Expression list: Physics Derivation Graph

List of Expressions

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ID	latex	name	description	used in derivation	symbols	sympy	dimensional consistency
0007455074	$ \vec{F} = m\ \vec{a} $ \vec{F} = m\ \vec{a}	Newton's second law	vector form relating force to mass and acceleration	Newton's Second Law of Motion: vector to scalar	0000005156: $ m $	empty str sent to sympy_to_latex_str = pdg_{0002423}	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
0203024440	$ 1 = \int_0^W a \sin\left(\frac{n \pi}{W} x\right) \psi(x)^* dx $ 1 = \int_0^W a \sin\left(\frac{n \pi}{W} x\right) \psi(x)^* dx			particle in a 1D box	0000009489: $ \psi $ 0000003141: $ \pi $ 0000001464: $ x $ 0000001592: $ n $ 0000002523: $ W $ 0000009139: $ a $	1 = \int\limits_{0}^{pdg_{0002523}} pdg_{0009139} \sin{\left(\frac{pdg_{0001464} pdg_{0001592} pdg_{0003141}}{pdg_{0002523}} \right)} \overline{\operatorname{pdg}_{0009489}{\left(pdg_{0001464} \right)}}\, dpdg_{0001464}	ERROR for dim with 0203024440
0404050504	$ \lambda = \frac{v}{f} $ \lambda = \frac{v}{f}			frequency relations	0000001115: $ \lambda $ 0000004201: $ f $ 0000001357: $ v $	pdg_{0001115} = \frac{pdg_{0001357}}{pdg_{0004201}}	ERROR for dim with 0404050504
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 $ \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		https://physicsderivationgraph.blogspot.com/2020/09/evaluating-definite-integrals-for.html	particle in a 1D box	0000003141: $ \pi $ 0000001592: $ n $ 0000004037: $ x $ 0000002523: $ W $ 0000009139: $ a $	\frac{1}{pdg_{0009139}^{2}} = \frac{pdg_{0002523}}{2} - \frac{pdg_{0002523} \sin{\left(\frac{2 pdg_{0001592} pdg_{0003141} pdg_{0004037}}{pdg_{0002523}} \right)}}{4 pdg_{0001592} pdg_{0003141}}	ERROR for dim with 0439492440
0934990943	$ k = \frac{2 \pi}{v T} $ k = \frac{2 \pi}{v T}			frequency relations	0000005321: $ k $ 0000009491: $ T $ 0000003141: $ \pi $ 0000001357: $ v $	pdg_{0005321} = \frac{2 pdg_{0003141}}{pdg_{0001357} pdg_{0009491}}	ERROR for dim with 0934990943
0948572140	$ \int \cos(a x) dx = \frac{1}{a}\sin(a x) $ \int \cos(a x) dx = \frac{1}{a}\sin(a x)			particle in a 1D box	0000001464: $ x $ 0000009199: $ dx $ 0000009139: $ a $	\int \cos{\left(pdg_{0001464} pdg_{0009139} \right)}\, dpdg_{0009199} = \frac{\sin{\left(pdg_{0001464} pdg_{0009139} \right)}}{pdg_{0009139}}	ERROR for dim with 0948572140
1010393913	$ \langle \psi\| \hat{A}^+ \|\psi \rangle = \langle a \rangle^* $ \langle \psi\| \hat{A}^+ \|\psi \rangle = \langle a \rangle^*		https://docs.sympy.org/latest/modules/stats.html	quantum basics Hermitian operators have realvalued observables	0000009329: $ \|\psi \rangle $ 0000009139: $ a $ 0000004065: $ \langle \psi\| $ 0000005598: $ \hat{A} $	\operatorname{Bra}{\left(pdg_{0004065} \right)} \operatorname{Dagger}{\left(\operatorname{Operator}{\left(pdg_{0005598} \right)} \right)} \operatorname{Ket}{\left(pdg_{0009329} \right)} = TypeError in get_sympy_as_latex_per_expr_id: 'Exp1' object is not callable	TypeError: unable to parse conjugate(E(Symbol('pdg0009139'))) as SymPy; error='Exp1' object is not callable
1010393944	$ x = \langle\psi_{\alpha}\| a_{\beta} \|\psi_{\beta} \rangle $ x = \langle\psi_{\alpha}\| a_{\beta} \|\psi_{\beta} \rangle			quantum basics orthogonality	0000001464: $ x $ 0000002090: $ \| \psi_{\beta} \rangle $ 0000004679: $ \langle \psi_{\alpha} \| $ 0000007752: $ a_{\beta} $	pdg_{0001464} = pdg_{0007752} \operatorname{Bra}{\left(pdg_{0004679} \right)} \operatorname{Ket}{\left(pdg_{0002090} \right)}	ERROR for dim with 1010393944
1010923823	$ k W = n \pi $ k W = n \pi			particle in a 1D box	0000005321: $ k $ 0000001592: $ n $ 0000002523: $ W $ 0000003141: $ \pi $	pdg_{0002523} pdg_{0005321} = pdg_{0001592} pdg_{0003141}	ERROR for dim with 1010923823
1020010291	$ 0 = a \sin(k W) $ 0 = a \sin(k W)			particle in a 1D box	0000005321: $ k $ 0000002523: $ W $ 0000009139: $ a $	0 = pdg_{0009139} \sin{\left(pdg_{0002523} pdg_{0005321} \right)}	ERROR for dim with 1020010291
1020394900	$ p = h/\lambda $ p = h/\lambda			derivation of Schrodinger Equation	0000004413: $ h $ 0000001134: $ p $ 0000001115: $ \lambda $	pdg_{0001134} = \frac{pdg_{0004413}}{pdg_{0001115}}	ERROR for dim with 1020394900
1020394902	$ E = h f $ E = h f			derivation of Schrodinger Equation	0000004931: $ E $ 0000004413: $ h $ 0000004201: $ f $	pdg_{0004931} = pdg_{0004201} pdg_{0004413}	ERROR for dim with 1020394902
1020854560	$ I = (A + B)(A + B)^* $ I = (A + B)(A + B)^*			double intensity when phase is coherent (optics)	0000007882: $ I $ 0000004453: $ A $ 0000004698: $ B $	pdg_{0007882} = \left(pdg_{0004453} + pdg_{0004698}\right) \left(\overline{pdg_{0004453}} + \overline{pdg_{0004698}}\right)	ERROR for dim with 1020854560
1029039903	$ p = m v $ p = m v			derivation of Schrodinger Equation	0000001134: $ p $ 0000005156: $ m $ 0000001357: $ v $	pdg_{0001134} = pdg_{0001357} pdg_{0005156}	ERROR for dim with 1029039903
1029039904	$ p^2 = m^2 v^2 $ p^2 = m^2 v^2			derivation of Schrodinger Equation	0000001134: $ p $ 0000005156: $ m $ 0000001357: $ v $	pdg_{0001134}^{2} = pdg_{0001357}^{2} pdg_{0005156}^{2}	ERROR for dim with 1029039904
1038566242	$ \sinh x = \frac{\exp(x) - \exp(-x)}{2} $ \sinh x = \frac{\exp(x) - \exp(-x)}{2}			hyperbolic trigonometric identities	0000001464: $ x $	\sinh{\left(pdg_{0001464} \right)} = \frac{e^{pdg_{0001464}}}{2} - \frac{e^{- pdg_{0001464}}}{2}	ERROR for dim with 1038566242
1075552184	$ \|\vec{a}\| = a $ \|\vec{a}\| = a			Newton's Second Law of Motion: vector to scalar	0000009140: $ a $	=	sympy_lhs not provided for expression
1085150613	$ C_V = \left(\frac{\partial U}{\partial T}\right)_V $ C_V = \left(\frac{\partial U}{\partial T}\right)_V		definition of heat capacity at constant volume	first law of thermodynamics	0000005786: $ U $ 0000006682: $ C_V $ 0000007343: $ T $	pdg_{0006682} = \frac{d}{d pdg_{0007343}} pdg_{0005786}	ERROR for dim with 1085150613
1087417579	$ 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) $ 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)			angle of maximum distance for projectile motion	0000001575: $ \theta $ 0000001649: $ g $ 0000002467: $ t_f $ 0000005153: $ v_0 $	0 = - \frac{pdg_{0001649} pdg_{0002467}^{2}}{2} + pdg_{0002467} pdg_{0005153} \sin{\left(pdg_{0001575} \right)}	ERROR for dim with 1087417579
1114820451	$ W_{\rm by\ system} = \Delta KE $ W_{\rm by\ system} = \Delta KE	Work is change in energy		velocity at distance r of object dropped from infinity	0000006191: $ W_{\rm by\ system} $ 0000005734: $ \Delta KE $	pdg_{0006191} = pdg_{0005734}	ERROR for dim with 1114820451
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} $ {\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}			hyperbolic trigonometric identities	0000001464: $ x $	\tanh^{2}{\left(pdg_{0001464} \right)} + \operatorname{sech}^{2}{\left(pdg_{0001464} \right)} = \frac{\left(e^{pdg_{0001464}} - e^{- pdg_{0001464}}\right)^{2}}{\left(e^{pdg_{0001464}} + e^{- pdg_{0001464}}\right)^{2}} + \frac{4}{\left(e^{pdg_{0001464}} + e^{- pdg_{0001464}}\right)^{2}}	ERROR for dim with 1128605625
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} = \frac{(9.80665 m/s^2) (6.378110^6 m)^2}{6.6743010^{-11}m^3 kg^{-1} s^{-2}}			mass of the Earth	0000005458: $ m_{\rm Earth} $	pdg_{0005458} = 6.3781	ERROR for dim with 1132941271
1143343287	$ G \frac{m_{\rm Earth}}{r_{\rm Earth}} = \frac{1}{2} v_{\rm escape}^2 $ G \frac{m_{\rm Earth}}{r_{\rm Earth}} = \frac{1}{2} v_{\rm escape}^2			escape velocity	0000005458: $ m_{\rm Earth} $ 0000008656: $ v_{\rm escape} $ 0000003236: $ r_{\rm Earth} $ 0000006277: $ G $	\frac{pdg_{0005458} pdg_{0006277}}{pdg_{0003236}} = \frac{pdg_{0008656}^{2}}{2}	ERROR for dim with 1143343287
1158485859	$ \frac{-\hbar^2}{2m} \nabla^2 = {\cal H} $ \frac{-\hbar^2}{2m} \nabla^2 = {\cal H}			derivation of Schrodinger Equation	0000001054: $ \hbar $ 0000005156: $ m $	- \frac{nabla^{2} pdg_{0001054}^{2}}{2 pdg_{0005156}} = calH	ERROR for dim with 1158485859
1166310428	$ 0 dt = d v_x $ 0 dt = d v_x			equations of motion in 2D (calculus)	0000005005: $ d v_x $	0 = pdg_{0005005}	ERROR for dim with 1166310428
1172039918	$ I_{\rm coherent} = 4 \|A\|^2 $ I_{\rm coherent} = 4 \|A\|^2			double intensity when phase is coherent (optics)	0000008251: $ I_{\rm coherent} $ 0000004453: $ A $	pdg_{0008251} = 4 \left\|{pdg_{0004453}}\right\|^{2}	ERROR for dim with 1172039918
1189963325	$ F_{\rm gravity} \propto \frac{1}{r^2} $ F_{\rm gravity} \propto \frac{1}{r^2}			Newton's Law of Gravitation	0000002530: $ r $	empty str sent to sympy_to_latex_str \propto \frac{1}{pdg_{0000002530}^{2}}	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
1190768176	$ \kappa_T = \frac{-nRT}{V} \left( \frac{ \partial }{\partial P}\left(\frac{1}{P}\right) \right)_T $ \kappa_T = \frac{-nRT}{V} \left( \frac{ \partial }{\partial P}\left(\frac{1}{P}\right) \right)_T			coefficient of isothermal compressibility using the equation of state for an ideal gas	0000008134: $ P $ 0000008179: $ R $ 0000007586: $ V $ 0000002834: $ n $ 0000004645: $ \kappa_T $ 0000007343: $ T $	pdg_{0004645} = - \frac{pdg_{0002834} pdg_{0007343} pdg_{0008179} \frac{d}{d pdg_{0008134}} \frac{1}{pdg_{0008134}}}{pdg_{0007586}}	ERROR for dim with 1190768176
1191796961	$ \frac{1}{2} g t_f = v_0 \sin(\theta) $ \frac{1}{2} g t_f = v_0 \sin(\theta)			angle of maximum distance for projectile motion	0000001575: $ \theta $ 0000001649: $ g $ 0000002467: $ t_f $ 0000005153: $ v_0 $	\frac{pdg_{0001649} pdg_{0002467}}{2} = pdg_{0005153} \sin{\left(pdg_{0001575} \right)}	ERROR for dim with 1191796961
1201689765	$ x'^2 + y'^2 + z'^2 = c^2 t'^2 $ x'^2 + y'^2 + z'^2 = c^2 t'^2		describes a spherical wavefront for an observer in a moving frame of reference	Lorentz transformation	0000001888: $ y' $ 0000004567: $ c $ 0000005456: $ x' $ 0000004306: $ z' $ 0000004989: $ t' $	pdg_{0001888}^{2} + pdg_{0004306}^{2} + pdg_{0005456}^{2} = pdg_{0004567}^{2} pdg_{0004989}^{2}	ERROR for dim with 1201689765
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 $ \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			particle in a 1D box	0000003141: $ \pi $ 0000009199: $ dx $ 0000001592: $ n $ 0000004037: $ x $ 0000002523: $ W $ 0000009139: $ a $	\frac{1}{pdg_{0009139}^{2}} = \int\limits_{0}^{pdg_{0002523}} \left(\frac{pdg_{0009199}}{2} - \frac{\int\limits_{0}^{pdg_{0002523}} \cos{\left(\frac{2 pdg_{0001592} pdg_{0003141} pdg_{0004037}}{pdg_{0002523}} \right)}\, dpdg_{0004037}}{2}\right)\, dpdg_{0004037}	ERROR for dim with 1202310110
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 $ \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx			particle in a 1D box	0000003141: $ \pi $ 0000001592: $ n $ 0000004037: $ x $ 0000002523: $ W $ 0000009139: $ a $	\frac{1}{pdg_{0009139}^{2}} = \frac{pdg_{0002523}}{2} - \frac{\int\limits_{0}^{pdg_{0002523}} \cos{\left(\frac{2 pdg_{0001592} pdg_{0003141} pdg_{0004037}}{pdg_{0002523}} \right)}\, dpdg_{0004037}}{2}	ERROR for dim with 1202312210
1203938249	$ a_{\beta} \langle \psi_{\alpha} \| \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} \| \psi_{\beta} \rangle $ a_{\beta} \langle \psi_{\alpha} \| \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} \| \psi_{\beta} \rangle			quantum basics orthogonality	0000002090: $ \| \psi_{\beta} \rangle $ 0000004679: $ \langle \psi_{\alpha} \| $ 0000007752: $ a_{\beta} $	pdg_{0007752} \operatorname{Bra}{\left(pdg_{0004679} \right)} \operatorname{Ket}{\left(pdg_{0002090} \right)} = tokenize.TokenError in get_sympy_as_latex_per_expr_id: ('EOF in multi-line statement', (2, 0))	TokenError: unable to parse Symbol('pdg0007752')Bra('pdg0004679')Ket('pdg0002090')) as SymPy; error=('EOF in multi-line statement', (2, 0))
1219718533	$ \|\vec{F}\| = F $ \|\vec{F}\| = F			Newton's Second Law of Motion: vector to scalar	0000006777: $ \vec{F} $ 0000004202: $ F $	Eq(Abs(Symbol('pdg0004202')),Symbol('pdg0004202')) \left\|{pdg_{0004202}}\right\| = pdg_{0004202}	ERROR for dim with 1219718533
1248277773	$ \cos(2 x) = 1 - 2 (\sin(x))^2 $ \cos(2 x) = 1 - 2 (\sin(x))^2			Euler equation: trig square root	0000001464: $ x $	\cos{\left(2 pdg_{0001464} \right)} = 1 - 2 \sin^{2}{\left(pdg_{0001464} \right)}	ERROR for dim with 1248277773
1259826355	$ d = (v - a t) t + \frac{1}{2} a t^2 $ d = (v - a t) t + \frac{1}{2} a t^2			equations of motion in 1D with constant acceleration - SUVAT (algebra)	0000001467: $ t $ 0000009140: $ a $ 0000001943: $ d $ 0000001357: $ v $	pdg_{0001943} = \frac{pdg_{0001467}^{2} pdg_{0009140}}{2} + pdg_{0001467} \left(pdg_{0001357} - pdg_{0001467} pdg_{0009140}\right)	ERROR for dim with 1259826355
1265150401	$ d = \frac{2 v_0 + a t}{2} t $ d = \frac{2 v_0 + a t}{2} t			equations of motion in 1D with constant acceleration - SUVAT (algebra)	0000001467: $ t $ 0000009140: $ a $ 0000005153: $ v_0 $ 0000001943: $ d $	pdg_{0001943} = pdg_{0001467} \left(\frac{pdg_{0001467} pdg_{0009140}}{2} + pdg_{0005153}\right)	ERROR for dim with 1265150401
1292735067	$ F_{\rm gravity} = G \frac{m_1 m_2}{r^2} $ F_{\rm gravity} = G \frac{m_1 m_2}{r^2}			Newton's Law of Gravitation Kepler's Third Law: period squared propto distance cubed	0000006277: $ G $ 0000005022: $ m_1 $ 0000002530: $ r $ 0000002867: $ F_{\rm gravity} $ 0000004851: $ m_2 $	pdg_{0002867} = \frac{pdg_{0004851} pdg_{0005022} pdg_{0006277}}{pdg_{0002530}^{2}}	ERROR for dim with 1292735067
1293913110	$ 0 = b $ 0 = b			particle in a 1D box	0000001939: $ b $	0 = pdg_{0001939}	ERROR for dim with 1293913110
1293923844	$ \lambda = v T $ \lambda = v T			frequency relations	0000001115: $ \lambda $ 0000009491: $ T $ 0000001357: $ v $	pdg_{0001115} = pdg_{0001357} pdg_{0009491}	ERROR for dim with 1293923844
1306360899	$ x = v_{0, x} t + x_0 $ x = v_{0, x} t + x_0			equations of motion in 2D (calculus)	0000001572: $ x_0 $ 0000001467: $ t $ 0000002958: $ v_{0, x} $ 0000004037: $ x $	pdg_{0004037} = pdg_{0001467} pdg_{0002958} + pdg_{0001572}	ERROR for dim with 1306360899
1310571337	$ \theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster} $ \theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster}			optics: Law of refraction to Brewster's angle	0000004928: $ \theta_{\rm Brewster} $	pdg_{0004928} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
1311403394	$ \alpha = \frac{1}{V} \frac{nR}{P} \left( \frac{\partial T}{\partial T} \right)_P $ \alpha = \frac{1}{V} \frac{nR}{P} \left( \frac{\partial T}{\partial T} \right)_P			coefficient of thermal expansion using the equation of state for an ideal gas	0000008134: $ P $ 0000008179: $ R $ 0000007586: $ V $ 0000002834: $ n $ 0000004686: $ \alpha $ 0000007343: $ T $	pdg_{0004686} = \frac{pdg_{0002834} pdg_{0008179} \frac{d}{d pdg_{0007343}} pdg_{0007343}}{pdg_{0007586} pdg_{0008134}}	ERROR for dim with 1311403394
1314464131	$ \vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} $ \vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}			Maxwell equations to electric field wave equation	0000001467: $ t $	pdg_{0001467} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
1314864131	$ \vec{ \nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E} $ \vec{ \nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}			Maxwell equations to electric field wave equation	0000001467: $ t $ 0000007940: $ \epsilon_0 $ 0000004326: $ \vec{E} $ 0000002069: $ \vec{H} $	\operatorname{cross}{\left(nabla,pdg_{0002069} \right)} = pdg_{0007940} \frac{d}{d pdg_{0001467}} pdg_{0004326}	ERROR for dim with 1314864131
1330874553	$ v_{\rm escape} = \sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}} $ v_{\rm escape} = \sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}			escape velocity	0000005458: $ m_{\rm Earth} $ 0000008656: $ v_{\rm escape} $ 0000003236: $ r_{\rm Earth} $ 0000006277: $ G $	pdg_{0008656} = \sqrt{2} \sqrt{\frac{pdg_{0005458} pdg_{0006277}}{pdg_{0003236}}}	ERROR for dim with 1330874553
1357848476	$ A = \|A\| \exp(i \theta) $ A = \|A\| \exp(i \theta)			double intensity when phase is coherent (optics)	0000004621: $ i $ 0000004453: $ A $ 0000001575: $ \theta $	pdg_{0004453} = e^{pdg_{0001575} pdg_{0004621}} \left\|{pdg_{0004453}}\right\|	ERROR for dim with 1357848476
1395858355	$ x = \langle \psi_{\alpha}\| a_{\alpha} \|\psi_{\beta}\rangle $ x = \langle \psi_{\alpha}\| a_{\alpha} \|\psi_{\beta}\rangle			quantum basics orthogonality	0000001464: $ x $ 0000002427: $ a_{\alpha} $ 0000002090: $ \| \psi_{\beta} \rangle $ 0000004679: $ \langle \psi_{\alpha} \| $	pdg_{0001464} = pdg_{0002427} \operatorname{Bra}{\left(pdg_{0004679} \right)} \operatorname{Ket}{\left(pdg_{0002090} \right)}	ERROR for dim with 1395858355
1405465835	$ y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0 $ y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0			equations of motion in 2D (calculus) projectile path in 2D is parabolic	0000001467: $ t $ 0000001649: $ g $ 0000005647: $ y $ 0000009107: $ v_y $ 0000001469: $ y_0 $	pdg_{0005647} = - \frac{pdg_{0001467}^{2} pdg_{0001649}}{2} + pdg_{0001467} pdg_{0009107} + pdg_{0001469}	ERROR for dim with 1405465835
1457415749	$ \frac{1}{R_{\rm total}} = \frac{1}{R_1} + \frac{1}{R_2} $ \frac{1}{R_{\rm total}} = \frac{1}{R_1} + \frac{1}{R_2}	total resistance for two resistors in parallel		total electrical resistance for circuit with two resistors in parallel	0000001908: $ R_{\rm total} $ 0000008697: $ R_1 $ 0000003461: $ R_2 $	\frac{1}{pdg_{0001908}} = \frac{1}{pdg_{0008697}} + \frac{1}{pdg_{0003461}}	ERROR for dim with 1457415749
1525861537	$ I = \|A\|^2 + \|B\|^2 + A B^* + B A^* $ I = \|A\|^2 + \|B\|^2 + A B^* + B A^*			double intensity when phase is coherent (optics)	0000007882: $ I $ 0000004453: $ A $ 0000004698: $ B $	pdg_{0007882} = pdg_{0004453} \overline{pdg_{0004698}} + pdg_{0004698} \overline{pdg_{0004453}} + \left\|{pdg_{0004453}}\right\|^{2} + \left\|{pdg_{0004698}}\right\|^{2}	ERROR for dim with 1525861537
1528310784	$ \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} $ \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}			Lorentz transformation	0000004567: $ c $ 0000001790: $ \gamma $ 0000001357: $ v $	pdg_{0001790} = \frac{1}{\sqrt{- \frac{pdg_{0001357}^{2}}{pdg_{0004567}^{2}} + 1}}	ERROR for dim with 1528310784
1541916015	$ \theta = \frac{\pi}{4} $ \theta = \frac{\pi}{4}			angle of maximum distance for projectile motion	0000001575: $ \theta $ 0000003141: $ \pi $	pdg_{0001575} = \frac{pdg_{0003141}}{4}	ERROR for dim with 1541916015
1556389363	$ E_{\rm Rydberg} = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2} $ E_{\rm Rydberg} = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}		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	0000001054: $ \hbar $ 0000003141: $ \pi $ 0000002515: $ m_e $ 0000001999: $ e $ 0000007940: $ \epsilon_0 $ 0000009838: $ E_{\rm Rydberg} $	pdg_{0009838} = \frac{pdg_{0001999}^{4} pdg_{0002515}}{32 pdg_{0001054}^{2} pdg_{0003141}^{2} pdg_{0007940}^{2}}	ERROR for dim with 1556389363
1559688463	$ \left(\frac{T_{\rm geostationary\ orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r_{\rm geostationary\ orbit} $ \left(\frac{T_{\rm geostationary\ orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r_{\rm geostationary\ orbit}			radius for satellite in geostationary orbit	0000005595: $ T_{\rm geostationary\ orbit} $ 0000007110: $ r_{\rm geostationary\ orbit} $ 0000003141: $ \pi $ 0000005458: $ m_{\rm Earth} $ 0000006277: $ G $	\frac{\sqrt[3]{2} \sqrt[3]{\frac{pdg_{0005458} pdg_{0005595}^{2} pdg_{0006277}}{pdg_{0003141}^{2}}}}{2} = pdg_{0007110}	ERROR for dim with 1559688463
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) $ \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)			Lorentz transformation	0000001467: $ t $ 0000001357: $ v $ 0000001790: $ \gamma $ 0000004567: $ c $ 0000005647: $ y $ 0000006728: $ z $ 0000004037: $ x $	- 2 pdg_{0001357} pdg_{0001467} pdg_{0001790}^{2} pdg_{0004037} + pdg_{0004037}^{2} \left(pdg_{0001790}^{2} - \frac{pdg_{0004567}^{2} \left(1 - pdg_{0001790}^{2}\right)^{2}}{pdg_{0001357}^{2} pdg_{0001790}^{2}}\right) + pdg_{0005647}^{2} + pdg_{0006728}^{2} - \frac{2 pdg_{0001467} pdg_{0004037} pdg_{0004567}^{2} \left(1 - pdg_{0001790}^{2}\right)}{pdg_{0001357}} = pdg_{0001467}^{2} \left(- pdg_{0001357}^{2} pdg_{0001790}^{2} + pdg_{0001790}^{2} pdg_{0004567}^{2}\right)	ERROR for dim with 1586866563
1590774089	$ dW = F dx $ dW = F dx			escape velocity work and force and energy	0000009199: $ dx $ 0000004202: $ F $ 0000009398: $ dW $	pdg_{0009398} = pdg_{0004202} pdg_{0009199}	ERROR for dim with 1590774089
1636453295	$ \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = - \nabla^2 \vec{E} $ \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = - \nabla^2 \vec{E}			Maxwell equations to electric field wave equation	0000004326: $ \vec{E} $	\operatorname{cross}{\left(nabla,\operatorname{cross}{\left(nabla,pdg_{0004326} \right)} \right)} = - nabla^{2} pdg_{0004326}	ERROR for dim with 1636453295
1638282134	$ \vec{p}_{\rm before} = \vec{p}_{\rm after} $ \vec{p}_{\rm before} = \vec{p}_{\rm after}			Compton's equation for scattering	0000005493: $ \vec{p}_{\rm after} $ 0000001302: $ \vec{p}_{\rm before} $	pdg_{0001302} = pdg_{0005493}	ERROR for dim with 1638282134
1639827492	$ - c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1 $ - c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1			Lorentz transformation	0000004567: $ c $ 0000001790: $ \gamma $ 0000001357: $ v $	- \frac{pdg_{0004567}^{2} \left(1 - pdg_{0001790}^{2}\right)}{pdg_{0001357}^{2} pdg_{0001790}^{2}} = 1	ERROR for dim with 1639827492
1648958381	$ \nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right) $ \nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right)		https://physicsderivationgraph.blogspot.com/2020/09/representing-laplace-operator-nabla-in.html	derivation of Schrodinger Equation	0000001054: $ \hbar $ 0000001467: $ t $ 0000009489: $ \psi $ 0000004621: $ i $ 0000002046: $ \vec{p} $ 0000009472: $ \vec{r} $	nabla^{2} \operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)} = \frac{pdg_{0004621} \operatorname{pdg}_{0002046}{\left(\operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)} \right)}}{pdg_{0001054}}	ERROR for dim with 1648958381
1650441634	$ y_0 = 0 $ y_0 = 0		define coordinate system such that initial height is at origin	angle of maximum distance for projectile motion	0000001469: $ y_0 $	pdg_{0001469} = 0	ERROR for dim with 1650441634
1676472948	$ 0 = v_x - v_{0, x} $ 0 = v_x - v_{0, x}			equations of motion in 2D (calculus)	0000002958: $ v_{0, x} $ 0000005505: $ v_x $	0 = - pdg_{0002958} + pdg_{0005505}	ERROR for dim with 1676472948
1702349646	$ -g dt = d v_y $ -g dt = d v_y			equations of motion in 2D (calculus)	0000001649: $ g $ 0000005674: $ d v_y $	- dt pdg_{0001649} = pdg_{0005674}	ERROR for dim with 1702349646
1772416655	$ \frac{E_2 - E_1}{t} = v F - F v $ \frac{E_2 - E_1}{t} = v F - F v			time invariant force conserves energy	0000004550: $ E_2 $ 0000001467: $ t $ 0000005579: $ E_1 $	\frac{pdg_{0004550} - pdg_{0005579}}{pdg_{0001467}} = 0	ERROR for dim with 1772416655
1772973171	$ -\frac{k}{m} x = -A \omega^2 \cos(\omega t) $ -\frac{k}{m} x = -A \omega^2 \cos(\omega t)			equation of motion for a spring	0000005156: $ m $ 0000009491: $ T $ 0000002321: $ \omega $	- \frac{k x}{pdg_{0005156}} = - A pdg_{0002321}^{2} \cos{\left(pdg_{0002321} pdg_{0009491} \right)}	ERROR for dim with 1772973171
1784114349	$ \sqrt{\frac{k}{m}} = \omega $ \sqrt{\frac{k}{m}} = \omega			equation of motion for a spring	0000001356: $ k $ 0000005156: $ m $ 0000002321: $ \omega $	\sqrt{\frac{pdg_{0001356}}{pdg_{0005156}}} = pdg_{0002321}	ERROR for dim with 1784114349
1809909100	$ \frac{E_2 - E_1}{t} = 0 $ \frac{E_2 - E_1}{t} = 0			time invariant force conserves energy	0000004550: $ E_2 $ 0000001467: $ t $ 0000005579: $ E_1 $	\frac{pdg_{0004550} - pdg_{0005579}}{pdg_{0001467}} = 0	ERROR for dim with 1809909100
1811867899	$ T^2 = \frac{d_1+d_2}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1} $ T^2 = \frac{d_1+d_2}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1}			Kepler's Third Law: period squared propto distance cubed	0000003141: $ \pi $ 0000002798: $ d_2 $ 0000006277: $ G $ 0000005022: $ m_1 $ 0000002530: $ r $ 0000009491: $ T $	pdg_{0009491}^{2} = \frac{4 pdg_{0002530}^{2} pdg_{0002798} pdg_{0003141}^{2}}{pdg_{0005022} pdg_{0006277}}	ERROR for dim with 1811867899
1815398659	$ U = Q + W $ U = Q + W			first law of thermodynamics	0000005786: $ U $ 0000001088: $ W $ 0000009432: $ Q $	pdg_{0005786} = pdg_{0001088} + pdg_{0009432}	ERROR for dim with 1815398659
1819663717	$ a_x = \frac{d}{dt} v_x $ a_x = \frac{d}{dt} v_x			equations of motion in 2D (calculus)	0000001467: $ t $ 0000005505: $ v_x $ 0000007159: $ a_x $	pdg_{0007159} = \frac{d}{d pdg_{0001467}} pdg_{0005505}	ERROR for dim with 1819663717
1840080113	$ KE_2 = 0 $ KE_2 = 0		object is not moving at $x=\infty$	escape velocity	0000001552: $ j $	pdg_{0001552} = 0	ERROR for dim with 1840080113
1857710291	$ 0 = a \sin(n \pi) $ 0 = a \sin(n \pi)			particle in a 1D box	0000001592: $ n $ 0000003141: $ \pi $ 0000009139: $ a $	0 = pdg_{0009139} \sin{\left(pdg_{0001592} pdg_{0003141} \right)}	ERROR for dim with 1857710291
1858578388	$ \nabla^2 E( \vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E( \vec{r})\exp(i \omega t) $ \nabla^2 E( \vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E( \vec{r})\exp(i \omega t)		https://physicsderivationgraph.blogspot.com/2020/09/representing-laplace-operator-nabla-in.html	electric field wave equation: from time dependent to time independent	0000006238: $ E $ 0000001467: $ t $ 0000004621: $ i $ 0000002321: $ \omega $ 0000007940: $ \epsilon_0 $ 0000006197: $ \mu_0 $ 0000009472: $ \vec{r} $	nabla^{2} \operatorname{pdg}_{0006238}{\left(pdg_{0009472} \right)} e^{pdg_{0001467} pdg_{0002321} pdg_{0004621}} = - pdg_{0002321}^{2} pdg_{0006197} pdg_{0007940} \operatorname{pdg}_{0006238}{\left(pdg_{0009472} \right)} e^{pdg_{0001467} pdg_{0002321} pdg_{0004621}}	ERROR for dim with 1858578388
1858772113	$ k = \frac{n \pi}{W} $ k = \frac{n \pi}{W}			particle in a 1D box	0000005321: $ k $ 0000001592: $ n $ 0000002523: $ W $ 0000003141: $ \pi $	pdg_{0005321} = \frac{pdg_{0001592} pdg_{0003141}}{pdg_{0002523}}	ERROR for dim with 1858772113
1888494137	$ -\sqrt{\frac{k}{m}} = \omega $ -\sqrt{\frac{k}{m}} = \omega			equation of motion for a spring	0000001356: $ k $ 0000005156: $ m $ 0000002321: $ \omega $	- \sqrt{\frac{pdg_{0001356}}{pdg_{0005156}}} = pdg_{0002321}	ERROR for dim with 1888494137
1916173354	$ -\gamma^2 v^2 + c^2 \gamma^2 = c^2 $ -\gamma^2 v^2 + c^2 \gamma^2 = c^2			Lorentz transformation	0000004567: $ c $ 0000001790: $ \gamma $ 0000001357: $ v $	- pdg_{0001357}^{2} pdg_{0001790}^{2} + pdg_{0001790}^{2} pdg_{0004567}^{2} = pdg_{0004567}^{2}	ERROR for dim with 1916173354
1928085940	$ Z^* = \|Z\| \exp( -i \theta ) $ Z^* = \|Z\| \exp( -i \theta )			double intensity when phase is coherent (optics)	0000003192: $ Z $	pdg_{0003192} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
1931103031	$ \frac{k}{m} = \omega^2 $ \frac{k}{m} = \omega^2			equation of motion for a spring	0000001356: $ k $ 0000005156: $ m $ 0000002321: $ \omega $	\frac{pdg_{0001356}}{pdg_{0005156}} = pdg_{0002321}^{2}	ERROR for dim with 1931103031
1934748140	$ \int \|\psi(x)\|^2 dx = 1 $ \int \|\psi(x)\|^2 dx = 1			particle in a 1D box	0000001464: $ x $ 0000009489: $ \psi $ 0000009199: $ dx $	\int \left\|{\operatorname{pdg}_{0009489}{\left(pdg_{0001464} \right)}}\right\|^{2}\, dpdg_{0009199} = 1	ERROR for dim with 1934748140
