Learning objectives
The object is to supply the students with knowledge and understanding of sequences and series of functions, ordinary differential equations and systems.
Amain aim is to give them the competence to treat and apply these instruments.
Objects of the course are also : judgement independence, strong written and oral communication skills, learning ability, in accordance with the specific objects of the mathematics degree.
Prerequisites
Analisi matematica I e Analisi Matematica 2A
Course unit content
Sequences and series of functions.
Ordinary differential equations and systems.
Full programme
Sequences and series of functions. Pointwise and uniform convergence, total convergence of series of functions. Theorems about exchange of limits under uniform convergence, convergence and continuity, convergence and derivatives, convergence and integrals. Dini's theorem about monotonic convergence.
Power series and Taylor series: convergence radius and its characterizations. Trigonometrical series: Fourier coefficients, Bessel inequality, pointwise and total convergence.
The sup norm in spaces of bounded functions. Density of polynomials in C([a,b]).
Ordinary differential equations (ODEs) and systems of ODEs. The Cauchy problem. Local existence and uniqueness theorem. Extensions of the solutions, and conditions for existence in the large. Qualitative study of solutions and continuous dependence on the initial data. Solutions of some type of first order ODEs. Linear equations and systems, variation of constants formula for non-homogeneous problems. Evolution operators. Linear equations and systems with constant coefficients, exponential of a square matrix.
Bibliography
M. Fusco, P:Marcellini, C. Sbordone, Analisi Matematica due, ed. Liguori.
M. Bramanti, C.D. Pagani, S.Salsa, Analisi matematica 2, ed. Zanichelli.
G. Prodi, Lezioni di Analisi Matematica II, ed. Tecnico Scientifica.
Teaching methods
The teaching consists in frontal lessons where both theoretical and applicable aspects are expounded. The exercises are worked out with the collaboration of the students and are programmed in order that students can solve autonomously the problems arising from the theoretical lessons.
Assessment methods and criteria
The final exam consists in a written test followed by an oral discussion. In both the tests the student has to demonstrate knowledge, comprehension and to be able to connect knowledge and comprehension about the subjects of the course.
The written test consists in some open questions.
The marks, ranging from 0 to 30, will be attributed considering the accuracy of the exposition and the operating methods.
The test will be considered sufficient if the average of the total amount of the written and oral tests is greater or equal to 18.
Other information
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2030 agenda goals for sustainable development
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