So I (just learned that I) can get the area of any closed area $S$ with boundary $\partial S$ on the flat plane through Stokes' (or in the simplified case Green's) theorem by using a vector field with constant curl: $$ \iint_{S} (\vec{\nabla} \times \vec{F}) \cdot d\vec{A} = \int_{\partial S} \vec{F} \cdot d\vec{s} $$ in 2D with the vector field $$ \vec{F} = \begin{bmatrix} -y \\ x \end{bmatrix} ; \ \ \ \vec{\nabla} \times \vec{F} = 2 $$ gives me $$ A = \frac{1}{2} \int_{\partial S} \vec{F} \cdot d\vec{s} = \frac{1}{2} \int_{\partial S} x\ dy - y\ dx $$ Now my question: Can I use the same approach to cactulate areas of regions on non-planar surfaces (e.g. a sphere) by using a vector field that is constant along their surfaces? (is that in principle a valid approach?)
And for that matter: is it even possible to define a vector field with constant curl relative to a sphere or even an arbitrary surface? Or do things like maybe the hairy ball theorem make this impossible?
A problem I could think of is that also by Stokes' theorem the integral of all the curl on a closed surface like a sphere must be zero, so can the curl of a vector field not be constant everywhere on a closed surface? Maybe you could still get a field with a singularity? Or maybe it is not constant everywhere but you can still extract the area? (Sry I'm bad at maths)
Many thanks!