8. You have built a dome structure whose surface area is given by: $S = x^2 + y^2 + z^2 = 9$. The trajectory of air molecules across the dome's surface is given by: $F(x, y, z) = 2ycosz \hat{i} + e^xsinz \hat{j} + xe^y \hat{k}$. Use Stokes' Theorem to compute the deflection of the wind along the dome's surface: $\iint_S curl \, F \cdot ds$. Hint: Recall the trig identity $sin^2x = 1/2 (1 - cos \, 2x)$ and $\int 1 - cos \, 2x = \frac{1}{2}sin \, 2x = x - \frac{1}{2}sin \, 2x$
