Chapter 3
Symmetry of Second Rank Tensors,
Cross Product
Abstract This chapter deals with the symmetry of second rank tensors and the
definition of the cross product of two vectors. In general, a second rank tensor con-
tains a part which is symmetric and a part which is antisymmetric with respect to
the interchange of its indices. For 3D, there exists a dual relation between the an-
tisymmetric part of the second rank tensor and a vector. The symmetric part of the
tensor is further decomposed into its isotropic part involving the trace of the tensor
and the symmetric traceless part. Fourth rank projection tensors are defined which,
when applied on an arbitrary second rank tensor, project onto its isotropic, antisym-
metric and symmetric traceless parts. The properties of dyadics, viz. second rank
tensors composed of the components of two vectors, are discussed. The dual relation
between its antisymmetric part and a vector corresponds to the definition of the cross
product or vector product, various physical applications are presented.
3.1 Symmetry
3.1.1 Symmetric and Antisymmetric Parts
An arbitrary tensorAof rank 2 can be decomposed into itssymmetricandantisym-
metricpartsAsymandAasyaccording to
Asymμν =1
2
(
Aμν+Aνμ)
, Aasyμν=1
2
(
Aμν−Aνμ)
. (3.1)
Clearly, the interchange of subscripts implies
Asymμν =Asymνμ, Aasyμν=−Aasyνμ. (3.2)In three dimensions,AsymandAasyhave 6 and 3 independent components.
© Springer International Publishing Switzerland 2015
S. Hess,Tensors for Physics, Undergraduate Lecture Notes in Physics,
DOI 10.1007/978-3-319-12787-3_3
33