I will not say, as is common in reviewers of books on calculus, that this is the best book of calculus that there is. Indeed, calculus is a subject with so many textbooks that it can be said that there is no best textbook, but that each person can find one that suits his/her needs.

Nonetheless, Courant's book is an old text, around 70 years old. It belongs to these classics of science that were influential and held its own as a source of common knowledge. Why?

I believe that the answer to this question is simple: Courant's book has the perfect balance between theory and applications. It does not use too much pedantry in its exposition, is full of examples (for the student to do and also some worked-out), ranging from simple to very difficult, and yet it proves everything that is important in a way that no mathematician can complain. Indeed, the authors leaves the most difficult demonstrations to appendixes that can be found in each chapter, so the reader that doesn't want to enter into the complications of the proofs can skip them. And the book is written in a conversational style, that much probably influenced the book that, in my humble opinion, is the best that can be found treating the subjects it treats (so I also have my favourite calculus text: Spivak's Calculus!).

There are two volumes, the first one dealing mainly with calculus of one variable and the second with multivariate and complex analysis. It contains the core of the mathematical theory useful for physicists and engineers and has this that is amazing: it develops the theory and always gives good physical examples. Indeed, a whole course of theoretical physics is contained in this book, almost hidden.

So, if someone is reading this review and is in doubt whether the book is good or not, I can say, with the experience of having read a long list of calculus texts, that the book is good and is worth-while. It is useful to the mathematician and to the engineer, to the philosopher and to the physicist, and serves extremely well both as a text book for class study, self-study and for reference. If you are worried that the treatment is dated, I can say that, although today the most common treatment of, say, multivariate calculus is through linear algebra, that leaves the subject much cleaner, Courant's work still is of value in that it explains everything in as simple way as possible, mantaining always ahead the objectives of each section. It is essentially a book of applications of analysis and if you read and work the examples, you will turn yourself into an expert both in theory and application and will be able to follow easily any work that has classical analysis as prerequisite.

Great classical book!

This two-volume text, originally written in German while Courant was still at Gottingen, is very much better for a serious student than most introductory texts on analysis. Most introductory texts have a flavor of having been written by geniuses for idiots; in this book, Courant treats the student as being his peer in intellect and interest, lacking only knowledge. This makes it an excellent book even for somebody reasonably familiar with the calculus. Although it covers the material from a strictly classical viewpoint, the text and the examples provide enough thinking material to help the student understand the motivation that led to measure theory, Lebesgue-Stieltjes integration, and algebraic topology; the wellsprings of these in classical analysis are seldom explained in modern math courses. So I can recommend it to any senior planning to do graduate work in math, or to any first-year graduate student in math. And of course, it can be well used as a first calculus text for students who are prepared to think and put in effort on the subject.

Courant himself, of course, was a great mathematician, although I don't personally consider him one of the greatest mathematicians of the 20th century; he was a better leader and inspirer of others than a creator of new mathematics. But among other things, he served as David Hilbert's personal assistant for two years, and this gave him superb judgment about what's important and what isn't. This shows throughout the book.

It also helps that the translator into English was E. J. McShane. McShane is less well-known than he perhaps deserves to be, because he was a truly first-rate mathematical researcher (in analysis) himself. This, together with the fact that McShane spent a year or two at Gottingen while Courant was still leading the Mathematics Institute at Gottingen, and came to know Courant well, allowed McShane to translate Courant's text with great understanding of

Courant's way of thinking.

My own copy of this text, bought more than 50 years ago, is in tatters, because I still haul it out and re-read pieces of it to connect my thinking when I'm groping.

I used this book in an Honors Calculus course decades ago, and it's still a useful reference. Unlike most calculus books, this is one from which you can learn real mathematics by self-study. It is not only solid on calculational techniques, but is also an introduction to real analysis, and to good mathematical reasoning and proof technique. Courant was a famous applied mathematician, and he introduced and developed the concepts in a way that is very well motivated and clear (not very common in mathematics texts these days).

Different calculus textbooks will go in and out of fashion as professors try to overcome the poor preparation of their students, but Courant's book will endure as long as there are students who really want to understand thoroughly what they are doing.

