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Update 2022-03-15 10:07

master
Jean-Sébastien 2 years ago
parent
4808df71e6
commit
55f0de8197
100 changed files with 1076 additions and 975 deletions
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1665 
   <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"/>
 1665 
   <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"/>
1666 
   <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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   <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"/>
1667 
 </svg></a>
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 </svg></a>
1668 
 </p>
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 </p>
1669 
-<div class="alteqlabels" id="org16d29dc">
1669 
+<div class="alteqlabels" id="org3066755">
1670 
 <ul class="org-ul">
 1670 
 <ul class="org-ul">
1671 
 <li>Gr4(1.79)</li>
 1671 
 <li>Gr4(1.79)</li>
1672 
 </ul>
 1672 
 </ul>
1687 
 <div id="outline-container-c_m_cs_cyl_div" class="outline-6">
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 <div id="outline-container-c_m_cs_cyl_div" class="outline-6">
1688 
 <h6 id="c_m_cs_cyl_div"><a href="#c_m_cs_cyl_div">Divergence</a></h6>
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 <h6 id="c_m_cs_cyl_div"><a href="#c_m_cs_cyl_div">Divergence</a></h6>
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1693 
   <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="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"/>
1694 
   <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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   <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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+<div class="alteqlabels" id="orge14058b">
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 <ul class="org-ul">
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 <ul class="org-ul">
1699 
 <li>Gr4(2.21)</li>
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 <li>Gr4(2.21)</li>
1700 
 </ul>
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 </ul>
1715 
 <div id="outline-container-c_m_cs_cyl_curl" class="outline-6">
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1722 
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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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1724 
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1725 
-<div class="alteqlabels" id="orgcd134fe">
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+<div class="alteqlabels" id="orgeec607c">
1726 
 <ul class="org-ul">
 1726 
 <ul class="org-ul">
1727 
 <li>Gr4(2.21)</li>
 1727 
 <li>Gr4(2.21)</li>
1728 
 </ul>
 1728 
 </ul>
1771 
 </div>
 1771 
 </div>
1772 
 <div id="postamble" class="status">
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 <div id="postamble" class="status">
1773 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1773 
 <p class="author">Author: Jean-Sébastien Caux</p>
1774 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1774 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1775 
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build/c_m_cs_hyp.html View File

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-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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1634 
 <p class="author">Author: Jean-Sébastien Caux</p>
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 <p class="author">Author: Jean-Sébastien Caux</p>
1635 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1635 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1636 
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 </details>
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 </details>
1623 
 The usual Cartesian coordinates relate to spherical coordinates
 1623 
 The usual Cartesian coordinates relate to spherical coordinates
1624 
 according to
 1624 
 according to
1625 
 </p>
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 </p>
1626 
-<div class="eqlabel" id="org348038f">
1626 
+<div class="eqlabel" id="org309b2b3">
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1628 
 <a id="sph_xyz"></a><a href="./c_m_cs_sph.html#sph_xyz"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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1629 
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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"/>
1630 
   <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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   <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"/>
1631 
 </svg></a>
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1632 
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1633 
-<div class="alteqlabels" id="org5f5a1a3">
1633 
+<div class="alteqlabels" id="org9ad4ca2">
1634
1634
1635 
 </div>
 1635 
 </div>
1636
1636
1655 
 where the explicit relation between spherical and
 1655 
 where the explicit relation between spherical and
1656 
 Cartesian unit vectors is
 1656 
 Cartesian unit vectors is
1657 
 </p>
 1657 
 </p>
1658 
-<div class="eqlabel" id="orgbdd14bc">
1658 
+<div class="eqlabel" id="orgbf81903">
1659 
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 1659 
 <p>
1660 
 <a id="sph_uv"></a><a href="./c_m_cs_sph.html#sph_uv"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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1661 
   <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"/>
 1661 
   <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"/>
1662 
   <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"/>
 1662 
   <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"/>
1663 
 </svg></a>
 1663 
 </svg></a>
1664 
 </p>
 1664 
 </p>
1665 
-<div class="alteqlabels" id="orga250579">
1665 
+<div class="alteqlabels" id="org437eb02">
1666
1666
1667 
 </div>
 1667 
 </div>
1668
1668
1685 
 <p>
 1685 
 <p>
1686 
 An infinitesimal displacement \(d{\bf l}\) can be written as
 1686 
 An infinitesimal displacement \(d{\bf l}\) can be written as
1687 
 </p>
 1687 
 </p>
1688 
-<div class="eqlabel" id="org6494f4b">
1688 
+<div class="eqlabel" id="orgda909cf">
1689 
 <p>
 1689 
 <p>
1690 
 <a id="sph_dl"></a><a href="./c_m_cs_sph.html#sph_dl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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1691 
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 1691 
   <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"/>
1692 
   <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"/>
 1692 
   <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"/>
1693 
 </svg></a>
 1693 
 </svg></a>
1694 
 </p>
 1694 
 </p>
1695 
-<div class="alteqlabels" id="orgaafdc77">
1695 
+<div class="alteqlabels" id="org02f05bf">
1696
1696
1697 
 </div>
 1697 
 </div>
1698
1698
1708 
 <p>
 1708 
 <p>
1709 
 Infinitesimal volume element:
 1709 
 Infinitesimal volume element:
1710 
 </p>
 1710 
 </p>
1711 
-<div class="eqlabel" id="orga40a4ab">
1711 
+<div class="eqlabel" id="orgd2cfc4a">
1712 
 <p>
 1712 
 <p>
1713 
 <a id="sph_dtau"></a><a href="./c_m_cs_sph.html#sph_dtau"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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1714 
   <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"/>
 1714 
   <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"/>
1715 
   <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"/>
 1715 
   <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"/>
1716 
 </svg></a>
 1716 
 </svg></a>
1717 
 </p>
 1717 
 </p>
1718 
-<div class="alteqlabels" id="orgc7eb689">
1718 
+<div class="alteqlabels" id="org0a4c199">
1719
1719
1720 
 </div>
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1721
1721
1736 
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 <div id="outline-container-c_m_cs_sph_grad" class="outline-6">
1737 
 <h6 id="c_m_cs_sph_grad"><a href="#c_m_cs_sph_grad">Gradient</a></h6>
 1737 
 <h6 id="c_m_cs_sph_grad"><a href="#c_m_cs_sph_grad">Gradient</a></h6>
1738 
 <div class="outline-text-6" id="text-c_m_cs_sph_grad">
 1738 
 <div class="outline-text-6" id="text-c_m_cs_sph_grad">
1739 
-<div class="eqlabel" id="orgb45a655">
1739 
+<div class="eqlabel" id="orge44fa24">
1740 
 <p>
 1740 
 <p>
1741 
 <a id="sph_grad"></a><a href="./c_m_cs_sph.html#sph_grad"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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1742 
   <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"/>
 1742 
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1743 
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 <h6 id="c_m_cs_sph_lap"><a href="#c_m_cs_sph_lap">Laplacian</a></h6>
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 <p class="author">Author: Jean-Sébastien Caux</p>
1854 
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 <p class="author">Author: Jean-Sébastien Caux</p>
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1645 
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 <p class="author">Author: Jean-Sébastien Caux</p>
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1647 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1647 
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1617
1617
1618 
-<div id="outline-container-orgc12bd11" class="outline-6">
1619 
-<h6 id="orgc12bd11"><a href="#orgc12bd11">Divergence of gradient</a></h6>
1620 
-<div class="outline-text-6" id="text-orgc12bd11">
1618 
+<div id="outline-container-orgc162c65" class="outline-6">
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1620 
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1621 
 <p>
 1621 
 <p>
1622 
 \({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the <b>Laplacian</b> of the scalar field \(T\).
 1622 
 \({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the <b>Laplacian</b> of the scalar field \(T\).
1623 
 The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components
 1623 
 The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components
1626 
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1628
1629 
-<div id="outline-container-org0ea7ae4" class="outline-6">
1630 
-<h6 id="org0ea7ae4"><a href="#org0ea7ae4">Curl of a gradient</a></h6>
1631 
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1629 
+<div id="outline-container-org2f7ef92" class="outline-6">
1630 
+<h6 id="org2f7ef92"><a href="#org2f7ef92">Curl of a gradient</a></h6>
1631 
+<div class="outline-text-6" id="text-org2f7ef92">
1632 
 <p>
 1632 
 <p>
1633 
 This always vanishes.
 1633 
 This always vanishes.
1634 
 </p>
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 </p>
1635 
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 </div>
1636 
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1637
1637
1638 
-<div id="outline-container-org78d85e1" class="outline-6">
1639 
-<h6 id="org78d85e1"><a href="#org78d85e1">Gradient of the divergence</a></h6>
1640 
-<div class="outline-text-6" id="text-org78d85e1">
1638 
+<div id="outline-container-orgceccef1" class="outline-6">
1639 
+<h6 id="orgceccef1"><a href="#orgceccef1">Gradient of the divergence</a></h6>
1640 
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1641 
 <p>
 1641 
 <p>
1642 
 \({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics.  No special name.
 1642 
 \({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics.  No special name.
1643 
 </p>
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1644 
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1646
1646
1647 
-<div id="outline-container-orge7f86bf" class="outline-6">
1648 
-<h6 id="orge7f86bf"><a href="#orge7f86bf">Divergence of a curl</a></h6>
1649 
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1647 
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1648 
+<h6 id="orgc07e1dc"><a href="#orgc07e1dc">Divergence of a curl</a></h6>
1649 
+<div class="outline-text-6" id="text-orgc07e1dc">
1650 
 <p>
 1650 
 <p>
1651 
 This always vanishes.
 1651 
 This always vanishes.
1652 
 </p>
 1652 
 </p>
1653 
 </div>
 1653 
 </div>
1654 
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 1654 
 </div>
1655
1655
1656 
-<div id="outline-container-orgbb8199b" class="outline-6">
1657 
-<h6 id="orgbb8199b"><a href="#orgbb8199b">Curl of curl</a></h6>
1658 
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1659 
-<div class="eqlabel" id="orga59fc73">
1656 
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1657 
+<h6 id="org1a67d52"><a href="#org1a67d52">Curl of curl</a></h6>
1658 
+<div class="outline-text-6" id="text-org1a67d52">
1659 
+<div class="eqlabel" id="org5d4ff40">
1660 
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+<div class="alteqlabels" id="orgd2518b8">
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1694 
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 </div>
1695 
 <div id="postamble" class="status">
 1695 
 <div id="postamble" class="status">
1696 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1696 
 <p class="author">Author: Jean-Sébastien Caux</p>
1697 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1697 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1698 
 <p class="validation"></p>
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1645 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1645 
 <p class="author">Author: Jean-Sébastien Caux</p>
1646 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1646 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1647 
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1318
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1353 
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1641 
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 1641 
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1642 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1642 
 <p class="author">Author: Jean-Sébastien Caux</p>
1643 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1643 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1644 
 <p class="validation"></p>
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1310 
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1311 
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1312 
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1313 
-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
1314 
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 1313 
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1318
1314
1319 
 </li>
 1315 
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1353 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
 1349 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
1354
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 1351 
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1352 
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1353 
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1354 
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1357 
 </ul>
 1357 
 </ul>
1358 
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 1358 
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1664 
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 1664 
 </div>
1665 
 <div id="postamble" class="status">
 1665 
 <div id="postamble" class="status">
1666 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1666 
 <p class="author">Author: Jean-Sébastien Caux</p>
1667 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1667 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1668 
 <p class="validation"></p>
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1669 
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6 
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 <title>Pre-Quantum Electrodynamics</title>
 7 
 <title>Pre-Quantum Electrodynamics</title>
1310 
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 1310 
 </summary>
1311 
 <ul>
 1311 
 <ul>
1312 
 <li>
 1312 
 <li>
1313 
-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
1314 
-
1315 
-</li>
1316 
-<li>
1317 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
 1313 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
1318
1314
1319 
 </li>
 1315 
 </li>
1353 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
 1349 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
1354
1350
1355 
 </li>
 1351 
 </li>
1352 
+<li>
1353 
+<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
1354 
+
1355 
+</li>
1356
1356
1357 
 </ul>
 1357 
 </ul>
1358 
 </details>
 1358 
 </details>
1622 
 <p>
 1622 
 <p>
1623 
 <b>Gradient of a product</b>:
 1623 
 <b>Gradient of a product</b>:
1624 
 </p>
 1624 
 </p>
1625 
-<div class="eqlabel" id="orged10256">
1625 
+<div class="eqlabel" id="org824336d">
1626 
 <p>
 1626 
 <p>
1627 
 <a id="grad_prod"></a><a href="./c_m_dc_pr.html#grad_prod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1627 
 <a id="grad_prod"></a><a href="./c_m_dc_pr.html#grad_prod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1628 
   <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"/>
 1628 
   <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"/>
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"/>
 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"/>
1630 
 </svg></a>
 1630 
 </svg></a>
1631 
 </p>
 1631 
 </p>
1632 
-<div class="alteqlabels" id="orgfc8793a">
1632 
+<div class="alteqlabels" id="org2bf23af">
1633 
 <ul class="org-ul">
 1633 
 <ul class="org-ul">
1634 
 <li>Gr (3)</li>
 1634 
 <li>Gr (3)</li>
1635 
 <li>W (1-111)</li>
 1635 
 <li>W (1-111)</li>
1649 
 <p>
 1649 
 <p>
1650 
 <b>Gradient of a scalar product</b>:
 1650 
 <b>Gradient of a scalar product</b>:
1651 
 </p>
 1651 
 </p>
1652 
-<div class="eqlabel" id="org4b4bdf3">
1652 
+<div class="eqlabel" id="org41d1a68">
1653 
 <p>
 1653 
 <p>
1654 
 <a id="grad_sprod"></a><a href="./c_m_dc_pr.html#grad_sprod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1654 
 <a id="grad_sprod"></a><a href="./c_m_dc_pr.html#grad_sprod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1655 
   <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"/>
 1655 
   <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"/>
1656 
   <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"/>
 1656 
   <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"/>
1657 
 </svg></a>
 1657 
 </svg></a>
1658 
 </p>
 1658 
 </p>
1659 
-<div class="alteqlabels" id="org02913cc">
1659 
+<div class="alteqlabels" id="org466aa7e">
1660 
 <ul class="org-ul">
 1660 
 <ul class="org-ul">
1661 
 <li>Gr (4)</li>
 1661 
 <li>Gr (4)</li>
1662 
 <li>W (1-112)</li>
 1662 
 <li>W (1-112)</li>
1676 
 <p>
 1676 
 <p>
1677 
 <b>Divergence of a product</b>:
 1677 
 <b>Divergence of a product</b>:
1678 
 </p>
 1678 
 </p>
1679 
-<div class="eqlabel" id="org1df402a">
1679 
+<div class="eqlabel" id="org1ad95d1">
1680 
 <p>
 1680 
 <p>
1681 
 <a id="div_prod"></a><a href="./c_m_dc_pr.html#div_prod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1681 
 <a id="div_prod"></a><a href="./c_m_dc_pr.html#div_prod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1682 
   <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"/>
 1682 
   <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"/>
1683 
   <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"/>
 1683 
   <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"/>
1684 
 </svg></a>
 1684 
 </svg></a>
1685 
 </p>
 1685 
 </p>
1686 
-<div class="alteqlabels" id="orgfa5050d">
1686 
+<div class="alteqlabels" id="org93785a5">
1687 
 <ul class="org-ul">
 1687 
 <ul class="org-ul">
1688 
 <li>Gr (5)</li>
 1688 
 <li>Gr (5)</li>
1689 
 <li>W (1-115)</li>
 1689 
 <li>W (1-115)</li>
1703 
 <p>
 1703 
 <p>
1704 
 <b>Divergence of a cross product</b>:
 1704 
 <b>Divergence of a cross product</b>:
1705 
 </p>
 1705 
 </p>
1706 
-<div class="eqlabel" id="org5048e1e">
1706 
+<div class="eqlabel" id="org4333997">
1707 
 <p>
 1707 
 <p>
1708 
 <a id="div_xprod"></a><a href="./c_m_dc_pr.html#div_xprod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1708 
 <a id="div_xprod"></a><a href="./c_m_dc_pr.html#div_xprod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1709 
   <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"/>
 1709 
   <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"/>
1710 
   <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"/>
 1710 
   <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"/>
1711 
 </svg></a>
 1711 
 </svg></a>
1712 
 </p>
 1712 
 </p>
1713 
-<div class="alteqlabels" id="orgd39dff6">
1713 
+<div class="alteqlabels" id="org2626fe4">
1714 
 <ul class="org-ul">
 1714 
 <ul class="org-ul">
1715 
 <li>Gr (6)</li>
 1715 
 <li>Gr (6)</li>
1716 
 <li>W (1-116)</li>
 1716 
 <li>W (1-116)</li>
1730 
 <p>
 1730 
 <p>
1731 
 <b>Curl of a product</b>:
 1731 
 <b>Curl of a product</b>:
1732 
 </p>
 1732 
 </p>
1733 
-<div class="eqlabel" id="orgbd31452">
1733 
+<div class="eqlabel" id="orgfa1c2f8">
1734 
 <p>
 1734 
 <p>
1735 
 <a id="curl_prod"></a><a href="./c_m_dc_pr.html#curl_prod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1735 
 <a id="curl_prod"></a><a href="./c_m_dc_pr.html#curl_prod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1736 
   <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"/>
 1736 
   <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"/>
1737 
   <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"/>
 1737 
   <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"/>
1738 
 </svg></a>
 1738 
 </svg></a>
1739 
 </p>
 1739 
 </p>
1740 
-<div class="alteqlabels" id="org825aae4">
1740 
+<div class="alteqlabels" id="orgb82ff5c">
1741 
 <ul class="org-ul">
 1741 
 <ul class="org-ul">
1742 
 <li>Gr (7)</li>
 1742 
 <li>Gr (7)</li>
1743 
 <li>W (1-118)</li>
 1743 
 <li>W (1-118)</li>
1757 
 <p>
 1757 
 <p>
1758 
 <b>Curl of a cross product</b>:
 1758 
 <b>Curl of a cross product</b>:
1759 
 </p>
 1759 
 </p>
1760 
-<div class="eqlabel" id="org9c6b6a4">
1760 
+<div class="eqlabel" id="org9d0ffb0">
1761 
 <p>
 1761 
 <p>
1762 
 <a id="curl_xprod"></a><a href="./c_m_dc_pr.html#curl_xprod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1762 
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 1768 
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1770 
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 <li>W (1-119)</li>
1805 
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 1805 
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1807 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1807 
 <p class="author">Author: Jean-Sébastien Caux</p>
1808 
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1808 
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 <p class="author">Author: Jean-Sébastien Caux</p>
1642 
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1653 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1653 
 <p class="author">Author: Jean-Sébastien Caux</p>
1654 
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1654 
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1318
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 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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 </details>
1636 
 More generally,
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 More generally,
1637 
 </p>
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 </p>
1638
1638
1639 
-<div class="eqlabel" id="org492d0c0">
1639 
+<div class="eqlabel" id="org501b97b">
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1644 
 </svg></a>
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1645 
 </p>
 1645 
 </p>
1646 
-<div class="alteqlabels" id="org9b8bed8">
1646 
+<div class="alteqlabels" id="org2eebb69">
1647 
 <ul class="org-ul">
 1647 
 <ul class="org-ul">
1648 
 <li>Gr (1.100)</li>
 1648 
 <li>Gr (1.100)</li>
1649 
 </ul>
 1649 
 </ul>
1662 
 Since
 1662 
 Since
1663 
 </p>
 1663 
 </p>
1664
1664
1665 
-<div class="eqlabel" id="orga3b7a91">
1665 
+<div class="eqlabel" id="orga23d9bb">
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1667 
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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"/>
1670 
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1671 
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 </p>
1672 
-<div class="alteqlabels" id="org98ac626">
1672 
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 1673 
 <ul class="org-ul">
1674 
 <li>Gr (1.101)</li>
 1674 
 <li>Gr (1.101)</li>
1675 
 </ul>
 1675 
 </ul>
1685 
 <p>
 1685 
 <p>
1686 
 we have that
 1686 
 we have that
1687 
 </p>
 1687 
 </p>
1688 
-<div class="eqlabel" id="org596ba00">
1688 
+<div class="eqlabel" id="orgc9c91cd">
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1690 
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1695 
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1697 
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1698 
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1724 
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 1724 
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1726 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1726 
 <p class="author">Author: Jean-Sébastien Caux</p>
1727 
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1727 
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1728 
 <p class="validation"></p>
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1659 
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1660 
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1660 
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 <p class="author">Author: Jean-Sébastien Caux</p>
1645 
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1645 
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 <p class="author">Author: Jean-Sébastien Caux</p>
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-<p class="date">Created: 2022-03-07 Mon 20:38</p>
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 the circuit. This current is really driven by an electric field having a component
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 the circuit. This current is really driven by an electric field having a component
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 is that the electromotive force is proportional
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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"/>
 1686 
   <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"/>
1687 
   <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"/>
 1687 
   <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"/>
1688 
 </svg></a>
 1688 
 </svg></a>
1689 
 </p>
 1689 
 </p>
1690 
-<div class="alteqlabels" id="orgffe5f68">
1690 
+<div class="alteqlabels" id="orga00ad3a">
1691 
 <ul class="org-ul">
 1691 
 <ul class="org-ul">
1692 
 <li>Gr (7.14)</li>
 1692 
 <li>Gr (7.14)</li>
1693 
 </ul>
 1693 
 </ul>
1702 
 \]
 1702 
 \]
1703 
 so we obtain
 1703 
 so we obtain
1704 
 </p>
 1704 
 </p>
1705 
-<div class="core div" id="org80fe686">
1705 
+<div class="core div" id="org35a9412">
1706 
 <p>
 1706 
 <p>
1707 
 <b>Faraday's law</b> (integral form <i>N.B.: for a stationary loop</i>)
 1707 
 <b>Faraday's law</b> (integral form <i>N.B.: for a stationary loop</i>)
1708 
 </p>
 1708 
 </p>
1709 
-<div class="eqlabel" id="orgfa5af5c">
1709 
+<div class="eqlabel" id="org6646a96">
1710 
 <p>
 1710 
 <p>
1711 
 <a id="Fl_int"></a><a href="./emd_Fl_Fl.html#Fl_int"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1711 
 <a id="Fl_int"></a><a href="./emd_Fl_Fl.html#Fl_int"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1712 
   <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"/>
 1712 
   <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 
   <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"/>
 1713 
   <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 
 </svg></a>
 1714 
 </svg></a>
1715 
 </p>
 1715 
 </p>
1716 
-<div class="alteqlabels" id="orga84de5d">
1716 
+<div class="alteqlabels" id="org4c06533">
1717 
 <ul class="org-ul">
 1717 
 <ul class="org-ul">
1718 
 <li>Gr (7.15)</li>
 1718 
 <li>Gr (7.15)</li>
1719 
 </ul>
 1719 
 </ul>
1737 
 \]
 1737 
 \]
1738 
 we obtain
 1738 
 we obtain
1739 
 </p>
 1739 
 </p>
1740 
-<div class="core div" id="org2c503e3">
1741 
-<div class="eqlabel" id="org9023002">
1740 
+<div class="core div" id="orgfef0799">
1741 
+<div class="eqlabel" id="org07be4b3">
1742 
 <p>
 1742 
 <p>
1743 
 <a id="Fl"></a><a href="./emd_Fl_Fl.html#Fl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1743 
 <a id="Fl"></a><a href="./emd_Fl_Fl.html#Fl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1744 
   <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"/>
 1744 
   <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"/>
1745 
   <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"/>
 1745 
   <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"/>
1746 
 </svg></a>
 1746 
 </svg></a>
1747 
 </p>
 1747 
 </p>
1748 
-<div class="alteqlabels" id="orgb7f3817">
1748 
+<div class="alteqlabels" id="orgd458eb7">
1749 
 <ul class="org-ul">
 1749 
 <ul class="org-ul">
1750 
 <li>Gr (7.16)</li>
 1750 
 <li>Gr (7.16)</li>
1751 
 </ul>
 1751 
 </ul>
1790 
 </div>
 1790 
 </div>
1791 
 <div id="postamble" class="status">
 1791 
 <div id="postamble" class="status">
1792 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1792 
 <p class="author">Author: Jean-Sébastien Caux</p>
1793 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1793 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1794 
 <p class="validation"></p>
 1794 
 <p class="validation"></p>
1795 
 </div>
 1795 
 </div>
1796
1796

+ 12
- 12
build/emd_Fl_e.html View File

1 
 <!DOCTYPE html>
 1 
 <!DOCTYPE html>
2 
 <html lang="en">
 2 
 <html lang="en">
3 
 <head>
 3 
 <head>
4 
-<!-- 2022-03-07 Mon 20:38 -->
4 
+<!-- 2022-03-15 Tue 08:10 -->
5 
 <meta charset="utf-8">
 5 
 <meta charset="utf-8">
6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
 6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
7 
 <title>Pre-Quantum Electrodynamics</title>
 7 
 <title>Pre-Quantum Electrodynamics</title>
1310 
 </summary>
 1310 
 </summary>
1311 
 <ul>
 1311 
 <ul>
1312 
 <li>
 1312 
 <li>
1313 
-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
1314 
-
1315 
-</li>
1316 
-<li>
1317 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
 1313 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
1318
1314
1319 
 </li>
 1315 
 </li>
1353 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
 1349 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
1354
1350
1355 
 </li>
 1351 
 </li>
1352 
+<li>
1353 
+<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
1354 
+
1355 
+</li>
1356
1356
1357 
 </ul>
 1357 
 </ul>
1358 
 </details>
 1358 
 </details>
1639 
 \]
 1639 
 \]
1640 
 Generalization to volume currents:
 1640 
 Generalization to volume currents:
1641 
 </p>
 1641 
 </p>
1642 
-<div class="eqlabel" id="org78f90c5">
1642 
+<div class="eqlabel" id="org9c43b0a">
1643 
 <p>
 1643 
 <p>
1644 
 <a id="W_intAJ"></a><a href="./emd_Fl_e.html#W_intAJ"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1644 
 <a id="W_intAJ"></a><a href="./emd_Fl_e.html#W_intAJ"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1645 
   <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"/>
 1645 
   <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"/>
1646 
   <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"/>
 1646 
   <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"/>
1647 
 </svg></a>
 1647 
 </svg></a>
1648 
 </p>
 1648 
 </p>
1649 
-<div class="alteqlabels" id="org0277370">
1649 
+<div class="alteqlabels" id="org1b4fa12">
1650 
 <ul class="org-ul">
 1650 
 <ul class="org-ul">
1651 
 <li>Gr (7.31)</li>
 1651 
 <li>Gr (7.31)</li>
1652 
 </ul>
 1652 
 </ul>
1681 
 \]
 1681 
 \]
1682 
 We can integrate over all space:  after neglecting boundary terms (assuming fields fall to zero at infinity), we are left with
 1682 
 We can integrate over all space:  after neglecting boundary terms (assuming fields fall to zero at infinity), we are left with
1683 
 </p>
 1683 
 </p>
1684 
-<div class="core div" id="org9994672">
1685 
-<div class="eqlabel" id="org2cf22c6">
1684 
+<div class="core div" id="orgc92d34b">
1685 
+<div class="eqlabel" id="org7ec4181">
1686 
 <p>
 1686 
 <p>
1687 
 <a id="W_intBsq"></a><a href="./emd_Fl_e.html#W_intBsq"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1687 
 <a id="W_intBsq"></a><a href="./emd_Fl_e.html#W_intBsq"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1688 
   <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"/>
 1688 
   <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"/>
1689 
   <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"/>
 1689 
   <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"/>
1690 
 </svg></a>
 1690 
 </svg></a>
1691 
 </p>
 1691 
 </p>
1692 
-<div class="alteqlabels" id="orgde5daac">
1692 
+<div class="alteqlabels" id="org073f8ff">
1693 
 <ul class="org-ul">
 1693 
 <ul class="org-ul">
1694 
 <li>Gr (7.34)</li>
 1694 
 <li>Gr (7.34)</li>
1695 
 </ul>
 1695 
 </ul>
1718 
 which are equations <a href="./ems_es_e.html#W_vcd">W_vcd</a>, <a href="./ems_es_e.html#W_intEsq">W_intEsq</a>, <a href="./emd_Fl_e.html#W_intAJ">W_intAJ</a> and <a href="./emd_Fl_e.html#W_intBsq">W_intBsq</a>.
