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Analysis Book for a Beginner

  1. Apr 30, 2010 #1
    I'm basically through with the AP Calculus BC curriculum, and I think I have a good grasp on calculus as a set of tools. However, I don't think I've got a great grasp on calculus as a concept [we haven't done any proofs this year and I know very very little about Delta-Epsilon proofs.]

    I was looking at this book to work on over the summer; opinions?


    Actually, while I'm at it, I may as well ask for a recommendation for a book for someone who's finished the AP Physics B curriculum. I'm more interested in particle physics and quantum mechanics than kinematics, but I understand that those fields require some things I probably don't know. That said, any book recommendations to start me off?
  2. jcsd
  3. May 1, 2010 #2


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    I personally think you are better off learning linear algebra, multivariable/vector calculus and differential equations before jumping into analysis. However, if learning analysis is something that just sounds plain fun to you for a summer challenge, then who am I to hold you back? In such a case I would recommend a used copy of the 2nd (or perhaps 3rd) edition of "analysis, with an introduction to proof" by Lay.


    He does a good job of teaching the basics of logic, methods of proof, etc. at the beginning of the book. The book is clear and covers the highlights of the topic. He also has answers to some of the questions, which can be a help. It also can be had cheap, so if you hate the book you aren't out much.

    Ross' book is supposed to be pretty good, and it has solutions to some problems as well, but I just have never picked it up so I have no opinion. It is certainly used at many universities these days (google and you will find it everywhere!)

    good luck

    Last edited by a moderator: Apr 25, 2017
  4. May 2, 2010 #3
    Anyone else want to comment on Ross vs. Lay? Ross is about $20 more expensive but looks a little friendlier + apparently has exercises and solutions that Lay doesn't.
  5. May 2, 2010 #4
    Seeing as how no one else has, I'm going to put in the obligatory vote for Spivak's Calculus. It's really more of an analysis book, but it's incredible. The problems are extremely hard, however, so it would really be a challenge, but if you're up to it, go for it.
  6. May 2, 2010 #5
    Do you say this knowing that I have a very poor background in proofs, including basically no experience with delta-epsilon limits? Spivak's Calculus appeals to me because people speak of it as a bridge between the very-challenging and easy, but I'm not sure if I'm supposed to be on that bridge.

    Also, this is a bit embarrassing, but as far as I can tell, Spivak's Calculus doesn't include a solution manual? So (sigh)...how will I know if I'm doing it right? :uhh:

    EDIT: One popular Amazon review kind of backs me up on this. Part of it says:

    That doesn't sound like it would be right for what I'm doing?
  7. May 3, 2010 #6
    I have used Ross for my elementary analysis class. I think it's a wonderful textbook to both use it in class, or to self-study. The book has a lot of examples and descriptions to understand how the proofs are really written. Exercises weren't very complicated (although not necessarily easy), and about half of them come with solutions or hints in the back. Only prerequisite for this textbook is really just one-year of calculus, and it might be better to read this textbook right after you've seen infinite sequences and series because this textbook BEGINS with those topics (right after the preliminary stuff).

    I think AP calc at your school probably focused mostly on computations (i.e. the stuff you're tested on the AP exam), and not much on the theory and concepts. This book should teach you the stuff you didn't learn in AP calc, and I think you'll enjoy reading this book.

    One more thing: While the book is targeted for students who have little or no experience in writing proofs, it might be helpful to have a little bit of understanding of proofs and logic. You don't need to be an expert in this, but it might be helpful to know a few key things like basic set theory, logic (e.g. quantifies, truth tables, negations, a conditional statement is equivalent to its contrapositive, etc), and inductions. I don't know if you can/want to find a book in this... you might be able to find something online for free.
  8. May 4, 2010 #7
    Unless someone says otherwise, I think I'm getting the Ross book. Thanks for the opinions (although if there's someone out there who's looked at both Ross and Spivak - say something!)
  9. May 4, 2010 #8


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    I've self studied through Spivak, and the way he presents the material is absolutely amazing. He does so in an informal manner, in that way you don't get bored as you would be if you're reading something like Apostol's text. And no one said it's going to be easy, but after going through it, PMA doesn't seem as monstrous anymore.

  10. May 5, 2010 #9
    I think I will get Spivak after all. Since UChicago's Honor Calculus sequence uses the book (and I'll be heading there come September!) it would probably be the best way to go. I hope I am up for the challenge; I'm betting you guys will see me posting on these fora often over the next few months. Thanks for all the input.
  11. May 5, 2010 #10


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    Spivak does have a solution manual:

    http://www.mathpop.com/bookhtms/cal.htm [Broken]

    The web site indicates that sales are "restricted" because the book is used as a text in some universities. Perhaps if you write to the publisher explaining your situation, they will sell you a copy.
    Last edited by a moderator: May 4, 2017
  12. May 5, 2010 #11
    Actually, since I'll be taking a course this Fall that may use the book, my situation is probably not totally one that merits a solution manual. I think I'll just use these fora for no price at all :)
    Last edited by a moderator: May 4, 2017
  13. May 5, 2010 #12
    Last edited by a moderator: May 4, 2017
  14. May 25, 2010 #13
    Just wanna say, having tried working through the book's first chapter: don't get Spivak if you don't have prior proof experience :x
  15. May 25, 2010 #14


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    No offense, but in the beginning you were debating whether to get Ross or Lay. They're full blown analysis books, and they all rely on proofs. Spivak is kind of the bridge between cookbook math and the "real" stuff.
  16. May 25, 2010 #15
    I am using Spivak for study right now, and although I love the presentation, some of the errors (in the 3rd edition at least) are sort of aggravating and I tried emailing the publisher (in other words, Spivak) for an errata list and there is apparently none. I'm not crazy when I say that there are errors: http://www.math.toronto.edu/~drorbn/classes/0405/157AnalysisI/SpivakComments/SpivakComments.html

    but I'm starting to think they are there (and not just the ones in the link, I suspect there are more) just to keep you on your toes.
  17. May 25, 2010 #16


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    if you're smart enough to point out the errors, then it shouldn't be THAT big of a deal
  18. May 25, 2010 #17
    Yeah, my post was intended more as a warning about Spivak - not necessarily a recommendation of Ross, since I don't know what that book's like.

    Are there any books/online guides on proof-writing anyone wants to pull out? I read Spivak's proofs and they don't seem particularly rigorous, although I have no doubt that they are.
  19. May 26, 2010 #18


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    No what I meant was that you were looking for a full blown analysis textbook when you don't even have the right foundation yet.
  20. May 26, 2010 #19


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    How is Ross more an analysis book than Spivak?
  21. May 26, 2010 #20


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    I'm kind of wondering the same about Lay vs Spivak. From Google books Spivak looks like it is at the same level as Lay. Except Lay spends the first two chapters on logic, methods of proof, set theory, etc. I personally found that proof by contradiction, contrapositive, etc., only made sense after I had seen it from a logic point of view. The chapters gradually get you up to speed on how to write proofs. Also, every section has practice exercises spread throughout, with complete solutions prior to the end-of-section exercises. So the reader can get some good practice as he reads each chapter, and before attempting exercises. A reasonable number of the end-of-section exercises have solutions/hints in the back of the book - almost half of the problems in the early sections, and gradually decreasing in number and completeness as you move through the book and presumably become more fluent in writing proofs.

    The content of Spivak's Prologue doesn't show up until ~page 80 in Lay (chapter 3), so Lay prepares the reader better. Perhaps Lay does cover some things a little more abstract than Spivak (?) But Lay is certainly way below Rudin's undergrad analysis book. So I would guess Lay and Spivak are somewhat equivalent, although I must admit spivak looked more inspiring.

    But this is all academic. I really think the OP is better off learning about multivariable/vector calculus, linear algebra, ODEs, probability theory, etc., prior to looking into analysis. Perhaps then the motivation for why we may need to worry about sticky situations can actually matter sometimes. And then, learning logic/proofs is a must before jumping into analysis, as the OP found. I'm sure there are many good books on this, and folks around here can probably recommend the best, but I am only familiar with Lay's approach to teaching proofs.

  22. May 27, 2010 #21
    Well, the course that I *think* I'll place into at my University this Fall [the prerequisites are just a good knowledge of AP BC-type Calculus] uses Spivak, so I figured that Spivak would be a good choice. On the other hand, I'm wondering if the course teaches things better.
  23. May 31, 2010 #22
    Hello Everyone!

