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Integration and Laplacian in polar coordinates

  1. Apr 13, 2014 #1
    1. The problem statement, all variables and given/known data
    I have a function y that is axisymmetric, so that y=y(r).

    I want to solve for r such that ∇2y(r) = Z.

    Can anyone tell me if I'm following the right procedure? I'm not sure since there are two "∂/∂r"s present...


    2. Relevant equations
    2 = (1/r)(∂/∂r)(r*(∂/∂r)) + (1/r2)*(∂2/∂θ2)
    --> The terms involving theta become zero since y is a function of r only


    3. The attempt at a solution
    2y = (1/r)(∂/∂r)(r*(∂y/∂r)) = Z
    (∂/∂r)(r*(∂y/∂r)) = Zr
    integrate with respect to r: r(∂y/∂r) = (1/2)Z*r2 + C
    (∂y/∂r) = (1/2)Zr + C/r
    integrate with respect to r again: y = (1/4)Z*r2 + Cln(r) + K
     
  2. jcsd
  3. Apr 13, 2014 #2

    Dick

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    Homework Helper

    Looks fine to me. If you substitute that back into the original equation you do get Z, right? That's a good way to check.
     
    Last edited: Apr 13, 2014
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