# Geometry I

lmat1141  2019-2020  Louvain-la-Neuve

Geometry I
Note from June 29, 2020
Although we do not yet know how long the social distancing related to the Covid-19 pandemic will last, and regardless of the changes that had to be made in the evaluation of the June 2020 session in relation to what is provided for in this learning unit description, new learnig unit evaluation methods may still be adopted by the teachers; details of these methods have been - or will be - communicated to the students by the teachers, as soon as possible.
7 credits
45.0 h + 30.0 h
Q2
Teacher(s)
Language
French
Prerequisites
Euclidean geometry : affine and euclidean space, quadrics .
Differential geometry :  plane and skew curves ; local theory of surfaces in 3-dimensional space.
Main themes
Euclidean geometry : affine and euclidean space, quadrics .
Differential geometry :  plane and skew curves ; local theory of surfaces in 3-dimensional space.
Aims
 At the end of this learning unit, the student is able to : 1 Contribution of the course to learning outcomes in the Bachelor in Mathematics programme. By the end of this activity, students will have made progress in: -recognise and understand a basic foundation of mathematics. --Choose and use the basic tools of calculation to solve mathematical problems. --Recognise the fundamental concepts of important current mathematical theories. --Establish the main connections between these theories, analyse them and explain them through the use of examples. - identify, by use of the abstract and experimental approach specific to the exact sciences, the unifying features of different situations and experiments in mathematics or in closely related fields (probability and statistics, physics, computing). - show evidence of abstract thinking and of a critical spirit. Argue within the context of the axiomatic method Recognise the key arguments and the structure of a proof. Construct and draw up a proof independently. Evaluate the rigour of a mathematical or logical argument and identify any possible flaws in it. Distinguish between the intuition and the validity of a result and the different levels of rigorous understanding of this same result. Learning outcomes specific to the course. By the end of this activity, students will be able to: - Determine loci in affine and euclidean spaces and represent them graphically - Determine and characterize affine maps and isometries. - Classify quadrics, especially in dimension 2 and 3. Determine their geometric invariants : adapted frame, asymptotic directions and use them to represent graphically the quadric. - Compute and interpret  differential invariants of a curve as tangent vector, curvature vector, Frenet frame, length of a curve. - Compute and interpret  differential invariants of a surface as tangent plane, fundamental form, normal, principal and total  curvature, area of a surface.

The contribution of this Teaching Unit to the development and command of the skills and learning outcomes of the programme(s) can be accessed at the end of this sheet, in the section entitled “Programmes/courses offering this Teaching Unit”.
Content
The course has two parts. The first one, more algebraic, is focused on Euclidian and Affine Geometry, with also the classification of quadrics. In the second part, using tools of differential calculus, we study curves and surfaces.
Teaching methods
This course extends the skills acquired in the introductory algebra and analysis courses by situating the various concepts studied in these courses in the context of plane geometry and solid geometry. Students will be encouraged to develop geometric intuition and to express it in the formal language of algebra or analysis. Conversely, they will have to be able to interpret analytic or algebraic results in a geometrical way, and to approach problems from different points of view.
Learning activities consist of lectures, exercise sessions and tutorial sessions.
The tutorial sessions give students individual help and follow-up in their learning
The three activities are given in presential sessions.
Evaluation methods
Bibliography
Syllabus disponible sur moodle avec références bibliographiques.
Teaching materials
• syllabus sur moodle
Faculty or entity

#### Programmes / formations proposant cette unité d'enseignement (UE)

Title of the programme
Sigle
Credits
Prerequisites
Aims
Bachelor in Physics

Bachelor in Mathematics