\documentclass{article}
\usepackage{iftex}
\usepackage{ifthen}
\usepackage{epsfig}
\usepackage[utf8]{inputenc}
\usepackage{xcolor}
\ifPDFTeX
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  \doxynewunicodechar{⁻}{${}^{-}$}% Superscript minus
  \doxynewunicodechar{²}{${}^{2}$}% Superscript two
  \doxynewunicodechar{³}{${}^{3}$}% Superscript three

\fi
\pagestyle{empty}
\begin{document}
$\gamma_{ij}$
\pagebreak

$K_{ij}$
\pagebreak

$\Theta=0$
\pagebreak

$\Theta$
\pagebreak

$\Delta_{\theta\phi}u+\lambda u=\sigma$
\pagebreak

\[  \Delta_{\theta\phi}h - 2h = \Psi^4 |DF| h^2\Theta + 
  \Delta_{\theta\phi}h - 2h
\]
\pagebreak

$h=h(\theta,\phi)$
\pagebreak

$\Psi$
\pagebreak

$|DF|:= (\gamma^{ij}D_iFD_jF)^{1/2} $
\pagebreak

$F$
\pagebreak

$ F:=r-h(\theta,\phi) $
\pagebreak

$D_i$
\pagebreak

$F:= r - h(\theta,\phi)$
\pagebreak

$h$
\pagebreak

$\theta$
\pagebreak

$\phi$
\pagebreak

$\xi$
\pagebreak

$\cos(j \theta)$
\pagebreak

$\sin(j \theta)$
\pagebreak

$\cos(2j \theta)$
\pagebreak

$\sin(2j \theta)$
\pagebreak

$\cos((2j+1) \theta)$
\pagebreak

$\sin((2j+1) \theta)$
\pagebreak

$P_{2j}^m(\cos\theta)$
\pagebreak

$P_{2j+1}^m(\cos\theta)$
\pagebreak

$P_l^m(\cos\theta)$
\pagebreak

$(\cos(m\phi),\ \sin(m\phi))$
\pagebreak

$\varphi$
\pagebreak

$B_i(\theta_j)$
\pagebreak

$B_i$
\pagebreak

$\theta_j$
\pagebreak

$B_i(\phi_k)$
\pagebreak

$\phi_k$
\pagebreak

$\frac{\partial}{\partial \xi}$
\pagebreak

$\frac{1}{\xi}$
\pagebreak

$\frac{\partial}{\partial \theta}$
\pagebreak

$\frac{1}{\sin \theta}$
\pagebreak

$\sin \theta$
\pagebreak

$\cos \theta$
\pagebreak

$Y^l_m$
\pagebreak

${\tt rot\_phi = 0}$
\pagebreak

${\tt rot\_phi} = \pi$
\pagebreak

$K^{ij}$
\pagebreak

$A^{ij}$
\pagebreak

$\nabla^i N^j + \nabla^j N^i-\frac{2}{3}
\nabla_k N^k \delta^{ij}$
\pagebreak

$2NK^{ij}$
\pagebreak

$\beta^i$
\pagebreak

$\vec{N}$
\pagebreak

$\nabla N$
\pagebreak

$\nabla \Psi$
\pagebreak

$= \frac{1}{2}-2\frac{a}{r}$
\pagebreak

$= \frac{1}{2}$
\pagebreak

$= \frac{1}{2}+\frac{a}{r}$
\pagebreak

$\frac{J}{M}=a$
\pagebreak

$r_0$
\pagebreak

$\frac{1}{2}\sqrt{M^2-a^2}$
\pagebreak

$\Omega = \frac{a}{2M\sqrt{M^2-a^2}}$
\pagebreak

$\Psi^4 = 
1+\frac{2M}{r}+\frac{3M^2+a^2\cos^2\theta}{2r^2}+\frac{2Mr_0^2}{r^3}+
\frac{r_0^4}{r^4}$
\pagebreak

$N = \left(1-\frac{2MR}{\Sigma}+\frac{4a^2M^2R^2\sin^2\theta}
{\Sigma^2\left(R^2+a^2\right)+2a^2\Sigma R\sin^2\theta}\right)^{\frac{1}{2}}$
\pagebreak

$N^\phi = \frac{2aMR}{\Sigma\left(R^2+a^2\right)+2a^2MR\sin^2\theta}$
\pagebreak

$R = r+\frac{M^2-a^2}{4r}+M$
\pagebreak

$\Sigma = R^2+a^2-2MR$
\pagebreak

\[*\Delta N = -\frac{2}{\Psi}\nabla_i \Psi \nabla^i N 
  + N \Psi^4 K_{ij} K^{ij}
\]
\pagebreak

\[\Delta \Psi = - \frac{\Psi^5} {8} K_{ij}K^{ij}
\]
\pagebreak

$\partial_r \Psi =-\frac{1}{2 {\tt rayon}} 
f\left(\theta, \phi\right)$
\pagebreak

$\vec{\beta}$
\pagebreak

\[*\Delta \beta^i +\frac{1}{3} \nabla^i \left(\nabla_j \beta^j\right)
  = -6A^{ij}\frac{\nabla_j \Psi}{\Psi} + 2 K^{ij}\nabla_j N
\]
\pagebreak

$\vec{\beta} = -\Omega \vec{m} - \vec{V}$
\pagebreak

$\vec{V}$
\pagebreak

$\vec{m} = \frac{\partial}{\partial \phi}$
\pagebreak

${\mathcal{C}}^2$
\pagebreak

\[(\Delta \Psi = - \frac{\Psi^5} {8} K_{ij}K^{ij}
\]
\pagebreak

$\partial_r \Psi=-\frac{1}{2 {\tt rayon}} 
  f\left(\theta, \phi\right)$
\pagebreak

$M = -\frac{1}{2 \pi} \oint_{\infty} \nabla^i \Psi {\mathrm d} S_i$
\pagebreak

$M = \frac{1}{4 \pi} \oint_{\infty} \nabla^i N {\mathrm d} S_i$
\pagebreak

$ J = -\frac{1}{8 \pi} \oint_{\infty} K_j^i m^j {\mathrm d}S_i$
\pagebreak

$ J = -\frac{1}{8 \pi} \oint_{H} \Psi^6 K_j^i m^j {\mathrm d}S_i$
\pagebreak

$\Phi = \log \Psi$
\pagebreak

\[\Delta N_a = -\frac{2}{\Psi}\nabla_i \Psi_a \nabla^i N + N \Psi^4 K_{ij}K_a^{ij}
\]
\pagebreak

$_a$
\pagebreak

\[*\Delta \Psi_a = -\frac {\Psi^5}{8} K_{ij}K_a^{ij}
\]
\pagebreak

$\partial_r \Psi = -\frac{1}{2 r_0} f\left(\theta, \phi\right)$
\pagebreak

\[*\Delta \beta_a^i +\frac{1}{3} \nabla^i \left(\nabla_j \beta_a^j\right) = 
  -6A^{ij}\frac{\nabla_i \Psi_a}{\Psi} + 2K^{ij}\nabla_j N_a
\]
\pagebreak

$\Omega$
\pagebreak

$\vec{N} = -\Omega \vec{m}$
\pagebreak

$Psi$
\pagebreak

$\Omega = 0$
\pagebreak

$\beta$
\pagebreak

$ J = -\sum_{a= 1, 2} \frac{1}{8 \pi} \oint_{H_a} \Psi^6 K_j^i m^j {\mathrm d}S_i$
\pagebreak

$H_a$
\pagebreak

$1/r$
\pagebreak

$J^r$
\pagebreak

$J^t$
\pagebreak

$J^p$
\pagebreak

$M_{\rm ADM} - M_{\rm bar,1} - M_{\rm bar,2}$
\pagebreak

\[  ds^2 = - \alpha^2 dt^2 + \gamma_{ij} ( dx^i + \beta^i dt )
                                       ( dx^j + \beta^j dt )
\]
\pagebreak

$\ A^{ij}$
\pagebreak

$\eta_{ik} \eta_{jl} A^{ij} A^{kl}$
\pagebreak

$A_{ij}$
\pagebreak

$\tilde \gamma_{ij}$
\pagebreak

$W = F\left(r\right) + G \left(r\right) w$
\pagebreak

$W$
\pagebreak

$\mathcal{C}^1$
\pagebreak

$w$
\pagebreak

$G$
\pagebreak

$r$
\pagebreak

$= {\rm STD} \Longrightarrow F=0 \, \& \, G=1$
\pagebreak

$= {\rm W\_BETA} \Longrightarrow F=1 \, \& \, G = r$
\pagebreak

$= {\rm W\_BETA\_INF} \Longrightarrow F = 1+ \frac{1}{3}r^2\ln r \, \& \, G = r$
\pagebreak

$= {\rm H\_BETA} \Longrightarrow F=1 \, \&\, G=1$
\pagebreak

$= {\rm LAMBDA_RN} \Longrightarrow F=-\ln r \, \& \, G=1$
\pagebreak

$= {\rm NU_RN} \Longrightarrow F=\ln r \, \& \, G=1$
\pagebreak

$F'$
\pagebreak

$G'$
\pagebreak

${\tt Cmp} ^{\tt int}$
\pagebreak

${\tt Cmp} ^{\tt double}$
\pagebreak

$\partial/\partial r$
\pagebreak

$1/r \partial/\partial \theta$
\pagebreak

$1/(r\sin\theta) \partial/\partial \phi$
\pagebreak

$\partial/\partial x$
\pagebreak

$x=r\sin\theta \cos\phi$
\pagebreak

$\partial/\partial y$
\pagebreak

$y=r\sin\theta \sin\phi$
\pagebreak

$\partial/\partial z$
\pagebreak

$z=r\cos\theta$
\pagebreak

$x_1 = - x_2$
\pagebreak

$y_1 = - y_2$
\pagebreak

$z_1 = z_2$
\pagebreak

$(r, \theta, \phi)$
\pagebreak

$\Phi$
\pagebreak

$r^{-n}$
\pagebreak

$r^{\tt puis} $
\pagebreak

$Y_l^m$
\pagebreak

$\partial / \partial r$
\pagebreak

$r^2 \partial/ \partial r$
\pagebreak

$1/r \partial / \partial \theta$
\pagebreak

$r \partial/ \partial \theta$
\pagebreak

$1/(r\sin\theta) \partial / \partial \phi$
\pagebreak

$r/\sin\theta \partial/ \partial \phi$
\pagebreak

$r^2 \partial/ \partial x$
\pagebreak

$r^2 \partial/ \partial y$
\pagebreak

$r^2 \partial/ \partial z$
\pagebreak

$\partial/\partial x_i$
\pagebreak

$x_i = (x, y, z)$
\pagebreak

$r^2 \partial/ \partial x_i$
\pagebreak

$\Delta u$
\pagebreak

$r^2 \,  \Delta u$
\pagebreak

$r^4 \, \Delta u$
\pagebreak

$r\sin\theta$
\pagebreak

$\int u \, r^2 \sin\theta \,  dr\, d\theta \, d\phi$
\pagebreak

$a_k(\theta, \phi)$
\pagebreak

\[\sum_{k=0}^n {a_k(\theta, \phi) \over r^k}
\]
\pagebreak

$r \rightarrow \infty$
\pagebreak

$\sigma$
\pagebreak

$\Delta u = \sigma$
\pagebreak

$r^2 \sigma$
\pagebreak

$r^4 \sigma$
\pagebreak

$\sigma_{\rm regu} = \sigma - \sigma_{\rm div}$
\pagebreak

$1/(\gamma-1)$
\pagebreak

$\gamma$
\pagebreak

$|\Delta u - \sigma|$
\pagebreak

$|\sigma|$
\pagebreak

$C^n$
\pagebreak

$C^1$
\pagebreak

$\frac{1}{r^i}$
\pagebreak

${\tt puis} \leq i \leq {\tt puis+nbre}$
\pagebreak

$l \leq {\tt lmax}$
\pagebreak

\[  ds^2 = - N^2  dt^2 + \gamma_{ij} ( dx^i + \beta^i dt )
              (dx^j + \beta^j dt )
\]
\pagebreak

\[   T_{\alpha\beta} = E n_\alpha n_\beta + P_\alpha n_\beta + n_\alpha P_\beta + S_{\alpha\beta}
\]
\pagebreak

\[  ds^2 = - N^2 dt^2 + A^2 (dr^2 + r^2 d\theta^2)
           + B^2 r^2 \sin^2\theta (d\varphi - N^\varphi dt)^2
\]
\pagebreak

$t=\mathrm{const}$
\pagebreak

$R^3$
\pagebreak

\[   ds^2 = - N^2 dt^2 + A^2 (dr^2 + r^2 d\theta^2)
               + B^2 r^2 \sin^2\theta (d\varphi - N^\varphi dt)^2
\]
\pagebreak

\[  ds^2 = - N^2 dt^2 + A^2 (dr^2 + r^2 d\theta^2)
            + B^2 r^2 \sin^2\theta (d\varphi - N^\varphi dt)^2
\]
\pagebreak

\[  ds^2 = copy Carlos metric
\]
\pagebreak

$P^i$
\pagebreak

$S_{ij}$
\pagebreak

$N^\varphi$
\pagebreak

$A^2 K_{ij} K^{ij}$
\pagebreak

\[ A^2 K_{ij} K^{ij} = {B^2 \over 2 N^2} \, r^2\sin^2\theta \,  
   \left[ \left( {\partial N^\varphi \over \partial r} \right) ^2
     + {1\over r^2} \left( {\partial N^\varphi \over 
        \partial \theta} \right) ^2 \right] \ . 
\]
\pagebreak

$k_1$
\pagebreak

$k_2$
\pagebreak

\[ A^2 K_{ij} K^{ij} = 2 A^2 (k_1^2 + k_2^2) \ . 
\]
\pagebreak

$N^\phi$
\pagebreak

$\tilde N^\varphi = N^\varphi r\sin\theta$
\pagebreak

$\nu = \ln N$
\pagebreak

$\zeta = \ln(AN)$
\pagebreak

$\tilde G = (NB-1) r\sin\theta$
\pagebreak

$W^i$
\pagebreak

\[ N^i = {7\over 8} W^i - {1\over 8} 
        \left(\nabla^i\chi+\nabla^iW^kx_k\right)
\]
\pagebreak

$\chi$
\pagebreak

$N^i$
\pagebreak

${\bar I}_{zz}$
\pagebreak

${\bar I}_{ij}$
\pagebreak

$Q = -3/2 {\bar I}_{zz}$
\pagebreak

$\lambda$
\pagebreak

$\sigma_q$
\pagebreak

\[    \Delta_2 u = \sigma_m + \sigma_q
\]
\pagebreak

\[    \int_0^{2\pi} \int_0^{+\infty} \sigma(r, \theta)
            \, r \, dr \, d\theta = 0       \ ,
\]
\pagebreak

$\sigma = \sigma_m + \sigma_q$
\pagebreak

\[    \lambda = - { \int_0^{2\pi} \int_0^{+\infty} \sigma_m(r, \theta)
            \, r \, dr \, d\theta       \over
        \int_0^{2\pi} \int_0^{+\infty} \sigma_q(r, \theta)
            \, r \, dr \, d\theta } \ .
\]
\pagebreak

$\sigma' = \sigma_m + \lambda \sigma_q$
\pagebreak

$\sigma_m$
\pagebreak

$M$
\pagebreak

$a$
\pagebreak

$(e_i)$
\pagebreak

\[ \Gamma^i_{\ jk} := \langle e^i, \nabla_{e_k} \, e_j \rangle 
\]
\pagebreak

$\Delta^i_{\ jk}$
\pagebreak

$\Gamma^i_{\ jk}$
\pagebreak

${\bar \Gamma}^i_{\ jk}$
\pagebreak

\[ \Gamma^i_{\ jk} := \langle e^i, \nabla_{e_k} \, e_j \rangle
\]
\pagebreak

$\nabla T$
\pagebreak

$T$
\pagebreak

$e_i$
\pagebreak

$T^{ij}$
\pagebreak

$T = T^{ij} \; e_i \otimes e_j$
\pagebreak

\[      {\rm div}\, T = \nabla_k T^{ik} \; e_i
\]
\pagebreak

$\nabla$
\pagebreak

\[      {\rm div} T = \nabla_k T^{ik} \; e_i
\]
\pagebreak

$ \frac{\partial}{\partial \xi} $
\pagebreak

$ \frac{\partial^2}{\partial \xi^2} $
\pagebreak

$ \xi $
\pagebreak

$\xi -1$
\pagebreak

$ \xi^2 $
\pagebreak

$(\xi -1)^2$
\pagebreak

$ \xi - 1$
\pagebreak

$ \xi \frac{\partial}{\partial \xi} $
\pagebreak

$ (\xi -1)  \frac{\partial}{\partial \xi} $
\pagebreak

$ \frac{1}{\xi} \frac{\partial}{\partial \xi} $
\pagebreak

$ \frac{1}{\xi-1} \frac{\partial}{\partial \xi} $
\pagebreak

$ \xi^2 \frac{\partial^2}{\partial \xi^2} $
\pagebreak

$ (\xi - 1)^2 \frac{\partial^2}{\partial \xi^2} $
\pagebreak

$ \xi \frac{\partial^2}{\partial \xi^2} $
\pagebreak

$ (\xi - 1) \frac{\partial^2}{\partial \xi^2} $
\pagebreak

$ \xi^2 \frac{\partial}{\partial \xi} $
\pagebreak

$ (\xi-1)^2 \frac{\partial}{\partial \xi} $
\pagebreak

$ \xi^3 \frac{\partial}{\partial \xi} $
\pagebreak

$ (\xi -1)^3  \frac{\partial}{\partial \xi} $
\pagebreak

$ \xi^3 \frac{\partial^2}{\partial \xi^2} $
\pagebreak

$ (\xi - 1)^3 \frac{\partial^2}{\partial \xi^2} $
\pagebreak

$ \xi^4 \frac{\partial^2}{\partial \xi^2} $
\pagebreak

$ (\xi - 1)^4 \frac{\partial^2}{\partial \xi^2} $
\pagebreak

$m_0$
\pagebreak

\[   e(n) = {\kappa \over \gamma-1} n^\gamma + \mu_0 \, n \ , \qquad \qquad (1)
\]
\pagebreak

$\mu_0$
\pagebreak

\[   \mu(n) := {de\over dn} = {\kappa \gamma \over \gamma-1} n^{\gamma-1}
        + \mu_0 \ .\qquad \qquad (2)
\]
\pagebreak

$p = \mu n - e$
\pagebreak

\[   p(n) = \kappa n^\gamma  \ . \qquad \qquad (3)
\]
\pagebreak

\[   H(n) := c^2 \ln \left( {e+p \over m_0 c^2\, n} \right)   \ . \qquad \qquad (4)
\]
\pagebreak

\[   H = c^2 \ln \left( {\mu \over m_0 c^2} \right) \ .  \qquad \qquad (5)
\]
\pagebreak

\[   n(H) = \left[ {\gamma-1\over \gamma} {m_0 c^2 \over \kappa}
              \left( \exp(H) - {\mu_0\over m_0 c^2} \right)
        \right] ^{1/(\gamma-1)}  \ .    \qquad \qquad (6)
\]
\pagebreak

$\epsilon$
\pagebreak

\[  \epsilon(n) = {\kappa \over \gamma-1} n^\gamma  \ . \qquad\qquad (1)
\]
\pagebreak

\[  \mu(n) := {d\epsilon\over dn} = {\kappa \gamma \over \gamma-1} n^{\gamma-1}
      \ . \qquad\qquad (2)
\]
\pagebreak

$p = \mu n - \epsilon$
\pagebreak

\[  p(n) = \kappa n^\gamma  \ .  \qquad\qquad (3)
\]
\pagebreak

\[  h(n) := {\epsilon + p \over m_0 n}   \ .  \qquad\qquad (4)
\]
\pagebreak

\[  h = {\mu \over m_0}  \ . \qquad\qquad (5)
\]
\pagebreak

\[  n(h) = \left[ {\gamma-1\over \gamma} {m_0 \over \kappa} h
     \right] ^{1/(\gamma-1)}  \ .  \qquad\qquad (6)
\]
\pagebreak

$e = \rho_0 \,  c^2$
\pagebreak

$\rho = \rho_0$
\pagebreak

$m$
\pagebreak

$g_s$
\pagebreak

$g_s = 2$
\pagebreak

$c^2$
\pagebreak

$H = c^2 \ln\left( {e+p \over m_B c^2 n} \right) $
\pagebreak

$m_B$
\pagebreak

$n_{\rm nuc} := 0.1 \ {\rm fm}^{-3}$
\pagebreak

$\rho_{\rm nuc} c^2$
\pagebreak

$\rho_{\rm nuc} := 1.66\ 10^{17} \ {\rm kg/m}^3$
\pagebreak

$d\ln n/d\ln H$
\pagebreak

$d\ln e/d\ln H$
\pagebreak

$d\ln p/d\ln H$
\pagebreak

$\kappa$
\pagebreak

$\rho_{\rm nuc} c^2 / n_{\rm nuc}^\gamma$
\pagebreak

$m_B = 1.66\ 10^{-27} \ {\rm kg}$
\pagebreak

$m_B c^2$
\pagebreak

$\gamma-1$
\pagebreak

$(\gamma-1) / (\gamma \kappa) m_0$
\pagebreak

$\mu_0/m_0$
\pagebreak

$\ln (\mu_0/m_0)$
\pagebreak

$d\ln n/d\ln h$
\pagebreak

$d\ln e/d\ln h$
\pagebreak

$d\ln p/d\ln h$
\pagebreak

$\rho_0$
\pagebreak

$H > {\tt ent_c}$
\pagebreak

$h > {\tt ent_c}$
\pagebreak

$B_{60}^{3/4}$
\pagebreak

${\rm fm}^{-3}$
\pagebreak

$60\ {\rm MeV\, fm}^{-3}$
\pagebreak

$\epsilon_{\rm fit}$
\pagebreak

$c_s^2 = 1/3(1+\epsilon_{\rm fit})$
\pagebreak

$B_{60}$
\pagebreak

${\rm MeV\,  fm^{-3}}$
\pagebreak

$0.1{\rm \  fm}^{-3}$
\pagebreak

$(4+\epsilon_{\rm fit})/(1+\epsilon_{\rm fit})$
\pagebreak

$ (\gamma - 1 -x_{\rm nd}) / \gamma / x_{\rm nd}$
\pagebreak

$u^\alpha_{\rm n}$
\pagebreak

$u^\alpha_{\rm p}$
\pagebreak

$f_1$
\pagebreak

$f_2$
\pagebreak

\[{\rm d}{\cal E}=f^1{\rm d}n_1+f^2{\rm d}n_2+\alpha{\rm d}\Delta^2
\label{eeosbfdefent}
\]
\pagebreak

$H_1 = \ln\left( \frac{f^1}{m_1c^2} \right)$
\pagebreak

$m_1$
\pagebreak

$H_2$
\pagebreak

$\Delta^2$
\pagebreak

$\Gamma_\Delta = -g_{\alpha\beta} u^\alpha_{\rm n} u^\beta_{\rm p}$
\pagebreak

$m_2$
\pagebreak

$\cal{E}$
\pagebreak

$n_1$
\pagebreak

$n_2$
\pagebreak

\[\Gamma_\Delta = -g_{\alpha\beta} u^\alpha_{\rm n} u^\beta_{\rm p}
\label{e:defgamamdelta}
\]
\pagebreak

\[ \label{eeosbfpolye}
  {\cal{E}} = \frac{1}{2}\kappa_1 n_1^{\gamma_1} + m_1 c^2 \, n_1 
         \  + \frac{1}{2}\kappa_2 n_2^{\gamma_2} + m_2 c^2 \, n_2 
         \  + \kappa_3 n_1^{\gamma_3} n_2^{\gamma_4} 
         \  + \Delta^2 \beta n_1^{\gamma_5} n_2^{\gamma_6}\ .  
\]
\pagebreak

\[\mu_1 := {{\rm d}{\cal{E}} \over {\rm d}n_1} = \frac{1}{2}\gamma_1\kappa_1
        n_1^{\gamma_1-1} + m_1 c^2 + \gamma_3 \kappa_3 
        n_1^{\gamma_3-1} n_2^{\gamma_4} + \Delta^2 \gamma_5 \beta 
        n_1^{\gamma_5-1} n_2^{\gamma_6}\ , 
\]
\pagebreak

\[\mu_2 := {{\rm d}{\cal{E}} \over {\rm d}n_2} = \frac{1}{2}\gamma_2\kappa_2
        n_2^{\gamma_2-1} + m_2 c^2 + \gamma_4 \kappa_3 
        n_1^{\gamma_3} n_2^{\gamma_4-1} + \Delta^2 \gamma_6 \beta 
        n_1^{\gamma_5} n_2^{\gamma_6-1} \ .
\]
\pagebreak

$p = \mu_1 n_1 + \mu_2 n_2 - {\cal E}$
\pagebreak

\[  p = \frac{1}{2} (\gamma_1 -1)\kappa_1 n_1^{\gamma_1} +
 \frac{1}{2}(\gamma_2-1)\kappa_2 n_2^{\gamma_2} + (\gamma_3 +\gamma_4
 -1)\kappa_3 n_1^{\gamma_3}n_2^{\gamma_4} + \Delta^2 \beta \left(
 (\gamma_5 + \gamma_6 - 1) n_1^{\gamma_5} n_2^{\gamma_6} \right) \ .  
\]
\pagebreak

\[\label{eeosbfentanal} 
  H_i := c^2 \ln \left( \frac{f_i}{m_ic^2} \right)   \ .  
\]
\pagebreak

$\gamma_1 = \gamma_2 = 2$
\pagebreak

$\gamma_3 = \gamma_4 = \gamma_5 =
\gamma_6 = 1$
\pagebreak

$\cal E$
\pagebreak

\[\Delta = \left( \vec{v}_n - \vec{v}_p \right)^2
\label{e:defdeltan}
\]
\pagebreak

\[ \label{eeosbfnewte}
  {\cal{E}} = \frac{1}{2}\kappa_1 n_1^{\gamma_1} 
         \  + \frac{1}{2}\kappa_2 n_2^{\gamma_2} 
         \  + \kappa_3 n_1^{\gamma_3} n_2^{\gamma_4} 
         \  + \Delta^2 \beta n_1^{\gamma_5} n_2^{\gamma_6}\ .  
\]
\pagebreak

\[\mu_1 := {{\rm d}{\cal{E}} \over {\rm d}n_1} = \frac{1}{2}\gamma_1\kappa_1
        n_1^{\gamma_1-1} + \gamma_3 \kappa_3 
        n_1^{\gamma_3-1} n_2^{\gamma_4} + \Delta^2 \gamma_5 \beta 
        n_1^{\gamma_5-1} n_2^{\gamma_6}\ , 
\]
\pagebreak

\[\mu_2 := {{\rm d}{\cal{E}} \over {\rm d}n_2} = \frac{1}{2}\gamma_2\kappa_2
        n_2^{\gamma_2-1} + \gamma_4 \kappa_3 
        n_1^{\gamma_3} n_2^{\gamma_4-1} + \Delta^2 \gamma_6 \beta 
        n_1^{\gamma_5} n_2^{\gamma_6-1} \ .
\]
\pagebreak

\[h_i = \frac{\mu_i}{m_i}
\]
\pagebreak

$(\Delta^2, \mu_n, \mu_p )$
\pagebreak

$\mu_n$
\pagebreak

$\mu_p$
\pagebreak

$\Delta^{2}$
\pagebreak

$H_1$
\pagebreak

$\Delta^2 $
\pagebreak

$^2$
\pagebreak

$n_{\rm nuc}^2c^2$
\pagebreak

$\times$
\pagebreak

$K^{11}=2\frac{\partial{\cal{E}}}{\partial{n_1^2}}$
\pagebreak

$x^2=n_1n_2\Gamma_\Delta$
\pagebreak

$K^{12}=\frac{\partial {\cal E}}{\partial (n_1n_2\Gamma_\Delta)}$
\pagebreak

$K^{22} = 2\frac{\partial {\cal E}}{\partial n_2^2}$
\pagebreak

$K^{11}$
\pagebreak

$K^{22}$
\pagebreak

$x^2=n_1n_2\Gamma_\Delta^2$
\pagebreak

$K^{12}$
\pagebreak

$\gamma_1$
\pagebreak

$\gamma_2$
\pagebreak

$\gamma_3$
\pagebreak

$\gamma_4$
\pagebreak

$\gamma_5$
\pagebreak

$\gamma_6$
\pagebreak

$\kappa_1$
\pagebreak

$\kappa_2$
\pagebreak

$\kappa_3$
\pagebreak

$\gamma_1-1$
\pagebreak

$\gamma_2-1$
\pagebreak

$\gamma_3+\gamma_4-1$
\pagebreak

$\gamma_5+\gamma_6-1$
\pagebreak

$\gamma_3 = \gamma_4 = \gamma_5 = \gamma_6 = 1$
\pagebreak

$\gamma_1 \not= 2$
\pagebreak

$\gamma_2 \not= 2$
\pagebreak

$\gamma_3 = \gamma_5 = 1$
\pagebreak

$\gamma_4 = \gamma_6 = 1$
\pagebreak

$\gamma_i$
\pagebreak

$i\in 3\dots 6$
\pagebreak

$\rho_{\rm nuc} c^2 $
\pagebreak

$ \log H_n $
\pagebreak

$ H_n = \ln \left( \frac{\mu_n}{m_n} \right) $
\pagebreak

$ \log H_p $
\pagebreak

$ H_p = \ln \left( \frac{\mu_p}{m_p} \right) $
\pagebreak

$ \Delta^{2} $
\pagebreak

$ \log \Psi $
\pagebreak

$ \frac {\partial \log \Psi } {\partial \log H_n}$
\pagebreak

$ \frac {\partial \log \Psi } {\partial \log H_p}$
\pagebreak

$ \frac {\partial^2 \log \Psi} {\partial \log H_n \partial \log H_p} $
\pagebreak

$ \frac {\partial^2 \log \Psi} {\partial \log H_n \partial \log H_n} $
\pagebreak

$ \frac {\partial^2 \log \Psi} {\partial \log H_p \partial \log H_p} $
\pagebreak

$ \frac {\partial \log \Psi}{\partial \Delta^{2}}  $
\pagebreak

$ \frac {\partial^2 \log \Psi} {\partial \log H_n \partial \Delta^{2}} $
\pagebreak

$ \frac {\partial^2 \log \Psi} {\partial \log H_p \partial \Delta^{2}} $
\pagebreak

$ \frac {\partial^3 \log \Psi} {\partial \log H_p \partial \log H_n  \partial \Delta^{2}} $
\pagebreak

$ K_{nn} $
\pagebreak

$ K_{np} $
\pagebreak

$ K_{pp} $
\pagebreak

$ \Delta^2$
\pagebreak

$ dp = (e+p) dh $
\pagebreak

$n_{\rm nuc}:=0.1\ {\rm fm}^{-3}$
\pagebreak

$(n, B)$
\pagebreak

$\log p$
\pagebreak

$\log h$
\pagebreak

$B$
\pagebreak

$\frac{d \log p}{d \log h}$
\pagebreak

$\frac{d \log p}{d \log B}$
\pagebreak

$\frac{d^2 \log p}{d \log h d \log B}$
\pagebreak

$d\ln p/d\ln n$
\pagebreak

$({\rm g/cm^3})^{1-\gamma_0}$
\pagebreak

$\rho_1$
\pagebreak

$m0$
\pagebreak

$\log H$
\pagebreak

$d\log P/d\log H$
\pagebreak

$\log n_b$
\pagebreak

$d\log P/d\log nb$
\pagebreak

$\psi0$
\pagebreak

${T^\mu}_\nu$
\pagebreak

$S^\varphi_\varphi$
\pagebreak

${S^i}_j$
\pagebreak

$J_\mathrm{euler}$
\pagebreak

$J^i$
\pagebreak

$J_\mathrm{euler}(\varphi)=r \sin\theta\, J^\varphi$
\pagebreak

$E+S_i^i$
\pagebreak

$\rightarrow \rho$
\pagebreak

$\phi=0$
\pagebreak

$\theta=\pi/2$
\pagebreak

$\phi=\pi/2$
\pagebreak

$\phi=\pi$
\pagebreak

$\theta=0$
\pagebreak

$(\theta', \phi')$
\pagebreak

$Q = {\bar Q } - 4/3 (1/4 + b) M^3 $
\pagebreak

$ {\bar Q } $
\pagebreak

$ {\bar Q }$
\pagebreak

$bM^2$
\pagebreak

$F(\Omega)$
\pagebreak

\[F(\Omega) = u^t u_\varphi \ . 
\]
\pagebreak

$\Omega(r,\theta)$
\pagebreak

$\int_{\Omega_{\rm c}}^\Omega F(\Omega') \, d\Omega' $
\pagebreak

\[F(\Omega) - {B^2 r^2\sin^2\theta (\Omega - N^\varphi)
    \over N^2 - B^2 r^2 \sin^2(\Omega-N^\varphi)^2} = 0
\]
\pagebreak

\[F(\Omega) = u^t u_\varphi \ .
\]
\pagebreak

$\varphi \varphi$
\pagebreak

$\cal M$
\pagebreak

$\sigma = \frac{2{\cal M}M}{QJ}$
\pagebreak

${\bar Q }$
\pagebreak

$B_o$
\pagebreak

\[ \label{eetoilemetrique}
    ds^2 = - (N^2 - N_i N^i) dt^2 - 2 N_i \,  dt\,  dx^i
        + A^2  \,  {\tilde h}_{ij} \,  dx^i dx^j
\]
\pagebreak

${\tilde h}_{ij}$
\pagebreak

$A^2$
\pagebreak

$\tilde h_{ij}$
\pagebreak

$1/c^2$
\pagebreak

$\phi=3\pi/2$
\pagebreak

$\phi=2k\pi/np$
\pagebreak

$\Psi_0$
\pagebreak

$\tilde K^{ij} = A^2 K^{ij}$
\pagebreak

$\tilde K^{ij}  = A^2 K^{ij}$
\pagebreak

$A^2 K_{ij}^{\rm auto} K^{ij}_{\rm auto}$
\pagebreak

$A^2 K_{ij}^{\rm auto} K^{ij}_{\rm comp}$
\pagebreak

$B^i/N$
\pagebreak

$\psi$
\pagebreak

$A^2 K_{ij}^{\rm auto} K^{ij}_{\rm comp} $
\pagebreak

\[  X_G := \int A^3 \Gamma_{\rm n} \,  n \,  X \, d^3x \ ,  
\]
\pagebreak

$\Gamma_{\rm n}$
\pagebreak

$V^i$
\pagebreak

$h_{ij} V^i W^j = A^2 f_{ij} V^i W^j$
\pagebreak

${\tilde K_{ij}}$
\pagebreak

${\tilde K_{ij}} = B^{-2} K_{ij}$
\pagebreak

$ h^{ij} $
\pagebreak

$\hat{A}^{ij}$
\pagebreak

$ r=1 $
\pagebreak

$ \psi $
\pagebreak

$ gamij $
\pagebreak

$ K_{ij}$
\pagebreak

$ N $
\pagebreak

$ \beta^{i} $
\pagebreak

$ N \psi $
\pagebreak

$\xi = 1$
\pagebreak

$\phi'$
\pagebreak

$\theta'$
\pagebreak

$F_l(\theta', \phi')$
\pagebreak

$G_l(\theta', \phi')$
\pagebreak

$(\theta, \phi)$
\pagebreak

$\phi=\pi/4$
\pagebreak

$\theta=0,\ \pi/4,\ \pi/2$
\pagebreak

$\theta=0,\ \pi/4$
\pagebreak

$\theta \to -\theta$
\pagebreak

$(x,y) \to (-x,-y)$
\pagebreak

$(r,\theta,\phi)$
\pagebreak

$\xi\in [0,1]$
\pagebreak

$ 1 \le {\tt l} \le {\tt nz}-2$
\pagebreak

$\xi\in [-1,1]$
\pagebreak

$[0,\pi/2]$
\pagebreak

$[0,\pi]$
\pagebreak

$[0,\pi[$
\pagebreak

$\pi$
\pagebreak

$[0,2\pi[$
\pagebreak

$\xi\in[0,1]$
\pagebreak

$\xi\in[-1,1]$
\pagebreak

$u=1/r$
\pagebreak

$\theta\in[0,\pi/2]$
\pagebreak

$\theta\in[0,\pi]$
\pagebreak

$\phi\in[0,\pi[$
\pagebreak

$ \phi \mapsto \phi + \pi$
\pagebreak

$\phi\in[0,2\pi[$
\pagebreak

$A_{ij}^{\rm tot}$
\pagebreak

\[    p(n, s_b) = \kappa n^\gamma e^{(\gamma-1)s_b}\ .\qquad (1)
\]
\pagebreak

\[    e(n, s_b) = \frac{\kappa}{\gamma - 1} n^\gamma e^{(\gamma-1)s_b} + m_0\, n\ .
    \qquad (2) \]
\pagebreak

$H = c^2 \ln\left( {e+p \over m_B c^2 n} (to be modified) \right) $
\pagebreak

$k_B$
\pagebreak

$s_b$
\pagebreak

$H = \log ( e + P ) / n_B$
\pagebreak

$s_B$
\pagebreak

$\partial P/\partial H$
\pagebreak

$\partial P/\partial s_B$
\pagebreak

$\partial^2 P/\partial s_B \partial H$
\pagebreak

$ \Omega$
\pagebreak

$ \varphi $
\pagebreak

$ N_{auto} $
\pagebreak

$ N_{comp} $
\pagebreak

$ \Psi_{auto} $
\pagebreak

$ \Psi_{comp} $
\pagebreak

$ \overline\nabla_i N $
\pagebreak

$ \overline\nabla_i \Psi $
\pagebreak

$ \beta^i_{auto} $
\pagebreak

$ \beta^i_{comp} $
\pagebreak

$ A^{ij}_{auto} $
\pagebreak

$ A^{ij}_{comp} $
\pagebreak

$ A^{ij}*2N $
\pagebreak

$ A^{ij}A_{ij} $
\pagebreak

$ \gamma_{ij} $
\pagebreak

$ \gamma_{ij} = \Psi^4 \tilde\gamma_{ij} $
\pagebreak

$ A^{ij} $
\pagebreak

$ A^{ij} = \Psi^4 \left( K^{ij} - \frac{1}{3} K \gamma^{ij} \right) $
\pagebreak

$ \tilde\gamma_{ij} = \Psi^{-4} \gamma_{ij} $
\pagebreak

$\nabla_i N$
\pagebreak

$\nabla_i \Psi$
\pagebreak

$ \Psi $
\pagebreak

\[ \Psi := \left( \frac{\det\gamma_{ij}}{\det f_{ij}} 
     \right) ^{1/12} \]
\pagebreak

$ \bf n + \bf s $
\pagebreak

$ \beta $
\pagebreak

$ \partial_r N + a N = 0 $
\pagebreak

$ \omega $
\pagebreak

$ N = a $
\pagebreak

$  V^i $
\pagebreak

$ \Psi^4 $
\pagebreak

$ \beta^i $
\pagebreak

$ \gamma $
\pagebreak

$ K^{ij} $
\pagebreak

$ N$
\pagebreak

$M = \frac{1}{4 \pi} \oint_{\infty} D^i N {\mathrm d} S_i$
\pagebreak

$h^{ij}$
\pagebreak

$(\xi, \theta', \phi')$
\pagebreak

$x=r\sin\theta\cos\phi, \ldots$
\pagebreak

$(\partial/\partial r,1/r\partial/\partial \theta,
        1/(r\sin\theta)\partial/\partial \phi)$
\pagebreak

$(\partial/\partial x,\partial/\partial y,\partial/\partial z)$
\pagebreak

$\sin\theta$
\pagebreak

$\cos\theta$
\pagebreak

$\sin\phi$
\pagebreak

$\cos\phi$
\pagebreak

$(\partial/\partial r,1/r\partial/\partial \theta,
 1/(r\sin\theta)\partial/\partial \phi)$
\pagebreak

$r=R_l(\xi, \theta', \phi')$
\pagebreak

$R(\theta, \varphi)$
\pagebreak

$0\le {\tt l} \le {\tt nzet-1}$
\pagebreak

$\partial/ \partial \xi$
\pagebreak

$-\partial/ \partial u = \xi^2 \partial/ \partial \xi$
\pagebreak

$\partial/ \partial r$
\pagebreak

$-\partial/ \partial u = r^2 \partial/ \partial r$
\pagebreak

$1/r \partial/ \partial \theta$
\pagebreak

$1/u \partial/ \partial \theta = r \partial/ \partial \theta$
\pagebreak

$1/(r\sin\theta) \partial/ \partial \phi$
\pagebreak

$1/(u\sin\theta) \partial/ \partial \phi = 
    r/\sin\theta \partial/ \partial \phi$
\pagebreak

$\partial/ \partial xi$
\pagebreak

$\partial/ \partial \ln r$
\pagebreak

$\partial/ \partial \theta$
\pagebreak

$1/\sin\theta \partial/ \partial \varphi$
\pagebreak

$r\to \infty$
\pagebreak

$ F(r,\theta,\varphi) = \int_r^\infty f(r',\theta,\varphi) \, dr' $
\pagebreak

$\tan\theta$
\pagebreak

$ a \Delta\psi + {\bf b} \cdot \nabla \psi = \sigma$
\pagebreak

$\psi(0)=0$
\pagebreak

$\Delta_{\theta\varphi} u + \lambda u = \sigma$
\pagebreak

$\Delta_{\theta\varphi} u := \frac{\partial^2 u}
 {\partial \theta^2} + \frac{1}{\tan \theta} \frac{\partial u}
 {\partial \theta} +\frac{1}{\sin^2 \theta}\frac{\partial^2 u}
 {\partial \varphi^2}$
\pagebreak

\[{\partial^2 u\over\partial r^2} + 
    {1\over r} {\partial u \over \partial r} + 
    {1\over r^2} {\partial^2 u\over\partial \theta^2} = 
    \sigma \ . 
\]
\pagebreak

$f^{J+1}$
\pagebreak

$f^J$
\pagebreak

$f^{J-1}$
\pagebreak

$\diamond u = \sigma$
\pagebreak

$r=R(\xi, \theta', \phi')$
\pagebreak

$\theta=\theta'$
\pagebreak

$\phi=\phi'$
\pagebreak

$r=\alpha \xi + \beta$
\pagebreak

$ u={1\over r} = \alpha \xi + \beta$
\pagebreak

$\alpha$
\pagebreak

$r = \alpha [\xi + A(\xi) F(\theta', \phi') + B(\xi) G(\theta', \phi')]
        + \beta$
\pagebreak

$ u={1\over r} = \alpha [\xi + A(\xi) F(\theta', \phi')] + \beta$
\pagebreak

$A(\xi)$
\pagebreak

$B(\xi)$
\pagebreak

$A(\xi) = 3 \xi^4 - 2 \xi^6$
\pagebreak

$B(\xi) = (5\xi^3 - 3\xi^5)/2$
\pagebreak

$A(\xi) = (\xi^3 - 3\xi + 2)/4$
\pagebreak

$B(\xi) = (-\xi^3 + 3\xi +2)/4$
\pagebreak

$F(\theta', \phi')$
\pagebreak

$G(\theta', \phi')$
\pagebreak

$\ln r=\alpha \xi + \beta$
\pagebreak

$\xi/R$
\pagebreak

$(\xi-1)/U$
\pagebreak

$1/(\partial R/\partial\xi) = \partial \xi /\partial r$
\pagebreak

$-1/(\partial U/\partial\xi) = - \partial \xi /\partial u$
\pagebreak

$\partial R/\partial\theta'$
\pagebreak

$-\partial U/\partial\theta'$
\pagebreak

${1\over\sin\theta} \partial R/\partial\varphi'$
\pagebreak

$-{1\over\sin\theta}\partial U/\partial\varphi'$
\pagebreak

$1/R \times (\partial R/\partial\theta')$
\pagebreak

$-1/U \times (\partial U/\partial\theta)$
\pagebreak

$1/(R\sin\theta) \times (\partial R/\partial\varphi')$
\pagebreak

$-1/(U\sin\theta) \times (\partial U/\partial\varphi')$
\pagebreak

$1/R^2 \times (\partial R/\partial\theta')$
\pagebreak

$-1/U^2 \times (\partial U/\partial\theta')$
\pagebreak

$1/(R^2\sin\theta) \times (\partial R/\partial\varphi')$
\pagebreak

$-1/(U^2\sin\theta) \times (\partial U/\partial\varphi')$
\pagebreak

$\partial^2 R/\partial\xi^2$
\pagebreak

$-\partial^2 U/\partial\xi^2 $
\pagebreak

$1/R^2 \times [ 1/\sin(\theta)\times \partial /\partial\theta'
  (\sin\theta \partial R /\partial\theta') + 1/\sin^2\theta
  \partial^2 R /\partial{\varphi'}^2] $
\pagebreak

$- 1/U^2 \times [ 1/\sin(\theta)\times \partial /\partial\theta'
  (\sin\theta \partial U /\partial\theta') + 1/\sin^2\theta
  \partial^2 U /\partial{\varphi'}^2] $
\pagebreak

$\partial^2 R/\partial\xi\partial\theta'$
\pagebreak

$-\partial^2 U/\partial\xi\partial\theta'$
\pagebreak

$1/\sin\theta \times \partial^2 R/\partial\xi\partial\varphi'$
\pagebreak

$-1/\sin\theta \times \partial^2 U/\partial\xi\partial\varphi' $
\pagebreak

$1/R^2 \partial^2 R/\partial{\theta'}^2$
\pagebreak

$-1/U^2 \partial^2 U/\partial{\theta'}^2$
\pagebreak

$r=R_l(\xi, {\theta'}_j, {\phi'}_k)$
\pagebreak

$\psi^J$
\pagebreak

$\psi^{J-1}$
\pagebreak

$\ell$
\pagebreak

$A'(\xi)$
\pagebreak

$A''(\xi)$
\pagebreak

$A(\xi)/\xi$
\pagebreak

$A(\xi)/\xi^2$
\pagebreak

$A(\xi)/(\xi-1)$
\pagebreak

$A(\xi)/(\xi-1)^2$
\pagebreak

$B'(\xi)$
\pagebreak

$B''(\xi)$
\pagebreak

$B(\xi)/\xi$
\pagebreak

$B(\xi)/\xi^2$
\pagebreak

$1/(\partial R/\partial \xi) R/\xi$
\pagebreak

$1/(\partial R/\partial \xi) R/(\xi + \beta/\alpha)$
\pagebreak

$1/(\partial U/\partial \xi) U/(\xi-1)$
\pagebreak

$[ R/ (\alpha \xi + \beta) ]^2 (\partial R/\partial \xi) / \alpha$
\pagebreak

$\partial U/\partial \xi / \alpha$
\pagebreak

${\rm tab}(k,j) = r(\phi_k,\theta_j)$
\pagebreak

$F(\theta',\phi')$
\pagebreak

$G(\theta',\phi')$
\pagebreak

$(\theta_*, \phi_*)$
\pagebreak

$\mu$
\pagebreak

$a \tilde\Delta u = \sigma + R(u)$
\pagebreak

$\tilde\Delta$
\pagebreak

$\tilde\Delta u^{J+1} = s^J$
\pagebreak

\[  s^J = 1/a_l^{\rm max} \{ {\tt source} + R(u^J) + (a_l^{\rm max}-a)
         [ \lambda s^{J-1} + (1-\lambda) s^{J-2} ] \} \ ,  
\]
\pagebreak

