Permutation Tensor

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,

$$\epsilon_{ijk} \equiv \hat {\bf x}_i\cdot(\hat {\bf x}_j\times\hat {\bf x}_k)=[\hat{\bf x}_i,\hat{\bf x}_j,\hat{\bf x}_k],$$ (1)

the pseudotensor is a generalization to an arbitrary Basis defined by

$$\epsilon_{\alpha\beta\cdots\mu} = \sqrt{\vert g\vert} \,[\alpha, \beta, \ldots, \mu]$$ (2)

$$\epsilon^{\alpha\beta\cdots\mu} = {[\alpha, \beta, \ldots, \mu]\over \sqrt{\vert g\vert}},$$ (3)

where

$$[\alpha, \beta, \ldots, \mu]=\cases{1 & the arguments are an ... ...an odd permutation\cr 0 & two or more arguments are equal,\cr}$$ (4)

and $g\equiv {\rm det}(g_{\alpha\beta})$, where $g_{\alpha\beta}$ is the Metric Tensor. $\epsilon({\bf x}_1,\ldots,{\bf x}_n)$ is Nonzero Iff the Vectors are Linearly Independent.

See also Permutation Symbol, Scalar Triple Product