International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 |
International Tables for Crystallography (2013). Vol. D, ch. 1.1, pp. 13-14
Section 1.1.4.5.3.1. Tensors of rank 2^{a}Institut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France |
A bilinear form is said to be antisymmetric if Its components satisfy the relations The associated matrix, T, is therefore also antisymmetric: The number of independent components is equal to , 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: with and .
Example. As shown in Section 1.1.3.7.2, the components of the vector product of two vectors, x and y, are really the independent components of an antisymmetric tensor of rank 2. The magnetic quantities, B, H (Section 1.1.4.3.2), the tensor representing the pyromagnetic effect (Section 1.1.1.3) etc. are axial tensors.