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Need help with EM/Jackson problem

  1. Jan 26, 2005 #1
    In Classical Electrodynamics - 2nd Ed., J.D. Jackson on page 50 there is a problem I need help with. Its problem 1.2 which states

    I can't seem to get started on this one. Note that [tex]D(\alpha;x,y,z)[/tex] is the product of three Gaussian functions i.e.

    [tex]D(\alpha;x,y,z)[/tex] = G(x)G(y)G(z)

    The product UVW reminds me of a Jacobian but I'm not quite sure how. :tongue:

    Any thoughts/solutions/answers? I want to know the answer more than I want to be walked through it with hints (I have hundreds of more problems to work through besides this one which I gave up on). Thanks.

    Pete
     
  2. jcsd
  3. Jan 27, 2005 #2

    dextercioby

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    HINT:( :tongue2: )
    [tex] \iiint_{R^{3}} \lim_{\alpha\rightarrow 0} [D(\alpha;x,y,z)] \ dx \ dy \ dz [/tex]

    [tex]=\iiint_{R^{3}} \lim_{\alpha\rightarrow 0} [D(\alpha;x-x',y-y',z-z')] \ d(x-x') \ d(y-y') \ d(z-z') =1 [/tex]

    Now make the change of coordinates...

    Daniel.
     
    Last edited: Jan 27, 2005
  4. Jan 29, 2005 #3

    reilly

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    Pete -- Think of a delta function as the inverse of a volume element, so if the volume at a point changes by dx dy dz -> dx dy dz/UVW, the delta function must transform as advertised.

    More detail: as alpha-> zero, the Gaussian gets very narrow, and the transformation from (x,y,z) to (u,v,w) reduces to a constant one. So, the factor x*x + y*y + z*z in the original Gaussian, becomes, from dx=du/U, (u/U)**2 + (v/V)**2 + (w/W) **2. The rest is a bit of algebra.

    All a delta function cares about is its own point -- the rest be damned.

    Regards,
    R
     
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