Forms

Basic Definitions

A p-form is a type (0,p) tensor \omega = \omega_{\alpha_1\cdots \alpha_p} dx^{\alpha_1} \otimes \cdots \otimes dx^{\alpha_p} that is totally antisymmetric:

    \[ \omega_{\alpha_1 \cdots \alpha_p} = \omega_{[\alpha_1 \cdots \alpha_p]} \]

The square brackets denote antisymmetrization; for example, \omega_{[\alpha\beta]} \equiv (\omega_{\alpha\beta} - \omega_{\beta\alpha})/2. The exterior product (wedge product) of a p-form and a q-form is a p+q form:

    \[ (\omega \wedge \mu)_{\alpha_1\cdots \alpha_p \beta_1\cdots \beta_q} \equiv \frac{(p+q)!}{p!q!} \omega_{[\alpha_1\cdots \alpha_p} \mu_{\beta_1\cdots \beta_q]} \]

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

    \[ (\omega\wedge\mu)_{\alpha\beta} = 2\omega_{[\alpha}\mu_{\beta]} = \omega_\alpha \mu_\beta - \omega_\beta \mu_\alpha \]

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

    \[ (d\omega)_{\beta\alpha_1\cdots \alpha_p} \equiv (p+1)\partial_{[\beta} \omega_{\alpha_1\cdots \alpha_p]} \]

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

If \omega = \omega_\alpha dx^\alpha and \mu = \mu_\alpha dx^\alpha are both 1-forms, then (\omega\wedge \mu)_{\alpha\beta} = 2 \omega_{[\alpha}\mu_{\beta]} or, in index-free notation,

    \[ \omega\wedge\mu = \omega\otimes \mu - \mu\otimes \omega \]

Let x^1, x^2, etc. denote right handed coordinates on an n-dimensional manifold. Thus dx^\alpha are 1-forms and we have dx^\alpha \wedge dx^\beta = dx^\alpha\otimes dx^\beta - dx^\beta\otimes dx^\alpha. Extending to a multiple product of 1-forms,

    \[ dx^{\alpha_1}\wedge dx^{\alpha_2} \wedge \cdots \wedge dx^{\alpha_p} = p!\, dx^{[\alpha_1}\otimes dx^{\alpha_2} \otimes \cdots \otimes dx^{\alpha_p]} \]

Expand the general p-form \omega in the coordinate basis:

    \[ \omega = \omega_{\alpha_1\cdots \alpha_p} dx^{\alpha_1}\otimes \cdots \otimes dx^{\alpha_p} =\frac{1}{p!} \omega_{\alpha_1\cdots \alpha_p} dx^{\alpha_1}\wedge \cdots \wedge dx^{\alpha_p} \]

For an n-form in particular,

    \[ \omega = \omega_{1\cdots n} \, dx^1 \wedge \cdots \wedge dx^n \]

The integral of the n-form is defined by

    \[ \int \omega = \int \omega_{1\cdots n}\, dx^1 \cdots dx^n \equiv \int \omega_{1\cdots n} \, d^nx \]

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^10 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 \omega,

    \[ \int_{\mathcal M} d\omega = \int_{\partial{\mathcal M}} \omega \]

LEVI-CIVITA TENSOR IN 4D

Let {\mathcal M} be a 4-dimensional spacetime manifold with metric g_{\mu\nu} and signature -+++. The natural volume element is the Levi-Civita tensor defined by

    \[ \epsilon \equiv \sqrt{-g} dx^0\wedge \cdots \wedge dx^3 \]

where g is the determinant of the metric and the coordinates are x^\mu with \mu = 0,1,2,3. This volume element has components

    \[ \epsilon_{\mu\nu\alpha\beta} = \left\{\begin{array}{ll} \pm\sqrt{-g} & if\ \mu\nu\alpha\beta\ is\ an\ even/odd\ permutation\ of\ 0123 \\ 0 & otherwise \end{array} \]

