- #1
Feodalherren
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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 let's 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?!