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Last update: G_M (15.05.2012)
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Last update: doc. RNDr. Roman Lávička, Ph.D. (25.09.2020)
The exam will be written. The student will receive credit for homework. |
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Last update: prof. RNDr. Vladimír Souček, DrSc. (10.10.2012)
L. Krump, V. Souček, J. A. Těšínský: Matematická analýza na varietách, skriptum, Karolinum, 2008 (2. vydání). O. Kowalski: Úvod do Riemannovy geometrie, skriptum, Karolinum, 1975 (2. vydání). M. Fecko: Diferenciálna geometria Lieovy grupy pre fyzikov, IRIS, Bratislava, 2008. |
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Last update: doc. RNDr. Roman Lávička, Ph.D. (25.09.2019)
Requirements to the exam correspond to the syllabus to the extent to which topics were covered during lectures and tutorials. |
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Last update: G_M (15.05.2012)
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. |