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@ -1345,58 +1345,63 @@ matrix equation (Ax=b) or eigenvalue problem.</p>
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<p>From the "Frequency domain" section of <code>meanas.fdmath</code>, we have</p>
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<div class="arithmatex">\[
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\begin{aligned}
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\tilde{E}_{l, \vec{r}} &= \tilde{E}_{\vec{r}} e^{-\imath \omega l \Delta_t} \\
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\tilde{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &= \tilde{H}_{\vec{r} + \frac{1}{2}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t} \\
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\tilde{J}_{l, \vec{r}} &= \tilde{J}_{\vec{r}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t} \\
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\tilde{M}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &= \tilde{M}_{\vec{r} + \frac{1}{2}} e^{-\imath \omega l \Delta_t} \\
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\tilde{E}_{l, \vec{r}} &= \tilde{E}_{\vec{r}} e^{-\imath \omega l \Delta_t} \\
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\tilde{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &= \tilde{H}_{\vec{r} + \frac{1}{2}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t} \\
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\tilde{J}_{l, \vec{r}} &= \tilde{J}_{\vec{r}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t} \\
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\tilde{M}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &= \tilde{M}_{\vec{r} + \frac{1}{2}} e^{-\imath \omega l \Delta_t} \\
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\hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{\vec{r}})
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-\Omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} &= -\imath \Omega \tilde{J}_{\vec{r}} e^{\imath \omega \Delta_t / 2} \\
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\Omega &= 2 \sin(\omega \Delta_t / 2) / \Delta_t
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-\Omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} &= -\imath \Omega \tilde{J}_{\vec{r}} e^{\imath \omega \Delta_t / 2} \\
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\Omega &= 2 \sin(\omega \Delta_t / 2) / \Delta_t
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\end{aligned}
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\]</div>
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<p>resulting in</p>
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<div class="arithmatex">\[
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\begin{aligned}
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\tilde{\partial}_t &\Rightarrow -\imath \Omega e^{-\imath \omega \Delta_t / 2}\\
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\hat{\partial}_t &\Rightarrow -\imath \Omega e^{ \imath \omega \Delta_t / 2}\\
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\tilde{\partial}_t &\Rightarrow -\imath \Omega e^{-\imath \omega \Delta_t / 2}\\
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\hat{\partial}_t &\Rightarrow -\imath \Omega e^{ \imath \omega \Delta_t / 2}\\
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\end{aligned}
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\]</div>
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<p>Maxwell's equations are then</p>
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<div class="arithmatex">\[
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\begin{aligned}
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\tilde{\nabla} \times \tilde{E}_{\vec{r}} &=
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\tilde{\nabla} \times \tilde{E}_{\vec{r}} &=
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\imath \Omega e^{-\imath \omega \Delta_t / 2} \hat{B}_{\vec{r} + \frac{1}{2}}
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- \hat{M}_{\vec{r} + \frac{1}{2}} \\
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\hat{\nabla} \times \hat{H}_{\vec{r} + \frac{1}{2}} &=
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\hat{\nabla} \times \hat{H}_{\vec{r} + \frac{1}{2}} &=
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-\imath \Omega e^{ \imath \omega \Delta_t / 2} \tilde{D}_{\vec{r}}
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+ \tilde{J}_{\vec{r}} \\
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\tilde{\nabla} \cdot \hat{B}_{\vec{r} + \frac{1}{2}} &= 0 \\
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\hat{\nabla} \cdot \tilde{D}_{\vec{r}} &= \rho_{\vec{r}}
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\tilde{\nabla} \cdot \hat{B}_{\vec{r} + \frac{1}{2}} &= 0 \\
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\hat{\nabla} \cdot \tilde{D}_{\vec{r}} &= \rho_{\vec{r}}
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\end{aligned}
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\]</div>
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<p>With <span class="arithmatex">\(\Delta_t \to 0\)</span>, this simplifies to</p>
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<div class="arithmatex">\[
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\begin{aligned}
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\tilde{E}_{l, \vec{r}} &\to \tilde{E}_{\vec{r}} \\
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\tilde{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &\to \tilde{H}_{\vec{r} + \frac{1}{2}} \\
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\tilde{J}_{l, \vec{r}} &\to \tilde{J}_{\vec{r}} \\
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\tilde{M}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &\to \tilde{M}_{\vec{r} + \frac{1}{2}} \\
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\Omega &\to \omega \\
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\tilde{\partial}_t &\to -\imath \omega \\
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\hat{\partial}_t &\to -\imath \omega \\
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\tilde{E}_{l, \vec{r}} &\to \tilde{E}_{\vec{r}} \\
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\tilde{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &\to \tilde{H}_{\vec{r} + \frac{1}{2}} \\
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\tilde{J}_{l, \vec{r}} &\to \tilde{J}_{\vec{r}} \\
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\tilde{M}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &\to \tilde{M}_{\vec{r} + \frac{1}{2}} \\
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\Omega &\to \omega \\
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\tilde{\partial}_t &\to -\imath \omega \\
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\hat{\partial}_t &\to -\imath \omega \\
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\end{aligned}
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\]</div>
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<p>and then</p>
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<div class="arithmatex">\[
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\begin{aligned}
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\tilde{\nabla} \times \tilde{E}_{\vec{r}} &=
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\tilde{\nabla} \times \tilde{E}_{\vec{r}} &=
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\imath \omega \hat{B}_{\vec{r} + \frac{1}{2}}
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- \hat{M}_{\vec{r} + \frac{1}{2}} \\
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\hat{\nabla} \times \hat{H}_{\vec{r} + \frac{1}{2}} &=
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\hat{\nabla} \times \hat{H}_{\vec{r} + \frac{1}{2}} &=
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-\imath \omega \tilde{D}_{\vec{r}}
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+ \tilde{J}_{\vec{r}} \\
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\end{aligned}
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\]</div>
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<div class="arithmatex">\[
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\hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{\vec{r}})
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-\omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} = -\imath \omega \tilde{J}_{\vec{r}} \\
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@ -2064,6 +2069,7 @@ mask selecting the total-field region, then the TFSF source is the commutator</p
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<div class="arithmatex">\[
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\frac{A Q - Q A}{-i \omega} E.
