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 in an \({n}\)-dimensional vector space 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
\(\displaystyle \left(\varphi\wedge\psi\right)_{\mu_{1}\cdots\mu_{j+k}}=\cfrac{1}{j!k!}\varphi_{\nu_{1}\cdots\nu_{j}}\psi_{\nu_{j+1}\cdots\nu_{j+k}}\delta_{\mu_{1}\cdots\mu_{j+k}}^{\nu_{1}\cdots\nu_{j+k}}.\)
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_{k+1}\dots\mu_{n}}&=\frac{\sqrt{\left|\det\left(g\right)\right|}}{k!}\varphi^{\mu_{1}\cdots\mu_{k}}\varepsilon_{\mu_{1}\cdots\mu_{n}}\\\Rightarrow\left(*\varphi\right)_{\mu_{1}\dots\mu_{n-k}}&=\frac{\left(-1\right)^{s}}{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}}.\end{aligned}
In particular, for an \({n}\)-form and a \({0}\)-form we have
\begin{aligned}*\varphi&=\frac{\left(-1\right)^{s}}{\sqrt{\left|\det\left(g\right)\right|}}\varphi_{1\cdots n},\\\left(*\varphi\right)_{1\dots n}&=\sqrt{\left|\det\left(g\right)\right|}\varphi.
\end{aligned}
| Δ Recall that some texts (including the first print edition of this book) define the Hodge star by the relation \({A\wedge C=\left\langle *A,C\right\rangle \Omega}\), in which case these formulas are modified by a factor \({(-1)^{s}}\). |