Dimitrienko Yu.I. Tensor Analysis and Nonlinear Tensor Functions

Springer-Science+Business Media, B.V., 2002. — 680 p. — ISBN: 978-94-017-3221-5.Tensor calculus appeared in its present-day form thanks to Ricci, who, first of all, suggested mathematical methods for operations on systems with indices at the close of the XIX century. Although these systems had been detected before, namely in investigations of non-Euclidean geometry by Gauss, Riemann, Christoffel and of elastic bodies by Cauchy, Euler, Lagrange, Poisson (see paragraph 'Sources of Tensor Calculus'), it was Ricci who developed the convenient compact system of symbols and concepts, which is widely used nowadays in different fields of mechanics, physics, chemistry, crystallophysics and other sciences.
At present tensor calculus goes on developing: advance directions appear and some concepts, introduced before, are re-interpreted. That is why, in spite of existing works on tensors (see References), there is an actual need of expounding these questions. To illustrate the above, we give one example. The following questions: 'May a second-order tensor be represented visually or graphically as well as a vector in three-dimensional space?' and 'What is a dyad?' - can cause difficulties even for readers experienced in studying of tensors.
The present book is intended for a reader beginning to study methods of tensor calculus. That is why the introduction of the book gives the well-known concept of a vector as a geometric object in three-dimensional space. On the basis of the concept, the author suggests a geometric definition of a tensor. This definition allows us to see a tensor and main operations on tensors. And only after this acquaintance with tensors, there is a formal generalized definition of a tensor in an arbitrary linear n-dimensional space. According to the definition, a tensor is introduced as an element of a factor-space relative to the special equivalence. The book presents this approach in a mathematically rigorous form (the preceding works did not take into account the role of zero vectors in the equivalence relation). It should be noted that this approach introduces the notion of a tensor as an individual object, while other existing definitions introduce not a tensor itself but only concepts related to a tensor: tensor components, or linear transformations (for a second-order tensor), or bilinear functionals etc. The principal idea, that a tensor is an individual object, is the basis of the present book.
The book is constructed by the mathematical principle: there are definitions, theorems, proofs and exercises at the end of each paragraph . The beginning and the end of each proof are denoted by symbols ▼ and ▲, respectively.
The indexless form of tensors is preferable in the book, that allows us to formulate different relationships in mechanics and physics compactly without overloading a physical essence of phenomena. At the same time, there are corresponding component and matrix representations of tensor relationships, when they are appropriate