  Local action principles on a manifold $\M$ are invariant (if at all) only
under diffeomorphisms that preserve the boundary of $\M$. Suppose, however,
that we wish to study only part of a system described by such a principle;
namely, the part that lies in a bounded region $R$ of spacetime where $R$ is
specified in some diffeomorphism invariant manner. In this case, a description
of the physics within $R$ should be invariant under {\it all} diffeomorphisms
regardless of whether they preserve the boundary of this region. The following
letter shows that physics in such a region can be described by an action
principle that $i$) is invariant under both diffeomorphisms which preserve the
boundary of $R$ and those that do not, $ii$) leaves the dynamics of the part of
the system {\it outside} the region $R$ completely undetermined, and $iii$) can
be constructed without first solving the original equations of motion.
