integral - Homework 8(Due Thursday June 20 1 Let S be the...

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Homework 8 (Due Thursday June 20) 1. Let S be the surface z = x (1 x ) y (1 y ) for 0 x 1 and 0 y 1 . A picture is given below. Evaluate the integral ˜ S x k · dA , where dA is the upward pointing normal vector. [Hint: Express the vector field x k as curl F for some vector field F (you might have to play around with a few possibilities), and then use Stoke’s Theorem.] 2. Use Stokes theorem to evaluate ´ F · dr where F ( x, y, z ) = e x i + e x j + e z k and C is the boundary of the part of the plane 2 x + y + 2 z = 2 in the first octant: 1
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3. Use Stoke’s Theorem to evaluate ´ C F · dr where F ( x, y, z ) = x 2 y i + 1 3 x 3 j + xy k and
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