Mathematical methods for materials science

In this course, we will briefly review the origins of important mathematical concepts, the main results and theorems, and train on how to apply them in a concrete way in relevant core problems found in engineering and materials science. We will review the concepts in class and have exercise sessions to deepen understanding and apply them to engineering problems.

We will also have a few sessions where the concepts are shown and manipulated computationnally, to enhance intuition and understanding.

This class is hence also a good review of some aspects of materials science core concepts such as diffusion, wave propagation, materials structure and crystallography, mechanical properties, statistical and quantum mechanics, with an emphasis on setting up a problem mathematically and solving it.

Note that this course is not a mathematics class focused on theory and demonstrating theorems, but rather on mathematical methods to express and solve engineering problems. It is particularly suited for students who feel they need to learn better how to apply mathematical concepts to practical problems. It can also be interesting to revisit and bring practical mathematical skills up to speed for an engineering education at the Master and PhD level. Finally, it wil refresh important tools to visualize and treat mathematcial problems computationnally.

The concepts that we will revisit include:

-        Usual functions and differentiation: Taylor expansion, manipulation of log, exponential, hyperbolics etc.. : examples in thermally activited phenomena, optics and semiconductor physics.

-        Complex numbers: examples from Optical waves propagation to rheology.

-        Integral calculations and Fourier / Laplace transforms: examples in crystallography, quantum mechanics, visco-elastic materials, diffusion...

-        Differential equations: examples in diffusion, wave equation, etc..

-        Probability and Statistics: examples in Thermodynamics, and statistical and solid state physics.

-        Linear algebra and Matrices: review basic concepts and go deeper in Hilbert spaces and self-adjoint operators as a support to the solid state physics class taught in parallel that semester. Examples in mechanical properties of materials and quantum mechanics / solid states physics.