  The vorticity of a vector field on 3-dimensional Euclidean space is usually
given by the curl of the vector field. In this paper, we extend this concept to
n-dimensional compact and oriented Riemannian manifold. We analyse many
properties of this operation. We prove that a vector field on a compact
Riemannian manifold admits a unique Helmholtz decomposition and establish that
every smooth vector field on an open neighbourhood of a compact Riemannian
manifold admits a Stokes' type identity.
