数学
代数公式
复数共轭
$$ | z |^2 = z \cdot \bar{z} $$
$$ \overline{ z_1 + z_2 } = \overline{z_1} + \overline{z_2} $$
$$ \overline{ z_1 - z_2 } = \overline{z_1} - \overline{z_2} $$
$$ \overline{ z_1 \cdot z_2 } = \overline{z_1} \cdot \overline{z_2} $$
$$ \overline{\left( \frac{z_1}{z_2} \right)} = \frac{\overline{z_1}}{\overline{z_2}} $$
三角函数
欧拉公式
$$e^{i\theta} = \cos \theta + i \sin \theta$$
$$\sin \theta = \frac{e^{i \theta} - e^{- i \theta}}{2i}$$
$$\cos \theta = \frac{e^{i \theta} + e^{- i \theta}}{2}$$
诱导公式
基本关系：$\sin^2 a + \cos^2 a = 1$, $\tan a = \frac{\sin a}{\cos a}$, $\cot a = \frac{\cos a}{\sin a}$, $\sec a = \frac{1}{\cos a}$, $\csc a = \frac{1}{\sin a}$.
扩展关系：$\sec^2 a - \tan^2 a = 1$, $\csc^2 a - \cot^2 a = 1$.
| $\sin$ | $\cos$ | $\tan$ | $\cot$ | $\sec$ | $\csc$ | |
|---|---|---|---|---|---|---|
| $- a$ | $- \sin a$ | $\cos a$ | $- \tan a$ | $ - \cot a$ | $\sec a$ | $- \csc a$ |
| $\frac{\pi}{2} \pm a$ | $\cos a$ | $\mp \sin a$ | $\mp \cot a$ | $\mp \tan a$ | $\mp \csc a$ | $\sec a$ |
| $\pi \pm a$ | $\mp \sin a$ | $- \cos a$ | $\pm \tan a$ | $\pm \cot a$ | $- \sec a$ | $\mp \csc a$ |
| $\frac{3\pi}{2} \pm a$ | $- \cos a$ | $\pm \sin a$ | $\mp \cot a$ | $\mp \tan a$ | $\pm \csc a$ | $- \sec a$ |
| $2\pi \pm a$ | $\pm \sin a$ | $\cos a$ | $\pm \tan a$ | $\pm \cot a$ | $\sec a$ | $\pm \csc a$ |
两角和差公式
$$\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b$$
$$\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b$$
$$\tan(a \pm b) = \frac{\tan a \pm \tan b}{1 \mp \tan a \tan b}$$
$$\cot (a \pm b) = \frac{\cot a \cot b \mp 1}{\cot b \pm \cot a}$$
半角公式
$$\sin^2 \frac{a}{2} = \frac{1 - \cos a}{2}$$
$$\cos^2 \frac{a}{2} = \frac{1 + \cos a}{2}$$
$$\tan^2 \frac{a}{2} = \frac{1 - \cos a}{1 + \cos a}$$
万能公式
$$\sin a = \frac{2 \tan \frac{a}{2}}{1 + \tan^2 \frac{a}{2}}$$
$$\cos a = \frac{1 - \tan^2 \frac{a}{2}}{1 + \tan^2 \frac{a}{2}}$$
$$\tan a = \frac{2 \tan \frac{a}{2}}{1 - \tan^2 \frac{a}{2}}$$
三倍角公式
$$\sin 3a = 3 \sin a - 4 \sin^3 a$$
$$\cos 3 a = 4 \cos^3 a - 3 \cos a$$
$$\tan 3a = \frac{3 \tan a - \tan^3 a}{1 - 3 \tan^2 a} = \tan a \tan (\frac{\pi}{3} + a) \tan (\frac{\pi}{3} - a)$$
$$\cot 3a = \frac{\cot^3 a - 3 \cot a}{3 \cot^2 a - 1}$$
降冪公式
| $\sin^2 a = \frac{1}{2} (1 - \cos 2a)$ | $\sin^3 a = \frac{1}{4} (3 \sin a - \sin 3a)$ | $\sin^4 a = \frac{1}{8} (3 - 4 \cos 2a + \cos 4a)$ |
|---|---|---|
| $\cos^2 a = \frac{1}{2} (1 + \cos 2a)$ | $\cos^3 a = \frac{1}{4} (3 \cos a + \cos 3a)$ | $\cos^4 a = \frac{1}{8} (3 + 4 \cos 2a + \cos 4a)$ |
和差化积
| $\sin a + \sin b = 2 \sin \frac{a + b}{2} \cos \frac{a - b}{2}$ | $\sin a - \sin b = 2 \sin \frac{a - b}{2} \cos \frac{a + b}{2}$ | $\sin^2 a - \sin^2 b = \sin (a + b) \sin (a - b)$ |
|---|---|---|
| $\cos a + \cos b = 2 \cos \frac{a + b}{2} \cos \frac{a - b}{2}$ | $\cos a - \cos b = - 2 \sin \frac{a + b}{2} \sin \frac{a - b}{2}$ | $\cos^2 a - \cos^2 b = - \sin (a + b) \sin (a - b)$ |
| $\tan a + \tan b = \frac{\sin (a + b)}{\cos a \cos b}$ | $\tan a - \tan b = \frac{\sin (a - b)}{\cos a \cos b}$ | $\cos^2 a - \sin^2 b = \cos (a + b) \cos (a - b)$ |
| $\cot a + \cot b = \frac{\sin (a + b)}{\sin a \sin b}$ | $\cot a - \cot b = - \frac{\sin (a - b)}{\sin a \sin b}$ | $\sin^2 - \cos^2 b = - \cos (a + b) \cos (a - b)$ |
积化和差
$$\sin a \sin b = - \frac{1}{2} [ \cos (a + b) - \cos (a - b) ]$$
$$\sin a \cos b = \frac{1}{2} [ \sin (a + b) + \sin (a - b) ]$$
$$\cos a \cos b = \frac{1}{2} [ \cos (a + b) + \cos (a - b) ]$$
$$\cos a \sin b = \frac{1}{2} [ \sin (a + b) - \sin (a - b) ]$$
微积分
基本法则
- 常数：$ (c)’ = 0 $；$ (c u)’ = c u’ $
- 四则运算：$ (u \pm v)’ = u’ \pm v’ $；$ (u v)’ = u’ v + u v’ $；$ \left( \frac{u}{v} \right)’ = \frac{u’ v - u v’}{v^2} $
- 复合函数：$ f’_x \left( u(x) \right) = f’_u (u) u’_x $
- 反函数：$ x’_y = \frac{1}{y’_x} $
- 隐函数：若 $ F(x, y) = 0 $，则 $ y’_x = - \frac{F’_x}{F’_y} $
- 参数函数：若 $ x = \varphi(t), y = \psi(t) $，则 $ y = \psi \left( \varphi^{ - 1} (x) \right) $ 和 $ y’_x = \frac{y’_t}{x’_t} $
导数公式
| $f(x)$ | $f’(x)$ | $f(x)$ | $f’(x)$ | $f(x)$ | $f’(x)$ |
|---|---|---|---|---|---|
| $x^n$ | $n x^{n -1}$ | $a^x$ | $a^x \ln a$ | $e^x$ | $e^x$ |
| $x^x$ | $x^x (1 + \ln x)$ | $\log_a x$ | $\frac{1}{x \ln a}$ | $\ln x$ | $\frac{1}{x}$ |
| $\sin x$ | $\cos x$ | $\arcsin x$ | $\frac{1}{\sqrt{1 - x^2}}$ | $\sinh x$ | $\cosh x$ |
| $\cos x$ | $- \sin x$ | $\arccos x$ | $- \frac{1}{\sqrt{1 - x^2}}$ | $\cosh x$ | $\sinh x$ |
| $\tan x$ | $\frac{1}{\cos^2 x}$ | $\arctan x$ | $\frac{1}{1 + x^2}$ | $\tanh x$ | $\frac{1}{\cosh^2 x}$ |
| $\cot x$ | $- \frac{1}{\sin^2 x}$ | $\text{arccot}~x$ | $- \frac{1}{1 + x^2}$ | $\coth x$ | $- \frac{1}{\sinh^2 x}$ |
| $\sec x$ | $\frac{\sin x}{\cos^2 x}$ | $\text{arcsec}~x$ | $\frac{1}{x \sqrt{x^2 - 1}}$ | $\text{sech}~x$ | $- \frac{\sinh x}{\cosh^2 x}$ |
| $\csc x$ | $- \frac{\cos x}{\sin^2 x}$ | $\text{arccsc}~x$ | $- \frac{1}{x \sqrt{x^2 - 1}}$ | $\text{csch}~x$ | $- \frac{\cosh x}{\sinh^2 x}$ |
积分号的微分
$$ \frac{\text{d}}{\text{d}y} \int_{\phi(y)}^{\psi(y)} f(x, y) \text{d}x = f[\psi(y), y]\psi’(y) - f[\phi(y), y]\phi’(y) + \int_{\phi(y)}^{\psi(y)} f_y’(x, y) \text{d}x $$
$$ \text{特别地，}\quad \frac{\text{d}}{\text{d}y} \int_a^y f(x) \text{d}x = f(y) \quad\text{和}\quad \frac{\text{d}}{\text{d}y} \int_a^{\infty} f(x, y) \text{d}x = \int_a^{\infty} f_y’(x, y) \text{d}x $$
幂级数展开
$$\frac{1}{1 - x} = 1 + x + x^2 + \cdots + x^n + \cdots, |x| < 1$$
$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots + (- 1)^n \frac{x^{2n + 1}}{(2n + 1)!} + \cdots, |x| < \infty$$
$$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots + (- 1)^n \frac{x^{2n}}{(2n)!} + \cdots, |x| < \infty$$
$$e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!} + \cdots, |x| < \infty$$
$$\ln (1 + x) = x - \frac{x^2}{3} + \frac{x^3}{3} - \cdots + (- 1)^{n + 1} \frac{x^n}{n} + \cdots, - 1 < x \leq 1$$
泰勒展开
$$f(x) = f(x_0) + f’(x_0) (x - x_0) + \frac{f’’(x_0)}{2!} (x - x_0)^2 + \cdots + \frac{f^{(n)}(x_0)}{n!} (x - x_0)^n + R_n(x)$$
积分变换
傅里叶变换
定义
$$F(\omega) = \int_{-\infty}^{+\infty} f(t) e^{-j \omega t} dt$$
$$f(t) = \frac{1}{2\pi} \int_{-\infty}^{+\infty} F(\omega) e^{j \omega t} d\omega$$
性质
| 性质 | 像原函数 | 像函数 |
|---|---|---|
| 线性 | $a f_1(t) + b f_2(t)$ | $a F_1(\omega) + b F_2(\omega)$ |
| 位移 | $f(t-t_0)$ $e^{j \omega_0 t} f(t)$ |
$e^{-j \omega t_0} F(\omega)$ $F(\omega - \omega_0)$ |
| 相似 | $f(a t)$ | $\frac{1}{\vert a \vert} F(\frac{\omega}{a})$ |
| 对称 | $F(\pm t)$ | $2\pi f(\mp \omega)$ |
| 微分 | $f^{(n)} (t)$ $(-j t)^n f(t)$ |
$(j \omega)^n F(\omega)$ $F^{(n)}(\omega)$ |
| 积分 | $\int_{-\infty}^t \int_{-\infty}^t \cdots \int_{-\infty}^t f(t) dt \cdots dt dt$ | $\frac{1}{(j \omega)^n} F(\omega)$ |
| 卷积 | $(f_1 \ast f_2) (t) = \int_{-\infty}^{+\infty} f_1(\tau) f_2(t - \tau) d\tau$ $f_1 (t) \cdot f_2 (t)$ |
$F_1(\omega) F_2(\omega)$ $\frac{1}{2\pi} (F_1 \ast F_2) (\omega)$ |
| 逆时 | $f(-t)$ | $\overline{F(\omega)}$ |
| 互相关 | $(f_1 \circ f_2)(t) = \int_{-\infty}^{+\infty} \overline{f_1(\tau)} f_2(t + \tau) d\tau$ | $\overline{F_1(\omega)} F_2(\omega)$ |
能量积分
$$\int_{-\infty}^{+\infty} \overline{f_1(t)} f_2(t) dt = \frac{1}{2\pi} \int_{-\infty}^{+\infty} \overline{F_1 (\omega)} F_2 (\omega) d\omega$$
$$\int_{-\infty}^{+\infty} \vert f (t) \vert ^2 dt = \frac{1}{2\pi} \int_{-\infty}^{+\infty} \vert F(\omega) \vert ^2 d\omega$$
常见变换对
| 像原函数 | 像函数 |
|---|---|
| 矩形单脉冲 $f(t) = \begin{cases} E, & \vert t \vert \leq \frac{\tau}{2} \newline 0, & \text{other} \newline \end{cases}$ | $2E \frac{\sin \frac{\omega \tau}{2}}{\omega}$ |
| 指数衰减函数 $f(t) = \begin{cases} 0, & t < 0 \newline e^{-\beta t}, & t \geq 0 \end{cases}, (\beta > 0)$ | $\frac{1}{\beta + j \omega}$ |
| 单位函数 $f(t) = u(t)$ | $\frac{1}{j \omega} + \pi \delta(\omega)$ |
| 三角形脉冲 $f(t) = \begin{cases} \frac{2A}{\tau} (\frac{\tau}{2} + t), & -\frac{\tau}{2} \leq t < 0 \newline \frac{2A}{\tau} (\frac{\tau}{2} - t), & 0 \leq t < \frac{\tau}{2} \end{cases}$ | $\frac{4A}{\tau \omega^2} (1 - \cos \frac{\omega \tau}{2})$ |
| 钟形脉冲 $f(t) = A e^{-\beta t^2}, (\beta > 0)$ | $\sqrt(\frac{\pi}{\beta}) A e^{- \frac{\omega^2}{4 \beta}}$ |
| 傅里叶核 $f(t) = \frac{\sin \omega_0 t}{\pi t}$ | $F(\omega) = \begin{cases} 1, & \vert \omega \vert \leq \omega_0 \newline 0, & \text{other} \end{cases}$ |
| 高斯分布函数 $f(t) = \frac{1}{\sqrt{2\pi \sigma}} e^{-\frac{t^2}{2\sigma^2}}$ | $e^{- \frac{\sigma^2 \omega^2}{2}}$ |
| 单位脉冲函数 $f(t) = \delta(t)$ | $1$ |
| 周期性脉冲函数 $f(t) = \sum_{n=-\infty}^{+\infty} \delta(t - n T)$ | $\frac{2\pi}{T} \sum_{n=-\infty}^{+\infty} \delta(\omega - \frac{2n\pi}{T})$ |
| 符号函数 $f(t) = \mathrm{sgn~} t = 2 u(t) - 1$ | $\frac{2}{j \omega}$ |
| $f(t) = \cos \omega_0 t$ | $\pi [ \delta(\omega + \omega_0) + \delta(\omega - \omega_0) ]$ |
| $f(t) = \sin \omega_0 t$ | $j \pi [ \delta(\omega + \omega_0) - \delta(\omega - \omega_0) ]$ |
| $u(t - c)$ | $\frac{1}{j \omega} e^{-j \omega c} + \pi \delta(\omega)$ |
| $u(t) \cdot t^n$ | $\frac{n!}{(j \omega)^{n + 1}} + \pi j^n \delta^{(n)} (\omega)$ |
| $u(t) \sin at$ | $\frac{a}{a^2 - \omega^2} + \frac{\pi}{2 j} [ \delta(\omega - a) - \delta(\omega + a) ]$ |
| $u(t) \cos at$ | $\frac{j \omega}{a^2 - \omega^2} + \frac{\pi}{2} [ \delta(\omega - a) + \delta(\omega + a) ]$ |
| $u(t) e^{j a t}$ | $\frac{1}{j (\omega - a)} + \pi \delta(\omega - a)$ |
| $u(t - c) e^{j a t}$ | $\frac{1}{j(\omega - a)} e^{-j (\omega - a) c} + \pi \delta(\omega - a)$ |
| $u(t) e^{j a t} t^n$ | $\frac{n!}{[j (\omega - a)]^{n + 1}} + \pi j^n \delta^{(n)}(\omega - a)$ |
| $e^{a \vert t \vert}, \Re (a) < 0$ | $ - \frac{2a}{\omega^2 + a^2}$ |
| $\delta(t - c)$ | $e^{-j \omega c}$ |
| $\delta^{(n)}(t)$ | $(j \omega)^n$ |
| $\delta^{(n)}(t - c)$ | $(j \omega)^n e^{- j \omega c}$ |
| $1$ | $2 \pi \delta(\omega)$ |
| $t^n$ | $2 \pi j^n \delta^{(n)} (\omega)$ |
| $e^{j a t}$ | $2 \pi \delta(\omega - a)$ |
| $t^n e^{j a t}$ | $2 \pi j^n \delta^{(n)} (\omega - a)$ |
| $\frac{1}{a^2 + t^2}, \Re (a) < 0$ | $- \frac{\pi}{a} e^{a \vert \omega \vert}$ |
| $\frac{t}{(a^2 + t^2)^2}, \Re (a) < 0$ | $\frac{j \omega \pi}{2a} e^{a \vert \omega \vert}$ |
| $\frac{\sin bt}{a^2 + t^2}, \Re (a) < 0, b \in \mathscr{R}$ | $ - \frac{\pi}{a} e^{a \vert \omega - b \vert}$ |
| $\frac{\cos bt}{a^2 + t^2}, \Re (a) < 0, b \in \mathscr{R}$ | $ - \frac{\pi}{2a} [ e^{a \vert \omega - b \vert} + e^{a \vert \omega + b \vert} ]$ |
| $\frac{e^{j b t}}{a^2 + t^2}, \Re (a) < 0, b \in \mathscr{R}$ | $ - \frac{\pi}{2 a j} [ e^{a \vert \omega - b \vert} - e^{a \vert \omega + b \vert} ]$ |
| $\frac{\sinh at}{\sinh \pi t}, - \pi < a < \pi$ | $\frac{\sin a}{\cosh \omega + \cos a}$ |
| $\frac{\sinh at}{\cosh \pi t}, - \pi < a < \pi$ | $-2j \frac{\sin \frac{a}{2} \sinh \frac{\omega}{2}}{\cosh \omega + \cos a}$ |
| $\frac{\cosh at}{\cosh \pi t}, - \pi < a < \pi$ | $2 \frac{\cos \frac{a}{2} \cosh \frac{\omega}{2}}{\cosh \omega + \cos a}$ |
| $\frac{1}{\cosh at}$ | $\frac{\pi}{a} \frac{1}{\cosh \frac{\pi \omega}{2a}}$ |
| $\sin at^2$ | $\sqrt{\frac{\pi}{a}} \cos \Big( \frac{\omega^2}{4a} + \frac{\pi}{4} \Big)$ |
| $\cos at^2$ | $\sqrt{\frac{\pi}{a}} \cos \Big( \frac{\omega^2}{4a} - \frac{\pi}{4} \Big)$ |
| $\frac{1}{t} \sin at$ | $\begin{cases} \pi, & \vert \omega \vert \leq a \newline 0, & \vert \omega \vert > a \end{cases}$ |
| $\frac{1}{t^2} \sin^2 at$ | $\begin{cases} \pi \Big( a - \frac{\vert \omega \vert}{2} \Big), & \vert \omega \vert \leq 2a \newline 0, & \vert \omega \vert > 2a \end{cases}$ |
| $\frac{\sin at}{\sqrt{\vert t \vert}}$ | $j \sqrt{\frac{\pi}{2}} \Big( \frac{1}{\sqrt{\vert \omega + a \vert}} - \frac{1}{\sqrt{\vert \omega - a \vert}} \Big)$ |
| $\frac{\cos at}{\sqrt{\vert t \vert}}$ | $\sqrt{\frac{\pi}{2}} \Big( \frac{1}{\sqrt{\vert \omega + a \vert}} + \frac{1}{\sqrt{\vert \omega - a \vert}} \Big)$ |
| $\frac{1}{\sqrt{\vert t \vert}}$ | $\sqrt{\frac{2\pi}{\vert \omega \vert}}$ |
| $e^{- at^2}, \Re(a) > 0$ | $\sqrt{\frac{\pi}{a}} e^{- \frac{\omega^2}{4a}}$ |
| $\vert t \vert$ | $- \frac{2}{\omega^2}$ |
| $\frac{1}{\vert t \vert}$ | $\frac{\sqrt{2\pi}}{\vert \omega \vert}$ |
坐标系
单位矢量
柱坐标系
对于柱坐标系 $(\rho, \varphi, z)$，有
$$\begin{cases}
x = \rho \cos \varphi \newline
y = \rho \sin \varphi \newline
z = z
\end{cases} \qquad (0 \leqslant \rho < \infty, 0 \leqslant \varphi < 2 \pi, - \infty < z < \infty)$$
单位矢量为
$$\begin{cases}
\vec{e}_{\rho} = \cos \varphi \vec{i} + \sin \varphi \vec{j} \newline
\vec{e}_{\varphi} = - \sin \varphi \vec{i} + \cos \varphi \vec{j} \newline
\vec{e}_z = \vec{k} \newline
\end{cases}$$
其偏导数为
$$\frac{\partial \vec{e}_{\rho}}{\partial \rho} = \frac{\partial \vec{e}_{\varphi}}{\partial \rho} = \frac{\partial \vec{e}_z}{\partial \rho} = 0$$
$$\frac{\partial \vec{e}_{\rho}}{\partial \varphi} = \vec{e}_{\varphi}, \frac{\partial \vec{e}_{\varphi}}{\partial \varphi} = - \vec{e}_{\rho}, \frac{\vec{e}_z}{\partial \varphi} = 0$$
$$\frac{\partial \vec{e}_{\rho}}{\partial z} = \frac{\partial \vec{e}_{\varphi}}{\partial z} = \frac{\partial \vec{e}_z}{\partial z} = 0$$
球坐标系
对于球坐标系 $(r, \theta, \varphi)$，有
$$\begin{cases}
x = r \sin \theta \cos \varphi \newline
y = r \sin \theta \sin \varphi \newline
z = r \cos \theta \newline
\end{cases} \qquad (0 \leqslant r < \infty, 0 \leqslant \varphi < 2 \pi, 0 \leqslant \theta \leqslant \pi)$$
单位矢量为
$$\begin{cases}
\vec{e}_r = \sin \theta \cos \varphi \vec{i} + \sin \theta \sin \varphi \vec{j} + \cos \theta \vec{k} \newline
\vec{e}_{\theta} = \cos \theta \cos \varphi \vec{i} + \cos \theta \sin \varphi \vec{j} - \sin \theta \vec{k} \newline
\vec{e}_{\varphi} = - \sin \varphi \vec{i} + \cos \varphi \vec{j} \newline
\end{cases}$$
其偏导数为
$$\frac{\partial \vec{e}_r}{\partial r} = \frac{\partial \vec{e}_{\theta}}{\partial r} = \frac{\partial \vec{e}_{\varphi}}{\partial r} = 0$$
$$\frac{\partial \vec{e}_r}{\partial \theta} = \vec{e}_{\theta}, \frac{\partial \vec{e}_{\theta}}{\partial \theta} = - \vec{e}_r, \frac{\partial \vec{e}_{\varphi}}{\partial \theta} = 0$$
$$\frac{\partial \vec{e}_r}{\partial \varphi} = \sin \theta \vec{e}_{\varphi}, \frac{\partial \vec{e}_{\theta}}{\partial \varphi} = \cos \theta \vec{e}_{\varphi}, \frac{\partial \vec{e}_{\varphi}}{\partial \varphi} = - \sin \theta \vec{e}_r - \cos \theta \vec{e}_{\theta}$$
