Product Rule for Curl

Theorem
Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions..

Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$.

Let $\mathbf f := \tuple {\map {f_x} {\mathbf x}, \map {f_y} {\mathbf x}, \map {f_z} {\mathbf x} }: \R^3 \to \R^3$ be a vector-valued function on $\R^3$.

Let $\map g {x, y, z}: \R^3 \to \R$ be a real-valued function on $\R^3$.

Let $\nabla \times \mathbf f$ denote the curl of $f$.

Then:
 * $\nabla \times \paren {g \, \mathbf f} = \map g {\nabla \times \mathbf f} + \paren {\nabla g} \times \mathbf f$