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In Spivak's Calculus on Manifolds, he defines the integral of a $k$-form over a $k$-chain, and proves a version of Stokes' theorem for this situation, before moving on to discuss the integral of a differential form over a manifold. Other books, such as Introduction to Smooth Manifolds by Lee and Analysis on Manifolds by Munkres, seem to skip this step. They define the integral of a $k$-form over a $k$-manifold without mentioning or proving theorems about $k$-chains. (Am I correct about that?)

What is the advantage (if any) of Spivak's approach? To me, $k$-chains appear to be an unnecessary weird definition in a subject that already has too many unfamiliar definitions.

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    $\begingroup$ Most likely, for the same reason I explained here. $\endgroup$ Commented Jun 16, 2020 at 3:34
  • $\begingroup$ @MoisheKohan Thanks for that explanation, that was enlightening. $\endgroup$ Commented Aug 29, 2020 at 19:36

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In Lee the integral is only defined for an n-form on an n-manifold. If you want to integrate a lower dimensional k-form (k < n), you need to use a submanifold of dimension k and pull the form back with inclusion. I believe one purpose of chains is to avoid talking about submanifolds and orientations of manifold boundaries. Instead We can integrate lower dimensional forms in a higher dimensional space using singular chains.

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