Analysis Analysis after Spivak

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1. Apr 14, 2015

Totally

I'm using gap year to prepare for B.S. in electrical engineering. Currently I'm solving through Spivak's "Calculus", Lang's "Introduction to Linear Algebra" and Velleman's "How To Prove It." I have three books on analysis, Rudin's "Principles of Mathematical Analysis", Abbott's "Understanding Analysis" and Needham's "Visual Complex Analysis."

Now, as I understand real analysis should come before complex. So the question is between Abbott and Rudin. Namely, is analysis background from Spivak good enough to jump into Rudin and skip Abbott? I know Spivak isn't a real analysis book, but more like a bridge between analysis and calculus. I've also heard that Rudin can be a headache to read so I'm wondering if I should tackle it or go through Abbott first, which seems more "user friendly."

Although Spivak is fixing the situation, my calculus isn't very consistent because until now I've been cherry picking topics that seemed more interesting and pouring time in learning those instead of climbing consistently. For example, I have no problem with things like Gaussian quadrature or Lagrange multipliers but I'm by no means solid on series.

2. Apr 14, 2015

axmls

Rudin should be possible after Spivak. I did Spivak the summer after I took calculus II, and after Spivak, I started Rudin (but other coursework got in the way of me getting appreciably far). I'd say if you have a good grasp on Spivak, you should be able to do Rudin.

The only thing I'd warn you about is that Spivak's exposition is miles ahead of Rudin's. Rudin is straight to the point, and doesn't often motivate what he's doing, whereas with Spivak, you always know the reason he's proving something (and his writing style is quite entertaining). Rudin has a reputation of being a bit "dry."

Furthermore, Spivak doesn't discuss any topology, which Rudin does, so you'll basically have to start that from scratch (but it's similar to things you already know).

It's important not to cherry pick concepts from math. Especially in a textbook--usually everything that is written is written for a reason: it will be used later on. As an electrical engineering major, you may need to know your Taylor series for your physics classes.

3. Apr 14, 2015

Totally

Thats what happens when your first exposition to calculus doesn't come from a mathematician The consistency is lax at best, although I still enjoyed that teaching style.
Anyways, in that case, I'll jump into Rudin after I'm done and use Abbott as a supplementary if I get completely lost. I guess Rudin must be really impressive if it gets so much praise despite being, as you say, dry - and that is by far the softest expression about his writing style that I've heard so far.

4. Apr 14, 2015

axmls

Well, Rudin is just very concise. He'll leave some nontrivial holes in some of the steps that the reader is expected to fill in. Whereas Spivak can spend a page or two talking about the importance of a theorem, Rudin's book has about 3 theorems per page in some sections. Theorem, proof, theorem, proof, theorem, proof, corollary, theorem...

My advice is to take it slow and don't move on until you're sure you understand each section.

5. Apr 15, 2015

Totally

Sounds somewhat similar to Apostol, I don't think I dislike that. Thank you for your help!

6. Apr 15, 2015

jasonRF

As an EE, I would say that complex analysis is much more useful than real analysis for most (not all) electrical engineers. If you want to lean real analysis then go for it. But with Spivak under your belt you probably know about uniform convergence and such, so your should be prepared for the traditional approach to complex analysis. I have never read Needham's book, although flipping through it in the library at work it looks like a beautiful, non-traditional geometric approach to the subject.

If your goal is to prepare for EE, I hope you are also doing things like writing software in some language or another, or playing with microcontrollers, or building transistor amplifiers, etc. EE is much more than math, although there are disciplines in EE (controls, communications and information theory, signal processing) that can get very mathematical at the graduate level.

jason

7. Apr 21, 2015

mathwonk

real analysis is not a prerequisite for complex. real analysis deals with really bad functions and complex deals with really nice ones. you should know your path integrals for complex, unless the complex book you choose book treats them thoroughly.

i thoroughly dislike rudin's book and do not think it is useful for most people to learn from. so if you start it and get bogged down, it is not your fault and there is no reason at all to press on, it is just not a good source in my opinion.