Complements of Mathematics

Period of duration of course
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Course info
Number of course hours
80
Number of hours of lecturers of reference
80
Number of hours of supplementary teaching
0
CFU 12
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Modalità esame

Written and oral exam

Lecturer

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Lecturer

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Prerequisiti

First year students in Mathematics and Physics

Programma

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.

 

 

Obiettivi formativi

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. 

Riferimenti bibliografici

Book 

COMPLEMENTI DI MATEMATICA

AMBROSIO LUIGIMANTEGAZZA CARLORICCI FULVIO

delle edizioni della Normale