    I am a new member of this forum. I do want to post my own thread about this, but I cannot figure out how to. For the time being, I will stick to this thread as this is the closest it gets to my query.

    I too am a 16 year old, and my interests are somewhat similar to Karkas's with an addition of Astrophysics. I, since I was 3 have been interested in Physics, more specifically, the ways of the Universe. As a matter of fact, as a toddler, the pictures in an Isaac Asimov's encyclopedia interested me to such an extent that it played a major part in how I taught myself to read.

    All went well till I was 11. I knew about supernova's, dark matter, the bending of light and other such things, you'd find in basic encyclopedias pretty throughly. My enthusiasm was to such an extent that I usually used to get thrown out of class for two reasons primarily- for asking too many questions, or for answering to many, enthusiastically despite repeated warnings from the teacher about not asking out of my turn.

    Then, I joined a boarding school.

    This is when I saw a catastrophic change in my choice of passions. Owing to a few bad grades in Physics and the general requirement of the school to excel in as many activities you can, I began focusing more and more on sports, music and studies in general.

    Though I do not regret the amount I achieved, being an above average student and learning to play the piano along with being a district champion in athletics, I lost touch with the subject, which I today, regard as the most important thing to ever happen to me- The Universe.

    I am on student exchange at the moment, and having enough time to do according to my wishes without worrying about the academic pressure, I watched a BBC Documentary about called Atom last week.

    Such was its effect that, it has rekindled in me a passion, I'd once lost. I've been watching as many documentaries as I can till today, having just finished one. And I feel that I really want to learn about Physics all again.

    Each passing day finds me more and more interested in the subject, but it as a whole but two specific topics of it, or I should say three. Nuclear, Astro and Quantum.

    I really dont want to lose this interest again, and for that I want to educate myself as much as I can.

    Having gone through the previous posts, I figured one thing out. That you have to have an understanding of Calculus among many other things. For this, I will start with my Calculus tomorrow itself. I am supposed to be taught Calculus this year anyways, but owing to my rocketing interest and the sense of conciousness that I cannot make a further leap into the subjects without a better understanding, I plan to pre-pone my plans a bit.

    Can you suggest a book which will come in Handy. A book which I can rely on to educate myself a bit more than my class? Any answers would be very welcome.

    I hope you excuse me for this horribly wrong post, but I feel very strongly for the subject and couldn't think of any other way to tell you more about myself.

    Any replies would be immensely appreciated.

    Thank You! :)
  24. Jun 1, 2010 #23
    You should check to see what your class will be using. If you are confident, the Spivak book being discussed in this thread is excellent. However, you may find it approaches things differently (more theoretically) than your upcoming course. This is good, but only if you have the headspace for both...

    There are also excellent free calculus video lectures here:

    Good luck - the universe is worth exploring!
  25. Jun 1, 2010 #24
    Thank You Sankaku for the suggestion. I did check on the book on the internet and I got the idea that it is more for college students. Well, I'm 16 right now and we haven't started doing Calculus in school yet. Its scheduled to be done much later this year.

    Secondly, if you could suggest a book which covers the Universe and the Quantum World in a way I, a 16 year old beginner, can understand. I'm looking for something which is not too elementary but at the same time, not too complicated either, but instead gets complicated gradually teaching side by side at the same time.

    If any of you know of such a book. After looking so much on the internet, I'm starting to doubt even the notion of their existence! :P

    Any help would be greatly appreciated! :)

  26. Jun 2, 2010 #25
    That you find Spivak difficult can be a good thing, since that's (often) a hallmark of a good book. I would imagine the course would clarify some things for you and, perhaps more importantly, force you to move at a brisker pace. Moving too slowly and trying to understand too much is something I've found to be a problem when self-studying. However, it really depends on the course syllabus, the professor, and the background of the class in which you will be.

    I took a quick look at the FAQ about the Honors Calculus sequence. For my analysis course, we used The Way of Analysis by Strichartz. It's wordy, but that's an added recommendation for your situation, and it does include a section in the beginning advising how students should understand proofs. That said, you've already invested in Spivak, and it does appear to be a good book. As others have mentioned, feel free to ask any questions you have.
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