International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D, ch. 1.1, p. 7
Section 1.1.3.2. Behaviour under a change of basis^{a}Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France |
A multilinear form is, by definition, invariant under a change of basis. Let us consider, for example, the trilinear form (1.1.3.1). If we change the system of coordinates, the components of vectors x, y, z become
Let us put these expressions into the trilinear form (1.1.3.1):
Now we can equally well make the components of the tensor appear in the new basis:
As the decomposition is unique, one obtains
One thus deduces the rule for transforming the components of a tensor q times covariant and r times contravariant: they transform like the product of q covariant components and r contravariant components.
This transformation rule can be taken inversely as the definition of the components of a tensor of rank .
Example. The operator O representing a symmetry operation has the character of a tensor. In fact, under a change of basis, O transforms into O′: so that Now the matrices A and B are inverses of one another: The symmetry operator is a tensor of rank 2, once covariant and once contravariant.