Change of variance of the components of a tensor

Authier, A.

doi:10.1107/97809553602060000628

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

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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. Authier^a ^*

^a Institut 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) 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) 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 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: $[m \,\,{\rm d}^{2}x^{i}/ {\rm d}t^2 = g^{ij}\partial _{j}W.]$

References

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