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. 9
Section 1.1.3.6. Change of variance of the components of a tensor^{a}Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France |
Equation (1.1.2.17) describing the behaviour of the quantities under a change of basis shows that they are the components of a tensor of rank 2, the metric tensor. In the same way, equation (1.1.2.19) shows that the 's transform under a change of basis like the product of two contravariant coordinates. The coefficients and are the components of a unique tensor, in one case doubly contravariant, in the other case doubly covariant. In a general way, the Euclidean tensors (constructed in a space where one has defined the scalar product) are geometrical entities that can have covariant, contravariant or mixed components.
Let us take a tensor product We know that It follows that is a tensor product of two vectors expressed in the dual space:
One can thus pass from the doubly covariant form to the doubly contravariant form of the tensor by means of the relation
This result is general: to change the variance of a tensor (in practice, to raise or lower an index), it is necessary to make the contracted product of this tensor using or , according to the case. For instance,
Let us consider, for example, the force, F, which is a tensor quantity (tensor of rank 1). One can define it: