# Homework Help: Gaussian theorem question

1. Feb 4, 2013

### r19ecua

1. The problem statement, all variables and given/known data
Suppose the one-dimensional field A = Kx * ax 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.

2. Relevant equations

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

Direction on the left is applied to the integral on its right.
3. The attempt at a solution

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

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

For the top side
$\int_{-2}^2$$\int_1^4x$dxdy az

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

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

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

2. Feb 4, 2013

### 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.