### Duration

25h Th, 15h Pr

### Number of credits

Bachelor in physics | 4 crédits |

### Lecturer

### Language(s) of instruction

French language

### Organisation and examination

Teaching in the second semester

### Schedule

### Units courses prerequisite and corequisite

Prerequisite or corequisite units are presented within each program

### Learning unit contents

This course is an introduction to the geometry of affine spaces in general and Euclidean affine spaces in particular.

At the secondary school level, students have learned the geometry of Euclidean affine spaces of dimension 2 and 3 (although they were not named so). Geometric objects in these spaces

such as points, lines, planes, circles, spheres, angles, distances and their properties are thus familiar.

However, modern science requires to be able to work in higher dimensional spaces. The first part of the course will thus be devoted to the extension of geometric concepts from dimension 2 and 3 to such more general spaces.

Clearly, in such frameworks, intuition is not enough anymore to solve problems or even to set correct definitions. Our approach of geometry will thus rely on algebraic concepts of linear algebra.

In the second part of the course, we will study some aspects of the theory of curves and surfaces.

Specifically, we will study the following topics (among others), after a short reminder of concepts concerning general vector spaces (see the course Algèbre):

- General affine spaces;
- Euclidean vector spaces;
- Euclidean affine spaces;
- Affine transformations;
- Theory of curves, mostly in dimension 2 and 3;

### Learning outcomes of the learning unit

At the end of the lectures, the students should know the basic concepts of affine and Euclidean geometry in finite dimension. They should be able to apply them to solve problems in geometry, in particular using tools of analytic geometry.

They should know the basics of theory of curves.

They will use the relevant tools of linear algebra, in particular the elementary theory of vector spaces.

They should be able to handle some abstract mathematical concepts that appear in modern physics.

The basics concepts introduced in this course will be generalized for applications in classical or quantum mechanics, relativity and many more areas of Physics.

### Prerequisite knowledge and skills

Good knowledge of mathematics from secondary school is expected.

Naturally, being trained to abstraction and mathematical reasoning is an advantage.

Some topics of the lectures of algebra will be used in geometry (for instance matrix calculus, determinants, ranks, systems of equations, vector spaces,...

Some concepts of the course Analyse I will also be used (derivatives, usual functions)

### Planned learning activities and teaching methods

The theory is explained on the blackboard and using data projector. Students are encouraged to ask questions and to participate.

The practical sessions are mainly dedicated to solve exercises corresponding to the theory considered during the lecture sessions. These sessions are also useful to obtain extra informations or enlightenments on the concepts presented during the lecture sessions.

I strongly suggest that the students form small groups to discuss the topics of geometry and exchange their knowledge. They should then make lists of particular points that they do not understand and ask for explanations, either from me or from the person in charge of the exercises.

This can be done by making an appointment or at the end of the lectures or exercise sessions.

It is indeed not normal not to understand particular points of the lectures, and is it most likely that I will not go through this point once again during the lectures, but rather use it to explain another one. It is very unlikely that the situation will become better without an action taken by the student...

### Mode of delivery (face to face, distance learning, hybrid learning)

Blended learning

*Additional information:*

The lectures are given in face to face mode, but videos of the explanations are also available on the relevant platform of ULiege.

### Recommended or required readings

Detailed lecture notes are available on e-campus

These lecture notes will be adapted in January 2023.

A printed version will be available at the beginning of the second semester (upon request), at very low price. Please contact the secretary of the Department of Mathematics, D. Bartholomeus (Building B37, Office 0/28).

**Exam(s) in session**

Any session

- In-person

written exam ( open-ended questions ) AND oral exam

*Additional information:*

The final examination consists of two parts, one of them is the assessment of theoric knowledge, the other concerns the applications/exercises.

The part concerning exercises is a written exam.The questions involve techniques that are explained during the lectures and exercises sessions.

In the part concerning the theory, students will be asked to explain a concept developed during the lectures but also includes direct applications of it. In order to avoid stress for the exam, a list a *major *questions that will be asked during the exam will be provided at the end of the lectures.

It is a written exam for students in physics and oral exam for students in mathematics.

Pay attention : it is expected that the students be able to prove the theorems, except otherwise stated.

If both grades are greater than or equal to 6/20 (without rounding), the final grade is an arithmetic mean of the results obtained by the students for the two parts of the exam. Otherwise, the grade will be strictly less than 8/20

### Work placement(s)

### Organisational remarks and main changes to the course

### Contacts

Feel free to contact me for any question, preferably by e-mail
(P.Mathonet@uliege.be )
to make an appointment or for very short questions, or come to my office
(Building B37, Grande Traverse 12 - Sart Tilman, office 0/27).

You can also try to call me on the phone 04/366 94 80.

For questions regarding lectures and exercises, feel also free to contact the person in charge of the exercises (building B37).