PřF:F6420 Diff. and int. calculus - Course Information

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

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. Michal Lenc, Ph.D.
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

Tue 15:00–16:50 F4,03017, Tue 17:00–18:50 F4,03017

Prerequisites

F3063 Integration and series
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

• Physics (programme PřF, N-FY)

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.

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.

• Enrolment Statistics (Spring 2013, recent)
  
• Permalink: https://is.muni.cz/course/sci/spring2013/F6420