MIT OpenCourseWare http ocw mit edu 18 02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use visit http ocw mit edu terms V5 Simply Connected Regions 1 The Extended Green s Theorem In the work on Green s theorem so far it has been assumed that the region R has as its boundary a single simple closed curve But this isn t necessary Suppose the region has a boundary composed of several simple closed curves like the ones pictured We suppose these boundary curves C1 C all lie within the domain where F is continuously differentiable Most importantly all the curves must be directed so that the normal n points away from R Extended Green s Theorem With the curve orientations as shown In other words Green s theorem also applies to regions with several boundary curves provided that we take the line integral over the complete boundary with each part of the boundary oriented so the normal n points outside R Proof We use subdivision the idea is adequately conveyed by an example Consider a region with three boundary curves as shown The three cuts illustrated divide up R into two regions R1 and R2 each bounded by a single simple closed curve and Green s theorem in the usual form can be applied to each piece Letting B1 and B2 be the boundary curves shown we have therefore 2 f B1 F dr u curl F dA h2 F dr IS curl F dA Add these two equations together The right sides add up to the right side of 1 The left sides add up to the left side of 1 for m 2 since over each of the three cuts there are two line integrals taken in opposite directions which therefore cancel each other out 2 Simply connected and multiply connected regions Though Green s theorem is still valid for a region with holes like the ones we just considered the relation curl F 0 F V f is not The reason for this is as follows We are trying to show that curl F 0 for any closed curve lying in R We expect to be able to use Green s theorem But if the region has a hole like the one pictured we cannot apply Green s theorem to the curve C because the interior of C is not entirely contained in R To see what a delicate affair this is consider the earlier Example 2 in Section V2 The field G there satisfies curl G 0 everywhere but the origin The region R is the xy plane with 0 O removed But G is not a gradient field because fc G dr 0 around a circle C surrounding the origin This is clearer if we use Green s theorem in normal form Section V4 If the flow field satisfies div F 0 everywhere except at one point that doesn t I I 1 V VECTOR INTEGRAL CALCULUS 2 guarantee that the flux through every closed curve will be 0 For the spot where div F is undefined might be a source through which fluid is being added to the flow In order to be able to prove under reasonable hypotheses that curl F 0 F V f we define our troubles away by assuming that R is the sort of region where the difficulties described above cannot occur i e we assume that R has no holes such regions are called simply connected Definition A two dimensional region D of the plane consisting of one connected piece is called simply connected if it has this property whenever a simple closed curve C lies entirely in D then its interior also lies entirely in D As examples the xy plane the right half plane where x 2 0 and the unit circle with its interior are all simply connected regions But the xy plane minus the origin is not simplyconnected since any circle surrounding the origin lies in D yet its interior does not As indicated one can think of a simply connected region as one without holes Regions with holes are said to be multiply connected or not simply connected Theorem Let F M i N j be continuously differentiable in a simply connected region D of the xy plane Then in D 3 curl F 0 F Vf for some f x y in terms of components Proof Since a field is a gradient field if its line integral around any closed path is 0 it suffices to show curl F 0 F dr 0 for every closed curve C in D We prove 4 in two steps Assume first that C is a simple closed curve let R be its interior Then since D is simplyconnected R will lie entirely inside D Therefore F will be continuously differentiable in R and we can use Green s theorem Next consider the general case where C is closed but not simple i e it intersects itself Then C can be broken into smaller simple closed curves for which the above argument will be valid A formal argument would be awkward to give but the examples illustrate In both cases the path starts and ends a t P and k d L d k d L d r In both cases C2 is a simple closed path and also Cl C3 is a simple closed path Since D is simply connected the interiors automatically lie in D so that by the first part of the argument V5 SIMPLY CONNECTED REGIONS Adding these up we get The above argument works if C intersects itself a finite number of times If C intersects itself infinitely often we would have to resort to approximations to C we skip this case We pause now to summarize compactly the central result both in the language of vector fields and in the equivalent language of differentials Curl Theorem Let F Mi Nj be a continuously differentiable vector field in a simply connected region D of the xy plane Then the following four statements are equivalent if any one is true for F in D so are the other three 1 F dr is path independent 1 J6 M dx N dy is path independent for any two points P Q in D for any simple closed curve C lying in D 3 F V f 3 M dx for some f in D 4 curl F 0 in D N dy df 4 My N for some f in D in D Remarks We summarize below what still holds true even if one or more of the hypotheses doesn t hold D is not simply connected or the field F is not differentiable everywhere in D 1 Statements 1 2 and 3 are equivalent even if F is only continuous D need not be simply connected 2 Statements 1 2 and 3 each implies 4 if if F is continuously differentiable D need not be simply connected But 4 implies 1 2 3 only if D is simply connected Example 1 Is F xy i x2j Solution We have curl F x a gradient …
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