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Cursus: WISB342
WISB342
Differentieerbare variëteiten
Cursus informatie
CursuscodeWISB342
Studiepunten (EC)7,5
Inhoud
A manifold is an abstraction which generalizes the concept of embedded surface in R^3 and is the basic object studied in differential geometry. The underlying idea is similar to how cartographers describe the earth: there is a map, i.e., a plane representation, for every part of Earth and if two maps represent the same location or have an overlap, there is a unique (smooth) way to identify the overlapping points on both maps. Similarly, a manifold should look locally like R^n, i.e. there are maps which identify parts of the manifold with the flat space R^n and if two maps describe overlapping regions, there is a unique smooth way to identify the overlapping points. Most of the notions from calculus on R^n are local in nature and hence can be transported to manifolds. Further, some nonlocal constructions, such as integration, can be performed on manifolds using patching arguments.
 This course will cover the following concepts:
  • definition and examples of manifolds
  • quotients and Lie groups,
  • tangent and cotangent spaces as well as vector bundles,
  • vector fields and forms, as well as sections of vector bundles,
  • submanifolds,
  • diffeomorphisms,
  • distributions,
  • tensor and exterior algebras,
  • exterior derivative and de Rham cohomology,
  • integration and Stoke’s theorem.
 The course will also cover the following important results relating the concepts above:
  • implicit and inverse function theorems,
  • Cartan identities and Cartan calculus,
  • Frobenius theorem,
  • Stoke’s theorem
     
Kennis en inzicht :

The students should learn the contents of the course, namely
  • the definition of a manifold as well as ways to obtain several examples e.g.,
  • by finding parametrizations,
  • as regular level sets of functions,
  • as quotients of other manifolds by group actions.
  • tangent and cotangent spaces as well as vector bundles,
  • vector fields and forms, as well as sections of vector bundles,
  • submanifolds,
  • diffeomorphisms,
  • implicit and inverse function theorems,
  • distributions and Frobenius theorem,
  • tensor and exterior algebras, exterior derivative and de Rham cohomology,
  • Cartan calculus,
  • integration and Stoke’s theorem
Vaardigheden:

At the end of the course, the successful student will have demonstrated their abilities to:
  • use the inverse and implicit function theorems on manifolds,
  • perform computations with vector fields and forms using Cartan calculus,
  • use Stoke’s theorem,
  • compute the de Rham cohomology of simple spaces.
 
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