SubjectsSubjects(version: 845)
Course, academic year 2019/2020
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Introduction to Analysis on Manifolds - NMAG335
Title in English: Úvod do analýzy na varietách
Guaranteed by: Mathematical Institute of Charles University (32-MUUK)
Faculty: Faculty of Mathematics and Physics
Actual: from 2019
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech
Teaching methods: full-time
Guarantor: doc. RNDr. Roman Lávička, Ph.D.
Class: M Bc. OM
M Bc. OM > Zaměření MA
M Bc. OM > Zaměření MSTR
M Bc. OM > Povinně volitelné
Classification: Mathematics > Geometry
Incompatibility : NGEM002
Interchangeability : NGEM002
In complex pre-requisite: NMAG349
Annotation -
Last update: G_M (15.05.2012)
One of the basic courses in the area of general differential geometry. A recommended course for specialization Mathematical Structures within General Mathematics.
Course completion requirements -
Last update: prof. RNDr. Vladimír Souček, DrSc. (13.10.2017)

'Zápočet' is given after a successful result of a written test at the end of the semester. This is at the same time a necessary condition to advance for the examination. The exam is in a written form.

Literature - Czech
Last update: prof. RNDr. Vladimír Souček, DrSc. (10.10.2012)

L. Krump, V. Souček, J. A. Těšínský: Matematická analýza na varietách, skriptum, Karolinum, 2008 (2. vydání).

O. Kowalski: Úvod do Riemannovy geometrie, skriptum, Karolinum, 1975 (2. vydání).

M. Fecko: Diferenciálna geometria Lieovy grupy pre fyzikov, IRIS, Bratislava, 2008.

Syllabus -
Last update: G_M (15.05.2012)

1. Topological manifold (charts, transition functions, atlas), smooth manifolds (differential structure), basic examples of manifolds.

2. Smooth maps between manifolds, smooth functions, diffeomorphisms; tangent vectors in a point, tangent space in a point, coordinates on tangent space, geometrical interpretation of vectors; tangent map to a smooth map, coordinate description, Jacobians.

3. A summary of properties of tensor algebra of a vector space; outer algebra of a vector space, basic properties of outer multiplication; symmetric algerba of a vector space, orientation of a vector space, volume of a paralleliped using outer product and the Gramm matrix.

4. Tensor fields on a manifold, Riemann (pseudo)-metric on a manifold, Minkowski spacetime, algebra of differnetial forms as a modul over the ring of functions, orientation of a manifold; de Rham differential in coordinates and without coordinates, exact and closed forms, de Rham complex, de Rham cohomology, Poincare lemma; inverse image of tensor fields and forms by a smooth map, coordinate description, basic properties.

5. Manifolds with a boundary, its tangent space, differential forms on manifolds with boundary, orientation.

6. Integration of forms on a manifold with boundary, Stokes theorem.

7. Volume form on a (pseudo)-Riemannian manifold, integration of functions on such manifolds, local computations.

If possible: 8. Lie derivative of vector fields, contraction of forms by vector fileds, connection with the de Rham differential.

 
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