The Levi-Civita symbol, represented as ε, is a three-dimensional array (it is not a tensor because its components do not change with a change in coordinate system), each element of which is 1, -1, or 0 depending on the whether the permutations of its elements are even, odd, or neither; in other words, whether the cyclic order is increasing or decreasing (for example, (1,2,3) and (3,1,2) are even permutations while (3,2,1) and (2,1,3) are odd).

For example, in three dimensions, the Levi-Civita symbol is equal to

The Levi-Civita symbol is anti-symmetric, meaning when any two indices are changed, its sign alternates. It is also related to the Kronecker delta by

The Levi-Civita symbol is useful for defining determinants of matrices, and by extension the cross product, in Einstein notation. For example:

  • Determinant of a 3×3 matrix:
  • Determinant of a n×n matrix:
  • Vector cross product:
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