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

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

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


上一题 - 荒原之梦   答 案   下一题 - 荒原之梦

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

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

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

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

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


上一题 - 荒原之梦   答 案   下一题 - 荒原之梦

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

三元函数方向导数的计算(B013)

问题

若已知函数 $f(x, y, z)$ 在点 $\left(x_{0}, y_{0}, z_{0}\right)$ 处可微, 且 $(\cos \alpha, \cos \beta, \cos \gamma)$ 是 $\boldsymbol{l}$ 方向的方向余弦.
那么,该函数在该点沿任何方向 $\boldsymbol{l}$ 的方向导数 $\left.\frac{\partial f}{\partial \boldsymbol{l}}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}$ 都存在,则 $\left.\frac{\partial f}{\partial \boldsymbol{l}}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}$ $=$ $?$

选项

[A].   $\left.\frac{\partial f}{\partial \boldsymbol{l}}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}$ $=$ $f_{x}\left(x_{0}, y_{0}, z_{0}\right) \cos \alpha$ $+$ $f_{y}\left(x_{0}, y_{0}, z_{0}\right) \cos \beta$ $+$ $f_{z}\left(x_{0}, y_{0}, z_{0}\right) \cos \gamma$

[B].   $\left.\frac{\partial f}{\partial \boldsymbol{l}}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}$ $=$ $f_{x}\left(x_{0}, y_{0}, z_{0}\right) \cos \gamma$ $+$ $f_{y}\left(x_{0}, y_{0}, z_{0}\right) \cos \beta$ $+$ $f_{z}\left(x_{0}, y_{0}, z_{0}\right) \cos \alpha$

[C].   $\left.\frac{\partial f}{\partial \boldsymbol{l}}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}$ $=$ $f_{x}\left(x_{0}, y_{0}, z_{0}\right) \cos \alpha$ $-$ $f_{y}\left(x_{0}, y_{0}, z_{0}\right) \cos \beta$ $-$ $f_{z}\left(x_{0}, y_{0}, z_{0}\right) \cos \gamma$

[D].   $\left.\frac{\partial f}{\partial \boldsymbol{l}}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}$ $=$ $f_{x}\left(x_{0}, y_{0}, z_{0}\right) \sin \alpha$ $+$ $f_{y}\left(x_{0}, y_{0}, z_{0}\right) \sin \beta$ $+$ $f_{z}\left(x_{0}, y_{0}, z_{0}\right) \sin \gamma$


上一题 - 荒原之梦   答 案   下一题 - 荒原之梦

$\left.\frac{\partial f}{\partial \boldsymbol{l}}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}$ $=$ $f_{x}\left(x_{0}, y_{0}, z_{0}\right) \cos \alpha$ $+$ $f_{y}\left(x_{0}, y_{0}, z_{0}\right) \cos \beta$ $+$ $f_{z}\left(x_{0}, y_{0}, z_{0}\right) \cos \gamma$

二元函数方向导数的计算(B013)

问题

若已知函数 $f(x, y)$ 在点 $\left(x_{0}, y_{0}\right)$ 处可微, 且 $(\cos \alpha, \cos \beta)$ 是 $\boldsymbol{l}$ 方向的方向余弦.
那么,该函数在该点沿任何方向 $\boldsymbol{l}$ 的方向导数 $\left.\frac{\partial f}{\partial \boldsymbol{l}}\right|_{\left(x_{0}, y_{0}\right)}$ 都存在,则 $\left.\frac{\partial f}{\partial \boldsymbol{l}}\right|_{\left(x_{0}, y_{0}\right)}$ $=$ $?$

选项

[A].   $\left.\frac{\partial f}{\partial \boldsymbol{l}}\right|_{\left(x_{0}, y_{0}\right)}$ $=$ $f_{x}\left(x_{0}, y_{0}\right) \cos \alpha$ $-$ $f_{y}\left(x_{0}, y_{0}\right) \cos \beta$

[B].   $\left.\frac{\partial f}{\partial \boldsymbol{l}}\right|_{\left(x_{0}, y_{0}\right)}$ $=$ $f_{x}\left(x_{0}, y_{0}\right) \cos \beta$ $+$ $f_{y}\left(x_{0}, y_{0}\right) \cos \alpha$

[C].   $\left.\frac{\partial f}{\partial \boldsymbol{l}}\right|_{\left(x_{0}, y_{0}\right)}$ $=$ $f_{x}\left(x_{0}, y_{0}\right) \sin \alpha$ $+$ $f_{y}\left(x_{0}, y_{0}\right) \sin \beta$

[D].   $\left.\frac{\partial f}{\partial \boldsymbol{l}}\right|_{\left(x_{0}, y_{0}\right)}$ $=$ $f_{x}\left(x_{0}, y_{0}\right) \cos \alpha$ $+$ $f_{y}\left(x_{0}, y_{0}\right) \cos \beta$


上一题 - 荒原之梦   答 案   下一题 - 荒原之梦

$\left.\frac{\partial f}{\partial \boldsymbol{l}}\right|_{\left(x_{0}, y_{0}\right)}$ $=$ $f_{x}\left(x_{0}, y_{0}\right) \cos \alpha$ $+$ $f_{y}\left(x_{0}, y_{0}\right) \cos \beta$

方向导数的定义/方向导数的存在性证明(B013)

问题

已知 $\boldsymbol{l}$ 为平面上以点 $\left(x_{0}, y_{0}\right)$ 为起点, 以 $(\cos \alpha, \cos \beta)$ 为方向向量的射线, 若将函数 $z$ $=$ $f(x, y)$ 限制在射线 $\boldsymbol{l}$ 上, 则,以下哪个选项对应的极限成立,可以说明函数 $z$ 在点 $\left(x_{0}, y_{0}\right)$ 处沿射线 $\boldsymbol{l}$ 方向的方向导数 $\frac{\partial f}{\partial \boldsymbol{l}} \left(x_{0}, y_{0}\right)$ 存在?

选项

[A].   $\lim _{t \rightarrow 0^{+}}$ $\frac{f\left(x_{0}+t \cos \alpha, y_{0}+t \cos \beta\right)-f\left(x_{0}, y_{0}\right)}{t}$, $t$ $\geqslant$ $1$

[B].   $\lim _{t \rightarrow 0^{+}}$ $\frac{f\left(x_{0}+t \cos \alpha, y_{0}+t \cos \beta\right)-f\left(x_{0}, y_{0}\right)}{t}$, $t$ $\geqslant$ $0$

[C].   $\lim _{t \rightarrow 0^{+}}$ $\frac{f\left(x_{0}+\cos \alpha, y_{0}+\cos \beta\right)-f\left(x_{0}, y_{0}\right)}{t}$, $t$ $\geqslant$ $0$

[D].   $\lim _{t \rightarrow 0^{-}}$ $\frac{f\left(x_{0}+t \cos \alpha, y_{0}+t \cos \beta\right)-f\left(x_{0}, y_{0}\right)}{t}$, $t$ $\geqslant$ $0$


上一题 - 荒原之梦   答 案   下一题 - 荒原之梦

$\lim _{t \rightarrow 0^{+}}$ $\frac{f\left(x_{0}+t \cos \alpha, y_{0}+t \cos \beta\right)-f\left(x_{0}, y_{0}\right)}{t}$, $t$ $\geqslant$ $0$

三元空间曲面上某点处的法线方程(B013)

问题

设曲面 $\Sigma$ 的方程为 $z$ $=$ $f(x, y, z)$, 则在 $\Sigma$ 上的点 $\left(x_{0}, y_{0}, z_{0}\right)$ 处的法线方程是多少?

