equations.json

[
	{
		"family" : "algebra",
		"name" : "Great Orthogonality Theorem",
		"latex" : ["\\frac{d^\\alpha}{|G|}\\sum_{g\\in G}{\\Gamma^\\alpha(g)}_{ij}^*{\\Gamma^\\beta(g)}_{mn}=\\delta_{\\alpha\\beta}\\delta_{im}\\delta_{jn}"]
	},
	
	
	{
		"family" : "calculus",
		"name" : "Cyclical Rule",
		"latex" : ["\\pen{\\frac{\\partial x}{\\partial y}}_z\\pen{\\frac{\\partial y}{\\partial z}}_x\\pen{\\frac{\\partial z}{\\partial x}}_y=-1"]
	},
	{
		"family" : "calculus",
		"name" : "d'Alembertian",
		"latex" : ["\\Box = \\partial^\\mu\\partial_\\mu = \\frac 1{c^2}\\frac{\\partial^2}{\\partial t^2}-\\nabla^2"]
	},
	{
		"family" : "calculus",
		"name" : "Derivative",
		"latex" : ["\\frac{df}{dx}=\\lim_{h\\to 0}\\frac{f(x+h)-f(x)} h"]
	},
	{
		"family" : "calculus",
		"name" : "Divergence Theorem",
		"latex" : ["\\int_V\\nabla\\cdot\\bvec Fd^3r=\\oint_{\\partial V}\\bvec F\\cdot d\\bvec S"]
	},
	{
		"family" : "calculus",
		"name" : "Fourier Transform",
		"latex" : [
			"f(\\bvec r,t)=\\int\\frac{d^3k}{(2\\pi)^3}\\int_{-\\infty}^\\infty\\frac{d\\omega}{2\\pi}\\hat f(\\bvec k,\\omega)e^{i(\\bvec k\\cdot\\bvec r-\\omega t)}",
		"\\hat f(\\bvec{k},\\omega)=\\int d^3r\\int_{-\\infty}^\\infty dtf(\\bvec r,t)e^{-i(\\bvec k\\cdot\\bvec r-\\omega t)}"
		]
	},
	{
		"family" : "calculus",
		"name" : "Green's Identities",
		"latex" : [
			"\\int_V(\\psi\\nabla^2\\phi+\\nabla\\psi\\cdot\\nabla\\phi)d^3r=\\oint_{\\partial V}\\phi\\nabla\\psi\\cdot d\\bvec S",
			"\\int_V(\\phi\\nabla^2\\psi-\\psi\\nabla^2\\phi)d^3r=\\oint_{\\partial V}(\\phi\\nabla\\psi-\\psi\\nabla\\phi)\\cdot d\\bvec S"
		]
	},
	{
		"family" : "calculus",
		"name" : "Stokes' Theorem",
		"latex" : [
			"\\int_{\\partial\\Omega}\\omega=\\int_\\Omega\\omega",
			"\\int_S\\nabla\\times\\bvec{F}\\cdot d\\bvec S=\\oint_{\\partial S}\\bvec F\\cdot d\\bvec l"
		]
	},
	
	
	{
		"family" : "chemistry",
		"name" : "Arrhenius Equation",
		"latex" : ["k=Ae^{-\\frac{E_a}{RT}}"]
	},
	{
		"family" : "chemistry",
		"name" : "Henderson-Hasselbalch Equation",
		"latex" : ["\\text{pH}=\\text pK_a+\\log_{10}\\pen{\\frac{[A^-]}{[HA]}}"]
	},
	{
		"family" : "chemistry",
		"name" : "pH",
		"latex" : ["\\text{pH}=-\\log_{10}[\\text H_3\\text O^+]"]
	},
	
	
	{
		"family" : "classical-mechanics",
		"name" : "Action",
		"latex" : ["S=\\int_{t_1}^{t_2}dtL"]
	},
	{
		"family" : "classical-mechanics",
		"name" : "Center of Mass",
		"latex" : ["\\bvec r_{\\text{cm}}=\\frac{\\sum m_i\\bvec r_i}{\\sum m_i}"]
	},
	{
		"family" : "classical-mechanics",
		"name" : "Euler-Lagrange Equations",
		"latex" : ["\\frac{\\partial L}{\\partial q_i}-\\frac d{dt}\\frac{\\partial L}{\\partial\\dot{q}_i}=0"]
	},
	{
		"family" : "classical-mechanics",
		"name" : "Gravitational Field",
		"latex" : ["\\bvec g(\\bvec r)=-G\\int\\rho(\\bvec r')\\frac{\\bvec r-\\bvec r'}{\\abs{\\bvec r-\\bvec r}^3}d\\bvec r'"]
	},
	{
		"family" : "classical-mechanics",
		"name" : "Hamilton's Equations",
		"latex" : [
			"\\dot p_i=-\\frac{\\partial H}{\\partial q_i}",
			"\\dot q_i=\\frac{\\partial H}{\\partial p_i}"
		]
	},
	{
		"family" : "classical-mechanics",
		"name" : "Kinematic Equations",
		"latex" : [
			"x=x_0+v_0t+\\frac 1 2at^2",
			"v^2=v_0^2+2a(x-x_0)",
			"x-x_0=\\pen{\\frac{v_0+v} 2}t=v_{\\text{avg}}t"
		]
	},
	{
		"family" : "classical-mechanics",
		"name" : "Lagrangian",
		"latex" : ["L=T-U"]
	},
	{
		"family" : "classical-mechanics",
		"name" : "Liouville's Theorem",
		"latex" : ["\\frac{\\partial\\rho}{\\partial t}=-\\{\\rho,H\\}"]
	},
	{
		"family" : "classical-mechanics",
		"name" : "Moment of Inertia",
		"latex" : [
			"I=\\sum m_ir_i^2=\\int r^2dm",
			"\\text{Rod (Center): }I=\\frac 1{12}ML^2",
			"\\text{Rod (End): }I=\\frac 1{3}ML^2",
			"\\text{Ring: }I=MR^2",
			"\\text{Disk: }I=\\frac 1 2MR^2",
			"\\text{Sphere: }I=\\frac 2{3}MR^2",
			"\\text{Ball: }I=\\frac 2{5}MR^2"
		]
	},
	{
		"family" : "classical-mechanics",
		"name" : "Newton's Law of Gravitation",
		"latex" : ["\\bvec{F}_{12}=-G\\frac{m_1m_2}{\\abs{\\bvec r_{12}}^2}\\hat r_{12}"]
	},
	{
		"family" : "classical-mechanics",
		"name" : "Parallel Axis Theorem",
		"latex" : ["I=I+{\\text{cm}}+Md^2"]
	},
	{
		"family" : "classical-mechanics",
		"name" : "Poisson Bracket",
		"latex" : ["\\{f,g\\}=\\sum_i\\pen{\\frac{\\partial f}{\\partial q_i}\\frac{\\partial g}{\\partial p_i}-\\frac{\\partial f}{\\partial p_i}\\frac{\\partial g}{\\partial q_i}}"]
	},
	{
		"family" : "classical-mechanics",
		"name" : "Radial Acceleration",
		"latex" : ["a_{\\text{rad}}=\\frac{v^2} r=\\omega^2r"]
	},
	{
		"family" : "classical-mechanics",
		"name" : "Reduced Mass",
		"latex" : ["\\mu=\\frac{m_1m_2}{m_1+m_2}"]
	},
	
