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ID latex name description used in derivation symbols sympy dimensional consistency lean 4
0007455074 \( \vec{F} = m\ \vec{a} \)

\vec{F} = m\ \vec{a}
Newton's second law vector form relating force to mass and acceleration

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

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}

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

\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}

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)

\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

\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

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

pdg_{0002523} pdg_{0005321} = pdg_{0001592} pdg_{0003141}

ERROR for dim with 1010923823
1020010291 \( 0 = a \sin(k W) \)

0 = a \sin(k W)

0 = pdg_{0009139} \sin{\left(pdg_{0002523} pdg_{0005321} \right)}

ERROR for dim with 1020010291
1020394900 \( p = h/\lambda \)

p = h/\lambda

pdg_{0001134} = \frac{pdg_{0004413}}{pdg_{0001115}}

ERROR for dim with 1020394900
1020394902 \( E = h f \)

E = h f

pdg_{0004931} = pdg_{0004201} pdg_{0004413}

ERROR for dim with 1020394902
1020854560 \( I = (A + B)(A + B)^* \)

I = (A + B)(A + 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

pdg_{0001134} = pdg_{0001357} pdg_{0005156}

ERROR for dim with 1029039903
1029039904 \( p^2 = m^2 v^2 \)

p^2 = m^2 v^2

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}

\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

=

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

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)

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

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}

\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 \( m_{\rm Earth} = \frac{(9.80665 m/s^2) (6.3781*10^6 m)^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}} \)

m_{\rm Earth} = \frac{(9.80665 m/s^2) (6.3781*10^6 m)^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}}

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

\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}

- \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

0 = pdg_{0005005}

ERROR for dim with 1166310428
1172039918 \( I_{\rm coherent} = 4 |A|^2 \)

I_{\rm coherent} = 4 |A|^2

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}

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

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)

\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

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

\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

\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

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

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

\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

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

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}

pdg_{0002867} = \frac{pdg_{0004851} pdg_{0005022} pdg_{0006277}}{pdg_{0002530}^{2}}

ERROR for dim with 1292735067
1293913110 \( 0 = b \)

0 = b

0 = pdg_{0001939}

ERROR for dim with 1293913110
1293923844 \( \lambda = v T \)

\lambda = v T

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

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}

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

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}

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}

\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}}}

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)

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

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

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

\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^*

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}}}

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}

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

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}

\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)

- 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

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}

\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}

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

- \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

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

pdg_{0001469} = 0

ERROR for dim with 1650441634
1676472948 \( 0 = v_x - v_{0, x} \)

0 = v_x - v_{0, x}

0 = - pdg_{0002958} + pdg_{0005505}

ERROR for dim with 1676472948
1702349646 \( -g dt = d v_y \)

-g dt = 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

\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)

- \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

\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

\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}

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

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

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 $x=\infty$

pdg_{0001552} = 0

ERROR for dim with 1840080113
1857710291 \( 0 = a \sin(n \pi) \)

0 = a \sin(n \pi)

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

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}

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

- \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

- 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 )

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

\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

\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

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

pdg_{0002958} pdg_{0004711} = pdg_{0009199}

ERROR for dim with 1963253044
1967582749 \( t = \frac{v - v_0}{a} \)

t = \frac{v - v_0}{a}

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'

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

- 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}

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}}

\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

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

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

\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)

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

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)

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' )

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)

\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

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]}

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}

\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)

- 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

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)

- \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}]

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

\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

\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

\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

\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)

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

pdg_{0008849} = 0

ERROR for dim with 2267521164
2271186630 \( V = I_{\rm total} R_{\rm total} \)

V = I_{\rm total} R_{\rm total}

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)

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}

\frac{pdg_{0005458} pdg_{0006277}}{pdg_{0003236}^{2}} = pdg_{0007557}

ERROR for dim with 2308660627
2334518266 \( m a = -k x \)

m a = -k 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

\int 0\, dpdg_{0001467} = \int 1\, dpdg_{0005005}

ERROR for dim with 2366691988
2378095808 \( x_f = x_0 + d \)

x_f = x_0 + 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

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)

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

\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)^+

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

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

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)

