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Differential Geometry of Curves and Surfaces - NGEM012

Title:	Diferenciální geometrie křivek a ploch
Guaranteed by:	Mathematical Institute of Charles University (32-MUUK)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2018
Semester:	summer
E-Credits:	5
Hours per week, examination:	summer s.:2/2, C+Ex [HT]
Capacity:	unlimited
Min. number of students:	unlimited
4EU+:	no
Virtual mobility / capacity:	no
State of the course:	cancelled
Language:	Czech
Teaching methods:	full-time
Teaching methods:	full-time

Guarantor:	prof. RNDr. Vladimír Souček, DrSc.
Classification:	Mathematics > Geometry
Pre-requisite :	{Math. Analysis 1a, 1b}
Interchangeability :	NMAG204
Is incompatible with:	NMAG204
Is interchangeable with:	NMAG204

Opinion survey results Examination dates Schedule Noticeboard

Annotation -

Theory of curves in Euclidean space, Frenet formulae, curvatures. Surfaces in the three dimensional Euclidean space, the first and second fundamental forms, main curvatures of surface, Gauss and mean curvature. Examples. Christoffel symbols, Gauss and Codazzi equations, covariant derivation on surface, parallel transport, geodesics, geodesic curvature. Lobatschewski plane and its geometry.

Last update: KRYSL/MFF.CUNI.CZ (13.05.2008)

Aim of the course -

Teaching of differential geometry of curves and surfaces.

Last update: KRYSL/MFF.CUNI.CZ (13.05.2008)

Literature -

[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.

Last update: KRYSL/MFF.CUNI.CZ (13.05.2008)

Teaching methods -

Lecture and exercises.

Last update: T_MUUK (23.04.2010)

Requirements to the exam - Czech

Ke zkοušce je možno přistoupit jen po získání zápočtu.

Last update: Šír Zbyněk, doc. RNDr., Ph.D. (01.05.2011)

Syllabus -

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.

Last update: KRYSL/MFF.CUNI.CZ (13.05.2008)