# MATHEMATICAL ANALYSIS

# MATHEMATICAL ANALYSIS

Knowledge of linear algebra and of differential and integral calculus for real functions of one or several variables (see the courses of Calculus and of Linear Algebra and Geometry of the 1st year).

Gaining knowledge and understanding of:

Ordinary differential equations.

Sequences and series of functions.

Metric spaces and continuous functions.

Applications of differential and integral calculus.

1. Ordinary differential equations:

Local existence and uniqueness of solutions for the initial-value problem for first order equations and systems. Maximal and global solutions.

Linear systems. Global existence and uniqueness for the Cauchy problem. General integral. General integral of the associated homogeneous system. Fundamental system of solutions. Fundamental matrix. Method of the variation of constants.

2. Sequences and series of functions:

Pointwise and uniform convergence. Total convergence.

Uniform convergence and continuity. Uniform convergence and differentiation.

Power series. Taylor series.

3. Metric spaces and continuous functions:

Balls. Open and closed sets.Bounded sets. Closure of a set.

Continuous mappings between metric spaces.

Convergent and Cauchy sequences in metric spaces. Closed sets and sequences. Complete metric spaces. Completeness of function spaces.

The contraction mapping theorem.

Normed vector spaces. Banach spaces.

Compact metric spaces and continuous mappings. Uniform continuity.

Connected metric spaces and continuous mappings.

4. Applications of the differential calculus for real functions of several real variables:

Differential and integral calculus over curves and surfaces.

Constrained minima and maxima. Implicit functions.

1)T.M. Apostol: Calcolo, Bollati Boringhieri.

2)N.Fusco-P.Marcellini-C.Sbordone: Analisi matematica due, Liguori.

3)E.Giusti: Analisi Matematica 2, Bollati Boringhieri

1983 (prima edizione), 1989 (seconda edizione).

4)E.Giusti: Esercizi e Complementi di Analisi Matematica, Volume secondo, Bollati Boringhieri.

5)G. Prodi: Analisi Matematica, Bollati Boringhieri.

6)R.B.Reisel: Elementary Theory of metric spaces, Springer

Lectures and exercises

Written and oral test.

It is strongly recommended to the student a constant and timely presence in attending the lectures.