010-104-2003-2-final

# 010-104-2003-2-final - > o 2 eK 1r 3r 4 Z 9< s2 j h...

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Unformatted text preview: > ; o 2 eK 2003 12 13{ 1r 3r 4 Z 9 < : s2: j h ]_ \ s&` "r r(8& 120&). HH x h 1. (15&) Compute h c f ds, where f (x, y) = x2 + 3y 2 - xy and c(t) is the circular arc of radius 3 centered at (0, 0) in the xy-plane from (0, 3) to (-3, 0) through the second quadrant. 2. 7 (15&) Let c(t) = (cos3 t, sin3 t, t), 0 t h . 2 Let F be a vector field defined by F(x, y, z) = (sin z, cos z, - (xy) 3 ). Evaluate the integral c 1 F ds. 3. 4. (15&) Evaluate h c 4 2xyzdx + x2 zdy + x2 ydz, t 3 where c(t) = (cos t, 1 + sin3 2 , 1 + t), 0 t . (15&) Let T be a surface defined implicitly by F (x, y, z) = 0 for (x, y) in h a domain D of R2 . Show that F dS = z F D dA T 5. 6. (15&) Find the flux of the vector field F = (x, y, xz) outward (away from h the z-axis) through the surface S = {(x, y, z) | z = x2 + y 2 and z 4}. (15&) Find the area of the region bounded by the cycloid h x = a( - sin ), y = a(1 - cos ), a > 0, 0 2 and the x-axis using Green's Theorem. 7. (15&) Let S be the capped cylindrical surface which is the union of two h surface S1 and S2 , where S1 is the set of (x, y, z) with x2 + y 2 = 1, 0 z 1, and S2 is the set of (x, y, z) with x2 + y 2 + (z - 1)2 = 1, z 1. Let F(x, y, z) = (zx + z 2 y + x, z 3 yx + y, z 4 x2 ). Compute S ( F) dS. 8. (15&) Let F be a vector field defined by F = h r , where r = (x, y, z). |r|3 Let S be the cube centered at the origin with the edge of length 1. Prove that the flux of the vector field F outward through the cube S is 4, that is, F dS = 4. S ...
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