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

Γ

的一般式方程为

{\begin{cases} F (x, y, z) = 0, \\ G (x, y, z) = 0 \end{cases}

, 则在曲线

Γ

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].

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}

)

)

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}

)

)

[B].

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}

)

)

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}

)

)

[C].

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}

)

)

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].

n_{1}

=

(

F_{x x}^{''}

(

x_{0}, y_{0}, z_{0}

)

F_{y y}^{''}

(

x_{0}, y_{0}, z_{0}

)

F_{z z}^{''}

(x_{0}, y_{0}, z_{0}

)

)

n_{2}

=

(

G_{x x}^{''}

(

x_{0}, y_{0}, z_{0}

)

G_{y y}^{''}

(

x_{0}, y_{0}, z_{0}

)

G_{z z}^{''}

(x_{0}, y_{0}, z_{0}

)

)

答案

$n_{1}$ $=$ $($ $F_{x}^{'}$ $($ $x_{0}, y_{0}, z_{0}$ $)$ , $F_{y}^{'}$ $($ $x_{0}, y_{0}, z_{0}$ $)$ , $F_{z}^{'}$ $(x_{0}, y_{0}, z_{0}$ $)$ $)$
$n_{2}$ $=$ $($ $G_{x}^{'}$ $($ $x_{0}, y_{0}, z_{0}$ $)$ , $G_{y}^{'}$ $($ $x_{0}, y_{0}, z_{0}$ $)$ , $G_{z}^{'}$ $(x_{0}, y_{0}, z_{0}$ $)$ $)$

问题

选项

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