Can you think of a way to apply Stokes' theorem to find the area of a surface in $\mathbb{R^3}$ by constructing some vector field $F(x, y, z)$ and taking its line integral over the boundary of the surface?
If $n$ is the normal vector to the surface, then I need to find some vector field $F(x,y,z)$ such that $\langle \operatorname{curl} F, n\rangle = |n|$, right?
Or $\langle \operatorname{curl} F, n\rangle = 1$ ? Why? I'm confused with this! Can somebody help me? Thanks!