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