Irreducible Cartesian Tensors

Abstract

This paper considers certain simple and practically useful properties of Cartesian tensors in three‐dimensional space which are irreducible under the three‐dimensional rotation group. Ordinary tensor algebra is emphasized throughout and particular use is made of natural tensors having the least rank consistent with belonging to a particular irreducible representation of the rotation group. An arbitrary tensor of rank n may be reduced by first deriving from the tensor all its linearly independent tensors in natural form, and then by embedding these lower‐rank tensors in the tensor space of rank n. An explicit reduction of third‐rank tensors is given as well as a convenient specification of fourth‐ and fifth‐rank isotropic tensors. A particular classification of the natural tensors is through a Cartesian parentage scheme, which is developed. Some applications of isotropic tensors are given.