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Reading course on differential topology:
This page contains notes I wrote for a short reading course on de Rham cohomology and assorted topics in differential topology. I do not offer any guarantee that this will be useful, readable, or accessible to anyone; read at your own risk. Be warned that I intentionally avoid using standard terminology. I have not intentionally included any mistakes, but they may be present nonetheless.
Part 1: Introduction to de Rham cohomology (covector fields and d)
Part 2: Introduction to de Rham cohomology (2-covector fields)
Part 3: Introduction to de Rham cohomology (wedge of differential forms and Leibniz rule)
Part 4: Vector fields and derivations
Part 5: Vector fields, integral curves, and integral surfaces
Parts 1–3 are essentially a guided and expanded version of pages 1–4 of Bott–Tu, covering the basics of differential forms and de Rham cohomology (although the term "cohomology" does not appear in these notes!). Parts 4 and 5 ask the student to prove the fundamental theorems underlying vector fields as derivations, the existence and uniqueness of integral curves, the bracket of vector fields, and Frobenius' theorem on the integrability of distributions.
These notes will only be useful if you work out all the exercises as you go, setting the notes aside for as long as it takes you to solve them. Skimming through and promising yourself that you could do the exercises would be a complete waste of your time.
Be aware: These notes were written for a student about to start graduate math courses. Therefore although I try to maintain a down-to-earth perspective, these notes expect a high level of mathematical maturity. If you get lost, don't be discouraged! Come back in the summer before grad school, and you'll be surprised how much easier you find it.
Minimum background: Stanford students who have taken 62CM (previously called 52H) should be well-positioned to read these notes, possibly supplemented by my notes on wedge products (Sections 1 and 2). For other students, Parts 1–3 can in theory be done with a very solid computational understanding of multivariable calculus (say with A/A+ grades in 51, 52, and 113), though they may appear unmotivated without more context. Parts 4 and 5 would require experience with proofs at least at the level of 120, 171, or the 60XM series. You do not need to know what a manifold is.
For students elsewhere, to benefit from these notes you'd likely be at least a junior or senior honors math major. In any case, there is no reason to read this unless you have had a theoretical/honors multivariable calculus course, as well as a theoretical linear algebra course (including the wedge product of vector spaces, which is rarely covered in a first or second linear algebra course). I regret that I cannot respond to questions about these notes from anyone except current Stanford students.