Tiziana DURANTE | CALCULUS 2

cod. 0612700135

CALCULUS 2

	0612700135
	DEPARTMENT OF INFORMATION AND ELECTRICAL ENGINEERING AND APPLIED MATHEMATICS
	EQF6
	COMPUTER ENGINEERING
	2023/2024

CURRICULUM NETWORKS

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2022
	SPRING SEMESTER

TEACHING UNITS

SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
MAT/05	4	32	LESSONS	BASIC COMPULSORY SUBJECTS
MAT/05	2	16	EXERCISES	BASIC COMPULSORY SUBJECTS

PROFESSORS

	TIZIANA DURANTE T Curriculum
	VITTORIO ZAMPOLI Curriculum

	Objectives
	THE COURSE AIMS AT THE ACQUISITION OF FURTHER ELEMENTS OF MATHEMATICAL ANALYSIS AND COMPLEX ANALYSIS. THE OBJECTIVES OF THE COURSE ARE THE ACQUISITION OF RESULTS AND PROOF TECHNIQUES, AND THE ABILITY TO USE ITS CALCULATION TOOLS. KNOWLEDGE AND UNDERSTANDING FUNCTION SEQUENCES AND SERIES. FOURIER SERIES. MULTIVARIABLE REAL FUNCTIONS. ORDINARY DIFFERENTIAL EQUATIONS. COMPLEX FUNCTIONS OF COMPLEX VARIABLES. MATH SOFTWARE TOOLS. APPLYING KNOWLEDGE AND UNDERSTANDING APPLY THEOREMS AND RULES IN PROBLEMS SOLVING. COMPUTE THE FOURIER SERIES EXPANSION OF A FUNCTION. SOLVE ORDINARY LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS. SOLVE EXERCISES OF COMPLEX ANALYSIS. DETERMINE THE MOST APPROPRIATE METHODS TO SOLVE EFFICIENTLY A MATHEMATICAL PROBLEM.

THE COURSE AIMS AT THE ACQUISITION OF FURTHER ELEMENTS OF MATHEMATICAL ANALYSIS AND COMPLEX ANALYSIS. THE OBJECTIVES OF THE COURSE ARE THE ACQUISITION OF RESULTS AND PROOF TECHNIQUES, AND THE ABILITY TO USE ITS CALCULATION TOOLS.

KNOWLEDGE AND UNDERSTANDING
FUNCTION SEQUENCES AND SERIES. FOURIER SERIES. MULTIVARIABLE REAL FUNCTIONS. ORDINARY DIFFERENTIAL EQUATIONS. COMPLEX FUNCTIONS OF COMPLEX VARIABLES. MATH SOFTWARE TOOLS.

APPLYING KNOWLEDGE AND UNDERSTANDING
APPLY THEOREMS AND RULES IN PROBLEMS SOLVING. COMPUTE THE FOURIER SERIES EXPANSION OF A FUNCTION. SOLVE ORDINARY LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS. SOLVE EXERCISES OF COMPLEX ANALYSIS. DETERMINE THE MOST APPROPRIATE METHODS TO SOLVE EFFICIENTLY A MATHEMATICAL PROBLEM.

	Prerequisites
	FOR A SUCCESSFUL ACHIEVEMENT OF THE PROPOSED GOALS AND, IN PARTICULAR, FOR A PROPER UNDERSTANDING OF THE CONTENTS SCHEDULED FOR THE TEACHING, STUDENTS ARE REQUIRED TO MASTER THE FOLLOWING PARTICULARLY USEFUL KNOWLEDGE ABOUT: DIFFERENTIAL AND INTEGRAL CALCULUS FOR FUNCTIONS OF ONE VARIABLE, NUMERICAL SEQUENCES AND SERIES. MANDATORY PREPARATORY TEACHINGS MATHEMATICAL ANALYSIS I

