Series
1. Introduction, Finite and Infinite Series
2. Sums of non negative numbers
3. Rearrangement of series: absolute convergence, semi-convergence
Illustrations: decimal representation of natural numbers, of real numbers, notion of numerical errors
Topology of Normed Vector Spaces
1. Norms, Comparison of Norms
2. Sequences in Normed Spaces and Convergence
3. Topology: open and closed sets, closure, density
4. Limits, continuity of functions, compactness
5. Linear Applications between Normed Vector Spaces
Illustrations: Iterative Methods to solve linear systems, matrix condition number
Several Variable Differential Calculus
1. Derivatives in several variable calculus
2. Partial and directional derivatives
3. Taylor expansion
4. The inverse function theorem, the implicit function theorem
Illustration: Newton method to solve a system of nonlinear equations.
Objectifs
At the end of this module, the student will have understood and be able to explain (main concepts):
• The notion of numerical series and convergence of series
• The notion of differentiability of a function of multiple variables, partial derivatives
• The concept of norms, convergence of sequences in vectorial spaces, basic topological concepts: open and closed sets, compactness
The student will be able to:
Use the main mathematical results to
• Study the convergence of numerical series with majoration and comparison methods
• Study the differentiability of a function of multiple (real) variables, carry out an asymptotic expansion
• Manipulate norms of vectors and matrices, study the convergence of a sequence or function limits in vectors spaces.
Pré-requis
Basic Calculus, Basic Linear algebra
Évaluation
L’évaluation des acquis d’apprentissage est réalisée en continu tout le long du semestre. En fonction des enseignements, elle peut prendre différentes formes : examen écrit, oral, compte-rendu, rapport écrit, évaluation par les pairs…
