Spivak's Calculus

  1. This will be a pretty straight-forward question. Is Spivak's Calculus comprehensive enough for one embarking on undergraduate studies in Physics? Or is there some Maths left out that one would need to have a basic understanding of? If yes, what other book(s) would you reccomend, if you were to restrict yourself to only one or two additional ones?

    Thanks in advance.
  2. jcsd
  3. Not only is Spivak comprehensive, it is over the top. Spivak's book is "rigorous" and "advanced." I can not suggest it as first contact with differential and integral calculus. Real analysis is supposed to be for real analysis, not calculus.

    Spivak wrote a book called The Hitchiker's Guide to Calculus which is a prelude to his Calculus. You could read these two books at once if you like. That may be a good idea(also get his Calculus on Manifolds text).

    Most standard calculus books do not have a good balance between rigor and intuition. However, I think that this book comes close:
  4. It wouldn't be my first contact with Calculus, it would be me brushing up my forgotten skills. Does this make a difference to your answer?
  5. jasonRF

    jasonRF 855
    Science Advisor
    Gold Member

    I have not read Spivak's Calculus, but I do know it is basically a book on analysis of functions of a single real variable. There are a lot of topics a physicist should know that are not in that book. As a bare minimum, a physicist should know multivariable and vector calculus, linear algebra, and differential equations. Partial differential equations, complex analysis, numerical analysis, probability and statistics are also very good for a physicist to know.

    If you are limited to just a couple of more books, there are several good "mathematical physics" type of books that come to mind: Boas has written one (do a search in the forum), and I also like Riley, Hobson and Bence. Prof. Nearing has written an e-book that also has some nice insight:


    If you finish with Spivak and want to continue along those lines, Calculus volume II by Apostol has a lot of good material on linear algebra, differential equations, multivariable and vector calculus, probability and numerical methods. It is also takes a rigorous approach to the subject.


  6. Thanks, I'll look into what you suggested. There's just so many books in English and I'm not used to having to choose between them. Back where I come from, we had Maths for 4 years in high school and I'm sure we've covered everything I'd need to know as far as Physics is concerned. And in our final year when we were preparing for the final exams, where we had to recount everything we did in those 4 years, we had a book that covered ALL topics taught thereto. Problem is, the book was very succint and did not include a lot of proofs as we covered that in classes. But since I can't find my notes from back then, I'd have to brush up on all of that by getting a more comprehensive and detailed book. I figure Spivak's Calculus could be one of those. However, it itself is already 500 pages long and it seems ridiculous that I'd have to go through 2000 pages (as I'd have to go through other books covering topics not included in Calculus) just to get back on that level as I was before. Still, I'd do it if I there was no other way, but that's why I'd like to limit myself to 3 books at the most.

    So would that Apostol's book cover what is lacking in Spivak's book or would there still be topics left untouched after finishing those two?

    *I guess I could also go find a book my own language if all else fails, but I'd like to get acquainted with the English terminology a bit more.
  7. Oh, I see. Try
    Principals of Mathematical Analysis- Rudin
    Real Variables- Ash
    to concisely review the theoretical aspects of calculus.

    To solve calculation problems, go on the internet: http://www.math.temple.edu/~cow/

    Or get Apostol's Mathematical Analysis for a more comprehensive review.
  8. Thanks, I'll look that up, as well. What would be left then? Trigonometry, probabilities, algebra, vectors, ...? Which book would cover that or which umbrella would those topics fall under? Like I said, we did everything as we went along and didn't really differentiate Maths into Algebra/Calculus/Trigonometry (well, we did it of course by topics, but we didn't say "this year we're going to be dealing with Algebra", we just always had Maths as an all-encompassing subject in our minds).
  9. I'm using Spivak's calculus right now. Spivak's calculus a analysis book, its very rigorous but only covers basic cal. Its not a good book to learn calculus, its a book to learn proofs. Though I'm using it for a 1st year math course.

    If you learned calculus before and plan to do math/physics its a good book work on. You might want to get Stewart calculus (if its application based)
  10. jasonRF

    jasonRF 855
    Science Advisor
    Gold Member

    I just re-read your post - the first time I obviously didn't read carefully enough. Sorry! By "embarking on undergraduate studies in physics" I am guessing you mean that in the near-term you will be starting an undergraduate program?

    If this is the case, then having a solid understanding of high-school algebra, trigonometry, and a strong calculus background will get you a long way. I am presuming that the physics program will have you taking a bunch of math classes along the way - all the topics I listed before are those that you will learn during your undergraduate studies.

    Working through something like Spivak is useful, but only after you are comfortable solving more routine calculus problems (taking derivatives/integrals of various types of functions, etc.). If you are not comfortable solving routine calculus problems, then I would look for a used copy of an old edition of one of the standard Calculus books. I used Thomas and Finney and I thought it was good; likewise, Anton has written a nice calculus book. These types of calculus books tend to have appendices that cover the most important topics in algebra and trigonometry. You may be able to find such a book online as well.

