F3063 Integration of forms

Faculty of Science
Autumn 2007
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 7:00–9:50 F3,03015, Fri 13:00–14:50 F3,03015
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 contains the theory of riemannian integral of differential forms, i.e. covariant tensor fields, on subsets of n-dimensional euclidean space. Using differential forms as integrated objects, the definition of the integral is obtained by the more natural way than the "classical" one. It includes classical line integrals and surface integrals. Its fundamental result - the general Stokes theorem - includes all classical integral theorems as special cases. The practical calculus of integrals with physical meaning is emphasized.
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 (in Czech)
Výuka: přednáška, klasické cvičení Zkouška: písemná (dvě části: (a) příklady, (b) test) a ústní
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.
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 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, Autumn 2019, Spring 2021, Spring 2022, Spring 2023, Spring 2025.
  • Enrolment Statistics (Autumn 2007, recent)
  • Permalink: https://is.muni.cz/course/sci/autumn2007/F3063