PlanetMath: Levi-Civita permutation symbol

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Levi-Civita permutation symbol

(Definition)

Definition 1 Let $k_i \in \{1,\cdots, n\}$ for all $i=1,\cdots ,n$ . The Levi-Civita permutation symbols $\varepsilon_{k_1\cdots k_n}$ and $\varepsilon^{k_1\cdots k_n}$ are defined as

$\displaystyle \varepsilon_{k_1\cdots k_m}=\varepsilon^{k_1\cdots k_m}= \left\{ ... ....e., when\,} k_i = k_j, \ \mbox {for some } \ i\neq j. \ \end{array} \right.$

The Levi-Civita permutation symbol is a special case of the generalized Kronecker delta symbol. Using this fact one can write the Levi-Civita permutation symbol as the determinant of an $n\times n$ matrix consisting of traditional delta symbols. See the entry on the generalized Kronecker symbol for details.

When using the Levi-Civita permutation symbol and the generalized Kronecker delta symbol, the Einstein summation convention is usually employed. In the below, we shall also use this convention.

Properties

When , we have for all in $\{1,2\}$ ,

$\displaystyle \varepsilon_{ij} \varepsilon^{mn}$ $\displaystyle =$ $\displaystyle \delta_i^m \delta_j^n - \delta_i^n \delta_j^m,$ (1)

$\displaystyle \varepsilon_{ij} \varepsilon^{in}$ $\displaystyle =$ $\displaystyle \delta_j^n,$ (2)

$\displaystyle \varepsilon_{ij} \varepsilon^{ij}$ $\displaystyle =$ $\displaystyle 2.$ (3)
When , we have for all in $\{1,2,3\}$ ,

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

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

Let us prove these properties. The proofs are instructional since they demonstrate typical argumentation methods for manipulating the permutation symbols.

Proof. For equation 1, let us first note that both sides are antisymmetric with respect of and . 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 and . (Both sides are then one). Since the equation is anti-symmetric in and , any set of values for these can be reduced the above case (which holds). The equation thus holds for all values of and . Using equation 1, we have for equation 2

$\displaystyle \varepsilon_{ij}\varepsilon^{in}$	$\displaystyle =$	$\displaystyle \delta_i^i \delta_j^n - \delta^n_i \delta^i_j$
	$\displaystyle =$	$\displaystyle 2 \delta_j^n - \delta^n_j$
	$\displaystyle =$	$\displaystyle \delta_j^n.$

Here we used the Einstein summation convention with

going from

. 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

and

such that both permutation symbols on the left are nonzero. Then, with

fixed, there are only two ways to choose

and

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. The last property follows since

and for any distinct indices

in $\{1,2,3\}$ , we have $\varepsilon_{ijk} \varepsilon^{ijk}=1$ (no summation). $\Box$

Examples and Applications.

The determinant of an $n\times n$ matrix $A=(a_{ij})$ can be written as
$\displaystyle \det A = \varepsilon_{i_1\cdots i_n} a_{1i_1} \cdots a_{ni_n},$
where each should be summed over $1,\ldots, n$ .
If and are vectors in $\mathbbmss{R}^3$ (represented in some right hand oriented orthonormal basis), then the th component of their cross product equals
$\displaystyle (A\times B)^i = \varepsilon^{ijk} A^j B^k.$
For instance, the first component of $A\times B$ is . From the above expression for the cross product, it is clear that $A\times B = -B\times A$ . Further, if is a vector like and , then the triple scalar product equals
$\displaystyle A\cdot(B\times C) = \varepsilon^{ijk} A^i B^j C^k.$
From this expression, it can be seen that the triple scalar product is antisymmetric when exchanging any adjacent arguments. For example, $A\cdot(B\times C)= -B\cdot(A\times C)$ .
Suppose is a vector field defined on some open set of $\mathbbmss{R}^3$ with Cartesian coordinates . Then the th component of the curl of equals
$\displaystyle (\nabla \times F)^i(x) = \varepsilon^{ijk}\frac{\partial}{\partial x^j} F^k(x).$