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

**Study level:**

Intermediate studies

**Grading scale:**

0-5

**Language:**

English, Finnish

**Responsible organisation:**

Department of Mathematics and Statistics

**Curriculum periods:**

2020-2021, 2021-2022, 2022-2023, 2023-2024

## 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

## Additional information

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

**Grading scale:**

0-5

**Language:**

English, Finnish

x

### Exam (5–7 cr)

**Type:**

Exam

**Grading scale:**

0-5

**Language:**

English, Finnish