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simpl_conctd_reg - MIT OpenCourseWare http/ocw.mit.edu...

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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 200 7 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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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 h2 F . dr = IS,, curl F dA B1 ' I I--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 = 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 ap-
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