Answer
$\iint_{S} curl F \cdot dS=\int_{C} F \cdot dr =0$
Work Step by Step
Stoke's Theorem states that $\iint_{S} curl F \cdot dS=\int_{C} F \cdot dr $
Let us divide the sphere into the upper and lower hemispheres, say $S_1, S_2$ respectively.
This means that $C$ is a circle in the xy plane oriented in the counter-clockwise direction.
Therefore, $\iint_{S_1} curl F \cdot dS=\int_{C} F \cdot dr $
Now, we have: $\iint_{S_2} curl F \cdot dS=\int_{-C} F \cdot dr=-\int_{C} F \cdot dr $
Thus, $\iint_{S} curl F \cdot dS=\iint_{S_1} curl F \cdot dS+\iint_{S_2} curl F \cdot dS=\int_{C} F \cdot dr-\int_{C} F \cdot dr=0$
