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 calculusWeek 2 : Definition and examples of smooth manifoldsWeek 3 : Smooth maps between manifolds, submanifoldsWeek 4 : Tangent spaces and vector fieldsWeek 5 : Lie brackets and Frobenius theoremWeek 6 : Lie groups and Lie algebrasWeek 7 : Tensors and differential formsWeek 8 : Exterior derivativeWeek 9 : OrientationWeek 10 : Manifolds with boundaryWeek 11 : Integration on manifoldsWeek 12 : Stokes Theorem