Chapter

Chapter 1.1 Introduction to the properties of tensors

Physical properties of crystals

First Online Edition (2006)

Part 1. Tensorial aspects of physical properties

  1. A. Authier

Published Online: 1 JAN 2006

DOI: 10.1107/97809553602060000628

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Authier, A. 2006. Introduction to the properties of tensors. International Tables for Crystallography. D:1:1.1:3–33.

Author Information

  1. Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France

Publication History

  1. Published Online: 1 JAN 2006

Abstract

This chapter introduces the notion of tensors in physics (field tensors and material property tensors), starting from the matrix of physical properties. The basic properties of vector spaces and the elementary mathematical properties of tensors are then recalled. The most important part of the chapter, Section 1.1.4, is devoted to the symmetry properties of tensors: Neumann’s principle, Curie laws and the reduction of the components of polar and axial tensors of rank up to 4 for the 32 crystallographic groups. The relations between the thermodynamic functions and the physical property tensors are developed in Section 1.1.5.

Keywords:

  • Curie laws;
  • Neumann’s principle;
  • Onsager relations;
  • Voigt matrix;
  • adiabatic coefficients;
  • antisymmetric tensors;
  • axial tensors;
  • axial vectors;
  • bilinear forms;
  • contracted product;
  • contraction;
  • contravariant;
  • covariant;
  • dielectric constant;
  • dielectric polarization;
  • dielectric susceptibility;
  • direct inspection method;
  • dual space;
  • electric field;
  • electric polarization;
  • electrostriction;
  • enantiomorphic groups;
  • extensive quantities;
  • intensive quantities;
  • intrinsic symmetry;
  • linear forms;
  • magnetic induction;
  • material tensors;
  • matrix of physical properties;
  • matrix method;
  • metric tensor;
  • outer product;
  • permutation tensor;
  • photoelastic effect;
  • physical property tensors;
  • piezoelectric constants;
  • piezoelectric effect;
  • piezoelectric tensor;
  • piezoelectricity;
  • piezo-optic effect;
  • piezo-optic tensor;
  • polar tensors;
  • polar vectors;
  • pseudoscalar;
  • pseudovectors;
  • pyroelectric coefficients;
  • pyroelectric effect;
  • pyroelectricity;
  • reciprocal basis;
  • reciprocal lattice;
  • reciprocal space;
  • reduction of tensor components;
  • representation surface;
  • rotation matrix;
  • symmetric tensors;
  • tensor derivatives;
  • tensor product;
  • tensors;
  • thermal expansion;
  • uniaxial crystals;
  • vector product;
  • vector spaces