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

Faculty of Science
Spring 2009
Extent and Intensity
2/2/0. 4 credit(s). Type of Completion: z (credit).
Teacher(s)
prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Michael Krbek, Ph.D. (seminar tutor)
Guaranteed by
prof. RNDr. Michal Lenc, Ph.D.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: Mgr. Michael Krbek, Ph.D.
Timetable
Mon 10:00–11:50 F3,03015, Tue 7:00–8:50 F3,03015
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
One of disciplines of an advanced course of mathematical analysis for students of physics, interested in mathematical physics. The attention is devoted to generalization of concepts of differential and integral calculus on euclidean spaces to more general underlying structures -- differential manifolds. Together with the correct presentation of mathematical concepts the applications in mathematical physics are emphasized.

The mail goal of the discipline is to give students a review and understanding of such fundamental concepts as differentiaable manifold, vector and tensor fields on differential manifolds, differential forms and basic operatons with them, as well as the possibilities of application of these concepts in theoretical physics.

Absolving the discipline student obtains following knowledge and skills:

* Understending a concept of differentiable manifold as a more general underlying space for physical theories in comparison with "standardly used" euclidean spaces.
* Practical skills with the use of charts and atlases on differentiable manifolds, transformation of coordinates.
* Understanding of fundamental geometrical objects on differentiable manifolds as vector fields and differential forms.
* Practical skills in calculus of vector fields and differential forms in differentiable manifolds.
* Undestanding a general concept of integral of a differential form on a manifold and its practical use.
* A review of possibilities of the use of given geometrical concepts in physical theories.
Syllabus
  • 1. Fundaments of topology, topological manifolds, homeomorphisms.
  • 2. Atlases, differential manifolds, diffeomorphisms.
  • 3. Atlases: practical calculations and examples.
  • 4. Tensor algebra.
  • 5. Tensor fields on manifolds, tensor bundles.
  • 6. Tensor bundles - practical calculations.
  • 7. Induced diffeomorphisms on tensor bundles.
  • 8. Lie derivative.
  • 9. Linear connection.
  • 10. Physical applications-basis manifolds of GTR.
  • 11. Decomposition of unity
  • 12. Integrals of differential forms on diferential manifolds, Stokes theorem.
  • 13. 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
Teaching: lectures and exercises
Exam: credit/no-credit, written test
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 2005, Spring 2007, Spring 2011, spring 2012 - acreditation, Spring 2013, Spring 2015, Spring 2017, Spring 2019, Spring 2021, Spring 2024.
  • Enrolment Statistics (Spring 2009, recent)
  • Permalink: https://is.muni.cz/course/sci/spring2009/F6420