Complements of Mathematics
Period of duration of course
First year students in Mathematics and Physics
Set theory. Cardinality. Axiom of choice.
Construction of integer, rational and real numbers. Completeness of R. Limits of sequences. Limit points, upper and lower limits. Cauchy criterion, infinite limits.
Numerical series and convergence criteria.
Topology of Rn.
Metric spaces, completeness and the contraction theorem.
Sequences and series of functions, pointwise and uniform convergence.
Power series. Limits of derivatives and integrals.
Functions of several real variables. Partial and directional derivatives. Differential. Total
differential theorem. Hessian matrix.
A few hints on functions from Rn to Rm, Jacobian matrix, differentiation of composite functions.
Implicit functions. Regular curves. Conservative vector fields and potentials.
Differential equations and Cauchy problems. Local existence and uniqueness theorem. Linear differential equations and systems.
Goal of the lectures is to introduce students to basic topics but under an advanced and stimulating viewpoint, offering relevant additional elements with respect to traditional courses.