多元函数微分学
基本概念
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邻域 \(B(M_0,r) = \{M\mid \rho(M,M_0)<r\}\)
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内点，核\(E^\circ\)、外点 \((E^c)^\circ\)、边界点，边界 \(\partial E\)
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内点 \(+\) 非孤立边界点 \(=\) 聚点，导集\(E'\)
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\(E=E^\circ \Rightarrow\) 开集 \(\leftrightarrow\) 闭集
多变量函数的极限
二重极限
\[\lim_{M\to M_0} f(M)=\lim_{(x,y)\to (x_0,y_0)}f(x,y)=a\]
累次极限
\[\lim_{x\to x_0}\lim_{y\to y_0}f(x,y) = a\]
性质
连续性
\[\lim_{(x,y)\to (x_0,y_0)}f(x,y) = f(x_0,y_0)\]
可偏导
\[f_x'(x_0,y_0) = \lim_{\Delta x\to 0}\dfrac{f(x_0+\Delta x, y_0)-f(x_0,y_0)}{\Delta x}\]
\[f_y'(x_0,y_0) = \lim_{\Delta y\to 0}\dfrac{f(x_0, y_0+\Delta y)-f(x_0,y_0)}{\Delta y}\]
可微
\[\Delta f-\left[f_x'(x_0,y_0)\Delta x + f_y'(x_0,y_0)\Delta y\right]=o(\sqrt{\Delta x^2+\Delta y^2})\]
或
\[\lim_{\rho \to 0}\dfrac{\Delta f-f_x'(x_0,y_0)\Delta x-f_y'(x_0,y_0)\Delta y}{\rho} = 0\]
其中 \(\Delta f = f(x_0+\Delta x,y_0+\Delta y)-f(x_0,y_0)\)
隐函数
多元隐函数
\(F(x_0,y_0,z_0) = 0, z = f(x,y)\)
\(\dfrac{\partial z}{\partial x} = -\dfrac{F_x'}{F_z'},\ \dfrac{\partial z}{\partial y} = -\dfrac{F_y'}{F_z'},\ F_z'\neq 0\)
隐映射存在定理
\(F(x_0,y_0,u_0,v_0) = 0, G(x_0,y_0,u_0,v_0) = 0\)，且在某一邻域存在连续的偏导数
\[
J_0 = \dfrac{\partial (F,G)}{\partial (u,v)}\bigg|_{P_0}=\begin{vmatrix}\dfrac{\partial F}{\partial u} & \dfrac{\partial F}{\partial v}\\\\
\dfrac{\partial G}{\partial u} & \dfrac{\partial G}{\partial v}\end{vmatrix}_{P_0}\neq 0
\]
则 \(\begin{cases}F(x,y,u,v)=0\\G(x,y,u,v)=0\end{cases}\) 可唯一确定隐映射 \(\begin{cases}u=u(x,y)\\v=v(x,y)\end{cases}\)
\[
\frac{\partial u}{\partial x} = -\frac{1}{J} \frac{\partial (F, G)}{\partial (x, v)} = -\frac{1}{J} \begin{vmatrix}
F_x & F_v \\
G_x & G_v
\end{vmatrix}
\]
\[
\frac{\partial u}{\partial y} = -\frac{1}{J} \frac{\partial (F, G)}{\partial (y, v)} = -\frac{1}{J} \begin{vmatrix}
F_y & F_v \\
G_y & G_v
\end{vmatrix}
\]
\[
\frac{\partial v}{\partial x} = -\frac{1}{J} \frac{\partial (F, G)}{\partial (u, x)} = -\frac{1}{J} \begin{vmatrix}
F_u & F_x \\
G_u & G_x
\end{vmatrix}
\]
\[
\frac{\partial v}{\partial y} = -\frac{1}{J} \frac{\partial (F, G)}{\partial (u, y)} = -\frac{1}{J} \begin{vmatrix}
F_u & F_y \\
G_u & G_y
\end{vmatrix}
\]
逆映射存在定理
\[
J_0 = \left.\dfrac{\partial (u, v)}{\partial (x, y)}\right|_{P_0} \neq 0,
\]
则
\[
\begin{cases}
u = u(x, y) \\
v = v(x, y)
\end{cases}
\]
在 \(B(u_0, v_0)\) 存在逆映射 \(\begin{cases}x = x(u, v) \\y = y(u, v)\end{cases}\) 满足 \(\begin{cases}x_0 = x(u_0, v_0) \\y_0 = y(u_0, v_0)\end{cases}\)
且有连续偏导数
\[
\begin{matrix}
\dfrac{\partial x}{\partial u} = \dfrac{1}{J} \cdot \dfrac{\partial v}{\partial y} & \dfrac{\partial x}{\partial v} = -\dfrac{1}{J} \cdot \dfrac{\partial u}{\partial y}\\\\
