应充分利用曲线的方程简化被积函数, 代,Py QxLD\uf0a2\uf0a2\uf03d \uf02d\uf03d\uf0a2函数在 所围的区域内具有一阶连续偏导数, 满足格林公式使用条件.224LLxdyydxyxdyydxx\uf02d\uf03d\uf02d\uf02b\uf0f2\ue000\uf0f2\ue000cL\uf05c\uf03d\uf0f2\ue000 \ue000\uf0f2( , ),P x y Q x y( , )LC函数在 和所围的区域内具有一阶连续偏导数, 满足格林公式使用条件.2LDxdyydxdxdy\uf070\uf03d\uf0a2\uf02d\uf03d\uf0f2\ue000\uf0f2\uf0f2221: 4xLy\uf02b\uf03d 格林公式: \uf0f2\uf0f2\uf0f2D\uf02b\uf03d\uf0b6\uf0b6\uf02d\uf0b6\uf0b6LQdyPdxdxdyyPxQ)( 取,,xQyP\uf03d\uf02d\uf03d 得 \uf0f2\uf0f2\uf0f2D\uf02d\uf03dLydxxdydxdy2闭区域D的面积 \uf0f2\uf02d\uf03dLydxxdyA21. 取,, 0xQP\uf03dQ\uf03d\uf03d 得 \uf0f2\uf03d\uf03dLxdy\uf0f2\uf02dA 取, 0,\uf03d\uf02dyP 得 LydxA (3)计算平面面积 曲线 AMO由函数 axy\uf02d\uf03d1], 0 [,axx\uf0ce表示, 例 6 计算抛物线围成的面积。 ) 0()(2\uf03e\uf03d\uf02baaxyx与 x轴所解 ONA为直线0\uf03dy。 \uf0f2\uf02d\uf03d\uf05cLydxxdyA2\uf0f2\uf0f2\uf02d\uf02b\uf02d\uf03dAMOONAydxxdyydxxdy21211)0 ,(aANM\uf0f2\uf02d\uf03dAMOydxxdy2 \uf0f2\uf02dAMOydxxdy21dxxaxdxaxaxa)() 12(210\uf02d\uf02d\uf02d\uf03d \uf0f220146aaxdxa\uf03d\uf03d\uf0f2) 0 ,(aANM例 6 计算抛物线围成的面积。 ) 0()(2\uf03e\uf03d\uf02baaxyx与 x轴所 Gyxo\uf0f2\uf02b1 LQdyPdx则称曲线积分\uf0f2否则称为与路径有关。\uf02bLQdyPdx在G内与路径无关, \uf0f2\uf02b2LQdyPdx1 L2 L\uf0d7B\uf0d7 A如果在区域G内有\uf03d 8.2.2 平面上曲线积分与路径无关的条件 平面上曲线积分与路径无关的等价条件定理8.2.2 设是单连通域,)y , x(Q),y , x(P在 内D具有一阶连续偏导数,(1) 沿中任意光滑闭曲线, 有0ddLP x\uf0f2 \ue000, 曲线积分Q y\uf02b\uf03d。(2) 对中任一分段光滑曲线(3)yQxPdd \uf02b),(yxuyQxPyxudd),(d\uf02b\uf03d(4) 在内每一点都有PQyx\uf0b6\uf0b6\uf0b6\uf0b6\uf03d。\uf0f2\uf02bLyQxPdd与路径无关, 只与起止点有关. 函数则以下四个条件等价:在内是某一函数的全微分,即DDDLLDD 说明: 积分与路径无关时, 曲线积分可记为证明 (1) (2)设21, LL\uf0f2\uf0f2\uf02b\uf03d1ddLyQxP\uf02d2\uf02b\uf02b\uf03d1ddLLyQxPAB1L2 L\uf0f2\uf02b\uf03d2ddLyQxP\uf0f2\uf02b\uf05c1ddLyQxP为D内任意两条由A到B的有向分段光滑曲线则(根据条件(1) )\uf0f2\uf02b\uf03dBAyQxPdd\uf0f2\uf02bAByQxPdd0\uf03d12ddddLLP xQ yP xQ y\uf02b\uf02d\uf02b\uf0f2\uf0f2\uf0f22ddLP xQ y\uf02d\uf02b\uf02b 证明 (2)在D内取定点(3)(0yx),0A因曲线积分\uf0f2\uf02b\uf03d),((),00dd),(yyxxyQxPyxu),(),(yxuyxxuux\uf02d\uf044,\uf02b\uf044)y\uf03d\uf044则),(yxP\uf03dxuxuxx\uf0b6\uf044\uf044\uf03d\uf0b6\uf0b6\uf05c\uf0aeu\uf0b6\uf0440lim),(limx\uf0ae\uf0440yxxP\uf044\uf02b\uf03d\uf071\uf0f2\uf02b,\uf02b\uf03d)((ddyxxxyQxP\uf0f2P\uf044)yx\uf02b,\uf03d),((dyxxxxPxyx\uf044\uf044\uf02b\uf03d),(\uf071同理可证y),,(yxQ\uf03d因此有yQxPuddd\uf02b\uf03d和任一点,与路径无关,),(yxxC\uf044\uf02b),(yxB),(00yxA构造函数),(yxB 证明 (3) (4)),(yxu设存在函数使得Q\uf02byxPuddd\uf03d则)y, x(Qyu),y , x(Pxu\uf03d\uf0b6\uf0b6\uf0b6\uf03d\uf0b6\uf0b6\uf0b6在D内具有连续的偏导数,xyuyxu\uf0b6\uf0b6\uf0b6\uf03d\uf0b6\uf0b6\uf0b622所以从而在D内每一点都有xQyP\uf0b6\uf0b6\uf03d\uf0b6\uf0b6xyuxQ,yxuyP\uf0b6\uf0b6\uf0b6\uf03d\uf0b6\uf0b6\uf0b6\uf0b6\uf03d\uf0b6\uf05c22QP, 证明 (4)(1)设L为D中任一分段光滑闭曲线,DD \uf0cc\uf0a2(如图) ,上因此在 D\uf0a2xQyP\uf0b6\uf0b6\uf0ba\uf0b6\uf0b6利用格林公式 , 得ydx)xQxQ(ydQxdPLDd\uf0b6\uf0b6\uf02d\uf0b6\uf0b6\uf03d\uf02b\uf0f2\uf0f2\uf0f2 \uf0a2DD\uf0a2L0\uf03d所围区域为证毕(1) 沿中任意光滑闭曲线, 有0ddLP x\uf0f2 \ue000Q y\uf02b\uf03d。