1 矢量分析与场论

1.1 矢量代数

高数笔记

1.1.1 矢量的运算

矢量的概念
这里只考虑三维的矢量（行向量）.
- $\bm a = (a_1, a_2, a_3) = a_1 \bm i + a_2 \bm j + a_3 \bm k$ .
- $\vqty{\bm a} = \sqrt{\bm a \bm a\trans} = \sqrt{a_1^2 + a_2^2 + a_3^2}$ .
- $\bm e_\bm a = \dfrac{\bm a}{\vqty{\bm a}} = \dfrac{\bm a}{a}$ .
矢量的加减
- $\bm a + \bm b = (a_1 + b_1, a_2 + b_2, a_3 + b_3)$ .
- $-\bm a = (-a_1, -a_2, -a_3)$ .
- $\bm a - \bm b = (a_1 - b_1, a_2 - b_2, a_3 - b_3)$ .
矢量的乘法
- $k \bm a = (k a_1, k a_2, k a_3)$ .
- $\bm a \bm\cdot \bm b = a_1 b_1 + a_2 b_2 + a_3 b_3$ .
- $\bm a \cp \bm b = (a_2 b_3 - a_3 b_2, a_3 b_1 - a_1 b_3, a_1 b_2 - a_2 b_1)$ .
- $(\bm a \bm b\bm c) = (\bm a, \bm b, \bm c) = (\bm a \cp \bm b) \bm\cdot \bm c$ .

1.1.2 矢量的性质

基本性质（并不完整）
- 加法：零元、负元、结合律、交换律.
- 数乘：结合律、第一第二分配律、消去律.
- 点积：交换律、数乘结合律、加法分配律.
点乘
- $\bm a \bm\cdot \bm b = \bm a \bm b\trans = \vqty{\bm a} \cdot \vqty{\bm b} \cos\theta$ .
- $\vqty{\bm a} = \sqrt{\bm a \bm a \trans} = \sqrt{\bm a \bm\cdot \bm a}$ .
- $\bm a \perp \bm b \RLA \bm a \bm\cdot \bm b = 0$ .
叉乘
- 叉乘性质
  - $\bm a \cp \bm b = -\bm b \cp \bm a$ .
  - $\vqty{\bm a \cp \bm b} = \vqty{\bm a} \cdot \vqty{\bm b} \cdot \vqty{\sin\theta}$ .
  - $\bm a \parallel \bm b \RLA \bm a \cp \bm b = \bm 0$ .
- 其它表示
  - $\bm a \cp \bm b = \begin{vmatrix} \bm i & \bm j & \bm k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$ .
  - $\bm a \cp \bm b = \bm a \begin{pmatrix} 0 & -b_3 & b_2 \\ b_3 & 0 & -b_1\\ -b_2 & b_1 & 0 \end{pmatrix}$ .
  - $\bm a\trans \cp \bm b\trans = \begin{pmatrix} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0 \end{pmatrix} \bm b\trans$ .
混合积
- $(\bm a \bm b \bm c) = (\bm a \cp \bm b) \bm\cdot \bm c = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}$ .
- $(\bm a \bm b \bm c) = (\bm b \bm c \bm a) = (\bm c \bm a \bm b) = -(\bm a \bm c \bm b) = -(\bm c \bm b \bm a) = -(\bm b \bm a \bm c)$ .
- $\RLA (\bm a \bm b \bm c) = 0 \RLA \exist \alpha, \beta, \gamma \colon \alpha \bm a + \beta \bm b + \gamma \bm c = \bm 0$ .
其它性质
- $\bm a$ $\bm a \cdot \bm b = \bm a \cdot \bm c$ $\bm a \cp \bm b = \bm a \cp \bm c$ $\bm b = \bm c$ .
- $\bm a$ $\bm b$ $\bm b_\parallel = \dfrac{\bm a \cdot \bm b}{\bm a \cdot \bm a} \bm a$ $\bm b_\perp = \bm b - \bm b_\parallel$ .

1.1.3 矢量恒等式

拉格朗日恒等式
1. $(\bm a \cp \bm b) \cp \bm c = \bm b (\bm a \bm\cdot \bm c) - \bm a (\bm b \bm\cdot \bm c)$ .
2. $\bm a \cp (\bm b \cp \bm c) = \bm b (\bm a \bm\cdot \bm c) - \bm c (\bm a \bm\cdot \bm b)$ .
$\bm a \cp (\bm b \cp \bm c) + \bm b \cp (\bm c \cp \bm a) + \bm c \cp (\bm a \cp \bm b) = \bm 0$ .
四向量恒等式
1. $(\bm a \cp \bm b) \bm\cdot (\bm c \cp \bm d) = (\bm a \bm\cdot \bm c) (\bm b \bm\cdot \bm d) - (\bm a \bm\cdot \bm d) (\bm b \bm\cdot \bm c)$ .
2. $(\bm a \cp \bm b) \cp (\bm c \cp \bm d) = \bm c [\bm a \vdot (\bm b \cp \bm d)] - \bm d [\bm a \vdot (\bm b \cp \bm c)]$ .
3. $\bm a \cp \bqty{\bm b \cp \pqty{\bm c \cp \bm d}} = (\bm a \cp \bm c) (\bm b \vdot \bm d) - (\bm a \cp \bm d) (\bm b \vdot \bm c)$ .

