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I Question about divergence theorem and delta dirac function

  1. Feb 13, 2019 at 6:00 AM #1
    upload_2019-2-13_19-59-47.png
    How do you prove 1.85 is valid for all closed surface containing the origin? (i.e. the line integral = 4pi for any closed surface including the origin)
     
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  3. Feb 13, 2019 at 7:30 AM #2

    jedishrfu

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    I would say by inspection, if you note that for a non-spherical surface the R would in fact be a function: ##R(r,\theta,\phi)## and that ##R(r,\theta,\phi)## factors cancel out using the rules of algebra, leaving an integral of angles only.

    Since the ##R(r,\theta,\phi)## factors cancel, then you can choose R to be a spherical surface with no loss in generality.

    Perhaps, @Mark44 or @fresh_42 have a more rigorous set of steps.
     
  4. Feb 13, 2019 at 8:20 AM #3

    PeroK

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    You can use the divergence theorem. Take any volume that includes the origin. To avoid the problem at the origin you could remove a small spherical cavity at the origin. The total surface integral is zero and the surface integral of the small cavity is ##-4\pi##.
     
    Last edited: Feb 13, 2019 at 11:22 AM
  5. Feb 13, 2019 at 10:47 AM #4
    upload_2019-2-14_0-47-2.png
    How to get 1.100 from 1.99? I can't find the derivation in the book.....
     
  6. Feb 13, 2019 at 11:24 AM #5

    PeroK

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    That's just a substitution ##\vec{r}## to ##\vec{r} - \vec{r'}##.
     
  7. Feb 13, 2019 at 11:34 AM #6
    How do you do the substitution? Why is it ok to let ##\vec{r}## = ##\vec{r} - \vec{r'}## ? They are not equal..
     
  8. Feb 13, 2019 at 12:58 PM #7

    PeroK

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    In general, if ##f'(x) = g(x)##, then ##f'(x-a) = g(x-a)##. It's a simple application of the chain rule.
     
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