So, reading both Baby Rudin (Principles of Mathematical Analysis) and Munkres (Analysis on Manifolds), I start to wonder the difference between this two approaches. In Rudin's, he defines the Stokes Theorem in chains: Let $\Psi$ a $k$-chain of class $C''$ in an open set $V \subset \mathbb{R}^n$ and $\omega$ a $(k - 1)$-form of class $C'$ in $V$. Then: $$\int_{\Psi} d\omega = \int_{\partial \Psi} \omega.$$ While in Munkres, the definition is on a oriented manifold: Let $k>1$ and let $M$ be a compact oriented $k$-manifold in $\mathbb{R}^n$; give $\partial M$ the induced orientation if $\partial M \neq \emptyset$. Let $\omega$ be a $(k - 1)$-form defined in an open set of $\mathbb{R}^n$ containing $M$.
Then, if $\partial M = \emptyset$, $$\int_{\partial M} \omega = 0.$$ If $\partial M\neq\emptyset$, $$\int_M d\omega = \int_{\partial M} \omega.$$ I don't have the mathematical maturity yet to understand both statements completely (I have a litle knowledge of what a manifold and a differential form is, but I don't know very well what a chain happens to be), but, in case of future studies, what is the difference between these? Which one is more general?
Searching about integration on chains, I came across some definitions of algebraic topology, such as homology and simplexes. What is the relation between the Stokes theorem and these concepts?
Sorry about the long post and such "disjoint" questions, but this different approaches poked up my curiosity. Thanks in advance!
The definition in Rudin's book is a bit general since it is known by Whitehead in 1939 that any smooth manifolds can be triangulated, i.e. $M$ is the union of several simplices which only intersects at boundaries.
In fact, if we want to define the integration on a general manifold(without mentioning the underlying Euclidean space), we need to take coordinate charts and integrate the differential form in each of these charts, and in particular we could pick the simplices with an embedding into the manifold and integrate the differential form on this simplex by pulling it back to the local chart and doing the integration just as usual, and sum them up to obtain a "global" integration.