Editing Cross product (section)

===Index notation for tensors===
The cross product can alternatively be defined in terms of the [[Levi-Civita symbol#Levi-Civita tensors|Levi-Civita tensor]] ''E<sub>ijk</sub>'' and a dot product ''η<sup>mi</sup>'', which are useful in converting vector notation for tensor applications:

:<math>\mathbf{c} = \mathbf{a \times b} \Leftrightarrow\ c^m = \sum_{i=1}^3 \sum_{j=1}^3 \sum_{k=1}^3 \eta^{mi} E_{ijk} a^j b^k</math>

where the [[Indexed family|indices]] <math>i,j,k</math> correspond to vector components.  This characterization of the cross product is often expressed more compactly using the [[Einstein summation convention]] as
:<math>\mathbf{c} = \mathbf{a \times b} \Leftrightarrow\ c^m = \eta^{mi} E_{ijk} a^j b^k</math>

in which repeated indices are summed over the values 1 to 3.

In a positively-oriented orthonormal basis ''η<sup>mi</sup>'' = δ<sup>''mi''</sup> (the [[Kronecker delta]]) and <math> E_{ijk} = \varepsilon_{ijk}</math>  (the  [[Levi-Civita symbol]]). In that case, this representation is another form of the skew-symmetric representation of the cross product:

:<math>[\varepsilon_{ijk} a^j] = [\mathbf{a}]_\times.</math>

In [[classical mechanics]]: representing the cross product by using the Levi-Civita symbol can cause mechanical symmetries to be obvious when physical systems are [[isotropic]]. (An example: consider a particle in a [[Hooke's law]] potential in three-space, free to oscillate in three dimensions; none of these dimensions are "special" in any sense, so symmetries lie in the cross-product-represented angular momentum, which are made clear by the abovementioned Levi-Civita representation).{{Citation needed|date=November 2009}}