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

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. Authiera*

aInstitut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France
Correspondence e-mail: aauthier@wanadoo.fr

1.1.3.6. Change of variance of the components of a tensor

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1.1.3.6.1. Tensor nature of the metric tensor

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Equation (1.1.2.17)[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 (1.1.2.19)[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.

1.1.3.6.2. How to change the variance of the components of a tensor

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Let us take a tensor product [t\hskip1pt^{ij}= x^{i}y\hskip1pt^{j}.]We know that [x^{i}= g^{ik}x_{k} \ \ {\rm and} \ \ y\hskip1pt^{j}= g\hskip1pt^{jl }y_{l}.]It follows that [t\hskip1pt^{ij}= g^{ik}g\hskip1pt^{jl }x_{k}y_{l}.][x_{k}y_{l}] is a tensor product of two vectors expressed in the dual space: [x_{k}y_{l }= t_{kl}.]

One can thus pass from the doubly covariant form to the doubly contravariant form of the tensor by means of the relation [t\hskip1pt^{ij}= g^{ik}g\hskip1pt^{jl }t_{kl}.]

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 [g^{ij}] or [g_{ij}], according to the case. For instance, [t^{l}_{k}= g\hskip1pt^{jl }t_{l k}; \quad t\hskip1pt_{k}^{ij} = g_{kl}t\hskip1pt^{ijl}.]

Remark.[g^i_{j}= g^{ik}g_{kj}= \delta ^i_{j}.]This is a property of the metric tensor.

1.1.3.6.3. Examples of the use in physics of different representations of the same quantity

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Let us consider, for example, the force, F, which is a tensor quantity (tensor of rank 1). One can define it:

  • (i) by the fundamental law of dynamics: [{\bf F} = m {\boldGamma},\quad \hbox{with }F^{i}= m \,\,\hbox{d}^{2}x^{i}/ {\rm d}t^{2},]where m is the mass and [\boldGamma] is the acceleration. The force appears here in a contravariant form.

  • (ii) as the derivative of the energy, W: [F_{i}= \partial W/\partial x^{i}= \partial _{i}W.]

    The force appears here in covariant form. In effect, we shall see in Section 1.1.3.8.1[link] that to form a derivative with respect to a variable contravariant augments the covariance by unity. The general expression of the law of dynamics is therefore written with the energy as follows: [m \,\,{\rm d}^{2}x^{i}/ {\rm d}t^2 = g^{ij}\partial _{j}W.]








































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