L2 - CUPGE Maths

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This course will approach, in the continuity of the analysis course of S2, the notion of series with terms of any sign. The Riemann integral will be defined and applied to treat differential equations, especially linear ones. The integration part will be extended to generalized integrals.

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This course will introduce probability spaces, the notions 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 notions of symmetric group, determinants and will deal with the reduction of endomorphisms in finite dimension (up to Jordan form) and its applications. It is a first step towards spectral analysis.

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This unit concerns the study of the mechanics of rigid solids. It is the natural continuation of the unit devoted to the kinematics and statics of rigid solids in L1. In this unit we will place ourselves in a dynamic framework and apply the Fundamental Principle of Dynamics. The writing of this principle requires the knowledge of the torsor of the external actions, studied in L1, but also the knowledge of the dynamic torsor. The latter can be calculated with the help of the kinetic torsor which, for a rigid solid, uses the notion of moment of inertia. The main applications studied in this unit concern the rigid solid or simple cases of articulated systems of rigid solids. In addition, we will study the particular case of contact and friction actions (Coulomb friction) and we will approach the Kinetic Energy Theorem.

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This course is the first step in the teaching of electromagnetism at the university. Electrostatics, stationary currents and magnetostatics are covered.

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In this course, we will introduce an overview of algebraic structures (ring, ideal, body) before tackling the algebra K[X] and defining the arithmetic on polynomials by making parallels with the arithmetic of integers seen in L1. Computational parts on polynomial functions and rational fractions will be treated (explicit factorizations/decompositions).

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This module completes and formalizes the notions of thermodynamics introduced by the EU Thermodynamics 1, by deepening several aspects: thermodynamic potentials defined from Legendre transformations, thermodynamics of open systems, phase transitions of the pure body and irreversible processes, with incursions at the microscopic level in order to give an overview of the physical foundations of the theory.

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The first-semester course reviews the grammar 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, plane engine, bike parts, electronic device) through themed lessons and videos in the field of mechanical engineering.

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

S4

Grammatical 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 heat engines, and on emerging technologies(drones, driverless vehicles, 3D-printing).

Students are also expected to produce a CV in English and practice writing emails in a formal style, so as to be prepared for internship or job-seeking situations where fluency in English will either be necessary or an additional skill.

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

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This course will cover the concepts of sequences and series of functions and the various convergences. The integer and Fourier series will also be developed.

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In this course, an introduction to the topology of R^n, the basic notions of differential calculus of R^n functions in R and optimization will be covered. 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 deal with isometries, duality, quadratic forms and endomorphisms.

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Introduction in this course of the main statistical concepts (graphic representation of data, indicators 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 and more particularly to matrices. The notions 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 notions of magnetostatics and to establish the relations of 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 volumetric or filiform circuits. The second part is devoted to the properties of fields and potentials in variable regime. After introducing Faraday's law describing induction phenomena, we establish Maxwell's time-dependent equations. An energetic treatment allows us to define the electric and magnetic energies, as well as the Poynting vector. We apply these concepts to different examples such as electromechanical conversion or induction heating via eddy currents. A last chapter is devoted to the equations of propagation of fields and potentials, and to their application in vacuum-like systems, as well as in perfect conductors and insulators. The notion of skin depth is also introduced.

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This course will allow students to discover the various careers in mathematics through presentations of job opportunities, thematic conferences and round tables.

For students who benefit from a pre-professionalization contract, the UE accompanies their activity in the institution, by bringing some elements intended to enrich their observation and to give them some distance. It is also a question of preparing the written work that they will have to submit.

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