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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. $\endgroup$ – Cheerful Parsnip 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. $\endgroup$ – Cheerful Parsnip Aug 19 '15 at 12:47

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