# MATHEMATICAL ANALYSIS 3

Degree course:
Corso di First cycle degree in MATHEMATICS
Academic year when starting the degree:
2017/2018
Year:
2
Academic year in which the course will be held:
2018/2019
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:

Mathematical Analysis 1 and 2, Linear algebra and geometry, geometry 1.

Final Examination:
Orale

The exam is divided into two parts
Written exam: duration 3 hours with exercises (4/5) on the topics developed during the course in order to verify the level of skills acquired.
Oral exam: after passing the written test to assess the level of knowledge reached.

Assessment:
Voto Finale

The course is a natural continuation of the course in mathematical analysis 2. It aims to deepen the study and modern classical analysis begun in the previous year.
The student will acquire a working knowledge of advanced analysis methods, the statements and major demonstrations, he will increase his skills and he will be able to solve exercises, even theoretical, related to the topics.

1) Sequence and series of functions. Uniform and total convergence. Theorem of double limit. Uniform convergence and differentiability. A nowhere differentiable continuous function. Weierstrass approximation theorem. Space of continuous functions on compact sets. Equicontinouos and equibounded sets. Ascoli-Arzelà theorem
2) Complements on ordinary differential equations, Peano existence theorem, Extension of solutions. Qualitative study of differential equation. Applications to geometric and physical problems.
3) Sigma-algebra and measure. Measurable functions. Integral of positive functions. Monotone convergence theorem. Fatou’s lemma. Integrable functions. Dominate convergence theorem. Lebesgue measure in R and R^n. Measure on algebra and semi-algebra. Caratheodory exstension theorem. Distribution function of measures in R. Product measure. Fubini and Tonelli theorems. Integral depending on a parameter.
Outline of Gamma and Beta functions. Stirling formula.
4) Curves and surfaces. Length of a curve and surface area (formulas). Integration on curves. Differential forms. Integration of differential forms. Exact forms and closed forms. Necessary and sufficient conditions. Applications to differential equations. The Gauss-Green formula.

1) Sequence and series of functions. Uniform and total convergence. Theorem of double limit. Uniform convergence and differentiability. A nowhere differentiable continuous function. Weierstrass approximation theorem. Space of continuous functions on compact sets. Equicontinouos and equibounded sets. Ascoli-Arzelà theorem
2) Complements on ordinary differential equations, Peano existence theorem, Extension of solutions. Qualitative study of differential equation. Applications to geometric and physical problems.
3) Sigma-algebra and measure. Measurable functions. Integral of positive functions. Monotone convergence theorem. Fatou’s lemma. Integrable functions. Dominate convergence theorem. Lebesgue measure in R and R^n. Measure on algebra and semi-algebra. Caratheodory exstension theorem. Distribution function of measures in R. Product measure. Fubini and Tonelli theorems. Integral depending on a parameter.
Outline of Gamma and Beta functions. Stirling formula.
4) Curves and surfaces. Length of a curve and surface area (formulas). Integration on curves. Differential forms. Integration of differential forms. Exact forms and closed forms. Necessary and sufficient conditions. Applications to differential equations. The Gauss-Green formula.

W. Rudin, Principi di analisi matematica, Mc Graw Hill.
De Marco, Analisi Due, Zanichelli
H. Royden, Real Analysis, Prentice Hall
E. Giusti, Analisi Matematica 2, Boringhieri

Classroom lessons and homework exercises

Office hours
By appointment