F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2024
- 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
- Mgr. Michael Krbek, 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
- Mon 19. 2. to Sun 26. 5. Fri 8:00–9:50 F3,03015
- Timetable of Seminar Groups:
- 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
- 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 main 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 topic 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: L. - Teacher's information
- https://www.physics.muni.cz/~krbek/variety.shtml
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 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:
- 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
- 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. - 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
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2019
- 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 18. 2. to Fri 17. 5. Mon 18:00–19:50 F4,03017
- Timetable of Seminar Groups:
- 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.
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2017
- 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 20. 2. to Mon 22. 5. Mon 15:00–16:50 F4,03017
- Timetable of Seminar Groups:
- 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.
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2015
- 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
- Mon 18:00–19:50 F4,03017, Thu 18:00–19:50 F3,03015
- 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.
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2013
- 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.
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2011
- Extent and Intensity
- 2/2/0. 4 credit(s). Type of Completion: z (credit).
- Teacher(s)
- prof. RNDr. Jana Musilová, CSc. (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. - Timetable
- Wed 9:00–10:50 F4,03017, Wed 16:00–17:50 F3,03015
- 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
- Biophysics (programme PřF, M-FY)
- Physics (programme PřF, M-FY)
- Physics (programme PřF, N-FY)
- Upper Secondary School Teacher Training in Physics (programme PřF, M-FY)
- Course objectives
- One of disciplines of an advanced course of mathematical analysis for students of physics, interested in mathematical physics. The 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 differentiaable manifold, vector and tensor fields on differential manifolds, differential forms and basic operatons with them, as well as the possibilities of application of these concepts in theoretical physics.
Absolving the discipline student obtains following knowledge and skills:
* Understending a concept of differentiable manifold as a more general underlying space for physical theories in comparison with "standardly used" euclidean spaces.
* Practical skills with the use of charts and atlases on differentiable manifolds, transformation of coordinates.
* Understanding of fundamental geometrical objects on differentiable manifolds as vector fields and differential forms.
* Practical skills in calculus of vector fields and differential forms in differentiable manifolds.
* Undestanding 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)
- Study Materials
The course is taught once in two years.
General note: S.
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2009
- Extent and Intensity
- 2/2/0. 4 credit(s). Type of Completion: z (credit).
- Teacher(s)
- prof. RNDr. Jana Musilová, CSc. (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. - Timetable
- Mon 10:00–11:50 F3,03015, Tue 7:00–8:50 F3,03015
- 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
- Biophysics (programme PřF, M-FY)
- Physics (programme PřF, M-FY)
- Physics (programme PřF, N-FY)
- Upper Secondary School Teacher Training in Physics (programme PřF, M-FY)
- Course objectives
- One of disciplines of an advanced course of mathematical analysis for students of physics, interested in mathematical physics. The 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 differentiaable manifold, vector and tensor fields on differential manifolds, differential forms and basic operatons with them, as well as the possibilities of application of these concepts in theoretical physics.
Absolving the discipline student obtains following knowledge and skills:
* Understending a concept of differentiable manifold as a more general underlying space for physical theories in comparison with "standardly used" euclidean spaces.
* Practical skills with the use of charts and atlases on differentiable manifolds, transformation of coordinates.
* Understanding of fundamental geometrical objects on differentiable manifolds as vector fields and differential forms.
* Practical skills in calculus of vector fields and differential forms in differentiable manifolds.
* Undestanding 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.
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2007
- Extent and Intensity
- 2/2/0. 4 credit(s). Type of Completion: z (credit).
- Teacher(s)
- Mgr. Pavla Musilová, Ph.D. (lecturer)
prof. RNDr. Jana Musilová, CSc. (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. - Timetable
- Mon 7:00–8:50 Fs1 6/1017
- Timetable of Seminar Groups:
- 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
- Biophysics (programme PřF, M-FY)
- Physics (programme PřF, M-FY)
- Physics (programme PřF, N-FY)
- Upper Secondary School Teacher Training in Physics (programme PřF, M-FY)
- Course objectives
- Předmět pokročilého kursu matematické analýzy pro fyziky, vhodný pro zájemce o problematiku matematické fyziky. Zabývá se především zobecněním pojmů diferenciálního a integrálního počtu na euklidovských prostorech na obecnější podkladové struktury -- diferencovatelné variety. Spolu s korektním výkladem matematických pojmů je důraz kladen na jejich aplikace v matematické fyzice.
