VC3.2

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By definition of gradient,we have

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

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

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

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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