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Cours scientifiques - FMA_3F002_EP : 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 in particular the important results of the inverse function and implicit function theorems. If time permits we will conclude with an abstract theory of optimization, with and without constraints.

Prerequisite: Real analysis (MAA102); topology of normed vector spaces and multivariable calculus (MAA202)

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

Vous devez avoir validé l'équation suivante : UE FMA_2S007_EP

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

Vos modalités d'acquisition :

Evaluation will be based on:

  • Coursework: Short tests and/or homework assignments
  • Two exams: Midterm and Final (2-hours long written open-book exam)

The final numerical grade (out of 20) will be computed by taking the best score among:

  • Coursework (50%) + Midterm (25%) + Final (25%)
  • Midterm (50%) + Final (50%)

and then it will be converted into the official letter grade according to a table published by the instructor on the Moodle page of the course at the beginning of the semester.

In case of failure, there will be a remedial exam (2-hours long written closed-book exam).

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:

    - topological and metric spaces, neighborhoods, basis, countability and separation axioms

    - continuity, construction of topologies, limit of sequences

    - completeness, fixed-point theorem, Baire’s theorem

    - compactness

    - connectedness

    - space of continuous functions, uniform convergence, equicontinuity, Arzelà-Ascoli theorem, Stone-Weierstrass theorem

    - normed spaces, Banach spaces, continuous linear maps, uniform boundedness principle, open mapping theorem, closed graph theorem

    - differentiable maps, partial derivatives

    - mean-value theorem, high-order derivatives, Taylor formula

    - inverse function theorem, implicit function theorem

    - optimization

    Veuillez patienter