Customer Reviews:
Worth a look.......2007-01-30
This work has an honored place on my bookshelf. A colleague
recommended it to me when I was in school and I bought a copy after
looking at it in the school's library. It sits next to my copy of
"The Feynman Lectures in Physics". These are works you go to for
insight. I like Courant's mixture of physical examples with the
mathematics.
After encountering Courant's book for the first time, I remember
wondering why the first volume wasn't used as the textbook for the
typical year and a half of basic calculus. Then, as now, I can only
conclude that teachers probably think it's not watered down enough for
the students. Maybe it's a blessing in disguise to come across
Courant after you've been taught calculus from an uninspiring "modern"
text.
Everyone's needs are different, so take all reviews with a grain of
salt. As a working scientist/engineer, my primary use of the calculus
is as a tool to get things done, so I'm typically more interested in
learning the mechanics than getting a deep understanding like a
mathematician would. Courant works for this, yet still allows one
to dig in deeper when desired. It's still an awfully good book, even
if it is 70 years old.
great calculus book.......2006-04-05
I agree whole heartedly with the other reviews here, and merely wish to second their opinions. I bought my copy in 1960 and would never part with it. I still have crystal clear images in my mind from courant's exposition of the 1-1 correspondence between points of the real line and infinite decimals, his comment that all absolute convergence tests result from comparing with a known convergent object - either the geometric series or an integral, and his preface emphasizing "the chief task" of one who would pursue the study of Science: acquiring a firm grasp of the application of general principles in particular cases.
I have studied and taught calculus, advanced calculus, real and complex analysis, Riemann surfaces, differential equations, and differential manifolds both real and complex, for over 40 years, but anyone who reads thoroughly these 2 volumes and masters them will know more calculus than I do.
I am a pure mathematician, and I believe these volumes are highly recommended not just to physicists and engineers, but to anyone who would master their subject. I also love the book of Spivak, but after teaching from them together one summer, and comparing proofs, I concluded that Spivak himself probably learned the subject from Courant.
Classical German calculus.......2005-12-08
Courant knows the art of writing a good preface. He attacks "diffuseness" and "pedantry" and aims at "exhibiting the close connection between analysis and its applications" and "to give due credit to intuition as the source of mathematical truth". The book also has a tone that is unusual today: Courant speaks to us the way a dignified, open-hearted professor speaks to an intelligent student. No rambling pretensions; just to-the-point, good mathematics. This is the perfect solid-as-they-come, timeless book on the calculus, and most likely it will never be surpassed in this domain. One must be warned, however, that this is a very serious book and reader-friendliness has lower priority than technical coherence and brilliance of formal organisation. The likely reader will know calculus already and use Courant for masterful, concise exposition of standard topics as well as a wealth of topics that have been watered out of most current calculus curricula (e.g., evolutes, involutes, envelopes, curvature, geodesics, centres of mass, the gamma function, the catenary, the cycloid, the lemniscate, the brachistochrone problem, Kepler's laws, Maxwell's equations, the zeta function, etc.). Everybody knows that all the usual calculus books, "reform" or not, are pathetic. But what is even worse is that there are no good alternatives even if one is prepared to dig deep into the library shelves in hope of finding an author who has not sacrificed his intellectual dignity at the altar of royalties. Take for example Serge Lang's books "A First Course in Calculus" and "Short Calculus". Lang is of course the virtual definition of the mainstream of respectable mathematics. Nevertheless, these books are soaked with the common formalistic attitude. In fact, as if his books had not finished the job, Lang adds an appendix to both books called "Physics and Mathematics", which very explicitly drives a wedge between physics and intuition and mathematics. Courant is a good antidote to such modern nonsense.
Best Calculus book.......2003-12-10
This is the best calculus text for aspiring physicists as well as applied mathematics students. However, don't know why Amazon sent the book with different front cover to me. It's not the one shown in the picture but rather a black cover--exactly the one seen on barnesandnoble with the same isbn. Although the covers are different, the content is the same.
Classical book.......2002-06-08
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!
Amazon.com
A 1996 revision of a timeless classic originally published in 1941. Highly recommended for any serious student, teacher or scholar of mathematics.