	Contents
	TEACHING UNIT 1: FUNCTIONS SEQUENCE AND SERIES (LECTURE/ PRACTICE/LABORATORY HOURS 3/3/0) - 1 (2 HOURS LECTURES/PRACTICE): Definitions. Pointwise and uniform convergence. Examples and counterexamples. - 2 (2 HOURS LECTURES): Theorem on the uniform limit of continuous function. Theorem on the integration of uniformly convergent sequences. Function series. Pointwise, uniform and total convergence. Power series. Definitions. Convergence radius and convergence set. Cauchy-Hadamard theorem. D’Alembert theorem. - 3 (2 HOURS PRACTICE): Examples of uniform convergence. Calculation of the convergence radius and convergence set. KNOWLEDGE AND UNDERSTANDING ABILITY: Understanding of the basic concepts. APPLYING KNOWLEDGE AND UNDERSTANDING: Ability to apply understanding to a wider context related to the applications. TEACHING UNIT 2: FOURIER SERIES. (LECTURE/ PRACTICE/LABORATORY HOURS 3/3/0) - 4 (2 HOURS LECTURES): Trigonometric polynomials. Fourier series. Fourier coefficients. Examples. Complex form. - 5 (2 HOURS LECTURES/PRACTICE): Properties of the periodic functions. Pointwise convergence theorem. Uniform convergence theorem. Quadratic convergence. Riemann- Lebesgue’s theorem. Parseval identity. Exercises on Fourier series. - 6 (2 ORE PRACTICE): Calculation of the sum of numerical series by means of the Parseval identity and the convergence. Exercises: series expansions and convergence. KNOWLEDGE AND UNDERSTANDING ABILITY: Understanding of the basic concepts. APPLYING KNOWLEDGE AND UNDERSTANDING: Ability in Fourier analysis. TEACHING UNIT 3: FUNCTIONS OF SEVERAL VARIABLES (LECTURE/ PRACTICE/LABORATORY HOURS 4/6/0) - 7 (2 HOURS LECTURES/PRACTICE): Definitions. Limits and continuity. Weierstrass theorem. Exercises. - 8 (2 HOURS LECTURES): Partial differentiation. Gradient. Examples. Schwarz theorem. Differentiability and theorem of the total differential. - 9 (2 HOURS PRACTICE): Directional derivatives. Exercises. Derivative of the composite functions. Exercises. - 10 (2 HOURS LECTURES/PRACTICE): Local minima and maxima. Necessary condition of the first order and sufficient condition of the second order. Exercises. - 11 (2 HOURS PRACTICE): Exercises on local minima and maxima. KNOWLEDGE AND UNDERSTANDING ABILITY: Understanding of functions of several variables. APPLYING KNOWLEDGE AND UNDERSTANDING: Ability in the calculus of partial derivatives, gradient and local minima and maxima. TEACHING UNIT 4: ORDINARY DIFFERENTIAL EQUATIONS. (LECTURE/ PRACTICE/LABORATORY HOURS 4/6/0) - 12 (2 HOURS LECTURES): Definitions. Particular and general integral. Examples. Cauchy problem. Existence and uniqueness theorem. - 13 (2 HOURS PRACTICE): First order differential equations. - 14 (2 HOURS LECTURES): Linear equations. - 15 (2 HOURS PRACTICE): Solutions methods and exercises on linear differential equations with constant coefficients. - 16 (2 HOURS PRACTICE): Exercises on higher order equations. Overview on solutions methods in differential equations. KNOWLEDGE AND UNDERSTANDING ABILITY: Understanding of the differential equations and the Cauchy problem. APPLYING KNOWLEDGE AND UNDERSTANDING: Ability to solve differential equations. TEACHING UNIT 5: COMPLEX ANALYSIS. (LECTURE/ PRACTICE/LABORATORY HOURS 8/8/0) - 17 (2 HOURS LECTURES): Complex functions. Holomorphic functions. - 18 (2 HOURS LECTURES): Outline of the curves. Line integrals. Cauchy-Riemann conditions. -19 (2 HOURS PRATICE): Exercises on line integrals. -20 (2 HOURS LECTURES/PRACTICE): Elementary functions in the complex field. Singular points. -21 (2 HOURS PRACTICE/LECTURES): Singularity classification. Exercises. Cauchy’s theorem and Cauchy’s integral formula. Taylor and Laurent series. -22 (2 HOURS LECTURES/PRACTICE): Residues and residue theorem. Examples on the calculus of residues. -23 (2 HOURS PRACTICE): Examples on the calculus of residues and exercises on line integrals by means of residue theorem. -24 (2 HOURS LECTURES/PRACTICE): Residues and applications to the calculus of integrals of real functions. Theorem of the integrale average. Liouville’s theorem. Fundamental theorem of algebra. Singularity to infinity. Residues. Examples. KNOWLEDGE AND UNDERSTANDING ABILITY: Understanding of the basic concepts in the complex analysis. APPLYING KNOWLEDGE AND UNDERSTANDING: Ability in the calculus of residues and line integrals. TOTAL LECTURE/ PRACTICE/LABORATORY HOURS 22/26/0

