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Surface Integral of a Sphere (non-divergence)

  1. Nov 11, 2008 #1
    1. The problem statement, all variables and given/known data

    Evaluate: [tex]\int[/tex][tex]\int[/tex]G(r)dA

    Where G = z
    S: x2 + y2 + z2 = 9 [tex]z \geq 0[/tex]

    2. Relevant equations

    x = r sinu cosv
    y = r sinu sin v
    z = r cos u

    3. The attempt at a solution

    r(u,v) = (r sinu cosv)i + (r sinu sinv)j + (r cosu)k
    ru = (r cosu cosv)i + (-r cos u sinv)j + (-r sinu)k
    rv = (-r sinu sinv)i + (r sinu cosv)j + 0k

    dA = |ru x rv|

    I am not sure if I am approaching this correctly or if I am way off base. My next step was to complete the dot product of z with dA but this does not seem right and I can't find any good examples in my text.

    Thank you in advance.
  2. jcsd
  3. Nov 11, 2008 #2


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    You are doing it ok. There's a simpler way to get dA. You know that dV in spherical coordinates is just r^2*sin(u)*du*dv*dr, right? dA over a sphere is just that without the dr. But you should get the same thing by finding the norm of your cross product.
    Last edited: Nov 11, 2008
  4. Nov 12, 2008 #3


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    r= 3 in this problem and you don't use "the dot product of z with dA" because neither is a vector! Just multiply and integrate.
  5. Nov 12, 2008 #4
    Thank you very much for the help! I believe that I have figured it out now.
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