# Ingegneria Alimentare - Food Engineering | ADVANCED MATHEMATICS

## Ingegneria Alimentare - Food Engineering ADVANCED MATHEMATICS

 0622800001 DIPARTIMENTO DI INGEGNERIA INDUSTRIALE EQF7 FOOD ENGINEERING 2022/2023

 OBBLIGATORIO YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2019 AUTUMN SEMESTER
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/05 9 90 LESSONS SUPPLEMENTARY COMPULSORY SUBJECTS
 MARIA DI DOMENICO
Objectives
Knowledge and understanding:
Knowledge and understanding of the fundamental and advanced concepts of the analysis of complex functions of complex variables (properties, derivative and integral), Fourier series, distributions, 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 basics about of a mathematical software.

Applying knowledge and understanding – engineering analysis:
Ability to apply the studied theorems and rules to troubleshooting. Knowing how to identify and formulate problems and to obtain solutions using a mathematical software.

Applied knowledge and understanding applied - engineering design:
Knowing how to use a mathematical software for design calculations.

Communication skills – transversal skills:
Ability to expose orally, with appropriate terminology, the topics of the course. Ability to work in groups, solving exercises in collaborative mode by using a computer.

Learning skills – transversal skills:
Ability to apply knowledge in different situations than those presented in the course and ability to refine own knowledge.
Prerequisites
•INTEGRAL CALCULUS OF ONE VARIABLE FUNCTIONS, INTEGRAL ON CURVES AND DIFFERENTIAL FORMS
•BASIC OF COMPLEX NUMBERS
•LINEAR ALGEBRA
•NUMERICAL AND FUNCTION SERIES
•FUNCTION OF SEVERAL VARIABLES,
•ORDINARY DIFFERENTIAL EQUATIONS
Contents
1) FOURIER SERIES (4H TEO; 4H ES)
DEFINITION AND PROPERTIES. EXAMPLES. POINTWISE AND UNIFORM CONVERGENCE

2) FOURIER TRANSFORM (5H TH; 5 ES)
DEFINITION OF FOURIER TRANSFORM. PROPERTIES FOURIER TRANSFORM OF THE CONVOLUTION. INVERSE TRANSFORM THEOREM

3) LAPLACE TRANSFORM (8H TH; 7 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. GENERALIZATION OF INITIAL-VALUE THEOREM. GENERALIZATION OF FINAL-VALUE THEOREM. APPLICATION TO DIFFERENTIAL EQUATIONS.

4) ELEMENTS OF TENSORIAL CALCULUS (2H TEO)
DEFINITION OF A SECOND ORDER TENSOR. SYMMETRY AND ANTISYMMETRY. TENSORIAL ALGEBRA

5) BALANCE EQUATIONS (9H TEO)
LOGICAL DERIVATION OF THE BALANCE EQUATIONS. GLOBAL AND LOCAL FORMULATION. EULERIAN AND LAGRANGIAN FORMULATION. LAW OF CONSERVATION OF MASS: CONTINUITY EQUATION. LAW OF CONSERVATION OF MOMENTUM. LAW OF ANGULAR CONSERVATION. LAW OF ENERGY CONSERVATION.

6) PARTIAL DIFFERENTIAL EQUATIONS (10H TH; 7 ES)
CLASSIFICATION OF PDES. SECOND ORDER PDE. CHARACTERISTICS. HEAT, WAVE, AND LAPLACE EQUATIONS. BOUNDARY VALUE PROBLEMS. SOLUTIONS 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.

7) COMPLEX FUNCTIONS OF COMPLEX VARIABLES (18H TH; 8H ES)
RECALLS ABOUT COMPLEX PLANE AND ELEMENTARY FUNCTIONS. DERIVATIVE IN THE COMPLEX PLANE. ANALYTIC FUNCTIONS. CAUCHY RIEMANN EQUATIONS. ELEMENTARY FUNCTIONS IN THE COMPLEX PLANE. SINGULARITIES. INTEGRATION ON COMPLEX CURVES. CAUCHY THEOREM. CAUCHY INTEGRAL FORMULA. MORERA THEOREM. FUNDAMENTAL THEOREM OF ALGEBRA. TAYLOR AND LAURENT SERIES. RESIDUE CALCULUS AND ITS APPLICATION TO REAL FUNCTIONS INTEGRATION.

8) SOFTWARE OF MATHEMATICAL CALCULUS (3H TH)
BASIC CALCULATION. NUMERICAL CALCULATION AND SYMBOLIC COMPUTATION. PLOT.
Teaching Methods
THE COURSE CONSISTS IN A TOTAL AMOUNT OF 90 HOURS WHICH ARE WORTH 9 CREDITS. IN PARTICULAR, TEACHING INCLUDES THEORETICAL LESSONS (56H), CLASSROOM EXERCISES (31 H) AND ACTIVITIES AT THE PERSONAL COMPUTER (3 H) TO PRESENT THE SOFTWARE AND TO MAKE EXERCISES AND DOUBLE-CHECKING RESULTS.
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. THE WRITTEN TEST CONSISTS OF SOME QUESTIONS TO BE ANSWERED IN THREE HOURS. TO PASS THE TEST, THE STUDENT HAS TO BE ABLE TO SOLVE COMPLEX INTEGRAL ON CURVES AND TO PROPERLY USE THE RESIDUE CALCULUS; TO COMPUTE LAURENT AND FOURIER SERIER; TO BE ABLE TO FIND FOURIER TRANSFORM OF A GIVEN FUNCTION AND TO SOLVE A DIFFERENTIAL EQUATION BY USING THE LAPLACE TRANSFORM AND cTO 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. THE WRITTEN TEST CAN BE REPLACED BY MEANS OF TWO INTERMEDIATE TESTS, ONE DURING THE COURSE AND THE SENCOND RIGHT AFTER ITS END.
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.
KNOWLEDGE OF THE RESIDUAL THEORY AND THE CORRECT APPLICATION OF THE VARIOUS METHODS FOR COMPLEX AND REAL INTEGRALS, 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. IT IS POSSIBLE TO START THE ORAL EXAM BY MEANS OF SHORT PRESENTATION AIMED AT DEEPENING A SPECIFIC TOPIC, TREATED DURING THE COURSE.
Texts

Testi
[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.