VC3.44

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Here \phi =x^{2}yz+2xz^{2}\,

Now,\nabla \phi =[i{\frac  {\partial }{\partial }}+j{\frac  {\partial }{\partial y}}+k{\frac  {\partial }{\partial z}}](x^{2}yz+4xz^{2})\,

=(2xyz+4z^{2})i+x^{2}zj+(x^{2}y+8xz)k=8i-j-10k\, at the given point.

The unit vector in the direction of 2i-j-2k\,={\hat  a}={\frac  {2i-j-2k}{{\sqrt  {4+1+4}}}}={\frac  {1}{3}}(2i-j-2k)\,

So the required directional derivative at(1,-2,-1)=\nabla \phi \cdot {\hat  a}=(8i-j-10k)\cdot {\frac  {2i-j-2k}{3}}={\frac  {16+1+20}{3}}={\frac  {37}{3}}\,

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