LeviCivita symbol
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the LeviCivita symbol represents a collection of numbers; defined from the sign of a permutation of the natural numbers 1, 2, …, n, for some positive integer n. It is named after the Italian mathematician and physicist Tullio LeviCivita. Other names include the permutation symbol, antisymmetric symbol, or alternating symbol, which refer to its antisymmetric property and definition in terms of permutations.
The standard letters to denote the LeviCivita symbol are the Greek lower case epsilon ε or ϵ, or less commonly the Latin lower case e. Index notation allows one to display permutations in a way compatible with tensor analysis:
where each index i_{1}, i_{2}, …, i_{n} takes values 1, 2, …, n. There are n^{n} indexed values of ε_{i1i2…in}, which can be arranged into an ndimensional array. The key definitive property of the symbol is total antisymmetry in all the indices. When any two indices are interchanged, equal or not, the symbol is negated:
If any two indices are equal, the symbol is zero. When all indices are unequal, we have:
where p (called the parity of the permutation) is the number of interchanges of indices necessary to unscramble i_{1}, i_{2}, …, i_{n} into the order 1, 2, …, n, and the factor (−1)^{p} is called the sign or signature of the permutation. The value ε_{12…n} must be defined, else the particular values of the symbol for all permutations are indeterminate. Most authors choose ε_{12…n} = +1, which means the LeviCivita symbol equals the sign of a permutation when the indices are all unequal. This choice is used throughout this article.
The term "ndimensional LeviCivita symbol" refers to the fact that the number of indices on the symbol n matches the dimensionality of the relevant vector space in question, which may be Euclidean or nonEuclidean, pure space or spacetime. The values of the LeviCivita symbol are independent of any metric tensor and coordinate system. Also, the specific term "symbol" emphasizes that it is not a tensor because of how it transforms between coordinate systems, however it can be interpreted as a tensor density.
The LeviCivita symbol allows the determinant of a square matrix, and the cross product of two vectors in threedimensional Euclidean space, to be expressed in index notation.
Definition
The common dimensionalities of the LeviCivita symbol are in three and four dimensions, and to some extent two dimensions, so it is useful to see these definitions before the general one in any number of dimensions.
Two dimensions
The twodimensional LeviCivita symbol is defined by:
The values can be arranged into a 2 × 2 antisymmetric matrix:
Use of the twodimensional symbol is relatively uncommon, although in certain specialized topics like supersymmetry^{[1]} and twistor theory^{[2]} it appears in the context of 2spinors. The three and higherdimensional LeviCivita symbols are used more commonly.
Three dimensions
In three dimensions, the LeviCivita symbol is defined as follows:^{[3]}
That is, ε_{ijk} 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. In three dimensions only, the cyclic permutations of (1, 2, 3) are all even permutations, similarly the anticyclic permutations are all odd permutations. This means in 3d it is sufficient to take cyclic or anticyclic permutations of (1,2,3) and easily obtain all the even or odd permutations.
Analogous to 2dimensional matrices, the values of the 3dimensional LeviCivita symbol can be arranged into a 3 × 3 × 3 array:
where i is the depth, j the row and k the column.
Some examples:
Four dimensions
In four dimensions, the LeviCivita symbol is defined as:
These values can be arranged into a 4 × 4 × 4 × 4 array, although in 4 dimensions and higher this is difficult to draw.
Some examples:
Generalization to n dimensions
The LeviCivita symbol can be generalized to n dimensions:^{[4]}
Thus, it is the sign of the permutation in the case of a permutation, and zero otherwise.
Using the capital pi notation ∏ for ordinary multiplication of numbers, an explicit expression for the symbol is:
where the product is totally antisymmetric in all indices, and the sign function (denoted by "sgn") extracts the sign of each difference discarding the absolute value. The formula is valid for all index values, and for any n (when n = 0 or 1, this is the empty product). However, computing the formula above naively is O(n^{2}) in time complexity, whereas the sign can be computed via the disjoint cycles of a permutation in only O(n log(n)) cost.
Properties
A tensor whose components in an orthonormal basis are given by the LeviCivita symbol (a tensor of covariant rank n) is sometimes called a 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. As the LeviCivita symbol is a pseudotensor, the result of taking a cross product is a pseudovector, not a vector.^{[5]}
Under a general coordinate change, the components of the permutation tensor are 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.^{[5]}
In indexfree tensor notation, the LeviCivita symbol is replaced by the concept of the Hodge dual.
In a context where tensor index notation is used to manipulate tensor components, the LeviCivita symbol may be written with its indices as either subscripts or superscripts with no change in meaning, as might be convenient. Thus, one could write
In these examples, superscripts should be considered equivalent with subscripts.
Summation symbols can be eliminated by using Einstein notation, where an index repeated between two or more terms indicates summation over that index. For example,
 .
In the following examples, Einstein notation is used.
Two dimensions
In two dimensions, when all i, j, m, n each take the values 1 and 2,^{[3]}

( 1 )

( 2 )

( 3 )
Three dimensions
Index and symbol values
In three dimensions, when all i, j, k, m, n each take values 1, 2, and 3:^{[3]}

( 4 )

( 5 )

( 6 )
Product
The LeviCivita symbol is related to the Kronecker delta. In three dimensions, the relationship is given by the following equations (vertical lines denote the determinant):^{[4]}
A special case of this result is (4):
sometimes called the "contracted epsilon identity".
In Einstein notation, the duplication of the i index implies the sum on i. The previous is then denoted ε_{ijk}ε_{imn} = δ_{jm}δ_{kn} − δ_{jn}δ_{km}.
n dimensions
Index and symbol values
In n dimensions, when all i_{1}, …,i_{n}, j_{1}, …, j_{n} take values 1, 2, …, n:

