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This arrangement corresponds to the traditional organization of most calculus courses, but I feel that it will only diminish the value of the book for students who have seen a small amount of calculus previously, and for bright students with a reasonable background.

Joseph Lipman also told me of this proof, together with the similar trick for the proof of the last theorem in the Appendix to Chapter 11, which went unproved in the first edition. In the preface to the third edition I noted that it was 13 years between the first and second editions, and then another 14 years before the third, expressing the hope that the next edition would appear sooner.There are in all about 160 new problems, many of which are intermediate in difficulty between the few routine problems at the beginning of each chapter and the more difficult ones that occur later. Finally, there is a Suggested Reading list, to which the problems often refer, and a glossary of symbols.

I am content to hope that some other students will be able to use the book to such good purpose, and with such enthusiasm. Considerable attention is paid to motivating the discussion, showing why each result is important (though mainly in the pure mathematics context, applications of calculus being mainly found in the problems at the end of each chapter). Michael David Spivak is a mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Perish Press. It is well structured, clear and precises, it made me love the subject because for the first time I saw it presented for the logical wonder it is, and as an assortment of formulas to be used. Another large change is merely a rearrangement of old material: “The Cos­ mopolitan Integral,” previously a second Appendix to Chapter 13, is now an Appendix to the chapter on “Integration in Elementary Terms” (previously Chap­ ter 18, now Chapter 19); moreover, those problems from that chapter which used the material from that Appendix now appear as problems in the newly placed Appendix.Moreover, the Appendix to Chapter 12 has been extended to treat vector operations on vector-valued curves.

All that is necessary is a solid understanding of high school pre-calculus and mathematical curiousity. Davies told me the trick for Problem 11-66, which previously was proved only in Problem 20-8 [21-8 in the third edition], and Marina Ratner suggested several interesting problems, especially ones on uniform continuity and infinite series. This is always safe— after all, the class is unlikely to rise up in a body and protest publicly—but the students themselves, it seems to me, deserve the right to assign credit for the thor­ oughness with which they absorbed an impressive amount of mathematics. I have often been told that the title of this book should really be something like “An Introduction to Analysis,” because the book is usually used in courses where the students have already learned the mechanical aspects of calculus—such courses are standard in Europe, and they are becoming more common in the United States.Frederick Gordon pointed out several serious mistakes in the original problems, and supplied some non-trivial corrections, as well as the neat proof of Theorem 12-2, which took two Lemmas and two pages in the first edition. who were always eager to increase the appeal of the book, while recognizing the audience for which it was intended. The inadequacies which preliminary editions always involve were gallantly en­ dured by a rugged group of freshmen in the honors mathematics course at Brandeis University during the academic year 1965-1966. The proofs and exercises are the most elegant that I have seen in any calculus text, as is Spivak's hallmark. I expect to solve a good amount of hard calculus problems so I would choose whichever has more problems.

For the moment we consider only addition: this operation is performed on a pair of numbers—the sum a + b exists for any two given numbers a and b (which may possibly be the same number, of course). I do understand its novelty, and I especially sense its charm from the perspective of a self-learner. Like putting an everyday object under a microscope: very interesting, but now there are many more interesting objects I want to put under a microscope. It also includes challenging exercises and proofs to help students develop critical thinking skills. There are now separate Appendices for many topics that were previously slighted: polar coordinates, uniform continuity, parameterized curves, Riemann sums, and the use of integrals for evaluating lengths, volumes and surface areas.

The reason for this is suggested by the final item of the reading list: where the author suggests that one could learn as much from all of the other items on the reading list by simply reading the “Oeuvres complètes de Niels Henrik Abel”, Abel being the same mathematician who lends his name to the 'Nobel' in mathematics, the Abel Prize.

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