The n-dimensional Euclidean space.
Differential calculus for (scalar- and vector-valued) functions of several variables.
Free and constrained optimization problems.
Multiple integrals.
Curves. curvilinear integrals.
Parametric surfaces, surface and flow integrals.
Function spaces. Sequences and series of functions, power series.
Fourier series.
Suggested textbook (theoretical part)
C. Pagani, S. Salsa, Analisi matematica Vol. 2, Zanichelli 2009.
Suggested textbooks (exercises)
S. Salsa, A. Squellati, Esercizi di Analisi Matematica 2, Zanichelli, 2011.
P. Marcellini, C. Sbordone, Esercitazioni di Analisi Matematica 2, prima e seconda parte, Liguori, 2017.
Other textbooks (theoretical part)
N. Fusco, P. Marcellini, C. Sbordone, Analisi matematica 2, Liguori, 2016
Learning Objectives
Knowledge: differential and integral calculus for functions of several variables; free and constrained optimisation for functions of several variable; curves and surfaces; Gauss-Green formulas and the divergence theorem; sequences and series of functions; Fourier series.
Skills: autonomy in proposing, articulating and rigorously supporting arguments for the resolution of problems related to the listed subjects (see Knowledge); confident use of symbols and main results; control of errors.
Abilities/capacities: consolidated communication skills in writing.
Prerequisites
Differential and integral calculus for single-variable functions. Numerical sequences and infinite series. Basics on Ordinary Differential Equations. Linear algebra and analytic geometry.
Prerequisite (formal): "Analisi Matematica I". Recommended: "Geometria".
Teaching Methods
Lessons, in the absence of a rigid separation between theory and practice.
Further information
The teachers will use Moodle platform as a help for the course, and all students will be invited to register to the Moodle course page. On this page the teacher will upload, along the course, exercises that will be subsequently solved and explained during the lectures.
Type of Assessment
The final examination is formed of two written parts. The first part consist in solving some exercises such as, for instance: computation of integrals of functions of two or three variables; optimisation of functions of several variables; study of a given curve of surface; study of a series of functions.
In the second part the student is required to answer some theory questions concerning the content of the course. The second part is reserved to those students who passed the first part. The final mark is an appropriate mean of the marks of the two parts.
During the course, if possible, there will be intermediate tests which will give direct access to the second part of the exam.
Course program
The n-dimensional Euclidean space. Scalar product, Euclidean norm, Cauchy-Schwartz inequality, subadditivity of the norm. Vector product. Topology: open, closed and compact sets.
Functions of several variables. Limits; continuity. Partial derivatives, gradient, directional derivatives. Differentiability. Second derivatives, Hessian matrix. Optimisation: point of relative and absolute maximum and minimum. Techniques to identify free and constrained maxima and minima.
Integral calculus for functions of several variables (only the cases of two and three variables). Definition and fundamental properties of the integral. Integrability of some classes of functions. Normal domains and reduction formulas for multiple integrals. Change of variables in multiple integrals; polar coordinates.
Parametric curves. Support and orientation. Regular curves, tangent vector. Length of a curve and curvilinear integrals. The Gauss-Green formulas.
Surfaces in the three-dimensional space. Regular surfaces; tangent plane and normal vector. Area of a surface and surface integrals. The divergence theorem.
Sequences and series of functions. Point-wise and uniform convergence. Total convergence of series of functions. Power series. Fourier series.