Theorems:

Theorems which the student is expected to state and apply: the inverse function theorem, the existence of partitions of unity, the existence and uniqueness of flows of vector fields and their properties, the general theorem of Stokes. Theorems which the student is expected to be able to prove: the theorem on rank, the existence of a Riemann metric on a manifold, the theorem on embedding of a closed manifold into $\mathbb{R}^n$.

The material on differential topology is generally covered in Math 124, which assumes as undergraduate preparation a course on analysis on manifolds at the level of Spivak's ``Calculus on manifolds''. Students who do not have this background should normally enroll during the first year in Math 73, which furnishes the necessary prerequisites for Math 124.