VC3.9

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\nabla u=i{\frac  {\partial }{\partial x}}(3x^{2}y)+j{\frac  {\partial }{\partial y}}(3x^{2}y)+k{\frac  {\partial }{\partial z}}(3x^{2}y)\,

=i(6xy)+j(3x^{2})+k(0)=6xyi+3x^{2}j\,

\nabla v=i{\frac  {\partial }{\partial x}}(xz^{2}-2y)+j{\frac  {\partial }{\partial y}}(xz^{2}-2y)+k{\frac  {\partial }{\partial z}}(xz^{2}-2y)\,

=z^{2}i-2j+2xzk\,

Therefore,(\nabla u)\cdot (\nabla v)=(6xyi+3x^{2}j)\cdot (z^{2}i-2j+2xzk)\,

=6xyz^{2}-6x^{2}\,

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