Spivak or Apostol?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

Both books give you an introduction to calculus, being: set and number theory, some induction, series and notation. Big difference can be the order in which topics are introduced, for Apostol integration comes first due to historical reasons, while for Spivak derivatives comes first.

For me, Apostol provides an interesting framework to learn about how calculus' ideas evolved historically, as well volume 1 introduces lineal algebra, providing a more gentle progressions towards multivariable calculus and differential equations in volume 2. On the hand, Spivak goes deeper into proofs with rigour, which is essential for every mathematician or even physicist. Therefore, I decided to use Apostol as my "guide" book and Spivak to delve deeper into concepts in which I'm interested.

Just look at the index of both books: Spivak and Apostol.