Some time ago while thinking about life and such, I thought to myself does recasting electromagnetism in bundle theory make certain calculations easier? To be precise are there theorems in differential geometry and fiber bundle theory that will make some calculations in electromagnetism easier?
Motivation:
A few days ago I realized that a theorem in algebraic topology helped me calculate a homotopy group by calculating a homology. I am still not sure how I could have directly calculated the homotopy group but the theorem made the calculation easier, simpler and doable.
This got me thinking maybe I have been ignoring modern mathematics theorems at my own peril. There are some simpler theorems in calculus that have applications in electromagnetism like Stokes's and Gauss's theorem, but I was wondering if some ideas grew out of differential geometry and fiber bundle language that made calculating things significantly easier in electromagnetism. I am thinking of things like the scalar potential, vector potential, electric field, and magnetic field.
I am just wondering. I'm inclined to believe that people who go out of their way to define everything in this language are probably using some new and exciting theorems to calculate things. At this stage in my life I am marginally competent with some electromagnetism I guess some U(1) theory as well but I am probably not well versed with broader gauge theories but I am wondering again are there theorems that make calculations easier in this language? I know very very basic fiber language. In fact, probably just the definitions of base space, projection map, fiber, section etc, but I have not had the pleasure of delving deep to see the theorems and such so I don't know.