Ingegneria Meccanica | CALCULUS II

cod. 0612300002

CALCULUS II

	0612300002
	DIPARTIMENTO DI INGEGNERIA INDUSTRIALE
	EQF6
	MECHANICAL ENGINEERING
	2022/2023

PERCORSO IN COMMON

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

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
1	MATEMATICA II (MODULO MAT/05)
	MAT/05	3	30	LESSONS	BASIC COMPULSORY SUBJECTS
2	MATEMATICA II (MODULO MAT/07)
	MAT/07	6	60	LESSONS	BASIC COMPULSORY SUBJECTS

	Objectives
	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 MAIN EDUCATIONAL OBJECTIVITIES ARE: - KNOWLEDGE AND UNDERSTANDING UNDERSTANDING THE TERMINOLOGY USED IN MATHEMATICAL ANALYSIS; KNOWLEDGE OF DEMONSTRATION METHODS; KNOWLEDGE OF THE FUNDAMENTAL CONCEPTS OF MATHEMATICAL ANALYSIS (SEQUENCES AND SERIES OF REAL-VALUED FUNCTIONS, REAL-VALUED FUNCTIONS OF SEVERAL VARIABLES, DIFFERENTIAL EQUATIONS, MULTIPLE INTEGRALS, CURVES AND CURVILINEAR INTEGRALS, SURFACES AND SURFACE INTEGRALS) - 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, MULTIPLE INTEGRALS, CURVILINEAR AND SURFACE INTEGRALS, LINEAR SYSTEMS OF EQUATIONS. - 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, TO ORALLY PRESENT A TOPIC RELATED TO MATHEMATICS. - LEARNING SKILLS THE STUDENT HAS TO DEVELOP THE LEARNING SKILLS THAT WILL BE NECESSARY FOR INSERTING HIM IN THE FOLLOWING STUDIES WITH A HIGH AUTONOMY OF STUDY, AND CRITICALLY FACE MORE GENERAL PROBLEMS.

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 MAIN EDUCATIONAL OBJECTIVITIES ARE:

- KNOWLEDGE AND UNDERSTANDING
UNDERSTANDING THE TERMINOLOGY USED IN MATHEMATICAL ANALYSIS; KNOWLEDGE OF DEMONSTRATION METHODS; KNOWLEDGE OF THE FUNDAMENTAL CONCEPTS OF MATHEMATICAL ANALYSIS (SEQUENCES AND SERIES OF REAL-VALUED FUNCTIONS, REAL-VALUED FUNCTIONS OF SEVERAL VARIABLES, DIFFERENTIAL EQUATIONS, MULTIPLE INTEGRALS, CURVES AND CURVILINEAR INTEGRALS, SURFACES AND SURFACE INTEGRALS)

- 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, MULTIPLE INTEGRALS, CURVILINEAR AND SURFACE INTEGRALS, LINEAR SYSTEMS OF EQUATIONS.

- 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, TO ORALLY PRESENT A TOPIC RELATED TO MATHEMATICS.

- LEARNING SKILLS
THE STUDENT HAS TO DEVELOP THE LEARNING SKILLS THAT WILL BE NECESSARY FOR INSERTING HIM IN THE FOLLOWING STUDIES WITH A HIGH AUTONOMY OF STUDY, AND CRITICALLY FACE MORE GENERAL PROBLEMS.

	Prerequisites
	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 CALCULUS.

	Contents
	LINEAR ALGEBRA (12 HOURS) VECTORS AND MATRICES. LINEAR SYSTEMS OF EQUATIONS. EINGENVALUES AND EINGENVECTORS. ANALYTIC GEOMETRY (20 HOURS) GEOMETRY IN 2 DIMENSIONS. TRANSLATIONS AND ROTATIONS. CONICS. GROMETRY IN 3 DIMENSIONS. QUADRICS. FUNCTIONS OF SEVERAL VARIABLES (12 HOURS) TOPOLOGY. LIMITS 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 CURVES AND CURVILINEAR INTEGRALS (6 HOURS) REGULAR CURVES. LENGTH OF A CURVE. CURVILINEAR INTEGRAL OF A FUNCTION. DIFFERENTIAL FORMS (6 HOURS) VECTOR FIELDS. CURVILINEAR INTEGRAL OF A LINEAR DIFFERENTIAL FORM. CLOSED AND EXACT FORMS. CRITERIA OF EXACTNESS. INTEGRALS OF FUNCTIONS OF SEVERAL VARIABLES (14 HOURS) PROPERTIES. APPLICATION TO AREAS AND VOLUMES. REDUCTION FORMULAS. CHANGE OF VARIABLES. SURFACES AND SURFACE INTEGRALS (6 HOURS) AREA OF A SURFACE AND SURFACE INTEGRALS. DIVERGENCE THEOREM. STOKES FORMULA.

Contents

LINEAR ALGEBRA (12 HOURS)
VECTORS AND MATRICES. LINEAR SYSTEMS OF EQUATIONS. EINGENVALUES AND EINGENVECTORS.

ANALYTIC GEOMETRY (20 HOURS)
GEOMETRY IN 2 DIMENSIONS. TRANSLATIONS AND ROTATIONS. CONICS. GROMETRY IN 3 DIMENSIONS. QUADRICS.

FUNCTIONS OF SEVERAL VARIABLES (12 HOURS)
TOPOLOGY. LIMITS 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

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

DIFFERENTIAL FORMS (6 HOURS)
VECTOR FIELDS. CURVILINEAR INTEGRAL OF A LINEAR DIFFERENTIAL FORM. CLOSED AND EXACT FORMS. CRITERIA OF EXACTNESS.