- Syllabus
- 1. Fundaments of topology, topological manifolds, homeomorphisms. 2. Atlases, differential manifolds, diffeomorphisms. 3. Tensor algebra. 4. Tensors on manifolds, tensor bundles. 5. Induced diffeomorphisms on tensor bundles, Lie derivative. 6. Linear connection. 7. Physical applications-basis manifolds of GTR. 8. Integrals of differential forms on diferential manifolds, decomposition of unity, Stokes theorem. 9. 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 (in Czech)
- Výuka: přednáška a cvičení. Zápočet: písemná kontrola.
- Language of instruction
- Czech
- Further comments (probably available only in Czech)
- The course is taught once in two years.
General note: S.
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2005
- Extent and Intensity
- 2/2/0. 4 credit(s). Type of Completion: z (credit).
- Teacher(s)
- Mgr. Pavla Musilová, Ph.D. (lecturer)
prof. RNDr. Jana Musilová, CSc. (lecturer) - Guaranteed by
- prof. RNDr. Michal Lenc, Ph.D.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: Mgr. Pavla Musilová, Ph.D. - Timetable
- Thu 8:00–9:50 F23-106, Thu 10:00–11:50 F23-106
- 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
- Biophysics (programme PřF, M-FY)
- Physics (programme PřF, M-FY)
- Physics (programme PřF, N-FY)
- Upper Secondary School Teacher Training in Physics (programme PřF, M-FY)
- Course objectives
- Předmět pokročilého kursu matematické analýzy pro fyziky, vhodný pro zájemce o problematiku matematické fyziky. Zabývá se především zobecněním pojmů diferenciálního a integrálního počtu na euklidovských prostorech na obecnější podkladové struktury -- diferencovatelné variety. Spolu s korektním výkladem matematických pojmů je důraz kladen na jejich aplikace v matematické fyzice.
- Syllabus
- 1. Fundaments of topology, topological manifolds, homeomorphisms. 2. Atlases, differential manifolds, diffeomorphisms. 3. Tensor algebra. 4. Tensors on manifolds, tensor bundles. 5. Induced diffeomorphisms on tensor bundles, Lie derivative. 6. Linear connection. 7. Physical applications-basis manifolds of GTR. 8. Integrals of differential forms on diferential manifolds, decomposition of unity, Stokes theorem. 9. 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 (in Czech)
- Výuka: přednáška a cvičení. Zápočet: písemná kontrola.
- Language of instruction
- Czech
- Further comments (probably available only in Czech)
- The course is taught once in two years.
General note: S.
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2003
- Extent and Intensity
- 2/2/0. 4 credit(s). Type of Completion: z (credit).
- Teacher(s)
- prof. RNDr. Jana Musilová, CSc. (lecturer)
prof. RNDr. Jana Musilová, CSc. (seminar tutor)
Mgr. Pavla Musilová, Ph.D. (seminar tutor) - Guaranteed by
- doc. RNDr. Aleš Lacina, CSc.
Department of Plasma Physics and Technology – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc. - Timetable of Seminar Groups
- F6420/01: No timetable has been entered into IS. P. Musilová
- 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
- Biophysics (programme PřF, M-FY)
- Physics (programme PřF, M-FY)
- Physics (programme PřF, N-FY)
- Upper Secondary School Teacher Training in Physics (programme PdF, M-FY)
- Upper Secondary School Teacher Training in Physics (programme PřF, M-FY)
- Course objectives
- Předmět pokročilého kursu matematické analýzy pro fyziky, vhodný pro zájemce o problematiku matematické fyziky. Zabývá se především zobecněním pojmů diferenciálního a integrálního počtu na euklidovských prostorech na obecnější podkladové struktury -- diferencovatelné variety. Spolu s korektním výkladem matematických pojmů je důraz kladen na jejich aplikace v matematické fyzice.
- Syllabus
- 1. Fundaments of topology, topological manifolds, homeomorphisms. 2. Atlases, differential manifolds, diffeomorphisms. 3. Tensor algebra. 4. Tensors on manifolds, tensor bundles. 5. Induced diffeomorphisms on tensor bundles, Lie derivative. 6. Linear connection. 7. Physical applications-basis manifolds of GTR. 8. Integrals of differential forms on diferential manifolds, decomposition of unity, Stokes theorem. 9. 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 (in Czech)
- Výuka: přednáška a cvičení. Zápočet: písemná kontrola.
