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

cod. 0612500002

MATHEMATICS II

	0612500002
	DEPARTMENT OF CIVIL ENGINEERING
	EQF6
	CIVIL AND ENVIRONMENTAL ENGINEERING
	2022/2023

PERCORSO COMUNE

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2022
	SPRING SEMESTER

TEACHING UNITS

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

PROFESSORS

	VINCENZO TIBULLO T Curriculum
	MICHELE CIARLETTA Curriculum

	Objectives
	EXPECTED LEARNING RESULTS AND COMPETENCE TO ACQUIRE ACQUISITION OF FURTHER BASIC CONCEPTS OF MATHEMATICAL ANALYSIS, CALCULUS OF TWO VARIABLES AND MULTIVARIATE FUNCTIONS, LINEAR ALGEBRA AND RELATED PHYSICAL AND ENGINEERING APPLICATIONS. KNOWLEDGE AND UNDERSTANDING ACQUISITION OF SKILLS IN THE MATHEMATICAL LANGUAGE, IN THE BASIC MATHEMATICAL CONCEPTS AND IN THEIR GRAPHICAL REPRESENTATION WITH PARTICULAR REGARD TO THE FOLLOWING TOPICS: VECTOR SPACES; LINEAR APPLICATIONS AND MATRICES; DETERMINANTS; LINEAR SYSTEMS; SPECTRAL THEORY; LINEAR AND QUADRATIC FORMS; SEQUENCES AND SERIES OF FUNCTIONS; MULTIVARIATE FUNCTIONS; DIFFERENTIAL EQUATIONS; CURVES AND LINE INTEGRALS OF FUNCTIONS; DIFFERENTIAL FORMS AND THEIR LINE INTEGRALS; MULTIPLE INTEGRALS; SURFACES AND SURFACE INTEGRALS. WIDER CAPABILITY OF UNDERSTANDING AND ACQUISITION OF MATHEMATICAL LANGUAGE. APPLYING KNOWLEDGE AND UNDERSTANDING APPLYING ACQUIRED KNOWLEDGE TO PERFORM CALCULATIONS WITH VECTORS AND MATRICES; DETERMINE DIMENSION AND BASES OF A VECTOR SPACE; REPRESENT LINEAR APPLICATIONS, LINEAR AND QUADRATIC FORMS WITH NUMERIC VECTORS AND MATRICES; CALCULATE RANK AND DETERMINANT; SOLVE LINEAR SYSTEMS; CALCULATE EIGENVALUES AND EIGENVECTORS, DETERMINE THE SIGNATURE OF A QUADRATIC FORM; ESTABLISH THE CONVERGENCE OF SEQUENCES AND SERIES OF FUNCTIONS AND CALCULATE SIMPLE SUMS OF SERIES; USE THE DIFFERENTIAL CALCULUS IN SEVERAL VARIABLES; SOLVE MAXIMUM AND MINIMUM PROBLEMS; SOLVE DIFFERENTIAL EQUATIONS; CALCULATE THE LENGTH OF A CURVE AND LINE INTEGRALS OF FUNCTIONS AND DIFFERENTIAL FORMS, CALCULATE MULTIPLE INTEGRALS, AREAS AND SURFACE INTEGRALS. ABILITY TO FORMULATE IN MATHEMATICAL TERMS AND SOLVE SIMPLE PROBLEMS OF APPLIED SCIENCES AND IN PARTICULAR OF ENGINEERING. MAKING JUDGEMENTS ABILITY TO CHOOSE THE MATHEMATICAL MODELS AND METHODS SUITABLE FOR THE VARIOUS SITUATIONS AND TO VERIFY THE VALIDITY OF THE RESULTS OBTAINED FROM THE QUALITATIVE AND QUANTITATIVE POINT OF VIEW. COMMUNICATION ABILITY TO PRESENT, WITH PROPER TECHNICAL LANGUAGE AND WITH PROPER GRAPHIC REPRESENTATION, THE NOTIONS AND MATHEMATICAL METHODS ACQUIRED, EVEN BY INTEGRATING THE KNOWLEDGE ACQUIRED WITH THAT TYPICAL OF OTHER DISCIPLINES. LEARNING SKILLS CONSOLIDATION OF THE KNOWLEDGE AND SKILLS ACQUIRED TO LEARN MORE ADVANCED MATHEMATICAL TOPICS AND CONTENT OF OTHER SCIENTIFIC DISCIPLINES USING MATHEMATICAL INSTRUMENTS WITHOUT DIFFICULTY.

EXPECTED LEARNING RESULTS AND COMPETENCE TO ACQUIRE
ACQUISITION OF FURTHER BASIC CONCEPTS OF MATHEMATICAL ANALYSIS, CALCULUS OF TWO VARIABLES AND MULTIVARIATE FUNCTIONS, LINEAR ALGEBRA AND RELATED PHYSICAL AND ENGINEERING APPLICATIONS.

KNOWLEDGE AND UNDERSTANDING
ACQUISITION OF SKILLS IN THE MATHEMATICAL LANGUAGE, IN THE BASIC MATHEMATICAL CONCEPTS AND IN THEIR GRAPHICAL REPRESENTATION WITH PARTICULAR REGARD TO THE FOLLOWING TOPICS: VECTOR SPACES; LINEAR APPLICATIONS AND MATRICES; DETERMINANTS; LINEAR SYSTEMS; SPECTRAL THEORY; LINEAR AND QUADRATIC FORMS; SEQUENCES AND SERIES OF FUNCTIONS; MULTIVARIATE FUNCTIONS; DIFFERENTIAL EQUATIONS; CURVES AND LINE INTEGRALS OF FUNCTIONS; DIFFERENTIAL FORMS AND THEIR LINE INTEGRALS; MULTIPLE INTEGRALS; SURFACES AND SURFACE INTEGRALS.
WIDER CAPABILITY OF UNDERSTANDING AND ACQUISITION OF MATHEMATICAL LANGUAGE.

APPLYING KNOWLEDGE AND UNDERSTANDING
APPLYING ACQUIRED KNOWLEDGE TO PERFORM CALCULATIONS WITH VECTORS AND MATRICES; DETERMINE DIMENSION AND BASES OF A VECTOR SPACE; REPRESENT LINEAR APPLICATIONS, LINEAR AND QUADRATIC FORMS WITH NUMERIC VECTORS AND MATRICES; CALCULATE RANK AND DETERMINANT; SOLVE LINEAR SYSTEMS; CALCULATE EIGENVALUES AND EIGENVECTORS, DETERMINE THE SIGNATURE OF A QUADRATIC FORM; ESTABLISH THE CONVERGENCE OF SEQUENCES AND SERIES OF FUNCTIONS AND CALCULATE SIMPLE SUMS OF SERIES; USE THE DIFFERENTIAL CALCULUS IN SEVERAL VARIABLES; SOLVE MAXIMUM AND MINIMUM PROBLEMS; SOLVE DIFFERENTIAL EQUATIONS; CALCULATE THE LENGTH OF A CURVE AND LINE INTEGRALS OF FUNCTIONS AND DIFFERENTIAL FORMS, CALCULATE MULTIPLE INTEGRALS, AREAS AND SURFACE INTEGRALS.
ABILITY TO FORMULATE IN MATHEMATICAL TERMS AND SOLVE SIMPLE PROBLEMS OF APPLIED SCIENCES AND IN PARTICULAR OF ENGINEERING.

MAKING JUDGEMENTS
ABILITY TO CHOOSE THE MATHEMATICAL MODELS AND METHODS SUITABLE FOR THE VARIOUS SITUATIONS AND TO VERIFY THE VALIDITY OF THE RESULTS OBTAINED FROM THE QUALITATIVE AND QUANTITATIVE POINT OF VIEW.

COMMUNICATION
ABILITY TO PRESENT, WITH PROPER TECHNICAL LANGUAGE AND WITH PROPER GRAPHIC REPRESENTATION, THE NOTIONS AND MATHEMATICAL METHODS ACQUIRED, EVEN BY INTEGRATING THE KNOWLEDGE ACQUIRED WITH THAT TYPICAL OF OTHER DISCIPLINES.

