Formulas for derivatives of global flows and pullbacks

In this paper, we compute time derivatives of pullbacks, scalar parameter derivatives of global flows, and scalar parameter derivatives of pullbacks. In the process, we prove reductive formulas for pullbacks involving natural lifts, as well as an explicit formula for the global flow of a tangent lift perturbed by a vertical lift. The analysis is distinguished by the fact that it is carried out for time-varying vector fields that satisfy minimal regularity conditions.