- Language of instruction
- Czech
- Further comments (probably available only in Czech)
- The course is taught once in two years.
General note: S.
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2001
- Extent and Intensity
- 2/2/0. 4 credit(s). Type of Completion: z (credit).
- Teacher(s)
- prof. RNDr. Jana Musilová, CSc. (lecturer)
prof. RNDr. Jana Musilová, CSc. (seminar tutor)
Mgr. Pavla Musilová, Ph.D. (seminar tutor) - Guaranteed by
- doc. RNDr. Aleš Lacina, CSc.
Department of Plasma Physics and Technology – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc. - Prerequisites (in Czech)
- F3063 Integration of forms
- 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
- Biophysics (programme PřF, M-FY)
- Physics (programme PřF, M-FY)
- Physics (programme PřF, N-FY)
- Upper Secondary School Teacher Training in Physics (programme PdF, M-FY)
- Upper Secondary School Teacher Training in Physics (programme PřF, M-FY)
- Course objectives
- Pokročilý kurs matematické analýzy pro fyziky, vhodný pro zájemce o problematiku matematické fyziky. Základy topologie, topologické variety, homeomorfismy. Atlasy, diferencovatelné variety, difeomorfismy. Tenzorová algebra. Tenzory na varietách, tenzorová rozvrstvení. Indukované difeomorfismy tenzorových prostorů, Lieovy derivace. Lineární konexe. Fyzikální aplikace-základní variety OTR. Integrování diferenciálních forem na diferencovatelných varietách, rozklad jednotky, Stokesův teorém. Klasické integrální věty, fyzikální aplikace.
- Language of instruction
- Czech
- Further comments (probably available only in Czech)
- The course is taught once in two years.
The course is taught: every week.
General note: L.
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2000
- Extent and Intensity
- 2/2/0. 4 credit(s). Type of Completion: z (credit).
- Teacher(s)
- prof. RNDr. Jana Musilová, CSc. (lecturer)
prof. RNDr. Jana Musilová, CSc. (seminar tutor)
Mgr. Pavla Musilová, Ph.D. (seminar tutor) - Guaranteed by
- doc. RNDr. Aleš Lacina, CSc.
Department of Plasma Physics and Technology – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc. - Prerequisites (in Czech)
- F3063 Integration of forms
- 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
- Biophysics (programme PřF, M-FY)
- Physics (programme PřF, M-FY)
- Physics (programme PřF, N-FY)
- Upper Secondary School Teacher Training in Physics (programme PdF, M-FY)
- Upper Secondary School Teacher Training in Physics (programme PřF, M-FY)
- Syllabus
- Pokročilý kurs matematické analýzy pro fyziky, vhodný pro zájemce o problematiku matematické fyziky. Základy topologie, topologické variety, homeomorfismy. Atlasy, diferencovatelné variety, difeomorfismy. Tenzorová algebra. Tenzory na varietách, tenzorová rozvrstvení. Indukované difeomorfismy tenzorových prostorů, Lieovy derivace. Lineární konexe. Fyzikální aplikace-základní variety OTR. Integrování diferenciálních forem na diferencovatelných varietách, rozklad jednotky, Stokesův teorém. Klasické integrální věty, fyzikální aplikace.
- Language of instruction
- Czech
- Further comments (probably available only in Czech)
- The course is taught once in two years.
The course is taught: every week.
General note: L.
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2025
The course is not taught in Spring 2025
- 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
- Mgr. Michael Krbek, 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 - 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
- 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 main 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 topic 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)
- The course is taught once in two years.
The course is taught: every week.
General note: L. - Teacher's information
- https://www.physics.muni.cz/~krbek/variety.shtml
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2023
The course is not taught in Spring 2023
- 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 - 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
- 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 main 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 topic 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)
- The course is taught once in two years.
The course is taught: every week.
General note: S. - Teacher's information
- https://www.physics.muni.cz/~krbek/variety.shtml
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2022
The course is not taught in Spring 2022
- 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 - 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
- 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. - 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)
- The course is taught once in two years.
The course is taught: every week.
General note: S. - Teacher's information
- https://www.physics.muni.cz/~krbek/variety.shtml
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2020
The course is not taught in Spring 2020
- 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 - 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. - Learning outcomes
- Students will be able to - understand basic notions in differentiable manifolds - apply differential forms - use Riemann geometry in geometrical and physical situations - solve problems
- 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
- Teaching methods
- lectures and tutorials
- 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.
The course is taught: every week.
General note: S.
