  This paper gives quantitative global estimates between a time dependent flow
on a Riemannian manifold $\left( M\right) $ and the flow of a vector field
constructed by truncating the formal Magnus expansion for the logarithm of the
flow. As a corollary, we also find quantitative estimates between the
composition of the flows of two given time independent vector fields on $M$ and
the flow of a truncated version of the Baker-Cambel-Hausdorff-Dynkin expansion
associated to the two given vector fields.
