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Course, academic year 2023/2024
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Introduction to Analysis on Manifolds - NGEM002
Title: Úvod do analýzy na varietách Mathematical Institute of Charles University (32-MUUK) Faculty of Mathematics and Physics from 2018 winter 6 winter s.:2/2, C+Ex [HT] unlimited unlimited no no cancelled Czech full-time full-time
Guarantor: Mgr. Lukáš Krump, Ph.D. Mathematics > Geometry NMAG335 NMAG335 NALG018 NMAG335, NMAA038
 Annotation - ---CzechEnglish
Last update: T_MUUK (23.05.2003)
One of the basic courses in the area of general differential geometry. Notions from algebra and real analysis are amalgamated and developed in a new, geometrical direction. We define the notions of tensor and exterior algebras, differential forms in R^n and their integrals over k-dimensional surfaces in R^n. We define also smooth manifolds with border, tangent vectors, vector and tensor fields, integral of a differential form on a manifold and the highlight is the proof of the general Stokes theorem. Last but not least is the notion of integral of a function on a Riemannian manifold.
 Syllabus - ---CzechEnglish
Last update: T_MUUK (23.05.2003)

Minimal syllabus:

1. Topological manifold (charts, transition functions, atlas), smooth manifolds (differential structure), basic examples of manifolds. 2. Smooth maps between manifolds, smooth functions, diffeomorphisms; tangent vectors in a point, tangent space in a point, coordinates on tangent space, geometrical interpretation of vectors; tangent map to a smooth map, coordinate description, Jacobians. 3. A summary of properties of tensor algebra of a vector space; outer algebra of a vector space, basic properties of outer multiplication; symmetric algerba of a vector space, orientation of a vector space, volume of a paralleliped using outer product and the Gramm matrix. 4. Tensor fields on a manifold, Riemann (pseudo)-metric on a manifold, Minkowski spacetime, algebra of differnetial forms as a modul over the ring of functions, orientation of a manifold; de Rham differential in coordinates and without coordinates, exact and closed forms, de Rham complex, de Rham cohomology, Poincare lemma; inverse image of tensor fields and forms by a smooth map, coordinate description, basic properties. 5. Manifolds with a boundary, its tangent space, differential forms on manifolds with boundary, orientation. 6. Integration of forms on a manifold with boundary, Stokes theorem. 7. Volume form on a (pseudo)-Riemannian manifold, integration of functions on such manifolds, local computations. If possible: 8. Lie derivative of vector fields, contraction of forms by vector fileds, connection with the de Rham differential.

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