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 $ \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			Lorentz transformation	0000001467: $ t $ 0000001357: $ v $ 0000001790: $ \gamma $ 0000004567: $ c $ 0000005647: $ y $ 0000006728: $ z $ 0000004037: $ x $	pdg_{0001357}^{2} pdg_{0001467}^{2} pdg_{0001790}^{2} - 2 pdg_{0001357} pdg_{0001467} pdg_{0001790}^{2} pdg_{0004037} + pdg_{0001790}^{2} pdg_{0004037}^{2} + pdg_{0005647}^{2} + pdg_{0006728}^{2} = pdg_{0001467}^{2} pdg_{0001790}^{2} pdg_{0004567}^{2} + \frac{2 pdg_{0001467} pdg_{0004037} pdg_{0004567}^{2} \left(1 - pdg_{0001790}^{2}\right)}{pdg_{0001790}} + \frac{pdg_{0004037}^{2} pdg_{0004567}^{2} \left(1 - pdg_{0001790}^{2}\right)}{pdg_{0001790}^{2}}	ERROR for dim with 1935543849
1963253044	$ v_{0, x} dt = dx $ v_{0, x} dt = dx			equations of motion in 2D (calculus)	0000004711: $ dt $ 0000002958: $ v_{0, x} $ 0000009199: $ dx $	pdg_{0002958} pdg_{0004711} = pdg_{0009199}	ERROR for dim with 1963253044
1967582749	$ t = \frac{v - v_0}{a} $ t = \frac{v - v_0}{a}			equations of motion in 1D with constant acceleration - SUVAT (algebra)	0000001467: $ t $ 0000009140: $ a $ 0000005153: $ v_0 $ 0000001357: $ v $	pdg_{0001467} = \frac{pdg_{0001357} - pdg_{0005153}}{pdg_{0009140}}	ERROR for dim with 1967582749
1974334644	$ \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t' $ \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'			Lorentz transformation	0000004037: $ x $ 0000001467: $ t $ 0000001357: $ v $ 0000001790: $ \gamma $ 0000004989: $ t' $	pdg_{0001467} pdg_{0001790} + \frac{\operatorname{pdg}_{0004037}{\left(1 - pdg_{0001790}^{2} \right)}}{pdg_{0001357} pdg_{0001790}} = pdg_{0004989}	ERROR for dim with 1974334644
1977955751	$ -g = \frac{d}{dt} v_y $ -g = \frac{d}{dt} v_y			equations of motion in 2D (calculus)	0000009107: $ v_y $ 0000001649: $ g $ 0000001467: $ t $	- pdg_{0001649} = \frac{d}{d pdg_{0001467}} pdg_{0009107}	ERROR for dim with 1977955751
1994296484	$ v_{\rm satellite}^2 = G \frac{m_{\rm Earth}}{r} $ v_{\rm satellite}^2 = G \frac{m_{\rm Earth}}{r}			radius for satellite in geostationary orbit	0000005458: $ m_{\rm Earth} $ 0000004082: $ v_{\rm satellite} $ 0000002530: $ r $ 0000006277: $ G $	pdg_{0004082}^{2} = \frac{pdg_{0005458} pdg_{0006277}}{pdg_{0002530}}	ERROR for dim with 1994296484
2005061870	$ v(r) = \sqrt{\frac{2 G m_2}{r}} $ v(r) = \sqrt{\frac{2 G m_2}{r}}			velocity at distance r of object dropped from infinity	0000006277: $ G $ 0000002530: $ r $ 0000004851: $ m_2 $ 0000001357: $ v $	\operatorname{pdg}_{0001357}{\left(pdg_{0002530} \right)} = \sqrt{2} \sqrt{\frac{pdg_{0004851} pdg_{0006277}}{pdg_{0002530}}}	ERROR for dim with 2005061870
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) $ \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)		https://physicsderivationgraph.blogspot.com/2020/09/representing-laplace-operator-nabla-in.html	electric field wave equation: from time dependent to time independent	0000006238: $ E $ 0000001467: $ t $ 0000004621: $ i $ 0000002321: $ \omega $ 0000007940: $ \epsilon_0 $ 0000006197: $ \mu_0 $ 0000009472: $ \vec{r} $	nabla^{2} \operatorname{pdg}_{0006238}{\left(pdg_{0009472} \right)} e^{pdg_{0001467} pdg_{0002321} pdg_{0004621}} = \frac{partial pdg_{0006197} pdg_{0007940} \operatorname{pdg}_{0006238}{\left(pdg_{0009472} \right)} e^{pdg_{0001467} pdg_{0002321} pdg_{0004621}}}{pdg_{0001467}^{2}}	ERROR for dim with 2029293929
2042298788	$ 0 = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}} + \frac{1}{2} m v_{\rm escape}^2 $ 0 = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}} + \frac{1}{2} m v_{\rm escape}^2			escape velocity	0000005458: $ m_{\rm Earth} $ 0000008656: $ v_{\rm escape} $ 0000005156: $ m $ 0000006277: $ G $ 0000003236: $ r_{\rm Earth} $	0 = \frac{pdg_{0005156} pdg_{0008656}^{2}}{2} - \frac{pdg_{0005156} pdg_{0005458} pdg_{0006277}}{pdg_{0003236}}	ERROR for dim with 2042298788
2051901211	$ \frac{V}{R_1} = I_1 $ \frac{V}{R_1} = I_1			total electrical resistance for circuit with two resistors in parallel	0000008697: $ R_1 $ 0000003978: $ I_1 $ 0000006599: $ V $	\frac{pdg_{0006599}}{pdg_{0008697}} = pdg_{0003978}	ERROR for dim with 2051901211
2061086175	$ W_{\rm to\ system} = -G m_1 m_2 \left(\frac{-1}{r} - \frac{-1}{\infty}\right) $ W_{\rm to\ system} = -G m_1 m_2 \left(\frac{-1}{r} - \frac{-1}{\infty}\right)			velocity at distance r of object dropped from infinity	0000002530: $ r $ 0000005022: $ m_1 $ 0000006277: $ G $ 0000004851: $ m_2 $ 0000009372: $ W_{\rm to\ system} $	pdg_{0009372} = - pdg_{0005022} pdg_{0006277} \operatorname{pdg}_{0004851}{\left(- \frac{1}{pdg_{0002530}} \right)}	ERROR for dim with 2061086175
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^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0			Lorentz transformation	0000001790: $ \gamma $	pdg_{0001790} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
2086924031	$ 0 = - \frac{1}{2} g t_f + v_0 \sin(\theta) $ 0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)			angle of maximum distance for projectile motion	0000001575: $ \theta $ 0000001649: $ g $ 0000002467: $ t_f $ 0000005153: $ v_0 $	0 = - \frac{pdg_{0001649} pdg_{0002467}}{2} + pdg_{0005153} \sin{\left(pdg_{0001575} \right)}	ERROR for dim with 2086924031
2096918413	$ x = \gamma ( \gamma x - \gamma v t + v t' ) $ x = \gamma ( \gamma x - \gamma v t + v t' )			Lorentz transformation	0000001467: $ t $ 0000001357: $ v $ 0000001790: $ \gamma $ 0000004989: $ t' $ 0000004037: $ x $	pdg_{0004037} = \operatorname{pdg}_{0001790}{\left(- pdg_{0001357} pdg_{0001467} pdg_{0001790} + pdg_{0001357} pdg_{0004989} + pdg_{0001790} pdg_{0004037} \right)}	ERROR for dim with 2096918413
2103023049	$ \sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) $ \sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)			identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation hyperbolic trigonometric identities Euler equation: trigonometric relations	0000001464: $ x $ 0000004621: $ i $	\sin{\left(pdg_{0001464} \right)} = \frac{e^{pdg_{0001464} pdg_{0004621}} - e^{- pdg_{0001464} pdg_{0004621}}}{2 pdg_{0004621}}	ERROR for dim with 2103023049
2113211456	$ f = 1/T $ f = 1/T			frequency relations frequency and period	0000004201: $ f $ 0000009491: $ T $	pdg_{0004201} = \frac{1}{pdg_{0009491}}	ERROR for dim with 2113211456
2114909846	$ \theta_A = \frac{[A_{\rm adsorption}]}{[S_0]} $ \theta_A = \frac{[A_{\rm adsorption}]}{[S_0]}			Langmuir Adsorption	0000001791: $ \theta_A $ 0000004940: $ [A_{\rm adsorption}] $ 0000003037: $ [S_0] $	pdg_{0001791} = \frac{pdg_{0004940}}{pdg_{0003037}}	ERROR for dim with 2114909846
2121790783	$ \tanh^2(x) = \frac{ \left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2} $ \tanh^2(x) = \frac{ \left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2}			hyperbolic trigonometric identities	0000001464: $ x $	\tanh^{2}{\left(pdg_{0001464} \right)} = \frac{\left(e^{pdg_{0001464}} - e^{- pdg_{0001464}}\right)^{2}}{\left(e^{pdg_{0001464}} + e^{- pdg_{0001464}}\right)^{2}}	ERROR for dim with 2121790783
2123139121	$ -\exp(-i x) = -\cos(x)+i \sin(x) $ -\exp(-i x) = -\cos(x)+i \sin(x)			Euler equation: trigonometric relations	0000001464: $ x $ 0000004621: $ i $	- e^{- pdg_{0001464} pdg_{0004621}} = pdg_{0004621} \sin{\left(pdg_{0001464} \right)} - \cos{\left(pdg_{0001464} \right)}	ERROR for dim with 2123139121
2131616531	$ T f = 1 $ T f = 1			frequency relations frequency and period	0000004201: $ f $ 0000009491: $ T $	pdg_{0004201} pdg_{0009491} = 1	ERROR for dim with 2131616531
2148049269	$ -\frac{k}{m} A \cos(\omega t) = -A \omega^2 \cos(\omega t) $ -\frac{k}{m} A \cos(\omega t) = -A \omega^2 \cos(\omega t)			equation of motion for a spring	0000005156: $ m $ 0000009491: $ T $ 0000002321: $ \omega $	- \frac{A k \cos{\left(pdg_{0002321} pdg_{0009491} \right)}}{pdg_{0005156}} = - A pdg_{0002321}^{2} \cos{\left(pdg_{0002321} pdg_{0009491} \right)}	ERROR for dim with 2148049269
2168306601	$ [S_0] = \left(\frac{k_{\rm desorption}}{k_{\rm adsorption}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}] $ [S_0] = \left(\frac{k_{\rm desorption}}{k_{\rm adsorption}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}]			Langmuir Adsorption	0000009046: $ p_A $ 0000008379: $ k_{\rm desorption} $ 0000006850: $ k_{\rm adsorption} $ 0000004940: $ [A_{\rm adsorption}] $ 0000003037: $ [S_0] $	pdg_{0003037} = pdg_{0004940} \left(1 + \frac{pdg_{0008379}}{pdg_{0006850} pdg_{0009046}}\right)	ERROR for dim with 2168306601
2186083170	$ \frac{KE_2 - KE_1}{t} = v F $ \frac{KE_2 - KE_1}{t} = v F			time invariant force conserves energy	0000001352: $ KE_2 $ 0000001467: $ t $ 0000001357: $ v $ 0000004202: $ F $ 0000001955: $ KE_1 $	\frac{pdg_{0001352} - pdg_{0001955}}{pdg_{0001467}} = pdg_{0001357} pdg_{0004202}	ERROR for dim with 2186083170
2217103163	$ \frac{m_1 d_1}{d_2} = m_2 $ \frac{m_1 d_1}{d_2} = m_2			Kepler's Third Law: period squared propto distance cubed	0000002798: $ d_2 $ 0000005022: $ m_1 $ 0000004851: $ m_2 $ 0000007652: $ d_1 $	\frac{pdg_{0005022} pdg_{0007652}}{pdg_{0002798}} = pdg_{0004851}	ERROR for dim with 2217103163
2236639474	$ (A + B)(A + B)^* = \|A + B\|^2 $ (A + B)(A + B)^* = \|A + B\|^2			double intensity when phase is coherent (optics)	0000004453: $ A $ 0000004698: $ B $	\left(pdg_{0004453} + pdg_{0004698}\right)^{2} = \left\|{pdg_{0004453} + pdg_{0004698}}\right\|^{2}	ERROR for dim with 2236639474
2257410739	$ \left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alpha $ \left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alpha			first law of thermodynamics	0000005786: $ U $ 0000007343: $ T $	\frac{d}{d pdg_{0007343}} pdg_{0005786} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
2258485859	$ {\cal H} \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) $ {\cal H} \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)			derivation of Schrodinger Equation	0000001054: $ \hbar $ 0000001467: $ t $ 0000009489: $ \psi $ 0000006799: $ {\cal H} $ 0000004621: $ i $ 0000009472: $ \vec{r} $	pdg_{0006799} \operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)} = pdg_{0001054} pdg_{0004621} \frac{\partial}{\partial pdg_{0001467}} \operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)}	ERROR for dim with 2258485859
2267521164	$ PE_2 = 0 $ PE_2 = 0		object goes to $\infty$ away from gravitational source	escape velocity	0000008849: $ PE_2 $	pdg_{0008849} = 0	ERROR for dim with 2267521164
2271186630	$ V = I_{\rm total} R_{\rm total} $ V = I_{\rm total} R_{\rm total}			total electrical resistance for circuit with two resistors in parallel	0000001908: $ R_{\rm total} $ 0000009647: $ I_{\rm total} $ 0000006599: $ V $	pdg_{0006599} = pdg_{0001908} pdg_{0009647}	ERROR for dim with 2271186630
2297105551	$ d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta) $ d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)			angle of maximum distance for projectile motion	0000001575: $ \theta $ 0000001649: $ g $ 0000005153: $ v_0 $ 0000001943: $ d $	pdg_{0001943} = \frac{2 pdg_{0005153}^{2} \sin{\left(pdg_{0001575} \right)} \cos{\left(pdg_{0001575} \right)}}{pdg_{0001649}}	ERROR for dim with 2297105551
2308660627	$ G \frac{m_{\rm Earth}}{r_{\rm Earth}^2} = g_{\rm Earth} $ G \frac{m_{\rm Earth}}{r_{\rm Earth}^2} = g_{\rm Earth}			mass of the Earth	0000005458: $ m_{\rm Earth} $ 0000003236: $ r_{\rm Earth} $ 0000007557: $ g_{\rm Earth} $ 0000006277: $ G $	\frac{pdg_{0005458} pdg_{0006277}}{pdg_{0003236}^{2}} = pdg_{0007557}	ERROR for dim with 2308660627
2334518266	$ m a = -k x $ m a = -k x			equation of motion for a spring	0000001356: $ k $ 0000009140: $ a $ 0000005156: $ m $ 0000004037: $ x $	pdg_{0005156} pdg_{0009140} = - pdg_{0001356} pdg_{0004037}	ERROR for dim with 2334518266
2366691988	$ \int 0 dt = \int d v_x $ \int 0 dt = \int d v_x			equations of motion in 2D (calculus)	0000005005: $ d v_x $ 0000001467: $ t $	\int 0\, dpdg_{0001467} = \int 1\, dpdg_{0005005}	ERROR for dim with 2366691988
2378095808	$ x_f = x_0 + d $ x_f = x_0 + d			angle of maximum distance for projectile motion	0000001572: $ x_0 $ 0000003652: $ x_f $ 0000001943: $ d $	pdg_{0003652} = pdg_{0001572} + pdg_{0001943}	ERROR for dim with 2378095808
2394240499	$ x = a_{\beta} \langle \psi_{\alpha} \| \psi_{\beta} \rangle $ x = a_{\beta} \langle \psi_{\alpha} \| \psi_{\beta} \rangle			quantum basics orthogonality	0000001464: $ x $ 0000002090: $ \| \psi_{\beta} \rangle $ 0000004679: $ \langle \psi_{\alpha} \| $ 0000007752: $ a_{\beta} $	pdg_{0001464} = pdg_{0007752} \operatorname{Bra}{\left(pdg_{0004679} \right)} \operatorname{Ket}{\left(pdg_{0002090} \right)}	ERROR for dim with 2394240499
2394853829	$ \exp(-i x) = \cos(-x)+i \sin(-x) $ \exp(-i x) = \cos(-x)+i \sin(-x)			Euler equation: trigonometric relations	0000001464: $ x $ 0000004621: $ i $	e^{- pdg_{0001464} pdg_{0004621}} = - pdg_{0004621} \sin{\left(pdg_{0001464} \right)} + \cos{\left(pdg_{0001464} \right)}	ERROR for dim with 2394853829
2394935831	$ ( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} \| \psi_{\beta} \rangle = 0 $ ( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} \| \psi_{\beta} \rangle = 0			quantum basics orthogonality	0000002427: $ a_{\alpha} $ 0000002090: $ \| \psi_{\beta} \rangle $ 0000004679: $ \langle \psi_{\alpha} \| $ 0000007752: $ a_{\beta} $	\left(- pdg_{0002427} + pdg_{0007752}\right) \operatorname{Bra}{\left(pdg_{0004679} \right)} \operatorname{Ket}{\left(pdg_{0002090} \right)} = 0	ERROR for dim with 2394935831
2394935835	$ \left(\langle\psi\| \hat{A} \|\psi \rangle \right)^+ = \left(\langle a \rangle\right)^+ $ \left(\langle\psi\| \hat{A} \|\psi \rangle \right)^+ = \left(\langle a \rangle\right)^+			quantum basics Hermitian operators have realvalued observables	0000004065: $ \langle \psi\| $	pdg_{0004065} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
2395958385	$ \nabla^2 \psi \left( \vec{r},t \right) = \frac{-p^2}{\hbar} \psi( \vec{r},t) $ \nabla^2 \psi \left( \vec{r},t \right) = \frac{-p^2}{\hbar} \psi( \vec{r},t)		https://physicsderivationgraph.blogspot.com/2020/09/representing-laplace-operator-nabla-in.html	derivation of Schrodinger Equation	0000001054: $ \hbar $ 0000001467: $ t $ 0000009489: $ \psi $ 0000001134: $ p $ 0000009472: $ \vec{r} $	nabla^{2} \operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)} = - \frac{pdg_{0001134}^{2} \operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)}}{pdg_{0001054}}	ERROR for dim with 2395958385
2404934990	$ \langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2 $ \langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2			variance relation	0000001464: $ x $	pdg_{0001464} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
2405307372	$ \sin(2 x) = 2 \sin(x) \cos(x) $ \sin(2 x) = 2 \sin(x) \cos(x)			identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation angle of maximum distance for projectile motion	0000001464: $ x $	\sin{\left(2 pdg_{0001464} \right)} = 2 \sin{\left(pdg_{0001464} \right)} \cos{\left(pdg_{0001464} \right)}	ERROR for dim with 2405307372
2417941373	$ - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2 $ - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2			Lorentz transformation	0000004567: $ c $ 0000001790: $ \gamma $ 0000001357: $ v $	- \frac{pdg_{0004567}^{2} \left(1 - pdg_{0001790}^{2}\right)^{2}}{pdg_{0001357}^{2} pdg_{0001790}^{2}} = 1 - pdg_{0001790}^{2}	ERROR for dim with 2417941373
2431507955	$ PE_2 = -F x_2 $ PE_2 = -F x_2			time invariant force conserves energy	0000004202: $ F $ 0000005467: $ x_2 $ 0000008849: $ PE_2 $	pdg_{0008849} = - pdg_{0004202} pdg_{0005467}	ERROR for dim with 2431507955
2461349007	$ - \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y $ - \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y			equations of motion in 2D (calculus)	0000001467: $ t $ 0000001649: $ g $ 0000009431: $ v_{0, y} $ 0000005647: $ y $ 0000001469: $ y_0 $	- \frac{pdg_{0001467}^{2} pdg_{0001649}}{2} + pdg_{0001467} pdg_{0009431} + pdg_{0001469} = pdg_{0005647}	ERROR for dim with 2461349007
2472653783	$ \alpha = \frac{1}{T} $ \alpha = \frac{1}{T}			coefficient of thermal expansion using the equation of state for an ideal gas	0000004686: $ \alpha $ 0000007343: $ T $	pdg_{0004686} = \frac{1}{pdg_{0007343}}	ERROR for dim with 2472653783
2484824786	$ F = m g $ F = m g			mass of the Earth	0000001649: $ g $ 0000005156: $ m $ 0000004202: $ F $	pdg_{0004202} = pdg_{0001649} pdg_{0005156}	ERROR for dim with 2484824786
2494533900	$ \langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2 $ \langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2			variance relation	0000001464: $ x $	pdg_{0001464} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
2501591100	$ \exp(i \pi) + 1 = 0 $ \exp(i \pi) + 1 = 0			Euler equation to exp(i pi) + 1 = 0	0000004621: $ i $ 0000003141: $ \pi $	e^{pdg_{0003141} pdg_{0004621}} + 1 = 0	ERROR for dim with 2501591100
2503972039	$ 0 = KE_{\rm escape} + PE_{\rm Earth\ surface} $ 0 = KE_{\rm escape} + PE_{\rm Earth\ surface}			escape velocity	0000006431: $ PE_{\rm Earth\ surface} $ 0000005332: $ KE_{\rm escape} $	0 = pdg_{0005332} + pdg_{0006431}	ERROR for dim with 2503972039
2519058903	$ \sin(2 \theta) = 2 \sin(\theta) \cos(\theta) $ \sin(2 \theta) = 2 \sin(\theta) \cos(\theta)			angle of maximum distance for projectile motion	0000001575: $ \theta $	\sin{\left(2 pdg_{0001575} \right)} = 2 \sin{\left(pdg_{0001575} \right)} \cos{\left(pdg_{0001575} \right)}	ERROR for dim with 2519058903
2542420160	$ c^2 \gamma^2 - v^2 \gamma^2 = c^2 $ c^2 \gamma^2 - v^2 \gamma^2 = c^2			Lorentz transformation	0000004567: $ c $ 0000001790: $ \gamma $ 0000001357: $ v $	- pdg_{0001357}^{2} pdg_{0001790}^{2} + pdg_{0001790}^{2} pdg_{0004567}^{2} = pdg_{0004567}^{2}	ERROR for dim with 2542420160
2575937347	$ n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} ) $ n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} )			optics: Law of refraction to Brewster's angle	0000002243: $ \theta_{\rm refracted} $ 0000002941: $ n_1 $ 0000004928: $ \theta_{\rm Brewster} $ 0000001958: $ n_2 $	pdg_{0002941} \sin{\left(pdg_{0004928} \right)} = pdg_{0001958} \sin{\left(pdg_{0002243} \right)}	ERROR for dim with 2575937347
2613006036	$ \frac{PV}{T} = nR $ \frac{PV}{T} = nR			coefficient of thermal expansion using the equation of state for an ideal gas	0000008134: $ P $ 0000008179: $ R $ 0000007586: $ V $ 0000002834: $ n $ 0000007343: $ T $	\frac{pdg_{0007586} pdg_{0008134}}{pdg_{0007343}} = pdg_{0002834} pdg_{0008179}	ERROR for dim with 2613006036
2617541067	$ \left(\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r $ \left(\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r			radius for satellite in geostationary orbit	0000008762: $ T_{\rm orbit} $ 0000003141: $ \pi $ 0000005458: $ m_{\rm Earth} $ 0000002530: $ r $ 0000006277: $ G $	\frac{\sqrt[3]{2} \sqrt[3]{\frac{pdg_{0005458} pdg_{0006277} pdg_{0008762}^{2}}{pdg_{0003141}^{2}}}}{2} = pdg_{0002530}	ERROR for dim with 2617541067
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) $ \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)			derivation of Schrodinger Equation	0000001054: $ \hbar $	pdg_{0001054} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
2700934933	$ 2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right) $ 2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)			double intensity when phase is coherent (optics)	0000001464: $ x $ 0000004621: $ i $ 0000008586: $ \phi $ 0000001575: $ \theta $	2 \cos{\left(pdg_{0001464} \right)} = e^{pdg_{0004621} \left(pdg_{0001575} - pdg_{0008586}\right)} + e^{- pdg_{0004621} \left(pdg_{0001575} - pdg_{0008586}\right)}	ERROR for dim with 2700934933
2715678478	$ I R_{\rm total} = I R_1 + I R_2 $ I R_{\rm total} = I R_1 + I R_2			total electrical resistance for circuit with two resistors in series	0000001908: $ R_{\rm total} $ 0000004501: $ I $ 0000003461: $ R_2 $ 0000008697: $ R_1 $	pdg_{0001908} pdg_{0004501} = pdg_{0003461} pdg_{0004501} + pdg_{0004501} pdg_{0008697}	ERROR for dim with 2715678478
2719691582	$ \|A\| = \|B\| $ \|A\| = \|B\|		in a loop	double intensity when phase is coherent (optics)	0000004453: $ A $ 0000004698: $ B $	\left\|{pdg_{0004453}}\right\| = \left\|{pdg_{0004698}}\right\|	ERROR for dim with 2719691582
2741489181	$ a_y = -g $ a_y = -g			equations of motion in 2D (calculus)	0000001649: $ g $ 0000007055: $ a_y $	pdg_{0007055} = - pdg_{0001649}	ERROR for dim with 2741489181
2750380042	$ v_{\rm escape} = -\sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}} $ v_{\rm escape} = -\sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}			escape velocity	0000005458: $ m_{\rm Earth} $ 0000008656: $ v_{\rm escape} $ 0000003236: $ r_{\rm Earth} $ 0000006277: $ G $	pdg_{0008656} = - \sqrt{2} \sqrt{\frac{pdg_{0005458} pdg_{0006277}}{pdg_{0003236}}}	ERROR for dim with 2750380042
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) $ \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)			hyperbolic trigonometric identities	0000001464: $ x $	- \sinh^{2}{\left(pdg_{0001464} \right)} + \cosh^{2}{\left(pdg_{0001464} \right)} = 1	ERROR for dim with 2762326680
2768857871	$ \frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1} $ \frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1}			optics: Law of refraction to Brewster's angle	0000002941: $ n_1 $ 0000004928: $ \theta_{\rm Brewster} $ 0000001958: $ n_2 $	\frac{\sin{\left(pdg_{0004928} \right)}}{\cos{\left(pdg_{0004928} \right)}} = \frac{pdg_{0001958}}{pdg_{0002941}}	ERROR for dim with 2768857871
2770069250	$ \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t} $ \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}			time invariant force conserves energy	0000001352: $ KE_2 $ 0000001467: $ t $ 0000005579: $ E_1 $ 0000004550: $ E_2 $ 0000008849: $ PE_2 $ 0000001955: $ KE_1 $ 0000004093: $ PE_1 $	\frac{pdg_{0004550} - pdg_{0005579}}{pdg_{0001467}} = \frac{pdg_{0001352} - pdg_{0001955}}{pdg_{0001467}} + \frac{- pdg_{0004093} + pdg_{0008849}}{pdg_{0001467}}	ERROR for dim with 2770069250
2809345867	$ \frac{V}{R_{\rm total}} = I_{\rm total} $ \frac{V}{R_{\rm total}} = I_{\rm total}			total electrical resistance for circuit with two resistors in parallel	0000001908: $ R_{\rm total} $ 0000009647: $ I_{\rm total} $ 0000006599: $ V $	\frac{pdg_{0006599}}{pdg_{0001908}} = pdg_{0009647}	ERROR for dim with 2809345867
2848934890	$ \langle a \rangle^* = \langle a \rangle $ \langle a \rangle^* = \langle a \rangle			quantum basics Hermitian operators have realvalued observables	0000009139: $ a $	pdg_{0009139} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
2857430695	$ a = \frac{v_2 - v_1}{t} $ a = \frac{v_2 - v_1}{t}	acceleration		time invariant force conserves energy	0000001467: $ t $ 0000009140: $ a $ 0000002473: $ v_1 $ 0000004770: $ v_2 $	pdg_{0009140} = \frac{- pdg_{0002473} + pdg_{0004770}}{pdg_{0001467}}	ERROR for dim with 2857430695
2858549874	$ - \frac{1}{2} g t^2 + v_{0, y} t = y - y_0 $ - \frac{1}{2} g t^2 + v_{0, y} t = y - y_0			equations of motion in 2D (calculus)	0000001467: $ t $ 0000001649: $ g $ 0000009431: $ v_{0, y} $ 0000005647: $ y $ 0000001469: $ y_0 $	- \frac{pdg_{0001467}^{2} pdg_{0001649}}{2} + pdg_{0001467} pdg_{0009431} = - pdg_{0001469} + pdg_{0005647}	ERROR for dim with 2858549874
2883079365	$ r_{\rm Schwarzschild} c^2 = 2 G m $ r_{\rm Schwarzschild} c^2 = 2 G m			Schwarzschild radius for non-rotating black hole	0000006277: $ G $ 0000004518: $ r_{\rm Schwarzschild} $ 0000005156: $ m $ 0000004567: $ c $	pdg_{0004518} pdg_{0004567}^{2} = 2 pdg_{0005156} pdg_{0006277}	ERROR for dim with 2883079365
2897612567	$ v = \alpha c \sqrt{ \frac{m_e}{A m_p} } $ v = \alpha c \sqrt{ \frac{m_e}{A m_p} }			upper limit on velocity in condensed matter	0000002077: $ v $ 0000001370: $ \alpha $ 0000002515: $ m_e $ 0000004567: $ c $ 0000003285: $ A $ 0000005916: $ m_p $	pdg_{0002077} = pdg_{0001370} pdg_{0004567} \sqrt{\frac{pdg_{0002515}}{pdg_{0003285} pdg_{0005916}}}	ERROR for dim with 2897612567
2902772962	$ \tanh(x) = \frac{\frac{1}{2}\left( \exp(x)-\exp(-x) \right)}{\cosh(x)} $ \tanh(x) = \frac{\frac{1}{2}\left( \exp(x)-\exp(-x) \right)}{\cosh(x)}			hyperbolic trigonometric identities	0000001464: $ x $	\tanh{\left(pdg_{0001464} \right)} = \frac{\frac{e^{pdg_{0001464}}}{2} - \frac{e^{- pdg_{0001464}}}{2}}{\cosh{\left(pdg_{0001464} \right)}}	ERROR for dim with 2902772962
2906548078	$ T^2 = \frac{r}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1} $ T^2 = \frac{r}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1}			Kepler's Third Law: period squared propto distance cubed	0000003141: $ \pi $ 0000002798: $ d_2 $ 0000007652: $ d_1 $ 0000006277: $ G $ 0000005022: $ m_1 $ 0000002530: $ r $ 0000009491: $ T $	pdg_{0009491}^{2} = \frac{4 pdg_{0002530}^{3} pdg_{0002798} pdg_{0003141}^{2}}{pdg_{0005022} pdg_{0006277} \left(pdg_{0002798} + pdg_{0007652}\right)}	ERROR for dim with 2906548078
2907404069	$ W_{\rm by\ system} = W_{\rm to\ system} $ W_{\rm by\ system} = W_{\rm to\ system}			velocity at distance r of object dropped from infinity	0000006191: $ W_{\rm by\ system} $ 0000009372: $ W_{\rm to\ system} $	pdg_{0006191} = pdg_{0009372}	ERROR for dim with 2907404069
2924222857	$ v_{\rm initial} = v(r=\infty) $ v_{\rm initial} = v(r=\infty)			velocity at distance r of object dropped from infinity	0000001934: $ v_{\rm initial} $ 0000001357: $ v $	pdg_{0001934} = pdg_{0001357}	ERROR for dim with 2924222857
2944838499	$ \psi(x) = a \sin(\frac{n \pi}{W} x) $ \psi(x) = a \sin(\frac{n \pi}{W} x)			particle in a 1D box	0000009489: $ \psi $ 0000003141: $ \pi $ 0000001464: $ x $ 0000001592: $ n $ 0000004037: $ x $ 0000002523: $ W $ 0000009139: $ a $	\operatorname{pdg}_{0009489}{\left(pdg_{0001464} \right)} = pdg_{0009139} \sin{\left(\frac{pdg_{0001592} pdg_{0003141} pdg_{0004037}}{pdg_{0002523}} \right)}	ERROR for dim with 2944838499
2977457786	$ 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}} = v_{\rm escape}^2 $ 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}} = v_{\rm escape}^2			escape velocity	0000005458: $ m_{\rm Earth} $ 0000008656: $ v_{\rm escape} $ 0000003236: $ r_{\rm Earth} $ 0000006277: $ G $	\frac{2 pdg_{0005458} pdg_{0006277}}{pdg_{0003236}} = pdg_{0008656}^{2}	ERROR for dim with 2977457786
2983053062	$ x = \gamma (x' + v t') $ x = \gamma (x' + v t')			Lorentz transformation	0000001357: $ v $ 0000001790: $ \gamma $ 0000005456: $ x' $ 0000004989: $ t' $ 0000004037: $ x $	pdg_{0004037} = pdg_{0001790} \left(pdg_{0001357} pdg_{0004989} + pdg_{0005456}\right)	ERROR for dim with 2983053062
2998709778	$ v_{\rm initial} = 0 $ v_{\rm initial} = 0			velocity at distance r of object dropped from infinity	0000001934: $ v_{\rm initial} $	pdg_{0001934} = 0	ERROR for dim with 2998709778
2999795755	$ c^2 \gamma^2 = v^2 \gamma^2 + c^2 $ c^2 \gamma^2 = v^2 \gamma^2 + c^2			Lorentz transformation	0000001357: $ v $ 0000001790: $ \gamma $ 0000004567: $ c $	pdg_{0001790}^{2} pdg_{0004567}^{2} = pdg_{0001357}^{2} pdg_{0001790}^{2} + pdg_{0004567}^{2}	ERROR for dim with 2999795755
3004158505	$ \frac{T^2}{r} F_{\rm gravity} = \left( \frac{4 \pi^2 m r}{T^2} \right)\frac{T^2}{r} $ \frac{T^2}{r} F_{\rm gravity} = \left( \frac{4 \pi^2 m r}{T^2} \right)\frac{T^2}{r}			Newton's Law of Gravitation	0000008762: $ T_{\rm orbit} $ 0000003141: $ \pi $ 0000005156: $ m $ 0000002530: $ r $ 0000002867: $ F_{\rm gravity} $	\frac{pdg_{0002867} pdg_{0008762}^{2}}{pdg_{0002530}} = 4 pdg_{0003141}^{2} pdg_{0005156}	ERROR for dim with 3004158505
3046191961	$ v_{\rm Earth\ orbit} = \frac{C_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}} $ v_{\rm Earth\ orbit} = \frac{C_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}}			speed of Earth around Sun	0000007427: $ v_{\rm Earth\ orbit} $ 0000005344: $ t_{\rm Earth\ orbit} $ 0000001534: $ C_{\rm Earth\ orbit} $	pdg_{0007427} = \frac{pdg_{0001534}}{pdg_{0005344}}	ERROR for dim with 3046191961
3060393541	$ I_{\rm incoherent} = 2\|A\|^2 $ I_{\rm incoherent} = 2\|A\|^2			double intensity when phase is coherent (optics)	0000002435: $ I_{\rm incoherent} $ 0000004453: $ A $	pdg_{0002435} = 2 \left\|{pdg_{0004453}}\right\|^{2}	ERROR for dim with 3060393541
3061811650	$ n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} ) $ n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} )			optics: Law of refraction to Brewster's angle	0000002941: $ n_1 $ 0000004928: $ \theta_{\rm Brewster} $ 0000001958: $ n_2 $	pdg_{0002941} \sin{\left(pdg_{0004928} \right)} = pdg_{0001958} \cos{\left(pdg_{0004928} \right)}	ERROR for dim with 3061811650
3080027960	$ v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}} $ v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}}			speed of Earth around Sun	0000007427: $ v_{\rm Earth\ orbit} $ 0000005344: $ t_{\rm Earth\ orbit} $ 0000003141: $ \pi $ 0000006081: $ r_{\rm Earth\ orbit} $	pdg_{0007427} = \frac{2 pdg_{0003141} pdg_{0006081}}{pdg_{0005344}}	ERROR for dim with 3080027960
3085575328	$ I = \|A\|^2 + \|B\|^2 + \|A\| \|B\| \exp(i (\theta - \phi)) + \|A\| \|B\| \exp(-i (\theta - \phi)) $ I = \|A\|^2 + \|B\|^2 + \|A\| \|B\| \exp(i (\theta - \phi)) + \|A\| \|B\| \exp(-i (\theta - \phi))			double intensity when phase is coherent (optics)	0000007882: $ I $ 0000004453: $ A $ 0000004621: $ i $ 0000001575: $ \theta $ 0000004698: $ B $ 0000008586: $ \phi $	pdg_{0007882} = \left\|{pdg_{0004453}}\right\|^{2} + \left\|{pdg_{0004698}}\right\|^{2} + e^{- pdg_{0004621} \left(pdg_{0001575} - pdg_{0008586}\right)} \left\|{pdg_{0004453} pdg_{0004698} \left\|{e^{pdg_{0004621} \left(pdg_{0001575} - pdg_{0008586}\right)} \left\|{pdg_{0004698}}\right\| + \left\|{pdg_{0004453}}\right\|}\right\|}\right\|	ERROR for dim with 3085575328
3121234211	$ \frac{k}{2\pi} = \lambda $ \frac{k}{2\pi} = \lambda			derivation of Schrodinger Equation	0000005321: $ k $ 0000001115: $ \lambda $ 0000003141: $ \pi $	\frac{pdg_{0005321}}{2 pdg_{0003141}} = pdg_{0001115}	ERROR for dim with 3121234211
3121234212	$ p = \frac{h k}{2\pi} $ p = \frac{h k}{2\pi}			derivation of Schrodinger Equation	0000004413: $ h $ 0000005321: $ k $ 0000001134: $ p $ 0000003141: $ \pi $	pdg_{0001134} = \frac{pdg_{0004413} pdg_{0005321}}{2 pdg_{0003141}}	ERROR for dim with 3121234212
3121513111	$ k = \frac{2 \pi}{\lambda} $ k = \frac{2 \pi}{\lambda}			derivation of Schrodinger Equation frequency relations	0000005321: $ k $ 0000001115: $ \lambda $ 0000003141: $ \pi $	pdg_{0005321} = \frac{2 pdg_{0003141}}{pdg_{0001115}}	ERROR for dim with 3121513111
3131111133	$ T = 1 / f $ T = 1 / f			frequency relations frequency and period	0000009491: $ T $ 0000004201: $ f $	pdg_{0009491} = \frac{1}{pdg_{0004201}}	ERROR for dim with 3131111133
3131211131	$ \omega = 2 \pi f $ \omega = 2 \pi f			derivation of Schrodinger Equation frequency relations	0000004201: $ f $ 0000003141: $ \pi $ 0000002321: $ \omega $	pdg_{0002321} = 2 pdg_{0003141} pdg_{0004201}	ERROR for dim with 3131211131
3132131132	$ \omega = \frac{2\pi}{T} $ \omega = \frac{2\pi}{T}			frequency relations Kepler's Third Law: period squared propto distance cubed	0000009491: $ T $ 0000003141: $ \pi $ 0000002321: $ \omega $	pdg_{0002321} = \frac{2 pdg_{0003141}}{pdg_{0009491}}	ERROR for dim with 3132131132
3147472131	$ \frac{\omega}{2 \pi} = f $ \frac{\omega}{2 \pi} = f			derivation of Schrodinger Equation	0000004201: $ f $ 0000003141: $ \pi $ 0000002321: $ \omega $	\frac{pdg_{0002321}}{2 pdg_{0003141}} = pdg_{0004201}	ERROR for dim with 3147472131
3169580383	$ \vec{a} = \frac{d\vec{v}}{dt} $ \vec{a} = \frac{d\vec{v}}{dt}		acceleration is the change in speed over a duration	equations of motion in 2D (calculus)	0000001467: $ t $ 0000006373: $ \vec{v} $ 0000002423: $ \vec{a} $	pdg_{0002423} = \frac{d}{d pdg_{0001467}} pdg_{0006373}	ERROR for dim with 3169580383
3176662571	$ F_{\rm centripetal} = F_{\rm gravity} $ F_{\rm centripetal} = F_{\rm gravity}		applicable to any satellite orbit	radius for satellite in geostationary orbit Kepler's Third Law: period squared propto distance cubed	0000002867: $ F_{\rm gravity} $ 0000001687: $ F_{\rm centripetal} $	pdg_{0002867} = pdg_{0001687}	ERROR for dim with 3176662571
3182633789	$ \gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 $ \gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1			Lorentz transformation	0000004567: $ c $ 0000001790: $ \gamma $ 0000001357: $ v $	pdg_{0001790}^{2} - \frac{pdg_{0004567}^{2} \left(1 - pdg_{0001790}^{2}\right)^{2}}{pdg_{0001357}^{2} pdg_{0001790}^{2}} = 1	ERROR for dim with 3182633789
3214170322	$ v(r=\infty) = 0 $ v(r=\infty) = 0			velocity at distance r of object dropped from infinity	0000001357: $ v $	pdg_{0001357} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
3253234559	$ x = \frac{v_2^2 - v_1^2}{2 a} $ x = \frac{v_2^2 - v_1^2}{2 a}			work and force and energy	0000009140: $ a $ 0000002473: $ v_1 $ 0000004770: $ v_2 $ 0000004037: $ x $	pdg_{0004037} = \frac{- pdg_{0002473}^{2} + pdg_{0004770}^{2}}{2 pdg_{0009140}}	ERROR for dim with 3253234559
3274926090	$ t = \frac{x - x_0}{v_{0, x}} $ t = \frac{x - x_0}{v_{0, x}}			projectile path in 2D is parabolic	0000001572: $ x_0 $ 0000001467: $ t $ 0000002958: $ v_{0, x} $ 0000004037: $ x $	pdg_{0001467} = \frac{- pdg_{0001572} + pdg_{0004037}}{pdg_{0002958}}	ERROR for dim with 3274926090
3285732911	$ (\cos(x))^2 = 1-(\sin(x))^2 $ (\cos(x))^2 = 1-(\sin(x))^2			Euler equation: trig square root	0000001464: $ x $	\cos^{2}{\left(pdg_{0001464} \right)} = 1 - \sin^{2}{\left(pdg_{0001464} \right)}	ERROR for dim with 3285732911
3291685884	$ E = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2} $ E = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}			upper limit on velocity in condensed matter	0000001054: $ \hbar $ 0000003141: $ \pi $ 0000002515: $ m_e $ 0000002241: $ E $ 0000001999: $ e $ 0000007940: $ \epsilon_0 $	pdg_{0002241} = \frac{pdg_{0001999}^{4} pdg_{0002515}}{32 pdg_{0001054}^{2} pdg_{0003141}^{2} pdg_{0007940}^{2}}	ERROR for dim with 3291685884
3331824625	$ \exp(i \pi) = -1 $ \exp(i \pi) = -1			Euler equation to exp(i pi) + 1 = 0	0000004621: $ i $ 0000003141: $ \pi $	e^{pdg_{0003141} pdg_{0004621}} = -1	ERROR for dim with 3331824625
3350830826	$ Z Z^* = \|Z\|^2 $ Z Z^* = \|Z\|^2			double intensity when phase is coherent (optics)	0000003192: $ Z $	pdg_{0003192} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
3360172339	$ W = KE_2 - KE_1 $ W = KE_2 - KE_1			work and force and energy	0000001352: $ KE_2 $ 0000001955: $ KE_1 $ 0000006789: $ W $	pdg_{0006789} = pdg_{0001352} - pdg_{0001955}	ERROR for dim with 3360172339
3364286646	$ m_{\rm Earth} = 5.97210^{24} kg $ m_{\rm Earth} = 5.97210^{24} kg			mass of the Earth	0000005458: $ m_{\rm Earth} $	pdg_{0005458} = 5.972 \cdot 10^{24} kg	ERROR for dim with 3364286646
3366703541	$ a = \frac{v - v_0}{t} $ a = \frac{v - v_0}{t}		acceleration is the average change in speed over a duration	equations of motion in 1D with constant acceleration - SUVAT (algebra)	0000001467: $ t $ 0000009140: $ a $ 0000005153: $ v_0 $ 0000001357: $ v $	pdg_{0009140} = \frac{pdg_{0001357} - pdg_{0005153}}{pdg_{0001467}}	ERROR for dim with 3366703541
3411994811	$ v_{\rm average} = \frac{d}{t} $ v_{\rm average} = \frac{d}{t}			Newton's Law of Gravitation equations of motion in 1D with constant acceleration - SUVAT (algebra)	0000006709: $ v_{\rm average} $ 0000001467: $ t $ 0000001943: $ d $	pdg_{0006709} = \frac{pdg_{0001943}}{pdg_{0001467}}	ERROR for dim with 3411994811
3417126140	$ \tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 } $ \tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 }			optics: Law of refraction to Brewster's angle	0000002941: $ n_1 $ 0000004928: $ \theta_{\rm Brewster} $ 0000001958: $ n_2 $	\tan{\left(pdg_{0004928} \right)} = \frac{pdg_{0001958}}{pdg_{0002941}}	ERROR for dim with 3417126140
3426941928	$ x = \gamma ( \gamma (x - v t) + v t' ) $ x = \gamma ( \gamma (x - v t) + v t' )			Lorentz transformation	0000001467: $ t $ 0000001357: $ v $ 0000001790: $ \gamma $ 0000004989: $ t' $ 0000004037: $ x $	pdg_{0004037} = pdg_{0001790} \left(pdg_{0001357} pdg_{0004989} + pdg_{0001790} \left(- pdg_{0001357} pdg_{0001467} + pdg_{0004037}\right)\right)	ERROR for dim with 3426941928
3462972452	$ v = v_0 + a t $ v = v_0 + a t			equations of motion in 1D with constant acceleration - SUVAT (algebra)	0000001467: $ t $ 0000009140: $ a $ 0000005153: $ v_0 $ 0000001357: $ v $	pdg_{0001357} = pdg_{0001467} pdg_{0009140} + pdg_{0005153}	ERROR for dim with 3462972452
3464107376	$ \alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p $ \alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p		definition of expansion coefficient	coefficient of thermal expansion using the equation of state for an ideal gas first law of thermodynamics	0000004686: $ \alpha $ 0000007586: $ V $ 0000007343: $ T $	pdg_{0004686} = \frac{\frac{d}{d pdg_{0007343}} pdg_{0007586}}{pdg_{0007586}}	ERROR for dim with 3464107376
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) $ \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)			identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation	0000001464: $ x $ 0000004621: $ i $	\sin{\left(pdg_{0001464} \right)} \cos{\left(pdg_{0001464} \right)} = \frac{\left(\frac{e^{pdg_{0001464} pdg_{0004621}}}{2} + \frac{e^{- pdg_{0001464} pdg_{0004621}}}{2}\right) \left(e^{pdg_{0001464} pdg_{0004621}} - e^{- pdg_{0001464} pdg_{0004621}}\right)}{2 pdg_{0004621}}	ERROR for dim with 3470587782
3472836147	$ r_{\rm Earth\ orbit} = 1.496\ 10^8 {\rm km} $ r_{\rm Earth\ orbit} = 1.496\ 10^8 {\rm km}			speed of Earth around Sun	0000006081: $ r_{\rm Earth\ orbit} $	pdg_{0006081} = 1.496	ERROR for dim with 3472836147
3485125659	$ x_f = v_0 t_f \cos(\theta) + x_0 $ x_f = v_0 t_f \cos(\theta) + x_0			angle of maximum distance for projectile motion	0000003652: $ x_f $ 0000001572: $ x_0 $ 0000002467: $ t_f $ 0000001575: $ \theta $ 0000005153: $ v_0 $	pdg_{0003652} = pdg_{0001572} + pdg_{0002467} pdg_{0005153} \cos{\left(pdg_{0001575} \right)}	ERROR for dim with 3485125659
3485475729	$ \nabla^2 E( \vec{r}) = - \frac{\omega^2}{c^2} E( \vec{r}) $ \nabla^2 E( \vec{r}) = - \frac{\omega^2}{c^2} E( \vec{r})		https://physicsderivationgraph.blogspot.com/2020/09/representing-laplace-operator-nabla-in.html	electric field wave equation: from time dependent to time independent	0000006238: $ E $ 0000004567: $ c $ 0000002321: $ \omega $ 0000009472: $ \vec{r} $	nabla^{2} \operatorname{pdg}_{0006238}{\left(pdg_{0009472} \right)} = - \frac{pdg_{0002321}^{2} \operatorname{pdg}_{0006238}{\left(pdg_{0009472} \right)}}{pdg_{0004567}^{2}}	ERROR for dim with 3485475729
3488423948	$ k_{\rm adsorption} p_A [S] = k_{\rm desorption} [A_{\rm adsorption}] $ k_{\rm adsorption} p_A [S] = k_{\rm desorption} [A_{\rm adsorption}]			Langmuir Adsorption	0000009046: $ p_A $ 0000008379: $ k_{\rm desorption} $ 0000009067: $ [S] $ 0000006850: $ k_{\rm adsorption} $ 0000004940: $ [A_{\rm adsorption}] $	pdg_{0006850} pdg_{0009046} pdg_{0009067} = pdg_{0004940} pdg_{0008379}	ERROR for dim with 3488423948
3497828859	$ V = \frac{n R T}{P} $ V = \frac{n R T}{P}			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	0000008134: $ P $ 0000008179: $ R $ 0000007586: $ V $ 0000002834: $ n $ 0000007343: $ T $	pdg_{0007586} = \frac{pdg_{0002834} pdg_{0007343} pdg_{0008179}}{pdg_{0008134}}	ERROR for dim with 3497828859
3507029294	$ k_{\rm adsorption} p_A [S] = r_{\rm desorption} $ k_{\rm adsorption} p_A [S] = r_{\rm desorption}			Langmuir Adsorption	0000009067: $ [S] $ 0000001966: $ r_{\rm desorption} $ 0000009046: $ p_A $ 0000006850: $ k_{\rm adsorption} $	pdg_{0006850} pdg_{0009046} pdg_{0009067} = pdg_{0001966}	ERROR for dim with 3507029294
3512166162	$ W = F x $ W = F x			work and force and energy	0000004202: $ F $ 0000006789: $ W $ 0000004037: $ x $	pdg_{0006789} = pdg_{0004037} pdg_{0004202}	ERROR for dim with 3512166162
3547519267	$ S = k_{\rm Boltzmann} \ln \Omega $ S = k_{\rm Boltzmann} \ln \Omega		assumes equally probable microstates	first law of thermodynamics	0000003434: $ \Omega $ 0000001157: $ k_{\rm Boltzmann} $ 0000001394: $ S $	pdg_{0001394} = pdg_{0001157} \log{\left(pdg_{0003434} \right)}	ERROR for dim with 3547519267
3566149658	$ W_{\rm to\ system} = \int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx $ W_{\rm to\ system} = \int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx			velocity at distance r of object dropped from infinity	0000002530: $ r $ 0000005022: $ m_1 $ 0000006277: $ G $ 0000004037: $ x $ 0000004851: $ m_2 $ 0000009372: $ W_{\rm to\ system} $	pdg_{0009372} = \int\limits_{\infty}^{pdg_{0002530}} \left(- \frac{pdg_{0004851} pdg_{0005022} pdg_{0006277}}{pdg_{0004037}^{2}}\right)\, dpdg_{0004037}	ERROR for dim with 3566149658
3585845894	$ \langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2 $ \langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2			variance relation	0000001464: $ x $	pdg_{0001464} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
3591237106	$ \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v $ \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v			time invariant force conserves energy	0000001352: $ KE_2 $ 0000001467: $ t $ 0000001357: $ v $ 0000004202: $ F $ 0000004550: $ E_2 $ 0000001955: $ KE_1 $	\frac{pdg_{0004550} - pg_{5579}}{pdg_{0001467}} = - pdg_{0001357} pdg_{0004202} + \frac{pdg_{0001352} - pdg_{0001955}}{pdg_{0001467}}	ERROR for dim with 3591237106
3599953931	$ [S_0] = [S] + [A_{\rm adsorption}] $ [S_0] = [S] + [A_{\rm adsorption}]			Langmuir Adsorption	0000004940: $ [A_{\rm adsorption}] $ 0000009067: $ [S] $ 0000003037: $ [S_0] $	pdg_{0003037} = pdg_{0004940} + pdg_{0009067}	ERROR for dim with 3599953931
3605073197	$ \kappa_T = \frac{-nRT}{V} \left( \frac{-1}{P^2}\right) $ \kappa_T = \frac{-nRT}{V} \left( \frac{-1}{P^2}\right)			coefficient of isothermal compressibility using the equation of state for an ideal gas	0000008134: $ P $ 0000008179: $ R $ 0000007586: $ V $ 0000002834: $ n $ 0000004645: $ \kappa_T $ 0000007343: $ T $	pdg_{0004645} = \frac{pdg_{0002834} pdg_{0007343} pdg_{0008179}}{pdg_{0007586} pdg_{0008134}^{2}}	ERROR for dim with 3605073197
3607070319	$ d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right) $ d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)			angle of maximum distance for projectile motion	0000001649: $ g $ 0000005153: $ v_0 $ 0000001943: $ d $ 0000003141: $ \pi $	pdg_{0001943} = \frac{pdg_{0005153}^{2} \sin{\left(\frac{pdg_{0003141}}{2} \right)}}{pdg_{0001649}}	ERROR for dim with 3607070319
3614055652	$ v = \frac{2 \pi r}{T_{\rm orbit}} $ v = \frac{2 \pi r}{T_{\rm orbit}}			radius for satellite in geostationary orbit	0000008762: $ T_{\rm orbit} $ 0000002530: $ r $ 0000003141: $ \pi $ 0000001357: $ v $	pdg_{0001357} = \frac{2 pdg_{0002530} pdg_{0003141}}{pdg_{0008762}}	ERROR for dim with 3614055652
3649797559	$ F_{\rm centripetal} = m_2 d_2 \omega^2 $ F_{\rm centripetal} = m_2 d_2 \omega^2			Kepler's Third Law: period squared propto distance cubed	0000002798: $ d_2 $ 0000001687: $ F_{\rm centripetal} $ 0000004851: $ m_2 $ 0000002321: $ \omega $	pdg_{0001687} = pdg_{0002321}^{2} pdg_{0002798} pdg_{0004851}	ERROR for dim with 3649797559
3650370389	$ \frac{T^2}{r} F_{\rm gravity} = 4 \pi^2 m $ \frac{T^2}{r} F_{\rm gravity} = 4 \pi^2 m			Newton's Law of Gravitation	0000008762: $ T_{\rm orbit} $ 0000003141: $ \pi $ 0000005156: $ m $ 0000002530: $ r $ 0000002867: $ F_{\rm gravity} $	\frac{pdg_{0002867} pdg_{0008762}^{2}}{pdg_{0002530}} = 4 pdg_{0003141}^{2} pdg_{0005156}	ERROR for dim with 3650370389
3660957533	$ \cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right) $ \cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)			double intensity when phase is coherent (optics)	0000001464: $ x $ 0000004621: $ i $ 0000008586: $ \phi $ 0000001575: $ \theta $	\cos{\left(pdg_{0001464} \right)} = \frac{e^{pdg_{0004621} \left(pdg_{0001575} - pdg_{0008586}\right)}}{2} + \frac{e^{- pdg_{0004621} \left(pdg_{0001575} - pdg_{0008586}\right)}}{2}	ERROR for dim with 3660957533
3676159007	$ v_{0, x} \int dt = \int dx $ v_{0, x} \int dt = \int dx			equations of motion in 2D (calculus)	0000001464: $ x $ 0000002958: $ v_{0, x} $ 0000001467: $ t $	pdg_{0002958} \int 1\, dpdg_{0001467} = \int 1\, dpdg_{0001464}	ERROR for dim with 3676159007
3736177473	$ r_{\rm adsorption} = k_{\rm adsorption} p_A [S] $ r_{\rm adsorption} = k_{\rm adsorption} p_A [S]			Langmuir Adsorption	0000009067: $ [S] $ 0000006687: $ r_{\rm adsorption} $ 0000009046: $ p_A $ 0000006850: $ k_{\rm adsorption} $	pdg_{0006687} = pdg_{0006850} pdg_{0009046} pdg_{0009067}	ERROR for dim with 3736177473
3781109867	$ T^2 = \frac{r^3 4 \pi^2}{(d_1+d_2) \frac{m_1}{d_2}G} $ T^2 = \frac{r^3 4 \pi^2}{(d_1+d_2) \frac{m_1}{d_2}G}			Kepler's Third Law: period squared propto distance cubed	0000003141: $ \pi $ 0000002798: $ d_2 $ 0000007652: $ d_1 $ 0000006277: $ G $ 0000005022: $ m_1 $ 0000002530: $ r $ 0000009491: $ T $	pdg_{0009491}^{2} = \frac{4 pdg_{0002530}^{3} pdg_{0002798} pdg_{0003141}^{2}}{pdg_{0005022} pdg_{0006277} \left(pdg_{0002798} + pdg_{0007652}\right)}	ERROR for dim with 3781109867
3806977900	$ E_2 - E_1 = 0 $ E_2 - E_1 = 0			time invariant force conserves energy	0000004550: $ E_2 $ 0000005579: $ E_1 $	pdg_{0004550} - pdg_{0005579} = 0	ERROR for dim with 3806977900
3829492824	$ \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) = \cos(x) $ \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) = \cos(x)			Euler equation: trigonometric relations	0000001464: $ x $ 0000004621: $ i $	\frac{e^{pdg_{0001464} pdg_{0004621}}}{2} + \frac{e^{- pdg_{0001464} pdg_{0004621}}}{2} = \cos{\left(pdg_{0001464} \right)}	ERROR for dim with 3829492824
3846041519	$ PE_{\rm Earth\ surface} = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}} $ PE_{\rm Earth\ surface} = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}}			escape velocity	0000005458: $ m_{\rm Earth} $ 0000005156: $ m $ 0000006277: $ G $ 0000006431: $ PE_{\rm Earth\ surface} $ 0000003236: $ r_{\rm Earth} $	pdg_{0006431} = - \frac{pdg_{0005156} pdg_{0005458} pdg_{0006277}}{pdg_{0003236}}	ERROR for dim with 3846041519
3868998312	$ {\rm sech}^2\ x = \frac{4}{\left(\exp(x)+\exp(-x)\right)^2} $ {\rm sech}^2\ x = \frac{4}{\left(\exp(x)+\exp(-x)\right)^2}			hyperbolic trigonometric identities	0000001464: $ x $	\operatorname{sech}^{2}{\left(pdg_{0001464} \right)} = \frac{4}{\left(e^{pdg_{0001464}} + e^{- pdg_{0001464}}\right)^{2}}	ERROR for dim with 3868998312
3896798826	$ m_2 d_2 \omega^2 = G \frac{m_1 m_2}{r^2} $ m_2 d_2 \omega^2 = G \frac{m_1 m_2}{r^2}			Kepler's Third Law: period squared propto distance cubed	0000002798: $ d_2 $ 0000002321: $ \omega $ 0000006277: $ G $ 0000005022: $ m_1 $ 0000002530: $ r $ 0000004851: $ m_2 $	pdg_{0002321}^{2} pdg_{0002798} pdg_{0004851} = \frac{pdg_{0004851} pdg_{0005022} pdg_{0006277}}{pdg_{0002530}^{2}}	ERROR for dim with 3896798826
3906710072	$ G \frac{m_{\rm Earth}}{r} = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2} $ G \frac{m_{\rm Earth}}{r} = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2}			radius for satellite in geostationary orbit	0000008762: $ T_{\rm orbit} $ 0000003141: $ \pi $ 0000005458: $ m_{\rm Earth} $ 0000002530: $ r $ 0000006277: $ G $	\frac{pdg_{0005458} pdg_{0006277}}{pdg_{0002530}} = \frac{4 pdg_{0002530}^{2} pdg_{0003141}^{2}}{pdg_{0008762}^{2}}	ERROR for dim with 3906710072
3920616792	$ T_{\rm geostationary orbit} = 24\ {\rm hours} $ T_{\rm geostationary orbit} = 24\ {\rm hours}		this applies for geostationary orbits	radius for satellite in geostationary orbit	0000005595: $ T_{\rm geostationary\ orbit} $	pdg_{0005595} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
3924948349	$ a_{\beta} \langle \psi_{\alpha} \| \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} \| \psi_{\beta} \rangle = 0 $ a_{\beta} \langle \psi_{\alpha} \| \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} \| \psi_{\beta} \rangle = 0			quantum basics orthogonality	0000007752: $ a_{\beta} $	pdg_{0007752} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
3935058307	$ v = \sqrt{ \frac{m_e}{m} \frac{e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} } $ v = \sqrt{ \frac{m_e}{m} \frac{e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} }			upper limit on velocity in condensed matter	0000001054: $ \hbar $ 0000003141: $ \pi $ 0000002515: $ m_e $ 0000002077: $ v $ 0000009863: $ m $ 0000001999: $ e $ 0000007940: $ \epsilon_0 $	pdg_{0002077} = \frac{\sqrt{2} \sqrt{\frac{pdg_{0001999}^{4} pdg_{0002515}}{pdg_{0001054}^{2} pdg_{0003141}^{2} pdg_{0007940}^{2} pdg_{0009863}}}}{8}	ERROR for dim with 3935058307
3942849294	$ \exp(i x)-\exp(-i x) = 2 i \sin(x) $ \exp(i x)-\exp(-i x) = 2 i \sin(x)			Euler equation: trigonometric relations	0000001464: $ x $ 0000004621: $ i $	e^{pdg_{0001464} pdg_{0004621}} - e^{- pdg_{0001464} pdg_{0004621}} = 2 pdg_{0004621} \sin{\left(pdg_{0001464} \right)}	ERROR for dim with 3942849294
3943939590	$ x = a_{\alpha} \langle \psi_{\alpha}\| \psi_{\beta}\rangle $ x = a_{\alpha} \langle \psi_{\alpha}\| \psi_{\beta}\rangle			quantum basics orthogonality	0000002427: $ a_{\alpha} $	pdg_{0002427} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
3947269979	$ \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} $ \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}			Maxwell equations to electric field wave equation	0000001467: $ t $	pdg_{0001467} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
3948571256	$ \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}E \psi( \vec{r},t) $ \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}E \psi( \vec{r},t)			derivation of Schrodinger Equation	0000006238: $ E $ 0000001054: $ \hbar $ 0000001467: $ t $ 0000009489: $ \psi $ 0000004621: $ i $ 0000009472: $ \vec{r} $	\frac{\partial}{\partial pdg_{0001467}} \operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)} = - \frac{pdg_{0004621} pdg_{0006238} \operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)}}{pdg_{0001054}}	ERROR for dim with 3948571256
3948574224	$ \psi( \vec{r},t) = \psi_0 \exp\left(i\left( \vec{k}\cdot\vec{r} - \omega t \right) \right) $ \psi( \vec{r},t) = \psi_0 \exp\left(i\left( \vec{k}\cdot\vec{r} - \omega t \right) \right)			derivation of Schrodinger Equation	0000001467: $ t $ 0000009489: $ \psi $ 0000008330: $ \psi_0 $ 0000004621: $ i $ 0000002321: $ \omega $ 0000002718: $ \exp $ 0000005321: $ k $ 0000009472: $ \vec{r} $	\operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)} = pdg_{0008330} \operatorname{pdg}_{0002718}{\left(\operatorname{pdg}_{0004621}{\left(- pdg_{0001467} pdg_{0002321} + \operatorname{dot}{\left(pdg_{0005321},pdg_{0009472} \right)} \right)} \right)}	ERROR for dim with 3948574224
3948574226	$ \psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right) $ \psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)			derivation of Schrodinger Equation	0000001054: $ \hbar $ 0000001467: $ t $ 0000009489: $ \psi $ 0000008330: $ \psi_0 $ 0000004621: $ i $ 0000001134: $ p $ 0000002321: $ \omega $ 0000002718: $ \exp $ 0000009472: $ \vec{r} $	\operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)} = pdg_{0008330} \operatorname{pdg}_{0002718}{\left(\operatorname{pdg}_{0004621}{\left(- pdg_{0001467} pdg_{0002321} + \frac{pdg_{0001134} pdg_{0009472}}{pdg_{0001054}} \right)} \right)}	ERROR for dim with 3948574226
3948574228	$ \psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right) $ \psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)			derivation of Schrodinger Equation	0000006238: $ E $ 0000001054: $ \hbar $ 0000001467: $ t $ 0000009489: $ \psi $ 0000008330: $ \psi_0 $ 0000004621: $ i $ 0000001134: $ p $ 0000002718: $ \exp $ 0000009472: $ \vec{r} $	\operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)} = pdg_{0008330} \operatorname{pdg}_{0002718}{\left(\operatorname{pdg}_{0004621}{\left(\frac{pdg_{0001134} pdg_{0009472}}{pdg_{0001054}} - \frac{pdg_{0001467} pdg_{0006238}}{pdg_{0001054}} \right)} \right)}	ERROR for dim with 3948574228
3948574230	$ \psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right) $ \psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)			derivation of Schrodinger Equation	0000006238: $ E $ 0000001054: $ \hbar $ 0000001467: $ t $ 0000009489: $ \psi $ 0000008330: $ \psi_0 $ 0000004621: $ i $ 0000001134: $ p $ 0000002718: $ \exp $ 0000009472: $ \vec{r} $	\operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)} = pdg_{0008330} \operatorname{pdg}_{0002718}{\left(\frac{pdg_{0004621} \left(pdg_{0001134} pdg_{0009472} - pdg_{0001467} pdg_{0006238}\right)}{pdg_{0001054}} \right)}	ERROR for dim with 3948574230
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) $ \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)			derivation of Schrodinger Equation	0000006238: $ E $ 0000001054: $ \hbar $ 0000001467: $ t $ 0000009489: $ \psi $ 0000008330: $ \psi_0 $ 0000004621: $ i $ 0000001134: $ p $ 0000002718: $ \exp $ 0000009472: $ \vec{r} $	\frac{\partial}{\partial pdg_{0001467}} \operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)} = pdg_{0008330} \frac{\partial}{\partial pdg_{0001467}} \operatorname{pdg}_{0002718}{\left(\operatorname{pdg}_{0004621}{\left(\frac{pdg_{0001134} pdg_{0009472}}{pdg_{0001054}} - \frac{pdg_{0001467} pdg_{0006238}}{pdg_{0001054}} \right)} \right)}	ERROR for dim with 3948574233
3951205425	$ \vec{p}_{\rm after} = \vec{p}_{1} $ \vec{p}_{\rm after} = \vec{p}_{1}			Compton's equation for scattering	0000006029: $ \vec{p}_1 $ 0000005493: $ \vec{p}_{\rm after} $	pdg_{0005493} = pdg_{0006029}	ERROR for dim with 3951205425
4072200527	$ \frac{m_{\rm satellite} v_{\rm satellite}^2}{r} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2} $ \frac{m_{\rm satellite} v_{\rm satellite}^2}{r} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}			radius for satellite in geostationary orbit	0000005458: $ m_{\rm Earth} $ 0000004082: $ v_{\rm satellite} $ 0000003569: $ m_{\rm satellite} $ 0000002530: $ r $ 0000006277: $ G $	\frac{pdg_{0003569} pdg_{0004082}^{2}}{pdg_{0002530}} = \frac{pdg_{0003569} pdg_{0005458} pdg_{0006277}}{pdg_{0002530}^{2}}	ERROR for dim with 4072200527
4075539836	$ A A^* = \|A\|^2 $ A A^* = \|A\|^2			double intensity when phase is coherent (optics)	0000004453: $ A $	pdg_{0004453} \overline{pdg_{0004453}} = \left\|{pdg_{0004453}}\right\|^{2}	ERROR for dim with 4075539836
4087145886	$ V = I R $ V = I R	Ohm's law	https://en.wikipedia.org/wiki/Ohm%27s_law	total electrical resistance for circuit with two resistors in parallel total electrical resistance for circuit with two resistors in series	0000004501: $ I $ 0000006458: $ R $ 0000006599: $ V $	pdg_{0006599} = pdg_{0004501} pdg_{0006458}	ERROR for dim with 4087145886
4107032818	$ E_{\rm Rydberg} = E $ E_{\rm Rydberg} = E			upper limit on velocity in condensed matter	0000002241: $ E $ 0000009838: $ E_{\rm Rydberg} $	pdg_{0009838} = pdg_{0002241}	ERROR for dim with 4107032818
4128500715	$ V = I_1 R_1 $ V = I_1 R_1			total electrical resistance for circuit with two resistors in parallel	0000008697: $ R_1 $ 0000003978: $ I_1 $ 0000006599: $ V $	pdg_{0006599} = pdg_{0003978} pdg_{0008697}	ERROR for dim with 4128500715
4139999399	$ x - \gamma^2 x = - \gamma^2 v t + \gamma v t' $ x - \gamma^2 x = - \gamma^2 v t + \gamma v t'			Lorentz transformation	0000001467: $ t $ 0000001357: $ v $ 0000001790: $ \gamma $ 0000004989: $ t' $ 0000004037: $ x $	- pdg_{0001790}^{2} pdg_{0004037} + pdg_{0004037} = - pdg_{0001357} pdg_{0001467} pdg_{0001790}^{2} + pdg_{0001357} pdg_{0001790} pdg_{0004989}	ERROR for dim with 4139999399
