The main goal is to complete the introduction of the basic topics of mathematical analysis for functions of several variables. In particular, the theory of multiple integration and of curve and surface integration is developed. The problem of constrained maxima and minima is sketched. Furthermore, numerical series and series of functions (power series and Fourier series in particular) are studied. Finally, we show how to determine the general solution of a system of linear differential equations of order 2.

The knowledge that will be obtained through this course includes basic tools
of theory of numerical series and series of functions and of the multivariate
integral calculus. In particular, by the end of the course, the student will be able to
handle series of numbers and functions, and will be familiar with the basic tools of
Fourier analysis with discrete frequencies. Moreover, the student will be able to set
out (and, in some cases, also carry out explicitly) the computation of quantities such
as areas and volumes, moments of inertia and centroids of complex planar and solid figures.

Working knowledge is required of mathematical notions and tools introduced during the first year courses. In particular, this course relies upon the basic notions of differential and integral calculus in one variable, linear
algebra, geometry of curves and surfaces, differential calculus in several variables.

Multiple integrals and integral calculus in several variables.
Length of a curve and area of a surface. Curve and surface integration. Green, Gauss and Stokes Theorems. Conservative vector fields. Constrained maxima and minima.
(2 cfu)
Numerical series and series of functions. Taylor expansions and power series. Fourier series. Systems of linear differential equations. (2 cfu)
Exercises (2cfu)

Exercises will be carried out by the teacher at the blackboard. They will cover the topics of the lectures. (2 cfu)

Handbook of teacher’s lessons

A. Bacciotti "Integrali in piu' variabili, Serie" Edizioni CELID, Torino

Other books for personal work and exercices

A. Bacciotti, F. Ricci "Lezioni di Analisi Matematica II", Levrotto e Bella, Torino

C. Canuto, A. Tabacco, Analisi Matematica II, Springer

S. Lancelotti "Lezioni di Analisi Matematica II", Edizioni Celid, Torino

N. Fusco, F. Marcellini, C. Sbordone, Elementi di Analisi Matematica due, Liguori Ed. M. Bramanti, C.D. Pagani, S. Salsa: Matematica, Calcolo infinitesimale e algebra lineare, seconda edizione, Zanichelli

L. Pandolfi, Lezioni di Analisi Matematica 2 (can be freely downloaded from "Portale della didattica")

A. Bacciotti, P. Boieri, D. Farina: Esercizi di calcolo differenziale e integrale in più variabili, Società editrice Esculapio.

F. Marcellini, C. Sbordone, Esercitazioni di matematica, secondo volume prima e seconda parte, Liguori Ed.

S. Salsa - A. Squellati: Esercizi di Analisi Matematica 2, Parte prima, seconda, terza, Masson.

S. Lancelotti "Esercizi di Analisi Matematica II", Edizioni Celid,
Torino

M. Bramanti "Esercitazioni di Analisi Matematica II", Esculapio

C. Canuto, A. Tabacco, Mathematical Analysis II, Springer 2010.
Further course material as lectures on-line, lecture notes, proposed and solved exercises
will be given through the ‘portale della didattica’ website

The final exam consists of two parts, the first part written, the second one oral.
In the written part the solution of practical exercises is required.
The oral part consists in a colloquium about the written part.It will include in addition theoretical questions.
In order to be admitted to the oral part of the exam, students should score 15/30 or higher in the written part.
During the exam, no course material is allowed. It is forbidden to use electronic devices, mobile phones
and scientific calculators.