	
	{
		"family" : "complex-analysis",
		"name" : "Kramers-Kronig Relations",
		"latex" : [
			"\\chi_\\text{Re}(\\omega)=\\frac 1{\\pi}\\mathcal P\\int_{-\\infty}^\\infty d\\omega'\\frac{\\chi_\\text{Im}(\\omega')}{\\omega'-\\omega}",
			"\\chi_\\text{Im}(\\omega)=-\\frac 1{\\pi}\\mathcal P\\int_{-\\infty}^\\infty d\\omega'\\frac{\\chi_\\text{Re}(\\omega')}{\\omega'-\\omega}"
		]
	},
	{
		"family" : "complex-analysis",
		"name" : "Residue",
		"latex" : [
			"\\text{Res}(f,c)=\\frac 1{2\\pi i}\\oint_\\gamma f(z)dz",
			"\\text{Res}(f,c)=\\lim_{z\\to c}(z-c)f(z)\\text{ (Simple Pole)}"
		]
	},
	{
		"family" : "complex-analysis",
		"name" : "Residue Theorem",
		"latex" : ["\\oint_\\gamma f(z)dz=2\\pi i\\sum_kI(\\gamma,a_k)\\text{Res}(f,a_k)"]
	},
	
	
	{
		"family" : "constants",
		"name" : "Avogadro's Number",
		"latex" : ["N_A=\\num{6.02214e23}"]
	},
	{
		"family" : "constants",
		"name" : "Bohr Radius",
		"latex" : [
			"a_0=\\SI{5.29177211e-11}{\\meter}\\text{ (SI)}"
		]
	},
	{
		"family" : "constants",
		"name" : "Boltzmann's Constant",
		"latex" : [
			"k_B=\\SI{1.38064852e-23}{\\joule\\per\\kelvin}\\text{ (SI)}"
		]
	},
	{
		"family" : "constants",
		"name" : "Coulomb's Constant",
		"latex" : [
			"k=\\SI{8.98755179e9}{\\newton\\meter\\squared\\per\\coulomb\\squared}\\text{ (SI)}"
		]
	},
	{
		"family" : "constants",
		"name" : "Electron Mass",
		"latex" : [
			"m_e=\\SI{9.10938356e-31}{\\kilogram}\\text{ (SI)}"
		]
	},
	{
		"family" : "constants",
		"name" : "Elementary Charge",
		"latex" : [
			"e=\\SI{1.60217662e-19}{\\coulomb}\\text{ (SI)}"
		]
	},
	{
		"family" : "constants",
		"name" : "Fine Structure Constant",
		"latex" : [
			"\\alpha=\\num{\\frac 1{137}}"
		]
	},
	{
		"family" : "constants",
		"name" : "Gravitational Constant",
		"latex" : [
			"G=\\SI{6.67408e-11}{\\meter\\cubed\\per\\kilogram\\second\\squared}\\text{ (SI)}"
		]
	},
	{
		"family" : "constants",
		"name" : "Planck's Constant",
		"latex" : [
			"h=\\SI{6.62607004e-34}{\\joule\\second}\\text{ (SI)}",
			"\\hbar=\\SI{1.05457180e-34}{\\joule\\second\\per\\radian}\\text{ (SI)}"
		]
	},
	{
		"family" : "constants",
		"name" : "Speed of Light",
		"latex" : [
			"c=\\SI{2.99792458e8}{\\meter\\per\\second}\\text{ (SI)}"
		]
	},
	{
		"family" : "constants",
		"name" : "Vacuum Permeability",
		"latex" : [
			"\\mu_0=\\SI{4\\pi e-7}{\\henry\\per\\meter}\\text{ (SI)}"
		]
	},
	{
		"family" : "constants",
		"name" : "Vacuum Permittivity",
		"latex" : [
			"\\epsilon_0=\\SI{8.85418782}{\\farad\\per\\meter}\\text{ (SI)}"
		]
	},
	