4147472132	$ E = \frac{h \omega}{2 \pi} $ E = \frac{h \omega}{2 \pi}			derivation of Schrodinger Equation	0000004931: $ E $ 0000004413: $ h $ 0000003141: $ \pi $ 0000002321: $ \omega $	pdg_{0004931} = \frac{pdg_{0002321} pdg_{0004413}}{2 pdg_{0003141}}	ERROR for dim with 4147472132
4158986868	$ a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt} $ a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}			equations of motion in 2D (calculus)	0000001467: $ t $	pdg_{0001467} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
4166155526	$ {\rm sech}\ x = \frac{2}{\exp(x)+\exp(-x)} $ {\rm sech}\ x = \frac{2}{\exp(x)+\exp(-x)}			hyperbolic trigonometric identities	0000001464: $ x $	\operatorname{sech}{\left(pdg_{0001464} \right)} = \frac{2}{e^{pdg_{0001464}} + e^{- pdg_{0001464}}}	ERROR for dim with 4166155526
4180845508	$ v_{\rm Earth\ orbit} = 29.8 \frac{{\rm km}}{{\rm sec}} $ v_{\rm Earth\ orbit} = 29.8 \frac{{\rm km}}{{\rm sec}}			speed of Earth around Sun	0000007427: $ v_{\rm Earth\ orbit} $	pdg_{0007427} = 29.8	ERROR for dim with 4180845508
4182362050	$ Z = \|Z\| \exp( i \theta ) $ Z = \|Z\| \exp( i \theta )		Z \in \mathbb{C}	double intensity when phase is coherent (optics)	0000004621: $ i $ 0000003192: $ Z $ 0000001575: $ \theta $	pdg_{0003192} = e^{pdg_{0001575} pdg_{0004621}} \left\|{pdg_{0003192}}\right\|	ERROR for dim with 4182362050
4188580242	$ T^2 = \frac{r^3 4 \pi^2}{\left(m_1+\left(\frac{m_1}{d_2}d_1\right)\right)G} $ T^2 = \frac{r^3 4 \pi^2}{\left(m_1+\left(\frac{m_1}{d_2}d_1\right)\right)G}			Kepler's Third Law: period squared propto distance cubed	0000003141: $ \pi $ 0000002798: $ d_2 $ 0000007652: $ d_1 $ 0000006277: $ G $ 0000005022: $ m_1 $ 0000002530: $ r $ 0000009491: $ T $	pdg_{0009491}^{2} = \frac{4 pdg_{0002530}^{3} pdg_{0003141}^{2}}{pdg_{0006277} \left(pdg_{0005022} + \frac{pdg_{0005022} pdg_{0007652}}{pdg_{0002798}}\right)}	ERROR for dim with 4188580242
4192519596	$ B = \|B\| \exp(i \phi) $ B = \|B\| \exp(i \phi)			double intensity when phase is coherent (optics)	0000004621: $ i $ 0000008586: $ \phi $ 0000004698: $ B $	pdg_{0004698} = e^{pdg_{0004621} pdg_{0008586}} \left\|{pdg_{0004698}}\right\|	ERROR for dim with 4192519596
4245712581	$ v = \frac{2 \pi r}{t} $ v = \frac{2 \pi r}{t}			radius for satellite in geostationary orbit	0000001467: $ t $ 0000002530: $ r $ 0000003141: $ \pi $ 0000001357: $ v $	pdg_{0001357} = \frac{2 pdg_{0002530} pdg_{0003141}}{pdg_{0001467}}	ERROR for dim with 4245712581
4267808354	$ F_{\rm gravity} = m \frac{v^2}{r} $ F_{\rm gravity} = m \frac{v^2}{r}			Newton's Law of Gravitation	0000002867: $ F_{\rm gravity} $ 0000005156: $ m $ 0000002530: $ r $ 0000001357: $ v $	pdg_{0002867} = \frac{pdg_{0001357}^{2} pdg_{0005156}}{pdg_{0002530}}	ERROR for dim with 4267808354
4268085801	$ x_0 + d = v_0 t_f \cos(\theta) + x_0 $ x_0 + d = v_0 t_f \cos(\theta) + x_0			angle of maximum distance for projectile motion	0000001943: $ d $ 0000001572: $ x_0 $ 0000002467: $ t_f $ 0000001575: $ \theta $ 0000005153: $ v_0 $	pdg_{0001572} + pdg_{0001943} = pdg_{0001572} + pdg_{0002467} pdg_{0005153} \cos{\left(pdg_{0001575} \right)}	ERROR for dim with 4268085801
4270680309	$ \frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t} $ \frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}			time invariant force conserves energy	0000001352: $ KE_2 $ 0000001467: $ t $ 0000002473: $ v_1 $ 0000005156: $ m $ 0000004770: $ v_2 $ 0000001955: $ KE_1 $	\frac{pdg_{0001352} - pdg_{0001955}}{pdg_{0001467}} = \frac{pdg_{0005156} \left(- pdg_{0002473}^{2} + pdg_{0004770}^{2}\right)}{2 pdg_{0001467}}	ERROR for dim with 4270680309
4275004561	$ c^2 = 2 G \frac{m}{r_{\rm Schwarzschild}} $ c^2 = 2 G \frac{m}{r_{\rm Schwarzschild}}			Schwarzschild radius for non-rotating black hole	0000006277: $ G $ 0000004518: $ r_{\rm Schwarzschild} $ 0000005156: $ m $ 0000004567: $ c $	pdg_{0004567}^{2} = \frac{2 pdg_{0005156} pdg_{0006277}}{pdg_{0004518}}	ERROR for dim with 4275004561
4287102261	$ x^2 + y^2 + z^2 = c^2 t^2 $ x^2 + y^2 + z^2 = c^2 t^2		describes a spherical wavefront	Lorentz transformation	0000001467: $ t $ 0000004567: $ c $ 0000005647: $ y $ 0000006728: $ z $ 0000004037: $ x $	pdg_{0004037}^{2} + pdg_{0005647}^{2} + pdg_{0006728}^{2} = pdg_{0001467}^{2} pdg_{0004567}^{2}	ERROR for dim with 4287102261
4298359835	$ E = \frac{1}{2}m v^2 $ E = \frac{1}{2}m v^2			derivation of Schrodinger Equation	0000004931: $ E $ 0000005156: $ m $ 0000001357: $ v $	pdg_{0004931} = \frac{pdg_{0001357}^{2} pdg_{0005156}}{2}	ERROR for dim with 4298359835
4298359845	$ E = \frac{1}{2m}m^2 v^2 $ E = \frac{1}{2m}m^2 v^2			derivation of Schrodinger Equation	0000004931: $ E $ 0000005156: $ m $ 0000001357: $ v $	pdg_{0004931} = \frac{pdg_{0001357}^{2} pdg_{0005156}}{2}	ERROR for dim with 4298359845
4298359851	$ E = \frac{p^2}{2m} $ E = \frac{p^2}{2m}			derivation of Schrodinger Equation	0000004931: $ E $ 0000001134: $ p $ 0000005156: $ m $	pdg_{0004931} = \frac{pdg_{0001134}^{2}}{2 pdg_{0005156}}	ERROR for dim with 4298359851
4301729661	$ [S_0] = \frac{[A_{\rm adsorption}]}{\left( \frac{k_{\rm adsorption}}{k_{\rm desorption}} \right) p_A} + [A_{\rm adsorption}] $ [S_0] = \frac{[A_{\rm adsorption}]}{\left( \frac{k_{\rm adsorption}}{k_{\rm desorption}} \right) p_A} + [A_{\rm adsorption}]			Langmuir Adsorption	0000009046: $ p_A $ 0000008379: $ k_{\rm desorption} $ 0000006850: $ k_{\rm adsorption} $ 0000004940: $ [A_{\rm adsorption}] $ 0000003037: $ [S_0] $	pdg_{0003037} = pdg_{0004940} + \frac{pdg_{0004940} pdg_{0008379}}{pdg_{0006850} pdg_{0009046}}	ERROR for dim with 4301729661
4303372136	$ E_1 = KE_1 + PE_1 $ E_1 = KE_1 + PE_1			time invariant force conserves energy escape velocity	0000001955: $ KE_1 $ 0000005579: $ E_1 $ 0000004093: $ PE_1 $	pdg_{0005579} = pdg_{0001955} + pdg_{0004093}	ERROR for dim with 4303372136
4341171256	$ i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{p^2}{2 m} \psi( \vec{r},t) $ i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{p^2}{2 m} \psi( \vec{r},t)			derivation of Schrodinger Equation	0000001054: $ \hbar $ 0000001467: $ t $ 0000009489: $ \psi $ 0000004621: $ i $ 0000001134: $ p $ 0000005156: $ m $ 0000009472: $ \vec{r} $	pdg_{0001054} pdg_{0004621} \frac{\partial}{\partial pdg_{0001467}} \operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)} = \frac{pdg_{0001134}^{2} \operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)}}{2 pdg_{0005156}}	ERROR for dim with 4341171256
4348571256	$ \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi( \vec{r},t) $ \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi( \vec{r},t)			derivation of Schrodinger Equation	0000001054: $ \hbar $ 0000001467: $ t $ 0000009489: $ \psi $ 0000004621: $ i $ 0000001134: $ p $ 0000005156: $ m $ 0000009472: $ \vec{r} $	\frac{\partial}{\partial pdg_{0001467}} \operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)} = - \frac{pdg_{0001134}^{2} pdg_{0004621} \operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)}}{2 pdg_{0001054} pdg_{0005156}}	ERROR for dim with 4348571256
4370074654	$ t = t_f $ t = t_f			angle of maximum distance for projectile motion	0000002467: $ t_f $ 0000001467: $ t $	pdg_{0001467} = pdg_{0002467}	ERROR for dim with 4370074654
4393258808	$ F_{\rm centripetal} = m r \omega^2 $ F_{\rm centripetal} = m r \omega^2			Kepler's Third Law: period squared propto distance cubed	0000005156: $ m $ 0000001687: $ F_{\rm centripetal} $ 0000002530: $ r $ 0000002321: $ \omega $	pdg_{0001687} = pdg_{0002321}^{2} pdg_{0002530} pdg_{0005156}	ERROR for dim with 4393258808
4393670960	$ W_{\rm to\ system} = \frac{G m_1 m_2}{r} $ W_{\rm to\ system} = \frac{G m_1 m_2}{r}			velocity at distance r of object dropped from infinity	0000006277: $ G $ 0000005022: $ m_1 $ 0000002530: $ r $ 0000004851: $ m_2 $ 0000009372: $ W_{\rm to\ system} $	pdg_{0009372} = \frac{pdg_{0004851} pdg_{0005022} pdg_{0006277}}{pdg_{0002530}}	ERROR for dim with 4393670960
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{ \nabla}\cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right) = \frac{i}{\hbar} \vec{ \nabla}\cdot\left( \vec{p} \psi( \vec{r},t) \right)			derivation of Schrodinger Equation	0000001054: $ \hbar $ 0000001467: $ t $ 0000009489: $ \psi $ 0000004621: $ i $ 0000002046: $ \vec{p} $ 0000009472: $ \vec{r} $	AttributeError in get_sympy_as_latex_per_expr_id: 'Symbol' object has no attribute 'dot' = \frac{nabla pdg_{0002046} pdg_{0004621} \operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)}}{pdg_{0001054}}	AttributeError: unable to parse Symbol('nabla').dot(Symbol('nabla')( Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467')))) as SymPy; error='Symbol' object has no attribute 'dot'
4428528271	$ F_{\rm spring} = -k x $ F_{\rm spring} = -k x	Hooke's law	https://en.wikipedia.org/wiki/Hooke%27s_law	equation of motion for a spring	0000001356: $ k $ 0000004183: $ F_{\rm spring} $ 0000004037: $ x $	pdg_{0004183} = - pdg_{0001356} pdg_{0004037}	ERROR for dim with 4428528271
4447113478	$ \int dW = G m_1 m_2 \int_{ r_{\rm Earth} }^{\infty} \frac{1}{x^2} dx $ \int dW = G m_1 m_2 \int_{ r_{\rm Earth} }^{\infty} \frac{1}{x^2} dx			escape velocity	0000006789: $ W $ 0000005022: $ m_1 $ 0000006277: $ G $ 0000003236: $ r_{\rm Earth} $ 0000004851: $ m_2 $ 0000004037: $ x $	\int 1\, dpdg_{0006789} = pdg_{0004851} pdg_{0005022} pdg_{0006277} \int\limits_{pdg_{0003236}}^{infty} \frac{1}{pdg_{0004037}^{2}}\, dpdg_{0004037}	ERROR for dim with 4447113478
4501377629	$ \tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} $ \tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}			optics: Law of refraction to Brewster's angle	0000004928: $ \theta_{\rm Brewster} $	\tan{\left(pdg_{0004928} \right)} = \frac{\sin{\left(pdg_{0004928} \right)}}{\cos{\left(pdg_{0004928} \right)}}	ERROR for dim with 4501377629
4504256452	$ B^* = \|B\| \exp(-i \phi) $ B^* = \|B\| \exp(-i \phi)			double intensity when phase is coherent (optics)	0000004621: $ i $ 0000008586: $ \phi $ 0000004698: $ B $	\overline{pdg_{0004698}} = e^{- pdg_{0004621} pdg_{0008586}} \left\|{pdg_{0004698}}\right\|	ERROR for dim with 4504256452
4560648264	$ v = \sqrt{ \frac{K + (4/3) G}{\rho} } $ v = \sqrt{ \frac{K + (4/3) G}{\rho} }			upper limit on velocity in condensed matter	0000001466: $ K $ 0000002077: $ v $ 0000003935: $ \rho $ 0000003033: $ G $	pdg_{0002077} = \sqrt{\frac{pdg_{0001466} + \frac{4 pdg_{0003033}}{3}}{pdg_{0003935}}}	ERROR for dim with 4560648264
4580545876	$ d = v t - a t^2 + \frac{1}{2} a t^2 $ d = v t - a t^2 + \frac{1}{2} a t^2			equations of motion in 1D with constant acceleration - SUVAT (algebra)	0000001467: $ t $ 0000009140: $ a $ 0000001943: $ d $ 0000001357: $ v $	pdg_{0001943} = pdg_{0001357} pdg_{0001467} - \frac{pdg_{0001467}^{2} pdg_{0009140}}{2}	ERROR for dim with 4580545876
4585828572	$ \epsilon_0 \mu_0 = \frac{1}{c^2} $ \epsilon_0 \mu_0 = \frac{1}{c^2}			electric field wave equation: from time dependent to time independent	0000007940: $ \epsilon_0 $ 0000004567: $ c $ 0000006197: $ \mu_0 $	pdg_{0006197} pdg_{0007940} = \frac{1}{pdg_{0004567}^{2}}	ERROR for dim with 4585828572
4585932229	$ \cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) $ \cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)			identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation double intensity when phase is coherent (optics) hyperbolic trigonometric identities Euler equation: trigonometric relations	0000001464: $ x $ 0000004621: $ i $	\cos{\left(pdg_{0001464} \right)} = \frac{e^{pdg_{0001464} pdg_{0004621}}}{2} + \frac{e^{- pdg_{0001464} pdg_{0004621}}}{2}	ERROR for dim with 4585932229
4593428198	$ v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{3.16\ 10^7 {\rm seconds}} $ v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{3.16\ 10^7 {\rm seconds}}			speed of Earth around Sun	0000007427: $ v_{\rm Earth\ orbit} $ 0000006081: $ r_{\rm Earth\ orbit} $ 0000003141: $ \pi $	pdg_{0007427} = 0.632911392405063 pdg_{0003141} pdg_{0006081}	ERROR for dim with 4593428198
4598294821	$ \exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2 $ \exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2			Euler equation: trig square root	0000001464: $ x $ 0000004621: $ i $	e^{2 pdg_{0001464} pdg_{0004621}} = 2 pdg_{0004621} \sin{\left(pdg_{0001464} \right)} \cos{\left(pdg_{0001464} \right)} - \sin^{2}{\left(pdg_{0001464} \right)} + \cos^{2}{\left(pdg_{0001464} \right)}	ERROR for dim with 4598294821
4627284246	$ F_{\rm centripetal} = \frac{m_{\rm satellite} v_{\rm satellite}^2}{r} $ F_{\rm centripetal} = \frac{m_{\rm satellite} v_{\rm satellite}^2}{r}			radius for satellite in geostationary orbit	0000004082: $ v_{\rm satellite} $ 0000001687: $ F_{\rm centripetal} $ 0000002530: $ r $ 0000003569: $ m_{\rm satellite} $	pdg_{0001687} = \frac{pdg_{0003569} pdg_{0004082}^{2}}{pdg_{0002530}}	ERROR for dim with 4627284246
4638429483	$ \exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x)) $ \exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))			Euler equation: trig square root	0000001464: $ x $ 0000004621: $ i $	e^{2 pdg_{0001464} pdg_{0004621}} = \left(pdg_{0004621} \sin{\left(pdg_{0001464} \right)} + \cos{\left(pdg_{0001464} \right)}\right)^{2}	ERROR for dim with 4638429483
4648451961	$ v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1) $ v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)			time invariant force conserves energy	0000002473: $ v_1 $ 0000004770: $ v_2 $	- pdg_{0002473}^{2} + pdg_{0004770}^{2} = \left(- pdg_{0002473} + pdg_{0004770}\right) \left(pdg_{0002473} + pdg_{0004770}\right)	ERROR for dim with 4648451961
4662369843	$ x' = \gamma (x - v t) $ x' = \gamma (x - v t)			Lorentz transformation	0000001467: $ t $ 0000001357: $ v $ 0000001464: $ x $ 0000001790: $ \gamma $ 0000005456: $ x' $	pdg_{0005456} = pdg_{0001790} \left(- pdg_{0001357} pdg_{0001467} + pdg_{0001464}\right)	ERROR for dim with 4662369843
4664063894	$ F \propto m_1 $ F \propto m_1			Newton's Law of Gravitation	0000005022: $ m_1 $ 0000004202: $ F $	pdg_{0000004202} \propto pdg_{0000005022}	inconsistent dimensions
4669290568	$ PE_1 = -F x_1 $ PE_1 = -F x_1			time invariant force conserves energy	0000004202: $ F $ 0000003852: $ x_1 $ 0000004093: $ PE_1 $	pdg_{0004093} = - pdg_{0003852} pdg_{0004202}	ERROR for dim with 4669290568
4689334676	$ \theta_A = \frac{K_{\rm equilibrium}\ p_A}{1+K_{\rm equilibrium}\ p_A} $ \theta_A = \frac{K_{\rm equilibrium}\ p_A}{1+K_{\rm equilibrium}\ p_A}			Langmuir Adsorption	0000001791: $ \theta_A $ 0000004933: $ K_{\rm equilibrium} $ 0000009046: $ p_A $	pdg_{0001791} = \frac{pdg_{0004933} pdg_{0009046}}{pdg_{0004933} pdg_{0009046} + 1}	ERROR for dim with 4689334676
4742644828	$ \exp(i x)+\exp(-i x) = 2 \cos(x) $ \exp(i x)+\exp(-i x) = 2 \cos(x)			Euler equation: trigonometric relations	0000001464: $ x $ 0000004621: $ i $	e^{pdg_{0001464} pdg_{0004621}} + e^{- pdg_{0001464} pdg_{0004621}} = 2 \cos{\left(pdg_{0001464} \right)}	ERROR for dim with 4742644828
4748157455	$ a t = v - v_0 $ a t = v - v_0			equations of motion in 1D with constant acceleration - SUVAT (algebra)	0000001467: $ t $ 0000009140: $ a $ 0000005153: $ v_0 $ 0000001357: $ v $	pdg_{0001467} pdg_{0009140} = pdg_{0001357} - pdg_{0005153}	ERROR for dim with 4748157455
4778077984	$ t_f = \frac{2 v_0 \sin(\theta)}{g} $ t_f = \frac{2 v_0 \sin(\theta)}{g}			angle of maximum distance for projectile motion	0000001575: $ \theta $ 0000001649: $ g $ 0000002467: $ t_f $ 0000005153: $ v_0 $	pdg_{0002467} = \frac{2 pdg_{0005153} \sin{\left(pdg_{0001575} \right)}}{pdg_{0001649}}	ERROR for dim with 4778077984
4784793837	$ \frac{KE_2 - KE_1}{t} = m v a $ \frac{KE_2 - KE_1}{t} = m v a			time invariant force conserves energy	0000001352: $ KE_2 $ 0000001467: $ t $ 0000001357: $ v $ 0000009140: $ a $ 0000005156: $ m $ 0000001955: $ KE_1 $	\frac{pdg_{0001352} - pdg_{0001955}}{pdg_{0001467}} = pdg_{0001357} pdg_{0005156} pdg_{0009140}	ERROR for dim with 4784793837
4798787814	$ a t + v_0 = v $ a t + v_0 = v			equations of motion in 1D with constant acceleration - SUVAT (algebra)	0000001467: $ t $ 0000009140: $ a $ 0000005153: $ v_0 $ 0000001357: $ v $	pdg_{0001467} pdg_{0009140} + pdg_{0005153} = pdg_{0001357}	ERROR for dim with 4798787814
4800170179	$ F = m g_{\rm Earth} $ F = m g_{\rm Earth}			mass of the Earth	0000005156: $ m $ 0000007557: $ g_{\rm Earth} $ 0000004202: $ F $	pdg_{0004202} = pdg_{0005156} pdg_{0007557}	ERROR for dim with 4800170179
4805233006	$ i \sin(i x) = \frac{1}{2}\left(\exp(x) - \exp(-x) \right) $ i \sin(i x) = \frac{1}{2}\left(\exp(x) - \exp(-x) \right)				0000001464: $ x $ 0000004621: $ i $	pdg_{0004621} \sin{\left(pdg_{0001464} pdg_{0004621} \right)} = \frac{e^{pdg_{0001464}}}{2} - \frac{e^{- pdg_{0001464}}}{2}	ERROR for dim with 4805233006
4811121942	$ W = \frac{1}{2} m v_2^2 - \frac{1}{2} m v_1^2 $ W = \frac{1}{2} m v_2^2 - \frac{1}{2} m v_1^2			work and force and energy	0000002473: $ v_1 $ 0000005156: $ m $ 0000006789: $ W $ 0000004770: $ v_2 $	pdg_{0006789} = - \frac{pdg_{0002473}^{2} pdg_{0005156}}{2} + \frac{pdg_{0004770}^{2} pdg_{0005156}}{2}	ERROR for dim with 4811121942
4820320578	$ F_{\rm gravity} = F_{\rm centripetal} $ F_{\rm gravity} = F_{\rm centripetal}			Newton's Law of Gravitation	0000002867: $ F_{\rm gravity} $ 0000001687: $ F_{\rm centripetal} $	pdg_{0002867} = pdg_{0001687}	ERROR for dim with 4820320578
4827492911	$ \cos(2 x)+(\sin(x))^2 = 1 - (\sin(x))^2 $ \cos(2 x)+(\sin(x))^2 = 1 - (\sin(x))^2			Euler equation: trig square root	0000001464: $ x $	\sin^{2}{\left(pdg_{0001464} \right)} + \cos{\left(2 pdg_{0001464} \right)} = 1 - \sin^{2}{\left(pdg_{0001464} \right)}	ERROR for dim with 4827492911
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} $ {\rm sech}^2\ x + \tanh^2(x) = \frac{4+\left(\exp(2x)-1-1+\exp(-2x)\right)}{\left(\exp(x)+\exp(-x)\right)^2}			hyperbolic trigonometric identities	0000001464: $ x $	\tanh^{2}{\left(pdg_{0001464} \right)} + \operatorname{sech}^{2}{\left(pdg_{0001464} \right)} = \frac{e^{2 pdg_{0001464}} + 2 + e^{- 2 pdg_{0001464}}}{\left(e^{pdg_{0001464}} + e^{- pdg_{0001464}}\right)^{2}}	ERROR for dim with 4830221561
4838429483	$ \exp(2 i x) = \cos(2 x)+i \sin(2 x) $ \exp(2 i x) = \cos(2 x)+i \sin(2 x)			Euler equation: trig square root	0000001464: $ x $ 0000004621: $ i $	e^{2 pdg_{0001464} pdg_{0004621}} = pdg_{0004621} \sin{\left(2 pdg_{0001464} \right)} + \cos{\left(2 pdg_{0001464} \right)}	ERROR for dim with 4838429483
4843995999	$ \frac{1}{2 i}\left(\exp(i x)-\exp(-i x) \right) = \sin(x) $ \frac{1}{2 i}\left(\exp(i x)-\exp(-i x) \right) = \sin(x)			Euler equation: trigonometric relations	0000001464: $ x $ 0000004621: $ i $	\frac{e^{pdg_{0001464} pdg_{0004621}} - e^{- pdg_{0001464} pdg_{0004621}}}{2 pdg_{0004621}} = \sin{\left(pdg_{0001464} \right)}	ERROR for dim with 4843995999
4857472413	$ 1 = \int \psi(x)\psi(x)^* dx $ 1 = \int \psi(x)\psi(x)^* dx			particle in a 1D box	0000009199: $ dx $	pdg_{0009199} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
4857475848	$ \frac{1}{a^2} = \frac{W}{2} $ \frac{1}{a^2} = \frac{W}{2}			particle in a 1D box	0000002523: $ W $ 0000009139: $ a $	\frac{1}{pdg_{0009139}^{2}} = \frac{pdg_{0002523}}{2}	ERROR for dim with 4857475848
4858693811	$ \frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2} = r^3 $ \frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2} = r^3			radius for satellite in geostationary orbit	0000008762: $ T_{\rm orbit} $ 0000003141: $ \pi $ 0000005458: $ m_{\rm Earth} $ 0000002530: $ r $ 0000006277: $ G $	\frac{pdg_{0005458} pdg_{0006277} pdg_{0008762}^{2}}{4 pdg_{0003141}^{2}} = pdg_{0002530}^{3}	ERROR for dim with 4858693811
4866160902	$ \frac{V}{R_{\rm total}} = \frac{V}{R_1} + \frac{V}{R_2} $ \frac{V}{R_{\rm total}} = \frac{V}{R_1} + \frac{V}{R_2}			total electrical resistance for circuit with two resistors in parallel	0000001908: $ R_{\rm total} $ 0000008697: $ R_1 $ 0000003461: $ R_2 $ 0000006599: $ V $	\frac{pdg_{0006599}}{pdg_{0001908}} = \frac{pdg_{0006599}}{pdg_{0008697}} + \frac{pdg_{0006599}}{pdg_{0003461}}	ERROR for dim with 4866160902
4872163189	$ \tanh(x) = \frac{\sinh(x)}{\cosh(x)} $ \tanh(x) = \frac{\sinh(x)}{\cosh(x)}			hyperbolic trigonometric identities	0000001464: $ x $	\tanh{\left(pdg_{0001464} \right)} = \frac{\sinh{\left(pdg_{0001464} \right)}}{\cosh{\left(pdg_{0001464} \right)}}	ERROR for dim with 4872163189
4872970974	$ \frac{PE_2 - PE_1}{t} = -F v $ \frac{PE_2 - PE_1}{t} = -F v			time invariant force conserves energy	0000001467: $ t $ 0000001357: $ v $ 0000004202: $ F $ 0000008849: $ PE_2 $ 0000004093: $ PE_1 $	\frac{- pdg_{0004093} + pdg_{0008849}}{pdg_{0001467}} = - pdg_{0001357} pdg_{0004202}	ERROR for dim with 4872970974
4878728014	$ \sin(i x) = \frac{1}{2i}\left(\exp(-x) - \exp(x) \right) $ \sin(i x) = \frac{1}{2i}\left(\exp(-x) - \exp(x) \right)			hyperbolic trigonometric identities	0000001464: $ x $ 0000004621: $ i $	\sin{\left(pdg_{0001464} pdg_{0004621} \right)} = \frac{- e^{pdg_{0001464}} + e^{- pdg_{0001464}}}{2 pdg_{0004621}}	ERROR for dim with 4878728014
4923339482	$ i x = \log(y) $ i x = \log(y)			Euler equation proof	0000001464: $ x $ 0000004621: $ i $ 0000001452: $ y $	pdg_{0001464} pdg_{0004621} = \frac{\log{\left(pdg_{0001452} \right)}}{\log{\left(10 \right)}}	ERROR for dim with 4923339482
4928007622	$ KE_1 = \frac{1}{2} m v_1^2 $ KE_1 = \frac{1}{2} m v_1^2			time invariant force conserves energy work and force and energy	0000005156: $ m $ 0000001955: $ KE_1 $ 0000002473: $ v_1 $	pdg_{0001955} = \frac{pdg_{0002473}^{2} pdg_{0005156}}{2}	ERROR for dim with 4928007622
4928239482	$ \log(y) = i x $ \log(y) = i x			Euler equation proof	0000001464: $ x $ 0000001452: $ y $ 0000004621: $ i $	\frac{\log{\left(pdg_{0001452} \right)}}{\log{\left(10 \right)}} = pdg_{0001464} pdg_{0004621}	ERROR for dim with 4928239482
4938429482	$ \exp(-i x) = \cos(x)+i \sin(-x) $ \exp(-i x) = \cos(x)+i \sin(-x)			Euler equation: trigonometric relations	0000001464: $ x $ 0000004621: $ i $	e^{- pdg_{0001464} pdg_{0004621}} = - pdg_{0004621} \sin{\left(pdg_{0001464} \right)} + \cos{\left(pdg_{0001464} \right)}	ERROR for dim with 4938429482
4938429483	$ \exp(i x) = \cos(x)+i \sin(x) $ \exp(i x) = \cos(x)+i \sin(x)			Euler equation to exp(i pi) + 1 = 0 Euler equation proof Euler equation: trig square root Euler equation: trigonometric relations	0000001464: $ x $ 0000004621: $ i $	e^{pdg_{0001464} pdg_{0004621}} = pdg_{0004621} \sin{\left(pdg_{0001464} \right)} + \cos{\left(pdg_{0001464} \right)}	ERROR for dim with 4938429483
4938429484	$ \exp(-i x) = \cos(x)-i \sin(x) $ \exp(-i x) = \cos(x)-i \sin(x)			Euler equation: trigonometric relations	0000001464: $ x $ 0000004621: $ i $	e^{- pdg_{0001464} pdg_{0004621}} = - pdg_{0004621} \sin{\left(pdg_{0001464} \right)} + \cos{\left(pdg_{0001464} \right)}	ERROR for dim with 4938429484
4939880586	$ V_{\rm total} = I R_{\rm total} $ V_{\rm total} = I R_{\rm total}			total electrical resistance for circuit with two resistors in series	0000001908: $ R_{\rm total} $ 0000004501: $ I $ 0000004691: $ V_{\rm total} $	pdg_{0004691} = pdg_{0001908} pdg_{0004501}	ERROR for dim with 4939880586
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) $ \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)			derivation of Schrodinger Equation	0000006238: $ E $ 0000001054: $ \hbar $ 0000001467: $ t $ 0000009489: $ \psi $ 0000008330: $ \psi_0 $ 0000004621: $ i $ 0000002046: $ \vec{p} $ 0000009472: $ \vec{r} $	nabla \operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)} = \frac{pdg_{0002046} pdg_{0004621} pdg_{0008330} e^{\frac{pdg_{0004621} \left(- pdg_{0001467} pdg_{0006238} + pdg_{0002046} pdg_{0009472}\right)}{pdg_{0001054}}}}{pdg_{0001054}}	ERROR for dim with 4943571230
4947831649	$ \frac{1}{2} m_1 v_{\rm final}^2 = W_{\rm to\ system} $ \frac{1}{2} m_1 v_{\rm final}^2 = W_{\rm to\ system}			velocity at distance r of object dropped from infinity	0000008909: $ v_{\rm final} $ 0000005022: $ m_1 $ 0000009372: $ W_{\rm to\ system} $	\frac{pdg_{0005022} pdg_{0008909}^{2}}{2} = pdg_{0009372}	ERROR for dim with 4947831649
4948763856	$ 2 a d + v_0^2 = v^2 $ 2 a d + v_0^2 = v^2			equations of motion in 1D with constant acceleration - SUVAT (algebra)	0000009140: $ a $ 0000005153: $ v_0 $ 0000001943: $ d $ 0000001357: $ v $	2 pdg_{0001943} pdg_{0009140} + pdg_{0005153}^{2} = pdg_{0001357}^{2}	ERROR for dim with 4948763856
4948934890	$ \langle \psi\| \hat{A} \|\psi \rangle = \langle a \rangle^* $ \langle \psi\| \hat{A} \|\psi \rangle = \langle a \rangle^*			quantum basics Hermitian operators have realvalued observables	0000004065: $ \langle \psi\| $	pdg_{0004065} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
4949359835	$ \langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2 $ \langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2			variance relation	0000001464: $ x $	pdg_{0001464} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
4968680693	$ \tan( x ) = \frac{ \sin( x )}{\cos( x )} $ \tan( x ) = \frac{ \sin( x )}{\cos( x )}			optics: Law of refraction to Brewster's angle	0000001464: $ x $	\tan{\left(pdg_{0001464} \right)} = \frac{\sin{\left(pdg_{0001464} \right)}}{\cos{\left(pdg_{0001464} \right)}}	ERROR for dim with 4968680693
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) $ \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)		https://physicsderivationgraph.blogspot.com/2020/09/representing-laplace-operator-nabla-in.html	electric field wave equation: from time dependent to time independent	0000006238: $ E $ 0000001467: $ t $ 0000004621: $ i $ 0000002321: $ \omega $ 0000007940: $ \epsilon_0 $ 0000006197: $ \mu_0 $ 0000009472: $ \vec{r} $	nabla^{2} \operatorname{pdg}_{0006238}{\left(pdg_{0009472} \right)} e^{pdg_{0001467} pdg_{0002321} pdg_{0004621}} = pdg_{0002321} pdg_{0004621} pdg_{0006197} pdg_{0007940} \frac{\partial}{\partial pdg_{0001467}} \operatorname{pdg}_{0006238}{\left(pdg_{0009472} \right)} e^{pdg_{0001467} pdg_{0002321} pdg_{0004621}}	ERROR for dim with 4985825552
5002539602	$ dU = C_V dT + \pi_T dV $ dU = C_V dT + \pi_T dV			first law of thermodynamics	0000005480: $ \pi_T $ 0000006682: $ C_V $	dU = dT pdg_{0006682} + dV pdg_{0005480}	ERROR for dim with 5002539602
5085809757	$ \frac{k_{\rm adsorption}}{k_{\rm desorption}} = \frac{[A_{\rm adsorption}]}{p_A [S]} $ \frac{k_{\rm adsorption}}{k_{\rm desorption}} = \frac{[A_{\rm adsorption}]}{p_A [S]}			Langmuir Adsorption	0000009046: $ p_A $ 0000008379: $ k_{\rm desorption} $ 0000009067: $ [S] $ 0000006850: $ k_{\rm adsorption} $ 0000004940: $ [A_{\rm adsorption}] $	\frac{pdg_{0006850}}{pdg_{0008379}} = \frac{pdg_{0004940}}{pdg_{0009046} pdg_{0009067}}	ERROR for dim with 5085809757
5125940051	$ I = \|A\|^2 + B B^* + A B^* + B A^* $ I = \|A\|^2 + B B^* + A B^* + B A^*			double intensity when phase is coherent (optics)	0000007882: $ I $ 0000004453: $ A $ 0000004698: $ B $	pdg_{0007882} = pdg_{0004453} \overline{pdg_{0004698}} + pdg_{0004698} \overline{pdg_{0004453}} + pdg_{0004698} \overline{pdg_{0004698}} + \left\|{pdg_{0004453}}\right\|^{2}	ERROR for dim with 5125940051
5128670694	$ m_1 d_1 = m_2 d_2 $ m_1 d_1 = m_2 d_2			Kepler's Third Law: period squared propto distance cubed	0000002798: $ d_2 $ 0000005022: $ m_1 $ 0000004851: $ m_2 $ 0000007652: $ d_1 $	pdg_{0005022} pdg_{0007652} = pdg_{0002798} pdg_{0004851}	ERROR for dim with 5128670694
5136652623	$ E = KE + PE $ E = KE + PE	mechanical energy is the sum of the potential plus kinetic energies		time invariant force conserves energy	0000004931: $ E $ 0000004930: $ PE $ 0000004929: $ KE $	pdg_{0004931} = pdg_{0004929} + pdg_{0004930}	ERROR for dim with 5136652623