    Best of luck.

  11. I personally used Spivak's calculus to self learn single variable calculus during the summer myself. I thought that it was a pretty good book, easy to learn from, but I don't really have any other calculus books to compare to.
  12. Thanks again, everyone! So let's say I have access to Spivak's Calculus, Apostol's Calculus and Stewart's Calculus (5th ed.). Which one of those would be best in terms of covering as broad of an area as possible with going deep enough for me to follow undergraduate all of the undergraduate physics courses? Am I right in thinking that by going through the latter two books the only maths field left untouched would be geometry and probabilities?
  13. Landau

    Landau 905
    Science Advisor

    Definitely Stewart. To a mathematician, I would recommend Spivak or Apostol. But for a physicist, these are way more narrow (because they go way deeper) than Stewart. For undergraduate physics you don't need the sophistication, so your time is better spent on a broader area, such as vector calculus, line integrals, etc., which are covered in Stewart but not in the others. These subjects are vital to every physicist.
  14. Ryker, I got Spivak two weeks ago due to it's reputation & from the first few chapters I find the main body of the text to be very interesting. It's a bit different to other Calc texts I've seen. However, as far as the end of chapter questions go they assume too much of the student.

    Forget about your basic calculus, the questions ask so much that without a great deal of previous exposure to mathematical logic you'd have to read a few other books to supplement it - but I can't seem to find any.... By chapter three it asks you to derive the Lagrange Interpolation formula, and the fact that the use of Lagrangians in advanced physics takes a lot of time to learn I can't imagine how anything Lagrange has done is within reach of someone without extensive study.I've given up on Spivak until I can find other texts that will train me to comprehend Spivak.

    After reading this page I assume you're looking for the math required by Undergrad Physics, Spivak really is a cut above what's required by most courses. My advice to you is to get a really good college level introductory engineering book if you're unsure about your mathematical background, i.e. Trig & Algebra, & once you've gotten past the calculus portion of that book should you consider college level Physics. It's a little known secret that intro engineering books contain all the math needed for your intro physics.

    If you know all the math that one would learn in a pre-calculus course & know basic integration/differentiation I guarantee that you'd get pretty far in a physics book such as "University Physics" by Young and Freedman, (or the Halliday & Resnick text - which I do not recommend). If learning/re-learning Calc concurrently you'd have no problem, I suggest Stewart or Thomas calculus. These books are looked upon as being basic but they still contain a heck of a lot of helpful material that you need to know cold before moving on.

    I'm currently studying Mechanics out of Kleppner & an old Berkley Mechanics text, I also recommend them if you know a bit of basic physics, but the Young and Freedman is way better for a first exposure, it's got a lot of quality stuff inside it.

    Hope all this helps somewhat :)
  15. jgens

    jgens 1,621
    Gold Member

    I happen to disagree with this analysis. I've worked through most of Spivak's text and I would argue that if you really understand the material then you should be able to complete most of the problems in the text. While most of Spivak's problems require some critical thought on the student's part (opposed to the strictly formulaic approach cheerily adopted by authors like Stewart), there are relatively few problems which are too difficult for a student that understands the material, especially given the sheer volume of hints that he provides. Granted, in any given chapter, there are certainly problems which students may be unable to solve without guidance, these are few and far between.
  16. I agree.

    Be careful who you get your recommendation from. Mathematicians will have a different opinion regarding Spivak then physicists will.

    You'll be pushing epsilons and deltas around in Spivak, this skill has no use in physics. While you're busy trying to prove that a sequence converges, you'll be missing out on the computational parts of calculus that ARE relevant to physics.

    I'm a math major. I loved Spivak. :smile:
  17. fluidistic

    fluidistic 3,336
    Gold Member

    At my university, for the freshman's year, Spivak's book is the book used for math majors and physics majors. We didn't use exclusively this book, but it was the main book.
    We weren't supposed to solve all of the problems included in the book, but a bunch of them, yes.
    So yes, Spivak is comprehensible for freshmen as long as they make an appropriate effort.
  18. Landau

    Landau 905
    Science Advisor

    Lagrangians in analytical mechanics and the Lagrange Interpolation formula have *nothing* to do with each other. The idea of the latter is really basic and should be within the reach of anyone who knows calculus (of course, deriving it yourself can be tough, but your "physics Lagrangian" argument is bogus).
  19. It wasn't really an argument Landau... more like a bad joke... ;)
  20. Landau

    Landau 905
    Science Advisor

    OK, my mistake ;)
  21. On another note, Lagrangians(unlike the more geometric Hamiltonians) are very simple. A high school student would be able to understand "kinetic energy minus potential energy" and the principal of least action(Feynman learned this principal from his HS physics teacher) should be comprehendable to physics students.
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