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ID latex name description used in derivation symbols sympy dimensional consistency lean 4
2431507955 \(PE_2\) = \(-F x_2\)
PE_2 = -F x_2
TODO

sympy not provided for expression
2461349007 \(- \frac{1}{2} g t^2 + v_{0, y} t + y_0\) = \(y\)
- \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y
TODO

sympy not provided for expression
2472653783 \(\alpha\) = \(\frac{1}{T}\)
\alpha = \frac{1}{T}
TODO

sympy not provided for expression
2484824786 \(F\) = \(m g\)
F = m g
TODO

sympy not provided for expression
2494533900 \(\langle x^2\rangle -\langle x \rangle^2\) = \(\langle x^2 \rangle-\langle x \rangle^2\)
\langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
TODO

sympy not provided for expression
2501591100 \(\exp(i \pi) + 1\) = \(0\)
\exp(i \pi) + 1 = 0
TODO

sympy not provided for expression
2503972039 \(0\) = \(KE_{\rm escape} + PE_{\rm Earth\ surface}\)
0 = KE_{\rm escape} + PE_{\rm Earth\ surface}
TODO

sympy not provided for expression
2519058903 \(\sin(2 \theta)\) = \(2 \sin(\theta) \cos(\theta)\)
\sin(2 \theta) = 2 \sin(\theta) \cos(\theta)
TODO

sympy not provided for expression
2542420160 \(c^2 \gamma^2 - v^2 \gamma^2\) = \(c^2\)
c^2 \gamma^2 - v^2 \gamma^2 = c^2
TODO

sympy not provided for expression
2575937347 \(n_1 \sin( \theta_{\rm Brewster} )\) = \(n_2 \sin( \theta_{\rm refracted} )\)
n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} )
TODO

sympy not provided for expression
2613006036 \(\frac{PV}{T}\) = \(nR\)
\frac{PV}{T} = nR
TODO

sympy not provided for expression
2617541067 \(\left(\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3}\) = \(r\)
\left(\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r
TODO

sympy not provided for expression
2648958382 \(\nabla^2 \psi \left( \vec{r},t \right)\) = \(\frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) \right)\)
\nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) \right)
TODO

sympy not provided for expression
2700934933 \(2 \cos(x)\) = \(\left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)\)
2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
TODO

sympy not provided for expression
2715678478 \(I R_{\rm total}\) = \(I R_1 + I R_2\)
I R_{\rm total} = I R_1 + I R_2
TODO

sympy not provided for expression
2719691582 \(|A|\) = \(|B|\)
|A| = |B|
in a loop TODO

sympy not provided for expression
2741489181 \(a_y\) = \(-g\)
a_y = -g
TODO

sympy not provided for expression
2750380042 \(v_{\rm escape}\) = \(-\sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}\)
v_{\rm escape} = -\sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}
TODO

sympy not provided for expression
2762326680 \(\cosh^2 x - \sinh^2 x\) = \(\frac{1}{4}\left( \exp(2x)+1+1+\exp(-2x) - \left(\exp(2x)-1-1+\exp(-2x)\right) \right)\)
\cosh^2 x - \sinh^2 x = \frac{1}{4}\left( \exp(2x)+1+1+\exp(-2x) - \left(\exp(2x)-1-1+\exp(-2x)\right) \right)
TODO

sympy not provided for expression
2768857871 \(\frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}\) = \(\frac{n_2}{n_1}\)
\frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1}
TODO

sympy not provided for expression
2770069250 \(\frac{E_2 - E_1}{t}\) = \(\frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}\)
\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}
TODO

sympy not provided for expression
2809345867 \(\frac{V}{R_{\rm total}}\) = \(I_{\rm total}\)
\frac{V}{R_{\rm total}} = I_{\rm total}
TODO

sympy not provided for expression
2848934890 \(\langle a \rangle^*\) = \(\langle a \rangle\)
\langle a \rangle^* = \langle a \rangle
TODO

sympy not provided for expression
2857430695 \(a\) = \(\frac{v_2 - v_1}{t}\)
a = \frac{v_2 - v_1}{t}
acceleration TODO

sympy not provided for expression
2858549874 \(- \frac{1}{2} g t^2 + v_{0, y} t\) = \(y - y_0\)
- \frac{1}{2} g t^2 + v_{0, y} t = y - y_0
TODO

sympy not provided for expression
2883079365 \(r_{\rm Schwarzschild} c^2\) = \(2 G m\)
r_{\rm Schwarzschild} c^2 = 2 G m
TODO

sympy not provided for expression
2897612567 \(v\) = \(\alpha c \sqrt{ \frac{m_e}{A m_p} }\)
v = \alpha c \sqrt{ \frac{m_e}{A m_p} }
TODO

sympy not provided for expression
2902772962 \(\tanh(x)\) = \(\frac{\frac{1}{2}\left( \exp(x)-\exp(-x) \right)}{\cosh(x)}\)
\tanh(x) = \frac{\frac{1}{2}\left( \exp(x)-\exp(-x) \right)}{\cosh(x)}
TODO

sympy not provided for expression
2906548078 \(T^2\) = \(\frac{r}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1}\)
T^2 = \frac{r}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1}
TODO

sympy not provided for expression
2907404069 \(W_{\rm by\ system}\) = \(W_{\rm to\ system}\)
W_{\rm by\ system} = W_{\rm to\ system}
TODO

sympy not provided for expression
2924222857 \(v_{\rm initial}\) = \(v(r=\infty)\)
v_{\rm initial} = v(r=\infty)
TODO

sympy not provided for expression
2944838499 \(\psi(x)\) = \(a \sin(\frac{n \pi}{W} x)\)
\psi(x) = a \sin(\frac{n \pi}{W} x)
TODO

sympy not provided for expression
2977457786 \(2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}\) = \(v_{\rm escape}^2\)
2 G \frac{m_{\rm Earth}}{r_{\rm Earth}} = v_{\rm escape}^2
TODO

sympy not provided for expression
2983053062 \(x\) = \(\gamma (x' + v t')\)
x = \gamma (x' + v t')
TODO

sympy not provided for expression
2998709778 \(v_{\rm initial}\) = \(0\)
v_{\rm initial} = 0
TODO

sympy not provided for expression
2999795755 \(c^2 \gamma^2\) = \(v^2 \gamma^2 + c^2\)
c^2 \gamma^2 = v^2 \gamma^2 + c^2
TODO

sympy not provided for expression
3004158505 \(\frac{T^2}{r} F_{gravitational}\) = \(\left( \frac{4 \pi^2 m r}{T^2} \right)\frac{T^2}{r}\)
\frac{T^2}{r} F_{gravitational} = \left( \frac{4 \pi^2 m r}{T^2} \right)\frac{T^2}{r}
TODO

sympy not provided for expression
3046191961 \(v_{\rm Earth\ orbit}\) = \(\frac{C_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}}\)
v_{\rm Earth\ orbit} = \frac{C_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}}
TODO

sympy not provided for expression
3060393541 \(I_{\rm incoherent}\) = \(2|A|^2\)
I_{\rm incoherent} = 2|A|^2
TODO

sympy not provided for expression
3061811650 \(n_1 \sin( \theta_{\rm Brewster} )\) = \(n_2 \cos( \theta_{\rm Brewster} )\)
n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} )
TODO

sympy not provided for expression
3080027960 \(v_{\rm Earth\ orbit}\) = \(\frac{2 \pi r_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}}\)
v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}}
TODO

sympy not provided for expression
3085575328 \(I\) = \(|A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))\)
I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))
TODO

sympy not provided for expression
3121234211 \(\frac{k}{2\pi}\) = \(\lambda\)
\frac{k}{2\pi} = \lambda
TODO

sympy not provided for expression
3121234212 \(p\) = \(\frac{h k}{2\pi}\)
p = \frac{h k}{2\pi}
TODO

sympy not provided for expression
3121513111 \(k\) = \(\frac{2 \pi}{\lambda}\)
k = \frac{2 \pi}{\lambda}
TODO

sympy not provided for expression
3131111133 \(T\) = \(1 / f\)
T = 1 / f
TODO

sympy not provided for expression
3131211131 \(\omega\) = \(2 \pi f\)
\omega = 2 \pi f
TODO

sympy not provided for expression
3132131132 \(\omega\) = \(\frac{2\pi}{T}\)
\omega = \frac{2\pi}{T}
TODO

sympy not provided for expression
3147472131 \(\frac{\omega}{2 \pi}\) = \(f\)
\frac{\omega}{2 \pi} = f
TODO

sympy not provided for expression
3169580383 \(\vec{a}\) = \(\frac{d\vec{v}}{dt}\)
\vec{a} = \frac{d\vec{v}}{dt}
acceleration is the change in speed over a duration TODO

sympy not provided for expression
3176662571 \(F_{\rm centripetal}\) = \(F_{\rm gravity}\)
F_{\rm centripetal} = F_{\rm gravity}
applicable to any satellite orbit TODO

sympy not provided for expression
3182633789 \(\gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4}\) = \(1\)
\gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1
TODO

sympy not provided for expression
3214170322 \(v(r=\infty)\) = \(0\)
v(r=\infty) = 0
TODO

sympy not provided for expression
3253234559 \(x\) = \(\frac{v_2^2 - v_1^2}{2 a}\)
x = \frac{v_2^2 - v_1^2}{2 a}
TODO

sympy not provided for expression
3274926090 \(t\) = \(\frac{x - x_0}{v_{0, x}}\)
t = \frac{x - x_0}{v_{0, x}}
TODO

sympy not provided for expression
3285732911 \((\cos(x))^2\) = \(1-(\sin(x))^2\)
(\cos(x))^2 = 1-(\sin(x))^2
TODO

sympy not provided for expression
3291685884 \(E\) = \(\frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}\)
E = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}
TODO

sympy not provided for expression
3331824625 \(\exp(i \pi)\) = \(-1\)
\exp(i \pi) = -1
TODO

sympy not provided for expression
3350830826 \(Z Z^*\) = \(|Z|^2\)
Z Z^* = |Z|^2
TODO

sympy not provided for expression
3360172339 \(W\) = \(KE_2 - KE_1\)
W = KE_2 - KE_1
TODO

sympy not provided for expression
3364286646 \(m_{\rm Earth}\) = \(5.972*10^{24} kg\)
m_{\rm Earth} = 5.972*10^{24} kg
TODO

sympy not provided for expression
3366703541 \(a\) = \(\frac{v - v_0}{t}\)
a = \frac{v - v_0}{t}
acceleration is the average change in speed over a duration TODO

sympy not provided for expression
3411994811 \(v_{\rm average}\) = \(\frac{d}{t}\)
v_{\rm average} = \frac{d}{t}
TODO

sympy not provided for expression
3417126140 \(\tan( \theta_{\rm Brewster} )\) = \(\frac{ n_2 }{ n_1 }\)
\tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 }
TODO

sympy not provided for expression
3426941928 \(x\) = \(\gamma ( \gamma (x - v t) + v t' )\)
x = \gamma ( \gamma (x - v t) + v t' )
TODO

sympy not provided for expression
3462972452 \(v\) = \(v_0 + a t\)
v = v_0 + a t
TODO

sympy not provided for expression
3464107376 \(\alpha\) = \(\frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p\)
\alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p
definition of expansion coefficient TODO

sympy not provided for expression
3470587782 \(\sin(x) \cos(x)\) = \(\frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)
\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)
TODO

sympy not provided for expression
3472836147 \(r_{\rm Earth\ orbit}\) = \(1.496\ 10^8 {\rm km}\)
r_{\rm Earth\ orbit} = 1.496\ 10^8 {\rm km}
TODO

sympy not provided for expression
3485125659 \(x_f\) = \(v_0 t_f \cos(\theta) + x_0\)
x_f = v_0 t_f \cos(\theta) + x_0
TODO

sympy not provided for expression
3485475729 \(\nabla^2 E( \vec{r})\) = \(- \frac{\omega^2}{c^2} E( \vec{r})\)
\nabla^2 E( \vec{r}) = - \frac{\omega^2}{c^2} E( \vec{r})
https://physicsderivationgraph.blogspot.com/2020/09/representing-laplace-operator-nabla-in.html TODO

sympy not provided for expression
3488423948 \(k_{\rm adsorption} p_A [S]\) = \(k_{\rm desorption} [A_{\rm adsorption}]\)
k_{\rm adsorption} p_A [S] = k_{\rm desorption} [A_{\rm adsorption}]
TODO

sympy not provided for expression
3497828859 \(V\) = \(\frac{n R T}{P}\)
V = \frac{n R T}{P}
TODO

sympy not provided for expression
3507029294 \(k_{\rm adsorption} p_A [S]\) = \(r_{\rm desorption}\)
k_{\rm adsorption} p_A [S] = r_{\rm desorption}
TODO

sympy not provided for expression
3512166162 \(W\) = \(F x\)
W = F x
TODO

sympy not provided for expression
3547519267 \(S\) = \(k_{\rm Boltzmann} \ln \Omega\)
S = k_{\rm Boltzmann} \ln \Omega
assumes equally probable microstates TODO

sympy not provided for expression
3566149658 \(W_{\rm to\ system}\) = \(\int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx\)
W_{\rm to\ system} = \int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx
TODO

sympy not provided for expression
3585845894 \(\langle \left(x-\langle x \rangle\right)^2 \rangle\) = \(\langle x^2 \rangle-\langle x \rangle^2\)
\langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
TODO

sympy not provided for expression
3591237106 \(\frac{E_2 - E_1}{t}\) = \(\frac{(KE_2 - KE_1)}{t} - F v\)
\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v
TODO

sympy not provided for expression
3599953931 \([S_0]\) = \([S] + [A_{\rm adsorption}]\)
[S_0] = [S] + [A_{\rm adsorption}]
TODO

sympy not provided for expression
3605073197 \(\kappa_T\) = \(\frac{-nRT}{V} \left( \frac{-1}{P^2}\right)\)
\kappa_T = \frac{-nRT}{V} \left( \frac{-1}{P^2}\right)
TODO

sympy not provided for expression
3607070319 \(d\) = \(\frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)\)
d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)
TODO

sympy not provided for expression
3614055652 \(v\) = \(\frac{2 \pi r}{T_{\rm orbit}}\)
v = \frac{2 \pi r}{T_{\rm orbit}}
TODO

