Topology of R^n and functions of several variables

This course will cover an introduction to the topology of R^n, the basic concepts of differential calculus of functions from R^n to R, and optimization. Parametric curves will also be discussed.

Topology on R^n.

Standard norms 1, 2, and infinity, and equivalence of these norms. Concepts of open and closed sets, neighborhoods. Definition of continuity of a function of several variables from R^n to R^p, continuity in terms of open sets and neighborhoods.

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

Functions of several variables. (The concept of differentiability will only be covered in L3.)

Directional derivatives, partial derivatives. Representation, contour lines. Gradient of a real-valued function, DL1 if partial derivatives are continuous. Finite increment inequality.

Hessian, DL2, Schwarz's theorem.

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

Least squares methods

Parametric curves

Derivatives of composite functions. Definition, kinematic perspective, examples, representation. Tangent vector, length of C^1 curves, reparameterization. Local study of curves.

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

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

Recommended prerequisites: L1 math

Hourly volumes:

            CM: 24

            TD: 25.5

            TP:

            Land: