\chapter{曲面的表示与逼近}
\section{曲面的表示}
\subsection{曲面的显式表示}
设有界闭区域$D \subset \realnum^2$，函数$f:D \to \ndreal$连续，我们称函数$f$的图像
\[G(f) = \{(x, y, f(x, y)) \in \realnum^3 \mid (x, y) \in D\}\]
为一张曲面，它展布在$D$上，称这曲面是由显式方程
\[z = f(x, y), (x, y) \in D\]
所确定的。

\subsection{曲面的隐式表示}
设三元函数$F$定义在区域$D \subset \realnum^3$，区域$D$中所有满足方程
\[F(x, y, z) = 0\tag{$\ast$} \label{曲面隐式表示}\]
的点集组成一张曲面，称为由方程\eqref{曲面隐式表示}所确定的隐式曲面。

\subsection{曲面的参数表示}
设$\bvec{f}: D \to \realnum^3$，$D \subset \realnum^2$是平面区域。则集合
\[S = \{(x, y, z) \mid (x, y, z) = \bvec{f}(u, v), (u, v) \in D\} = f(D)\]
称为$\realnum^3$空间中的一个曲面，$\bvec{f}(u, v)$称为曲面的$S$的参数表示。

\section{曲面的法向与切平面}
\subsection{有显式表示的曲面的切平面与法向量}
设曲面$S$有显式表示$z = f(x, y)$，令$z_0 = f(x_0, y_0)$，则$P = (x_0, y_0, z_0) \in S$，且$S$在$P$点的切平面方程为
\[z = z_0 + \frac{\partial f}{\partial x} (x_0, y_0) (x - x_0) + \frac{\partial f}{\partial y} (x_0, y_0) (y - y_0)\]
该平面的法向量
\[\bvec{n} = \pm \left(\frac{\partial f}{\partial x}(x_0, y_0), \frac{\partial f}{\partial y}(x_0, y_0), -1\right)\]
因此切平面方程还可以写成向量内积的形式：
\[\brak{\bvec{n}, \bvec{r} - \bvec{r}_0} = 0\]
其中
\[\bvec{r} = (x, y, z), \bvec{r}_0 = (x_0, y_0, z_0)\eqper\]

\subsection{隐式曲面的切平面与法向量}
设曲面$S$有隐式表示$F(x, y, z) = 0$，取$P = (x_0, y_0, z_0) \in S$，即$F(x_0, y_0, z_0) = 0$，不妨令$F \in C^1$且$\dfrac{\partial F}{\partial z}(x_0, y_0, z_0) \neq 0$，则根据隐函数定理，$P$点附近$S$有显示表示$z = z(x, y)$，切平面方程为
\[z = z_0 + \frac{\partial z}{\partial x}(x_0, y_0) (x - x_0) + \frac{\partial z}{\partial y}(x_0, y_0) (y - y_0)\]
其中
\[\frac{\partial z}{\partial x} = -\frac{\dfrac{\partial F}{\partial x}}{\dfrac{\partial F}{\partial z}}, \frac{\partial z}{\partial y} = -\frac{\dfrac{\partial F}{\partial y}}{\dfrac{\partial F}{\partial z}}\]
带入切平面方程，整理得
\[\frac{\partial F}{\partial x}(P) (x - x_0) + \frac{\partial F}{\partial y}(P) (y - y_0) + \frac{\partial F}{\partial z}(P) (z - z_0) = 0\]
因此法向量
\[\bvec{n} = \pm \gra F(P)\]
写成向量内积的形式
\[\brak{\bvec{n}, \bvec{r} - \bvec{r}_0} = 0\eqper\]

\subsection{参数曲面的切平面与法向量}
设曲面$S$有参数表示$\bvec{r} = \bvec{r}(u, v)$，$(u, v) \in D$。写成分量形式：$x = x(u, v), y = y(u, v), z = z(u, v)$。

