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I am looking for an intro book to learn about diff forms, maybe undergrad. Reading sentences like "Let M be a smooth manifold. A differential form of degree k is a smooth section of the kth exterior power of the cotangent bundle of M." somehow is not doing it for me. Any recommendation?

Frank
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  • What background are you coming from? Do you want to study the topic in general, or for a specific purpose (say physics)? – Nathaniel Bubis Aug 27 '12 at 15:41
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    Indeed, unless you explain what your background is, this is essentially shotting in the dark! Also, presumably you have looked already at a few textbooks: telling us which those were and why excatly you found them unsatisfactory might be a good idea. – Mariano Suárez-Álvarez Aug 27 '12 at 15:51
  • I'd like to study the topic in general. I have a master's in computing science, with many hours of maths. I have strong notions of differentiation/integration, I studied Hilbert spaces for QM, know about Lebesgue integral... Ah, and I've heard diff forms are an alternative to tensor calculus, which I'm interested in (I've studied tensor calculus for GR). – Frank Aug 27 '12 at 15:52
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    The topic in general is not much of a topic, really. Have you studied differential geometry? Something about manifolds? – Mariano Suárez-Álvarez Aug 27 '12 at 15:54
  • @Mariano: yes I've been exposed to manifolds and differential geometry. – Frank Aug 27 '12 at 16:41
  • First learn the linear algebra of exterior product, and then learn what a cotangent bundle is (which you should already know if you know any differential geometry). – Eric O. Korman Aug 27 '12 at 17:39
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    There is a short, but great article by Terence Tao in the "Princeton Companion to Mathematics". If there's a copy available at your local library, you could have a look. The article does a good job explaining why it is worthwhile studying differential forms. Without that motivation, following Eric's suggestion would, at least for me, be a little dry. – Gregor Botero Aug 27 '12 at 17:44
  • It should be possible to introduce this topic at about the level of second-year calculus. But I don't know if a book has been written at that level. – Michael Hardy Aug 27 '12 at 17:46
  • I found the original paper by Cartan, Sur certaines expressions différentielles et le problème de Pfaff, from the Wikipedia entry. I'm reading that. – Frank Aug 27 '12 at 18:15
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    Reading Cartan is probably the worst possible idea. Do what everyone else does and read the book by Warner. – Mariano Suárez-Álvarez Aug 27 '12 at 21:08

1 Answers1

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If you are a physicist, read the first chapters of "Gravitation" by Misner, Thorne and Wheeler. Or, to get some rigor: "Differential forms, a complement to vector calculus" by Weintraub. Or both.

If you are a mathematician, I recommend "From Calculus to Cohomology", by Madsen and Tornehave.

If you are both, like me, read all three. Those books made me an enthusiast on differential forms, and all their possible generalizations (like connections).

geodude
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