Multi-Variable Calculus Assignment

due Friday, April 27, 2001

Exercises

1. Compute the following multi-variable derivatives:

   1. Find div(yz, zx, xy).

   2. Find curl(yz, zx, xy).

   3. Find div(curl(yz, zx, xy)).

   4. Find grad(arctan(y/x)).

   5. Find div(grad(arctan(y/x))).

   6. Find (for f a general scalar function): div(grad(f)).

   7. Find for n = 2 (general f): curl(grad(f)).

   8. Find for n = 3 (general f): curl(grad(f)).

2. Attempt to apply Green's Theorem to evaluate the path integral over the circle of radius a with center at the origin, when traversed once counter-clockwise, of each of the following vector fields:

   1. F(x, y) = (0, x).

   2. F(x, y) = (x - y, x + y).

   3. F(x, y) = (-y/r, x/r), where r = SQRT{x^{2} + y^{2}}.

   4. F(x, y) = (-y/r^{2} , x/r^{2}).

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