nLab Levi-Civita tensor

Levi-Civita tensor is a totally antisymmetric tensor of rank $k$ in $k$-dimensional vector space.

In the space $k^n$ where $k$ is the ground field, in the basis basis $e_i = (0,0,\ldots, 1,0,\ldots, 0)$ where $1$ is in the $i$-th position, one defines the components of the Levi-Civita tensor $\epsilon_{i_1,\ldots,i_n}$ to be zero if at least two indices are the same, $1$ if $(i_1,\ldots,i_n)$ is an even permutation of $(1,\ldots, n)$ and $-1$ if it is an odd permutation of $(1,\ldots,n)$. This has a generalization for an arbitrary Riemannian manifold. There are many useful identities in calculating with Levi-Civita tensor.

• wikipedia Levi-Civita symbol
• Weinstein’s wolfram world: Permutation Tensor
• Mathematics Wiki Levi-Civita symbol