The book under review is the second volume of a two-volume set. (For Volume I see [the authors, Real analysis. Foundations and functions of one variable. Translated from the 3rd Hungarian edition. New York, NY: Springer (2015; Zbl 1325.26001)]) This second volume is mainly concerned with multivariate calculus and with series of numbers and of functions. Both volumes were published originally in Hungarian and now in English translation.
This book develops a thorough treatment of multivariate derivatives, viewing them both as linear transformations and as partial derivatives. There are also provided rigorous proofs of several basic results, including the inverse function theorem, implicit function theorem, and Lagrange multipliers. The author also develops multiple integrals on Jordan-measurable sets as well as line and surface integrals. The book includes valuable information on the theory of infinite series, convergence of sequences, Cesàro and Abel summation, multiplication of series, and convergence criteria. The final part is devoted to applications in numerical analysis and number theory.
The content of this volume is divided into the following eight chapters: 1. \({\mathbb R}\rightarrow {\mathbb R}\) functions; 2. Functions from \({\mathbb R}^p\) to \({\mathbb R}^q\); 3. The Jordan measure; 4. Integrals of multivariable functions I; 5. Integrals of multivariable functions II; 6. Infinite series; 7. Sequences and series of functions; 8. Miscellaneous topics. The material is introduced gradually and the abstract notions and properties are illustrated with about 600 exercises. The book is consistent in addressing the classical analysis of real functions of several variables. This volume will appeal to students in pure and applied mathematics, as well as scientists looking to acquire a firm footing in mathematical analysis.