin 3 d \\
\vdots \\
\sin N d \\
\end{pmatrix}
</math>
ãèšç®ãããšã第''k'' è¡ç®ã«ã€ããŠ
:<math>
\sin (k-1) d + \left(-2+ \frac {\omega^2} {\omega _0^2}\right) \sin kd + \sin (k+1) d
= 2 \sin kd \left(2\cos d - 2 + \frac {\omega^2} {\omega _0^2}\right)
</math>
ãšãªãè¡åããããåŸã®å€ããsin ''kd'' ×(å®æ°) ã®åœ¢ãããŠããããšããããã
<!-- ãã£ãŠã -->
<!-- ï¿œegin{align} -->
<!-- ï¿œrac {\omega^2} {\omega _0^2}& = 2 - 2\cos d \ -->
<!-- & = 4 \sin ^2 ï¿œrac d 2 -->
<!-- \end{align} -->
<!-- ãã£ãŠã -->
<!-- ï¿œe -->
<!-- \omega^2 = 4 \omega _0^2 \sin ^2 ï¿œrac d 2 -->
<!-- \ee (?) -->
確ãã«ãã®ãã¯ãã«ã¯ãäžããããè¡åã®åºæãã¯ãã«ãšãªãã
==é£ç¶æ¥µéãžã®ç§»è¡==
åç¯ã§''N'' è¡''N'' åã®å€§ããªè¡åã®åºæãã¯ãã«ãç°¡åã«æ±ããããããšãèŠããå®éã«ã¯ãã®ããšã¯äžã§èŠãè¡åã®æ§è³ªã«ãã£ãŠããããã®æ§è³ªãå
·äœçã«èŠãããã«ãç²åã®æ°ãããããŠå€ããç²åãé£ç¶çã«ååžããŠãããšèŠãå Žåãèããã
2é埮å
:<math>
\frac {\partial^2 {u}}{\partial^2 x } (x)
</math>
ãé¢æ£çãªéã«çŽãããšãèããã''x'' ãé¢æ£åããŠ''x''<sub>''i'' - 1</sub> , ''x<sub>i</sub>'' , ''x''<sub>''i'' + 1</sub> ãªã©ãšãããšããè¿äŒŒçã«
:<math>\begin{align}
u'(x+h) &\sim \frac {u(x _{i+1)}-u(x _{i})} h\\
u'(x) &\sim \frac {u(x _{i})-u(x _{i-1})} h
\end{align}</math>
ãšæžããããšã«æ³šç®ãããšã
:<math>\begin{align}
u''(x) &\sim \frac {u'(x+h) -u'(x) } h\\
&\sim \frac 1 h \left( \frac {u(x _{i+1)}-u(x _{i})} h-\frac {u(x _{i})-u(x _{i-1})} h \right) \\
&= \frac {u(x_{i+1}) - 2u(x_i) + u(x_{i-1})} {h^2}
\end{align}</math>
ãšãªããååã® ''u''<sub>''i'' + 1</sub> - 2''u<sub>i</sub>'' + ''u''<sub>''i'' - 1</sub> ã¯éåæ¹çšåŒ(3)ã®å³èŸºã«ãçŸããŠãããããã2é埮åãè¡šããŠããããšãåããã
<!-- 埮å°ãªç¯å²ã¯ã©ããã? -->
<!-- mã質éå¯åºŠã«ãããšhã¯1ã€æ¶ããããã©ããš1ã€ã¯? -->
<!-- k to ã€ã³ã°çã®å®çŸ©? -->
åŒ(3)ã«ä»£å
¥ãããšã''v'' ãããå®æ°ãšããŠ
:<math>
\left(\frac 1 {v^2} \frac {\partial^2 {}}{\partial^2 t } - \frac {\partial^2 {}}{\partial^2 x } \right) u(x,t) = 0
</math>
ãåŸãããããã®æ¹çšåŒãæ³¢åæ¹çšåŒãšåŒã¶ã
<!-- ããã§ã¯ãååŠçãªç©äœã®éåãéã㊠-->
<!-- ãã®æ¹çšåŒãåŸããããã以å€ã«ãæ³¢åæ¹çšåŒã -->
<!-- åŸãããã€ãã®æ¹æ³ãç¥ãããŠããã -->
<!-- (ãšã¯ããæµäœååŠãå
ã蟿ãã°å€å
žååŠã...ã) -->
åŸã«åããããšã ããæ³¢åæ¹çšåŒã¯ç©äœã®éåãéããŠãšãã«ã®ãŒãäŒæ¬ããŠè¡ãæ§åãè¡šãæ¹çšåŒãšãªã£ãŠãããããããå
ã¯ããã®æ¹çšåŒã®æ§è³ªãèŠãŠè¡ãã
{{DEFAULTSORT:ãããšããšã¯ãšããµããããããã}}
[[Category:æ¯åãšæ³¢å|ãµãããããããã®ãããšã]] | 2005-06-05T03:06:21Z | 2024-03-16T02:58:43Z | [
"ãã³ãã¬ãŒã:Pathnav"
] | https://ja.wikibooks.org/wiki/%E6%8C%AF%E5%8B%95%E3%81%A8%E6%B3%A2%E5%8B%95/%E8%A4%87%E6%95%B0%E7%B2%92%E5%AD%90%E3%81%AE%E6%8C%AF%E5%8B%95 |
2,110 | ç ç® èŠåç®ã»èªäžç®ã»äŒç¥šç® | æ°åŠ>ç ç®>èŠåç®ã»èªäžç®ã»äŒç¥šç®
èŠåç®ãšã¯ãå æžç®ãçç®ã®ãããªåœ¢ã§äžŠã¹ãã
ã®ãããªãã®ã§èšç®ããçš®ç®ã§ããã ããããå·Šã«äœãæžããŠããªãæ°åã¯è¶³ããã-ããæžããŠããæ°åã¯åŒããåèšãåºãã
èªäžç®ãšã¯ãèŠåç®ãè©Šéšå®ãèªäžãããããèšç®ããçš®ç®ã§ããã ãã®æãç¬ç¹ã®çšèªã䜿ãããã
ãé¡ããŸããŠã¯xxxåä¹ãxxxåä¹ãã»ã»ã»xxxåã§ã¯ãã
ãšããå
·åã§ããã
åŒãç®ãããå Žåã¯ããåŒããŠã¯xxxåä¹ããšèšãã
ãã®å Žåã次ã«ãå ããŠxxxåä¹ããšèšããŸã§ã¯å
šãŠåŒãç¶ããã
7æ¡(äœçŸäž)ã®èšç®ãããŠãããšãã«æ¥ã«3æ¡(äœçŸ)ã«ãªããšããèªäžç®ãªãã§ã¯ã®åŒã£æããããã
ããããçŸäžãšçŸã®ãããªçŽããããæ°ã®åºå¥ãã€ããããã«ã倧ããæ¹(ãã®å Žåã¯çŸäž)ã®æ°ãèšãåã«ã倧ãããçãä»ããããšãããã
ãçŸäžåäºåä¹ã倧ããçŸå
«åäžåäžåäºåä¹ã
