# L1in2years Descartes

### Semester 1: 4 ects

• Mathematical language and reasoning - beginning: 0 ects
• Elementary Mathematical Tools - beginning: 0 ects
• Introduction to Computer Science: 4 ects
• Optics and Images - beginning: 0 ects
• Scientific English 1 - beginning: 0 ects
• Methodology - beginning: 0 ects
• Mathematics Gateway: 0 ects

### Semester 2: 22 ects

• Mathematical Language and Reasoning - end: 6 ects
• Elementary Mathematical Tools - end : 6 ects
• Computer implementation: 3 ects
• Optics and images - end: 4 ects
• Scientific English 1 - continued: 0 ects
• Personal and Professional Student Project 1 (PPPE 1) - beginning: 3 ects

### Semester 3 : 10 ects

• Linear Algebra - start: 0 ects
• Programming 1 - start: 0 ects
• Electricity - beginning: 0 ects
• Forces and Statics: 4 ects
• Point Mechanics - beginning: 0 ects
• Methodology - end: 3 ects
• Scientific English 1 - end: 3 ects

### Semester 4 : 24 ects

• Linear Algebra - end : 6 ects
• Programming 1 - end : 6 ects
• Electricity - end : 3 ects
• Point Mechanics - end : 3 ects
• PPPE 1 - end : 0 ects

Options :

• Analysis 1: 6 ects
• Computer Operation: 6 ects
• Physics / Waves : 3 ects
• Physics / Thermodynamics 1 : 3 ects
• Forces and Dynamics: 6 ects

PES option for further studies

• Complementary mathematics: 0 ects
• Complementary physics: 0 ects
• Internship in a company: 0 ects

Common sense is the most widely shared thing in the world.
René Descartes

## Description

• ### Basic mathematical tools

Vector calculus in R2 and R3 and geometry in the plane and in space: definition of a vector from two points, operations on vectors (sum, multiplication by a scalar, change of reference frame (in exercise), scalar product, norm of a vector, projection of a vector, vector product, definition and properties.

Equations of lines and planes: Cartesian equations and parametric representations. The notions of distance (distance from a point to a line, to a plane,...), transformations of the plane or space and area calculations are not included in the program.

Usual functions :

• Derivative of a function at a point, geometric interpretation, instantaneous speed (notion of limit of the rate of increase)
• rate of increase), derivative function, derivative of a compound function, calculation of partial derivatives (notations in class, the rest in exercises for
• in class, the rest in exercises to manipulate them and work on the derivatives of compound functions)
• Study of the neprian logarithm, exponential, trigonometric and reciprocal trigonometric functions
• (arcsinus, arccosinus, arctangent): domain of definition, parity, priodicity, derivative, monotonicity, table of variation, graphic representation
• Primitives of usual functions and composite functions like u0(t), u(t) , u0(t)eu(t),...

Complex numbers

• Algebraic writing, trigonometric writing, exponential writing
• Solving second degree equations with real coefficients

Differential equations

• Linear differential equations of the first order with constant coefficients :

• Solving y0(t) + ay(t) = 0 on an interval of R
• Solving y0(t) + ay(t) = b(t) by the method of variation of the constant
• Second order linear differential equations with constant coefficients
• Responsible of the UE: Olivier GUES
• Control of knowledge : Continuous control (Continuous control + 2 mid-term exams + 2 hours written exam)
• ### Mathematical language and reasoning

1. Basic notions in logic and reasoning: negation of a proposition, logical connectors and, or, quantifiers, implication, equivalence, contraposed.
Formal translation of elementary statements into natural language, formal translation of classical properties of functions.
Different types of reasoning and mathematical proofs: reasoning by contraposition, proof by recurrence, reasoning by the absurd
2. Vocabulary of set theory: inclusion, equality of two sets, double inclusion, intersection, reunion, complement, Morgan's laws, set of parts of a set, Cartesian product
3. Functions, applications: domain of definition, composition of applications, direct image, reciprocal image, injection, surjection, bijection, reciprocal application
4. Order relations: majoring, minoring, largest element, smallest element, upper bound, lower bound.
Functions and order relation: increasing, decreasing, majoring, minoring functions. Examples of order relations, divisibility relation in N, gcd, lcm.
Sequences and order relation: increasing, decreasing, increasing, decreasing sequences
5. Equivalence relations: definition of an equivalence relation, equivalence classes, quotient set
Elementary examples of equivalence relations.
Example of Z/nZ: reminders on Euclidean division of integers, Bezout, definition of Z/nZ and operations on Z/nZ
• Responsible for this course : Emmanuel BEFFARA
• Control of knowledge : Continuous control (Continuous control + 2 mid-term exams + 2 hours written exam)
• ### Methodology

To inform students about their brain functioning and the effects of context during learning, revision, during an exam, so that they can :