\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

- \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

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

- \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}

pdg_{0004686} = \frac{1}{pdg_{0007343}}

ERROR for dim with 2472653783
2484824786 \( F = m g \)

F = m g

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

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

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}

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)

\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

- 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} )

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

\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

\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)

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)

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

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

\left|{pdg_{0004453}}\right| = \left|{pdg_{0004698}}\right|

ERROR for dim with 2719691582
2741489181 \( a_y = -g \)

a_y = -g

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}}}

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)

- \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}

\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}

\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}

\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

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

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

- \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

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} }

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)}

\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}

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}

pdg_{0006191} = pdg_{0009372}

ERROR for dim with 2907404069
2924222857 \( v_{\rm initial} = v(r=\infty) \)

v_{\rm initial} = v(r=\infty)

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)

\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

\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')

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

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

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}

\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}}

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

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} )

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}}

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))

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

\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}

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}

pdg_{0005321} = \frac{2 pdg_{0003141}}{pdg_{0001115}}

ERROR for dim with 3121513111
3131111133 \( T = 1 / f \)

T = 1 / f

pdg_{0009491} = \frac{1}{pdg_{0004201}}

ERROR for dim with 3131111133
3131211131 \( \omega = 2 \pi f \)

\omega = 2 \pi f

pdg_{0002321} = 2 pdg_{0003141} pdg_{0004201}

ERROR for dim with 3131211131
3132131132 \( \omega = \frac{2\pi}{T} \)

\omega = \frac{2\pi}{T}

pdg_{0002321} = \frac{2 pdg_{0003141}}{pdg_{0009491}}

ERROR for dim with 3132131132
3147472131 \( \frac{\omega}{2 \pi} = f \)

\frac{\omega}{2 \pi} = f

\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

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

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

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

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}

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}}

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

\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}

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

e^{pdg_{0003141} pdg_{0004621}} = -1

ERROR for dim with 3331824625
3350830826 \( Z Z^* = |Z|^2 \)

Z Z^* = |Z|^2

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

pdg_{0006789} = pdg_{0001352} - pdg_{0001955}

ERROR for dim with 3360172339
3364286646 \( m_{\rm Earth} = 5.972*10^{24} kg \)

m_{\rm Earth} = 5.972*10^{24} kg

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

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}

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 }

\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' )

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

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

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)

\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}

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

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

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}]

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}

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}

pdg_{0006850} pdg_{0009046} pdg_{0009067} = pdg_{0001966}

ERROR for dim with 3507029294
3512166162 \( W = F x \)

W = F 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

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

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

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

\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}]

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)

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)

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}}

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

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

\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)

\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

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]

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}

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

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)

\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}}

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}

\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}

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}

\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

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

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} }

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)

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

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}

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)

\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)

\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)

\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)

\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)

\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)

\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}

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}

\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

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

pdg_{0006599} = pdg_{0004501} pdg_{0006458}

ERROR for dim with 4087145886
4107032818 \( E_{\rm Rydberg} = E \)

E_{\rm Rydberg} = E

pdg_{0009838} = pdg_{0002241}

ERROR for dim with 4107032818
4128500715 \( V = I_1 R_1 \)

V = I_1 R_1

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'

- 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}

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}

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)}

\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}}

pdg_{0007427} = 29.8

ERROR for dim with 4180845508
4182362050 \( Z = |Z| \exp( i \theta ) \)

Z = |Z| \exp( i \theta )
Z \in \mathbb{C}

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}

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)

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}

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}

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

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}

\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}}

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

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

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

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}

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}]

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

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)

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)

\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

pdg_{0001467} = pdg_{0002467}

ERROR for dim with 4370074654
4393258808 \( F_{\rm centripetal} = m r \omega^2 \)

F_{\rm centripetal} = m r \omega^2

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}

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)

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

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

\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} )}

\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)

\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} }

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

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}

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)

\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}}

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

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}

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))

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)

- 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)

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

pdg_{0000004202} \propto pdg_{0000005022}

inconsistent dimensions
4669290568 \( PE_1 = -F x_1 \)

PE_1 = -F x_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}

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)

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

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}

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

\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

pdg_{0001467} pdg_{0009140} + pdg_{0005153} = pdg_{0001357}

ERROR for dim with 4798787814
4800170179 \( F = m g_{\rm Earth} \)