 1718 
 which are equations <a href="./ems_es_e.html#W_vcd">W_vcd</a>, <a href="./ems_es_e.html#W_intEsq">W_intEsq</a>, <a href="./emd_Fl_e.html#W_intAJ">W_intAJ</a> and <a href="./emd_Fl_e.html#W_intBsq">W_intBsq</a>.
1719 
 </p>
 1719 
 </p>
1720
1720
1721 
-<div class="example div" id="orgd2854e0">
1721 
+<div class="example div" id="org60417d4">
1722 
 <p>
 1722 
 <p>
1723 
 <b>Example: energy in coaxial cable</b>
 1723 
 <b>Example: energy in coaxial cable</b>
1724 
 </p>
 1724 
 </p>
1768 
 </div>
 1768 
 </div>
1769 
 <div id="postamble" class="status">
 1769 
 <div id="postamble" class="status">
1770 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1770 
 <p class="author">Author: Jean-Sébastien Caux</p>
1771 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1771 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1772 
 <p class="validation"></p>
 1772 
 <p class="validation"></p>
1773 
 </div>
 1773 
 </div>
1774
1774

+ 15
- 15
build/emd_Fl_i.html View File

1 
 <!DOCTYPE html>
 1 
 <!DOCTYPE html>
2 
 <html lang="en">
 2 
 <html lang="en">
3 
 <head>
 3 
 <head>
4 
-<!-- 2022-03-07 Mon 20:38 -->
4 
+<!-- 2022-03-15 Tue 08:10 -->
5 
 <meta charset="utf-8">
 5 
 <meta charset="utf-8">
6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
 6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
7 
 <title>Pre-Quantum Electrodynamics</title>
 7 
 <title>Pre-Quantum Electrodynamics</title>
1310 
 </summary>
 1310 
 </summary>
1311 
 <ul>
 1311 
 <ul>
1312 
 <li>
 1312 
 <li>
1313 
-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
1314 
-
1315 
-</li>
1316 
-<li>
1317 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
 1313 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
1318
1314
1319 
 </li>
 1315 
 </li>
1353 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
 1349 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
1354
1350
1355 
 </li>
 1351 
 </li>
1352 
+<li>
1353 
+<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
1354 
+
1355 
+</li>
1356
1356
1357 
 </ul>
 1357 
 </ul>
1358 
 </details>
 1358 
 </details>
1641 
 \]
 1641 
 \]
1642 
 and we can write the mutual inductance as the <b>Neumann formula</b>,
 1642 
 and we can write the mutual inductance as the <b>Neumann formula</b>,
1643 
 </p>
 1643 
 </p>
1644 
-<div class="eqlabel" id="org766e684">
1644 
+<div class="eqlabel" id="org554c244">
1645 
 <p>
 1645 
 <p>
1646 
 <a id="Newmann_M"></a><a href="./emd_Fl_i.html#Newmann_M"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1646 
 <a id="Newmann_M"></a><a href="./emd_Fl_i.html#Newmann_M"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1647 
   <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"/>
 1647 
   <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"/>
1648 
   <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"/>
 1648 
   <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"/>
1649 
 </svg></a>
 1649 
 </svg></a>
1650 
 </p>
 1650 
 </p>
1651 
-<div class="alteqlabels" id="orga92dd51">
1651 
+<div class="alteqlabels" id="orgf48cffa">
1652 
 <ul class="org-ul">
 1652 
 <ul class="org-ul">
1653 
 <li>Gr (7.22)</li>
 1653 
 <li>Gr (7.22)</li>
1654 
 </ul>
 1654 
 </ul>
1665 
 Two things:
 1665 
 Two things:
1666 
 first, \(M_{21}\) is purely geometrical.  Second,
 1666 
 first, \(M_{21}\) is purely geometrical.  Second,
1667 
 </p>
 1667 
 </p>
1668 
-<div class="eqlabel" id="org577027c">
1668 
+<div class="eqlabel" id="org12f955a">
1669 
 <p>
 1669 
 <p>
1670 
 <a id="Msym"></a><a href="./emd_Fl_i.html#Msym"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1670 
 <a id="Msym"></a><a href="./emd_Fl_i.html#Msym"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1671 
   <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"/>
 1671 
   <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"/>
1672 
   <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"/>
 1672 
   <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"/>
1673 
 </svg></a>
 1673 
 </svg></a>
1674 
 </p>
 1674 
 </p>
1675 
-<div class="alteqlabels" id="org21aff84">
1675 
+<div class="alteqlabels" id="orga59d4b0">
1676 
 <ul class="org-ul">
 1676 
 <ul class="org-ul">
1677 
 <li>Gr (7.23)</li>
 1677 
 <li>Gr (7.23)</li>
1678 
 </ul>
 1678 
 </ul>
1687 
 \]
 1687 
 \]
1688 
 </p>
 1688 
 </p>
1689
1689
1690 
-<div class="example div" id="org9334fe0">
1690 
+<div class="example div" id="orgdd5f18b">
1691 
 <p>
 1691 
 <p>
1692 
 <b>Example: solenoid in solenoid</b>
 1692 
 <b>Example: solenoid in solenoid</b>
1693 
 </p>
 1693 
 </p>
1735 
 \]
 1735 
 \]
1736 
 Changing current also induces EMF in the source loop itself:
 1736 
 Changing current also induces EMF in the source loop itself:
1737 
 </p>
 1737 
 </p>
1738 
-<div class="eqlabel" id="org4807c27">
1738 
+<div class="eqlabel" id="org06772cd">
1739 
 <p>
 1739 
 <p>
1740 
 <a id="PLI"></a><a href="./emd_Fl_i.html#PLI"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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 <a id="PLI"></a><a href="./emd_Fl_i.html#PLI"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1741 
   <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="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"/>
1742 
   <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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   <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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-<div class="alteqlabels" id="orga19e4d1">
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+<div class="alteqlabels" id="org5ce4f36">
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 <ul class="org-ul">
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 <ul class="org-ul">
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 <li>Gr (7.25)</li>
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 <li>Gr (7.25)</li>
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 </ul>
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 </ul>
1765 
 </p>
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 </p>
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-<div class="example div" id="org940f5ab">
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+<div class="example div" id="org678dce1">
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 <p>
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 <p>
1770 
 <b>Example: self-inductance of toroidal coil</b>
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 <b>Example: self-inductance of toroidal coil</b>
1771 
 </p>
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 </p>
1804 
 Think of <i>back EMF</i>.
 1804 
 Think of <i>back EMF</i>.
1805 
 </p>
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 </p>
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-<div class="example div" id="orgbeea66c">
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+<div class="example div" id="org42aa8b9">
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 <p>
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 <p>
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 <b>Example: circuit</b>
 1809 
 <b>Example: circuit</b>
1810 
 </p>
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 </p>
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 </div>
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 </div>
1854 
 <div id="postamble" class="status">
 1854 
 <div id="postamble" class="status">
1855 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1855 
 <p class="author">Author: Jean-Sébastien Caux</p>
1856 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1856 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1857 
 <p class="validation"></p>
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 <p class="validation"></p>
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 </div>
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 </summary>
1311 
 <ul>
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 <ul>
1312 
 <li>
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1313 
-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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-
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-</li>
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-<li>
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 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
1318
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 </li>
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 </li>
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 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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+<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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 </details>
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-<div class="example div" id="org56facb7">
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+<div class="example div" id="orgcfd6427">
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 <p>
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 <p>
1642 
 <b>Example: loop with time-dependent flux</b>
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 <b>Example: loop with time-dependent flux</b>
1643 
 </p>
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 </p>
1664 
 </div>
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 </div>
1665
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-<div class="example div" id="org367fcfc">
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+<div class="example div" id="orgfc45720">
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 <p>
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 <p>
1669 
 <b>Example: wheel with charged rim traversed by flux</b>
 1669 
 <b>Example: wheel with charged rim traversed by flux</b>
1670 
 </p>
 1670 
 </p>
1705 
 <i>slow enough</i> phenomena.
 1705 
 <i>slow enough</i> phenomena.