$a_l^{\rm max} := \max(a)$
\pagebreak

$u^J$
\pagebreak

$u^{J-1}$
\pagebreak

$s^{J-1}$
\pagebreak

$ \partial x / \partial \ln r$
\pagebreak

$ A(i, j)$
\pagebreak

$[ku+1+i-j, j]$
\pagebreak

$\mathrm {max}(1, j-ku) \leq i \leq
\mathrm{min} (n, j+kl)$
\pagebreak

${\tt Mtbl}^{\tt int}$
\pagebreak

${\tt Mtbl}^{\tt double}$
\pagebreak

$(\xi, {\theta'}_j, {\phi'}_k)$
\pagebreak

$\xi = 1 $
\pagebreak

$\xi = -1 $
\pagebreak

$\xi = -1$
\pagebreak

${\partial \over \partial \xi} $
\pagebreak

${\partial^2\over \partial \xi^2} $
\pagebreak

${1 \over \xi} $
\pagebreak

${1 \over \xi-1} $
\pagebreak

${1 \over \xi^2}$
\pagebreak

${1 \over (\xi-1)^2}$
\pagebreak

$\xi \, Id$
\pagebreak

$(\xi-1) \, Id $
\pagebreak

${1 \over (\xi-1)}$
\pagebreak

$(\xi-1) \, Id$
\pagebreak

$(\xi-1)^2 \, Id$
\pagebreak

${\partial \over \partial \theta}$
\pagebreak

${\partial^2 \over \partial \theta^2}$
\pagebreak

$Id\over\sin\theta$
\pagebreak

$Id\over\cos\theta$
\pagebreak

$\cos\theta \, Id$
\pagebreak

$\sin\theta \, Id$
\pagebreak

${\partial \over \partial \phi}$
\pagebreak

${\partial^2 \over \partial \phi^2}$
\pagebreak

$\cos\phi \, Id$
\pagebreak

$\sin\phi \, Id$
\pagebreak

$u$
\pagebreak

$\Delta - m^2$
\pagebreak

$m > 0$
\pagebreak

$\Delta + m^2$
\pagebreak

$ a r^2 \partial^2 / \partial r^2 + b r \partial / \partial r + c$
\pagebreak

$ a \partial^2 / \partial r^2 + b \partial / \partial r + c$
\pagebreak

$\Delta + \frac{2}{r} 
*\frac{\partial}{\partial r} 
  + \frac{2}{r^2} $
\pagebreak

$ l \not= 0 $
\pagebreak

$\frac{\partial^2}{\partial r^2} + \frac{2}{r} \frac{\partial}
  {\partial r} - \frac{2}{r^2} $
\pagebreak

$\Delta + \frac{4}{r} 
*\frac{\partial}{\partial r} 
  + \frac{6}{r^2} $
\pagebreak

$l \geq 2$
\pagebreak

$b$
\pagebreak

$c$
\pagebreak

$l$
\pagebreak

$\left(\Delta - m^2\right)$
\pagebreak

$\left(\Delta - 2 \partial_r / r\right)$
\pagebreak

$\left(\Delta - 2 \partial_r / r - m^2\right)$
\pagebreak

$\left(\Delta + m^2\right)$
\pagebreak

$a r^2 \partial^2/\partial r^2 + 
b r \partial /\partial r + c$
\pagebreak

$a \partial^2/\partial r^2 + 
b \partial /\partial r + c$
\pagebreak

$\Delta - 2\partial /\partial r$
\pagebreak

$\Delta + \frac{2}{r} \frac{\partial}{\partial r} 
+ \frac{2 - l(l+1)}{r^2} $
\pagebreak

$\frac{\partial^2}{\partial r^2} + \frac{2}{r} 
\frac{\partial}{\partial r} - \frac{2}{r^2} $
\pagebreak

$\eta $
\pagebreak

$\frac{\partial^2}{\partial r^2} + 
\frac{2}{r} \frac{\partial}{\partial r} - \frac{l(l-1)}{r^2} $
\pagebreak

$\frac{\partial^2}{\partial r^2} + 
\frac{2}{r} \frac{\partial}{\partial r} - \frac{(l+1)(l+2)}{r^2} $
\pagebreak

$ \eta $
\pagebreak

$\Delta + \frac{4}{r} \frac{\partial}{\partial r} 
+ \frac{6 - l(l+1)}{r^2} $
\pagebreak

${\tt Scalar}^{\tt int}$
\pagebreak

${\tt Scalar}^{\tt double}$
\pagebreak

$\partial/\partial \theta$
\pagebreak

$1/\sin\theta \partial/\partial \phi$
\pagebreak

$\partial/\partial radial $
\pagebreak

$\partial/\partial \rho $
\pagebreak

$ r^{\rm dzpuis} $
\pagebreak

$\partial / \partial \theta$
\pagebreak

$\partial / \partial \ln r$
\pagebreak

$\partial / \partial \rho $
\pagebreak

$1/\sin\theta \partial / \partial \phi$
\pagebreak

$\Delta_{\theta\varphi}$
\pagebreak

$\Delta_{\theta\varphi} f = \frac{\partial^2 f}
{\partial \theta^2} + \frac{1}{\tan \theta} \frac{\partial f}
{\partial \theta} +\frac{1}{\sin^2 \theta}\frac{\partial^2 f}
{\partial \varphi^2}$
\pagebreak

$ \forall n\leq N,\, b_n = \sigma(n/N ) a_n$
\pagebreak

$ \sigma(x) = \exp\left( -\ln (10^\alpha ) x^{2p} \right) $
\pagebreak

$ \sigma(x) = \exp\left( \alpha x^{p} \right) $
\pagebreak

$ \forall \ell \leq \ell_{\rm max},\, \forall m,\, b_{\ell m} = \sigma(\ell/\ell_{\rm max} ) a_{\ell m}$
\pagebreak

$ \sigma(x) $
\pagebreak

$\ell_{\rm max}$
\pagebreak

$C^k$
\pagebreak

$\sum_{j=0}^k {\alpha_j \over r^{\ell+n+j}}$
\pagebreak

${\tt puis}  \leq i \leq {\tt puis+nbre}$
\pagebreak

\[  \sum_{k=0}^n {a_k(\theta, \phi) \over r^k}
\]
\pagebreak

$r^3 \sigma$
\pagebreak

$ fact\_dir.\Psi + fact\_neu.\frac{\partial 
\Phi}{\partial r}$
\pagebreak

$t=J \to J+1$
\pagebreak

$ \frac{\partial \phi}{\partial r} + \frac{n}{r} 
\phi = \sigma$
\pagebreak

$ \theta_+ = 0$
\pagebreak

$(\theta, \varphi)$
\pagebreak

$ q_{ab} $
\pagebreak

$ H_{ab} $
\pagebreak

$ L_a $
\pagebreak

$ J_{ab} $
\pagebreak

$\sqrt{\det q_{ab}} $
\pagebreak

$ R = q^{ab}R_{ab} $
\pagebreak

$ Q_{ab}$
\pagebreak

$ Pi $
\pagebreak

$ J $
\pagebreak

$ S_a $
\pagebreak

$ Q_{ab} $
\pagebreak

$ Pi  $
\pagebreak

$ s_a $
\pagebreak

\[{\cal A} = \int h^2 \sqrt{\det q_{ab}} \sin \theta {\rm d}\theta
{\rm d}\varphi \]
\pagebreak

\[{\cal J} = \int h^2 L_i \phi^i \sqrt{\det q_{ab}} \sin \theta {\rm d}\theta
{\rm d}\varphi \]
\pagebreak

