If I have a volume V enclosed by a surface S, and $\nabla \times X$ is given on the surface, what information does that give me about X on S. Is there a method of showing that X = 0 on S? (in the context of the divergence theorem?)
Note that if $C$ is any constant vector, $\nabla \times (X+C) = \nabla \times X$. If the surface is compact and $X$ is continuous, it is bounded on the surface, so if we take $|C|$ large enough there is no point on the surface where $X+C=0$. So there is certainly nothing that $\nabla \times X$ can tell you that would imply that $X=0$ somewhere on $S$.