- #1

- 603

- 6

## Homework Statement

Verify the divergence theorem if [itex]\textbf{F} = <1-x^{2}, -y^{2}, z >[/itex] for a solid cylinder of radius 1 that lies between the planes z=0 and z=2.

## Homework Equations

Divergence theorem

## The Attempt at a Solution

I can do the triple integral part no problem. Where I run into issues is the surface integral part.

Parametrizing a cylinder is done by <rcost,rsint,z>, correct?

So looking at the top part I want it oriented in the positive k direction to get flow OUT of the cylinder - hence S: <cost,sint,2> because it lies in the plane z=2.

Similarly for the bottom S: <cost,-sint,0>

Now for the side, the side never has any k components so S: <cost,sint,0>

Now lets look at the top again. If I take the derivatives with repect to Z and try to cross them I end up with 0 net flow in every direction, it does not agree with the triple integral and isn't correct.

I have a feeling that I'm not parametrizing my cylinder correct, I remember it being a special case.

Another thing that I tried was parametrizing the top as <cost,sint,z>

Then the dS vector becomes

<cost,sint,0>

Now it's showing no component in the z direction - clearly it should have a component in the z direction?!