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 contains 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 antisymmetric 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, antisymmetric 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.