mm3 - , 0). Q4]. ..[5 points] Determine (giving reasons)...

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Calculus IV [2443–004] Midterm III For full credit, give reasons for all your answers. Q1]. ..[15 points] Evaluate the following triple integral by first sketching the region of integration, and then converting it to a spherical coordinates integral. Z 1 - 1 Z 0 - 1 - y 2 Z 2 - x 2 - y 2 x 2 + y 2 z dzdxdy Q2]. ..[20 points] Write down the equation in the statement of Green’s theorem, indicating what the various parts of it stand for. Compute R C F · d r directly, where F = h- x 2 y 2 ,xy i and C is the positively oriented boundary of the region bounded by the y -axis, the line y = 1, and the curve y = x . Use Green’s theorem to compute the path integral above by a second method. Compare your answers. Q3]. ..[20 points] State the fundamental theorem for path integrals. Let F = h ye yz cos( xy ) , ze yz sin( xy ) + xe yz cos( xy ) , ye yz sin( xy ) i . Show that curl ( F ) = 0 . Find a function f so that F = f . Use the fundamental theorem to give a quick computation of the path integral R C F · d r where C is the straight line curve from (0 ,e,π/ 2) to ( π, 1 / 2
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Unformatted text preview: , 0). Q4]. ..[5 points] Determine (giving reasons) whether the following vector eld has positive, negative or zero divergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
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This note was uploaded on 01/12/2010 for the course MATH 241 taught by Professor Teleman,c during the Winter '08 term at University of California, Berkeley.

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