	
	{
		"family" : "coordinates",
		"name" : "Cylindrical Coordinates",
		"latex" : [
			"x=r\\cos\\phi",
			"y=r\\sin\\phi",
			"z=z",
			"dxdydz=rdrd\\phi dz",
			"\\nabla f=\\frac{\\partial f}{\\partial r}\\hat r+\\frac 1 r\\frac{\\partial f}{\\partial\\phi}\\hat{\\phi}+\\frac{\\partial f}{\\partial z}\\hat{z}",
			"\\nabla^2f=\\frac 1 r\\frac{\\partial f}{\\partial r}\\pen{r\\frac{\\partial f}{\\partial r}}+\\frac 1{r^2}\\frac{\\partial^2f}{\\partial\\phi^2}+\\frac{\\partial^2f}{\\partial z^2}",
			"\\nabla\\cdot\\bvec a=\\frac 1 r\\frac{\\partial}{\\partial r}\\pen{rA_r}+\\frac 1 r\\frac{\\partial A_\\phi}{\\partial\\phi}+\\frac{\\partial A_z}{\\partial z}",
			"\\nabla\\times\\bvec a=\\pen{\\frac 1 r\\frac{\\partial A_z}{\\partial\\phi}-\\frac{\\partial A_\\phi}{\\partial z}}\\hat r+\\pen{\\frac{\\partial A_r}{\\partial z}-\\frac{\\partial A_z}{\\partial r}}\\hat{\\phi}+\\frac 1 r\\pen{\\frac{\\partial}{\\partial r}\\pen{rA_\\phi}-\\frac{\\partial A_r}{\\partial\\phi}}\\hat{z}"
		]
	},
	{
		"family" : "coordinates",
		"name" : "Polar Coordinates",
		"latex" : [
			"x=r\\cos\\phi",
			"y=r\\sin\\phi",
			"dxdy=rdrd\\phi",
			"\\nabla f=\\frac{\\partial f}{\\partial r}\\hat r+\\frac 1 r\\frac{\\partial f}{\\partial\\phi}\\hat{\\phi}",
			"\\nabla^2f=\\frac 1 r\\frac{\\partial f}{\\partial r}\\pen{r\\frac{\\partial f}{\\partial r}}+\\frac 1{r^2}\\frac{\\partial^2f}{\\partial\\phi^2}"
		]
	},
	{
		"family" : "coordinates",
		"name" : "Spherical Coordinates",
		"latex" : [
			"x=r\\sin\\theta\\cos\\phi",
			"y=r\\sin\\theta\\sin\\phi",
			"z=r\\cos\\theta",
			"dxdydz=r^2\\sin\\theta drd\\theta d\\phi",
			"\\nabla f=\\frac{\\partial f}{\\partial r}\\hat r+\\frac 1 r\\frac{\\partial f}{\\partial\\theta}\\hat{\\theta}+\\frac 1{r\\sin\\theta}\\frac{\\partial f}{\\partial\\phi}\\hat{\\phi}",
			"\\nabla^2f=\\frac 1{r^2}\\frac{\\partial}{\\partial r}\\pen{r^2\\frac{\\partial f}{\\partial r}}+\\frac 1{r^2\\sin\\theta}\\frac{\\partial}{\\partial\\theta}\\pen{\\sin\\theta\\frac{\\partial f}{\\partial\\theta}}+\\frac 1{r^2\\sin^2\\theta}{\\frac{\\partial^2f}{\\partial\\phi^2}}",
			"\\nabla\\cdot\\bvec a=\\frac 1{r^2}\\frac{\\partial}{\\partial r}\\pen{r^2A_r}+\\frac 1{r\\sin\\theta}\\frac{\\partial}{\\partial\\theta}\\pen{\\sin\\theta A_\\theta}+\\frac 1{r\\sin\\theta}\\frac{\\partial A_\\phi}{\\partial\\phi}",
			"\\nabla\\times\\bvec a=\\frac 1{r\\sin\\theta}\\pen{\\frac{\\partial}{\\partial\\theta}\\pen{A_\\phi\\sin\\theta}-\\frac{\\partial A_\\theta}{\\partial\\phi}}\\hat r+\\frac 1 r\\pen{\\frac 1{\\sin\\theta}\\frac{\\partial A_r}{\\partial\\phi}-\\frac{\\partial}{\\partial r}\\pen{rA_\\phi}}\\hat{\\theta}+\\frac 1 r\\pen{\\frac{\\partial}{\\partial r}\\pen{rA_\\theta}-\\frac{\\partial A_r}{\\partial\\theta}}\\hat{\\phi}"
		]
	},
	
	
	{
		"family" : "differential-equations",
		"name" : "Diffusion Equation",
		"latex" : ["\\frac{\\partial u}{\\partial t}-\\nabla\\cdot\\benclose{D(u,\\bvec r)\\nabla u(\\bvec r,t)}=0"]
	},
	{
		"family" : "differential-equations",
		"name" : "Heat Equation",
		"latex" : ["\\frac{\\partial u}{\\partial t}-D\\nabla^2u=0"]
	},
	{
		"family" : "differential-equations",
		"name" : "Wave Equation",
		"latex" : ["\\nabla^2u=\\frac 1{v^2}\\frac{\\partial^2u}{\\partial t^2}"]
	},
	
