Topology of R^n and functions of several variables

  • ECTS

    5 credits

  • Component

    Faculty of Science

Description

In this course, an introduction to the topology of R^n, the basic notions of differential calculus of R^n functions in R and optimization will be covered. Parametric curves will also be covered.

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Objectives

Topology on R^n.

Standard 1,2 and infinite norms and equivalence of these norms. Notions of open and closed, neighborhoods. Definition of the continuity of a function of several variables from R^n into R^p , continuity in terms of openings and neighborhoods.

Limits of sequences and compactness in R^n, characterization of closures by sequences.

Functions of several variables. (the notion of differentiability will be seen only in L3)

Directional derivatives, partial derivatives. Representation, level curves. Gradient of a real-valued function, DL1 if continuous partial derivatives. Inequality of finite increments.

Hessian, DL2, Schwarz Theorem.

Optimization of functions from R^n into R:Free extrema: Notion of critical point. Local extremum, definition and necessary condition. Necessary and sufficient conditions for local extrema. Examples

Least squares methods

Parametric curves

Derivatives of compound functions. Definition, kinematic point of view, examples, representation. Tangent vector, length of C^1 curves, reparameterization. Local study of curves

Derivations of functions with values in C (exponential, sum, product, quotient)

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Necessary pre-requisites

A course in analysis of functions of one real variable in L1 (HAX103X)

 

Recommended prerequisites: L1 math

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Additional information

Hourly volumes:

            CM : 24

            TD : 25,5

            TP:

            Terrain:

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