Maria DI DOMENICO | ADVANCED MATHEMATICS

cod. 0622800001

ADVANCED MATHEMATICS

	0622800001
	DEPARTMENT OF INDUSTRIAL ENGINEERING
	EQF7
	FOOD ENGINEERING
	2024/2025

PERCORSO COMUNE

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2024
	AUTUMN SEMESTER

TEACHING UNITS

SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
MAT/07	5	50	LESSONS	SUPPLEMENTARY COMPULSORY SUBJECTS
MAT/07	4	40	EXERCISES	SUPPLEMENTARY COMPULSORY SUBJECTS

PROFESSORS

	IVANA BOCHICCHIO T Curriculum
	MARIA DI DOMENICO Curriculum

	Objectives
	KNOWLEDGE AND UNDERSTANDING: KNOWLEDGE AND UNDERSTANDING OF THE FUNDAMENTAL AND ADVANCED CONCEPTS OF VECTORIAL AND TENSORIAL CALCULUS, OF FOURIER SERIES, FOURIER TRANSFORMS, LAPLACE TRANSFORM AND ANTI-TRANSFORM. KNOWLEDGE AND UNDERSTANDING OF THE FUNDAMENTAL CONCEPTS OF PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS. KNOWLEDGE AND UNDERSTANDING OF THE BASIC TENETS OF THERMO-MECHANICS OF CONTINUA. APPLYING KNOWLEDGE AND UNDERSTANDING – ENGINEERING ANALYSIS: ABILITY TO MODEL A PHYSICAL PROBLEM ACCORDING TO THE PRINCIPLES OF THERMO-MECHANICS OF CONTINUOUS SYSTEMS USING A MATHEMATICAL-PHYSICS APPROACH. KNOWING HOW APPLY THE THEOREMS AND THE STUDIED RULES TO SOLVE PHYSICAL PROBLEMS. APPLIED KNOWLEDGE AND UNDERSTANDING APPLIED - ENGINEERING DESIGN: KNOWING HOW TO IDENTIFY THE CORRECT PHYSICAL MEANING OF EACH TERM IN AN ENGINEERING PROBLEM. COMMUNICATION SKILLS – TRANSVERSAL SKILLS: ABILITY TO EXPOSE ORALLY, WITH APPROPRIATE TERMINOLOGY, THE TOPICS OF THE COURSE. ABILITY TO WORK IN GROUPS. LEARNING SKILLS – TRANSVERSAL SKILLS: ABILITY TO APPLY KNOWLEDGE IN DIFFERENT SITUATIONS THAN THOSE PRESENTED IN THE COURSE AND ABILITY TO REFINE OWN KNOWLEDGE.

KNOWLEDGE AND UNDERSTANDING:
KNOWLEDGE AND UNDERSTANDING OF THE FUNDAMENTAL AND ADVANCED CONCEPTS OF VECTORIAL AND TENSORIAL
CALCULUS, OF FOURIER SERIES, FOURIER TRANSFORMS, LAPLACE TRANSFORM AND ANTI-TRANSFORM. KNOWLEDGE AND UNDERSTANDING OF THE FUNDAMENTAL CONCEPTS OF PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS.
KNOWLEDGE AND UNDERSTANDING OF THE BASIC TENETS OF THERMO-MECHANICS OF CONTINUA.
APPLYING KNOWLEDGE AND UNDERSTANDING – ENGINEERING ANALYSIS:
ABILITY TO MODEL A PHYSICAL PROBLEM ACCORDING TO THE PRINCIPLES OF THERMO-MECHANICS OF CONTINUOUS SYSTEMS USING A MATHEMATICAL-PHYSICS APPROACH. KNOWING HOW APPLY THE THEOREMS AND THE STUDIED RULES TO SOLVE
PHYSICAL PROBLEMS.
APPLIED KNOWLEDGE AND UNDERSTANDING APPLIED - ENGINEERING DESIGN:
KNOWING HOW TO IDENTIFY THE CORRECT PHYSICAL MEANING OF EACH TERM IN AN ENGINEERING PROBLEM.
COMMUNICATION SKILLS – TRANSVERSAL SKILLS:
ABILITY TO EXPOSE ORALLY, WITH APPROPRIATE TERMINOLOGY, THE TOPICS OF THE COURSE. ABILITY TO WORK IN GROUPS.
LEARNING SKILLS – TRANSVERSAL SKILLS:
ABILITY TO APPLY KNOWLEDGE IN DIFFERENT SITUATIONS THAN THOSE PRESENTED IN THE COURSE AND ABILITY TO REFINE OWN KNOWLEDGE.

