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problems1212

# problems1212 - A Let G = y 2 z sin(xy x2 y 2 and let f(x y...

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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 ( x (0) ,y (0) ,z (0) ) = ( 1 , 1 , 0 ) and x (0) = 1. Find y (0) and z (0).
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