Learning outcomes of the course unit
Knowledge and understanding: At the end of the course, the student will have to know the definitions and the fundamental results concerning sequences, series of functions and in particular power series, ordinary differential equations (ODE), and will have to be able to understand how these go into problems resolution.
Ability to apply knowledge and understanding: Thanks to the exercises carried out in the classroom, the student will learn how to apply the theoretical knowledge acquired to the solution of concrete problems. Furthermore, the student must be able to apply the acquired knowledge to produce rigorous demonstrations of mathematical results not identical to those already known but clearly related to them, for the resolution of problems even on average elaborate ones, and to understand the relationships with the material learned in others courses, as well as formulating problems in mathematical form for their analysis and resolution.
Autonomy of judgment: The student must be able to evaluate the consistency and correctness of the demonstrations produced during the written exam, constructing and developing logical arguments with a clear identification of assumptions and conclusions; he will have to recognize correct proofs and identify fallacious reasoning.
Communication skills. The student must be able to communicate in a clear and precise manner, appropriate for an intermediate-stage mathematician, mathematical contents related to the program carried out. The lectures and direct confrontation with the teacher will favor the acquisition of a specific and appropriate scientific lexicon.
Learning ability. After having attended the course, the student will be able to autonomously deepen his knowledge in the areas covered, starting from the basic and fundamental knowledge provided by the course. He / she will be able to autonomously consult specialized texts, even outside the topics covered in detail during the lessons, in order to effectively address complex problems inherent to these topics.
Calculus for functions of one real variable; linear algebra; differential and integral calculus in several variables.
Course contents summary
Sequences and function series. Ordinary Differential Equations
- Sequences and series of functions
Function sequences: pointwise and uniform convergence. Integral limit interchange. Function and power series: radius of convergence; uniform convergence, continuity and integration. -
The Cauchy problem. Regularity. Existence and uniqueness of solutions. Continuous dependence on data. Integration of some ordinary differential equations. Systems of First Order Linear Differential Equations. Qualitative behavior of solutions of ODE’s.
E. Acerbi, G. Buttazzo: Secondo corso di Analisi Matematica. Pitagora, Bologna, 2016.
Also, one may use any other good book on Analysis in several variables, as, e.g.,
N. Fusco, P. Marcellini, C. Sbordone: Lezioni di analisi matematica due, Zanichelli Matematica, Bologna, 2020.
W. Fleming: Functions of several variables. Second edition. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1977.
Lectures are held in online form, using the Teams program (see the Elly page of the course for more details and links), encompassing both theoretical and applied aspects.
Moreover, exercises are solved by students with the guidance of the teacher, so as to verify the degree of comprehension and knowledge of the students.
Assessment methods and criteria
The final exam consists of a written and an oral session. Books, lecture notes, calculators, etc... are not allowed. Students are admitted to the oral sessions only if they pass the written examination (with a mark greater or equal than 15/30). In the written examination some open questions are asked (usually in 2 or 3 groups within 1,5/2 hours).
The students should exhibit calculus skills and mastery of different subjects taught in the course. Marks are given to each question, according to theoretical correctness, precision of execution, precision of exposition.
The oral examination consists of a discussion (to be given during the same exams session) about the written examination and of questions to verify the level of comprehension of the theoretical parts of the course. The oral and written scores are averaged to give the student the overall result for the exam.