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The course descriptions is : A theoretical course in calculus; emphasizing proofs and techniques. Trigonometric identities. Limits and continuity; least upper bounds, intermediate and extreme value theorems. Derivatives, mean value and inverse function theorems. Integrals; fundamental theorem; elementary transcendental functions. Techniques of integration. Taylor's theorem; sequences and series; uniform convergence and power series.

This course is using Spivak's calculus, but I'm also looking for another proof based book accompany with Spivak, thx

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  • $\begingroup$ I have used four different books for analysis including Spivak's Calculus on Manifolds. My recommendation would be to use Kosmala's A Friendly Introduction to Analysis. It is almost purely epsilon-delta based, whereas Spivak will be using open balls. $\endgroup$
    – drawnonward
    Commented May 5, 2014 at 4:55
  • $\begingroup$ Does it have plenty of exercise? How is its difficulty comparing to spivak? $\endgroup$
    – ElleryL
    Commented May 5, 2014 at 5:00
  • $\begingroup$ Yes, there are many exercises and examples. I have never read Spivak's Calculus, only Calculus on Manifolds, so I can't compare. However, I do believe Spivak is an authority when it comes to calculus. I know manifolds wasn't an easy read. $\endgroup$
    – drawnonward
    Commented May 5, 2014 at 5:04
  • $\begingroup$ One or more of the books listed here might be useful. $\endgroup$
    – Dave L. Renfro
    Commented May 5, 2014 at 21:04

2 Answers 2

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The Theory of The Calculus is called Mathematical Analysis ...Elementary Analysis for the R to R functions....so any of the very many books on Analysis should serve your purpose.

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  • $\begingroup$ More advanced Analysis is usually approached "Topologically" Spivak's Calculus on Manifolds tries to do that...shame he does not know where to put apostrophies! $\endgroup$
    – kozenko
    Commented May 5, 2014 at 5:02
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You might try the standards:

  1. Apostol - Calculus Vol I
  2. Hardy - A Course of Pure Mathematics
  3. Piskunov - Differential and Integral Calculus

You might also try Rosenlicht's "Introduction to Analysis". It's a cheap, awesome Dover book that presents basic analysis at a good level. Not exactly parallel to Spivak, but you might find the presentation enlightening paired with Spivak.

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