 | | |
| MATH0002-3 | Mathematical Analysis I
|
|
| Duration : | 50h Th, 40h Pr |
|
| Number of credits : |
|
|
| Lecturer : | Eric Delhez |
|
Language(s) of instruction :
|
| French language |
|
Course contents :
|
| The course provides an introduction to the main tools of calculus for the engineering science.
The following subjects are covered :
- Functions of a real variable : limit, continuity, derivative, graph, indefinite integral, Riemann integral,...
- Ordinary differential equations
- Functions of several real variables : limit, continuity, differentiation, extrema, change of variables
- Sequences and series : numerical sequences and series, sequences and series of functions, power series.
- Lebesgue Integral theory : line, surface and volume integrals, parametric integrals.
Each of these topics is introduced through applications showing how the corresponding tools can be used to tackle practical problems, while keeping a rigorous theoretical presentation of the concepts. |
|
Learning outcomes of the course :
|
| At the end of the course, the student will master the basic theoretical concepts of mathematical analysis and will be able to use the corresponding tools of calculus in both abstract mathematical contexts and in simple applications from the engineering world. He/She will be capable of using the mathematical language to formulate, analyze and solve simple original problems by resorting with rigor and discernment to the tools of calculus.
The student will also be capable of following and understanding abstract reasonings (demonstrations), reproducing them in a structured way, giving proper rigorous justifications of the different logical steps and producing original abstract reasonings closely resembling those presented to him/her. |
|
Prerequisites and co-requisites/ Recommended optional programme components :
|
| The course relies on knowledge of the basic concepts and mathematical tools introduced at secondary school (program of mathematical at 6h/week of the Belgium's French Community). In particular, the students are expected to be skilled in algebraic manipulations, including complex numbers, evaluation of limits, derivatives and indefinite integrals of usual algebraic and transcendental functions (trigonometric and inverse trigonometric functions, logarithm, exponential).
The students will also be familiar with the concepts of necessary condition, sufficient condition, necessary and sufficient condition and with the basic logical reasonings behind mathematical demonstrations.
Some of the concepts introduced in the courses of algebra (determinant, linear independence, abstract vector space) and geometry (parameterization of curves and surfaces, normal and tangent vectors, polar and cylindrical coordinate system) are used at some places. The students will therefore benefit from attending also MATH0003-1 and MATH0013-1. |
|
Planned learning activities and teaching methods :
|
| The course includes both ex-cathedra lectures (50 h) and exercise sessions (40 h).
- The new concepts are introduced during the lectures with references to practical or theoretical issues. The main theoretical results are then derived and are used to introduce and justify the tools of calculus.
- During the exercise sessions, the focus is on the development of the technical skills of the students, first in a pure mathematical context, then in simple academic problems. In the same time, the theoretical concepts are illustrated and clarified.
These two activities are complementary and form a coherent approach of the subject. Mastering the techniques used to solve the exercise requires a good knowledge of the underlying theory. Conversely, the illustrations developed during the exercise sessions help to understand the abstract concepts.
In order to benefit from the various learning activities, the students will work regularly in order to keep abreast. The introduction of concepts and derivation of new theoretical results occurs through a gradual approach in which the different elements are presented sequentially and rely on each other. Attending a session requires the understanding of the concepts introduced at the previous sessions.
Volontary learning activites are organized during the academic year.
- Question and answer sessions are planned at various key moments of the academic year. These provide good opportunities to meet the professor and assistants and ask them all possible questions about both theoretical and practical aspects.
- Formative assessments are proposed at the end of each of the main chapters. The questions are similar to those of real exams. Through these assessments, the students can better understand the level of understanding that they are expected to reach. Participation is voluntary. The marks are never taken into account in the final evaluation.
|
|
Mode of delivery (face-to-face ; distance-learning) :
|
| Face-to-face learning. |
|
Recommended or required readings :
|
| Analyse Mathématique, E.J.M. DELHEZ (2 volumes in french). Lecture notes distributed by the AEE with full coverage of the theory and exercices. |
|
Assessment methods and criteria :
|
| A written test in January on the theory and exercises from chapters 1, 2 and 3. A mark greater than or equal to 10/20 gives right to an exemption for the final test. Marks less than 10/20 are discarded.
A written test and an oral test are organized in May/June.
- The written test is about the exercises. Students with a mark larger than or equal to 10/20 at the January test can be exempted from the exercises of chapters 1 to 3.
- At the oral test, each student has to answer one main question about one of the theoretical topics from chapters 3 to 5. He/she can then use the official notes to prepare his/her answer at the blackboard which he/she will then explain and develop orally. The test is more about the understanding of the concepts and the reasoning than on the memorization. Mastering the theoretical concepts and knowledge of the main theorems (and hypotheses) is required to develop a rigorous and precise reasoning.
The final mark is a weighted average of the marks obtained at the written test (60 %) and the oral test (40 %). For those students who passed the January test with a mark of 10/20 or larger, this mark weigths 30 % and the written test of May/June accounts for only 30 %.
A written test is organized in August/September (retakes) and covers all the topics.
At written tests, students are never asked to reproduce complete demonstrations. Questions requiring abstract reasonings and use of concepts are however included. Also, a good mastering of the theoretical concepts and results is necessary to solve the problems and exercises. In particular, theoretical justification can be asked. |
|
Organizational remarks :
|
| The course takes place during both the first and second quadrimesters at the rate of one half day per week.
Ex-cathedra lectures are given in front of the full group of students. In order to promote a better interaction, the group is then split into smaller groups for the exercise sessions.
The schedule and organization details are available at http://www.ulg.ac.be/mathgen.
(http://www.ulg.ac.be/mathgen/cours/analyse/analyse.html) |
|
Contacts :
|
| Prof. Eric J.M. DELHEZ
Institut de Mathématique, B37
Tél. 04/366.94.19
E.Delhez@ulg.ac.be
List of assistants and their contact details available at http://www.ulg.ac.be/mathgen. |
|
|
| |
