Aim of the course is to provide an introduction to the study of the main equations of mathematical physics, using almost exclusively classical tools of mathematical analysis.

Lectures

Oral exam

Differential and integral calculus in multiple dimensions. Elements of classical mechanics.

Reminders on vectorial calculus, gradient, curl and divergence. Divergence theorem. Stokes's theorem. Green's formuals. Orthogonal curvilinear coordinate systems. Transport equations. Partial differential equations of the second order. Classification. Elliptic equations. Laplace equation, the mean value theorem, the maximum principle. Introduction to complex analysis (analytic functions, Cauchy-Riemann formulas). Dirichlet and Neumann problems for the circle. Parabolic equations. Heat diffusion. Exact solutions and the method of similarity. Heat diffusion: resolution of the Cauchy problem using the one-dimensional Fourier method. Initial-boundary value problem for the heat equation: the method of separation of variables. Hyperbolic equations. The wave equation. Vibrations of membranes. Introduction to the mechanics of fluids.

Enrico Persico, INTRODUZIONE ALLA FISICA MATEMATICA, Bologna : Zanichelli, 1971, third ed.