# Forms

## Contents

#### 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.