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expression ID Latex list of symbols name notes used in derivation
2344320475 E_2
7939947931 KE_2
5803210729 PE_2
4213426349 E_1
1092872200 KE_1
8061701434 PE_1
6838659900 KE_2
5011888122 v_2
6964468708 KE_1
3105350101 v_1
5239755033 v_1
4319470443 v_2
4522137851 PE_2
4755369593 x_2
6749533119 PE_1
1552869972 x_1
6353701615 \theta_{\rm refracted}
9029795851 \theta_{\rm Brewster}
6239815585 C_{\rm Earth\ orbit}
4202292449 r_{\rm Earth\ orbit}
7708501762 C_{\rm Earth\ orbit}
9601500174 v_{\rm Earth\ orbit}
4470433702 t_{\rm Earth\ orbit}
4583868070 B
8120663858 y_f
4162188238 t_f
5194141542 x_f
6463266449 t_f
8111389082 x
6701855578 v_2
3183197515 v_1
9072369552 m_{\rm Earth}
7140470627 m
8416464049 KE_{\rm escape}
8871333437 PE_{\rm Earth\ surface}
9370882921 KE_{\rm escape}
6498985149 v_{\rm escape}
2135482543 m
8020058613 r
1238593037 c
9933742680 r_{\rm Schwarzschild}
9830343096 V_1
9174439158 R_1
1608399874 V_2
1484794622 R_2
1323602089 I_1
5258419993 R_1
5463275819 I_2
9746066299 R_2
7798615279 I_{\rm total}
7410124465 R_{\rm total}
3594626260 F_{\rm gravity}
4153613253 m_{\rm Earth}
3486213448 m_{\rm satellite}
7819443873 r
1333474099 F_{\rm centripetal}
2114570475 m_{\rm satellite}
9789485295 v_{\rm satellite}
5208737840 T_{\rm geostationary\ orbit}
7053449926 r_{\rm geostationary\ orbit}
1193980495 m_{\rm Earth}
9903988330 m
6535639720 r_{\rm Earth}
0000040490 a^2
0000999900 b/(2 a)
0001030901 \cos(x)
0001111111 (\sin(x))^2
0001209482 2 \pi
0001304952 \hbar
0001334112 W
0001921933 2 i
0002239424 2
0002338514 \vec{p}_{2}
0002342425 m/m
0002393922 x
0002424922 a
0002436656 i \hbar
0002449291 b/(2 a)
0002838490 b/(2 a)
0002919191 \sin(-x)
0002929944 1/2
0002940021 2 \pi
0003232242 t
0003413423 \cos(-x)
0003747849 -1
0003838111 2
0003919391 x
0003949052 -x
0003949921 \hbar
0003954314 dx
0003981813 -\sin(x)
0004089571 2 x
0004264724 y
0004307451 (b/(2 a))^2
0004582412 x
0004829194 2
0004831494 a
0004849392 x
0004858592 h
0004934845 x
0004948585 a
0005395034 a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
0005626421 t
0005749291 f
0005938585 \frac{-\hbar^2}{2m}
0006466214 (\sin(x))^2
0006544644 t
0006563727 x
0006644853 c/a
0006656532 e
0007471778 2(\sin(x))^2
0007563791 i
0007636749 x
0007894942 (\sin(x))^2
0008837284 T
0008842811 \cos(2 x)
0009458842 \psi(x)
0009484724 \frac{n \pi}{W}x
0009485857 a^2\frac{2}{W}
0009485858 \frac{2n\pi}{W}
0009492929 v du
0009587738 \psi
0009877781 y
0203024440 1 = \int_0^W a \sin\left(\frac{n \pi}{W} x\right) \psi(x)^* dx
0404050504 \lambda = \frac{v}{f}
0439492440 \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) \right|_0^W evaluating-definite-integrals-for.html
0934990943 k = \frac{2 \pi}{v T}
0948572140 \int \cos(a x) dx = \frac{1}{a}\sin(a x)
1010393913 \langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^* stats.html
1010393944 x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle
1010923823 k W = n \pi
1020010291 0 = a \sin(k W)
1020394900 p = h/\lambda
1020394902 E = h f
1029039903 p = m v
1029039904 p^2 = m^2 v^2
1158485859 \frac{-\hbar^2}{2m} \nabla^2 = {\cal H}
1202310110 \frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
1202312210 \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
1203938249 a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
1248277773 \cos(2 x) = 1 - 2 (\sin(x))^2
1293913110 0 = b
1293923844 \lambda = v T
1314464131 \vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
1314864131 \vec{ \nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}
1395858355 x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle
1636453295 \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = - \nabla^2 \vec{E}
1638282134 \vec{p}_{\rm before} = \vec{p}_{\rm after}
1648958381 \nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right) representing-laplace-operator-nabla-in.html
1857710291 0 = a \sin(n \pi)
1858578388 \nabla^2 E( \vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E( \vec{r})\exp(i \omega t) representing-laplace-operator-nabla-in.html
1858772113 k = \frac{n \pi}{W}
1934748140 \int |\psi(x)|^2 dx = 1
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) representing-laplace-operator-nabla-in.html
2103023049 \sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)
2113211456 f = 1/T
