第八周。
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14多变量函数的微分学.tex
148
14多变量函数的微分学.tex
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\end{proof}
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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。
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\section{向量值函数的微分}
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\begin{definition}
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如果映射\boldf 满足存在Jacobian $J \boldf (\bvec{x}_0)$且满足
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\[\boldf (\bvec{x}_0 + \Delta \bvec{x}) - \boldf (\bvec{x}_0) = J \boldf (\bvec{x}_0) \Delta \bvec{x} + o\left(\norm{\Delta \bvec{x}}\right)\]
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其中
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\[J\boldf (\bvec{x}_0) =
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\begin{bmatrix}
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D_1 f_1(\bvec{x}_0) & \cdots & D_n f_1(\bvec{x}_0)\\
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\vdots & \ddots & \vdots\\
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D_1 f_m(\bvec{x}_0) & \cdots & D_n f_m(\bvec{x}_0)
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\end{bmatrix}
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=
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\begin{bmatrix}
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\gra f_1 (\bvec{x}_0)\\
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\vdots\\
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\gra f_m (\bvec{x}_0)
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\end{bmatrix}\]
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此时\boldf 在$\bvec{x}_0$点的微分记为
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\[\dif \boldf (\bvec{x}_0) = J \boldf (\bvec{x}_0) \Delta \bvec{x} \eqper\]
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\end{definition}
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\begin{theorem}
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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。
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\end{theorem}
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\section{复合求导}
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\begin{theorem}
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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
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\[J(\bvec{g} \circ \bvec{f}) = J \bvec{g}(\boldf (\bvec{x}_0)) J \boldf(\bvec{x}_0)\eqper\]
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\end{theorem}
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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
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\[\begin{bmatrix}
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\dfrac{\partial u_1}{\partial x_1} & \dfrac{\partial u_1}{\partial x_2} & \cdots & \dfrac{\partial u_1}{\partial x_n}\\[1em]
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\dfrac{\partial u_2}{\partial x_1} & \dfrac{\partial u_2}{\partial x_2} & \cdots & \dfrac{\partial u_2}{\partial x_n}\\[1ex]
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\vdots & \vdots & \ddots & \vdots\\
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\dfrac{\partial u_k}{\partial x_1} & \dfrac{\partial u_k}{\partial x_2} & \cdots & \dfrac{\partial u_k}{\partial x_n}
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\end{bmatrix}
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=
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\begin{bmatrix}
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\dfrac{\partial u_1}{\partial y_1} & \dfrac{\partial u_1}{\partial y_2} & \cdots & \dfrac{\partial u_1}{\partial y_m}\\[1em]
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\dfrac{\partial u_2}{\partial y_1} & \dfrac{\partial u_2}{\partial y_2} & \cdots & \dfrac{\partial u_2}{\partial y_m}\\[1ex]
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\vdots & \vdots & \ddots & \vdots\\
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\dfrac{\partial u_k}{\partial y_1} & \dfrac{\partial u_k}{\partial y_2} & \cdots & \dfrac{\partial u_k}{\partial y_m}
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\end{bmatrix}
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\begin{bmatrix}
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\dfrac{\partial y_1}{\partial x_1} & \dfrac{\partial y_1}{\partial x_2} & \cdots & \dfrac{\partial y_1}{\partial x_n}\\[1em]
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\dfrac{\partial y_2}{\partial x_1} & \dfrac{\partial y_2}{\partial x_2} & \cdots & \dfrac{\partial y_2}{\partial x_n}\\[1ex]
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\vdots & \vdots & \ddots & \vdots\\
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\dfrac{\partial y_m}{\partial x_1} & \dfrac{\partial y_m}{\partial x_2} & \cdots & \dfrac{\partial y_m}{\partial x_n}
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\end{bmatrix}\eqper\]
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\section{隐函数定理}
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\begin{theorem}[隐函数定理]
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设开集$D \subset \realnum^2$，函数$F: D \to \realnum$满足条件：
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\begin{enumerate}[label=(\roman{*})]
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\item $F \in C^1(D)$；
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\item 点$(x_0, y_0) \in D$使得$F(x_0, y_0) = 0$；
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\item $\dfrac{\partial F(x_0, y_0)}{\partial y} \neq 0$，
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\end{enumerate}
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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
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\begin{enumerate}
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\item 对任意的$\abs{x - x_0} < \delta$，$f(x_0) = y_0$，有$F(x, f(x)) = 0$；
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\item $f \in C^1(x_0 - \delta, x_0 + \delta)$；
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\item 对$x \in (x_0 - \delta, x_0 + \delta)$，$y = f(x)$，有
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\[\deriv{f}(x) = -\frac{\dfrac{\partial F}{\partial x}(x, y)}{\dfrac{\partial F}{\partial y}(x, y)}\eqper\]
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\end{enumerate}
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\end{theorem}
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\begin{theorem}
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设开集$D \subset \realnum^{n + 1}$，$F: D \to \realnum$，满足条件：
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\begin{enumerate}[label=(\roman{*})]
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\item $F \in C^{(1)}(D)$；
