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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.

 Integration of forms

Differential forms of degree k are integrated over k dimensional chains. If [itex]k=0[/itex], this is just evaluation of functions at points. Other values of [itex]k=1, 2, 3, ...[/itex] 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 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.