Anna CANALE | Mathematics II

cod. 0612400002

MATHEMATICS II

	0612400002
	DEPARTMENT OF INDUSTRIAL ENGINEERING
	EQF6
	ELECTRONIC ENGINEERING
	2024/2025

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

	ANNA CANALE T Curriculum
	FABRIZIO PUGLIESE 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. LEARNING SKILL THE STUDENT SHOULD BE ABLE TO SHOW ABILITY IN INDEPENDENTLY DEALING WITH PROBLEMS NOT EXPRESSLY TREATED IN THE COURSE OF HIS STUDIES USING THE TOOLS, KNOWLEDGE AND SKILLS ACQUIRED.

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.
LEARNING SKILL
THE STUDENT SHOULD BE ABLE TO SHOW ABILITY IN INDEPENDENTLY DEALING WITH PROBLEMS NOT EXPRESSLY TREATED IN THE COURSE OF HIS STUDIES USING THE TOOLS, KNOWLEDGE AND SKILLS ACQUIRED.

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

	Contents
	SECTION MATHEMATICAL ANALYSIS (LESSON / EXERCISES : HOURS 30/30) SEQUENCES AND SERIES OF FUNCTIONS. POINTWISE AND UNIFORM CONVERGENCE. EXAMPLES AND COUNTEREXAMPLES. THEOREM ON THE CONTINUITY OF THE LIMIT. CAUCHYCRITERION 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. ( 6 / 4 ) FUNCTIONS OF SEVERAL VARIABLES. LIMIT AND CONTINUITY. WEIERSTRASS THEOREM.PARTIAL DERIVATIVES. THE SCHWARZ THEOREM. GRADIENT. DIFFERENTIABILITY. THEOREM OF THE TOTAL DIFFERENTIAL. COMPOSITE FUNCTIONS. DERIVATION OF COMPOSITE FUNCTIONS. DIRECTIONAL DERIVATIVES. MAXIMA AND MINIMA. NECESSARY AND SUFFICIENT CONDITIONS. TAYLOR'S FORMULA. ( 8 / 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. RESOLUTION METHODS. ( 6 / 8) 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. LINE INTEGRAL OF A FUNCTION. (2 / 2 ) DIFFERENTIAL FORMS: DEFINITIONS. CLOSED AND EXACT FORMS. CRITERIA OF EXACTNESS. RELATION BETWEEN EXACTNESS AND CLOSURE. PRIMITIVES. CURVILINEAR INTEGRAL OF A LINEAR DIFFERENTIAL FORM. ( 2 / 3 ) SURFACES AND SURFACE INTEGRALS. PROPERTIES. CHANGE OF PARAMETRIC REPRESENTATIONS. AREA OF A SURFACE AND SURFACE INTEGRALS. DIVERGENCE THEOREM. STOKES FORMULA. (2 / 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 PROD UCT. EUCLIDEAN VECTORIAL SPACE. NORM. CAUCHY-SCHWARZ INEQUALITY. ANGLE BETWEEN VECTORS. ORTHONORMAL BASIS. (5 / 7) MATRICES AND LINEAR SYSTEMS: MATRICES. DETERMINANT AND RANK. SOLVING LINEARSYSTEMS: 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 ANDPROPERTIES. ALGEBRAIC AND GEOMETRIC MULTIPLICITY. DIAGONALIZATION.DIAGONALIZATION OF SYMMETRIC MATRIX. POSITIVE MATRIX ( 3 / 4)

Contents

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

SEQUENCES AND SERIES OF FUNCTIONS. POINTWISE AND UNIFORM CONVERGENCE. EXAMPLES AND COUNTEREXAMPLES. THEOREM ON THE CONTINUITY OF THE LIMIT. CAUCHYCRITERION 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. ( 6 / 4 )

