Mathematical Analysis 2
A.Y. 2025/2026
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
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
1. Elements of set theory and metric spaces
Euclidean spaces. Scalar products in a vectorial space and Cauchy-Schwarz inequality. Normed vectorial spaces. Metric induced by a norm. Topology in a metric space. Neighborhoods and classification of points and sets (open, closed), and their properties. Compact set: necessary conditions, characterization. Heine- Borel Theorem. Limit of sequences in metric spaces. Bolzano-Weierstrass Theorem. Complete metric spaces, Banach spaces. Compact spaces are complete. Functions in metric spaces: continuity. Weierstrass theoremand generalizations. Uniformly contiunous functions and
Heine-Cantor Theorem. Lipschitz Funcations. The contraction mapping theorem. Connected sets in Rn.
2. Differentail calculus for functions on Rn
Directional and partial derivatives. The gradient vector and its meaning. Differentiability: necessary conditions and sufficient conditions. The mean value therem for functions on Rn,
Vectorial functions: the Jacobian matrix. Composition of differentiable functions.
Higher order partial derivatives. Hessian matrix and the Svhwarz Lemma. The Tayor formulas.
Free omptimization for real valued functions and the Fermat Lemma. The role of the Hessian matrix in the classification of extremal points.
The implicit function theorem. The local inversin Theorem. Local and global diffeomorphism. The constraint optimization: the Lagrange multipliers Theorem.
Euclidean spaces. Scalar products in a vectorial space and Cauchy-Schwarz inequality. Normed vectorial spaces. Metric induced by a norm. Topology in a metric space. Neighborhoods and classification of points and sets (open, closed), and their properties. Compact set: necessary conditions, characterization. Heine- Borel Theorem. Limit of sequences in metric spaces. Bolzano-Weierstrass Theorem. Complete metric spaces, Banach spaces. Compact spaces are complete. Functions in metric spaces: continuity. Weierstrass theoremand generalizations. Uniformly contiunous functions and
Heine-Cantor Theorem. Lipschitz Funcations. The contraction mapping theorem. Connected sets in Rn.
2. Differentail calculus for functions on Rn
Directional and partial derivatives. The gradient vector and its meaning. Differentiability: necessary conditions and sufficient conditions. The mean value therem for functions on Rn,
Vectorial functions: the Jacobian matrix. Composition of differentiable functions.
Higher order partial derivatives. Hessian matrix and the Svhwarz Lemma. The Tayor formulas.
Free omptimization for real valued functions and the Fermat Lemma. The role of the Hessian matrix in the classification of extremal points.
The implicit function theorem. The local inversin Theorem. Local and global diffeomorphism. The constraint optimization: the Lagrange multipliers Theorem.
Prerequisites for admission
1. All topics that have been developed in Mathematical Analysis 1: number fields, sequences of real numbers and series, limits of functions, differential and integral calculus for real functions in one variable.
2. Basics in linear algebra (matrices, determinants, linear systems).
3. Basics in analytic geometry (lines and conics in two dimensions).
2. Basics in linear algebra (matrices, determinants, linear systems).
3. Basics in analytic geometry (lines and conics in two dimensions).
Teaching methods
Teaching will be held by presenting theory and exercises at the blackboard in the classroom, possibly by two different teachers, with the students attending. Tutoring activities will be proposed: some tutors will presents or comments exercises previously proposed to students.
Teaching Resources
N.Fusco, P.Marcellini, C.Sbordone, "Analisi Matematica due", Liguori ed. oppure Zanichelli
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.
B.Gelbaum, J.Olmsted, "Counterexamples in Analysis", Holden-Day.
W.Rudin, "Principles of Mathematical Analysis", McGraw-Hill
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.
B.Gelbaum, J.Olmsted, "Counterexamples in Analysis", Holden-Day.
W.Rudin, "Principles of Mathematical Analysis", McGraw-Hill
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: 6
Practicals: 24 hours
Lessons: 36 hours
Lessons: 36 hours
Shifts:
Professor:
Tarsi Cristina
Turno 1
Professor:
Messina FrancescaTurno 2
Professor:
Calzi MattiaProfessor(s)