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Practical Analysis in One Variable

Donald Estep
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
[Reviewed by
Mihaela Poplicher
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Published as part of the "Undergraduate Texts in Mathematics" series, this book is intended by the author as a textbook for two kinds of courses: the honors calculus sequence, whose typical students are engineering majors, most of whom have advanced placement in calculus; and the introductory course in real analysis taken by mathematics majors who have completed the calculus sequence. The material was used successfully in these types of courses at Georgia Tech, Colorado State University; some parts of the book have also been tested at Chalmers University of Technology in Sweden.

The book contains most of the classical topics in real analysis, but they are presented in the context of approximating solutions of physical models, a fundamental problem in applied mathematics. This approach is due mostly to the author's research interests in applied mathematics and numerical analysis. As Donald Estep notes in the Preface: "This book attempts to place the basic ideas of real analysis and numerical analysis together in an applied setting that is both accessible and motivational to young students of all technical persuasions."

Practical Analysis in One Variable is divided in three parts: "Numbers and Functions, Sequences and Limits" (containing 15 chapters), "Differential and Integral Calculus" (with 16 chapters), and "You Want Analysis? You've Got Your Analysis Right Here" (10 chapters).

Although the book is not a sequence of theorems followed by proofs (as in a standard text) it does contain quite rigorous proofs, usually placed as motivations, followed by the statements of the theorems.

The first part of the book starts (shockingly, at least for me) with a chapter on "Mathematical Modeling" and sections such as "The Dinner Soup Model" and "The Muddy Yard Model". This approach probably attracts students' attention; both of these models are solved later in the book. The first part has chapters dedicated to the sets of numbers: natural numbers, rational numbers, and real numbers (with motivations for the need of introducing them). It also has chapters on all kinds of functions (but not exponentials and logarithms!), on sequences and limits, inverse functions, fixed points and contraction maps. Most of these chapters also include several mathematical models. Somewhat surprising, in this first part of the book there is a chapter on Lipschitz Continuity (continuity of a function over an interval is studied in some detail in the last part of the book.)

The second part of the book treats the usual topics in differentiation and integration, but continues to motivate the discussion of most topics using many mathematical models. The concept of "Differentiability on Intervals" is discussed after the introduction of the "Strong Differentiability on Intervals" and "Uniform Strong Differentiability". The chapter "Rocket Propulsion and the Logarithm" begins with "A Model of Rocket Propulsion" as a motivation for the study of the properties of logarithm and logarithmic functions, and continues with an in-depth study of the logarithm, including "Derivatives and Integrals Involving the Logarithm" and "Solving the Model of Rocket Propulsion."

Also different from most other books is the placement of the chapter on "Constant Relative Rate of Change and the Exponential" after the chapter on logarithm, and the definition of the exponential function as the inverse of the logarithm (in most books, the exponential is discussed first and the logarithm is introduced as the inverse of the exponential.) The chapter on "Modeling with Differential Equations" appears before the one on "Antidifferentiation" and ends (appropriately!) with "Solving Galileo's Model of a Free-Falling Object." Also included in this second part of the book are: a chapter on "A Mass-Spring System and the Trigonometric Functions" and a chapter on "Fixed Point Iteration and Newton's Method."

The third part of the book, as suggested in its title, includes chapters on somewhat more advanced topics. The titles of a few chapters give a very good idea of the material included here: "Sequences of Functions", "Delicate Limits and Gross Behavior" (including a section on "L'Hospital's Rule"), "The Weierstrass Approximation Theorem", "The Picard Iteration", "The Forward Euler Method." The discussion in this last part of the book is not so much based on mathematical models, and many of the chapters have sections on "Unanswered Questions" including bibliography for further, more in-depth study.

The book ends with "A Conclusion or an Introduction?" where the author confesses: "It did not seem right to finish with a conclusion since this book is only an introduction to the vast subject of analysis. Instead, we finish by discussing where to go after reading this book." A discussion of "further reading" as well as a (long, for students at least) bibliography follow.

In addition to the book itself, one might also want to look at the author's website, where "Supplemental Material and Corrections" can be found.

I confess that when I first started reading this book I was intrigued by the new approach of real analysis but did not quite see what it might be good for. In the end, however, I was convinced that it could be a very good text book, especially in courses taken mostly by engineering majors: I am sure these students would find the approach of the book attractive and motivating.

Mihaela Poplicher is assistant professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is