PřF:F3063 Integration of forms - Course Information
F3063 Integration of forms
Faculty of ScienceAutumn 2008
- Extent and Intensity
- 3/2/0. 4 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).
- Teacher(s)
- prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Lenka Czudková, 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: prof. RNDr. Jana Musilová, CSc. - Timetable
- Thu 13:00–15:50 F4,03017
- Timetable of Seminar Groups:
F3063/02: Tue 9:00–10:50 F4,03017, L. Czudková - Prerequisites
- M1100 Mathematical Analysis I && M2100 Mathematical Analysis II
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus. - 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
- The discipline is a part of the fundamental course of mathematical analysis for students of physics. It includes the theory of riemannian integration of differential forms, i.e. covariant antisymmetric tensor fields, on subsets of euclidean spaces of arbitrary dimensions. Using differential forms as integrated objects one can define the concept of integral by a more natural way than is that obtained by standardly used methods. (On the other hand, the mentioned more modern approach ought to be considered as "classical one", the frequently used approach being rather "conservative".) The concept of integral is general, including classical path and surface integrals. A basic result of the theory is the general Stokes theorem containing all classical integral theorems as special cases. The attention is also payed to the calculus -- practical calculations of integrals especially those with a physical meaning. The following main goals are followed:
* To show students more general and more effective approach to riemannian integration theory.
* To connect the general mathematical theory with practical gometric and physical applications and to show their close relation.
Absolving the course a student obtains following abilities and skills:
* General understanding of problems concerning vector and tensor fields as geometric (invariant) objects on euclidean spaces.
* Practical skills with vector and tensor calculus in spaces of general dimensions (algebra) and vector and tensor fields on euclidean spaces (analysis).
* Understanding of the concept of a differential form, practical skills with angebraic and analytic manipulations with differential forms (exterior product, exterior derivative, pullback).
* Understanding of the concept of riemannian integral of a differential k-form on a k-dimensional integration region in n-dimensional euclidean space (k lower or equal to n), understanding of the concept of path and surface integrals as special cases of the general concept.
* Knowledge of the general Stokes theorem (transforming the integration of a (k-1)-formy on the boundary of a k-dimensional integration region M to integration of the exterior derivative of this form on M), its proof and its classical applications (Green theorem, classical Stokes theorem, Gauss-Ostrogradsky theorem).
* Practical skills in calculus of integrals, including the use of general Stokes theorem, calculation of work of a vector field, flow of a vector field through a surface, etc.
* Practical use of the general Stokes theorem for obtaining differential (local) physical laws from their integral (global) versions, e.g. equations of fluid mechanics, Maxwell equations in classical electrodynamics, etc.
* Practical calculations of n-dimensional volumes integrating volume elements (curves, surfaces, bodies). - Syllabus
- 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions.
- 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem.
- 3. Generalization of the integral - decomposition of unity. 4. Spaces of covariant tensors.
- 5. Vector and tensor fields, differential forms.
- 6. Exterior product, exterior derivative.
- 7. Pullback.
- 8. Integral of a differential form on singular cubes.
- 9. General Stokes theorem.
- 10. Integrals of the first and second type, classical versions of Stokes theorem.
- 11. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects.
- 12. Appliactions-work of a force field along a curve, flux of a vector field through a surface.
- 13. Volume element. Areas and volumes.
- Literature
- KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info
- SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
- NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info
- Assessment methods
- Teaching: lectures, consultative exercises Exam: written test (two parts: (a) solving problems, (b) test) and oral exam.
- Language of instruction
- Czech
- Follow-Up Courses
- Further comments (probably available only in Czech)
- The course can also be completed outside the examination period.
The course is taught annually.
- Enrolment Statistics (Autumn 2008, recent)
- Permalink: https://is.muni.cz/course/sci/autumn2008/F3063