Differentiation of functions of several real variables

Description

Subject objectives: In this basic course, the theoretical basis and the basic skills of differential calculus in the area of real multivariate functions will be provided, a tool of the utmost relevance within the Mathematics Degree. Particularly, in the context of functions defined in a subset of R^n into R^m, students will learn how to generalize the related concepts and results already known for real functions of one variable, as well as the main analogies and differences, towards the analysis of the former class of functions. Contents: 1. Limits and continuity of functions from a subset of R^n into R^m [3 h CLE] - Directional, iterated and subset limits of scalar functions - Continuity of scalar functions - The case of vector valued functions 2. First order differentiability of scalar functions of several real variables [5h CLE] - Partial and directional derivatives - Differential of a function and differentiable functions - Necessary and sufficient conditions of differentiability - Gradient vector 3. First order differentiability of vector valued functions of several real variables [4h CLE] - Jacobian matrix - Differentiation rules 4. The mean value theorem [2h CLE] - The mean value theorem for scalar functions - A generalization to vector fields 5. Higher order differentiability [4h CLE] - Second order differential and Hessian matrix - Symmetry of the second-order differential - Class k functions 6. Taylor's theorem and its consequences [4h CLE] - Taylor polynomials - Approximation of regular functions by polynomials: Taylor's theorem - Computation of the extrema of a scalar function 7. The implicit function theorem and the inverse function theorem [4h CLE] - Solving equations locally: the implicit function theorem - Inverting functions locally: the inverse function theorem 8. Applications of the implicit function theorem and the inverse function theorem [2h CLE] - Change of variables - Computation of constrained extrema of a scalar function - Geometrical problems