Academic Year 2018/2019 - 1° Year

FIS/01 - 6 CFU - 1° Semester

Adequate knowledge and skills in the field of mathematical methods applied to physics, as a tool for the treatment and modeling of geophysical problems.

lectures accompanied by exercises

**Differential and integral calculus for functions of several variables**

Functions of several variables: limits and continuity - Differentiation of functions of several variables: partial and directional derivates - Differential and differentiable functions - Higher order derivatives and lemma Schwartz - Differential operators: gradient, divergence, curl, laplacian - Implicit functions - Bound and free maximum and minimum of several variable functions - Integral calculus for functions of one variable: Peano-Jordan measure and Lebesgue measure - Riemann integral - Indefinite integral - Fundamental theorem of calculus - Improper integral - Integral calculus for functions of several variables: double and triple integrals - Change of variables - Reduction formulas - Integrals depending on a parameter: Leibinz rule - 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 Leibnitz criterion - AQbsolutely convergent series - Series of functions - Pointwise and uniform convergence - Taylor series and Mac Laurin - Mac Laurin expansion of some elementary functions - Power series - Multipole expansion of Newtonian type potentials - 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 - Physical applications: 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 sheet 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