
\def\eqlte{\buildrel\scriptstyle{\rm\vrule width 0pt depth .2em lte}\over=}

$$\dot{} \equiv {\pd\over\pd t}\label{dot}$$

$$\nabla \equiv \left({\pd\over\pd x}, {\pd\over\pd y}, {\pd\over\pd z}\right)\label{nabla}$$

$${}_{,i} = {\pd\over\pd x_i}\label{,i}$$

$$\vec a\cdot\vec b = (a_1b_1, a_2b_2, a_3c_3)\label{a.b}$$

$$\vec a\times\vec b = (a_2b_3-a_3b_2, a_3b_1-a_1b_3, a_1b_2-a_2b_1)\label{axb}$$

$$\dot a = {2\over n}\vec a\cdot\hat T\label{gauss}$$

$$\dot a = -{2\over na}{\pd{\cal R}\over\pd\lambda}\label{lagrange}$$

$$\dot A^2 = H_0^2\left({\Omega_{\rm m}\over A} + {\Omega_{\rm r}\over A^2} + \Omega_\Lambda A^2 + 1 - \Omega_{\rm m} - \Omega_{\rm r} - \Omega_{\Lambda}\right)\label{fridman}$$

$$A_{ij}' = {A_{ij}-D_{ij}-O_{ij}\over F_{ij}-D_{ij}'-O_{ij}'}\label{A_ij_}$$

$$A_{21}n_2 + (B_{21}n_2-B_{12}n_1){1\over 4\pi}\int I_f\varphi_{12}\d f + C_{21}n_2 -C_{12}n_1 = 0\label{A21}$$

$$A_V \doteq 3\,E(B\!-\!V)\label{A_V}$$

$$\nabla\cdot\vec B = 0\label{divB}$$

$${\pd\vec B\over\pd t} = -\nabla\times\vec E\label{dotB}$$

$$B_f = {2hf^3\over c^2}{1\over\e^{hf\over kT}-1}\label{B_f}$$

$$\cos c = \cos a\cos b + \sin a\sin b\cos\gamma\label{cosc}$$

$$C_{\rm J} = {1\over 2}(\dot x^2+\dot y^2+\dot z^2) - {1\over 2}n^2(x^2+y^2) - {GM_1\over r_1} - {GM_2\over r_2}\label{C_J}$$

%$${\d e\over\d t} = {\eta\over na}(a_R\sin f + a_T\cos f + a_T\cos E)\label{de_dt}$$

$$E = M + e\cos E\label{kepler_eq}$$

$$E_n = -{k_{\rm e}q^2\over 2r_0}{1\over n^2}\label{E_n}$$

$$2\langle E_{\rm k}\rangle = -\langle E_{\rm g}\rangle\label{virial}$$

$$\nabla\cdot\vec E = {1\over\varepsilon}\rho_{\rm e}\label{divE}$$

$${\pd\vec E\over\pd t} = \mu\varepsilon\nabla\times\vec B - \mu\vec j\label{dotE}$$

$$f(x) = \sum_{i=0}^\infty {f^{(i)}(x_0)\over i!}(x-x_0)^i\label{taylor}$$

$$f_{;i} = f_{,i}\label{f_;j}$$

$$f^i_{\phantom{i};j} = f^i_{\phantom{i},j} + f^k\Gamma^i_{\phantom{i}kj}\label{f^i_;j}$$

$$f^{ij}_{\phantom{ij};k} = f^{ij}_{\phantom{ij},k} + f^{lj}\Gamma^i_{\phantom{i}lk} + f^{il}\Gamma^j_{\phantom{j}lk}\label{f^ij_;k}$$

$$f_i = g_{ik}f^k\label{f_i}$$

%$$f^i = g^{ik}f_k\label{f^i}$$

$$f_{\rm G} = {1\over\sqrt{2\pi}\sigma}\e^{-{(x-\mu)^2\over 2\sigma^2}}\label{f_G}$$

