Symmetric Second Rank Tensors

Abstract

This chapter deals with properties and applications of symmetric second rank tensors which are composed of isotropic and symmetric traceless parts. A principle axes representation is considered and the cases of isotropic, uniaxial and biaxial tensors are discussed. Applications comprise the moment of inertia tensor, the radius of gyration tensor, the molecular polarizability tensor, the dielectric tensor and birefringence, electric and magnetic torques. Geometric interpretations of symmetric tensors are possible via bilinear forms or via a linear mapping. The scalar invariants are discussed. The consequences of a Hamilton-Cayley theorem for triple and quadruple products of symmetric traceless tensors are presented. A volume conserving affine mapping of a sphere onto an ellipsoid is considered.