Concerning the "reminder" on differential forms, which we used in Lecture 5: this is stuff you should probably already know. The book of Range gives a short account. I have added some reference material with more detail (and proofs), which I think are quite good. They come from "Geometry of differential forms" by Morita and "Calculus on Manifolds" by Spivak.
Note that the notation used by Spivak, and myself on Lecture 5, namely Lamda(V) to denote the alternating forms on V, is not really standard: it conflicts with the algebraic meaning of the exterior algebra, which is a quotient of the tensor algebra. What is really meant is Lamda(V*), sometimes written Omega(V*), and generalizes to the somewhat standard Omega(T*M): the space of forms, i.e. alternating tensors on the tangent space at each point. In fancy words, a section of the exterior product of the cotangent bundle.