Teachings

Select Academic Year:     2017/2018 2018/2019 2019/2020 2020/2021 2021/2022 2022/2023
Professor
GIUSEPPE VIGLIALORO (Tit.)
Period
First Semester 
Teaching style
Convenzionale 
Lingua Insegnamento
ITALIANO 



Informazioni aggiuntive

Course Curriculum CFU Length(h)
[70/72]  CIVIL ENGINEERING [72/00 - Ord. 2013]  PERCORSO COMUNE 9 90
[70/73]  ENVIRONMENTAL AND LAND ENGINEERING [73/00 - Ord. 2020]  PERCORSO COMUNE 9 90
[70/77]  CHEMICAL ENGINEERING [77/00 - Ord. 2020]  PERCORSO COMUNE 9 90

Objectives

1. Knowledge and understanding. At the end of the course the student will have knowledge of topics concerning real sequences, the infinitesimal calculus of real functions of a real variable and ordinary differential equations.
2. Applying knowledge and understanding. The student will be introduced to the main applications of the theoretical notions of the course, concerning both the solution of mathematical problems and the study of physical problems.
3. Autonomy of judgment. The student will learn how to classify single problems of differential and integral calculus, as well as differential equations, in the right class and to apply the most convenient method.
4. Communicative Skills. Students will acquire the ability to communicate what they learn and elaborate and also to express and argue the choice of methodology with respect to another to solve a mathematical problem.
5. Learning skills. Thanks to the notions acquired in this course, the student will be able to perfect his knowledge of higher mathematics and its applications to engineering.

Objectives

The objectives of the Mathematical Analysis Course 1 are defined in accordance with the information contained in the SUd CdS frame and more in detail can be described as follows:

Knowledge and understanding: At the end of the course the student will have knowledge of topics concerning real sequences, the infinitesimal calculus of real functions of a real variable and ordinary differential equations.

Ability to Apply Knowledge and Understanding: Students will be able to understand and interpret mathematical problems whose resolution is related to the knowledge of infinitesimal calculus as well as to real sequences and differential equations.

Autonomy of judgment: theoretical and applied notions will enable the student to understand which mathematical models and techniques are most appropriate for describing natural phenomena.

Communicative Skills: Students will acquire the ability to communicate what they learn and elaborate and also to express and argue the choice of methodology with respect to another to solve a mathematical problem.

Learning Abilities: Students will learn methodologies and tools such as: infinitesimal analysis and its direct applications to optimization problems and mathematical modeling in the general sense.

Prerequisites

Good knowledge of algebra, trigonometry and elementary analytic geometry

Contents

Introduction to set theory. Sets of Natural, Integer, Rational Numbers. Real Numbers: definitions, algebraic operations, distance. Subsets of real numbers. Least upper bound, greatest lower bound; maximum and minimum, accumulation, isolated, internal , external, boundary points; bounded, open, closed sets.

Real functions. Domain of definition, graph of elementary functions. Bounded, periodic, symmetric, monotonic, composite and inverse functions. Maxima and minima.

Limit theory. Basic limit theorems.

Continuous functions. The definition of continuity and basic theorems. Types of discontinuities. Weierstrass Theorem. The intermediate value theorem.

Differential calculus. Definition of derivative. The algebra of derivatives. The derivative of polynomials, of rational , exponential , logarithm, trigonometric functions. Geometric interpretation. Higher derivatives. The chain rule. Estreme values of a function. Increasing and decreasing functions, The mean value, Rolle Cauchy, DeL’Hopital theorems. Second derivative test for extrema. Convex and concave functions. Graph. Taylor and MacLaurin formula.

Integration. Antiderivative. Definite integral.Partitions of intervals. Definition of integral by upper and lower integrals. The area of a set. Theory and techniques of integrations. Fundamental theorem of integral calculus. Improper integrals.

Differential equations. Physical motivations. Theminology and notations. First order differential equation: separable equations, linear, Bernoulli and Clairaut equations. Cauchy theorem for the existence and uniqueness of the solution. Linear equations of order n. Wronskian determinant, Liouville theorem. Linear equations with constant coefficients: Lagrange method and special methods for determining a particular solution of the nonhomogeneous equation.


