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Title: Introduction to Calculus and Analysis, Volume 1 (Classics in Mathematics) by Richard Courant, Fritz John ISBN: 3-540-65058-X Publisher: Springer Verlag Pub. Date: January, 1999 Format: Paperback Volumes: 1 List Price(USD): $49.95 |
Average Customer Rating: 4.88 (8 reviews)
Rating: 5
Summary: simply the Best Calculus Book
Comment: An intuitive, rigorous and a beautifully conceptual approach to calculus is what distinguishes this book from the thousands of run-of-the-mill "Calculus I" textbooks published every year.
This is not surprising because 1) Courant and John were both important German-born mathematicians, both schooled in that great mathematical mecca, Gottingen, both making fundamental contributions to many classical branches of pure and applied mathematics. Courant is an especially important mathematician since he not only studied under the greats Minkowski and Hilbert - even serving as the latter's assistant - but founded the Courant Institute of Mathematical Sciences in New York, modelled on the Gottingen Mathematical Institute. 2) That typical German thoroughness and emphasis on the mastery of the "fundamental concepts", so dear to German textbooks, is evident in all sections of the book, particularly in the introductory material on the number continuum, functions, continuity etc.
The exercises at the end of chapters are substantial and excellent, and help to develop proof skills in students as well as a subtle mathematical intuition.
Mathematics is best learnt by studying books written by important mathematicians. Classic books like these should always serve to prove the truth of Abel's dictum that to master mathematics one should 'study the masters and not the pupils'.
Rating: 5
Summary: You must have this.
Comment: My review of the first volume pretty much applies here as well. How many *calculus* texts have an introduction to complex variables, and the theory of analytic functions? This is the only one I've ever seen, and I don't think anyone else could make it more enriching than Courant. Useful material on vector calculus, the theory of matrices, and even introductory material on the *calculus of variations* (something we usually don't see at *all* in the undergrad curriculum) is included. It is refreshing to have an instructor like Courant, who doesn't assume we can't follow higher mathematical roads, but also doesn't sit at the other end of the spectrum, just waving a wand and "poof, here is the result".
Courant also published a standard reference work (also two volumes, I believe) on Mathematical Physics. While the level of mathematics required is post-grad, I was still able to read sizeable sections of it without getting lost.
We can only hope Dover decides to publish Courant's works one day, to make them a little more affordable. But still, you can buy both volumes of Courant's intro to calculus for about the same price as a modern calculus text that waters down the material, and on top of that, provides inadequate explanation for the material it does cover.
Rating: 5
Summary: Superior as an introductory calculus text!
Comment: 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. Anyone with a background in symbolic logic will instantly be able to follow the proof methods, and most discrete math courses have a section on proofs. In any event, ignorance of proof methods will not detract much from the book's value. 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. In addition, 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.
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Title: What Is Mathematics?: An Elementary Approach to Ideas and Methods by Richard Courant, Herbert Robbins, Ian Stewart ISBN: 0195105192 Publisher: Oxford University Press Pub. Date: May, 1996 List Price(USD): $21.50 |
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Title: Differential and Integral Calculus: Volume II (Wiley Classics Library) by Richard Courant ISBN: 0471608408 Publisher: John Wiley & Sons Pub. Date: February, 1988 List Price(USD): $89.95 |
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Title: A Course of Pure Mathematics by G.H. Hardy ISBN: 0521092272 Publisher: Cambridge University Press Pub. Date: 04 March, 1993 List Price(USD): $35.00 |
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Title: Options Markets by John C. Cox, Mark Rubinstein ISBN: 0136382053 Publisher: Pearson Education POD Pub. Date: March, 1985 List Price(USD): $63.40 |
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Title: Introductory Statistics for Business and Economics, 4th Edition by Thomas H. Wonnacott, Ronald J. Wonnacott ISBN: 047161517X Publisher: Wiley Text Books Pub. Date: 02 January, 1990 List Price(USD): $100.95 |
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