So I have a problem which (in one case) leads me to the following vector Laplace equation:
$\nabla^2 \mathbf{A} = 0$
with $\mathbf{A}$ the magnetic vector potential, whereon I have imposed the Coulomb gauge $\nabla\cdot\mathbf{A} = 0$. The problem contains cylindrical symmetry, which is why I want to solve this in cylindrical coordinates. Boundary conditions are to follow from solving the homogeneous Helmholtz equation and demanding continuity at a cylindrical surface.
However, I need to solve the Laplace equation first, which I don't know how to do. If the coordinates were cartesian, the vector equation would just be equivalent to three scalar equations, but now it's not so easy. Does anyone feel like explaining to me how I would best go about solving this vector Laplace equation in cylindrical coordinates? Greatly appreciated!