【答案】1 

【解析】由题易知，\(\frac{\partial u}{\partial x}=y^{2}z^{3}\)，\(\frac{\partial u}{\partial y}=2xyz^{3}\)，\(\frac{\partial u}{\partial z}=3xy^{2}z^{2}\) 

则在\(x = 1,y = 1,z = 1\)处有\(\left(\frac{\partial u}{\partial x},\frac{\partial u}{\partial y},\frac{\partial u}{\partial z}\right)=(1,2,3)\) 

对于向量\(\boldsymbol{\bar{n}}=(2,2,-1)\)，归一化可得\(\boldsymbol{\bar{n}}_{0}=\left(\frac{2}{3},\frac{2}{3},-\frac{1}{3}\right)\) 

故\(\left. \frac{\partial u}{\partial \boldsymbol{n}}\right\vert _{(1,1,1)}=\left(\frac{\partial u}{\partial x},\frac{\partial u}{\partial y},\frac{\partial u}{\partial z}\right)\cdot \boldsymbol{\bar{n}}_{0}=(1,2,3)\cdot \left(\frac{2}{3},\frac{2}{3},-\frac{1}{3}\right)=1\cdot \frac{2}{3}+2\cdot \frac{2}{3}+3\cdot \left(-\frac{1}{3}\right)=1\)