证明

1.1.4 矢量微积分

$\bm a = \bm a(t), \bm b = \bm b(t), f = f(t)$ $\lambda, \mu$ $\bm k$ 为常向量.

求导
1. $\ddv{t} (\lambda \bm a + \mu \bm b) = \lambda \ddv{\bm a}{t} + \mu \ddv{\bm b}{t}$ .
2. $\ddv{t} (f \bm a) = \ddv{f}{t} \bm a + f \ddv{\bm a}{t}$ .
3. $\ddv{t} (\bm a \vdot \bm b) = \ddv{\bm a}{t} \vdot \bm b + \bm a \vdot \ddv{\bm b}{t}$ .
4. $\ddv{t} (\bm a \cp \bm b) = \ddv{\bm a}{t} \cp \bm b + \bm a \cp \ddv{\bm b}{t}$ .
5. $\ddv{t} \bm a(f) = \ddv{\bm a}{f} \cdot \ddv{f}{t}$ .
积分
1. $\dint (\lambda \bm a + \mu \bm b) \dt = \lambda \dint \bm a \dt + \mu \dint \bm b \dt$ .
2. $\dint \bm k \vdot \bm a \dt = \bm k \vdot \dint \bm a \dt$ .
3. $\dint \bm k \cp \bm a \dt = \bm k \cp \dint \bm a \dt$ .

1.1.5 矢量的旋转

$\bm k$ $\bm v$ $\theta$ $\bm v'$ $\bm v_\text{rot}$ ），则

\begin{aligned} v^{'} & = \cos θ v + (1 - \cos θ) (v \cdot k) k + \sin θ k \times v \\ = v + (1 - \cos θ) k \times (k \times v) + \sin θ k \times v . \end{aligned}

证明

备注有关空间旋转的更多内容可参考笔记平面旋转与空间旋转.

1.2 常用坐标系

1.2.1 正交坐标系

基本符号
- $(u_1, u_2, u_3)$ .
- $\bm a_{u_1}, \bm a_{u_2}, \bm a_{u_3}$ .
  $\bm e_{u_1}, \bm e_{u_2}, \bm e_{u_3}$ .
- $h_i = h_i(u_1, u_2, u_3) = \ddv{l_{u_i}}{u_i}$ .
  又称为度量系数或尺度因子.
空间度量
- $\begin{cases} \dd{l_{u_1}} = h_1 \dd{u_1}, \\ \dd{l_{u_2}} = h_2 \dd{u_2}, \\ \dd{l_{u_3}} = h_3 \dd{u_3}. \end{cases}$
- $\begin{cases} \dd{\bm S_1} = \bm a_{u_1} (h_2 h_3 \dd{u_2} \dd{u_3}), \\ \dd{\bm S_2} = \bm a_{u_2} (h_3 h_1 \dd{u_3} \dd{u_1}), \\ \dd{\bm S_3} = \bm a_{u_3} (h_1 h_2 \dd{u_1} \dd{u_2}). \end{cases}$
- $\dd{V} = h_1 h_2 h_3 \dd{u_1} \dd{u_2} \dd{u_3}$ .

1.2.2 直角坐标系

基本符号
- $(x, y, z)$ .
- $\bm a_x, \bm a_y, \bm a_z$ .
- $\bm r = x \bm a_x + y \bm a_y + z \bm a_z$ .
空间度量
- $\begin{cases} \dd{l_x} = \dx, \\ \dd{l_y} = \dy, \\ \dd{l_z} = \dz. \end{cases}$
- $\begin{cases} \dd{S_x} = \dydz, \\ \dd{S_y} = \dzdx, \\ \dd{S_z} = \dxdy. \end{cases}$
- $\dd{V} = \dxdydz$ .
坐标变量关系
- $\ra$ $\begin{cases} x = \rho\cos\varphi, \\ y = \rho\sin\varphi, \\ z = z. \end{cases}$
- $\ra$ $\begin{cases} x = r \sin\theta \cos\varphi, \\ y = r \sin\theta \sin\varphi, \\ z = r \cos\theta. \end{cases}$
单位矢量关系
- $\ra$ $\begin{bmatrix} \bm a_x \\ \bm a_y \\ \bm a_z \end{bmatrix} = \begin{bmatrix} \cos\varphi & -\sin\varphi & 0 \\ \sin\varphi & \cos\varphi & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \bm a_\rho \\ \bm a_\varphi \\ \bm a_z \end{bmatrix}$ .
- $\ra$ $\begin{bmatrix} \bm a_x \\ \bm a_y \\ \bm a_z \end{bmatrix} = \begin{bmatrix} \sin\theta \cos\varphi & \cos\theta \cos\varphi & -\sin\varphi \\ \sin\theta \sin\varphi & \cos\theta \sin\varphi & \cos\varphi \\ \cos\theta & -\sin\theta & 0 \end{bmatrix} \begin{bmatrix} \bm a_r \\ \bm a_\theta \\ \bm a_\varphi \end{bmatrix}$ .
单位矢量的偏导、散度与旋度均为零.

