I'm a looking for a good book to self-study differential forms. Particularly, I'm looking for a book that is as similar as possible to Bert Mendelson's "Introduction to topology" (i.e. a book that procede by following a: "Definition, theorem, proof" style). In addition, the book that I'm looking for should be as much self consistent as possibile. I'm a first year graduate student in nuclear engineering. My prerequisites are a good understanding of (multivariate and vector) calculus, linear algebra, and a little of functional analysis, Lebesgue integration theory, PDE. I know nothing about differential geometry, but to my (very) poor understanding differential forms and concept like manifolds and so on are linked to each other.
Differential forms are things that live on manifolds. So, to learn about differential forms, you should really also learn about manifolds. To this end, the best recommendation I can give is Loring Tu's An Introduction to Manifolds. Tu develops the basic theory of manifolds and differential forms and closes with a exposition of de Rham cohomology, which allows one to extract topological information about a manifold from the behavior of the differential forms on it.
Quite underrated, yet very practical-oriented is the book by D.G. Edelen "Applied Exterior Calculus". You can also try to check G. de Rham "Differentiable manifolds". Finally, for mathematical physics way of thinking, you can try M. Nakahara "Geometry, Topology and Physics", or T. Frankel "Geometry of Physics". All these books contain a large part about differential forms.
Look no further than Bernard Schutz "Geometrical Methods of Mathematical Physics" (CUP 1980) for an intro which starts at the beginning. Differential Forms are merely infinitesimal increments of coordinate axes - which transform inversely to tangents to these axes - and linear combinations thereof. In an x,y,z coordinate system an example would be 3dx+dy+4dz. Their inverse transformation nature is signified by writing their indices UPSTAIRS rather than downstairs. A usual 1st example of a differential form is the differential of a function.