Levi-Civita symbol

In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol represents a collection of numbers; defined from the sign of a permutation of the natural numbers $1, 2, \dots, n$ , for some positive integer $n$ . It is named after the Italian mathematician and physicist Tullio Levi-Civita. Other names include the permutation symbol, antisymmetric symbol, or alternating symbol, which refer to its antisymmetric property and definition in terms of permutations.

The standard letters to denote the Levi-Civita symbol are the Greek lower case epsilon $ε$ or $ϵ$ , or less commonly the Latin lower case $e$ . Index notation allows one to display permutations in a way compatible with tensor analysis:

where each index $i 1, i 2, \dots, i n$ takes values $1, 2, \dots, n$ . There are $n n$ indexed values of $ε i 1 i 2 \dots i n$ , which can be arranged into an $n$ -dimensional array. The key definitive property of the symbol is total antisymmetry in all the indices. When any two indices are interchanged, equal or not, the symbol is negated:

...
Wikipedia