VC3.5

Here, $grad u=[i\frac{\partial}{\partial x}+j\frac{\partial}{\partial y}+k\frac{\partial}{\partial z}](x+y+z)\,$

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

= $i+j+k\,$ [Equation 1]

$grad v=i\frac{\partial}{\partial x}(x^2+y^2+z^2)+j\frac{\partial}{\partial y}(x^2+y^2+z^2)+k\frac{\partial}{\partial z}(x^2+y^2+z^2)\,$

= $2xi+2yj+2zk\,$ [Equation 2]

$grad w=i\frac{\partial}{\partial x}(xy+yz+zx)+j\frac{\partial}{\partial y}(xy+yz+zx)+k\frac{\partial}{\partial z}(xy+yz+zx)\,$

= $(y+z)i+(z+x)j+(x+y)k\,$ [Equation 3]

Therefore, $grad u\cdot (grad v\times grad w)=[grad u,grad v,grad w]\,$

= $\begin{vmatrix} 1 & 1 & 1 \\ 2x & 2y & 2z \\ y+z & z+x & x+y \end{vmatrix}\,$

= $2\begin{vmatrix} 1 & 1 & 1 \\ x & y & z \\ y+z & z+x & x+y \end{vmatrix}\,$ , Performing $R_3\longrightarrow R_3+R_2\,$

= $2(x+y+z)\begin{vmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ y+z & z+x & x+y \end{vmatrix}\,$

= $0\,$ [As first two rows are identical]