第八周。

14多变量函数的微分学.tex

@@ -141,3 +141,151 @@
 \end{proof}
 
 总结起来，偏导数在$\bvec{a}$点都连续可以推出函数在$\bvec{a}$点可微，进而可以推出函数在$\bvec{a}$点连续，也可以推出函数在$\bvec{a}$点所有方向导数都存在。
+
+\section{向量值函数的微分}
+\begin{definition}
+    如果映射\boldf 满足存在Jacobian $J \boldf (\bvec{x}_0)$且满足
+    \[\boldf (\bvec{x}_0 + \Delta \bvec{x}) - \boldf (\bvec{x}_0) = J \boldf (\bvec{x}_0) \Delta \bvec{x} + o\left(\norm{\Delta \bvec{x}}\right)\]
+    其中
+    \[J\boldf (\bvec{x}_0) = 
+    \begin{bmatrix}
+        D_1 f_1(\bvec{x}_0) & \cdots & D_n f_1(\bvec{x}_0)\\
+        \vdots & \ddots & \vdots\\
+        D_1 f_m(\bvec{x}_0) & \cdots & D_n f_m(\bvec{x}_0)
+    \end{bmatrix}
+    =
+    \begin{bmatrix}
+        \gra f_1 (\bvec{x}_0)\\
+        \vdots\\
+        \gra f_m (\bvec{x}_0)
+    \end{bmatrix}\]
+    此时\boldf 在$\bvec{x}_0$点的微分记为
+    \[\dif \boldf (\bvec{x}_0) = J \boldf (\bvec{x}_0) \Delta \bvec{x} \eqper\]
+\end{definition}
+
+\begin{theorem}
+    若映射\boldf 在开集$D$上存在Jacobian $J \boldf$，且$J \boldf$的各元素在点$\bvec{x}_0$处都连续，则映射\boldf 在点$\bvec{x}_0$处可微。
+\end{theorem}
+
+\section{复合求导}
+\begin{theorem}
+    设$D \in \ndreal$，$\bvec{f}: D \to \realnum^m$，$\bvec{g}: \Omega \to \realnum^k$，$\bvec{f}(D) \subset \Omega \subset \realnum^m$。如果\boldf 在$\bvec{x}_0 \in D\interior$上可微，$\bvec{g}$在$\boldf(\bvec{x}_0)$上可微，那么复合映射$\bvec{g} \circ \boldf$在点$\bvec{x}_0$处可微，且
+    \[J(\bvec{g} \circ \bvec{f}) = J \bvec{g}(\boldf (\bvec{x}_0)) J \boldf(\bvec{x}_0)\eqper\]
+\end{theorem}
+
+如果我们记$\bvec{u} = \bvec{g}(\bvec{y}), \bvec{y} = \bvec{f}(\bvec{x})$，那么$\bvec{g} \circ \bvec{f}$的Jacobin可以写为
+\[\begin{bmatrix}
+    \dfrac{\partial u_1}{\partial x_1} & \dfrac{\partial u_1}{\partial x_2} & \cdots & \dfrac{\partial u_1}{\partial x_n}\\[1em]
+    \dfrac{\partial u_2}{\partial x_1} & \dfrac{\partial u_2}{\partial x_2} & \cdots & \dfrac{\partial u_2}{\partial x_n}\\[1ex]
+    \vdots & \vdots & \ddots & \vdots\\
+    \dfrac{\partial u_k}{\partial x_1} & \dfrac{\partial u_k}{\partial x_2} & \cdots & \dfrac{\partial u_k}{\partial x_n}
+\end{bmatrix}
+=
+\begin{bmatrix}
+    \dfrac{\partial u_1}{\partial y_1} & \dfrac{\partial u_1}{\partial y_2} & \cdots & \dfrac{\partial u_1}{\partial y_m}\\[1em]
+    \dfrac{\partial u_2}{\partial y_1} & \dfrac{\partial u_2}{\partial y_2} & \cdots & \dfrac{\partial u_2}{\partial y_m}\\[1ex]
+    \vdots & \vdots & \ddots & \vdots\\
+    \dfrac{\partial u_k}{\partial y_1} & \dfrac{\partial u_k}{\partial y_2} & \cdots & \dfrac{\partial u_k}{\partial y_m}
+\end{bmatrix}
+\begin{bmatrix}
