Levi-Civita symbol

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The Levi-Civita symbol, usually denoted as εijk, is a notational convenience (similar to the Kronecker delta δij). 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

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

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 εijk...r and depending on n indices taking values from 1 to n. It is determined by being antisymmetric in the indices and by ε123...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 εijk.

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

Notes

1. 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.