  A coupled system of non-linear partial differential equations is presented
which describes non-perturbatively the evolution of deformations of a
relativistic membrane of arbitrary dimension, $D$, in an arbitrary background
spacetime. These equations can be considered from a formal point of view as
higher dimensional analogs of the Raychaudhuri equations for point particles to
which they are shown to reduce when $D=1$. For $D=1$ or $D=2$ (a string), there
are no constraints on the initial data. If $D>2$, however, there will be
constraints with a corresponding complication of the evolution problem. The
consistent evolution of the constraints is guaranteed by an integrability
condition which is satisfied when the equations of motion are satisfied.
Explicit calculations are performed for membranes described by the Nambu
action.
