Giuseppina Gerarda BARBIERI | MATHEMATICS II

cod. 0612200005

MATHEMATICS II

	0612200005
	DIPARTIMENTO DI INGEGNERIA INDUSTRIALE
	EQF6
	CHEMICAL ENGINEERING
	2019/2020

CURRICULUM INGEGNERIA ALIMENTARE

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2016
	SECONDO SEMESTRE

TEACHING UNITS

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

PROFESSORS

GIUSEPPINA GERARDA BARBIERI T Curriculum

	Objectives
	KNOWLEDGE AND UNDERSTANDING: UNDERSTANDING THE TERMINOLOGY USED IN MATHEMATICAL ANALYSIS; KNOWLEDGE OF DEMONSTRATION METHODS; KNOWLEDGE OF THE FUNDAMENTAL CONCEPTS OF MATHEMATICAL ANALYSIS. KNOWLEDGE RELATED TO: INTEGRAL FUNCTIONS OF A VARIABLE, NUMERICAL SERIES, SEQUENCES AND SERIES OF FUNCTIONS, FUNCTIONS OF SEVERAL VARIABLES, DIFFERENTIAL EQUATIONS, MULTIPLE INTEGRATION, CURVES AND CURVES INTEGRALS, SURFACES AND SURFACE INTEGRALS, VECTOR FIELDS. APPLYING KNOWLEDGE AND UNDERSTANDING – ENGINEERING ANALYSIS APPLYING THE THEOREMS AND THE RULES STUDIED TO SOLVE PROBLEMS. BUILDING METHODS AND TROUBLESHOOTING PROCEDURES. KNOW HOW TO PROCESS AND COMMUNICATE INFORMATION USING A FORMAL LINGUISTIC LOG. APPLYING KNOWLEDGE OF THE CONCEPTS AND METHODS OF CALCULUS AND MATHEMATICAL TOOLS TO SOLVE DIFFERENTIAL EQUATIONS, INTEGRAL CURVES, INTEGRAL INTEGRAL AND SURFACE INTEGRALS, PERFORM SERIES AND INTEGRAL CALCULATIONS, CALCULATE MAXIMUM AND MINIMUM FUNCTIONS OF TWO VARIABLES. APPLYING KNOWLEDGE TO DEVELOP DEMONSTRATIONS OF CERTAIN THEOREMS CONSISTENTLY APPLYING KNOWLEDGE AND UNDERSTANDING – ENGINEERING DESIGN APPLYING KNOWLEDGE TO FIND THE MOST APPROPRIATE METHODS TO SOLVE A MATH PROBLEM. BE ABLE TO FIND OPTIMIZATIONS IN THE PROCESS OF SOLVING A MATH PROBLEM. MAKING JUDGMENTS - ENGINEERING PRACTICE: APPLYING THE ACQUIRED KNOWLEDGE TO CONTEXTS DIFFERENT FROM THOSE PRESENTED DURING THE COURSE COMMUNICATION SKILLS – TRANSVERSAL SKILLS: LEARN MORE ABOUT THE TOPICS COVERED BY TEACHING MATERIALS OTHER THAN THOSE PROPOSED DURING THE COURSE. LEARNING SKILLS – TRANSVERSAL SKILLS: LEARN HOW TO DECIPHER THE TOPICS DISCUSSED USING TEACHING MATERIALS OTHER THAN THOSE PROPOSED DURING THE COURSE. DEVELOP A POSITIVE ATTITUDE TOWARD MATHS BASED ON RESPECT FOR TRUTH AND AVAILABILITY TO SEEK MOTIVATION AND TO CLARIFY ITS VALIDITY.

KNOWLEDGE AND UNDERSTANDING:
UNDERSTANDING THE TERMINOLOGY USED IN MATHEMATICAL ANALYSIS; KNOWLEDGE OF DEMONSTRATION METHODS; KNOWLEDGE OF THE FUNDAMENTAL CONCEPTS OF MATHEMATICAL ANALYSIS. KNOWLEDGE RELATED TO: INTEGRAL FUNCTIONS OF A VARIABLE, NUMERICAL SERIES, SEQUENCES AND SERIES OF FUNCTIONS, FUNCTIONS OF SEVERAL VARIABLES, DIFFERENTIAL EQUATIONS, MULTIPLE INTEGRATION, CURVES AND CURVES INTEGRALS, SURFACES AND SURFACE INTEGRALS, VECTOR FIELDS.

