4.6.3 Cross-products and the Levi-Civita symbol

In this manual, cross products are sometimes written with the notation

$${\bf A} \times {\bf B} = {\bf A} \wedge {\bf B}.$$ (4.66)

This notation is consistent with many math and physics texts. Its use is helpful for those situations when the usual $\times$ symbol can be mistaken for the spatial variable x.

When writing the components of a vector cross-product, it is often useful to employ the Levi-Civita symbol $\epsilon_{ijk}$

$$({\bf A} \wedge {\bf B})_{k} = \epsilon_{ijk} \, A^{i} \, B^{j},$$ (4.67)

where repeated indices are summed over the spatial directions. The Levi-Civita symbol $\epsilon_{ijk}$ is defined by

$$\epsilon_{ijk} = \left\{ \begin{array}{lll} 0, & \mbox{if any two... ...& \mbox{if $i,j,k$\space is an odd permutation of $1,2,3$ .} \end{array}\right.$$ (4.68)

This symbol is anti-symmetric on each pair of indices.