Mathematical analysis 2 (2018/2019)

Course code
4S00031
Credits
12
Coordinator
Sisto Baldo
MAT/05 - MATHEMATICAL ANALYSIS
Language of instruction
Italian
Teaching is organised as follows:
Activity Credits Period Academic staff Timetable
Teoria 1 6 I semestre Giandomenico Orlandi

Teoria 2 2 I semestre Sisto Baldo

Esercitazioni 4 I semestre Virginia Agostiniani

Learning outcomes

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 formalizie and solve moderately difficult problems on the arguments of the course.

Syllabus

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.

Assessment methods and criteria

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 if they pass a mid-term test at the beginning of december. 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.

 Activity Author Title Publisher Year ISBN Note Teoria 1 Giusti E. Analisi Matematica 2 Boringhieri 1983 CL 7491948 Teoria 1 C.D. Pagani, S. Salsa Analisi Matematica 2 (Edizione 2) Zanichelli 2016 9788808630786 Teoria 1 Robert A. Adams, Christofer Essex Calcolo Differenziale 2 - Funzioni di più variabili (Edizione 5) AMBROSIANA 2014 978-8808-18468-9 Teoria 1 James Stewart Calcolo: funzioni di più variabili (Edizione 3) Apogeo 2002 8873037488 Teoria 1 Kenneth R. Davidson, Allan P. Donsig Real Analysis and applications: theory in practice Springer 2010 978-0443042089 Teoria 2 Giuseppe De Marco Analisi 2. Secondo corso di analisi matematica per l'università Lampi di Stampa (Decibel Zanichelli) 1999 8848800378 Teoria 2 V. Barutello, M. Conti, D.L. Ferrario, S. Terracini, G. Verzini Analisi matematica. Dal calcolo all'analisi Vol. 2 Apogeo 2007 88-503-242 Teoria 2 Adams, R. Calcolo differenziale (vol. 2). Funzioni di più variabili. Ambrosiana 2003 8840812687 Esercitazioni Giuseppe De Marco Analisi 2. Secondo corso di analisi matematica per l'università Lampi di Stampa (Decibel Zanichelli) 1999 8848800378 Esercitazioni Adams, R. Calcolo differenziale (vol. 2). Funzioni di più variabili. Ambrosiana 2003 8840812687