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 1.1.3.6.2. 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

#### 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 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,

Remark.This is a property of the metric tensor.