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

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

Section Differential operators

A. Authiera*

aInstitut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France
Correspondence e-mail: 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.

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