A course in mathematical analysis. Part 2. Book 2: Integral calculus of functions of several variables. Integral calculus on manifolds. Exterior differential forms. (Q2784251)

This is the fourth volume of a four-volume textbook covering a basic four-semester course ``Mathematical Analysis'' delivered by the author for the first and second years undergraduate students at the Mathematical Department of the Novosibirsk State University. (See Zbl 1031.26001, Zbl 1031.26002 and Zbl 1031.26003 for reviews of the first three volumes.) One volume corresponds to one semester. The textbook is written in very detail and follows Russian traditions of teaching the calculus which goes back to [\textit{G. M. Fikhtengol'ts}, ``Foundations of analysis'', Vol. 1-3. (Russian) (1949; Zbl 0041.37807; Zbl 0033.10703; Zbl 0034.31903)]. On the other hand it represents the current state of teaching the mathematical analysis at the Novosibirsk State University: in the book under review, many subjects are treated from original points of view. We will give brief comments to the contents. NEWLINENEWLINENEWLINEChapter 13. Calculus of functions of several variables (the theory of multiple integrals): \S 1. Integration of step-functions. \S 2. The definition and simplest properties of the Lebesgue integral. \S 3. Examples of systems with integration. \S 4. Theorems about passage to the limit under the integral sign. \S 5. Measurable functions and sets. \S 6. Measurable sets and functions in \(\mathbb R^n\). \S 7. Fubini theorem and its corollaries. \S 8. Change of variable in a multiple integral. In this chapter the author introduces the notions of the integral of a function in several variables and of the measure of a set. First he defines the notion of the integral for simple functions, then he extends the class of functions for which the notion of the integral is already defined. He shows that, using different methods of extension, we will arrive at different notions of integral, in particular, to the notions of the Riemann or Lebesgue integrals. The author uses the ``one-step extension scheme'' proposed by M.~Stone and constructs the notion of Lebesgue integral in more detail. The notion of the measure of a set is obtained as a partial case of the notion of the integral. The following basic theorems of the theory of multiple integrals are proven: convergence and differentiation theorems, Fubini and Tonelli theorems about reduction of a multiple integral to iterated integrals. NEWLINENEWLINENEWLINEChapter 14. Fourier series and Fourier transform: \S 1. Fourier series. Definition and preliminary results. \S 2. The general notion of an orthogonal system of functions. \S 3. Main theorems on pointwise convergence of the Fourier series. \S 4. Expansion of functions of bounded variation into Fourier series. \S 5. Fourier transform. The author introduces the notion of the trigonometric Fourier series and proves the Riemann--Lebesgue theorem on the Fourier coefficients of an integrable function, Dirichlet and Dini theorems on pointwise convergence of Fourier series, Parseval equality for square integrable functions, a sufficient condition for Fourier series to be uniformly convergent. The general notion of a Hilbert space is introduced, \(L_2([-\pi,\pi])\) is proven to be a Hilbert space, the trigonometric functions are shown to be a complete set of orthogonal functions in \(L_2([-\pi,\pi])\), convergence of Fourier series in \(L_2([-\pi,\pi])\) is studied. The notion of the Fourier transform is introduced and its basic properties, including the inversion formula, are proven. NEWLINENEWLINENEWLINEChapter 15. Integral calculus on manifolds. Exterior differential forms: \S 1. Multilinear functions and multivectors. \S 2. Calculus of exterior differential forms. \S 3. Supplementary information on smooth submanifolds of the space \(\mathbb R^n\). \S 4. Area of a \(k\)-manifold. \S 5. Exterior differential forms on a manifold. \S 6. Generalized Stokes integral formula. In this chapter the author introduces the notion of an exterior differential form and basic operations over such forms, including their multiplication, pullback, exterior differentiation, and integration. A generalized Stokes' theorem is proven and its applications are discussed, including the fixed-point theorem due to L.~Brouwer. NEWLINENEWLINENEWLINEEach chapter is followed by a list of exercises (81 exercises in total). Most of them are original ones and are selected in such a way as to provide students with the possibility of going deeper into the theory under discussion. NEWLINENEWLINENEWLINEThe textbook is intended for undergraduate students and university teachers.