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...