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:

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

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.

An Illustrated Handbook