CRISTIAN TACELLI | Mathematics III
CRISTIAN TACELLI Mathematics III
cod. 0612400003
MATHEMATICS III
| 0612400003 | |
| DEPARTMENT OF INDUSTRIAL ENGINEERING | |
| EQF6 | |
| ELECTRONIC ENGINEERING | |
| 2024/2025 |
| OBBLIGATORIO | |
| YEAR OF COURSE 2 | |
| YEAR OF DIDACTIC SYSTEM 2018 | |
| AUTUMN SEMESTER |
| SSD | CFU | HOURS | ACTIVITY | |
|---|---|---|---|---|
| MAT/05 | 6 | 60 | LESSONS |
| Objectives | |
|---|---|
| THE COURSE PROVIDES THE BASIC KNOWLEDGE ABOUT COMPLEX FUNCTIONS OF COMPLEX VARIABLES, FOURIER SERIES, FOURIER TRANSFORM, LAPLACE TRANSFORM, PARTIAL DIFFERENTIAL EQUATIONS. KNOWLEDGE AND UNDERSTANDING AT THE END OF THE COURSE THE STU DENT WILL KNOW: - THE MAIN NOTIONS AND RESULTS ABOUT COMPLEX FU NCTIONS OF COMPLEX VARIABLE - THE MAIN PROPERTIES AND THEOREMS ABOUT FOURIER SERIES, FOURIER AND LAPLACE TRANSFORMS - THE METHOD S FOR THE CALCULUS OF TRANSFORMS AND INVERSE TRANSFORMS - THE PROOF TECHNIQUES. APPLYING KNOWLEDGE AND UNDERSTANDING AT THE END OF THE COURSE THE STUDENT WIL L BE ABLE TO: - APPLY THEOREMS AND RULES IN PROBLEMS SOLVING - SOLVE EXERCISES OF COMPLEX ANALYSIS - COMPUTE THE FOURIER SERIES EXPANSIO N OF A FUNCTION - CALCULATE FOURIER AND LAPLACE TRANSFORMS AND INVE RSE LAPLACE TRANSFORMS - SOLVE ORDINARY DIFFERENTIAL EQUATIONS, SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS AND INTEGRO-DIFFERENTIAL EQUATIONS BY USING TRANSFORMS - SOLVE BOUNDARY VALUE PROBLEMS FOR THE LAPLACE EQUATION, THE HEAT EQUATION AND THE VIBRATING STRING EQUATION - IDENTIFY THE BEST METHODS TO EFFIC IENTLY SOLVE A MATHEMATICAL PROBLEM - EXPLAIN THE RESOLUTION OF EXERCISES IN THE WRITTEN PROOF - EXPLAIN VERBALLY THE LEARNED KNOWLEDGE - APPLY THE ACQUIRED KNOWLEDGE TO DIFFERE NT CONTEXTS FROM THOSE PRESENTED DURING THE LESSONS - DEEPEN THE TOPICS D EALT BY USING MATERIALS OTHER THAN THE PROPOSED ONES. Judgment autonomy: Know how to identify the most appropriate methods to efficiently solve a mathematical problem. Communication skills: Know how to report in written form on problem solving. being able to report orally on the knowledge learned. Learning ability: Knowing how to apply the knowledge acquired to examples other than those presented during the course, and to delve deeper into the topics covered using materials other than those proposed. |
| Prerequisites | |
|---|---|
| KNOWLEDGE OF INTEGRATION OF FUNCTIONS OF ONE VARIABLE, INTEGRALS ON CURVES, INTEGRALS OF DIFFERENTIAL FORMS, SERIES EXPANSIONS, FUNCTIONS OF SEVERAL VARIABLES, ORDINARY DIFFERENTIAL EQUATIONS. MANDATORY PREPARATORY TEACHINGS: MATEMA TICA II |
| Contents | |
|---|---|
| COMPLEX FUNCTIONS OF COMPLEX VARIABLES (8H LECTURES, 8H EXERCISE). HOLOMORPHIC FUNCTIONS AND THEIR PROPERTIES. THE CAUCHY-RIEMANN CONDITIONS. ELEMENTARY FUNCTIONS. SINGULAR POINTS. CAUCHY’S THEOREM AND CAUCHY’S INTEGRAL FORMULAS. MORERA THEOREM. MEAN VALUE THEOREM. LIOUVILLE THEOREM. TAYLOR’S AND LAURENT SERIES. CLASSIFICATION OF SINGULAR POINTS. RESIDUES AND THE RESIDUE THEOREM. FOURIER SERIES (6H LECTURES, 6H EXERCISE). EULER-FOURIER COEFFICIENTS. BESSEL INEQUALITY. PUNCTUAL AND UNIFORM CONVERGENCE THEOREMS. SERIES INTEGRATION AND DERIVATION. FOURIER TRANSFO