$\bar{A} =\nabla (xy + yz + zx)$ $= \left ( \overrightarrow {i} \dfrac {\partial} {\partial x} + \overrightarrow {j} \dfrac {\partial}{\partial y} + \overrightarrow {k} \dfrac {\partial} {\partial z} \right ) (xy + yz + zx)$ $=\overrightarrow {i} \dfrac {\partial}{\partial x} (xy + yz+ zx) + \overrightarrow {j} \dfrac {\partial }{\partial y} (xy + yz + zx) + \overrightarrow {k} \dfrac {\partial }{\partial z} (xy + yz +zx)$ $= \overrightarrow {i}\cdot (y+0+z)+ \overrightarrow {j} \cdot (x+z+0)+ \overrightarrow{k} \cdot (0+y+x)$ $\therefore \bar{A} = (y+z) \overrightarrow {i} + (x+z) \overrightarrow{j}+ (y+x)\overrightarrow {k}$ $\therefore \nabla\cdot \overrightarrow {A} = \left ( \overrightarrow {i} \dfrac {\partial}{\partial x} + \overrightarrow {j} \dfrac {\partial}{\partial y} + \overrightarrow {k} \dfrac {\partial}{\partial z} \right ) \cdot \Big [ (y+z)\overrightarrow {i} + (x+z)\overrightarrow{j} + (y+x)\overrightarrow {k} \Big]$ $= \dfrac {\partial }{\partial x} (y+z) + \dfrac {\partial}{\partial y} (x+z)+ \dfrac {\partial}{\partial z} (y+x)$ $= 0+0+0$ $= 0$ $\nabla \times \bar{A} = \begin{vmatrix} \overrightarrow {i} &\overrightarrow {j} &\overrightarrow {k} \ \partial / \partial x & \partial / \partial y &\partial / \partial z \ y+z & x+z &y+x \end{vmatrix}$ $= \overrightarrow {i} \left [ \dfrac {\partial }{\partial y} (y+x) - \dfrac {\partial }{\partial z} (x+z) \right ] - \overrightarrow {j} \left [ \dfrac {\partial} {\partial x} (y+x) - \dfrac {\partial} {\partial z} (y+z) \right ] + \left [\dfrac {\partial }{\partial x} (x+z) - \dfrac {\partial}{\partial y} (y+z) \right ] $ $= \overrightarrow {i} \Big [ (1+0) - (0+1) \Big ] - \overrightarrow {j} \Big [ (0+1) - (0+1) \Big ] \overrightarrow {k} \Big [ (1+0) - (1+0) \Big ]$ $=0$