Curl of Curl is Gradient of Divergence minus Laplacian

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 $\mathbf f: \R^3 \to \R^3$ be a vector-valued function on $\R^3$:


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

Then:
 * $\nabla \times \left({\nabla \times \mathbf f}\right) = \nabla \left({\nabla \cdot \mathbf f}\right) - \nabla^2 \mathbf f$

where:
 * $\nabla \times \mathbf f$ denotes the curl of $\mathbf f$
 * $\nabla \cdot \mathbf f$ denotes the divergence of $\mathbf f$
 * $\nabla \left({\nabla \cdot \mathbf f}\right)$ denotes the gradient of the divergence of $\mathbf f$
 * $\nabla^2 \mathbf f$ denotes the Laplacian of $\mathbf f$.