Greens Functions

  • #1
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[SOLVED] Greens Functions

Homework Statement


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]

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.
 

Answers and Replies

  • #2
Dick
Science Advisor
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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:
  • #3
Dick
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I.e. just switch the roles of psi and phi?
 
  • #4
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Thanks for the help! This will also eliminate the middle term anyways too, since psi will be zero there on the boundary, right?
 
  • #5
Dick
Science Advisor
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Thanks for the help! This will also eliminate the middle term anyways too, since psi will be zero there on the boundary, right?
Right.
 
  • #6
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thanks a lot for the help Dick
 

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