
Course unit
MATHEMATICAL ANALYSIS
SCP4063594, A.A. 2017/18
Information concerning the students who enrolled in A.Y. 2016/17
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Educational activities in elective or integrative disciplines 
MAT/05 
Mathematical Analysis 
9.0 
Course unit organization
Period 
First semester 
Year 
2nd Year 
Teaching method 
frontal 
Type of hours 
Credits 
Teaching hours 
Hours of Individual study 
Shifts 
Practice 
3.0 
34 
41.0 
No turn 
Lecture 
6.0 
48 
102.0 
No turn 
Examination board
Board 
From 
To 
Members of the board 
3 Commissione a.a. 2017/2018 
01/10/2017 
30/09/2018 
TREU
GIULIA
(Presidente)
CESARONI
ANNALISA
(Membro Effettivo)
MANNUCCI
PAOLA
(Membro Effettivo)
PARMEGGIANI
GEMMA
(Membro Effettivo)

Prerequisites:

The topics of the courses of Linear Algebra and Calculus 1. 
Target skills and knowledge:

We will deal with multivariate differential and integral calculus, sequences and series of functions, ordinary differential equations. The students will learn both the theoretical foundations and the practical skills that will let them to solve applicative problems. 
Examination methods:

Written examination. 
Assessment criteria:

Every question of each exercise contributes for a certain specified amount to the highest grade of 33/30 (corresponding to 30 cum laude).
The correctness, accuracy and completeness of the answers given to the different exercises are the criteria for the evaluation. 
Course unit contents:

Sequences and series of functions. Pointwise and uniform convergence of sequences of real functions. The uniform limit of a sequence of continuous functions. Pointwise, uniform and total convergence of series of real functions. Power series, convergence radius. Taylor's series. Analitic functions. Differential calculus for real functions of n real variables. Basics of topology in the ndimensional euclidean space. Open, closed, compact and connected sets. Limits. Algebra of limits. Continuity of a function at a point and in set. Continuity of composed fuction. Weierstrass theorem and connection theorem. Partial and directional derivatives. Higher order derivatives, Hessian matrix, Schwartz theorem. Differential of a function at a point. Chain rule. The problem of finding maxima and minima of a function. First order necessary condition for unconstrained problems. Sufficient conditions. Implicit function theorem. The problem of finding maxima and minima of a function: the case with constraints. Lagrange multipliers theorem. Lebesgue measure theory. The σalgebra of Lebesgue measurable sets. Measurable and integrable functions.Definition of the integral of a function on a measurable set. Properties of the integral. FubiniTonelli's theorem (reduction formula) and the change of variables formula. Ordinary differential equations. 
Planned learning activities and teaching methods:

There will be 82 hours of lectures, at least onethird of them dedicated to numerical and theoretical exercises. 
Additional notes about suggested reading:

Mathematical Analysis. In the MOODLE page of Statistical Science, in the spase devoted to the course, there are the lecture notes, the exercises of past examinations and other staff. The theacher will provide the password to access to the theaching staff. 
Textbooks (and optional supplementary readings) 

P. Marcellini e C. Sbordone, Esercitazioni di Matematica, II vol. Parti prima e seconda. : Liguori, .

N. FUSCO, P. MARCELLINI, C. SBORDONE, Analisi due. : Liguori, .

Michiel Bertsch, Roberta Dal Passo, Lorenzo Giacomelli, Analisi Matematica. : McGraw Hill, .


