CODE 114430 2024/2025 6 cfu anno 1 INGEGNERIA MECCANICA 8784 (L-9) - LA SPEZIA MAT/05 LA SPEZIA 2° Semester

## OVERVIEW

The course provides the basic tools of Mathematical Analysis concerning functions of one real variable.

## AIMS AND CONTENT

### LEARNING OUTCOMES

Rigorous treatment of Mathematical Analysis, focusing on differential and integral calculus of functions of real variables. In particular, students will have to acquire good skills regarding the following topics: Indefinite integrals. Applications to Physics and Geometry. Definite integrals. Geometric motivation. Partitions, ladder functions, integral of ladder functions. Properties of the integral of ladder functions. Definition of the integral according to Riemann. Integrability criterion. Example of a non-R-integrable function. Integrability of continuous functions. Integrability of the function of monotone functions. Properties of the integral: linearity, monotonicity, additivity. Integrability of f+, f_ and |f|. The integral as a limit of Riemann sums. The oriented integral and its properties. The integral mean theorem. Integral functions and improper integrals. Differential equations. Order of an equation, normal form. Geometric interpretation of first order equations: direction fields. Differential equations with separable variables. Functions of multiple variables.

### 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. integral , Existence and Uniqueness theorems for differential problems);
2. to give physical and geometric interpretation of the basic concepts of Mathematical Analysis;
3. to set up problem solving with an intuitive approach;
4. to select the suitable mathematical tools in problem solving;
5. to solve problems with deductive reasoning.

### PREREQUISITES

Differential calculus for functions of one real variable, i.e. the content of Mathematical Analysis Module 1 course.

### TEACHING METHODS

The course consists of 52 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.

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

### SYLLABUS/CONTENT

Antiderivatives. Riemann integral : definition and basic properties. Mean value Theorem. The Fundamental Theorem and Formula of Integral Calculus. Areas of plane regions. Changes of variable: the method of substitution. Integration by parts. Integrals of rational and trigonometric functions. Integral functions. Improper Integral. Ordinary differential equations : the  method of separation of variables and Existence and Uniqueness Theorems for the Cauchy Problem.

Ordinary linear differential equations: structure of the general solution in the Homogeneous and Nonhomogeneus case. Solving methods for n order linear equations with constant coefficients and first order linear equations with continuous coefficients.

Functions of several variables. Level sets. Continuity and differentiability. Directional and partial derivatives. Derivatives of higher order. Schwartz Theorem. Quadratic
forms. Unconstrained optimization. Necessary first order condition and sufficient second order condition.

Theory
C. Canuto – A. Tabacco: Analisi Matematica 1. Pearson. 2021.
C. Canuto, A. Tabacco, Analisi Matematica 2, 2a edizione, Springer-Verlag Italia, 2014
M.Bramanti - C.D.Pagani - S. Salsa: Analisi Matematica 1. Zanichelli, 2008.
T. Zolezzi: Dispense di analisi matematica I e II.
F. Parodi – T. Zolezzi: Appunti di analisi matematica. ECIG, 2002
R. Adams : Calcolo differenziale I. Funzioni di una variabile reale. Casa ed. Ambrosiana, 1992.

Exercises
M. Baronti – F. De Mari – R. van der Putten – I. Venturi: Calculus Problems. Springer 2016
M. Pavone: Temi svolti di analisi matematica I.
Marcellini-Sbordone: Esercitazioni di matematica, I volume
S. Salsa – A. Squellati: Esercizi di Matematica, volume 1.

## LESSONS

### LESSONS START

The lessons start on Febbraury 2025

### Class schedule

The timetable for this course is available here: Portale EasyAcademy

## 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 / three 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 professor and the University referent at least 15 days before each examination to agree on exhamination methods that, in compliance with the course objectives, take into account the individual learning requirements.
To enroll the exam you must register by the deadline on the website https://servizionline.unige.it/studenti/esami/prenotazione

### 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.