= Solution
{c}
Apply the <product rule> inside the Cartesian expression for the <curl>:
$$
\begin{aligned}
\nabla\times(\zeta\mathbf v)
&=\sum_i\mathbf e_i\times\partial_i(\zeta\mathbf v)\\
&=\sum_i\mathbf e_i\times\bigl((\partial_i\zeta)\mathbf v
+\zeta\partial_i\mathbf v\bigr)\\
&=\left(\sum_i(\partial_i\zeta)\mathbf e_i\right)\times\mathbf v
+\zeta\sum_i\mathbf e_i\times\partial_i\mathbf v.
\end{aligned}
$$
The first sum is the <gradient> of the <scalar field> and the second is the <curl> of the <vector field>. Thus the <curl product rule for a scalar field> is
$$
\boxed{\nabla\times(\zeta\mathbf v)
=\zeta\nabla\times\mathbf v+\nabla\zeta\times\mathbf v}.
$$
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