	Prerequisites
	THE COURSE PRESUPPOSES THE KNOWLEDGE OF: BASIC OF COMPLEX NUMBERS; INTEGRAL CALCULUS OF ONE VARIABLE FUNCTIONS; NUMERICAL AND FUNCTION SERIES; FUNCTION OF SEVERAL VARIABLES; ORDINARY DIFFERENTIAL EQUATIONS; LINEAR ALGEBRA

	Contents
	1) ELEMENTS OF TENSORIAL CALCULUS (18H TEO) SCALAR AND VECTOR FUNCTIONS. OPERATIONS ON VECTORS: SCALAR PRODUCT, VECTOR PRODUCT AND MIXED PRODUCT. CARTESIAN REPRESENTATION OF VECTORS. DOUBLE VECTOR PRODUCT AND VECTOR DIVISION. RESULTANT AND RESULTING MOMENT OF APPLIED VECTORS. SECOND ORDER TENSORS: INVERSE TENSORS, TRANSPOSED TENSORS, SYMMETRIC AND ANTISYMMETRIC TENSORS, ORTHOGONAL TENSORS. MATRICES AND OPERATIONS WITH MATRICES. TENSORIAL ALGEBRA. DIFFERENTIAL OPERATORS. 2) FOURIER SERIES (4H TEO; 3H ES) DEFINITION AND PROPERTIES. EXAMPLES. POINTWISE AND UNIFORM CONVERGENCE 3) FOURIER TRANSFORM (4H TH; 3 ES) DEFINITION OF FOURIER TRANSFORM. PROPERTIES. FOURIER TRANSFORM OF THE CONVOLUTION. INVERSE TRANSFORM THEOREM 4) LAPLACE TRANSFORM (6H TH; 4 ES) DEFINITION OF THE LAPLACE TRANSFORM. LAPLACE TRANSFORMS OF SOME ELEMENTARY FUNCTIONS. SUFFICIENT CONDITIONS FOR EXISTENCE OF LAPLACE TRANSFORMS. SOME IMPORTANT PROPERTIES OF LAPLACE TRANSFORMS. LAPLACE TRANSFORM OF DERIVATIVES. LAPLACE TRANSFORM OF INTEGRALS. MULTIPLICATION BY T TO THE POWER N. DIVISION BY T. PERIODIC FUNCTIONS. BEHAVIOR OF F (S) AS S APPROACH TO INFINITY. INITIAL-VALUE THEOREM. FINAL-VALUE THEOREM. APPLICATION TO DIFFERENTIAL EQUATIONS. 5) INTRODUCTION TO THERMOMECHANICS OF CONTINUOUS SYSTEM (20H TEO) MOTION OF A CONTINUOUS SYSTEM. GLOBAL AND LOCAL FORMULATION: EULERIAN AND LAGRANGIAN FORMULATION. MATERIAL DERIVATIVE. INTEGRAL FORMULATIONS OF THE GENERAL PRINCIPLES OF MECHANICS 6) PARTIAL DIFFERENTIAL EQUATIONS (16H TH; 12 ES) INTRODUCTION. CLASSIFICATION OF PDES. SECOND ORDER PDES. CLASSICAL EQUATIONS OF MATHEMATICAL PHISICS: HEAT, WAVE, AND LAPLACE EQUATIONS. BOUNDARY VALUE PROBLEMS. SOLUTIONS OF LINEAR PDES BY USING THE LAPLACE TRANSFORM. SEPARATION OF VARIABLES. HEAT EQUATION IN FOURIER AND MAXWEL—CATTANEO TEORY. SOLUTION OF HEAT EQUATIONS IN BOUNDED AND UNBOUNDED DOMAIN.