Contents

TEACHING UNIT 1: FUNCTIONS SEQUENCE AND SERIES
(LECTURE/ PRACTICE/LABORATORY HOURS 3/3/0)
- 1 (2 HOURS LECTURES/PRACTICE): Definitions. Pointwise and uniform convergence. Examples and counterexamples.
- 2 (2 HOURS LECTURES): Theorem on the uniform limit of continuous function. Theorem on the integration of uniformly convergent sequences. Function series. Pointwise, uniform and total convergence. Power series. Definitions. Convergence radius and convergence set. Cauchy-Hadamard theorem. D’Alembert theorem.
- 3 (2 HOURS PRACTICE): Examples of uniform convergence. Calculation of the convergence radius and convergence set.

KNOWLEDGE AND UNDERSTANDING ABILITY: Understanding of the basic concepts.
APPLYING KNOWLEDGE AND UNDERSTANDING: Ability to apply understanding to a wider context related to the applications.

TEACHING UNIT 2: FOURIER SERIES.
(LECTURE/ PRACTICE/LABORATORY HOURS 3/3/0)
- 4 (2 HOURS LECTURES): Trigonometric polynomials. Fourier series. Fourier coefficients. Examples. Complex form.
- 5 (2 HOURS LECTURES/PRACTICE): Properties of the periodic functions. Pointwise convergence theorem. Uniform convergence theorem. Quadratic convergence. Riemann- Lebesgue’s theorem. Parseval identity. Exercises on Fourier series.
- 6 (2 ORE PRACTICE): Calculation of the sum of numerical series by means of the Parseval identity and the convergence. Exercises: series expansions and convergence.

KNOWLEDGE AND UNDERSTANDING ABILITY: Understanding of the basic concepts.
APPLYING KNOWLEDGE AND UNDERSTANDING: Ability in Fourier analysis.

TEACHING UNIT 3: FUNCTIONS OF SEVERAL VARIABLES
(LECTURE/ PRACTICE/LABORATORY HOURS 4/6/0)

- 7 (2 HOURS LECTURES/PRACTICE): Definitions. Limits and continuity. Weierstrass theorem. Exercises.
- 8 (2 HOURS LECTURES): Partial differentiation. Gradient. Examples. Schwarz theorem. Differentiability and theorem of the total differential.
- 9 (2 HOURS PRACTICE): Directional derivatives. Exercises. Derivative of the composite functions. Exercises.
- 10 (2 HOURS LECTURES/PRACTICE): Local minima and maxima. Necessary condition of the first order and sufficient condition of the second order. Exercises.
- 11 (2 HOURS PRACTICE): Exercises on local minima and maxima.

KNOWLEDGE AND UNDERSTANDING ABILITY: Understanding of functions of several variables.
APPLYING KNOWLEDGE AND UNDERSTANDING: Ability in the calculus of partial derivatives, gradient and local minima and maxima.

TEACHING UNIT 4: ORDINARY DIFFERENTIAL EQUATIONS.
(LECTURE/ PRACTICE/LABORATORY HOURS 4/6/0)

- 12 (2 HOURS LECTURES): Definitions. Particular and general integral. Examples. Cauchy problem. Existence and uniqueness theorem.
- 13 (2 HOURS PRACTICE): First order differential equations.
- 14 (2 HOURS LECTURES): Linear equations.
- 15 (2 HOURS PRACTICE): Solutions methods and exercises on linear differential equations with constant coefficients.
- 16 (2 HOURS PRACTICE): Exercises on higher order equations. Overview on solutions methods in differential equations.

KNOWLEDGE AND UNDERSTANDING ABILITY: Understanding of the differential equations and the Cauchy problem.
APPLYING KNOWLEDGE AND UNDERSTANDING: Ability to solve differential equations.

