Course Information

F3063 Integration of forms

Extent and Intensity

2/2/0. 4 credit(s). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Michael Krbek, Ph.D. (seminar tutor)

Guaranteed by

prof. RNDr. Jana Musilová, CSc.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.
Supplier department: Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science

Timetable

Mon 17. 2. to Sat 24. 5. Wed 13:00–14:50 F3,03015

• Timetable of Seminar Groups:

F3063/01: Mon 17. 2. to Sat 24. 5. Thu 18:00–19:50 F3,03015, M. Krbek

Prerequisites

( M1100 Mathematical Analysis I && M2100 Mathematical Analysis II )||( M1100F Mathematical Analysis I && M2100F Mathematical Analysis II )
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It includes the theory of Riemannian integration of differential forms, i.e. covariant antisymmetric tensor fields, on subsets of euclidean spaces of arbitrary dimensions. Using differential forms as integrated objects one can define the concept of integral by a more natural way than is that obtained by standardly used methods. (On the other hand, the mentioned more modern approach ought to be considered as "classical one", the frequently used approach being rather "conservative".) The concept of integral is general, including classical path and surface integrals. A basic result of the theory is the general Stokes theorem containing all classical integral theorems as special cases. The attention is also payed to the calculus -- practical calculations of integrals especially those with a physical meaning. The following main goals are followed:

* To show students more general and more effective approach to Riemannian integration theory.
* To connect the general mathematical theory with practical geometric and physical applications and to show their close relation.

Learning outcomes

Absolving the course a student obtains following abilities and skills:

* General understanding of problems concerning vector and tensor fields as geometric (invariant) objects on euclidean spaces.
* Practical skills with vector and tensor calculus in spaces of general dimensions (algebra) and vector and tensor fields on euclidean spaces (analysis).
* Understanding of the concept of a differential form, practical skills with algebraic and analytic manipulations with differential forms (exterior product, exterior derivative, pullback).
* Understanding of the concept of Riemannian integral of a differential k-form on a k-dimensional integration region in n-dimensional euclidean space (k lower or equal to n), understanding of the concept of path and surface integrals as special cases of the general concept.
* Knowledge of the general Stokes theorem (transforming the integration of a (k-1)-form on the boundary of a k-dimensional integration region M to integration of the exterior derivative of this form on M), its proof and its classical applications (Green theorem, classical Stokes theorem, Gauss-Ostrogradsky theorem).
* Practical skills in calculus of integrals, including the use of general Stokes theorem, calculation of work of a vector field, flow of a vector field through a surface, etc.
* Practical use of the general Stokes theorem for obtaining differential (local) physical laws from their integral (global) versions, e.g. equations of fluid mechanics, Maxwell equations in classical electrodynamics, etc.
* Practical calculations of n-dimensional volumes integrating volume elements (curves, surfaces, bodies).
*

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions.
• 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem.
• 3. Spaces of covariant tensors. Algebraic operations.
• 4. Antisymmetric tensors, exterior product.
• 5. Vector and tensor fields, differential forms.
• 6. Exterior derivative.
• 7. Induced mappings -- tangent mapping.
• 8. Induced mappings -- pullback of forms.
• 9. Integral of a differential form on singular cubes.
• 10. General Stokes theorem.
• 11. Integrals of the first and second type, classical versions of Stokes theorem.
• 12. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. Volume element. Areas and volumes.
• 13. Applications-work of a force field along a curve, flux of a vector field through a surface.
• 14. Practical calculations.

Literature

required literature
• MUSILOVÁ, Jana and Pavla MUSILOVÁ. Matematika : pro porozumění i praxi : netradiční výklad tradičních témat vysokoškolské matematiky. První vydání. Brno: VUTIUM, 2017, xv, 365. ISBN 9788021455030. info

recommended literature
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info

Teaching methods

Lectures: theoretical explanation with practical examples. Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems.

Assessment methods

Teaching: lectures, exercises.
Exam: written and oral part. In the written part students demonstrate the ability of solving typical moderately extensive and complex problems.

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

Study Materials
The course can also be completed outside the examination period.
The course is taught once in two years.
General note: S.

F3063 Integration of forms

Extent and Intensity

2/2/0. 4 credit(s). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Michael Krbek, Ph.D. (seminar tutor)

Guaranteed by

prof. RNDr. Jana Musilová, CSc.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.
Supplier department: Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science

Timetable

Wed 11:00–12:50 F4,03017

• Timetable of Seminar Groups:

F3063/01: Thu 17:00–18:50 F3,03015, M. Krbek

Prerequisites

( M1100 Mathematical Analysis I && M2100 Mathematical Analysis II )||( M1100F Mathematical Analysis I && M2100F Mathematical Analysis II )
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It includes the theory of Riemannian integration of differential forms, i.e. covariant antisymmetric tensor fields, on subsets of euclidean spaces of arbitrary dimensions. Using differential forms as integrated objects one can define the concept of integral by a more natural way than is that obtained by standardly used methods. (On the other hand, the mentioned more modern approach ought to be considered as "classical one", the frequently used approach being rather "conservative".) The concept of integral is general, including classical path and surface integrals. A basic result of the theory is the general Stokes theorem containing all classical integral theorems as special cases. The attention is also payed to the calculus -- practical calculations of integrals especially those with a physical meaning. The following main goals are followed:

* To show students more general and more effective approach to Riemannian integration theory.
* To connect the general mathematical theory with practical geometric and physical applications and to show their close relation.

Learning outcomes

Absolving the course a student obtains following abilities and skills:

* General understanding of problems concerning vector and tensor fields as geometric (invariant) objects on euclidean spaces.
* Practical skills with vector and tensor calculus in spaces of general dimensions (algebra) and vector and tensor fields on euclidean spaces (analysis).
* Understanding of the concept of a differential form, practical skills with algebraic and analytic manipulations with differential forms (exterior product, exterior derivative, pullback).
* Understanding of the concept of Riemannian integral of a differential k-form on a k-dimensional integration region in n-dimensional euclidean space (k lower or equal to n), understanding of the concept of path and surface integrals as special cases of the general concept.
* Knowledge of the general Stokes theorem (transforming the integration of a (k-1)-form on the boundary of a k-dimensional integration region M to integration of the exterior derivative of this form on M), its proof and its classical applications (Green theorem, classical Stokes theorem, Gauss-Ostrogradsky theorem).
* Practical skills in calculus of integrals, including the use of general Stokes theorem, calculation of work of a vector field, flow of a vector field through a surface, etc.
* Practical use of the general Stokes theorem for obtaining differential (local) physical laws from their integral (global) versions, e.g. equations of fluid mechanics, Maxwell equations in classical electrodynamics, etc.
* Practical calculations of n-dimensional volumes integrating volume elements (curves, surfaces, bodies).
*

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions.
• 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem.
• 3. Spaces of covariant tensors. Algebraic operations.
• 4. Antisymmetric tensors, exterior product.
• 5. Vector and tensor fields, differential forms.
• 6. Exterior derivative.
• 7. Induced mappings -- tangent mapping.
• 8. Induced mappings -- pullback of forms.
• 9. Integral of a differential form on singular cubes.
• 10. General Stokes theorem.
• 11. Integrals of the first and second type, classical versions of Stokes theorem.
• 12. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. Volume element. Areas and volumes.
• 13. Applications-work of a force field along a curve, flux of a vector field through a surface.
• 14. Practical calculations.

Literature

required literature
• MUSILOVÁ, Jana and Pavla MUSILOVÁ. Matematika : pro porozumění i praxi : netradiční výklad tradičních témat vysokoškolské matematiky. První vydání. Brno: VUTIUM, 2017, xv, 365. ISBN 9788021455030. info

recommended literature
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info

Teaching methods

Lectures: theoretical explanation with practical examples. Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems.

Assessment methods

Teaching: lectures, exercises.
Exam: written test (consisting of two parts: (a) solving problems, (b) test), oral part. In the part (a) students demonstrate the ability of solving extensive and complex problems of linear algebra and geometry. In the part (b) they demonstrate the understanding of important concepts and theorems, as well as the good orientation in the discipline.

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

Study Materials
The course can also be completed outside the examination period.
The course is taught once in two years.

F3063 Integration of forms

Extent and Intensity

2/2/0. 4 credit(s). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Michael Krbek, Ph.D. (seminar tutor)

Guaranteed by

prof. RNDr. Jana Musilová, CSc.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.
Supplier department: Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science

Timetable

Mon 15:00–16:50 F4,03017, Tue 10:00–11:50 F2 6/2012

Prerequisites

( M1100 Mathematical Analysis I && M2100 Mathematical Analysis II )||( M1100F Mathematical Analysis I && M2100F Mathematical Analysis II )
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It includes the theory of Riemannian integration of differential forms, i.e. covariant antisymmetric tensor fields, on subsets of euclidean spaces of arbitrary dimensions. Using differential forms as integrated objects one can define the concept of integral by a more natural way than is that obtained by standardly used methods. (On the other hand, the mentioned more modern approach ought to be considered as "classical one", the frequently used approach being rather "conservative".) The concept of integral is general, including classical path and surface integrals. A basic result of the theory is the general Stokes theorem containing all classical integral theorems as special cases. The attention is also payed to the calculus -- practical calculations of integrals especially those with a physical meaning. The following main goals are followed:

* To show students more general and more effective approach to Riemannian integration theory.
* To connect the general mathematical theory with practical geometric and physical applications and to show their close relation.

Learning outcomes

Absolving the course a student obtains following abilities and skills:

* General understanding of problems concerning vector and tensor fields as geometric (invariant) objects on euclidean spaces.
* Practical skills with vector and tensor calculus in spaces of general dimensions (algebra) and vector and tensor fields on euclidean spaces (analysis).
* Understanding of the concept of a differential form, practical skills with algebraic and analytic manipulations with differential forms (exterior product, exterior derivative, pullback).
* Understanding of the concept of Riemannian integral of a differential k-form on a k-dimensional integration region in n-dimensional euclidean space (k lower or equal to n), understanding of the concept of path and surface integrals as special cases of the general concept.
* Knowledge of the general Stokes theorem (transforming the integration of a (k-1)-form on the boundary of a k-dimensional integration region M to integration of the exterior derivative of this form on M), its proof and its classical applications (Green theorem, classical Stokes theorem, Gauss-Ostrogradsky theorem).
* Practical skills in calculus of integrals, including the use of general Stokes theorem, calculation of work of a vector field, flow of a vector field through a surface, etc.
* Practical use of the general Stokes theorem for obtaining differential (local) physical laws from their integral (global) versions, e.g. equations of fluid mechanics, Maxwell equations in classical electrodynamics, etc.
* Practical calculations of n-dimensional volumes integrating volume elements (curves, surfaces, bodies).
*

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions.
• 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem.
• 3. Spaces of covariant tensors. Algebraic operations.
• 4. Antisymmetric tensors, exterior product.
• 5. Vector and tensor fields, differential forms.
• 6. Exterior derivative.
• 7. Induced mappings -- tangent mapping.
• 8. Induced mappings -- pullback of forms.
• 9. Integral of a differential form on singular cubes.
• 10. General Stokes theorem.
• 11. Integrals of the first and second type, classical versions of Stokes theorem.
• 12. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. Volume element. Areas and volumes.
• 13. Applications-work of a force field along a curve, flux of a vector field through a surface.
• 14. Practical calculations.

Literature

required literature
• MUSILOVÁ, Jana and Pavla MUSILOVÁ. Matematika : pro porozumění i praxi : netradiční výklad tradičních témat vysokoškolské matematiky. První vydání. Brno: VUTIUM, 2017, xv, 365. ISBN 9788021455030. info

recommended literature
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info

Teaching methods

Lectures: theoretical explanation with practical examples. Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems.

