ANALYSIS II
Course objectives
Overall objectives: to acquire the main tools of Mathematical Analysis concerning the functions of several real variables. Specific objectives: Knowledge and understanding: students passing the exam at the end of the course will have a deep knowledge of the main concepts of mathematical analysis related to functions of several variables, with particular attention to differential calculus, to integration theory, to integrability of differential forms, to main theorems, such as divergence and Stokes ones. Apply knowledge and understanding: Students passing the exam at the end of the course will be able to apply various techniques of differential and integral calculus for functions of several variables. In particular, they will be able to compute integrals of functions of two and three variables, and to find extremals, to integrate differential forms or to compute a surface area. Critical and judgmental skills: the student will be able to analyze analogies and relationships between the new topics and those related to the theory of the functions of one real variable (acquired in Calculus I) and to the general theory of metric spaces (acquired in the course of Analysis I); it will also have a first approach to the measure theory that will be explored in the subsequent course of Real Analysis.
Channel 1
ANNALISA MALUSA
Lecturers' profile
Program - Frequency - Exams
Course program
Parametric curves
Functions, limits and continuity.
Differential calculus for scalar functions of several variables.
Differential calculus for vector-valued functions.
Differential calculus for complex functions.
Curved integrals, differential forms and vector fields.
Double integrals and Gauss-Green formulas.
Complex integrals, singularities and residuals.
Surface integrals, Stokes' theorem and divergence theorem
Prerequisites
Basic knowledge of the analysis taught during the calculus course (i.e. differential and integral calculus for real functions of one real variable). It is not mandatory, but it is strongly recommended, to have a good control of the main notions of topology for metric spaces, convergence of series and sequences of functions, power series.
Books
On the e-learning platform of the course both course notes and teaching materials concerning exercises will be made available.
Exam mode
The exam aims at evaluate the knowledge of students through a written test (consisting in solving problems of the same type as those carried out in the classes and proposed in the exercise sheets available on the e-learning platform) and an oral test (consisting in the presentation of some concepts and results illustrated in class).
The e-learning platform will provide detailed information on the topics covered in class and the applications that are expected to be known during the exam.
The written test will last about three hours and can be replaced by two intermediate tests, both lasting two hours, the first of which will take place at mid-term and the second immediately at the end of the term. The first intermediate test will focus mainly on the differential calculus for functions of several variables, the second on the integral calculus.
To pass the exam, students must obtain a mark of not less than 18/30, which can be obtained by proving their knowledge of basic concepts and calculus techniques. To achieve a score of 30/30 with honors, the student must instead demonstrate that she/he has acquired excellent knowledge of all the topics covered during the course and be able to connect them in a logical and consistent way.
Bibliography
Fusco Marcellini Sbordone: Lezioni di Analisi matematica due. Zanichelli
Pagani Salsa: Analisi Matematica 2, Zanichelli
Lanzara Montefusco: Esercizi svolti di Analisi Vettoriale (con elementi di teoria)
Lesson mode
The classes will be given using a blackboard
ANNALISA MALUSA
Lecturers' profile
Program - Frequency - Exams
Course program
Parametric curves
Functions, limits and continuity.
Differential calculus for scalar functions of several variables.
Differential calculus for vector-valued functions.
Differential calculus for complex functions.
Curved integrals, differential forms and vector fields.
Double integrals and Gauss-Green formulas.
Complex integrals, singularities and residuals.
Surface integrals, Stokes' theorem and divergence theorem
Prerequisites
Basic knowledge of the analysis taught during the calculus course (i.e. differential and integral calculus for real functions of one real variable). It is not mandatory, but it is strongly recommended, to have a good control of the main notions of topology for metric spaces, convergence of series and sequences of functions, power series.
Books
On the e-learning platform of the course both course notes and teaching materials concerning exercises will be made available.
Exam mode
The exam aims at evaluate the knowledge of students through a written test (consisting in solving problems of the same type as those carried out in the classes and proposed in the exercise sheets available on the e-learning platform) and an oral test (consisting in the presentation of some concepts and results illustrated in class).
The e-learning platform will provide detailed information on the topics covered in class and the applications that are expected to be known during the exam.
The written test will last about three hours and can be replaced by two intermediate tests, both lasting two hours, the first of which will take place at mid-term and the second immediately at the end of the term. The first intermediate test will focus mainly on the differential calculus for functions of several variables, the second on the integral calculus.