LEARNING SKILLS
CONSOLIDATION OF THE KNOWLEDGE AND SKILLS ACQUIRED TO LEARN MORE ADVANCED MATHEMATICAL TOPICS AND CONTENT OF OTHER SCIENTIFIC DISCIPLINES USING MATHEMATICAL INSTRUMENTS WITHOUT DIFFICULTY.

	Prerequisites
	PREPARATORY TO THE TEACHING IS THE TEACHING OF MATHEMATICS I. IN PARTICULAR, STUDENT MUST HAVE THE BASIC KNOWLEDGE OF MATHEMATICAL ANALYSIS, WITH PARTICULAR REFERENCE TO: ALGEBRAIC EQUATIONS AND INEQUALITIES, THE STUDY OF A REAL-VALUED FUNCTION, SEQUENCES AND NUMERICAL SERIES, LIMITS OF A REAL-VALUED FUNCTION, CONTINUITY AND DIFFERENTIABILITY OF A REAL-VALUED FUNCTION, FUNDAMENTAL THEOREMS OF DIFFERENTIAL AND INTEGRAL CALCULUS.

Prerequisites

PREPARATORY TO THE TEACHING IS THE TEACHING OF MATHEMATICS I. IN PARTICULAR, STUDENT MUST HAVE THE BASIC KNOWLEDGE OF MATHEMATICAL ANALYSIS, WITH PARTICULAR REFERENCE TO:
ALGEBRAIC EQUATIONS AND INEQUALITIES, THE STUDY OF A REAL-VALUED FUNCTION, SEQUENCES AND NUMERICAL SERIES, LIMITS OF A REAL-VALUED FUNCTION, CONTINUITY AND DIFFERENTIABILITY OF A REAL-VALUED FUNCTION, FUNDAMENTAL THEOREMS OF DIFFERENTIAL AND INTEGRAL CALCULUS.

	Contents
	SEQUENCES OF FUNCTIONS (4 HOURS) POINTWISE AND UNIFORM CONVERGENCE. MAIN THEOREMS (THE CONTINUITY OF THE LIMIT, PASSAGE TO THE LIMIT UNDER THE INTEGRAL SIGN, AND UNDER THE SIGN OF THE DERIVATIVE). CAUCHY CRITERION FOR UNIFORM CONVERGENCE. SERIES OF FUNCTIONS (4 HOURS) POINTWISE, UNIFORM, TOTAL CONVERGENCE. POWER SERIES. MAIN THEOREMS (CAUCHY-HADAMARD, D'ALEMBERT, INTEGRATION AND DERIVATION FOR THE SERIES). FUNCTIONS OF SEVERAL VARIABLES (12 HOURS) LIMIT AND CONTINUITY. PARTIAL AND DIRECTIONAL DERIVATIVES. MAIN THEOREMS (SCHWARZ, TOTAL DIFFERENTIAL, DERIVATION OF COMPOSITE FUNCTIONS). GRADIENT. DIFFERENTIABILITY. MAXIMA AND MINIMA. ORDINARY DIFFERENTIAL EQUATIONS (14 HOURS) PARTICULAR INTEGRAL AND GENERAL SOLUTION. THE CAUCHY PROBLEM. LOCAL AND GLOBAL EXISTENCE AND UNIQUENESS THEOREM. MAIN FIRST-ORDER DIFFERENTIAL EQUATIONS. N-TH ORDER LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS INTEGRALS OF FUNCTIONS OF SEVERAL VARIABLES (14 HOURS) PROPERTIES. APPLICATION TO AREAS AND VOLUMES. REDUCTION FORMULAS. CHANGE OF VARIABLES. CURVES AND CURVILINEAR INTEGRALS (6 HOURS) REGULAR CURVES. LENGTH OF A CURVE. CURVILINEAR INTEGRAL OF A FUNCTION. DIFFERENTIAL FORMS (10 HOURS) VECTOR FIELDS. CURVILINEAR INTEGRAL OF A LINEAR DIFFERENTIAL FORM. CLOSED AND EXACT FORMS. CRITERIA OF EXACTNESS. SURFACES AND SURFACE INTEGRALS (6 HOURS) AREA OF A SURFACE AND SURFACE INTEGRALS. DIVERGENCE THEOREM. STOKES FORMULA. LINEAR ALGEBRA (14 HOURS) VECTORS AND MATRICES. MAIN COMPUTATIONS. LINEAR DEPENDENT AND INDEPENDENT VECTORS. LINEAR SYSTEMS OF EQUATIONS. EINGENVALUES AND EINGENVECTORS. ANALYTIC GEOMETRY (6 HOURS) LINES AND PLANES IN 2 AND 3-DIMENSIONAL SPACES.