选项

[A].   $\frac{x+x_{0}}{\left.F_{x}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{y+y_{0}}{\left.F_{y}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{z+z_{0}}{\left.F_{z}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$

[B].   $\frac{x-x_{0}}{\left.F_{x}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{y-y_{0}}{\left.F_{y}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{z-z_{0}}{\left.F_{z}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$

[C].   $\left.F_{x}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(x-x_{0}\right)$ $+$ $\left.F_{y}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(y-y_{0}\right)$ $+$ $\left.F_{x}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(z-z_{0}\right)$ $=$ $0$

[D].   $\frac{x-x_{0}}{\left.F_{x x}^{\prime \prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{y-y_{0}}{\left.F_{y y}^{\prime \prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{z-z_{0}}{\left.F_{z z}^{\prime \prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$


上一题 - 荒原之梦   答 案   下一题 - 荒原之梦

$\frac{x-x_{0}}{\left.F_{x}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{y-y_{0}}{\left.F_{y}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{z-z_{0}}{\left.F_{z}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$

三元空间曲面上某点处的切平面方程(B013)

问题

设曲面 $\Sigma$ 的方程为 $z$ $=$ $f(x, y, z)$, 则在 $\Sigma$ 上的点 $\left(x_{0}, y_{0}, z_{0}\right)$ 处的切平面方程是多少?

选项

[A].   $\left.F_{x}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(x-x_{0}\right)$ $+$ $\left.F_{y}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(y-y_{0}\right)$ $+$ $\left.F_{x}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(z-z_{0}\right)$ $=$ $1$

[B].   $\left.F_{x}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(x-x_{0}\right)$ $+$ $\left.F_{y}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(y-y_{0}\right)$ $+$ $\left.F_{x}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(z-z_{0}\right)$ $=$ $0$

[C].   $\frac{x-x_{0}}{\left.F_{x}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{y-y_{0}}{\left.F_{y}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{z-z_{0}}{\left.F_{z}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$

[D].   $\left.F_{x}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(x-x_{0}\right)$ $-$ $\left.F_{y}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(y-y_{0}\right)$ $-$ $\left.F_{x}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(z-z_{0}\right)$ $=$ $0$


上一题 - 荒原之梦   答 案   下一题 - 荒原之梦

$\left.F_{x}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(x-x_{0}\right)$ $+$ $\left.F_{y}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(y-y_{0}\right)$ $+$ $\left.F_{x}^{\prime}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(z-z_{0}\right)$ $=$ $0$

二元空间曲面上某点处的法线方程(B013)

问题

设曲面 $\Sigma$ 的方程为 $z$ $=$ $f(x, y)$, 则在 $\Sigma$ 上的点 $\left(x_{0}, y_{0}, z_{0}\right)$ 处的法线方程是多少?

选项

[A].   $\frac{x-x_{0}}{\left.\frac{\partial z}{\partial x}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{y-y_{0}}{\left.\frac{\partial z}{\partial y}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{z-z_{0}}{1}$

[B].   $\frac{x-x_{0}}{\left.\frac{\partial z}{\partial x}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{y-y_{0}}{\left.\frac{\partial z}{\partial y}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{z-z_{0}}{-1}$

[C].   $\left.\frac{\partial z}{\partial x}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(x-x_{0}\right)$ $+$ $\left.\frac{\partial z}{\partial y}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(y-y_{0}\right)$ $-$ $\left(z-z_{0}\right)$ $=$ $0$

[D].   $\frac{x+x_{0}}{\left.\frac{\partial z}{\partial x}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{y+y_{0}}{\left.\frac{\partial z}{\partial y}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{z+z_{0}}{-1}$


上一题 - 荒原之梦   答 案   下一题 - 荒原之梦

$\frac{x-x_{0}}{\left.\frac{\partial z}{\partial x}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{y-y_{0}}{\left.\frac{\partial z}{\partial y}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{z-z_{0}}{-1}$

二元空间曲面上某点处的切平面方程(B013)

问题

设曲面 $\Sigma$ 的方程为 $z$ $=$ $f(x, y)$, 则在 $\Sigma$ 上的点 $\left(x_{0}, y_{0}, z_{0}\right)$ 处的切平面方程是多少?

