- LMAT1121 Analyse mathématique 1 (or another Real Analysis course in one variable),
- LMAT1131 Algèbre linéaire (or another Linear Algebra course on linear and quadratic functions).
The course presents the Differential and Integral Calculus in several variables, and its mathematical foundations.
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 of the Differential and Integral Calculus in several variables:
' by defining, giving properties and illustrating by examples and counter-examples the convergence of a sequence of vectors, the notions of compact set, continuity, differentiability and integrability of functions of several variables, and by connecting theses properties among themselves using theorems and counter-examples,
' by explaining and illustrating the connection between different notions of derivatives of functions of several variables,
' by stating and illustrating Weierstass's theorem and the mean value theorem,
' by stating and graphically illustrating the necessary conditions in optimisation problems (with or without constraints),
' by extending proofs in one variable to the setting of several variables.
- Solve problems using analytical tools:
' by connecting a function of several variables to its graphical representations (graph, level sets, traces),
' by verifying the continuity, differentiability or integrability of a function,
' by solving optimisation problems in several variables,
' by approximating a function of several variables by its Taylor polynomial, and by exploiting such an information to deduce some graphical interpretation,
' by computing limits, derivatives and integrals of functions of several variables.
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.
The tutorial sessions give students individual help and attentions in their learning.
The three activities are given in presential sessions.
Differential and Integral Calculus in several variables :
- vector spaces, sequences, and topology,
- continuity,
- differentiability,
- Taylor expansion,
- optimisation problems with and without constraints,
- integral calculus.
Lecture notes: printed version available at DUC and on-line version available on the website http://www.duc.be/.
