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Cours scientifiques - MAA302 : Topology and Differential Calculus

Domaine > Mathématiques.

Descriptif

The first part of the course MAA302 is devoted to the theory of topological and metric spaces in an abstract setting, including the basic notions of continuity, completeness, compactness, and connectedness. We then shift our focus towards the space of continuous functions on a compact set, with the important theorems of Arzèla-Ascoli and Stone-Weierstrass, as well as towards Banach spaces, including the following fundamental results in functional analysis: the uniform boundedness principle, the open mapping theorem, and the closed graph theorem. The final part of the course is devoted to differential calculus on Banach spaces, studying the important results of the inverse function and implicit function theorems, and concluding with an abstract theory of optimization, with and without constraints.

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

Vous devez avoir validé l'équation suivante : UE MAA207

Format des notes

Numérique sur 20

Littérale/grade américain

Pour les étudiants du diplôme Programmes d'échange internationaux

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

Le rattrapage est autorisé (Note de rattrapage conservée écrêtée à une note seuil de 10)
    L'UE est acquise si Note finale >= 9
    • Crédits ECTS acquis : 5 ECTS

    La note obtenue rentre dans le calcul de votre GPA.

    Programme détaillé

    This course will cover the following topics:

    - Metric spaces, and general topological spaces
    - Compactness, completeness and connectedness
    - Normed vector spaces and Banach spaces
    - Functions spaces
    - Differentiability in Banach spaces, differential of a function
    - Higher order differentials and Taylor Formulas
    - Local inversion theorem and Implicit functions theorem
    - Optimization in an open set, first and second order extremality conditions I Constrained optimization and Lagrange multipliers

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