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Definition
A formal solution assigns allowed jets; a holonomic one is the actual jet of a section.
For a relation , a formal solution is a smooth section whose image lies in . It is holonomic if for a section of , using its actual first jet.
Locally a formal solution consists of values and assigned derivatives . Holonomicity requires ; pointwise membership in alone does not enforce this compatibility.
A first-order differential relation for sections of a smooth bundle is a subset of its first jet bundle. A solution is a smooth section of with for every .
Let be a smooth fiber bundle. A smooth global section of is a smooth map satisfying
Equivalently, for every , and this choice varies smoothly with . A local section over an open set is a smooth map with , where is the inclusion. A fiber bundle always has local sections near each base point, but it need not have a global section.
Let be a smooth fiber bundle. Two smooth local sections , defined near , have the same first jet at if
where and are their differentials. The equivalence class is denoted .
The first jet bundle consists of all such classes, with projections
Its smooth structure is defined by the following jet charts. In local bundle coordinates , assign to the coordinates
The derivative coordinates range freely over real matrices; changes of jet coordinates are the smooth transformations obtained by the chain rule. These charts define the smooth bundle structures and , not merely smooth projections of an unspecified structure.