I would like to prove <br />
\nabla \varphi = lim _{\int dv \rightarrow 0} \frac{\int \varphi d \vec \delta }{\int dv} <br />
with Gauss's theorem. According to Gauss's theorem, we know <br />
\int _{s}  \varphi d \vec \delta = \int _{v} \nabla \varphi dv <br />
Now if I want to prove it, I should write
<br />
\int _{v}  \nabla \varphi dv = \nabla \varphi  \int _{v} dv  <br />
Is this correct? How is that possible?