VC3.2

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By definition of gradient,we have

grad f=[i\frac{\partial}{\partial x}+j\frac{\partial}{\partial y}+k\frac{\partial}{\partial z}](3x^2y-y^3z^2)\,

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

= 6xyi+3(x^2-y^2z^2)j-2y^3zk\,

Therefore at the point (1,-2,-1): grad \ f = 6(1)(-2)i+3 \left [ (1)^2-(-2)^2(-1)^2 \right ] j -2 \left [ (-2)^3(-1) \right ] k \,

= - 12i - 9j - 16k \,

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