L2 - CUPGE Mathematics

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This course will build on the S2 analysis course by covering the concepts of series with terms of any sign. Riemann integrals will be defined and applied to solve differential equations, particularly linear ones. The integration section will be expanded to include generalized integrals.

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This course will introduce probability spaces, the concepts of probability and independence, and will define discrete and density random variables with an emphasis on modeling.

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This course will cover the concepts of symmetric groups and determinants, and will address the reduction of endomorphisms in finite dimensions (up to Jordan form) and its applications. It is a first step toward spectral analysis.

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This course unit concerns the study of rigid body mechanics. It is the natural continuation of the course unit devoted to the kinematics and statics of rigid bodies in L1. In this course unit, we will place ourselves in a dynamic framework and apply the Fundamental Principle of Dynamics. Writing this principle requires knowledge of the tensor of external actions, studied in L1, as well as knowledge of the dynamic tensor. The latter can be calculated using the kinetic tensor, which involves the concept of moment of inertia for a rigid solid. The main applications studied in this course concern rigid solids or simple cases of articulated systems of rigid solids. In addition, we will study the special case of contact and friction actions (Coulomb friction) and we will discuss the kinetic energy theorem.

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This course is the first step in teaching electromagnetism at university. It covers electrostatics, steady currents, and magnetostatics.

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In this course, we will introduce an overview of algebraic structures (rings, ideals, fields) before tackling the algebra K[X] and defining arithmetic on polynomials, drawing parallels with the arithmetic of integers seen in L1. Computational aspects of polynomial functions and rational fractions will be covered (explicit factorizations/decompositions).

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This module complements and formalizes the concepts of thermodynamics introduced in the Thermodynamics 1 course, exploring several aspects in greater depth: thermodynamic potentials defined using Legendre transformations, thermodynamics of open systems, phase transitions of pure substances and irreversible processes, with forays into the microscopic level to provide an overview of the physical foundations of the theory.

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The first semester course reviews the grammar concepts essential for oral and written communication (tenses and aspect, asking questions, comparisons and superlatives, passive voice) as well as essential general vocabulary (numbers, measurements, shapes); It also includes an introduction to technical vocabulary (basic building materials, airplane engines, bike parts, electronic devices) through lessons and videos on topics related to mechanical engineering.

Finally, numerous activities are offered to promote oral expression skills (presentation vocabulary, simulations, role-playing, and tabletop games) so that students are able to describe the specific features, functions, and uses of a technical device of their choice during an oral presentation in groups of two.

S4

Grammar aspects are limited to a review of modal auxiliaries.

The vocabulary is much more focused on the various elements involved in the design and operation of different types of combustion engines and on emerging technologies (drones, driverless vehicles, 3D printing).

Students must also produce a CV in English and practice writing emails in a formal style, so that they are prepared for situations when looking for internships or jobs where English proficiency will either be necessary or an additional skill.

Practicing expression is always the main objective, with an individual oral presentation at the end of the semester on their second-year mechanics project.

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This course will cover the concepts of sequences and series of functions and various types of convergence. Entire series and Fourier series will also be discussed.

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This course will cover an introduction to the topology of R^n, the basic concepts of differential calculus of functions from R^n to R, and optimization. Parametric curves will also be discussed.

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This course is an introduction to bilinear algebra and will cover Euclidean and Hermitian spaces. It will cover everything related to isometries, duality, quadratic forms, and endomorphisms.

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Introduction to the main statistical concepts in this course (graphical representation of data, measures of central tendency and dispersion, relationship between two variables, confidence intervals).

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This course will cover numerical methods applied to linear algebra, with a particular focus on matrices. The concepts of conditioning, matrix decompositions and iterative methods, and eigenvalue computation will be introduced.

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The first part of this course aims to consolidate the concepts of magnetostatics and establish the relationships between the electromagnetic field at the interface of a plane of charges or current. We also introduce the expression of Laplace forces (force and moment) acting on volume or wire circuits. The second part is devoted to the properties of fields and potentials in variable regimes. After introducing Faraday's law describing induction phenomena, we establish Maxwell's time-dependent equations. An energy treatment allows us to define electrical and magnetic energies, as well as the Poynting vector. We apply these concepts to various examples, such as electromechanical conversion and induction heating via eddy currents. The final chapter is devoted to the propagation equations of fields and potentials, and their application in systems assimilated to a vacuum, as well as in perfect conductors and insulators. The concept of skin depth is also introduced.

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This course will introduce students to the various careers available in mathematics through presentations on career opportunities, themed lectures, and roundtable discussions.

For students who have a pre-professional AED contract, the EU supports their activities at the institution by providing them with information designed to enrich their observations and give them perspective. This also involves preparing them to write the paper they will be required to submit.

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