Contents

1) ELEMENTS OF TENSORIAL CALCULUS (18H TEO)
SCALAR AND VECTOR FUNCTIONS. OPERATIONS ON VECTORS: SCALAR PRODUCT,
VECTOR PRODUCT AND MIXED PRODUCT. CARTESIAN REPRESENTATION OF VECTORS.
DOUBLE VECTOR PRODUCT AND VECTOR DIVISION. RESULTANT AND RESULTING
MOMENT OF APPLIED VECTORS. SECOND ORDER TENSORS: INVERSE TENSORS,
TRANSPOSED TENSORS, SYMMETRIC AND ANTISYMMETRIC TENSORS, ORTHOGONAL
TENSORS. MATRICES AND OPERATIONS WITH MATRICES. TENSORIAL ALGEBRA.
DIFFERENTIAL OPERATORS.
2) FOURIER SERIES (4H TEO; 3H ES)
DEFINITION AND PROPERTIES. EXAMPLES. POINTWISE AND UNIFORM CONVERGENCE

3) FOURIER TRANSFORM (4H TH; 3 ES)
DEFINITION OF FOURIER TRANSFORM. PROPERTIES. FOURIER TRANSFORM OF THE
CONVOLUTION. INVERSE TRANSFORM THEOREM

4) LAPLACE TRANSFORM (6H TH; 4 ES)
DEFINITION OF THE LAPLACE TRANSFORM. LAPLACE TRANSFORMS OF SOME
ELEMENTARY FUNCTIONS. SUFFICIENT CONDITIONS FOR EXISTENCE OF LAPLACE
TRANSFORMS. SOME IMPORTANT PROPERTIES OF LAPLACE TRANSFORMS. LAPLACE
TRANSFORM OF DERIVATIVES. LAPLACE TRANSFORM OF INTEGRALS. MULTIPLICATION BY
T TO THE POWER N. DIVISION BY T. PERIODIC FUNCTIONS. BEHAVIOR OF F (S) AS S
APPROACH TO INFINITY. INITIAL-VALUE THEOREM. FINAL-VALUE THEOREM. APPLICATION
TO DIFFERENTIAL EQUATIONS.
5) INTRODUCTION TO THERMOMECHANICS OF CONTINUOUS SYSTEM (20H TEO)
MOTION OF A CONTINUOUS SYSTEM. GLOBAL AND LOCAL FORMULATION: EULERIAN AND
LAGRANGIAN FORMULATION. MATERIAL DERIVATIVE. INTEGRAL FORMULATIONS OF THE
GENERAL PRINCIPLES OF MECHANICS

6) PARTIAL DIFFERENTIAL EQUATIONS (16H TH; 12 ES)
INTRODUCTION. CLASSIFICATION OF PDES. SECOND ORDER PDES. CLASSICAL EQUATIONS
OF MATHEMATICAL PHISICS: HEAT, WAVE, AND LAPLACE EQUATIONS. BOUNDARY VALUE
PROBLEMS. SOLUTIONS OF LINEAR PDES BY USING THE LAPLACE TRANSFORM.
SEPARATION OF VARIABLES. HEAT EQUATION IN FOURIER AND MAXWEL—CATTANEO
TEORY. SOLUTION OF HEAT EQUATIONS IN BOUNDED AND UNBOUNDED DOMAIN.

	Teaching Methods
	THE COURSE CONSISTS IN A TOTAL AMOUNT OF 90 HOURS WHICH ARE WORTH 9 CREDITS. IN PARTICULAR, TEACHING INCLUDES THEORETICAL LESSONS (68H) AND CLASSROOM EXERCISES (22 H). THE LESSONS WILL ALLOW TO ACQUIRE THE KNOWLEDGE OF TENSOR CALCULUS, OF THEORY OF PARTIAL DIFFERENTIAL EQUATIONS AND OF BALANCE EQUATIONS WITH PARTICULAR REFERENCE TO THE CLASSICAL EQUATIONS OF CONTINUOUS THERMO-MECHANICS SUCH AS, FOR EXAMPLE, THE HEAT EQUATION IN THE FOURIER AND CATTANEO THEORY. THE EXERCISES WILL ALLOW TO DEVELOP THE ABILITY TO APPLY THE THEORETICAL CONCEPTS TO IDENTIFY AND SOLVE PARTIAL DIFFERENTIAL EQUATIONS. ATTENDANCE AT LECTURES IS STRONGLY RECCOMENDED.

