Self-Studying: Theoretical Calculus vs Analysis

[bacte2013]

Dear Physics Forum advisers,

I am a sophomore in US with double majors in mathematics and microbiology; my current computational/mathematical biology research got my interested in the mathematics, particularly the Analysis and Algebra, and led me to start with calculus II (computational aspect) and discrete mathematics. My coursework plan is to take multi-variable calculus on summer and introductory analysis & theoretical linear algebra & mathematical statistics on Fall 2015. My calculus II course uses the specialized course packet and I have been using "Calculus with Analytic Geometry" by George Simmons to supplement it. However, I want to learn more about the single-variable calculus and proofs behind it because I am really interested in them and computational aspect does not satisfy me. Since I do not have heavy course load on this semester, I have a lot of time to devote on self-studying the mathematics.

My plan is to either start with "advanced" calculus books like Apostol vol.I, Courant vol.I, Peter Lax, and Spivak OR introductory analysis books like Rudin (PMA), Zorich, Apostol (mathematical analysis), Strichartz, Abott, Ross, and Pugh. I am fairly good with proof methodology which I learned from my current discrete mathematics course and "How to Prove It" by Velleman. Should I jump right into those analysis books or should I start with those advanced calculus books? I already finished with Simmons book and course packet except for the series & sequence chapters.

My future plan is to attend a mathematics graduate program in either applied math (specifically the biological science) or pure math (specifically algebra or analysis).

Thank you very much for your time, and I look forward to your advice and input!
PK

 

[micromass]

There are better books than the ones you listed. Some of my recommendations:

Analysis by its history: https://www.amazon.com/dp/0387945512/?tag=pfamazon01-20
This is a truly unique and excellent book. It approaches analysis from a historic perspective and makes you understand why we do the things we do. It contains some really beautiful things. I still skim through it now and then because it's truly wonderful.

Bloch: https://www.amazon.com/dp/0387721762/?tag=pfamazon01-20
This books proves everything. And I really do mean everything. It starts from natural numbers and goes all the way to rigorous analysis. Not even things like decimal representation is accepted without proof.

That said, you might find your proof abilities are lacking. Just doing Velleman really isn't enough. But if you persist, you'll get the proofs down very quickly!

 

[KCC2]

Jump right into analysis if that's what you're interested in. You don't really need to know advanced calculus topics in order to understand single-variable analysis. The best subject to self-study is the subject you're the most interested in. I don't know too much about different analysis books, but Rudin may be a bit tough. Rudin will be great later when you want to revisit the material.