Minimal prerequisite to understand De Rham Cohomology To understand the concept of De Rham Cohomology formally, what is the minimum background required?  
That is, what are the critical concepts? For example, a list might be something like: tangent bundle, cotangent bundle, differential forms, integration on manifolds
I have a somewhat short amount of time to write an expository paper on De Rham Cohomology for a seminar course, and I want to get a feel for the necessary prerequisite components.
 A: You've made a pretty good start with your list.
For the definition of deRham cohomology, you need the cochain complex of differential forms
$$
\cdots\rightarrow\Omega^n\xrightarrow{d}\Omega^{n+1}\xrightarrow{d}\cdots
$$
However, the definition of the deRham groups won't make a heck of a lot of sense unless your audience already possesses some familiarity with homology theory, say the singular homology and cohomology groups.
For the statement of deRham's theorem, and why it "makes sense", you need singular cohomology, the pairing $(\omega,c)\mapsto \int_c\omega$ (i.e., integration over chains), and Stokes' theorem.
There are two well-known proofs of deRham's theorem. deRham's original proof is presented in the book by Singer and Thorpe Lecture Notes on Elementary Topology and Geometry. Later on Weil gave another proof; Weil's proof fits in better with modern concepts (e.g., homological algebra, sheaves). Sternberg had a very nice set of slides on this, but they don't seem to be available on the web any more.
