Sequences and series of functions.
Functions of several variables: limits, derivatives, integrals.
Implicit function Theorem.
Curves and surfaces; curvilinear integrals and surface integrals. Gauss-Green formulas and the divergence theorem.
Ordinary differential equations of the first order and of higher order, linear and non-linear.
C. Pagani, S. Salsa, Analisi matematica 2, ed. Zanichelli
Learning Objectives
The course aims at providing the students with knowledge and understanding in Mathematical Analysis at a level more advanced than that of the course of Mathematical Analysis I. A specific purposes is to let the students develop basic technical expertise, and critical thinking, that are needed in modelling and solving mathematical problems in different settings. Special attention will be paid to help the students develop communication skills necessary for teamwork. The course covers topics and encourages learning ability that are needed, or strongly suggested, to pursue studies in mathematics or in any scientific subject
Prerequisites
Real numbers. Sequences. Functions of one real varaible. Limits of functions. Differential calculus. Integral calculus. Series.
Teaching Methods
Lectures: presentation of the theory described in the course program, with teacher-student direct interaction, to ensure a full understanding of the subject.
Training sessions: training of the students to modelling and solving a wide selection of problems of mathematical analysis.
The training sessions are aimed at:
-- helping the students develop communication skills and apply theoretical notions;
-- improve their independence in judgement.
Supplementary exercise sheets and samples of final examinations are distributed.
Type of Assessment
Intermediate and final written examination: A selection of exercises is proposed. The problems are designed to assess the ability of the students to apply their skills to problem modelling and solving. In the evaluation, special attention is paid to the correctness of the solution procedure, as well as to the originality and effectiveness of the methods adopted.
Oral examination: A number of questions are posed. The oral examination is designed to evaluate the degree of understanding of the theory presented in the course. In the assessment, special attention is paid to communication ability, critical thinking and appropriate use of mathematical language.
Course program
Sequences of functions; pointwise and uniform convergence. Theorems on continuity, differentiation and integration for sequences of functions. Series of functions; pointwise, uniform and total convergence. Power series; Taylor expansion; analytic functions and their properties. Functions of several variable; continuity; partial and directional derivatives; differentiability. Optimization for functions of several variables; critical points and local maxima and minima. Basic measure theory in the n-dimensional Euclidean space. Multiple integrals; integrability criteria; riduction formulae for multiple integrals. Ordinari differential equations of the first order; the Cauchy problem; theorems on existence and uniqueness of the solution for the Cauchy problem (local and global). Techniques for solving first-order equations. Linear ordinary differential equations of higher-order; algebraic description of the space of the solutions. Study of some special type of linear equations. Implicit function theorem. Curves and surfaces in the 2- and 3-dimensional space; curvilinear and surface integrals; differential forms. Divergence Theorem and Gauss-Green formulas.