I'm studying Spivak's Calculus on Manifolds. There the integration of $k$-forms on $k$-chains is defined and Stokes' theorem on chains is also proved, but they are never used for other theorems. I also cannot imagine where to use it in applications or maths. Why do we need to define $k$-chains? Where can I use it?
A related question, but not answered, is here:
What's the advantage of defining the integral of a $k$-form over a $k$-chain?