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Physics Derivation Graph: 988 Expressions
expression ID	Latex	list of symbols	name	dimension check	notes	used in derivation
0000040490	a^2	a		TODO		particle in a 1D box
0000999900	b/(2 a)	b a		TODO		quadratic equation derivation
0001030901	\cos(x)	x		TODO		Euler equation: trigonometric relations
0001111111	(\sin(x))^2	x		TODO		Euler equation: trig square root
0001209482	2 \pi	\pi		TODO		derivation of Schrodinger Equation
0001304952	\hbar	\hbar		TODO		derivation of Schrodinger Equation
0001334112	W	W		TODO		particle in a 1D box
0001921933	2 i	i		TODO		Euler equation: trigonometric relations
0002239424	2			TODO		derivation of Schrodinger Equation
0002338514	\vec{p}_{2}	\vec{p}_2		TODO		Compton's equation for scattering
0002342425	m/m	m		TODO		derivation of Schrodinger Equation
0002393922	x	x		TODO		Euler equation: trigonometric relations
0002424922	a	a		TODO		quadratic equation derivation
0002436656	i \hbar	\hbar i		TODO		derivation of Schrodinger Equation
0002449291	b/(2 a)	b a		TODO		quadratic equation derivation
0002838490	b/(2 a)	b a		TODO		quadratic equation derivation
0002919191	\sin(-x)	x		TODO		Euler equation: trigonometric relations
0002929944	1/2			TODO		particle in a 1D box
0002940021	2 \pi	\pi		TODO		derivation of Schrodinger Equation
0003232242	t	t		TODO		electric field wave equation: from time dependent to time independent
0003413423	\cos(-x)	x		TODO		Euler equation: trigonometric relations
0003747849	-1			TODO		Euler equation: trigonometric relations
0003838111	2			TODO		Euler equation: trig square root
0003919391	x	x		TODO		Euler equation: trigonometric relations
0003949052	-x	x		TODO		Euler equation: trigonometric relations
0003949921	\hbar	\hbar		TODO		derivation of Schrodinger Equation
0003954314	dx	dx		TODO		Euler equation proof
0003981813	-\sin(x)	x		TODO		Euler equation: trigonometric relations
0004089571	2 x	x		TODO		Euler equation: trig square root
0004264724	y	y		TODO		Euler equation proof
0004307451	(b/(2 a))^2	b a		TODO		quadratic equation derivation
0004582412	x	x		TODO		Euler equation: trig square root
0004829194	2			TODO		Euler equation: trigonometric relations
0004831494	a	a		TODO		particle in a 1D box
0004849392	x	x		TODO		Euler equation: trigonometric relations
0004858592	h	h		TODO		quadratic equation derivation
0004934845	x	x		TODO		particle in a 1D box
0004948585	a	a		TODO		particle in a 1D box
0005395034	a_{\alpha} \langle \psi_{\alpha} \| \psi_{\beta} \rangle	\| \psi_{\beta} \rangle a_{\alpha} \langle \psi_{\alpha} \|		TODO		quantum basics orthogonality
0005626421	t	t		TODO		Maxwell equations to electric field wave equation
0005749291	f	f		TODO		frequency relations
0005938585	\frac{-\hbar^2}{2m}	\hbar m		TODO		derivation of Schrodinger Equation
0006466214	(\sin(x))^2	x		TODO		Euler equation: trig square root
0006544644	t	t		TODO		derivation of Schrodinger Equation
0006563727	x	x		TODO		Euler equation proof
0006644853	c/a	a c		TODO		quadratic equation derivation
0006656532	e	\exp		TODO		Euler equation proof
0007471778	2(\sin(x))^2	x		TODO		Euler equation: trig square root
0007563791	i	i		TODO		Euler equation proof
0007636749	x	x		TODO		Euler equation proof
0007894942	(\sin(x))^2	x		TODO		Euler equation: trig square root
0008837284	T	T		TODO		frequency relations
0008842811	\cos(2 x)	x		TODO		Euler equation: trig square root
0009458842	\psi(x)	x \psi		TODO		particle in a 1D box
0009484724	\frac{n \pi}{W}x	x n \pi W		TODO		particle in a 1D box
0009485857	a^2\frac{2}{W}	a W		TODO		particle in a 1D box
0009485858	\frac{2n\pi}{W}	n \pi W		TODO		particle in a 1D box
0009492929	v du	u v		TODO		integration by parts
0009587738	\psi	\psi		TODO		particle in a 1D box
0009877781	y	y		TODO		Euler equation proof
0203024440	1 = \int_0^W a \sin\left(\frac{n \pi}{W} x\right) \psi(x)^* dx	a \pi \psi x n W		TODO		particle in a 1D box
0404050504	\lambda = \frac{v}{f}	\lambda f v		TODO		frequency relations
0439492440	\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) \right\|_0^W	a \pi x n W		TODO	evaluating-definite-integrals-for.html	particle in a 1D box
0934990943	k = \frac{2 \pi}{v T}	k \pi T v		TODO		frequency relations
0948572140	\int \cos(a x) dx = \frac{1}{a}\sin(a x)	dx x a		TODO		particle in a 1D box
1010393913	\langle \psi\| \hat{A}^+ \|\psi \rangle = \langle a \rangle^*	\|\psi \rangle a \hat{A} \langle \psi\|		TODO	stats.html	quantum basics Hermitian operators have realvalued observables
1010393944	x = \langle\psi_{\alpha}\| a_{\beta} \|\psi_{\beta} \rangle	x \| \psi_{\beta} \rangle a_{\beta} \langle \psi_{\alpha} \|		TODO		quantum basics orthogonality
1010923823	k W = n \pi	k n \pi W		TODO		particle in a 1D box
1020010291	0 = a \sin(k W)	k a W		TODO		particle in a 1D box
1020394900	p = h/\lambda	\lambda p h		TODO		derivation of Schrodinger Equation
1020394902	E = h f	h f E		TODO		derivation of Schrodinger Equation
1020854560	I = (A + B)(A + B)^*	A I B		TODO		double intensity when phase is coherent (optics)
1025759423	y	y		TODO		time invariant force conserves energy
1029039903	p = m v	m p v		TODO		derivation of Schrodinger Equation
1029039904	p^2 = m^2 v^2	m p v		TODO		derivation of Schrodinger Equation
1036530438	d_2	d_2		TODO		Kepler's Third Law: period squared propto distance cubed
1038566242	\sinh x = \frac{\exp(x) - \exp(-x)}{2}	x		TODO		hyperbolic trigonometric identities
1085150613	C_V = \left(\frac{\partial U}{\partial T}\right)_V	U T C_V		TODO	definition of heat capacity at constant volume	first law of thermodynamics
1087417579	0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)	v_0 t_f \theta g		TODO		angle of maximum distance for projectile motion
1092872200	KE_1	KE_1		TODO		time invariant force conserves energy
1100332145	R	R		TODO		total electrical resistance for circuit with two resistors in parallel
1114820451	W_{\rm by\ system} = \Delta KE	W_{\rm by\ system} \Delta KE	Work is change in energy	TODO		velocity at distance r of object dropped from infinity
1128605625	{\rm sech}^2\ x + \tanh^2(x) = \frac{4}{\left(\exp(x)+\exp(-x)\right)^2} + \frac{\left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2}	x		TODO		hyperbolic trigonometric identities
1132941271	m_{\rm Earth} = \frac{(9.80665 m/s^2) (6.378110^6 m)^2}{6.6743010^{-11}m^3 kg^{-1} s^{-2}}	m_{\rm Earth}		TODO		mass of the Earth
1143343287	G \frac{m_{\rm Earth}}{r_{\rm Earth}} = \frac{1}{2} v_{\rm escape}^2	G r_{\rm Earth} m_{\rm Earth} v_{\rm escape}		TODO		escape velocity
1158485859	\frac{-\hbar^2}{2m} \nabla^2 = {\cal H}	\hbar m		TODO		derivation of Schrodinger Equation
1166310428	0 dt = d v_x	d v_x dt		TODO		equations of motion in 2D (calculus)
1172039918	I_{\rm coherent} = 4 \|A\|^2	A I_{\rm coherent}		TODO		double intensity when phase is coherent (optics)
1190768176	\kappa_T = \frac{-nRT}{V} \left( \frac{ \partial }{\partial P}\left(\frac{1}{P}\right) \right)_T	n R T \kappa_T V P		TODO		coefficient of isothermal compressibility using the equation of state for an ideal gas
1191796961	\frac{1}{2} g t_f = v_0 \sin(\theta)	v_0 g t_f \theta		TODO		angle of maximum distance for projectile motion
1193980495	m_{\rm Earth}	m_{\rm Earth}		TODO		mass of the Earth
1201689765	x'^2 + y'^2 + z'^2 = c^2 t'^2	c t' x' y' z'		TODO	describes a spherical wavefront for an observer in a moving frame of reference	Lorentz transformation
1202310110	\frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx	dx a \pi x n W		TODO		particle in a 1D box
1202312210	\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx	a \pi x n W		TODO		particle in a 1D box
1203938249	a_{\beta} \langle \psi_{\alpha} \| \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} \| \psi_{\beta} \rangle	\| \psi_{\beta} \rangle a_{\beta} \langle \psi_{\alpha} \|		TODO		quantum basics orthogonality
1238593037	c	c		TODO		Schwarzschild radius for non-rotating black hole
1248277773	\cos(2 x) = 1 - 2 (\sin(x))^2	x		TODO		Euler equation: trig square root
1258245373	E	E		TODO		time invariant force conserves energy
1259826355	d = (v - a t) t + \frac{1}{2} a t^2	v a t d		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
1265150401	d = \frac{2 v_0 + a t}{2} t	v_0 a t d		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
1268845856	[A_{\rm adsorption}]	[A_{\rm adsorption}]		TODO		Langmuir Adsorption
1277713901	d	d		TODO		speed of Earth around Sun
1292735067	F_{gravitational} = G \frac{m_1 m_2}{r^2}	G m_1 F_{\rm gravity} m_2 r		TODO		Newton's Law of Gravitation Kepler's Third Law: period squared propto distance cubed
1293913110	0 = b	b		TODO		particle in a 1D box
1293923844	\lambda = v T	\lambda T v		TODO		frequency relations
1306360899	x = v_{0, x} t + x_0	x v_{0, x} t x_0		TODO		equations of motion in 2D (calculus)
1310571337	\theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster}	\theta_{\rm Brewster}		TODO		optics: Law of refraction to Brewster's angle
1311403394	\alpha = \frac{1}{V} \frac{nR}{P} \left( \frac{\partial T}{\partial T} \right)_P	n R T \alpha V P		TODO		coefficient of thermal expansion using the equation of state for an ideal gas
1314464131	\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}	t		TODO		Maxwell equations to electric field wave equation
1314864131	\vec{ \nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}	\vec{H} \vec{E} \epsilon_0 t		TODO		Maxwell equations to electric field wave equation
1323602089	I_1	I_1		TODO		total electrical resistance for circuit with two resistors in parallel
1330874553	v_{\rm escape} = \sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}	G m_{\rm Earth} r_{\rm Earth} v_{\rm escape}		TODO		escape velocity
1333474099	F_{\rm centripetal}	F_{\rm centripetal}		TODO		radius for satellite in geostationary orbit
1357848476	A = \|A\| \exp(i \theta)	A \theta i		TODO		double intensity when phase is coherent (optics)
1377431959	R	R		TODO		total electrical resistance for circuit with two resistors in parallel
1395858355	x = \langle \psi_{\alpha}\| a_{\alpha} \|\psi_{\beta}\rangle	x \| \psi_{\beta} \rangle a_{\alpha} \langle \psi_{\alpha} \|		TODO		quantum basics orthogonality
1405465835	y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0	y g t v_y y_0		TODO		equations of motion in 2D (calculus) projectile path in 2D is parabolic
1413137236	m_1	m_1		TODO		escape velocity
1439089569	v_{0, x}	v_{0, x}		TODO		equations of motion in 2D (calculus)
1451839362	t	t		TODO		equation of motion for a spring
1457415749	\frac{1}{R_{\rm total}} = \frac{1}{R_1} + \frac{1}{R_2}	R_{\rm total} R_1 R_2	total resistance for two resistors in parallel	TODO		total electrical resistance for circuit with two resistors in parallel
1484794622	R_2	R_2		TODO		total electrical resistance for circuit with two resistors in series
1511199318	Z	Z		TODO		double intensity when phase is coherent (optics)
1512581563	x	x		TODO		optics: Law of refraction to Brewster's angle
1525861537	I = \|A\|^2 + \|B\|^2 + A B^* + B A^*	A I B		TODO		double intensity when phase is coherent (optics)
1528310784	\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}	c \gamma v		TODO		Lorentz transformation
1541916015	\theta = \frac{\pi}{4}	\pi \theta		TODO		angle of maximum distance for projectile motion
1552869972	x_1	x_1		TODO		time invariant force conserves energy
1556389363	E_{\rm Rydberg} = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}	e m_e \hbar \epsilon_0 \pi E_{\rm Rydberg}		TODO	the bonding energy in condensed phases is given by the Rydberg energy on the order of several e	upper limit on velocity in condensed matter
1559688463	\left(\frac{T_{\rm geostationary\ orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r_{\rm geostationary\ orbit}	G T_{\rm geostationary\ orbit} m_{\rm Earth} r_{\rm geostationary\ orbit} \pi		TODO		radius for satellite in geostationary orbit
1571582377	F_{gravitational} \propto \frac{1}{r^2}	F_{\rm gravity} r		TODO		Newton's Law of Gravitation
1586866563	\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)	y x z t c \gamma v		TODO		Lorentz transformation
1590774089	dW = F dx	dx dW F		TODO		escape velocity work and force and energy
1608399874	V_2	V_2		TODO		total electrical resistance for circuit with two resistors in series
1614343171	dt	dt		TODO		equations of motion in 2D (calculus)
1616666229	v_{\rm final}	v_{\rm final}		TODO		velocity at distance r of object dropped from infinity
1635147226	m_2	m_2		TODO		Kepler's Third Law: period squared propto distance cubed
1636453295	\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = - \nabla^2 \vec{E}	\vec{E}		TODO		Maxwell equations to electric field wave equation
1638282134	\vec{p}_{\rm before} = \vec{p}_{\rm after}	\vec{p}_{\rm before} \vec{p}_{\rm after}		TODO		Compton's equation for scattering
1639827492	- c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1	c \gamma v		TODO		Lorentz transformation
1648958381	\nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right)	\vec{p} \hbar \psi t \vec{r} i		TODO	representing-laplace-operator-nabla-in.html	derivation of Schrodinger Equation
