Product of Two Levi-Civita Tensors
Product of Two Levi-Civita Tensors Harun Basmacı, Ankara University July 2022 Abstract In this teeny-weeny piece of paper, I will be trying to derive a formula for the product of two Levi-Civita tensors by using basic trigonometric properties- basically expressing them as four Kronecker delta. Also, I will demonstrate the expression I finally get, whether it works or not. 1 Derivation Suppose $\hat{e}_i, \hat{e}_j,$ and $\hat{e}_k$ are unit vectors, therefore \begin{equation} \hat{e}_i = \langle 1,0,0 \rangle \end{equation} \begin{equation} \hat{e}_j = \langle 0,1,0 \rangle \end{equation} \begin{equation} \hat{e}_k = \langle 0,0,1 \rangle. \end{equation} We know that \begin{equation}\epsilon_{ijk} = \hat{e}_i \cdot (\hat{e}_j \times \hat{e}_k)\end{equation} \begin{equation} =\hat{e}_i \cdot (\lVert{\hat{e}_j}\rVert \lVert{\hat{e}_k}\rVert \sin{\theta_{jk}}\hat{e}_i)\end{equation} ($\theta_{jk}$ represents the angle between the vectors $\hat{e}_j$ and $\hat{e}_k$) \begin{e