LMAT1122 Analyse mathématique 2 (ou un cours fondamental d'analyse des fonctions vectorielles).
The prerequisite(s) for this Teaching Unit (Unité d’enseignement – UE) for the programmes/courses that offer this Teaching Unit are specified at the end of this sheet.
Inversion theorems, convergence of sequences and series of functions (including Fourier series) and the divergence theorem
Contribution du cours aux acquis d'apprentissage du programme de bachelier en mathématique. A la fin de cette activité, l'étudiant aura progressé dans :
By the end of the course, the student should have progressed in obtaining the following skills:
- Recognise and understand a basic foundation of mathematics.
- Choose and use the basic tools of calculation to solve mathematical problems.
- Recognise the fundamental concepts of important current mathematical theories.
- Establish the main connections between these theories, analyse them and explain them through the use of examples.
- Identify, by use of the abstract and experimental approach specific to the exact sciences, the unifying features of different situations and experiments in mathematics or in closely related fields (probability and statistics, physics, computing).
- Show evidence of abstract thinking and of a critical spirit.
- Identify, by use of the abstract and experimental approach specific to the exact sciences, the unifying features of different situations and experiments in mathematics or in closely related fields (probability and statistics, physics, computing).
- Show evidence of abstract thinking and of a critical spirit.
- Recognise the key arguments and the structure of a proof.
- Evaluate the rigour of a mathematical or logical argument and identify any possible flaws in it.
- Distinguish between the intuition and the validity of a result and the different levels of rigorous understanding of this same result.
- Write a mathematical text according to the conventions of the discipline.
Acquis d'apprentissage spécifiques au cours. A la fin de cette activité, l'étudiant sera capable de :
By the end of the course, the student should be able to :
- Recognise the concepts, tools and methods in the study of sequences of functions:
' by stating, proving and illustrating existence and global uniqueness conditions,
' by stating different notions of convergence of sequences of functions, and illustrating their connections by proofs and counter-examples,
' by stating, proving and illustrating conditions involving the continuity, differentiability or integrability of sequences and series of functions,
' by applying those notions in the study of Fourier series,
' by stating, proving and illustrating the divergence theorem.
- Solve problems using analytical tools:
' by determining whether a system of equations has a solution, and how the solutions locally depend on some parameter,
' by studying the convergence of a sequence of functions,
' by determining the continuity, differentiability and integrability properties of the limit of a sequence or a series of functions.
The contribution of this Teaching Unit to the development and command of the skills and learning outcomes of the programme(s) can be accessed at the end of this sheet, in the section entitled “Programmes/courses offering this Teaching Unit”.
Learning will be assessed by a final examination. The questions will ask students to:
- reproduce the subject matter, especially definitions, theorems, proofs, and examples
- select and apply methods from the course to solve problems and exercises
- adapt methods of demonstration from the course to new situations
-summarise and compare topics and concepts.
Assessment will focus on
- knowledge, understanding and application of the different mathematical methods and topics from the course
- precision of calculations
- rigour of arguments, proofs and reasons
- quality of construction of answers.
Learning activities consist of lectures, exercise sessions and tutorial sessions.
The lectures aim to introduce fundamental concepts, to explain them by showing examples and by determining their results, to show their reciprocal connections and their connections with other courses in the programme for the Bachelor in Mathematics.
The exercise sessions aim to teach how to select and use calculation methods and how to construct proofs.
These activities are given in presential sessions.
Differential and Integral Calculus in several variables
- resolution of equations
- sequences and series of functions
- integral convergence theorems
- divergence theorem
- Fourier series
Lecture notes available on the iCampus website.
