structural:summarization, information:classification
Relation with measures

On a general differentiable manifold (without additional structure), differential forms cannot be integrated over subsets of the manifold; this distinction is key to the distinction between differential forms, which are integrated over chains or oriented submanifolds, and measures, which are integrated over subsets. The simplest example is attempting to integrate the 1 -form dx over the interval. Assuming the usual distance (and thus measure) on the real line, this integral is either 1 or − 1, depending on orientation: ∫ 0 1 d x = 1 {\displaystyle \textstyle {\int _{0}^{1}dx=1}}, while ∫ 1 0 d x = − ∫ 0 1 d x = − 1 {\displaystyle \textstyle {\int _{1}^{0}dx=-\int _{0}^{1}dx=-1}}. By contrast, the integral of the measure | dx | on the interval is unambiguously 1 (i.e. the integral of the constant function 1 with respect to this measure is 1). Similarly, under a change of coordinates a differential n -form changes by the Jacobian determinant J, while a measure changes by the absolute value of the Jacobian determinant, | J |, which further reflects the issue of orientation. For example, under the map x ↦ − x on the line, the differential form dx pulls back to − dx; orientation has reversed; while the Lebesgue measure, which here we denote | dx |, pulls back to | dx |; it does not change.

In the presence of the additional data of an orientation, it is possible to integrate n -forms (top-dimensional forms) over the entire manifold or over compact subsets; integration over the entire manifold corresponds to integrating the form over the fundamental class of the manifold,. Formally, in the presence of an orientation, one may identify n -forms with densities on a manifold; densities in turn define a measure, and thus can be integrated (Folland 1999, Section 11.4, pp. 361 – 362).

On an orientable but not oriented manifold, there are two choices of orientation; either choice allows one to integrate n -forms over compact subsets, with the two choices differing by a sign. On non-orientable manifold, n -forms and densities cannot be identified —notably, any top-dimensional form must vanish somewhere (there are no volume forms on non-orientable manifolds), but there are nowhere-vanishing densities— thus while one can integrate densities over compact subsets, one cannot integrate n -forms. One can instead identify densities with top-dimensional pseudoforms.

Even in the presence of an orientation, there is in general no meaningful way to integrate k -forms over subsets for k < n because there is no consistent way to use the ambient orientation to orient k -dimensional subsets. Geometrically, a k -dimensional subset can be turned around in place, yielding the same subset with the opposite orientation; for example, the horizontal axis in a plane can be rotated by 180 degrees. Compare the Gram determinant of a set of k vectors in an n -dimensional space, which, unlike the determinant of n vectors, is always positive, corresponding to a squared number. An orientation of a k -submanifold is therefore extra data not derivable from the ambient manifold.

On a Riemannian manifold, one may define a k -dimensional Hausdorff measure for any k (integer or real), which may be integrated over k -dimensional subsets of the manifold. A function times this Hausdorff measure can then be integrated over k -dimensional subsets, providing a measure-theoretic analog to integration of k -forms. The n -dimensional Hausdorff measure yields a density, as above.