微积分全英微积分全英 (66).pdf

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1、14.3 Independence of PathMain contents 1.The definition of independence of path 2.Independence of path theorem 3.Calculation of curve integral and potential function 4.Summary1.The definition of independence of pathGG is simply connected,means that G has no“holes”.A set G is connected if any two poi

2、nts in G can be joined by a piecewise smooth curve lying entirely in G.xyOA connected setDef.The definition of independence of pathG+1ddLyQxP+=2ddLyQxPBIf in region G,we have AL1L2Def.G is simply connected.Then the curve integral+LyQxPddisindependent of path in G,and dependent otherwise.xyO2.Indepen

3、dence of path theoremTheorem:PQyx=Suppose that P(x,y)and Q(x,y)have first order continuous partial derivative in an open simply connected region G.Then the necessary and sufficient condition for curve integral to be independent of path(or the curve integral along an arbitrary closed curve in G is ze

4、ro)is that the equation holds in G.ddLP xQ y+Notes：(1)Open set is simply connected;(2)(,)and(,)have the first order continuouspartial derivative in.PQyx=Criteria for independence of pathddLP xQ y+(3)Exist(,)in such that dddu x yDuP xQ y=+=+Four equivalent propositions for independence of path(1)dd0,

5、closed curve CP xQ yCD+=+=(4)In,.PQDyx=(2)is independent of the path in LPdxQdyD+Conditions:If ,(,)have the first order continuous partialderivative in a simply connected domain D,the followingfour propositions holds:Prove:(1)(2)(3)(4)(1)vector,is conservative.3.Calculation of curve integral and pot

6、ential functionIf PQyx 1100(,)(,)(,)d(,)dB xyA xyP x yxQ x yy+),(01yxC),(11yxB 100(,)dyyQ xyy=D(x0,y1)101(,)dxxP x yx+or1100(,)(,)(,)d(,)dB xyA xyP x yxQ x yy+thenOxy),(00yxA 100(,)dxxP x yx=101(,)dyyQ xyy+ddLP xQ y+Find(2+2)+2+4,where is the curve of =sin2from(0,0)to 1,1.Example 1xyO 2315=224(2)d()

7、dxxyxxyy+120dxx=140(1)dyy+24(2)xQxyxx=+=2(22)xPxxyyy=+=PQyx=(0,0)(1,1)1,1(B )0,1(I=Example 22(,2)Pxyyyxy=)()(yQy xxx x=2(,),P x yxy=(,)()Q x yy x=xQyP =Integral is independent of path()2y xxy=If integral 2+is independent of path where()hasacontinuousderivativeand 0=0,find(0,0)(1,1)2+.xyO100dx=12=(1,

8、0)10dy y+()2yxxy =2()xxC=+0C=2()xx=)1,1(1)(1,1)2(0,0)d()dxyxy xy+(0)0,=(1,1)(0202,)ddyyxxxy=+Example 2Example 2xyO(2)1,1(1,1)2(0,0)d()dxyxy xy+)1,0(100dyy=1201 dxx+0=1202x+12=(1,1)22(0,0)ddxyxyxy=+Calculation of potential functionxQyP =Potential functions Full differential 00(,)(,)(,)(,)d(,)dB xyA x

9、yu x yP x yxQ x yy=+According to four equivalent propositionsd=ddu P xQ y+LyQxPddExample 3xyO)0,(x(x,y)Isfull differential?()d(2)dyyexxxeyy+=holds in xoy plane,222yxexy+=yyxexxeyxuyyxyd)2(d)(),(),()0,0(+=yyxeyyd)2(0 +xxexd)(00+=(1)Is+(2)full differential?If yes,find its potential function.so this ex

10、pression is a full differential.and the plane is simply connected.A potential function isExample 3Since function u satisfiesPxexuy=+=yy2)(=thenSo,22(,)2yxu x ye xyC=+()dyuexx=+=22yxe x+()y+2yxeya function of y(2)uy=()ye x y+=2()2 d yy yyC=+2yuxeyQy=22d2yxxey=+22(,)2yxu x yxey=+()d(2)dyyexxxeyy+(dd)y

11、yexxey=+d()yxe=(d2 d)x xy y+22d2xy+),(yxu(3)Example 3Find the general solution of differential equation=2+3+1+.Example 421,1dyyxdxx+=+A.Constant variation method:B.Formula method:.1Cyx=+().1C xyx=+34().34xxC xC=+11211,dxdxxxyex edxC+=+First-order linear equationExample 423()(1)0,xxy dxx dy+=1,PQyx=A

12、.Curve integral method:2300(,)()(1),xyu x yxx dxx dy=+B.Use the law of differential:23()0,dyxdyydxx dxx dx+=34()034xxdyd xydd+=，34()0.34xxd yxy+=It is a full differential equation.Example 4C.Indefinite integral method:23,uxxyx=+23()xxy dx+34(),34xxxyC y=+(),uxC yy=+and1,uxy=+()1,xC yx+=+()1,C y=(),C

13、 yy=So the general solution is 34.34xxyxyC+=4.SummaryG1ddLP xQ y+2ddLP xQ y=+B1.If in region G,we have AL1L2xyO(1)dd0,closed curve CP xQ yCD+=+=(3)Exist(,)in such that dddu x yDuP xQ y=+=+(4)In,.PQDyx=(2)is independent of the path in LPdxQdyD+4.Summary2.If a line integral is independent of path,then we often write it as ddCP xQ y+(,)(,)ddc da bP xQ y+indicating only the initial point and the terminal point for the path C.(,)a b(,)c dWe know there is a function satisfying .Then we have(,)f x yddfP xQ y=+(,)(,)(,)(,)dd(,)(,)(,).c dc da ba bP xQ yf x yf c df a b+=Independence of Path