# Levi-Civita symbol

Is the Levi-Civita symbol a tensor?

R. A. Sharipov afirm (In "Quick Introduction to Tensor Analysis", page 30) that "...the Levi-Civita symbol is NOT a tensor..."

$\epsilon_{jkq}=\epsilon^{jkq}=\left\{\begin{array}{rl} 0, & \mbox{if among$j$,$k$,$q$there are at least two equal numbers} \\ 1, & \mbox{if$(j,k,q)$is even permutation of numbers$(1,2,3)$} \\ -1, & \mbox{if$(j,k,q)$is odd permutation of numbers} \end{array}\right.$

What does that phrase mean?

Thanks!

• If you define a rank-2 tensor $A$ as an object that transforms as $A_{ij}' = a_{ik} a_{jl} A_{kl}$ then a pseudotensor transforms as $A_{ij}' = J a_{ik} a_{jl} A_{kl}$ where $J = \det A$. The permutation tensor is a rank-3 pseudotensor, see "Classical Mechanics" by Goldstein for the general form. – Biswajit Banerjee Apr 28 '14 at 4:02
• But I need to understand the words of Sharipov in the context of his article Sharipov . ¿Does he want to say: "Despite its form or notation will not be a tensor? – Esteban May 5 '14 at 1:31