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

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

	Teaching Methods
	COMPULSORY ATTENDANCE. LECTURES ARE IN 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 EXAM IS COMPOSED BY A WRITTEN TEST AND AN ORAL INTERVIEW. TO PASS THE EXAM THE STUDENT HAS TO PASS BOTH THE WRITTEN AND ORAL TEST. THERE WILL BE IN-COURSE WRITTEN TESTS, ON THE LATEST TOPICS SEEN DURING THE LECTURES. THE STUDENTS WHO PASS THOSE TESTS WILL BE EXONERATED FROM THE WRITTEN TEST. WRITTEN TEST: IT CONSISTS IN SOLVING PROBLEMS SIMILAR TO THE ONES WHICH HAVE BEEN STUDIED DURING THE COURSE. IN MORE DETAILS, THERE WILL BE THE FOLLOWING TYPES OF EXERCISES: STUDY OF A PARAMETRIC LINEAR SYSTEM OR STUDY OF EINGENVALUES AND EIGENVECTORS (AT MOST 5 POINTS); SOLUTION OF A CAUCHY PROBLEM (AT MOST 3 POINTS); STUDY OF MAXIMA AND MINIMA OF A FUNCTION (AT MOST 3 POINTS); STUDY OF A CURVILINEAR INTEGRAL OF A DIFFERENTIAL FORM (AT MOST 4 POINT); COMPUTATION OF AN INTEGRAL OF FUNCTION OF 2 VARIABLES (AT MOST 5 POINTS); COMPUTATION OF A SURFACE INTEGRAL (AT MOST 5 POINTS); STUDY OF A CONIC OR A QUADRIC (AT MOST 5 POINTS). SCORES ARE EXPRESSED ON A SCALE FROM 1 TO 30 AND IS THE SUM OF THE SCORES OBTAINED IN EACH EXERCISES ABOVE. THOSE SCORES DEPEND ON THE STUDENT'S ABILITY IN APPLYING OWN KNOWLEDGE. TO PASS THE EXAM A MINIMUM SCORE OF 18 IS REQUIRED. ORAL INTERVIEW: IT IS SUBJECTED TO BEING SELECTED IN THE PREVIOUS WRITTEN TEST. IT NORMALLY TAKES 20 MINUTES. IT AIMS AT EVALUATING THE KNOWLEDGE OF ALL TOPICS, AND COVERS DEFINITIONS, THEOREM PROOFS, EXERCISE SOLVING. FINAL EVALUATION: THE FINAL MARK, EXPRESSED ON A SCALE FROM 18 TO 30 (POSSIBLY WITH LAUDEM), DEPENDS ON THE MARK OF THE WRITTEN EXAM, WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE ORAL INTERVIEW.

Verification of learning

THE EXAM IS COMPOSED BY A WRITTEN TEST AND AN ORAL INTERVIEW. TO PASS THE EXAM THE STUDENT HAS TO PASS BOTH THE WRITTEN AND ORAL TEST.

THERE WILL BE IN-COURSE WRITTEN TESTS, ON THE LATEST TOPICS SEEN DURING THE LECTURES. THE STUDENTS WHO PASS THOSE TESTS WILL BE EXONERATED FROM THE WRITTEN TEST.

WRITTEN TEST: IT CONSISTS IN SOLVING PROBLEMS SIMILAR TO THE ONES WHICH HAVE BEEN STUDIED DURING THE COURSE. IN MORE DETAILS, THERE WILL BE THE FOLLOWING TYPES OF EXERCISES: STUDY OF A PARAMETRIC LINEAR SYSTEM OR STUDY OF EINGENVALUES AND EIGENVECTORS (AT MOST 5 POINTS); SOLUTION OF A CAUCHY PROBLEM (AT MOST 3 POINTS); STUDY OF MAXIMA AND MINIMA OF A FUNCTION (AT MOST 3 POINTS); STUDY OF A CURVILINEAR INTEGRAL OF A DIFFERENTIAL FORM (AT MOST 4 POINT); COMPUTATION OF AN INTEGRAL OF FUNCTION OF 2 VARIABLES (AT MOST 5 POINTS); COMPUTATION OF A SURFACE INTEGRAL (AT MOST 5 POINTS); STUDY OF A CONIC OR A QUADRIC (AT MOST 5 POINTS). SCORES ARE EXPRESSED ON A SCALE FROM 1 TO 30 AND IS THE SUM OF THE SCORES OBTAINED IN EACH EXERCISES ABOVE. THOSE SCORES DEPEND ON THE STUDENT'S ABILITY IN APPLYING OWN KNOWLEDGE. TO PASS THE EXAM A MINIMUM SCORE OF 18 IS REQUIRED.

ORAL INTERVIEW: IT IS SUBJECTED TO BEING SELECTED IN THE PREVIOUS WRITTEN TEST. IT NORMALLY TAKES 20 MINUTES. IT AIMS AT EVALUATING THE KNOWLEDGE OF ALL TOPICS, AND COVERS DEFINITIONS, THEOREM PROOFS, EXERCISE SOLVING.

FINAL EVALUATION: THE FINAL MARK, EXPRESSED ON A SCALE FROM 18 TO 30 (POSSIBLY WITH LAUDEM), DEPENDS ON THE MARK OF THE WRITTEN EXAM, WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE ORAL INTERVIEW.

	Texts
	THEORY - N. FUSCO, P. MARCELLINI, C. SBORDONE, “ANALISI MATEMATICA 2 “, LIGUORI EDITORE (2016) - E. GIUSTI, "ANALISI MATEMATICA 2", BOLLATI BORINGHIERI (1989) EXERCISES - P. MARCELLINI - C. SBORDONE, “ESERCITAZIONI DI MATEMATICA Vol. 2° PRIMA E SECONDA PARTE“, LIGUORI EDITORE (2016)