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Good Calculus Books?

  1. May 8, 2013 #1

    I'm in high school and I'm wrapping up calc this year. Does anyone know of any good math books (particularly focused on calculus- single or multivariable) for me to read this summer?

  2. jcsd
  3. May 8, 2013 #2
    Since you have already been introduced to how to use calculus and the general ideas involved, it is a good time to get a solid grounding in the theory of calculus: I would recommend Spivak's Calculus. To really understand multivariable calculus, you should understand basic linear algebra. So you should pick up a textbook on basic linear algebra as well. Or you can challenge yourself into learning the necessary linear algebra at the same time as multivariable calculus, and pick up Hubbard's Vector Calculus, Linear Algebra and Differential Forms . If you like challenges and learning new things, you should find this book a lot of fun.
  4. May 8, 2013 #3
    Thanks for the reply,

    Is Spivak's Calculus a textbook? Also, I'd like to read about linear algebra and multivariable calculus separately, as I'm taking them as separate courses next year. Do you know of anything good that concentrates on these subjects separately?

    By the way, I'd like to add to my original post that I'm interested in reading almost anything math-related, as long as it has some realistic applications (for example, I'm not really interested in number theory, it seems kind of pointless to me).
  5. May 8, 2013 #4


    Staff: Mentor

    Number theory used to be thought of as "pure" mathematics (i.e., not applied), but no more. Computer encryption algorithms are heavily dependent on number theory.
  6. May 8, 2013 #5


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    If you are not interested in pure mathematics, I doubt you will like books like Spivak's single and multivariable calculus texts.
  7. May 9, 2013 #6
    I really don't know why one would recommend Spivak to a Calculus major in highschool... I remember years ago I googled "best calculus book" and ended up ordering Spivak's book all doe eyed to get it and start working it out... might as well have just bought a real analysis book. It's way too hard for anyone who isn't advanced in their mathematical maturity or who isn't extremely gifted in mathematics. I'd probably recommend Stewart... it's used universally, is slightly more gauged towards engineering in general, but it's still a good book to be used with outside resources like this forum.
  8. May 9, 2013 #7
    Spivak is written for motivated students who already finished calculus I and II. It is absolutely not written for people new to calculus. That said, Spivak really is a very difficult book with very difficult exercises. It's only meant for those who are up for a tough challenge.

    So yes, maybe Spivak is too difficult for the OP, and since he's not interested in pure math, he might not be interested in Spivak. But he might still check the book out to see for himself.

    I think Stewart is a horrible book. I absolutely don't recommend it. I do recommend the excellent book by Lang: "a first course in calculus". The OP already finished a calculus sequence, so maybe it's too easy for him.
  9. May 9, 2013 #8
    I agree that the Langs book are good, I used it for linear algebra (if we're both assuming it's the yellow-covered one with the publisher Springer that does all those graduate work-books on topology and geometry). I said Stewart because that's what I used and I got a pretty good understanding of fundamental calculus from it.
  10. May 9, 2013 #9
    I actually used Stewart's Calculus textbook for my math class this year (well, technically it wasn't a class). I think I'm going to try to read Spivak's book.
  11. May 9, 2013 #10
    What didn't you like about it? Is it because the problems were too easy?

    By the way, when I said that I wasn't interested in pure math, I mostly meant number theory. I love doing challenging integrals and I realize that this has no application to the real world. However, I am not particularly interested in number theory and I doubt that I will enjoy studying abstract algebra.
    Last edited: May 9, 2013
  12. May 10, 2013 #11
    I rather liked Rogawski's book, Early Transcendentals. Pretty straight forward with good explanations of concepts, and the problem sets have a decent range of difficulty.
  13. May 10, 2013 #12
    If you want a harder (more theoretical) textbook that has applications as well, I'd recommend Apostol's Calculus. It's sufficiently rigorous, has some great problems, and really balances technique and theory. You can get an international edition of this book on abebooks.com for about $15. The book also has a nice treatment of linear algebra. There is a second volume that goes further into linear algebra and then multivariable calculus.

