第十周。
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93
15曲面的表示与逼近.tex
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15曲面的表示与逼近.tex
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\chapter{曲面的表示与逼近}
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\section{曲面的表示}
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\subsection{曲面的显式表示}
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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
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\[G(f) = \{(x, y, f(x, y)) \in \realnum^3 \mid (x, y) \in D\}\]
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为一张曲面，它展布在$D$上，称这曲面是由显式方程
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\[z = f(x, y), (x, y) \in D\]
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所确定的。
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\subsection{曲面的隐式表示}
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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
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\[F(x, y, z) = 0\tag{$\ast$} \label{曲面隐式表示}\]
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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。
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\subsection{曲面的参数表示}
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设$\bvec{f}: D \to \realnum^3$，$D \subset \realnum^2$是平面区域。则集合
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\[S = \{(x, y, z) \mid (x, y, z) = \bvec{f}(u, v), (u, v) \in D\} = f(D)\]
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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。
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\section{曲面的法向与切平面}
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\subsection{有显式表示的曲面的切平面与法向量}
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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
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\[z = z_0 + \frac{\partial f}{\partial x} (x_0, y_0) (x - x_0) + \frac{\partial f}{\partial y} (x_0, y_0) (y - y_0)\]
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该平面的法向量
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\[\bvec{n} = \pm \left(\frac{\partial f}{\partial x}(x_0, y_0), \frac{\partial f}{\partial y}(x_0, y_0), -1\right)\]
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因此切平面方程还可以写成向量内积的形式：
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\[\brak{\bvec{n}, \bvec{r} - \bvec{r}_0} = 0\]
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其中
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\[\bvec{r} = (x, y, z), \bvec{r}_0 = (x_0, y_0, z_0)\eqper\]
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\subsection{隐式曲面的切平面与法向量}
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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
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\[z = z_0 + \frac{\partial z}{\partial x}(x_0, y_0) (x - x_0) + \frac{\partial z}{\partial y}(x_0, y_0) (y - y_0)\]
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其中
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\[\frac{\partial z}{\partial x} = -\frac{\dfrac{\partial F}{\partial x}}{\dfrac{\partial F}{\partial z}}, \frac{\partial z}{\partial y} = -\frac{\dfrac{\partial F}{\partial y}}{\dfrac{\partial F}{\partial z}}\]
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带入切平面方程，整理得
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\[\frac{\partial F}{\partial x}(P) (x - x_0) + \frac{\partial F}{\partial y}(P) (y - y_0) + \frac{\partial F}{\partial z}(P) (z - z_0) = 0\]
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因此法向量
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\[\bvec{n} = \pm \gra F(P)\]
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写成向量内积的形式
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\[\brak{\bvec{n}, \bvec{r} - \bvec{r}_0} = 0\eqper\]
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\subsection{参数曲面的切平面与法向量}
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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：$x = x(u, v), y = y(u, v), z = z(u, v)$。
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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$\bvec{n} \perp \dfrac{\partial \bvec{r}}{\partial v}(u_0, v_0)$。
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综合起来，得到
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\[\bvec{n} \parallel \dfrac{\partial \bvec{r}}{\partial u}(u_0, v_0) \times \dfrac{\partial \bvec{r}}{\partial v}(u_0, v_0)\]
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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
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\[\brak{\dfrac{\partial \bvec{r}}{\partial u}(u_0, v_0) \times \dfrac{\partial \bvec{r}}{\partial v}(u_0, v_0), \bvec{r} - \bvec{r}_0}\]
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利用行列式也可以得到切平面方程
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\[\begin{vmatrix}
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x - x_0 & y - y_0 & z - z_0\\
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D_u x(u_0, v_0) & D_u y(u_0, v_0) & D_u z(u_0, v_0)\\
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D_v x(u_0, v_0) & D_v y(y_0, v_0) & D_v z(u_0, v_0)
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\end{vmatrix}
