Books

We aren't going to assume much about set theory in this course, but, in case you are curious, here are a few references.

Introduction to Smooth Manifolds, by John M. Lee.
This is our text, and it will be a good source to read about things for the details of proofs and for the appendices about basic background
Introduction to Differentiable Manifolds and Riemannian Geometry, by William Boothby
A more concise introduction that also includes some basic material on Riemannian geometry.
Calculus on Manifolds, by Michael D. Spivak.
A short (less than 150 pages) paperback that is a classic. It covers a lot of introductory material about vector spaces, norms, and so on, but his treatment of differential forms and Stokes' Theorem is more complicated than it needs to be.
A Comprehensive Introduction to Differential Geometry, by Michael D. Spivak.
A long (5 volumes!) treatise that covers everything we will do in our class and much more. The Volume 1, which can be read alone, covers a lot of what we will do in a very engaging style and in more detail than Calculus on Manifolds.
Topology from the Differentiable Viewpoint, by John W. Milnor.
An even shorter (less than 65 pages) paperback that introduces the all basic ideas of smooth manifolds and smooth maps in a very clear and elegant way. The author doesn't treat a lot of the other topics that we need to treat, but for an introduction to thinking about smooth manifolds, it can't be beat.

Miscellaneous Notes

From time to time, I'll put some extra reading here and alert you to its existence.

A proof of Sard's Theorem
From time to time, it will be useful to know that the set of critical values of a smooth map is 'very small'. To be precise, it has measure zero in a precise sense that we will discuss in class. (There's also a proof in Milnor's Topology from the Differentiable Viewpoint, in case you want to look there.)