+    \dfrac{\partial y_1}{\partial x_1} & \dfrac{\partial y_1}{\partial x_2} & \cdots & \dfrac{\partial y_1}{\partial x_n}\\[1em]
+    \dfrac{\partial y_2}{\partial x_1} & \dfrac{\partial y_2}{\partial x_2} & \cdots & \dfrac{\partial y_2}{\partial x_n}\\[1ex]
+    \vdots & \vdots & \ddots & \vdots\\
+    \dfrac{\partial y_m}{\partial x_1} & \dfrac{\partial y_m}{\partial x_2} & \cdots & \dfrac{\partial y_m}{\partial x_n}
+\end{bmatrix}\eqper\]
+
+\section{隐函数定理}
+\begin{theorem}[隐函数定理]
+    设开集$D \subset \realnum^2$，函数$F: D \to \realnum$满足条件：
+    \begin{enumerate}[label=(\roman{*})]
+        \item $F \in C^1(D)$；
+        \item 点$(x_0, y_0) \in D$使得$F(x_0, y_0) = 0$；
+        \item $\dfrac{\partial F(x_0, y_0)}{\partial y} \neq 0$，
+    \end{enumerate}
+    则存在$\delta, \eta > 0$以及唯一的函数$f: (x_0 - \delta, x_0 + \delta) \to (y_0 - \eta, y_0 + \eta)$具有性质
+    \begin{enumerate}
+        \item 对任意的$\abs{x - x_0} < \delta$，$f(x_0) = y_0$，有$F(x, f(x)) = 0$；
+        \item $f \in C^1(x_0 - \delta, x_0 + \delta)$；
+        \item 对$x \in (x_0 - \delta, x_0 + \delta)$，$y = f(x)$，有
+        \[\deriv{f}(x) = -\frac{\dfrac{\partial F}{\partial x}(x, y)}{\dfrac{\partial F}{\partial y}(x, y)}\eqper\]
+    \end{enumerate}
+\end{theorem}
+
+\begin{theorem}
+    设开集$D \subset \realnum^{n + 1}$，$F: D \to \realnum$，满足条件：
+    \begin{enumerate}[label=(\roman{*})]
+        \item $F \in C^{(1)}(D)$；
+        \item 点$(\bvec{x}_0, y_0) \in D$使得$F(\bvec{x}_0, y_0) = 0$；
+        \item $\dfrac{\partial F(\bvec{x}_0, y_0)}{\partial y} \neq 0$，
+    \end{enumerate}
+    则存在$\delta, \eta > 0$以及唯一的函数$f: B_\delta (\bvec{x}_0) \to (y_0 - \eta, y_0 + \eta)$具有性质
+    \begin{enumerate}
+        \item 对任意的$\norm{\bvec{x} - \bvec{x}_0} < \delta$，$f(\bvec{x}_0) = y_0$，有$F(\bvec{x}, f(\bvec{x})) = 0$；
+        \item $f \in C^1 (B_\delta (\bvec{x}_0))$；
+        \item 对$\bvec{x} \in B_\delta (\bvec{x}_0)$，$y = f(\bvec{x})$，有
+        \[D_i f(x) = -\frac{\dfrac{\partial F}{\partial x_i}(\bvec{x}, y)}{\dfrac{\partial F}{\partial y}(\bvec{x}, y)}, i = 1, 2, \dots, n\eqper\]
+    \end{enumerate}
+\end{theorem}
+
+\section{隐映射定理}
+我们先引入几个记号。设想有$m$个方程形成的方程组
+\[\begin{cases}
+    F_1(x_1, \dots, x_n, y_1, \dots, y_m) = 0,\\
+    \qquad \dots\dots\\
+    F_m(x_1, \dots, x_n, y_1, \dots, y_m) = 0
+\end{cases}\label{隐映射定理1}\tag{1}\]
+如果这个方程组是一个合适的约束，那么我们可以期望从中解出$y_1, \dots, y_m$，使得其中的每一个都是$x_1, \dots, x_n$的函数，即
+\[\begin{cases}
+    y_1 = f_1(x_1, \dots, x_n)\\
+    \qquad \dots\dots\\
+    y_m = f_m(x_1, \dots, x_n)
+\end{cases}\label{隐映射定理2}\tag{2}\]
