Encyclopedia > Differential form

  Article Content

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 <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.

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.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Johann Karl Friedrich Rosenkranz

... he was quite blind. Throughout his long professorial career, and in all his numerous publications he remained, in spite of occasional deviations on particular points, ...

 
 
 
This page was created in 34.9 ms