Curl Operator Distributes over Addition

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


 * $\mathbf f := \left({f_x \left({\mathbf x}\right), f_y \left({\mathbf x}\right), f_z \left({\mathbf x}\right)}\right)$


 * $\mathbf g := \left({g_x \left({\mathbf x}\right), g_y \left({\mathbf x}\right), g_z \left({\mathbf x}\right)}\right)$

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

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