F = m g_{\rm Earth}

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)

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

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}

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

\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}

\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)

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)

\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

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}

\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

\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}

\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)}

\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

\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)

\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)

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

pdg_{0001955} = \frac{pdg_{0002473}^{2} pdg_{0005156}}{2}

ERROR for dim with 4928007622
4928239482 \( \log(y) = i x \)

\log(y) = i x

\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)

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)

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)

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}

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)

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}

\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

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^*

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

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 )}

\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

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

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]}

\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^*

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

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

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)

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

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}

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

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)

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

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)}

\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}

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}

pdg_{0001943} = \frac{pdg_{0005153}^{2}}{pdg_{0001649}}

ERROR for dim with 5353282496
5373931751 \( t = t_f \)

t = t_f

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

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

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)

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}

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

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)

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}

- 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

\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}

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

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)

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)

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

pdg_{0005480} = \frac{d}{d pdg_{0007586}} pdg_{0005786}

ERROR for dim with 5634116660
5646314683 \( m = A m_p \)

m = A 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}

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}}

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)

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

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)

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)

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

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}

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

- 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

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

\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}

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}}

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}

pdg_{0006191} = pdg_{0005340}

ERROR for dim with 5850144586
5857434758 \( \int a dx = a x \)

\int a dx = a x

\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

\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

- \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}

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

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

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

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

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)

\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

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

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}

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)

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))

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)}

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)

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}

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)

\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

\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

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}

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

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

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

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}

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}

- 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

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}}

\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

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^*)

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}

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

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}

\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

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}

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

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

\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} }

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

pdg_{0002435} = 2 \left|{pdg_{0004453}}\right|^{2}

ERROR for dim with 6529793063
6555185548 \( A^* = |A| \exp(-i \theta) \)

A^* = |A| \exp(-i \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

\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

- 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

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})

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

pdg_{0009647} = pdg_{0003978} + pdg_{0004856}

ERROR for dim with 6753224061
6774684564 \( \theta = \phi \)

\theta = \phi
for coherent waves

pdg_{0001575} = pdg_{0008586}

ERROR for dim with 6774684564
6783009163 \( r_{\rm adsorption} = r_{\rm desorption} \)

r_{\rm adsorption} = r_{\rm desorption}

pdg_{0006687} = pdg_{0001966}

ERROR for dim with 6783009163
6785303857 \( C = 2 \pi r \)

C = 2 \pi r

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}

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}

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} )

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

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

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

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}

\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)

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

pdg_{0000004202} \propto pdg_{0000005156}

inconsistent dimensions
6925244346 \( \alpha = \frac{PV}{T} \frac{1}{VP} \)

\alpha = \frac{PV}{T} \frac{1}{VP}

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

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}

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}]

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}}

pdg_{0007427} = 0.632911392405063 pdg_{0003141}

ERROR for dim with 6998364753
7002609475 \( \frac{V}{R_2} = I_2 \)

\frac{V}{R_2} = I_2

\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

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

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

pdg_{0001888} = pdg_{0005647}

ERROR for dim with 7057864873
7107090465 \( B B^* = |B|^2 \)

B B^* = |B|^2

pdg_{0004698} \overline{pdg_{0004698}} = \left|{pdg_{0004698}}\right|^{2}

ERROR for dim with 7107090465
7112613117 \( m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}} \)

m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}}

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}

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}

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

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

pdg_{0001908} = pdg_{0003461} + pdg_{0008697}

ERROR for dim with 7217021879
7222189955 \( F \propto m_2 \)

F \propto m_2

pdg_{0000004202} \propto pdg_{0000004851}

inconsistent dimensions
7233558441 \( d = v_0 t_f \cos(\theta) \)

d = v_0 t_f \cos(\theta)

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}

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)

\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}

\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

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}

\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}

\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}

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

\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

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}]

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)

\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

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

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}

\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})

\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})

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})

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

\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

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

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})

- 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})

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)

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)

\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}

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

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

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} )

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

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)

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}

\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 )

- 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)

- \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}

\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'

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}}

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

=

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

\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}}

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}

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

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}

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}

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}

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}}

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

\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

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

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

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}

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^*

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} }

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

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}]}

\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}

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

- 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

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

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}

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

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 )

