Mathematical Sciences for Artificial Intelligence

Elements of topology in R^N. Continuous functions. Directional derivatives, differentiability, plane tangent, derivation formula of composite functions. Theorem of the total differential, second derivatives and Schwarz's theorem. Taylor formula in several variables. Necessary and sufficient conditions for critical points to be local maxima or minima. Implicit function theorem. Lagrange multipliers. Curves, parameterizations and support of a curve. Regular curves. Velocity vector and tangent vector. Length of a curve. Integrals of a scalar function on curves. Work of a vector field. Rotor of a vector field. Irrotational and conservative vector fields . Linear differential forms and primitive functions. Simply connected open sets. Relationship between exact and closed differential forms. Closed differential forms in set with a hole. Improper integrals; comparison test for nonnegative integrands; absolute or conditional convergence. Comparison between improper integrals and series. Continuity and differentiability of integrals depending on a parameter. Lebesgue measure and Lebesgue integral in R^N. Improper integrals. Double and triple integrals and reduction formulas. Change of variables. Guldino's formula for the volume of a solid of rotation. Convergence of sequences of functions. Series of functions and power series. Smooth surfaces. Tangent plane, unit normal vector. Surface edge and orientation of the edge. Surface area. Guldino's formula for the area of a surface of rotation. Surface integrals. Flow of a vector field through a surface. Gauss-Green formula. Divergence theorem and Stokes theorem in the plane and in the space. Reminders of linear differential equations with constant coefficients. Equations with separable variables, Bernoulli equations, Euler equations, autonomous equation. Cauchy problem: existence and uniqueness in small. Maximal solution. Qualitative study of differential equations.

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 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).