I'm wondering if anyone knows a good introductory book where differential geometry is naturally brought into Maxwell's equation? Some textbooks are too straightforward to describe it in geometric language without any interpretation. I think it is because I didn't find the right book and may it be included in classical field theory? Can anyone recommend some literature on classical field theory for me? Textbooks or monographs at a graduate student level are better.
W. E. Thirring gives a treatment of Maxwell's equation (and other classical field theories) in terms of differential forms, in the second volume of his course in mathematical physics. I'm not sure whether you are looking for something in this vein, but it would be worth to look at.
One of my favourite sources for this is the Misner-Thorne-Wheeler book "Gravitation" (which lives up to its name :)).
I recommand two books or chapters from the following books. The first one is
Chocquet Bruhat, Y., DeWitt-Morette, C. Dillard-Bleick, M., Analysis, manifolds and physics. Elsevier, Amsterdam 2000, and
Lounesto, P., Clifford Algebras and Spinors, London Math. Society.
In these books, the Hodge star for indefinite bilinear form is defined and the Maxwell equations via this involution are formulated.In the second one, there should be some connections to the Coulomb law, mentioned, and also not only the formulation via the de Rham complex, but also via spinors. Very nice book (as a continuation of the first and the second one) is that one of Frankel, T., The Geometry of Physics: An Introduction, where people, more motivated from Physics, can see the Maxwell electromagnetism from the perspective of their gauge nature, and also connections of Quantum mechanics to the gauge approach to electrodymanics.
A propos, I studied from Thirring, but the book when I was a student seemed to be a bit advances, in the sense may be to much things were left for the excercixes. But it's a nice analytical book.
(Landau, Lifschitz were to "physical" for me. They are more applied, but they seem not to take care of the math background, that's the life...but fo sure deserve also an attention.)
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