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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."
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