二元函数的梯度（B013）

问题

若已知函数 $f(x, y)$ 在平面区域 $D$ 内具有一阶连续偏导数，则对于每一点 $\left(x_{0}, y_{0}\right) \in D$, 该函数在点 $\left(x_{0}, y_{0}\right)$ 处的梯度 $\operatorname{grad} f\left(x_{0}, y_{0}\right)$ $=$ $?$

选项

[A].   $\operatorname{grad} f\left(x_{0}, y_{0} \right)$ $=$ $f_{x}\left(x_{0}, y_{0}\right)$ $+$ $f_{y}\left(x_{0}, y_{0}\right)$

[B].   $\operatorname{grad} f\left(x_{0}, y_{0} \right)$ $=$ $f_{x}\left(x_{0}, y_{0}\right) \boldsymbol{i}$ $+$ $f_{y}\left(x_{0}, y_{0}\right) \boldsymbol{j}$

[C].   $\operatorname{grad} f\left(x_{0}, y_{0} \right)$ $=$ $f_{x}\left(x_{0}, y_{0}\right) \boldsymbol{i}$ $\times$ $f_{y}\left(x_{0}, y_{0}\right) \boldsymbol{j}$

[D].   $\operatorname{grad} f\left(x_{0}, y_{0} \right)$ $=$ $f_{x}\left(x_{0}, y_{0}\right) \boldsymbol{i}$ $-$ $f_{y}\left(x_{0}, y_{0}\right) \boldsymbol{j}$

   答 案   

$\operatorname{grad} f\left(x_{0}, y_{0}\right)$ $=$ $f_{x}\left(x_{0}, y_{0}\right) \boldsymbol{i}$ $+$ $f_{y}\left(x_{0}, y_{0}\right) \boldsymbol{j}$