Levi-Civita symbol Guide, Meaning , Facts, Information and Description

In mathematics, and in particular in tensor calculus, the Levi-Civita symbol, also called the permutation symbol, is defined as follows:

It is named after Tullio Levi-Civita. It is used in many areas of mathematics and physics. For example, in linear algebra, the cross product of two vectorss can be written as:

or more simply:

This can be further simplified by using Einstein notation.

The Levi-Civita symbol can be generalized to higher dimensions:

See even permutation or symmetric group for a definition of 'even permutation' and 'odd permutation'

The tensor whose components are given by the Levi-Civita symbol (a tensor of covariant rank n) is sometimes called the permutation tensor. It is actually a pseudotensor because it get a minus sign under orthogonal transformation of jacobian determinant -1 (i.e. a rotation composed with a reflection).

A related symbol is the Kronecker delta.

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