TEACHING UNIT 5: COMPLEX ANALYSIS.
(LECTURE/ PRACTICE/LABORATORY HOURS 8/8/0)

- 17 (2 HOURS LECTURES): Complex functions. Holomorphic functions.
- 18 (2 HOURS LECTURES): Outline of the curves. Line integrals. Cauchy-Riemann conditions.
-19 (2 HOURS PRATICE): Exercises on line integrals.
-20 (2 HOURS LECTURES/PRACTICE): Elementary functions in the complex field. Singular points.
-21 (2 HOURS PRACTICE/LECTURES): Singularity classification. Exercises. Cauchy’s theorem and Cauchy’s integral formula. Taylor and Laurent series.
-22 (2 HOURS LECTURES/PRACTICE): Residues and residue theorem. Examples on the calculus of residues.
-23 (2 HOURS PRACTICE): Examples on the calculus of residues and exercises on line integrals by means of residue theorem.
-24 (2 HOURS LECTURES/PRACTICE): Residues and applications to the calculus of integrals of real functions. Theorem of the integrale average. Liouville’s theorem. Fundamental theorem of algebra. Singularity to infinity. Residues. Examples.

KNOWLEDGE AND UNDERSTANDING ABILITY: Understanding of the basic concepts in the complex analysis.
APPLYING KNOWLEDGE AND UNDERSTANDING: Ability in the calculus of residues and line integrals.

TOTAL LECTURE/ PRACTICE/LABORATORY HOURS 22/26/0

	Teaching Methods
	THE TEACHING CONSISTS OF FRONTAL LECTURES FOR A TOTAL OF 20 HOURS AND CLASSROOM EXERCISE SESSIONS FOR A TOTAL OF 28 HOURS. ATTENDANCE TO THE COURSE IS MANDATORY AND IT IS CERTIFIABLE EXCLUSIVELY BY USING THE PERSONAL BADGE. IN ORDER TO PARTICIPATE TO THE FINAL ASSESSMENT AND TO GAIN THE CREDITS CORRESPONDING TO THE COURSE, THE STUDENT MUST HAVE ATTENDED AT LEAST 70% OF THE HOURS OF ASSISTED TEACHING ACTIVITIES.

Teaching Methods

THE TEACHING CONSISTS OF FRONTAL LECTURES FOR A TOTAL OF 20 HOURS AND CLASSROOM EXERCISE SESSIONS FOR A TOTAL OF 28 HOURS.
ATTENDANCE TO THE COURSE IS MANDATORY AND IT IS CERTIFIABLE EXCLUSIVELY BY USING THE PERSONAL BADGE. IN ORDER TO PARTICIPATE TO THE FINAL ASSESSMENT AND TO GAIN THE CREDITS CORRESPONDING TO THE COURSE, THE STUDENT MUST HAVE ATTENDED AT LEAST 70% OF THE HOURS OF ASSISTED TEACHING ACTIVITIES.

	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. C. D’APICE, T. DURANTE, R. MANZO, VERSO L’ESAME DI MATEMATICA II, MAGGIOLI, 2015. C. D’APICE, R. MANZO, VERSO L’ESAME DI MATEMATICA III, MAGGIOLI, 2015. SUPPLEMENTARY TEACHING MATERIAL WILL BE AVAILABLE ON THE UNIVERSITY E-LEARNING PLATFORM (HTTP://ELEARNING.UNISA.IT) ACCESSIBLE TO STUDENTS USING THEIR OWN UNIVERSITY CREDENTIALS.

Texts

WRITTEN NOTES GIVEN BY THE TEACHER.
N. FUSCO, P. MARCELLINI, C. SBORDONE, ELEMENTI DI ANALISI MATEMATICA DUE, LIGUORI EDITORE.
C. D’APICE, T. DURANTE, R. MANZO, VERSO L’ESAME DI MATEMATICA II, MAGGIOLI, 2015.
C. D’APICE, R. MANZO, VERSO L’ESAME DI MATEMATICA III, MAGGIOLI, 2015.

SUPPLEMENTARY TEACHING MATERIAL WILL BE AVAILABLE ON THE UNIVERSITY E-LEARNING PLATFORM (HTTP://ELEARNING.UNISA.IT) ACCESSIBLE TO STUDENTS USING THEIR OWN UNIVERSITY CREDENTIALS.

	More Information
	THE COURSE IS HELD IN ITALIAN. THE MID-TERM TEST IS AIMED EXCLUSIVELY AT STUDENTS ATTENDING THE COURSE.

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Tiziana DURANTE | CALCULUS 2