Introduction to Curves and Surfaces

Course guarantor:

Ladislav Hlavatý

Lecturer:

Ladislav Hlavatý

Tutor:

Ladislav Hlavatý

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:

Syllabus of lectures:

Outline of the lecture:

1. Examples and definition of curves

2. Plane curves, natural equation of a curve

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. Transformation properties of the first fundamental form

Outline of the exercises:

1. Curvature and length of the curve

2. Curvature and area of a surface

3. Metric tensor

Syllabus of tutorials:

Outline of the exercises:

1. Curvature and length of the curve

2. Curvature and area of a surface

3. Metric tensor

Study Objective:

Knowledge:

To provide the simplest examples of manifolds and their properties.

Acquired skills:

Solve mathematical problems defined on manifolds.

Study materials:

Key references:

[1] L. Hlavatý, Úvod do křivek a ploch (in Czech)

www.fjfi.cvut.cz > katedra fyziky > studentský servis > Doprovod přednášek > Úvod do křivek a ploch

Recommended references:

[2] B. Hostinský, Diferenciální geometrie křivek a ploch, Přírodovědecké nakladatelství v Praze, 1949 (in Czech)

[3] W. Kuehnel, Diferential Geometry, AMS2006

[4] T. Banchoff, S Lovett , Diferential Geometry of Curves and Surfaces, CRC Press 2016

Note:

Time-table for winter semester 2024/2025:

Time-table is not available yet

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans:

• Aplikovaná algebra a analýza (elective course)
• Aplikované matematicko-stochastické metody (elective course)
• Fyzikální inženýrství - Fyzika plazmatu a termojaderné fúze (elective course)
• Jaderná a částicová fyzika (elective course)
• Matematické inženýrství - Matematická fyzika (elective course)
• Matematické inženýrství - Matematické modelování (elective course)
• Nuclear and Particle Physics (elective course)
• Mathematical Engineering - Mathematical Physics (elective course)
• Physical Engineering - Plasma Physics and Thermonuclear Fusion (elective course)