As all of us know, in divergence theorem we have This:
this means instead of integrating a divergence of a vector over a volume, you can integrate that vector over the surface.
But I'm wondering what can we do with This:
I'm looking for it's concept and formula.
EDIT: in other words I want to know how we use divergence theorem when we have only one partial derivative for 3D vector and what is its intuition.
Consider $\vec{A}=(f_x(x,y,z),0,0)$ then divergence $${\rm div}\,\vec{A} =\frac{\partial f_x}{\partial x}$$
Then using divergence theorem gives $$\iiint \frac{\partial f_x}{\partial x}\, dV=\oint f_x (\vec{i}\cdot \vec{n}) dS$$
$f_x$ is dumb function here, we can take $f_x=\vec{F}=f_x \vec{i}+f_y \vec{j}+f_z \vec{k}$ and obtain $$\iiint \frac{\partial \vec{F}}{\partial x}\,dV=\oint \vec{F} (\vec{i}\cdot \vec{n}) \, dS$$