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Update 2022-02-08 17:21

master
Jean-Sébastien 2 years ago
parent
077433c40a
commit
3454aba504
100 changed files with 870 additions and 548 deletions
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build/a.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
 <html lang="en">
3 3 
 <head>
4 
-<!-- 2022-02-08 Tue 06:55 -->
4 
+<!-- 2022-02-08 Tue 17:21 -->
5 5 
 <meta charset="utf-8">
6 6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -272,6 +272,10 @@ Table of contents
272 272 
 </summary>
273 273 
 <ul>
274 274 
 <li>
275 
+<a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
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+
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+</li>
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+<li>
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 <a href="./in_t_c.html#in_t_c">Contextual colors</a><span class="headline-id">in.t.c</span>
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 </li>
 
@@ -736,7 +740,7 @@ Table of contents
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 </li>
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 <li>
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-<a href="./emsm_esm_d.html#emsm_esm_d">Dielectrics</a><span class="headline-id">emsm.esm.d</span>
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+<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
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 </li>
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 <li>
 
@@ -1629,7 +1633,7 @@ Table of contents
1629 1633
1630 1634 
 <hr><div id="postamble" class="status">
1631 1635 
 <p class="author">Author: Jean-Sébastien Caux</p>
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-<p class="date">Created: 2022-02-08 Tue 06:55</p>
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+<p class="date">Created: 2022-02-08 Tue 17:21</p>
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1634 1638 
 </div>
1635 1639

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@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
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 <html lang="en">
3 3 
 <head>
4 
-<!-- 2022-02-08 Tue 06:55 -->
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+<!-- 2022-02-08 Tue 17:21 -->
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 <meta charset="utf-8">
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 <meta name="viewport" content="width=device-width, initial-scale=1">
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -272,6 +272,10 @@ Table of contents
272 272 
 </summary>
273 273 
 <ul>
274 274 
 <li>
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+<a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
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+
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+</li>
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+<li>
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 <a href="./in_t_c.html#in_t_c">Contextual colors</a><span class="headline-id">in.t.c</span>
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 </li>
 
@@ -736,7 +740,7 @@ Table of contents
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 </li>
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 <li>
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-<a href="./emsm_esm_d.html#emsm_esm_d">Dielectrics</a><span class="headline-id">emsm.esm.d</span>
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+<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
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 </li>
742 746 
 <li>
 
@@ -1712,7 +1716,7 @@ Wiley, 1986 [2nd edition]
1712 1716
1713 1717 
 <hr><div id="postamble" class="status">
1714 1718 
 <p class="author">Author: Jean-Sébastien Caux</p>
1715 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
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+<p class="date">Created: 2022-02-08 Tue 17:21</p>
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1717 1721 
 </div>
1718 1722

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build/c.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
 <html lang="en">
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 <head>
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-<!-- 2022-02-08 Tue 06:55 -->
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+<!-- 2022-02-08 Tue 17:21 -->
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 <meta charset="utf-8">
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 <meta name="viewport" content="width=device-width, initial-scale=1">
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -272,6 +272,10 @@ Table of contents
272 272 
 </summary>
273 273 
 <ul>
274 274 
 <li>
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+<a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
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+</li>
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+<li>
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 <a href="./in_t_c.html#in_t_c">Contextual colors</a><span class="headline-id">in.t.c</span>
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 </li>
 
@@ -736,7 +740,7 @@ Table of contents
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-<a href="./emsm_esm_d.html#emsm_esm_d">Dielectrics</a><span class="headline-id">emsm.esm.d</span>
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+<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
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@@ -1629,7 +1633,7 @@ Table of contents
1629 1633
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 <hr><div id="postamble" class="status">
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 <p class="author">Author: Jean-Sébastien Caux</p>
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-<p class="date">Created: 2022-02-08 Tue 06:55</p>
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+<p class="date">Created: 2022-02-08 Tue 17:21</p>
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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 </div>
1635 1639

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build/c_m.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
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+<!-- 2022-02-08 Tue 17:21 -->
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -272,6 +272,10 @@ Table of contents
272 272 
 </summary>
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 <ul>
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 <li>
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+<a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
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+<li>
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 <a href="./in_t_c.html#in_t_c">Contextual colors</a><span class="headline-id">in.t.c</span>
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@@ -736,7 +740,7 @@ Table of contents
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-<a href="./emsm_esm_d.html#emsm_esm_d">Dielectrics</a><span class="headline-id">emsm.esm.d</span>
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@@ -1635,7 +1639,7 @@ Table of contents
1635 1639
1636 1640 
 <hr><div id="postamble" class="status">
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 <p class="author">Author: Jean-Sébastien Caux</p>
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-<p class="date">Created: 2022-02-08 Tue 06:55</p>
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+<p class="date">Created: 2022-02-08 Tue 17:21</p>
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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 </div>
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build/c_m_cs.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
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 <html lang="en">
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-<!-- 2022-02-08 Tue 06:55 -->
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+<!-- 2022-02-08 Tue 17:21 -->
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -272,6 +272,10 @@ Table of contents
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 </summary>
273 273 
 <ul>
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 <li>
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+<a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
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+
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+<li>
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 <a href="./in_t_c.html#in_t_c">Contextual colors</a><span class="headline-id">in.t.c</span>
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 </li>
 
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 <hr><div id="postamble" class="status">
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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 </div>
1637 1641

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- 67
build/c_m_cs_cyl.html View File 

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-<!-- 2022-02-08 Tue 06:55 -->
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+<!-- 2022-02-08 Tue 17:21 -->
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -272,6 +272,10 @@ Table of contents
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 </summary>
273 273 
 <ul>
274 274 
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+<a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
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+
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+</li>
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+<li>
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@@ -1622,47 +1626,39 @@ Table of contents
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 \((r, \phi, z)\).  Relation to Cartesian coordinates:
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 </p>
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-\[
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+\begin{equation}
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 x = r \cos \phi, y = r \sin \phi, z = z
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 \label{Gr(1.74)}
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-\]
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 The unit vectors are
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 \hat{\bf r} = \cos \phi ~\hat{\bf x} + \sin \phi~\hat{\bf y},
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 \hat{\boldsymbol \phi} = -\sin \theta ~\hat{\bf x} + \cos \phi~\hat{\bf y},
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 \hat{\bf z} = \hat{\bf z}.
1641 1642 
 \label{Gr(1.75)}
1642 
-\]
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 Infinitesimal displacement:
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 </p>
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 d{\bf l} = dr ~\hat{\bf r} + r d\phi~\hat{\boldsymbol \phi} + dz ~\hat{\bf z}.
1652 1651 
 \label{Gr(1.77)}
1653 
-\]
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 Volume element:
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 </p>
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 d\tau = r dr d\phi dz
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 \label{Gr(1.78)}
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 Range of parameters:  \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in ]-\infty, \infty[\).
 