1706 
 </p>
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 </p>
1707
1707
1708 
-<div class="example div" id="orgb2a276c">
1708 
+<div class="example div" id="org1d9bce6">
1709 
 <p>
 1709 
 <p>
1710 
 <b>Example: field from wire with time-dependent current</b>
 1710 
 <b>Example: field from wire with time-dependent current</b>
1711 
 </p>
 1711 
 </p>
1764 
 </div>
 1764 
 </div>
1765 
 <div id="postamble" class="status">
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 <div id="postamble" class="status">
1766 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1766 
 <p class="author">Author: Jean-Sébastien Caux</p>
1767 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1767 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1768 
 <p class="validation"></p>
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-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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 </details>
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 </div>
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 </div>
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 <div id="postamble" class="status">
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 <div id="postamble" class="status">
1642 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1642 
 <p class="author">Author: Jean-Sébastien Caux</p>
1643 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1643 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1644 
 <p class="validation"></p>
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-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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-
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1318
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 </ul>
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 </ul>
1358 
 </details>
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 </details>
1616 
 <p>
 1616 
 <p>
1617 
 Full set of equations for the electromagnetic field:
 1617 
 Full set of equations for the electromagnetic field:
1618 
 </p>
 1618 
 </p>
1619 
-<div class="core div" id="orgce899b3">
1619 
+<div class="core div" id="orgee05fb0">
1620 
 <p>
 1620 
 <p>
1621 
 <b>Maxwell's equations</b> <i>(in vacuum)</i>
 1621 
 <b>Maxwell's equations</b> <i>(in vacuum)</i>
1622 
 </p>
 1622 
 </p>
1623 
-<div class="eqlabel" id="org6517c73">
1623 
+<div class="eqlabel" id="orgf090588">
1624 
 <p>
 1624 
 <p>
1625 
 <a id="Max_vac"></a><a href="./emd_Me_Me.html#Max_vac"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1625 
 <a id="Max_vac"></a><a href="./emd_Me_Me.html#Max_vac"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1626 
   <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"/>
 1626 
   <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"/>
1627 
   <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"/>
 1627 
   <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"/>
1628 
 </svg></a>
 1628 
 </svg></a>
1629 
 </p>
 1629 
 </p>
1630 
-<div class="alteqlabels" id="orgfa86c68">
1630 
+<div class="alteqlabels" id="org538c235">
1631
1631
1632 
 </div>
 1632 
 </div>
1633
1633
1644 
 <p>
 1644 
 <p>
1645 
 Complement:
 1645 
 Complement:
1646 
 </p>
 1646 
 </p>
1647 
-<div class="core div" id="orgba4da4a">
1647 
+<div class="core div" id="org14efa58">
1648 
 <p>
 1648 
 <p>
1649 
 Force law <a href="./ems_ms_lf_pc.html#LorFo">LorFo</a>
 1649 
 Force law <a href="./ems_ms_lf_pc.html#LorFo">LorFo</a>
1650 
 \[
 1650 
 \[
1667 
 <p>
 1667 
 <p>
1668 
 Better way of writing:  all fields on left, all sources on right,
 1668 
 Better way of writing:  all fields on left, all sources on right,
1669 
 </p>
 1669 
 </p>
1670 
-<div class="core div" id="org8fce2a6">
1671 
-<div class="eqlabel" id="org384b902">
1670 
+<div class="core div" id="orge7a8009">
1671 
+<div class="eqlabel" id="orgaf03f6a">
1672 
 <p>
 1672 
 <p>
1673 
 <a id="Max_vac_s"></a><a href="./emd_Me_Me.html#Max_vac_s"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1673 
 <a id="Max_vac_s"></a><a href="./emd_Me_Me.html#Max_vac_s"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1674 
   <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"/>
 1674 
   <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"/>
1675 
   <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"/>
 1675 
   <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"/>
1676 
 </svg></a>
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 </svg></a>
1677 
 </p>
 1677 
 </p>
1678 
-<div class="alteqlabels" id="orgb8b5d0a">
1678 
+<div class="alteqlabels" id="orgc1dd627">
1679 
 <ul class="org-ul">
 1679 
 <ul class="org-ul">
1680 
 <li>Gr (7.42)</li>
 1680 
 <li>Gr (7.42)</li>
1681 
 </ul>
 1681 
 </ul>
1713 
 </div>
 1713 
 </div>
1714 
 <div id="postamble" class="status">
 1714 
 <div id="postamble" class="status">
1715 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1715 
 <p class="author">Author: Jean-Sébastien Caux</p>
1716 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1716 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1717 
 <p class="validation"></p>
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 <p class="validation"></p>
1718 
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 <title>Pre-Quantum Electrodynamics</title>
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 <title>Pre-Quantum Electrodynamics</title>
1310 
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 </summary>
1311 
 <ul>
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 <ul>
1312 
 <li>
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 <li>
1313 
-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
1314 
-
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-</li>
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-<li>
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 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
 1313 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
1318
1314
1319 
 </li>
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 </li>
1353 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
 1349 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
1354
1350
1355 
 </li>
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 </li>
1352 
+<li>
1353 
+<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
1354 
+
1355 
+</li>
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1357 
 </ul>
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 </ul>
1358 
 </details>
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 </details>
1622 
 \]
 1622 
 \]
1623 
 The extra term would thus be eliminated if we were to put
 1623 
 The extra term would thus be eliminated if we were to put
1624 
 </p>
 1624 
 </p>
1625 
-<div class="core div" id="org76622f8">
1626 
-<div class="eqlabel" id="org47af899">
1625 
+<div class="core div" id="org9029eb1">
1626 
+<div class="eqlabel" id="org7937f05">
1627 
 <p>
 1627 
 <p>
1628 
 <a id="AmpMax"></a><a href="./emd_Me_dc.html#AmpMax"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1628 
 <a id="AmpMax"></a><a href="./emd_Me_dc.html#AmpMax"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1629 
   <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"/>
 1629 
   <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"/>
1630 
   <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"/>
 1630 
   <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"/>
1631 
 </svg></a>
 1631 
 </svg></a>
1632 
 </p>
 1632 
 </p>
1633 
-<div class="alteqlabels" id="org4b2df9a">
1633 
+<div class="alteqlabels" id="orgc4ec53c">
1634 
 <ul class="org-ul">
 1634 
 <ul class="org-ul">
1635 
 <li>Gr (7.36)</li>
 1635 
 <li>Gr (7.36)</li>
1636 
 </ul>
 1636 
 </ul>
1659 
 <p>
 1659 
 <p>
1660 
 Maxwell baptized this term the
 1660 
 Maxwell baptized this term the
1661 
 </p>
 1661 
 </p>
1662 
-<div class="core div" id="org1ac2c47">
1662 
+<div class="core div" id="orgf5b87fa">
1663 
 <p>
 1663 
 <p>
1664 
 <b>Displacement current</b>
 1664 
 <b>Displacement current</b>
1665 
 </p>
 1665 
 </p>
1666 
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 1675 
 <li>Gr (7.37)</li>
1676 
 </ul>
 1676 
 </ul>
1724 
 </div>
 1724 
 </div>
1725 
 <div id="postamble" class="status">
 1725 
 <div id="postamble" class="status">
1726 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1726 
 <p class="author">Author: Jean-Sébastien Caux</p>
1727 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1727 
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1728 
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1312 
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1354
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 </details>
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 </details>
1630 
 \]
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 \]
1631 
 But:  try same with \((iv)\):
 1631 
 But:  try same with \((iv)\):
1632 
 </p>
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 </p>
1633 
-<div class="eqlabel" id="org83e3419">
1633 
+<div class="eqlabel" id="org76390cc">
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 <li>Gr (7.35)</li>
1643 
 </ul>
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 </ul>
1682 
 </div>
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 </div>
1683 
 <div id="postamble" class="status">
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1684 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1684 
 <p class="author">Author: Jean-Sébastien Caux</p>
1685 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1685 
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1686 
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1666 
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 Prerequisites
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 Prerequisites
