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Gauss Divergence Theorem

  1. May 11, 2010 #1
    1. The problem statement, all variables and given/known data

    Verify Gauss Divergence Theorem ∭∇.F dxdydz=∬F. (N)dA
    Where the closed surface S is the sphere x^2+y^2+z^2=9 and the vector field F = xz^2i+x^2yj+y^2zk


    3. The attempt at a solution

    I have tried to solve the left hand side which appear to be (972*pi)/5
    However, I cant seems to solve the right hand side to get the same answer.
    I substitute x = 3sin(theta)cos(phi), y=3sin(theta)sin(phi), z=3cos(theta)
    Therefore N=9sin^2(theta)cos(phi)i+9sin^2(theta)cos(phi)j+9cos(theta)
    and F = F=27sin(θ)cos^2(θ)cos(φ)+27sin^3(θ)cos^2(φ)sin(φ)+27sin^2(θ)cos(θ)sin^2(φ)
    then I used ∫(0-2pi)∫(0-pi) F. (N) dθdφ
    I got the final answer as (324*pi)/5 which does not match with left hand side.
    Hope anyone can help here plz. Thanks!
     
  2. jcsd
  3. May 11, 2010 #2

    vela

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    Your N is wrong. Describe to me what you think N is.
     
  4. May 11, 2010 #3
    N is the normal vector?

    where r(θ,φ) = (3sinθcosφ, 3sinθsinφ, 3cosθ)

    and N = rθ X rφ

    Thus, N=9sin^2(θ)cos(φ)i+9sin^2(θ)cos(φ)j+9cos(θ)sin(θ)k
     
  5. May 11, 2010 #4

    vela

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    Yes, it's a normal vector. More important, it's the unit normal vector. Since you're using a sphere, it will just be the radial unit vector.

    Edit: Oh, I see what you're doing. That's not just the normal vector but the normal vector scaled by the part of the area element. I think you're just cranking out the integral wrong, but let me try calculate it here to make sure.
     
    Last edited: May 11, 2010
  6. May 11, 2010 #5
    Do you mean the range values when i am integrating? I dont understand what to use. Isnt it 0-pi for the inner integral and 0-2pi for the outer integral?
     
  7. May 11, 2010 #6

    vela

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    The y-component of your N should have sin(φ), not cos(φ).
     
  8. May 11, 2010 #7

    vela

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    Yeah, you did everything right except you made a mistake on the y-component of N. If you fix that, you should get the right answer. It worked out for me.
     
  9. May 11, 2010 #8
    OK! I got it! Tnx for the help very much!! I didnt realise this mistake and was really stress over it.. Thanks again for the help and I got the answer :D
     
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