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   <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
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   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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++ \frac{\partial T}{\partial z} ~\hat{\bf z}
1700 1691 
 \tag{cylgrad}\label{cylgrad}
1701 
-\]
1702 
-</p>
1692 
+\end{equation}
1703 1693 
 </div>
1704 1694 
 </div>
1705 1695
1706 1696 
 <div id="outline-container-c_m_cs_cyl_div" class="outline-6">
1707 1697 
 <h6 id="c_m_cs_cyl_div"><a href="#c_m_cs_cyl_div">Divergence</a></h6>
1708 1698 
 <div class="outline-text-6" id="text-c_m_cs_cyl_div">
1709 
-<div class="eqlabel" id="org9646f8e">
1699 
+<div class="eqlabel" id="orgf9d0262">
1710 1700 
 <p>
1711 1701 
 <a id="cyl_div"></a><a href="./c_m_cs_cyl.html#cyl_div"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1712 1702 
   <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
1713 1703 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
1714 1704 
 </svg></a>
1715 1705 
 </p>
1716 
-<div class="alteqlabels" id="orgef4d1a0">
1706 
+<div class="alteqlabels" id="orgd456be1">
1717 1707 
 <ul class="org-ul">
1718 1708 
 <li>Gr4(2.21)</li>
1719 1709 
 </ul>
 
@@ -1722,32 +1712,26 @@ Range of parameters:  \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
1722 1712
1723 1713 
 </div>
1724 1714
1725 
-<p>
1726 
-\[
1727 
-{\boldsymbol ∇} ⋅ {\bf v} = \frac{1}{r} \frac{\partial}{\partial r} (r v_r)
1728 
-</p>
1729 
-<ul class="org-ul">
1730 
-<li>\frac{1}{r} \frac{∂ v_{φ}}{∂ φ} + \frac{\partial v_z}{\partial z}.</li>
1731 
-</ul>
1732 
-<p>
1715 
+\begin{equation}
1716 
+{\boldsymbol \nabla} \cdot {\bf v} = \frac{1}{r} \frac{\partial}{\partial r} (r v_r)
1717 
++ \frac{1}{r} \frac{\partial v_{\phi}}{\partial \phi} + \frac{\partial v_z}{\partial z}.
1733 1718 
 \tag{cyl_div}
1734 1719 
 \label{cyl_div}
1735 
-\]
1736 
-</p>
1720 
+\end{equation}
1737 1721 
 </div>
1738 1722 
 </div>
1739 1723
1740 1724 
 <div id="outline-container-c_m_cs_cyl_curl" class="outline-6">
1741 1725 
 <h6 id="c_m_cs_cyl_curl"><a href="#c_m_cs_cyl_curl">Curl</a></h6>
1742 1726 
 <div class="outline-text-6" id="text-c_m_cs_cyl_curl">
1743 
-<div class="eqlabel" id="org174b722">
1727 
+<div class="eqlabel" id="orgeb922d3">
1744 1728 
 <p>
1745 1729 
 <a id="cyl_curl"></a><a href="./c_m_cs_cyl.html#cyl_curl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1746 1730 
   <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
1747 1731 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
1748 1732 
 </svg></a>
1749 1733 
 </p>
1750 
-<div class="alteqlabels" id="org9f83d45">
1734 
+<div class="alteqlabels" id="org66ae846">
1751 1735 
 <ul class="org-ul">
1752 1736 
 <li>Gr4(2.21)</li>
1753 1737 
 </ul>
 
@@ -1756,36 +1740,24 @@ Range of parameters:  \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
1756 1740
1757 1741 
 </div>
1758 1742
1759 
-<p>
1760 
-\[
1761 
-{\boldsymbol ∇} × {\bf v} = \left( \frac{1}{r} \frac{\partial v_z}{\partial \phi} - \frac{∂ v_{φ}}{∂ z}\right) ~\hat{\bf r}
1762 
-</p>
1763 
-<ul class="org-ul">
1764 
-<li>\left( \frac{\partial v_r}{\partial z} - \frac{\partial v_z}{\partial r} \right) ~\hat{\boldsymbol \phi}</li>
1765 
-<li>\frac{1}{r} \left( \frac{\partial}{\partial r} (r v_{φ}) - \frac{\partial v_r}{\partial \phi} \right) ~\hat{\bf z}</li>
1766 
-</ul>
1767 
-<p>
1743 
+\begin{align}
1744 
+{\boldsymbol \nabla} \times {\bf v} = \left( \frac{1}{r} \frac{\partial v_z}{\partial \phi} - \frac{\partial v_{\phi}}{\partial z}\right) ~\hat{\bf r}
1745 
++ \left( \frac{\partial v_r}{\partial z} - \frac{\partial v_z}{\partial r} \right) ~\hat{\boldsymbol \phi} \nonumber \\
1746 
++ \frac{1}{r} \left( \frac{\partial}{\partial r} (r v_{\phi}) - \frac{\partial v_r}{\partial \phi} \right) ~\hat{\bf z}
1768 1747 
 \tag{cyl_curl}
1769 1748 
 \label{cyl_curl}
1770 
-\]
1771 
-</p>
1749 
+\end{align}
1772 1750 
 </div>
1773 1751 
 </div>
1774 1752
1775 1753 
 <div id="outline-container-c_m_cs_cyl_lap" class="outline-6">
1776 1754 
 <h6 id="c_m_cs_cyl_lap"><a href="#c_m_cs_cyl_lap">Laplacian</a></h6>
1777 1755 
 <div class="outline-text-6" id="text-c_m_cs_cyl_lap">
1778 
-<p>
1779 
-\[
1780 
-{\boldsymbol ∇}^2 T = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial T}{\partial r} \right)
1781 
-</p>
1782 
-<ul class="org-ul">
1783 
-<li>\frac{1}{r^2} \frac{\partial^2 T}{\partial \phi^2} + \frac{\partial^2 T}{\partial z^2}</li>
1784 
-</ul>
1785 
-<p>
1756 
+\begin{equation}
1757 
+{\boldsymbol \nabla}^2 T = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial T}{\partial r} \right)
1758 
++ \frac{1}{r^2} \frac{\partial^2 T}{\partial \phi^2} + \frac{\partial^2 T}{\partial z^2}
1786 1759 
 \label{Gr(1.82)}
1787 
-\]
1788 
-</p>
1760 
+\end{equation}
1789 1761 
 </div>
1790 1762 
 </div>
1791 1763 
 </div>
 
@@ -1795,7 +1767,7 @@ Range of parameters:  \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
1795 1767
1796 1768 
 <hr><div id="postamble" class="status">
1797 1769 
 <p class="author">Author: Jean-Sébastien Caux</p>
1798 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1770 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1799 1771 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1800 1772 
 </div>
1801 1773

+ 7
- 3
build/c_m_cs_hyp.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
 <html lang="en">
3 3 
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4 
-<!-- 2022-02-08 Tue 06:55 -->
4 
+<!-- 2022-02-08 Tue 17:21 -->
5 5 
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6 6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -272,6 +272,10 @@ Table of contents
272 272 
 </summary>
273 273 
 <ul>
274 274 
 <li>
275 
+<a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
276 
+
277 
+</li>
278 
+<li>
275 279 
 <a href="./in_t_c.html#in_t_c">Contextual colors</a><span class="headline-id">in.t.c</span>
276 280
277 281 
 </li>
 
@@ -736,7 +740,7 @@ Table of contents
736 740
737 741 
 </li>
738 742 
 <li>
739 
-<a href="./emsm_esm_d.html#emsm_esm_d">Dielectrics</a><span class="headline-id">emsm.esm.d</span>
743 
+<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
740 744
741 745 
 </li>
742 746 
 <li>
 
@@ -1624,7 +1628,7 @@ Table of contents
1624 1628
1625 1629 
 <hr><div id="postamble" class="status">
1626 1630 
 <p class="author">Author: Jean-Sébastien Caux</p>
1627 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1631 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1628 1632 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1629 1633 
 </div>
1630 1634

+ 25
- 45
build/c_m_cs_sph.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
 <html lang="en">
3 3 
 <head>
4 
-<!-- 2022-02-08 Tue 06:55 -->
4 
+<!-- 2022-02-08 Tue 17:21 -->
5 5 
 <meta charset="utf-8">
6 6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -272,6 +272,10 @@ Table of contents
272 272 
 </summary>
273 273 
 <ul>
274 274 
 <li>
275 
+<a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
276 
+
277 
+</li>
278 
+<li>
275 279 
 <a href="./in_t_c.html#in_t_c">Contextual colors</a><span class="headline-id">in.t.c</span>
276 280
277 281 
 </li>
 
