Tensor derivatives

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. 10

Section 1.1.3.8. Tensor derivatives

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

1.1.3.8. Tensor derivatives

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1.1.3.8.1. Interpretation of the coefficients of the matrix – change of coordinates

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We have under a change of axes: $[x^{\prime i}= A^i_{j}x\hskip1pt^{j}.]$ This shows that the new components, $[x'^{i}]$ , can be considered linear functions of the old components, $[x\hskip1pt^{j}]$ , and one can write $[A^i_{j}= \partial x'^{i}/\partial x\hskip1pt^{j}= \partial _{j} x'^{i}.]$ It should be noted that the covariance has been increased.

1.1.3.8.2. 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 $[\partial /\partial x^{i}]$ .

1.1.3.8.3. Differential operators

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If one applies the operator nabla to a scalar φ, one obtains $[\hbox{grad } \varphi = {\boldnabla }\varphi.]$ This is a covariant vector in reciprocal space.

Now let us form the tensor product of $[{\boldnabla }]$ by a vector v of variable components. We then have $[{\boldnabla} \otimes {\bf v} = {\partial v\hskip1pt^{j}\over\partial x^{i}}{\bf e}_{i}\otimes {\bf e}\hskip1pt^{j}.]$

The quantities $[\partial _{i}v\hskip1pt^{j}]$ form a tensor of rank 2. If we contract it, we obtain the divergence of v: $[\hbox{div } {\bf v} = \partial _{i}v^{i}.]$ Taking the vector product, we get $[\hbox{curl } {\bf v} = {\boldnabla } \wedge {\bf v}.]$ The curl is then an axial vector .

1.1.3.8.4. Development of a vector function in a Taylor series

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Let $[{\bf u}({\bf r})]$ be a vector function. Its development as a Taylor series is written $[u^{i}({\bf r} + \hbox{d}{\bf r}) = u^{i}({\bf r}) + {\partial u^i \over \partial x^j}\,\, \hbox{d}x\hskip1pt^{j} + {\textstyle{1 \over 2}} {\partial^2 u^i \over \partial x^j \partial x^k}\,\, \hbox{d}x\hskip1pt^{j}\,\, \hbox{d}x^{k} + \ldots . \eqno (1.1.3.5)]$ The coefficients of the expansion, $[\partial u^{i}/\partial x\hskip1pt^{j}]$ , $[\partial ^{2}u^{i}/\partial x\hskip1pt^{j}\partial x^{k}, \ldots]$ are tensors of rank $[2, 3, \ldots]$ .

An example is given by the relation between displacement and electric field: $[D^{i}= \varepsilon^i _{j}E\hskip1pt^{j}+ \chi ^{i}_{jk}E\hskip1pt^{j}E^{k}+ \ldots ]$ (see Sections 1.6.2 and 1.7.2 ).

We see that the linear relation usually employed is in reality a development that is arrested at the first term. The second term corresponds to nonlinear optics. In general, it is very small but is not negligible in ferroelectric crystals in the neighbourhood of the ferroelectric–paraelectric transition. Nonlinear optics are studied in Chapter 1.7 .

References

International Tables for Crystallography (2006). Vol. D. ch. 1.1, p. 10