I am trying to solve the following problem. Suppose I have a field $\Phi(r)$, which is the solution to a partial differential equation:
$\mathcal{L}\Phi(r) = s(r)$, as long as $r \neq r_0$
Here $\mathcal{L}$ is a differential operator (linear) and $s(r)$ is a given source field. On top, there is a very simple boundary condition: there exists a single point $r_0$ for which
$\Phi(r_0) = 0$
How can I derive the following gradient:
$\frac{d\Phi(r)}{dr_0}$
In other words, how does $\Phi(r)$ change when I move the point for which the boundary condition applies. I can't get a grip on this problem, so any help or referral to a textbook that might help would be much appreciated!
Michiel