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Brewster}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba9025853427 :a_node:feedSymbol('pdg0004928')\theta_{\rm Brewster}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba9029795851 :a_node:feed9040079362Symbol('pdg0004201')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baf :a_node:feed9053099840Symbol('pdg0004501')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baI :a_node:feed9070454719a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_0^2Pow(Symbol('pdg0005153'), Integer(2)) :a_node:feed9072369552Symbol('pdg0005458')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bam_{\rm Earth} :a_node:feed9110536742a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba2 xMul(Integer(2), Symbol('pdg0001464')) :a_node:feed9174439158Symbol('pdg0008697')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baR_1 :a_node:feed9305761407Symbol('pdg0001357')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav :a_node:feed9346215480Symbol('pdg0008762')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baT_{\rm orbit} :a_node:feed9350663581Symbol('pdg0003141')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\pi :a_node:feed9350720370Symbol('pdg0005156')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bam :a_node:feed9355039511Symbol('pdg0001649')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bag :a_node:feed9370882921Symbol('pdg0005332')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baKE_{\rm escape} :a_node:feedMul(Mul(Integer(1), Pow(Symbol('pdg0002798'), Integer(-1))), Pow(Mul(Integer(1), Pow(Symbol('pdg0002798'), Integer(-1))), Integer(-1)))9524810853a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{1/d_2}{1/d_2} :a_node:feed9565166889Symbol('pdg0009491')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baT :a_node:feed9590696981a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba9.80665Float('9.8066499999999994', precision=53) :a_node:feed9594072504Symbol('pdg0004851')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bam_2 :a_node:feedSymbol('pdg0007427')v_{\rm Earth\ orbit}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba9601500174 :a_node:feed9623791270Symbol('pdg0004037')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bad :a_node:feed9645178657a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baa tMul(Symbol('pdg0009140'), Symbol('pdg0001467')) :a_node:feed9674924517a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baK >> GStrictGreaterThan(Symbol('pdg0001466'), Symbol('pdg0003033')) :a_node:feed9746066299Symbol('pdg0003461')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baR_2 :a_node:feed9753878784Symbol('pdg0001357')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav :a_node:feed9761485403Symbol('pdg0003192')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baZ :a_node:feed9789485295Symbol('pdg0004082')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_{\rm satellite} :a_node:feed9794128647Symbol('pdg0005458')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bam_1 :a_node:feed9830343096Symbol('pdg0008257')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baV_1 :a_node:feed9881106100Symbol('pdg0009140')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baa :a_node:feed9884115626Symbol('pdg0002530')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bar :a_node:feed9885190237Symbol('pdg0004621')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bai :a_node:feed9903988330Symbol('pdg0005156')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bam :a_node:feed9906920183Symbol('pdg0001464')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax :a_node:feedSymbol('pdg0004518')r_{\rm Schwarzschild}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba9933742680 :a_node:feedMul(Float('6.6742999999999997', precision=53), Pow(Integer(10), Mul(Integer(-1), Integer(11))))9956609318a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba6.67430*10^{-11} :a_node:expressionm\ \vec{a}0007455074vector form relating force to mass and accelerationNewton's second lawSymbol('pdg0002423')=2026-02-27_22-25-57-571724a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\vec{F}https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion#Second_law :a_node:derivationconvert vector representation to scalar00089966672026-02-27_21-32-14-900164a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baNewton's Second Law of Motion: vector to scalar :a_node:step00019160832026-02-27_23-19-05-499790a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:expressiona1075552184=2026-02-28_12-20-11-381958a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba|\vec{a}| :a_node:expression1219718533FSymbol('pdg0004202')Abs(Symbol('pdg0004202'))=Eq(Abs(Symbol('pdg0004202')),Symbol('pdg0004202'))2026-02-28_13-00-31-662004a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba|\vec{F}| :a_node:step35360945102026-02-28_13-01-58-465558a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step15266800352026-02-28_13-02-39-741214a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:derivationcurl curl identitya84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0000000005 :a_node:derivationderivation of Schrodinger Equationa84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0000000006 :a_node:inference_rule1379304934magnitude of vectorsAbsolute value of both sides2026-02-28_22-09-27-241975a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba011 :a_node:expression5272284986|m\ \vec{a}|magnitude of vector representation of Newton's second law=Eq(Abs(Symbol('pdg0000006777')),Abs(Mul(Symbol('pdg0000009863'),Symbol('pdg0000002423')))Abs(Symbol('pdg0000006777'))Abs(Mul(Symbol('pdg0000009863'),Symbol('pdg0000002423'))2026-02-28_22-39-04-598575a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba|\vec{F}| :a_node:step51054236062026-02-28_22-42-34-002083a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step85414427222026-02-28_22-52-39-213458a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step83198762012026-02-28_22-53-36-058780a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:derivation64703188272026-03-01_14-06-23-459141a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://fermatslasttheorem.blogspot.com/2006/02/eulers-formula.htmlEuler's equation from MacLaurin series :a_node:expression9605409442MacLaurin series with expanded terms{\rm the\ function\ is\ infinitely\ differentiable\ at\ x = 0.}a Maclaurin series is a special case of a Taylor series where the expansion point is always x=0.f(0)+f'(x)\ x + \frac{f''(x)\ x^2}{2!} + \frac{f'''(x)\ x^3}{3!} + ...2026-03-01_14-13-41-223833a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baf(x)https://en.wikipedia.org/wiki/Taylor_series= :a_node:expression8637447837MacLaurin series as summation{\rm the\ function\ is\ infinitely\ differentiable\ at\ x = 0.}a Maclaurin series is a special case of a Taylor series where the expansion point is always x=0.\sum_{n=0}^{\infty} \frac{f^n(0)}{n!} x^n2026-03-01_14-18-02-534007a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baf(x)https://en.wikipedia.org/wiki/Taylor_series= :a_node:step61536265902026-03-01_14-19-41-297707a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step40274845512026-03-01_14-44-00-184445a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step12153014212026-03-01_14-46-58-323422a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:expression7764834870Taylor series as summation{\rm the\ function\ is\ infinitely\ differentiable\ at}\ x = a.\sum_{n=0}^{\infty} \frac{f^n(a)}{n!} (x-a)^n2026-03-01_15-05-53-683114a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baf(x)https://en.wikipedia.org/wiki/Taylor_series= :a_node:step63230487632026-03-01_16-09-19-320345a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:derivation0000000007a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baelectric field wave equation: from time dependent to time independent :a_node:scalar:symbol89372945100infinity0arbitraryvariable00\inftyany02026-03-02_12-37-08-600877a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0https://en.wikipedia.org/wiki/Infinity0 :a_node:feed6941869627Symbol('pdg0000004930')2026-03-04_19-11-22-742121a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baPE :a_node:derivationfrequency relationsa84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0000000008 :a_node:feed7408927653Symbol('pdg0000004929')2026-03-04_19-14-28-935250a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baKE :a_node:expressionm6923310769Symbol('pdg0000005156')Symbol('pdg0000004202')\propto{\rm acceleration\ is\ constant}2026-03-05_18-48-25-744569a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF :a_node:expressionm_14664063894Symbol('pdg0000005022')Symbol('pdg0000004202')\propto{\rm acceleration\ is\ constant}2026-03-05_18-51-09-464234a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF :a_node:expressionm_27222189955Symbol('pdg0000004851')Symbol('pdg0000004202')\propto{\rm acceleration\ is\ constant}2026-03-05_18-52-02-944988a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF :a_node:expression1189963325\frac{1}{r^2}Pow(Symbol('pdg0000002530'),-2){\rm acceleration\ is\ constant}\propto2026-03-05_18-54-47-046540a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF_{\rm gravity} :a_node:expression8617866819\frac{m_1\ m_2}{r^2}Symbol('pdg4482727458')\proptoMul(Mul(Symbol('pdg0000005022'),Symbol('pdg0000004851')),Pow(Symbol('pdg0000002530'),-2))2026-03-05_19-42-58-652653a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF_{\rm gravity} :a_node:scalar:symbolforce due to gravity14482727458-20real0variable0F_{\rm gravity}any2026-03-05_19-44-14-234372a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba10 :a_node:scalar:symbolconstant4133185897realconstant of solution to differential equation1variable00c_1any002026-03-24_14-51-53-850428a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba00 :a_node:scalar:symbolconstant1777778008realconstant of solution to differential equation1variable00c_2any002026-03-24_14-52-01-350760a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba00 :a_node:scalar:symbol25073440571maximum velocity-1realvariable00v_{\rm max}any02026-03-24_14-53-40-523859a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba00 :a_node:scalar:symbol60475826511maximum acceleration-2realvariable00a_{\rm max}any02026-03-24_14-53-45-252147a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba00 :a_node:scalar:symbolangle23578093480real0variable00\alphaany02026-03-24_14-54-41-010905a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba00 :a_node:scalar:symbolangle39231268730real0variable00\betaany02026-03-24_14-54-49-671153a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba00 :a_node:operation2288488773determine the phase of a complex value1Complex argument2026-03-24_19-08-54-051997a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://en.wikipedia.org/wiki/Argument_(complex_analysis)\arg :a_node:derivationintegration by partsa84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0000000009 :a_node:derivationparticle in a 1D boxa84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0000000010 :a_node:derivationquadratic equation derivationa84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://en.wikipedia.org/wiki/Quadratic_formula#Derivations_of_the_formula0000000011 :a_node:derivation0000000012a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baquantum basics Hermitian operators have realvalued observables :a_node:derivationquantum basics orthogonalitya84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0000000013 :a_node:derivation0000000014a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bavariance relation :a_node:derivation0000000016a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baidentity sin(2 x) = 2 sin(x) cos(x) using Euler's equation :a_node:derivation0000000017a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baEuler equation to exp(i pi) + 1 = 0 :a_node:derivationtime invariant force conserves energya84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0000000018 :a_node:derivation0000129143a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baescape velocity :a_node:derivation0000142831a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://en.wikipedia.org/wiki/Derivation_of_the_Schwarzschild_solutionSchwarzschild radius for non-rotating black hole :a_node:derivation0000146432a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://hj.hi.is/EE2/HD1lausn.pdfcoefficient of thermal expansion using the equation of state for an ideal gas :a_node:derivation0000187793a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baequations of motion in 2D (calculus) :a_node:derivation0000201726a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baangle of maximum distance for projectile motion :a_node:derivationfrom https://www.youtube.com/watch?v=fJYdFIZlD8kNewton's Law of Gravitationa84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0000207210 :a_node:derivationradius for satellite in geostationary orbita84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://en.wikipedia.org/wiki/Geostationary_orbit#Derivation_of_geostationary_altitude0000282755 :a_node:derivation0000332170a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://en.wikipedia.org/wiki/Equations_of_motionequations of motion in 1D with constant acceleration - SUVAT (algebra) :a_node:derivation0000374317a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bavelocity at distance r of object dropped from infinityhttps://www.youtube.com/watch?v=5F1XcTjpJs4 - Derivation of Gravitational Potential Energy by Rhett Allain :a_node:derivation0000375160a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://hj.hi.is/EE2/HD1lausn.pdfcoefficient of isothermal compressibility using the equation of state for an ideal gas :a_node:derivationspeed of Earth around Sunfrom \cite{1999_Tipler_Llewellyn}, page 9a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0000387954 :a_node:derivationfirst law of thermodynamicshttps://www.youtube.com/watch?v=QTiqF-HtkS0 and https://www.youtube.com/watch?v=3Yls-t3B49Ua84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0000513999 :a_node:derivation0000522862a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baoptics: Law of refraction to Brewster's angle\cite{2001_HRW}; see figure 34-27 on page 824 :a_node:derivationmass of the Eartha84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0000527822 :a_node:derivation0000539398a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138badouble intensity when phase is coherent (optics) :a_node:derivationLorentz transformationsource: \cite{1999_Tipler_Llewellyn}, page 21; see also https://en.wikipedia.org/wiki/Lorentz_transformation and https://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformationsa84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0000551770 :a_node:derivationupper limit on velocity in condensed matterhttps://arxiv.org/pdf/2004.04818.pdfa84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0000608598 :a_node:derivationequation of motion for a springa84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0000670255 :a_node:derivation0000681943a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttp://www.dfcd.net/articles/derivations/resistors.htmltotal electrical resistance for circuit with two resistors in parallel :a_node:derivationhyperbolic trigonometric identitiesa84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttp://www.physics.miami.edu/~nearing/mathmethods/mathematical_methods-one.pdf0000713234 :a_node:derivationLangmuir Adsorptionfrom https://arxiv.org/pdf/2210.12150.pdfa84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0000764666 :a_node:derivation0000820976a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion#Third_lawKepler's Third Law: period squared propto distance cubed :a_node:derivationfrequency and periodsubset of the frequency relation derivation. 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:a_node:step0005860845a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0005937834a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0006398792a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0006672580a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0007068872a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba$y(t_f) = y_f = 0$ :a_node:step0007996351a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0008131060a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0008352636a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0008794366a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0009270409a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0009551659a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba 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:a_node:step0008612576a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0008712986a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0008915037a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0009260286a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0009512128a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0009585552a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0001336657a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138babased on the comparison of the $t^2$ terms :a_node:step0001337934a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baequation 1-13 on page 21 in \cite{1999_Tipler_Llewellyn} :a_node:step0002712078a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0003002960a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0003151962a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0003201871a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0003244831a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0003464414a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0004044426a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0004052521a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0004777578a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0004875843a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138babased on the comparison of the $x^2$ terms :a_node:step0005029881a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138basolve for $\gamma$ :a_node:step0005211114a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba 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the comparison of the $(x t)$ terms :a_node:step0008057586a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0008199201a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0008488825a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0009486255a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0009520931a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0009869359a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baequation 1-14 on page 21 in \cite{1999_Tipler_Llewellyn} :a_node:step0001346919a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0001452028a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0002951905a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba :a_node:step0003423374a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba 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:a_node:symbol:vectorwavenumber-1arbitraryarbitraryreal000falseany0variablehttps://en.wikipedia.org/wiki/Wavenumber0\vec{k}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba00000073940 :a_node:symbol:vectorradius vector1arbitraryarbitraryreal000falseany0variable0\vec{r}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba00000094720 :a_node:symbol:vectormomentum before collisionarbitraryfalsereal1-1arbitraryvariable00\vec{p}_{\rm before}https://en.wikipedia.org/wiki/Momentum0any0a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba00000013021 :a_node:symbol:vectormagnetic field0arbitraryarbitraryreal000falseany0variablehttps://en.wikipedia.org/wiki/Magnetic_field0\vec{H}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba00000020690 :a_node:symbol:vectormomentum 21arbitraryarbitraryfalsereal0-10any0variablehttps://en.wikipedia.org/wiki/Momentum0\vec{p}_2a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba00000020971 :a_node:symbol:vectoracceleration1arbitraryarbitraryfalsereal0-20any0variable0\vec{a}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba00000024230 :a_node:symbol:vectorposition1arbitraryarbitraryvector000falseany0variable0\vec{x}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba00000029110 :a_node:symbol:vectormomentum of electronarbitraryfalsereal1-1arbitraryvariable00\vec{p}_{\rm electron}https://en.wikipedia.org/wiki/Momentum0any0a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba00000042991 :a_node:symbol:vectorelectric field0arbitraryarbitrarycomplex000falseany0variablehttps://en.wikipedia.org/wiki/Electric_field0\vec{E}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba00000043260 :a_node:symbol:vectormomentum after collisionarbitraryreal10arbitraryfalsevariable-1\vec{p}_{\rm after}0https://en.wikipedia.org/wiki/Momentum0any0a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba00000054931 :a_node:relationLHS = RHS0002222545equals["\\equals[2]{ #1 = #2}"]a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=["real","vector","matrix","complex"] :a_node:relationinequality0006856691less than or equal to2026-02-03_20-08-04-843996a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\leq :a_node:relation0008891337less thaninequality2026-02-03_20-08-33-749616a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\lt :a_node:relation0001247576not equal to2026-02-03_20-08-52-797783a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\neq :a_node:relationgreater than00049142812026-02-03_20-09-04-620259a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\gt :a_node:relation0003129032greater than or equal to2026-02-03_20-09-20-297941a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\geq :a_node:relation0005574373less thaninequality2026-02-03_20-09-36-391116a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba< :a_node:relation0002860537greater thaninequality2026-02-03_20-09-52-307605a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba> :a_node:relation0007513412approximately equal to2026-02-03_20-10-16-403563a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\approx :a_node:relation0002179795proportional to2026-02-03_20-10-42-459034a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\propto :a_node:inference_ruleLHS of expr 1 equals LHS of expr 2210000111355a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baLHS of Eq.~\ref{eq:#1} is equal to LHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.0 :a_node:inference_ruleRHS of expr 1 equals RHS of expr 2210000111863a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baRHS of Eq.~\ref{eq:#1} is equal to RHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.0 :a_node:inference_rule0000111469X cross both sides by111a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baTake cross product of $#1$ and Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.linear algebra :a_node:inference_rule0000111145X dot both sides111a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baTake inner product of $#1$ with Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.linear algebra :a_node:inference_ruleadd X to both sides110000111530a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baAdd $#1$ to both sides of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.1 :a_node:inference_ruleadd expr 1 to expr 2210000111980a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baAdd Eq.~\ref{eq:#1} to Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#2}.0 :a_node:inference_ruleadd zero to LHS110000111242a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baAdd zero to LHS of Eq.~\ref{eq:#2}, where $0=#1$; yields Eq.~\ref{eq:#3}.1 :a_node:inference_ruleadd zero to RHS110000111717a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baAdd zero to RHS of Eq.~\ref{eq:#2}, where $0=#1$; yields Eq.~\ref{eq:#3}.1 :a_node:inference_ruleapply divergence0000111463multi-variable calculus1a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baApply divergence to both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.01 :a_node:inference_ruleExample: apply sin(x) to 'a = b + c'apply function to both sides of expression110000111490a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baApply function $#1$ with argument $#2$ to Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}2 :a_node:inference_rule0000111531apply gradient to scalar functionmulti-variable calculus1a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baApply gradient to both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.01 :a_node:inference_rule0000111390apply operator to braquantum: Dirac notation1a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baApply operator in Eq.~\ref{eq:#1} to bra; yields Eq.~\ref{eq:#2}.01 :a_node:inference_rule0000111946apply operator to ketquantum: Dirac notation1a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baApply operator in Eq.~\ref{eq:#1} to ket; yields Eq.~\ref{eq:#2}.01 :a_node:inference_ruleassume N dimensions010000111791a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baAssume $#1$ dimensions; decompose vector to be Eq.~\ref{eq:#2}.1 :a_node:inference_rule0000111368both sides cross X111a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baTake cross product of Eq.~\ref{eq:#2} and $#1$; yields Eq.~\ref{eq:#3}linear algebra :a_node:inference_rule0000111567both sides dot X111a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baTake inner product of Eq.~\ref{eq:#2} with $#1$; yields Eq.~\ref{eq:#3}linear algebra :a_node:inference_ruleboundary condition110000111278a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baBoundary condition: Eq.~\ref{eq:#2} when Eq.~\ref{eq#1}.0 :a_node:inference_ruleboundary condition for expression110000111802a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baA boundary condition for Eq.~\ref{eq:#1} is Eq.~\ref{eq:#2}0 :a_node:inference_rulechange five variables in expression110000111410a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baChange of variable $#1$ to $#2$ and $#3$ to $#4$ and $#5$ to $#6$ and $#7$ to $#8$ and $#9$ to $#10$ in Eq.~\ref{eq:#11}; yields Eq.~\ref{eq:#12}.10 :a_node:inference_rulechange four variables in expression110000111777a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baChange of variable $#1$ to $#2$ and $#3$ to $#4$ and $#5$ to $#6$ and $#7$ to $#8$ in Eq.~\ref{eq:#9}; yields Eq.~\ref{eq:#10}.8 :a_node:inference_rulechange six variables in expression110000111471a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baChange of variable $#1$ to $#2$ and $#3$ to $#4$ and $#5$ to $#6$ and $#7$ to $#8$ and $#9$ to $#10$ and $#11$ to $#12$ in Eq.~\ref{eq:#13}; yields Eq.~\ref{eq:#14}.12 :a_node:inference_rulechange three variables in expression110000111236a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baChange of variable $#1$ to $#2$ and $#3$ to $#4$ and $#5$ to $#6$ in Eq.~\ref{eq:#7}; yields Eq.~\ref{eq:#8}.6 :a_node:inference_rulechange two variables in expression110000111984a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baChange variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\ref{eq:#5}; yields Eq.~\ref{eq:#6}.4 :a_node:inference_rulechange variable X to Y110000111886a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baChange variable $#1$ to $#2$ in Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.2 :a_node:inference_ruleclaim LHS equals RHS100000111345a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baThus we see that LHS of Eq.~\ref{eq:#1} is equal to RHS.0 :a_node:inference_ruleclaim expr 1 equals expr 2200000111550a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baThus we see that Eq.~\ref{eq:#1} is equivalent to Eq.~\ref{eq:#2}.0 :a_node:inference_rulecombine like terms110000111728a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baCombine like terms in Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.0 :a_node:inference_rule0000111484conjugate both sides011a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baConjugate both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.complex values :a_node:inference_rule0000111996conjugate function X111a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baConjugate $#1$ in Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.complex values :a_node:inference_rule0000111696conjugate transpose both sidescomplex valued linear algebra1a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baConjugate transpose of both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.01 :a_node:inference_ruledeclare assumption010000111104a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baEq.