# Forms

##### Basic Definitions

A p-form is a type (0,p) tensor that is totally antisymmetric:

The square brackets denote antisymmetrization; for example, . The exterior product (wedge product) of a p-form and a q-form is a p+q form:

If and are both 1-forms,

The exterior derivative of a p-form is a p+1 form defined by

Any derivative operator can be used in place of the coordinate derivative . Note that . If is a p-form and is a q-form,

where the plus sign applies if p is even, the minus sign applies if p is odd.

##### INTEGRATION

If and are both 1-forms, then or, in index-free notation,

Let , , etc. denote right handed coordinates on an n-dimensional manifold. Thus are 1-forms and we have . Extending to a multiple product of 1-forms,

Expand the general p-form in the coordinate basis:

For an n-form in particular,

The integral of the n-form is defined by

A volume element is a nonvanishing n-form . It can be used to define the integral of a function by

##### STOKES’ THEOREM

Let be a right handed coordinate system on an n-dimensional manifold with boundary. Let define the exterior. Then is a right handed coordinate system on the n-1 dimensional boundary. Stokes theorem says that for an n-1 form ,

##### LEVI-CIVITA TENSOR IN 4D

Let be a 4-dimensional spacetime manifold with metric and signature . The natural volume element is the Levi-Civita tensor defined by

where is the determinant of the metric and the coordinates are with . This volume element has components

and it satisfies

(1)

For an element of , the 3-dimensional boundary of , the natural volume element is defined by

where is the outward pointing unit normal. If follows that

where depending on whether is spacelike or timelike.

##### LEVI-CIVITA TENSOR IN 3D

Let be a 3-dimensional spacetime manifold with metric and signature . The natural volume element is the Levi-Civita tensor defined by

where is the determinant of the metric and the coordinates are with . This volume element has components

and it satisfies

(2)

##### STOKES’ THEOREM AGAIN

In terms of natural volume elements, Stokes’ theorem reduces to

where is the covariant derivative compatible with the metric and is a contravariant vector. The integral on the right-hand side must be summed over all elements of the boundary. This relation is written in coordinates as

where is the determinant of the induced metric on .

##### STOKES’ THEOREM IN TWO DIMENSIONS

Let the dimension of the manifold be n=2. Assume the metric has signature ++. Let denote the Levi–Civita tensor on and write the vector field in terms of a covector field as . In this case Stokes’ theorem reduces to

This is what Jackson calls “Stokes’ theorem”. In this context, the two dimensional manifold is a surface embedded in three dimensional space and the two dimensional Levi-Civita tensor is equal to the three dimensional Levi-Civita tensor with one index contracted with the unit normal to the surface. The result above relates the curl of a vector field, integrated over a surface, to the circulation on the boundary.