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TABLE OF CONTENTS
EQUATION LIST
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Coupled Conductors and Transformers
#physics #coupled #conductors #transformers
$$M_{12}=M_{21}:=M=k\sqrt{L_1L_2}=\frac{N_1\Phi_1}{I_2}=\frac{N_2\Phi_2}{I_1}\sim N_1N_2 \\ \frac{V_1}{V_2}=\frac{I_2}{I_1}=-\frac{i\omega M}{i\omega L_2+R_{\rm load}}\approx-\sqrt{\frac{L_1}{L_2}}=-\frac{N_1}{N_2} \\ \Phi_{12}=M_{12}I_2 \\ \Phi_{21}=M_{21}I_1$$
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Electric Oscillations
#physics #oscillations
$$\text{Impedance} \\ Z=R+ix \\ \text{Series connection} \\ V=IZ, Z_{\rm tot}=\sum_i Z_i~,~~L_{\rm tot}=\sum_i L_i~,~~ \frac{1}{C_{\rm tot}}=\sum_i\frac{1}{C_i}~,~~Q=\frac{Z_0}{R}~,~~ Z=R(1+iQ\delta) \\ \text{Parallel connection} \\ \frac{1}{Z_{\rm tot}}=\sum_i\frac{1}{Z_i}~,~~ \frac{1}{L_{\rm tot}}=\sum_i\frac{1}{L_i}~,~~ C_{\rm tot}=\sum_i C_i~,~~Q=\frac{R}{Z_0}~,~~ Z=\frac{R}{1+iQ\delta}$$
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Harmonic Oscillations
#physics #harmonic #oscillations
$$\Psi(t)=\hat{\Psi}(t)e^{i(\omega t \pm \phi)}=\hat{\Psi}(t)\cos (\omega t \pm \phi) \\ \sum_{i} \hat{\Psi_{i}}\cos(\alpha_{i} \pm \omega t) =\hat{\Phi}\cos (\beta \pm \omega t) \\ \tan (\beta)=\frac{\sum_{i} \hat{\Psi_{i}} \sin (\alpha_{i})}{\sum_{i} \hat{\Psi_{i}} \cos (\alpha_{i})} \\ \hat{\Phi}^{2} = \sum_{i} \hat{\Psi_{i}^{2}} + 2 \sum_{j > i} \sum_{i} \hat{\Psi_{i}} \hat{\Psi_{j}} \cos (\alpha_{i} - \alpha_{j}) \\ \int x(t) dt=\frac{x(t)}{i \omega} \\ \frac{d^{n}(x(t))}{d t^{n}}=(i \omega)^{n} x(t)$$
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Lorentz Transformation Latex
#physics #relativity #lorentz #transformation
$$(\vec{x}^{'}, t^{'}) = (\vec{x}^{'}(\vec{x},t), t^{'}(\vec{x},t)) \\ \vec{x}^{'} = \vec{x} +\frac{(\gamma-1)(\vec{x}\vec{v}) \vec{v}}{|v|^{2}} - \gamma \vec{v}t \\ t^{'} = \frac{\gamma(t-\vec{x}\vec{v})}{c^{2}} \\ \gamma = \frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}} \\ \frac{\partial^{2}}{\partial{x^{2}}} + \frac{\partial^{2}}{\partial{y^{2}}} + \frac{\partial^{2}}{\partial{z^{2}}} - \frac{1}{c^{2}} \frac{\partial^{2}}{\partial{t^{2}}} = \frac{\partial^{2}}{\partial{{x^{'}}^{2}}} + \frac{\partial^{2}}{\partial{{y^{'}}^{2}}} + \frac{\partial^{2}}{\partial{{z^{'}}^{2}}} - \frac{1}{c^{2}} \frac{\partial^{2}}{\partial{{t^{'}}^{2}}}$$
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Maxwell Equations Integral
#physics #maxwell #electricity #magnetism
$$\oiint (\vec{D}\cdot \vec{n}) \mathrm{d}^{2}A=Q_{\text{free,included}}\\ \oiint (\vec{B}\cdot \vec{n}) \mathrm{d}^{2}A=0 \\ \oint \vec{E} \mathrm{d}\vec{s}=-\frac{\mathrm{d}\Phi}{\mathrm{d}t}\\ \oint \vec{H} \mathrm{d}\vec{s}=I_{\text{free,included}}+\frac{\mathrm{d}\Psi }{\mathrm{d}t}$$
