Code: 02GMF1 
Geometric Methods in Physics 1 
Lecturer: doc. Ing. Libor Šnobl Ph.D. 
Weekly load: 2+2 
Completion: A, EX 
Department: 14102 
Credits: 4 
Semester: W 
 Description:

Foundations of geometric methods in physics on manifolds. Differential forms.
 Contents:

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.
 Seminar contents:

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.
 Recommended literature:

Key references:
[1] L. W. Tu: Differential Geometry: Connections, Curvature, and Characteristic Classes (Graduate Texts in Mathematics), Springer 2017
[2] Th. Frankel, The Geometry of Physics: An Introduction, Cambridge University Press 2011
Recommended references:
[3] M. Nakahara: Geometry, Topology and Physics, IOP Publishing, Bristol 1998
 Keywords:
 differentiable manifold, vector fields, pform, integration of forms
Abbreviations used:
Semester:
 W ... winter semester (usually October  February)
 S ... spring semester (usually March  June)
 W,S ... both semesters
Mode of completion of the course:
 A ... Assessment (no grade is given to this course but credits are awarded. You will receive only P (Passed) of F (Failed) and number of credits)
 GA ... Graded Assessment (a grade is awarded for this course)
 EX ... Examination (a grade is awarded for this course)
 A, EX ... Examination (the award of Assessment is a precondition for taking the Examination in the given subject, a grade is awarded for this course)
Weekly load (hours per week):
 P ... lecture
 C ... seminar
 L ... laboratory
 R ... proseminar
 S ... seminar