Customer Reviews

Elementary Analysis: The Theory of Calculus

5 star:		(16)
4 star:		(4)
3 star:		(5)
2 star:		(0)
1 star:		(1)

 

 

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.................

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.

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.

I am using this book in an undergraduate analysis class. I have to say that the book is very good. The author takes time to prove all the theorems and there are about 10-15 exercises for each section with random solutions in the back. Sometimes he skimps on the proofs and at times it is hard to follow his logic. But overall I would recommend this for anyone interested in analysis who has a good background in calculus. He starts with basic properties of numbers but then jumps pretty quickly into sequences, series and the rest of calculus. Put your thinking caps on for this one!

I used this book in a first undergraduate course on real analysis at Berkeley. The proofs are nice and straightforward but the subject matter is lacking. A better introduction to topological notions (i.e. compactness, connectedness, and continuity) would be appreciated. More on integration would be nice. All in all, the book just needs more material. Would probably augment a more in-depth book (such as Rudin or Protter/Morrey) nicely.

If you had a rigorous, proof-heavy calculus course, you could probably skip this book and go straight to something more advanced, like Apostol or Rudin. On the other hand, if your calculus class skimped on proofs, preferred a cookbook approach, or if the word "limit" doesn't automatically conjure up images of deltas and epsilons, then this is the analysis book for you.

While Ross assumes a working knowledge of calculus and basic proof-writing skills, he does NOT assume the reader has encountered too many of the results before. Thus, he includes numerous examples and exercises to familiarize the reader the definitions. The theorems aren't proven in the simplest way possible, but rather in a more intuitive way with few gaps. Sure, the proofs would be a lot shorter if he used some basic results from topology, but that would be a bit distracting for someone who wasn't very familiar with the material or with proof-writing in general.

In summary, this is an excellent introductory book for newcomers to analysis. If you learned calculus from Stewart, go here first. If you learned calculus from Apostol or Courant, this is a good companion to Rudin.

I used this book in my junior year.It will be helpful to read this book if you have taken some sort of "proofs" class before. This book jumps straight into sequences and later on into series. So if you have had exposure to these concepts in some elementary calculus courses, then you will ease into the book very easily.This is a real math book, and so the book starts with axioms, then some definitions and then theorems and proofs. Ken also includes some sections on metric spaces and point-set topology, and shows how real analysis and the latter are inter related.However, it is not necessary to have had any point-set topology to follow the proofs.To get a full appreciation of the subject matter, it is a must to do the exercises, and Ken provides partial proofs in the back, ample examples in each section. This book is dull, if you'll let it be.There were times when I struggled with the matter, especially in the point-set topology sections, but in the end it paid off. I give it five stars. Money well spent!

This book is very understandable and the presentation is very clear. There are many worked out examples to illustrate the theory , and many of the excersises come with complete solutions. Considering how difficult the topic of real analysis can be for university students, I strongly recommend this book for anyone that need to take an introductory real analysis text, it is also a good prep. text for anyone that plans to study the Rudin text on real analyis. I have been a full time university math tutor since 1994 and I think this is one of the few texts that actually "tutors" the student into understanding the foundations of real analysis.

I used this book for the analysis sequence at cal poly for my undergraduate coursework. This is one of the best books i've read. In addition to the standard material for a two-quarter course, it concluded some nice topological supplementary like compactness, open/closed sets and continuity that got me interested in general topology. Included in the back are the much appreciated hints for the exercises. Excellent and radical approach to Riemann-Stielje Integrals in the integration section. Good for an intro. to proofs in general.

I went through a lot of math books that made me wonder if the author was purposely making it hard or if they were just bad at explaining things.
I didn't have that experience with this book. This book is well written and easy to understand (given its subject). Highly recommended for anyone learning Advanced Calculus 1 for the first time.