@@ -736,7 +740,7 @@ Table of contents
736 740
737 741 
 </li>
738 742 
 <li>
739 
-<a href="./emsm_esm_d.html#emsm_esm_d">Dielectrics</a><span class="headline-id">emsm.esm.d</span>
743 
+<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
740 744
741 745 
 </li>
742 746 
 <li>
 
@@ -1688,70 +1692,46 @@ Infinitesimal surface element:  depends on situation.
1688 1692 
 <div id="outline-container-c_m_cs_sph_grad" class="outline-6">
1689 1693 
 <h6 id="c_m_cs_sph_grad"><a href="#c_m_cs_sph_grad">Gradient</a></h6>
1690 1694 
 <div class="outline-text-6" id="text-c_m_cs_sph_grad">
1691 
-<p>
1692 
-\[
1693 
-{\boldsymbol ∇} T = \frac{\partial T}{\partial r} \hat{\boldsymbol r} + \frac{1}{r} \frac{\partial T}{\partial \theta} \hat{\boldsymbol \theta}
1694 
-</p>
1695 
-<ul class="org-ul">
1696 
-<li>\frac{1}{r\sin \theta} \frac{\partial T}{\partial \phi} \hat{\boldsymbol \phi}.</li>
1697 
-</ul>
1698 
-<p>
1695 
+\begin{equation}
1696 
+{\boldsymbol \nabla} T = \frac{\partial T}{\partial r} \hat{\boldsymbol r} + \frac{1}{r} \frac{\partial T}{\partial \theta} \hat{\boldsymbol \theta}
1697 
++ \frac{1}{r\sin \theta} \frac{\partial T}{\partial \phi} \hat{\boldsymbol \phi}.
1699 1698 
 \label{Gr(1.70)}
1700 
-\]
1701 
-</p>
1699 
+\end{equation}
1702 1700 
 </div>
1703 1701 
 </div>
1704 1702
1705 1703 
 <div id="outline-container-c_m_cs_sph_div" class="outline-6">
1706 1704 
 <h6 id="c_m_cs_sph_div"><a href="#c_m_cs_sph_div">Divergence</a></h6>
1707 1705 
 <div class="outline-text-6" id="text-c_m_cs_sph_div">
1708 
-<p>
1709 
-\[
1710 
-{\boldsymbol ∇} ⋅ {\bf v} = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 v_r) + \frac{1}{r\sin \theta} \frac{\partial}{\partial \theta} (sinθ v_{θ})
1711 
-</p>
1712 
-<ul class="org-ul">
1713 
-<li>\frac{1}{r \sin \theta} \frac{∂ v_{φ}}{∂ φ}</li>
1714 
-</ul>
1715 
-<p>
1706 
+\begin{equation}
1707 
+{\boldsymbol \nabla} \cdot {\bf v} = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 v_r) + \frac{1}{r\sin \theta} \frac{\partial}{\partial \theta} (\sin\theta v_{\theta})
1708 
++ \frac{1}{r \sin \theta} \frac{\partial v_{\phi}}{\partial \phi}
1716 1709 
 \label{Gr(1.71)}
1717 
-\]
1718 
-</p>
1710 
+\end{equation}
1719 1711 
 </div>
1720 1712 
 </div>
1721 1713
1722 1714 
 <div id="outline-container-c_m_cs_sph_curl" class="outline-6">
1723 1715 
 <h6 id="c_m_cs_sph_curl"><a href="#c_m_cs_sph_curl">Curl</a></h6>
1724 1716 
 <div class="outline-text-6" id="text-c_m_cs_sph_curl">
1725 
-<p>
1726 
-\[
1727 
-{\boldsymbol ∇} × {\bf v} = \frac{1}{r\sin \theta} \left[ \frac{\partial}{\partial \theta} (sin θ v_{φ}) - \frac{∂ v_{θ}}{∂ φ} \right] \hat{\bf r}
1728 
-</p>
1729 
-<ul class="org-ul">
1730 
-<li>\frac{1}{r} \left[ \frac{1}{\sin \theta} \frac{\partial v_r}{\partial \phi} - \frac{\partial}{\partial r} (r v_{φ}) \right] \hat{\boldsymbol \theta}</li>
1731 
-<li>\frac{1}{r} \left[ \frac{\partial}{\partial r} (r v_{θ}) - \frac{\partial v_r}{\partial \theta} \right] \hat{\boldsymbol \phi}</li>
1732 
-</ul>
1733 
-<p>
1717 
+\begin{equation}
1718 
+{\boldsymbol \nabla} \times {\bf v} = \frac{1}{r\sin \theta} \left[ \frac{\partial}{\partial \theta} (\sin \theta v_{\phi}) - \frac{\partial v_{\theta}}{\partial \phi} \right] \hat{\bf r}
1719 
++ \frac{1}{r} \left[ \frac{1}{\sin \theta} \frac{\partial v_r}{\partial \phi} - \frac{\partial}{\partial r} (r v_{\phi}) \right] \hat{\boldsymbol \theta}
1720 
++ \frac{1}{r} \left[ \frac{\partial}{\partial r} (r v_{\theta}) - \frac{\partial v_r}{\partial \theta} \right] \hat{\boldsymbol \phi}
1734 1721 
 \label{Gr(1.72)}
1735 
-\]
1736 
-</p>
1722 
+\end{equation}
1737 1723 
 </div>
1738 1724 
 </div>
1739 1725
1740 1726 
 <div id="outline-container-c_m_cs_sph_lap" class="outline-6">
1741 1727 
 <h6 id="c_m_cs_sph_lap"><a href="#c_m_cs_sph_lap">Laplacian</a></h6>
1742 1728 
 <div class="outline-text-6" id="text-c_m_cs_sph_lap">
1743 
-<p>
1744 
-\[
1745 
-{\boldsymbol ∇}^2 T = \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial T}{\partial r}\right)
1746 
-</p>
1747 
-<ul class="org-ul">
1748 
-<li>\frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( sin θ \frac{\partial T}{\partial \theta}\right)</li>
1749 
-<li>\frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 T}{\partial \phi^2}</li>
1750 
-</ul>
1751 
-<p>
1729 
+\begin{equation}
1730 
+{\boldsymbol \nabla}^2 T = \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial T}{\partial r}\right)
1731 
++ \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial T}{\partial \theta}\right)
1732 
++ \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 T}{\partial \phi^2}
1752 1733 
 \label{Gr(1.73)}
1753 
-\]
1754 
-</p>
1734 
+\end{equation}
1755 1735 
 </div>
1756 1736 
 </div>
1757 1737 
 </div>
 
@@ -1761,7 +1741,7 @@ Infinitesimal surface element:  depends on situation.
1761 1741
1762 1742 
 <hr><div id="postamble" class="status">
1763 1743 
 <p class="author">Author: Jean-Sébastien Caux</p>
1764 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1744 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1765 1745 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1766 1746 
 </div>
1767 1747

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7 7 
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@@ -272,6 +272,10 @@ Table of contents
272 272 
 </summary>
273 273 
 <ul>
274 274 
 <li>
275 
+<a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
276 
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277 
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+<li>
275 279 
 <a href="./in_t_c.html#in_t_c">Contextual colors</a><span class="headline-id">in.t.c</span>
276 280
277 281 
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@@ -736,7 +740,7 @@ Table of contents
736 740
737 741 
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738 742 
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-<a href="./emsm_esm_d.html#emsm_esm_d">Dielectrics</a><span class="headline-id">emsm.esm.d</span>
743 
+<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
740 744
741 745 
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742 746 
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@@ -1634,7 +1638,7 @@ Table of contents
1634 1638
1635 1639 
 <hr><div id="postamble" class="status">
1636 1640 
 <p class="author">Author: Jean-Sébastien Caux</p>
1637 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1641 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1638 1642 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1639 1643 
 </div>
1640 1644

