METODI MATEMATICI APPLICATI ALLA FISICA

Differential and integral calculus for functions of several variables
Functions of several variables: limits and continuity - Differentiation of functions of several variables: partial derivative and directional - Differential and differentiable functions - higher order derivatives and lemma Schwartz - differential operators: gradient, divergence, curl, Laplace - implicit functions - bound and free ends of a function of several variables. integral calculus for functions of one variable: Peano-Jordan measure and Lebesgue measure - Riemann integral - indefinite integral - fundamental theorem of calculus - improper integrals. integral calculus for functions of several variables: double and triple integrals - change of variables - Reduction formulas - integrals depending on a parameter: rule Leibinz. Notes on line integrals and surface: linear and quadratic differential forms - divergence theorem - Theorem Stokes - Green identity.
Numerical Serie and series of functions
Numerical series - general theorems on numerical series - various examples of series - the convergence criteria of the positive series - series for alternating and criterion Leibnitz - absolutely convergent series. Series of functions - Pointwise and uniform convergence - Taylor series and Mac Laurin - development of Mac Laurin of some elementary functions - power series. Multipole expansion of the potential of the Newtonian type - Legendre polynomials.
Elementi of Fourier analysis
Fourier Series - convergence of the Fourier series - uniqueness theorem - Examples and applications of Fourier series - transformed and its fundamental properties - transform of the convolution of functions - Laplace transform as a special case of the Fourier transform - Some Fourier and of considerable Laplace.
Ordinary differential equations (ODE)
General information on differential equations - the Cauchy problem - differential equations of the first order - the first order differential equations with separable variables - Cauchy's theorem on the existence and uniqueness of the solution - linear equations of the first order and second order - applications to physics: free oscillations, damped and forced.
Fundamental equations of the theory of elasticity
Volume and surface forces - Efforts and deformations: elastic moduli - stress tensor - tensor of solid deformation - relationship between the stresses and strains: the law of Hooke- the equation of motion of elastic solids - Waves longitudinal and transverse in solids - Waves in fluids.
Differential equazions PDE (PDE)
General information on partial differential equations - linear second order PDE and their classification - Laplace equation and Poisson: theorem of uniqueness - Formula Green -Functions harmonics and their property - The mean value theorem - - Potential masses extended in space. wave equation: D'Alembert solution - equation of vibrating strings: endless rope and over - Fourier method. Heat equation: the principal-solution of the Cauchy - unlimited and limited Sbarra problem - Solution by Laplace transform - Numerical methods for PDE solution.

The italicized topics covered in the discussion as optional studies.

A. Avantaggiati Institutions of Mathematics, C.E.A.

M. Bramanti, C.D. Pagani, S. Salsa, Mathematical Analysis Vol. 1 and 2, Zanichelli

Guido Cosenza: Mathematical methods of physics, Bollati Basic Books

Giampaolo Cicogna: Mathematical methods of physics, Springer