Which Book to Start With for Mathematical Self-Study: Rudin or Spivak?

[a7d07c8114]

I want to do some self-study both over the summer and during the school year, and I've chosen my books. They are:

Rudin, Principles of Mathematical Analysis
Spivak, Calculus + Calculus on Manifolds

I will likely be taking a yearlong sequence starting next fall in analysis using Rudin, either that or a yearlong sequence in algebra using Artin (doing both simultaneously isn't an option at the math department here, apparently it backfires on everyone who tries it). I'd like to use my self-study to really prepare for analysis and exploring further topics like topology and differential geometry, also starting next fall through the spring. Overpreparation is completely acceptable.

My question: in what order should I read these books to get the most out of both?

My mathematical background, in case it matters:

My calculus background includes an honors sequence in multivariate calculus and ODE's. Unfortunately I can't name a book for you, because everything was taught out of my professor's custom notes, but the approach was very rigorous. The multivariate calculus wasn't on the level of Spivak, but made sure the calculus and the linear algebra was both taught completely to avoid handwaving what would normally be skipped in a regular calculus class. I am also reasonably proficient with mathematical formalism and proof-writing/reading, and did very well in an introduction to mathematical reasoning course in the fall. For a qualitative picture, I'm comfortable with the early chapters of Rudin I have read so far. Though I'm lost in Spivak's unfamiliar notation in the later chapters of Calculus on Manifolds, he seems to introduce everything very clearly (and there seems to be a consensus on this), so I'm not too worried. V.I. Arnold's Ordinary Differential Equations, on the other hand, is more intimidating.

Edit: Also, is anyone familiar with Elias Zakon's Mathematical Analysis? (Available for free for self-study at http://www.trillia.com/) Since I'll probably work with Rudin in the fall, it seems that maybe I'd get more out of seeing two different approaches to analysis. For that matter, is anyone familiar with the other three books on the site as well?

 

[slider142]

If your calculus education was suitably rigorous with proofs and linear algebra, you may find Spivak's "Calculus" to be a good review. Try reading this one first, or alternate with appropriate chapters in Rudin for a wider view of the topic. Rudin will introduce you to some new tools for modern analysis and topology, which should keep you occupied and ground your understanding of the pure basis for calculus.
Spivak's "Calculus on Manifolds" will indeed give very precise definitions leading up to the notation of the last chapter, which provides an extremely elegant and concise introduction to the language of differential geometry. This can be learned independently of the previous two texts as long as you have a good grounding in calculus and linear algebra.

 

[Robert1986]

When I took Analysis, I sort of supplemented the course with Spivak.

 

[a7d07c8114]

Thank you both for your replies. I have decided to pursue Spivak's Calculus on Manifolds first, followed by Calculus with Rudin to supplement if time permits. (I should have fun over summer break, and Calculus on Manifolds looks the most entertaining.)

 

[Mike706]

Calculus on manifolds covers the material of chapters 9 and 10 of Rudin, so if you do plan on doing both anyways you may wish to reconsider the order. Not that I'm implying that you need to read Rudin first, but that seems to be the logical order.