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I want to get used to differential forms. Thus I would like to solve a bunch of problems, especially on integration of differential forms.

So I need a collection of problems with answers/solutions, starting from really elementary ones. No theorem proving, just straightforward calculations.

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This is a textbook, not a problem book, but perhaps try the exercises in Hubbard & Hubbard, Vector Calculus, Linear Algebra, and Differential Forms. – Neal Jul 1 '12 at 16:45
I would recommend the book of Do Carmo "Differential Forms and Applications". Personally I first learn differential form from this book, and I did all the exercises which I benefit a lot. – Paul Jul 1 '12 at 18:35
up vote 5 down vote accepted

You may like Chapter 10 "Differential Forms, Integral Formulae, De-Rham Cohomology" of Mishchenko, Solovyev, Fomenko "Problems in Differential Geometry and Topology" (English translation, Mir Publishers, 1985). I have not seen a newer version of it that may be even better: A.T. Fomenko, A.S. Mischenko, Y.P. Solovyev: Selected problems in differential geometry and topology. Cambridge Scientific Publishers, Cambridge, 2006

Update 1. There is also a very interesting collection of problems by Prof. W.-H. Steeb. See Ch.4 Differential Forms and Applications in "Problems and Solutions in Differential Geometry"

Update 2. A comprehensive set of problems on differential geometry can be found in Analysis and Algebra on Differentiable Manifolds: A Workbook for Students, by P. M. Gadea, J. Munoz Masqué, see Ch.2 "Tensor Fields and Differential Forms".

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