	
	{
		"family" : "differential-geometry",
		"name" : "Bianchi Identity",
		"latex" : ["\\nabla_{[\\lambda}R_{\\rho\\sigma]\\mu\\nu}=0"]
	},
	{
		"family" : "differential-geometry",
		"name" : "Christoffel Symbol",
		"latex" : ["\\Gamma_{\\mu\\nu}^\\lambda=\\frac 1 2g^{\\lambda\\sigma}(\\partial_\\mu g_{\\nu\\sigma}+\\partial_\\nu g_{\\sigma\\mu}-\\partial_\\sigma g_{\\mu\\nu})"]
	},
	{
		"family" : "differential-geometry",
		"name" : "Covariant Derivative",
		"latex" : [
			"\\nabla_\\mu V^\\nu=\\partial_\\mu V^\\nu+\\Gamma_{\\mu\\lambda}^\\nu V^\\lambda",
			"\\nabla_\\mu \\omega_\\nu=\\partial_\\mu \\omega_\\nu-\\Gamma_{\\mu\\nu}^\\lambda \\omega_\\lambda"
		]
	},
	{
		"family" : "differential-geometry",
		"name" : "Directional Covariant Derivative",
		"latex" : ["\\frac{D}{d\\lambda}=\\frac{dx^\\mu}{d\\lambda}\\nabla_\\mu"]
	},
	{
		"family" : "differential-geometry",
		"name" : "Einstein Tensor",
		"latex" : ["G_{\\mu\\nu}=R_{\\mu\\nu}-\\tfrac 1 2Rg_{\\mu\\nu}"]
	},
	{
		"family" : "differential-geometry",
		"name" : "Einstein's Equation",
		"latex" : ["R_{\\mu\\nu}-\\tfrac 1 2Rg_{\\mu\\nu}+\\Lambda g_{\\mu\\nu}=8\\pi GT_{\\mu\\nu}"]
	},
	{
		"family" : "differential-geometry",
		"name" : "Exterior Derivative",
		"latex" : ["(dA)_{\\mu_1\\dots\\mu_{p+1}} = (p+1)\\partial_{[\\mu_1}A_{\\mu_2\\dots\\mu_{p+1}]}"]
	},
	{
		"family" : "differential-geometry",
		"name" : "Geodesic Equation",
		"latex" : [
			"\\frac{d^2x^\\mu}{d\\lambda^2}+\\Gamma_{\\rho\\sigma}^\\mu\\frac{dx^\\rho}{d\\lambda}\\frac{dx^\\sigma}{d\\lambda}=0",
			"K_\\mu\\frac{dx^\\mu}{d\\lambda}=\\text{constant}"
		]
	},
	{
		"family" : "differential-geometry",
		"name" : "Hodge Star Operator",
		"latex" : ["(\\star\\omega)_{\\mu_1\\dots\\mu_{n-p}}=\\frac 1{p!}{\\epsilon^{\\nu_1\\dots\\nu_p}}_{\\mu_1\\dots\\mu_{n-p}}\\omega_{\\nu_1\\dots\\nu_p}"]
	},
	{
		"family" : "differential-geometry",
		"name" : "Killing's Equation",
		"latex" : ["\\nabla_{(\\mu}K_{\\nu)}=0"]
	},
	{
		"family" : "differential-geometry",
		"name" : "Parallel Transport",
		"latex" : [
			"\\frac{D}{d\\lambda}T=0",
			"\\frac{d}{d\\lambda}V^\\mu+\\Gamma_{\\sigma\\rho}^\\mu\\frac{dx^\\sigma}{d\\lambda}V^\\rho=0"
		]
	},
	{
		"family" : "differential-geometry",
		"name" : "Ricci Scalar",
		"latex" : ["R=g^{\\mu\\nu}R_{\\mu\\nu}"]
	},
	{
		"family" : "differential-geometry",
		"name" : "Ricci Tensor",
		"latex" : ["R_{\\mu\\nu}={R^\\lambda}_{\\mu\\lambda\\nu}"]
	},
	{
		"family" : "differential-geometry",
		"name" : "Riemann Tensor",
		"latex" : ["{R^\\rho}_{\\sigma\\mu\\nu}=\\partial_\\mu\\Gamma_{\\nu\\sigma}^\\rho-\\partial_\\nu\\Gamma_{\\mu\\sigma}^\\rho+\\Gamma_{\\mu\\lambda}^\\rho\\Gamma_{\\nu\\sigma}^\\lambda-\\Gamma_{\\nu\\lambda}^\\rho\\Gamma_{\\mu\\sigma}^\\lambda"]
	},
	{
		"family" : "differential-geometry",
		"name" : "Vector Transformation Laws",
		"latex" : [
			"V^{\\mu'} = \\frac{\\partial x^{\\mu'}}{\\partial x^\\mu}V^\\mu",
			"V_{\\mu'} = \\frac{\\partial x^\\mu}{\\partial x^{\\mu'}}V_\\mu"
		]
	},
	{
		"family" : "differential-geometry",
		"name" : "Weyl Tensor",
		"latex" : ["C_{\\rho\\sigma\\mu\\nu}=R_{\\rho\\sigma\\mu\\nu}-\\frac 2{n-2}(g_{\\rho[\\mu}R_{\\nu]\\sigma}-g_{\\sigma[\\mu}R_{\\nu]\\rho})+\\frac 2{(n-1)(n-2)}g_{\\rho[\\mu}g_{\\nu]\\sigma}R"]
	},
	