sympy not provided for expression
3649797559 \(F_{\rm centripetal}\) = \(m_2 d_2 \omega^2\)
F_{\rm centripetal} = m_2 d_2 \omega^2
TODO

sympy not provided for expression
3650370389 \(\frac{T^2}{r} F_{gravitational}\) = \(4 \pi^2 m\)
\frac{T^2}{r} F_{gravitational} = 4 \pi^2 m
TODO

sympy not provided for expression
3660957533 \(\cos(x)\) = \(\frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)\)
\cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
TODO

sympy not provided for expression
3676159007 \(v_{0, x} \int dt\) = \(\int dx\)
v_{0, x} \int dt = \int dx
TODO

sympy not provided for expression
3736177473 \(r_{\rm adsorption}\) = \(k_{\rm adsorption} p_A [S]\)
r_{\rm adsorption} = k_{\rm adsorption} p_A [S]
TODO

sympy not provided for expression
3781109867 \(T^2\) = \(\frac{r^3 4 \pi^2}{(d_1+d_2) \frac{m_1}{d_2}G}\)
T^2 = \frac{r^3 4 \pi^2}{(d_1+d_2) \frac{m_1}{d_2}G}
TODO

sympy not provided for expression
3806977900 \(E_2 - E_1\) = \(0\)
E_2 - E_1 = 0
TODO

sympy not provided for expression
3829492824 \(\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\) = \(\cos(x)\)
\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) = \cos(x)
TODO

sympy not provided for expression
3846041519 \(PE_{\rm Earth\ surface}\) = \(-G \frac{m_{\rm Earth} m}{r_{\rm Earth}}\)
PE_{\rm Earth\ surface} = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}}
TODO

sympy not provided for expression
3868998312 \({\rm sech}^2\ x\) = \(\frac{4}{\left(\exp(x)+\exp(-x)\right)^2}\)
{\rm sech}^2\ x = \frac{4}{\left(\exp(x)+\exp(-x)\right)^2}
TODO

sympy not provided for expression
3896798826 \(m_2 d_2 \omega^2\) = \(G \frac{m_1 m_2}{r^2}\)
m_2 d_2 \omega^2 = G \frac{m_1 m_2}{r^2}
TODO

sympy not provided for expression
3906710072 \(G \frac{m_{\rm Earth}}{r}\) = \(\frac{4 \pi^2 r^2}{T_{\rm orbit}^2}\)
G \frac{m_{\rm Earth}}{r} = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2}
TODO

sympy not provided for expression
3920616792 \(T_{\rm geostationary orbit}\) = \(24\ {\rm hours}\)
T_{\rm geostationary orbit} = 24\ {\rm hours}
this applies for geostationary orbits TODO

sympy not provided for expression
3924948349 \(a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle\) = \(0\)
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
TODO

sympy not provided for expression
3935058307 \(v\) = \(\sqrt{ \frac{m_e}{m} \frac{e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} }\)
v = \sqrt{ \frac{m_e}{m} \frac{e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} }
TODO

sympy not provided for expression
3942849294 \(\exp(i x)-\exp(-i x)\) = \(2 i \sin(x)\)
\exp(i x)-\exp(-i x) = 2 i \sin(x)
TODO

sympy not provided for expression
3943939590 \(x\) = \(a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle\)
x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle
TODO

sympy not provided for expression
3947269979 \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E}\) = \(-\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\)
\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
TODO

sympy not provided for expression
3948571256 \(\frac{\partial}{\partial t} \psi( \vec{r},t)\) = \(\frac{-i}{\hbar}E \psi( \vec{r},t)\)
\frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}E \psi( \vec{r},t)
TODO

sympy not provided for expression
3948574224 \(\psi( \vec{r},t)\) = \(\psi_0 \exp\left(i\left( \vec{k}\cdot\vec{r} - \omega t \right) \right)\)
\psi( \vec{r},t) = \psi_0 \exp\left(i\left( \vec{k}\cdot\vec{r} - \omega t \right) \right)
TODO

sympy not provided for expression
3948574226 \(\psi( \vec{r},t)\) = \(\psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)\)
\psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)
TODO

sympy not provided for expression
3948574228 \(\psi( \vec{r},t)\) = \(\psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)\)
\psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
TODO

sympy not provided for expression
3948574230 \(\psi( \vec{r},t)\) = \(\psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)
\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)
TODO

sympy not provided for expression
3948574233 \(\frac{\partial}{\partial t} \psi( \vec{r},t)\) = \(\psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)\)
\frac{\partial}{\partial t} \psi( \vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
TODO

sympy not provided for expression
3951205425 \(\vec{p}_{\rm after}\) = \(\vec{p}_{1}\)
\vec{p}_{\rm after} = \vec{p}_{1}
TODO

sympy not provided for expression
4072200527 \(\frac{m_{\rm satellite} v_{\rm satellite}^2}{r}\) = \(G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}\)
\frac{m_{\rm satellite} v_{\rm satellite}^2}{r} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}
TODO

sympy not provided for expression
4075539836 \(A A^*\) = \(|A|^2\)
A A^* = |A|^2
TODO

sympy not provided for expression
4087145886 \(V\) = \(I R\)
V = I R
Ohm's law https://en.wikipedia.org/wiki/Ohm%27s_law TODO

sympy not provided for expression
4107032818 \(E_{\rm Rydberg}\) = \(E\)
E_{\rm Rydberg} = E
TODO

sympy not provided for expression
4128500715 \(V\) = \(I_1 R_1\)
V = I_1 R_1
TODO

sympy not provided for expression
4139999399 \(x - \gamma^2 x\) = \(- \gamma^2 v t + \gamma v t'\)
x - \gamma^2 x = - \gamma^2 v t + \gamma v t'
TODO

sympy not provided for expression
4147472132 \(E\) = \(\frac{h \omega}{2 \pi}\)
E = \frac{h \omega}{2 \pi}
TODO

sympy not provided for expression
4158986868 \(a_x \hat{x} + a_y \hat{y}\) = \(\frac{d\vec{v}}{dt}\)
a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}
TODO

sympy not provided for expression
4166155526 \({\rm sech}\ x\) = \(\frac{2}{\exp(x)+\exp(-x)}\)
{\rm sech}\ x = \frac{2}{\exp(x)+\exp(-x)}
TODO

sympy not provided for expression
4180845508 \(v_{\rm Earth\ orbit}\) = \(29.8 \frac{{\rm km}}{{\rm sec}}\)
v_{\rm Earth\ orbit} = 29.8 \frac{{\rm km}}{{\rm sec}}
TODO

sympy not provided for expression
4182362050 \(Z\) = \(|Z| \exp( i \theta )\)
Z = |Z| \exp( i \theta )
Z \in \mathbb{C} TODO

sympy not provided for expression
4188580242 \(T^2\) = \(\frac{r^3 4 \pi^2}{\left(m_1+\left(\frac{m_1}{d_2}d_1\right)\right)G}\)
T^2 = \frac{r^3 4 \pi^2}{\left(m_1+\left(\frac{m_1}{d_2}d_1\right)\right)G}
TODO

sympy not provided for expression
4192519596 \(B\) = \(|B| \exp(i \phi)\)
B = |B| \exp(i \phi)
TODO

sympy not provided for expression
4245712581 \(v\) = \(\frac{2 \pi r}{t}\)
v = \frac{2 \pi r}{t}
TODO

sympy not provided for expression
4267808354 \(F_{gravitational}\) = \(m \frac{v^2}{r}\)
F_{gravitational} = m \frac{v^2}{r}
TODO

sympy not provided for expression
4268085801 \(x_0 + d\) = \(v_0 t_f \cos(\theta) + x_0\)
x_0 + d = v_0 t_f \cos(\theta) + x_0
TODO

sympy not provided for expression
4270680309 \(\frac{KE_2 - KE_1}{t}\) = \(\frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}\)
\frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}
TODO

sympy not provided for expression
4275004561 \(c^2\) = \(2 G \frac{m}{r_{\rm Schwarzschild}}\)
c^2 = 2 G \frac{m}{r_{\rm Schwarzschild}}
TODO

sympy not provided for expression
4287102261 \(x^2 + y^2 + z^2\) = \(c^2 t^2\)
x^2 + y^2 + z^2 = c^2 t^2
describes a spherical wavefront TODO

sympy not provided for expression
4298359835 \(E\) = \(\frac{1}{2}m v^2\)
E = \frac{1}{2}m v^2
TODO

sympy not provided for expression
4298359845 \(E\) = \(\frac{1}{2m}m^2 v^2\)
E = \frac{1}{2m}m^2 v^2
TODO

sympy not provided for expression
4298359851 \(E\) = \(\frac{p^2}{2m}\)
E = \frac{p^2}{2m}
TODO

sympy not provided for expression
4301729661 \([S_0]\) = \(\frac{[A_{\rm adsorption}]}{\left( \frac{k_{\rm adsorption}}{k_{\rm desorption}} \right) p_A} + [A_{\rm adsorption}]\)
[S_0] = \frac{[A_{\rm adsorption}]}{\left( \frac{k_{\rm adsorption}}{k_{\rm desorption}} \right) p_A} + [A_{\rm adsorption}]
TODO

sympy not provided for expression
4303372136 \(E_1\) = \(KE_1 + PE_1\)
E_1 = KE_1 + PE_1
TODO

sympy not provided for expression
4341171256 \(i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)\) = \(\frac{p^2}{2 m} \psi( \vec{r},t)\)
i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{p^2}{2 m} \psi( \vec{r},t)
TODO

sympy not provided for expression
4348571256 \(\frac{\partial}{\partial t} \psi( \vec{r},t)\) = \(\frac{-i}{\hbar}\frac{p^2}{2 m} \psi( \vec{r},t)\)
\frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi( \vec{r},t)
TODO

sympy not provided for expression
4370074654 \(t\) = \(t_f\)
t = t_f
TODO

sympy not provided for expression
4393258808 \(F_{\rm centripetal}\) = \(m r \omega^2\)
F_{\rm centripetal} = m r \omega^2
TODO

sympy not provided for expression
4393670960 \(W_{\rm to\ system}\) = \(\frac{G m_1 m_2}{r}\)
W_{\rm to\ system} = \frac{G m_1 m_2}{r}
TODO

sympy not provided for expression
4394958389 \(\vec{ \nabla}\cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right)\) = \(\frac{i}{\hbar} \vec{ \nabla}\cdot\left( \vec{p} \psi( \vec{r},t) \right)\)
\vec{ \nabla}\cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right) = \frac{i}{\hbar} \vec{ \nabla}\cdot\left( \vec{p} \psi( \vec{r},t) \right)
TODO

sympy not provided for expression
4428528271 \(F_{\rm{spring}}\) = \(-k x\)
F_{\rm{spring}} = -k x
Hooke's law https://en.wikipedia.org/wiki/Hooke%27s_law TODO

sympy not provided for expression
4447113478 \(\int dW\) = \(G m_1 m_2 \int_{ r_{\rm Earth} }^{\infty} \frac{1}{x^2} dx\)
\int dW = G m_1 m_2 \int_{ r_{\rm Earth} }^{\infty} \frac{1}{x^2} dx
TODO

sympy not provided for expression
4501377629 \(\tan( \theta_{\rm Brewster} )\) = \(\frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}\)
\tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}
TODO

sympy not provided for expression
4504256452 \(B^*\) = \(|B| \exp(-i \phi)\)
B^* = |B| \exp(-i \phi)
TODO

sympy not provided for expression
4560648264 \(v\) = \(\sqrt{ \frac{K + (4/3) G}{\rho} }\)
v = \sqrt{ \frac{K + (4/3) G}{\rho} }
TODO

sympy not provided for expression
4580545876 \(d\) = \(v t - a t^2 + \frac{1}{2} a t^2\)
d = v t - a t^2 + \frac{1}{2} a t^2
TODO

sympy not provided for expression
4585828572 \(\epsilon_0 \mu_0\) = \(\frac{1}{c^2}\)
\epsilon_0 \mu_0 = \frac{1}{c^2}
TODO

sympy not provided for expression
4585932229 \(\cos(x)\) = \(\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)
\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)
TODO

sympy not provided for expression
4593428198 \(v_{\rm Earth\ orbit}\) = \(\frac{2 \pi r_{\rm Earth\ orbit}}{3.16\ 10^7 {\rm seconds}}\)
v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{3.16\ 10^7 {\rm seconds}}
TODO

sympy not provided for expression
4598294821 \(\exp(2 i x)\) = \((\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2\)
\exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
TODO

sympy not provided for expression
4627284246 \(F_{\rm centripetal}\) = \(\frac{m_{\rm satellite} v_{\rm satellite}^2}{r}\)
F_{\rm centripetal} = \frac{m_{\rm satellite} v_{\rm satellite}^2}{r}
TODO

sympy not provided for expression
4638429483 \(\exp(2 i x)\) = \((\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))\)
\exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
TODO

sympy not provided for expression
4648451961 \(v_2^2 - v_1^2\) = \((v_2 + v_1)(v_2 - v_1)\)
v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)
TODO

sympy not provided for expression
4662369843 \(x'\) = \(\gamma (x - v t)\)
x' = \gamma (x - v t)
TODO

sympy not provided for expression
4669290568 \(PE_1\) = \(-F x_1\)
PE_1 = -F x_1
TODO

sympy not provided for expression
4689334676 \(\theta_A\) = \(\frac{K_{\rm equilibrium}\ p_A}{1+K_{\rm equilibrium}\ p_A}\)
\theta_A = \frac{K_{\rm equilibrium}\ p_A}{1+K_{\rm equilibrium}\ p_A}
TODO

sympy not provided for expression
4742644828 \(\exp(i x)+\exp(-i x)\) = \(2 \cos(x)\)
\exp(i x)+\exp(-i x) = 2 \cos(x)
TODO

sympy not provided for expression
4748157455 \(a t\) = \(v - v_0\)
a t = v - v_0
TODO

sympy not provided for expression
4778077984 \(t_f\) = \(\frac{2 v_0 \sin(\theta)}{g}\)
t_f = \frac{2 v_0 \sin(\theta)}{g}
TODO

sympy not provided for expression
4784793837 \(\frac{KE_2 - KE_1}{t}\) = \(m v a\)
\frac{KE_2 - KE_1}{t} = m v a
TODO

