Product Rule for Curl

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

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

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

Let $g \left({x, y, x}\right): \R^3 \to \R$ be a real-valued function on $\R^3$.

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

Then:
 * $\nabla \times \left({g \, \mathbf f}\right) = g \left({\nabla \times \mathbf f}\right) + \left({\nabla g}\right) \times \mathbf f$