Levi-Civita symbol

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Symbol of Levi-Civita.

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

${displaystyle epsilon _{ijk}=left{{begin{matrix}+1&{mbox{si }}(i,j,k){mbox{ es }}(1,2,3),(2,3,1){mbox{ o }}(3,1,2)\-1&{mbox{si }}(i,j,k){mbox{ es }}(3,2,1),(1,3,2){mbox{ o }}(2,1,3)\0&{mbox{de otro modo }}i=j{text{ o }}j=k{text{ o }}k=iend{matrix}}right.}$

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 vectors can be written as:

${displaystyle mathbf {atimes b} ={begin{vmatrix}mathbf {e_{1}} &mathbf {e_{2}} &mathbf {e_{3}} \a_{1}&a_{2}&a_{3}\b_{1}&b_{2}&b_{3}\end{vmatrix}}=sum _{i=1}^{3}left(sum _{j,k=1}^{3}epsilon _{ijk}a_{j}b_{k}right)mathbf {e} _{i}}$

or more simply:

${displaystyle mathbf {atimes b} =mathbf {c} c_{i}=sum _{j,k=1}^{3}epsilon _{ijk}a_{j}b_{k}}$

This last expression can be further simplified using Einstein's notation, a convention in which the summation symbol can be omitted. The tensor whose components are given by the Levi-Civita symbol (a covariant tensor of rank 3) is sometimes called the permutation tensor.

Definition

The most common dimensions of the Levi-Civita symbol are the third and fourth, and to some extent the second, so it is useful to look at these definitions before generalizing to any number of dimensions.

Two Dimensions

The two-dimensional Levi-Civita symbol is defined by:

${displaystyle epsilon _{ij}=left{{begin{matrix}+1&{mbox{si }}(i,j){mbox{ es }}(1,2)\-1&{mbox{si }}(i,j){mbox{ es }}(2,1)\0&{mbox{de otro modo }}i=jend{matrix}}right.}$

Three Dimensions

The Levi-Civita symbol in three dimensions is defined as follows:

${displaystyle epsilon _{ijk}=left{{begin{matrix}+1&{mbox{si }}(i,j,k){mbox{ es }}(1,2,3),(2,3,1){mbox{ o }}(3,1,2)\-1&{mbox{si }}(i,j,k){mbox{ es }}(3,2,1),(1,3,2){mbox{ o }}(2,1,3)\0&{mbox{de otro modo }}i=j{mbox{ o }}j=k{mbox{ o }}k=iend{matrix}}right.}$

Four Dimensions

${displaystyle epsilon _{ijkl}=left{{begin{matrix}+1&{mbox{si }}(i,j,k,l){mbox{ es }}(1,2,3,4),(1,3,4,2),(1,4,2,3),(2,1,4,3),(2,3,1,4),(2,4,3,1),(3,1,2,4),(3,2,4,1),(3,4,1,2),(4,1,3,2),(4,2,1,3){mbox{ o }}(4,3,2,1)\-1&{mbox{si }}(i,j,k,l){mbox{ es }}(1,2,4,3),(1,3,2,4),(1,4,3,2),(2,1,3,4),(2,3,4,1),(2,4,1,3),(3,1,4,2),(3,2,1,4),(3,4,2,1),(4,1,2,3),(4,2,3,1){mbox{ o }}(4,3,1,2)\0&{mbox{de otro modo }}i=jlor j=klor k=llor l=ilor k=ilor j=kend{matrix}}right.}$

Generalization to n Dimensions

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

${displaystyle epsilon _{ijkldots }=left{{begin{matrix}+1&{mbox{si }}(i,j,k,l,dots){mbox{ es una permutación par de }}(1,2,3,4,dots)\-1&{mbox{si }}(i,j,k,l,dots){mbox{ es una permutación impar de }}(1,2,3,4,dots)\0&{mbox{si dos índices son iguales}}end{matrix}}right.}$

A permutation of the series 1,2,3,4... is even if it can be reduced to the initial ordered series through an even number of exchanges of positions, otherwise it is odd. See even permutation or symmetric group for a formal definition of 'even permutation' and of 'odd permutation'.

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