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# 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.