1.2.3 圆柱坐标系

基本符号
- $(\rho, \varphi, z)$ .
- $\bm a_\rho, \bm a_\varphi, \bm a_z$ .
- $\bm r = \bm e_\rho \rho + \bm e_z z$ .
空间度量
- $\begin{cases} \dd{l_\rho} = \dd{\rho}, \\ \dd{l_\varphi} = \rho \dd{\varphi}, \\ \dd{l_z} = \dz. \end{cases}$
- $\begin{cases} \dd{S_\rho} = \rho \ddf{\varphi}{z}, \\ \dd{S_\varphi} = \ddf{r}{z}, \\ \dd{S_z} = \rho\ddf{\rho}{\varphi}. \end{cases}$
- $\dd{V} = \rho\dddf{\rho}{\varphi}{z}$ .
坐标变量关系
- $\ra$ $\begin{cases} \rho = \sqrt{x^2 + y^2}, \\ \varphi = \arctan\dfrac{y}{x}, \\ z = z. \end{cases}$
- $\ra$ $\begin{cases} \rho = r \sin\theta, \\ \varphi = \varphi, \\ z = r \cos\theta. \end{cases}$
单位矢量关系
- $\ra$ $\begin{bmatrix} \bm a_\rho \\ \bm a_\varphi \\ \bm a_z \end{bmatrix} = \begin{bmatrix} \cos\varphi & \sin\varphi & 0 \\ -\sin\varphi & \cos\varphi & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \bm a_x \\ \bm a_y \\ \bm a_z \end{bmatrix}$ .
- $\ra$ $\begin{bmatrix} \bm a_\rho \\ \bm a_\varphi \\ \bm a_z \end{bmatrix} = \begin{bmatrix} \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \\ \cos\theta & -\sin\theta & 0 \end{bmatrix} \begin{bmatrix} \bm a_r \\ \bm a_\theta \\ \bm a_\varphi \end{bmatrix}$ .
单位矢量导数（未标注的均为零）
- $\varphi$ $\begin{cases} \ddv{\bm a_\rho}{\varphi} = \bm a_\varphi, \\ \ddv{\bm a_\varphi}{\varphi} = -\bm a_\rho. \\ % \ddv{\bm a_z}{\varphi} = 0. \end{cases}$
- $\div \bm a_\rho = \dfrac{1}{\rho}$ .
- $\curl \bm a_\varphi = \dfrac{1}{\rho} \bm a_z$ .

1.2.4 球极坐标系

基本符号
- $(r, \theta, \varphi)$ .
- $\bm a_r, \bm a_\theta, \bm a_\varphi$ .
- $\bm r = r \bm a_r$ .
空间度量
- $\begin{cases} \dd{l_r} = \dr, \\ \dd{l_\theta} = r \dd{\theta}, \\ \dd{l_\varphi} = r \sin\theta \dd{\varphi}. \end{cases}$
- $\begin{cases} \dd{S_r} = r^2 \sin\theta \ddf{\theta}{\varphi}, \\ \dd{S_\theta} = r \sin\theta \ddf{r}{\varphi}, \\ \dd{S_\varphi} = r \ddf{r}{\theta}. \end{cases}$
- $\dd{V} = r^2 \sin\theta \dddf{r}{\theta}{\varphi}$ .
坐标变量关系
- $\ra$ $\begin{cases} r = \sqrt{x^2 + y^2 + z^2}, \\ \theta = \arctan\dfrac{\sqrt{x^2 + y^2}}{z}, \\ \varphi = \arctan\dfrac{y}{x}. \end{cases}$
- $\ra$ $\begin{cases} r = \sqrt{\rho^2 + z^2}, \\ \theta = \arctan\dfrac{\rho}{z}, \\ \varphi = \varphi. \end{cases}$
单位矢量关系
- $\ra$ $\begin{bmatrix} \bm a_r \\ \bm a_\theta \\ \bm a_\varphi \end{bmatrix} = \begin{bmatrix} \sin\theta \cos\varphi & \sin\theta \sin\varphi & \cos\theta \\ \cos\theta \cos\varphi & \cos\theta \sin\varphi & -\sin\theta \\ -\sin\varphi & \cos\varphi & 0 \end{bmatrix} \begin{bmatrix} \bm a_x \\ \bm a_y \\ \bm a_z \end{bmatrix}$ .
- $\ra$ $\begin{bmatrix} \bm a_r \\ \bm a_\theta \\ \bm a_\varphi \end{bmatrix} = \begin{bmatrix} \sin\theta & 0 & \cos\theta \\ \cos\theta & 0 & -\sin\theta \\ 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} \bm a_\rho \\ \bm a_\varphi \\ \bm a_z \end{bmatrix}$ .
单位矢量导数（未标注的均为零）
- $\theta$ $\begin{cases} \dpdv{\bm a_r}{\theta} = \bm a_\theta, \\ \dpdv{\bm a_\theta}{\theta} = -\bm a_r, \\ \dpdv{\bm a_\varphi}{\theta} = \bm 0. \end{cases}$
- $\varphi$ $\begin{cases} \dpdv{\bm a_r}{\varphi} = \bm a_\varphi \sin\theta, \\ \dpdv{\bm a_\theta}{\varphi} = \bm a_\varphi \cos\theta, \\ \dpdv{\bm a_\varphi}{\varphi} = - \bm a_r \sin\theta - \bm a_\theta \cos\theta. \end{cases}$
- $\begin{cases} \div \bm a_r = \dfrac{2}{r}, \\ \div \bm a_\theta = \dfrac{\cot\theta}{r}, \\ \div \bm a_\varphi = 0. \end{cases}$
- $\begin{cases} \curl \bm a_r = \bm 0, \\ \curl \bm a_\theta = \dfrac{1}{r} \bm a_\varphi, \\ \curl \bm a_\varphi = \dfrac{\cot\theta}{r} \bm a_r - \dfrac{1}{r} \bm a_\theta. \end{cases}$
$\bm R_1 = (r_1, \theta_1, \varphi_1)$ $\bm R_2 = (r_2, \theta_2, \varphi_2)$ 夹角的余弦为
$\cos γ = \cos θ_{1} \cos θ_{2} + \sin θ_{1} \sin θ_{2} \cos (φ_{1} - φ_{2}) .$

