Elementary logic and Algebra
Propositional calculus, quantifiers. Arguments ad absurdo, by recursion.
Set and function terminology, sets N, Z and Q: arithmetic and combinatorics,
Polynomials: Euclidian division.
Properties of the set R
Interval, neighbourhood, upper bound. Sequences: limit (Cauchy criterion), rate of convergence, recursion un+1 = f(un). Numerical functions of the real variable: limits and continuity, differentiability, finite increments formula, monotony and inverse functions, Taylor formulas and inequalities, finite expansions, usual functions.
The field of complex numbers, usual complex functions (exponentials ...).
Linear algebra
Vector spaces, linear maps, basis and dimension. Matrices, determinants, linear systems. Eigenvalues and eigenvectors, characteristic polynomial, diagonalization. Application to differential systems and equations.
Analysis
Rational functions and their decomposition, Computation of primitives: integral defined on a closed bounded interval, numerical methods. Taylor formula with integral remainder. Vector valued function of the real variable in R2 and R3 (excluding metric properties). Parametric curves in R2 or R3. First and second order linear differential equations Path integral
Numerical series
Functions of the real variable: sequences and series of functions, entire series, applications to Fourier series. Simple, absolute, uniform and normal convergences. Integrals over a real interval, integrals depending on a parameter. Examples and applications (Fourier, Laplace).
Numerical and vectorial analysis
Differential calculus: multivariable functions. Partial derivatives and linear tangent application. Taylor formula of order 2: application to local extrema. Multiple integrals (functions of 2 or 3 variables). Computation via successive integrations and change of variables formula.
Finite dimensional euclidean spaces
Scalar products, norms, orthonormal basis and orthonormalization. Adjoint, hermitian, unitary and normal operators. Introduction to the space L2. Orthonormal basis in L2, Legendre polynomials, basis of trigonometric functions. Applications to Fourier series. Fourier transformation : Plancherel equality.
International Unit System, Dimensional analysis.
Mechanics
Kinematics: trajectories, velocity, acceleration, motion of rigid bodies, change of reference frame.
Newtonian dynamics: first, second and third laws, inertial and non-inertial reference frames, conservation laws, forces and potentials, gravitational field, central forces, small oscillations.
Fluids: pressure, hydrostatics, Euler and Lagrange variables of a continuum, continuity equation, Euler equation of motion.
Thermodynamics: first law, internal energy, work, heat. Reversible and irreversible processes, second law, Carnot cycles. Equations of state, change of phase, ideal gases, chemical potentials, chemical reactions, equilibrium equations, affinity.
Electricity & Magnetism
Electrostatics: electric charge, Coulomb's law, electric field, potential, Gauss' law, equilibrium of conductors, capacitance.
Magnetostatics: magnetic field, Ampère's laws, Faraday's law of induction.
Electric currents: electric current, Ohm's law, conductivity, Kirchhoff's laws, time varying currents, free and forced oscillations, condensers, inductance, complex impedance, resonant circuits.
Maxwell equations: Lorentz force, plane electromagnetic waves, radiation, light waves, reflexion, refraction, Huyghens principle, diffraction, interference phenomena.
Atomic & molecular physics
Quantum mechanics: Planck's law, Bohr's atom, de Broglie's relation, uncertainty principle, wave function, Schrödinger's equation, stationary states, quantization of energy.
Structure of matter : hydrogen atom, periodic table of the elements, molecules, solid state, elementary statistical physics.