#### Objective

: The aim of this course is to familiarize students with the functions of two or three variables and with vector functions. It aims also to familiarize them with the notion of partial derivatives of order 1and 2, then to find the extrema of functions of two variables. It prepares also students to deal with vector functions of n variables and differentials of order 1.
Pedagogical Objectives:
1. Knowledge of the elementary topology of IR2 and of IR3.
2. Knowledge of the usual norms on IR2 and IR3.
3. Manipulate the functions of 2 or 3 variables.
4. Manipulate vector function of a real variable.
5. Find the limit and study the continuity of a function of 2 or 3 variables.
6. Compute the partial derivatives of order 1 and 2.
7. Compute the directional derivatives.
8. Find the local extremum of a function of 2 variables.
9. Compute the differential of a vector function of 2 variables.
10. Compute the differential of a vector function of 2 variables using the Jacoby matrix.
11. Solve the equation f(x,y)=0 using implicit function theorem.

#### Content

1. Convergence in IRn (n=2 et n=3) : Norm sup and Euclidian norm in IRn, Neighborhoods, convex subsets.
2. Vector function of a real variable : limit, continuity, derivability, derivatives of superior order, Taylor formula.
3. Real functions of many real variables (n=2, n=3) : limit, continuity, partial first derivatives, directional derivatives, gradient, derivative of a composed function, derivative of superior order, tangent plane, differentiability of order 1, mean-value theorem, Taylor formula, extremum.
4. Equation of the form f(x,y)=0 and implicit functions.
5. Mappings from IRn into IRm (n=2,3 ; m=2,3) : Continuity, partial derivatives, Jacoby matrix of a composed function.
Comment: The notions in this course are in majority known by the students for functions of one real variable. Thus, this course constitute a generalization of already known notions.
It prepares students for the second year courses, especially the calculus of triple integrals.