(4) 在内每一点都有PQyx\uf0b6\uf0b6\uf0b6\uf0b6\uf03d。DLD yyx说明: 根据定理2, 若在某区域内,xQyP\uf0b6\uf0b6\uf03d\uf0b6\uf0b6则2) 求曲线积分时, 可利用格林公式简化计算,3) 可用积分法求在域D内的原函数:Dyx\uf0ce),(00及动点,),(Dyx\uf0ced )y , x(Qxd )y , x(P)y , x(u)y , xy ,x0()(\uf02b\uf03d\uf0f20\uf0f2\uf03dxxxd )y , x(P00或\uf0f2\uf03dyyyd )y ,0x(Q)y , x(u00 y0 x则原函数为\uf0f2\uf02byyyd )y , x(Q0xx\uf0f2\uf02bxd ) y , x(P0若积分路径不是闭曲线, 可添加辅助线;Pdxdu\uf03d取定点1) 计算曲线积分时, 可选择方便的积分路径;Qdy\uf02b yA xoL例7 计算,d)(d)3(22yxyxyxL\uf02d\uf02b\uf02b\uf0f2其中L 为上半24xxy\uf02d\uf03d从 O (0, 0) 到 A (4, 0) .解区域为D , 则为了使用格林公式, 添加辅助线段,AOD它与L所围原式yxyxyxAOLd)(d)3(22\uf02d\uf02b\uf02b\uf03d\uf0f2\uf02b( 4)d d\uf02dDx y\uf02d\uf03d\uf0f2\uf0f28 \uf02b\uf070\uf0f2\uf02d\uf02b\uf02b\uf02bOAyxyxyxd)(d)3(22\uf0f2\uf02b402dxx圆周364\uf03d 例8 验证yyxxyxdd22\uf02b是某个函数的全微分, 并求出这个函数. 证 设,,22yxQyxP\uf03d\uf03d则xQyxyP\uf0b6\uf0b6\uf03d\uf03d\uf0b6\uf0b62由定理2 可知, 存在函数 u (x , y) , 使2\uf02b\uf03dyyxxyxuddd2\uf0f2x0\uf02b\uf03d),() 0 , 0 (22dd),(yxyyxxyxyxu。) 0 , 0 (。),(yx) 0 ,(x\uf0f2\uf0d7\uf03dxxd0yyxyd02\uf0f2\uf03dyyxyd02\uf0f2\uf02b2221yx\uf03d 例9 验证22ddyxxyyx\uf02b\uf02d在右半平面内存在原函数 , 并求出它. 证令2222,yxxQy2x2yP\uf02b\uf03d\uf02bx\uf02d\uf03d则)0()(222\uf03e\uf0b6\uf0b6\uf03d\uf02b\uf02d\uf03d\uf0b6\uf0b6xxQyxyyP由定理 2 可知存在原函数\uf0f2\uf02b\uf02d\uf03d),() 0 , 1 (22dd),(yxyxxyyxyxu\uf0f2\uf0d7\uf02d\uf03dxx1d0) 0(arctan\uf03e\uf03dxxyoxy\uf0f2\uf02b\uf02byyxyx022d) 0 ,( x) 0 , 1 (),(yx) 0(\uf03ex oxy) 0 ,( x) 0 , 1 (),(yx\uf0f2\uf02b\uf02d\uf03d),() 0 , 1 (22dd),(yxyxxyyxyxu\uf0f2\uf02b\uf03dyyy021dyxyyarctan1arctanarctan\uf02d\uf02b\uf03dyxarctan2\uf02d\uf03d\uf070\uf0f2\uf02b\uf02dxyxxy122d或), 1 (y) 0(arctan\uf03e\uf03dxxy 例10 设质点在力场作用下沿曲线 L :xycos2\uf070\uf03d由)2, 0(\uf070A移动到, ) 0,2(\uf070B求力场所作的功Wy解)dd(2\uf0f2\uf02d\uf03dLyxxyrk令,,22rxkQrykP\uf02d\uf03d\uf03d则有) 0()(22422\uf0b9\uf02b\uf02d\uf03d\uf0b6\uf0b6yxryxkyPxQ\uf0b6\uf0b6\uf03d可见, 在不含原点的单连通区域内积分与路径无关.. )(22yxr\uf02b\uf03d其中LBAox),(2xyrkF\uf02d\uf03dsFWLd\uf0f2\uf0d7\uf03d :AB)dd(2yxxyr\uf02dkWAB\uf02d\uf03d\uf0f2\uf071d\uf071\uf071\uf070)cos(sin2022\uf02b\uf03d \uf0f2k) 02:(\uf071sin2,cos2\uf0ae\uf03d\uf03d\uf070\uf071\uf070\uf071\uf070yxk2\uf070\uf03d取圆弧LBAyox