sympy not provided for expression
4798787814 \(a t + v_0\) = \(v\)
a t + v_0 = v
TODO

sympy not provided for expression
4800170179 \(F\) = \(m g_{\rm Earth}\)
F = m g_{\rm Earth}
TODO

sympy not provided for expression
4805233006 \(i \sin(i x)\) = \(\frac{1}{2}\left(\exp(x) - \exp(-x) \right)\)
i \sin(i x) = \frac{1}{2}\left(\exp(x) - \exp(-x) \right)
TODO

sympy not provided for expression
4811121942 \(W\) = \(\frac{1}{2} m v_2^2 - \frac{1}{2} m v_1^2\)
W = \frac{1}{2} m v_2^2 - \frac{1}{2} m v_1^2
TODO

sympy not provided for expression
4820320578 \(F_{gravitational}\) = \(F_{centripetal}\)
F_{gravitational} = F_{centripetal}
TODO

sympy not provided for expression
4827492911 \(\cos(2 x)+(\sin(x))^2\) = \(1 - (\sin(x))^2\)
\cos(2 x)+(\sin(x))^2 = 1 - (\sin(x))^2
TODO

sympy not provided for expression
4830221561 \({\rm sech}^2\ x + \tanh^2(x)\) = \(\frac{4+\left(\exp(2x)-1-1+\exp(-2x)\right)}{\left(\exp(x)+\exp(-x)\right)^2}\)
{\rm sech}^2\ x + \tanh^2(x) = \frac{4+\left(\exp(2x)-1-1+\exp(-2x)\right)}{\left(\exp(x)+\exp(-x)\right)^2}
TODO

sympy not provided for expression
4838429483 \(\exp(2 i x)\) = \(\cos(2 x)+i \sin(2 x)\)
\exp(2 i x) = \cos(2 x)+i \sin(2 x)
TODO

sympy not provided for expression
4843995999 \(\frac{1}{2 i}\left(\exp(i x)-\exp(-i x) \right)\) = \(\sin(x)\)
\frac{1}{2 i}\left(\exp(i x)-\exp(-i x) \right) = \sin(x)
TODO

sympy not provided for expression
4857472413 \(1\) = \(\int \psi(x)\psi(x)^* dx\)
1 = \int \psi(x)\psi(x)^* dx
TODO

sympy not provided for expression
4857475848 \(\frac{1}{a^2}\) = \(\frac{W}{2}\)
\frac{1}{a^2} = \frac{W}{2}
TODO

sympy not provided for expression
4858693811 \(\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2}\) = \(r^3\)
\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2} = r^3
TODO

sympy not provided for expression
4866160902 \(\frac{V}{R_{\rm total}}\) = \(\frac{V}{R_1} + \frac{V}{R_2}\)
\frac{V}{R_{\rm total}} = \frac{V}{R_1} + \frac{V}{R_2}
TODO

sympy not provided for expression
4872163189 \(\tanh(x)\) = \(\frac{\sinh(x)}{\cosh(x)}\)
\tanh(x) = \frac{\sinh(x)}{\cosh(x)}
TODO

sympy not provided for expression
4872970974 \(\frac{PE_2 - PE_1}{t}\) = \(-F v\)
\frac{PE_2 - PE_1}{t} = -F v
TODO

sympy not provided for expression
4878728014 \(\sin(i x)\) = \(\frac{1}{2i}\left(\exp(-x) - \exp(x) \right)\)
\sin(i x) = \frac{1}{2i}\left(\exp(-x) - \exp(x) \right)
TODO

sympy not provided for expression
4923339482 \(i x\) = \(\log(y)\)
i x = \log(y)
TODO

sympy not provided for expression
4928007622 \(KE_1\) = \(\frac{1}{2} m v_1^2\)
KE_1 = \frac{1}{2} m v_1^2
TODO

sympy not provided for expression
4928239482 \(\log(y)\) = \(i x\)
\log(y) = i x
TODO

sympy not provided for expression
4938429482 \(\exp(-i x)\) = \(\cos(x)+i \sin(-x)\)
\exp(-i x) = \cos(x)+i \sin(-x)
TODO

sympy not provided for expression
4938429483 \(\exp(i x)\) = \(\cos(x)+i \sin(x)\)
\exp(i x) = \cos(x)+i \sin(x)
TODO

sympy not provided for expression
4938429484 \(\exp(-i x)\) = \(\cos(x)-i \sin(x)\)
\exp(-i x) = \cos(x)-i \sin(x)
TODO

sympy not provided for expression
4939880586 \(V_{\rm total}\) = \(I R_{\rm total}\)
V_{\rm total} = I R_{\rm total}
TODO

sympy not provided for expression
4943571230 \(\vec{ \nabla} \psi( \vec{r},t)\) = \(\frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)
\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)
TODO

sympy not provided for expression
4947831649 \(\frac{1}{2} m_1 v_{\rm final}^2\) = \(W_{\rm to\ system}\)
\frac{1}{2} m_1 v_{\rm final}^2 = W_{\rm to\ system}
TODO

sympy not provided for expression
4948763856 \(2 a d + v_0^2\) = \(v^2\)
2 a d + v_0^2 = v^2
TODO

sympy not provided for expression
4948934890 \(\langle \psi| \hat{A} |\psi \rangle\) = \(\langle a \rangle^*\)
\langle \psi| \hat{A} |\psi \rangle = \langle a \rangle^*
TODO

sympy not provided for expression
4949359835 \(\langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2\) = \(\langle x^2 \rangle-\langle x \rangle^2\)
\langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
TODO

sympy not provided for expression
4968680693 \(\tan( x )\) = \(\frac{ \sin( x )}{\cos( x )}\)
\tan( x ) = \frac{ \sin( x )}{\cos( x )}
TODO

sympy not provided for expression
4985825552 \(\nabla^2 E( \vec{r})\exp(i \omega t)\) = \(i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E( \vec{r})\exp(i \omega t)\)
\nabla^2 E( \vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E( \vec{r})\exp(i \omega t)
https://physicsderivationgraph.blogspot.com/2020/09/representing-laplace-operator-nabla-in.html TODO

sympy not provided for expression
5002539602 \(dU\) = \(C_V dT + \pi_T dV\)
dU = C_V dT + \pi_T dV
TODO

sympy not provided for expression
5085809757 \(\frac{k_{\rm adsorption}}{k_{\rm desorption}}\) = \(\frac{[A_{\rm adsorption}]}{p_A [S]}\)
\frac{k_{\rm adsorption}}{k_{\rm desorption}} = \frac{[A_{\rm adsorption}]}{p_A [S]}
TODO

sympy not provided for expression
5125940051 \(I\) = \(|A|^2 + B B^* + A B^* + B A^*\)
I = |A|^2 + B B^* + A B^* + B A^*
TODO

sympy not provided for expression
5128670694 \(m_1 d_1\) = \(m_2 d_2\)
m_1 d_1 = m_2 d_2
TODO

sympy not provided for expression
5136652623 \(E\) = \(KE + PE\)
E = KE + PE
mechanical energy is the sum of the potential plus kinetic energies TODO

sympy not provided for expression
5144263777 \(v^2\) = \(v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)\)
v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)
TODO

sympy not provided for expression
5148266645 \(t'\) = \(\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t\)
t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t
TODO

sympy not provided for expression
5177311762 \(v\) = \(\frac{2 \pi r}{T}\)
v = \frac{2 \pi r}{T}
TODO

sympy not provided for expression
5323719091 \(i \sinh x\) = \(\frac{1}{2i} \left( \exp(-x) - \exp(x) \right)\)
i \sinh x = \frac{1}{2i} \left( \exp(-x) - \exp(x) \right)
TODO

sympy not provided for expression
5345738321 \(F\) = \(m a\)
F = m a
Newton's second law of motion https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion#Newton's_second_law TODO

sympy not provided for expression
5349669879 \(\tanh(x)\) = \(\frac{ \exp(x)-\exp(-x)}{\exp(x)+\exp(-x)}\)
\tanh(x) = \frac{ \exp(x)-\exp(-x)}{\exp(x)+\exp(-x)}
TODO

sympy not provided for expression
5349866551 \(\vec{v}\) = \(v_x \hat{x} + v_y \hat{y}\)
\vec{v} = v_x \hat{x} + v_y \hat{y}
TODO

sympy not provided for expression
5353282496 \(d\) = \(\frac{v_0^2}{g}\)
d = \frac{v_0^2}{g}
TODO

sympy not provided for expression
5373931751 \(t\) = \(t_f\)
t = t_f
TODO

sympy not provided for expression
5379546684 \(y_f\) = \(- \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0\)
y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
TODO

sympy not provided for expression
5404822208 \(v_{\rm escape}\) = \(\sqrt{2 G \frac{m}{r}}\)
v_{\rm escape} = \sqrt{2 G \frac{m}{r}}
escape velocity TODO

sympy not provided for expression
5415824175 \(x(t)\) = \(A \cos(\omega t)\)
x(t) = A \cos(\omega t)
TODO

sympy not provided for expression
5426308937 \(v\) = \(\frac{d}{t}\)
v = \frac{d}{t}
TODO

sympy not provided for expression
5438722682 \(x\) = \(v_0 t \cos(\theta) + x_0\)
x = v_0 t \cos(\theta) + x_0
TODO

sympy not provided for expression
5514556106 \(E_2 - E_1\) = \((KE_2 - KE_1) + (PE_2 - PE_1)\)
E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)
TODO

sympy not provided for expression
5530148480 \(\vec{p}_{1}-\vec{p}_{2}\) = \(\vec{p}_{electron}\)
\vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}
TODO

sympy not provided for expression
5542528160 \(\int dW\) = \(F \int_0^x dx\)
\int dW = F \int_0^x dx
TODO

sympy not provided for expression
5563580265 \(F_{\rm gravity}\) = \(G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}\)
F_{\rm gravity} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}
TODO

sympy not provided for expression
5586102077 \(r\) = \(d_1 + d_2\)
r = d_1 + d_2
TODO

sympy not provided for expression
5596822289 \(W_{\rm to\ system}\) = \(-G m_1 m_2 \left(\left.\frac{-1}{x}\right|^r_{\infty}\right)\)
W_{\rm to\ system} = -G m_1 m_2 \left(\left.\frac{-1}{x}\right|^r_{\infty}\right)
TODO

sympy not provided for expression
5611024898 \(d\) = \(\frac{1}{2 a} (v^2 - v_0^2)\)
d = \frac{1}{2 a} (v^2 - v_0^2)
TODO

sympy not provided for expression
5634116660 \(\pi_T\) = \(\left(\frac{\partial U}{\partial V}\right)_T\)
\pi_T = \left(\frac{\partial U}{\partial V}\right)_T
definition of internal pressure at constant temperature TODO

sympy not provided for expression
5646314683 \(m\) = \(A m_p\)
m = A m_p
TODO

sympy not provided for expression
5658865948 \(T^2\) = \(\frac{r^3 4 \pi^2}{(m_1+m_2)G}\)
T^2 = \frac{r^3 4 \pi^2}{(m_1+m_2)G}
TODO

sympy not provided for expression
5693047217 \(v_{\rm final}\) = \(-\sqrt{\frac{2 G m_2}{r}}\)
v_{\rm final} = -\sqrt{\frac{2 G m_2}{r}}
TODO

sympy not provided for expression
5727578862 \(\frac{d^2}{dx^2} \psi(x)\) = \(-k^2 \psi(x)\)
\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)
TODO

sympy not provided for expression
5732331610 \(W\) = \(G m_1 m_2 \left( \frac{1}{x} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)\)
W = G m_1 m_2 \left( \frac{1}{x} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)
2022-03-25 BHP: Conversion between Latex and Sympy is incomplete TODO

sympy not provided for expression
5733146966 \(KE_2 - KE_1\) = \(\frac{1}{2} m \left(v_2^2 - v_1^2\right)\)
KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)
TODO

sympy not provided for expression
5733721198 \(d\) = \(\frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)\)
d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)
TODO

sympy not provided for expression
5763749235 \(-c^2 + c^2 \gamma^2\) = \(v^2 \gamma^2\)
-c^2 + c^2 \gamma^2 = v^2 \gamma^2
TODO

sympy not provided for expression
5779256336 \(W_{\rm by\ system}\) = \(KE_{\rm final} - KE_{\rm initial}\)
W_{\rm by\ system} = KE_{\rm final} - KE_{\rm initial}
TODO

sympy not provided for expression
5781981178 \(x^2 - y^2\) = \((x+y)(x-y)\)
x^2 - y^2 = (x+y)(x-y)
difference of squares https://en.wikipedia.org/wiki/Difference_of_two_squares TODO

sympy not provided for expression
5789289057 \(v\) = \(\alpha c \sqrt{ \frac{m_e}{2 m} }\)
v = \alpha c \sqrt{ \frac{m_e}{2 m} }
equation 4 in the PDF TODO

sympy not provided for expression
5832984291 \((\sin(x))^2 + (\cos(x))^2\) = \(1\)
(\sin(x))^2 + (\cos(x))^2 = 1
TODO

sympy not provided for expression
5838268428 \(\alpha c\) = \(\frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar}\)
\alpha c = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar}
TODO

sympy not provided for expression
5846639423 \(v_{\rm final}\) = \(\sqrt{\frac{2 G m_2}{r}}\)
v_{\rm final} = \sqrt{\frac{2 G m_2}{r}}
TODO

sympy not provided for expression
5850144586 \(W_{\rm by\ system}\) = \(KE_{\rm final}\)
W_{\rm by\ system} = KE_{\rm final}
TODO

sympy not provided for expression
5857434758 \(\int a dx\) = \(a x\)
\int a dx = a x
TODO

sympy not provided for expression
5866629429 \({\rm sech}^2\ x + \tanh^2(x)\) = \(1\)
{\rm sech}^2\ x + \tanh^2(x) = 1
TODO

sympy not provided for expression
5868688585 \(\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right)\) = \(\frac{p^2}{2m} \psi( \vec{r},t)\)
\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = \frac{p^2}{2m} \psi( \vec{r},t)
https://physicsderivationgraph.blogspot.com/2020/09/representing-laplace-operator-nabla-in.html TODO

sympy not provided for expression
5900595848 \(k\) = \(\frac{\omega}{v}\)
k = \frac{\omega}{v}
TODO

sympy not provided for expression
5902985919 \(\vec{F}\) = \(G \frac{m_1 m_2}{x^2} \hat{x}\)
\vec{F} = G \frac{m_1 m_2}{x^2} \hat{x}
Newton's law of universal gravitation TODO

sympy not provided for expression
5928285821 \(x^2 + 2 x (b/(2 a)) + (b/(2 a))^2\) = \((x + (b/(2 a)))^2\)
x^2 + 2 x (b/(2 a)) + (b/(2 a))^2 = (x + (b/(2 a)))^2
TODO

sympy not provided for expression
5928292841 \(x^2 + (b/a)x + (b/(2 a))^2\) = \(-c/a + (b/(2 a))^2\)
x^2 + (b/a)x + (b/(2 a))^2 = -c/a + (b/(2 a))^2
TODO

sympy not provided for expression
5938459282 \(x^2 + (b/a)x\) = \(-c/a\)
x^2 + (b/a)x = -c/a
TODO

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5945893986 \(\frac{d^2 x}{dt^2}\) = \(-A \omega^2 \cos(\omega t)\)
\frac{d^2 x}{dt^2} = -A \omega^2 \cos(\omega t)
TODO

sympy not provided for expression
5958392859 \(x^2 + (b/a)x+(c/a)\) = \(0\)
x^2 + (b/a)x+(c/a) = 0
TODO

sympy not provided for expression
5959282914 \(x^2 + x(b/a) + (b/(2 a))^2\) = \((x+(b/(2 a)))^2\)
x^2 + x(b/a) + (b/(2 a))^2 = (x+(b/(2 a)))^2
TODO

sympy not provided for expression
5962145508 \(\alpha\) = \(\frac{nR}{VP}\)
\alpha = \frac{nR}{VP}
TODO

sympy not provided for expression
5978756813 \(W\) = \(G m_{\rm Earth} m \left( 0 - \frac{-1}{ r_{\rm Earth}} \right)\)
W = G m_{\rm Earth} m \left( 0 - \frac{-1}{ r_{\rm Earth}} \right)
TODO

sympy not provided for expression
5982958248 \(x\) = \(-\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))\)
x = -\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))
TODO

sympy not provided for expression
5982958249 \(x+(b/(2 a))\) = \(-\sqrt{(b/(2 a))^2 - (c/a)}\)
x+(b/(2 a)) = -\sqrt{(b/(2 a))^2 - (c/a)}
TODO

sympy not provided for expression
5985371230 \(\vec{ \nabla} \psi( \vec{r},t)\) = \(\frac{i}{\hbar} \vec{p} \psi( \vec{r},t)\)
\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi( \vec{r},t)
TODO

sympy not provided for expression
6026694087 \(F_{centripetal}\) = \(m \frac{v^2}{r}\)
F_{centripetal} = m \frac{v^2}{r}
TODO

sympy not provided for expression
6031385191 \(\sinh^2 x\) = \(\left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)\)
\sinh^2 x = \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)
TODO

sympy not provided for expression
6055078815 \(\left(\frac{\partial U}{\partial T}\right)_p\) = \(C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T \left( \frac{\partial V}{\partial T} \right)_p\)
\left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T \left( \frac{\partial V}{\partial T} \right)_p
constant pressure TODO

sympy not provided for expression
6061695358 \(V_2\) = \(I R_2\)
V_2 = I R_2
TODO

sympy not provided for expression
6083821265 \(v_0 \cos(\theta)\) = \(v_{0, x}\)
v_0 \cos(\theta) = v_{0, x}
TODO

sympy not provided for expression
6091977310 \(KE_{\rm initial}\) = \(\frac{1}{2} m_1 v_{\rm initial}^2\)
KE_{\rm initial} = \frac{1}{2} m_1 v_{\rm initial}^2
TODO

sympy not provided for expression
6131764194 \(W\) = \(G m_{\rm Earth} m \left( \frac{1}{x^2} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)\)
W = G m_{\rm Earth} m \left( \frac{1}{x^2} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)
https://physicsderivationgraph.blogspot.com/2020/09/evaluating-definite-integrals-for.html TODO

sympy not provided for expression
6134836751 \(v_{0, x}\) = \(v_x\)
v_{0, x} = v_x
TODO

sympy not provided for expression
6175547907 \(v_{\rm average}\) = \(\frac{v + v_0}{2}\)
v_{\rm average} = \frac{v + v_0}{2}
TODO

sympy not provided for expression
6204539227 \(-g t + v_{0, y}\) = \(\frac{dy}{dt}\)
-g t + v_{0, y} = \frac{dy}{dt}
TODO

sympy not provided for expression
6240206408 \(I_{\rm incoherent}\) = \(|A|^2 + |B|^2\)
I_{\rm incoherent} = |A|^2 + |B|^2
TODO

sympy not provided for expression
6240546932 \(\frac{1}{K_{equilibrium}}\) = \(\frac{k_{\rm desorption}}{k_{\rm adsorption}}\)
\frac{1}{K_{equilibrium}} = \frac{k_{\rm desorption}}{k_{\rm adsorption}}
TODO

sympy not provided for expression
6268336290 \(F_{gravitational}\) = \(\frac{m}{r}\left(\frac{2\pi r}{T}\right)^2\)
F_{gravitational} = \frac{m}{r}\left(\frac{2\pi r}{T}\right)^2
TODO

sympy not provided for expression
6306552185 \(I\) = \((A + B)(A^* + B^*)\)
I = (A + B)(A^* + B^*)
TODO

sympy not provided for expression
6348260313 \(C_{\rm Earth\ orbit}\) = \(2 \pi r_{\rm Earth\ orbit}\)
C_{\rm Earth\ orbit} = 2 \pi r_{\rm Earth\ orbit}
TODO

sympy not provided for expression
6397683463 \(V \alpha\) = \(\left( \frac{\partial V}{\partial T} \right)_p\)
V \alpha = \left( \frac{\partial V}{\partial T} \right)_p
TODO

sympy not provided for expression
6404535647 \(\cosh x\) = \(\frac{\exp(x) + \exp(-x)}{2}\)
\cosh x = \frac{\exp(x) + \exp(-x)}{2}
TODO

sympy not provided for expression
6421241247 \(d\) = \(v t - \frac{1}{2} a t^2\)
d = v t - \frac{1}{2} a t^2
TODO

sympy not provided for expression
6450985774 \(n_1 \sin( \theta_1 )\) = \(n_2 \sin( \theta_2 )\)
n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )
Law of Refraction eq 34-44 on page 819 in \cite{2001_HRW} TODO

sympy not provided for expression
6457044853 \(v - a t\) = \(v_0\)
v - a t = v_0
TODO

sympy not provided for expression
6457999644 \(\frac{[S_0]}{[A_{\rm adsorption}]}\) = \(\frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1\)
\frac{[S_0]}{[A_{\rm adsorption}]} = \frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1
TODO

sympy not provided for expression
6504442697 \(v\) = \(\sqrt{ \frac{K}{\rho} }\)
v = \sqrt{ \frac{K}{\rho} }
TODO

sympy not provided for expression
6529793063 \(I_{\rm incoherent}\) = \(|A|^2 + |A|^2\)
I_{\rm incoherent} = |A|^2 + |A|^2
TODO

sympy not provided for expression
6555185548 \(A^*\) = \(|A| \exp(-i \theta)\)
A^* = |A| \exp(-i \theta)
TODO

sympy not provided for expression
6556875579 \(\frac{I_{\rm coherent}}{I_{\rm incoherent}}\) = \(2\)
\frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2
TODO

sympy not provided for expression
6572039835 \(-g t + v_{0, y}\) = \(v_y\)
-g t + v_{0, y} = v_y
TODO

sympy not provided for expression
6715248283 \(PE\) = \(-F x\)
PE = -F x
potential energy https://en.wikipedia.org/wiki/Potential_energy TODO

sympy not provided for expression
6742123016 \(\vec{p}_{electron}\cdot\vec{p}_{electron}\) = \(( \vec{p}_{1}\cdot\vec{p}_{1})+( \vec{p}_{2}\cdot\vec{p}_{2})-2( \vec{p}_{1}\cdot\vec{p}_{2})\)
\vec{p}_{electron}\cdot\vec{p}_{electron} = ( \vec{p}_{1}\cdot\vec{p}_{1})+( \vec{p}_{2}\cdot\vec{p}_{2})-2( \vec{p}_{1}\cdot\vec{p}_{2})
TODO

sympy not provided for expression
6753224061 \(I_{\rm total}\) = \(I_1 + I_2\)
I_{\rm total} = I_1 + I_2
TODO

sympy not provided for expression
6774684564 \(\theta\) = \(\phi\)
\theta = \phi
for coherent waves TODO

sympy not provided for expression
6783009163 \(r_{\rm adsorption}\) = \(r_{\rm desorption}\)
r_{\rm adsorption} = r_{\rm desorption}
TODO

sympy not provided for expression
6785303857 \(C\) = \(2 \pi r\)
C = 2 \pi r
TODO

sympy not provided for expression
6800170830 \(r_{\rm Schwarzschild}\) = \(\frac{2 G m}{c^2}\)
r_{\rm Schwarzschild} = \frac{2 G m}{c^2}
TODO

sympy not provided for expression
6829281943 \(F_{\rm centripetal}\) = \(G \frac{m_1 m_2}{r^2}\)
F_{\rm centripetal} = G \frac{m_1 m_2}{r^2}
TODO

sympy not provided for expression
6831637424 \(\sin( 90^{\circ} - \theta_{\rm Brewster} )\) = \(\cos( \theta_{\rm Brewster} )\)
\sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} )
TODO

sympy not provided for expression
6831694380 \(a\) = \(\frac{d^2 x}{dt^2}\)
a = \frac{d^2 x}{dt^2}
acceleration TODO

sympy not provided for expression
6870322215 \(KE_{\rm escape}\) = \(\frac{1}{2} m v_{\rm escape}^2\)
KE_{\rm escape} = \frac{1}{2} m v_{\rm escape}^2
TODO

sympy not provided for expression
6885625907 \(\exp(i \pi)\) = \(-1 + i 0\)
\exp(i \pi) = -1 + i 0
TODO

sympy not provided for expression
6892595652 \(\frac{1}{2} m_1 v_{\rm final}^2\) = \(\frac{G m_1 m_2}{r}\)
\frac{1}{2} m_1 v_{\rm final}^2 = \frac{G m_1 m_2}{r}
TODO

sympy not provided for expression
6908055431 \(x(t)\) = \(A \cos\left(\frac{k}{m} t\right)\)
x(t) = A \cos\left(\frac{k}{m} t\right)
TODO

sympy not provided for expression
6925244346 \(\alpha\) = \(\frac{PV}{T} \frac{1}{VP}\)
\alpha = \frac{PV}{T} \frac{1}{VP}
TODO

sympy not provided for expression
6935745841 \(F\) = \(G \frac{m_1 m_2}{x^2}\)
F = G \frac{m_1 m_2}{x^2}
Newton's law of universal gravitation https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation#Modern_form TODO

sympy not provided for expression
6946088325 \(v\) = \(\frac{C}{t}\)
v = \frac{C}{t}
TODO

sympy not provided for expression
6955192897 \(r_{\rm desorption}\) = \(k_{\rm desorption} [A_{\rm adsorption}]\)
r_{\rm desorption} = k_{\rm desorption} [A_{\rm adsorption}]
TODO

sympy not provided for expression
6998364753 \(v_{\rm Earth\ orbit}\) = \(\frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}\)
v_{\rm Earth\ orbit} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}
TODO

sympy not provided for expression
7002609475 \(\frac{V}{R_2}\) = \(I_2\)
\frac{V}{R_2} = I_2
TODO

sympy not provided for expression
7010294143 \(T_{\rm orbit}^2 G m_{\rm Earth}\) = \(4 \pi^2 r^3\)
T_{\rm orbit}^2 G m_{\rm Earth} = 4 \pi^2 r^3
TODO

sympy not provided for expression
7011114072 \(d\) = \(\frac{(v_0 + a t) + v_0}{2} t\)
d = \frac{(v_0 + a t) + v_0}{2} t
TODO

sympy not provided for expression
7057864873 \(y'\) = \(y\)
y' = y
frame of reference is moving only along x direction TODO

sympy not provided for expression
7107090465 \(B B^*\) = \(|B|^2\)
B B^* = |B|^2
TODO

sympy not provided for expression
7112613117 \(m_{\rm Earth}\) = \(\frac{(9.80665 m/s^2) r_{\rm Earth}^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}}\)
m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}}
TODO

sympy not provided for expression
7112646057 \(v_{\rm final}^2\) = \(\frac{2 G m_2}{r}\)
v_{\rm final}^2 = \frac{2 G m_2}{r}
TODO

sympy not provided for expression
7175416299 \(t_{\rm Earth\ orbit}\) = \(1 {\rm year}\)
t_{\rm Earth\ orbit} = 1 {\rm year}
TODO

sympy not provided for expression
7215099603 \(v^2\) = \(v_0^2 + 2 a t v_0 + a^2 t^2\)
v^2 = v_0^2 + 2 a t v_0 + a^2 t^2
TODO

sympy not provided for expression
7217021879 \(R_{\rm total}\) = \(R_1 + R_2\)
R_{\rm total} = R_1 + R_2
TODO

sympy not provided for expression
7233558441 \(d\) = \(v_0 t_f \cos(\theta)\)
d = v_0 t_f \cos(\theta)
TODO

sympy not provided for expression
7252338326 \(v_y\) = \(\frac{dy}{dt}\)
v_y = \frac{dy}{dt}
TODO

sympy not provided for expression
7267155233 \(\frac{PE_2 - PE_1}{t}\) = \(-F \left( \frac{x_2 - x_1}{t} \right)\)
\frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)
TODO

sympy not provided for expression
7267424860 \(\frac{1}{\theta_A}\) = \(\frac{1+(K_{\rm equilibrium}\ p_A)}{K_{\rm equilibrium}\ p_A}\)
\frac{1}{\theta_A} = \frac{1+(K_{\rm equilibrium}\ p_A)}{K_{\rm equilibrium}\ p_A}
TODO

sympy not provided for expression
7354529102 \(y\) = \(- \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0\)
y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0
TODO

