Geometry I, Part 1

Duration

25h Th, 15h Pr

Number of credits

Lecturer

Pierre Mathonet

Language(s) of instruction

French language

Organisation and examination

Teaching in the second semester

Schedule

Schedule online

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;
• Introduction to the theory of surfaces.

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 and be familiar with the theory of surfaces in dimension 3.
The 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, partial derivatives,...)

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)

The students are supposed to attend the lectures of theory and practice.
See de Schedule on the program celcat.

Recommended or required readings

Detailed lecture notes are available on my web page
http://www.geodiff.ulg.ac.be,
These lecture notes will be adapted in January 2020.

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).
 

Assessment methods and criteria

The final examination consists of two parts, one of them is the assessment of theoric knowledge, the other i concerns the applications/exercises. 
The written part is devoted to the resolution of problems and exercises, concerning the topics developed during the lectures and exercise 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. It will be organized as a written or oral exam. 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.
The exact practical information will be discussed during the lectures.
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)

Organizational remarks

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).

Adaptation of teaching commitments following the COVID-19 pandemic for the May-June 2020 session

Teaching methods implemented : distance-learning

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Assessment subjects

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Assessment methods

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Contacts

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Adaptation of teaching commitments following the COVID-19 pandemic for the Aug-Sept 2020 session

Assessment subjects

La matière du cours théorique est la même que celle de la session de mai-juin Une liste de questions pour l'examen oral initialement prévu avait été envoyée aux étudiants. Elle permet toujours de se préparer pour l'examen, en découpant le cours, mais aucune question de restitution pure ne pourra être demandée, l'examen se déroulant à distance (et donc à livre ouvert). Elles seront remplacées par des questions de réflexion concernant la matière. La matière des exercices (les "savoir faire") est celle abordée aux travaux pratiques, que ce soit en présentiel ou à distance.

Assessment methods

- L'examen sera écrit et réalisé à distance. - Il se déroulera de la façon suivante :
a) Au moment prévu à l'horaire, chaque étudiant recevra sous format électronique (fichier pdf) un questionnaire d'examen.
b) Il y aura bien sûr plusieurs questionnaires.
c) Les réponses à ce questionnaire devront être transmises électroniquement (par exemple, scan ou photos des pages rédigées) dans un délai imparti.Les étudiants sont invités à s'informer sur les meilleures méthodes pour scanner leurs copies.
- Je ne m'attends pas à ce que vous n'utilisiez pas de machine à calculer (la tentation serait trop forte si vous avez un doute sur la valeur de cos(pi/3)...). Il vous sera cependant demandé de fournir des explications de vos développements, en les justifiant au maximum.
- La théorie se fait forcément à livre ouvert, donc il n'y a pas de restitution pure. Cette partie est remplacée par des questions/exercices de nature théorique. Il reste donc important de bien comprendre/retravailler la matière enseignée tant au cours théorique qu'aux séances d'exercices.
- La répartition des points Exercices/théorie pourra être modifiée, avec plus de poids sur les exercices.

Contacts

P.Mathonet@uliege.be ou nzenaidi@uliege.be

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