# Having trouble interpretting surface differential form

This comes up in a physics note: ($$S$$ is a surface, $$\vec{S}$$ is a normal vector, $$\nabla V$$ is a gradient of $$V$$, and $$\frac{\partial}{\partial n}$$ is normal derivative) $$\nabla V \cdot {\rm d}\vec{S}=\frac{\partial V}{\partial n}{\rm d}S$$ It's probably super obvious, but I have very shaky background in differentials and mostly just "shut up and calculate." I don't even know what the equation is supposed to mean. What exactly is $$dS$$ here (have done surface integral before but never ask this)? In case this is somehow ill-defined, I just want to know why is $$\int_{S}\nabla V \cdot {\rm d}\vec{S}=\int_{S}\frac{\partial V}{\partial n}{\rm d}S$$ super obvious?

I think the notation $$\vec{S}$$ by itself is awkward, but... here $${\rm d}S$$ is the area form in the surface $$S$$, so that $$\int_S{\rm d}S = {\rm area}(S),$$and $${\rm d}\vec{S} = \vec{n}\,{\rm d}S$$, where $$\vec{n}$$ is a unit normal field along $$S$$. For example, the flux of a vector field $$\vec{F}$$ tangent to $$\Bbb R^3$$ along $$S$$ is $$\int_S \vec{F} \cdot {\rm d}\vec{S} = \int_S \vec{F}\cdot \vec{n}\,{\rm d}S.$$If $$V$$ is a scalar field on $$\Bbb R^3$$, $$\partial V/\partial \vec{n}$$ is the directional derivative of $$V$$ in the direction of $$\vec{n}$$, and it equals $$\nabla V \cdot \vec{n}$$. It is really just a matter of understanding the notation. We have $$\int_S \nabla V\cdot {\rm d}\vec{S} = \int_S \nabla V\cdot \vec{n}\,{\rm d}S = \int_S \frac{\partial V}{\partial\vec{n}}\,{\rm d}S.$$