# Matematica | MATHEMATICAL ANALYSIS III

## Matematica MATHEMATICAL ANALYSIS III

 0512300008 DIPARTIMENTO DI MATEMATICA EQF6 MATHEMATICS 2022/2023

 OBBLIGATORIO YEAR OF COURSE 2 YEAR OF DIDACTIC SYSTEM 2018 AUTUMN SEMESTER
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/05 5 40 LESSONS COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS MAT/05 3 36 EXERCISES COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
 LYOUBOMIRA SOFTOVA PALAGACHEVA T
ExamDate
APPELLO ANALISI MATEMATICA III21/02/2023 - 09:00
APPELLO ANALISI MATEMATICA III21/02/2023 - 09:00
Objectives
IN THIS COURSE THERE ARE INTRODUCED THE BASIC PORPERTIES OF DIFFERENTIAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES. THE ATTENTION IS FOCUSED ON OPTIMIZATION PROBLEMS. THE BASIC TECHNIQUE FOR SOLVING ELEMENTARY ORDINARY DIFFERENTIAL EQUATIONS ARE GIVEN
ABILITY TO APPLY AWARENESS:
ONE OF THE MAIN PURPOSE OF THE COURSE IS TO ACHIEVE THE STUDENT TO SOLVE OPTIMIZATION PROBLEMS USING THE TECHNICQUE ASSIMILATED.
Prerequisites
BASIC PROPERTIES OF FUNCTIONS OF A REAL VARIABLE: CONTINUITY, DIFFERENTIABILITY, INTEGRABILITY.
Contents
METRIC SPACES, TOPOLOGY AND CONVERGENCE. SEQUENCES IN METRIC SPACES. BANACH THEOREM. (4H LECTURES+4 EXERCISES)

FUNCTIONAL SEQUENCES. POINTWISE AND UNIFORM CONVERGENCE. SERIES OF FUNCTIONS, CONVERGENCE CRITERIA. CONTINUITY, DERIVABILITY AND INTEGRABILITY THEOREMS. POWER SERIES, CONVERGENCE RADIUS. TAYLOR SERIES. FOURIER SERIES. ORTHONORMAL SYSTEMS. UNIFORM AND POINTWISE CONVERGENCE. DIRICHLET THEOREM. (12H LECTURES + 10H EXERCISES)

TOPOLOGY IN RN. FUNCTIONS OF MORE VARIABLES, LIMITS AND CONTINUITY, DERIVABILITY AND DERIVATION RULES. DIFFERENTIABILITY, GRADIENT FORMULA. SCHWARTZ THEOREM. FORMULA OF TAYLOR WITH REST OF LAGRANGE AND PEANO. MINIMUM AND MAXIMUM OF FUNCTIONS OF MORE VARIABLES. . STATIONARY POINTS AND STUDY OF NATURE THROUGH THE FERMAT THEOREM AND THE PROPERTIES OF THE QUADRATIC FORMS, SUFFICIENT CONDITION. LINEAR LEAST SQUARES. (14H LECTURES + 10H EXERCISES)

FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS, EQUATIONS WITH SEPARABLE VARIABLES, LINEAR AND BERNOULLI EQUATIONS, HOMOGENEOUS EQUATIONS. CAUCHY PROBLEM. THEOREM OF EXISTENCE AND UNIQUENESS OF THE SOLUTION OF THE CAUCHY PROBLEM. QUALITATIVE STUDY OF SOLUTIONS OF FIRST ORDER EQUATIONS.
LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS OF HIGHER ORDER. DETERMINANT OF WRONSKI. MATHEMATICAL MODELS. (10H LECTURES + 12H EXERCISES)
Teaching Methods
FRONTAL LECTURES AND EXERCISES
Verification of learning
THE LEARNING VERIFICATION TAKES PLACE THROUGH AN ORAL EXAM AND INCLUDES A WRITTEN EXAM, TO INTEGRATE THE ORAL EXAM. IN PARTICULAR, ON THE BASIS OF METHODOLOGIES, INSTRUMENTS AND CONTENT GIVEN DURING THE LECTURES, THE STUDENT HAVE TO SHOW THAT HE IS ABLE TO UNDERSTAND THE PROBLEM, FIND THE CORRECT MATHEMATICAL-QUANTITATIVE INTERPRETATION, RECOGNIZE THE APPROPRIATE METHOD, UNDERSTAND THE ANSWERS DEDUCED BY THE METHOD AND ITS INFERENCES.
Texts
C. PAGANI, S. SALSA, ANALISI MATEMATICA 1, PP. 496, ZANICHELLI, 2015;
C. PAGANI, S. SALSA, ANALISI MATEMATICA 2, PP. 560, ZANICHELLI, 2016;
M. BRAMANTI, C. PAGANI, S. SALSA, ANALISI MATEMATICA 2, PP. 504, ZANICHELLI, 2009.
M. AMAR, A.M. BERSANI, ESERCIZI DI ANALISI MATEMATICA, PROGETTO LEONARDO, BOLOGNA, 2004
M. BRAMANTI, ESERCITAZIONI DI ANALISI MATEMATICA 2, ESCULAPIO, 2012;
L. MOSCHINI, LEZIONI DI ANALISI MATEMATICA II, ESCULAPIO, 2021;
L. MOSCHINI, ESERCIZI DI ANALISI MATEMATICA II, ESCULAPIO, 2021;