Mathematics

Objective: The first part of this course focuses on the real functions of several real variables and particularly on optimization without constraints and under constraints of equality and inequality. Existence theorems, necessary conditions and sufficient conditions of optimality are studied. The second part of the course is devoted to the study of differential systems: linear and non-linear systems, notion of equilibrium and stability of equilibria.

Outline:

I – Reminders

1. Diagonalization of matrices
2. Quadratic forms

II - Real functions of several real variables

1. Topology of R^n
2. Functions of n variables: limit, continuity, partial derivatives, differentiability, partial derivatives seconds ...
3. Concave functions, convex functions. Unconstrainedextrema search
4. Extrema under constraints. An existence theorem: the Weierstrass optimization theorem
5. Extrema under constraints of equality. The Lagrange multipliers
6. Extrema under equality and inequality constraints. The Kuhn and Tucker theorem

III- Dynamic systems

1. Linear differential systems
2. Equilibrium and stability
3. Non-linear differential systems
4. Equilibrium and stability

Grading: 2 continuous assessments during the semester and a final exam at the end of the semester.

Bibliography:

• Hayek N. & Leca J.P : Mathématiques pour l’Économie. Analyse-Algèbre. 5ème édition, Dunod ;
• Michel Ph. : Cours de Mathématiques pour Économistes. Economica.