Teacher

prof. aggr. Maria Antonietta LEPELLERE

Credits

9 CFU

Objectives

Foundations in Mathematical Analysis; maturity in the rational argument, consciousness and ability in Calculus, in formulation and resolution of problems; education of a scientific and rational mind structure, critic and creative, able to modelize situations and phenomena with the required rigor.

Acquired skills:

- acquisition of the concept of limit, directional derivative, gradient and differential, their properties and applications;
- to be able to solve free and constrained optimization problems;
- to be able to handle with the properties of implicit functions and differentiable manifolds;
- knowledge of multiple integrals and of integration methods;
- knowledge of vector fields and their integral and differential relations;
- knowledge of various techniques for the treatment of the differential equations and systems of differential equations.

Contents

Functional Analysis. Metric spaces and normed spaces. Complete metric spaces and Banach spaces. Limits, continuity and contractions. Fixed point theorem of Banach-Cacciopoli and applications. Stability of contractions and their use in numerical calculation. Norm of a linear operator. Vector spaces with scalar product and Hilbert spaces. Trigonometric series and Fourier series. Pointwise convergence of Fourier series and Gibbs phenomenon. (15 hours)

Differential calculus. Real functions of n variables: graphs and level sets (the case n = 2). Limits and continuity for real functions of several variables. Partial derivatives, differentiable functions. The gradient vector. Relationship between differentiability and continuity. Directional derivatives. Tangent plane to the graph of a function of two variables. Differentiable function. The formula for the gradient. Directions of maximum and minimum growth of a function. ETeorema successive derivatives of Schwarz. Taylor's formula to the second order. Second differential. The Hessian matrix. Theorem of Fermat. Study of the nature of the critical points. Implicitly defined functions and the theorem of Dini. Extreme bound. The method of Lagrange multipliers. Extreme bound with inequality constraints: Kuhn-Tucker theorem. Convex functions. Envelopes to one or two parameters. (20 hours)

Differential Equations and Systems of Differential Equations. Cauchy problem. Integral equation of Volterra. Theorem of existence and uniqueness of the Cauchy problem. Theorem of Peano. Prolungability of solutions and Theorem of existence and uniqueness. Only existence theorem in big. Study of systems of linear equations. Solution space. Exponential matrix and its properties. Use of the exponential matrix to solve the homogeneous systems of linear equations. Wronskian matrix and determinant. Inhomogeneous systems. Calculation of particular solution of inhomogeneous, method of similarity and variation of parameters. Study of stability of autonomous systems. Nonlinear dynamical systems: linearisation, stability and Asymptotic stability of equilibrium points. Applications to mechanical vibrations (or linear), vibration damped, damped oscillations, resonance and beats.

Outline of linear partial differential equations: the equation of the vibrating string, the warmth equation, the Laplace equation. Method of separation of variables. (20 hours)

Integral calculus. Measure and Lebesgue integral. Elementary sets and their extent. Misurability of open sets of limited and closed limited sets. Misurability of bounded sets. Measurable functions and properties. Lebesgue integral. Dominated Convergence Theorem of Lebesgue. Theorem of Beppo-Levi. Reduction Fubini theorem. Radon-Nikodym theorem and Change of Variables. Polar coordinates in the plane, cylindrical, spherical and toric coordinates in the space. (20 hours)

Vector fields. Differential and integral calculus of curves. Length of a curve. Changes of parameters. Line integrals. Field lines. Gradient, divergence and rotor. Line integral of a vector field. Labour and circuitry. Conservative fields and potentials. Irrotational fields. Simply connected sets. Differential Forms. Formula of Gauss Green in the plan. Area and surface integrals. Surface Integral  of a vector field. Flow. Divergence theorem. Theorem of Stokes. (15 hours)

References:

- M. Bramanti, C. Pagani, S. Salsa, Analisi Matematica 2, Zanichelli

- Educational materials provided by the teacher

Exams

Written and oral