In this book (the twelfth edition) every economic subject is very well explained without difficult mathemayical concepts. People who like the mathematical approach can find something in the appendix. There are a lot of examples about real facts happened to real companies (Microsoft,...) or organization (OPEC...). The logic approach and the examples are very usefull for a beginner that first of all has to understand the main concepts rather than struggling in a difficult language or math.

The authors have explained all principles with remarkable ease taking numerous to the point examples.

I agree with the above review and would simply like to add my own thoughts. The book illustrates the fascinating way in which mathematics, society, religion, politics and of course physics have affected each other (it goes both ways!) through out the ages. Furthermore, the author nicely illustrates the processes by which people think and how those processes have also changed through the ages (i.e., The Age of Reason versus The Renisance). This book left me with real insights as to the nature and limitations of the current state of mathematics and physics. Things are not as they seem, my friend! Lastly, the author displays an appreciation for the humor and irony of the history which makes this book hard to put down at times. I never thought a math/history book could be a "page turner"... Read it.

In most mathematics classes, students are presented with a completed edifice, and given a floor plan to help them navigate the halls. While this approach works for many people, others need a little more basic information. In this book, Morris Kline builds the building, starting with the mud and straw of the bricks.

"Mathematics in Western Culture" shows that the history of mathematics is one of hundreds of years of people sitting in the sand, drawing shapes and lines, scratching their heads, and trying to figure things out. This is not necessarily Dr. Kline's intention for the book, but this is certainly one of the many messages to be derived from it.

A fascinating, exciting book which makes mathematics more understandable and accessible.

If you are interested in mathematics and are/aren't a mathematician, if you are interested in American/German jewish history of the 20th century, if you are interested in science during WW2, you will love this book - I did!

This is an introductory textbook and to be considered as a first book in economics. It starts at an elementary stage with full explanations of all major points of standard economic theory. The point-by-point summary at the end of each chapter reinforces the reader on the concept learnt. The captions and notes below each diagram are full and self-sufficient. It's better to have the study guide along side for more practice on economic thoughts into daily life.

This is an interesting and wide ranging book. In the main it presents, develops and explains it's ideas very well, although I did not always find it, as one reviewer, a mister Albert Einstein described it, "easily understandable". I have two minor complaints about this book:

1) Print quality
For no apparent reason the text size varies occasionally, and in places the printing is slightly blurred, so that sometimes the subscripts and superscripts on formulae are illegible. Perhaps they skimped on typesetting costs by photoreproducing formulae from the original printing?

2) Incompleteness
If you bought this book because the front cover says "...representation of the fundamental concepts and methods of the whole field of mathematics" (another A.E. quote) you may be disappointed to find this is not the case. Trigonometry, for example, is not discussed, except where it crops up in other topics such as applying calculus to trig functions.

I give this book 5 stars because it is a classic. I believe, however, that it is too sketchy to be useful for the beginner as it is advertised. For chapter 1, for example, on number theory, I recommend Hardy's "Introduction to the Theory of Numbers." For the second chapter, on the number systems, I recommend a book like Birkhoff and MacLane's "Modern Algebra." It's difficult to write a survey of mathematics textbook without being sketchy and Courant isn't up to the task. In addition, the bibliography at the end of the book is fairly outdated, although the two books I mentioned above are included there. I also wish Courant would have provided more information on the evolution of mathematical concepts and ideas. This is something Kline does in his "History of Mathematical Thought." I find this information vital in answering the question "what is mathematics?" If you really want to get a good idea of what mathematics is you should start with a general history of mathematics like Kline's book and quickly move on to Greek mathematics. Even a small understanding of Euclid's axiomatic method will help you understand modern day mathematics and why mathemticians do what they do the way they do it. Having said that, I plan on making more use of Courant's book later on in my mathematics career.

