(960, #1) Let F(x, y, z) = < 3 x, x y, 2 x z> and E be the
cube bounded by the planes x= 0, x = 1, y = 0, y = 1, z = 0, z
= 1. Verify that the divergence theorem is true for this case
by calculating the two integrals that appear there.
Solution: Computing the integral
appears, at first, to be tedious because it has to be calculated on the 6 different faces of the cube. However, since the faces are parallel to the coordinate planes each of the integrals reduces to the form
where n is the appropriate normal. tabulating the results
we have:
| z = 1, x = 0..1, y = 0..1 |  | |
| z = 0, x = 0..1, y = 0..1 |  | |
| y = 1, x = 0..1, z = 0 ..1 |  | |
| y = 0, x = 0..1, z = 0..1 |  | |
| x = 1, y = 0..1, z = 0..1 |  | |
| x = 0, y = 0..1, z = 0..1 | 
|
Consequently,
This agrees with the right side of the divergence theorem equation:
