35354 - Mathematical Physics and Statistical Processing of Observations M

Course contents

Continuum Mechanics

Linear Algebra and Tensors

Linear operators, matrices and representation of a linear operator on an assigned basis, transpose operators, product between two operators; identity operator; adjugate operator; inverse operator; some special relations of linear algebra in dimension 3; Levi-Civita symbol; scalar product between operators; trace of an operator; skew-symmetric and symmetric operators; dual vector associated to a skew-symmetric operator; operators expressed as a sum of a symmetric and a skew-symmetric operators; rotation operators and their properties; characteristic polynomial associated to an operator; tensorial product and its properties; eigenvalues and eigenvectors of an operator; similarity transformation; diagonalization of a symmetric matrix: principal invariants of a matrix; positive and negative definite matrices; Sylvester theorem, Hamilton-Cayley theorem, polar decomposition theorem.

Deformation and kinematics in continuum mechanics

Deformation gradient tensor;  the Cauchy-Green deformation tensor and the Green-Saint Venant deformation tensor;  angular deformation, linear, superficial and volumetric coefficient of dilatation; eulerian and lagrangian reference frame ; eulerian and lagrangian velocity and acceleration; tensor of the deformation velocity; vorticity tensor; gradient of velocity in a rigid motion.

  

Forces and stress tensor

Body forces, superficial forces, contact forces and specific stress, Cauchy theorem and stress tensor, symmetry of the stress tensor, boundary conditions.

Balance and conservation laws

Gauss-Green  and  transport theorems, balance laws, continuity equation; momentum balance law, angular momentum balance law, virtual work principle and power of the internal forces, mathematical structure of balance laws in eulerian and lagrangian variables; first and second Piola-Kirchoff stress tensors; Galilean invariance

Constitutive Equations

General principles of modern theory of constitutive equations, principle of material objectivity, entropy principle; some examples: Thermoelastic and elastic bodies,  Fourier-Navier-Stokes fluids,  ideal  fluids and Bernoulli theorem, ideal incompressible fluids and static solutions.

Hyperbolic systems and non-linear wave propagation

Hyperbolicity: definitions, properties and examples; linear  hyperbolic equations,  methods of characteristics and wave equation; examples of elasticity and Euler fluids. Burger equation and non-linear hyperbolic systems, Riemann problem, classical and weak solutions, shock and rarefaction waves, mathematical model of traffic and solution of the traffic light problem.

Probability theory  and Statistics

Probability theory

Definition and main properties: random variables and vectors of random variables; mean, variance, moments and covariance matrix; independence of random variables.Models of random variables: normal random variables; normal multivariate random variables; linear combination of normal random variables; Chi-square distribution; Student-t distribution, Fisher distribution. Limit laws in probability theory: sequences of random variables and convergence; weak law of large numbers; central limit theorem.

Statistics and statistical inference

Sampling; sample mean and sample variance, estimators; confidence regions for mean and variance of a normal population; efficiency of the estimators; hypothesis testing; linear regression: estimate of the regression parameters, distribution of the estimators and statistical inference of the regression parameters.

Readings/Bibliography

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

Office hours

See the website of Tommaso Ruggeri