The Levi-Civita symbol, sometimes referred to as the Levi-Civita tensor is a [pseudo-tensor](Tensors%20and%20Multilinear%20Algebra%20(index).md#Pseudo-tensors) that [arises](Levi-Civita%20Symbol.md#The%20Levi-Civita%20symbol%20as%20a%20signum%20of%20a%20permutation) from the permutation of tensor indices in $n$ dimensions. These permutations are donated by elements of the pseudo-tensor which are expressed as $\epsilon_{i_1i_2...i_n} = \epsilon^{i_1,i_2,...,i_n} = 0,+1,-1$ Notice that $\epsilon_{i_1i_2...i_n} = \epsilon^{i_1,i_2,...,i_n},$ meaning that the Levi-Civita symbol has equivalent [covariant](Covariant.md) and [contravariant](Contravariant.md) forms. Whether it is $0,$ $+1,$ or $-1$ is the [sign of the permutation.](Levi-Civita%20Symbol.md#The%20Levi-Civita%20symbol%20as%20a%20signum%20of%20a%20permutation) # The Levi-Civita symbol as a signum of a permutation The Levi-civita symbol comes about as the [signum](Signum.md) (or sign) of a [permutation](Permutations.md). In the simplest case we may consider, $\epsilon_{ij}$ where the subscripts are the indices of a sequence consisting of only a pair of elements. If $(i,j)=(1,2)$, $\epsilon_{ij}=+1$, if $(j,i)=(2,1)$, $\epsilon_{ij}=-1$, and if $(i,j)=(1,1)$ or $(i,j)=(2,2)$, then $\epsilon_{ij}=0$, corresponding with the positive, negative, and $0$ outputs of the [signum](Signum.md) function for the trivial permutation of two elements. This is generalized into $n$ numbers of permutation elements as. $\epsilon_{i_1,i_2,...,i_n}=\epsilon^{i_1,i_2,...,i_n}=\begin{cases} \mathrm{sgn}(P_{i_1,i_2...,i_n}^{1,2...,n})\\ 1 \end{cases}$ where superscript of the permutation, $P_{i_1,i_2...,i_n}^{1,2...,n}$, indicates a permutation of an ordered set. # As matrix elements # Useful properties in 3 dimensions 1. $\epsilon_{imn}\epsilon_{jmn}=2\delta_{ji}$ # Cyclic Property #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/MultiLinearAlgebra/Pseudotensors