Answer
$0$
Work Step by Step
The stoke's Theorem states that $\iint_{S} curl F \cdot dS=\int_{C} F \cdot dr $
We will have to divide the sphere into upper and lower hemispheres let us say $S_1, S_2$ respectively.
This implies that $C$ shows a circle in $xy$- plane oriented in counter -clockwise direction.
Thus, we have: $\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, we have $\iint_{S} curl F \cdot dS=\iint_{S_1} curl \ F \cdot dS+\iint_{S_2} curl \ F \cdot dS$
or, $\int_{C} F \cdot dr-\int_{C} F \cdot dr=0$
