
Course unit
MATHEMATICAL ANALYSIS
SCP4063594, A.A. 2019/20
Information concerning the students who enrolled in A.Y. 2018/19
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.2019/20 
01/10/2019 
30/09/2020 
TREU
GIULIA
(Presidente)
CESARONI
ANNALISA
(Membro Effettivo)
MONTEFALCONE
FRANCESCOPAOLO
(Membro Effettivo)
PARMEGGIANI
GEMMA
(Membro Effettivo)

Prerequisites:

Students are requested to know the topics covered by of the courses of Linear Algebra and Calculus 1. 
Target skills and knowledge:

Students will acquire the practical skills related to differential and integral calculus in several variables, to sequences and series of functions and to some types of ordinary differential equations.
Students will also acquire the theoretical foundations of the topics indicated above. This will allow them a methodologically rigorous use of the tools and will help to train their analytical and critical skills. 
Examination methods:

The acquired knowledge will be verified by a a written test. The test has to be completed in two hours and thirty minutes.
The test includes
1) two or three theoretical questions in which the students are asked to correctly report definitions, statements and some simple demonstrations of theorems presented in class;
2) a theoretical question asking the students to elaborate the basic concepts presented in the course;
3) three or four exercises in which it is asked to correctly apply, also from the methodological point of view, the tools presented in the course. 
Assessment criteria:

Each question of each exercise contributes for a certain specified amount to the maximum score of 33/30 (corresponding to 30 and honors).
The correctness, accuracy and completeness of the answers are criteria for a positive evaluation.
In particular, the knowledge of course topics, the acquisition of methodologies, the ability to apply the acquired tools and analytical skills will be evaluated. 
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:

The entire course consists in 82 hours of lectures, at least one third of which is dedicated to exercises.
During the lessons some interactive activities are carried out, if the topics allow it. Such activities, for example, can consist in solving short exercises in pairs or in small groups.
Some online tests will be offered on a regular basis. Students can perform these tests to check their level of learning. Some questions will stimulate a deeper reflection on the topics of the course and will also encourage collaboration among the students.
Students may always ask clarification questions or ask the teacher to develope deeper the topics of the lecture. 
Additional notes about suggested reading:

During the first lesson the teacher will illustrate the reference texts in order to guide the students in the optimal use of them.
Lecture notes, exercises taken from previous examinations, other exercises and any other useful educational material will be uploaded in the Moodle platform of the Departement of Statistical Sciences.
Login to the Moodle platform requires a password that will be communicated by the teacher. 
Textbooks (and optional supplementary readings) 

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

E. Acerbi, G. Buttazzo, Secondo corso di Analisi Matematica. : Pitagora Editrice Bologna, .

Bertsch, Michiel; Dal_Passo, Roberta, Analisi matematicaMichiel Bertsch, Roberta Dal Passo, Lorenzo Giacomelli. Milano: McGraw Hill, 2011.

Innovative teaching methods: Teaching and learning strategies
 Interactive lecturing
 Questioning
 Auto correcting quizzes or tests for periodic feedback or exams
 Active quizzes for Concept Verification Tests and class discussions
 Video shooting made by the teacher/the students
 Use of online videos
 Loading of files and pages (web pages, Moodle, ...)
Innovative teaching methods: Software or applications used
 Moodle (files, quizzes, workshops, ...)
 One Note (digital ink)
 Kaltura (desktop video shooting, file loading on MyMedia Unipd)
 Latex
 Mathematica
Sustainable Development Goals (SDGs)

