Natural bundle

In mathematics, a natural bundle is any fiber bundle associated to the "s"-frame bundle formula_1 for some formula_2. It turns out that its transition functions depend functionally on local changes of coordinates in the base manifold formula_3 together with their partial derivatives up to order at most formula_4. 
The concept of a natural bundle was introduced by Albert Nijenhuis as a modern reformulation of the classical concept of an arbitrary bundle of geometric objects.
An example of natural bundle (of first order) is the tangent bundle formula_5 of a manifold formula_3.