8. Let F = Mi+Nj+Pk be continuously differentiable, S := \{(x, y, z) \in \mathbb{R}^3 : x^2+y^2 +z^2 = 1\} be the unit sphere, S? = \{(x, y, z) \in S : z \ge 0\} and S? = \{(x, y, z) \in S : z \le 0\} be upper and lower hemispheres, and B := \{(x, y, z) : x^2 + y^2 + z^2 < 1\} be the unit ball.
(a) Use the fact S = S? \cup S? and Stoke's theorem, and show \int_S(\nabla \times F) \cdot nd\sigma = 0.