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Second Semester 
Teaching style
Lingua Insegnamento

Informazioni aggiuntive

Course Curriculum CFU Length(h)
[60/64]  MATHEMATICS [64/00 - Ord. 2017]  PERCORSO COMUNE 10 80


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 establishing the pointwise or uniform convergence of a sequence of functions. 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 judgments

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

Being able to consult and interpret textbooks to build and expand one's own knowledge. Being able to use the sources of information as a tool in problem-solving activities. Being able, if necessary, to use the sources of information in order to go slightly beyond a strict interpretation of the course boundaries.


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


Introduction into the theory of ordinary differential equations.

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.

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.

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.

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

The course is composed of 80 lecture hours.

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 subsequent lectures are organized as follows: one part of the activity consists of an exposition of the course contents aimed at achieving objective 1 (knowledge and understanding); a second part consists in the description of the algorithmic procedures related to objective 2 (applying knowledge and understanding). Students are frequently invited to tackle mathematical problems proposed by the lecturer with specific reference to the course contents, the material developed in the preceding lectures, the experience achieved in the preceding editions of the course, and the students feedback. Problems are aimed at the following targets:

a) supporting the achievement of objective 1 (knowledge and understanding) by means of an explicit contextualization of the theoretical notions;

b) supporting the achievement of objective 2 (applying knowledge and understanding) by an effective usage of the algorithmic procedures;

c) supporting the achievement of objective 3 (making judgements): indeed, in order to solve the proposed problems, students must take rational and autonomous choices;

d) supporting the achievement of objective 5 (learning skills): indeed, students are specifically invited to find help in the suggested textbooks.

The third kind of activity consists in the discussion, led by the lecturer, and in the resolution of the suggested problems, with active participation of the interested students, having the following purposes:

1) confirm knowledge (objective 1) and applicative abilities (objective 2);

2) improve the ability in making judgments (objective 3) by comparing one's own ideas with those of other students and the lecturer;

3) improve communication skills (objective 4) .

It may well happen that two or three of the different activities described so far take place within the same hour.

Lectures are always aimed at discourage a purely syntactic-symbolic (mnemonic) attitude, and are frequently supported by graphical representations (partly obtained by hand, and partly created by specific software). CAS (Computer Algebra Systems) are also used to symbolically solve ODEs and to get the Fourier expansion of some functions.

Students are allowed to ask for a deeper exposition of specific subjects (related to the course contents). If necessary, students may also ask for a presentation of some prerequisite.

Anonymous means of communication with the lecturer (in written form and/or by means of interactive presentation software) took place in order to keep the subsequent lectures always in tune with the students needs.

The lecturer is always available by e-mail (greco@unica.it)

Verification of learning

On the occasion of the final evaluation, the student is asked some questions chosen in the realm of the course contents. Questions are usually of the same kind and the same level of difficulty as those asked in class and answered and discussed in the subsequent lectures. A collection of model-questions is available at http://people.unica.it/antoniogreco/materiale-didattico/. The student must answer in a mixed oral/written form, making use of a blackboard in order to:

A. Support the complex arguments that are typical of the mathematical thinking by means of the conventional symbols as well as with graphical representations (when possible).

B. Show the ability to manipulate formal expressions by a correct application of the governing rules.

The final score ranges from 18/30 to 30/30 cum laude. Ability in the computation of limits and in the differential calculus, knowledge of the graphs of the most elementary functions, and in some minor respect the ability in the computation of simple integrals and in the determination of the character of simple numerical series are necessary conditions to pass the exam, thus getting a final score not less than 18/30. A more thorough achievement of the educational goals listed above corresponds to a proportionally higher score. Particularly brilliant students may be required to solve more involved problems, corresponding to the highest cognitive level of Bloom's taxonomy, in order to get the maximum final score of 30/30 cum laude.

The final score takes the following elements into account:

1. The underlying logic in the solution of the proposed problem;

2. The correctness of the adopted procedure;

3. The correspondence of the proposed solution with the expected abilities;

4. The usage of a convenient language, and in particular the correct usage of terms such as "if", "that is to say" and "then";

5. The ability to take part into a discussion, and the ability to locate and correct possible mistakes without getting lost.

Initiative, critical and personal thinking are appreciated, as well as the creative usage of the student's knowledge.

Keep in mind that students differs very much one from the other because each student is characterized by a peculiar combination of the elements listed above. By contrast, the final score consists of a single numerical value and therefore cannot maintain any memory of each concurring element. Consequently, very different students may well achieve an identical score: to make an example, think for instance about a very careful student (element no. 1) which is shy and uncertain when relating to other people (element no. 5), to be compared to an impulsive student, very brilliant and clear in supporting her claims.


The lectures are based on the textbook by

Fusco, Marcellini, and Sbordone:
Lezioni di Analisi Matematica Due, Zanichelli.

The textbook is equivalent to:

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


Further reading:

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

Analisi Matematica, vol. 2, Boringhieri.

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

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

Principles of Mathematical Analysis, McGraw-Hill.

More Information

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

Additional material is available at http://people.unica.it/antoniogreco/materiale-didattico/

The present report conforms to the Regolamento didattico and to the SUA. An effort was made to comply with the recommendations of the DISCENTIA project and the guidelines of PQA (the Committee for Quality Control). Educational goals and criteria for final evaluation are specified, respectively, on the basis of the Dublin descriptors and Bloom's taxonomy.


Regolamento didattico: http://corsi.unica.it/matematica/regolamenti/laurea-triennale/
Scheda SUA: http://corsi.unica.it/matematica/assicurazione-della-qualita/documenti-sua-e-riesame-cds/
DISCENTIA project: http://sites.unica.it/qualita/discentia/
PQA: http://people.unica.it/pqa/
Bloom's taxonomy: https://cft.vanderbilt.edu/guides-sub-pages/blooms-taxonomy/
Dublin descriptors, in short: http://www.quadrodeititoli.it/descrittori.aspx?descr=172&IDL=2
More on the Dublin descriptors is found in the original paper "A Framework for Qualifications of the European Higher Education Area" by the Bologna Working Group (2005) available online at ECA (European Consortium for Accreditation): http://ecahe.eu/ On the same site see also "The Origin of the Dublin Descriptors" by Dr. M. Leegwater

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