An integrable differential ideal is an ideal 
in the ring of smooth forms
on a manifold 
satisfying 
whenever 
, where 
is the exterior derivative.
As an ideal, 
is closed under addition, scalar multiplication, and wedge
product with an arbitrary form.
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 integrable differential ideal 
on 
, a smooth map 
is called integral 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, for a smooth function 
,
the ordinary differential equation 
on 
corresponds to 
on 
.
