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, p. 9
Section 1.1.3.6.1. Tensor nature of the metric tensor^{a}Institut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France |
Equation (1.1.2.17) describing the behaviour of the quantities under a change of basis shows that they are the components of a tensor of rank 2, the metric tensor. In the same way, equation (1.1.2.19) shows that the 's transform under a change of basis like the product of two contravariant coordinates. The coefficients and are the components of a unique tensor, in one case doubly contravariant, in the other case doubly covariant. In a general way, the Euclidean tensors (constructed in a space where one has defined the scalar product) are geometrical entities that can have covariant, contravariant or mixed components.