DeepNLP Equation: Equation Database

Amplitude of a driven oscillation

#physics #amplitude #oscillation

$$A = \frac{{F_0 }}{{\sqrt {m^2 \left( {\omega _0^2 - \omega ^2 } \right)^2 + b^2 \omega ^2 } }}$$

Angular frequency for a damped oscillation

#physics #angular #damped oscillation #oscillation

$$\omega ' = \omega _0 \sqrt {1 - \left( {\frac{b}{{2m\omega _0 }}} \right)^2 } = \omega _0 \sqrt {1 - \frac{1}{{4Q^2 }}}$$

Coupled Conductors and Transformers

#physics #coupled #conductors #transformers

$$M_{12}=M_{21}:=M=k\sqrt{L_1L_2}=\frac{N_1\Phi_1}{I_2}=\frac{N_2\Phi_2}{I_1}\sim N_1N_2 \\ \frac{V_1}{V_2}=\frac{I_2}{I_1}=-\frac{i\omega M}{i\omega L_2+R_{\rm load}}\approx-\sqrt{\frac{L_1}{L_2}}=-\frac{N_1}{N_2} \\ \Phi_{12}=M_{12}I_2 \\ \Phi_{21}=M_{21}I_1$$

Cylindrical Waves

#physics #cylindrical waves

$$\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}-\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial u}{\partial r}\right)=0 \\ u(r,t)=\frac{\hat{u}}{\sqrt{r}}\cos(k(r\pm vt))$$

Einstein Field Equations

#physics

$$G^{\alpha\beta}:=R^{\alpha\beta}-\frac{1}{2}g^{\alpha\beta}R \\ G_{\alpha\beta}=\frac{8\pi \kappa}{c^{2}}T_{\alpha\beta}$$

Electric Oscillations

#physics #oscillations

$$\text{Impedance} \\ Z=R+ix \\ \text{Series connection} \\ V=IZ, Z_{\rm tot}=\sum_i Z_i~,~~L_{\rm tot}=\sum_i L_i~,~~ \frac{1}{C_{\rm tot}}=\sum_i\frac{1}{C_i}~,~~Q=\frac{Z_0}{R}~,~~ Z=R(1+iQ\delta) \\ \text{Parallel connection} \\ \frac{1}{Z_{\rm tot}}=\sum_i\frac{1}{Z_i}~,~~ \frac{1}{L_{\rm tot}}=\sum_i\frac{1}{L_i}~,~~ C_{\rm tot}=\sum_i C_i~,~~Q=\frac{R}{Z_0}~,~~ Z=\frac{R}{1+iQ\delta}$$

General Relativity

#physics #relativity

$$\frac{\mathrm{d}^{2} x^{\alpha}}{\mathrm{d} s^{2}}+\Gamma^{\alpha}_{\beta\gamma}\frac{\mathrm{d} x^{\beta}}{\mathrm{d} s}\frac{\mathrm{d} x^{\gamma}}{\mathrm{d} s}=0$$

Harmonic Oscillations

#physics #harmonic #oscillations

$$\Psi(t)=\hat{\Psi}(t)e^{i(\omega t \pm \phi)}=\hat{\Psi}(t)\cos (\omega t \pm \phi) \\ \sum_{i} \hat{\Psi_{i}}\cos(\alpha_{i} \pm \omega t) =\hat{\Phi}\cos (\beta \pm \omega t) \\ \tan (\beta)=\frac{\sum_{i} \hat{\Psi_{i}} \sin (\alpha_{i})}{\sum_{i} \hat{\Psi_{i}} \cos (\alpha_{i})} \\ \hat{\Phi}^{2} = \sum_{i} \hat{\Psi_{i}^{2}} + 2 \sum_{j > i} \sum_{i} \hat{\Psi_{i}} \hat{\Psi_{j}} \cos (\alpha_{i} - \alpha_{j}) \\ \int x(t) dt=\frac{x(t)}{i \omega} \\ \frac{d^{n}(x(t))}{d t^{n}}=(i \omega)^{n} x(t)$$

Lorentz Transformation Latex

#physics #relativity #lorentz #transformation

Maxwell Equations Differential

#physics #maxwell #electricity #magnetism

$$\nabla \cdot \vec{D}=\rho_{free} \\ \nabla \cdot \vec{B}=0 \\ \nabla \times \vec{E}=-\frac{\partial{\vec{B}}}{\partial{t}} \\ \nabla \times \vec{H}=\vec{J}_{free}+\frac{\partial{\vec{D}}}{\partial{t}}$$

Maxwell Equations Integral

#physics #maxwell #electricity #magnetism

$$\oiint (\vec{D}\cdot \vec{n}) \mathrm{d}^{2}A=Q_{\text{free,included}}\\ \oiint (\vec{B}\cdot \vec{n}) \mathrm{d}^{2}A=0 \\ \oint \vec{E} \mathrm{d}\vec{s}=-\frac{\mathrm{d}\Phi}{\mathrm{d}t}\\ \oint \vec{H} \mathrm{d}\vec{s}=I_{\text{free,included}}+\frac{\mathrm{d}\Psi }{\mathrm{d}t}$$

Mechanic Oscillations

#physics #mechanic #oscillations

$$m\ddot{x}=F(t)-k\dot{x}-Cx \\ F(t)=\hat{F}\cos(\omega t) \\ -m\omega^2 x=F-Cx-ik\omega x \\ \omega_0^2=C/m \\ x=\frac{F}{m(\omega_0^2-\omega^2)+ik\omega} \\ \dot{x}=\frac{F}{i\sqrt{Cm}\delta+k} \\ \delta=\frac{\omega}{\omega_0}-\frac{\omega_0}{\omega} \\ Z=F/\dot{x} \\ Q=\frac{\sqrt{Cm}}{k}$$

Pendulums

#physics #pendulums

$$T=1/f \\ T=2\pi\sqrt{m/C} \\ T=2\pi\sqrt{I/\tau} \\ T=2\pi\sqrt{I/\kappa} \\ T=2\pi\sqrt{l/g}$$

Plane Waves

#physics #plane waves

$$u(\vec{x},t)=2^n\hat{u}\cos(\omega t)\sum_{i=1}^n\sin(k_ix_i) \\ u(\vec{x},t)=\hat{u}\cos(\vec{k}\cdot\vec{x}\pm\omega t+\varphi) \\ \frac{f}{f_0}=\frac{v_{\rm f}-v_{\rm obs}}{v_{\rm f}}$$

Red and Blue Shift

#physics #red and blue shift

$$\vec{e_{v}}\vec{e_{r}}=\cos(\Phi) \\ \frac{f^{'}}{f}=\gamma(1-\frac{v\cos(\Phi)}{c}) \\ \frac{\Delta f}{f}=\frac{\kappa M}{rc^{2}} \\ \frac{\lambda_{0}}{\lambda_{1}}=\frac{R_{0}}{R_{1}}$$

Riemannian Tensor

#physics #riemannian rensor

$$R^{\mu}_{v \alpha \beta}T^{v}=\triangledown_{\alpha}\triangledown_{\beta}T^{\mu}-\triangledown_{\beta}\triangledown_{\alpha}T^{\mu} \\ \triangledown_{j}a^{i}=\partial_{j}a^{i}+\Gamma^{i}_{jk}a^{k} \\ \triangledown_{j}a_{i}=\partial_{j}a_{i}-\Gamma^{k}_{ij}a_{k} \\ \Gamma^{i}_{jk}=\frac{\partial^{2} \bar{x}^{l}}{\partial{x^{j}}\partial{x^{k}}}\frac{\partial{x^{i}}}{\partial{\bar{x}^{l}}}$$

Spherical Waves

#physics #spherical waves

$$\frac{1}{v^2}\frac{\partial^2 (ru)}{\partial t^2}-\frac{\partial^2 (ru)}{\partial r^2}=0 \\ u(r,t)=C_1\frac{f(r-vt)}{r}+C_2\frac{g(r+vt)}{r}$$

Stress-energy Tensor

#physics #stress-energy tensor

$$T_{\mu v}=(\varrho c^{2}+p)u_{p}u_{v}+pg_{\mu v}+\frac{1}{c^{2}}(F^{\mu}_{\alpha}F^{\alpha v} + \frac{1}{4}g^{\mu v}F^{\alpha\beta}F_{\alpha\beta}) \\ \triangledown_{v} T_{\mu v}=0 \\ F_{\alpha\beta}=\frac{\partial{A_{\beta}}}{\partial{x^{\alpha}}} - \frac{\partial{A_{\alpha}}}{\partial{x^{\beta}}} \\ \frac{d p_{\alpha}}{d \tau}=qF_{\alpha\beta}u^{\beta}$$

The general solution of Wave Equation

#physics #general solution #waves

$$\frac{\partial^2 u(x,t)}{\partial t^2}=\sum_{m=0}^{N}\left(b_m\frac{\partial ^m}{\partial x^m}\right)u(x,t) \\ u(x,t)=\int\limits_{-\infty}^{\infty}\left(a(k){\rm e}^{i(kx-\omega_1(k)t)}+ b(k){\rm e}^{i(kx-\omega_2(k)t)}\right)dk \\ u(x,t)=A{\rm e}^{i(kx-\omega t)} \\ \omega_j=\omega_j(k)$$

Wave Equation

#physics #wave

$$\nabla^2u-\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2}-\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}=0 \\ v=f\lambda \\ k\lambda=2\pi \\ \omega=2\pi f \\ v_{\rm g}=\frac{d\omega}{dk}=v_{\rm ph}+k\frac{dv_{\rm ph}}{dk}= v_{\rm ph}\left(1-\frac{k}{n}\frac{dn}{dk}\right) \\ v=\sqrt{\kappa/\varrho}$$

Waves In Long Conductors

#physics #waves #oscillations

$$Z_0=\sqrt{\frac{dL}{dx}\frac{dx}{dC}} \\ v=\sqrt{\frac{dx}{dL}\frac{dx}{dC}}$$

Angular frequency for a damped oscillation

#Physics #Angular frequency

$$\omega ' = \omega _0 \sqrt {1 - \left( {\frac{b}{{2m\omega _0 }}} \right)^2 } = \omega _0 \sqrt {1 - \frac{1}{{4Q^2 }}}$$

Bending of light Fermat's principle

#physics #optics #Fermat's principle

$$\delta\int\limits_1^2 dt=\delta\int\limits_1^2\frac{n(s)}{c}ds=0\Rightarrow \delta\int\limits_1^2 n(s)ds=0$$

Bending of light Snell's law

#physics #optics #Snell's law

$$n_i\sin(\theta_i)=n_t\sin(\theta_t) \\ \frac{n_2}{n_1}=\frac{\lambda_1}{\lambda_2}=\frac{v_1}{v_2} \\ n^2=1+\frac{n_{\rm e}e^2}{\varepsilon_0m}\sum_j\frac{f_j}{\omega_{0,j}^2-\omega^2-i\delta\omega} \\ v_{\rm g}=c/(1+(n_{\rm e}e^2/2\varepsilon_0m\omega^2)) $$

Birefringence and Dichroism

#physics #optics #girefringence #dichroism

$$\text{NA}$$

Diffraction

#physics #optics #diffraction

$$\frac{I(\theta)}{I_0}=\left(\frac{\sin(u)}{u}\right)^2\cdot \left(\frac{\sin(Nv)}{\sin(v)}\right)^2 \\ u=\pi b\sin(\theta)/\lambda \\ v=\pi d\sin(\theta)/\lambda$$

Displacement of a driven oscillator

#physics #displacement

$$x = A\cos \left( {\omega t + \delta } \right)$$

Fabry Perot Interferometer

#physics #optics #fabry perot

$$T+R+A=1$$

Fresnel Equations

#physics #optics #fresnel

$$r_\parallel=\frac{\tan(\theta_i-\theta_t)}{\tan(\theta_i+\theta_t)} \\ r_\perp =\frac{\sin(\theta_t-\theta_i)}{\sin(\theta_t+\theta_i)} \\ t_\parallel=\frac{2\sin(\theta_t)\cos(\theta_i)}{\sin(\theta_t+\theta_i)\cos(\theta_t-\theta_i)} \\ t_\perp =\frac{2\sin(\theta_t)\cos(\theta_i)}{\sin(\theta_t+\theta_i)}$$

Green functions for the initial-value problem

#physics #green functions

$$u(x,t)=\int\limits_{-\infty}^\infty f(x')Q(x,x',t)dx'+ \int\limits_{-\infty}^\infty g(x')P(x,x',t)dx'$$

Mirrors

#physics #optics #mirrors

$$\frac{1}{f}=\frac{1}{v}+\frac{1}{b}=\frac{2}{R}+\frac{h^2}{2}\left(\frac{1}{R}-\frac{1}{v}\right)^2$$

Non-linear Wave Equation

#physics #non-linear wave

$$\frac{d^2x}{dt^2}-\varepsilon\omega_0(1-\beta x^2)\frac{dx}{dt}+\omega_0^2x=0 \\ \frac{d}{dt}\left\{\frac{1}{2}\left(\frac{dx}{dt}\right)^2+\frac{1}{2} \omega_0^2x^2\right\}= \varepsilon\omega_0(1-\beta x^2)\left(\frac{dx}{dt}\right)^2 \\ \frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}-\underbrace{au\frac{\partial u}{\partial x}}_{\rm non-lin}+ \underbrace{b^2\frac{\partial ^3u}{\partial x^3}}_{\rm dispersive}=0 \\ u(x-ct)=\frac{-d}{\cosh^2(e(x-ct))}$$

Optical System Matrix Methods

#physics #optics #Matrix Methods

$$\left(\begin{array}{c}n_2\alpha_2\\y_2\end{array}\right)=M \left(\begin{array}{c}n_1\alpha_1\\y_1\end{array}\right) \\ {\rm Tr}(M)=1$$

Optics Magnification

#physics #optics #magnification

$$N=-\frac{b}{v} \\ N_{\alpha}=-\frac{\alpha_{\rm syst}}{\alpha_{\rm none}}$$

Optics Polarization

#physics #optics #polarization

$$P=\frac{I_{\rm p}}{I_{\rm p}+I_{\rm u}}=\frac{I_{\rm max}-I_{\rm min}}{I_{\rm max}+I_{\rm min}} \\ I(\theta)=I(0)\cos^2(\theta)$$

Optics Principal Planes

#physics #optics #principal planes

$$h_1=n\frac{m_{11}-1}{m_{12}} \\ h_2=n\frac{m_{22}-1}{m_{12}}$$

Prisms and Dispersion

#physics #optics #prisms #dispersion

$$\delta=\theta_i+\theta_{i'}+\alpha \\ n=\frac{\sin(\frac{1}{2}(\delta_{\rm min}+\alpha))}{\sin(\frac{1}{2}\alpha)} \\ D=\frac{d\delta}{d\lambda}=\frac{d\delta}{dn}\frac{dn}{d\lambda} \\ \frac{d\delta}{dn}=\frac{2\sin(\frac{1}{2}\alpha)}{\cos(\frac{1}{2}(\delta_{\rm min}+\alpha))}$$

Reflection and Transmission

#physics #optics #reflection #transmission

$$r_\parallel\equiv\left(\frac{E_{0r}}{E_{0i}}\right)_\parallel \\ r_\perp\equiv\left(\frac{E_{0r}}{E_{0i}}\right)_\perp \\ t_\parallel\equiv\left(\frac{E_{0t}}{E_{0i}}\right)_\parallel \\ t_\perp\equiv\left(\frac{E_{0t}}{E_{0i}}\right)_\perp$$

The Stationary Phase Method Wave Equation

#physics #stationary phase

$$\int\limits_{-\infty}^\infty a(k){\rm e}^{i(kx-\omega(k)t)}dk\approx \sum_{i=1}^{N}\sqrt{\frac{2\pi}{\frac{d^2\omega(k_i)}{dk_i^2}}} \exp\left[-i\mbox{$\frac{1}{4}$}\pi+i(k_ix-\omega(k_i)t)\right]$$

Thermodynamics Ideal Mixtures

#physics #thermodynamics #ideal mixtures

$$U_{\rm mixture}=\sum_i n_i U^0_i \\ H_{\rm mixture}=\sum_i n_i H^0_i \\ S_{\rm mixture}=n\sum_i x_i S^0_i+\Delta S_{\rm mix} \\ \Delta S_{\rm mix}=-nR\sum\limits_i x_i\ln(x_i)$$

Thermodynamics Statistical Basis

#physics #thermodynamics

$$P=N!\prod_i\frac{g_i^{n_i}}{n_i!} \\ n_i=\frac{N}{Z}g_i\exp\left(-\frac{W_i}{kT}\right) \\ Z=\sum\limits_ig_i\exp(-W_i/kT) \\ Z=\frac{V(2\pi mkT)^{3/2}}{h^3} \\ \text{Entropy in Thermodynamic Equilibrium} \\ S=\frac{U}{T}+kN\ln\left(\frac{Z}{N}\right)+kN\approx\frac{U}{T}+k\ln\left(\frac{Z^N}{N!}\right) \\ \text{Ideal gas} \\ S=kN+kN\ln\left(\frac{V(2\pi mkT)^{3/2}}{Nh^3}\right)$$

Waveguides and resonating cavities

#physics #waveguides

$$\begin{array}{ll} \vec{n}\cdot(\vec{D}_2-\vec{D}_1)=\sigma~~&~~\vec{n}\times(\vec{E}_2-\vec{E}_1)=0\\ \vec{n}\cdot(\vec{B}_2-\vec{B}_1)=0~~&~~\vec{n}\times(\vec{H}_2-\vec{H}_1)=\vec{K} \end{array} \\ \vec{E}(\vec{x},t)=\vec{\cal E}(x,y){e}^{i(kz-\omega t)} \\ \vec{B}(\vec{x},t)=\vec{\cal{B}}(x,y){e}^{i(kz-\omega t)}$$

Angular Momentum

#physics #mechanics

$$M = I\omega$$

Average Acceleration

#physics #mechanics #Average Acceleration

$$a_{av} = \frac{{\Delta \upsilon }}{{\Delta t}}$$

Bulk Modulus

#physics #mechanics #bulk modulus

$$B = - \frac{P}{{\Delta V}/V}$$

Centripetal Acceleration

#physics #mechanics

$$a=\frac{v^{2}}{r}$$

Displacement of a slightly damped oscillator

#physics #oscillations

$$x = A_0 \exp \left( { - \frac{b}{{2m}}t} \right)\cos \left( {\omega 't + \delta } \right)$$

Energy change in a damped oscillation

#physics #oscillations

$$\frac{{\Delta E}}{E} = - \frac{b}{m}T \\ E = E_0 \exp \left( { - \frac{b}{m}t} \right) = E_0 \exp \left( { - \frac{t}{\tau }} \right)$$

Energy transmitted by a harmonic wave

#physics #oscillations

$$\Delta E = \frac{1}{2}\mu \omega ^2 A^2 \Delta x = \frac{1}{2}\mu \omega ^2 A^2 \upsilon \Delta t$$

Harmonic Wave Function

#physics #wave

$$y(x,t) = A\sin \left[ {2\pi \left( {\frac{x}{\lambda } - \frac{t}{T}} \right)} \right] = A\sin \left[ {k(x - \upsilon t)} \right]$$

Isobaric Processes

#physics #thermodynamics #isobaric processes

$$\text{Isobaric Processes} \\ H_2-H_1=\int_1^2 C_pdT \\ \text{Reversible isobaric process} \\ H_2-H_1=Q_{\rm rev}$$

Mechanics Compressibility

#physics #mechanics #compressibility

$$k = \frac{1}{B} = - \frac{{\Delta V}/V}{P}$$

Phase Transitions

#physics #thermodynamics #phase transitions

$$\Delta S_m=S_m^\alpha - S_m^\beta=\frac{r_{\beta\alpha}}{T_0} \\ S_m=\left(\frac{\partial G_m}{\partial T}\right)_{p} \\ \frac{dp}{dT}=\frac{S_m^\alpha-S_m^\beta}{V_m^\alpha-V_m^\beta}= \frac{r_{\beta\alpha}}{(V_m^\alpha-V_m^\beta)T} \\ p=p_0{\rm e}^{-r_{\beta\alpha/RT}} \\ r_{\beta\alpha}=r_{\alpha\beta} \\ r_{\beta\alpha}=r_{\gamma\alpha}-r_{\gamma\beta} \\ \frac{dp}{dT}=\frac{S_m^\alpha-S_m^\beta}{V_m^\alpha-V_m^\beta}= \frac{r_{\beta\alpha}}{(V_m^\alpha-V_m^\beta)T} \\ p=p_0{\rm e}^{-r_{\beta\alpha/RT}}$$

Phase constant of a driven oscillation

#physics #oscillation

$$\tan \delta = \frac{{b\omega }}{{m\left( {\omega _0^2 - \omega ^2 } \right)}}$$

Reversible Adiabatic Processes

#physics #thermodynamics

$$\text{adiabatic processes} \\ W=U_1-U_2 \\ \text{reversible adiabatic processes} \\ \gamma=C_p/C_V$$

State functions Maxwell Relations

#physics #thermodynamics

The Carnot Cycle

#physics #thermodynamics #the carnot cycle

$$\eta=1-\frac{|Q_2|}{|Q_1|}=1-\frac{T_2}{T_1}:=\eta_{\rm C} \\ \xi=\frac{|Q_2|}{W}=\frac{|Q_2|}{|Q_1|-|Q_2|}=\frac{T_2}{T_1-T_2}$$

Thermal Heat Capacity

#physics #thermodynamics #heat capacity

$$C_p-C_V=T\left(\frac{\partial p}{\partial T}\right)_{V}\cdot\left(\frac{\partial V}{\partial T}\right)_{p}=-T\left(\frac{\partial V}{\partial T}\right)_{p}^2\left(\frac{\partial p}{\partial V}\right)_{T}\geq0 \\ \displaystyle C_X=T\left(\frac{\partial S}{\partial T}\right)_{X} \\ \displaystyle C_p=\left(\frac{\partial H}{\partial T}\right)_{p} \\ \displaystyle C_V=\left(\frac{\partial U}{\partial T}\right)_{V} \\ C_{mp}-C_{mV}=R$$

Thermodynamic Potential

#physics #thermodynamics

$$dG=-SdT+Vdp+\sum_i\mu_idn_i \\ \displaystyle\mu=\left(\frac{\partial G}{\partial n_i}\right)_{p,T,n_j} \\ V=\sum_{i=1}^c n_i\left(\frac{\partial V}{\partial n_i}\right)_{n_j,p,T}:=\sum_{i=1}^c n_i V_i \\ \begin{aligned} V_m&=&\sum_i x_i V_i\\ 0&=&\sum_i x_i dV_i\end{aligned}$$

Thermodynamics Conservation of Energy

#physics #thermodynamics

$$Q=\Delta U+W \\ d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt} Q=dU+d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt}W \\ d\hspace{-1ex}\rule[1.25ex]{2mm}{0.4pt}W=pdV \\ Q=\Delta H+W_{\rm i}+\Delta E_{\rm kin}+\Delta E_{\rm pot}$$

Thermodynamics Definitions

#physics #thermodynamics

$$f(x,y,z)=0 \\ dz=\left(\frac{\partial z}{\partial x}\right)_{y}dx+\left(\frac{\partial z}{\partial y}\right)_{x}dy \\ \left(\frac{\partial x}{\partial y}\right)_{z}\cdot\left(\frac{\partial y}{\partial z}\right)_{x}\cdot\left(\frac{\partial z}{\partial x}\right)_{y}=-1 \\ \varepsilon^m F(x,y,z)=F(\varepsilon x,\varepsilon y,\varepsilon z) \\ mF(x,y,z)=x\frac{\partial F}{\partial x}+y\frac{\partial F}{\partial y}+z\frac{\partial F}{\partial z}$$

Thermodynamics Nernst Law

#physics #thermodynamics

$$\lim_{T\rightarrow0}\left(\frac{\partial S}{\partial X}\right)_{T}=0$$

Throttle Processes

#physics #thermodynamics #throttle processes

$$\text{Isobaric Processes} \\ H_2-H_1=\int_1^2 C_pdT \\ \text{Reversible Isobaric Process} \\ H_2-H_1=Q_{\rm rev}$$

Black Body Radiation

#physics #quantum #Black Body Radiation

$$w(f)=\frac{8\pi hf^3}{c^3}\frac{1}{{\rm e}^{hf/kT}-1} \\ w(\lambda)=\frac{8\pi hc}{\lambda^5}\frac{1}{{\rm e}^{hc/\lambda kT}-1} \\ P=A\sigma T^4 \\ T\lambda_{\rm max}=k_{\rm W}$$

Harmonic Oscillator Quantum

#physics #quantum

$$H=\frac{p^2}{2m}+\frac{1}{2} m\omega^2 x^2= \frac{1}{2} \hbar\omega+\omega A^\dagger A \\ A=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x+\frac{ip}{\sqrt{2m\omega}} \\ A^\dagger=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x-\frac{ip}{\sqrt{2m\omega}} \\ HAu_E=(E-\hbar\omega)Au_E \\ u_n=\frac{1}{\sqrt{n!}}\left(\frac{A^\dagger}{\sqrt{\hbar}}\right)^nu_0 \\ u_0=\sqrt[4]{\frac{m\omega}{\pi\hbar}}\exp\left(-\frac{m\omega x^2}{2\hbar}\right) \\ E_n=( \frac{1}{2} +n)\hbar\omega$$

Mechanics Continuity Equation

#physics #mechanics #continuity equation

$$I_{V} = \upsilon A$$

Mechanics Displacement

#physics #mechanics #displacement

$${\Delta x}=x-x_{0}=v_{0}t+\frac{1}{2}at^{2}$$

Mechanics Stress

#physics #mechanics #stress

$${\text{Stress}} = \frac{F}{A}$$

Newton Second Law Force

#physics #mechanics #newton

$$F = ma$$

Operators in Quantum Physics

#physics #quantum

$$\int\psi_1^*A\psi_2d^3V=\int\psi_2(A\psi_1)^*d^3V \\ A\Psi=a\Psi \\ \Psi=\sum\limits_nc_nu_n \\ \frac{dA}{dt}=\frac{\partial A}{\partial t}+\frac{[A,H]}{i\hbar}$$

Parity

#physics #quantum

$${\cal P}\psi(x)=\psi(-x) \\ \psi(x)= \underbrace{\frac{1}{2} (\psi(x)+\psi(-x))}_{\rm even:~\hbox{$\psi^+$}}+ \underbrace{\frac{1}{2} (\psi(x)-\psi(-x))}_{\rm odd:~\hbox{$\psi^-$}} \\ \psi^+= \frac{1}{2} (1+{\cal P})\psi(x,t) \\ \psi^-= \frac{1}{2} (1-{\cal P})\psi(x,t)$$

Quantum Wave Functions

#physics #quantum #wave

$$\Phi(k,t)=\frac{1}{\sqrt{h}}\int\Psi(x,t){\rm e}^{-ikx}dx \\ \Psi(x,t)=\frac{1}{\sqrt{h}}\int\Phi(k,t){\rm e}^{ikx}dk \\ v_{\rm g}=p/m \\ E=\hbar\omega \\ \left\langle f(t) \right\rangle=\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int\Psi^* f\Psi d^3V \\ \left\langle f_p(t) \right\rangle=\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int\Phi^*f\Phi d^3V_p \\ \left\langle f(t) \right\rangle=\left\langle \Phi|f|\Phi \right\rangle \\ \left\langle \Phi|\Phi \right\rangle=\left\langle \Psi|\Psi \right\rangle=1$$

Shear Modulus

#physics #mechanics #shear modulus

$$M_{s} = \frac{F_{s} / A}{ {\Delta x} /L}=\frac{F_{s}/A}{\tan \theta}M_{s} = \frac{F_{s}/A}{\Delta x/L} = \frac{F_{s}/A}{\tan \theta}$$

Simple Harmonic Motion Acceleration

#physics #mechanics

$$a = - \omega ^2 x = - \omega ^2 r\sin (\omega t)$$

Speed of Sound Waves in a Fluid

#physics #mechanics

$$\upsilon = \sqrt {\frac{B}{\rho }}$$

The Compton Effect

#physics #quantum #Compton Effect

$$\lambda'=\lambda+\frac{h}{mc}(1-\cos\theta)=\lambda+\lambda_{\rm C}(1-\cos\theta)$$