FUNCTIONS OF SEVERAL VARIABLES. LIMIT AND CONTINUITY. WEIERSTRASS THEOREM.PARTIAL DERIVATIVES. THE SCHWARZ THEOREM. GRADIENT. DIFFERENTIABILITY. THEOREM OF THE TOTAL DIFFERENTIAL. COMPOSITE FUNCTIONS. DERIVATION OF COMPOSITE FUNCTIONS. DIRECTIONAL DERIVATIVES. MAXIMA AND MINIMA. NECESSARY AND SUFFICIENT CONDITIONS. TAYLOR'S FORMULA. ( 8 / 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. RESOLUTION METHODS. ( 6 / 8)

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. LINE INTEGRAL OF A FUNCTION. (2 / 2 )

DIFFERENTIAL FORMS: DEFINITIONS. CLOSED AND EXACT FORMS. CRITERIA OF EXACTNESS. RELATION BETWEEN EXACTNESS AND CLOSURE. PRIMITIVES. CURVILINEAR INTEGRAL OF A LINEAR DIFFERENTIAL FORM. ( 2 / 3 )

SURFACES AND SURFACE INTEGRALS. PROPERTIES. CHANGE OF PARAMETRIC REPRESENTATIONS. AREA OF A SURFACE AND SURFACE INTEGRALS. DIVERGENCE THEOREM. STOKES FORMULA. (2 / 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 PROD UCT. EUCLIDEAN VECTORIAL SPACE. NORM. CAUCHY-SCHWARZ INEQUALITY. ANGLE BETWEEN VECTORS. ORTHONORMAL BASIS. (5 / 7)
MATRICES AND LINEAR SYSTEMS: MATRICES. DETERMINANT AND RANK. SOLVING LINEARSYSTEMS: 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 ANDPROPERTIES. ALGEBRAIC AND GEOMETRIC MULTIPLICITY. DIAGONALIZATION.DIAGONALIZATION OF SYMMETRIC MATRIX. POSITIVE MATRIX ( 3 / 4)

	Teaching Methods
	LECTURES

	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 provides for exercises and knowledge of the topics contained in the program. 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. The exam 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), is attributed when the student demonstrates a complete and in-depth knowledge of all the topics. The student reaches the level of excellence if he/she proves to be able to deal independently with problems not expressly dealt during the lessons.

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 provides for exercises and knowledge of the topics contained in the program.
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.
The exam 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), is attributed when the student demonstrates a complete and in-depth knowledge of all the topics.
The student reaches the level of excellence if he/she proves to be able to deal independently with problems not expressly dealt during the lessons.

	Texts
	WRITTEN NOTES GIVEN BY THE TEACHER. N. FUSCO, P. MARCELLINI, C. SBORDONE, ELEMENTI DI ANALISI MATEMATICA DUE, LIGUORI EDITORE. N. FUSCO, P. MARCELLINI, C. SBORDONE, ANALISI MATEMATICA DUE, LIGUORI EDITORE. P. MARCELLINI, C. SBORDONE, ESERCITAZIONI DI MATEMATICA, VOLUME II, PARTE PRIMA E SECONDA, LIGUORI EDITORE. C. D’APICE - T. DURANTE - R. MANZO, VERSO L’ESAME DI MATEMATICA 2, MAGGIOLI EDITORE. F. BOTTACIN, ALGEBRA LINEARE E GEOMETRIA, ESCULAPIO EDITORE

Texts

WRITTEN NOTES GIVEN BY THE TEACHER.
N. FUSCO, P. MARCELLINI, C. SBORDONE, ELEMENTI DI ANALISI MATEMATICA DUE, LIGUORI EDITORE.
N. FUSCO, P. MARCELLINI, C. SBORDONE, ANALISI MATEMATICA DUE, LIGUORI EDITORE.
P. MARCELLINI, C. SBORDONE, ESERCITAZIONI DI MATEMATICA, VOLUME II, PARTE PRIMA E SECONDA, LIGUORI EDITORE.
C. D’APICE - T. DURANTE - R. MANZO, VERSO L’ESAME DI MATEMATICA 2, MAGGIOLI EDITORE.

F. BOTTACIN, ALGEBRA LINEARE E GEOMETRIA, ESCULAPIO EDITORE

	More Information
	COMPULSORY ATTENDANCE. TEACHING IN ITALIAN.

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Anna CANALE | Mathematics II