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@@ -1,7 +1,7 @@
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@@ -272,6 +272,10 @@ Table of contents
272 272 
 </summary>
273 273 
 <ul>
274 274 
 <li>
275 
+<a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
276 
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277 
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278 
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275 279 
 <a href="./in_t_c.html#in_t_c">Contextual colors</a><span class="headline-id">in.t.c</span>
276 280
277 281 
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@@ -736,7 +740,7 @@ Table of contents
736 740
737 741 
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738 742 
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-<a href="./emsm_esm_d.html#emsm_esm_d">Dielectrics</a><span class="headline-id">emsm.esm.d</span>
743 
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740 744
741 745 
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742 746 
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@@ -1636,7 +1640,7 @@ v_x &amp; v_y &amp; v_z \end{array} \right| \nonumber \\
1636 1640
1637 1641 
 <hr><div id="postamble" class="status">
1638 1642 
 <p class="author">Author: Jean-Sébastien Caux</p>
1639 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1643 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1640 1644 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1641 1645 
 </div>
1642 1646

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@@ -272,6 +272,10 @@ Table of contents
272 272 
 </summary>
273 273 
 <ul>
274 274 
 <li>
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275 279 
 <a href="./in_t_c.html#in_t_c">Contextual colors</a><span class="headline-id">in.t.c</span>
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-<a href="./emsm_esm_d.html#emsm_esm_d">Dielectrics</a><span class="headline-id">emsm.esm.d</span>
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+<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
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 <li>
 
@@ -1620,9 +1624,9 @@ Table of contents
1620 1624 
 <div class="outline-text-5" id="text-c_m_dc_d2">
1621 1625 
 </div>
1622 1626
1623 
-<div id="outline-container-orgc3b89bb" class="outline-6">
1624 
-<h6 id="orgc3b89bb"><a href="#orgc3b89bb">Divergence of gradient</a></h6>
1625 
-<div class="outline-text-6" id="text-orgc3b89bb">
1627 
+<div id="outline-container-org5fd48d1" class="outline-6">
1628 
+<h6 id="org5fd48d1"><a href="#org5fd48d1">Divergence of gradient</a></h6>
1629 
+<div class="outline-text-6" id="text-org5fd48d1">
1626 1630 
 <p>
1627 1631 
 \({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the <b>Laplacian</b> of the scalar field \(T\).
1628 1632 
 The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components
 
@@ -1631,36 +1635,36 @@ given by the Laplacian of the corresponding vector elements.
1631 1635 
 </div>
1632 1636 
 </div>
1633 1637
1634 
-<div id="outline-container-org5caf3f5" class="outline-6">
1635 
-<h6 id="org5caf3f5"><a href="#org5caf3f5">Curl of a gradient</a></h6>
1636 
-<div class="outline-text-6" id="text-org5caf3f5">
1638 
+<div id="outline-container-org41a3448" class="outline-6">
1639 
+<h6 id="org41a3448"><a href="#org41a3448">Curl of a gradient</a></h6>
1640 
+<div class="outline-text-6" id="text-org41a3448">
1637 1641 
 <p>
1638 1642 
 This always vanishes.
1639 1643 
 </p>
1640 1644 
 </div>
1641 1645 
 </div>
1642 1646
1643 
-<div id="outline-container-org4f9116a" class="outline-6">
1644 
-<h6 id="org4f9116a"><a href="#org4f9116a">Gradient of the divergence</a></h6>
1645 
-<div class="outline-text-6" id="text-org4f9116a">
1647 
+<div id="outline-container-org6c4b5a7" class="outline-6">
1648 
+<h6 id="org6c4b5a7"><a href="#org6c4b5a7">Gradient of the divergence</a></h6>
1649 
+<div class="outline-text-6" id="text-org6c4b5a7">
1646 1650 
 <p>
1647 1651 
 \({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics.  No special name.
1648 1652 
 </p>
1649 1653 
 </div>
1650 1654 
 </div>
1651 1655
1652 
-<div id="outline-container-org2c1c081" class="outline-6">
1653 
-<h6 id="org2c1c081"><a href="#org2c1c081">Divergence of a curl</a></h6>
1654 
-<div class="outline-text-6" id="text-org2c1c081">
1656 
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1657 
+<h6 id="org68b62ab"><a href="#org68b62ab">Divergence of a curl</a></h6>
1658 
+<div class="outline-text-6" id="text-org68b62ab">
1655 1659 
 <p>
1656 1660 
 This always vanishes.
1657 1661 
 </p>
1658 1662 
 </div>
1659 1663 
 </div>
1660 1664
1661 
-<div id="outline-container-orgc977e97" class="outline-6">
1662 
-<h6 id="orgc977e97"><a href="#orgc977e97">Curl of curl</a></h6>
1663 
-<div class="outline-text-6" id="text-orgc977e97">
1665 
+<div id="outline-container-org62690be" class="outline-6">
1666 
+<h6 id="org62690be"><a href="#org62690be">Curl of curl</a></h6>
1667 
+<div class="outline-text-6" id="text-org62690be">
1664 1668 
 <p>
1665 1669 
 \[
1666 1670 
 {\boldsymbol \nabla} \times ({\boldsymbol \nabla} \times {\bf v}) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v}) - {\boldsymbol \nabla}^2 {\bf v}
 
@@ -1674,7 +1678,7 @@ This always vanishes.
1674 1678
1675 1679 
 <hr><div id="postamble" class="status">
1676 1680 
 <p class="author">Author: Jean-Sébastien Caux</p>
1677 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1681 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1678 1682 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1679 1683 
 </div>
1680 1684

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@@ -1,7 +1,7 @@
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@@ -272,6 +272,10 @@ Table of contents
272 272 
 </summary>
273 273 
 <ul>
274 274 
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+<a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
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+
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275 279 
 <a href="./in_t_c.html#in_t_c">Contextual colors</a><span class="headline-id">in.t.c</span>
276 280
277 281 
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@@ -736,7 +740,7 @@ Table of contents
736 740
737 741 
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738 742 
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739 
-<a href="./emsm_esm_d.html#emsm_esm_d">Dielectrics</a><span class="headline-id">emsm.esm.d</span>
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+<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
740 744
741 745 
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742 746 
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@@ -1635,7 +1639,7 @@ The <b>del/grad/nabla operator</b> is defined as
1635 1639
1636 1640 
 <hr><div id="postamble" class="status">
1637 1641 
 <p class="author">Author: Jean-Sébastien Caux</p>
1638 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1642 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1639 1643 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1640 1644 
 </div>
1641 1645

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@@ -1,7 +1,7 @@
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@@ -272,6 +272,10 @@ Table of contents
272 272 
 </summary>
273 273 
 <ul>
274 274 
 <li>
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+<a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
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276 280
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@@ -736,7 +740,7 @@ Table of contents
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741 745 
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742 746 
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@@ -1632,7 +1636,7 @@ Table of contents
1632 1636
1633 1637 
 <hr><div id="postamble" class="status">
1634 1638 
 <p class="author">Author: Jean-Sébastien Caux</p>
1635 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1639 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1636 1640 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1637 1641 
 </div>
1638 1642

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@@ -1,7 +1,7 @@
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@@ -272,6 +272,10 @@ Table of contents
272 272 
 </summary>
273 273 
 <ul>
274 274 
 <li>
275 
+<a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
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+<li>
275 279 
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276 280
277 281 
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@@ -736,7 +740,7 @@ Table of contents
736 740
737 741 
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738 742 
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-<a href="./emsm_esm_d.html#emsm_esm_d">Dielectrics</a><span class="headline-id">emsm.esm.d</span>
743 
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740 744
741 745 
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742 746 
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@@ -1656,7 +1660,7 @@ is a vector called the <b>gradient</b> of \(T\).
1656 1660
1657 1661 
 <hr><div id="postamble" class="status">
1658 1662 
 <p class="author">Author: Jean-Sébastien Caux</p>
1659 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1663 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1660 1664 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1661 1665 
 </div>
1662 1666

