Most of all, however, I am indebted to my friend Ted Shifrin, who has been using the book for the text in his renowned course at the University of Georgia for all these years, and who prodded and helped me to finally make this needed revision. As a general thing - it's been a long time since I read this textbook but I have good memories of it. As well as being a fine description of the basics of analysis (mostly real, with a toe in the deep water of complex functions), it is an excelent book to ease the transition from mathematics as taught at school level to the rigours of university mathematics.

Yes, there are various resources available to supplement this textbook, including solution manuals, study guides, and online resources such as video lectures and practice problems. But seriously--if you're looking for practical application or pragmatic knowledge then this book isn't exactly for you.But if you are looking to learn mathematics--in this case, calculus--then get your ass over this book and start engaging the grey matter. This is a beautiful exposition of the theory of calculus from the axiomatic approach with full motivation and proof of the major theorems and some interesting facts that are seldom presented at this level. Property P2 ought to represent a distinguishing characteristic of the number 0, and it is comforting to note that we are already in a position to prove this. Has a lot of explainig clear graphics,give examples of bizarre functions for clearing concepts,give a original introduction to complex variable by convergent complex power series,makes a formal costruction of the real numbers field, and make at this level unusual excursions in more advanced results as the demostration of the irracionality of pi or the demostration of the trancendence of e. Fortunately, this fact need not be listed separately, since it follows from the property Pl already listed.

Michael David Spivak is a mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Perish Press. 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. The answer section contains solutions to about half the examples from an assortment of problems that should provided a good test of technical competence. If you like--or think you might like--the mathematician's ever-refining pursuit for rigour, for watertight argument, for abstract thought--then you'll love this book.The assertion that 1 This edition differs from the third mainly in the inclusion of additional problems, as well as a complete update of the Suggested Reading, together with some changes of exposition, mainly in Chapters 5 and 20. They range from elementary ones useful for practicing the mechanical aspects of the subject to extremely challenging and abstract ones that connect the material in the book to more advanced concepts. Derivatives and Integrals 9 Derivatives 149 10 Differentiation 168 11 Significance of the Derivative 188 Appendix. A favor tiene el enfoque matemático que no desestima las aplicaciones ingenieriles, pero prioriza la teoría formal pura.

Indeed, if a number x satisfies a +x = a for any one number a, then x = 0 (and consequently this equation also holds for all numbers a). It is used in calculus courses, particularly those with a pure mathematics emphasis, at many universities.Considering that this book is a bit old and consequently uses unusual notations, the explanations are extremely good and the author's approach reminds me of the best Leonard Euler's texts. The proofs and exercises are the most elegant that I have seen in any calculus text, as is Spivak's hallmark. His aim is to present calculus as the first real encounter with it is the place to learn how logical reasoning combined with fundamental concepts can be developed into a rigorous mathematical theory rather than a bunch of tools and techniques learned by rote. The equality of (2) and (3) is a direct consequence of P1, although this may not be apparent at first sight (one must let b 4- c play the role of b in Pl, and d the role of c). Since the foundations of analysis provided the arena in which modern modes of math­ ematical thinking developed, calculus ought to be the place in which to expect, rather than avoid, the strengthening of insight with logic.