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Cours scientifiques - FMA_3S006_EP : Differential Geometry

Domaine > Mathématiques.

Descriptif

This course provides an overview of the classical differential geometry of curves and surfaces.
More precisely, we will study the local theory of (regular, parametrized) curves (curvature, torsion), regular surfaces, and the local theory (first and second fundamental forms) and intrinsic geometry (Theorema Egregium and Gauss-Bonnet theorem) of the latter.

Weekly exercise sessions form an integral part of the course.
The instructor will provide lecture notes covering the material seen in class.

Prerequisites: some basic linear algebra, as seen is any undergraduate class (such as MAA101 and 201), and familiarity with multivariable calculus (differential of functions from ℝ^n to ℝ^2, as seen for example in MAA202). The most important notions will be briefly reviewed during the first lecture.

Objectifs pédagogiques

This course provides an overview of the classical differential geometry of curves and surfaces.

Pour les étudiants du diplôme Programmes d'échange internationaux

Some basic linear algebra, as seen is any undergraduate class (such as MAA101 and 201), and familiarity with multivariable calculus (differential of functions from ℝ^n to ℝ^2, as seen for example in MAA202).

Pour les étudiants du diplôme Bachelor of Science de l'Ecole polytechnique

Vous devez avoir validé l'équation suivante : UE FMA_2F002_EP

Some basic linear algebra, as seen is any undergraduate class (such as MAA101 and 201), and familiarity with multivariable calculus (differential of functions from ℝ^n to ℝ^2, as seen for example in MAA202).

Règle d'exclusion : UE APM_3S008_EP

Format des notes

Numérique sur 20

Littérale/grade américain

Pour les étudiants du diplôme Programmes d'échange internationaux

Vos modalités d'acquisition :

The assessment will consist of two parts: 
- A continuous assessment (coursework) based on short tests and/or homework assignments
- A 2 hours long written final exam (closed book).

The final numerical grade will be the the arithmetic mean of the grade of the coursework and that of the final exam, all of which will be on a scale from 0 to 20.

It will be converted to the official final letter grade according to a table published by the instructor at the beginning of the course.

Pour les étudiants du diplôme Bachelor of Science de l'Ecole polytechnique

Vos modalités d'acquisition :

The assessment will consist of two parts: 
- A continuous assessment (coursework) based on short tests and/or homework assignments
- A 2 hours long written final exam (closed book).

The final numerical grade will be the the arithmetic mean of the grade of the coursework and that of the final exam, all of which will be on a scale from 0 to 20.

It will be converted to the official final letter grade according to a table published by the instructor at the beginning of the course.

Le rattrapage est autorisé (Note de rattrapage conservée écrêtée à une note seuil de 11)
    L'UE est acquise si Note finale >= 9
    • Crédits ECTS acquis : 4 ECTS

    La note obtenue rentre dans le calcul de votre GPA.

    Programme détaillé

    - parametrized curves: length, curvature, torsion
    - Frenet frames, Frenet formulas, fundamental theorem of space curves
    - global results on curves
    - surfaces in three-dimensional space, calculus on surfaces
    - first fundamental form, isometries
    - Gauss map, Weingarten map
    - second fundamental form, curvatures, classification of points
    - Gauss's Theorema Egregium, Codazzi-Mainardi equations
    - geodesics theory and applications
    - Gauss-Bonnet Theorem

    Mots clés

    Differential geometry, curves, surfaces, curvature

    Méthodes pédagogiques

    Blackboard classes, tutorials

    Support pédagogique multimédia

    Oui

    Veuillez patienter