B5551 - ANALISI MATEMATICA 2B

Learning outcomes

At the end of the course, the student has acquired further and more advanced knowledge of Mathematical Analysis. They are familiar with measure theory and the Lebesgue integral, series of functions, line integrals, and differential 1-forms. The student is able to apply this knowledge to the solution of simple practical problems arising in pure and applied sciences.

Course contents

Lebesgue Integral:

Outer measure; Carathéodory measurability. Monotonicity and sub-additivity. Outer measure and Lebesgue measure in 𝑅ⁿ.

Measurable functions; simple functions. Definition of the integral (in the Lebesgue sense); integrability. General properties of the integral. Tonelli’s and Fubini’s Theorems (in ℝⁿ and on arbitrary sets). Change of variables theorem for the Lebesgue integral. Beppo Levi’s Monotone Convergence Theorem. Fatou’s Lemma; Lebesgue’s Dominated Convergence Theorem.

Line Integrals and 1-Forms:

Basic notions of parametrized paths. Rectifiability and length of a path. Length of 𝐶¹-curves. Differential 1-forms and vector fields: line integrals. Conservative and irrotational vector fields (or exact and closed 1-forms and their relations). Fundamental Theorem of Calculus for line integrals. Star-shaped domains and Poincaré’s Theorem.

Brief Overview of Surface Integrals in R^3, the Divergence Theorem (in R^2 and R^3), and Stokes’ Theorem (in R^3).

 Sequences and Series of Functions:

Sequences of functions; pointwise and uniform convergence. Uniform convergence preserves: Riemann integrability and continuity. Cauchy criterion for uniform convergence. Total convergence of function series. 𝐶¹-type convergence. Power series. Cauchy-Hadamard Theorem. Differentiation and integration of power series. Brief overview: real analytic functions.

Readings/Bibliography

Notes will be available on Virtuale [http://virtuale.unibo.it/] .

To study in depth the topics of the course, students can consult:

E. Lanconelli: Lezioni di Analisi Matematica 2, prima  e seconda parte, ed. Pitagora

M. Bertsch, A. Dall'Aglio, L. Giacomelli: Epsilon 2. Secondo corso di analisi matematica. Ed. Mc Graw Hill

N. Fusco, P. Marcellini, C. Sbordone: Lezioni di Analisi Matematica due, ed. Zanichelli

G.C. Barozzi, G. Dore, E. Obrecht: Elementi di Analisi Matematica, vol. 2, ed. Zanichelli

Textbooks about exercises:

M. Bramanti, Esercitazioni di Analisi Matematica 2, ed. Esculapio

P. Marcellini, C. Sbordone: Esercitazioni di Analisi Matematica due, parte I e parte II, ed. Zanichelli

Teaching methods

The course is structured in lectures in the classroom which illustrate the fundamental concepts relating to the properties of real functions of several real variables and of the integral calculus. The lessons are integrated with examples and counterexamples related to the fundamental concepts illustrated. In addition, many exercises are performed.

Assessment methods

The examination consists of a preliminary written test and an oral one.

The written test lasts 2 hours and 30', consists of exercises related to the arguments of the course. In order to participate to the written test the student must register at least three days before the test through AlmaEsami [https://almaesami.unibo.it/] .

The written test remains valid for the oral exam in the same examination period.

The test about the theory follows the written test; it mainly concerns the theoretical aspects of the course. The student must show to know the concepts explained during the course (in particular definitions, theorems and their proofs) and how to connect them.

Teaching tools

Tutoring, if assigned

Office hours

See the website of Giovanni Cupini