2 Volume Set, Differential and Integral Calculus

Rating:
Summary: This book really is a classic
Review: 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.

Rating:
Summary: Classical book
Review: 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!

Rating:
Summary: Best Calculus book
Review: 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.

Rating:
Summary: What a wonderful book!
Review: 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.