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JOURNALS // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika // Archive

Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2021 Number 3, Pages 69–74 (Mi vmumm4407)

Mechanics

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

D. V. Georgievskii

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: 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.

Key words: vector, symmetric tensor of the second rank, invariant, deviator, spherical tensor, invariant correspondence.

UDC: 514.86 + 539.3

Received: 11.11.2020


 English version:
Moscow University Mathematics Bulletin, Moscow University Mеchanics Bulletin, 2021, 76:3, 83–87

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© Steklov Math. Inst. of RAS, 2025