坐标系转换
矢量在直角坐标系 $(v_x, v_y, v_z)$ 和任意坐标系 $(v_{\xi}, v_{\eta}, v_{\zeta})$ 中可以分别表达为
$$\vec{v} = v_x \vec{i} + v_y \vec{j} + v_z \vec{k} = v_{\xi} \vec{e}_{\xi} + v_{\eta} \vec{e}_{\eta} + v_{\zeta} \vec{e}_{\zeta}$$
柱坐标系与直角坐标系
由柱坐标系 $(v_{\rho}, v_{\varphi}, v_z)$ 到直角坐标系 $(v_x, v_y, v_z)$ ：
$$\begin{cases}
v_x = v_{\rho} \cos \varphi - v_{\varphi} \sin \varphi \newline
v_y = v_{\rho} \sin \varphi + v_{\varphi} \cos \varphi \newline
v_z = v_z \newline
\end{cases}$$
由直角坐标系 $(v_x, v_y, v_z)$ 到柱坐标系 $(v_{\rho}, v_{\varphi}, v_z)$ ：
$$\begin{cases}
v_{\rho} = v_x \cos \varphi +v_y \sin \varphi \newline
v_{\varphi} = - v_x \sin \varphi + v_y \cos \varphi \newline
v_z = v_z \newline
\end{cases}$$
球坐标系与直角坐标系
由球坐标系 $(v_r, v_{\theta}, v_{\varphi})$ 到直角坐标系 $(v_x, v_y, v_z)$ ：
$$\begin{cases}
v_x = v_r \sin \theta \cos \varphi + v_{\theta} \cos \theta \cos \varphi - v_{\varphi} \sin \varphi \newline
v_y = v_r \sin \theta \sin \varphi + v_{\theta} \cos \theta \sin \varphi + v_{\varphi} \cos \varphi \newline
v_z = v_r \cos \theta - v_{\theta} \sin \theta \newline
\end{cases}$$
由直角坐标系 $(v_x, v_y, v_z)$ 到球坐标系 $(v_r, v_{\theta}, v_{\varphi})$ ：
$$\begin{cases}
v_r = v_x \sin \theta \cos \varphi + v_y \sin \theta \sin \varphi + v_z \cos \theta \newline
v_{\theta} = v_x \cos \theta \cos \varphi + v_y \cos \theta \sin \varphi - v_z \sin \theta \newline
v_{\varphi} = - v_x \sin \varphi + v_y \cos \varphi \newline
\end{cases}$$
各种算子
取 $U$ 为一个标函数，$\mathbf{V}$ 为一个矢函数。
直角坐标系
哈密顿算子：$\nabla = \vec{i} \frac{\partial}{\partial x} + \vec{j} \frac{\partial}{\partial y} + \vec{k} \frac{\partial}{\partial z}$
梯度：$\text{grad } U = \nabla U = \vec{i} \frac{\partial U}{\partial x} + \vec{j} \frac{\partial U}{\partial y} + \vec{k} \frac{\partial U}{\partial z}$
散度：$\text{div } \mathbf{V} = \nabla \cdot \mathbf{V} = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z}$
旋度：$\text{rot } \mathbf{V} = \nabla \times \mathbf{V} = \left( \frac{\partial v_z}{\partial y} - \frac{\partial v_y}{\partial z} \right) \vec{i} + \left( \frac{\partial v_x}{\partial z} - \frac{\partial v_z}{\partial x} \right) \vec{j} + \left( \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y} \right) \vec{k} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \newline \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \newline v_x & v_y & v_z \newline \end{vmatrix}$
拉普拉斯算子：$\Delta U = \text{div } \text{grad } U = \frac{\partial^2 U}{\partial x^2} + \frac{\partial^2 U}{\partial y^2} + \frac{\partial^2 U}{\partial z^2}$ 和 $\Delta \mathbf{V} = \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right) ( v_x \vec{i} + v_y \vec{j} + v_z \vec{k} )$
算子混合运算：
$$\text{div } \text{rot } \mathbf{V} = \nabla \cdot (\nabla \times \mathbf{V}) = 0$$
$$\text{rot } \text{grad } U = \nabla \times (\nabla U) = 0$$
$$\text{div } \text{grad } (\alpha_1 U_1 + \alpha_2 U_2) = \alpha_1 \text{div } \text{grad } U_1 + \alpha_2 \text{div } \text{grad } U_2$$
$$\text{div } \text{grad } (U_1 U_2) = U_1 \text{div } \text{grad } U_2 + U_2 \text{div } \text{gard } U_1 + 2 \text{grad } U_1 \cdot \text{grad } U_2$$
$$\text{grad } \text{div } \mathbf{V} - \text{rot } \text{rot } \mathbf{V} = \nabla (\nabla \cdot \mathbf{V}) - \nabla \times (\nabla \times \mathbf{V}) = \Delta \mathbf{V}$$
柱坐标系
哈密顿算子：$\widetilde{\nabla} = \vec{e}_{\rho} \frac{\partial}{\partial \rho} + \vec{e}_{\varphi} \frac{1}{\rho} \frac{\partial}{\partial \varphi} + \vec{e}_z \frac{\partial}{\partial z}$
梯度：$\text{grad } U = \widetilde{\nabla} U = \vec{e}_{\rho} \frac{\partial U}{\partial \rho} + \vec{e}_{\varphi} \frac{1}{\rho} \frac{\partial U}{\partial \varphi} + \vec{e}_z \frac{\partial U}{\partial z}$
散度：$\text{div } \mathbf{V} = \widetilde{\nabla} \cdot \mathbf{V} = \frac{1}{\rho} \frac{\partial (\rho v_{\rho})}{\partial \rho} + \frac{1}{\rho} \frac{\partial v_{\varphi}}{\partial \varphi} + \frac{\partial v_z}{\partial z}$
旋度：$\text{rot } \mathbf{V} = \widetilde{\nabla} \times \mathbf{V} = \left( \frac{1}{\rho} \frac{\partial v_z}{\partial \varphi} - \frac{\partial v_{\varphi}}{\partial z} \right) \vec{e}_{\rho} + \left( \frac{\partial v_{\rho}}{\partial z} - \frac{\partial v_z}{\partial \rho} \right) \vec{e}_{\varphi} + \left( \frac{1}{\rho} \frac{\partial (\rho v_{\varphi})}{\partial \rho} - \frac{1}{\rho} \frac{\partial v_{\rho}}{\partial \varphi} \right) \vec{e}_z$
拉普拉斯算子：$\Delta U = \text{div } \text{grad } U = \frac{1}{\rho} \frac{\partial}{\partial \rho} \left( \rho \frac{\partial U}{\partial \rho} \right) + \frac{1}{\rho^2} \frac{\partial^2 U}{\partial \varphi^2} + \frac{\partial^2 U}{\partial z^2}$
球坐标系
哈密顿算子：$\widetilde{\widetilde{\nabla}} = \vec{e}_r \frac{\partial}{\partial r} + \vec{e}_{\theta} \frac{1}{r} \frac{\partial}{\partial \theta} + \vec{e}_{\varphi} \frac{1}{r \sin \theta} \frac{\partial}{\partial \varphi}$
梯度：$\text{grad } U = \widetilde{\widetilde{\nabla}} U = \vec{e}_r \frac{\partial U}{\partial r} + \vec{e}_{\theta} \frac{1}{r} \frac{\partial U}{\partial \theta} + \vec{e}_{\varphi} \frac{1}{r \sin \theta} \frac{\partial U}{\partial \varphi}$
散度：$\text{div } U = \widetilde{\widetilde{\nabla}} \cdot \mathbf{V} = \frac{1}{r^2} \left[ \frac{\partial}{\partial r} (r^2 v_r) \right] + \frac{1}{r \sin \theta} \left[ \frac{\partial}{\partial \theta} (\sin \theta v_{\theta}) \right] + \frac{1}{r \sin \theta} \frac{\partial v_{\varphi}}{\partial \varphi}$
旋度：$\text{rot } \mathbf{V} = \widetilde{\widetilde{\nabla}} \times \mathbf{V} = \left[ \frac{1}{r \sin \theta} \left( \frac{\partial}{\partial \theta} (\sin \theta v_{\varphi}) - \frac{\partial v_{\theta}}{\partial \varphi} \right) \right] \vec{e}_r + \left[ \frac{1}{r \sin \theta} \frac{\partial v_r}{\partial \varphi} - \frac{1}{r} \frac{\partial}{\partial r} (r v_{\varphi}) \right] \vec{e}_{\theta} + \left[ \frac{1}{r} \frac{\partial}{\partial r} (r v_{\theta}) - \frac{1}{r} \frac{\partial v_r}{\partial \theta} \right] \vec{e}_{\varphi}$
拉普拉斯算子：$\Delta U = \text{div } \text{grad } U = \frac{1}{r^2} \left[ \frac{\partial}{\partial r} \left( r^2 \frac{\partial U}{\partial r} \right) \right] + \frac{1}{r \sin \theta} \left[ \frac{\partial}{\partial \theta} \left( \sin \theta \frac{1}{r} \frac{\partial U}{\partial \theta} \right) \right] + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 U}{\partial \varphi^2}$
物理
地震学
弹性参数转换
纵波波速 $\alpha$ 和横波波速 $\beta$：
$$\alpha = \sqrt{ \frac{ \lambda + 2\mu }{ \rho} }$$
$$\beta = \sqrt{ \frac{\mu}{\rho} }$$
泊松比 $\nu$：
$$\nu = \frac{\lambda}{ 2(\mu + \lambda) }$$
弹性常数 $c_{jklm}$，体积模量 $\kappa$ 和弹性模量 $E$：
$$c_{jklm} = \lambda \delta_{jk}\delta_{lm} + \mu (\delta_{jl}\delta_{km} + \delta_{jm}\delta_{kl})$$
$$\kappa = \lambda + \frac{2}{3} \mu$$
$$E = \frac{\mu (3\lambda + 2\mu)}{\lambda + \mu}$$
其中，$\rho$ 为密度，$\lambda$ 和 $\mu$ 为拉梅参数。
波动方程
声波
在各向同性介质中，声波波动方程有以下几种形式：
- 二阶标量方程：$$ \nabla^2 p - \frac{1}{c^2} \frac{\partial^2 p}{\partial t^2} = f \quad \text{或} \quad \rho \frac{\partial^2 p}{\partial t^2} - \nabla \cdot \left( c^2 \nabla p \right) = f $$