To pass the exam, students must obtain a mark of not less than 18/30, which can be obtained by proving their knowledge of basic concepts and calculus techniques. To achieve a score of 30/30 with honors, the student must instead demonstrate that she/he has acquired excellent knowledge of all the topics covered during the course and be able to connect them in a logical and consistent way.
Bibliography
Fusco Marcellini Sbordone: Lezioni di Analisi matematica due. Zanichelli
Pagani Salsa: Analisi Matematica 2, Zanichelli
Lanzara Montefusco: Esercizi svolti di Analisi Vettoriale (con elementi di teoria)
Lesson mode
The classes will be given using a blackboard
Channel 2
ADRIANA GARRONI
Lecturers' profile
Program - Frequency - Exams
Course program
Real functions of several real variables: limits and continuity, derivation and differentiability, Taylor's formula.
Convex functions. Vector valued functions. Inverse function theorem, Implicit function Theorem.
Maxima and minima of functions of several real variables. Curves. Differential forms.
Lebesgue measure and Lebesgue integral. Passage to the limit under Lebesgue integral. Fubini-Tonelli Theorem. Multiple integrals. Changes of variables.
Divergence Theorem, Stokes Theorem.
Prerequisites
The course requires familiarity with the theory of real functions of one real variable, developed in the course of Analysis I, and with the general theory of metric spaces, developed within the course of Elementi di Analisi reale. This knowledge is essential.
Books
On the elearning page of the course will be made available both notes of the course and educational material related to the exercises.
Fleming: Functions of several variables. Springer
Giusti: Analisi Matematica 2. Bollati Boringhieri
Frequency
Not mandatory but strongly suggested
Exam mode
The exam consists of a written test (consisting in solving problems of the same type as those carried out in the exercises and proposed in the exercise sheets available on the e-learning platform) and an oral test (consisting in the presentation of some concepts and results illustrated in the classes).
The e-learning platform will provide detailed information on the topics covered during classes and the applications that are expected to be known during the exam.
The written test will last about three hours and can be replaced by two intermediate tests, both lasting two hours, the first of which will take place at half course and the second immediately at the end of the course. The first intermediate test will focus mainly on the topics of differential calculus for functions of several variables, the second on integral calculus.
To pass the exam it is necessary to obtain a grade of at least 18/30, by demonstrating knowledge of the basic concepts and techniques. To achieve a score of 30/30 cum laude, the student must instead demonstrate to have acquired an excellent knowledge of all the topics covered during the course and to be able to link them in a logical and coherent way.
Lesson mode
Lessons in class
ADRIANA GARRONI
Lecturers' profile
Program - Frequency - Exams
Course program
Real functions of several real variables: limits and continuity, derivation and differentiability, Taylor's formula.
Convex functions. Vector valued functions. Inverse function theorem, Implicit function Theorem.
Maxima and minima of functions of several real variables. Curves. Differential forms.
Lebesgue measure and Lebesgue integral. Passage to the limit under Lebesgue integral. Fubini-Tonelli Theorem. Multiple integrals. Changes of variables.
Divergence Theorem, Stokes Theorem.
Prerequisites
The course requires familiarity with the theory of real functions of one real variable, developed in the course of Analysis I, and with the general theory of metric spaces, developed within the course of Elementi di Analisi reale. This knowledge is essential.
Books
On the elearning page of the course will be made available both notes of the course and educational material related to the exercises.
Fleming: Functions of several variables. Springer
Giusti: Analisi Matematica 2. Bollati Boringhieri
Frequency
Not mandatory but strongly suggested
Exam mode
The exam consists of a written test (consisting in solving problems of the same type as those carried out in the exercises and proposed in the exercise sheets available on the e-learning platform) and an oral test (consisting in the presentation of some concepts and results illustrated in the classes).
The e-learning platform will provide detailed information on the topics covered during classes and the applications that are expected to be known during the exam.
The written test will last about three hours and can be replaced by two intermediate tests, both lasting two hours, the first of which will take place at half course and the second immediately at the end of the course. The first intermediate test will focus mainly on the topics of differential calculus for functions of several variables, the second on integral calculus.
To pass the exam it is necessary to obtain a grade of at least 18/30, by demonstrating knowledge of the basic concepts and techniques. To achieve a score of 30/30 cum laude, the student must instead demonstrate to have acquired an excellent knowledge of all the topics covered during the course and to be able to link them in a logical and coherent way.
Lesson mode
Lessons in class
- Lesson code10599698
- Academic year2024/2025
- CourseMathematics
- CurriculumMatematica per le applicazioni
- Year2nd year
- Semester1st semester
- SSDMAT/05
- CFU9
- Subject areaFormazione Teorica