sympy not provided for expression
7376526845 \(\sin(\theta)\) = \(\frac{v_{0, y}}{v_0}\)
\sin(\theta) = \frac{v_{0, y}}{v_0}
TODO

sympy not provided for expression
7391837535 \(\cos(\theta)\) = \(\frac{v_{0, x}}{v_0}\)
\cos(\theta) = \frac{v_{0, x}}{v_0}
TODO

sympy not provided for expression
7455581657 \(v_{0, x}\) = \(\frac{dx}{dt}\)
v_{0, x} = \frac{dx}{dt}
TODO

sympy not provided for expression
7466829492 \(\vec{ \nabla} \cdot \vec{E}\) = \(0\)
\vec{ \nabla} \cdot \vec{E} = 0
TODO

sympy not provided for expression
7513513483 \(\gamma^2 (c^2 - v^2)\) = \(c^2\)
\gamma^2 (c^2 - v^2) = c^2
TODO

sympy not provided for expression
7517073655 \([S_0]\) = \(\left(\frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}]\)
[S_0] = \left(\frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}]
TODO

sympy not provided for expression
7564894985 \(\int \cos\left(\frac{2n\pi}{W} x\right) dx\) = \(\frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\)
\int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)
TODO

sympy not provided for expression
7572664728 \(\cos(2 x) + 2 (\sin(x))^2\) = \(1\)
\cos(2 x) + 2 (\sin(x))^2 = 1
TODO

sympy not provided for expression
7573835180 \(PE_{\rm Earth\ surface}\) = \(-W\)
PE_{\rm Earth\ surface} = -W
the potential energy at the surface of the Earth is equal to the work needed to get it from the center of the Earth to the surface TODO

sympy not provided for expression
7575738420 \(\left(\sin\left(\frac{n \pi}{W}x\right) \right)^2\) = \(\frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}\)
\left(\sin\left(\frac{n \pi}{W}x\right) \right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}
TODO

sympy not provided for expression
7575859295 \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E}\) = \(\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})
TODO

sympy not provided for expression
7575859300 \(\epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{ \nabla} \times \vec{E} )_k\) = \(\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
\epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{ \nabla} \times \vec{E} )_k = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})
TODO

sympy not provided for expression
7575859302 \(\epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n\) = \(\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
\epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})
TODO

sympy not provided for expression
7575859304 \(\epsilon^{i,j,k} \epsilon_{n,j,k}\) = \(\delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}\)
\epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}
https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors TODO

sympy not provided for expression
7575859306 \(\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n\) = \(\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})
https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors TODO

sympy not provided for expression
7575859308 \(\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right)\) = \(\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})
https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors TODO

sympy not provided for expression
7575859310 \(\hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n\) = \(\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
\hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})
TODO

sympy not provided for expression
7575859312 \(\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\) = \(\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})
TODO

sympy not provided for expression
7621705408 \(I\) = \(|A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)\)
I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)
TODO

sympy not provided for expression
7652131521 \(\frac{dx}{dt}\) = \(-A \omega \sin (\omega t)\)
\frac{dx}{dt} = -A \omega \sin (\omega t)
TODO

sympy not provided for expression
7672365885 \(F_{gravitational}\) = \(\frac{4 \pi^2 m r}{T^2}\)
F_{gravitational} = \frac{4 \pi^2 m r}{T^2}
TODO

sympy not provided for expression
7675171493 \(V_1\) = \(I R_1\)
V_1 = I R_1
TODO

sympy not provided for expression
7676652285 \(KE_2\) = \(\frac{1}{2} m v_2^2\)
KE_2 = \frac{1}{2} m v_2^2
TODO

sympy not provided for expression
7696214507 \(n_1 \sin( \theta_{\rm Brewster} )\) = \(n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} )\)
n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} )
TODO

sympy not provided for expression
7701249282 \(v_u\) = \(\alpha c \sqrt{ \frac{m_e}{m_p} }\)
v_u = \alpha c \sqrt{ \frac{m_e}{m_p} }
when A = 1 TODO

sympy not provided for expression
7729413831 \(a_x \hat{x} + a_y \hat{y}\) = \(\frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)\)
a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)
TODO

sympy not provided for expression
7731226616 \({\rm sech}\ x\) = \(\frac{1}{\cosh x}\)
{\rm sech}\ x = \frac{1}{\cosh x}
TODO

sympy not provided for expression
7734996511 \(PE_2 - PE_1\) = \(-F ( x_2 - x_1 )\)
PE_2 - PE_1 = -F ( x_2 - x_1 )
TODO

sympy not provided for expression
7735731560 \(\cosh^2 x - \sinh^2 x\) = \(\frac{1}{4}\left( \exp(2x)+1+1+\exp(-2x) - \left(\exp(2x)-1-1-\exp(-2x)\right) \right)\)
\cosh^2 x - \sinh^2 x = \frac{1}{4}\left( \exp(2x)+1+1+\exp(-2x) - \left(\exp(2x)-1-1-\exp(-2x)\right) \right)
TODO

sympy not provided for expression
7735737409 \(\frac{KE_2 - KE_1}{t}\) = \(m v \frac{ v_2 - v_1 }{t}\)
\frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}
TODO

sympy not provided for expression
7741202861 \(x\) = \(\gamma^2 x - \gamma^2 v t + \gamma v t'\)
x = \gamma^2 x - \gamma^2 v t + \gamma v t'
TODO

sympy not provided for expression
7749253510 \(W\) = \(G \frac{m_{\rm Earth} m }{ r_{\rm Earth}}\)
W = G \frac{m_{\rm Earth} m }{ r_{\rm Earth}}
TODO

sympy not provided for expression
7826132469 \(\left(\frac{\partial U}{\partial T}\right)_p\) = \(C_V + \pi_T V \alpha\)
\left(\frac{\partial U}{\partial T}\right)_p = C_V + \pi_T V \alpha
TODO

sympy not provided for expression
7837519722 \(v\) = \(\sqrt{f} \sqrt{\frac{E}{m}}\)
v = \sqrt{f} \sqrt{\frac{E}{m}}
TODO

sympy not provided for expression
7846240076 \(m_{\rm Earth}\) = \(\frac{(9.80665 m/s^2) r_{\rm Earth}^2}{G}\)
m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{G}
TODO

sympy not provided for expression
7875206161 \(E_2\) = \(KE_2 + PE_2\)
E_2 = KE_2 + PE_2
TODO

sympy not provided for expression
7882872592 \(W_{\rm to\ system}\) = \(\int_{\infty}^r \vec{F}\cdot d\vec{r}\)
W_{\rm to\ system} = \int_{\infty}^r \vec{F}\cdot d\vec{r}
TODO

sympy not provided for expression
7906112355 \(\gamma^2\) = \(\frac{c^2}{c^2 - \gamma^2}\)
\gamma^2 = \frac{c^2}{c^2 - \gamma^2}
TODO

sympy not provided for expression
7917051060 \(\vec{p}_{electron}\) = \(\vec{p}_{1}-\vec{p}_{2}\)
\vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}
TODO

sympy not provided for expression
7924063906 \(K_{equilibrium}\) = \(\frac{k_{\rm adsorption}}{k_{\rm desorption}}\)
K_{equilibrium} = \frac{k_{\rm adsorption}}{k_{\rm desorption}}
TODO

sympy not provided for expression
7928111771 \(\frac{1}{\theta_A}\) = \(\frac{1}{K_{\rm equilibrium} p_A} + 1\)
\frac{1}{\theta_A} = \frac{1}{K_{\rm equilibrium} p_A} + 1
TODO

sympy not provided for expression
7939765107 \(v^2\) = \(v_0^2 + 2 a d\)
v^2 = v_0^2 + 2 a d
TODO

sympy not provided for expression
8046208134 \(I_{\rm coherent}\) = \(|A|^2 + |A|^2 + |A| |A| 2\)
I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2
TODO

sympy not provided for expression
8049905441 \(\Delta KE\) = \(KE_{\rm final} - KE_{\rm initial}\)
\Delta KE = KE_{\rm final} - KE_{\rm initial}
change in kinetic energy TODO

sympy not provided for expression
8059639673 \(v^2\) = \(\frac{4 \pi^2 r^2}{T_{\rm orbit}^2}\)
v^2 = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2}
TODO

sympy not provided for expression
8065128065 \(I\) = \(A A^* + B B^* + A B^* + B A^*\)
I = A A^* + B B^* + A B^* + B A^*
TODO

sympy not provided for expression
8090924099 \(v\) = \(\sqrt{ \left( f\frac{E}{a^3} \right) \frac{1}{\rho} }\)
v = \sqrt{ \left( f\frac{E}{a^3} \right) \frac{1}{\rho} }
TODO

sympy not provided for expression
8106885760 \(\alpha\) = \(\frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c}\)
\alpha = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c}
fine structure constant definition TODO

sympy not provided for expression
8131665171 \(\frac{1}{\theta_A}\) = \(\frac{[S_0]}{[A_{\rm adsorption}]}\)
\frac{1}{\theta_A} = \frac{[S_0]}{[A_{\rm adsorption}]}
TODO

sympy not provided for expression
8139187332 \(\vec{p}_{1}\) = \(\vec{p}_{2}+\vec{p}_{electron}\)
\vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}
TODO

sympy not provided for expression
8145337879 \(-g t dt + v_{0, y} dt\) = \(dy\)
-g t dt + v_{0, y} dt = dy
TODO

sympy not provided for expression
8198310977 \(0\) = \(- \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0\)
0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
TODO

sympy not provided for expression
8228733125 \(a_y\) = \(\frac{d}{dt} v_y\)
a_y = \frac{d}{dt} v_y
TODO

sympy not provided for expression
8257621077 \(\vec{p}_{\rm before}\) = \(\vec{p}_{1}\)
\vec{p}_{\rm before} = \vec{p}_{1}
TODO

sympy not provided for expression
8269198922 \(2 a d\) = \(v^2 - v_0^2\)
2 a d = v^2 - v_0^2
TODO

sympy not provided for expression
8283354808 \(I_{\rm coherent}\) = \(|A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )\)
I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
TODO

sympy not provided for expression
8311458118 \(\vec{p}_{\rm after}\) = \(\vec{p}_{2}+\vec{p}_{electron}\)
\vec{p}_{\rm after} = \vec{p}_{2}+\vec{p}_{electron}
TODO

sympy not provided for expression
8332931442 \(\exp(i \pi)\) = \(\cos(\pi)+i \sin(\pi)\)
\exp(i \pi) = \cos(\pi)+i \sin(\pi)
TODO

sympy not provided for expression
8357234146 \(KE\) = \(\frac{1}{2} m v^2\)
KE = \frac{1}{2} m v^2
kinetic energy https://en.wikipedia.org/wiki/Kinetic_energy TODO

sympy not provided for expression
8360117126 \(\gamma\) = \(\frac{-1}{\sqrt{1-\frac{v^2}{c^2}}}\)
\gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}}
not a physically valid result in this context TODO

sympy not provided for expression
8361238989 \(a_{centripetal}\) = \(\frac{v^2}{r}\)
a_{centripetal} = \frac{v^2}{r}
TODO

sympy not provided for expression
8368984890 \(\kappa_T\) = \(\frac{-1}{V} \left( \frac{ \partial }{\partial P}\left(\frac{nRT}{P}\right) \right)_T\)
\kappa_T = \frac{-1}{V} \left( \frac{ \partial }{\partial P}\left(\frac{nRT}{P}\right) \right)_T
TODO

sympy not provided for expression
8396997949 \(I\) = \(| A + B |^2\)
I = | A + B |^2
intensity of two waves traveling opposite directions on same path TODO

sympy not provided for expression
8399484849 \(\langle x^2 - 2 x \langle x \rangle + \langle x \rangle^2 \rangle\) = \(\langle x^2 \rangle-\langle x \rangle^2\)
\langle x^2 - 2 x \langle x \rangle + \langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
TODO

sympy not provided for expression
8405272745 \(W_{\rm to\ system}\) = \(-G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx\)
W_{\rm to\ system} = -G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx
TODO

sympy not provided for expression
8418527415 \(\sin(i x)\) = \(i \sinh(x)\)
\sin(i x) = i \sinh(x)
TODO

sympy not provided for expression
8435841627 \(P V\) = \(n R T\)
P V = n R T
https://en.wikipedia.org/wiki/Ideal_gas_law TODO

sympy not provided for expression
8460820419 \(v_x\) = \(\frac{dx}{dt}\)
v_x = \frac{dx}{dt}
TODO

sympy not provided for expression
8483686863 \(\sin(2 x)\) = \(\frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x) \right)\)
\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x) \right)
TODO

sympy not provided for expression
8484544728 \(-a k^2\sin(k x) + -b k^2\cos(k x)\) = \(-a k^2 \sin(kx) + -b k^2 \cos(k x)\)
-a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)
TODO

sympy not provided for expression
8485757728 \(a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x)\) = \(-a k^2 \sin(kx) + -b k^2 \cos(kx)\)
a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)
TODO

sympy not provided for expression
8485867742 \(\frac{2}{W}\) = \(a^2\)
\frac{2}{W} = a^2
TODO

sympy not provided for expression
8486706976 \(v_{0, x} t + x_0\) = \(x\)
v_{0, x} t + x_0 = x
TODO

sympy not provided for expression
8489593958 \(d(u v)\) = \(u dv + v du\)
d(u v) = u dv + v du
TODO

sympy not provided for expression
8489593960 \(d(u v) - v du\) = \(u dv\)
d(u v) - v du = u dv
TODO

sympy not provided for expression
8489593962 \(u dv\) = \(d(u v) - v du\)
u dv = d(u v) - v du
TODO

sympy not provided for expression
8489593964 \(\int u dv\) = \(u v - \int v du\)
\int u dv = u v - \int v du
TODO

sympy not provided for expression
8494839423 \(\nabla^2 \vec{E}\) = \(\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\)
\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
TODO

sympy not provided for expression
8495187962 \(\theta_{\rm Brewster}\) = \(\arctan{ \left( \frac{ n_1 }{ n_2 } \right) }\)
\theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }
TODO

sympy not provided for expression
8497631728 \(I\) = \(|A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )\)
I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )
TODO