1650441634	y_0 = 0	y_0		TODO	define coordinate system such that initial height is at origin	angle of maximum distance for projectile motion
1676472948	0 = v_x - v_{0, x}	v_x v_{0, x}		TODO		equations of motion in 2D (calculus)
1702349646	-g dt = d v_y	d v_y g		TODO		equations of motion in 2D (calculus)
1716984328	i x	x i		TODO		hyperbolic trigonometric identities
1742775076	Z	Z		TODO		double intensity when phase is coherent (optics)
1772416655	\frac{E_2 - E_1}{t} = v F - F v	E_2 E_1 t F v		TODO		time invariant force conserves energy
1772973171	-\frac{k}{m} x = -A \omega^2 \cos(\omega t)	m \omega T		TODO		equation of motion for a spring
1784114349	\sqrt{\frac{k}{m}} = \omega	m k \omega		TODO		equation of motion for a spring
1809909100	\frac{E_2 - E_1}{t} = 0	E_2 E_1 t		TODO		time invariant force conserves energy
1811867899	T^2 = \frac{d_1+d_2}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1}	T d_2 G m_1 d_1 r \pi		TODO		Kepler's Third Law: period squared propto distance cubed
1815398659	U = Q + W	U W Q		TODO		first law of thermodynamics
1819663717	a_x = \frac{d}{dt} v_x	v_x a_x t		TODO		equations of motion in 2D (calculus)
1823570358	C	C		TODO		radius for satellite in geostationary orbit
1840080113	KE_2 = 0	j		TODO	object is not moving at $x=\infty$	escape velocity
1848400430	F \propto m	m		TODO		Newton's Law of Gravitation
1857710291	0 = a \sin(n \pi)	a n \pi		TODO		particle in a 1D box
1858578388	\nabla^2 E( \vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E( \vec{r})\exp(i \omega t)	E \mu_0 t \omega \vec{r} \epsilon_0 i		TODO	representing-laplace-operator-nabla-in.html	electric field wave equation: from time dependent to time independent
1858772113	k = \frac{n \pi}{W}	k n \pi W		TODO		particle in a 1D box
1888494137	-\sqrt{\frac{k}{m}} = \omega	m k \omega		TODO		equation of motion for a spring
1894894315	Z	Z		TODO		double intensity when phase is coherent (optics)
1916173354	-\gamma^2 v^2 + c^2 \gamma^2 = c^2	c \gamma v		TODO		Lorentz transformation
1928085940	Z^* = \|Z\| \exp( -i \theta )	Z		TODO		double intensity when phase is coherent (optics)
1931103031	\frac{k}{m} = \omega^2	m k \omega		TODO		equation of motion for a spring
1934748140	\int \|\psi(x)\|^2 dx = 1	dx x \psi		TODO		particle in a 1D box
1935543849	\gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2	y x t z c \gamma v		TODO		Lorentz transformation
1945487024	p_A [S]	[S] p_A		TODO		Langmuir Adsorption
1963253044	v_{0, x} dt = dx	dx v_{0, x} dt		TODO		equations of motion in 2D (calculus)
1967582749	t = \frac{v - v_0}{a}	v_0 a t v		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
1974334644	\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'	x t t' \gamma v		TODO		Lorentz transformation
1977955751	-g = \frac{d}{dt} v_y	v_y g t		TODO		equations of motion in 2D (calculus)
1994296484	v_{\rm satellite}^2 = G \frac{m_{\rm Earth}}{r}	G r m_{\rm Earth} v_{\rm satellite}		TODO		radius for satellite in geostationary orbit
2005061870	v(r) = \sqrt{\frac{2 G m_2}{r}}	G m_2 r v		TODO		velocity at distance r of object dropped from infinity
2016063530	t	t		TODO		time invariant force conserves energy
2029293929	\nabla^2 E( \vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r})\exp(i \omega t)	E \mu_0 t \omega \vec{r} \epsilon_0 i		TODO	representing-laplace-operator-nabla-in.html	electric field wave equation: from time dependent to time independent
2042298788	0 = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}} + \frac{1}{2} m v_{\rm escape}^2	m G v_{\rm escape} r_{\rm Earth} m_{\rm Earth}		TODO		escape velocity
2051901211	\frac{V}{R_1} = I_1	R_1 V I_1		TODO		total electrical resistance for circuit with two resistors in parallel
2061086175	W_{\rm to\ system} = -G m_1 m_2 \left(\frac{-1}{r} - \frac{-1}{\infty}\right)	G m_1 W_{\rm to\ system} m_2 r		TODO		velocity at distance r of object dropped from infinity
2064205392	A	A		TODO		double intensity when phase is coherent (optics)
2076171250	-\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0	\gamma		TODO		Lorentz transformation
2081689540	t	t		TODO		time invariant force conserves energy
2086924031	0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)	v_0 g \theta t_f		TODO		angle of maximum distance for projectile motion
2091584724	g_{\rm Earth}	g_{\rm Earth}		TODO		mass of the Earth
2096918413	x = \gamma ( \gamma x - \gamma v t + v t' )	x t t' \gamma v		TODO		Lorentz transformation
2103023049	\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)	x i		TODO		hyperbolic trigonometric identities Euler equation: trigonometric relations identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation
2113211456	f = 1/T	T f		TODO		frequency and period frequency relations
2114570475	m_{\rm satellite}	m_{\rm satellite}		TODO		radius for satellite in geostationary orbit
2114909846	\theta_A = \frac{[A_{\rm adsorption}]}{[S_0]}	\theta_A [S_0] [A_{\rm adsorption}]		TODO		Langmuir Adsorption
2121790783	\tanh^2(x) = \frac{ \left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2}	x		TODO		hyperbolic trigonometric identities
2123139121	-\exp(-i x) = -\cos(x)+i \sin(x)	x i		TODO		Euler equation: trigonometric relations
2131616531	T f = 1	T f		TODO		frequency and period frequency relations
2135482543	m	m		TODO		escape velocity
2148049269	-\frac{k}{m} A \cos(\omega t) = -A \omega^2 \cos(\omega t)	\omega m T		TODO		equation of motion for a spring
2168306601	[S_0] = \left(\frac{k_{\rm desorption}}{k_{\rm adsorption}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}]	k_{\rm adsorption} p_A [S_0] [A_{\rm adsorption}] k_{\rm desorption}		TODO		Langmuir Adsorption
2186083170	\frac{KE_2 - KE_1}{t} = v F	KE_1 t KE_2 F v		TODO		time invariant force conserves energy
2217103163	\frac{m_1 d_1}{d_2} = m_2	m_1 m_2 d_1 d_2		TODO		Kepler's Third Law: period squared propto distance cubed
2226340358	\gamma v	\gamma v		TODO		Lorentz transformation
2232825726	g_{\rm Earth}	g_{\rm Earth}		TODO		mass of the Earth
2236639474	(A + B)(A + B)^* = \|A + B\|^2	A B		TODO		double intensity when phase is coherent (optics)
2242144313	a	a		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
2257410739	\left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alpha	U T		TODO		first law of thermodynamics
2258485859	{\cal H} \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)	{\cal H} \hbar \psi t \vec{r} i		TODO		derivation of Schrodinger Equation
2267521164	PE_2 = 0	PE_2		TODO	object goes to $\infty$ away from gravitational source	escape velocity
2271186630	V = I_{\rm total} R_{\rm total}	R_{\rm total} V I_{\rm total}		TODO		total electrical resistance for circuit with two resistors in parallel
2293352649	\theta - \phi	\theta \phi		TODO		double intensity when phase is coherent (optics)
2297105551	d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)	v_0 g \theta d		TODO		angle of maximum distance for projectile motion
2308660627	G \frac{m_{\rm Earth}}{r_{\rm Earth}^2} = g_{\rm Earth}	G r_{\rm Earth} g_{\rm Earth} m_{\rm Earth}		TODO		mass of the Earth
2334518266	m a = -k x	x m k a		TODO		equation of motion for a spring
2344320475	E_2	E_2		TODO		time invariant force conserves energy
2346150725	r	r		TODO		speed of Earth around Sun
2346952973	m	m		TODO		Newton's Law of Gravitation
2366691988	\int 0 dt = \int d v_x	d v_x t		TODO		equations of motion in 2D (calculus)
2378095808	x_f = x_0 + d	x_0 x_f d		TODO		angle of maximum distance for projectile motion
2394240499	x = a_{\beta} \langle \psi_{\alpha} \| \psi_{\beta} \rangle	x \| \psi_{\beta} \rangle a_{\beta} \langle \psi_{\alpha} \|		TODO		quantum basics orthogonality
2394853829	\exp(-i x) = \cos(-x)+i \sin(-x)	x i		TODO		Euler equation: trigonometric relations
2394935831	( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} \| \psi_{\beta} \rangle = 0	\| \psi_{\beta} \rangle a_{\beta} a_{\alpha} \langle \psi_{\alpha} \|		TODO		quantum basics orthogonality
2394935835	\left(\langle\psi\| \hat{A} \|\psi \rangle \right)^+ = \left(\langle a \rangle\right)^+	\langle \psi\|		TODO		quantum basics Hermitian operators have realvalued observables
2395958385	\nabla^2 \psi \left( \vec{r},t \right) = \frac{-p^2}{\hbar} \psi( \vec{r},t)	\hbar \psi t p \vec{r}		TODO	representing-laplace-operator-nabla-in.html	derivation of Schrodinger Equation
2396787389	r_{\rm Earth}	r_{\rm Earth}		TODO		escape velocity
2397692197	a^3	a		TODO		upper limit on velocity in condensed matter
2403773761	t	t		TODO		angle of maximum distance for projectile motion
2404934990	\langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2	x		TODO		variance relation
2405307372	\sin(2 x) = 2 \sin(x) \cos(x)	x		TODO		angle of maximum distance for projectile motion identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation
2417941373	- c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2	c \gamma v		TODO		Lorentz transformation
2431507955	PE_2 = -F x_2	PE_2 F x_2		TODO		time invariant force conserves energy
2461349007	- \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y	y v_{0, y} g t y_0		TODO		equations of motion in 2D (calculus)
2472653783	\alpha = \frac{1}{T}	T \alpha		TODO		coefficient of thermal expansion using the equation of state for an ideal gas
2484824786	F = m g	m g F		TODO		mass of the Earth
2494533900	\langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2	x		TODO		variance relation
2501591100	\exp(i \pi) + 1 = 0	\pi i		TODO		Euler equation to e^(i pi) + 1 = 0
2503972039	0 = KE_{\rm escape} + PE_{\rm Earth\ surface}	PE_{\rm Earth\ surface} KE_{\rm escape}		TODO		escape velocity
2510804451	2/g	g		TODO		angle of maximum distance for projectile motion
2519058903	\sin(2 \theta) = 2 \sin(\theta) \cos(\theta)	\theta		TODO		angle of maximum distance for projectile motion
2542420160	c^2 \gamma^2 - v^2 \gamma^2 = c^2	c \gamma v		TODO		Lorentz transformation
2575937347	n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} )	\theta_{\rm Brewster} \theta_{\rm refracted} n_1 n_2		TODO		optics: Law of refraction to Brewster's angle
2613006036	\frac{PV}{T} = nR	n R T V P		TODO		coefficient of thermal expansion using the equation of state for an ideal gas
2617541067	\left(\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r	G m_{\rm Earth} r T_{\rm orbit} \pi		TODO		radius for satellite in geostationary orbit
2648958382	\nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) \right)	\hbar		TODO		derivation of Schrodinger Equation
2660368546	r	r		TODO		Schwarzschild radius for non-rotating black hole
2674546234	m_{\rm Earth}	m_{\rm Earth}		TODO		escape velocity
2685587762	\frac{r_{\rm Earth}^2}{G}	G r_{\rm Earth}		TODO		mass of the Earth
2698469612	V	V		TODO		total electrical resistance for circuit with two resistors in series
2700934933	2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)	x \theta i \phi		TODO		double intensity when phase is coherent (optics)
2715678478	I R_{\rm total} = I R_1 + I R_2	R_{\rm total} R_1 R_2 I		TODO		total electrical resistance for circuit with two resistors in series
2719691582	\|A\| = \|B\|	A B		TODO	in a loop	double intensity when phase is coherent (optics)
2741489181	a_y = -g	a_y g		TODO		equations of motion in 2D (calculus)
2750380042	v_{\rm escape} = -\sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}	G m_{\rm Earth} r_{\rm Earth} v_{\rm escape}		TODO		escape velocity
2754264786	2			TODO		radius for satellite in geostationary orbit
2762326680	\cosh^2 x - \sinh^2 x = \frac{1}{4}\left( \exp(2x)+1+1+\exp(-2x) - \left(\exp(2x)-1-1+\exp(-2x)\right) \right)	x		TODO		hyperbolic trigonometric identities
2764966428	m_2	m_2		TODO		escape velocity
2768857871	\frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1}	\theta_{\rm Brewster} n_1 n_2		TODO		optics: Law of refraction to Brewster's angle
2770069250	\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}	E_2 KE_1 E_1 PE_2 t KE_2 PE_1		TODO		time invariant force conserves energy
2773628333	\theta_1	\theta_1		TODO		optics: Law of refraction to Brewster's angle
2809345867	\frac{V}{R_{\rm total}} = I_{\rm total}	R_{\rm total} V I_{\rm total}		TODO		total electrical resistance for circuit with two resistors in parallel
2848934890	\langle a \rangle^* = \langle a \rangle	a		TODO		quantum basics Hermitian operators have realvalued observables
2857430695	a = \frac{v_2 - v_1}{t}	v_1 a v_2 t	acceleration	TODO		time invariant force conserves energy
2858549874	- \frac{1}{2} g t^2 + v_{0, y} t = y - y_0	y v_{0, y} g t y_0		TODO		equations of motion in 2D (calculus)
2867848403	I	I		TODO		total electrical resistance for circuit with two resistors in parallel
2883079365	r_{\rm Schwarzschild} c^2 = 2 G m	c m G r_{\rm Schwarzschild}		TODO		Schwarzschild radius for non-rotating black hole
2897612567	v = \alpha c \sqrt{ \frac{m_e}{A m_p} }	m_e c A \alpha v m_p		TODO		upper limit on velocity in condensed matter
2902772962	\tanh(x) = \frac{\frac{1}{2}\left( \exp(x)-\exp(-x) \right)}{\cosh(x)}	x		TODO		hyperbolic trigonometric identities