Assessment methods

Teaching: lectures, exercises.
Exam: written test (consisting of two parts: (a) solving problems, (b) test), oral part. In the part (a) students demonstrate the ability of solving extensive and complex problems of linear algebra and geometry. In the part (b) they demonstrate the understanding of important concepts and theorems, as well as the good orientation in the discipline.

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

Study Materials
The course can also be completed outside the examination period.
The course is taught annually.

F3063 Integration of forms

Extent and Intensity

2/2/0. 4 credit(s). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Michael Krbek, Ph.D. (seminar tutor)

Guaranteed by

prof. RNDr. Jana Musilová, CSc.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.
Supplier department: Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science

Timetable

Mon 1. 3. to Fri 14. 5. Mon 15:00–16:50 F1 6/1014

• Timetable of Seminar Groups:

F3063/01: Mon 1. 3. to Fri 14. 5. Thu 16:00–17:50 F4,03017, M. Krbek

Prerequisites

( M1100 Mathematical Analysis I && M2100 Mathematical Analysis II )||( M1100F Mathematical Analysis I && M2100F Mathematical Analysis II )
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It includes the theory of Riemannian integration of differential forms, i.e. covariant antisymmetric tensor fields, on subsets of euclidean spaces of arbitrary dimensions. Using differential forms as integrated objects one can define the concept of integral by a more natural way than is that obtained by standardly used methods. (On the other hand, the mentioned more modern approach ought to be considered as "classical one", the frequently used approach being rather "conservative".) The concept of integral is general, including classical path and surface integrals. A basic result of the theory is the general Stokes theorem containing all classical integral theorems as special cases. The attention is also payed to the calculus -- practical calculations of integrals especially those with a physical meaning. The following main goals are followed:

* To show students more general and more effective approach to Riemannian integration theory.
* To connect the general mathematical theory with practical geometric and physical applications and to show their close relation.

Learning outcomes

Absolving the course a student obtains following abilities and skills:

* General understanding of problems concerning vector and tensor fields as geometric (invariant) objects on euclidean spaces.
* Practical skills with vector and tensor calculus in spaces of general dimensions (algebra) and vector and tensor fields on euclidean spaces (analysis).
* Understanding of the concept of a differential form, practical skills with algebraic and analytic manipulations with differential forms (exterior product, exterior derivative, pullback).
* Understanding of the concept of Riemannian integral of a differential k-form on a k-dimensional integration region in n-dimensional euclidean space (k lower or equal to n), understanding of the concept of path and surface integrals as special cases of the general concept.
* Knowledge of the general Stokes theorem (transforming the integration of a (k-1)-form on the boundary of a k-dimensional integration region M to integration of the exterior derivative of this form on M), its proof and its classical applications (Green theorem, classical Stokes theorem, Gauss-Ostrogradsky theorem).
* Practical skills in calculus of integrals, including the use of general Stokes theorem, calculation of work of a vector field, flow of a vector field through a surface, etc.
* Practical use of the general Stokes theorem for obtaining differential (local) physical laws from their integral (global) versions, e.g. equations of fluid mechanics, Maxwell equations in classical electrodynamics, etc.
* Practical calculations of n-dimensional volumes integrating volume elements (curves, surfaces, bodies).
*

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions.
• 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem.
• 3. Spaces of covariant tensors. Algebraic operations.
• 4. Antisymmetric tensors, exterior product.
• 5. Vector and tensor fields, differential forms.
• 6. Exterior derivative.
• 7. Induced mappings -- tangent mapping.
• 8. Induced mappings -- pullback of forms.
• 9. Integral of a differential form on singular cubes.
• 10. General Stokes theorem.
• 11. Integrals of the first and second type, classical versions of Stokes theorem.
• 12. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. Volume element. Areas and volumes.
• 13. Applications-work of a force field along a curve, flux of a vector field through a surface.
• 14. Practical calculations.

Literature

required literature
• MUSILOVÁ, Jana and Pavla MUSILOVÁ. Matematika : pro porozumění i praxi : netradiční výklad tradičních témat vysokoškolské matematiky. První vydání. Brno: VUTIUM, 2017, xv, 365. ISBN 9788021455030. info

recommended literature
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info

Teaching methods

Lectures: theoretical explanation with practical examples. Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems.

Assessment methods

Teaching: lectures, exercises.
Exam: written test (consisting of two parts: (a) solving problems, (b) test), oral part. In the part (a) students demonstrate the ability of solving extensive and complex problems of linear algebra and geometry. In the part (b) they demonstrate the understanding of important concepts and theorems, as well as the good orientation in the discipline.

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

Study Materials
The course can also be completed outside the examination period.
The course is taught annually.

F3063 Integration of forms

Extent and Intensity

2/2/0. 4 credit(s). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Michael Krbek, Ph.D. (seminar tutor)

Guaranteed by

prof. RNDr. Jana Musilová, CSc.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.
Supplier department: Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science

Timetable

Wed 12:00–13:50 F4,03017

• Timetable of Seminar Groups:

F3063/01: Wed 14:00–15:50 F1 6/1014, M. Krbek

Prerequisites

( M1100 Mathematical Analysis I && M2100 Mathematical Analysis II )||( M1100F Mathematical Analysis I && M2100F Mathematical Analysis II )
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It includes the theory of Riemannian integration of differential forms, i.e. covariant antisymmetric tensor fields, on subsets of euclidean spaces of arbitrary dimensions. Using differential forms as integrated objects one can define the concept of integral by a more natural way than is that obtained by standardly used methods. (On the other hand, the mentioned more modern approach ought to be considered as "classical one", the frequently used approach being rather "conservative".) The concept of integral is general, including classical path and surface integrals. A basic result of the theory is the general Stokes theorem containing all classical integral theorems as special cases. The attention is also payed to the calculus -- practical calculations of integrals especially those with a physical meaning. The following main goals are followed:

* To show students more general and more effective approach to Riemannian integration theory.
* To connect the general mathematical theory with practical geometric and physical applications and to show their close relation.

Learning outcomes

Absolving the course a student obtains following abilities and skills:

* General understanding of problems concerning vector and tensor fields as geometric (invariant) objects on euclidean spaces.
* Practical skills with vector and tensor calculus in spaces of general dimensions (algebra) and vector and tensor fields on euclidean spaces (analysis).
* Understanding of the concept of a differential form, practical skills with algebraic and analytic manipulations with differential forms (exterior product, exterior derivative, pullback).
* Understanding of the concept of Riemannian integral of a differential k-form on a k-dimensional integration region in n-dimensional euclidean space (k lower or equal to n), understanding of the concept of path and surface integrals as special cases of the general concept.
* Knowledge of the general Stokes theorem (transforming the integration of a (k-1)-form on the boundary of a k-dimensional integration region M to integration of the exterior derivative of this form on M), its proof and its classical applications (Green theorem, classical Stokes theorem, Gauss-Ostrogradsky theorem).
* Practical skills in calculus of integrals, including the use of general Stokes theorem, calculation of work of a vector field, flow of a vector field through a surface, etc.
* Practical use of the general Stokes theorem for obtaining differential (local) physical laws from their integral (global) versions, e.g. equations of fluid mechanics, Maxwell equations in classical electrodynamics, etc.
* Practical calculations of n-dimensional volumes integrating volume elements (curves, surfaces, bodies).
*

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions.
• 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem.
• 3. Spaces of covariant tensors. Algebraic operations.
• 4. Antisymmetric tensors, exterior product.
• 5. Vector and tensor fields, differential forms.
• 6. Exterior derivative.
• 7. Induced mappings -- tangent mapping.
• 8. Induced mappings -- pullback of forms.
• 9. Integral of a differential form on singular cubes.
• 10. General Stokes theorem.
• 11. Integrals of the first and second type, classical versions of Stokes theorem.
• 12. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. Volume element. Areas and volumes.
• 13. Applications-work of a force field along a curve, flux of a vector field through a surface.
• 14. Practical calculations.

Literature

required literature
• MUSILOVÁ, Jana and Pavla MUSILOVÁ. Matematika : pro porozumění i praxi : netradiční výklad tradičních témat vysokoškolské matematiky. První vydání. Brno: VUTIUM, 2017, xv, 365. ISBN 9788021455030. info

recommended literature
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info

Teaching methods

Lectures: theoretical explanation with practical examples. Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems.

Assessment methods

Teaching: lectures, exercises.
Exam: written test (consisting of two parts: (a) solving problems, (b) test), oral part. In the part (a) students demonstrate the ability of solving extensive and complex problems of linear algebra and geometry. In the part (b) they demonstrate the understanding of important concepts and theorems, as well as the good orientation in the discipline.

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

Study Materials
The course can also be completed outside the examination period.
The course is taught annually.

F3063 Integration of forms

Extent and Intensity

2/2/0. 4 credit(s) (plus 2 credits for an exam). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Michael Krbek, Ph.D. (seminar tutor)

Guaranteed by

prof. RNDr. Jana Musilová, CSc.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.
Supplier department: Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science

Timetable

Mon 17. 9. to Fri 14. 12. Wed 12:00–13:50 F4,03017

• Timetable of Seminar Groups:

F3063/01: Mon 17. 9. to Fri 14. 12. Thu 9:00–10:50 F4,03017

Prerequisites

M1100 Mathematical Analysis I && M2100 Mathematical Analysis II
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It includes the theory of Riemannian integration of differential forms, i.e. covariant antisymmetric tensor fields, on subsets of euclidean spaces of arbitrary dimensions. Using differential forms as integrated objects one can define the concept of integral by a more natural way than is that obtained by standardly used methods. (On the other hand, the mentioned more modern approach ought to be considered as "classical one", the frequently used approach being rather "conservative".) The concept of integral is general, including classical path and surface integrals. A basic result of the theory is the general Stokes theorem containing all classical integral theorems as special cases. The attention is also payed to the calculus -- practical calculations of integrals especially those with a physical meaning. The following main goals are followed:

* To show students more general and more effective approach to Riemannian integration theory.
* To connect the general mathematical theory with practical geometric and physical applications and to show their close relation.

Learning outcomes

Absolving the course a student obtains following abilities and skills:

* General understanding of problems concerning vector and tensor fields as geometric (invariant) objects on euclidean spaces.
* Practical skills with vector and tensor calculus in spaces of general dimensions (algebra) and vector and tensor fields on euclidean spaces (analysis).
* Understanding of the concept of a differential form, practical skills with algebraic and analytic manipulations with differential forms (exterior product, exterior derivative, pullback).
* Understanding of the concept of Riemannian integral of a differential k-form on a k-dimensional integration region in n-dimensional euclidean space (k lower or equal to n), understanding of the concept of path and surface integrals as special cases of the general concept.
* Knowledge of the general Stokes theorem (transforming the integration of a (k-1)-form on the boundary of a k-dimensional integration region M to integration of the exterior derivative of this form on M), its proof and its classical applications (Green theorem, classical Stokes theorem, Gauss-Ostrogradsky theorem).
* Practical skills in calculus of integrals, including the use of general Stokes theorem, calculation of work of a vector field, flow of a vector field through a surface, etc.
* Practical use of the general Stokes theorem for obtaining differential (local) physical laws from their integral (global) versions, e.g. equations of fluid mechanics, Maxwell equations in classical electrodynamics, etc.
* Practical calculations of n-dimensional volumes integrating volume elements (curves, surfaces, bodies).
*

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions.
• 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem.
• 3. Spaces of covariant tensors. Algebraic operations.
• 4. Antisymmetric tensors, exterior product.
• 5. Vector and tensor fields, differential forms.
• 6. Exterior derivative.
• 7. Induced mappings -- tangent mapping.
• 8. Induced mappings -- pullback of forms.
• 9. Integral of a differential form on singular cubes.
• 10. General Stokes theorem.
• 11. Integrals of the first and second type, classical versions of Stokes theorem.
• 12. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. Volume element. Areas and volumes.
• 13. Applications-work of a force field along a curve, flux of a vector field through a surface.
• 14. Practical calculations.

