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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).

$ \varepsilon_{a_1 a_2 a_3 \ldots a_n} = \begin{cases} +1 & \text{if }(a_1 , a_2 , a_3 , \ldots , a_n) \text{ is an even permutation} \\ -1 & \text{if }(a_1 , a_2 , a_3 , \ldots , a_n) \text{ is an odd permutation} \\ 0 & \text{otherwise} \end{cases} $

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

$ \varepsilon_{ijk} = \begin{cases} +1 & \text{if } (i,j,k) \text{ is } (1,2,3), (2,3,1), \text{ or } (3,1,2), \\ -1 & \text{if } (i,j,k) \text{ is } (3,2,1), (1,3,2), \text{ or } (2,1,3), \\ \;\;\,0 & \text{if } i = j, \text{ or } j = k, \text{ or } k = i \end{cases} $

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

$ \varepsilon_{ijk} \varepsilon_{inm} = \delta_{jm} \delta_{kn} - \delta_{jn} \delta_{km} $

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: $ | \mathbf{A} | = \varepsilon_{ijk} a_{1i} a_{2j} a_{3k} $
  • Determinant of a n×n matrix: $ | \mathbf{A} | = \varepsilon_{i_1\cdots i_n} a_{1i_1} \cdots a_{ni_n} $
  • Vector cross product: $ \mathbf{v} \times \mathbf{u}= \varepsilon_{ijk }v^j u^k \mathbf{e}_i $
  • Scalar triple product: $ \vec{ \mathbf{v} } \, \times \, \vec{ \mathbf{w} } \, \cdot \, \vec{ \mathbf{u} } = \varepsilon_{ijk} v^i w^j u^k $
  • Curl: $ \nabla \times \mathbf{F} = \varepsilon^{ijk}\frac{\partial}{\partial x^j} F^k(x,y,z) $
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