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

International Tables for Crystallography (2006). Vol. D, ch. 1.1, p. 9

Section Tensor nature of the metric tensor

A. Authiera*

aInstitut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France
Correspondence e-mail: Tensor nature of the metric tensor

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Equation ([link] describing the behaviour of the quantities [g_{ij} = {\bf e}_{i} \cdot {\bf e}_{j}] 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 ([link] shows that the [g^{ij}]'s transform under a change of basis like the product of two contravariant coordinates. The coefficients [g^{ij}] and [g_{ij}] 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.

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