VC3.16

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{\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)]\,

=i(0)-j(0)+k(0)=0\,

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