Integral and differential calculus for functions of several variables. Ordinary differential equation of the first order and of higher order, linear and non-linear. Implicit function Theorem. Curves and surfaces; curvilinear integrals and surface integrals. Gauss-Green formulas and the divergence Theorem. Numerical series. Sequences and series of functions.
Textbook:
N. Fusco, P. Marcellini, C. Sbordone, Analisi matematica due, ed. Liguori
Additional textbooks ì:
E. Giusti, Analisi matematica 2, ed. Boringhieri.
M. Bramanti, C. Pagani, S. Salsa, Analisi matematica 2, ed. Zanichelli
E. Giusti, Esercizi e complementi di analisi matematica, vol. II, ed. Boringhieri
Learning Objectives
Knowledges:
The course aims to provide basic knowledges in the theory of differential and integral calculus for functions of several variables, of ordinary differential equations and of sequences and series of functions.
Expertises:
Students will be able to solve exercises, even of theoretical type, relevant to the topics of the course, and will be acquainted with the main techniques of proof of the theoretical statements viewed in the course.
Prerequisites
Differential and integral calculus for ral functions of 1 real variable. Basic theory of numerical sequences.
Teaching Methods
CFU: 12
Overall amount of hours for the course: 300
Overall amount of hours of individual activity: 170
Overall amount of hours in class (theory and exercises): 130
Overall amount of hours for tests during the course: 9
Office located at: Dipartimento di Matematica "Ulisse Dini"
Viale Morgagni, 67/A
50134 - Firenze (FI)
Type of Assessment
The exam consists of a written and an oral part; the access to the oral part is obtained if the written part is successful.
Several exercises will be proposed in the written part. The exercises aim to evaluate the ability to apply the theoretical and technical knowledge acquired by the student during the course. Special attention, in the evaluation, will be devoted to the correctness of the process, the originality and the effectiveness of the methods adopted for the solution. Some of the exercises, of theoretical nature, will test the acquisition of the main techniques of proof of the theorems presented during the year.
In the oral exam the questions will concerns both the theorems and the exercises. The aim is to test the knowledge and the degree of understanding of the theory presented in class. The competence in the critical presentation of the theory and the use of an suitable mathematical language will be both appreciated.
Course program
Numerical series. Sequences of functions; pointwise and uniform convergence. Theorems concerning continuity, integration and differentiation for sequences of functions. Series of functions; pointwise, uniform and total convergence, and relations among them. 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 point and techniques to identify local maximum and minimum points among critical points. Measure theory in the n-dimensional Euclidean space. Multiple integrals; integrability criteria; reduction formulae for multiple integrals. Ordinari differential equations of the first order; the Cauchy problem; theorems of existence and uniqueness of the solution for the Cauchy problem (local and global). Resolution techniques for 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.