MATHEMATICAL ANALYSIS 3

Degree course: 
Corso di First cycle degree in MATHEMATICS
Academic year when starting the degree: 
2015/2016
Year: 
2
Academic year in which the course will be held: 
2016/2017
Course type: 
Compulsory subjects, characteristic of the class
Credits: 
8
Period: 
First Semester
Standard lectures hours: 
64
Detail of lecture’s hours: 
Lesson (64 hours)
Requirements: 

Prerequisites
Basic knowledge of analysis for functions of several variables. Lebesgue integration theory. Foundations on differential equations.

Final Examination: 
Orale
Assessment: 
Voto Finale

Aims and outcomes
The course aims at introducing students to the fundamental methods and techniques of advanced Mathematics. The student will be able to analyze, model and solve advanced problems in various disciplines:

Program

1) Complements on ordinary differential equations
a) Peano existence theorem
b) Extension of solutions and Gronwall lemma
c) Qualitative study, boundary value problems
d) Applications to geometric and physical problems.
e) Systems of differential equations and applications to partial differential equations.
f) Introduction to the calculus of variations.

2) Complements of integral calculus
a) Integrals depending on a parameter. Continuity, differentiability.
b) Applications to geometric and physical problems
c) Outline of Gamma and Beta functions.
d) Stirling formula.

3) Metric and normed spaces.
a) Completion theorem.
b) Space of continuous functions on compact sets.
c) Equicontinouos and equibounded sets. Ascoli-Arzelà theorem.
d) Stone-Weierstrass theorem

4) Complements of calculus.
a) A nowhere differentiable continuous function.
b) Power series
c) The inverse function theorem and the theorem of rank.

5) Curves and surfaces
a) Length of a curve and surface area (formulas)
b) Integration on curves.

6) Differential forms
a) Integration of differential forms
b) Exact forms and closed forms
c) Necessary and sufficient conditions.
d) Applications to differential equations.
e) The Gauss-Green formula

Teaching methods
Classroom lessons

Textbooks and references
W. Rudin, Principles of Mathematical Analysis, Mc Graw Hill.
DE Marco, Analisi Due, Zanichelli
E. Giusti, Analisi Matematica 2, Boringhieri

Final examination
Written and oral examination

Office hours
By appointment
Frontal lectures: 64 hours.

Textbooks

W. Rudin, Principles of Mathematical Analysis, Mc Graw Hill.
DE Marco, Analisi Due, Zanichelli
E. Giusti, Analisi Matematica 2, Boringhieri

Final examination
Written and oral exam.

Borrowers