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

International Tables for Crystallography (2013). Vol. D, ch. 1.1, p. 6

Section 1.1.2.4.1. Covariant coordinates

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France
Correspondence e-mail: aauthier@wanadoo.fr

1.1.2.4.1. Covariant coordinates

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Using the developments (1.1.2.1)[link] and (1.1.2.5)[link], the scalar products of a vector x and of the basis vectors [{\bf e}_{i}] can be written [x_{i} = {\bf x} \cdot {\bf e}_{i} = x\hskip 1pt^{j}{\bf e}_{j} \cdot {\bf e}_{i} = x\hskip 1pt^{j}g_{ij}. \eqno(1.1.2.9)]The n quantities [x_{i}] are called covariant components, and we shall see the reason for this a little later. The relations (1.1.2.9)[link] can be considered as a system of equations of which the components [x\hskip 1pt^{j}] are the unknowns. One can solve it since [\Delta (g_{ij}) \neq 0] (see the end of Section 1.1.2.2[link]). It follows that [x\hskip 1pt^{j} = x_{i}g^{ij} \eqno(1.1.2.10)]with [g^{ij}g_{jk} = \delta_{k}^{i}. \eqno(1.1.2.11)]

The table of the [g^{ij}]'s is the inverse of the table of the [g_{ij}]'s. Let us now take up the development of x with respect to the basis [{\bf e}_{i}]: [{\bf x} = x^{i}{\bf e}_{i}.]

Let us replace [x^{i}] by the expression (1.1.2.10)[link]: [{\bf x} = x_{j}g^{ij}{\bf e}_{i}, \eqno(1.1.2.12)]and let us introduce the set of n vectors [{\bf e}\hskip 1pt^{j} = g^{ij}{\bf e}_{i} \eqno(1.1.2.13)]which span the space [E^{n}\,\,(j = 1, \ldots, n)]. This set of n vectors forms a basis since (1.1.2.12)[link] can be written with the aid of (1.1.2.13)[link] as [{\bf x} = x_{j}{\bf e}\hskip 1pt^{j}. \eqno(1.1.2.14)]

The [x_{j}]'s are the components of x in the basis [{\bf e}\hskip 1pt^{j}]. This basis is called the dual basis. By using (1.1.2.11)[link] and (1.1.2.13)[link], one can show in the same way that [{\bf e}_{j} = g_{ij}{\bf e}\hskip 1pt^{j}. \eqno(1.1.2.15)]

It can be shown that the basis vectors [{\bf e}\hskip 1pt^{j}] transform in a change of basis like the components [x\hskip 1pt^{j}] of the physical space. They are therefore contravariant. In a similar way, the components [x_{j}] of a vector x with respect to the basis [{\bf e}\hskip 1pt^{j}] transform in a change of basis like the basis vectors in direct space, [{\bf e}_{j}]; they are therefore covariant: [\left. \matrix{{\bf e}\hskip 1pt^{j} = B\hskip1pt_{k}^{j} {\bf e}'^{k}\semi &{\bf e}'^{k} = A_{j}^{k} {\bf e}\hskip 1pt^{j}\cr x_{i} = A\hskip1pt_{i}^{j} x'_{j}\semi &x'_{j} = B_{j}^{i}x_{i}.\cr}\right\} \eqno(1.1.2.16)]








































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