# MATA6350 Differential geometry of curves and surfaces (5–7 cr)

Study level:
Intermediate studies
0-5
Language:
English, Finnish
Responsible organisation:
Department of Mathematics and Statistics
Curriculum periods:
2020-2021, 2021-2022, 2022-2023

## Description

Local and global properties of curves from the point of view of differential geometry. For instance: parametrization of curves, curvature and torsion of curves, the local canonical form, the Jordan curve theorem, isoperimetric inequality.

Basic theory of surfaces, for instance: different ways to define and express surfaces, tangents and derivations, the first fundamental form, area and different notions of curvature, Gauss’ Theorema egregium. Possibly also geodesics and the Gauss-Bonnet theorem.

## Learning outcomes

After the completion of the course the student
• can examine the length and parametrization of curves
• masters the definitions of curvature and torsion and can apply these
• knows the local canonical form of curves
• knows the contents and the significance of the Jordan curve theorem and the isoperimetric inequality.
• can examine the properties of surfaces using different expressions for surfaces
• knows the first fundamental form of surfaces
• can determine areas and various curvatures of surfaces
• knows the contents and the significance Gauss’ Theorema egregium

28h lectures, and exercises

## Description of prerequisites

Vector Analysis 1 and 2

## Study materials

M. Abate, F. Tovena: Curves and Surfaces, Chapters 1 - 4  (at least)

## Literature

• M. Abate, F. Tovena: Curves and Surfaces, Springer-Verlag Mailand, 2012; ISBN: 978-88-470-1940-9

## Completion methods

### Method 1

Evaluation criteria:
Course exam and exercises
Select all marked parts

### Method 2

Evaluation criteria:
Final exam
Select all marked parts
Parts of the completion methods
x

### Participation in teaching (5–7 cr)

Type:
Participation in teaching
0-5
Language:
English, Finnish
No published teaching
x

Type:
Exam