2906548078	T^2 = \frac{r}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1}	T d_2 G m_1 d_1 r \pi		TODO		Kepler's Third Law: period squared propto distance cubed
2907404069	W_{\rm by\ system} = W_{\rm to\ system}	W_{\rm by\ system} W_{\rm to\ system}		TODO		velocity at distance r of object dropped from infinity
2924222857	v_{\rm initial} = v(r=\infty)	v v_{\rm initial}		TODO		velocity at distance r of object dropped from infinity
2944838499	\psi(x) = a \sin(\frac{n \pi}{W} x)	a \pi x \psi x n W		TODO		particle in a 1D box
2957211007	m^3 kg^{-1} s^{-2}			TODO		mass of the Earth
2977457786	2 G \frac{m_{\rm Earth}}{r_{\rm Earth}} = v_{\rm escape}^2	G r_{\rm Earth} m_{\rm Earth} v_{\rm escape}		TODO		escape velocity
2983053062	x = \gamma (x' + v t')	x t' \gamma v x'		TODO		Lorentz transformation
2998709778	v_{\rm initial} = 0	v_{\rm initial}		TODO		velocity at distance r of object dropped from infinity
2999795755	c^2 \gamma^2 = v^2 \gamma^2 + c^2	c \gamma v		TODO		Lorentz transformation
3004158505	\frac{T^2}{r} F_{gravitational} = \left( \frac{4 \pi^2 m r}{T^2} \right)\frac{T^2}{r}	m \pi F_{\rm gravity} T_{\rm orbit} r		TODO		Newton's Law of Gravitation
3031116098	R_2	R_2		TODO		total electrical resistance for circuit with two resistors in parallel
3041762466	i	i		TODO
3046191961	v_{\rm Earth\ orbit} = \frac{C_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}}	t_{\rm Earth\ orbit} v_{\rm Earth\ orbit} C_{\rm Earth\ orbit}		TODO		speed of Earth around Sun
3060393541	I_{\rm incoherent} = 2\|A\|^2	I_{\rm incoherent} A		TODO		double intensity when phase is coherent (optics)
3061811650	n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} )	\theta_{\rm Brewster} n_1 n_2		TODO		optics: Law of refraction to Brewster's angle
3080027960	v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}}	t_{\rm Earth\ orbit} v_{\rm Earth\ orbit} \pi r_{\rm Earth\ orbit}		TODO		speed of Earth around Sun
3085575328	I = \|A\|^2 + \|B\|^2 + \|A\| \|B\| \exp(i (\theta - \phi)) + \|A\| \|B\| \exp(-i (\theta - \phi))	\theta A I B \phi i		TODO		double intensity when phase is coherent (optics)
3088463019	m_2	m_2		TODO		radius for satellite in geostationary orbit
3105350101	v_1	v_1		TODO		time invariant force conserves energy
3121234211	\frac{k}{2\pi} = \lambda	k \lambda \pi		TODO		derivation of Schrodinger Equation
3121234212	p = \frac{h k}{2\pi}	k p \pi h		TODO		derivation of Schrodinger Equation
3121513111	k = \frac{2 \pi}{\lambda}	k \lambda \pi		TODO		frequency relations derivation of Schrodinger Equation
3131111133	T = 1 / f	T f		TODO		frequency and period frequency relations
3131211131	\omega = 2 \pi f	\omega \pi f		TODO		frequency relations derivation of Schrodinger Equation
3132131132	\omega = \frac{2\pi}{T}	\omega \pi T		TODO		frequency relations Kepler's Third Law: period squared propto distance cubed
3147472131	\frac{\omega}{2 \pi} = f	\omega \pi f		TODO		derivation of Schrodinger Equation
3166466250	m_1	m_1		TODO		velocity at distance r of object dropped from infinity
3169580383	\vec{a} = \frac{d\vec{v}}{dt}	\vec{a} \vec{v} t		TODO	acceleration is the change in speed over a duration	equations of motion in 2D (calculus)
3176662571	F_{\rm centripetal} = F_{\rm gravity}	F_{\rm centripetal} F_{\rm gravity}		TODO	applicable to any satellite orbit	radius for satellite in geostationary orbit Kepler's Third Law: period squared propto distance cubed
3182633789	\gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1	c \gamma v		TODO		Lorentz transformation
3182907803	x_0	x_0		TODO		equations of motion in 2D (calculus)
3183197515	v_1	v_1		TODO		work and force and energy
3214170322	v(r=\infty) = 0	v		TODO		velocity at distance r of object dropped from infinity
3219318145	\frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}			TODO		speed of Earth around Sun
3236313290	d	d		TODO		radius for satellite in geostationary orbit
3246378279	m	m		TODO		mass of the Earth
3253234559	x = \frac{v_2^2 - v_1^2}{2 a}	x a v_1 v_2		TODO		work and force and energy
3268645065	x	x		TODO		Euler equation to e^(i pi) + 1 = 0
3270039798	2			TODO		equations of motion in 2D (calculus)
3273630811	x	x		TODO		angle of maximum distance for projectile motion
3274176452	v_{\rm initial}	v_{\rm initial}		TODO		velocity at distance r of object dropped from infinity
3274926090	t = \frac{x - x_0}{v_{0, x}}	x v_{0, x} t x_0		TODO		projectile path in 2D is parabolic
3285732911	(\cos(x))^2 = 1-(\sin(x))^2	x		TODO		Euler equation: trig square root
3291685884	E = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}	e m_e \hbar E \epsilon_0 \pi		TODO		upper limit on velocity in condensed matter
3331824625	\exp(i \pi) = -1	\pi i		TODO		Euler equation to e^(i pi) + 1 = 0
3342155559	m	m		TODO		radius for satellite in geostationary orbit
3350802342	KE_{\rm initial}	KE_{\rm initial}		TODO		velocity at distance r of object dropped from infinity
3350830826	Z Z^* = \|Z\|^2	Z		TODO		double intensity when phase is coherent (optics)
3353418803	x	x		TODO		mass of the Earth
3360172339	W = KE_2 - KE_1	KE_2 KE_1 W		TODO		work and force and energy
3364286646	m_{\rm Earth} = 5.972*10^{24} kg	m_{\rm Earth}		TODO		mass of the Earth
3366703541	a = \frac{v - v_0}{t}	v_0 a t v		TODO	acceleration is the average change in speed over a duration	equations of motion in 1D with constant acceleration - SUVAT (algebra)
3398368564	F	F		TODO		radius for satellite in geostationary orbit
3411994811	v_{\rm average} = \frac{d}{t}	t v_{\rm average} d		TODO		Newton's Law of Gravitation equations of motion in 1D with constant acceleration - SUVAT (algebra)
3412946408	v^2 \gamma^2	\gamma v		TODO		Lorentz transformation
3417126140	\tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 }	\theta_{\rm Brewster} n_1 n_2		TODO		optics: Law of refraction to Brewster's angle
3426941928	x = \gamma ( \gamma (x - v t) + v t' )	x t t' \gamma v		TODO		Lorentz transformation
3433441359	V	V		TODO		total electrical resistance for circuit with two resistors in parallel
3448601530	\frac{T^2}{r}	r T		TODO		Newton's Law of Gravitation
3462972452	v = v_0 + a t	v_0 a t v		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
3464107376	\alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p	T V \alpha		TODO	definition of expansion coefficient	first law of thermodynamics coefficient of thermal expansion using the equation of state for an ideal gas
3470587782	\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)	x i		TODO		identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation
3472836147	r_{\rm Earth\ orbit} = 1.496\ 10^8 {\rm km}	r_{\rm Earth\ orbit}		TODO		speed of Earth around Sun
3485125659	x_f = v_0 t_f \cos(\theta) + x_0	\theta v_0 x_0 t_f x_f		TODO		angle of maximum distance for projectile motion
3485475729	\nabla^2 E( \vec{r}) = - \frac{\omega^2}{c^2} E( \vec{r})	c \omega E \vec{r}		TODO	representing-laplace-operator-nabla-in.html	electric field wave equation: from time dependent to time independent
3486213448	m_{\rm satellite}	m_{\rm satellite}		TODO		radius for satellite in geostationary orbit
3488423948	k_{\rm adsorption} p_A [S] = k_{\rm desorption} [A_{\rm adsorption}]	k_{\rm adsorption} [S] p_A [A_{\rm adsorption}] k_{\rm desorption}		TODO		Langmuir Adsorption
3495403335	x	x		TODO		Lorentz transformation
3497828859	V = \frac{n R T}{P}	n R T V P		TODO		coefficient of thermal expansion using the equation of state for an ideal gas coefficient of isothermal compressibility using the equation of state for an ideal gas
3507029294	k_{\rm adsorption} p_A [S] = r_{\rm desorption}	r_{\rm desorption} [S] p_A k_{\rm adsorption}		TODO		Langmuir Adsorption
3512166162	W = F x	x W F		TODO		work and force and energy
3531380618	v(r)	r v		TODO		velocity at distance r of object dropped from infinity
3547519267	S = k_{\rm Boltzmann} \ln \Omega	k_{Boltzmann} S \Omega		TODO	assumes equally probable microstates	first law of thermodynamics
3566149658	W_{\rm to\ system} = \int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx	G x m_1 W_{\rm to\ system} m_2 r		TODO		velocity at distance r of object dropped from infinity
3585845894	\langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2	x		TODO		variance relation
3591237106	\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v	E_2 KE_1 t KE_2 F v		TODO		time invariant force conserves energy
3594626260	F_{\rm gravity}	F_{\rm gravity}		TODO		radius for satellite in geostationary orbit
3599953931	[S_0] = [S] + [A_{\rm adsorption}]	[S_0] [S] [A_{\rm adsorption}]		TODO		Langmuir Adsorption
3605073197	\kappa_T = \frac{-nRT}{V} \left( \frac{-1}{P^2}\right)	n R T \kappa_T V P		TODO		coefficient of isothermal compressibility using the equation of state for an ideal gas
3607070319	d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)	v_0 g \pi d		TODO		angle of maximum distance for projectile motion
3614055652	v = \frac{2 \pi r}{T_{\rm orbit}}	T_{\rm orbit} \pi r v		TODO		radius for satellite in geostationary orbit
3634715785	m	m		TODO		equation of motion for a spring
3649797559	F_{\rm centripetal} = m_2 d_2 \omega^2	m_2 F_{\rm centripetal} \omega d_2		TODO		Kepler's Third Law: period squared propto distance cubed
3650370389	\frac{T^2}{r} F_{gravitational} = 4 \pi^2 m	m \pi F_{\rm gravity} T_{\rm orbit} r		TODO		Newton's Law of Gravitation
3650814381	F_{gravitational} \propto \frac{m_1 m_2}{r^2}	m_1 m_2 F_{\rm gravity} r		TODO		Newton's Law of Gravitation
3652511721	v	v		TODO		work and force and energy
3660957533	\cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)	x \theta i \phi		TODO		double intensity when phase is coherent (optics)
3663007361	2			TODO		Schwarzschild radius for non-rotating black hole
3676159007	v_{0, x} \int dt = \int dx	x v_{0, x} t		TODO		equations of motion in 2D (calculus)
3685779219	\sqrt{f} \approx 2	f		TODO		upper limit on velocity in condensed matter
3722461713	t	t		TODO		radius for satellite in geostationary orbit
3723096423	6.3781*10^6			TODO		mass of the Earth
3731774096	KE	KE		TODO		velocity at distance r of object dropped from infinity
3736177473	r_{\rm adsorption} = k_{\rm adsorption} p_A [S]	[S] r_{\rm adsorption} p_A k_{\rm adsorption}		TODO		Langmuir Adsorption
3749492596	E	E		TODO		time invariant force conserves energy
3781109867	T^2 = \frac{r^3 4 \pi^2}{(d_1+d_2) \frac{m_1}{d_2}G}	T d_2 G m_1 d_1 r \pi		TODO		Kepler's Third Law: period squared propto distance cubed
3806977900	E_2 - E_1 = 0	E_2 E_1		TODO		time invariant force conserves energy
3809726424	PE	PE		TODO		time invariant force conserves energy
3829492824	\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) = \cos(x)	x i		TODO		Euler equation: trigonometric relations
3846041519	PE_{\rm Earth\ surface} = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}}	m G PE_{\rm Earth\ surface} r_{\rm Earth} m_{\rm Earth}		TODO		escape velocity
3846345263	T_{\rm orbit}	T_{\rm orbit}		TODO		radius for satellite in geostationary orbit
3868998312	{\rm sech}^2\ x = \frac{4}{\left(\exp(x)+\exp(-x)\right)^2}	x		TODO		hyperbolic trigonometric identities
3876446703	m	m		TODO		Newton's Law of Gravitation
3896798826	m_2 d_2 \omega^2 = G \frac{m_1 m_2}{r^2}	d_2 G m_1 \omega m_2 r		TODO		Kepler's Third Law: period squared propto distance cubed
3906710072	G \frac{m_{\rm Earth}}{r} = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2}	\pi G m_{\rm Earth} T_{\rm orbit} r		TODO		radius for satellite in geostationary orbit
3911081515	-1			TODO		Langmuir Adsorption
3920616792	T_{\rm geostationary orbit} = 24\ {\rm hours}	T_{\rm geostationary\ orbit}		TODO	this applies for geostationary orbits	radius for satellite in geostationary orbit
3921072591	m_1	m_{\rm Earth}		TODO		mass of the Earth
3924948349	a_{\beta} \langle \psi_{\alpha} \| \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} \| \psi_{\beta} \rangle = 0	a_{\beta}		TODO		quantum basics orthogonality
3935058307	v = \sqrt{ \frac{m_e}{m} \frac{e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} }	\epsilon_0 m_e \hbar m v e \pi		TODO		upper limit on velocity in condensed matter
3939572542	KE_{\rm final}	KE_{\rm final}		TODO		velocity at distance r of object dropped from infinity
3942849294	\exp(i x)-\exp(-i x) = 2 i \sin(x)	x i		TODO		Euler equation: trigonometric relations
3943939590	x = a_{\alpha} \langle \psi_{\alpha}\| \psi_{\beta}\rangle	a_{\alpha}		TODO		quantum basics orthogonality
3947269979	\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}	t		TODO		Maxwell equations to electric field wave equation
3948571256	\frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}E \psi( \vec{r},t)	E \hbar \psi t i \vec{r}		TODO		derivation of Schrodinger Equation
3948574224	\psi( \vec{r},t) = \psi_0 \exp\left(i\left( \vec{k}\cdot\vec{r} - \omega t \right) \right)	k \psi_0 \psi t \exp \omega i \vec{r}		TODO		derivation of Schrodinger Equation
3948574226	\psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)	\hbar \psi_0 \psi t \exp \omega i p \vec{r}		TODO		derivation of Schrodinger Equation
3948574228	\psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)	E \hbar \psi_0 \psi t \exp i p \vec{r}		TODO		derivation of Schrodinger Equation
