International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D, ch. 1.1, pp. 3-4
Section 1.1.1.2. Notion of tensor in physics^{a}Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France |
Each of the quantities mentioned in the preceding section is represented by a mathematical expression. Some are direction independent and are represented by scalars: specific mass, specific heat, volume, pressure, entropy, temperature, quantity of electricity, electric potential. Others are direction dependent and are represented by vectors: force, electric field, electric displacement, the gradient of a scalar quantity. Still others cannot be represented by scalars or vectors and are represented by more complicated mathematical expressions. Magnetic quantities are represented by axial vectors (or pseudovectors), which are a particular kind of tensor (see Section 1.1.4.5.3). A few examples will show the necessity of using tensors in physics and Section 1.1.3 will present elementary mathematical properties of tensors.
Remark. Of the four examples given above, the first three (thermal expansion, dielectric constant, stressed rod) are related to physical property tensors (also called material tensors), which are characteristic of the medium and whose components have the same value everywhere in the medium if the latter is homogeneous, while the fourth one (expansion in Taylor series of a field of vectors) is related to a field tensor whose components vary at every point of the medium. This is the case, for instance, for the strain and for the stress tensors (see Sections 1.3.1 and 1.3.2 ).
References
Brillouin, L. (1949). Les tenseurs en mécanique et en élasticité. Paris: Masson & Cie.Voigt, W. (1910). Lehrbuch der Kristallphysik. Leipzig: Teubner. 2nd ed. (1929); photorep. (1966). New York: Johnson Reprint Corp. and Leipzig: Teubner.