You should try to understand differential forms, if you have those then Maxwells equations in vaccuum really aren't strange any more.
But Differential forms aren't that strange really. They are the mathematical objects that allow you to integrate along surfaces and curves. Of course their theory hadn't been developed when Maxwell wrote them. And Maxwell was very much concerned with EM in matter, which mixes the properties of the EM fields and materials, and that can thus be expected to get a bit messy.
But I don't see what you are striving for when you say "really" understand them. I think this is a psychological category, rather than a hard criterion. Can you apply the formalism to calculate consequences? That's the main issue. Maybe you can have a better or worse intuition about the consequences, but that is often mainly due to practice. You can't expect to correctly intuit all possible consequences of a system as rich as EM.
> Young man, in mathematics you don't understand things. You just get used to them. -- John von Neumann
I don't think the equations of electrodynamics themselves are difficult to understand and accept at an intuitive level, as one can easily describe them in plain words.
And a bit more deeply than just differential forms and the exterior derivative: connections on vector bundles and their curvature. Deeper still: connections on a principal bundle and the induced connections on associated bundles.
But Differential forms aren't that strange really. They are the mathematical objects that allow you to integrate along surfaces and curves. Of course their theory hadn't been developed when Maxwell wrote them. And Maxwell was very much concerned with EM in matter, which mixes the properties of the EM fields and materials, and that can thus be expected to get a bit messy.
But I don't see what you are striving for when you say "really" understand them. I think this is a psychological category, rather than a hard criterion. Can you apply the formalism to calculate consequences? That's the main issue. Maybe you can have a better or worse intuition about the consequences, but that is often mainly due to practice. You can't expect to correctly intuit all possible consequences of a system as rich as EM.
> Young man, in mathematics you don't understand things. You just get used to them. -- John von Neumann