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

Faculty of Science
Spring 2021
Extent and Intensity
2/2/0. 4 credit(s). Type of Completion: z (credit).
Teacher(s)
Mgr. Michael Krbek, Ph.D. (lecturer)
Mgr. Michael Krbek, Ph.D. (seminar tutor)
Guaranteed by
prof. RNDr. Jana Musilová, CSc.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: Mgr. Michael Krbek, Ph.D.
Supplier department: Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Timetable
Mon 1. 3. to Fri 14. 5. Fri 13:00–14:50 F3,03015
  • Timetable of Seminar Groups:
F6420/01: Mon 1. 3. to Fri 14. 5. Fri 15:00–16:50 F3,03015, M. Krbek
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 the advanced course of mathematical analysis for students of physics, interested in mathematical physics. 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 differentiable manifold, vector and tensor fields on differential manifolds, differential forms and basic operations with them, as well as the possibilities of application of these concepts in theoretical physics.

The student shall obtain the following knowledge and skills:

* Understanding the concept of differentiable manifold as a more general underlying space for physical theories in comparison with "standardly used" euclidean spaces.
* Practical skills in the use of charts and atlases on differentiable manifolds, transformation of coordinates.
* Understanding of fundamental geometrical objects on differentiable manifolds such as vector fields and differential forms.
* Practical skills in the calculus of vector fields and differential forms on differentiable manifolds.
* Understanding 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.
Learning outcomes
Students will be able to - understand basic notions in differentiable manifolds - apply differential forms - describe distributions using differential forms - use the theory for the description of symmetries - formulate Riemannian geometry in the orthonormal frame
Syllabus
  • 1. Fundamentals 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
    recommended literature
  • CHERN, Shiing-Shen, Wei-huan CHEN and Kai Shue LAM. Lectures on differential geometry. Singapore: World Scientific, 1998, x, 356 s. ISBN 981-02-3494-5. info
    not specified
  • 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
Teaching methods
lectures and tutorials
Assessment methods
Teaching: lectures and exercises Grading is based on the presentation of a selected topis from a list of topics or after consultation with the lecturer based on the student's suggestions
Language of instruction
Czech
Further comments (probably available only in Czech)
Study Materials
The course is taught once in two years.
General note: S.
Teacher's information
https://www.physics.muni.cz/~krbek/variety.shtml
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 2009, Spring 2011, spring 2012 - acreditation, Spring 2013, Spring 2015, Spring 2017, Spring 2019, Spring 2024.
  • Enrolment Statistics (Spring 2021, recent)
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