Divergence Operator Distributes over Addition

Theorem
Let $\mathbf V \left({x_1, x_2, \ldots, x_n}\right)$ be a vector space of $n$ dimensions.

Let $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ be the standard ordered basis of $\mathbf V$.

Let $\mathbf f$ and $\mathbf g: \mathbf V \to \mathbf V$ be vector-valued functions on $\mathbf V$:


 * $\mathbf f := \left({f_1 \left({\mathbf x}\right), f_2 \left({\mathbf x}\right), \ldots, f_n \left({\mathbf x}\right)}\right)$


 * $\mathbf g := \left({g_1 \left({\mathbf x}\right), g_2 \left({\mathbf x}\right), \ldots, g_n \left({\mathbf x}\right)}\right)$

Let $\nabla \cdot \mathbf f$ denote the divergence of $f$.

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