010.104-2004-2-final

# 010.104-2004-2-final - o 2 eK> 1r 3r 4 Z 9< s2...

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

This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online