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Calculus: Apostol or Spivak?

  1. Feb 16, 2010 #1
    Hey guys. Seeing that the Calculus textbooks of Apostol and Spivak have been mentioned and recommended numerous times in this forum, I was wondering which one is better?

    Currently, I am senior taking calculus. I feel that my calculus class is too easy and will not help me in college where I would like to major in either math or physics. Recently, I have been doing many precalculus exercises/problems given that my previous math classes were also quite elementary.

    Anyhow, which textbook is better?
  2. jcsd
  3. Feb 16, 2010 #2
    Apostol and Spivak might as well be introduction to analysis books. In my opinion. It really depends on what you want to do. If you want to learn more about the theory of mathematics. In which case, Apostol is fine though a little dry. Spivak's writing style is a bit more engaging, but his book only covers single variable.

    If you would rather learn about math methods for solving problems of certain kinds, I would recommend an undergraduate math methods book, such as Mathematical Methods in the Physical Sciences by Mary Boas.
  4. Feb 16, 2010 #3
    I learned from neither of those; rather, I learned from "Baby Rudin." It is, in my opinion, better than either of those books and is representative of higher mathematics texts. The book is entitled "Principles of Mathematical Analysis" by Walter Rudin. It covers the real and complex number systems, some topology (the parts useful in analysis), derivatives, and integrals (both Riemann and Lebesgue) in roughly 300 pages.
  5. Feb 17, 2010 #4
    Neither is 'better,' they are both good for different reasons. Spivak is cheaper, though! As naele said, Spivak's style is a bit more natural, but some people thrive on a drier book like Apostol. Apostol starts with Integration, which can be a bit disconcerting when used along side the more common sequence taught in the US.

    See if you can find a copy at a library and get a feel for the style of each.
  6. Feb 17, 2010 #5
    I never liked Apostol's calculus text since it was terribly dry and I didn't particularly care for his treatment of integration before differentiation. Spivak is excellent for just the problems alone. The text itself is great, but if you're planning to major in math or physics, you might want to look into a slightly more advanced introductory analysis text. Spivak will teach you how to work with epsilon's and delta's and his book is for all intents and purposes a basic analysis text. However, Spivak hesitates to build up naive set theory and refuses to introduce metric topology for the sake of keeping the exposition elementary. Consequently, you might find out in a later analysis course that some of Spivak's theorems have much more intuitive and often simpler proofs when just a bit more theory is developed.

    My recommendation is to take a look at Apostol's mathematical analysis text. According to Apostol himself, the exposition is at the advanced calculus level, so the text won't be as terse as say, Rudin's Principles of Mathematical Analysis. This doesn't mean that Apostol sacrifices any mathematical rigor. His proofs are completely rigorous and clear, but he will fill in the details in the more difficult proofs, which some might see as a detriment (but it's not like anyone said you couldn't try proving each theorem on your own). Apostols starts with an axiomatic approach to the real and complex number systems and then covers naive set theory. He then develops point-set topology in R^n and then briefly discusses general metric topology. I like this approach since working in Euclidean space captures most of the intuition behind many results in basic analysis anyways. Then he covers limits and continuity in general metric spaces, and then moves onto one-variable differentiation and includes a brief discussion of partial differentiation and complex differentiation at the end. He then develops standard topics in one-variable theory before launching into analysis in R^n. I think his chapters on analysis in R^n are superb though I admit I have little use for his chapters on the Lebesgue integral since he doesn't adopt a measure-theoretical approach.

    Anyways, I think that you'll find a reasonable amount of overlap between Spivak's calculus and just about any other introductory analysis text, especially if you work through Spivak's exercises. Hence I would recommend trying a slightly more advanced text and see how that goes. Spivak's ordering of topics is also a bit strange, but I guess I didn't complain when I was learning from it.
  7. Feb 17, 2010 #6
    Either book is fine if you are up to it. Spivak is, in general, a lot cheaper than Apostol. Provided,
    you WILL need to buy the The Hitchhiker's Guide to Calculus to accompany it as well as Calculus on Manifolds for multivariate calculus. The good thing about Spivak is that you WILL be reading him later in your career(his volumes on differential geometry is a must), therefore, it would not hurt to be familiar with his writing style.

    The good thing about these rigorous calculus books is that it allows you to skip your first Analysis book("baby rudin").
  8. Feb 18, 2010 #7
    Thank you for the much appreciated feedback.

    I guess I should have ask, how is each one different? And what are other advantages and disadvantages that have not already been mentioned in this thread?

    Also, I have been reading similar threads regarding my question, and basically it comes to a matter of choice and personal taste. Is this true given that both books are of high quality that it doesn't matter which one I choose?

    Again thanks for the feedback.
  9. Feb 4, 2012 #8
    Come on guys, he's a high school senior seeing calculus for the first time. Rudin is a bit beyond his current level. Go to a good used book store and find an old edition of a reputable book, pre-calculator era. Thomas, Swokowski, Stein, etc. Get a firm grasp of the fundamentals before you proceed. Take the AP exam and get a good enough grade to place into an honors calc 1 course. Do not skip calc 1 in college. I've seen dozens of smart students struggle with calc 2 in college because high school calculus won't prepare you.

    P.S. I prefer Apostol to Spivak
  10. Feb 4, 2012 #9
    For beginning calculus, I think spivak is even better than rudin. Spivak's book has a lot of instructive and interesting exercises. Rudin's multivariable calculus section is abysmal.. Although, the sections on series and metric space topology stuff in the rudin book is a what makes his book nice
  11. Feb 4, 2012 #10
    Eh...Rudin is horrible for a first introduction to analysis, and is absolutely out of the question as a calculus text. If he's interested in learning real analysis, there are better texts out there.
  12. Feb 4, 2012 #11
    Rudins book is only good for those who already know most of the material. It contains some interesting stuff, but it's not for the beginning student.

    Spivak is an extremely good book.
  13. Feb 4, 2012 #12


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    people who already know the material can appreciate any of them, a person learning it for the first time usually prefers spivak.
  14. Feb 4, 2012 #13


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    Apostol's book is more comprehensive and covers a lot more ground than Spivak's. But I would recommend Spivak for a first look. You'll probably encounter the more advanced Spivak book Calculus on Manifolds at some point, which, despite its small size, covers a remarkable amount of material, and his 3-volume tome on Differential Geometry is indeed a staple of the field.

    But it's still worth-while to take a look at Apostol after Spivak. In a lot of ways, multi-variate calculus isn't that much harder than single variable calculus, and lets you tackle a lot more situations.

    I'd recommend waiting a bit to try to read "baby Rudin". Eventually, you'll want to, and you'll probably keep it around as a reference for the rest of your life, but it's hard going for someone in high-school.

    Spivak's exposition is very clear. Don't skip the appendix, it's one of the best parts of the book.
  15. Feb 5, 2012 #14
    OP: Feb17-10, 03:29 AM

    alan2: Y, 03:15 PM

    Is this a server error or am I the only one noticing that this thread is a bit old?

    Just throwing this in while everyone is commenting: I don't get how Apostol is "dry" o_O Maybe I should try Spivak, I guess, and see how it compares.
  16. Feb 5, 2012 #15


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    You do realize that by the date, the OP is probably a sophomore in college by now?
  17. Feb 5, 2012 #16
    I think Apostol's calculus is just written in a more advanced style. The material is pretty dense and the details are often left to the reader. On the other hand, Spivak includes lengthy expositions which are particularly helpful if you are self studying, and his enthusiasm is on full display making it a joy to read.
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