A prolongation of a differential equation on a jet bundle is a higher-order differential
equation obtained by adjoining differential consequences of the original equations.
If 
, its 
th prolongation 
requires the original equations and
all of their total derivatives through order
to vanish. A local section
solves the original differential equation
exactly when its prolonged jets satisfy every finite prolongation.
The infinite prolongation 
is the inverse limit of the finite prolongations.
It carries the restricted Cartan distribution,
whose integral manifolds describe the prolonged
graphs of solutions, and is the geometric object underlying a diffiety.
