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Problem with taking the divergence

  1. Apr 22, 2010 #1
    hi, can someone tell me how [itex]\nabla\dot(\frac{\widehat{r}}{r^2})=4\pi\delta^3(r)[/itex]

    thanks
     
  2. jcsd
  3. Apr 22, 2010 #2

    berkeman

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    Staff: Mentor

    How would you normally proceed with taking the divergence? What coordinate system is this in?
     
  4. Apr 25, 2010 #3
    i would use spherical coord system. using the corresponding form of div. but this gives 0 as the r^2 in the formula is cancelled but the r^2 in the div eqn
     
  5. Apr 25, 2010 #4

    gabbagabbahey

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    Homework Helper
    Gold Member

    Sure, it gives zero everywhere, except at r=0. Remember, [itex]\frac{1}{r^2}[/itex] is undefined at [itex]r=0[/itex]:wink:

    What does the divergence theorem tell you about the volume integral

    [tex]\int_{\mathcal{V}}\left(\mathbf{\nabla}\cdot\frac{\hat{\mathbf{r}}}{r^2}\right)dV[/tex]

    in two cases:

    (1) When [itex]\mathcal{V}[/itex] is any volume enclosing the origin?
    (2) When [itex]\mathcal{V}[/itex] is any volume not enclosing the origin?

    Compare these results, along with the fact that [tex]\mathbf{\nabla}\cdot\frac{\hat{\mathbf{r}}}{r^2}[/itex] is zero everywhere except at [itex]r=0[/itex], where it is undefined, to the properties defining the 3D Dirac Delta function.
     
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