Product Rule for Divergence

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 $\mathbf f := \left({f_1 \left({\mathbf x}\right), f_2 \left({\mathbf x}\right), \ldots, f_n \left({\mathbf x}\right)}\right): \mathbf V \to \mathbf V$ be a vector-valued function on $\mathbf V$.

Let $g \left({x_1, x_2, \ldots, x_n}\right): \mathbf V \to \R$ be a real-valued function on $\mathbf V$.

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

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