2123139121 -\exp(-i x) = -\cos(x)+i \sin(x)
2131616531 T f = 1
2258485859 {\cal H} \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)
2394240499 x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle
2394853829 \exp(-i x) = \cos(-x)+i \sin(-x)
2394935831 ( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
2394935835 \left(\langle\psi| \hat{A} |\psi \rangle \right)^+ = \left(\langle a \rangle\right)^+
2395958385 \nabla^2 \psi \left( \vec{r},t \right) = \frac{-p^2}{\hbar} \psi( \vec{r},t) representing-laplace-operator-nabla-in.html
2404934990 \langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
2494533900 \langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
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)
2848934890 \langle a \rangle^* = \langle a \rangle
2944838499 \psi(x) = a \sin(\frac{n \pi}{W} x)
3121234211 \frac{k}{2\pi} = \lambda
3121234212 p = \frac{h k}{2\pi}
3121513111 k = \frac{2 \pi}{\lambda}
3131111133 T = 1 / f
3131211131 \omega = 2 \pi f
3132131132 \omega = \frac{2\pi}{T}
3147472131 \frac{\omega}{2 \pi} = f
3285732911 (\cos(x))^2 = 1-(\sin(x))^2
3485475729 \nabla^2 E( \vec{r}) = - \frac{\omega^2}{c^2} E( \vec{r}) representing-laplace-operator-nabla-in.html
3585845894 \langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
3829492824 \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) = \cos(x)
3924948349 a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
3942849294 \exp(i x)-\exp(-i x) = 2 i \sin(x)
3943939590 x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle
3947269979 \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
3948571256 \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}E \psi( \vec{r},t)
3948574224 \psi( \vec{r},t) = \psi_0 \exp\left(i\left( \vec{k}\cdot\vec{r} - \omega t \right) \right)
3948574226 \psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)
3948574228 \psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
3948574230 \psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)
3948574233 \frac{\partial}{\partial t} \psi( \vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
3951205425 \vec{p}_{\rm after} = \vec{p}_{1}
4147472132 E = \frac{h \omega}{2 \pi}
4298359835 E = \frac{1}{2}m v^2
4298359845 E = \frac{1}{2m}m^2 v^2
4298359851 E = \frac{p^2}{2m}
4341171256 i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{p^2}{2 m} \psi( \vec{r},t)
4348571256 \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi( \vec{r},t)
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)
4585828572 \epsilon_0 \mu_0 = \frac{1}{c^2}
4585932229 \cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)
4598294821 \exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
4638429483 \exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
4742644828 \exp(i x)+\exp(-i x) = 2 \cos(x)
4827492911 \cos(2 x)+(\sin(x))^2 = 1 - (\sin(x))^2
4838429483 \exp(2 i x) = \cos(2 x)+i \sin(2 x)
4843995999 \frac{1}{2 i}\left(\exp(i x)-\exp(-i x) \right) = \sin(x)
4857472413 1 = \int \psi(x)\psi(x)^* dx
4857475848 \frac{1}{a^2} = \frac{W}{2}
4923339482 i x = \log(y)
4928239482 \log(y) = i x
4938429482 \exp(-i x) = \cos(x)+i \sin(-x)
4938429483 \exp(i x) = \cos(x)+i \sin(x)
4938429484 \exp(-i x) = \cos(x)-i \sin(x)
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)
4948934890 \langle \psi| \hat{A} |\psi \rangle = \langle a \rangle^*
4949359835 \langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
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) representing-laplace-operator-nabla-in.html
5530148480 \vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}
5727578862 \frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)
5832984291 (\sin(x))^2 + (\cos(x))^2 = 1
5857434758 \int a dx = a x
5868688585 \frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = \frac{p^2}{2m} \psi( \vec{r},t) representing-laplace-operator-nabla-in.html
5900595848 k = \frac{\omega}{v}
5928285821 x^2 + 2 x (b/(2 a)) + (b/(2 a))^2 = (x + (b/(2 a)))^2
5928292841 x^2 + (b/a)x + (b/(2 a))^2 = -c/a + (b/(2 a))^2
5938459282 x^2 + (b/a)x = -c/a
5958392859 x^2 + (b/a)x+(c/a) = 0
5959282914 x^2 + x(b/a) + (b/(2 a))^2 = (x+(b/(2 a)))^2
5982958248 x = -\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))
5982958249 x+(b/(2 a)) = -\sqrt{(b/(2 a))^2 - (c/a)}
5985371230 \vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi( \vec{r},t)
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})
7466829492 \vec{ \nabla} \cdot \vec{E} = 0
7564894985 \int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)
7572664728 \cos(2 x) + 2 (\sin(x))^2 = 1
7575738420 \left(\sin\left(\frac{n \pi}{W}x\right) \right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}
7575859295 \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859300 \epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{ \nabla} \times \vec{E} )_k = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859302 \epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859304 \epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} Covariance_and_contravariance_of_vectors
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}) Covariance_and_contravariance_of_vectors
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}) Covariance_and_contravariance_of_vectors
7575859310 \hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859312 \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7917051060 \vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}
8139187332 \vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}
8257621077 \vec{p}_{\rm before} = \vec{p}_{1}
8311458118 \vec{p}_{\rm after} = \vec{p}_{2}+\vec{p}_{electron}
8399484849 \langle x^2 - 2 x \langle x \rangle + \langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
8484544728 -a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)
8485757728 a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)
8485867742 \frac{2}{W} = a^2
8489593958 d(u v) = u dv + v du
8489593960 d(u v) - v du = u dv
8489593962 u dv = d(u v) - v du
8489593964 \int u dv = u v - \int v du
8494839423 \nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
8572657110 1 = \int |\psi(x)|^2 dx
8572852424 \vec{E} = E( \vec{r},t)
8575746378 \int \frac{1}{2} dx = \frac{1}{2} x
8575748999 \frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x) \right) = -k^2 \left(a \sin(kx) + b \cos(kx) \right)
8576785890 1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
8577275751 0 = a \sin(0) + b\cos(0)
8582885111 \psi(x) = a \sin(kx) + b \cos(kx)
8582954722 x^2 + 2 x h + h^2 = (x + h)^2
8849289982 \psi(x)^* = a \sin(\frac{n \pi}{W} x)
8889444440 1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right) \right)^2 dx
9059289981 \psi(x) = a \sin(k x)
9285928292 ax^2 + bx + c = 0
9291999979 \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t}
9294858532 \hat{A}^+ = \hat{A}
9385938295 (x+(b/(2 a)))^2 = -(c/a) + (b/(2 a))^2
9393939991 \psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
9393939992 \psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
9394939493 \nabla^2 E( \vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r},t)
9429829482 \frac{d}{dx} y = -\sin(x) + i\cos(x)
9482113948 \frac{dy}{y} = i dx
9482438243 (\cos(x))^2 = \cos(2 x) + (\sin(x))^2
9482923849 \exp(i x) = y
9482928242 \cos(2 x) = (\cos(x))^2 - (\sin(x))^2
9482928243 \cos(2 x) + (\sin(x))^2 = (\cos(x))^2
9482943948 \log(y) = i dx
9482984922 \frac{d}{dx} y = (i\sin(x) + \cos(x)) i
9483928192 \cos(2 x) + i\sin(2 x) = (\cos(x))^2 + 2 i \cos(x) \sin(x) - (\sin(x))^2
9485384858 \nabla^2 E( \vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E( \vec{r})\exp(i \omega t)
9485747245 \sqrt{\frac{2}{W}} = a
9485747246 -\sqrt{\frac{2}{W}} = a
9492920340 y = \cos(x)+i \sin(x)
9495857278 \psi(x=W) = 0
9499428242 E( \vec{r},t) = E( \vec{r})\exp(i \omega t)
9582958293 x = \sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))
9582958294 x+(b/(2 a)) = \sqrt{(b/(2 a))^2 - (c/a)}
9585727710 \psi(x=0) = 0
9596004948 x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle
9848292229 dy = y i dx
9848294829 \frac{d}{dx} y = y i
9858028950 \frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
9889984281 2 (\sin(x))^2 = 1 - \cos(2 x)
9919999981 \rho = 0
9958485859 \frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)
9988949211 (\sin(x))^2 = \frac{1 - \cos(2 x)}{2}
9991999979 \vec{ \nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}
9999998870 \frac{ \vec{p}}{\hbar} = \vec{k}
9999999870 \frac{p}{\hbar} = k
9999999960 \hbar = h/(2 \pi)
9999999961 \frac{E}{\hbar} = \omega
9999999962 p = \hbar k
9999999965 E = \omega \hbar
9999999968 x = \frac{-b-\sqrt{b^2-4ac}}{2 a}
