CALCULUS II

ROBERTO CAPONE CALCULUS II

0612300002
DIPARTIMENTO DI INGEGNERIA INDUSTRIALE
EQF6
MECHANICAL ENGINEERING
2020/2021

OBBLIGATORIO
YEAR OF COURSE 1
YEAR OF DIDACTIC SYSTEM 2018
SECONDO SEMESTRE
CFUHOURSACTIVITY
990LESSONS


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.
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.

THE AFOREMENTIONED BASIC KNOWLEDGES ARE COMPULSORY: EACH PART OF THE EXAM WILL BE FORBIDDEN IF THE EXAM OF MATHEMATICS I DOESN'T HAVE BEEN PREVIOUSLY PASSED.
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. QUADRATIC FORMS. EINGENVALUES AND EINGENVECTORS.

ANALYTIC GEOMETRY (6 HOURS)
LINES AND PLANES IN N-DIMENSIONAL SPACE.
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 STUDIED DURING THE COURSE. SCORES ARE EXPRESSED ON A SCALE FROM 1 TO 30; THEY DEPEND ON THE STUDENT'S ABILITY IN APPLYING OWN KNOWLEDGE. In order 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)
More Information
LECTURES ARE IN ITALIAN.
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