FUNCTIONS WITH MULTIPLE VARIABLES A
Learning outcomes of the course unit
The purpose of this course is to provide a rigorous outline of the
calculus of functions of several real variables.
Funzioni di una variabile A
Funzioni di una variabile B
Course contents summary
Metric spaces and normed vector spaces.
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.
Differential calculus for functions of several variables.
Directional and partial derivatives. Differentiable functions. Composition, the chain rule.Differential of the inverse function. Curves. Mean value theorem.
Functions of class C^1. Higher order derivatives, mixed derivatives.
Taylor's theorem. Relative extrema for functions of several varibles.
The inverse function theorem. The implicit function theorem.
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
forms are exact.
G. Prodi, Lezioni di Analisi Matematica II, ETS Editrice;
W. Rudin, Principi di Analisi matematica, McGraw-Hill.
E. Giusti, Analisi Matematica 2, Boringhieri