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9999999962, p = \hbar k 9999999965, E = \omega \hbar 9999999968, x = \frac{-b-\sqrt{b^2-4ac}}{2 a} 9999999969, x = \frac{-b+\sqrt{b^2-4ac}}{2 a} 9999999975, \langle \psi| \hat{A} |\psi \rangle = \langle a \rangle 9999999981, \vec{ \nabla} \cdot \vec{E} = \rho/\epsilon_0 0203024440, 1 = \int_0^W a \sin\left(\frac{n \pi}{W} x\right) \psi(x)^* dx 0404050504, \lambda = \frac{v}{f} 0439492440, \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) \right|_0^W 0934990943, k = \frac{2 \pi}{v T} 0948572140, \int \cos(a x) dx = \frac{1}{a}\sin(a x) 1010393913, \langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^* 1010393944, x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle 1010923823, k W = n \pi 1020010291, 0 = a \sin(k W) 1020394900, p = h/\lambda 1020394902, E = h f 1020854560, I = (A + B)(A + B)^* 1029039903, p = m v 1029039904, p^2 = m^2 v^2 1038566242, \sinh x = \frac{\exp(x) - \exp(-x)}{2} 1085150613, C_V = \left(\frac{\partial U}{\partial T}\right)_V 1087417579, 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) 1114820451, W_{\rm by\ system} = \Delta KE 1128605625, {\rm sech}^2\ x + \tanh^2(x) = \frac{4}{\left(\exp(x)+\exp(-x)\right)^2} + \frac{\left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2} 1132941271, m_{\rm Earth} = \frac{(9.80665 m/s^2) (6.3781*10^6 m)^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}} 1143343287, G \frac{m_{\rm Earth}}{r_{\rm Earth}} = \frac{1}{2} v_{\rm escape}^2 1158485859, \frac{-\hbar^2}{2m} \nabla^2 = {\cal H} 1166310428, 0 dt = d v_x 1172039918, I_{\rm coherent} = 4 |A|^2 1190768176, \kappa_T = \frac{-nRT}{V} \left( \frac{ \partial }{\partial P}\left(\frac{1}{P}\right) \right)_T 1191796961, \frac{1}{2} g t_f = v_0 \sin(\theta) 1201689765, x'^2 + y'^2 + z'^2 = c^2 t'^2 1202310110, \frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx 1202312210, \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx 1203938249, a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle 1248277773, \cos(2 x) = 1 - 2 (\sin(x))^2 1259826355, d = (v - a t) t + \frac{1}{2} a t^2 1265150401, d = \frac{2 v_0 + a t}{2} t 1292735067, F_{gravitational} = G \frac{m_1 m_2}{r^2} 1293913110, 0 = b 1293923844, \lambda = v T 1306360899, x = v_{0, x} t + x_0 1310571337, \theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster} 1311403394, \alpha = \frac{1}{V} \frac{nR}{P} \left( \frac{\partial T}{\partial T} \right)_P 1314464131, \vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} 1314864131, \vec{ \nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E} 1330874553, v_{\rm escape} = \sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}} 1357848476, A = |A| \exp(i \theta) 1395858355, x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle 1405465835, y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0 1457415749, \frac{1}{R_{\rm total}} = \frac{1}{R_1} + \frac{1}{R_2} 1525861537, I = |A|^2 + |B|^2 + A B^* + B A^* 1528310784, \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} 1541916015, \theta = \frac{\pi}{4} 1556389363, E_{\rm Rydberg} = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2} 1559688463, \left(\frac{T_{\rm geostationary\ orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r_{\rm geostationary\ orbit} 1586866563, \left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right) 1590774089, dW = F dx 1636453295, \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = - \nabla^2 \vec{E} 1638282134, \vec{p}_{\rm before} = \vec{p}_{\rm after} 1639827492, - c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1 1648958381, \nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right) 1650441634, y_0 = 0 1676472948, 0 = v_x - v_{0, x} 1702349646, -g dt = d v_y 1772416655, \frac{E_2 - E_1}{t} = v F - F v 1772973171, -\frac{k}{m} x = -A \omega^2 \cos(\omega t) 1784114349, \sqrt{\frac{k}{m}} = \omega 1809909100, \frac{E_2 - E_1}{t} = 0 1811867899, T^2 = \frac{d_1+d_2}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1} 1815398659, U = Q + W 1819663717, a_x = \frac{d}{dt} v_x 1840080113, KE_2 = 0 1857710291, 0 = a \sin(n \pi) 1858578388, \nabla^2 E( \vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E( \vec{r})\exp(i \omega t) 1858772113, k = \frac{n \pi}{W} 1888494137, -\sqrt{\frac{k}{m}} = \omega 1916173354, -\gamma^2 v^2 + c^2 \gamma^2 = c^2 1928085940, Z^* = |Z| \exp( -i \theta ) 1931103031, \frac{k}{m} = \omega^2 1934748140, \int |\psi(x)|^2 dx = 1 1935543849, \gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2 1963253044, v_{0, x} dt = dx 1967582749, t = \frac{v - v_0}{a} 1974334644, \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t' 1977955751, -g = \frac{d}{dt} v_y 1994296484, v_{\rm satellite}^2 = G \frac{m_{\rm Earth}}{r} 2005061870, v(r) = \sqrt{\frac{2 G m_2}{r}} 2029293929, \nabla^2 E( \vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r})\exp(i \omega t) 2042298788, 0 = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}} + \frac{1}{2} m v_{\rm escape}^2 2051901211, \frac{V}{R_1} = I_1 2061086175, W_{\rm to\ system} = -G m_1 m_2 \left(\frac{-1}{r} - \frac{-1}{\infty}\right) 2076171250, -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0 2086924031, 0 = - \frac{1}{2} g t_f + v_0 \sin(\theta) 2096918413, x = \gamma ( \gamma x - \gamma v t + v t' ) 2103023049, \sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) 2113211456, f = 1/T 2114909846, \theta_A = \frac{[A_{\rm adsorption}]}{[S_0]} 2121790783, \tanh^2(x) = \frac{ \left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2} 2123139121, -\exp(-i x) = -\cos(x)+i \sin(x) 2131616531, T f = 1 2148049269, -\frac{k}{m} A \cos(\omega t) = -A \omega^2 \cos(\omega t) 2168306601, [S_0] = \left(\frac{k_{\rm desorption}}{k_{\rm adsorption}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}] 2186083170, \frac{KE_2 - KE_1}{t} = v F 2217103163, \frac{m_1 d_1}{d_2} = m_2 2236639474, (A + B)(A + B)^* = |A + B|^2 2257410739, \left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alpha 2258485859, {\cal H} \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) 2267521164, PE_2 = 0 2271186630, V = I_{\rm total} R_{\rm total} 2297105551, d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta) 2308660627, G \frac{m_{\rm Earth}}{r_{\rm Earth}^2} = g_{\rm Earth} 2334518266, m a = -k x 2366691988, \int 0 dt = \int d v_x 2378095808, x_f = x_0 + d 2394240499, x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle 2394853829, \exp(-i x) = \cos(-x)+i \sin(-x) 2394935831, ( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0 2394935835, \left(\langle\psi| \hat{A} |\psi \rangle \right)^+ = \left(\langle a \rangle\right)^+ 2395958385, \nabla^2 \psi \left( \vec{r},t \right) = \frac{-p^2}{\hbar} \psi( \vec{r},t) 2404934990, \langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2 2405307372, \sin(2 x) = 2 \sin(x) \cos(x) 2417941373, - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2 2431507955, PE_2 = -F x_2 2461349007, - \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y 2472653783, \alpha = \frac{1}{T} 2484824786, F = m g 2494533900, \langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2 2501591100, \exp(i \pi) + 1 = 0 2503972039, 0 = KE_{\rm escape} + PE_{\rm Earth\ surface} 2519058903, \sin(2 \theta) = 2 \sin(\theta) \cos(\theta) 2542420160, c^2 \gamma^2 - v^2 \gamma^2 = c^2 2575937347, n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} ) 2613006036, \frac{PV}{T} = nR 2617541067, \left(\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r 2648958382, \nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) \right) 2700934933, 2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right) 2715678478, I R_{\rm total} = I R_1 + I R_2 2719691582, |A| = |B| 2741489181, a_y = -g 2750380042, v_{\rm escape} = -\sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}} 2762326680, \cosh^2 x - \sinh^2 x = \frac{1}{4}\left( \exp(2x)+1+1+\exp(-2x) - \left(\exp(2x)-1-1+\exp(-2x)\right) \right) 2768857871, \frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1} 2770069250, \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t} 2809345867, \frac{V}{R_{\rm total}} = I_{\rm total} 2848934890, \langle a \rangle^* = \langle a \rangle 2857430695, a = \frac{v_2 - v_1}{t} 2858549874, - \frac{1}{2} g t^2 + v_{0, y} t = y - y_0 2883079365, r_{\rm Schwarzschild} c^2 = 2 G m 2897612567, v = \alpha c \sqrt{ \frac{m_e}{A m_p} } 2902772962, \tanh(x) = \frac{\frac{1}{2}\left( \exp(x)-\exp(-x) \right)}{\cosh(x)} 2906548078, T^2 = \frac{r}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1} 2907404069, W_{\rm by\ system} = W_{\rm to\ system} 2924222857, v_{\rm initial} = v(r=\infty) 2944838499, \psi(x) = a \sin(\frac{n \pi}{W} x) 2977457786, 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}} = v_{\rm escape}^2 2983053062, x = \gamma (x' + v t') 2998709778, v_{\rm initial} = 0 2999795755, c^2 \gamma^2 = v^2 \gamma^2 + c^2 3004158505, \frac{T^2}{r} F_{gravitational} = \left( \frac{4 \pi^2 m r}{T^2} \right)\frac{T^2}{r} 3046191961, v_{\rm Earth\ orbit} = \frac{C_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}} 3060393541, I_{\rm incoherent} = 2|A|^2 3061811650, n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} ) 3080027960, v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}} 3085575328, I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi)) 3121234211, \frac{k}{2\pi} = \lambda 3121234212, p = \frac{h k}{2\pi} 3121513111, k = \frac{2 \pi}{\lambda} 3131111133, T = 1 / f 3131211131, \omega = 2 \pi f 3132131132, \omega = \frac{2\pi}{T} 3147472131, \frac{\omega}{2 \pi} = f 3169580383, \vec{a} = \frac{d\vec{v}}{dt} 3176662571, F_{\rm centripetal} = F_{\rm gravity} 3182633789, \gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 3214170322, v(r=\infty) = 0 3253234559, x = \frac{v_2^2 - v_1^2}{2 a} 3274926090, t = \frac{x - x_0}{v_{0, x}} 3285732911, (\cos(x))^2 = 1-(\sin(x))^2 3291685884, E = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2} 3331824625, \exp(i \pi) = -1 3350830826, Z Z^* = |Z|^2 3360172339, W = KE_2 - KE_1 3364286646, m_{\rm Earth} = 5.972*10^{24} kg 3366703541, a = \frac{v - v_0}{t} 3411994811, v_{\rm average} = \frac{d}{t} 3417126140, \tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 } 3426941928, x = \gamma ( \gamma (x - v t) + v t' ) 3462972452, v = v_0 + a t 3464107376, \alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p 3470587782, \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) 3472836147, r_{\rm Earth\ orbit} = 1.496\ 10^8 {\rm km} 3485125659, x_f = v_0 t_f \cos(\theta) + x_0 3485475729, \nabla^2 E( \vec{r}) = - \frac{\omega^2}{c^2} E( \vec{r}) 3488423948, k_{\rm adsorption} p_A [S] = k_{\rm desorption} [A_{\rm adsorption}] 3497828859, V = \frac{n R T}{P} 3507029294, k_{\rm adsorption} p_A [S] = r_{\rm desorption} 3512166162, W = F x 3547519267, S = k_{\rm Boltzmann} \ln \Omega 3566149658, W_{\rm to\ system} = \int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx 3585845894, \langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2 3591237106, \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v 3599953931, [S_0] = [S] + [A_{\rm adsorption}] 3605073197, \kappa_T = \frac{-nRT}{V} \left( \frac{-1}{P^2}\right) 3607070319, d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right) 3614055652, v = \frac{2 \pi r}{T_{\rm orbit}} 3649797559, F_{\rm centripetal} = m_2 d_2 \omega^2 3650370389, \frac{T^2}{r} F_{gravitational} = 4 \pi^2 m 3660957533, \cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right) 3676159007, v_{0, x} \int dt = \int dx 3736177473, r_{\rm adsorption} = k_{\rm adsorption} p_A [S] 3781109867, T^2 = \frac{r^3 4 \pi^2}{(d_1+d_2) \frac{m_1}{d_2}G} 3806977900, E_2 - E_1 = 0 3829492824, \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) = \cos(x) 3846041519, PE_{\rm Earth\ surface} = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}} 3868998312, {\rm sech}^2\ x = \frac{4}{\left(\exp(x)+\exp(-x)\right)^2} 3896798826, m_2 d_2 \omega^2 = G \frac{m_1 m_2}{r^2} 3906710072, G \frac{m_{\rm Earth}}{r} = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2} 3920616792, T_{\rm geostationary orbit} = 24\ {\rm hours} 3924948349, a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0 3935058307, v = \sqrt{ \frac{m_e}{m} \frac{e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} } 3942849294, \exp(i x)-\exp(-i x) = 2 i \sin(x) 3943939590, x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle 3947269979, \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} 3948571256, \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}E \psi( \vec{r},t) 3948574224, \psi( \vec{r},t) = \psi_0 \exp\left(i\left( \vec{k}\cdot\vec{r} - \omega t \right) \right) 3948574226, \psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right) 3948574228, \psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right) 3948574230, \psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right) 3948574233, \frac{\partial}{\partial t} \psi( \vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right) 3951205425, \vec{p}_{\rm after} = \vec{p}_{1} 4072200527, \frac{m_{\rm satellite} v_{\rm satellite}^2}{r} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2} 4075539836, A A^* = |A|^2 4087145886, V = I R 4107032818, E_{\rm Rydberg} = E 4128500715, V = I_1 R_1 4139999399, x - \gamma^2 x = - \gamma^2 v t + \gamma v t' 4147472132, E = \frac{h \omega}{2 \pi} 4158986868, a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt} 4166155526, {\rm sech}\ x = \frac{2}{\exp(x)+\exp(-x)} 4180845508, v_{\rm Earth\ orbit} = 29.8 \frac{{\rm km}}{{\rm sec}} 4182362050, Z = |Z| \exp( i \theta ) 4188580242, T^2 = \frac{r^3 4 \pi^2}{\left(m_1+\left(\frac{m_1}{d_2}d_1\right)\right)G} 4192519596, B = |B| \exp(i \phi) 4245712581, v = \frac{2 \pi r}{t} 4267808354, F_{gravitational} = m \frac{v^2}{r} 4268085801, x_0 + d = v_0 t_f \cos(\theta) + x_0 4270680309, \frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t} 4275004561, c^2 = 2 G \frac{m}{r_{\rm Schwarzschild}} 4287102261, x^2 + y^2 + z^2 = c^2 t^2 4298359835, E = \frac{1}{2}m v^2 4298359845, E = \frac{1}{2m}m^2 v^2 4298359851, E = \frac{p^2}{2m} 4301729661, [S_0] = \frac{[A_{\rm adsorption}]}{\left( \frac{k_{\rm adsorption}}{k_{\rm desorption}} \right) p_A} + [A_{\rm adsorption}] 4303372136, E_1 = KE_1 + PE_1 4341171256, i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{p^2}{2 m} \psi( \vec{r},t) 4348571256, \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi( \vec{r},t) 4370074654, t = t_f 4393258808, F_{\rm centripetal} = m r \omega^2 4393670960, W_{\rm to\ system} = \frac{G m_1 m_2}{r} 4394958389, \vec{ \nabla}\cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right) = \frac{i}{\hbar} \vec{ \nabla}\cdot\left( \vec{p} \psi( \vec{r},t) \right) 4428528271, F_{\rm{spring}} = -k x 4447113478, \int dW = G m_1 m_2 \int_{ r_{\rm Earth} }^{\infty} \frac{1}{x^2} dx 4501377629, \tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} 4504256452, B^* = |B| \exp(-i \phi) 4560648264, v = \sqrt{ \frac{K + (4/3) G}{\rho} } 4580545876, d = v t - a t^2 + \frac{1}{2} a t^2 4585828572, \epsilon_0 \mu_0 = \frac{1}{c^2} 4585932229, \cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) 4593428198, v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{3.16\ 10^7 {\rm seconds}} 4598294821, \exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2 4627284246, F_{\rm centripetal} = \frac{m_{\rm satellite} v_{\rm satellite}^2}{r} 4638429483, \exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x)) 4648451961, v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1) 4662369843, x' = \gamma (x - v t) 4669290568, PE_1 = -F x_1 4689334676, \theta_A = \frac{K_{\rm equilibrium}\ p_A}{1+K_{\rm equilibrium}\ p_A} 4742644828, \exp(i x)+\exp(-i x) = 2 \cos(x) 4748157455, a t = v - v_0 4778077984, t_f = \frac{2 v_0 \sin(\theta)}{g} 4784793837, \frac{KE_2 - KE_1}{t} = m v a 4798787814, a t + v_0 = v 4800170179, F = m g_{\rm Earth} 4805233006, i \sin(i x) = \frac{1}{2}\left(\exp(x) - \exp(-x) \right) 4811121942, W = \frac{1}{2} m v_2^2 - \frac{1}{2} m v_1^2 4820320578, F_{gravitational} = F_{centripetal} 4827492911, \cos(2 x)+(\sin(x))^2 = 1 - (\sin(x))^2 4830221561, {\rm sech}^2\ x + \tanh^2(x) = \frac{4+\left(\exp(2x)-1-1+\exp(-2x)\right)}{\left(\exp(x)+\exp(-x)\right)^2} 4838429483, \exp(2 i x) = \cos(2 x)+i \sin(2 x) 4843995999, \frac{1}{2 i}\left(\exp(i x)-\exp(-i x) \right) = \sin(x) 4857472413, 1 = \int \psi(x)\psi(x)^* dx 4857475848, \frac{1}{a^2} = \frac{W}{2} 4858693811, \frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2} = r^3 4866160902, \frac{V}{R_{\rm total}} = \frac{V}{R_1} + \frac{V}{R_2} 4872163189, \tanh(x) = \frac{\sinh(x)}{\cosh(x)} 4872970974, \frac{PE_2 - PE_1}{t} = -F v 4878728014, \sin(i x) = \frac{1}{2i}\left(\exp(-x) - \exp(x) \right) 4923339482, i x = \log(y) 4928007622, KE_1 = \frac{1}{2} m v_1^2 4928239482, \log(y) = i x 4938429482, \exp(-i x) = \cos(x)+i \sin(-x) 4938429483, \exp(i x) = \cos(x)+i \sin(x) 4938429484, \exp(-i x) = \cos(x)-i \sin(x) 4939880586, V_{\rm total} = I R_{\rm total} 4943571230, \vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right) 4947831649, \frac{1}{2} m_1 v_{\rm final}^2 = W_{\rm to\ system} 4948763856, 2 a d + v_0^2 = v^2 4948934890, \langle \psi| \hat{A} |\psi \rangle = \langle a \rangle^* 4949359835, \langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2 4968680693, \tan( x ) = \frac{ \sin( x )}{\cos( x )} 4985825552, \nabla^2 E( \vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E( \vec{r})\exp(i \omega t) 5002539602, dU = C_V dT + \pi_T dV 5085809757, \frac{k_{\rm adsorption}}{k_{\rm desorption}} = \frac{[A_{\rm adsorption}]}{p_A [S]} 5125940051, I = |A|^2 + B B^* + A B^* + B A^* 5128670694, m_1 d_1 = m_2 d_2 5136652623, E = KE + PE 5144263777, v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right) 5148266645, t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t 5177311762, v = \frac{2 \pi r}{T} 5323719091, i \sinh x = \frac{1}{2i} \left( \exp(-x) - \exp(x) \right) 5345738321, F = m a 5349669879, \tanh(x) = \frac{ \exp(x)-\exp(-x)}{\exp(x)+\exp(-x)} 5349866551, \vec{v} = v_x \hat{x} + v_y \hat{y} 5353282496, d = \frac{v_0^2}{g} 5373931751, t = t_f 5379546684, y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0 5404822208, v_{\rm escape} = \sqrt{2 G \frac{m}{r}} 5415824175, x(t) = A \cos(\omega t) 5426308937, v = \frac{d}{t} 5438722682, x = v_0 t \cos(\theta) + x_0 5514556106, E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1) 5530148480, \vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron} 5542528160, \int dW = F \int_0^x dx 5563580265, F_{\rm gravity} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2} 5586102077, r = d_1 + d_2 5596822289, W_{\rm to\ system} = -G m_1 m_2 \left(\left.\frac{-1}{x}\right|^r_{\infty}\right) 5611024898, d = \frac{1}{2 a} (v^2 - v_0^2) 5634116660, \pi_T = \left(\frac{\partial U}{\partial V}\right)_T 5646314683, m = A m_p 5658865948, T^2 = \frac{r^3 4 \pi^2}{(m_1+m_2)G} 5693047217, v_{\rm final} = -\sqrt{\frac{2 G m_2}{r}} 5727578862, \frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x) 5732331610, W = G m_1 m_2 \left( \frac{1}{x} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right) 5733146966, KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right) 5733721198, d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right) 5763749235, -c^2 + c^2 \gamma^2 = v^2 \gamma^2 5779256336, W_{\rm by\ system} = KE_{\rm final} - KE_{\rm initial} 5781981178, x^2 - y^2 = (x+y)(x-y) 5789289057, v = \alpha c \sqrt{ \frac{m_e}{2 m} } 5832984291, (\sin(x))^2 + (\cos(x))^2 = 1 5838268428, \alpha c = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar} 5846639423, v_{\rm final} = \sqrt{\frac{2 G m_2}{r}} 5850144586, W_{\rm by\ system} = KE_{\rm final} 5857434758, \int a dx = a x 5866629429, {\rm sech}^2\ x + \tanh^2(x) = 1 5868688585, \frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = \frac{p^2}{2m} \psi( \vec{r},t) 5900595848, k = \frac{\omega}{v} 5902985919, \vec{F} = G \frac{m_1 m_2}{x^2} \hat{x} 5928285821, x^2 + 2 x (b/(2 a)) + (b/(2 a))^2 = (x + (b/(2 a)))^2 5928292841, x^2 + (b/a)x + (b/(2 a))^2 = -c/a + (b/(2 a))^2 5938459282, x^2 + (b/a)x = -c/a 5945893986, \frac{d^2 x}{dt^2} = -A \omega^2 \cos(\omega t) 5958392859, x^2 + (b/a)x+(c/a) = 0 5959282914, x^2 + x(b/a) + (b/(2 a))^2 = (x+(b/(2 a)))^2 5962145508, \alpha = \frac{nR}{VP} 5978756813, W = G m_{\rm Earth} m \left( 0 - \frac{-1}{ r_{\rm Earth}} \right) 5982958248, x = -\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a)) 5982958249, x+(b/(2 a)) = -\sqrt{(b/(2 a))^2 - (c/a)} 5985371230, \vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) 6026694087, F_{centripetal} = m \frac{v^2}{r} 6031385191, \sinh^2 x = \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right) 6055078815, \left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T \left( \frac{\partial V}{\partial T} \right)_p 6061695358, V_2 = I R_2 6083821265, v_0 \cos(\theta) = v_{0, x} 6091977310, KE_{\rm initial} = \frac{1}{2} m_1 v_{\rm initial}^2 6131764194, W = G m_{\rm Earth} m \left( \frac{1}{x^2} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right) 6134836751, v_{0, x} = v_x 6175547907, v_{\rm average} = \frac{v + v_0}{2} 6204539227, -g t + v_{0, y} = \frac{dy}{dt} 6240206408, I_{\rm incoherent} = |A|^2 + |B|^2 6240546932, \frac{1}{K_{equilibrium}} = \frac{k_{\rm desorption}}{k_{\rm adsorption}} 6268336290, F_{gravitational} = \frac{m}{r}\left(\frac{2\pi r}{T}\right)^2 6306552185, I = (A + B)(A^* + B^*) 6348260313, C_{\rm Earth\ orbit} = 2 \pi r_{\rm Earth\ orbit} 6397683463, V \alpha = \left( \frac{\partial V}{\partial T} \right)_p 6404535647, \cosh x = \frac{\exp(x) + \exp(-x)}{2} 6421241247, d = v t - \frac{1}{2} a t^2 6450985774, n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 ) 6457044853, v - a t = v_0 6457999644, \frac{[S_0]}{[A_{\rm adsorption}]} = \frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1 6504442697, v = \sqrt{ \frac{K}{\rho} } 6529793063, I_{\rm incoherent} = |A|^2 + |A|^2 6555185548, A^* = |A| \exp(-i \theta) 6556875579, \frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2 6572039835, -g t + v_{0, y} = v_y 6715248283, PE = -F x 6742123016, \vec{p}_{electron}\cdot\vec{p}_{electron} = ( \vec{p}_{1}\cdot\vec{p}_{1})+( \vec{p}_{2}\cdot\vec{p}_{2})-2( \vec{p}_{1}\cdot\vec{p}_{2}) 6753224061, I_{\rm total} = I_1 + I_2 6774684564, \theta = \phi 6783009163, r_{\rm adsorption} = r_{\rm desorption} 6785303857, C = 2 \pi r 6800170830, r_{\rm Schwarzschild} = \frac{2 G m}{c^2} 6829281943, F_{\rm centripetal} = G \frac{m_1 m_2}{r^2} 6831637424, \sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} ) 6831694380, a = \frac{d^2 x}{dt^2} 6870322215, KE_{\rm escape} = \frac{1}{2} m v_{\rm escape}^2 6885625907, \exp(i \pi) = -1 + i 0 6892595652, \frac{1}{2} m_1 v_{\rm final}^2 = \frac{G m_1 m_2}{r} 6908055431, x(t) = A \cos\left(\frac{k}{m} t\right) 6925244346, \alpha = \frac{PV}{T} \frac{1}{VP} 6935745841, F = G \frac{m_1 m_2}{x^2} 6946088325, v = \frac{C}{t} 6955192897, r_{\rm desorption} = k_{\rm desorption} [A_{\rm adsorption}] 6998364753, v_{\rm Earth\ orbit} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}} 7002609475, \frac{V}{R_2} = I_2 7010294143, T_{\rm orbit}^2 G m_{\rm Earth} = 4 \pi^2 r^3 7011114072, d = \frac{(v_0 + a t) + v_0}{2} t 7057864873, y' = y 7107090465, B B^* = |B|^2 7112613117, m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}} 7112646057, v_{\rm final}^2 = \frac{2 G m_2}{r} 7175416299, t_{\rm Earth\ orbit} = 1 {\rm year} 7215099603, v^2 = v_0^2 + 2 a t v_0 + a^2 t^2 7217021879, R_{\rm total} = R_1 + R_2 7233558441, d = v_0 t_f \cos(\theta) 7252338326, v_y = \frac{dy}{dt} 7267155233, \frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right) 7267424860, \frac{1}{\theta_A} = \frac{1+(K_{\rm equilibrium}\ p_A)}{K_{\rm equilibrium}\ p_A} 7354529102, y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0 7376526845, \sin(\theta) = \frac{v_{0, y}}{v_0} 7391837535, \cos(\theta) = \frac{v_{0, x}}{v_0} 7455581657, v_{0, x} = \frac{dx}{dt} 7466829492, \vec{ \nabla} \cdot \vec{E} = 0 7513513483, \gamma^2 (c^2 - v^2) = c^2 7517073655, [S_0] = \left(\frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}] 7564894985, \int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) 7572664728, \cos(2 x) + 2 (\sin(x))^2 = 1 7573835180, PE_{\rm Earth\ surface} = -W 7575738420, \left(\sin\left(\frac{n \pi}{W}x\right) \right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} 7575859295, \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) 7575859300, \epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{ \nabla} \times \vec{E} )_k = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) 7575859302, \epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) 7575859304, \epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} 7575859306, \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) 7575859308, \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) 7575859310, \hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) 7575859312, \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) 7621705408, I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi) 7652131521, \frac{dx}{dt} = -A \omega \sin (\omega t) 7672365885, F_{gravitational} = \frac{4 \pi^2 m r}{T^2} 7675171493, V_1 = I R_1 7676652285, KE_2 = \frac{1}{2} m v_2^2 7696214507, n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} ) 7701249282, v_u = \alpha c \sqrt{ \frac{m_e}{m_p} } 7729413831, a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right) 7731226616, {\rm sech}\ x = \frac{1}{\cosh x} 7734996511, PE_2 - PE_1 = -F ( x_2 - x_1 ) 7735731560, \cosh^2 x - \sinh^2 x = \frac{1}{4}\left( \exp(2x)+1+1+\exp(-2x) - \left(\exp(2x)-1-1-\exp(-2x)\right) \right) 7735737409, \frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t} 7741202861, x = \gamma^2 x - \gamma^2 v t + \gamma v t' 7749253510, W = G \frac{m_{\rm Earth} m }{ r_{\rm Earth}} 7826132469, \left(\frac{\partial U}{\partial T}\right)_p = C_V + \pi_T V \alpha 7837519722, v = \sqrt{f} \sqrt{\frac{E}{m}} 7846240076, m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{G} 7875206161, E_2 = KE_2 + PE_2 7882872592, W_{\rm to\ system} = \int_{\infty}^r \vec{F}\cdot d\vec{r} 7906112355, \gamma^2 = \frac{c^2}{c^2 - \gamma^2} 7917051060, \vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2} 7924063906, K_{equilibrium} = \frac{k_{\rm adsorption}}{k_{\rm desorption}} 7928111771, \frac{1}{\theta_A} = \frac{1}{K_{\rm equilibrium} p_A} + 1 7939765107, v^2 = v_0^2 + 2 a d 8046208134, I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2 8049905441, \Delta KE = KE_{\rm final} - KE_{\rm initial} 8059639673, v^2 = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2} 8065128065, I = A A^* + B B^* + A B^* + B A^* 8090924099, v = \sqrt{ \left( f\frac{E}{a^3} \right) \frac{1}{\rho} } 8106885760, \alpha = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c} 8131665171, \frac{1}{\theta_A} = \frac{[S_0]}{[A_{\rm adsorption}]} 8139187332, \vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron} 8145337879, -g t dt + v_{0, y} dt = dy 8198310977, 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0 8228733125, a_y = \frac{d}{dt} v_y 8257621077, \vec{p}_{\rm before} = \vec{p}_{1} 8269198922, 2 a d = v^2 - v_0^2 8283354808, I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 ) 8311458118, \vec{p}_{\rm after} = \vec{p}_{2}+\vec{p}_{electron} 8332931442, \exp(i \pi) = \cos(\pi)+i \sin(\pi) 8357234146, KE = \frac{1}{2} m v^2 8360117126, \gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}} 8361238989, a_{centripetal} = \frac{v^2}{r} 8368984890, \kappa_T = \frac{-1}{V} \left( \frac{ \partial }{\partial P}\left(\frac{nRT}{P}\right) \right)_T 8396997949, I = | A + B |^2 8399484849, \langle x^2 - 2 x \langle x \rangle + \langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2 8405272745, W_{\rm to\ system} = -G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx 8418527415, \sin(i x) = i \sinh(x) 8435841627, P V = n R T 8460820419, v_x = \frac{dx}{dt} 8483686863, \sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x) \right) 8484544728, -a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x) 8485757728, a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx) 8485867742, \frac{2}{W} = a^2 8486706976, v_{0, x} t + x_0 = x 8489593958, d(u v) = u dv + v du 8489593960, d(u v) - v du = u dv 8489593962, u dv = d(u v) - v du 8489593964, \int u dv = u v - \int v du 8494839423, \nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} 8495187962, \theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) } 8497631728, I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi ) 8515803375, z' = z 8532702080, \cosh^2 x = \left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right) 8552710882, KE_{\rm final} = \frac{1}{2} m_1 v_{\rm final}^2 8558338742, E_2 = E_1 8563535636, \cosh^2 x - \sinh^2 x = \left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right) - \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right) 8572657110, 1 = \int |\psi(x)|^2 dx 8572852424, \vec{E} = E( \vec{r},t) 8575746378, \int \frac{1}{2} dx = \frac{1}{2} x 8575748999, \frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x) \right) = -k^2 \left(a \sin(kx) + b \cos(kx) \right) 8576785890, 1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx 8577275751, 0 = a \sin(0) + b\cos(0) 8582885111, \psi(x) = a \sin(kx) + b \cos(kx) 8582954722, x^2 + 2 x h + h^2 = (x + h)^2 8584698994, -g \int dt = \int d v_y 8588429722, \sin( 90^{\circ} - x ) = \cos( x ) 8602221482, \langle \cos(\theta - \phi) \rangle = 0 8602512487, \vec{a} = a_x \hat{x} + a_y \hat{y} 8604483515, dW = G \frac{m_1 m_2}{x^2} dx 8651044341, \cos(i x) = \frac{1}{2} \left( \exp(-x) + \exp(x) \right) 8655294002, a = -\frac{k}{m}x 8661803554, F = G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2} 8688588981, a^3 \rho = m 8699789241, 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right) 8706092970, d = \left(\frac{v + v_0}{2}\right)t 8721295221, t_{\rm Earth\ orbit} = 3.16 10^7 {\rm seconds} 8730201316, \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t' 8747785338, \cos(i x) = \cosh(x) 8750379055, 0 = \frac{d}{dt} v_x 8808860551, -g \int t dt + v_{0, y} \int dt = \int dy 8849289982, \psi(x)^* = a \sin(\frac{n \pi}{W} x) 8889444440, 1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right) \right)^2 dx 8908736791, \rho = \frac{m}{a^3} 8922441655, d = \frac{v_0^2}{g} \sin(2 \theta) 8945218208, \theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ} 8946383937, v_{\rm escape}^2 = 2 G \frac{m}{r} 8949329361, v_0 \sin(\theta) = v_{0, y} 8953094349, W = m a x 8960645192, KE_2 + PE_2 = KE_1 + PE_1 8991236357, \frac{d^2 x}{dt^2} = -\frac{k}{m} x 9031609275, x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t' 9059289981, \psi(x) = a \sin(k x) 9063568209, V_{\rm total} = V_1 + V_2 9070394000, m_2 d_2 \frac{4 \pi^2}{T^2} = G \frac{m_1 m_2}{r^2} 9081138616, W_{\rm by\ system} = \frac{1}{2} m_1 v_{\rm final}^2 9112191201, y_f = 0 9152823411, \frac{1}{T^2} = \frac{1}{d_2 4 \pi^2} G \frac{m_1}{r^2} 9170048197, T^2 = d_2 4 \pi^2 \frac{r^2}{G m_1} 9180861128, 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right) 9191880568, Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta ) 9226945488, F = \frac{m v^2}{r} 9243879541, V = I_2 R_2 9262596735, d = 2 \pi r 9285928292, ax^2 + bx + c = 0 9291999979, \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} 9294858532, \hat{A}^+ = \hat{A} 9337785146, v = \frac{x_2 - x_1}{t} 9341391925, \vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y} 9356924046, \frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t} 9376481176, K = f \frac{E}{a^3} 9385938295, (x+(b/(2 a)))^2 = -(c/a) + (b/(2 a))^2 9393939991, \psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right) 9393939992, \psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right) 9394939493, \nabla^2 E( \vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r},t) 9397152918, v = \frac{v_1 + v_2}{2} 9407192813, G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2} = m g_{\rm Earth} 9409776983, x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t' 9412953728, v_{\rm escape}^2 = 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}} 9413609246, \cosh^2 x - \sinh^2 x = 1 9413699705, W = m a \frac{v_2^2 - v_1^2}{2 a} 9429829482, \frac{d}{dx} y = -\sin(x) + i\cos(x) 9440616166, m_{\rm Earth} = \frac{g_{\rm Earth} r_{\rm Earth}^2}{G} 9482113948, \frac{dy}{y} = i dx 9482438243, (\cos(x))^2 = \cos(2 x) + (\sin(x))^2 9482923849, \exp(i x) = y 9482928242, \cos(2 x) = (\cos(x))^2 - (\sin(x))^2 9482928243, \cos(2 x) + (\sin(x))^2 = (\cos(x))^2 9482943948, \log(y) = i dx 9482984922, \frac{d}{dx} y = (i\sin(x) + \cos(x)) i 9483928192, \cos(2 x) + i\sin(2 x) = (\cos(x))^2 + 2 i \cos(x) \sin(x) - (\sin(x))^2 9485384858, \nabla^2 E( \vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E( \vec{r})\exp(i \omega t) 9485747245, \sqrt{\frac{2}{W}} = a 9485747246, -\sqrt{\frac{2}{W}} = a 9492920340, y = \cos(x)+i \sin(x) 9495857278, \psi(x=W) = 0 9499428242, E( \vec{r},t) = E( \vec{r})\exp(i \omega t) 9510328252, KE_{\rm initial} = 0 9562264720, [S] = \frac{k_{\rm desorption} [A_{\rm adsorption}]}{k_{\rm adsorption} p_A} 9582958293, x = \sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a)) 9582958294, x+(b/(2 a)) = \sqrt{(b/(2 a))^2 - (c/a)} 9585727710, \psi(x=0) = 0 9596004948, x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle 9640720571, v = \frac{e^2}{4 \pi \epsilon_0 \hbar} \sqrt{\frac{m_e}{2 m}} 9658195023, d = v_0 t + \frac{1}{2} a t^2 9703482302, G \frac{m_{\rm Earth} m}{r_{\rm Earth}} = \frac{1}{2} m v_{\rm escape}^2 9707028061, a_x = 0 9718685793, \kappa_T = \frac{1}{P} 9749777192, 0 = KE_1 + PE_1 9756089533, \sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} ) 9759901995, v - v_0 = a t 9781951738, \kappa_T = \frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T 9805063945, \gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2 9838128064, d_2 \frac{4 \pi^2}{T^2} = G \frac{m_1}{r^2} 9847143017, \kappa_T = \frac{-PV}{V} \left( \frac{-1}{P^2}\right) 9848292229, dy = y i dx 9848294829, \frac{d}{dx} y = y i 9854442418, v = \sqrt{\frac{E}{m}} 9858028950, \frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx 9862900242, y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0 9882526611, v_{0, x} t = x - x_0 9889984281, 2 (\sin(x))^2 = 1 - \cos(2 x) 9894826550, 2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \left(\exp(i x)+\exp(-i x) \right) 9897284307, \frac{d}{t} = \frac{v + v_0}{2} 9919999981, \rho = 0 9941599459, dU = \left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV 9958485859, \frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) 9973952056, -g t = v_y - v_{0, y} 9988949211, (\sin(x))^2 = \frac{1 - \cos(2 x)}{2} 9991999979, \vec{ \nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t} 9999998870, \frac{ \vec{p}}{\hbar} = \vec{k} 9999999870, \frac{p}{\hbar} = k 9999999960, \hbar = h/(2 \pi) 9999999961, \frac{E}{\hbar} = \omega

feed count: 1
0000040490, a^2 0000999900, b/(2 a) 0001030901, \cos(x) 0001111111, (\sin(x))^2 0001209482, 2 \pi 0001304952, \hbar 0001334112, W 0001921933, 2 i 0002239424, 2 0002338514, \vec{p}_{2} 0002342425, m/m 0002393922, x 0002424922, a 0002436656, i \hbar 0002449291, b/(2 a) 0002838490, b/(2 a) 0002919191, \sin(-x) 0002929944, 1/2 0002940021, 2 \pi 0003232242, t 0003413423, \cos(-x) 0003747849, -1 0003838111, 2 0003919391, x 0003949052, -x 0003949921, \hbar 0003954314, dx 0003981813, -\sin(x) 0004089571, 2 x 0004264724, y 0004307451, (b/(2 a))^2 0004582412, x 0004829194, 2 0004831494, a 0004849392, x 0004858592, h 0004934845, x 0004948585, a 0005395034, a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle 0005626421, t 0005749291, f 0005938585, \frac{-\hbar^2}{2m} 0006466214, (\sin(x))^2 0006544644, t 0006563727, x 0006644853, c/a 0006656532, e 0007471778, 2(\sin(x))^2 0007563791, i 0007636749, x 0007894942, (\sin(x))^2 0008837284, T 0008842811, \cos(2 x) 0009458842, \psi(x) 0009484724, \frac{n \pi}{W}x 0009485857, a^2\frac{2}{W} 0009485858, \frac{2n\pi}{W} 0009492929, v du 0009587738, \psi 0009877781, y 1025759423, y 1036530438, d_2 1092872200, KE_1 1100332145, R 1193980495, m_{\rm Earth} 1238593037, c 1258245373, E 1268845856, [A_{\rm adsorption}] 1277713901, d 1323602089, I_1 1333474099, F_{\rm centripetal} 1377431959, R 1413137236, m_1 1439089569, v_{0, x} 1451839362, t 1484794622, R_2 1511199318, Z 1512581563, x 1552869972, x_1 1571582377, F_{gravitational} \propto \frac{1}{r^2} 1608399874, V_2 1614343171, dt 1616666229, v_{\rm final} 1635147226, m_2 1716984328, i x 1742775076, Z 1823570358, C 1848400430, F \propto m 1894894315, Z 1945487024, p_A [S] 2016063530, t 2064205392, A 2081689540, t 2091584724, g_{\rm Earth} 2114570475, m_{\rm satellite} 2135482543, m 2226340358, \gamma v 2232825726, g_{\rm Earth} 2242144313, a 2293352649, \theta - \phi 2344320475, E_2 2346150725, r 2346952973, m 2396787389, r_{\rm Earth} 2397692197, a^3 2403773761, t 2510804451, 2/g 2660368546, r 2674546234, m_{\rm Earth} 2685587762, \frac{r_{\rm Earth}^2}{G} 2698469612, V 2754264786, 2 2764966428, m_2 2773628333, \theta_1 2867848403, I 2957211007, m^3 kg^{-1} s^{-2} 3031116098, R_2 3041762466, i 3088463019, m_2 3105350101, v_1 3166466250, m_1 3182907803, x_0 3183197515, v_1 3219318145, \frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}} 3236313290, d 3246378279, m 3268645065, x 3270039798, 2 3273630811, x 3274176452, v_{\rm initial} 3342155559, m 3350802342, KE_{\rm initial} 3353418803, x 3398368564, F 3412946408, v^2 \gamma^2 3433441359, V 3448601530, \frac{T^2}{r} 3486213448, m_{\rm satellite} 3495403335, x 3531380618, v(r) 3594626260, F_{\rm gravity} 3634715785, m 3650814381, F_{gravitational} \propto \frac{m_1 m_2}{r^2} 3652511721, v 3663007361, 2 3685779219, \sqrt{f} \approx 2 3722461713, t 3723096423, 6.3781*10^6 3731774096, KE 3749492596, E 3809726424, PE 3846345263, T_{\rm orbit} 3876446703, m 3911081515, -1 3921072591, m_1 3939572542, KE_{\rm final} 3967985562, 2 4057686137, C 4147101187, KE 4153613253, m_{\rm Earth} 4162188238, t_f 4167526462, v_{0, y} 4188639044, x 4202292449, r_{\rm Earth\ orbit} 4213426349, E_1 4218009993, x 4264859781, F \propto m_1 4319470443, v_2 4319544433, 1/3 4437214608, Z 4470433702, t_{\rm Earth\ orbit} 4490788873, F \propto m_2 4522137851, PE_2 4583868070, B 4587046017, KE 4651061153, m_2 4755369593, x_2 4829590294, t_f 4830480629, x 4901237716, 1 4935235303, x 4961662865, x 5011888122, v_2 5021965469, KE 5050429607, G \frac{m_{\rm Earth} m}{r_{\rm Earth}} 5074423401, V 5075406409, PE 5089196493, F 5181421075, R_1 5194141542, x_f 5208737840, T_{\rm geostationary\ orbit} 5239755033, v_1 5258419993, R_1 5284610349, \gamma^2 5359471792, \frac{m_{\rm satellite}}{r} 5398681502, v 5398681503, v 5426418187, [A_{\rm adsorption}] 5453995431, \arctan{ x } 5463275819, I_2 5516739892, -1 5542390646, 2 a 5585739998, I 5591692598, KE_1 5620558729, v_0 5623794884, A + B 5632428182, \cos( \theta_{\rm Brewster} ) 5667870149, \theta 5669500954, v^2 \gamma^2 5684907106, \frac{1}{d_2 4 \pi^2} 5770088141, r 5775658332, 2 5778176146, t 5781435087, g 5787469164, 1 - \gamma^2 5799753649, 2 5803210729, PE_2 5846177002, t 5868731041, v_0 5890617067, R 5904227750, m 5960438249, E_1 6023986360, x 6038673136, v 6050070428, v_{0, x} 6098638221, y_0 6158970683, PE_1 6238632840, r T_{\rm orbit}^2 6239815585, C_{\rm Earth\ orbit} 6259833695, A 6281834543, m_1 6296166842, P 6346902704, 1 6353701615, \theta_{\rm refracted} 6383056612, KE 6408214498, c^2 6410818363, \theta 6417359412, v_0 6463266449, t_f 6498985149, v_{\rm escape} 6529120965, B 6535639720, r_{\rm Earth} 6546594355, R_{\rm total} 6554292307, t 6599829782, v_{\rm final} 6672141531, dt 6681646197, v 6701855578, v_2 6729698807, v_0 6732786762, t 6749533119, PE_1 6838659900, KE_2 6964468708, KE_1 6974054946, \frac{1}{2} g t_f 6976493023, x 7049769409, 2 7053449926, r_{\rm geostationary\ orbit} 7083390553, t 7140470627, m 7154592211, \theta_2 7159989263, i x 7191277455, R 7194432406, r_{\rm Schwarzschild} 7214442790, x 7263534144, c^2 7321695558, \theta_{\rm Brewster} 7326066466, G 7337056406, \gamma^2 x 7375348852, \theta_{\rm Brewster} 7410124465, R_{\rm total} 7410526982, 2/m_1 7445388869, -1 7453225570, x 7473576008, \frac{-1}{A \cos(\omega t)} 7476820482, C 7497687256, V 7556442438, 4 \pi^2 7560908617, m 7564010952, -1 7587034465, \theta 7607271250, \theta 7630953440, \frac{K_{\rm equilibrium} p_A}{K_{\rm equilibrium} p_A} 7708501762, C_{\rm Earth\ orbit} 7743841045, \gamma^2 7774819339, R 7798615279, I_{\rm total} 7816982139, m/s^2 7819443873, r 7844317489, I 7857757625, n_1 7905984866, m_1 7912578203, v 7924842770, T 7935917166, r_{\rm Earth} 7939947931, KE_2 8014566709, \gamma^2 v t 8020058613, r 8044416349, d_2 8061701434, PE_1 8066819515, v 8072682558, x_0 8111389082, x 8120663858, y_f 8122039815, \frac{d_1+d_2}{d_1+d_2} 8135396036, t 8162179726, k_{\rm adsorption} p_A 8173074178, x 8362338572, v_{\rm escape} 8406170337, y 8416464049, KE_{\rm escape} 8571466509, c^2 - \gamma^2 8607458157, dt 8642992037, 2 8710504862, A 8717193282, dt 8854422847, dT 8857931498, c 8865085668, t 8871333437, PE_{\rm Earth\ surface} 8880467139, 2 8916428651, m 9025853427, \theta_{\rm Brewster} 9029795851, \theta_{\rm Brewster} 9040079362, f 9053099840, I 9070454719, v_0^2 9072369552, m_{\rm Earth} 9110536742, 2 x 9174439158, R_1 9305761407, v 9346215480, T_{\rm orbit} 9350663581, \pi 9350720370, m 9355039511, g 9370882921, KE_{\rm escape} 9524810853, \frac{1/d_2}{1/d_2} 9565166889, T 9590696981, 9.80665 9594072504, m_2 9601500174, v_{\rm Earth\ orbit} 9623791270, d 9645178657, a t 9674924517, K >> G 9746066299, R_2 9753878784, v 9761485403, Z 9789485295, v_{\rm satellite} 9794128647, m_1 9830343096, V_1 9881106100, a 9884115626, r 9885190237, i 9903988330, m 9906920183, x 9933742680, r_{\rm Schwarzschild} 9956609318, 6.67430*10^{-11}

output count: 2
