Descriptif
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.
Objectifs pédagogiques
At the end of this course, students will be able to
- carry out integrals over multiple variables, including with iterated bounds, in Cartesian, cylindrical and spherical coordinates
- compute the total differential of a function of several variables, successfully distinguishing ordinary and partial derivatives
- compute the gradient of a scalar field and the divergence/curl of a vector field in Cartesian, cylindrical and spherical coordinates
- apply Green's theorem in the plane, which includes evaluating the integral of a field along a non-trivial contour
- apply the divergence theorem and Stokes theorem in three dimensions, and explain under which hypotheses they can be applied
- expand a periodic function in a Fourier series, discuss the conditions under which the series converges, and evaluate the Fourier coefficients either in the cosine/sine basis or in the complex exponential basis
- apply Parseval's theorem to compute infinite sums
- solve partial differential equations such as the wave or heat equations on a finite domain using Fourier series
Diplôme(s) concerné(s)
Parcours de rattachement
Pour les étudiants du diplôme Bachelor of Science de l'Ecole polytechnique
Vous devez avoir validé l'équation suivante : UE MEC_1F000_EP
MEC100
Format des notes
Numérique sur 20Littérale/grade américainPour les étudiants du diplôme Bachelor of Science de l'Ecole polytechnique
Vos modalités d'acquisition :
The course is evaluated as follows
- continuous assessment (30% of final grade): 3, 20-minute in-class tests
- final written exam (closed book, 2 hours, 70% of final grade)
Remedials consist of a 1-hour, closed book, written exam.
Le rattrapage est autorisé (Note de rattrapage conservée écrêtée à une note seuil de 10)La note obtenue rentre dans le calcul de votre GPA.
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