$ \phi $
\pagebreak

$ phi $
\pagebreak

\[R_{s} \]
\pagebreak

\[ M = \frac{1}{2 R_{s}} \sqrt{R_{s}^{4} + 4{\cal J}^{2}} \]
\pagebreak

$ \theta_+ $
\pagebreak

$ \theta_- $
\pagebreak

$ \sigma_{ab} $
\pagebreak

$\beta^i/N$
\pagebreak

$ Q = \psi^2 N $
\pagebreak

$\psi^4$
\pagebreak

$\tilde \gamma^{ij}$
\pagebreak

$\tilde K^{ij}$
\pagebreak

$K_{ij} K^{ij}$
\pagebreak

$K_{ij}^{\rm auto} K^{ij}_{\rm auto}$
\pagebreak

$K_{ij}^{\rm auto} K^{ij}_{\rm comp}$
\pagebreak

$lnq_auto$
\pagebreak

$lnq$
\pagebreak

$beta_auto$
\pagebreak

$beta$
\pagebreak

$Q$
\pagebreak

$logn$
\pagebreak

$\tilde \gamma$
\pagebreak

$\tilde K^{ij} \tilde K_{ij}$
\pagebreak

$ \chi_{auto} = (N\Psi)_{auto} $
\pagebreak

$ \chi_{comp} = (N\Psi)_{comp} $
\pagebreak

$\Psi_{auto}$
\pagebreak

$\Psi_{comp}$
\pagebreak

$\hat{A}_{ij}\hat{A}^{ij}$
\pagebreak

$\hat{A}_{ij}^{\rm auto}\hat{A}^{ij}_{\rm auto}$
\pagebreak

$\hat{A}_{ij}^{\rm auto}\hat{A}^{ij}_{\rm comp}$
\pagebreak

$\chi_{auto}$
\pagebreak

$\chi_{comp}$
\pagebreak

$\beta_{auto}$
\pagebreak

$\beta^i_{auto}$
\pagebreak

$\beta^i_{comp}$
\pagebreak

$\chi_{auto} = (N\Psi)_{auto}$
\pagebreak

$\chi_{comp} = (N\Psi)_{comp}$
\pagebreak

$\Psi^4$
\pagebreak

$\hat{A}^{ij}\hat{A}_{ij}$
\pagebreak

$ \partial_j \alpha $
\pagebreak

$ \eta^{ik} \partial_k \beta^j $
\pagebreak

$ \partial_j \psi $
\pagebreak

$ A^{ij}$
\pagebreak

$A_{ij}^{\rm auto}$
\pagebreak

$\tilde A^{ij}$
\pagebreak

${\rm km}^2$
\pagebreak

$ r_{\rm mean} = \sqrt{\mathcal{A}} / 4\pi$
\pagebreak

$0 < l <$
\pagebreak

$\Psi^2$
\pagebreak

$Q = \Psi^2 N$
\pagebreak

$ln(\Psi)$
\pagebreak

$\gamma_{ij}v^i v^j$
\pagebreak

$f_{ij}$
\pagebreak

$\tilde{\gamma}_{ij}$
\pagebreak

$\tilde{A}_{ij}$
\pagebreak

$\tilde{A}_{ij} A^{ij}$
\pagebreak

$\tilde{\gamma}^{ij}=f^{ij}+h^{ij}$
\pagebreak

$D_j h^{ij}$
\pagebreak

$Q=\Psi^2 N$
\pagebreak

$ e = \sqrt{1 - \left( \frac{R^c_e}{R^c_p} \right)^2} $
\pagebreak

$R^c_e$
\pagebreak

$R^c_p$
\pagebreak

\[F(\Omega) - {\gamma_{\varphi \varphi} (\beta^\varphi + \Omega)
    \over N^2 - \gamma_{\varphi \varphi}(\beta^\varphi + \Omega)^2} = 0
\]
\pagebreak

$(e_r,e_\theta,e_\varphi)$
\pagebreak

${}^t T^{ij}$
\pagebreak

\[        T^{ij} = {}^t T^{ij} + \nabla^i W^j + \nabla^j W^i  
        \qquad\mbox{with}\quad \nabla_j {}^t T^{ij} = 0 
*\]
\pagebreak

$\nabla_i$
\pagebreak

$\eta$
\pagebreak

$(T^{r\theta}, T^{r\varphi})$
\pagebreak

\[    T^{r\theta} =  {1\over r} \left( {\partial \eta \over \partial\theta} -
    {1\over\sin\theta} {\partial \mu \over \partial\varphi} \right) 
*\]
\pagebreak

\[    T^{r\varphi} =  {1\over r} \left( {1\over\sin\theta} 
                {\partial \eta \over \partial\varphi}
                + {\partial \mu \over \partial\theta} \right)
*\]
\pagebreak

\[    T^{r\theta} =  {1\over r} \left( {\partial \eta \over \partial\theta} -
     {1\over\sin\theta} {\partial \mu \over \partial\varphi} \right) 
*\]
\pagebreak

$T^{\theta\theta}, 
  T^{\varphi\varphi}$
\pagebreak

$T^{\theta\varphi}$
\pagebreak

\[  \frac{1}{2}\left(T^{\theta\theta} - T^{\varphi\varphi} \right) 
  = \frac{\partial^2 W}{\partial\theta^2} - \frac{1}{\tan
  \theta} \frac{\partial W}{\partial \theta} - \frac{1}{\sin^2 \theta} 
  \frac{\partial^2 W}{\partial \varphi^2} - 2\frac{\partial}{\partial \theta} 
  \left( \frac{1}{\sin \theta} \frac{\partial X}{\partial \varphi} \right) ,
*\]
\pagebreak

\[   T^{\theta\varphi} = \frac{\partial^2 X}{\partial\theta^2} - \frac{1}{\tan
  \theta} \frac{\partial X}{\partial \theta} - \frac{1}{\sin^2 \theta} 
  \frac{\partial^2 X}{\partial \varphi^2} + 2\frac{\partial}{\partial \theta} 
  \left( \frac{1}{\sin \theta} \frac{\partial W}{\partial \varphi} \right) .
*\]
\pagebreak

$ T = T^{\theta\theta} + T^{\varphi\varphi} $
\pagebreak

$\ell \geq 2$
\pagebreak

\[A = \frac{\partial X}{\partial r} - \frac{\mu}{r^2}.
\]
\pagebreak

$ \tilde{B}$
\pagebreak

$ h^{rr}, \eta, W$
\pagebreak

\[\tilde{B} = (\ell + 2) \frac{\partial W}{\partial r} + \ell(\ell + 2)
\frac{W}{r} - \frac{2\eta}{r^2} + \frac{(\ell +2)T}{2r(\ell + 1)}
+ \frac{1}{2(\ell + 1)} \frac{\partial T}{\partial r} - \frac{h^{rr}}
{(\ell + 1)r}.
\]
\pagebreak

$ \tilde{C}$
\pagebreak

\[\tilde{C} = - (\ell - 1) \frac{\partial W}{\partial r} + (\ell + 1)(\ell - 1)
\frac{W}{r} - \frac{2\eta}{r^2} + \frac{(\ell - 1)T}{2r\ell}
- \frac{1}{2 \ell } \frac{\partial T}{\partial r} - \frac{h^{rr}}
{\ell r}.
\]
\pagebreak

$ T^{rr} $
\pagebreak

$ \eta / r$
\pagebreak

$\mu / r $
\pagebreak

$\tilde{B}$
\pagebreak

$\tilde{C}$
\pagebreak

\begin{eqnarray*}\frac{\partial \tilde{\mu}}{\partial r}  + \frac{3\tilde{\mu}}{r} + \left( 
\Delta_{\theta\varphi } + 2\right) X &=& 0;\\
\frac{\partial X}{\partial r} - \frac{\tilde{\mu}}{r} &=& A. \end{eqnarray*}
\pagebreak

$\tilde{\mu} = \mu / r$
\pagebreak

$\tilde{\mu}$
\pagebreak

\begin{eqnarray*}\frac{\partial T^{rr}}{r} + \frac{3T^{rr}}{r} +\frac{1}{r}
 \Delta_{\theta\varphi } \tilde{\eta} &=& \frac{h}{r};\\
\frac{\partial \tilde{\eta}}{\partial r} + \frac{3\tilde{\eta}}{r} -
\frac{T^{rr}}{2r} + \left( \Delta_{\theta\varphi } + 2\right) 
\frac{W}{r} &=& -\frac{h}{2r};\\
(\ell + 2) \frac{\partial W}{\partial r} + \ell(\ell + 2)
\frac{W}{r} - \frac{2\tilde{\eta}}{r} + \frac{(\ell +2)T}{2r(\ell + 1)}
+ \frac{1}{2(\ell + 1)} \frac{\partial T}{\partial r} - \frac{T^{rr}}
{(\ell + 1)r} &=& \tilde{B} - \frac{1}{2(\ell +1)} \frac{\partial h}
{\partial r} - \frac{\ell +2}{\ell +1} \frac{h}{2r}.\end{eqnarray*}
\pagebreak

$\tilde{\eta} = \eta / r$
\pagebreak

$\tilde{\eta}$
\pagebreak

$\ell=0,1$
\pagebreak

$\frac {\mu}  {r} $
\pagebreak

$ h^{rr} $
\pagebreak

$ \tilde{B} $
\pagebreak

$= S^{ij}$
\pagebreak

\[     \Delta h^{ij} = S^{ij}.
*\]
\pagebreak

$h^{ij}=0$
\pagebreak

$h^{rr} = \frac{\chi}{r^2}$
\pagebreak

$h^{rr}$
\pagebreak

$h^{r\theta}$
\pagebreak

$h^{r\varphi}$
\pagebreak

$h^{\theta\theta}$
\pagebreak

$h^{\theta\varphi}$
\pagebreak

$h^{\varphi\varphi}$
\pagebreak

\[     \Delta h^{ij} = S^{ij}
*\]
\pagebreak

${\tt Tbl}^{\tt int}$
\pagebreak

${\tt Tbl}^{\tt double}$
\pagebreak

${\tt Tbl_val}^{\tt int}$
\pagebreak

${\tt Tbl_val}^{\tt double}$
\pagebreak

$\tau^{i_1\ldots i_p}_{j_1\ldots j_q}$
\pagebreak

$ \tau^{i_1\ldots i_p}_{j_1\ldots j_q} = \gamma^{\frac{n}{2}} 
T^{i_1\ldots i_p}_{j_1\ldots j_q}$
\pagebreak

