Answer
$\iint_S curl F \cdot dS=0$
Work Step by Step
The Divergence Theorem states that $\iint_S \overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_Ediv \overrightarrow{F}dV $
Here, $S$ is a closed surface and $E$ is the region inside that surface.
$div F=\dfrac{\partial a}{\partial x}+\dfrac{\partial b}{\partial y}+\dfrac{\partial c}{\partial z}$
Now, we have $\iint_S curl\overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_E div (curl F)dV $
$\iint_S curl\overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_E div (curl F)dV=\iiint_E (0) dV $
This implies that $div (curl F)=0$
Hence, it has been verified that $\iint_S curl F \cdot dS=0$