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 Objectives
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 Objectives
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 <p class="author">Author: Jean-Sébastien Caux</p>
1665 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
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 <p>
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 <p>
1617 
 The angular momentum of EM fields is directly given by
 1617 
 The angular momentum of EM fields is directly given by
1618 
 </p>
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 </p>
1619 
-<div class="main div" id="org7d69a07">
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+<div class="main div" id="org9f96756">
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 <p>
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 <p>
1621 
 <b>Angular momentum of EM fields</b>
 1621 
 <b>Angular momentum of EM fields</b>
1622 
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 </p>
1623 
-<div class="eqlabel" id="org0855fcd">
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+<div class="eqlabel" id="org2a11d69">
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 <p class="author">Author: Jean-Sébastien Caux</p>
1664 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1664 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
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 \]
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 \]
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 Since this is true for any volume, we have (re)derived the
 1636 
 Since this is true for any volume, we have (re)derived the
1637 
 </p>
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 </p>
1638 
-<div class="core div" id="org75f48b5">
1638 
+<div class="core div" id="org3f0dec4">
1639 
 <p>
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 <p>
1640 
 <b>Continuity equation</b> <a href="./ems_ms_ce.html#conteq">conteq</a>
 1640 
 <b>Continuity equation</b> <a href="./ems_ms_ce.html#conteq">conteq</a>
1641 
 \[
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 \[
1675 
 </div>
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 <div id="postamble" class="status">
1677 
 <p class="author">Author: Jean-Sébastien Caux</p>
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 <p class="author">Author: Jean-Sébastien Caux</p>
1678 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1678 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
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1313 
-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
1314 
-
1315 
-</li>
1316 
-<li>
1317 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
 1313 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
1318
1314
1319 
 </li>
 1315 
 </li>
1353 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
 1349 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
1354
1350
1355 
 </li>
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 </li>
1352 
+<li>
1353 
+<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
1354 
+
1355 
+</li>
1356
1356
1357 
 </ul>
 1357 
 </ul>
1358 
 </details>
 1358 
 </details>
1628 
 <p>
 1628 
 <p>
1629 
 This is thus simply a conservation law for momentum, with
 1629 
 This is thus simply a conservation law for momentum, with
1630 
 </p>
 1630 
 </p>
1631 
-<div class="main div" id="org60dbc44">
1631 
+<div class="main div" id="orgc119c5d">
1632 
 <p>
 1632 
 <p>
1633 
 <b>Momentum density in the EM fields</b>
 1633 
 <b>Momentum density in the EM fields</b>
1634 
 </p>
 1634 
 </p>
1635 
-<div class="eqlabel" id="org7808fcd">
1635 
+<div class="eqlabel" id="org4c00e17">
1636 
 <p>
 1636 
 <p>
1637 
 <a id="gExB"></a><a href="./emd_ce_mom.html#gExB"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1637 
 <a id="gExB"></a><a href="./emd_ce_mom.html#gExB"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1638 
   <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"/>
 1638 
   <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"/>
1639 
   <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"/>
 1639 
   <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"/>
1640 
 </svg></a>
 1640 
 </svg></a>
1641 
 </p>
 1641 
 </p>
1642 
-<div class="alteqlabels" id="org1473c1c">
1642 
+<div class="alteqlabels" id="org62e68b8">
1643
1643
1644 
 </div>
 1644 
 </div>
1645
1645
1655 
 <p>
 1655 
 <p>
1656 
 In a region in which the mechanical momentum is not changing due to external influences, we then have the
 1656 
 In a region in which the mechanical momentum is not changing due to external influences, we then have the
1657 
 </p>
 1657 
 </p>
1658 
-<div class="main div" id="orgf7ee289">
1658 
+<div class="main div" id="org55398a2">
1659 
 <p>
 1659 
 <p>
1660 
 <b>Continuity equation for EM momentum</b>
 1660 
 <b>Continuity equation for EM momentum</b>
1661 
 </p>
 1661 
 </p>
1662 
-<div class="eqlabel" id="orgd7db956">
1662 
+<div class="eqlabel" id="org02bb382">
1663 
 <p>
 1663 
 <p>
1664 
 <a id="contg"></a><a href="./emd_ce_mom.html#contg"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1664 
 <a id="contg"></a><a href="./emd_ce_mom.html#contg"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1665 
   <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"/>
 1665 
   <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"/>
1666 
   <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"/>
 1666 
   <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"/>
1667 
 </svg></a>
 1667 
 </svg></a>
1668 
 </p>
 1668 
 </p>
1669 
-<div class="alteqlabels" id="orge8fbc29">
1669 
+<div class="alteqlabels" id="orgf040d5c">
1670
1670
1671 
 </div>
 1671 
 </div>
1672
1672
1699 
 </div>
 1699 
 </div>
1700 
 <div id="postamble" class="status">
 1700 
 <div id="postamble" class="status">
1701 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1701 
 <p class="author">Author: Jean-Sébastien Caux</p>
1702 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1702 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1703 
 <p class="validation"></p>
 1703 
 <p class="validation"></p>
1704 
 </div>
 1704 
 </div>
1705
1705

+ 15
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build/emd_ce_mst.html View File

1 
 <!DOCTYPE html>
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2 
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3 
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 <head>
4 
-<!-- 2022-03-07 Mon 20:38 -->
4 
+<!-- 2022-03-15 Tue 08:10 -->
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 <meta charset="utf-8">
6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
 6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
7 
 <title>Pre-Quantum Electrodynamics</title>
 7 
 <title>Pre-Quantum Electrodynamics</title>
1310 
 </summary>
 1310 
 </summary>
1311 
 <ul>
 1311 
 <ul>
1312 
 <li>
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1313 
-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
1314 
-
1315 
-</li>
1316 
-<li>
1317 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
1318
1314
1319 
 </li>
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 </li>
1353 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
 1349 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
1354
1350
1355 
 </li>
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 </li>
1352 
+<li>
1353 
+<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
1354 
+
1355 
+</li>
1356
1356
1357 
 </ul>
 1357 
 </ul>
1358 
 </details>
 1358 
 </details>
1667 
 <p>
 1667 
 <p>
1668 
 This expression can be greatly simplified by introducing the
 1668 
 This expression can be greatly simplified by introducing the
1669 
 </p>
 1669 
 </p>
1670 
-<div class="main div" id="orgaa8145a">
1670 
+<div class="main div" id="org6a32dc6">
1671 
 <p>
 1671 
 <p>
1672 
 <b>Maxwell stress tensor</b>
 1672 
 <b>Maxwell stress tensor</b>
1673 
 </p>
 1673 
 </p>
1674 
-<div class="eqlabel" id="org0e3a6cc">
1674 
+<div class="eqlabel" id="org46015d8">
1675 
 <p>
 1675 
 <p>
1676 
 <a id="MaxST"></a><a href="./emd_ce_mst.html#MaxST"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1676 
 <a id="MaxST"></a><a href="./emd_ce_mst.html#MaxST"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1677 
   <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"/>
 1677 
   <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"/>
1678 
   <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"/>
 1678 
   <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"/>
1679 
 </svg></a>
 1679 
 </svg></a>
1680 
 </p>
 1680 
 </p>
1681 
-<div class="alteqlabels" id="org4826577">
1681 
+<div class="alteqlabels" id="org9675080">
1682
1682
1683 
 </div>
 1683 
 </div>
1684
1684
1699 
 <p>
 1699 
 <p>
1700 
 We then obtain the
 1700 
 We then obtain the
1701 
 </p>
 1701 
 </p>
1702 
-<div class="main div" id="org66e8efe">
1702 
+<div class="main div" id="org0171e33">
1703 
 <p>
 1703 
 <p>
1704 
 <b>EM force per unit volume</b>
 1704 
 <b>EM force per unit volume</b>
1705 
 </p>
 1705 
 </p>
1706 
-<div class="eqlabel" id="org49902c2">
1706 
+<div class="eqlabel" id="org5306def">
1707 
 <p>
 1707 
 <p>
1708 
 <a id="fT"></a><a href="./emd_ce_mst.html#fT"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1708 
 <a id="fT"></a><a href="./emd_ce_mst.html#fT"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1709 
   <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"/>
 1709 
   <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"/>
1710 
   <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"/>
 1710 
   <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"/>
1711 
 </svg></a>
 1711 
 </svg></a>
1712 
 </p>
 1712 
 </p>
1713 
-<div class="alteqlabels" id="orge0c52b1">
1713 
+<div class="alteqlabels" id="orgc0f4f98">
