1. May 25, 2012

caltdel

Hey all,

This is my first post on here, but I'm a long-time lurker :D

I'm looking to start learning Calculus(and Mathematics, in general) and was wondering if Apostol Volume ! is good for a beginner. I've been teaching myself Mathematics for about half a year now and I've mastered the Pre-Calculus(Robert Blitzer) and I'm currently studying Courant's What is Mathematics? to get a larger Idea of what exactly Math is before I dive into Calculus. I chose Apostol because it seems to give a more pure/proof-based approach to Calculus. It's my assumption that it's far easier to learn the applications after I learn the theory than vice versa. However, I've seen on different threads that Apostol may not be for the faint of heart and was wondering exactly how hard you think it would be for one with no knowledge of Calculus, but strong grounding in Pre-Calculus. Some suggest it would be best to study regular single variable Calculus (akin to that found on MIT OCW) before diving into Apostol, but is it entirely necessary or would it just make the process easier?

NB: Pre-Calculus brushed over proving statements via mathematical induction and spent the last chapter introduces Calculus(Finding Limits through tables and graphs, and by using the properties of limits, Continuity, and introduction to derivatives)

ctrl+alt+del

2. May 25, 2012

micromass

Staff Emeritus
Apostol is of course a great book. But reading Apostol without any knowledge of calculus is hard. Basically, you have to learn two things: formal mathematics and calculus. Neither is very easy.

This is why it is actually recommended to first read an easier book on calculus, and then reading the hard book to understand the formal, proof-based mathematics as well.

A very nice, easier, book is "A first course in calculus" by Serge Lang. Perhaps this would suit you better.

Anyway, I haven't said that it's impossible to read Apostol at this stage. All I'm saying is that I would have found it quite difficult at your stage of education.

3. May 25, 2012

jbunniii

If you want to learn rigorous calculus then it would be a rather long detour to work through a non-rigorous book first. I don't recommend doing that.

So it comes down to two questions: (1) are you sufficiently prepared for a rigorous book and (2) which rigorous book?

Did your pre-calculus studies involve any proofs, especially epsilon-delta proofs? If so, then the answer to (1) is yes, and you should be in good shape to tackle any of the standard rigorous texts: Apostol, Spivak, or Courant to name three. Which of these you choose is largely a matter of taste and there are many threads in this forum discussing their merits. If at all possible, try to borrow several of these and decide for yourself.

If you have not done much in the way of epsilon delta proofs, then you have a big learning curve to get started. I personally think Spivak would be the superior choice in that case as he is very careful to motivate and deconstruct what is going on. But be warned that his exercises are the toughest out of the three books I listed above.

Apostol covers more ground than Spivak, but the exposition isn't nearly as friendly. I would recommend going with Apostol only if you're already very comfortable with proofs.

4. Jun 3, 2012

caltdel

Hey jbunniii,

I'm familiar with proofs, though I'm not sure if it's to a sufficient level. I'm familiar with mathematical induction, deduction, and reductio ad absurdum. I just purchased Apostol I and II and I'm looking through the pages. This is certainly going to be hard but I'll just give it the full beans and see where it goes. I read somewhere that it doesn't give enough in the way of geometrical intuition. Being the newb that I am, I have no idea what that is. Could you please let me know if it's important and if it's true?
Thanks again

5. Jun 3, 2012

jbunniii

I don't think it's true at all. I'm looking at the first few pages of Volume I right now, and they're full of geometric (and historical) motivation for both differentiation and integration. And flipping through the book, I see many figures that give geometric insights into what is going on in the equations.

If you want to see a calculus/analysis book that is devoid of geometric insight and figures, see Rudin's "Principles of Mathematical Analysis."

Good luck with your reading! Try to do as many exercises as you can, as this is how you really learn the material. And if you get stuck, or just want to check your answers, this forum is a great place to do that.

6. Jun 3, 2012

7. Jun 7, 2012

caltdel

Thanks! I'm enjoying it already. Maybe when I finish the Single Variable volume I can start with the Feynman Lectures :D