# 72760 - Mathematical Physics And Statistical Processing Of Observations M

## Learning outcomes

With the acquisition of the contents of this Course, the student has the knowledge of the thermo - mechanical of continuous media with particular regarding to the modern theory of constitutive equations. He also has basic knowledge of partial differential systems of nonlinear hyperbolic type and is capable of handling the statistical data of the observations .

## Course contents

PRIORI KNOWLEDGE

A prior knowledge and understanding of differential and integral calculus , the basics of Euclidean geometry , linear algebra and Rational Mechanics s is required to attend with profit this course.

In addition, students should master the topics of Analysis, Geometry and Rational Mechanics.

Fluent spoken and written Italian is a necessary pre-requisite: all lectures and tutorials, and all study material will be in Italian

COURSECONTENTS

Elements of Linear Algebra:
matrix operators; representation of an operator in an assigned basis; Operator transposed; product of two operators; operator identity '; complementary operator; inverse operator; identity 'remarkable matrix computations; Levi-Civita symbol; scalar product between operators; trace of an operator; symmetric and antisymmetric operators; dual vector associated with an antisymmetric operator; expression of an operator as the sum of a symmetric and an antisymmetric operator; Rotation operators and property; characteristic polynomial of an operator; tensor product and property '; eigenvalues and eigenvectors of an operator; similarity transformations; diagonalization of a symmetric matrix; principal invariants of a matrix; operators defined of sign; Sylvester theorem; Cayley-Hamilton theorem; polar theorem.

Deformation and kinematics in continuum mechanics
Deformation gradient operator; Deformation operators of Cauchy-Green and Green-Saint Venant; Eulerian and Lagrangian points of view;

Balance equations and conservation laws.

Gauss-Green; transport theorem; balance equations and conservation laws; classical and weak solutions solutions; continuity equation '; Momentum equation; Cauchy Theorem and stress tensor; symmetry of the stress tensor; boundary conditions; principle of virtual work and power of the internal forces; Lagrangian formulation of the balance equations; Piola-Kirchhoff first and second tensor; Galilean invariance.

Theory of constitutive equations

Introductory general principles and considerations; principle of material indifference; entropy principle; Examples: thermoelastic bodies, perfect fluids and Bernoulli's theorem, perfect incompressible fluids, fluids Navier-Stokes-Fourier, non-Newtonian fluids. The principle of entropy restrictions
Restrictions of the entropy principle in the case elasticity 'non-linear and in the case of Newtonian fluids.

Heat rigid conductor
Fourier heat equation for a rigid conductor, Paradox of speed 'instant, Maxwell Cattaneo equation.

hyperbolic systems of wave propagation

linear systems, quasi-linear, semi-linear; classification of partial differential equations; wave equation; the problem of the vibrating string; hyperbolic systems and characteristic velocities; strictly hyperbolic systems; method of characteristics; the Riemann problem; shock waves and rarefaction; the problem of traffic cars;

Statistical Observations of treatment
The foundations of probability '; continuous random variables; numerical characteristics of a random variable (mean, mode, median, variance, skewness, kurtosis); the laws of distribution, normal distribution and its properties'; distributions chi-square, Student and Fisher. The fundamentals of statistics. Sampling distributions; estimation theory; optimum of an estimator; the method of least squares; confidence intervals for the mean and variance in the case of normal variables.

Tommaso Ruggeri, Introduzione alla Termomeccanica dei Continui, Ed. Monduzzi, Bologna;

Lectures

## Assessment methods

The exam is done through a final exam, which ensures the acquisition of knowledge and skills expected by an oral exam.

The oral exam that consists of 3 theory questions that the student answers by writing what he knows in a sheet of paper (20 min). Immediately after the student discusses what is written and integrates with the teacher the questions (about 20 min).

To obtain a passing grade, students are required to at least demonstrate a knowledge of the key concepts of the subject

Higher grades will be awarded to students who demonstrate an organic understanding of the subject, a high ability for critical application, and a clear and concise presentation of the contents