Answer
$-1$
Work Step by Step
Stokes' Theorem states that $\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr $
Now, $\iint_{S} curl \space F \cdot dS=\iint_{D} -(-2z)(-1) -(-2x) (-1) -2y dA$
Substitute $1-x-y=z$
$\iint_{S} curl \space F \cdot dS=\iint_{D} -2+2x+2y-2x-2y dA$
Since, $D$ is the triangle formed by the vertices $(0,0), (1,0)$ and $(0,1)$
Now, $\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr =-2 \iint_{D} dA=-2 \times \dfrac{1}{2}=-1$
