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Curl of the Curl of Vector Functions 📂Mathematical Physics

Curl of the Curl of Vector Functions

Formulas

The curl of the curl of a vector function is as follows.

∇×(∇×A)=∇(∇⋅A)−∇2A \nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}

Explanation

The first term, ∇(∇⋅A)\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 ∇j=∂∂xj\nabla _{j} = \dfrac{\partial }{\partial x_{j}}, then,

∇×(∇×A)=ϵijkei∇j(∇×A)k=ϵijkei∇j(ϵklm∇lAm)=ϵijkϵklmei∇j∇lAm=(δilδjm−δimδjl)ei∇j∇lAm=δilδjmei∇j∇lAm−δimδjlei∇j∇lAm=ei∇i∇jAj−∇j∇jeiAi=ei∇i(∇⋅A)−∇j∇jA=∇(∇⋅A)−∇⋅∇A \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 ϵijkϵklm=(δilδjm−δimδjl)\epsilon_{ijk}\epsilon_{klm}=(\delta_{il}\delta_{jm} - \delta_{im} \delta_{jl}).

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