Portale di Ateneo - En.Unibs.it

Differential calculus for functions of several variables

Limits, continuity, partial and directional derivatives, differentiability. Extremal points, classification of stationary points, test by Hessian matrix and Hessian determinant. Lagrange multipliers.

Curves and line integrals

Definition of curve parametrization, length, arc length. Vector-valued functions of several variables. Curvilinear integrals of the first and second species. Vector-valued functions, potentials, gradients.

Integration and differentiation of vector-valued functions

Differentiability of vector fields; derivation of composite functions. Multiple integrals: definitions, reduction formulas, change of variables. Gauss-Green formula in the plan. Operators rotor, gradient and divergence. Surfaces, area of a surface, surface integrals. Stokes' theorem. Gauss' divergence theorem.

Sequences and series of functions

Sequences: pointwise and uniform convergence. Series of functions: pointwise, uniform, total convergence; derivation and integration term by term. Power series and Taylor series expansions.

Fourier series
Trigonometric series. Fourier series: definition, mean convergence, pointwise and uniform convergence.

Ordinary Differential Equations
The Cauchy problem for first order equations, local and global solutions. Differential equations of higher order.