# Mathematical analysis 2 (2016/2017)

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 3 I sem. Sisto Baldo
Teoria 2 5 I sem. Giandomenico Orlandi
Esercitazioni 4 I sem. Antonio Marigonda

### Lesson timetable

I sem.
Activity Day Time Type Place Note
Teoria 1 Tuesday 11:30 AM - 1:30 PM lesson Lecture Hall F from Oct 4, 2016  to Oct 4, 2016
Esercitazioni Monday 1:30 PM - 4:30 PM lesson Not defined from Oct 18, 2016  to Oct 18, 2016

### 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.

### 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.
The oral test will concentrate mainly but not exclusively on the theory.

 Activity Author Title Publisher Year ISBN Note Esercitazioni Giuseppe De Marco Analisi 2. Secondo corso di analisi matematica per l'università Lampi di Stampa (Decibel Zanichelli) 1999 8848800378 Esercitazioni M. Conti, D. L. Ferrario, S. Terracini, G. Verzini Analisi matematica. Dal calcolo all'analisi, Vol. 1 (Edizione 1) Apogeo 2006 88-503-221 Esercitazioni V. Barutello, M. Conti, D.L. Ferrario, S. Terracini, G. Verzini Analisi matematica. Dal calcolo all'analisi Vol. 2 Apogeo 2007 88-503-242 Esercitazioni Conti M., Ferrario D.L., Terracini S,. Verzini G. Analisi matematica. Dal calcolo all'analisi. Volume 1. Apogeo Esercitazioni Conti F. et al. Analisi Matematica, teoria e applicazioni McGraw-Hill, Milano 2001 8838660026 Esercitazioni Giuseppe de Marco Analisi uno. Primo corso di analisi matematica. Teoria ed esercizi Zanichelli 1996 8808243125 Esercitazioni Giuseppe de Marco Analisi Zero, presentazione rigorosa di alcuni concetti base di matematica per i corsi universitari (Edizione 3) Edizione Decibel/Zanichelli 1997 978-8808-19831-0

#### Statistics about transparency requirements (Attuazione Art. 2 del D.M. 31/10/2007, n. 544)

 Outcomes Exams Outcomes Percentages Average Standard Deviation Positive 55.55% 23 3 Rejected -- Absent 44.44% Ritirati -- Canceled --
 18 19 20 21 22 23 24 25 26 27 28 29 30 30 e Lode 12.5% 12.5% 10.0% 10.0% 7.5% 7.5% 2.5% 12.5% 5.0% 10.0% 0.0% 0.0% 7.5% 2.5%

Data from AA 2016/2017 based on 72 students. I valori in percentuale sono arrotondati al numero intero più vicino.