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

2021/2022
Belonging course:
Disciplinary sector:
MATHEMATICAL ANALYSIS (MAT/05)
Language:
Italian
Credits:
9
Year of study:
2
Teachers:
Cycle:
First Semester
Hours of front activity:
72

Italian

### Course description

Objectives of the course:
The aim of the course is to provide basic knowledge concerning the theory of functions of several real variables (continuity, differentiability, integrability), of ordinary differential equations and their applications to concrete problems. Eventually, the student will have to prove to have understood both theoretical and applicative parts of the course and to be able to use those methods for modeling and solving economic, financial and corporate problems.

Expected learning outcomes
Knowledge and understanding: The student should be able to prove knowledge of mathematical tools and ability to identify those suitable for modeling and solving economic, financial and corporate issues.
Applying knowledge and understanding: the student should be able to apply the acquired 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.

Making judgements: the student should be able to use the acquired knowledge also in an autonomous way, by also applying them to specific issues related to economic problems.
Communication: the student should be able to answer in a clear and thorough way to the questions of the written examination and to those of the oral examination.
Lifelong learning skills: the student should be able to show a good learning ability, by widening, for example, his/her knowledge with use of relevant bibliographic references.

### Prerequisites

It is necessary to have passed the exam in Mathematics I (of the first year). In fact, it is essential to have acquired basic knowledge such as: elementary functions (in particular linear function, power, exponential and logarithmic function), analysis of a real function of a real variable, calculation of derivatives of a function of one variable and rules of derivation, integration of functions, elements of linear algebra.

### Syllabus

Functions of two or several variables. Domain of functions of two variables. Limits and Continuity. Partial Derivatives. Higher Order Derivatives: Hessian matrix. Schwarz Theorem. Gradient vector and Differentiability. Composite functions. Directional Derivatives. (12 hours)

Local and Global Maxima and Minima. Convex and concave functions. Constrained maxima and minima: method of Lagrange multipliers. Applications. (12 hours)

Hints on integrals of functions of one variable. Double integrals: normal domains ; change of variable. (12 hours)

Curves: introduction. Tangent vector to a Curve. Orientation of a curve. Rectifiable curve: length of a curve. Integration on curves. (12 hours)

Ordinary Differential Equations: Existence and Uniqueness: the Cauchy problem,
Linear First Order Equations, Linear Second Order Equations, Separable Equations, Bernoulli Equations. Systems of Differential Equations. (18 hours)
Mathematical models: examples. (6 hours)

### Teaching Methods

The course includes frontal lessons, during which the themes of the program are discussed, and exercises in attendance.

### Textbooks

Texts
(the differential equations are in the first part of Esercitazioni di Analisi Matematica due; the method of Lagrange multipliers for constrained maxima and minima is in the second part)

• N.Fusco, P.Marcellini, C.Sbordone: Elementi di Analisi Matematica due, Zanichelli Editore, 2020.
• P. Marcellini, C. Sbordone: Esercitazioni di Analisi Matematica due - prima parte e seconda parte – Zanichelli Editore, 2017
• C.P. Simon - L.E. Blume, Mathematics for Economists

### Learning assessment

Assessment methods.
The assessment is based on written examination (the use of notes, books and informatics devices -smartphone, tablet, pc, ecc.- is not allowed) and an oral interview. The written test is composed of exercises in order to assess the achievement by the student of the learning objectives and aims to verify the acquisition, by the student, of the mathematical tools necessary to determine relative and constrained maxima / minima of functions of two variables, calculate partial and directional derivatives, calculate double integrals and integration on curves, solve differential equations.. The oral exam focuses on the theoretical topics dealt with during the course and it is designed to evaluate the student's ability to express and formalize mathematical concepts. The vote of the examination is expressed in scale from 0 to 30, and it is results of written and oral examination. The laude can be assigned if the student shows, in his/her answers, a particular ability to deepen the topics mentioned in the examination’s questions.