Levi-Civita symbol

The Levi-Civita symbol, usually denoted as &epsilon;ijk equals one if ijk = xyz or any permutation that keeps the same cyclic order, or minus one if the order is different, or zero if any two of the indices are the same. The Levi-Civita symbol also is known as the alternating tensor or the completely antisymmetric tensor with three indices in three dimensions.

The completely antisymmetric tensor with N indices in N-dimensions has only one independent component, and is denoted in two, three and four dimensions as &epsilon;ij, &epsilon;ijk, &epsilon;ijkl. Consequently, in three dimensions the completely antisymmetric tensor is entirely specified by stating &epsilon;123 = 1 in Cartesian coordinates.