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@@ -272,6 +272,10 @@ Table of contents
272 272 
 </summary>
273 273 
 <ul>
274 274 
 <li>
275 
+<a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
276 
+
277 
+</li>
278 
+<li>
275 279 
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276 280
277 281 
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@@ -736,7 +740,7 @@ Table of contents
736 740
737 741 
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738 742 
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739 
-<a href="./emsm_esm_d.html#emsm_esm_d">Dielectrics</a><span class="headline-id">emsm.esm.d</span>
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+<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
740 744
741 745 
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742 746 
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@@ -1629,7 +1633,7 @@ Six product rules (on inside front cover of Gr).
1629 1633
1630 1634 
 <hr><div id="postamble" class="status">
1631 1635 
 <p class="author">Author: Jean-Sébastien Caux</p>
1632 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1636 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1633 1637 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1634 1638 
 </div>
1635 1639

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@@ -1,7 +1,7 @@
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7 7 
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@@ -272,6 +272,10 @@ Table of contents
272 272 
 </summary>
273 273 
 <ul>
274 274 
 <li>
275 
+<a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
276 
+
277 
+</li>
278 
+<li>
275 279 
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276 280
277 281 
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@@ -736,7 +740,7 @@ Table of contents
736 740
737 741 
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738 742 
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739 
-<a href="./emsm_esm_d.html#emsm_esm_d">Dielectrics</a><span class="headline-id">emsm.esm.d</span>
743 
+<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
740 744
741 745 
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742 746 
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@@ -1631,7 +1635,7 @@ Table of contents
1631 1635
1632 1636 
 <hr><div id="postamble" class="status">
1633 1637 
 <p class="author">Author: Jean-Sébastien Caux</p>
1634 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1638 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1635 1639 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1636 1640 
 </div>
1637 1641

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@@ -1,7 +1,7 @@
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -272,6 +272,10 @@ Table of contents
272 272 
 </summary>
273 273 
 <ul>
274 274 
 <li>
275 
+<a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
276 
+
277 
+</li>
278 
+<li>
275 279 
 <a href="./in_t_c.html#in_t_c">Contextual colors</a><span class="headline-id">in.t.c</span>
276 280
277 281 
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@@ -736,7 +740,7 @@ Table of contents
736 740
737 741 
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738 742 
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739 
-<a href="./emsm_esm_d.html#emsm_esm_d">Dielectrics</a><span class="headline-id">emsm.esm.d</span>
743 
+<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
740 744
741 745 
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742 746 
 <li>
 
@@ -1643,7 +1647,7 @@ Consequences:  for any smooth differentiable function \(f(x)\),
1643 1647
1644 1648 
 <hr><div id="postamble" class="status">
1645 1649 
 <p class="author">Author: Jean-Sébastien Caux</p>
1646 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1650 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1647 1651 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1648 1652 
 </div>
1649 1653

+ 21
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@@ -1,7 +1,7 @@
1 1 
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -272,6 +272,10 @@ Table of contents
272 272 
 </summary>
273 273 
 <ul>
274 274 
 <li>
275 
+<a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
276 
+
277 
+</li>
278 
+<li>
275 279 
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276 280
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@@ -736,7 +740,7 @@ Table of contents
736 740
737 741 
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739 
-<a href="./emsm_esm_d.html#emsm_esm_d">Dielectrics</a><span class="headline-id">emsm.esm.d</span>
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+<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
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@@ -1618,43 +1622,37 @@ Table of contents
1618 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
1619 1623 
 </svg></a><span class="headline-id">c.m.dd.3d</span></h5>
1620 1624 
 <div class="outline-text-5" id="text-c_m_dd_3d">
1621 
-<p>
1622 
-\[
1625 
+\begin{equation}
1623 1626 
 \delta^{(3)} ({\bf r} - {\bf r}') = \delta (x - x') \delta (y - y') \delta (z - z')
1624 1627 
 \label{Gr(1.96)}
1625 
-\]
1626 
-</p>
1628 
+\end{equation}
1627 1629
1628 
-<p>
1629 
-\[
1630 
+\begin{equation}
1630 1631 
 \int d\tau f({\bf r}) \delta^{(3)} ({\bf r} - {\bf a}) = f({\bf a})
1631 1632 
 \label{Gr(1.97)}
1632 
-\]
1633 
-</p>
1633 
+\end{equation}
1634 1634
1635 1635 
 <p>
1636 1636 
 Resolution of divergence of \(\hat{\bf r}/r^2\) paradox:
1637 1637 
 </p>
1638 1638
1639 
-<p>
1640 
-\[
1639 
+\begin{equation}
1641 1640 
 {\boldsymbol \nabla} \cdot \left(\frac{\hat{\bf r}}{r^2} \right)= 4\pi \delta^{(3)} ({\bf r}).
1642 1641 
 \label{Gr(1.99)}
1643 
-\]
1644 
-</p>
1642 
+\end{equation}
1645 1643
1646 1644 
 <p>
1647 1645 
 More generally,
1648 1646 
 </p>
1649 1647
1650 
-<div class="eqlabel" id="org99c1900">
1648 
+<div class="eqlabel" id="org276d185">
1651 1649 
 <p>
1652 1650 
 <a id="divdel"></a><a href="./c_m_dd_3d.html#divdel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1653 1651 
   <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
1654 1652 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
1655 1653 
 </svg></a>
1656 1654 
 </p>
1657 
-<div class="alteqlabels" id="orge7a14d6">
1655 
+<div class="alteqlabels" id="org676b181">
1658 1656 
 <ul class="org-ul">
1659 1657 
 <li>Gr (1.100)</li>
1660 1658 
 </ul>
 
@@ -1662,43 +1660,37 @@ More generally,
1662 1660 
 </div>
1663 1661
1664 1662 
 </div>
1665 
-<p>
1666 
-\[
1663 
+\begin{equation}
1667 1664 
 \boxed{
1668 1665 
 {\boldsymbol \nabla} \cdot \left(\frac{{\bf r} - {\bf r}'}{|{\bf r} - {\bf r}'|^3} \right)= 4\pi \delta^{(3)} ({\bf r}).
1669 1666 
 }
1670 1667 
 \tag{divdel}\label{divdel}
1671 
-\]
1672 
-</p>
1668 
+\end{equation}
1673 1669
1674 1670 
 <p>
1675 1671 
 Since
1676 1672 
 </p>
1677 1673
1678 
-<p>
1679 
-\[
1674 
+\begin{equation}
1680 1675 
 {\boldsymbol \nabla}_1 \left(\frac{1}{r_{12}}\right) = -\frac{\hat{\bf r}_{12}}{r_{12}^2}
1681 1676 
 \label{Gr(1.101)}
1682 
-\]
1683 
-</p>
1677 
+\end{equation}
1684 1678
1685 1679 
 <p>
1686 1680 
 we have that
1687 1681 
 </p>
1688 1682
1689 
-<p>
1690 
-\[
1683 
+\begin{equation}
1691 1684 
 {\boldsymbol \nabla}^2 \left( \frac{1}{|{\bf r} - {\bf r}'|} \right) = -4\pi \delta^{(3)} ({\bf r} - {\bf r}')
1692 1685 
 \label{Gr(1.102)}
1693 
-\]
1694 
-</p>
1686 
+\end{equation}
1695 1687 
 </div>
1696 1688 
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1697 1689
1698 1690
1699 1691 
 <hr><div id="postamble" class="status">
1700 1692 
 <p class="author">Author: Jean-Sébastien Caux</p>
1701 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1693 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1702 1694 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1703 1695 
 </div>
1704 1696

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@@ -272,6 +272,10 @@ Table of contents
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273 273 
 <ul>
274 274 
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+<a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
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@@ -736,7 +740,7 @@ Table of contents
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@@ -1622,12 +1626,10 @@ Table of contents
1622 1626 
 Try to calculate this directly:
1623 1627 
 </p>
1624 1628
1625 
-<p>
1626 
-\[
1629 
+\begin{equation}
1627 1630 
 {\boldsymbol \nabla} \cdot \frac{\hat{\bf r}}{r^2} = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{1}{r^2}\right) =^{?} 0
1628 1631 
 \label{Gr(1.84)}
1629 
-\]
1630 
-</p>
1632 
+\end{equation}
1631 1633
1632 1634 
 <p>
1633 1635 
 Now apply the divergence theorem.
 