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}

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)

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

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

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}

a_{c*(e*(n*(t*(r*(i*(p*(e*(t*(a*l)))))))))} = \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

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

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

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

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)

\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

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}

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)

\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)

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)

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

\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

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

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

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

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

\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}

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) }

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 )

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

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)

\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

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

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)

- \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

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)

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

\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)

\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

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)

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)

\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

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

- dt g = pdg_{0005674}

ERROR for dim with 8584698994
8588429722 \( \sin( 90^{\circ} - x ) = \cos( x ) \)

\sin( 90^{\circ} - x ) = \cos( 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

\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

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

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}

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.

=

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)

\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

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}

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

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)

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

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}

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

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)

\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

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

- 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)

\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

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

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)

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}

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}

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}

pdg_{0005153} \sin{\left(pdg_{0001575} \right)} = pdg_{0009431}

ERROR for dim with 8949329361
8953094349 \( W = m a x \)

W = m a 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

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

\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'

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)

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

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}

\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

pdg_{0006191} = \frac{pdg_{0005022} pdg_{0008909}^{2}}{2}

ERROR for dim with 9081138616
9112191201 \( y_f = 0 \)

y_f = 0

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}

\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}

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)

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 )

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

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

pdg_{0006599} = pdg_{0003461} pdg_{0004856}

ERROR for dim with 9243879541
9262596735 \( d = 2 \pi r \)

d = 2 \pi r

pdg_{0001943} = 2 pdg_{0002530} pdg_{0003141}

ERROR for dim with 9262596735
9285928292 \( ax^2 + bx + c = 0 \)

ax^2 + bx + c = 0

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}

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}

\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

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}

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}

\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

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

\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)

\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)

\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)

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

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}

\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'

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}}

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

- \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}

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)

\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}

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

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

\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

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

\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

\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

\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

\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

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)

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

\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

- \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)

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

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)

\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

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}

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))

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)}

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

\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

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.

=

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}}

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

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

\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

pdg_{0007159} = 0

ERROR for dim with 9707028061
9718685793 \( \kappa_T = \frac{1}{P} \)

\kappa_T = \frac{1}{P}

pdg_{0004645} = \frac{1}{pdg_{0008134}}

ERROR for dim with 9718685793
9749777192 \( 0 = KE_1 + PE_1 \)

0 = KE_1 + 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} )

\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

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

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

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}

\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)

pdg_{0004645} = \frac{1}{pdg_{0008134}}

ERROR for dim with 9847143017
9848292229 \( dy = y i dx \)

dy = y i 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

\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}}

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

\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

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

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)

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)

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}

\frac{pdg_{0001943}}{pdg_{0001467}} = \frac{pdg_{0001357}}{2} + \frac{pdg_{0005153}}{2}

ERROR for dim with 9897284307
9919999981 \( \rho = 0 \)

\rho = 0

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)

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)

- \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}

- 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}

\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}

\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}

\frac{pdg_{0002046}}{pdg_{0001054}} = pdg_{0007394}

ERROR for dim with 9999998870
9999999870 \( \frac{p}{\hbar} = k \)

\frac{p}{\hbar} = k

\frac{pdg_{0001134}}{pdg_{0001054}} = pdg_{0005321}

ERROR for dim with 9999999870
9999999960 \( \hbar = h/(2 \pi) \)

\hbar = h/(2 \pi)

pdg_{0001054} = \frac{pdg_{0004413}}{2 pdg_{0003141}}

ERROR for dim with 9999999960
9999999961 \( \frac{E}{\hbar} = \omega \)

\frac{E}{\hbar} = \omega

\frac{pdg_{0004931}}{pdg_{0001054}} = pdg_{0002321}

ERROR for dim with 9999999961
9999999962 \( p = \hbar k \)

p = \hbar k

pdg_{0001134} = pdg_{0001054} pdg_{0005321}

ERROR for dim with 9999999962
9999999965 \( E = \omega \hbar \)

E = \omega \hbar

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}

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}

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

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

nabla pdg_{0004326} = \frac{pdg_{0003935}}{pdg_{0007940}}

ERROR for dim with 9999999981

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