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Divergence of 1/r^2; delta dirac's role

  1. Sep 26, 2012 #1
    1. The problem statement, all variables and given/known data
    Given [itex]\nabla\frac{1}{r}[/itex], show [itex]\nabla\bullet\nabla\frac{1}{r}[/itex] = -4πδ(r), where δ(r) is the delta dirac function.


    3. The attempt at a solution
    I've used divergence theorem and also solved the equation itself, so I know that outright solving is zero and the divergence theorem gives -4π. But I'm not sure how to show the presence, or rather, how I get the delta dirac function. I understand its role and but not necessarily why it needs to be there. Any help on direction on this one would be greatly appreciated.
     
  2. jcsd
  3. Sep 26, 2012 #2

    Ray Vickson

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    Think of ##\nabla \frac{1}{r}## as an electrical field ##\mathbf{E}##, so that ##\nabla \cdot \mathbf{E}## is its divergence. Now use Gauss' Law
    http://www.deepspace.ucsb.edu/wp-content/uploads/2010/08/033_Chapter-22-Flux-and-Gauss-Law-PML.pdf .

    RGV
     
  4. Sep 26, 2012 #3
    Thank you for your prompt reply - this also makes it a little clearer. Seeing as this is a general mathematical property, is it simply enough to state this or is there some more rigorous proof needed?

    i.e., I see now why at 0 it has the delta dirac, but not why ithe -4pi portion. This is the result of the integral - how does it fit back into the original divergence? Please pardon my ignorance - it's been a while.
     
  5. Sep 26, 2012 #4

    Ray Vickson

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    Just read the link; it's all there.

    As for "rigour", well, that is another matter altogether, and it concerns the undeniable fact that δ(r) is not really a legitimate function at all---it is a so-called "distribution", or "generalized function". There are lots of different approaches to defining distributions and to proving results about them. The first few pages of http://www.math.umn.edu/~olver/pd_/gf.pdf [Broken]
    deal with the basic issues involved in the 1-dimensional case. Some of the same basic ideas carry over to the 3-dimensional case you are dealing with. For more material, Google "generalized functions".

    RGV
     
    Last edited by a moderator: May 6, 2017
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