Ingegneria Civile per l'Ambiente ed il Territorio | Mathematics II

cod. 0612500002

MATHEMATICS II

	0612500002
	DIPARTIMENTO DI INGEGNERIA CIVILE

	CIVIL AND ENVIRONMENTAL ENGINEERING
	2015/2016

PERCORSO COMUNE

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2012
	SECONDO SEMESTRE

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/05	9	90	LESSONS	BASIC COMPULSORY SUBJECTS

PROFESSORS

	FRANCESCO SAVERIO TORTORIELLO T Curriculum
	MICHELE CIARLETTA Curriculum

	Objectives
	THE COURSE AIMS AT THE ACQUISITION OF BASIC ELEMENTS OF MATHEMATICAL ANALYSIS: INTEGRALS OF FUNCTIONS OF ONE VARIABLE, NUMERICAL SERIES, SEQUENCES AND SERIES OF FUNCTIONS, FUNCTIONS OF SEVERAL VARIABLES, DIFFERENTIAL EQUATIONS, INTEGRALS OF FUNCTIONS OF SEVERAL VARIABLES, CURVES AND CURVILINEAR INTEGRALS, SURFACES.LEARNING OUTCOMES OF COURSE CONSIST OF THE ACHIEVEMENT OF RESULTS AND DEMONSTRATION TECHNIQUES, AS WELL AS THE ABILITY TO USE COMPUTATIONAL INSTRUMENTS.THE COURSE’S MAIN AIM IS TO STRENGTHEN BASIC MATHEMATICAL KNOWLEDGE AND TO DEVELOP AND TO PROVIDE USEFUL TOOLS FOR A SCIENTIFIC APPROACH TO THE PROBLEMS AND PHENOMENA THAT STUDENTS ENCOUNTER IN PURSUIT OF THEIR STUDIES. THE THEORETICAL PART OF THE COURSE WILL BE PRESENTED IN A RIGOROUS BUT CONCISE WAY AND IT WILL BE ACCOMPANIED BY A PARALLEL EXERCISE ACTIVITIES DESIGNED TO FOSTER THE UNDERSTANDING OF CONCEPTS. KNOWLEDGE AND UNDERSTANDING: UNDERSTANDING THE TERMINOLOGY USED IN MATHEMATICAL ANALYSIS KNOWLEDGE OF DEMONSTRATION METHODS KNOWLEDGE OF THE FUNDAMENTAL CONCEPTS OF MATHEMATICAL ANALYSIS APPLYING KNOWLEDGE AND UNDERSTANDING: KNOWING HOW TO APPLY THEOREMS AND RULES DESIGNED TO SOLVE PROBLEMS KNOWING HOW TO CONSISTENTLY BUILD SOME DEMONSTRATIONS KNOWING HOW TO BUILD METHODS AND PROCEDURES FOR THE RESOLUTION OF PROBLEMS KNOWING HOW TO SOLVE SIMPLE DIFFERENTIAL EQUATIONS KNOWING HOW TO SOLVE SIMPLE INTEGRALS AND DOUBLE INTEGRALS MAKING JUDGEMENTS: KNOWING HOW TO IDENTIFY THE MOST APPROPRIATE METHODS TO EFFICIENTLY SOLVE A MATHEMATICAL PROBLEM TO BE ABLE TO FIND SOME OPTIMIZATIONS IN THE SOLVING PROCESS OF A MATHEMATICAL PROBLEM COMMUNICATION SKILLS: ABILITY TO WORK IN GROUPS ABILITY TO ORALLY PRESENT A TOPIC RELATED TO MATHEMATICS LEARNING SKILLS: SKILL OF APPLYING THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE COURSE SKILL TO DEEPEN THE TOPICS DEALT WITH BY USING MATERIALS DIFFERENT FROM THOSE PROPOSED.

