Tiziana DURANTE | ANALISI MATEMATICA 2

cod. 0612700113

ANALISI MATEMATICA 2

	0612700113
	DIPARTIMENTO DI INGEGNERIA DELL'INFORMAZIONE ED ELETTRICA E MATEMATICA APPLICATA
	EQF6
	COMPUTER ENGINEERING
	2018/2019

PERCORSO COMUNE

	OBBLIGATORIO
	YEAR OF COURSE 2
	YEAR OF DIDACTIC SYSTEM 2017
	PRIMO SEMESTRE

TEACHING UNITS

SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
MAT/05	4	32	LESSONS	BASIC COMPULSORY SUBJECTS
MAT/05	2	16	EXERCISES	BASIC COMPULSORY SUBJECTS

PROFESSORS

	TIZIANA DURANTE T Curriculum
	ROSANNA MANZO Curriculum

	Objectives
	LEARNING OUTCOMES: THE COURSE AIMS AT THE ACQUISITION OF FURTHER ELEMENTS OF MATHEMATICAL ANALYSIS. THE OBJECTIVES OF THE COURSE ARE THE ACQUISITION OF RESULTS AND PROOF TECHNIQUES, AND THE ABILITY TO USE ITS CALCULATION TOOLS. KNOWLEDGE AND UNDERSTANDING: THE TEACHING AIMS AT ACQUIRING THE FOLLOWING ELEMENTS OF MATHEMATICAL ANALYSIS: FUNCTION SEQUENCES AND SERIES, FOURIER SERIES, MULTIVARIABLE FUNCTIONS, DIFFERENTIAL EQUATIONS, COMPLEX FUNCTIONS OF COMPLEX VARIABLES. THE SPECIFIC LEARNING OUTCOMES ACTUALLY CONSIST IN THE ACHIEVEMENT OF RESULTS AND PROOF TECHNIQUES, AS WELL AS IN THE ABILITY TO SOLVE EXERCISES AND TO DEAL, IN A CONSTRUCTIVE MANNER, WITH ADVANCED TEXTBOOKS FOR A SUFFICIENTLY INDEPENDENT APPROACH TO THE PROBLEM SOLVING. APPLYING KNOWLEDGE AND UNDERSTANDING: BEING ABLE TO APPLY THEOREMS AND RULES IN PROBLEMS SOLVING. BEING ABLE TO COMPUTE THE FOURIER SERIES EXPANSION OF A FUNCTION. BEING ABLE TO CALCULATE PARTIAL AND DIRECTIONAL DERIVATIVES, LOCAL MINIMA AND MAXIMA FOR MULTIVARIABLE FUNCTIONS. BEING ABLE TO SOLVE DIFFERENTIAL EQUATIONS. BEING ABLE TO SOLVE EXERCISES OF COMPLEX ANALYSIS. BEING ABLE TO IDENTIFY THE BEST METHODS TO EFFICIENTLY SOLVE A MATHEMATICAL PROBLEM. BEING ABLE TO APPLY THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE LESSONS.

LEARNING OUTCOMES:
THE COURSE AIMS AT THE ACQUISITION OF FURTHER ELEMENTS OF MATHEMATICAL ANALYSIS. THE OBJECTIVES OF THE COURSE ARE THE ACQUISITION OF RESULTS AND PROOF TECHNIQUES, AND THE ABILITY TO USE ITS CALCULATION TOOLS.
KNOWLEDGE AND UNDERSTANDING:
THE TEACHING AIMS AT ACQUIRING THE FOLLOWING ELEMENTS OF MATHEMATICAL ANALYSIS: FUNCTION SEQUENCES AND SERIES, FOURIER SERIES, MULTIVARIABLE FUNCTIONS, DIFFERENTIAL EQUATIONS, COMPLEX FUNCTIONS OF COMPLEX VARIABLES. THE SPECIFIC LEARNING OUTCOMES ACTUALLY CONSIST IN THE ACHIEVEMENT OF RESULTS AND PROOF TECHNIQUES, AS WELL AS IN THE ABILITY TO SOLVE EXERCISES AND TO DEAL, IN A CONSTRUCTIVE MANNER, WITH ADVANCED TEXTBOOKS FOR A SUFFICIENTLY INDEPENDENT APPROACH TO THE PROBLEM SOLVING.
APPLYING KNOWLEDGE AND UNDERSTANDING:
BEING ABLE TO APPLY THEOREMS AND RULES IN PROBLEMS SOLVING.
BEING ABLE TO COMPUTE THE FOURIER SERIES EXPANSION OF A FUNCTION.
BEING ABLE TO CALCULATE PARTIAL AND DIRECTIONAL DERIVATIVES, LOCAL MINIMA AND MAXIMA FOR MULTIVARIABLE FUNCTIONS.
BEING ABLE TO SOLVE DIFFERENTIAL EQUATIONS.
BEING ABLE TO SOLVE EXERCISES OF COMPLEX ANALYSIS.
BEING ABLE TO IDENTIFY THE BEST METHODS TO EFFICIENTLY SOLVE A MATHEMATICAL PROBLEM.
BEING ABLE TO APPLY THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE LESSONS.

	Prerequisites
	PREREQUISITES: FOR A SUCCESSFUL ACHIEVEMENT OF THE PROPOSED GOALS AND, IN PARTICULAR, FOR A PROPER UNDERSTANDING OF THE CONTENTS SCHEDULED FOR THE TEACHING, STUDENTS ARE REQUIRED TO MASTER THE FOLLOWING PARTICULARLY USEFUL KNOWLEDGE ABOUT: DIFFERENTIAL AND INTEGRAL CALCULUS FOR FUNCTIONS OF ONE VARIABLE, NUMERICAL SEQUENCES AND SERIES. MANDATORY PREPARATORY TEACHINGS: MATHEMATICAL ANALYSIS 1, PHYSICS 1, PROGRAMMING BASICS.

Prerequisites

PREREQUISITES:
FOR A SUCCESSFUL ACHIEVEMENT OF THE PROPOSED GOALS AND, IN PARTICULAR, FOR A PROPER UNDERSTANDING OF THE CONTENTS SCHEDULED FOR THE TEACHING, STUDENTS ARE REQUIRED TO MASTER THE FOLLOWING PARTICULARLY USEFUL KNOWLEDGE ABOUT: DIFFERENTIAL AND INTEGRAL CALCULUS FOR FUNCTIONS OF ONE VARIABLE, NUMERICAL SEQUENCES AND SERIES.
MANDATORY PREPARATORY TEACHINGS:
MATHEMATICAL ANALYSIS 1, PHYSICS 1, PROGRAMMING BASICS.

	Contents
	FUNCTION SEQUENCES AND SERIES: FUNCTION SEQUENCES. POINTWISE AND UNIFORM CONVERGENCE. EXAMPLES AND COUNTEREXAMPLES. UNIFORM LIMIT OF CONTINUOUS FUNCTION. INTEGRATION OF UNIFORMLY CONVERGENT SEQUENCES. THEOREM OF PASSAGE OF LIMIT UNDER DERIVATIVE SIGN. FUNCTION SERIES. DEFINITIONS. POINTWISE, UNIFORM AND TOTAL CONVERGENCE. TERM BY TERM DIFFERENTIATION AND INTEGRABILITY. POWER SERIES. DEFINITIONS. CONVERGENCE SET. CAUCHY-HADAMARD THEOREM. D’ALEMBERT THEOREM. (LECTURE/EXERCISE HOURS 4/4) FOURIER SERIES: FOURIER SERIES. DEFINITIONS. EXAMPLES. BESSEL INEQUALITY. POINTWISE CONVERGENCE THEOREM. UNIFORM CONVERGENCE THEOREM. SERIES INTEGRATION. SERIES DERIVATION. (3/4) MULTIVARIABLE FUNCTIONS. DEFINITIONS. LIMITS AND CONTINUITY. WEIERSTRASS THEOREM. CANTOR THEOREM. PARTIAL DIFFERENTIATION. SCHWARZ THEOREM. GRADIENT. THE CHAIN RULE. DIRECTIONAL DERIVATIVES. TAYLOR FORMULA AND HIGHER ORDER DERIVATIVES. QUADRATIC FORMS. DEFINITE, SEMI-DEFINITE AND INDEFINITE MATRICES LOCAL MINIMA AND MAXIMA. VECTORIAL FUNCTIONS. (4/6) DIFFERENTIAL EQUATIONS: DEFINITIONS. PARTICULAR AND GENERAL INTEGRAL. EXAMPLES. CAUCHY PROBLEM. EXISTENCE AND UNIQUENESS THEOREM. FIRST ORDER DIFFERENTIAL EQUATIONS. LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS. SOLUTIONS METHODS. (4/6) COMPLEX ANALYSIS: COMPLEX FUNCTIONS OF COMPLEX VARIABLES. LINE INTEGRALS OF COMPLEX FUNCTIONS OF COMPLEX VARIABLES. HOLOMORPHIC FUNCTIONS AND THEIR PROPERTIES. THE CAUCHY-RIEMANN CONDITIONS. ELEMENTARY FUNCTIONS IN THE COMPLEX FILED. SINGULAR POINTS. CAUCHY’S THEOREM AND CAUCHY’S INTEGRAL FORMULAS. MORERA THEOREM. LIOUVILLE THEOREM. TAYLOR’S AND LAURENT SERIES AND CLASSIFICATION OF SINGULAR POINTS. RESIDUES, THE RESIDUE THEOREM AND ITS APPLICATION TO THE EVALUATION OF INTEGRALS OF REAL FUNCTIONS. THE DIRAC DELTA FUNCTION. (5/8)

Contents

FUNCTION SEQUENCES AND SERIES:
FUNCTION SEQUENCES. POINTWISE AND UNIFORM CONVERGENCE. EXAMPLES AND COUNTEREXAMPLES. UNIFORM LIMIT OF CONTINUOUS FUNCTION. INTEGRATION OF UNIFORMLY CONVERGENT SEQUENCES. THEOREM OF PASSAGE OF LIMIT UNDER DERIVATIVE SIGN. FUNCTION SERIES. DEFINITIONS. POINTWISE, UNIFORM AND TOTAL CONVERGENCE. TERM BY TERM DIFFERENTIATION AND INTEGRABILITY. POWER SERIES. DEFINITIONS. CONVERGENCE SET. CAUCHY-HADAMARD THEOREM. D’ALEMBERT THEOREM.
(LECTURE/EXERCISE HOURS 4/4)

FOURIER SERIES:
FOURIER SERIES. DEFINITIONS. EXAMPLES. BESSEL INEQUALITY. POINTWISE CONVERGENCE THEOREM. UNIFORM CONVERGENCE THEOREM. SERIES INTEGRATION. SERIES DERIVATION. (3/4)

MULTIVARIABLE FUNCTIONS.
DEFINITIONS. LIMITS AND CONTINUITY. WEIERSTRASS THEOREM. CANTOR THEOREM. PARTIAL DIFFERENTIATION. SCHWARZ THEOREM. GRADIENT. THE CHAIN RULE. DIRECTIONAL DERIVATIVES. TAYLOR FORMULA AND HIGHER ORDER DERIVATIVES. QUADRATIC FORMS. DEFINITE, SEMI-DEFINITE AND INDEFINITE MATRICES LOCAL MINIMA AND MAXIMA. VECTORIAL FUNCTIONS.
(4/6)

DIFFERENTIAL EQUATIONS:
DEFINITIONS. PARTICULAR AND GENERAL INTEGRAL. EXAMPLES. CAUCHY PROBLEM. EXISTENCE AND UNIQUENESS THEOREM. FIRST ORDER DIFFERENTIAL EQUATIONS. LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS. SOLUTIONS METHODS.
(4/6)

