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

International Tables for Crystallography (2013). Vol. D, ch. 1.1, pp. 13-14

Section Tensors of rank 2

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France
Correspondence e-mail: Tensors of rank 2

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A bilinear form is said to be antisymmetric if [T({\bf x},{\bf y}) = - T({\bf y},{\bf x}).]Its components satisfy the relations [t_{ij}= - t_{ji}.]The associated matrix, T, is therefore also antisymmetric: [T = - T^{T}= \pmatrix { 0 & t_{12} & t_{13}\cr - t_{12} & 0 & t_{23}\cr - t_{13} & - t_{23} & 0\cr}.]The number of independent components is equal to [(n^{2} - n)/2], where n is the number of dimensions of the space. It is equal to 3 in a three-dimensional space, and one can consider these components as those of a pseudovector or axial vector. It must never be forgotten that under a change of basis the components of an axial vector transform like those of a tensor of rank 2.

Every tensor can be decomposed into the sum of two tensors, one symmetric and the other one antisymmetric: [T = S + A,]with [ S = ( T + T^{T})/2] and [A = (T - T^{T})/2].

Example.  As shown in Section[link], the components of the vector product of two vectors, x and y, [z_{k}= \varepsilon _{ijk}x^{i}y\hskip1pt^{j},]are really the independent components of an antisymmetric tensor of rank 2. The magnetic quantities, B, H (Section[link]), the tensor representing the pyromagnetic effect (Section[link]) etc. are axial tensors.

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