Given a smooth map on smooth manifolds, we know that the differential maps
However, we don’t know that this is a “nice” mapping between tangent spaces.
F is not guaranteed to be either injective or surjective.
So if we have a vector space then we don’t know if we can get a vector field using since maybe it would not map a vector to .
Definition
Two vector fields and are called F-related if
Conditions
and are -related if and only if for every smooth (real-valued) function defined on an open subset of ,