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17-1 - Math 200 Spring 2010 Handout#28 Section 17-1 Greens...

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Math 200, Spring 2010 Handout #28 Section 17-1 Green’s Theorem Green’s Theorem A simple curv e is one that does not intersect itself, except possibly at its endpoints. A positively oriented closed curve is one whose enclosed region lies to the left of the curve as it is traversed. Green’s Theorem: Let C be a simple closed curve with a positive orientation. If P and Q have continuous partial derivatives on the region D enclosed by C , then C D Q P Pdx Qdy dA x y + = ∫∫ Note: Other notation used for the line integral on a positively oriented closed curve: C Pdx Qdy + integralloop The symbol C integralloop is used to denote a positive orientation of the closed curve C . D Pdx Qdy + The symbol D is used to represent the positively oriented boundary curve of region D . If , P Q = F , then ( ) ( ) ( ) ( ) ( ) ( ) , , , , b C a C dx dy d P x t y t Q x t y t dt Pdx Qdy dt dt = = + F s i i integralloop integralloop Note: Green’s Theorem can be thought of as a Fundamental Theorem of Calculus for double integrals. Comparing D D Q P dA Pdx Qdy x y = + ∫∫ with ( ) ( ) ( ) b a F x dx F b F a = , in both cases, the left side involves an integral of a derivative expression and, in both cases, the right side involves the values on the original functions ( P , Q , and F ) only on the boundary of the domain.

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