选项

[A].   $\left.\frac{\partial z}{\partial x}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(x-x_{0}\right)$ $-$ $\left.\frac{\partial z}{\partial y}\right|_{\left(x_{0}, y_{0}, x_{2}\right)}\left(y-y_{0}\right)$ $+$ $\left(z-z_{0}\right)$ $=$ $0$

[B].   $\left.\frac{\partial z}{\partial x}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(x-x_{0}\right)$ $+$ $\left.\frac{\partial z}{\partial y}\right|_{\left(x_{0}, y_{0}, x_{2}\right)}\left(y-y_{0}\right)$ $-$ $\left(z-z_{0}\right)$ $=$ $1$

[C].   $\left.\frac{\partial z}{\partial x}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(x-x_{0}\right)$ $+$ $\left.\frac{\partial z}{\partial y}\right|_{\left(x_{0}, y_{0}, x_{2}\right)}\left(y-y_{0}\right)$ $-$ $\left(z-z_{0}\right)$ $=$ $0$

[D].   $\left.\frac{\partial z}{\partial x}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(x+x_{0}\right)$ $+$ $\left.\frac{\partial z}{\partial y}\right|_{\left(x_{0}, y_{0}, x_{2}\right)}\left(y+y_{0}\right)$ $-$ $\left(z+z_{0}\right)$ $=$ $0$


上一题 - 荒原之梦   答 案   下一题 - 荒原之梦

$\left.\frac{\partial z}{\partial x}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(x-x_{0}\right)$ $+$ $\left.\frac{\partial z}{\partial y}\right|_{\left(x_{0}, y_{0}, x_{2}\right)}\left(y-y_{0}\right)$ $-$ $\left(z-z_{0}\right)$ $=$ $0$

空间曲线的法平面方程:基于一般式方程(B013)

问题

若已知空间曲线 $\Gamma$ 的一般式方程为 $\left\{\begin{array}{l} F(x, y, z)=0, \\ G(x, y, z)=0 \end{array}\right.$, 则在曲线 $\Gamma$ 上的点 $(x_{0}, y_{0}, z_{0})$ 处,曲面 $F(x, y, z)$ $=$ $0$ 和 $G(x, y, z)$ $=$ $0$ 的两个法向量 $n_{1}$ 和 $n_{2}$ 分别为:

$\boldsymbol{n}_{1}$ $=$ $\left(F_{x}^{\prime}\left(x_{0}, y_{0}, z_{0} \right), F_{y}^{\prime}\left(x_{0}, y_{0}, z_{0} \right), F_{z}^{\prime}\left(x_{0}, y_{0}, z_{0}\right)\right)$
$\boldsymbol{n}_{1}$ $=$ $\left(G_{x}^{\prime}\left(x_{0}, y_{0}, z_{0} \right), G_{y}^{\prime}\left(x_{0}, y_{0}, z_{0} \right), G_{z}^{\prime}\left(x_{0}, y_{0}, z_{0}\right)\right)$

该点处的切向量为:
$\boldsymbol{\tau}$ $=$ $\boldsymbol{n}_{1} \times \boldsymbol{n}_{2}$

若记切向量 $\boldsymbol{\tau}$ $=$ $(A, B, C)$,
则曲线 $\Gamma$ 在点 $(x_{0}, y_{0}, z_{0})$ 处的法平面方程是多少?

选项

[A].   $\frac{x-x_{0}}{A}$ $=$ $\frac{y-y_{0}}{B}$ $=$ $\frac{z-z_{0}}{C}$

[B].   $A$ $\left(x-x_{0}\right)$ $-$ $B$ $\left(y-y_{0}\right)$ $-$ $C$ $\left(z-z_{0}\right)$ $=$ $0$

[C].   $A$ $\left(x+x_{0}\right)$ $+$ $B$ $\left(y+y_{0}\right)$ $+$ $C$ $\left(z+z_{0}\right)$ $=$ $0$

[D].   $A$ $\left(x-x_{0}\right)$ $+$ $B$ $\left(y-y_{0}\right)$ $+$ $C$ $\left(z-z_{0}\right)$ $=$ $0$


上一题 - 荒原之梦   答 案   下一题 - 荒原之梦

$A$ $\left(x-x_{0}\right)$ $+$ $B$ $\left(y-y_{0}\right)$ $+$ $C$ $\left(z-z_{0}\right)$ $=$ $0$

空间曲线的切线方程:基于一般式方程(B013)

