Differential Form - Integration

Integration

Differential forms of degree k are integrated over k dimensional chains. If k = 0, this is just evaluation of functions at points. Other values of k = 1, 2, 3, ... correspond to line integrals, surface integrals, volume integrals etc. Simply, a chain parametrizes a domain of integration as a collection of cells (images of cubes or other domains D) that are patched together; to integrate, one pulls back the form on each cell of the chain to a form on the cube (or other domain) and integrates there, which is just integration of a function on as the pulled back form is simply a multiple of the volume form For example, given a path integrating a form on the path is simply pulling back the form to a function on (properly, to a form ) and integrating the function on the interval.

Let

be a differential form and S a differentiable k-manifold over which we wish to integrate, where S has the parameterization

for u in the parameter domain D. Then (Rudin 1976) defines the integral of the differential form over S as

where

is the determinant of the Jacobian. The Jacobian exists because S is differentiable.

More generally, a -form can be integrated over an -dimensional submanifold, for, to obtain a -form. This comes up, for example, in defining the pushforward of a differential form by a smooth map by attempting to integrate over the fibers of .