9999999969 x = \frac{-b+\sqrt{b^2-4ac}}{2 a}
9999999975 \langle \psi| \hat{A} |\psi \rangle = \langle a \rangle
9999999981 \vec{ \nabla} \cdot \vec{E} = \rho/\epsilon_0
4961662865 x
9110536742 2 x
8483686863 \sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x) \right)
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)
8642992037 2
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)
8699789241 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
9180861128 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)
2405307372 \sin(2 x) = 2 \sin(x) \cos(x)
3268645065 x
9350663581 \pi
8332931442 \exp(i \pi) = \cos(\pi)+i \sin(\pi)
6885625907 \exp(i \pi) = -1 + i 0
3331824625 \exp(i \pi) = -1
4901237716 1
2501591100 \exp(i \pi) + 1 = 0
5136652623 E = KE + PE mechanical energy is the sum of the potential plus kinetic energies
7875206161 E_2 = KE_2 + PE_2
4303372136 E_1 = KE_1 + PE_1
5514556106 E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)
8357234146 KE = \frac{1}{2} m v^2 kinetic energy Kinetic_energy
7676652285 KE_2 = \frac{1}{2} m v_2^2
4928007622 KE_1 = \frac{1}{2} m v_1^2
5733146966 KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)
6554292307 t
4270680309 \frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}
5781981178 x^2 - y^2 = (x+y)(x-y) difference of squares Difference_of_two_squares
4648451961 v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)
9356924046 \frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}
9397152918 v = \frac{v_1 + v_2}{2} average velocity
2857430695 a = \frac{v_2 - v_1}{t} acceleration
7735737409 \frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}
4784793837 \frac{KE_2 - KE_1}{t} = m v a
5345738321 F = m a Newton's second law of motion Newton%27s_laws_of_motion#Newton's_second_law
2186083170 \frac{KE_2 - KE_1}{t} = v F
6715248283 PE = -F x potential energy Potential_energy
2431507955 PE_2 = -F x_2
4669290568 PE_1 = -F x_1
7734996511 PE_2 - PE_1 = -F ( x_2 - x_1 )
2016063530 t
7267155233 \frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)
9337785146 v = \frac{x_2 - x_1}{t} average velocity
4872970974 \frac{PE_2 - PE_1}{t} = -F v
2081689540 t
2770069250 \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}
3591237106 \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v
1772416655 \frac{E_2 - E_1}{t} = v F - F v
1809909100 \frac{E_2 - E_1}{t} = 0
5778176146 t
3806977900 E_2 - E_1 = 0
5960438249 E_1
8558338742 E_2 = E_1 conservation of energy Conservation_of_energy
8945218208 \theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ} based on figure 34-27 on page 824 in \cite{2001_HRW}
9025853427 \theta_{\rm Brewster}
1310571337 \theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster}
6450985774 n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 ) Law of Refraction eq 34-44 on page 819 in \cite{2001_HRW}
2575937347 n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} )
7696214507 n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} )
8588429722 \sin( 90^{\circ} - x ) = \cos( x )
7375348852 \theta_{\rm Brewster}
1512581563 x
6831637424 \sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} )
3061811650 n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} )
7857757625 n_1
9756089533 \sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} )
5632428182 \cos( \theta_{\rm Brewster} )
2768857871 \frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1}
4968680693 \tan( x ) = \frac{ \sin( x )}{\cos( x )}
7321695558 \theta_{\rm Brewster}
9906920183 x
4501377629 \tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}
3417126140 \tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 }
5453995431 \arctan{ x }
6023986360 x
8495187962 \theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }
9040079362 f
9565166889 T
5426308937 v = \frac{d}{t}
7476820482 C
1277713901 d
6946088325 v = \frac{C}{t}
6785303857 C = 2 \pi r
6348260313 C_{\rm Earth\ orbit} = 2 \pi r_{\rm Earth\ orbit}
3046191961 v_{\rm Earth\ orbit} = \frac{C_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}}
3080027960 v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}}
7175416299 t_{\rm Earth\ orbit} = 1 {\rm year}
3219318145 \frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}
8721295221 t_{\rm Earth\ orbit} = 3.16 10^7 {\rm seconds}
4593428198 v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{3.16\ 10^7 {\rm seconds}}
