010-104-2003-2-final - > ; o 2 eK 2003 12 13{ 1r 3r...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

Ask a homework question - tutors are online