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

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.
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 vectorvalued 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, GaussGreen 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, CauchyLipschitz'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.
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 midterm 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  Giuseppe De Marco  Analisi 2. Secondo corso di analisi matematica per l'università  Lampi di Stampa (Decibel Zanichelli)  1999  8848800378  
Teoria 1  V. Barutello, M. Conti, D.L. Ferrario, S. Terracini, G. Verzini  Analisi matematica. Dal calcolo all'analisi Vol. 2  Apogeo  2007  88503242  
Teoria 1  Adams, R.  Calcolo differenziale (vol. 2). Funzioni di più variabili.  Ambrosiana  2003  8840812687  
Teoria 2  Robert A. Adams, Christofer Essex  Calcolo Differenziale 2  Funzioni di più variabili (Edizione 5)  AMBROSIANA  2014  9788808184689  
Teoria 2  James Stewart  Calcolo: funzioni di più variabili (Edizione 3)  Apogeo  2002  8873037488  
Teoria 2  Tom M. Apostol  Calcolo, vol. 3  Boringhieri  xx  
Teoria 2  Kenneth R. Davidson, Allan P. Donsig  Real Analysis and applications: theory in practice  Springer  2010  9780443042089  
Esercitazioni  Giuseppe De Marco  Analisi 2. Secondo corso di analisi matematica per l'università  Lampi di Stampa (Decibel Zanichelli)  1999  8848800378  
Esercitazioni  G. De Marco  Analisi due  Zanichelli (decibel)  1999  8808012158  
Esercitazioni  M. Conti, D. L. Ferrario, S. Terracini, G. Verzini  Analisi matematica. Dal calcolo all'analisi, Vol. 1 (Edizione 1)  Apogeo  2006  88503221  
Esercitazioni  V. Barutello, M. Conti, D.L. Ferrario, S. Terracini, G. Verzini  Analisi matematica. Dal calcolo all'analisi Vol. 2  Apogeo  2007  88503242  
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  McGrawHill, 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  9788808198310  
Esercitazioni  M. Squassina, S. Zuccher  Introduzione all'Analisi Qualitativa delle Equazioni Differenziali Ordinarie. 332 pagine, 365 figure.  Apogeo Editore  2008  9788850310845 
Title  Format (Language, Size, Publication date) 
Differenziazione delle funzioni a valori vettoriali  pdf (it, 330 KB, 21/10/17) 
Dispensa di Esercitazioni  pdf (it, 2506 KB, 19/01/18) 
Fogli di esercizi da 0 a 9  pdf (it, 278 KB, 10/12/17) 
Soluzioni appelli di Analisi 2  pdf (it, 2933 KB, 07/10/17) 