Lecture 22 Review curlF N M Example F y x curl y x 1 1 2 Imagine a closed curve c oriented counter clockwise with an inside region of R Green s theorem states Consequences of Green s theorem 1 curlF 0 F r 0 for any c F is a gradient of the vector eld i e F f for some f 2 Proves path independence Example F 0 x F r x y Compute the curl curlF N M X 0 1 Compute the double integral 1 A Area R Compute the line integral F r 0 1 cost sint cost t cost cos t t sint cos t t cos t t How to gure out cos t t we have to remember that cos2t cos t sin t 2cos t 1 cos t 1 cos2t So why does it work F P Q P x P A and Q y Q A P x P x P x P x P x P x P x c t y x t x and c t y x t x P x P t y t and P x P x P t y t Therefore P x P t y t P t y t But then we change all of the t s to x s and combine the integrals P x P x y P x y x P x y y x P A Green s theorem Union of boxes P A P A Area of the union of boxes
View Full Document
Unlocking...