Differential k-Form

A differential -form is a tensor of tensor rank that is antisymmetric under exchange of any pair of indices. The number of algebraically independent components in dimensions is given by the binomial coefficient . In particular, a one-form (often simply called a "differential") is a quantity

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where , ..., , , ..., are the components of a covariant tensor. Changing variables from to gives

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where

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which is the covariant transformation law.

A -alternating multilinear form on a vector space corresponds to an element of , the th exterior power of the dual vector space to . A differential -form on a manifold is a bundle section of the vector bundle , the th exterior power of the cotangent bundle. Hence, it is possible to write a -form in coordinates by

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where ranges over all increasing subsets of elements from , and the are functions.

An important operation on differential forms, the exterior derivative, is used in the celebrated Stokes' theorem. The exterior derivative of a form is a -form. In fact, by definition, if is the coordinate function, thought of as a zero-form, then .

Another important operation on forms is the wedge product, or exterior product. If is a -form and is -form, then is a form. Also, a -form can be contracted with an -vector, i.e., a bundle section of , to give a -form, or if , an -vector. If the manifold has a metric, then there is an operation dual to the exterior product, called the interior product.

In higher dimensions, there are more kinds of differential forms. For instance, on the tangent space to there is the zero-form 1, two one-forms and , and one two-form . A one-form can be written uniquely as . In four dimensions, is a two-form that cannot be written as .

The minimum number of terms necessary to write a form is sometimes called the rank of the form, usually in the case of a two-form. When a form has rank one, it is called decomposable. Another meaning for rank of a form is its rank as a tensor, in which case a -form can be described as an antisymmetric tensor of rank , in fact of type . The rank of a form can also mean the dimension of its form envelope, in which case the rank is an integer-valued function. With the latter definition of rank, a -form is decomposable iff it has rank .

When is the dimension of a manifold , then is also the dimension of the tangent space . Consequently, an -form always has rank one, and for , a -form must be zero. Hence, an -form is called a top-dimensional form. A top-dimensional form can be form-integrated without using a metric. Consequently, a -form can be integrated on a -dimensional submanifold. Differential forms are a vector space (with a C-infty topology) and therefore have a dual space. Submanifolds represent an element of the dual via integration, so it is common to say that they are in the dual space of forms, which is the space of currents. With a metric, the Hodge star operator defines a map from -forms to -forms such that .

When is a smooth map, it pushes forward manifold tangent vectors from to according to the Jacobian . Hence, a differential form on pulls back to a differential form on .

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The pullback map is a linear map which commutes with the exterior derivative,

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