Levi-Civita symbol - Citizendia

The Levi-Civita symbol, also called the permutation symbol or antisymmetric symbol, is a mathematical symbol used in particular in tensor calculus. In several fields of Mathematics the term permutation is used with different but closely related meanings Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually It is named after the Italian mathematician and physicist Tullio Levi-Civita. The' Italian people' are a Southern European Ethnic group located primarily in Italy, Switzerland, France and by virtue of a wide-ranging Tullio Levi-Civita ( March 29, 1873 — December 29, 1941) (pronounced /'levi ˈʧivita/ was an Italian Mathematician,

1 Definition
- 1.1 Relation to Kronecker delta
- 1.2 Generalization to n dimensions
2 Properties
- 2.1 Proofs
3 Examples
4 Notation
5 References

Definition

Visualization of a Levi-Civita symbol.

In three dimensions, the Levi-Civita symbol is defined as follows:

$\varepsilon_{ijk} = \begin{cases} +1 & \mbox{if } (i,j,k) \mbox{ is } (1,2,3), (3,1,2) \mbox{ or } (2,3,1), \\ -1 & \mbox{if } (i,j,k) \mbox{ is } (3,2,1), (1,3,2) \mbox{ or } (2,1,3), \\ 0 & \mbox{otherwise: }i=j \mbox{ or } j=k \mbox{ or } k=i, \end{cases}$

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. In Mathematics, the Permutations of a Finite set (ie the bijective mappings from the set to itself fall into two classes of equal size the even

An alternative notation for the three dimensional Levi-Civita symbol without case differentiation is

$\varepsilon_{ijk} = -[(i-j)^2%3][(i-k)^2%3][(j-k)^2%3][(j-(i%3)-\frac{1}{2})^2-\frac{5}{4}]$

with $%$ representing the modulo operator and $i,j,k \in \lbrace1,2,3\rbrace$ .

For example, in linear algebra, the determinant of a 3×3 matrix A can be written

$\det A = \sum_{i,j,k=1}^3 \varepsilon_{ijk} a_{1i} a_{2j} a_{3k}$

(and similarly for a square matrix of general size, see below)

and the cross product of two vectors can be written as a determinant:

$\mathbf{a \times b} = \begin{vmatrix} \mathbf{e_1} & \mathbf{e_2} & \mathbf{e_3} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ \end{vmatrix} = \sum_{i,j,k=1}^3 \varepsilon_{ijk} \mathbf{e_i} a_j b_k$

or more simply:

$\mathbf{a \times b} = \mathbf{c},\ c_i = \sum_{j,k=1}^3 \varepsilon_{ijk} a_j b_k.$

According to the Einstein notation, the summation symbol may be omitted. Linear algebra is the branch of Mathematics concerned with In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which In Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notational

The tensor whose components are given by the Levi-Civita symbol (a tensor of covariant rank n) is sometimes called the permutation tensor. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually It is actually a pseudotensor because under an orthogonal transformation of jacobian determinant −1 (i. In Physics and Mathematics, a pseudotensor is usually a quantity that transforms like a Tensor under a Proper rotation, but gains an additional In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant. 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. In Physics and Mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation but gains an

Relation to Kronecker delta

The Levi-Civita symbol is related to the Kronecker delta. In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two In three dimensions, the relationship is given by the following equations:

$\varepsilon_{ijk}\varepsilon_{lmn} = \det \begin{bmatrix} \delta_{il} & \delta_{im}& \delta_{in}\\ \delta_{jl} & \delta_{jm}& \delta_{jn}\\ \delta_{kl} & \delta_{km}& \delta_{kn}\\ \end{bmatrix}$

$= \delta_{il}\left( \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km}\right) - \delta_{im}\left( \delta_{jl}\delta_{kn} - \delta_{jn}\delta_{kl} \right) + \delta_{in} \left( \delta_{jl}\delta_{km} - \delta_{jm}\delta_{kl} \right) \,$

$\sum_{i=1}^3 \varepsilon_{ijk}\varepsilon_{imn} = \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km}$ ("contracted epsilon identity")

(In Einstein notation, the duplication of the i index implies the sum on i. The previous is then noted: $\varepsilon_{ijk}\varepsilon_{imn} = \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km}$ )

$\sum_{i,j=1}^3 \varepsilon_{ijk}\varepsilon_{ijn} = 2\delta_{kn}$

Generalization to n dimensions

The Levi-Civita symbol can be generalized to higher dimensions:

$\varepsilon_{ijkl\dots} = \left\{ \begin{matrix} +1 & \mbox{if }(i,j,k,l,\dots) \mbox{ is an even permutation of } (1,2,3,4,\dots) \\ -1 & \mbox{if }(i,j,k,l,\dots) \mbox{ is an odd permutation of } (1,2,3,4,\dots) \\ 0 & \mbox{if any two labels are the same} \end{matrix} \right.$

Thus, it is the sign of the permutation in the case of a permutation, and zero otherwise. In Mathematics, the Permutations of a Finite set (ie the bijective mappings from the set to itself fall into two classes of equal size the even

Furthermore, for any n the property

$\sum_{i,j,k,\dots=1}^n \varepsilon_{ijk\dots}\varepsilon_{ijk\dots} = n!$

follows from the facts that (a) every permutation is either even or odd, (b) (+1)² = (-1)² = 1, and (c) the permutations of any n-element set number exactly n!.