Teaching Methods

THE COURSE CONSISTS IN A TOTAL AMOUNT OF 90 HOURS WHICH ARE WORTH 9 CREDITS.
IN PARTICULAR, TEACHING INCLUDES THEORETICAL LESSONS (68H) AND CLASSROOM
EXERCISES (22 H). THE LESSONS WILL ALLOW TO ACQUIRE THE KNOWLEDGE OF TENSOR
CALCULUS, OF THEORY OF PARTIAL DIFFERENTIAL EQUATIONS AND OF BALANCE
EQUATIONS WITH PARTICULAR REFERENCE TO THE CLASSICAL EQUATIONS OF
CONTINUOUS THERMO-MECHANICS SUCH AS, FOR EXAMPLE, THE HEAT EQUATION IN THE
FOURIER AND CATTANEO THEORY. THE EXERCISES WILL ALLOW TO DEVELOP THE
ABILITY TO APPLY THE THEORETICAL CONCEPTS TO IDENTIFY AND SOLVE PARTIAL
DIFFERENTIAL EQUATIONS.
ATTENDANCE AT LECTURES IS STRONGLY RECCOMENDED.

	Verification of learning
	THE ASSESSMENT OF THE ACHIEVEMENT OF THE OBJECTIVES WILL BE DONE BY MEANS OF A WRITTEN TEST AND AN ORAL INTERVIEW. IN PARTICULAR, THE EXAM IS DEVOTED TO VERIFY THE KNOWLEDGE AND THE UNDERSTANDING OF THE CONCEPTS PRESENTED IN THE COURSE, THE COMMAND OF THE PHYSICAL-MATHEMATICAL LANGUAGE, THE ABILITY TO IDENTIFY AND APPLY THE MOST APPROPRIATE AND EFFICIENT METHODS IN SOLVING EXERCISES. THE WRITTEN TEST AND THE ORAL TEST WILL TAKE PLACE ON CALENDARIZED DAYS. THE WRITTEN TEST CONSISTS OF SOME QUESTIONS TO BE ANSWERED IN THREE HOURS. TO PASS THE TEST, THE STUDENT HAS TO BE ABLE TO COMPUTE LAURENT AND FOURIER SERIER; TO FIND FOURIER TRANSFORM OF A GIVEN FUNCTION, TO SOLVE A DIFFERENTIAL EQUATION BY USING THE LAPLACE TRANSFORM AND TO SOLVE A STANDARD PDE (ELLIPTIC, PARABOLIC OR HYPERBOLIC) BY USING THE METHOD OF SEPARATION OF VARIABLES. THE STUDENT IS ADMITTED TO ORAL INTERVIEWS IF 18/30 MARKS HAVE BEEN REACHED IN THE WRITTEN TEST. IN THE ORAL INTERVIEW TYPICALLY THE WRITTEN TEST IS DISCUSSED AND THE STUDENT IS REQUIRED TO PROVE THEOREMS. IT WILL BE EVALUATED THE DEGREE OF MATURITY ACQUIRED ON THE CONTENT, THE QUALITY OF ORAL EXPOSITION AND THE AUTONOMY OF JUDGMENT SHOWN. THE ABILITY TO DEFINE THE INTRODUCED TRANSFORMATIONS AND THE ABILITY TO INTERPRET STANDARD PDE SOLUTIONS IN LIMITED AND UNLIMITED SPACE DOMAINS IS ESSENTIAL TO ACHIEVE SUFFICIENT RESULTS. THE STUDENT ACHIEVES THE LEVEL OF EXCELLENCE IF HE KNOWS HOW TO DEAL WITH UNUSUAL PROBLEMS THAT ARE NOT EXPLICITLY FACED DURING THE CLASS.

