L2 - CUPGE Math

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Building on the analysis course from the second semester, this course will cover the concepts of series with terms of arbitrary sign. The Riemann integral will be defined and applied to solve differential equations, particularly linear ones. The section on integration 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 continuous 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 to finite dimensions (up to Jordan form) and its applications. It serves as an introduction to spectral analysis.

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This unit focuses on the mechanics of rigid bodies. It is the natural continuation of the unit on the kinematics and statics of rigid bodies covered in the first year of undergraduate studies. In this unit, we will adopt a dynamic framework and apply the Fundamental Principle of Dynamics. Deriving this principle requires knowledge of the torque of external forces, which was studied in the first year, as well as knowledge of the dynamic torque. The latter can be calculated using the kinetic torque, which, for a rigid body, involves the concept of moment of inertia. The main applications studied in this unit concern rigid bodies or simple cases of articulated systems of rigid bodies. In addition, we will study the special case of contact and friction forces (Coulomb friction) and discuss the Kinetic Energy Theorem.

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This course is the first step in university-level electromagnetism instruction. It covers electrostatics, steady-state currents, and magnetostatics.

See the syllabus in the “More Info” tab

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In this course, we will provide an overview of algebraic structures (rings, ideals, fields) before introducing the algebra K[X] and defining polynomial arithmetic, drawing parallels with the arithmetic of integers covered in L1. We will also cover computational aspects of polynomial functions and rational fractions (factorization and explicit expansions).

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This module supplements and formalizes the thermodynamic concepts introduced in the Thermodynamics 1 course unit, exploring several aspects in greater depth: thermodynamic potentials defined using Legendre transformations, the thermodynamics of open systems, phase transitions in 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 covers the grammatical 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, bicycle parts, electronic devices) through themed lessons and videos in the field of mechanical engineering.

Finally, a wide range of activities are offered to develop oral communication skills (presentation vocabulary, simulations, role-playing, and board games) so that students will be able to describe—during an oral presentation in pairs—the specific features, functions, and uses of a piece of technical equipment of their choice.

S4

The grammatical aspects are limited to a review of modal auxiliaries.

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

Students must also submit a resume in English and practice writing formal emails, so that they are prepared for situations involving internship or job searches where proficiency in English will be either required or considered an added asset.

The focus is always on practical application, culminating in 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, as well as various types of convergence. Integral series and Fourier series will also be discussed.

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This course will provide 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 covered.

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This course is an introduction to bilinear algebra and will cover Euclidean and Hermitian spaces. It will address topics such as isometries, duality, quadratic forms, and endomorphisms.

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This course introduces the main statistical concepts (graphical representation of data, measures of central tendency and dispersion, the relationship between two variables, and 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, iterative methods, and eigenvalue calculation will be introduced.

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The first part of this course aims to reinforce the concepts of magnetostatics and to establish the relationships governing the behavior of the electromagnetic field at the interface of a plane of charges or currents. We also introduce the expression for Laplace’s forces (force and torque) acting on volumetric or filiform circuits. The second part is devoted to the properties of fields and potentials in a time-varying regime. After introducing Faraday’s law, which describes induction phenomena, we derive the time-dependent Maxwell’s equations. An energy-based approach allows us to define the electric and magnetic energies, as well as the Poynting vector. We apply these concepts to various examples, such as electromechanical conversion or induction heating via eddy currents. A final chapter is devoted to the equations of propagation of fields and potentials, and their application in systems approximated as 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 in mathematics through presentations on career opportunities, thematic lectures, and roundtable discussions.

For students with an AED contract as part of a pre-professional training program, the course unit supports their work at the school by providing resources designed to enrich their observations and help them gain perspective. It also aims to prepare them for the written assignment they will be required to submit.

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