Levi-Civita permutation symbol

Definition 1.

Let ki{1,,n} for all i=1,,n. The Levi-Civita permutation symbols εk1kn and εk1kn are defined as

εk1km=εk1km={+1𝑤ℎ𝑒𝑛{lkl} is an even permutation (of {1,,n}),-1𝑤ℎ𝑒𝑛{lkl} is an odd permutation,0otherwise, i.e., when ki=kj,for some ij.

The Levi-Civita permutation symbol is a special case of the generalized Kronecker delta symbol. Using this fact one can write the Levi-Civita permutation symbol as the determinantMathworldPlanetmath of an n×n matrix consisting of traditional delta symbols. See the entry on the generalized Kronecker symbolMathworldPlanetmath for details.

When using the Levi-Civita permutation symbol and the generalized Kronecker delta symbol, the Einstein summation convention is usually employed. In the below, we shall also use this convention.


  • When n=2, we have for all i,j,m,n in {1,2},

    εijεmn = δimδjn-δinδjm, (1)
    εijεin = δjn, (2)
    εijεij = 2. (3)
  • When n=3, we have for all i,j,k,m,n in {1,2,3},

    εjmnεimn = 2δji, (4)
    εijkεijk = 6. (5)

Let us prove these properties. The proofs are instructional since they demonstrate typical argumentation methods for manipulating the permutation symbols.

Proof. For equation 1, let us first note that both sides are antisymmetric with respect of ij and mn. We therefore only need to consider the case ij and mn. By substitution, we see that the equation holds for ε12ε12, i.e., for i=m=1 and j=n=2. (Both sides are then one). Since the equation is anti-symmetric in ij and mn, any set of values for these can be reduced the above case (which holds). The equation thus holds for all values of ij and mn. Using equation 1, we have for equation 2

εijεin = δiiδjn-δinδji
= 2δjn-δjn
= δjn.

Here we used the Einstein summation convention with i going from 1 to 2. Equation 3 follows similarly from equation 2. To establish equation 4, let us first observe that both sides vanish when ij. Indeed, if ij, then one can not choose m and n such that both permutation symbols on the left are nonzero. Then, with i=j fixed, there are only two ways to choose m and n from the remaining two indices. For any such indices, we have εjmnεimn=(εimn)2=1 (no summation), and the result follows. The last property follows since 3!=6 and for any distinct indices i,j,k in {1,2,3}, we have εijkεijk=1 (no summation).

Examples and Applications.

  • The determinant of an n×n matrix A=(aij) can be written as


    where each il should be summed over 1,,n.

  • If A=(A1,A2,A3) and B=(B1,B2,B3) are vectors in 3 (represented in some right hand oriented orthonormal basisMathworldPlanetmath), then the ith componentPlanetmathPlanetmathPlanetmath of their cross productMathworldPlanetmath equals


    For instance, the first component of A×B is A2B3-A3B2. From the above expression for the cross product, it is clear that A×B=-B×A. Further, if C=(C1,C2,C3) is a vector like A and B, then the triple scalar product equals


    From this expression, it can be seen that the triple scalar product is antisymmetric when exchanging any adjacentPlanetmathPlanetmathPlanetmath arguments. For example, A(B×C)=-B(A×C).

  • Suppose F=(F1,F2,F3) is a vector fieldMathworldPlanetmath defined on some open set of 3 with Cartesian coordinatesMathworldPlanetmath x=(x1,x2,x3). Then the ith component of the curl of F equals

Title Levi-Civita permutation symbol
Canonical name LeviCivitaPermutationSymbol
Date of creation 2013-03-22 13:31:29
Last modified on 2013-03-22 13:31:29
Owner matte (1858)
Last modified by matte (1858)
Numerical id 13
Author matte (1858)
Entry type Definition
Classification msc 05A10
Related topic KroneckerDelta
Related topic GeneralizedKroneckerDeltaSymbol