Harley Flanders, "Differential Forms with Applications to Physical Sciences". QuiteAuthor spends a mathslot of time to develop the understanding of forms, exterior derivatives, determinants, star operator, push-y textforwrds and pull-backs. But goodDevelopes 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, IMHObut 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 One of my favourites. Less axiomatic than previous textReally 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 still rigorousit 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. I liked what I read from itIn 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 I did not read it too muchstill good.