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Unformatted text preview: (25) Let v F = a x,y,z A , and consider the portion D of the cylinder x 2 + y 2 < 1 which lies above the xy-plane and inside the sphere x 2 + y 2 + z 2 = 2. a) (15) Calculate the Fux of v F out of D by directly evaluating surface integrals. (do not use the divergence theorem) b) (10) Use the result of part (a) to ±nd the volume of D . (Justify your answer carefully). Problem 5. (25) Consider the surface S given by the graph z = x 2 − y 2 over the unit disk x 2 + y 2 < 1. Let C be the boundary of S oriented counterclockwise when viewed from above. a) (10) ²ind the work done by v F = xy ˆ k around C by directly evaluating a line integral. b) (15) Verify your answer by applying Stokes’ theorem....
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This note was uploaded on 04/28/2009 for the course MATH 18.02 taught by Professor Auroux during the Fall '08 term at MIT.
- Fall '08