    I highly recommend them.
  14. May 11, 2013 #13
    Thanks for the suggestions. I'm also interested in reading a little about complex analysis, because I'm taking it next year. Any suggestions for books about that?
  15. May 11, 2013 #14
    I doubt you actually mean a very rigorous complex analysis course since that requires quite some math prereqs. But here are two nice books which do not require much knowledge:

    Last edited by a moderator: May 6, 2017
  16. May 11, 2013 #15
    +1 for visual complex analysis. If you ever want a similar book for abstract algebra (specifically group theory), then there is a comparable book called Visual Group Theory.
  17. May 11, 2013 #16
    You're right, it probably won't be very rigorous because I haven't taken real analysis. I'm going to take linear alg/multivariable at the same time, the head of the math department thought that would be OK.
    Last edited by a moderator: May 6, 2017
  18. May 11, 2013 #17
    Be sure you know about partial derivatives and line integrals before the complex analysis course. If you do, you'll probably be ok.
  19. May 11, 2013 #18


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    Spivak and Apostol are very good but very hard. They will suit only a small percentage of people. If they appeal to you after inspection and experimentation, go for it. If you would like other options, I like the Lectures on freshman calculus by cruse and granberg. Lang's books are good for clear explanations of basic ideas but insufficient in examples for mastery. There are many editions of Stewart, some of the earlier ones of which I liked, maybe the second. The first edition of Edwards and Penney was also nice, as are the early editions of Thomas from the 1950's, (very non theoretical, engineering oriented).

    I recommend you go to a university library and browse the calculus section to see what appeals to you.

    See if you can understand this pdf file I am attaching. It contains actual proofs of the three main theorems not usually proved in differential calculus, but it will take work to understand them. You need to know that a real number is represented by an infinite decimal, and you need to know the definition of continuity.

    These notes prove what Spivak calls "three hard theorems" in an early chapter of his book. After teaching for almost 50 years, I have decided they are hard partly because Mike makes them look hard by adopting the abstract axiomatic and set theoretic approach that is so common today.

    There is no difficulty in giving rigorous proofs that are closer to intuition and might be more easily understood, but we never try because we just follow the tradition of the textbooks that are commonly used today. I have tried to offer an alternative. However my notes are very brief, and you may or may not need help filing in the details.

    E.g. a lemma I omit that you need is that, if a function f is continuous at p, then there is some interval centered at p on which f is bounded. This is an easy consequence of the definition of continuity, but it will be a good test of whether you have understood that definition.

    Attached Files:

    Last edited: May 11, 2013
  20. May 12, 2013 #19
    I can't open your attachment
  21. May 12, 2013 #20


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    apparently it hasn't been "approved" by the server. i will try to append a simple text version.

    I've tried to give elementary proofs of the “three big theorems” on continuity used in elementary calculus (taken together they say the continuous image of a closed bounded interval is again a closed bounded interval). I suggest this can be presented at least in a typical first year honors calculus class. First they should know the epsilon – delta definition of continuity.

    1) Intermediate value theorem: Assume f continuous on [0,1], and assume f(0)<0<f(1). We claim that f(x) = 0 at some x between 0 and 1. By looking at the values f(0), f(0.1), f(0.2),…,f(.9), f(1.0), there is some integer a1 between 0 and 9 so that f(.a1) ≤ 0 ≤ f(.a1+.1). If one of these two values is zero we stop.

    If not, then there is some integer a2 so that f(.a1a2) ≤ 0 ≤ f(.a1a2 +.01).

    Either we find at some stage a finite decimal x where f(x) = 0, or else we find a sequence of decimals xn = .a1a2….an, and xn+1 = = .a1a2….an + 1/10^n, so that f(xn) < 0 < f(xn+1) for all n, and |xn – xn+1| < 1/10^n.

    Since both sequences {xn} and {xn + 1/10^n} converge to the same decimal x = a1a2a3……., and since all f(xn)<0 while all
    f(xn +1/10^n) >0, it follows that 0 ≤ f(x) ≤ 0. QED.

    Here are two more such arguments along the same lines.

    2) Every function f continuous on [0,1] is bounded there.
    proof: if not then it is unbounded on some interval of form
    [.a1, .a1+.1],
    hence also on some interval of form [.a1a2, .a1a2 +.01].

    Continuing we find an infinite decimal x = .a1a2a3.... in [0,1], such that f is unbounded on every interval containing x. But if f is continuous at x, then f is bounded on some neighborhood of x. QED.

    3) Claim: A continuous f takes on a maximum on [0,1].
    proof: By theorem 1 above (IVT) the set of values of f form an interval, and by theorem 2), they form a bounded interval. If that interval is not closed on the right it has form say (c,d), but then the continuous function 1/(f(x)-d) would be unbounded on [0,1]. QED.
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