问题

若已知空间曲线 $\Gamma$ 的一般式方程为 $\left\{\begin{array}{l} F(x, y, z)=0, \\ G(x, y, z)=0 \end{array}\right.$, 则在曲线 $\Gamma$ 上的点 $(x_{0}, y_{0}, z_{0})$ 处,曲面 $F(x, y, z)$ $=$ $0$ 和 $G(x, y, z)$ $=$ $0$ 的两个法向量 $n_{1}$ 和 $n_{2}$ 分别为:

$\boldsymbol{n}_{1}$ $=$ $\left(F_{x}^{\prime}\left(x_{0}, y_{0}, z_{0} \right), F_{y}^{\prime}\left(x_{0}, y_{0}, z_{0} \right), F_{z}^{\prime}\left(x_{0}, y_{0}, z_{0}\right)\right)$
$\boldsymbol{n}_{1}$ $=$ $\left(G_{x}^{\prime}\left(x_{0}, y_{0}, z_{0} \right), G_{y}^{\prime}\left(x_{0}, y_{0}, z_{0} \right), G_{z}^{\prime}\left(x_{0}, y_{0}, z_{0}\right)\right)$

该点处的切向量为:
$\boldsymbol{\tau}$ $=$ $\boldsymbol{n}_{1} \times \boldsymbol{n}_{2}$

若记切向量 $\boldsymbol{\tau}$ $=$ $(A, B, C)$,
则曲线 $\Gamma$ 在点 $(x_{0}, y_{0}, z_{0})$ 处的切线方程是多少?

选项

[A].   $\frac{x-x_{0}}{\sqrt{A}}$ $=$ $\frac{y-y_{0}}{\sqrt{B}}$ $=$ $\frac{z-z_{0}}{\sqrt{C}}$

[B].   $\frac{x+x_{0}}{A}$ $=$ $\frac{y+y_{0}}{B}$ $=$ $\frac{z+z_{0}}{C}$

[C].   $\frac{x-x_{0}}{A}$ $=$ $\frac{y-y_{0}}{B}$ $=$ $\frac{z-z_{0}}{C}$

[D].   $\frac{A}{x-x_{0}}$ $=$ $\frac{B}{y-y_{0}}$ $=$ $\frac{C}{z-z_{0}}$


上一题 - 荒原之梦   答 案   下一题 - 荒原之梦

$\frac{x-x_{0}}{A}$ $=$ $\frac{y-y_{0}}{B}$ $=$ $\frac{z-z_{0}}{C}$

空间曲线的切向量:基于一般式方程(B013)

问题

若已知空间曲线 $\Gamma$ 的一般式方程为 $\left\{\begin{array}{l} F(x, y, z)=0, \\ G(x, y, z)=0 \end{array}\right.$, 则在曲线 $\Gamma$ 上的点 $(x_{0}, y_{0}, z_{0})$ 处,曲面 $F(x, y, z)$ $=$ $0$ 和 $G(x, y, z)$ $=$ $0$ 的两个法向量 $n_{1}$ 和 $n_{2}$ 分别为:
$\boldsymbol{n}_{1}$ $=$ $\left(F_{x}^{\prime}\left(x_{0}, y_{0}, z_{0} \right), F_{y}^{\prime}\left(x_{0}, y_{0}, z_{0} \right), F_{z}^{\prime}\left(x_{0}, y_{0}, z_{0}\right)\right)$
$\boldsymbol{n}_{1}$ $=$ $\left(G_{x}^{\prime}\left(x_{0}, y_{0}, z_{0} \right), G_{y}^{\prime}\left(x_{0}, y_{0}, z_{0} \right), G_{z}^{\prime}\left(x_{0}, y_{0}, z_{0}\right)\right)$,

则曲线 $\Gamma$ 在点 $(x_{0}, y_{0}, z_{0})$ 处的切向量 $\boldsymbol{\tau}$ $=$ $?$

选项

[A].   $\boldsymbol{\tau}$ $=$ $\boldsymbol{n}_{1} \times \boldsymbol{n}_{2}$

[B].   $\boldsymbol{\tau}$ $=$ $\boldsymbol{n}_{1} + \boldsymbol{n}_{2}$

[C].   $\boldsymbol{\tau}$ $=$ $\boldsymbol{n}_{1} – \boldsymbol{n}_{2}$

[D].   $\boldsymbol{\tau}$ $=$ $\boldsymbol{n}_{1} \div \boldsymbol{n}_{2}$


上一题 - 荒原之梦   答 案   下一题 - 荒原之梦

$\boldsymbol{\tau}$ $=$ $\boldsymbol{n}_{1} \times \boldsymbol{n}_{2}$

形成空间曲线的空间曲面的法向量:基于一般式方程(B013)

