  The mesochronic velocity is the average of the velocity field along
trajectories generated by the same velocity field over a time interval of
finite duration. In this paper we classify initial conditions of trajectories
evolving in incompressible vector fields according to the character of motion
of material around the trajectory. In particular, we provide calculations that
can be used to determine the number of expanding directions and the presence of
rotation from the characteristic polynomial of the Jacobian matrix of
mesochronic velocity. In doing so, we show that (a) the mesochronic velocity
can be used to characterize dynamical deformation of three-dimensional volumes,
(b) the resulting mesochronic analysis is a finite-time extension of the
Okubo--Weiss--Chong analysis of incompressible velocity fields, (c) the
two-dimensional mesochronic analysis from Mezic et al. \emph{A New Mixing
Diagnostic and Gulf Oil Spill Movement}, Science 330, (2010), 486-489, extends
to three-dimensional state spaces. Theoretical considerations are further
supported by numerical computations performed for a dynamical system arising in
fluid mechanics, the unsteady Arnold--Beltrami--Childress (ABC) flow.
