VC3.16

From Exampleproblems

$\mathrm{curl}F=\begin{vmatrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ z & x & y \end{vmatrix}\,$

= $i[\frac{\partial y}{\partial y}-\frac{\partial x}{\partial z}]-j[\frac{\partial y}{\partial x}-\frac{\partial z}{\partial z}]+k[\frac{\partial x}{\partial x}-\frac{\partial z}{\partial y}]=i+j+k\,$

Hence $\mathrm{curl}\mathrm{curl}F=\mathrm{curl} (i+j+k)\,$

= $\begin{vmatrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ 1 & 1 & 1 \end{vmatrix}\,$

= $i[\frac{\partial}{\partial y}(1)-\frac{\partial}{\partial z}(1)]-j[\frac{\partial}{\partial x}(1)-\frac{\partial}{\partial z}(1)]+k[\frac{\partial}{\partial x}(1)-\frac{\partial}{\partial y}(1)]\,$