ãšããå
·åã§ããã
äŒç¥šç®ãšã¯ãäžèŸºã ãçããŠãã暪13cm瞊8cmã®åå(ãããäŒç¥šãšãã)ã䜿ã£ãŠèšç®ããçš®ç®ã§ããã äŒç¥šã«ã¯1æã«ã€ã5ã€ã®æ°åãæžãããŠãããè£ã¯çœçŽã§ããããããã1ææ¯ã«è¶³ãã®ã§ã¯ãªãã1æã®ãã¡1ã€ç®ãªã1ã€ç®ã3ã€ç®ãªã3ã€ç®ã®æ°åå士ã足ãã
äžã®å³ã§åããã ããããå®éã«ã¯ç¶Žã£ãŠããã®ã§å·Šæã§ããããå³æã§èšç®ããªããã°ãªããªãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "æ°åŠ>ç ç®>èŠåç®ã»èªäžç®ã»äŒç¥šç®",
"title": ""
},
{
"paragraph_id": 1,
"tag": "p",
"text": "èŠåç®ãšã¯ãå æžç®ãçç®ã®ãããªåœ¢ã§äžŠã¹ãã",
"title": "èŠåç®"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ã®ãããªãã®ã§èšç®ããçš®ç®ã§ããã ããããå·Šã«äœãæžããŠããªãæ°åã¯è¶³ããã-ããæžããŠããæ°åã¯åŒããåèšãåºãã",
"title": "èŠåç®"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "èªäžç®ãšã¯ãèŠåç®ãè©Šéšå®ãèªäžãããããèšç®ããçš®ç®ã§ããã ãã®æãç¬ç¹ã®çšèªã䜿ãããã",
"title": "èªäžç®"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ãé¡ããŸããŠã¯xxxåä¹ãxxxåä¹ãã»ã»ã»xxxåã§ã¯ãã",
"title": "èªäžç®"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ãšããå
·åã§ããã",
"title": "èªäžç®"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "åŒãç®ãããå Žåã¯ããåŒããŠã¯xxxåä¹ããšèšãã",
"title": "èªäžç®"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ãã®å Žåã次ã«ãå ããŠxxxåä¹ããšèšããŸã§ã¯å
šãŠåŒãç¶ããã",
"title": "èªäžç®"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "7æ¡(äœçŸäž)ã®èšç®ãããŠãããšãã«æ¥ã«3æ¡(äœçŸ)ã«ãªããšããèªäžç®ãªãã§ã¯ã®åŒã£æããããã",
"title": "èªäžç®"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "ããããçŸäžãšçŸã®ãããªçŽããããæ°ã®åºå¥ãã€ããããã«ã倧ããæ¹(ãã®å Žåã¯çŸäž)ã®æ°ãèšãåã«ã倧ãããçãä»ããããšãããã",
"title": "èªäžç®"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "ãçŸäžåäºåä¹ã倧ããçŸå
«åäžåäžåäºåä¹ã",
"title": "èªäžç®"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "ãšããå
·åã§ããã",
"title": "èªäžç®"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "äŒç¥šç®ãšã¯ãäžèŸºã ãçããŠãã暪13cm瞊8cmã®åå(ãããäŒç¥šãšãã)ã䜿ã£ãŠèšç®ããçš®ç®ã§ããã äŒç¥šã«ã¯1æã«ã€ã5ã€ã®æ°åãæžãããŠãããè£ã¯çœçŽã§ããããããã1ææ¯ã«è¶³ãã®ã§ã¯ãªãã1æã®ãã¡1ã€ç®ãªã1ã€ç®ã3ã€ç®ãªã3ã€ç®ã®æ°åå士ã足ãã",
"title": "äŒç¥šç®"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "äžã®å³ã§åããã ããããå®éã«ã¯ç¶Žã£ãŠããã®ã§å·Šæã§ããããå³æã§èšç®ããªããã°ãªããªãã",
"title": "äŒç¥šç®"
}
] | æ°åŠïŒç ç®ïŒèŠåç®ã»èªäžç®ã»äŒç¥šç® | [[æ°åŠ]]ïŒ[[ç ç®]]ïŒèŠåç®ã»èªäžç®ã»äŒç¥šç®
==èŠåç®==
èŠåç®ãšã¯ãå æžç®ãçç®ã®ãããªåœ¢ã§äžŠã¹ãã
45
23
-62
-81
59
74
-18
36
97
---
ã®ãããªãã®ã§èšç®ããçš®ç®ã§ããã
ããããå·Šã«äœãæžããŠããªãæ°åã¯è¶³ããã-ããæžããŠããæ°åã¯åŒããåèšãåºãã
==èªäžç®==
èªäžç®ãšã¯ãèŠåç®ãè©Šéšå®ãèªäžãããããèšç®ããçš®ç®ã§ããã
ãã®æãç¬ç¹ã®çšèªã䜿ãããã
ãé¡ããŸããŠã¯xxxåä¹ãxxxåä¹ãã»ã»ã»xxxåã§ã¯ãã
ãšããå
·åã§ããã
åŒãç®ãããå Žåã¯ããåŒããŠã¯xxxåä¹ããšèšãã
ãã®å Žåã'''次ã«ãå ããŠxxxåä¹ããšèšããŸã§ã¯å
šãŠåŒãç¶ããã'''
7æ¡(äœçŸäž)ã®èšç®ãããŠãããšãã«æ¥ã«3æ¡(äœçŸ)ã«ãªããšããèªäžç®ãªãã§ã¯ã®åŒã£æããããã
ããããçŸäžãšçŸã®ãããªçŽããããæ°ã®åºå¥ãã€ããããã«ã倧ããæ¹ïŒãã®å Žåã¯çŸäžïŒã®æ°ãèšãåã«ã倧ãããçãä»ããããšãããã
ãçŸäžåäºåä¹ã倧ããçŸå
«åäžåäžåäºåä¹ã
ãšããå
·åã§ããã
==äŒç¥šç®==
äŒç¥šç®ãšã¯ãäžèŸºã ãçããŠãã暪13cm瞊8cmã®åå(ãããäŒç¥šãšãã)ã䜿ã£ãŠèšç®ããçš®ç®ã§ããã
äŒç¥šã«ã¯1æã«ã€ã5ã€ã®æ°åãæžãããŠãããè£ã¯çœçŽã§ããããããã1ææ¯ã«è¶³ãã®ã§ã¯ãªãã1æã®ãã¡1ã€ç®ãªã1ã€ç®ã3ã€ç®ãªã3ã€ç®ã®æ°åå士ã足ãã
