ECTS - University of Pisa
| Modules | Area | Type | Hours | Teacher(s) | |
| MATHEMATICAL PHYSICS FOR GEOSCIENCES | FIS/03 | LEZIONI | 48 |
|
Alla fine del corso lo studente sarà in grado di dimostrare una conoscenza generale di: i) formalismo matematico delle funzioni a più variabili; ii) fondamenti di analisi tensoriale; iii) equazioni differenziali ordinarie e alle derivate parziali.
The student who successfully completes the course will be able to demonstrate general knowledge of: i) the mathematical formalism of multi-variable functions; ii) tensor analysis and iii) ordinary and partial differential equations.
Nell'esame finale allo studente verrà chiesto di risolvere semplici esercizi e/o di discutere gli argomenti presentati a lezione.
During the final exam, the student will be asked to solve simple exercises, or to discuss topics presented during the lectures.
Alla fine del corso lo studente sarà in grado di applicazione il formalismo matematico insegnato a lezione in vari contesti quali elettrostatica, euqaioni d'onda e meccanica del continuo.
The student who successfully completes the course will be able to apply the mathematical formalism to various physical contexts such as electrostatics, waves and continuum mechanics.
Durante l'esame finale, allo studente verrà richiesto di risolvere alcuni semplici esercizi e/o di discutere gli argomenti presentati a lezione.
During the final exam, the student will be asked to solve simple exercises, or to discuss topics presented during the lectures.
N/A
N/A
N/A
N/A
Conoscenza dell'analisi di base: calcolo differenziale ed integrale a singola variable.
Basic knowledge of mathematical analysis: differential and integral calculus of single-variable functions.
Lezioni frontali con qualche esercitazione pratica alla lavagna.
Teaching will mainly consist of frontal lectures, with few practical exercitations.
Part 1: Linear Algebra. Scalars, vectors, and tensors. Basic operations with vectors: quick reminder. Linear applications. Matrix algebra. Change of coordinates. Isometries. Tensors in physics: the case of the strain tensor. Visualization of the strain tensor. Invariants. Diagonalization of a matrix. Connection with the invariants. The vector product. The Levi-Civita tensor and the Kronecker delta. Flux through an area.
Part 2: Ordinary differential equations. Classical motion under gravity and friction. Laminar regime: Stokes drag. Rayleigh drag. The Malthus and Verhulst growth laws. The 1D harmonic oscillator. The 1D damped harmonic oscillator. The 1D undamped and forced harmonic oscillator. The method of variation of parameters. The method of undetermined coefficients. Beats. Resonant behavior. The 1D damped and forced harmonic oscillator. Constant force. Simple periodic forcing. The case of a general external force: the superposition principle. General periodic forcing: the Fourier series. Fully general forcing: the Fourier transform. The Green’s function.
Part 3: Partial differential equations. Prelude: multivariable calculus. Partial derivatives, gradients, and the Jacobian matrix. The multivariable chain rule. Extrema of multivariable functions: the Hessian and the Taylor expansion for multivariable functions. Curl and divergence. Stationary points of a function of two variables. Second-order Taylor expansion. Coulomb potential. Landau gauge. The 1D d’Alembert equation. The d’Alembert solution (free space and no external forces). Boundary conditions: separation of variables, normal modes, and standing waves.
Part 1: Linear Algebra. Scalars, vectors, and tensors. Basic operations with vectors: quick reminder. Linear applications. Matrix algebra. Change of coordinates. Isometries. Tensors in physics: the case of the strain tensor. Visualization of the strain tensor. Invariants. Diagonalization of a matrix. Connection with the invariants. The vector product. The Levi-Civita tensor and the Kronecker delta. Flux through an area.
Part 2: Ordinary differential equations. Classical motion under gravity and friction. Laminar regime: Stokes drag. Rayleigh drag. The Malthus and Verhulst growth laws. The 1D harmonic oscillator. The 1D damped harmonic oscillator. The 1D undamped and forced harmonic oscillator. The method of variation of parameters. The method of undetermined coefficients. Beats. Resonant behavior. The 1D damped and forced harmonic oscillator. Constant force. Simple periodic forcing. The case of a general external force: the superposition principle. General periodic forcing: the Fourier series. Fully general forcing: the Fourier transform. The Green’s function.
Part 3: Partial differential equations. Prelude: multivariable calculus. Partial derivatives, gradients, and the Jacobian matrix. The multivariable chain rule. Extrema of multivariable functions: the Hessian and the Taylor expansion for multivariable functions. Curl and divergence. Stationary points of a function of two variables. Second-order Taylor expansion. Coulomb potential. Landau gauge. The 1D d’Alembert equation. The d’Alembert solution (free space and no external forces). Boundary conditions: separation of variables, normal modes, and standing waves.
Testi consigliati
- N.S.Piskunov “Calcolo Differenziale ed Integrale” – 2010 – Editori Riuniti.
- J.Stewart “Essential calculus: early transcendentals” – 2012 – Brooks Cole.
- Ulteriore materiale didattico verrò fornito durante le lezioni.
Suggested readings
- N.S.Piskunov “Calcolo Differenziale ed Integrale” – 2010 – Editori Riuniti.
- J.Stewart “Essential calculus: early transcendentals” – 2012 – Brooks Cole.
- Further material will be provided during the lectures.
La frequenza non è obbligatoria, ma raccomandata.
Attendance is not mandatory, but recommended.
Esame orale.
Oral exam.
