What are the prerequisites for Michael Spivak's monumental A Comprehensive Introduction to Differential Geometry? In particular for volume 1? Are these 5 volumes self-consistent in the sense that a knowledge of the prerequisites of Vol.1 is sufficient to tackle all the volumes?
[Math] What are the prerequisites for Michael Spivak’s monumental A Comprehensive Introduction to Differential Geometry
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Best Answer
Roughly:
Your calculus background should certainly involve real proofs of things like the intermediate value theorem, and the extreme value theorem. Your multivariable course should have proven the implicit and inverse function theorems. And if you'd heard of Sard's theorem (Milnor's Topology from the Differentiable Viewpoint might be a good reference), that'd do no harm either.
To be honest: I'd recommend reading (and doing most of the exercises) in Barrett O'Neil's book "Elementary Differential Geometry" as a first step. It's all for surfaces in 3-space, but it'll ground you in the main ideas so that much of Spivak will just seem like reasonably natural generalizations of what you've already learned.
Oh...and all this is for Volume 1. Later volumes certainly rely on a bit more abstract algebra.