Modern Mathematical Concepts For Engineers - Part I: From Infinitesimal Calculus To Measure Theory

This textbook provides a comprehensive exploration of mathematical modeling, emphasizing the limitations of pointwise analysis and highlighting the great potential of measure theory. Measure theory not only offers better representations of phenomena, such as distributions and continuous probabilities, but also facilitates the analysis of classical partial differential equation problems. Additionally, the book introduces new classes of problems that go beyond traditional pointwise and infinitesimal analysis, integrating random and deterministic aspects into a unified framework.The narrative follows Bernard Namier, a young engineering apprentice, and Laurent Corps, a retired mathematics expert who tutors Bernard to tackle a challenging project assigned at work. Through their interwoven dialogue and the fictitious texts they study, readers are immersed in mathematical modeling as applied to engineering problems, with insights that extend to broader applications. This unique structure bridges two educational traditions: the abstract, Bourbaki-style approach common in France and the progressive, example-based learning prevalent in the Anglosphere. As a result, readers will master core concepts through rigorous proofs and applied examples and gain confidence in handling complex systems.