RM (4H LECTURES, 6H EXERCISE). DEFINITION AND PROPERTIES. THE RELATIONSHIP BETWEEN DERIVATION AND MULTIPLICATION BY MONOMIALS. CONVOLUTION TRANSFORM. INVERSION FORMULA. LAPLACE TRANSFORM (5H LECTURES , 10H EXERCISE). DEFINITION AND PROPERTIES. BEHAVIOR OF LAPLACE TRANSFORM AT INFINITY. INITIAL AND FINAL VALUE THEOREM. LAPLACE TRANSFORMS OF DERIVATIVES. MULTIPLICATION BY POWERS OF T. LAPLACE TRANSFORM OF INTEGRALS. DIVISION BY T. PERIODIC FUNCTIONS. CONVOLUTION TRANSFORM. INVERSE LAPLACE TRANSFORM. INVERSION FORMULAS. APPLICATION TO ORDINARY DIFFERENTIAL EQUATIONS, SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS AND INTEGRO-DIFFERENTIAL EQUATIONS. PARTIAL DIFF ERENTIAL EQUATIONS (2H LECTURES, 5H EXERCISE). INTRODUCTION. HEAT, WAVE AND LAPLACE EQUATIONS. BOUNDARY VALUE PROBLEMS. SOLUTIONS USING SEPARATION OF VARIABLES AND LAPLACE TRANSFORMS. |
| Teaching Methods | |
|---|---|
| FRONTAL LECTURES FOR A TOTAL OF 25 HOURS AND CLASSROOM EXERCISE SESSIONS FOR A TOTAL OF 35 HOURS. |
| Verification of learning | |
|---|---|
| THE FINAL EXAM IS DESIGNED TO EVALUATE AS A WHOLE: THE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED DURING THE COURSE. THE MASTERY OF THE MATHEMATICAL LANGUAGE IN THE WRITTEN AND ORAL TEST. THE SKILL OF PROVING THEOREMS. THE SKILL OF SOLVING EXERCISES. THE ABILITY TO IDENTIFY AND APPLY THE BEST AND MORE EFFICIENT METHOD IN EXERCISES SOLVING. THE ABILITY TO USE THE ACQUIRED KNOWLEDGE. THE EXAM CONSISTS OF A WRITTEN TEST AND AN ORAL EXAMINATION. WRITTEN TEST: THE WRITTEN TEST CONSISTS IN SOLVING TYPICAL PROBLEMS PRESENTED IN THE COURSE AND THE TOOLS USED IN SOLVING THE EXERCISES AND THE CLARITY OF ARGUMENTATION WILL BE TAKEN INTO CONSIDERATION IN THE EVALUATION. 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 STUDENTS ON THIS TOPICS AT THE WRITTEN TEST. THE WRITTEN TEST WILL BE EVALUATED IN THIRTIETHS. THE ORAL INTERVIEW IS DEVOTED TO EVALUATE THE DEGREE OF KNOWLEDGE AND MASTERY IN ALL THE TOPICS OF THE COURSE, AS DEFINITIONS, AS PROOFS OF THEOREMS AND IN SOLVING EXERCISES. THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY WITH LAUDE), DEPENDS ON THE GLOBAL VALUTATION OF THE STUDENT. THE MINIMUM REQUIREMENT FOR PASSING THE WRITTEN TEST IS THE ACHIEVEMENT OF A TOTAL SCORE OF 16/30 AND THE MINIMUM KNOWLEDGE (DEFINITIONS AND BASIC RULES) ON EVERY SINGLE EXERCISE. THE MINIMUM REQUIREMENT FOR PASSING THE ORAL TEST IS TO HAVE BASIC KNOWLEDGE ON EACH TOPIC AND THE ABILITY TO CONNECT ONE TO ANOTHER, PROVIDE STATEMENTS AND MOTIVATIONS OF THE THEOREMS COVERED IN THE COURSE AND PROVIDE PROOF OF PART OF THEM. |
| Texts | |
|---|---|
| WRITTEN NOTES GIVEN BY THE TEACHER. C. D’APICE, R. MANZO, VERSO L’ESAME DI M ATEMATICA 3, MAGGIOLI, 2015. COMPLEMENTARY REFERENCES TEXTS MURRAY R. SPIEGEL, VARIABILI COMPL ESSE, COLLANA SCHAUM’S. MURRAY R. SPIEGEL, ANALISI DI FOURIER, COLLANA SCHAUM’S. MURRAY R. SPIEGEL, TRASFORMATE DI LAPLACE, COLLANA SCH AUM’S. PAUL DUCHATEAU, D. ZACHMANN, PARTIAL DIFFERENTIAL EQUATIONS , SCHAUM’S OUTLINES SERIES. |
| More Information | |
|---|---|
| ATTENDANCE TO THE COURSE IS COMPULSORY. TEACHING IN ITALIAN. |
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