CODE 60503 2023/2024 6 cfu anno 2 INGEGNERIA NAUTICA 8721 (L-9) - LA SPEZIA MAT/05 Italian LA SPEZIA 1° Semester Per sostenere l'esame di questo insegnamento è necessario aver sostenuto i seguenti esami: INGEGNERIA E DESIGN PER LA NAUTICA 10134 (coorte 2022/2023) MATHEMATICS + GEOMETRY 56973 2022 Questo insegnamento è un modulo di: AULAWEB

## OVERVIEW

The course is directed to second year students which are familiar with basic notions in Calculus and Analytical Geometry.

## AIMS AND CONTENT

### LEARNING OUTCOMES

Provide basic concepts and more specific mathematical tools to better understand the contents of some nautical engineering courses .

### AIMS AND LEARNING OUTCOMES

The knowledge of mathematical basic tools useful in physical problems modelling.  The skill of setting up and solving problems by using intuitive and deductive reasoning as well as recognizing and using the suitable mathematical tools in solving problems in a physical setting. At the end of the course the student will be able

1. to state the concepts ( theorems and definitions ) introduced during the course ( f.i. level set, partial derivatives, optimization, line integral, integral in R^2 and R^3 );

2. to give physical and geometric interpretation of the basic concepts of Mathematical Analysis;

3. to select the suitable mathematical tools in problem solving;

4. to solve problems with deductive reasoning.

### PREREQUISITES

Basic notions in Calculus and Analytical Geometry.

### TEACHING METHODS

The course consists of 552 hours of lectures and practices. In the lectures the topics of the syllabus are explained with definitions and theorems and some proofs which can be useful for the comprehension of the topics and to develop the logical and deductive skills. Every theoretical topic is explained with easy examples and some exercises. In the practices, many exercises are solved with the aim of going into the knowledge of theoretical topics treated in the lectures and preparing the student for the exam. Some guided practices will be held to help the student to valuate one's preparation. Several intermediate tests are provided.

Students  have several exercises at their disposal on Aulaweb.

### SYLLABUS/CONTENT

Euclidean spaces. Toplology in R^n. Functions of several variables. Level sets. Continuity and differentiability. Directional and partial derivatives. Derivatives of higher order. Schwartz Theorem. Taylor approximation with Peano's and Lagrange's reminder. Quadratic forms. Unconstrained optimization. Necessary first order condition and sufficient second order condition. Implicit function theorem. Change of coordinates. Constrained optimization.

Systems of nonlinear differential equations. Cauchy problem. Existence and uniqueness of the solution.

Double and triple integrals. Normal domains in R^2. Integration formulas and theorem of change of variables.

Lines and Surfaces in R^n and line and surface integral.

Vector fields. Irrotational and conservative vector fields. Gauss - Green formulas and Divergence Theorem.

C. Canuto, A. Tabacco, "Analisi Matematica II", Springer, 2014.

M. Bramanti, C. Pagani, S. Salsa. “Analisi matematica 2”, Zanichelli, 2009.

S. Salsa, A. Squellati. “Esercizi di Analisi matematica 2”, Zanichelli 2011.

## TEACHERS AND EXAM BOARD

### Exam Board

STEFANO VIGNOLO (President)

VALENTINA BERTELLA (President Substitute)

ROBERTUS VAN DER PUTTEN (President Substitute)

## LESSONS

### LESSONS START

https://corsi.unige.it/8721/p/studenti-orario

### Class schedule

MATHEMATICAL ANALYSIS 2

## EXAMS

### EXAM DESCRIPTION

The exam consists in a written and oral examination. The written examination consists in two problems concerning the topics treated. The students have two hours at their disposal. After the written examination, the students who obtained a grade higher than 13/30 may take the oral examination. Two intermediate examinations will be held which may substitute the written examination.

Students with SLD, disability or other regularly certified special educational needs are advised to contact the instructor at the beginning of the course to agree on teaching and examination methods that, in compliance with the course objectives, take into account the individual learning requirements.

### ASSESSMENT METHODS

The aim of the examination is verifying the skills acquired by the student. The problems proposed in the examination call for the choice and the application of suitable mathematical tools, besides their  solution needs the skill of constructing a logical connection applying theoretical topics treated. The student must solve the exercises justifying the most important passages recalling  theorems and definitions and underlying the physical and geometric interpretation of the problem.

The final evaluation depends also on the quality of the written exposition and on the ability of reasoning.

### Exam schedule

Data Ora Luogo Degree type Note
15/01/2024 14:00 LA SPEZIA Scritto
09/02/2024 09:30 LA SPEZIA Scritto
12/06/2024 09:30 LA SPEZIA Scritto
15/07/2024 09:30 LA SPEZIA Scritto
06/09/2024 14:00 LA SPEZIA Scritto

### FURTHER INFORMATION

The course requires knowledge of the content of Mathematical Analysis 1 and Elements of Mathematics for Engineering