Levi-Civita symbol
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
The Levi-Civita symbol changes sign whenever two of the indices are interchanged, that is, it is antisymmetric.
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 εijk.
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.
It give rise to an n-dimensional completely antisymmetric (or alternating) pseudotensor.
The Levi-Civita symbol equals the sign of the permutation (ijk).
Likewise, the generalized symbol equals the sign of the permutation (ijk...r)[1] or, equivalently, the determinant of the corresponding unit vectors. Therefore the symbols are also called (Levi-Civita) permutation symbols.
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, while an odd permutation requires an odd number.