 | | |
| MATH0476-1 | Geometry, (2nd semester)
|
|
| Duration : | 30h Th, 15h Pr |
|
| Number of credits : |
|
|
| Lecturer : | Pierre Mathonet |
|
Language(s) of instruction :
|
| French language |
|
Organisation and examination :
|
| Teaching in the second semester |
|
Course 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) :
- General vector spaces;
- General affine spaces;
- Euclidean vector spaces;
- Euclidean affine spaces;
- Affine transformations;
- Theory of curves in dimension 2 and 3;
- Introduction to the theory of surfaces.
We will study these geometric concepts in a rigorous manner with or without the help of coordinates and analytic geometry.
The useful concepts of linear algebra (matrices, determinants, linear systems) will also be reviewed. |
|
Learning outcomes of the course :
|
| 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 know 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. |
|
Prerequisites and co-requisites/ Recommended optional programme components :
|
| Basic knowledge from secondary school is expected.
Naturally, being trained to abstraction and mathematical reasoning is an advantage.
Some topics of the lectures "Mathématiques générales" by F. Bastin will be used in geometry (for instance matrix calculus, integration theory, derivatives, chain rule, changes of variables...). |
|
Planned learning activities and teaching methods :
|
| The theory is explained on the blackboard and/or 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 Miss Kreusch.
This can be done by making an appointment or at the end of the lectures or exercise sessions.
It is indeed not usual 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.
The schedule will be determined by C. Becco (Dept. of Physics), and will be communicated at the beginning of the concerned semester. |
|
Recommended or required readings :
|
| The lecture notes of geometry of Professor M. Rigo are available (in french) on his website :
http://www.discmath.ulg.ac.be/notes.html
They can also be obtained at low price from the secretary of the Department of Mathematics at the beginning of the second semester.
Please contact D. Bartholomeus (Building B37, office 0/28) to order a copy.
I am also writing my lecture notes for them moment. They should be (partially at least) available at the beginning of the second semester on my web page
http://www.geodiff.ulg.ac.be,
or sent by e-mail to the students. |
|
Assessment methods and criteria :
|
| The final examination consists of two parts, a written one and an oral one.
The written part is devoted to the resolution of problems and exercises, concerning the topics developed during the lectures and exercise sessions.
The oral part is devoted to the theory 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.
Pay attention : it is expected that the students be able to prove the theorems, except otherwise stated.
The final result is an arithmetic mean of the results obtained by the students for the two parts of the exam. |
|
Work placement(s) :
|
| |
|
Organizational remarks :
|
| |
|
Contacts :
|
| Feel free to contact me for any question, preferably by e-mail
(P.Mathonet@ulg.ac.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, from October 2012).
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 Miss M. Kreusch (building B37, office 1/20). |
|
|
| Items online : |
|
| Useful material |
| As already mentioned, lectures notes and other useful material will be available soon on my web page. |
|
|