$\delta_{ij}$
\pagebreak

$x^k$
\pagebreak

$x_k$
\pagebreak

$\Delta + \lambda \nabla\nabla$
\pagebreak

$\Delta N^i +\lambda \nabla^i \nabla_k N^k = S^i$
\pagebreak

$\lambda \not= 1$
\pagebreak

\[ N^i = \frac{1}{2}\frac{\lambda+2}{\lambda+1}W^i-\frac{1}{2}
\frac{\lambda}{\lambda+1}\left(\nabla^i\chi+\nabla^iW^kx_k\right)
\]
\pagebreak

$\Delta W^i = S^i$
\pagebreak

$\Delta \chi = -x_kS^k$
\pagebreak

$\Delta N^i +\lambda \nabla^i
\nabla_k N^k = S^i$
\pagebreak

\[N^i = \frac{1}{2}\frac{\lambda+2}{\lambda+1}W^i-\frac{1}{2}
\frac{\lambda}{\lambda+1}\left(\nabla^i\chi+\nabla^iW^kx_k\right)
\]
\pagebreak

\[ \Delta N^i = S^i-\lambda \nabla^i \chi 
\]
\pagebreak

\[  \Delta \chi = \frac{1}{\lambda+1}\nabla_k S^k
\]
\pagebreak

\[ \Delta \chi = \frac{1}{\lambda+1}\nabla_k S^k
\]
\pagebreak

$1/(\gamma - 1)$
\pagebreak

$L^1$
\pagebreak

$e^i$
\pagebreak

$T_i$
\pagebreak

$T=T_i \; e^i$
\pagebreak

$\nabla_j T_i$
\pagebreak

\[    \nabla T = \nabla_j T_i \; e^i \otimes e^j
\]
\pagebreak

\[     {\rm div}\,  T = \nabla_k T^{ik} \; e_i
\]
\pagebreak

$ \gamma^{ij} $
\pagebreak

$ K_{ij} $
\pagebreak

$ \mathbf{\gamma} $
\pagebreak

\[R + K^2 - K_{ij}K^{ij} = 16\pi E 
\]
\pagebreak

$\mbox{\boldmath{$n $}}$
\pagebreak

\[D_j K_i^{\ j} - D_i K = 8 \pi J_i
\]
\pagebreak

$ J_i $
\pagebreak

\[    \frac{\partial K_{ij}}{\partial t} - 
\pounds_{\mbox{\boldmath{$\beta $}}} K_{ij} =  - D_i D_j N 
 + N \left[ R_{ij} - 2 K_{ik} K^k_{\ j} + K K_{ij} 
 + 4\pi \left( (S-E)\gamma_{ij} - 2 S_{ij} \right) 
 \right]
\]
\pagebreak

$ S_{ij} $
\pagebreak

$ f_{ij} $
\pagebreak

$ N\Psi $
\pagebreak

$ \tilde\gamma^{ij} $
\pagebreak

$ f^{ij} $
\pagebreak

$\tilde\gamma^{ij} = f^{ij} + h^{ij} $
\pagebreak

$ \hat{A}^{ij} $
\pagebreak

$ \hat{A}^{ij} = \Psi^{10} \left( K^{ij} - \frac{1}{3} K \gamma^{ij} \right) $
\pagebreak

$ \hat{A}^{ij} = {\cal D}^i X^j + {\cal D}^j X^i - \frac{2}{3} 
{\cal D}_k X^k f^{ij} + \hat{A}^{ij}_{TT} $
\pagebreak

$ X^i $
\pagebreak

$ \hat{A}^{ij}_{TT} $
\pagebreak

$ \tilde\gamma_{ij} $
\pagebreak

$ H^i = {\cal D}_j \tilde\gamma^{ij} $
\pagebreak

$\det\tilde\gamma^{ij} = f^{-1} $
\pagebreak

$N\Psi$
\pagebreak

$ \hat{A}{ij} $
\pagebreak

\[ \Psi := \left( \frac{\det\gamma_{ij}}{\det f_{ij}} \right) ^{1/12} \]
\pagebreak

$ \hat{S}^i = \Psi^6 S^i $
\pagebreak

$ \bar{h}^{ij} $
\pagebreak

$\tilde{B} $
\pagebreak

$ {\tilde u}^{ij} = \partial h^{ij} /\partial t $
\pagebreak

$\tilde\gamma_{ij}$
\pagebreak

$ K = K_i^{\ i} $
\pagebreak

$ \partial K / \partial t $
\pagebreak

$\chi $
\pagebreak

$ \hat{E} = \Psi^6 E $
\pagebreak

$\Psi_{\rm new}$
\pagebreak

$ S^* = \Psi^6 S $
\pagebreak

$(N\Psi)_{\rm new}$
\pagebreak

$\beta^i_{\rm new}$
\pagebreak

$ S^{ij} $
\pagebreak

$\det\tilde\gamma^{ij} = \det f^{ij} $
\pagebreak

$\delta < \delta_{crit}$
\pagebreak

$\delta > \delta_{crit}$
\pagebreak

$A/m^2$
\pagebreak

$0\leq i \leq N-1$
\pagebreak

${\tt Valeur}^{\tt int}$
\pagebreak

${\tt Valeur}^{\tt double}$
\pagebreak

$\partial / \partial \xi$
\pagebreak

$\partial^2 / \partial \xi^2$
\pagebreak

$1 / \xi$
\pagebreak

$1 / \xi^2$
\pagebreak

$\partial^2 / \partial \theta^2$
\pagebreak

$1 / \sin(\theta)$
\pagebreak

$1 / \cos(\theta)$
\pagebreak

$\cos(\theta) \, Id$
\pagebreak

$\sin(\theta) \, Id$
\pagebreak

$\partial / \partial \phi$
\pagebreak

$\partial^2 / \partial \phi^2$
\pagebreak

$\cos(\phi) \, Id$
\pagebreak

$\sin(\phi) \, Id$
\pagebreak

$ \partial^2_theta - \cos\theta/\sin\theta \partial_\theta$
\pagebreak

$1/\sin(\theta) \, \partial / \partial \phi$
\pagebreak

${1 \over \xi}$
\pagebreak

${1 \over \xi-1}$
\pagebreak

${\xi}$
\pagebreak

$l=L(\theta', \phi') \qquad (1) $
\pagebreak

$\xi = X(\theta', \phi') \qquad (2)$
\pagebreak

$L({\theta'}_j, {\phi'}_k)$
\pagebreak

$X({\theta'}_j, {\phi'}_k)$
\pagebreak

$\vec{V}: \quad \vec{V} = 
\vec{\nabla} \phi + \vec{\nabla} \wedge \vec{\psi}$
\pagebreak

$\vec{W} =  \vec{\nabla} \wedge 
  \vec{\psi}$
\pagebreak

$\vec{V}: \quad \vec{V} = \vec{\nabla} \phi + \vec{\nabla} 
*\wedge \vec{\psi}$
\pagebreak

$(V^\theta, V^\varphi)$
\pagebreak

\[ V^\theta =   {\partial \eta \over \partial\theta} -
     {1\over\sin\theta} {\partial \mu \over \partial\varphi} 
\]
\pagebreak

\[ V^\varphi =  {1\over\sin\theta} 
            {\partial \eta \over \partial\varphi}
            + {\partial \mu \over \partial\theta} 
\]
\pagebreak

\[ V^\theta =  {\partial \eta \over \partial\theta} -
  {1\over\sin\theta} {\partial \mu \over \partial\varphi} 
\]
\pagebreak

$A$
\pagebreak

\[    A = {\partial \eta \over \ partial r} + { \eta \over r} - {V^r \over r}
\]
\pagebreak

$V^r$
\pagebreak

$\vec{W}$
\pagebreak

\[ V^\theta =  {\partial \eta \over \partial\theta} -
  {1\over\sin\theta} {\partial \mu \over \partial\varphi}
\]
\pagebreak

\[    A = {\partial \eta \over \partial r} + { \eta \over r} - {V^r \over r}
\]
\pagebreak

$V^\theta$
\pagebreak

$V^\varphi$
\pagebreak

\[ V^\theta =  {\partial \eta \over \partial\theta} - 
    {1\over\sin\theta} {\partial \mu \over \partial\varphi}
\]
\pagebreak

$ D^i V^j + D^j V^i $
\pagebreak

$ D_i V_j + D_j V_i $
\pagebreak

$ D_i $
\pagebreak

$ D^i V^j + D^j V^i - \frac{2}{3} D_k V^k \, \gamma^{ij} $
\pagebreak

$ D_i V_j + D_j V_i - \frac{2}{3} D^k V_k \, \gamma_{ij}$
\pagebreak

$\Delta N^i +\lambda \nabla^i 
\nabla_k N^k = S^i$
\pagebreak

$ \mu $
\pagebreak

$ \oint_S V^i ds_i $
\pagebreak

$ V^i $
\pagebreak

$ ds_i $
\pagebreak

$V_r$
\pagebreak

$= \vec{V}$
\pagebreak

\[   \Delta \vec{W} = \vec{V}
\]
\pagebreak

$\vec{W}=0$
\pagebreak

\[     {\partial \eta \over \ partial r} + { \eta \over r} 
                    - {V^r \over r} = A
\]
\pagebreak

\[    {\partial V^r \over \partial r} + {2 V^r \over r} 
        + {1 \over r} \Delta_{\vartheta \varphi} \eta = 0
\]
\pagebreak

\[    {\partial \eta \over \partial r} + {\eta \over r} - {V^r \over r} = A
\]
\pagebreak

\[    {\partial V^r \over \partial r} + {2 V^r \over r} 
        + {1 \over r}\Delta_{\vartheta \varphi}\eta = 0
\]
\pagebreak

\end{document}