问题

若已知空间曲线 $\Gamma$ 的一般式方程为 $\left\{\begin{array}{l} F(x, y, z)=0, \\ G(x, y, z)=0 \end{array}\right.$, 则在曲线 $\Gamma$ 上的点 $(x_{0}, y_{0}, z_{0})$ 处,曲面 $F(x, y, z)$ $=$ $0$ 和 $G(x, y, z)$ $=$ $0$ 的两个法向量 $n_{1}$ 和 $n_{2}$ 分别是多少?

选项

[A].   $\boldsymbol{n}_{1}$ $=$ $($ $F_{x x}^{\prime \prime}$ $($ $x_{0}, y_{0}, z_{0}$ $)$, $F_{y y}^{\prime \prime}$ $($ $x_{0}, y_{0}, z_{0}$ $)$, $F_{z z}^{\prime \prime}$ $(x_{0}, y_{0}, z_{0}$ $)$ $)$
$\boldsymbol{n}_{2}$ $=$ $($ $G_{x x}^{\prime \prime}$ $($ $x_{0}, y_{0}, z_{0}$ $)$, $G_{y y}^{\prime \prime}$ $($ $x_{0}, y_{0}, z_{0}$ $)$, $G_{z z}^{\prime \prime}$ $(x_{0}, y_{0}, z_{0}$ $)$ $)$


[B].   $\boldsymbol{n}_{1}$ $=$ $($ $- F_{x}^{\prime}$ $($ $x_{0}, y_{0}, z_{0}$ $)$, $- F_{y}^{\prime}$ $($ $x_{0}, y_{0}, z_{0}$ $)$, $- F_{z}^{\prime}$ $(x_{0}, y_{0}, z_{0}$ $)$ $)$
$\boldsymbol{n}_{2}$ $=$ $($ $- G_{x}^{\prime}$ $($ $x_{0}, y_{0}, z_{0}$ $)$, $- G_{y}^{\prime}$ $($ $x_{0}, y_{0}, z_{0}$ $)$, $- G_{z}^{\prime}$ $(x_{0}, y_{0}, z_{0}$ $)$ $)$


[C].   $\boldsymbol{n}_{1}$ $=$ $($ $F_{x}$ $($ $x_{0}, y_{0}, z_{0}$ $)$, $F_{y}$ $($ $x_{0}, y_{0}, z_{0}$ $)$, $F_{z}$ $(x_{0}, y_{0}, z_{0}$ $)$ $)$
$\boldsymbol{n}_{2}$ $=$ $($ $G_{x}$ $($ $x_{0}, y_{0}, z_{0}$ $)$, $G_{y}$ $($ $x_{0}, y_{0}, z_{0}$ $)$, $G_{z}$ $(x_{0}, y_{0}, z_{0}$ $)$ $)$


[D].   $\boldsymbol{n}_{1}$ $=$ $($ $F_{x}^{\prime}$ $($ $x_{0}, y_{0}, z_{0}$ $)$, $F_{y}^{\prime}$ $($ $x_{0}, y_{0}, z_{0}$ $)$, $F_{z}^{\prime}$ $(x_{0}, y_{0}, z_{0}$ $)$ $)$
$\boldsymbol{n}_{2}$ $=$ $($ $G_{x}^{\prime}$ $($ $x_{0}, y_{0}, z_{0}$ $)$, $G_{y}^{\prime}$ $($ $x_{0}, y_{0}, z_{0}$ $)$, $G_{z}^{\prime}$ $(x_{0}, y_{0}, z_{0}$ $)$ $)$



