1. Let S be the surface Find

I know the answer is supposed to be but can't seem to get there. I've tried evaluating it with the divergence theorem, but nothing I do seems to work correctly. I know that is equal to and I have , which would make . Then the integral becomes .

Am I doing this right? I can't figure out where I'm messing up.

These next two I have the same issues with:

2. Let S be the surface . Find the upward flux of the vector field across S.

I think that the divergence is 0, and wouldn't that make the whole integral 0? But I'm told the correct answer is -1. Why is this?

3. Let S be the portion of the cylinder given by .

Orient S by normal vectors pointing away from the z-axis and compute the flux of across S.

In this case, wouldn't the divergence also be 0, since ?

But I know the answer to be .

I'm obviously missing the same sort of thing in all these problems so any help would be appreciated very much!