Levi-Civita Symbol

There are two other names:

Alternating Tensor
Permutation Symbol

The Levi-Civita symbol in two dimensions:

$\epsilon_{ij} = \begin{bmatrix} \epsilon_{11} & \epsilon_{12} \\ \epsilon_{21} & \epsilon_{22} \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$

The Levi-Civita symbol in three dimensions:

$\epsilon_{ijk} = \begin{bmatrix} \begin{bmatrix} \epsilon_{111} & \epsilon_{121} & \epsilon_{131} \\ \epsilon_{211} & \epsilon_{221} & \epsilon_{231} \\ \epsilon_{311} & \epsilon_{321} & \epsilon_{331} \end{bmatrix} & \begin{bmatrix} \epsilon_{112} & \epsilon_{122} & \epsilon_{132} \\ \epsilon_{212} & \epsilon_{222} & \epsilon_{232} \\ \epsilon_{312} & \epsilon_{322} & \epsilon_{332} \end{bmatrix} & \begin{bmatrix} \epsilon_{113} & \epsilon_{123} & \epsilon_{133} \\ \epsilon_{213} & \epsilon_{223} & \epsilon_{233} \\ \epsilon_{313} & \epsilon_{323} & \epsilon_{333} \end{bmatrix} \end{bmatrix}$

$= \begin{bmatrix} \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{bmatrix} & \begin{bmatrix} 0 & 0 & -1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix} & \begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \end{bmatrix}$

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