  We define a new notion of fiber-wise linear differential operator on the
total space of a vector bundle $E$. Our main result is that fiber-wise linear
differential operators on $E$ are equivalent to (polynomial) derivations of an
appropriate line bundle over $E^\ast$. We believe this might represent a first
step towards a definition of multiplicative (resp. infinitesimally
multiplicative) differential operators on a Lie groupoid (resp. a Lie
algebroid). We also discuss the linearization of a differential operator around
a submanifold.