3472836147 r_{\rm Earth\ orbit} = 1.496\ 10^8 {\rm km}
6998364753 v_{\rm Earth\ orbit} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}
4180845508 v_{\rm Earth\ orbit} = 29.8 \frac{{\rm km}}{{\rm sec}}
4662369843 x' = \gamma (x - v t)
2983053062 x = \gamma (x' + v t')
3426941928 x = \gamma ( \gamma (x - v t) + v t' )
2096918413 x = \gamma ( \gamma x - \gamma v t + v t' )
7741202861 x = \gamma^2 x - \gamma^2 v t + \gamma v t'
7337056406 \gamma^2 x
4139999399 x - \gamma^2 x = - \gamma^2 v t + \gamma v t'
3495403335 x
9031609275 x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'
8014566709 \gamma^2 v t
9409776983 x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'
2226340358 \gamma v
8730201316 \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t' first term was multiplied by \gamma/\gamma
1974334644 \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'
5148266645 t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t
4287102261 x^2 + y^2 + z^2 = c^2 t^2 describes a spherical wavefront
1201689765 x'^2 + y'^2 + z'^2 = c^2 t'^2 describes a spherical wavefront for an observer in a moving frame of reference
7057864873 y' = y frame of reference is moving only along x direction
8515803375 z' = z frame of reference is moving only along x direction
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
4057686137 C
2346150725 r
9753878784 v
8135396036 t
2773628333 \theta_1
7154592211 \theta_2
3749492596 E
1258245373 E
4147101187 KE
3809726424 PE
6383056612 KE
5075406409 PE
5398681503 v
5398681502 v
9305761407 v
4218009993 x
4188639044 x
8173074178 x
1025759423 y
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
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)
3182633789 \gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1
1916173354 -\gamma^2 v^2 + c^2 \gamma^2 = c^2
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
5284610349 \gamma^2
2417941373 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2
5787469164 1 - \gamma^2
1639827492 - c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1
5669500954 v^2 \gamma^2
5763749235 -c^2 + c^2 \gamma^2 = v^2 \gamma^2
6408214498 c^2
2999795755 c^2 \gamma^2 = v^2 \gamma^2 + c^2
3412946408 v^2 \gamma^2
2542420160 c^2 \gamma^2 - v^2 \gamma^2 = c^2
7743841045 \gamma^2
7513513483 \gamma^2 (c^2 - v^2) = c^2
8571466509 c^2 - \gamma^2
7906112355 \gamma^2 = \frac{c^2}{c^2 - \gamma^2}
1528310784 \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
8360117126 \gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}} not a physically valid result in this context
3366703541 a = \frac{v - v_0}{t} acceleration is the average change in speed over a duration
7083390553 t
4748157455 a t = v - v_0
6417359412 v_0
4798787814 a t + v_0 = v
3462972452 v = v_0 + a t
3411994811 v_{\rm average} = \frac{d}{t}
6175547907 v_{\rm average} = \frac{v + v_0}{2}
9897284307 \frac{d}{t} = \frac{v + v_0}{2}
8865085668 t
8706092970 d = \left(\frac{v + v_0}{2}\right)t
7011114072 d = \frac{(v_0 + a t) + v_0}{2} t
1265150401 d = \frac{2 v_0 + a t}{2} t
9658195023 d = v_0 t + \frac{1}{2} a t^2
5799753649 2
7215099603 v^2 = v_0^2 + 2 a t v_0 + a^2 t^2
5144263777 v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)
7939765107 v^2 = v_0^2 + 2 a d
6729698807 v_0
9759901995 v - v_0 = a t
2242144313 a
1967582749 t = \frac{v - v_0}{a}
5733721198 d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)
5611024898 d = \frac{1}{2 a} (v^2 - v_0^2)
5542390646 2 a
8269198922 2 a d = v^2 - v_0^2
9070454719 v_0^2
4948763856 2 a d + v_0^2 = v^2
9645178657 a t
6457044853 v - a t = v_0
1259826355 d = (v - a t) t + \frac{1}{2} a t^2
4580545876 d = v t - a t^2 + \frac{1}{2} a t^2
6421241247 d = v t - \frac{1}{2} a t^2
4182362050 Z = |Z| \exp( i \theta ) Z \in \mathbb{C}
1928085940 Z^* = |Z| \exp( -i \theta )
9191880568 Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )
3350830826 Z Z^* = |Z|^2
8396997949 I = | A + B |^2 intensity of two waves traveling opposite directions on same path
4437214608 Z
5623794884 A + B
2236639474 (A + B)(A + B)^* = |A + B|^2
1020854560 I = (A + B)(A + B)^*
6306552185 I = (A + B)(A^* + B^*)
8065128065 I = A A^* + B B^* + A B^* + B A^*
9761485403 Z
8710504862 A
4075539836 A A^* = |A|^2
6529120965 B
1511199318 Z
7107090465 B B^* = |B|^2
5125940051 I = |A|^2 + B B^* + A B^* + B A^*
