In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita 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 Levi-Civita. 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 Levi-Civita 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 i1, i2, …, in takes values 1, 2, …, n. There are nn indexed values of , which can be arranged into an n-dimensional 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 i1, i2, …, in into the order 1, 2, …, n, and the factor (−1)p is called the sign or signature of the permutation. The value ε12…n must be defined, else the particular values of the symbol for all permutations are indeterminate. Most authors choose ε12…n = +1, which means the Levi-Civita symbol equals the sign of a permutation when the indices are all unequal. This choice is used throughout this article.
The term "n-dimensional Levi-Civita 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 non-Euclidean, pure space or spacetime. The values of the Levi-Civita symbol are independent of any metric tensor and coordinate system.
- 1 Definition
- 2 Properties
- 3 Applications and examples
- 4 Tensor density
- 5 Levi-Civita tensors
- 6 See also
- 7 References
- 8 External links
The common dimensionalities of the Levi-Civita symbol are in 3d and 4d, and to some extent 2d, so it is useful to see these definitions before the general one in any number of dimensions.
The two-dimensional Levi-Civita symbol is defined by:
The values can be arranged into a 2 × 2 antisymmetric matrix:
Use of the 2d symbol is relatively uncommon, although in certain specialized topics like supersymmetry and twistor theory it appears in the context of 2-spinors. The 3d and higher-dimensional Levi-Civita symbols are used more commonly.
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. In three dimensions, and not higher, 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 2d matrices, the values of the 3d Levi-Civita symbol can be arranged into a 3×3×3 array:
where i is the depth, j the row and k the column.
In four dimensions, the Levi-Civita symbol is defined as:
These values can be arranged into a 4×4×4×4 array, although in 4d and higher this is difficult to draw.
Generalization to n dimensions
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 true for all index values, and for any n (when n = 1 or 0, this is the empty product). However, it is seldom used in practice since interchanging indices is quicker.
A tensor whose components in an orthonormal basis are given by the Levi-Civita 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 Levi-Civita symbol is a pseudotensor, the result of taking a cross product is a pseudovector, not a vector.
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 Levi-Civita 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.
In index-free tensor notation, the Levi-Civita symbol is replaced by the concept of the Hodge dual.
In a context of where tensor index notation is used to manipulate tensor components, the Levi-Civita 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.
- Index and symbol values
In three dimensions, when all i, j, k, m, n each take values 1, 2, and 3:
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:
- Index and symbol values
In n dimensions, when all i1,...,in, j1,...,jn take values 1, 2,..., n:
follows from the facts that
- every permutation is either even or odd,
- (+1)2 = (−1)2 = 1, and
- the number of permutations of any n-element set number is exactly n!.
In general, for n dimensions, one can write the product of two Levi-Civita symbols as:
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 , 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.
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
Similarly the determinant of an n × n matrix A = (aij) can be written as
where each ir should be summed over 1,..., n, or equivalently:
where now each ir and each jr should be summed over 1,.., n. More generally, we have the identity
Vector cross product
Cross product (two vectors)
hence also using the Levi-Civita symbol, and more simply:
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 = (c1, c2, c3) is another 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)
In any arbitrary curvilinear coordinate system and even in the absence of a metric on the manifold, the Levi-Civita 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,
Notice that these are numerically identical. In particular, the sign is the same.
On a pseudo-Riemannian manifold, one may define coordinate-invariant covariant and contravariant tensor fields whose coordinate representations agree with the Levi-Civita symbol wherever the coordinate system is such that the basis of the tangent space is orthonormal with respect to the metric. These tensors should not be confused with each other, nor should they be confused with the tensor density fields mentioned above. The covariant Levi-Civita tensor in a given coordinate system is
where 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
but the contravariant Levi-Civita tensor is
Notice the minus sign. The following are identities.
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- Permutation Tensor - mathworld.wolfram