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I intend to read "Physics for Mathematicians" by Spivak, and he says that vols. 01 and 02 of "A Comprehensive Introduction to Differential Geometry" are necessary to understand the book.

Are those the only books that contain the necessary knowledge to read it or there are other books that I can study from to read Spivak's Physics book?

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Presumably, any solid introduction to smooth manifolds and differential geometry will do. I'm currently lending my copy of Physics for Mathematicians, Mechanics I to a friend, so I can't say for sure.

I might recommend fellow Math SE user John Lee's Introduction to Smooth Manifolds. However, I would still recommend Spivak here, especially since he wrote the physics book.

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