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60 of 62 people found the following review helpful:
5.0 out of 5 stars
What a wonderful book!, July 22, 2001
This review is from: Differential and Integral Calculus, Vol. 1 (Volume 1) (Paperback)
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.
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20 of 20 people found the following review helpful:
5.0 out of 5 stars
Worth a look, January 29, 2007
This review is from: Differential and Integral Calculus, Vol. 1 (Volume 1) (Paperback)
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.
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19 of 19 people found the following review helpful:
5.0 out of 5 stars
Classical German calculus, December 7, 2005
This review is from: Differential and Integral Calculus, Vol. 1 (Volume 1) (Paperback)
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.
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