Divergence Theorem and Flux

Note: I've attached images of my work at the bottem of this post.

I've calculated the flux through a given surface by using The

Divergence Theorem and by using the regular flux method. These

methods give different results, however.

This leads me to assume one of the following is true:

Note: N is the unit vector outter normal to S (probably didn't have

to be said, but I didnt include it in my question on the paper!)

1) My math is wrong in one or both of the cases

2) One of these methods doesn't apply here

3) I didn't fully complete the regular flux method and I didnt

account for the s4 side (as depicted in the picture in the second

image) but I should have.

The equation used is: NdS = +/- (-dF/dx - dF/dy + K)

Of course for the sides that are along the cordinate axis

it is easiar to compute F*N because N in that case would

be along one of the cordinate axes and simply be a unit vector

along the axis.

I believe that the surface is entirely smooth (1-piece)

and can be evaluated using one instance of that equation.

Also I thought the question is asking for the Flux through

(i.e. only the flux out of the top surface) This is my

original assumption. However...

I'm fairly sure that my original assumption was wrong and that I

need to compute all the various sides (diagram shown in the second

link). Of course, most of these sides will result in 0 flux due to

the dot product of perpendicular vectors. But this is computed

through the divergence theorem easily enough. So I think that the

way I did it with the divergence theorem is the correct way. Yes?

Calculated Flux through just the upper surface

http://img163.imageshack.us/img163/6020/question0011.th.jpg

Calculated via the Divergence Theorem

http://img28.imageshack.us/img28/1120/question10review0001.th.jpg

Cheers