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?