1. reflect on their knowledge, on the effectiveness of their learning and revision methods, on their motivations
2. understand the reasoning they use to solve complex problems
3. deconstruct (when necessary) some of the misconceptions about their abilities, efforts and work;
4. managing the stress of certain assessment situations;
5. develop effective working methods (knowledge and know-how), taking into account the functioning of the memory, and postures (life skills) adapted to improve their learning and the development of the competencies targeted by the training they have chosen. Apply this learning to revisit fundamental disciplinary or interdisciplinary notions for L1.
• Responsible of the UE : Renaud HARDRE
• Control of knowledge : Continuous control (50% Logbook ; 25% Report 1 ; 25% Report 2)
• ### Methodology

To inform students about their brain functioning and the effects of context during learning, revision, during an exam, so that they can :

1. reflect on their knowledge, on the effectiveness of their learning and revision methods, on their motivations
2. understand the reasoning they use to solve complex problems
3. deconstruct (when necessary) some of the misconceptions about their abilities, efforts and work;
4. managing the stress of certain assessment situations;
5. develop effective working methods (knowledge and know-how), taking into account the functioning of the memory, and postures (life skills) adapted to improve their learning and the development of the competencies targeted by the training they have chosen. Apply this learning to revisit fundamental disciplinary or interdisciplinary notions for L1.
• Responsible of the UE : Renaud HARDRE
• Control of knowledge : Continuous control (50% Logbook ; 25% Report 1 ; 25% Report 2)
• ### Introduction to Computer Science

The objective of this teaching unit is to discover different aspects of computer science, both in its information processing aspects and in the study of computation. The course is based on the study of a complex concrete problem that is decomposed into simpler tasks. During this decomposition, several aspects are studied.

• Responsible of the UE : Benjamin MONMEGE
• Knowledge control methods: Final grade = 0,2*Continuous control + 0,3*Partial + 0,5* Final exam (2h)
• ### Introduction to Computer Science

The objective of this teaching unit is to discover different aspects of computer science, both in its information processing aspects and in the study of computation. The course is based on the study of a complex concrete problem that is decomposed into simpler tasks. During this decomposition, several aspects are studied.

• Responsible of the UE : Benjamin MONMEGE
• Knowledge control methods: Final grade = 0,2*Continuous control + 0,3*Partial + 0,5* Final exam (2h)
• ### Personal and Professional Student Project

To discover the training courses and the professional fields concerned accessible at the end of the portal and the licence. To engage in personal reflection based on one's own project - Representation of the fields, the sector of activity, the profession.

• Methodology of documentary research.
• Development of the map of professions and methodology of interview grids.
• Development of the personal and professional project
• Discovery of the fields of study after L1
• Individual interview at the end of the semester on the student's project
• ### Scientific English

The aim will be to use active pedagogical methods to practice the English language, both orally and in writing, through activities such as drawing up a protocol or writing a report. Thus, we will build a practice of physics around the uncertainties in the measurements with a specialized and scientific vocabulary.

• ### How the computers work

This unit covers the essentials for understanding the principles of computer operation.

• ### Electricity

• Fundamental quantities in electricity, networks (1,5 H)
• Kirchhoff's laws, Thevenin's theorem (4 H).
• L and C energy storage components, transient regime: Study of the charge and discharge of a capacitor through a resistor. The parallel study of an RL circuit is done in TD. (2,5 H)
• Practical work: The practical work will introduce each part of the course (6H)
• ### Point mechanics

Objectives:

• To identify a system assimilated to a material point and to make a balance of the forces applying to this system.
• Describe the motion of a material point: notion of position, velocity and acceleration. Choose a coordinate system and express the velocity and acceleration.
• Write the laws of dynamics for a material point, mastering the concepts of acceleration, force, energy, work and power.
• Know how to solve these laws in simple cases (uniformly accelerated rectilinear, parabolic, circular motions), making the link between analytical approach, graphical representation and numerical application.

Program :

• Domain of the Mechanics of the material point. Framework and validity of classical mechanics, definition of the material point.
• Rectilinear motions. Description of motion: position, velocity and acceleration for rectilinear motions. Fundamental Principles of Dynamics and Reciprocal Actions. Kinetic Energy Theorem, notions of kinetic energy, power and work of a force. Potential and mechanical energies, conservative systems.
• Kinematics of any motion. Cartesian coordinates, cylindrical coordinates and Frenet reference frame: respective expressions for the position, velocity and acceleration of a material point.
• Dynamics of any motion. Implementation and projection of the Fundamental Principle of Dynamics according to the coordinate system. Generalization of the work and power of a force, of the potential energy, implementation of the Kinetic Energy Theorem.
• Responsible for the course: Jean-Marc VIREY
• Examination methods: 2 exams and a 2-hour written exam (continuous examination spread out over the semester)
• ### Computer implementation

The goal of this course is to illustrate the notions seen in Introduction to Computer Science through simple programs in Python. The focus will be on the notions of algorithm and data representation. Examples from other disciplines of the René Descartes portal will be developed.