------- ------- ------- ------- -------
| 123 |->| 678 |->| 135 |->| 246 |->| 111 | =1,293
| 234 |->| 789 |->| 357 |->| 468 |->| 222 | =2,070
| 345 |->| 890 |->| 579 |->| 680 |->| 333 | =2,827
| 456 |->| 901 |->| 791 |->| 802 |->| 444 | =3,394
| 567 |->| 12 |->| 913 |->| 24 |->| 555 | =2,071
------- ------- ------- ------- -------
1æç® 2æç® 3æç® 4æç® 5æç® çã
äžã®å³ã§åããã ããããå®éã«ã¯ç¶Žã£ãŠããã®ã§å·Šæã§ããããå³æã§èšç®ããªããã°ãªããªãã
[[Category:ç ç®|ã¿ãšããããã¿ãããããŠãã²ãããã]] | 2005-06-05T08:13:44Z | 2024-03-18T17:49:02Z | [] | https://ja.wikibooks.org/wiki/%E7%8F%A0%E7%AE%97_%E8%A6%8B%E5%8F%96%E7%AE%97%E3%83%BB%E8%AA%AD%E4%B8%8A%E7%AE%97%E3%83%BB%E4%BC%9D%E7%A5%A8%E7%AE%97 |
2,114 | æ¯åãšæ³¢å/æ³¢åæ¹çšåŒã®æ§è³ª | æ³¢åæ¹çšåŒã¯å埮åæ¹çšåŒã§ããã®ã§ãããã解ãããã«å¢çæ¡ä»¶ãå®ããã°ãªããªãã1次å
ã®æ³¢åæ¹çšåŒ
ãèãããšã Ο = x + v t , η = x â v t {\displaystyle \xi =x+vt,\eta =x-vt} ãšãããšãã
ãçšãããšã
ããã
ãšãªãããã®è§£ã¯ã
ã§äžãããã(f , g ã¯ä»»æã®é¢æ°)ããã®è§£ã®ãã¡ãx + v t ã«äŸåããé¢æ°ã¯é床v 㧠-x æ¹åã«ç§»åããæ³¢ã«å¯Ÿå¿ããx - v t ã«äŸåããé¢æ°ã¯é床v ã§x æ¹åã«ç§»åããæ³¢ã«å¯Ÿå¿ããã
ãã®é¢æ°ãå®å
šã«æ±ºããã«ã¯äŸãã°ãæ³¢ãã€ãããç©äœã®t = 0 ã§ã®äœçœ®ãšé床ãå
šãŠã®ç¹x ã§ç¥ãããŠããã°ãããäŸãã°ã
ãã€ãé床ã¯t = 0 ãã€å
šãŠã®x ã§0ãšããããšãã
ã«ä»£å
¥ãããšã
ãåŸãããæå»t ã§ã®é¢æ°u ã®å€ã¯ã
ãšãªãã
æéäŸåæ§ãäœçœ®ã«ãããã«æ±ºãŸãæ³¢ããå®åšæ³¢ãšåŒã¶ã(?)ãã®ãšãã
ã®ããã«ã解ã®x , t ã«å¯ŸããäŸåæ§ãåé¢ã§ããããããæ³¢åæ¹çšåŒã«ä»£å
¥ãããšã
ãšå€åœ¢ã§ãããããã§ãæåŸã®åŒã®å·ŠèŸºã¯t ã ãã®é¢æ°ã§ãããå³èŸºã®åŒã¯x ã ãã®é¢æ°ã§ããã®ã§ãã©ã¡ãã®å€ãå®æ°ã«çããã¯ãã§ããããã®å®æ°ãã-Ï /v ãšãããšã
ãšãªãã解
ãåŸã(A , B ã¯ä»»æå®æ°)ãäžæ¹ãX ã«ã€ããŠãåæ§ã«
ãåŸãããšãã§ãã解
ãåŸã(A , B ã¯ä»»æå®æ°)ã
ç¹ã«ãx = 0, x = l ã§u (t , x ) = 0 ãšãªãå Žåãèãããããã¯ãç©äœã®ç«¯ãåºå®ãããŠããå Žåã«å¯Ÿå¿ããã®ã§åºå®ç«¯ãšåŒã°ããããã®ãšããx = 0 ã§u = 0 ããB = 0 ãåŸãããããŸãã
ããã
(n ã¯æŽæ°)ãšãªãã
ãåŸããããn = 0 ã¯å
šãæ³¢ãèµ·ããŠããªãç¶æ³ã«å¯Ÿå¿ããn = 1 ã¯ç¯ã1ã€ã ãã®æ³¢ãèµ·ããŠããç¶æ³ã«å¯Ÿå¿ãããn > 1 ã¯ãç¯ãn åã®æ³¢ã«å¯Ÿå¿ããã
å
šãŠã®ç¹ã®æéäŸåæ§ãåäžãªã®ã§T (t ) ã決ããã«ã¯ããäžç¹ã§ã®æ¯åã®ããæå»ã§ã®äœçœ®ãšé床ãäžããã°ãã(å®éã«ã¯ããæå»ã§äž¡æ¹ãäžããå¿
èŠã¯ãªããéãæå»ã§1ã€ãã€äžããŠããã)ãäŸãã°ãt = 0, x = l /2 ã§ãu = 0, â u â t = a {\displaystyle {\frac {\partial {u}}{\partial {t}}}=a} (a ã¯å®æ°)ãäžãããããšãããšã
ã«ã€ããŠãB = 0, ÏA = aãŸãã¯A = a /ÏãåŸãããããã£ãŠããã®æ¹çšåŒã®è§£ã¯
ãšãªãã
2次å
å¹³é¢äžã§ãããæ¹åãx æ¹åãšåãããããšåçŽãªæ¹åãy 軞ãšåããx 軞ãšy 軞ãã€ããããŠãæ¹çšåŒãå€ãããªãããšã«æ³šç®ãããšãæ³¢åæ¹çšåŒã¯
ãšãªãã
2次å
å¹³é¢äžã§ã®åºå®ç«¯ã®å®åšæ³¢ã¯ã2ã€ã®æŽæ°ã䜿ã£ãŠè¡šããããããš(å€æ°åé¢)ã2ã€ã®æŽæ°ãm,nãšãããšãã®m = 1,n=1ã®æãªã©ã®å³ã
3次å
å¹³é¢äžã§ãããæ¹åã«x 軞ãããããšåçŽãªæ¹åã«y 軞ãåããããããé ã«å³æã®èŠªæã人差ãæãäžæã«å¯Ÿå¿ããããã«z 軞ãåããããããã®è»žãã€ããããŠãæ¹çšåŒãå€ãããªãããšã«æ³šç®ãããšãæ³¢åæ¹çšåŒã¯
ãšãªãã
(f ã¯r , t ã ãã®é¢æ°ãÎ ã¯ã©ãã©ã·ã¢ã³ã)(?)ãã®ãšããäžããããæ³¢åæ¹çšåŒã¯ã
ãšãªããããã㧠r f (r , t ) ã«ã€ããŠã¯ãã®åŒã¯éåžžã®1次å
ã®æ³¢åæ¹çšåŒã«å¯Ÿå¿ããããã£ãŠãã®æ¹çšåŒã®è§£ãšããŠ
(u , v ã¯ä»»æã®é¢æ°)ãåŸãããããã¯ç察称ãªæ³¢ãè¡šããããšãããçé¢æ³¢ãšåŒã°ããã
å
ã®å Žåã§èãããšåããããããå
ã®é床cã¯ãè§é床ãåšæ³¢æ°ãšã¯ç¡é¢ä¿ã§ããã
ãªãããã®ãæ³¢ã«ãããé床AÏããäœçžé床ãšãããäœçžé床ã¯ãæ
å ±ãäŒããé床ã§ã¯ãªãã
å®éã«æ
å ±ãäŒããããé床ã®ããšã矀é床ãšããã