1525861537 I = |A|^2 + |B|^2 + A B^* + B A^*
1742775076 Z
7607271250 \theta
4192519596 B = |B| \exp(i \phi)
2064205392 A
1894894315 Z
1357848476 A = |A| \exp(i \theta)
6555185548 A^* = |A| \exp(-i \theta)
4504256452 B^* = |B| \exp(-i \phi)
7621705408 I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)
3085575328 I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))
4935235303 x
2293352649 \theta - \phi
3660957533 \cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
3967985562 2
2700934933 2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
8497631728 I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )
6774684564 \theta = \phi for coherent waves
2719691582 |A| = |B| in a loop
8283354808 I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
8046208134 I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2
1172039918 I_{\rm coherent} = 4 |A|^2
8602221482 \langle \cos(\theta - \phi) \rangle = 0 incoherent light source
6240206408 I_{\rm incoherent} = |A|^2 + |B|^2
6529793063 I_{\rm incoherent} = |A|^2 + |A|^2
3060393541 I_{\rm incoherent} = 2|A|^2
6556875579 \frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2
3169580383 \vec{a} = \frac{d\vec{v}}{dt} acceleration is the change in speed over a duration
8602512487 \vec{a} = a_x \hat{x} + a_y \hat{y} decompose acceleration into two components
4158986868 a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}
5349866551 \vec{v} = v_x \hat{x} + v_y \hat{y}
7729413831 a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)
1819663717 a_x = \frac{d}{dt} v_x
8228733125 a_y = \frac{d}{dt} v_y
9707028061 a_x = 0
2741489181 a_y = -g
3270039798 2
8880467139 2
8750379055 0 = \frac{d}{dt} v_x
1977955751 -g = \frac{d}{dt} v_y
6672141531 dt
1702349646 -g dt = d v_y
8584698994 -g \int dt = \int d v_y
9973952056 -g t = v_y - v_{0, y}
4167526462 v_{0, y}
6572039835 -g t + v_{0, y} = v_y
7252338326 v_y = \frac{dy}{dt}
6204539227 -g t + v_{0, y} = \frac{dy}{dt}
1614343171 dt
8145337879 -g t dt + v_{0, y} dt = dy
8808860551 -g \int t dt + v_{0, y} \int dt = \int dy
2858549874 - \frac{1}{2} g t^2 + v_{0, y} t = y - y_0
6098638221 y_0
2461349007 - \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y
8717193282 dt
1166310428 0 dt = d v_x
2366691988 \int 0 dt = \int d v_x
1676472948 0 = v_x - v_{0, x}
1439089569 v_{0, x}
6134836751 v_{0, x} = v_x
8460820419 v_x = \frac{dx}{dt}
7455581657 v_{0, x} = \frac{dx}{dt}
8607458157 dt
1963253044 v_{0, x} dt = dx
3676159007 v_{0, x} \int dt = \int dx
9882526611 v_{0, x} t = x - x_0
3182907803 x_0
8486706976 v_{0, x} t + x_0 = x
1306360899 x = v_{0, x} t + x_0
7049769409 2
9341391925 \vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}
6410818363 \theta
7391837535 \cos(\theta) = \frac{v_{0, x}}{v_0}
7376526845 \sin(\theta) = \frac{v_{0, y}}{v_0}
5868731041 v_0
6083821265 v_0 \cos(\theta) = v_{0, x}
5438722682 x = v_0 t \cos(\theta) + x_0
5620558729 v_0
8949329361 v_0 \sin(\theta) = v_{0, y}
1405465835 y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0
9862900242 y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0
6050070428 v_{0, x}
3274926090 t = \frac{x - x_0}{v_{0, x}}
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
8406170337 y
2403773761 t
5379546684 y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
5373931751 t = t_f
9112191201 y_f = 0
8198310977 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
1650441634 y_0 = 0 define coordinate system such that initial height is at origin
1087417579 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)
4829590294 t_f
2086924031 0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)
6974054946 \frac{1}{2} g t_f
1191796961 \frac{1}{2} g t_f = v_0 \sin(\theta)
2510804451 2/g
4778077984 t_f = \frac{2 v_0 \sin(\theta)}{g}
3273630811 x
6732786762 t
3485125659 x_f = v_0 t_f \cos(\theta) + x_0
4370074654 t = t_f
2378095808 x_f = x_0 + d
4268085801 x_0 + d = v_0 t_f \cos(\theta) + x_0
8072682558 x_0
7233558441 d = v_0 t_f \cos(\theta)
2297105551 d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)
7587034465 \theta
7214442790 x
2519058903 \sin(2 \theta) = 2 \sin(\theta) \cos(\theta)
8922441655 d = \frac{v_0^2}{g} \sin(2 \theta)
5667870149 \theta
1541916015 \theta = \frac{\pi}{4}
3607070319 d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)
5353282496 d = \frac{v_0^2}{g}
