Tables for
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.1, p. 10

Section Vector product

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France
Correspondence e-mail: Vector product

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Consider the so-called permutation tensor of rank 3 (it is actually an axial tensor – see Section[link]) defined by [\left\{ \matrix{\varepsilon _{ijk} = + 1 &\hbox{ if the permutation } ijk \hbox{ is even}\hfill\cr \varepsilon _{ijk} = - 1 &\hbox{ if the permutation } ijk \hbox{ is odd}\hfill\cr \varepsilon _{ijk} = 0\hfill &\hbox{ if at least two of the three indices are equal}\cr} \right.]and let us form the contracted product [z_{k}= \textstyle{1\over 2} \varepsilon_{ijk}p^{ij}=\varepsilon_{ijk}x^iy^j. \eqno(]It is easy to check that[\left\{\matrix{z_{1} = x^{2} y^{3} - y^{2} x^{3}\cr z_{2} = x^{3} y^{1} - y^{3} x^{1}\cr z_{3} = x^{1} y^{2} - y^{2} x^{1}.\cr}\right.]

One recognizes the coordinates of the vector product.

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