公式速查表 Formula Sheets

I数学基础 · Mathematical Tools

1.1 矢量代数 Vector Algebra

名称 Name	表达式 Formula
模 Magnitude magnitude	\(\displaystyle \lvert\vec{A}\rvert=\sqrt{A_x^2+A_y^2+A_z^2}\)
单位矢量 Unit vector	\(\displaystyle \hat{A}=\frac{\vec{A}}{\lvert\vec{A}\rvert}\)
点积（代数） Dot product (algebraic)	\(\displaystyle \vec{A}\cdot\vec{B}=A_xB_x+A_yB_y+A_zB_z\)
点积（几何） Dot product (geometric)	\(\displaystyle \vec{A}\cdot\vec{B}=AB\cos\theta\)
叉积（大小） Cross product magnitude	\(\displaystyle \lvert\vec{A}\times\vec{B}\rvert=AB\sin\theta\)
叉积（行列式） Cross product (determinant)	\(\displaystyle \vec{A}\times\vec{B}=\begin{vmatrix}\hat{\imath}&\hat{\jmath}&\hat{k}\\ A_x&A_y&A_z\\ B_x&B_y&B_z\end{vmatrix}\)
BAC–CAB 恒等式 BAC–CAB identity	\(\displaystyle \vec{A}\times(\vec{B}\times\vec{C})=\vec{B}(\vec{A}\cdot\vec{C})-\vec{C}(\vec{A}\cdot\vec{B})\)
混合三重积 Scalar triple product	\(\displaystyle V=\vec{A}\cdot(\vec{B}\times\vec{C})\)

1.2 微积分速查 Calculus Quick Reference

名称 Name	表达式 Formula
导数定义 Derivative definition	\(\displaystyle \dv{f}{x}=\lim_{\Delta x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}\)
乘积法则 Product rule	\(\displaystyle (fg)'=f'g+fg'\)
商法则 Quotient rule	\(\displaystyle \left(\frac{f}{g}\right)'=\frac{f'g-fg'}{g^2}\)
链式法则 Chain rule	\(\displaystyle \dv{f}{t}=\pdv{f}{x}\dv{x}{t}\)
分部积分 Integration by parts	\(\displaystyle \int u\,dv=uv-\int v\,du\)
泰勒展开 Taylor series	\(\displaystyle f(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\cdots\)
二项近似 Binomial approx.	\(\displaystyle (1+x)^n\approx 1+nx\;\;(|x|\ll 1)\)
指数 / 三角 Exponential / trig	\(\displaystyle e^x\approx 1+x+\tfrac{x^2}{2},\;\;\sin x\approx x-\tfrac{x^3}{6},\;\;\cos x\approx 1-\tfrac{x^2}{2}\)
梯度 / 散度 / 旋度 Grad / Div / Curl	\(\displaystyle \vec{F}=-\nabla U,\;\;\nabla\cdot\vec{F},\;\;\nabla\times\vec{F}\)

II运动学 · Kinematics

2.1 一维匀加速运动 1D Constant Acceleration

名称 Name	表达式 Formula
速度 Velocity	\(\displaystyle v=v_0+at\)
位移 Displacement	\(\displaystyle x=x_0+v_0t+\tfrac{1}{2}at^2\)
时间消去 Time-eliminated	\(\displaystyle v^2=v_0^2+2a\,\Delta x\)
平均速度 Average velocity	\(\displaystyle \bar{v}=\frac{v_0+v}{2}\)

2.2 抛体运动 Projectile Motion

名称 Name	表达式 Formula
位置 Position	\(\displaystyle x(t)=v_0\cos\theta\cdot t,\quad y(t)=v_0\sin\theta\cdot t-\tfrac{1}{2}gt^2\)
轨迹方程 Trajectory	\(\displaystyle y=x\tan\theta-\frac{gx^2}{2v_0^2\cos^2\theta}\)
射程 Range (level)	\(\displaystyle R=\frac{v_0^2\sin 2\theta}{g}\)
最大高度 Maximum height	\(\displaystyle H=\frac{v_0^2\sin^2\theta}{2g}\)
飞行时间 Time of flight	\(\displaystyle T=\frac{2v_0\sin\theta}{g}\)

2.3 圆周 / 极坐标 Circular & Polar Motion

名称 Name	表达式 Formula
线速度–角速度 Linear vs angular	\(\displaystyle v=r\omega\)
向心加速度 Centripetal accel.	\(\displaystyle a_c=\frac{v^2}{r}=\omega^2 r\)
极坐标速度 Velocity in polar	\(\displaystyle \vec{v}=\dot{r}\,\hat{r}+r\dot{\theta}\,\hat{\theta}\)
极坐标加速度 Acceleration in polar	\(\displaystyle \vec{a}=(\ddot{r}-r\dot{\theta}^2)\,\hat{r}+(r\ddot{\theta}+2\dot{r}\dot{\theta})\,\hat{\theta}\)
匀角加速 Constant α	\(\displaystyle \omega=\omega_0+\alpha t,\;\;\theta=\theta_0+\omega_0 t+\tfrac{1}{2}\alpha t^2\)

