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Mathematical analysis 2

A.Y. 2019/2020

Learning objectives

The course is devoted to carry on the illustration of basic concepts of Mathematical Analysis, previously started with the course of Mathematical Analysis 1. That is done not bounding the teaching to the calculus techniques, but opening it to an incisive learning. Mean objects are Riemann integration theory for functions of one real variable, differential calculus for functions of several real variables with application to free optimization, study of sequences and series of functions and basic background on ordinary differential equations together with related integration techniques.

Expected learning outcomes

We expect the student to absorb the fully basic notions, that have been taught, at the suitably incisive level relatively to his scholarship, taking into account that the given one is a course in Mathematical Analysis, not simply in Calculus. Assume the student will had taken successfully the course: not only he will get a suitable manual skill in calculus, but he will be also able to deal with problems, in the context of the taught subjects, that, even set in stabilized models, cannot be solved by a passive application of standard rules. In particular:

· when dealing with the integration of ordinary differential equations or the free optimization, he will be able to get suitable estimates when the exact results are not available or not requested;

· he will be able to manage the limit processes for sequences of functions with respect to the most significant properties of regularity.

· when dealing with the integration of ordinary differential equations or the free optimization, he will be able to get suitable estimates when the exact results are not available or not requested;

· he will be able to manage the limit processes

**Lesson period:**
Second semester

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### CORSO A

Responsible

Lesson period

Second semester

**Course syllabus**

The Riemann integral for functions of one real variable. Improper integrals. Rectifiable curves. Functions of several variables and differentiability. Unconstrained extrema. Sequences of functions: pointwise and uniform convergence. Series of functions, power series. First order differential equations: Cauchy problem, existence and uniqueness of solutions. Higher order differential equations. Linear ODE's.

**Prerequisites for admission**

1. All subjects that have been developed in Mathematical Analysis 1.

2. Basics in linear algebra (matrices, determinants, linear systems).

3. Basics in anlaytic geometry (lines and conics in two dimensions).

2. Basics in linear algebra (matrices, determinants, linear systems).

3. Basics in anlaytic geometry (lines and conics in two dimensions).

**Teaching methods**

Teaching will be held by handling theory and exercises at the board in front of the students, possibly by two different teachers.

**Teaching Resources**

B. Gelbaum, J. Olmsted, Counterexamples in Analysis, Holden-Day C.Maderna, Analisi Matematica 2, Città Studi Ed. W. Rudin, Principles of Mathematical Analysis, McGraw P.M. Soardi. Analisi Matematica. Città Studi Notes of the teacher and suggested exercise

**Assessment methods and Criteria**

The final examination consists of a written test and an oral colloquium.

- During the written exam, the student must solve some problems in the format of open questions, providing a full explanation for some of the given answers. The aim is to assess the student's ability to solve problems in the subjects that have been taught . The duration of the written exam will be proportional to the number of proposed problems, also taking into account the nature and complexity of the problems themselves (however, the duration will not exceed three hours). The outcomes of the tests will be available in the SIFA service through the UNIMIA portal.

- The colloquium can be taken only if the written component does not fall below a suitable level. In the oral exam, the student will be required to illustrate results presented during the course as well as to solve problems in the context of the presented subjects, in order to evaluate her/his knowledge and comprehension of the covered subjects as well as the ability in applying them. The duration of the colloquium depends on the reaction time of the student to the proposed questions (the expected average is 45 minutes).

The complete final examination is passed if the full level of the two parts (the written and the oral ones) is sufficiently high. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.

- During the written exam, the student must solve some problems in the format of open questions, providing a full explanation for some of the given answers. The aim is to assess the student's ability to solve problems in the subjects that have been taught . The duration of the written exam will be proportional to the number of proposed problems, also taking into account the nature and complexity of the problems themselves (however, the duration will not exceed three hours). The outcomes of the tests will be available in the SIFA service through the UNIMIA portal.

- The colloquium can be taken only if the written component does not fall below a suitable level. In the oral exam, the student will be required to illustrate results presented during the course as well as to solve problems in the context of the presented subjects, in order to evaluate her/his knowledge and comprehension of the covered subjects as well as the ability in applying them. The duration of the colloquium depends on the reaction time of the student to the proposed questions (the expected average is 45 minutes).

The complete final examination is passed if the full level of the two parts (the written and the oral ones) is sufficiently high. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 8

Practicals: 40 hours

Lessons: 32 hours

Lessons: 32 hours

Professors:
Calanchi Marta, Tarsi Cristina

### CORSO B

Responsible

Lesson period

Second semester

**Course syllabus**

The Riemann integral for functions of one real variable. Improper integrals. Rectifiable curves. Functions of several variables and differentiability. Unconstrained extrema. Sequences of functions: pointwise and uniform convergence. Series of functions, power series. First order differential equations: Cauchy problem, existence and uniqueness of solutions. Higher order differential equations. Linear ODE's.

**Prerequisites for admission**

1. All subjects that have been developed in Mathematical Analysis 1.

2. Basics in linear algebra (matrices, determinants, linear systems).

3. Basics in anlaytic geometry (lines and conics in two dimensions).

2. Basics in linear algebra (matrices, determinants, linear systems).

3. Basics in anlaytic geometry (lines and conics in two dimensions).

**Teaching methods**

Teaching will be held by handling theory and exercises at the board in front of the students, possibly by two different teachers.

**Teaching Resources**

B. Gelbaum, J. Olmsted, Counterexamples in Analysis, Holden-Day

C.Maderna, Analisi Matematica 2, Città Studi Ed.

W. Rudin, Principles of Mathematical Analysis, McGraw

P.M. Soardi. Analisi Matematica. Città Studi

Notes of the teacher and suggested exercises available at the web page of the teacher.

C.Maderna, Analisi Matematica 2, Città Studi Ed.

W. Rudin, Principles of Mathematical Analysis, McGraw

P.M. Soardi. Analisi Matematica. Città Studi

Notes of the teacher and suggested exercises available at the web page of the teacher.

**Assessment methods and Criteria**

The final examination consists of a written test and an oral colloquium.

- During the written exam, the student must solve some problems in the format of open questions, providing a full explanation for some of the given answers. The aim is to assess the student's ability to solve problems in the subjects that have been taught . The duration of the written exam will be proportional to the number of proposed problems, also taking into account the nature and complexity of the problems themselves (however, the duration will not exceed three hours). The outcomes of the tests will be available in the SIFA service through the UNIMIA portal.

- The colloquium can be taken only if the written component does not fall below a suitable level. In the oral exam, the student will be required to illustrate results presented during the course as well as to solve problems in the context of the presented subjects, in order to evaluate her/his knowledge and comprehension of the covered subjects as well as the ability in applying them. The duration of the colloquium depends on the reaction time of the student to the proposed questions (the expected average is 45 minutes).

The complete final examination is passed if the full level of the two parts (the written and the oral ones) is sufficiently high. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.

- During the written exam, the student must solve some problems in the format of open questions, providing a full explanation for some of the given answers. The aim is to assess the student's ability to solve problems in the subjects that have been taught . The duration of the written exam will be proportional to the number of proposed problems, also taking into account the nature and complexity of the problems themselves (however, the duration will not exceed three hours). The outcomes of the tests will be available in the SIFA service through the UNIMIA portal.

- The colloquium can be taken only if the written component does not fall below a suitable level. In the oral exam, the student will be required to illustrate results presented during the course as well as to solve problems in the context of the presented subjects, in order to evaluate her/his knowledge and comprehension of the covered subjects as well as the ability in applying them. The duration of the colloquium depends on the reaction time of the student to the proposed questions (the expected average is 45 minutes).

The complete final examination is passed if the full level of the two parts (the written and the oral ones) is sufficiently high. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 8

Practicals: 40 hours

Lessons: 32 hours

Lessons: 32 hours

Professors:
Messina Francesca, Zanco Clemente