Knowledge of differentiable manifolds has become very important in a large
number of areas of mathematics and of its applications. In fact, much of
advanced calculus and analysis is based on the study of differentiable
manifolds. For example, topics such as line and surface integrals,
divergence and curl of vector fields and Stoke's and Green's theorems are
most naturally described using manifold theory.
We will give a careful introduction to differentiable manifolds,
illustrating each new definition and theorem with the study of spheres,
tori, real and complex projective spaces and matrix groups. We will talk
about tangent spaces, vector fields, differential forms and integral curves. We will
conclude with Stokes' theorem on manifolds.
A course guide can be found here.