Thread: Vector Form of Green's Theorem

I understand that the line integral means the area underneath the surface to the curve.

I also see how the line integral calculates work done by a particle traveling along path C through the vector field F.

I'm given that and I know that 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.

suppose F is a flow the integral is the net outcome of the flow surrounded by the curve. In the 3D version you get Gauss's theorem of the static electric field, then the integral is the net flow of the field, and the right hand is the source of the flow. So the equation means "the net outcome of the flow equals to the total amount of sources surrounded"