Cours de Mathématiques - Préparation à l’agrégation - Licence - 6e édition

Book in French — English title: A Course in Mathematics: Agrégation Preparation – Undergraduate Level – 6th Edition

Publishing a mathematics course today may seem a tall order. Having long shied away from the challenge, I took advantage of the recent changes to the syllabus of the classes préparatoires to write up my notes properly. I then could not resist the urge to make this work available to the teaching and student community.

This course is deliberately concise. My aim is to offer the reader, in a single volume, the whole (and indeed more) of what can be taught in one year to a good student in the classes préparatoires or at undergraduate level. It is in no sense a book intended for teaching oneself mathematics, and the reader will find no exercises in it and very few worked examples. On the other hand, all the definitions and all the results of the course are included, almost always with complete proofs. It is therefore a book that can serve as a tool for revision and as a reference throughout the year. Students will turn to it to find, in another form, the lectures they have already attended, to deepen that teaching through supplementary material that frequently features in written or oral examination questions, or for quick revision before tests, examinations and competitive entrance examinations. There is no shortage of books of exercises and problems, from every publisher, that allow readers to bring all their sagacity and research skills to bear and to complement, through examples and applications, the results set out here.

We set out to write a course in mathematics, not a course in Mathématiques Spéciales. For this reason, we have not followed the syllabus of the MP classes to the letter. On the one hand, we wanted this book to be easily usable by undergraduate students and by candidates for the CAPES and the Agrégation; on the other, there is a well-known proverb in the education system: syllabuses come and go, mathematics (resp. physics, engineering science, French, philosophy, etc.) remains. That is why, on subjects treated in a somewhat unusual way at this level in the current classes préparatoires syllabus, such as integration, we have presented two points of view, that of integrable functions as well as that of improper integrals, at the cost, of course, of some repetition. In addition, we have deliberately included a number of supplementary topics that therefore lie outside any syllabus, either for their own mathematical interest and intrinsic beauty (Jordan reduction, general topology, convexity in normed vector spaces, applications of the Baire theorem), or because they set out somewhat old-fashioned techniques that still delight competitive-examination examiners (envelopes, orthogonal trajectories, pedal curves, etc.). We hope that syllabus purists will forgive us.