PHY_1S000_EP - Maths in Practice: Vector and Fourier Analysis (2024-2025)

The general purpose of this course is to introduce from a physicist's perspectives a number of mathematical tools that are necessary to follow the course PHY104: Electromagnetism and Light in S2 and subsequent Physics courses in Year 2. The course is divided in two parts, one devoted to notions of vector analysis and the other to Fourier analysis.

• Vector analysis:

The final objective of this part of the course is Stokes theorem, which is concerned with the integration of differential vector and scalar operators over general spaces embedded in two or three-dimensional Euclidean spaces R² and R³. It plays a fundamental role in Electromagnetism through Gauss's law and Ampère's law, but also in Fluid Mechanics. To reach this stage, students are introduced to multivariable calculus, differentials, vector differential operators such as the gradient, divergence and curl, and line integrals. We then study Green's theorem in the plane as a warm up towards the divergence theorem and Stokes theorem, which conclude the first part.

• Fourier analysis

Fourier analysis is another cornerstone of modern physics. It initially rests on the study of periodic phenomena, which repeat themselves in space or time and can be described using so-called Fourier series of periodic functions. Our final objective in the second part of the course is to examplify the uses of such Fourier series to solve important partial differential equations, such as the wave equation and the heat equation. We conclude with a first glimpse into Fourier transforms for non-periodic functions.