1、Section 11.21The Relation between Line Integrals of Second Type and Double IntegralsSimply connected domainMultiply connected domainPositive direction of(C)23The Relation between Line Integral of Second Type and Double IntegralGreen,George(1793-1841),English mathematician and physicist Greens Theore
2、mTheorem Suppose there is a closed bounded domain(1)(,),(,)().P x y Q x yC bounded by a piecewise smooth simple curve(C),and functions 2R Then the following relation holds:,CQPdP x y dxQ x y dyxy indicates that the integration is C where in the positive direction of(C).4The Proof of Greens TheoremPr
3、oofi)Suppose that the domain()can be expressed as12()(),.xyxxycydand12()(),yxyyxaxbThen 21()()()byxayxPPddxdyyy 21,(),().baP x yxP x yxdx On the other hand,()()(,)CPP x y dxdy (,)(,),ACBBDAP x y dxP x y dx 21,(),().baP x yxP x yxdx 12,(),()baabP x yxdxP x yxdx()(,)CP x y dx 5The Proof of Greens Theo
4、remProof(continued)Similarly,we have21()()()dxycxyQQddydxxx 21(),(),dcQ xyyQ xyydy and(,)(,),DACCBDQ x y dyQ x y dy 21(),(),.dcQ xyyQ xyydy 12(),(),cddcQ xyy dyQ xyy dy()(,)CQ x y dy ,.CQPdP x y dxQ x y dyxy Therefore,6The Proof of Greens TheoremProof(continued)ii)Suppose that the domain(s)can be di
5、vided into some subdomainsdomain is type of domain in condition i).Since the sum of line integral of second typeover the common edges of these subdomainsis zero,we can obtain QPdxy ,.CP x y dxQ x y dy ()(1,2,3)ii by a line parallels to y axis or parallels to x axis,each7The Relation between Line Int
6、egral of Second Type and Double IntegralCorollary Greens formula still holds if ()is a multiply connected domain bounded by finite number of piecewise smooth simple closedcurves.12.CCQPdPdxQdyPdxQdyxy 8Use Greens Theorem to Find the Line Integral of the Second TypeSolution then,by Greens Theorem,and
7、(),Qxy Since()Pxy where(C)is22221.xyab()()(),CIxy dxxy dy Example FindWe have()()2QPIdxdydxdyxy 2.ab 22()CIxy dxx ydy ()QPdxy ()4xy dxdy 22004cossinadr rdr cos,sin0,02x ry rr a 0.Example Find where(C)is222.xya22(),CIxy dxx ydy then,by Greens Theorem,we haveSolution and2,Qx y Since 2Pxy Use Greens Th
8、eorem to Find the Line Integral of the Second Type9Example Evaluate where(C)is(1)the circle with positive direction.(2)the above half circle form A(R,0)to B(-R,0).22(),CIxy dyx ydx 222xyR22yRx10Find Area of a Region by Line Integrals of Second typeNote If(,)Q x yx,we can calculate the area of(,)P x
9、yy anda region(s),which is bounded by(C),by()()1.2Cdxdyxdyydx Example Find the area of the starlike shape33cos,sin.xat ybtSolution Area()12Cxdyydx 2220133cossin.28abttdtab 24242013cossin3sincos2abttdtabttdt 11Path IndependenceIf A and B are two points in an open region D in space,the workFrd done in
10、 moving a particle from A to B by a field F defined on D usually depends on the path taken.,where(C)is show in the figure.12223()04sin(sin).3Cy dxaada For example,find the integral then2(),CIf the path from A to B is22()00.aCay dxdx If we take path then1(),C2()Cy dx Path IndependenceFor some kinds o
11、f line integrals of the second type,the value of the integral depends only on the initial point A and the terminal point B and is independence of the path of integration.Question When will the line integral of the second type independent on the path taken?,where(C)isFor example,find the integral 322
12、()23Cyx dyx y dx show in the figure.If the path from O to B isthen1(),COB1():Cyx 22():Cyx xyA3()COAOB 13226()02371Cyx dyx y dxx dxIf the path from O to B isthen2(),C13224()02351Cyx dyx y dxx dxIf the path from O to B isthen3(),C1322()023021Cyx dyx y dxydyPath IndependenceProof We will prove these st
13、atements in the order 1231.Theorem 12.2.6u is called a potential function of F=(P,Q)on D.14Path IndependenceProof(continued)12Suppose that A and B are two points in(),We travel from A to B by taking two arbitrary curvesIf these(),inside.AQBdenote by APBandtwo curves do not intersect,then by proposit
14、ion 1 .APBBQAAQBPdxQdyPdxQdyPdxQdy Therefore 0.APBBQAAPBQAPdxQdyPdxQdyPdxQdy If these two curves have a intersection other than A and B,we also havethe conclusion.15Path IndependenceProof(continued)and 00(,)()A xy We take any point 00(,)(,)(,).x yxyu x yPdxQdy form the line integral with variable up
