Elementary Analysis: The Theory of Calculus by Ross

[micromass]

• Author: Kenneth Ross
• Title: Elementary Analysis: The Theory of Calculus
• Amazon link: https://www.amazon.com/dp/1441928111/?tag=pfamazon01-20
• Prerequisities: Proofs, Calculus
• Level: Undergrad

Table of Contents:

[LIST]
[*] Preface
[*] Introduction
[LIST]
[*] The Set N of Natural Numbers
[*] The Set Q of Rational Numbers
[*] The Set R of Real Numbers
[*] The Completeness Axiom
[*] The Symbols +\infty and -\infty
[*] A Development of R
[/LIST]
[*] Sequences
[LIST]
[*] Limits of Sequences
[*] A Discussion about Proofs
[*] Limit Theorems for Sequences
[*] Monotone Sequences and Cauchy Sequences
[*] Subsequences
[*] lim sup's and lim inf's
[*] Some Topological Concepts in Metric Spaces
[*] Series
[*] Alternating Series and Integral Tests
[*] Decimal Expansions of Real Numbers
[/LIST]
[*] Continuity
[LIST]
[*] Continuous Functions
[*] Properties of Continuous Functions 
[*] Uniform Continuity
[*] Limits of Functions
[*] More on Metric Spaces: Continuity
[*] More on Metric Spaces: Connectedness
[/LIST]
[*] Sequences and Series of Functions
[LIST]
[*] Power Series
[*] Uniform Convergence
[*] More on Uniform Convergence
[*] Differentiation and Integration of Power Series
[*] Weierstrass's Approximation Theorem
[*] Differentiation
[*] Basic Properties of the Derivative
[*] The Mean Value Theorem
[*] L'Hospital's Rule
[*] Taylor's Theorem
[/LIST]
[*] Integration
[LIST]
[*] The Riemann Integral
[*] Properties of the Riemann Integral
[*] Fundamental Theorem of Calculus
[*] Riemann-Stieltjes Integrals
[*] Improper Integrals
[*] A Discussion of Exponents and Logarithms
[/LIST]
[*] Appendix on Set Notation
[*] Selected Hints and Answers
[*] References
[*] Symbols Index
[*] Index 
[/LIST]

 

[SrVishi]

I've used a good amount of this book a fair while back, so I'll go off what I remember, though I believe my memory of this book is entirely what I thought about it as I was reading through it. It's a good intro book. It's pretty soft on the reader, assuming they have had an exposure to proofs and some set theory (both for a set-theoretic background and extra help with having a sufficient mathematical maturity), though I would expect an individual going into an analysis class to already have these, at least to a fair degree. Now, again, I think this is a good book. It is not the most advanced book, primarily because of the amount of topics it covers, but also because the author usually explains things thoroughly. I think this book is perhaps comparable to Abbott's Understanding Analysis (which is an incredible first book or supplement for the struggling student in analysis), perhaps a slightly higher level, just because he formalizes more concepts than Abbott does (i.e Ross explicitly states the field axioms, but from what I recall, Abbott doesn't in his book, along with some other topics as well.) However, I'm sure there are some people who might even use Abbott first or as a supplement to this book. Now, there are a few minor things I don't like, such as Ross sometimes complicating an argument or proof (happened rarely) which seemed a little disproportionate to the rest of the book, but there are a few issues which stop this from being a 'complete' introduction to undergraduate analysis. Mainly, it's the selection of topics. Again, he explains what he covers very well, but he doesn't cover all that much. There are no sections on multivariate analysis nor Lebesgue theory. I guess that's okay since this is an introductory book, but this is just another reminder that that's exactly what it is, and that a student is overwhelmingly obligated to read another more advanced analysis book afterwards if they want to go onto graduate school (they should be able to handle Apostol or maybe Pugh afterwards). One thing that really bugs me about his selection of topics is how he has a section on the construction of the reals, but doesn't even go through with it. Now there is a semi construction of it in the exercises of that section, but why the hell not at least construct the whole thing by guiding the student in the exercises? It was such a wasted opportunity. Next, he doesn't even go into constructing exponential and logarithmic functions formally, instead leaving that topic as a "discussion" at the end of the book. That made me kind of angry, as it is indeed an important subject and he just disregarded it. All in all, I guess it is a sufficient book, though if one were going to start with a book of this level, I would suggest reading instead Understanding analysis, and then moving on to Apostol, Pugh, or whatever higher level undergraduate book they may find best for them. Actually, another book that is of nearly the same style, and perhaps more appropriate for the student who has recently left the calculus sequence (yet with sufficient proof knowledge) would be A First Course in Real Analysis by Protter and Morrey. This books covers loads more material (except for Lebesgue theory, sadly), is rigorous, yet explains things very clearly. I would recommend this over Ross' book.

