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It is often said that differential forms (sections of an exterior power of the cotangent bundle) are the things that you can integrate. But unless I'm being thoroughly dense differential forms are not the only things that you can integrate, c.f. the arclength form (on a 2d manifold) $ds=\sqrt{dx^2+dy^2}$, the unsigned 1-d forms $|f(x,y)dx+g(x,y)dy|$, or the unsigned area forms $|h(x,y)dx\wedge dy|$.

My question is:

Where do the arclength form $ds=\sqrt{dx^2+dy^2}$, the unsigned 1-d forms |f(x,y)dx+g(x,y)dy|, and the unsigned area forms $|h(x,y)dx\wedge dy|$ live relative to the differentials $dx$ and $dy$, which I understand to live in the cotangent bundle of some 2-dimensional manifold?

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That's funny; I thought measurable functions were the things you can integrate... – Qiaochu Yuan Aug 22 '10 at 21:36
@Qiaochu: evidently, there's more than one kind of thing you can integrate. – Pete L. Clark Aug 22 '10 at 22:11
The notation used in the right hand side of «$ds=\sqrt{dx^2+dy^2}$» is just a notation; in particular, it is not something that is built out of $dx$ and $dy$... – Mariano Suárez-Alvarez Aug 23 '10 at 2:00
@Mariano, I understand ds as a continuous function on the tangent space at a point. My limited understanding tells me that it is a non-linear form because c ds(v)=ds(c v) for positive constants c. I suspect that if you apply 'positively' homogeneous function of degree 1 in n variables to (dx_1, dx_2,..., dx_n), you would get a form. – Vladimir Sotirov Aug 24 '10 at 18:22
I guess these non-linear forms are taken from David Bachmann's book "A Geometric Approach to Differential Forms", aren't they? – shuhalo Feb 22 '12 at 9:57
up vote 23 down vote accepted

The answer to "what kinds of things can you integrate" depends on the context.

  • Measurable functions are things you can integrate over measure spaces, which includes in particular measurable subsets of R^n.
  • Differential forms are things you can integrate over oriented smooth manifolds -- the key thing about them is that their integrals are invariant under smooth, orientation-preserving changes of coordinates.
  • Densities are things that can be integrated in a coordinate-independent way on any smooth manifold, regardless of whether it has an orientation or not.
  • Coming full circle, every Riemannian manifold (i.e., smooth manifold endowed with a Riemannian metric) has a naturally-defined density dV, so in that context you can integrate measurable functions again: the integral of the function f is defined to be the integral of the density f dV.

All three of the expressions you asked about are examples of densities. For details, see my book Introduction to Smooth Manifolds, pp. 375-382.

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