5144263777	$ v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right) $ v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)			equations of motion in 1D with constant acceleration - SUVAT (algebra)	0000001357: $ v $	pdg_{0001357} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
5148266645	$ t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t $ t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t			Lorentz transformation	0000001790: $ \gamma $	pdg_{0001790} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
5177311762	$ v = \frac{2 \pi r}{T} $ v = \frac{2 \pi r}{T}			Newton's Law of Gravitation	0000008762: $ T_{\rm orbit} $ 0000002530: $ r $ 0000003141: $ \pi $ 0000001357: $ v $	pdg_{0001357} = \frac{2 pdg_{0002530} pdg_{0003141}}{pdg_{0008762}}	ERROR for dim with 5177311762
5272284986	$ \|\vec{F}\| = \|m\ \vec{a}\| $ \|\vec{F}\| = \|m\ \vec{a}\|	magnitude of vector representation of Newton's second law		Newton's Second Law of Motion: vector to scalar	0000009863: $ m $ 0000006777: $ \vec{F} $ 0000002423: $ \vec{a} $	Eq(Abs(Symbol('pdg0000006777')),Abs(Mul(Symbol('pdg0000009863'),Symbol('pdg0000002423'))) \left\|{pdg_{0000006777}}\right\| = tokenize.TokenError in get_sympy_as_latex_per_expr_id: ('EOF in multi-line statement', (2, 0))	TokenError: unable to parse Abs(Mul(Symbol('pdg0000009863'),Symbol('pdg0000002423')) as SymPy; error=('EOF in multi-line statement', (2, 0))
5323719091	$ i \sinh x = \frac{1}{2i} \left( \exp(-x) - \exp(x) \right) $ i \sinh x = \frac{1}{2i} \left( \exp(-x) - \exp(x) \right)			hyperbolic trigonometric identities	0000001464: $ x $ 0000004621: $ i $	pdg_{0004621} \sinh{\left(pdg_{0001464} \right)} = \frac{- e^{pdg_{0001464}} + e^{- pdg_{0001464}}}{2 pdg_{0004621}}	ERROR for dim with 5323719091
5345738321	$ F = m a $ F = m a	Newton's second law of motion	https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion#Newton's_second_law	Newton's Second Law of Motion: vector to scalar time invariant force conserves energy Newton's Law of Gravitation mass of the Earth equation of motion for a spring work and force and energy	0000009140: $ a $ 0000005156: $ m $ 0000004202: $ F $	pdg_{0004202} = pdg_{0005156} pdg_{0009140}	ERROR for dim with 5345738321
5349669879	$ \tanh(x) = \frac{ \exp(x)-\exp(-x)}{\exp(x)+\exp(-x)} $ \tanh(x) = \frac{ \exp(x)-\exp(-x)}{\exp(x)+\exp(-x)}			hyperbolic trigonometric identities	0000001464: $ x $	\tanh{\left(pdg_{0001464} \right)} = \frac{e^{pdg_{0001464}} - e^{- pdg_{0001464}}}{e^{pdg_{0001464}} + e^{- pdg_{0001464}}}	ERROR for dim with 5349669879
5349866551	$ \vec{v} = v_x \hat{x} + v_y \hat{y} $ \vec{v} = v_x \hat{x} + v_y \hat{y}			equations of motion in 2D (calculus)	0000001700: $ \hat{y} $ 0000005505: $ v_x $ 0000009107: $ v_y $ 0000008339: $ \hat{x} $ 0000006373: $ \vec{v} $	pdg_{0006373} = pdg_{0001700} pdg_{0009107} + pdg_{0005505} pdg_{0008339}	ERROR for dim with 5349866551
5353282496	$ d = \frac{v_0^2}{g} $ d = \frac{v_0^2}{g}			angle of maximum distance for projectile motion	0000001649: $ g $ 0000005153: $ v_0 $ 0000001943: $ d $	pdg_{0001943} = \frac{pdg_{0005153}^{2}}{pdg_{0001649}}	ERROR for dim with 5353282496
5373931751	$ t = t_f $ t = t_f			angle of maximum distance for projectile motion	0000002467: $ t_f $ 0000001467: $ t $	pdg_{0001467} = pdg_{0002467}	ERROR for dim with 5373931751
5379546684	$ y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0 $ y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0			angle of maximum distance for projectile motion	0000007092: $ y_f $ 0000001649: $ g $ 0000002467: $ t_f $ 0000001575: $ \theta $ 0000005153: $ v_0 $ 0000001469: $ y_0 $	pdg_{0007092} = pdg_{0001469} - \frac{pdg_{0001649} pdg_{0002467}^{2}}{2} + pdg_{0002467} pdg_{0005153} \sin{\left(pdg_{0001575} \right)}	ERROR for dim with 5379546684
5404822208	$ v_{\rm escape} = \sqrt{2 G \frac{m}{r}} $ v_{\rm escape} = \sqrt{2 G \frac{m}{r}}	escape velocity		escape velocity Schwarzschild radius for non-rotating black hole	0000006277: $ G $ 0000005156: $ m $ 0000002530: $ r $ 0000008656: $ v_{\rm escape} $	pdg_{0008656} = \sqrt{2} \sqrt{\frac{pdg_{0005156} pdg_{0006277}}{pdg_{0002530}}}	ERROR for dim with 5404822208
5415824175	$ x(t) = A \cos(\omega t) $ x(t) = A \cos(\omega t)			equation of motion for a spring	0000001467: $ t $ 0000009885: $ A $ 0000002321: $ \omega $	x{\left(pdg_{0001467} \right)} = pdg_{0009885} \cos{\left(pdg_{0001467} pdg_{0002321} \right)}	ERROR for dim with 5415824175
5426308937	$ v = \frac{d}{t} $ v = \frac{d}{t}			radius for satellite in geostationary orbit speed of Earth around Sun	0000001467: $ t $ 0000001943: $ d $ 0000001357: $ v $	pdg_{0001357} = \frac{pdg_{0001943}}{pdg_{0001467}}	ERROR for dim with 5426308937
5438722682	$ x = v_0 t \cos(\theta) + x_0 $ x = v_0 t \cos(\theta) + x_0			equations of motion in 2D (calculus) angle of maximum distance for projectile motion	0000001467: $ t $ 0000001572: $ x_0 $ 0000001575: $ \theta $ 0000005153: $ v_0 $ 0000004037: $ x $	pdg_{0004037} = pdg_{0001467} pdg_{0005153} \cos{\left(pdg_{0001575} \right)} + pdg_{0001572}	ERROR for dim with 5438722682
5514556106	$ E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1) $ E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)			time invariant force conserves energy	0000001352: $ KE_2 $ 0000005579: $ E_1 $ 0000004550: $ E_2 $ 0000008849: $ PE_2 $ 0000001955: $ KE_1 $ 0000004093: $ PE_1 $	pdg_{0004550} - pdg_{0005579} = pdg_{0001352} - pdg_{0001955} - pdg_{0004093} + pdg_{0008849}	ERROR for dim with 5514556106
5530148480	$ \vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron} $ \vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}			Compton's equation for scattering	0000002097: $ \vec{p}_2 $ 0000004299: $ \vec{p}_{\rm electron} $ 0000006029: $ \vec{p}_1 $	- pdg_{0002097} + pdg_{0006029} = pdg_{0004299}	ERROR for dim with 5530148480
5542528160	$ \int dW = F \int_0^x dx $ \int dW = F \int_0^x dx			work and force and energy	0000004037: $ x $ 0000006789: $ W $ 0000004202: $ F $	\int 1\, dpdg_{0006789} = pdg_{0004202} \int\limits_{0}^{pdg_{0004037}} 1\, dpdg_{0004037}	ERROR for dim with 5542528160
5563580265	$ F_{\rm gravity} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2} $ F_{\rm gravity} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}			radius for satellite in geostationary orbit	0000005458: $ m_{\rm Earth} $ 0000003569: $ m_{\rm satellite} $ 0000002530: $ r $ 0000006277: $ G $ 0000002867: $ F_{\rm gravity} $	pdg_{0002867} = \frac{pdg_{0003569} pdg_{0005458} pdg_{0006277}}{pdg_{0002530}^{2}}	ERROR for dim with 5563580265
5586102077	$ r = d_1 + d_2 $ r = d_1 + d_2			Kepler's Third Law: period squared propto distance cubed	0000002798: $ d_2 $ 0000002530: $ r $ 0000007652: $ d_1 $	pdg_{0002530} = pdg_{0002798} + pdg_{0007652}	ERROR for dim with 5586102077
5596822289	$ W_{\rm to\ system} = -G m_1 m_2 \left(\left.\frac{-1}{x}\right\|^r_{\infty}\right) $ W_{\rm to\ system} = -G m_1 m_2 \left(\left.\frac{-1}{x}\right\|^r_{\infty}\right)			velocity at distance r of object dropped from infinity	0000005022: $ m_1 $ 0000006277: $ G $ 0000004851: $ m_2 $ 0000009372: $ W_{\rm to\ system} $	pdg_{0009372} = - pdg_{0004851} pdg_{0005022} pdg_{0006277}	ERROR for dim with 5596822289
5611024898	$ d = \frac{1}{2 a} (v^2 - v_0^2) $ d = \frac{1}{2 a} (v^2 - v_0^2)			equations of motion in 1D with constant acceleration - SUVAT (algebra) work and force and energy	0000009140: $ a $ 0000005153: $ v_0 $ 0000001943: $ d $ 0000001357: $ v $	pdg_{0001943} = \frac{pdg_{0001357}^{2} - pdg_{0005153}^{2}}{2 pdg_{0009140}}	ERROR for dim with 5611024898
5634116660	$ \pi_T = \left(\frac{\partial U}{\partial V}\right)_T $ \pi_T = \left(\frac{\partial U}{\partial V}\right)_T		definition of internal pressure at constant temperature	first law of thermodynamics	0000005480: $ \pi_T $ 0000005786: $ U $ 0000007586: $ V $	pdg_{0005480} = \frac{d}{d pdg_{0007586}} pdg_{0005786}	ERROR for dim with 5634116660
5646314683	$ m = A m_p $ m = A m_p			upper limit on velocity in condensed matter	0000003285: $ A $ 0000009863: $ m $ 0000005916: $ m_p $	pdg_{0009863} = pdg_{0003285} pdg_{0005916}	ERROR for dim with 5646314683
5658865948	$ T^2 = \frac{r^3 4 \pi^2}{(m_1+m_2)G} $ T^2 = \frac{r^3 4 \pi^2}{(m_1+m_2)G}			Kepler's Third Law: period squared propto distance cubed	0000003141: $ \pi $ 0000005022: $ m_1 $ 0000006277: $ G $ 0000002530: $ r $ 0000009491: $ T $ 0000004851: $ m_2 $	pdg_{0009491}^{2} = \frac{4 pdg_{0002530}^{3} pdg_{0003141}^{2}}{pdg_{0006277} \left(pdg_{0004851} + pdg_{0005022}\right)}	ERROR for dim with 5658865948
5693047217	$ v_{\rm final} = -\sqrt{\frac{2 G m_2}{r}} $ v_{\rm final} = -\sqrt{\frac{2 G m_2}{r}}			velocity at distance r of object dropped from infinity	0000008909: $ v_{\rm final} $ 0000002530: $ r $ 0000004851: $ m_2 $ 0000006277: $ G $	pdg_{0008909} = - \sqrt{2} \sqrt{\frac{pdg_{0004851} pdg_{0006277}}{pdg_{0002530}}}	ERROR for dim with 5693047217
5727578862	$ \frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x) $ \frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)			particle in a 1D box	0000009199: $ dx $	pdg_{0009199} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
5732331610	$ W = G m_1 m_2 \left( \frac{1}{x} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right) $ W = G m_1 m_2 \left( \frac{1}{x} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)		2022-03-25 BHP: Conversion between Latex and Sympy is incomplete	escape velocity	0000006277: $ G $	pdg_{0006277} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
5733146966	$ KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right) $ KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)			time invariant force conserves energy	0000001352: $ KE_2 $ 0000002473: $ v_1 $ 0000005156: $ m $ 0000004770: $ v_2 $ 0000001955: $ KE_1 $	pdg_{0001352} - pdg_{0001955} = \frac{pdg_{0005156} \left(- pdg_{0002473}^{2} + pdg_{0004770}^{2}\right)}{2}	ERROR for dim with 5733146966
5733721198	$ d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right) $ d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)			equations of motion in 1D with constant acceleration - SUVAT (algebra)	0000009140: $ a $ 0000005153: $ v_0 $ 0000001943: $ d $ 0000001357: $ v $	pdg_{0001943} = \frac{\left(pdg_{0001357} - pdg_{0005153}\right) \left(pdg_{0001357} + pdg_{0005153}\right)}{2 pdg_{0009140}}	ERROR for dim with 5733721198
5763749235	$ -c^2 + c^2 \gamma^2 = v^2 \gamma^2 $ -c^2 + c^2 \gamma^2 = v^2 \gamma^2			Lorentz transformation	0000001357: $ v $ 0000001790: $ \gamma $ 0000004567: $ c $	pdg_{0001790}^{2} pdg_{0004567}^{2} - pdg_{0004567}^{2} = pdg_{0001357}^{2} pdg_{0001790}^{2}	ERROR for dim with 5763749235
5779256336	$ W_{\rm by\ system} = KE_{\rm final} - KE_{\rm initial} $ W_{\rm by\ system} = KE_{\rm final} - KE_{\rm initial}			velocity at distance r of object dropped from infinity	0000005340: $ KE_{\rm final} $ 0000006191: $ W_{\rm by\ system} $ 0000004121: $ KE_{\rm initial} $	pdg_{0006191} = - pdg_{0004121} + pdg_{0005340}	ERROR for dim with 5779256336
5781981178	$ x^2 - y^2 = (x+y)(x-y) $ x^2 - y^2 = (x+y)(x-y)	difference of squares	https://en.wikipedia.org/wiki/Difference_of_two_squares	time invariant force conserves energy	0000001464: $ x $ 0000001452: $ y $	- pdg_{0001452}^{2} + pdg_{0001464}^{2} = \left(- pdg_{0001452} + pdg_{0001464}\right) \left(pdg_{0001452} + pdg_{0001464}\right)	ERROR for dim with 5781981178
5789289057	$ v = \alpha c \sqrt{ \frac{m_e}{2 m} } $ v = \alpha c \sqrt{ \frac{m_e}{2 m} }		equation 4 in the PDF	upper limit on velocity in condensed matter	0000002077: $ v $ 0000001370: $ \alpha $ 0000002515: $ m_e $ 0000004567: $ c $ 0000009863: $ m $	pdg_{0002077} = \frac{\sqrt{2} pdg_{0001370} pdg_{0004567} \sqrt{\frac{pdg_{0002515}}{pdg_{0009863}}}}{2}	ERROR for dim with 5789289057
5832984291	$ (\sin(x))^2 + (\cos(x))^2 = 1 $ (\sin(x))^2 + (\cos(x))^2 = 1			Euler equation: trig square root	0000001464: $ x $	\sin^{2}{\left(pdg_{0001464} \right)} + \cos^{2}{\left(pdg_{0001464} \right)} = 1	ERROR for dim with 5832984291
5838268428	$ \alpha c = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar} $ \alpha c = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar}			upper limit on velocity in condensed matter	0000001054: $ \hbar $ 0000001370: $ \alpha $ 0000003141: $ \pi $ 0000004567: $ c $ 0000001999: $ e $ 0000007940: $ \epsilon_0 $	pdg_{0001370} pdg_{0004567} = \frac{pdg_{0001999}^{2}}{4 pdg_{0001054} pdg_{0003141} pdg_{0007940}}	ERROR for dim with 5838268428
5846639423	$ v_{\rm final} = \sqrt{\frac{2 G m_2}{r}} $ v_{\rm final} = \sqrt{\frac{2 G m_2}{r}}			velocity at distance r of object dropped from infinity	0000008909: $ v_{\rm final} $ 0000002530: $ r $ 0000004851: $ m_2 $ 0000006277: $ G $	pdg_{0008909} = \sqrt{2} \sqrt{\frac{pdg_{0004851} pdg_{0006277}}{pdg_{0002530}}}	ERROR for dim with 5846639423
5850144586	$ W_{\rm by\ system} = KE_{\rm final} $ W_{\rm by\ system} = KE_{\rm final}			velocity at distance r of object dropped from infinity	0000005340: $ KE_{\rm final} $ 0000006191: $ W_{\rm by\ system} $	pdg_{0006191} = pdg_{0005340}	ERROR for dim with 5850144586
5857434758	$ \int a dx = a x $ \int a dx = a x			particle in a 1D box	0000001464: $ x $ 0000009139: $ a $	\int pdg_{0009139}\, dpdg_{0001464} = pdg_{0001464} pdg_{0009139}	ERROR for dim with 5857434758
5866629429	$ {\rm sech}^2\ x + \tanh^2(x) = 1 $ {\rm sech}^2\ x + \tanh^2(x) = 1			hyperbolic trigonometric identities	0000001464: $ x $	\tanh^{2}{\left(pdg_{0001464} \right)} + \operatorname{sech}^{2}{\left(pdg_{0001464} \right)} = 1	ERROR for dim with 5866629429
5868688585	$ \frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = \frac{p^2}{2m} \psi( \vec{r},t) $ \frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = \frac{p^2}{2m} \psi( \vec{r},t)		https://physicsderivationgraph.blogspot.com/2020/09/representing-laplace-operator-nabla-in.html	derivation of Schrodinger Equation	0000001054: $ \hbar $ 0000001467: $ t $ 0000009489: $ \psi $ 0000001134: $ p $ 0000005156: $ m $ 0000009472: $ \vec{r} $	- \frac{nabla^{2} pdg_{0001054}^{2} \operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)}}{2 pdg_{0005156}} = \frac{pdg_{0001134}^{2} \operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)}}{2 pdg_{0005156}}	ERROR for dim with 5868688585
5900595848	$ k = \frac{\omega}{v} $ k = \frac{\omega}{v}			frequency relations	0000002321: $ \omega $ 0000005321: $ k $ 0000001357: $ v $	pdg_{0005321} = \frac{pdg_{0002321}}{pdg_{0001357}}	ERROR for dim with 5900595848
5902985919	$ \vec{F} = G \frac{m_1 m_2}{x^2} \hat{x} $ \vec{F} = G \frac{m_1 m_2}{x^2} \hat{x}	Newton's law of universal gravitation		velocity at distance r of object dropped from infinity	0000004202: $ F $ 0000005022: $ m_1 $ 0000006277: $ G $ 0000004851: $ m_2 $ 0000004037: $ x $	pdg_{0004202} = \frac{pdg_{0004851} pdg_{0005022} pdg_{0006277}}{pdg_{0004037}}	ERROR for dim with 5902985919
5928285821	$ x^2 + 2 x (b/(2 a)) + (b/(2 a))^2 = (x + (b/(2 a)))^2 $ x^2 + 2 x (b/(2 a)) + (b/(2 a))^2 = (x + (b/(2 a)))^2			quadratic equation derivation	0000001464: $ x $ 0000001939: $ b $ 0000009139: $ a $	pdg_{0001464}^{2} + \frac{pdg_{0001464} pdg_{0001939}}{pdg_{0009139}} + \frac{pdg_{0001939}^{2}}{4 pdg_{0009139}^{2}} = \left(pdg_{0001464} + \frac{pdg_{0001939}}{2 pdg_{0009139}}\right)^{2}	ERROR for dim with 5928285821
5928292841	$ x^2 + (b/a)x + (b/(2 a))^2 = -c/a + (b/(2 a))^2 $ x^2 + (b/a)x + (b/(2 a))^2 = -c/a + (b/(2 a))^2			quadratic equation derivation	0000001464: $ x $ 0000004231: $ c $ 0000001939: $ b $ 0000009139: $ a $	pdg_{0001464}^{2} + \frac{pdg_{0001464} pdg_{0001939}}{pdg_{0009139}} + \frac{pdg_{0001939}^{2}}{4 pdg_{0009139}^{2}} = \frac{pdg_{0001939}^{2}}{4 pdg_{0009139}^{2}} - \frac{pdg_{0004231}}{pdg_{0009139}}	ERROR for dim with 5928292841
5938459282	$ x^2 + (b/a)x = -c/a $ x^2 + (b/a)x = -c/a			quadratic equation derivation	0000001464: $ x $ 0000004231: $ c $ 0000001939: $ b $ 0000009139: $ a $	pdg_{0001464}^{2} + \frac{pdg_{0001464} pdg_{0001939}}{pdg_{0009139}} = - \frac{pdg_{0004231}}{pdg_{0009139}}	ERROR for dim with 5938459282
5945893986	$ \frac{d^2 x}{dt^2} = -A \omega^2 \cos(\omega t) $ \frac{d^2 x}{dt^2} = -A \omega^2 \cos(\omega t)			equation of motion for a spring	0000001467: $ t $ 0000009885: $ A $ 0000002321: $ \omega $	\frac{d^{2} x}{dt^{2}} = - pdg_{0002321}^{2} pdg_{0009885} \cos{\left(pdg_{0001467} pdg_{0002321} \right)}	ERROR for dim with 5945893986
5958392859	$ x^2 + (b/a)x+(c/a) = 0 $ x^2 + (b/a)x+(c/a) = 0			quadratic equation derivation	0000001464: $ x $ 0000004231: $ c $ 0000009139: $ a $	pdg_{0001464}^{2} + pdg_{0001464} + \frac{pdg_{0004231}}{pdg_{0009139}} = 0	ERROR for dim with 5958392859
5959282914	$ x^2 + x(b/a) + (b/(2 a))^2 = (x+(b/(2 a)))^2 $ x^2 + x(b/a) + (b/(2 a))^2 = (x+(b/(2 a)))^2			quadratic equation derivation	0000001464: $ x $ 0000001939: $ b $ 0000009139: $ a $	pdg_{0001464}^{2} + \frac{pdg_{0001464} pdg_{0001939}}{pdg_{0009139}} + \frac{pdg_{0001939}^{2}}{4 pdg_{0009139}^{2}} = \left(pdg_{0001464} + \frac{pdg_{0001939}}{2 pdg_{0009139}}\right)^{2}	ERROR for dim with 5959282914
5962145508	$ \alpha = \frac{nR}{VP} $ \alpha = \frac{nR}{VP}			coefficient of thermal expansion using the equation of state for an ideal gas	0000008134: $ P $ 0000008179: $ R $ 0000007586: $ V $ 0000002834: $ n $ 0000004686: $ \alpha $	pdg_{0004686} = \frac{pdg_{0002834} pdg_{0008179}}{pdg_{0007586} pdg_{0008134}}	ERROR for dim with 5962145508
5978756813	$ W = G m_{\rm Earth} m \left( 0 - \frac{-1}{ r_{\rm Earth}} \right) $ W = G m_{\rm Earth} m \left( 0 - \frac{-1}{ r_{\rm Earth}} \right)			escape velocity	0000005458: $ m_{\rm Earth} $ 0000005156: $ m $ 0000006789: $ W $ 0000006277: $ G $ 0000003236: $ r_{\rm Earth} $	pdg_{0006789} = \frac{pdg_{0005156} pdg_{0005458} pdg_{0006277}}{pdg_{0003236}}	ERROR for dim with 5978756813
5982958248	$ x = -\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a)) $ x = -\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))			quadratic equation derivation	0000001464: $ x $ 0000004231: $ c $ 0000001939: $ b $ 0000009139: $ a $	pdg_{0001464} = - \frac{pdg_{0001939}}{2 pdg_{0009139}} - \sqrt{\frac{pdg_{0001939}^{2}}{4 pdg_{0009139}^{2}} - \frac{pdg_{0004231}}{pdg_{0009139}}}	ERROR for dim with 5982958248
5982958249	$ x+(b/(2 a)) = -\sqrt{(b/(2 a))^2 - (c/a)} $ x+(b/(2 a)) = -\sqrt{(b/(2 a))^2 - (c/a)}			quadratic equation derivation	0000001464: $ x $ 0000004231: $ c $ 0000001939: $ b $ 0000009139: $ a $	pdg_{0001464} + \frac{pdg_{0001939}}{2 pdg_{0009139}} = - \sqrt{\frac{pdg_{0001939}^{2}}{4 pdg_{0009139}^{2}} - \frac{pdg_{0004231}}{pdg_{0009139}}}	ERROR for dim with 5982958249
5985371230	$ \vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) $ \vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi( \vec{r},t)			derivation of Schrodinger Equation	0000001054: $ \hbar $ 0000001467: $ t $ 0000009489: $ \psi $ 0000004621: $ i $ 0000002046: $ \vec{p} $ 0000009472: $ \vec{r} $	nabla \operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)} = \frac{pdg_{0002046} pdg_{0004621} \operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)}}{pdg_{0001054}}	ERROR for dim with 5985371230
6026694087	$ F_{\rm centripetal} = m \frac{v^2}{r} $ F_{\rm centripetal} = m \frac{v^2}{r}			Newton's Law of Gravitation	0000005156: $ m $ 0000001687: $ F_{\rm centripetal} $ 0000002530: $ r $	pdg_{0001687} = \frac{pdg_{0005156} v^{2}}{pdg_{0002530}}	ERROR for dim with 6026694087
6031385191	$ \sinh^2 x = \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right) $ \sinh^2 x = \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)			hyperbolic trigonometric identities	0000001464: $ x $	\sinh^{2}{\left(pdg_{0001464} \right)} = \left(\frac{e^{pdg_{0001464}}}{2} - \frac{e^{- pdg_{0001464}}}{2}\right)^{2}	ERROR for dim with 6031385191
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 $ \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		constant pressure	first law of thermodynamics	0000005786: $ U $ 0000007343: $ T $	\frac{d}{d pdg_{0007343}} pdg_{0005786} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
6061695358	$ V_2 = I R_2 $ V_2 = I R_2			total electrical resistance for circuit with two resistors in series	0000008721: $ V_2 $ 0000004501: $ I $ 0000003461: $ R_2 $	pdg_{0008721} = pdg_{0003461} pdg_{0004501}	ERROR for dim with 6061695358
6083821265	$ v_0 \cos(\theta) = v_{0, x} $ v_0 \cos(\theta) = v_{0, x}			equations of motion in 2D (calculus)	0000001575: $ \theta $ 0000002958: $ v_{0, x} $ 0000005153: $ v_0 $	pdg_{0005153} \cos{\left(pdg_{0001575} \right)} = pdg_{0002958}	ERROR for dim with 6083821265
6091977310	$ KE_{\rm initial} = \frac{1}{2} m_1 v_{\rm initial}^2 $ KE_{\rm initial} = \frac{1}{2} m_1 v_{\rm initial}^2			velocity at distance r of object dropped from infinity	0000005022: $ m_1 $ 0000004121: $ KE_{\rm initial} $ 0000001934: $ v_{\rm initial} $	pdg_{0004121} = \frac{pdg_{0001934}^{2} pdg_{0005022}}{2}	ERROR for dim with 6091977310
6131764194	$ W = G m_{\rm Earth} m \left( \frac{1}{x^2} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right) $ W = G m_{\rm Earth} m \left( \frac{1}{x^2} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)		https://physicsderivationgraph.blogspot.com/2020/09/evaluating-definite-integrals-for.html	escape velocity	0000005458: $ m_{\rm Earth} $ 0000006277: $ G $ 0000005156: $ m $ 0000004037: $ x $	W = \frac{pdg_{0005156} pdg_{0005458} pdg_{0006277}}{pdg_{0004037}^{2}}	ERROR for dim with 6131764194
6134836751	$ v_{0, x} = v_x $ v_{0, x} = v_x			equations of motion in 2D (calculus)	0000002958: $ v_{0, x} $ 0000005505: $ v_x $	pdg_{0002958} = pdg_{0005505}	ERROR for dim with 6134836751
6175547907	$ v_{\rm average} = \frac{v + v_0}{2} $ v_{\rm average} = \frac{v + v_0}{2}			equations of motion in 1D with constant acceleration - SUVAT (algebra)	0000006709: $ v_{\rm average} $ 0000005153: $ v_0 $ 0000001357: $ v $	pdg_{0006709} = \frac{pdg_{0001357}}{2} + \frac{pdg_{0005153}}{2}	ERROR for dim with 6175547907
6204539227	$ -g t + v_{0, y} = \frac{dy}{dt} $ -g t + v_{0, y} = \frac{dy}{dt}			equations of motion in 2D (calculus)	0000009431: $ v_{0, y} $ 0000005647: $ y $ 0000001467: $ t $ 0000006277: $ G $	- pdg_{0001467} pdg_{0006277} + pdg_{0009431} = \frac{d}{d pdg_{0001467}} pdg_{0005647}	ERROR for dim with 6204539227
6240206408	$ I_{\rm incoherent} = \|A\|^2 + \|B\|^2 $ I_{\rm incoherent} = \|A\|^2 + \|B\|^2			double intensity when phase is coherent (optics)	0000002435: $ I_{\rm incoherent} $ 0000004453: $ A $ 0000004698: $ B $	pdg_{0002435} = \left\|{pdg_{0004453}}\right\|^{2} + \left\|{pdg_{0004698}}\right\|^{2}	ERROR for dim with 6240206408
6240546932	$ \frac{1}{K_{equilibrium}} = \frac{k_{\rm desorption}}{k_{\rm adsorption}} $ \frac{1}{K_{equilibrium}} = \frac{k_{\rm desorption}}{k_{\rm adsorption}}			Langmuir Adsorption	0000004933: $ K_{\rm equilibrium} $ 0000008379: $ k_{\rm desorption} $ 0000006850: $ k_{\rm adsorption} $	\frac{1}{pdg_{0004933}} = \frac{pdg_{0008379}}{pdg_{0006850}}	ERROR for dim with 6240546932
6268336290	$ F_{\rm gravity} = \frac{m}{r}\left(\frac{2\pi r}{T}\right)^2 $ F_{\rm gravity} = \frac{m}{r}\left(\frac{2\pi r}{T}\right)^2			Newton's Law of Gravitation	0000008762: $ T_{\rm orbit} $ 0000003141: $ \pi $ 0000002530: $ r $ 0000002867: $ F_{\rm gravity} $ 0000004851: $ m_2 $	pdg_{0002867} = \frac{4 pdg_{0002530} pdg_{0003141}^{2} pdg_{0004851}}{pdg_{0008762}^{2}}	ERROR for dim with 6268336290
6306552185	$ I = (A + B)(A^* + B^) $ I = (A + B)(A^ + B^*)			double intensity when phase is coherent (optics)	0000007882: $ I $ 0000004453: $ A $ 0000004698: $ B $	pdg_{0007882} = \left(pdg_{0004453} + pdg_{0004698}\right) \left(\overline{pdg_{0004453}} + \overline{pdg_{0004698}}\right)	ERROR for dim with 6306552185
6348260313	$ C_{\rm Earth\ orbit} = 2 \pi r_{\rm Earth\ orbit} $ C_{\rm Earth\ orbit} = 2 \pi r_{\rm Earth\ orbit}			speed of Earth around Sun	0000006081: $ r_{\rm Earth\ orbit} $ 0000001534: $ C_{\rm Earth\ orbit} $ 0000003141: $ \pi $	pdg_{0001534} = 2 pdg_{0003141} pdg_{0006081}	ERROR for dim with 6348260313
6397683463	$ V \alpha = \left( \frac{\partial V}{\partial T} \right)_p $ V \alpha = \left( \frac{\partial V}{\partial T} \right)_p			first law of thermodynamics	0000004686: $ \alpha $ 0000007586: $ V $ 0000007343: $ T $	pdg_{0004686} pdg_{0007586} = \frac{d}{d pdg_{0007343}} pdg_{0007586}	ERROR for dim with 6397683463
6404535647	$ \cosh x = \frac{\exp(x) + \exp(-x)}{2} $ \cosh x = \frac{\exp(x) + \exp(-x)}{2}			hyperbolic trigonometric identities	0000001464: $ x $	\cosh{\left(pdg_{0001464} \right)} = \frac{e^{pdg_{0001464}}}{2} + \frac{e^{- pdg_{0001464}}}{2}	ERROR for dim with 6404535647
6421241247	$ d = v t - \frac{1}{2} a t^2 $ d = v t - \frac{1}{2} a t^2			equations of motion in 1D with constant acceleration - SUVAT (algebra)	0000001467: $ t $ 0000009140: $ a $ 0000001943: $ d $ 0000001357: $ v $	pdg_{0001943} = pdg_{0001357} pdg_{0001467} - \frac{pdg_{0001467}^{2} pdg_{0009140}}{2}	ERROR for dim with 6421241247
6450985774	$ n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 ) $ n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )	Law of Refraction	eq 34-44 on page 819 in \cite{2001_HRW}	optics: Law of refraction to Brewster's angle	0000003509: $ \theta_1 $ 0000007545: $ \theta_2 $ 0000002941: $ n_1 $ 0000001958: $ n_2 $	pdg_{0002941} \sin{\left(pdg_{0003509} \right)} = pdg_{0001958} \sin{\left(pdg_{0007545} \right)}	ERROR for dim with 6450985774
6457044853	$ v - a t = v_0 $ v - a t = v_0			equations of motion in 1D with constant acceleration - SUVAT (algebra)	0000001467: $ t $ 0000009140: $ a $ 0000005153: $ v_0 $ 0000001357: $ v $	pdg_{0001357} - pdg_{0001467} pdg_{0009140} = pdg_{0005153}	ERROR for dim with 6457044853
6457999644	$ \frac{[S_0]}{[A_{\rm adsorption}]} = \frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1 $ \frac{[S_0]}{[A_{\rm adsorption}]} = \frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1			Langmuir Adsorption	0000004940: $ [A_{\rm adsorption}] $ 0000004933: $ K_{\rm equilibrium} $ 0000009046: $ p_A $ 0000003037: $ [S_0] $	\frac{pdg_{0003037}}{pdg_{0004940}} = 1 + \frac{1}{pdg_{0004933} pdg_{0009046}}	ERROR for dim with 6457999644
6504442697	$ v = \sqrt{ \frac{K}{\rho} } $ v = \sqrt{ \frac{K}{\rho} }			upper limit on velocity in condensed matter	0000002077: $ v $ 0000003935: $ \rho $	pdg_{0002077} = \sqrt{\frac{K}{pdg_{0003935}}}	ERROR for dim with 6504442697
6529793063	$ I_{\rm incoherent} = \|A\|^2 + \|A\|^2 $ I_{\rm incoherent} = \|A\|^2 + \|A\|^2			double intensity when phase is coherent (optics)	0000002435: $ I_{\rm incoherent} $ 0000004453: $ A $	pdg_{0002435} = 2 \left\|{pdg_{0004453}}\right\|^{2}	ERROR for dim with 6529793063
6555185548	$ A^* = \|A\| \exp(-i \theta) $ A^* = \|A\| \exp(-i \theta)			double intensity when phase is coherent (optics)	0000004621: $ i $ 0000004453: $ A $ 0000001575: $ \theta $	\overline{pdg_{0004453}} = e^{- pdg_{0001575} pdg_{0004621}} \left\|{pdg_{0004453}}\right\|	ERROR for dim with 6555185548
6556875579	$ \frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2 $ \frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2			double intensity when phase is coherent (optics)	0000008251: $ I_{\rm coherent} $ 0000002435: $ I_{\rm incoherent} $	\frac{pdg_{0008251}}{pdg_{0002435}} = 2	ERROR for dim with 6556875579
6572039835	$ -g t + v_{0, y} = v_y $ -g t + v_{0, y} = v_y			equations of motion in 2D (calculus)	0000009107: $ v_y $ 0000001649: $ g $ 0000001467: $ t $ 0000009431: $ v_{0, y} $	- pdg_{0001467} pdg_{0001649} + pdg_{0009431} = pdg_{0009107}	ERROR for dim with 6572039835
