Good book about differential forms 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. 
 A: 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. 
A: 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.
