On the invariant correspondence between the symmetric second-rank tensors and the vector systems

We discuss 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. The purpose of these representations is a convenient geometric interpretation of certain mechanical properties of objects described by high-rank tensors. We propose an invariant correspondence of symmetric tensors of the second rank in three-dimensional space and pairs of vectors from the same space. On the basis of this correspondence, a geometric interpretation of the action of an isotropic symmetric tensor function of a tensor argument is given.