Literature

required literature
• MUSILOVÁ, Jana and Pavla MUSILOVÁ. Matematika : pro porozumění i praxi : netradiční výklad tradičních témat vysokoškolské matematiky. První vydání. Brno: VUTIUM, 2017, xv, 365. ISBN 9788021455030. info

recommended literature
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info

Teaching methods

Lectures: theoretical explanation with practical examples. Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems.

Assessment methods

Teaching: lectures, exercises.
Exam: written test (consisting of two parts: (a) solving problems, (b) test), oral part. In the part (a) students demonstrate the ability of solving extensive and complex problems of linear algebra and geometry. In the part (b) they demonstrate the understanding of important concepts and theorems, as well as the good orientation in the discipline.

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

The course can also be completed outside the examination period.
The course is taught annually.

F3063 Integration of forms

Extent and Intensity

2/2/0. 4 credit(s) (plus 2 credits for an exam). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Michael Krbek, Ph.D. (seminar tutor)

Guaranteed by

prof. RNDr. Jana Musilová, CSc.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.
Supplier department: Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science

Timetable

Mon 18. 9. to Fri 15. 12. Tue 8:00–9:50 F3,03015, Wed 12:00–13:50 F1 6/1014

Prerequisites

M1100 Mathematical Analysis I && M2100 Mathematical Analysis II
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It includes the theory of Riemannian integration of differential forms, i.e. covariant antisymmetric tensor fields, on subsets of euclidean spaces of arbitrary dimensions. Using differential forms as integrated objects one can define the concept of integral by a more natural way than is that obtained by standardly used methods. (On the other hand, the mentioned more modern approach ought to be considered as "classical one", the frequently used approach being rather "conservative".) The concept of integral is general, including classical path and surface integrals. A basic result of the theory is the general Stokes theorem containing all classical integral theorems as special cases. The attention is also payed to the calculus -- practical calculations of integrals especially those with a physical meaning. The following main goals are followed:

* To show students more general and more effective approach to Riemannian integration theory.
* To connect the general mathematical theory with practical geometric and physical applications and to show their close relation.

Absolving the course a student obtains following abilities and skills:

* General understanding of problems concerning vector and tensor fields as geometric (invariant) objects on euclidean spaces.
* Practical skills with vector and tensor calculus in spaces of general dimensions (algebra) and vector and tensor fields on euclidean spaces (analysis).
* Understanding of the concept of a differential form, practical skills with algebraic and analytic manipulations with differential forms (exterior product, exterior derivative, pullback).
* Understanding of the concept of Riemannian integral of a differential k-form on a k-dimensional integration region in n-dimensional euclidean space (k lower or equal to n), understanding of the concept of path and surface integrals as special cases of the general concept.
* Knowledge of the general Stokes theorem (transforming the integration of a (k-1)-form on the boundary of a k-dimensional integration region M to integration of the exterior derivative of this form on M), its proof and its classical applications (Green theorem, classical Stokes theorem, Gauss-Ostrogradsky theorem).
* Practical skills in calculus of integrals, including the use of general Stokes theorem, calculation of work of a vector field, flow of a vector field through a surface, etc.
* Practical use of the general Stokes theorem for obtaining differential (local) physical laws from their integral (global) versions, e.g. equations of fluid mechanics, Maxwell equations in classical electrodynamics, etc.
* Practical calculations of n-dimensional volumes integrating volume elements (curves, surfaces, bodies).
*

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions.
• 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem.
• 3. Spaces of covariant tensors. Algebraic operations.
• 4. Antisymmetric tensors, exterior product.
• 5. Vector and tensor fields, differential forms.
• 6. Exterior derivative.
• 7. Induced mappings -- tangent mapping.
• 8. Induced mappings -- pullback of forms.
• 9. Integral of a differential form on singular cubes.
• 10. General Stokes theorem.
• 11. Integrals of the first and second type, classical versions of Stokes theorem.
• 12. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. Volume element. Areas and volumes.
• 13. Applications-work of a force field along a curve, flux of a vector field through a surface.
• 14. Practical calculations.

Literature

required literature
• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info

recommended literature
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info

Teaching methods

Lectures: theoretical explanation with practical examples. Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems.

Assessment methods

Teaching: lectures, exercises, 2 tests in the 1st and 2nd third of the semester
Exam: written test (consisting of two parts: (a) solving problems, (b) test), oral part. In the part (a) students demonstrate the ability of solving extensive and complex problems of linear algebra and geometry. In the part (b) they demonstrate the understanding of important concepts and theorems, as well as the good orientation in the discipline.

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

Study Materials
The course can also be completed outside the examination period.
The course is taught annually.

F3063 Integration of forms

Extent and Intensity

2/2/0. 4 credit(s) (plus 2 credits for an exam). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Michael Krbek, Ph.D. (seminar tutor)

Guaranteed by

prof. RNDr. Jana Musilová, CSc.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.
Supplier department: Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science

Timetable

Mon 19. 9. to Sun 18. 12. Tue 16:00–17:50 MS1,01016, Wed 12:00–13:50 FLenc,03028

Prerequisites

M1100 Mathematical Analysis I && M2100 Mathematical Analysis II
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It includes the theory of Riemannian integration of differential forms, i.e. covariant antisymmetric tensor fields, on subsets of euclidean spaces of arbitrary dimensions. Using differential forms as integrated objects one can define the concept of integral by a more natural way than is that obtained by standardly used methods. (On the other hand, the mentioned more modern approach ought to be considered as "classical one", the frequently used approach being rather "conservative".) The concept of integral is general, including classical path and surface integrals. A basic result of the theory is the general Stokes theorem containing all classical integral theorems as special cases. The attention is also payed to the calculus -- practical calculations of integrals especially those with a physical meaning. The following main goals are followed:

* To show students more general and more effective approach to Riemannian integration theory.
* To connect the general mathematical theory with practical geometric and physical applications and to show their close relation.

Absolving the course a student obtains following abilities and skills:

* General understanding of problems concerning vector and tensor fields as geometric (invariant) objects on euclidean spaces.
* Practical skills with vector and tensor calculus in spaces of general dimensions (algebra) and vector and tensor fields on euclidean spaces (analysis).
* Understanding of the concept of a differential form, practical skills with algebraic and analytic manipulations with differential forms (exterior product, exterior derivative, pullback).
* Understanding of the concept of Riemannian integral of a differential k-form on a k-dimensional integration region in n-dimensional euclidean space (k lower or equal to n), understanding of the concept of path and surface integrals as special cases of the general concept.
* Knowledge of the general Stokes theorem (transforming the integration of a (k-1)-form on the boundary of a k-dimensional integration region M to integration of the exterior derivative of this form on M), its proof and its classical applications (Green theorem, classical Stokes theorem, Gauss-Ostrogradsky theorem).
* Practical skills in calculus of integrals, including the use of general Stokes theorem, calculation of work of a vector field, flow of a vector field through a surface, etc.
* Practical use of the general Stokes theorem for obtaining differential (local) physical laws from their integral (global) versions, e.g. equations of fluid mechanics, Maxwell equations in classical electrodynamics, etc.
* Practical calculations of n-dimensional volumes integrating volume elements (curves, surfaces, bodies).
*

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions.
• 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem.
• 3. Spaces of covariant tensors. Algebraic operations.
• 4. Antisymmetric tensors, exterior product.
• 5. Vector and tensor fields, differential forms.
• 6. Exterior derivative.
• 7. Induced mappings -- tangent mapping.
• 8. Induced mappings -- pullback of forms.
• 9. Integral of a differential form on singular cubes.
• 10. General Stokes theorem.
• 11. Integrals of the first and second type, classical versions of Stokes theorem.
• 12. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. Volume element. Areas and volumes.
• 13. Applications-work of a force field along a curve, flux of a vector field through a surface.
• 14. Practical calculations.

Literature

required literature
• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info

recommended literature
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info

Teaching methods

Lectures: theoretical explanation with practical examples. Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems.

Assessment methods

Teaching: lectures, exercises.
Exam: written test (consisting of two parts: (a) solving problems, (b) test), oral part. In the part (a) students demonstrate the ability of solving extensive and complex problems of linear algebra and geometry. In the part (b) they demonstrate the understanding of important concepts and theorems, as well as the good orientation in the discipline.

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

Study Materials
The course can also be completed outside the examination period.
The course is taught annually.

F3063 Integration of forms

Extent and Intensity

2/2/0. 4 credit(s) (plus 2 credits for an exam). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Michael Krbek, Ph.D. (seminar tutor)

Guaranteed by

prof. RNDr. Jana Musilová, CSc.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.
Supplier department: Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science

Timetable

Mon 8:00–9:50 F2 6/2012, Thu 15:00–16:50 Fs1 6/1017

Prerequisites

M1100 Mathematical Analysis I && M2100 Mathematical Analysis II
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It includes the theory of Riemannian integration of differential forms, i.e. covariant antisymmetric tensor fields, on subsets of euclidean spaces of arbitrary dimensions. Using differential forms as integrated objects one can define the concept of integral by a more natural way than is that obtained by standardly used methods. (On the other hand, the mentioned more modern approach ought to be considered as "classical one", the frequently used approach being rather "conservative".) The concept of integral is general, including classical path and surface integrals. A basic result of the theory is the general Stokes theorem containing all classical integral theorems as special cases. The attention is also payed to the calculus -- practical calculations of integrals especially those with a physical meaning. The following main goals are followed:

* To show students more general and more effective approach to Riemannian integration theory.
* To connect the general mathematical theory with practical geometric and physical applications and to show their close relation.

Absolving the course a student obtains following abilities and skills:

* General understanding of problems concerning vector and tensor fields as geometric (invariant) objects on euclidean spaces.
* Practical skills with vector and tensor calculus in spaces of general dimensions (algebra) and vector and tensor fields on euclidean spaces (analysis).
* Understanding of the concept of a differential form, practical skills with algebraic and analytic manipulations with differential forms (exterior product, exterior derivative, pullback).
* Understanding of the concept of Riemannian integral of a differential k-form on a k-dimensional integration region in n-dimensional euclidean space (k lower or equal to n), understanding of the concept of path and surface integrals as special cases of the general concept.
* Knowledge of the general Stokes theorem (transforming the integration of a (k-1)-form on the boundary of a k-dimensional integration region M to integration of the exterior derivative of this form on M), its proof and its classical applications (Green theorem, classical Stokes theorem, Gauss-Ostrogradsky theorem).
* Practical skills in calculus of integrals, including the use of general Stokes theorem, calculation of work of a vector field, flow of a vector field through a surface, etc.
* Practical use of the general Stokes theorem for obtaining differential (local) physical laws from their integral (global) versions, e.g. equations of fluid mechanics, Maxwell equations in classical electrodynamics, etc.
* Practical calculations of n-dimensional volumes integrating volume elements (curves, surfaces, bodies).
*

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions.
• 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem.
• 3. Spaces of covariant tensors. Algebraic operations.
• 4. Antisymmetric tensors, exterior product.
• 5. Vector and tensor fields, differential forms.
• 6. Exterior derivative.
• 7. Induced mappings -- tangent mapping.
• 8. Induced mappings -- pullback of forms.
• 9. Integral of a differential form on singular cubes.
• 10. General Stokes theorem.
• 11. Integrals of the first and second type, classical versions of Stokes theorem.
• 12. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. Volume element. Areas and volumes.
• 13. Applications-work of a force field along a curve, flux of a vector field through a surface.
• 14. Practical calculations.

Literature

required literature
• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info

recommended literature
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info

Teaching methods

Lectures: theoretical explanation with practical examples. Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems.

Assessment methods

Teaching: lectures, exercises.
Exam: written test (consisting of two parts: (a) solving problems, (b) test), oral part. In the part (a) students demonstrate the ability of solving extensive and complex problems of linear algebra and geometry. In the part (b) they demonstrate the understanding of important concepts and theorems, as well as the good orientation in the discipline.

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

Study Materials
The course can also be completed outside the examination period.
The course is taught annually.

F3063 Integration of forms

Extent and Intensity

2/2/0. 4 credit(s) (plus 2 credits for an exam). Type of Completion: zk (examination).

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: prof. RNDr. Jana Musilová, CSc.
Supplier department: Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science

Timetable

Wed 15:00–16:50 F3,03015, Thu 17:00–18:50 F1 6/1014

Prerequisites

M1100 Mathematical Analysis I && M2100 Mathematical Analysis II
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It includes the theory of Riemannian integration of differential forms, i.e. covariant antisymmetric tensor fields, on subsets of euclidean spaces of arbitrary dimensions. Using differential forms as integrated objects one can define the concept of integral by a more natural way than is that obtained by standardly used methods. (On the other hand, the mentioned more modern approach ought to be considered as "classical one", the frequently used approach being rather "conservative".) The concept of integral is general, including classical path and surface integrals. A basic result of the theory is the general Stokes theorem containing all classical integral theorems as special cases. The attention is also payed to the calculus -- practical calculations of integrals especially those with a physical meaning. The following main goals are followed:

* To show students more general and more effective approach to Riemannian integration theory.
* To connect the general mathematical theory with practical geometric and physical applications and to show their close relation.

Absolving the course a student obtains following abilities and skills:

* General understanding of problems concerning vector and tensor fields as geometric (invariant) objects on euclidean spaces.
* Practical skills with vector and tensor calculus in spaces of general dimensions (algebra) and vector and tensor fields on euclidean spaces (analysis).
* Understanding of the concept of a differential form, practical skills with algebraic and analytic manipulations with differential forms (exterior product, exterior derivative, pullback).
* Understanding of the concept of Riemannian integral of a differential k-form on a k-dimensional integration region in n-dimensional euclidean space (k lower or equal to n), understanding of the concept of path and surface integrals as special cases of the general concept.
* Knowledge of the general Stokes theorem (transforming the integration of a (k-1)-form on the boundary of a k-dimensional integration region M to integration of the exterior derivative of this form on M), its proof and its classical applications (Green theorem, classical Stokes theorem, Gauss-Ostrogradsky theorem).
* Practical skills in calculus of integrals, including the use of general Stokes theorem, calculation of work of a vector field, flow of a vector field through a surface, etc.
* Practical use of the general Stokes theorem for obtaining differential (local) physical laws from their integral (global) versions, e.g. equations of fluid mechanics, Maxwell equations in classical electrodynamics, etc.
* Practical calculations of n-dimensional volumes integrating volume elements (curves, surfaces, bodies).
*

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions.
• 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem.
• 3. Spaces of covariant tensors. Algebraic operations.
• 4. Antisymmetric tensors, exterior product.
• 5. Vector and tensor fields, differential forms.
• 6. Exterior derivative.
• 7. Induced mappings -- tangent mapping.
• 8. Induced mappings -- pullback of forms.
• 9. Integral of a differential form on singular cubes.
• 10. General Stokes theorem.
• 11. Integrals of the first and second type, classical versions of Stokes theorem.
• 12. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. Volume element. Areas and volumes.
• 13. Applications-work of a force field along a curve, flux of a vector field through a surface.
• 14. Practical calculations.

Literature

required literature
• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info

recommended literature
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info

Teaching methods

Lectures: theoretical explanation with practical examples. Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems.

Assessment methods

Teaching: lectures, exercises.
Exam: written test (consisting of two parts: (a) solving problems, (b) test), oral part. In the part (a) students demonstrate the ability of solving extensive and complex problems of linear algebra and geometry. In the part (b) they demonstrate the understanding of important concepts and theorems, as well as the good orientation in the discipline.

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

Study Materials
The course can also be completed outside the examination period.
The course is taught annually.

F3063 Integration of forms

Extent and Intensity

2/2/0. 4 credit(s) (plus 2 credits for an exam). Type of Completion: zk (examination).

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: prof. RNDr. Jana Musilová, CSc.
Supplier department: Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science

Timetable

Mon 8:00–9:50 F4,03017, Thu 8:00–9:50 Fs1 6/1017

Prerequisites

M1100 Mathematical Analysis I && M2100 Mathematical Analysis II
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It includes the theory of Riemannian integration of differential forms, i.e. covariant antisymmetric tensor fields, on subsets of euclidean spaces of arbitrary dimensions. Using differential forms as integrated objects one can define the concept of integral by a more natural way than is that obtained by standardly used methods. (On the other hand, the mentioned more modern approach ought to be considered as "classical one", the frequently used approach being rather "conservative".) The concept of integral is general, including classical path and surface integrals. A basic result of the theory is the general Stokes theorem containing all classical integral theorems as special cases. The attention is also payed to the calculus -- practical calculations of integrals especially those with a physical meaning. The following main goals are followed:

* To show students more general and more effective approach to Riemannian integration theory.
* To connect the general mathematical theory with practical geometric and physical applications and to show their close relation.

Absolving the course a student obtains following abilities and skills:

* General understanding of problems concerning vector and tensor fields as geometric (invariant) objects on euclidean spaces.
* Practical skills with vector and tensor calculus in spaces of general dimensions (algebra) and vector and tensor fields on euclidean spaces (analysis).
* Understanding of the concept of a differential form, practical skills with algebraic and analytic manipulations with differential forms (exterior product, exterior derivative, pullback).
* Understanding of the concept of Riemannian integral of a differential k-form on a k-dimensional integration region in n-dimensional euclidean space (k lower or equal to n), understanding of the concept of path and surface integrals as special cases of the general concept.
* Knowledge of the general Stokes theorem (transforming the integration of a (k-1)-form on the boundary of a k-dimensional integration region M to integration of the exterior derivative of this form on M), its proof and its classical applications (Green theorem, classical Stokes theorem, Gauss-Ostrogradsky theorem).
* Practical skills in calculus of integrals, including the use of general Stokes theorem, calculation of work of a vector field, flow of a vector field through a surface, etc.
* Practical use of the general Stokes theorem for obtaining differential (local) physical laws from their integral (global) versions, e.g. equations of fluid mechanics, Maxwell equations in classical electrodynamics, etc.
* Practical calculations of n-dimensional volumes integrating volume elements (curves, surfaces, bodies).
*

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions.
• 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem.
• 3. Spaces of covariant tensors. Algebraic operations.
• 4. Antisymmetric tensors, exterior product.
• 5. Vector and tensor fields, differential forms.
• 6. Exterior derivative.
• 7. Induced mappings -- tangent mapping.
• 8. Induced mappings -- pullback of forms.
• 9. Integral of a differential form on singular cubes.
• 10. General Stokes theorem.
• 11. Integrals of the first and second type, classical versions of Stokes theorem.
• 12. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. Volume element. Areas and volumes.
• 13. Applications-work of a force field along a curve, flux of a vector field through a surface.
• 14. Practical calculations.

Literature

required literature
• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info

recommended literature
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info

Teaching methods

Lectures: theoretical explanation with practical examples. Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems.

Assessment methods

Teaching: lectures, exercises.
Exam: written test (consisting of two parts: (a) solving problems, (b) test), oral part. In the part (a) students demonstrate the ability of solving extensive and complex problems of linear algebra and geometry. In the part (b) they demonstrate the understanding of important concepts and theorems, as well as the good orientation in the discipline.

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

Study Materials
The course can also be completed outside the examination period.
The course is taught annually.

F3063 Integration of forms

Extent and Intensity

2/2/0. 4 credit(s) (plus 2 credits for an exam). Type of Completion: zk (examination).

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: prof. RNDr. Jana Musilová, CSc.
Supplier department: Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science

Timetable

Thu 8:00–9:50 F1 6/1014, Thu 14:00–15:50 F4,03017

Prerequisites

M1100 Mathematical Analysis I && M2100 Mathematical Analysis II
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)
• Physics (programme PřF, N-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It includes the theory of riemannian integration of differential forms, i.e. covariant antisymmetric tensor fields, on subsets of euclidean spaces of arbitrary dimensions. Using differential forms as integrated objects one can define the concept of integral by a more natural way than is that obtained by standardly used methods. (On the other hand, the mentioned more modern approach ought to be considered as "classical one", the frequently used approach being rather "conservative".) The concept of integral is general, including classical path and surface integrals. A basic result of the theory is the general Stokes theorem containing all classical integral theorems as special cases. The attention is also payed to the calculus -- practical calculations of integrals especially those with a physical meaning. The following main goals are followed:

* To show students more general and more effective approach to riemannian integration theory.
* To connect the general mathematical theory with practical gometric and physical applications and to show their close relation.

Absolving the course a student obtains following abilities and skills:

* General understanding of problems concerning vector and tensor fields as geometric (invariant) objects on euclidean spaces.
* Practical skills with vector and tensor calculus in spaces of general dimensions (algebra) and vector and tensor fields on euclidean spaces (analysis).
* Understanding of the concept of a differential form, practical skills with angebraic and analytic manipulations with differential forms (exterior product, exterior derivative, pullback).
* Understanding of the concept of riemannian integral of a differential k-form on a k-dimensional integration region in n-dimensional euclidean space (k lower or equal to n), understanding of the concept of path and surface integrals as special cases of the general concept.
* Knowledge of the general Stokes theorem (transforming the integration of a (k-1)-formy on the boundary of a k-dimensional integration region M to integration of the exterior derivative of this form on M), its proof and its classical applications (Green theorem, classical Stokes theorem, Gauss-Ostrogradsky theorem).
* Practical skills in calculus of integrals, including the use of general Stokes theorem, calculation of work of a vector field, flow of a vector field through a surface, etc.
* Practical use of the general Stokes theorem for obtaining differential (local) physical laws from their integral (global) versions, e.g. equations of fluid mechanics, Maxwell equations in classical electrodynamics, etc.
* Practical calculations of n-dimensional volumes integrating volume elements (curves, surfaces, bodies).
*

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions.
• 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem.
• 3. Spaces of covariant tensors. Algebraic operations.
• 4. Antisymmetric tensors, exterior product.
• 5. Vector and tensor fields, differential forms.
• 6. Exterior derivative.
• 7. Induced mappings -- tangent mapping.
• 8. Induced mappings -- pullback of forms.
• 9. Integral of a differential form on singular cubes.
• 10. General Stokes theorem.
• 11. Integrals of the first and second type, classical versions of Stokes theorem.
• 12. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. Volume element. Areas and volumes.
• 13. Appliactions-work of a force field along a curve, flux of a vector field through a surface.
• 14. Practical calculations.

Literature

required literature
• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info

recommended literature
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info

Teaching methods

Lectures: theoretical explanation with practical examples. Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems.

Assessment methods

Teaching: lectures, consultative exercises Exam: written test (two parts: (a) solving problems, (b) test) and oral exam.

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

Study Materials
The course can also be completed outside the examination period.
The course is taught annually.

F3063 Integration of forms

Extent and Intensity

2/2/0. 4 credit(s) (plus 2 credits for an exam). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Lenka Czudková, Ph.D. (seminar tutor)

Guaranteed by

prof. RNDr. Michal Lenc, Ph.D.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.

Timetable

Tue 11:00–12:50 F4,03017, Wed 8:00–9:50 F1 6/1014

Prerequisites

M1100 Mathematical Analysis I && M2100 Mathematical Analysis II
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)
• Physics (programme PřF, N-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It includes the theory of riemannian integration of differential forms, i.e. covariant antisymmetric tensor fields, on subsets of euclidean spaces of arbitrary dimensions. Using differential forms as integrated objects one can define the concept of integral by a more natural way than is that obtained by standardly used methods. (On the other hand, the mentioned more modern approach ought to be considered as "classical one", the frequently used approach being rather "conservative".) The concept of integral is general, including classical path and surface integrals. A basic result of the theory is the general Stokes theorem containing all classical integral theorems as special cases. The attention is also payed to the calculus -- practical calculations of integrals especially those with a physical meaning. The following main goals are followed:

* To show students more general and more effective approach to riemannian integration theory.
* To connect the general mathematical theory with practical gometric and physical applications and to show their close relation.

Absolving the course a student obtains following abilities and skills:

* General understanding of problems concerning vector and tensor fields as geometric (invariant) objects on euclidean spaces.
* Practical skills with vector and tensor calculus in spaces of general dimensions (algebra) and vector and tensor fields on euclidean spaces (analysis).
* Understanding of the concept of a differential form, practical skills with angebraic and analytic manipulations with differential forms (exterior product, exterior derivative, pullback).
* Understanding of the concept of riemannian integral of a differential k-form on a k-dimensional integration region in n-dimensional euclidean space (k lower or equal to n), understanding of the concept of path and surface integrals as special cases of the general concept.
* Knowledge of the general Stokes theorem (transforming the integration of a (k-1)-formy on the boundary of a k-dimensional integration region M to integration of the exterior derivative of this form on M), its proof and its classical applications (Green theorem, classical Stokes theorem, Gauss-Ostrogradsky theorem).
* Practical skills in calculus of integrals, including the use of general Stokes theorem, calculation of work of a vector field, flow of a vector field through a surface, etc.
* Practical use of the general Stokes theorem for obtaining differential (local) physical laws from their integral (global) versions, e.g. equations of fluid mechanics, Maxwell equations in classical electrodynamics, etc.
* Practical calculations of n-dimensional volumes integrating volume elements (curves, surfaces, bodies).
*

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions.
• 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem.
• 3. Spaces of covariant tensors. Algebraic operations.
• 4. Antisymmetric tensors, exterior product.
• 5. Vector and tensor fields, differential forms.
• 6. Exterior derivative.
• 7. Induced mappings -- tangent mapping.
• 8. Induced mappings -- pullback of forms.
• 9. Integral of a differential form on singular cubes.
• 10. General Stokes theorem.
• 11. Integrals of the first and second type, classical versions of Stokes theorem.
• 12. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. Volume element. Areas and volumes.
• 13. Appliactions-work of a force field along a curve, flux of a vector field through a surface.
• 14. Practical calculations.

Literature

required literature
• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info

recommended literature
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info

Teaching methods

Lectures: theoretical explanation with practical examples. Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems.

Assessment methods

Teaching: lectures, consultative exercises Exam: written test (two parts: (a) solving problems, (b) test) and oral exam.

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

Study Materials
The course can also be completed outside the examination period.
The course is taught annually.

F3063 Integration of forms

Extent and Intensity

4/2/0. 6 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).

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: prof. RNDr. Jana Musilová, CSc.

Timetable

Mon 13:00–14:50 F3,03015, Mon 17:00–18:50 F4,03017, Thu 14:00–15:50 F2 6/2012

Prerequisites

M1100 Mathematical Analysis I && M2100 Mathematical Analysis II
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)
• Physics (programme PřF, M-FY)
• Physics (programme PřF, N-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It includes the theory of riemannian integration of differential forms, i.e. covariant antisymmetric tensor fields, on subsets of euclidean spaces of arbitrary dimensions. Using differential forms as integrated objects one can define the concept of integral by a more natural way than is that obtained by standardly used methods. (On the other hand, the mentioned more modern approach ought to be considered as "classical one", the frequently used approach being rather "conservative".) The concept of integral is general, including classical path and surface integrals. A basic result of the theory is the general Stokes theorem containing all classical integral theorems as special cases. The attention is also payed to the calculus -- practical calculations of integrals especially those with a physical meaning. The following main goals are followed:

* To show students more general and more effective approach to riemannian integration theory.
* To connect the general mathematical theory with practical gometric and physical applications and to show their close relation.

Absolving the course a student obtains following abilities and skills:

* General understanding of problems concerning vector and tensor fields as geometric (invariant) objects on euclidean spaces.
* Practical skills with vector and tensor calculus in spaces of general dimensions (algebra) and vector and tensor fields on euclidean spaces (analysis).
* Understanding of the concept of a differential form, practical skills with angebraic and analytic manipulations with differential forms (exterior product, exterior derivative, pullback).
* Understanding of the concept of riemannian integral of a differential k-form on a k-dimensional integration region in n-dimensional euclidean space (k lower or equal to n), understanding of the concept of path and surface integrals as special cases of the general concept.
* Knowledge of the general Stokes theorem (transforming the integration of a (k-1)-formy on the boundary of a k-dimensional integration region M to integration of the exterior derivative of this form on M), its proof and its classical applications (Green theorem, classical Stokes theorem, Gauss-Ostrogradsky theorem).
* Practical skills in calculus of integrals, including the use of general Stokes theorem, calculation of work of a vector field, flow of a vector field through a surface, etc.
* Practical use of the general Stokes theorem for obtaining differential (local) physical laws from their integral (global) versions, e.g. equations of fluid mechanics, Maxwell equations in classical electrodynamics, etc.
* Practical calculations of n-dimensional volumes integrating volume elements (curves, surfaces, bodies).
* Understanding of fundamentals of convergence theory of number series as well as series of functions.
* Understanding the difference of definition of convergence and uniform convergence of series of functions.
* Practical calculations concerning the convergence of power series and Fourier series.

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions.
• 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem.
• 3. Spaces of covariant tensors. Algebraic operations.
• 4. Vector and tensor fields, differential forms.
• 5. Exterior derivative. Pullback.
• 6. Integral of a differential form on singular cubes.
• General Stokes theorem.
• 8. Integrals of the first and second type, classical versions of Stokes theorem.
• 9. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. Volume element. Areas and volumes.
• 10. Appliactions-work of a force field along a curve, flux of a vector field through a surface.
• 11. Fundamentals of theory of convergence of infinite number series.
• 12. Series of functions, convergence, uniform convergence.
• 13. Power and Fourier series.
• 14. Applications of infinite series theory: Solving differential equations, approximation of functions, physical applications.

Literature

• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info
• DOŠLÁ, Zuzana and Vítězslav NOVÁK. Nekonečné řady. 2. vyd. Brno: Masarykova univerzita, 2007, iv, 113. ISBN 9788021043343. info

Teaching methods

Lectures: theoretical explanation with practical examples. Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems.

Assessment methods

Teaching: lectures, consultative exercises Exam: written test (two parts: (a) solving problems, (b) test) and oral exam.

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

Study Materials
The course can also be completed outside the examination period.
The course is taught annually.

F3063 Integration of forms

Extent and Intensity

4/2/0. 6 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Lenka Czudková, Ph.D. (seminar tutor)

Guaranteed by

prof. RNDr. Michal Lenc, Ph.D.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.

Timetable

Wed 7:00–10:50 F3,03015, Fri 8:00–9:50 F3,03015

Prerequisites

M1100 Mathematical Analysis I && M2100 Mathematical Analysis II
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)
• Physics (programme PřF, M-FY)
• Physics (programme PřF, N-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It includes the theory of riemannian integration of differential forms, i.e. covariant antisymmetric tensor fields, on subsets of euclidean spaces of arbitrary dimensions. Using differential forms as integrated objects one can define the concept of integral by a more natural way than is that obtained by standardly used methods. (On the other hand, the mentioned more modern approach ought to be considered as "classical one", the frequently used approach being rather "conservative".) The concept of integral is general, including classical path and surface integrals. A basic result of the theory is the general Stokes theorem containing all classical integral theorems as special cases. The attention is also payed to the calculus -- practical calculations of integrals especially those with a physical meaning. The following main goals are followed:

* To show students more general and more effective approach to riemannian integration theory.
* To connect the general mathematical theory with practical gometric and physical applications and to show their close relation.

Absolving the course a student obtains following abilities and skills:

* General understanding of problems concerning vector and tensor fields as geometric (invariant) objects on euclidean spaces.
* Practical skills with vector and tensor calculus in spaces of general dimensions (algebra) and vector and tensor fields on euclidean spaces (analysis).
* Understanding of the concept of a differential form, practical skills with angebraic and analytic manipulations with differential forms (exterior product, exterior derivative, pullback).
* Understanding of the concept of riemannian integral of a differential k-form on a k-dimensional integration region in n-dimensional euclidean space (k lower or equal to n), understanding of the concept of path and surface integrals as special cases of the general concept.
* Knowledge of the general Stokes theorem (transforming the integration of a (k-1)-formy on the boundary of a k-dimensional integration region M to integration of the exterior derivative of this form on M), its proof and its classical applications (Green theorem, classical Stokes theorem, Gauss-Ostrogradsky theorem).
* Practical skills in calculus of integrals, including the use of general Stokes theorem, calculation of work of a vector field, flow of a vector field through a surface, etc.
* Practical use of the general Stokes theorem for obtaining differential (local) physical laws from their integral (global) versions, e.g. equations of fluid mechanics, Maxwell equations in classical electrodynamics, etc.
* Practical calculations of n-dimensional volumes integrating volume elements (curves, surfaces, bodies).
* Understanding of fundamentals of convergence theory of number series as well as series of functions.
* Understanding the difference of definition of convergence and uniform convergence of series of functions.
* Practical calculations concerning the convergence of power series and Fourier series.

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions.
• 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem.
• 3. Spaces of covariant tensors. Algebraic operations.
• 4. Vector and tensor fields, differential forms.
• 5. Exterior derivative. Pullback.
• 6. Integral of a differential form on singular cubes.
• General Stokes theorem.
• 8. Integrals of the first and second type, classical versions of Stokes theorem.
• 9. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. Volume element. Areas and volumes.
• 10. Appliactions-work of a force field along a curve, flux of a vector field through a surface.
• 11. Fundamentals of theory of convergence of infinite number series.
• 12. Series of functions, convergence, uniform convergence.
• 13. Power and Fourier series.
• 14. Applications of infinite series theory: Solving differential equations, approximation of functions, physical applications.

Literature

• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info
• DOŠLÁ, Zuzana and Vítězslav NOVÁK. Nekonečné řady. 2. vyd. Brno: Masarykova univerzita, 2007, iv, 113. ISBN 9788021043343. info

Teaching methods

Lectures: theoretical explanation with practical examples. Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems.

Assessment methods

Teaching: lectures, consultative exercises Exam: written test (two parts: (a) solving problems, (b) test) and oral exam.

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

Study Materials
The course can also be completed outside the examination period.
The course is taught annually.

F3063 Integration of forms

Extent and Intensity

3/2/0. 4 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Lenka Czudková, Ph.D. (seminar tutor)

Guaranteed by

prof. RNDr. Michal Lenc, Ph.D.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.

Timetable

Thu 13:00–15:50 F4,03017

• Timetable of Seminar Groups:

F3063/01: Thu 8:00–9:50 F2 6/2012, L. Czudková
F3063/02: Tue 9:00–10:50 F4,03017, L. Czudková

Prerequisites

M1100 Mathematical Analysis I && M2100 Mathematical Analysis II
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)
• Physics (programme PřF, M-FY)
• Physics (programme PřF, N-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It includes the theory of riemannian integration of differential forms, i.e. covariant antisymmetric tensor fields, on subsets of euclidean spaces of arbitrary dimensions. Using differential forms as integrated objects one can define the concept of integral by a more natural way than is that obtained by standardly used methods. (On the other hand, the mentioned more modern approach ought to be considered as "classical one", the frequently used approach being rather "conservative".) The concept of integral is general, including classical path and surface integrals. A basic result of the theory is the general Stokes theorem containing all classical integral theorems as special cases. The attention is also payed to the calculus -- practical calculations of integrals especially those with a physical meaning. The following main goals are followed:

* To show students more general and more effective approach to riemannian integration theory.
* To connect the general mathematical theory with practical gometric and physical applications and to show their close relation.

Absolving the course a student obtains following abilities and skills:

* General understanding of problems concerning vector and tensor fields as geometric (invariant) objects on euclidean spaces.
* Practical skills with vector and tensor calculus in spaces of general dimensions (algebra) and vector and tensor fields on euclidean spaces (analysis).
* Understanding of the concept of a differential form, practical skills with angebraic and analytic manipulations with differential forms (exterior product, exterior derivative, pullback).
* Understanding of the concept of riemannian integral of a differential k-form on a k-dimensional integration region in n-dimensional euclidean space (k lower or equal to n), understanding of the concept of path and surface integrals as special cases of the general concept.
* Knowledge of the general Stokes theorem (transforming the integration of a (k-1)-formy on the boundary of a k-dimensional integration region M to integration of the exterior derivative of this form on M), its proof and its classical applications (Green theorem, classical Stokes theorem, Gauss-Ostrogradsky theorem).
* Practical skills in calculus of integrals, including the use of general Stokes theorem, calculation of work of a vector field, flow of a vector field through a surface, etc.
* Practical use of the general Stokes theorem for obtaining differential (local) physical laws from their integral (global) versions, e.g. equations of fluid mechanics, Maxwell equations in classical electrodynamics, etc.
* Practical calculations of n-dimensional volumes integrating volume elements (curves, surfaces, bodies).

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions.
• 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem.
• 3. Generalization of the integral - decomposition of unity. 4. Spaces of covariant tensors.
• 5. Vector and tensor fields, differential forms.
• 6. Exterior product, exterior derivative.
• 7. Pullback.
• 8. Integral of a differential form on singular cubes.
• 9. General Stokes theorem.
• 10. Integrals of the first and second type, classical versions of Stokes theorem.
• 11. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects.
• 12. Appliactions-work of a force field along a curve, flux of a vector field through a surface.
• 13. Volume element. Areas and volumes.

Literature

• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info

Assessment methods

Teaching: lectures, consultative exercises Exam: written test (two parts: (a) solving problems, (b) test) and oral exam.

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

The course can also be completed outside the examination period.
The course is taught annually.

F3063 Integration of forms

Extent and Intensity

3/2/0. 4 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Lenka Czudková, Ph.D. (seminar tutor)

Guaranteed by

prof. RNDr. Michal Lenc, Ph.D.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.

Timetable

Thu 7:00–9:50 F3,03015, Fri 13:00–14:50 F3,03015

Prerequisites

M1100 Mathematical Analysis I && M2100 Mathematical Analysis II
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)
• Physics (programme PřF, M-FY)
• Physics (programme PřF, N-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It contains the theory of riemannian integral of differential forms, i.e. covariant tensor fields, on subsets of n-dimensional euclidean space. Using differential forms as integrated objects, the definition of the integral is obtained by the more natural way than the "classical" one. It includes classical line integrals and surface integrals. Its fundamental result - the general Stokes theorem - includes all classical integral theorems as special cases. The practical calculus of integrals with physical meaning is emphasized.

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions. 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem. 3. Generalization of the integral - decomposition of unity. 4. Spaces of covariant tensors. 5. Vector and tensor fields, differential forms. 6. Exterior product, exterior derivative. 7. Pullback. 8. Integral of a differential form on singular cubes. 9. General Stokes theorem. 10. Integrals of the first and second type, classical versions of Stokes theorem. 11. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. 12. Appliactions-work of a force field along a curve, flux of a vector field through a surface. 13. Volume element. Areas and volumes.

Literature

• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info

Assessment methods (in Czech)

Výuka: přednáška, klasické cvičení Zkouška: písemná (dvě části: (a) příklady, (b) test) a ústní

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

The course can also be completed outside the examination period.
The course is taught annually.

F3063 Integration of forms

Extent and Intensity

3/2/0. 4 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Lenka Czudková, Ph.D. (seminar tutor)

Guaranteed by

prof. RNDr. Michal Lenc, Ph.D.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.

Timetable

Wed 11:00–13:50 F3,03015, Thu 8:00–9:50 F4,03017

Prerequisites

M1100 Mathematical Analysis I && M2100 Mathematical Analysis II
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)
• Physics (programme PřF, M-FY)
• Physics (programme PřF, N-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It contains the theory of riemannian integral of differential forms, i.e. covariant tensor fields, on subsets of n-dimensional euclidean space. Using differential forms as integrated objects, the definition of the integral is obtained by the more natural way than the "classical" one. It includes classical line integrals and surface integrals. Its fundamental result - the general Stokes theorem - includes all classical integral theorems as special cases. The practical calculus of integrals with physical meaning is emphasized.

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions. 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem. 3. Generalization of the integral - decomposition of unity. 4. Spaces of covariant tensors. 5. Vector and tensor fields, differential forms. 6. Exterior product, exterior derivative. 7. Pullback. 8. Integral of a differential form on singular cubes. 9. General Stokes theorem. 10. Integrals of the first and second type, classical versions of Stokes theorem. 11. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. 12. Appliactions-work of a force field along a curve, flux of a vector field through a surface. 13. Volume element. Areas and volumes.

Literature

• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info

Assessment methods (in Czech)

Výuka: přednáška, klasické cvičení Zkouška: písemná (dvě části: (a) příklady, (b) test) a ústní

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

The course can also be completed outside the examination period.
The course is taught annually.

F3063 Integration of forms

Extent and Intensity

3/2/0. 4 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Lenka Czudková, Ph.D. (seminar tutor)

Guaranteed by

prof. RNDr. Michal Lenc, Ph.D.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.

Timetable

Wed 10:00–11:50 F4,03017, Thu 7:00–9:50 F4,03017

Prerequisites

M1100 Mathematical Analysis I && M2100 Mathematical Analysis II
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)
• Physics (programme PřF, M-FY)
• Physics (programme PřF, N-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It contains the theory of riemannian integral of differential forms, i.e. covariant tensor fields, on subsets of n-dimensional euclidean space. Using differential forms as integrated objects, the definition of the integral is obtained by the more natural way than the "classical" one. It includes classical line integrals and surface integrals. Its fundamental result - the general Stokes theorem - includes all classical integral theorems as special cases. The practical calculus of integrals with physical meaning is emphasized.

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions. 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem. 3. Generalization of the integral - decomposition of unity. 4. Spaces of covariant tensors. 5. Vector and tensor fields, differential forms. 6. Exterior product, exterior derivative. 7. Pullback. 8. Integral of a differential form on singular cubes. 9. General Stokes theorem. 10. Integrals of the first and second type, classical versions of Stokes theorem. 11. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. 12. Appliactions-work of a force field along a curve, flux of a vector field through a surface. 13. Volume element. Areas and volumes.

Literature

• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info

Assessment methods (in Czech)

Výuka: přednáška, klasické cvičení Zkouška: písemná (dvě části: (a) příklady, (b) test) a ústní

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

The course can also be completed outside the examination period.
The course is taught annually.

F3063 Integration of forms

Extent and Intensity

3/2/0. 4 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Lenka Czudková, Ph.D. (seminar tutor)

Guaranteed by

prof. RNDr. Michal Lenc, Ph.D.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.

Timetable

Thu 7:00–9:50 F3,03015

• Timetable of Seminar Groups:

F3063/01: Thu 16:00–17:50 F1 6/1014, L. Czudková
F3063/02: Thu 18:00–19:50 F3,03015, L. Czudková

Prerequisites

M1100 Mathematical Analysis I && M2100 Mathematical Analysis II
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)
• Physics (programme PřF, M-FY)
• Physics (programme PřF, N-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It contains the theory of riemannian integral of differential forms, i.e. covariant tensor fields, on subsets of n-dimensional euclidean space. Using differential forms as integrated objects, the definition of the integral is obtained by the more natural way than the "classical" one. It includes classical line integrals and surface integrals. Its fundamental result - the general Stokes theorem - includes all classical integral theorems as special cases. The practical calculus of integrals with physical meaning is emphasized.

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions. 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem. 3. Generalization of the integral - decomposition of unity. 4. Spaces of covariant tensors. 5. Vector and tensor fields, differential forms. 6. Exterior product, exterior derivative. 7. Pullback. 8. Integral of a differential form on singular cubes. 9. General Stokes theorem. 10. Integrals of the first and second type, classical versions of Stokes theorem. 11. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. 12. Appliactions-work of a force field along a curve, flux of a vector field through a surface. 13. Volume element. Areas and volumes.

Literature

• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info

Assessment methods (in Czech)

Výuka: přednáška, klasické cvičení Zkouška: písemná (dvě části: (a) příklady, (b) test) a ústní

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

The course can also be completed outside the examination period.
The course is taught annually.

F3063 Integration of forms

Extent and Intensity

3/2/0. 4 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Tomáš Radlička, Ph.D. (seminar tutor)
Mgr. Lenka Czudková, Ph.D. (seminar tutor)

Guaranteed by

prof. RNDr. Michal Lenc, Ph.D.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.

Prerequisites

M1100 Mathematical Analysis I && M2100 Mathematical Analysis II
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)
• Physics (programme PřF, M-FY)
• Physics (programme PřF, N-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It contains the theory of riemannian integral of differential forms, i.e. covariant tensor fields, on subsets of n-dimensional euclidean space. Using differential forms as integrated objects, the definition of the integral is obtained by the more natural way than the "classical" one. It includes classical line integrals and surface integrals. Its fundamental result - the general Stokes theorem - includes all classical integral theorems as special cases. The practical calculus of integrals with physical meaning is emphasized.

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions. 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem. 3. Generalization of the integral - decomposition of unity. 4. Spaces of covariant tensors. 5. Vector and tensor fields, differential forms. 6. Exterior product, exterior derivative. 7. Pullback. 8. Integral of a differential form on singular cubes. 9. General Stokes theorem. 10. Integrals of the first and second type, classical versions of Stokes theorem. 11. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. 12. Appliactions-work of a force field along a curve, flux of a vector field through a surface. 13. Volume element. Areas and volumes.

Literature

• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info

Assessment methods (in Czech)

Výuka: přednáška, klasické cvičení Zkouška: písemná (dvě části: (a) příklady, (b) test) a ústní

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

The course can also be completed outside the examination period.
The course is taught annually.
The course is taught every week.

F3063 Integration of forms

Extent and Intensity

3/2/0. 4 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Tomáš Radlička, Ph.D. (seminar tutor)
Mgr. Ing. Jitka Janová, 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

M1100 Mathematical Analysis I && M2100 Mathematical Analysis II
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)
• Physics (programme PřF, M-FY)
• Physics (programme PřF, N-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It contains the theory of riemannian integral of differential forms, i.e. covariant tensor fields, on subsets of n-dimensional euclidean space. Using differential forms as integrated objects, the definition of the integral is obtained by the more natural way than the "classical" one. It includes classical line integrals and surface integrals. Its fundamental result - the general Stokes theorem - includes all classical integral theorems as special cases. The practical calculus of integrals with physical meaning is emphasized.

