Physique mathématique I [SPHYB210]

Learning outcomes

Mathematical tools necessary for the physics lectures and research work.

Objectives

The objective of the course is to provide the student with the mathematical tools he will need to follow courses in theoretical physics and to carry on personal research works.

Content

The following mathematical tools are addressed

- Tensorial Algebra

- Vectorial Analysis

- The Dirac 'delta' function

- Linear response theory

- Group theory in physics and representation

Table of contents

Part I : Mathematical Physics

1)Tensor algebra Orthogonal transformations Scalar and vectorial fields Tensors et pseudo-tensors of arbitrary rank, examples Dyadic notations

2)Vector analysis Gradient, divergence and curl operators Continuity equation Divergence formula (Ostrogradski theorem) Stokes formula Scalar potential and vector potential Helmholtz theorem Orthogonal curvilinear coordinates Differential operators in orthogonal curvilinear coordinates Cylindrical and spherical coordinates

3)Dirac delta fonction Definition Properties Derivative

4)On the solution of mathematical physics equations using intergal transforms Fourier transforms: definition and properties Signal propagation along a coaxial electrical line Solution of the heat equation Solution of the diffusion equation in one and three dimensional spaces Magnetic field due to an arbitrary distribution of stationary currents

5)Linear response theory Definitions Dynamical susceptibility, examples and properties Kramers-Krönig relations

6) Introduction to the Green function

Part II : Group theory

1) Mathematical group

2) Representation of a group

3) Irreducible representation

4) Application to molecular vibrations