PDE-Note-01

B 站链接：西工大-马啸-姜礼尚《数学物理方程讲义》
马老师的课件在视频简介中

基础知识回顾

场论

梯度

定义

设 $u$ 是数量函数，则梯度

$$\begin{equation*} \text{grad }u = \nabla u = \left(\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial u}{\partial z}\right) \end{equation*}$$

其中，定义算子：

$$\begin{equation*} \nabla = \left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right) \end{equation*}$$

性质

1. 若 $u, v$ 是数量函数, 则

$$\begin{align*} \nabla(u + v) &= \nabla u + \nabla v \\ \nabla(u \cdot v) &= u \nabla v + v \nabla u \\ \nabla(u^2) &= 2u(\nabla u). \end{align*}$$

2. 若 $\vec{r} = (x, y, z), \phi = \phi(x, y, z)$, 则

$$\begin{align*} \mathrm{d}\phi & = \phi_x \mathrm{d}x + \phi_y \mathrm{d}y + \phi_z \mathrm{d}z \\ & = (\phi_x, \phi_y, \phi_z) \cdot (\mathrm{d}x, \mathrm{d}y, \mathrm{d}z) \\ & = \mathrm{d}\vec{r} \cdot \nabla\phi. \end{align*}$$

3. 若 $f = f(u), u = u(x, y, z)$, 则

$$\begin{align*} \nabla f & = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}) = (\frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial x}, \frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial y}, \frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial z}) \\ & = \frac{\partial f}{\partial u} () \\ & = f'(u)\nabla u. \end{align*}$$

散度

定义

$\vec{A}(x, y, z) = (P(x, y, z), Q(x, y, z), R(x, y, z))$ 为空间区域 $V$ 上的向量函数, 对于 $V$ 上任意点 $(x, y, z)$, 定义散度为：

$$\begin{equation*} \text{div }f = \nabla \cdot f = \frac{\partial f_x}{\partial x} + \frac{\partial f_y}{\partial y} + \frac{\partial f_z}{\partial z} \end{equation*}$$

设 $\vec{n} = (\cos \alpha, \cos \beta, \cos \gamma)$ 为曲面的单位法向量, 则 $\mathrm{d}\vec{S} = \vec{n}\, \mathrm{d}S$ 称为面积元素向量，则
Gauss 公式可以写成如下形式：

$$\begin{equation*} \iiint_V \nabla \cdot \vec{A}\, \mathrm{d}V = \iint_S \vec{A} \cdot \mathrm{d}\vec{S} \end{equation*}$$

性质

1. 若 $u, v$ 是向量函数, 则：

$$\begin{align*} \nabla \cdot (u + v) = \nabla \cdot u + \nabla \cdot v \end{align*}$$

2. 若 $\phi$ 是数量函数, $F$ 是向量函数, 则

$$\begin{align*} \nabla \cdot (\phi F) = \phi \nabla \cdot F + F \cdot \nabla \phi \end{align*}$$

3. 若 $\phi = \phi(x, y, z)$ 是一数量函数, 则

$$\begin{align*} \nabla \cdot \nabla \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2} = \Delta \phi \end{align*}$$

旋度

定义

$\vec{A}(x, y, z) = (P(x, y, z), Q(x, y, z), R(x, y, z))$ 为空间区域 $V$ 上的向量函数, 对于 $V$ 上任意点 $(x, y, z)$, 定义旋度为：

$$\begin{align*} \text{rot} \vec{A} &= \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) \\ & = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{vmatrix} = \nabla \times \vec{A} \end{align*}$$

Green 公式

有 $P(x, y), Q(x, y) \in C^1(\Omega) \cap C^2(\Omega)$, 则:

$$\begin{align*} \iint\limits_\Omega \begin{vmatrix} \frac{\partial}{\partial x} & \frac{\partial}{\partial y} \\ P & Q \end{vmatrix} \mathrm{d} x \mathrm{d} y = \int\limits_{\partial\Omega} P\mathrm{d}x + Q\mathrm{d}y \\ \end{align*}$$

