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Differential and Integral Calculus, Vol. One

5.0 out of 5 stars 11 customer reviews
ISBN-13: 978-4871878388
ISBN-10: 4871878384
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Product Details

  • Paperback: 668 pages
  • Publisher: Ishi Press (June 13, 2010)
  • Language: English
  • ISBN-10: 4871878384
  • ISBN-13: 978-4871878388
  • Product Dimensions: 5.5 x 1.5 x 8.5 inches
  • Shipping Weight: 2.1 pounds (View shipping rates and policies)
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (11 customer reviews)
  • Amazon Best Sellers Rank: #1,362,501 in Books (See Top 100 in Books)

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By Victor A. Vyssotsky on July 22, 2001
Format: 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.
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Format: 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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Format: 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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Format: Paperback
This is one of the best calculus books ever written, I bought my copy in 1960 for $6.95. For a while recently the price had gone up to an unaffordable $150+. Now it is back down to a reasonable, indeed bargain level, of around $30. Everyone serious about learning calculus well, should buy this book at that price. While teaching from Spivak's wonderful book, I became convinced that he himself had probably learned from Courant.

After over 50 years I have never forgotten the beautiful explanation of how to faithfully represent real numbers as infinite decimals from pages 8-10, or the amazing algorithm given in a footnote! at the bottom of page 27, for deriving an explicit formula for the sum of the kth powers of the first n positive integers.

The explanation of how essentially all absolute convergence tests for series work on pages 377-382 is similarly succinct and
clear. I.e. they all proceed by comparing the size of the given series with another one whose convergence is known, typically either with a geometric series or with a convergent integral. That's it!

Pages 386-398 make absolutely clear the behavior of uniform convergence of functions and their properties with respect
to differentiation and integration of the limits.

Volume 2 even introduces complex variables and calculus of variations. The indeterminacy of the complex logarithm is made absolutely clear on page 543 of volume 2.

Material is so clearly explained in this book that I have basically never forgotten any of it. I cannot recommend it highly enough. But notice I did not say it is easy, just clear. You still have to think hard about it. But you will benefit. Unlike most modern calculus books, there are no useless or wasted pages or words here. I also agree with other reviewers that even the preface is inspiring and useful.
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