# What is the theory of non-linear forms (as contrasted to the theory of differential forms)?

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