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

Remarks:

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 Levi-Civita symbol is used in the definition of the Levi-Civita tensor that has components denoted as εijk.

The symbol can be generalized to n-dimensions, to become the n-index symbol εijk...r completely antisymmetric in its indices, and with ε123...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.

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