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

International Tables for Crystallography (2006). Vol. D, ch. 1.1, p. 6

Section Covariant coordinates

A. Authiera*

aInstitut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France
Correspondence e-mail: Covariant coordinates

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Using the developments ([link] and ([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(]The n quantities [x_{i}] are called covariant components, and we shall see the reason for this a little later. The relations ([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[link]). It follows that [x\hskip 1pt^{j} = x_{i}g^{ij} \eqno(]with [g^{ij}g_{jk} = \delta_{k}^{i}. \eqno(]

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

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 ([link] and ([link], one can show in the same way that [{\bf e}_{j} = g_{ij}{\bf e}\hskip 1pt^{j}. \eqno(]

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(]

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