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\begin{document}
\thispagestyle{empty}

\Large
\vspace*{1cm}
$\psi$ $\epsilon$ ${\rm d} \psi$ ${\rm d} \epsilon$

\newpage
\vspace*{1cm}
\begin{equation}
F_1 = F_2 = \frac{Gm_1m_2}{r^2}
\end{equation}

\newpage
\vspace*{1cm}
\begin{equation}
F(r) = \frac{GM_{\rm E}m_2}{r^2}
\end{equation}

\newpage
\vspace*{1cm}
\begin{equation}
T = 2\pi \sqrt{\frac{r^3}{GM_{\rm E}}}
\end{equation}

\newpage
\vspace*{1cm}
\begin{equation}
\vec{df}(\vec{x}) = \frac{Gmdm'(\vec{x}-\vec{x}')}{||\vec{x}-\vec{x}'||^3} 
\end{equation}

\newpage
\vspace*{1cm}
\begin{equation}
\vec{dg}(\vec{x}) = \frac{df(\vec{x})}{m} = \frac{Gdm'(\vec{x}-\vec{x}')}{||\vec{x}-\vec{x}'||^2}
\end{equation}

\newpage
\vspace*{1cm}
\begin{equation}
\vec{g}(\vec{x}) = \int_{\rm mass} \frac{G\rho(\vec{x}')(\vec{x}'-\vec{x})}{||\vec{x}-\vec{x}'||^3} d^3x'
\end{equation}

\newpage
\vspace*{1cm}
\begin{equation}
\nabla \Phi(\vec{x}) = \partial_x\Phi\vec{e}_x + \partial_y\Phi\vec{e}_y + \partial_z\Phi\vec{e}_z
\end{equation}

\vspace*{1cm}
\begin{equation}
\nabla \Phi (\vec{x}) = \vec{g}(\vec{x}) 
\end{equation}

\newpage
\vspace*{1cm}
\begin{equation}
\Phi(\vec{x}) = \int_{\rm mass} \frac{G\rho(\vec{x}')}{||\vec{x}-\vec{x}'||} d^3x'
\end{equation}

\newpage
\vspace*{1cm}
\begin{equation}
\Phi(\vec{x}) = \frac{GM}{d}
\end{equation}

\newpage
\vspace*{1cm}
\begin{equation}
\nabla^2 \Phi(\vec{x}) = \partial_x^2\Phi + \partial_y^2\Phi + \partial_z^2\Phi
\end{equation}

\vspace*{1cm}
\begin{equation}
\nabla^2 \Phi(\vec{x}) = -4\pi G\rho(\vec{x})
\end{equation}

\vspace*{1cm}
\begin{equation}
\int_S \hat{n} \cdot \vec{g} dS = - 4\pi G \int_V \rho dV
\end{equation}


\vspace*{1cm}
\begin{equation}
\frac{GM}{||\vec{x}-\vec{x}'||}
\end{equation}

\vspace*{1cm}
\begin{equation}
\nabla^2 \Phi(\vec{x}) = 0
\end{equation}

\vspace*{1cm}
\begin{equation}
\Phi_1 = \Phi_2
\end{equation}

%\vspace*{1cm}
\begin{equation}
\hat{n} \cdot \nabla \Phi_2 - \hat{n}\cdot\nabla \Phi = -4\pi G \delta \rho
\end{equation}

\newpage 
\vspace*{1cm}
\begin{equation}
\Phi(\vec{x}) = \int_{\rm mass} \frac{G\rho(\vec{x}')}{||\vec{x}-\vec{x}'||} d^3x' + \frac{1}{2}(x^2+y^2)\omega^2
\end{equation}

\vspace*{1cm}
\begin{equation}
\nabla^2 \Phi(\vec{x}) = -4\pi G\rho(\vec{x}) + 2\omega^2
\end{equation}

\vspace*{1cm}
\begin{equation}
\nabla^2 \Phi(\vec{x}) = 2\omega^2
\end{equation}

\newpage
\vspace*{1cm}
\begin{eqnarray}
x & = & r \sin\theta \cos \varphi\\
y & = & r \sin\theta \sin \varphi\\
z & = & r \cos \theta
\end{eqnarray}

\vspace*{1cm}
\begin{equation}
\nabla^2 \Phi = \frac{1}{r} \partial_r(r^2\partial_r \Phi) + \frac{1}{r^2\sin\theta} \partial_\theta(\sin\theta\partial_\theta\Phi) + \frac{1}{r^2\sin^2\theta}\partial^2_\varphi\Phi
\end{equation}

\vspace*{1cm}
\begin{equation}
\Phi(r,\theta,\varphi) = T(\theta)L(\varphi)R(r)
\end{equation}

\vspace*{1cm}
\begin{equation}
R(r) = \left\{ r^l r^{-(l+1} \right.
\end{equation}
\begin{equation}
L(\varphi) = \left\{ e^{im\varphi} e^{-im\varphi} \right.
\end{equation}
\begin{equation}
T(\cos\theta) = P_l (\cos\theta)
\end{equation}

$P_l$: "Legendre polynomials

\begin{equation}
\Phi (r, \theta,\varphi) = \left\{ r^l r^{-(l+1)} \right\} \times Y_l^m (\theta,\varphi)
\end{equation}

\begin{equation}
Y_l^m(\theta,\varphi) = \sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}} P_l^m(\cos\theta)e^{im\varphi}
\end{equation}

\begin{equation}
Y_l^m \equiv (-1)^m Y_l^{(-m)*}
\end{equation}

\newpage
\begin{equation}
\rho(\vec{x'}) = \sum_{l',m'} \rho_{l'}^{m'}(r')Y_{l'}^{m'}(\theta',\varphi')
\end{equation}

\begin{equation}
\Phi(\vec{x}) = 4\pi G \sum_{l,m} \frac{Y_l^m(\theta,\varphi)}{2l+1} \frac{1}{r^{l+1}} \left[ \int (r')^{l+2} \rho_l^m(r')dr'\right]
\end{equation}
 
\begin{eqnarray}
\Phi_{\rm T} & = & \left[\frac{GM}{r} + \frac{1}{3}\Omega^2r^2\right] - \left[ \frac{1}{3}\Omega^2r^2 + \frac{GM}{r^3}a^2J_2\right]P_2(\cos\theta) \\
& &  - \frac{GM}{r} \sum_{l=2}^\infty \sum_{m=-l; (l,m)\neq(2,0)}^l \Phi_l^m Y_l^m(\theta,\varphi) \left(\frac{a}{r}\right)^l 
\end{eqnarray}

\begin{equation}
N = \left. \frac{\Delta \Phi}{g_{\rm T}}\right|_{r=a}
\end{equation}

\begin{equation}
\Delta \Phi = - \frac{GM}{r} \sum_{l=2}^\infty \sum_{m=-l; (l,m)\neq(2,0)}^l \Phi_l^m Y_l^m(\theta,\varphi) \left(\frac{a}{r}\right)^l 
\end{equation}

\begin{equation}
N = -a \sum_{l,m} \Phi_l^m Y_l^m
\end{equation}

\begin{equation}
N = - \frac{a}{2 \pi g_{\rm T}|_a} \int \Delta g(\theta',\varphi')f(\gamma) \sin\theta'd\theta'd\varphi'
\end{equation}
 
\begin{equation}
\Phi = \frac{GM}{r} \left( 1 + \sum_{n=2}^{n_{max}} \left(\frac{a}{r}\right)^n
\sum_{m=0}^n P_{nm}(\sin\phi) \left[C_{nm}\cos m\lambda + S_{nm} \sin m \lambda \right] \right)
\end{equation}

\newpage
\vspace*{1cm}
{\Large 
\begin{tabular}{lll}\hline
Effect& Source & Magnitude \\ 
      &        & (1 mGal = 10$^{-5}$ m $\cdot$ s$^{-2}$) \\ \hline
Main & Total mass of Earth, Oblate shape, Earth Rotation & 980,000 - 990,000 mGal \\
Elevation & Distance from centre of Earth & 0.3 mGal/m \\
Crustal & Density variations in rocks within the Earth's crust & $<$500 mGal [usually 10's mGal] \\
Tides & Sun and moon & 	$<$1 mGal \\ \hline
\end{tabular}
}
\newpage
\begin{equation}
   \vec{a}_0 = \frac{GM_b}{s^2}\frac{\vec{s}}{s}
\end{equation}
\begin{tabular}{lp{21cm}}
$\vec{a}_0$:& homogeneous centrifugal force in body 1 due to revolution around barycenter;\\
$M_b$: &mass of the body 2;\\
$\vec{s}$:& distance between the centers of the two bodies.\\
\end{tabular}

\begin{equation}
  \vec{b} = \vec{a}_p - \vec{a}_0 = \frac{GM_b}{d^2}\cdot\frac{\vec{d}}{d} - \frac{GM_b}{s^2}\frac{\vec{s}}{s}
\end{equation}
$\vec{d}$: distance between point $P$ and the center of body 2.

\newpage
\begin{equation}
  \vec{b} = \nabla V 
\end{equation}

\begin{equation}
 \nabla V = \frac{\partial f}{\partial r}\hat{\vec{r}} + \frac{1}{r} \frac{\partial V}{\partial \vartheta} \hat{\vec{\vartheta}} + \frac{1}{r\sin\vartheta} \frac{\partial V}{\partial \varphi} \hat{\vec{\varphi}}
\end{equation}

\begin{equation}
  V = GM_b \left(\frac{1}{d} - \frac{1}{s} - \frac{r \cos \psi}{s^2}\right)
\end{equation}
$\psi$: geocentric zenith angle

\begin{equation}
  V = \frac{GM_b}{s} \cdot \sum_{l=2}^\infty \left(\frac{r}{s}\right)^l P_l (\cos \psi)
\end{equation}

\begin{tabular}{lcc}\hline
Body & $r/s$ & $l_{\rm max}$ \\ \hline
Moon & $1.6\cdot10^{-2}$ & 6 \\
Sun  & $4.5\cdot10^{-5}$ & 3 \\
Planets & & 2 \\ \hline
\end{tabular}

\newpage
\begin{equation}
P_l(\cos\psi) = \frac{1}{2l+1}\sum_{m=0}^l P_{lm}(\cos\vartheta) \cdot P_{lm}(\cos\vartheta_b)\cdot \cos(m\varphi-m\varphi_b)
\end{equation}

\begin{tabular}{ll}
$\vartheta$: & geocentric spherical polar distance of the point;\\
$\vartheta_b$: &geocentric spherical polar distance of the second body;\\
$\varphi$: &geocentric spherical longitude of the point; \\
$\varphi_b$: &geocentric spherical longitude of the second body.\\ 
$P_{lm}$: & Fully normalized spherical harmonics\\
\end{tabular}

{\Large
\begin{equation}
 V(t) = GM_b \sum_{l=2}^{l_{\rm max}} \frac{r^l}{s^{l+1}(t)} \frac{1}{2l+1} \sum_{m=0}^l P_{lm} (\cos \vartheta) \cdot P_{lm}(\cos\vartheta_b(t))\cdot \cos(m\varphi-m\varphi_b(t))
\end{equation}
with $s$, $\vartheta_b$, and $\varphi_b$ time dependent.
}

\newpage
\begin{eqnarray}
 V(t) & = & D \sum_{l=1}^{l_{\rm max}} \sum_{m=0}^{l} \left(\frac{r}{a}\right)^l \Gamma(\vartheta) \cdot P_{lm}(\cos\vartheta)\\
     &   & \cdot \sum_i [ C_i^{lm}(t) \cos(\alpha_i(t)) + S_i^{lm}(t) \sin (\alpha_i(t))]
\end{eqnarray}

\begin{center}
\begin{tabular}{lp{18cm}}
$D$, $\Gamma$: & Normalization constants \\
$a$: & semi-major axis of the reference ellipsoid\\
$C_i$, $S_i$: & time dependent coefficients of the catalogue; given as \\
& $C_i^{lm}(t) = C0_i^{lm} + t \cdot C1_1^{lm}$ and \\
& $S_i^{lm}(t) = S0_i^{lm} + t \cdot S1_1^{lm}$.\\
\end{tabular}
\end{center}

\begin{equation}
 \alpha_i(t) = m\cdot \lambda + \sum_{j=1}^{j_{\rm max}} k_{ij} \cdot arg_j(t); \; k_{i1} = m
\end{equation}

\begin{center}
\begin{tabular}{lp{18cm}}
$k_{ij}$: & integer coefficients, given in catalog\\
$arg_j(t)$: & astronomical arguments, can be computed from ephemerides (polynomials)\\
\end{tabular}
\end{center}

\newpage
{\large
\begin{tabular}{lrrcrl} \hline
catalogue & no. of & no. of & max. & truncation & ephemerides \\ 
          & waves  & coeff. & degree& [m$^2$/s$^2$] & \\\hline
Doodson (1921) & 378 & 378 & 3 & $1.0\cdot 10^{-4}$& BN\\
Cartwright et al. (1971, 1973) & 505 & 1,010 & 3 & $0.4\cdot 10^{-4}$& BN\\
B\"ullesfeld (1985) & 656 & 656 & 4 & $0.2\cdot 10^{-4}$& BN\\
Tamura (1987) & 1,200 & 1,326 & 4 & $0.4\cdot 10^{-5}$& DE118/LE62\\
Xi (1989) & 2,934 & 2,934 & 4 & $0.9\cdot 10^{-6}$& BN\\
Tamura (1993) & 2,060 & 3,046 & 4 & $0.4\cdot 10^{-5}$& DE200/LE200\\
Roosbeck (1996) & 6,499 & 7,202 & 5 & $0.8\cdot 10^{-7}$& CT/BR\\
Hartmann \& Wenzel (1995a,b) & 12,935 & 19,271 & 6 & $0.1\cdot 10^{-9}$& DE200/LE200\\\hline
\end{tabular}
}

\newpage
\vspace*{1cm}
\begin{equation}
  \delta N = \frac{V}{g}
\end{equation}

\begin{tabular} {lrp{21cm}} \hline
Love/ & $l=2$ & definition \\
Shida &       & \\\hline
h & 0.612 & ratio of height of solid body tide to the deforming potential \\
k & 0.303 & ratio of additional tidal potential to the deforming potential \\
l & 0.04  & ratio of horizontal displacement of the crust to that of the equlibrium tide of a fluid Earth\\\hline
\end{tabular}

\newpage
\begin{equation}
   \vec{x}(t) = \vec{x}_0 + \sum_{k=1}^K \delta \vec{x}_k(t)
\end{equation}

\newpage

\vspace*{1cm}
Geoid is defined as the surface
\begin{equation}
  r = a + \delta r(\theta,\phi)
\end{equation}
with $a$ radius of the Earth,  
\begin{equation}
  \overline{\delta r(\theta,\phi)} = 0
\end{equation}
and
\begin{equation}
   V_{T} (r=a+\delta r,\theta,\phi) = {\rm const.}
\end{equation}  

\newpage

\vspace*{1cm}
\begin{equation}
   \Delta V = V_T - {\rm all }P_2{\rm-terms}  
\end{equation}

\begin{equation}
    N = \left. \frac{\Delta V}{g_T}\right|_{r=a}
\end{equation}

\vspace{0.5cm}
Stoke's formula:
\begin{equation}
    N(\theta,\phi) = - \frac{a}{2\pi g_T|_a} \int \Delta g(\theta',\phi') f(\gamma) \sin \theta' d\theta'd\phi'
\end{equation}
with $\gamma$ angle between ($\theta$,$\phi$) and ($\theta'$,$\phi'$) and
\begin{eqnarray}
   f(\gamma) & = & \left[ \frac{1}{2}{\rm csc} \frac{\gamma}{2} -1 - \cos \gamma\right] \\ 
             &   & + 3\left[1 - \cos \gamma - 2 \sin \frac{\gamma}{2} - \cos \gamma \ln \left\{ \sin \frac{\gamma}{2} + \sin^2 \frac{\gamma}{2} \right\} \right]
\end{eqnarray}


\newpage
\vspace*{1cm}
\begin{equation}
 \sigma = \left( \begin{array}{ccc}
                  \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
                  \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\
                  \sigma_{zx} & \sigma_{zy} & \sigma_{zz} 
                 \end{array} \right)
\end{equation}

\newpage
\vspace*{1cm}
\begin{equation} \label{e-allgemeinM}
       \sigma_{ij}(\xi_k,t) = 
         f_{ij} (x_k(\eta_l,t-s); \xi_k)|_{s \in [0,\infty] \; \eta_l \in B}.
\end{equation}
$B$: body\\
$\sigma_{ij}$: stress tensor\\
$f_{ij}$: time-independent functional;\\
$\xi_k$, $\eta_l$: positions;\\
$x_k:$ displacement; \\
$t$: time

\vspace{1cm}
Elastic, isotrop, linear material (Hooke's law):
\begin{equation} \label{e-hookeE}
      \sigma_{ij} = 2\mu \epsilon_{ij} + \lambda \epsilon_{kk} \delta_{ij}
\end{equation}
$\epsilon_{ij}$: strain tensor; \\
$\mu$, $\lambda$: Lame's constants.

\newpage
\vspace*{1cm}
Linear, visco-elastic body:
\begin{equation} \label{e-lin-visko}
     \sigma_{ij}(t) = 2 \mu(0) \epsilon_{ij}(t) + \lambda(0) \epsilon_{kk}(t) \delta_{ij}
                 + \int_{0}^{\infty} \left\{ 2 \mu(s) \epsilon_{ij}(t-s) +
                                             \lambda(s) \epsilon_{kk}(t-s) 
                   \delta_{ij} \right\} ds
\end{equation}
\newpage
\vspace*{1cm}
\begin{equation}
 \epsilon = \left( \begin{array}{ccc}
                  \epsilon_{xx} & \epsilon_{xy} & \epsilon_{xz} \\
                  \epsilon_{yx} & \epsilon_{yy} & \epsilon_{yz} \\
                  \epsilon_{zx} & \epsilon_{zy} & \epsilon_{zz} 
                 \end{array} \right)
\end{equation}

\vspace*{1cm}
\begin{equation} \label{e-infiE}
      \epsilon_{ij} = \frac{1}{2} 
                      \left( \frac{\partial u_i}{\partial x_j} +
                          \frac{\partial u_j}{\partial x_i} \right) 
\end{equation}
$u_i$: displacement;\\
$x_j$: position.

\newpage
\vspace*{1cm}
{\bf Volume force} $F_i$:
\begin{equation}
      F_i = \lim_{\Delta V \to 0} \frac{\Delta F_i^*}{\Delta V}
\end{equation}
$F_i$: volume force; \\
$\Delta F_i^*$: total force acting on volume $\Delta V$.

\vspace{1cm}
{\bf Surface force} vector $(p_{ni})dS$ (with index $n$ indicating that the \\ surface element $dS$ has the outward pointing normal $n_i$):
\begin{equation} \label{e-defflaeche}
     p_{ni} = \lim_{\Delta S \to 0} \frac{p_{ni}^*}{\Delta S}
\end{equation}
$p_{ni}^*$: components of the total force acting on area $\Delta S$.\\
$p_{ni}$ has the unit force/area (stress).

\newpage
\vspace*{1cm}
Momentum balance:
\begin{equation} \label{e-feldE}
       \rho \left(\frac{\partial v_i}{\partial x_k} v_k + \frac{\partial v_i}{\partial t} \right) =
       \frac{\partial \sigma_{ij}}{\partial x_j} + f_i.
\end{equation}
$\rho$: density;\\
$v_i$: velocity;\\
$t$: time;\\
$f_i$: forces.

\vspace{0.5cm}
Small gradients of velocity:
\begin{equation} \label{e-feldEfest}
       \rho \frac{\partial^2 u_i}{\partial t^2} =
       \frac{\partial \sigma_{ij}}{\partial x_j} + f_i
\end{equation}
$u_i$: displacement.

\vspace{0.5cm}
Boundary condition:
\begin{equation} \label{e-randbedingung}
       p_i = \sigma_{ji} n_j.
\end{equation}

\newpage

\begin{eqnarray}
    G_r(a,\vartheta) & = & \frac{a}{M}\sum_{n=0}^\infty h'_n P_n(\cos \vartheta)\\
    G_\vartheta(a,\vartheta) & = & \frac{a}{M}\sum_{n=0}^\infty \ell'_n \pdiff{\vartheta}P_n(\cos \vartheta)\\
    G_\varphi(a,\vartheta) & = & \frac{G}{a}\sum_{n=0}^\infty (k'_n+1) P_n(\cos \vartheta)
\end{eqnarray}

\newpage

\vspace{0.5cm}
\begin{equation}
\Delta t = \int_S \frac{1}{v} ds - \int_G \frac{1}{c} dg
\end{equation}

\vspace{0.5cm}
\begin{equation}
\ell = \int_S \frac{c}{v}ds -\int_G dg
\end{equation}

\vspace{0.5cm}
\begin{equation}
\ell = \int_S \frac{c}{v}ds -\int_S ds + \int_S ds -\int_G dg
\end{equation}

\vspace{0.5cm}
\begin{equation}
\ell = \int_S (n-1) ds + S - G
\end{equation}

\vspace{0.5cm}
\begin{tabular}{ll}
$\Delta t$: & time delay\\
$v$: & propagation velocity \\
$c$: & speed of light in vacuum \\
$S$: & actual path \\
$G$: & straight line; path in vacuum\\
$\ell$: & excess path length \\
$n$: & refractive index in the atmosphere\\
\end{tabular}


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\end{document}
