Curl of curl identity

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).

It’s 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). Treat $\nabla$ as a vector, apply BAC-CAB, then recognize the $\nabla \cdot \nabla$ as the Laplacian.

Deriving the wave equation

This is the standard tool for getting wave equations out of 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

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

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

A computational shortcut

It’s also the practical way to compute ${\nabla }^2\mathbf{F}$ in curvilinear coordinates (cylindrical, spherical). In Cartesian, the Laplacian of a vector is 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 that come up constantly:

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

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