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Unformatted text preview: x 2 + y 2 + z 2 = 2. Let S 1 be the spherical cap forming the upper surface, and let S 2 be the cone forming the lower surface. Orient S 1 and S 2 “upwards”, so the normal vector has a positive ˆ kcomponent. a) (10) Calculate the Fux of v F through S 1 . b) (10) Calculate the Fux of v F through S 2 . c) (10) Verify your answers by using the divergence theorem to ±nd the total Fux of v F out of D . Problem 5. (25) Let v F = x ˆ , and consider be the triangle with vertices P = (1 , , 0), Q = (0 , 1 , 0), and R = (0 , , 1). Denote by C 1 ,C 2 ,C 3 the sides of the triangle, oriented as shown on the picture. a) (10) Compute the work of v F along each of C 1 , C 2 and C 3 . y 1 Q x 1 P z 1 R C 1 C 2 C 3 b) (15) Verify your answers by applying Stokes’ theorem to the triangle PQR....
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 Fall '08
 Auroux
 Vector Calculus, Stokes' theorem, Surface integral

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