MATHEMATICAL ANALYSIS 2

Degree course: 
Corso di First cycle degree in MATHEMATICS
Academic year when starting the degree: 
2017/2018
Year: 
1
Academic year in which the course will be held: 
2017/2018
Course type: 
Basic compulsory subjects
Credits: 
8
Period: 
Second semester
Standard lectures hours: 
56
Detail of lecture’s hours: 
Lesson (56 hours)
Requirements: 

Mathematical Analysis 1:

Basic knowledge of high school algebra, trigonometry and analytic geometry.

Mathematical Analysis 2:

Mathematical analysis I, Linear algebra.

Final Examination: 
Orale

Mathematical Analysis 1:

The exam is divided into three parts:
-a written test consisting in three to five exercises covering the main topics studied in the course, which will test the ability of students in applying the computational techniques learned in class;
-a second written test covering the theoretical aspects of the course, and consisting in giving statements and proofs of a few theorems seen in class, which will test the understanding of the underlying theory and the ability to reproduce rigorous proof;
- an oral part, which follow immediately the second written test, consisting in the discussion of the two written tests.
Each part will be evaluated with a grade in the range 0 to 30, and the final grade will be the average, if greater than or equal to 18, of the grades of the three parts. Access to the second part of the exam is conditioned to having obtained at least 14/30 in the first part.

Mathematical Analysis 2:

Written and oral examination

Assessment: 
Voto Finale

Mathemtical Analysis 1:

The course aims at introducing students to the fundamental methods and techniques of Mathematics, and in particular to the differential and integral calculus of one real variable and to sequences and series. A further goal is to train students in applying analytic techniques to other sciences.
Students will know the fundamental of differential and integral calculus of real functions of one real variable. In particular they will be able to study the qualitative graph of elementary function, to solve elementary ordinary differential equations, to determine the convergence of sequences and series; they will be able to state and prove the basics theorems of Analysis.

Mathemtical Analysis 2:

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.

Mathematical Analysis 1:

Number sets. Real and complex numbers. Topology of the real line. Limits of sequences and series.. Limits of functions and continuity. Continuous functions and topology. Differentiable functions and their properties. Differentiation rules. Fundamental theorems of the differential calculus. Taylor formula and second order conditions to determine the nature of a stationary point. Convexity. Study of the graph of a function. Riemann integral: definition of integrable function and properties of the integral. Classes of integrable functions. The fundamental theorem of calculus. Integration methods and computation of definite integrals. Improper integrals.

Mathematical Analysis 2:

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.

Mathematical Analysis 1:

Dispense di Analisi Matematica 1 disponibili alla pagina:
http://www.matapp.unimib.it/~demichele/libro.pdf
E. Giusti, Analisi Matematica 1, Boringhieri
E. Giusti, Complementi di Analisi Matematica 1, Boringhieri
W. Rudin, Principi di Analisi Matematica, McGraw Hill
M. Bramanti, C. Pagani, S, Salsa, Analisi Matematica 1, Zanichelli

Mathematical Analysis 2:

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

Mathematical Analysis 1:

Frontal lectures: 56 hours. Exercise sessions: 24 hours.

Mathematical Analysis 2:

Frontal lectures

Office hours

Mathematical Analysis 1: By appointment.

Mathematical Analysis 2: By appointment.