~\ref{eq:#1} is an assumption.0 :a_node:inference_ruledeclare final expression100000111341a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baEq.~\ref{eq:#1} is one of the final equations.0 :a_node:inference_rulehttps://en.wikipedia.org/wiki/Ansatzdeclare guess solution110000111237a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baJudicious choice as a guessed solution to Eq.~\ref{eq:#1} is Eq.~\ref{eq:#2},0 :a_node:inference_rule000011129901declare identitya84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baEq.~\ref{eq:#1} is an identity.0 :a_node:inference_ruledeclare initial expression010000111981a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baEq.~\ref{eq:#1} is an initial equation.0 :a_node:inference_rule0000111649differentiate with respect todifferential equations1a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baDifferentiate Eq.~\ref{eq:#2} with respect to $#1$; yields Eq.~\ref{eq:#3}.11 :a_node:inference_rule0000111474distribute conjugate to factors011a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baDistribute conjugate to factors in Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.complex values :a_node:inference_rule0000111890distribute conjugate transpose to factorscomplex valued linear algebra1a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baDistribute conjugate transpose to factors in Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.01 :a_node:inference_ruledivide both sides by110000111975a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baDivide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.1 :a_node:inference_ruledivide expr 1 by expr 2210000111421a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baDivide Eq.~\ref{eq:#1} by Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.0 :a_node:inference_ruledrop non-dominant term110000111782a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baBased on the assumption $#1$, drop non-dominant term in Eq.~\ref{#2}; yeilds Eq.~\ref{#3}1 :a_node:inference_rule0000111662evaluate definite integralmulti-variable calculus1a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baEvaluate definite integral Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.01 :a_node:inference_rule000011156111expand LHSa84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baExpand the LHS of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.0 :a_node:inference_rule000011154611expand RHSa84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baExpand the RHS of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.0 :a_node:inference_ruleexpand integrand0000111581multi-variable calculus1a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baExpand integrand of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.01 :a_node:inference_rule0000111166expand magnitude to conjugate111a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baExpand $#1$ in Eq.~\ref{eq:#2} with conjugate; yields Eq.~\ref{eq:#3}.complex values :a_node:inference_rule000011176821expr 1 is equivalent to expr 2 under the conditiona84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baEq.~\ref{eq:#1} is equivalent to Eq.~\ref{eq:#2} under the condition in Eq.~\ref{eq:#3}.0 :a_node:inference_rule000011169821expr 1 is true under condition expr 2a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baEq.~\ref{eq:#1} is valid when Eq.~\ref{eq:#2} occurs; yields Eq.~\ref{eq:#3}.0 :a_node:inference_rule000011143211factor out Xa84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baFactor $#1$ from Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.1 :a_node:inference_rulefactor out X from LHS110000111613a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baFactor $#1$ from the LHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.1 :a_node:inference_rulefactor out X from RHS110000111260a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baFactor $#1$ from the RHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.1 :a_node:inference_rule000011132911function is evena84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba$#1$ is even with respect to $#2$, so replace $#1$ with $#3$ in Eq.~\ref{eq:#4}; yields Eq.~\ref{eq:#5}.3 :a_node:inference_rule000011152211function is odda84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba$#1$ is odd with respect to $#2$, so replace $#1$ with $#3$ in Eq.~\ref{eq:#4}; yields Eq.~\ref{eq:#5}.3 :a_node:inference_rule0000111344indefinite integral overmulti-variable calculus1a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baIndefinite integral of both sides of Eq.~\ref{eq:#2} over $#1$; yields Eq.~\ref{eq:#3}.11 :a_node:inference_rule0000111137indefinite integrate LHS overmulti-variable calculus1a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baIndefinite integral of LHS of Eq.~\ref{eq:#2} over $#1$; yields Eq.~\ref{eq:#3}.11 :a_node:inference_rule0000111132indefinite integrate RHS overmulti-variable calculus1a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baIndefinite integral of RHS of Eq.~\ref{eq:#2} over $#1$; yields Eq.~\ref{eq:#3}.11 :a_node:inference_rule0000111608indefinite integrationmulti-variable calculus1a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baIndefinite integral of both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.01 :a_node:inference_rulenot clear how to specify limits for an arbitrary number of integralsintegrate0000111408multi-variable calculus1a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baIntegrate Eq.~ref{eq:#1}; yields Eq.~ref{eq:#2}.01 :a_node:inference_rule0000111937integrate over from tomulti-variable calculus1a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baIntegrate Eq.~\ref{eq:#4} over $#1$ from lower limit $#2$ to upper limit $#3$; yields Eq.~\ref{eq:#5}.31 :a_node:inference_rule000011172111make expr powera84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baMake Eq.~\ref{eq:#2} the power of $#1$; yields Eq.~\ref{eq:#3}.1 :a_node:inference_rulemaximum of expression110000111773a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baThe maximum of Eq.~\ref{eq:#2} with respect to $#1$ is Eq.~\ref{eq:#3}1 :a_node:inference_rulemultiply LHS by unity110000111173a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baMultiply LHS of Eq.~\ref{eq:#2} by 1, which in this case is $#1$; yields Eq.~\ref{eq:#3}1 :a_node:inference_rulemultiply RHS by unity110000111646a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baMultiply RHS of Eq.~\ref{eq:#2} by 1, which in this case is $#1$; yields Eq.~\ref{eq:#3}1 :a_node:inference_rulemultiply both sides by110000111182a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baMultiply both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.1 :a_node:inference_rulemultiply expr 1 by expr 2210000111253a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baMultiply Eq.~\ref{eq:#1} by Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.0 :a_node:inference_rulenormalization condition010000111493a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baNormalization condition is Eq.~\ref{eq:#1}.0 :a_node:inference_rule0000111680partially differentiate with respect todifferential equations1a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baPartially differentiate Eq.~\ref{eq:#2} with respect to $#1$; yields Eq.~\ref{eq:#3}.11 :a_node:inference_ruleraise both sides to power110000111483a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baRaise both sides of Eq.~\ref{eq:#2} to $#1$; yields Eq.~\ref{eq:#3}.1 :a_node:inference_rulereplace constant with value110000111715a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baReplace constant $#1$ with value $#2$ and units $#3$ in Eq.~\ref{eq:#4}; yields Eq.~\ref{eq:#5}3 :a_node:inference_rule0000111935011replace curl with LeviCevita summation contravarianta84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baReplace curl in Eq.~\ref{eq:#1} with Levi-Cevita contravariant; yields Eq.~\ref{eq:#2}.linear algebra :a_node:inference_rulereplace scalar with vector110000111215a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baReplace scalar variables in Eq.~\ref{eq:#1} with equivalent vector variables; yields Eq.~\ref{eq:#2}.0 :a_node:inference_rule0000111894011replace summation notation with vector notationa84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baReplace summation notation in Eq.~\ref{eq:#1} with vector notation; yields Eq.~\ref{eq:#2}.linear algebra :a_node:inference_rule0000111891select imaginary parts011a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baSelect imaginary parts of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.complex values :a_node:inference_ruleselect real parts0000111198011a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baSelect real parts of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.complex values :a_node:inference_ruleseparate three vector components130000111552a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baSeparate three vector components in Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2} and Eq.~\ref{eq:#3} and Eq.~\ref{eq:#4}.0 :a_node:inference_ruleseparate two vector components120000111270a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baSeparate two vector components in Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2} and Eq.~\ref{eq:#3}0 :a_node:inference_rulegeometry argument in 2D000011129512separate vector into two trigonometric ratiosa84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baSeparate vector in Eq.~\ref{eq:#2} into components related by angle $#1$; yields Eq.~\ref{eq:#3} and Eq.~\ref{eq:#4}.1 :a_node:inference_rule000011145711simplifya84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baSimplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.0 :a_node:inference_rule000011154311solve for Xa84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baSolve Eq.~\ref{eq:#2} for $#1$; yields Eq.~\ref{eq:#3}.1 :a_node:inference_rulerelated to raise both sides to powersquare root both sides120000111524a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baTake the square root of both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2} and Eq.~\ref{eq:#3}.0 :a_node:inference_rulesubstitute LHS of expr 1 into expr 2210000111556a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baSubstitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.0 :a_node:inference_rule000011135161substitute LHS of five expressions into expressiona84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baSubstitute LHS of Eq.~\ref{eq:#1} and LHS of Eq.~\ref{eq:#2} and LHS of Eq.~\ref{eq:#3} and LHS of Eq.~\ref{eq:#4} and LHS of Eq.~\ref{eq:#5} into Eq.~\ref{eq:#6}; yields Eq.~\ref{eq:#7}.0 :a_node:inference_rule000011179751substitute LHS of four expressions into expressiona84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baSubstitute LHS of Eq.~\ref{eq:#1} and LHS of Eq.~\ref{eq:#2} and LHS of Eq.~\ref{eq:#3} and LHS of Eq.~\ref{eq:#4} into Eq.~\ref{eq:#5}; yields Eq.~\ref{eq:#6}.0 :a_node:inference_rule000011138171substitute LHS of six expressions into expressiona84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baSubstitute LHS of Eq.~\ref{eq:#1} and LHS of Eq.~\ref{eq:#2} and LHS of Eq.~\ref{eq:#3} and LHS of Eq.~\ref{eq:#4} and LHS of Eq.~\ref{eq:#5} and LHS of Eq.~\ref{eq:#6} into Eq.~\ref{eq:#7}; yields Eq.~\ref{eq:#8}.0 :a_node:inference_rule000011124641substitute LHS of three expressions into expressiona84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baSubstitute LHS of Eq.~\ref{eq:#1} and LHS of Eq.~\ref{eq:#2} and LHS of Eq.~\ref{eq:#3} into Eq.~\ref{eq:#4}; yields Eq.~\ref{eq:#5}.0 :a_node:inference_rule000011173231substitute LHS of two expressions into expressiona84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baSubstitute LHS of Eq.~\ref{eq:#1} and LHS of Eq.~\ref{eq:#2} into Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.0 :a_node:inference_rulesubstitute RHS of expr 1 into expr 2210000111634a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baSubstitute RHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.0 :a_node:inference_rulesubtract X from both sides110000111282a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baSubtract $#1$ from both sides of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.1 :a_node:inference_rulecomplement of 'add expr X to expr Y'subtract expr 1 from expr 2210000111222a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baSubtract Eq.~\ref{eq:#1} from Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#2}.0 :a_node:inference_rule000011137311sum exponentsa84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baSum exponents on LHS and RHS of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.0 :a_node:inference_rulesum exponents LHS110000111566a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baSum exponents on LHS of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.0 :a_node:inference_rulesum exponents RHS110000111409a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baSum exponents on RHS of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.0 :a_node:inference_ruleswap LHS with RHS110000111268a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baSwap LHS of Eq.~\ref{eq:#1} with RHS; yields Eq.~\ref{eq:#2}.0 :a_node:inference_rule0000111776take curl of both sides011a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baApply curl to both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.linear algebra :a_node:operation0002222764["real","vector","matrix","complex"]addition["\\addition[2]{ #1 + #2}"]a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba+2 :a_node:operation["\\cosine[1]{\\cos #1}"]00022229401["real"]cosinea84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\cos :a_node:operation["\\times[2]{ #1 \\times #2}"]00022222782["vector"]cross producta84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\times :a_node:operation["\\curl[1]{ \\nabla \\times #1}"]0002222896\vec{\nabla} \timescurla84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba1["real"] :a_node:operation["\\integralDefinite[4]{ \\int_{#1}^{#2} #3 #4}"]00022222804["real","vector","matrix","complex"]definite integrala84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\int :a_node:operation0002222591divergence\vec{\nabla} \cdot["\\nabla[1]{ \\nabla \\cdot #1}"]a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba1["real"] :a_node:operation0002222891["real","vector","matrix","complex"]division["\\divisionOneLine[2]{ #1 / #2}","\\divisionFrac[2]{ \\frac{#1}{#2}"]a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba/2 :a_node:operation["\\dotproduct[2]{ #1 \\dot #2}"]00022224552["vector"]dot producta84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\circ :a_node:operation["\\elementwiseAddition[2]{ #1 + #2}"]0002222435["vector","matrix"]element-wise additiona84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba+2 :a_node:operation0002222329["list"]function["\\function[1]{ f #1}"]a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baf1 :a_node:operation0002222105gradient\vec{\nabla}["\\nabla[1]{ \\nabla #1}"]a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba1["real"] :a_node:operation0002222657indefinite intergral2["real","vector","matrix","complex"]["\\integralIndefinite[2]{ \\int #1 #2}"]a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\int :a_node:operation0002222829["real","vector","matrix","complex"]multiplication["\\multiplication[2]{ #1 #2}"]a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba*2 :a_node:operation00022221441["real"]sine["\\sine[1]{ \\sin #1}"]a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\sin :a_node:operation0002222439spatial vector differential\vec{\nabla}["\\spatialVectorDifferential[1]{ \\vec{ \\nabla} #1}"]a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba2["real"] :a_node:operation0002222427["real","vector","matrix","complex"]subtraction["\\subtraction[2]{ #1 - #2}"]a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba-2 :a_node:operation["\\summation[4]{ \\sum_{#1}^{#2} #3}"]00022223484["real","vector","matrix","complex"]summationa84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\sum :a_node:operation0001094924multiply two terms2multiplication2026-02-03_20-12-30-058202a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\cdot :a_node:operation00025327891tangent2026-02-03_20-13-19-805244a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://en.wikipedia.org/wiki/Trigonometric_functions\tan :a_node:operation0006521282natural logarithm2026-02-03_20-15-32-860258a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\ln1 :a_node:operation00099937661minimum2026-02-03_20-16-37-928305a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\min :a_node:operation00049581311maximum2026-02-03_20-16-53-150054a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\max :a_node:operation00078437441square root2026-02-03_20-20-22-114327a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://en.wikipedia.org/wiki/Square_root\sqrt{} :a_node:operation00012992881determinant2026-02-03_20-21-57-077304a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://en.wikipedia.org/wiki/Determinant\det :a_node:operation00028418811hyperbolic sine2026-02-03_20-23-06-671617a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://en.wikipedia.org/wiki/Hyperbolic_function\sinh :a_node:operation00058188521hyperbolic cosine2026-02-03_20-23-38-602387a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://en.wikipedia.org/wiki/Hyperbolic_function\cosh :a_node:operation00043527521hyperbolic tangent2026-02-03_20-24-13-692494a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://en.wikipedia.org/wiki/Hyperbolic_function\tanh :a_node:operation00020901031hyperbolic cotangent2026-02-03_20-24-52-089811a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://en.wikipedia.org/wiki/Hyperbolic_function\coth :a_node:operation00055607431arc sine2026-02-03_20-26-07-143879a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://en.wikipedia.org/wiki/Inverse_trigonometric_functions\arcsin :a_node:operation00061518181arc cosine2026-02-03_20-26-54-956683a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://en.wikipedia.org/wiki/Inverse_trigonometric_functions\arccos :a_node:operation00011302111arc tangent2026-02-03_20-27-23-678769a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://en.wikipedia.org/wiki/Inverse_trigonometric_functions\arctan :a_node:operation00033826391cotangent2026-02-03_20-28-22-739327a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://en.wikipedia.org/wiki/Trigonometric_functions\cot :a_node:operation00067399651secant2026-02-03_20-28-53-229154a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://en.wikipedia.org/wiki/Trigonometric_functions\sec :a_node:operation00097808581cosecant2026-02-03_20-29-18-464628a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://en.wikipedia.org/wiki/Trigonometric_functions\csc :a_node:operationdivision00044262382divide2026-02-03_20-29-56-886659a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\div :a_node:operation0002495151factorial2026-02-03_20-30-31-862476a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://en.wikipedia.org/wiki/Factorial!1 :a_node:operation00089400072indefinite surface integral2026-02-03_20-31-16-620576a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\oint :a_node:operation0003403771default base 102log, arbitrary base2026-02-03_20-31-51-328478a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\log :a_node:operation00086326871exponential function2026-02-03_20-32-36-874402a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bahttps://en.wikipedia.org/wiki/Exponential_function\exp :a_node:operation00035308442modulo2026-02-03_20-33-22-143156a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\mod{} :a_node:expression3829492824\cos(x)cos(Symbol('pdg0001464'))Add(Mul(Rational(1, 2), exp(Mul(Symbol('pdg0001464'), Symbol('pdg0004621')))), Mul(Rational(1, 2), exp(Mul(Integer(-1), Symbol('pdg0001464'), Symbol('pdg0004621')))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)= :a_node:expression-G \frac{m_{\rm Earth} m}{r_{\rm Earth}}3846041519Symbol('pdg0006431')PE_{\rm Earth\ surface}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Integer(-1), Pow(Symbol('pdg0003236'), Integer(-1)), Symbol('pdg0005156'), Symbol('pdg0005458'), Symbol('pdg0006277')) :a_node:expression\frac{4}{\left(\exp(x)+\exp(-x)\right)^2}3868998312=Pow(sech(Symbol('pdg0001464')), Integer(2))Mul(Integer(4), Pow(Add(exp(Symbol('pdg0001464')), exp(Mul(Integer(-1), Symbol('pdg0001464')))), Integer(-2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba{\rm sech}^2\ x :a_node:expression3896798826G \frac{m_1 m_2}{r^2}=Mul(Pow(Symbol('pdg0002321'), Integer(2)), Symbol('pdg0002798'), Symbol('pdg0004851'))Mul(Pow(Symbol('pdg0002530'), Integer(-2)), Symbol('pdg0004851'), Symbol('pdg0005022'), Symbol('pdg0006277'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bam_2 d_2 \omega^2 :a_node:expression\frac{4 \pi^2 r^2}{T_{\rm orbit}^2}3906710072Mul(Integer(4), Pow(Symbol('pdg0002530'), Integer(2)), Pow(Symbol('pdg0003141'), Integer(2)), Pow(Symbol('pdg0008762'), Integer(-2)))G \frac{m_{\rm Earth}}{r}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Pow(Symbol('pdg0002530'), Integer(-1)), Symbol('pdg0005458'), Symbol('pdg0006277')) :a_node:expression392061679224\ {\rm hours}this applies for geostationary orbitsSymbol('pdg0005595')T_{\rm geostationary orbit}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba= :a_node:expression03924948349Symbol('pdg0007752')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baa_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle= :a_node:expression\sqrt{ \frac{m_e}{m} \frac{e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} }3935058307Mul(Rational(1, 8), Pow(Integer(2), Rational(1, 2)), Pow(Mul(Pow(Symbol('pdg0001054'), Integer(-2)), Pow(Symbol('pdg0001999'), Integer(4)), Symbol('pdg0002515'), Pow(Symbol('pdg0003141'), Integer(-2)), Pow(Symbol('pdg0007940'), Integer(-2)), Pow(Symbol('pdg0009863'), Integer(-1))), Rational(1, 2)))Symbol('pdg0002077')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav= :a_node:expression39428492942 i \sin(x)Mul(Integer(2), Symbol('pdg0004621'), sin(Symbol('pdg0001464')))\exp(i x)-\exp(-i x)a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Add(exp(Mul(Symbol('pdg0001464'), Symbol('pdg0004621'))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('pdg0001464'), Symbol('pdg0004621'))))) :a_node:expressiona_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle3943939590Symbol('pdg0002427')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax= :a_node:expression-\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}3947269979Symbol('pdg0001467')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\vec{ \nabla} \times \vec{ \nabla} \times \vec{E}= :a_node:expression3948571256\frac{-i}{\hbar}E \psi( \vec{r},t)Derivative(Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467')), Tuple(Symbol('pdg0001467'), Integer(1)))Mul(Integer(-1), Pow(Symbol('pdg0001054'), Integer(-1)), Symbol('pdg0004621'), Symbol('pdg0006238'), Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{\partial}{\partial t} \psi( \vec{r},t)= :a_node:expression\psi_0 \exp\left(i\left( \vec{k}\cdot\vec{r} - \omega t \right) \right)3948574224=Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467'))Mul(Symbol('pdg0008330'), Function('pdg0002718')(Function('pdg0004621')(Add(Mul(Integer(-1), Symbol('pdg0001467'), Symbol('pdg0002321')), Function('dot')(Symbol('pdg0005321'), Symbol('pdg0009472'))))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\psi( \vec{r},t) :a_node:expression\psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)3948574226=Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467'))Mul(Symbol('pdg0008330'), Function('pdg0002718')(Function('pdg0004621')(Add(Mul(Integer(-1), Symbol('pdg0001467'), Symbol('pdg0002321')), Mul(Pow(Symbol('pdg0001054'), Integer(-1)), Symbol('pdg0001134'), Symbol('pdg0009472'))))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\psi( \vec{r},t) :a_node:expression\psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)3948574228=Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467'))Mul(Symbol('pdg0008330'), Function('pdg0002718')(Function('pdg0004621')(Add(Mul(Pow(Symbol('pdg0001054'), Integer(-1)), Symbol('pdg0001134'), Symbol('pdg0009472')), Mul(Integer(-1), Pow(Symbol('pdg0001054'), Integer(-1)), Symbol('pdg0001467'), Symbol('pdg0006238'))))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\psi( \vec{r},t) :a_node:expression\psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)3948574230=Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467'))Mul(Symbol('pdg0008330'), Function('pdg0002718')(Mul(Pow(Symbol('pdg0001054'), Integer(-1)), Symbol('pdg0004621'), Add(Mul(Symbol('pdg0001134'), Symbol('pdg0009472')), Mul(Integer(-1), Symbol('pdg0001467'), Symbol('pdg0006238'))))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\psi( \vec{r},t) :a_node:expression3948574233\psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)Derivative(Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467')), Tuple(Symbol('pdg0001467'), Integer(1)))Mul(Symbol('pdg0008330'), Derivative(Function('pdg0002718')(Function('pdg0004621')(Add(Mul(Pow(Symbol('pdg0001054'), Integer(-1)), Symbol('pdg0001134'), Symbol('pdg0009472')), Mul(Integer(-1), Pow(Symbol('pdg0001054'), Integer(-1)), Symbol('pdg0001467'), Symbol('pdg0006238'))))), Tuple(Symbol('pdg0001467'), Integer(1))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{\partial}{\partial t} \psi( \vec{r},t)= :a_node:expression3951205425\vec{p}_{1}Symbol('pdg0006029')Symbol('pdg0005493')=a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\vec{p}_{\rm after} :a_node:expression4072200527G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}Mul(Pow(Symbol('pdg0002530'), Integer(-1)), Symbol('pdg0003569'), Pow(Symbol('pdg0004082'), Integer(2)))Mul(Pow(Symbol('pdg0002530'), Integer(-2)), Symbol('pdg0003569'), Symbol('pdg0005458'), Symbol('pdg0006277'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{m_{\rm satellite} v_{\rm satellite}^2}{r}= :a_node:expression|A|^24075539836=Mul(Symbol('pdg0004453'), conjugate(Symbol('pdg0004453')))Pow(Abs(Symbol('pdg0004453')), Integer(2))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baA A^* :a_node:expression4087145886Ohm's lawI RMul(Symbol('pdg0004501'), Symbol('pdg0006458'))https://en.wikipedia.org/wiki/Ohm%27s_lawSymbol('pdg0006599')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baV= :a_node:expression4107032818ESymbol('pdg0002241')Symbol('pdg0009838')=a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baE_{\rm Rydberg} :a_node:expression4128500715I_1 R_1Mul(Symbol('pdg0003978'), Symbol('pdg0008697'))Symbol('pdg0006599')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baV= :a_node:expression- \gamma^2 v t + \gamma v t'4139999399=Add(Mul(Integer(-1), Pow(Symbol('pdg0001790'), Integer(2)), Symbol('pdg0004037')), Symbol('pdg0004037'))Add(Mul(Integer(-1), Symbol('pdg0001357'), Symbol('pdg0001467'), Pow(Symbol('pdg0001790'), Integer(2))), Mul(Symbol('pdg0001357'), Symbol('pdg0001790'), Symbol('pdg0004989')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax - \gamma^2 x :a_node:expression\frac{h \omega}{2 \pi}4147472132Mul(Rational(1, 2), Symbol('pdg0002321'), Pow(Symbol('pdg0003141'), Integer(-1)), Symbol('pdg0004413'))Symbol('pdg0004931')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baE= :a_node:expression4158986868\frac{d\vec{v}}{dt}Symbol('pdg0001467')a_x \hat{x} + a_y \hat{y}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba= :a_node:expression4166155526\frac{2}{\exp(x)+\exp(-x)}sech(Symbol('pdg0001464'))=Mul(Integer(2), Pow(Add(exp(Symbol('pdg0001464')), exp(Mul(Integer(-1), Symbol('pdg0001464')))), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba{\rm sech}\ x :a_node:expression29.8 \frac{{\rm km}}{{\rm sec}}4180845508Symbol('pdg0007427')v_{\rm Earth\ orbit}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Float('29.800000000000001', precision=53) :a_node:expression4182362050|Z| \exp( i \theta )Z \in \mathbb{C}Mul(exp(Mul(Symbol('pdg0001575'), Symbol('pdg0004621'))), Abs(Symbol('pdg0003192')))Symbol('pdg0003192')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baZ= :a_node:expression4188580242\frac{r^3 4 \pi^2}{\left(m_1+\left(\frac{m_1}{d_2}d_1\right)\right)G}Pow(Symbol('pdg0009491'), Integer(2))Mul(Integer(4), Pow(Symbol('pdg0002530'), Integer(3)), Pow(Symbol('pdg0003141'), Integer(2)), Pow(Symbol('pdg0006277'), Integer(-1)), Pow(Add(Symbol('pdg0005022'), Mul(Pow(Symbol('pdg0002798'), Integer(-1)), Symbol('pdg0005022'), Symbol('pdg0007652'))), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baT^2= :a_node:expression4192519596|B| \exp(i \phi)Mul(exp(Mul(Symbol('pdg0004621'), Symbol('pdg0008586'))), Abs(Symbol('pdg0004698')))Symbol('pdg0004698')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baB= :a_node:expression4245712581\frac{2 \pi r}{t}Mul(Integer(2), Pow(Symbol('pdg0001467'), Integer(-1)), Symbol('pdg0002530'), Symbol('pdg0003141'))Symbol('pdg0001357')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav= :a_node:expression4267808354m \frac{v^2}{r}Symbol('pdg0002867')=Mul(Pow(Symbol('pdg0001357'), Integer(2)), Pow(Symbol('pdg0002530'), Integer(-1)), Symbol('pdg0005156'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF_{\rm gravity} :a_node:expression4268085801v_0 t_f \cos(\theta) + x_0=Add(Symbol('pdg0001572'), Symbol('pdg0001943'))Add(Symbol('pdg0001572'), Mul(Symbol('pdg0002467'), Symbol('pdg0005153'), cos(Symbol('pdg0001575'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax_0 + d :a_node:expression\frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}4270680309Mul(Rational(1, 2), Pow(Symbol('pdg0001467'), Integer(-1)), Symbol('pdg0005156'), Add(Mul(Integer(-1), Pow(Symbol('pdg0002473'), Integer(2))), Pow(Symbol('pdg0004770'), Integer(2))))\frac{KE_2 - KE_1}{t}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Pow(Symbol('pdg0001467'), Integer(-1)), Add(Symbol('pdg0001352'), Mul(Integer(-1), Symbol('pdg0001955')))) :a_node:expression42750045612 G \frac{m}{r_{\rm Schwarzschild}}Pow(Symbol('pdg0004567'), Integer(2))Mul(Integer(2), Pow(Symbol('pdg0004518'), Integer(-1)), Symbol('pdg0005156'), Symbol('pdg0006277'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bac^2= :a_node:expressionc^2 t^24287102261describes a spherical wavefront=Add(Pow(Symbol('pdg0004037'), Integer(2)), Pow(Symbol('pdg0005647'), Integer(2)), Pow(Symbol('pdg0006728'), Integer(2)))Mul(Pow(Symbol('pdg0001467'), Integer(2)), Pow(Symbol('pdg0004567'), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax^2 + y^2 + z^2 :a_node:expression4298359835\frac{1}{2}m v^2Mul(Rational(1, 2), Pow(Symbol('pdg0001357'), Integer(2)), Symbol('pdg0005156'))Symbol('pdg0004931')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baE= :a_node:expression4298359845\frac{1}{2m}m^2 v^2Mul(Rational(1, 2), Pow(Symbol('pdg0001357'), Integer(2)), Symbol('pdg0005156'))Symbol('pdg0004931')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baE= :a_node:expression4298359851\frac{p^2}{2m}Mul(Rational(1, 2), Pow(Symbol('pdg0001134'), Integer(2)), Pow(Symbol('pdg0005156'), Integer(-1)))Symbol('pdg0004931')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baE= :a_node:expression4301729661\frac{[A_{\rm adsorption}]}{\left( \frac{k_{\rm adsorption}}{k_{\rm desorption}} \right) p_A} + [A_{\rm adsorption}]Symbol('pdg0003037')=Add(Symbol('pdg0004940'), Mul(Symbol('pdg0004940'), Pow(Symbol('pdg0006850'), Integer(-1)), Symbol('pdg0008379'), Pow(Symbol('pdg0009046'), Integer(-1))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba[S_0] :a_node:expression4303372136KE_1 + PE_1Add(Symbol('pdg0001955'), Symbol('pdg0004093'))Symbol('pdg0005579')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baE_1= :a_node:expression4341171256\frac{p^2}{2 m} \psi( \vec{r},t)Mul(Symbol('pdg0001054'), Symbol('pdg0004621'), Derivative(Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467')), Tuple(Symbol('pdg0001467'), Integer(1))))Mul(Rational(1, 2), Pow(Symbol('pdg0001134'), Integer(2)), Pow(Symbol('pdg0005156'), Integer(-1)), Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bai \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)= :a_node:expression4348571256\frac{-i}{\hbar}\frac{p^2}{2 m} \psi( \vec{r},t)Derivative(Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467')), Tuple(Symbol('pdg0001467'), Integer(1)))Mul(Integer(-1), Rational(1, 2), Pow(Symbol('pdg0001054'), Integer(-1)), Pow(Symbol('pdg0001134'), Integer(2)), Symbol('pdg0004621'), Pow(Symbol('pdg0005156'), Integer(-1)), Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{\partial}{\partial t} \psi( \vec{r},t)= :a_node:expressiont_f4370074654Symbol('pdg0002467')Symbol('pdg0001467')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bat= :a_node:expression4393258808m r \omega^2Symbol('pdg0001687')=Mul(Pow(Symbol('pdg0002321'), Integer(2)), Symbol('pdg0002530'), Symbol('pdg0005156'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF_{\rm centripetal} :a_node:expression4393670960\frac{G m_1 m_2}{r}Symbol('pdg0009372')=Mul(Pow(Symbol('pdg0002530'), Integer(-1)), Symbol('pdg0004851'), Symbol('pdg0005022'), Symbol('pdg0006277'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baW_{\rm to\ system} :a_node:expression4394958389\frac{i}{\hbar} \vec{ \nabla}\cdot\left( \vec{p} \psi( \vec{r},t) \right)Symbol('nabla').dot(Symbol('nabla')( Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467'))))Mul(Mul(Pow(Symbol('pdg0001054'), Integer(-1)), Symbol('pdg0004621')), Symbol('nabla')*(Mul(Symbol('pdg0002046'), Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467')))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\vec{ \nabla}\cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right)= :a_node:expression-k x4428528271Hooke's lawhttps://en.wikipedia.org/wiki/Hooke%27s_lawSymbol('pdg0004183')=Mul(Integer(-1), Symbol('pdg0001356'), Symbol('pdg0004037'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF_{\rm spring} :a_node:expression4447113478G m_1 m_2 \int_{ r_{\rm Earth} }^{\infty} \frac{1}{x^2} dx=Integral(Integer(1), Tuple(Symbol('pdg0006789')))Mul(Symbol('pdg0004851'), Symbol('pdg0005022'), Symbol('pdg0006277'), Integral(Pow(Symbol('pdg0004037'), Integer(-2)), Tuple(Symbol('pdg0004037'), Symbol('pdg0003236'), Symbol('infty'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\int dW :a_node:expression\frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}4501377629Mul(sin(Symbol('pdg0004928')), Pow(cos(Symbol('pdg0004928')), Integer(-1)))tan(Symbol('pdg0004928'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\tan( \theta_{\rm Brewster} )= :a_node:expression|B| \exp(-i \phi)4504256452conjugate(Symbol('pdg0004698'))Mul(exp(Mul(Integer(-1), Symbol('pdg0004621'), Symbol('pdg0008586'))), Abs(Symbol('pdg0004698')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baB^*= :a_node:expression\sqrt{ \frac{K + (4/3) G}{\rho} }4560648264Pow(Mul(Pow(Symbol('pdg0003935'), Integer(-1)), Add(Symbol('pdg0001466'), Mul(Rational(4, 3), Symbol('pdg0003033')))), Rational(1, 2))Symbol('pdg0002077')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav= :a_node:expressionv t - a t^2 + \frac{1}{2} a t^24580545876Add(Mul(Symbol('pdg0001357'), Symbol('pdg0001467')), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('pdg0001467'), Integer(2)), Symbol('pdg0009140')))Symbol('pdg0001943')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bad= :a_node:expression4585828572\frac{1}{c^2}=Mul(Symbol('pdg0006197'), Symbol('pdg0007940'))Pow(Symbol('pdg0004567'), Integer(-2))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\epsilon_0 \mu_0 :a_node:expression4585932229\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)cos(Symbol('pdg0001464'))=Add(Mul(Rational(1, 2), exp(Mul(Symbol('pdg0001464'), Symbol('pdg0004621')))), Mul(Rational(1, 2), exp(Mul(Integer(-1), Symbol('pdg0001464'), Symbol('pdg0004621')))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\cos(x) :a_node:expression\frac{2 \pi r_{\rm Earth\ orbit}}{3.16\ 10^7 {\rm seconds}}4593428198Symbol('pdg0007427')v_{\rm Earth\ orbit}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Float('0.63291139240506322', precision=53), Symbol('pdg0003141'), Symbol('pdg0006081')) :a_node:expression4598294821(\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2=exp(Mul(Integer(2), Symbol('pdg0001464'), Symbol('pdg0004621')))Add(Mul(Integer(2), Symbol('pdg0004621'), sin(Symbol('pdg0001464')), cos(Symbol('pdg0001464'))), Mul(Integer(-1), Pow(sin(Symbol('pdg0001464')), Integer(2))), Pow(cos(Symbol('pdg0001464')), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\exp(2 i x) :a_node:expression4627284246\frac{m_{\rm satellite} v_{\rm satellite}^2}{r}Symbol('pdg0001687')=Mul(Pow(Symbol('pdg0002530'), Integer(-1)), Symbol('pdg0003569'), Pow(Symbol('pdg0004082'), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF_{\rm centripetal} :a_node:expression4638429483(\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))=exp(Mul(Integer(2), Symbol('pdg0001464'), Symbol('pdg0004621')))Pow(Add(Mul(Symbol('pdg0004621'), sin(Symbol('pdg0001464'))), cos(Symbol('pdg0001464'))), Integer(2))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\exp(2 i x) :a_node:expression4648451961(v_2 + v_1)(v_2 - v_1)=Add(Mul(Integer(-1), Pow(Symbol('pdg0002473'), Integer(2))), Pow(Symbol('pdg0004770'), Integer(2)))Mul(Add(Mul(Integer(-1), Symbol('pdg0002473')), Symbol('pdg0004770')), Add(Symbol('pdg0002473'), Symbol('pdg0004770')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_2^2 - v_1^2 :a_node:expression4662369843\gamma (x - v t)Mul(Symbol('pdg0001790'), Add(Mul(Integer(-1), Symbol('pdg0001357'), Symbol('pdg0001467')), Symbol('pdg0001464')))Symbol('pdg0005456')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax'= :a_node:expression4669290568-F x_1Mul(Integer(-1), Symbol('pdg0003852'), Symbol('pdg0004202'))Symbol('pdg0004093')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baPE_1= :a_node:expression4689334676\frac{K_{\rm equilibrium}\ p_A}{1+K_{\rm equilibrium}\ p_A}Symbol('pdg0001791')=Mul(Symbol('pdg0004933'), Symbol('pdg0009046'), Pow(Add(Mul(Symbol('pdg0004933'), Symbol('pdg0009046')), Integer(1)), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\theta_A :a_node:expression47426448282 \cos(x)Mul(Integer(2), cos(Symbol('pdg0001464')))\exp(i x)+\exp(-i x)a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Add(exp(Mul(Symbol('pdg0001464'), Symbol('pdg0004621'))), exp(Mul(Integer(-1), Symbol('pdg0001464'), Symbol('pdg0004621')))) :a_node:expression4748157455v - v_0Mul(Symbol('pdg0001467'), Symbol('pdg0009140'))Add(Symbol('pdg0001357'), Mul(Integer(-1), Symbol('pdg0005153')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baa t= :a_node:expression\frac{2 v_0 \sin(\theta)}{g}4778077984Mul(Integer(2), Pow(Symbol('pdg0001649'), Integer(-1)), Symbol('pdg0005153'), sin(Symbol('pdg0001575')))Symbol('pdg0002467')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bat_f= :a_node:expressionm v a4784793837Mul(Symbol('pdg0001357'), Symbol('pdg0005156'), Symbol('pdg0009140'))\frac{KE_2 - KE_1}{t}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Pow(Symbol('pdg0001467'), Integer(-1)), Add(Symbol('pdg0001352'), Mul(Integer(-1), Symbol('pdg0001955')))) :a_node:expression4798787814vSymbol('pdg0001357')=Add(Mul(Symbol('pdg0001467'), Symbol('pdg0009140')), Symbol('pdg0005153'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baa t + v_0 :a_node:expression4800170179m g_{\rm Earth}Mul(Symbol('pdg0005156'), Symbol('pdg0007557'))Symbol('pdg0004202')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF= :a_node:expression4805233006\frac{1}{2}\left(\exp(x) - \exp(-x) \right)=Mul(Symbol('pdg0004621'), sin(Mul(Symbol('pdg0001464'), Symbol('pdg0004621'))))Add(Mul(Rational(1, 2), exp(Symbol('pdg0001464'))), Mul(Integer(-1), Rational(1, 2), exp(Mul(Integer(-1), Symbol('pdg0001464')))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bai \sin(i x) :a_node:expression\frac{1}{2} m v_2^2 - \frac{1}{2} m v_1^24811121942Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('pdg0002473'), Integer(2)), Symbol('pdg0005156')), Mul(Rational(1, 2), Pow(Symbol('pdg0004770'), Integer(2)), Symbol('pdg0005156')))Symbol('pdg0006789')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baW= :a_node:expressionF_{\rm centripetal}4820320578Symbol('pdg0001687')Symbol('pdg0002867')=a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF_{\rm gravity} :a_node:expression48274929111 - (\sin(x))^2Add(Integer(1), Mul(Integer(-1), Pow(sin(Symbol('pdg0001464')), Integer(2))))\cos(2 x)+(\sin(x))^2a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Add(Pow(sin(Symbol('pdg0001464')), Integer(2)), cos(Mul(Integer(2), Symbol('pdg0001464')))) :a_node:expression4830221561\frac{4+\left(\exp(2x)-1-1+\exp(-2x)\right)}{\left(\exp(x)+\exp(-x)\right)^2}Add(Pow(tanh(Symbol('pdg0001464')), Integer(2)), Pow(sech(Symbol('pdg0001464')), Integer(2)))Mul(Pow(Add(exp(Symbol('pdg0001464')), exp(Mul(Integer(-1), Symbol('pdg0001464')))), Integer(-2)), Add(exp(Mul(Integer(2), Symbol('pdg0001464'))), Integer(2), exp(Mul(Integer(-1), Integer(2), Symbol('pdg0001464')))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba{\rm sech}^2\ x + \tanh^2(x)= :a_node:expression4838429483\cos(2 x)+i \sin(2 x)=exp(Mul(Integer(2), Symbol('pdg0001464'), Symbol('pdg0004621')))Add(Mul(Symbol('pdg0004621'), sin(Mul(Integer(2), Symbol('pdg0001464')))), cos(Mul(Integer(2), Symbol('pdg0001464'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\exp(2 i x) :a_node:expression4843995999\sin(x)sin(Symbol('pdg0001464'))Mul(Rational(1, 2), Pow(Symbol('pdg0004621'), Integer(-1)), Add(exp(Mul(Symbol('pdg0001464'), Symbol('pdg0004621'))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('pdg0001464'), Symbol('pdg0004621'))))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{1}{2 i}\left(\exp(i x)-\exp(-i x) \right)= :a_node:expression\int \psi(x)\psi(x)^* dx4857472413Symbol('pdg0009199')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba1= :a_node:expression\frac{W}{2}4857475848=Pow(Symbol('pdg0009139'), Integer(-2))Mul(Rational(1, 2), Symbol('pdg0002523'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{1}{a^2} :a_node:expression4858693811r^3Mul(Rational(1, 4), Pow(Symbol('pdg0003141'), Integer(-2)), Symbol('pdg0005458'), Symbol('pdg0006277'), Pow(Symbol('pdg0008762'), Integer(2)))Pow(Symbol('pdg0002530'), Integer(3))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2}= :a_node:expression\frac{V}{R_1} + \frac{V}{R_2}4866160902Add(Mul(Symbol('pdg0006599'), Pow(Symbol('pdg0008697'), Integer(-1))), Mul(Pow(Symbol('pdg0003461'), Integer(-1)), Symbol('pdg0006599')))\frac{V}{R_{\rm total}}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Pow(Symbol('pdg0001908'), Integer(-1)), Symbol('pdg0006599')) :a_node:expression\frac{\sinh(x)}{\cosh(x)}4872163189tanh(Symbol('pdg0001464'))=Mul(sinh(Symbol('pdg0001464')), Pow(cosh(Symbol('pdg0001464')), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\tanh(x) :a_node:expression-F v4872970974Mul(Integer(-1), Symbol('pdg0001357'), Symbol('pdg0004202'))\frac{PE_2 - PE_1}{t}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Pow(Symbol('pdg0001467'), Integer(-1)), Add(Mul(Integer(-1), Symbol('pdg0004093')), Symbol('pdg0008849'))) :a_node:expression4878728014\frac{1}{2i}\left(\exp(-x) - \exp(x) \right)=sin(Mul(Symbol('pdg0001464'), Symbol('pdg0004621')))Mul(Rational(1, 2), Pow(Symbol('pdg0004621'), Integer(-1)), Add(Mul(Integer(-1), exp(Symbol('pdg0001464'))), exp(Mul(Integer(-1), Symbol('pdg0001464')))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\sin(i x) :a_node:expression4923339482\log(y)Mul(Symbol('pdg0001464'), Symbol('pdg0004621'))log(Symbol('pdg0001452'), Integer(10))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bai x= :a_node:expression4928007622\frac{1}{2} m v_1^2Mul(Rational(1, 2), Pow(Symbol('pdg0002473'), Integer(2)), Symbol('pdg0005156'))Symbol('pdg0001955')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baKE_1= :a_node:expression4928239482i x=log(Symbol('pdg0001452'), Integer(10))Mul(Symbol('pdg0001464'), Symbol('pdg0004621'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\log(y) :a_node:expression4938429482\cos(x)+i \sin(-x)=exp(Mul(Integer(-1), Symbol('pdg0001464'), Symbol('pdg0004621')))Add(Mul(Symbol('pdg0004621'), sin(Mul(Integer(-1), Symbol('pdg0001464')))), cos(Symbol('pdg0001464')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\exp(-i x) :a_node:expression4938429483\cos(x)+i \sin(x)=exp(Mul(Symbol('pdg0001464'), Symbol('pdg0004621')))Add(Mul(Symbol('pdg0004621'), sin(Symbol('pdg0001464'))), cos(Symbol('pdg0001464')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\exp(i x) :a_node:expression4938429484\cos(x)-i \sin(x)=exp(Mul(Integer(-1), Symbol('pdg0001464'), Symbol('pdg0004621')))Add(Mul(Integer(-1), Symbol('pdg0004621'), sin(Symbol('pdg0001464'))), cos(Symbol('pdg0001464')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\exp(-i x) :a_node:expression4939880586I R_{\rm total}Symbol('pdg0004691')=Mul(Symbol('pdg0001908'), Symbol('pdg0004501'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baV_{\rm total} :a_node:expression4943571230\frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)Symbol('nabla')*(Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467')))Mul(Mul(Pow(Symbol('pdg0001054'), Integer(-1)), Symbol('pdg0004621')), Mul(Symbol('pdg0002046'), Mul(Symbol('pdg0008330'), exp(Mul(Mul(Pow(Symbol('pdg0001054'), Integer(-1)), Symbol('pdg0004621')), Add(Mul(Integer(-1), Symbol('pdg0006238'), Symbol('pdg0001467')), Mul(Symbol('pdg0002046'), Symbol('pdg0009472'))))))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\vec{ \nabla} \psi( \vec{r},t)= :a_node:expression4947831649W_{\rm to\ system}Symbol('pdg0009372')Mul(Rational(1, 2), Symbol('pdg0005022'), Pow(Symbol('pdg0008909'), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{1}{2} m_1 v_{\rm final}^2= :a_node:expressionv^24948763856=Add(Mul(Integer(2), Symbol('pdg0001943'), Symbol('pdg0009140')), Pow(Symbol('pdg0005153'), Integer(2)))Pow(Symbol('pdg0001357'), Integer(2))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba2 a d + v_0^2 :a_node:expression4948934890\langle a \rangle^*Symbol('pdg0004065')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\langle \psi| \hat{A} |\psi \rangle= :a_node:expression\langle x^2 \rangle-\langle x \rangle^24949359835Symbol('pdg0001464')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2= :a_node:expression4968680693\frac{ \sin( x )}{\cos( x )}tan(Symbol('pdg0001464'))=Mul(sin(Symbol('pdg0001464')), Pow(cos(Symbol('pdg0001464')), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\tan( x ) :a_node:expression4985825552https://physicsderivationgraph.blogspot.com/2020/09/representing-laplace-operator-nabla-in.htmli \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E( \vec{r})\exp(i \omega t)Mul(Pow(Symbol('nabla'), Integer(2)), Function('pdg0006238')(Symbol('pdg0009472')), exp(Mul(Symbol('pdg0001467'), Symbol('pdg0002321'), Symbol('pdg0004621'))))Mul(Symbol('pdg0002321'), Symbol('pdg0004621'), Symbol('pdg0006197'), Symbol('pdg0007940'), Derivative(Mul(Function('pdg0006238')(Symbol('pdg0009472')), exp(Mul(Symbol('pdg0001467'), Symbol('pdg0002321'), Symbol('pdg0004621')))), Tuple(Symbol('pdg0001467'), Integer(1))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\nabla^2 E( \vec{r})\exp(i \omega t)= :a_node:expression5002539602C_V dT + \pi_T dVSymbol('dU')Add(Mul(Symbol('dT'), Symbol('pdg0006682')), Mul(Symbol('dV'), Symbol('pdg0005480')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138badU= :a_node:expression5085809757\frac{[A_{\rm adsorption}]}{p_A [S]}Mul(Symbol('pdg0006850'), Pow(Symbol('pdg0008379'), Integer(-1)))Mul(Symbol('pdg0004940'), Pow(Symbol('pdg0009046'), Integer(-1)), Pow(Symbol('pdg0009067'), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{k_{\rm adsorption}}{k_{\rm desorption}}= :a_node:expression|A|^2 + B B^* + A B^* + B A^*5125940051Add(Mul(Symbol('pdg0004453'), conjugate(Symbol('pdg0004698'))), Mul(Symbol('pdg0004698'), conjugate(Symbol('pdg0004453'))), Mul(Symbol('pdg0004698'), conjugate(Symbol('pdg0004698'))), Pow(Abs(Symbol('pdg0004453')), Integer(2)))Symbol('pdg0007882')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baI= :a_node:expressionm_2 d_25128670694=Mul(Symbol('pdg0005022'), Symbol('pdg0007652'))Mul(Symbol('pdg0002798'), Symbol('pdg0004851'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bam_1 d_1 :a_node:expression5136652623KE + PEAdd(Symbol('pdg0004929'), Symbol('pdg0004930'))mechanical energy is the sum of the potential plus kinetic energiesSymbol('pdg0004931')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baE= :a_node:expressionv_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)5144263777Symbol('pdg0001357')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav^2= :a_node:expression\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t5148266645Symbol('pdg0001790')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bat'= :a_node:expression5177311762\frac{2 \pi r}{T}Mul(Integer(2), Symbol('pdg0002530'), Symbol('pdg0003141'), Pow(Symbol('pdg0008762'), Integer(-1)))Symbol('pdg0001357')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav= :a_node:expression5323719091\frac{1}{2i} \left( \exp(-x) - \exp(x) \right)=Mul(Symbol('pdg0004621'), sinh(Symbol('pdg0001464')))Mul(Rational(1, 2), Pow(Symbol('pdg0004621'), Integer(-1)), Add(Mul(Integer(-1), exp(Symbol('pdg0001464'))), exp(Mul(Integer(-1), Symbol('pdg0001464')))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bai \sinh x :a_node:expressionm a5345738321Newton's second law of motionMul(Symbol('pdg0005156'), Symbol('pdg0009140'))https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion#Newton's_second_lawSymbol('pdg0004202')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF= :a_node:expression5349669879\frac{ \exp(x)-\exp(-x)}{\exp(x)+\exp(-x)}tanh(Symbol('pdg0001464'))=Mul(Add(exp(Symbol('pdg0001464')), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('pdg0001464'))))), Pow(Add(exp(Symbol('pdg0001464')), exp(Mul(Integer(-1), Symbol('pdg0001464')))), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\tanh(x) :a_node:expressionv_x \hat{x} + v_y \hat{y}5349866551Symbol('pdg0006373')=Add(Mul(Symbol('pdg0001700'), Symbol('pdg0009107')), Mul(Symbol('pdg0005505'), Symbol('pdg0008339')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\vec{v} :a_node:expression5353282496\frac{v_0^2}{g}Mul(Pow(Symbol('pdg0001649'), Integer(-1)), Pow(Symbol('pdg0005153'), Integer(2)))Symbol('pdg0001943')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bad= :a_node:expressiont_f5373931751Symbol('pdg0002467')Symbol('pdg0001467')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bat= :a_node:expression- \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_05379546684Add(Symbol('pdg0001469'), Mul(Integer(-1), Rational(1, 2), Symbol('pdg0001649'), Pow(Symbol('pdg0002467'), Integer(2))), Mul(Symbol('pdg0002467'), Symbol('pdg0005153'), sin(Symbol('pdg0001575'))))Symbol('pdg0007092')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bay_f= :a_node:expression5404822208\sqrt{2 G \frac{m}{r}}escape velocitySymbol('pdg0008656')=Mul(Pow(Integer(2), Rational(1, 2)), Pow(Mul(Pow(Symbol('pdg0002530'), Integer(-1)), Symbol('pdg0005156'), Symbol('pdg0006277')), Rational(1, 2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_{\rm escape} :a_node:expressionA \cos(\omega t)5415824175Function('x')(Symbol('pdg0001467'))Mul(Symbol('pdg0009885'), cos(Mul(Symbol('pdg0001467'), Symbol('pdg0002321'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax(t)= :a_node:expression5426308937\frac{d}{t}Mul(Pow(Symbol('pdg0001467'), Integer(-1)), Symbol('pdg0001943'))Symbol('pdg0001357')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav= :a_node:expressionv_0 t \cos(\theta) + x_05438722682Add(Mul(Symbol('pdg0001467'), Symbol('pdg0005153'), cos(Symbol('pdg0001575'))), Symbol('pdg0001572'))Symbol('pdg0004037')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax= :a_node:expression5514556106(KE_2 - KE_1) + (PE_2 - PE_1)=Add(Symbol('pdg0004550'), Mul(Integer(-1), Symbol('pdg0005579')))Add(Symbol('pdg0001352'), Mul(Integer(-1), Symbol('pdg0001955')), Mul(Integer(-1), Symbol('pdg0004093')), Symbol('pdg0008849'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baE_2 - E_1 :a_node:expression5530148480\vec{p}_{electron}Symbol('pdg0004299')\vec{p}_{1}-\vec{p}_{2}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Add(Mul(Integer(-1), Symbol('pdg0002097')), Symbol('pdg0006029')) :a_node:expression5542528160F \int_0^x dx=Integral(Integer(1), Tuple(Symbol('pdg0006789')))Mul(Symbol('pdg0004202'), Integral(Integer(1), Tuple(Symbol('pdg0004037'), Integer(0), Symbol('pdg0004037'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\int dW :a_node:expression5563580265G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}Symbol('pdg0002867')=Mul(Pow(Symbol('pdg0002530'), Integer(-2)), Symbol('pdg0003569'), Symbol('pdg0005458'), Symbol('pdg0006277'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF_{\rm gravity} :a_node:expression5586102077d_1 + d_2Add(Symbol('pdg0002798'), Symbol('pdg0007652'))Symbol('pdg0002530')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bar= :a_node:expression5596822289-G m_1 m_2 \left(\left.