and it satisfies

(1)   \begin{eqnarray*} \epsilon_{\alpha_1\alpha_2\alpha_3\alpha_4} \epsilon^{\alpha_1\alpha_2\alpha_3\alpha_4} & = & -24 \, \\ \epsilon_{\alpha_1\alpha_2\alpha_3\alpha_4} \epsilon^{\alpha_1\alpha_2\alpha_3\beta} & = & -6 \, \delta_{\alpha_4}^{\beta_4} \\ \epsilon_{\alpha_1\alpha_2\alpha_3\alpha_4} \epsilon^{\alpha_1\alpha_2\beta_3\beta_4} & = & -4 \, \delta_{[\alpha_3}^{\beta_3} \delta_{\alpha_4]}^{\beta_4} \\ \epsilon_{\alpha_1\alpha_2\alpha_3\alpha_4} \epsilon^{\alpha_1\beta_2\beta_3\beta_4} & = & -6 \, \delta_{[\alpha_2}^{\beta^2}\delta_{\alpha_3}^{\beta_3} \delta_{\alpha_4]}^{\beta_4} \\ \epsilon_{\alpha_1\alpha_2\alpha_3\alpha_4} \epsilon^{\beta_1\beta_2\beta_3\beta_4} & = & -24 \, \delta_{[\alpha_1}^{\beta_1}\delta_{\alpha_2}^{\beta_2} \delta_{\alpha_3}^{\beta_3}\delta_{\alpha_4]}^{\beta_4} \end{eqnarray*}

For an element of \partial{\mathcal M}, the 3-dimensional boundary of {\mathcal M}, the  natural volume element  is defined by

    \[ \tilde\epsilon_{\alpha_2\cdots \alpha_n} = n^{\alpha_1}\epsilon_{\alpha_1\cdot \alpha_n} \]

where n^\alpha is the outward pointing unit normal. If follows that

    \[  \epsilon_{\alpha_1\cdots \alpha_n} = 4 (n\cdot n) n_{[\alpha_1}\tilde\epsilon_{\alpha_2\cdots \alpha_n]} \]

where n\cdot n = \pm 1 depending on whether n^\alpha is spacelike or timelike.

LEVI-CIVITA TENSOR IN 3D

Let {\Sigma} be a 3-dimensional spacetime manifold with metric g_{ij} and signature +++. The natural volume element is the Levi-Civita tensor defined by

    \[ \epsilon \equiv \sqrt{g} dx^1\wedge \cdots \wedge dx^3 \]

where g is the determinant of the metric and the coordinates are x^i with i = 1,2,3. This volume element has components

    \[ \epsilon_{ijk} = \left\{\begin{array}{ll} \pm\sqrt{g} & if\ ijk \ is\ an\ even/odd\ permutation\ of\ 123 \\ 0 & otherwise \end{array} \]

and it satisfies

(2)   \begin{eqnarray*} \epsilon^{ijk}\epsilon_{\ell mn} & = & 6 \delta^i_{[\ell} \delta^j_m \delta^k_{n]} \\ \epsilon^{ijk}\epsilon_{imn} & = & 2\delta^j_{[m} \delta^k_{n]} \\ \epsilon^{ijk}\epsilon_{ijn} & = & 2\delta^k_n \\ \epsilon^{ijk}\epsilon_{ijk} & = & 6  \end{eqnarray*}

STOKES’ THEOREM AGAIN

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

    \[ \int_{{\mathcal M}} \epsilon \, \nabla_\alpha V^\alpha = (n\cdot n)\int_{\partial{\mathcal M}} \tilde\epsilon \, n_\alpha V^\alpha \]

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

    \[ \int_{{\mathcal M}} d^nx \sqrt{-g} \nabla_\alpha V^\alpha = (n\cdot n) \int_{\partial{\mathcal M}} d^{n-1}x \sqrt{|h|} n_\alpha V^\alpha \]

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_{\mu\nu} denote the Levi–Civita tensor on {\mathcal M} and write the vector field V^\alpha in terms of a covector field W_\alpha as V^\alpha = \epsilon^{\alpha\beta}W_\beta. In this case Stokes’ theorem reduces to

    \[ \int_{{\mathcal M}} d^2x \sqrt{g} \epsilon^{\alpha\beta} \nabla_\alpha W_\beta = \int_{\partial{\mathcal M}} dx \sqrt{h} n_\alpha \epsilon^{\alpha\beta} W_\beta \]

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.