Curl of the Curl of Vector Functions
Formulas
The curl of the curl of a vector function is as follows.
$$ \nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A} $$
Explanation
The first term, $\nabla(\nabla \cdot \mathbf{A})$, is the divergence of the gradient, which doesn’t have a specific name. The second term is important enough to have a name. $\nabla \cdot \nabla$ is called the Laplacian, specifically, the Laplacian of a vector function.
There isn’t any special meaning to the curl of a curl, it’s just important to know that it can be expressed as two other types of second-order derivatives.
Proof
The summation sign $\sum$ is omitted using Einstein notation. Calculated using the Levi-Civita symbol, it follows as: if we say $\nabla _{j} = \dfrac{\partial }{\partial x_{j}}$, then,
$$ \begin{align*} \nabla \times ( \nabla \times \mathbf{A}) &= \epsilon_{ijk} \mathbf{e}_{i} \nabla_{j} (\nabla \times \mathbf{A})_{k} \\ &= \epsilon_{ijk} \mathbf{e}_{i} \nabla_{j} (\epsilon_{klm} \nabla_{l} A_{m}) \\ &= {\color{blue}\epsilon_{ijk}\epsilon_{klm}}\mathbf{e}_{i} \nabla_{j} \nabla_{l} A_{m} \\ &= {\color{blue}(\delta_{il}\delta_{jm} - \delta_{im} \delta_{jl}) }\mathbf{e}_{i} \nabla _{j} \nabla _{l} A_{m} \\ &= \delta_{il}\delta_{jm}\mathbf{e}_{i} \nabla _{j} \nabla _{l} A_{m} - \delta_{im} \delta_{jl} \mathbf{e}_{i} \nabla _{j} \nabla _{l} A_{m} \\ &= \mathbf{e}_{i}\nabla_{i} \nabla_{j} A_{j} - \nabla_{j} \nabla_{j} \mathbf{e}_{i} A_{i} \\ &= \mathbf{e}_{i}\nabla_{i} (\nabla \cdot \mathbf{A}) - \nabla_{j} \nabla_{j} \mathbf{A} \\ &= \nabla(\nabla \cdot \mathbf{A}) - \nabla \cdot \nabla \mathbf{A} \end{align*} $$
The fourth equality holds because of $\epsilon_{ijk}\epsilon_{klm}=(\delta_{il}\delta_{jm} - \delta_{im} \delta_{jl})$.
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