FUNCTIONS WITH MULTIPLE VARIABLES B
Learning outcomes of the course unit
Provide basic knowledge of Mathematical Analysis
Functions of one variable B
Course contents summary
Pointwise convergence, uniform convergence, Cauchy criterion, limit-swap theorem for a uniformly convergent sequence, relations between
convergence and continuity, derivability, integrability, completeness
of spaces of continuous functions and bounded functions, Ascoli-Arzela`
theorem, Bernstein polynomials, density of polynomials in
C([a,b]), series of functions, total convergence, series of powers,
convergence radius, Cauchy-Hadamard theorem, Abel's
theorem, trigonometric polynomials, approximation of a function with
trigonometric polynomials, Fourier coefficients, Bessel's
inequality, pointwise and total convergence of the Fourier series,
integration of the Fourier series.
Normal form, systems, Cauchy problem, Cauchy-Lipschitz theorem, Gronwall's lemma, sufficient conditions for global existence, method of separation of
variables, qualitative study of solutions, comparison
theorem, systems of linear differential equations of the first
order in normal form, linearly independent solutions of
homogeneous systems, evolution operator, method of variation
of constants, linear equations of order n and associated
system, characteristic polynomials of constant coefficient
equations, general solution, exponential of a matrix.
Peano-Jordan measure theory in R^n and multiple integrals, integral reduction theorem, change of variables in integrals theorem, Gauss-Green formulae in dimension 2 and their consequences, divergence theorem.
G. Prodi, Lezioni di Analisi Matematica. Parte II, ETS PISA;
W. Rudin, Principi di Analisi Matematica, MCGRAW-HILL.
Teaching methods: classroom lectures and classroom exercises
Assessment method: the exam consists of a written test and an oral test.