9999999962, p = \hbar k 9999999965, E = \omega \hbar 9999999968, x = \frac{-b-\sqrt{b^2-4ac}}{2 a} 9999999969, x = \frac{-b+\sqrt{b^2-4ac}}{2 a} 9999999975, \langle \psi| \hat{A} |\psi \rangle = \langle a \rangle 9999999981, \vec{ \nabla} \cdot \vec{E} = \rho/\epsilon_0 0203024440, 1 = \int_0^W a \sin\left(\frac{n \pi}{W} x\right) \psi(x)^* dx 0404050504, \lambda = \frac{v}{f} 0439492440, \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) \right|_0^W 0934990943, k = \frac{2 \pi}{v T} 0948572140, \int \cos(a x) dx = \frac{1}{a}\sin(a x) 1010393913, \langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^* 1010393944, x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle 1010923823, k W = n \pi 1020010291, 0 = a \sin(k W) 1020394900, p = h/\lambda 1020394902, E = h f 1020854560, I = (A + B)(A + B)^* 1029039903, p = m v 1029039904, p^2 = m^2 v^2 1038566242, \sinh x = \frac{\exp(x) - \exp(-x)}{2} 1085150613, C_V = \left(\frac{\partial U}{\partial T}\right)_V 1087417579, 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) 1114820451, W_{\rm by\ system} = \Delta KE 1128605625, {\rm sech}^2\ x + \tanh^2(x) = \frac{4}{\left(\exp(x)+\exp(-x)\right)^2} + \frac{\left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2} 1132941271, m_{\rm Earth} = \frac{(9.80665 m/s^2) (6.3781*10^6 m)^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}} 1143343287, G \frac{m_{\rm Earth}}{r_{\rm Earth}} = \frac{1}{2} v_{\rm escape}^2 1158485859, \frac{-\hbar^2}{2m} \nabla^2 = {\cal H} 1166310428, 0 dt = d v_x 1172039918, I_{\rm coherent} = 4 |A|^2 1190768176, \kappa_T = \frac{-nRT}{V} \left( \frac{ \partial }{\partial P}\left(\frac{1}{P}\right) \right)_T 1191796961, \frac{1}{2} g t_f = v_0 \sin(\theta) 1201689765, x'^2 + y'^2 + z'^2 = c^2 t'^2 1202310110, \frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx 1202312210, \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx 1203938249, a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle 1248277773, \cos(2 x) = 1 - 2 (\sin(x))^2 1259826355, d = (v - a t) t + \frac{1}{2} a t^2 1265150401, d = \frac{2 v_0 + a t}{2} t 1292735067, F_{gravitational} = G \frac{m_1 m_2}{r^2} 1293913110, 0 = b 1293923844, \lambda = v T 1306360899, x = v_{0, x} t + x_0 1310571337, \theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster} 1311403394, \alpha = \frac{1}{V} \frac{nR}{P} \left( \frac{\partial T}{\partial T} \right)_P 1314464131, \vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} 1314864131, \vec{ \nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E} 1330874553, v_{\rm escape} = \sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}} 1357848476, A = |A| \exp(i \theta) 1395858355, x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle 1405465835, y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0 1457415749, \frac{1}{R_{\rm total}} = \frac{1}{R_1} + \frac{1}{R_2} 1525861537, I = |A|^2 + |B|^2 + A B^* + B A^* 1528310784, \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} 1541916015, \theta = \frac{\pi}{4} 1556389363, E_{\rm Rydberg} = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2} 1559688463, \left(\frac{T_{\rm geostationary\ orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r_{\rm geostationary\ orbit} 1586866563, \left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right) 1590774089, dW = F dx 1636453295, \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = - \nabla^2 \vec{E} 1638282134, \vec{p}_{\rm before} = \vec{p}_{\rm after} 1639827492, - c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1 1648958381, \nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right) 1650441634, y_0 = 0 1676472948, 0 = v_x - v_{0, x} 1702349646, -g dt = d v_y 1772416655, \frac{E_2 - E_1}{t} = v F - F v 1772973171, -\frac{k}{m} x = -A \omega^2 \cos(\omega t) 1784114349, \sqrt{\frac{k}{m}} = \omega 1809909100, \frac{E_2 - E_1}{t} = 0 1811867899, T^2 = \frac{d_1+d_2}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1} 1815398659, U = Q + W 1819663717, a_x = \frac{d}{dt} v_x 1840080113, KE_2 = 0 1857710291, 0 = a \sin(n \pi) 1858578388, \nabla^2 E( \vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E( \vec{r})\exp(i \omega t) 1858772113, k = \frac{n \pi}{W} 1888494137, -\sqrt{\frac{k}{m}} = \omega 1916173354, -\gamma^2 v^2 + c^2 \gamma^2 = c^2 1928085940, Z^* = |Z| \exp( -i \theta ) 1931103031, \frac{k}{m} = \omega^2 1934748140, \int |\psi(x)|^2 dx = 1 1935543849, \gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2 1963253044, v_{0, x} dt = dx 1967582749, t = \frac{v - v_0}{a} 1974334644, \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t' 1977955751, -g = \frac{d}{dt} v_y 1994296484, v_{\rm satellite}^2 = G \frac{m_{\rm Earth}}{r} 2005061870, v(r) = \sqrt{\frac{2 G m_2}{r}} 2029293929, \nabla^2 E( \vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r})\exp(i \omega t) 2042298788, 0 = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}} + \frac{1}{2} m v_{\rm escape}^2 2051901211, \frac{V}{R_1} = I_1 2061086175, W_{\rm to\ system} = -G m_1 m_2 \left(\frac{-1}{r} - \frac{-1}{\infty}\right) 2076171250, -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0 2086924031, 0 = - \frac{1}{2} g t_f + v_0 \sin(\theta) 2096918413, x = \gamma ( \gamma x - \gamma v t + v t' ) 2103023049, \sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) 2113211456, f = 1/T 2114909846, \theta_A = \frac{[A_{\rm adsorption}]}{[S_0]} 2121790783, \tanh^2(x) = \frac{ \left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2} 2123139121, -\exp(-i x) = -\cos(x)+i \sin(x) 2131616531, T f = 1 2148049269, -\frac{k}{m} A \cos(\omega t) = -A \omega^2 \cos(\omega t) 2168306601, [S_0] = \left(\frac{k_{\rm desorption}}{k_{\rm adsorption}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}] 2186083170, \frac{KE_2 - KE_1}{t} = v F 2217103163, \frac{m_1 d_1}{d_2} = m_2 2236639474, (A + B)(A + B)^* = |A + B|^2 2257410739, \left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alpha 2258485859, {\cal H} \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) 2267521164, PE_2 = 0 2271186630, V = I_{\rm total} R_{\rm total} 2297105551, d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta) 2308660627, G \frac{m_{\rm Earth}}{r_{\rm Earth}^2} = g_{\rm Earth} 2334518266, m a = -k x 2366691988, \int 0 dt = \int d v_x 2378095808, x_f = x_0 + d 2394240499, x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle 2394853829, \exp(-i x) = \cos(-x)+i \sin(-x) 2394935831, ( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0 2394935835, \left(\langle\psi| \hat{A} |\psi \rangle \right)^+ = \left(\langle a \rangle\right)^+ 2395958385, \nabla^2 \psi \left( \vec{r},t \right) = \frac{-p^2}{\hbar} \psi( \vec{r},t) 2404934990, \langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2 2405307372, \sin(2 x) = 2 \sin(x) \cos(x) 2417941373, - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2 2431507955, PE_2 = -F x_2 2461349007, - \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y 2472653783, \alpha = \frac{1}{T} 2484824786, F = m g 2494533900, \langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2 2501591100, \exp(i \pi) + 1 = 0 2503972039, 0 = KE_{\rm escape} + PE_{\rm Earth\ surface} 2519058903, \sin(2 \theta) = 2 \sin(\theta) \cos(\theta) 2542420160, c^2 \gamma^2 - v^2 \gamma^2 = c^2 2575937347, n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} ) 2613006036, \frac{PV}{T} = nR 2617541067, \left(\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r 2648958382, \nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) \right) 2700934933, 2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right) 2715678478, I R_{\rm total} = I R_1 + I R_2 2719691582, |A| = |B| 2741489181, a_y = -g 2750380042, v_{\rm escape} = -\sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}} 2762326680, \cosh^2 x - \sinh^2 x = \frac{1}{4}\left( \exp(2x)+1+1+\exp(-2x) - \left(\exp(2x)-1-1+\exp(-2x)\right) \right) 2768857871, \frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1} 2770069250, \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t} 2809345867, \frac{V}{R_{\rm total}} = I_{\rm total} 2848934890, \langle a \rangle^* = \langle a \rangle 2857430695, a = \frac{v_2 - v_1}{t} 2858549874, - \frac{1}{2} g t^2 + v_{0, y} t = y - y_0 2883079365, r_{\rm Schwarzschild} c^2 = 2 G m 2897612567, v = \alpha c \sqrt{ \frac{m_e}{A m_p} } 2902772962, \tanh(x) = \frac{\frac{1}{2}\left( \exp(x)-\exp(-x) \right)}{\cosh(x)} 2906548078, T^2 = \frac{r}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1} 2907404069, W_{\rm by\ system} = W_{\rm to\ system} 2924222857, v_{\rm initial} = v(r=\infty) 2944838499, \psi(x) = a \sin(\frac{n \pi}{W} x) 2977457786, 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}} = v_{\rm escape}^2 2983053062, x = \gamma (x' + v t') 2998709778, v_{\rm initial} = 0 2999795755, c^2 \gamma^2 = v^2 \gamma^2 + c^2 3004158505, \frac{T^2}{r} F_{gravitational} = \left( \frac{4 \pi^2 m r}{T^2} \right)\frac{T^2}{r} 3046191961, v_{\rm Earth\ orbit} = \frac{C_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}} 3060393541, I_{\rm incoherent} = 2|A|^2 3061811650, n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} ) 3080027960, v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}} 3085575328, I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi)) 3121234211, \frac{k}{2\pi} = \lambda 3121234212, p = \frac{h k}{2\pi} 3121513111, k = \frac{2 \pi}{\lambda} 3131111133, T = 1 / f 3131211131, \omega = 2 \pi f 3132131132, \omega = \frac{2\pi}{T} 3147472131, \frac{\omega}{2 \pi} = f 3169580383, \vec{a} = \frac{d\vec{v}}{dt} 3176662571, F_{\rm centripetal} = F_{\rm gravity} 3182633789, \gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 3214170322, v(r=\infty) = 0 3253234559, x = \frac{v_2^2 - v_1^2}{2 a} 3274926090, t = \frac{x - x_0}{v_{0, x}} 3285732911, (\cos(x))^2 = 1-(\sin(x))^2 3291685884, E = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2} 3331824625, \exp(i \pi) = -1 3350830826, Z Z^* = |Z|^2 3360172339, W = KE_2 - KE_1 3364286646, m_{\rm Earth} = 5.972*10^{24} kg 3366703541, a = \frac{v - v_0}{t} 3411994811, v_{\rm average} = \frac{d}{t} 3417126140, \tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 } 3426941928, x = \gamma ( \gamma (x - v t) + v t' ) 3462972452, v = v_0 + a t 3464107376, \alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p 3470587782, \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) 3472836147, r_{\rm Earth\ orbit} = 1.496\ 10^8 {\rm km} 3485125659, x_f = v_0 t_f \cos(\theta) + x_0 3485475729, \nabla^2 E( \vec{r}) = - \frac{\omega^2}{c^2} E( \vec{r}) 3488423948, k_{\rm adsorption} p_A [S] = k_{\rm desorption} [A_{\rm adsorption}] 3497828859, V = \frac{n R T}{P} 3507029294, k_{\rm adsorption} p_A [S] = r_{\rm desorption} 3512166162, W = F x 3547519267, S = k_{\rm Boltzmann} \ln \Omega 3566149658, W_{\rm to\ system} = \int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx 3585845894, \langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2 3591237106, \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v 3599953931, [S_0] = [S] + [A_{\rm adsorption}] 3605073197, \kappa_T = \frac{-nRT}{V} \left( \frac{-1}{P^2}\right) 3607070319, d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right) 3614055652, v = \frac{2 \pi r}{T_{\rm orbit}} 3649797559, F_{\rm centripetal} = m_2 d_2 \omega^2 3650370389, \frac{T^2}{r} F_{gravitational} = 4 \pi^2 m 3660957533, \cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right) 3676159007, v_{0, x} \int dt = \int dx 3736177473, r_{\rm adsorption} = k_{\rm adsorption} p_A [S] 3781109867, T^2 = \frac{r^3 4 \pi^2}{(d_1+d_2) \frac{m_1}{d_2}G} 3806977900, E_2 - E_1 = 0 3829492824, \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) = \cos(x) 3846041519, PE_{\rm Earth\ surface} = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}} 3868998312, {\rm sech}^2\ x = \frac{4}{\left(\exp(x)+\exp(-x)\right)^2} 3896798826, m_2 d_2 \omega^2 = G \frac{m_1 m_2}{r^2} 3906710072, G \frac{m_{\rm Earth}}{r} = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2} 3920616792, T_{\rm geostationary orbit} = 24\ {\rm hours} 3924948349, a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0 3935058307, v = \sqrt{ \frac{m_e}{m} \frac{e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} } 3942849294, \exp(i x)-\exp(-i x) = 2 i \sin(x) 3943939590, x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle 3947269979, \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} 3948571256, \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}E \psi( \vec{r},t) 3948574224, \psi( \vec{r},t) = \psi_0 \exp\left(i\left( \vec{k}\cdot\vec{r} - \omega t \right) \right) 3948574226, \psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right) 3948574228, \psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right) 3948574230, \psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right) 3948574233, \frac{\partial}{\partial t} \psi( \vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right) 3951205425, \vec{p}_{\rm after} = \vec{p}_{1} 4072200527, \frac{m_{\rm satellite} v_{\rm satellite}^2}{r} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2} 4075539836, A A^* = |A|^2 4087145886, V = I R 4107032818, E_{\rm Rydberg} = E 4128500715, V = I_1 R_1 4139999399, x - \gamma^2 x = - \gamma^2 v t + \gamma v t' 4147472132, E = \frac{h \omega}{2 \pi} 4158986868, a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt} 4166155526, {\rm sech}\ x = \frac{2}{\exp(x)+\exp(-x)} 4180845508, v_{\rm Earth\ orbit} = 29.8 \frac{{\rm km}}{{\rm sec}} 4182362050, Z = |Z| \exp( i \theta ) 4188580242, T^2 = \frac{r^3 4 \pi^2}{\left(m_1+\left(\frac{m_1}{d_2}d_1\right)\right)G} 4192519596, B = |B| \exp(i \phi) 4245712581, v = \frac{2 \pi r}{t} 4267808354, F_{gravitational} = m \frac{v^2}{r} 4268085801, x_0 + d = v_0 t_f \cos(\theta) + x_0 4270680309, \frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t} 4275004561, c^2 = 2 G \frac{m}{r_{\rm Schwarzschild}} 4287102261, x^2 + y^2 + z^2 = c^2 t^2 4298359835, E = \frac{1}{2}m v^2 4298359845, E = \frac{1}{2m}m^2 v^2 4298359851, E = \frac{p^2}{2m} 4301729661, [S_0] = \frac{[A_{\rm adsorption}]}{\left( \frac{k_{\rm adsorption}}{k_{\rm desorption}} \right) p_A} + [A_{\rm adsorption}] 4303372136, E_1 = KE_1 + PE_1 4341171256, i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{p^2}{2 m} \psi( \vec{r},t) 4348571256, \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi( \vec{r},t) 4370074654, t = t_f 4393258808, F_{\rm centripetal} = m r \omega^2 4393670960, W_{\rm to\ system} = \frac{G m_1 m_2}{r} 4394958389, \vec{ \nabla}\cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right) = \frac{i}{\hbar} \vec{ \nabla}\cdot\left( \vec{p} \psi( \vec{r},t) \right) 4428528271, F_{\rm{spring}} = -k x 4447113478, \int dW = G m_1 m_2 \int_{ r_{\rm Earth} }^{\infty} \frac{1}{x^2} dx 4501377629, \tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} 4504256452, B^* = |B| \exp(-i \phi) 4560648264, v = \sqrt{ \frac{K + (4/3) G}{\rho} } 4580545876, d = v t - a t^2 + \frac{1}{2} a t^2 4585828572, \epsilon_0 \mu_0 = \frac{1}{c^2} 4585932229, \cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) 4593428198, v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{3.16\ 10^7 {\rm seconds}} 4598294821, \exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2 4627284246, F_{\rm centripetal} = \frac{m_{\rm satellite} v_{\rm satellite}^2}{r} 4638429483, \exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x)) 4648451961, v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1) 4662369843, x' = \gamma (x - v t) 4669290568, PE_1 = -F x_1 4689334676, \theta_A = \frac{K_{\rm equilibrium}\ p_A}{1+K_{\rm equilibrium}\ p_A} 4742644828, \exp(i x)+\exp(-i x) = 2 \cos(x) 4748157455, a t = v - v_0 4778077984, t_f = \frac{2 v_0 \sin(\theta)}{g} 4784793837, \frac{KE_2 - KE_1}{t} = m v a 4798787814, a t + v_0 = v 4800170179, F = m g_{\rm Earth} 4805233006, i \sin(i x) = \frac{1}{2}\left(\exp(x) - \exp(-x) \right) 4811121942, W = \frac{1}{2} m v_2^2 - \frac{1}{2} m v_1^2 4820320578, F_{gravitational} = F_{centripetal} 4827492911, \cos(2 x)+(\sin(x))^2 = 1 - (\sin(x))^2 4830221561, {\rm sech}^2\ x + \tanh^2(x) = \frac{4+\left(\exp(2x)-1-1+\exp(-2x)\right)}{\left(\exp(x)+\exp(-x)\right)^2} 4838429483, \exp(2 i x) = \cos(2 x)+i \sin(2 x) 4843995999, \frac{1}{2 i}\left(\exp(i x)-\exp(-i x) \right) = \sin(x) 4857472413, 1 = \int \psi(x)\psi(x)^* dx 4857475848, \frac{1}{a^2} = \frac{W}{2} 4858693811, \frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2} = r^3 4866160902, \frac{V}{R_{\rm total}} = \frac{V}{R_1} + \frac{V}{R_2} 4872163189, \tanh(x) = \frac{\sinh(x)}{\cosh(x)} 4872970974, \frac{PE_2 - PE_1}{t} = -F v 4878728014, \sin(i x) = \frac{1}{2i}\left(\exp(-x) - \exp(x) \right) 4923339482, i x = \log(y) 4928007622, KE_1 = \frac{1}{2} m v_1^2 4928239482, \log(y) = i x 4938429482, \exp(-i x) = \cos(x)+i \sin(-x) 4938429483, \exp(i x) = \cos(x)+i \sin(x) 4938429484, \exp(-i x) = \cos(x)-i \sin(x) 4939880586, V_{\rm total} = I R_{\rm total} 4943571230, \vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right) 4947831649, \frac{1}{2} m_1 v_{\rm final}^2 = W_{\rm to\ system} 4948763856, 2 a d + v_0^2 = v^2 4948934890, \langle \psi| \hat{A} |\psi \rangle = \langle a \rangle^* 4949359835, \langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2 4968680693, \tan( x ) = \frac{ \sin( x )}{\cos( x )} 4985825552, \nabla^2 E( \vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E( \vec{r})\exp(i \omega t) 5002539602, dU = C_V dT + \pi_T dV 5085809757, \frac{k_{\rm adsorption}}{k_{\rm desorption}} = \frac{[A_{\rm adsorption}]}{p_A [S]} 5125940051, I = |A|^2 + B B^* + A B^* + B A^* 5128670694, m_1 d_1 = m_2 d_2 5136652623, E = KE + PE 5144263777, v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right) 5148266645, t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t 5177311762, v = \frac{2 \pi r}{T} 5323719091, i \sinh x = \frac{1}{2i} \left( \exp(-x) - \exp(x) \right) 5345738321, F = m a 5349669879, \tanh(x) = \frac{ \exp(x)-\exp(-x)}{\exp(x)+\exp(-x)} 5349866551, \vec{v} = v_x \hat{x} + v_y \hat{y} 5353282496, d = \frac{v_0^2}{g} 5373931751, t = t_f 5379546684, y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0 5404822208, v_{\rm escape} = \sqrt{2 G \frac{m}{r}} 5415824175, x(t) = A \cos(\omega t) 5426308937, v = \frac{d}{t} 5438722682, x = v_0 t \cos(\theta) + x_0 5514556106, E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1) 5530148480, \vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron} 5542528160, \int dW = F \int_0^x dx 5563580265, F_{\rm gravity} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2} 5586102077, r = d_1 + d_2 5596822289, W_{\rm to\ system} = -G m_1 m_2 \left(\left.\frac{-1}{x}\right|^r_{\infty}\right) 5611024898, d = \frac{1}{2 a} (v^2 - v_0^2) 5634116660, \pi_T = \left(\frac{\partial U}{\partial V}\right)_T 5646314683, m = A m_p 5658865948, T^2 = \frac{r^3 4 \pi^2}{(m_1+m_2)G} 5693047217, v_{\rm final} = -\sqrt{\frac{2 G m_2}{r}} 5727578862, \frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x) 5732331610, W = G m_1 m_2 \left( \frac{1}{x} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right) 5733146966, KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right) 5733721198, d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right) 5763749235, -c^2 + c^2 \gamma^2 = v^2 \gamma^2 5779256336, W_{\rm by\ system} = KE_{\rm final} - KE_{\rm initial} 5781981178, x^2 - y^2 = (x+y)(x-y) 5789289057, v = \alpha c \sqrt{ \frac{m_e}{2 m} } 5832984291, (\sin(x))^2 + (\cos(x))^2 = 1 5838268428, \alpha c = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar} 5846639423, v_{\rm final} = \sqrt{\frac{2 G m_2}{r}} 5850144586, W_{\rm by\ system} = KE_{\rm final} 5857434758, \int a dx = a x 5866629429, {\rm sech}^2\ x + \tanh^2(x) = 1 5868688585, \frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = \frac{p^2}{2m} \psi( \vec{r},t) 5900595848, k = \frac{\omega}{v} 5902985919, \vec{F} = G \frac{m_1 m_2}{x^2} \hat{x} 5928285821, x^2 + 2 x (b/(2 a)) + (b/(2 a))^2 = (x + (b/(2 a)))^2 5928292841, x^2 + (b/a)x + (b/(2 a))^2 = -c/a + (b/(2 a))^2 5938459282, x^2 + (b/a)x = -c/a 5945893986, \frac{d^2 x}{dt^2} = -A \omega^2 \cos(\omega t) 5958392859, x^2 + (b/a)x+(c/a) = 0 5959282914, x^2 + x(b/a) + (b/(2 a))^2 = (x+(b/(2 a)))^2 5962145508, \alpha = \frac{nR}{VP} 5978756813, W = G m_{\rm Earth} m \left( 0 - \frac{-1}{ r_{\rm Earth}} \right) 5982958248, x = -\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a)) 5982958249, x+(b/(2 a)) = -\sqrt{(b/(2 a))^2 - (c/a)} 5985371230, \vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) 6026694087, F_{centripetal} = m \frac{v^2}{r} 6031385191, \sinh^2 x = \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right) 6055078815, \left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T \left( \frac{\partial V}{\partial T} \right)_p 6061695358, V_2 = I R_2 6083821265, v_0 \cos(\theta) = v_{0, x} 6091977310, KE_{\rm initial} = \frac{1}{2} m_1 v_{\rm initial}^2 6131764194, W = G m_{\rm Earth} m \left( \frac{1}{x^2} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right) 6134836751, v_{0, x} = v_x 6175547907, v_{\rm average} = \frac{v + v_0}{2} 6204539227, -g t + v_{0, y} = \frac{dy}{dt} 6240206408, I_{\rm incoherent} = |A|^2 + |B|^2 6240546932, \frac{1}{K_{equilibrium}} = \frac{k_{\rm desorption}}{k_{\rm adsorption}} 6268336290, F_{gravitational} = \frac{m}{r}\left(\frac{2\pi r}{T}\right)^2 6306552185, I = (A + B)(A^* + B^*) 6348260313, C_{\rm Earth\ orbit} = 2 \pi r_{\rm Earth\ orbit} 6397683463, V \alpha = \left( \frac{\partial V}{\partial T} \right)_p 6404535647, \cosh x = \frac{\exp(x) + \exp(-x)}{2} 6421241247, d = v t - \frac{1}{2} a t^2 6450985774, n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 ) 6457044853, v - a t = v_0 6457999644, \frac{[S_0]}{[A_{\rm adsorption}]} = \frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1 6504442697, v = \sqrt{ \frac{K}{\rho} } 6529793063, I_{\rm incoherent} = |A|^2 + |A|^2 6555185548, A^* = |A| \exp(-i \theta) 6556875579, \frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2 6572039835, -g t + v_{0, y} = v_y 6715248283, PE = -F x 6742123016, \vec{p}_{electron}\cdot\vec{p}_{electron} = ( \vec{p}_{1}\cdot\vec{p}_{1})+( \vec{p}_{2}\cdot\vec{p}_{2})-2( \vec{p}_{1}\cdot\vec{p}_{2}) 6753224061, I_{\rm total} = I_1 + I_2 6774684564, \theta = \phi 6783009163, r_{\rm adsorption} = r_{\rm desorption} 6785303857, C = 2 \pi r 6800170830, r_{\rm Schwarzschild} = \frac{2 G m}{c^2} 6829281943, F_{\rm centripetal} = G \frac{m_1 m_2}{r^2} 6831637424, \sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} ) 6831694380, a = \frac{d^2 x}{dt^2} 6870322215, KE_{\rm escape} = \frac{1}{2} m v_{\rm escape}^2 6885625907, \exp(i \pi) = -1 + i 0 6892595652, \frac{1}{2} m_1 v_{\rm final}^2 = \frac{G m_1 m_2}{r} 6908055431, x(t) = A \cos\left(\frac{k}{m} t\right) 6925244346, \alpha = \frac{PV}{T} \frac{1}{VP} 6935745841, F = G \frac{m_1 m_2}{x^2} 6946088325, v = \frac{C}{t} 6955192897, r_{\rm desorption} = k_{\rm desorption} [A_{\rm adsorption}] 6998364753, v_{\rm Earth\ orbit} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}} 7002609475, \frac{V}{R_2} = I_2 7010294143, T_{\rm orbit}^2 G m_{\rm Earth} = 4 \pi^2 r^3 7011114072, d = \frac{(v_0 + a t) + v_0}{2} t 7057864873, y' = y 7107090465, B B^* = |B|^2 7112613117, m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}} 7112646057, v_{\rm final}^2 = \frac{2 G m_2}{r} 7175416299, t_{\rm Earth\ orbit} = 1 {\rm year} 7215099603, v^2 = v_0^2 + 2 a t v_0 + a^2 t^2 7217021879, R_{\rm total} = R_1 + R_2 7233558441, d = v_0 t_f \cos(\theta) 7252338326, v_y = \frac{dy}{dt} 7267155233, \frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right) 7267424860, \frac{1}{\theta_A} = \frac{1+(K_{\rm equilibrium}\ p_A)}{K_{\rm equilibrium}\ p_A} 7354529102, y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0 7376526845, \sin(\theta) = \frac{v_{0, y}}{v_0} 7391837535, \cos(\theta) = \frac{v_{0, x}}{v_0} 7455581657, v_{0, x} = \frac{dx}{dt} 7466829492, \vec{ \nabla} \cdot \vec{E} = 0 7513513483, \gamma^2 (c^2 - v^2) = c^2 7517073655, [S_0] = \left(\frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}] 7564894985, \int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) 7572664728, \cos(2 x) + 2 (\sin(x))^2 = 1 7573835180, PE_{\rm Earth\ surface} = -W 7575738420, \left(\sin\left(\frac{n \pi}{W}x\right) \right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} 7575859295, \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) 7575859300, \epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{ \nabla} \times \vec{E} )_k = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) 7575859302, \epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) 7575859304, \epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} 7575859306, \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) 7575859308, \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) 7575859310, \hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) 7575859312, \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) 7621705408, I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi) 7652131521, \frac{dx}{dt} = -A \omega \sin (\omega t) 7672365885, F_{gravitational} = \frac{4 \pi^2 m r}{T^2} 7675171493, V_1 = I R_1 7676652285, KE_2 = \frac{1}{2} m v_2^2 7696214507, n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} ) 7701249282, v_u = \alpha c \sqrt{ \frac{m_e}{m_p} } 7729413831, a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right) 7731226616, {\rm sech}\ x = \frac{1}{\cosh x} 7734996511, PE_2 - PE_1 = -F ( x_2 - x_1 ) 7735731560, \cosh^2 x - \sinh^2 x = \frac{1}{4}\left( \exp(2x)+1+1+\exp(-2x) - \left(\exp(2x)-1-1-\exp(-2x)\right) \right) 7735737409, \frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t} 7741202861, x = \gamma^2 x - \gamma^2 v t + \gamma v t' 7749253510, W = G \frac{m_{\rm Earth} m }{ r_{\rm Earth}} 7826132469, \left(\frac{\partial U}{\partial T}\right)_p = C_V + \pi_T V \alpha 7837519722, v = \sqrt{f} \sqrt{\frac{E}{m}} 7846240076, m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{G} 7875206161, E_2 = KE_2 + PE_2 7882872592, W_{\rm to\ system} = \int_{\infty}^r \vec{F}\cdot d\vec{r} 7906112355, \gamma^2 = \frac{c^2}{c^2 - \gamma^2} 7917051060, \vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2} 7924063906, K_{equilibrium} = \frac{k_{\rm adsorption}}{k_{\rm desorption}} 7928111771, \frac{1}{\theta_A} = \frac{1}{K_{\rm equilibrium} p_A} + 1 7939765107, v^2 = v_0^2 + 2 a d 8046208134, I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2 8049905441, \Delta KE = KE_{\rm final} - KE_{\rm initial} 8059639673, v^2 = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2} 8065128065, I = A A^* + B B^* + A B^* + B A^* 8090924099, v = \sqrt{ \left( f\frac{E}{a^3} \right) \frac{1}{\rho} } 8106885760, \alpha = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c} 8131665171, \frac{1}{\theta_A} = \frac{[S_0]}{[A_{\rm adsorption}]} 8139187332, \vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron} 8145337879, -g t dt + v_{0, y} dt = dy 8198310977, 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0 8228733125, a_y = \frac{d}{dt} v_y 8257621077, \vec{p}_{\rm before} = \vec{p}_{1} 8269198922, 2 a d = v^2 - v_0^2 8283354808, I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 ) 8311458118, \vec{p}_{\rm after} = \vec{p}_{2}+\vec{p}_{electron} 8332931442, \exp(i \pi) = \cos(\pi)+i \sin(\pi) 8357234146, KE = \frac{1}{2} m v^2 8360117126, \gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}} 8361238989, a_{centripetal} = \frac{v^2}{r} 8368984890, \kappa_T = \frac{-1}{V} \left( \frac{ \partial }{\partial P}\left(\frac{nRT}{P}\right) \right)_T 8396997949, I = | A + B |^2 8399484849, \langle x^2 - 2 x \langle x \rangle + \langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2 8405272745, W_{\rm to\ system} = -G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx 8418527415, \sin(i x) = i \sinh(x) 8435841627, P V = n R T 8460820419, v_x = \frac{dx}{dt} 8483686863, \sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x) \right) 8484544728, -a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x) 8485757728, a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx) 8485867742, \frac{2}{W} = a^2 8486706976, v_{0, x} t + x_0 = x 8489593958, d(u v) = u dv + v du 8489593960, d(u v) - v du = u dv 8489593962, u dv = d(u v) - v du 8489593964, \int u dv = u v - \int v du 8494839423, \nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} 8495187962, \theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) } 8497631728, I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi ) 8515803375, z' = z 8532702080, \cosh^2 x = \left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right) 8552710882, KE_{\rm final} = \frac{1}{2} m_1 v_{\rm final}^2 8558338742, E_2 = E_1 8563535636, \cosh^2 x - \sinh^2 x = \left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right) - \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right) 8572657110, 1 = \int |\psi(x)|^2 dx 8572852424, \vec{E} = E( \vec{r},t) 8575746378, \int \frac{1}{2} dx = \frac{1}{2} x 8575748999, \frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x) \right) = -k^2 \left(a \sin(kx) + b \cos(kx) \right) 8576785890, 1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx 8577275751, 0 = a \sin(0) + b\cos(0) 8582885111, \psi(x) = a \sin(kx) + b \cos(kx) 8582954722, x^2 + 2 x h + h^2 = (x + h)^2 8584698994, -g \int dt = \int d v_y 8588429722, \sin( 90^{\circ} - x ) = \cos( x ) 8602221482, \langle \cos(\theta - \phi) \rangle = 0 8602512487, \vec{a} = a_x \hat{x} + a_y \hat{y} 8604483515, dW = G \frac{m_1 m_2}{x^2} dx 8651044341, \cos(i x) = \frac{1}{2} \left( \exp(-x) + \exp(x) \right) 8655294002, a = -\frac{k}{m}x 8661803554, F = G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2} 8688588981, a^3 \rho = m 8699789241, 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right) 8706092970, d = \left(\frac{v + v_0}{2}\right)t 8721295221, t_{\rm Earth\ orbit} = 3.16 10^7 {\rm seconds} 8730201316, \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t' 8747785338, \cos(i x) = \cosh(x) 8750379055, 0 = \frac{d}{dt} v_x 8808860551, -g \int t dt + v_{0, y} \int dt = \int dy 8849289982, \psi(x)^* = a \sin(\frac{n \pi}{W} x) 8889444440, 1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right) \right)^2 dx 8908736791, \rho = \frac{m}{a^3} 8922441655, d = \frac{v_0^2}{g} \sin(2 \theta) 8945218208, \theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ} 8946383937, v_{\rm escape}^2 = 2 G \frac{m}{r} 8949329361, v_0 \sin(\theta) = v_{0, y} 8953094349, W = m a x 8960645192, KE_2 + PE_2 = KE_1 + PE_1 8991236357, \frac{d^2 x}{dt^2} = -\frac{k}{m} x 9031609275, x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t' 9059289981, \psi(x) = a \sin(k x) 9063568209, V_{\rm total} = V_1 + V_2 9070394000, m_2 d_2 \frac{4 \pi^2}{T^2} = G \frac{m_1 m_2}{r^2} 9081138616, W_{\rm by\ system} = \frac{1}{2} m_1 v_{\rm final}^2 9112191201, y_f = 0 9152823411, \frac{1}{T^2} = \frac{1}{d_2 4 \pi^2} G \frac{m_1}{r^2} 9170048197, T^2 = d_2 4 \pi^2 \frac{r^2}{G m_1} 9180861128, 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right) 9191880568, Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta ) 9226945488, F = \frac{m v^2}{r} 9243879541, V = I_2 R_2 9262596735, d = 2 \pi r 9285928292, ax^2 + bx + c = 0 9291999979, \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} 9294858532, \hat{A}^+ = \hat{A} 9337785146, v = \frac{x_2 - x_1}{t} 9341391925, \vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y} 9356924046, \frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t} 9376481176, K = f \frac{E}{a^3} 9385938295, (x+(b/(2 a)))^2 = -(c/a) + (b/(2 a))^2 9393939991, \psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right) 9393939992, \psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right) 9394939493, \nabla^2 E( \vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r},t) 9397152918, v = \frac{v_1 + v_2}{2} 9407192813, G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2} = m g_{\rm Earth} 9409776983, x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t' 9412953728, v_{\rm escape}^2 = 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}} 9413609246, \cosh^2 x - \sinh^2 x = 1 9413699705, W = m a \frac{v_2^2 - v_1^2}{2 a} 9429829482, \frac{d}{dx} y = -\sin(x) + i\cos(x) 9440616166, m_{\rm Earth} = \frac{g_{\rm Earth} r_{\rm Earth}^2}{G} 9482113948, \frac{dy}{y} = i dx 9482438243, (\cos(x))^2 = \cos(2 x) + (\sin(x))^2 9482923849, \exp(i x) = y 9482928242, \cos(2 x) = (\cos(x))^2 - (\sin(x))^2 9482928243, \cos(2 x) + (\sin(x))^2 = (\cos(x))^2 9482943948, \log(y) = i dx 9482984922, \frac{d}{dx} y = (i\sin(x) + \cos(x)) i 9483928192, \cos(2 x) + i\sin(2 x) = (\cos(x))^2 + 2 i \cos(x) \sin(x) - (\sin(x))^2 9485384858, \nabla^2 E( \vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E( \vec{r})\exp(i \omega t) 9485747245, \sqrt{\frac{2}{W}} = a 9485747246, -\sqrt{\frac{2}{W}} = a 9492920340, y = \cos(x)+i \sin(x) 9495857278, \psi(x=W) = 0 9499428242, E( \vec{r},t) = E( \vec{r})\exp(i \omega t) 9510328252, KE_{\rm initial} = 0 9562264720, [S] = \frac{k_{\rm desorption} [A_{\rm adsorption}]}{k_{\rm adsorption} p_A} 9582958293, x = \sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a)) 9582958294, x+(b/(2 a)) = \sqrt{(b/(2 a))^2 - (c/a)} 9585727710, \psi(x=0) = 0 9596004948, x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle 9640720571, v = \frac{e^2}{4 \pi \epsilon_0 \hbar} \sqrt{\frac{m_e}{2 m}} 9658195023, d = v_0 t + \frac{1}{2} a t^2 9703482302, G \frac{m_{\rm Earth} m}{r_{\rm Earth}} = \frac{1}{2} m v_{\rm escape}^2 9707028061, a_x = 0 9718685793, \kappa_T = \frac{1}{P} 9749777192, 0 = KE_1 + PE_1 9756089533, \sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} ) 9759901995, v - v_0 = a t 9781951738, \kappa_T = \frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T 9805063945, \gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2 9838128064, d_2 \frac{4 \pi^2}{T^2} = G \frac{m_1}{r^2} 9847143017, \kappa_T = \frac{-PV}{V} \left( \frac{-1}{P^2}\right) 9848292229, dy = y i dx 9848294829, \frac{d}{dx} y = y i 9854442418, v = \sqrt{\frac{E}{m}} 9858028950, \frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx 9862900242, y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0 9882526611, v_{0, x} t = x - x_0 9889984281, 2 (\sin(x))^2 = 1 - \cos(2 x) 9894826550, 2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \left(\exp(i x)+\exp(-i x) \right) 9897284307, \frac{d}{t} = \frac{v + v_0}{2} 9919999981, \rho = 0 9941599459, dU = \left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV 9958485859, \frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) 9973952056, -g t = v_y - v_{0, y} 9988949211, (\sin(x))^2 = \frac{1 - \cos(2 x)}{2} 9991999979, \vec{ \nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t} 9999998870, \frac{ \vec{p}}{\hbar} = \vec{k} 9999999870, \frac{p}{\hbar} = k 9999999960, \hbar = h/(2 \pi) 9999999961, \frac{E}{\hbar} = \omega
9999999962, p = \hbar k 9999999965, E = \omega \hbar 9999999968, x = \frac{-b-\sqrt{b^2-4ac}}{2 a} 9999999969, x = \frac{-b+\sqrt{b^2-4ac}}{2 a} 9999999975, \langle \psi| \hat{A} |\psi \rangle = \langle a \rangle 9999999981, \vec{ \nabla} \cdot \vec{E} = \rho/\epsilon_0 0203024440, 1 = \int_0^W a \sin\left(\frac{n \pi}{W} x\right) \psi(x)^* dx 0404050504, \lambda = \frac{v}{f} 0439492440, \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) \right|_0^W 0934990943, k = \frac{2 \pi}{v T} 0948572140, \int \cos(a x) dx = \frac{1}{a}\sin(a x) 1010393913, \langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^* 1010393944, x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle 1010923823, k W = n \pi 1020010291, 0 = a \sin(k W) 1020394900, p = h/\lambda 1020394902, E = h f 1020854560, I = (A + B)(A + B)^* 1029039903, p = m v 1029039904, p^2 = m^2 v^2 1038566242, \sinh x = \frac{\exp(x) - \exp(-x)}{2} 1085150613, C_V = \left(\frac{\partial U}{\partial T}\right)_V 1087417579, 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) 1114820451, W_{\rm by\ system} = \Delta KE 1128605625, {\rm sech}^2\ x + \tanh^2(x) = \frac{4}{\left(\exp(x)+\exp(-x)\right)^2} + \frac{\left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2} 1132941271, m_{\rm Earth} = \frac{(9.80665 m/s^2) (6.3781*10^6 m)^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}} 1143343287, G \frac{m_{\rm Earth}}{r_{\rm Earth}} = \frac{1}{2} v_{\rm escape}^2 1158485859, \frac{-\hbar^2}{2m} \nabla^2 = {\cal H} 1166310428, 0 dt = d v_x 1172039918, I_{\rm coherent} = 4 |A|^2 1190768176, \kappa_T = \frac{-nRT}{V} \left( \frac{ \partial }{\partial P}\left(\frac{1}{P}\right) \right)_T 1191796961, \frac{1}{2} g t_f = v_0 \sin(\theta) 1201689765, x'^2 + y'^2 + z'^2 = c^2 t'^2 1202310110, \frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx 1202312210, \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx 1203938249, a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle 1248277773, \cos(2 x) = 1 - 2 (\sin(x))^2 1259826355, d = (v - a t) t + \frac{1}{2} a t^2 1265150401, d = \frac{2 v_0 + a t}{2} t 1292735067, F_{gravitational} = G \frac{m_1 m_2}{r^2} 1293913110, 0 = b 1293923844, \lambda = v T 1306360899, x = v_{0, x} t + x_0 1310571337, \theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster} 1311403394, \alpha = \frac{1}{V} \frac{nR}{P} \left( \frac{\partial T}{\partial T} \right)_P 1314464131, \vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} 1314864131, \vec{ \nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E} 1330874553, v_{\rm escape} = \sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}} 1357848476, A = |A| \exp(i \theta) 1395858355, x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle 1405465835, y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0 1457415749, \frac{1}{R_{\rm total}} = \frac{1}{R_1} + \frac{1}{R_2} 1525861537, I = |A|^2 + |B|^2 + A B^* + B A^* 1528310784, \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} 1541916015, \theta = \frac{\pi}{4} 1556389363, E_{\rm Rydberg} = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2} 1559688463, \left(\frac{T_{\rm geostationary\ orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r_{\rm geostationary\ orbit} 1586866563, \left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right) 1590774089, dW = F dx 1636453295, \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = - \nabla^2 \vec{E} 1638282134, \vec{p}_{\rm before} = \vec{p}_{\rm after} 1639827492, - c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1 1648958381, \nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right) 1650441634, y_0 = 0 1676472948, 0 = v_x - v_{0, x} 1702349646, -g dt = d v_y 1772416655, \frac{E_2 - E_1}{t} = v F - F v 1772973171, -\frac{k}{m} x = -A \omega^2 \cos(\omega t) 1784114349, \sqrt{\frac{k}{m}} = \omega 1809909100, \frac{E_2 - E_1}{t} = 0 1811867899, T^2 = \frac{d_1+d_2}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1} 1815398659, U = Q + W 1819663717, a_x = \frac{d}{dt} v_x 1840080113, KE_2 = 0 1857710291, 0 = a \sin(n \pi) 1858578388, \nabla^2 E( \vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E( \vec{r})\exp(i \omega t) 1858772113, k = \frac{n \pi}{W} 1888494137, -\sqrt{\frac{k}{m}} = \omega 1916173354, -\gamma^2 v^2 + c^2 \gamma^2 = c^2 1928085940, Z^* = |Z| \exp( -i \theta ) 1931103031, \frac{k}{m} = \omega^2 1934748140, \int |\psi(x)|^2 dx = 1 1935543849, \gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2 1963253044, v_{0, x} dt = dx 1967582749, t = \frac{v - v_0}{a} 1974334644, \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t' 1977955751, -g = \frac{d}{dt} v_y 1994296484, v_{\rm satellite}^2 = G \frac{m_{\rm Earth}}{r} 2005061870, v(r) = \sqrt{\frac{2 G m_2}{r}} 2029293929, \nabla^2 E( \vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r})\exp(i \omega t) 2042298788, 0 = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}} + \frac{1}{2} m v_{\rm escape}^2 2051901211, \frac{V}{R_1} = I_1 2061086175, W_{\rm to\ system} = -G m_1 m_2 \left(\frac{-1}{r} - \frac{-1}{\infty}\right) 2076171250, -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0 2086924031, 0 = - \frac{1}{2} g t_f + v_0 \sin(\theta) 2096918413, x = \gamma ( \gamma x - \gamma v t + v t' ) 2103023049, \sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) 2113211456, f = 1/T 2114909846, \theta_A = \frac{[A_{\rm adsorption}]}{[S_0]} 2121790783, \tanh^2(x) = \frac{ \left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2} 2123139121, -\exp(-i x) = -\cos(x)+i \sin(x) 2131616531, T f = 1 2148049269, -\frac{k}{m} A \cos(\omega t) = -A \omega^2 \cos(\omega t) 2168306601, [S_0] = \left(\frac{k_{\rm desorption}}{k_{\rm adsorption}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}] 2186083170, \frac{KE_2 - KE_1}{t} = v F 2217103163, \frac{m_1 d_1}{d_2} = m_2 2236639474, (A + B)(A + B)^* = |A + B|^2 2257410739, \left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alpha 2258485859, {\cal H} \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) 2267521164, PE_2 = 0 2271186630, V = I_{\rm total} R_{\rm total} 2297105551, d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta) 2308660627, G \frac{m_{\rm Earth}}{r_{\rm Earth}^2} = g_{\rm Earth} 2334518266, m a = -k x 2366691988, \int 0 dt = \int d v_x 2378095808, x_f = x_0 + d 2394240499, x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle 2394853829, \exp(-i x) = \cos(-x)+i \sin(-x) 2394935831, ( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0 2394935835, \left(\langle\psi| \hat{A} |\psi \rangle \right)^+ = \left(\langle a \rangle\right)^+ 2395958385, \nabla^2 \psi \left( \vec{r},t \right) = \frac{-p^2}{\hbar} \psi( \vec{r},t) 2404934990, \langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2 2405307372, \sin(2 x) = 2 \sin(x) \cos(x) 2417941373, - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2 2431507955, PE_2 = -F x_2 2461349007, - \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y 2472653783, \alpha = \frac{1}{T} 2484824786, F = m g 