Let $S$ be closed, oriented surface in $\mathbb{R}^3$ with unit normal vector ${\bf n}$. Let ${\bf c}\in \mathbb{R}^3$ be a ${\it constant}$ vector field. Show that the surface integral $$\iint_S{\bf c\cdot n} dS$$ is identically zero.
Because in the comment, Ivo Terek hinted on the curl of the constant vector field. Here is my updated thought: $$\iint_S{\bf c\cdot n} dS=\oint_{Curve} curl(c)\cdot n dt=\oint_{Curve} 0\cdot n dt$$ Since the curve starts and ends at the same point, say $a$ by integration property, $$\oint_{a}^a curl(c)\cdot n =0$$