Can divergence free vector fields be approximated by smooth ones?

Question

If $M$ is a compact Riemannian manifold, is the space of $C^{\infty}$ divergence-free vector fields dense in the space of $C^r$ divergence-free vector fields, in the $C^r$ topology ($r\geq 1$)? How about if we consider divergence-free vector fields compactly supported on $\mathbb R^n$?

Answer 1

The flat case is pretty obvious. Let $(\rho_\epsilon)_{\epsilon\rightarrow0}$ be a sequence of $C^\infty$-functions which approximate the Dirac mass. If $V$ is a compactly supported, divergence-free $C^r$-vector field, then $V_\epsilon:=\rho_\epsilon*V$ is divergence-free, $C^\infty$, and converges towards $V$ in the $C^r$-topology.