Exam_exam_4_(2)

# Exam_exam_4_(2) - are all the vectors you can make using 0...

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MATH 241 – Exam 4 – Boyle Friday May 8, 1998 Show your work. Put a box around the result of a computation. No calculators, no book, no notes. 1. (25 points) Let F be the vector field in the plane defined by the rule F ( x, y ) = (1 + ye xy + 3 x 2 y, xe xy + x 3 + 2 y ) and let r ( t ) be the oriented curve from the origin to the point (1 , 1) given by the rule r ( t ) = ( t 2 , t 3 ) , 0 t 1 . Compute R C F · dr . 2. (25 points) Let Σ be the unit sphere, with the outward normal orientation, and let F be the vector field F ( x, y, z ) = ( x 3 , y 3 , z 3 ). Compute the following integral: Z Z Σ F · n dS . 3. (25 points) Let D denote the unit box with the bottom side missing (the eight corners of the box
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Unformatted text preview: are all the vectors you can make using 0 and 1 as coordinate entries). Let D have the orientation given by outward normal vectors, and let C be the boundary of D with the induced orientation. For the vector ±eld F = ( M, N, P ) = (2 x, x 2 y + z, xyz ), compute R C Mdx + Ndy + P dz . 4. (25 points) Let Σ be the paraboloid z = x 2 + y 2 which lies between the planes z = 0 and z = 1. Compute R R Σ z dS ....
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