sympy not provided for expression
8515803375 \(z'\) = \(z\)
z' = z
frame of reference is moving only along x direction TODO

sympy not provided for expression
8532702080 \(\cosh^2 x\) = \(\left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right)\)
\cosh^2 x = \left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right)
TODO

sympy not provided for expression
8552710882 \(KE_{\rm final}\) = \(\frac{1}{2} m_1 v_{\rm final}^2\)
KE_{\rm final} = \frac{1}{2} m_1 v_{\rm final}^2
TODO

sympy not provided for expression
8558338742 \(E_2\) = \(E_1\)
E_2 = E_1
conservation of energy https://en.wikipedia.org/wiki/Conservation_of_energy TODO

sympy not provided for expression
8563535636 \(\cosh^2 x - \sinh^2 x\) = \(\left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right) - \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)\)
\cosh^2 x - \sinh^2 x = \left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right) - \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)
TODO

sympy not provided for expression
8572657110 \(1\) = \(\int |\psi(x)|^2 dx\)
1 = \int |\psi(x)|^2 dx
TODO

sympy not provided for expression
8572852424 \(\vec{E}\) = \(E( \vec{r},t)\)
\vec{E} = E( \vec{r},t)
TODO

sympy not provided for expression
8575746378 \(\int \frac{1}{2} dx\) = \(\frac{1}{2} x\)
\int \frac{1}{2} dx = \frac{1}{2} x
TODO

sympy not provided for expression
8575748999 \(\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x) \right)\) = \(-k^2 \left(a \sin(kx) + b \cos(kx) \right)\)
\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x) \right) = -k^2 \left(a \sin(kx) + b \cos(kx) \right)
TODO

sympy not provided for expression
8576785890 \(1\) = \(\int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx\)
1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
TODO

sympy not provided for expression
8577275751 \(0\) = \(a \sin(0) + b\cos(0)\)
0 = a \sin(0) + b\cos(0)
TODO

sympy not provided for expression
8582885111 \(\psi(x)\) = \(a \sin(kx) + b \cos(kx)\)
\psi(x) = a \sin(kx) + b \cos(kx)
TODO

sympy not provided for expression
8582954722 \(x^2 + 2 x h + h^2\) = \((x + h)^2\)
x^2 + 2 x h + h^2 = (x + h)^2
TODO

sympy not provided for expression
8584698994 \(-g \int dt\) = \(\int d v_y\)
-g \int dt = \int d v_y
TODO

sympy not provided for expression
8588429722 \(\sin( 90^{\circ} - x )\) = \(\cos( x )\)
\sin( 90^{\circ} - x ) = \cos( x )
TODO

sympy not provided for expression
8602221482 \(\langle \cos(\theta - \phi) \rangle\) = \(0\)
\langle \cos(\theta - \phi) \rangle = 0
incoherent light source TODO

sympy not provided for expression
8602512487 \(\vec{a}\) = \(a_x \hat{x} + a_y \hat{y}\)
\vec{a} = a_x \hat{x} + a_y \hat{y}
decompose acceleration into two components TODO

sympy not provided for expression
8604483515 \(dW\) = \(G \frac{m_1 m_2}{x^2} dx\)
dW = G \frac{m_1 m_2}{x^2} dx
TODO

sympy not provided for expression
8651044341 \(\cos(i x)\) = \(\frac{1}{2} \left( \exp(-x) + \exp(x) \right)\)
\cos(i x) = \frac{1}{2} \left( \exp(-x) + \exp(x) \right)
TODO

sympy not provided for expression
8655294002 \(a\) = \(-\frac{k}{m}x\)
a = -\frac{k}{m}x
TODO

sympy not provided for expression
8661803554 \(F\) = \(G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2}\)
F = G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2}
TODO

sympy not provided for expression
8688588981 \(a^3 \rho\) = \(m\)
a^3 \rho = m
TODO

sympy not provided for expression
8699789241 \(2 \sin(x) \cos(x)\) = \(\frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)\)
2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
TODO

sympy not provided for expression
8706092970 \(d\) = \(\left(\frac{v + v_0}{2}\right)t\)
d = \left(\frac{v + v_0}{2}\right)t
TODO

sympy not provided for expression
8721295221 \(t_{\rm Earth\ orbit}\) = \(3.16 10^7 {\rm seconds}\)
t_{\rm Earth\ orbit} = 3.16 10^7 {\rm seconds}
TODO

sympy not provided for expression
8730201316 \(\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t\) = \(t'\)
\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'
first term was multiplied by \gamma/\gamma TODO

sympy not provided for expression
8747785338 \(\cos(i x)\) = \(\cosh(x)\)
\cos(i x) = \cosh(x)
TODO

sympy not provided for expression
8750379055 \(0\) = \(\frac{d}{dt} v_x\)
0 = \frac{d}{dt} v_x
TODO

sympy not provided for expression
8808860551 \(-g \int t dt + v_{0, y} \int dt\) = \(\int dy\)
-g \int t dt + v_{0, y} \int dt = \int dy
TODO

sympy not provided for expression
8849289982 \(\psi(x)^*\) = \(a \sin(\frac{n \pi}{W} x)\)
\psi(x)^* = a \sin(\frac{n \pi}{W} x)
TODO

sympy not provided for expression
8889444440 \(1\) = \(\int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right) \right)^2 dx\)
1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right) \right)^2 dx
TODO

sympy not provided for expression
8908736791 \(\rho\) = \(\frac{m}{a^3}\)
\rho = \frac{m}{a^3}
geometry TODO

sympy not provided for expression
8922441655 \(d\) = \(\frac{v_0^2}{g} \sin(2 \theta)\)
d = \frac{v_0^2}{g} \sin(2 \theta)
TODO

sympy not provided for expression
8945218208 \(\theta_{\rm Brewster} + \theta_{\rm refracted}\) = \(90^{\circ}\)
\theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ}
based on figure 34-27 on page 824 in \cite{2001_HRW} TODO

sympy not provided for expression
8946383937 \(v_{\rm escape}^2\) = \(2 G \frac{m}{r}\)
v_{\rm escape}^2 = 2 G \frac{m}{r}
TODO

sympy not provided for expression
8949329361 \(v_0 \sin(\theta)\) = \(v_{0, y}\)
v_0 \sin(\theta) = v_{0, y}
TODO

sympy not provided for expression
8953094349 \(W\) = \(m a x\)
W = m a x
TODO

sympy not provided for expression
8960645192 \(KE_2 + PE_2\) = \(KE_1 + PE_1\)
KE_2 + PE_2 = KE_1 + PE_1
TODO

sympy not provided for expression
8991236357 \(\frac{d^2 x}{dt^2}\) = \(-\frac{k}{m} x\)
\frac{d^2 x}{dt^2} = -\frac{k}{m} x
TODO

sympy not provided for expression
9031609275 \(x (1 - \gamma^2 )\) = \(- \gamma^2 v t + \gamma v t'\)
x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'
TODO

sympy not provided for expression
9059289981 \(\psi(x)\) = \(a \sin(k x)\)
\psi(x) = a \sin(k x)
TODO

sympy not provided for expression
9063568209 \(V_{\rm total}\) = \(V_1 + V_2\)
V_{\rm total} = V_1 + V_2
TODO

sympy not provided for expression
9070394000 \(m_2 d_2 \frac{4 \pi^2}{T^2}\) = \(G \frac{m_1 m_2}{r^2}\)
m_2 d_2 \frac{4 \pi^2}{T^2} = G \frac{m_1 m_2}{r^2}
TODO

sympy not provided for expression
9081138616 \(W_{\rm by\ system}\) = \(\frac{1}{2} m_1 v_{\rm final}^2\)
W_{\rm by\ system} = \frac{1}{2} m_1 v_{\rm final}^2
TODO

sympy not provided for expression
9112191201 \(y_f\) = \(0\)
y_f = 0
TODO

sympy not provided for expression
9152823411 \(\frac{1}{T^2}\) = \(\frac{1}{d_2 4 \pi^2} G \frac{m_1}{r^2}\)
\frac{1}{T^2} = \frac{1}{d_2 4 \pi^2} G \frac{m_1}{r^2}
TODO

sympy not provided for expression
9170048197 \(T^2\) = \(d_2 4 \pi^2 \frac{r^2}{G m_1}\)
T^2 = d_2 4 \pi^2 \frac{r^2}{G m_1}
TODO

sympy not provided for expression
9180861128 \(2 \sin(x) \cos(x)\) = \(\frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)\)
2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)
TODO

sympy not provided for expression
9191880568 \(Z Z^*\) = \(|Z| |Z| \exp( -i \theta ) \exp( i \theta )\)
Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )
TODO

sympy not provided for expression
9226945488 \(F\) = \(\frac{m v^2}{r}\)
F = \frac{m v^2}{r}
Centripetal force https://en.wikipedia.org/wiki/Centripetal_force TODO

sympy not provided for expression
9243879541 \(V\) = \(I_2 R_2\)
V = I_2 R_2
TODO

sympy not provided for expression
9262596735 \(d\) = \(2 \pi r\)
d = 2 \pi r
TODO

sympy not provided for expression
9285928292 \(ax^2 + bx + c\) = \(0\)
ax^2 + bx + c = 0
TODO

sympy not provided for expression
9291999979 \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E}\) = \(-\mu_0\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t}\)
\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t}
TODO

sympy not provided for expression
9294858532 \(\hat{A}^+\) = \(\hat{A}\)
\hat{A}^+ = \hat{A}
TODO

sympy not provided for expression
9337785146 \(v\) = \(\frac{x_2 - x_1}{t}\)
v = \frac{x_2 - x_1}{t}
average velocity TODO

sympy not provided for expression
9341391925 \(\vec{v}_0\) = \(v_{0, x} \hat{x} + v_{0, y} \hat{y}\)
\vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}
TODO

sympy not provided for expression
9356924046 \(\frac{KE_2 - KE_1}{t}\) = \(m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}\)
\frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}
TODO

sympy not provided for expression
9376481176 \(K\) = \(f \frac{E}{a^3}\)
K = f \frac{E}{a^3}
proportionality coefficient fvaries in the range 1-4 for a majority of elemental solids TODO

sympy not provided for expression
9385938295 \((x+(b/(2 a)))^2\) = \(-(c/a) + (b/(2 a))^2\)
(x+(b/(2 a)))^2 = -(c/a) + (b/(2 a))^2
TODO

sympy not provided for expression
9393939991 \(\psi(x)\) = \(-\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)\)
\psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
TODO

sympy not provided for expression
9393939992 \(\psi(x)\) = \(\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)\)
\psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
TODO

sympy not provided for expression
9394939493 \(\nabla^2 E( \vec{r},t)\) = \(\mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r},t)\)
\nabla^2 E( \vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r},t)
TODO

sympy not provided for expression
9397152918 \(v\) = \(\frac{v_1 + v_2}{2}\)
v = \frac{v_1 + v_2}{2}
average velocity TODO

sympy not provided for expression
9407192813 \(G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2}\) = \(m g_{\rm Earth}\)
G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2} = m g_{\rm Earth}
TODO

sympy not provided for expression
9409776983 \(x (1 - \gamma^2 ) + \gamma^2 v t\) = \(\gamma v t'\)
x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'
TODO

sympy not provided for expression
9412953728 \(v_{\rm escape}^2\) = \(2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}\)
v_{\rm escape}^2 = 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}
TODO

sympy not provided for expression
9413609246 \(\cosh^2 x - \sinh^2 x\) = \(1\)
\cosh^2 x - \sinh^2 x = 1
TODO

sympy not provided for expression
9413699705 \(W\) = \(m a \frac{v_2^2 - v_1^2}{2 a}\)
W = m a \frac{v_2^2 - v_1^2}{2 a}
TODO

sympy not provided for expression
9429829482 \(\frac{d}{dx} y\) = \(-\sin(x) + i\cos(x)\)
\frac{d}{dx} y = -\sin(x) + i\cos(x)
TODO

sympy not provided for expression
9440616166 \(m_{\rm Earth}\) = \(\frac{g_{\rm Earth} r_{\rm Earth}^2}{G}\)
m_{\rm Earth} = \frac{g_{\rm Earth} r_{\rm Earth}^2}{G}
TODO

sympy not provided for expression
9482113948 \(\frac{dy}{y}\) = \(i dx\)
\frac{dy}{y} = i dx
TODO

sympy not provided for expression
9482438243 \((\cos(x))^2\) = \(\cos(2 x) + (\sin(x))^2\)
(\cos(x))^2 = \cos(2 x) + (\sin(x))^2
TODO

sympy not provided for expression
9482923849 \(\exp(i x)\) = \(y\)
\exp(i x) = y
TODO

sympy not provided for expression
9482928242 \(\cos(2 x)\) = \((\cos(x))^2 - (\sin(x))^2\)
\cos(2 x) = (\cos(x))^2 - (\sin(x))^2
TODO

sympy not provided for expression
9482928243 \(\cos(2 x) + (\sin(x))^2\) = \((\cos(x))^2\)
\cos(2 x) + (\sin(x))^2 = (\cos(x))^2
TODO

sympy not provided for expression
9482943948 \(\log(y)\) = \(i dx\)
\log(y) = i dx
TODO

sympy not provided for expression
9482984922 \(\frac{d}{dx} y\) = \((i\sin(x) + \cos(x)) i\)
\frac{d}{dx} y = (i\sin(x) + \cos(x)) i
TODO

sympy not provided for expression
9483928192 \(\cos(2 x) + i\sin(2 x)\) = \((\cos(x))^2 + 2 i \cos(x) \sin(x) - (\sin(x))^2\)
\cos(2 x) + i\sin(2 x) = (\cos(x))^2 + 2 i \cos(x) \sin(x) - (\sin(x))^2
TODO

sympy not provided for expression
9485384858 \(\nabla^2 E( \vec{r})\exp(i \omega t)\) = \(- \frac{\omega^2}{c^2} E( \vec{r})\exp(i \omega t)\)
\nabla^2 E( \vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E( \vec{r})\exp(i \omega t)
TODO

sympy not provided for expression
9485747245 \(\sqrt{\frac{2}{W}}\) = \(a\)
\sqrt{\frac{2}{W}} = a
TODO

sympy not provided for expression
9485747246 \(-\sqrt{\frac{2}{W}}\) = \(a\)
-\sqrt{\frac{2}{W}} = a
TODO

