The Levi-Civita symbol

While Eq. (10.6) is convenient for working out the cross product of explicit numerical vectors, as in the previous problem, it becomes quite cumbersome when the vectors contain symbolic entries and when multiple products are involved. A more efficient way of writing the cross product is facilitated by the Levi-Cicita symbol ε_ijk in three dimensions

$${\epsilon }_{ijk}=\{\begin{array}{ll}+1& \text{if }(i,j,k,)=(1,2,3),(2,3,1),(3,1,2)\\ -1& \text{if }(i,j,k)=(2,1,3),(3,2,1),(1,3,2)\\ 0& \text{otherwise}\end{array}$$

Using this symbol, the cross product can be written, using Einstein summation convention as

for example the 1st component is

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