Synopses & Reviews
This book gives a treatment of exterior differential systems including both the general theory and various applications. Topics include: a review of exterior algebra, simple exterior differential systems, the generation of integral manifolds through the solution of a succession of initial- value problems, involution, linear differential systems, tableau and torsion, the characteristic variety of a differential system, prolongation, the Algebra of a linear Pfaffian system, and an introduction to Spencer Theory. Much emphasis is placed on the general theory while many examples are given.
Synopsis
This book gives a treatment of exterior differential systems. It will in- clude both the general theory and various applications. An exterior differential system is a system of equations on a manifold defined by equating to zero a number of exterior differential forms. When all the forms are linear, it is called a pfaffian system. Our object is to study its integral manifolds, i. e., submanifolds satisfying all the equations of the system. A fundamental fact is that every equation implies the one obtained by exterior differentiation, so that the complete set of equations associated to an exterior differential system constitutes a differential ideal in the algebra of all smooth forms. Thus the theory is coordinate-free and computations typically have an algebraic character; however, even when coordinates are used in intermediate steps, the use of exterior algebra helps to efficiently guide the computations, and as a consequence the treatment adapts well to geometrical and physical problems. A system of partial differential equations, with any number of inde- pendent and dependent variables and involving partial derivatives of any order, can be written as an exterior differential system. In this case we are interested in integral manifolds on which certain coordinates remain independent. The corresponding notion in exterior differential systems is the independence condition: certain pfaffian forms remain linearly indepen- dent. Partial differential equations and exterior differential systems with an independence condition are essentially the same object.
Table of Contents
Contents: Introduction.- Preliminaries.- Basic Theorems.- Cartan-Kähler Theory.- Linear Differential Systems.- The Characteristic Variety.- Prolongation Theory.- Examples.- Applications of Commutative Algebra and Algebraic Geometry to the Study of Exterior Differential Systems.- Partial Differential Equations.- Linear Differential Operators.- Bibliography.- Index.