Levi-Civita Symbol

The Levi-Civita symbol is a mathematical object, denoted by the Greek letter epsilon, that represents a collection of numbers defined from the sign of a permutation of the numbers 1 through n. It equals plus 1 for an even permutation of its indices, minus 1 for an odd permutation, and 0 whenever any two indices repeat, and swapping any two of its indices flips its sign, a property called total antisymmetry. Named after the Italian mathematician Tullio Levi-Civita, it is used in linear algebra, tensor analysis and differential geometry to write determinants, vector cross products and the curl of a vector field compactly in index notation. Because its transformation properties differ slightly from an ordinary tensor, it is technically classified as a tensor density rather than a true tensor.