Why Do Divergence Theorem and Regular Flux Method Yield Different Results?

[BatmanACC]

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
"[URL
http://img163.imageshack.us/img163/6020/question0011.th.jpg
[/URL]
Calculated via the Divergence Theorem

http://img28.imageshack.us/img28/1120/question10review0001.th.jpg
[/URL]

 

[Nino MMP]

Diagram of the surface[URL http://img5.imageshack.us/img5/4887/question2review0001.th.jpg [/URL]Yes, I believe that the way you did it with the divergence theorem is the correct way. The regular flux method assumes that the surface is flat and that the normal vector is constant over the entire surface. However, in your case, since the surface is curved, the normal vector is not constant and thus the regular flux method does not apply here.