空间中的曲线与曲面

多变量函数的微分学

张瑞
中国科学技术大学数学科学学院

空间中的曲线与曲面

参数曲线

定义 1.
映射

\[f(t)=(x(t),y(t),z(t)) , t\in[\alpha,\beta] \]

完全刻画了质点的运动轨迹。常写成向径式

\[\vec r(t)=(x(t),y(t),z(t)) , t\in[\alpha,\beta] \]

或

\[\begin{cases} x=x(t) \\ y=y(t) \\ z=z(t) \end{cases} , x\in[\alpha,\beta] \]

称为空间曲线的参数方程

  • 空间曲线$L: \vec r=\vec r(t)$，若$x'(t)$, $y'(t)$, $z'(t)$在$[\alpha,\beta]$上连续，且不同时为$0$，则称$L$为光滑曲线。
  • 若$L$自身不相交，即$\vec r(t_1)\neq\vec r(t_2), \forall t_1\neq t_2$，则称$L$为简单曲线或Jordan曲线。
  • 若$L$首尾相接，即$\vec r(\alpha)=\vec r(\beta)$，则称$L$为闭曲线。

切线

  • 若$\vec r(t)$表示一个质点的运动轨迹，则质点在$t_0$到$t$时刻的平均速度(向量)为
    \[\frac{\vec r(t)-\vec r(t_0)}{t-t_0} \]
    在$t_0$时刻的瞬时速度(向量)就是极限
    \[\lim_{t\to t_0}\frac{\vec r(t)-\vec r(t_0)}{t-t_0} \]
  • 几何上也是类似的。在$t_0$时刻的瞬时变化率为向量
    \[\lim_{t\to t_0}\frac{\vec r(t)-\vec r(t_0)}{t-t_0} \]

根据向量函数运算规则，可以得到

\[\begin{aligned} & \lim_{t \to t_0}\frac{\vec r(t)-\vec r(t_0)}{t-t_0} \\ %\vec r'(t_0) =&\lim_{t\to t_0}\frac{\big(x(t)-x(t_0),y(t)-y(t_0),z(t)-z(t_0)\big)}{t-t_0} \\ =&(\lim_{t\to t_0}\frac{x(t)-x(t_0)}{t-t_0}, \lim_{t\to t_0}\frac{y(t)-y(t_0)}{t-t_0}, \lim_{t\to t_0}\frac{z(t)-z(t_0)}{t-t_0}) \\ =&(x'(t_0), y'(t_0), z'(t_0)) \end{aligned} \]
  • 这个极限记为$\vec r'(t_0)$,
    \[\vec r'(t_0) =\displaystyle \lim_{t \to t_0}\frac{\vec r(t)-\vec r(t_0)}{t-t_0} =(x'(t_0), y'(t_0), z'(t_0)) \]
  • 与1维情形类似，这个极限表示了曲线在$M_0$处的切线方向，称为切向量。
  • 它指向参数增加的方向。
  • 若$L: \vec r(t)$是光滑曲线，则$\vec r'(t_0)\neq \vec 0$。
  • 曲线$L$在$M_0$处的切线方程
    \[\frac{x-x(t_0)}{x'(t_0)}=\frac{y-y(t_0)}{y'(t_0)}=\frac{z-z(t_0)}{z'(t_0)} \]
  • 在$M_0$处的法平面方程
    \[x'(t_0)(x-x(t_0))+y'(t_0)(y-y(t_0))+z'(t_0)(z-z(t_0))=0 \]
  • $\vec r(t)$的微分
    \[\begin{aligned} d \vec r =&(dx,dy,dz) \\ =&(x'(t)dt, y'(t)dt, z'(t)dt)=\vec r'(t) dt \end{aligned} \]

若$\vec r(t), t\in[\alpha,\beta]$表示质点的位移(向量)，

  • 则
    \[\begin{aligned} |\vec r'(t)|dt =&\sqrt{x'(t)^2+y'(t)^2+z'(t)^2}dt \\ =&\sqrt{dx^2+dy^2+dz^2} =|d\vec r(t)| \end{aligned} \]
    表示一瞬时的路程(标量)。
  • 由“微元分析法”，
    \[\int_{\alpha}^{\beta} |d\vec r| =\int_{\alpha}^{\beta}|\vec r'(t)|dt\ =\int_{\alpha}^{\beta}\sqrt{x'(t)^2+y'(t)^2+z'(t)^2}dt \]
    就是质点走过的路程，也就是曲线的长度。在后面会再讨论。

