53 Worksheet 11_16 Solutions

53 Worksheet 11_16 Solutions - Math 53: Multivariable...

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Math 53: Multivariable Calculus Solutions for Worksheet 11/16/11: (It’s not easy being) Green’s Theorem Exercise 0.1. Calculate I C xdx + ydy, where C is any simple closed curve. Solution. Let C be any simple closed curve, and let R be the region it bounds. In the notation of Green’s theorem, we have ∂P ∂y = 0 and ∂Q ∂x = 0. Green’s theorem tells us that I C xdx + ydy = ZZ R (0 - 0) dA = ZZ R 0 dA = 0 , since the integral of the function 0 over any region is 0. (How to do without Green’s theorem: note that the vector field we’re integrating, namely F ( x,y ) = x i + y j , is conservative, as it is the gradient of f ( x,y ) = x 2 2 + y 2 2 . The integral of any conservative vector field around a closed curve is 0.) ± Exercise 0.2. Use Green’s theorem to evaluate R C F · d r , where F = h y 2 cos x,x 2 + 2 y sin x i and C is the triangle from (0 , 0) to (2 , 6) to (2 , 0) to (0 , 0). Solution.
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53 Worksheet 11_16 Solutions - Math 53: Multivariable...

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