HW11 - Math 1920 - Spring 09 - HW 11 16.4 - Problem 8 We...

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16.4 - Problem 8 We have ∂M ∂x = 1, ∂M ∂y = 1, ∂N ∂x = - 2 x and ∂N ∂y = - 2 y . So by Green’s theorem, the flux is ( R is the region inside the curve) Flux = Z C M d y - N d x = Z Z R ∂M ∂x + ∂N ∂y d z d y = Z 1 0 Z x 0 (1 - 2 y )d y d x = 1 6 and the circulation is Circulation = Z C M d x + N d y = Z Z R ∂N ∂x - ∂M ∂y d x d y = Z 1 0 Z x 0 ( - 2 x - 1)d y d x = - 7 6 16.4 - Problem 16 This time we have ∂M ∂y = - 2, ∂N ∂x = 2 So by Green’s theorem, the work/circulation is ( R is the region inside the curve) Work = Z C M d x + N d y = Z Z R ∂N ∂x - ∂M ∂y d x d y = Z Z R 4d y d x = 4 × Area of disc = 16 π 16.4 - Problem 18 Here ∂M ∂y = 3, ∂N ∂x = 2 and so by Green’s theorem, the work/circulation is ( R is the region inside the curve) Work = Z C 3 y d x +2 x d y = Z Z R ∂N ∂x - ∂M ∂y d x d y = Z Z R (2 - 3)d y d x = - Z π 0 Z sin x 0 d y d x = - 2 16.4 - Problem 24 The curve is given by the parametrization r ( t
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This note was uploaded on 12/25/2009 for the course MATH 1920 taught by Professor Pantano during the Spring '06 term at Cornell University (Engineering School).

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HW11 - Math 1920 - Spring 09 - HW 11 16.4 - Problem 8 We...

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