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MATH-F214
Compléments de mathématiques
Course teacher(s)
Ignace LORIS (Coordinator)ECTS credits
5
Language(s) of instruction
french
Course content
0) Legendre transformation, differential and implicit derivation
1) Dynamical systems (modelling, phase plane, linear and nonlinear systems, Hamiltonian systems, Liouville theorem)
2) Series and integral solutions of differential equations (change of variable, symmetries, Frobenius method, integral solutions)
3) Hilbert spaces (square integrable functions, orthonormal basis, linear operators, eigenvalues and eigenvectors, commuting operators)
4) Fourier series (periodic functions, lattice and reciprocal lattice (also in 3D), properties, convergence, real form, forbidden symmetries, numerical computation)
5) Fourier transformation (definition, properties, interpretation, gaussian function, inversion formula, sampling, relation with Fourier series, Heisenberg uncertainty relation)
6) Convolution (definition, interpretation, properties, link with probability, Dirac delta, convolution in imaging)
7) Partial differential equations and diffusion (heat equation, fundamental solution, solution on the line, half-line, plane, separation of variables)
8) Hermite polynomials and quantum harmonic oscillator (Hermite polynomials and functions, properties, quantum harmonic oscillator)
9) Spherical harmonic functions (laplacian in spherical coordinates, spherical harmonics, eigenfunctions of spherical laplacian, parity and recurrence relations, Legendre polynomials)
11) Hydrogen atom (Laguerre polynomials, Schrödinger equation, central potential, complete solution)
1) Dynamical systems (modelling, phase plane, linear and nonlinear systems, Hamiltonian systems, Liouville theorem)
2) Series and integral solutions of differential equations (change of variable, symmetries, Frobenius method, integral solutions)
3) Hilbert spaces (square integrable functions, orthonormal basis, linear operators, eigenvalues and eigenvectors, commuting operators)
4) Fourier series (periodic functions, lattice and reciprocal lattice (also in 3D), properties, convergence, real form, forbidden symmetries, numerical computation)
5) Fourier transformation (definition, properties, interpretation, gaussian function, inversion formula, sampling, relation with Fourier series, Heisenberg uncertainty relation)
6) Convolution (definition, interpretation, properties, link with probability, Dirac delta, convolution in imaging)
7) Partial differential equations and diffusion (heat equation, fundamental solution, solution on the line, half-line, plane, separation of variables)
8) Hermite polynomials and quantum harmonic oscillator (Hermite polynomials and functions, properties, quantum harmonic oscillator)
9) Spherical harmonic functions (laplacian in spherical coordinates, spherical harmonics, eigenfunctions of spherical laplacian, parity and recurrence relations, Legendre polynomials)
11) Hydrogen atom (Laguerre polynomials, Schrödinger equation, central potential, complete solution)
Objectives (and/or specific learning outcomes)
At the end of this teaching unit, a student will be able to
1) comprenhend and manipulate Legendre transformations and the differential
2) model a temporal evolution with a dynamical system, solve a linear dynamical system, understand the phase plane and Liouville theorem
3) understand the Frobenius method
4) verify if a function is an eigenfunction of a linear operator
5) write the Fourier series of a simple function, draw a lattice and reciprocal lattice in 2D
6) manipulate some Fourier integrals and draw some Fourier transforms in 2D
7) understand the role of convolution in experimental science and probability
8) understand the mathematical description of diffusion
9) manipulate Hermite polynomials
10) use spherical harmonic functions and Legendre polynomials
11) separate variables in the Schrödinger equation in spherical coordinates
In the context of the course subjects, a student will be able to interpret and produce mathematical content (such as texts, diagrams, and formulas) using a variety of notations (lowercase and uppercase letters, italics, boldface, Latin and Greek letters, numbers, symbols, etc.), presented in common fonts and sizes.
1) comprenhend and manipulate Legendre transformations and the differential
2) model a temporal evolution with a dynamical system, solve a linear dynamical system, understand the phase plane and Liouville theorem
3) understand the Frobenius method
4) verify if a function is an eigenfunction of a linear operator
5) write the Fourier series of a simple function, draw a lattice and reciprocal lattice in 2D
6) manipulate some Fourier integrals and draw some Fourier transforms in 2D
7) understand the role of convolution in experimental science and probability
8) understand the mathematical description of diffusion
9) manipulate Hermite polynomials
10) use spherical harmonic functions and Legendre polynomials
11) separate variables in the Schrödinger equation in spherical coordinates
In the context of the course subjects, a student will be able to interpret and produce mathematical content (such as texts, diagrams, and formulas) using a variety of notations (lowercase and uppercase letters, italics, boldface, Latin and Greek letters, numbers, symbols, etc.), presented in common fonts and sizes.
Prerequisites and Corequisites
Required and Corequired knowledge and skills
General mathematics (cartesian coordinates, functions, derivatives, integrals, matrices, determinants)
Required and corequired courses
Cours ayant celui-ci comme co-requis
Teaching methods and learning activities
Theoretical courses and exercises
References, bibliography, and recommended reading
Syllabus for sale at PUB and available on UV (Moodle)
Course notes
- Syllabus
- Université virtuelle
Contribution to the teaching profile
– Acquire, assimilate and exploit basic knowledge of mathematics, physics, chemistry, biology and geo-sciences
– Develop transversal knowledge
– Collect, analyse and synthesize knowledge
– Identify problems and formulate scientific questions
– Solve problems
– demonstrate intellectual openness
Other information
Contacts
Prof. Ignace Loris: Ignace.Loris@ulb.be, local 2.O.7.107, Teams, ...
Campus
Plaine
Evaluation
Method(s) of evaluation
- written examination
written examination
- Open question with short answer
- Open question with developed answer
- Closed question with multiple choices (MCQ)
- Visual question
- Closed question True or False (T/F)
One integrated written exam of theory and exercices. Exceptionally (pandemic, open session, ...) the written exam could be replaced by an oral exam.
Mark calculation method (including weighting of intermediary marks)
No partial marks. One mark out of 20.
Language(s) of evaluation
- french
- (if applicable english )