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 chains. 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.
Referenced By
Chern-Simons | Chern-Simons 1-form | Chern-Simons 3-form | Chern-Simons 5-form | Chern-Simons form | Chern-Simons theory | Gauge curvature | Gauge field | Gauge theory | Gauge transformation | List of differential geometry topics | List of mathematical topics (D-F) | List of mathematical topics (F-Z) | Poisson bracket | Quantum Yang-Mills theory | Symplectic geometry | Symplectic space
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