On the Invariant Correspondence between the Symmetric Second-Rank Tensors and the Vector Systems

Abstract

The possibilities of various representations of high-rank tensors in three-dimensional space using lower-rank tensors, in particular, the representations of second-rank tensors by vector fields, is discussed. The purpose of these representations is a convenient geometric interpretation of certain mechanical properties of objects described by high-rank tensors. An invariant correspondence between symmetric second-rank tensors in three-dimensional space and pairs of vectors from the same space is proposed. On the basis of this correspondence, a geometric interpretation of the action of an isotropic symmetric tensor function of a tensor argument is given.