Studying at the University of Verona
Here you can find information on the organisational aspects of the Programme, lecture timetables, learning activities and useful contact details for your time at the University, from enrolment to graduation.
Study Plan
This information is intended exclusively for students already enrolled in this course.If you are a new student interested in enrolling, you can find information about the course of study on the course page:
Laurea in Matematica applicata - Enrollment from 2025/2026The Study Plan includes all modules, teaching and learning activities that each student will need to undertake during their time at the University.
Please select your Study Plan based on your enrollment year.
1° Year
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2° Year activated in the A.Y. 2017/2018
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3° Year activated in the A.Y. 2018/2019
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Legend | Type of training activity (TTA)
TAF (Type of Educational Activity) All courses and activities are classified into different types of educational activities, indicated by a letter.
Mathematical analysis 2 (2017/2018)
Teaching code
4S00031
Credits
12
Language
Italian
Scientific Disciplinary Sector (SSD)
MAT/05 - MATHEMATICAL ANALYSIS
The teaching is organized as follows:
Teoria 2
Esercitazioni
Teoria 1
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.
Program
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
Activity | Author | Title | Publishing house | Year | ISBN | Notes |
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Teoria 2 | Robert A. Adams, Christofer Essex | Calcolo Differenziale 2 - Funzioni di più variabili (Edizione 5) | AMBROSIANA | 2014 | 978-8808-18468-9 | |
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 | 978-0443042089 | |
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 | 88-08-01215-8 | |
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 | |
Esercitazioni | M. Squassina, S. Zuccher | Introduzione all'Analisi Qualitativa delle Equazioni Differenziali Ordinarie. 332 pagine, 365 figure. | Apogeo Editore | 2008 | 9788850310845 | |
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 | 88-503-242 | |
Teoria 1 | Adams, R. | Calcolo differenziale (vol. 2). Funzioni di più variabili. | Ambrosiana | 2003 | 8840812687 |
Examination Methods
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.
Teaching materials e documents
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Differenziazione delle funzioni a valori vettoriali (it, 330 KB, 10/21/17)
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Dispensa di Esercitazioni (it, 2506 KB, 1/19/18)
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Fogli di esercizi da 0 a 9 (it, 278 KB, 12/10/17)
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Soluzioni appelli di Analisi 2 (it, 2933 KB, 10/7/17)