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Mechanic Oscillations
#physics #mechanic #oscillations
$$m\ddot{x}=F(t)-k\dot{x}-Cx \\ F(t)=\hat{F}\cos(\omega t) \\ -m\omega^2 x=F-Cx-ik\omega x \\ \omega_0^2=C/m \\ x=\frac{F}{m(\omega_0^2-\omega^2)+ik\omega} \\ \dot{x}=\frac{F}{i\sqrt{Cm}\delta+k} \\ \delta=\frac{\omega}{\omega_0}-\frac{\omega_0}{\omega} \\ Z=F/\dot{x} \\ Q=\frac{\sqrt{Cm}}{k}$$
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Riemannian Tensor
#physics #riemannian rensor
$$R^{\mu}_{v \alpha \beta}T^{v}=\triangledown_{\alpha}\triangledown_{\beta}T^{\mu}-\triangledown_{\beta}\triangledown_{\alpha}T^{\mu} \\ \triangledown_{j}a^{i}=\partial_{j}a^{i}+\Gamma^{i}_{jk}a^{k} \\ \triangledown_{j}a_{i}=\partial_{j}a_{i}-\Gamma^{k}_{ij}a_{k} \\ \Gamma^{i}_{jk}=\frac{\partial^{2} \bar{x}^{l}}{\partial{x^{j}}\partial{x^{k}}}\frac{\partial{x^{i}}}{\partial{\bar{x}^{l}}}$$
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Stress-energy Tensor
#physics #stress-energy tensor
$$T_{\mu v}=(\varrho c^{2}+p)u_{p}u_{v}+pg_{\mu v}+\frac{1}{c^{2}}(F^{\mu}_{\alpha}F^{\alpha v} + \frac{1}{4}g^{\mu v}F^{\alpha\beta}F_{\alpha\beta}) \\ \triangledown_{v} T_{\mu v}=0 \\ F_{\alpha\beta}=\frac{\partial{A_{\beta}}}{\partial{x^{\alpha}}} - \frac{\partial{A_{\alpha}}}{\partial{x^{\beta}}} \\ \frac{d p_{\alpha}}{d \tau}=qF_{\alpha\beta}u^{\beta}$$
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The general solution of Wave Equation
#physics #general solution #waves
$$\frac{\partial^2 u(x,t)}{\partial t^2}=\sum_{m=0}^{N}\left(b_m\frac{\partial ^m}{\partial x^m}\right)u(x,t) \\ u(x,t)=\int\limits_{-\infty}^{\infty}\left(a(k){\rm e}^{i(kx-\omega_1(k)t)}+ b(k){\rm e}^{i(kx-\omega_2(k)t)}\right)dk \\ u(x,t)=A{\rm e}^{i(kx-\omega t)} \\ \omega_j=\omega_j(k)$$
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Wave Equation
#physics #wave
$$\nabla^2u-\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2}-\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}=0 \\ v=f\lambda \\ k\lambda=2\pi \\ \omega=2\pi f \\ v_{\rm g}=\frac{d\omega}{dk}=v_{\rm ph}+k\frac{dv_{\rm ph}}{dk}= v_{\rm ph}\left(1-\frac{k}{n}\frac{dn}{dk}\right) \\ v=\sqrt{\kappa/\varrho}$$
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Bending of light Snell's law
#physics #optics #Snell's law
$$n_i\sin(\theta_i)=n_t\sin(\theta_t) \\ \frac{n_2}{n_1}=\frac{\lambda_1}{\lambda_2}=\frac{v_1}{v_2} \\ n^2=1+\frac{n_{\rm e}e^2}{\varepsilon_0m}\sum_j\frac{f_j}{\omega_{0,j}^2-\omega^2-i\delta\omega} \\ v_{\rm g}=c/(1+(n_{\rm e}e^2/2\varepsilon_0m\omega^2)) $$
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Fresnel Equations
#physics #optics #fresnel