15、per limit we denote this integral by 232,under proposition 00(,)(,),x yxyPdxQdy ,.uuPQxyWe will prove that duPdxQdyorIn fact,by the definition of the partial derivative of u,we have0(,)(,)lim,xuu xx yu x yxx while 0000(,)(,)(,)(,)(,)(,).xx yx yxyxyu xx yu x yPdxQdyPdxQdy 16Path Independencewe have2,
16、By proposition (,)(,)xx yxxx yxPdxQdyPdxThen (,)(,)(,)(,).xx yx yu xx yu x yPdxQdy (,),01.P xx yx ,we have(,)P x yThus,by the continuity of 00(,)(,)limlim(,)(,).xxuu xx yu x yP xx yP x yxx Then the proposition has been proved.(,).uQ x yy Similarly,we haveProof(continued)17Path IndependenceProof(cont
17、inued)31Suppose that(C)is an arbitrary piecewise smooth simple closed curve,whose equations are(),(),xx tyy t()(),()().xxyy()tThen and ()(),()()(),()()CPdxQdyP x ty tx tQ x ty ty tdt (),()du x ty tdtdt (),()0.u x ty t u dxu dydtx dty dt Finish.Path Independence18Theorem Let ()be a simply connected d
18、omain in the plane,11110000(,)(,)(,)1100(,)(,)(,)(,)x yx yx yx yPdx Qdyu x yu x yu x y If the line integral1100(,)(,)x yx yIPdx Qdy ,P QC is independent of the path,thenTheorem Let()be a simply connected domain in the plane,P Q,(,)().PQx yyx then The three propositions in last theorem are true ,PQCy
19、x 19Path IndependenceDefinition Path Independence and Conservative FieldLet F be a field defined on an open region D in space and suppose thatFrd A to B is the same over all paths from A to B.Then the integraldone in moving from for any two points A and B in D,the workFrBAd is path independent in D
20、and the field is conservative on D.Under conditions normally met in practice,a field F is conservative if and only if it is the gradient field of a scalar function f;that is if and only The function f is then called a potential functionFf if for some f.for F and we have the following equivalent stat
21、ements:20Path IndependenceExample Find the integral(1,1)(0,0)()().xy dxdy Solution then,we have(,),Q x yyxSince(,)P x yxyand1.PQyx Therefore,the integral does not dependent on the path taken,then wetake path(0,0)(0,1)(1,1),1100(0,)(,1)IQy dyP xdy1100(1)ydyxdx1110.22Finish.21Path IndependenceMethods
22、for finding potential functionsLet()be a simply connected domain in the plane,A(M)=(P,Q)be a vector field,.PQCyx If there exists a function of two variables,u(x,y)such that(,)uQ x yy (,),uP x yx or(,)(,)duP x y dxQ x y dythen function u(x,y)is called the potential function of A(M),and the vector fie
23、ld A(M)is called a potential field.a primitive function of the total differential Pdx+Qdy How to find the potential functions?22Path IndependenceExample Verify that the vector field 222(36,33)Axxyyxis a potential field and find its potential function.Solution Method 1(by line integrals)Method 2(by p
24、artial integrals)Method 3(combining terms into total differentials)23Path Independencesuch that2(2).()xy dxydyduxy Example Does there have a function (,),u x yIf so,find it.Solution 3322,.()()PyQyyxyxxy thenand2(,),()yQ x yxy Let 22(,)()xyP x yxy are both continuous,PQyxIn the domain of 0 xy0,xyorTh
25、en there must exist a functionand equal.such that(,),u x y2(2)?()xy dxydyduxy 24Path IndependenceSolution(continued)(,)2(1,0)(2)(,)()x yxy dxydyu x yxy 10lnlnyexxxyxy 2101()xyydxdyxxy(,)ln.yu x yxyCxy ln1xxyxy thenSo,lnyxyxy Finish.25Path IndependenceExample Find the integral(1,0)122(0,1),xdxydyIxy
26、and222()CxdxydyIxy where(C)is221.xy26Path Independence(0)0,1.2 Solution then2(,),P x yxy(,)(),Q x yyx Let 2()2,Pxyxyyy()(),Qyxyxxx Since the integral is independent on the path,we have.PQyx we have2().xxC That is()2yxxy and then Since 0.C Therefore,is independent 2()Lxy dxyx dy Example Suppose that
27、the integralon the path taken,where has continuous derivative,and Find(0)0,(1,1)2(0,0)().xy dxyx dy (1,1)2(0,0)()xy dxyx dy (1,1)22(0,0)xy dxx ydy Finish.ReviewGreens formulaUsing Greens formula to find the line integral of the second typeConditions for path independence of a line integralPotential function and potential field(methods for finding potential functions)27
