Levi-Civita symbol

In three dimensions, ${ϵ}_{ijk}$ is zero when indices repeat and is the sign of the permutation $(i,j,k)$ otherwise.

The determinant transformation formula for the Levi-Civita symbol shows that it is invariant under every proper rotation matrix, for which $\det R=1$. Under an orthogonal reflection its sign reverses. This distinguishes proper-rotation isotropy from the pseudotensor transformation convention when both orientations are included.

Contracting one index gives

For three-dimensional vectors,The corresponding identity in ${\mathbb{R}}^n$ is

Contracting two Levi-Civita symbols gives