**Sferik koordinatalar sistemasi** — uch oʻlchamli koordinatalar sistemasi boʻlib, fazodagi nuqtaning vaziyati uchta kattalik bilan ({\displaystyle r,\theta ,\varphi }) bilan aniqlanadi. Bu yerda {\displaystyle {\displaystyle r}} — koordinatalar boshigacha boʻlgan masofa, {\displaystyle {\displaystyle \theta }} va {\displaystyle {\displaystyle \varphi }}— mos holda zenit va azimutal burchaklar.

Zenit va azimut tushunchalari astronomiyada keng qoʻllaniladi. Zenit — ixtiyoriy tanlangan nuqta (kuzatish nuqtasi) dan vertikal yuqoriga yoʻnalgan boʻlib, fundamental tekislikda yotadi. Astronomiyada fundamental tekislik sifatida ekvator yotgan tekislik yoki ekliptika tekisligi olinadi. Azimut — fundamental tekislikdagi ixtiyoriy tanlangan nur bilan boshlangʻich kuzatish nuqtasi orasidagi burchak.

1-rasm. Ixtiyoriy nuqtaning fazodagi vaziyatini uchta dekart koordinatalari yoki uchta sferik koordinatalar orqali aniqlash mumkin.

Boshqa koordinata sistemalariga oʻtish
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### Dekart koordinatalar sistemasi

Agar nuqtaning sferik koordinatalari {\displaystyle (r,\;\theta ,\;\varphi )} berilgan boʻlsa, dekart koordinatalariga oʻtish uchun quyidagi formulalardan foydalaniladi:

{\displaystyle {\begin{cases}x=r\sin \theta \cos \varphi ,\\y=r\sin \theta \sin \varphi ,\\z=r\cos \theta .\end{cases}}}
Dekart koordinatalaridan sferik koordinatalarga oʻtish uchun esa:

{\displaystyle {\begin{cases}r={\sqrt {x^{2}+y^{2}+z^{2}}},\\\theta =\arccos {\dfrac {z}{\sqrt {x^{2}+y^{2}+z^{2}}}}=\mathrm {arctg} {\dfrac {\sqrt {x^{2}+y^{2}}}{z}},\\\varphi =\mathrm {arctg} {\dfrac {y}{x}}.\end{cases}}}
Sferik koordinatalarga oʻtish yakobiani:

{\displaystyle {\begin{alignedat}{2}J&={\frac {\partial (x,y,z)}{\partial (r,\theta ,\varphi )}}={\begin{vmatrix}\sin \theta \cos \varphi &r\cos \theta \cos \varphi &-r\sin \theta \sin \varphi \\\sin \theta \sin \varphi &r\cos \theta \sin \varphi &r\sin \theta \cos \varphi \\\cos \theta &-r\sin \theta &0\end{vmatrix}}=\\&=\cos \theta (r^{2}\cos \varphi ^{2}\cos \theta \sin \theta +r^{2}\sin ^{2}\varphi \cos \theta \sin \theta )+r\sin \theta (r\sin ^{2}\theta \cos ^{2}\varphi +r\sin ^{2}\theta \sin ^{2}\varphi )=\\&=r^{2}\cos ^{2}\theta \sin \theta +r^{2}\sin ^{2}\theta \sin \theta =\\&=r^{2}\sin \theta .\end{alignedat}}}
Shunday qilib, dekart koordinatalaridan sferik koordinatalarga oʻtishdagi hajm elementi quyidagi koʻrinishga ega boʻladi:

{\displaystyle \mathrm {d} V=\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z=J(r,\theta ,\varphi )\,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi =r^{2}\sin \theta \,\,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi }
### Silindrik koordinatalar sistemasi

Agar nuqtaning silindrik koordinatalari berilgan boʻlsa, sferik koordinatalarga oʻtish uchun quyidagi formulalardan foydalaniladi:

{\displaystyle {\begin{cases}\rho =r\sin \theta \\\varphi =\varphi \\z=r\cos \theta \end{cases}}}
Yoki aksincha, sferik koordinatalardan silindrik koordinatalarga oʻtish uchun quyidagi formulalardan foydalaniladi:

{\displaystyle {\begin{cases}r={\sqrt {\rho ^{2}+z^{2}}},\\\theta =\mathrm {arctg} {\dfrac {\rho }{z}},\\\varphi =\varphi .\end{cases}}}
Silindrik koordinatalardan sferik koordinatalarga oʻtish yakobiani :

{\displaystyle J=r}

Sferik koordinatalar sistemasida differensiallash va integrallash
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{\displaystyle (r,\theta ,\varphi )} nuqtadan {\displaystyle (r+\mathrm {d} r,\,\theta +\mathrm {d} \theta ,\,\varphi +\mathrm {d} \varphi )} nuqtaga oʻtkazilgan vektor {\displaystyle \mathrm {d} \mathbf {r} } ning uzunligi quyidagiga teng:

{\displaystyle \mathrm {d} \mathbf {r} =\mathrm {d} r\,{\boldsymbol {\hat {r}}}+r\,\mathrm {d} \theta \,{\boldsymbol {\hat {\theta }}}+r\sin {\theta }\,\mathrm {d} \varphi \,\mathbf {\boldsymbol {\hat {\varphi }}} ,}
bu yerda

{\displaystyle {\boldsymbol {\hat {r}}}=\sin \theta \cos \varphi {\boldsymbol {\hat {\imath }}}+\sin \theta \sin \varphi {\boldsymbol {\hat {\jmath }}}+\cos \theta {\boldsymbol {\hat {k}}}}
{\displaystyle {\boldsymbol {\hat {\theta }}}=\cos \theta \cos \varphi {\boldsymbol {\hat {\imath }}}+\cos \theta \sin \varphi {\boldsymbol {\hat {\jmath }}}-\sin \theta {\boldsymbol {\hat {k}}}}
{\displaystyle {\boldsymbol {\hat {\varphi }}}=-\sin \varphi {\boldsymbol {\hat {\imath }}}+\cos \varphi {\boldsymbol {\hat {\jmath }}}}
Sferik koordinatalar ortogonal hisoblanadi. Shu sababli ularning metrik tenzori diagonal koʻrinishda boʻladi:

{\displaystyle g\_{ij}={\begin{pmatrix}1&0&0\\0&r^{2}&0\\0&0&r^{2}\sin ^{2}\theta \end{pmatrix}},\quad g^{ij}={\begin{pmatrix}1&0&0\\0&{\dfrac {1}{r^{2}}}&0\\0&0&{\dfrac {1}{r^{2}\sin ^{2}\theta }}\end{pmatrix}}}
* {\displaystyle \det(g\_{ij})=r^{4}\sin ^{2}\theta .\ }
* Yoy uzunligi differensialining kvadrati:

{\displaystyle ds^{2}=dr^{2}+r^{2}\,d\theta ^{2}+r^{2}\sin ^{2}\theta \,d\varphi ^{2}.}
* Lame koeffitsiyentlari:

{\displaystyle H\_{r}=1,\quad H\_{\theta }=r,\quad H\_{\varphi }=r\sin \theta .}
* Kristoffel belgilari {\displaystyle \{r,\;\theta ,\;\varphi \}}:

{\displaystyle \Gamma \_{22}^{1}=-r,\quad \Gamma \_{33}^{1}=-r\sin ^{2}\theta ,}
{\displaystyle \Gamma \_{21}^{2}=\Gamma \_{12}^{2}=\Gamma \_{13}^{3}=\Gamma \_{31}^{3}={\frac {1}{r}},}
{\displaystyle \Gamma \_{33}^{2}=-\cos \theta \sin \theta ,\quad \Gamma \_{23}^{3}=\Gamma \_{32}^{3}=\mathrm {ctg} \,\theta .}
Sferik koordinatalar sistemasida birlik vektorlar

Sferik koordinatalar sistemasida masofa
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Fazodagi vaziyati sferik koordinatalar sistemasida berilgan ikki nuqtaning joylashuvi quyidagicha boʻlsin:

{\displaystyle {\begin{aligned}{\mathbf {r} }&=(r,\theta ,\varphi ),\\{\mathbf {r} '}&=(r',\theta ',\varphi ')\end{aligned}}}
U holda ushbu nuqtalar orasidagi masofani quyidagi formula orqali hisoblash mumkin:

{\displaystyle {\begin{aligned}{\mathbf {D} }&={\sqrt {r^{2}+r'^{2}-2rr'(\sin {\theta }\sin {\theta '}\cos {(\varphi -\varphi ')}+\cos {\theta }\cos {\theta '})}}\end{aligned}}}

Harakat tenglamasi
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Nuqtaning vaziyati sferik koordinatalarda quyidagi koʻrinishda berilgan boʻlsin:

{\displaystyle \mathbf {r} =r\mathbf {\hat {r}} .}
U holda uning tezligi:

{\displaystyle \mathbf {v} ={\dot {r}}\mathbf {\hat {r}} +r\,{\dot {\theta }}\,{\hat {\boldsymbol {\theta }}}+r\,{\dot {\varphi }}\sin \theta \,\mathbf {\hat {\boldsymbol {\varphi }}} ,}
hamda tezlanishi:

{\displaystyle {\begin{aligned}\mathbf {a} ={}&\left({\ddot {r}}-r\,{\dot {\theta }}^{2}-r\,{\dot {\varphi }}^{2}\sin ^{2}\theta \right)\mathbf {\hat {r}} \\&{}+\left(r\,{\ddot {\theta }}+2{\dot {r}}\,{\dot {\theta }}-r\,{\dot {\varphi }}^{2}\sin \theta \cos \theta \right){\hat {\boldsymbol {\theta }}}\\&{}+\left(r{\ddot {\varphi }}\,\sin \theta +2{\dot {r}}\,{\dot {\varphi }}\,\sin \theta +2r\,{\dot {\theta }}\,{\dot {\varphi }}\,\cos \theta \right){\hat {\boldsymbol {\varphi }}}.\end{aligned}}}
ga teng boʻladi.

Burchak momenti:

{\displaystyle \mathbf {L} =m\mathbf {r} \times \mathbf {v} =mr^{2}({\dot {\theta }}\,{\hat {\boldsymbol {\varphi }}}-{\dot {\varphi }}\sin \theta \,\mathbf {\hat {\boldsymbol {\theta }}} ).}
{\displaystyle \varphi } oʻzgarmas boʻlganda yoki {\displaystyle \theta ={\frac {\pi }{2}}} boʻlganda, moddiy nuqtaning harakat tenglamasi qutb koordinatalar sistemasiga oʻtadi.

{\displaystyle \mathbf {L} =-i\hbar ~\mathbf {r} \times \nabla =i\hbar \left({\frac {\hat {\boldsymbol {\theta }}}{\sin(\theta )}}{\frac {\partial }{\partial \phi }}-{\hat {\boldsymbol {\phi }}}{\frac {\partial }{\partial \theta }}\right).}

Yana qarang
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* Eyler burchaklari
* Gipersferik koordinatalar

Manbalar
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* Morse PM, Feshbach H*. Methods of Theoretical Physics, Part I*. New York: McGraw-Hill, 1953 — 658 bet. ISBN 0-07-043316-X.
* Margenau H, Murphy GM*. The Mathematics of Physics and Chemistry*. New York: D. van Nostrand, 1956 — 177–178 bet.
* Korn GA, Korn TM*. Mathematical Handbook for Scientists and Engineers*. New York: McGraw-Hill, 1961 — 174–175 bet. ASIN B0000CKZX7.
* Sauer R, Szabó I*. Mathematische Hilfsmittel des Ingenieurs*. New York: Springer Verlag, 1967 — 95–96 bet.
* Moon P, Spencer DE „Spherical Coordinates (r, θ, ψ)",*. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions*, corrected 2nd ed. 3rd print, New York: Springer-Verlag, 1988 — 24–27 (Table 1.05) bet. ISBN 978-0-387-18430-2.
* Duffett-Smith P, Zwart J*. Practical Astronomy with your Calculator or Spreadsheet, 4th Edition*. New York: Cambridge University Press, 2011 — 34 bet. ISBN 978-0521146548.
* Python tilida yozilgan programma

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