1.3 哈密顿算子

1.3.1 正交坐标系

定义
- $\grad f = \bm a_n \DV{f}{n} = \bm a_n \d\dv{f}{n}$ .
- $\div \bm F = \d\lim_{\Delta V \to 0} \dfrac{1}{\Delta V} \oint_S \bm F \bm\cdot \dd{\bm S}$ .
- $\curl \bm F = \bm a_{\Delta \bm S} \d\lim_{\Delta S \to \bm 0} \dfrac{1}{\Delta S} \oint_l \bm F \bm\cdot \dd{\bm l}$ .
推论
- $\grad f = \d\dfrac{\bm a_{u_1}}{h_1} \pdv{f}{u_1} + \dfrac{\bm a_{u_2}}{h_2} \pdv{f}{u_2} + \dfrac{\bm a_{u_3}}{h_3} \pdv{f}{u_3}$ .
- $\div \bm F = \dfrac{1}{h_1 h_2 h_3} \pqty{\d\pdv{u_1} F_1 h_2 h_3 + \pdv{u_2} h_1 F_2 h_3 + \pdv{u_3} h_1 h_2 F_3}$ .
- $\curl \bm F = \dfrac{1}{h_1 h_2 h_3} \begin{vmatrix} h_1 \bm a_{u_1} & h_2 \bm a_{u_2} & h_3 \bm a_{u_3} \\ \dpdv{u_1} & \dpdv{u_2} & \dpdv{u_3} \\ h_1 F_1 & h_2 F_2 & h_3 F_3 \end{vmatrix}$ .
$P = (x, y, z)$ $P' = (x', y', z')$ $R$ $\grad\dfrac{1}{R} = -\dfrac{\bm R}{R^3} = -\grad' \dfrac{1}{R}$ .

1.3.2 直角坐标系

$\grad f = \bm a_x \dpdv{f}{x} + \bm a_y \dpdv{f}{y} + \bm a_z \dpdv{f}{z}$ .
$\d\div \bm F = \pdv{F_x}{x} + \pdv{F_y}{y} + \pdv{F_z}{z}$ .
$\curl \bm F = \begin{vmatrix} \bm a_x & \bm a_y & \bm a_z \\ \d\pdv{x} & \dpdv{y} & \dpdv{z} \\ F_x & F_y & F_z \\ \end{vmatrix}$ .

1.3.3 圆柱坐标系

$\grad f = \bm a_\rho \dpdv{f}{\rho} + \dfrac{\bm a_\varphi}{\rho} \dpdv{f}{\varphi} + \bm a_z \dpdv{f}{z}$ .
$\div \bm F = \dfrac{1}{\rho} \d\pdv{(\rho F_\rho)}{\rho} + \dfrac{1}{\rho} \pdv{F_\varphi}{\varphi} + \pdv{F_z}{z}$ .
$\curl \bm F = \dfrac{1}{\rho} \begin{vmatrix} \bm a_\rho & \rho \bm a_\varphi & \bm a_z \\ \dpdv{\rho} & \dpdv{\varphi} & \dpdv{z} \\ F_\rho & \rho F_\varphi & F_z \end{vmatrix}$ .

