MATHEMATICAL ANALYSIS IV

Matematica MATHEMATICAL ANALYSIS IV

0512300011
DIPARTIMENTO DI MATEMATICA
EQF6
MATHEMATICS
2021/2022

OBBLIGATORIO
YEAR OF COURSE 2
YEAR OF DIDACTIC SYSTEM 2018
SPRING SEMESTER
CFUHOURSACTIVITY
864LESSONS
Objectives
IIN THE COURSE THERE ARE INTRODUCED THE BASIC PORPERTIES OF MULTIPLE INTEGRALS. MAIN ATTENTION IS FOCUSED ON INTEGRATION OF FUNCTIONS OF TWO VARIABLES, DIVERGENCE THEOREM AND GAUSS-GREEN FORMULA. DIFFERENTIAL FORMS, STOKES FORMULA AND OPTIMIZATION PROBLEMS USING LAGRANGE MULTIPLIERS.

ONE OF THE MAIN PURPOSE OF THE COURSE IS TO DEVELOPE THE ABILITY OF THE STUDENT TO SOLVE PROBLEMS USING THE TECHNICQUE OF THE CALCULUS OF FUNCTIONS OF MORE VARIABLES.
Prerequisites
BASIC PROPERTIES OF FUNCTIONS OF SEVERAL REAL VARIABLE: CONTINUITY, DIFFERENTIABILITY.
Contents
CURVES, RECTIFABLE CURVES, NATURAL PARAMETER. CURVILINEAR INTEGRALS, APPLICATIONS. MASS AND CENTER OF GRAVITY OF A MATERIAL WIRE. (8H LESSONS + 4H EXERCISES)

VECTOR-VALUED FUNCTIONS OF SEVERAL VARIABLES, GENERALITY. SURFACES, CONTINUITY, DIFFERENTIABILITY. REGULAR SURFACES, IMPLICIT FUNCTIONS, DINI'S THEOREMS.
LAGRANGE MULTIPLIERS.
(6H LESSONS + 2H EXERCISES)

LINEAR DIFFERENTIAL FORMS. VECTOR FIELDS. CURVILINEAR INTEGRALS OF LINEAR DIFFERENTIAL FORMS. CONSERVATION FIELDS, COLCULUS OF THE POTENTIAL. SOLENOIDAL FIELDS. CLOSED AND EXACT DIFFERENTIAL FORMS. (10H LESSONS + 4H EXERCISES)

DOUBLE INTEGRALS, DEFINITION. DOUBLE INTEGRALS ON NORMAL DOMAINS. REDUCTION FORMULAS. CHANGE OF VARIABLES. GENERALIZED DOUBLE INTEGRALS. TRIPLE INTEGRALS. CHANGE OF VARIABLES. APPLICATIONS, MASS CALCULATION, CENTER OF GRAVITY AND MOMENT OF INERTIA OF A NON-HOMOGENEOUS PLATE AND BODY.
MULTIPLE INTEGRALS, GENERALIZED MULTIPLE INTEGRALS. EULER'S FUNCTIONS GAMMA AND BETTA. (16H LESSONS + 4H EXERCISES)


SURFACES AND INTEGRALS OVER SURFACES. REGULAR SURFACES. LOCAL COORDINATES AND REPARAMETRIZATION.
TANGENT PLANE AND NORMAL VERSOR. CALCULUS OF THE AREA OF SURFACES.
SURFACE INTEGRALS. FORMULAS OF GAUSS-GREEN AND STOKES. DIVERGENCE THEOREM. (8H LESSONS + 2H EXERCISES)
Teaching Methods
FRONTAL LECTURES. PRACTICE LECTURES.
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 LESSONS, THE STUDENT MUST DEMONSTRATE 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
S. SALSA, A. SQUELLATI, ESERCIZI DI ANALISI MATEMATICA VOL. 2, ZANICHELLI, 2011
P. MARCELLINI, C. SBORDONE, ESERCITAZIONI DI ANALISI MATEMATICA 2, ZANICHELLI, 2017
M. BRAMANTI, ESERCITAZIONI DI ANALISI MATEMATICA 2, ESCULAPIO, 2012
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