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Which one of those is the best for a person interested in pure mathematics and who wants to have a deep understanding of calculus? Apostol or Spivak? Could you guys tell me some differences between the approaches of them? What about the exercises? I would like to be challenged, but in a constructive way.

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    They are both masterful texts, but Spivak has by far more interesting and more complete exercises. It is, however, only a single-variable text, whereas Apostol covers multivariable (and linear algebra and some numerical stuff and some probability). There are other good options for serious multivariable + linear algebra after Spivak, however. One is the popular book by Hubbard & Hubbard; another—which for understandable reasons I am more partial to—is my own text on *Multivariable Mathematics*. – 2017-02-16

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I don't have experience with Spivak, but Apostol has provided me with a great supplement to the lecture notes of my teacher. The exercises can however be very challenging.

My personal recommendation would be to start with Stephen Abbott's Understanding Analysis and once you've laid a foundation to move over to Apostol.

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    I think you're thinking of Apostol's analysis (or "advanced calculus") text, rather than his two-volume calculus text? – 2017-02-16
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    Yes, I am. By "deep understanding of calculus?" I'm assuming that the poster already knows basic calculus and wants to move on to understanding the foundations. – 2017-02-16
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    Not how I read it. :) – 2017-02-17
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    While I don't think Abbottt's text is a big jump from Apostol for understanding, the original post would seem to suggest lack of familiarity with calculus since Spivak and Apostol are the "intro" calculus books. – 2017-07-01
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While both books have complete proofs and a good emphasis on theory, Spivak's book is better as an introduction to rigorous math because many of its problems are more difficult and theoretically oriented than Apostol's. (I assume you mean his Calculus). Spivak's book also has a solution manual, which is very useful when you're studying on your own.

On the other hand, Apostol actually covers more material, even just within Volume 1. Volume 2 of Apostol is actually one of the best introductions to multivariable calculus. Apostol's book also has a greater variety of exercises involving applications of calculus to physics.

If you have time, learning from both books is a good idea. Otherwise if your main focus is pure math, then I would recommend Spivak.

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    I see... I'm really charmed by Apostol's approach of introducing integrals before everthing else, though. Do you think that Spivak's exercises quality overcomes even that? Aren't Apostol's exercises challenging as well? To be fair, I'm not fond of the idea that Apostol's book has a certain practical approach (for example, applications of calculus to physics, as you mentioned), which makes me face a huge dilemma. – 2017-02-17
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    Personally, I prefer Apostol's writing to Spivak's. Apostol's exercises are also often challenging and interesting, but less so than Spivak's. The main difference is that Apostol has fewer exercises of purely theoretical interest. For someone going into math, Spivak's book is probably better. The exception would be if a reader found the exercises to be too difficult, in which case he should increase his ability first by learning more math, which could mean calculus or other things. Having exercises on physical applications is a plus, because it helps you build intuition that's also useful... – 2017-02-17
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    for math. While Apostol's approach of doing integration first is interesting and different, I don't think it makes a significant difference one way or the other to its overall suitability as a textbook. Lastly, Apostol's "practical approach" at no point means he ignores the mathematical theory. It's just that the level of focus on theory in the exercises doesn't match Spivak. Apostol is in fact much better on theory than the vast majority of calculus textbooks. If I can put it this way, I'd say Spivak's goal is 80% to teach you how to think about math, and 20% to teach you calculus. With... – 2017-02-18
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    Apostol, the balance is 50-50. And in Stewart, it's 0-100. But the lack of theory in Stewart means that within that 100, you learn a lot less than in Apostol's 50. – 2017-02-18