F3063 Integration of forms

Faculty of Science
Autumn 2019
Extent and Intensity
2/2/0. 4 credit(s). Type of Completion: zk (examination).
Teacher(s)
prof. RNDr. Jana Musilová, CSc. (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: prof. RNDr. Jana Musilová, CSc.
Supplier department: Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Timetable
Wed 12:00–13:50 F4,03017
  • Timetable of Seminar Groups:
F3063/01: Wed 14:00–15:50 F1 6/1014, M. Krbek
Prerequisites
( M1100 Mathematical Analysis I && M2100 Mathematical Analysis II )||( M1100F Mathematical Analysis I && M2100F 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 geometric and physical applications and to show their close relation.
Learning outcomes
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 algebraic 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)-form 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. Spaces of covariant tensors. Algebraic operations.
  • 4. Antisymmetric tensors, exterior product.
  • 5. Vector and tensor fields, differential forms.
  • 6. Exterior derivative.
  • 7. Induced mappings -- tangent mapping.
  • 8. Induced mappings -- pullback of forms.
  • 9. Integral of a differential form on singular cubes.
  • 10. General Stokes theorem.
  • 11. Integrals of the first and second type, classical versions of Stokes theorem.
  • 12. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. Volume element. Areas and volumes.
  • 13. Applications-work of a force field along a curve, flux of a vector field through a surface.
  • 14. Practical calculations.
Literature
    required literature
  • MUSILOVÁ, Jana and Pavla MUSILOVÁ. Matematika : pro porozumění i praxi : netradiční výklad tradičních témat vysokoškolské matematiky. První vydání. Brno: VUTIUM, 2017, xv, 365. ISBN 9788021455030. info
    recommended literature
  • SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
  • KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info
  • NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info
Teaching methods
Lectures: theoretical explanation with practical examples. Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems.
Assessment methods
Teaching: lectures, exercises.
Exam: written test (consisting of two parts: (a) solving problems, (b) test), oral part. In the part (a) students demonstrate the ability of solving extensive and complex problems of linear algebra and geometry. In the part (b) they demonstrate the understanding of important concepts and theorems, as well as the good orientation in the discipline.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
Study Materials
The course can also be completed outside the examination period.
The course is taught annually.
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 1999, Autumn 2010 - only for the accreditation, Autumn 2000, Autumn 2001, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, spring 2012 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Spring 2021, Spring 2022, Spring 2023, Spring 2025.
  • Enrolment Statistics (Autumn 2019, recent)
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