# Levi-Civita symbol

## Contents

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

The Levi-Civita symbol changes sign whenever two of the indices are interchanged, that is, it is antisymmetric. In different words, the Levi-Civita symbol with three indices equals the *sign* of the permutation (*ijk*).^{[1]}

The symbol has been generalized to *n* dimensions, denoted as ε_{ijk...r} and depending on *n* indices taking values from 1 to *n*.
It is determined by being antisymmetric in the indices and by ε_{123...n} = 1. The generalized symbol equals the sign of the permutation (*ijk...r*) or, equivalently, the determinant of the corresponding unit vectors. Therefore the symbols also are called (Levi-Civita) *permutation symbols*.

### Levi-Civita tensor

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

The generalized symbol gives rise to an *n*-dimensional completely antisymmetric (or alternating) pseudotensor.

## Notes

- ↑ The sign of a permutation is 1 for even, −1 for odd permutations and 0 if two indices are equal. An
*even*permutation is a sequence (*ijk...r*) that can be restored to (123...*n*) using an even number of interchanges of pairs, while an odd permutation requires an odd number.