Hi all, I have a tricky problem in pertubation theory.(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Perturbation theory to solve diff eq?

Can you offer guidance or do you also need help?

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