There are many books which treat classical electrodynamics on smooth manifolds using the language of differential forms and connections. I think this is a wonderful approach, especially fruitful when one considers other gauge theories or simply wants to become a mathematical physicist.
However, these wonderful books don't cover some "easy" topics, present in standard introductory courses as Jackson or Griffiths;
- To describe point charges or infinitely thin wires, distributional densities are needed.
- Many examples use the manifolds with corners (as triangles or cubes).
- I don't know a book on gauge theory that introduces Fourier theory and Green's functions and applies it to solve Poisson equation in "standard" problems (as in Jackson and Griffiths).
- In presence of infinitely thin conductors (again, this is related to 1.), the electric field becomes discontinuous, what is not handled by the usual smooth approach with differential forms.
These non-smooth issues look severe to the beginner and they happen quite often in physics. They usually (always?) vanish when handled appropriately (as introducing Stokes' theorem for manifolds with corners or developing variational calculus with piecewise-smooth curves in classical mechanics).
I wonder if you could recommend a book or lecture notes on electrodynamics concerned with such smoothness issues.