上一题 - 荒原之梦   答 案   下一题 - 荒原之梦

$\boldsymbol{n}_{1}$ $=$ $($ $F_{x}^{\prime}$ $($ $x_{0}, y_{0}, z_{0}$ $)$, $F_{y}^{\prime}$ $($ $x_{0}, y_{0}, z_{0}$ $)$, $F_{z}^{\prime}$ $(x_{0}, y_{0}, z_{0}$ $)$ $)$
$\boldsymbol{n}_{2}$ $=$ $($ $G_{x}^{\prime}$ $($ $x_{0}, y_{0}, z_{0}$ $)$, $G_{y}^{\prime}$ $($ $x_{0}, y_{0}, z_{0}$ $)$, $G_{z}^{\prime}$ $(x_{0}, y_{0}, z_{0}$ $)$ $)$

空间曲线的切向量:基于参数方程(B013)

问题

若已知空间曲线 $\Gamma$ 的参数方程为 $\left\{\begin{array}{l}x=x(t), \\ y=y(t) \\ z=z(t)\end{array}\right.$, 则曲线 $\Gamma$ 在点 $(x_{0}, y_{0}, z_{0})$(对应参数 $t$ $=$ $t_{0}$)处的切向量为多少?

选项

[A].   $\tau$ $=$ $\left\{x \left(t_{0} \right), y \left(t_{0}\right), z \left(t_{0}\right)\right\}$

[B].   $\tau$ $=$ $\left\{x^{\prime}\left(t_{0} \right), y^{\prime}\left(t_{0}\right), z^{\prime}\left(t_{0}\right)\right\}$

[C].   $\tau$ $=$ $\left\{x^{\prime \prime}\left(t_{0} \right), y^{\prime \prime}\left(t_{0}\right), z^{\prime \prime}\left(t_{0}\right)\right\}$

[D].   $\tau$ $=$ $-$ $\left\{x^{\prime}\left(t_{0} \right), y^{\prime}\left(t_{0}\right), z^{\prime}\left(t_{0}\right)\right\}$


上一题 - 荒原之梦   答 案   下一题 - 荒原之梦

$\tau$ $=$ $\left\{x^{\prime}\left(t_{0} \right), y^{\prime}\left(t_{0}\right), z^{\prime}\left(t_{0}\right)\right\}$

空间曲线的法平面方程:基于参数方程(B013)

问题

若已知空间曲线 $\Gamma$ 的参数方程为 $\left\{\begin{array}{l}x=x(t), \\ y=y(t) \\ z=z(t)\end{array}\right.$, 则曲线 $\Gamma$ 在点 $(x_{0}, y_{0}, z_{0})$(对应参数 $t$ $=$ $t_{0}$)处的法平面方程为多少?

选项

[A].   $x^{\prime}\left(t_{0} \right)$ $\left(x+x_{0} \right)$ $-$ $y^{\prime}\left(t_{0} \right)$ $\left(y+y_{0} \right)$ $-$ $z^{\prime}\left(t_{0} \right)$ $\left(z+z_{0} \right)$ $=$ $0$

[B].   $x^{\prime}\left(t_{0} \right)$ $\left(x-x_{0} \right)$ $+$ $y^{\prime}\left(t_{0} \right)$ $\left(y-y_{0} \right)$ $+$ $z^{\prime}\left(t_{0} \right)$ $\left(z-z_{0} \right)$ $=$ $0$

[C].   $x^{\prime}\left(t_{0} \right)$ $\left(x-x_{0} \right)$ $\times$ $y^{\prime}\left(t_{0} \right)$ $\left(y-y_{0} \right)$ $\times$ $z^{\prime}\left(t_{0} \right)$ $\left(z-z_{0} \right)$ $=$ $1$

[D].   $x \left(t_{0} \right)$ $\left(x-x_{0} \right)$ $+$ $y \left(t_{0} \right)$ $\left(y-y_{0} \right)$ $+$ $z \left(t_{0} \right)$ $\left(z-z_{0} \right)$ $=$ $0$


上一题 - 荒原之梦   答 案   下一题 - 荒原之梦

$x^{\prime}\left(t_{0} \right)$ $\left(x-x_{0} \right)$ $+$ $y^{\prime}\left(t_{0} \right)$ $\left(y-y_{0} \right)$ $+$ $z^{\prime}\left(t_{0} \right)$ $\left(z-z_{0} \right)$ $=$ $0$


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