$$f_{\rm L} = {\gamma\over\pi}{1\over(x-\mu)^2 + \gamma^2}\label{f_L}$$

$$f_{\rm P} = {\lambda^k\e^{-\lambda}\over k!}\label{f_P}$$

$$f_{\rm Ny} = {1\over2\delta}\label{f_Ny}$$

$$\int_V\nabla\cdot\vec f\d V = \oint_S\vec f\cdot\d\vec S\label{gauss2}$$

$$\int_S\nabla\times\vec f\cdot\d\vec S = \oint_s\vec f\cdot\d\vec s\label{stokes}$$

$$F(\omega) = \int_{-\infty}^\infty f(t)\e^{\i\omega t}\d t\label{fourier}$$

$$\vec F = m\ddot{\vec r}\label{F}$$

$$\vec F = -2\oldvec\omega\times\vec v\label{F_C}$$

$$\vec F = (Q_{\rm abs}+Q_{\rm sca}){1\over c}\Phi S\left(\hat r - {\vec v\cdot\hat r\over c}\hat r - {\vec v\over c}\right)\label{F_rad}$$

$$\vec F = -{2\over 3}{1\over c}\int \epsilon\sigma T^4\d\vec S\label{F_ye}$$

$$F = {2GMm\over r^3}R\label{tides}$$

$$g_{ik} = \vec e_i\vec e_k\label{g_ik}$$

$$g_{ik} = \pmatrix{
-c^2 & 0 & 0 & 0 \cr
0    & 1 & 0 & 0 \cr
0    & 0 & 1 & 0 \cr
0    & 0 & 0 & 1 \cr
}\label{minkowski}$$

$$g_{ik} = \pmatrix{
-c^2(1-{r_{\rm g}/r}) & 0                      & 0   & 0                  \cr
0                     & (1-{r_{\rm g}/r})^{-1} & 0   & 0                  \cr
0                     & 0                      & r^2 & 0                  \cr
0                     & 0                      & 0   & (r\sin\vartheta)^2 \cr
}\label{schwarzschild}$$

$$g_{ik} = \pmatrix{
-c^2 & 0                & 0      & 0                  \cr
0    & A^2(1-Kr^2)^{-1} & 0      & 0                  \cr
0    & 0                & (Ar)^2 & 0                  \cr
0    & 0                & 0      & (Ar\sin\vartheta)^2 \cr
}\label{flrw}$$

$$g^{ik} = (g_{ik})^{-1}\label{g^ik}$$

%$$g_{ik;j} = 0\label{g_ik;j}$$

$$H = {p^2\over 2m} - {GMm\over q}\label{H_2body}$$

$$H = {p^2\over 2m} - {k_{\rm e}e^2\over q}\label{H_hydrogen}$$

%$${\d i\over\d t} = {1\over na\eta}{r\over a}a_W\cos(f+\omega)\label{di_dt}$$

$$\eqalignno{
{\rm H} + {\rm H} &\leftrightarrows {\rm H}^{*} & \label{collision}\cr
{\rm H} + \gamma &\leftrightarrows {\rm H}^{*} & \label{excitation}\cr
{\rm H} + \gamma &\leftrightarrows {\rm p}^{+} + {\rm e}^- & \label{ionisation}\cr
{\rm H}^- + \gamma &\leftrightarrows {\rm H} + {\rm e}^{-} & \label{H-}\cr
}$$

$$I(\oldvec\alpha) = I_0\left[1+\Re\left(\mu\e^{i k\oldvec\alpha\cdot\vec B}\right)\right]\label{I_alpha}$$

$$I_f \equiv {\d E\over\d t\d S\cos\vartheta\d\Omega\d f}\label{I_f}$$

$${1\over c}{\pd I_f\over\pd t} + \hat k\cdot\nabla I_f = j_f\rho - \kappa_f\rho I_f\quad\hbox{pro }\forall\hat k\label{rte}$$

$$I_f = I_f(0)\e^{-\kappa_f\rho x} + S_f(1-\e^{-\kappa_f\rho x})\label{rte2}$$

$$I = {U\over R}\label{I}$$

$$\dot I = {U\over L}\label{dotI}$$

$$\int I\d t = UC\label{intI}$$

$$\sum_{i=1}^N I_i = 0\label{sumI}$$

$$j_f\varrho = {hf\over 4\pi}n_2A_{21}\varphi_{12}\label{j_f}$$

$$j_f\varrho = \kappa_f^{\rm sca}\rho {1\over 4\pi}\int I_f\d\Omega\label{j_f2}$$