COMPLEX ANALYSIS:
COMPLEX FUNCTIONS OF COMPLEX VARIABLES. LINE INTEGRALS OF COMPLEX FUNCTIONS OF COMPLEX VARIABLES. HOLOMORPHIC FUNCTIONS AND THEIR PROPERTIES. THE CAUCHY-RIEMANN CONDITIONS. ELEMENTARY FUNCTIONS IN THE COMPLEX FILED. SINGULAR POINTS. CAUCHY’S THEOREM AND CAUCHY’S INTEGRAL FORMULAS. MORERA THEOREM. LIOUVILLE THEOREM. TAYLOR’S AND LAURENT SERIES AND CLASSIFICATION OF SINGULAR POINTS. RESIDUES, THE RESIDUE THEOREM AND ITS APPLICATION TO THE EVALUATION OF INTEGRALS OF REAL FUNCTIONS. THE DIRAC DELTA FUNCTION.
(5/8)

	Teaching Methods
	THE TEACHING CONSISTS OF FRONTAL LECTURES FOR A TOTAL OF 20 HOURS AND CLASSROOM EXERCISE SESSIONS FOR A TOTAL OF 28 HOURS. ATTENDANCE TO THE COURSE IS MANDATORY, AND IT IS CERTIFIABLE EXCLUSIVELY BY USING THE PERSONAL BADGE. IN ORDER TO PARTICIPATE TO THE FINAL ASSESSMENT AND TO GAIN THE CREDITS CORRESPONDING TO THE COURSE, THE STUDENT MUST HAVE ATTENDED AT LEAST 70% OF THE HOURS OF ASSISTED TEACHING ACTIVITIES.

Teaching Methods

THE TEACHING CONSISTS OF FRONTAL LECTURES FOR A TOTAL OF 20 HOURS AND CLASSROOM EXERCISE SESSIONS FOR A TOTAL OF 28 HOURS.
ATTENDANCE TO THE COURSE IS MANDATORY, AND IT IS CERTIFIABLE EXCLUSIVELY BY USING THE PERSONAL BADGE. IN ORDER TO PARTICIPATE TO THE FINAL ASSESSMENT AND TO GAIN THE CREDITS CORRESPONDING TO THE COURSE, THE STUDENT MUST HAVE ATTENDED AT LEAST 70% OF THE HOURS OF ASSISTED TEACHING ACTIVITIES.

	Verification of learning
	WITH REGARD TO THE LEARNING OUTCOMES OF THE TEACHING, THE FINAL EXAM AIMS TO EVALUATE: THE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED DURING THE THEORETICAL LECTURES AND THE CLASSROOM EXERCISE SESSIONS; THE MASTERY OF THE MATHEMATICAL LANGUAGE IN WRITTEN AND ORAL TESTS; THE SKILL OF PROVING THEOREMS; THE SKILL OF SOLVING EXERCISES; THE ABILITY TO IDENTIFY AND APPLY THE BEST AND EFFICIENT METHODS IN EXERCISES SOLVING; THE ABILITY TO APPLY THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE LESSONS. THE EXAM NECESSARY TO ASSESS THE ACHIEVEMENT OF THE LEARNING OBJECTIVES CONSISTS IN A WRITTEN TEST, PRELIMINARY WITH RESPECT TO THE ORAL EXAMINATION, AND IN AN ORAL INTERVIEW. THE WRITTEN TEST CONSISTS IN SOLVING PROBLEMS IMPLEMENTED ON THE BASIS OF WHAT HAS BEEN PROPOSED IN THE FRAMEWORK OF THE THEORETICAL LECTURES AND EXERCISE SESSIONS. SUCH A WRITTEN TEST, THAT THE STUDENT WILL HAVE TO FACE IN TOTAL AUTONOMY, WILL LAST 3 HOURS. IN ITS EVALUATION, THE RESOLUTION METHODS WILL BE TAKEN INTO ACCOUNT TOGETHER WITH THE CLARITY AND COMPLETENESS OF EXPOSITION. THERE WILL BE A MID-TERM TEST CONCERNING THE TOPICS ALREADY PRESENTED IN THE COURSE, WHICH IN CASE OF A SUFFICIENT MARK, WILL EXEMPT THE STUDENT ON THESE TOPICS AT THE FINAL WRITTEN TEST. IN THE CASE OF PRODUCTION OF A SUFFICIENT PROOF, IT WILL BE EVALUATED THROUGH QUALITATIVE SCALES (RANGES OF MARKS). THE INTERVIEW IS DEVOTED TO EVALUATE THE DEGREE OF KNOWLEDGE OF ALL THE TOPICS OF THE TEACHING, AND WILL COVER DEFINITIONS, THEOREMS PROOFS, EXERCISES SOLVING. THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY CUM LAUDE), WILL DEPEND ON THE RANGE OF MARKS OF THE WRITTEN TEST, WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE ORAL INTERVIEW.

