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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.

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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.

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  • $\begingroup$ Bott and Tu's book is also quite good. $\endgroup$ Aug 19 '15 at 11:28
  • $\begingroup$ @GrumpyParsnip Though it is great, I hesitate to recommend that book to someone without a firm grounding in basic manifold theory. Tu says in the introduction to the book I recommend that he intended it as a prelude to Bott and Tu, to fill in the necessary background. $\endgroup$
    – Potato
    Aug 19 '15 at 12:45
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    $\begingroup$ I agree completely. $\endgroup$ Aug 19 '15 at 12:47
  • $\begingroup$ Actually, Tu himself writes in the preface of his An Introduction to Manifolds that it's intended to bridge the gap between a background in undergrad real analysis and abstract algebra and the background necessary for Bott and Tu. $\endgroup$ Sep 10 '20 at 16:08
  • $\begingroup$ I think it might be useful for future reference to also make mention of Lee's Introduction to Smooth Manifolds. Both Lee's and Tu's books require some background in point-set topology, but Tu adds the necessary background in an appendix, while Lee references his other book, on topological manifolds, and Munkres's Topology. $\endgroup$ Sep 10 '20 at 16:12
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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.

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