sympy not provided for expression
9492920340 \(y\) = \(\cos(x)+i \sin(x)\)
y = \cos(x)+i \sin(x)
TODO

sympy not provided for expression
9495857278 \(\psi(x=W)\) = \(0\)
\psi(x=W) = 0
2022-03-25 BHP: Conversion between Latex and Sympy is incomplete TODO

sympy not provided for expression
9499428242 \(E( \vec{r},t)\) = \(E( \vec{r})\exp(i \omega t)\)
E( \vec{r},t) = E( \vec{r})\exp(i \omega t)
TODO

sympy not provided for expression
9510328252 \(KE_{\rm initial}\) = \(0\)
KE_{\rm initial} = 0
TODO

sympy not provided for expression
9562264720 \([S]\) = \(\frac{k_{\rm desorption} [A_{\rm adsorption}]}{k_{\rm adsorption} p_A}\)
[S] = \frac{k_{\rm desorption} [A_{\rm adsorption}]}{k_{\rm adsorption} p_A}
TODO

sympy not provided for expression
9582958293 \(x\) = \(\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))\)
x = \sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))
TODO

sympy not provided for expression
9582958294 \(x+(b/(2 a))\) = \(\sqrt{(b/(2 a))^2 - (c/a)}\)
x+(b/(2 a)) = \sqrt{(b/(2 a))^2 - (c/a)}
TODO

sympy not provided for expression
9585727710 \(\psi(x=0)\) = \(0\)
\psi(x=0) = 0
TODO

sympy not provided for expression
9596004948 \(x\) = \(\langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle\)
x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle
TODO

sympy not provided for expression
9640720571 \(v\) = \(\frac{e^2}{4 \pi \epsilon_0 \hbar} \sqrt{\frac{m_e}{2 m}}\)
v = \frac{e^2}{4 \pi \epsilon_0 \hbar} \sqrt{\frac{m_e}{2 m}}
TODO

sympy not provided for expression
9658195023 \(d\) = \(v_0 t + \frac{1}{2} a t^2\)
d = v_0 t + \frac{1}{2} a t^2
TODO

sympy not provided for expression
9703482302 \(G \frac{m_{\rm Earth} m}{r_{\rm Earth}}\) = \(\frac{1}{2} m v_{\rm escape}^2\)
G \frac{m_{\rm Earth} m}{r_{\rm Earth}} = \frac{1}{2} m v_{\rm escape}^2
TODO

sympy not provided for expression
9707028061 \(a_x\) = \(0\)
a_x = 0
TODO

sympy not provided for expression
9718685793 \(\kappa_T\) = \(\frac{1}{P}\)
\kappa_T = \frac{1}{P}
TODO

sympy not provided for expression
9749777192 \(0\) = \(KE_1 + PE_1\)
0 = KE_1 + PE_1
TODO

sympy not provided for expression
9756089533 \(\sin( \theta_{\rm Brewster} )\) = \(\frac{n_2}{n_1} \cos( \theta_{\rm Brewster} )\)
\sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} )
TODO

sympy not provided for expression
9759901995 \(v - v_0\) = \(a t\)
v - v_0 = a t
TODO

sympy not provided for expression
9781951738 \(\kappa_T\) = \(\frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T\)
\kappa_T = \frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T
definition of isothermal compressibility TODO

sympy not provided for expression
9805063945 \(\gamma^2 (x - v t)^2 + y^2 + z^2\) = \(c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2\)
\gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2
TODO

sympy not provided for expression
9838128064 \(d_2 \frac{4 \pi^2}{T^2}\) = \(G \frac{m_1}{r^2}\)
d_2 \frac{4 \pi^2}{T^2} = G \frac{m_1}{r^2}
TODO

sympy not provided for expression
9847143017 \(\kappa_T\) = \(\frac{-PV}{V} \left( \frac{-1}{P^2}\right)\)
\kappa_T = \frac{-PV}{V} \left( \frac{-1}{P^2}\right)
TODO

sympy not provided for expression
9848292229 \(dy\) = \(y i dx\)
dy = y i dx
TODO

sympy not provided for expression
9848294829 \(\frac{d}{dx} y\) = \(y i\)
\frac{d}{dx} y = y i
TODO

sympy not provided for expression
9854442418 \(v\) = \(\sqrt{\frac{E}{m}}\)
v = \sqrt{\frac{E}{m}}
TODO

sympy not provided for expression
9858028950 \(\frac{1}{a^2}\) = \(\int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx\)
\frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
TODO

sympy not provided for expression
9862900242 \(y\) = \(- \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0\)
y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0
TODO

sympy not provided for expression
9882526611 \(v_{0, x} t\) = \(x - x_0\)
v_{0, x} t = x - x_0
TODO

sympy not provided for expression
9889984281 \(2 (\sin(x))^2\) = \(1 - \cos(2 x)\)
2 (\sin(x))^2 = 1 - \cos(2 x)
TODO

sympy not provided for expression
9894826550 \(2 \sin(x) \cos(x)\) = \(\frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \left(\exp(i x)+\exp(-i x) \right)\)
2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \left(\exp(i x)+\exp(-i x) \right)
TODO

sympy not provided for expression
9897284307 \(\frac{d}{t}\) = \(\frac{v + v_0}{2}\)
\frac{d}{t} = \frac{v + v_0}{2}
TODO

sympy not provided for expression
9919999981 \(\rho\) = \(0\)
\rho = 0
TODO

sympy not provided for expression
9941599459 \(dU\) = \(\left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV\)
dU = \left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV
based on U(p, T, V) = U(T, V) TODO

sympy not provided for expression
9958485859 \(\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right)\) = \(i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)\)
\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)
TODO

sympy not provided for expression
9973952056 \(-g t\) = \(v_y - v_{0, y}\)
-g t = v_y - v_{0, y}
TODO

sympy not provided for expression
9988949211 \((\sin(x))^2\) = \(\frac{1 - \cos(2 x)}{2}\)
(\sin(x))^2 = \frac{1 - \cos(2 x)}{2}
TODO

sympy not provided for expression
9991999979 \(\vec{ \nabla} \times \vec{E}\) = \(-\mu_0\frac{\partial \vec{H}}{\partial t}\)
\vec{ \nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}
TODO

sympy not provided for expression
9999998870 \(\frac{ \vec{p}}{\hbar}\) = \(\vec{k}\)
\frac{ \vec{p}}{\hbar} = \vec{k}
TODO

sympy not provided for expression
9999999870 \(\frac{p}{\hbar}\) = \(k\)
\frac{p}{\hbar} = k
TODO

sympy not provided for expression
9999999960 \(\hbar\) = \(h/(2 \pi)\)
\hbar = h/(2 \pi)
TODO

sympy not provided for expression
9999999961 \(\frac{E}{\hbar}\) = \(\omega\)
\frac{E}{\hbar} = \omega
TODO

sympy not provided for expression
9999999962 \(p\) = \(\hbar k\)
p = \hbar k
TODO

sympy not provided for expression
9999999965 \(E\) = \(\omega \hbar\)
E = \omega \hbar
TODO

sympy not provided for expression
9999999968 \(x\) = \(\frac{-b-\sqrt{b^2-4ac}}{2 a}\)
x = \frac{-b-\sqrt{b^2-4ac}}{2 a}
TODO

sympy not provided for expression
9999999969 \(x\) = \(\frac{-b+\sqrt{b^2-4ac}}{2 a}\)
x = \frac{-b+\sqrt{b^2-4ac}}{2 a}
TODO

sympy not provided for expression
9999999975 \(\langle \psi| \hat{A} |\psi \rangle\) = \(\langle a \rangle\)
\langle \psi| \hat{A} |\psi \rangle = \langle a \rangle
TODO

sympy not provided for expression
9999999981 \(\vec{ \nabla} \cdot \vec{E}\) = \(\rho/\epsilon_0\)
\vec{ \nabla} \cdot \vec{E} = \rho/\epsilon_0
TODO

sympy not provided for expression
0203024440 \(1\) = \(\int_0^W a \sin\left(\frac{n \pi}{W} x\right) \psi(x)^* dx\)
1 = \int_0^W a \sin\left(\frac{n \pi}{W} x\right) \psi(x)^* dx
TODO

sympy not provided for expression
0404050504 \(\lambda\) = \(\frac{v}{f}\)
\lambda = \frac{v}{f}
TODO

sympy not provided for expression
0439492440 \(\frac{1}{a^2}\) = \(\frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) \right|_0^W\)
\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) \right|_0^W
https://physicsderivationgraph.blogspot.com/2020/09/evaluating-definite-integrals-for.html TODO

sympy not provided for expression
0934990943 \(k\) = \(\frac{2 \pi}{v T}\)
k = \frac{2 \pi}{v T}
TODO

sympy not provided for expression
0948572140 \(\int \cos(a x) dx\) = \(\frac{1}{a}\sin(a x)\)
\int \cos(a x) dx = \frac{1}{a}\sin(a x)
TODO

sympy not provided for expression
1010393913 \(\langle \psi| \hat{A}^+ |\psi \rangle\) = \(\langle a \rangle^*\)
\langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*
https://docs.sympy.org/latest/modules/stats.html TODO

sympy not provided for expression
1010393944 \(x\) = \(\langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle\)
x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle
TODO

sympy not provided for expression
1010923823 \(k W\) = \(n \pi\)
k W = n \pi
TODO

sympy not provided for expression
1020010291 \(0\) = \(a \sin(k W)\)
0 = a \sin(k W)
TODO

sympy not provided for expression
1020394900 \(p\) = \(h/\lambda\)
p = h/\lambda
TODO

sympy not provided for expression
1020394902 \(E\) = \(h f\)
E = h f
TODO

sympy not provided for expression
1020854560 \(I\) = \((A + B)(A + B)^*\)
I = (A + B)(A + B)^*
TODO

sympy not provided for expression
1029039903 \(p\) = \(m v\)
p = m v
TODO

sympy not provided for expression
1029039904 \(p^2\) = \(m^2 v^2\)
p^2 = m^2 v^2
TODO

sympy not provided for expression
1038566242 \(\sinh x\) = \(\frac{\exp(x) - \exp(-x)}{2}\)
\sinh x = \frac{\exp(x) - \exp(-x)}{2}
TODO

sympy not provided for expression
1085150613 \(C_V\) = \(\left(\frac{\partial U}{\partial T}\right)_V\)
C_V = \left(\frac{\partial U}{\partial T}\right)_V
definition of heat capacity at constant volume TODO

sympy not provided for expression
1087417579 \(0\) = \(- \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)\)
0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)
TODO

sympy not provided for expression
1114820451 \(W_{\rm by\ system}\) = \(\Delta KE\)
W_{\rm by\ system} = \Delta KE
Work is change in energy TODO

sympy not provided for expression
1128605625 \({\rm sech}^2\ x + \tanh^2(x)\) = \(\frac{4}{\left(\exp(x)+\exp(-x)\right)^2} + \frac{\left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2}\)
{\rm sech}^2\ x + \tanh^2(x) = \frac{4}{\left(\exp(x)+\exp(-x)\right)^2} + \frac{\left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2}
TODO

sympy not provided for expression
1132941271 \(m_{\rm Earth}\) = \(\frac{(9.80665 m/s^2) (6.3781*10^6 m)^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}}\)
m_{\rm Earth} = \frac{(9.80665 m/s^2) (6.3781*10^6 m)^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}}
TODO

sympy not provided for expression
1143343287 \(G \frac{m_{\rm Earth}}{r_{\rm Earth}}\) = \(\frac{1}{2} v_{\rm escape}^2\)
G \frac{m_{\rm Earth}}{r_{\rm Earth}} = \frac{1}{2} v_{\rm escape}^2
TODO

sympy not provided for expression
1158485859 \(\frac{-\hbar^2}{2m} \nabla^2\) = \({\cal H}\)
\frac{-\hbar^2}{2m} \nabla^2 = {\cal H}
TODO

sympy not provided for expression
1166310428 \(0 dt\) = \(d v_x\)
0 dt = d v_x
TODO

sympy not provided for expression
1172039918 \(I_{\rm coherent}\) = \(4 |A|^2\)
I_{\rm coherent} = 4 |A|^2
TODO

sympy not provided for expression
1190768176 \(\kappa_T\) = \(\frac{-nRT}{V} \left( \frac{ \partial }{\partial P}\left(\frac{1}{P}\right) \right)_T\)
\kappa_T = \frac{-nRT}{V} \left( \frac{ \partial }{\partial P}\left(\frac{1}{P}\right) \right)_T
TODO

sympy not provided for expression
1191796961 \(\frac{1}{2} g t_f\) = \(v_0 \sin(\theta)\)
\frac{1}{2} g t_f = v_0 \sin(\theta)
TODO

sympy not provided for expression
1201689765 \(x'^2 + y'^2 + z'^2\) = \(c^2 t'^2\)
x'^2 + y'^2 + z'^2 = c^2 t'^2
describes a spherical wavefront for an observer in a moving frame of reference TODO

sympy not provided for expression
1202310110 \(\frac{1}{a^2}\) = \(\int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx\)
\frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
TODO

sympy not provided for expression
1202312210 \(\frac{1}{a^2}\) = \(\frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx\)
\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
TODO

sympy not provided for expression
1203938249 \(a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle\) = \(a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle\)
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
TODO

sympy not provided for expression
1248277773 \(\cos(2 x)\) = \(1 - 2 (\sin(x))^2\)
\cos(2 x) = 1 - 2 (\sin(x))^2
TODO

sympy not provided for expression
1259826355 \(d\) = \((v - a t) t + \frac{1}{2} a t^2\)
d = (v - a t) t + \frac{1}{2} a t^2
TODO

sympy not provided for expression
1265150401 \(d\) = \(\frac{2 v_0 + a t}{2} t\)
d = \frac{2 v_0 + a t}{2} t
TODO

sympy not provided for expression
1292735067 \(F_{gravitational}\) = \(G \frac{m_1 m_2}{r^2}\)
F_{gravitational} = G \frac{m_1 m_2}{r^2}
TODO

sympy not provided for expression
1293913110 \(0\) = \(b\)
0 = b
TODO

sympy not provided for expression
1293923844 \(\lambda\) = \(v T\)
\lambda = v T
TODO

