Mathematical Analysis 3
A.Y. 2024/2025
Learning objectives
The course aims to complete the basic Knowledge of Mathematical Analysis for the students of the three-year degree in Physics. In particular the course will provide the student with the knowledge and the mastery of the following concepts, widely used in Physics:
· implicit functions and constrained maxima and minima;
· differential forms, exact and closed forms;
· Lebesgue theory for functions of several variables;
· surfaces and surface integral;
· Gauss-Green formulas, divergence and Stokes theorem.
· implicit functions and constrained maxima and minima;
· differential forms, exact and closed forms;
· Lebesgue theory for functions of several variables;
· surfaces and surface integral;
· Gauss-Green formulas, divergence and Stokes theorem.
Expected learning outcomes
At the end of the course the students will have acquired the following knowledge and skills.
1. Knowledge and understanding of the following subjects:
· implicit functions;
· constrained maxima and minima;
· differential forms, exact and closed forms;
· Lebesgue theory for functions of several variables;
· surfaces and surface integral;
· Gauss-Green formulas, divergence and Stokes theorem.
2. Ability to present and discuss in a critical way the concepts studied;
3. Ability to solve problems by using the results and the tools, they have learnt.
In particular they will be able
· to study implicit functions
· to minimize functions of several variables with constraints
· to calculate integrals in several variables;
· to understand whether a differential form is exact and to determine a primitive;
· to apply the divergence and Stokes theorems in the plane and in the space
4. Ability to use the tools they have studied in different framework.
1. Knowledge and understanding of the following subjects:
· implicit functions;
· constrained maxima and minima;
· differential forms, exact and closed forms;
· Lebesgue theory for functions of several variables;
· surfaces and surface integral;
· Gauss-Green formulas, divergence and Stokes theorem.
2. Ability to present and discuss in a critical way the concepts studied;
3. Ability to solve problems by using the results and the tools, they have learnt.
In particular they will be able
· to study implicit functions
· to minimize functions of several variables with constraints
· to calculate integrals in several variables;
· to understand whether a differential form is exact and to determine a primitive;
· to apply the divergence and Stokes theorems in the plane and in the space
4. Ability to use the tools they have studied in different framework.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
CORSO A
Responsible
Lesson period
First semester
Course syllabus
Implicit functions.
Maxima and minima with constraints; Lagrange multipliers.
Lebesgue measure and integral in R^n; multiple integrals.
Curves and curve integrals.
Linear differential forms.
Surfaces and surface integral.
Maxima and minima with constraints; Lagrange multipliers.
Lebesgue measure and integral in R^n; multiple integrals.
Curves and curve integrals.
Linear differential forms.
Surfaces and surface integral.
Prerequisites for admission
1. Differential and integral calculus for real functions of a real variable.
2. Sequences and numerical series.
3. Basic knowledge of Analytical Geometry and Linear Algebra.
4. Differential calculus for vectorial functions of several real variables.
5. Free optimization for functions of several variables.
6. Differential equations: some techniques of resolution.
7. Cauchy problems: the main results for existence and / or uniqueness of the solution.
8. Sequences and series of functions.
2. Sequences and numerical series.
3. Basic knowledge of Analytical Geometry and Linear Algebra.
4. Differential calculus for vectorial functions of several real variables.
5. Free optimization for functions of several variables.
6. Differential equations: some techniques of resolution.
7. Cauchy problems: the main results for existence and / or uniqueness of the solution.
8. Sequences and series of functions.
Teaching methods
Attendance mode: strongly recommended;
Mode of delivery: lectures and exercise classes.
Mode of delivery: lectures and exercise classes.
Teaching Resources
Textbook:
N. Fusco, P. Marcellini, C. Sbordone, Analisi Matematica due, Liguori ed.
Exercise book:
C. Maderna, G. Molteni, M. Vignati, Esercizi scelti di Analisi matematica 2 e 3, Città Studi ed.
Other textbooks:
G. Molteni, M. Vignati, Analisi matematica 3, Città Studi ed.
C. Maderna, P.M. Soardi, Lezioni di Analisi matematica II, Città Studi ed.
N. Fusco, P. Marcellini, C. Sbordone, Analisi Matematica due, Liguori ed.
Exercise book:
C. Maderna, G. Molteni, M. Vignati, Esercizi scelti di Analisi matematica 2 e 3, Città Studi ed.
Other textbooks:
G. Molteni, M. Vignati, Analisi matematica 3, Città Studi ed.
C. Maderna, P.M. Soardi, Lezioni di Analisi matematica II, Città Studi ed.
Assessment methods and Criteria
The exam consists of a written and an oral part. The written part consists of a written test
about the arguments discussed during the exercise classes. The duration of the written exam
will be proportional to the number of exercises assigned, also taking into account the nature
and complexity of the exercises themselves (however, the duration will not exceed three hours). The oral exam can be taken only if the written component has been successfully passed.
