= Solution
For constant $b$, the <curl product rule for a scalar field> gives
$$
\nabla\times[(b\cdot r)b]=\nabla(b\cdot r)\times b=b\times b=0.
$$
Using part (b) and the order of the <cross product>,
$$
\iint_SF\times dS=-\iint_Sn\times F\,dS
=-\iiint_V\nabla\times F\,dV
=\boxed{0.}
$$
Here $dS$ inside a vector product denotes the outward vector area element $n\,dS$. Closedness is the condition that permits replacing the entire surface by the boundary integral of a volume.
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