  The exponential and Cayley map on SE(3) are the prevailing coordinate maps
used in Lie group integration schemes for rigid body and flexible body systems.
Such geometric integrators are the Munthe-Kaas and generalized-alpha schemes,
which involve the differential and its directional derivative of the respective
coordinate map. Relevant closed form expressions, which were reported over the
last two decades, are scattered in the literature, and some are reported
without proof. This paper provides a reference summarizing all relevant closed
form relations along with the relevant proofs. including the right-trivialized
differential of the exponential and Cayley map and their directional
derivatives (resembling the Hessian). The latter gives rise to an implicit
generalized-alpha scheme for rigid/flexible multibody systems in terms of the
Cayley map with improved computational efficiency.
