CRISTIAN TACELLI | MATHEMATICAL ANALYSIS IV

cod. 0512300011

MATHEMATICAL ANALYSIS IV

	0512300011
	DEPARTMENT OF MATHEMATICS
	EQF6
	MATHEMATICS
	2024/2025

PERCORSO STANDARD EDUCATIONAL PATH

	OBBLIGATORIO
	YEAR OF COURSE 2
	YEAR OF DIDACTIC SYSTEM 2018
	SPRING SEMESTER

TEACHING UNITS

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

	Objectives
	THE COURSE PROVIDES THE BASIC KNOWLEDGE ABOUT INTEGRAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES. KNOWLEDGE AND UNDERSTANDING: THE COURSE PROVIDES THE BASIC KNOWLEDGE ABOUT INTRODUCTION TO THE THEORY OF MULTIPLE INTEGRALS. FURTHERMORE, THE SAME COURSE INTRODUCES KNOWLEDGE OF THE DIFFERENTIAL FORMS OF VECTOR FIELDS AND OF THE PROBLEMS OF MAXIMUM AND MINIMUM WITH CONSTRAINTS. ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING: THE GOAL OF THE COURSE IS TO MAKE THE STUDENT ABLE TO SOLVING PROBLEMS OF MAXIMUM AND MINIMUM AND TO BE ABLE TO CALCULATE INTEGRALS ON DOMAINS OF SPACE AND PLANE, ON CURVES AND SURFACES AND FINALLY TO RECOGNIZE THE PROPERTIES OF A VECTOR FIELD BY APPLYING THE THEORETICAL KNOWLEDGE ACQUIRED.

THE COURSE PROVIDES THE BASIC KNOWLEDGE ABOUT INTEGRAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES.
KNOWLEDGE AND UNDERSTANDING:
THE COURSE PROVIDES THE BASIC KNOWLEDGE ABOUT INTRODUCTION TO THE THEORY OF MULTIPLE INTEGRALS. FURTHERMORE, THE SAME COURSE INTRODUCES KNOWLEDGE OF THE DIFFERENTIAL FORMS OF VECTOR FIELDS AND OF THE PROBLEMS OF MAXIMUM AND MINIMUM WITH CONSTRAINTS.
ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING: THE GOAL OF THE COURSE IS TO MAKE THE STUDENT ABLE TO SOLVING PROBLEMS OF MAXIMUM AND MINIMUM AND TO BE ABLE TO CALCULATE INTEGRALS ON DOMAINS OF SPACE AND PLANE, ON CURVES AND SURFACES AND FINALLY TO RECOGNIZE THE PROPERTIES OF A VECTOR FIELD BY APPLYING THE THEORETICAL KNOWLEDGE ACQUIRED.

	Prerequisites
	KNOWLEDGE OF THE THEORY AND APPLICATIONS OF INTEGRAL AND DIFFERENTIAL CALCULUS FOR FUNCTIONS OF ONE VARIABLE.