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of Sciencespring 2018
The course is not taught in spring 2018
- 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 - 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.
The course is taught: every week.
General note: S.
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2016
The course is not taught in Spring 2016
- 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 - 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.
The course is taught: every week.
General note: S.
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2014
The course is not taught in Spring 2014
- 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 - 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.
The course is taught: every week.
General note: S.
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2012
The course is not taught in Spring 2012
- Extent and Intensity
- 2/2/0. 4 credit(s). Type of Completion: z (credit).
- Teacher(s)
- prof. RNDr. Jana Musilová, CSc. (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 - 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 an advanced course of mathematical analysis for students of physics, interested in mathematical physics. The 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 differentiaable manifold, vector and tensor fields on differential manifolds, differential forms and basic operatons with them, as well as the possibilities of application of these concepts in theoretical physics.
Absolving the discipline student obtains following knowledge and skills:
* Understending a concept of differentiable manifold as a more general underlying space for physical theories in comparison with "standardly used" euclidean spaces.
* Practical skills with the use of charts and atlases on differentiable manifolds, transformation of coordinates.
* Understanding of fundamental geometrical objects on differentiable manifolds as vector fields and differential forms.
* Practical skills in calculus of vector fields and differential forms in differentiable manifolds.
* Undestanding 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.
The course is taught: every week.
General note: S.
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2010
The course is not taught in Spring 2010
- Extent and Intensity
- 2/2/0. 4 credit(s). Type of Completion: z (credit).
- Teacher(s)
- prof. RNDr. Jana Musilová, CSc. (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. - 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
- Biophysics (programme PřF, M-FY)
- Physics (programme PřF, M-FY)
- Physics (programme PřF, N-FY)
- Upper Secondary School Teacher Training in Physics (programme PřF, M-FY)
- Course objectives
- One of disciplines of an advanced course of mathematical analysis for students of physics, interested in mathematical physics. The 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 differentiaable manifold, vector and tensor fields on differential manifolds, differential forms and basic operatons with them, as well as the possibilities of application of these concepts in theoretical physics.
Absolving the discipline student obtains following knowledge and skills:
* Understending a concept of differentiable manifold as a more general underlying space for physical theories in comparison with "standardly used" euclidean spaces.
* Practical skills with the use of charts and atlases on differentiable manifolds, transformation of coordinates.
* Understanding of fundamental geometrical objects on differentiable manifolds as vector fields and differential forms.
* Practical skills in calculus of vector fields and differential forms in differentiable manifolds.
* Undestanding 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.
The course is taught: every week.
General note: S.
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2008
The course is not taught in Spring 2008
- Extent and Intensity
- 2/2/0. 4 credit(s). Type of Completion: z (credit).
- Teacher(s)
- Mgr. Pavla Musilová, Ph.D. (lecturer)
prof. RNDr. Jana Musilová, CSc. (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. - 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
- Biophysics (programme PřF, M-FY)
- Physics (programme PřF, M-FY)
- Physics (programme PřF, N-FY)
- Upper Secondary School Teacher Training in Physics (programme PřF, M-FY)
- Course objectives
- Předmět pokročilého kursu matematické analýzy pro fyziky, vhodný pro zájemce o problematiku matematické fyziky. Zabývá se především zobecněním pojmů diferenciálního a integrálního počtu na euklidovských prostorech na obecnější podkladové struktury -- diferencovatelné variety. Spolu s korektním výkladem matematických pojmů je důraz kladen na jejich aplikace v matematické fyzice.
- Syllabus
- 1. Fundaments of topology, topological manifolds, homeomorphisms. 2. Atlases, differential manifolds, diffeomorphisms. 3. Tensor algebra. 4. Tensors on manifolds, tensor bundles. 5. Induced diffeomorphisms on tensor bundles, Lie derivative. 6. Linear connection. 7. Physical applications-basis manifolds of GTR. 8. Integrals of differential forms on diferential manifolds, decomposition of unity, Stokes theorem. 9. 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 (in Czech)
- Výuka: přednáška a cvičení. Zápočet: písemná kontrola.
- Language of instruction
- Czech
- Further comments (probably available only in Czech)
- The course is taught once in two years.
The course is taught: every week.
General note: S.
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2006
The course is not taught in Spring 2006
- Extent and Intensity
- 2/2/0. 4 credit(s). Type of Completion: z (credit).