+为了缩短记号，可令
+\[\bvec{F} = \begin{bmatrix}
+    F_1\\ \vdots\\ F_m
+\end{bmatrix}, 
+\boldf = \begin{bmatrix}
+    f_1\\ \vdots\\ f_m
+\end{bmatrix}\]
+那么\eqref{隐映射定理1}式可以写为
+\[\bvec{F}(\bvec{x}, \bvec{y}) = \bvec{0}\]
+\eqref{隐映射定理2}式可以写为
+\[\bvec{y} = \boldf (\bvec{x})\eqper\]
+
+我们设$\bvec{F}$定义在开集$D \subset \realnum^{m + n}$，那么在$m \times (n + m)$矩阵
+\[J \bvec{F} = \begin{bmatrix}
+    \dfrac{\partial F_1}{\partial x_1} & \cdots & \dfrac{\partial F_1}{\partial x_n} & \dfrac{\partial F_1}{\partial y_1} & \cdots & \dfrac{\partial F_1}{y_m}\\[1ex]
+    \vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\
+    \dfrac{\partial F_m}{\partial x_1} & \cdots & \dfrac{\partial F_m}{\partial x_n} & \dfrac{\partial F_m}{\partial y_1} & \cdots & \dfrac{\partial F_m}{y_m}
+\end{bmatrix}\]
+中作分块$J\bvec{F} = \begin{bmatrix}
+    J_x \bvec{F} & J_y \bvec{F}
+\end{bmatrix}$，
+其中
+\[J_x \bvec{F} = \begin{bmatrix}
+    \dfrac{\partial F_1}{\partial x_1} & \cdots & \dfrac{\partial F_1}{\partial x_n}\\
+    \vdots & \ddots & \vdots\\
+    \dfrac{\partial F_m}{\partial x_1} & \cdots & \dfrac{\partial F_m}{\partial x_n}
+\end{bmatrix},
+J_y \bvec{F} = \begin{bmatrix}
+    \dfrac{\partial F_1}{\partial y_1} & \cdots & \dfrac{\partial F_1}{\partial y_m}\\
+    \vdots & \ddots & \vdots\\
+    \dfrac{\partial F_m}{\partial y_1} & \cdots & \dfrac{\partial F_m}{\partial y_m}
+\end{bmatrix}\]
+其中$J_y \bvec{F}$是$m$阶方阵。
+
+\begin{theorem}[隐映射定理]
+    设开集$D \subset \realnum^{n + m}$，映射$\bvec{F}: D \to \realnum^m$，满足下列条件：
+    \begin{enumerate}[label=(\roman{*})]
+        \item $\bvec{F} \in C^1(D)$；
+        \item 点$(\bvec{x}_0, \bvec{y}_0) \in D$使得$\bvec{F}(\bvec{x}_0, \bvec{y}_0) = \bvec{0}$；
+        \item $\det[J_y \bvec{F}(\bvec{x}_0, \bvec{y}_0)] \neq 0$，
+    \end{enumerate}
+    则存在$\delta, \eta > 0$以及唯一的函数$\boldf: B_\delta (\bvec{x}_0) \to B_\eta (\bvec{y}_0)$具有性质
+    \begin{enumerate}
+        \item 对任意的$\norm{\bvec{x} - \bvec{x}_0} < \delta$，$\bvec{f}(\bvec{x}_0) = \bvec{y}_0$，有$\bvec{F}(\bvec{x}, f(\bvec{x})) = \bvec{0}$；
+        \item $\bvec{f} \in C^1 (B_\delta (\bvec{x}_0), \realnum^m)$；
+        \item 对$\bvec{x} \in B_\delta (\bvec{x}_0)$，$\bvec{y} = \bvec{f}(\bvec{x})$，有
+        \[J\bvec{f}(\bvec{x}) = -(J_y \bvec{F}(\bvec{x}, \bvec{y}))^{-1} J_x \bvec{F}(\bvec{x}, \bvec{y})\eqper\]
+    \end{enumerate}
+\end{theorem}

高等微积分.tex

@@ -16,6 +16,7 @@
 \usepackage{extarrows}
 \usepackage{physics}
 \usepackage{mathrsfs}
+\usepackage{nicematrix}
 % \usepackage{mathptmx}
 \usetikzlibrary{arrows.meta}