In index-free tensor notation, the Levi-Civita symbol is replaced by the concept of the Hodge dual. In Mathematics, the Hodge star operator or Hodge dual is a significant Linear map introduced in general by W
In general $n$ dimensions one can write the product of two Levi-Civita symbols as:

$\varepsilon_{ijk\dots}\varepsilon_{mnl\dots} = \det \begin{bmatrix} \delta_{im} & \delta_{in} & \delta_{il} & \dots \\ \delta_{jm} & \delta_{jn} & \delta_{jl} & \dots \\ \delta_{km} & \delta_{kn} & \delta_{kl} & \dots \\ \vdots & \vdots & \vdots \\ \end{bmatrix}$ .

Now we can contract $m$ indices. This will add a factor of $m!$ to the determinant and we need to omit the relevant Kronecker delta.

Properties

(in these examples, superscripts should be considered equivalent with subscripts)

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

$\varepsilon_{ij} \varepsilon^{mn}$

=

$\delta_i^m \delta_j^n - \delta_i^n \delta_j^m$ , (1)

$\varepsilon_{ij} \varepsilon^{in}$

=

$\delta_j^n$ , (2)

$\varepsilon_{ij} \varepsilon^{ij}=2$ . (3)

2. When $n = 3$ , we have for all $i, j, k, m, n$ in ${1,2,3},$

$\varepsilon_{jmn} \varepsilon^{imn}=2\delta^i_j$ , (4)

$\varepsilon_{ijk} \varepsilon^{ijk}=6$ , (5)

$\varepsilon_{ijk} \varepsilon^{imn}=\delta^{m}_j\delta^{n}_k - \delta^{n}_j\delta^{m}_k$ . (6)

Proofs

For equation 1, both sides are antisymmetric with respect of $i j$ and $m n$ . We therefore only need to consider the case $i\neq j$ and $m\neq n$ . By substitution, we see that the equation holds for $\varepsilon_{12} \varepsilon^{12}$ , i. e. , for $i = m = 1$ and $j = n = 2$ . (Both sides are then one). Since the equation is antisymmetric in $i j$ and $m n$ , any set of values for these can be reduced to the above case (which holds). The equation thus holds for all values of $i j$ and $m n$ . Using equation 1, we have for equation 2 $\varepsilon_{ij}\varepsilon^{in}$ $=$ $\delta_i^i \delta_j^n - \delta^n_i \delta^i_j$

=

$2 \delta_j^n - \delta^n_j$

=

$\delta_j^n$ .

Here we used the Einstein summation convention with $i$ going from $1$ to $2$ . In Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notational Equation 3 follows similarly from equation 2. To establish equation 4, let us first observe that both sides vanish when $i\neq j$ . Indeed, if $i\neq j$ , 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 $\varepsilon_{jmn} \varepsilon^{imn} = (\varepsilon^{imn})^2 = 1$ (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 $\varepsilon_{ijk} \varepsilon^{ijk}=1$ (no summation).

Examples

1. The determinant of an $n\times n$ matrix $A = (a i j)$ can be written as

$\det A = \varepsilon_{i_1\cdots i_n} a_{1i_1} \cdots a_{ni_n},$

where each $i l$ should be summed over $1,\ldots, n.$

Equivalently, it may be written as

$\det A = \frac{1}{n!} \varepsilon_{i_1\cdots i_n} \varepsilon_{j_1\cdots j_n} a_{i_1 j_1} \cdots a_{i_n j_n},$

where now each $i l$ and each $j l$ should be summed over $1,\ldots, n$ .

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 $i$ th component of their cross product equals

$(A\times B)^i = \varepsilon^{ijk} A^j B^k.$

For instance, the first component of $A\times B$ is $A 2 B 3 - A 3 B 2$ . From the above expression for the cross product, it is clear that $A\times B = -B\times A$ . Further, if $C = (C 1, C 2, C 3)$ is a vector like $A$ and $B$ , then the triple scalar product equals

$A\cdot(B\times C) = \varepsilon^{ijk} A^i B^j C^k.$

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

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 $i$ th component of the curl of $F$ equals

$(\nabla \times F)^i(x) = \varepsilon^{ijk}\frac{\partial}{\partial x^j} F^k(x).$

Notation

Main article: Symmetric tensor#Properties

A shorthand notation for anti-symmetrization is denoted by a pair of square brackets. In Mathematics, a symmetric tensor is a Tensor that is invariant under a Permutation of its vector arguments For example, for an n x n matrix, M,

$M_{[ab]} = \frac{1}{2}\varepsilon_{abc} \varepsilon^{cde} M_{de } = \, \frac{1}{2}(M_{ab} - M_{ba})$

and for a rank 3 tensor T,

$T_{[abc]} = \, \frac{1}{3!}(T_{abc}-T_{acb}+T_{bca}-T_{bac}+T_{cab}-T_{cba})$

References

Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W. H. Freeman, New York; ISBN 0-7167-0344-0. (See section 3. 5 for a review of tensors in general relativity). General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916

This article incorporates material from Levi-Civita permutation symbol on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative online Mathematics Encyclopedia.

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