Elementary Analysis: The Theory of Calculus [Hardcover]

Kenneth A. Ross (Author)

Book Description

ISBN-10: 038790459X | ISBN-13: 978-0387904597 | Publication Date: March 3, 1980

Designed for students having no previous experience with rigorous proofs, this text can be used immediately after standard calculus courses. It is highly recommended for anyone planning to study advanced analysis, as well as for future secondary school teachers. A limited number of concepts involving the real line and functions on the real line are studied, while many abstract ideas, such as metric spaces and ordered systems, are avoided completely. A thorough treatment of sequences of numbers is used as a basis for studying standard calculus topics, and optional sections invite students to study such topics as metric spaces and Riemann-Stieltjes integrals.

Editorial Reviews

Review

From the reviews:

K.A. Ross

Elementary Analysis

The Theory of Calculus

"This book is intended for the student who has a good, but naïve, understanding of elementary calculus and now wishes to gain a thorough understanding of a few basic concepts in analysis, such as continuity, convergence of sequences and series of numbers, and convergence of sequences and series of functions. There are many nontrivial examples and exercises, which illuminate and extend the material. The author has tried to write in an informal but precise style, stressing motivation and methods of proof, and, in this reviewer’s opinion, has succeeded admirably."—MATHEMATICAL REVIEWS

"This book occupies a niche between a calculus course and a full-blown real analysis course. … I think the book should be viewed as a text for a bridge or transition course that happens to be about analysis … . Lots of counterexamples. Most calculus books get the proof of the chain rule wrong, and Ross not only gives a correct proof but gives an example where the common mis-proof fails." (Allen Stenger, The Mathematical Association of America, June, 2008)

Product Details

• Hardcover: 272 pages
• Publisher: Springer (March 3, 1980)
• Language: English
• ISBN-10: 038790459X
• ISBN-13: 978-0387904597
• Product Dimensions: 9.3 x 6.3 x 0.9 inches
• Shipping Weight: 1.4 pounds (View shipping rates and policies)
• Average Customer Review: 4.3 out of 5 stars  See all reviews (26 customer reviews)
• Amazon Best Sellers Rank: #42,228 in Books (See Top 100 in Books)

More About the Author

Discover books, learn about writers, read author blogs, and more.

Customer Reviews

Most Helpful Customer Reviews

103 of 113 people found the following review helpful:

5.0 out of 5 stars The Book Needs To Be Judged For What It's INTENDED For.................., December 26, 2006

By 

Andrew V. Locascio "OldSearcher" (New York,USA) - See all my reviews
(REAL NAME)   

This review is from: Elementary Analysis: The Theory of Calculus (Hardcover)

