Exterior forms as anti-symmetric arrays

In terms of a basis \({\beta^{\mu}}\) of \({V^{*}}\), we can write a \({k}\)-form \({\varphi}\) as

\(\displaystyle \varphi=\frac{1}{k!}\sum_{\mu_{1},\dotsc,\mu_{k}}\varphi_{\mu_{1}\dots\mu_{k}}\beta^{\mu_{1}}\wedge\dotsb\wedge\beta^{\mu_{k}}.\)

Δ The above way of writing the components is not unique, and others are in common use, the main alternative omitting the factorial.

The advantage of the expression above is that, with our isomorphism convention, the component array can be identified with the anti-symmetric covariant tensor component array in the same basis:

\(\displaystyle \varphi\mapsto\frac{1}{k!}\varphi_{\mu_{1}\dots\mu_{k}}\sum_{\pi}\textrm{sign}\left(\pi\right)\bigotimes_{i}\beta^{\pi\left(i\right)}=\varphi_{\mu_{1}\dots\mu_{k}}\beta^{\mu_{1}}\otimes\cdots\otimes\beta^{\mu_{k}}\)

Here we have dropped the summation sign in favor of the Einstein summation convention, and the last equality follows from the anti-symmetry of the component array. This means that as with tensors, in terms of the \({k}\)-form as a multilinear mapping we have

\(\displaystyle \varphi_{\mu_{1}\dots\mu_{k}}=\varphi\left(e_{\mu_{1}},\dotsb,e_{\mu_{k}}\right).\)

In particular, for an \({n}\)-form we have

\begin{aligned}\varphi & =\varphi_{1\dots n}\beta^{1}\wedge\dotsb\wedge\beta^{n}\\
& =\varphi_{\mu_{1}\dots\mu_{n}}\beta^{\mu_{1}}\otimes\cdots\otimes\beta^{\mu_{n}}\\
\Rightarrow\varphi\left(e_{\mu_{1}},\dotsb,e_{\mu_{n}}\right) & =\varphi_{\mu_{1}\dots\mu_{n}}.
\end{aligned}

Since exterior forms are built from only the dual space \({V^{*}}\), in this context we will also use the symbol \({\Omega}\) to refer to the unit \({n}\)-form. In an arbitrary basis we can then write

\begin{aligned}\Omega & =\sqrt{\left|\det\left(g\right)\right|}\:\beta^{1}\wedge\dotsb\wedge\beta^{n}\\
& =\sqrt{\left|\det\left(g\right)\right|}\:\varepsilon_{1\ldots n}\beta^{1}\wedge\dotsb\wedge\beta^{n}\\
& =\sqrt{\left|\det\left(g\right)\right|}\:\varepsilon_{\mu_{1}\ldots\mu_{n}}\beta^{\mu_{1}}\otimes\cdots\otimes\beta^{\mu_{n}},
\end{aligned}

where \({\epsilon_{\mu_{1}\ldots\mu_{n}}\equiv\sqrt{\left|\det\left(g\right)\right|}\:\varepsilon_{\mu_{1}\ldots\mu_{n}}}\) is therefore the array of a tensor, sometimes called the Levi-Civita tensor.

The component array expression for the exterior product of a \({j}\)-form \({\varphi}\) and a \({k}\)-form \({\psi}\) is then

\(\displaystyle \left(\varphi\wedge\psi\right)_{\mu_{1}\cdots\mu_{j+k}}=\cfrac{1}{j!k!}\varphi_{\nu_{1}\cdots\nu_{j}}\psi_{\nu_{j+1}\cdots\nu_{j+k}}\delta_{\mu_{1}\cdots\mu_{j+k}}^{\nu_{1}\cdots\nu_{j+k}},\)

where the generalized Kronecker delta

\(\displaystyle \delta_{\mu_{1}\cdots\mu_{k}}^{\nu_{1}\cdots\nu_{k}}\equiv\sum_{\pi}\textrm{sign}\left(\pi\right)\delta_{\mu_{1}}^{\nu_{\pi\left(1\right)}}\cdots\delta_{\mu_{k}}^{\nu_{\pi\left(k\right)}}\)

gives the sign of the permutation of upper versus lower indices and vanishes if they are not permutations or have a repeated index. In particular, for two 1-forms we have

\(\displaystyle \left(\varphi\wedge\psi\right)_{\mu\nu}=\varphi_{\mu}\psi_{\nu}-\varphi_{\nu}\psi_{\mu}.\)

Δ A potential source of confusion is that using abstract index notation one may write \({\varphi_{a}\wedge\psi_{b}}\), but \({\left(\varphi_{a}\wedge\psi_{b}\right)v^{a}w^{b}\neq\varphi_{a}v^{a}\wedge\psi_{b}w^{b}=\varphi_{a}v^{a}\psi_{b}w^{b}}\).

The component expression for the inner product of two \({k}\)-forms is

\begin{aligned}\left\langle \varphi,\psi\right\rangle _{\textrm{form}} & =\frac{1}{k!}\varphi_{\mu_{1}\dots\mu_{k}}\psi^{\mu_{1}\dots\mu_{k}},\end{aligned}

and that of the Hodge star of a \({k}\)-form is

\begin{aligned}\left(*\varphi\right)_{\mu_{1}\dots\mu_{n-k}} & =\frac{1}{k!\sqrt{\left|\det\left(g\right)\right|}}\varepsilon^{\nu_{1}\cdots\nu_{n}}\varphi_{\nu_{1}\cdots\nu_{k}}g_{\mu_{1}\nu_{k+1}}\cdots g_{\mu_{n-k}\nu_{n}}\\\Rightarrow\left(*\varphi\right)_{\mu_{k+1}\dots\mu_{n}} & =\frac{\left(-1\right)^{s}\sqrt{\left|\det\left(g\right)\right|}}{k!}\varphi^{\mu_{1}\cdots\mu_{k}}\varepsilon_{\mu_{1}\cdots\mu_{n}}.\end{aligned}

In particular, for an \({n}\)-form and a \({0}\)-form we have

\begin{aligned}*\varphi & =\frac{1}{\sqrt{\left|\det\left(g\right)\right|}}\varphi_{1\cdots n},\\
\left(*\varphi\right)_{1\dots n} & =\left(-1\right)^{s}\sqrt{\left|\det\left(g\right)\right|}\varphi.
\end{aligned}

Δ Recall that some authors define the Hodge star by the relation \({A\wedge*B=\left\langle A,B\right\rangle \Omega}\), in which case these formulas are modified by a factor \({(-1)^{s}}\).

An Illustrated Handbook