Teacher

prof. Fabio ZANOLIN

Credits

12 CFU

Language

Italian

Objectives

Basic concepts from mathematical logic and set theory. Sets of natural numbers, integers, rational numbers, real numbers and complex numbers, with their main properties. The completeness of the reals. Ordered sets. Maximum, minimum, "inf" and "sup" of a subset of the reals and their properties. Sequences and series and their limits.  Monotone sequences. Limits and continuity of real functions. Introduction and development of the basic concepts of differential and integral calculus for real valued functions defined on subsets of the real line. Introduction to ordinary differential equations (ODEs) with a special emphasis to their applications to physical and mechanical problems. The Cauchy problem for ordinary differential equations.Solvability of certain special ODEs using some integration techniques: first order ODEs with separated variables, linear equations.

Acquired skills

- Foundations of Mathematical logic. What does it mean the word "proof".
- Numerical sets. Completeness of the set of real numbers.
- Learning of the main concept in Mathematical Analysis: limits.
- Learning of the basic tools for functions of one real variable: derivation and integration.
- The Cauchy problem for ordinary differential equations: meaning of "local solution"; physical and mechanical motivations.

Lectures and exercises (topics and specific content)

Mathematical logic, set theory, numerical sets:  mathematical logic, set theory; the sets N, Z, Q, R, C; equivalence, quotient set; cardinality (24 hours).
Numerical sequences and series: limit, minimum and maximum limit of a numerical sequence; Cauchy sequences; converging numerical series; main convergence criterions (20 hours).
Functions of one real variable: functions of one real variable (60 hours).
On the Cauchy problem: ordinary differential equations with saparate variables, linear of the first order, linear with constant coefficients (16 hours).
Exercises (24 hours).

References

- Primo corso di Analisi Matematica, Acerbi-Buttazzo, Pitagora ed.

Type of exam

Written and oral