1.3.4 球极坐标系

$\grad f = \bm a_r \dpdv{f}{r} + \dfrac{\bm a_\theta}{r} \dpdv{f}{\theta} + \dfrac{\bm a_\varphi}{r\sin\theta} \dpdv{f}{\varphi}$ .
$\div \bm F = \dfrac{1}{r^2} \d\pdv{(r^2 F_r)}{r} + \dfrac{1}{r \sin\theta} \pdv{\sin\theta F_\theta}{\theta} + \dfrac{1}{r\sin\theta} \pdv{F_\varphi}{\varphi}$ .
$\curl \bm F = \dfrac{1}{r^2 \sin\theta} \begin{vmatrix} \bm a_r & r \bm a_\theta & r\sin\theta \bm a_\varphi \\ \dpdv{r} & \dpdv{\theta} & \dpdv{\varphi} \\ F_r & r F_\theta & r\sin\theta F_\varphi \end{vmatrix}$ .

1.3.5 记号与拓展

可以将哈密顿算子写成向量的形式，但此时数乘与点乘均不可交换. 以直角坐标系为例：

$\d \grad = \pqty{\pdv{x}, \pdv{y}, \pdv{z}}$ .
左乘
- $\d \grad f = \pqty{\pdv{f}{x}, \pdv{f}{y}, \pdv{f}{z}}$ .
- $\d\div \bm F = \pdv{F_1}{x} + \pdv{F_2}{y} + \pdv{F_3}{z}$ .
- $\d \curl \bm F = \pqty{\pdv{F_3}{y} - \pdv{F_2}{z}, \pdv{F_1}{z} - \pdv{F_3}{x}, \pdv{F_2}{x} - \pdv{F_1}{y}}$ .
右乘
- $\d f \grad = \pqty{f\pdv{x}, f\pdv{y}, f\pdv{z}}$ .
- $\d \bm F \bm\cdot \grad = F_1 \pdv{x} + F_2 \pdv{y} + F_3 \pdv{z}$ .
- $\d \bm F \cp \grad = \pqty{F_3 \pdv{y} - F_2 \pdv{z}, F_1 \pdv{z} - F_3 \pdv{x}, F_2 \pdv{x} - F_1 \pdv{y}}$ .
备注
- $\div \bm F \ne \bm F \bm\cdot \grad$ .
- $\div \grad f := \div (\grad f) = (\div \grad) f$ .
- $\curl \grad f := \curl (\grad f) = (\curl \grad) f \equiv \bm 0$ .
约定
- 梯度、散度、旋度的优先级高于数乘、点乘与叉乘，并且从右向左计算.
- $\curl \curl \bm F := \curl (\curl \bm F) \ne (\curl \grad) \cp \bm F \equiv \bm 0$ .
- $\curl \bm a \cp \bm b := (\curl \bm a) \cp \bm b \ne \curl (\bm a \cp \bm b)$ .
向量的梯度
$\begin{matrix} \nabla F = \frac{\partial (F_{1}, F_{2}, F_{3})}{\partial (x, y, z)} = [\begin{matrix} \frac{\partial F_{1}}{\partial x} & \frac{\partial F_{1}}{\partial y} & \frac{\partial F_{1}}{\partial z} \\ \frac{\partial F_{2}}{\partial x} & \frac{\partial F_{2}}{\partial y} & \frac{\partial F_{3}}{\partial z} \\ \frac{\partial F_{3}}{\partial x} & \frac{\partial F_{3}}{\partial y} & \frac{\partial F_{3}}{\partial z} \end{matrix}] . \end{matrix}$

1.4 哈密顿算子的性质

以下均假设混合偏导连续.

1.4.1 哈密顿算子与叉乘

Hamilton 算子在右边
1. $\bm a \cp (\bm b \cp \grad) = \bm b (\bm a \bm\cdot \grad) - (\bm a \bm\cdot \bm b) \grad$ .
2. $(\bm a \cp \bm b) \cp \grad = \bm b (\bm a \bm\cdot \grad) - \bm a (\bm b \bm\cdot \grad)$ .
Hamilton 算子在中间
1. $\bm a \cp (\curl \bm b) = \bm a \grad \bm b - (\bm a \bm\cdot \grad) \bm b$ . 🌙
2. $(\bm a \cp \grad) \cp \bm b = \bm a (\div \bm b) - \bm a \grad \bm b$ . 🌙
Hamilton 算子在左边
1. $\curl (\bm a \cp \bm b) = \bm a (\div \bm b) - (\bm a \bm\cdot \grad) \bm b - \bm b (\div \bm a) + (\bm b \bm\cdot \grad) \bm a$ . 🌙
2. $(\curl \bm a) \cp \bm b = (\bm b \bm\cdot \grad) \bm a - \bm b \grad \bm a$ .
上述公式的推论
1. $\bm a \cp (\curl \bm b) + (\bm a \cp \grad) \cp \bm b = \bm a (\div \bm b) - (\bm a \bm\cdot \grad) \bm b$ .
2. $\curl (\bm a \times \bm b) = (\bm a \cp \grad + \curl \bm a) \cp \bm b - (\bm b \cp \grad + \curl \bm b) \cp \bm a$ .