3948574230	\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)	E \hbar \psi_0 \psi t \exp i p \vec{r}		TODO		derivation of Schrodinger Equation
3948574233	\frac{\partial}{\partial t} \psi( \vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)	E \hbar \psi_0 \psi t \exp i p \vec{r}		TODO		derivation of Schrodinger Equation
3951205425	\vec{p}_{\rm after} = \vec{p}_{1}	\vec{p}_1 \vec{p}_{\rm after}		TODO		Compton's equation for scattering
3967985562	2			TODO		double intensity when phase is coherent (optics)
4057686137	C	C		TODO		speed of Earth around Sun
4072200527	\frac{m_{\rm satellite} v_{\rm satellite}^2}{r} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}	G m_{\rm satellite} m_{\rm Earth} r v_{\rm satellite}		TODO		radius for satellite in geostationary orbit
4075539836	A A^* = \|A\|^2	A		TODO		double intensity when phase is coherent (optics)
4087145886	V = I R	V I R	Ohm's law	TODO	Ohm%27s_law	total electrical resistance for circuit with two resistors in parallel total electrical resistance for circuit with two resistors in series
4107032818	E_{\rm Rydberg} = E	E_{\rm Rydberg} E		TODO		upper limit on velocity in condensed matter
4128500715	V = I_1 R_1	R_1 V I_1		TODO		total electrical resistance for circuit with two resistors in parallel
4139999399	x - \gamma^2 x = - \gamma^2 v t + \gamma v t'	x t t' \gamma v		TODO		Lorentz transformation
4147101187	KE	KE		TODO		time invariant force conserves energy
4147472132	E = \frac{h \omega}{2 \pi}	\omega h \pi E		TODO		derivation of Schrodinger Equation
4153613253	m_{\rm Earth}	m_{\rm Earth}		TODO		radius for satellite in geostationary orbit
4158986868	a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}	t		TODO		equations of motion in 2D (calculus)
4162188238	t_f	t_f		TODO		angle of maximum distance for projectile motion
4166155526	{\rm sech}\ x = \frac{2}{\exp(x)+\exp(-x)}	x		TODO		hyperbolic trigonometric identities
4167526462	v_{0, y}	v_{0, y}		TODO		equations of motion in 2D (calculus)
4180845508	v_{\rm Earth\ orbit} = 29.8 \frac{{\rm km}}{{\rm sec}}	v_{\rm Earth\ orbit}		TODO		speed of Earth around Sun
4182362050	Z = \|Z\| \exp( i \theta )	Z \theta i		TODO	Z \in \mathbb{C}	double intensity when phase is coherent (optics)
4188580242	T^2 = \frac{r^3 4 \pi^2}{\left(m_1+\left(\frac{m_1}{d_2}d_1\right)\right)G}	T d_2 G m_1 d_1 r \pi		TODO		Kepler's Third Law: period squared propto distance cubed
4188639044	x	x		TODO		time invariant force conserves energy
4192519596	B = \|B\| \exp(i \phi)	B i \phi		TODO		double intensity when phase is coherent (optics)
4202292449	r_{\rm Earth\ orbit}	r_{\rm Earth\ orbit}		TODO		speed of Earth around Sun
4213426349	E_1	E_1		TODO		time invariant force conserves energy
4218009993	x	x		TODO		time invariant force conserves energy
4245712581	v = \frac{2 \pi r}{t}	\pi r t v		TODO		radius for satellite in geostationary orbit
4264859781	F \propto m_1	m_1		TODO		Newton's Law of Gravitation
4267808354	F_{gravitational} = m \frac{v^2}{r}	m F_{\rm gravity} r v		TODO		Newton's Law of Gravitation
4268085801	x_0 + d = v_0 t_f \cos(\theta) + x_0	\theta v_0 x_0 t_f d		TODO		angle of maximum distance for projectile motion
4270680309	\frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}	v_1 m KE_1 t KE_2 v_2		TODO		time invariant force conserves energy
4275004561	c^2 = 2 G \frac{m}{r_{\rm Schwarzschild}}	c m r_{\rm Schwarzschild} G		TODO		Schwarzschild radius for non-rotating black hole
4287102261	x^2 + y^2 + z^2 = c^2 t^2	y x t z c		TODO	describes a spherical wavefront	Lorentz transformation
4298359835	E = \frac{1}{2}m v^2	v m E		TODO		derivation of Schrodinger Equation
4298359845	E = \frac{1}{2m}m^2 v^2	v m E		TODO		derivation of Schrodinger Equation
4298359851	E = \frac{p^2}{2m}	m p E		TODO		derivation of Schrodinger Equation
4301729661	[S_0] = \frac{[A_{\rm adsorption}]}{\left( \frac{k_{\rm adsorption}}{k_{\rm desorption}} \right) p_A} + [A_{\rm adsorption}]	k_{\rm adsorption} p_A [S_0] [A_{\rm adsorption}] k_{\rm desorption}		TODO		Langmuir Adsorption
4303372136	E_1 = KE_1 + PE_1	KE_1 E_1 PE_1		TODO		time invariant force conserves energy escape velocity
4319470443	v_2	v_2		TODO		time invariant force conserves energy
4319544433	1/3			TODO		radius for satellite in geostationary orbit
4341171256	i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{p^2}{2 m} \psi( \vec{r},t)	m \hbar \psi t i p \vec{r}		TODO		derivation of Schrodinger Equation
4348571256	\frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi( \vec{r},t)	m \hbar \psi t \vec{r} p i		TODO		derivation of Schrodinger Equation
4370074654	t = t_f	t_f t		TODO		angle of maximum distance for projectile motion
4393258808	F_{\rm centripetal} = m r \omega^2	m F_{\rm centripetal} \omega r		TODO		Kepler's Third Law: period squared propto distance cubed
4393670960	W_{\rm to\ system} = \frac{G m_1 m_2}{r}	G m_1 W_{\rm to\ system} m_2 r		TODO		velocity at distance r of object dropped from infinity
4394958389	\vec{ \nabla}\cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right) = \frac{i}{\hbar} \vec{ \nabla}\cdot\left( \vec{p} \psi( \vec{r},t) \right)	\vec{p} \hbar \psi t \vec{r} i		TODO		derivation of Schrodinger Equation
4428528271	F_{\rm{spring}} = -k x	x F_{\rm{spring}} k	Hooke's law	TODO	Hooke%27s_law	equation of motion for a spring
4437214608	Z	Z		TODO		double intensity when phase is coherent (optics)
4447113478	\int dW = G m_1 m_2 \int_{ r_{\rm Earth} }^{\infty} \frac{1}{x^2} dx	G x m_1 r_{\rm Earth} W m_2		TODO		escape velocity
4470433702	t_{\rm Earth\ orbit}	t_{\rm Earth\ orbit}		TODO		speed of Earth around Sun
4490788873	F \propto m_2	m_2		TODO		Newton's Law of Gravitation
4501377629	\tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}	\theta_{\rm Brewster}		TODO		optics: Law of refraction to Brewster's angle
4504256452	B^* = \|B\| \exp(-i \phi)	B i \phi		TODO		double intensity when phase is coherent (optics)
4522137851	PE_2	PE_2		TODO		time invariant force conserves energy
4560648264	v = \sqrt{ \frac{K + (4/3) G}{\rho} }	K G v \rho		TODO		upper limit on velocity in condensed matter
4580545876	d = v t - a t^2 + \frac{1}{2} a t^2	v a t d		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
4583868070	B	B		TODO		double intensity when phase is coherent (optics)
4585828572	\epsilon_0 \mu_0 = \frac{1}{c^2}	c \epsilon_0 \mu_0		TODO		electric field wave equation: from time dependent to time independent
4585932229	\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)	x i		TODO		double intensity when phase is coherent (optics) hyperbolic trigonometric identities Euler equation: trigonometric relations identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation
4587046017	KE	KE		TODO		velocity at distance r of object dropped from infinity
4593428198	v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{3.16\ 10^7 {\rm seconds}}	v_{\rm Earth\ orbit} \pi r_{\rm Earth\ orbit}		TODO		speed of Earth around Sun
4598294821	\exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2	x i		TODO		Euler equation: trig square root
4627284246	F_{\rm centripetal} = \frac{m_{\rm satellite} v_{\rm satellite}^2}{r}	m_{\rm satellite} F_{\rm centripetal} r v_{\rm satellite}		TODO		radius for satellite in geostationary orbit
4638429483	\exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))	x i		TODO		Euler equation: trig square root
4648451961	v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)	v_1 v_2		TODO		time invariant force conserves energy
4651061153	m_2	m_2		TODO		mass of the Earth
4662369843	x' = \gamma (x - v t)	t x \gamma v x'		TODO		Lorentz transformation
4669290568	PE_1 = -F x_1	x_1 F PE_1		TODO		time invariant force conserves energy
4689334676	\theta_A = \frac{K_{\rm equilibrium}\ p_A}{1+K_{\rm equilibrium}\ p_A}	K_{\rm equilibrium} \theta_A p_A		TODO		Langmuir Adsorption
4742644828	\exp(i x)+\exp(-i x) = 2 \cos(x)	x i		TODO		Euler equation: trigonometric relations
4748157455	a t = v - v_0	v_0 a t v		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
4755369593	x_2	x_2		TODO		time invariant force conserves energy
4778077984	t_f = \frac{2 v_0 \sin(\theta)}{g}	v_0 t_f \theta g		TODO		angle of maximum distance for projectile motion
4784793837	\frac{KE_2 - KE_1}{t} = m v a	m KE_1 t KE_2 a v		TODO		time invariant force conserves energy
4798787814	a t + v_0 = v	v_0 a t v		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
4800170179	F = m g_{\rm Earth}	m F g_{\rm Earth}		TODO		mass of the Earth
4805233006	i \sin(i x) = \frac{1}{2}\left(\exp(x) - \exp(-x) \right)	x i		TODO
4811121942	W = \frac{1}{2} m v_2^2 - \frac{1}{2} m v_1^2	v_1 m W v_2		TODO		work and force and energy
4820320578	F_{gravitational} = F_{centripetal}	F_{\rm centripetal} F_{\rm gravity}		TODO		Newton's Law of Gravitation
4827492911	\cos(2 x)+(\sin(x))^2 = 1 - (\sin(x))^2	x		TODO		Euler equation: trig square root
4829590294	t_f	t_f		TODO		angle of maximum distance for projectile motion
4830221561	{\rm sech}^2\ x + \tanh^2(x) = \frac{4+\left(\exp(2x)-1-1+\exp(-2x)\right)}{\left(\exp(x)+\exp(-x)\right)^2}	x		TODO		hyperbolic trigonometric identities
4830480629	x	x		TODO		radius for satellite in geostationary orbit
4838429483	\exp(2 i x) = \cos(2 x)+i \sin(2 x)	x i		TODO		Euler equation: trig square root
4843995999	\frac{1}{2 i}\left(\exp(i x)-\exp(-i x) \right) = \sin(x)	x i		TODO		Euler equation: trigonometric relations
4857472413	1 = \int \psi(x)\psi(x)^* dx	dx		TODO		particle in a 1D box
4857475848	\frac{1}{a^2} = \frac{W}{2}	a W		TODO		particle in a 1D box
4858693811	\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2} = r^3	G m_{\rm Earth} r T_{\rm orbit} \pi		TODO		radius for satellite in geostationary orbit
4866160902	\frac{V}{R_{\rm total}} = \frac{V}{R_1} + \frac{V}{R_2}	R_{\rm total} R_1 V R_2		TODO		total electrical resistance for circuit with two resistors in parallel
4872163189	\tanh(x) = \frac{\sinh(x)}{\cosh(x)}	x		TODO		hyperbolic trigonometric identities
4872970974	\frac{PE_2 - PE_1}{t} = -F v	PE_2 t PE_1 F v		TODO		time invariant force conserves energy
4878728014	\sin(i x) = \frac{1}{2i}\left(\exp(-x) - \exp(x) \right)	x i		TODO		hyperbolic trigonometric identities
4901237716	1			TODO		Euler equation to e^(i pi) + 1 = 0
4923339482	i x = \log(y)	y x i		TODO		Euler equation proof
4928007622	KE_1 = \frac{1}{2} m v_1^2	v_1 m KE_1		TODO		time invariant force conserves energy work and force and energy
4928239482	\log(y) = i x	y x i		TODO		Euler equation proof
4935235303	x	x		TODO		double intensity when phase is coherent (optics)
4938429482	\exp(-i x) = \cos(x)+i \sin(-x)	x i		TODO		Euler equation: trigonometric relations
4938429483	\exp(i x) = \cos(x)+i \sin(x)	x i		TODO		Euler equation proof Euler equation to e^(i pi) + 1 = 0 Euler equation: trig square root Euler equation: trigonometric relations
4938429484	\exp(-i x) = \cos(x)-i \sin(x)	x i		TODO		Euler equation: trigonometric relations
4939880586	V_{\rm total} = I R_{\rm total}	R_{\rm total} V_{\rm total} I		TODO		total electrical resistance for circuit with two resistors in series
4943571230	\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)	E \vec{p} \hbar \psi_0 \psi t \vec{r} i		TODO		derivation of Schrodinger Equation
4947831649	\frac{1}{2} m_1 v_{\rm final}^2 = W_{\rm to\ system}	m_1 v_{\rm final} W_{\rm to\ system}		TODO		velocity at distance r of object dropped from infinity
4948763856	2 a d + v_0^2 = v^2	v a v_0 d		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
4948934890	\langle \psi\| \hat{A} \|\psi \rangle = \langle a \rangle^*	\langle \psi\|		TODO		quantum basics Hermitian operators have realvalued observables
4949359835	\langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2	x		TODO		variance relation
4961662865	x	x		TODO		identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation
4968680693	\tan( x ) = \frac{ \sin( x )}{\cos( x )}	x		TODO		optics: Law of refraction to Brewster's angle
4985825552	\nabla^2 E( \vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E( \vec{r})\exp(i \omega t)	E \mu_0 t \omega \vec{r} \epsilon_0 i		TODO	representing-laplace-operator-nabla-in.html	electric field wave equation: from time dependent to time independent
5002539602	dU = C_V dT + \pi_T dV	C_V \pi_T		TODO		first law of thermodynamics
5011888122	v_2	v_2		TODO		time invariant force conserves energy
5021965469	KE	KE		TODO		escape velocity
5050429607	G \frac{m_{\rm Earth} m}{r_{\rm Earth}}	G m r_{\rm Earth} m_{\rm Earth}		TODO		escape velocity
5074423401	V	V		TODO		first law of thermodynamics
5075406409	PE	PE		TODO		time invariant force conserves energy
5085809757	\frac{k_{\rm adsorption}}{k_{\rm desorption}} = \frac{[A_{\rm adsorption}]}{p_A [S]}	k_{\rm adsorption} [S] p_A [A_{\rm adsorption}] k_{\rm desorption}		TODO		Langmuir Adsorption
5089196493	F	F		TODO		radius for satellite in geostationary orbit
5125940051	I = \|A\|^2 + B B^* + A B^* + B A^*	A I B		TODO		double intensity when phase is coherent (optics)
