ADEJUMOBI, M. Tensory a jejich aplikace v mechanice [online]. Brno: Vysoké učení technické v Brně. Fakulta strojního inženýrství. 2020.

Posudek vedoucího

Tomáš, Jiří

The present thesis consists of the theoretical and application parts. The first part contains an algebraic introduction of tensors together with the list of operations over them like vector space operations, contraction, raising and lowering of indices. Nevertheless, the opeartion of tensor product is absent. On the other hand, the metric tensor should be presented after giving the definition of a manifold, tangent and cotangent bundle and tensor field and precede the definition of the operation of raising and lowering of indices. The next sections give the explanation of the concepts of Levi Civita connection, parallel transport and geodesic curve. The application part starts with Gauss and Stokes theorem and continues with the elementary concepts of the continuum mechanics like configuration and deformation. In its center, the elementary kinds of deformation tensors are presented together with the stress tensor, which regards surface forces (tractions). Finally, there are disscussed the principle stresses. As for the assesment, there are some conceptional flaws, namely in the lay out of the sections. Besides the inadequate placement of the metric tensor disscussed above I am missing the explicit tensor character of Gauss and Stokes theorems, in the first row for the divergency and curl operators. The placement of this part to the beginning of the thesis is also not optimal. Further, the eigen value problem and polar decomposition should be presented in the very end in connection with the principal stresses. Finally, I am missing some illustrative, original pictures. On the other hand, the author succesfully managed some advanced concepts of modern differential geometry like manifold, connection, parallel transport and geodesics and correctly indicated the relation between metric tensor and the first fundamental form from the classical differential geometry. All of those concepts are intrinsic and quite difficult. Taking into account the reasons above and the rather short time for the elaboration I suggest the assesment C.

Posudek oponenta

Doupovec, Miroslav

The aim of the thesis under review is to describe a survey of basic concepts and properties of tensors including basic operations over them and to present some applications in mechanics. As noted in the conclusion, the main idea of the author was to introduce tensors as certain extensions of vectors. Of course, this is very good idea. Unfortunately, such approach was not presented successfully in the present thesis. Moreover, the thesis has not character of a standard mathematical text. • Chapter 2 (which is dedicated to tensors) does not contain any correct definition of a tensor. Sentences from 2.3 „Tensor is a mathematical object that has n indices and also obeys certain transformation rules“ or „Tensor is an object that transforms like a tensor“ are not definitions of a tensor. The standard and correct way is to introduce a tensor of type (k,l) as certain multilinear map over some vector space V. I also miss the definition a vector space, linear form, bilinear form, quadratic form etc. • Some notions are being used before their definition (e.g. vector field at page 23, 24). • One of the most important operations with tensors is the tensor product. But this is not defined in Chapter 2. • Many definitions and conclusions are not precise, some of them are even confusing. For example, $A$ in formula (1.3) at page 16 means a quadratic form (on the left hand side) and a matrix (on the right hand side). • The author cites some classical results from Wikipedia (not from standard textbooks). • At page 26 it is written „Having defined the general concept of tensor over an n-dimensional vector space…“. This is not true: the thesis does not contain any such definition. • Contraction (section 2.4.3) can be defined more generally (not only with respect to the last indices) • 4.1.1 is the well known Gauss-Ostrogradsky Theorem from integral calculus, which requires no use of tensors. So I do not understand why this classical theorem is presented in the thesis as an application of tensors. The same is true for Stoke‘s Theorem 4.2.1. On the other hand, in Chapter „4: Application of tensors…“ I expected the general tensor form of Stoke‘s Theorem from Differential Geometry for the integration of differential forms. Using such a point of view, the classical theorems (Stokes, Gauss-Ostrogradsky, Green) could be obtained as consequences of the general Stoke‘s Theorem. The thesis contains also many other imperfections. By my opinion, the objectives were fulfilled partially. That is why, my overall classification is „E: sufficient“.

eVSKP id 124597