The final syllabus for the …
The final syllabus for the course is the chapters 1-5 and 7-8 in Spivak, and the supplementary notes about de Rham Cohomology by Bjørn Jahren with the following modifications:
Given an oriented manifold with boundary, you need to know the definition of the induced orientation of the boundary. This definition is given in problem 16 of chapter 3 and this definition is included in the syllabus (the definition of induced orientation is repeated in chapter 8 on p. 260).
In chapter 5 I have not lectured proposition 15 and this is not part of the syllabus.
Proposition 14, chapter 7 (the Frobenius integrability theorem) and the following corollary (Corollary 15) is not part of the syllabus.
I have not said anything about volume elements and therefore the stuff about this (page 258-259 in chapter 8) is not part of the syllabus.
In the notes by Jahren I have not explained the ring structure of the de Rham Cohomology for the complex projective spaces (p. 13). So this is not part of the syllabus. Also exercise 1 and 2 in the end of the notes are not part of the syllabus.
The content of Theorem 9, 10, 11, Corollary 15 and Theorem16 and 17 in Spivak chapter 8, are stated and given alternative proofs in the notes by Jahren, and you can read the exposition of this stuff from these notes instead of the one given in Spivak .