方程

数学

$$y = f(x) = A\vec{x} = nn(x)$$

$$y = L\Big(f(x)\Big)$$

基础代数

$$1+1=2$$

$$a *b = b*a$$

$$(a*b)*c = a*(b*c)$$

基础几何

$$\angle{A} + \angle{B}+ \angle{C} = 180\degree$$

测试

$$a^2 +b^2 = c^2$$

三角函数

$$\vec{a} \cdot \vec{b} = |\vec{a} ||\vec{b} |cos\theta$$

$$\vec{a} \times \vec{b} = |\vec{a} ||\vec{b} |sin\theta = Area$$

极限

$$e = \lim_{n\rightarrow\infty}(1+\frac{1}{n})^n$$

导数

$$\vec{v} = \frac{\vec{\Delta{s}}}{\Delta{t}} = \frac{d\vec{s}}{dt}$$

$$\vec{a} = \frac{\vec{\Delta{v}}}{\Delta{t}}= \frac{dv}{dt} = \frac{d^2\vec{s}}{dt^2}$$

$$gradient = \frac {dh}{ds}$$

梯度散度旋度

$$\nabla{f} = \Big( \frac{\partial{f}}{\partial{x}},\frac{\partial{f}}{\partial{y}},\frac{\partial{f}}{\partial{z}}\Big)$$

$$\nabla \cdot \vec{F} = \frac{\partial{\vec{F}}}{\partial{x}} + \frac{\partial{\vec{F}}}{\partial{y}} + \frac{\partial{\vec{F}}}{\partial{z}}$$

拉普拉斯方程

$$\nabla \nabla \cdot f = \frac{\partial{f^2}}{\partial^2{x}} + \frac{\partial{f^2}}{\partial^2{y}} + \frac{\partial{f^2}}{\partial^2{z}} = 0$$

级数

$$f(x) \approx \sum_{n=0}^{m} a_nx^n$$

$$f(x) \approx f(x_0) + \frac{f'(x_0)}{1!}(x-x_0)+ \frac{f''(x_0)}{2!}(x-x_0)^2 + \cdot \cdot \cdot + \frac {f^{(n)}(x_0)} {n!}(x-x_0)^n$$

$$e^x \approx 1 + x + \frac{x^2}{2!}+ \frac{x^3}{3!} + \cdot\cdot\cdot$$

$$cosx \approx 1 - \frac{x^2}{2!}+ \frac{x^4}{4!}- \frac{x^6}{6!} + \cdot\cdot\cdot$$

傅里叶级数

人口增长

$$\begin{cases} \frac{dx}{dt} = rx&(1) \\ r = k \times (1-\frac{x}{x_m})&(2) \end{cases}$$

守恒

$$m_1\vec{v_1} +m_2\vec{v_2} = m_1\vec{v'_1} +m_2\vec{v'_2}$$

$$2H_2 + O_2 = 2H_2O$$

欧拉方程

$$e^{i\pi} + 1 = 0$$

$$e^{ix}=cosx + isinx$$

质能方程

$$E = mc^2$$

黎曼等式

$$\zeta(s) = \sum_{n=1}^{\infty }\frac{1}{n^s}$$

$$\zeta(2) = \frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdot \cdot \cdot = \frac{\pi^2}{6}$$

卷积

$$f(x) *g(x) = \int_{-\infty}^{+\infty}f(\tau)g(x-\tau)d\tau$$

$$F\Big(f(x)*g(x)\Big)=F\Big(f(x)\Big)F\Big(g(x)\Big)$$

渲染方程

$$L_o(p_o,\omega_o) = L_e(p_o, \omega_o) + \int_{\Omega^+ }^{}f(p_i\rightarrow p_o,\omega_i\rightarrow \omega_o)L_i(p_i,\omega_i)cos\theta d\omega_i$$

马尔科夫链

大数定理

$$\overline{a} = \frac{1}{n}\sum_{i=1}^n{a_i} \to \mu = E(a_i) as n \to \infty$$

场与通量

$$F=G\frac{Mm}{r^2}$$

$$F=\frac{1}{4\pi \varepsilon_0} \frac{qq'}{r^2}$$

$$E = \frac{\Phi cos\theta}{A} = \frac{\Phi cos\theta}{4\pi r^2}$$

波动方程

热方程

高斯方程

$$\iiint_\Omega \frac {\partial U} {\partial x} + \frac {\partial V} {\partial y} + \frac {\partial W} {\partial z} dv = \oiint_{\varSigma} Udydz + Vdxdz + Wdxdy$$

$$\iiint_\Omega( \nabla \cdot \vec {F}d \omega) = \oiint_{\varSigma} \vec{F} \cdot \vec{n} d \sigma$$

Stokes’s 定理

$$\int_c (\nabla \cdot f) d \vec{r}= f(p_1) - f(p_0)$$

$$\oint_{L^+} \vec{F} d \vec{r} = \iint_D \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} dx dy = \iint_D (\nabla \times \vec {F}) ds$$

$$\oint_{L^+} \vec{F} d \vec{n} = \iint_D \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} dx dy = \iint_D (\nabla \cdot \vec {F}) ds$$

$$\int_{\partial{\Omega}}\omega = \int_{\Omega}d\omega$$

麦克斯韦方程组

$$\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$$

$$\nabla \cdot \vec{B} = 0$$

$$\nabla \times \vec{E} = -\frac{\partial{\vec{B}}}{\partial{t}}$$

$$\nabla \times \vec{B} = \mu_0\vec{J} + \mu_0\epsilon_0 \frac{\partial{\vec{E}}}{\partial{t}} \ (\mu_0\epsilon_0 = \frac{1}{c^2})$$

薛定谔方程

$$i \hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi$$

闵可夫斯基时空观

$$\Delta s^2 = -(c\Delta t)^2 + (\Delta p)^2 = -(c\Delta t')^2 + (\Delta p')^2 < 0$$

泛函

$$A[f] = \int_{x_1}^{x_2}L\Big(x, f(x), f'(x)\Big)dx$$

拉格朗日量

$$\mathcal {L} = T - V = C$$

$$\frac{d}{dt}\Big(\frac{\partial \mathcal{L}}{\partial\dot{q_i}}\Big) - \frac{\partial \mathcal {L}}{\partial q_i} = 0$$

熵

$$S = KlogW$$

$$H = -\sum_{i=1}^{n}p_ilog_2{p_i}$$

广义相对论方程

最小作用量原理