Curl Operator Distributes over Addition

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 $\mathbf f$ and $\mathbf g: \R^3 \to \R^3$ be vector-valued functions on $\R^3$:


 * $\mathbf f := \tuple {\map {f_x} {\mathbf x}, \map {f_y} {\mathbf x}, \map {f_z} {\mathbf x} }$


 * $\mathbf g := \tuple {\map {g_x} {\mathbf x}, \map {g_y} {\mathbf x}, \map {g_z} {\mathbf x} }$

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

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