$$r_\parallel=\frac{\tan(\theta_i-\theta_t)}{\tan(\theta_i+\theta_t)} \\ r_\perp =\frac{\sin(\theta_t-\theta_i)}{\sin(\theta_t+\theta_i)} \\ t_\parallel=\frac{2\sin(\theta_t)\cos(\theta_i)}{\sin(\theta_t+\theta_i)\cos(\theta_t-\theta_i)} \\ t_\perp =\frac{2\sin(\theta_t)\cos(\theta_i)}{\sin(\theta_t+\theta_i)}$$
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Non-linear Wave Equation
#physics #non-linear wave
$$\frac{d^2x}{dt^2}-\varepsilon\omega_0(1-\beta x^2)\frac{dx}{dt}+\omega_0^2x=0 \\ \frac{d}{dt}\left\{\frac{1}{2}\left(\frac{dx}{dt}\right)^2+\frac{1}{2} \omega_0^2x^2\right\}= \varepsilon\omega_0(1-\beta x^2)\left(\frac{dx}{dt}\right)^2 \\ \frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}-\underbrace{au\frac{\partial u}{\partial x}}_{\rm non-lin}+ \underbrace{b^2\frac{\partial ^3u}{\partial x^3}}_{\rm dispersive}=0 \\ u(x-ct)=\frac{-d}{\cosh^2(e(x-ct))}$$
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Prisms and Dispersion
#physics #optics #prisms #dispersion
$$\delta=\theta_i+\theta_{i'}+\alpha \\ n=\frac{\sin(\frac{1}{2}(\delta_{\rm min}+\alpha))}{\sin(\frac{1}{2}\alpha)} \\ D=\frac{d\delta}{d\lambda}=\frac{d\delta}{dn}\frac{dn}{d\lambda} \\ \frac{d\delta}{dn}=\frac{2\sin(\frac{1}{2}\alpha)}{\cos(\frac{1}{2}(\delta_{\rm min}+\alpha))}$$
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Reflection and Transmission
#physics #optics #reflection #transmission
$$r_\parallel\equiv\left(\frac{E_{0r}}{E_{0i}}\right)_\parallel \\ r_\perp\equiv\left(\frac{E_{0r}}{E_{0i}}\right)_\perp \\ t_\parallel\equiv\left(\frac{E_{0t}}{E_{0i}}\right)_\parallel \\ t_\perp\equiv\left(\frac{E_{0t}}{E_{0i}}\right)_\perp$$
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Thermodynamics Statistical Basis
#physics #thermodynamics
$$P=N!\prod_i\frac{g_i^{n_i}}{n_i!} \\ n_i=\frac{N}{Z}g_i\exp\left(-\frac{W_i}{kT}\right) \\ Z=\sum\limits_ig_i\exp(-W_i/kT) \\ Z=\frac{V(2\pi mkT)^{3/2}}{h^3} \\ \text{Entropy in Thermodynamic Equilibrium} \\ S=\frac{U}{T}+kN\ln\left(\frac{Z}{N}\right)+kN\approx\frac{U}{T}+k\ln\left(\frac{Z^N}{N!}\right) \\ \text{Ideal gas} \\ S=kN+kN\ln\left(\frac{V(2\pi mkT)^{3/2}}{Nh^3}\right)$$
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Waveguides and resonating cavities
#physics #waveguides
$$\begin{array}{ll} \vec{n}\cdot(\vec{D}_2-\vec{D}_1)=\sigma~~&~~\vec{n}\times(\vec{E}_2-\vec{E}_1)=0\\ \vec{n}\cdot(\vec{B}_2-\vec{B}_1)=0~~&~~\vec{n}\times(\vec{H}_2-\vec{H}_1)=\vec{K} \end{array} \\ \vec{E}(\vec{x},t)=\vec{\cal E}(x,y){e}^{i(kz-\omega t)} \\ \vec{B}(\vec{x},t)=\vec{\cal{B}}(x,y){e}^{i(kz-\omega t)}$$
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Phase Transitions
#physics #thermodynamics #phase transitions