1714
1714
1715 
 </div>
 1715 
 </div>
1716
1716
1726 
 <p>
 1726 
 <p>
1727 
 where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
 1727 
 where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
1728 
 </p>
 1728 
 </p>
1729 
-<div class="main div" id="org887bc5d">
1729 
+<div class="main div" id="org28ab853">
1730 
 <p>
 1730 
 <p>
1731 
 <b>Total force on charges in volume</b>
 1731 
 <b>Total force on charges in volume</b>
1732 
 </p>
 1732 
 </p>
1733 
-<div class="eqlabel" id="orgd12201e">
1733 
+<div class="eqlabel" id="orgb821a87">
1734 
 <p>
 1734 
 <p>
1735 
 <a id="totFo"></a><a href="./emd_ce_mst.html#totFo"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1735 
 <a id="totFo"></a><a href="./emd_ce_mst.html#totFo"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1736 
   <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"/>
 1736 
   <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"/>
1737 
   <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"/>
 1737 
   <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"/>
1738 
 </svg></a>
 1738 
 </svg></a>
1739 
 </p>
 1739 
 </p>
1740 
-<div class="alteqlabels" id="org0dc7e2f">
1740 
+<div class="alteqlabels" id="orgdcbb5c3">
1741
1741
1742 
 </div>
 1742 
 </div>
1743
1743
1770 
 </div>
 1770 
 </div>
1771 
 <div id="postamble" class="status">
 1771 
 <div id="postamble" class="status">
1772 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1772 
 <p class="author">Author: Jean-Sébastien Caux</p>
1773 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1773 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1774 
 <p class="validation"></p>
 1774 
 <p class="validation"></p>
1775 
 </div>
 1775 
 </div>
1776
1776

+ 25
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build/emd_ce_poy.html View File

1 
 <!DOCTYPE html>
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2 
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+<!-- 2022-03-15 Tue 08:10 -->
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 <meta charset="utf-8">
6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
 6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
7 
 <title>Pre-Quantum Electrodynamics</title>
 7 
 <title>Pre-Quantum Electrodynamics</title>
1310 
 </summary>
 1310 
 </summary>
1311 
 <ul>
 1311 
 <ul>
1312 
 <li>
 1312 
 <li>
1313 
-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
1314 
-
1315 
-</li>
1316 
-<li>
1317 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
 1313 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
1318
1314
1319 
 </li>
 1315 
 </li>
1353 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
 1349 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
1354
1350
1355 
 </li>
 1351 
 </li>
1352 
+<li>
1353 
+<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
1354 
+
1355 
+</li>
1356
1356
1357 
 </ul>
 1357 
 </ul>
1358 
 </details>
 1358 
 </details>
1617 
 Earlier:  work done to assemble a static charge distribution:
 1617 
 Earlier:  work done to assemble a static charge distribution:
1618 
 \[
 1618 
 \[
1619 
 W_e = \frac{\varepsilon_0}{2} \int d\tau ~E^2
 1619 
 W_e = \frac{\varepsilon_0}{2} \int d\tau ~E^2
1620 
-\tag{\ref{eq:Energy_as_int_E2}}
1621 
 \]
 1620 
 \]
1622 
 Work necessary to get currents going:
 1621 
 Work necessary to get currents going:
1623 
 \[
 1622 
 \[
1637 
 Really, we're looking at a small volume element \(d\tau\) carrying charge \(\rho d\tau\), moving
 1636 
 Really, we're looking at a small volume element \(d\tau\) carrying charge \(\rho d\tau\), moving
1638 
 at velocity \({\bf v}\) such that \({\bf J} = \rho {\bf v}\).  Thus,
 1637 
 at velocity \({\bf v}\) such that \({\bf J} = \rho {\bf v}\).  Thus,
1639 
 </p>
 1638 
 </p>
1640 
-<div class="eqlabel" id="orgc9958b2">
1639 
+<div class="eqlabel" id="org935dce9">
1641 
 <p>
 1640 
 <p>
1642 
 <a id="dWdt_intEJ"></a><a href="./emd_ce_poy.html#dWdt_intEJ"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1641 
 <a id="dWdt_intEJ"></a><a href="./emd_ce_poy.html#dWdt_intEJ"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1643 
   <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"/>
 1642 
   <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"/>
1644 
   <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"/>
 1643 
   <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"/>
1645 
 </svg></a>
 1644 
 </svg></a>
1646 
 </p>
 1645 
 </p>
1647 
-<div class="alteqlabels" id="orgf9bbccb">
1646 
+<div class="alteqlabels" id="orgad7d22c">
1648 
 <ul class="org-ul">
 1647 
 <ul class="org-ul">
1649 
 <li>Gr (8.6)</li>
 1648 
 <li>Gr (8.6)</li>
1650 
 </ul>
 1649 
 </ul>
1684 
 Substituting this in <a href="./emd_ce_poy.html#dWdt_intEJ">dWdt_intEJ</a> and using the divergence theorem,
 1683 
 Substituting this in <a href="./emd_ce_poy.html#dWdt_intEJ">dWdt_intEJ</a> and using the divergence theorem,
1685 
 we obtain
 1684 
 we obtain
1686 
 </p>
 1685 
 </p>
1687 
-<div class="main div" id="orgead023b">
1686 
+<div class="main div" id="orgdbc3e5c">
1688 
 <p>
 1687 
 <p>
1689 
 <b>Poynting's theorem</b>
 1688 
 <b>Poynting's theorem</b>
1690 
 </p>
 1689 
 </p>
1691 
-<div class="eqlabel" id="org1b1ef48">
1690 
+<div class="eqlabel" id="orgc64c555">
1692 
 <p>
 1691 
 <p>
1693 
 <a id="👉Thm"></a><a href="./emd_ce_poy.html#👉Thm"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1692 
 <a id="👉Thm"></a><a href="./emd_ce_poy.html#👉Thm"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1694 
   <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"/>
 1693 
   <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"/>
1695 
   <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"/>
 1694 
   <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"/>
1696 
 </svg></a>
 1695 
 </svg></a>
1697 
 </p>
 1696 
 </p>
1698 
-<div class="alteqlabels" id="org9c51283">
1697 
+<div class="alteqlabels" id="orgf1ac980">
1699 
 <ul class="org-ul">
 1698 
 <ul class="org-ul">
1700 
 <li>Gr (8.9)</li>
 1699 
 <li>Gr (8.9)</li>
1701 
 </ul>
 1700 
 </ul>
1721 
 <p>
 1720 
 <p>
1722 
 Energy per unit time, per unit area carried by EM fields: given by the
 1721 
 Energy per unit time, per unit area carried by EM fields: given by the
1723 
 </p>
 1722 
 </p>
1724 
-<div class="core div" id="orgafa4bdd">
1723 
+<div class="core div" id="orga8e7c87">
1725 
 <p>
 1724 
 <p>
1726 
 <b>Poynting vector</b>
 1725 
 <b>Poynting vector</b>
1727 
 </p>
 1726 
 </p>
1728 
-<div class="eqlabel" id="org0aaf227">
1727 
+<div class="eqlabel" id="org7fb23da">
1729 
 <p>
 1728 
 <p>
1730 
 <a id="PoyntingVec"></a><a href="./emd_ce_poy.html#PoyntingVec"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1729 
 <a id="PoyntingVec"></a><a href="./emd_ce_poy.html#PoyntingVec"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1731 
   <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"/>
 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"/>
1732 
   <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"/>
 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"/>
1733 
 </svg></a>
 1732 
 </svg></a>
1734 
 </p>
 1733 
 </p>
1735 
-<div class="alteqlabels" id="org05edf23">
1734 
+<div class="alteqlabels" id="org8150e29">
1736 
 <ul class="org-ul">
 1735 
 <ul class="org-ul">
1737 
 <li>Gr (8.10)</li>
 1736 
 <li>Gr (8.10)</li>
1738 
 </ul>
 1737 
 </ul>
1751 
 <p>
 1750 
 <p>
1752 
 We can thus express Poynting's theorem more compactly:
 1751 
 We can thus express Poynting's theorem more compactly:
1753 
 </p>
 1752 
 </p>
1754 
-<div class="core div" id="orgf7b3c73">
1753 
+<div class="core div" id="org41e1182">
1755 
 <p>
 1754 
 <p>
1756 
 <b>Poynting's theorem</b> (integral form)
 1755 
 <b>Poynting's theorem</b> (integral form)
1757 
 </p>
 1756 
 </p>
1758 
-<div class="eqlabel" id="org610ce5e">
1757 
+<div class="eqlabel" id="orgfa25d4a">
1759 
 <p>
 1758 
 <p>
1760 
 <a id="PoyntingThm_int"></a><a href="./emd_ce_poy.html#PoyntingThm_int"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1759 
 <a id="PoyntingThm_int"></a><a href="./emd_ce_poy.html#PoyntingThm_int"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1761 
   <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"/>
 1760 
   <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"/>
1762 
   <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"/>
 1761 
   <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"/>
1763 
 </svg></a>
 1762 
 </svg></a>
1764 
 </p>
 1763 
 </p>
1765 
-<div class="alteqlabels" id="org462a7a9">
1764 
+<div class="alteqlabels" id="org264bb7a">
1766 
 <ul class="org-ul">
 1765 
 <ul class="org-ul">
1767 
 <li>Gr (8.11)</li>
 1766 
 <li>Gr (8.11)</li>
1768 
 </ul>
 1767 
 </ul>
1781 
 <p>
 1780 
 <p>
1782 
 where we have defined the total
 1781 
 where we have defined the total
1783 
 </p>
 1782 
 </p>
1784 
-<div class="core div" id="org49394fc">
1783 
+<div class="core div" id="orga6dad30">
1785 
 <p>
 1784 
 <p>
1786 
 <b>Energy in electromagnetic fields</b>
 1785 
 <b>Energy in electromagnetic fields</b>
1787 
 </p>
 1786 
 </p>
1788 
-<div class="eqlabel" id="org2fd18f3">
1787 
+<div class="eqlabel" id="org16b17ef">
1789 
 <p>
 1788 
 <p>
1790 
 <a id="Uem"></a><a href="./emd_ce_poy.html#Uem"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1789 
 <a id="Uem"></a><a href="./emd_ce_poy.html#Uem"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1791 
   <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"/>
 1790 
   <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"/>
1792 
   <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"/>
 1791 
   <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"/>
1793 
 </svg></a>
 1792 
 </svg></a>
1794 
 </p>
 1793 
 </p>
1795 
-<div class="alteqlabels" id="orge0e6bd0">
1794 
+<div class="alteqlabels" id="orgdf4c40f">
