Gradient of Dot Product

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

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

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

Proof
Then:

and similarly:

Next:

and: