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Advanced calculus. Reprint of the 2nd edition 2006 originally published by Thomson Brooks/Cole. (English) Zbl 1229.26001
Pure and Applied Undergraduate Texts 5. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4791-6/hbk). xviii, 590 p. (2009).
This volume is intended as a text for courses that furnish the backbone of the student’s undergraduate education in mathematical analysis. The author’s main goal is to rigorously present the fundamental concepts within the context of illuminating examples and stimulating exercises.
This book is self-contained and starts with the introduction of basic tools using the completeness axiom. Chapter 2 is concerned with basic notions such as monotonicity, linearity, and sum and product properties of convergent sequences. In the next chapter, limits of functions and basic properties of continuous functions are studied. The notion of differentiability is first introduced in chapter 4. Next, the author studies some basic elementary functions including the logarithmic and trigonometric functions. Chapter 6 is devoted to the basic concept of Riemann integration. Further topics on integration are developed in the next chapter. The study of the approximation of functions by Taylor polynomials is the subject of chapter 8, while chapter 9 deals with the convergence of sequences of functions. The study of functions of several variables is developed in chapter 10. Topological properties are carefully analyzed in chapter 11. The next chapter deals with some basic properties of metric spaces. I mention here the contraction mapping principle, which is a strong tool for the solvability-theory of nonlinear scalar differential equations. The material related to differentiation of functions of several variables is covered in chapters 13 and 14, while mappings between Euclidean spaces that have continuous differentiable component functions are studied in chapter 15. The inverse function theorem and the implicit function theorem are the focus of chapters 16 and 17, respectively. In chapter 18, the integral is first defined for bounded functions defined on generalized rectangles. In chapter 19, Fubini’s theorem on iterated integration is proved and the change of variables theorem for the integral of functions of several variables is established. The last chapter deals with the study of line and surface integrals. Throughout this volume, special care is given to the quality of the proofs.
This is a well-written and well-structured book with clearly explained proofs and a good supply of exercises, some of them are quite challenging. It is this reviewer’s opinion that the volume should be an excellent and useful tool for undergraduate students.

MSC:
26-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions
26Axx Functions of one variable
26Bxx Functions of several variables
97I10 Comprehensive works on analysis education
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