A differential ideal is an ideal 
in the ring of smooth forms
on a manifold 
. That is, it is closed under addition, scalar multiplication,
and wedge product with an arbitrary form. The ideal 
is called integrable if, whenever 
, then also 
, where 
is the exterior derivative.
For example, in 
, the ideal
 |
(1)
|
where the 
are arbitrary smooth functions, is an integrable differential ideal. However, if
the second term were of the form 
, then the ideal would not be integrable because
it would not contain 
.
Given an integral differential ideal 
on 
, a smooth map 
is called integrable if the pullback
of every form 
vanishes on 
, i.e., 
. In coordinates, an integral manifold solves a system
of partial differential equations.
For example, using 
above, a map 
from an open set
in 
is integral if
 |
(2)
|
 |
(3)
|
 |
(4)
|
 |
(5)
|
Conversely, any system of partial differential equations can be expressed as an integrable differential ideal on a jet
bundle. For instance, 
on 
corresponds to 
on 
.
