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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.

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Your understanding is correct. As integration of forms is (or can be) defined by a Riemann integral, there is no theoretical difference between the two. The language of differential forms provides a notational advantage; forms are a lifesaver when we want to do analysis on manifolds, and they allow us to state things like Stokes' theorem in a concise manner. –  Gunnar Magnusson Sep 11 '11 at 15:32
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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. –  user02138 Nov 14 '12 at 16:36
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1 Answer

up vote 2 down vote accepted
  • 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.
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Could you explain, why the Riemann integral is prefered over the Lebesgue integral in this case - or rather, why it is enforced by the structure of the theory? –  Martin Oct 23 '11 at 9:15
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As far I as can see, the Riemann integral is prefered simply because this approach could be used in the vast majority of interesting applications (since objects we deal with in differential geometry are also quite good and regular). There is no enforcement by the theory of differential forms, but we construct the theory of differential forms using the Riemann integral. I believe one could construct analogous theory of some kind of weird $n$-forms using the Lebesgue integral, but I don't know if such a theory already exists and if it's needed. –  Ivan Polekhin Dec 1 '11 at 7:15
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@Martin, Ivan: there is indeed a theory built on Lebesgue integration, called the theory of currents. The Riemann integral and the Lebesgue integral are the same when you consider bounded continuous functions... and in a large part of differential geometry, we consider smooth objects. Nevertheless, once we do analysis on manifolds and need to consider a weaker notion of convergence, we are led to the ideas of currents, singular signed measures and things like this coming from the theory of Lebesgue integrals. –  Sam Lisi Apr 5 '12 at 8:43
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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. –  Sam Lisi Apr 5 '12 at 8:47
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