%$$J_\alpha(x) = \sum_{i=0}^\infty {(-1)^i\over i!\,\Gamma(i+\alpha+1)}\left(x\over 2\right)^{2i+\alpha}\label{bessel}$$

$$\vec L = \vec r\times\vec p\label{vecL}$$

$$\dot{\vec L} = \vec M\label{dotL}$$

$${\d L_R\over\d R} = 4\pi R^2\!\varrho\,\varepsilon\label{dLR_dR}$$

$${\d M_R\over\d R} = 4\pi R^2\varrho\label{dMR_dR}$$

$$\vec M = \vec r\times\vec F\label{vecM}$$

$$N = \sqrt{S}\label{N}$$

$$N = \sqrt{N_{\rm star}^2 + N_{\rm sky}^2 + N_{\rm dark}^2 + N_{\rm readout}^2 + N_{\rm discrete}^2}\label{N2}$$

$$N = \int N_i\d\oldvec\Gamma\label{N3}$$

$$N_i = {g_i\over\e^{E_i-\mu\over kT}-1}\label{bose-einstein}$$

$$N_i = {g_i\over\e^{E_i-\mu\over kT}+1}\label{fermi-dirac}$$

$${N_i\over N_j} \eqlte {g_i\over g_j}\e^{-{E_{ij}\over kT}}\label{Ni_Nj}$$

$$f(v) \eqlte \sqrt{2\over\pi} \left({m\over kT}\right)^{\!3/2} v^2\e^{-{mv^2\over 2kT}}\label{maxwell-boltzmann}$$

$$p \eqlte {1\over 3}aT^4\label{p_rad}$$

$$p = {1\over c}\int I\cos^2\!\vartheta\,\d\Omega\label{p_rad2}$$

$$p = {1\over 3V}\int N_iv|\vec p|\d\oldvec\Gamma\label{p}$$

$${\d p\over\d R} = -{GM_R\varrho\over R^2}\label{dp_dR}$$

$${\d p\over\d R} = -{G\over R^2} \left(M_R+4\pi R^3{p\over c^2}\right) \left(\rho+{p\over c^2}\right) \left({1-{r_{\rm g}\over R}}\right)^{-1}\label{tov}$$

$${\d p\over\d T} = {l_{\rm h}\over T(V'-V)}\label{clausius-clapeyron}$$

$$P_{20}(\cos\vartheta) = {1\over 2}(3\cos^2\vartheta-1)\label{P_20}$$

$$p(B|A) = p(A|B){p(B)\over p(A)}\label{bayes}$$

$$p = \int \Psi\Psi^*\d V\label{p_Psi}$$

$$\eqalignno{
{\rm p} + {\rm p} &\rightarrow {\rm D} + {\rm e}^+ + \nu_{\rm e} & \label{pp1}\cr
{\rm e}^+ + {\rm e}^- &\rightarrow 2\gamma & \label{pp2}\cr
{\rm D} + {\rm p} &\rightarrow {}^3{\rm He} + \gamma & \label{pp3}\cr
{}^3{\rm He} + {}^3{\rm He} &\rightarrow \alpha + 2{\rm p} & \label{pp4}\cr
}$$

$$\dot p = -{\pd H\over\pd q}\label{dotp}$$

$$\dot q = {\pd H\over\pd p}\label{dotq}$$

$$\delta Q + \mu\d N = \d U + \delta W\label{1st}$$

$$r_{\rm H} = a\left({m\over 3M}\right)^{\!1/3}\label{r_H}$$

$${\cal R}_{21} = -{Gm_1\over r_{12}} + {Gm_1\over r_1^3}\vec r_1\cdot\vec r_2\label{calR}$$

