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\section{Linear Magnetic Inversion}\label{sec:forward magnetic} 
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For the magnetic inversion we use the anomaly of the magnetic flux 
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density~\index{magnetic flux density} of the Earth. 
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The controlling material parameter is the susceptibility~\index{susceptibility} 
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$k$ of the rock. 
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With magnetization $M$ and inducing magnetic field anomaly $H^s$, the magnetic 
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flux density anomaly $B^s$ is given as 
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\begin{equation}\label{ref:MAG:EQU:1} 
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B_i = \mu_0 \cdot ( H^s_i + M_i ) 
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\end{equation} 
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where $\mu_0 = 4 \pi \cdot 10^{7} \frac{Vs}{Am}$. 
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In this forward model we make the simplifying assumption that the magnetization 
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is proportional to the known geomagnetic flux density $B^b$: 
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\begin{equation}\label{ref:MAG:EQU:4} 
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\mu_0 \cdot M_i = k \cdot B^b_i \;. 
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\end{equation} 
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Values for the magnetic flux density can be obtained by the International 
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Geomagnetic Reference Field (IGRF)~\cite{IGRF} 
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(or the Australian Geomagnetic Reference Field (AGRF)~\cite{AGRF}). 
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In most cases it is reasonable to assume that that the background field is 
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constant across the domain. 
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The magnetic field anomaly $H^s$ can be represented by the gradient of a 
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magnetic scalar potential\index{scalar potential!magnetic} $\psi$. 
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We use the form 
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\begin{equation}\label{ref:MAG:EQU:6} 
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\mu_0 \cdot H^s_i =  \psi_{,i} 
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\end{equation} 
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With this notation one gets from Equations~(\ref{ref:MAG:EQU:1}) and~(\ref{ref:MAG:EQU:4}): 
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\begin{equation}\label{ref:MAG:EQU:7} 
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B_i =  \psi_{,i} + k \cdot B^b_i 
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\end{equation} 
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As the $B^s$ magnetic flux density anomaly we obtain the PDE 
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\begin{equation}\label{ref:MAG:EQU:8} 
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 \psi_{,ii} =  (k B^b_i)_{,i} 
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\end{equation} 
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which needs to be solved for a given susceptibility $k$. 
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The magnetic scalar potential is set to zero at the top of the domain 
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$\Gamma_{0}$. 
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On all other faces the normal component of the magnetic flux density anomaly 
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$B_i$ is set to zero, i.e. $n_i \psi_{,i} = k \cdot n_i B^b_i$ with outer 
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normal field $n_i$. 
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From the magnetic scalar potential we can calculate the magnetic flux density 
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anomaly via Equation~(\ref{ref:MAG:EQU:8}) to calculate the defect to the given 
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data. 
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If $B^{(s)}_i$ is a measurement of the magnetic flux density anomaly for 
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survey $s$ and $\omega^{(s)}_i$ is a weighting factor the data defect 
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$J^{mag}(k)$ in the notation of Chapter~\ref{chapter:ref:inversion cost function} is given as 
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\begin{equation}\label{ref:MAG:EQU:9} 
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J^{mag}(k) = \frac{1}{2}\sum_{s} \int_{\Omega} ( \omega^{(s)}_i \cdot (B_{i} B^{(s)}_i) ) ^2 dx 
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\end{equation} 
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Summation over $i$ is performed. 
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The cost function kernel\index{cost function!kernel} is given as 
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\begin{equation}\label{ref:MAG:EQU:10} 
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K^{mag}(\psi_{,i},k) = \frac{1}{2}\sum_{s} ( \omega^{(s)}_i \cdot (k \cdot B^b_i  \psi_{,i}  B^{(s)}_i) ) ^2 
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\end{equation} 
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Notice that if magnetic flux density is measured in air one can ignore the 
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$k\cdot B^b_i$ as the susceptibility is zero. 
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In practice the magnetic flux density $b^{(s)}$ is measured along a certain 
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direction $d^{(s)}_i$ with a standard error deviation $\sigma^{(s)}$ at 
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certain locations in the domain. 
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In this case one sets $B^{(s)}_i=b^{(s)} \cdot d^{(s)}_i$ and the weighting 
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factors $\omega^{(s)}$ as 
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\begin{equation}\label{ref:MAG:EQU:11} 
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\omega^{(s)}_i 
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= \left\{ 
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\begin{array}{lcl} 
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f \cdot \frac{d^{(s)}_i}{\sigma^{(s)}} & & \mbox{data are available} \\ 
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& \mbox{ where } & \\ 
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0 & & \mbox{ otherwise } \\ 
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\end{array} 
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\right. 
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\end{equation} 
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where it is assumed that $d^{(s)}_i \cdot d^{(s)}_i =1$. With the objective to control the 
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gradient of the cost function the scaling factor $f$ is chosen in the way that 
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\begin{equation}\label{ref:MAG:EQU:12} 
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\sum_{s} \int_{\Omega} ( \omega^{(s)}_i B^{(s)}_i ) 
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\cdot ( \omega^{(s)}_j \frac{1}{L_j} ) \cdot L^2 \cdot 
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( B^b_n \frac{1}{L_n} ) 
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\cdot k' \; 
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dx =\alpha 
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\end{equation} 
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where $\alpha$ defines a scaling factor which is typically set to one and $L$ is defined by equation~(\ref{ref:EQU:REG:6b}). 
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$k'$ is considering the 
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derivative of the density with respect to the level set function. 
