Divergence theorem over a hemisphere

[jonwell]

I was told this problem could be done with divergence theorem, instead of as a surface integral, by adding the unit disc on the bottom, doing the calculation, then subtracting it again.

Homework Statement

Homework Equations

The Attempt at a Solution

for del . f I get i + j = 2. Which makes the integral equal twice the volume of the hemisphere, or 4/3 pi. Now I'm supposed to subtract the unit disc, but I get pi when I calculate that surface, which leaves me with 1/3 pi. The answer should be 7/6 pi.

Thanks :)

 

[Dick]

The outward pointing normal to the disk is -k. So F.(-k) is -1 and the contribution from the disk is -pi. The total integral from the divergence theorem is 4pi/3 (as you said). I think that makes the integral over the upper hemisphere 7pi/3, doesn't it? Not 7pi/6?

 

[jonwell]

Ya, I got that a couple times too (after I remembered the orientation), however the answer I was provided is 7/6 pi. That could be wrong. I'll re-do the surface integral the other way and see what I come up with I guess.

 

[Dick]

Good idea! Let me know what you get.

 

[jonwell]

well, I'm still getting 7pi/3, so I'm going to assume that the two of us combined are smarter than the given answer ;) Thanks for your help!