differentialrelationer

Differentialrelationer is not a standard term in the mathematical literature, but in a wiki-style article it can be used to denote a formal object or entity that encodes differential relations between smooth manifolds. In this sense, a differentialrelationer represents the data needed to specify when a map between manifolds satisfies a prescribed differential constraint.

A common formal framework for differential relations uses jet bundles. Let X and Y be smooth manifolds and let J^r(X, Y) denote the r-jet bundle. A differential relation of order r is a subset R of J^r(X, Y); a map f: X → Y is a solution if its r-jet j^r f(x) lies in R for all x in X. A differentialrelationer, then, is an object that encapsulates such a relation R, including its defining equations, geometric structure, and symmetries. When R is a smooth submanifold of J^r(X, Y) and is invariant under diffeomorphisms of X, the relation is called geometric.

Key operations associated with a differentialrelationer include prolongation to higher jet orders, pullback by smooth maps, and sometimes extension along submanifolds. These operations generate a hierarchy of related relations (R, prolong(R), etc.) and are central to the study of partial differential relations and the h-principle, where formal solutions are compared with actual realizations.

Examples range from first-order PDE constraints to more complex systems encoded by algebraic or differential equations on jet coordinates. The concept is closely tied to jet bundle geometry, geometric PDE theory, and the study of solution spaces under symmetry and locality considerations.

Note: The term appears as a descriptive label rather than a standard, widely adopted technical term. See also differential relation, jet bundle, h-principle, prolongation.