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Course, academic year 2022/2023
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Geometry - NMAG204
Title: Geometrie
Guaranteed by: Mathematical Institute of Charles University (32-MUUK)
Faculty: Faculty of Mathematics and Physics
Actual: from 2020
Semester: summer
E-Credits: 4
Hours per week, examination: summer s.:2/1, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
Virtual mobility / capacity: no
State of the course: not taught
Language: Czech
Teaching methods: full-time
Guarantor: prof. RNDr. Jan Rataj, CSc.
Class: M Bc. MMIT > Povinně volitelné
M Bc. OM
M Bc. OM > Povinné
M Bc. OM > 2. ročník
Classification: Mathematics > Geometry
Pre-requisite : {One 1st year Analysis course}
Incompatibility : NGEM012, NMAG211
Interchangeability : NGEM012, NMAG211
Is incompatible with: NMAG211
Is pre-requisite for: NMPG349
Is interchangeable with: NGEM012, NMAG211
Annotation -
Last update: G_M (15.05.2012)
Lecture on Differential geometry for students of General Mathematics. Surfaces in the three dimensional Euclidean space, the first and second fundamental forms, main curvatures of surface, Gauss and mean curvature, geodesics, geodesic curvature.
Aim of the course -
Last update: G_M (24.04.2012)

Teaching of differential geometry of curves and surfaces.

Course completion requirements - Czech
Last update: prof. RNDr. Jan Rataj, CSc. (11.02.2020)

Podmínkou udělení zápočtu je odevzdání 6 průběžně zadávaných domácích úkolů. Charakter zápočtu neumožňuje jeho opakování. Podmínkou připuštění ke zkoušce je udělený zápočet. Zkouška probíhá písemnou formou a má dvě části, početní a teoretickou.

Literature -
Last update: G_M (24.04.2012)

[1] do Carmo, M., P., Differential geometry of curves and surfaces, Prentice Hall, 1976.

[2] Klingenberg W., A., Course in differential geometry, GTM 51, Springer 1978.

[3] Bures, J., Hrubcik, K., Diferencialni geometrie krivek a ploch, Karolinum, Praha, 1998.

Teaching methods -
Last update: G_M (24.04.2012)

Lecture and exercises.

Requirements to the exam - Czech
Last update: doc. RNDr. Zbyněk Šír, Ph.D. (17.02.2018)

Ke zkοušce je možno přistoupit jen po získání zápočtu. Zkouška probíhá písemnou formou a má dvě části, početní a teoretickou. Je nutno získat předepsaný počet bodů z každé části.

Syllabus -
Last update: G_M (24.04.2012)

A. INTRODUCTION

1. Motivation. The Euclidean space and its properties.

2. Differentiation in R^n. Tangent space, differential of a mapping.

B. CURVES

3. Definition and basic properties. Curvature and torsion. The Frenet frame, Frenet formulae and its applications.

4. Curves in plane and space.

C. SURFACES

5. Definition and basic properties. The first fundamental form.

6. Second fundamental form, Weingarten's mapping.

7. Curves on a surface, principal curvatures, Gauss and mean curvature.

8. Principal and asymptotic directions and curves, isometric surfaces.

9. Intrinsic geometry of a surface, geodetic curves.

10. Introduction to hyperbolic geometry.

 
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