Teaching code

4S00031

Academic staff

Giandomenico Orlandi, Sara Galasso

Coordinator

Giandomenico Orlandi

Credits

12

Language

Italian

Scientific Disciplinary Sector (SSD)

MAT/05 - MATHEMATICAL ANALYSIS

Period

Semester 1 dal Oct 3, 2022 al Jan 27, 2023.

Lessons timetable Moodle Seminars 0

Learning objectives

Topics treated in this course are: Calculus for functions of several variables, sequences and series of functions, ordinary differential equations, Lebesgue measure and integral. Emphasis will be given to examples and applications. At the end of the course, students must possess adequate skills of synthesis and abstraction. They must recognize and produce rigorous proofs. They must be able to formalize and solve moderately difficult problems on the arguments of the course.

Prerequisites and basic notions

Linear algebra - Calculus 1

Program

Some content will be available online in case of COVID-19 or for students with specific learning needs. In addition, a number of the lessons/all the lessons (see the course
schedule) will be held in-class.
i) Calculus in several variables. Neighborhoods in several variables, continuity in several variables, directional derivatives, differential of functions in several variables, Theorem of Total Differential, gradient of scalar functions, Jacobian matrix for vector-valued functions, level curves of scalar functions. Parametrized surfaces, tangent and normal vectors, changes of coordinates. Higher order derivatives and differentials, Hessian matrix, Schwarz's Theorem, Taylor's Series.
(ii) Optimization problems for functions in several variables. Critical points, free optimization, constrained optimization, Lagrange's Multiplier Theorem, Implicit and inverse function theorem, Contraction Principle.
(iii) Integral of functions in several variables. Fubini and Tonelli theorems, integral on curves, change of variables formula.
(iv) Integral of scalar function on surfaces, vector fields, conservatice vector fields, scalar potentials, curl and divergence of a vector fields, introduction to differential forms, closed and exact forms, Poincare lemma, Gauss-Green formulas.
(v) Flux through surfaces, Stokes' Theorem, Divergence Theorem
(vi) Introduction to metric spaces and normed spaces, spaces of functions, sequence of functions, uniform convergence, function series, total convergence, derivation and integration of a series of functions.
(vii) Introduction to Lebesgue's Measure Theory. Measurable sets and functions, stability of measurable functions, simple functions, approximation results, Lebesgue integral. Monotone Convergence Theorem, Fatou's Lemma, Dominated convergence Theorem and their consequences.
(viii) Ordinary differential equation, existence and uniqueness results, Cauchy-Lipschitz's Theorem. Extension of a solution, maximal solution, existence and uniqueness results for systems of ODE, linear ODE of order n, Variation of the constants method,
other resolutive formulas.
(ix) Fourier's series for periodic functions, convergence results, application to solutions of some PDE.

Bibliography

Didactic methods

In presence teaching. Online multimedia resources on the e-learning platform of the course.

Learning assessment procedures

The final exam consists of a written test followed, in case of a positive result, by an oral test. The written test consists of some exercises on the program: students are exonerated from the first part of the test in the winter session of February if they pass a mid-term test at the beginning of December.

Evaluation criteria

The written test evaluates the ability of students at solving problems pertaining to the syllabus of the course, and also their skills in the analysis, synthesis and abstraction of questions stated either in the natural language or in the specific language of mathematics. The written test is graded on a scale from 0 to 30 points (best), with a pass mark of 18/30..
The oral test will concentrate mainly but not exclusively on the theory. It aims at verifying the ability of students at constructing correct and rigorous proofs and their skills in analysis, synthesis and abstraction. The oral exam is graded on a scale from -5 to +5 point, which are added to the marks earned in the written test. Finally, homework exercises will be graded from 0 to 2 points.

Criteria for the composition of the final grade

final grade=mark of the written test + mark of the oral exam + mark for the homework.

Exam language

italiano