2021/2022

2021/2022

DM270

MAT/05 (MATHEMATICAL ANALYSIS)

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

ARTIFICIAL INTELLIGENCE

PERCORSO COMUNE

1°

Annual (04/10/2021 - 17/06/2022)

12

108 lesson hours

English

WRITTEN AND ORAL TEST

RONDI LUCA (titolare) - 12 ECTS

There are no particular prerequisites except the basic notions of mathematics that can be acquired in any secondary high school. All the topics of the course are developed from the very beginning and students are not required to know anything about them in advance.

The aim of the course is to provide the basic mathematical tools, both from a conceptual and from a calculus point of view, which are essential to successfully attend a university undergraduate program in a scientific area. The course should also provide the required mathematics prerequisites for the other courses of the program.

At the end of the course, students should prove to have a sufficient knowledge of basic mathematics, that includes the main properties of sets, of the main number sets, in particular of real numbers, of functions between sets, of elementary functions, of combinatorics and of complex numbers. Also, they should know the basic results in the theory of differential and integral calculus for functions of one real variable, of numerical series, of topology for Euclidean spaces, of differential calculus in several variables, of multiple integrals and of ordinary differential equations.

Finally, at the end of the course students should be able to apply the theoretical results to solve elementary problems and exercises and in particular they should be able to tackle the following kinds of problems: computation of limits of sequences or functions, analysis of the continuity of a function, computation of derivates, study of the qualitative graph of a function, computation of the Taylor polynomial and expansion, computation of definite and indefinite integrals, study of the character of numerical series, analysis of the continuity of a function of several variables, computation of partial derivatives, analysis of the differentiability of a function of several variables, determination of tangent spaces to the graph or level sets of a function of several variables, solution of maximum and minimum problems, analysis of free and constrained maximum and minimum points, computation of multiple integrals, computation of areas and volumes, use of polar, cylindrical and spherical coordinates, analysis of the Cauchy problem for ordinary differential equations: existence and uniqueness of solutions, solution of the Cauchy problem for some classes of ordinary differential equations.

1. Real numbers and real functions.
The set of real numbers. Maximum, minimum, supremum, infimum. Elementary properties of functions. Elementary functions. Basics of combinatorics. Complex numbers.

2. Limits of sequences.
Definitions and first properties. Bounded sequences. Operations with limits. Comparison theorems. Monotone sequences. Undetermined forms. Special limits.

3. Limits of functions and continuous functions.
Definition and first properties of limits of functions and of continuous functions. Types of discontinuities. Limits and continuity of the composition of functions. Some important theorems on continuous functions.

4. Derivatives and study of functions.
Definition of derivatives. Computation of derivatives. Theorems of Fermat, Rolle, Lagrange and Cauchy and their consequences. Second and higher order derivatives. Applications to the study of functions. L'Hopital theorem and Taylor formula.

5. Integration.
Definite integrals and method of exhaustion. Definition of integrable functions and classes of integrable functions. Properties of the definite integrals. Indefinite integrals. Fundamental theorem of integral calculus. Integration methods. Integration by parts and by substitution. Integration of rational functions.

6. Numerical series.
Definition of numerical series and of their character. Harmonic and geometric series. Convergence criteria for series.

7. Topology in Euclidean spaces.
Definition of distance. Definition of neighbourhoods, open sets, closed sets. Definition of the interior, closure and boundary of a set.

8. Differential calculus in several variables.
Partial derivatives and directional derivatives. Differentiable functions. Jacobian matrix and gradient. Tangent space to the graph. Continuity of partial derivatives implies differentiability. Differential of the composition of maps (chain rule). Higher order derivatives and Hessian matrix.

9. Maximum and minimum problems.
Necessary and sufficient conditions for local maxima and minima. Implicit functions theorem. Geometric properties of the gradient and of level sets. Constrained minimum problems. Lagrange multipliers.

10. Multiple integrals.
Definition of Riemann integral in several variables and its characterisations. Peano-Jordan measure. Fubini theorem and
applications. Change of variables. Polar coordinates in the plane and spherical and cylindrical coordinates in the space. Guldino theorem.

11. Ordinary differential equations.
Definition of Cauchy problem. Local existence and uniqueness theorem. Global uniqueness theorem and maximal solutions. Methods for solving some classes of first order scalar equations. Higher order linear equations.

Approximately, Chapter 1-6 will be covered in the first semester, Chapter 7-11 will be covered in the second semester.

The final program will we published at the end of classes on the course web page

Lectures given by the teacher of the course, approximately 6 hours per week. A significant part of the lectures will be devoted to examples and exercise sessions. Attending the lectures is strongly suggested.

Weekly homework: every week a sheet of exercises on the topics discussed in the lectures will be provided on the course web page. Students are encouraged to try to solve all these suggested exercises.

Two-hours tutoring class per week: tutors will be available to discuss the previous week exercises and to answer questions and clarify doubts. Participation to this activity is strongly encouraged.

During classes period, the teacher will have office hours two hours per week at a fixed schedule. He will also be available at other times by appointment.

Textbooks:
C. Canuto and A. Tabacco, Mathematical Analysis I, second edition, Springer, 2015
C. Canuto and A. Tabacco, Mathematical Analysis II, second edition, Springer, 2015

or their Italian versions:

C. Canuto and A. Tabacco, Analisi Matematica 1, Pearson, 2021
C. Canuto and A. Tabacco, Analisi Matematica 2, Pearson, 2021

Other suitable textbooks:

P. Marcellini and C. Sbordone, Elementi di Analisi Matematica uno, Liguori, 2002
N. Fusco, P. Marcellini and C. Sbordone, Elementi di Analisi Matematica due, Liguori, 2001

Exercises will be provided on the course web page

The exam consists of a written test. Upon request by the exam committee, the written exam may be complemented by an oral part.
For this oral exam no further preparation with respect to the written test is required.

The written test is a closed books test: no notes, books, calculators or similar instruments, items with a photocamera or able to connect to the internet are not allowed.

Students are required to provide an ID card with a photograph.

In the written test, students should solve some exercises on the topics of the course and answer to some questions of theoretical character on the program of the course. For some of the exercises or questions, only the solutions or the answers will be required, without any detailed explanation. Other exercises or questions will require a fully detailed solution or answer. For these exercises, both the correctness of the answer and the justification of it are evaluated.

To sit at a written test, students are required to register through the university online system, within the deadline provided.

There are three exam periods: June-July, September, January-February. In each of them there will be two exam sessions.

Partial and final scores will be communicated to the students by e-mail.

The exam is passed if the final score is greater than or equal to 18/30. Students are allowed, within a few days, to reject a positive final score and to retake the written test to improve their scores.

The written test lasts at most 3 hours and is divided into two parts: Part 1 on the topics of the first semester and Part 2 on the topics of the second semester. Students have the following options:

- take the whole exam, that is, Part 1 and Part 2 at the same time, on any exam session
- take only Part 1 of the exam in the first session of an exam period (half time allowed); if the score on Part 1 is greater than or equal to 15/30, they can take Part 2 of the exam in the second session of the same exam period (half time allowed). If also the score of Part 2 is greater than or equal to 15/30 and the mean of Part 1 and Part 2 is greater than or equal to 18/30, the exam is passed, with the mean as final score. If they fail to pass Part 2 or the mean is below 18/30, they need to retake both parts in the next exam period.

First year students can also take Part 1 of the exam during the two exam sessions in January-February. If they obtain a score greater than or equal to 15/30, they can take Part 2 of the exam in any of the two exam sessions in June-July.

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