Differential form
In differential geometry, 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.
Differential forms of degree k are integrated over k dimensional chains. If <math>k=0</math>, this is just evaluation of functions at points.
Other values of <math>k=1, 2, 3, ...</math> correspond to line integrals, surface integrals, volume integrals etc.
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 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.
Integration of forms
Operations on forms