A very interesting exposition of some of the main branches and ideas of mathematics. This is a book for beginners and experts, students and professors. The authors exposes number theory, algebra, geometry, topology and calculus. (For the last topic I recomend the great book of Courant and Fritz, Introduction to Calculus and Analysis.) The mathematical concepts are introduced and motivated by real problems, it seems to me very applied and connected to physics. I have been learning much things with this book. It is very interesting and I recomend for all people that want to read about mathematics.

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'.

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.

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.

"Hilbert" is justly famous as one of the best mathematical biographies around. Constance Reid, who also wrote a biography of Hilbert's student Courant, initially ran into some resistance from Hilbert's associates when she started work on this book. Max Born was not keen on the idea of a woman, who was neither German nor a mathematician, writing a study of Hilbert's life. Born was enthusiastic about the final product, however, and it has become a classic.

Hilbert took over from Poincare the title of the most famous mathematician in the world. His mathematical achievements are numerous and varied; Reid does a good job of providing an overview of the impact Hilbert had on many different fields, and of his style; his strengths and weaknesses. There is a good deal of coverage of the famous twenty-three Hilbert problems, presented to the Second International Congress of Mathematicians in Paris in 1900, including a large section of the talk Hilbert gave.

Reid paints a vivid picture of the mathematical circle at Gottingen, a luminous collection of talents. Minkowski and Hilbert were close friends; Klein was the director of the institute there; Emmy Noether was there; Hurwitz; Zermelo; Landau; the list is long and impressive. It's all the more sad to read about the way the Institute was destroyed by the Nazis in the name of racial purity. Almost without exception the leading mathematicians emigrated, one by one, to America. Hilbert, who had retired in 1930 (retirement at age 68 was mandatory) was forced to watch as the work of decades was dismantled. The last years, of age, fading memory and the privations of war, are mercifully given less than a dozen pages.

Hilbert's life leads from the great days of the mid-nineteenth century to the Nazis and the atomic bomb. Reid has done a wonderful job of capturing the feel of Germany over his long life, and the mathematic impact and importance of his work. A compulsory book for those interested in modern mathematical history.

Very well written. Gives a great feel for who Hilbert was as a person. It also does a good job of placing his achievments amoung the other mathematians of his time. I have attempted reading a few biography's of other mathematicians that focus on the math and not the person. They are almost unreadable. This book does not make this mistake. I have just bought "Foundations of Geometry" because of this book. I highly recommend it. PS - 4 stars is as high a rating as I give. I like having room to move for the truely fantastic.

David Hilbert was arguably one of the greatest mathematicians
ever!. He contributed to several branches of mathematics,
including functional analysis, mathematical physics,
calculus of variations, and algebraic number theory.
(Everyone knows what a Hilbert space is right!)

At the turn of the 20th century, Hilbert enumerated
23 unsolved problems of mathematics that he considered worthy
of further investigation. To this day, very few of these, including
the 10th problem, on the finite solvability of Diophantine
equations, have been resolved! (thanks to
Yuri Matiyasevich, Martin Davis and Julia Robinson!).
Besides, Hilbert was also a character (read the section
when Norbert Weiner of cybernetics fame, came to give
a talk at Gottingen, and .... :-)).

Incidentally the author Constance Reid is the sister of
Julia Robinson (of Hilbert's 10th problem fame!),
hence there can no one better to write about
Hilbert!. Besides Constance Reid is a well known chronicler
of mathematicians lives (this one is a tour de force and
her best!).

No one can can call himself/herself a mathematician without
having Reid's book on his/her bookshelf. Strongly
recommended!

I brought this book ( volume 2 also as well ) because of its " fame ", but when I read it, it has several draw backs. First, may be the original vesion is in German, so even with good translation, it seem does not fit in the usual English style we get used to .Also the topics it choose is too few and also the area covered is too narrow and not well co-ordinated. For example, the whole volumme I is almost dedicated to Calculus of variation only. In volume 2, the whole book is dedicted to differentiation equations. But that is not the greatest drawback. The most bad point is that the book just presents formulae after formulae, equations after equations, without giving examples of how to use it,and also no exercise for me to practice. Compared the the timeless classic " A course of mordern analysis " by Whittaker and Watson, it is definitely at a lower level. This book cannot be described as " classics ".