Multi-Variable Calculus Assignment

due Friday, April 27, 2001

Exercises

1. Compute the following multi-variable derivatives:

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

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

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

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

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

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

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

(h) 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:

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

(b) F (x, y) = (x − y, x + y).

(c) F (x, y) = (−y/r, x/r), where r = \sqrt{x^{2} + y^{2}}.

(d) F (x, y) = (−y/r^{2}, x/r^{2}).