010.104-2004-2-final - o ; 2 eK > 2004 12 11{ 1r 3r...

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Unformatted text preview: o ; 2 eK > 2004 12 11{ 1r 3r 4 Z 9 < : s2: ]_ \ s&` "r r(8& 120&). j h HH x h 1. 2. 3. (10&) Find the circulation of the field F = (x2 + y)i + 4zj + x2 k counter h clockwise around the curve C which is given by x2 + y 2 = 4, z = 2. (15&) Calculate the path integral of f (x, y) = x + 1 y along a path given h 2 in polar coordinates by r = 1 + cos , 0 2. (15&) Calculate h 3(x2 + y + z)dS, S where S is the unit sphere, x2 + y 2 + z 2 = 1. 4. (15&) Let T be the torus obtained by rotating the circle z 2 +(y -a)2 = b2 , h a > b about the z-axis. Then T is the image of the square 0 , 2 in the -plane under the mapping F : R2 R3 defined by x = (a + b cos ) cos , y = (a + b cos ) sin , z = b sin . Calculate the area of T . 5. (20&) Find the following integrals using the Green's theorem. h (a) C (x + y 2 )dx + (y + x2 )dy where C is the boundary of the square [-1, 1] [-1, 1] oriented in the counterclockwise direction. (y + ex )dx + (2x2 + cos y)dy where C is the boundary of the triangle with vertices (0, 0), (1, 1), (2, 0) oriented in the counterclockwise direction. C (b) 6. 7. (15&) Calculate the area of the region enclosed by h x(t) = cos3 t, y(t) = sin3 t (0 t 2). (15&) Let S be a surface consisting of two surface S1 and S2 where h S1 : x2 + y 2 = 1, 0 z 1 S2 : x2 + y 2 + (z - 1)2 = 1, z 1. Let F be a vector field defined by F = (xz + yz 2 + x)i + (xyz 3 + y)j + x2 z 4 k. Compute S curl F dS where the orientation of S is defined so as to satisfy n k > 0. 8. (15&) Let F be a vector field defined by h F = grad 1 , |r - (1, 0, 0)| where r = (x, y, z). Find the flux of the vector field F outward through y2 z2 x2 the ellipsoid 2 + 2 + 2 = 1, a > 1. a b c ...
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