- Teacher(s)
- Mgr. Pavla Musilová, Ph.D. (lecturer)
prof. RNDr. Jana Musilová, CSc. (lecturer) - Guaranteed by
- prof. RNDr. Michal Lenc, Ph.D.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: Mgr. Pavla Musilová, Ph.D. - 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
- Biophysics (programme PřF, M-FY)
- Physics (programme PřF, M-FY)
- Physics (programme PřF, N-FY)
- Upper Secondary School Teacher Training in Physics (programme PřF, M-FY)
- Course objectives
- Předmět pokročilého kursu matematické analýzy pro fyziky, vhodný pro zájemce o problematiku matematické fyziky. Zabývá se především zobecněním pojmů diferenciálního a integrálního počtu na euklidovských prostorech na obecnější podkladové struktury -- diferencovatelné variety. Spolu s korektním výkladem matematických pojmů je důraz kladen na jejich aplikace v matematické fyzice.
- Syllabus
- 1. Fundaments of topology, topological manifolds, homeomorphisms. 2. Atlases, differential manifolds, diffeomorphisms. 3. Tensor algebra. 4. Tensors on manifolds, tensor bundles. 5. Induced diffeomorphisms on tensor bundles, Lie derivative. 6. Linear connection. 7. Physical applications-basis manifolds of GTR. 8. Integrals of differential forms on diferential manifolds, decomposition of unity, Stokes theorem. 9. 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 (in Czech)
- Výuka: přednáška a cvičení. Zápočet: písemná kontrola.
- Language of instruction
- Czech
- Further comments (probably available only in Czech)
- The course is taught once in two years.
The course is taught: every week.
General note: S.
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2004
The course is not taught in Spring 2004
- Extent and Intensity
- 2/2/0. 4 credit(s). Type of Completion: z (credit).
- Teacher(s)
- prof. RNDr. Jana Musilová, CSc. (lecturer)
prof. RNDr. Jana Musilová, CSc. (seminar tutor)
Mgr. Pavla Musilová, 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. - 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
- Biophysics (programme PřF, M-FY)
- Physics (programme PřF, M-FY)
- Physics (programme PřF, N-FY)
- Upper Secondary School Teacher Training in Physics (programme PřF, M-FY)
- Course objectives
- Předmět pokročilého kursu matematické analýzy pro fyziky, vhodný pro zájemce o problematiku matematické fyziky. Zabývá se především zobecněním pojmů diferenciálního a integrálního počtu na euklidovských prostorech na obecnější podkladové struktury -- diferencovatelné variety. Spolu s korektním výkladem matematických pojmů je důraz kladen na jejich aplikace v matematické fyzice.
- Syllabus
- 1. Fundaments of topology, topological manifolds, homeomorphisms. 2. Atlases, differential manifolds, diffeomorphisms. 3. Tensor algebra. 4. Tensors on manifolds, tensor bundles. 5. Induced diffeomorphisms on tensor bundles, Lie derivative. 6. Linear connection. 7. Physical applications-basis manifolds of GTR. 8. Integrals of differential forms on diferential manifolds, decomposition of unity, Stokes theorem. 9. 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 (in Czech)
- Výuka: přednáška a cvičení. Zápočet: písemná kontrola.
- Language of instruction
- Czech
- Further comments (probably available only in Czech)
- The course is taught once in two years.
The course is taught: every week.
General note: S.
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2002
The course is not taught in Spring 2002
- Extent and Intensity
- 2/2/0. 4 credit(s). Type of Completion: z (credit).
- Teacher(s)
- prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Pavla Musilová, Ph.D. (seminar tutor) - Guaranteed by
- doc. RNDr. Aleš Lacina, CSc.
Department of Plasma Physics and Technology – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc. - Prerequisites (in Czech)
- F3063 Integration of forms
- 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
- Biophysics (programme PřF, M-FY)
- Physics (programme PřF, M-FY)
- Physics (programme PřF, N-FY)
- Upper Secondary School Teacher Training in Physics (programme PdF, M-FY)
- Upper Secondary School Teacher Training in Physics (programme PřF, M-FY)
- Course objectives
- Pokročilý kurs matematické analýzy pro fyziky, vhodný pro zájemce o problematiku matematické fyziky. Základy topologie, topologické variety, homeomorfismy. Atlasy, diferencovatelné variety, difeomorfismy. Tenzorová algebra. Tenzory na varietách, tenzorová rozvrstvení. Indukované difeomorfismy tenzorových prostorů, Lieovy derivace. Lineární konexe. Fyzikální aplikace-základní variety OTR. Integrování diferenciálních forem na diferencovatelných varietách, rozklad jednotky, Stokesův teorém. Klasické integrální věty, fyzikální aplikace.
