  For a submanifold M in a Euclidean space, the tangential component x^T of the
position vector field x of M is the most natural vector field tangent to the
Euclidean submanifold, called the canonical vector field of M. In this article,
first we prove that the canonical vector field of every Euclidean submanifold
is always conservative. Then we initiate the study of Euclidean submanifolds
with incompressible canonical vector fields. In particular, we obtain the
necessary and sufficient conditions for the canonical vector field of a
Euclidean submanifold to be incompressible. Further, we provide examples of
Euclidean submanifolds with incompressible canonical vector field. Moreover, we
classify planar curves, surfaces of revolution and hypercylinders with
incompressible canonical vector fields.
