Levi-Civita Symbol - Definition - Three Dimensions

Three Dimensions

In three dimensions, the Levi-Civita symbol is defined as follows:

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

i.e. is 1 if (i, j, k) is an even permutation of (1,2,3), −1 if it is an odd permutation, and 0 if any index is repeated. Some authors (e.g. Roger Penrose) use the distinct symbols and to emphasize that they are not directly related by index raising or lowering operations.

The formula for the three-dimensional Levi-Civita symbol is:

\varepsilon_{ijk} = \frac{\left( i-j \right)\left( j-k \right)\left( k-i \right)}{2}

which is a product of differences in the indices; where each difference corresponds to a cyclic permutation of their alphabetical ordering.

Values of the Levi-Civita symbol for a right-handed coordinate system. Corresponding visualization of the Levi-Civita symbol for a left-handed coordinate system. Empty cubes mean 0, red ones +1, and blue ones −1. Visualization of the Levi-Civita symbol as a 3×3×3 array.


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