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Divergence and curl rule simplification?

  1. Oct 22, 2008 #1
    Just for reference, i got this question from reading an online ebook:
    The bottom equation on page 24 is where i these equations came up.

    I have been reading some stuff and i keep coming across an annotation which looks exactly
    like a divergence symbol except backwards.
    V dot (del sign)

    Specifically, it was used as follows:
    [j(x') dot del']del'(1/(abs(x-x')))

    where del is the upside triangle symbol and x and x' are just two different variables.
    del' operates only on the x' symbol.

    Also, can anyone point me out to any online reference which shows how to simplify
    vector identities which include integrals. Example:

    1. The problem statement, all variables and given/known data

    Simplify Kcurl(curl(integral(d^3x'*j(x')/abs(x-x'))))
    where, again, x and x' are two independent variables and d^3x' just represent the volume
    differential (not a variable!)

    2. Relevant equations
    curl(curl(a)) = grad(div(a))-laplace(a)

    3. The attempt at a solution
    No clue where to start. But their solution was as follows:
    = -K*(integral(d^3x'j(x')laplace(1/(abs(x-x'))))
    +K*(integral(d^3x'[j(x') dot del']grad'(1/(abs(x-x')))))

    where del' or grad' just means do in terms of x' variable.

    I got lost when they started using grad' and dot del' in the second integral, even though
    they had used a curl in terms of x, not x'.
  2. jcsd
  3. Oct 22, 2008 #2
    Read this page: http://en.wikipedia.org/wiki/Del

    That should take care of your first concern. Apropos your second concern, the integral seems to be acting like a scalar.
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