Chiara ESPOSITO | Mathematics II

cod. 0612400002

MATHEMATICS II

	0612400002
	DIPARTIMENTO DI INGEGNERIA INDUSTRIALE
	EQF6
	ELECTRONIC ENGINEERING
	2021/2022

PERCORSO IN COMMON

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2018
	SPRING SEMESTER

TEACHING UNITS

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

PROFESSORS

	CRISTIAN TACELLI T Curriculum
	CHIARA ESPOSITO Curriculum

	Objectives
	THE COURSE DEALS WITH FUNCTIONS OF SEVERAL VARIABLES, LINE INTEGRALS, MULTIPLE INTEGRALLS, DIFFERENTIAL EQUATIONS, SEQUENCES AND SERIES OF FUNCTIONS, LINEAR ALGEBRA, ANALYTIC GEOMETRY. THE AIM IS TO REACH A GOOD LEVEL OF UNDERSTANDING AND KNOWLEDGE OF THE TOPICS AND TO BE ABLE TO APPLY THE METHODS AND RESULTS TO DIFFERENT FIELDS. MOREOVER THE AIM IS TO STRENGTHEN THE BASIC MATHEMATICAL KNOWLEDGE, TO DEVELOP AND TO PROVIDE USEFUL TOOLS FOR A SCIENTIFIC APPROACH TO THE PROBLEMS STUDIED. EXERCISE ACTIVITIES WILL CONSOLIDATE UNDERSTANDING OF THE CONCEPTS.

THE COURSE DEALS WITH FUNCTIONS OF SEVERAL VARIABLES, LINE INTEGRALS, MULTIPLE INTEGRALLS, DIFFERENTIAL EQUATIONS, SEQUENCES AND SERIES OF FUNCTIONS, LINEAR ALGEBRA, ANALYTIC GEOMETRY.
THE AIM IS TO REACH A GOOD LEVEL OF UNDERSTANDING AND KNOWLEDGE OF THE TOPICS AND TO BE ABLE TO APPLY THE METHODS AND RESULTS TO DIFFERENT FIELDS. MOREOVER THE AIM IS TO STRENGTHEN THE BASIC MATHEMATICAL KNOWLEDGE, TO DEVELOP AND TO PROVIDE USEFUL TOOLS FOR A SCIENTIFIC APPROACH TO THE PROBLEMS STUDIED.
EXERCISE ACTIVITIES WILL CONSOLIDATE UNDERSTANDING OF THE CONCEPTS.

	Prerequisites
	GOOD KNOWLEDGE OF THE SUBJECTS CONTAINED IN THE COURSE MATEMATICA I.

	Contents
	SECTION MATHEMATICAL ANALYSIS (LESSON / EXERCISES : HOURS 30/30) NUMERICAL SERIES AND CONVERGENCE. GEOMETRIC SERIE, HARMONICSERIE. SERIES WITH POSITIVE TERMS AND CONVERGENCE CRITERIA: THE CRITERIA OF COMPARISON, THE CRITERIA OF RATIO, THE CRITERIA OF ROOT. ( 2 / 2 ). SEQUENCES AND SERIES OF FUNCTIONS. 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. 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. ( 6 / 4 ). FUNCTIONS OF SEVERAL VARIABLES. LIMIT AND CONTINUITY. WEIERSTRASS 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. TAYLOR'S FORMULA AND HIGHER ORDER DIFFERENTIALS. QUADRATIC FORMS. SQUARE MATRICES DEFINED, SEMIDEFINITE AND INDEFINITE. MAXIMA AND MINIMA. ( 6 / 6 ). DIFFERENTIAL EQUATIONS:DEFINITIONS.THE CAUCHY PROBLEM. LOCAL AND GLOBAL EXISTENCE AND UNIQUENESS THEOREM. DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. LINEAR DIFFERENTIAL EQUATIONS. LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS. WRONSKIAN AND ITS PROPERTIES. RESOLUTION METHODS. ( 5 / 7). INTEGRALS OF FUNCTIONS OF SEVERAL VARIABLES. PROPERTIES. APPLICATION TO AREAS AND VOLUMES. REDUCTION FORMULAS. CHANGE OF VARIABLES. ( 4 / 5) CURVES AND LINE INTEGRALS. REGULAR CURVES. LENGTH OF A CURVE. RECTIFIABILITY THEOREM. LINE INTEGRAL OF A FUNCTION. ( 2 / 2 ). DIFFERENTIAL FORMS: DEFINITIONS. VECTOR FIELDS. CURVILINEAR INTEGRAL OF A LINEAR DIFFERENTIAL FORM. CLOSED AND EXACT FORMS. CRITERIA OF EXACTNESS. RELATION BETWEEN EXACTNESS AND CLOSURE. ( 2 / 2 ). SURFACES AND SURFACE INTEGRALS. 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. (3 / 2 ). SECTION LINEAR ALGEBRA AND GEOMETRY (LESSON / EXERCISES : HOURS 12/18 VECTORIAL SPACES: VECTORIAL SPACES AND SUBSPACES. LINEAR DEPENDENCE AND INDEPENDENCE. BASIS AND COMPONENTS. DIMENSION. SUM AND INTERSECTION OF SUBSPACES. INNER PRODUCT. EUCLIDEAN VECTORIAL SPACE. NORM. CAUCHY-SCHWARZ INEQUALITY. ANGLE BETWEEN VECTORS. ORTHONORMAL BASIS. (5 / 7) MATRICES AND LINEAR SYSTEMS: MATRICES. DETERMINANT AND RANK. SOLVING LINEAR SYSTEMS: ROUCHE'-CAPELLI THEOREM, CRAMER THEOREM, GAUSS METHOD. (2/ 5 ). LINDAR MAPS. KERNEL AND RANGE. DIMENSION'S THEOREM. (2/2) EIGENVALUES AND DIAGONALIZATION: CHARACTERISTIC POLYNOMIAL. EIGENSPACE AND PROPERTIES. ALGEBRAIC AND GEOMETRIC MULTIPLICITY. DIAGONALIZATION. DIAGONALIZATION OF SYMMETRIC MATRIX. POSITIVE MATRIX ( 3 / 4).

Contents

SECTION MATHEMATICAL ANALYSIS (LESSON / EXERCISES : HOURS 30/30)

NUMERICAL SERIES AND CONVERGENCE. GEOMETRIC SERIE, HARMONICSERIE. SERIES WITH POSITIVE TERMS AND CONVERGENCE CRITERIA: THE CRITERIA OF COMPARISON, THE CRITERIA OF RATIO, THE CRITERIA OF ROOT. ( 2 / 2 ).

SEQUENCES AND SERIES OF FUNCTIONS. 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. 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. ( 6 / 4 ).

FUNCTIONS OF SEVERAL VARIABLES. LIMIT AND CONTINUITY. WEIERSTRASS 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. TAYLOR'S FORMULA AND HIGHER ORDER DIFFERENTIALS. QUADRATIC FORMS. SQUARE MATRICES DEFINED, SEMIDEFINITE AND INDEFINITE. MAXIMA AND MINIMA. ( 6 / 6 ).

DIFFERENTIAL EQUATIONS:DEFINITIONS.THE CAUCHY PROBLEM. LOCAL AND GLOBAL EXISTENCE AND UNIQUENESS THEOREM. DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. LINEAR DIFFERENTIAL EQUATIONS. LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS. WRONSKIAN AND ITS PROPERTIES. RESOLUTION METHODS. ( 5 / 7).

INTEGRALS OF FUNCTIONS OF SEVERAL VARIABLES. PROPERTIES. APPLICATION TO AREAS AND VOLUMES. REDUCTION FORMULAS. CHANGE OF VARIABLES. ( 4 / 5)

CURVES AND LINE INTEGRALS. REGULAR CURVES. LENGTH OF A CURVE. RECTIFIABILITY THEOREM. LINE INTEGRAL OF A FUNCTION.
( 2 / 2 ).