- Language of instruction
- Czech
- Further comments (probably available only in Czech)
- The course is taught once in two years.
The course is taught: every week.
General note: L.
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of Sciencespring 2012 - acreditation
The information about the term spring 2012 - acreditation is not made public
- Extent and Intensity
- 2/2/0. 4 credit(s). Type of Completion: z (credit).
- Teacher(s)
- prof. RNDr. Jana Musilová, CSc. (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 - 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 an advanced course of mathematical analysis for students of physics, interested in mathematical physics. The 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 differentiaable manifold, vector and tensor fields on differential manifolds, differential forms and basic operatons with them, as well as the possibilities of application of these concepts in theoretical physics.
Absolving the discipline student obtains following knowledge and skills:
* Understending a concept of differentiable manifold as a more general underlying space for physical theories in comparison with "standardly used" euclidean spaces.
* Practical skills with the use of charts and atlases on differentiable manifolds, transformation of coordinates.
* Understanding of fundamental geometrical objects on differentiable manifolds as vector fields and differential forms.
* Practical skills in calculus of vector fields and differential forms in differentiable manifolds.
* Undestanding 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.
The course is taught: every week.
General note: S.
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2011 - only for the accreditation
- Extent and Intensity
- 2/2/0. 4 credit(s). Type of Completion: z (credit).
- Teacher(s)
- prof. RNDr. Jana Musilová, CSc. (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. - 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
- Biophysics (programme PřF, M-FY)
- Physics (programme PřF, M-FY)
- Physics (programme PřF, N-FY)
- Upper Secondary School Teacher Training in Physics (programme PřF, M-FY)
- Course objectives
- One of disciplines of an advanced course of mathematical analysis for students of physics, interested in mathematical physics. The 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 differentiaable manifold, vector and tensor fields on differential manifolds, differential forms and basic operatons with them, as well as the possibilities of application of these concepts in theoretical physics.
Absolving the discipline student obtains following knowledge and skills:
* Understending a concept of differentiable manifold as a more general underlying space for physical theories in comparison with "standardly used" euclidean spaces.
* Practical skills with the use of charts and atlases on differentiable manifolds, transformation of coordinates.
* Understanding of fundamental geometrical objects on differentiable manifolds as vector fields and differential forms.
* Practical skills in calculus of vector fields and differential forms in differentiable manifolds.
* Undestanding 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.
The course is taught: every week.
General note: S.
F6420 Differential and integral calculus on differential manifolds and its applications in physics
Faculty of ScienceSpring 2008 - for the purpose of the accreditation
The course is not taught in Spring 2008 - for the purpose of the accreditation
- Extent and Intensity
- 2/2/0. 4 credit(s). Type of Completion: z (credit).
- Teacher(s)
- Mgr. Pavla Musilová, Ph.D. (lecturer)
prof. RNDr. Jana Musilová, CSc. (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. - 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
- Biophysics (programme PřF, M-FY)
- Physics (programme PřF, M-FY)
- Physics (programme PřF, N-FY)
- Upper Secondary School Teacher Training in Physics (programme PřF, M-FY)
- Course objectives
- Předmět pokročilého kursu matematické analýzy pro fyziky, vhodný pro zájemce o problematiku matematické fyziky. Zabývá se především zobecněním pojmů diferenciálního a integrálního počtu na euklidovských prostorech na obecnější podkladové struktury -- diferencovatelné variety. Spolu s korektním výkladem matematických pojmů je důraz kladen na jejich aplikace v matematické fyzice.
- Syllabus
- 1. Fundaments of topology, topological manifolds, homeomorphisms. 2. Atlases, differential manifolds, diffeomorphisms. 3. Tensor algebra. 4. Tensors on manifolds, tensor bundles. 5. Induced diffeomorphisms on tensor bundles, Lie derivative. 6. Linear connection. 7. Physical applications-basis manifolds of GTR. 8. Integrals of differential forms on diferential manifolds, decomposition of unity, Stokes theorem. 9. 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 (in Czech)
- Výuka: přednáška a cvičení. Zápočet: písemná kontrola.
- Language of instruction
- Czech
- Further comments (probably available only in Czech)
- The course is taught once in two years.
The course is taught: every week.
General note: S.
- Enrolment Statistics (recent)