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
This course aims at providing basic expertise in the theory of differential and integral calculus for functions of several variables, of ordinary differential equations and of sequences and series of functions.
Students will have to be able to solve exercises, even of theoretical type, on the topics of the course, and will have to be acquainted with the main techniques of proof pertaining the subject of the course.
Prerequisites
Expertise on the topics of the course of Mathemtical Analysis 1
Teaching Methods
CFU: 12
Overall amount of hours for the course: 300
Overall amount of hours of individual study: 180
Overall amount of class hours (theory and exercises): 120
Overall amount of hours for tests during the course: 9
Contact:
Office located at: Dipartimento di Matematica e Informatica "Ulisse Dini"
Viale Morgagni, 67
50134 - Firenze
e-mail cianchi@unifi.it
Type of Assessment
The exam consists of a written test and of an oral interview. The written test can be replaced by intermediate tests during the course.
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.