高等数学微积分课件8.3.2曲线积分与路径无关

Asseenbefore,thelineintegralusuallydependsonpaths.Example1()Lxydxxydy2()Lxydxxydy011L2L13512Whenpathindependence?Conservativefield:theworkdoneonlydependsontheinitialandterminalpointsrFLWdABFIfindependence,LPdxQdyD1LACBD2LAEBDABCE1L2L1LPdxQdy2LPdxQdy▹Pdx+QdyL1ò-Pdx+Qdy=0L2òThismeans：ForeachclosedcurveLinD，0LPdxQdyLDWehave▹Pdx+QdyL1ò-Pdx+Qdy=0L2ò▹Pdx+QdyL1ò+Pdx+Qdy-L2ò=0Wenowknowthatthelineintegralispathindependence,inDthenForanyclosedcurveCinD.LPdxQdy0CPdxQdyWefurtherprovethatinthiscase,thereisafunctioncalledpotentialfunctionz=f(x,y)suchthatdz=Pdx+Qdy,inthiscase,ffPQxyLPdxQdyLffdxdyxy¶f¶x,¶f¶yidx,dyLò,dzdfxyPdxQdyIfthenffdxdyxyTheoremTheFundamentalTheoremforLineintegrals(())|rrbaLfdtf:()(:rr)Lttab(())r(())rffbaSimilartoNewton-Leibniztheorem()()|()()bbaafffxdxxbafABPf.suppose(,,)fxyzisafunctionofthreevariablesrLfd()()r(r)batftdt{,,}{,,}badxdydzdtxyzdtdftffdt()badxdydzdtxdtydfftzdtf[(r())]badtdtdft()|r()batf(())r(())rffbaChainRuleN-Ltheorem(())(())rrrLdbfaffMeans:Inthiscase,thelineintegralonlydependsontheinitialandterminalpointsofthelineL.(())(())rrrBAdbfaffWecouldnotethisbyABSummary:thelineintegralispathindependenceinD,thenitisequivalenttothat0foranyclosedcurveCinD.CPdxQdyPdxQdyORthereisafunctionz=f(x,y)suchthatdz=Pdx+QdyWhetherornotitiseasytodetermineagivenlineintegralispathindependenceinD?Thereisaneasyway!!Let’sbacktotheGreen’stheorem,if0foranyclosedcurveCinD.CPdxQdyThen'forclosedcurveCin0,anyD.DCCQPdPdxQxdyxydyThismeansi,nDQPxyUnbelievablecriteria:inDQPxythereisafunctionz=f(x,y)suchthatdzPdxQdyforanyclosedcurveCinD,0CPdxQdypathindependenceinD(,)()()()()BLABAPdxQdydzzBzAfBfA1()LxydxxydyExample:isnotpathindependencesince()()1PxyQxyxxxyyExample222sin(cos3)Lxydxxyydy2:(:01)Lyxx01LSolutionHardifalongL!!2sinPxy22cos3Qxyy2cosPxyyQxPathindependence,soChooseanotherpath!01L222sin(cos3)Lxydxxyydy0y0dy102sin0xdx1220(1cos3)yydy1x0dxsin11Example:Findingpotentialfunctions32(2)(34)xydxxydy32Pxy234QxySolution23PyyQx32(2)(34)xydxxydyisthetotaldifferentialofsomefunctionsf(x,y)32(2)(34)(,)xydxxydydfxy32(2)(34)(,)xydxxydydfxy(,)(0,0)fxyf(,)(0,0)xy32(2)(34)xydxxydy(,)xy0y02xxdxxx0dy0dx20(34)yxydy2x34xyySoanyfunctionoftheform23(,)4fxyxxyyCisapotentialfunction.Wecanchooseanypointastheinitialpoint,Anotherwaytofindpotentialfunctions23((34))2(,)xydxdyxyxdfy(,)fydxdydfxxyf32fxyx32fxydx23xxyyThekeyistofindsincey234fxyy32fxydx23xxyy234fxyyso4'y4yyC23andsof,4xyxxyyCEND例3.7()()Lxydxxydy解1PyQx曲线积分与路径无关可沿折线积分1y2x0dy0dx(2,3)(1,0)()()xydxxydy20(1)xdx31(2)ydy4(2,3)1设(,)fxy起点11(,)Axy终点22(,)BxyAB则2211(,)(,)rrxyLxyddff设(,,)fxyz起点111(,,)Axyz终点222(,,)Bxyz则2211(,)(,)xyxffy222111(,,)(,,)rrxyzLxyzddff222111(,,)(,,)xyzxyffz推论曲线积分与路径无关的充分条件势场（梯度场、保守场）的曲线积分与路径无关。保守力场F沿任一闭曲线所作的功为零。Fr0LdW物理解释：LD例3.5引力场的功引力场：3(,,)FrGmMxyzrp.140是势场：F=f势函数：r(r)fGmM222(,,)GmMxyzzfxy功：W(2,2,1)(1,2,1)rrLddff(2,2,1)(1,2,1)ff11()36GmM(1,2,1)(2,2,1)我们已经知道：势场是曲线积分与路径无关的充分条件那么，什么是曲线积分与路径无关的必要条件？定理3.4势场也是曲线积分与路径无关的必要条件定理3.4曲线积分与路径无关的必要条件读书FrLd在D内与路径无关F=fF={,}xyfffF是势场（梯度场，保守场）证取定一点0(,)MabD0MDab(,)MxDy(,)Mxy由于曲线积分与路径无关作“上限函数”(,)fxy,)(,)((,)(,)xyabPxydxQxydy(,)fxy,)(,)((,)(,)xyabPxydxQxydy以下证明：F=f即：xfP,)(,)((,)(,)xyabPxydxQxydyyfQ(,))(,[]axybPdxQdyPx(,))(,[]axybPdxQdyQy回忆：定积分中的上限函数：()()xadfxdxfxdx我们现在要证明的是类似的性质：即：()()xafxxdx是f(x)的原函数(,)(,)[],abxyPdxQdyPx(,))(,[]axybPdxQdyQy0MDab(,)Mxy在M左侧取一点：(,)Ncy(,)Ncy(,)fxy((,),)axbyPdxQdy((,),)acbyPdxQdy((,),)cxyyPdxQdy:NMyy(:)xcx(,)fxy((,),)acbyPdxQdy((,),)cxyyPdxQdy这一项与x无关沿直线NM的积分0MDab(,)Mxy(,)Ncy(,)fxy((,),)acbyPdxQdy(,)0cxPxydxQ(,)fxxy0x(,)cxPxydx(,)Pxy0dy同理可证：(,))(,[]axybPdxQdyQy推论3.1设F={P,Q}在连通开区域D上连续，则FrLd在D内与路径无关的充要条件是：F是势场：存在函数f(x,y)，使Ff下面我们寻求判定一个矢量场是势场的方便的条件什么判定一个矢量场是势场的方便的条件？如果一个矢量场F={P,Q}是势场，P、Q应该满足什么条件?设F(,)fxy{(,),(,)}xyfxyfxy则xPfyQfxyPfyyxQfxxyPfyyxQfx假设fxy和fyx连续则PQyxyxfQx所以在D上这是判定F={P,Q}是势场的一个方便的必要条件。xyPfy问题是：这个等式是不是判定F={P,Q}是势场的一个充分条件？PQyx如果是充分条件，则意味着沿D内的任何闭曲线L的曲线积分都为零LD答案是否定的！反例？即：PQyx0LPdxQdy例3.3例3.3我们注意到：22yPxy22xQxy的定义域不是单连通的：D问题是不是出在这里？(0,0)DDEFEDFDD是单连通区域DEEDD非单连通单连通区域：复单连通区域：wang如果D是单连通的区域，是否有PQyx0LPdxQdy答案是肯定的！定理3.5读书若D是单连通区域PQyxFrLd与路径无关或0LPdxQdy证D对D内任意闭曲线LL1D因为D是单连通的，L所围的区域1DD在D1上，格林公式的条件满足所以，由格林公式LPdxQdy()DQPdxdyxy0Ddxdy0当曲线积分与路径无关时，可沿平行于坐标轴的折线积分00(,)(,)xxyyPdxQdy0M(,)Mxy0x0yxy0(,)Qxy000:(:)MQyyxxx0yy0dyxx0dx0:(:)QMxxyyy0xx0dxyy0dy0M(,)Mxy0x0yxy0yy0dyxx0dx00(,)(,)xxyyPdxQdy00(,)xxPxydx0(,)yyQxydy00(,)(,)xxyyPdxQdy0(,)xxPxydx00(,)yyQxydy0M(,)Mxy0x0yxy0xx0dxyy0dy当曲线积分LPdxQdy与路径无关时L0M(,)Mxy我们可以不沿指定的L积分一般选择沿折线积分若F={P,Q}是势场：Ff则{,}ffxy,fP