VC3.9

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

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

\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^2i-2j+2xzk\,

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

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

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