sympy not provided for expression
1306360899 \(x\) = \(v_{0, x} t + x_0\)
x = v_{0, x} t + x_0
TODO

sympy not provided for expression
1310571337 \(\theta_{\rm refracted}\) = \(90^{\circ} - \theta_{\rm Brewster}\)
\theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster}
TODO

sympy not provided for expression
1311403394 \(\alpha\) = \(\frac{1}{V} \frac{nR}{P} \left( \frac{\partial T}{\partial T} \right)_P\)
\alpha = \frac{1}{V} \frac{nR}{P} \left( \frac{\partial T}{\partial T} \right)_P
TODO

sympy not provided for expression
1314464131 \(\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t}\) = \(\epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\)
\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
TODO

sympy not provided for expression
1314864131 \(\vec{ \nabla} \times \vec{H}\) = \(\epsilon_0 \frac{\partial }{\partial t}\vec{E}\)
\vec{ \nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}
TODO

sympy not provided for expression
1330874553 \(v_{\rm escape}\) = \(\sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}\)
v_{\rm escape} = \sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}
TODO

sympy not provided for expression
1357848476 \(A\) = \(|A| \exp(i \theta)\)
A = |A| \exp(i \theta)
TODO

sympy not provided for expression
1395858355 \(x\) = \(\langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle\)
x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle
TODO

sympy not provided for expression
1405465835 \(y\) = \(- \frac{1}{2} g t^2 + v_{0, y} t + y_0\)
y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0
TODO

sympy not provided for expression
1457415749 \(\frac{1}{R_{\rm total}}\) = \(\frac{1}{R_1} + \frac{1}{R_2}\)
\frac{1}{R_{\rm total}} = \frac{1}{R_1} + \frac{1}{R_2}
total resistance for two resistors in parallel TODO

sympy not provided for expression
1525861537 \(I\) = \(|A|^2 + |B|^2 + A B^* + B A^*\)
I = |A|^2 + |B|^2 + A B^* + B A^*
TODO

sympy not provided for expression
1528310784 \(\gamma\) = \(\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\)
\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
TODO

sympy not provided for expression
1541916015 \(\theta\) = \(\frac{\pi}{4}\)
\theta = \frac{\pi}{4}
TODO

sympy not provided for expression
1556389363 \(E_{\rm Rydberg}\) = \(\frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}\)
E_{\rm Rydberg} = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}
the bonding energy in condensed phases is given by the Rydberg energy on the order of several e TODO

sympy not provided for expression
1559688463 \(\left(\frac{T_{\rm geostationary\ orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3}\) = \(r_{\rm geostationary\ orbit}\)
\left(\frac{T_{\rm geostationary\ orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r_{\rm geostationary\ orbit}
TODO

sympy not provided for expression
1586866563 \(\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right)\) = \(t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)\)
\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)
TODO

sympy not provided for expression
1590774089 \(dW\) = \(F dx\)
dW = F dx
TODO

sympy not provided for expression
1636453295 \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E}\) = \(- \nabla^2 \vec{E}\)
\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = - \nabla^2 \vec{E}
TODO

sympy not provided for expression
1638282134 \(\vec{p}_{\rm before}\) = \(\vec{p}_{\rm after}\)
\vec{p}_{\rm before} = \vec{p}_{\rm after}
TODO

sympy not provided for expression
1639827492 \(- c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2}\) = \(1\)
- c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1
TODO

sympy not provided for expression
1648958381 \(\nabla^2 \psi \left( \vec{r},t \right)\) = \(\frac{i}{\hbar} \vec{p} \cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right)\)
\nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right)
https://physicsderivationgraph.blogspot.com/2020/09/representing-laplace-operator-nabla-in.html TODO

sympy not provided for expression
1650441634 \(y_0\) = \(0\)
y_0 = 0
define coordinate system such that initial height is at origin TODO

sympy not provided for expression
1676472948 \(0\) = \(v_x - v_{0, x}\)
0 = v_x - v_{0, x}
TODO

sympy not provided for expression
1702349646 \(-g dt\) = \(d v_y\)
-g dt = d v_y
TODO

sympy not provided for expression
1772416655 \(\frac{E_2 - E_1}{t}\) = \(v F - F v\)
\frac{E_2 - E_1}{t} = v F - F v
TODO

sympy not provided for expression
1772973171 \(-\frac{k}{m} x\) = \(-A \omega^2 \cos(\omega t)\)
-\frac{k}{m} x = -A \omega^2 \cos(\omega t)
TODO

sympy not provided for expression
1784114349 \(\sqrt{\frac{k}{m}}\) = \(\omega\)
\sqrt{\frac{k}{m}} = \omega
TODO

sympy not provided for expression
1809909100 \(\frac{E_2 - E_1}{t}\) = \(0\)
\frac{E_2 - E_1}{t} = 0
TODO

sympy not provided for expression
1811867899 \(T^2\) = \(\frac{d_1+d_2}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1}\)
T^2 = \frac{d_1+d_2}{d_1+d_2} d_2 4 \pi^2 \frac{r^2}{G m_1}
TODO

sympy not provided for expression
1815398659 \(U\) = \(Q + W\)
U = Q + W
TODO

sympy not provided for expression
1819663717 \(a_x\) = \(\frac{d}{dt} v_x\)
a_x = \frac{d}{dt} v_x
TODO

sympy not provided for expression
1840080113 \(KE_2\) = \(0\)
KE_2 = 0
object is not moving at $x=\infty$ TODO

sympy not provided for expression
1857710291 \(0\) = \(a \sin(n \pi)\)
0 = a \sin(n \pi)
TODO

sympy not provided for expression
1858578388 \(\nabla^2 E( \vec{r})\exp(i \omega t)\) = \(- \omega^2 \mu_0 \epsilon_0 E( \vec{r})\exp(i \omega t)\)
\nabla^2 E( \vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E( \vec{r})\exp(i \omega t)
https://physicsderivationgraph.blogspot.com/2020/09/representing-laplace-operator-nabla-in.html TODO

sympy not provided for expression
1858772113 \(k\) = \(\frac{n \pi}{W}\)
k = \frac{n \pi}{W}
TODO

sympy not provided for expression
1888494137 \(-\sqrt{\frac{k}{m}}\) = \(\omega\)
-\sqrt{\frac{k}{m}} = \omega
TODO

sympy not provided for expression
1916173354 \(-\gamma^2 v^2 + c^2 \gamma^2\) = \(c^2\)
-\gamma^2 v^2 + c^2 \gamma^2 = c^2
TODO

sympy not provided for expression
1928085940 \(Z^*\) = \(|Z| \exp( -i \theta )\)
Z^* = |Z| \exp( -i \theta )
TODO

sympy not provided for expression
1931103031 \(\frac{k}{m}\) = \(\omega^2\)
\frac{k}{m} = \omega^2
TODO

sympy not provided for expression
1934748140 \(\int |\psi(x)|^2 dx\) = \(1\)
\int |\psi(x)|^2 dx = 1
TODO

sympy not provided for expression
1935543849 \(\gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2\) = \(c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2\)
\gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2
TODO

sympy not provided for expression
1963253044 \(v_{0, x} dt\) = \(dx\)
v_{0, x} dt = dx
TODO

sympy not provided for expression
1967582749 \(t\) = \(\frac{v - v_0}{a}\)
t = \frac{v - v_0}{a}
TODO

sympy not provided for expression
1974334644 \(\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v}\) = \(t'\)
\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'
TODO

sympy not provided for expression
1977955751 \(-g\) = \(\frac{d}{dt} v_y\)
-g = \frac{d}{dt} v_y
TODO

sympy not provided for expression
1994296484 \(v_{\rm satellite}^2\) = \(G \frac{m_{\rm Earth}}{r}\)
v_{\rm satellite}^2 = G \frac{m_{\rm Earth}}{r}
TODO

sympy not provided for expression
2005061870 \(v(r)\) = \(\sqrt{\frac{2 G m_2}{r}}\)
v(r) = \sqrt{\frac{2 G m_2}{r}}
TODO

sympy not provided for expression
2029293929 \(\nabla^2 E( \vec{r})\exp(i \omega t)\) = \(\mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r})\exp(i \omega t)\)
\nabla^2 E( \vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r})\exp(i \omega t)
https://physicsderivationgraph.blogspot.com/2020/09/representing-laplace-operator-nabla-in.html TODO

sympy not provided for expression
2042298788 \(0\) = \(-G \frac{m_{\rm Earth} m}{r_{\rm Earth}} + \frac{1}{2} m v_{\rm escape}^2\)
0 = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}} + \frac{1}{2} m v_{\rm escape}^2
TODO

sympy not provided for expression
2051901211 \(\frac{V}{R_1}\) = \(I_1\)
\frac{V}{R_1} = I_1
TODO

sympy not provided for expression
2061086175 \(W_{\rm to\ system}\) = \(-G m_1 m_2 \left(\frac{-1}{r} - \frac{-1}{\infty}\right)\)
W_{\rm to\ system} = -G m_1 m_2 \left(\frac{-1}{r} - \frac{-1}{\infty}\right)
TODO

sympy not provided for expression
2076171250 \(-\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v}\) = \(0\)
-\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0
TODO

sympy not provided for expression
2086924031 \(0\) = \(- \frac{1}{2} g t_f + v_0 \sin(\theta)\)
0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)
TODO

sympy not provided for expression
2096918413 \(x\) = \(\gamma ( \gamma x - \gamma v t + v t' )\)
x = \gamma ( \gamma x - \gamma v t + v t' )
TODO

sympy not provided for expression
2103023049 \(\sin(x)\) = \(\frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\)
\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)
TODO

sympy not provided for expression
2113211456 \(f\) = \(1/T\)
f = 1/T
TODO

sympy not provided for expression
2114909846 \(\theta_A\) = \(\frac{[A_{\rm adsorption}]}{[S_0]}\)
\theta_A = \frac{[A_{\rm adsorption}]}{[S_0]}
TODO

sympy not provided for expression
2121790783 \(\tanh^2(x)\) = \(\frac{ \left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2}\)
\tanh^2(x) = \frac{ \left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2}
TODO

sympy not provided for expression
2123139121 \(-\exp(-i x)\) = \(-\cos(x)+i \sin(x)\)
-\exp(-i x) = -\cos(x)+i \sin(x)
TODO

sympy not provided for expression
2131616531 \(T f\) = \(1\)
T f = 1
TODO

sympy not provided for expression
2148049269 \(-\frac{k}{m} A \cos(\omega t)\) = \(-A \omega^2 \cos(\omega t)\)
-\frac{k}{m} A \cos(\omega t) = -A \omega^2 \cos(\omega t)
TODO

sympy not provided for expression
2168306601 \([S_0]\) = \(\left(\frac{k_{\rm desorption}}{k_{\rm adsorption}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}]\)
[S_0] = \left(\frac{k_{\rm desorption}}{k_{\rm adsorption}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}]
TODO

sympy not provided for expression
2186083170 \(\frac{KE_2 - KE_1}{t}\) = \(v F\)
\frac{KE_2 - KE_1}{t} = v F
TODO

sympy not provided for expression
2217103163 \(\frac{m_1 d_1}{d_2}\) = \(m_2\)
\frac{m_1 d_1}{d_2} = m_2
TODO

sympy not provided for expression
2236639474 \((A + B)(A + B)^*\) = \(|A + B|^2\)
(A + B)(A + B)^* = |A + B|^2
TODO

sympy not provided for expression
2257410739 \(\left(\frac{\partial U}{\partial T}\right)_p\) = \(C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alpha\)
\left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alpha
TODO

sympy not provided for expression
2258485859 \({\cal H} \psi \left( \vec{r},t \right)\) = \(i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)\)
{\cal H} \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)
TODO

sympy not provided for expression
2267521164 \(PE_2\) = \(0\)
PE_2 = 0
object goes to $\infty$ away from gravitational source TODO

sympy not provided for expression
2271186630 \(V\) = \(I_{\rm total} R_{\rm total}\)
V = I_{\rm total} R_{\rm total}
TODO

sympy not provided for expression
2297105551 \(d\) = \(v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)\)
d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)
TODO

sympy not provided for expression
2308660627 \(G \frac{m_{\rm Earth}}{r_{\rm Earth}^2}\) = \(g_{\rm Earth}\)
G \frac{m_{\rm Earth}}{r_{\rm Earth}^2} = g_{\rm Earth}
TODO

sympy not provided for expression
2334518266 \(m a\) = \(-k x\)
m a = -k x
TODO

sympy not provided for expression
2366691988 \(\int 0 dt\) = \(\int d v_x\)
\int 0 dt = \int d v_x
TODO

sympy not provided for expression
2378095808 \(x_f\) = \(x_0 + d\)
x_f = x_0 + d
TODO

sympy not provided for expression
2394240499 \(x\) = \(a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle\)
x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle
TODO

sympy not provided for expression
2394853829 \(\exp(-i x)\) = \(\cos(-x)+i \sin(-x)\)
\exp(-i x) = \cos(-x)+i \sin(-x)
TODO

sympy not provided for expression
2394935831 \(( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle\) = \(0\)
( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
TODO

sympy not provided for expression
2394935835 \(\left(\langle\psi| \hat{A} |\psi \rangle \right)^+\) = \(\left(\langle a \rangle\right)^+\)
\left(\langle\psi| \hat{A} |\psi \rangle \right)^+ = \left(\langle a \rangle\right)^+
TODO

sympy not provided for expression
2395958385 \(\nabla^2 \psi \left( \vec{r},t \right)\) = \(\frac{-p^2}{\hbar} \psi( \vec{r},t)\)
\nabla^2 \psi \left( \vec{r},t \right) = \frac{-p^2}{\hbar} \psi( \vec{r},t)
https://physicsderivationgraph.blogspot.com/2020/09/representing-laplace-operator-nabla-in.html TODO

sympy not provided for expression
2404934990 \(\langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2\) = \(\langle x^2 \rangle-\langle x \rangle^2\)
\langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
TODO

sympy not provided for expression
2405307372 \(\sin(2 x)\) = \(2 \sin(x) \cos(x)\)
\sin(2 x) = 2 \sin(x) \cos(x)
TODO

sympy not provided for expression
2417941373 \(- c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4}\) = \(1 - \gamma^2\)
- c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2
TODO

sympy not provided for expression

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