或者由 $\mathrm{d}x = -\cos(\vec{n}, y)\mathrm{d}S, \mathrm{d}y = \cos(\vec{n}, x)\mathrm{d}S$，有：

$$\begin{align*} \iint\limits_\Omega \left(\frac{\partial Q}{\partial x} + \frac{\partial P}{\partial y}\right)\mathrm{d}x\mathrm{d}y = \int\limits_{\partial\Omega} Q\cos(n, x) + P\cos(n, y)\mathrm{d}S \end{align*}$$

设 $P, Q, R \in C^1(V), S = \partial V$，则：

$$\begin{align*} \iiint\limits_V \left(\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}\right) \mathrm{d}x\mathrm{d}y\mathrm{d}z = \iint P\mathrm{d}y\mathrm{d}z + \iint Q\mathrm{d}x\mathrm{d}z + \iint R\mathrm{d}x\mathrm{d}y \end{align*}$$

Gauss 公式

Gauss 公式可写为如下向量形式:

$$\begin{align*} \iiint\limits_V \text{div} \vec{A} \mathrm{d}V = \iint\limits_{\partial V} \vec{A} \cdot \mathrm{d}\vec{S} = \iint\limits_{\partial V} \vec{A} \cdot \vec{n} \mathrm{d}S \end{align*}$$

其中，$\vec{n} = (\cos(n, x), \cos(n, y), \cos(n, z))$

Gauss 公式常用变体

1. 取 $\vec{A} = \nabla u$

$$\begin{align*} \iiint\limits_V \Delta u \mathrm{d}V = \iint\limits_{\partial V} \nabla u \cdot \mathrm{d}\vec{S} = \iint\limits_{\partial V} \nabla u \cdot \mathrm{d}\vec{n}\mathrm{d}S = \iint\limits_S \frac{\partial u}{\partial n} \mathrm{d}S \end{align*}$$

2. 格林第一公式

• 取 $\vec{A} = u \nabla v$：

$$\begin{align*} \iiint\limits_V u \Delta v \mathrm{d}V = \iint\limits_{\partial V} u\frac{\partial\vec{v}}{\partial n}\mathrm{d}S - \iiint\limits_V \nabla u\cdot \nabla v \mathrm{d}V \end{align*}$$

• 同理，取 $\vec{A} = u \nabla v$ 可得，

$$\begin{align*} \iiint\limits_V v \Delta u\mathrm{d}V = \iint\limits_{\partial V} v \frac{\partial\vec{u}}{\partial n}\mathrm{d}S - \iiint\limits_V \nabla u \cdot \nabla v\mathrm{d}V \end{align*}$$

3. 格林第二公式

将格林第一公式的上下两式相减即得:

$$\begin{align*} \iiint\limits_V v\Delta u - u\Delta v \mathrm{d}V &= \iint\limits_{\partial V} v\frac{\partial u}{\partial n} - u\frac{\partial v}{\partial n}\mathrm{d}S \end{align*}$$

Stokes 公式

旋度定理

设 $S$ 是分片光滑的有向曲面，$S$ 的边界为有向闭曲线 $\Gamma$，即 $\Gamma = \partial S$，且 $\mathrm{d} \vec{S}$ 的正向与 $S$ 的正向一致。设函数 $P(x, y, z)$, $Q(x, y, z)$, $R(x, y, z)$ 都是定义在 曲面 $S$ 连同其边界 $\Gamma$ 上且都具有一阶连续偏导数的函数，则有：

$$\begin{align*} \iint\limits_S \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right)\mathrm{d}y\mathrm{d}z + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right)\mathrm{d}z\mathrm{d}x + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\mathrm{d}x\mathrm{d}y &= \oint\limits_\Gamma P\mathrm{d}x + Q\mathrm{d}y + R\mathrm{d}z \end{align*}$$

可以形式化地写成：

$$\begin{align*} \iint\limits_S \begin{vmatrix} \cos \alpha & \cos \beta & \cos \gamma \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{vmatrix}\mathrm{d}S = \iint\limits_S \begin{vmatrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{vmatrix}\mathrm{d}S = \oint\limits_\Gamma P\mathrm{d}x + Q\mathrm{d}y + R\mathrm{d}z \end{align*}$$

用微分算符可写成:

$$\begin{align*} \iint\limits_S \nabla \times \vec{F} \cdot \mathrm{d}\vec{S} = \oint\limits_{\partial S} \vec{F} \cdot \mathrm{d}\vec{r} \end{align*}$$