三元函数的梯度(B013)

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


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


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