  We extend the Gauss formula relating the extrinsic connection to the
intrinsic connection to a formula for the extrinsic, Euclidean Laplacian to the
intrinsic Laplacian of a vector field on a hypersurface. As a byproduct, we
derive a formula for the Laplacian of a $1$-form on a surface of revolution in
terms of the Lie derivatives. Physically, these formulas are motivated by the
study of the formulation of the incompressible Navier-Stokes equations on a
Riemannian manifold.