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\subsection{Usage} 
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\LG{Add example} 
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\begin{classdesc}{MagneticModel}{domain, w, B, background_field, 
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\optional{, useSphericalCoordinates=False} 
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\optional{, fixPotentialAtBottom=False}, 
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\optional{, tol=1e8}} 
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opens a magnetic forward model over the \Domain \member{domain} with 
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weighting factors \member{w} ($=\omega^{(s)}$) and measured magnetic flux 
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density anomalies \member{B} ($=B^{(s)}$). 
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The weighting factors and the measured magnetic flux density anomalies must be vectors. 
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\member{background_field} defines the background magnetic flux density $B^b$ 
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as a vector with north, east and vertical components. 
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If \member{useSphericalCoordinates} is \True spherical coordinates are used. 
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\member{tol} sets the tolerance for the solution of the PDE~(\ref{ref:MAG:EQU:8}). 
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If \member{fixPotentialAtBottom} is set to \True, the gravitational potential 
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at the bottom is set to zero in addition to the potential on the top. 
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\end{classdesc} 
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\begin{methoddesc}[MagneticModel]{rescaleWeights}{ 
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\optional{scale=1.} 
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\optional{k_scale=1.}} 
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rescale the weighting factors such condition~(\ref{ref:MAG:EQU:12}) holds where 
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\member{scale} sets the scale $\alpha$ 
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and \member{k_scale} sets $k'$. This method should be called before any inversion is started 
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in order to make sure that all components of the cost function are appropriately scaled. 
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\end{methoddesc} 
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\subsection{Gradient Calculation} 
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This section briefly explains how the gradient 
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$\frac{\partial J^{mag}}{\partial k}$ of the cost function $J^{mag}$ with 
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respect to the susceptibility $k$ is calculated. We follow the concept as outlined in section~\ref{chapter:ref:inversion cost function:gradient}. 
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The magnetic potential $\psi$ from PDE~(\ref{ref:MAG:EQU:8}) is solved in weak form: 
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\begin{equation}\label{ref:MAG:EQU:201} 
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\int_{\Omega} q_{,i} \psi_{,i} \; dx = \int_{\Omega} k \cdot q_{,i} B^b_i \; dx 
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\end{equation} 
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for all $q$ with $q=0$ on $\Gamma_{0}$. 
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In the following we set $\Psi[k]=\psi$ for a given susceptibility $k$ as 
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solution of the variational problem~(\ref{ref:MAG:EQU:201}). 
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If $\Gamma_{k}$ denotes the region of the domain where the susceptibility is 
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known and for a given direction $p$ with $p=0$ on $\Gamma_{k}$ one has 
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\begin{equation}\label{ref:MAG:EQU:201aa} 
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\int_{\Omega} \frac{\partial J^{mag}}{\partial k} \cdot p \; dx = \int_{\Omega} 
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\sum_{s} (\omega^{(s)}_j 
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( B^{(s)}_jB_{j})) \cdot ( \omega^{(s)}_i ( \Psi[p]_{,i}  p \cdot B^b_i ) ) \; dx 
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\end{equation} 
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with 
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\begin{equation}\label{ref:MAG:EQU:202c} 
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Y_i[\psi]= \sum_{s} (\omega^{(s)}_j 
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(B^{(s)}_j  B_{j}) ) \cdot \omega^{(s)}_i 
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\end{equation} 
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This is written as 
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\begin{equation}\label{ref:MAG:EQU:202cc} 
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\int_{\Omega} \frac{\partial J^{mag}}{\partial k} \cdot p \; dx = \int_{\Omega} 
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Y_i[\psi] \Psi[p]_{,i}  p \cdot Y_i[\psi]B^b_i \; dx 
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\end{equation} 
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We then set $Y^*[\psi]$ as the solution of the equation 
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\begin{equation}\label{ref:MAG:EQU:202d} 
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\int_{\Omega} r_{,i} Y^*[\psi]_{,i} \; dx = \int_{\Omega} r_{,i} ,Y_i[\psi] \; dx \mbox{ for all } p \mbox{ with } r=0 \mbox{ on } \Gamma_{0} 
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\end{equation} 
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with $Y^*[\psi]=0$ on $\Gamma_{0}$. With $r=\Psi[p]$ we get 
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\begin{equation}\label{ref:MAG:EQU:202dd} 
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\int_{\Omega} \Psi[p]_{,i} Y^*[\psi]_{,i} \; dx = \int_{\Omega} \Psi[p]_{,i} ,Y_i[\psi] \; dx 
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\end{equation} 
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and from Equation~(\ref{ref:MAG:EQU:201}) with $q=Y^*[\psi]$ we get 
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\begin{equation}\label{ref:MAG:EQU:20e} 
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\int_{\Omega} Y^*[\psi]_{,i} \Psi[p]_{,i} \; dx = \int_{\Omega} p \cdot Y^*[\psi]_{,i} B^b_i \; dx 
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\end{equation} 
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which leads to 
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\begin{equation}\label{ref:MAG:EQU:20ee} 
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\int_{\Omega} \Psi[p]_{,i} ,Y_i[\psi] \; dx = \int_{\Omega} p \cdot Y^*[\psi]_{,i} B^b_i \; dx 
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\end{equation} 
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and finally 
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\begin{equation}\label{ref:MAG:EQU:201a} 
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\int_{\Omega} \frac{\partial J^{mag}}{\partial k} \cdot p \; dx = \int_{\Omega} 
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p \cdot (Y_i[\psi]  Y^*[\psi]_{,i}) B^b_i \; dx 
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\end{equation} 
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or 
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\begin{equation}\label{ref:MAG:EQU:201b} 
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\frac{\partial J^{mag}}{\partial k} = (Y^*[\psi]_{,i}Y_i[\psi]) B^b_i 
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\end{equation} 
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