 LeviCivita symbol

Not to be confused with LeviCivita connection.
The LeviCivita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical symbol used in particular in tensor calculus. It is named after the Italian mathematician and physicist Tullio LeviCivita.
Contents
Definition
In three dimensions, the LeviCivita symbol is defined as follows:
i.e. 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.
The formula for the three dimensional LeviCivita symbol is:The formula in four dimensions is:
For example, in linear algebra, the determinant of a 3×3 matrix A can be written
(and similarly for a square matrix of general size, see below)
and the cross product of two vectors can be written as a determinant:
or more simply:
According to the Einstein notation, the summation symbols may be omitted.
The tensor whose components in an orthonormal basis are given by the LeviCivita 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 acquires a minus sign. Because the LeviCivita symbol is a pseudotensor, the result of taking a cross product is a pseudovector, not a vector.
Note that under a general coordinate change, the components of the permutation tensor get multiplied by the jacobian of the transformation matrix. This implies that in coordinate frames different from the one in which the tensor was defined, its components can differ from those of the LeviCivita symbol by an overall factor. If the frame is orthonormal, the factor will be ±1 depending on whether the orientation of the frame is the same or not.
Relation to Kronecker delta
The LeviCivita symbol is related to the Kronecker delta. In three dimensions, the relationship is given by the following equations:


 ("contracted epsilon identity")
In Einstein notation, the duplication of the i index implies the sum on i. The previous is then denoted:
Generalization to n dimensions
The LeviCivita symbol can be generalized to higher dimensions:
Thus, it is the sign of the permutation in the case of a permutation, and zero otherwise.
The generalized formula is:
where n is the dimension (rank) and G(n) is the Barnes Gfunction.
For any n the propertyfollows from the facts that (a) every permutation is either even or odd, (b) (+1)^{2} = (1)^{2} = 1, and (c) the permutations of any nelement set number exactly n!.
In indexfree tensor notation, the LeviCivita symbol is replaced by the concept of the Hodge dual.
In general, for n dimensions, one can write the product of two LeviCivita symbols as:
 .
Properties
(in these examples, superscripts should be considered equivalent with subscripts)
1. In two dimensions, when all i,j,m,n are in {1,2},
2. In three dimensions, when all i,j,k,m,n are in {1,2,3},
3. In n dimensions, when all i_{1},...,i_{n},j_{1},...,j_{n} are in {1,...,n},:
Proofs
For equation 1, both sides are antisymmetric with respect of ij and mn. We therefore only need to consider the case and . By substitution, we see that the equation holds for ε_{12}ε^{12}, 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 ε_{ij}ε^{in} =

 =

 = .
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 . Indeed, if , 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 ε_{jmn}ε^{imn} = (ε^{imn})^{2} = 1 (no summation), and the result follows. Property (5) follows since 3! = 6 and for any distinct indices i,j,k in {1,2,3}, we have ε_{ijk}ε^{ijk} = 1 (no summation).
Examples
1. The determinant of an matrix A = (a_{ij}) can be written as
where each i_{l} should be summed over
Equivalently, it may be written as
where now each i_{l} and each j_{l} should be summed over .
2. If A = (A^{1},A^{2},A^{3}) and B = (B^{1},B^{2},B^{3}) are vectors in R^{3} (represented in some right hand oriented orthonormal basis), then the ith component of their cross product equals
For instance, the first component of is A^{2}B^{3} − A^{3}B^{2}. From the above expression for the cross product, it is clear that . Further, if C = (C^{1},C^{2},C^{3}) 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 adjacent arguments. For example, .
3. Suppose F = (F^{1},F^{2},F^{3}) is a vector field defined on some open set of R^{3} with Cartesian coordinates x = (x^{1},x^{2},x^{3}). Then the ith component of the curl of F equals
Notation
Main article: Antisymmetric tensorA shorthand notation for antisymmetrization is denoted by a pair of square brackets. For example, in arbitrary dimensions, for a rank 2 covariant tensor M,
and for a rank 3 covariant tensor T,
In three dimensions, these are equivalent to
While in four dimensions, these are equivalent to
More generally, in n dimensions
Tensor density
In any arbitrary curvilinear coordinate system and even in the absence of a metric on the manifold, the LeviCivita symbol as defined above may be considered to be a tensor density field in two different ways. It may be regarded as a contravariant tensor density of weight +1 or as a covariant tensor density of weight 1. In four dimensions,
Notice that the value, and in particular the sign, does not change.
Ordinary tensor
In the presence of a metric tensor field, one may define an ordinary contravariant tensor field which agrees with the LeviCivita symbol at each event whenever the coordinate system is such that the metric is orthonormal at that event. Similarly, one may also define an ordinary covariant tensor field which agrees with the LeviCivita symbol at each event whenever the coordinate system is such that the metric is orthonormal at that event. These ordinary tensor fields should not be confused with each other, nor should they be confused with the tensor density fields mentioned above. One of these ordinary tensor fields may be converted to the other by raising or lowering the indices with the metric as is usual, but a minus sign is needed if the metric signature contains an odd number of negatives. For example, in Minkowski space (the four dimensional spacetime of special relativity)
Notice the minus sign.
See also
References
 Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W.H. Freeman, New York; ISBN 0716703440. (See section 3.5 for a review of tensors in general relativity).
This article incorporates material from LeviCivita permutation symbol on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.Categories: Linear algebra
 Tensors
 Permutations

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