Curl of Curl is Gradient of Divergence minus Laplacian

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


 * $\mathbf f := \tuple {\map {f_x} {\mathbf x}, \map {f_y} {\mathbf x}, \map {f_z} {\mathbf x} }$

Then:
 * $\nabla \times \paren {\nabla \times \mathbf f} = \map \nabla {\nabla \cdot \mathbf f} - \nabla^2 \mathbf f$

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