Let \(V\) be a \(n\)-dimensional vector space. The wedge product of two alternating tensors \(\omega\in \Lambda^k(V^*)\) and \(\eta\in \Lambda^l(V^*)\) is defined by
\begin{equation*} \omega \wedge \eta:= \frac{(k+l)!}{k!l!} \Alt(\omega\otimes \eta), \end{equation*}where \(\Alt\) denotes the alternation operator and \(\otimes \) the tensor product .
Remarks
- the wedge product is bilinear, associative and anticommutative, i.e. \begin{equation*} \eta \wedge \omega = (-1)^{kl}\omega\wedge \eta. \end{equation*}
- For elementary alternating tensors we have \begin{equation*} \varepsilon^I\wedge \varepsilon^J=\varepsilon^{IJ}, \end{equation*} where \(IJ\) is obtained by concatenating \(I\) and \(J\). Especially, we have for \(I=(i_1,\ldots, i_k)\) \begin{equation*} \varepsilon^I=\varepsilon^{i_1}\wedge \cdots \wedge \varepsilon^{i_k}. \end{equation*}
- The both remarks above uniquely determine the wedge product.
- For any covectors \(\omega^1,\ldots, \omega^k\) and vectors \(v_1,\ldots, v_k\) we have \begin{equation*} \omega^1\wedge \cdots \wedge \omega^k(v_1,\ldots, v_k)=\det(\omega^j(v_i)) \hat{} \end{equation*}
- Compared to the symmetric product we add a silly factor. This factor is motivated by the remark above and therefor this definition of the wedge product is called determinant convention. There are reasons to omit this factor. In this case the definition of the wedge product is called Alt convention.