问题
若已知函数 $f(x, y, z)$ 在平面区域 $D$ 内具有一阶连续偏导数，则对于每一点 $\left(x_{0}, y_{0}, z_{0}\right) \in D$, 该函数在点 $\left(x_{0}, y_{0}, z_{0}\right)$ 处的梯度 $\operatorname{grad} f\left(x_{0}, y_{0}, z_{0}\right)$ $=$ $?$选项
[A]. $\operatorname{grad} f\left(x_{0}, y_{0}, z_{0}\right)$ $=$ $f_{x}\left(x_{0}, y_{0}, z_{0}\right) \boldsymbol{i}$ $\times$ $f_{y}\left(x_{0}, y_{0}, z_{0}\right) \boldsymbol{j}$ $\times$ $f_{z}\left(x_{0}, y_{0}, z_{0}\right) \boldsymbol{k}$[B]. $\operatorname{grad} f\left(x_{0}, y_{0}, z_{0}\right)$ $=$ $f_{x}\left(x_{0}, y_{0}, z_{0}\right) \boldsymbol{i}$ $-$ $f_{y}\left(x_{0}, y_{0}, z_{0}\right) \boldsymbol{j}$ $-$ $f_{z}\left(x_{0}, y_{0}, z_{0}\right) \boldsymbol{k}$
[C]. $\operatorname{grad} f\left(x_{0}, y_{0}, z_{0}\right)$ $=$ $f_{x}\left(x_{0}, y_{0}, z_{0}\right)$ $+$ $f_{y}\left(x_{0}, y_{0}, z_{0}\right)$ $+$ $f_{z}\left(x_{0}, y_{0}, z_{0}\right)$
[D]. $\operatorname{grad} f\left(x_{0}, y_{0}, z_{0}\right)$ $=$ $f_{x}\left(x_{0}, y_{0}, z_{0}\right) \boldsymbol{i}$ $+$ $f_{y}\left(x_{0}, y_{0}, z_{0}\right) \boldsymbol{j}$ $+$ $f_{z}\left(x_{0}, y_{0}, z_{0}\right) \boldsymbol{k}$
$\operatorname{grad} f\left(x_{0}, y_{0}, z_{0}\right)$ $=$ $f_{x}\left(x_{0}, y_{0}, z_{0}\right) \boldsymbol{i}$ $+$ $f_{y}\left(x_{0}, y_{0}, z_{0}\right) \boldsymbol{j}$ $+$ $f_{z}\left(x_{0}, y_{0}, z_{0}\right) \boldsymbol{k}$