空间曲面

$D$为$\mathbb{R}^2$中的区域，定义在$D$上的一个二元向量值函数

\[\vec r(u,v)=(x(u,v),y(u,v),z(u,v)) , (u,v)\in D \]

确定了空间中的一个曲面，称为曲面的向径式方程。

它等价于曲面的参数方程

\[\begin{cases} x=x(u,v) \\ y=y(u,v) \\ z=z(u,v) \end{cases} , (u,v)\in D \]

例 1. 球面方程

\[\begin{cases} x(\theta, \phi)=R\sin\theta\cos\phi , \\ y(\theta, \phi)=R\sin\theta\sin\phi , \\ z(\theta, \phi)=R\cos\theta \end{cases} \theta\in[0,\pi], \phi\in[0,2\pi] \]

当$\theta$固定时，得到纬线；当$\phi$固定时，得到经线。

\begin{tikzpicture} [scale=0.5] \begin{axis} [ view={110}{20}, axis lines=middle, width=10cm, ztick=\empty,ytick=\empty,xtick=\empty, %xlabel=$x$, %zlabel={$z$} ] \addplot3[ domain=0:90, samples = 60, samples y=0, %thick, ] ({cos(x)}, {sin(x)}, 0); \addplot3[ domain=0:90, samples = 60, samples y=0, %thick, ] (0, {cos(x)}, {sin(x)}); \addplot3[ domain=0:90, samples = 60, samples y=0, %thick, ] ({cos(x)}, 0, {sin(x)}); \addplot3[line legend] coordinates { (-0.1,0,0) (1.3,0,0) } node[left] {$x$}; \addplot3[->] coordinates { (0,-0.1,0) (0,1.3,0) } node[right] {$y$}; \addplot3[->] coordinates { (0,0,-0.1) (0,0,1.3) } node[above] {$z$}; % \addplot3[dashed, blue] coordinates { (0,0,0) ({sin(45)},{cos(45)},0) }; \addplot3[ domain=0:90, samples = 60, samples y=0, dashed, blue, ] ({sin(x)*cos(45)}, {sin(x)*sin(45)}, {cos(x)}); \addplot3[ domain=0:90, samples = 60, samples y=0, blue, ] ({sin(30)*cos(x)}, {sin(30)*sin(x)}, {cos(30)}); \addplot3[-latex] coordinates { (0,0,0) ({sin(30)*cos(45)}, {sin(30)*sin(45)}, {cos(30)}) } node[above] {$P$} node[pos=0.5, below] {$\rho$}; \addplot3[-latex] coordinates { ({sin(30)*cos(45)}, {sin(30)*sin(45)+0.1}, {cos(30)}) ({sin(30)*cos(45)}, {sin(30)*sin(45)}, {cos(30)}) } node[pos=0, right] {$(\rho,\theta,\phi)$}; \addplot3[ domain=0:30, samples = 5, samples y=0, blue, -latex, ] ({0.2*sin(x)*cos(45)}, {0.2*sin(x)*sin(45)}, {0.2*cos(x)}) node[pos=0.5, above] {$\theta$}; \addplot3[ domain=0:45, samples = 5, samples y=0, blue, -latex, ] ({0.2*sin(90)*cos(x)}, {0.2*sin(90)*sin(x)}, {0.2*cos(90)}) node[pos=0.5, below] {$\phi$}; \addplot3[red] coordinates { ({sin(30)*cos(45)}, {sin(30)*sin(45)}, {cos(30)}) ({sin(30)*cos(45)}, {sin(30)*sin(45)}, 0) } node[pos=1, below] {$P_1$} node[pos=0.5, right] {$z$}; \addplot3[red] coordinates { ({sin(30)*cos(45)}, {sin(30)*sin(45)}, 0) (0, {sin(30)*sin(45)}, 0) } %({sin(30)*cos(45)}, 0, 0) } node[pos=1, above] {$P_2$} node[pos=0.5, below] {$x$}; \addplot3[red] coordinates { (0, {sin(30)*sin(45)}, 0) (0, 0, 0) } node[pos=1, above left, black] {$O$} node[pos=0.5, below] {$y$}; \end{axis} \end{tikzpicture}

