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I am stucking with theorem 3.13 p67 Calculus on manifold of M Spivak. I don't understand how he obtains $\int_{g(U)} \phi. f=\int_{U}(\phi. f)(g)|\det g'|$. And which books should i read about partition of unity and proof of changing variable in multiple integral apart from Spivak one? Thanks in advanced for you help!

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Please be more specific. You might want to take a look at Analysis on Manifolds by Munkres. –  wj32 Oct 24 '12 at 10:25

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