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Unformatted text preview: 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 . 18.02 Lecture 24. – Tue, Nov 6, 2007 Simply connected regions. [slightly different from the actual notations used] Recall Green’s theorem: if C is a closed curve around R counterclockwise then line integrals can be expressed as double integrals: F dr = curl( F ) dA, F n ˆ ds = div( F ) dA, · · C R C R where curl( M ˆ ı + N ˆ j ) = N x − M y , div( P ˆ ı + Q ˆ j ) = P x + Q y . For Green’s theorem to hold, F must be defined on the entire region R enclosed by C . Example: (same as in pset): F = − y 2 ˆ ı + + y x 2 ˆ j , C = unit circle counterclockwise, then curl( F ) = x ∂ ( x ∂ ( − y ) = = 0 . So, if we look at both sides of Green’s theorem: ∂x x 2 + y 2 ) − ∂y x 2 + y 2 ··· F r = 2 π (from pset) , curl dA = 0 ? d F dA = · C R R The problem is that R includes 0, where F is not defined. Definition: a region R in the plane is simply connected if, given any closed curve in R, its interior region is entirely contained in R. Examples shown. So: Green’s theorem applies safely when the domain in which F is defined and differentiable is simply connected: then we automatically know that, if F is defined on C , then it’s also defined in the region bounded by C . In the above example, can’t apply Green to the unit circle, because the domain of definition of F is not simply connected. Still, we can apply Green’s theorem to...
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This note was uploaded on 05/04/2011 for the course MATH 18.02 taught by Professor Auroux during the Spring '08 term at MIT.
 Spring '08
 Auroux
 Multivariable Calculus

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