@@ -1635,16 +1637,14 @@ Integrate over a sphere of radius \(R\) centered at the origin (Prob.  1.38b):
1635 1637 
 </p>
1636 1638
1637 1639
1638 
-<p>
1639 
-\[
1640 
+\begin{equation}
1640 1641 
 \oint \frac{\hat{\bf r}}{r^2} \cdot d{\bf a} = \int \left(\frac{1}{R^2} \hat{\bf r} \right) \cdot \left( R^2 \sin \theta d\theta d\phi ~\hat{\bf r} \right)
1641 1642 
 = \int_0^{\pi} d\theta \sin \theta \int_0^{2\pi} d\phi = 4\pi
1642 1643 
 \label{Gr(1.85)}
1643 
-\]
1644 
-</p>
1644 
+\end{equation}
1645 1645
1646 1646 
 <p>
1647 
-Problem:  in (1.84), we've divided by zero when \(r = 0\).
1647 
+Problem:  in Gr(1.84), we've divided by zero when \(r = 0\).
1648 1648 
 </p>
1649 1649 
 </div>
1650 1650 
 </div>
 
@@ -1653,7 +1653,7 @@ Problem:  in (1.84), we've divided by zero when \(r = 0\).
1653 1653
1654 1654 
 <hr><div id="postamble" class="status">
1655 1655 
 <p class="author">Author: Jean-Sébastien Caux</p>
1656 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1656 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1657 1657 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1658 1658 
 </div>
1659 1659

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@@ -272,6 +272,10 @@ Table of contents
272 272 
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273 273 
 <ul>
274 274 
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275 279 
 <a href="./in_t_c.html#in_t_c">Contextual colors</a><span class="headline-id">in.t.c</span>
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277 281 
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737 741 
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742 746 
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@@ -1634,7 +1638,7 @@ Table of contents
1634 1638
1635 1639 
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1636 1640 
 <p class="author">Author: Jean-Sébastien Caux</p>
1637 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1641 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1638 1642 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1639 1643 
 </div>
1640 1644

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@@ -272,6 +272,10 @@ Table of contents
272 272 
 </summary>
273 273 
 <ul>
274 274 
 <li>
275 
+<a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
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275 279 
 <a href="./in_t_c.html#in_t_c">Contextual colors</a><span class="headline-id">in.t.c</span>
276 280
277 281 
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@@ -736,7 +740,7 @@ Table of contents
736 740
737 741 
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738 742 
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739 
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743 
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740 744
741 745 
 </li>
742 746 
 <li>
 
@@ -1632,7 +1636,7 @@ Table of contents
1632 1636
1633 1637 
 <hr><div id="postamble" class="status">
1634 1638 
 <p class="author">Author: Jean-Sébastien Caux</p>
1635 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1639 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1636 1640 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1637 1641 
 </div>
1638 1642

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@@ -272,6 +272,10 @@ Table of contents
272 272 
 </summary>
273 273 
 <ul>
274 274 
 <li>
275 
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275 279 
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276 280
277 281 
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@@ -736,7 +740,7 @@ Table of contents
736 740
737 741 
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738 742 
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739 
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741 745 
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742 746 
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@@ -1641,7 +1645,7 @@ Table of contents
1641 1645
1642 1646 
 <hr><div id="postamble" class="status">
1643 1647 
 <p class="author">Author: Jean-Sébastien Caux</p>
1644 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1648 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1645 1649 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1646 1650 
 </div>
1647 1651

+ 7
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@@ -272,6 +272,10 @@ Table of contents
272 272 
 </summary>
273 273 
 <ul>
274 274 
 <li>
275 
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275 279 
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276 280
277 281 
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736 740
737 741 
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738 742 
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739 
-<a href="./emsm_esm_d.html#emsm_esm_d">Dielectrics</a><span class="headline-id">emsm.esm.d</span>
743 
+<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
740 744
741 745 
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742 746 
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@@ -1635,7 +1639,7 @@ This is know either as <b>Gauss' theorem</b>, <b>Green's theorem</b> or the <b>d
1635 1639
1636 1640 
 <hr><div id="postamble" class="status">
1637 1641 
 <p class="author">Author: Jean-Sébastien Caux</p>
1638 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1642 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1639 1643 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1640 1644 
 </div>
1641 1645

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@@ -272,6 +272,10 @@ Table of contents
272 272 
 </summary>
273 273 
 <ul>
274 274 
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275 
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275 279 
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276 280
277 281 
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@@ -736,7 +740,7 @@ Table of contents
736 740
737 741 
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738 742 
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739 
-<a href="./emsm_esm_d.html#emsm_esm_d">Dielectrics</a><span class="headline-id">emsm.esm.d</span>
743 
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740 744
741 745 
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742 746 
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@@ -1642,7 +1646,7 @@ or in other words
1642 1646
1643 1647 
 <hr><div id="postamble" class="status">
1644 1648 
 <p class="author">Author: Jean-Sébastien Caux</p>
1645 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1649 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1646 1650 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1647 1651 
 </div>
1648 1652

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7 7 
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272 272 
 </summary>
273 273 
 <ul>
274 274 
 <li>
275 
+<a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
276 
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277 
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278 
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275 279 
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276 280
277 281 
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736 740
737 741 
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738 742 
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739 
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743 
+<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
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741 745 
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742 746 
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@@ -1620,9 +1624,9 @@ Table of contents
1620 1624 
 <div class="outline-text-5" id="text-c_m_ic_lsv">
1621 1625 
 </div>
1622 1626
1623 
-<div id="outline-container-org860e5de" class="outline-6">
1624 
-<h6 id="org860e5de"><a href="#org860e5de">Line Integrals</a></h6>
1625 
-<div class="outline-text-6" id="text-org860e5de">
1627 
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1628 
+<h6 id="org609a3df"><a href="#org609a3df">Line Integrals</a></h6>
1629 
+<div class="outline-text-6" id="text-org609a3df">
1626 1630 
 <p>
1627 1631 
 \[
1628 1632 
 {\int_{\bf a}^{\bf b}}_{\cal P} {\bf v} \cdot d{\bf l}
 
@@ -1651,9 +1655,9 @@ Integral over a closed loop:
1651 1655 
 </div>
1652 1656 
 </div>
1653 1657
1654 
-<div id="outline-container-orged48130" class="outline-6">
1655 
-<h6 id="orged48130"><a href="#orged48130">Surface Integrals</a></h6>
1656 
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1658 
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1659 
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1660 
+<div class="outline-text-6" id="text-org8f657b7">
1657 1661 
 <p>
1658 1662 
 \[
1659 1663 
 \int_{\cal S} {\bf v} \cdot d{\bf a}
 