6715248283	$ PE = -F x $ PE = -F x	potential energy	https://en.wikipedia.org/wiki/Potential_energy	time invariant force conserves energy	0000004930: $ PE $ 0000004202: $ F $ 0000004037: $ x $	pdg_{0004930} = - pdg_{0004037} pdg_{0004202}	ERROR for dim with 6715248283
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}_{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})			Compton's equation for scattering	0000004299: $ \vec{p}_{\rm electron} $	pdg_{0004299} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
6753224061	$ I_{\rm total} = I_1 + I_2 $ I_{\rm total} = I_1 + I_2			total electrical resistance for circuit with two resistors in parallel	0000004856: $ I_2 $ 0000003978: $ I_1 $ 0000009647: $ I_{\rm total} $	pdg_{0009647} = pdg_{0003978} + pdg_{0004856}	ERROR for dim with 6753224061
6774684564	$ \theta = \phi $ \theta = \phi		for coherent waves	double intensity when phase is coherent (optics)	0000008586: $ \phi $ 0000001575: $ \theta $	pdg_{0001575} = pdg_{0008586}	ERROR for dim with 6774684564
6783009163	$ r_{\rm adsorption} = r_{\rm desorption} $ r_{\rm adsorption} = r_{\rm desorption}			Langmuir Adsorption	0000001966: $ r_{\rm desorption} $ 0000006687: $ r_{\rm adsorption} $	pdg_{0006687} = pdg_{0001966}	ERROR for dim with 6783009163
6785303857	$ C = 2 \pi r $ C = 2 \pi r			Newton's Law of Gravitation radius for satellite in geostationary orbit speed of Earth around Sun	0000002530: $ r $ 0000003034: $ C $ 0000003141: $ \pi $	pdg_{0003034} = 2 pdg_{0002530} pdg_{0003141}	ERROR for dim with 6785303857
6800170830	$ r_{\rm Schwarzschild} = \frac{2 G m}{c^2} $ r_{\rm Schwarzschild} = \frac{2 G m}{c^2}			Schwarzschild radius for non-rotating black hole	0000006277: $ G $ 0000004518: $ r_{\rm Schwarzschild} $ 0000005156: $ m $ 0000004567: $ c $	pdg_{0004518} = \frac{2 pdg_{0005156} pdg_{0006277}}{pdg_{0004567}^{2}}	ERROR for dim with 6800170830
6829281943	$ F_{\rm centripetal} = G \frac{m_1 m_2}{r^2} $ F_{\rm centripetal} = G \frac{m_1 m_2}{r^2}			Kepler's Third Law: period squared propto distance cubed	0000001687: $ F_{\rm centripetal} $ 0000005022: $ m_1 $ 0000002530: $ r $ 0000006277: $ G $ 0000004851: $ m_2 $	pdg_{0001687} = \frac{pdg_{0004851} pdg_{0005022} pdg_{0006277}}{pdg_{0002530}^{2}}	ERROR for dim with 6829281943
6831637424	$ \sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} ) $ \sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} )			optics: Law of refraction to Brewster's angle	0000004928: $ \theta_{\rm Brewster} $	pdg_{0004928} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
6831694380	$ a = \frac{d^2 x}{dt^2} $ a = \frac{d^2 x}{dt^2}	acceleration		equation of motion for a spring	0000009140: $ a $	a = \frac{d^{2} x}{dt^{2}}	ERROR for dim with 6831694380
6870322215	$ KE_{\rm escape} = \frac{1}{2} m v_{\rm escape}^2 $ KE_{\rm escape} = \frac{1}{2} m v_{\rm escape}^2			escape velocity	0000008656: $ v_{\rm escape} $ 0000005156: $ m $ 0000005332: $ KE_{\rm escape} $	pdg_{0005332} = \frac{pdg_{0005156} pdg_{0008656}^{2}}{2}	ERROR for dim with 6870322215
6885625907	$ \exp(i \pi) = -1 + i 0 $ \exp(i \pi) = -1 + i 0			Euler equation to exp(i pi) + 1 = 0	0000004621: $ i $ 0000003141: $ \pi $	e^{pdg_{0003141} pdg_{0004621}} = -1	ERROR for dim with 6885625907
6892595652	$ \frac{1}{2} m_1 v_{\rm final}^2 = \frac{G m_1 m_2}{r} $ \frac{1}{2} m_1 v_{\rm final}^2 = \frac{G m_1 m_2}{r}			velocity at distance r of object dropped from infinity	0000008909: $ v_{\rm final} $ 0000005022: $ m_1 $ 0000002530: $ r $ 0000006277: $ G $ 0000004851: $ m_2 $	\frac{pdg_{0005022} pdg_{0008909}^{2}}{2} = \frac{pdg_{0004851} pdg_{0005022} pdg_{0006277}}{pdg_{0002530}}	ERROR for dim with 6892595652
6908055431	$ x(t) = A \cos\left(\frac{k}{m} t\right) $ x(t) = A \cos\left(\frac{k}{m} t\right)			equation of motion for a spring	0000005156: $ m $ 0000001467: $ t $ 0000009885: $ A $	x{\left(pdg_{0001467} \right)} = pdg_{0009885} \cos{\left(\frac{k pdg_{0001467}}{pdg_{0005156}} \right)}	ERROR for dim with 6908055431
6923310769	$ F \propto m $ F \propto m			Newton's Law of Gravitation	0000004202: $ F $ 0000005156: $ m $	pdg_{0000004202} \propto pdg_{0000005156}	inconsistent dimensions
6925244346	$ \alpha = \frac{PV}{T} \frac{1}{VP} $ \alpha = \frac{PV}{T} \frac{1}{VP}			coefficient of thermal expansion using the equation of state for an ideal gas	0000004686: $ \alpha $ 0000008134: $ P $ 0000007586: $ V $ 0000007343: $ T $	pdg_{0004686} = \frac{pdg_{0007586} pdg_{0008134}}{pdg_{0007343}}	ERROR for dim with 6925244346
6935745841	$ F = G \frac{m_1 m_2}{x^2} $ F = G \frac{m_1 m_2}{x^2}	Newton's law of universal gravitation	https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation#Modern_form	escape velocity radius for satellite in geostationary orbit mass of the Earth	0000004202: $ F $ 0000005022: $ m_1 $ 0000006277: $ G $ 0000004851: $ m_2 $ 0000004037: $ x $	pdg_{0004202} = \frac{pdg_{0004851} pdg_{0005022} pdg_{0006277}}{pdg_{0004037}^{2}}	ERROR for dim with 6935745841
6946088325	$ v = \frac{C}{t} $ v = \frac{C}{t}			speed of Earth around Sun	0000001467: $ t $ 0000003034: $ C $ 0000001357: $ v $	pdg_{0001357} = \frac{pdg_{0003034}}{pdg_{0001467}}	ERROR for dim with 6946088325
6955192897	$ r_{\rm desorption} = k_{\rm desorption} [A_{\rm adsorption}] $ r_{\rm desorption} = k_{\rm desorption} [A_{\rm adsorption}]			Langmuir Adsorption	0000004940: $ [A_{\rm adsorption}] $ 0000001966: $ r_{\rm desorption} $ 0000008379: $ k_{\rm desorption} $	pdg_{0001966} = pdg_{0004940} pdg_{0008379}	ERROR for dim with 6955192897
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} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}			speed of Earth around Sun	0000007427: $ v_{\rm Earth\ orbit} $ 0000003141: $ \pi $	pdg_{0007427} = 0.632911392405063 pdg_{0003141}	ERROR for dim with 6998364753
7002609475	$ \frac{V}{R_2} = I_2 $ \frac{V}{R_2} = I_2			total electrical resistance for circuit with two resistors in parallel	0000004856: $ I_2 $ 0000003461: $ R_2 $ 0000006599: $ V $	\frac{pdg_{0006599}}{pdg_{0003461}} = pdg_{0004856}	ERROR for dim with 7002609475
7010294143	$ T_{\rm orbit}^2 G m_{\rm Earth} = 4 \pi^2 r^3 $ T_{\rm orbit}^2 G m_{\rm Earth} = 4 \pi^2 r^3			radius for satellite in geostationary orbit	0000008762: $ T_{\rm orbit} $ 0000003141: $ \pi $ 0000005458: $ m_{\rm Earth} $ 0000002530: $ r $ 0000006277: $ G $	pdg_{0005458} pdg_{0006277} pdg_{0008762}^{2} = 4 pdg_{0002530}^{3} pdg_{0003141}^{2}	ERROR for dim with 7010294143
7011114072	$ d = \frac{(v_0 + a t) + v_0}{2} t $ d = \frac{(v_0 + a t) + v_0}{2} t			equations of motion in 1D with constant acceleration - SUVAT (algebra)	0000001467: $ t $ 0000009140: $ a $ 0000005153: $ v_0 $ 0000001943: $ d $	pdg_{0001943} = pdg_{0001467} \left(\frac{pdg_{0001467} pdg_{0009140}}{2} + pdg_{0005153}\right)	ERROR for dim with 7011114072
7057864873	$ y' = y $ y' = y		frame of reference is moving only along x direction	Lorentz transformation	0000005647: $ y $ 0000001888: $ y' $	pdg_{0001888} = pdg_{0005647}	ERROR for dim with 7057864873
7107090465	$ B B^* = \|B\|^2 $ B B^* = \|B\|^2			double intensity when phase is coherent (optics)	0000004698: $ B $	pdg_{0004698} \overline{pdg_{0004698}} = \left\|{pdg_{0004698}}\right\|^{2}	ERROR for dim with 7107090465
7112613117	$ m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{6.6743010^{-11}m^3 kg^{-1} s^{-2}} $ m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{6.6743010^{-11}m^3 kg^{-1} s^{-2}}			mass of the Earth	0000005458: $ m_{\rm Earth} $	pdg_{0005458} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
7112646057	$ v_{\rm final}^2 = \frac{2 G m_2}{r} $ v_{\rm final}^2 = \frac{2 G m_2}{r}			velocity at distance r of object dropped from infinity	0000008909: $ v_{\rm final} $ 0000002530: $ r $ 0000004851: $ m_2 $ 0000006277: $ G $	pdg_{0008909}^{2} = \frac{2 pdg_{0004851} pdg_{0006277}}{pdg_{0002530}}	ERROR for dim with 7112646057
7175416299	$ t_{\rm Earth\ orbit} = 1 {\rm year} $ t_{\rm Earth\ orbit} = 1 {\rm year}			speed of Earth around Sun	0000005344: $ t_{\rm Earth\ orbit} $	pdg_{0005344} = 1	ERROR for dim with 7175416299
7215099603	$ v^2 = v_0^2 + 2 a t v_0 + a^2 t^2 $ v^2 = v_0^2 + 2 a t v_0 + a^2 t^2			equations of motion in 1D with constant acceleration - SUVAT (algebra)	0000001467: $ t $ 0000009140: $ a $ 0000005153: $ v_0 $ 0000001357: $ v $	pdg_{0001357}^{2} = pdg_{0001467}^{2} pdg_{0009140}^{2} + 2 pdg_{0001467} pdg_{0005153} pdg_{0009140} + pdg_{0005153}^{2}	ERROR for dim with 7215099603
7217021879	$ R_{\rm total} = R_1 + R_2 $ R_{\rm total} = R_1 + R_2			total electrical resistance for circuit with two resistors in series	0000001908: $ R_{\rm total} $ 0000008697: $ R_1 $ 0000003461: $ R_2 $	pdg_{0001908} = pdg_{0003461} + pdg_{0008697}	ERROR for dim with 7217021879
7222189955	$ F \propto m_2 $ F \propto m_2			Newton's Law of Gravitation	0000004851: $ m_2 $ 0000004202: $ F $	pdg_{0000004202} \propto pdg_{0000004851}	inconsistent dimensions
7233558441	$ d = v_0 t_f \cos(\theta) $ d = v_0 t_f \cos(\theta)			angle of maximum distance for projectile motion	0000001575: $ \theta $ 0000002467: $ t_f $ 0000005153: $ v_0 $ 0000001943: $ d $	pdg_{0001943} = pdg_{0002467} pdg_{0005153} \cos{\left(pdg_{0001575} \right)}	ERROR for dim with 7233558441
7252338326	$ v_y = \frac{dy}{dt} $ v_y = \frac{dy}{dt}			equations of motion in 2D (calculus)	0000009107: $ v_y $ 0000005647: $ y $ 0000001467: $ t $	pdg_{0009107} = \frac{d}{d pdg_{0001467}} pdg_{0005647}	ERROR for dim with 7252338326
7267155233	$ \frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right) $ \frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)			time invariant force conserves energy	0000001467: $ t $ 0000004202: $ F $ 0000008849: $ PE_2 $ 0000005467: $ x_2 $ 0000003852: $ x_1 $ 0000004093: $ PE_1 $	\frac{- pdg_{0004093} + pdg_{0008849}}{pdg_{0001467}} = - \frac{pdg_{0004202} \left(- pdg_{0003852} + pdg_{0005467}\right)}{pdg_{0001467}}	ERROR for dim with 7267155233
7267424860	$ \frac{1}{\theta_A} = \frac{1+(K_{\rm equilibrium}\ p_A)}{K_{\rm equilibrium}\ p_A} $ \frac{1}{\theta_A} = \frac{1+(K_{\rm equilibrium}\ p_A)}{K_{\rm equilibrium}\ p_A}			Langmuir Adsorption	0000001791: $ \theta_A $ 0000004933: $ K_{\rm equilibrium} $ 0000009046: $ p_A $	\frac{1}{pdg_{0001791}} = \frac{pdg_{0004933} pdg_{0009046} + 1}{pdg_{0004933} pdg_{0009046}}	ERROR for dim with 7267424860
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 = - \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			projectile path in 2D is parabolic	0000001649: $ g $ 0000002958: $ v_{0, x} $ 0000009431: $ v_{0, y} $ 0000005647: $ y $ 0000001572: $ x_0 $ 0000004037: $ x $ 0000001469: $ y_0 $	pdg_{0005647} = pdg_{0001469} - \frac{pdg_{0001649}^{2} \left(- pdg_{0001572} + pdg_{0004037}\right)^{2}}{2 pdg_{0002958}^{2}} + \frac{pdg_{0009431} \left(- pdg_{0001572} + pdg_{0004037}\right)}{pdg_{0002958}}	ERROR for dim with 7354529102
7376526845	$ \sin(\theta) = \frac{v_{0, y}}{v_0} $ \sin(\theta) = \frac{v_{0, y}}{v_0}			equations of motion in 2D (calculus)	0000009431: $ v_{0, y} $ 0000001575: $ \theta $ 0000005153: $ v_0 $	\sin{\left(pdg_{0001575} \right)} = \frac{pdg_{0005153}}{pdg_{0009431}}	ERROR for dim with 7376526845
7391837535	$ \cos(\theta) = \frac{v_{0, x}}{v_0} $ \cos(\theta) = \frac{v_{0, x}}{v_0}			equations of motion in 2D (calculus)	0000002958: $ v_{0, x} $ 0000001575: $ \theta $ 0000005153: $ v_0 $	\cos{\left(pdg_{0001575} \right)} = \frac{pdg_{0005153}}{pdg_{0002958}}	ERROR for dim with 7391837535
7455581657	$ v_{0, x} = \frac{dx}{dt} $ v_{0, x} = \frac{dx}{dt}			equations of motion in 2D (calculus)	0000009199: $ dx $ 0000002958: $ v_{0, x} $ 0000001467: $ t $	pdg_{0002958} = \frac{d}{d pdg_{0001467}} pdg_{0009199}	ERROR for dim with 7455581657
7466829492	$ \vec{ \nabla} \cdot \vec{E} = 0 $ \vec{ \nabla} \cdot \vec{E} = 0			Maxwell equations to electric field wave equation	0000006238: $ E $	\operatorname{dot}{\left(pdg_{0006238},nabla \right)} = 0	ERROR for dim with 7466829492
7513513483	$ \gamma^2 (c^2 - v^2) = c^2 $ \gamma^2 (c^2 - v^2) = c^2			Lorentz transformation	0000004567: $ c $ 0000001790: $ \gamma $ 0000001357: $ v $	pdg_{0001790}^{2} \left(- pdg_{0001357}^{2} + pdg_{0004567}^{2}\right) = pdg_{0004567}^{2}	ERROR for dim with 7513513483
7517073655	$ [S_0] = \left(\frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}] $ [S_0] = \left(\frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}]			Langmuir Adsorption	0000004940: $ [A_{\rm adsorption}] $ 0000004933: $ K_{\rm equilibrium} $ 0000009046: $ p_A $ 0000003037: $ [S_0] $	pdg_{0003037} = pdg_{0004940} \left(1 + \frac{1}{pdg_{0004933} pdg_{0009046}}\right)	ERROR for dim with 7517073655
7564894985	$ \int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) $ \int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)			particle in a 1D box	0000004037: $ x $ 0000001592: $ n $ 0000002523: $ W $ 0000003141: $ \pi $	\int \cos{\left(\frac{2 pdg_{0001592} pdg_{0003141} pdg_{0004037}}{pdg_{0002523}} \right)}\, dpdg_{0004037} = \frac{pdg_{0002523} \sin{\left(\frac{2 pdg_{0001592} pdg_{0003141} pdg_{0004037}}{pdg_{0002523}} \right)}}{2 pdg_{0001592} pdg_{0003141}}	ERROR for dim with 7564894985
7572664728	$ \cos(2 x) + 2 (\sin(x))^2 = 1 $ \cos(2 x) + 2 (\sin(x))^2 = 1			Euler equation: trig square root	0000004037: $ x $	2 \sin^{2}{\left(pdg_{0004037} \right)} + \cos{\left(2 pdg_{0004037} \right)} = 1	ERROR for dim with 7572664728
7573835180	$ PE_{\rm Earth\ surface} = -W $ PE_{\rm Earth\ surface} = -W		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	0000006431: $ PE_{\rm Earth\ surface} $ 0000006789: $ W $	pdg_{0006431} = - pdg_{0006789}	ERROR for dim with 7573835180
7575738420	$ \left(\sin\left(\frac{n \pi}{W}x\right) \right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} $ \left(\sin\left(\frac{n \pi}{W}x\right) \right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}			particle in a 1D box	0000003141: $ \pi $ 0000001464: $ x $ 0000001592: $ n $ 0000002523: $ W $ 0000004037: $ x $	\sin^{2}{\left(\frac{pdg_{0001592} pdg_{0003141} pdg_{0004037}}{pdg_{0002523}} \right)} = \frac{1}{2} - \frac{\cos{\left(\frac{2 pdg_{0001464} pdg_{0001592} pdg_{0003141}}{pdg_{0002523}} \right)}}{2}	ERROR for dim with 7575738420
7575859295	$ \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) $ \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})			curl curl identity Maxwell equations to electric field wave equation	0000006238: $ E $	\operatorname{cross}{\left(pdg_{0006238},\operatorname{cross}{\left(nabla,nabla \right)} \right)} = \operatorname{nabla}{\left(- nabla^{2} pdg_{0006238} + \operatorname{dot}{\left(pdg_{0006238},nabla \right)} \right)}	ERROR for dim with 7575859295
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}) $ \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})			curl curl identity	0000001552: $ j $	pdg_{0001552} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
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}) $ \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})			curl curl identity	0000006238: $ E $ 0000001567: $ i $ 0000001552: $ j $ 0000007930: $ m $ 0000001592: $ n $ 0000009690: $ k $ 0000008349: $ \hat{x}_i $ 0000004326: $ \vec{E} $	nabla^{pdg_{0007930}} nabla_{j} pdg_{0006238}^{pdg_{0001592}} pdg_{0008349} \varepsilon_{pdg_{0001567} pdg_{0001552} pdg_{0009690}} \varepsilon_{pdg_{0001592} pdg_{0001552} pdg_{0009690}} = nabla \left(- nabla^{2} pdg_{0004326} + nabla pdg_{0004326}\right)	ERROR for dim with 7575859302
7575859304	$ \epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} $ \epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}		https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors	curl curl identity	0000007984: $ i $ 0000001552: $ j $ 0000007930: $ m $ 0000008304: $ l $ 0000001592: $ n $ 0000009690: $ k $	\varepsilon_{pdg_{0001592} pdg_{0001552} pdg_{0009690}} \varepsilon_{pdg_{0007984} pdg_{0001552} pdg_{0009690}} = - \delta_{h pdg_{0007930}} \delta_{pdg_{0008304} pdg_{0009690}} + \delta_{pdg_{0001552} pdg_{0008304}} \delta_{pdg_{0007930} pdg_{0009690}}	ERROR for dim with 7575859304
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}) $ \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})		https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors	curl curl identity	0000006238: $ E $ 0000001552: $ j $ 0000007930: $ m $ 0000008304: $ l $ 0000001592: $ n $ 0000009690: $ k $ 0000008349: $ \hat{x}_i $ 0000004326: $ \vec{E} $	nabla^{pdg_{0007930}} nabla_{pdg0001552} pdg_{0006238}^{pdg_{0001592}} pdg_{0008349} \left(- \delta_{h pdg_{0007930}} \delta_{pdg_{0008304} pdg_{0009690}} + \delta_{pdg_{0001552} pdg_{0008304}} \delta_{pdg_{0007930} pdg_{0009690}}\right) = nabla \left(- nabla^{2} pdg_{0004326} + nabla pdg_{0004326}\right)	ERROR for dim with 7575859306
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}) $ \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})		https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors	curl curl identity	0000006238: $ E $ 0000001552: $ j $ 0000007930: $ m $ 0000008304: $ l $ 0000001592: $ n $ 0000009690: $ k $ 0000008349: $ \hat{x}_i $	0 = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
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}) $ \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})			curl curl identity	0000006238: $ E $ 0000002380: $ \hat{x}_m $ 0000007930: $ m $ 0000001592: $ n $ 0000001434: $ \hat{x}_n $ 0000004326: $ \vec{E} $	- nabla^{pdg_{0007930}} nabla_{m} pdg_{0001434} pdg_{0006238}^{pdg_{0001592}} + nabla^{pdg_{0007930}} nabla_{n} pdg_{0002380} pdg_{0006238}^{pdg_{0001592}} = \operatorname{nabla}{\left(- nabla^{2} pdg_{0004326} + nabla pdg_{0004326} \right)}	ERROR for dim with 7575859310
7575859312	$ \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) $ \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})			curl curl identity		nabla = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
7621705408	$ I = \|A\|^2 + \|B\|^2 + \|A\| \|B\| \exp(-i \theta) \exp(i \phi) + \|A\| \|B\| \exp(i \theta) \exp(-i \phi) $ I = \|A\|^2 + \|B\|^2 + \|A\| \|B\| \exp(-i \theta) \exp(i \phi) + \|A\| \|B\| \exp(i \theta) \exp(-i \phi)			double intensity when phase is coherent (optics)	0000007882: $ I $ 0000004453: $ A $ 0000004621: $ i $ 0000001575: $ \theta $ 0000004698: $ B $ 0000008586: $ \phi $	pdg_{0007882} = e^{pdg_{0001575} pdg_{0004621}} e^{- pdg_{0004621} pdg_{0008586}} \left\|{pdg_{0004453} pdg_{0004698} \left\|{\left\|{pdg_{0004453}}\right\| + e^{- pdg_{0001575} pdg_{0004621}} e^{pdg_{0004621} pdg_{0008586}} \left\|{pdg_{0004698}}\right\|}\right\|}\right\| + \left\|{pdg_{0004453}}\right\|^{2} + \left\|{pdg_{0004698}}\right\|^{2}	ERROR for dim with 7621705408
7652131521	$ \frac{dx}{dt} = -A \omega \sin (\omega t) $ \frac{dx}{dt} = -A \omega \sin (\omega t)			equation of motion for a spring	0000002321: $ \omega $ 0000001467: $ t $ 0000009885: $ A $ 0000004037: $ x $	\frac{d}{d pdg_{0001467}} pdg_{0004037} = - pdg_{0002321} pdg_{0009885} \sin{\left(pdg_{0001467} pdg_{0002321} \right)}	ERROR for dim with 7652131521
7672365885	$ F_{\rm gravity} = \frac{4 \pi^2 m r}{T^2} $ F_{\rm gravity} = \frac{4 \pi^2 m r}{T^2}			Newton's Law of Gravitation	0000008762: $ T_{\rm orbit} $ 0000003141: $ \pi $ 0000002530: $ r $ 0000002867: $ F_{\rm gravity} $ 0000004851: $ m_2 $	pdg_{0002867} = \frac{4 pdg_{0002530} pdg_{0003141}^{2} pdg_{0004851}}{pdg_{0008762}^{2}}	ERROR for dim with 7672365885
7675171493	$ V_1 = I R_1 $ V_1 = I R_1			total electrical resistance for circuit with two resistors in series	0000008257: $ V_1 $ 0000004501: $ I $ 0000008697: $ R_1 $	pdg_{0008257} = pdg_{0004501} pdg_{0008697}	ERROR for dim with 7675171493
7676652285	$ KE_2 = \frac{1}{2} m v_2^2 $ KE_2 = \frac{1}{2} m v_2^2			time invariant force conserves energy work and force and energy	0000001352: $ KE_2 $ 0000005156: $ m $ 0000004770: $ v_2 $	pdg_{0001352} = \frac{pdg_{0004770}^{2} pdg_{0005156}}{2}	ERROR for dim with 7676652285
7696214507	$ n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} ) $ n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} )			optics: Law of refraction to Brewster's angle	0000002941: $ n_1 $ 0000004928: $ \theta_{\rm Brewster} $ 0000001958: $ n_2 $	pdg_{0002941} \sin{\left(pdg_{0004928} \right)} = - pdg_{0001958} \sin{\left(pdg_{0004928} - 90 \right)}	ERROR for dim with 7696214507
7701249282	$ v_u = \alpha c \sqrt{ \frac{m_e}{m_p} } $ v_u = \alpha c \sqrt{ \frac{m_e}{m_p} }		when A = 1	upper limit on velocity in condensed matter	0000001370: $ \alpha $ 0000002515: $ m_e $ 0000004567: $ c $ 0000004635: $ v_u $ 0000005916: $ m_p $	pdg_{0004635} = pdg_{0001370} pdg_{0004567} \sqrt{\frac{pdg_{0002515}}{pdg_{0005916}}}	ERROR for dim with 7701249282
7729413831	$ a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right) $ a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)			equations of motion in 2D (calculus)	0000001700: $ \hat{y} $ 0000001467: $ t $ 0000005505: $ v_x $ 0000009107: $ v_y $ 0000007055: $ a_y $ 0000008339: $ \hat{x} $ 0000007159: $ a_x $	pdg_{0001700} pdg_{0007055} + pdg_{0007159} pdg_{0008339} = \frac{\partial}{\partial pdg_{0001467}} \left(pdg_{0001700} pdg_{0009107} + pdg_{0005505} pdg_{0008339}\right)	ERROR for dim with 7729413831
7731226616	$ {\rm sech}\ x = \frac{1}{\cosh x} $ {\rm sech}\ x = \frac{1}{\cosh x}			hyperbolic trigonometric identities	0000001464: $ x $	\operatorname{sech}{\left(pdg_{0001464} \right)} = \frac{1}{\cosh{\left(pdg_{0001464} \right)}}	ERROR for dim with 7731226616
7734996511	$ PE_2 - PE_1 = -F ( x_2 - x_1 ) $ PE_2 - PE_1 = -F ( x_2 - x_1 )			time invariant force conserves energy	0000004202: $ F $ 0000008849: $ PE_2 $ 0000005467: $ x_2 $ 0000003852: $ x_1 $ 0000004093: $ PE_1 $	- pdg_{0004093} + pdg_{0008849} = - pdg_{0004202} \left(- pdg_{0003852} + pdg_{0005467}\right)	ERROR for dim with 7734996511
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) $ \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)				0000001464: $ x $	- \sinh^{2}{\left(pdg_{0001464} \right)} + \cosh^{2}{\left(pdg_{0001464} \right)} = 1 + \frac{e^{- 2 pdg_{0001464}}}{2}	ERROR for dim with 7735731560
7735737409	$ \frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t} $ \frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}			time invariant force conserves energy	0000001352: $ KE_2 $ 0000001467: $ t $ 0000002473: $ v_1 $ 0000001357: $ v $ 0000005156: $ m $ 0000004770: $ v_2 $ 0000001955: $ KE_1 $	\frac{pdg_{0001352} - pdg_{0001955}}{pdg_{0001467}} = \frac{pdg_{0001357} pdg_{0005156} \left(- pdg_{0002473} + pdg_{0004770}\right)}{pdg_{0001467}}	ERROR for dim with 7735737409
7741202861	$ x = \gamma^2 x - \gamma^2 v t + \gamma v t' $ x = \gamma^2 x - \gamma^2 v t + \gamma v t'			Lorentz transformation	0000001467: $ t $ 0000001357: $ v $ 0000001790: $ \gamma $ 0000004989: $ t' $ 0000004037: $ x $	pdg_{0004037} = - pdg_{0001357} pdg_{0001467} pdg_{0001790}^{2} + pdg_{0001357} pdg_{0001790} pdg_{0004989} + pdg_{0001790}^{2} pdg_{0004037}	ERROR for dim with 7741202861
7749253510	$ W = G \frac{m_{\rm Earth} m }{ r_{\rm Earth}} $ W = G \frac{m_{\rm Earth} m }{ r_{\rm Earth}}			escape velocity	0000005458: $ m_{\rm Earth} $ 0000005156: $ m $ 0000006789: $ W $ 0000006277: $ G $ 0000003236: $ r_{\rm Earth} $	pdg_{0006789} = \frac{pdg_{0005156} pdg_{0005458} pdg_{0006277}}{pdg_{0003236}}	ERROR for dim with 7749253510
7764834870	$ f(x) = \sum_{n=0}^{\infty} \frac{f^n(a)}{n!} (x-a)^n $ f(x) = \sum_{n=0}^{\infty} \frac{f^n(a)}{n!} (x-a)^n	Taylor series as summation		Euler's equation from MacLaurin series	8937294510: $ \infty $ 0000001592: $ n $ 0000009139: $ a $ 0000001464: $ x $	=	sympy_lhs not provided for expression
7826132469	$ \left(\frac{\partial U}{\partial T}\right)_p = C_V + \pi_T V \alpha $ \left(\frac{\partial U}{\partial T}\right)_p = C_V + \pi_T V \alpha			first law of thermodynamics	0000005786: $ U $ 0000007343: $ T $	\frac{d}{d pdg_{0007343}} pdg_{0005786} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
7837519722	$ v = \sqrt{f} \sqrt{\frac{E}{m}} $ v = \sqrt{f} \sqrt{\frac{E}{m}}			upper limit on velocity in condensed matter	0000002077: $ v $ 0000009863: $ m $ 0000002241: $ E $ 0000006235: $ f $	pdg_{0002077} = \sqrt{pdg_{0006235}} \sqrt{\frac{pdg_{0002241}}{pdg_{0009863}}}	ERROR for dim with 7837519722
7846240076	$ m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{G} $ m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{G}			mass of the Earth	0000005458: $ m_{\rm Earth} $ 0000003236: $ r_{\rm Earth} $ 0000006277: $ G $	pdg_{0005458} = \frac{9 pdg_{0003236}^{2}}{pdg_{0006277}}	ERROR for dim with 7846240076
7875206161	$ E_2 = KE_2 + PE_2 $ E_2 = KE_2 + PE_2			time invariant force conserves energy escape velocity	0000004550: $ E_2 $ 0000001352: $ KE_2 $ 0000008849: $ PE_2 $	pdg_{0004550} = pdg_{0001352} + pdg_{0008849}	ERROR for dim with 7875206161
7882872592	$ W_{\rm to\ system} = \int_{\infty}^r \vec{F}\cdot d\vec{r} $ W_{\rm to\ system} = \int_{\infty}^r \vec{F}\cdot d\vec{r}			velocity at distance r of object dropped from infinity	0000002530: $ r $ 0000009372: $ W_{\rm to\ system} $ 0000006777: $ \vec{F} $	pdg_{0009372} = \int\limits_{\infty}^{pdg_{0002530}} \left(\operatorname{Dot}{\left(pdg_{0006777},pdg_{0002530} \right)}\right)\, dpdg_{0002530}	ERROR for dim with 7882872592
7906112355	$ \gamma^2 = \frac{c^2}{c^2 - \gamma^2} $ \gamma^2 = \frac{c^2}{c^2 - \gamma^2}			Lorentz transformation	0000001790: $ \gamma $ 0000004567: $ c $	pdg_{0001790}^{2} = \frac{pdg_{0004567}^{2}}{- pdg_{0001790}^{2} + pdg_{0004567}^{2}}	ERROR for dim with 7906112355
7917051060	$ \vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2} $ \vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}			Compton's equation for scattering	0000002097: $ \vec{p}_2 $ 0000004299: $ \vec{p}_{\rm electron} $ 0000006029: $ \vec{p}_1 $	pdg_{0004299} = - pdg_{0002097} + pdg_{0006029}	ERROR for dim with 7917051060
7924063906	$ K_{equilibrium} = \frac{k_{\rm adsorption}}{k_{\rm desorption}} $ K_{equilibrium} = \frac{k_{\rm adsorption}}{k_{\rm desorption}}			Langmuir Adsorption	0000004933: $ K_{\rm equilibrium} $ 0000008379: $ k_{\rm desorption} $ 0000006850: $ k_{\rm adsorption} $	pdg_{0004933} = \frac{pdg_{0006850}}{pdg_{0008379}}	ERROR for dim with 7924063906
7928111771	$ \frac{1}{\theta_A} = \frac{1}{K_{\rm equilibrium} p_A} + 1 $ \frac{1}{\theta_A} = \frac{1}{K_{\rm equilibrium} p_A} + 1			Langmuir Adsorption	0000001791: $ \theta_A $ 0000004933: $ K_{\rm equilibrium} $ 0000009046: $ p_A $	\frac{1}{pdg_{0001791}} = 1 + \frac{1}{pdg_{0004933} pdg_{0009046}}	ERROR for dim with 7928111771
7939765107	$ v^2 = v_0^2 + 2 a d $ v^2 = v_0^2 + 2 a d			equations of motion in 1D with constant acceleration - SUVAT (algebra)	0000009140: $ a $ 0000005153: $ v_0 $ 0000001943: $ d $ 0000001357: $ v $	pdg_{0001357}^{2} = 2 pdg_{0001943} pdg_{0009140} + pdg_{0005153}^{2}	ERROR for dim with 7939765107
8046208134	$ I_{\rm coherent} = \|A\|^2 + \|A\|^2 + \|A\| \|A\| 2 $ I_{\rm coherent} = \|A\|^2 + \|A\|^2 + \|A\| \|A\| 2			double intensity when phase is coherent (optics)	0000008251: $ I_{\rm coherent} $ 0000004453: $ A $	pdg_{0008251} = 4 \left\|{pdg_{0004453}}\right\|^{2}	ERROR for dim with 8046208134
8049905441	$ \Delta KE = KE_{\rm final} - KE_{\rm initial} $ \Delta KE = KE_{\rm final} - KE_{\rm initial}	change in kinetic energy		velocity at distance r of object dropped from infinity	0000005340: $ KE_{\rm final} $ 0000005734: $ \Delta KE $ 0000004121: $ KE_{\rm initial} $	pdg_{0005734} = - pdg_{0004121} + pdg_{0005340}	ERROR for dim with 8049905441
