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Reading through Jackson: Gauss Theorem

  1. May 23, 2012 #1
    1. The problem statement, all variables and given/known data

    I'm reading through Jackson and ran into the following:

    An application of Gauss's theorem to

    ∇'[itex]^{2}[/itex]G=-4πδ(x-x')

    shows that

    [itex]\oint[/itex]([itex]\partial[/itex]G/[itex]\partial[/itex]n')da'= -4∏

    where G is a Green function given by 1/|x-x'| + F, and F is a function whose Laplacian is zero.

    (Sec. 1.10, Formal Solution of Electrostaic Boundary Value Problem)


    2. Relevant equations

    Divergence theorem?
    Gauss's theorem?

    3. The attempt at a solution

    I don't see how to arrive at the surface integral. This looks a bit like an application of the divergence theorem because of the surface integral term. It also looks something like Gauss's law in differential form. Is this what the author means by applying Gauss's theorem?
     
  2. jcsd
  3. May 23, 2012 #2

    vela

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    Nothing too mysterious going on. Gauss's law tells us
    $$\int_V \nabla\cdot (\nabla G)\,dv = \oint_S [(\nabla G)\cdot\hat{n}]\,dS$$ The integrand of the surface integral is simply the directional derivative in the ##\hat{n}## direction, which is equal to ∂G/∂n, where n is the coordinate along the direction of ##\hat{n}##.
     
  4. May 23, 2012 #3
    Thanks -- what was I thinking... G is a 1/r potential...
     
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