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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$\begingroup$ I have three books that are all rigorous. Carmo's "Differential Forms and Applications", Edwards "Advanced Calculus: A Differential Forms Approach" and Bachman's "A Geometric Approach to Differential Forms." $\endgroup$– Gregory GrantAug 19, 2015 at 11:04
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1$\begingroup$ R.W.R Darling's book Differential Forms and Connections was a great intro for me when I began graduate studies. Find it here amazon.co.uk/Differential-Forms-Connections-R-Darling/dp/… $\endgroup$– AutolatryAug 19, 2015 at 11:07
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$\begingroup$ Possible duplicate of Good intro to differential forms $\endgroup$– Danilo Gregorin AfonsoOct 31, 2017 at 17:57
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3 Answers
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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, 2015 at 11:28
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1$\begingroup$ @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. $\endgroup$– PotatoAug 19, 2015 at 12:45
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$\begingroup$ 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, 2020 at 16:08
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$\begingroup$ 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, 2020 at 16:12
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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.
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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.