APPLYING KNOWLEDGE AND UNDERSTANDING – ENGINEERING ANALYSIS
APPLYING THE THEOREMS AND THE RULES STUDIED TO SOLVE PROBLEMS. BUILDING METHODS AND TROUBLESHOOTING PROCEDURES. KNOW HOW TO PROCESS AND COMMUNICATE INFORMATION USING A FORMAL LINGUISTIC LOG. APPLYING KNOWLEDGE OF THE CONCEPTS AND METHODS OF CALCULUS AND MATHEMATICAL TOOLS TO SOLVE DIFFERENTIAL EQUATIONS, INTEGRAL CURVES, INTEGRAL INTEGRAL AND SURFACE INTEGRALS, PERFORM SERIES AND INTEGRAL CALCULATIONS, CALCULATE MAXIMUM AND MINIMUM FUNCTIONS OF TWO VARIABLES. APPLYING KNOWLEDGE TO DEVELOP DEMONSTRATIONS OF CERTAIN THEOREMS CONSISTENTLY

APPLYING KNOWLEDGE AND UNDERSTANDING – ENGINEERING DESIGN
APPLYING KNOWLEDGE TO FIND THE MOST APPROPRIATE METHODS TO SOLVE A MATH PROBLEM. BE ABLE TO FIND OPTIMIZATIONS IN THE PROCESS OF SOLVING A MATH PROBLEM.

MAKING JUDGMENTS - ENGINEERING PRACTICE:
APPLYING THE ACQUIRED KNOWLEDGE TO CONTEXTS DIFFERENT FROM THOSE PRESENTED DURING THE COURSE

COMMUNICATION SKILLS – TRANSVERSAL SKILLS:
LEARN MORE ABOUT THE TOPICS COVERED BY TEACHING MATERIALS OTHER THAN THOSE PROPOSED DURING THE COURSE.

LEARNING SKILLS – TRANSVERSAL SKILLS:
LEARN HOW TO DECIPHER THE TOPICS DISCUSSED USING TEACHING MATERIALS OTHER THAN THOSE PROPOSED DURING THE COURSE. DEVELOP A POSITIVE ATTITUDE TOWARD MATHS BASED ON RESPECT FOR TRUTH AND AVAILABILITY TO SEEK MOTIVATION AND TO CLARIFY ITS VALIDITY.

	Prerequisites
	Calculus I

	Contents
	INTEGRATION OF FUNCTIONS OF A SINGLE VARIABLE: DEFINITION OF ANTIDERIVATIVE AND INDEFINITE INTEGRAL. BASIC INTEGRALS. RULES AND METHOD OF INTEGRATION. INTEGRAL OF RATIONAL FUNCTIONS. DEFINITE INTEGRAL AND GEOMETRICAL MEANING. FUNDAMENTAL THEOREM OF CALCULUS. (HOURS 6/8/) SEQUENCES OF FUNCTIONS: POINTWISE AND UNIFORM CONVERGENCE.THEOREM ON THE CONTINUITY OF THE LIMIT. CAUCHY CRITERION FOR UNIFORM CONVERGENCE. THEOREMS OF PASSAGE TO THE LIMIT UNDER THE INTEGRAL SIGN AND UNDER THE SIGN OF THE DERIVATIVE. FUNCTIONS SERIES: POINTWISE, UNIFORM AND TOTAL CONVERGENCE. CAUCHY CRITERIA. DERIVATION AND INTEGRATION FOR SERIES. POWER SERIES: SET AND RADIUS OF CONVERGENCE, CAUCHY-HADAMARD THEOREM. ABEL'S THEOREM. (HOURS 5/5/-) DIFFERENTIAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES: LIMIT AND CONTINUITY. WEIERSTRASS THEOREM. PARTIAL DERIVATIVES, GRADIENT, DIRECTIONAL DERIVATIVES. SCHWARZ THEOREM. DIFFERENTIABILITY. SUFFICIENT CONDITIONS FOR DIFFERENTIABILITY. THE CHAIN RULE FOR COMPUTING THE DIFFERENTIAL OF THE COMPOSITION OF TWO FUNCTIONS. TAYLOR'S FORMULA QUADRATIC FORMS. FREE AND CONSTRAINED OPTIMIZATION. (HOURS 7/9/-) ORDINARY DIFFERENTIAL EQUATIONS: PARTICULAR INTEGRAL AND GENERAL SOLUTION. THE CAUCHY PROBLEM,LOCAL AND GLOBAL EXISTENCE AND UNIQUENESS THEOREM. PROLONGATION OF SOLUTIONS. LINEAR DIFFERENTIAL EQUATIONS. RESOLUTION METHODS. SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS: A FEW BASIC FACTS. (HOURS 6/8/-) MULTIPLE INTEGRALS: APPLICATIONS TO AREAS AND VOLUMES. THE PAPPUS-GULDINO THEOREM. FUBINI THEOREM. CHANGE OF VARIABLES. (HOURS 5/7/-) CURVES: REGULAR CURVES. LENGTH OF A CURVE. LINE INTEGRAL. DIFFERENTIAL FORMS: INTEGRAL OF A DIFFERENTIAL FORM. CLOSED AND EXACT FORMS. CRITERIA OF EXACTNESS. CLOSED FORMS IN A SIMPLY CONNECTED SET. (HOURS 6/6/-) SURFACES: AREA OF A SURFACE AND SURFACE INTEGRALS. THE PAPPUS-GULDINO THEOREM. DIVERGENCE THEOREM AND STOKES FORMULA.(HOURS 5/7/-)

