Gradient of Dot Product

Definition
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 $\nabla \mathbf f$ denote the gradient of $f$.

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

where:
 * $\mathbf f \times \mathbf g$ denotes vector cross product
 * $\mathbf f \cdot \mathbf g$ denotes dot product

Proof
Then:

and similarly:

Next:

and: