Geometric Methods in Physics 1
Code  Completion  Credits  Range  Language 

02GMF1  Z,ZK  4  2+2  Czech 
 Lecturer:
 Libor Šnobl (guarantor)
 Tutor:
 Libor Šnobl (guarantor)
 Supervisor:
 Department of Physics
 Synopsis:

Foundations of analysis on manifolds. Differential forms. Integration, Stokes theorem.
 Requirements:

The course of theoretical physics (02TEF1, 02TEF2)
 Syllabus of lectures:

1. Manifolds.
2. Tangent vectors, tangent spaces.
3. Tangent bundle, vector fields as its sections, integral curves, vector fields as derivation on the space of smooth functions, commutator.
4. Covectors, pforms, corresponding fibre bundles.
5. Differential forms, wedge product, outer derivation, closed and exact forms.
6. Induced maps of tensorial objects.
7. Lie derivative.
8. Geometric formulation of Hamilton´s mechanics, symplectic form, Hamiltonian vector fields, Poisson brackets, integrals of motion.
9. Orientation on a manifold, decomposition of a unit, integration of forms, Stokes´ theorem on pchains.
10. Metrics, affine connection and and curvature.
 Syllabus of tutorials:

Solving problems on the following topics:
1. Manifolds.
2. Tangent vectors, tangent spaces.
3. Vector fields.
4. Covectors, forms.
5. Differential forms, wedge product, outer derivation.
6. Induced maps of tensorial objects.
7. Lie derivative.
8. Geometric formulation of Hamilton´s mechanics.
9. Integration of forms, Stokes´ theorem
10. Metrics and curvature.
 Study Objective:

Knowledge:
Foundations of analysis on manifolds
Skills:
Application of geometrical methods in theoretical physics
 Study materials:

Key references:
[1] L. W. Tu: Differential Geometry: Connections, Curvature, and Characteristic Classes (Graduate Texts in Mathematics), Springer 2017
Recommended references:
[2] Th. Frankel, The Geometry of Physics: An Introduction, Cambridge University Press 2011
[3] M. Nakahara: Geometry, Topology and Physics, IOP Publishing, Bristol 1998
 Note:
 Timetable for winter semester 2020/2021:
 Timetable is not available yet
 Timetable for summer semester 2020/2021:
 Timetable is not available yet
 The course is a part of the following study plans:

 BS Matematické inženýrství  Matematická fyzika (compulsory course of the specialization)