Differential form

In differential geometry, a differential form of degree k is a smooth section of the kth 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.

1-forms are a particularly useful basic concept in the coordinate-free treatment of tensors. In this context, they can be defined as real-valued linear functions of vectors, and they can be seen to create a dual space with regard to the vector space of the vectors they are defined over. An older name for 1-forms in this context is "covariant vectors".

Integration of forms

Differential forms of degree k are integrated over k dimensional chainss. If , this is just evaluation of functions at points. Other values of correspond to line integrals, surface integrals, volume integrals etc.

See also Stokes' theorem.

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. d²=0, see de Rham cohomology for more details.

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.