固定一个$v$值，让$u$变化，则$\vec r(u,v)$在曲面上画出一条曲线，称为u曲线。 同样有v曲线。

\begin{tikzpicture}[x=2cm, y=2cm, global scale=1., >=latex,thick] \coordinate (orig) at (0,0); \draw[thick,->] (orig) -- (2.2,0) node[right] {$y$};% y軸 \draw[thick,->] (orig) -- (0,1.5) node[above] {$z$};% z軸 \draw[thick,->] (orig) -- (-130:0.5) node[below] {$x$};% x軸 \draw[fill=blue, fill opacity=0.5, draw=orange, thick] (150:0.5) to[out=20, in=120] coordinate [pos=0.25] (p11) coordinate [pos=0.5] (p12) coordinate [pos=0.75] (p13) (-10:1.5) to[out=80, in=170] coordinate [pos=0.25] (p21) coordinate [pos=0.5] (p22) coordinate [pos=0.75] (p23) (15:2.1) to[out=140, in=-20] coordinate [pos=0.25] (p31) coordinate [pos=0.5] (p32) coordinate [pos=0.75] (p33) (75:1.3) to[out=-130, in=80] coordinate [pos=0.25] (p41) coordinate [pos=0.5] (p42) coordinate [pos=0.75] (p43) cycle; % draw v-curve \draw[dashed, red] (p31) to[out=-150, in=80] (p13); \draw[dashed, red, name path=v1] (p32) to[out=-150, in=80] (p12); \draw[dashed, red] (p33) to[out=-150, in=80] (p11); % draw u-curve \draw[red] (p23) to[out=140, in=20] (p41); \draw[red, name path=u1] (p22) to[out=140, in=20] (p42); \draw[red] (p21) to[out=120, in=20] (p43); \path[name intersections={of=u1 and v1}]; \coordinate (uv) at (intersection-1); \fill (uv) circle (1pt); \draw[->] (20:0.5) node [below] {$v$ curve} -- (35:0.8); \draw[->] (60:1.5) node [above] {$u$ curve} -- (35:1.3); % 画 u-v 平面 \begin{scope}[xshift = -4cm] \draw[thick,->] (-0.2,0) -- (1.2,0) node[right] {$u$};% x軸 \draw[thick,->] (0,-0.1) -- (0,1.1) node[above] {$v$};% y軸 \draw [orange, name path=domain] plot [smooth cycle] coordinates {(0.1,0.5) (0.6,0.1) (1.1,0.55) (0.55,1.0) }; \foreach \i in {0.25, 0.5, 0.75} { \path[name path=v1] (\i,0)--(\i,1.5); \path[name intersections={of=v1 and domain}]; \draw[dashed, red] (intersection-1) -- (intersection-2); \path[name path=u1] (0,\i)--(1.5,\i); \path[name intersections={of=u1 and domain}]; \draw[red] (intersection-1) -- (intersection-2); } \end{scope} % 画一个箭头 \draw[->, ultra thick] (130:1.2) to[out=50, in=170] node[above]{$\vec r(u,v)$} (110:1); \end{tikzpicture}

若$\vec r(u,v)\in C^1(D)$，即$x(u,v)$, $y(u,v)$, $z(u,v)$均为$D$上的$C^1$函数，$M_0=\vec r(u_0, v_0)$为曲面上的一点

  • 过$M_0$的$u$曲线的参数方程为
    \[\begin{cases} x=x(u,v_0) \\ y=y(u, v_0) \\ z=z(u, v_0) \end{cases} \]
    因此，$M_0$处u曲线的切向量就是
    \[(x'_u(u_0,v_0), y'_u(u_0,v_0), z'_u(u_0,v_0))=\vec r'_u(u_0, v_0) \]
  • 类似可以得到，$M_0$处v曲线的切向量是$\vec r'_v(u_0,v_0)$

任取曲面上过$M_0$ 点的光滑曲线$L$，则

  • $L$的原像为平面区域$D$中，过$(u_0,v_0)$的一条曲线。设这条曲线有参数方程

    \[u=u(t), v=v(t) \]

    其中$u_0=u(t_0)$, $v_0=v(t_0)$。

  • 则$L$具有参数方程

    \[\begin{cases} x=x(u(t),v(t)) \\ y=y(u(t),v(t)) \\ z=z(u(t),v(t)) \end{cases} \]

