Let \(V\) be a \(n\)-dimensional vector space. Then the elementary alternating tensor \(\varepsilon^I\) for a multi-index \(I=(i_1,\ldots, i_k)\in \{1,\ldots,n\}^k\) is a covariant tensors defined by
\begin{equation*} \varepsilon^I(v_1,\ldots,v_k)=\det\begin{pmatrix} \varepsilon^{i_1}(v_1) & \cdots & \varepsilon^{i_1}(v_k) \\ \vdots & \ddots & \vdots \\ \varepsilon^{i_k}& \cdots & \varepsilon^{i_k}(v_k)\end{pmatrix}, \end{equation*}where \(\varepsilon^i\) denotes the \(i\)-th dual basis element .
Remarks
- For \(\mathbb{R}^n\) and the standard dual basis \(\varepsilon^{12\ldots n}\) is the determinant function .
- Elementary alternating tensors are used to denote the basis for the space of alternating tensors
- using the wedge product we get \begin{equation*} \varepsilon^I=\varepsilon^{i_1}\wedge \cdots \wedge \varepsilon^{i_k}. \end{equation*}