Contents

INTEGRATION OF FUNCTIONS OF A SINGLE VARIABLE: DEFINITION OF ANTIDERIVATIVE AND INDEFINITE INTEGRAL. BASIC INTEGRALS. RULES AND METHOD OF INTEGRATION. INTEGRAL OF RATIONAL FUNCTIONS. DEFINITE INTEGRAL AND GEOMETRICAL MEANING. FUNDAMENTAL THEOREM OF CALCULUS.     (HOURS 6/8/)

SEQUENCES OF FUNCTIONS: POINTWISE AND UNIFORM CONVERGENCE.THEOREM ON THE CONTINUITY OF THE LIMIT. CAUCHY CRITERION FOR UNIFORM CONVERGENCE. THEOREMS OF PASSAGE TO THE LIMIT UNDER THE INTEGRAL SIGN AND UNDER THE SIGN OF THE DERIVATIVE. FUNCTIONS SERIES: POINTWISE, UNIFORM AND TOTAL CONVERGENCE. CAUCHY CRITERIA. DERIVATION AND INTEGRATION FOR SERIES.  POWER SERIES: SET  AND RADIUS OF CONVERGENCE, CAUCHY-HADAMARD THEOREM. ABEL'S THEOREM.  (HOURS 5/5/-)

DIFFERENTIAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES: LIMIT AND CONTINUITY. WEIERSTRASS THEOREM. PARTIAL DERIVATIVES, GRADIENT, DIRECTIONAL DERIVATIVES. SCHWARZ THEOREM. DIFFERENTIABILITY. SUFFICIENT CONDITIONS FOR DIFFERENTIABILITY. THE CHAIN RULE FOR COMPUTING THE DIFFERENTIAL  OF THE COMPOSITION OF TWO  FUNCTIONS. TAYLOR'S FORMULA QUADRATIC FORMS. FREE AND CONSTRAINED  OPTIMIZATION. (HOURS 7/9/-)

ORDINARY DIFFERENTIAL EQUATIONS: PARTICULAR INTEGRAL AND GENERAL SOLUTION. THE CAUCHY PROBLEM,LOCAL AND GLOBAL EXISTENCE AND UNIQUENESS THEOREM. PROLONGATION OF SOLUTIONS. LINEAR DIFFERENTIAL EQUATIONS. RESOLUTION METHODS. SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS: A FEW BASIC FACTS. (HOURS 6/8/-)

MULTIPLE INTEGRALS: APPLICATIONS TO AREAS AND VOLUMES. THE PAPPUS-GULDINO THEOREM. FUBINI THEOREM. CHANGE OF VARIABLES. (HOURS 5/7/-)

CURVES: REGULAR CURVES. LENGTH OF A CURVE. LINE INTEGRAL.  DIFFERENTIAL FORMS: INTEGRAL OF A DIFFERENTIAL FORM. CLOSED AND EXACT FORMS. CRITERIA OF EXACTNESS. CLOSED FORMS IN A SIMPLY CONNECTED SET. (HOURS 6/6/-)

SURFACES: AREA OF A SURFACE AND SURFACE INTEGRALS. THE PAPPUS-GULDINO THEOREM. DIVERGENCE THEOREM AND STOKES FORMULA.(HOURS 5/7/-)

	Teaching Methods
	THE COURSE INCLUDES LECTURES (ABOUT 40H), CLASSROOM EXERCISES (ABOUT 50H) FOR A TOTAL AMOUNT OF 90 HOURS WHICH ARE WORTH 9 CREDITS. LECTURE ATTENDANCE IS STRONGLY RECOMMENDED

	Verification of learning
	AN EXAM CONSISTS OF A WRITTEN TEST AND AN ORAL INTERVIEW. THE WRITTEN TEST TAKES 150 MINUTES AND IT CONSISTS OF SOLVING TYPICAL PROBLEMS PRESENTED DURING THE COURSE (THERE ARE EXAMPLES AVAILABLE ON THE WEBSITE). MORE PRECISELY, THE STUDENT MUST DEMONSTRATE TO BE ABLE TO SOLVE PROBLEMS THAT INVOLVE ORDINARY DIFFERENTIAL EQUATIONS, FUNCTION SERIES, OPTIMIZATION, MULTIPLE INTEGRALS, AND CURVES OR SURFACES.TO GAIN ACCESS TO THE ORAL EXAM, THE STUDENT MUST REACH AT LEAST 18 POINTS. ORAL INTERVIEW: IT AIMS AT EVALUATING THE KNOWLEDGE OF ALL TOPICS, AND COVERS DEFINITIONS, THEOREM PROOFS, EXERCISE SOLVING. FINAL EVALUATION: THE FINAL MARK DEPENDS ON THE MARK OF THE WRITTEN EXAM WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE ORAL INTERVIEW. THE STUDENT REACHES THE LEVEL OF EXCELLENCE ONCE HE PROVES TO BE ABLE TO FACE PROBLEMS NOT EXPRESSLY DEALT WITH DURING THE COURSE. THERE WILL BE TWO WRITTEN TESTS DURING THE COURSE RELATED TO THE TWO HALVES OF THE PROGRAMME.THE FINAL SCORE WILL BE GIVEN BY THE AVERAGE OF THE TESTS. IF THE FINAL SCORE EXCEEDS 18/30, THE STUDENT MAY BE CONSIDERED EXEMPT FROM THE WRITTEN TEST. THE ABILITY TO COMPUTE SIMPLE DOUBLE INTEGRALS AND TO SOLVE SIMPLE ORDINARY DIFFERENTIAL EQUATIONS AND KNOWLEDGE OF FUNDAMENTALS OF DIFFERENTIAL CALCULUS OF TWO VARIABLES, AND OF POWER SERIES IS ESSENTIAL TO ACHIEVE SUFFICIENT RESULTS. THE STUDENT REACHES THE LEVEL OF EXCELLENCE ONCE HE PROVES TO BE ABLE TO FACE PROBLEMS NOT EXPRESSLY DEALT WITH DURING THE COURSE.

Verification of learning

AN EXAM CONSISTS OF A WRITTEN TEST AND AN ORAL INTERVIEW.
THE WRITTEN TEST TAKES 150 MINUTES AND IT CONSISTS OF SOLVING TYPICAL PROBLEMS PRESENTED DURING THE COURSE (THERE ARE EXAMPLES AVAILABLE ON THE WEBSITE). MORE PRECISELY, THE STUDENT MUST DEMONSTRATE TO BE ABLE TO SOLVE PROBLEMS THAT INVOLVE ORDINARY DIFFERENTIAL EQUATIONS, FUNCTION SERIES, OPTIMIZATION, MULTIPLE INTEGRALS, AND CURVES OR SURFACES.TO GAIN ACCESS TO THE ORAL EXAM, THE STUDENT MUST REACH AT LEAST 18 POINTS.
ORAL INTERVIEW: IT AIMS AT EVALUATING THE KNOWLEDGE OF ALL TOPICS, AND COVERS DEFINITIONS, THEOREM PROOFS, EXERCISE SOLVING.
FINAL EVALUATION: THE FINAL MARK DEPENDS ON THE MARK OF THE WRITTEN EXAM WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE ORAL INTERVIEW. THE STUDENT REACHES THE LEVEL OF EXCELLENCE ONCE HE PROVES TO BE ABLE TO FACE PROBLEMS NOT EXPRESSLY DEALT WITH DURING THE COURSE.
THERE WILL BE TWO WRITTEN TESTS DURING THE COURSE RELATED TO THE TWO HALVES OF THE PROGRAMME.THE FINAL SCORE WILL BE GIVEN BY THE AVERAGE OF THE TESTS. IF THE FINAL SCORE EXCEEDS 18/30, THE STUDENT MAY BE CONSIDERED EXEMPT FROM THE WRITTEN TEST.

THE ABILITY TO COMPUTE SIMPLE DOUBLE INTEGRALS AND TO SOLVE SIMPLE ORDINARY DIFFERENTIAL EQUATIONS AND KNOWLEDGE OF FUNDAMENTALS OF DIFFERENTIAL CALCULUS OF TWO VARIABLES, AND OF POWER SERIES IS ESSENTIAL TO ACHIEVE SUFFICIENT RESULTS.
THE STUDENT REACHES THE LEVEL OF EXCELLENCE ONCE HE PROVES TO BE ABLE TO FACE PROBLEMS NOT EXPRESSLY DEALT WITH DURING THE COURSE.

	Texts
	N. FUSCO, P. MARCELLINI, C. SBORDONE, ANALISI MATEMATICA DUE, LIGUORI EDITORE; P. MARCELLINI, C.SBORDONE, ESERCITAZIONI DI ANALISI MATEMATICA UNO E DUE, LIGUORI EDITORE; C. D'APICE, T. DURANTE, R. MANZO, VERSO L'ESAME DI MATEMATICA 2, MAGGIOLI (2015); EDUCATIONAL CONTENTS ON ELEARNING PLATFORM.

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Giuseppina Gerarda BARBIERI | MATHEMATICS II