	
	{
		"family" : "electromagnetism",
		"name" : "Ampere's Law",
		"latex" : [
			"\\nabla\\times\\bvec B=\\mu_0\\bvec j",
			"\\oint_C\\bvec B\\cdot d\\bvec l=\\mu_0I_{\\text{en}}"
		]
	},
	{
		"family" : "electromagnetism",
		"name" : "Biot-Savart Law",
		"latex" : [
			"\\bvec B(\\bvec r)=\\frac{\\mu_0}{4\\pi}\\int_C\\frac{Id\\bvec l\\times(\\bvec r-\\bvec l)}{\\abs{\\bvec r-\\bvec l}^3}",
			"\\bvec B(\\bvec r)=\\frac{\\mu_0}{4\\pi}\\int\\frac{\\bvec j\\times(\\bvec r-\\bvec l)}{\\abs{\\bvec r-\\bvec l}^3}dV"
		]
	},
	{
		"family" : "electromagnetism",
		"name" : "Claussius-Mossotti Formula",
		"latex" : ["\\frac{n\\alpha}3=\\frac{\\epsilon_r-1}{\\epsilon_r+2}"]
	},
	{
		"family" : "electromagnetism",
		"name" : "Coulomb's Law",
		"latex" : ["\\bvec{F}_{12}=\\frac 1{4\\pi\\epsilon_0}\\frac{q_1q_2}{\\abs{\\bvec r_{12}}^2}\\hat r_{12}"]
	},
	{
		"family" : "electromagnetism",
		"name" : "Dielectric Constant",
		"latex" : ["\\epsilon_r=\\frac\\epsilon{\\epsilon_0}"]
	},
	{
		"family" : "electromagnetism",
		"name" : "Electric Field",
		"latex" : ["\\bvec E(\\bvec r)=\\frac 1{4\\pi\\epsilon_0}\\int\\rho(\\bvec r')\\frac{\\bvec r-\\bvec r'}{\\abs{\\bvec r-\\bvec r}^3}d\\bvec r'"]
	},
	{
		"family" : "electromagnetism",
		"name" : "Electromagnetic Tensor",
		"latex" : ["F_{\\mu\\nu} = \\begin{pmatrix}0 & E_x & E_y & E_z \\\\ -E_x & 0 & -B_z & B_y \\\\ -E_y & B_z & 0 & -B_x \\\\ -E_z & -B_y & B_x & 0\\end{pmatrix}"]
	},
	{
		"family" : "electromagnetism",
		"name" : "Faraday's Law",
		"latex" : ["\\mathcal E=-\\frac{d\\Phi_B}{dt}"]
	},
	{
		"family" : "electromagnetism",
		"name" : "Gauss's Law",
		"latex" : [
			"\\nabla\\cdot\\bvec E=\\frac\\rho{\\epsilon_0}",
			"\\oint_S\\bvec E\\cdot d\\bvec S=\\frac{Q_{\\text{en}}}{\\epsilon_0}"
		]
	},
	{
		"family" : "electromagnetism",
		"name" : "Laplace's Equation",
		"latex" : ["\\nabla^2\\phi=0"]
	},
	{
		"family" : "electromagnetism",
		"name" : "Lorentz Force",
		"latex" : ["\\bvec F=q(\\bvec E+\\bvec v\\times\\bvec B)"]
	},
	{
		"family" : "electromagnetism",
		"name" : "Magnetic Force",
		"latex" : ["\\bvec F=q\\bvec v\\times\\bvec B=I\\bvec l\\times\\bvec B"]
	},
	{
		"family" : "electromagnetism",
		"name" : "Maxwell's Equations",
		"latex" : [
			"\\nabla\\cdot\\bvec E=\\frac{\\rho}{\\epsilon_0}\\Leftrightarrow\\oint_S\\bvec E\\cdot d\\bvec S=\\frac{Q_{\\text{en}}}{\\epsilon_o}",
			"\\nabla\\cdot\\bvec B=0\\Leftrightarrow\\oint_S\\bvec B\\cdot d\\bvec S=0",
			"\\nabla\\times\\bvec E=-\\frac{\\partial\\bvec B}{\\partial t}\\Leftrightarrow\\oint_C\\bvec E\\cdot d\\bvec l=-\\frac{d\\Phi_B}{dt}",
			"\\nabla\\times\\bvec B=\\mu_0\\bvec j+\\frac 1{c^2}\\frac{\\partial\\bvec E}{\\partial t}\\Leftrightarrow\\oint_C\\bvec B\\cdot d\\bvec l=\\mu_0\\int_S\\bvec j\\cdot d\\bvec{S}+\\frac 1{c^2}\\frac{d\\Phi_E}{dt}"
		]
	},
	{
		"family" : "electromagnetism",
		"name" : "Motional EMF",
		"latex" : ["\\mathcal E=\\oint(\\bvec v\\times\\bvec B)\\cdot d\\bvec l"]
	},
	{
		"family" : "electromagnetism",
		"name" : "Parallel Capacitors",
		"latex" : [
			"C_\\text{eq}=\\sum_iC_i",
			"\\Delta V=\\text{constant}"
		]
	},
	{
		"family" : "electromagnetism",
		"name" : "Parallel Resistors",
		"latex" : [
			"R_\\text{eq}=\\pen{\\sum_i\\frac 1{R_i}}^{-1}",
			"\\Delta V=\\text{constant}"
		]
	},
	{
		"family" : "electromagnetism",
		"name" : "Poisson's Equation",
		"latex" : ["\\nabla^2\\phi=-\\frac\\rho{\\epsilon_0}"]
	},
	{
		"family" : "electromagnetism",
		"name" : "Poynting Vector",
		"latex" : [
			"\\bvec{S}=\\frac 1{\\mu_0}\\bvec E\\times\\bvec B",
			"\\bvec{S}=\\bvec E\\times\\bvec h^*"
		]
	},
	{
		"family" : "electromagnetism",
		"name" : "Series Capacitors",
		"latex" : [
			"C_\\text{eq}=\\pen{\\sum_i\\frac 1{C_i}}^{-1}",
			"Q=\\text{constant}"
		]
	},
	{
		"family" : "electromagnetism",
		"name" : "Series Resistors",
		"latex" : [
			"R_\\text{eq}=\\sum_iR_i",
			"I=\\text{constant}"
		]
	},
	{
		"family" : "electromagnetism",
		"name" : "Skin Depth",
		"latex" : ["\\delta(\\omega)=\\sqrt{\\frac 2{\\mu\\sigma\\omega}}"]
	},
	
	
	{
		"family" : "fluid-mechanics",
		"name" : "Bernoulli's Equation",
		"latex" : ["p_1+\\rho gy_1+\\frac 1 2\\rho v_1^2=p_2+\\rho gy_2+\\frac 1 2\\rho v_2^2"]
	},
	{
		"family" : "fluid-mechanics",
		"name" : "Continuity Equation",
		"latex" : ["\\frac{\\partial\\rho}{\\partial t}+\\nabla\\cdot\\bvec j=\\sigma"]
	},
	
