Mathematical analysis 2

This book is intended primarily for first-year undergraduate students enrolled in the LMD (Bachelor–Master–Doctorate) program in the natural sciences, as well as for second-year students in applied sciences. It is designed to serve as a reference for students pursuing engineering studies and may also be useful to those studying economics, agronomy, medicine, or biology. The present volume contains the second part of a mathematical analysis course that we have taught for many years to first-year university students in mathematics and computer science, physical sciences, science and technology, life and natural sciences, and engineering. The structure of this book has evolved gradually through years of teaching. After each lecture, we collected and refined our notes, incorporating the observations and pedagogical insights gained from the teaching process. Our aim is to provide students and instructors with a mathematics textbook that closely reflects the material presented on the blackboard, while incorporating many of the remarks and explanations that naturally accompany classroom instruction. The topics covered in this volume follow the official syllabus of the Analysis II course established by the Ministry of Higher Education and Scientific Research (MESRS). Organized into four chapters, this second volume covers integral calculus, ordinary differential equations, functions of several variables, and plane curves. The material is presented progressively and rigorously, beginning with the fundamental concepts of the Riemann integral for real-valued functions, with particular emphasis on the connection between theoretical foundations and practical applications. We then develop the principal methods for solving first- and second-order ordinary differential equations, followed by the essential properties of functions of several variables, including limits, continuity, differentiability, and their consequences for local behavior and the determination of extrema. The volume concludes with an introduction to the study of parametric curves in the affine plane. Each topic is introduced through precise definitions, consistently illustrated and motivated by examples, generally chosen to be simple and classical, so that mathematical intuition can be developed from elementary situations. The main properties are subsequently formulated as theorems, lemmas, and propositions. Their proofs follow a pedagogical approach designed to make the arguments accessible to undergraduate students while maintaining mathematical rigor. Each chapter contains a complete, concise, and self-contained presentation of the course material, followed by 30 fully solved exercises. The book therefore provides a total of 120 exercises with detailed solutions to help students consolidate their knowledge and master the material. In addition, an appendix contains 40 supplementary exercises with partial solutions, encouraging independent study and enabling students to assess their understanding of the theory and their ability to apply it through clear, coherent, and rigorous mathematical reasoning. The selection of exercises emphasizes a thorough understanding of the fundamental concepts. Most problems are accessible to students and generally avoid unnecessarily sophisticated tricks or techniques. Our objective is to ensure steady progress throughout the course and to help students develop the skills needed to solve nearly all the proposed exercises. More challenging problems are also included to deepen their knowledge and familiarize them with the essential tools of mathematical analysis. Throughout the preparation of this volume, we have endeavored to provide students with a solid foundation for exploring the fascinating world of mathematical analysis. We hope that this book will serve as a reliable textbook for introductory courses in mathematical analysis. We recommend it to both the students for whom it is intended and the instructors who teach these courses. Bekkai Messirdi bmessirdi@yahoo.fr Abdellah Gherbi gherbi_abdellah@esgee-oran.dz Sofiane Messirdi messirdi.sofiane@hotmail.fr Sanaa Messirdi blahasanaa@yahoo.fr