Book Description
For more than two thousand years a familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today, unfortunately, the traditional place of mathematics in education is in grave danger. The teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but does not lead to real understanding or to greater intellectual independence. This new edition of Richard Courant's and Herbert Robbins's classic work seeks to address this problem. Its goal is to put the meaning back into mathematics. Written for beginners and scholars, for students and teachers, for philosophers and engineers, What is Mathematics?, Second Edition is a sparkling collection of mathematical gems that offers an entertaining and accessible portrait of the mathematical world. Covering everything from natural numbers and the number system to geometrical constructions and projective geometry, from topology and calculus to matters of principle and the Continuum Hypothesis, this fascinating survey allows readers to delve into mathematics as an organic whole rather than an empty drill in problem solving. With chapters largely independent of one another and sections that lead upward from basic to more advanced discussions, readers can easily pick and choose areas of particular interest without impairing their understanding of subsequent parts. Brought up to date with a new chapter by Ian Stewart, What is Mathematics?, Second Edition offers new insights into recent mathematical developments and describes proofs of the Four-Color Theorem and Fermat's Last Theorem, problems that were still open when Courant and Robbins wrote this masterpiece, but ones that have since been solved. Formal mathematics is like spelling and grammar--a matter of the correct application of local rules. Meaningful mathematics is like journalism--it tells an interesting story. But unlike some journalism, the story has to be true. The best mathematics is like literature--it brings a story to life before your eyes and involves you in it, intellectually and emotionally. What is Mathematics is like a fine piece of literature--it opens a window onto the world of mathematics for anyone interested to view.
Customer Reviews:
Outstanding writing.......2007-08-11
Whether or not you are a mathematician or if it is just a hobby, this book is fantastic!!! You can open it up to any page and just begin reading...there is no need to start beginning to end. A great work to read as time allows!!!
enlightenment !.......2007-06-10
We all have done math and have had math classes,but this book will open your eyes as to what math really is.It is a language that can show you the beauty of the world.
A clear and concise classic........2007-05-16
Perfect for someone like myself starting advanced mathematics from scratch.
Now I Get It - A Concise Overview.......2007-01-09
This book has been around now for a few years to say the least. It is intriguing and captivating in its concise approach to explain and answer the question of just what Mathematics is. If you have a hankering to understand what Mathematics does or perhaps why it even exists, this book is pretty informative and interesting reading. One feature I liked about it is its use of many examples to propel the thought process into that zone of understanding within us. When you are in the zone many basic concepts used in both rudimentary and complex problems become apparent frequently leading one to say "Now I get it."
Unity and beauty.......2006-12-15
This is an excellent display of the beauty and unity of mathematics. Take geometry for example. Euclidean geometry is pretty but ruler-and-compass constructions have limitations. Their precise nature can only be understood by recasting geometry in terms of algebra through coordinate systems. This change of perspectives enables us to see clearly that it is impossible to trisect an angle or double a cube but that it is possible to construct a circle tangent to any three given circles (Apollonius's problem). Some fools think of this as a linear development ending with the triumph of algebra, but our authors know better, letting the pendulum swing back, through the beautiful topic of circle inversion, which can be carried out with compasses alone. This leads to a beautiful geometric solution of Apollonius's problem and construction of linkage devises that turn circular motion into rectilinear motion, desirable for engineering applications. Then there is of course a thorough discussion of projective geometry; this is a perfect topic for a book like this because of its great beauty and deep connections with many other areas of mathematics. Lines look like lines projectively, but circles don't have to look like circles, so projective geometry is in a way a "ruler only" geometry. Indeed, it adds to our discussion of constructions by revealing, as a simple consequence of the invariance of the cross-ratio, the theorem on the complete quadrilateral, which can be used, for example, to give a ruler-only construction of the continuation of a line beyond an obstacle. So the fact that we still have rulers in projective geometry gives a new fruitful perspective on old things. And the the fact that we don't have compasses since circles project to conic sections is also rewarding since it means that the theory of conic sections is unified by the projective viewpoint; in particular, this should imply that there are ruler-only constructions of conic sections and indeed we find a beautiful way of carrying out such constructions by projective pencils. Projective geometry shines even more as we study the projective disc model of hyperbolic geometry, but unfortunately our excursion into non-Euclidean geometry is very brief. The chapters on the calculus are also very nice and include for example a derivation of Leibniz' formula for pi (by series expansion of the arctangent) and Bernoulli's law-of-refraction solution of the brachistochrone problem, thus beautifully bringing together calculus, geometry, mechanics and optics, and the very last section of the book is yet another triumphant success for the unity of mathematics: chapter 1 was on elementary number theory and we now end with a heuristic proof of a result hinted at then, the prime number theorem (prime counting function is asymptotically equal to x/log(x)). First, log(n!) is asymptotically equal to n(log(n)) (proved by rectangular estimation of the integral of log(n!)), and log(n!) contains information about primes as follows. How many times does p divide n!=n(n-1)(n-2)...? It divides every p:th number once, every (p^2):th number once more, etc., so the answer is (n/p)+(n/p^2)+...=n/(p-1), thus, writing n! as its prime factorisation, we have n!=product of p^(n/(p-1)), and taking the logarithm we get log(n!)=sum of (n/(p-1))log(p). But log(n!) "=" n(log(n)) by the above, so log(n)=sum of log(p)/(p-1)=integral of prime density times log(x)/(x-1). Differentiating both sides and solving for prime density gives (x-1)/(x(log(x)), which has asymptotical antiderivative x/log(x). QED.
