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## MATHEMATICAL ANALYSIS 2

CODE 72288 2022/2023 6 cfu during the 2nd year of 10800 INGEGNERIA MECCANICA - ENERGIA E PRODUZIONE (L-9) - SAVONA MAT/05 Italian SAVONA 1° Semester This unit is a module of: AULAWEB

## OVERVIEW

The main goal of this course is to present the basic topics in the mathematical analysis of functions of several variables. In particular, differential calculus in several variables, the theory of multiple and line integration is treated.

## AIMS AND CONTENT

### LEARNING OUTCOMES

The knowledge of the mathematical tools introduced and the skill of using them to solve problems eventually in a physical setting. The skill of  setting up a problem correctly and the development of intuitive and deductive skills in solving problems.

### 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.

### TEACHING METHODS

The course consists of 36 hours of lectures and 24 hours of 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 in R^n and line integral.

Vector fields. Irrotational and conservative vector fields. Gauss - Green formulas and Divergence Theorem in R^2.

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

CLAUDIO CARMELI (President)

ROBERTUS VAN DER PUTTEN

OTTAVIO CALIGARIS (President Substitute)

DANILO PERCIVALE (President Substitute)

## LESSONS

### LESSONS START

https://corsi.unige.it/10800/p/studenti-orario

### Class schedule

All class schedules are posted on the EasyAcademy portal.

## EXAMS

### EXAM DESCRIPTION

A written examination which consists in two problems concernng the topics treted. The students have two hours at their disposal. After the written examination, the board of examiners might call the student for an oral examination.

Two intermediate examinations will be held.

### 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

Date Time Location Type Notes
10/01/2023 10:00 SAVONA Scritto
31/01/2023 10:00 SAVONA Scritto
12/06/2023 09:30 SAVONA Scritto
12/07/2023 09:30 SAVONA Scritto
14/09/2023 09:30 SAVONA Scritto

### FURTHER INFORMATION

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