Sequences of real number. Limits of real sequences and theorems. Sub-sequences and convergence criteria

Contents

Introduction to set theory. Sets of Natural, Integer, Rational Numbers. Real Numbers: definitions, algebraic operations, distance. Subsets of real numbers. Least upper bound, greatest lower bound; maximum and minimum, accumulation, isolated, internal , external, boundary points; bounded, open, closed sets.

Real functions. Domain of definition, graph of elementary functions. Bounded, periodic, symmetric, monotonic, composite and inverse functions. Maxima and minima.

Limit theory. Basic limit theorems.

Continuous functions. The definition of continuity and basic theorems. Types of discontinuities. Weierstrass Theorem. The intermediate value theorem.

Differential calculus. Definition of derivative. The algebra of derivatives. The derivative of polynomials, of rational , exponential , logarithm, trigonometric functions. Geometric interpretation. Higher derivatives. The chain rule. Estreme values of a function. Increasing and decreasing functions, The mean value, Rolle Cauchy, DeL’Hopital theorems. Second derivative test for extrema. Convex and concave functions. Graph. Taylor and MacLaurin formula.

Integration. Antiderivative. Definite integral.Partitions of intervals. Definition of integral by upper and lower integrals. The area of a set. Theory and techniques of integrations. Fundamental theorem of integral calculus. Improper integrals.

Differential equations. Physical motivations. Theminology and notations. First order differential equation: separable equations, linear, Bernoulli and Clairaut equations. Cauchy theorem for the existence and uniqueness of the solution. Linear equations of order n. Wronskian determinant, Liouville theorem. Linear equations with constant coefficients: Lagrange method and special methods for determining a particular solution of the nonhomogeneous equation.
Sequences of real number
Limits of sequence. Theorems.

Teaching Methods

Compatibly with the teaching method established in the Manifesto Accademico 2020-21, as consequence of the COVID-19 emergency, the tools used for the lectures will be both the blackboard and tablet with projection system via classroom screen and via internet streaming.

Frontal lectures (theory): 72 hours
Frontal lectures (exercises): 18 hours

Teaching Methods

Compatibly with the teaching method established in the Manifesto Accademico 2020-21, as consequence of the COVID-19 emergency, the tools used for the lectures will be both the blackboard and tablet with projection system via classroom screen and via internet streaming.

Frontal lectures (theory): 72 hours
Frontal lectures (exercises): 18 hours

Verification of learning

Compatibly with the modality of exams established in the Manifesto degli Studi 2020-21, as consequence of the the COVID-19 emergency, the exams will be held either in the presence or on the Teams platform or on an alternative telematic platform previously agreed between the teacher and the student.

The exam consists of a written test in which the following topics are proposed: general analysis of a real function of a real variable, integral calculus and applications, differential equations and numeric sequences. The student will have to demonstrate that he has understood and learnt the techniques for handling and exposing each of the topics discussed and to know how to apply the various methodologies linked to the techniques of resolution. The exam score is awarded by a vote between 1 and 30. The test consists of 5 exercises, 4 of which are mandatory (with score 7.5 each) and 1 optional (with score 3) and the score is determined according to the following rule: sum of the scores obtained in the individual exercises. In evaluating the examination, the final vote determination takes into account the logic followed by the student for each proposed exercise, the calculation strategy chosen in terms of the hypothesis of the problem, the clarity of the exposition and the reasoning.

Texts

Marco Bramanti, Carlo D. Pagani, Sandro Salsa: Analisi matematica 1. Zanichelli,
Sandro Salsa, Annamaria Squellati: Esercizi di Analisi matematica 1, Zanichelli.

Texts

P. Marcellini, C. Sbordone.
Elementi di Analisi Matematica 1
Liguori Editore

P. Marcellini, C. Sbordone
Esercitazioni di matematica.
vol 1, parte 1 e 2.
Liguori Editore

More Information

More information is available in the personal web page of the Prefessor

More Information

More information are available in the personal web page of the Prefessor

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