  • $L$在$M_0$处的切向量为

    \[\vec \tau=(x'(t_0), y'(t_0), z'(t_0)) \]
\[\begin{aligned} \vec \tau =&(x'(t_0), y'(t_0), z'(t_0)) \\ =&(x'_u u'(t_0)+x'_v v'(t_0), \\ &\ y'_u u'(t_0)+y'_v v'(t_0), \\ &\ z'_u u'(t_0)+z'_v v'(t_0)) \\ =&(x'_u ,y'_u ,z'_u) u'(t_0)+(x'_v ,y'_v ,z'_v) v'(t_0) \\ =&\vec r'_u(u_0,v_0) u'(t_0)+\vec r'_v(u_0,v_0) v'(t_0) \end{aligned} \]

这个表明，曲面上过$M_0$的任何一条曲线的切向量为$r'_u(u_0,v_0)$与$r'_v(u_0,v_0)$的线性组合。

因此切向量在$r'_u(u_0,v_0)$与$r'_v(u_0,v_0)$张成的平面上。

定义 2.
曲面过$M_0$, 由$\vec r'_u(u_0,v_0)$与$\vec r'_v(u_0,v_0)$张成的平面, 称为曲面在$M_0$处的切平面

它的法向量是

\[\begin{aligned} &\vec r'_u(u_0,v_0)\times \vec r'_v(u_0,v_0) \\ =&\begin{vmatrix} \vec i & \vec j & \vec k \\ x'_u & y'_u & z'_u \\ x'_v & y'_v & z'_v \\ \end{vmatrix} \\ =&\left(\frac{\partial (y,z)}{\partial (u,v)}, \frac{\partial (z,x)}{\partial (u,v)}, \frac{\partial (x,y)}{\partial (u,v)}\right) \end{aligned} \]
\begin{tikzpicture}[x=2cm, y=2cm, global scale=0.8, >=latex,thick] \coordinate (orig) at (0,0); \draw[thick,->] (orig) -- (2.2,0) node[right] {$y$};% y軸 \draw[thick,->] (orig) -- (0,1.5) node[above] {$z$};% z軸 \draw[thick,->] (orig) -- (-130:0.5) node[below] {$x$};% x軸 \draw[fill=blue, fill opacity=0.5, draw=orange, thick] (150:0.5) to[out=20, in=120] coordinate [pos=0.25] (p11) coordinate [pos=0.5] (p12) coordinate [pos=0.75] (p13) (-10:1.5) to[out=80, in=170] coordinate [pos=0.25] (p21) coordinate [pos=0.5] (p22) coordinate [pos=0.75] (p23) (15:2.1) to[out=140, in=-20] coordinate [pos=0.25] (p31) coordinate [pos=0.5] (p32) coordinate [pos=0.75] (p33) (75:1.3) to[out=-130, in=80] coordinate [pos=0.25] (p41) coordinate [pos=0.5] (p42) coordinate [pos=0.75] (p43) cycle; % draw v-curve \draw[dashed, red] (p31) to[out=-150, in=80] (p13); \draw[dashed, red, name path=v1] (p32) to[out=-150, in=80] (p12); \draw[dashed, red] (p33) to[out=-150, in=80] (p11); % draw u-curve \draw[red] (p23) to[out=140, in=20] (p41); \draw[red, name path=u1] (p22) to[out=140, in=20] (p42); \draw[red] (p21) to[out=120, in=20] (p43); \path[name intersections={of=u1 and v1}]; \coordinate (uv) at (intersection-1); \draw[blue, fill opacity=0.5, fill=yellow] (uv) -- ++(-20:0.5) -- ++(42:0.5) -- ++ (-20:-0.5) --cycle; \draw[->, red] (uv) -- +(-20:0.7) node[right] {$\vec r'_u$}; \draw[->, red, dashed] (uv) -- +(42:0.7) node[above] {$\vec r'_v$}; \draw[->, green] (uv) -- +(70:0.5) node[above] {$\vec n$}; \end{tikzpicture}

定义 3.
若曲面的参数方程中$x(u,v)$, $y(u,v)$, $z(u,v)$均为$D$上的$C^1$函数， 且$D$中处处有$\vec r'_u\times \vec r'_v\neq 0$，称曲面为光滑曲面。

由二元函数

\[z=f(x,y), (x,y)\in D \]

表达的曲面通常称为显式曲面。

显式曲面可以写成参数方程

\[\vec r=\vec r(x,y)=(x,y,f(x,y)) , \quad (x,y)\in D \]

则

\[\begin{aligned} r'_x=(1,0,f'_x), \quad r'_y=(0,1,f'_y) \end{aligned} \]

可以得到切平面的法向量

\[\begin{aligned} \left| \begin{array}{ccc} \vec i & \vec j & \vec k \\ 1 & 0 & f'_x \\ 0 & 1 & f'_y \end{array} \right|=(-f'_x,-f'_y,1) \end{aligned} \]

映射的微分

若$\vec r(u,v)$在$(u,v)$处可微，则微分为

\[\begin{aligned} d \vec r=&(dx, dy, dz) \\ =&(\frac{\partial x}{\partial u} du+\frac{\partial x}{\partial v} dv,\frac{\partial y}{\partial u} du+\frac{\partial y}{\partial v} dv,\frac{\partial z}{\partial u} du+\frac{\partial z}{\partial v} dv) \\ =&(\frac{\partial x}{\partial u}, \frac{\partial y}{\partial u}, \frac{\partial z}{\partial u})du+(\frac{\partial x}{\partial v}, \frac{\partial y}{\partial v}, \frac{\partial z}{\partial v})dv \\ =&\vec r'_u du+\vec r'_v dv \end{aligned} \]

且$(\Delta u,\Delta v)\to(0,0)$时，有

\[|\vec r(u+du,v+dv)-\vec r(u,v)-d\vec r| = o(\sqrt{du^2+dv^2}) \]

面积微元

\begin{tikzpicture}[x=2cm, y=2cm, global scale=1, >=latex,thick] \coordinate (orig) at (0,0); \draw[thick,->] (orig) -- (2.2,0) node[right] {$y$};% y軸 \draw[thick,->] (orig) -- (0,1.5) node[above] {$z$};% z軸 \draw[thick,->] (orig) -- (-130:0.5) node[below] {$x$};% x軸 \path%[ draw=orange, thick] %fill=blue, opacity=0.5, (150:0.5) to[out=20, in=120] coordinate [pos=0.25] (p11) coordinate [pos=0.5] (p12) coordinate [pos=0.75] (p13) (-10:1.5) to[out=80, in=170] coordinate [pos=0.25] (p21) coordinate [pos=0.5] (p22) coordinate [pos=0.75] (p23) (15:2.1) to[out=140, in=-20] coordinate [pos=0.25] (p31) coordinate [pos=0.5] (p32) coordinate [pos=0.75] (p33) (75:1.3) to[out=-130, in=80] coordinate [pos=0.25] (p41) coordinate [pos=0.5] (p42) coordinate [pos=0.75] (p43) cycle; % draw v-curve \draw[dashed, red, name path=v3] (p31) to[out=-150, in=80] (p13); %\draw[dashed, red, name path=v2] (p32) to[out=-150, in=80] (p12); \draw[dashed, red, name path=v1] (p33) to[out=-150, in=80] (p11); % draw u-curve \draw[red, name path=u3] (p23) to[out=140, in=20] (p41); %\draw[red, name path=u2] (p22) to[out=140, in=20] (p42); \draw[red, name path=u1] (p21) to[out=120, in=20] (p43); \path[name intersections={of=u3 and v1}]; \coordinate (uv3) at (intersection-1); \path[name intersections={of=u3 and v3}]; \coordinate (uv2) at (intersection-1); \path[name intersections={of=u1 and v3}]; \coordinate (uv1) at (intersection-1); \path[name intersections={of=u1 and v1}]; \coordinate (uv) at (intersection-1); \fill[blue, opacity=0.5] (uv) to [out=0, in=140] (uv1) to [out=60,in=-140] (uv2) to [out=144, in=-10] (uv3) to [out=-150, in=52] cycle; \fill (uv) circle (1pt); \draw[dashed, blue, fill opacity=0.5, fill=yellow] (uv) -- ++(0:1) node[above left, opacity=1]{$N'$} -- ++(52:0.75) node[above, opacity=1]{$P'$} -- ++ (0:-1) node[above, opacity=1]{$Q'$}--cycle; \draw[->, red] (uv) -- node[above] {$\vec r'_u$} +(0:1.1); \draw[->, red, dashed] (uv) -- node[above, pos=0.2] {$\vec r'_v$} +(52:0.87); \node at (uv) [below] {$M$}; \node at (uv1) [below] {$N$}; \node at (uv2) [below] {$P$}; %\node at (uv3) [below] {$Q$}; %\draw[->, green] (uv) -- +(70:0.5) node[above] {$\vec n$}; % 画 u-v 平面 \begin{scope}[xshift = -4cm] \draw[thick,->] (-0.2,0) -- (1.2,0) node[right] {$u$};% x軸 \draw[thick,->] (0,-0.1) -- (0,1.1) node[above] {$v$};% y軸 \draw [orange, name path=domain] plot [smooth cycle] coordinates {(0.1,0.5) (0.3,0.1) (1.1,0.45) (0.55,1.0) }; \foreach \i/\j/\k in {0.3/u/v, 0.7/u+du/v+dv} { \path[name path=v1] (\i,0)--(\i,1.5); \path[name intersections={of=v1 and domain}]; \draw[dashed, red] (intersection-2) -- (\i,0) node[below] {$\j$}; %(intersection-2); \path[name path=u1] (0,\i)--(1.5,\i); \path[name intersections={of=u1 and domain}]; \draw[red] (intersection-1) -- (0,\i) node[left] {$\k$}; %(intersection-2); } \fill[opacity=0.5, blue] (0.3,0.3) rectangle (0.7,0.7); \end{scope} % 画一个箭头 \draw[->, ultra thick] (130:1.2) to[out=50, in=170] node[above]{$\vec r(u,v)$} (110:1); \end{tikzpicture}