ãªãããŸããããæ³¢åããè€æ°åã®æ£åŒŠæ³¢ãã足ãåããããåŒãç®ãããããªããšãæ°åŒã§è¡šçŸã§ãªããšå Žåããã®ãããªæ³¢åããåæ£ã®ãããæ³¢åãšããã
ã€ãŸããåæ£ã®ããæ³¢åã®ãæ
å ±ãäŒããããé床ã®ããšãã矀é床ãšããã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "æ³¢åæ¹çšåŒã¯å埮åæ¹çšåŒã§ããã®ã§ãããã解ãããã«å¢çæ¡ä»¶ãå®ããã°ãªããªãã1次å
ã®æ³¢åæ¹çšåŒ",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 1,
"tag": "p",
"text": "ãèãããšã Ο = x + v t , η = x â v t {\\displaystyle \\xi =x+vt,\\eta =x-vt} ãšãããšãã",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 2,
"tag": "p",
"text": "ãçšãããšã",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 3,
"tag": "p",
"text": "ããã",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 4,
"tag": "p",
"text": "ãšãªãããã®è§£ã¯ã",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 5,
"tag": "p",
"text": "ã§äžãããã(f , g ã¯ä»»æã®é¢æ°)ããã®è§£ã®ãã¡ãx + v t ã«äŸåããé¢æ°ã¯é床v 㧠-x æ¹åã«ç§»åããæ³¢ã«å¯Ÿå¿ããx - v t ã«äŸåããé¢æ°ã¯é床v ã§x æ¹åã«ç§»åããæ³¢ã«å¯Ÿå¿ããã",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 6,
"tag": "p",
"text": "ãã®é¢æ°ãå®å
šã«æ±ºããã«ã¯äŸãã°ãæ³¢ãã€ãããç©äœã®t = 0 ã§ã®äœçœ®ãšé床ãå
šãŠã®ç¹x ã§ç¥ãããŠããã°ãããäŸãã°ã",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 7,
"tag": "p",
"text": "ãã€ãé床ã¯t = 0 ãã€å
šãŠã®x ã§0ãšããããšãã",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 8,
"tag": "p",
"text": "ã«ä»£å
¥ãããšã",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 9,
"tag": "p",
"text": "ãåŸãããæå»t ã§ã®é¢æ°u ã®å€ã¯ã",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 10,
"tag": "p",
"text": "ãšãªãã",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 11,
"tag": "p",
"text": "æéäŸåæ§ãäœçœ®ã«ãããã«æ±ºãŸãæ³¢ããå®åšæ³¢ãšåŒã¶ã(?)ãã®ãšãã",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 12,
"tag": "p",
"text": "ã®ããã«ã解ã®x , t ã«å¯ŸããäŸåæ§ãåé¢ã§ããããããæ³¢åæ¹çšåŒã«ä»£å
¥ãããšã",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 13,
"tag": "p",
"text": "ãšå€åœ¢ã§ãããããã§ãæåŸã®åŒã®å·ŠèŸºã¯t ã ãã®é¢æ°ã§ãããå³èŸºã®åŒã¯x ã ãã®é¢æ°ã§ããã®ã§ãã©ã¡ãã®å€ãå®æ°ã«çããã¯ãã§ããããã®å®æ°ãã-Ï /v ãšãããšã",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 14,
"tag": "p",
"text": "ãšãªãã解",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 15,
"tag": "p",
"text": "ãåŸã(A , B ã¯ä»»æå®æ°)ãäžæ¹ãX ã«ã€ããŠãåæ§ã«",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 16,
"tag": "p",
"text": "ãåŸãããšãã§ãã解",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 17,
"tag": "p",
"text": "ãåŸã(A , B ã¯ä»»æå®æ°)ã",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 18,
"tag": "p",
"text": "ç¹ã«ãx = 0, x = l ã§u (t , x ) = 0 ãšãªãå Žåãèãããããã¯ãç©äœã®ç«¯ãåºå®ãããŠããå Žåã«å¯Ÿå¿ããã®ã§åºå®ç«¯ãšåŒã°ããããã®ãšããx = 0 ã§u = 0 ããB = 0 ãåŸãããããŸãã",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 19,
"tag": "p",
"text": "ããã",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 20,
"tag": "p",
"text": "(n ã¯æŽæ°)ãšãªãã",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 21,
"tag": "p",
"text": "ãåŸããããn = 0 ã¯å
šãæ³¢ãèµ·ããŠããªãç¶æ³ã«å¯Ÿå¿ããn = 1 ã¯ç¯ã1ã€ã ãã®æ³¢ãèµ·ããŠããç¶æ³ã«å¯Ÿå¿ãããn > 1 ã¯ãç¯ãn åã®æ³¢ã«å¯Ÿå¿ããã",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 22,