Verification of learning

THE ASSESSMENT OF THE ACHIEVEMENT OF THE OBJECTIVES WILL BE DONE BY MEANS
OF A WRITTEN TEST AND AN ORAL INTERVIEW. IN PARTICULAR, THE EXAM IS DEVOTED
TO VERIFY THE KNOWLEDGE AND THE UNDERSTANDING OF THE CONCEPTS PRESENTED IN
THE COURSE, THE COMMAND OF THE PHYSICAL-MATHEMATICAL LANGUAGE, THE
ABILITY TO IDENTIFY AND APPLY THE MOST APPROPRIATE AND EFFICIENT METHODS IN
SOLVING EXERCISES. THE WRITTEN TEST AND THE ORAL TEST WILL TAKE PLACE ON
CALENDARIZED DAYS. THE WRITTEN TEST CONSISTS OF SOME QUESTIONS TO BE
ANSWERED IN THREE HOURS. TO PASS THE TEST, THE STUDENT HAS TO BE ABLE TO
COMPUTE LAURENT AND FOURIER SERIER; TO FIND FOURIER TRANSFORM OF A GIVEN
FUNCTION, TO SOLVE A DIFFERENTIAL EQUATION BY USING THE LAPLACE TRANSFORM
AND TO SOLVE A STANDARD PDE (ELLIPTIC, PARABOLIC OR HYPERBOLIC) BY USING THE
METHOD OF SEPARATION OF VARIABLES. THE STUDENT IS ADMITTED TO ORAL
INTERVIEWS IF 18/30 MARKS HAVE BEEN REACHED IN THE WRITTEN TEST.
IN THE ORAL INTERVIEW TYPICALLY THE WRITTEN TEST IS DISCUSSED AND THE STUDENT
IS REQUIRED TO PROVE THEOREMS. IT WILL BE EVALUATED THE DEGREE OF MATURITY
ACQUIRED ON THE CONTENT, THE QUALITY OF ORAL EXPOSITION AND THE AUTONOMY
OF JUDGMENT SHOWN. THE ABILITY TO DEFINE THE INTRODUCED TRANSFORMATIONS
AND THE ABILITY TO INTERPRET STANDARD PDE SOLUTIONS IN LIMITED AND UNLIMITED
SPACE DOMAINS IS ESSENTIAL TO ACHIEVE SUFFICIENT RESULTS.
THE STUDENT ACHIEVES THE LEVEL OF EXCELLENCE IF HE KNOWS HOW TO DEAL WITH
UNUSUAL PROBLEMS THAT ARE NOT EXPLICITLY FACED DURING THE CLASS.

	Texts
	[1] MURRAY R. SPIEGEL, SCHAUMS OUTLINE OF FOURIER ANALYSIS WITH APPLICATIONS TO BOUNDARY VALUE PROBLEMS, COLLANA - SCHAUM'S [2] MURRAY SPIEGEL, SCHAUMS OUTLINE OF LAPLACE TRANSFORM COLLANA - SCHAUM'S. [3] A.N. TIKHONOV AND A.A. SAMARSKII, EQUATIONS OF MATHEMATICAL PHYSICS – DOVER [2] T. MANACORDA, INTRODUZIONE ALLE TERMOMECCANICA DEI CONTINUI- QUADERNI UMI [4] A. MORRO E T. RUGGERI, PROPAGAZIONE DEL CALORE ED EQUAZIONI COSTITUTIVE [5] MURRAY R. SPIEGEL, VARIABILI COMPLESSE, COLLANA - SCHAUM'S.

Texts

[1] MURRAY R. SPIEGEL, SCHAUMS OUTLINE OF FOURIER ANALYSIS WITH APPLICATIONS
TO BOUNDARY VALUE PROBLEMS, COLLANA - SCHAUM'S
[2] MURRAY SPIEGEL, SCHAUMS OUTLINE OF LAPLACE TRANSFORM COLLANA - SCHAUM'S.
[3] A.N. TIKHONOV AND A.A. SAMARSKII, EQUATIONS OF MATHEMATICAL PHYSICS – DOVER
[2] T. MANACORDA, INTRODUZIONE ALLE TERMOMECCANICA DEI CONTINUI- QUADERNI
UMI
[4] A. MORRO E T. RUGGERI, PROPAGAZIONE DEL CALORE ED EQUAZIONI COSTITUTIVE
[5] MURRAY R. SPIEGEL, VARIABILI COMPLESSE, COLLANA - SCHAUM'S.

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Maria DI DOMENICO | ADVANCED MATHEMATICS