Introduction to Calculus and Analysis I

by Richard Courant, Fritz John, Richard Courant & Fritz John

Book Overview

From the reviews: "Volume 1 covers a basic course in real analysis of one variable and Fourier series. It is well-illustrated, well-motivated and very well-provided with a multitude of unusually useful and accessible exercises. (...) There are three aspects of Courant and John in which it outshines (some) contemporaries: (i) the extensive historical references, (ii) the chapter on numerical methods, and (iii) the two chapters on physics and geometry...

• Format: Paperback
• Language: English
• ISBN: 354065058X
• ISBN13: 9783540650584
• Release Date: December 1998
• Publisher: Springer
• Length: 661 Pages
• Weight: 2.55 lbs.
• Dimensions: 1.6" x 6.3" x 9.3"

Customer Reviews

Solutions to problems and exercises

Springer have reprinted the original 1960s Wiley editions of "Introduction to Calculus and Analysis" volumes I and II by Courant and John in three new volumes under their "Classics in Mathematics" title: "Introduction to Calculus and Analysis I (pages 1-661)" (ISBN: 3-540-65058-X), "Introduction to Calculus and Analysis II/1, Chapters 1-4 (pages 1-542)" (ISBN: 3-540-66569-2), and "Introduction to Calculus and Analysis II/2, Chapters 5-8 (pages 543-954)" (ISBN: 3-540-66570-6). The back section of Volume II/2 (pages 821-939) has solutions to the exercises in both the books comprising volume II, that is "Introduction to Calculus and Analysis II/1" and "Introduction to Calculus and Analysis II/2". Note that when Volume I of the original Courant and John "Introduction to Calculus and Analysis" was published in the 1960s by Wiley, an accompanying solutions manual for Volume I was prepared by Prof. Albert A. Blank. When Volume II was published by Wiley, Prof. Blank's solutions were incorporated into the back of Volume II (in other words, Volume II comes with the answers to the questions at the back of the book... or in the back of Volume II/2 in the case of this Springer "Classics in Mathematics" reprint.) However, the Springer reprint of Wiley's Volume I lacks solutions to the exercises in the textbook. If you buy Volume I, do a check on the Internet for an old 1960s copy of Prof. Albert Blank's "Problems in Calculus and Analysis", which is the original solutions manual to Courant's Volume I.

More than an introduction

Those books (volumes 1-2) can be seen as a new edition of Courant's classical Differential and Integral Calculus, volumes 1-2 (that can still be used for general calculus courses). The first volume was written while Courant was still alive, and the second was postumous. I believe that they are the best work to start understanding analysis. Indeed, for the general scientist (as a physicist) it contains all the theory needed for any application. The book is not easy reading though. Much of the text can be understood on first reading, but there are pretty profound sections, mostly on the appendixes, that turn the book genuinely onto a book of analysis. The second volume requires some mathematical maturity, and I doubt whether it is suitable for beginners, but it is simply the best book of multivariate calculus that I know - and it is really difficult to think of a better presentation. Courant was a giant, and his concept of mathematics shines in every page of those books (although he did not see the publication of the second volume, his hand can be seen in every page). For the serious mathematician, a must-have. For the beginner, the best way to get in love. Courant and John don't lie, they give every proof and guide you most gently in this complicated garden called mathematics. I'd give it aleph stars if it was possible.

Superior as an introductory calculus text!

I don't use the word "superior" lightly, but this book definitely warrants it. Courant was a first rate teacher and mathematician, and his brilliance shows in his exposition. The main obstacle to some readers may be that Courant does not follow the "cookbook calculus" approach that seems so rampant today, but actually bothers to prove his results. He does, however, reserve most of the more difficult proofs for the appendices at the end of the chapter, which is most appreciated. The result is an exciting read, yet rigorous. The reader is very well prepared for future courses in mathematical analysis, and even has a leg up on real analysis. While Courant's insistence on proof does mean that the student needs to have a basic grounding in proof methods, this is usually a standard part of the undergraduate curriclum. Courant rightly recognizes that calculus should be taught in a logical, yet rigorous presentation from the beginning. The absence of this in modern texts mean that students learn how to manipulate formulas, but have no idea what makes the results they are assuming true. The "mechanics" of calculus and analysis, the most crucial thing to be learn, is missed. In particular, I enjoyed his presentation of integration *before* differentiation, which goes against the grain of basic calc texts, yet is historically and pedagogically correct. Integration actually paves the way for differentiation, and gives more motivation for the FTC. Most texts on real analysis work in that order anyway, as an understanding of Lebesgue measure and integration is crucial to understanding the process of differentiation. In addition, I don't think I have ever before or since seen such a careful explanation of the theory of the logarithm or exponential functions. Again, the presentation makes it work, as just introducing the "exponential function", then a little later, the "log function" as the "inverse" of the exponential function is, to put it mildly, artificial and distasteful. The natural progression from the definite integral definition of the logarithm to the exponential function is displayed in its full glory. In short, Courant manages to present some of the most crucial results of calculus and basic analysis without boring the reader to tears with arcane details, or worse, leaving the reader hanging on important theorems and ideas. This is a balance only a great mathematician could strike, and it is clear why this book remains a classic after almost 60 years. Note: The second volume of this work covers the multivariable portion of calculus, and will be more difficult to follow without prior exposure to the subject. However, the introductions to the theory of matrices and the calculus of variations are very readable, and it is recommended that the reader take the time to peruse them. Also, don't miss the material on special functions, lightly touched on in the first volume, but explained in fuller detail in the second.

A very good course in Calculus and Analysis

Courant and John have done an excellent job in making me a Calculus-and-Analysis fan. The book is very well written, well motivated and easy to understand - even for me, as an absolute beginner. This doesn't mean it hasn't got any difficult parts, but because the books is so well written, it becomes even more challenging. One of the best books on Calculus and Analysis, even on Mathematics in general. It makes you want to throw your poor high school textbooks in the waste bin.

A Classic in Mathematical Exposition

Richard Courant was a master of mathematical exposition, and this is one of his best works. In keeping with Courant's philosphy, this book is free from the excessive abstraction often found even in introductory calculus textbooks. Nevertheless it does not gloss over difficulties in the material, and is in no sense an easy book. This book a complete rewrite of Courant's original "Calculus" which first appeared in German. An especially good chapter is the one on the "Theory of Plane Curves."