1590774089 dW = F dx
5542528160 \int dW = F \int_0^x dx
3512166162 W = F x
6935745841 F = G \frac{m_1 m_2}{x^2} Newton's law of universal gravitation Newton%27s_law_of_universal_gravitation#Modern_form
8604483515 dW = G \frac{m_1 m_2}{x^2} dx
4447113478 \int dW = G m_1 m_2 \int_{ r_{\rm Earth} }^{\infty} \frac{1}{x^2} dx
5732331610 W = G m_1 m_2 \left( \frac{1}{x} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)
8953094349 W = m a x
9623791270 d
3652511721 v
3253234559 x = \frac{v_2^2 - v_1^2}{2 a}
9413699705 W = m a \frac{v_2^2 - v_1^2}{2 a}
4811121942 W = \frac{1}{2} m v_2^2 - \frac{1}{2} m v_1^2
3360172339 W = KE_2 - KE_1
1413137236 m_1
2764966428 m_2
6131764194 W = G m_{\rm Earth} m \left( \frac{1}{x^2} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right) evaluating-definite-integrals-for.html
5978756813 W = G m_{\rm Earth} m \left( 0 - \frac{-1}{ r_{\rm Earth}} \right)
7749253510 W = G \frac{m_{\rm Earth} m }{ r_{\rm Earth}}
8960645192 KE_2 + PE_2 = KE_1 + PE_1
2267521164 PE_2 = 0 object goes to $\infty$ away from gravitational source
1840080113 KE_2 = 0 object is not moving at $x=\infty$
9749777192 0 = KE_1 + PE_1
5591692598 KE_1
6158970683 PE_1
7573835180 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
3846041519 PE_{\rm Earth\ surface} = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}}
5021965469 KE
6681646197 v
6870322215 KE_{\rm escape} = \frac{1}{2} m v_{\rm escape}^2
2042298788 0 = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}} + \frac{1}{2} m v_{\rm escape}^2
2503972039 0 = KE_{\rm escape} + PE_{\rm Earth\ surface}
5050429607 G \frac{m_{\rm Earth} m}{r_{\rm Earth}}
9703482302 G \frac{m_{\rm Earth} m}{r_{\rm Earth}} = \frac{1}{2} m v_{\rm escape}^2
1143343287 G \frac{m_{\rm Earth}}{r_{\rm Earth}} = \frac{1}{2} v_{\rm escape}^2
5775658332 2
2977457786 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}} = v_{\rm escape}^2
9412953728 v_{\rm escape}^2 = 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}
1330874553 v_{\rm escape} = \sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}
2750380042 v_{\rm escape} = -\sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}
2674546234 m_{\rm Earth}
2396787389 r_{\rm Earth}
5404822208 v_{\rm escape} = \sqrt{2 G \frac{m}{r}} escape velocity
3663007361 2
8946383937 v_{\rm escape}^2 = 2 G \frac{m}{r}
8362338572 v_{\rm escape}
2660368546 r
4275004561 c^2 = 2 G \frac{m}{r_{\rm Schwarzschild}}
7194432406 r_{\rm Schwarzschild}
2883079365 r_{\rm Schwarzschild} c^2 = 2 G m
7263534144 c^2
6800170830 r_{\rm Schwarzschild} = \frac{2 G m}{c^2}
4087145886 V = I R Ohm's law Ohm%27s_law
2698469612 V
7774819339 R
7675171493 V_1 = I R_1
7497687256 V
5890617067 R
6061695358 V_2 = I R_2
9063568209 V_{\rm total} = V_1 + V_2
4939880586 V_{\rm total} = I R_{\rm total}
2715678478 I R_{\rm total} = I R_1 + I R_2
7844317489 I
7217021879 R_{\rm total} = R_1 + R_2
2867848403 I
7191277455 R
4128500715 V = I_1 R_1
5585739998 I
1377431959 R
9243879541 V = I_2 R_2
3031116098 R_2
7002609475 \frac{V}{R_2} = I_2
5181421075 R_1
2051901211 \frac{V}{R_1} = I_1
6753224061 I_{\rm total} = I_1 + I_2
9053099840 I
1100332145 R
2271186630 V = I_{\rm total} R_{\rm total}
6546594355 R_{\rm total}
2809345867 \frac{V}{R_{\rm total}} = I_{\rm total}
4866160902 \frac{V}{R_{\rm total}} = \frac{V}{R_1} + \frac{V}{R_2}
3433441359 V
1457415749 \frac{1}{R_{\rm total}} = \frac{1}{R_1} + \frac{1}{R_2} total resistance for two resistors in parallel
3398368564 F
9794128647 m_1
3088463019 m_2
9226945488 F = \frac{m v^2}{r} Centripetal force Centripetal_force
5089196493 F
3342155559 m
7912578203 v
4627284246 F_{\rm centripetal} = \frac{m_{\rm satellite} v_{\rm satellite}^2}{r}
4830480629 x
5563580265 F_{\rm gravity} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}
3176662571 F_{\rm centripetal} = F_{\rm gravity} applicable to any satellite orbit
4072200527 \frac{m_{\rm satellite} v_{\rm satellite}^2}{r} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}
1823570358 C
3236313290 d
9262596735 d = 2 \pi r
4245712581 v = \frac{2 \pi r}{t}
3722461713 t
9346215480 T_{\rm orbit}
3614055652 v = \frac{2 \pi r}{T_{\rm orbit}}
2754264786 2
8059639673 v^2 = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2}
5359471792 \frac{m_{\rm satellite}}{r}
1994296484 v_{\rm satellite}^2 = G \frac{m_{\rm Earth}}{r}
