Learning objectives
<br />The purpose of this course is to provide a rigorous outline of the <br />calculus of functions of several real variables.
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. Basic <br />topological notions in euclidean spaces. Normed vector spaces. Metric spaces. <br />Basic topological notions in metric spaces. Compactness, connectedness. <br />Equivalent metrics and equivalent norms. Limits and continuity of functions. <br />Lipschitz condition. Sequences and series in normed spaces. Completeness, <br />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. <br />Differential forms and vector fileds. Line integrals. <br />Closed and exact differential forms. On a star-shaped domain, all closed <br />forms are exact. <br /><br />
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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