$$\Delta S_m=S_m^\alpha - S_m^\beta=\frac{r_{\beta\alpha}}{T_0} \\ S_m=\left(\frac{\partial G_m}{\partial T}\right)_{p} \\ \frac{dp}{dT}=\frac{S_m^\alpha-S_m^\beta}{V_m^\alpha-V_m^\beta}= \frac{r_{\beta\alpha}}{(V_m^\alpha-V_m^\beta)T} \\ p=p_0{\rm e}^{-r_{\beta\alpha/RT}} \\ r_{\beta\alpha}=r_{\alpha\beta} \\ r_{\beta\alpha}=r_{\gamma\alpha}-r_{\gamma\beta} \\ \frac{dp}{dT}=\frac{S_m^\alpha-S_m^\beta}{V_m^\alpha-V_m^\beta}= \frac{r_{\beta\alpha}}{(V_m^\alpha-V_m^\beta)T} \\ p=p_0{\rm e}^{-r_{\beta\alpha/RT}}$$
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State functions Maxwell Relations
#physics #thermodynamics
$$\left(\frac{\partial T}{\partial V}\right)_{S}=-\left(\frac{\partial p}{\partial S}\right)_{V}~,~~\left(\frac{\partial T}{\partial p}\right)_{S}=\left(\frac{\partial V}{\partial S}\right)_{p}~,~~ \left(\frac{\partial p}{\partial T}\right)_{V}=\left(\frac{\partial S}{\partial V}\right)_{T}~,~~\left(\frac{\partial V}{\partial T}\right)_{p}=-\left(\frac{\partial S}{\partial p}\right)_{T} \\ TdS=C_VdT+T\left(\frac{\partial p}{\partial T}\right)_{V}dV~~\mbox{and}~~TdS=C_pdT-T\left(\frac{\partial V}{\partial T}\right)_{p}dp$$
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Thermal Heat Capacity
#physics #thermodynamics #heat capacity
$$C_p-C_V=T\left(\frac{\partial p}{\partial T}\right)_{V}\cdot\left(\frac{\partial V}{\partial T}\right)_{p}=-T\left(\frac{\partial V}{\partial T}\right)_{p}^2\left(\frac{\partial p}{\partial V}\right)_{T}\geq0 \\ \displaystyle C_X=T\left(\frac{\partial S}{\partial T}\right)_{X} \\ \displaystyle C_p=\left(\frac{\partial H}{\partial T}\right)_{p} \\ \displaystyle C_V=\left(\frac{\partial U}{\partial T}\right)_{V} \\ C_{mp}-C_{mV}=R$$
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Thermodynamic Potential
#physics #thermodynamics
$$dG=-SdT+Vdp+\sum_i\mu_idn_i \\ \displaystyle\mu=\left(\frac{\partial G}{\partial n_i}\right)_{p,T,n_j} \\ V=\sum_{i=1}^c n_i\left(\frac{\partial V}{\partial n_i}\right)_{n_j,p,T}:=\sum_{i=1}^c n_i V_i \\ \begin{aligned} V_m&=&\sum_i x_i V_i\\ 0&=&\sum_i x_i dV_i\end{aligned}$$
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Thermodynamics Definitions
#physics #thermodynamics
$$f(x,y,z)=0 \\ dz=\left(\frac{\partial z}{\partial x}\right)_{y}dx+\left(\frac{\partial z}{\partial y}\right)_{x}dy \\ \left(\frac{\partial x}{\partial y}\right)_{z}\cdot\left(\frac{\partial y}{\partial z}\right)_{x}\cdot\left(\frac{\partial z}{\partial x}\right)_{y}=-1 \\ \varepsilon^m F(x,y,z)=F(\varepsilon x,\varepsilon y,\varepsilon z) \\ mF(x,y,z)=x\frac{\partial F}{\partial x}+y\frac{\partial F}{\partial y}+z\frac{\partial F}{\partial z}$$
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Harmonic Oscillator Quantum
#physics #quantum
