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Math book strong in theory deduction and proofs?

  1. Nov 8, 2008 #1
    My situation:

    I did good in high-school, learned algebra, functions, exp/log functions, limits/continuity, calculus, vectors, trigonometry, diff equations of 1st and 2nd order, and perhaps a few others things I left out. Came out with an A in math and physics, so far so good. Now in college I need a student job and I will try private teaching in math and physics. Problem is that we only borrowed the books in high-school so now I need to purchase some so I can prepare properly for sessions, knowing all theory 100%, perhaps even learn some of the proofs in the process.

    So what I'm basically looking for is one or more books that covers the above mentioned subjects, i.e. a pretty wide range of late-high-school, early-college stuff. I would appreciate a book that does logical deduction on the theory. Also I'd love to have the various proofs available to me (proof of pythgoras, a*c=-1, various log calculation rules, differential quotients etc.). I'm not particularily looking for a super duper rigorous book, though I am asking for a somewhat rigous book, not a "calculus for dummies" type of book. However I find it very hard to find such a book, the market seems immense and confusing.

    Candidates I have found so far:

    Calculus - A Complete Course (Robert A. Adams) - http://www.amazon.co.uk/exec/obidos/ASIN/1405839244/ref=ord_cart_shr?_encoding=UTF8&m=A3P5ROKL5A1OLE
    Pros:Contains just about every subject on my list; a complete package, inexpensive especially considering what I get (link to table of contents: http://vig.pearsoned.co.uk/catalog/academic/product/0,1144,0321270002-TOC,00.html)
    Cons:I'm a bit concerned about the depth of the book, how detailed the theory deduction is, if there are any proofs in it etc. I don't don't know if my suspicions are founded though.

    Precalculus (Michael Sullivan) 7th Edition 2004 AND perhaps Algebra and Trigonometry (Michael Sullivan) 8th Edition 2007 - http://www.amazon.co.uk/Precalculus...1_121?ie=UTF8&s=books&qid=1226168009&sr=1-121 and http://www.amazon.co.uk/Algebra-Tri...=sr_1_1?ie=UTF8&s=books&qid=1226170803&sr=1-1
    Pros:From the look of it, probably more rigorous than "calculus", the little feedback I've found from these two books is positive
    Cons:..but to counter that I've found some negative feedback on one of his other books, College Algebra (https://www.amazon.com/College-Algebra-7th-Michael-Sullivan/dp/0131430920), hard to find any info/feedback or even a table of contents, a great deal more expensive than "Calculus: A Complete Course.

    I'm leaning towards the "Calculus: A Complete Course" because it's a relatively inexpensive package and seems to contain a lot of material, but I'm not sure though.

    I very much want more suggestions, but prefer if you have first-hand knowledge the particular book in question.

    Thanks :)
  2. jcsd
  3. Nov 8, 2008 #2


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    Apostle's Calculus
  4. Nov 8, 2008 #3
    You mean this? http://www.amazon.co.uk/Calculus-On...n-Linear-Algebra/dp/0471000051/ref=pd_sim_b_1

    Looking from the table of contents and first couple of pages, it looks good. I like it's style, looks rigorous enough as well. It's very expensive though, especially for just 666 pages. But I won't rule it out just because of the price, a brilliant textbook is priceless. I gonna wait for some more recommendations to come in before I purchase though. Thanks for the tip :)

    I just bumped into this article btw: http://math-blog.com/2007/05/13/the-most-enlightening-calculus-books/

    Here I'm seeing recommendations for this Michael Spivak's "Calculus" as well. Unfortunately I can only find the the second edition on the US amazon, only the 1st edition on UK amazon (I live in Europe): http://www.amazon.co.uk/Calculus-Michael-Spivak/dp/0521867444

    But why the hell Apostol's Calculus must be 100£ when it's from 1975 is beyond me...

    In the case of me buying Apostol's Calculus I'd still need to cover the other subjects with some other books.
  5. Nov 8, 2008 #4
  6. Nov 8, 2008 #5


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    here is one of the best calculus books in existence:

    Differential And Integral Calculus Volume 1
    Courant, R.