Book Description
Since the first volume of this work came out in Germany in 1937, this book, together with its first volume, has remained standard in the field. Courant and Hilbert's treatment restores the historically deep connections between physical intuition and mathematical development, providing the reader with a unified approach to mathematical physics. The present volume represents Richard Courant's final revision of 1961.
Customer Reviews:
state of the art in the 50's.......2006-10-17
Beware the german edition is the prewar edition, and the english edition is a
complete overhaul to include modern material at the time. This is not really a
textbook, but a sourcebook of "issues" in partial differential equations. Some
problems have been forgotten now due to the focussing on real, linear equations
and weak solutions (as described, for example, in the books by Hormander). It is
a gem for who wants some taste of the full breadth of theory of PDE's.
Book Description
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. 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. (...)" The Mathematical Gazette (75.1991.471)
Customer Reviews:
a superb book.......2006-04-05
This is a rewrite of the great book by Courant, and it does justice to its origin. I prefer the somewhat more charming original book of Courant myself, but I have taught from this one too and learned something more.
Since the original Courant costs $120 for the 2 volume set, this volume at $33 is a bargain, so snap it up. This is 10 times as valuable as most current $130 calculus books.
More than an introduction.......2005-12-15
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.
Absolutely beautiful!.......2005-01-24
I give 5 stars to this book because in contrast with the majority of the calculus textbooks it gives the reader the perfect combination between rigor and intuiton. Another thing that I also like a lot is the fact that volume 2 has solutions to almost all the excercises, which is great because some of the problems are very difficult. I really think this book is a "must have".
simply the Best Calculus Book.......2002-08-12
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'.
You must have this........2002-05-28
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.
Book Description
S. R. S. Varadhan is recognized as a top expert in probability theory. This volume presents topics in probability theory covered during a first-year graduate course given by Varadhan at the Courant Institute of Mathematical Sciences. The necessary background material in measure theory is developed, including the standard topics, such as extension theorem, construction of measures, integration, product spaces, Radon-Nikodym theorem, and conditional expectation.
In the first part of the book, characteristic functions are introduced, followed by the study of weak convergence of probability distributions. Then both the weak and strong limit theorems for sums of independent random variables are proved, including the weak and strong laws of large numbers, central limit theorems, laws of the iterated logarithm, and the Kolmogorov three series theorem. The first part concludes with infinitely divisible distributions and limit theorems for sums of uniformly infinitesimal independent random variables.
The second part of the book mainly deals with dependent random variables, particularly martingales and Markov chains. Topics include standard results regarding discrete parameter martingales and Doob's inequalities. The standard topics in Markov chains are treated, i.e., transience, and null and positive recurrence. A varied collection of examples is given to demonstrate the connection between martingales and Markov chains.
Additional topics covered in the book include stationary Gaussian processes, ergodic theorems, dynamic programming, optimal stopping, and filtering. A large number of examples and exercises is included. The book is a suitable text for a first-year graduate course in probability.
Titles in this series are copublished with the Courant Institute of Mathematical Sciences at New York University.