8059639673	$ v^2 = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2} $ v^2 = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2}			radius for satellite in geostationary orbit	0000008762: $ T_{\rm orbit} $ 0000002530: $ r $ 0000003141: $ \pi $ 0000001357: $ v $	pdg_{0001357}^{2} = \frac{4 pdg_{0002530}^{2} pdg_{0003141}^{2}}{pdg_{0008762}^{2}}	ERROR for dim with 8059639673
8065128065	$ I = A A^* + B B^* + A B^* + B A^* $ I = A A^* + B B^* + A B^* + B A^*			double intensity when phase is coherent (optics)	0000007882: $ I $ 0000004453: $ A $ 0000004698: $ B $	pdg_{0007882} = pdg_{0004453} \overline{pdg_{0004453}} + pdg_{0004453} \overline{pdg_{0004698}} + pdg_{0004698} \overline{pdg_{0004453}} + pdg_{0004698} \overline{pdg_{0004698}}	ERROR for dim with 8065128065
8090924099	$ v = \sqrt{ \left( f\frac{E}{a^3} \right) \frac{1}{\rho} } $ v = \sqrt{ \left( f\frac{E}{a^3} \right) \frac{1}{\rho} }			upper limit on velocity in condensed matter	0000002077: $ v $ 0000005854: $ a $ 0000003935: $ \rho $ 0000002241: $ E $ 0000006235: $ f $	pdg_{0002077} = \sqrt{\frac{pdg_{0002241} pdg_{0006235}}{pdg_{0003935} pdg_{0005854}^{3}}}	ERROR for dim with 8090924099
8106885760	$ \alpha = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c} $ \alpha = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c}		fine structure constant definition	upper limit on velocity in condensed matter	0000001054: $ \hbar $ 0000001370: $ \alpha $ 0000003141: $ \pi $ 0000004567: $ c $ 0000001999: $ e $ 0000007940: $ \epsilon_0 $	pdg_{0001370} = \frac{pdg_{0001999}^{2}}{4 pdg_{0001054} pdg_{0003141} pdg_{0004567} pdg_{0007940}}	ERROR for dim with 8106885760
8131665171	$ \frac{1}{\theta_A} = \frac{[S_0]}{[A_{\rm adsorption}]} $ \frac{1}{\theta_A} = \frac{[S_0]}{[A_{\rm adsorption}]}			Langmuir Adsorption	0000001791: $ \theta_A $ 0000004940: $ [A_{\rm adsorption}] $ 0000003037: $ [S_0] $	\frac{1}{pdg_{0001791}} = \frac{pdg_{0003037}}{pdg_{0004940}}	ERROR for dim with 8131665171
8139187332	$ \vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron} $ \vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}			Compton's equation for scattering	0000002097: $ \vec{p}_2 $ 0000004299: $ \vec{p}_{\rm electron} $ 0000006029: $ \vec{p}_1 $	pdg_{0006029} = pdg_{0002097} + pdg_{0004299}	ERROR for dim with 8139187332
8145337879	$ -g t dt + v_{0, y} dt = dy $ -g t dt + v_{0, y} dt = dy			equations of motion in 2D (calculus)	0000004711: $ dt $ 0000001467: $ t $ 0000001649: $ g $ 0000009431: $ v_{0, y} $ 0000005842: $ dy $	- pdg_{0001467} pdg_{0001649} pdg_{0004711} + pdg_{0004711} pdg_{0009431} = pdg_{0005842}	ERROR for dim with 8145337879
8198310977	$ 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0 $ 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0			angle of maximum distance for projectile motion	0000001649: $ g $ 0000002467: $ t_f $ 0000001575: $ \theta $ 0000005153: $ v_0 $ 0000001469: $ y_0 $	0 = pdg_{0001469} - \frac{pdg_{0001649} pdg_{0002467}^{2}}{2} + pdg_{0002467} pdg_{0005153} \sin{\left(pdg_{0001575} \right)}	ERROR for dim with 8198310977
8228733125	$ a_y = \frac{d}{dt} v_y $ a_y = \frac{d}{dt} v_y			equations of motion in 2D (calculus)	0000009107: $ v_y $ 0000007055: $ a_y $ 0000001467: $ t $	pdg_{0007055} = \frac{d}{d pdg_{0001467}} pdg_{0009107}	ERROR for dim with 8228733125
8257621077	$ \vec{p}_{\rm before} = \vec{p}_{1} $ \vec{p}_{\rm before} = \vec{p}_{1}			Compton's equation for scattering	0000006029: $ \vec{p}_1 $ 0000001302: $ \vec{p}_{\rm before} $	pdg_{0001302} = pdg_{0006029}	ERROR for dim with 8257621077
8269198922	$ 2 a d = v^2 - v_0^2 $ 2 a d = v^2 - v_0^2			equations of motion in 1D with constant acceleration - SUVAT (algebra)	0000009140: $ a $ 0000005153: $ v_0 $ 0000001943: $ d $ 0000001357: $ v $	2 pdg_{0001943} pdg_{0009140} = pdg_{0001357}^{2} - pdg_{0005153}^{2}	ERROR for dim with 8269198922
8283354808	$ I_{\rm coherent} = \|A\|^2 + \|B\|^2 + \|A\| \|B\| 2 \cos( 0 ) $ I_{\rm coherent} = \|A\|^2 + \|B\|^2 + \|A\| \|B\| 2 \cos( 0 )			double intensity when phase is coherent (optics)	0000008251: $ I_{\rm coherent} $ 0000004453: $ A $ 0000004698: $ B $	pdg_{0008251} = \left\|{pdg_{0004453}}\right\|^{2} + 2 \left\|{pdg_{0004453}}\right\| \left\|{pdg_{0004698}}\right\| + \left\|{pdg_{0004698}}\right\|^{2}	ERROR for dim with 8283354808
8311458118	$ \vec{p}_{\rm after} = \vec{p}_{2}+\vec{p}_{electron} $ \vec{p}_{\rm after} = \vec{p}_{2}+\vec{p}_{electron}			Compton's equation for scattering	0000002097: $ \vec{p}_2 $ 0000004299: $ \vec{p}_{\rm electron} $ 0000005493: $ \vec{p}_{\rm after} $	pdg_{0005493} = pdg_{0002097} + pdg_{0004299}	ERROR for dim with 8311458118
8332931442	$ \exp(i \pi) = \cos(\pi)+i \sin(\pi) $ \exp(i \pi) = \cos(\pi)+i \sin(\pi)			Euler equation to exp(i pi) + 1 = 0	0000004621: $ i $ 0000003141: $ \pi $	e^{pdg_{0003141} pdg_{0004621}} = pdg_{0004621} \sin{\left(pdg_{0003141} \right)} + \cos{\left(pdg_{0003141} \right)}	ERROR for dim with 8332931442
8357234146	$ KE = \frac{1}{2} m v^2 $ KE = \frac{1}{2} m v^2	kinetic energy	https://en.wikipedia.org/wiki/Kinetic_energy	time invariant force conserves energy escape velocity velocity at distance r of object dropped from infinity	0000005156: $ m $ 0000001357: $ v $ 0000004929: $ KE $	pdg_{0004929} = \frac{pdg_{0001357}^{2} pdg_{0005156}}{2}	ERROR for dim with 8357234146
8360117126	$ \gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}} $ \gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}}		not a physically valid result in this context	Lorentz transformation	0000004567: $ c $ 0000001790: $ \gamma $ 0000001357: $ v $	pdg_{0001790} = - \frac{1}{\sqrt{- \frac{pdg_{0001357}^{2}}{pdg_{0004567}^{2}} + 1}}	ERROR for dim with 8360117126
8361238989	$ a_{\rm centripetal} = \frac{v^2}{r} $ a_{\rm centripetal} = \frac{v^2}{r}			Newton's Law of Gravitation	0000002530: $ r $ 0000001357: $ v $	a_{c(e(n(t(r(i(p(e(t(al)))))))))} = \frac{pdg_{0001357}^{2}}{pdg_{0002530}}	ERROR for dim with 8361238989
8368984890	$ \kappa_T = \frac{-1}{V} \left( \frac{ \partial }{\partial P}\left(\frac{nRT}{P}\right) \right)_T $ \kappa_T = \frac{-1}{V} \left( \frac{ \partial }{\partial P}\left(\frac{nRT}{P}\right) \right)_T			coefficient of isothermal compressibility using the equation of state for an ideal gas	0000008134: $ P $ 0000008179: $ R $ 0000007586: $ V $ 0000002834: $ n $ 0000004645: $ \kappa_T $ 0000007343: $ T $	pdg_{0004645} = - \frac{\frac{\partial}{\partial pdg_{0008134}} \frac{pdg_{0002834} pdg_{0007343} pdg_{0008179}}{pdg_{0008134}}}{pdg_{0007586}}	ERROR for dim with 8368984890
8396997949	$ I = \| A + B \|^2 $ I = \| A + B \|^2		intensity of two waves traveling opposite directions on same path	double intensity when phase is coherent (optics)	0000007882: $ I $ 0000004453: $ A $ 0000004698: $ B $	pdg_{0007882} = \left\|{pdg_{0004453} + pdg_{0004698}}\right\|^{2}	ERROR for dim with 8396997949
8399484849	$ \langle x^2 - 2 x \langle x \rangle + \langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2 $ \langle x^2 - 2 x \langle x \rangle + \langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2			variance relation	0000001464: $ x $	pdg_{0001464} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
8405272745	$ W_{\rm to\ system} = -G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx $ W_{\rm to\ system} = -G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx			velocity at distance r of object dropped from infinity	0000002530: $ r $ 0000005022: $ m_1 $ 0000006277: $ G $ 0000004037: $ x $ 0000004851: $ m_2 $ 0000009372: $ W_{\rm to\ system} $	pdg_{0009372} = - pdg_{0004851} pdg_{0005022} pdg_{0006277} \int\limits_{\infty}^{pdg_{0002530}} \frac{1}{pdg_{0004037}^{2}}\, dpdg_{0004037}	ERROR for dim with 8405272745
8418527415	$ \sin(i x) = i \sinh(x) $ \sin(i x) = i \sinh(x)			hyperbolic trigonometric identities	0000001464: $ x $ 0000004621: $ i $	\sin{\left(pdg_{0001464} pdg_{0004621} \right)} = pdg_{0004621} \sinh{\left(pdg_{0001464} \right)}	ERROR for dim with 8418527415
8435841627	$ P V = n R T $ P V = n R T		https://en.wikipedia.org/wiki/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	0000008134: $ P $ 0000008179: $ R $ 0000007586: $ V $ 0000002834: $ n $ 0000007343: $ T $	pdg_{0007586} pdg_{0008134} = pdg_{0002834} pdg_{0007343} pdg_{0008179}	ERROR for dim with 8435841627
8460820419	$ v_x = \frac{dx}{dt} $ v_x = \frac{dx}{dt}			equations of motion in 2D (calculus)	0000009199: $ dx $ 0000001467: $ t $ 0000005505: $ v_x $	pdg_{0005505} = \frac{d}{d pdg_{0001467}} pdg_{0009199}	ERROR for dim with 8460820419
8483686863	$ \sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x) \right) $ \sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x) \right)			identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation	0000001464: $ x $ 0000004621: $ i $	\sin{\left(2 pdg_{0001464} \right)} = \frac{e^{2 pdg_{0001464} pdg_{0004621}} - e^{- 2 pdg_{0001464} pdg_{0004621}}}{2 pdg_{0004621}}	ERROR for dim with 8483686863
8484544728	$ -a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x) $ -a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)			particle in a 1D box	0000004037: $ x $	pdg_{0004037} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
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) $ 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)			particle in a 1D box	0000009199: $ dx $	pdg_{0009199} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
8485867742	$ \frac{2}{W} = a^2 $ \frac{2}{W} = a^2			particle in a 1D box	0000002523: $ W $ 0000009139: $ a $	\frac{2}{pdg_{0002523}} = pdg_{0009139}^{2}	ERROR for dim with 8485867742
8486706976	$ v_{0, x} t + x_0 = x $ v_{0, x} t + x_0 = x			equations of motion in 2D (calculus)	0000001572: $ x_0 $ 0000001467: $ t $ 0000002958: $ v_{0, x} $ 0000004037: $ x $	pdg_{0001467} pdg_{0002958} + pdg_{0001572} = pdg_{0004037}	ERROR for dim with 8486706976
8489593958	$ d(u v) = u dv + v du $ d(u v) = u dv + v du			integration by parts	0000004221: $ u $	pdg_{0004221} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
8489593960	$ d(u v) - v du = u dv $ d(u v) - v du = u dv			integration by parts	0000004221: $ u $	pdg_{0004221} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
8489593962	$ u dv = d(u v) - v du $ u dv = d(u v) - v du			integration by parts	0000004221: $ u $	pdg_{0004221} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
8489593964	$ \int u dv = u v - \int v du $ \int u dv = u v - \int v du			integration by parts	0000005177: $ v $ 0000004221: $ u $	\int pdg_{0004221}\, dpdg_{0005177} = pdg_{0004221} pdg_{0005177} - \int pdg_{0005177}\, dpdg_{0004221}	ERROR for dim with 8489593964
8494839423	$ \nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} $ \nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}			electric field wave equation: from time dependent to time independent Maxwell equations to electric field wave equation	0000007940: $ \epsilon_0 $ 0000001467: $ t $ 0000006197: $ \mu_0 $ 0000004326: $ \vec{E} $	nabla^{2} pdg_{0004326} = \frac{partial pdg_{0004326} pdg_{0006197} pdg_{0007940}}{pdg_{0001467}^{2}}	ERROR for dim with 8494839423
8495187962	$ \theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) } $ \theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }			optics: Law of refraction to Brewster's angle	0000002941: $ n_1 $ 0000004928: $ \theta_{\rm Brewster} $ 0000001958: $ n_2 $	pdg_{0004928} = \operatorname{atan}{\left(\frac{pdg_{0002941}}{pdg_{0001958}} \right)}	ERROR for dim with 8495187962
8497631728	$ I = \|A\|^2 + \|B\|^2 + \|A\| \|B\| 2 \cos( \theta - \phi ) $ I = \|A\|^2 + \|B\|^2 + \|A\| \|B\| 2 \cos( \theta - \phi )			double intensity when phase is coherent (optics)	0000007882: $ I $ 0000004453: $ A $ 0000001575: $ \theta $ 0000004698: $ B $ 0000008586: $ \phi $	pdg_{0007882} = 2 \cos{\left(pdg_{0001575} - pdg_{0008586} \right)} \left\|{pdg_{0004453}}\right\| \left\|{pdg_{0004698}}\right\| + \left\|{pdg_{0004453}}\right\|^{2} + \left\|{pdg_{0004698}}\right\|^{2}	ERROR for dim with 8497631728
8515803375	$ z' = z $ z' = z		frame of reference is moving only along x direction	Lorentz transformation	0000004306: $ z' $ 0000006728: $ z $	pdg_{0004306} = pdg_{0006728}	ERROR for dim with 8515803375
8532702080	$ \cosh^2 x = \left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right) $ \cosh^2 x = \left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right)			hyperbolic trigonometric identities	0000001464: $ x $	\cosh^{2}{\left(pdg_{0001464} \right)} = \left(\frac{e^{pdg_{0001464}}}{2} + \frac{e^{- pdg_{0001464}}}{2}\right)^{2}	ERROR for dim with 8532702080
8552710882	$ KE_{\rm final} = \frac{1}{2} m_1 v_{\rm final}^2 $ KE_{\rm final} = \frac{1}{2} m_1 v_{\rm final}^2			velocity at distance r of object dropped from infinity	0000008909: $ v_{\rm final} $ 0000005340: $ KE_{\rm final} $ 0000005022: $ m_1 $	pdg_{0005340} = \frac{pdg_{0005022} pdg_{0008909}^{2}}{2}	ERROR for dim with 8552710882
8558338742	$ E_2 = E_1 $ E_2 = E_1	conservation of energy	https://en.wikipedia.org/wiki/Conservation_of_energy	time invariant force conserves energy escape velocity	0000004550: $ E_2 $ 0000005579: $ E_1 $	pdg_{0004550} = pdg_{0005579}	ERROR for dim with 8558338742
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) $ \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)			hyperbolic trigonometric identities	0000001464: $ x $	- \sinh^{2}{\left(pdg_{0001464} \right)} + \cosh^{2}{\left(pdg_{0001464} \right)} = \left(\frac{e^{pdg_{0001464}}}{2} + \frac{e^{- pdg_{0001464}}}{2}\right)^{2} - \frac{\left(e^{pdg_{0001464}} - e^{- pdg_{0001464}}\right)^{2}}{4}	ERROR for dim with 8563535636
8572657110	$ 1 = \int \|\psi(x)\|^2 dx $ 1 = \int \|\psi(x)\|^2 dx			particle in a 1D box	0000001464: $ x $ 0000009489: $ \psi $	1 = \int \left\|{\operatorname{pdg}_{0009489}{\left(pdg_{0001464} \right)}}\right\|^{2}\, dpdg_{0001464}	ERROR for dim with 8572657110
8572852424	$ \vec{E} = E( \vec{r},t) $ \vec{E} = E( \vec{r},t)			electric field wave equation: from time dependent to time independent	0000006238: $ E $ 0000001467: $ t $ 0000004326: $ \vec{E} $ 0000009472: $ \vec{r} $	pdg_{0004326} = \operatorname{pdg}_{0006238}{\left(pdg_{0009472},pdg_{0001467} \right)}	ERROR for dim with 8572852424
8575746378	$ \int \frac{1}{2} dx = \frac{1}{2} x $ \int \frac{1}{2} dx = \frac{1}{2} x			particle in a 1D box	0000001464: $ x $	\int \frac{1}{2}\, dpdg_{0001464} = \frac{pdg_{0001464}}{2}	ERROR for dim with 8575746378
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) $ \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)			particle in a 1D box	0000001464: $ x $ 0000005321: $ k $ 0000001939: $ b $ 0000009199: $ dx $ 0000009139: $ a $	\frac{d^{2} \left(pdg_{0001939} \cos{\left(pdg_{0001464} pdg_{0005321} \right)} + pdg_{0009139} \sin{\left(pdg_{0001464} pdg_{0005321} \right)}\right)}{pdg_{0009199}^{2}} = - pdg_{0005321}^{2} \left(pdg_{0001939} \cos{\left(pdg_{0001464} pdg_{0005321} \right)} + pdg_{0009139} \sin{\left(pdg_{0001464} pdg_{0005321} \right)}\right)	ERROR for dim with 8575748999
8576785890	$ 1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx $ 1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx			particle in a 1D box	0000003141: $ \pi $ 0000001464: $ x $ 0000001592: $ n $ 0000002523: $ W $ 0000009139: $ a $	1 = \int\limits_{0}^{pdg_{0002523}} pdg_{0009139}^{2} \left(\frac{1}{2} - \frac{\cos{\left(\frac{2 pdg_{0001464} pdg_{0001592} pdg_{0003141}}{pdg_{0002523}} \right)}}{2}\right)\, dpdg_{0001464}	ERROR for dim with 8576785890
8577275751	$ 0 = a \sin(0) + b\cos(0) $ 0 = a \sin(0) + b\cos(0)			particle in a 1D box	0000001939: $ b $ 0000009139: $ a $	0 = pdg_{0001939}	ERROR for dim with 8577275751
8582885111	$ \psi(x) = a \sin(kx) + b \cos(kx) $ \psi(x) = a \sin(kx) + b \cos(kx)			particle in a 1D box	0000009489: $ \psi $ 0000005321: $ k $ 0000001939: $ b $ 0000009139: $ a $ 0000004037: $ x $	\operatorname{pdg}_{0009489}{\left(pdg_{0004037} \right)} = pdg_{0001939} \cos{\left(pdg_{0004037} pdg_{0005321} \right)} + pdg_{0009139} \sin{\left(pdg_{0004037} pdg_{0005321} \right)}	ERROR for dim with 8582885111
8582954722	$ x^2 + 2 x h + h^2 = (x + h)^2 $ x^2 + 2 x h + h^2 = (x + h)^2			quadratic equation derivation	0000001464: $ x $ 0000003410: $ h $	pdg_{0001464}^{2} + 2 pdg_{0001464} pdg_{0003410} + pdg_{0003410}^{2} = \left(pdg_{0001464} + pdg_{0003410}\right)^{2}	ERROR for dim with 8582954722
8584698994	$ -g \int dt = \int d v_y $ -g \int dt = \int d v_y			equations of motion in 2D (calculus)	0000005674: $ d v_y $	- dt g = pdg_{0005674}	ERROR for dim with 8584698994
8588429722	$ \sin( 90^{\circ} - x ) = \cos( x ) $ \sin( 90^{\circ} - x ) = \cos( x )			optics: Law of refraction to Brewster's angle	0000001464: $ x $	- \sin{\left(pdg_{0001464} - 90 \right)} = \cos{\left(pdg_{0001464} \right)}	ERROR for dim with 8588429722
8602221482	$ \langle \cos(\theta - \phi) \rangle = 0 $ \langle \cos(\theta - \phi) \rangle = 0		incoherent light source	double intensity when phase is coherent (optics)	0000008586: $ \phi $ 0000001575: $ \theta $	\cos{\left(pdg_{0001575} - pdg_{0008586} \right)} = 0	ERROR for dim with 8602221482
8602512487	$ \vec{a} = a_x \hat{x} + a_y \hat{y} $ \vec{a} = a_x \hat{x} + a_y \hat{y}		decompose acceleration into two components	equations of motion in 2D (calculus)	0000001700: $ \hat{y} $ 0000007055: $ a_y $ 0000008339: $ \hat{x} $ 0000007159: $ a_x $ 0000002423: $ \vec{a} $	pdg_{0002423} = pdg_{0001700} pdg_{0007055} + pdg_{0007159} pdg_{0008339}	ERROR for dim with 8602512487
8604483515	$ dW = G \frac{m_1 m_2}{x^2} dx $ dW = G \frac{m_1 m_2}{x^2} dx			escape velocity	0000005022: $ m_1 $ 0000006277: $ G $ 0000009199: $ dx $ 0000004037: $ x $ 0000004851: $ m_2 $ 0000009398: $ dW $	pdg_{0009398} = \frac{pdg_{0004851} pdg_{0005022} pdg_{0006277} pdg_{0009199}}{pdg_{0004037}^{2}}	ERROR for dim with 8604483515
8617866819	$ F_{\rm gravity} \propto \frac{m_1\ m_2}{r^2} $ F_{\rm gravity} \propto \frac{m_1\ m_2}{r^2}			Newton's Law of Gravitation	4482727458: $ F_{\rm gravity} $ 0000005022: $ m_1 $ 0000004851: $ m_2 $ 0000002530: $ r $	pdg_{4482727458} \propto \frac{pdg_{0000004851} pdg_{0000005022}}{pdg_{0000002530}^{2}}	inconsistent dimensions
8637447837	$ f(x) = \sum_{n=0}^{\infty} \frac{f^n(0)}{n!} x^n $ f(x) = \sum_{n=0}^{\infty} \frac{f^n(0)}{n!} x^n	MacLaurin series as summation	a Maclaurin series is a special case of a Taylor series where the expansion point is always x=0.	Euler's equation from MacLaurin series	8937294510: $ \infty $ 0000001592: $ n $ 0000001464: $ x $	=	sympy_lhs not provided for expression
8651044341	$ \cos(i x) = \frac{1}{2} \left( \exp(-x) + \exp(x) \right) $ \cos(i x) = \frac{1}{2} \left( \exp(-x) + \exp(x) \right)			hyperbolic trigonometric identities	0000001464: $ x $ 0000004621: $ i $	\cos{\left(pdg_{0001464} pdg_{0004621} \right)} = \frac{e^{pdg_{0001464}}}{2} + \frac{e^{- pdg_{0001464}}}{2}	ERROR for dim with 8651044341
8655294002	$ a = -\frac{k}{m}x $ a = -\frac{k}{m}x			equation of motion for a spring	0000001356: $ k $ 0000009140: $ a $ 0000005156: $ m $ 0000004037: $ x $	pdg_{0009140} = - \frac{pdg_{0001356} pdg_{0004037}}{pdg_{0005156}}	ERROR for dim with 8655294002
8661803554	$ F = G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2} $ F = G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2}			mass of the Earth	0000004202: $ F $ 0000005458: $ m_{\rm Earth} $ 0000005156: $ m $ 0000006277: $ G $ 0000003236: $ r_{\rm Earth} $	pdg_{0004202} = \frac{pdg_{0005156} pdg_{0005458} pdg_{0006277}}{pdg_{0003236}^{2}}	ERROR for dim with 8661803554
8688588981	$ a^3 \rho = m $ a^3 \rho = m			upper limit on velocity in condensed matter	0000005854: $ a $ 0000009863: $ m $ 0000003935: $ \rho $	pdg_{0003935} pdg_{0005854}^{3} = pdg_{0009863}	ERROR for dim with 8688588981
8699789241	$ 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right) $ 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)			identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation	0000001464: $ x $ 0000004621: $ i $	2 \sin{\left(pdg_{0001464} \right)} \cos{\left(pdg_{0001464} \right)} = \frac{e^{2 pdg_{0001464} pdg_{0004621}} - e^{- 2 pdg_{0001464} pdg_{0004621}}}{2 pdg_{0004621}}	ERROR for dim with 8699789241
8706092970	$ d = \left(\frac{v + v_0}{2}\right)t $ d = \left(\frac{v + v_0}{2}\right)t			equations of motion in 1D with constant acceleration - SUVAT (algebra)	0000001467: $ t $ 0000005153: $ v_0 $ 0000001943: $ d $ 0000001357: $ v $	pdg_{0001943} = pdg_{0001467} \left(\frac{pdg_{0001357}}{2} + \frac{pdg_{0005153}}{2}\right)	ERROR for dim with 8706092970
8721295221	$ t_{\rm Earth\ orbit} = 3.16 10^7 {\rm seconds} $ t_{\rm Earth\ orbit} = 3.16 10^7 {\rm seconds}			speed of Earth around Sun	0000005344: $ t_{\rm Earth\ orbit} $	pdg_{0005344} = 3	ERROR for dim with 8721295221
8730201316	$ \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t' $ \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'		first term was multiplied by \gamma/\gamma	Lorentz transformation	0000001790: $ \gamma $	pdg_{0001790} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
8747785338	$ \cos(i x) = \cosh(x) $ \cos(i x) = \cosh(x)			hyperbolic trigonometric identities	0000001464: $ x $ 0000004621: $ i $	\cos{\left(pdg_{0001464} pdg_{0004621} \right)} = \cosh{\left(pdg_{0001464} \right)}	ERROR for dim with 8747785338
8750379055	$ 0 = \frac{d}{dt} v_x $ 0 = \frac{d}{dt} v_x			equations of motion in 2D (calculus)	0000001467: $ t $ 0000005505: $ v_x $	0 = \frac{d}{d pdg_{0001467}} pdg_{0005505}	ERROR for dim with 8750379055
8808860551	$ -g \int t dt + v_{0, y} \int dt = \int dy $ -g \int t dt + v_{0, y} \int dt = \int dy			equations of motion in 2D (calculus)	0000005647: $ y $ 0000001649: $ g $ 0000001467: $ t $ 0000009431: $ v_{0, y} $	- pdg_{0001649} \int pdg_{0001467}\, dpdg_{0001467} + pdg_{0009431} \int 1\, dpdg_{0001467} = \int 1\, dpdg_{0005647}	ERROR for dim with 8808860551
8849289982	$ \psi(x)^* = a \sin(\frac{n \pi}{W} x) $ \psi(x)^* = a \sin(\frac{n \pi}{W} x)			particle in a 1D box	0000009489: $ \psi $ 0000003141: $ \pi $ 0000001592: $ n $ 0000009139: $ a $ 0000002523: $ W $ 0000004037: $ x $	\overline{\operatorname{pdg}_{0009489}{\left(pdg_{0004037} \right)}} = pdg_{0009139} \sin{\left(\frac{pdg_{0001592} pdg_{0003141} pdg_{0004037}}{pdg_{0002523}} \right)}	ERROR for dim with 8849289982
8889444440	$ 1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right) \right)^2 dx $ 1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right) \right)^2 dx			particle in a 1D box	0000003141: $ \pi $ 0000001464: $ x $ 0000001592: $ n $ 0000002523: $ W $ 0000009139: $ a $	1 = \int\limits_{0}^{pdg_{0002523}} pdg_{0009139}^{2} \sin^{2}{\left(\frac{pdg_{0001464} pdg_{0001592} pdg_{0003141}}{pdg_{0002523}} \right)}\, dpdg_{0001464}	ERROR for dim with 8889444440
8908736791	$ \rho = \frac{m}{a^3} $ \rho = \frac{m}{a^3}		geometry	upper limit on velocity in condensed matter	0000005854: $ a $ 0000009863: $ m $ 0000003935: $ \rho $	pdg_{0003935} = \frac{pdg_{0009863}}{pdg_{0005854}^{3}}	ERROR for dim with 8908736791
8922441655	$ d = \frac{v_0^2}{g} \sin(2 \theta) $ d = \frac{v_0^2}{g} \sin(2 \theta)			angle of maximum distance for projectile motion	0000001575: $ \theta $ 0000001649: $ g $ 0000005153: $ v_0 $ 0000001943: $ d $	pdg_{0001943} = \frac{pdg_{0005153}^{2} \sin{\left(2 pdg_{0001575} \right)}}{pdg_{0001649}}	ERROR for dim with 8922441655
8945218208	$ \theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ} $ \theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ}		based on figure 34-27 on page 824 in \cite{2001_HRW}	optics: Law of refraction to Brewster's angle	0000004928: $ \theta_{\rm Brewster} $	pdg_{0004928} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
8946383937	$ v_{\rm escape}^2 = 2 G \frac{m}{r} $ v_{\rm escape}^2 = 2 G \frac{m}{r}			Schwarzschild radius for non-rotating black hole	0000006277: $ G $ 0000005156: $ m $ 0000002530: $ r $ 0000008656: $ v_{\rm escape} $	pdg_{0008656}^{2} = \frac{2 pdg_{0005156} pdg_{0006277}}{pdg_{0002530}}	ERROR for dim with 8946383937
8949329361	$ v_0 \sin(\theta) = v_{0, y} $ v_0 \sin(\theta) = v_{0, y}			equations of motion in 2D (calculus)	0000001575: $ \theta $ 0000009431: $ v_{0, y} $ 0000005153: $ v_0 $	pdg_{0005153} \sin{\left(pdg_{0001575} \right)} = pdg_{0009431}	ERROR for dim with 8949329361
8953094349	$ W = m a x $ W = m a x			work and force and energy	0000005156: $ m $ 0000009140: $ a $ 0000006789: $ W $ 0000004037: $ x $	pdg_{0006789} = pdg_{0004037} pdg_{0005156} pdg_{0009140}	ERROR for dim with 8953094349
8960645192	$ KE_2 + PE_2 = KE_1 + PE_1 $ KE_2 + PE_2 = KE_1 + PE_1			escape velocity	0000001552: $ j $ 0000001955: $ KE_1 $ 0000004093: $ PE_1 $ 0000008849: $ PE_2 $	pdg_{0001552} + pdg_{0008849} = pdg_{0001955} + pdg_{0004093}	ERROR for dim with 8960645192
8991236357	$ \frac{d^2 x}{dt^2} = -\frac{k}{m} x $ \frac{d^2 x}{dt^2} = -\frac{k}{m} x			equation of motion for a spring	0000001356: $ k $ 0000005156: $ m $ 0000004037: $ x $	\frac{d^{2} pdg_{0004037}}{dt^{2}} = - \frac{pdg_{0001356} pdg_{0004037}}{pdg_{0005156}}	ERROR for dim with 8991236357
9031609275	$ x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t' $ x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'			Lorentz transformation	0000001790: $ \gamma $	pdg_{0001790} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
9059289981	$ \psi(x) = a \sin(k x) $ \psi(x) = a \sin(k x)			particle in a 1D box	0000001464: $ x $	pdg_{0001464} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
9063568209	$ V_{\rm total} = V_1 + V_2 $ V_{\rm total} = V_1 + V_2			total electrical resistance for circuit with two resistors in series	0000008257: $ V_1 $ 0000004691: $ V_{\rm total} $ 0000008721: $ V_2 $	pdg_{0004691} = pdg_{0008257} + pdg_{0008721}	ERROR for dim with 9063568209
9070394000	$ m_2 d_2 \frac{4 \pi^2}{T^2} = G \frac{m_1 m_2}{r^2} $ m_2 d_2 \frac{4 \pi^2}{T^2} = G \frac{m_1 m_2}{r^2}			Kepler's Third Law: period squared propto distance cubed	0000003141: $ \pi $ 0000002798: $ d_2 $ 0000006277: $ G $ 0000005022: $ m_1 $ 0000002530: $ r $ 0000009491: $ T $ 0000004851: $ m_2 $	\frac{4 pdg_{0002798} pdg_{0003141}^{2} pdg_{0004851}}{pdg_{0009491}^{2}} = \frac{pdg_{0004851} pdg_{0005022} pdg_{0006277}}{pdg_{0002530}^{2}}	ERROR for dim with 9070394000
9081138616	$ W_{\rm by\ system} = \frac{1}{2} m_1 v_{\rm final}^2 $ W_{\rm by\ system} = \frac{1}{2} m_1 v_{\rm final}^2			velocity at distance r of object dropped from infinity	0000008909: $ v_{\rm final} $ 0000006191: $ W_{\rm by\ system} $ 0000005022: $ m_1 $	pdg_{0006191} = \frac{pdg_{0005022} pdg_{0008909}^{2}}{2}	ERROR for dim with 9081138616
9112191201	$ y_f = 0 $ y_f = 0			angle of maximum distance for projectile motion	0000007092: $ y_f $	pdg_{0007092} = 0	ERROR for dim with 9112191201
9152823411	$ \frac{1}{T^2} = \frac{1}{d_2 4 \pi^2} G \frac{m_1}{r^2} $ \frac{1}{T^2} = \frac{1}{d_2 4 \pi^2} G \frac{m_1}{r^2}			Kepler's Third Law: period squared propto distance cubed	0000003141: $ \pi $ 0000002798: $ d_2 $ 0000006277: $ G $ 0000005022: $ m_1 $ 0000002530: $ r $ 0000009491: $ T $	\frac{1}{pdg_{0009491}^{2}} = \frac{pdg_{0005022} pdg_{0006277}}{4 pdg_{0002530}^{2} pdg_{0002798} pdg_{0003141}^{2}}	ERROR for dim with 9152823411
9170048197	$ T^2 = d_2 4 \pi^2 \frac{r^2}{G m_1} $ T^2 = d_2 4 \pi^2 \frac{r^2}{G m_1}			Kepler's Third Law: period squared propto distance cubed	0000003141: $ \pi $ 0000002798: $ d_2 $ 0000006277: $ G $ 0000005022: $ m_1 $ 0000002530: $ r $ 0000009491: $ T $	pdg_{0009491}^{2} = \frac{4 pdg_{0002530}^{2} pdg_{0002798} pdg_{0003141}^{2}}{pdg_{0005022} pdg_{0006277}}	ERROR for dim with 9170048197