"tag": "p",
"text": "å
šãŠã®ç¹ã®æéäŸåæ§ãåäžãªã®ã§T (t ) ã決ããã«ã¯ããäžç¹ã§ã®æ¯åã®ããæå»ã§ã®äœçœ®ãšé床ãäžããã°ãã(å®éã«ã¯ããæå»ã§äž¡æ¹ãäžããå¿
èŠã¯ãªããéãæå»ã§1ã€ãã€äžããŠããã)ãäŸãã°ãt = 0, x = l /2 ã§ãu = 0, â u â t = a {\\displaystyle {\\frac {\\partial {u}}{\\partial {t}}}=a} (a ã¯å®æ°)ãäžãããããšãããšã",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 23,
"tag": "p",
"text": "ã«ã€ããŠãB = 0, ÏA = aãŸãã¯A = a /ÏãåŸãããããã£ãŠããã®æ¹çšåŒã®è§£ã¯",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 24,
"tag": "p",
"text": "ãšãªãã",
"title": "1次å
ã®æ³¢åæ¹çšåŒ"
},
{
"paragraph_id": 25,
"tag": "p",
"text": "2次å
å¹³é¢äžã§ãããæ¹åãx æ¹åãšåãããããšåçŽãªæ¹åãy 軞ãšåããx 軞ãšy 軞ãã€ããããŠãæ¹çšåŒãå€ãããªãããšã«æ³šç®ãããšãæ³¢åæ¹çšåŒã¯",
"title": "2次å
å¹³é¢äžã®æ³¢"
},
{
"paragraph_id": 26,
"tag": "p",
"text": "ãšãªãã",
"title": "2次å
å¹³é¢äžã®æ³¢"
},
{
"paragraph_id": 27,
"tag": "p",
"text": "2次å
å¹³é¢äžã§ã®åºå®ç«¯ã®å®åšæ³¢ã¯ã2ã€ã®æŽæ°ã䜿ã£ãŠè¡šããããããš(å€æ°åé¢)ã2ã€ã®æŽæ°ãm,nãšãããšãã®m = 1,n=1ã®æãªã©ã®å³ã",
"title": "2次å
å¹³é¢äžã®æ³¢"
},
{
"paragraph_id": 28,
"tag": "p",
"text": "3次å
å¹³é¢äžã§ãããæ¹åã«x 軞ãããããšåçŽãªæ¹åã«y 軞ãåããããããé ã«å³æã®èŠªæã人差ãæãäžæã«å¯Ÿå¿ããããã«z 軞ãåããããããã®è»žãã€ããããŠãæ¹çšåŒãå€ãããªãããšã«æ³šç®ãããšãæ³¢åæ¹çšåŒã¯",
"title": "3次å
空éäžã®æ³¢"
},
{
"paragraph_id": 29,
"tag": "p",
"text": "ãšãªãã",
"title": "3次å
空éäžã®æ³¢"
},
{
"paragraph_id": 30,
"tag": "p",
"text": "(f ã¯r , t ã ãã®é¢æ°ãÎ ã¯ã©ãã©ã·ã¢ã³ã)(?)ãã®ãšããäžããããæ³¢åæ¹çšåŒã¯ã",
"title": "3次å
空éäžã®æ³¢"
},
{
"paragraph_id": 31,
"tag": "p",
"text": "ãšãªããããã㧠r f (r , t ) ã«ã€ããŠã¯ãã®åŒã¯éåžžã®1次å
ã®æ³¢åæ¹çšåŒã«å¯Ÿå¿ããããã£ãŠãã®æ¹çšåŒã®è§£ãšããŠ",
"title": "3次å
空éäžã®æ³¢"
},
{
"paragraph_id": 32,
"tag": "p",
"text": "(u , v ã¯ä»»æã®é¢æ°)ãåŸãããããã¯ç察称ãªæ³¢ãè¡šããããšãããçé¢æ³¢ãšåŒã°ããã",
"title": "3次å
空éäžã®æ³¢"
},
{
"paragraph_id": 33,
"tag": "p",
"text": "",
"title": "3次å
空éäžã®æ³¢"
},
{
"paragraph_id": 34,
"tag": "p",
"text": "å
ã®å Žåã§èãããšåããããããå
ã®é床cã¯ãè§é床ãåšæ³¢æ°ãšã¯ç¡é¢ä¿ã§ããã",
"title": "äœçžé床ãšçŸ€é床"
},
{
"paragraph_id": 35,
"tag": "p",
"text": "ãªãããã®ãæ³¢ã«ãããé床AÏããäœçžé床ãšãããäœçžé床ã¯ãæ
å ±ãäŒããé床ã§ã¯ãªãã",
"title": "äœçžé床ãšçŸ€é床"
},
{
"paragraph_id": 36,
"tag": "p",
"text": "å®éã«æ
å ±ãäŒããããé床ã®ããšã矀é床ãšããã",
"title": "äœçžé床ãšçŸ€é床"
},
{
"paragraph_id": 37,
"tag": "p",
"text": "ãªãããŸããããæ³¢åããè€æ°åã®æ£åŒŠæ³¢ãã足ãåããããåŒãç®ãããããªããšãæ°åŒã§è¡šçŸã§ãªããšå Žåããã®ãããªæ³¢åããåæ£ã®ãããæ³¢åãšããã",
"title": "äœçžé床ãšçŸ€é床"
},
{
"paragraph_id": 38,
"tag": "p",
"text": "ã€ãŸããåæ£ã®ããæ³¢åã®ãæ
å ±ãäŒããããé床ã®ããšãã矀é床ãšããã",
"title": "äœçžé床ãšçŸ€é床"
}
] | null | {{Pathnav|ã¡ã€ã³ããŒãž|èªç¶ç§åŠ|ç©çåŠ|æ¯åãšæ³¢å|frame=1}}
== 1次å
ã®æ³¢åæ¹çšåŒ ==
=== æ³¢åæ¹çšåŒã®äžè¬è§£ ===
æ³¢åæ¹çšåŒã¯å埮åæ¹çšåŒã§ããã®ã§ãããã解ãããã«å¢çæ¡ä»¶ãå®ããã°ãªããªãã1次å
ã®æ³¢åæ¹çšåŒ
:<math>
\left(\frac 1 {v^2} \frac{\partial^2{{}}}{\partial{t}^2} - \frac{\partial^2{{}}}{\partial{x}^2} \right) u(x,t) = 0
</math>
ãèãããšã<math>\xi = x + vt, \eta = x -vt</math> ãšãããšãã
:<math>
\begin{align}
