F6420 Differential and integral calculus on differential manifolds and its applications in physics

Faculty of Science
Spring 2005
Extent and Intensity
2/2/0. 4 credit(s). Type of Completion: z (credit).
Teacher(s)
Mgr. Pavla Musilová, Ph.D. (lecturer)
prof. RNDr. Jana Musilová, CSc. (lecturer)
Guaranteed by
prof. RNDr. Michal Lenc, Ph.D.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: Mgr. Pavla Musilová, Ph.D.
Timetable
Thu 8:00–9:50 F23-106, Thu 10:00–11:50 F23-106
Prerequisites
F3063 Integration of forms
Differential and integral calculus of functions of multiple variables (Riemann integral), fundamentals of tensor algebra, integral of differential forms on euclidean spaces.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
Předmět pokročilého kursu matematické analýzy pro fyziky, vhodný pro zájemce o problematiku matematické fyziky. Zabývá se především zobecněním pojmů diferenciálního a integrálního počtu na euklidovských prostorech na obecnější podkladové struktury -- diferencovatelné variety. Spolu s korektním výkladem matematických pojmů je důraz kladen na jejich aplikace v matematické fyzice.
Syllabus
  • 1. Fundaments of topology, topological manifolds, homeomorphisms. 2. Atlases, differential manifolds, diffeomorphisms. 3. Tensor algebra. 4. Tensors on manifolds, tensor bundles. 5. Induced diffeomorphisms on tensor bundles, Lie derivative. 6. Linear connection. 7. Physical applications-basis manifolds of GTR. 8. Integrals of differential forms on diferential manifolds, decomposition of unity, Stokes theorem. 9. Classical integral theorems, physical applications.
Literature
  • KRUPKA, Demeter. Úvod do analýzy na varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1986, 96 s. info
  • NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info
  • SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
Assessment methods (in Czech)
Výuka: přednáška a cvičení. Zápočet: písemná kontrola.
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught once in two years.
General note: S.
The course is also listed under the following terms Spring 2011 - only for the accreditation, Spring 2000, Spring 2001, Spring 2003, Spring 2007, Spring 2009, Spring 2011, spring 2012 - acreditation, Spring 2013, Spring 2015, Spring 2017, Spring 2019, Spring 2021, Spring 2024.
  • Enrolment Statistics (Spring 2005, recent)
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