9180861128	$ 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right) $ 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)			identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation	0000001464: $ x $ 0000004621: $ i $	2 \sin{\left(pdg_{0001464} \right)} \cos{\left(pdg_{0001464} \right)} = \frac{e^{2 pdg_{0001464} pdg_{0004621}} - e^{- 2 pdg_{0001464} pdg_{0004621}}}{2 pdg_{0004621}}	ERROR for dim with 9180861128
9191880568	$ Z Z^* = \|Z\| \|Z\| \exp( -i \theta ) \exp( i \theta ) $ Z Z^* = \|Z\| \|Z\| \exp( -i \theta ) \exp( i \theta )			double intensity when phase is coherent (optics)	0000003192: $ Z $	pdg_{0003192} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
9226945488	$ F = \frac{m v^2}{r} $ F = \frac{m v^2}{r}	Centripetal force	https://en.wikipedia.org/wiki/Centripetal_force	radius for satellite in geostationary orbit	0000005156: $ m $ 0000002530: $ r $ 0000001357: $ v $ 0000004202: $ F $	pdg_{0004202} = \frac{pdg_{0001357}^{2} pdg_{0005156}}{pdg_{0002530}}	ERROR for dim with 9226945488
9243879541	$ V = I_2 R_2 $ V = I_2 R_2			total electrical resistance for circuit with two resistors in parallel	0000004856: $ I_2 $ 0000003461: $ R_2 $ 0000006599: $ V $	pdg_{0006599} = pdg_{0003461} pdg_{0004856}	ERROR for dim with 9243879541
9262596735	$ d = 2 \pi r $ d = 2 \pi r			radius for satellite in geostationary orbit	0000002530: $ r $ 0000001943: $ d $ 0000003141: $ \pi $	pdg_{0001943} = 2 pdg_{0002530} pdg_{0003141}	ERROR for dim with 9262596735
9285928292	$ ax^2 + bx + c = 0 $ ax^2 + bx + c = 0			quadratic equation derivation	0000001464: $ x $ 0000004231: $ c $ 0000001939: $ b $ 0000009139: $ a $	pdg_{0001464}^{2} pdg_{0009139} + pdg_{0001464} pdg_{0001939} + pdg_{0004231} = 0	ERROR for dim with 9285928292
9291999979	$ \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} $ \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t}			Maxwell equations to electric field wave equation	0000001467: $ t $ 0000006197: $ \mu_0 $ 0000004326: $ \vec{E} $ 0000002069: $ \vec{H} $	nabla^{2} pdg_{0004326} = - nabla pdg_{0006197} \frac{d}{d pdg_{0001467}} pdg_{0002069}	ERROR for dim with 9291999979
9294858532	$ \hat{A}^+ = \hat{A} $ \hat{A}^+ = \hat{A}			quantum basics Hermitian operators have realvalued observables	0000005598: $ \hat{A} $	\operatorname{Dagger}{\left(\operatorname{Operator}{\left(pdg_{0005598} \right)} \right)} = \operatorname{Operator}{\left(pdg_{0005598} \right)}	ERROR for dim with 9294858532
9337785146	$ v = \frac{x_2 - x_1}{t} $ v = \frac{x_2 - x_1}{t}	average velocity		time invariant force conserves energy	0000003852: $ x_1 $ 0000001467: $ t $ 0000005467: $ x_2 $ 0000001357: $ v $	pdg_{0001357} = \frac{- pdg_{0003852} + pdg_{0005467}}{pdg_{0001467}}	ERROR for dim with 9337785146
9341391925	$ \vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y} $ \vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}			equations of motion in 2D (calculus)	0000001700: $ \hat{y} $ 0000002958: $ v_{0, x} $ 0000009431: $ v_{0, y} $ 0000008339: $ \hat{x} $ 0000006091: $ \vec{v}_0 $	pdg_{0006091} = pdg_{0001700} pdg_{0009431} + pdg_{0002958} pdg_{0008339}	ERROR for dim with 9341391925
9356924046	$ \frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t} $ \frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}			time invariant force conserves energy	0000001352: $ KE_2 $ 0000001467: $ t $ 0000002473: $ v_1 $ 0000005156: $ m $ 0000004770: $ v_2 $ 0000001955: $ KE_1 $	\frac{pdg_{0001352} - pdg_{0001955}}{pdg_{0001467}} = \frac{pdg_{0005156} \left(- pdg_{0002473} + pdg_{0004770}\right) \left(\frac{pdg_{0002473}}{2} + \frac{pdg_{0004770}}{2}\right)}{pdg_{0001467}}	ERROR for dim with 9356924046
9376481176	$ K = f \frac{E}{a^3} $ K = f \frac{E}{a^3}		proportionality coefficient fvaries in the range 1-4 for a majority of elemental solids	upper limit on velocity in condensed matter	0000005854: $ a $ 0000002241: $ E $ 0000006235: $ f $	K = \frac{pdg_{0002241} pdg_{0006235}}{pdg_{0005854}^{3}}	ERROR for dim with 9376481176
9385938295	$ (x+(b/(2 a)))^2 = -(c/a) + (b/(2 a))^2 $ (x+(b/(2 a)))^2 = -(c/a) + (b/(2 a))^2			quadratic equation derivation	0000001464: $ x $ 0000004231: $ c $ 0000001939: $ b $ 0000009139: $ a $	\left(pdg_{0001464} + \frac{pdg_{0001939}}{2 pdg_{0009139}}\right)^{2} = \frac{pdg_{0001939}^{2}}{4 pdg_{0009139}^{2}} - \frac{pdg_{0004231}}{pdg_{0009139}}	ERROR for dim with 9385938295
9393939991	$ \psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right) $ \psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)			particle in a 1D box	0000009489: $ \psi $ 0000003141: $ \pi $ 0000001464: $ x $ 0000001592: $ n $ 0000002523: $ W $	\operatorname{pdg}_{0009489}{\left(pdg_{0001464} \right)} = - \sqrt{2} \sqrt{\frac{1}{pdg_{0002523}}} \sin{\left(\frac{pdg_{0001464} pdg_{0001592} pdg_{0003141}}{pdg_{0002523}} \right)}	ERROR for dim with 9393939991
9393939992	$ \psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right) $ \psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)			particle in a 1D box	0000009489: $ \psi $ 0000003141: $ \pi $ 0000001464: $ x $ 0000001592: $ n $ 0000002523: $ W $	\operatorname{pdg}_{0009489}{\left(pdg_{0001464} \right)} = \sqrt{2} \sqrt{\frac{1}{pdg_{0002523}}} \sin{\left(\frac{pdg_{0001464} pdg_{0001592} pdg_{0003141}}{pdg_{0002523}} \right)}	ERROR for dim with 9393939992
9394939493	$ \nabla^2 E( \vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r},t) $ \nabla^2 E( \vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r},t)			electric field wave equation: from time dependent to time independent	0000006238: $ E $ 0000001467: $ t $ 0000007940: $ \epsilon_0 $ 0000006197: $ \mu_0 $ 0000009472: $ \vec{r} $	nabla^{2} \operatorname{pdg}_{0006238}{\left(pdg_{0009472},pdg_{0001467} \right)} = \frac{partial pdg_{0006197} pdg_{0007940} \operatorname{pdg}_{0006238}{\left(pdg_{0009472},pdg_{0001467} \right)}}{pdg_{0001467}^{2}}	ERROR for dim with 9394939493
9397152918	$ v = \frac{v_1 + v_2}{2} $ v = \frac{v_1 + v_2}{2}	average velocity		time invariant force conserves energy	0000002473: $ v_1 $ 0000004770: $ v_2 $ 0000001357: $ v $	pdg_{0001357} = \frac{pdg_{0002473}}{2} + \frac{pdg_{0004770}}{2}	ERROR for dim with 9397152918
9407192813	$ G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2} = m g_{\rm Earth} $ G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2} = m g_{\rm Earth}			mass of the Earth	0000007557: $ g_{\rm Earth} $ 0000005458: $ m_{\rm Earth} $ 0000005156: $ m $ 0000006277: $ G $ 0000003236: $ r_{\rm Earth} $	\frac{pdg_{0005156} pdg_{0005458} pdg_{0006277}}{pdg_{0003236}^{2}} = pdg_{0005156} pdg_{0007557}	ERROR for dim with 9407192813
9409776983	$ x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t' $ x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'			Lorentz transformation	0000001790: $ \gamma $	pdg_{0001790} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
9412953728	$ v_{\rm escape}^2 = 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}} $ v_{\rm escape}^2 = 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}			escape velocity	0000005458: $ m_{\rm Earth} $ 0000008656: $ v_{\rm escape} $ 0000003236: $ r_{\rm Earth} $ 0000006277: $ G $	pdg_{0008656}^{2} = \frac{2 pdg_{0005458} pdg_{0006277}}{pdg_{0003236}}	ERROR for dim with 9412953728
9413609246	$ \cosh^2 x - \sinh^2 x = 1 $ \cosh^2 x - \sinh^2 x = 1			hyperbolic trigonometric identities	0000001464: $ x $	- \sinh^{2}{\left(pdg_{0001464} \right)} + \cosh^{2}{\left(pdg_{0001464} \right)} = 1	ERROR for dim with 9413609246
9413699705	$ W = m a \frac{v_2^2 - v_1^2}{2 a} $ W = m a \frac{v_2^2 - v_1^2}{2 a}			work and force and energy	0000002473: $ v_1 $ 0000005156: $ m $ 0000006789: $ W $ 0000004770: $ v_2 $	pdg_{0006789} = pdg_{0005156} \left(- \frac{pdg_{0002473}^{2}}{2} + \frac{pdg_{0004770}^{2}}{2}\right)	ERROR for dim with 9413699705
9429829482	$ \frac{d}{dx} y = -\sin(x) + i\cos(x) $ \frac{d}{dx} y = -\sin(x) + i\cos(x)			Euler equation proof	0000001464: $ x $ 0000001452: $ y $ 0000004621: $ i $	\frac{d}{d pdg_{0001464}} pdg_{0001452} = pdg_{0004621} \cos{\left(pdg_{0001464} \right)} - \sin{\left(pdg_{0001464} \right)}	ERROR for dim with 9429829482
9440616166	$ m_{\rm Earth} = \frac{g_{\rm Earth} r_{\rm Earth}^2}{G} $ m_{\rm Earth} = \frac{g_{\rm Earth} r_{\rm Earth}^2}{G}			mass of the Earth	0000005458: $ m_{\rm Earth} $ 0000003236: $ r_{\rm Earth} $ 0000007557: $ g_{\rm Earth} $ 0000006277: $ G $	pdg_{0005458} = \frac{pdg_{0003236}^{2} pdg_{0007557}}{pdg_{0006277}}	ERROR for dim with 9440616166
9482113948	$ \frac{dy}{y} = i dx $ \frac{dy}{y} = i dx			Euler equation proof	0000004621: $ i $	pdg_{0004621} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
9482438243	$ (\cos(x))^2 = \cos(2 x) + (\sin(x))^2 $ (\cos(x))^2 = \cos(2 x) + (\sin(x))^2			Euler equation: trig square root	0000001464: $ x $	\cos^{2}{\left(pdg_{0001464} \right)} = \sin^{2}{\left(pdg_{0001464} \right)} + \cos{\left(2 pdg_{0001464} \right)}	ERROR for dim with 9482438243
9482923849	$ \exp(i x) = y $ \exp(i x) = y			Euler equation proof	0000001464: $ x $ 0000004621: $ i $ 0000001452: $ y $	e^{pdg_{0001464} pdg_{0004621}} = pdg_{0001452}	ERROR for dim with 9482923849
9482928242	$ \cos(2 x) = (\cos(x))^2 - (\sin(x))^2 $ \cos(2 x) = (\cos(x))^2 - (\sin(x))^2			Euler equation: trig square root	0000001464: $ x $	\cos{\left(2 pdg_{0001464} \right)} = - \sin^{2}{\left(pdg_{0001464} \right)} + \cos^{2}{\left(pdg_{0001464} \right)}	ERROR for dim with 9482928242
9482928243	$ \cos(2 x) + (\sin(x))^2 = (\cos(x))^2 $ \cos(2 x) + (\sin(x))^2 = (\cos(x))^2			Euler equation: trig square root	0000001464: $ x $	\sin^{2}{\left(pdg_{0001464} \right)} + \cos{\left(2 pdg_{0001464} \right)} = \cos^{2}{\left(pdg_{0001464} \right)}	ERROR for dim with 9482928243
9482943948	$ \log(y) = i dx $ \log(y) = i dx			Euler equation proof	0000009199: $ dx $ 0000001452: $ y $ 0000004621: $ i $	\frac{\log{\left(pdg_{0001452} \right)}}{\log{\left(10 \right)}} = pdg_{0004621} pdg_{0009199}	ERROR for dim with 9482943948
9482984922	$ \frac{d}{dx} y = (i\sin(x) + \cos(x)) i $ \frac{d}{dx} y = (i\sin(x) + \cos(x)) i			Euler equation proof	0000001464: $ x $ 0000001452: $ y $ 0000004621: $ i $	\frac{d}{d pdg_{0001464}} pdg_{0001452} = pdg_{0004621} \left(pdg_{0004621} \sin{\left(pdg_{0001464} \right)} + \cos{\left(pdg_{0001464} \right)}\right)	ERROR for dim with 9482984922
9483928192	$ \cos(2 x) + i\sin(2 x) = (\cos(x))^2 + 2 i \cos(x) \sin(x) - (\sin(x))^2 $ \cos(2 x) + i\sin(2 x) = (\cos(x))^2 + 2 i \cos(x) \sin(x) - (\sin(x))^2			Euler equation: trig square root	0000001464: $ x $ 0000004621: $ i $	pdg_{0004621} \sin{\left(2 pdg_{0001464} \right)} + \cos{\left(2 pdg_{0001464} \right)} = 2 pdg_{0004621} \sin{\left(pdg_{0001464} \right)} \cos{\left(pdg_{0001464} \right)} - \sin^{2}{\left(pdg_{0001464} \right)} + \cos^{2}{\left(pdg_{0001464} \right)}	ERROR for dim with 9483928192
9485384858	$ \nabla^2 E( \vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E( \vec{r})\exp(i \omega t) $ \nabla^2 E( \vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E( \vec{r})\exp(i \omega t)			electric field wave equation: from time dependent to time independent	0000006238: $ E $ 0000001467: $ t $ 0000004621: $ i $ 0000004567: $ c $ 0000002321: $ \omega $ 0000002718: $ \exp $ 0000009472: $ \vec{r} $	nabla^{2} \operatorname{pdg}_{0002718}{\left(pdg_{0001467} pdg_{0002321} pdg_{0004621} \right)} \operatorname{pdg}_{0006238}{\left(pdg_{0009472} \right)} = - \frac{pdg_{0002321}^{2} \operatorname{pdg}_{0002718}{\left(pdg_{0001467} pdg_{0002321} pdg_{0004621} \right)} \operatorname{pdg}_{0006238}{\left(pdg_{0009472} \right)}}{pdg_{0004567}^{2}}	ERROR for dim with 9485384858
9485747245	$ \sqrt{\frac{2}{W}} = a $ \sqrt{\frac{2}{W}} = a			particle in a 1D box	0000002523: $ W $ 0000009139: $ a $	\sqrt{2} \sqrt{\frac{1}{pdg_{0002523}}} = pdg_{0009139}	ERROR for dim with 9485747245
9485747246	$ -\sqrt{\frac{2}{W}} = a $ -\sqrt{\frac{2}{W}} = a			particle in a 1D box	0000002523: $ W $ 0000009139: $ a $	- \sqrt{2} \sqrt{\frac{1}{pdg_{0002523}}} = pdg_{0009139}	ERROR for dim with 9485747246
9492920340	$ y = \cos(x)+i \sin(x) $ y = \cos(x)+i \sin(x)			Euler equation proof	0000001464: $ x $ 0000001452: $ y $ 0000004621: $ i $	pdg_{0001452} = pdg_{0004621} \sin{\left(pdg_{0001464} \right)} + \cos{\left(pdg_{0001464} \right)}	ERROR for dim with 9492920340
9495857278	$ \psi(x=W) = 0 $ \psi(x=W) = 0		2022-03-25 BHP: Conversion between Latex and Sympy is incomplete	particle in a 1D box	0000002523: $ W $	pdg_{0002523} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
9499428242	$ E( \vec{r},t) = E( \vec{r})\exp(i \omega t) $ E( \vec{r},t) = E( \vec{r})\exp(i \omega t)			electric field wave equation: from time dependent to time independent	0000006238: $ E $ 0000001467: $ t $ 0000004621: $ i $ 0000002321: $ \omega $ 0000002718: $ \exp $ 0000009472: $ \vec{r} $	\operatorname{pdg}_{0006238}{\left(pdg_{0009472},pdg_{0001467} \right)} = \operatorname{pdg}_{0002718}{\left(pdg_{0001467} pdg_{0002321} pdg_{0004621} \right)} \operatorname{pdg}_{0006238}{\left(pdg_{0009472} \right)}	ERROR for dim with 9499428242
9510328252	$ KE_{\rm initial} = 0 $ KE_{\rm initial} = 0			velocity at distance r of object dropped from infinity	0000004121: $ KE_{\rm initial} $	pdg_{0004121} = 0	ERROR for dim with 9510328252
9562264720	$ [S] = \frac{k_{\rm desorption} [A_{\rm adsorption}]}{k_{\rm adsorption} p_A} $ [S] = \frac{k_{\rm desorption} [A_{\rm adsorption}]}{k_{\rm adsorption} p_A}			Langmuir Adsorption	0000009046: $ p_A $ 0000008379: $ k_{\rm desorption} $ 0000009067: $ [S] $ 0000006850: $ k_{\rm adsorption} $ 0000004940: $ [A_{\rm adsorption}] $	pdg_{0009067} = \frac{pdg_{0004940} pdg_{0008379}}{pdg_{0006850} pdg_{0009046}}	ERROR for dim with 9562264720
9582958293	$ x = \sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a)) $ x = \sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))			quadratic equation derivation	0000001464: $ x $ 0000004231: $ c $ 0000001939: $ b $ 0000009139: $ a $	pdg_{0001464} = - \frac{pdg_{0001939}}{2 pdg_{0009139}} + \sqrt{\frac{pdg_{0001939}^{2}}{4 pdg_{0009139}^{2}} - \frac{pdg_{0004231}}{pdg_{0009139}}}	ERROR for dim with 9582958293
9582958294	$ x+(b/(2 a)) = \sqrt{(b/(2 a))^2 - (c/a)} $ x+(b/(2 a)) = \sqrt{(b/(2 a))^2 - (c/a)}			quadratic equation derivation	0000001464: $ x $ 0000004231: $ c $ 0000001939: $ b $ 0000009139: $ a $	pdg_{0001464} + \frac{pdg_{0001939}}{2 pdg_{0009139}} = \sqrt{\frac{pdg_{0001939}^{2}}{4 pdg_{0009139}^{2}} - \frac{pdg_{0004231}}{pdg_{0009139}}}	ERROR for dim with 9582958294
9585727710	$ \psi(x=0) = 0 $ \psi(x=0) = 0			particle in a 1D box	0000001464: $ x $ 0000009489: $ \psi $	\operatorname{pdg}_{0009489}{\left(pdg_{0001464} = 0 \right)} = 0	ERROR for dim with 9585727710
9596004948	$ x = \langle\psi_{\alpha}\| \hat{A} \|\psi_{\beta}\rangle $ x = \langle\psi_{\alpha}\| \hat{A} \|\psi_{\beta}\rangle			quantum basics orthogonality	0000001464: $ x $ 0000002090: $ \| \psi_{\beta} \rangle $ 0000004679: $ \langle \psi_{\alpha} \| $ 0000005598: $ \hat{A} $	pdg_{0001464} = pdg_{0005598} \operatorname{Bra}{\left(pdg_{0004679} \right)} \operatorname{Ket}{\left(pdg_{0002090} \right)}	ERROR for dim with 9596004948
9605409442	$ f(x) = f(0)+f'(x)\ x + \frac{f''(x)\ x^2}{2!} + \frac{f'''(x)\ x^3}{3!} + ... $ f(x) = f(0)+f'(x)\ x + \frac{f''(x)\ x^2}{2!} + \frac{f'''(x)\ x^3}{3!} + ...	MacLaurin series with expanded terms	a Maclaurin series is a special case of a Taylor series where the expansion point is always x=0.	Euler's equation from MacLaurin series	0000001464: $ x $	=	sympy_lhs not provided for expression
9640720571	$ v = \frac{e^2}{4 \pi \epsilon_0 \hbar} \sqrt{\frac{m_e}{2 m}} $ v = \frac{e^2}{4 \pi \epsilon_0 \hbar} \sqrt{\frac{m_e}{2 m}}			upper limit on velocity in condensed matter	0000001054: $ \hbar $ 0000003141: $ \pi $ 0000002515: $ m_e $ 0000002077: $ v $ 0000009863: $ m $ 0000001999: $ e $ 0000007940: $ \epsilon_0 $	pdg_{0002077} = \frac{\sqrt{2} pdg_{0001999}^{2} \sqrt{\frac{pdg_{0002515}}{pdg_{0009863}}}}{8 pdg_{0001054} pdg_{0003141} pdg_{0007940}}	ERROR for dim with 9640720571
9658195023	$ d = v_0 t + \frac{1}{2} a t^2 $ d = v_0 t + \frac{1}{2} a t^2			equations of motion in 1D with constant acceleration - SUVAT (algebra)	0000001467: $ t $ 0000009140: $ a $ 0000005153: $ v_0 $ 0000001943: $ d $	pdg_{0001943} = \frac{pdg_{0001467}^{2} pdg_{0009140}}{2} + pdg_{0001467} pdg_{0005153}	ERROR for dim with 9658195023
9703482302	$ G \frac{m_{\rm Earth} m}{r_{\rm Earth}} = \frac{1}{2} m v_{\rm escape}^2 $ G \frac{m_{\rm Earth} m}{r_{\rm Earth}} = \frac{1}{2} m v_{\rm escape}^2			escape velocity	0000005458: $ m_{\rm Earth} $ 0000008656: $ v_{\rm escape} $ 0000005156: $ m $ 0000006277: $ G $ 0000003236: $ r_{\rm Earth} $	\frac{pdg_{0005156} pdg_{0005458} pdg_{0006277}}{pdg_{0003236}} = \frac{pdg_{0005156} pdg_{0008656}^{2}}{2}	ERROR for dim with 9703482302
9707028061	$ a_x = 0 $ a_x = 0			equations of motion in 2D (calculus)	0000007159: $ a_x $	pdg_{0007159} = 0	ERROR for dim with 9707028061
9718685793	$ \kappa_T = \frac{1}{P} $ \kappa_T = \frac{1}{P}			coefficient of isothermal compressibility using the equation of state for an ideal gas	0000004645: $ \kappa_T $ 0000008134: $ P $	pdg_{0004645} = \frac{1}{pdg_{0008134}}	ERROR for dim with 9718685793
9749777192	$ 0 = KE_1 + PE_1 $ 0 = KE_1 + PE_1			escape velocity	0000001955: $ KE_1 $ 0000004093: $ PE_1 $	0 = pdg_{0001955} + pdg_{0004093}	ERROR for dim with 9749777192
9756089533	$ \sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} ) $ \sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} )			optics: Law of refraction to Brewster's angle	0000002941: $ n_1 $ 0000004928: $ \theta_{\rm Brewster} $ 0000001958: $ n_2 $	\sin{\left(pdg_{0004928} \right)} = \frac{pdg_{0001958} \cos{\left(pdg_{0004928} \right)}}{pdg_{0002941}}	ERROR for dim with 9756089533
9759901995	$ v - v_0 = a t $ v - v_0 = a t			equations of motion in 1D with constant acceleration - SUVAT (algebra)	0000001467: $ t $ 0000009140: $ a $ 0000005153: $ v_0 $ 0000001357: $ v $	pdg_{0001357} - pdg_{0005153} = pdg_{0001467} pdg_{0009140}	ERROR for dim with 9759901995
9781951738	$ \kappa_T = \frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T $ \kappa_T = \frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T		definition of isothermal compressibility	coefficient of isothermal compressibility using the equation of state for an ideal gas first law of thermodynamics	0000004645: $ \kappa_T $ 0000008134: $ P $ 0000007586: $ V $	pdg_{0004645} = - \frac{\frac{d}{d pdg_{0008134}} pdg_{0007586}}{pdg_{0007586}}	ERROR for dim with 9781951738
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 $ \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			Lorentz transformation	0000001467: $ t $ 0000001357: $ v $ 0000001790: $ \gamma $ 0000004567: $ c $ 0000005647: $ y $ 0000006728: $ z $ 0000004037: $ x $	pdg_{0001790}^{2} \left(- pdg_{0001357} pdg_{0001467} + pdg_{0004037}\right)^{2} + pdg_{0005647}^{2} + pdg_{0006728}^{2} = pdg_{0001790}^{2} pdg_{0004567}^{2} \left(pdg_{0001467} + \frac{pdg_{0004037} \left(1 - pdg_{0001790}^{2}\right)}{pdg_{0001357} pdg_{0001790}^{2}}\right)^{2}	ERROR for dim with 9805063945
9838128064	$ d_2 \frac{4 \pi^2}{T^2} = G \frac{m_1}{r^2} $ d_2 \frac{4 \pi^2}{T^2} = G \frac{m_1}{r^2}			Kepler's Third Law: period squared propto distance cubed	0000003141: $ \pi $ 0000002798: $ d_2 $ 0000006277: $ G $ 0000005022: $ m_1 $ 0000002530: $ r $ 0000009491: $ T $	\frac{4 pdg_{0002798} pdg_{0003141}^{2}}{pdg_{0009491}^{2}} = \frac{pdg_{0005022} pdg_{0006277}}{pdg_{0002530}^{2}}	ERROR for dim with 9838128064
9847143017	$ \kappa_T = \frac{-PV}{V} \left( \frac{-1}{P^2}\right) $ \kappa_T = \frac{-PV}{V} \left( \frac{-1}{P^2}\right)			coefficient of isothermal compressibility using the equation of state for an ideal gas	0000004645: $ \kappa_T $ 0000008134: $ P $	pdg_{0004645} = \frac{1}{pdg_{0008134}}	ERROR for dim with 9847143017
9848292229	$ dy = y i dx $ dy = y i dx			Euler equation proof	0000004621: $ i $ 0000001452: $ y $ 0000005842: $ dy $ 0000009199: $ dx $	pdg_{0005842} = pdg_{0001452} pdg_{0004621} pdg_{0009199}	ERROR for dim with 9848292229
9848294829	$ \frac{d}{dx} y = y i $ \frac{d}{dx} y = y i			Euler equation proof	0000001464: $ x $ 0000001452: $ y $ 0000004621: $ i $	\frac{d}{d pdg_{0001464}} pdg_{0001452} = pdg_{0001452} pdg_{0004621}	ERROR for dim with 9848294829
9854442418	$ v = \sqrt{\frac{E}{m}} $ v = \sqrt{\frac{E}{m}}			upper limit on velocity in condensed matter	0000002077: $ v $ 0000009863: $ m $ 0000002241: $ E $	pdg_{0002077} = \sqrt{\frac{pdg_{0002241}}{pdg_{0009863}}}	ERROR for dim with 9854442418
9858028950	$ \frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx $ \frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx			particle in a 1D box	0000003141: $ \pi $ 0000001464: $ x $ 0000001592: $ n $ 0000002523: $ W $ 0000009139: $ a $	\frac{1}{pdg_{0009139}^{2}} = \int\limits_{0}^{pdg_{0002523}} \left(\frac{1}{2} - \frac{\cos{\left(\frac{2 pdg_{0001464} pdg_{0001592} pdg_{0003141}}{pdg_{0002523}} \right)}}{2}\right)\, dpdg_{0001464}	ERROR for dim with 9858028950
9862900242	$ y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0 $ y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0			equations of motion in 2D (calculus) angle of maximum distance for projectile motion	0000001467: $ t $ 0000001649: $ g $ 0000005647: $ y $ 0000001575: $ \theta $ 0000005153: $ v_0 $ 0000001469: $ y_0 $	pdg_{0005647} = - \frac{pdg_{0001467}^{2} pdg_{0001649}}{2} + pdg_{0001467} pdg_{0005153} \sin{\left(pdg_{0001575} \right)} + pdg_{0001469}	ERROR for dim with 9862900242
9882526611	$ v_{0, x} t = x - x_0 $ v_{0, x} t = x - x_0			equations of motion in 2D (calculus) projectile path in 2D is parabolic	0000001572: $ x_0 $ 0000001467: $ t $ 0000002958: $ v_{0, x} $ 0000004037: $ x $	pdg_{0001467} pdg_{0002958} = - pdg_{0001572} + pdg_{0004037}	ERROR for dim with 9882526611
9889984281	$ 2 (\sin(x))^2 = 1 - \cos(2 x) $ 2 (\sin(x))^2 = 1 - \cos(2 x)			Euler equation: trig square root	0000001464: $ x $	2 \sin^{2}{\left(pdg_{0001464} \right)} = 1 - \cos{\left(2 pdg_{0001464} \right)}	ERROR for dim with 9889984281
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) $ 2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \left(\exp(i x)+\exp(-i x) \right)			identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation	0000001464: $ x $ 0000004621: $ i $	2 \sin{\left(pdg_{0001464} \right)} \cos{\left(pdg_{0001464} \right)} = \frac{\left(e^{pdg_{0001464} pdg_{0004621}} - e^{- pdg_{0001464} pdg_{0004621}}\right) \left(e^{pdg_{0001464} pdg_{0004621}} + e^{- pdg_{0001464} pdg_{0004621}}\right)}{2 pdg_{0004621}}	ERROR for dim with 9894826550
9897284307	$ \frac{d}{t} = \frac{v + v_0}{2} $ \frac{d}{t} = \frac{v + v_0}{2}			equations of motion in 1D with constant acceleration - SUVAT (algebra)	0000001467: $ t $ 0000005153: $ v_0 $ 0000001943: $ d $ 0000001357: $ v $	\frac{pdg_{0001943}}{pdg_{0001467}} = \frac{pdg_{0001357}}{2} + \frac{pdg_{0005153}}{2}	ERROR for dim with 9897284307
9919999981	$ \rho = 0 $ \rho = 0			Maxwell equations to electric field wave equation	0000003935: $ \rho $	pdg_{0003935} = 0	ERROR for dim with 9919999981
9941599459	$ dU = \left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV $ dU = \left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV		based on U(p, T, V) = U(T, V)	first law of thermodynamics	0000005786: $ U $ 0000007343: $ T $	dU = \frac{d}{d pdg_{0007343}} pdg_{0005786}	ERROR for dim with 9941599459
9958485859	$ \frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) $ \frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)			derivation of Schrodinger Equation	0000001054: $ \hbar $ 0000001467: $ t $ 0000009489: $ \psi $ 0000005156: $ m $ 0000009472: $ \vec{r} $	- \frac{nabla^{2} pdg_{0001054}^{2} \operatorname{pdg}_{0009489}{\left(pdg_{0009472},pdg_{0001467} \right)}}{2 pdg_{0005156}} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
9973952056	$ -g t = v_y - v_{0, y} $ -g t = v_y - v_{0, y}			equations of motion in 2D (calculus)	0000009431: $ v_{0, y} $ 0000001649: $ g $ 0000001467: $ t $ 0000005153: $ v_0 $	- pdg_{0001467} pdg_{0001649} = - pdg_{0005153} + pdg_{0009431}	ERROR for dim with 9973952056
9988949211	$ (\sin(x))^2 = \frac{1 - \cos(2 x)}{2} $ (\sin(x))^2 = \frac{1 - \cos(2 x)}{2}			particle in a 1D box Euler equation: trig square root	0000001464: $ x $	\sin^{2}{\left(pdg_{0001464} \right)} = \frac{1}{2} - \frac{\cos{\left(2 pdg_{0001464} \right)}}{2}	ERROR for dim with 9988949211
9991999979	$ \vec{ \nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t} $ \vec{ \nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}			Maxwell equations to electric field wave equation	0000006238: $ E $ 0000001467: $ t $ 0000006197: $ \mu_0 $ 0000002069: $ \vec{H} $	\operatorname{cross}{\left(pdg_{0006238},nabla \right)} = - pdg_{0006197} \frac{d}{d pdg_{0001467}} pdg_{0002069}	ERROR for dim with 9991999979
9999998870	$ \frac{ \vec{p}}{\hbar} = \vec{k} $ \frac{ \vec{p}}{\hbar} = \vec{k}			derivation of Schrodinger Equation	0000001054: $ \hbar $ 0000002046: $ \vec{p} $ 0000007394: $ \vec{k} $	\frac{pdg_{0002046}}{pdg_{0001054}} = pdg_{0007394}	ERROR for dim with 9999998870
9999999870	$ \frac{p}{\hbar} = k $ \frac{p}{\hbar} = k			derivation of Schrodinger Equation	0000001054: $ \hbar $ 0000005321: $ k $ 0000001134: $ p $	\frac{pdg_{0001134}}{pdg_{0001054}} = pdg_{0005321}	ERROR for dim with 9999999870
9999999960	$ \hbar = h/(2 \pi) $ \hbar = h/(2 \pi)			derivation of Schrodinger Equation	0000001054: $ \hbar $ 0000004413: $ h $ 0000003141: $ \pi $	pdg_{0001054} = \frac{pdg_{0004413}}{2 pdg_{0003141}}	ERROR for dim with 9999999960
9999999961	$ \frac{E}{\hbar} = \omega $ \frac{E}{\hbar} = \omega			derivation of Schrodinger Equation	0000004931: $ E $ 0000001054: $ \hbar $ 0000002321: $ \omega $	\frac{pdg_{0004931}}{pdg_{0001054}} = pdg_{0002321}	ERROR for dim with 9999999961
9999999962	$ p = \hbar k $ p = \hbar k			derivation of Schrodinger Equation	0000001054: $ \hbar $ 0000005321: $ k $ 0000001134: $ p $	pdg_{0001134} = pdg_{0001054} pdg_{0005321}	ERROR for dim with 9999999962
9999999965	$ E = \omega \hbar $ E = \omega \hbar			derivation of Schrodinger Equation	0000004931: $ E $ 0000001054: $ \hbar $ 0000002321: $ \omega $	pdg_{0004931} = pdg_{0001054} pdg_{0002321}	ERROR for dim with 9999999965
9999999968	$ x = \frac{-b-\sqrt{b^2-4ac}}{2 a} $ x = \frac{-b-\sqrt{b^2-4ac}}{2 a}			quadratic equation derivation	0000001464: $ x $ 0000004231: $ c $ 0000001939: $ b $ 0000009139: $ a $	pdg_{0001464} = \frac{- pdg_{0001939} - \sqrt{pdg_{0001939}^{2} - 4 pdg_{0004231} pdg_{0009139}}}{2 pdg_{0009139}}	ERROR for dim with 9999999968
9999999969	$ x = \frac{-b+\sqrt{b^2-4ac}}{2 a} $ x = \frac{-b+\sqrt{b^2-4ac}}{2 a}			quadratic equation derivation	0000001464: $ x $ 0000004231: $ c $ 0000001939: $ b $ 0000009139: $ a $	pdg_{0001464} = \frac{- pdg_{0001939} + \sqrt{pdg_{0001939}^{2} - 4 pdg_{0004231} pdg_{0009139}}}{2 pdg_{0009139}}	ERROR for dim with 9999999969
9999999975	$ \langle \psi\| \hat{A} \|\psi \rangle = \langle a \rangle $ \langle \psi\| \hat{A} \|\psi \rangle = \langle a \rangle			quantum basics Hermitian operators have realvalued observables	0000004065: $ \langle \psi\| $	pdg_{0004065} = empty str sent to sympy_to_latex_str	SyntaxError: unable to parse as SymPy; error=invalid syntax (<string>, line 0)
9999999981	$ \vec{ \nabla} \cdot \vec{E} = \rho/\epsilon_0 $ \vec{ \nabla} \cdot \vec{E} = \rho/\epsilon_0			Maxwell equations to electric field wave equation	0000003935: $ \rho $ 0000007940: $ \epsilon_0 $ 0000004326: $ \vec{E} $	nabla pdg_{0004326} = \frac{pdg_{0003935}}{pdg_{0007940}}	ERROR for dim with 9999999981

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