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. Sequences and series of functions.
E. Giusti, Analisi matematica 2, ed. Boringhieri.
N. Fusco, P. Marcellini, C. Sbordone, Analisi matematica due, ed. Liguori
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
None
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
Contact:
Office located at: Dipartimento di Matematica "Ulisse Dini"
Viale Morgagni, 65
50134 - Firenze (FI)
tel. 055 2751464
e-mail colesant@math.unifi.it
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 succesful. The written part can be replaced by tests made during the course.
Course program
Sequences of functions; pointwise and uniform convergence. Theorems concerning continuity, intergration and differentiation for sequences of functions. Series of functions; pointwise, uniform and total convergence, and relations amnong 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; riduction formulae for multiple integrals. Ordinari differential equations of the first order; the Cauchy problem; theorems of existnce 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.