problems1212 - A Let G = y 2 z sin xy x 2 y 2 and let f x y...

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Unformatted text preview: A. Let G = ( y 2 z, sin( xy ) , x 2 + y 2 ) and let f ( x, y, z ) be a function satisfying ∇ f = ( e − x 2 , sin( y ) , z 3 ) . Calculate ∂ ∂x ( f ( G ) ) . B. Compute the area of the region where 1 ≤ xy 2 ≤ 2 and 1 ≤ x 2 y 3 ≤ 2. Hint: use a change of coordinates. C. Let C be the semicircle ( x − 2) 2 + ( y − 3) 2 = 4, y ≥ 3 (oriented counterclockwise), and let F = ( 2 x + y 2 , 2 xy + 3 x ) . Compute integraltext C F · d r . Hint: compute part of it directly and the rest using the Fundamental Theorem of Line integrals. D. Find the center of mass of the hemispherical shell z = radicalbig 1 − x 2 − y 2 assuming it has mass density function 1 − z . E. Compute the flux of F = ( y 2 z, x − ye z , e z + z ) through the hemisphere x 2 + y 2 + z 2 = 4, z ≥ 0. Hint: use the divergence theorem to relate this to the flux through a different surface. F. A particle moves along the intersection of the surfaces x 2 + y 2 + z 2 = 2 and z = x 2 − y 2 . Suppose that (...
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This note was uploaded on 05/04/2011 for the course PHYSICS 7A taught by Professor Lanzara during the Spring '08 term at Berkeley.

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