Introduction to Calculus and Analysis, Vol. 1 (Classics in Mathematics) [Hardcover]

Richard Courant (Author), Fritz John (Author)

Book Description

ISBN-10: 0387971513 | ISBN-13: 978-0387971513 | Publication Date: October 2, 1989

From the Preface: (...) The book is addressed to students on various levels, to mathematicians, scientists, engineers. It does not pretend to make the subject easy by glossing over difficulties, but rather tries to help the genuinely interested reader by throwing light on the interconnections and purposes of the whole. Instead of obstructing the access to the wealth of facts by lengthy discussions of a fundamental nature we have sometimes postponed such discussions to appendices in the various chapters. Numerous examples and problems are given at the end of various chapters. Some are challenging, some are even difficult; most of them supplement the material in the text. In an additional pamphlet more problems and exercises of a routine character will be collected, and moreover, answers or hints for the solutions will be given. This first volume of concerned primarily with functions of a single variable, whereas the second volume will discuss the more ramified theories of calculus (...).

Editorial Reviews

Review

The mathematical Gazette (75.1991.471): "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 There are three aspects of Courant and John in which it outshines (some) contemporaries: (i) the extensive historical references, (ii) the chapter on numerical methods, and (iii) the two chapters on physics and geometry. The exercises in Courant and John are put together purposefully, and either look numerically interesting, or are intuitively significant, or lead to applications. It is the best text known to the reviewer for anyone trying to make an analysis course less abstract. (...)" --This text refers to the Paperback edition.

About the Author

Biography of Richard Courant Richard Courant was born in 1888 in a small town of what is now Poland, and died in New Rochelle, N.Y. in 1972. He received his doctorate from the legendary David Hilbert in Göttingen, where later he founded and directed its famed mathematics Institute, a Mecca for mathematicians in the twenties. In 1933 the Nazi government dismissed Courant for being Jewish, and he emigrated to the United States. He found, in New York, what he called "a reservoir of talent" to be tapped. He built, at New York University, a new mathematical Sciences Institute that shares the philosophy of its illustrious predecessor and rivals it in worldwide influence. For Courant mathematics was an adventure, with applications forming a vital part. This spirit is reflected in his books, in particular in his influential calculus text, revised in collaboration with his brilliant younger colleague, Fritz John. (P.D. Lax) Biography of Fritz John Fritz John was born on June 14, 1910, in Berlin. After his school years in Danzig (now Gdansk, Poland), he studied in Göttingen and received his doctorate in 1933, just when the Nazi regime came to power. As he was half-Jewish and his bride Aryan, he had to flee Germany in 1934. After a year in Cambridge, UK, he accepted a position at the University of Kentucky, and in 1946 joined Courant, Friedrichs and Stoker in building up New York University the institute that later became the Courant Institute of Mathematical Sciences. He remained there until his death in New Rochelle on February 10, 1994. John's research and the books he wrote had a strong impact on the development of many fields of mathematics, foremost in partial differential equations. He also worked on Radon transforms, illposed problems, convex geometry, numerical analysis, elasticity theory. In connection with his work in latter field, he and Nirenberg introduced the space of the BMO-functions (bounded mean oscillations). Fritz John's work exemplifies the unity of mathematics as well as its elegance and its beauty. (J. Moser) --This text refers to the Paperback edition.

Product Details

• Hardcover: 661 pages
• Publisher: Springer (October 2, 1989)
• Language: English
• ISBN-10: 0387971513
• ISBN-13: 978-0387971513
• Product Dimensions: 9 x 6 x 1 inches
• Shipping Weight: 1.4 pounds
• Average Customer Review: 4.8 out of 5 stars  See all reviews (11 customer reviews)
• Amazon Best Sellers Rank: #347,188 in Books (See Top 100 in Books)

Customer Reviews

Most Helpful Customer Reviews

48 of 49 people found the following review helpful:

5.0 out of 5 stars Solutions to problems and exercises, March 4, 2008

By 

purveyor - See all my reviews

This review is from: Introduction to Calculus and Analysis, Vol. 1 (Classics in Mathematics) (Paperback)

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.

45 of 46 people found the following review helpful:

5.0 out of 5 stars Superior as an introductory calculus text!, May 27, 2002

By 

Justin Bost (Salisbury, NC) - See all my reviews

This review is from: Introduction to Calculus and Analysis, Vol. 1 (Classics in Mathematics) (Paperback)

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.

34 of 34 people found the following review helpful:

5.0 out of 5 stars More than an introduction, December 15, 2005

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Guilherme (São Paulo, SP, Brasil) - See all my reviews

This review is from: Introduction to Calculus and Analysis, Vol. 1 (Classics in Mathematics) (Paperback)

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.