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, p-forms, 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 p-chains.
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, p-form, integration of forms

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