Let $(M^n,g)$ be a Riemannian manifold, and let $V$ be a $C^{\infty}$ vector field on $M$. Is it possible to locally approximate $V$ by gradient vector fields $\nabla f\_i$, such that the $L^{\infty}$-norm 
$$\sup\_{B\_x(r)}|V-\nabla f\_i|\rightarrow 0?$$
for $r$ small enough. Thank you very much.



local approximation of a vector field on a Riemannian manifold
If you're asking the $f\_i$ to be smooth then the answer is no, unless $V$ is a gradient (in which case the statement is trivial).


Consider for example the Euclidean space $R^N$, with $B=B\_r(0)$ the standard ball centered at the origin. Assume by contradiction that there is such an approximating sequence for smooth $f\_i$'s. Then, by simply adding/subtracting constants, you could normalize $f\_i(0)=0$. The hypothesis $\nabla f\_i\to V$ (which is bounded) implies uniform Lipschitz bounds (write $|\nabla f\_i|\leq |V|+|\nabla f\_i-V|$), the Arzelà-Ascoli theorem shows that $\{f\_i\}\_i$ is relatively compact in $C(B)$. Extracting a converging subsequence if needed, you immediately get uniform convergence $f\_i\to f$ for some $f\in C(B)$. Since we assumed that the derivatives converge to start with (no extraction needed) we also get $\nabla f\_i\to\nabla f=V$.