III动力学 · Dynamics

3.1 牛顿定律与常见力 Newton's Laws & Forces

名称 Name	表达式 Formula
第二定律 Second law	\(\displaystyle \vec{F}_{net}=m\vec{a}=\dv{\vec{p}}{t}\)
第三定律 Third law	\(\displaystyle \vec{F}_{AB}=-\vec{F}_{BA}\)
静摩擦 Static friction	\(\displaystyle f_s\le \mu_s N\)
动摩擦 Kinetic friction	\(\displaystyle f_k=\mu_k N\)
胡克定律 Hooke's law	\(\displaystyle F=-kx\)
线性阻力 Linear drag	\(\displaystyle \vec{F}_d=-b\vec{v}\)
二次阻力 Quadratic drag	\(\displaystyle F_d=\tfrac{1}{2}\rho C_d A\,v^2\)
收尾速度 Terminal velocity	\(\displaystyle v_t=\frac{mg}{b}\) (linear); \(\displaystyle v_t=\sqrt{\frac{2mg}{\rho C_d A}}\) (quadratic)

IV功与能 · Work & Energy

名称 Name	表达式 Formula
功 Work	\(\displaystyle W=\int \vec{F}\cdot d\vec{r}\)
功率 Power	\(\displaystyle P=\dv{W}{t}=\vec{F}\cdot\vec{v}\)
动能 Kinetic energy	\(\displaystyle K=\tfrac{1}{2}mv^2\)
动能定理 Work–energy theorem	\(\displaystyle W_{net}=\Delta K\)
总机械能 Total mechanical energy	\(\displaystyle E=K+U\)
非保守力功 Non-conservative work	\(\displaystyle W_{nc}=\Delta E\)
力–势能 Force from potential	\(\displaystyle \vec{F}=-\nabla U,\;\;F_x=-\dv{U}{x}\)
重力势能（近地表） Gravitational PE	\(\displaystyle U=mgh\)
弹簧势能 Spring PE	\(\displaystyle U=\tfrac{1}{2}kx^2\)
引力势能（通用） Gravitational PE (general)	\(\displaystyle U=-\frac{GMm}{r}\)

V动量 · Momentum

名称 Name	表达式 Formula
动量 Momentum	\(\displaystyle \vec{p}=m\vec{v}\)
冲量 Impulse	\(\displaystyle \vec{J}=\int\vec{F}\,dt=\Delta\vec{p}\)
动量守恒 Conservation	\(\displaystyle \sum\vec{p}_i=\sum\vec{p}_f\;\;\;(F_{ext}=0)\)
质心 Position of CM	\(\displaystyle \vec{r}_{cm}=\frac{1}{M}\sum_i m_i\vec{r}_i\)
质心运动 CM motion	\(\displaystyle M\vec{a}_{cm}=\vec{F}_{ext},\;\;\vec{P}_{tot}=M\vec{v}_{cm}\)
一维弹性碰撞 1D elastic (target at rest)	\(\displaystyle v_1'=\frac{m_1-m_2}{m_1+m_2}v_1,\;\;v_2'=\frac{2m_1}{m_1+m_2}v_1\)
一维弹性碰撞（一般） 1D elastic (general)	\(\displaystyle v_1'=\frac{(m_1-m_2)v_1+2m_2 v_2}{m_1+m_2},\;\;v_2'=\frac{2m_1 v_1+(m_2-m_1)v_2}{m_1+m_2}\)
恢复系数 Coefficient of restitution	\(\displaystyle e=\frac{v_{2f}-v_{1f}}{v_{1i}-v_{2i}}\)
齐奥尔科夫斯基公式 Rocket equation	\(\displaystyle \Delta v=v_{ex}\ln\!\frac{m_0}{m_f}\)

VI转动 · Rotation

6.1 转动动力学 Rotational Dynamics

名称 Name	表达式 Formula
力矩 Torque	\(\displaystyle \vec{\tau}=\vec{r}\times\vec{F}\)
转动第二定律 Newton II (rot.)	\(\displaystyle \vec{\tau}_{net}=I\vec{\alpha}=\dv{\vec{L}}{t}\)
转动动能 Rotational KE	\(\displaystyle K_{rot}=\tfrac{1}{2}I\omega^2\)
角动量 Angular momentum	\(\displaystyle \vec{L}=\vec{r}\times\vec{p}=I\vec{\omega}\)
平行轴定理 Parallel axis theorem	\(\displaystyle I=I_{cm}+Md^2\)
垂直轴定理 Perpendicular axis (planar)	\(\displaystyle I_z=I_x+I_y\)
无滑滚动 Rolling (no slip)	\(\displaystyle v=R\omega,\;\;a=R\alpha\)
滚动总动能 Rolling total KE	\(\displaystyle K=\tfrac{1}{2}Mv^2+\tfrac{1}{2}I_{cm}\omega^2\)
回转频率 Precession rate	\(\displaystyle \Omega_p=\frac{\tau}{L}=\frac{Mgr}{I\omega}\)

6.2 常见转动惯量 Moments of Inertia (about CM, axis through CM)

物体 Object	\(I\)
质点 Point mass (at radius r)	\(\displaystyle I=Mr^2\)
细圆环 Hoop/ring (⊥ to plane)	\(\displaystyle I=MR^2\)
圆盘 / 实心柱 Disk / solid cylinder (axial)	\(\displaystyle I=\tfrac{1}{2}MR^2\)
实心球 Solid sphere	\(\displaystyle I=\tfrac{2}{5}MR^2\)
薄球壳 Thin spherical shell	\(\displaystyle I=\tfrac{2}{3}MR^2\)
匀质杆 Rod (axis through center)	\(\displaystyle I=\tfrac{1}{12}ML^2\)
匀质杆 Rod (axis through end)	\(\displaystyle I=\tfrac{1}{3}ML^2\)
实心柱 Disk (axis through diameter)	\(\displaystyle I=\tfrac{1}{4}MR^2+\tfrac{1}{12}ML^2\)
矩形板 Rectangular slab (⊥ through center)	\(\displaystyle I=\tfrac{1}{12}M(a^2+b^2)\)

VII振动 · Oscillations

名称 Name	表达式 Formula
简谐运动方程 SHM equation	\(\displaystyle \ddot{x}+\omega^2 x=0\)
通解 General solution	\(\displaystyle x(t)=A\cos(\omega t+\phi)\)
弹簧振子 Spring–mass	\(\displaystyle \omega=\sqrt{\frac{k}{m}},\;\;T=2\pi\sqrt{\frac{m}{k}}\)
单摆（小角度） Simple pendulum	\(\displaystyle T=2\pi\sqrt{\frac{L}{g}}\)
物理摆 Physical pendulum	\(\displaystyle T=2\pi\sqrt{\frac{I}{Mgd}}\)
总能量 SHM energy	\(\displaystyle E=\tfrac{1}{2}kA^2\)
最大速度/加速度 Peak v / a	\(\displaystyle v_{max}=A\omega,\;\;a_{max}=A\omega^2\)
阻尼方程 Damped oscillator	\(\displaystyle m\ddot{x}+b\dot{x}+kx=0\)
阻尼频率 Underdamped freq.	\(\displaystyle \omega_d=\sqrt{\omega_0^2-\gamma^2},\;\;\gamma=\frac{b}{2m}\)
共振振幅 Resonance amplitude	\(\displaystyle A=\frac{F_0/m}{\sqrt{(\omega_0^2-\omega^2)^2+(2\gamma\omega)^2}}\)
品质因数 Quality factor	\(\displaystyle Q=\frac{\omega_0}{2\gamma}\)

VIII引力 · Gravitation

名称 Name	表达式 Formula
万有引力 Newton's law	\(\displaystyle F=\frac{GMm}{r^2}\)
引力场 Gravitational field	\(\displaystyle \vec{g}=-\frac{GM}{r^2}\,\hat{r}\)
表面重力 Surface gravity	\(\displaystyle g=\frac{GM}{R^2}\)
引力势能 Gravitational PE	\(\displaystyle U=-\frac{GMm}{r}\)
圆轨道速度 Circular orbit speed	\(\displaystyle v_{orbit}=\sqrt{\frac{GM}{r}}\)
逃逸速度 Escape speed	\(\displaystyle v_{esc}=\sqrt{\frac{2GM}{r}}\)
轨道总能量 Circular orbit energy	\(\displaystyle E=-\frac{GMm}{2r},\;\;U=2E,\;\;K=-E\)
开普勒第三定律 Kepler's 3rd law	\(\displaystyle T^2=\frac{4\pi^2}{GM}\,a^3\)
活力公式 Vis-viva equation	\(\displaystyle v^2=GM\!\left(\frac{2}{r}-\frac{1}{a}\right)\)
开普勒第二定律 Kepler's 2nd (areal)	\(\displaystyle \dv{A}{t}=\frac{L}{2m}\)

IX分析力学 · Analytical Mechanics

名称 Name	表达式 Formula
作用量 Action	\(\displaystyle S=\int_{t_1}^{t_2} L\,dt\)
拉格朗日量 Lagrangian	\(\displaystyle L=T-V\)
共轭动量 Conjugate momentum	\(\displaystyle p_i=\pdv{L}{\dot{q}_i}\)
欧拉–拉格朗日方程 Euler–Lagrange eq.	\(\displaystyle \dv{t}\pdv{L}{\dot{q}_i}-\pdv{L}{q_i}=0\)
哈密顿量 Hamiltonian	\(\displaystyle H=\sum_i p_i\dot{q}_i-L\)
哈密顿方程 Hamilton's equations	\(\displaystyle \dot{q}_i=\pdv{H}{p_i},\;\;\dot{p}_i=-\pdv{H}{q_i}\)
守恒量对应 Conservation symmetry	\(\displaystyle \pdv{L}{q_i}=0\Rightarrow p_i=\text{const.}\) (Noether)

X相对论 · Special Relativity

10.1 时空与变换 Spacetime & Transformations

名称 Name	表达式 Formula
洛伦兹因子 Lorentz factor	\(\displaystyle \gamma=\frac{1}{\sqrt{1-v^2/c^2}}\)
时间膨胀 Time dilation	\(\displaystyle \Delta t=\gamma\,\Delta\tau\)
长度收缩 Length contraction	\(\displaystyle L=\frac{L_0}{\gamma}\)
洛伦兹变换 Lorentz transform	\(\displaystyle x'=\gamma(x-vt),\;\;t'=\gamma\!\left(t-\frac{vx}{c^2}\right)\)
速度合成 Velocity addition	\(\displaystyle u'=\frac{u-v}{1-uv/c^2}\)
不变间隔 Invariant interval	\(\displaystyle (\Delta s)^2=(c\,\Delta t)^2-(\Delta x)^2-(\Delta y)^2-(\Delta z)^2\)
相对论多普勒 Relativistic Doppler	\(\displaystyle f'=f\sqrt{\frac{1\mp\beta}{1\pm\beta}},\;\;\beta=v/c\)

10.2 相对论动量与能量 Relativistic Dynamics

名称 Name	表达式 Formula
四维速度 Four-velocity	\(\displaystyle u^\mu=\gamma(c,\vec{v})\)
相对论动量 Relativistic momentum	\(\displaystyle \vec{p}=\gamma m\vec{v}\)
总能量 Total energy	\(\displaystyle E=\gamma mc^2\)
静能 Rest energy	\(\displaystyle E_0=mc^2\)
动能 Kinetic energy	\(\displaystyle K=(\gamma-1)mc^2\)
能量–动量关系 Energy–momentum	\(\displaystyle E^2=(pc)^2+(mc^2)^2\)
四动量 Four-momentum	\(\displaystyle p^\mu=(E/c,\,\vec{p})\)

XI流体 · Fluid Mechanics

名称 Name	表达式 Formula
压强 Pressure	\(\displaystyle P=\frac{F}{A}\)
静水压 Hydrostatic pressure	\(\displaystyle P=P_0+\rho g h\)
阿基米德浮力 Buoyant force	\(\displaystyle F_b=\rho_f V g\)
连续性方程 Continuity (incompressible)	\(\displaystyle A_1 v_1=A_2 v_2\)
伯努利方程 Bernoulli	\(\displaystyle P+\tfrac{1}{2}\rho v^2+\rho g y=\text{const}\)
托里拆利 Torricelli	\(\displaystyle v=\sqrt{2gh}\)
雷诺数 Reynolds number	\(\displaystyle Re=\frac{\rho v L}{\eta}\)
泊肃叶定律 Poiseuille (laminar pipe)	\(\displaystyle Q=\frac{\pi R^4\,\Delta P}{8\eta L}\)
斯托克斯阻力 Stokes drag	\(\displaystyle F=6\pi\eta R v\)

XII常用常数 · Useful Constants

常数 Constant	数值 Value
重力加速度 Standard gravity, \(g\)	\(\displaystyle 9.8\,\unit{m/s^2}\)
万有引力常量 Gravitational constant, \(G\)	\(\displaystyle 6.674\times10^{-11}\,\unit{N\,m^2/kg^2}\)
光速 Speed of light, \(c\)	\(\displaystyle 3.0\times10^{8}\,\unit{m/s}\)
电子质量 Electron mass, \(m_e\)	\(\displaystyle 9.11\times10^{-31}\,\unit{kg}\)
元电荷 Elementary charge, \(e\)	\(\displaystyle 1.602\times10^{-19}\,\unit{C}\)
水密度 Water density (4 °C)	\(\displaystyle 1.0\times10^{3}\,\unit{kg/m^3}\)
标准大气压 Standard atmosphere, \(1\,\mathrm{atm}\)	\(\displaystyle 1.013\times10^{5}\,\unit{Pa}\)