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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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).