Curl of curl identity

The curl-of-curl identity says that for any twice-differentiable vector field $\mathbf{F}$,

$\nabla \times (\nabla \times \mathbf{F})=\nabla (\nabla \cdot \mathbf{F})-{\nabla }^2\mathbf{F}.$

Here $\nabla (\nabla \cdot \mathbf{F})$ is the Gradient of the Divergence (a vector), and ${\nabla }^2\mathbf{F}$ is the Laplacian applied componentwise (in Cartesian coordinates).

The identity is the vector analog of the algebraic identity $\mathbf{A}\times (\mathbf{B}\times \mathbf{C})=\mathbf{B}(\mathbf{A}\cdot \mathbf{C})-\mathbf{C}(\mathbf{A}\cdot \mathbf{B})$ (the BAC-CAB rule) — formally treat $\nabla$ as a vector and apply BAC-CAB, then recognize the $\nabla \cdot \nabla$ as the Laplacian.

Why it matters for Maxwell’s equations

The identity is the standard tool for deriving wave equations from Maxwell’s equations. Take Faraday’s law in source-free vacuum:

$\nabla \times \mathbf{E}=-{\mu }_0\frac{\partial \mathbf{H}}{\partial t}.$

Take the curl of both sides:

$\nabla \times (\nabla \times \mathbf{E})=-{\mu }_0\frac{\partial }{\partial t}(\nabla \times \mathbf{H}).$

Now use Ampère-Maxwell in source-free space, $\nabla \times \mathbf{H}={ϵ}_0{\partial }_t\mathbf{E}$:

$\nabla \times (\nabla \times \mathbf{E})=-{\mu }_0{ϵ}_0\frac{{\partial }^2\mathbf{E}}{\partial t^2}.$

Apply the curl-curl identity to the left side, and use $\nabla \cdot \mathbf{E}=0$ (Gauss’s law in vacuum) to kill the $\nabla (\nabla \cdot \mathbf{E})$ term:

$-{\nabla }^2\mathbf{E}=-{\mu }_0{ϵ}_0\frac{{\partial }^2\mathbf{E}}{\partial t^2},$

so

${\nabla }^2\mathbf{E}={\mu }_0{ϵ}_0\frac{{\partial }^2\mathbf{E}}{\partial t^2}.$

This is the electromagnetic wave equation in free space. The same procedure starting from Ampère-Maxwell gives the wave equation for $\mathbf{H}$. The propagation speed is $1/\sqrt{{\mu }_0{ϵ}_0}=c$ — the speed of light.

A computational shortcut

The identity is also the practical way to compute ${\nabla }^2\mathbf{F}$ in curvilinear coordinates (cylindrical, spherical). In Cartesian, the Laplacian of a vector is just the componentwise Laplacian of the scalars. In curvilinear coordinates, the position-dependent unit vectors mean componentwise differentiation isn’t right — but $\nabla (\nabla \cdot \mathbf{F})$ and $\nabla \times (\nabla \times \mathbf{F})$ each have well-defined curvilinear formulas. Rearrange:

${\nabla }^2\mathbf{F}=\nabla (\nabla \cdot \mathbf{F})-\nabla \times (\nabla \times \mathbf{F}).$

Companion identities

Two related “operator identities” used constantly in vector calculus:

• $\nabla \cdot (\nabla \times \mathbf{F})=0$ — divergence of a curl is zero.
• $\nabla \times (\nabla \varphi )=0$ — curl of a gradient is zero.

These three together (curl-of-curl, div-of-curl, curl-of-grad) cover every second-order combination of $\nabla$ acting on scalars and vectors that you’ll meet.