Topology of R^n and Functions of Several Variables

This course will provide 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 covered.

Topology on R^n.

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

Bounds of Sequences and Compactness in R^n; Characterization of Closed Sets via Sequences.

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

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

Hessian, DL2, Schwarz's Theorem.

Optimization of Functions of R^n in R: Free Extremes: The Concept of a Critical Point. Local Extremum, Definition and Necessary Condition. Necessary and Sufficient Conditions for Local Extremes. Examples

Least-Squares Methods

Parameterized Curves

Derivatives of composite functions. Definition, kinematic perspective, examples, representation. Tangent vector, length of C¹ curves, reparameterization. Local analysis of curves

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

An L1 Course on the Analysis of Functions of a Real Variable (HAX103X)

Recommended prerequisites: First-year math

Hourly volumes:

            CM: 24

            TD: 25.5

            Practical Work:

            Lot: