ADVANCED MATHEMATICS

International Teaching ADVANCED MATHEMATICS

0622800001
DIPARTIMENTO DI INGEGNERIA INDUSTRIALE
EQF7
FOOD ENGINEERING
2021/2022



OBBLIGATORIO
YEAR OF COURSE 1
YEAR OF DIDACTIC SYSTEM 2019
PRIMO SEMESTRE
CFUHOURSACTIVITY
990LESSONS
Objectives
KNOWLEDGE AND UNDERSTANDING:
KNOWLEDGE AND UNDERSTANDING OF THE FUNDAMENTAL AND ADVANCED CONCEPTS OF THE ANALYSIS OF COMPLEX FUNCTIONS OF A COMPLEX VARIABLE (PROPERTIES, DERIVATIVE AND INTEGRAL), 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 SOME OF THE BASIC TOOLS OF THE SOFTWARE MATHEMATICA.

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 ALSO USING ADVANCED CALCULUS TOOLS.

APPLYING KNOWLEDGE AND UNDERSTANDING – ENGINEERING DESIGN
KNOWING HOW TO APPLY COMPLEX AND FOURIER ANALYSIS TO SEEK SOLUTIONS FOR DESIGN PROBLEMS.

COMMUNICATION SKILLS – TRANSVERSAL SKILLS:
ABILITY TO EXPOSE ORALLY A TOPIC OF THE COURSE. ABILITY TO WORK IN GROUPS, SOLVING PROBLEMS IN A COLLABORATIVE WAY.

LEARNING SKILLS – TRANSVERSAL SKILLS:
KNOWING HOW TO CREATE INTERDISCIPLINARY LINKS AND APPLY THE KNOWLEDGE AND TECHNIQUES ACQUIRED IN CONTEXTS DIFFERENT FROM THOSE PRESENTED DURING THE COURSE.

INVESTIGATION SKILLS - TRANSVERSAL SKILLS
The acquired knowledge should allow to target the problems arising from engineering by means of suitable mathematical tools, either already available, or possibly introducing new ones.
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) COMPLEX FUNCTIONS OF COMPLEX VARIABLES (20H TH; 10H 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.
2) FOURIER SERIES (5H TEO; 5H ES)
DEFINITION AND PROPERTIES. EXAMPLES. POINTWISE AND UNIFORM CONVERGENCE
3)DISTRIBUTIONS (4H TH; 3H ES): DEFINITION, CONVERGENCE, EXAMPLES, DERIVATION
4) FOURIER TRANSFORM (5H TH; 5 ES)
DEFINITION OF FOURIER TRANSFORM. PROPERTIES FOURIER TRANSFORM OF THE CONVOLUTION. INVERSE TRANSFORM THEOREM
5) LAPLACE TRANSFORM (8H TH; 7 ES)
DEFINITION OF THE LAPLACE TRANSFORM. LAPLACE TRANSFORMS OF SOME ELEMENTARY FUNCTIONS. SECTIONAL OR PIECEWISE CONTINUITY. FUNCTIONS OF EXPONENTIAL ORDER. SUFFICIENT CONDITIONS FOR EXISTENCE OF LAPLACE TRANSFORMS. SOME IMPORTANT PROPERTIES OF LAPLACE TRANSFORMS. LINEARITY PROPERTY. FIRST TRANSLATION OR SHIFTING PROPERTY. SECOND TRANSLATION OR SHIFTING PROPERTY. CHANGE OF SCALE PROPERTY. 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.
6) PARTIAL DIFFERENTIAL EQUATIONS (9H TH; 6 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 BOUNDED AND UNBOUNDED DOMAIN. STEADY STATES PROBLEMS.
7) 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 (51H), CLASSROOM EXERCISES (36 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 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. 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
MARCO CODEGONE, METODI MATEMATICI PER L'INGENGERIA, ZANICHELLI.
MURRAY R. SPIEGEL, VARIABILI COMPLESSE, COLLANA - SCHAUM'S.
MURRAY R. SPIEGEL, SCHAUMS OUTLINE OF FOURIER ANALYSIS WITH APPLICATIONS TO BOUNDARY VALUE PROBLEMS, COLLANA - SCHAUM'S.
MURRAY SPIEGEL, SCHAUMS OUTLINE OF LAPLACE TRANSFORM COLLANA - SCHAUM'S.
NOTES OF THE COURSE.
  BETA VERSION Data source ESSE3