Élie Cartan, Exterior Differential Systems and its Applications. Translated form French (original) (raw)

This book is the reproduction, quite radically changed, a course taught during the first semester 1936-1937 at the Faculty of Sciences of the University of Paris. The first part of this work is devoted to discussion of the theory of systems of differential equations in total, which was the subject of several memoirs, already old, published mainly in the Annals of Ecole Normale Superieure 1 between years 1902 and 1908, this theory was the basis for my theory of the structure of infinite transformation groups, in the sense of S. Lie 2. It has since been generalized by various authors, especially by E. Kähler 3 , who has extended to any system of differential equations Exterior. I adopt in this work the notation advocated by E. Kähler of designating by dω, and called exterior differential of an exterior differential form ω of any degree, what I called earlier by ω ′ and what I called the exterior derivative of the form ω. After a first chapter, purely algebraic, the exterior forms and Exterior systems of equations, Chapter II is devoted to exterior differential forms (symbolic forms of E. Goursat 4) and the operation of exterior differentiation. Chapter III introduces the concept of closed exterior differential system and characteristic system, outlines the theory of completely integrable systems with applications to the classical problem of Pfaff 5. Chapter IV introduces the concepts of integral element, character and gender, devoted also to two fundamental theorems of existence.