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= 0\]
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与法向量
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\[\bvec{n} = \dfrac{\partial \bvec{r}}{\partial u}(u_0, v_0) \times \dfrac{\partial \bvec{r}}{\partial v}(u_0, v_0) =
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\begin{vmatrix}
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\bvec{e}_1 & \bvec{e}_2 & \bvec{e}_3\\
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D_u x & D_u y & D_u z\\
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D_v x & D_v y & D_v z
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\end{vmatrix}\eqper\]
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同时，记$D_u \bvec{r} \times D_v \bvec{r} = (A, B, C)$，由定义
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\[A = \begin{vmatrix}
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D_u y & D_u z\\
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D_v y & D_v z
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\end{vmatrix},
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B = \begin{vmatrix}
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D_u z & D_u x\\
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D_v z & D_v x
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\end{vmatrix},
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C = \begin{vmatrix}
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D_u x & D_u y\\
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D_v x & D_v y
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\end{vmatrix}\]
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引入第一基本量记号
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\[E = \brak{D_u \bvec{r}, D_u \bvec{r}}, F = \brak{D_u \bvec{r}, D_v \bvec{r}}, G = \brak{D_v \bvec{r}, D_v \bvec{r}}\eqper\]
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\section{曲线的切向量}
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对曲线$\bvec{r} = \bvec{r}(t)$，在$\bvec{r}_0 = \bvec{r}(t_0)$处的切向量为
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\[\deriv{\bvec{r}}(t_0) = \left(\deriv{x}(t_0), \deriv{y}(t_0) \deriv{z}(t_0)\right)\eqper\]
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对曲线
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\(\left\{\begin{aligned}
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& F(x, y, z) = 0\\
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& G(x, y, z) = 0
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\end{aligned}\right.\)
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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$\gra F(\bvec{r}_0) \times \gra G(\bvec{r}_0)$。
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73
16多重积分.tex
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16多重积分.tex
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\chapter{多重积分}
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\section{矩形区域上的积分}
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\begin{definition}[二重积分]
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设$I$是$\realnum^2$中的闭矩形，$I = [a, b] \times [c, d]$。作$[a, b]$的分割
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\[\pi_x: a = x_0 < x_1 < \dots < x_n = b\]
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又作$[c, d]$的分割
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\[\pi_y: c = y_0 < y_1 < \dots < y_m = d\]
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两组平行线把$I$分割成$k = n \times m$个子矩形
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\[[x_{i - 1}, x_i] \times [y_{j - 1}, y_j], i = 1, 2, \dots, n, j = 1, 2, \dots, m\]
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这$k$个子矩形的全体组成$I$的一个分割$\pi = \pi_x \times \pi_y$，用一定的次序重排这$k$个子矩形，将它们编号为$I_1, I_2, \dots, I_k$，在每一个$I_i$中任取一点$\xi_i(i = 1, 2, \dots, k)$，作积分和（也称Riemann和）
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\[\sum_{i = 1}^k f(\xi_i) \sigma(I_i)\]
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记
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\[\norm{\pi} = \max\{\diam (I_1), \dots, \diam(I_k)\}\]
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这里$\diam(I_i)$是矩形$I_i$对角线的长度，称$\norm{\pi}$为分割$\pi$的宽度；$\sigma(I_i)$表示矩形$I_i$的面积。
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如果存在数$A$使得对任意给定的$\varepsilon > 0$，有$\delta > 0$，凡是$\norm{\pi} < \delta$时，不论值点$\xi_i$在子矩形$I_i$中如何选择，都有
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\[\abs{\sum_{i = 1}^k f(\xi_i) \sigma(I_i) - A} < \varepsilon\]
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则称函数$f$在矩形$I$上可积，并将$A$写作
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\[\iint \limits_I f(x, y) \dif x \dif y \qquad \text{或者} \qquad \int \limits_I f \dif \sigma\]
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称之为$f$在矩形$I$上的二重积分，或简称$f$在$I$上的积分。这里$f$称为被积函数，$I$称为积分区域。
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\end{definition}
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\begin{theorem}[可积函数的有界性]
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如果$f$在$I$上可积，那么$f$必在$I$上有界。
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\end{theorem}
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\begin{theorem}[可积函数线性性质]
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设$f, g \in R(D)$，$\alpha, \beta \in \realnum$，则$\alpha f + \beta g \in R(D)$，且
