# 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},
\end{aligned}

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).
\end{aligned}

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.