I was amazed by the power of integration on forms when I learned that the Stokes' theorem can be written in a beautiful way (don't assume that I know more than this fact itself): $ \int_{\Omega}d\omega=\int_{\partial\Omega}\omega. $ from which Green's theorem, the divergence theorem, and the fundamental theorem of calculus follow.
I learned the definition (it might not be the most general one) from Loring W. Tu's An Introduction to Manifolds:
Let $\omega=f(x)dx^1\wedge \cdots\wedge dx^n$ be a $C^{\infty}$ $n$-form on an open subset $U\subset{\mathbb R}^n$, with standard coordinates $x^1,\cdots,x^n$. Its integral over a subset $A\subset U$ is defined to be the Riemann integral of $f(x)$: $ \int_{A}\omega=\int_{A}f(x)dx^1\wedge\cdots\wedge dx^n:=\int_{A}f(x)dx^1\cdots dx^n $ if the Riemann integral exists.
As I understand, since "integration on forms" is defined by the Riemann integral, it does not provide a new kind of integrals (e.g. Lebesgue integrals in measure theory, Itō integrals in stochastic analysis, etc.). Instead of doing so, it provides a new view of Riemann integral, in which for example $f(x)dx$ has its new meaning, $1$-form.
Here are my questions:
- Is what I understand above correct? Or what's the fundamental difference between these two kinds of integrals?
- Can I say that this new integration provides a new way to prove the theorems in Riemann integral theory?
- [EDIT: Is the above definition the only way to define "integration on forms"?]
I feel that my questions might be vague. Any suggestions to improve it will be really appreciated.