MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable CalculusFall 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, pro-vided 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 exam-ple. 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 F dr = u,curl F dA h2F . dr = IS,,curl FdA B1 ' II--1 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 = Vf 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 ap-ply 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't2 V. VECTOR INTEGRAL CALCULUS 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 = Vf, 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 Clies 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 simply- connected, 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 = Mi + Nj 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 Cin D. We prove (4) in two steps. Assume first that Cis a simple closed curve; let R be its interior. Then since D is simply- connected, 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 Cis closed but not simple--i.e., it intersects itself. Then Ccan 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 at P, and k~.d~=L~~.d~+k~.d~+L~~.dr. In both cases, C2is a simple closed path, and also Cl +C3is 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 = Vf for some f in D 3.' M dx + N dy = df for some f in D 4. curl F = 0 in D 4.' My = N, 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 a gradient field? Solution. We have curl F = x # 0, so the theorem says it is
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