Exterior forms as multilinear mappings

An exterior form (AKA \({k}\)-form, alternating form) is defined to be an element of \({\Lambda^{k}V^{*}}\). Just as we formed the isomorphism \({\otimes\varphi_{i}\mapsto\Pi\varphi_{i}}\) to view covariant tensors as multilinear mappings on \({V}\), we can view \({k}\)-forms as alternating multilinear mappings on \({V}\). Restricting attention to the exterior product of \({k}\) 1-forms \({\bigwedge\varphi_{i}}\), we define the isomorphism

\begin{aligned}\bigwedge_{i=1}^{k}\varphi_{i} & \mapsto\sum_{\pi}\textrm{sign}\left(\pi\right)\prod_{i=1}^{k}\varphi_{\pi\left(i\right)}\\
& =\sum_{i_{1},i_{2},\dotsc,i_{k}}\varepsilon^{i_{1}i_{2}\dots i_{k}}\varphi_{i_{1}}\varphi_{i_{2}}\dotsm\varphi_{i_{k}}=\varepsilon^{I}\varphi_{I},

where we recall the section on combinatorial notations.

The above isomorphism extends the interpretation of forms acting on vectors as yielding a projection. Specifically, if the parallelepiped \({\varphi^{\sharp}=\bigwedge\varphi_{i}^{\sharp}}\) has volume \({V}\), then \({\varphi(v_{1},\ldots v_{k})/V}\) is the volume of the projection of the parallelepiped \({v=\bigwedge v_{i}}\) onto \({\varphi^{\sharp}}\).

Extending this to arbitrary forms \({\varphi\in\Lambda^{j}V^{*}}\) and \({\psi\in\Lambda^{k}V^{*}}\), we have

\begin{aligned} & \left(\varphi\wedge\psi\right)\left(v_{1},\dotsc,v_{j+k}\right)\\
& \mapsto\cfrac{1}{j!k!}\sum_{\pi}\textrm{sign}\left(\pi\right)\varphi\left(v_{\pi\left(1\right)},\dotsc,v_{\pi\left(j\right)}\right)\psi\left(v_{\pi\left(j+1\right)},\dotsc,v_{\pi\left(j+k\right)}\right).

Just as with tensors, this isomorphism is canonical but not unique; but in the case of exterior forms, other isomorphisms are in common use. The main alternative isomorphism inserts a term \({1/k!}\) in the first relation above, which results in \({1/j!k!}\) being replaced by \({1/\left(j+k\right)!}\) in the second. Note that this alternative is inconsistent with the interpretation of exterior products as parallelepipeds.

Δ It is important to understand which convention a given author is using. The first convention above is common in physics, and we will adhere to it in this book.

An Illustrated Handbook