1. Dene parabolic coordinates (u, v, ) by x = uv cos , y = uv sin , z = 1 2 u 2 v2 , where u > 0, v > 0, and 0 < 2. (a) Find the scale factors hu , hv , h . (b) Find the unit vectors eu , ev , e . (c) Show that the parabolic coordinate system is orthogonal. (d) Write down the expression for the volume element dV . (e) Let f (u, v, ) = u4 v 3 and F(u, v, ) = cos 2 e . Find expressions for (i) (ii) (iii) (iv) rf ; r F; r F; r2 f in terms of u, v, and . 2. Let (x, y, z) = 2 x sin(y)i + (x2 cos(y) 2ye z )j + y 2 e z k. (a) Show that F is a conservative vector eld. (b) Find a scalar function (x, y, z) such that F = r . (c) Evaluate Z F ds C where C is the curve traced out by the path c(t) = (cos t, cos t, sin2 t), 0 t . 2 (d) Determine the work done by F to move a particle around the parallelogram P with vertices ( 1, 1), (0, 2), (3, 1), (4, 2). 3. Using Gauss' Divergence Theorem, evaluate ZZ F dS @ where F (x, y, z) = 2i + y 2 j + z 2 k and is the solid sphere x2 + y 2 + (z 4. Show that the volume of the ellipsoid x2 y 2 z 2 + 2 + 2 = 1, a > 0, b > 0, c > 0 a2 b c is 4abc . 3 3)2 9