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