5128670694	m_1 d_1 = m_2 d_2	m_1 m_2 d_1 d_2		TODO		Kepler's Third Law: period squared propto distance cubed
5136652623	E = KE + PE	KE PE E	mechanical energy is the sum of the potential plus kinetic energies	TODO		time invariant force conserves energy
5144263777	v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)	v		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
5148266645	t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t	\gamma		TODO		Lorentz transformation
5177311762	v = \frac{2 \pi r}{T}	T_{\rm orbit} \pi r v		TODO		Newton's Law of Gravitation
5181421075	R_1	R_1		TODO		total electrical resistance for circuit with two resistors in parallel
5194141542	x_f	x_f		TODO		angle of maximum distance for projectile motion
5208737840	T_{\rm geostationary\ orbit}	T_{\rm geostationary\ orbit}		TODO		radius for satellite in geostationary orbit
5239755033	v_1	v_1		TODO		time invariant force conserves energy
5258419993	R_1	R_1		TODO		total electrical resistance for circuit with two resistors in parallel
5284610349	\gamma^2	\gamma		TODO		Lorentz transformation
5323719091	i \sinh x = \frac{1}{2i} \left( \exp(-x) - \exp(x) \right)	x i		TODO		hyperbolic trigonometric identities
5345738321	F = m a	m a F	Newton's second law of motion	TODO	Newton%27s_laws_of_motion#Newton's_second_law	Newton's Law of Gravitation work and force and energy mass of the Earth time invariant force conserves energy equation of motion for a spring
5349669879	\tanh(x) = \frac{ \exp(x)-\exp(-x)}{\exp(x)+\exp(-x)}	x		TODO		hyperbolic trigonometric identities
5349866551	\vec{v} = v_x \hat{x} + v_y \hat{y}	\hat{y} v_x v_y \hat{x} \vec{v}		TODO		equations of motion in 2D (calculus)
5353282496	d = \frac{v_0^2}{g}	v_0 g d		TODO		angle of maximum distance for projectile motion
5359471792	\frac{m_{\rm satellite}}{r}	m_{\rm satellite} r		TODO		radius for satellite in geostationary orbit
5373931751	t = t_f	t_f t		TODO		angle of maximum distance for projectile motion
5379546684	y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0	y_f g \theta v_0 y_0 t_f		TODO		angle of maximum distance for projectile motion
5398681502	v	v		TODO		work and force and energy
5398681503	v	v		TODO		time invariant force conserves energy
5404822208	v_{\rm escape} = \sqrt{2 G \frac{m}{r}}	G m r v_{\rm escape}	escape velocity	TODO		Schwarzschild radius for non-rotating black hole escape velocity
5415824175	x(t) = A \cos(\omega t)	A \omega t		TODO		equation of motion for a spring
5426308937	v = \frac{d}{t}	d t v		TODO		radius for satellite in geostationary orbit speed of Earth around Sun
5426418187	[A_{\rm adsorption}]	[A_{\rm adsorption}]		TODO		Langmuir Adsorption
5438722682	x = v_0 t \cos(\theta) + x_0	x \theta t v_0 x_0		TODO		equations of motion in 2D (calculus) angle of maximum distance for projectile motion
5453995431	\arctan{ x }	x		TODO		optics: Law of refraction to Brewster's angle
5463275819	I_2	I_2		TODO		total electrical resistance for circuit with two resistors in parallel
5514556106	E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)	E_2 KE_1 E_1 PE_2 KE_2 PE_1		TODO		time invariant force conserves energy
5516739892	-1			TODO		Langmuir Adsorption
5530148480	\vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}	\vec{p}_2 \vec{p}_1 \vec{p}_{\rm electron}		TODO		Compton's equation for scattering
5542390646	2 a	a		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
5542528160	\int dW = F \int_0^x dx	x W F		TODO		work and force and energy
5563580265	F_{\rm gravity} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}	G F_{\rm gravity} m_{\rm satellite} m_{\rm Earth} r		TODO		radius for satellite in geostationary orbit
5585739998	I	I		TODO		total electrical resistance for circuit with two resistors in parallel
5586102077	r = d_1 + d_2	r d_1 d_2		TODO		Kepler's Third Law: period squared propto distance cubed
5591692598	KE_1	KE_1		TODO		escape velocity
5596822289	W_{\rm to\ system} = -G m_1 m_2 \left(\left.\frac{-1}{x}\right\|^r_{\infty}\right)	G m_2 W_{\rm to\ system} m_1		TODO		velocity at distance r of object dropped from infinity
5611024898	d = \frac{1}{2 a} (v^2 - v_0^2)	v a v_0 d		TODO		work and force and energy equations of motion in 1D with constant acceleration - SUVAT (algebra)
5620558729	v_0	v_0		TODO		equations of motion in 2D (calculus)
5623794884	A + B	A B		TODO		double intensity when phase is coherent (optics)
5632428182	\cos( \theta_{\rm Brewster} )	\theta_{\rm Brewster}		TODO		optics: Law of refraction to Brewster's angle
5634116660	\pi_T = \left(\frac{\partial U}{\partial V}\right)_T	U V \pi_T		TODO	definition of internal pressure at constant temperature	first law of thermodynamics
5646314683	m = A m_p	m_p A m		TODO		upper limit on velocity in condensed matter
5658865948	T^2 = \frac{r^3 4 \pi^2}{(m_1+m_2)G}	T G m_1 r m_2 \pi		TODO		Kepler's Third Law: period squared propto distance cubed
5667870149	\theta	\theta		TODO		angle of maximum distance for projectile motion
5669500954	v^2 \gamma^2	\gamma v		TODO		Lorentz transformation
5684907106	\frac{1}{d_2 4 \pi^2}	\pi d_2		TODO		Kepler's Third Law: period squared propto distance cubed
5693047217	v_{\rm final} = -\sqrt{\frac{2 G m_2}{r}}	v_{\rm final} m_2 r G		TODO		velocity at distance r of object dropped from infinity
5727578862	\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)	dx		TODO		particle in a 1D box
5732331610	W = G m_1 m_2 \left( \frac{1}{x} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)	G		TODO	2022-03-25 BHP: Conversion between Latex and Sympy is incomplete	escape velocity
5733146966	KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)	v_1 m KE_1 KE_2 v_2		TODO		time invariant force conserves energy
5733721198	d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)	v a v_0 d		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
5763749235	-c^2 + c^2 \gamma^2 = v^2 \gamma^2	c \gamma v		TODO		Lorentz transformation
5770088141	r	r		TODO		radius for satellite in geostationary orbit
5775658332	2			TODO		escape velocity
5778176146	t	t		TODO		time invariant force conserves energy
5779256336	W_{\rm by\ system} = KE_{\rm final} - KE_{\rm initial}	W_{\rm by\ system} KE_{\rm initial} KE_{\rm final}		TODO		velocity at distance r of object dropped from infinity
5781435087	g	g		TODO		mass of the Earth
5781981178	x^2 - y^2 = (x+y)(x-y)	y x	difference of squares	TODO	Difference_of_two_squares	time invariant force conserves energy
5787469164	1 - \gamma^2	\gamma		TODO		Lorentz transformation
5789289057	v = \alpha c \sqrt{ \frac{m_e}{2 m} }	m_e m c \alpha v		TODO	equation 4 in the PDF	upper limit on velocity in condensed matter
5799753649	2			TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
5803210729	PE_2	PE_2		TODO		time invariant force conserves energy
5832984291	(\sin(x))^2 + (\cos(x))^2 = 1	x		TODO		Euler equation: trig square root
5838268428	\alpha c = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar}	e \hbar c \alpha \epsilon_0 \pi		TODO		upper limit on velocity in condensed matter
5846177002	t			TODO		equation of motion for a spring
5846639423	v_{\rm final} = \sqrt{\frac{2 G m_2}{r}}	v_{\rm final} m_2 r G		TODO		velocity at distance r of object dropped from infinity
5850144586	W_{\rm by\ system} = KE_{\rm final}	W_{\rm by\ system} KE_{\rm final}		TODO		velocity at distance r of object dropped from infinity
5857434758	\int a dx = a x	x a		TODO		particle in a 1D box
5866629429	{\rm sech}^2\ x + \tanh^2(x) = 1	x		TODO		hyperbolic trigonometric identities
5868688585	\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = \frac{p^2}{2m} \psi( \vec{r},t)	m \hbar \psi t p \vec{r}		TODO	representing-laplace-operator-nabla-in.html	derivation of Schrodinger Equation
5868731041	v_0	v_0		TODO		equations of motion in 2D (calculus)
5890617067	R	R		TODO		total electrical resistance for circuit with two resistors in series
5900595848	k = \frac{\omega}{v}	k \omega v		TODO		frequency relations
5902985919	\vec{F} = G \frac{m_1 m_2}{x^2} \hat{x}	G x m_1 F m_2	Newton's law of universal gravitation	TODO		velocity at distance r of object dropped from infinity
5904227750	m	m		TODO		velocity at distance r of object dropped from infinity
5928285821	x^2 + 2 x (b/(2 a)) + (b/(2 a))^2 = (x + (b/(2 a)))^2	b x a		TODO		quadratic equation derivation
5928292841	x^2 + (b/a)x + (b/(2 a))^2 = -c/a + (b/(2 a))^2	b x a c		TODO		quadratic equation derivation
5938459282	x^2 + (b/a)x = -c/a	b x a c		TODO		quadratic equation derivation
5945893986	\frac{d^2 x}{dt^2} = -A \omega^2 \cos(\omega t)	\omega t A		TODO		equation of motion for a spring
5958392859	x^2 + (b/a)x+(c/a) = 0	x a c		TODO		quadratic equation derivation
5959282914	x^2 + x(b/a) + (b/(2 a))^2 = (x+(b/(2 a)))^2	b x a		TODO		quadratic equation derivation
5960438249	E_1	E_1		TODO		time invariant force conserves energy
5962145508	\alpha = \frac{nR}{VP}	n R \alpha V P		TODO		coefficient of thermal expansion using the equation of state for an ideal gas
5978756813	W = G m_{\rm Earth} m \left( 0 - \frac{-1}{ r_{\rm Earth}} \right)	m G r_{\rm Earth} W m_{\rm Earth}		TODO		escape velocity
5982958248	x = -\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))	b x a c		TODO		quadratic equation derivation
5982958249	x+(b/(2 a)) = -\sqrt{(b/(2 a))^2 - (c/a)}	b x a c		TODO		quadratic equation derivation
5985371230	\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi( \vec{r},t)	\vec{p} \hbar \psi t \vec{r} i		TODO		derivation of Schrodinger Equation
6023986360	x	x		TODO		optics: Law of refraction to Brewster's angle
6026694087	F_{centripetal} = m \frac{v^2}{r}	m F_{\rm centripetal} r		TODO		Newton's Law of Gravitation
6031385191	\sinh^2 x = \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)	x		TODO		hyperbolic trigonometric identities
6038673136	v	v		TODO		velocity at distance r of object dropped from infinity
6050070428	v_{0, x}	v_{0, x}		TODO		projectile path in 2D is parabolic
6055078815	\left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T \left( \frac{\partial V}{\partial T} \right)_p	U T		TODO	constant pressure	first law of thermodynamics
6061695358	V_2 = I R_2	R_2 I V_2		TODO		total electrical resistance for circuit with two resistors in series
6083821265	v_0 \cos(\theta) = v_{0, x}	v_0 \theta v_{0, x}		TODO		equations of motion in 2D (calculus)
6091977310	KE_{\rm initial} = \frac{1}{2} m_1 v_{\rm initial}^2	m_1 KE_{\rm initial} v_{\rm initial}		TODO		velocity at distance r of object dropped from infinity
6098638221	y_0	y_0		TODO		equations of motion in 2D (calculus)
6131764194	W = G m_{\rm Earth} m \left( \frac{1}{x^2} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)	G m x m_{\rm Earth}		TODO	evaluating-definite-integrals-for.html	escape velocity
6134836751	v_{0, x} = v_x	v_x v_{0, x}		TODO		equations of motion in 2D (calculus)
6158970683	PE_1	PE_1		TODO		escape velocity
6175547907	v_{\rm average} = \frac{v + v_0}{2}	v_0 v_{\rm average} v		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
6204539227	-g t + v_{0, y} = \frac{dy}{dt}	G y v_{0, y} t		TODO		equations of motion in 2D (calculus)
6238632840	r T_{\rm orbit}^2	r T_{\rm orbit}		TODO		radius for satellite in geostationary orbit
6239815585	C_{\rm Earth\ orbit}	C_{\rm Earth\ orbit}		TODO		speed of Earth around Sun
6240206408	I_{\rm incoherent} = \|A\|^2 + \|B\|^2	I_{\rm incoherent} A B		TODO		double intensity when phase is coherent (optics)
6240546932	\frac{1}{K_{equilibrium}} = \frac{k_{\rm desorption}}{k_{\rm adsorption}}	K_{\rm equilibrium} k_{\rm desorption} k_{\rm adsorption}		TODO		Langmuir Adsorption
6259833695	A	A		TODO		upper limit on velocity in condensed matter
6268336290	F_{gravitational} = \frac{m}{r}\left(\frac{2\pi r}{T}\right)^2	F_{\rm gravity} r T_{\rm orbit} m_2 \pi		TODO		Newton's Law of Gravitation
6281834543	m_1	m_1		TODO		velocity at distance r of object dropped from infinity
6296166842	P	P		TODO		coefficient of isothermal compressibility using the equation of state for an ideal gas
6306552185	I = (A + B)(A^* + B^*)	A I B		TODO		double intensity when phase is coherent (optics)
6346902704	1			TODO		Langmuir Adsorption
6348260313	C_{\rm Earth\ orbit} = 2 \pi r_{\rm Earth\ orbit}	\pi C_{\rm Earth\ orbit} r_{\rm Earth\ orbit}		TODO		speed of Earth around Sun
6353701615	\theta_{\rm refracted}	\theta_{\rm refracted}		TODO		optics: Law of refraction to Brewster's angle
6383056612	KE	KE		TODO		time invariant force conserves energy
6397683463	V \alpha = \left( \frac{\partial V}{\partial T} \right)_p	T V \alpha		TODO		first law of thermodynamics
6404535647	\cosh x = \frac{\exp(x) + \exp(-x)}{2}	x		TODO		hyperbolic trigonometric identities
6408214498	c^2	c		TODO		Lorentz transformation
6410818363	\theta	\theta		TODO		equations of motion in 2D (calculus)
6417359412	v_0	v_0		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
6421241247	d = v t - \frac{1}{2} a t^2	v a t d		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
6450985774	n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )	\theta_1 \theta_2 n_1 n_2	Law of Refraction	TODO	eq 34-44 on page 819 in \cite{2001_HRW}	optics: Law of refraction to Brewster's angle