I don't understand people that constantly knock this book.The vicious barrage of critisms levied against this text is usually by arrogant math majors at top level schools.Thier attitude is basically that,"If Rudin is too hard for you,you are too dumb to learn this,get over it." You know,the first edition of Rudin was written over 4 decades ago, when calculus was usually first exposed to high school students on a regular basis and eplison-delta proofs were not uncommon in a college level calculus course.Therefore,after a meaty,theoretical calculus course that taught limits,derivatives and integrals carefully in addition to related rates,differential equations and the applications that today's watered-down calculus courses laughingly consider mathematics,those students of past generations were READY for something brutally terse like Rudin.The sad truth is that in today's pathetically dumbed down mathematics eduation system in the US-where high schools are happy if they can get students to use thier CALCULATORS to add and subtract correctly-Rudin or Apostol are simply way past the preparation level of any but the best students after calculus.The need for a "bridge" course that gave students the minimum exposure to a hard core approach to calculus was realized in the early 1980's-and Ross' book is still,to me,the best of the lot.Not only does Ross explain basic concepts well such as limits,convergence and the Riemann integral-he does something most textbooks on analysis and calculus sadly lack and to me is essential for a beginner:tons and TONS of worked examples given immediately after a definition.Proving theorums in rigorous mathematics-and real analysis in particular-is to a large degree the generalization of concrete examples.Ross's examples are wonderfully chosen and illustrate each concept wonderfully-after studying each example and then working the problems at the end of each section-which are terrific and just the right level for a beginner-the perfect foundation will be laid for further study in analysis in Rudin,Pugh or Apostol.(In many ways,while we're on the subject-I feel Charles Chapman Pugh's REAL MATHEMATICAL ANALYSIS has made Rudin obsolete.Pugh's book is just as challenging,just as complete as Rudin's-but it is a WHOLE lot more user friendly.To me,this is the perfect next step after Ross.) The more advanced texts given above sadly do not provide examples.Using Ross to supply those examples as collateral reading for either an honors calculus course or a real analysis course would be a VERY helpful strategy for the education of beginners in analysis.Lastly-the book is exactly what the title says it it:The complete structure of calculus laid bare.For students looking towards graduate school in mathematics,many of them have a great deal of difficulty mastering calculus,even after advanced study in real analysis,due to the fact that the abstract view they've acquired clouds the forest for the trees.Ross will assist them greatly in seeing what is essential in the foundations of calculus and how it connects to the more advanced perspectives on metric and topological spaces.
I'm tired of people knocking this book.I say those who knock it haven't really LOOKED at it and it's intended audience.If you REALLY want to complain-the tragedy in this country is that the educational system has collapsed to the point where a book like Ross is NECESSARY to train math majors.In an ideal world,Ross would be used as a CALCULUS text to suppliment a more applied approach and math majors would then go on to study Pugh thier sophomore years and finish thier PHDs in math at the age of 22.Sadly,that's not the world we live in anymore.So until someone decides to put the guts back into calculus,we'll still need books like Ross and Spivak's CALCULUS and Estep's PRACTICAL ANALYSIS IN ONE VARIABLE.Otherwise-none of us have a chance at an education in a college system that doesn't give a damn about educating.We should be grateful such texts exist.We should thank Ross and all the other mathematicians who don't buy into the "too dumb to waste analysis on" BS.
Buy this book.And be sure to thank those of us who haven't locked themselves in thier Ivy Tower drinking camomile tea and given up on the Cattle.................

20 of 22 people found the following review helpful:

4.0 out of 5 stars Excellent introduction to analysis, August 7, 2004

By 

D. A Flory (Houston, TX USA) - See all my reviews

This review is from: Elementary Analysis: The Theory of Calculus (Hardcover)

Of the many analysis books I have seen, I think this is one of the best for the student approaching the subject for the first time.

It is mathematically rigourous, yet develops the major concepts of analysis in a leisurely (in the good sense of the word) way with interesting and sometimes unusual examples.

Beginners will especially appreciate the quality exercises and the solution guide in the back.

The style of this book is a bit similar to Spivak's *Calculus* in that the author is a bit wordy. I find Ross' presentation more direct and less pretentious than Spivak--and far less intimidating.

This is definitely the best introductory analysis book I know of for self-study. A student who masters the material in this book will be well prepared to tackle Rudin and other classic works in real analysis.

19 of 21 people found the following review helpful:

5.0 out of 5 stars Raymond Cheng, January 15, 2000

By 

"kwrcheng" (Calgary, Alberta, Canada) - See all my reviews

This review is from: Elementary Analysis: The Theory of Calculus (Hardcover)

The book is rigorously written and is extremely good for math majors. I don't think this book is very suitable for non-math majors however, since they might think it's too dull. The book does not go on and on like some math textbooks with non-essential talk. It gets into the material right the way. The proofs have been carefully chosen so that they're as simple and as elegant as possible. Topology is treated in optional sections, and the focus of the book is sequences. Indeed, the treatment of sequences is very thorough. Also, many notions are also defined in terms of sequences. However, proofs that this definition and the usual delta-epsilon definition are equivalent is given. The style of writing is clear, concise, and avoids uncessaary discussion. Proofs are given out in full and are seldom left to the readers as an exercise. In keeping with the style of this book, historical facts and references are not provided. I think this book should be a must-have for all math undergrads.