证明

1.4.2 哈密顿算子与梯度

标量的梯度——加减乘除与复合
1. $\grad (\lambda f + \mu g) = \lambda\grad f + \mu \grad g$ .
2. $\grad (fg) = f\grad g + g\grad f$ .
3. $\grad(\dfrac{f}{g}) = \dfrac{g \grad f - f \grad g}{g^2}$ .
4. $\grad (\phi(f)) = \phi'(f) \grad f$ .
标量的梯度——向量点乘与散度
1. $\grad (\bm a \bm\cdot \bm b) = \bm a \grad \bm b + \bm b \grad \bm a$ . 🌙
2. $\grad (\bm a \bm\cdot \bm b) = \bm a \times (\curl \bm b) + \bm b \times (\curl \bm a) + (\bm a \bm\cdot \grad) \bm b + (\bm b \bm\cdot \grad) \bm a$ .
3. $\grad (\bm a \bm\cdot \bm b) = \bm a (\div \bm b) + \bm b (\div \bm a) - (\bm a \cp \grad) \cp \bm b - (\bm b \cp \grad) \cp \bm a$ .
4. $\grad (\div \bm a) = \Delta \bm a + \curl(\curl \bm a)$ .
$\grad (\lambda \bm a + \mu \bm b) = \lambda \grad \bm a + \mu \grad \bm b$ .

证明

1.4.3 哈密顿算子与散度

$\div(\lambda \bm a + \mu \bm b) = \lambda(\div \bm a) + \mu(\div \bm b)$ .
$\div(f\bm a) = f\div \bm a + \grad f \bm\cdot \bm a$ .
$\div (\bm a \cp \bm b) = (\curl \bm a) \bm\cdot \bm b - \bm a \bm\cdot (\curl \bm b)$ .
$\div(g \grad f) = \grad g \bm\cdot \grad f + g\Delta f$ .
$\div (\curl \bm a) = 0$ .（旋度场无源）
$\div (f \curl \bm a) = (\grad f) \bm\cdot (\curl \bm a)$ .

证明

1.4.4 哈密顿算子与旋度

$\curl (\lambda \bm a + \mu \bm b) = \lambda (\curl \bm a) + \mu(\curl \bm b)$ .
$\curl(f\bm a) = f \curl \bm a + \grad f \times \bm a$ .
$\curl (\bm a \times \bm b) = \bm a (\div \bm b) - \bm b (\div \bm a) - (\bm a \bm\cdot \grad) \bm b + (\bm b \bm\cdot \grad) \bm a$ .
$\curl (\grad f) = \bm 0$ .（梯度场无旋）
$\curl (g \grad f) = (\grad g) \cp (\grad f)$ .
$\curl (\curl \bm a) = \grad(\div \bm a) - \Delta \bm a$ .

证明

1.4.5 哈密顿算子与积分

1.5 拉普拉斯算子

1.5.1 一般定义

$\Delta f = \grad^2 f = \div \grad f$ .
$\Delta \bm a = \grad (\div \bm a) - \curl (\curl \bm a)$ .
$\Delta \dfrac{1}{\bm r - \bm r'} = -4\pi \delta(\bm r - \bm r')$ .

1.5.2 二维空间

$\Delta f = \dfrac{1}{h_1 h_2} \dsum \d\pdv{u_1}\pqty{\dfrac{h_2}{h_1} \pdv{f}{u_1}}$ .
$\d \Delta f = \pdv[2]{f}{x} + \pdv[2]{f}{y}$ .
$\dfrac{1}{\rho} \d\pdv{\rho} \pqty{\rho\pdv{f}{\rho}} + \dfrac{1}{\rho^2} \pdv[2]{f}{\varphi}$ .

1.5.3 三维空间

$\Delta f = \dfrac{1}{h_1 h_2 h_3} \d\sum {\d \pdv{u_1} \pqty{\dfrac{h_2 h_3}{h_1} \pdv{f}{u_1}}}$ .
$\Delta f = \d \pdv[2]{f}{x} + \pdv[2]{f}{y} + \pdv[2]{f}{z}$ .
$\Delta f = \dfrac{1}{\rho} \d\pdv{\rho} \pqty{\rho \pdv{f}{\rho}} + \dfrac{1}{\rho^2} \pdv[2]{f}{\varphi} + \pdv[2]{f}{z}$ .
$\Delta f = \dfrac{1}{r^2} \d\pdv{r} \pqty{r^2 \pdv{f}{r}} + \dfrac{1}{r^2 \sin\theta} \pdv{\theta} \pqty{\sin\theta \pdv{f}{\theta}} + \dfrac{1}{r^2\sin^2\theta} \pdv[2]{f}{\varphi}$ .