( 7 )

( 8 )

( 9 )
where the exclamation mark (!) denotes the factorial, and δα…
β… is the generalized Kronecker delta. For any n, the property
follows from the facts that
 every permutation is either even or odd,
 (+1)^{2} = (−1)^{2} = 1, and
 the number of permutations of any nelement set number is exactly n!.
Product
In general, for n dimensions, one can write the product of two LeviCivita symbols as:
 .
Proofs
For (1), both sides are antisymmetric with respect of ij and mn. We therefore only need to consider the case i ≠ j and m ≠ n. 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.
Here we used the Einstein summation convention with i going from 1 to 2. Next, (3) follows similarly from (2).
To establish (5), notice that both sides vanish when i ≠ j. Indeed, if i ≠ j, 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
(no summation), and the result follows.
Then (6) follows since 3! = 6 and for any distinct indices i, j, k taking values 1, 2, 3, we have
 (no summation, distinct i, j, k)
Applications and examples
Determinants
In linear algebra, the determinant of a 3 × 3 square matrix A = [a_{ij}] can be written^{[6]}
Similarly the determinant of an n × n matrix A = [a_{ij}] can be written as^{[5]}
where each i_{r} should be summed over 1, …, n, or equivalently:
where now each i_{r} and each j_{r} should be summed over 1, …, n. More generally, we have the identity^{[5]}
Vector cross product
Cross product (two vectors)
If a = (a^{1}, a^{2}, a^{3}) and b = (b^{1}, b^{2}, b^{3}) are vectors in ℝ^{3} (represented in some righthanded coordinate system using an orthonormal basis), their cross product can be written as a determinant:^{[5]}
hence also using the LeviCivita symbol, and more simply:
In Einstein notation, the summation symbols may be omitted, and the ith component of their cross product equals^{[4]}
The first component is
then by cyclic permutations of 1, 2, 3 the others can be derived immediately, without explicitly calculating them from the above formulae:
Triple scalar product (three vectors)
From the above expression for the cross product, we have:
 .
If c = (c^{1}, c^{2}, c^{3}) is a third vector, then the triple scalar product equals
From this expression, it can be seen that the triple scalar product is antisymmetric when exchanging any pair of arguments. For example,
 .
Curl (one vector field)
If F = (F^{1}, F^{2}, F^{3}) is a vector field defined on some open set of ℝ^{3} as a function of position x = (x^{1}, x^{2}, x^{3}) (using Cartesian coordinates). Then the ith component of the curl of F equals^{[4]}
which follows from the cross product expression above, substituting components of the gradient vector operator (nabla).
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 n dimensions using the generalized Kronecker delta,^{[7]}^{[8]}
Notice that these are numerically identical. In particular, the sign is the same.
LeviCivita tensors
On a pseudoRiemannian manifold, one may define coordinateinvariant covariant and contravariant tensor fields whose coordinate representations agree with the LeviCivita symbol wherever the coordinate system is such that the basis of the tangent space is orthonormal with respect to the metric and matches a selected orientation. These tensors should not be confused with each other, nor should they be confused with the tensor density fields mentioned above. The covariant LeviCivita tensor (also known as the Riemannian volume form) in a given coordinate system is
where g_{ab} is the representation of the metric in that coordinate system. This tensor may be converted to the contravariant tensor by raising the indices with the metric as usual, but a minus sign is needed if the metric signature contains an odd number of negatives.
where s is the number of negatives in the signature. This results in the following:
Example: Minkowski Space
In Minkowski space (the fourdimensional spacetime of special relativity), the covariant LeviCivita tensor is
but the contravariant LeviCivita tensor is
Notice the minus sign. The following are identities.
See also
Notes
 ↑ Labelle, P. (2010). Supersymmetry. Demystified. McGrawHill. pp. 57–58. ISBN 9780071636414.
 ↑ Hadrovich, F. "Twistor Primer". Retrieved 20130903.
 1 2 3 Tyldesley, J. R. (1973). An introduction to Tensor Analysis: For Engineers and Applied Scientists. Longman. ISBN 0582443555.
 1 2 3 4 Kay, D. C. (1988). Tensor Calculus. Schaum’s Outlines. McGraw Hill. ISBN 0070334846.
 1 2 3 4 5 Riley, K. F.; Hobson, M. P.; Bence, S. J. (2010). Mathematical Methods for Physics and Engineering. Cambridge University Press. ISBN 9780521861533.
 ↑ Lipcshutz, S.; Lipson, M. (2009). Linear Algebra. Schaum’s Outlines (4th ed.). McGraw Hill. ISBN 9780071543521.
 ↑ Murnaghan, F. D. (1925), "The generalized Kronecker symbol and its application to the theory of determinants", Amer. Math. Monthly, 32: 233–241, doi:10.2307/2299191
 ↑ Lovelock, David; Rund, Hanno (1989). Tensors, Differential Forms, and Variational Principles. Courier Dover Publications. p. 113. ISBN 0486658406.
References
 Wheeler, J. A.; Misner, C.; Thorne, K. S. (1973). Gravitation. W. H. Freeman & Co. pp. 85–86, §3.5. ISBN 0716703440.
 Neuenschwander, D. E. (2015). Tensor Calculus for Physics. Johns Hopkins University Press. pp. 11, 29, 95. ISBN 9781421415659.
External links
This article incorporates material from LeviCivita permutation symbol on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.