Università Roma Tre

1. Functions of two or more variables: continuity, differentiability.

Algebraic structure, inner product in Rn, Cauchy–Schwartz inequality. Norm and metric in Rn. Convergent sequences. Open and closed sets, closure and boundary of a set. Open coverings, every open covering admits a countable sub-covering. Compactness: closet and bounded sets possess the finite covering property; characterization through sequences. Limits of functions, continuity; characterization through sequences. Weierstrass and Heine-Cantor theorems. Connectedness. Path-connected sets, the mean value theorem.
Parlai and directional derivatives. Linear functions, differentiability, gradient vector. Differentiability implies partial derivability (but not viceversa), C1 functions are differentiable. Geometric meaning of the differential, the gradient, the tangent plane.
The mean value theorem. Functions with vanishing gradient are locally costant. C1 functions are locally Lipschitz. Derivation along a differentiable path.
Higher-order derivatives, the Schwartz theorem. Hessian matrix. Symmetric matrices and associated quadratic forms: definitedness, semi-definitedness and the sign of the eigenvalues. Second-order Taylor formula. Local maximum and minimum points, necessary/sufficient conditions.
Vectorial functions. Continuity, differentiability, Jacobian matrix. The chain rule.

2. Integrals depending on a parameter

Dominatedness and continuous dependence, limit under integral sign. Dominatedness and derivability under integral sign. Applications: computation of the Dirichlet integral, the Gamma function, the Stirling formula. The convolution operation, Young inequality. Regularizing kernels, locally-uniform approximability of continuous functions through C∞ functions. The Weierstrass approximation theorem. Convolutione between 2π-periodic functions, uniform approximation of continuous 2π-periodic functions through trigonometric polynomials.

3. Metric spaces, the contraction mapping theorem

Metric spaces, open/closed sets, convergence, characterization of closed sets through sequences. Completeness. Banach spaces. C(K,Rn), endowed with the uniform convergence norm, is a Banach space. Density of trigonometric polynomials in the space of 2π-periodic functions endowed of the uniform convergence metric. Fourier coefficients and series, Bessel inequality. Approximability through Fourier series. Regularity and decay rate of Fourier coefficients.
Identity principle, a function with absolutely convergent Fourier series can be approximated by its Fourier series. Approximability of Fourier series for periodic and C1 functions.
Continuous functions between metric spaces, contractions. The contraction mapping theorem.
Systems of ordinary differential equations, existence and local uniqueness for the Cauchy problem. Continuous dependence by the data. Extension, global existence; systems of gradient type, conservative systems, Hamiltonian. Linear systems, fundamental matrix, Wronskian, the variation constant formula.