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\item 点$(\bvec{x}_0, y_0) \in D$使得$F(\bvec{x}_0, y_0) = 0$；
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\item $\dfrac{\partial F(\bvec{x}_0, y_0)}{\partial y} \neq 0$，
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\end{enumerate}
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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
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\begin{enumerate}
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\item 对任意的$\norm{\bvec{x} - \bvec{x}_0} < \delta$，$f(\bvec{x}_0) = y_0$，有$F(\bvec{x}, f(\bvec{x})) = 0$；
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\item $f \in C^1 (B_\delta (\bvec{x}_0))$；
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\item 对$\bvec{x} \in B_\delta (\bvec{x}_0)$，$y = f(\bvec{x})$，有
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\[D_i f(x) = -\frac{\dfrac{\partial F}{\partial x_i}(\bvec{x}, y)}{\dfrac{\partial F}{\partial y}(\bvec{x}, y)}, i = 1, 2, \dots, n\eqper\]
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\end{enumerate}
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\end{theorem}
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\section{隐映射定理}
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我们先引入几个记号。设想有$m$个方程形成的方程组
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\[\begin{cases}
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F_1(x_1, \dots, x_n, y_1, \dots, y_m) = 0,\\
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\qquad \dots\dots\\
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F_m(x_1, \dots, x_n, y_1, \dots, y_m) = 0
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\end{cases}\label{隐映射定理1}\tag{1}\]
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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
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\[\begin{cases}
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y_1 = f_1(x_1, \dots, x_n)\\
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\qquad \dots\dots\\
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y_m = f_m(x_1, \dots, x_n)
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\end{cases}\label{隐映射定理2}\tag{2}\]
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为了缩短记号，可令
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\[\bvec{F} = \begin{bmatrix}
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F_1\\ \vdots\\ F_m
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\end{bmatrix},
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\boldf = \begin{bmatrix}
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f_1\\ \vdots\\ f_m
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\end{bmatrix}\]
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那么\eqref{隐映射定理1}式可以写为
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\[\bvec{F}(\bvec{x}, \bvec{y}) = \bvec{0}\]
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\eqref{隐映射定理2}式可以写为
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\[\bvec{y} = \boldf (\bvec{x})\eqper\]
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我们设$\bvec{F}$定义在开集$D \subset \realnum^{m + n}$，那么在$m \times (n + m)$矩阵
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\[J \bvec{F} = \begin{bmatrix}
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\dfrac{\partial F_1}{\partial x_1} & \cdots & \dfrac{\partial F_1}{\partial x_n} & \dfrac{\partial F_1}{\partial y_1} & \cdots & \dfrac{\partial F_1}{y_m}\\[1ex]
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\vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\
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\dfrac{\partial F_m}{\partial x_1} & \cdots & \dfrac{\partial F_m}{\partial x_n} & \dfrac{\partial F_m}{\partial y_1} & \cdots & \dfrac{\partial F_m}{y_m}
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\end{bmatrix}\]
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中作分块$J\bvec{F} = \begin{bmatrix}
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J_x \bvec{F} & J_y \bvec{F}
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\end{bmatrix}$，
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其中
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\[J_x \bvec{F} = \begin{bmatrix}
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\dfrac{\partial F_1}{\partial x_1} & \cdots & \dfrac{\partial F_1}{\partial x_n}\\
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\vdots & \ddots & \vdots\\
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\dfrac{\partial F_m}{\partial x_1} & \cdots & \dfrac{\partial F_m}{\partial x_n}
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\end{bmatrix},
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J_y \bvec{F} = \begin{bmatrix}
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\dfrac{\partial F_1}{\partial y_1} & \cdots & \dfrac{\partial F_1}{\partial y_m}\\
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\vdots & \ddots & \vdots\\
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\dfrac{\partial F_m}{\partial y_1} & \cdots & \dfrac{\partial F_m}{\partial y_m}
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\end{bmatrix}\]
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其中$J_y \bvec{F}$是$m$阶方阵。
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\begin{theorem}[隐映射定理]
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设开集$D \subset \realnum^{n + m}$，映射$\bvec{F}: D \to \realnum^m$，满足下列条件：
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\begin{enumerate}[label=(\roman{*})]
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\item $\bvec{F} \in C^1(D)$；
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\item 点$(\bvec{x}_0, \bvec{y}_0) \in D$使得$\bvec{F}(\bvec{x}_0, \bvec{y}_0) = \bvec{0}$；
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\item $\det[J_y \bvec{F}(\bvec{x}_0, \bvec{y}_0)] \neq 0$，
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\end{enumerate}
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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
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\begin{enumerate}
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\item 对任意的$\norm{\bvec{x} - \bvec{x}_0} < \delta$，$\bvec{f}(\bvec{x}_0) = \bvec{y}_0$，有$\bvec{F}(\bvec{x}, f(\bvec{x})) = \bvec{0}$；
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\item $\bvec{f} \in C^1 (B_\delta (\bvec{x}_0), \realnum^m)$；
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\item 对$\bvec{x} \in B_\delta (\bvec{x}_0)$，$\bvec{y} = \bvec{f}(\bvec{x})$，有
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\[J\bvec{f}(\bvec{x}) = -(J_y \bvec{F}(\bvec{x}, \bvec{y}))^{-1} J_x \bvec{F}(\bvec{x}, \bvec{y})\eqper\]
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\end{enumerate}
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\end{theorem}
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