Levi-Civita symbol

Levi-Civita symbol (From Wikipedia, the free encyclopedia) The Levi-Civita symbol, also called the permutation symbol or antisymmetric
symbol, is a mathematical symbol used in particular in tensor calculus. It
is named after the Italian mathematician and physicist Tullio Levi-Civita.
Definition In three dimensions, the Levi-Civita symbol is defined as follows: [pic] i.e. it is 1 if (i, j, k) is an even permutation of (1,2,3), -1 if it is an
odd permutation, and 0 if any index is repeated. For example, in linear algebra, the determinant of a 3×3 matrix A can be
written [pic] (and similarly for a square matrix of general size, see below) and the cross product of two vectors can be written as a determinant: [pic] or more simply: [pic] According to the Einstein notation, the summation symbol may be omitted. 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 under an orthogonal transformation of
jacobian determinant -1 (i.e., a rotation composed with a reflection), it
gets a -1. Because the Levi-Civita symbol is a pseudotensor, the result of
taking a cross product is a pseudovector, not a vector.
Relation to Kronecker delta The Levi-Civita symbol is related to the Kronecker delta. In three
dimensions, the relationship is given by the following equations: [pic] [pic] [pic]("contracted epsilon identity") [pic]
Generalization to n dimensions The Levi-Civita symbol can be generalized to higher dimensions: [pic] Thus, it is the sign of the permutation in the case of a permutation, and
zero otherwise. Furthermore, it can be shown that [pic] is always fulfilled in n dimensions. In index-free tensor notation, the
Levi-Civita symbol is replaced by the concept of the Hodge dual.
In general n dimensions one can write the product of two Levi-Civita
symbols as: [pic]. Now we can contract m indexes, this will add a m! factor to the determinant
and we need to omit the relevant Kronecker delta.
Properties (superscripts should be considered equivalent with subscripts) 1. When n = 2, we have for all i,j,m,n in {1,2}, [pic]= [pic], (1) [pic]= [pic], (2) [pic]. (3) 2. When n = 3, we have for all i,j,k,m,n in {1,2,3}, [pic](4) [pic](5) Proofs For equation 1, both sides are antisymmetric with respect of ij and mn. We
therefore only need to consider the case [pic]and [pic]. By substitution,
we see that the equation holds for [pic], i.e., for i = m = 1 and j = n =
2. (Both sides are then one). Since the equation is antisymmetric in ij and
mn, any set of values for these can be reduced to the above case (which
holds). The equation thus holds for all values of ij and mn. Using equation
1, we have for equation 2: [pic]= [pic]= [pic]= [pic]. 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 [pic]. Indeed, if [pic], 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
[pic](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
[pic](no summation). Examples 1. The determinant of an [pic]matrix A = (aij) can be written as [pic] where each il should be summed over [pic] Equivalently, it may be written as [pic] where now each il and each jl should be summed over [pic]. 2. If A = (A1,A2,A3) and B = (B1,B2,B3) are vectors in R3 (represented in
some right hand oriented orthonormal basis), then the ith component of
their cross product equals [pic] For instance, the first component of [pic]is A2B3 - A3B2. From the above
expression for the cross product, it is clear that [pic]. Further, if C =
(C1,C2,C3) is a vector like A and B, then the triple scalar product equals [pic] From this expression, it can be seen that the triple scalar product is
antisymmetric when exchanging any adjacent arguments. For example, [pic]. 3. Suppose F = (F1,F2,F3) is a vector field defined on some open set of R3
with Cartesian coordinates x = (x1,x2,x3). Then the ith component of the
curl of F equals [pic]
Notation A shorthand notation for anti-symmetrization is denoted by a pair of square
brackets. For example, for an n x n matrix, M, [pic] and for a rank 3 tensor T, [pic]