## Exercise 9a

1.
Stewart, Section 14.8, Exercises 1, 2, 10, 19.

2.
Show by direct calculation that the conclusion of Stokes Theorem does not hold for .

3.
Let ,
(a)
Find the gradient of . Find the curl of .
(b)
Show that , everywhere except on the non-negative part of the -axis, where

and are strictly positive constants.
(c)
Let be the curve oriented in the anti-clockwise direction when viewed from above. Use Stokes theorem to show that if is any oriented surface with oriented boundary , and does not intersect the non-negative part of the , then

4.
(Optional) Suppose is a vector field with , everywhere in -space. Define in terms of by

where are arbitrary constants. Show by direct calculation that everywhere in -space.

Math 13 Winter 1999
1999-03-03