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Greens Functions

  1. May 2, 2008 #1
    [SOLVED] Greens Functions

    1. The problem statement, all variables and given/known data
    show:
    [tex]\int\int\int_{D}\vec{F}\cdot\vec{G}dV = 0[/tex]
    where:
    [tex]\vec{F}=\nabla\phi[/tex]
    [tex]\vec{G}=\nabla\psi[/tex]
    [tex]\nabla\cdot\vec{F}=0[/tex]
    [tex]\psi|_{\partial D}=0[/tex]

    3. The attempt at a solution
    This looks like a problem for Greens first theorem:

    [tex]\int\int\int_{D}\phi\nabla^{2}\psi dV = \int\int_{\partial D}\phi\nabla\psi dS - \int\int\int_{D}\nabla\psi\cdot\nabla\phi dV[/tex]

    The very right term is clearly the integral that I'm looking for. So, i will set it to look like the requested answer. Also, I know that
    [tex]\psi|_{\partial D}=0[/tex]
    meaning that I can also throw out the second term because that term wants me to integrate the gradient of psi over the surface, while I know that psi is 0 over the surface. So, I am left with this:

    [tex]\int\int\int_{D}\phi\nabla^{2}\psi dV = - \int\int\int_{D}\vec{F}\cdot\vec{G} dV[/tex]

    So, this means that the term on the left mus equal zero. Does anyone know how I can show this? Psi is not zero through the domain, and the problem doesn't specify that it is a harmonic potential (although I suppose it could be). Could someone please help me with this step? Any help at all is greatly appreciated.
     
  2. jcsd
  3. May 2, 2008 #2

    Dick

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    Uh, phi is given to be harmonic, not psi. Since the divergence of it's gradient is zero. Why don't you move the laplacian operator over to phi?
     
    Last edited: May 2, 2008
  4. May 2, 2008 #3

    Dick

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    I.e. just switch the roles of psi and phi?
     
  5. May 3, 2008 #4
    Thanks for the help! This will also eliminate the middle term anyways too, since psi will be zero there on the boundary, right?
     
  6. May 3, 2008 #5

    Dick

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    Right.
     
  7. May 3, 2008 #6
    thanks a lot for the help Dick
     
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