VC3.18

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

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

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

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