6457044853	v - a t = v_0	v_0 a t v		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
6457999644	\frac{[S_0]}{[A_{\rm adsorption}]} = \frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1	K_{\rm equilibrium} [S_0] p_A [A_{\rm adsorption}]		TODO		Langmuir Adsorption
6463266449	t_f	t_f		TODO		angle of maximum distance for projectile motion
6498985149	v_{\rm escape}	v_{\rm escape}		TODO		escape velocity
6504442697	v = \sqrt{ \frac{K}{\rho} }	v \rho		TODO		upper limit on velocity in condensed matter
6529120965	B	B		TODO		double intensity when phase is coherent (optics)
6529793063	I_{\rm incoherent} = \|A\|^2 + \|A\|^2	I_{\rm incoherent} A		TODO		double intensity when phase is coherent (optics)
6535639720	r_{\rm Earth}	r_{\rm Earth}		TODO		mass of the Earth
6546594355	R_{\rm total}	R_{\rm total}		TODO		total electrical resistance for circuit with two resistors in parallel
6554292307	t	t		TODO		time invariant force conserves energy
6555185548	A^* = \|A\| \exp(-i \theta)	A \theta i		TODO		double intensity when phase is coherent (optics)
6556875579	\frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2	I_{\rm incoherent} I_{\rm coherent}		TODO		double intensity when phase is coherent (optics)
6572039835	-g t + v_{0, y} = v_y	v_y v_{0, y} g t		TODO		equations of motion in 2D (calculus)
6599829782	v_{\rm final}	v_{\rm final}		TODO		velocity at distance r of object dropped from infinity
6672141531	dt	dt		TODO		equations of motion in 2D (calculus)
6681646197	v	v		TODO		escape velocity
6701855578	v_2	v_2		TODO		work and force and energy
6715248283	PE = -F x	x PE F	potential energy	TODO	Potential_energy	time invariant force conserves energy
6729698807	v_0	v_0		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
6732786762	t	t		TODO		angle of maximum distance for projectile motion
6742123016	\vec{p}_{electron}\cdot\vec{p}_{electron} = ( \vec{p}_{1}\cdot\vec{p}_{1})+( \vec{p}_{2}\cdot\vec{p}_{2})-2( \vec{p}_{1}\cdot\vec{p}_{2})	\vec{p}_{\rm electron}		TODO		Compton's equation for scattering
6749533119	PE_1	PE_1		TODO		time invariant force conserves energy
6753224061	I_{\rm total} = I_1 + I_2	I_2 I_1 I_{\rm total}		TODO		total electrical resistance for circuit with two resistors in parallel
6774684564	\theta = \phi	\theta \phi		TODO	for coherent waves	double intensity when phase is coherent (optics)
6783009163	r_{\rm adsorption} = r_{\rm desorption}	r_{\rm desorption} r_{\rm adsorption}		TODO		Langmuir Adsorption
6785303857	C = 2 \pi r	\pi C r		TODO		Newton's Law of Gravitation radius for satellite in geostationary orbit speed of Earth around Sun
6800170830	r_{\rm Schwarzschild} = \frac{2 G m}{c^2}	r_{\rm Schwarzschild} m c G		TODO		Schwarzschild radius for non-rotating black hole
6829281943	F_{\rm centripetal} = G \frac{m_1 m_2}{r^2}	G m_1 F_{\rm centripetal} m_2 r		TODO		Kepler's Third Law: period squared propto distance cubed
6831637424	\sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} )	\theta_{\rm Brewster}		TODO		optics: Law of refraction to Brewster's angle
6831694380	a = \frac{d^2 x}{dt^2}		acceleration	TODO		equation of motion for a spring
6838659900	KE_2	KE_2		TODO		time invariant force conserves energy
6870322215	KE_{\rm escape} = \frac{1}{2} m v_{\rm escape}^2	m KE_{\rm escape} v_{\rm escape}		TODO		escape velocity
6885625907	\exp(i \pi) = -1 + i 0	\pi i		TODO		Euler equation to e^(i pi) + 1 = 0
6892595652	\frac{1}{2} m_1 v_{\rm final}^2 = \frac{G m_1 m_2}{r}	m_1 G v_{\rm final} m_2 r		TODO		velocity at distance r of object dropped from infinity
6908055431	x(t) = A \cos\left(\frac{k}{m} t\right)	A m t		TODO		equation of motion for a spring
6925244346	\alpha = \frac{PV}{T} \frac{1}{VP}	T V P \alpha		TODO		coefficient of thermal expansion using the equation of state for an ideal gas
6935745841	F = G \frac{m_1 m_2}{x^2}	G x m_1 F m_2	Newton's law of universal gravitation	TODO	Newton%27s_law_of_universal_gravitation#Modern_form	mass of the Earth radius for satellite in geostationary orbit escape velocity
6946088325	v = \frac{C}{t}	t C v		TODO		speed of Earth around Sun
6955192897	r_{\rm desorption} = k_{\rm desorption} [A_{\rm adsorption}]	r_{\rm desorption} k_{\rm desorption} [A_{\rm adsorption}]		TODO		Langmuir Adsorption
6964468708	KE_1	KE_1		TODO		time invariant force conserves energy
6974054946	\frac{1}{2} g t_f	g t_f		TODO		angle of maximum distance for projectile motion
6976493023	x	x		TODO		hyperbolic trigonometric identities
6998364753	v_{\rm Earth\ orbit} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}	v_{\rm Earth\ orbit} \pi		TODO		speed of Earth around Sun
7002609475	\frac{V}{R_2} = I_2	I_2 V R_2		TODO		total electrical resistance for circuit with two resistors in parallel
7010294143	T_{\rm orbit}^2 G m_{\rm Earth} = 4 \pi^2 r^3	G m_{\rm Earth} r T_{\rm orbit} \pi		TODO		radius for satellite in geostationary orbit
7011114072	d = \frac{(v_0 + a t) + v_0}{2} t	v_0 a t d		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
7049769409	2			TODO		equations of motion in 2D (calculus)
7053449926	r_{\rm geostationary\ orbit}	r_{\rm geostationary\ orbit}		TODO		radius for satellite in geostationary orbit
7057864873	y' = y	y y'		TODO	frame of reference is moving only along x direction	Lorentz transformation
7083390553	t	t		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
7107090465	B B^* = \|B\|^2	B		TODO		double intensity when phase is coherent (optics)
7112613117	m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}}	m_{\rm Earth}		TODO		mass of the Earth
7112646057	v_{\rm final}^2 = \frac{2 G m_2}{r}	v_{\rm final} m_2 r G		TODO		velocity at distance r of object dropped from infinity
7140470627	m	m		TODO		escape velocity
7154592211	\theta_2	\theta_2		TODO		optics: Law of refraction to Brewster's angle
7159989263	i x	x i		TODO		hyperbolic trigonometric identities
7175416299	t_{\rm Earth\ orbit} = 1 {\rm year}	t_{\rm Earth\ orbit}		TODO		speed of Earth around Sun
7191277455	R	R		TODO		total electrical resistance for circuit with two resistors in parallel
7194432406	r_{\rm Schwarzschild}	r_{\rm Schwarzschild}		TODO		Schwarzschild radius for non-rotating black hole
7214442790	x	x		TODO		angle of maximum distance for projectile motion
7215099603	v^2 = v_0^2 + 2 a t v_0 + a^2 t^2	v_0 a t v		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
7217021879	R_{\rm total} = R_1 + R_2	R_{\rm total} R_1 R_2		TODO		total electrical resistance for circuit with two resistors in series
7233558441	d = v_0 t_f \cos(\theta)	v_0 t_f \theta d		TODO		angle of maximum distance for projectile motion
7252338326	v_y = \frac{dy}{dt}	y v_y t		TODO		equations of motion in 2D (calculus)
7263534144	c^2	c		TODO		Schwarzschild radius for non-rotating black hole
7267155233	\frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)	PE_2 x_2 t x_1 F PE_1		TODO		time invariant force conserves energy
7267424860	\frac{1}{\theta_A} = \frac{1+(K_{\rm equilibrium}\ p_A)}{K_{\rm equilibrium}\ p_A}	K_{\rm equilibrium} \theta_A p_A		TODO		Langmuir Adsorption
7321695558	\theta_{\rm Brewster}	\theta_{\rm Brewster}		TODO		optics: Law of refraction to Brewster's angle
7326066466	G	G		TODO		mass of the Earth
7337056406	\gamma^2 x	x \gamma		TODO		Lorentz transformation
7354529102	y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0	y v_{0, y} x v_{0, x} g y_0 x_0		TODO		projectile path in 2D is parabolic
7375348852	\theta_{\rm Brewster}	\theta_{\rm Brewster}		TODO		optics: Law of refraction to Brewster's angle
7376526845	\sin(\theta) = \frac{v_{0, y}}{v_0}	v_{0, y} v_0 \theta		TODO		equations of motion in 2D (calculus)
7391837535	\cos(\theta) = \frac{v_{0, x}}{v_0}	v_{0, x} \theta v_0		TODO		equations of motion in 2D (calculus)
7410124465	R_{\rm total}	R_{\rm total}		TODO		total electrical resistance for circuit with two resistors in parallel
7410526982	2/m_1	m_1		TODO		velocity at distance r of object dropped from infinity
7445388869	-1			TODO		Kepler's Third Law: period squared propto distance cubed
7453225570	x	x		TODO		hyperbolic trigonometric identities
7455581657	v_{0, x} = \frac{dx}{dt}	dx v_{0, x} t		TODO		equations of motion in 2D (calculus)
7466829492	\vec{ \nabla} \cdot \vec{E} = 0	E		TODO		Maxwell equations to electric field wave equation
7473576008	\frac{-1}{A \cos(\omega t)}	t		TODO		equation of motion for a spring
7476820482	C	C		TODO		speed of Earth around Sun
7497687256	V	V		TODO		total electrical resistance for circuit with two resistors in series
7513513483	\gamma^2 (c^2 - v^2) = c^2	c \gamma v		TODO		Lorentz transformation
7517073655	[S_0] = \left(\frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}]	K_{\rm equilibrium} [S_0] p_A [A_{\rm adsorption}]		TODO		Langmuir Adsorption
7556442438	4 \pi^2	\pi		TODO		radius for satellite in geostationary orbit
7560908617	m	m		TODO		mass of the Earth
7564010952	-1			TODO		Langmuir Adsorption
7564894985	\int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)	x n \pi W		TODO		particle in a 1D box
7572664728	\cos(2 x) + 2 (\sin(x))^2 = 1	x		TODO		Euler equation: trig square root
7573835180	PE_{\rm Earth\ surface} = -W	W PE_{\rm Earth\ surface}		TODO	the potential energy at the surface of the Earth is equal to the work needed to get it from the center of the Earth to the surface	escape velocity
7575738420	\left(\sin\left(\frac{n \pi}{W}x\right) \right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}	\pi x x n W		TODO		particle in a 1D box
7575859295	\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})	E		TODO		curl curl identity Maxwell equations to electric field wave equation
7575859300	\epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{ \nabla} \times \vec{E} )_k = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})	j		TODO		curl curl identity
7575859302	\epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})	i E j \vec{E} \hat{x}_i k n m		TODO		curl curl identity
7575859304	\epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}	j l i k n m		TODO	Covariance_and_contravariance_of_vectors	curl curl identity
7575859306	\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})	E j \vec{E} l \hat{x}_i k n m		TODO	Covariance_and_contravariance_of_vectors	curl curl identity
7575859308	\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})	E j l k \hat{x}_i n m		TODO	Covariance_and_contravariance_of_vectors	curl curl identity
7575859310	\hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})	E \hat{x}_m \hat{x}_n \vec{E} n m		TODO		curl curl identity
7575859312	\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})			TODO		curl curl identity
7587034465	\theta	\theta		TODO		angle of maximum distance for projectile motion
7607271250	\theta	\theta		TODO		double intensity when phase is coherent (optics)
7621705408	I = \|A\|^2 + \|B\|^2 + \|A\| \|B\| \exp(-i \theta) \exp(i \phi) + \|A\| \|B\| \exp(i \theta) \exp(-i \phi)	\theta A I B \phi i		TODO		double intensity when phase is coherent (optics)
7630953440	\frac{K_{\rm equilibrium} p_A}{K_{\rm equilibrium} p_A}	K_{\rm equilibrium} p_A		TODO		Langmuir Adsorption
7652131521	\frac{dx}{dt} = -A \omega \sin (\omega t)	x \omega A t		TODO		equation of motion for a spring
7672365885	F_{gravitational} = \frac{4 \pi^2 m r}{T^2}	r F_{\rm gravity} T_{\rm orbit} m_2 \pi		TODO		Newton's Law of Gravitation
7675171493	V_1 = I R_1	V_1 R_1 I		TODO		total electrical resistance for circuit with two resistors in series
7676652285	KE_2 = \frac{1}{2} m v_2^2	KE_2 m v_2		TODO		time invariant force conserves energy work and force and energy
7696214507	n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} )	\theta_{\rm Brewster} n_1 n_2		TODO		optics: Law of refraction to Brewster's angle
7701249282	v_u = \alpha c \sqrt{ \frac{m_e}{m_p} }	m_e v_u c \alpha m_p		TODO	when A = 1	upper limit on velocity in condensed matter
7708501762	C_{\rm Earth\ orbit}	C_{\rm Earth\ orbit}		TODO		speed of Earth around Sun
7729413831	a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)	\hat{y} v_x a_x t a_y v_y \hat{x}		TODO		equations of motion in 2D (calculus)
7731226616	{\rm sech}\ x = \frac{1}{\cosh x}	x		TODO		hyperbolic trigonometric identities
7734996511	PE_2 - PE_1 = -F ( x_2 - x_1 )	PE_2 x_2 x_1 F PE_1		TODO		time invariant force conserves energy
7735731560	\cosh^2 x - \sinh^2 x = \frac{1}{4}\left( \exp(2x)+1+1+\exp(-2x) - \left(\exp(2x)-1-1-\exp(-2x)\right) \right)	x		TODO
7735737409	\frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}	v_1 m KE_1 t KE_2 v_2 v		TODO		time invariant force conserves energy
7741202861	x = \gamma^2 x - \gamma^2 v t + \gamma v t'	x t t' \gamma v		TODO		Lorentz transformation
7743841045	\gamma^2	\gamma		TODO		Lorentz transformation
7749253510	W = G \frac{m_{\rm Earth} m }{ r_{\rm Earth}}	m G r_{\rm Earth} W m_{\rm Earth}		TODO		escape velocity
7774819339	R	R		TODO		total electrical resistance for circuit with two resistors in series
7798615279	I_{\rm total}	I_{\rm total}		TODO		total electrical resistance for circuit with two resistors in parallel
