Is Poisson's equation rotationally invariant? Is Laplace's equation rotationally invariant? If so, how can I see this?
in reply to comments, my problem is that:
In my notes, when calculating the Green's function, the first step taken is to state that the problem is rotationally symmetric, so the Green's function $G(x⃗ ;x_ 0)=G(|x −\vec x_0|)$. Why is this valid?
An equation is said to be rotationally invariant if a rotated solution is again a solution. Laplace's equation is rotationally invariant; you can see this e.g. by expressing the Laplacian in spherical coordinates. Poisson's equation is generally not rotationally invariant because the right-hand side is generally not rotationally invariant. However, the Green's function is the solution of Poisson's equation when the right-hand side is a delta distribution, and since both the Laplacian and the delta distribution are rotationally invariant, so is the Green's function.