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Book for a first proof-oriented calculus course

  1. May 14, 2012 #1
    Could anyone give any insight on Tom Apostol's Calculus Vol. 1 and Spivak's Calculus related to a proof-oriented calculus course covering the following topics: Axioms for the real numbers, Riemann integral, limits and continuous functions, derivatives of functions of one variable, fundamental theorem of calculus, Taylor's theorem, and infinite series, power series, and elementary functions? Pros/Cons of both? The course requires Apostol's but I would consider working through Spivak too if his treatment of this topics is better than Apostol's. Any link to a relevant thread is appreciated. Another two questions: Is Apostol's Vol. 2 at the same level of Spivak's Calculus on Manifolds? Is Apostol's coverage of Linear Algebra a sound basis for Lang's Linear Algebra?
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  3. May 15, 2012 #2


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    spivak is more fun than apostol, but apostol may be a tiny bit more scholarly. I.e. I liked spivak as a student, but later I liked apostol. if you are a student, i recommend spivak.
  4. May 15, 2012 #3
    As for the multivariable books, Spivak is much more condensed and is at a higher level than Apostol. Namely, Spivak does vector calculus with differential forms, while Apostol does not.
  5. May 22, 2012 #4
    I liked Apostol volume 1 a lot, and probably more than Spivak. If you have no experience with proofs though, you might like Spivak more. I really didn't like Apostol volume 2 however. He treats too many subjects in too short of a span, and you are probably better off learning linear algebra thoroughly with Lang then with Axler (or some similar progression).

    As for vector calc, Spivak (Calculus on Manifolds) is pretty sophisticated, and you should probably do some more linear algebra (more than whats in apostol, that's for sure) and some real analysis before you tackle it. It is at a much higher level than Apostol Volume 2.

    Summary: Both Apostol and Spivak are great for calculus (as mathwonk said they differ in tone), but in my opinion, Apostol volume 2 is not that great at anything.
  6. May 22, 2012 #5


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    A multivariable calculus book that I like that is at a higher level than your run-off-the-mill calculus books is the one by Williamson, Crowell, and Trotter. I believe it's called Calculus of Vector Functions. Get the 3rd edition [ or older ], and not the 4th [ which I believe is renamed as Multivariable Mathematics ]. There are used ones for literally less than 5 bucks on Amazon.
  7. May 22, 2012 #6
    Spivak > Apostol

    I never liked Apostols book that much
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