Topology of R^n and functions of several variables

This course covers an introduction to the topology of R^n, the basics of differential calculus for R^n functions in R and optimization. Parametric curves will also be covered.

Topology on R^n.

Standard 1,2 and infinite norms and their equivalence. 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 only be seen in L3).

Directional derivatives, partial derivatives. Representation, contour lines. Gradient of a real-valued function, DL1 if partial derivatives are continuous. Inequality of finite increments.

Hessian, DL2, Schwarz Theorem.

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

Least squares methods

Parametric curves

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

Derivations of C-valued functions (exponential, sum, product, quotient)

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

Recommended prerequisites: L1 maths

Hourly volumes :

            CM: 24

            TD: 25.5

            TP :

            Terrain :