The oral part consists of an oral examination that focuses mainly on theoretical topics of the
course (knowledge of the major theoretical results, connections between different parts of
programs, demonstrations of proven results in class) and on the written examination.
The solution of some exercises may also be required during the oral part of the examination.
The complete final examination is passed if all the two parts (written and oral) are successfully
passed. Final marks are given using the numerical range 0-30, and will be communicated immediately
after the oral examination.
about the arguments discussed during the exercise classes. The duration of the written exam
will be proportional to the number of exercises assigned, also taking into account the nature
and complexity of the exercises themselves (however, the duration will not exceed three hours). The oral exam can be taken only if the written component has been successfully passed.
The oral part consists of an oral examination that focuses mainly on theoretical topics of the
course (knowledge of the major theoretical results, connections between different parts of
programs, demonstrations of proven results in class) and on the written examination.
The solution of some exercises may also be required during the oral part of the examination.
The complete final examination is passed if all the two parts (written and oral) are successfully
passed. 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: 6
Practicals: 24 hours
Lessons: 32 hours
Lessons: 32 hours
Professors:
Calanchi Marta, Calzi Mattia
CORSO B
Responsible
Lesson period
First semester
Course syllabus
Implicit functions theorem.
Constrained optimization; Lagrange multipliers.
Lebesgue measure and integral in R^n; multiple integrals.
Curves and curve integrals.
Linear differential forms.
Surfaces and surface integral.
Constrained optimization; Lagrange multipliers.
Lebesgue measure and integral in R^n; multiple integrals.
Curves and curve integrals.
Linear differential forms.
Surfaces and surface integral.
Prerequisites for admission
1. Differential and integral calculus for real-valued functions of one real variable.
2. Sequences and numerical series.
3. Basic knowledge of Analytical Geometry and Linear Algebra.
4. Differential calculus for vector-valued functions of several real variables.
5. Free optimization for functions of several variables.
6. Ordinary differential equations: some techniques of resolution.
7. Cauchy problems: the main results for existence and / or uniqueness of the solution.
8. Sequences and series of functions.
2. Sequences and numerical series.
3. Basic knowledge of Analytical Geometry and Linear Algebra.
4. Differential calculus for vector-valued functions of several real variables.
5. Free optimization for functions of several variables.
6. Ordinary differential equations: some techniques of resolution.
7. Cauchy problems: the main results for existence and / or uniqueness of the solution.
8. Sequences and series of functions.
Teaching methods
The course is delivered through lectures and exercise classes.
Attendance is strongly recommended.
Attendance is strongly recommended.
Teaching Resources
Textbook:
N. Fusco, P. Marcellini, C. Sbordone, Analisi Matematica due, Liguori ed.
Exercise book:
C. Maderna, G. Molteni, M. Vignati, Esercizi scelti di Analisi matematica 2 e 3, Città Studi ed.
Other reference books:
G. Molteni, M. Vignati, Analisi matematica 3, Città Studi ed.
C. Maderna, P.M. Soardi, Lezioni di Analisi matematica II, Città Studi ed.
N. Fusco, P. Marcellini, C. Sbordone, Analisi Matematica due, Liguori ed.
Exercise book:
C. Maderna, G. Molteni, M. Vignati, Esercizi scelti di Analisi matematica 2 e 3, Città Studi ed.
Other reference books:
G. Molteni, M. Vignati, Analisi matematica 3, Città Studi ed.
C. Maderna, P.M. Soardi, Lezioni di Analisi matematica II, Città Studi ed.
Assessment methods and Criteria
The exam consists of a written and an oral part. The written part consists of a written test about the arguments discussed during the exercise classes. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). Admission to the oral exam will be granted only upon passing the written exam.
The oral exam focuses mainly on theoretical topics of the course (knowledge of the major theoretical results, connections between different parts of the program, knowledge and understanding of the proofs presented in class) and on the written examination.
The solution of some exercises may also be required during the oral part of the examination.
The complete final examination is passed if the two parts (written and oral) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
The oral exam focuses mainly on theoretical topics of the course (knowledge of the major theoretical results, connections between different parts of the program, knowledge and understanding of the proofs presented in class) and on the written examination.
The solution of some exercises may also be required during the oral part of the examination.
The complete final examination is passed if the two parts (written and oral) are successfully passed. 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: 6
Practicals: 24 hours
Lessons: 32 hours
Lessons: 32 hours
Professors:
Cavalletti Fabio, Stuvard Salvatore
Professor(s)
Reception:
Please, request an appointment via email
Room 1005, Department of Mathematics, Via Cesare Saldini 50, first floor or via Zoom conference call