Levi-Civita symbol

$\varepsilon_{ijk}=\pm1$ for an even/odd permutation of $1,2,3$ , else $0$ -- the index-notation cross product.

The Levi-Civita symbol $\varepsilon_{ijk}$ equals $+1$ for an even permutation of $(1,2,3)$ , $-1$ for an odd permutation, and $0$ whenever any two indices repeat. It packages the cross product and determinant into a single index-notation object: $(\mathbf{u}\times\mathbf{v})_i = \varepsilon_{ijk}u_jv_k$ .

Examples

$\varepsilon_{123}=1$ , $\varepsilon_{213}=-1$ , $\varepsilon_{112}=0$ .