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.