If $D$ is a region to which Green's theorem applies, $\mathbf{n}$ is the outward unit normal vector to $D$, and $$ \mathbf{F}=M(x, y) \mathbf{i}+N(x, y) \mathbf{j} $$ is a $C^{1}$ vector field on $D$, then $$ \oint_{\partial D} \mathbf{F} \cdot \mathbf{n} d s=\iint_{D} \nabla \cdot \mathbf{F} d A $$ I am encountering this theorem in my first vector calculus course, so I had weird confusion in understanding of this equation. First of all, I am writing, what I understand from this equation:
If we calculate the divergence of F at every infinitesimal volume in the region D and we calculate the Flux through the boundary ${\partial D}$ ,then they turns out to be same.
But in LHS of the equation we are considering the value of field F on the boundaries, since we are integrating over ${\partial D}$ only whereas in RHS, we are considering the value of F at each point inside the region to calculate the divergence over the region. So how are the value of F inside the whole region and at just the boundaries can be related in this way?
How can a line integral of projection of field F on an outward normal n over the boundary can represent integral of the divergence of all infinitesimal volumes in the region?(I am sorry, if I hadn't made my statements precise,Line integrals and Divergence are totally new concepts for me)