$$\langle{\cal R}_{21}\rangle \doteq C_0 + C_1(e_1^2+e_2^2) + C_3e_1e_2\cos(\varpi_1-\varpi_2)\label{calR2}$$

$$R_{ik} - {1\over2}Rg_{ik} + \Lambda g_{ik} = {8\pi G\over c^4}T_{ik}\label{einstein}$$

$$R^l_{\phantom{l}ijk} = {f_{i;jk}-f_{i;kj}\over f_l}\label{R^l_ijk}$$

$$R_{ik} = R^l_{\phantom{l}ilk}\label{R_ik}$$

$$R = g^{ik}R_{ik}\label{R}$$

$$R_z(\varphi) \equiv \pmatrix{
\cos\varphi & -\sin\varphi & 0 \cr
\sin\varphi & \cos\varphi  & 0 \cr
0           & 0            & 1 \cr
}\label{R_z}$$

$$\vec r' = R_z(\Omega)\times R_x(i)\times R_z(\omega)\vec r\label{vecr_}$$

$$\ddot{\vec r} + G(M+m){\vec r\over r^3} = 0\label{2body}$$

$${\rm Re} = {xv\over\nu}\label{Re}$$

$$S_{\rm adu} = \int {\Phi_{\kern-1pt f} St_{\rm exp}\over hf} TQ\eta\,\d f\label{S_adu}$$

$$\vec S = \nu\nabla\vec v\label{vecS}$$

$$\dot{\vec S} = E\nabla\vec v\label{dotvecS}$$

$$S_f \equiv {j_f\over\kappa_f}\label{S_f}$$

$$S_f \eqlte B_f\label{S_f2}$$

%$$\d S = {\delta Q\over T}\label{dS}$$

$$t' = \gamma\left(t + {\beta\over c}x\right)\label{t'}$$

$${\d T\over\d R}\Bigr|_{\rm rad} = -{3\kappa\varrho L_R\over 16\pi acR^2 T^3}\label{dT_dR_rad}$$

$${\d T\over\d R}\Bigr|_{\rm ad} = {1\over C_p\varrho}{\d p\over\d R}\label{dT_dR_ad}$$

$$\left|{\d T\over\d R}\right|_{\rm ad} < \left|{\d T\over\d R}\right|_{\rm rad}\label{schwarzschild2}$$

$$T^{ik} = \pmatrix{
\rho c^2 & 0 & 0 & 0 \cr
0        & p & 0 & 0 \cr
0        & 0 & p & 0 \cr
0        & 0 & 0 & p \cr
}\label{T^ik}$$

%$$T^{ik} = \rho u^iu^j\label{T^ik2}$$

$$u = {1\over c}\int I\d\Omega\label{u}$$

$${\pd u\over\pd t} = \chi\nabla\cdot\nabla u\label{hde}$$

$${\pd^2 u\over\pd t^2} = c^2\nabla\cdot\nabla u\label{wave}$$

$$u = A\e^{\i(\vec k\cdot\vec r-\omega t)}\label{wave2}$$

$$U = -{GM\over r}\sum_{\ell=0}^n\sum_{m=-\ell}^\ell\left({R\over r}\right)^{\!\ell}\!\! P_{\ell m}(\cos\vartheta)\left[C_{\ell m}\cos(m\varphi)+S_{\ell m}\sin(m\varphi)\right]\label{U}$$