2494533900, \langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2 2501591100, \exp(i \pi) + 1 = 0 2503972039, 0 = KE_{\rm escape} + PE_{\rm Earth\ surface} 2519058903, \sin(2 \theta) = 2 \sin(\theta) \cos(\theta) 2542420160, c^2 \gamma^2 - v^2 \gamma^2 = c^2 2575937347, n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} ) 2613006036, \frac{PV}{T} = nR 2617541067, \left(\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r 2648958382, \nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) \right) 2700934933, 2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right) 2715678478, I R_{\rm total} = I R_1 + I R_2 2719691582, |A| = |B| 2741489181, a_y = -g 2750380042, v_{\rm escape} = -\sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}} 2762326680, \cosh^2 x - \sinh^2 x = \frac{1}{4}\left( \exp(2x)+1+1+\exp(-2x) - \left(\exp(2x)-1-1+\exp(-2x)\right) \right) 2768857871, \frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1} 2770069250, \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t} 2809345867, \frac{V}{R_{\rm total}} = I_{\rm total} 2848934890, \langle a \rangle^* = \langle a \rangle 2857430695, a = \frac{v_2 - v_1}{t} 2858549874, - \frac{1}{2} g t^2 + v_{0, y} t = y - y_0 2883079365, r_{\rm Schwarzschild} c^2 = 2 G m 2897612567, v = \alpha c \sqrt{ \frac{m_e}{A m_p} } 2902772962, \tanh(x) = \frac{\frac{1}{2}\left( \exp(x)-\exp(-x) \right)}{\cosh(x)} 2906548078, T^2 = \frac{r}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1} 2907404069, W_{\rm by\ system} = W_{\rm to\ system} 2924222857, v_{\rm initial} = v(r=\infty) 2944838499, \psi(x) = a \sin(\frac{n \pi}{W} x) 2977457786, 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}} = v_{\rm escape}^2 2983053062, x = \gamma (x' + v t') 2998709778, v_{\rm initial} = 0 2999795755, c^2 \gamma^2 = v^2 \gamma^2 + c^2 3004158505, \frac{T^2}{r} F_{gravitational} = \left( \frac{4 \pi^2 m r}{T^2} \right)\frac{T^2}{r} 3046191961, v_{\rm Earth\ orbit} = \frac{C_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}} 3060393541, I_{\rm incoherent} = 2|A|^2 3061811650, n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} ) 3080027960, v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}} 3085575328, I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi)) 3121234211, \frac{k}{2\pi} = \lambda 3121234212, p = \frac{h k}{2\pi} 3121513111, k = \frac{2 \pi}{\lambda} 3131111133, T = 1 / f 3131211131, \omega = 2 \pi f 3132131132, \omega = \frac{2\pi}{T} 3147472131, \frac{\omega}{2 \pi} = f 3169580383, \vec{a} = \frac{d\vec{v}}{dt} 3176662571, F_{\rm centripetal} = F_{\rm gravity} 3182633789, \gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 3214170322, v(r=\infty) = 0 3253234559, x = \frac{v_2^2 - v_1^2}{2 a} 3274926090, t = \frac{x - x_0}{v_{0, x}} 3285732911, (\cos(x))^2 = 1-(\sin(x))^2 3291685884, E = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2} 3331824625, \exp(i \pi) = -1 3350830826, Z Z^* = |Z|^2 3360172339, W = KE_2 - KE_1 3364286646, m_{\rm Earth} = 5.972*10^{24} kg 3366703541, a = \frac{v - v_0}{t} 3411994811, v_{\rm average} = \frac{d}{t} 3417126140, \tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 } 3426941928, x = \gamma ( \gamma (x - v t) + v t' ) 3462972452, v = v_0 + a t 3464107376, \alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p 3470587782, \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) 3472836147, r_{\rm Earth\ orbit} = 1.496\ 10^8 {\rm km} 3485125659, x_f = v_0 t_f \cos(\theta) + x_0 3485475729, \nabla^2 E( \vec{r}) = - \frac{\omega^2}{c^2} E( \vec{r}) 3488423948, k_{\rm adsorption} p_A [S] = k_{\rm desorption} [A_{\rm adsorption}] 3497828859, V = \frac{n R T}{P} 3507029294, k_{\rm adsorption} p_A [S] = r_{\rm desorption} 3512166162, W = F x 3547519267, S = k_{\rm Boltzmann} \ln \Omega 3566149658, W_{\rm to\ system} = \int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx 3585845894, \langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2 3591237106, \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v 3599953931, [S_0] = [S] + [A_{\rm adsorption}] 3605073197, \kappa_T = \frac{-nRT}{V} \left( \frac{-1}{P^2}\right) 3607070319, d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right) 3614055652, v = \frac{2 \pi r}{T_{\rm orbit}} 3649797559, F_{\rm centripetal} = m_2 d_2 \omega^2 3650370389, \frac{T^2}{r} F_{gravitational} = 4 \pi^2 m 3660957533, \cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right) 3676159007, v_{0, x} \int dt = \int dx 3736177473, r_{\rm adsorption} = k_{\rm adsorption} p_A [S] 3781109867, T^2 = \frac{r^3 4 \pi^2}{(d_1+d_2) \frac{m_1}{d_2}G} 3806977900, E_2 - E_1 = 0 3829492824, \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) = \cos(x) 3846041519, PE_{\rm Earth\ surface} = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}} 3868998312, {\rm sech}^2\ x = \frac{4}{\left(\exp(x)+\exp(-x)\right)^2} 3896798826, m_2 d_2 \omega^2 = G \frac{m_1 m_2}{r^2} 3906710072, G \frac{m_{\rm Earth}}{r} = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2} 3920616792, T_{\rm geostationary orbit} = 24\ {\rm hours} 3924948349, a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0 3935058307, v = \sqrt{ \frac{m_e}{m} \frac{e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} } 3942849294, \exp(i x)-\exp(-i x) = 2 i \sin(x) 3943939590, x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle 3947269979, \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} 3948571256, \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}E \psi( \vec{r},t) 3948574224, \psi( \vec{r},t) = \psi_0 \exp\left(i\left( \vec{k}\cdot\vec{r} - \omega t \right) \right) 3948574226, \psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right) 3948574228, \psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right) 3948574230, \psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right) 3948574233, \frac{\partial}{\partial t} \psi( \vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right) 3951205425, \vec{p}_{\rm after} = \vec{p}_{1} 4072200527, \frac{m_{\rm satellite} v_{\rm satellite}^2}{r} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2} 4075539836, A A^* = |A|^2 4087145886, V = I R 4107032818, E_{\rm Rydberg} = E 4128500715, V = I_1 R_1 4139999399, x - \gamma^2 x = - \gamma^2 v t + \gamma v t' 4147472132, E = \frac{h \omega}{2 \pi} 4158986868, a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt} 4166155526, {\rm sech}\ x = \frac{2}{\exp(x)+\exp(-x)} 4180845508, v_{\rm Earth\ orbit} = 29.8 \frac{{\rm km}}{{\rm sec}} 4182362050, Z = |Z| \exp( i \theta ) 4188580242, T^2 = \frac{r^3 4 \pi^2}{\left(m_1+\left(\frac{m_1}{d_2}d_1\right)\right)G} 4192519596, B = |B| \exp(i \phi) 4245712581, v = \frac{2 \pi r}{t} 4267808354, F_{gravitational} = m \frac{v^2}{r} 4268085801, x_0 + d = v_0 t_f \cos(\theta) + x_0 4270680309, \frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t} 4275004561, c^2 = 2 G \frac{m}{r_{\rm Schwarzschild}} 4287102261, x^2 + y^2 + z^2 = c^2 t^2 4298359835, E = \frac{1}{2}m v^2 4298359845, E = \frac{1}{2m}m^2 v^2 4298359851, E = \frac{p^2}{2m} 4301729661, [S_0] = \frac{[A_{\rm adsorption}]}{\left( \frac{k_{\rm adsorption}}{k_{\rm desorption}} \right) p_A} + [A_{\rm adsorption}] 4303372136, E_1 = KE_1 + PE_1 4341171256, i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{p^2}{2 m} \psi( \vec{r},t) 4348571256, \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi( \vec{r},t) 4370074654, t = t_f 4393258808, F_{\rm centripetal} = m r \omega^2 4393670960, W_{\rm to\ system} = \frac{G m_1 m_2}{r} 4394958389, \vec{ \nabla}\cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right) = \frac{i}{\hbar} \vec{ \nabla}\cdot\left( \vec{p} \psi( \vec{r},t) \right) 4428528271, F_{\rm{spring}} = -k x 4447113478, \int dW = G m_1 m_2 \int_{ r_{\rm Earth} }^{\infty} \frac{1}{x^2} dx 4501377629, \tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} 4504256452, B^* = |B| \exp(-i \phi) 4560648264, v = \sqrt{ \frac{K + (4/3) G}{\rho} } 4580545876, d = v t - a t^2 + \frac{1}{2} a t^2 4585828572, \epsilon_0 \mu_0 = \frac{1}{c^2} 4585932229, \cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) 4593428198, v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{3.16\ 10^7 {\rm seconds}} 4598294821, \exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2 4627284246, F_{\rm centripetal} = \frac{m_{\rm satellite} v_{\rm satellite}^2}{r} 4638429483, \exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x)) 4648451961, v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1) 4662369843, x' = \gamma (x - v t) 4669290568, PE_1 = -F x_1 4689334676, \theta_A = \frac{K_{\rm equilibrium}\ p_A}{1+K_{\rm equilibrium}\ p_A} 4742644828, \exp(i x)+\exp(-i x) = 2 \cos(x) 4748157455, a t = v - v_0 4778077984, t_f = \frac{2 v_0 \sin(\theta)}{g} 4784793837, \frac{KE_2 - KE_1}{t} = m v a 4798787814, a t + v_0 = v 4800170179, F = m g_{\rm Earth} 4805233006, i \sin(i x) = \frac{1}{2}\left(\exp(x) - \exp(-x) \right) 4811121942, W = \frac{1}{2} m v_2^2 - \frac{1}{2} m v_1^2 4820320578, F_{gravitational} = F_{centripetal} 4827492911, \cos(2 x)+(\sin(x))^2 = 1 - (\sin(x))^2 4830221561, {\rm sech}^2\ x + \tanh^2(x) = \frac{4+\left(\exp(2x)-1-1+\exp(-2x)\right)}{\left(\exp(x)+\exp(-x)\right)^2} 4838429483, \exp(2 i x) = \cos(2 x)+i \sin(2 x) 4843995999, \frac{1}{2 i}\left(\exp(i x)-\exp(-i x) \right) = \sin(x) 4857472413, 1 = \int \psi(x)\psi(x)^* dx 4857475848, \frac{1}{a^2} = \frac{W}{2} 4858693811, \frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2} = r^3 4866160902, \frac{V}{R_{\rm total}} = \frac{V}{R_1} + \frac{V}{R_2} 4872163189, \tanh(x) = \frac{\sinh(x)}{\cosh(x)} 4872970974, \frac{PE_2 - PE_1}{t} = -F v 4878728014, \sin(i x) = \frac{1}{2i}\left(\exp(-x) - \exp(x) \right) 4923339482, i x = \log(y) 4928007622, KE_1 = \frac{1}{2} m v_1^2 4928239482, \log(y) = i x 4938429482, \exp(-i x) = \cos(x)+i \sin(-x) 4938429483, \exp(i x) = \cos(x)+i \sin(x) 4938429484, \exp(-i x) = \cos(x)-i \sin(x) 4939880586, V_{\rm total} = I R_{\rm total} 4943571230, \vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right) 4947831649, \frac{1}{2} m_1 v_{\rm final}^2 = W_{\rm to\ system} 4948763856, 2 a d + v_0^2 = v^2 4948934890, \langle \psi| \hat{A} |\psi \rangle = \langle a \rangle^* 4949359835, \langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2 4968680693, \tan( x ) = \frac{ \sin( x )}{\cos( x )} 4985825552, \nabla^2 E( \vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E( \vec{r})\exp(i \omega t) 5002539602, dU = C_V dT + \pi_T dV 5085809757, \frac{k_{\rm adsorption}}{k_{\rm desorption}} = \frac{[A_{\rm adsorption}]}{p_A [S]} 5125940051, I = |A|^2 + B B^* + A B^* + B A^* 5128670694, m_1 d_1 = m_2 d_2 5136652623, E = KE + PE 5144263777, v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right) 5148266645, t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t 5177311762, v = \frac{2 \pi r}{T} 5323719091, i \sinh x = \frac{1}{2i} \left( \exp(-x) - \exp(x) \right) 5345738321, F = m a 5349669879, \tanh(x) = \frac{ \exp(x)-\exp(-x)}{\exp(x)+\exp(-x)} 5349866551, \vec{v} = v_x \hat{x} + v_y \hat{y} 5353282496, d = \frac{v_0^2}{g} 5373931751, t = t_f 5379546684, y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0 5404822208, v_{\rm escape} = \sqrt{2 G \frac{m}{r}} 5415824175, x(t) = A \cos(\omega t) 5426308937, v = \frac{d}{t} 5438722682, x = v_0 t \cos(\theta) + x_0 5514556106, E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1) 5530148480, \vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron} 5542528160, \int dW = F \int_0^x dx 5563580265, F_{\rm gravity} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2} 5586102077, r = d_1 + d_2 5596822289, W_{\rm to\ system} = -G m_1 m_2 \left(\left.\frac{-1}{x}\right|^r_{\infty}\right) 5611024898, d = \frac{1}{2 a} (v^2 - v_0^2) 5634116660, \pi_T = \left(\frac{\partial U}{\partial V}\right)_T 5646314683, m = A m_p 5658865948, T^2 = \frac{r^3 4 \pi^2}{(m_1+m_2)G} 5693047217, v_{\rm final} = -\sqrt{\frac{2 G m_2}{r}} 5727578862, \frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x) 5732331610, W = G m_1 m_2 \left( \frac{1}{x} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right) 5733146966, KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right) 5733721198, d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right) 5763749235, -c^2 + c^2 \gamma^2 = v^2 \gamma^2 5779256336, W_{\rm by\ system} = KE_{\rm final} - KE_{\rm initial} 5781981178, x^2 - y^2 = (x+y)(x-y) 5789289057, v = \alpha c \sqrt{ \frac{m_e}{2 m} } 5832984291, (\sin(x))^2 + (\cos(x))^2 = 1 5838268428, \alpha c = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar} 5846639423, v_{\rm final} = \sqrt{\frac{2 G m_2}{r}} 5850144586, W_{\rm by\ system} = KE_{\rm final} 5857434758, \int a dx = a x 5866629429, {\rm sech}^2\ x + \tanh^2(x) = 1 5868688585, \frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = \frac{p^2}{2m} \psi( \vec{r},t) 5900595848, k = \frac{\omega}{v} 5902985919, \vec{F} = G \frac{m_1 m_2}{x^2} \hat{x} 5928285821, x^2 + 2 x (b/(2 a)) + (b/(2 a))^2 = (x + (b/(2 a)))^2 5928292841, x^2 + (b/a)x + (b/(2 a))^2 = -c/a + (b/(2 a))^2 5938459282, x^2 + (b/a)x = -c/a 5945893986, \frac{d^2 x}{dt^2} = -A \omega^2 \cos(\omega t) 5958392859, x^2 + (b/a)x+(c/a) = 0 5959282914, x^2 + x(b/a) + (b/(2 a))^2 = (x+(b/(2 a)))^2 5962145508, \alpha = \frac{nR}{VP} 5978756813, W = G m_{\rm Earth} m \left( 0 - \frac{-1}{ r_{\rm Earth}} \right) 5982958248, x = -\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a)) 5982958249, x+(b/(2 a)) = -\sqrt{(b/(2 a))^2 - (c/a)} 5985371230, \vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) 6026694087, F_{centripetal} = m \frac{v^2}{r} 6031385191, \sinh^2 x = \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right) 6055078815, \left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T \left( \frac{\partial V}{\partial T} \right)_p 6061695358, V_2 = I R_2 6083821265, v_0 \cos(\theta) = v_{0, x} 6091977310, KE_{\rm initial} = \frac{1}{2} m_1 v_{\rm initial}^2 6131764194, W = G m_{\rm Earth} m \left( \frac{1}{x^2} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right) 6134836751, v_{0, x} = v_x 6175547907, v_{\rm average} = \frac{v + v_0}{2} 6204539227, -g t + v_{0, y} = \frac{dy}{dt} 6240206408, I_{\rm incoherent} = |A|^2 + |B|^2 6240546932, \frac{1}{K_{equilibrium}} = \frac{k_{\rm desorption}}{k_{\rm adsorption}} 6268336290, F_{gravitational} = \frac{m}{r}\left(\frac{2\pi r}{T}\right)^2 6306552185, I = (A + B)(A^* + B^*) 6348260313, C_{\rm Earth\ orbit} = 2 \pi r_{\rm Earth\ orbit} 6397683463, V \alpha = \left( \frac{\partial V}{\partial T} \right)_p 6404535647, \cosh x = \frac{\exp(x) + \exp(-x)}{2} 6421241247, d = v t - \frac{1}{2} a t^2 6450985774, n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 ) 6457044853, v - a t = v_0 6457999644, \frac{[S_0]}{[A_{\rm adsorption}]} = \frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1 6504442697, v = \sqrt{ \frac{K}{\rho} } 6529793063, I_{\rm incoherent} = |A|^2 + |A|^2 6555185548, A^* = |A| \exp(-i \theta) 6556875579, \frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2 6572039835, -g t + v_{0, y} = v_y 6715248283, PE = -F x 6742123016, \vec{p}_{electron}\cdot\vec{p}_{electron} = ( \vec{p}_{1}\cdot\vec{p}_{1})+( \vec{p}_{2}\cdot\vec{p}_{2})-2( \vec{p}_{1}\cdot\vec{p}_{2}) 6753224061, I_{\rm total} = I_1 + I_2 6774684564, \theta = \phi 6783009163, r_{\rm adsorption} = r_{\rm desorption} 6785303857, C = 2 \pi r 6800170830, r_{\rm Schwarzschild} = \frac{2 G m}{c^2} 6829281943, F_{\rm centripetal} = G \frac{m_1 m_2}{r^2} 6831637424, \sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} ) 6831694380, a = \frac{d^2 x}{dt^2} 6870322215, KE_{\rm escape} = \frac{1}{2} m v_{\rm escape}^2 6885625907, \exp(i \pi) = -1 + i 0 6892595652, \frac{1}{2} m_1 v_{\rm final}^2 = \frac{G m_1 m_2}{r} 6908055431, x(t) = A \cos\left(\frac{k}{m} t\right) 6925244346, \alpha = \frac{PV}{T} \frac{1}{VP} 6935745841, F = G \frac{m_1 m_2}{x^2} 6946088325, v = \frac{C}{t} 6955192897, r_{\rm desorption} = k_{\rm desorption} [A_{\rm adsorption}] 6998364753, v_{\rm Earth\ orbit} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}} 7002609475, \frac{V}{R_2} = I_2 7010294143, T_{\rm orbit}^2 G m_{\rm Earth} = 4 \pi^2 r^3 7011114072, d = \frac{(v_0 + a t) + v_0}{2} t 7057864873, y' = y 7107090465, B B^* = |B|^2 7112613117, m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}} 7112646057, v_{\rm final}^2 = \frac{2 G m_2}{r} 7175416299, t_{\rm Earth\ orbit} = 1 {\rm year} 7215099603, v^2 = v_0^2 + 2 a t v_0 + a^2 t^2 7217021879, R_{\rm total} = R_1 + R_2 7233558441, d = v_0 t_f \cos(\theta) 7252338326, v_y = \frac{dy}{dt} 7267155233, \frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right) 7267424860, \frac{1}{\theta_A} = \frac{1+(K_{\rm equilibrium}\ p_A)}{K_{\rm equilibrium}\ p_A} 7354529102, y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0 7376526845, \sin(\theta) = \frac{v_{0, y}}{v_0} 7391837535, \cos(\theta) = \frac{v_{0, x}}{v_0} 7455581657, v_{0, x} = \frac{dx}{dt} 7466829492, \vec{ \nabla} \cdot \vec{E} = 0 7513513483, \gamma^2 (c^2 - v^2) = c^2 7517073655, [S_0] = \left(\frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}] 7564894985, \int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) 7572664728, \cos(2 x) + 2 (\sin(x))^2 = 1 7573835180, PE_{\rm Earth\ surface} = -W 7575738420, \left(\sin\left(\frac{n \pi}{W}x\right) \right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} 7575859295, \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) 7575859300, \epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{ \nabla} \times \vec{E} )_k = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) 7575859302, \epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) 7575859304, \epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} 7575859306, \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) 7575859308, \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) 7575859310, \hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) 7575859312, \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) 7621705408, I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi) 7652131521, \frac{dx}{dt} = -A \omega \sin (\omega t) 7672365885, F_{gravitational} = \frac{4 \pi^2 m r}{T^2} 7675171493, V_1 = I R_1 7676652285, KE_2 = \frac{1}{2} m v_2^2 7696214507, n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} ) 7701249282, v_u = \alpha c \sqrt{ \frac{m_e}{m_p} } 7729413831, a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right) 7731226616, {\rm sech}\ x = \frac{1}{\cosh x} 7734996511, PE_2 - PE_1 = -F ( x_2 - x_1 ) 7735731560, \cosh^2 x - \sinh^2 x = \frac{1}{4}\left( \exp(2x)+1+1+\exp(-2x) - \left(\exp(2x)-1-1-\exp(-2x)\right) \right) 7735737409, \frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t} 7741202861, x = \gamma^2 x - \gamma^2 v t + \gamma v t' 7749253510, W = G \frac{m_{\rm Earth} m }{ r_{\rm Earth}} 7826132469, \left(\frac{\partial U}{\partial T}\right)_p = C_V + \pi_T V \alpha 7837519722, v = \sqrt{f} \sqrt{\frac{E}{m}} 7846240076, m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{G} 7875206161, E_2 = KE_2 + PE_2 7882872592, W_{\rm to\ system} = \int_{\infty}^r \vec{F}\cdot d\vec{r} 7906112355, \gamma^2 = \frac{c^2}{c^2 - \gamma^2} 7917051060, \vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2} 7924063906, K_{equilibrium} = \frac{k_{\rm adsorption}}{k_{\rm desorption}} 7928111771, \frac{1}{\theta_A} = \frac{1}{K_{\rm equilibrium} p_A} + 1 7939765107, v^2 = v_0^2 + 2 a d 8046208134, I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2 8049905441, \Delta KE = KE_{\rm final} - KE_{\rm initial} 8059639673, v^2 = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2} 8065128065, I = A A^* + B B^* + A B^* + B A^* 8090924099, v = \sqrt{ \left( f\frac{E}{a^3} \right) \frac{1}{\rho} } 8106885760, \alpha = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c} 8131665171, \frac{1}{\theta_A} = \frac{[S_0]}{[A_{\rm adsorption}]} 8139187332, \vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron} 8145337879, -g t dt + v_{0, y} dt = dy 8198310977, 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0 8228733125, a_y = \frac{d}{dt} v_y 8257621077, \vec{p}_{\rm before} = \vec{p}_{1} 8269198922, 2 a d = v^2 - v_0^2 8283354808, I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 ) 8311458118, \vec{p}_{\rm after} = \vec{p}_{2}+\vec{p}_{electron} 8332931442, \exp(i \pi) = \cos(\pi)+i \sin(\pi) 8357234146, KE = \frac{1}{2} m v^2 8360117126, \gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}} 8361238989, a_{centripetal} = \frac{v^2}{r} 8368984890, \kappa_T = \frac{-1}{V} \left( \frac{ \partial }{\partial P}\left(\frac{nRT}{P}\right) \right)_T 8396997949, I = | A + B |^2 8399484849, \langle x^2 - 2 x \langle x \rangle + \langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2 8405272745, W_{\rm to\ system} = -G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx 8418527415, \sin(i x) = i \sinh(x) 8435841627, P V = n R T 8460820419, v_x = \frac{dx}{dt} 8483686863, \sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x) \right) 8484544728, -a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x) 8485757728, a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx) 8485867742, \frac{2}{W} = a^2 8486706976, v_{0, x} t + x_0 = x 8489593958, d(u v) = u dv + v du 8489593960, d(u v) - v du = u dv 8489593962, u dv = d(u v) - v du 8489593964, \int u dv = u v - \int v du 8494839423, \nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} 8495187962, \theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) } 8497631728, I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi ) 8515803375, z' = z 8532702080, \cosh^2 x = \left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right) 8552710882, KE_{\rm final} = \frac{1}{2} m_1 v_{\rm final}^2 8558338742, E_2 = E_1 8563535636, \cosh^2 x - \sinh^2 x = \left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right) - \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right) 8572657110, 1 = \int |\psi(x)|^2 dx 8572852424, \vec{E} = E( \vec{r},t) 8575746378, \int \frac{1}{2} dx = \frac{1}{2} x 8575748999, \frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x) \right) = -k^2 \left(a \sin(kx) + b \cos(kx) \right) 8576785890, 1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx 8577275751, 0 = a \sin(0) + b\cos(0) 8582885111, \psi(x) = a \sin(kx) + b \cos(kx) 8582954722, x^2 + 2 x h + h^2 = (x + h)^2 8584698994, -g \int dt = \int d v_y 8588429722, \sin( 90^{\circ} - x ) = \cos( x ) 8602221482, \langle \cos(\theta - \phi) \rangle = 0 8602512487, \vec{a} = a_x \hat{x} + a_y \hat{y} 8604483515, dW = G \frac{m_1 m_2}{x^2} dx 8651044341, \cos(i x) = \frac{1}{2} \left( \exp(-x) + \exp(x) \right) 8655294002, a = -\frac{k}{m}x 8661803554, F = G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2} 8688588981, a^3 \rho = m 8699789241, 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right) 8706092970, d = \left(\frac{v + v_0}{2}\right)t 8721295221, t_{\rm Earth\ orbit} = 3.16 10^7 {\rm seconds} 8730201316, \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t' 8747785338, \cos(i x) = \cosh(x) 8750379055, 0 = \frac{d}{dt} v_x 8808860551, -g \int t dt + v_{0, y} \int dt = \int dy 8849289982, \psi(x)^* = a \sin(\frac{n \pi}{W} x) 8889444440, 1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right) \right)^2 dx 8908736791, \rho = \frac{m}{a^3} 8922441655, d = \frac{v_0^2}{g} \sin(2 \theta) 8945218208, \theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ} 8946383937, v_{\rm escape}^2 = 2 G \frac{m}{r} 8949329361, v_0 \sin(\theta) = v_{0, y} 8953094349, W = m a x 8960645192, KE_2 + PE_2 = KE_1 + PE_1 8991236357, \frac{d^2 x}{dt^2} = -\frac{k}{m} x 9031609275, x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t' 9059289981, \psi(x) = a \sin(k x) 9063568209, V_{\rm total} = V_1 + V_2 9070394000, m_2 d_2 \frac{4 \pi^2}{T^2} = G \frac{m_1 m_2}{r^2} 9081138616, W_{\rm by\ system} = \frac{1}{2} m_1 v_{\rm final}^2 9112191201, y_f = 0 9152823411, \frac{1}{T^2} = \frac{1}{d_2 4 \pi^2} G \frac{m_1}{r^2} 9170048197, T^2 = d_2 4 \pi^2 \frac{r^2}{G m_1} 9180861128, 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right) 9191880568, Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta ) 9226945488, F = \frac{m v^2}{r} 9243879541, V = I_2 R_2 9262596735, d = 2 \pi r 9285928292, ax^2 + bx + c = 0 9291999979, \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} 9294858532, \hat{A}^+ = \hat{A} 9337785146, v = \frac{x_2 - x_1}{t} 9341391925, \vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y} 9356924046, \frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t} 9376481176, K = f \frac{E}{a^3} 9385938295, (x+(b/(2 a)))^2 = -(c/a) + (b/(2 a))^2 9393939991, \psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right) 9393939992, \psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right) 9394939493, \nabla^2 E( \vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r},t) 9397152918, v = \frac{v_1 + v_2}{2} 9407192813, G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2} = m g_{\rm Earth} 9409776983, x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t' 9412953728, v_{\rm escape}^2 = 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}} 9413609246, \cosh^2 x - \sinh^2 x = 1 9413699705, W = m a \frac{v_2^2 - v_1^2}{2 a} 9429829482, \frac{d}{dx} y = -\sin(x) + i\cos(x) 9440616166, m_{\rm Earth} = \frac{g_{\rm Earth} r_{\rm Earth}^2}{G} 9482113948, \frac{dy}{y} = i dx 9482438243, (\cos(x))^2 = \cos(2 x) + (\sin(x))^2 9482923849, \exp(i x) = y 9482928242, \cos(2 x) = (\cos(x))^2 - (\sin(x))^2 9482928243, \cos(2 x) + (\sin(x))^2 = (\cos(x))^2 9482943948, \log(y) = i dx 9482984922, \frac{d}{dx} y = (i\sin(x) + \cos(x)) i 9483928192, \cos(2 x) + i\sin(2 x) = (\cos(x))^2 + 2 i \cos(x) \sin(x) - (\sin(x))^2 9485384858, \nabla^2 E( \vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E( \vec{r})\exp(i \omega t) 9485747245, \sqrt{\frac{2}{W}} = a 9485747246, -\sqrt{\frac{2}{W}} = a 9492920340, y = \cos(x)+i \sin(x) 9495857278, \psi(x=W) = 0 9499428242, E( \vec{r},t) = E( \vec{r})\exp(i \omega t) 9510328252, KE_{\rm initial} = 0 9562264720, [S] = \frac{k_{\rm desorption} [A_{\rm adsorption}]}{k_{\rm adsorption} p_A} 9582958293, x = \sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a)) 9582958294, x+(b/(2 a)) = \sqrt{(b/(2 a))^2 - (c/a)} 9585727710, \psi(x=0) = 0 9596004948, x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle 9640720571, v = \frac{e^2}{4 \pi \epsilon_0 \hbar} \sqrt{\frac{m_e}{2 m}} 9658195023, d = v_0 t + \frac{1}{2} a t^2 9703482302, G \frac{m_{\rm Earth} m}{r_{\rm Earth}} = \frac{1}{2} m v_{\rm escape}^2 9707028061, a_x = 0 9718685793, \kappa_T = \frac{1}{P} 9749777192, 0 = KE_1 + PE_1 9756089533, \sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} ) 9759901995, v - v_0 = a t 9781951738, \kappa_T = \frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T 9805063945, \gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2 9838128064, d_2 \frac{4 \pi^2}{T^2} = G \frac{m_1}{r^2} 9847143017, \kappa_T = \frac{-PV}{V} \left( \frac{-1}{P^2}\right) 9848292229, dy = y i dx 9848294829, \frac{d}{dx} y = y i 9854442418, v = \sqrt{\frac{E}{m}} 9858028950, \frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx 9862900242, y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0 9882526611, v_{0, x} t = x - x_0 9889984281, 2 (\sin(x))^2 = 1 - \cos(2 x) 9894826550, 2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \left(\exp(i x)+\exp(-i x) \right) 9897284307, \frac{d}{t} = \frac{v + v_0}{2} 9919999981, \rho = 0 9941599459, dU = \left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV 9958485859, \frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) 9973952056, -g t = v_y - v_{0, y} 9988949211, (\sin(x))^2 = \frac{1 - \cos(2 x)}{2} 9991999979, \vec{ \nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t} 9999998870, \frac{ \vec{p}}{\hbar} = \vec{k} 9999999870, \frac{p}{\hbar} = k 9999999960, \hbar = h/(2 \pi) 9999999961, \frac{E}{\hbar} = \omega

note after step (latex):