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 How to change the 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: 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.

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