International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D, ch. 1.1, p. 5
Section 1.1.2.2. Metric tensor^{a}Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France |
We shall limit ourselves to a Euclidean space for which we have defined the scalar product. The analytical expression of the scalar product of two vectors and is Let us put The nine components are called the components of the metric tensor. Its tensor nature will be shown in Section 1.1.3.6.1. Owing to the commutativity of the scalar product, we have
The table of the components is therefore symmetrical. One of the definition properties of the scalar product is that if for all x, then . This is translated as
In order that only the trivial solution exists, it is necessary that the determinant constructed from the 's is different from zero: This important property will be used in Section 1.1.2.4.1.