%$$U_{\rm Y} = -g^2{\e^{-\alpha m r}\over r}\label{yukawa}$$

$$\nabla\cdot\nabla U = 4\pi G\varrho\label{poisson}$$

$$\sum_{i=1}^N U_i = 0\label{sumU}$$

$${\pd\vec v\over\pd t} + \vec v\cdot\nabla\vec v = -{1\over\rho}\nabla p + {1\over\rho}\nabla\cdot\vec S\label{navier-stokes}$$

$$x' = \gamma(x - Vt)\label{x'}$$

$${X\over1-X} =  {(2\pi m_{\rm e}kT)^{3/2}\over h^3}{n\over n_{\rm e}n_{\rm p}}\e^{-{E_{\rm i}\over kT}}\label{saha}$$

$${\pd^2 x^i\over\pd\tau^2} + {\pd x^j\over\pd\tau}{\pd x^k\over\pd\tau}\Gamma^i_{\phantom{i}kj} = 0\label{geodetic}$$

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$$\eqalignno{
\alpha + \alpha &\leftrightarrows {}^8{\rm Be}^* & \label{alpha_alpha}\cr
{}^8{\rm Be}^* + \alpha &\rightarrow {}^{12}{\rm C} & \label{Be8_alpha}
}$$

$$\sin\alpha = \sin\beta{\sin a\over\sin b}\label{sinalpha}$$

$$\beta \equiv {v\over c}\label{beta}$$

$$\gamma \equiv {1\over\sqrt{1-{v^2\over c^2}}}\label{gamma}$$

%$$\Gamma(n+1) = n!\label{Gamma}$$

$$\Gamma^i_{\phantom{i}kj} = {1\over 2}g^{im}\left(g_{mk,j} + g_{mj,k} - g_{kj,m}\right)\label{christoffel}$$

$$\langle\Delta x\rangle\langle\Delta p\rangle \ge {\hbar\over2}\label{heisenberg}$$

$$\kappa_f\varrho = {hf\over 4\pi}(n_1B_{12}-n_2B_{21})\varphi_{12}\label{kappa_f}$$

$$\kappa_f^{\rm abs} = {3(1-A)\over 4r\rho_{\rm d}}\label{kappa_f2}$$

%$$\kappa_f^{\rm sca} = {3A\over 2r\rho_{\rm d}}\label{kappa_f3}$$

$$\lambda' = \lambda\left[1+{g(y'-y)\over c^2}\right]\label{lambda_g}$$

$$\mu = {\int I(\oldvec\alpha')\e^{\i k\oldvec\alpha'\!\cdot\vec B}\d\alpha'\over I_0}\label{mu}$$

%$$\mu = \mu_0 + N_{\rm A}kT\ln X\label{mu2}$$

$$\nu = \alpha c_{\rm s}z\label{alpha}$$

$${\pd\varrho\over\pd t} + \vec v\cdot\nabla\varrho = -\varrho\nabla\cdot\vec v\label{drho_dt}$$

$$\d\tau_f \equiv \kappa_f\rho\d x\label{dtau}$$

$$\Phi = \int I\cos\vartheta\d\Omega\label{Phi1}$$

$$\Phi(1-A) - K\nabla u\cdot\hat n - 4\epsilon\sigma u^4 \eqlte 0\label{Phi2}$$

$$\chi^2 \equiv \sum_{i=1}^n \left({y_i-f(x_i)\over\sigma_i}\right)^{\!2}\label{chi2}$$

$$\i\hbar{\pd\Psi\over\pd t} = {\hbar^2\over 2m}\nabla\cdot\nabla\Psi - {k_{\rm e}Qq\over r}\Psi\label{schrodinger}$$

$$\Psi_{n\ell m} = \sum_{n=0}^\infty \sum_{\ell=0}^n \sum_{m=-\ell}^\ell K \e^{-{r\over nr_0}} \left(2r\over nr_0\right)^{\!\ell} L^{2\ell+1}_{n-\ell-1}\!\left({2r\over nr_0}\right) Y_{\ell m}(\vartheta,\varphi)\label{psi_nlm}$$

%$$\omega_{\rm p} = {3\over 2}{GM\over r^3}{I_{33}-I_{11}\over I_{33}}{\cos i\over\omega_{\rm Z}}\label{omega_p}$$

%$$\omega'-\omega = {6\pi G(M+m)\over c^2a(1-e^2)}\label{omega_r}$$

\endinput

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