$$H=\frac{p^2}{2m}+\frac{1}{2} m\omega^2 x^2= \frac{1}{2} \hbar\omega+\omega A^\dagger A \\ A=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x+\frac{ip}{\sqrt{2m\omega}} \\ A^\dagger=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x-\frac{ip}{\sqrt{2m\omega}} \\ HAu_E=(E-\hbar\omega)Au_E \\ u_n=\frac{1}{\sqrt{n!}}\left(\frac{A^\dagger}{\sqrt{\hbar}}\right)^nu_0 \\ u_0=\sqrt[4]{\frac{m\omega}{\pi\hbar}}\exp\left(-\frac{m\omega x^2}{2\hbar}\right) \\ E_n=( \frac{1}{2} +n)\hbar\omega$$
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Parity
#physics #quantum
$${\cal P}\psi(x)=\psi(-x) \\ \psi(x)= \underbrace{\frac{1}{2} (\psi(x)+\psi(-x))}_{\rm even:~\hbox{$\psi^+$}}+ \underbrace{\frac{1}{2} (\psi(x)-\psi(-x))}_{\rm odd:~\hbox{$\psi^-$}} \\ \psi^+= \frac{1}{2} (1+{\cal P})\psi(x,t) \\ \psi^-= \frac{1}{2} (1-{\cal P})\psi(x,t)$$
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Quantum Wave Functions
#physics #quantum #wave
$$\Phi(k,t)=\frac{1}{\sqrt{h}}\int\Psi(x,t){\rm e}^{-ikx}dx \\ \Psi(x,t)=\frac{1}{\sqrt{h}}\int\Phi(k,t){\rm e}^{ikx}dk \\ v_{\rm g}=p/m \\ E=\hbar\omega \\ \left\langle f(t) \right\rangle=\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int\Psi^* f\Psi d^3V \\ \left\langle f_p(t) \right\rangle=\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int\Phi^*f\Phi d^3V_p \\ \left\langle f(t) \right\rangle=\left\langle \Phi|f|\Phi \right\rangle \\ \left\langle \Phi|\Phi \right\rangle=\left\langle \Psi|\Psi \right\rangle=1$$
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The Schrödinger Equation
#physics #quantum
$$-\dfrac{\hbar^{2}}{2m}\bigtriangledown ^{2} \psi +U\psi=E\psi = i\hbar \dfrac{\partial \psi}{\partial t} \\ H=p^2/2m+U, H\psi=E\psi \\ \psi(x,t)=\left(\sum+\int dE\right)c(E)u_E(x)\exp\left(-\frac{iEt}{\hbar}\right) \\ \displaystyle J=\frac{\hbar}{2im}(\psi^*\nabla\psi-\psi\nabla\psi^*) \\ \displaystyle\frac{\partial P(x,t)}{\partial t}=-\nabla J(x,t)$$
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The Uncertainty Principle
#physics #quantum
$$(\Delta A)^2=\left\langle \psi|A_{\rm op}-\left\langle A \right\rangle|^2\psi \right\rangle=\left\langle A^2 \right\rangle-\left\langle A \right\rangle^2 \\ \Delta A\cdot\Delta B\geq \frac{1}{2} |\left\langle \psi|[A,B]|\psi \right\rangle| \\ \Delta E\cdot\Delta t\geq\hbar \\ \Delta p_x\cdot\Delta x\geq \frac{1}{2} \hbar \\ \Delta L_x\cdot\Delta L_y\geq \frac{1}{2} \hbar L_z$$
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Fourier Series
#math #fourier series
$$y(x)=c_{0}+\sum^{M}_{m=1}c_{m}\cos mx+\sum^{M^{'}}_{m=1}s_{m}\sin mx \\ c_{0}=\frac{1}{2\pi}\int^{\pi}_{-\pi}y(x) \mathrm{d} x \\ c_{m}=\frac{1}{\pi}\int^{\pi}_{-\pi}y(x) \cos mx \mathrm{d} x \\ s_{m}=\frac{1}{\pi}\int^{\pi}_{-\pi}y(x) \sin mx \mathrm{d} x$$
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