SM/0027 - MATHEMATICAL ANALYSIS 3

Academic Year 2018/2019

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Objectives

1. Knowledge and understanding

Knowledge and understanding of the fundamentals of the theory of ordinary differential equations, togehter with some of its abstract developements, such as the representation of a solution as a limit (of a sequence, or a sum). Knowledge and understanding of the fundamentals of Lebesgue integration theory.

2. Applying knowledge and understanding

Ability in the usage of the theory of ordinary differential equations for finding the general solution of a given equation and the particular solution of a simple initial-value problem. Ability in the determination of the Fourier, as well as the Taylor expansion of a simple given function. Ability in the usage of the standard theorems for finding the limit of an integral.

3. Making judgements

To develop criticism, ability in personal thinking, and initiative, with reference to the specific problems the lectures deal with. Ability in judging the soundness of a proof, or more generally of a mathematical argument, as well as the well-posedness of a definition.

4. Communication skills

Ability in communicating information, ideas, problems and solutions by means of the current terminology, in order to effectively interact with other students and teachers, in the present, and prospectively with colleagues or in a possible teaching activity. Ability in supporting a claim by means of a logical argument (a proof).

5. Learning skills

By attending the lectures and by means of a personal study on the main textbooks available, as well as by interacting with other students and the teacher, the student develops a suitable methodology for the interpretation and the investigation of subsequent problems they will encounter in the forthcoming lectures and in the subsequent job activities.

Prerequisites

Prerequisites

Differential and integral calculus for functions of one or several real variables. Numerical sequences and series.

Contents

Introduction into the theory of ordinary differential equations (24 hours).

Basic notions on ordinary differential equations. Concept of a solution and test of the correctedness of a proposed solution. General solution. Solution of a first-order equation by separation of variables. Linear differential equations, and the structure of the solution space. Euler's equation. Initial-value problems. Local existence and uniqueness of a solution.


Sequences and series of functions (24 hours).

Pointwise convergence. Uniform convergence. Cauchy's criterion. Continuity, differentiability, and the integral of the limiting function. Power series. Interval of convergence. Analytical nature of the sum. Basic examples. Fourier series: the origin; sufficient conditions for convergence; Bessel's inequality; Parseval equality.


Elementary function spaces (8 hours).

The spaces C^k([a,b]), their norm, their metric, and completeness. The p-norm in R^N. Banch's contracting mapping theorem.


Introduction into the theory of Lebesgue measure and integration (24 hours).

The construction of Lebesgue measure. Countable additivity. The concept of a measurable function. Definition of the Lebesgue integral and comparison with the Riemann integral. Passing to the limit under the integral sign: the bounded convergence theorem, the dominated convergence theorem, the monotone convergence theorem, Fatou's lemma. Differentiation under the integral sign. Theorems of Fubini and Tonelli.

Teaching Methods

In the first lecture, students are given information on the course contents, the evaluation methods, textbooks and lecture notes, together with suggestions on how to proficiently carry on their study, with reference to the course web site.

The course is composed of lectures (40 hours), homeworks, discussion of exercises, and laboratory (symbolic solution of differential equations using a CAS - Computer Algebra System; representation of abstract concepts by means of: computer graphics, interactive applets, or simple home-made models; short presentations given by interested students) (40 hours).

Students are invited to actively attend the course, also by taking part to the discussions, and to ask questions and propose strategies to solve mathematical problems arising from the lectures.

Verification of learning

The final examination is aimed at measuring the level of achievement of the course objectives listed above. According to the Regulation of Teaching Activities approved on May 30, 2017, the verification of learning is based on an oral examination. The student is asked some questions chosen in the realm of the course contents. Answers are to be given in part in a written form (usually on a blackboard), in order to evaluate the abilities requiring manipulation of simbolic as well as numeric expressions.

The final score ranges from 18/30 to 30/30 cum laude. Active participation to lectures and problem-solving sessions may increase the score of 1-2 points. Initiative, critical and personal thinking are appreciated.

The final score also depends on the ability to take part into a discussion, and on the ability to locate and correct possible mistakes without getting lost.

Usually, questions are similar to the ones posed and answered during the problem-solving sessions. Questions are aimed at evaluating the achievement of the knowledge and the abilities indicated in the course objectives. In order to pass the exam, thus getting a final score not less than 18/30, the student must have a basic knowledge of the proposed topics. A more thorough achievement of the objectives listed above corresponds to a proportionally higher score. In order to get the maximum final score of 30/30 cum laude, the student must show excellent knowledge of all the proposed topics, and can be required to attack more difficul tasks.

Texts

The lectures are based on the textbook by

Fusco, Marcellini, and Sbordone:
Analisi Matematica Due, Liguori.

Additional exercises can be found in

Marcellini, Sbordone:
Esercitazioni di Matematica, vol. 2, parte prima e parte seconda, Liguori.

Lecture notes are available at

http://people.unica.it/antoniogreco/didattica/materiale-didattico/

Further reading:

Barutello, Conti, Ferrario, Terracini, Verzini:
Analisi Matematica 2, Apogeo.

Giusti:
Analisi Matematica, vol. 2, Boringhieri.

Kline:
Mathematical Thought from Ancient to Modern Times, vol. 1 and vol. 2, Oxford University Press.

Pagani, Salsa:
Analisi Matematica, vol. 2, Masson/Zanichelli.

Rudin:
Principles of Mathematical Analysis, McGraw-Hill.

More Information

For information on specific learning disabilities (SLD) see http://corsi.unica.it/matematica/info-dsa/