7816982139	m/s^2			TODO		mass of the Earth
7819443873	r	r		TODO		radius for satellite in geostationary orbit
7826132469	\left(\frac{\partial U}{\partial T}\right)_p = C_V + \pi_T V \alpha	U T		TODO		first law of thermodynamics
7837519722	v = \sqrt{f} \sqrt{\frac{E}{m}}	m E v f		TODO		upper limit on velocity in condensed matter
7844317489	I	I		TODO		total electrical resistance for circuit with two resistors in series
7846240076	m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{G}	G r_{\rm Earth} m_{\rm Earth}		TODO		mass of the Earth
7857757625	n_1	n_1		TODO		optics: Law of refraction to Brewster's angle
7875206161	E_2 = KE_2 + PE_2	E_2 KE_2 PE_2		TODO		time invariant force conserves energy escape velocity
7882872592	W_{\rm to\ system} = \int_{\infty}^r \vec{F}\cdot d\vec{r}	\vec{F} r W_{\rm to\ system}		TODO		velocity at distance r of object dropped from infinity
7905984866	m_1	m_1		TODO		Newton's Law of Gravitation
7906112355	\gamma^2 = \frac{c^2}{c^2 - \gamma^2}	c \gamma		TODO		Lorentz transformation
7912578203	v	v		TODO		radius for satellite in geostationary orbit
7917051060	\vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}	\vec{p}_2 \vec{p}_1 \vec{p}_{\rm electron}		TODO		Compton's equation for scattering
7924063906	K_{equilibrium} = \frac{k_{\rm adsorption}}{k_{\rm desorption}}	K_{\rm equilibrium} k_{\rm desorption} k_{\rm adsorption}		TODO		Langmuir Adsorption
7924842770	T	T		TODO		coefficient of thermal expansion using the equation of state for an ideal gas
7928111771	\frac{1}{\theta_A} = \frac{1}{K_{\rm equilibrium} p_A} + 1	K_{\rm equilibrium} \theta_A p_A		TODO		Langmuir Adsorption
7935917166	r_{\rm Earth}	r_{\rm Earth}		TODO		mass of the Earth
7939765107	v^2 = v_0^2 + 2 a d	d a v_0 v		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
7939947931	KE_2	KE_2		TODO		time invariant force conserves energy
8014566709	\gamma^2 v t	\gamma t v		TODO		Lorentz transformation
8020058613	r	r		TODO		escape velocity
8044416349	d_2	d_2		TODO		Kepler's Third Law: period squared propto distance cubed
8046208134	I_{\rm coherent} = \|A\|^2 + \|A\|^2 + \|A\| \|A\| 2	A I_{\rm coherent}		TODO		double intensity when phase is coherent (optics)
8049905441	\Delta KE = KE_{\rm final} - KE_{\rm initial}	\Delta KE KE_{\rm initial} KE_{\rm final}	change in kinetic energy	TODO		velocity at distance r of object dropped from infinity
8059639673	v^2 = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2}	T_{\rm orbit} \pi r v		TODO		radius for satellite in geostationary orbit
8061701434	PE_1	PE_1		TODO		time invariant force conserves energy
8065128065	I = A A^* + B B^* + A B^* + B A^*	A I B		TODO		double intensity when phase is coherent (optics)
8066819515	v	v		TODO		velocity at distance r of object dropped from infinity
8072682558	x_0	x_0		TODO		angle of maximum distance for projectile motion
8090924099	v = \sqrt{ \left( f\frac{E}{a^3} \right) \frac{1}{\rho} }	f \rho E a v		TODO		upper limit on velocity in condensed matter
8106885760	\alpha = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c}	\epsilon_0 \hbar c \alpha e \pi		TODO	fine structure constant definition	upper limit on velocity in condensed matter
8111389082	x	x		TODO		work and force and energy
8120663858	y_f	y_f		TODO		angle of maximum distance for projectile motion
8122039815	\frac{d_1+d_2}{d_1+d_2}	d_1 d_2		TODO		Kepler's Third Law: period squared propto distance cubed
8131665171	\frac{1}{\theta_A} = \frac{[S_0]}{[A_{\rm adsorption}]}	\theta_A [S_0] [A_{\rm adsorption}]		TODO		Langmuir Adsorption
8135396036	t	t		TODO		speed of Earth around Sun
8139187332	\vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}	\vec{p}_2 \vec{p}_1 \vec{p}_{\rm electron}		TODO		Compton's equation for scattering
8145337879	-g t dt + v_{0, y} dt = dy	v_{0, y} g t dt dy		TODO		equations of motion in 2D (calculus)
8162179726	k_{\rm adsorption} p_A	p_A k_{\rm adsorption}		TODO		Langmuir Adsorption
8173074178	x	x		TODO		time invariant force conserves energy
8198310977	0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0	g \theta v_0 y_0 t_f		TODO		angle of maximum distance for projectile motion
8228733125	a_y = \frac{d}{dt} v_y	a_y v_y t		TODO		equations of motion in 2D (calculus)
8257621077	\vec{p}_{\rm before} = \vec{p}_{1}	\vec{p}_{\rm before} \vec{p}_1		TODO		Compton's equation for scattering
8269198922	2 a d = v^2 - v_0^2	v a v_0 d		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
8283354808	I_{\rm coherent} = \|A\|^2 + \|B\|^2 + \|A\| \|B\| 2 \cos( 0 )	A I_{\rm coherent} B		TODO		double intensity when phase is coherent (optics)
8311458118	\vec{p}_{\rm after} = \vec{p}_{2}+\vec{p}_{electron}	\vec{p}_2 \vec{p}_{\rm electron} \vec{p}_{\rm after}		TODO		Compton's equation for scattering
8332931442	\exp(i \pi) = \cos(\pi)+i \sin(\pi)	\pi i		TODO		Euler equation to e^(i pi) + 1 = 0
8357234146	KE = \frac{1}{2} m v^2	v m KE	kinetic energy	TODO	Kinetic_energy	time invariant force conserves energy velocity at distance r of object dropped from infinity escape velocity
8360117126	\gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}}	c \gamma v		TODO	not a physically valid result in this context	Lorentz transformation
8361238989	a_{centripetal} = \frac{v^2}{r}	r v		TODO		Newton's Law of Gravitation
8362338572	v_{\rm escape}	v_{\rm escape}		TODO		Schwarzschild radius for non-rotating black hole
8368984890	\kappa_T = \frac{-1}{V} \left( \frac{ \partial }{\partial P}\left(\frac{nRT}{P}\right) \right)_T	n R T \kappa_T V P		TODO		coefficient of isothermal compressibility using the equation of state for an ideal gas
8396997949	I = \| A + B \|^2	A I B		TODO	intensity of two waves traveling opposite directions on same path	double intensity when phase is coherent (optics)
8399484849	\langle x^2 - 2 x \langle x \rangle + \langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2	x		TODO		variance relation
8405272745	W_{\rm to\ system} = -G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx	G x m_1 W_{\rm to\ system} m_2 r		TODO		velocity at distance r of object dropped from infinity
8406170337	y	y		TODO		angle of maximum distance for projectile motion
8416464049	KE_{\rm escape}	KE_{\rm escape}		TODO		escape velocity
8418527415	\sin(i x) = i \sinh(x)	x i		TODO		hyperbolic trigonometric identities
8435841627	P V = n R T	n R T V P		TODO	Ideal_gas_law	coefficient of thermal expansion using the equation of state for an ideal gas coefficient of isothermal compressibility using the equation of state for an ideal gas
8460820419	v_x = \frac{dx}{dt}	v_x dx t		TODO		equations of motion in 2D (calculus)
8483686863	\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x) \right)	x i		TODO		identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation
8484544728	-a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)	x		TODO		particle in a 1D box
8485757728	a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)	dx		TODO		particle in a 1D box
8485867742	\frac{2}{W} = a^2	a W		TODO		particle in a 1D box
8486706976	v_{0, x} t + x_0 = x	x v_{0, x} t x_0		TODO		equations of motion in 2D (calculus)
8489593958	d(u v) = u dv + v du	u		TODO		integration by parts
8489593960	d(u v) - v du = u dv	u		TODO		integration by parts
8489593962	u dv = d(u v) - v du	u		TODO		integration by parts
8489593964	\int u dv = u v - \int v du	u v		TODO		integration by parts
8494839423	\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}	\epsilon_0 \vec{E} \mu_0 t		TODO		electric field wave equation: from time dependent to time independent Maxwell equations to electric field wave equation
8495187962	\theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }	\theta_{\rm Brewster} n_1 n_2		TODO		optics: Law of refraction to Brewster's angle
8497631728	I = \|A\|^2 + \|B\|^2 + \|A\| \|B\| 2 \cos( \theta - \phi )	\theta A I B \phi		TODO		double intensity when phase is coherent (optics)
8515803375	z' = z	z z'		TODO	frame of reference is moving only along x direction	Lorentz transformation
8532702080	\cosh^2 x = \left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right)	x		TODO		hyperbolic trigonometric identities
8552710882	KE_{\rm final} = \frac{1}{2} m_1 v_{\rm final}^2	m_1 v_{\rm final} KE_{\rm final}		TODO		velocity at distance r of object dropped from infinity
8558338742	E_2 = E_1	E_2 E_1	conservation of energy	TODO	Conservation_of_energy	time invariant force conserves energy escape velocity
8563535636	\cosh^2 x - \sinh^2 x = \left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right) - \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)	x		TODO		hyperbolic trigonometric identities
8571466509	c^2 - \gamma^2	c \gamma		TODO		Lorentz transformation
8572657110	1 = \int \|\psi(x)\|^2 dx	x \psi		TODO		particle in a 1D box
8572852424	\vec{E} = E( \vec{r},t)	E \vec{E} t \vec{r}		TODO		electric field wave equation: from time dependent to time independent
8575746378	\int \frac{1}{2} dx = \frac{1}{2} x	x		TODO		particle in a 1D box
8575748999	\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x) \right) = -k^2 \left(a \sin(kx) + b \cos(kx) \right)	k dx a b x		TODO		particle in a 1D box
8576785890	1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx	a \pi x n W		TODO		particle in a 1D box
8577275751	0 = a \sin(0) + b\cos(0)	b a		TODO		particle in a 1D box
8582885111	\psi(x) = a \sin(kx) + b \cos(kx)	k a x b \psi		TODO		particle in a 1D box
8582954722	x^2 + 2 x h + h^2 = (x + h)^2	x h		TODO		quadratic equation derivation
8584698994	-g \int dt = \int d v_y	d v_y		TODO		equations of motion in 2D (calculus)
8588429722	\sin( 90^{\circ} - x ) = \cos( x )	x		TODO		optics: Law of refraction to Brewster's angle
8602221482	\langle \cos(\theta - \phi) \rangle = 0	\theta \phi		TODO	incoherent light source	double intensity when phase is coherent (optics)
8602512487	\vec{a} = a_x \hat{x} + a_y \hat{y}	\hat{y} a_x a_y \hat{x} \vec{a}		TODO	decompose acceleration into two components	equations of motion in 2D (calculus)
8604483515	dW = G \frac{m_1 m_2}{x^2} dx	dx G m_1 x dW m_2		TODO		escape velocity
8607458157	dt	dt		TODO		equations of motion in 2D (calculus)
8642992037	2			TODO		identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation
8651044341	\cos(i x) = \frac{1}{2} \left( \exp(-x) + \exp(x) \right)	x i		TODO		hyperbolic trigonometric identities
8655294002	a = -\frac{k}{m}x	x m k a		TODO		equation of motion for a spring
8661803554	F = G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2}	m G r_{\rm Earth} F m_{\rm Earth}		TODO		mass of the Earth
8688588981	a^3 \rho = m	a m \rho		TODO		upper limit on velocity in condensed matter
8699789241	2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)	x i		TODO		identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation
8706092970	d = \left(\frac{v + v_0}{2}\right)t	v v_0 t d		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
8710504862	A	A		TODO		double intensity when phase is coherent (optics)
8717193282	dt	dt		TODO		equations of motion in 2D (calculus)
8721295221	t_{\rm Earth\ orbit} = 3.16 10^7 {\rm seconds}	t_{\rm Earth\ orbit}		TODO		speed of Earth around Sun
8730201316	\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'	\gamma		TODO	first term was multiplied by \gamma/\gamma	Lorentz transformation
8747785338	\cos(i x) = \cosh(x)	x i		TODO		hyperbolic trigonometric identities
8750379055	0 = \frac{d}{dt} v_x	v_x t		TODO		equations of motion in 2D (calculus)
8808860551	-g \int t dt + v_{0, y} \int dt = \int dy	y v_{0, y} g t		TODO		equations of motion in 2D (calculus)
8849289982	\psi(x)^* = a \sin(\frac{n \pi}{W} x)	a \pi x \psi n W		TODO		particle in a 1D box
8854422847	dT	T		TODO		first law of thermodynamics
8857931498	c	c		TODO		upper limit on velocity in condensed matter
8865085668	t	t		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
8871333437	PE_{\rm Earth\ surface}	PE_{\rm Earth\ surface}		TODO		escape velocity
8880467139	2			TODO		equations of motion in 2D (calculus)
8889444440	1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right) \right)^2 dx	a \pi x n W		TODO		particle in a 1D box
8908736791	\rho = \frac{m}{a^3}	a m \rho		TODO	geometry	upper limit on velocity in condensed matter
8916428651	m	m		TODO		Kepler's Third Law: period squared propto distance cubed
8922441655	d = \frac{v_0^2}{g} \sin(2 \theta)	v_0 g \theta d		TODO		angle of maximum distance for projectile motion
8945218208	\theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ}	\theta_{\rm Brewster}		TODO	based on figure 34-27 on page 824 in \cite{2001_HRW}	optics: Law of refraction to Brewster's angle
8946383937	v_{\rm escape}^2 = 2 G \frac{m}{r}	G m r v_{\rm escape}		TODO		Schwarzschild radius for non-rotating black hole
8949329361	v_0 \sin(\theta) = v_{0, y}	v_{0, y} v_0 \theta		TODO		equations of motion in 2D (calculus)
8953094349	W = m a x	x m W a		TODO		work and force and energy
8960645192	KE_2 + PE_2 = KE_1 + PE_1	j KE_1 PE_2 PE_1		TODO		escape velocity
8991236357	\frac{d^2 x}{dt^2} = -\frac{k}{m} x	x m k		TODO		equation of motion for a spring
