What are the two sides that contribute to the divergence theorem?

[r19ecua]

Homework Statement

Suppose the one-dimensional field A = Kx * a_x exists in a region. Illustrate the validity of the Gaussian theorem by evaluating its volume and surface integrals inside and on the rectangular parallelepiped bounded by the surfaces: x=1,x=4,y=2,y=-2,z=0 and z=3, for a given A.

Homework Equations

(Right) $\int_0^3$$\int_1^4x$dxdz a_y + (left) $\int_0^3$$\int_1^4x$dxdz -(a_y) + (top) $\int_{-2}^2$$\int_1^4x$dxdy a_z + (bottom) $\int_{-2}^2$$\int_1^4x$dxdy -(a_z) + (front) $\int_0^3$$\int_{-2}^2$dydz (a_x) + (back) $\int_0^3$$\int_{-2}^2$dydz -(a_x)


Direction on the left is applied to the integral on its right.

The Attempt at a Solution

For the Right side
$\int_0^3$$\int_1^4x$dxdz a_y
My answer to this integral is 45/2

For the left side
$\int_0^3$$\int_1^4x$dxdz -(a_y)
My answer to this integral is -45/2

For the top side
$\int_{-2}^2$$\int_1^4x$dxdy a_z
My answer is 30

For the bottom
$\int_{-2}^2$$\int_1^4x$dxdy -(a_z)
-30

For the front
$\int_0^3$$\int_{-2}^2$dydz (a_x)
36

For the back
$\int_0^3$$\int_{-2}^2$dydz -(a_x)
-36

When I add these up, I get zero... however, when I use the divergence theorem I get 36.

This answer is suppose to equal the answer I get via the divergence theorem formula. I'm confused :(

 

[rude man]

There are only 2 sides that "see" the field A head-on. Mathematically, there are only two sides for which ∫A*ds ≠ 0 where ds is a vector element of area on any side. Which sides are those?

Now integrate A over those two sides, remembering that the dot-product A*ds will be positive for one side and negative for the other. In other words, the normal to any closed surface always points out of the surface.