1.5.4 三维矢量

$\Delta \bm F = \bm a_x \Delta F_x + \bm a_y \Delta F_y + \bm a_z \Delta F_z$ .
$\d \Delta \bm F = \pqty{ \Delta F_\rho - \dfrac{2}{\rho^2} \pdv{F_\varphi}{\varphi} - \dfrac{F_\rho}{\rho^2} } \bm a_\rho + \pqty{ \Delta F_\varphi + \dfrac{2}{\rho^2} \pdv{F_\rho}{\varphi} - \dfrac{F_\varphi}{\rho^2} } \bm a_\varphi + \Delta F_z \cdot \bm a_z$ .
球坐标系：
$\begin{aligned} Δ F & = [Δ F_{r} - \frac{2}{r^{2}} (F_{r} + \cot θ F_{θ} + \csc θ \frac{\partial F_{φ}}{\partial φ} + \frac{\partial A_{θ}}{\partial θ})] a_{r} + \\ [Δ F_{θ} - \frac{1}{r^{2}} (\csc^{2} θ F_{θ} - 2 \frac{\partial F_{r}}{\partial θ} + 2 \cot θ \csc θ \frac{\partial F_{φ}}{\partial φ})] a_{θ} + \\ [Δ F_{φ} - \frac{1}{r^{2}} (\csc^{2} θ F_{φ} - 2 \csc θ \frac{\partial F_{r}}{\partial φ} - 2 \cot θ \csc θ \frac{\partial F_{θ}}{\partial φ})] . \end{aligned}$

证明

对于圆柱坐标系，

\begin{aligned} F & = F_{ρ} a_{ρ} + F_{φ} a_{φ} + F_{z} a_{z}, \\ \frac{\partial F}{\partial φ} & = (\frac{\partial F_{ρ}}{\partial φ} - F_{φ}) a_{ρ} + (\frac{\partial F_{φ}}{\partial φ} + F_{ρ}) a_{φ} + \frac{\partial F_{z}}{\partial φ} a_{z}, \\ \frac{\partial^{2} F}{\partial φ^{2}} & = (\frac{\partial^{2} F_{ρ}}{\partial φ^{2}} - 2 \frac{\partial F_{φ}}{\partial φ} - F_{ρ}) a_{ρ} + (\frac{\partial^{2} F_{φ}}{\partial φ^{2}} + 2 \frac{\partial F_{ρ}}{\partial φ} - F_{φ}) a_{φ} + \frac{\partial^{2} F_{z}}{\partial φ^{2}} a_{z}, \\ Δ F & = \frac{1}{ρ} \frac{\partial}{\partial ρ} (ρ \frac{\partial F}{\partial ρ}) + \frac{1}{ρ^{2}} \frac{\partial^{2} F}{\partial φ^{2}} + \frac{\partial^{2} F}{\partial z^{2}} \\ = \frac{1}{ρ} \frac{\partial F}{\partial ρ} + \frac{\partial^{2} F}{\partial ρ^{2}} + \frac{1}{ρ^{2}} \frac{\partial^{2} F}{\partial φ^{2}} + \frac{\partial^{2} F}{\partial z^{2}} \\ = (\frac{1}{ρ} \frac{\partial F_{ρ}}{\partial ρ} + \frac{\partial^{2} F_{ρ}}{\partial ρ^{2}} + \frac{1}{ρ^{2}} \frac{\partial^{2} F_{ρ}}{\partial φ^{2}} + \frac{\partial^{2} F_{ρ}}{\partial z^{2}} - \frac{2}{ρ^{2}} \frac{\partial F_{φ}}{\partial φ} - \frac{F_{ρ}}{ρ^{2}}) a_{ρ} + \\ (\frac{1}{ρ} \frac{\partial F_{φ}}{\partial ρ} + \frac{\partial^{2} F_{φ}}{\partial ρ^{2}} + \frac{1}{ρ^{2}} \frac{\partial^{2} F_{φ}}{\partial φ^{2}} + \frac{\partial^{2} F_{φ}}{\partial z^{2}} + \frac{2}{ρ^{2}} \frac{\partial F_{ρ}}{\partial φ} - \frac{F_{φ}}{ρ^{2}}) a_{φ} + \\ (\frac{1}{ρ} \frac{\partial F_{z}}{\partial ρ} + \frac{\partial^{2} F_{z}}{\partial ρ^{2}} + \frac{1}{ρ^{2}} \frac{\partial^{2} F_{z}}{\partial φ^{2}} + \frac{\partial^{2} F_{z}}{\partial z^{2}}) a_{z} \\ = (Δ F_{ρ} - \frac{2}{ρ^{2}} \frac{\partial F_{φ}}{\partial φ} - \frac{F_{ρ}}{ρ^{2}}) a_{ρ} + (Δ F_{φ} + \frac{2}{ρ^{2}} \frac{\partial F_{ρ}}{\partial φ} - \frac{F_{φ}}{ρ^{2}}) a_{φ} + Δ F_{z} \cdot a_{z} . \end{aligned}

对于球坐标系，这里采用另一种思路，

\begin{aligned} Δ F & = \nabla (\nabla \cdot F) - \nabla \times (\nabla \times F) \\ = [Δ F_{r} - \frac{2}{r^{2}} (F_{r} + \cot θ F_{θ} + \csc θ \frac{\partial F_{φ}}{\partial φ} + \frac{\partial A_{θ}}{\partial θ})] a_{r} + \\ [Δ F_{θ} - \frac{1}{r^{2}} (\csc^{2} θ F_{θ} - 2 \frac{\partial F_{r}}{\partial θ} + 2 \cot θ \csc θ \frac{\partial F_{φ}}{\partial φ})] a_{θ} + \\ [Δ F_{φ} - \frac{1}{r^{2}} (\csc^{2} θ F_{φ} - 2 \csc θ \frac{\partial F_{r}}{\partial φ} - 2 \cot θ \csc θ \frac{\partial F_{θ}}{\partial φ})] . \end{aligned}

证毕.

