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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.

  • 1
    One should be careful here. Depending on the source, differentials may commute in the Riemann integral, but not as differential forms. That is, $dx \wedge dy = - dy \wedge dx$ is always true, but some authors take $dx \ dy = dy \ dx$, as the order of integration is irrelevant.2012-11-14

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  • Yes, your are right.
  • I think it's better to say that the theorems in Riemann integral theory (Green's theorem, Gauss–Ostrogradsky theorem etc.) are rather theorems on integrating of differential forms.
  • I don't think that your definition is comprehensive, but generally there is only one way to define integration of forms and it's to reduce integration of a form to Riemann integral.
  • 1
    I would say that the deepest observation from Stokes' theorem is that in an$n$dimensional oriented manifold, a k-dimensional oriented submanifold is, in some sense, dual to a k-form (by integrating the k form over the submanifold). You also have duality between k-forms and n-k-forms (by wedging them together and integrating over the total manifold). From this, you are led to consider a form that ``represents'' a submanifold. This is the basic idea behind Poincaré duality, a key idea in topology and geometry. A beautiful but difficult exposition of some of these ideas is in Bott & Tu.2012-04-05