 

[bacte2013]

I've used a good amount of this book a fair while back, so I'll go off what I remember, though I believe my memory of this book is entirely what I thought about it as I was reading through it. It's a good intro book. It's pretty soft on the reader, assuming they have had an exposure to proofs and some set theory (both for a set-theoretic background and extra help with having a sufficient mathematical maturity), though I would expect an individual going into an analysis class to already have these, at least to a fair degree. Now, again, I think this is a good book. It is not the most advanced book, primarily because of the amount of topics it covers, but also because the author usually explains things thoroughly. I think this book is perhaps comparable to Abbott's Understanding Analysis (which is an incredible first book or supplement for the struggling student in analysis), perhaps a slightly higher level, just because he formalizes more concepts than Abbott does (i.e Ross explicitly states the field axioms, but from what I recall, Abbott doesn't in his book, along with some other topics as well.) However, I'm sure there are some people who might even use Abbott first or as a supplement to this book. Now, there are a few minor things I don't like, such as Ross sometimes complicating an argument or proof (happened rarely) which seemed a little disproportionate to the rest of the book, but there are a few issues which stop this from being a 'complete' introduction to undergraduate analysis. Mainly, it's the selection of topics. Again, he explains what he covers very well, but he doesn't cover all that much. There are no sections on multivariate analysis nor Lebesgue theory. I guess that's okay since this is an introductory book, but this is just another reminder that that's exactly what it is, and that a student is overwhelmingly obligated to read another more advanced analysis book afterwards if they want to go onto graduate school (they should be able to handle Apostol or maybe Pugh afterwards). One thing that really bugs me about his selection of topics is how he has a section on the construction of the reals, but doesn't even go through with it. Now there is a semi construction of it in the exercises of that section, but why the hell not at least construct the whole thing by guiding the student in the exercises? It was such a wasted opportunity. Next, he doesn't even go into constructing exponential and logarithmic functions formally, instead leaving that topic as a "discussion" at the end of the book. That made me kind of angry, as it is indeed an important subject and he just disregarded it. All in all, I guess it is a sufficient book, though if one were going to start with a book of this level, I would suggest reading instead Understanding analysis, and then moving on to Apostol, Pugh, or whatever higher level undergraduate book they may find best for them. Actually, another book that is of nearly the same style, and perhaps more appropriate for the student who has recently left the calculus sequence (yet with sufficient proof knowledge) would be A First Course in Real Analysis by Protter and Morrey. This books covers loads more material (except for Lebesgue theory, sadly), is rigorous, yet explains things very clearly. I would recommend this over Ross' book.

I also recommend the "Analysis I-II" by Terrence Tao. He starts by building up the comprehensive and detailed construction of the real numbers and set theory. Proofs are also covered in detail, including what proof technique will be best for proving certain theorems and also providing valuable hints about proving on your own. I actually started studying the analysis by studying the first few chapters on Analysis I and jumping to Apostol and Pugh. Tao's brilliant yet clear exposition and treatment of the real number system and set theory made the transition very smooth. Another books I recommend are "Real Numbers and Real Analysis" by Ethan Bloch and "Introduction to Analysis" by Mattuck.

By the way, Tao also keeps the ongoing errata list on his blog.

 

[SrVishi]

I also recommend the "Analysis I-II" by Terrence Tao. He starts by building up the comprehensive and detailed construction of the real numbers and set theory. Proofs are also covered in detail, including what proof technique will be best for proving certain theorems and also providing valuable hints about proving on your own. I actually started studying the analysis by studying the first few chapters on Analysis I and jumping to Apostol and Pugh. Tao's brilliant yet clear exposition and treatment of the real number system and set theory made the transition very smooth. Another books I recommend are "Real Numbers and Real Analysis" by Ethan Bloch and "Introduction to Analysis" by Mattuck.

By the way, Tao also keeps the ongoing errata list on his blog.

I would love to read Tao's books but I can't seem to find them online. Whenever I go to his website, it seems to be a broken link!

 

[UnivMathProdigy]

Oh, boy. I remember this book from when I was starting to learn advanced math (proofs) and had no training in proof-writing. In fact, when I took Real Analysis in grad school, I wasn't even sure I had undergraduate-level analysis back then. It turns out I did; it was under the name "Advanced Calculus" and silly me couldn't make the connection.

Now that I have some type of analysis background (one semester in Real Analysis and one in Functional Analysis), I'm willing to give this book another try.

 

[PeroK]

I bought this book 35 years ago and still have it today. This was my introduction to rigorous maths. How difficult some of those concepts seemed back then. And how exciting when the light went on one day and it all started to make sense.

The strength of the book is its clarity in presenting material that is probably very different from anything the student has come across before. Other books I considered at the time seemed to assume a general familiarity with rigorous concepts, which I didn't have.

For me it represented a whole new way of thinking. Where would I have been without it!

A nostalgic rather than objective review perhaps.