The Schrödinger Equation

#physics #quantum

$$-\dfrac{\hbar^{2}}{2m}\bigtriangledown ^{2} \psi +U\psi=E\psi = i\hbar \dfrac{\partial \psi}{\partial t} \\ H=p^2/2m+U, H\psi=E\psi \\ \psi(x,t)=\left(\sum+\int dE\right)c(E)u_E(x)\exp\left(-\frac{iEt}{\hbar}\right) \\ \displaystyle J=\frac{\hbar}{2im}(\psi^*\nabla\psi-\psi\nabla\psi^*) \\ \displaystyle\frac{\partial P(x,t)}{\partial t}=-\nabla J(x,t)$$

The Tunnel Effect

#physics #quantum

$$\psi(x)=a^{-1/2}\sin(kx) \\ E_n=n^2h^2/8a^2m \\ \psi_1=A{\rm e}^{ikx}+B{\rm e}^{-ikx} \\ \psi_2=C{\rm e}^{ik'x}+D{\rm e}^{-ik'x} \\ \psi_3=A'{\rm e}^{ikx} \\ k'^2=2m(W-W_0)/\hbar^2 \ k^2=2mW \\ T=|A'|^2/|A|^2$$

The Uncertainty Principle

#physics #quantum

$$(\Delta A)^2=\left\langle \psi|A_{\rm op}-\left\langle A \right\rangle|^2\psi \right\rangle=\left\langle A^2 \right\rangle-\left\langle A \right\rangle^2 \\ \Delta A\cdot\Delta B\geq \frac{1}{2} |\left\langle \psi|[A,B]|\psi \right\rangle| \\ \Delta E\cdot\Delta t\geq\hbar \\ \Delta p_x\cdot\Delta x\geq \frac{1}{2} \hbar \\ \Delta L_x\cdot\Delta L_y\geq \frac{1}{2} \hbar L_z$$

Velocity

#physics #mechanics #velocity

$$\upsilon = \upsilon _0 + at$$

Viscous Flow

#physics #mechanics #viscous flow

$$F = \eta \frac{{\upsilon A}}{z}$$

Youngs Modulus

#physics #mechanics #youngs modulus

$$\Upsilon = \frac{F/A}{\Delta L/L} =\frac{\text{Stress}}{\text{Strain}}$$

Fourier Series

#math #fourier series

$$y(x)=c_{0}+\sum^{M}_{m=1}c_{m}\cos mx+\sum^{M^{'}}_{m=1}s_{m}\sin mx \\ c_{0}=\frac{1}{2\pi}\int^{\pi}_{-\pi}y(x) \mathrm{d} x \\ c_{m}=\frac{1}{\pi}\int^{\pi}_{-\pi}y(x) \cos mx \mathrm{d} x \\ s_{m}=\frac{1}{\pi}\int^{\pi}_{-\pi}y(x) \sin mx \mathrm{d} x$$

Fourier Transforms

#math #fourier transforms

$$y(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} \hat{y}(\omega) e^{i\omega t} \mathrm{d} \omega \\ \hat{y}(\omega)=\int_{-\infty}^{\infty} y(t) e^{-i\omega t} \mathrm{d} t$$

Bellman Equation

#machine learning

$$v_{\pi}(s)=\sum_{a}\pi(a|s)\sum_{s^{'},r}p(s^{'},r|s,a)[r+\gamma v_{\pi}(s^{'})]$$

Jensen-Shannon Divergence JS-Divergence

#machine learning

$$JS(P||Q)=\frac{1}{2}KL(P||\frac{(P+Q)}{2})+\frac{1}{2}KL(Q||\frac{(P+Q)}{2})$$

KL-Divergence

#machine learning

$$KL(P||Q)=\sum_{x}P(x)\log(\frac{P(x)}{Q(x)})$$

Kullback-Leibler Divergence

#machine learning #kl divergence

$$KL(P||Q)=\sum_{x}P(x)\log(\frac{P(x)}{Q(x)})$$

Support Vector Machine SVM

#machine learning #svm

$$\max_{w,b} \frac{2}{||w||} \\ s.t.\ y_{i}(w^{T}x_{i} + b) \geq 1, i=1,2,...,m \\ L(w,b,\alpha)=\frac{1}{2}||w||^2 + \sum^{m}_{i=1}a_{i}(1-y_{i}(w^{T}x_{i} + b))$$

Equation Database

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EQUATION LIST

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