\frac{\partial{{}}}{\partial{x}} &= \frac{\partial{\xi}}{\partial{x}} \frac{\partial{{}}}{\partial{\xi}} + \frac{\partial{\eta}}{\partial{x}} \frac{\partial{{}}}{\partial{\eta}}\\
&= \frac{\partial{{}}}{\partial{\xi}} + \frac{\partial{{}}}{\partial{\eta}},\\
\frac{\partial{{}}}{\partial{t}} &= \frac{\partial{\xi}}{\partial{t}} \frac{\partial{{}}}{\partial{\xi}} + \frac{\partial{\eta}}{\partial{t}} \frac{\partial{{}}}{\partial{\eta}}\\
&= v\left(\frac{\partial{{}}}{\partial{\xi}} - \frac{\partial{{}}}{\partial{\eta}}\right)
\end{align}
</math>
ãçšãããšã
:<math>
\begin{align}
\frac 1 {v^2} \frac{\partial^2{{}}}{\partial{t}^2} - \frac{\partial^2{{}}}{\partial{x}^2}
&=\frac 1 {v^2} v^2 \left(\frac{\partial{{}}}{\partial{\xi}} - \frac{\partial{{}}}{\partial{\eta}}\right)^2
- \left(\frac{\partial{{}}}{\partial{\xi}} + \frac{\partial{{}}}{\partial{\eta}}\right)^2\\
&= \left(\frac{\partial{{}}}{\partial{\xi}} - \frac{\partial{{}}}{\partial{\eta}}\right)^2 - \left(\frac{\partial{{}}}{\partial{\xi}} + \frac{\partial{{}}}{\partial{\eta}}\right)^2\\
&= -4 \frac{\partial{{}}}{\partial{\xi}} \frac{\partial{{}}}{\partial{\eta}}
\end{align}
</math>
ããã
:<math>
-4 \frac{\partial{{}}}{\partial{\xi}} \frac{\partial{{}}}{\partial{\eta}} u(x,t) = 0
</math>
ãšãªãããã®è§£ã¯ã
:<math>
\begin{align}
u(x,t) &= f(\xi ) + g(\eta)\\
& = f(x+vt ) + g(x-vt)\\
\end{align}
</math>
ã§äžããããïŒ''f'' , ''g'' ã¯ä»»æã®é¢æ°ïŒããã®è§£ã®ãã¡ã''x'' + ''v t'' ã«äŸåããé¢æ°ã¯é床''v'' 㧠-''x'' æ¹åã«ç§»åããæ³¢ã«å¯Ÿå¿ãã''x'' - ''v t'' ã«äŸåããé¢æ°ã¯é床''v'' ã§''x'' æ¹åã«ç§»åããæ³¢ã«å¯Ÿå¿ããã
ãã®é¢æ°ãå®å
šã«æ±ºããã«ã¯äŸãã°ãæ³¢ãã€ãããç©äœã®''t'' = 0 ã§ã®äœçœ®ãšé床ãå
šãŠã®ç¹''x'' ã§ç¥ãããŠããã°ãããäŸãã°ã
:<math>u(x,0) = a(x) = \begin{cases}1&-l<x<l \\ 0 & \text{otherwise}\end{cases}</math>
ãã€ãé床ã¯''t'' = 0 ãã€å
šãŠã®''x'' ã§0ãšããããšãã
:<math>
u (x,0) = f(x)+g(x),\qquad
\frac{\partial{{}}}{\partial{t}} u(x,0) = v (f(x) - g(x) )
</math>
ã«ä»£å
¥ãããšã
:<math>
f(x) = g(x) = \frac 12 u(x,0) = \frac 12 a(x)
</math>
ãåŸãããæå»''t'' ã§ã®é¢æ°''u'' ã®å€ã¯ã
:<math>
u = \frac 12 ( a(x+vt) + a(x-vt) )
</math>
ãšãªãã
*å³
=== å®åšæ³¢ ===
æéäŸåæ§ãäœçœ®ã«ãããã«æ±ºãŸãæ³¢ããå®åšæ³¢ãšåŒã¶ã(?)ãã®ãšãã
:<math>
u(x,t) = X(x) T(t)
</math>
ã®ããã«ã解ã®''x'' , ''t'' ã«å¯ŸããäŸåæ§ãåé¢ã§ããããããæ³¢åæ¹çšåŒã«ä»£å
¥ãããšã
:<math>
\begin{align}
\left(\frac 1 {v^2} \frac{\partial^2{{}}}{\partial{t}^2} - \frac{\partial^2{{}}}{\partial{x}^2} \right) u(x,t) &= 0\\
X\frac 1 {v^2} \frac{\partial^2{{}}}{\partial{t}^2} T - T \frac{\partial^2{{}}}{\partial{x}^2} X &= 0\\
\frac 1 T \frac 1 {v^2} \frac{\partial^2{{}}}{\partial{t}^2} T - \frac 1 X \frac{\partial^2{{}}}{\partial{x}^2} X &= 0\\
\frac 1 T \frac 1 {v^2} \frac{\partial^2{{}}}{\partial{t}^2} T &= \frac 1 X \frac{\partial^2{{}}}{\partial{x}^2} X
\end{align}
</math>
ãšå€åœ¢ã§ãããããã§ãæåŸã®åŒã®å·ŠèŸºã¯''t'' ã ãã®é¢æ°ã§ãããå³èŸºã®åŒã¯''x'' ã ãã®é¢æ°ã§ããã®ã§ãã©ã¡ãã®å€ãå®æ°ã«çããã¯ãã§ããããã®å®æ°ãã-ω<sup>2</sup> /''v''<sup>2</sup> ãšãããšã
:<math>
\begin{align}
\frac 1 T \frac{\partial^2{{}}}{\partial{t}^2} T &= -\omega^2\\
\frac{\partial^2{{}}}{\partial{t}^2} T + \omega^2 T&= 0\\
