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

International Tables for Crystallography (2006). Vol. D, ch. 1.1, p. 7

Section Linear forms

A. Authiera*

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

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A linear form in the space [E_{n}] is written [T({\bf x}) = t_{i}x^{i},]where [T({\bf x})] is independent of the chosen basis and the [t_{i}]'s are the coordinates of T in the dual basis. Let us consider now a bilinear form in the product space [E_{n} \otimes F_{p}] of two vector spaces with n and p dimensions, respectively: [T({\bf x},{\bf y}) = t_{ij}x^{i}y\hskip1pt^{j}.]

The np quantities [t_{ij}]'s are, by definition, the components of a tensor of rank 2 and the form [T({\bf x},{\bf y})] is invariant if one changes the basis in the space [E_{n} \otimes F_{p}]. The tensor [t_{ij}] is said to be twice covariant. It is also possible to construct a bilinear form by replacing the spaces [E_{n}] and [F_{p}] by their respective conjugates [E^{n}] and [F^{p}]. Thus, one writes [T({\bf x},{\bf y}) = t_{ij}x^{i}y\hskip1pt^{j} = t\hskip1.5pt_{i}^{j} x^{i}y_{j} = t\hskip1pt^{i}_{j}x_{i}y\hskip1pt^{j} = t^{ij}x_{i}y_{j},]where [t^{ij}] is the doubly contravariant form of the tensor, whereas [t\hskip1pt_{i}^{j}] and [t^{i}_{j}] are mixed, once covariant and once contravariant.

We can generalize by defining in the same way tensors of rank 3 or higher by using trilinear or multilinear forms. A vector is a tensor of rank 1, and a scalar is a tensor of rank 0.

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