Answer
(a)
Work Step by Step
If ${\bf{F}}$ is perpendicular to the unit normal vector ${\bf{n}}$ at every point, ${\bf{F}}\cdot{\bf{n}} = 0$. This implies that ${\bf{F}}$ is tangent to $S$ at every point.
Since $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_S^{} {\bf{F}}\cdot{\rm{d}}{\bf{S}} = \mathop \smallint \limits_{}^{} \mathop \smallint \limits_S^{} \left( {{\bf{F}}\cdot{\bf{n}}} \right){\rm{d}}S$, so
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_S^{} {\bf{F}}\cdot{\rm{d}}{\bf{S}}$ is zero.
Therefore, the answer is (a).
