Practical Analysis in One Variable (Undergraduate Texts in Mathematics)

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. The essentials of real analysis are presented in the context of a fundamental problem of applied mathematics, which is to approximate the solution of a physical model. The framework of existence, uniqueness, and methods to approximate solutions of model equations is sufficiently broad to introduce and motivate all the basic ideas of real analysis. The book includes background and review material, numerous examples, visualizations and alternate explanations of some key ideas, and a variety of exercises ranging from simple computations to analysis and estimates to computations on a computer. The book can be used in an honor calculus sequence typically taken by freshmen planning to major in engineering, mathematics, and science, or in an introductory course in rigorous real analysis offered to mathematics majors. Donald Estep is Professor of Mathematics at Colorado State University. He is the author of Computational Differential Equations, with K. Eriksson, P. Hansbo and C. Johnson (Cambridge University Press 1996) and Estimating the Error of Numerical Solutions of Systems of Nonlinear Reaction-Diffusion Equations with M. Larson and R. Williams (A.M.S. Memoirs, 2000), and recently co-edited Collected Lectures on the Preservation of Stability under Discretization, with Simon Tavener (S.I.A.M., 2002), as well as numerous research articles. His research interests include computational error estimation and adaptive finite element methods, numerical solution of evolutionary problems, and computational investigation of physical models.

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. The essentials of real analysis are presented in the context of a fundamental problem of applied mathematics, which is to approximate the solution of a physical model. The framework of existence, uniqueness, and methods to approximate solutions of model equations is sufficiently broad to introduce and motivate all the basic ideas of real analysis. The book includes background and review material, numerous examples, visualizations and alternate explanations of some key ideas, and a variety of exercises ranging from simple computations to analysis and estimates to computations on a computer. The book can be used in an honor calculus sequence typically taken by freshmen planning to major in engineering, mathematics, and science, or in an introductory course in rigorous real analysis offered to mathematics majors.

Donald Estep is Professor of Mathematics at Colorado State University. He is the author of Computational Differential Equations, with K. Eriksson, P. Hansbo and C. Johnson (Cambridge University Press 1996) and Error of Numerical Solutions of Systems of Nonlinear Reaction-Diffusion Equations with M. Larson and R. Williams (A.M.S. 2000), and recently co-edited Collected Lectures on the Preservation of Stability under Discretization, with Simon Tavener (S.I.A.M., 2002), as well as numerous research articles. His research interests include computational error estimation and adaptive finite element methods, numerical solution of evolutionary problems, and computational investigation of physical models.

Background I was an eighteen-year-old freshman when I began studying analysis. I had arrived at Columbia University ready to major in physics or perhaps engineering. But my seduction into mathematics began immediately with Lipman Bers'calculus course, which stood supreme in a year of exciting classes. Then after the course was over, Professor Bers called me into his o?ce and handed me a small blue book called Principles of Mathematical Analysis by W. Rudin. He told me that if I could read this book over the summer,understandmostofit,andproveitbydoingmostoftheproblems, then I might have a career as a mathematician. So began twenty years of struggle to master the ideas in “Little Rudin. ” I began because of a challenge to my ego but this shallow reason was quickly forgotten as I learned about the beauty and the power of analysis that summer. Anyone who recalls taking a “serious” mathematics course for the?rst time will empathize with my feelings about this new world into which I fell. In school, I restlessly wandered through complex analysis, analyticnumbertheory,andpartialdi?erentialequations,beforeeventually settling in numerical analysis. But underlying all of this indecision was an ever-present and ever-growing appreciation of analysis. An appreciation thatstillsustainsmyintellectevenintheoftencynicalworldofthemodern academic professional. But developing this appreciation did not come easy to me, and the p- sentation in this book is motivated by my struggles to understand the viii Preface most basic concepts of analysis. To paraphrase J. Students of analysis are often beset with frustration. They ask "Why did you bound that quantity with that other quantity?" The typical answer, "Because it works out in the proof!" is certainly true, yet wholly unsatisfactory for the student.This book begins with models, real-world problems, that originally motivated the development of analysis. The student easily grasps how, and more importantly why, quantities are bounded. The days of staring at an algebraic form for hours are gone! (Well, mostly.)Instead of the normal calculus-style, simple-to-complex development of the material, Estep introduces concepts in the natural order of the real-world problems. For example, Lipschitz continuity is introduced early to solve obvious extensions to previous problems. The mathematical idea of continuity is progressively extended and provides much of the motivation for the second half of the book.By orienting on the problems solved by analysis, Estep avoids many of the bewildering difficulties encountered in traditional introductory treatments. This is the best introductory analysis book I've seen. I'm very surprised that it hasn't received more attention. This text places the basic ideas of real analysis and numerical analysis together in an applied setting that is both accessible and motivational to young students. The essentials of real analysis are presented in the context of a fundamental problem of applied mathematics, which is to approximate the solution of a physical model. The framework of existence, uniqueness, and methods to approximate solutions of model equations is sufficiently broad to introduce and motivate all the basic ideas of real analysis. The book includes background and review material, numerous examples, visualizations and alternate explanations of some key ideas, and a variety of exercises ranging from simple computations to analysis and estimates to computations on a computer. "The book is well suited for an honors calculus sequence typically taken by first-year undergraduates planning to major in engineering, mathematics, and science and for an introductory course in rigorous real analysis offered to mathematics majors."--Jacket We are making soup for dinner, and, following a recipe, we ask our roommate to go to the grocery store and buy 10 dollars worth of potatoes, carrots, and beef according to the proportions 3:2:1 by weight.