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Multiple integrals

  1. Dec 21, 2005 #1
    Hi, I posted a question some time ago and the suggestion was to use some form of the product rule but I still can't figure out what to do.

    Q. Let f(x,y,z) and g(x,y,z) be C^2 scalar functions. Let D be an elementary region in space and [tex]\partial D[/tex] be the closed surface that bounds D. Prove that

    [tex]\int\limits_{}^{} {\int\limits_{}^{} {\int\limits_D^{} {\nabla f \bullet \nabla g} dV = \int\limits_{}^{} {\int\limits_{\partial D}^{} {f\nabla g \bullet dS} - \int\limits_{}^{} {\int\limits_{}^{} {\int\limits_D^{} {f\nabla ^2 gdV} } } } } } [/tex]

    Can someone give me a hint as to where to start, like any relevant identies which could be of use? Any help is appreciated thanks.
     
  2. jcsd
  3. Dec 21, 2005 #2

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    The following identity should help:

    [tex]\nabla \cdot (f \vec A) = \vec A \cdot \nabla f + f \nabla \cdot \vec A[/tex]

    Which is just one of the 3D versions of the product rule. (You can substitute [itex]
    \nabla g[/itex] into this formula)
     
  4. Dec 21, 2005 #3
    Thanks for the help, I'll try to finish this one off.

    Edit: Hmm...I could've sworn that your post made mention of the divergence theorem before...nevermind, I'll keep that in mind.;)
     
    Last edited: Dec 22, 2005
  5. Dec 22, 2005 #4

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    Yea, sorry about that. I had originally told you how to do the problem step by step, but I realized that's not what this forum is for, so I changed it. But yes, the divergence theorem is necessary.
     
  6. Dec 22, 2005 #5
    Oh ok, it doesn't really matter too much now. The divergence theorem is the only connection I know of between surface and volume integrals so I probably would've used it eventually anyway.
     
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