# Università degli Studi di Napoli "Parthenope"  ## Teaching schedule

2018/2019
Belonging course:
Disciplinary sector:
MATHEMATICAL ANALYSIS (MAT/05)
Language:
Italian
Credits:
9
Year of study:
2
Teachers:
Dott.ssa GIOVA Raffaella
Cycle:
First Semester
Hours of front activity:
72

Italian

### Course description

The main goal of the course is the learning of the
most important topic of a second course in Analysis
including the number series.
Particular attention will be devoted to the study of examples and to problem solving in order to allow the students to be able to exploit the tools of the analysis.
Knoledge e comprehension: the students will have to
learn the fundamental tools of mathematical analysis, with particular regard to the logical understanding of definitions and theorems and the identification of examples and counter-examples.
Ability to Apply Knowledge and Understanding: The students will have to acquire the knowledge in solving the main problems regarding the study of functions of several variables. This will involve the ability to identify appropriate theoretical tools
suitable to the particular problem under study by applying correctly the tools of the infinitesimal calculus.
Autonomy of judgment: Students must be able to know how to establish the logical veracity of affirmations and properties regarding the functions of several variables.
Communication Skills: Students must be able to expose in a formally correct logical way the theorems concerning the theory of the functions of several variables, highlighting hypothesis and thesis and illustrating the results through examples and applications.
Learning Skills: The student must be able to update and deepen the discussed topics, also by identifying the appropriate tools among those available on the web.

### Prerequisites

In order to attend the course it is important to know the differential and integral calculus of functions of one variable and the basis of the linear algebra.

### Syllabus

Number series: Definitions, geoemtric series and armonic ones; convergence tests for positive series: comparison test, root test, ratio test, limit comparison. Absolute convergence test and Leibniz test.
Power series: Total convergence, properties of the limit function, power series and Taylor series, examples.
Differential calculus for functions of several variables: critical points, maximum and minimum points,
Hessian matrix, Taylor formula, Lagrange multipliers.
Ordinary Differential equations: Cauchy problem;
equations of first order, linear and with separable variables; linear equations of second order with constant coefficients.
Curves and integration on curves; parametric equations and cartesian ones; tangent vector, lenght of a curve.Examples.
Multiple integrals: basic definitions, integation on normal domains; use of polar cohordinates.

Differential forms and vector fields: closed and exact differential forms. Work of a vector field; gradient vector fields.
differential geometry of surfaces: basic definitions, tangent plane; divergence theorem and Stokes theorem.
Examples. Probability

Number series: Definitions, geoemtric series and armonic ones; convergence tests for positive series: comparison test, root test, ratio test, limit comparison. Absolute convergence test and Leibniz test.
Power series: Total convergence, properties of the limit function, power series and Taylor series, examples.
Differential calculus for functions of several variables: critical points, maximum and minimum points,
Hessian matrix, Taylor formula, Lagrange multipliers.
Ordinary Differential equations: Cauchy problem;
equations of first order, linear and with separable variables; linear equations of second order with constant coefficients.
Curves and integration on curves; parametric equations and cartesian ones; tangent vector, lenght of a curve.Examples.
Multiple integrals: basic definitions, integation on normal domains; use of polar cohordinates.

Exlusive Topics of the Course Matematical Analysis II:

Differential forms and vector fields: closed and exact differential forms. Work of a vector field; gradient vector fields.
differential geometry of surfaces: basic definitions, tangent plane; divergence theorem and Stokes theorem.
Examples.

Probability theory: combinatoric: permutation, permutation with repetition, k-permutation of n elements, combinations (with and without repetition).
Axiomatic definition of probabiity.
Conditional probability, Bayes rule, total probability law.
Random variables: discrete and continuous, density of a probability distribution; expected value, average,
first and second momentum; the variance; central limit theorem.
Examples: Normal distribution, exponential distribution, Poisson distribution, binomial distribution, Gaussian distribution, Uniform distribution

### Teaching Methods

Lectures with numerous exercises

### Textbooks

Analysis 2. Terence Tao (medaglia Field). Hindustan Book Agency.

### Learning assessment

The exam consists of a written test and an oral test. In the written test, lasting 60 minutes, exercises will be proposed concerning numerical series, optimizations of functions of several variables and differential equations. While the oral exam will assess the knowledge acquired with questions that will focus on the definitions and proofs of the main theorems concerning the program topics.