Given Poisson equation $\nabla^2\phi=f$, what's the relation between the contours of $\phi$ and $f$. Or are there any at all?
In fact, let $\vec n = \nabla\phi$, we have $\nabla\cdot\vec n=f$. $\vec n$ is a direction field with divergence given by $f$, and the gradient of $f$ is also a vector field. So an equivalent question is what's the relation between the two vector fields.
I am not a PDE theoretician, is there any good material that goes deeper into the geometric interpretation of PDE, including simple ones like Poisson and Helmholtz equations?