# MATHEMATICAL ANALYSIS II

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Versione italiana
2022/2023
Teacher
MASSIMILIANO DANIELE ROSINI
Credits
6
Didactic period
Primo Semestre
SSD
MAT/05

#### Training objectives

The course continues the study of calculus tools and mathematical analysis theory started with the course of Analisi Matematica I.A and I.B.

The main goal is to teach the basic elementary techniques of differential and integral calculus for functions of several real variables and resolution methods for simple ordinary differential equations. The acquisition of these mathematical tools is considered essential in order to face successfully the subsequent technical teachings.

The main acquired knowledge will be:

* theory of numerical series and various convergence tests;
* theory of pointwise and uniform convergence and passage to the limit properties for sequences and series of functions, in particular power series, Taylor's series, Fourier's series;
* critical and extremal points;
* elementary differential geometry of regular curves and surfaces in the plane and in space;
* definition and computation of curvilinear and surface integrals, applications of the theorems of Gauss-Green and of Stokes;
* introduction to the theory of Lebesgue measure and integral and its passage to the limit properties.

The basic acquired abilities (that are the capacity of applying the acquired knowledge) will be to be able to:

* study pointwise and uniform convergence for sequences and series of functions, in particular power series;
* compute the Taylor's series of a differentiable function and the Fourier series of a periodic function;
* determine and classify free or constrained critical points of a function of several variables;
* determine extremal values of a function on a given domain;
* determine tangent and normal versors for a regular parametrized curve;
* determine the tangent plane and the normal versors for a regular parametrized surface;
* compute curvilinear integrals of the first and of the second kind, surface integrals, flux integrals.

#### Prerequisites

All contents of the course of Mathematical Analysis I.B are preliminary. Including in particular:
* Elementary functions.
* Differential and integral calculus in one or more variables.
Also required is the knowledge of:
* Basic linear algebra: linear maps, matrices, determinants, vector product and scalar product.
* Basic geometry: straight lines, planes, conic sections.

#### Course programme

Part I:
- Numerical sequences
1 Introduction (1 hour)
2 Limits of successions (1 hour)
3 Bolzano-Weierstrass Theorem (1 hour)
4 Bridge theorem (1 hour)
5 Calculation of limits of numerical sequences (1 hour)
6 Asymptoticity (1 hour)
7 Convergence test (2 hours)
8 Arithmetic and geometric progressions (2 hours)
- Numerical series
9 Finite sums (2 hours)
10 Infinite Sums (2 hours)
11 Convergence test (2 hours)
12 Parameter dependent series (2 hours)
- Sequences of functions
13 Punctual and uniform convergence (2 hours)
- Cauchy's problem
14 Introduction (2 hours)
15 Local existence and uniqueness theorems (2 hours)
- Series of functions
16 General results (2 hours)
17 Taylor series (2 hours)
18 Fourier series (2 hours)

Part II:
- Curves
1 Continuous curve, regular (1 hour)
2 curves in polar form (1 hour)
3 Length of a curved arc (1 hour)
4 Curvilinear abscissa (1 hour)
5 Wholemeal line of first species (1 hour)
- Optimization
6 Free highs and lows. Critical points (1 hour)
8 Study of the nature of the critical points (1 hour)
9 Implicitly Defined Functions (1 hour)
10 Maximum and minimum constrained (1 hour)
- Vector fields
11 integral lines (1 hour)
12 Conservative and Potential Fields (1 hour)
13 Differential Operators (1 hour)
14 Work of a vector field (1 hour)
15 The language of differential forms (1 hour)
- Functions f: R ^ n -> R ^ m
16 Examples (1 hour)
17 Surfaces (1 hour)
18 Regular coordinate transformations (1 hour)
19 Transformations of the volume element (1 hour)
20 Transformations of differential operators (1 hour)
- Integral calculus for functions of several variables
21 Wholemeal surface (2 hours)
22 Flow of a vector field (2 hours)
23 The Gauss-Green formula in the plane (2 hours)
24 The divergence theorem (2 hours)
25 The rotor theorem (2 hours)

#### Didactic methods

Classroom lectures with presentation of the theoretical aspects, the applications and exercises.

#### Learning assessment procedures

The verification of the learning of the course contents takes place through an exam consisting of two written tests: one of exercises and one of theory. The student must present himself / herself with an identification document. During the written test of exercises, the use of ballpoint pens, pencils, discoloration, eraser, ruler, calculator and a handwritten form (1 A4 sheet on both sides) to be delivered at the end of the test is allowed.

- In the first test it is required to solve 4 exercises that will rotate on all the topics of the course. The estimated time for the test is 2 hours. The first written test is assigned a score which varies between 0 and 25. To be able to enter the second test, a score of at least 11 points must be obtained in the first test. The score of the first test is considered valid for one year. In case the student positively supports more first tests, the one with higher score will be considered.

- In the second test the student will be asked to answer some questions regarding definitions, examples, properties, formulas, theorems, proofs, or applications. The estimated time for the second test is 20 minutes. The performance of the second written test is assigned a score expressed out of thirty which varies between 0 and 7. The second test is passed if a score of at least 3 points is obtained. Failure to pass the second test does not cancel the score of the first test, but it will be necessary to take another of the written theory tests.

The final grade, expressed in thirtieths, will take into account the sum of the scores obtained in the two written tests and the active participation in the lessons and tutoring.

Passing the exam is proof of having acquired the knowledge and skills specified in the educational objectives of the course.

#### Reference texts

Reference text:
M. Bramanti, C. D. Pagani, S. Salsa, Calcolo infinitesimale e algebra lineare

Recommended texts for study:
F.G. Alessio: Analisi Matematica 2, Teoria con esercizi svolti (Esculapio)
G. Anichini, G. Conti, M. Spadini: Analisi matematica 2 (Pearson)
M. Amar, A.M. Bersani: Analisi Matematica II Esercizi e richiami di teoria (Edizioni La Dotta)
V. Barutello, M. Conti, D.L. Ferrario, S. Terracini, G. Verzini: Analisi Matematica. Con elementi di geometria e calcolo vettoriale: 2 (Apogeo)
M. Bertsch, R. Dal Passo, L. Giacomelli: Analisi Matematica (Mc Graw Hill)
M. Bramanti: Esercitazioni di Analisi Matematica 2 (Esculapio)
G. Catino, F. Punzo: Esercizi svolti di Analisi Matematica e Geometria 2 (Esculapio)
M. Bramanti, C. D. Pagani, S. Salsa: Analisi matematica 2 (Zanichelli )
W. Dambrosio: Analisi matematica Fare e comprendere (Zanichelli )
G. De Marco: Analisi Matematica II (Zanichelli )
G. De Marco: Esercizi di analisi Matematica II (Zanichelli)
N. Fusco, P. Marcellini, C. Sbordone: Lezioni di analisi matematica due (Zanichelli)
E. Giusti: Analisi Matematica II (Boringhieri)
E. Giusti: Esercizi e complementi di Analisi Matematica II (Boringhieri)
E.H. Lieb, M. Loss: Analysis (American Mathematical Society)
P. Marcellini, C. Sbordone: Esercitazioni di Analisi matematica Due (Zanichelli)
W. Rudin: Principi di Analisi Matematica (Mc Graw Hill)
S. Salsa, A. Squellati: Esercizi di Analisi Matematica II (Zanichelli)
E. Stein, R. Shakarchi: Real Analysis: Measure Theory, Integration, and Hilbert Spaces (Princeton University Press)