Levi-Civita symbol

The Levi-Civita symbol, usually denoted as ε_ijk, is a conventional abbreviation (similar to the Kronecker delta δ_ij). It equals either 1, −1, or 0 depending on the values (1, 2, or 3) taken by the indices i, j, and k. It 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 symbol changes sign whenever two of the indices are interchanged.

The Levi-Civita symbol is a special case (for n=3, because it involves three indices) of a more general notion:
   It equals the sign of the permutation (ijk). Therefore it is also called (Levi-Civita) permutation symbol.

The Levi-Civita symbol is used in the definiton of the Levi-Civita tensor that is also denoted as denoted as ε_ijk.

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

Both the symbol and the tensor are named after the Italian mathematician and physicist Tullio Levi-Civita.