\dfrac{\partial y}{\partial u} = -\dfrac{1}{J} \cdot \dfrac{\partial v}{\partial x} & \dfrac{\partial y}{\partial v} = \dfrac{1}{J} \cdot \dfrac{\partial u}{\partial x}
\end{matrix}
\]
Taylor 公式和极值
Taylor 公式
Taylor公式
设函数 \( f(x, y) \) 在 \( B(P_0(x_0, y_0)) \) 有 \( n+1 \) 阶连续偏导数，则 \(\exists \theta \in (0, 1) \):
\[
f(x_0 + h, y_0 + k) = \sum_{m=0}^{n} \frac{1}{m!} \left( h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y} \right)^m f(x_0, y_0) + R_n
\]
其中 Lagrange 型余项
\[
R_n = \frac{1}{(n+1)!} \left( h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y} \right)^{n+1} f(x_0 + \theta h, y_0 + \theta k)
\]
极值
驻点
\(f_x'(x_0,y_0)=0=f_y'(x_0,y_0)\)，则 \(f(x_0,y_0)\) 为驻点
极值点
设 \(f(x, y)\) 在 \(B(P_0(x_0, y_0))\) 的二阶偏导数连续，且
\(f'_x(x_0, y_0) = 0, \quad f'_y(x_0, y_0) = 0\)
记 Hesse 矩阵
\[ H = \begin{pmatrix}
f''_{xx} & f''_{xy} \\\\
f''_{yx} & f''_{yy}
\end{pmatrix}_{P_0} \]
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若 \(H\) 为正定矩阵，则 \(f(x_0, y_0)\) 为严格极小值
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若 \(H\) 为负定矩阵，则 \(f(x_0, y_0)\) 为严格极大值。
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若 \(H\) 为不定矩阵，则 \(f(x_0, y_0)\) 非极值。
条件极值（Lagrange乘数法）
\[L(x,y,\lambda) = f(x,y) + \lambda \varphi(x,y)\]
\[\begin{cases}
L_x'=f_x'+\lambda \varphi_x'=0\\
L_y'=f_y'+\lambda \varphi_y'=0\\
L_{\lambda}'=\varphi(x,y)=0
\end{cases}\]
向量场的微商
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Hamilton 算子 \(\nabla = \dfrac{\partial}{\partial x}\bm{i} + \dfrac{\partial}{\partial y}\bm{j} + \dfrac{\partial}{\partial z}\bm{k}\)
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Laplace 算子 \(\Delta = \nabla \cdot \nabla = \dfrac{\partial^2}{\partial x^2} + \dfrac{\partial^2}{\partial y^2} + \dfrac{\partial^2}{\partial z^2}\)
梯度、散度与旋度
设有函数 \(u=u(x,y,z)\)
- \(u\) 的梯度 \(|\textbf{grad }u| = |\nabla u| = |u_x'\bm{i} + u_y'\bm{j} + u_z'\bm{k}|\)
设有向量场 \(\bm{v}(x,y,z) = P\bm{i} + Q\bm{j} + R\bm{k}\)
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\(\bm{v}\) 的散度 \(\text{div }\bm{v}\stackrel{\text{def}}{=}\nabla \cdot \bm{v} = \dfrac{\partial P}{\partial x} + \dfrac{\partial Q}{\partial y} + \dfrac{\partial R}{\partial z}\)
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\(\bm{v}\) 的旋度 \(\textbf{rot }\bm{v}\stackrel{\text{def}}{=}\nabla \times \bm{v} =
\begin{vmatrix}
\bm{i} & \bm{j} & \bm{k}\\
\dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z}\\
P & Q & R
\end{vmatrix}\)
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\(\textbf{rot grad }\varphi = \nabla \times \nabla \varphi = \bm{0}\)
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\(\text{div }\textbf{rot }\bm{v} = \nabla \cdot \nabla \times \bm{a} = 0\)