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Perturbation theory to solve diff eq?

  1. Jan 13, 2012 #1
    Hi all, I have a tricky problem in pertubation theory.

    I have a function:

    [tex]
    f(\vec{r}) = P(\vec{r}) + \left( B(\vec{r}) + b(\vec{r}) \right)^2
    [/tex]

    where [tex] b(\vec{r}) [/tex] is a small perturbation and is equal to 0 when [tex] P(\vec{r}) = 0 [/tex]

    Now, to solve the equation
    [tex]
    \nabla f(\vec{r}) = 0
    [/tex]

    for b(r) is fairly straightforward by noting that
    [tex]
    P + (B + b)^2 = C = B^2
    [/tex]
    using the fact that P = 0 ==> b = 0. And so,
    [tex]
    b(\vec{r}) = \frac{-P}{2B}
    [/tex]
    by expanding the above equation and neglecting the [tex]b^2[/tex] term. Now, my question is how do you solve the inhomogeneous equation:
    [tex]
    \nabla f(\vec{r}) = \vec{A}(\vec{r})
    [/tex]
    for b(r) where A is a known vector field with 0 curl?
     
    Last edited: Jan 13, 2012
  2. jcsd
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