3906710072 G \frac{m_{\rm Earth}}{r} = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2}
6238632840 r T_{\rm orbit}^2
7010294143 T_{\rm orbit}^2 G m_{\rm Earth} = 4 \pi^2 r^3
7556442438 4 \pi^2
4858693811 \frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2} = r^3
4319544433 1/3
2617541067 \left(\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r
3920616792 T_{\rm geostationary orbit} = 24\ {\rm hours} this applies for geostationary orbits
3846345263 T_{\rm orbit}
5770088141 r
1559688463 \left(\frac{T_{\rm geostationary\ orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r_{\rm geostationary\ orbit}
9881106100 a
5781435087 g
2484824786 F = m g
3921072591 m_1
4651061153 m_2
3353418803 x
8661803554 F = G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2}
9407192813 G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2} = m g_{\rm Earth}
3246378279 m
2308660627 G \frac{m_{\rm Earth}}{r_{\rm Earth}^2} = g_{\rm Earth}
2685587762 \frac{r_{\rm Earth}^2}{G}
9440616166 m_{\rm Earth} = \frac{g_{\rm Earth} r_{\rm Earth}^2}{G}
9355039511 g
2232825726 g_{\rm Earth}
4800170179 F = m g_{\rm Earth}
2091584724 g_{\rm Earth}
9590696981 9.80665
7816982139 m/s^2
7846240076 m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{G}
7326066466 G
9956609318 6.67430*10^{-11}
2957211007 m^3 kg^{-1} s^{-2}
7112613117 m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}}
7935917166 r_{\rm Earth}
3723096423 6.3781*10^6
7560908617 m
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}}
3364286646 m_{\rm Earth} = 5.972*10^{24} kg
1815398659 U = Q + W
9941599459 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)
3547519267 S = k_{\rm Boltzmann} \ln \Omega assumes equally probable microstates
1085150613 C_V = \left(\frac{\partial U}{\partial T}\right)_V definition of heat capacity at constant volume
5634116660 \pi_T = \left(\frac{\partial U}{\partial V}\right)_T definition of internal pressure at constant temperature
5002539602 dU = C_V dT + \pi_T dV
8854422847 dT
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 constant pressure
3464107376 \alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p definition of expansion coefficient
5074423401 V
6397683463 V \alpha = \left( \frac{\partial V}{\partial T} \right)_p
2257410739 \left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alpha
7826132469 \left(\frac{\partial U}{\partial T}\right)_p = C_V + \pi_T V \alpha
9781951738 \kappa_T = \frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T definition of isothermal compressibility
8435841627 P V = n R T Ideal_gas_law
6296166842 P
3497828859 V = \frac{n R T}{P}
8368984890 \kappa_T = \frac{-1}{V} \left( \frac{ \partial }{\partial P}\left(\frac{nRT}{P}\right) \right)_T
1190768176 \kappa_T = \frac{-nRT}{V} \left( \frac{ \partial }{\partial P}\left(\frac{1}{P}\right) \right)_T
3605073197 \kappa_T = \frac{-nRT}{V} \left( \frac{-1}{P^2}\right)
9847143017 \kappa_T = \frac{-PV}{V} \left( \frac{-1}{P^2}\right)
9718685793 \kappa_T = \frac{1}{P}
1311403394 \alpha = \frac{1}{V} \frac{nR}{P} \left( \frac{\partial T}{\partial T} \right)_P
5962145508 \alpha = \frac{nR}{VP}
7924842770 T
2613006036 \frac{PV}{T} = nR
6925244346 \alpha = \frac{PV}{T} \frac{1}{VP}
2472653783 \alpha = \frac{1}{T}
4560648264 v = \sqrt{ \frac{K + (4/3) G}{\rho} }
9376481176 K = f \frac{E}{a^3} proportionality coefficient fvaries in the range 1-4 for a majority of elemental solids
9674924517 K >> G yfN-LaW1BQAJ
6504442697 v = \sqrt{ \frac{K}{\rho} }
8090924099 v = \sqrt{ \left( f\frac{E}{a^3} \right) \frac{1}{\rho} }
8908736791 \rho = \frac{m}{a^3} geometry
2397692197 a^3
8688588981 a^3 \rho = m
7837519722 v = \sqrt{f} \sqrt{\frac{E}{m}}
3685779219 \sqrt{f} \approx 2
9854442418 v = \sqrt{\frac{E}{m}}
1556389363 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
4107032818 E_{\rm Rydberg} = E
8106885760 \alpha = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c} fine structure constant definition
8857931498 c
5838268428 \alpha c = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar}
3291685884 E = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}
3935058307 v = \sqrt{ \frac{m_e}{m} \frac{e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} }
9640720571 v = \frac{e^2}{4 \pi \epsilon_0 \hbar} \sqrt{\frac{m_e}{2 m}}
5789289057 v = \alpha c \sqrt{ \frac{m_e}{2 m} }