- 一阶压力-速度方程组：$$ \begin{cases} p_{, t} = - \rho c^2 v_{i, i} \newline v_{i, t} = - \frac{1}{\rho} p_{, i} + f_i \newline \end{cases} $$
其中，$p$ 为压力场，$v$ 为速度场，$f$ 为力源项，$\rho$ 为介质密度，$c$ 为介质的声波波速。
弹性波
在各向同性介质中，弹性波波动方程有以下几种形式：
- 二阶矢量位移方程：$$ \rho \frac{\partial^2 \mathbf{u}(\mathbf{x}, t)}{\partial t^2} = (\lambda + 2 \mu) \nabla \nabla \cdot \mathbf{u} (\mathbf{x}, t) - \mu \nabla \times \nabla \times \mathbf{u} (\mathbf{x}, t) + \mathbf{f}(\mathbf{x}_s, t) $$
- 二阶位移方程组：$$ \rho u_{i,tt} = (\lambda u_{k,k})_{,i} + (\mu u_{i,j})_{,j} + (\mu u_{j,i})_{,j} + f_i $$
- 位移-应力方程组：$$ \begin{cases} \rho u_{i,tt} = \sigma_{ij,j} + f_i \newline \sigma_{ij} = \lambda u_{k,k} \delta_{ij} + \mu (u_{i,j} + u_{j,i}) \end{cases} $$
- 一阶速度-应力方程组：$$ \begin{cases} \rho v_{i,t} = \sigma_{ij,j} + f_i \newline \sigma_{ij,t} = \lambda v_{k,k} \delta_{ij} + \mu (v_{i,j} + v_{j,i}) \end{cases} $$
其中，$u$ 为位移场，$v$ 为速度场，$\sigma$ 为应力场，$f$ 为力源项，$\rho$ 为介质密度，$\lambda$ 和 $\mu$ 为介质的拉梅参数。
波的传播
波数 $k$ 和波速 $c$：
$$ k = \frac{f}{c} \qquad \text{或} \qquad k = \frac{\omega}{c} $$
其中，$f$ 为线频率，$\omega$ 为角频率。
坐标转换
地震学中常常涉及多分量记录在两组常用坐标系下的转换，即 RTZ 到 NEZ，和 NEZ 到 RTZ 的转换。这里，对这两组转换公式做以总结。$ \phi $ 为方位角，即正北方向按顺时针旋转到从震源到台站连线上时所转过的角度。
从 RTZ 到 NEZ
转换公式为：
$$ \begin{bmatrix} U_{NN} & U_{NE} & U_{NZ} \newline U_{EN} & U_{EE} & U_{EZ} \newline U_{ZN} & U_{ZE} & U_{ZZ} \newline \end{bmatrix} =
\begin{bmatrix} 0 & \cos\phi & - \sin\phi \newline 0 & \sin\phi & \cos\phi \newline 1 & 0 & 0 \newline \end{bmatrix}
\begin{bmatrix} U_{ZZ} & U_{ZR} & U_{ZT} \newline U_{RZ} & U_{RR} & U_{RT} \newline U_{TZ} & U_{TR} & U_{TT} \newline \end{bmatrix}
\begin{bmatrix} 0 & 0 & 1 \newline \cos\phi & \sin\phi & 0 \newline - \sin\phi & \cos\phi & 0 \newline \end{bmatrix} $$
则
$$ U_{NN} = \begin{bmatrix} 0 & \cos\phi & - \sin\phi \end{bmatrix}
\begin{bmatrix} U_{ZR} \cos\phi - U_{ZT} \sin\phi \newline U_{RR} \cos\phi - U_{RT} \sin\phi \newline U_{TR} \cos\phi - U_{TT} \sin\phi \newline \end{bmatrix}
= U_{RR} \cos^2\phi - U_{RT} \sin\phi \cos\phi - U_{TR} \cos\phi \sin\phi + U_{TT} \sin^2\phi $$
$$ U_{NE} = \begin{bmatrix} 0 & \cos\phi & - \sin\phi \end{bmatrix}
\begin{bmatrix} U_{ZR} \sin\phi + U_{ZT} \cos\phi \newline U_{RR} \sin\phi + U_{RT} \cos\phi \newline U_{TR} \sin\phi + U_{TT} \cos\phi \newline \end{bmatrix}
= U_{RR} \sin\phi \cos\phi + U_{RT} \cos^2\phi - U_{TR} \sin^2\phi - U_{TT} \cos\phi \sin\phi $$
$$ U_{NZ} = \begin{bmatrix} 0 & \cos\phi & - \sin\phi \end{bmatrix}
\begin{bmatrix} U_{ZZ} \newline U_{RZ} \newline U_{TZ} \newline \end{bmatrix}
= U_{RZ} \cos\phi - U_{TZ} \sin\phi $$
$$ U_{EN} = \begin{bmatrix} 0 & \sin\phi & \cos\phi \end{bmatrix}
\begin{bmatrix} U_{ZR} \cos\phi - U_{ZT} \sin\phi \newline U_{RR} \cos\phi - U_{RT} \sin\phi \newline U_{TR} \cos\phi - U_{TT} \sin\phi \newline \end{bmatrix}
= U_{RR} \cos\phi \sin\phi - U_{RT} \sin^2 \phi + U_{TR} \cos^2\phi - U_{TT} \sin\phi \cos\phi $$
$$ U_{EE} = \begin{bmatrix} 0 & \sin\phi & \cos\phi \end{bmatrix}
\begin{bmatrix} U_{ZR} \sin\phi + U_{ZT} \cos\phi \newline U_{RR} \sin\phi + U_{RT} \cos\phi \newline U_{TR} \sin\phi + U_{TT} \cos\phi \newline \end{bmatrix}
= U_{RR} \sin^2\phi + U_{RT} \cos\phi \sin\phi + U_{TR} \sin\phi \cos\phi + U_{TT} \cos^2\phi $$
$$ U_{EZ} = \begin{bmatrix} 0 & \sin\phi & \cos\phi \end{bmatrix}
\begin{bmatrix} U_{ZZ} \newline U_{RZ} \newline U_{TZ} \newline \end{bmatrix}
= U_{RZ} \sin\phi + U_{TZ} \cos\phi $$
$$ U_{ZN} = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}
\begin{bmatrix} U_{ZR} \cos\phi - U_{ZT} \sin\phi \newline U_{RR} \cos\phi - U_{RT} \sin\phi \newline U_{TR} \cos\phi - U_{TT} \sin\phi \newline \end{bmatrix}
= U_{ZR} \cos\phi - U_{ZT} \sin\phi $$
$$ U_{ZE} = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}
\begin{bmatrix} U_{ZR} \sin\phi + U_{ZT} \cos\phi \newline U_{RR} \sin\phi + U_{RT} \cos\phi \newline U_{TR} \sin\phi + U_{TT} \cos\phi \newline \end{bmatrix}
= U_{ZR} \sin\phi + U_{ZT} \cos\phi $$
$$ U_{ZZ} = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}
\begin{bmatrix} U_{ZZ} \newline U_{RZ} \newline U_{TZ} \newline \end{bmatrix}
= U_{ZZ} $$
从 NEZ 到 RTZ
转换公式为：
$$ \begin{bmatrix} U_{ZZ} & U_{ZR} & U_{ZT} \newline U_{RZ} & U_{RR} & U_{RT} \newline U_{TZ} & U_{TR} & U_{TT} \newline \end{bmatrix} =
\begin{bmatrix} 0 & 0 & 1 \newline \cos\phi & \sin\phi & 0 \newline - \sin\phi & \cos\phi & 0 \newline \end{bmatrix}
\begin{bmatrix} U_{NN} & U_{NE} & U_{NZ} \newline U_{EN} & U_{EE} & U_{EZ} \newline U_{ZN} & U_{ZE} & U_{ZZ} \newline \end{bmatrix}
\begin{bmatrix} 0 & \cos\phi & - \sin\phi \newline 0 & \sin\phi & \cos\phi \newline 1 & 0 & 0 \newline \end{bmatrix} $$
则
$$ U_{ZZ} = \begin{bmatrix} 0 & 0 & 1 \end{bmatrix}
\begin{bmatrix} U_{NZ} \newline U_{EZ} \newline U_{ZZ} \newline \end{bmatrix}
= U_{ZZ} $$
$$ U_{ZR} = \begin{bmatrix} 0 & 0 & 1 \end{bmatrix}
\begin{bmatrix} U_{NN} \cos\phi + U_{NE} \sin\phi \newline U_{EN} \cos\phi + U_{EE} \sin\phi \newline U_{ZN} \cos\phi + U_{ZE} \sin\phi \newline \end{bmatrix}
= U_{ZN} \cos\phi + U_{ZE} \sin\phi $$
$$ U_{ZT} = \begin{bmatrix} 0 & 0 & 1 \end{bmatrix}
\begin{bmatrix} - U_{NN} \sin\phi + U_{NE} \cos\phi \newline - U_{EN} \sin\phi + U_{EE} \cos\phi \newline - U_{ZN} \sin\phi + U_{ZE} \cos\phi \newline \end{bmatrix}
= - U_{ZN} \sin\phi + U_{ZE} \cos\phi $$
$$ U_{RZ} = \begin{bmatrix} \cos\phi & \sin\phi & 0 \end{bmatrix}
\begin{bmatrix} U_{NZ} \newline U_{EZ} \newline U_{ZZ} \newline \end{bmatrix}
= U_{NZ} \cos\phi + U_{EZ} \sin\phi $$
$$ U_{RR} = \begin{bmatrix} \cos\phi & \sin\phi & 0 \end{bmatrix}
\begin{bmatrix} U_{NN} \cos\phi + U_{NE} \sin\phi \newline U_{EN} \cos\phi + U_{EE} \sin\phi \newline U_{ZN} \cos\phi + U_{ZE} \sin\phi \newline \end{bmatrix}
= U_{NN} \cos^2\phi + U_{NE} \sin\phi \cos\phi + U_{EN} \cos\phi \sin\phi + U_{EE} \sin^2\phi $$
$$ U_{RT} = \begin{bmatrix} \cos\phi & \sin\phi & 0 \end{bmatrix}
\begin{bmatrix} - U_{NN} \sin\phi + U_{NE} \cos\phi \newline - U_{EN} \sin\phi + U_{EE} \cos\phi \newline - U_{ZN} \sin\phi + U_{ZE} \cos\phi \newline \end{bmatrix}
= - U_{NN} \sin\phi \cos\phi + U_{NE} \cos^2\phi - U_{EN} \sin^2\phi + U_{EE} \cos\phi \sin\phi $$
$$ U_{TZ} = \begin{bmatrix} - \sin\phi & \cos\phi & 0 \end{bmatrix}
\begin{bmatrix} U_{NZ} \newline U_{EZ} \newline U_{ZZ} \newline \end{bmatrix}
= - U_{NZ} \sin\phi + U_{EZ} \cos\phi $$
$$ U_{TR} = \begin{bmatrix} - \sin\phi & \cos\phi & 0 \end{bmatrix}
\begin{bmatrix} U_{NN} \cos\phi + U_{NE} \sin\phi \newline U_{EN} \cos\phi + U_{EE} \sin\phi \newline U_{ZN} \cos\phi + U_{ZE} \sin\phi \newline \end{bmatrix}
= - U_{NN} \cos\phi \sin\phi - U_{NE} \sin^2\phi + U_{EN} \cos^2\phi + U_{EE} \sin\phi \cos\phi $$
$$ U_{TT} = \begin{bmatrix} - \sin\phi & \cos\phi & 0 \end{bmatrix}
\begin{bmatrix} - U_{NN} \sin\phi + U_{NE} \cos\phi \newline - U_{EN} \sin\phi + U_{EE} \cos\phi \newline - U_{ZN} \sin\phi + U_{ZE} \cos\phi \newline \end{bmatrix}
= U_{NN} \sin^2\phi - U_{NE} \cos\phi \sin\phi - U_{EN} \sin\phi \cos\phi + U_{EE} \cos^2\phi $$
