Ebook: A Course in Real Analysis

Features

Covers both single variable functions and multivariable functions, making the book suitable for one- and two-semester courses
Provides a detailed axiomatic account of the real number system
Develops the Lebesgue integral on Rn from the beginning
Gives an in-depth description of the algebra and calculus of differential forms on surfaces in Rn
Offers an easy transition to the more advanced setting of differentiable manifolds by covering proofs of Stokes’s theorem and the divergence theorem at the concrete level of compact surfaces in Rn
Summarizes relevant results from elementary set theory and linear algebra
Contains over 90 figures that illustrate the essential ideas behind a concept or proof
Includes more than 1,600 exercises throughout the text, with selected solutions in an appendix
Solutions manual available upon qualifying course adoption

A Course in Real Analysis provides a rigorous treatment of the foundations of differential and integral calculus at the advanced undergraduate level. The book’s material has been extensively classroom tested in the author’s two-semester undergraduate course on real analysis at The George Washington University.

The first part of the text presents the calculus of functions of one variable. This part covers traditional topics, such as sequences, continuity, differentiability, Riemann integrability, numerical series, and the convergence of sequences and series of functions. It also includes optional sections on Stirling’s formula, functions of bounded variation, Riemann–Stieltjes integration, and other topics.

The second part focuses on functions of several variables. It introduces the topological ideas (such as compact and connected sets) needed to describe analytical properties of multivariable functions. This part also discusses differentiability and integrability of multivariable functions and develops the theory of differential forms on surfaces in Rn.

The third part consists of appendices on set theory and linear algebra as well as solutions to some of the exercises. A full solutions manual offers complete solutions to all exercises for qualifying instructors.

With clear proofs, detailed examples, and numerous exercises, this textbook gives a thorough treatment of the subject. It progresses from single variable to multivariable functions, providing a logical development of material that will prepare students for more advanced analysis-based courses.