# Calculus Brief calculus book to read before studying the analysis

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1. Jul 21, 2015

### bacte2013

Dear Physics Forum personnel,

I am going to start studying the analysis texts (Rudin-PMA, Apostol-MA, Pugh-RMA) on the first week of August. I have a good proof skills through working on Artin's Algebra and Hoffman/Kunze's Linear Algebra, but I unfortunately only took computational 1-variable calculus (Lang's A First Course in Calculus), and I did not took multivariable calculus, which I might take concurrently with Analysis I and Theoretical Linear Algebra on the upcoming Fall. I am looking for a brief text which explains the key ideas from both 1-variable and multivariable calculus, one I can read and jump directly into the analysis texts. Could you recommend one? Also will my lack of multivariable calculus be a problem when I tackle those analysis texts? I seem to understand at least the beginning chapters.

2. Jul 21, 2015

### cpsinkule

I'm not sure about Apostol or Pugh, but Rudin is one of the most advanced Analysis texts on the market. It's definitely not suitable for a first time Analysis course. Analysis is basically the theory of Calculus, so technically one could take analysis without knowing anything about calculus. You start with properties of the fields and then metric spaces and work your way to sequences, limits, series, derivatives, and finally integration working through all the theorems and lemma\corollaries which establish the tools of calculus rigorously. So, to answer your question, calculus is not necessary to study analysis (although it never hurts to have a little knowledge of it a-priori)

3. Jul 21, 2015

### micromass

Staff Emeritus
Uuh, yes it is. I don't know how you could possibly start an analysis book without knowing calculus.

4. Jul 21, 2015

### bacte2013

Thank your for the advice. Rudin-PMA, although not readable as Apostol and Pugh, is still quite interesting to me. I am quite familiar with the basic idea of 1-variable calculus but I am curious if my lack of knowledge in the multivariable calculus will hurt me. Also I am looking for a brief book that covers the key ideas and computation techniques of both 1-variable and multivariable calculus, which I would like to read before jumping into them. Is it also possible to learn the multivariable calculus from the analysis books like Hubbard/Hubbard, Lang, Rudin, Pugh, etc.?

5. Jul 21, 2015

### cpsinkule

Can you elaborate? Analysis, in it's very nature, does not assume calculus...that's the whole point of analysis, to make rigorous and prove the tools of calculus.

6. Jul 21, 2015

### cpsinkule

https://www.amazon.com/Calculus-Int...pebp=1437516938780&perid=05BV88V3VTD5VMW01752 is a great book. it covers single variable and multivariable, but it is very long, over 900 pages.

Last edited by a moderator: May 7, 2017
7. Jul 21, 2015

### micromass

Staff Emeritus
Sure. But by your logic you might as well start in first grade by giving them the ZFC axioms. The whole point of the ZFC axioms is that it doesn't assume math, and that it is used to show that math works. So the first-graders see the ZFC axioms, and then construct addition and multiplication on $\mathbb{N}$. Technically it works, right? Do you see why nobody does it that way?

8. Jul 21, 2015

### cpsinkule

I understand what you're saying. To be fair, a proof and logic based course would be far more important than a calculus course as a pre-requisite. Mathematical maturity is more important than knowing how to integrate when you take analysis. That's just my opinion, though.

9. Jul 21, 2015

### micromass

Staff Emeritus
Just a random question, are you from France?

10. Jul 21, 2015

### bacte2013

Thank you all for the advice. I do have a good proof skill. I am really afraid about my unconnected knowledge in the single-variable and lack of knowledge in the multi-variable.

11. Jul 21, 2015

### cpsinkule

No, why?

12. Jul 21, 2015

### micromass

Staff Emeritus
Because French education has the attitude you have, and that has not benefited the french at all.

13. Jul 21, 2015

### cpsinkule

Most introductory analysis courses don't even consider multi-variable analysis, so I wouldn't worry about not being familiar with that.
I'm just giving my personal perspective. My first analysis course used Rudin. My university's proof\logic course came in handy 10 fold over anything learned in a Calculus course. Like I said, it's just my opinion and clearly my experience doesn't equate to everyone else's.

14. Jul 23, 2015

### Rescy

Try Spivak Calculus, then delve into Rudin PMA

15. Jul 23, 2015

### bacte2013

Thank you very much for all advice! I actually decided to study Apsotol's Mathematical Analysis and Pugh's Real Mathematical Analysis. Those books are very readable to me, and I believe they cover everything from 1-variable calculus of Spivak and Apostol. Also I will be taking the Analysis I course on this Fall, which uses Rudin-PMA.

16. Jul 24, 2015

### MidgetDwarf

Hahahaha. Funniest thing I read in a while. Careful he may Napoleon you!