Verification of learning

WITH REGARD TO THE LEARNING OUTCOMES OF THE TEACHING, THE FINAL EXAM AIMS TO EVALUATE: THE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED DURING THE THEORETICAL LECTURES AND THE CLASSROOM EXERCISE SESSIONS; THE MASTERY OF THE MATHEMATICAL LANGUAGE IN WRITTEN AND ORAL TESTS; THE SKILL OF PROVING THEOREMS; THE SKILL OF SOLVING EXERCISES; THE ABILITY TO IDENTIFY AND APPLY THE BEST AND EFFICIENT METHODS IN EXERCISES SOLVING; THE ABILITY TO APPLY THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE LESSONS.
THE EXAM NECESSARY TO ASSESS THE ACHIEVEMENT OF THE LEARNING OBJECTIVES CONSISTS IN A WRITTEN TEST, PRELIMINARY WITH RESPECT TO THE ORAL EXAMINATION, AND IN AN ORAL INTERVIEW.
THE WRITTEN TEST CONSISTS IN SOLVING PROBLEMS IMPLEMENTED ON THE BASIS OF WHAT HAS BEEN PROPOSED IN THE FRAMEWORK OF THE THEORETICAL LECTURES AND EXERCISE SESSIONS. SUCH A WRITTEN TEST, THAT THE STUDENT WILL HAVE TO FACE IN TOTAL AUTONOMY, WILL LAST 3 HOURS. IN ITS EVALUATION, THE RESOLUTION METHODS WILL BE TAKEN INTO ACCOUNT TOGETHER WITH THE CLARITY AND COMPLETENESS OF EXPOSITION.
THERE WILL BE A MID-TERM TEST CONCERNING THE TOPICS ALREADY PRESENTED IN THE COURSE, WHICH IN CASE OF A SUFFICIENT MARK, WILL EXEMPT THE STUDENT ON THESE TOPICS AT THE FINAL WRITTEN TEST.
IN THE CASE OF PRODUCTION OF A SUFFICIENT PROOF, IT WILL BE EVALUATED THROUGH QUALITATIVE SCALES (RANGES OF MARKS).
THE INTERVIEW IS DEVOTED TO EVALUATE THE DEGREE OF KNOWLEDGE OF ALL THE TOPICS OF THE TEACHING, AND WILL COVER DEFINITIONS, THEOREMS PROOFS, EXERCISES SOLVING.
THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY CUM LAUDE), WILL DEPEND ON THE RANGE OF MARKS OF THE WRITTEN TEST, WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE ORAL INTERVIEW.

	Texts
	WRITTEN NOTES GIVEN BY THE TEACHER. C. D’APICE, T. DURANTE, R. MANZO, VERSO L’ESAME DI MATEMATICA II, MAGGIOLI, 2015. C. D’APICE, R. MANZO, VERSO L’ESAME DI MATEMATICA III, MAGGIOLI, 2015. N. FUSCO, P. MARCELLINI, C. SBORDONE, ELEMENTI DI ANALISI MATEMATICA DUE, LIGUORI EDITORE.

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Tiziana DURANTE | ANALISI MATEMATICA 2