On $\mathbb{R}^3$, consider a compactly supported $2$-form $$\omega = f_1 \, dx_2 \wedge dx_3 + f_2 \, dx_3 \wedge dx_1 + f_3 \, dx_1 \wedge dx_2.$$
Choose for $S$ the parametrization $h: \mathbb{R}^2 \to S$ defined by $$h(x_1, x_2) = (x_1, x_2, G(x_1, x_2)).$$
The I start to lost at why we are doing this:
Compute $$h^* dx_1 \wedge dx_2 = dx_1 \wedge dx_2$$ $$h^* dx_2 \wedge dx_3 = -\frac{\partial G}{\partial x_1} dx_1 \wedge dx_2$$ $$h^* dx_3 \wedge dx_1 = -\frac{\partial G}{\partial x_2} dx_1 \wedge dx_2$$
So I get totally lost here
And we emerge with the formula $$\int_s \omega = \int_{\mathbb{R}^2}(n_1 f_1 + n_2 f_2 + n_3 f_3) dx_1 dx_2,$$ where $$\vec{n} = (n_1, n_2, n_3) = (-\frac{\partial G}{\partial x_1},-\frac{\partial G}{\partial x_2}, 1).$$
Can someone give me some explanation/intuition here? Thank you.
They're pulling back oriented areas in 3d space to areas in the 2d coordinate plane. This allows you to convert the very complicated 3d surface integral into a simpler 2d integral on a flat domain.