I am familiar with the usual way of doing Classical Field Theory, but I am currently taking a course where the professor works with differential forms to teach the subject. I wonder if anyone knows where I can read more about this.
A somewhat advanced but still physics oriented exposition can be found in Section 5.4 of
- Geometry, Topology and Physics, Mikio Nakahara, CRC Press.
Another approach, still very physics oriented, but a bit longer and slightly less andvanced can be found at Chapter 18 of
- Modern Geometry - Methods and Applications: Part I, B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, Springer.
The book itself is very good, it builds everything from the start and I strongly recommend it for a first approach to geometry for physicists.
If you need something quick and to the point you might also like Section 2.8.2 of
- D-Branes, Clifford V. Johnson, CUP.
Even though the book talks about $D$-branes, the section I pointed you to is self contained and understandable from someone that does not have that kind of background.
There is also Section 8.8 of
- Quantum theory of Fields, Volume 1, Steven Weinberg, CUP.
which is really short but talks about $p$-form electrodynamics, so it might be interesting to look at.
Harley Flanders, "Differential Forms with Applications to Physical Sciences". Author spends a lot of time to develop the understanding of forms, exterior derivatives, determinants, star operator, push-forwrds and pull-backs. Developes the notion of integration on manifolds and the Generalised Stokes theorem. The style is Theorem-Proof+ some discussion. Does include specific sections to link it to Physics, but this feels like a book by a mathematician. I found it tough going and had to come back to it several times, but it did pay off.
David Lovelock and Hanno Rund, "Tensors, Differential Forms, and Variational Principles". One of my favourites. Really good for developing the grasp of tensor calculus. Helped me to develop a strong understanding of the covariant and Lie derivatives. In the later sections links to Physics of General Relativity and Variational Calculus. There is a chapter on differential forms. This chapter is in my opinion not the strongest one, but it links to other parts in the text which helps to place the differential forms into the conventional context of vector/tensor calculus as it is taught to Physics students. The style is more common to physics books definitions+discussion (few Theorem-Proof sections)
David Bachman, "A Geometric Approach to Differntial Forms", helped me to tie together bits from the two previous books. In particular helped me to understand the notion of the tangent and co-tangent vector spaces on manifolds. Less axiomatic and more illustrated than Flanders, but still good.