Differential form
A differential form of degree k is a smooth section of the k-th exterior power of the cotangent bundle of a manifold. At any point p on a manifold, a k-form gives a multilinear map from the k-th cartesian power of the tangent space at p to R.
For example, the differential of a smooth function on a manifold (a 0-form) is a 1-form.
Integration of forms
Differential forms of degree k are integrated over k dimensional chains. If , this is just evaluation of functions at points. Other values of <math>k=1, 2, 3, ... correspond to line integrals, surface integrals, volume integrals etc.
Operations on forms
The set of all k-forms on a manifold is a vector space. Furthermore, there are two other operations: wedge product and exterior derivative. The degree of a wedge product of two forms is the sum of degrees and the degree of the exterior derivative of a k-form is k+1. Forms which are exterior derivatives are called exact and forms, whose exterior derivatives are 0 are called closed. Exact forms are closed, so the vector spaces of k-forms along with the exterior derivative are a cochain complex. Closed forms modulo exact forms are called the de Rham cohomology groups. The wedge product endows the direct sum of these groups with a ring structure.
The fundamental relationship between the exterior derivative and integration is given by the general Stokes theorem, which also provides the duality between de Rham cohomology and the homology of chains.