Answer
$\iint_S curl F \cdot dS=0$
Work Step by Step
Divergence Theorem: $\iiint_Ediv \overrightarrow{F}dV=\iint_S \overrightarrow{F}\cdot d\overrightarrow{S} $
where, $div F=\dfrac{\partial P}{\partial x}+\dfrac{\partial Q}{\partial y}+\dfrac{\partial R}{\partial z}$
Now, we have $\iint_S curl\overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_E div (curl F)dV $
and $\iint_S curl\overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_E div (curl F)dV=\iiint_E (0) dV $
This gives:
$div (curl F)=0$
Hence, the result has been verified such that $\iint_S curl F \cdot dS=0$
