MATHEMATICAL ANALYSIS II
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- Versione italiana
- Academic year
- 2018/2019
- 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.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, normal and binormal versors, and curvature and torsion values 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
- 1) Numerical sequences and series, convergence, divergence, irregular.
2) Geometric series. Telescopic Series.
3) Series with non-negative terms: comparison and asymptotic comparison tests.
4) Test of the n-th root and of the ratio.
5) Comparison between series and integral. Generalized harmonic series.
6) Absolute convergence. Alternate series and Leibniz criterion.
7) Function sequences.
8) Uniform convergence.
9) Properties of uniform convergence for function sequences.
10) Series of functions. Total and/or uniform convergence for series of functions.
11) Properties of uniform convergence for series of functions.
12) Series of powers and their convergence domain.
13) Determination of the convergence radius.
14) Taylor series. Functions developable into a Taylor series.
15) Applications of the Taylor series to solve of differential equations and to compute integrals.
16) Cauchy problem for ordinary differential equations in normal form, integral formulation ofCauchy problem, Picard iterate.
17) Theorem of existence for Cauchy problems for ordinary differential equations in normal form in the hypothesis of Lipschitz.
18) Local maxima and minima (not bound) of functions of several variables. Critical points.
19) Hessian Matrix and second order approximation of functions of several variables. Classification of critical points.
20) Implicit functions and Dini's theorem.
21) Critical and extremal points constrained.
22) Lagrange multipliers method.
23) Scalar product and L^2-norm. Least squares approximation.
24) Trigonometric polynomials. Definition of the Fourier series. Bessel inequality.
25) Fourier series in the case of square, triangular, sawtooth waves.
26) Piecewise regular functions. Convolutions of periodic functions. Fejér and Dirichlet Kernels.
27) Point convergence theorem for the Fourier series of piecewise regular functions.
28) Curves parametrized in R ^ n. Regular curves. Straight line and tangent verse to a curve.
29) Equivalent parameterizations.
30) Length of a curve. Parameterization for arc length.
31) Curvilinear integrals of first species.
32) Vector fields and differential forms.
33) Curvilinear integrals of second kind.
34) Conservation fields and exact forms. Irrotational fields and closed forms.
35) Calculation of potentials. Simply connected domains.
36) Curvature and torsion. Tangent, normal and binormal versors of R^3 curves.
37) Formulas and equations for the Frenet mobile reference system.
38) Surfaces parametrized in space R^3. Tangent plane and normal versor.
39) Surface element. Surface integrals.
40) The area element of the surface of revolution obtained by rotating a curve. Guldino's theorem.
41) Oriented surfaces. Flow of a vector field.
42) Simple domains, regular regular domains in R^2 and R^3.
43) Gauss-Green formulas, divergence and rotor theorems in R^2.
44) The divergence and Stokes theorems in R^3.
45) Zero measure sets according to Lebesgue, approximation with simple step functions.
46) Lebesgue integral and Lebesgue measure (almost axiomatic).
47) Theorems of passage to the limit for the integral of Lebesgue. Didactic methods
- Classroom lectures with presentation at the blackboard and with the projector of the theoretical aspects, the applications and exercises.
Learning assessment procedures
- The learning assessment of the course content is performed through two written exams. The student must present himself with a recognized document. It is not allowed to consult texts or notes, use calculators, PC, tablet or smartphone.
- In the first test the student is asked to solve some problems and exercises related to the topics. The time required for such written test is 2 hours. A score expressed in thirtieths is assigned to the written test. In order to access the second test, a score of at least 15 points must be obtained in the first test. The score of the first test is considered valid for the duration of the session. If the student passes more than one first test during the same session, the one with the highest score will be considered.
- In the second test the student will be asked to present some aspects of content developed during the course, illustrating some definitions, examples, properties, formulas, theorems, demonstrations, or applications. More than the mnemonic knowledge of the topics, we want to evaluate the logical understanding of the concepts, the precision and rigour of the mathematical language used to describe them and the ability to grasp the relationship between abstract aspects and concrete applications. The time scheduled for the second test is 1 hour. An insufficient result of the second test does not cancel the score of the first test.
The final grade, expressed in thirtieths, will take into account the sum of the grades obtained in the two written tests and the active participation in the lessons and tutoring.
Passing the final exam is the proof that knowledge and abilities outlined in the training objectives of the course have been achieved. Reference texts
- Reference text:
M. Bramanti, C. D. Pagani, S. Salsa, Calcolo infinitesimale e algebra lineare
Recommended texts for study:
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)
G. De Marco: Analisi Matematica II (Zanichelli )
G. De Marco: Esercizi di analisi Matematica II (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)
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)
