17. Divergence, Curl and Potentials
d. Differential Identities
2. Second Order Differential Operators and Identities
d. The Curl of a Curl (\(\vec\nabla\times\vec\nabla\times\vec F\))
This is the least important of the second order differential operators. However, the following identity is used to help solve Maxwell's equations for electromagnetism.
If all second partial derivatives of the components of \(\vec F\)
are continuous functions, then
\[
\vec\nabla\times\vec\nabla\times\vec F
=\vec\nabla(\vec\nabla\cdot\vec F)-\nabla^2\vec F
\]
where the notation \(\nabla^2\vec F\) means to compute the Laplacian of each
component of \(\vec F\).
The general curl is \[ \vec\nabla\times\vec F =\left\langle \partial_yF_3-\partial_zF_2, \partial_zF_1-\partial_xF_3, \partial_xF_2-\partial_yF_1\right\rangle \] So the curl of the curl is \[\begin{aligned} \vec\nabla\times\vec\nabla\times\vec F &=\begin{vmatrix} \hat\imath & \hat\jmath & \hat k \\[2pt] \partial_x & \partial_y & \partial_z \\[2pt] \partial_yF_3-\partial_zF_2 & \partial_zF_1-\partial_xF_3 & \partial_xF_2-\partial_yF_1 \end{vmatrix} \\[8pt] &=\hat\imath [\partial_y(\partial_xF_2-\partial_yF_1)-\partial_z(\partial_zF_1-\partial_xF_3)] \\[2pt] &\quad-\hat\jmath [\partial_x(\partial_xF_2-\partial_yF_1)-\partial_z(\partial_yF_3-\partial_zF_2)] \\[2pt] &\quad+\hat k [\partial_x(\partial_zF_1-\partial_xF_3)-\partial_y(\partial_yF_3-\partial_zF_2)] \\[4pt] &=\hat\imath [\partial_x(\partial_yF_2+\partial_zF_3)-\partial_y^2F_1-\partial_z^2F_1)] \\[2pt] &\quad+\hat\jmath [\partial_y(\partial_xF_1+\partial_zF_3)-\partial_x^2F_2-\partial_z^2F_2)] \\[2pt] &\quad+\hat k [\partial_z(\partial_xF_1+\partial_yF_2)-\partial_x^2F_3-\partial_y^2F_3)] \end{aligned}\] provided the mixed partial derivatives are equal. On the other hand, \[\begin{aligned} \vec\nabla(\vec\nabla\cdot\vec F)-\nabla^2\vec F &=\hat\imath \partial_x(\vec\nabla\cdot\vec F) +\hat\jmath \partial_y(\vec\nabla\cdot\vec F) +\hat k \partial_z(\vec\nabla\cdot\vec F) \\[2pt] &\quad-\hat\imath \nabla^2 F_1 -\hat\jmath \nabla^2 F_2 -\hat k \nabla^2 F_3 \\[4pt] &=\hat\imath [\partial_x(\partial_xF_1+\partial_yF_2+\partial_zF_3) -\partial_x^2F_1-\partial_y^2F_1-\partial_z^2F_1)] \\[2pt] &\quad+\hat\jmath [\partial_y(\partial_xF_1+\partial_yF_2+\partial_zF_3) -\partial_x^2F_2-\partial_y^2F_2-\partial_z^2F_2)] \\[2pt] &\quad+\hat k [\partial_z(\partial_xF_1+\partial_yF_2+\partial_zF_3) -\partial_x^2F_3-\partial_y^2F_3-\partial_z^2F_4)] \\[4pt] &=\hat\imath [\partial_x(\partial_yF_2+\partial_zF_3)-\partial_y^2F_1-\partial_z^2F_1)] \\[2pt] &\quad+\hat\jmath [\partial_y(\partial_xF_1+\partial_zF_3)-\partial_x^2F_2-\partial_z^2F_2)] \\[2pt] &\quad+\hat k [\partial_z(\partial_xF_1+\partial_yF_2)-\partial_x^2F_3-\partial_y^2F_3)] \end{aligned}\] which is the same result.