Elementary Analysis: The Theory of Calculus (Undergraduate Texts in Mathematics)

This book is an amazing choice for a first introduction to analysis. The proofs are very well written and a lot of proofs include sketches that help the reader follow the author's reasoning, as opposed to, say, Baby Rudin (Principles of mathematical analysis). This book is very comparable to the one by Abbott and it actually has a lot of hints and solutions to problems in the back of the book (Abbott's book does not). Only downside to this book is that it does not cover a wide range of topics, but the book is not really intended to be comprehensive anyways. It does not include any multivariable analysis. Prerequisites are a solid grasp on elementary calculus and basic proof writing skills. 5 stars.

Great book

Excellent exercises and very self-study friendly. This book can teach you real analysis very kindly. Some proof-writing experience is necessary.

Consensus on this book changes depending on how you like to do things.Quick rundown. Real analysis (at the undergraduate level) is an axiomatization of all concepts of single-variable calculus.This is a class that changes in difficulty surprisingly according to how you learn it (took it at UC Berkeley, math 104 fall 2017). Our class used Principles of Mathematical Analysis by Rudin and it was a horrendous experience. Everything I write will be from that point of view.The good:Ross’ elementary analysis is a really forgiving textbook. He explains all results thoroughly (almost to a fault), but that’s good for people who are completely new to this. Topic-wise, he covers only the most basic parts of analysis, but he explains every theorem thoroughly and I really like how he starts with explaining the real numbers and keeps the scope of the book in real numbers. That’s one thing Ross does very well, he keeps the scope of the book very centered on the real number system and it’s properties (completeness, Archimedean property, Q as a dense subset, etc.) and it serves the reader very well. I thought it was a really easy read, but that’s probably because I’ve read through Rudin chapters 1-7 about 5 times. Not much to say content-wise, but it’s really solid. No one section is outstanding, but they are all of good quality and it feels like he’s trying to develop all concepts in equal weight.The bad:Not as much to say here, but I think he’s a little too verbose and not enough math going on. He has a tendency’s to almost converse as he’s writing the material to you and it makes it kind of bad as a reference text. If I need a theorem or part of a proof really quickly while I’m doing a physics problem or other math, I certainly do not want to read 3 pages explaining 2 theorems when the proof+theorems would take half the page at most. But that’s really a matter of opinion and what level you’re at. Obviously, if you’ve never seen analysis before, you want as verbose an explanation as possible so you have more guidance and less racking your brain for explanations. But if you’re already past that level, then it’s not as good a text for you. I also didn’t like that Ross didn’t develop topology of the real numbers initially. I feel like that belonged in chapter 2 after he developed properties of the real numbers. The idea of open and closed sets and compactness and boundedness are all really important and could have been used immediately to shorten up proofs and I think it’s generally a better way to have analysis students think. A point of view with topology is extremely valuable. Results from real analysis can be quickly generalized with little trouble and slight adjustment if defined topologically.Overall, I recommend for undergraduates who have never seen analysis before. It will have you sufficiently prepared for future math.

My grandson, was very happy with this Xmas present.

This is an excellent book to give you insight into how calculus was originally developed. It starts with the basic principles and builds up to the derivative and the integral. If one closely follows the information presented it allows you to look much deeper into the underlying basics so you don’t have to take things on faith so to speak. It would be good for anyone trying to learn calculus to study this book or at least I think so.

The book is obviously not Rudin, at times over explains things, but is a solid introduction to the topic.

I used the book for assigned problems, only. The internet is free and will be faster to understand than most. Get a solutions manual instead of wasting time and money on this book.