设$(u,v)\in D$，$M=\vec r(u,v)$。给变量$u$,$v$一个小的增量$du$,$dv$

则映射的像为曲边四边形$MNPQ$。 $MN'=\vec r'_udu$为圆弧$\overset{\frown}{MN}$的近似。 $MQ'=\vec r'_vdu$为圆弧$\overset{\frown}{MQ}$的近似。

函数的增量与微分分别为

\[\begin{aligned} \overrightarrow{MP}=&\vec r(u+du,v+dv)-\vec r(u,v)=\Delta \vec r, \\ \overrightarrow{MP'}=&\vec r'_udu+\vec r'_vdv=d\vec r \end{aligned} \]

则可得到

\[|\Delta \vec r-d\vec r|=o(\rho) , \rho=\sqrt{du^2+dv^2} \]

这样，曲边四边形$MNPQ$可以用切平面的平行四边形$MN'P'Q'$来近似，而后者的面积为

\[|\vec r'_udu\times \vec r'_vdv|=|\vec r'_u\times \vec r'_v|dudv \]

这就是面积微元

\[\begin{aligned} \Delta \vec r=&(\Delta x, \Delta y, \Delta z) \\ =&(x'_udu+x'_vdv+o(\rho), \\ &\ y'_udu+y'_vdv+o(\rho), \\ &\ z'_udu+z'_vdv+o(\rho)) \\ =&\vec r'_udu+\vec r'_vdv+(o(\rho),o(\rho),o(\rho)) \\ =&d\vec r+(o(\rho),o(\rho),o(\rho)) \end{aligned} \]

所以有

\[|\Delta \vec r-d\vec r|=o(\rho) \]

隐函数表示的曲面与曲线

  • 设$F(x,y,z)$为区域$D\subset\mathbb{R}^3$上的$C^1$函数。
  • 点$M_0(x_0,y_0,z_0)\in D$满足$F(x_0,y_0,z_0)=0$，且
    \[(F'_x,F'_y,F'_z)|_{M_0}\neq \vec 0 \]

根据隐函数存在定理，$F$在$M_0$附近确定了一个连续可微的二元函数 ($z=z(x,y)$ 或$y=y(x,z)$ 或$x=x(y,z)$)， 从而在$M_0$附近给出了一张曲面。

定义 4.
把由方程

\[F(x,y,z)=0 \]

所确定的曲面称为隐式曲面

设曲线$\Gamma$为曲面上过$M_0(x_0,y_0,z_0)$的任一条光滑曲线，

  • 它的参数方程是
    \[\vec r=\vec r(t)=(x(t),y(t),z(t)) , t\in[\alpha,\beta] \]
  • 其中$x_0=x(t_0), y_0=y(t_0), z_0=z(t_0)$
  • 由$\Gamma$在曲面上，则有
    \[F(x(t),y(t),z(t))=0 \]
  • 对$t$求导后，有
    \[F'_x(M_0)x(t_0)+F'_y(M_0)y(t_0)+F'_z(M_0)z(t_0)=0 \]
  • 也就意味着，向量$\vec n=(F'_x(M_0),F'_y(M_0),F'_z(M_0))$ 与曲线$\Gamma$在$M_0$的切向量$(x'(t_0), y'(t_0), z'(t_0))$垂直