DIFFERENTIAL FORMS: DEFINITIONS. VECTOR FIELDS. CURVILINEAR INTEGRAL OF A LINEAR DIFFERENTIAL FORM. CLOSED AND EXACT FORMS. CRITERIA OF EXACTNESS. RELATION BETWEEN EXACTNESS AND CLOSURE. ( 2 / 2 ).

SURFACES AND SURFACE INTEGRALS. 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. (3 / 2 ).

SECTION LINEAR ALGEBRA AND GEOMETRY (LESSON / EXERCISES : HOURS 12/18

VECTORIAL SPACES: VECTORIAL SPACES AND SUBSPACES. LINEAR DEPENDENCE AND INDEPENDENCE. BASIS AND COMPONENTS. DIMENSION. SUM AND INTERSECTION OF SUBSPACES.
INNER PRODUCT. EUCLIDEAN VECTORIAL SPACE. NORM. CAUCHY-SCHWARZ INEQUALITY. ANGLE BETWEEN VECTORS. ORTHONORMAL BASIS. (5 / 7)

MATRICES AND LINEAR SYSTEMS: MATRICES. DETERMINANT AND RANK. SOLVING LINEAR SYSTEMS: ROUCHE'-CAPELLI THEOREM, CRAMER THEOREM, GAUSS METHOD. (2/ 5 ).

LINDAR MAPS. KERNEL AND RANGE. DIMENSION'S THEOREM. (2/2)

EIGENVALUES AND DIAGONALIZATION: CHARACTERISTIC POLYNOMIAL. EIGENSPACE AND PROPERTIES. ALGEBRAIC AND GEOMETRIC MULTIPLICITY. DIAGONALIZATION. DIAGONALIZATION OF SYMMETRIC MATRIX. POSITIVE MATRIX
( 3 / 4).

	Teaching Methods
	LECTURES

	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 AND AN ORAL EXAMINATION. WRITTEN TEST: THE WRITTEN TEST CONSISTS IN SOLVING TYPICAL PROBLEMS PRESENTED IN THE COURSE AND THE TOOLS USED IN SOLVING THE EXERCISES AND THE CLARITY OF ARGUMENTATION WILL BE TAKEN INTO CONSIDERATION IN THE EVALUATION. 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 STUDENTS ON THIS TOPICS AT THE WRITTEN TEST. THE WRITTEN TEST WILL BE EVALUATED IN THIRTIETHS. THE ORAL INTERVIEW 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 VALUTATION OF THE STUDENT.

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 AND AN ORAL EXAMINATION.
WRITTEN TEST: THE WRITTEN TEST CONSISTS IN SOLVING TYPICAL PROBLEMS PRESENTED IN THE COURSE AND THE TOOLS USED IN SOLVING THE EXERCISES AND THE CLARITY OF ARGUMENTATION WILL BE TAKEN INTO CONSIDERATION IN THE EVALUATION. 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 STUDENTS ON THIS TOPICS AT THE WRITTEN TEST. THE WRITTEN TEST WILL BE EVALUATED IN THIRTIETHS.
THE ORAL INTERVIEW 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 VALUTATION OF THE STUDENT.

	Texts
	P.MARCELLINI, C.SBORDONE, ANALISI MATEMATICA UNO, LIGUORI EDITORE 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). G. ALBANO, LA PROVA SCRITTA DI GEOMETRIA:TRA TEORIA E PRATICA, CUES (2011). SEYMOUR LIPSCHUTZ, MARC LIPSON, ALGEBRA LINEARE, MCGRAW-HILL LECTURE NOTES.

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	COMPULSORY ATTENDANCE. TEACHING IN ITALIAN.

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Chiara ESPOSITO | Mathematics II