方程与法则

方程

数学

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

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

基础代数

1+1=21+1=2

ab=baa *b = b*a

(ab)c=a(bc)(a*b)*c = a*(b*c)

基础几何

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

测试

a2+b2=c2a^2 +b^2 = c^2

三角函数

ab=abcosθ\vec{a} \cdot \vec{b} = |\vec{a} ||\vec{b} |cos\theta

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

极限

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

导数

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

a=ΔvΔt=dvdt=d2sdt2\vec{a} = \frac{\vec{\Delta{v}}}{\Delta{t}}= \frac{dv}{dt} = \frac{d^2\vec{s}}{dt^2}

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

梯度散度旋度

f=(fx,fy,fz)\nabla{f} = \Big( \frac{\partial{f}}{\partial{x}},\frac{\partial{f}}{\partial{y}},\frac{\partial{f}}{\partial{z}}\Big)

F=Fx+Fy+Fz\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

f=f22x+f22y+f22z=0\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)n=0manxnf(x) \approx \sum_{n=0}^{m} a_nx^n

f(x)f(x0)+f(x0)1!(xx0)+f(x0)2!(xx0)2++f(n)(x0)n!(xx0)nf(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

ex1+x+x22!+x33!+e^x \approx 1 + x + \frac{x^2}{2!}+ \frac{x^3}{3!} + \cdot\cdot\cdot

cosx1x22!+x44!x66!+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

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

守恒

m1v1+m2v2=m1v1+m2v2m_1\vec{v_1} +m_2\vec{v_2} = m_1\vec{v'_1} +m_2\vec{v'_2}

2H2+O2=2H2O2H_2 + O_2 = 2H_2O

欧拉方程

eiπ+1=0e^{i\pi} + 1 = 0

eix=cosx+isinxe^{ix}=cosx + isinx

质能方程

E=mc2E = mc^2

黎曼等式

ζ(s)=n=11ns\zeta(s) = \sum_{n=1}^{\infty }\frac{1}{n^s}

ζ(2)=112+122+132+142+=π26\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)=+f(τ)g(xτ)dτf(x) *g(x) = \int_{-\infty}^{+\infty}f(\tau)g(x-\tau)d\tau

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

渲染方程

Lo(po,ωo)=Le(po,ωo)+Ω+f(pipo,ωiωo)Li(pi,ωi)cosθdωiL_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

a=1ni=1naiμ=E(ai)asn\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=GMmr2F=G\frac{Mm}{r^2}

F=14πε0qqr2F=\frac{1}{4\pi \varepsilon_0} \frac{qq'}{r^2}

E=ΦcosθA=Φcosθ4πr2E = \frac{\Phi cos\theta}{A} = \frac{\Phi cos\theta}{4\pi r^2}

波动方程

热方程

高斯方程

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

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

Stokes’s 定理

c(f)dr=f(p1)f(p0)\int_c (\nabla \cdot f) d \vec{r}= f(p_1) - f(p_0)

L+Fdr=DQxPydxdy=D(×F)ds\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

L+Fdn=DPx+Qydxdy=D(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

Ωω=Ωdω\int_{\partial{\Omega}}\omega = \int_{\Omega}d\omega

麦克斯韦方程组

E=ρϵ0\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}

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

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

×B=μ0J+μ0ϵ0Et (μ0ϵ0=1c2)\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

iψt=H^ψi \hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi

闵可夫斯基时空观

Δs2=(cΔt)2+(Δp)2=(cΔt)2+(Δp)2<0\Delta s^2 = -(c\Delta t)^2 + (\Delta p)^2 = -(c\Delta t')^2 + (\Delta p')^2 < 0

泛函

A[f]=x1x2L(x,f(x),f(x))dxA[f] = \int_{x_1}^{x_2}L\Big(x, f(x), f'(x)\Big)dx

拉格朗日量

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

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

S=KlogWS = KlogW

H=i=1npilog2piH = -\sum_{i=1}^{n}p_ilog_2{p_i}

广义相对论方程

最小作用量原理