    (Lincoln, LIN, United Kingdom)
    Bookseller Rating: Book Price:
    £ 1.33
  7. Nov 8, 2008 #6


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    heres the second volume:

    Differential and Integrasl Calculus Volume 2 (II)
    Courant R

    British Heart Foundation Books & Music
    (Harrogate, North Yorkshire, NYK, United Kingdom)
    Bookseller Rating: Book Price:
    £ 9.00
  8. Nov 9, 2008 #7
    Just in short why are they good, and do they contain proofs for the differential quotient of k*x, a*x, 1/x, lnx, x^a etc.?

    BTW a small problem is that I can only seem to get the two "Differential And Integral Calculus" books from US Amazon, but they're not available from UK Amazon. So I'm thinking of going with Apostol or Spivak instead. But looking at the table of contents and the firs couple of pages, I think I might prefer Apostol and Courant styles over Spivak. Can't be sure about this, just a feeling though. I'm a bit concerned about the language in Courant's because it's a translation from german, which might show, perhaps making Apostol or Spivak more accessable. Is this a legitimate concern?

    But actually I'm not exactly sure that's what I'm looking for. I'm a physics student, not a math student for a reason after all. All I want is a math book that derives theory from definitions as it goes and proves most of it's claims, it doesn't necessarily have to be a 100-year-old 100% rigorous math book, because I'm concerned if it's too "dry" and old-fashioned I might not get around to read it at all, considering this is something I'll have to read on the sideline while learning physics at the university (the next 12 months for me will cover advanced mechanics, thermo, lin.alg., advanced calculus, E&M and finally QM). If there is no way around Apostol, Spivak or Courant, then so be it. All I want is really just a book that clearly defines calculus, derives and proves various differential quotients, exp/log/trig functions theory, proves various log rules such as loga*logb=log(a+b), define and proves diff equations in a short comprehensible way, stuff like that.
    Last edited: Nov 9, 2008
  9. Nov 9, 2008 #8
    Ok I just borrowed an ealier edition of Calculus: A Complete Crouse (the 5th, the newest, is the 6th) from an electrical engineer from my dorm and looking through I'm now sure that it's NOT what I'm looking for. Too brief in style, it spreads itself too thin in my opinion

    So unless someone can come up for a new suggestion, I guess there is no way around Spivak, Courant or Apostol for a calculus book.

    But in that case, I still need a very good book on trigonometry and differential equations.

    Question, is there a real difference between "Differential and Integral Calculus" and "Introduction to Calculus and Analysis", both Courant? Which is best?

    And mathwonk you I've seen you mention that Spivak's "Calculus" is somewhat derived from Courants "Introduction to Calculus and Analysis", well how does Apostol's book fit into that and how does it compare?
  10. Nov 12, 2008 #9
  11. Nov 13, 2008 #10
    I bought Introduction to Calculus and Analysis Volume 1 by Richard Courant and John Fritz for around $45. I'm currently working through this book and I have also used some of Apostol's Calculus text. I prefer Courant's exposition and his often simpler, more intuitive proofs (don't be mistaken though, the book is rigorous).
  12. Nov 14, 2008 #11
    Ok thanks for the answer. Have you read Michael Spivak's Calculus, how do them two compare to it?
  13. Nov 14, 2008 #12


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    if you look more closely, you will see i even located cheap copies for you in the UK. In fact they are so cheap you cannot go wrong just buying them immediately.
    Last edited: Nov 15, 2008
  14. Nov 17, 2008 #13
    I haven't read Spivak's Calculus, so I can't say anything about it. But one thing about Courant's book is that is also contains almost 100 pages on applications to physics and geometry. It also gets into fourier series.
  15. Nov 30, 2008 #14


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    forgive me if i do not answer detailed questions on why i recommend what i do, as after doing this in detail for years now on the who wants to be thread, i take it for granted by now, my recommendation might be just investigated on its merits, rather than having me try to persuade you of its value to you.

    in general when you ask an expert for advice, and then when you get it, you keep begging to be convinced the advice is right, you give an impression of someone not serious enough to take charge of your own learning.
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