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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],
\end{displaymath} (1)

the pseudotensor is a generalization to an arbitrary Basis defined by
$\displaystyle \epsilon_{\alpha\beta\cdots\mu}$ $\textstyle =$ $\displaystyle \sqrt{\vert g\vert} \,[\alpha, \beta, \ldots, \mu]$ (2)
$\displaystyle \epsilon^{\alpha\beta\cdots\mu}$ $\textstyle =$ $\displaystyle {[\alpha, \beta, \ldots, \mu]\over \sqrt{\vert g\vert}},$ (3)


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

© 1996-9 Eric W. Weisstein