Customer Reviews:
The distilled core - an ideal first text for measure theoretic probability.......2007-04-13
This is an ideal text for first year graduate probability. Take Billingsley, which is several times bulkier, and distill the core concepts that one should away from the course, and that is Varadhan. Beautifully written, a joy to read, and occasionally even a useful reference to have on the desk to go back to when formalizing proofs for publication.
For brevity's sake though, quite a few ideas have been left to exercises, and I am not sure if this is a text that an uninitiated can learn from without having the luxury of being in a class. These are course notes that Varadhan used to teach the course himself, so probably the intended audience is students taking a course - not self study.
Book Description
Written by a leading specialist in the area of atmosphere/ocean science (AOS), the book presents an excellent introduction to this important topic. The goals of these lecture notes, based on courses presented by the author at the Courant Institute of Mathematical Sciences, are to introduce mathematicians to the fascinating and important area of atmosphere/ocean science (AOS) and, conversely, to develop a mathematical viewpoint on basic topics in AOS of interest to the disciplinary AOS community, ranging from graduate students to researchers. The lecture notes emphasize the serendipitous connections between applied mathematics and geophysical flows in the style of modern applied mathematics, where rigorous mathematical analysis as well as asymptotic, qualitative, and numerical modeling all interact to ease the understanding of physical phenomena. Reading these lecture notes does not require a previous course in fluid dynamics, although a serious reader should supplement these notes with material such as additional information on geophysical flows, as suggested in the preface.
The book is intended for graduate students and researchers working in interdisciplinary areas between mathematics and AOS. It is excellent for supplementary course reading or independent study.
Product Description
The theory of hyperbolic equations is a large subject, and its applications are many: fluid dynamics and aerodynamics, the theory of elasticity, optics, electromagnetic waves, direct and inverse scattering, and the general theory of relativity. This book is an introduction to most facets of the theory and is an ideal text for a second-year graduate course on the subject. The first part deals with the basic theory: the relation of hyperbolicity to the finite propagation of signals, the concept and role of characteristic surfaces and rays, energy, and energy inequalities. The structure of solutions of equations with constant coefficients is explored with the help of the Fourier and Radon transforms. The existence of solutions of equations with variable coefficients with prescribed initial values is proved using energy inequalities. The propagation of singularities is studied with the help of progressing waves. The second part describes finite difference approximations of hyperbolic equations, presents a streamlined version of the Lax-Phillips scattering theory, and covers basic concepts and results for hyperbolic systems of conservation laws, an active research area today. Four brief appendices sketch topics that are important or amusing, such as Huygens' principle and a theory of mixed initial and boundary value problems. A fifth appendix by Cathleen Morawetz describes a nonstandard energy identity and its uses. Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University.
Customer Reviews:
a good textbook!.......2007-10-17
A very thorough approach to French grammar, designed for the student who is almost fluent. Full of examples and rules, it gives you all of the support you will need in expanding your French abilities. There are a few exercises scattered within whose answers can be found at the end of the book. I would only reccommend it to someone at the 3rd year level of college French.
Book Description
With the returning Naismith award winner, and the entire roster returning from last year's National championship squad, anything short of a third consecutive trophy would disappoint UConn fans. While the road to the title was not as smooth as the two previous seasons, UConn's 2004 season may be the most rewarding championship of them all, one that all Huskies fans will cherish.
Excellence3 captures the Husky hysteria fans have come to expect from The Hartford Courant and their award-winning sports coverage. The book illustrates UConn's triumphs and turning points through recaps of all the key games and profiles on the team's star players. It's all here: the stunning home loss to Duke on national television, the continued mastery of rival power Tennessee, Diana Taurasi triumphant senior season culminating in another shelf full of national awards and accolades, and the team righting the ship in time for their impressive NCAA tournament march to UConn's third consecutive national championship, and fifth overall. Excellence3 is the best way to cherish the UConn Huskies unforgettable 2004 season.
Customer Reviews:
Great Tidbits of Information.......2006-02-22
Great tidbits of information on the Best Women's Basketball Program in existance, as well as one of the best teams that ever played.
Average customer rating:
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Introduction to Calculus and Analysis, Volume 1 (Classics in Mathematics)
A Course of Pure Mathematics (Cambridge Mathematical Library)
What Is Mathematics? An Elementary Approach to Ideas and Methods
Calculus
Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra (Second Edition)
