7. Use Stokes' Theorem to evaluate:
(a) 4 marks $$ \iint_S \text{curl} \vec{F} \cdot d\vec{S} $$ where $$ \vec{F}(x, y, z) = ze^y \vec{i} + x \cos y \vec{j} + xz \sin y \vec{k}, $$ S is the hemisphere $$ x^2 + y^2 + z^2 = 16, y \geq 0, $$ oriented in the direction of the positive y-axis.
(b) 4 marks $$ \oint_C \vec{F} \cdot d\vec{r}, $$ where C is oriented counterclockwise as viewed from above.
$$ \vec{F}(x, y, z) = (x + y^2) \vec{i} + (y + z^2) \vec{j} + (z + x^2) \vec{k}, $$ C is the triangle with vertices $$ (1, 0, 0), (0, 1, 0), $$ and $$ (0, 0, 1). $$