The permutation tensor, also called the Levi-Civita tensor or isotropic tensor of rank 3 (Goldstein 1980, p. 172), is a pseudotensor which is antisymmetric under the interchange of any two slots. Recalling the definition of the permutation symbol in terms of a scalar triple product of the Cartesian unit vectors,
![epsilon_(ijk)=x_i^^·(x_j^^xx_k^^)=[x_i^^,x_j^^,x_k^^],](/images/equations/PermutationTensor/NumberedEquation1.svg) |
(1)
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the pseudotensor is a generalization to an arbitrary basis defined by
 |  | ![sqrt(|g|)[alpha,beta,...,mu]](/images/equations/PermutationTensor/Inline3.svg) |
(2)
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 |  | ![([alpha,beta,...,mu])/(sqrt(|g|)),](/images/equations/PermutationTensor/Inline6.svg) |
(3)
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where
![[alpha,beta,...,mu]={1 the arguments are an even permutation; -1 the arguments are an odd permutation; 0 two or more arguments are equal,](/images/equations/PermutationTensor/NumberedEquation2.svg) |
(4)
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and 
,
where 
is the metric tensor. 
is nonzero iff the vectors are linearly
independent.
When viewed as a tensor, the permutation symbol is sometimes known as the Levi-Civita tensor. The permutation tensor 
of rank four is important in general
relativity, and has components defined as
 |
(5)
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(Weinberg 1972, p. 38). The rank four permutation tensor satisfies the identity
 |
(6)
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