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\]</div>
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<p>This vanishes in the interior of the total-field and scattered-field regions
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and is supported only at their shared boundary, where the mask discontinuity
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makes <code>A</code> and <code>Q</code> fail to commute. The returned current is therefore the
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@ -2416,11 +2422,15 @@ the <code>meanas.fdmath.types</code> submodule for details.</p>
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<div class="doc doc-contents ">
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<p>Wave operator
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$$ \nabla \times (\frac{1}{\mu} \nabla \times) - \Omega^2 \epsilon $$</p>
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<div class="arithmatex">\[ \nabla \times (\frac{1}{\mu} \nabla \times) - \Omega^2 \epsilon \]</div>
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</p>
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<div class="highlight"><pre><span></span><code>del x (1/mu * del x) - omega**2 * epsilon
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</code></pre></div>
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<p>for use with the E-field, with wave equation
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$$ (\nabla \times (\frac{1}{\mu} \nabla \times) - \Omega^2 \epsilon) E = -\imath \omega J $$</p>
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<div class="arithmatex">\[ (\nabla \times (\frac{1}{\mu} \nabla \times) - \Omega^2 \epsilon) E = -\imath \omega J \]</div>
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</p>
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<div class="highlight"><pre><span></span><code>(del x (1/mu * del x) - omega**2 * epsilon) E = -i * omega * J
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</code></pre></div>
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<p>To make this matrix symmetric, use the preconditioners from <code>e_full_preconditioners()</code>.</p>
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@ -2678,11 +2688,15 @@ The PMC is applied per-field-component (i.e. <code>pmc.size == epsilon.size</cod
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<div class="doc doc-contents ">
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<p>Wave operator
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$$ \nabla \times (\frac{1}{\epsilon} \nabla \times) - \omega^2 \mu $$</p>
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<div class="arithmatex">\[ \nabla \times (\frac{1}{\epsilon} \nabla \times) - \omega^2 \mu \]</div>
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</p>
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<div class="highlight"><pre><span></span><code>del x (1/epsilon * del x) - omega**2 * mu
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</code></pre></div>
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<p>for use with the H-field, with wave equation
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$$ (\nabla \times (\frac{1}{\epsilon} \nabla \times) - \omega^2 \mu) E = \imath \omega M $$</p>
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<div class="arithmatex">\[ (\nabla \times (\frac{1}{\epsilon} \nabla \times) - \omega^2 \mu) E = \imath \omega M \]</div>
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</p>
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<div class="highlight"><pre><span></span><code>(del x (1/epsilon * del x) - omega**2 * mu) E = i * omega * M
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</code></pre></div>
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@ -2855,25 +2869,30 @@ The PMC is applied per-field-component (i.e. <code>pmc.size == epsilon.size</cod
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<div class="doc doc-contents ">
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<p>Wave operator for <code>[E, H]</code> field representation. This operator implements Maxwell's
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equations without cancelling out either E or H. The operator is
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$$ \begin{bmatrix}
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-\imath \omega \epsilon & \nabla \times \
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\nabla \times & \imath \omega \mu
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\end{bmatrix} $$</p>
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equations without cancelling out either E or H. The operator is</p>
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<div class="arithmatex">\[
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\begin{bmatrix}
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-\imath \omega \epsilon & \nabla \times \\
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\nabla \times & \imath \omega \mu
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\end{bmatrix}
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\]</div>
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<div class="highlight"><pre><span></span><code>[[-i * omega * epsilon, del x ],
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[del x, i * omega * mu]]
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</code></pre></div>
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<p>for use with a field vector of the form <code>cat(vec(E), vec(H))</code>:
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$$ \begin{bmatrix}
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-\imath \omega \epsilon & \nabla \times \
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\nabla \times & \imath \omega \mu
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<p>for use with a field vector of the form <code>cat(vec(E), vec(H))</code>:</p>
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<div class="arithmatex">\[
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\begin{bmatrix}
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-\imath \omega \epsilon & \nabla \times \\
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\nabla \times & \imath \omega \mu
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\end{bmatrix}
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\begin{bmatrix} E \
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\begin{bmatrix} E \\
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H
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\end{bmatrix}
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= \begin{bmatrix} J \
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= \begin{bmatrix} J \\
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-M
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\end{bmatrix} $$</p>
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\end{bmatrix}
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\]</div>
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<p><span class="doc-section-title">Parameters:</span></p>
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@ -3408,6 +3427,7 @@ usual antisymmetry of the cross product,</p>
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<div class="arithmatex">\[
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H \times E = -(E \times H),
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\]</div>
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<p>once the same staggered field placement is used on both sides.</p>
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@ -3527,6 +3547,7 @@ Then the TFSF current operator is the commutator</p>
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<div class="arithmatex">\[
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\frac{A Q - Q A}{-i \omega}.
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\]</div>
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<p>Inside regions where <code>Q</code> is locally constant, <code>A</code> and <code>Q</code> commute and the
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source vanishes. Only cells at the TF/SF boundary contribute nonzero current,
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which is exactly the desired distributed source for injecting the chosen
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