	Contents
	- PEANO-JORDAN MEASURE AND RIEMANN INTEGRALS OF FUNCTIONS OF SEVERAL VARIABILE (THEORY 9 HOURS, EXERCISES 11 HOURS) INTEGRALS OF FUNCTIONS OF SEVERAL VARIABLES. PEANO-JORDAN MEASURE. MEASURE OF INTERVALS AND MULTI-INTERVALS. INDEPENDENCE OF THE MEASURE FROM THE PARTITION. FINITE ADDITIVITY PROPERTY ON MULTIINTERVALS AND CONSEQUENCES. MEASURE OF A BOUNDED SET. CHARACTERIZATIONS OF A MEASURABLE SET. FINITE ADDITIVITY PROPERTY AND CONSEQUENCES. MEASURABILITY OF THE CLOSURE AND INTERIOR OF MULTIINTERVALS. NECESSARY AND SUFFICIENT CONDITION ON THE BOUNDARY OF A SET FOR ITS MEASURABILITY. RIEMANN INTEGRAL. SIMPLE FUNCTIONS AND INTEGRAL OF SIMPLE FUNCTIONS. INTEGRABILITY OF A BOUNDED FUNCTION. FUBINI THEOREM. APPLICATION OF FUBINI'S THEOREM. NORMAL DOMAINS. REDUCTION FORMULAS. MONOTONICITY OF THE INTEGRAL. CHARACTERIZATION OF INTEGRABLE FUNCTIONS. RELATIONSHIP BETWEEN MEASURABILITY OF A SET AND INTEGRABILITY OF ITS CHARACTERISTIC FUNCTION. MEASURABILITY OF THE GRAPH OF AN INTEGRABLE FUNCTION. INTEGRABILITY OF A CONTINUOUS FUNCTION. FUBINI THEOREM. APPLICATION OF THE FUBINI THEOREM. MEASUREMENT OF THE CIRCLE, SPHERE, HYPERSPHERE. NORMAL DOMAINS IN DIMENSION 3. CHANGE OF VARIABLES THEOREM. EXAMPLE CHANGE OF VARIABLES. POLAR, CYLINDRICAL AND SPHERICAL COORDINATES. VOLUME OF A SOLID OF ROTATION. - CURVES AND SURFACE (THEORY 9 HOURS, EXERCISES 9 HOURS) DEFINITION OF CURVE. REGULAR CURVES. CLOSED, SIMPLE, REGULAR CURVES. EQUIVALENT CURVES. RELATIONSHIP BETWEEN THE RANG OF A REGULAR CURVE AND THE GRAPH OF A FUNCTION. RECTIFIABLE CURVES. LENGTH OF A CURVE. THEOREM THE LENGTH OF A CURVE. LENGTH OF EQUIVALENT CURVES. CURVILINEAR INTEGRAL. REGULAR SURFACES. NORMAL VECTOR AND PLANE TANGENT TO A SURFACE. COORDINATED CURVES. EQUIVALENT SURFACES. AREA OF A SURFACE AND SURFACE INTEGRALS. - IMPLIED FUNCTIONS AND LAGRANGE MULTIPLIERS (THEORY 9 HOURS, EXERCISES 9 HOURS) IMPLICIT FUNCTION THEOREM. CALCULATION OF THE FIRST AND SECOND DERIVATIVES AND OF THE MAXIMA AND MINIMA OF AN IMPLICITLY DETERMINED FUNCTION. IMPLICIT FUNCTION THEOREM IN THE GENERAL CASE. CONSTRAINED MAXIMA AND MINIMA OF A FUNCTION. LAGRANGE MULTIPLIER METHOD. LOCAL INVERTIBILITY THEOREM. - LINEAR DIFFERENTIAL FORM AND VECTOR FIELD (THEORY 9 HOURS, EXERCISES 11 HOURS) DEFINITION. EXACT DIFFERENTIAL FORMS. CURVILINEAR INTEGRAL OF A DIFFERENTIAL FORM. INTEGRAL OVER EQUIVALENT CURVES. CURVILINEAR INTEGRAL OF AN EXACT DIFFERENTIAL FORM. CLOSED DIFFERENTIAL FORMS. RELATIONSHIP BETWEEN EXACT AND CLOSE FORM CLOSED FORM IN STAR SETS. METHODS FOR CALCULATING THE PRIMITIVE. REGULAR SETS. GAUSS-GREEN FORMULAS IN TWO DIMENSIONS. DIVERGENCE THEOREM IN TWO DIMENSIONS. CLOSED FORM IN SIMPLY CONNECTED OPEN SET. BOUNDARY OF A SURFACES. ROTOR FLUX OF A VECTOR FIELD THROUGH A SURFACE. STOKES THEOREM.

Contents

- PEANO-JORDAN MEASURE AND RIEMANN INTEGRALS OF FUNCTIONS OF SEVERAL VARIABILE (THEORY 9 HOURS, EXERCISES 11 HOURS)
INTEGRALS OF FUNCTIONS OF SEVERAL VARIABLES. PEANO-JORDAN MEASURE. MEASURE OF INTERVALS AND MULTI-INTERVALS. INDEPENDENCE OF THE MEASURE FROM THE PARTITION. FINITE ADDITIVITY PROPERTY ON MULTIINTERVALS AND CONSEQUENCES. MEASURE OF A BOUNDED SET.
CHARACTERIZATIONS OF A MEASURABLE SET. FINITE ADDITIVITY PROPERTY AND CONSEQUENCES. MEASURABILITY OF THE CLOSURE AND INTERIOR OF MULTIINTERVALS. NECESSARY AND SUFFICIENT CONDITION ON THE BOUNDARY OF A SET FOR ITS MEASURABILITY.
RIEMANN INTEGRAL. SIMPLE FUNCTIONS AND INTEGRAL OF SIMPLE FUNCTIONS. INTEGRABILITY OF A BOUNDED FUNCTION.
FUBINI THEOREM. APPLICATION OF FUBINI'S THEOREM. NORMAL DOMAINS. REDUCTION FORMULAS.
MONOTONICITY OF THE INTEGRAL. CHARACTERIZATION OF INTEGRABLE FUNCTIONS. RELATIONSHIP BETWEEN MEASURABILITY OF A SET AND INTEGRABILITY OF ITS CHARACTERISTIC FUNCTION. MEASURABILITY OF THE GRAPH OF AN INTEGRABLE FUNCTION.
INTEGRABILITY OF A CONTINUOUS FUNCTION. FUBINI THEOREM.
APPLICATION OF THE FUBINI THEOREM. MEASUREMENT OF THE CIRCLE, SPHERE, HYPERSPHERE. NORMAL DOMAINS IN DIMENSION 3.
CHANGE OF VARIABLES THEOREM. EXAMPLE
CHANGE OF VARIABLES. POLAR, CYLINDRICAL AND SPHERICAL COORDINATES. VOLUME OF A SOLID OF ROTATION.

