Forms

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Basic Definitions

Test edit, again, and again

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

\omega_{a_1 \cdots a_p} = \omega_{[a_1 \cdots a_p]}

Often the indices are dropped. The exterior product (wedge product) of a p-form and a q-form is a p+q form:

(\omega \wedge \mu)_{a_1\cdots a_p b_1\cdots b_q} \equiv \frac{(p+q)!}{p!q!} \omega_{[a_1\cdots a_p} \mu_{b_1\cdots b_q]}

If \omega and \mu are both 1-forms,

(\omega\wedge\mu)_{ab} = 2\omega_{[a}\mu_{b]} = \omega_a \mu_b - \omega_b \mu_a

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

(d\omega)_{ba_1\cdots a_p} \equiv (p+1)\partial_{[b} \omega_{a_1\cdots a_p]}

Any derivative operator can be used in place of the coordinate derivative \partial_b. Note that dd\omega = 0. If \omega is a p-form and \mu is a q-form,

d(\omega\wedge\mu) = d\omega \wedge \mu \pm \omega\wedge d\mu

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

Integration

Let x^1, x^2, etc. denote right handed coordinates on an n-dimensional manifold. Note:

(dx^{\mu_1})_{[a_1} (dx^{\mu_2})_{a_2} \cdots (dx^{\mu_p})_{a_p]} = \frac{1}{p!} (dx^{\mu_1} \wedge dx^{\mu_2} \wedge \cdots \wedge dx^{\mu_p})_{a_1 a_2\cdots a_p}

Expand the p-form \alpha_{a_1\cdots a_p} in the coordinate basis:

\begin{array}{rcl} \alpha_{a_1\cdots a_p} & = & \alpha_{\mu_1\cdots \mu_p} (dx^{\mu_1})_{a_1} \cdots (dx^{\mu_p})_{a_p} \\ & = & \frac{1}{p!} \alpha_{\mu_1\cdots \mu_p} (dx^{\mu_1} \wedge \cdots \wedge dx^{\mu_p})_{a_1 \cdots a_p} \end{array}

Without indices,

\alpha = \frac{1}{p!} \alpha_{\mu_1\cdots \mu_p} \, dx^{\mu_1} \wedge \cdots \wedge dx^{\mu_p}

For an n-form in particular,

\alpha = \alpha_{12\cdots n} \, dx^1\wedge dx^2 \wedge \cdots \wedge dx^n

The integral of the n-form is defined by

\int \alpha = \int \alpha_{1\cdots n}\, dx^1 \cdots dx^n

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

\int \epsilon \, f

Stokes' Theorem

Let x^1,\ldots x^n be a right handed coordinate system on an n-dimensional manifold {\mathcal M} with boundary. Let x^1<0 define the interior and x^1>0 define the exterior. Then x^2\ldots x^n is a right handed coordinate system on the n-1 dimensional boundary. Stokes theorem says that for an n-1 form \alpha,

\int_{\mathcal M} d\alpha = \int_{\partial{\mathcal M}} \alpha

Volume Element

If {\mathcal M} is an n dimensional manifold with metric g_{\mu\nu}, there is a natural volume element defined by

\epsilon \equiv \sqrt{|g|} dx^1\wedge \cdots \wedge dx^n

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

\begin{array}{rcl} \epsilon_{1\cdots n} & = & \sqrt{|g|} \\ \epsilon^{a_1\cdots a_n} \epsilon_{a_1\cdots a_n} & = & \varepsilon n! \\ \epsilon^{a_1\cdots a_n} \epsilon_{b_1\cdots b_n} & = & \varepsilon n! \delta^{a_1}_{[b_1} \cdots \delta^{a_n}_{b_n]} \\ \epsilon^{a_1\cdots a_j a_{j+1}\cdots a_n} \epsilon_{a_1\cdots a_j b_{j+1} \cdots b_n} & = & \varepsilon (n-j)! j! \delta^{a_{j+1}}_{[b_{j+1}} \cdots \delta^{a_n}_{b_n]} \end{array}

where \varepsilon is the signature (the sign of g).


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

\tilde\epsilon_{a_2\cdots a_n} \equiv n^{a_1} \epsilon_{a_1\cdots a_n}

where n^a is the outward pointing unit normal. It follows that

\epsilon_{a_1\cdots a_n} = s n n_{[a_1} \tilde\epsilon_{a_2\cdots a_n]}

where s\equiv n^a n_a.

Stokes' Theorem Again

In terms of the natural volume elements, Stokes' theorem reduces to

\int_{{\mathcal M}} \epsilon \, \nabla_a V^a = s\int_{\partial{\mathcal M}} \tilde\epsilon \, n_a V^a

where \nabla_a is the covariant derivative compatible with the metric and V^a 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

\int_{{\mathcal M}} d^nx \sqrt{|g|} \nabla_a V^a = s \int_{\partial{\mathcal M}} d^{n-1}x \sqrt{|h|} n_a V^a

where h is the determinant of the induced metric on \partial{\mathcal M}.

Stokes' Theorem in Two Dimensions

Let the dimension of the manifold be n=2. Assume the metric has signature ++. Let \epsilon^{ab} denote the Levi--Civita tensor on {\mathcal M} and write the vector field V^a in terms of a covector field W_a as V^a = \epsilon^{ab}W_b. In this case Stokes' theorem reduces to

\int_{{\mathcal M}} d^2x \sqrt{g} \epsilon^{ab} \nabla_a W_b = \int_{\partial{\mathcal M}} dx \sqrt{h} n_a \epsilon^{ab} W_b

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.

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