9025853427	\theta_{\rm Brewster}	\theta_{\rm Brewster}		TODO		optics: Law of refraction to Brewster's angle
9029795851	\theta_{\rm Brewster}	\theta_{\rm Brewster}		TODO		optics: Law of refraction to Brewster's angle
9031609275	x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'	\gamma		TODO		Lorentz transformation
9040079362	f	f		TODO		frequency and period
9053099840	I	I		TODO		total electrical resistance for circuit with two resistors in parallel
9059289981	\psi(x) = a \sin(k x)	x		TODO		particle in a 1D box
9063568209	V_{\rm total} = V_1 + V_2	V_1 V_{\rm total} V_2		TODO		total electrical resistance for circuit with two resistors in series
9070394000	m_2 d_2 \frac{4 \pi^2}{T^2} = G \frac{m_1 m_2}{r^2}	T d_2 G m_1 r m_2 \pi		TODO		Kepler's Third Law: period squared propto distance cubed
9070454719	v_0^2	v_0		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
9072369552	m_{\rm Earth}	m_{\rm Earth}		TODO		escape velocity
9081138616	W_{\rm by\ system} = \frac{1}{2} m_1 v_{\rm final}^2	W_{\rm by\ system} v_{\rm final} m_1		TODO		velocity at distance r of object dropped from infinity
9110536742	2 x	x		TODO		identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation
9112191201	y_f = 0	y_f		TODO		angle of maximum distance for projectile motion
9152823411	\frac{1}{T^2} = \frac{1}{d_2 4 \pi^2} G \frac{m_1}{r^2}	T \pi d_2 G m_1 r		TODO		Kepler's Third Law: period squared propto distance cubed
9170048197	T^2 = d_2 4 \pi^2 \frac{r^2}{G m_1}	T d_2 G m_1 r \pi		TODO		Kepler's Third Law: period squared propto distance cubed
9174439158	R_1	R_1		TODO		total electrical resistance for circuit with two resistors in series
9180861128	2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)	x i		TODO		identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation
9191880568	Z Z^* = \|Z\| \|Z\| \exp( -i \theta ) \exp( i \theta )	Z		TODO		double intensity when phase is coherent (optics)
9226945488	F = \frac{m v^2}{r}	m r F v	Centripetal force	TODO	Centripetal_force	radius for satellite in geostationary orbit
9243879541	V = I_2 R_2	R_2 V I_2		TODO		total electrical resistance for circuit with two resistors in parallel
9262596735	d = 2 \pi r	\pi r d		TODO		radius for satellite in geostationary orbit
9285928292	ax^2 + bx + c = 0	b x a c		TODO		quadratic equation derivation
9291999979	\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t}	\vec{H} \vec{E} \mu_0 t		TODO		Maxwell equations to electric field wave equation
9294858532	\hat{A}^+ = \hat{A}	\hat{A}		TODO		quantum basics Hermitian operators have realvalued observables
9305761407	v	v		TODO		time invariant force conserves energy
9337785146	v = \frac{x_2 - x_1}{t}	x_1 x_2 t v	average velocity	TODO		time invariant force conserves energy
9341391925	\vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}	\hat{y} v_{0, y} v_{0, x} \hat{x} \vec{v}_0		TODO		equations of motion in 2D (calculus)
9346215480	T_{\rm orbit}	T_{\rm orbit}		TODO		radius for satellite in geostationary orbit
9350663581	\pi	\pi		TODO		Euler equation to e^(i pi) + 1 = 0
9350720370	m	m		TODO		velocity at distance r of object dropped from infinity
9355039511	g	g		TODO		mass of the Earth
9356924046	\frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}	v_1 m KE_1 t KE_2 v_2		TODO		time invariant force conserves energy
9370882921	KE_{\rm escape}	KE_{\rm escape}		TODO		escape velocity
9376481176	K = f \frac{E}{a^3}	a E f		TODO	proportionality coefficient fvaries in the range 1-4 for a majority of elemental solids	upper limit on velocity in condensed matter
9385938295	(x+(b/(2 a)))^2 = -(c/a) + (b/(2 a))^2	b x a c		TODO		quadratic equation derivation
9393939991	\psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)	\pi \psi x n W		TODO		particle in a 1D box
9393939992	\psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)	\pi \psi x n W		TODO		particle in a 1D box
9394939493	\nabla^2 E( \vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r},t)	E \mu_0 t \epsilon_0 \vec{r}		TODO		electric field wave equation: from time dependent to time independent
9397152918	v = \frac{v_1 + v_2}{2}	v_1 v_2 v	average velocity	TODO		time invariant force conserves energy
9407192813	G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2} = m g_{\rm Earth}	m g_{\rm Earth} G r_{\rm Earth} m_{\rm Earth}		TODO		mass of the Earth
9409776983	x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'	\gamma		TODO		Lorentz transformation
9412953728	v_{\rm escape}^2 = 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}	G m_{\rm Earth} r_{\rm Earth} v_{\rm escape}		TODO		escape velocity
9413609246	\cosh^2 x - \sinh^2 x = 1	x		TODO		hyperbolic trigonometric identities
9413699705	W = m a \frac{v_2^2 - v_1^2}{2 a}	v_1 m W a v_2		TODO		work and force and energy
9429829482	\frac{d}{dx} y = -\sin(x) + i\cos(x)	y x i		TODO		Euler equation proof
9440616166	m_{\rm Earth} = \frac{g_{\rm Earth} r_{\rm Earth}^2}{G}	G r_{\rm Earth} g_{\rm Earth} m_{\rm Earth}		TODO		mass of the Earth
9482113948	\frac{dy}{y} = i dx	i		TODO		Euler equation proof
9482438243	(\cos(x))^2 = \cos(2 x) + (\sin(x))^2	x		TODO		Euler equation: trig square root
9482923849	\exp(i x) = y	y x i		TODO		Euler equation proof
9482928242	\cos(2 x) = (\cos(x))^2 - (\sin(x))^2	x		TODO		Euler equation: trig square root
9482928243	\cos(2 x) + (\sin(x))^2 = (\cos(x))^2	x		TODO		Euler equation: trig square root
9482943948	\log(y) = i dx	y dx i		TODO		Euler equation proof
9482984922	\frac{d}{dx} y = (i\sin(x) + \cos(x)) i	y x i		TODO		Euler equation proof
9483928192	\cos(2 x) + i\sin(2 x) = (\cos(x))^2 + 2 i \cos(x) \sin(x) - (\sin(x))^2	x i		TODO		Euler equation: trig square root
9485384858	\nabla^2 E( \vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E( \vec{r})\exp(i \omega t)	E t c \exp \omega \vec{r} i		TODO		electric field wave equation: from time dependent to time independent
9485747245	\sqrt{\frac{2}{W}} = a	a W		TODO		particle in a 1D box
9485747246	-\sqrt{\frac{2}{W}} = a	a W		TODO		particle in a 1D box
9492920340	y = \cos(x)+i \sin(x)	y x i		TODO		Euler equation proof
9495857278	\psi(x=W) = 0	W		TODO	2022-03-25 BHP: Conversion between Latex and Sympy is incomplete	particle in a 1D box
9499428242	E( \vec{r},t) = E( \vec{r})\exp(i \omega t)	E t \exp \omega \vec{r} i		TODO		electric field wave equation: from time dependent to time independent
9510328252	KE_{\rm initial} = 0	KE_{\rm initial}		TODO		velocity at distance r of object dropped from infinity
9524810853	\frac{1/d_2}{1/d_2}	d_2		TODO		Kepler's Third Law: period squared propto distance cubed
9562264720	[S] = \frac{k_{\rm desorption} [A_{\rm adsorption}]}{k_{\rm adsorption} p_A}	k_{\rm adsorption} [S] p_A [A_{\rm adsorption}] k_{\rm desorption}		TODO		Langmuir Adsorption
9565166889	T	T		TODO		frequency and period
9582958293	x = \sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))	b x a c		TODO		quadratic equation derivation
9582958294	x+(b/(2 a)) = \sqrt{(b/(2 a))^2 - (c/a)}	b x a c		TODO		quadratic equation derivation
9585727710	\psi(x=0) = 0	x \psi		TODO		particle in a 1D box
9590696981	9.80665			TODO		mass of the Earth
9594072504	m_2	m_2		TODO		Newton's Law of Gravitation
9596004948	x = \langle\psi_{\alpha}\| \hat{A} \|\psi_{\beta}\rangle	x \| \psi_{\beta} \rangle \hat{A} \langle \psi_{\alpha} \|		TODO		quantum basics orthogonality
9601500174	v_{\rm Earth\ orbit}	v_{\rm Earth\ orbit}		TODO		speed of Earth around Sun
9623791270	d	x		TODO		work and force and energy
9640720571	v = \frac{e^2}{4 \pi \epsilon_0 \hbar} \sqrt{\frac{m_e}{2 m}}	\epsilon_0 m_e \hbar m v e \pi		TODO		upper limit on velocity in condensed matter
9645178657	a t	a t		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
9658195023	d = v_0 t + \frac{1}{2} a t^2	v_0 a t d		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
9674924517	K >> G	K G		TODO	yfN-LaW1BQAJ	upper limit on velocity in condensed matter
9703482302	G \frac{m_{\rm Earth} m}{r_{\rm Earth}} = \frac{1}{2} m v_{\rm escape}^2	m G v_{\rm escape} r_{\rm Earth} m_{\rm Earth}		TODO		escape velocity
9707028061	a_x = 0	a_x		TODO		equations of motion in 2D (calculus)
9718685793	\kappa_T = \frac{1}{P}	\kappa_T P		TODO		coefficient of isothermal compressibility using the equation of state for an ideal gas
9746066299	R_2	R_2		TODO		total electrical resistance for circuit with two resistors in parallel
9749777192	0 = KE_1 + PE_1	KE_1 PE_1		TODO		escape velocity
9753878784	v	v		TODO		speed of Earth around Sun
9756089533	\sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} )	\theta_{\rm Brewster} n_1 n_2		TODO		optics: Law of refraction to Brewster's angle
9759901995	v - v_0 = a t	v_0 t a v		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
9761485403	Z	Z		TODO		double intensity when phase is coherent (optics)
9781951738	\kappa_T = \frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T	\kappa_T V P		TODO	definition of isothermal compressibility	first law of thermodynamics coefficient of isothermal compressibility using the equation of state for an ideal gas
9789485295	v_{\rm satellite}	v_{\rm satellite}		TODO		radius for satellite in geostationary orbit
9794128647	m_1	m_{\rm Earth}		TODO		radius for satellite in geostationary orbit
9805063945	\gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2	y x t z c \gamma v		TODO		Lorentz transformation
9830343096	V_1	V_1		TODO		total electrical resistance for circuit with two resistors in series
9838128064	d_2 \frac{4 \pi^2}{T^2} = G \frac{m_1}{r^2}	T d_2 G m_1 r \pi		TODO		Kepler's Third Law: period squared propto distance cubed
9847143017	\kappa_T = \frac{-PV}{V} \left( \frac{-1}{P^2}\right)	\kappa_T V P		TODO		coefficient of isothermal compressibility using the equation of state for an ideal gas
9848292229	dy = y i dx	y dx dy i		TODO		Euler equation proof
9848294829	\frac{d}{dx} y = y i	y x i		TODO		Euler equation proof
9854442418	v = \sqrt{\frac{E}{m}}	m v E		TODO		upper limit on velocity in condensed matter
9858028950	\frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx	a \pi x n W		TODO		particle in a 1D box
9862900242	y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0	y g \theta t v_0 y_0		TODO		equations of motion in 2D (calculus) angle of maximum distance for projectile motion
9881106100	a	a		TODO		mass of the Earth
9882526611	v_{0, x} t = x - x_0	x v_{0, x} t x_0		TODO		equations of motion in 2D (calculus) projectile path in 2D is parabolic
9884115626	r	r		TODO		Kepler's Third Law: period squared propto distance cubed
9885190237	i	i		TODO		hyperbolic trigonometric identities
9889984281	2 (\sin(x))^2 = 1 - \cos(2 x)	x		TODO		Euler equation: trig square root
9894826550	2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \left(\exp(i x)+\exp(-i x) \right)	x i		TODO		identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation
9897284307	\frac{d}{t} = \frac{v + v_0}{2}	v v_0 t d		TODO		equations of motion in 1D with constant acceleration - SUVAT (algebra)
9903988330	m	m		TODO		mass of the Earth
9906920183	x	x		TODO		optics: Law of refraction to Brewster's angle
9919999981	\rho = 0	\rho		TODO		Maxwell equations to electric field wave equation
9933742680	r_{\rm Schwarzschild}	r_{\rm Schwarzschild}		TODO		Schwarzschild radius for non-rotating black hole
9941599459	dU = \left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV	U T		TODO	based on U(p, T, V) = U(T, V)	first law of thermodynamics
9956609318	6.67430*10^{-11}			TODO		mass of the Earth
9958485859	\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)	m \hbar \psi t \vec{r}		TODO		derivation of Schrodinger Equation
9973952056	-g t = v_y - v_{0, y}	v_{0, y} v_0 g t		TODO		equations of motion in 2D (calculus)
9988949211	(\sin(x))^2 = \frac{1 - \cos(2 x)}{2}	x		TODO		particle in a 1D box Euler equation: trig square root
9991999979	\vec{ \nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}	\vec{H} E \mu_0 t		TODO		Maxwell equations to electric field wave equation
9999998870	\frac{ \vec{p}}{\hbar} = \vec{k}	\hbar \vec{k} \vec{p}		TODO		derivation of Schrodinger Equation
9999999870	\frac{p}{\hbar} = k	\hbar k p		TODO		derivation of Schrodinger Equation
9999999960	\hbar = h/(2 \pi)	\hbar h \pi		TODO		derivation of Schrodinger Equation
9999999961	\frac{E}{\hbar} = \omega	\hbar \omega E		TODO		derivation of Schrodinger Equation
9999999962	p = \hbar k	\hbar k p		TODO		derivation of Schrodinger Equation
9999999965	E = \omega \hbar	\hbar \omega E		TODO		derivation of Schrodinger Equation
9999999968	x = \frac{-b-\sqrt{b^2-4ac}}{2 a}	b x a c		TODO		quadratic equation derivation
9999999969	x = \frac{-b+\sqrt{b^2-4ac}}{2 a}	b x a c		TODO		quadratic equation derivation
9999999975	\langle \psi\| \hat{A} \|\psi \rangle = \langle a \rangle	\langle \psi\|		TODO		quantum basics Hermitian operators have realvalued observables
9999999981	\vec{ \nabla} \cdot \vec{E} = \rho/\epsilon_0	\epsilon_0 \vec{E} \rho		TODO		Maxwell equations to electric field wave equation

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