Practical Analysis in One Variable [Hardcover]

Donald Estep (Author)

Book Description

ISBN-10: 0387954848 | ISBN-13: 978-0387954844 | Publication Date: October 1, 2002 | Edition: 1

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.

Editorial Reviews

Review

From the reviews:

MAA ONLINE

"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 textbook, especially in courses taken mostly by engineering majors: I am sure these students would find the approach to the book attractive and motivating."

 

D. Estep

Practical Analysis in One Variable

"A very good textbook, especially in courses taken mostly by engineering majors: I am sure these students would find the approach to the book attractive and motivating."—MAA ONLINE

"The author attempts to place in this book the basic ideas of real analysis and numerical analysis together in an applied setting that is both accessible and motivational to beginners. The book, not written in the widespread ‘theorem-proof’ style common in most mathematical textbooks, includes a lot of background and review material, numerous examples, visualizations and alternate explanations of some key ideas, and a big variety of exercises." (Peter Bundschuh, Zentralblatt MATH, Vol. 1038 (13), 2004)

"This book comes from a live human being, not from a publisher’s marketing group. … Estep has a very radical philosophy of teaching. For each topic, he frankly tells the reader why we want to do this, why we need to do it this way, and then he actually does it! Completely, correctly, and readably! … For the reader who wants to teach himself analysis, I can think of no better book for self study." (Reuben Hersh, SIAM Review, Vol. 45 (2), 2003)

"Estep combines the basic ideas of real analysis and numerical analysis in an applied framework. This framework is beautifully presented in the context of a fundamental approach to applied mathematical problem solving … . The book is written in an engaging manner … . Background and review material and numerous examples … are provided in a very appealing manner. Abstract concepts are carefully explained and supported with a wealth of examples and illustrations … . Summing Up: Highly recommended." (D. E. Bentil, CHOICE, July, 2003)

"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. … 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 will find the approach of the book attractive and motivating." (Mihaela Poplicher, MAA Online, July, 2003)

"This book is intended either for an honors calculus sequence or for the first real analysis course for mathematics majors who have completed the calculus sequence. … There is an abundance of exercises ranging from simple computations to estimates to computational projects. There is emphasis on providing explanation in solutions, and some exercises call for proofs of theorems. It should be an interesting book for either of the intended uses." (G. A. Heuer, Mathematical Reviews, 2003 h)

"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 thought 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." (Zentralblatt für Didaktik der Mathematik, Issue 1, 2003)

From the Back Cover

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.

Product Details

• Hardcover: 648 pages
• Publisher: Springer; 1 edition (October 1, 2002)
• Language: English
• ISBN-10: 0387954848
• ISBN-13: 978-0387954844
• Product Dimensions: 9 x 6.8 x 1.3 inches
• Shipping Weight: 2.3 pounds (View shipping rates and policies)
• Average Customer Review: 4.7 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #1,960,463 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

9 of 9 people found the following review helpful:

4.0 out of 5 stars Finally, the context of analysis laid bare, June 7, 2006

By 

Jeffrey D. Younger - See all my reviews
(REAL NAME)   

This review is from: Practical Analysis in One Variable (Hardcover)

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.

1 of 1 people found the following review helpful:

5.0 out of 5 stars Wow, what a GEM on analysis!, November 12, 2011

By 

Gaston Feeblebunny - See all my reviews

This review is from: Practical Analysis in One Variable (Undergraduate Texts in Mathematics) (Paperback)

I was amazed and pleasantly surprised when I found this book--on the shelves at my university's science and engineering library. It was one of the few math books (OK, unless you ask my wife) that I actually went and BOUGHT for my own personal library--even though it was probably going to be readily-available at the school's library. I liked it THAT much.

What surprised me wasn't only how GOOD it was, but that it took me so long to notice the book at all. In fact, though Estep's book was published in 2002, and my school acquired their copy in 2004, it seems I'm only the second person (in seven years) to borrow it. I'd never heard a reference to it (by way of comparison, for example) in the reviews of any other texts on analysis.

Now, I've looked at a LOT of (entry-level) books on (real) analysis. I think this one is a "keeper." You could probably name any book in the LC range QA299 to QA303, and I'd have *some* familiarity with it. Lang, Apostol and Rudin (Baby and Papa) of course, as well as Abbott, Strichartz (love his "Way of Analysis,"), Pugh, Thomson/Bruckner & Bruckner, you-name-it. Practical Analysis in One Variable by Estep is one I wanted on my own shelf. ALONG WITH Protter & Morrey's "First Course in Real Analysis" (get that one *instead* of Protter's "Basic Elements..."). Rosenlicht's "Intro to Analysis" (a thirteen-dollar Dover bargain) is another one I think any student of analysis won't regret buying.

Estep (a professor at Colorado State) covers all the essentials of what "real analysis" you actually SHOULD'VE learned in calculus--by re-covering functions, limits, continuity, polynomials, sequences, series, etc. in the first 15 chapters. The second section ("Differential and Integral Calculus") is 16 more chapters.

The last part (ten chapters) is titled "You Want Analysis? We've Got Your Analysis Right Here." No, really.

5.0 out of 5 stars Very good for self-study, July 5, 2011

By 

HuckleberryFinn - See all my reviews

Amazon Verified Purchase

This review is from: Practical Analysis in One Variable (Hardcover)

Nice book to develop mathematical rigor. Hardly any pre-requisites, I think, apart from high school math. Introduces concepts with motivation, which I personally found easier to follow.