Multivariable Calculus (Math 214) Assignment

due December 8, 1999

1. Let F be the vector field given by

   F(x, y, z) = [e^{x} ( cos y) z^{2}, - e^{x} ( sin y) z^{2}, 2 e^{x} ( cos y) z] .

   Find the path integral of F over the polygonal path consisting of the line segment from the origin to the point (2, -1, 3) followed by the line segment from that point to the point (0, 0, 1).

2. Find the surface integral of the vector field

   F(x, y, z) = ({x}/{rho^{2}}, {y}/{rho^{2}}, {z}/{rho^{2}}) , rho^{2} = x^{2} + y^{2} + z^{2}

   over the boundary surface of the domain

   4 < x^{2} + y^{2} + z^{2} < 9

   when that surface is oriented by its exterior normal.

3. What is the path integral of the vector field F(x, y) = (-y, x) around the boundary, traversed counterclockwise, of a convex region R in the plane?

4. Find a vector field G in 3-space for which

   curl G = (yz, zx, xy) .

5. If S is any piece of surface in space whose oriented boundary is the circle

   x^{2} + y^{2} = a^{2}, z = 0 ,

   traversed clockwise with respect to the standard orientation of the horizontal plane z = 0, find the integral over S of the vector field

   (yz, zx, xy) .