FUNCTIONS WITH MULTIPLE VARIABLES A
cod. 13467

Academic year 2007/08
2° year of course - First 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 />The purpose of this course is to provide a rigorous outline of the <br />calculus of functions of several real variables. <br /><br />

Prerequisites

Funzioni di una variabile A<br />Funzioni di una variabile B

Course unit content

<br />Metric spaces and normed vector spaces.<br />Euclidean inner product. Cauchy-Schwarz inequality and euclidean norm. Basictopological notions in euclidean spaces. Normed vector spaces. Metric spaces.Basic topological notions in metric spaces. Compactness, connectedness.Equivalent metrics and equivalent norms. Limits and continuity of functions.Lipschitz condition. Sequences and series in normed spaces. Completeness,Banach spaces. Spaces of continuous functions, uniform norm, uniform convergence.Linear transformations, operators norm. Fixed point theorem for contraction mappings.Applications and examples of the fixed point theorem. Neumann series. <br /><br /><br />Differential calculus for functions of several variables. <br />Directional and partial derivatives. Differentiable functions. Composition, the chain rule.Differential of the inverse function. Curves. Mean value theorem. <br />Functions of class C^1. Higher order derivatives, mixed derivatives.  <br />Taylor's theorem. Relative extrema for functions of several varibles.  <br />The inverse function theorem. The implicit function theorem. <br />Constrained extrema, the Lagrange multiplier rule. Differential forms and vector fileds. Line integrals.Closed and exact differential forms.  On a star-shaped domain, all closed <br />forms are exact. 

Full programme

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Bibliography

<br />G. Prodi, Lezioni di Analisi Matematica II, ETS Editrice;<br />W. Rudin, Principi di Analisi matematica, McGraw-Hill. <br />E. Giusti, Analisi Matematica 2, Boringhieri

Teaching methods

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Assessment methods and criteria

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Other information

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