A Pfaffian equation on an open set in 
is a first-order differential
equation
 |
where the coefficients 
are smooth functions
and 
is a Pfaffian form (Esteves and Kleiman 2003). An
integral manifold is a submanifold
on which 
vanishes.
Where 
is nonzero, the equation admits a local foliation by
integral hypersurfaces precisely when
 |
the one-form case of the Frobenius integrability theorem (Lee 2012). Here 
is the exterior derivative
and 
is the wedge product. In this case, there are locally
a nonvanishing integrating factor 
and a smooth function 
such that 
, and the level sets
of 
are the integral hypersurfaces. A collection of
Pfaffian equations is a Pfaffian
system.
