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Laplacian of 1/r is zero at orign

  1. Jan 14, 2010 #1
    Hey All,

    In my vector calculus class my lecturer was showing that the laplacian of 1/r is zero. He further said that since 1/r and its derivatives are not defined at the origin we state that the Laplacian of 1/r is zero for all values of r not equal to zero. He then says that this caveat is extremely important in advanced courses.

    Can someone please tell me in what sort of courses this is important, and if possible why ?

    Cheers,
    Thrillhouse
     
  2. jcsd
  3. Jan 14, 2010 #2

    HallsofIvy

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    ?? In any course where you want to be right! 1/r is defined and smooth for any non-zero value of r but is not defined and so not continuous or differentiable at r=0. The Laplacian of 1/r does not exist at r= 0. There might be some cases in which you can get away with smoothing the Laplacian by assuming taking the value of the limit as r goes to 0 to be the value at r= 0, but you should always show that that is possible in that particular case, not just assume it.
     
  4. Jan 14, 2010 #3

    D H

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    This is a physics course, so of course the math is a bit looser there than Halls might like.

    The Laplacian of 1/r is *not* zero at the origin. The Laplacian of 1/r is

    [tex]\nabla^2\frac 1 r = -\,\frac{\delta(r)}{r^2}[/tex]

    For example, see http://www.iop.org/EJ/article/0143-...quest-id=31cf705f-8110-49fa-930d-e5ba7a4b95a5. This becomes quite important in advanced (upper undergrad and above) physics classes.
     
    Last edited by a moderator: Apr 24, 2017
  5. Jan 15, 2010 #4
    If a function has laplacian zero everywhere in a domain, then the norm of the function inside the domain is always <= the maximum norm on the boundary. The Maximum Principle. But that fails for 1/r. For example, the unit ball, we have norm 1 on the boundary, but norm 2 at a point with r=1/2 .
     
  6. Jan 17, 2010 #5

    arildno

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    No, it is not, DH.

    It is a completely wrong statement, the Laplacian of 1/r is 0 wherever it is defined.
     
    Last edited by a moderator: Apr 24, 2017
  7. Jan 17, 2010 #6

    D H

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    Take your beef up with the author of the reference I gave in my previous post. Or with Eric Weisstein in either the CRC Concise Encyclopedia of Mathematics (http://books.google.com/books?id=_8TyhSqHUiEC&pg=PA1702#v=onepage&q=&f=false) or at MathWorld (http://mathworld.wolfram.com/Laplacian.html).

    And a few more:

    http://books.google.com/books?id=kR8TqQRxbeoC&pg=PA98#v=onepage&q=&f=false

    http://books.google.com/books?id=ujOfdJxv_IQC&pg=PA106#v=onepage&q=&f=false
     
  8. Jan 17, 2010 #7

    arildno

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    It is irrelevant.

    All of them are wrong.
     
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