Knowlpedia

Definition

Curl of a three-dimensional vector field

The oriented antisymmetric first derivatives of a vector field on a subset of R^3.

For a uu on an open subset of standard oriented R3\mathbb R^3, its curl is

∇×u=(∂yuz−∂zuy, ∂zux−∂xuz, ∂xuy−∂yux).\nabla\times u=(\partial_yu_z-\partial_zu_y,\ \partial_zu_x-\partial_xu_z,\ \partial_xu_y-\partial_yu_x).

For C2C^2 fields, commuting mixed partials proves div⁡(∇×u)=0\operatorname{div}(\nabla\times u)=0. Similarly ∇×∇f=0\nabla\times\nabla f=0 for C2C^2 scalars.

Cylindrical components

For r>0r>0, differentiating the cylindrical basis gives

(∇×u)r=r−1∂θuz−∂zuθ,(∇×u)θ=∂zur−∂ruz,(∇×u)z=r−1∂r(ruθ)−r−1∂θur.\begin{aligned} (\nabla\times u)_r&=r^{-1}\partial_\theta u_z-\partial_z u_\theta,\\ (\nabla\times u)_\theta&=\partial_z u_r-\partial_r u_z,\\ (\nabla\times u)_z&=r^{-1}\partial_r(ru_\theta)-r^{-1}\partial_\theta u_r. \end{aligned}
Cutting off a potential

The yields ∇×(χA)=∇χ×A+χ∇×A\nabla\times(\chi A)=\nabla\chi\times A+\chi\nabla\times A. The extra term is essential when localizing a divergence-free field through a vector potential.

References