Edit: bound the unit sphere surface by $ z \ge 0 , y \ge 0, x \ge 0 $
Given the surface $x^2+y^2+z^2=1$ of unit sphere, I need to compute the surface integral of
$\int \int_{D} F \cdot ds $ on part of the sphere where $ z \ge 0 , y \ge 0, x \ge 0 $ and where $F = [0,x,0]$ is a vector field.
Instead of computing the surface integral I wanted to use Stoke's theorem and compute a line integral over the quarter unit cycle in the $xy$ plane ($y \ge 0, x \ge 0$) that bounds the surface. I was able to guess a vector field $G = [xz,0,0]$ for which $\nabla \times G = [0,x,0] = F $
I thought that the line integral $ \oint G \cdot dr $ over the quarter unit circle in the $xy$ plane ($y \ge 0, x \ge 0$) equals to the surface integral. But since the line integral evaluated for $z=0$ and then $G = [xz,0,0] = [0,0,0]$ and the line integral trivially evaluates to zero.
What do I do wrong here ( I know that zero is not the correct result for the surface integral)?