# Forms

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## Contents |

#### Basic Definitions

Test edit, again, and again

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

Often the indices are dropped. 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

Let etc. denote right handed coordinates on an n-dimensional manifold. Note:

Expand the p-form in the coordinate basis:

Without indices,

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 interior and 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 ,

#### Volume Element

If is an dimensional manifold with metric , there is a natural volume element defined by

where is the determinant of the metric. This volume element satisfies the following:

where is the signature (the sign of ).

For a manifold with boundary, the natural volume element on the boundary is defined by

where is the outward pointing unit normal. It follows that

where .

#### Stokes' Theorem Again

In terms of the 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 . 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.