About Book 1
The book is based on the lectures delivered by the authors at the Physics Department of Moscow University. In the course of systematic exposition of the material, the authors gave prominence to the most important concepts and theorems, the most significant of the latter being called fundamental theorems. The authors tried to give the formulations of new concepts and theorems not long before their direct application. The sequence of presentation of the material corresponds to that accepted at the Physics Department of Moscow University. The chapter “Preliminary Information on the Main Concepts of Mathematical Analysis,” in particular, comes before the exposition of the systematic course because this chapter deals with some very important physical problems and discusses mathematical means for their solution. Thus, it becomes clear from the very beginning what problems and concepts constitute the subject matter of mathematical analysis. Our lecturing experience shows that such a preliminary elucidation of the range of problems treated in the course of the analysis makes it appreciably easier for the students to understand the basic mathematical concepts.
The authors paid due attention to the ever-increasing role of computations and methods of approximation; that is why they tried to use algorithmic form in proving theorems and carrying out calculations wherever it was possible. In Chap. 12, in particular, the main emphasis is laid on the algorithmic aspect of the methods of approximation, and only then are the methods substantiated.
In addition to the main part of the material, the authors consider it possible to include some sections given in small type.
About Book 2
This book is based on lectures delivered by the authors at Moscow State University over several years. As in Part 1, the authors aim to present the material systematically, highlighting the most important notations and theorems. In addition to the core curriculum, the book explores several important questions that play a significant role in various branches of modern mathematics and physics, such as the theory of measure and Lebesgue integrals, Hilbert spaces and self-adjoint linear operators, Fourier series regularisation, and the theory of differential forms in Euclidean spaces.
Some topics, such as the conditions for term-wise differentiation and integration of functional sequences and series, variable changes in multiple integrals, and Green’s and Stokes’ formulas, are treated more generally and under weaker assumptions than usual. The book also addresses computational mathematics, including approximate methods for evaluating multiple integrals and calculating function values from Fourier coefficients.
The material, along with Part 1 (Volume 1), constitutes a comprehensive university course in mathematical analysis. The text allows for some flexibility, as chapters on advanced topics such as Lebesgue integrals, Hilbert spaces, and related supplements can be skipped without affecting comprehension of the main material.
Translated from the Russian by Vladimir Shokurov
You can get the Part 1 here and here
You can get the Part 2 here and here
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