VC3.18

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i). We have r=xi+yj+zk\,

{\mathrm  {div}}r=\nabla \cdot r=[i{\frac  {\partial }{\partial x}}+j{\frac  {\partial }{\partial y}}+k{\frac  {\partial }{\partial z}}]\cdot r\,

=i\cdot {\frac  {\partial r}{\partial x}}+k\cdot {\frac  {\partial r}{\partial y}}+k\cdot {\frac  {\partial r}{\partial z}}\,

=i\cdot {\frac  {\partial }{\partial x}}(xi+yj+zk)+j\cdot {\frac  {\partial }{\partial y}}(xi+yj+zk)+k\cdot {\frac  {\partial }{\partial z}}(xi+yj+zk)\,

=i\cdot i+j\cdot j+k\cdot k=1+1+1=3\,

ii). {\mathrm  {curl}}r=\nabla \times r=[i{\frac  {\partial }{\partial x}}+j{\frac  {\partial }{\partial y}}+k{\frac  {\partial }{\partial z}}]\times r\,

=i\times {\frac  {\partial r}{\partial x}}+j\times {\frac  {\partial r}{\partial y}}+k\times {\frac  {\partial r}{\partial z}}\,

=i\times {\frac  {\partial }{\partial x}}(xi+yj+zk)+j\times {\frac  {\partial }{\partial y}}(xi+yj+zk)+k\times {\frac  {\partial }{\partial z}}(xi+yj+zk)\,

=i\times i+j\times j+k\times k=0\,

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