	
	{
		"family" : "general-relativity",
		"name" : "Schwarzchild Metric",
		"latex" : ["ds^2=-\\pen{1-\\frac{2GM} r}dt^2+\\pen{1-\\frac{2GM} r}^{-1}dr^2+r^2d\\Omega^2"]
	},
	
	
	{
		"family" : "geometry",
		"name" : "Solid Angle",
		"latex" : [
			"d\\Omega=\\sin\\theta d\\theta d\\phi",
			"d\\Omega^2=d\\theta^2+\\sin^2\\theta d\\phi^2"
		]
	},
	
	
	{
		"family" : "information-theory",
		"name" : "Shannon Entropy",
		"latex" : ["H=-\\sum_ip_i\\log p_i"]
	},
	
	
	{
		"family" : "linear-algebra",
		"name" : "Cauchy-Schwarz Inequality",
		"latex" : ["|\\langle\\bvec u,\\bvec v\\rangle|\\leq\\|\\bvec u\\|\\|\\bvec v\\|"]
	},
	
	
	{
		"family" : "optics",
		"name" : "Malus's Law",
		"latex" : ["I=I_{\\text{max}}\\cos^2\\phi"]
	},
	{
		"family" : "optics",
		"name" : "Snell's Law",
		"latex" : ["n_1\\sin\\theta_1=n_2\\sin\\theta2"]
	},
	
	
	{
		"family" : "precalculus",
		"name" : "Change of Base",
		"latex" : ["\\log_bx=\\frac{\\log_ax}{\\log_ab}"]
	},
	{
		"family" : "precalculus",
		"name" : "Half-Life",
		"latex" : ["t_{1/2}=\\tau\\ln 2"]
	},
	{
		"family" : "precalculus",
		"name" : "Quadratic Formula",
		"latex" : [
			"ax^2+bx+c=0",
			"x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}"
		]
	},
	
	
	{
		"family" : "probability-theory",
		"name" : "Bayes' Theorem",
		"latex" : ["P(A|B)=\\frac{P(B|A)P(A)}{P(B)}"]
	},
	{
		"family" : "probability-theory",
		"name" : "Conditional Probability",
		"latex" : ["P(A|B)=\\frac{P(A\\cap B)}{P(B)}"]
	},
	{
		"family" : "probability-theory",
		"name" : "Variance",
		"latex" : ["\\text{Var}(X)=\\text E[X^2]-\\text E[X]^2"]
	},
	
	
	{
		"family" : "quantum-mechanics",
		"name" : "Baker-Campbell-Hausdorff Formula",
		"latex" : ["e^{A+B}\\approx e^Ae^Be^{-\\frac 1 2\\ben{A,B}}"]
	},
	{
		"family" : "quantum-mechanics",
		"name" : "de Broglie Wavelength",
		"latex" : ["\\lambda=\\frac h p"]
	},
	{
		"family" : "quantum-mechanics",
		"name" : "Dissipator",
		"latex" : ["D(\\rho)=\\sum_{k=1}^{N^2-1}\\gamma_k\\pen{A_i\\rho A_k^\\dagger-\\frac 1 2A_k^\\dagger A_k\\rho-\\frac 1 2\\rho A_k^\\dagger A_k}"]
	},
	{
		"family" : "quantum-mechanics",
		"name" : "Ehrenfest's Theorem",
		"latex" : ["\\frac{d\\langle A\\rangle}{dt}=\\frac 1{i\\hbar}\\langle [A,H]\\rangle+\\left\\langle\\frac{\\partial A}{\\partial t}\\right\\rangle"]
	},
	{
		"family" : "quantum-mechanics",
		"name" : "Klein-Gordon Equation",
		"latex" : [
			"\\frac 1{c^2}\\ddot\\psi-\\nabla^2\\psi+\\frac{m^2c^2}{\\hbar^2}\\psi=0",
			"\\partial_\\mu\\partial^\\mu\\psi-m^2\\psi=0\\text{ (Natural Units)}"
		]
	},
	{
		"family" : "quantum-mechanics",
		"name" : "Kraus Operators",
		"latex" : ["K_{\\mu\\nu}(t)=\\sqrt{\\lambda_\\nu}\\langle\\mu|U(t)|\\nu\\rangle"]
	},
	{
		"family" : "quantum-mechanics",
		"name" : "Lindblad Equation",
		"latex" : [
			"\\frac{d\\rho}{dt}=-\\frac i\\hbar[H(t),\\rho(t)]+\\sum_{i,j=1}^{N^2-1}a_{ij}\\pen{F_i\\rho F_j^\\dagger-\\frac 1 2F_j^\\dagger F_i\\rho-\\frac 1 2\\rho F_j^\\dagger F_i}",
			"\\frac{d\\rho}{dt}=-\\frac i\\hbar[H(t),\\rho(t)]+\\sum_{k=1}^{N^2-1}\\gamma_k\\pen{A_k\\rho A_k^\\dagger-\\frac 1 2A_k^\\dagger A_k\\rho-\\frac 1 2\\rho A_k^\\dagger A_k}"
		]
	},
	{
		"family" : "quantum-mechanics",
		"name" : "Liouville-von Neumann Equation",
		"latex" : ["\\frac{d\\rho}{dt}=-\\frac i\\hbar[H(t),\\rho(t)]"]
	},
	{
		"family" : "quantum-mechanics",
		"name" : "Pauli Matrices",
		"latex" : [
			"\\sigma_x=\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix}",
			"\\sigma_y=\\begin{pmatrix}0&-i\\\\i&0\\end{pmatrix}",
			"\\sigma_z=\\begin{pmatrix}1&0\\\\0&-1\\end{pmatrix}"
		]
	},
	{
		"family" : "quantum-mechanics",
		"name" : "Photon Energy",
		"latex" : ["E=h\\nu=\\frac{hc}\\lambda"]
	},
	{
		"family" : "quantum-mechanics",
		"name" : "Photon Momentum",
		"latex" : ["p=\\frac E c=\\frac h\\lambda"]
	},
	{
		"family" : "quantum-mechanics",
		"name" : "Schrodinger Equation",
		"latex" : [
			"i\\hbar\\frac{\\partial\\Psi}{\\partial t}=H\\Psi(\\bvec r,t)",
			"\\pen{-\\frac{\\hbar^2}{2m}\\nabla^2+V(\\bvec r)}\\psi(\\bvec r)=E\\psi(\\bvec r)"
		]
	},
	