令$(u_0, v_0) \in D$对应$P = (x_0, y_0, z_0) \in S$。只考虑$u$变化时对应的曲线：$\bvec{r} = \bvec{r}(u, v_0)$，切向为$\dfrac{\partial \bvec{r}}{\partial u}(u, v_0)$；类似地只考虑$v$变化时对应的曲线切向为$\dfrac{\partial \bvec{r}}{\partial v}(u_0, v)$。这两个法向量都应在$S$在$(u_0, v_0)$的切平面内，因此$S$在$P$点切平面的法向量$\bvec{n}$满足$\bvec{n} \perp \dfrac{\partial \bvec{r}}{\partial u}(u_0, v_0)$与$\bvec{n} \perp \dfrac{\partial \bvec{r}}{\partial v}(u_0, v_0)$。

综合起来，得到
\[\bvec{n} \parallel \dfrac{\partial \bvec{r}}{\partial u}(u_0, v_0) \times \dfrac{\partial \bvec{r}}{\partial v}(u_0, v_0)\]

进一步假设$\dfrac{\partial \bvec{r}}{\partial u}(u_0, v_0) \times \dfrac{\partial \bvec{r}}{\partial v}(u_0, v_0) \neq \bvec{0}$即两向量不共线，则可取$S$在$P$点法向$\bvec{n} = \dfrac{\partial \bvec{r}}{\partial u}(u_0, v_0) \times \dfrac{\partial \bvec{r}}{\partial v}(u_0, v_0)$，得到切平面方程
\[\brak{\dfrac{\partial \bvec{r}}{\partial u}(u_0, v_0) \times \dfrac{\partial \bvec{r}}{\partial v}(u_0, v_0), \bvec{r} - \bvec{r}_0} = 0\]
利用行列式也可以得到切平面方程
\[\begin{vmatrix}
    x - x_0 & y - y_0 & z - z_0\\
    D_u x(u_0, v_0) & D_u y(u_0, v_0) & D_u z(u_0, v_0)\\
    D_v x(u_0, v_0) & D_v y(y_0, v_0) & D_v z(u_0, v_0)
\end{vmatrix}
= 0\]
与法向量
\[\bvec{n} = \dfrac{\partial \bvec{r}}{\partial u}(u_0, v_0) \times \dfrac{\partial \bvec{r}}{\partial v}(u_0, v_0) =
\begin{vmatrix}
    \bvec{e}_1 & \bvec{e}_2 & \bvec{e}_3\\
    D_u x & D_u y & D_u z\\
    D_v x & D_v y & D_v z
\end{vmatrix}\eqper\]

同时，记$D_u \bvec{r} \times D_v \bvec{r} = (A, B, C)$，由定义
\[A = \begin{vmatrix}
    D_u y & D_u z\\
    D_v y & D_v z
\end{vmatrix}, 
B = \begin{vmatrix}
    D_u z & D_u x\\
    D_v z & D_v x
\end{vmatrix}, 
C = \begin{vmatrix}
    D_u x & D_u y\\
    D_v x & D_v y
\end{vmatrix}\]
引入第一基本量记号
\[E = \brak{D_u \bvec{r}, D_u \bvec{r}}, F = \brak{D_u \bvec{r}, D_v \bvec{r}}, G = \brak{D_v \bvec{r}, D_v \bvec{r}}\eqper\]

\section{曲线的切向量}
对曲线$\bvec{r} = \bvec{r}(t)$，在$\bvec{r}_0 = \bvec{r}(t_0)$处的切向量为
\[\deriv{\bvec{r}}(t_0) = \left(\deriv{x}(t_0), \deriv{y}(t_0) \deriv{z}(t_0)\right)\eqper\]

对曲线
\(\left\{\begin{aligned}
    & F(x, y, z) = 0\\
    & G(x, y, z) = 0
\end{aligned}\right.\)
即曲面$F(x, y, z) = 0$与$G(x, y, z) = 0$的交线，在$\bvec{r}_0 = (x_0, y_0, z_0)$处的切线为$\gra F(\bvec{r}_0) \times \gra G(\bvec{r}_0)$。