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. 6
Section 1.1.2.4.1. Covariant coordinates^{a}Institut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France |
Using the developments (1.1.2.1) and (1.1.2.5), the scalar products of a vector x and of the basis vectors can be written The n quantities are called covariant components, and we shall see the reason for this a little later. The relations (1.1.2.9) can be considered as a system of equations of which the components are the unknowns. One can solve it since (see the end of Section 1.1.2.2). It follows that with
The table of the 's is the inverse of the table of the 's. Let us now take up the development of x with respect to the basis :
Let us replace by the expression (1.1.2.10): and let us introduce the set of n vectors which span the space . This set of n vectors forms a basis since (1.1.2.12) can be written with the aid of (1.1.2.13) as
The 's are the components of x in the basis . This basis is called the dual basis. By using (1.1.2.11) and (1.1.2.13), one can show in the same way that
It can be shown that the basis vectors transform in a change of basis like the components of the physical space. They are therefore contravariant. In a similar way, the components of a vector x with respect to the basis transform in a change of basis like the basis vectors in direct space, ; they are therefore covariant: