#
Mathematical Analysis 2

A.Y. 2020/2021

Learning objectives

The aim of the course is to provide basic notions and tools in the setting of the classical integral calculus for real functions of one as well as several real variables and of the differential calculus for functions of several real variables.

Expected learning outcomes

Capability to relate different aspects of the subject, and self-confidence in the use of the main techniques of Calculus.

**Lesson period:** Second semester
(In case of multiple editions, please check the period, as it may vary)

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### Analisi Matematica 2 (ediz.1)

Responsible

Lesson period

Second semester

Videos with lessons and exercises collections will be made available at the official web space of this course and they will cover the topic of each week. Live meetings during the classes hours may be scheduled. The details of these meetings will be available in the course web page of the Ariel platform, as well as the videos and every kind of material needed for the course.

The exams will be done by following the ways suggested on the web page of the university. The written exam will have the same structure as the one done in presence, possibly reduced in time and number of exercises.

The exams will be done by following the ways suggested on the web page of the university. The written exam will have the same structure as the one done in presence, possibly reduced in time and number of exercises.

**Course syllabus**

1.The Riemann integral in one variable. Antiderivation and indefinite integrals. Definition and elementary properties of the Riemann integral on a compact interval. A fundamental characterization of the class of integrable functions. The geometrical meaning of the defined integral. Functions defined by integrals and their properties. The Fundamental Theorem of Calculus and the Fundamental Formula of Calculus in one variable. Improper integrals: definitions, examples and comparison techniques. Relationships between integrals and numerical series.

2.Differential calculus in several real variables. Limits and continuity. Directional derivatives, gradients and the Jacobian matrix. Differentiability: necessary and/or sufficiente conditions. Tangent hyperplanes. Lagrange's theorem. Vector valued functions: differentiation and composition, the Chain Rule. Second order derivatives, the Hessian matrix and Schwartz's theorem. Taylor's formula. Unconstrained optimization: local extrema and saddle points. Eigenvalues and the Hessian matrix.

3.The Riemann integral in several real variables.. The Riemann integral over pluri-rectangles: definition and techniques for the computation. Brief discussion of Peano-Jordan measure. The Riemann integral on admissible domains: definition, integrability and the calculation of mutltiple integrals on simple domains. Sets of measure zero, almost everywhere continuous functions and their integrability. Diffeomorphisms of open sets in R^n. Integration in higher dimensions. Change of variables and special coordinate systems: polar, cylindrical and spherical. Improper multiple integrals. The Gaussian integral, the volume oft he unit ball in R^n, Euler's Gamma function.

2.Differential calculus in several real variables. Limits and continuity. Directional derivatives, gradients and the Jacobian matrix. Differentiability: necessary and/or sufficiente conditions. Tangent hyperplanes. Lagrange's theorem. Vector valued functions: differentiation and composition, the Chain Rule. Second order derivatives, the Hessian matrix and Schwartz's theorem. Taylor's formula. Unconstrained optimization: local extrema and saddle points. Eigenvalues and the Hessian matrix.

3.The Riemann integral in several real variables.. The Riemann integral over pluri-rectangles: definition and techniques for the computation. Brief discussion of Peano-Jordan measure. The Riemann integral on admissible domains: definition, integrability and the calculation of mutltiple integrals on simple domains. Sets of measure zero, almost everywhere continuous functions and their integrability. Diffeomorphisms of open sets in R^n. Integration in higher dimensions. Change of variables and special coordinate systems: polar, cylindrical and spherical. Improper multiple integrals. The Gaussian integral, the volume oft he unit ball in R^n, Euler's Gamma function.

**Prerequisites for admission**

It is strongly suggested that the student has already passed the following exams: "Analisi Matematica 1" and "Geometria 1".

**Teaching methods**

Frontal teaching. Problem sessions. Homeworks and their solution during the tutorial time.

**Teaching Resources**

References for different arguments will be indicated day-by-day, choosing

from the following list of textbooks:

C.Maderna, "Analisi Matematica 2" II ediz., CittàStudi ed., 2010.

C.Maderna, P.M.Soardi, "Lezioni di Analisi Matematica II", CittàStudi ed., 1997.

C.D.Pagani, S.Salsa, "Analisi Matematica, v.2", Zanichelli ed., 2016.

P.M.Soardi, "Analisi Matematica", CittàStudi ed., 2010.

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

W.Rudin, "Principles of Mathematical Analysis", McGraw-Hill

N.Fusco, P.Marcellini, C.Sbordone, "Analisi Matematica due", Liguori ed.

Moreover, a good training aimed to the written exam can be obtained working on the problems contained in:

http://users.mat.unimi.it/users/vignati/Esercizi-An2-Mat-gram.html

from the following list of textbooks:

C.Maderna, "Analisi Matematica 2" II ediz., CittàStudi ed., 2010.

C.Maderna, P.M.Soardi, "Lezioni di Analisi Matematica II", CittàStudi ed., 1997.

C.D.Pagani, S.Salsa, "Analisi Matematica, v.2", Zanichelli ed., 2016.

P.M.Soardi, "Analisi Matematica", CittàStudi ed., 2010.

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

W.Rudin, "Principles of Mathematical Analysis", McGraw-Hill

N.Fusco, P.Marcellini, C.Sbordone, "Analisi Matematica due", Liguori ed.

Moreover, a good training aimed to the written exam can be obtained working on the problems contained in:

http://users.mat.unimi.it/users/vignati/Esercizi-An2-Mat-gram.html

**Assessment methods and Criteria**

The final examination consists of two parts: a written exam and an oral exam.

- During the written exam, the student must solve some exercises in the format of open-ended and/or short answer questions, with the aim of assessing the student's ability to solve problems in Mathematical Analysis. The duration of the written exam is proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration does not exceed three hours). In place of a single written exam given during the first examination session, the student may choose instead to take 2 midterm exams. The outcomes of these tests will be available in the SIFA service through the UNIMIA portal.

- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student is required to illustrate results presented during the course and may be required to solve problems regarding Mathematical Analysis, in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.

The complete final examination is passed if both parts (written and oral) are successfully passed. Final marks are given using the numerical range 0-30, and are communicated immediately after the oral examination.

- During the written exam, the student must solve some exercises in the format of open-ended and/or short answer questions, with the aim of assessing the student's ability to solve problems in Mathematical Analysis. The duration of the written exam is proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration does not exceed three hours). In place of a single written exam given during the first examination session, the student may choose instead to take 2 midterm exams. The outcomes of these tests will be available in the SIFA service through the UNIMIA portal.

- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student is required to illustrate results presented during the course and may be required to solve problems regarding Mathematical Analysis, in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.

The complete final examination is passed if both parts (written and oral) are successfully passed. Final marks are given using the numerical range 0-30, and are communicated immediately after the oral examination.

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6

Practicals: 36 hours

Lessons: 27 hours

Lessons: 27 hours

Professors:
Tarsi Cristina, Vesely Libor

### Analisi Matematica 2 (ediz.2)

Responsible

Lesson period

Second semester

Videos with lessons and exercises collections will be made available at the official web space of this course and they will cover the topic of each week. Live meetings during the classes hours may be scheduled by using Zoom.us. The details of these meetings will be available in the course web page of the Ariel platform, as well as the videos and every kind of material needed for the course.

The exams will be done by following the ways suggested on the web page of the university. The written exam will have the same structure as the one done in presence, possibly reduced in time and number of exercises.

The exams will be done by following the ways suggested on the web page of the university. The written exam will have the same structure as the one done in presence, possibly reduced in time and number of exercises.

**Course syllabus**

1.The Riemann integral in one variable. Antiderivation and indefinite integrals. Definition and elementary properties of the Riemann integral on a compact interval. A fundamental characterization of the class of integrable functions. The geometrical meaning of the defined integral. Functions defined by integrals and their properties. The Fundamental Theorem of Calculus and the Fundamental Formula of Calculus in one variable. Improper integrals: definitions, examples and comparison techniques. Relationships between integrals and numerical series.

2.Differential calculus in several real variables. Limits and continuity. Directional derivatives, gradients and the Jacobian matrix. Differentiability: necessary and/or sufficiente conditions. Tangent hyperplanes. Lagrange's theorem. Vector valued functions: differentiation and composition, the Chain Rule. Second order derivatives, the Hessian matrix and Schwartz's theorem. Taylor's formula. Unconstrained optimization: local extrema and saddle points. Eigenvalues and the Hessian matrix.

3.The Riemann integral in several real variables.. The Riemann integral over pluri-rectangles: definition and techniques for the computation. Brief discussion of Peano-Jordan measure. The Riemann integral on admissible domains: definition, integrability and the calculation of mutltiple integrals on simple domains. Sets of measure zero, almost everywhere continuous functions and their integrability. Diffeomorphisms of open sets in R^n. Integration in higher dimensions. Change of variables and special coordinate systems: polar, cylindrical and spherical. Improper multiple integrals. The Gaussian integral, the volume oft he unit ball in R^n, Euler's Gamma function.

2.Differential calculus in several real variables. Limits and continuity. Directional derivatives, gradients and the Jacobian matrix. Differentiability: necessary and/or sufficiente conditions. Tangent hyperplanes. Lagrange's theorem. Vector valued functions: differentiation and composition, the Chain Rule. Second order derivatives, the Hessian matrix and Schwartz's theorem. Taylor's formula. Unconstrained optimization: local extrema and saddle points. Eigenvalues and the Hessian matrix.

3.The Riemann integral in several real variables.. The Riemann integral over pluri-rectangles: definition and techniques for the computation. Brief discussion of Peano-Jordan measure. The Riemann integral on admissible domains: definition, integrability and the calculation of mutltiple integrals on simple domains. Sets of measure zero, almost everywhere continuous functions and their integrability. Diffeomorphisms of open sets in R^n. Integration in higher dimensions. Change of variables and special coordinate systems: polar, cylindrical and spherical. Improper multiple integrals. The Gaussian integral, the volume oft he unit ball in R^n, Euler's Gamma function.

**Prerequisites for admission**

It is strongly suggested that the student has already passed the following exams: "Analisi Matematica 1" and "Geometria 1".

**Teaching methods**

Frontal teaching. Problem sessions. Homeworks and their solution during the tutorial time.

**Teaching Resources**

References for different arguments will be indicated day-by-day, choosing

from the following list of textbooks:

C.Maderna, "Analisi Matematica 2" II ediz., CittàStudi ed., 2010.

C.Maderna, P.M.Soardi, "Lezioni di Analisi Matematica II", CittàStudi ed., 1997.

C.D.Pagani, S.Salsa, "Analisi Matematica, v.2", Zanichelli ed., 2016.

P.M.Soardi, "Analisi Matematica", CittàStudi ed., 2010.

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

W.Rudin, "Principles of Mathematical Analysis", McGraw-Hill

N.Fusco, P.Marcellini, C.Sbordone, "Analisi Matematica due", Liguori ed.

Moreover, a good training aimed to the written exam can be obtained working on the problems contained in:

http://users.mat.unimi.it/users/vignati/Esercizi-An2-Mat-gram.html

from the following list of textbooks:

C.Maderna, "Analisi Matematica 2" II ediz., CittàStudi ed., 2010.

C.Maderna, P.M.Soardi, "Lezioni di Analisi Matematica II", CittàStudi ed., 1997.

C.D.Pagani, S.Salsa, "Analisi Matematica, v.2", Zanichelli ed., 2016.

P.M.Soardi, "Analisi Matematica", CittàStudi ed., 2010.

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

W.Rudin, "Principles of Mathematical Analysis", McGraw-Hill

N.Fusco, P.Marcellini, C.Sbordone, "Analisi Matematica due", Liguori ed.

Moreover, a good training aimed to the written exam can be obtained working on the problems contained in:

http://users.mat.unimi.it/users/vignati/Esercizi-An2-Mat-gram.html

**Assessment methods and Criteria**

The final examination consists of two parts: a written exam and an oral exam.

- During the written exam, the student must solve some exercises in the format of open-ended and/or short answer questions, with the aim of assessing the student's ability to solve problems in Mathematical Analysis. The duration of the written exam is proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration does not exceed three hours). In place of a single written exam given during the first examination session, the student may choose instead to take 2 midterm exams. The outcomes of these tests will be available in the SIFA service through the UNIMIA portal.

- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student is required to illustrate results presented during the course and may be required to solve problems regarding Mathematical Analysis, in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.

The complete final examination is passed if both parts (written and oral) are successfully passed. Final marks are given using the numerical range 0-30, and are communicated immediately after the oral examination.

- During the written exam, the student must solve some exercises in the format of open-ended and/or short answer questions, with the aim of assessing the student's ability to solve problems in Mathematical Analysis. The duration of the written exam is proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration does not exceed three hours). In place of a single written exam given during the first examination session, the student may choose instead to take 2 midterm exams. The outcomes of these tests will be available in the SIFA service through the UNIMIA portal.

- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student is required to illustrate results presented during the course and may be required to solve problems regarding Mathematical Analysis, in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.

The complete final examination is passed if both parts (written and oral) are successfully passed. Final marks are given using the numerical range 0-30, and are communicated immediately after the oral examination.

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6

Practicals: 36 hours

Lessons: 27 hours

Lessons: 27 hours

Professors:
Ciraolo Giulio, Zanco Clemente