Contents

SEQUENCES OF FUNCTIONS (4 HOURS)
POINTWISE AND UNIFORM CONVERGENCE. MAIN THEOREMS (THE CONTINUITY OF THE LIMIT, PASSAGE TO THE LIMIT UNDER THE INTEGRAL SIGN, AND UNDER THE SIGN OF THE DERIVATIVE). CAUCHY CRITERION FOR UNIFORM CONVERGENCE.

SERIES OF FUNCTIONS (4 HOURS)
POINTWISE, UNIFORM, TOTAL CONVERGENCE. POWER SERIES. MAIN THEOREMS (CAUCHY-HADAMARD, D'ALEMBERT, INTEGRATION AND DERIVATION FOR THE SERIES).

FUNCTIONS OF SEVERAL VARIABLES (12 HOURS)
LIMIT AND CONTINUITY. PARTIAL AND DIRECTIONAL DERIVATIVES. MAIN THEOREMS (SCHWARZ, TOTAL DIFFERENTIAL, DERIVATION OF COMPOSITE FUNCTIONS). GRADIENT. DIFFERENTIABILITY. MAXIMA AND MINIMA.

ORDINARY DIFFERENTIAL EQUATIONS (14 HOURS)
PARTICULAR INTEGRAL AND GENERAL SOLUTION. THE CAUCHY PROBLEM. LOCAL AND GLOBAL EXISTENCE AND UNIQUENESS THEOREM. MAIN FIRST-ORDER DIFFERENTIAL EQUATIONS. N-TH ORDER LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS

INTEGRALS OF FUNCTIONS OF SEVERAL VARIABLES (14 HOURS)
PROPERTIES. APPLICATION TO AREAS AND VOLUMES. REDUCTION FORMULAS. CHANGE OF VARIABLES.

CURVES AND CURVILINEAR INTEGRALS (6 HOURS)
REGULAR CURVES. LENGTH OF A CURVE. CURVILINEAR INTEGRAL OF A FUNCTION.

DIFFERENTIAL FORMS (10 HOURS)
VECTOR FIELDS. CURVILINEAR INTEGRAL OF A LINEAR DIFFERENTIAL FORM. CLOSED AND EXACT FORMS. CRITERIA OF EXACTNESS.

SURFACES AND SURFACE INTEGRALS (6 HOURS)
AREA OF A SURFACE AND SURFACE INTEGRALS. DIVERGENCE THEOREM. STOKES FORMULA.

LINEAR ALGEBRA (14 HOURS)
VECTORS AND MATRICES. MAIN COMPUTATIONS. LINEAR DEPENDENT AND INDEPENDENT VECTORS. LINEAR SYSTEMS OF EQUATIONS. EINGENVALUES AND EINGENVECTORS.

ANALYTIC GEOMETRY (6 HOURS)
LINES AND PLANES IN 2 AND 3-DIMENSIONAL SPACES.

	Teaching Methods
	TEACHING IS PROVIDED IN PRESENCE. THE LANGUAGE OF INSTRUCTION IS ITALIAN. THE COURSE CONSISTS OF THEORETICAL LECTURES, DEVOTED TO THE EXPLANATION OF ALL THE COURSE CONTENTS AND CLASSROOM PRACTICE, PROVIDING THE STUDENTS WITH THE MAIN TOOLS NEEDED TO PROBLEM-SOLVING ACTIVITIES.

