Multi-Variable Calculus Assignment

due Friday, May 4, 2001

Exercises

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. Find the curl of the vector field

   F(x, y, z)  =  (z^{2} x, x^{2} y, y^{2} z)  .

4. If S is any piece of surface in space whose oriented boundary is the boundary of the first quadrant portion of the disk

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

   traversed counter-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)   .

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