nLab
Levi-Civita tensor

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

In the space k nk^n where kk is the ground field, in the basis basis e i=(0,0,,1,0,,0)e_i = (0,0,\ldots, 1,0,\ldots, 0) where 11 is in the ii-th position, one defines the components of the Levi-Civita tensor ϵ i 1,,i n\epsilon_{i_1,\ldots,i_n} to be zero if at least two indices are the same, 11 if (i 1,,i n)(i_1,\ldots,i_n) is an even permutation of (1,,n)(1,\ldots, n) and 1-1 if it is an odd permutation of (1,,n)(1,\ldots,n). This has a generalization for an arbitrary Riemannian manifold. There are many useful identities in calculating with Levi-Civita tensor.

Last revised on June 13, 2011 at 10:28:14. See the history of this page for a list of all contributions to it.