From 946e806ccc5ce84841ef2508c0ad3f9938401e09 Mon Sep 17 00:00:00 2001 From: unlockable Date: Wed, 12 Apr 2023 19:56:48 +0800 Subject: [PATCH] =?UTF-8?q?=E7=AC=AC=E5=85=AB=E5=91=A8=E3=80=82?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 14多变量函数的微分学.tex | 148 +++++++++++++++++++++++++++++++++++++++ 高等微积分.tex | 1 + 2 files changed, 149 insertions(+) diff --git a/14多变量函数的微分学.tex b/14多变量函数的微分学.tex index d6edeec..93b9540 100644 --- a/14多变量函数的微分学.tex +++ b/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} \ No newline at end of file diff --git a/高等微积分.tex b/高等微积分.tex index 436aeee..c606208 100644 --- a/高等微积分.tex +++ b/高等微积分.tex @@ -16,6 +16,7 @@ \usepackage{extarrows} \usepackage{physics} \usepackage{mathrsfs} +\usepackage{nicematrix} % \usepackage{mathptmx} \usetikzlibrary{arrows.meta}