	
	{
		"family" : "series",
		"name" : "Cosine Series",
		"latex" : ["\\cos x=\\sum_{k=0}^\\infty\\frac{(-1)^k}{(2k)!}x^{2k}"]
	},
	{
		"family" : "series",
		"name" : "Exponential Series",
		"latex" : ["e^x=\\sum_{k=0}^\\infty\\frac{x^k}{k!}"]
	},
	{
		"family" : "series",
		"name" : "Geometric Series",
		"latex" : [
			"\\sum_{k=0}^nr^k=\\frac{1-r^{n+1}}{1-r}",
			"\\sum_{k=0}^\\infty r^k=\\frac 1{1-r}"
		]
	},
	{
		"family" : "series",
		"name" : "Laurent Series",
		"latex" : [
			"f(z)=\\sum_{k=-\\infty}^\\infty a_k(z-z_0)^k",
			"a_k=\\frac 1{2\\pi i}\\oint_\\gamma\\frac{f(z)}{(z-z_0)^{k+1}}dz"
		]
	},
	{
		"family" : "series",
		"name" : "Maclaurin Series",
		"latex" : ["f(x)=\\sum_{k=0}^\\infty\\frac{f^{(k)}(0)}{k!}x^k"]
	},
	{
		"family" : "series",
		"name" : "Sine Series",
		"latex" : ["\\sin x=\\sum_{k=0}^\\infty\\frac{(-1)^k}{(2k+1)!}x^{2k+1}"]
	},
	{
		"family" : "series",
		"name" : "Taylor Series",
		"latex" : ["f(x)=\\sum_{k=0}^\\infty\\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k"]
	},
	
	
	{
		"family" : "special-relativity",
		"name" : "Length Contraction",
		"latex" : ["l=\\frac 1\\gamma l_0=\\sqrt{1-\\frac{v^2}{c^2}}l_0"]
	},
	{
		"family" : "special-relativity",
		"name" : "Lorentz Factor",
		"latex" : ["\\gamma = \\frac 1{\\sqrt{1-v^2/c^2}}"]
	},
	{
		"family" : "special-relativity",
		"name" : "Relativistic Energy",
		"latex" : ["E=\\gamma mc^2=\\sqrt{\\pen{mc^2}^2+\\pen{pc}^2}"]
	},
	{
		"family" : "special-relativity",
		"name" : "Time Dilation",
		"latex" : ["\\Delta t=\\gamma \\Delta t_0=\\frac 1{\\sqrt{1-\\frac{v^2}{c^2}}}\\Delta t_0"]
	},
	
	
	{
		"family" : "statistical-mechanics",
		"name" : "Partition Function",
		"latex" : ["Z=\\sum_ie^{-\\beta E_i}=\\frac 1{h^3}\\int d^3qd^3pe^{-\\beta H(q,p)}"]
	},
	
	
	{
		"family" : "statistics",
		"name" : "Normal Distribution",
		"latex" : ["f(x)=\\frac 1{\\sigma\\sqrt{2\\pi}}e^{-\\frac{(x-\\overline x)^2}{2\\sigma^2}}"]
	},
	{
		"family" : "statistics",
		"name" : "Standard Deviation",
		"latex" : ["\\Delta X=\\sqrt{\\langle X^2\\rangle-\\langle X\\rangle^2}"]
	},
	
	
	{
		"family" : "thermodynamics",
		"name" : "Enthalpy",
		"latex" : [
			"H=U+PV",
			"dH=TdS+VdP"
		]
	},
	{
		"family" : "thermodynamics",
		"name" : "Entropy",
		"latex" : [
			"S=k\\ln\\Omega",
			"dS=\\frac{\\delta q}T"
		]
	},
	{
		"family" : "thermodynamics",
		"name" : "Gibbs Free Energy",
		"latex" : [
			"G=U-TS+PV",
			"dG=-SdT+VdP"
		]
	},
	{
		"family" : "statistical-mechanics",
		"name" : "Grand Potential",
		"latex" : [
			"\\Phi_G=U-TS-\\mu N",
			"d\\Phi_G=-PdV-SdT-Nd\\mu"
		]
	},
	{
		"family" : "thermodynamics",
		"name" : "Helmholtz Free Energy",
		"latex" : [
			"F=U-TS=-kT\\ln(Z)",
			"dF=-SdT-PdV"
		]
	},
	{
		"family" : "thermodynamics",
		"name" : "Ideal Gas Law",
		"latex" : ["PV=nRT=NkT"]
	},
	{
		"family" : "thermodynamics",
		"name" : "Maxwell Relations",
		"latex" : [
			"\\pen{\\frac{\\partial T}{\\partial V}}_S=-\\pen{\\frac{\\partial P}{\\partial S}}_V",
			"\\pen{\\frac{\\partial T}{\\partial P}}_S=\\pen{\\frac{\\partial V}{\\partial S}}_P",
			"\\pen{\\frac{\\partial S}{\\partial V}}_T=\\pen{\\frac{\\partial P}{\\partial T}}_V",
			"\\pen{\\frac{\\partial S}{\\partial P}}_T=-\\pen{\\frac{\\partial V}{\\partial T}}_P"
		]
	},
	{
		"family" : "thermodynamics",
		"name" : "Thermodynamic Potential Energy",
		"latex" : [
			"U=TS-PV+\\sum_i\\mu_iN_i",
			"dU=TdS-pdV+\\sum_i\\mu_idN_i"
		]
	},
	
