This kind of work has an advantage: it presents in one volume all the L1 program. It has one drawback: as 1st year university program touches everything, it is a real mathematical patchwork. It is everything, brought a little out of order. It is not really built as were the treaties of the seventies (or Cagnac Ramis, for example). In general, I have gained from reading a sense of unease. I had the sense to follow an oral statement during pressed by teachers taking turns to talk about their micro-specialty. This kind of work is somewhat similar style to courses at American universities. It discusses a little while wanting above all to be effective, even transform theorems definitions, to go faster and easier. Numerous annexes, called "supplements", are used to stimulate mathematical thinking of the reader. The intention is good, but the presentation is often confused. What characterizes this kind of work is the lack of unity. Lack of unity in the logs in the approaches, in deductions, in the references above, each author writing in his corner the portion that is allocated. This lack of unity comes from too many stakeholders and, perhaps, of a failure in the coordination of interventions and essays. But what bothered me most in this book, it is the obscure nature of many demonstrations, especially when compared with those advanced by other works. Another fault, major, is the impressive number of shells. For example, notes are absent (eg note 7 on page 9) or, even, supplements are nonexistent (eg complement the famous chapter 21, the authors maintain us to the third addition to the chapter 22). I much preferred - to stay in the current literature - the course of Deschamps and Warusfel (ISDM) published by Wiley. The plan of this book is no more clear (as it also follows to the letter the L1 program), but the mathematical content (quality of expression statements, clarity and elegance of the demonstrations) it is much higher. Confused plan because too attached, in principle, the content of the program only the first year, lack of unity in the writing, often confused demonstrations and large number of shells are the major flaws of this course. If we really want to address the higher mathematics by the right side (double educational and scientific point of view, that is to say from the point of view of clarity and rigor in the progression of the presentation ), it is better to fall back on older works, which may only be found on the used market. I think of the first two volumes of Cagnac, Ramis, Commeau (in Masson) or Ramis, Deschamps, Odoux (in Masson, too), or the Queysanne (for algebra) and Couty and Ezra (for analysis ), both Armand Colin (U collection). For the rest, the book in question retains an advantage: its corrected exercises. So for the price of 50 euros, the reader will have a course over a corrected workbook. This is not to overlook, either, knowing that the Warusfel Deschamps, mentioned above, has also corrected a lot of exercises ...