	Verification of learning
	THE EXAMINATION SHALL CONSIST OF A WRITTEN TEST AND AN ORAL TEST. THE WRITTEN TEST CONSISTS IN THE RESOLUTION OF SIX EXERCISES ON THE FOLLOWING TOPICS: - EXTREMES OF FUNCTION OF TWO VARIABLES, - ORDINARY DIFFERENTIAL EQUATIONS, - DIFFERENTIAL FORMS IN R^2, - INTEGRAL OF A FUNCTIONS OF TWO VARIABLES ON A DOMAIN OF R^2; - EIGENVALUES AND EIGENVECTORS OF LINEAR TRANSFORMATIONS IN R^3, - ANALYTICAL GEOMETRY IN SPACE. THE ASSESSMENT OF THE TEST VARIES FROM THE MINIMUM INDICATED BY BAND "E" TO THE MAXIMUM INDICATED BY BAND "A". TO BE ADMITTED TO THE ORAL EXAMINATION IT IS NECESSARY TO PERFORM SUFFICIENTLY AT LEAST TWO EXERCISES BETWEEN THE FIRST FOUR AND AT LEAST ONE OF THE FOLLOWING TWO. IN ORDER TO BE ADMITTED, IT IS NECESSARY TO HAVE AN ASSESSMENT AT LEAST EQUAL TO THE BAND "D". THE ORAL TEST IS INTENDED TO ASSESS THE KNOWLEDGE OF THE COURSE TOPICS AND COVERS DEFINITIONS, THEOREMS AND THEIR PROOFS AND OCCASIONALLY THE RESOLUTION OF SIMPLE EXERCISES. THE FINAL ASSESSMENT, EXPRESSED IN THIRTIES (POSSIBLY WITH HONOURS), DEPENDS ON THE EVALUATION OF THE WRITTEN AND ORAL TESTS.

Verification of learning

THE EXAMINATION SHALL CONSIST OF A WRITTEN TEST AND AN ORAL TEST.

THE WRITTEN TEST CONSISTS IN THE RESOLUTION OF SIX EXERCISES ON THE FOLLOWING TOPICS:
- EXTREMES OF FUNCTION OF TWO VARIABLES,
- ORDINARY DIFFERENTIAL EQUATIONS,
- DIFFERENTIAL FORMS IN R^2,
- INTEGRAL OF A FUNCTIONS OF TWO VARIABLES ON A DOMAIN OF R^2;
- EIGENVALUES AND EIGENVECTORS OF LINEAR TRANSFORMATIONS IN R^3,
- ANALYTICAL GEOMETRY IN SPACE.

THE ASSESSMENT OF THE TEST VARIES FROM THE MINIMUM INDICATED BY BAND "E" TO THE MAXIMUM INDICATED BY BAND "A". TO BE ADMITTED TO THE ORAL EXAMINATION IT IS NECESSARY TO PERFORM SUFFICIENTLY AT LEAST TWO EXERCISES BETWEEN THE FIRST FOUR AND AT LEAST ONE OF THE FOLLOWING TWO. IN ORDER TO BE ADMITTED, IT IS NECESSARY TO HAVE AN ASSESSMENT AT LEAST EQUAL TO THE BAND "D".

THE ORAL TEST IS INTENDED TO ASSESS THE KNOWLEDGE OF THE COURSE TOPICS AND COVERS DEFINITIONS, THEOREMS AND THEIR PROOFS AND OCCASIONALLY THE RESOLUTION OF SIMPLE EXERCISES.

THE FINAL ASSESSMENT, EXPRESSED IN THIRTIES (POSSIBLY WITH HONOURS), DEPENDS ON THE EVALUATION OF THE WRITTEN AND ORAL TESTS.

	Texts
	THEORY - N. FUSCO, P. MARCELLINI, C. SBORDONE, “ANALISI MATEMATICA 2 “, LIGUORI EDITORE (2016) - NOTES OF THE TEACHER EXERCISES - P. MARCELLINI - C. SBORDONE, “ESERCITAZIONI DI MATEMATICA VOL. 2° PRIMA E SECONDA PARTE“, LIGUORI EDITORE (2016)

	More Information
	COURSE SCHEDULES WILL BE PUBLISHED ON THE DEPARTMENT WEBSITE. THERE IS A WEEKLY RECEPTION ACTIVITY. WEEKLY TUTORING ACTIVITIES WILL BE HELD BY QUALIFIED PERSONNEL.

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