隐式曲面$F(x,y,z)=0$在点$M_0$处的法向量为

\[\vec n=(F'_x(M_0),F'_y(M_0),F'_z(M_0)) \]

同样，显式曲面$z=f(x,y)$可以写成隐式

\[F(x,y,z)=z-f(x,y)=0 \]

则它的法向量为$(-f'_x, -f'_y, 1)$，与前面得到的一致。

例 2. (例6.6.2) 求球面$x^2+y^2+z^2=R^2$在点$(x,y,z)$处的法向量。

解. 由隐式方程的法向量公式，有

\[\vec n=(2x,2y,2z)=2(x,y,z)=2\vec r \]

$z>0$时，写成显式$z=\sqrt{R^2-x^2-y^2}$，则有

\[\begin{aligned} \vec n=&(-z'_x, -z'_y,1) \\ =&(-\frac{-x}{\sqrt{R^2-x^2-y^2}},-\frac{-y}{\sqrt{R^2-x^2-y^2}},1) \\ =&(\frac{x}{z}, \frac{y}z, 1) \\ =&\frac1z (x,y,z) \end{aligned} \]

将球面写成参数方程

\[x=R\sin\theta\cos\phi, y=R\sin\theta\sin\phi, z=R\cos\theta \]

则有

\[\begin{aligned} \vec n =&\vec r'_\theta \times \vec r'_\phi =\begin{vmatrix} \vec i & \vec j & \vec k \\ R\cos\theta\cos\phi & R\cos\theta\sin\phi & -R\sin\theta \\ -R\sin\theta\sin\phi & R\sin\theta\cos\phi & 0 \end{vmatrix} \\ =&R\sin\theta(x,y,z) \end{aligned} \]

需要$\theta\neq 0,\pi$

直接从几何上看，切平面与半径垂直，则$(x,y,z)$就是法向量

空间中的曲线可以表示为两个空间中的曲面的交线，

\[\begin{cases} & F(x,y,z)=0 , \\ & G(x,y,z)=0 \end{cases} , (x,y,z)\in D\subset \mathbb{R}^3 \]

若$M_0$为曲线上的点，则曲线在点$M_0$处的切线与两个曲面在$M_0$的切平面的法向量都垂直，

\[\begin{aligned} &(F'_x,F'_y,F'_z)\times(G'_x,G'_y,G'_z)|_{M_0} \\ =&\begin{vmatrix} \vec i & \vec j& \vec k \\ F'_x(M_0) &F'_y(M_0) &F'_z(M_0) \\ G'_x(M_0) &G'_y(M_0) &G'_z(M_0) \\ \end{vmatrix} \end{aligned} \]

例 3. 如下的两个平面正交

\[\begin{aligned} \pi_1: x^2+y^2+z^2=ax \\ \pi_2: x^2+y^2+z^2=by \end{aligned} \]

例 4. 求曲线在点$(-2,1,6)$处的切线与法平面

\[\begin{cases} & 2x^2+3y^2+z^2=47 , \\ & x^2+2y^2=z \end{cases} \]

例 5. 证明：曲面$xyz=10$上任一点的切平面与三个坐标面构成的四面体的体积是一个常数

例3.

例 6. 证明曲面$xy=z^2$与$x^2+y^2+z^2=9$正交

例 7. 证明曲面$\displaystyle F\left(\frac{x-a}{z-c},\frac{y-b}{z-c}\right)=0$的所有切平面都通过一个定点。

例 8. 证明曲面$\displaystyle F(ax-by, cx-bz)=0$ 的所有切平面与一直线平行

目录

谢谢

例 9. 本节读完

例9.

例 10. 证明曲面$zy=z^2$与曲面$x^2+y^2+z^2=9$正交

例 11. 证明曲面

\[F(\frac{x-a}{z-c},\frac{y-b}{z-c})=0 \]

的所有切平面通过一个定点

例 12. 证明曲面

\[F(ax-by, cx-bz)=0 \]

的所有切平面与一直线平行

例 13. 求椭圆

\[\begin{cases} & x^2+y^2=1 \\ & x+y+z=1 \end{cases} \]

的长半轴与短半轴