Mathematical Sciences for Artificial Intelligence

Sequences and series of functions (in particular, power series). Metric spaces, elements of topology, completeness, and Banach/Hilbert spaces. Functions of several variables: limits, continuous functions, directional derivatives, differentiability, and the chain rule. Total differential theorem, second-order derivatives, and Schwarz's theorem. Taylor's formula in several variables. Free and constrained extrema. Lebesgue measure. Lebesgue integral. Passage to the limit under the integral sign: theorems of Beppo Levi (Monotone Convergence), Fatou, and Lebesgue (Dominated Convergence). Parameter-dependent integrals and differentiation under the integral sign. Reduction formulas (Fubini). Change of variables in double and triple integrals. Lᵖ spaces. Parametrized curves. Line integrals of a scalar function. Work of a vector field. Irrotational vector fields. Conservative vector fields. Closed and exact linear differential forms. Simply connected sets. Regular parametrized surfaces and surface integrals. Flux of a vector field.

The course requires familiarity with the main topics discussed in the lectures of "Analisi I" and "Algebra Lineare e Strutture Algebriche". No preparatory course is mandatory.

Adopted texts N. Fusco, P. Marcellini, C. Sbordone, Lezioni di analisi matematica due, Zanichelli 2020 P. Marcellini, C. Sbordone, Esercitazioni di Analisi Matematica Due (vlumi 1-2), Zanichelli 2017.

Participation to lectures is recommended, but not mandatory.

The exam consists of a written test (with problems similar to those seen in class) and an oral exam (about the topics and results seen in class). The written test will last about two hours and it can be alternatively passed (subject to the instructor's discretion) by taking both midterm two-hour tests. Passing mark is 18/30. The student must prove to have acquired a basic knowledge of the main topics of the course and must be able to solve the simplest exercises assigned. In order to get the top mark 30/30 cum laude, the student must prove to have acquired a very good knowledge of the topics treated during the course, to be able to organize them in a coherent way and to be able to solve the assigned exercises.

Bibliography E. Giusti, Analisi Matematica 2, Bollati Boringhieri 1989. F. Lanzara, E. Montefusco, Esercizi svolti di Analisi Vettoriale e complementi di teoria, Edizioni LaDotta 1999

Theoretical lessons (48 hours) and classroom tutorials (36 hours).