I am looking at a demonstration of how Green's theorem for a planar surface comes from Stoke's theorem for a general surface, for a surface in the x-y plane. However I do not understand one of the steps; I do not understand how the following two are equivalent:
$∬_A(\frac{∂F_y}{∂x}−\frac{∂F_x}{∂y})dxdy=∫_CF_xdx+F_ydy$
From the first expression, I would have thought you have:
$∬_A(\frac{∂F_y}{∂x}−\frac{∂F_x}{∂y})dxdy=∬_A\frac{∂F_y}{∂x}dxdy−∬_A\frac{∂F_x}{∂y}dxdy$
$=∫_CF_ydy-F_xdx$
I know that I am not understanding something about how to use intergals here. I am trying to adopt a method I saw for proving of Gauss' divergence theorem, where
$\int_{z_1}^{z_2}\int_{y_1}^{y_2}\int_{x_1}^{x_2}\frac{∂F_x}{∂x}dxdydz=\int_{z_1}^{z_2}\int_{y_1}^{y_2}F(x_2,y,z)-F(x_1,y,z)dydz$
and I think I am getting a bit confused.
Thank you in advance.