PX275 - F3 - levi-civita symbol

the levi-civita symbol is a tool to write things more concisely in tensor analysis
in 2D:

ε_{i j} = {\begin{cases} + 1 & if (i, j) = (1, 2) \\ - 1 & if (i, j) = (2, 1) \\ 0 & if i = j \end{cases}

ε = (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix})

ε_{i j} = {\begin{cases} + 1 & if (i, j, k) = (1, 2, 3) or (2, 3, 1) or (3, 1, 2) \\ - 1 & if (i, j, k) = (3, 2, 1) or (1, 3, 2) or (2, 1, 3) \\ 0 & if i = j or j = k or k = i \end{cases}

example

\begin{aligned} \vec{a} \times \vec{b} & = | \begin{array}{c} \hat{i} & \hat{j} & \hat{k} \\ a_{x} & a_{y} & a_{z} \\ b_{x} & b_{y} & b_{z} \end{array} | = | \begin{array}{c} x_{1} & x_{2} & x_{3} \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{array} | \\ = x_{1} (a_{2} b_{3} - b_{2} a_{3}) + x_{2} (a_{3} b_{1} - b_{3} a_{1}) + x_{3} (a_{1} b_{2} - b_{1} a_{2}) \\ = \sum_{i = 1}^{3} \sum_{j = 1}^{3} \sum_{k = 1}^{3} ε_{i j k} x_{i} a_{j} b_{k} \end{aligned}