• ### Computer implementation

The goal of this course is to illustrate the notions seen in Introduction to Computer Science through simple programs in Python. The focus will be on the notions of algorithm and data representation. Examples from other disciplines of the René Descartes portal will be developed.

• ### Optics and images

Common content with the "Optics" UE of the Marie Curie portal:

• Snell-Descartes relations and Fermat's principle
• Objects and images (definitions)
• Gaussian conditions, notion of rigorous and approximate stigmatism
• Simple optical systems (diopters and mirrors)
• Lenses and multi-lens systems (glasses, microscope, etc...)
• Notions of magnification and magnification

Specific content for this course (proposals to be discussed) :

• Rigorous demonstrations of conjugation relations under Gaussian conditions
• Prime
• Centered systems
• Geometric and chromatic aberrations
• Introduction to sampling (pixels, sensors, etc...)
• Numerical approach to ray tracing (TP?)
• Introduction to matrix optics
• ### Forces and statics

• Lecture on the fields of application, problems and methods of current mechanics.
• Lecture/DD on the equilibrium of undeformable solids, fluids and deformable solids: phenomena, nature and balance of forces, equations and resolution in simple cases.
• 2 sessions of practical work.
• ### Forces and statics

• Lecture on the fields of application, problems and methods of current mechanics.
• Lecture/DD on the equilibrium of undeformable solids, fluids and deformable solids: phenomena, nature and balance of forces, equations and resolution in simple cases.
• 2 sessions of practical work.
• ### Linear algebra

• Examples of solving linear systems

• Gauss pivot method
• Introduction of 2nd and 3rd order determinants and Cramer formulas
• Vector spaces

• Definition of a vector space
• Subvector space
• Free families, generating families, generated vector subspace, rank of a vector family, finite dimensional vector space
• Bases, dimension of a vector space
• Direct sum vector subspaces
• Direct sum decomposition of two subspaces, additional subspaces
• Linear applications

• Definition of a linear application
• Kernel, image, rank
• Injective, surjective, bijective linear application, endomorphism, isomorphism, automorphism - Matrix of a linear application with respect to two bases
• Matrices of f + g, 𝜆f and f 𝗈g.
• Rank of a matrix and equivalent definitions, invertible matrix, calculating the inverse of a matrix with Gauss pivot
• Change of basis, pass matrix, change of basis formula for coordinates of a vector, equivalent matrices, similar matrices
• Symmetric matrices, transpose of a matrix
• ### Linear algebra

• Examples of solving linear systems

• Gauss pivot method
• Introduction of 2nd and 3rd order determinants and Cramer formulas
• Vector spaces

• Definition of a vector space
• Subvector space
• Free families, generating families, generated vector subspace, rank of a vector family, finite dimensional vector space
• Bases, dimension of a vector space
• Direct sum vector subspaces
• Direct sum decomposition of two subspaces, additional subspaces
• Linear applications

• Definition of a linear application
• Kernel, image, rank
• Injective, surjective, bijective linear application, endomorphism, isomorphism, automorphism - Matrix of a linear application with respect to two bases
• Matrices of f + g, 𝜆f and f 𝗈g.
• Rank of a matrix and equivalent definitions, invertible matrix, calculating the inverse of a matrix with Gauss pivot
• Change of basis, pass matrix, change of basis formula for coordinates of a vector, equivalent matrices, similar matrices
• Symmetric matrices, transpose of a matrix
• ### Programming 1

This course deals with the fundamental principles of programming. It is based on the Java programming language, and therefore addresses in particular the concepts of object-oriented programming.

• ### Waves

• Concept of wave and mathematical formalization of the wave ;
• Progressive and regressive waves;
• Wave equations of the 1st and 2nd order;
• Sinusoidal waves
• Polarization, transverse and longitudinal waves;
• Complex notation;
• Addition of waves: standing waves, beats;
• Notion of harmonic (Fourier series ?);
• Relation of dispersion; Index of the medium, speed of propagation;
• Practical work envisaged: Ultrasound, vibrating rope (use of the stroboscopic lamp), springs, wave tank ... These 2-hour practical sessions can be thought of with the TD sessions
• ### Thermodynamics 1

History of thermodynamics; The material system; Energy exchanges and the First Principle: work, heat and internal energy; Evolution and the Second Principle: entropy.

• ### Strengths and dynamics

Objectives:

• To address some classical and not so classical applications of point mechanics.
• Oscillators, central force motions, shocks.
• Interactions between a fluid and a solid

Program:

• Mechanical oscillators: mass at the end of a spring, weighted pendulum. Harmonic, damped and forced oscillators.
• Motions with central forces: conservation of angular momentum, motion in the field of gravity, satellites.
• Shocks: conservation of momentum, energy. Elastic rectilinear shocks, soft shocks. Two body problem and barycentric reference frame.
• Forces and contact: applications of point mechanics to the mechanics of flight, sport...
• Lab 1: elastic oscillator
• TP 2: aerodynamic forces