\frac{-1}{x}\right|^r_{\infty}\right)Symbol('pdg0009372')=Mul(Integer(-1), Symbol('pdg0004851'), Symbol('pdg0005022'), Symbol('pdg0006277'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baW_{\rm to\ system} :a_node:expression\frac{1}{2 a} (v^2 - v_0^2)5611024898Mul(Rational(1, 2), Pow(Symbol('pdg0009140'), Integer(-1)), Add(Pow(Symbol('pdg0001357'), Integer(2)), Mul(Integer(-1), Pow(Symbol('pdg0005153'), Integer(2)))))Symbol('pdg0001943')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bad= :a_node:expression5634116660definition of internal pressure at constant temperature\left(\frac{\partial U}{\partial V}\right)_TSymbol('pdg0005480')=Derivative(Symbol('pdg0005786'), Tuple(Symbol('pdg0007586'), Integer(1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\pi_T :a_node:expression5646314683A m_pMul(Symbol('pdg0003285'), Symbol('pdg0005916'))Symbol('pdg0009863')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bam= :a_node:expression5658865948\frac{r^3 4 \pi^2}{(m_1+m_2)G}Pow(Symbol('pdg0009491'), Integer(2))Mul(Integer(4), Pow(Symbol('pdg0002530'), Integer(3)), Pow(Symbol('pdg0003141'), Integer(2)), Pow(Symbol('pdg0006277'), Integer(-1)), Pow(Add(Symbol('pdg0004851'), Symbol('pdg0005022')), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baT^2= :a_node:expression5693047217-\sqrt{\frac{2 G m_2}{r}}Symbol('pdg0008909')=Mul(Integer(-1), Pow(Integer(2), Rational(1, 2)), Pow(Mul(Pow(Symbol('pdg0002530'), Integer(-1)), Symbol('pdg0004851'), Symbol('pdg0006277')), Rational(1, 2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_{\rm final} :a_node:expression5727578862-k^2 \psi(x)Symbol('pdg0009199')\frac{d^2}{dx^2} \psi(x)a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba= :a_node:expression2022-03-25 BHP: Conversion between Latex and Sympy is incompleteG m_1 m_2 \left( \frac{1}{x} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)5732331610Symbol('pdg0006277')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baW= :a_node:expression5733146966\frac{1}{2} m \left(v_2^2 - v_1^2\right)=Add(Symbol('pdg0001352'), Mul(Integer(-1), Symbol('pdg0001955')))Mul(Rational(1, 2), Symbol('pdg0005156'), Add(Mul(Integer(-1), Pow(Symbol('pdg0002473'), Integer(2))), Pow(Symbol('pdg0004770'), Integer(2))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baKE_2 - KE_1 :a_node:expression\frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)5733721198Mul(Rational(1, 2), Pow(Symbol('pdg0009140'), Integer(-1)), Add(Symbol('pdg0001357'), Mul(Integer(-1), Symbol('pdg0005153'))), Add(Symbol('pdg0001357'), Symbol('pdg0005153')))Symbol('pdg0001943')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bad= :a_node:expression5763749235v^2 \gamma^2=Add(Mul(Pow(Symbol('pdg0001790'), Integer(2)), Pow(Symbol('pdg0004567'), Integer(2))), Mul(Integer(-1), Pow(Symbol('pdg0004567'), Integer(2))))Mul(Pow(Symbol('pdg0001357'), Integer(2)), Pow(Symbol('pdg0001790'), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba-c^2 + c^2 \gamma^2 :a_node:expression5779256336KE_{\rm final} - KE_{\rm initial}Symbol('pdg0006191')=Add(Mul(Integer(-1), Symbol('pdg0004121')), Symbol('pdg0005340'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baW_{\rm by\ system} :a_node:expression5781981178https://en.wikipedia.org/wiki/Difference_of_two_squares(x+y)(x-y)difference of squares=Add(Mul(Integer(-1), Pow(Symbol('pdg0001452'), Integer(2))), Pow(Symbol('pdg0001464'), Integer(2)))Mul(Add(Mul(Integer(-1), Symbol('pdg0001452')), Symbol('pdg0001464')), Add(Symbol('pdg0001452'), Symbol('pdg0001464')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax^2 - y^2 :a_node:expression5789289057equation 4 in the PDF\alpha c \sqrt{ \frac{m_e}{2 m} }Mul(Rational(1, 2), Pow(Integer(2), Rational(1, 2)), Symbol('pdg0001370'), Symbol('pdg0004567'), Pow(Mul(Symbol('pdg0002515'), Pow(Symbol('pdg0009863'), Integer(-1))), Rational(1, 2)))Symbol('pdg0002077')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav= :a_node:expression58329842911Integer(1)(\sin(x))^2 + (\cos(x))^2a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Add(Pow(sin(Symbol('pdg0001464')), Integer(2)), Pow(cos(Symbol('pdg0001464')), Integer(2))) :a_node:expression5838268428\frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar}=Mul(Symbol('pdg0001370'), Symbol('pdg0004567'))Mul(Rational(1, 4), Pow(Symbol('pdg0001054'), Integer(-1)), Pow(Symbol('pdg0001999'), Integer(2)), Pow(Symbol('pdg0003141'), Integer(-1)), Pow(Symbol('pdg0007940'), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\alpha c :a_node:expression5846639423\sqrt{\frac{2 G m_2}{r}}Symbol('pdg0008909')=Mul(Pow(Integer(2), Rational(1, 2)), Pow(Mul(Pow(Symbol('pdg0002530'), Integer(-1)), Symbol('pdg0004851'), Symbol('pdg0006277')), Rational(1, 2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_{\rm final} :a_node:expressionKE_{\rm final}5850144586Symbol('pdg0005340')Symbol('pdg0006191')=a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baW_{\rm by\ system} :a_node:expression5857434758a x=Integral(Symbol('pdg0009139'), Tuple(Symbol('pdg0001464')))Mul(Symbol('pdg0001464'), Symbol('pdg0009139'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\int a dx :a_node:expression58666294291Add(Pow(tanh(Symbol('pdg0001464')), Integer(2)), Pow(sech(Symbol('pdg0001464')), Integer(2)))Integer(1)a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba{\rm sech}^2\ x + \tanh^2(x)= :a_node:expression5868688585https://physicsderivationgraph.blogspot.com/2020/09/representing-laplace-operator-nabla-in.html\frac{p^2}{2m} \psi( \vec{r},t)Mul(Integer(-1), Rational(1, 2), Pow(Symbol('nabla'), Integer(2)), Pow(Symbol('pdg0001054'), Integer(2)), Pow(Symbol('pdg0005156'), Integer(-1)), Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467')))Mul(Rational(1, 2), Pow(Symbol('pdg0001134'), Integer(2)), Pow(Symbol('pdg0005156'), Integer(-1)), Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right)= :a_node:expression5900595848\frac{\omega}{v}Mul(Pow(Symbol('pdg0001357'), Integer(-1)), Symbol('pdg0002321'))Symbol('pdg0005321')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bak= :a_node:expression5902985919Newton's law of universal gravitationG \frac{m_1 m_2}{x^2} \hat{x}Symbol('pdg0004202')=Mul(Pow(Symbol('pdg0004037'), Integer(-1)), Symbol('pdg0004851'), Symbol('pdg0005022'), Symbol('pdg0006277'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\vec{F} :a_node:expression(x + (b/(2 a)))^25928285821Add(Pow(Symbol('pdg0001464'), Integer(2)), Mul(Symbol('pdg0001464'), Symbol('pdg0001939'), Pow(Symbol('pdg0009139'), Integer(-1))), Mul(Rational(1, 4), Pow(Symbol('pdg0001939'), Integer(2)), Pow(Symbol('pdg0009139'), Integer(-2))))Pow(Add(Symbol('pdg0001464'), Mul(Rational(1, 2), Symbol('pdg0001939'), Pow(Symbol('pdg0009139'), Integer(-1)))), Integer(2))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax^2 + 2 x (b/(2 a)) + (b/(2 a))^2= :a_node:expression5928292841-c/a + (b/(2 a))^2Add(Mul(Rational(1, 4), Pow(Symbol('pdg0001939'), Integer(2)), Pow(Symbol('pdg0009139'), Integer(-2))), Mul(Integer(-1), Symbol('pdg0004231'), Pow(Symbol('pdg0009139'), Integer(-1))))x^2 + (b/a)x + (b/(2 a))^2a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Add(Pow(Symbol('pdg0001464'), Integer(2)), Mul(Symbol('pdg0001464'), Symbol('pdg0001939'), Pow(Symbol('pdg0009139'), Integer(-1))), Mul(Rational(1, 4), Pow(Symbol('pdg0001939'), Integer(2)), Pow(Symbol('pdg0009139'), Integer(-2)))) :a_node:expression-c/a5938459282=Add(Pow(Symbol('pdg0001464'), Integer(2)), Mul(Symbol('pdg0001464'), Symbol('pdg0001939'), Pow(Symbol('pdg0009139'), Integer(-1))))Mul(Integer(-1), Symbol('pdg0004231'), Pow(Symbol('pdg0009139'), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax^2 + (b/a)x :a_node:expression5945893986-A \omega^2 \cos(\omega t)=Mul(Pow(Symbol('d'), Integer(2)), Pow(Symbol('dt'), Integer(-2)), Symbol('x'))Mul(Integer(-1), Pow(Symbol('pdg0002321'), Integer(2)), Symbol('pdg0009885'), cos(Mul(Symbol('pdg0001467'), Symbol('pdg0002321'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{d^2 x}{dt^2} :a_node:expression05958392859Integer(0)=Add(Pow(Symbol('pdg0001464'), Integer(2)), Symbol('pdg0001464'), Mul(Symbol('pdg0004231'), Pow(Symbol('pdg0009139'), Integer(-1))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax^2 + (b/a)x+(c/a) :a_node:expression5959282914(x+(b/(2 a)))^2Pow(Add(Symbol('pdg0001464'), Mul(Rational(1, 2), Symbol('pdg0001939'), Pow(Symbol('pdg0009139'), Integer(-1)))), Integer(2))x^2 + x(b/a) + (b/(2 a))^2a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Add(Pow(Symbol('pdg0001464'), Integer(2)), Mul(Symbol('pdg0001464'), Symbol('pdg0001939'), Pow(Symbol('pdg0009139'), Integer(-1))), Mul(Rational(1, 4), Pow(Symbol('pdg0001939'), Integer(2)), Pow(Symbol('pdg0009139'), Integer(-2)))) :a_node:expression\frac{nR}{VP}5962145508Symbol('pdg0004686')=Mul(Symbol('pdg0002834'), Pow(Symbol('pdg0007586'), Integer(-1)), Pow(Symbol('pdg0008134'), Integer(-1)), Symbol('pdg0008179'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\alpha :a_node:expressionG m_{\rm Earth} m \left( 0 - \frac{-1}{ r_{\rm Earth}} \right)5978756813Mul(Pow(Symbol('pdg0003236'), Integer(-1)), Symbol('pdg0005156'), Symbol('pdg0005458'), Symbol('pdg0006277'))Symbol('pdg0006789')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baW= :a_node:expression-\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))5982958248Add(Mul(Integer(-1), Rational(1, 2), Symbol('pdg0001939'), Pow(Symbol('pdg0009139'), Integer(-1))), Mul(Integer(-1), Pow(Add(Mul(Rational(1, 4), Pow(Symbol('pdg0001939'), Integer(2)), Pow(Symbol('pdg0009139'), Integer(-2))), Mul(Integer(-1), Symbol('pdg0004231'), Pow(Symbol('pdg0009139'), Integer(-1)))), Rational(1, 2))))Symbol('pdg0001464')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax= :a_node:expression5982958249-\sqrt{(b/(2 a))^2 - (c/a)}=Add(Symbol('pdg0001464'), Mul(Rational(1, 2), Symbol('pdg0001939'), Pow(Symbol('pdg0009139'), Integer(-1))))Mul(Integer(-1), Pow(Add(Mul(Rational(1, 4), Pow(Symbol('pdg0001939'), Integer(2)), Pow(Symbol('pdg0009139'), Integer(-2))), Mul(Integer(-1), Symbol('pdg0004231'), Pow(Symbol('pdg0009139'), Integer(-1)))), Rational(1, 2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax+(b/(2 a)) :a_node:expression5985371230\frac{i}{\hbar} \vec{p} \psi( \vec{r},t)Mul(Symbol('nabla'), Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467')))Mul(Pow(Symbol('pdg0001054'), Integer(-1)), Symbol('pdg0002046'), Symbol('pdg0004621'), Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\vec{ \nabla} \psi( \vec{r},t)= :a_node:expression6026694087m \frac{v^2}{r}Symbol('pdg0001687')=Mul(Pow(Symbol('pdg0002530'), Integer(-1)), Symbol('pdg0005156'), Pow(Symbol('v'), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF_{\rm centripetal} :a_node:expression6031385191\left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)=Pow(sinh(Symbol('pdg0001464')), Integer(2))Pow(Add(Mul(Rational(1, 2), exp(Symbol('pdg0001464'))), Mul(Integer(-1), Rational(1, 2), exp(Mul(Integer(-1), Symbol('pdg0001464'))))), Integer(2))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\sinh^2 x :a_node:expression6055078815constant pressureC_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T \left( \frac{\partial V}{\partial T} \right)_pDerivative(Symbol('pdg0005786'), Tuple(Symbol('pdg0007343'), Integer(1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\left(\frac{\partial U}{\partial T}\right)_p= :a_node:expression6061695358I R_2Mul(Symbol('pdg0003461'), Symbol('pdg0004501'))Symbol('pdg0008721')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baV_2= :a_node:expressionv_{0, x}6083821265Symbol('pdg0002958')=Mul(Symbol('pdg0005153'), cos(Symbol('pdg0001575')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_0 \cos(\theta) :a_node:expression6091977310\frac{1}{2} m_1 v_{\rm initial}^2Symbol('pdg0004121')=Mul(Rational(1, 2), Pow(Symbol('pdg0001934'), Integer(2)), Symbol('pdg0005022'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baKE_{\rm initial} :a_node:expression6131764194https://physicsderivationgraph.blogspot.com/2020/09/evaluating-definite-integrals-for.htmlG m_{\rm Earth} m \left( \frac{1}{x^2} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)Symbol('W')Mul(Pow(Symbol('pdg0004037'), Integer(-2)), Symbol('pdg0005156'), Symbol('pdg0005458'), Symbol('pdg0006277'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baW= :a_node:expressionv_x6134836751Symbol('pdg0005505')Symbol('pdg0002958')=a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_{0, x} :a_node:expression6175547907\frac{v + v_0}{2}Symbol('pdg0006709')=Add(Mul(Rational(1, 2), Symbol('pdg0001357')), Mul(Rational(1, 2), Symbol('pdg0005153')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_{\rm average} :a_node:expression6204539227\frac{dy}{dt}=Add(Mul(Integer(-1), Symbol('pdg0001467'), Symbol('pdg0006277')), Symbol('pdg0009431'))Derivative(Symbol('pdg0005647'), Tuple(Symbol('pdg0001467'), Integer(1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba-g t + v_{0, y} :a_node:expression6240206408|A|^2 + |B|^2Symbol('pdg0002435')=Add(Pow(Abs(Symbol('pdg0004453')), Integer(2)), Pow(Abs(Symbol('pdg0004698')), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baI_{\rm incoherent} :a_node:expression\frac{k_{\rm desorption}}{k_{\rm adsorption}}6240546932Mul(Pow(Symbol('pdg0006850'), Integer(-1)), Symbol('pdg0008379'))\frac{1}{K_{equilibrium}}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Pow(Symbol('pdg0004933'), Integer(-1)) :a_node:expression6268336290\frac{m}{r}\left(\frac{2\pi r}{T}\right)^2Symbol('pdg0002867')=Mul(Integer(4), Symbol('pdg0002530'), Pow(Symbol('pdg0003141'), Integer(2)), Symbol('pdg0004851'), Pow(Symbol('pdg0008762'), Integer(-2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF_{\rm gravity} :a_node:expression6306552185(A + B)(A^* + B^*)Mul(Add(Symbol('pdg0004453'), Symbol('pdg0004698')), Add(conjugate(Symbol('pdg0004453')), conjugate(Symbol('pdg0004698'))))Symbol('pdg0007882')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baI= :a_node:expression63482603132 \pi r_{\rm Earth\ orbit}Symbol('pdg0001534')C_{\rm Earth\ orbit}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Integer(2), Symbol('pdg0003141'), Symbol('pdg0006081')) :a_node:expression6397683463\left( \frac{\partial V}{\partial T} \right)_p=Mul(Symbol('pdg0004686'), Symbol('pdg0007586'))Derivative(Symbol('pdg0007586'), Tuple(Symbol('pdg0007343'), Integer(1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baV \alpha :a_node:expression6404535647\frac{\exp(x) + \exp(-x)}{2}cosh(Symbol('pdg0001464'))=Add(Mul(Rational(1, 2), exp(Symbol('pdg0001464'))), Mul(Rational(1, 2), exp(Mul(Integer(-1), Symbol('pdg0001464')))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\cosh x :a_node:expressionv t - \frac{1}{2} a t^26421241247Add(Mul(Symbol('pdg0001357'), Symbol('pdg0001467')), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('pdg0001467'), Integer(2)), Symbol('pdg0009140')))Symbol('pdg0001943')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bad= :a_node:expression6450985774n_2 \sin( \theta_2 )Law of RefractionMul(Symbol('pdg0001958'), sin(Symbol('pdg0007545')))eq 34-44 on page 819 in \cite{2001_HRW}n_1 \sin( \theta_1 )a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Symbol('pdg0002941'), sin(Symbol('pdg0003509'))) :a_node:expression6457044853v_0Symbol('pdg0005153')=Add(Symbol('pdg0001357'), Mul(Integer(-1), Symbol('pdg0001467'), Symbol('pdg0009140')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav - a t :a_node:expression6457999644\frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1Mul(Symbol('pdg0003037'), Pow(Symbol('pdg0004940'), Integer(-1)))Add(Integer(1), Mul(Pow(Symbol('pdg0004933'), Integer(-1)), Pow(Symbol('pdg0009046'), Integer(-1))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{[S_0]}{[A_{\rm adsorption}]}= :a_node:expression\sqrt{ \frac{K}{\rho} }6504442697Pow(Mul(Symbol('K'), Pow(Symbol('pdg0003935'), Integer(-1))), Rational(1, 2))Symbol('pdg0002077')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav= :a_node:expression6529793063|A|^2 + |A|^2Symbol('pdg0002435')=Mul(Integer(2), Pow(Abs(Symbol('pdg0004453')), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baI_{\rm incoherent} :a_node:expression|A| \exp(-i \theta)6555185548conjugate(Symbol('pdg0004453'))Mul(exp(Mul(Integer(-1), Symbol('pdg0001575'), Symbol('pdg0004621'))), Abs(Symbol('pdg0004453')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baA^*= :a_node:expression65568755792Mul(Pow(Symbol('pdg0002435'), Integer(-1)), Symbol('pdg0008251'))Integer(2)a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{I_{\rm coherent}}{I_{\rm incoherent}}= :a_node:expression6572039835v_ySymbol('pdg0009107')=Add(Mul(Integer(-1), Symbol('pdg0001467'), Symbol('pdg0001649')), Symbol('pdg0009431'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba-g t + v_{0, y} :a_node:expression6715248283potential energy-F xMul(Integer(-1), Symbol('pdg0004037'), Symbol('pdg0004202'))https://en.wikipedia.org/wiki/Potential_energySymbol('pdg0004930')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baPE= :a_node:expression( \vec{p}_{1}\cdot\vec{p}_{1})+( \vec{p}_{2}\cdot\vec{p}_{2})-2( \vec{p}_{1}\cdot\vec{p}_{2})6742123016Symbol('pdg0004299')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\vec{p}_{electron}\cdot\vec{p}_{electron}= :a_node:expressionI_1 + I_26753224061Symbol('pdg0009647')=Add(Symbol('pdg0003978'), Symbol('pdg0004856'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baI_{\rm total} :a_node:expression\phi6774684564for coherent wavesSymbol('pdg0008586')Symbol('pdg0001575')=a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\theta :a_node:expressionr_{\rm desorption}6783009163Symbol('pdg0001966')Symbol('pdg0006687')=a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bar_{\rm adsorption} :a_node:expression67853038572 \pi rMul(Integer(2), Symbol('pdg0002530'), Symbol('pdg0003141'))Symbol('pdg0003034')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baC= :a_node:expression6800170830\frac{2 G m}{c^2}Symbol('pdg0004518')r_{\rm Schwarzschild}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Integer(2), Pow(Symbol('pdg0004567'), Integer(-2)), Symbol('pdg0005156'), Symbol('pdg0006277')) :a_node:expression6829281943G \frac{m_1 m_2}{r^2}Symbol('pdg0001687')=Mul(Pow(Symbol('pdg0002530'), Integer(-2)), Symbol('pdg0004851'), Symbol('pdg0005022'), Symbol('pdg0006277'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF_{\rm centripetal} :a_node:expression\cos( \theta_{\rm Brewster} )6831637424Symbol('pdg0004928')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\sin( 90^{\circ} - \theta_{\rm Brewster} )= :a_node:expression6831694380\frac{d^2 x}{dt^2}accelerationSymbol('a')Mul(Pow(Symbol('d'), Integer(2)), Pow(Symbol('dt'), Integer(-2)), Symbol('x'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baa= :a_node:expression6870322215\frac{1}{2} m v_{\rm escape}^2Symbol('pdg0005332')=Mul(Rational(1, 2), Symbol('pdg0005156'), Pow(Symbol('pdg0008656'), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baKE_{\rm escape} :a_node:expression-1 + i 06885625907=exp(Mul(Symbol('pdg0003141'), Symbol('pdg0004621')))Integer(-1)a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\exp(i \pi) :a_node:expression\frac{G m_1 m_2}{r}6892595652Mul(Rational(1, 2), Symbol('pdg0005022'), Pow(Symbol('pdg0008909'), Integer(2)))Mul(Pow(Symbol('pdg0002530'), Integer(-1)), Symbol('pdg0004851'), Symbol('pdg0005022'), Symbol('pdg0006277'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{1}{2} m_1 v_{\rm final}^2= :a_node:expression6908055431A \cos\left(\frac{k}{m} t\right)Function('x')(Symbol('pdg0001467'))Mul(Symbol('pdg0009885'), cos(Mul(Symbol('k'), Symbol('pdg0001467'), Pow(Symbol('pdg0005156'), Integer(-1)))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax(t)= :a_node:expression\frac{PV}{T} \frac{1}{VP}6925244346Symbol('pdg0004686')=Mul(Pow(Symbol('pdg0007343'), Integer(-1)), Symbol('pdg0007586'), Symbol('pdg0008134'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\alpha :a_node:expressionG \frac{m_1 m_2}{x^2}https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation#Modern_form6935745841Mul(Pow(Symbol('pdg0004037'), Integer(-2)), Symbol('pdg0004851'), Symbol('pdg0005022'), Symbol('pdg0006277'))Newton's law of universal gravitationSymbol('pdg0004202')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF= :a_node:expression6946088325\frac{C}{t}Mul(Pow(Symbol('pdg0001467'), Integer(-1)), Symbol('pdg0003034'))Symbol('pdg0001357')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav= :a_node:expression6955192897k_{\rm desorption} [A_{\rm adsorption}]Symbol('pdg0001966')=Mul(Symbol('pdg0004940'), Symbol('pdg0008379'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bar_{\rm desorption} :a_node:expression\frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}6998364753Symbol('pdg0007427')v_{\rm Earth\ orbit}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Float('0.63291139240506322', precision=53), Symbol('pdg0003141')) :a_node:expression7002609475I_2Symbol('pdg0004856')=Mul(Pow(Symbol('pdg0003461'), Integer(-1)), Symbol('pdg0006599'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{V}{R_2} :a_node:expression70102941434 \pi^2 r^3Mul(Symbol('pdg0005458'), Symbol('pdg0006277'), Pow(Symbol('pdg0008762'), Integer(2)))Mul(Integer(4), Pow(Symbol('pdg0002530'), Integer(3)), Pow(Symbol('pdg0003141'), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baT_{\rm orbit}^2 G m_{\rm Earth}= :a_node:expression\frac{(v_0 + a t) + v_0}{2} t7011114072Mul(Symbol('pdg0001467'), Add(Mul(Rational(1, 2), Symbol('pdg0001467'), Symbol('pdg0009140')), Symbol('pdg0005153')))Symbol('pdg0001943')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bad= :a_node:expressiony7057864873frame of reference is moving only along x directionSymbol('pdg0005647')Symbol('pdg0001888')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bay'= :a_node:expression|B|^27107090465=Mul(Symbol('pdg0004698'), conjugate(Symbol('pdg0004698')))Pow(Abs(Symbol('pdg0004698')), Integer(2))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baB B^* :a_node:expression7112613117\frac{(9.80665 m/s^2) r_{\rm Earth}^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}}Symbol('pdg0005458')=a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bam_{\rm Earth} :a_node:expression7112646057\frac{2 G m_2}{r}=Pow(Symbol('pdg0008909'), Integer(2))Mul(Integer(2), Pow(Symbol('pdg0002530'), Integer(-1)), Symbol('pdg0004851'), Symbol('pdg0006277'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_{\rm final}^2 :a_node:expression71754162991 {\rm year}Integer(1)Symbol('pdg0005344')t_{\rm Earth\ orbit}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba= :a_node:expressionv_0^2 + 2 a t v_0 + a^2 t^27215099603Pow(Symbol('pdg0001357'), Integer(2))Add(Mul(Pow(Symbol('pdg0001467'), Integer(2)), Pow(Symbol('pdg0009140'), Integer(2))), Mul(Integer(2), Symbol('pdg0001467'), Symbol('pdg0005153'), Symbol('pdg0009140')), Pow(Symbol('pdg0005153'), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav^2= :a_node:expressionR_1 + R_27217021879Symbol('pdg0001908')=Add(Symbol('pdg0003461'), Symbol('pdg0008697'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baR_{\rm total} :a_node:expressionv_0 t_f \cos(\theta)7233558441Mul(Symbol('pdg0002467'), Symbol('pdg0005153'), cos(Symbol('pdg0001575')))Symbol('pdg0001943')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bad= :a_node:expression7252338326\frac{dy}{dt}Derivative(Symbol('pdg0005647'), Tuple(Symbol('pdg0001467'), Integer(1)))Symbol('pdg0009107')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_y= :a_node:expression-F \left( \frac{x_2 - x_1}{t} \right)7267155233Mul(Integer(-1), Pow(Symbol('pdg0001467'), Integer(-1)), Symbol('pdg0004202'), Add(Mul(Integer(-1), Symbol('pdg0003852')), Symbol('pdg0005467')))\frac{PE_2 - PE_1}{t}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Pow(Symbol('pdg0001467'), Integer(-1)), Add(Mul(Integer(-1), Symbol('pdg0004093')), Symbol('pdg0008849'))) :a_node:expression\frac{1+(K_{\rm equilibrium}\ p_A)}{K_{\rm equilibrium}\ p_A}7267424860=Pow(Symbol('pdg0001791'), Integer(-1))Mul(Pow(Symbol('pdg0004933'), Integer(-1)), Pow(Symbol('pdg0009046'), Integer(-1)), Add(Mul(Symbol('pdg0004933'), Symbol('pdg0009046')), Integer(1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{1}{\theta_A} :a_node:expression- \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_07354529102Add(Symbol('pdg0001469'), Mul(Integer(-1), Rational(1, 2), Pow(Symbol('pdg0001649'), Integer(2)), Pow(Symbol('pdg0002958'), Integer(-2)), Pow(Add(Mul(Integer(-1), Symbol('pdg0001572')), Symbol('pdg0004037')), Integer(2))), Mul(Pow(Symbol('pdg0002958'), Integer(-1)), Symbol('pdg0009431'), Add(Mul(Integer(-1), Symbol('pdg0001572')), Symbol('pdg0004037'))))Symbol('pdg0005647')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bay= :a_node:expression7376526845\frac{v_{0, y}}{v_0}sin(Symbol('pdg0001575'))=Mul(Symbol('pdg0005153'), Pow(Symbol('pdg0009431'), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\sin(\theta) :a_node:expression7391837535\frac{v_{0, x}}{v_0}cos(Symbol('pdg0001575'))=Mul(Pow(Symbol('pdg0002958'), Integer(-1)), Symbol('pdg0005153'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\cos(\theta) :a_node:expression\frac{dx}{dt}7455581657Symbol('pdg0002958')=Derivative(Symbol('pdg0009199'), Tuple(Symbol('pdg0001467'), Integer(1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_{0, x} :a_node:expression74668294920Integer(0)\vec{ \nabla} \cdot \vec{E}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Function('dot')(Symbol('pdg0006238'), Symbol('nabla')) :a_node:expressionc^27513513483Pow(Symbol('pdg0004567'), Integer(2))\gamma^2 (c^2 - v^2)a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Pow(Symbol('pdg0001790'), Integer(2)), Add(Mul(Integer(-1), Pow(Symbol('pdg0001357'), Integer(2))), Pow(Symbol('pdg0004567'), Integer(2)))) :a_node:expression7517073655\left(\frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}]Symbol('pdg0003037')=Mul(Symbol('pdg0004940'), Add(Integer(1), Mul(Pow(Symbol('pdg0004933'), Integer(-1)), Pow(Symbol('pdg0009046'), Integer(-1)))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba[S_0] :a_node:expression7564894985\frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)Integral(cos(Mul(Integer(2), Symbol('pdg0001592'), Pow(Symbol('pdg0002523'), Integer(-1)), Symbol('pdg0003141'), Symbol('pdg0004037'))), Tuple(Symbol('pdg0004037')))Mul(Rational(1, 2), Pow(Symbol('pdg0001592'), Integer(-1)), Symbol('pdg0002523'), Pow(Symbol('pdg0003141'), Integer(-1)), sin(Mul(Integer(2), Symbol('pdg0001592'), Pow(Symbol('pdg0002523'), Integer(-1)), Symbol('pdg0003141'), Symbol('pdg0004037'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\int \cos\left(\frac{2n\pi}{W} x\right) dx= :a_node:expression75726647281Integer(1)\cos(2 x) + 2 (\sin(x))^2a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Add(Mul(Integer(2), Pow(sin(Symbol('pdg0004037')), Integer(2))), cos(Mul(Integer(2), Symbol('pdg0004037')))) :a_node:expression-W7573835180the 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 surfaceSymbol('pdg0006431')PE_{\rm Earth\ surface}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Integer(-1), Symbol('pdg0006789')) :a_node:expression7575738420\frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}Pow(sin(Mul(Symbol('pdg0001592'), Pow(Symbol('pdg0002523'), Integer(-1)), Symbol('pdg0003141'), Symbol('pdg0004037'))), Integer(2))Add(Rational(1, 2), Mul(Integer(-1), Rational(1, 2), cos(Mul(Integer(2), Symbol('pdg0001464'), Symbol('pdg0001592'), Pow(Symbol('pdg0002523'), Integer(-1)), Symbol('pdg0003141')))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\left(\sin\left(\frac{n \pi}{W}x\right) \right)^2= :a_node:expression7575859295\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})Function('cross')(Symbol('pdg0006238'), Function('cross')(Symbol('nabla'), Symbol('nabla')))Function('nabla')(Add(Mul(Integer(-1), Pow(Symbol('nabla'), Integer(2)), Symbol('pdg0006238')), Function('dot')(Symbol('pdg0006238'), Symbol('nabla'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\vec{ \nabla} \times \vec{ \nabla} \times \vec{E}= :a_node:expression\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})7575859300Symbol('pdg0001552')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{ \nabla} \times \vec{E} )_k= :a_node:expression7575859302\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})Mul(LeviCivita(Symbol('pdg0001567'),Symbol('pdg0001552'),Symbol('pdg0009690')), Mul(LeviCivita(Symbol('pdg0001592'),Symbol('pdg0001552'),Symbol('pdg0009690')), Mul(Symbol('pdg0008349'), Mul(Symbol('nabla_{j}'), Mul(Pow(Symbol('pdg0006238'), Symbol('pdg0001592')), Pow(Symbol('nabla'), Symbol('pdg0007930')))))))Symbol('nabla')*(Add(Mul(Integer(-1), Pow(Symbol('nabla'), Integer(2)), Symbol('pdg0004326')), Mul(Symbol('nabla'), Symbol('pdg0004326'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n= :a_node:expression7575859304https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors\delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}Mul(LeviCivita(Symbol('pdg0007984'),Symbol('pdg0001552'),Symbol('pdg0009690')), LeviCivita(Symbol('pdg0001592'),Symbol('pdg0001552'),Symbol('pdg0009690')))Add(Mul(KroneckerDelta(Symbol('pdg0008304'),Symbol('pdg0001552')), KroneckerDelta(Symbol('pdg0007930'),Symbol('pdg0009690'))), Mul(Integer(-1), KroneckerDelta(Symbol('pdg0008304'),Symbol('pdg0009690')), KroneckerDelta(Symbol('pdg0007930'),Symbol('h'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\epsilon^{i,j,k} \epsilon_{n,j,k}= :a_node:expression7575859306https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})Mul(Mul(Symbol('pdg0008349'), Mul(Symbol('nabla_{pdg0001552}'), Mul(Pow(Symbol('pdg0006238'), Symbol('pdg0001592')), Pow(Symbol('nabla'), Symbol('pdg0007930'))))), Add(Mul(KroneckerDelta(Symbol('pdg0008304'),Symbol('pdg0001552')), KroneckerDelta(Symbol('pdg0007930'),Symbol('pdg0009690'))), Mul(Integer(-1), KroneckerDelta(Symbol('pdg0008304'),Symbol('pdg0009690')), KroneckerDelta(Symbol('pdg0007930'),Symbol('h')))))Symbol('nabla')*(Add(Mul(Integer(-1), Pow(Symbol('nabla'), Integer(2)), Symbol('pdg0004326')), Mul(Symbol('nabla'), Symbol('pdg0004326'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n= :a_node:expression7575859308https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})(Mul(KroneckerDelta(Symbol('pdg0008304'),Symbol('pdg0001552')),KroneckerDelta(Symbol('pdg0007930'),Symbol('pdg0009690'))) - Mul(KroneckerDelta(Symbol('pdg0008304'),Symbol('pdg0001552')),KroneckerDelta(Symbol('pdg0007930'),Symbol('pdg0009690'))))*Symbol('pdg0008349')*Symbol('nabla_{pdg0001552}')*Pow(Symbol('nabla'), Symbol('pdg0007930'))*Pow(Symbol('pdg0006238'), Symbol('pdg0001592'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\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)= :a_node:expression7575859310\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})Add(Mul(Integer(-1), Pow(Symbol('nabla'), Symbol('pdg0007930')), Symbol('nabla_{m}'), Symbol('pdg0001434'), Pow(Symbol('pdg0006238'), Symbol('pdg0001592'))), Mul(Pow(Symbol('nabla'), Symbol('pdg0007930')), Symbol('nabla_{n}'), Symbol('pdg0002380'), Pow(Symbol('pdg0006238'), Symbol('pdg0001592'))))Function('nabla')(Add(Mul(Integer(-1), Pow(Symbol('nabla'), Integer(2)), Symbol('pdg0004326')), Mul(Symbol('nabla'), Symbol('pdg0004326'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n= :a_node:expression\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})7575859312Symbol('nabla')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})= :a_node:expression|A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)7621705408Add(Mul(exp(Mul(Symbol('pdg0001575'), Symbol('pdg0004621'))), exp(Mul(Integer(-1), Symbol('pdg0004621'), Symbol('pdg0008586'))), Abs(Mul(Symbol('pdg0004453'), Symbol('pdg0004698'), Abs(Add(Abs(Symbol('pdg0004453')), Mul(exp(Mul(Integer(-1), Symbol('pdg0001575'), Symbol('pdg0004621'))), exp(Mul(Symbol('pdg0004621'), Symbol('pdg0008586'))), Abs(Symbol('pdg0004698')))))))), Pow(Abs(Symbol('pdg0004453')), Integer(2)), Pow(Abs(Symbol('pdg0004698')), Integer(2)))Symbol('pdg0007882')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baI= :a_node:expression7652131521-A \omega \sin (\omega t)=Derivative(Symbol('pdg0004037'), Tuple(Symbol('pdg0001467'), Integer(1)))Mul(Integer(-1), Symbol('pdg0002321'), Symbol('pdg0009885'), sin(Mul(Symbol('pdg0001467'), Symbol('pdg0002321'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{dx}{dt} :a_node:expression7672365885\frac{4 \pi^2 m r}{T^2}Symbol('pdg0002867')=Mul(Integer(4), Symbol('pdg0002530'), Pow(Symbol('pdg0003141'), Integer(2)), Symbol('pdg0004851'), Pow(Symbol('pdg0008762'), Integer(-2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF_{\rm gravity} :a_node:expression7675171493I R_1Mul(Symbol('pdg0004501'), Symbol('pdg0008697'))Symbol('pdg0008257')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baV_1= :a_node:expression7676652285\frac{1}{2} m v_2^2Mul(Rational(1, 2), Pow(Symbol('pdg0004770'), Integer(2)), Symbol('pdg0005156'))Symbol('pdg0001352')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baKE_2= :a_node:expression7696214507n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} )Mul(Symbol('pdg0002941'), sin(Symbol('pdg0004928')))Mul(Symbol('pdg0001958'), sin(Add(Integer(90), Mul(Integer(-1), Symbol('pdg0004928')))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ban_1 \sin( \theta_{\rm Brewster} )= :a_node:expression7701249282when A = 1\alpha c \sqrt{ \frac{m_e}{m_p} }Mul(Symbol('pdg0001370'), Symbol('pdg0004567'), Pow(Mul(Symbol('pdg0002515'), Pow(Symbol('pdg0005916'), Integer(-1))), Rational(1, 2)))Symbol('pdg0004635')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_u= :a_node:expression\frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)7729413831Derivative(Add(Mul(Symbol('pdg0001700'), Symbol('pdg0009107')), Mul(Symbol('pdg0005505'), Symbol('pdg0008339'))), Tuple(Symbol('pdg0001467'), Integer(1)))a_x \hat{x} + a_y \hat{y}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Add(Mul(Symbol('pdg0001700'), Symbol('pdg0007055')), Mul(Symbol('pdg0007159'), Symbol('pdg0008339'))) :a_node:expression7731226616\frac{1}{\cosh x}sech(Symbol('pdg0001464'))=Pow(cosh(Symbol('pdg0001464')), Integer(-1))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba{\rm sech}\ x :a_node:expression7734996511-F ( x_2 - x_1 )=Add(Mul(Integer(-1), Symbol('pdg0004093')), Symbol('pdg0008849'))Mul(Integer(-1), Symbol('pdg0004202'), Add(Mul(Integer(-1), Symbol('pdg0003852')), Symbol('pdg0005467')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baPE_2 - PE_1 :a_node:expression\frac{1}{4}\left( \exp(2x)+1+1+\exp(-2x) - \left(\exp(2x)-1-1-\exp(-2x)\right) \right)7735731560Add(Integer(1), Mul(Rational(1, 2), exp(Mul(Integer(-1), Integer(2), Symbol('pdg0001464')))))\cosh^2 x - \sinh^2 xa84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Add(Mul(Integer(-1), Pow(sinh(Symbol('pdg0001464')), Integer(2))), Pow(cosh(Symbol('pdg0001464')), Integer(2))) :a_node:expressionm v \frac{ v_2 - v_1 }{t}7735737409Mul(Symbol('pdg0001357'), Pow(Symbol('pdg0001467'), Integer(-1)), Symbol('pdg0005156'), Add(Mul(Integer(-1), Symbol('pdg0002473')), Symbol('pdg0004770')))\frac{KE_2 - KE_1}{t}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Pow(Symbol('pdg0001467'), Integer(-1)), Add(Symbol('pdg0001352'), Mul(Integer(-1), Symbol('pdg0001955')))) :a_node:expression\gamma^2 x - \gamma^2 v t + \gamma v t'7741202861Add(Mul(Integer(-1), Symbol('pdg0001357'), Symbol('pdg0001467'), Pow(Symbol('pdg0001790'), Integer(2))), Mul(Symbol('pdg0001357'), Symbol('pdg0001790'), Symbol('pdg0004989')), Mul(Pow(Symbol('pdg0001790'), Integer(2)), Symbol('pdg0004037')))Symbol('pdg0004037')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax= :a_node:expressionG \frac{m_{\rm Earth} m }{ r_{\rm Earth}}7749253510Mul(Pow(Symbol('pdg0003236'), Integer(-1)), Symbol('pdg0005156'), Symbol('pdg0005458'), Symbol('pdg0006277'))Symbol('pdg0006789')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baW= :a_node:expressionC_V + \pi_T V \alpha7826132469Derivative(Symbol('pdg0005786'), Tuple(Symbol('pdg0007343'), Integer(1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\left(\frac{\partial U}{\partial T}\right)_p= :a_node:expression\sqrt{f} \sqrt{\frac{E}{m}}7837519722Mul(Pow(Symbol('pdg0006235'), Rational(1, 2)), Pow(Mul(Symbol('pdg0002241'), Pow(Symbol('pdg0009863'), Integer(-1))), Rational(1, 2)))Symbol('pdg0002077')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav= :a_node:expression7846240076\frac{(9.80665 m/s^2) r_{\rm Earth}^2}{G}Symbol('pdg0005458')=Mul(Integer(9), Pow(Symbol('pdg0003236'), Integer(2)), Pow(Symbol('pdg0006277'), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bam_{\rm Earth} :a_node:expression7875206161KE_2 + PE_2Add(Symbol('pdg0001352'), Symbol('pdg0008849'))Symbol('pdg0004550')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baE_2= :a_node:expression7882872592\int_{\infty}^r \vec{F}\cdot d\vec{r}Symbol('pdg0009372')=Integral(Function('Dot')(Symbol('pdg0006777'), Symbol('pdg0002530')), Tuple(Symbol('pdg0002530'), oo, Symbol('pdg0002530')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baW_{\rm to\ system} :a_node:expression7906112355\frac{c^2}{c^2 - \gamma^2}=Pow(Symbol('pdg0001790'), Integer(2))Mul(Pow(Symbol('pdg0004567'), Integer(2)), Pow(Add(Mul(Integer(-1), Pow(Symbol('pdg0001790'), Integer(2))), Pow(Symbol('pdg0004567'), Integer(2))), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\gamma^2 :a_node:expression7917051060\vec{p}_{1}-\vec{p}_{2}Symbol('pdg0004299')=Add(Mul(Integer(-1), Symbol('pdg0002097')), Symbol('pdg0006029'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\vec{p}_{electron} :a_node:expression7924063906\frac{k_{\rm adsorption}}{k_{\rm desorption}}Symbol('pdg0004933')=Mul(Symbol('pdg0006850'), Pow(Symbol('pdg0008379'), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baK_{equilibrium} :a_node:expression\frac{1}{K_{\rm equilibrium} p_A} + 17928111771=Pow(Symbol('pdg0001791'), Integer(-1))Add(Integer(1), Mul(Pow(Symbol('pdg0004933'), Integer(-1)), Pow(Symbol('pdg0009046'), Integer(-1))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{1}{\theta_A} :a_node:expressionv_0^2 + 2 a d7939765107Pow(Symbol('pdg0001357'), Integer(2))Add(Mul(Integer(2), Symbol('pdg0001943'), Symbol('pdg0009140')), Pow(Symbol('pdg0005153'), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav^2= :a_node:expression8046208134|A|^2 + |A|^2 + |A| |A| 2Symbol('pdg0008251')=Mul(Integer(4), Pow(Abs(Symbol('pdg0004453')), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baI_{\rm coherent} :a_node:expressionKE_{\rm final} - KE_{\rm initial}8049905441change in kinetic energySymbol('pdg0005734')=Add(Mul(Integer(-1), Symbol('pdg0004121')), Symbol('pdg0005340'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\Delta KE :a_node:expression8059639673\frac{4 \pi^2 r^2}{T_{\rm orbit}^2}Pow(Symbol('pdg0001357'), Integer(2))Mul(Integer(4), Pow(Symbol('pdg0002530'), Integer(2)), Pow(Symbol('pdg0003141'), Integer(2)), Pow(Symbol('pdg0008762'), Integer(-2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav^2= :a_node:expressionA A^* + B B^* + A B^* + B A^*8065128065Add(Mul(Symbol('pdg0004453'), conjugate(Symbol('pdg0004453'))), Mul(Symbol('pdg0004453'), conjugate(Symbol('pdg0004698'))), Mul(Symbol('pdg0004698'), conjugate(Symbol('pdg0004453'))), Mul(Symbol('pdg0004698'), conjugate(Symbol('pdg0004698'))))Symbol('pdg0007882')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baI= :a_node:expression\sqrt{ \left( f\frac{E}{a^3} \right) \frac{1}{\rho} }8090924099Pow(Mul(Symbol('pdg0002241'), Pow(Symbol('pdg0003935'), Integer(-1)), Pow(Symbol('pdg0005854'), Integer(-3)), Symbol('pdg0006235')), Rational(1, 2))Symbol('pdg0002077')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav= :a_node:expressionfine structure constant definition\frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c}8106885760Symbol('pdg0001370')=Mul(Rational(1, 4), Pow(Symbol('pdg0001054'), Integer(-1)), Pow(Symbol('pdg0001999'), Integer(2)), Pow(Symbol('pdg0003141'), Integer(-1)), Pow(Symbol('pdg0004567'), Integer(-1)), Pow(Symbol('pdg0007940'), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\alpha :a_node:expression\frac{[S_0]}{[A_{\rm adsorption}]}8131665171=Pow(Symbol('pdg0001791'), Integer(-1))Mul(Symbol('pdg0003037'), Pow(Symbol('pdg0004940'), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{1}{\theta_A} :a_node:expression8139187332\vec{p}_{2}+\vec{p}_{electron}Symbol('pdg0006029')=Add(Symbol('pdg0002097'), Symbol('pdg0004299'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\vec{p}_{1} :a_node:expressiondy8145337879Symbol('pdg0005842')-g t dt + v_{0, y} dta84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Add(Mul(Integer(-1), Symbol('pdg0001467'), Symbol('pdg0001649'), Symbol('pdg0004711')), Mul(Symbol('pdg0004711'), Symbol('pdg0009431'))) :a_node:expression8198310977- \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0Integer(0)Add(Symbol('pdg0001469'), Mul(Integer(-1), Rational(1, 2), Symbol('pdg0001649'), Pow(Symbol('pdg0002467'), Integer(2))), Mul(Symbol('pdg0002467'), Symbol('pdg0005153'), sin(Symbol('pdg0001575'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0= :a_node:expression8228733125\frac{d}{dt} v_yDerivative(Symbol('pdg0009107'), Tuple(Symbol('pdg0001467'), Integer(1)))Symbol('pdg0007055')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baa_y= :a_node:expression\vec{p}_{1}8257621077Symbol('pdg0006029')Symbol('pdg0001302')\vec{p}_{\rm before}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba= :a_node:expressionv^2 - v_0^28269198922=Mul(Integer(2), Symbol('pdg0001943'), Symbol('pdg0009140'))Add(Pow(Symbol('pdg0001357'), Integer(2)), Mul(Integer(-1), Pow(Symbol('pdg0005153'), Integer(2))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba2 a d :a_node:expression8283354808|A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )Symbol('pdg0008251')=Add(Pow(Abs(Symbol('pdg0004453')), Integer(2)), Mul(Integer(2), cos(Integer(0)), Abs(Symbol('pdg0004453')), Abs(Symbol('pdg0004698'))), Pow(Abs(Symbol('pdg0004698')), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baI_{\rm coherent} :a_node:expression8311458118\vec{p}_{2}+\vec{p}_{electron}Symbol('pdg0005493')=Add(Symbol('pdg0002097'), Symbol('pdg0004299'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\vec{p}_{\rm after} :a_node:expression8332931442\cos(\pi)+i \sin(\pi)=exp(Mul(Symbol('pdg0003141'), Symbol('pdg0004621')))Add(Mul(Symbol('pdg0004621'), sin(Symbol('pdg0003141'))), cos(Symbol('pdg0003141')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\exp(i \pi) :a_node:expression\frac{1}{2} m v^2kinetic energy8357234146Mul(Rational(1, 2), Pow(Symbol('pdg0001357'), Integer(2)), Symbol('pdg0005156'))https://en.wikipedia.org/wiki/Kinetic_energySymbol('pdg0004929')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baKE= :a_node:expression8360117126not a physically valid result in this context\frac{-1}{\sqrt{1-\frac{v^2}{c^2}}}Symbol('pdg0001790')=Mul(Integer(-1), Pow(Add(Mul(Integer(-1), Pow(Symbol('pdg0001357'), Integer(2)), Pow(Symbol('pdg0004567'), Integer(-2))), Integer(1)), Rational(-1, 2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\gamma :a_node:expression8361238989\frac{v^2}{r}=Symbol('a_{c*(e*(n*(t*(r*(i*(p*(e*(t*(a*l)))))))))}')Mul(Pow(Symbol('pdg0001357'), Integer(2)), Pow(Symbol('pdg0002530'), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baa_{\rm centripetal} :a_node:expression8368984890\frac{-1}{V} \left( \frac{ \partial }{\partial P}\left(\frac{nRT}{P}\right) \right)_TSymbol('pdg0004645')=Mul(Integer(-1), Pow(Symbol('pdg0007586'), Integer(-1)), Derivative(Mul(Symbol('pdg0002834'), Symbol('pdg0007343'), Pow(Symbol('pdg0008134'), Integer(-1)), Symbol('pdg0008179')), Tuple(Symbol('pdg0008134'), Integer(1))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\kappa_T :a_node:expression8396997949intensity of two waves traveling opposite directions on same path| A + B |^2Pow(Abs(Add(Symbol('pdg0004453'), Symbol('pdg0004698'))), Integer(2))Symbol('pdg0007882')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baI= :a_node:expression\langle x^2 \rangle-\langle x \rangle^28399484849Symbol('pdg0001464')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\langle x^2 - 2 x \langle x \rangle + \langle x \rangle^2 \rangle= :a_node:expression8405272745-G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dxSymbol('pdg0009372')=Mul(Integer(-1), Symbol('pdg0004851'), Symbol('pdg0005022'), Symbol('pdg0006277'), Integral(Pow(Symbol('pdg0004037'), Integer(-2)), Tuple(Symbol('pdg0004037'), oo, Symbol('pdg0002530'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baW_{\rm to\ system} :a_node:expressioni \sinh(x)8418527415=sin(Mul(Symbol('pdg0001464'), Symbol('pdg0004621')))Mul(Symbol('pdg0004621'), sinh(Symbol('pdg0001464')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\sin(i x) :a_node:expression8435841627https://en.wikipedia.org/wiki/Ideal_gas_lawn R TMul(Symbol('pdg0007586'), Symbol('pdg0008134'))Mul(Symbol('pdg0002834'), Symbol('pdg0007343'), Symbol('pdg0008179'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baP V= :a_node:expression8460820419\frac{dx}{dt}Derivative(Symbol('pdg0009199'), Tuple(Symbol('pdg0001467'), Integer(1)))Symbol('pdg0005505')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_x= :a_node:expression8483686863\frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x) \right)=sin(Mul(Integer(2), Symbol('pdg0001464')))Mul(Rational(1, 2), Pow(Symbol('pdg0004621'), Integer(-1)), Add(exp(Mul(Integer(2), Symbol('pdg0001464'), Symbol('pdg0004621'))), Mul(Integer(-1), exp(Mul(Integer(-1), Integer(2), Symbol('pdg0001464'), Symbol('pdg0004621'))))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\sin(2 x) :a_node:expression-a k^2 \sin(kx) + -b k^2 \cos(k x)8484544728Symbol('pdg0004037')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba-a k^2\sin(k x) + -b k^2\cos(k x)= :a_node:expression-a k^2 \sin(kx) + -b k^2 \cos(kx)8485757728Symbol('pdg0009199')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baa \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x)= :a_node:expression8485867742a^2=Mul(Integer(2), Pow(Symbol('pdg0002523'), Integer(-1)))Pow(Symbol('pdg0009139'), Integer(2))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{2}{W} :a_node:expression8486706976xSymbol('pdg0004037')=Add(Mul(Symbol('pdg0001467'), Symbol('pdg0002958')), Symbol('pdg0001572'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_{0, x} t + x_0 :a_node:expression8489593958u dv + v duSymbol('pdg0004221')=a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bad(u v) :a_node:expression8489593960u dvSymbol('pdg0004221')=a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bad(u v) - v du :a_node:expression8489593962d(u v) - v duSymbol('pdg0004221')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bau dv= :a_node:expression8489593964u v - \int v du=Integral(Symbol('pdg0004221'), Tuple(Symbol('pdg0005177')))Add(Mul(Symbol('pdg0004221'), Symbol('pdg0005177')), Mul(Integer(-1), Integral(Symbol('pdg0005177'), Tuple(Symbol('pdg0004221')))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\int u dv :a_node:expression\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}8494839423=Mul(Pow(Symbol('nabla'), Integer(2)), Symbol('pdg0004326'))Mul(Symbol('partial'), Pow(Symbol('pdg0001467'), Integer(-2)), Symbol('pdg0004326'), Symbol('pdg0006197'), Symbol('pdg0007940'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\nabla^2 \vec{E} :a_node:expression\arctan{ \left( \frac{ n_1 }{ n_2 } \right) }8495187962Symbol('pdg0004928')\theta_{\rm Brewster}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=atan(Mul(Pow(Symbol('pdg0001958'), Integer(-1)), Symbol('pdg0002941'))) :a_node:expression|A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )8497631728Add(Mul(Integer(2), cos(Add(Symbol('pdg0001575'), Mul(Integer(-1), Symbol('pdg0008586')))), Abs(Symbol('pdg0004453')), Abs(Symbol('pdg0004698'))), Pow(Abs(Symbol('pdg0004453')), Integer(2)), Pow(Abs(Symbol('pdg0004698')), Integer(2)))Symbol('pdg0007882')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baI= :a_node:expressionz8515803375frame of reference is moving only along x directionSymbol('pdg0006728')Symbol('pdg0004306')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baz'= :a_node:expression8532702080\left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right)=Pow(cosh(Symbol('pdg0001464')), Integer(2))Pow(Add(Mul(Rational(1, 2), exp(Symbol('pdg0001464'))), Mul(Rational(1, 2), exp(Mul(Integer(-1), Symbol('pdg0001464'))))), Integer(2))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\cosh^2 x :a_node:expression8552710882\frac{1}{2} m_1 v_{\rm final}^2Symbol('pdg0005340')=Mul(Rational(1, 2), Symbol('pdg0005022'), Pow(Symbol('pdg0008909'), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baKE_{\rm final} :a_node:expression8558338742conservation of energyhttps://en.wikipedia.org/wiki/Conservation_of_energyE_1Symbol('pdg0005579')Symbol('pdg0004550')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baE_2= :a_node:expression\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)8563535636Add(Pow(Add(Mul(Rational(1, 2), exp(Symbol('pdg0001464'))), Mul(Rational(1, 2), exp(Mul(Integer(-1), Symbol('pdg0001464'))))), Integer(2)), Mul(Integer(-1), Rational(1, 4), Pow(Add(exp(Symbol('pdg0001464')), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('pdg0001464'))))), Integer(2))))\cosh^2 x - \sinh^2 xa84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Add(Mul(Integer(-1), Pow(sinh(Symbol('pdg0001464')), Integer(2))), Pow(cosh(Symbol('pdg0001464')), Integer(2))) :a_node:expression8572657110\int |\psi(x)|^2 dxInteger(1)Integral(Pow(Abs(Function('pdg0009489')(Symbol('pdg0001464'))), Integer(2)), Tuple(Symbol('pdg0001464')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba1= :a_node:expressionE( \vec{r},t)8572852424Symbol('pdg0004326')=Function('pdg0006238')(Symbol('pdg0009472'), Symbol('pdg0001467'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\vec{E} :a_node:expression8575746378\frac{1}{2} x=Integral(Rational(1, 2), Tuple(Symbol('pdg0001464')))Mul(Rational(1, 2), Symbol('pdg0001464'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\int \frac{1}{2} dx :a_node:expression8575748999-k^2 \left(a \sin(kx) + b \cos(kx) \right)Mul(Pow(Symbol('d'), Integer(2)), Pow(Symbol('pdg0009199'), Integer(-2)), Add(Mul(Symbol('pdg0001939'), cos(Mul(Symbol('pdg0001464'), Symbol('pdg0005321')))), Mul(Symbol('pdg0009139'), sin(Mul(Symbol('pdg0001464'), Symbol('pdg0005321'))))))Mul(Integer(-1), Pow(Symbol('pdg0005321'), Integer(2)), Add(Mul(Symbol('pdg0001939'), cos(Mul(Symbol('pdg0001464'), Symbol('pdg0005321')))), Mul(Symbol('pdg0009139'), sin(Mul(Symbol('pdg0001464'), Symbol('pdg0005321'))))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x) \right)= :a_node:expression8576785890\int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dxInteger(1)Integral(Mul(Pow(Symbol('pdg0009139'), Integer(2)), Add(Rational(1, 2), Mul(Integer(-1), Rational(1, 2), cos(Mul(Integer(2), Symbol('pdg0001464'), Symbol('pdg0001592'), Pow(Symbol('pdg0002523'), Integer(-1)), Symbol('pdg0003141')))))), Tuple(Symbol('pdg0001464'), Integer(0), Symbol('pdg0002523')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba1= :a_node:expression8577275751a \sin(0) + b\cos(0)Integer(0)Add(Mul(Symbol('pdg0001939'), cos(Integer(0))), Mul(Symbol('pdg0009139'), sin(Integer(0))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0= :a_node:expression8582885111a \sin(kx) + b \cos(kx)=Function('pdg0009489')(Symbol('pdg0004037'))Add(Mul(Symbol('pdg0001939'), cos(Mul(Symbol('pdg0004037'), Symbol('pdg0005321')))), Mul(Symbol('pdg0009139'), sin(Mul(Symbol('pdg0004037'), Symbol('pdg0005321')))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\psi(x) :a_node:expression(x + h)^28582954722=Add(Pow(Symbol('pdg0001464'), Integer(2)), Mul(Integer(2), Symbol('pdg0001464'), Symbol('pdg0003410')), Pow(Symbol('pdg0003410'), Integer(2)))Pow(Add(Symbol('pdg0001464'), Symbol('pdg0003410')), Integer(2))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax^2 + 2 x h + h^2 :a_node:expression8584698994\int d v_ySymbol('pdg0005674')=Mul(Integer(-1), Symbol('dt'), Symbol('g'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba-g \int dt :a_node:expression\cos( x )8588429722cos(Symbol('pdg0001464'))\sin( 90^{\circ} - x )a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=sin(Add(Integer(90), Mul(Integer(-1), Symbol('pdg0001464')))) :a_node:expression8602221482incoherent light source0cos(Add(Symbol('pdg0001575'), Mul(Integer(-1), Symbol('pdg0008586'))))Integer(0)a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\langle \cos(\theta - \phi) \rangle= :a_node:expressiona_x \hat{x} + a_y \hat{y}decompose acceleration into two components8602512487Symbol('pdg0002423')=Add(Mul(Symbol('pdg0001700'), Symbol('pdg0007055')), Mul(Symbol('pdg0007159'), Symbol('pdg0008339')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\vec{a} :a_node:expressionG \frac{m_1 m_2}{x^2} dx8604483515Mul(Pow(Symbol('pdg0004037'), Integer(-2)), Symbol('pdg0004851'), Symbol('pdg0005022'), Symbol('pdg0006277'), Symbol('pdg0009199'))Symbol('pdg0009398')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138badW= :a_node:expression8651044341\frac{1}{2} \left( \exp(-x) + \exp(x) \right)=cos(Mul(Symbol('pdg0001464'), Symbol('pdg0004621')))Add(Mul(Rational(1, 2), exp(Symbol('pdg0001464'))), Mul(Rational(1, 2), exp(Mul(Integer(-1), Symbol('pdg0001464')))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\cos(i x) :a_node:expression8655294002-\frac{k}{m}xMul(Integer(-1), Symbol('pdg0001356'), Symbol('pdg0004037'), Pow(Symbol('pdg0005156'), Integer(-1)))Symbol('pdg0009140')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baa= :a_node:expressionG \frac{m_{\rm Earth} m}{r_{\rm Earth}^2}8661803554Mul(Pow(Symbol('pdg0003236'), Integer(-2)), Symbol('pdg0005156'), Symbol('pdg0005458'), Symbol('pdg0006277'))Symbol('pdg0004202')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF= :a_node:expression8688588981mSymbol('pdg0009863')=Mul(Symbol('pdg0003935'), Pow(Symbol('pdg0005854'), Integer(3)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baa^3 \rho :a_node:expression\frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)8699789241=Mul(Integer(2), sin(Symbol('pdg0001464')), cos(Symbol('pdg0001464')))Mul(Rational(1, 2), Pow(Symbol('pdg0004621'), Integer(-1)), Add(exp(Mul(Integer(2), Symbol('pdg0001464'), Symbol('pdg0004621'))), Mul(Integer(-1), exp(Mul(Integer(-1), Integer(2), Symbol('pdg0001464'), Symbol('pdg0004621'))))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba2 \sin(x) \cos(x) :a_node:expression\left(\frac{v + v_0}{2}\right)t8706092970Mul(Symbol('pdg0001467'), Add(Mul(Rational(1, 2), Symbol('pdg0001357')), Mul(Rational(1, 2), Symbol('pdg0005153'))))Symbol('pdg0001943')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bad= :a_node:expression3.16 10^7 {\rm seconds}8721295221Integer(3)Symbol('pdg0005344')t_{\rm Earth\ orbit}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba= :a_node:expressionfirst term was multiplied by \gamma/\gammat'8730201316Symbol('pdg0001790')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t= :a_node:expression8747785338\cosh(x)cosh(Symbol('pdg0001464'))=cos(Mul(Symbol('pdg0001464'), Symbol('pdg0004621')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\cos(i x) :a_node:expression8750379055\frac{d}{dt} v_xInteger(0)Derivative(Symbol('pdg0005505'), Tuple(Symbol('pdg0001467'), Integer(1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0= :a_node:expression8808860551\int dyAdd(Mul(Integer(-1), Symbol('pdg0001649'), Integral(Symbol('pdg0001467'), Tuple(Symbol('pdg0001467')))), Mul(Symbol('pdg0009431'), Integral(Integer(1), Tuple(Symbol('pdg0001467')))))Integral(Integer(1), Tuple(Symbol('pdg0005647')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba-g \int t dt + v_{0, y} \int dt= :a_node:expression8849289982a \sin(\frac{n \pi}{W} x)=conjugate(Function('pdg0009489')(Symbol('pdg0004037')))Mul(Symbol('pdg0009139'), sin(Mul(Symbol('pdg0001592'), Pow(Symbol('pdg0002523'), Integer(-1)), Symbol('pdg0003141'), Symbol('pdg0004037'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\psi(x)^* :a_node:expression8889444440\int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right) \right)^2 dxInteger(1)Integral(Mul(Pow(Symbol('pdg0009139'), Integer(2)), Pow(sin(Mul(Symbol('pdg0001464'), Symbol('pdg0001592'), Pow(Symbol('pdg0002523'), Integer(-1)), Symbol('pdg0003141'))), Integer(2))), Tuple(Symbol('pdg0001464'), Integer(0), Symbol('pdg0002523')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba1= :a_node:expression8908736791\frac{m}{a^3}geometrySymbol('pdg0003935')=Mul(Pow(Symbol('pdg0005854'), Integer(-3)), Symbol('pdg0009863'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\rho :a_node:expression\frac{v_0^2}{g} \sin(2 \theta)8922441655Mul(Pow(Symbol('pdg0001649'), Integer(-1)), Pow(Symbol('pdg0005153'), Integer(2)), sin(Mul(Integer(2), Symbol('pdg0001575'))))Symbol('pdg0001943')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bad= :a_node:expression8945218208based on figure 34-27 on page 824 in \cite{2001_HRW}90^{\circ}Symbol('pdg0004928')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\theta_{\rm Brewster} + \theta_{\rm refracted}= :a_node:expression89463839372 G \frac{m}{r}=Pow(Symbol('pdg0008656'), Integer(2))Mul(Integer(2), Pow(Symbol('pdg0002530'), Integer(-1)), Symbol('pdg0005156'), Symbol('pdg0006277'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_{\rm escape}^2 :a_node:expressionv_{0, y}8949329361Symbol('pdg0009431')=Mul(Symbol('pdg0005153'), sin(Symbol('pdg0001575')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_0 \sin(\theta) :a_node:expressionm a x8953094349Mul(Symbol('pdg0004037'), Symbol('pdg0005156'), Symbol('pdg0009140'))Symbol('pdg0006789')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baW= :a_node:expressionKE_1 + PE_18960645192=Add(Symbol('pdg0001552'), Symbol('pdg0008849'))Add(Symbol('pdg0001955'), Symbol('pdg0004093'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baKE_2 + PE_2 :a_node:expression8991236357-\frac{k}{m} x=Mul(Pow(Symbol('d'), Integer(2)), Pow(Symbol('dt'), Integer(-2)), Symbol('pdg0004037'))Mul(Integer(-1), Symbol('pdg0001356'), Symbol('pdg0004037'), Pow(Symbol('pdg0005156'), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{d^2 x}{dt^2} :a_node:expression9031609275- \gamma^2 v t + \gamma v t'Symbol('pdg0001790')=a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax (1 - \gamma^2 ) :a_node:expression9059289981a \sin(k x)Symbol('pdg0001464')=a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\psi(x) :a_node:expressionV_1 + V_29063568209Symbol('pdg0004691')=Add(Symbol('pdg0008257'), Symbol('pdg0008721'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baV_{\rm total} :a_node:expressionG \frac{m_1 m_2}{r^2}9070394000Mul(Pow(Symbol('pdg0002530'), Integer(-2)), Symbol('pdg0004851'), Symbol('pdg0005022'), Symbol('pdg0006277'))m_2 d_2 \frac{4 \pi^2}{T^2}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Integer(4), Symbol('pdg0002798'), Pow(Symbol('pdg0003141'), Integer(2)), Symbol('pdg0004851'), Pow(Symbol('pdg0009491'), Integer(-2))) :a_node:expression9081138616\frac{1}{2} m_1 v_{\rm final}^2Symbol('pdg0006191')=Mul(Rational(1, 2), Symbol('pdg0005022'), Pow(Symbol('pdg0008909'), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baW_{\rm by\ system} :a_node:expression91121912010Integer(0)Symbol('pdg0007092')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bay_f= :a_node:expression\frac{1}{d_2 4 \pi^2} G \frac{m_1}{r^2}9152823411=Pow(Symbol('pdg0009491'), Integer(-2))Mul(Rational(1, 4), Pow(Symbol('pdg0002530'), Integer(-2)), Pow(Symbol('pdg0002798'), Integer(-1)), Pow(Symbol('pdg0003141'), Integer(-2)), Symbol('pdg0005022'), Symbol('pdg0006277'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{1}{T^2} :a_node:expression9170048197d_2 4 \pi^2 \frac{r^2}{G m_1}Pow(Symbol('pdg0009491'), Integer(2))Mul(Integer(4), Pow(Symbol('pdg0002530'), Integer(2)), Symbol('pdg0002798'), Pow(Symbol('pdg0003141'), Integer(2)), Pow(Symbol('pdg0005022'), Integer(-1)), Pow(Symbol('pdg0006277'), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baT^2= :a_node:expression\frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)9180861128=Mul(Integer(2), sin(Symbol('pdg0001464')), cos(Symbol('pdg0001464')))Mul(Rational(1, 2), Pow(Symbol('pdg0004621'), Integer(-1)), Add(exp(Mul(Integer(2), Symbol('pdg0001464'), Symbol('pdg0004621'))), Mul(Integer(-1), exp(Mul(Integer(-1), Integer(2), Symbol('pdg0001464'), Symbol('pdg0004621'))))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba2 \sin(x) \cos(x) :a_node:expression9191880568|Z| |Z| \exp( -i \theta ) \exp( i \theta )Symbol('pdg0003192')=a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baZ Z^* :a_node:expression9226945488\frac{m v^2}{r}Centripetal forceMul(Pow(Symbol('pdg0001357'), Integer(2)), Pow(Symbol('pdg0002530'), Integer(-1)), Symbol('pdg0005156'))https://en.wikipedia.org/wiki/Centripetal_forceSymbol('pdg0004202')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF= :a_node:expression9243879541I_2 R_2Mul(Symbol('pdg0003461'), Symbol('pdg0004856'))Symbol('pdg0006599')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baV= :a_node:expression92625967352 \pi rMul(Integer(2), Symbol('pdg0002530'), Symbol('pdg0003141'))Symbol('pdg0001943')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bad= :a_node:expression09285928292Integer(0)=Add(Mul(Pow(Symbol('pdg0001464'), Integer(2)), Symbol('pdg0009139')), Mul(Symbol('pdg0001464'), Symbol('pdg0001939')), Symbol('pdg0004231'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baax^2 + bx + c :a_node:expression9291999979-\mu_0\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t}Mul(Pow(Symbol('nabla'), Integer(2)), Symbol('pdg0004326'))Mul(Integer(-1), Symbol('nabla'), Symbol('pdg0006197'), Derivative(Symbol('pdg0002069'), Tuple(Symbol('pdg0001467'), Integer(1))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\vec{ \nabla} \times \vec{ \nabla} \times \vec{E}= :a_node:expression\hat{A}9294858532=Function('Dagger')(Function('Operator')(Symbol('pdg0005598')))Function('Operator')(Symbol('pdg0005598'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\hat{A}^+ :a_node:expression\frac{x_2 - x_1}{t}average velocity9337785146Mul(Pow(Symbol('pdg0001467'), Integer(-1)), Add(Mul(Integer(-1), Symbol('pdg0003852')), Symbol('pdg0005467')))Symbol('pdg0001357')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav= :a_node:expression9341391925v_{0, x} \hat{x} + v_{0, y} \hat{y}Symbol('pdg0006091')=Add(Mul(Symbol('pdg0001700'), Symbol('pdg0009431')), Mul(Symbol('pdg0002958'), Symbol('pdg0008339')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\vec{v}_0 :a_node:expressionm \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}9356924046Mul(Pow(Symbol('pdg0001467'), Integer(-1)), Symbol('pdg0005156'), Add(Mul(Integer(-1), Symbol('pdg0002473')), Symbol('pdg0004770')), Add(Mul(Rational(1, 2), Symbol('pdg0002473')), Mul(Rational(1, 2), Symbol('pdg0004770'))))\frac{KE_2 - KE_1}{t}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Pow(Symbol('pdg0001467'), Integer(-1)), Add(Symbol('pdg0001352'), Mul(Integer(-1), Symbol('pdg0001955')))) :a_node:expression9376481176f \frac{E}{a^3}proportionality coefficient fvaries in the range 1-4 for a majority of elemental solidsSymbol('K')Mul(Symbol('pdg0002241'), Pow(Symbol('pdg0005854'), Integer(-3)), Symbol('pdg0006235'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baK= :a_node:expression9385938295-(c/a) + (b/(2 a))^2=Pow(Add(Symbol('pdg0001464'), Mul(Rational(1, 2), Symbol('pdg0001939'), Pow(Symbol('pdg0009139'), Integer(-1)))), Integer(2))Add(Mul(Rational(1, 4), Pow(Symbol('pdg0001939'), Integer(2)), Pow(Symbol('pdg0009139'), Integer(-2))), Mul(Integer(-1), Symbol('pdg0004231'), Pow(Symbol('pdg0009139'), Integer(-1))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba(x+(b/(2 a)))^2 :a_node:expression9393939991-\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)=Function('pdg0009489')(Symbol('pdg0001464'))Mul(Integer(-1), Pow(Integer(2), Rational(1, 2)), Pow(Pow(Symbol('pdg0002523'), Integer(-1)), Rational(1, 2)), sin(Mul(Symbol('pdg0001464'), Symbol('pdg0001592'), Pow(Symbol('pdg0002523'), Integer(-1)), Symbol('pdg0003141'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\psi(x) :a_node:expression9393939992\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)=Function('pdg0009489')(Symbol('pdg0001464'))Mul(Pow(Integer(2), Rational(1, 2)), Pow(Pow(Symbol('pdg0002523'), Integer(-1)), Rational(1, 2)), sin(Mul(Symbol('pdg0001464'), Symbol('pdg0001592'), Pow(Symbol('pdg0002523'), Integer(-1)), Symbol('pdg0003141'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\psi(x) :a_node:expression\mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r},t)9394939493Mul(Symbol('partial'), Pow(Symbol('pdg0001467'), Integer(-2)), Symbol('pdg0006197'), Symbol('pdg0007940'), Function('pdg0006238')(Symbol('pdg0009472'), Symbol('pdg0001467')))\nabla^2 E( \vec{r},t)a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Pow(Symbol('nabla'), Integer(2)), Function('pdg0006238')(Symbol('pdg0009472'), Symbol('pdg0001467'))) :a_node:expression\frac{v_1 + v_2}{2}average velocity9397152918Add(Mul(Rational(1, 2), Symbol('pdg0002473')), Mul(Rational(1, 2), Symbol('pdg0004770')))Symbol('pdg0001357')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav= :a_node:expressionm g_{\rm Earth}9407192813Mul(Pow(Symbol('pdg0003236'), Integer(-2)), Symbol('pdg0005156'), Symbol('pdg0005458'), Symbol('pdg0006277'))Mul(Symbol('pdg0005156'), Symbol('pdg0007557'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baG \frac{m_{\rm Earth} m}{r_{\rm Earth}^2}= :a_node:expression9409776983\gamma v t'Symbol('pdg0001790')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax (1 - \gamma^2 ) + \gamma^2 v t= :a_node:expression2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}9412953728=Pow(Symbol('pdg0008656'), Integer(2))Mul(Integer(2), Pow(Symbol('pdg0003236'), Integer(-1)), Symbol('pdg0005458'), Symbol('pdg0006277'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_{\rm escape}^2 :a_node:expression94136092461Integer(1)\cosh^2 x - \sinh^2 xa84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Add(Mul(Integer(-1), Pow(sinh(Symbol('pdg0001464')), Integer(2))), Pow(cosh(Symbol('pdg0001464')), Integer(2))) :a_node:expressionm a \frac{v_2^2 - v_1^2}{2 a}9413699705Mul(Symbol('pdg0005156'), Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('pdg0002473'), Integer(2))), Mul(Rational(1, 2), Pow(Symbol('pdg0004770'), Integer(2)))))Symbol('pdg0006789')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baW= :a_node:expression9429829482-\sin(x) + i\cos(x)=Derivative(Symbol('pdg0001452'), Tuple(Symbol('pdg0001464'), Integer(1)))Add(Mul(Symbol('pdg0004621'), cos(Symbol('pdg0001464'))), Mul(Integer(-1), sin(Symbol('pdg0001464'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{d}{dx} y :a_node:expression9440616166\frac{g_{\rm Earth} r_{\rm Earth}^2}{G}Symbol('pdg0005458')=Mul(Pow(Symbol('pdg0003236'), Integer(2)), Pow(Symbol('pdg0006277'), Integer(-1)), Symbol('pdg0007557'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bam_{\rm Earth} :a_node:expression9482113948i dxSymbol('pdg0004621')=a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{dy}{y} :a_node:expression9482438243\cos(2 x) + (\sin(x))^2=Pow(cos(Symbol('pdg0001464')), Integer(2))Add(Pow(sin(Symbol('pdg0001464')), Integer(2)), cos(Mul(Integer(2), Symbol('pdg0001464'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba(\cos(x))^2 :a_node:expression9482923849ySymbol('pdg0001452')=exp(Mul(Symbol('pdg0001464'), Symbol('pdg0004621')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\exp(i x) :a_node:expression9482928242(\cos(x))^2 - (\sin(x))^2=cos(Mul(Integer(2), Symbol('pdg0001464')))Add(Mul(Integer(-1), Pow(sin(Symbol('pdg0001464')), Integer(2))), Pow(cos(Symbol('pdg0001464')), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\cos(2 x) :a_node:expression9482928243(\cos(x))^2Pow(cos(Symbol('pdg0001464')), Integer(2))\cos(2 x) + (\sin(x))^2a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Add(Pow(sin(Symbol('pdg0001464')), Integer(2)), cos(Mul(Integer(2), Symbol('pdg0001464')))) :a_node:expression9482943948i dx=log(Symbol('pdg0001452'), Integer(10))Mul(Symbol('pdg0004621'), Symbol('pdg0009199'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\log(y) :a_node:expression9482984922(i\sin(x) + \cos(x)) i=Derivative(Symbol('pdg0001452'), Tuple(Symbol('pdg0001464'), Integer(1)))Mul(Symbol('pdg0004621'), Add(Mul(Symbol('pdg0004621'), sin(Symbol('pdg0001464'))), cos(Symbol('pdg0001464'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{d}{dx} y :a_node:expression(\cos(x))^2 + 2 i \cos(x) \sin(x) - (\sin(x))^29483928192Add(Mul(Integer(2), Symbol('pdg0004621'), sin(Symbol('pdg0001464')), cos(Symbol('pdg0001464'))), Mul(Integer(-1), Pow(sin(Symbol('pdg0001464')), Integer(2))), Pow(cos(Symbol('pdg0001464')), Integer(2)))\cos(2 x) + i\sin(2 x)a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Add(Mul(Symbol('pdg0004621'), sin(Mul(Integer(2), Symbol('pdg0001464')))), cos(Mul(Integer(2), Symbol('pdg0001464')))) :a_node:expression9485384858- \frac{\omega^2}{c^2} E( \vec{r})\exp(i \omega t)Mul(Pow(Symbol('nabla'), Integer(2)), Function('pdg0002718')(Mul(Symbol('pdg0001467'), Symbol('pdg0002321'), Symbol('pdg0004621'))), Function('pdg0006238')(Symbol('pdg0009472')))Mul(Integer(-1), Pow(Symbol('pdg0002321'), Integer(2)), Pow(Symbol('pdg0004567'), Integer(-2)), Function('pdg0002718')(Mul(Symbol('pdg0001467'), Symbol('pdg0002321'), Symbol('pdg0004621'))), Function('pdg0006238')(Symbol('pdg0009472')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\nabla^2 E( \vec{r})\exp(i \omega t)= :a_node:expression9485747245aSymbol('pdg0009139')=Mul(Pow(Integer(2), Rational(1, 2)), Pow(Pow(Symbol('pdg0002523'), Integer(-1)), Rational(1, 2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\sqrt{\frac{2}{W}} :a_node:expression9485747246aSymbol('pdg0009139')=Mul(Integer(-1), Pow(Integer(2), Rational(1, 2)), Pow(Pow(Symbol('pdg0002523'), Integer(-1)), Rational(1, 2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba-\sqrt{\frac{2}{W}} :a_node:expression9492920340\cos(x)+i \sin(x)Add(Mul(Symbol('pdg0004621'), sin(Symbol('pdg0001464'))), cos(Symbol('pdg0001464')))Symbol('pdg0001452')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bay= :a_node:expression94958572782022-03-25 BHP: Conversion between Latex and Sympy is incomplete0Symbol('pdg0002523')=a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\psi(x=W) :a_node:expression9499428242E( \vec{r})\exp(i \omega t)=Function('pdg0006238')(Symbol('pdg0009472'), Symbol('pdg0001467'))Mul(Function('pdg0002718')(Mul(Symbol('pdg0001467'), Symbol('pdg0002321'), Symbol('pdg0004621'))), Function('pdg0006238')(Symbol('pdg0009472')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baE( \vec{r},t) :a_node:expression09510328252Symbol('pdg0004121')=Integer(0)a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baKE_{\rm initial} :a_node:expression\frac{k_{\rm desorption} [A_{\rm adsorption}]}{k_{\rm adsorption} p_A}9562264720Mul(Symbol('pdg0004940'), Pow(Symbol('pdg0006850'), Integer(-1)), Symbol('pdg0008379'), Pow(Symbol('pdg0009046'), Integer(-1)))Symbol('pdg0009067')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba[S]= :a_node:expression\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))9582958293Add(Mul(Integer(-1), Rational(1, 2), Symbol('pdg0001939'), Pow(Symbol('pdg0009139'), Integer(-1))), Pow(Add(Mul(Rational(1, 4), Pow(Symbol('pdg0001939'), Integer(2)), Pow(Symbol('pdg0009139'), Integer(-2))), Mul(Integer(-1), Symbol('pdg0004231'), Pow(Symbol('pdg0009139'), Integer(-1)))), Rational(1, 2)))Symbol('pdg0001464')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax= :a_node:expression9582958294\sqrt{(b/(2 a))^2 - (c/a)}=Add(Symbol('pdg0001464'), Mul(Rational(1, 2), Symbol('pdg0001939'), Pow(Symbol('pdg0009139'), Integer(-1))))Pow(Add(Mul(Rational(1, 4), Pow(Symbol('pdg0001939'), Integer(2)), Pow(Symbol('pdg0009139'), Integer(-2))), Mul(Integer(-1), Symbol('pdg0004231'), Pow(Symbol('pdg0009139'), Integer(-1)))), Rational(1, 2))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax+(b/(2 a)) :a_node:expression09585727710=Function('pdg0009489')(Equality(Symbol('pdg0001464'), Integer(0)))Integer(0)a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\psi(x=0) :a_node:expression\langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle9596004948Mul(Symbol('pdg0005598'), Function('Bra')(Symbol('pdg0004679')), Function('Ket')(Symbol('pdg0002090')))Symbol('pdg0001464')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax= :a_node:expression\frac{e^2}{4 \pi \epsilon_0 \hbar} \sqrt{\frac{m_e}{2 m}}9640720571Mul(Rational(1, 8), Pow(Integer(2), Rational(1, 2)), Pow(Symbol('pdg0001054'), Integer(-1)), Pow(Symbol('pdg0001999'), Integer(2)), Pow(Symbol('pdg0003141'), Integer(-1)), Pow(Symbol('pdg0007940'), Integer(-1)), Pow(Mul(Symbol('pdg0002515'), Pow(Symbol('pdg0009863'), Integer(-1))), Rational(1, 2)))Symbol('pdg0002077')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav= :a_node:expressionv_0 t + \frac{1}{2} a t^29658195023Add(Mul(Rational(1, 2), Pow(Symbol('pdg0001467'), Integer(2)), Symbol('pdg0009140')), Mul(Symbol('pdg0001467'), Symbol('pdg0005153')))Symbol('pdg0001943')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bad= :a_node:expression9703482302\frac{1}{2} m v_{\rm escape}^2Mul(Pow(Symbol('pdg0003236'), Integer(-1)), Symbol('pdg0005156'), Symbol('pdg0005458'), Symbol('pdg0006277'))Mul(Rational(1, 2), Symbol('pdg0005156'), Pow(Symbol('pdg0008656'), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baG \frac{m_{\rm Earth} m}{r_{\rm Earth}}= :a_node:expression97070280610Integer(0)Symbol('pdg0007159')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baa_x= :a_node:expression9718685793\frac{1}{P}Symbol('pdg0004645')=Pow(Symbol('pdg0008134'), Integer(-1))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\kappa_T :a_node:expressionKE_1 + PE_19749777192Integer(0)Add(Symbol('pdg0001955'), Symbol('pdg0004093'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0= :a_node:expression\frac{n_2}{n_1} \cos( \theta_{\rm Brewster} )9756089533Mul(Symbol('pdg0001958'), Pow(Symbol('pdg0002941'), Integer(-1)), cos(Symbol('pdg0004928')))sin(Symbol('pdg0004928'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\sin( \theta_{\rm Brewster} )= :a_node:expression9759901995a t=Add(Symbol('pdg0001357'), Mul(Integer(-1), Symbol('pdg0005153')))Mul(Symbol('pdg0001467'), Symbol('pdg0009140'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav - v_0 :a_node:expressiondefinition of isothermal compressibility\frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T9781951738Symbol('pdg0004645')=Mul(Integer(-1), Pow(Symbol('pdg0007586'), Integer(-1)), Derivative(Symbol('pdg0007586'), Tuple(Symbol('pdg0008134'), Integer(1))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\kappa_T :a_node:expression9805063945c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2Add(Mul(Pow(Symbol('pdg0001790'), Integer(2)), Pow(Add(Mul(Integer(-1), Symbol('pdg0001357'), Symbol('pdg0001467')), Symbol('pdg0004037')), Integer(2))), Pow(Symbol('pdg0005647'), Integer(2)), Pow(Symbol('pdg0006728'), Integer(2)))Mul(Pow(Symbol('pdg0001790'), Integer(2)), Pow(Symbol('pdg0004567'), Integer(2)), Pow(Add(Symbol('pdg0001467'), Mul(Pow(Symbol('pdg0001357'), Integer(-1)), Pow(Symbol('pdg0001790'), Integer(-2)), Symbol('pdg0004037'), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('pdg0001790'), Integer(2)))))), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\gamma^2 (x - v t)^2 + y^2 + z^2= :a_node:expression9838128064G \frac{m_1}{r^2}Mul(Pow(Symbol('pdg0002530'), Integer(-2)), Symbol('pdg0005022'), Symbol('pdg0006277'))d_2 \frac{4 \pi^2}{T^2}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Integer(4), Symbol('pdg0002798'), Pow(Symbol('pdg0003141'), Integer(2)), Pow(Symbol('pdg0009491'), Integer(-2))) :a_node:expression9847143017\frac{-PV}{V} \left( \frac{-1}{P^2}\right)Symbol('pdg0004645')=Pow(Symbol('pdg0008134'), Integer(-1))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\kappa_T :a_node:expression9848292229y i dxMul(Symbol('pdg0001452'), Symbol('pdg0004621'), Symbol('pdg0009199'))Symbol('pdg0005842')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bady= :a_node:expressiony i9848294829=Derivative(Symbol('pdg0001452'), Tuple(Symbol('pdg0001464'), Integer(1)))Mul(Symbol('pdg0001452'), Symbol('pdg0004621'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{d}{dx} y :a_node:expression9854442418\sqrt{\frac{E}{m}}Pow(Mul(Symbol('pdg0002241'), Pow(Symbol('pdg0009863'), Integer(-1))), Rational(1, 2))Symbol('pdg0002077')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav= :a_node:expression\int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx9858028950=Pow(Symbol('pdg0009139'), Integer(-2))Integral(Add(Rational(1, 2), Mul(Integer(-1), Rational(1, 2), cos(Mul(Integer(2), Symbol('pdg0001464'), Symbol('pdg0001592'), Pow(Symbol('pdg0002523'), Integer(-1)), Symbol('pdg0003141'))))), Tuple(Symbol('pdg0001464'), Integer(0), Symbol('pdg0002523')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{1}{a^2} :a_node:expression- \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_09862900242Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('pdg0001467'), Integer(2)), Symbol('pdg0001649')), Mul(Symbol('pdg0001467'), Symbol('pdg0005153'), sin(Symbol('pdg0001575'))), Symbol('pdg0001469'))Symbol('pdg0005647')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bay= :a_node:expressionx - x_09882526611=Mul(Symbol('pdg0001467'), Symbol('pdg0002958'))Add(Mul(Integer(-1), Symbol('pdg0001572')), Symbol('pdg0004037'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_{0, x} t :a_node:expression98899842811 - \cos(2 x)=Mul(Integer(2), Pow(sin(Symbol('pdg0001464')), Integer(2)))Add(Integer(1), Mul(Integer(-1), cos(Mul(Integer(2), Symbol('pdg0001464')))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba2 (\sin(x))^2 :a_node:expression\frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \left(\exp(i x)+\exp(-i x) \right)9894826550=Mul(Integer(2), sin(Symbol('pdg0001464')), cos(Symbol('pdg0001464')))Mul(Rational(1, 2), Pow(Symbol('pdg0004621'), Integer(-1)), Add(exp(Mul(Symbol('pdg0001464'), Symbol('pdg0004621'))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('pdg0001464'), Symbol('pdg0004621'))))), Add(exp(Mul(Symbol('pdg0001464'), Symbol('pdg0004621'))), exp(Mul(Integer(-1), Symbol('pdg0001464'), Symbol('pdg0004621')))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba2 \sin(x) \cos(x) :a_node:expression9897284307\frac{v + v_0}{2}=Mul(Pow(Symbol('pdg0001467'), Integer(-1)), Symbol('pdg0001943'))Add(Mul(Rational(1, 2), Symbol('pdg0001357')), Mul(Rational(1, 2), Symbol('pdg0005153')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{d}{t} :a_node:expression99199999810Symbol('pdg0003935')=Integer(0)a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\rho :a_node:expression\left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV9941599459based on U(p, T, V) = U(T, V)Symbol('dU')Derivative(Symbol('pdg0005786'), Tuple(Symbol('pdg0007343'), Integer(1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138badU= :a_node:expression9958485859i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)Mul(Mul(Mul(Integer(-1), Pow(Symbol('pdg0001054'), Integer(2))), Pow(Mul(Integer(2), Symbol('pdg0005156')), Integer(-1))), Mul(Mul(Symbol('nabla'), Symbol('nabla')), Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right)= :a_node:expressionv_y - v_{0, y}9973952056Mul(Integer(-1), Symbol('pdg0001467'), Symbol('pdg0001649'))Add(Mul(Integer(-1), Symbol('pdg0005153')), Symbol('pdg0009431'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba-g t= :a_node:expression9988949211\frac{1 - \cos(2 x)}{2}=Pow(sin(Symbol('pdg0001464')), Integer(2))Add(Rational(1, 2), Mul(Integer(-1), Rational(1, 2), cos(Mul(Integer(2), Symbol('pdg0001464')))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba(\sin(x))^2 :a_node:expression9991999979-\mu_0\frac{\partial \vec{H}}{\partial t}Function('cross')(Symbol('pdg0006238'), Symbol('nabla'))Mul(Integer(-1), Symbol('pdg0006197'), Derivative(Symbol('pdg0002069'), Tuple(Symbol('pdg0001467'), Integer(1))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\vec{ \nabla} \times \vec{E}= :a_node:expression\vec{k}9999998870Symbol('pdg0007394')\frac{ \vec{p}}{\hbar}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Pow(Symbol('pdg0001054'), Integer(-1)), Symbol('pdg0002046')) :a_node:expression9999999870kSymbol('pdg0005321')=Mul(Pow(Symbol('pdg0001054'), Integer(-1)), Symbol('pdg0001134'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{p}{\hbar} :a_node:expression9999999960h/(2 \pi)Symbol('pdg0001054')=Mul(Rational(1, 2), Pow(Symbol('pdg0003141'), Integer(-1)), Symbol('pdg0004413'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\hbar :a_node:expression\omega9999999961Symbol('pdg0002321')=Mul(Pow(Symbol('pdg0001054'), Integer(-1)), Symbol('pdg0004931'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{E}{\hbar} :a_node:expression9999999962\hbar kMul(Symbol('pdg0001054'), Symbol('pdg0005321'))Symbol('pdg0001134')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bap= :a_node:expression9999999965\omega \hbarMul(Symbol('pdg0001054'), Symbol('pdg0002321'))Symbol('pdg0004931')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baE= :a_node:expression\frac{-b-\sqrt{b^2-4ac}}{2 a}9999999968Mul(Rational(1, 2), Pow(Symbol('pdg0009139'), Integer(-1)), Add(Mul(Integer(-1), Symbol('pdg0001939')), Mul(Integer(-1), Pow(Add(Pow(Symbol('pdg0001939'), Integer(2)), Mul(Integer(-1), Integer(4), Symbol('pdg0004231'), Symbol('pdg0009139'))), Rational(1, 2)))))Symbol('pdg0001464')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax= :a_node:expression\frac{-b+\sqrt{b^2-4ac}}{2 a}9999999969Mul(Rational(1, 2), Pow(Symbol('pdg0009139'), Integer(-1)), Add(Mul(Integer(-1), Symbol('pdg0001939')), Pow(Add(Pow(Symbol('pdg0001939'), Integer(2)), Mul(Integer(-1), Integer(4), Symbol('pdg0004231'), Symbol('pdg0009139'))), Rational(1, 2))))Symbol('pdg0001464')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax= :a_node:expression9999999975\langle a \rangleSymbol('pdg0004065')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\langle \psi| \hat{A} |\psi \rangle= :a_node:expression9999999981\rho/\epsilon_0Mul(Symbol('pdg0003935'), Pow(Symbol('pdg0007940'), Integer(-1)))\vec{ \nabla} \cdot \vec{E}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Symbol('nabla'), Symbol('pdg0004326')) :a_node:expression0203024440\int_0^W a \sin\left(\frac{n \pi}{W} x\right) \psi(x)^* dxInteger(1)Integral(Mul(Symbol('pdg0009139'), sin(Mul(Symbol('pdg0001464'), Symbol('pdg0001592'), Pow(Symbol('pdg0002523'), Integer(-1)), Symbol('pdg0003141'))), conjugate(Function('pdg0009489')(Symbol('pdg0001464')))), Tuple(Symbol('pdg0001464'), Integer(0), Symbol('pdg0002523')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba1= :a_node:expression0404050504\frac{v}{f}Symbol('pdg0001115')=Mul(Symbol('pdg0001357'), Pow(Symbol('pdg0004201'), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\lambda :a_node:expression\frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) \right|_0^W0439492440https://physicsderivationgraph.blogspot.com/2020/09/evaluating-definite-integrals-for.html=Pow(Symbol('pdg0009139'), Integer(-2))Add(Mul(Rational(1, 2), Symbol('pdg0002523')), Mul(Integer(-1), Rational(1, 4), Pow(Symbol('pdg0001592'), Integer(-1)), Symbol('pdg0002523'), Pow(Symbol('pdg0003141'), Integer(-1)), sin(Mul(Integer(2), Symbol('pdg0001592'), Pow(Symbol('pdg0002523'), Integer(-1)), Symbol('pdg0003141'), Symbol('pdg0004037')))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{1}{a^2} :a_node:expression0934990943\frac{2 \pi}{v T}Mul(Integer(2), Pow(Symbol('pdg0001357'), Integer(-1)), Symbol('pdg0003141'), Pow(Symbol('pdg0009491'), Integer(-1)))Symbol('pdg0005321')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bak= :a_node:expression0948572140\frac{1}{a}\sin(a x)=Integral(cos(Mul(Symbol('pdg0001464'), Symbol('pdg0009139'))), Tuple(Symbol('pdg0009199')))Mul(Pow(Symbol('pdg0009139'), Integer(-1)), sin(Mul(Symbol('pdg0001464'), Symbol('pdg0009139'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\int \cos(a x) dx :a_node:expression\langle a \rangle^*https://docs.sympy.org/latest/modules/stats.html1010393913Bra('pdg0004065')*Dagger(Operator('pdg0005598'))*Ket('pdg0009329')conjugate(E(Symbol('pdg0009139')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\langle \psi| \hat{A}^+ |\psi \rangle= :a_node:expression\langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle1010393944Mul(Symbol('pdg0007752'), Function('Bra')(Symbol('pdg0004679')), Function('Ket')(Symbol('pdg0002090')))Symbol('pdg0001464')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax= :a_node:expression1010923823n \piMul(Symbol('pdg0002523'), Symbol('pdg0005321'))Mul(Symbol('pdg0001592'), Symbol('pdg0003141'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bak W= :a_node:expressiona \sin(k W)1020010291Integer(0)Mul(Symbol('pdg0009139'), sin(Mul(Symbol('pdg0002523'), Symbol('pdg0005321'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0= :a_node:expression1020394900h/\lambdaMul(Pow(Symbol('pdg0001115'), Integer(-1)), Symbol('pdg0004413'))Symbol('pdg0001134')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bap= :a_node:expressionh f1020394902Mul(Symbol('pdg0004201'), Symbol('pdg0004413'))Symbol('pdg0004931')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baE= :a_node:expression1020854560(A + B)(A + B)^*Mul(Add(Symbol('pdg0004453'), Symbol('pdg0004698')), conjugate(Add(Symbol('pdg0004453'), Symbol('pdg0004698'))))Symbol('pdg0007882')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baI= :a_node:expressionm v1029039903Mul(Symbol('pdg0001357'), Symbol('pdg0005156'))Symbol('pdg0001134')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bap= :a_node:expression1029039904m^2 v^2Pow(Symbol('pdg0001134'), Integer(2))Mul(Pow(Symbol('pdg0001357'), Integer(2)), Pow(Symbol('pdg0005156'), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bap^2= :a_node:expression1038566242\frac{\exp(x) - \exp(-x)}{2}sinh(Symbol('pdg0001464'))=Add(Mul(Rational(1, 2), exp(Symbol('pdg0001464'))), Mul(Integer(-1), Rational(1, 2), exp(Mul(Integer(-1), Symbol('pdg0001464')))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\sinh x :a_node:expressiondefinition of heat capacity at constant volume\left(\frac{\partial U}{\partial T}\right)_V1085150613Derivative(Symbol('pdg0005786'), Tuple(Symbol('pdg0007343'), Integer(1)))Symbol('pdg0006682')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baC_V= :a_node:expression1087417579- \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)Integer(0)Add(Mul(Integer(-1), Rational(1, 2), Symbol('pdg0001649'), Pow(Symbol('pdg0002467'), Integer(2))), Mul(Symbol('pdg0002467'), Symbol('pdg0005153'), sin(Symbol('pdg0001575'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0= :a_node:expression\Delta KE1114820451Work is change in energySymbol('pdg0005734')Symbol('pdg0006191')=a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baW_{\rm by\ system} :a_node:expression1128605625\frac{4}{\left(\exp(x)+\exp(-x)\right)^2} + \frac{\left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2}Add(Pow(tanh(Symbol('pdg0001464')), Integer(2)), Pow(sech(Symbol('pdg0001464')), Integer(2)))Add(Mul(Pow(Add(exp(Symbol('pdg0001464')), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('pdg0001464'))))), Integer(2)), Pow(Add(exp(Symbol('pdg0001464')), exp(Mul(Integer(-1), Symbol('pdg0001464')))), Integer(-2))), Mul(Integer(4), Pow(Add(exp(Symbol('pdg0001464')), exp(Mul(Integer(-1), Symbol('pdg0001464')))), Integer(-2))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba{\rm sech}^2\ x + \tanh^2(x)= :a_node:expression1132941271\frac{(9.80665 m/s^2) (6.3781*10^6 m)^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}}Symbol('pdg0005458')=Float('6.3780999999999999', precision=53)a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bam_{\rm Earth} :a_node:expression1143343287\frac{1}{2} v_{\rm escape}^2Mul(Pow(Symbol('pdg0003236'), Integer(-1)), Symbol('pdg0005458'), Symbol('pdg0006277'))Mul(Rational(1, 2), Pow(Symbol('pdg0008656'), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baG \frac{m_{\rm Earth}}{r_{\rm Earth}}= :a_node:expression{\cal H}1158485859Mul(Integer(-1), Rational(1, 2), Pow(Symbol('nabla'), Integer(2)), Pow(Symbol('pdg0001054'), Integer(2)), Pow(Symbol('pdg0005156'), Integer(-1)))Symbol('calH')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{-\hbar^2}{2m} \nabla^2= :a_node:expression1166310428d v_xSymbol('pdg0005005')Integer(0)a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0 dt= :a_node:expression4 |A|^21172039918Symbol('pdg0008251')=Mul(Integer(4), Pow(Abs(Symbol('pdg0004453')), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baI_{\rm coherent} :a_node:expression1190768176\frac{-nRT}{V} \left( \frac{ \partial }{\partial P}\left(\frac{1}{P}\right) \right)_TSymbol('pdg0004645')=Mul(Integer(-1), Symbol('pdg0002834'), Symbol('pdg0007343'), Pow(Symbol('pdg0007586'), Integer(-1)), Symbol('pdg0008179'), Derivative(Pow(Symbol('pdg0008134'), Integer(-1)), Tuple(Symbol('pdg0008134'), Integer(1))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\kappa_T :a_node:expression1191796961v_0 \sin(\theta)=Mul(Rational(1, 2), Symbol('pdg0001649'), Symbol('pdg0002467'))Mul(Symbol('pdg0005153'), sin(Symbol('pdg0001575')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{1}{2} g t_f :a_node:expressionc^2 t'^21201689765describes a spherical wavefront for an observer in a moving frame of reference=Add(Pow(Symbol('pdg0001888'), Integer(2)), Pow(Symbol('pdg0004306'), Integer(2)), Pow(Symbol('pdg0005456'), Integer(2)))Mul(Pow(Symbol('pdg0004567'), Integer(2)), Pow(Symbol('pdg0004989'), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax'^2 + y'^2 + z'^2 :a_node:expression\int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx1202310110=Pow(Symbol('pdg0009139'), Integer(-2))Integral(Add(Mul(Rational(1, 2), Symbol('pdg0009199')), Mul(Integer(-1), Rational(1, 2), Integral(cos(Mul(Integer(2), Symbol('pdg0001592'), Pow(Symbol('pdg0002523'), Integer(-1)), Symbol('pdg0003141'), Symbol('pdg0004037'))), Tuple(Symbol('pdg0004037'), Integer(0), Symbol('pdg0002523'))))), Tuple(Symbol('pdg0004037'), Integer(0), Symbol('pdg0002523')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{1}{a^2} :a_node:expression\frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx1202312210=Pow(Symbol('pdg0009139'), Integer(-2))Add(Mul(Rational(1, 2), Symbol('pdg0002523')), Mul(Integer(-1), Rational(1, 2), Integral(cos(Mul(Integer(2), Symbol('pdg0001592'), Pow(Symbol('pdg0002523'), Integer(-1)), Symbol('pdg0003141'), Symbol('pdg0004037'))), Tuple(Symbol('pdg0004037'), Integer(0), Symbol('pdg0002523')))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{1}{a^2} :a_node:expression1203938249a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangleSymbol('pdg0007752')*Bra('pdg0004679')*Ket('pdg0002090')Symbol('pdg0007752')*Bra('pdg0004679')*Ket('pdg0002090'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baa_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle= :a_node:expression12482777731 - 2 (\sin(x))^2=cos(Mul(Integer(2), Symbol('pdg0001464')))Add(Integer(1), Mul(Integer(-1), Integer(2), Pow(sin(Symbol('pdg0001464')), Integer(2))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\cos(2 x) :a_node:expression(v - a t) t + \frac{1}{2} a t^21259826355Add(Mul(Rational(1, 2), Pow(Symbol('pdg0001467'), Integer(2)), Symbol('pdg0009140')), Mul(Symbol('pdg0001467'), Add(Symbol('pdg0001357'), Mul(Integer(-1), Symbol('pdg0001467'), Symbol('pdg0009140')))))Symbol('pdg0001943')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bad= :a_node:expression\frac{2 v_0 + a t}{2} t1265150401Mul(Symbol('pdg0001467'), Add(Mul(Rational(1, 2), Symbol('pdg0001467'), Symbol('pdg0009140')), Symbol('pdg0005153')))Symbol('pdg0001943')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bad= :a_node:expression1292735067G \frac{m_1 m_2}{r^2}Symbol('pdg0002867')=Mul(Pow(Symbol('pdg0002530'), Integer(-2)), Symbol('pdg0004851'), Symbol('pdg0005022'), Symbol('pdg0006277'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF_{\rm gravity} :a_node:expression1293913110bSymbol('pdg0001939')Integer(0)a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0= :a_node:expression1293923844v TSymbol('pdg0001115')=Mul(Symbol('pdg0001357'), Symbol('pdg0009491'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\lambda :a_node:expression1306360899v_{0, x} t + x_0Add(Mul(Symbol('pdg0001467'), Symbol('pdg0002958')), Symbol('pdg0001572'))Symbol('pdg0004037')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax= :a_node:expression90^{\circ} - \theta_{\rm Brewster}1310571337Symbol('pdg0004928')\theta_{\rm refracted}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba= :a_node:expression1311403394\frac{1}{V} \frac{nR}{P} \left( \frac{\partial T}{\partial T} \right)_PSymbol('pdg0004686')=Mul(Symbol('pdg0002834'), Pow(Symbol('pdg0007586'), Integer(-1)), Pow(Symbol('pdg0008134'), Integer(-1)), Symbol('pdg0008179'), Derivative(Symbol('pdg0007343'), Tuple(Symbol('pdg0007343'), Integer(1))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\alpha :a_node:expression\epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}1314464131Symbol('pdg0001467')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t}= :a_node:expression1314864131\epsilon_0 \frac{\partial }{\partial t}\vec{E}Function('cross')(Symbol('nabla'), Symbol('pdg0002069'))Mul(Symbol('pdg0007940'), Derivative(Symbol('pdg0004326'), Tuple(Symbol('pdg0001467'), Integer(1))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\vec{ \nabla} \times \vec{H}= :a_node:expression1330874553\sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}Symbol('pdg0008656')=Mul(Pow(Integer(2), Rational(1, 2)), Pow(Mul(Pow(Symbol('pdg0003236'), Integer(-1)), Symbol('pdg0005458'), Symbol('pdg0006277')), Rational(1, 2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_{\rm escape} :a_node:expression1357848476|A| \exp(i \theta)Mul(exp(Mul(Symbol('pdg0001575'), Symbol('pdg0004621'))), Abs(Symbol('pdg0004453')))Symbol('pdg0004453')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baA= :a_node:expression\langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle1395858355Mul(Symbol('pdg0002427'), Function('Bra')(Symbol('pdg0004679')), Function('Ket')(Symbol('pdg0002090')))Symbol('pdg0001464')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax= :a_node:expression- \frac{1}{2} g t^2 + v_{0, y} t + y_01405465835Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('pdg0001467'), Integer(2)), Symbol('pdg0001649')), Mul(Symbol('pdg0001467'), Symbol('pdg0009107')), Symbol('pdg0001469'))Symbol('pdg0005647')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bay= :a_node:expression\frac{1}{R_1} + \frac{1}{R_2}1457415749Add(Pow(Symbol('pdg0008697'), Integer(-1)), Pow(Symbol('pdg0003461'), Integer(-1)))total resistance for two resistors in parallel\frac{1}{R_{\rm total}}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Pow(Symbol('pdg0001908'), Integer(-1)) :a_node:expression|A|^2 + |B|^2 + A B^* + B A^*1525861537Add(Mul(Symbol('pdg0004453'), conjugate(Symbol('pdg0004698'))), Mul(Symbol('pdg0004698'), conjugate(Symbol('pdg0004453'))), Pow(Abs(Symbol('pdg0004453')), Integer(2)), Pow(Abs(Symbol('pdg0004698')), Integer(2)))Symbol('pdg0007882')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baI= :a_node:expression1528310784\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}Symbol('pdg0001790')=Pow(Add(Mul(Integer(-1), Pow(Symbol('pdg0001357'), Integer(2)), Pow(Symbol('pdg0004567'), Integer(-2))), Integer(1)), Rational(-1, 2))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\gamma :a_node:expression\frac{\pi}{4}1541916015Symbol('pdg0001575')=Mul(Rational(1, 4), Symbol('pdg0003141'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\theta :a_node:expression1556389363 the bonding energy in condensed phases is given by the Rydberg energy on the order of several e\frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}Symbol('pdg0009838')=Mul(Rational(1, 32), Pow(Symbol('pdg0001054'), Integer(-2)), Pow(Symbol('pdg0001999'), Integer(4)), Symbol('pdg0002515'), Pow(Symbol('pdg0003141'), Integer(-2)), Pow(Symbol('pdg0007940'), Integer(-2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baE_{\rm Rydberg} :a_node:expressionr_{\rm geostationary\ orbit}1559688463Symbol('pdg0007110')Mul(Rational(1, 2), Pow(Integer(2), Rational(1, 3)), Pow(Mul(Pow(Symbol('pdg0003141'), Integer(-2)), Symbol('pdg0005458'), Pow(Symbol('pdg0005595'), Integer(2)), Symbol('pdg0006277')), Rational(1, 3)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\left(\frac{T_{\rm geostationary\ orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3}= :a_node:expression1586866563t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)Add(Mul(Integer(-1), Integer(2), Symbol('pdg0001357'), Symbol('pdg0001467'), Pow(Symbol('pdg0001790'), Integer(2)), Symbol('pdg0004037')), Mul(Pow(Symbol('pdg0004037'), Integer(2)), Add(Pow(Symbol('pdg0001790'), Integer(2)), Mul(Integer(-1), Pow(Symbol('pdg0001357'), Integer(-2)), Pow(Symbol('pdg0001790'), Integer(-2)), Pow(Symbol('pdg0004567'), Integer(2)), Pow(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('pdg0001790'), Integer(2)))), Integer(2))))), Pow(Symbol('pdg0005647'), Integer(2)), Pow(Symbol('pdg0006728'), Integer(2)), Mul(Integer(-1), Integer(2), Pow(Symbol('pdg0001357'), Integer(-1)), Symbol('pdg0001467'), Symbol('pdg0004037'), Pow(Symbol('pdg0004567'), Integer(2)), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('pdg0001790'), Integer(2))))))Mul(Pow(Symbol('pdg0001467'), Integer(2)), Add(Mul(Integer(-1), Pow(Symbol('pdg0001357'), Integer(2)), Pow(Symbol('pdg0001790'), Integer(2))), Mul(Pow(Symbol('pdg0001790'), Integer(2)), Pow(Symbol('pdg0004567'), Integer(2)))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\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)= :a_node:expressionF dx1590774089Mul(Symbol('pdg0004202'), Symbol('pdg0009199'))Symbol('pdg0009398')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138badW= :a_node:expression- \nabla^2 \vec{E}1636453295Function('cross')(Symbol('nabla'), Function('cross')(Symbol('nabla'), Symbol('pdg0004326')))Mul(Integer(-1), Pow(Symbol('nabla'), Integer(2)), Symbol('pdg0004326'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\vec{ \nabla} \times \vec{ \nabla} \times \vec{E}= :a_node:expression1638282134\vec{p}_{\rm after}Symbol('pdg0005493')Symbol('pdg0001302')\vec{p}_{\rm before}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba= :a_node:expression16398274921Mul(Integer(-1), Pow(Symbol('pdg0001357'), Integer(-2)), Pow(Symbol('pdg0001790'), Integer(-2)), Pow(Symbol('pdg0004567'), Integer(2)), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('pdg0001790'), Integer(2)))))Integer(1)a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba- c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2}= :a_node:expression1648958381https://physicsderivationgraph.blogspot.com/2020/09/representing-laplace-operator-nabla-in.html\frac{i}{\hbar} \vec{p} \cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right)Mul(Pow(Symbol('nabla'), Integer(2)), Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467')))Mul(Pow(Symbol('pdg0001054'), Integer(-1)), Symbol('pdg0004621'), Function('pdg0002046')(Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\nabla^2 \psi \left( \vec{r},t \right)= :a_node:expression1650441634define coordinate system such that initial height is at origin0Integer(0)Symbol('pdg0001469')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bay_0= :a_node:expression1676472948v_x - v_{0, x}Integer(0)Add(Mul(Integer(-1), Symbol('pdg0002958')), Symbol('pdg0005505'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0= :a_node:expressiond v_y1702349646Symbol('pdg0005674')Mul(Integer(-1), Symbol('dt'), Symbol('pdg0001649'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba-g dt= :a_node:expression1772416655v F - F vInteger(0)=Mul(Pow(Symbol('pdg0001467'), Integer(-1)), Add(Symbol('pdg0004550'), Mul(Integer(-1), Symbol('pdg0005579'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{E_2 - E_1}{t} :a_node:expression1772973171-A \omega^2 \cos(\omega t)=Mul(Integer(-1), Symbol('k'), Pow(Symbol('pdg0005156'), Integer(-1)), Symbol('x'))Mul(Integer(-1), Symbol('A'), Pow(Symbol('pdg0002321'), Integer(2)), cos(Mul(Symbol('pdg0002321'), Symbol('pdg0009491'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba-\frac{k}{m} x :a_node:expression\omega1784114349Symbol('pdg0002321')=Pow(Mul(Symbol('pdg0001356'), Pow(Symbol('pdg0005156'), Integer(-1))), Rational(1, 2))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\sqrt{\frac{k}{m}} :a_node:expression01809909100Integer(0)=Mul(Pow(Symbol('pdg0001467'), Integer(-1)), Add(Symbol('pdg0004550'), Mul(Integer(-1), Symbol('pdg0005579'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{E_2 - E_1}{t} :a_node:expression1811867899\frac{d_1+d_2}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1}Pow(Symbol('pdg0009491'), Integer(2))Mul(Integer(4), Pow(Symbol('pdg0002530'), Integer(2)), Symbol('pdg0002798'), Pow(Symbol('pdg0003141'), Integer(2)), Pow(Symbol('pdg0005022'), Integer(-1)), Pow(Symbol('pdg0006277'), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baT^2= :a_node:expression1815398659Q + WAdd(Symbol('pdg0001088'), Symbol('pdg0009432'))Symbol('pdg0005786')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baU= :a_node:expression1819663717\frac{d}{dt} v_xDerivative(Symbol('pdg0005505'), Tuple(Symbol('pdg0001467'), Integer(1)))Symbol('pdg0007159')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baa_x= :a_node:expression1840080113object is not moving at $x=\infty$0Integer(0)Symbol('pdg0001552')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baKE_2= :a_node:expression1857710291a \sin(n \pi)Integer(0)Mul(Symbol('pdg0009139'), sin(Mul(Symbol('pdg0001592'), Symbol('pdg0003141'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0= :a_node:expression1858578388https://physicsderivationgraph.blogspot.com/2020/09/representing-laplace-operator-nabla-in.html- \omega^2 \mu_0 \epsilon_0 E( \vec{r})\exp(i \omega t)Mul(Pow(Symbol('nabla'), Integer(2)), Function('pdg0006238')(Symbol('pdg0009472')), exp(Mul(Symbol('pdg0001467'), Symbol('pdg0002321'), Symbol('pdg0004621'))))Mul(Integer(-1), Pow(Symbol('pdg0002321'), Integer(2)), Symbol('pdg0006197'), Symbol('pdg0007940'), Function('pdg0006238')(Symbol('pdg0009472')), exp(Mul(Symbol('pdg0001467'), Symbol('pdg0002321'), Symbol('pdg0004621'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\nabla^2 E( \vec{r})\exp(i \omega t)= :a_node:expression1858772113\frac{n \pi}{W}Mul(Symbol('pdg0001592'), Pow(Symbol('pdg0002523'), Integer(-1)), Symbol('pdg0003141'))Symbol('pdg0005321')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bak= :a_node:expression\omega1888494137Symbol('pdg0002321')=Mul(Integer(-1), Pow(Mul(Symbol('pdg0001356'), Pow(Symbol('pdg0005156'), Integer(-1))), Rational(1, 2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba-\sqrt{\frac{k}{m}} :a_node:expression1916173354c^2Add(Mul(Integer(-1), Pow(Symbol('pdg0001357'), Integer(2)), Pow(Symbol('pdg0001790'), Integer(2))), Mul(Pow(Symbol('pdg0001790'), Integer(2)), Pow(Symbol('pdg0004567'), Integer(2))))Pow(Symbol('pdg0004567'), Integer(2))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba-\gamma^2 v^2 + c^2 \gamma^2= :a_node:expression|Z| \exp( -i \theta )1928085940Symbol('pdg0003192')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baZ^*= :a_node:expression\omega^21931103031=Mul(Symbol('pdg0001356'), Pow(Symbol('pdg0005156'), Integer(-1)))Pow(Symbol('pdg0002321'), Integer(2))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{k}{m} :a_node:expression11934748140Integer(1)=Integral(Pow(Abs(Function('pdg0009489')(Symbol('pdg0001464'))), Integer(2)), Tuple(Symbol('pdg0009199')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\int |\psi(x)|^2 dx :a_node:expression1935543849c^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^2Add(Mul(Pow(Symbol('pdg0001357'), Integer(2)), Pow(Symbol('pdg0001467'), Integer(2)), Pow(Symbol('pdg0001790'), Integer(2))), Mul(Integer(-1), Integer(2), Symbol('pdg0001357'), Symbol('pdg0001467'), Pow(Symbol('pdg0001790'), Integer(2)), Symbol('pdg0004037')), Mul(Pow(Symbol('pdg0001790'), Integer(2)), Pow(Symbol('pdg0004037'), Integer(2))), Pow(Symbol('pdg0005647'), Integer(2)), Pow(Symbol('pdg0006728'), Integer(2)))Add(Mul(Pow(Symbol('pdg0001467'), Integer(2)), Pow(Symbol('pdg0001790'), Integer(2)), Pow(Symbol('pdg0004567'), Integer(2))), Mul(Integer(2), Symbol('pdg0001467'), Pow(Symbol('pdg0001790'), Integer(-1)), Symbol('pdg0004037'), Pow(Symbol('pdg0004567'), Integer(2)), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('pdg0001790'), Integer(2))))), Mul(Pow(Symbol('pdg0001790'), Integer(-2)), Pow(Symbol('pdg0004037'), Integer(2)), Pow(Symbol('pdg0004567'), Integer(2)), Add(Integer(1), Mul(Integer(-1), Pow(Symbol('pdg0001790'), Integer(2))))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2= :a_node:expression1963253044dxSymbol('pdg0009199')=Mul(Symbol('pdg0002958'), Symbol('pdg0004711'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_{0, x} dt :a_node:expression1967582749\frac{v - v_0}{a}Mul(Pow(Symbol('pdg0009140'), Integer(-1)), Add(Symbol('pdg0001357'), Mul(Integer(-1), Symbol('pdg0005153'))))Symbol('pdg0001467')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bat= :a_node:expressiont'1974334644Symbol('pdg0004989')Add(Mul(Symbol('pdg0001467'), Symbol('pdg0001790')), Mul(Pow(Symbol('pdg0001357'), Integer(-1)), Pow(Symbol('pdg0001790'), Integer(-1)), Function('pdg0004037')(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('pdg0001790'), Integer(2)))))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v}= :a_node:expression\frac{d}{dt} v_y1977955751Mul(Integer(-1), Symbol('pdg0001649'))Derivative(Symbol('pdg0009107'), Tuple(Symbol('pdg0001467'), Integer(1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba-g= :a_node:expression1994296484G \frac{m_{\rm Earth}}{r}=Pow(Symbol('pdg0004082'), Integer(2))Mul(Pow(Symbol('pdg0002530'), Integer(-1)), Symbol('pdg0005458'), Symbol('pdg0006277'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_{\rm satellite}^2 :a_node:expression\sqrt{\frac{2 G m_2}{r}}2005061870Function('pdg0001357')(Symbol('pdg0002530'))Mul(Pow(Integer(2), Rational(1, 2)), Pow(Mul(Pow(Symbol('pdg0002530'), Integer(-1)), Symbol('pdg0004851'), Symbol('pdg0006277')), Rational(1, 2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav(r)= :a_node:expression2029293929https://physicsderivationgraph.blogspot.com/2020/09/representing-laplace-operator-nabla-in.html\mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r})\exp(i \omega t)Mul(Pow(Symbol('nabla'), Integer(2)), Function('pdg0006238')(Symbol('pdg0009472')), exp(Mul(Symbol('pdg0001467'), Symbol('pdg0002321'), Symbol('pdg0004621'))))Mul(Symbol('partial'), Pow(Symbol('pdg0001467'), Integer(-2)), Symbol('pdg0006197'), Symbol('pdg0007940'), Function('pdg0006238')(Symbol('pdg0009472')), exp(Mul(Symbol('pdg0001467'), Symbol('pdg0002321'), Symbol('pdg0004621'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\nabla^2 E( \vec{r})\exp(i \omega t)= :a_node:expression2042298788-G \frac{m_{\rm Earth} m}{r_{\rm Earth}} + \frac{1}{2} m v_{\rm escape}^2Integer(0)Add(Mul(Rational(1, 2), Symbol('pdg0005156'), Pow(Symbol('pdg0008656'), Integer(2))), Mul(Integer(-1), Pow(Symbol('pdg0003236'), Integer(-1)), Symbol('pdg0005156'), Symbol('pdg0005458'), Symbol('pdg0006277')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0= :a_node:expression2051901211I_1Symbol('pdg0003978')=Mul(Symbol('pdg0006599'), Pow(Symbol('pdg0008697'), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{V}{R_1} :a_node:expression2061086175-G m_1 m_2 \left(\frac{-1}{r} - \frac{-1}{\infty}\right)Symbol('pdg0009372')=Mul(Integer(-1), Symbol('pdg0005022'), Symbol('pdg0006277'), Function('pdg0004851')(Mul(Integer(-1), Pow(Symbol('pdg0002530'), Integer(-1)))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baW_{\rm to\ system} :a_node:expression02076171250Symbol('pdg0001790')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba-\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v}= :a_node:expression2086924031- \frac{1}{2} g t_f + v_0 \sin(\theta)Integer(0)Add(Mul(Integer(-1), Rational(1, 2), Symbol('pdg0001649'), Symbol('pdg0002467')), Mul(Symbol('pdg0005153'), sin(Symbol('pdg0001575'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0= :a_node:expression\gamma ( \gamma x - \gamma v t + v t' )2096918413Function('pdg0001790')(Add(Mul(Integer(-1), Symbol('pdg0001357'), Symbol('pdg0001467'), Symbol('pdg0001790')), Mul(Symbol('pdg0001357'), Symbol('pdg0004989')), Mul(Symbol('pdg0001790'), Symbol('pdg0004037'))))Symbol('pdg0004037')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax= :a_node:expression2103023049\frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)sin(Symbol('pdg0001464'))=Mul(Rational(1, 2), Pow(Symbol('pdg0004621'), Integer(-1)), Add(exp(Mul(Symbol('pdg0001464'), Symbol('pdg0004621'))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('pdg0001464'), Symbol('pdg0004621'))))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\sin(x) :a_node:expression1/T2113211456Pow(Symbol('pdg0009491'), Integer(-1))Symbol('pdg0004201')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baf= :a_node:expression2114909846\frac{[A_{\rm adsorption}]}{[S_0]}Symbol('pdg0001791')=Mul(Pow(Symbol('pdg0003037'), Integer(-1)), Symbol('pdg0004940'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\theta_A :a_node:expression2121790783\frac{ \left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2}=Pow(tanh(Symbol('pdg0001464')), Integer(2))Mul(Pow(Add(exp(Symbol('pdg0001464')), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('pdg0001464'))))), Integer(2)), Pow(Add(exp(Symbol('pdg0001464')), exp(Mul(Integer(-1), Symbol('pdg0001464')))), Integer(-2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\tanh^2(x) :a_node:expression2123139121-\cos(x)+i \sin(x)=Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('pdg0001464'), Symbol('pdg0004621'))))Add(Mul(Symbol('pdg0004621'), sin(Symbol('pdg0001464'))), Mul(Integer(-1), cos(Symbol('pdg0001464'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba-\exp(-i x) :a_node:expression21316165311Mul(Symbol('pdg0004201'), Symbol('pdg0009491'))Integer(1)a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baT f= :a_node:expression-A \omega^2 \cos(\omega t)2148049269Mul(Integer(-1), Symbol('A'), Symbol('k'), Pow(Symbol('pdg0005156'), Integer(-1)), cos(Mul(Symbol('pdg0002321'), Symbol('pdg0009491'))))Mul(Integer(-1), Symbol('A'), Pow(Symbol('pdg0002321'), Integer(2)), cos(Mul(Symbol('pdg0002321'), Symbol('pdg0009491'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba-\frac{k}{m} A \cos(\omega t)= :a_node:expression2168306601\left(\frac{k_{\rm desorption}}{k_{\rm adsorption}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}]Symbol('pdg0003037')=Mul(Symbol('pdg0004940'), Add(Integer(1), Mul(Pow(Symbol('pdg0006850'), Integer(-1)), Symbol('pdg0008379'), Pow(Symbol('pdg0009046'), Integer(-1)))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba[S_0] :a_node:expressionv F2186083170Mul(Symbol('pdg0001357'), Symbol('pdg0004202'))\frac{KE_2 - KE_1}{t}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Pow(Symbol('pdg0001467'), Integer(-1)), Add(Symbol('pdg0001352'), Mul(Integer(-1), Symbol('pdg0001955')))) :a_node:expression2217103163m_2Symbol('pdg0004851')=Mul(Pow(Symbol('pdg0002798'), Integer(-1)), Symbol('pdg0005022'), Symbol('pdg0007652'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{m_1 d_1}{d_2} :a_node:expression|A + B|^22236639474=Pow(Add(Symbol('pdg0004453'), Symbol('pdg0004698')), Integer(2))Pow(Abs(Add(Symbol('pdg0004453'), Symbol('pdg0004698'))), Integer(2))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba(A + B)(A + B)^* :a_node:expression2257410739C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alphaDerivative(Symbol('pdg0005786'), Tuple(Symbol('pdg0007343'), Integer(1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\left(\frac{\partial U}{\partial T}\right)_p= :a_node:expression2258485859i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)Mul(Symbol('pdg0006799'), Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467')))Mul(Symbol('pdg0001054'), Symbol('pdg0004621'), Derivative(Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467')), Tuple(Symbol('pdg0001467'), Integer(1))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba{\cal H} \psi \left( \vec{r},t \right)= :a_node:expression2267521164object goes to $\infty$ away from gravitational source0Integer(0)Symbol('pdg0008849')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baPE_2= :a_node:expressionI_{\rm total} R_{\rm total}2271186630Mul(Symbol('pdg0001908'), Symbol('pdg0009647'))Symbol('pdg0006599')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baV= :a_node:expressionv_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)2297105551Mul(Integer(2), Pow(Symbol('pdg0001649'), Integer(-1)), Pow(Symbol('pdg0005153'), Integer(2)), sin(Symbol('pdg0001575')), cos(Symbol('pdg0001575')))Symbol('pdg0001943')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bad= :a_node:expression2308660627g_{\rm Earth}Symbol('pdg0007557')Mul(Pow(Symbol('pdg0003236'), Integer(-2)), Symbol('pdg0005458'), Symbol('pdg0006277'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baG \frac{m_{\rm Earth}}{r_{\rm Earth}^2}= :a_node:expression2334518266-k xMul(Symbol('pdg0005156'), Symbol('pdg0009140'))Mul(Integer(-1), Symbol('pdg0001356'), Symbol('pdg0004037'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bam a= :a_node:expression\int d v_x2366691988=Integral(Integer(0), Tuple(Symbol('pdg0001467')))Integral(Integer(1), Tuple(Symbol('pdg0005005')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\int 0 dt :a_node:expression2378095808x_0 + dAdd(Symbol('pdg0001572'), Symbol('pdg0001943'))Symbol('pdg0003652')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax_f= :a_node:expressiona_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle2394240499Mul(Symbol('pdg0007752'), Function('Bra')(Symbol('pdg0004679')), Function('Ket')(Symbol('pdg0002090')))Symbol('pdg0001464')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax= :a_node:expression2394853829\cos(-x)+i \sin(-x)=exp(Mul(Integer(-1), Symbol('pdg0001464'), Symbol('pdg0004621')))Add(Mul(Symbol('pdg0004621'), sin(Mul(Integer(-1), Symbol('pdg0001464')))), cos(Mul(Integer(-1), Symbol('pdg0001464'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\exp(-i x) :a_node:expression23949358310Mul(Add(Mul(Integer(-1), Symbol('pdg0002427')), Symbol('pdg0007752')), Function('Bra')(Symbol('pdg0004679')), Function('Ket')(Symbol('pdg0002090')))Integer(0)a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle= :a_node:expression\left(\langle a \rangle\right)^+2394935835Symbol('pdg0004065')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\left(\langle\psi| \hat{A} |\psi \rangle \right)^+= :a_node:expression2395958385https://physicsderivationgraph.blogspot.com/2020/09/representing-laplace-operator-nabla-in.html\frac{-p^2}{\hbar} \psi( \vec{r},t)Mul(Pow(Symbol('nabla'), Integer(2)), Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467')))Mul(Integer(-1), Pow(Symbol('pdg0001054'), Integer(-1)), Pow(Symbol('pdg0001134'), Integer(2)), Function('pdg0009489')(Symbol('pdg0009472'), Symbol('pdg0001467')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\nabla^2 \psi \left( \vec{r},t \right)= :a_node:expression\langle x^2 \rangle-\langle x \rangle^22404934990Symbol('pdg0001464')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2= :a_node:expression24053073722 \sin(x) \cos(x)=sin(Mul(Integer(2), Symbol('pdg0001464')))Mul(Integer(2), sin(Symbol('pdg0001464')), cos(Symbol('pdg0001464')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\sin(2 x) :a_node:expression1 - \gamma^22417941373Mul(Integer(-1), Pow(Symbol('pdg0001357'), Integer(-2)), Pow(Symbol('pdg0001790'), Integer(-2)), Pow(Symbol('pdg0004567'), Integer(2)), Pow(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('pdg0001790'), Integer(2)))), Integer(2)))Add(Integer(1), Mul(Integer(-1), Pow(Symbol('pdg0001790'), Integer(2))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba- c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4}= :a_node:expression2431507955-F x_2Mul(Integer(-1), Symbol('pdg0004202'), Symbol('pdg0005467'))Symbol('pdg0008849')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baPE_2= :a_node:expressiony2461349007Symbol('pdg0005647')Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('pdg0001467'), Integer(2)), Symbol('pdg0001649')), Mul(Symbol('pdg0001467'), Symbol('pdg0009431')), Symbol('pdg0001469'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba- \frac{1}{2} g t^2 + v_{0, y} t + y_0= :a_node:expression2472653783\frac{1}{T}Symbol('pdg0004686')=Pow(Symbol('pdg0007343'), Integer(-1))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\alpha :a_node:expressionm g2484824786Mul(Symbol('pdg0001649'), Symbol('pdg0005156'))Symbol('pdg0004202')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF= :a_node:expression\langle x^2 \rangle-\langle x \rangle^22494533900Symbol('pdg0001464')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\langle x^2\rangle -\langle x \rangle^2= :a_node:expression02501591100Integer(0)=Add(exp(Mul(Symbol('pdg0003141'), Symbol('pdg0004621'))), Integer(1))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\exp(i \pi) + 1 :a_node:expression2503972039KE_{\rm escape} + PE_{\rm Earth\ surface}Integer(0)Add(Symbol('pdg0005332'), Symbol('pdg0006431'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0= :a_node:expression25190589032 \sin(\theta) \cos(\theta)=sin(Mul(Integer(2), Symbol('pdg0001575')))Mul(Integer(2), sin(Symbol('pdg0001575')), cos(Symbol('pdg0001575')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\sin(2 \theta) :a_node:expressionc^22542420160Pow(Symbol('pdg0004567'), Integer(2))c^2 \gamma^2 - v^2 \gamma^2a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Add(Mul(Integer(-1), Pow(Symbol('pdg0001357'), Integer(2)), Pow(Symbol('pdg0001790'), Integer(2))), Mul(Pow(Symbol('pdg0001790'), Integer(2)), Pow(Symbol('pdg0004567'), Integer(2)))) :a_node:expression2575937347n_2 \sin( \theta_{\rm refracted} )Mul(Symbol('pdg0002941'), sin(Symbol('pdg0004928')))Mul(Symbol('pdg0001958'), sin(Symbol('pdg0002243')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ban_1 \sin( \theta_{\rm Brewster} )= :a_node:expressionnR2613006036=Mul(Pow(Symbol('pdg0007343'), Integer(-1)), Symbol('pdg0007586'), Symbol('pdg0008134'))Mul(Symbol('pdg0002834'), Symbol('pdg0008179'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{PV}{T} :a_node:expressionr2617541067Symbol('pdg0002530')Mul(Rational(1, 2), Pow(Integer(2), Rational(1, 3)), Pow(Mul(Pow(Symbol('pdg0003141'), Integer(-2)), Symbol('pdg0005458'), Symbol('pdg0006277'), Pow(Symbol('pdg0008762'), Integer(2))), Rational(1, 3)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\left(\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3}= :a_node:expression\frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) \right)2648958382Symbol('pdg0001054')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\nabla^2 \psi \left( \vec{r},t \right)= :a_node:expression2700934933\left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)=Mul(Integer(2), cos(Symbol('pdg0001464')))Add(exp(Mul(Symbol('pdg0004621'), Add(Symbol('pdg0001575'), Mul(Integer(-1), Symbol('pdg0008586'))))), exp(Mul(Integer(-1), Symbol('pdg0004621'), Add(Symbol('pdg0001575'), Mul(Integer(-1), Symbol('pdg0008586'))))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba2 \cos(x) :a_node:expression2715678478I R_1 + I R_2=Mul(Symbol('pdg0001908'), Symbol('pdg0004501'))Add(Mul(Symbol('pdg0003461'), Symbol('pdg0004501')), Mul(Symbol('pdg0004501'), Symbol('pdg0008697')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baI R_{\rm total} :a_node:expressionin a loop2719691582|B|Abs(Symbol('pdg0004698'))Abs(Symbol('pdg0004453'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba|A|= :a_node:expression-g2741489181Mul(Integer(-1), Symbol('pdg0001649'))Symbol('pdg0007055')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baa_y= :a_node:expression2750380042-\sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}Symbol('pdg0008656')=Mul(Integer(-1), Pow(Integer(2), Rational(1, 2)), Pow(Mul(Pow(Symbol('pdg0003236'), Integer(-1)), Symbol('pdg0005458'), Symbol('pdg0006277')), Rational(1, 2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_{\rm escape} :a_node:expression2762326680\frac{1}{4}\left( \exp(2x)+1+1+\exp(-2x) - \left(\exp(2x)-1-1+\exp(-2x)\right) \right)Integer(1)\cosh^2 x - \sinh^2 xa84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Add(Mul(Integer(-1), Pow(sinh(Symbol('pdg0001464')), Integer(2))), Pow(cosh(Symbol('pdg0001464')), Integer(2))) :a_node:expression\frac{n_2}{n_1}2768857871Mul(sin(Symbol('pdg0004928')), Pow(cos(Symbol('pdg0004928')), Integer(-1)))Mul(Symbol('pdg0001958'), Pow(Symbol('pdg0002941'), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}= :a_node:expression\frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}2770069250=Mul(Pow(Symbol('pdg0001467'), Integer(-1)), Add(Symbol('pdg0004550'), Mul(Integer(-1), Symbol('pdg0005579'))))Add(Mul(Pow(Symbol('pdg0001467'), Integer(-1)), Add(Symbol('pdg0001352'), Mul(Integer(-1), Symbol('pdg0001955')))), Mul(Pow(Symbol('pdg0001467'), Integer(-1)), Add(Mul(Integer(-1), Symbol('pdg0004093')), Symbol('pdg0008849'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{E_2 - E_1}{t} :a_node:expression2809345867I_{\rm total}Symbol('pdg0009647')\frac{V}{R_{\rm total}}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Pow(Symbol('pdg0001908'), Integer(-1)), Symbol('pdg0006599')) :a_node:expression2848934890\langle a \rangleSymbol('pdg0009139')=a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\langle a \rangle^* :a_node:expression\frac{v_2 - v_1}{t}acceleration2857430695Mul(Pow(Symbol('pdg0001467'), Integer(-1)), Add(Mul(Integer(-1), Symbol('pdg0002473')), Symbol('pdg0004770')))Symbol('pdg0009140')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baa= :a_node:expression2858549874y - y_0Add(Mul(Integer(-1), Rational(1, 2), Pow(Symbol('pdg0001467'), Integer(2)), Symbol('pdg0001649')), Mul(Symbol('pdg0001467'), Symbol('pdg0009431')))Add(Mul(Integer(-1), Symbol('pdg0001469')), Symbol('pdg0005647'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba- \frac{1}{2} g t^2 + v_{0, y} t= :a_node:expression28830793652 G mMul(Integer(2), Symbol('pdg0005156'), Symbol('pdg0006277'))r_{\rm Schwarzschild} c^2a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Symbol('pdg0004518'), Pow(Symbol('pdg0004567'), Integer(2))) :a_node:expression\alpha c \sqrt{ \frac{m_e}{A m_p} }2897612567Mul(Symbol('pdg0001370'), Symbol('pdg0004567'), Pow(Mul(Symbol('pdg0002515'), Pow(Symbol('pdg0003285'), Integer(-1)), Pow(Symbol('pdg0005916'), Integer(-1))), Rational(1, 2)))Symbol('pdg0002077')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav= :a_node:expression2902772962\frac{\frac{1}{2}\left( \exp(x)-\exp(-x) \right)}{\cosh(x)}tanh(Symbol('pdg0001464'))=Mul(Add(Mul(Rational(1, 2), exp(Symbol('pdg0001464'))), Mul(Integer(-1), Rational(1, 2), exp(Mul(Integer(-1), Symbol('pdg0001464'))))), Pow(cosh(Symbol('pdg0001464')), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\tanh(x) :a_node:expression2906548078\frac{r}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1}Pow(Symbol('pdg0009491'), Integer(2))Mul(Integer(4), Pow(Symbol('pdg0002530'), Integer(3)), Symbol('pdg0002798'), Pow(Symbol('pdg0003141'), Integer(2)), Pow(Symbol('pdg0005022'), Integer(-1)), Pow(Symbol('pdg0006277'), Integer(-1)), Pow(Add(Symbol('pdg0002798'), Symbol('pdg0007652')), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baT^2= :a_node:expressionW_{\rm to\ system}2907404069Symbol('pdg0009372')Symbol('pdg0006191')=a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baW_{\rm by\ system} :a_node:expression2924222857v(r=\infty)Symbol('pdg0001357')Symbol('pdg0001934')=a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_{\rm initial} :a_node:expression2944838499a \sin(\frac{n \pi}{W} x)=Function('pdg0009489')(Symbol('pdg0001464'))Mul(Symbol('pdg0009139'), sin(Mul(Symbol('pdg0001592'), Pow(Symbol('pdg0002523'), Integer(-1)), Symbol('pdg0003141'), Symbol('pdg0004037'))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\psi(x) :a_node:expressionv_{\rm escape}^22977457786Mul(Integer(2), Pow(Symbol('pdg0003236'), Integer(-1)), Symbol('pdg0005458'), Symbol('pdg0006277'))Pow(Symbol('pdg0008656'), Integer(2))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}= :a_node:expression2983053062\gamma (x' + v t')Mul(Symbol('pdg0001790'), Add(Mul(Symbol('pdg0001357'), Symbol('pdg0004989')), Symbol('pdg0005456')))Symbol('pdg0004037')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax= :a_node:expression02998709778Symbol('pdg0001934')=Integer(0)a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_{\rm initial} :a_node:expression2999795755v^2 \gamma^2 + c^2=Mul(Pow(Symbol('pdg0001790'), Integer(2)), Pow(Symbol('pdg0004567'), Integer(2)))Add(Mul(Pow(Symbol('pdg0001357'), Integer(2)), Pow(Symbol('pdg0001790'), Integer(2))), Pow(Symbol('pdg0004567'), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bac^2 \gamma^2 :a_node:expression3004158505\left( \frac{4 \pi^2 m r}{T^2} \right)\frac{T^2}{r}Mul(Pow(Symbol('pdg0002530'), Integer(-1)), Symbol('pdg0002867'), Pow(Symbol('pdg0008762'), Integer(2)))Mul(Integer(4), Pow(Symbol('pdg0003141'), Integer(2)), Symbol('pdg0005156'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{T^2}{r} F_{\rm gravity}= :a_node:expression\frac{C_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}}3046191961Symbol('pdg0007427')v_{\rm Earth\ orbit}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Symbol('pdg0001534'), Pow(Symbol('pdg0005344'), Integer(-1))) :a_node:expression2|A|^23060393541Symbol('pdg0002435')=Mul(Integer(2), Pow(Abs(Symbol('pdg0004453')), Integer(2)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baI_{\rm incoherent} :a_node:expression3061811650n_2 \cos( \theta_{\rm Brewster} )Mul(Symbol('pdg0002941'), sin(Symbol('pdg0004928')))Mul(Symbol('pdg0001958'), cos(Symbol('pdg0004928')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ban_1 \sin( \theta_{\rm Brewster} )= :a_node:expression\frac{2 \pi r_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}}3080027960Symbol('pdg0007427')v_{\rm Earth\ orbit}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Integer(2), Symbol('pdg0003141'), Pow(Symbol('pdg0005344'), Integer(-1)), Symbol('pdg0006081')) :a_node:expression|A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))3085575328Add(Pow(Abs(Symbol('pdg0004453')), Integer(2)), Pow(Abs(Symbol('pdg0004698')), Integer(2)), Mul(exp(Mul(Integer(-1), Symbol('pdg0004621'), Add(Symbol('pdg0001575'), Mul(Integer(-1), Symbol('pdg0008586'))))), Abs(Mul(Symbol('pdg0004453'), Symbol('pdg0004698'), Abs(Add(Mul(exp(Mul(Symbol('pdg0004621'), Add(Symbol('pdg0001575'), Mul(Integer(-1), Symbol('pdg0008586'))))), Abs(Symbol('pdg0004698'))), Abs(Symbol('pdg0004453'))))))))Symbol('pdg0007882')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baI= :a_node:expression\lambda3121234211Symbol('pdg0001115')=Mul(Rational(1, 2), Pow(Symbol('pdg0003141'), Integer(-1)), Symbol('pdg0005321'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{k}{2\pi} :a_node:expression3121234212\frac{h k}{2\pi}Mul(Rational(1, 2), Pow(Symbol('pdg0003141'), Integer(-1)), Symbol('pdg0004413'), Symbol('pdg0005321'))Symbol('pdg0001134')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bap= :a_node:expression\frac{2 \pi}{\lambda}3121513111Mul(Integer(2), Pow(Symbol('pdg0001115'), Integer(-1)), Symbol('pdg0003141'))Symbol('pdg0005321')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bak= :a_node:expression31311111331 / fPow(Symbol('pdg0004201'), Integer(-1))Symbol('pdg0009491')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baT= :a_node:expression31312111312 \pi fSymbol('pdg0002321')=Mul(Integer(2), Symbol('pdg0003141'), Symbol('pdg0004201'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\omega :a_node:expression\frac{2\pi}{T}3132131132Symbol('pdg0002321')=Mul(Integer(2), Symbol('pdg0003141'), Pow(Symbol('pdg0009491'), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\omega :a_node:expressionf3147472131Symbol('pdg0004201')\frac{\omega}{2 \pi}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Rational(1, 2), Symbol('pdg0002321'), Pow(Symbol('pdg0003141'), Integer(-1))) :a_node:expression\frac{d\vec{v}}{dt}acceleration is the change in speed over a duration3169580383Symbol('pdg0002423')=Derivative(Symbol('pdg0006373'), Tuple(Symbol('pdg0001467'), Integer(1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\vec{a} :a_node:expression3176662571applicable to any satellite orbitF_{\rm gravity}Symbol('pdg0001687')Symbol('pdg0002867')=a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF_{\rm centripetal} :a_node:expression31826337891Add(Pow(Symbol('pdg0001790'), Integer(2)), Mul(Integer(-1), Pow(Symbol('pdg0001357'), Integer(-2)), Pow(Symbol('pdg0001790'), Integer(-2)), Pow(Symbol('pdg0004567'), Integer(2)), Pow(Add(Integer(1), Mul(Integer(-1), Pow(Symbol('pdg0001790'), Integer(2)))), Integer(2))))Integer(1)a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4}= :a_node:expression32141703220Symbol('pdg0001357')=a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav(r=\infty) :a_node:expression\frac{v_2^2 - v_1^2}{2 a}3253234559Mul(Rational(1, 2), Pow(Symbol('pdg0009140'), Integer(-1)), Add(Mul(Integer(-1), Pow(Symbol('pdg0002473'), Integer(2))), Pow(Symbol('pdg0004770'), Integer(2))))Symbol('pdg0004037')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax= :a_node:expression\frac{x - x_0}{v_{0, x}}3274926090Mul(Pow(Symbol('pdg0002958'), Integer(-1)), Add(Mul(Integer(-1), Symbol('pdg0001572')), Symbol('pdg0004037')))Symbol('pdg0001467')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bat= :a_node:expression32857329111-(\sin(x))^2=Pow(cos(Symbol('pdg0001464')), Integer(2))Add(Integer(1), Mul(Integer(-1), Pow(sin(Symbol('pdg0001464')), Integer(2))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba(\cos(x))^2 :a_node:expression\frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}3291685884Mul(Rational(1, 32), Pow(Symbol('pdg0001054'), Integer(-2)), Pow(Symbol('pdg0001999'), Integer(4)), Symbol('pdg0002515'), Pow(Symbol('pdg0003141'), Integer(-2)), Pow(Symbol('pdg0007940'), Integer(-2)))Symbol('pdg0002241')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baE= :a_node:expression-13331824625=exp(Mul(Symbol('pdg0003141'), Symbol('pdg0004621')))Integer(-1)a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\exp(i \pi) :a_node:expression3350830826|Z|^2Symbol('pdg0003192')=a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baZ Z^* :a_node:expression3360172339KE_2 - KE_1Add(Symbol('pdg0001352'), Mul(Integer(-1), Symbol('pdg0001955')))Symbol('pdg0006789')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baW= :a_node:expression33642866465.972*10^{24} kgSymbol('pdg0005458')=Mul(Float('5.9720000000000003e+24', precision=53), Symbol('kg'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bam_{\rm Earth} :a_node:expression3366703541acceleration is the average change in speed over a duration\frac{v - v_0}{t}Mul(Pow(Symbol('pdg0001467'), Integer(-1)), Add(Symbol('pdg0001357'), Mul(Integer(-1), Symbol('pdg0005153'))))Symbol('pdg0009140')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baa= :a_node:expression\frac{d}{t}3411994811Symbol('pdg0006709')=Mul(Pow(Symbol('pdg0001467'), Integer(-1)), Symbol('pdg0001943'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_{\rm average} :a_node:expression3417126140\frac{ n_2 }{ n_1 }Mul(Symbol('pdg0001958'), Pow(Symbol('pdg0002941'), Integer(-1)))tan(Symbol('pdg0004928'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\tan( \theta_{\rm Brewster} )= :a_node:expression\gamma ( \gamma (x - v t) + v t' )3426941928Mul(Symbol('pdg0001790'), Add(Mul(Symbol('pdg0001357'), Symbol('pdg0004989')), Mul(Symbol('pdg0001790'), Add(Mul(Integer(-1), Symbol('pdg0001357'), Symbol('pdg0001467')), Symbol('pdg0004037')))))Symbol('pdg0004037')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax= :a_node:expression3462972452v_0 + a tAdd(Mul(Symbol('pdg0001467'), Symbol('pdg0009140')), Symbol('pdg0005153'))Symbol('pdg0001357')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav= :a_node:expressiondefinition of expansion coefficient\frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p3464107376Symbol('pdg0004686')=Mul(Pow(Symbol('pdg0007586'), Integer(-1)), Derivative(Symbol('pdg0007586'), Tuple(Symbol('pdg0007343'), Integer(1))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\alpha :a_node:expression\frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)3470587782=Mul(sin(Symbol('pdg0001464')), cos(Symbol('pdg0001464')))Mul(Rational(1, 2), Pow(Symbol('pdg0004621'), Integer(-1)), Add(Mul(Rational(1, 2), exp(Mul(Symbol('pdg0001464'), Symbol('pdg0004621')))), Mul(Rational(1, 2), exp(Mul(Integer(-1), Symbol('pdg0001464'), Symbol('pdg0004621'))))), Add(exp(Mul(Symbol('pdg0001464'), Symbol('pdg0004621'))), Mul(Integer(-1), exp(Mul(Integer(-1), Symbol('pdg0001464'), Symbol('pdg0004621'))))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\sin(x) \cos(x) :a_node:expression34728361471.496\ 10^8 {\rm km}Symbol('pdg0006081')r_{\rm Earth\ orbit}a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Float('1.496', precision=53) :a_node:expressionv_0 t_f \cos(\theta) + x_03485125659Add(Symbol('pdg0001572'), Mul(Symbol('pdg0002467'), Symbol('pdg0005153'), cos(Symbol('pdg0001575'))))Symbol('pdg0003652')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax_f= :a_node:expressionhttps://physicsderivationgraph.blogspot.com/2020/09/representing-laplace-operator-nabla-in.html- \frac{\omega^2}{c^2} E( \vec{r})3485475729Mul(Integer(-1), Pow(Symbol('pdg0002321'), Integer(2)), Pow(Symbol('pdg0004567'), Integer(-2)), Function('pdg0006238')(Symbol('pdg0009472')))\nabla^2 E( \vec{r})a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Pow(Symbol('nabla'), Integer(2)), Function('pdg0006238')(Symbol('pdg0009472'))) :a_node:expressionk_{\rm desorption} [A_{\rm adsorption}]3488423948Mul(Symbol('pdg0004940'), Symbol('pdg0008379'))k_{\rm adsorption} p_A [S]a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Symbol('pdg0006850'), Symbol('pdg0009046'), Symbol('pdg0009067')) :a_node:expression3497828859\frac{n R T}{P}Mul(Symbol('pdg0002834'), Symbol('pdg0007343'), Pow(Symbol('pdg0008134'), Integer(-1)), Symbol('pdg0008179'))Symbol('pdg0007586')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baV= :a_node:expression3507029294r_{\rm desorption}Symbol('pdg0001966')k_{\rm adsorption} p_A [S]a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba=Mul(Symbol('pdg0006850'), Symbol('pdg0009046'), Symbol('pdg0009067')) :a_node:expressionF x3512166162Mul(Symbol('pdg0004037'), Symbol('pdg0004202'))Symbol('pdg0006789')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baW= :a_node:expressionk_{\rm Boltzmann} \ln \Omega3547519267assumes equally probable microstatesMul(Symbol('pdg0001157'), log(Symbol('pdg0003434')))Symbol('pdg0001394')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baS= :a_node:expression3566149658\int_{\infty}^r \frac{-G m_1 m_2}{x^2} dxSymbol('pdg0009372')=Integral(Mul(Integer(-1), Pow(Symbol('pdg0004037'), Integer(-2)), Symbol('pdg0004851'), Symbol('pdg0005022'), Symbol('pdg0006277')), Tuple(Symbol('pdg0004037'), oo, Symbol('pdg0002530')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baW_{\rm to\ system} :a_node:expression\langle x^2 \rangle-\langle x \rangle^23585845894Symbol('pdg0001464')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\langle \left(x-\langle x \rangle\right)^2 \rangle= :a_node:expression\frac{(KE_2 - KE_1)}{t} - F v3591237106=Mul(Pow(Symbol('pdg0001467'), Integer(-1)), Add(Symbol('pdg0004550'), Mul(Integer(-1), Symbol('pg5579'))))Add(Mul(Integer(-1), Symbol('pdg0001357'), Symbol('pdg0004202')), Mul(Pow(Symbol('pdg0001467'), Integer(-1)), Add(Symbol('pdg0001352'), Mul(Integer(-1), Symbol('pdg0001955')))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{E_2 - E_1}{t} :a_node:expression[S] + [A_{\rm adsorption}]3599953931Symbol('pdg0003037')=Add(Symbol('pdg0004940'), Symbol('pdg0009067'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba[S_0] :a_node:expression3605073197\frac{-nRT}{V} \left( \frac{-1}{P^2}\right)Symbol('pdg0004645')=Mul(Symbol('pdg0002834'), Symbol('pdg0007343'), Pow(Symbol('pdg0007586'), Integer(-1)), Pow(Symbol('pdg0008134'), Integer(-2)), Symbol('pdg0008179'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\kappa_T :a_node:expression\frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)3607070319Mul(Pow(Symbol('pdg0001649'), Integer(-1)), Pow(Symbol('pdg0005153'), Integer(2)), sin(Mul(Rational(1, 2), Symbol('pdg0003141'))))Symbol('pdg0001943')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bad= :a_node:expression\frac{2 \pi r}{T_{\rm orbit}}3614055652Mul(Integer(2), Symbol('pdg0002530'), Symbol('pdg0003141'), Pow(Symbol('pdg0008762'), Integer(-1)))Symbol('pdg0001357')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav= :a_node:expression3649797559m_2 d_2 \omega^2Symbol('pdg0001687')=Mul(Pow(Symbol('pdg0002321'), Integer(2)), Symbol('pdg0002798'), Symbol('pdg0004851'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baF_{\rm centripetal} :a_node:expression36503703894 \pi^2 mMul(Pow(Symbol('pdg0002530'), Integer(-1)), Symbol('pdg0002867'), Pow(Symbol('pdg0008762'), Integer(2)))Mul(Integer(4), Pow(Symbol('pdg0003141'), Integer(2)), Symbol('pdg0005156'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\frac{T^2}{r} F_{\rm gravity}= :a_node:expression3660957533\frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)cos(Symbol('pdg0001464'))=Add(Mul(Rational(1, 2), exp(Mul(Symbol('pdg0004621'), Add(Symbol('pdg0001575'), Mul(Integer(-1), Symbol('pdg0008586')))))), Mul(Rational(1, 2), exp(Mul(Integer(-1), Symbol('pdg0004621'), Add(Symbol('pdg0001575'), Mul(Integer(-1), Symbol('pdg0008586')))))))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba\cos(x) :a_node:expression\int dx3676159007=Mul(Symbol('pdg0002958'), Integral(Integer(1), Tuple(Symbol('pdg0001467'))))Integral(Integer(1), Tuple(Symbol('pdg0001464')))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_{0, x} \int dt :a_node:expression3736177473k_{\rm adsorption} p_A [S]Symbol('pdg0006687')=Mul(Symbol('pdg0006850'), Symbol('pdg0009046'), Symbol('pdg0009067'))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bar_{\rm adsorption} :a_node:expression3781109867\frac{r^3 4 \pi^2}{(d_1+d_2) \frac{m_1}{d_2}G}Pow(Symbol('pdg0009491'), Integer(2))Mul(Integer(4), Pow(Symbol('pdg0002530'), Integer(3)), Symbol('pdg0002798'), Pow(Symbol('pdg0003141'), Integer(2)), Pow(Symbol('pdg0005022'), Integer(-1)), Pow(Symbol('pdg0006277'), Integer(-1)), Pow(Add(Symbol('pdg0002798'), Symbol('pdg0007652')), Integer(-1)))a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baT^2= :a_node:expression03806977900=Add(Symbol('pdg0004550'), Mul(Integer(-1), Symbol('pdg0005579')))Integer(0)a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baE_2 - E_1 :a_node:scalar:symbol00000010542Reduced Planck's constant-1realconstant00\hbarany0a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba1https://en.wikipedia.org/wiki/Planck_constant#Value0 :a_node:scalar:symbol0000001088work done to a system0real2variable-2Wany00a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba1https://en.wikipedia.org/wiki/Work_(thermodynamics)0 :a_node:scalar:symbolwavelength00000011150real1variable00\lambdaany0a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0https://en.wikipedia.org/wiki/Wavelength0 :a_node:scalar:symbolmomentum00000011340real1variable-1pany00a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba1https://en.wikipedia.org/wiki/Momentum0 :a_node:scalar:symbol0000001157Boltzmann constantreal2constant-2k_{\rm Boltzmann}0-100a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba1https://en.wikipedia.org/wiki/Boltzmann_constantany :a_node:scalar:symbolkinetic energy00000013520real2variable-2KE_2any00a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba1https://en.wikipedia.org/wiki/Kinetic_energy0 :a_node:scalar:symbollinear stiffness, aka spring constant00000013560real0variable-2kany00a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba1https://en.wikipedia.org/wiki/Stiffness0 :a_node:scalar:symbolvelocity00000013570real1variable-1vany00a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0https://en.wikipedia.org/wiki/Velocity0 :a_node:scalar:symbol0000001370fine-structure constant0real0constant00\alphaany0a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0https://en.wikipedia.org/wiki/Fine-structure_constant0 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:a_node:scalar:symbolindex0000001592integer0variable00nany00a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba00 :a_node:scalar:symbol0000001649acceleration due to gravity0real1variable-2gany00a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0https://en.wikipedia.org/wiki/Gravity0 :a_node:scalar:symbol00000016871centripetal force-20real0variable0F_{\rm centripetal}anya84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba1https://en.wikipedia.org/wiki/Centripetal_force0 :a_node:scalar:symbolunit vector00000017000real0variable00\hat{y}any0a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0https://en.wikipedia.org/wiki/Unit_vector0 :a_node:scalar:symbolLorentz factor00000017900real0variable00\gammaany0a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba0https://en.wikipedia.org/wiki/Lorentz_factor0 :a_node:scalar:symbolthe fraction of the surface sites covered with 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:a_node:feed5868731041Symbol('pdg0005153')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_0 :a_node:feed5890617067Symbol('pdg0006458')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baR :a_node:feed5904227750Symbol('pdg0005156')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bam :a_node:feed5960438249Symbol('pdg0005579')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baE_1 :a_node:feed6023986360Symbol('pdg0001464')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bax :a_node:feed6038673136Symbol('pdg0001357')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav :a_node:feed6050070428Symbol('pdg0002958')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bav_{0, x} :a_node:feed6098638221Symbol('pdg0001469')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138bay_0 :a_node:feed6158970683Symbol('pdg0004093')a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138baPE_1 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HAS_STEP31 HAS_STEP4 HAS_STEP32 HAS_STEP24 HAS_STEP6 HAS_STEP19 HAS_STEP17 HAS_STEP23 HAS_STEP29 HAS_STEP1 HAS_STEP7 HAS_STEP9 HAS_STEP6 HAS_STEP1 HAS_STEP11 HAS_STEP3 HAS_STEP1 HAS_STEP7 HAS_STEP2 HAS_STEP5 HAS_STEP8 HAS_STEP4 HAS_STEP2 HAS_STEP13 HAS_STEP6 HAS_STEP12 HAS_STEP9 HAS_STEP3 HAS_STEP10 HAS_STEP2 HAS_STEP7 HAS_STEP5 HAS_STEP4 HAS_STEP8 HAS_STEP1 HAS_STEP11 HAS_STEP9 HAS_STEP6 HAS_STEP8 HAS_STEP1 HAS_STEP7 HAS_STEP11 HAS_STEP5 HAS_STEP10 HAS_STEP2 HAS_STEP3 HAS_STEP4 HAS_STEP12 HAS_STEP6 HAS_STEP8 HAS_STEP15 HAS_STEP3 HAS_STEP11 HAS_STEP9 HAS_STEP5 HAS_STEP4 HAS_STEP12 HAS_STEP13 HAS_STEP1 HAS_STEP14 HAS_STEP10 HAS_STEP7 HAS_STEP2 HAS_STEP6 HAS_STEP4 HAS_STEP8 HAS_STEP1 HAS_STEP2 HAS_STEP9 HAS_STEP7 HAS_STEP0.5 IS_COMPRISED_OF HAS_STEP0 HAS_INFERENCE_RULE HAS_OUTPUT0 IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF HAS_STEP1 HAS_INFERENCE_RULE HAS_OUTPUT0 HAS_STEP2 HAS_INFERENCE_RULE HAS_OUTPUT0 IS_COMPRISED_OF HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED4 IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF HAS_STEP3 HAS_INFERENCE_RULE HAS_INPUT0 HAS_OUTPUT0 HAS_STEP4 HAS_INFERENCE_RULE HAS_INPUT0 HAS_INPUT1 HAS_INPUT2 HAS_OUTPUT0 HAS_STEP5 HAS_INFERENCE_RULE HAS_INPUT0 IS_COMPRISED_OF IS_COMPRISED_OF HAS_STEP0 HAS_INFERENCE_RULE HAS_OUTPUT0 HAS_STEP1 HAS_INFERENCE_RULE HAS_OUTPUT0 HAS_STEP2 HAS_INFERENCE_RULE HAS_INPUT0 HAS_INPUT1 HAS_STEP3 HAS_FEED5 HAS_INFERENCE_RULE HAS_OUTPUT0 IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED0 IS_COMPRISED_OF HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED0 HAS_FEED1 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED0 HAS_FEED3 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED0 HAS_FEED3 HAS_FEED1 HAS_FEED4 HAS_FEED2 HAS_FEED0 HAS_FEED5 HAS_FEED3 HAS_FEED4 HAS_FEED5 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED1 HAS_FEED2 HAS_FEED2 HAS_FEED3 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED0 HAS_FEED1 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED0 HAS_INPUT0 HAS_OUTPUT0 HAS_FEED0 HAS_FEED1 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_INPUT0 HAS_OUTPUT0 HAS_FEED1 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED4 HAS_FEED5 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED4 HAS_FEED5 HAS_FEED0 HAS_FEED6 HAS_FEED7 HAS_FEED0 HAS_FEED1 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED0 HAS_FEED1 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED1 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED4 HAS_FEED5 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED4 HAS_FEED5 HAS_FEED0 HAS_FEED0 IS_COMPRISED_OF HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED0 HAS_FEED1 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED4 HAS_FEED5 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED0 HAS_FEED1 HAS_FEED0 HAS_FEED1 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED0 HAS_FEED1 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED4 HAS_FEED5 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED0 HAS_FEED1 HAS_FEED0 HAS_FEED1 HAS_FEED0 HAS_FEED1 HAS_FEED0 HAS_FEED1 HAS_FEED0 HAS_FEED1 HAS_FEED1 HAS_FEED2 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED1 HAS_FEED2 HAS_FEED3 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 HAS_FEED0 IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF IS_COMPRISED_OF 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