The goal of this course is to introduce the student to the basics of smooth manifold theory. The course will start with a brief outline of the prerequisites from topology and multi-variable calculus.
After the introducion of differentiable manifolds, a large class of examples, including Lie groups, will be presented. The course will culminate with a proof of Stokes' theorem on manifolds.
INTENDED AUDIENCE : Masters and PhD students in mathematics, physics, robotics and control theory, information theory and climate sciences.
PREREQUISITES : Real analysis, linear algebra and multi-variable calculus, topology.
INDUSTRY SUPPORT : Nil
Week 1 : Review of topology and multi-variable calculus
Week 2 : Definition and examples of smooth manifolds
Week 3 : Smooth maps between manifolds, submanifolds
Week 4 : Tangent spaces and vector fields
Week 5 : Lie brackets and Frobenius theorem
Week 6 : Lie groups and Lie algebras
Week 7 : Tensors and differential forms
Week 8 : Exterior derivative
Week 9 : Orientation
Week 10 : Manifolds with boundary
Week 11 : Integration on manifolds
Week 12 : Stokes Theorem