- CURVES AND SURFACE (THEORY 9 HOURS, EXERCISES 9 HOURS)
DEFINITION OF CURVE. REGULAR CURVES. CLOSED, SIMPLE, REGULAR CURVES. EQUIVALENT CURVES. RELATIONSHIP BETWEEN THE RANG OF A REGULAR CURVE AND THE GRAPH OF A FUNCTION.
RECTIFIABLE CURVES. LENGTH OF A CURVE. THEOREM THE LENGTH OF A CURVE. LENGTH OF EQUIVALENT CURVES. CURVILINEAR INTEGRAL.
REGULAR SURFACES. NORMAL VECTOR AND PLANE TANGENT TO A SURFACE. COORDINATED CURVES. EQUIVALENT SURFACES. AREA OF A SURFACE AND SURFACE INTEGRALS.

- IMPLIED FUNCTIONS AND LAGRANGE MULTIPLIERS (THEORY 9 HOURS, EXERCISES 9 HOURS)
IMPLICIT FUNCTION THEOREM. CALCULATION OF THE FIRST AND SECOND DERIVATIVES AND OF THE MAXIMA AND MINIMA OF AN IMPLICITLY DETERMINED FUNCTION. IMPLICIT FUNCTION THEOREM IN THE GENERAL CASE.
CONSTRAINED MAXIMA AND MINIMA OF A FUNCTION. LAGRANGE MULTIPLIER METHOD.
LOCAL INVERTIBILITY THEOREM.

- LINEAR DIFFERENTIAL FORM AND VECTOR FIELD (THEORY 9 HOURS, EXERCISES 11 HOURS)
DEFINITION. EXACT DIFFERENTIAL FORMS. CURVILINEAR INTEGRAL OF A DIFFERENTIAL FORM. INTEGRAL OVER EQUIVALENT CURVES.
CURVILINEAR INTEGRAL OF AN EXACT DIFFERENTIAL FORM.
CLOSED DIFFERENTIAL FORMS. RELATIONSHIP BETWEEN EXACT AND CLOSE FORM CLOSED FORM IN STAR SETS.
METHODS FOR CALCULATING THE PRIMITIVE.
REGULAR SETS. GAUSS-GREEN FORMULAS IN TWO DIMENSIONS. DIVERGENCE THEOREM IN TWO DIMENSIONS. CLOSED FORM IN SIMPLY CONNECTED OPEN SET.
BOUNDARY OF A SURFACES. ROTOR FLUX OF A VECTOR FIELD THROUGH A SURFACE. STOKES THEOREM.

	Teaching Methods
	LECTURES (40 HOURS) AND EXERCISES (36 HOURS)

	Verification of learning
	THE FINAL EXAM IS DESIGNED TO EVALUATE AS A WHOLE: THE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED DURING THE COURSE, THE MASTERY OF THE MATHEMATICAL LANGUAGE IN THE WRITTEN AND ORAL TEST, THE SKILL OF PROVING THEOREMS, THE SKILL OF SOLVING EXERCISES, THE ABILITY TO IDENTIFY AND APPLY THE BEST AND MORE EFFICIENT METHOD IN EXERCISES SOLVING, THE ABILITY TO USE THE ACQUIRED KNOWLEDGE. THE EXAM CONSISTS OF A WRITTEN TEST (3 HOURS) AND AN ORAL EXAMINATION. WRITTEN TEST: THE WRITTEN TEST CONSISTS IN SOLVING 4 TYPICAL PROBLEMS PRESENTED IN THE COURSE CONCERNING -) MULTIPLE INTEGRALS -) IMPLIED FUNCTIONS AND LAGRANGE MULTIPLIERS -) DIFFERENTIAL FORM -) FLOW THROW SURFACES THE TOOLS USED IN SOLVING THE EXERCISES AND THE CLARITY OF ARGUMENTATION WILL BE TAKEN INTO CONSIDERATION IN THE EVALUATION. THERE WILL BE THREE MID TERM TESTS CONCERNING THE TOPICS ALREADY PRESENTED IN THE COURSE, WHICH IN CASE OF A SUFFICIENT MARK, WILL EXEMPT THE STUDENTS ON THIS TOPICS AT THE WRITTEN TEST. THE WRITTEN TEST WILL BE EVALUATED IN THIRTIETHS. THE MINIMUM REQUIREMENT FOR PASSING THE WRITTEN TEST IS THE ACHIEVEMENT OF A TOTAL SCORE OF 16/30 AND THE MINIMUM KNOWLEDGE (DEFINITIONS AND BASIC RULES) ON EVERY SINGLE EXERCISE. THE ORAL TEST IS DEVOTED TO EVALUATE THE DEGREE OF KNOWLEDGE AND MASTERY IN ALL THE TOPICS OF THE COURSE, AS DEFINITIONS, AS PROOFS OF THEOREMS AND IN SOLVING EXERCISES. THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY WITH LAUDE), DEPENDS ON THE GLOBAL EVALUATION OF THE STUDENT. TO ACHIEVE LAUDE IT IS NECESSARY TO PASS THE WRITTEN TEST WITH A MINIMUM SCORE OF 29/30.

Verification of learning

THE FINAL EXAM IS DESIGNED TO EVALUATE AS A WHOLE: THE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED DURING THE COURSE, THE MASTERY OF THE MATHEMATICAL LANGUAGE IN THE WRITTEN AND ORAL TEST, THE SKILL OF PROVING THEOREMS, THE SKILL OF SOLVING EXERCISES, THE ABILITY TO IDENTIFY AND APPLY THE BEST AND MORE EFFICIENT METHOD IN EXERCISES SOLVING, THE ABILITY TO USE THE ACQUIRED KNOWLEDGE.

THE EXAM CONSISTS OF A WRITTEN TEST (3 HOURS) AND AN ORAL EXAMINATION.
WRITTEN TEST: THE WRITTEN TEST CONSISTS IN SOLVING 4 TYPICAL PROBLEMS PRESENTED IN THE COURSE CONCERNING
-) MULTIPLE INTEGRALS
-) IMPLIED FUNCTIONS AND LAGRANGE MULTIPLIERS
-) DIFFERENTIAL FORM
-) FLOW THROW SURFACES

THE TOOLS USED IN SOLVING THE EXERCISES AND THE CLARITY OF ARGUMENTATION WILL BE TAKEN INTO CONSIDERATION IN THE EVALUATION. THERE WILL BE THREE MID TERM TESTS CONCERNING THE TOPICS ALREADY PRESENTED IN THE COURSE, WHICH IN CASE OF A SUFFICIENT MARK, WILL EXEMPT THE STUDENTS ON THIS TOPICS AT THE WRITTEN TEST. THE WRITTEN TEST WILL BE EVALUATED IN THIRTIETHS.
THE MINIMUM REQUIREMENT FOR PASSING THE WRITTEN TEST IS THE ACHIEVEMENT OF A TOTAL SCORE OF 16/30 AND THE MINIMUM KNOWLEDGE (DEFINITIONS AND BASIC RULES) ON EVERY SINGLE EXERCISE.
THE ORAL TEST IS DEVOTED TO EVALUATE THE DEGREE OF KNOWLEDGE AND MASTERY IN ALL THE TOPICS OF THE COURSE, AS DEFINITIONS, AS PROOFS OF THEOREMS AND IN SOLVING EXERCISES.
THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY WITH LAUDE), DEPENDS ON THE GLOBAL EVALUATION OF THE STUDENT.
TO ACHIEVE LAUDE IT IS NECESSARY TO PASS THE WRITTEN TEST WITH A MINIMUM SCORE OF 29/30.

	Texts
	ANALISI MATEMATICA 2, C. PAGANI - S. SALSA, ZANICHELLI, 2016 ESERCIZI DI ANALISI MATEMATICA 2, S. SALSA-A. SQUELLATI, ZANICHELLI, 2011 ESERCITAZIONI DI ANALISI MATEMATICA 2, P. MARCELLINI - C. SBORDONE, ZANICHELLI, 2017 ESERCITAZIONI DI ANALISI MATEMATICA 2, M. BRAMANTI, PROGETTO LEONARDO, BOLOGNA, 2012 ESERCIZI DI ANALISI MATEMATICA, M. AMAR - A.M. BERSANI, PROGETTO LEONARDO, BOLOGNA, 2004

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	Email: ctacelli@unisa.it

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CRISTIAN TACELLI | MATHEMATICAL ANALYSIS IV

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CRISTIAN TACELLI | MATHEMATICAL ANALYSIS IV