@@ -1673,9 +1677,9 @@ Over a closed surface:
1673 1677 
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1674 1678 
 </div>
1675 1679
1676 
-<div id="outline-container-org4728536" class="outline-6">
1677 
-<h6 id="org4728536"><a href="#org4728536">Volume Integrals</a></h6>
1678 
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1680 
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1681 
+<h6 id="org2ec267e"><a href="#org2ec267e">Volume Integrals</a></h6>
1682 
+<div class="outline-text-6" id="text-org2ec267e">
1679 1683 
 <p>
1680 1684 
 \[
1681 1685 
 \int_{\cal V} T d\tau
 
@@ -1703,7 +1707,7 @@ d\tau = dx dy dz
1703 1707
1704 1708 
 <hr><div id="postamble" class="status">
1705 1709 
 <p class="author">Author: Jean-Sébastien Caux</p>
1706 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1710 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1707 1711 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1708 1712 
 </div>
1709 1713

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272 272 
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273 273 
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274 274 
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737 741 
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1618 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
1619 1623 
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1620 1624 
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1621 
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1623 1627 
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1624 1628 
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1625 1629 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
1626 1630 
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1627 1631 
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1632 
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1630 1634 
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1631 1635 
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@@ -1657,7 +1661,7 @@ the boundary shrinks to a point.
1657 1661
1658 1662 
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1659 1663 
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1660 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1664 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1661 1665 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1662 1666 
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1663 1667

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1651 
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1647 
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1651 
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1654 
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1637 
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 <b>scalar potential</b> \(V\):
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+\begin{equation}
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 {\boldsymbol \nabla} \times {\bf F} = 0 \Longleftrightarrow {\bf F} = -{\boldsymbol \nabla} V
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 \label{Gr(1.103)}
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@@ -1678,7 +1680,7 @@ The following conditions are equivalent:
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 Objectives
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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 <b>Electromotive force (or electromotance)</b>,
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 <b>Faraday's law</b> (differential form)
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   \[
 
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 \hspace{2cm} \mbox{(7.31 and 7.34)}
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 \paragraph{Example 7.13:}  coaxial cable (inner cylinder radius \(a\), outer \(b\)) carries current \(I\).
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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 \paragraph{Example 7.10:}
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 \paragraph{Example 7.11:}  find self-inductance of toroidal coil with
1714 1718 
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1740 1744 
 \paragraph{Example 7.12:}  circuit with inductance \(L\), resistor \(R\) and battery \({\cal E}_0\).
1741 1745 
 What is the current ?
 
@@ -1761,7 +1765,7 @@ where \(\tau \equiv L/R\) is the {\bf time constant} of the circuit.
1761 1765
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1768 
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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 {\bf Example 7.7:}
1648 1652 
 \({\bf B}(t)\) points up in circular region of radius \(R\).  What is the induced \({\bf E}(t)\) ?
 
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1658 1662 
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 {\bf Example 7.8:} wheel or radius \(b\) with line charge \(\lambda\) on the rim.
1664 1668 
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@@ -1692,7 +1696,7 @@ called the {\bf quasistatic} approximation, and works provided we deal with
1692 1696 
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1693 1697 
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1697 1701 
 {\bf Example 7.9:}  infinitely long straight wire carries \(I(t)\).  Find
1698 1702 
 induced \({\bf E}\) field as a function of distance \(s\) from wire.
 
@@ -1727,7 +1731,7 @@ Reason:  in this case, we've overstepped the quasistatic limit.  We need
1727 1731
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1730 
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1734 
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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1635 
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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 <p>
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 Full set of equations for the electromagnetic field:
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 {\bf Maxwell's equations} {\it (in vacuum)}
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 Complement:
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 {\bf Force law}
1643 1647 
 \[
 
@@ -1661,7 +1665,7 @@ take divergence of \((iv)\).
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1662 1666 
 Better way of writing:  all fields on left, all sources on right,
1663 1667 
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 \begin{align}
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   (i) &amp;{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0},
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   &amp;(iii) {\boldsymbol \nabla} \times {\bf E} + \frac{\partial {\bf B}}{\partial t} = 0, \\
 
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1681 
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 The extra term would thus be eliminated if we were to put
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 \[
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     {\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}
 
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 Maxwell baptized this term the
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 {\bf Displacement current}
1658 1662 
 \[
 
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 <p class="author">Author: Jean-Sébastien Caux</p>
1690 
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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1622 1626 
 The angular momentum of EM fields is directly given by
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 {\bf Angular momentum of EM fields}
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 \[
 
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1676 1680 
 This expression can be greatly simplified by introducing the
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 {\bf Maxwell stress tensor}
1681 1685 
 \[
 
@@ -1698,7 +1702,7 @@ The element \(T_{ij}\) represents the force per unit area in the $i$th direction
1698 1702 
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1699 1703 
 We then obtain
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 {\bf EM force per unit volume}
1704 1708 
 \[
 
@@ -1710,7 +1714,7 @@ We then obtain
1710 1714 
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1711 1715 
 where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
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 {\bf Total force on charges in volume}
1716 1720 
 \[
 
@@ -1726,7 +1730,7 @@ where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
1726 1730
1727 1731 
 <hr><div id="postamble" class="status">
1728 1732 
 <p class="author">Author: Jean-Sébastien Caux</p>
1729 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1733 
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1730 1734 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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@@ -1687,7 +1691,7 @@ so we get
1687 1691 
 Substituting this in \ref{Gr(8.6)} and using the divergence theorem,
1688 1692 
 we obtain
1689 1693 
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1690 
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1692 1696 
 {\bf Poynting's theorem}
1693 1697 
 \[
 
@@ -1712,7 +1716,7 @@ energy is carried by EM fields out of \({\cal V}\) across its boundary surface.
1712 1716 
 <p>
1713 1717 
 Energy per unit time, per unit area carried by EM fields:
1714 1718 
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 {\bf Poynting vector}
1718 1722 
 \[
 
@@ -1725,7 +1729,7 @@ Energy per unit time, per unit area carried by EM fields:
1725 1729 
 <p>
1726 1730 
 We can thus express Poynting's theorem more compactly:
1727 1731 
 </p>
1728 
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1732 
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1730 1734 
 {\bf Poynting's theorem}
1731 1735 
 \[
 
@@ -1738,7 +1742,7 @@ We can thus express Poynting's theorem more compactly:
1738 1742 
 <p>
1739 1743 
 where we have defined the total
1740 1744 
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1743 1747 
 {\bf Energy in electromagnetic fields}
1744 1748 
 \[
 
@@ -1761,7 +1765,7 @@ Then,
1761 1765 
 \]
1762 1766 
 so we get the
1763 1767 
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1766 1770 
 {\bf Poynting theorem (differential form)}
1767 1771 
 \[
 
@@ -1778,7 +1782,7 @@ and has a similar for to the continuity equation
1778 1782
1779 1783
1780 1784
1781 
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 <p>
1783 1787 
 \paragraph{Example 8.1}  Current in a wire:  Joule heating.  Energy per unit time delivered to wire:  from Poynting.
1784 1788 
 Assuming that the field is uniform, the electric field parallel to the wire is
 
@@ -1810,7 +1814,7 @@ and the value is as expected.
1810 1814
1811 1815 
 <hr><div id="postamble" class="status">
1812 1816 
 <p class="author">Author: Jean-Sébastien Caux</p>
1813 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1817 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1814 1818 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1815 1819 
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1624 1628 
 Prerequisites
1625 1629 
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1629 1633 
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1631 1635
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1634 1638 
 Objectives
1635 1639 
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@@ -1658,7 +1662,7 @@ Objectives
1658 1662
1659 1663 
 <hr><div id="postamble" class="status">
1660 1664 
 <p class="author">Author: Jean-Sébastien Caux</p>
1661 
-<p class="date">Created: 2022-02-08 Tue 06:55</p>
1665 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1662 1666 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1663 1667 
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1649 1653 
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1650 1654 
 or more succinctly:
1651 1655 
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1654 1658 
 {\bf Poynting vector of a monochromatic EM wave}
1655 1659 
 \[
 