\end{align}
</math>
ãšãªãã解
:<math>
T(t) = A \sin (\omega t ) + B \cos (\omega t)
</math>
ãåŸãïŒ''A'' , ''B'' ã¯ä»»æå®æ°ïŒãäžæ¹ã''X'' ã«ã€ããŠãåæ§ã«
:<math>
\frac{\partial^2{{}}}{\partial{x}^2} X + \frac 1 {v^2} \omega^2 X= 0
</math>
ãåŸãããšãã§ãã解
:<math>
X(x) = A \sin \left(\frac \omega v x \right) + B \cos \left(\frac \omega v x\right)
</math>
ãåŸãïŒ''A'' , ''B'' ã¯ä»»æå®æ°ïŒã
ç¹ã«ã''x'' = 0, ''x'' = ''l'' ã§''u'' (''t'' , ''x'' ) = 0 ãšãªãå Žåãèãããããã¯ãç©äœã®ç«¯ãåºå®ãããŠããå Žåã«å¯Ÿå¿ããã®ã§åºå®ç«¯ãšåŒã°ããã<!-- (note: 埮åæ¹çšåŒã§å¢çæ¡ä»¶ãäžããããåé¡ããåºæå€åé¡ãšåŒã¶ã)(?) -->ãã®ãšãã''x'' = 0 ã§''u'' = 0 ãã''B'' = 0 ãåŸãããããŸãã
:<math>
X(l) = A \sin \left(\frac \omega v l \right)
</math>
ããã
:<math>
\frac\omega v l = \pi n
</math>
ïŒ''n'' ã¯æŽæ°ïŒãšãªãã
:<math>
\omega = \frac {\pi nv} {l }
</math>
ãåŸãããã''n'' = 0 ã¯å
šãæ³¢ãèµ·ããŠããªãç¶æ³ã«å¯Ÿå¿ãã''n'' = 1 ã¯ç¯ã1ã€ã ãã®æ³¢ãèµ·ããŠããç¶æ³ã«å¯Ÿå¿ããã''n'' > 1 ã¯ãç¯ã''n'' åã®æ³¢ã«å¯Ÿå¿ããã
*å³
å
šãŠã®ç¹ã®æéäŸåæ§ãåäžãªã®ã§''T'' (''t'' ) ã決ããã«ã¯ããäžç¹ã§ã®æ¯åã®ããæå»ã§ã®äœçœ®ãšé床ãäžããã°ããïŒå®éã«ã¯ããæå»ã§äž¡æ¹ãäžããå¿
èŠã¯ãªããéãæå»ã§1ã€ãã€äžããŠãããïŒãäŸãã°ã''t'' = 0, ''x'' = ''l'' /2 ã§ã''u'' = 0,
<math>
\frac{\partial{u}}{\partial{t}} = a
</math>
ïŒ''a'' ã¯å®æ°ïŒãäžãããããšãããšã
:<math>
T(t) = A \sin (\omega t ) + B \cos (\omega t)
</math>
ã«ã€ããŠã''B'' = 0, ω''A'' = ''a''ãŸãã¯''A'' = ''a'' /ωãåŸãããããã£ãŠããã®æ¹çšåŒã®è§£ã¯
:<math>
u(t,x) = \frac a \omega \sin (\omega t) \sin \left(\frac {\omega _n} v x \right) ,\qquad \omega _n = \frac {\pi v n} {l }
</math>
ãšãªãã
==2次å
å¹³é¢äžã®æ³¢==
===2次å
空éäžã®æ³¢åæ¹çšåŒ===
2次å
å¹³é¢äžã§ãããæ¹åã''x'' æ¹åãšåãããããšåçŽãªæ¹åã''y'' 軞ãšåãã''x'' 軞ãš''y'' 軞ãã€ããããŠãæ¹çšåŒãå€ãããªãããšã«æ³šç®ãããšãæ³¢åæ¹çšåŒã¯
:<math>
\frac 1 {v^2}\frac{\partial^2{{}}}{\partial{t}^2} - \frac{\partial^2{{}}}{\partial{x}^2} - \frac{\partial^2{{}}}{\partial{y}^2} = 0
</math>
ãšãªãã
*TODO
2次å
å¹³é¢äžã§ã®åºå®ç«¯ã®å®åšæ³¢ã¯ã2ã€ã®æŽæ°ã䜿ã£ãŠè¡šããããããšïŒå€æ°åé¢ïŒã2ã€ã®æŽæ°ãm,nãšãããšãã®m = 1,n=1ã®æãªã©ã®å³ã
== 3次å
空éäžã®æ³¢ ==
=== 3次å
空éäžã®æ³¢åæ¹çšåŒ ===
3次å
å¹³é¢äžã§ãããæ¹åã«''x'' 軞ãããããšåçŽãªæ¹åã«''y'' 軞ãåããããããé ã«å³æã®èŠªæã人差ãæãäžæã«å¯Ÿå¿ããããã«''z'' 軞ãåããããããã®è»žãã€ããããŠãæ¹çšåŒãå€ãããªãããšã«æ³šç®ãããšãæ³¢åæ¹çšåŒã¯
:<math>
\frac 1 {v^2}\frac{\partial^2{{}}}{\partial{t}^2} - \frac{\partial^2{{}}}{\partial{x}^2} - \frac{\partial^2{{}}}{\partial{y}^2} - \frac{\partial^2{{}}}{\partial{z}^2} = 0
</math>
ãšãªãã
=== çé¢æ³¢ ===
:<math>
\Delta r = \frac 1 r \frac{\partial^2{{}}}{\partial{r}^2} (rf )
</math>
ïŒ''f'' ã¯''r'' , ''t'' ã ãã®é¢æ°ãΔ ã¯ã©ãã©ã·ã¢ã³ãïŒ(?)ãã®ãšããäžããããæ³¢åæ¹çšåŒã¯ã
:<math>
\frac 1 {v^2}\frac{\partial^2{{f }}}{\partial{t}^2} - \frac 1 r \frac{\partial^2{{}}}{\partial{r}^2} (rf ) = 0
</math>
ãšãªããããã㧠''r f'' (''r'' , ''t'' ) ã«ã€ããŠã¯ãã®åŒã¯éåžžã®1次å
ã®æ³¢åæ¹çšåŒã«å¯Ÿå¿ããããã£ãŠãã®æ¹çšåŒã®è§£ãšããŠ
:<math>
f(r,t) = \frac 1 r u(r+ vt ) + \frac 1 r v(r-vt)
</math>
ïŒ''u'' , ''v'' ã¯ä»»æã®é¢æ°ïŒãåŸãããããã¯ç察称ãªæ³¢ãè¡šããããšãããçé¢æ³¢ãšåŒã°ããã<!-- å€ã«åºãŠè¡ãæ³¢ãšäžã«ã¯ãã£ãŠæ¥ãæ³¢ã«ããããååããã£ããããª...ã -->
