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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?

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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)? –  nbubis 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-Alvarez Aug 27 '12 at 15:51
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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-Alvarez Aug 27 '12 at 15:54
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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 Bruns Aug 27 '12 at 17:44
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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-Alvarez Aug 27 '12 at 21:08

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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).

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I liked the Madsen and Tornehave book a lot. –  MTS Jul 23 '13 at 21:06
    
Thanks a lot! I'll try to read all 3. –  Frank Aug 12 '13 at 1:35

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