1.5.5 算子性质

标量函数
1. $\Delta (\lambda f + \mu g) = \lambda \Delta f + \mu \Delta g$ .
2. $\Delta (f g) = (\Delta f) g + 2 (\grad f) \bm\cdot (\grad g) + f (\Delta g)$ .
3. $\Delta (\phi(f)) = \phi''(f) (\grad f)^2 + \phi'(f) \Delta f$ .
向量函数
1. $\Delta (\lambda \bm a + \mu \bm b) = \lambda \Delta \bm a + \mu \Delta \bm b$ .
2. $\Delta \bm a = \grad (\div \bm a) - \curl (\curl \bm a)$ .
3. $\Delta \bm a = \bqty{\div (\grad \bm a)}\trans = \pqty{\grad^2 \bm a}\trans$ .

证明

1.6 场论定理

1.6.1 场论的三大定理

Green 公式
- $\d \intl_{\partial D} \bm r \bm\cdot \bm s \ds = \int\limits_{\partial D} P\dd{x} + Q\dd{y} = \iint\limits_D \pqty{ \pdv{Q}{x} - \pdv{P}{y} } \dxdy$ .
- $\d \intl_{\partial D} \bm r \bm\cdot \bm n \ds = \int\limits_{\partial D} P\dd{y} - Q\dd{x} = \iint\limits_D \pqty{ \pdv{P}{x} + \pdv{Q}{y} } \dxdy$ .
$\iintl_{\partial \Omega} \bm a \bm\cdot \dd{\bm S} = \iiintl_\Omega \div \bm a \dd{V}$ .
$\intl_{\partial \sum} \bm a \bm\cdot \dd{\bm s} = \iintl_{\sum} (\curl \bm a) \bm\cdot \dd{\bm S}$ .

1.6.2 两个格林恒等式

$\d \iintl_D \pqty{P \pdv{u}{x} + Q \pdv{u}{y}} \dxdy = \intl_{\partial D} Pu \dy - Qu \dx - \iintl_D \pqty{\pdv{P}{x} + \pdv{Q}{y}} u \dxdy$ .
Green 第一公式
- $\iiintl_\Omega \pqty{ \grad f \bm\cdot \grad g + f \Delta g } \dd{V} = \iintl_{\partial \Omega} f \pdv{g}{\bm n} \dd{S}$ .
- $\iiintl_\Omega \pqty{ \grad g \bm\cdot \grad f + g \Delta f } \dd{V} = \iintl_{\partial \Omega} g \pdv{f}{\bm n} \dd{S}$ .
$\iiintl_\Omega (f\Delta g - g\Delta f) \dd{V} = \iintl_{\partial \Omega} \pqty{ f\pdv{g}{\bm n} - g\pdv{f}{\bm n} } \dd{S}$ .

1.6.3 Helmholtz 定理

Helmholtz 定理

\begin{aligned} F & = - \nabla u + \nabla \times A \\ u (r) & = \frac{1}{4 π} \underset{V^{'}}{∭} \frac{\nabla^{'} \cdot F (r^{'})}{| r - r^{'} |} d V^{'} - \frac{1}{4 π} \iint_{S^{'}} \frac{e_{n} \cdot F (r^{'})}{| r - r^{'} |} d S^{'} \\ A (r) & = \frac{1}{4 π} \underset{V^{'}}{∭} \frac{\nabla^{'} \times F (r^{'})}{| r - r^{'} |} d V^{'} - \frac{1}{4 π} \iint_{S^{'}} \frac{e_{n} \times F (r^{'})}{| r - r^{'} |} d S^{'} \end{aligned}

Helmholtz 定理的解释
- $\bm F$ 单值且导数连续有界，则其由散度、旋度和边界条件唯一确定.
- $\bm F_t = -\grad u$ 由散度和在边界上的法向量唯一确定.
- $\bm F_c = \curl \bm A$ 由旋度和在边界上的切线分量唯一确定.
- ${\begin{cases} \nabla \cdot F_{t} = \nabla \cdot F, \\ \nabla \times F_{t} = 0. \end{cases} {\begin{cases} \nabla \cdot F_{c} = 0, \\ \nabla \times F_{c} = \nabla \times F . \end{cases}$
- $\vqty{\bm F} = \omicron\pqty{\vqty{\bm r - \bm r'}\inv}\ (\vqty{\bm r} \to \infty)$ ，则上述面积分为零.