= Curl product rule for a scalar field
{title2=$\nabla\times(\phi\mathbf F)=\phi\nabla\times\mathbf F+\nabla\phi\times\mathbf F$}
For a continuously differentiable <scalar field> $\phi$ and <vector field> $\mathbf F$, the <curl> satisfies
$$
\nabla\times(\phi\mathbf F)=\phi\nabla\times\mathbf F+\nabla\phi\times\mathbf F.
$$
Indeed, with fixed Cartesian <basis vectors> $\mathbf e_i$, the Cartesian expression for the <curl> and the <product rule> give
$$
\sum_i\mathbf e_i\times\partial_i(\phi\mathbf F)
=\sum_i(\partial_i\phi)\mathbf e_i\times\mathbf F
+\phi\sum_i\mathbf e_i\times\partial_i\mathbf F.
$$
The first sum is the <cross product> of the <gradient> of $\phi$ with $\mathbf F$; the second is $\phi$ times the <curl> of $\mathbf F$. The formula is useful when applying the <Stokes theorem> to a weighted <vector field>.
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