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions. 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem. 3. Generalization of the integral - decomposition of unity. 4. Spaces of covariant tensors. 5. Vector and tensor fields, differential forms. 6. Exterior product, exterior derivative. 7. Pullback. 8. Integral of a differential form on singular cubes. 9. General Stokes theorem. 10. Integrals of the first and second type, classical versions of Stokes theorem. 11. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. 12. Appliactions-work of a force field along a curve, flux of a vector field through a surface. 13. Volume element. 14. Volume of a riemannian manifold.

Literature

• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info

Assessment methods (in Czech)

Výuka: přednáška, klasické cvičení Zkouška: písemná (dvě části: (a) příklady, (b) test) a ústní

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

The course can also be completed outside the examination period.
The course is taught annually.
The course is taught every week.

F3063 Integration of forms

Extent and Intensity

3/2/0. 6 credit(s). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Pavla Musilová, Ph.D. (seminar tutor)

Guaranteed by

prof. RNDr. Michal Lenc, Ph.D.
Department of Plasma Physics and Technology – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.

Prerequisites

M1050 Diff. and integr. calculus && M2050 Diff. calc. and diff. equat.
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)
• Physics (programme PřF, M-FY)
• Physics (programme PřF, N-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It contains the theory of riemannian integral of differential forms, i.e. covariant tensor fields, on subsets of n-dimensional euclidean space. Using differential forms as integrated objects, the definition of the integral is obtained by the more natural way than the "classical" one. It includes classical line integrals and surface integrals. Its fundamental result - the general Stokes theorem - includes all classical integral theorems as special cases. The practical calculus of integrals with physical meaning is emphasized.

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions. 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem. 3. Generalization of the integral - decomposition of unity. 4. Spaces of covariant tensors. 5. Vector and tensor fields, differential forms. 6. Exterior product, exterior derivative. 7. Pullback. 8. Integral of a differential form on singular cubes. 9. General Stokes theorem. 10. Integrals of the first and second type, classical versions of Stokes theorem. 11. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. 12. Appliactions-work of a force field along a curve, flux of a vector field through a surface. 13. Volume element. 14. Volume of a riemannian manifold.

Literature

• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info

Assessment methods (in Czech)

Výuka: přednáška, klasické cvičení Zkouška: písemná (dvě části: (a) příklady, (b) test) a ústní

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

The course can also be completed outside the examination period.
The course is taught annually.
The course is taught every week.

F3063 Integration of forms

Extent and Intensity

3/2/0. 6 credit(s). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Michael Krbek, Ph.D. (seminar tutor)
Mgr. Pavla Musilová, Ph.D. (seminar tutor)

Guaranteed by

doc. RNDr. Zdeněk Bochníček, Dr.
Department of Plasma Physics and Technology – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.

Prerequisites (in Czech)

M1050 Diff. and integr. calculus && M2050 Diff. calc. and diff. equat.

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, B-FY)
• Physics (programme PřF, M-FY)
• Physics (programme PřF, N-FY)

Course objectives

Fundamental course of mathematical analysis for students of physics. Riemann integral: elements of topology, integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, decomposition of unity,transformation theorem. Integral of differential forms on euclidean spaces: Tensors, vector and tensor fields, differential forms on euclidean spaces, exterior product, exterior derivative, integral of a differential form on singular cubes, general Stokes theorem, volume element, integrals of the first and second type, classical versions of Stokes theorem, applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects, work of a force field along a curve, flux of a vector field through a surface, volume of a riemannian manifold.

Language of instruction

Czech

Further Comments

The course can also be completed outside the examination period.
The course is taught annually.
The course is taught every week.

F3063 Integration of forms

Extent and Intensity

3/2/0. 6 credit(s). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Michael Krbek, Ph.D. (seminar tutor)
Mgr. Pavla Musilová, Ph.D. (seminar tutor)

Guaranteed by

doc. RNDr. Zdeněk Bochníček, Dr.
Department of Plasma Physics and Technology – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.

Prerequisites (in Czech)

M1050 Diff. and integr. calculus && M2050 Diff. calc. and diff. equat.

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, B-FY)
• Physics (programme PřF, M-FY)
• Physics (programme PřF, N-FY)

Syllabus

• Fundamental course of mathematical analysis for students of physics. Riemann integral: elements of topology, integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, decomposition of unity,transformation theorem. Integral of differential forms on euclidean spaces: Tensors, vector and tensor fields, differential forms on euclidean spaces, exterior product, exterior derivative, integral of a differential form on singular cubes, general Stokes theorem, volume element, integrals of the first and second type, classical versions of Stokes theorem, applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects, work of a force field along a curve, flux of a vector field through a surface, volume of a riemannian manifold.

Language of instruction

Czech

Further Comments

The course is taught annually.
The course is taught every week.

F3063 Integration of forms

The course is not taught in Spring 2024

Extent and Intensity

2/2/0. 4 credit(s). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Michael Krbek, Ph.D. (seminar tutor)

Guaranteed by

prof. RNDr. Jana Musilová, CSc.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.
Supplier department: Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science

Prerequisites

( M1100 Mathematical Analysis I && M2100 Mathematical Analysis II )||( M1100F Mathematical Analysis I && M2100F Mathematical Analysis II )
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It includes the theory of Riemannian integration of differential forms, i.e. covariant antisymmetric tensor fields, on subsets of euclidean spaces of arbitrary dimensions. Using differential forms as integrated objects one can define the concept of integral by a more natural way than is that obtained by standardly used methods. (On the other hand, the mentioned more modern approach ought to be considered as "classical one", the frequently used approach being rather "conservative".) The concept of integral is general, including classical path and surface integrals. A basic result of the theory is the general Stokes theorem containing all classical integral theorems as special cases. The attention is also payed to the calculus -- practical calculations of integrals especially those with a physical meaning. The following main goals are followed:

* To show students more general and more effective approach to Riemannian integration theory.
* To connect the general mathematical theory with practical geometric and physical applications and to show their close relation.

Learning outcomes

Absolving the course a student obtains following abilities and skills:

* General understanding of problems concerning vector and tensor fields as geometric (invariant) objects on euclidean spaces.
* Practical skills with vector and tensor calculus in spaces of general dimensions (algebra) and vector and tensor fields on euclidean spaces (analysis).
* Understanding of the concept of a differential form, practical skills with algebraic and analytic manipulations with differential forms (exterior product, exterior derivative, pullback).
* Understanding of the concept of Riemannian integral of a differential k-form on a k-dimensional integration region in n-dimensional euclidean space (k lower or equal to n), understanding of the concept of path and surface integrals as special cases of the general concept.
* Knowledge of the general Stokes theorem (transforming the integration of a (k-1)-form on the boundary of a k-dimensional integration region M to integration of the exterior derivative of this form on M), its proof and its classical applications (Green theorem, classical Stokes theorem, Gauss-Ostrogradsky theorem).
* Practical skills in calculus of integrals, including the use of general Stokes theorem, calculation of work of a vector field, flow of a vector field through a surface, etc.
* Practical use of the general Stokes theorem for obtaining differential (local) physical laws from their integral (global) versions, e.g. equations of fluid mechanics, Maxwell equations in classical electrodynamics, etc.
* Practical calculations of n-dimensional volumes integrating volume elements (curves, surfaces, bodies).
*

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions.
• 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem.
• 3. Spaces of covariant tensors. Algebraic operations.
• 4. Antisymmetric tensors, exterior product.
• 5. Vector and tensor fields, differential forms.
• 6. Exterior derivative.
• 7. Induced mappings -- tangent mapping.
• 8. Induced mappings -- pullback of forms.
• 9. Integral of a differential form on singular cubes.
• 10. General Stokes theorem.
• 11. Integrals of the first and second type, classical versions of Stokes theorem.
• 12. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. Volume element. Areas and volumes.
• 13. Applications-work of a force field along a curve, flux of a vector field through a surface.
• 14. Practical calculations.

Literature

required literature
• MUSILOVÁ, Jana and Pavla MUSILOVÁ. Matematika : pro porozumění i praxi : netradiční výklad tradičních témat vysokoškolské matematiky. První vydání. Brno: VUTIUM, 2017, xv, 365. ISBN 9788021455030. info

recommended literature
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info

Teaching methods

Lectures: theoretical explanation with practical examples. Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems.

Assessment methods

Teaching: lectures, exercises.
Exam: written test (consisting of two parts: (a) solving problems, (b) test), oral part. In the part (a) students demonstrate the ability of solving extensive and complex problems of linear algebra and geometry. In the part (b) they demonstrate the understanding of important concepts and theorems, as well as the good orientation in the discipline.

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

The course can also be completed outside the examination period.
The course is taught once in two years.
The course is taught every week.
General note: S.

F3063 Integration of forms

The information about the term spring 2012 - acreditation is not made public

Extent and Intensity

2/2/0. 4 credit(s) (plus 2 credits for an exam). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Lenka Czudková, Ph.D. (seminar tutor)

Guaranteed by

prof. RNDr. Michal Lenc, Ph.D.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.
Supplier department: Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science

Prerequisites

M1100 Mathematical Analysis I && M2100 Mathematical Analysis II
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)
• Physics (programme PřF, N-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It includes the theory of riemannian integration of differential forms, i.e. covariant antisymmetric tensor fields, on subsets of euclidean spaces of arbitrary dimensions. Using differential forms as integrated objects one can define the concept of integral by a more natural way than is that obtained by standardly used methods. (On the other hand, the mentioned more modern approach ought to be considered as "classical one", the frequently used approach being rather "conservative".) The concept of integral is general, including classical path and surface integrals. A basic result of the theory is the general Stokes theorem containing all classical integral theorems as special cases. The attention is also payed to the calculus -- practical calculations of integrals especially those with a physical meaning. The following main goals are followed:

* To show students more general and more effective approach to riemannian integration theory.
* To connect the general mathematical theory with practical gometric and physical applications and to show their close relation.

Absolving the course a student obtains following abilities and skills:

* General understanding of problems concerning vector and tensor fields as geometric (invariant) objects on euclidean spaces.
* Practical skills with vector and tensor calculus in spaces of general dimensions (algebra) and vector and tensor fields on euclidean spaces (analysis).
* Understanding of the concept of a differential form, practical skills with angebraic and analytic manipulations with differential forms (exterior product, exterior derivative, pullback).
* Understanding of the concept of riemannian integral of a differential k-form on a k-dimensional integration region in n-dimensional euclidean space (k lower or equal to n), understanding of the concept of path and surface integrals as special cases of the general concept.
* Knowledge of the general Stokes theorem (transforming the integration of a (k-1)-formy on the boundary of a k-dimensional integration region M to integration of the exterior derivative of this form on M), its proof and its classical applications (Green theorem, classical Stokes theorem, Gauss-Ostrogradsky theorem).
* Practical skills in calculus of integrals, including the use of general Stokes theorem, calculation of work of a vector field, flow of a vector field through a surface, etc.
* Practical use of the general Stokes theorem for obtaining differential (local) physical laws from their integral (global) versions, e.g. equations of fluid mechanics, Maxwell equations in classical electrodynamics, etc.
* Practical calculations of n-dimensional volumes integrating volume elements (curves, surfaces, bodies).
* Understanding of fundamentals of convergence theory of number series as well as series of functions.
* Understanding the difference of definition of convergence and uniform convergence of series of functions.
* Practical calculations concerning the convergence of power series and Fourier series.

