MATHEMATICAL ANALYSIS 2

Degree course: 
Corso di First cycle degree in MATHEMATICS
Academic year when starting the degree: 
2019/2020
Year: 
1
Academic year in which the course will be held: 
2019/2020
Course type: 
Basic compulsory subjects
Credits: 
8
Period: 
Second semester
Standard lectures hours: 
68
Detail of lecture’s hours: 
Lesson (56 hours), Exercise (12 hours)
Requirements: 

Mathematical analysis I, Linear algebra.

Final Examination: 
Orale

The exam consists of two parts
Written exam: duration 3 hours with exercises (4/5) on the topics discussed in 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 the natural continuation of the first course in Mathematical Analysis and it aims to expand the study of classical and modern analysis. The students will acquire an operational knowledge of the methods of mathematical analysis, they will know the main statements and demonstrations, and will be able to solve exercises related to the topics discussed. They will also have a knowledge of demonstration techniques that can also be used for the independent demonstration of results related to those presented during the course.

1) Metric spaces, complete metric spaces, sequentially compact sets and their properties, continuous functions.
2) Contraction theorem.
3) Normed spaces, linear operators between normed spaces. Equivalent norms.
4) Functions from R ^ n in R ^ m. Continuity and differentiation. Partial derivatives, gradient and Jacobian matrix. Sufficient condition for differentiation. Chain rule.
5) Mean value theorem. Functions with null differential on connected sets.
6) Second partial derivatives.
7) Taylor’s theorem in several variables.
8) Implicit functions. Local theorem of existence and uniqueness.
9) Maximum and minimum. First order conditions.
10) Hessian matrix and sufficient conditions for extrema.
11) Extrema with constraints. The Lagrange Multipliers method.
12) Peano measure in R ^ n and measurable sets.
13) Integration on rectangles. Iterated integrals. Riemann integral on measurable sets.
14) First order differential equations and systems. Solutions.
15) The local existence and uniqueness theorem for the Cauchy problem.
16) Maximal solutions.
17) Sufficient conditions for existence on a interval.
18) Differential Equations of order n. Linear equations of order n. Independent solutions and solution space.
19) Linear equations of order n with constant coefficients.

1) Metric spaces, complete metric spaces, sequentially compact sets and their properties, continuous functions.
2) Contraction theorem.
3) Normed spaces, linear operators between normed spaces. Equivalent norms.
4) Functions from R ^ n in R ^ m. Continuity and differentiation. Partial derivatives, gradient and Jacobian matrix. Sufficient condition for differentiation. Chain rule.
5) Mean value theorem. Functions with null differential on connected sets.
6) Second partial derivatives.
7) Taylor’s theorem in several variables.
8) Implicit functions. Local theorem of existence and uniqueness.
9) Maximum and minimum. First order conditions.
10) Hessian matrix and sufficient conditions for extrema.
11) Extrema with constraints. The Lagrange Multipliers method.
12) Peano measure in R ^ n and measurable sets.
13) Integration on rectangles. Iterated integrals. Riemann integral on measurable sets.
14) First order differential equations and systems. Solutions.
15) The local existence and uniqueness theorem for the Cauchy problem.
16) Maximal solutions.
17) Sufficient conditions for existence on a interval.
18) Differential Equations of order n. Linear equations of order n. Independent solutions and solution space.
19) Linear equations of order n with constant coefficients

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

Frontal lectures and homework exercises

By e-mail appointment.