Recently, I was contemplating the beautiful formulation of electromagnetism (specifically Maxwell's equations) in terms of differential forms: $$F=\mathrm{d} A\implies \mathrm{d}F=0 \hspace{1cm}\text{and}\hspace{1cm} \mathrm{d}\star\mathrm{d}F=\mu_0 J $$ I started thinking about the history of this way of looking at things, and realized that I don't know much about it at all. My first question was therefore: Was it known already at the time of Maxwell (or soon after) that electromagnetism could be cast in this geometric form? How was this first introduced and who did it?
After consulting Maxwell's treatise, it became clear that at least Maxwell himself was not aware of this formulation. But maybe someone else immediately recognized the geometric formulation once Maxwell published his results...
In modern times, one is - at least as a physicist - usually first introduced to the field strength tensor $F$ through the covariant formulation of Maxwell's equation using tensor calculus, where it is defined as $F_{\mu\nu}=\partial_\mu A_\nu -\partial_\nu A_\mu$. When one then learns about differential forms etc. it is then obvious that $F=\mathrm{d}A$ and the geometric formulation follows quite naturally. However, was this also the case historically? Did 'they' come up with the tensor calculus formulation of $F$ first, and did they only then recognize the geometric description? Or was the geometric description discovered first? Another possibility is that it took the introduction of Einstein's general relativity for anyone to realize that fields can be interpreted in terms of geometry.
In conclusion, I am interested in a chronological description of the development of the different formulations of electromagnetism, with emphasis on the following points:
- Who first came up with the geometric formulation in terms of differential forms?
- Is it known at all how this person arrived at this?
- Was the geometric interpretation discovered before tensor calculus became popular, or only after it was know that $F_{\mu\nu}=\partial_\mu A_\nu -\partial_\nu A_\mu$? Was this after the introduction of GR, and was it at all influenced by Einstein's work?