Answer
$V(E)=\dfrac{1}{3} \iint_S \overrightarrow{F}\cdot d\overrightarrow{S}$
Work Step by Step
Divergence Theorem: $\iint_S \overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_Ediv \overrightarrow{F}dV $
Here, $S$ shows a closed surface. The region $E$ is inside that surface.
We have $div F=\dfrac{\partial P}{\partial x}+\dfrac{\partial q}{\partial y}+\dfrac{\partial r}{\partial z}=1+1+1=3$
Here, we have $\iint_S \overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_E (3)dV $
This implies that $\iiint_E dV=\dfrac{1}{3} \times \iint_S \overrightarrow{F}\cdot d\overrightarrow{S}$
Thus, we have $V(E)=\dfrac{1}{3} \iint_S \overrightarrow{F}\cdot d\overrightarrow{S}$ (Verified)
