# Perturbation theory to solve diff eq?

1. Jan 13, 2012

### vibe3

Hi all, I have a tricky problem in pertubation theory.

I have a function:

$$f(\vec{r}) = P(\vec{r}) + \left( B(\vec{r}) + b(\vec{r}) \right)^2$$

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

Now, to solve the equation
$$\nabla f(\vec{r}) = 0$$

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

Last edited: Jan 13, 2012