FUNCTIONS WITH MULTIPLE VARIABLES B
cod. 13472

Academic year 2008/09
2° year of course - Second semester
Professor
Academic discipline
Analisi matematica (MAT/05)
Field
Formazione analitica
Type of training activity
Characterising
60 hours
of face-to-face activities
7 credits
hub:
course unit
in - - -

Learning objectives

# <br />
Provide basic knowledge of Mathematical Analysis

Prerequisites

Functions of one variable B

Course unit content

<br />Pointwise convergence, uniform convergence, Cauchy criterion, limit-swap theorem for a uniformly convergent sequence, relations between <br />
convergence and continuity, derivability, integrability, completeness <br />
of spaces of continuous functions and bounded functions, Ascoli-Arzela` <br />
theorem, Bernstein polynomials, density of polynomials in <br />
C([a,b]), series of functions, total convergence, series of powers, <br />
convergence radius, Cauchy-Hadamard theorem, Abel's <br />
theorem, trigonometric polynomials, approximation of a function with <br />
trigonometric polynomials, Fourier coefficients, Bessel's <br />
inequality, pointwise and total convergence of the Fourier series, <br />
integration of the Fourier series. <br />
<br />
Differential equations. <br />
<br />
Normal form, systems, Cauchy problem, Cauchy-Lipschitz theorem, Gronwall's lemma, sufficient conditions for global existence, method of separation of <br />
variables, qualitative study of solutions, comparison <br />
theorem, systems of linear differential equations of the first <br />
order in normal form, linearly independent solutions of <br />
homogeneous systems, evolution operator, method of variation <br />
of constants, linear equations of order n and associated <br />
system, characteristic polynomials of constant coefficient <br />
equations, general solution, exponential of a matrix. <br />
<br />
Integrals: <br />
<br />
Peano-Jordan measure theory in R^n and multiple integrals, integral reduction theorem, change of variables in integrals theorem, Gauss-Green formulae <br />
in dimension 2 and their consequences, divergence theorem. <br />
<br />
 

Full programme

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Bibliography

G. Prodi, Lezioni di Analisi Matematica. Parte II, ETS PISA; <br />
<br />
W. Rudin, Principi di Analisi Matematica, MCGRAW-HILL.

Teaching methods

Teaching methods: classroom lectures and classroom exercises <br />
<br />
Methods of assessment: the examination consists of a written test and an oral test.

Assessment methods and criteria

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Other information

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