Mathematical analysis II
This is a very nice textbook on mathematical analysis, which will be useful to both the students and the lecturers. The list of chapters is as follows: Vol. I. 1) Some General Mathematical Concepts and Notation, 2) The Real Numbers, 3) Limits, 4) Continuous Functions, 5) Differential Calculus, 6) Integration, 7) Functions of Several Variables, 8) Differential Calculus in Several Variables, Some problems from the Midterm Examinations and Examination Topics. Vol. II. 9) Continuous Mappings (General Theory), 10) Differential Calculus from a General Viewpoint, 11) Multiple integrals, 12) Surfaces and Differential forms in Rn, 13) Line and Surface Integrals, 14) Elements of Vector Analysis and Field Theory, 15) Integration of Differential Forms on Manifolds, 16) Uniform Convergence and Basic Operations of Analysis, 17) Integrals Depending on the Parameter, 18) Fourier Series and Fourier Transform, 19) Asymptotic Expansions, Topics and Questions for Midterm Examinations and Examination Topics.
About style of explanation one can say that the definitions are motivated and precisely formulated. The proofs of theorems are in appropriate generality, presented in detail and without logical gaps. This is illustrated in many examples (many of them arise in applications) and each section ends with a list of problems and exercises, which extend and supplement the basic text. Finally, one can make several remarks on the approaches used. Real numbers are introduced axiomatically, the general concept of limits with respect to (filter) base is explained and used, e.g. in the definition of Riemann integral, and real powers of a positive number are introduced as limits of rational powers. Trigonometric functions are firstly introduced intuitively using the unit circle, then as sums of some power series and finally as inverse functions to functions arcsine and arccosine, which are defined after definition of the length of a curve. The multiple integral is firstly introduced as a Riemann integral over an n-dimensional interval (analogously as in 1-dimensional case). Then for the bounded set, D is defined as the integral over an interval containing D of the product of the given function and the characteristic function of the set D. The Lebesgue necessary and sufficient criterion for integrability is proved and frequently used. Line and surface integrals are introduced as integrals of differential forms over surfaces. Smooth and piecewise smooth surfaces are considered. Fundamental integral formulas (including the general Stokes formula) are proved. This material is explained in more detail in Chapter 15. In Chapter 14, vector versions of fundamental integral formulas are stated and vector fields having a potential are also studied.