Levi-Civita symbol

The Levi-Civita symbol, usually denoted as ε_ijk, is a notational convenience (similar to the Kronecker delta δ_ij). Its value is:

• equal to 1, if the indices are pairwise distinct and in cyclic order,
• equal to −1, if the indices are pairwise distinct but not in cyclic order, and
• equal to 0, if two of the indices are equal.

Thus

$\varepsilon _{ijk}={\begin{cases}\ \ 1&(ijk)=(123),(231),(312)\\-1&(ijk)=(132),(213),(321)\\\ \ 0&{\text{else}}\end{cases}}$

Remarks:

The Levi-Civita symbol—named after the Italian mathematician and physicist Tullio Levi-Civita—mainly occurs in differential geometry and mathematical physics where it is used to define the components of the (three-dimensional) Levi-Civita (pseudo)tensor that conventionally is also denoted by ε_ijk.

The symbol changes sign whenever two of the indices are interchanged.

The Levi-Civita symbol equals the sign of the permutation (ijk). Therefore it is also called (Levi-Civita) permutation symbol.

The symbol can be generalized to n-dimensions, to become the n-index symbol ε_ijk...r completely antisymmetric in its indices, and with ε_123...n = 1. More specifically, the symbol is has value 1 for even permutations of the n indices, value −1 for odd permutations, and value 0 otherwise.