== äœçžé床ãšçŸ€é床 ==
[[ãã¡ã€ã«:Wave group.gif|thumb|400px|æ°Žæ·±ãæ·±ãæ°Žã®è¡šé¢ã®æ°Žé¢æ³¢ã«ããããåšæ³¢æ°åæ£ãæã€æ³¢æïŒæ³¢çŸ€ïŒãè¡šãããã®ã<span style="border-bottom:solid 2px red;">èµ€ç¹ã¯'''äœçžé床'''</span>ã§åãã<span style="border-bottom:solid 2px lime;">ç·ç¹ã¯'''矀é床'''</span>ã§åããŠããããã®ããã«æ°Žæ·±ãæ·±ãå Žåã«ã¯ãæ°Žé¢ã§ã¯äœçžé床ã¯çŸ€é床ã®äºåã«ãªããå³ã®å·Šããå³ã«åãéãèµ€ç¹ã¯ç·ç¹ãäºåè¿œãè¶ãã<br>æ³¢æã®åŸæ¹(ã®ç·ç¹)ã§æ°ããæ³¢ãåºçŸããæ³¢æã®äžå¿ã«åãã£ãŠæ¯å¹
ã倧ãããªããæ³¢æã®åæ¹(ã®ç·ç¹)ã§æ¶ããŠããããã«èŠãããæ°Žé¢ã®éåæ³¢ã«ãããŠã¯ãã»ãšãã©ã®å Žåãæ°Žç²åã®é床ã¯äœçžé床ããããã£ãšå°ããã]]
å
ã®å Žåã§èãããšåããããããå
ã®é床cã¯ãè§é床ãåšæ³¢æ°ãšã¯ç¡é¢ä¿ã§ããã
ãªãããã®ãæ³¢ã«ãããé床AÏãã'''äœçžé床'''ãšãããäœçžé床ã¯ãæ
å ±ãäŒããé床ã§ã¯ãªãã
å®éã«æ
å ±ãäŒããããé床ã®ããšã'''矀é床'''ãšããã
ãªãããŸããããæ³¢åããè€æ°åã®æ£åŒŠæ³¢ãã足ãåããããåŒãç®ãããããªããšãæ°åŒã§è¡šçŸã§ãªããšå Žåããã®ãããªæ³¢åããåæ£ã®ãããæ³¢åãšããã
ã€ãŸããåæ£ã®ããæ³¢åã®ãæ
å ±ãäŒããããé床ã®ããšãã矀é床ãšããã
{{DEFAULTSORT:ãããšããšã¯ãšãã¯ãšãã»ããŠãããã®ãããã€}}
[[Category:æ¯åãšæ³¢å|ã¯ãšãã»ããŠãããã®ãããã€]]
[[ã«ããŽãª:埮åæ¹çšåŒ]] | 2005-06-08T11:10:02Z | 2024-03-16T02:59:26Z | [
"ãã³ãã¬ãŒã:Pathnav"
] | https://ja.wikibooks.org/wiki/%E6%8C%AF%E5%8B%95%E3%81%A8%E6%B3%A2%E5%8B%95/%E6%B3%A2%E5%8B%95%E6%96%B9%E7%A8%8B%E5%BC%8F%E3%81%AE%E6%80%A7%E8%B3%AA |
2,115 | ç©çæ°åŠI | æ¬é
ã¯ç©çåŠ ç©çæ°åŠI ã®è§£èª¬ã§ãã | [
{
"paragraph_id": 0,
"tag": "p",
"text": "æ¬é
ã¯ç©çåŠ ç©çæ°åŠI ã®è§£èª¬ã§ãã",
"title": ""
}
] | æ¬é
ã¯ç©çåŠ ç©çæ°åŠI ã®è§£èª¬ã§ãã 解æåŠ
1å€æ°ã®èšç®
å€å€æ°é¢æ°ã®åŸ®ç©å
æ°åã®åæ
ç·åœ¢ä»£æ°
è¡åã®å®çŸ©ãšç¹å¥ãªè¡å
éè¡åã®äžè¬åœ¢
2次圢åŒ
è¡åã®å¯Ÿè§å
埮åæ¹çšåŒ
埮åæ¹çšåŒã®å®çŸ©
埮åæ¹çšåŒã®è§£æ³
解ã®äžææ§
ç·åœ¢åŸ®åæ¹çšåŒ
ãã¯ãã«è§£æ
ãã¯ãã«é¢æ°ã®å®çŸ©
ãã³ãœã«ä»£æ°
å€å€æ°é¢æ°ã®ç©å
çŽäº€åº§æšç³»ã§ãªããšãã®èšç® è€çŽ 解æã®åãŸã§ãç©çæ°åŠIã®ç¯å²ãšããã | {{Pathnav|ã¡ã€ã³ããŒãž|èªç¶ç§åŠ|ç©çåŠ|frame=1|small=1}}
æ¬é
ã¯ç©çåŠ ç©çæ°åŠI ã®è§£èª¬ã§ãã
* [[ç©çæ°åŠI 解æåŠ|解æåŠ]]
** [[ç©çæ°åŠI 解æåŠ#1å€æ°ã®èšç®|1å€æ°ã®èšç®]]
** [[ç©çæ°åŠI 解æåŠ#å€å€æ°é¢æ°ã®åŸ®ç©å|å€å€æ°é¢æ°ã®åŸ®ç©å]]
** [[ç©çæ°åŠI 解æåŠ#æ°åã®åæ|æ°åã®åæ]]
* [[ç©çæ°åŠI ç·åœ¢ä»£æ°|ç·åœ¢ä»£æ°]]
** [[ç©çæ°åŠI ç·åœ¢ä»£æ°#è¡åã®å®çŸ©ãšç¹å¥ãªè¡å|è¡åã®å®çŸ©ãšç¹å¥ãªè¡å]]
** [[ç©çæ°åŠI ç·åœ¢ä»£æ°#éè¡åã®äžè¬åœ¢|éè¡åã®äžè¬åœ¢]]
** [[ç©çæ°åŠI ç·åœ¢ä»£æ°#2次圢åŒ|2次圢åŒ]]
** [[ç©çæ°åŠI ç·åœ¢ä»£æ°#è¡åã®å¯Ÿè§å|è¡åã®å¯Ÿè§å]]
* [[ç©çæ°åŠI 埮åæ¹çšåŒ|埮åæ¹çšåŒ]]
** [[ç©çæ°åŠI 埮åæ¹çšåŒ#埮åæ¹çšåŒã®å®çŸ©|埮åæ¹çšåŒã®å®çŸ©]]
** [[ç©çæ°åŠI 埮åæ¹çšåŒ#埮åæ¹çšåŒã®è§£æ³|埮åæ¹çšåŒã®è§£æ³]]
** [[ç©çæ°åŠI 埮åæ¹çšåŒ#解ã®äžææ§|解ã®äžææ§]]
** [[ç©çæ°åŠI 埮åæ¹çšåŒ#ç·åœ¢åŸ®åæ¹çšåŒ|ç·åœ¢åŸ®åæ¹çšåŒ]]
* [[ç©çæ°åŠI ãã¯ãã«è§£æ|ãã¯ãã«è§£æ]]
** [[ç©çæ°åŠI ãã¯ãã«è§£æ#ãã¯ãã«é¢æ°ã®å®çŸ©|ãã¯ãã«é¢æ°ã®å®çŸ©]]
** [[ç©çæ°åŠI ãã¯ãã«è§£æ#ãã³ãœã«ä»£æ°|ãã³ãœã«ä»£æ°]]
** [[ç©çæ°åŠI ãã¯ãã«è§£æ#å€å€æ°é¢æ°ã®ç©å|å€å€æ°é¢æ°ã®ç©å]]
** [[ç©çæ°åŠI ãã¯ãã«è§£æ#çŽäº€åº§æšç³»ã§ãªããšãã®èšç®|çŽäº€åº§æšç³»ã§ãªããšãã®èšç®]]
*è€çŽ 解æã®åãŸã§ã[[ç©çæ°åŠI]]ã®ç¯å²ãšããã
==é¢é£æç§æž==
*[[ç©çæ°åŠII]]
[[Category:ç©çåŠ|ãµã€ããããã1]]
[[Category:æ°åŠ|ãµã€ããããã1]] | 2005-06-09T12:57:32Z | 2024-03-16T06:15:11Z | [
"ãã³ãã¬ãŒã:Pathnav"
] | https://ja.wikibooks.org/wiki/%E7%89%A9%E7%90%86%E6%95%B0%E5%AD%A6I |