250029 VO Analysis 3 (2023S)

The course further develops analysis, in particular the theory of differentiation and integration in several variables. On the one hand, we will emphasize topological aspects of analysis, for which continuity of functions plays a key role. On the other hand, we will discuss geometrical aspects, an in particular identify subsets of R^n that are nice enough to allow for a geometrical study by analytical methods. The study of these so-called submanifolds will form the core of the course. An important topic will be integration over submanifolds, for which one first has to find the right objects to be integrated. This leads to differential forms which will be studied in detail and to a general version of Stokes' theorem, which generalizes the classical theorems of Green, Gauß and Stokes. These are fundamental for several parts of classical physics.

The prerequisites for the course include analysis in one dimension as well as in higher dimensions, in particular the module "Analysis 2" and multidimensional integrals as discussed in the module "Integration and Stochastics".