@@ -1665,7 +1669,7 @@ This has a transparent physical interpretation: the energy density \(u\) flows w
1665 1669 
 <p>
1666 1670 
 Similary, we get the
1667 1671 
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1669 1673 
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1670 1674 
 {\bf Momentum density of a monochromatic EM wave}
1671 1675 
 \[
 
@@ -1703,7 +1707,7 @@ The {\it radiation pressure} is the momentum transfer per unit area per unit of
1703 1707
1704 1708 
 <hr><div id="postamble" class="status">
1705 1709 
 <p class="author">Author: Jean-Sébastien Caux</p>
1706 
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1710 
+<p class="date">Created: 2022-02-08 Tue 17:21</p>
1707 1711 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1708 1712 
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1650 1654 
 Generalizing to propagation in the direction of an arbitrary wavevector
1651 1655 
 \({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the
1652 1656 
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 {\bf E and B fields for a monochromatic EM plane wave}
1656 1660 
 \[
 
@@ -1681,7 +1685,7 @@ or if you prefer explicit real parts (adding a possible phase shift \(\delta\)):
1681 1685
1682 1686 
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1683 1687 
 <p class="author">Author: Jean-Sébastien Caux</p>
1684 
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1688 
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1685 1689 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1686 1690 
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 Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\),
1648 1652 
 we get the
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 {\bf Wave equations for electric and magnetic fields in vacuum}
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1693 
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1694 1698 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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1633 
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1637 
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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@@ -1654,7 +1658,7 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
1654 1658 
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1655 1659 
 We therefore have the
1656 1660 
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 {\bf Polarization current density}
1660 1664 
 \[
 
@@ -1672,7 +1676,7 @@ the polarization current is the result of linear motion of charge when
1672 1676 
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1673 1677 
 associated to the conservation of bound charges:
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1678 1682 
   \({\boldsymbol J}_b\) but this is not the convention used here.
 
@@ -1695,7 +1699,7 @@ Changing magnetization does not lead to analogous accumulation of charge and cur
1695 1699 
 In view of this:  total charge density can be separated into 2 parts,
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 {\it free} and {\it bound}:
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@@ -1708,7 +1712,7 @@ In view of this:  total charge density can be separated into 2 parts,
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 {\it polarization}:
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@@ -1732,7 +1736,7 @@ Gauss's law:  can be rewritten
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 \]
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@@ -1776,7 +1780,7 @@ bound parts, since they don't involve \(\rho\) or \({\bf J}\).
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@@ -1826,7 +1830,7 @@ where \(\varepsilon \equiv \varepsilon_0(1 + \chi_e)\) and \(\mu \equiv \mu_0 (1
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1829 
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 {\bf Maxwell's equations {\it (in matter)}, integral form}
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1702 
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1636 
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1632 
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1636 
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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1728 
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1732 
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1729 1733 
 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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1656 1660 
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1657 1661 
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1658 1662 
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1662 1666 
 \[
 
@@ -1700,7 +1704,7 @@ I = \frac{1}{2} \varepsilon v E_0^2
1700 1704
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1702 1706 
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1703 
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1707 
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1704 1708 
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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 From now on we will orient the axes so that \({\boldsymbol k}_I\) lies in the \(xz\) plane. This means that \({\boldsymbol k}_R\) and \({\boldsymbol k}_T\) also lie in that plane. This is the
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 {\bf First law of reflection:}
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 the incident, reflected and transmitted wave vectors form a plane (called the plane of incidence) which also includes the normal to the surface.
 
@@ -1671,7 +1675,7 @@ Specializing (\ref{eq:RTObliquek}) to our notations, we have
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 with the incidence (\(\theta_I\)) and reflection (\(\theta_R\)) angles
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 and the angle of refraction (\(\theta_T\)) obey the following laws:
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 {\bf Law of reflection}
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 \[
 
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 Writing everything in terms of the incident amplitude, we get
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 {\bf Fresnel's equations for reflection and transmission amplitudes (parallel case)}
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 Behaviour: for \(\theta_I = 0\) we recover (\ref{Gr(9.82)}).
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 For grazing waves \(\theta_I \rightarrow \pi/2\) we have that \(\alpha \rightarrow \infty\) and the wave is totally reflected. The most interesting angle is the one at which \(\alpha = \beta\) and the reflected wave has zero amplitude. This is known as
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 {\bf Brewster's angle {\it (at which the reflected wave amplitude vanishes)}}
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   \[
 
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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@@ -1634,7 +1638,7 @@ while the equation for \(V\) becomes
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 \]
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 These can be written compactly upon introducing a new operator: the
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 {\bf d'Alembertian operator}
1640 1644 
 \[
 
@@ -1647,7 +1651,7 @@ These can be written compactly upon introducing a new operator: the
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 so we get the
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 {\bf Inhomogeneous Maxwell equations (Lorenz gauge)}
1653 1657 
 \[
 
@@ -1684,7 +1688,7 @@ we have by direct inspection
1684 1688
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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 Easiest:
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 \[
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     {\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}
 
@@ -1651,7 +1655,7 @@ Putting this into Faraday's law gives
1651 1655 
 \]
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 so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \nabla} V\)) so we get
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 \[
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     {\boldsymbol E} = -{\boldsymbol \nabla} V - \frac{\partial {\boldsymbol A}}{\partial t}
 
@@ -1664,7 +1668,7 @@ so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \
1664 1668 
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1665 1669 
 Using this potential representation for \({\boldsymbol E}\) and \({\boldsymbol B}\) automatically fulfills the two homogeneous Maxwell equations. For the inhomogeneous equations, substituting (\ref{eq:E_from_Potentials}) into Gauss's law gives
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 \[
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     {\boldsymbol \nabla}^2 V + \frac{\partial}{\partial t} {\boldsymbol \nabla} \cdot {\boldsymbol A} = -\frac{\rho}{\varepsilon_0}
 
@@ -1680,7 +1684,7 @@ whereas Amp{\`ere}-Maxwell becomes
1680 1684 
 \]
1681 1685 
 which becomes after simple rearrangement and use of the identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\),
1682 1686 
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   \left( {\boldsymbol ∇}^2 {\boldsymbol A} - μ_0 ε_0 \frac{∂^2 {\boldsymbol A}}{∂ t^2} \right)
 
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 <hr><div id="postamble" class="status">
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 <p class="author">Author: Jean-Sébastien Caux</p>
1706 
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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 <p class="author">Author: Jean-Sébastien Caux</p>
1635 
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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 Prerequisites
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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 defines an electrostatic problem, whose solution is in principle obtained
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 \begin{equation*}
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   {\bf E} ({\bf r}) = \frac{1}{4\pi\varepsilon_0} \int_{\mathbb{R}^3} d\tau' \rho({\bf r}') \frac{{\bf r} - {\bf r}'}{|{\bf r} - {\bf r}'|^3}
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 \end{equation*}
 
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 or (often simpler) by calculating the electrostatic potential, using either the
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 explicit construction (\ref{eq:V_from_rho})
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     V({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' \frac{\rho({\bf r}')}{|{\bf r} - {\bf r}'|}.
 
@@ -1650,7 +1654,7 @@ condition (\ref{Gr(2.20)}) can be expressed as the single
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 'local' (differential) condition (Poisson's equation) (\ref{eq:Poisson})
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 Laplace equation
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1693 
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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 <b>Earnshaw's theorem</b> <br>
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 Since solutions to Laplace's equation have no local minimum,
 
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 <p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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