Introduction to Curves and Surfaces

Course guarantor:

Lecturer:

Tutor:

Supervisor:

Department of Physics

Synopsis:

The goal of the lecture is an introduction to the differential geometry of simple manifolds - curves and two-dimensional surfaces. The basic concepts for the curves are introduced Frenets formulae are explained. In the surface theory we introduce first and second fundamental forms and mean and Gaussian curvature. Essential part of the lecture are the examples calculated by students

Requirements:

Differential calculus with several varaibles.

Syllabus of lectures:

1.Examples and definition of curves

2.Plane curves, natural equation

3.Space curves, curvature, torsion

4.Frenet formulas

5.Examples and definition of surfaces

6.The first fundamental form, lenght of a curve on the surface

7.The second fundamental form

8.Mean and Gauss curvature of the surface

9.Gauss Weingarten equations

10.Codazzi equation

11.Gauss theorema egregium

Syllabus of tutorials:

Curvature and length of a curve

Curvature of a surface

Metric tensor

Study Objective:

Knowledge:

To provide the simplest examples of manifolds and their properties

Skills:

solve mathematical problems defined on manifolds

Study materials:

Key references:

[1] M.P. do Carmo, Differential Geometry of Curves and Surfaces,

New Jersey : Prentice - Hall, 1976

Recommended references:

[2] A. Gray Modern Differential Geometry, Boca Raton : CRC Press, c1998

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

• Matematické inženýrství - Matematická fyzika (elective course)
• Mathematical Engineering - Mathematical Physics (elective course)