International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 |
International Tables for Crystallography (2013). Vol. D, ch. 1.1, pp. 9-10
Section 1.1.3.7. Outer product^{a}Institut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France |
The tensor defined by is called the outer product of vectors x and y. (Note: The symbol is different from the symbol for the vector product.) The analytical expression of this tensor of rank 2 is
The components of this tensor satisfy the properties It is an antisymmetric tensor of rank 2.
Consider the so-called permutation tensor of rank 3 (it is actually an axial tensor – see Section 1.1.4.5.3) defined by and let us form the contracted product It is easy to check that
One recognizes the coordinates of the vector product.
Expression (1.1.3.4) of the vector product shows that it is of a covariant nature. This is indeed correct, and it is well known that the vector product of two vectors of the direct lattice is a vector of the reciprocal lattice [see Section 1.1.4 of Volume B of International Tables for Crystallography (2008)].
The vector product is a very particular vector which it is better not to call a vector: sometimes it is called a pseudovector or an axial vector in contrast to normal vectors or polar vectors. The components of the vector product are the independent components of the antisymmetric tensor . In the space of n dimensions, one would write
The number of independent components of is equal to or 3 in the space of three dimensions and 6 in the space of four dimensions, and the independent components of are not the components of a vector in the space of four dimensions.
Let us also consider the behaviour of the vector product under the change of axes represented by the matrix
This is a symmetry with respect to a point that transforms a right-handed set of axes into a left-handed set and reciprocally. In such a change, the components of a normal vector change sign. Those of the vector product, on the contrary, remain unchanged, indicating – as one well knows – that the orientation of the vector product has changed and that it is not, therefore, a vector in the normal sense, i.e. independent of the system of axes.
References
International Tables for Crystallography (2008). Vol. B, Reciprocal Space, edited by U. Shmueli. Heidelberg: Springer.