It is used in many areas of mathematics and physics. For example, in linear algebra, the cross product of two vectors can be written as:
\begin{vmatrix} \mathbf{e_1} & \mathbf{e_2} & \mathbf{e_3} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ \end{vmatrix}= \sum_{i,j,k=1}^3 \epsilon_{ijk} \mathbf{e_i} a_j b_k </math> or more simply:
This can be further simplified by using Einstein notation.
The tensor whose components are given by the LeviCivita symbol (a tensor of covariant rank 3) is sometimes called the permutation tensor.
The LeviCivita symbol can be generalized to higher dimensions:
See even permutation or symmetric group for a definition of 'even permutation' and 'odd permutation'
A related symbol is the Kronecker delta.
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