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Flux Problem

  1. Oct 21, 2010 #1
    1. The problem statement, all variables and given/known data

    Find the flux of x^2 + y^2 + z^2 = 4a^2 through F = (x+yz)i + (y-zx)j + (z-e^x * siny)k
    directly without using divergence theorem.

    2. Relevant equations

    3. The attempt at a solution

    Using spherical coordinates I achieve N = 4asin(phi)cos(theta) i + 4asin(phi)sin(theta) j + 4asin(phi)cos(phi) k

    But when subbing in spherical to F, then F dot N, I can only simplify to a very messy integral of: -8a^3sin(phi) + 4a^2cos(phi)sin(phi) * e^(2sin(phi)cos(theta) * sin(2asin(phi)sin(theta)

    Any help would be appreciated :)
  2. jcsd
  3. Oct 21, 2010 #2


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    Is the radius of that sphere [itex]4a[/itex] or [itex]2a[/itex]?...Is that the unit normal to the surface?:wink:

  4. Oct 21, 2010 #3

    and yes radius is 2a
  5. Oct 21, 2010 #4


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    Your integrand should be

    [tex]8a^3\sin\phi - 4a^2\sin\phi\cos\phi e^{2a\sin\phi\cos\theta}\sin(2a\sin\phi\sin\theta)[/tex]
  6. Oct 21, 2010 #5
    okay I forgot the a on my e^(...) and I simply used -N so the switch of signs means nothing. Where do I go from here? Or is there another way to approach this problem? Stokes Theorem?
  7. Oct 21, 2010 #6


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    Of course, the divergence theorem is the easy and obvious way but you ruled that out.

    However, you almost have it. Notice that your second term is an odd function of theta. That means if you integrate theta from -pi to pi (which is just as good as 0 to 2pi) you will get zero. So all that is left is to do the double integral on the first term.

    [tex]\int_0^{2\pi}\int_0^\pi -8a^2 \sin\phi\ d\phi d\theta[/tex]

    which will give you the same answer as the trivial application of the divergence theorem.
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