Exterior forms as completely anti-symmetric tensors

An immediate result of this view of forms as multilinear mappings is that we can also view forms as completely anti-symmetric tensors under the identification of \({\prod\varphi_{i}}\) with \({\bigotimes\varphi_{i}}\). For example, for a 2-form we have the equivalent expressions \({\left(\varphi\wedge\psi\right)\left(v,w\right)\leftrightarrow\left(\varphi\otimes\psi-\psi\otimes\varphi\right)\left(v,w\right)\leftrightarrow\varphi\left(v\right)\psi\left(w\right)-\psi\left(v\right)\varphi\left(w\right)}\).

Note however that this identification does not lead to equality of the inner products defined on tensors and exterior forms; instead for two \({k}\)-forms we have

\({\left\langle \bigwedge\varphi_{i},\bigwedge\psi_{i}\right\rangle _{\textrm{form}}=\textrm{det}\left(\left\langle \varphi_{i},\psi_{j}\right\rangle \right),}\)

while as tensors we have

\({\left\langle \bigwedge\varphi_{i},\bigwedge\psi_{i}\right\rangle _{\textrm{tensor}}=\left\langle \varepsilon^{I}\varphi_{I},\varepsilon^{J}\varphi_{J}\right\rangle _{\textrm{tensor}}=k!\left(k-1\right)!\textrm{det}\left(\left\langle \varphi_{i},\psi_{j}\right\rangle \right).}\)

Fortunately, the tensor inner product is almost always expressed explicitly in terms of index contractions, so we will continue to use the \({\left\langle \;,\,\right\rangle }\) notation for the inner product of \({k}\)-forms.

An Illustrated Handbook