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Divergence theorem over a hemisphere

  1. Dec 7, 2008 #1
    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.

    1. The problem statement, all variables and given/known data


    2. Relevant equations


    3. 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 :)
  2. jcsd
  3. Dec 7, 2008 #2


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    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?
  4. Dec 7, 2008 #3
    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.
  5. Dec 7, 2008 #4


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    Good idea! Let me know what you get.
  6. Dec 7, 2008 #5
    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!
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