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions.
• 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem.
• 3. Spaces of covariant tensors. Algebraic operations.
• 4. Vector and tensor fields, differential forms.
• 5. Exterior derivative. Pullback.
• 6. Integral of a differential form on singular cubes.
• General Stokes theorem.
• 8. Integrals of the first and second type, classical versions of Stokes theorem.
• 9. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. Volume element. Areas and volumes.
• 10. Appliactions-work of a force field along a curve, flux of a vector field through a surface.
• 11. Fundamentals of theory of convergence of infinite number series.
• 12. Series of functions, convergence, uniform convergence.
• 13. Power and Fourier series.
• 14. Applications of infinite series theory: Solving differential equations, approximation of functions, physical applications.

Literature

• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info
• DOŠLÁ, Zuzana and Vítězslav NOVÁK. Nekonečné řady. 2. vyd. Brno: Masarykova univerzita, 2007, iv, 113. ISBN 9788021043343. info

Teaching methods

Lectures: theoretical explanation with practical examples. Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems.

Assessment methods

Teaching: lectures, consultative exercises Exam: written test (two parts: (a) solving problems, (b) test) and oral exam.

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

The course can also be completed outside the examination period.
The course is taught annually.
The course is taught every week.

F3063 Integration of forms

The information about the term Autumn 2011 - acreditation is not made public

Extent and Intensity

4/2/0. 6 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).

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: prof. RNDr. Jana Musilová, CSc.

Prerequisites

M1100 Mathematical Analysis I && M2100 Mathematical Analysis II
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)
• Physics (programme PřF, M-FY)
• Physics (programme PřF, N-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It includes the theory of riemannian integration of differential forms, i.e. covariant antisymmetric tensor fields, on subsets of euclidean spaces of arbitrary dimensions. Using differential forms as integrated objects one can define the concept of integral by a more natural way than is that obtained by standardly used methods. (On the other hand, the mentioned more modern approach ought to be considered as "classical one", the frequently used approach being rather "conservative".) The concept of integral is general, including classical path and surface integrals. A basic result of the theory is the general Stokes theorem containing all classical integral theorems as special cases. The attention is also payed to the calculus -- practical calculations of integrals especially those with a physical meaning. The following main goals are followed:

* To show students more general and more effective approach to riemannian integration theory.
* To connect the general mathematical theory with practical gometric and physical applications and to show their close relation.

Absolving the course a student obtains following abilities and skills:

* General understanding of problems concerning vector and tensor fields as geometric (invariant) objects on euclidean spaces.
* Practical skills with vector and tensor calculus in spaces of general dimensions (algebra) and vector and tensor fields on euclidean spaces (analysis).
* Understanding of the concept of a differential form, practical skills with angebraic and analytic manipulations with differential forms (exterior product, exterior derivative, pullback).
* Understanding of the concept of riemannian integral of a differential k-form on a k-dimensional integration region in n-dimensional euclidean space (k lower or equal to n), understanding of the concept of path and surface integrals as special cases of the general concept.
* Knowledge of the general Stokes theorem (transforming the integration of a (k-1)-formy on the boundary of a k-dimensional integration region M to integration of the exterior derivative of this form on M), its proof and its classical applications (Green theorem, classical Stokes theorem, Gauss-Ostrogradsky theorem).
* Practical skills in calculus of integrals, including the use of general Stokes theorem, calculation of work of a vector field, flow of a vector field through a surface, etc.
* Practical use of the general Stokes theorem for obtaining differential (local) physical laws from their integral (global) versions, e.g. equations of fluid mechanics, Maxwell equations in classical electrodynamics, etc.
* Practical calculations of n-dimensional volumes integrating volume elements (curves, surfaces, bodies).
* Understanding of fundamentals of convergence theory of number series as well as series of functions.
* Understanding the difference of definition of convergence and uniform convergence of series of functions.
* Practical calculations concerning the convergence of power series and Fourier series.

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions.
• 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem.
• 3. Spaces of covariant tensors. Algebraic operations.
• 4. Vector and tensor fields, differential forms.
• 5. Exterior derivative. Pullback.
• 6. Integral of a differential form on singular cubes.
• General Stokes theorem.
• 8. Integrals of the first and second type, classical versions of Stokes theorem.
• 9. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. Volume element. Areas and volumes.
• 10. Appliactions-work of a force field along a curve, flux of a vector field through a surface.
• 11. Fundamentals of theory of convergence of infinite number series.
• 12. Series of functions, convergence, uniform convergence.
• 13. Power and Fourier series.
• 14. Applications of infinite series theory: Solving differential equations, approximation of functions, physical applications.

Literature

• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info
• DOŠLÁ, Zuzana and Vítězslav NOVÁK. Nekonečné řady. 2. vyd. Brno: Masarykova univerzita, 2007, iv, 113. ISBN 9788021043343. info

Teaching methods

Lectures: theoretical explanation with practical examples. Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems.

Assessment methods

Teaching: lectures, consultative exercises Exam: written test (two parts: (a) solving problems, (b) test) and oral exam.

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

The course can also be completed outside the examination period.
The course is taught annually.
The course is taught every week.

F3063 Integration of forms

Extent and Intensity

4/2/0. 6 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).

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: prof. RNDr. Jana Musilová, CSc.

Prerequisites

M1100 Mathematical Analysis I && M2100 Mathematical Analysis II
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)
• Physics (programme PřF, M-FY)
• Physics (programme PřF, N-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It includes the theory of riemannian integration of differential forms, i.e. covariant antisymmetric tensor fields, on subsets of euclidean spaces of arbitrary dimensions. Using differential forms as integrated objects one can define the concept of integral by a more natural way than is that obtained by standardly used methods. (On the other hand, the mentioned more modern approach ought to be considered as "classical one", the frequently used approach being rather "conservative".) The concept of integral is general, including classical path and surface integrals. A basic result of the theory is the general Stokes theorem containing all classical integral theorems as special cases. The attention is also payed to the calculus -- practical calculations of integrals especially those with a physical meaning. The following main goals are followed:

* To show students more general and more effective approach to riemannian integration theory.
* To connect the general mathematical theory with practical gometric and physical applications and to show their close relation.

Absolving the course a student obtains following abilities and skills:

* General understanding of problems concerning vector and tensor fields as geometric (invariant) objects on euclidean spaces.
* Practical skills with vector and tensor calculus in spaces of general dimensions (algebra) and vector and tensor fields on euclidean spaces (analysis).
* Understanding of the concept of a differential form, practical skills with angebraic and analytic manipulations with differential forms (exterior product, exterior derivative, pullback).
* Understanding of the concept of riemannian integral of a differential k-form on a k-dimensional integration region in n-dimensional euclidean space (k lower or equal to n), understanding of the concept of path and surface integrals as special cases of the general concept.
* Knowledge of the general Stokes theorem (transforming the integration of a (k-1)-formy on the boundary of a k-dimensional integration region M to integration of the exterior derivative of this form on M), its proof and its classical applications (Green theorem, classical Stokes theorem, Gauss-Ostrogradsky theorem).
* Practical skills in calculus of integrals, including the use of general Stokes theorem, calculation of work of a vector field, flow of a vector field through a surface, etc.
* Practical use of the general Stokes theorem for obtaining differential (local) physical laws from their integral (global) versions, e.g. equations of fluid mechanics, Maxwell equations in classical electrodynamics, etc.
* Practical calculations of n-dimensional volumes integrating volume elements (curves, surfaces, bodies).
* Understanding of fundamentals of convergence theory of number series as well as series of functions.
* Understanding the difference of definition of convergence and uniform convergence of series of functions.
* Practical calculations concerning the convergence of power series and Fourier series.

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions.
• 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem.
• 3. Spaces of covariant tensors. Algebraic operations.
• 4. Vector and tensor fields, differential forms.
• 5. Exterior derivative. Pullback.
• 6. Integral of a differential form on singular cubes.
• General Stokes theorem.
• 8. Integrals of the first and second type, classical versions of Stokes theorem.
• 9. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. Volume element. Areas and volumes.
• 10. Appliactions-work of a force field along a curve, flux of a vector field through a surface.
• 11. Fundamentals of theory of convergence of infinite number series.
• 12. Series of functions, convergence, uniform convergence.
• 13. Power and Fourier series.
• 14. Applications of infinite series theory: Solving differential equations, approximation of functions, physical applications.

Literature

• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info
• DOŠLÁ, Zuzana and Vítězslav NOVÁK. Nekonečné řady. 2. vyd. Brno: Masarykova univerzita, 2007, iv, 113. ISBN 9788021043343. info

Teaching methods

Lectures: theoretical explanation with practical examples. Exercises: solving problems for understanding of basic concepts and theorems, contains also more complex problems.

Assessment methods

Teaching: lectures, consultative exercises Exam: written test (two parts: (a) solving problems, (b) test) and oral exam.

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

The course can also be completed outside the examination period.
The course is taught annually.
The course is taught every week.

F3063 Integration of forms

Extent and Intensity

3/2/0. 4 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).

Teacher(s)

prof. RNDr. Jana Musilová, CSc. (lecturer)
Mgr. Lenka Czudková, Ph.D. (seminar tutor)

Guaranteed by

prof. RNDr. Michal Lenc, Ph.D.
Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science
Contact Person: prof. RNDr. Jana Musilová, CSc.

Prerequisites

M1100 Mathematical Analysis I && M2100 Mathematical Analysis II
Mathematical Analysis: Differential calculus of functions of n-variables, n-dimensional Riemann integral. Algebra: Tensors and tensor calculus.

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Physics (programme PřF, B-FY)
• Physics (programme PřF, M-FY)
• Physics (programme PřF, N-FY)

Course objectives

The discipline is a part of the fundamental course of mathematical analysis for students of physics. It contains the theory of riemannian integral of differential forms, i.e. covariant tensor fields, on subsets of n-dimensional euclidean space. Using differential forms as integrated objects, the definition of the integral is obtained by the more natural way than the "classical" one. It includes classical line integrals and surface integrals. Its fundamental result - the general Stokes theorem - includes all classical integral theorems as special cases. The practical calculus of integrals with physical meaning is emphasized.

Syllabus

• 1. Fundamental concepts-brief repetition: elements of topology, differentiable functions. 2. Fundamental concepts-brief repetition: Riemann integral on n-dimensional euclidean spaces, integrable functions, Fubini theorem, transformation theorem. 3. Generalization of the integral - decomposition of unity. 4. Spaces of covariant tensors. 5. Vector and tensor fields, differential forms. 6. Exterior product, exterior derivative. 7. Pullback. 8. Integral of a differential form on singular cubes. 9. General Stokes theorem. 10. Integrals of the first and second type, classical versions of Stokes theorem. 11. Applications-geometrical and physical characteristic of 1-,2- and 3-dimensional objects. 12. Appliactions-work of a force field along a curve, flux of a vector field through a surface. 13. Volume element. Areas and volumes.

Literature

• KRUPKA, Demeter and Jana MUSILOVÁ. Integrální počet na euklidových prostorech a diferencovatelných varietách. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 320 s. info
• SPIVAK, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. 1st ed. Perseus Pr., 1996. ISBN 0805390219. info
• NAKAHARA, Mikio. Geometry, topology and physics. Bristol: Institute of physics publishing, 1990, xiii, 505. ISBN 0-85274-095-6. info

Assessment methods (in Czech)

Výuka: přednáška, klasické cvičení Zkouška: písemná (dvě části: (a) příklady, (b) test) a ústní

Language of instruction

Czech

Follow-Up Courses

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

Further comments (probably available only in Czech)

The course can also be completed outside the examination period.
The course is taught annually.
The course is taught every week.

• Enrolment Statistics (recent)