Chapter 5
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 prin-
ciple 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 bire-
fringence, electric and magnetic torques. Geometric interpretations of symmetric
tensors are possible via bilinear forms or via a linear mapping. The scalar invari-
ants are discussed. The consequences of a Hamilton-Cayley theorem for triple and
quadruple products of symmetric traceless tensors are presented. A volume conserv-
ing affine mapping of a sphere onto an ellipsoid is considered.
5.1 Isotropic and Symmetric Traceless Parts
Here the properties of symmetric tensors are discussed. For a tensorSthis means
Sμν=Sνμ.As mentioned before, cf. (3.3) and (3.4), such a tensor is equal to the sum of its
isotropic part, involving its traceSλλand its symmetric traceless part:
Sμν=1
3
Sλλδμν+Sμν, (5.1)with
Sμν =Sμν−1
3
Sλλδμν. (5.2)This decomposition is invariant under a rotation of the coordinate system. Notice that
Sνν =0, that is why this part of the tensor is calledsymmetric traceless. Frequently,
it is also referred to asirreduciblepart, because it can not be associated with a lower
rank tensor. Here, it is also calledanisotropicpart, because the symmetric traceless
partoftensorsusedinapplicationscharacterizestheanisotropyofphysicalproperties.
© Springer International Publishing Switzerland 2015
S. Hess,Tensors for Physics, Undergraduate Lecture Notes in Physics,
DOI 10.1007/978-3-319-12787-3_5
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