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\[\int \limits_D (\alpha f + \beta g) \dif \sigma = \alpha \int \limits_D f \dif \sigma + \beta \int \limits_D g \dif \sigma\eqper\]
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\end{theorem}
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\begin{theorem}
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若$f$和$g$在$I$上可积且$f \geq g$，那么
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\[\int \limits_I f \dif \sigma \geq \int \limits_I g \dif \sigma\eqper\]
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\end{theorem}
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\begin{theorem}
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设$D = D_1 \cup D_2$，且$D_1$与$D_2$没有公共内点，若$f(x, y) \in R(D)$，则$f(x, y) \in R(D_1), f(x, y) \in R(D_2)$，且
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\[\iint \limits_D f(x, y) \dif \sigma = \iint \limits_{D_1} f(x, y) \dif \sigma + \iint \limits_{D_2} f(x, y) \dif \sigma\eqper\]
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\end{theorem}
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\begin{theorem}
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若$f(x, y) \in R(D)$，则$\abs{f(x, y)} \in R(D)$，并且有
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\[\abs{\iint \limits_D f(x, y) \dif \sigma} \leq \iint \limits_D \abs{f(x, y)} \dif \sigma\eqper\]
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\end{theorem}
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\begin{theorem}
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若$f(x, y) \in R(D)$，则$f$有界，设$m \leq f(x, y) \leq M$，则有
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\[m \sigma(D) \leq \iint \limits_D f(x, y) \dif \sigma \leq M \sigma(D)\eqper\]
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\end{theorem}
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\begin{theorem}[积分中值定理]
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$D \subset \realnum^2$连通、有界闭，$\partial D$为零面积集，$f \in C(D)$，则存在$(\xi, \eta) \in D$，满足
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\[\iint \limits_D f(x, y) \sigma = f(\xi, \eta) \sigma(D)\eqper\]
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\end{theorem}
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\begin{theorem}
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$D \subset \realnum^2$为连通有界闭集，$\partial D$为零面积集，$g$在$D$上不变号，$f, g \in C(D)$。则存在$(\xi, \eta) \in D$，满足
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\[\iint \limits_D f(x, y) g(x, y) \dif \sigma = f(\xi, \eta) \iint \limits_D g(x, y) \dif \sigma\]
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\end{theorem}
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\begin{theorem}
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设$f \in R(D)$，$D$关于$OX$轴对称，则
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\begin{itemize}
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\item 若$f(x, y)$关于$y$为己函数，则$\displaystyle\iint \limits_D f(x, y) \dif \sigma = 0$;
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\item 若$f(x, y)$关于$y$为偶函数，记$D_1$为$D$位于$OX$轴上方的部分，则$\displaystyle\iint \limits_D f(x, y) \dif \sigma = 2 \iint \limits_{D_1} f(x, y) \dif \sigma$。
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\end{itemize}
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\end{theorem}
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\begin{theorem}[轮换不变性]
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若$D \subset \realnum^2$关于$x, y$是轮换对称的，即$(x, y) \in D \Leftrightarrow (y, x) \in D$，则
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\[\iint \limits_D f(x, y) \dif \sigma = \iint \limits_D f(y, x) \dif \sigma\eqper\]
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\end{theorem}
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@@ -63,9 +63,11 @@
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\newcommand{\closure}[1]{\overline{#1}}
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\newcommand{\ndreal}{\ensuremath{\realnum^n}}
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\newcommand{\boldf}{\ensuremath{\bvec{f}}}
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\renewcommand{\parallel}{\mathrel{/\mskip-2.5mu/}}
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\DeclareMathOperator{\sgn}{sgn}
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\DeclareMathOperator{\gra}{grad}
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\DeclareMathOperator{\diam}{diam}
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\title{{\Huge{\textbf{高等微积分}}}}
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\author{}
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@@ -98,4 +100,6 @@
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\include{12Fourier分析.tex}
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\include{13多变量函数的连续性.tex}
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\include{14多变量函数的微分学.tex}
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\include{15曲面的表示与逼近.tex}
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\include{16多重积分.tex}
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\end{document}
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