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Mathematical analysis II
02KXUJM, 02KXULI, 02KXUTR
A.A. 2023/24
Course Language
Inglese
Degree programme(s)
1st degree and Bachelor-level of the Bologna process in Ingegneria Meccanica (Mechanical Engineering) - Torino
1st degree and Bachelor-level of the Bologna process in Ingegneria Dell'Autoveicolo (Automotive Engineering) - Torino
1st degree and Bachelor-level of the Bologna process in Civil And Environmental Engineering - Torino
Course structure
| Teaching | Hours |
|---|---|
| Lezioni | 39 |
| Esercitazioni in aula | 21 |
Lecturers
| Teacher | Status | SSD | h.Les | h.Ex | h.Lab | h.Tut | Years teaching |
|---|---|---|---|---|---|---|---|
| Dovetta Simone | Ricercatore a tempo det. L.240/10 art.24-B | MATH-03/A | 39 | 21 | 0 | 0 | 3 |
Co-lectures
Context
| SSD | CFU | Activities | Area context | MAT/05 | 6 | A - Di base | Matematica, informatica e statistica |
|---|
2023/24
This course first completes the theory of functions of one variable which was developed in the Mathematical Analysis course, by introducing the student to basic concepts of numerical series, power series, Fourier series, and Laplace transform. The course then introduces basic topics in the mathematical analysis of functions of several variables including differential calculus in several variables, the theory of multiple integration, line, and surface integration.
The main goal of this course is to present the basic topics in the mathematical analysis of functions of several variables. In particular, differential calculus in several variables, the theory of multiple integration, line and surface integration. The course also presents the theory of numerical, power and Fourier series.
Expected acquired knowledge:
- Definition, properties, and convergence criteria for numerical series.
- Definition, properties, and convergence criteria for sequences and series of functions, power series, and Fourier series.
- Definition and main properties of Laplace transform.
- Properties and analysis of functions of several variables.
- Definition and properties of double, triple, line, and surface integrals.
- Conservative vector field; Green, Gauss, and Stokes theorems.
Expected acquired ablities:
- to classify the asymptotic behaviour (e.g., converge or the lack thereof) of numerical series, sequences of functions, and series of functions using the criteria introduced in class;
- to infer the set and type of convergence of sequences and series of functions;
- to compute the power series approximation of a function and the Fourier series approximation of a periodic function;
- to compute the Laplace transform of simple functions and compare their properties;
- to analyze a function of several variables; find and classify its critical points;
- to compute double, triple, line, and surface integrals;
- to reproduce, discuss, and explain theoretical results presented in class and apply them to solve simple problems.
- Understanding of the subjects of the course and computational skills in applying the mathematical tools presented in the course.
- Familiarity with the mathematical contents of engineering disciplines.
- Ability in building a logical sequence of arguments using the tools introduced in the course.
The topics covered in the Mathematical Analysis I and Linear Algebra and Geometry courses. In particular, notions and properties of limits, numerical sequences, differential and integral calculus for functions of one variable, differential equations, linear algebra, geometry of curves.
The topics contained in the courses of Mathematical Analysis I and Linear Algebra and Geometry. In particular, limits, sequences, differential and integral calculus for functions of one variable, differential equations, linear algebra, geometry of curves.
- Laplace transform.
- Numerical series: definition and convergence criteria. Power series and Taylor series. Fourier series.
- Review on vectors and elementary notions of topology of R^n. Functions of several variables, vector fields. Limits and continuity. Partial and directional derivatives, Jacobian matrix. Differentiability, gradient, and tangent plane. Second derivatives, Hessian matrix. Taylor polynomial. Critical points, unconstrained maxima and minima.
- Double and triple integrals, center of mass. Length of a curve and area of the region under the graph of a function. Line and surface integrals (for graphs of functions only). Circulation and flux of a vector field.
- Conservative vector fields. Green, Gauss, and Stokes theorems.
- Review on vectors and elements of topology of R^n. Functions of several variables, vector fields. Limits and continuity. Partial and directional derivatives, Jacobian matrix. Differentiability, gradient and tangent plane. Second derivatives, Hessian matrix. Taylor polynomial. Critical points, free extrema (15 hrs).
- Double and triple integrals, center of mass. Length of a curve and area of a graph. Line and surface integrals (graphs only), circulation and flux of a vector field. Conservative vector fields. Green, Gauss and Stokes theorems (25 hrs).
- Definition and convergence criteria for numerical series. Power series. Fourier series (20 hrs).
For further information, please contact the course responsible at giacomo.como@polito.it.
The course consists in theoretical lectures and practice classes. Theoretical lectures are devoted to the presentation of the topics, with definitions, properties, introductory examples, as well as a number of selected proofs which are believed to facilitate the learning process. The practice classes are devoted to the acquisition of the abilities through guided solution of problems.
Theoretical lessons: 40 hours. Exercises: 20 hours.
Theoretical lessons are devoted to the presentation of the topics, with definitions, properties and the proofs which are believed to facilitate the learning process. Every theoretical aspect is associated with introductory examples. The exercise hours are devoted to the analysis of the methods required to solve exercises with the aim of preparing the student to the exam.
The following textbook covers the topics of the course and will be used as a reference:
- C. Canuto, A. Tabacco, "Mathematical Analysis II", Springer`
Other material will be suggested in class and made avalaible thorugh the Portale della Didattica.
All the topics of the course can be found for instance in the following textbook:
- C. Canuto, A. Tabacco, "Mathematical Analysis II", Springer, 2015.
Exercises and exams of previous years are available on the Home Page of the course in the Teaching Portal.
Other possible material will be suggested in class and made available through the Teaching Portal.
Esercizi;
Exercises;
E' possibile sostenere l’esame in anticipo rispetto all’acquisizione della frequenza
You can take this exam before attending the course
Modalità di esame: Prova scritta (in aula); Prova orale facoltativa;
Exam: Written test; Optional oral exam;
...
The examination consists of a written test and, possibly, an oral test.
The written test consists of a number of multiple choice questions and one or more problems with open-ended answer. Both parts are aimed at checking the acquired knowledge and abilities. Questions cover both theoretical aspects and the solution of simple problems, including the evaluation of integrals and series. The problem section is more complex and allows for a more precise evaluation of the abilities acquired by the student.
The written test’s duration is two hours and it is closed-book: the use of any notes, books, exercise sheets, or pocket calculators is not allowed. The grade of the written test ranges from 0/30 to 30/30 and is the aggregate of the points assigned to the single answers to the multiple choice questions and the problems with open-ended answer. The maximum number of points that can be assigned to the answer of each question and problem are explicitly specified in the exam text.
The oral test takes place only if required either by the teacher or by the student, in the latter case only if the student’s grade in the written test is larger than or equal to 18/30. The oral test aims at evaluating more in depth the knowledge and abilities acquired by the student and possibly clarifying issues raised by the written test: potential questions in the oral test may include the topics and problems covered in written test, however they are not limited to them and may span the whole course contents. In case the oral test takes place, the final grade ranges from 0/30 to 30L/30 and depends both on the written test’s grade and the student’s performance in the oral test.
Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.
Exam: Written test; Optional oral exam;
The goal of the exam is to test the knowledge of the students on the topics included in the official program of the course and to verify their computational and theoretical skills in solving problems.
Marks range from 0 to 30 and the exam is successful if the mark is at least 18/30. The exam consists of a written part and an optional oral part. The written part consists of 7 exercises with closed answer and 1 exercise with open answer on the topics presented in the course. The aim of the exam is to certify the Expected Learning Outcomes (see above). Questions cover both computational and theoretical aspects, to evaluate the ability in building a logical sequence of arguments using the tools introduced in the course.
The exam lasts 100 minutes. Marks are given according to the following rules. Each exercise with closed answer assigns: 3 points if correct, 0 points if blank, -1 point if wrong. The exercise with open answer assigns a maximum of 9 points, that can be less depending on the presence and significance of mistakes. 1 additional point is reserved to the notational clarity and the rigour of the exposition and it allows to obtain a cum Laude mark. During the exam it is forbidden to use notes, books, exercise sheets and pocket calculators. The test results will be posted on the teaching portal.
Both students and the teacher can request an optional oral part that can alter both in the positive and in the negative the mark obtained in the written part. The optional oral part can only be requested in the same exam session of the written part. Students can request the optional oral part only if the mark they obtained in the written part is at least 18/30. The optional oral part concerns mainly the theoretical aspects of the course as definitions, statements of propositions and theorems and related proofs, and it requires a thorough knowledge of the whole matter.
In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.