VC3.33

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Let V=V_{1}i+V_{2}j+V_{3}k,r=xi+yj+zk\,

Now,V\times r={\begin{vmatrix}i&j&k\\V_{1}&V_{2}&V_{3}\\x&y&z\end{vmatrix}}\,

=(V_{2}z-V_{3}y)i+(V_{3}x-V_{1}z)j+(V_{1}y-V_{2}x)k\,

Therefore,\nabla \cdot (\nabla \times r)\,

={\frac  {\partial }{\partial x}}(V_{2}z-V_{3}y)+{\frac  {\partial }{\partial y}}(V_{3}x-V_{1}z)+{\frac  {\partial }{\partial z}}(V_{1}y-V_{2}x)\, by definition.

=z{\frac  {\partial V_{2}}{\partial x}}-y{\frac  {\partial V_{3}}{\partial x}}+x{\frac  {\partial V_{3}}{\partial y}}-z{\frac  {\partial V_{1}}{\partial z}}+y{\frac  {\partial V_{1}}{\partial z}}-x{\frac  {\partial V_{2}}{\partial z}}\,

=x[{\frac  {\partial V_{3}}{\partial y}}-{\frac  {\partial V_{2}}{\partial z}}]+y[{\frac  {\partial V_{1}}{\partial z}}-{\frac  {\partial V_{3}}{\partial x}}]+z[{\frac  {\partial V_{2}}{\partial x}}-y{\frac  {\partial V_{1}}{\partial y}}]\,

=(xi+yj+zk)\cdot [({\frac  {\partial V_{3}}{\partial y}}-{\frac  {\partial V_{2}}{\partial z}})i+({\frac  {\partial V_{1}}{\partial z}}-{\frac  {\partial V_{3}}{\partial x}})j+({\frac  {\partial V_{2}}{\partial x}}-y{\frac  {\partial V_{1}}{\partial y}})k]\,

=r\cdot (\nabla \times V)\,

=0\, if \nabla \times V=0\,

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