THE COURSE AIMS AT THE ACQUISITION OF BASIC ELEMENTS OF MATHEMATICAL ANALYSIS: INTEGRALS OF FUNCTIONS OF ONE VARIABLE, NUMERICAL SERIES, SEQUENCES AND SERIES OF FUNCTIONS, FUNCTIONS OF SEVERAL VARIABLES, DIFFERENTIAL EQUATIONS, INTEGRALS OF FUNCTIONS OF SEVERAL VARIABLES, CURVES AND CURVILINEAR INTEGRALS, SURFACES.LEARNING OUTCOMES OF COURSE CONSIST OF THE ACHIEVEMENT OF RESULTS AND DEMONSTRATION TECHNIQUES, AS WELL AS THE ABILITY TO USE COMPUTATIONAL INSTRUMENTS.THE COURSE’S MAIN AIM IS TO STRENGTHEN BASIC MATHEMATICAL KNOWLEDGE AND TO DEVELOP AND TO PROVIDE USEFUL TOOLS FOR A SCIENTIFIC APPROACH TO THE PROBLEMS AND PHENOMENA THAT STUDENTS ENCOUNTER IN PURSUIT OF THEIR STUDIES. THE THEORETICAL PART OF THE COURSE WILL BE PRESENTED IN A RIGOROUS BUT CONCISE WAY AND IT WILL BE ACCOMPANIED BY A PARALLEL EXERCISE ACTIVITIES DESIGNED TO FOSTER THE UNDERSTANDING OF CONCEPTS.
KNOWLEDGE AND UNDERSTANDING:
UNDERSTANDING THE TERMINOLOGY USED IN MATHEMATICAL ANALYSIS
KNOWLEDGE OF DEMONSTRATION METHODS
KNOWLEDGE OF THE FUNDAMENTAL CONCEPTS OF MATHEMATICAL ANALYSIS
APPLYING KNOWLEDGE AND UNDERSTANDING:
KNOWING HOW TO APPLY THEOREMS AND RULES DESIGNED TO SOLVE PROBLEMS
KNOWING HOW TO CONSISTENTLY BUILD SOME DEMONSTRATIONS
KNOWING HOW TO BUILD METHODS AND PROCEDURES FOR THE RESOLUTION OF PROBLEMS
KNOWING HOW TO SOLVE SIMPLE DIFFERENTIAL EQUATIONS
KNOWING HOW TO SOLVE SIMPLE INTEGRALS AND DOUBLE INTEGRALS
MAKING JUDGEMENTS:
KNOWING HOW TO IDENTIFY THE MOST APPROPRIATE METHODS TO EFFICIENTLY SOLVE A MATHEMATICAL PROBLEM
TO BE ABLE TO FIND SOME OPTIMIZATIONS IN THE SOLVING PROCESS OF A MATHEMATICAL PROBLEM
COMMUNICATION SKILLS:
ABILITY TO WORK IN GROUPS
ABILITY TO ORALLY PRESENT A TOPIC RELATED TO MATHEMATICS
LEARNING SKILLS:
SKILL OF APPLYING THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE COURSE
SKILL TO DEEPEN THE TOPICS DEALT WITH BY USING MATERIALS DIFFERENT FROM THOSE PROPOSED.

	Prerequisites
	FOR THE SUCCESSFUL ACHIEVEMENT OF THE GOALS, STUDENT MUST HAVE THE FOLLOWING PRE-REQUISITIONS: -KNOWLEDGE OF LINEAR ALGEBRA WITH PARTICULAR REFERENCE TO: MATRICES AND LINEAR SYSTEMS, VECTOR SPACES, LINEAR TRANSFORMATIONS AND DIAGONALIZATION, ANALYTICAL GEOMETRY -BASIC KNOWLEDGE OF MATHEMATICAL ANALYSIS, WITH PARTICULAR REFERENCE TO: ALGEBRAIC EQUATIONS AND INEQUALITIES, STUDY OF THE GRAPH OF A FUNCTION OF A REAL VARIABLE, SEQUENCES AND SERIES, LIMITS OF A FUNCTION, CONTINUITY AND DIFFERENTIABILITY OF A FUNCTION, FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS

Prerequisites

FOR THE SUCCESSFUL ACHIEVEMENT OF THE GOALS, STUDENT MUST HAVE THE FOLLOWING PRE-REQUISITIONS:
-KNOWLEDGE OF LINEAR ALGEBRA WITH PARTICULAR REFERENCE TO: MATRICES AND LINEAR SYSTEMS, VECTOR SPACES, LINEAR TRANSFORMATIONS AND DIAGONALIZATION, ANALYTICAL GEOMETRY
-BASIC KNOWLEDGE OF MATHEMATICAL ANALYSIS, WITH PARTICULAR REFERENCE TO: ALGEBRAIC EQUATIONS AND INEQUALITIES, STUDY OF THE GRAPH OF A FUNCTION OF A REAL VARIABLE, SEQUENCES AND SERIES, LIMITS OF A FUNCTION, CONTINUITY AND DIFFERENTIABILITY OF A FUNCTION, FUNDAMENTAL THEOREMS OF DIFFERENTIAL CALCULUS

	Contents
	INTEGRATION OF FUNCTIONS OF ONE VARIABLE:DEFINITION OF PRIMITIVE FUNCTION AND INDEFINITE INTEGRAL. IMMEDIATE INTEGRALS. RULES AND METHODS OF INTEGRATION. INTEGRAL OF RATIONAL FUNCTIONS. DEFINITE INTEGRAL AND ITS GEOMETRIC MEANING. MEAN VALUE THEOREM. INTEGRAL FUNCTION AND THE FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS.( HOURS LECTURE/PRACTICE/LABORATORY 6/6/-) NUMERICAL SERIES: INTRODUCTION TO NUMERICAL SERIES. CONVERGING, DIVERGING, AND INDETERMINATE SERIES. GEOMETRIC SERIE, HARMONICSERIE. SERIES WITH POSITIVE TERMS AND CONVERGENCE CRITERIA: THE CRITERIA OF COMPARISON, THE CRITERIA OF RATIO, THE CRITERIA OF ROOT. (HOURS 2/2/-) SEQUENCES AND SERIES OF FUNCTIONS:SUCCESSIONS. DEFINITIONS. POINTWISE AND UNIFORM CONVERGENCE . EXAMPLES AND COUNTEREXAMPLES. THEOREM ON THE CONTINUITY OF THE LIMIT. CAUCHY CRITERION FOR UNIFORM CONVERGENCE. THEOREMS OF PASSAGE TO THE LIMIT UNDER THE INTEGRAL SIGN. THEOREM OF PASSAGE TO THE LIMIT UNDER THE SIGN OF THE DERIVATIVE. SERIES OF FUNCTIONS. DEFINITIONS. POINTWISE, UNIFORM, TOTAL CONVERGENCE. CAUCHY CRITERIA. DERIVATION AND INTEGRATION FOR THE SERIES. POWER SERIES. DEFINITIONS. SET OF CONVERGENCE AND RADIUS OF CONVERGENCE. CAUCHY-HADAMARD THEOREM. D'ALEMBERT THEOREM. RADIUS OF CONVERGENCE OF THE DERIVED SERIES. UNIFORM AND CONVERGENCE. THEOREM OF INTEGRATION AND DERIVATION FOR THE SERIES. EXAMPLES AND COUNTEREXAMPLES.(HOURS 6/4/-) FUNCTIONS OF SEVERAL VARIABLES: DEFINITIONS. LIMIT AND CONTINUITY. WEIERSTRASS THEOREM. CANTOR'S THEOREM. PARTIAL DERIVATIVES. THE SCHWARZ THEOREM. GRADIENT. DIFFERENTIABILITY. THE THEOREM OF THE TOTAL DIFFERENTIAL. COMPOSITE FUNCTIONS. THEOREM DERIVATION OF COMPOSITE FUNCTIONS. DIFFERENTIABILITY OF COMPOSED FUNCTIONS. DIRECTIONAL DERIVATIVES. FUNCTIONS WITH ZERO GRADIENT IN A CONNECTED SET. FUNCTIONS DEFINED BY INTEGRALS. TAYLOR'S FORMULA AND HIGHER ORDER DIFFERENTIALS. QUADRATIC FORMS. SQUARE MATRICES DEFINED, SEMIDEFINITE AND INDEFINITE. MAXIMA AND MINIMA. VECTOR-VALUED FUNCTIONS.(HOURS 7/5/-) DIFFERENTIAL EQUATIONS:DEFINITIONS. PARTICULAR INTEGRAL AND GENERAL SOLUTION. EXAMPLES. THE CAUCHY PROBLEM. LOCAL EXISTENCE AND UNIQUENESS THEOREM. GLOBAL EXISTENCE AND UNIQUENESS THEOREM. PROLONGATION OF A SOLUTION. SOLUTIONS CEILINGS (NOTES). DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. LINEAR DIFFERENTIAL EQUATIONS. STRUCTURE OF THE SET OF SOLUTIONS. LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS. WRONSKIAN AND ITS PROPERTIES. RESOLUTION METHODS. (HOURS 6/7/-) INTEGRALS OF FUNCTIONS OF SEVERAL VARIABLES:DEFINITIONS. EXAMPLES. PROPERTIES. APPLICATION TO AREAS AND VOLUMES. THE FIRST THEOREM OF PAPPUS-GULDINO. REDUCTION FORMULAS. CHANGE OF VARIABLES. (HOURS 7/6/-) CURVES AND CURVILINEAR INTEGRALS: DEFINITION. REGULAR CURVES. LENGTH OF A CURVE. RECTIFIABILITYTHEOREM. CURVILINEAR INTEGRAL OF A FUNCTION. (HOURS 4/3/-) DIFFERENTIAL FORMS: DEFINITIONS. VECTOR FIELDS. CURVILINEAR INTEGRAL OF A LINEAR DIFFERENTIAL FORM. CLOSED AND EXACT FORMS. CRITERIA OF EXACTNESS. RELATION BETWEEN EXACTNESS AND CLOSURE. CLOSED FORMS INTO RECTANGLES OR OPEN STARRY. CLOSED FORMS IN OPEN SIMPLY CONNECTED SET. (HOURS 7/4/-) SURFACES AND SURFACE INTEGRALS:DEFINITIONS. EXAMPLES. PROPERTIES. CHANGE OF PARAMETRIC REPRESENTATIONS. AREA OF A SURFACE AND SURFACE INTEGRALS. SURFACES WITH BOUNDARY. THE SECOND THEOREM OF PAPPUS-GULDINO. DIVERGENCE THEOREM. STOKES FORMULA. (HOURS 5/3/-) TOTAL HOURS 50/40/-

Contents

INTEGRATION OF FUNCTIONS OF ONE VARIABLE:DEFINITION OF PRIMITIVE FUNCTION AND INDEFINITE INTEGRAL. IMMEDIATE INTEGRALS. RULES AND METHODS OF INTEGRATION. INTEGRAL OF RATIONAL FUNCTIONS. DEFINITE INTEGRAL AND ITS GEOMETRIC MEANING. MEAN VALUE THEOREM. INTEGRAL FUNCTION AND THE FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS.( HOURS LECTURE/PRACTICE/LABORATORY 6/6/-)
NUMERICAL SERIES: INTRODUCTION TO NUMERICAL SERIES. CONVERGING, DIVERGING, AND INDETERMINATE SERIES. GEOMETRIC SERIE, HARMONICSERIE. SERIES WITH POSITIVE TERMS AND CONVERGENCE CRITERIA: THE CRITERIA OF COMPARISON, THE CRITERIA OF RATIO, THE CRITERIA OF ROOT. (HOURS 2/2/-)
SEQUENCES AND SERIES OF FUNCTIONS:SUCCESSIONS. DEFINITIONS. POINTWISE AND UNIFORM CONVERGENCE . EXAMPLES AND COUNTEREXAMPLES. THEOREM ON THE CONTINUITY OF THE LIMIT. CAUCHY CRITERION FOR UNIFORM CONVERGENCE. THEOREMS OF PASSAGE TO THE LIMIT UNDER THE INTEGRAL SIGN. THEOREM OF PASSAGE TO THE LIMIT UNDER THE SIGN OF THE DERIVATIVE. SERIES OF FUNCTIONS. DEFINITIONS. POINTWISE, UNIFORM, TOTAL CONVERGENCE. CAUCHY CRITERIA. DERIVATION AND INTEGRATION FOR THE SERIES. POWER SERIES. DEFINITIONS. SET OF CONVERGENCE AND RADIUS OF CONVERGENCE. CAUCHY-HADAMARD THEOREM. D'ALEMBERT THEOREM. RADIUS OF CONVERGENCE OF THE DERIVED SERIES. UNIFORM AND CONVERGENCE. THEOREM OF INTEGRATION AND DERIVATION FOR THE SERIES. EXAMPLES AND COUNTEREXAMPLES.(HOURS 6/4/-)
FUNCTIONS OF SEVERAL VARIABLES: DEFINITIONS. LIMIT AND CONTINUITY. WEIERSTRASS THEOREM. CANTOR'S THEOREM. PARTIAL DERIVATIVES. THE SCHWARZ THEOREM. GRADIENT. DIFFERENTIABILITY. THE THEOREM OF THE TOTAL DIFFERENTIAL. COMPOSITE FUNCTIONS. THEOREM DERIVATION OF COMPOSITE FUNCTIONS. DIFFERENTIABILITY OF COMPOSED FUNCTIONS. DIRECTIONAL DERIVATIVES. FUNCTIONS WITH ZERO GRADIENT IN A CONNECTED SET. FUNCTIONS DEFINED BY INTEGRALS. TAYLOR'S FORMULA AND HIGHER ORDER DIFFERENTIALS. QUADRATIC FORMS. SQUARE MATRICES DEFINED, SEMIDEFINITE AND INDEFINITE. MAXIMA AND MINIMA. VECTOR-VALUED FUNCTIONS.(HOURS 7/5/-)
DIFFERENTIAL EQUATIONS:DEFINITIONS. PARTICULAR INTEGRAL AND GENERAL SOLUTION. EXAMPLES. THE CAUCHY PROBLEM. LOCAL EXISTENCE AND UNIQUENESS THEOREM. GLOBAL EXISTENCE AND UNIQUENESS THEOREM. PROLONGATION OF A SOLUTION. SOLUTIONS CEILINGS (NOTES). DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. LINEAR DIFFERENTIAL EQUATIONS. STRUCTURE OF THE SET OF SOLUTIONS. LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS. WRONSKIAN AND ITS PROPERTIES. RESOLUTION METHODS. (HOURS 6/7/-)
INTEGRALS OF FUNCTIONS OF SEVERAL VARIABLES:DEFINITIONS. EXAMPLES. PROPERTIES. APPLICATION TO AREAS AND VOLUMES. THE FIRST THEOREM OF PAPPUS-GULDINO. REDUCTION FORMULAS. CHANGE OF VARIABLES. (HOURS 7/6/-)
CURVES AND CURVILINEAR INTEGRALS: DEFINITION. REGULAR CURVES. LENGTH OF A CURVE. RECTIFIABILITYTHEOREM. CURVILINEAR INTEGRAL OF A FUNCTION. (HOURS 4/3/-)
DIFFERENTIAL FORMS: DEFINITIONS. VECTOR FIELDS. CURVILINEAR INTEGRAL OF A LINEAR DIFFERENTIAL FORM. CLOSED AND EXACT FORMS. CRITERIA OF EXACTNESS. RELATION BETWEEN EXACTNESS AND CLOSURE. CLOSED FORMS INTO RECTANGLES OR OPEN STARRY. CLOSED FORMS IN OPEN SIMPLY CONNECTED SET. (HOURS 7/4/-)
SURFACES AND SURFACE INTEGRALS:DEFINITIONS. EXAMPLES. PROPERTIES. CHANGE OF PARAMETRIC REPRESENTATIONS. AREA OF A SURFACE AND SURFACE INTEGRALS. SURFACES WITH BOUNDARY. THE SECOND THEOREM OF PAPPUS-GULDINO. DIVERGENCE THEOREM. STOKES FORMULA. (HOURS 5/3/-)
TOTAL HOURS 50/40/-

	Teaching Methods
	THE COURSE COVERS THEORETICAL LESSONS, DURING WHICH ALL COURSE CONTENTS WILL BE PRESENTED BY LECTURES, AND CLASSROOM EXERCISES DURING WHICH THE MAIN TOOLS NECESSARY FOR THE RESOLUTION OF EXERCISES RELATED TO TEACHING CONTENTS WILL BE PROVIDED.

	Verification of learning
	THE EVALUATION OF THE ACHIEVEMENT OF THE EXPECTED OUTCOMES WILL BE CARRIED OUT WITH A WRITTEN TEST AND AN ORAL INTERVIEW. TO PASS THE EXAM THE STUDENT MUST DEMONSTRATE THAT SHE/HE IS ABLE TO UNDERSTAND AND TO APPLY THE MAIN CONCEPTS EXPRESSED IN THE COURSE. THE FINAL MARK, OUT OF THIRTY (THE MAXIMUM IS THIRTY CUM LAUDE), DEPENDS ON THE MASTERING ABILITY OF THE COURSE CONTENTS, TAKING INTO ACCOUNT THE QUALITY OF THE WRITTEN, LAB AND ORAL PROOFS, IN PARTICULAR PROPERTY OF LANGUAGE WITH REFERENCE TO THE SPECIFIC TERMINOLOGY OF THE DISCIPLINE, KNOWLEDGE OF THE CONCEPTS AND ABILITY TO SOLVE EXERCISES, BY HANDS AND BY SOFTWARE.

Verification of learning

THE EVALUATION OF THE ACHIEVEMENT OF THE EXPECTED OUTCOMES WILL BE CARRIED OUT WITH A WRITTEN TEST AND AN ORAL INTERVIEW. TO PASS THE EXAM THE STUDENT MUST DEMONSTRATE THAT SHE/HE IS ABLE TO UNDERSTAND AND TO APPLY THE MAIN CONCEPTS EXPRESSED IN THE COURSE. THE FINAL MARK, OUT OF THIRTY (THE MAXIMUM IS THIRTY CUM LAUDE), DEPENDS ON THE MASTERING ABILITY OF THE COURSE CONTENTS, TAKING INTO ACCOUNT THE QUALITY OF THE WRITTEN, LAB AND ORAL PROOFS, IN PARTICULAR PROPERTY OF LANGUAGE WITH REFERENCE TO THE SPECIFIC TERMINOLOGY OF THE DISCIPLINE, KNOWLEDGE OF THE CONCEPTS AND ABILITY TO SOLVE EXERCISES, BY HANDS AND BY SOFTWARE.

	Texts
	GIUSTI, ANALISI MATEMATICA II, BOLLATI BORINHIERI N. FUSCO, P. MARCELLINI, C. SBORDONE, ANALISI MATEMATICA DUE, LIGUORI EDITORE C. D’APICE, T. DURANTE, R. MANZO, VERSO L’ESAME DI MATEMATICA II, CUES (2008). EDUCATIONAL CONTENTS ON E-LEARNING PLATFORM IWT. LECTURE NOTES.