VC3.5

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Here,gradu=[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]

gradv=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]

gradw=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,gradu\cdot (gradv\times gradw)=[gradu,gradv,gradw]\,

={\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]

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