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

International Tables for Crystallography (2013). Vol. D, ch. 1.1, p. 10

Section Generalization

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France
Correspondence e-mail: Generalization

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Consider a field of tensors [t\hskip1pt^{j}_{i}] that are functions of space variables. In a change of coordinate system, one has [t\hskip1pt^j_{i}= A^\alpha _{i}B^j_{\beta } t'^{\beta }_{\alpha }.]Differentiate with respect to [x^{k}]: [\eqalignno{{\partial t\hskip1pt^j_{i}\over \partial x^{k}} = \partial _{k}t\hskip1pt^j_{i} & = A^\alpha_{i}B^j_{\beta } {\partial t'^{\beta}_{\alpha }\over\partial x'^{\gamma }}{\partial x'^{\gamma }\over \partial x^{k}} &\cr \partial _{k}t\hskip1pt^j_{i} & =A^\alpha_{i}B^j_{\beta }A^\gamma_{k}\partial _{\gamma }t'^{\beta}_{\alpha }. &\cr}]It can be seen that the partial derivatives [\partial _{k}t\hskip1pt^j_{i}] behave under a change of axes like a tensor of rank 3 whose covariance has been increased by 1 with respect to that of the tensor [t\hskip1pt^j_{i}]. It is therefore possible to introduce a tensor of rank 1, [{\boldnabla}] (nabla), of which the components are the operators given by the partial derivatives [\partial /\partial x^{i}].

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