| Lectures and exercises |
hours |
| Topics |
Specific contents |
|
| Vector algebra |
Scalar and vector product. Kronecker and Levi-Civita symbols. Vector identities. |
3
|
| Vector calculus in three dimensions |
Scalar and vector fields. Directional derivative and gradient. Vector differential operators. Vector calculus identities. Derivation of the wave equation from Maxwell's equations. Gauss' and Stokes' theorems. Coordinate changes. Orthogonal curvilinear coordinates. |
8
|
| Linear systems with a finite number of degrees of freedom |
OHarmonic oscillator. Coupled oscillators. Normal modes and normal frequences. Systems of linear differential equations: Eigenvalues, eigenvectors, and general solution. System of N coupled oscillators. |
9
|
| Linear partial differential equations |
Continuum limit. Wave and diffusion equations. Linear second order partial differential equations with two independent variables: Classification and canonical forms. |
4
|
| Wave equation in one space dimension |
Separation of variables. Normal modes. Initial value problem and its general solution. Reflection. Resolution of problems with inhomogeneous boundary conditions. Well-posed problems. |
7
|
| Basic functional analysis |
Systems of orthonormal functions. Hilbert spaces and linear operators. Hermitian and self-adjoint operators. Eigenvalue problem. Sturm-Liouville problem. |
6
|
| Wave equation in three space dimensions |
Plane waves. Spherical waves. Separation of variables. Legendre polynomials and associated functions. Spherical harmonics. Bessel equation and Bessel functions. Definition and properties of the gamma function. |
8
|
| Distributions and Green's functions |
Basic distribution theory. Green's functions for linear differential equations. Causality. Dirac's delta in three dimensions. |
5
|
| Laplace equation |
Integral representation for the electrostatic potential. Green's function for the Laplace operator. Dirichlet's problem. Finite differences. Neumann's problem. |
6
|
| Diffusion equation |
Problems in one space dimension. Maximum-minimum theorem. Initial value problem and its general solution. Extreme values theorem. Problems in three space dimensions. |
4
|
| Total hours for lectures and exercises |
60 |
| for exercises only |
|
|
Further educational activities
|
hours
|
| Labs |
|
| Tutorials / Seminars |
|
| Workshops |
|
| Guided tours |
|
| |
|
| Total hours for further educational activities |
0 |
| Total hours |
60
|