Topics on tensor calculus with applications to mechanics (Q2404961)

These lecture notes, intended as a comprehensive book, are based on some special courses for the students of mechanics at the Faculty of Mathematics and Mechanics of the Lomonosov State University from Moscow. There are six chapters and 81 references. The first chapter provides the basics of linear algebra and functional analysis. The second chapter deals with polynomials with tensor coefficients and a generalized Bézout theorem is proved and used to obtain the Hamilton-Cayley theorem. Canonical representations for adjoint, normal, Hermitian and unitary tensors on the module of complex tensors of even rank are provided. The next chapter contains applications, namely various ways to construct linearly independent isotropic, gyrotropic, orthotropic and transversal-isotropic tensors. The fourth chapter is devoted to eigenvalue problems for rank-\(2p\) tensors and tensor-block matrices consisting of tensors of the same even rank. The next chapter presents several applications in mechanics, for example in the linear micropolar elasticity theory. The last chapter is a study of thin-body domains of the three-dimensional Euclidean space. This chapter is completed by exercises. In conclusion, this is an useful monograph for mechanics-oriented mathematicians and students.