I understand that the line integral $\displaystyle \int_Cf(x,y)ds$ means the area underneath the surface to the curve.

I also see how the line integral $\displaystyle \int_C\vec{F}\cdot{d}\vec{r}$ calculates work done by a particle traveling along path C through the vector field F.

I'm given that $\displaystyle \oint_C\vec{F}\cdot\vec{n}ds=\iint_Ddiv\vec{F}(x,y )dA$ and I know that $\displaystyle \vec{n}$ is the normal vector of path C. I'm just having trouble understanding what this line integral means. Not so much the proof in why these two things are equal, but what that line integral means.

Thank you.