Answer
$-1$
Work Step by Step
Stokes' Theorem can be defined as: $\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr $
Now, $$\iint_{S} curl \space F \cdot dS=\iint_{D} -(-2z)(-1) -(-2x) (-1) -2y dA$$
Consider $1-x-y=z$
$$\iint_{S} curl \ F \cdot dS=\iint_{D} -2+2x+2y-2x-2y dA$$
Since, $D$ is the triangle formed by the vertices $(0,0), (1,0)$ and, $(0,1)$, then we have:
$$\iint_{S} curl F \cdot dS \\=\iint_{C} F \cdot dr \\=-2 \iint_{D} dA \\=-2 \times \dfrac{1}{2} \\=-1$$