	
	{
		"family" : "trigonometry",
		"name" : "Double Angle",
		"latex" : [
			"\\sin(2\\theta)=2\\cos\\theta\\sin\\theta",
			"\\cos(2\\theta)=\\cos^2\\theta-\\sin^2\\theta",
			"\\tan(2\\theta)=\\frac{2\\tan\\theta}{1-\\tan^2\\theta}"
		]
	},
	{
		"family" : "trigonometry",
		"name" : "Euler's Equation",
		"latex" : ["e^{i\\theta}=\\cos\\theta+i\\sin\\theta"]
	},
	{
		"family" : "trigonometry",
		"name" : "Euler's Identity",
		"latex" : ["e^{i\\pi}+1=0"]
	},
	{
		"family" : "trigonometry",
		"name" : "Half Angle",
		"latex" : [
			"\\sin^2\\theta=\\frac{1-\\cos(2\\theta)} 2",
			"\\cos^2\\theta=\\frac{1+\\cos(2\\theta)} 2",
			"\\tan^2\\theta=\\frac{1-\\cos(2\\theta)}{1+\\cos(2\\theta)}"
		]
	},
	{
		"family" : "trigonometry",
		"name" : "Hyperbolic Functions",
		"latex" : [
			"\\sinh(x)=\\frac{e^x-e^{-x}} 2",
			"\\cosh(x)=\\frac{e^x+e^{-x}} 2",
			"\\sinh(x)=\\frac{e^x-e^{-x}}{e^x+e^{-x}}"
		]
	},
	{
		"family" : "trigonometry",
		"name" : "Hyperbolic Double Angle",
		"latex" : [
			"\\sinh(2x)=2\\cosh x\\sinh x",
			"\\cosh(2x)=\\cosh^2x+\\sinh^2x"
		]
	},
	{
		"family" : "trigonometry",
		"name" : "Hyperbolic Pythagorean Identity",
		"latex" : [
			"\\cosh^2(x)-\\sinh^2(x)=1",
			"\\sech^2(x)+\\tanh^2(x)=1",
			"\\coth^2(x)-\\csch^2(x)=1"
		]
	},
	{
		"family" : "trigonometry",
		"name" : "Law of Cosines",
		"latex" : ["c^2=a^2+b^2-2ab\\cos C"]
	},
	{
		"family" : "trigonometry",
		"name" : "Law of Sines",
		"latex" : ["\\frac{\\sin A} a=\\frac{\\sin B} b=\\frac{\\sin C}c"]
	},
	{
		"family" : "trigonometry",
		"name" : "Power Reduction",
		"latex" : [
			"\\sin^2\\theta=\\frac{1-\\cos(2\\theta)} 2",
			"\\cos^2\\theta=\\frac{1+\\cos(2\\theta)} 2",
			"\\tan^2\\theta=\\frac{1-\\cos(2\\theta)}{1+\\cos(2\\theta)}"
		]
	},
	{
		"family" : "trigonometry",
		"name" : "Product-to-Sum",
		"latex" : [
			"2\\cos a\\cos b=\\cos(a-b)+\\cos(a+b)",
			"2\\sin a\\sin b=\\cos(a-b)-\\cos(a+b)",
			"2\\sin a\\cos b=\\sin(a+b)+\\sin(a-b)"
		]
	},
	{
		"family" : "trigonometry",
		"name" : "Pythagorean Identity",
		"latex" : [
			"\\sin^2\\theta+\\cos^2\\theta=1",
			"\\sec^2\\theta-\\tan^2\\theta=1",
			"\\csc^2\\theta-\\cot^2\\theta=1"
		]
	},
	{
		"family" : "trigonometry",
		"name" : "Sum and Difference",
		"latex" : [
			"\\sin(a\\pm b)=\\sin a\\cos b\\pm\\cos a\\cos b",
			"\\cos(a\\pm b)=\\cos a\\cos b\\mp\\sin a\\sin b",
			"\\tan(a\\pm b)=\\frac{\\tan a\\pm\\tan b}{1\\mp\\tan a\\tan b}"
		]
	},
	{
		"family" : "trigonometry",
		"name" : "Sum-to-Product",
		"latex" : [
			"\\sin a+\\sin b=2\\sin\\pen{\\frac{a+b} 2}\\cos\\pen{\\frac{a-b} 2}",
			"\\sin a-\\sin b=2\\cos\\pen{\\frac{a+b} 2}\\sin\\pen{\\frac{a-b} 2}",
			"\\cos a+\\cos b=2\\cos\\pen{\\frac{a+b} 2}\\cos\\pen{\\frac{a-b} 2}",
			"\\cos a-\\cos b=-2\\sin\\pen{\\frac{a+b} 2}\\sin\\pen{\\frac{a-b} 2}"
		]
	},
	{
		"family" : "trigonometry",
		"name" : "Triangle Inequality",
		"latex" : ["\\|\\bvec x+\\bvec y\\|\\leq\\|\\bvec x\\|+\\|\\bvec y\\|"]
	},
	
	
	{
		"family" : "waves",
		"name" : "Doppler Effect",
		"latex" : [
			"f=\\pen{\\frac{c+v_r}{c+v_s}}f_0\\text{ (Non-Relativistic)}",
			"f=\\sqrt{\\frac{1-\\beta}{1+\\beta}}f_0\\text{ (Relativistic)}"
		]
	}
]