1796 
 <ul class="org-ul">
 1795 
 <ul class="org-ul">
1797 
 <li>Gr (8.5)</li>
 1796 
 <li>Gr (8.5)</li>
1798 
 </ul>
 1797 
 </ul>
1821 
 \]
 1820 
 \]
1822 
 so we get the
 1821 
 so we get the
1823 
 </p>
 1822 
 </p>
1824 
-<div class="core div" id="orgb7f6aa8">
1823 
+<div class="core div" id="orgfd64b94">
1825 
 <p>
 1824 
 <p>
1826 
 <b>Poynting theorem</b> (differential form)
 1825 
 <b>Poynting theorem</b> (differential form)
1827 
 </p>
 1826 
 </p>
1828 
-<div class="eqlabel" id="orgc43f9ba">
1827 
+<div class="eqlabel" id="org47e48c8">
1829 
 <p>
 1828 
 <p>
1830 
 <a id="PoyntingThm"></a><a href="./emd_ce_poy.html#PoyntingThm"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1829 
 <a id="PoyntingThm"></a><a href="./emd_ce_poy.html#PoyntingThm"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1831 
   <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"/>
 1830 
   <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"/>
1832 
   <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"/>
 1831 
   <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"/>
1833 
 </svg></a>
 1832 
 </svg></a>
1834 
 </p>
 1833 
 </p>
1835 
-<div class="alteqlabels" id="org50e67b6">
1834 
+<div class="alteqlabels" id="orge4a2af6">
1836 
 <ul class="org-ul">
 1835 
 <ul class="org-ul">
1837 
 <li>Gr (8.14)</li>
 1836 
 <li>Gr (8.14)</li>
1838 
 </ul>
 1837 
 </ul>
1855
1854
1856
1855
1857
1856
1858 
-<div class="example div" id="org76ab6ea">
1857 
+<div class="example div" id="org5f795de">
1859 
 <p>
 1858 
 <p>
1860 
 <b>Example:  Joule heating</b>
 1859 
 <b>Example:  Joule heating</b>
1861 
 </p>
 1860 
 </p>
1881 
 \]
 1880 
 \]
1882 
 Poynting:
 1881 
 Poynting:
1883 
 \[
 1882 
 \[
1884 
-{\boldsymbol S} = \frac{1}{\mu_0} \frac{V}{L} \frac{\mu_0 I}{2\pi a} \hat{\boldsymbol x} \times \hat{\boldsymbol \varphi} = -\frac{VI}{2\pi a L} \hat{\boldsymbol s}
1883 
+{\boldsymbol S} = \frac{1}{\mu_0} \frac{V}{L} \frac{\mu_0 I}{2\pi a} \hat{\boldsymbol x} \times \hat{\boldsymbol \varphi} = -\frac{VI}{2\pi a L} \hat{\boldsymbol r}
1885 
 \]
 1884 
 \]
1886 
 and points radially inwards.  Energy per unit time passing surface of wire:
 1885 
 and points radially inwards.  Energy per unit time passing surface of wire:
1887 
 \[
 1886 
 \[
1912 
 </div>
 1911 
 </div>
1913 
 <div id="postamble" class="status">
 1912 
 <div id="postamble" class="status">
1914 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1913 
 <p class="author">Author: Jean-Sébastien Caux</p>
1915 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1914 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1916 
 <p class="validation"></p>
 1915 
 <p class="validation"></p>
1917 
 </div>
 1916 
 </div>
1918
1917

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build/emd_emw.html View File

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+<!-- 2022-03-15 Tue 08:10 -->
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 <meta name="viewport" content="width=device-width, initial-scale=1">
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7 
 <title>Pre-Quantum Electrodynamics</title>
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 <title>Pre-Quantum Electrodynamics</title>
1310 
 </summary>
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 </summary>
1311 
 <ul>
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 <ul>
1312 
 <li>
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 <li>
1313 
-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
1314 
-
1315 
-</li>
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-<li>
1317 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
 1313 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
1318
1314
1319 
 </li>
 1315 
 </li>
1353 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
 1349 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
1354
1350
1355 
 </li>
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 </li>
1352 
+<li>
1353 
+<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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+
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+</li>
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1357 
 </ul>
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 </ul>
1358 
 </details>
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 </details>
1614 
 </svg></a><span class="headline-id">emd.emw</span></h3>
 1614 
 </svg></a><span class="headline-id">emd.emw</span></h3>
1615
1615
1616 
 <div class="outline-text-3" id="text-emd_emw">
 1616 
 <div class="outline-text-3" id="text-emd_emw">
1617 
-<details class="prereq" id="org42aca70">
1618 
-<summary id="org3fdd75c">
1617 
+<details class="prereq" id="org81c4de7">
1618 
+<summary id="org94e2b41">
1619 
 Prerequisites
 1619 
 Prerequisites
1620 
 </summary>
 1620 
 </summary>
1621 
 <ul class="org-ul">
 1621 
 <ul class="org-ul">
1624 
 </ul>
 1624 
 </ul>
1625 
 </details>
 1625 
 </details>
1626
1626
1627 
-<details class="objectives" id="org8aeb437">
1628 
-<summary id="org16d834d">
1627 
+<details class="objectives" id="org04214f1">
1628 
+<summary id="org840c379">
1629 
 Objectives
 1629 
 Objectives
1630 
 </summary>
 1630 
 </summary>
1631 
 <ul class="org-ul">
 1631 
 <ul class="org-ul">
1632 
 <li>Understand how to obtain the wave equation from Maxwell's equations</li>
 1632 
 <li>Understand how to obtain the wave equation from Maxwell's equations</li>
1633 
 <li>Understand monochromatic plane waves in vacuum</li>
 1633 
 <li>Understand monochromatic plane waves in vacuum</li>
1634 
 <li>Understand their energy and momentum</li>
 1634 
 <li>Understand their energy and momentum</li>
1635 
-<li>Understand EM waves in linear media</li>
1636 
-<li>Understand reflection and transmission of EM waves at an interface</li>
1637 
-<li>Know the laws of reflection and refraction</li>
1638 
-<li>Know Fresnel's equations (parallel polarization)</li>
1639 
-<li>Know Brewster's angle</li>
1640 
-<li>Understand EM waves in conductors</li>
1641 
-<li>Understand simple waveguides and coaxial cables</li>
1642 
 </ul>
 1635 
 </ul>
1643 
 </details>
 1636 
 </details>
1644 
 </div>
 1637 
 </div>
1666 
 </div>
 1659 
 </div>
1667 
 <div id="postamble" class="status">
 1660 
 <div id="postamble" class="status">
1668 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1661 
 <p class="author">Author: Jean-Sébastien Caux</p>
1669 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1662 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1670 
 <p class="validation"></p>
 1663 
 <p class="validation"></p>
1671 
 </div>
 1664 
 </div>
1672
1665

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-<!-- 2022-03-07 Mon 20:38 -->
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+<!-- 2022-03-15 Tue 08:10 -->
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 <meta name="viewport" content="width=device-width, initial-scale=1">
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 <meta name="viewport" content="width=device-width, initial-scale=1">
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 <title>Pre-Quantum Electrodynamics</title>
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 <title>Pre-Quantum Electrodynamics</title>
1310 
 </summary>
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 </summary>
1311 
 <ul>
 1311 
 <ul>
1312 
 <li>
 1312 
 <li>
1313 
-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
1314 
-
1315 
-</li>
1316 
-<li>
1317 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
 1313 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
1318
1314
1319 
 </li>
 1315 
 </li>
1353 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
 1349 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
1354
1350
1355 
 </li>
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 </li>
1352 
+<li>
1353 
+<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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+
1355 
+</li>
1356
1356
1357 
 </ul>
 1357 
 </ul>
1358 
 </details>
 1358 
 </details>
1644 
 \]
 1644 
 \]
1645 
 or more succinctly:
 1645 
 or more succinctly:
1646 
 </p>
 1646 
 </p>
1647 
-<div class="main div" id="orgee78283">
1647 
+<div class="main div" id="orge6fd2ac">
1648 
 <p>
 1648 
 <p>
1649 
 <b>Poynting vector of a monochromatic EM wave</b>
 1649 
 <b>Poynting vector of a monochromatic EM wave</b>
1650 
 </p>
 1650 
 </p>
1651 
-<div class="eqlabel" id="orgdedd25b">
1651 
+<div class="eqlabel" id="orgc1c96f7">
1652 
 <p>
 1652 
 <p>
1653 
 <a id="Poynting_mpw"></a><a href="./emd_emw_ep.html#Poynting_mpw"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1653 
 <a id="Poynting_mpw"></a><a href="./emd_emw_ep.html#Poynting_mpw"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1654 
   <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 
   <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"/>
1655 
   <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 
   <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"/>
1656 
 </svg></a>
 1656 
 </svg></a>
1657 
 </p>
 1657 
 </p>
1658 
-<div class="alteqlabels" id="orgdfa9b5e">
1658 
+<div class="alteqlabels" id="org907a483">
1659 
 <ul class="org-ul">
 1659 
 <ul class="org-ul">
1660 
 <li>Gr (9.57)</li>
 1660 
 <li>Gr (9.57)</li>
1661 
 </ul>
 1661 
 </ul>
1678 
 <p>
 1678 
 <p>
1679 
 Similary, we get the
 1679 
 Similary, we get the
1680 
 </p>
 1680 
 </p>
1681 
-<div class="main div" id="org108cc57">
1681 
+<div class="main div" id="orga2199bc">
1682 
 <p>
 1682 
 <p>
1683 
 <b>Momentum density of a monochromatic EM wave</b>
 1683 
 <b>Momentum density of a monochromatic EM wave</b>
1684 
 \[
 1684 
 \[
1729 
 </div>
 1729 
 </div>
1730 
 <div id="postamble" class="status">
 1730 
 <div id="postamble" class="status">
1731 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1731 
 <p class="author">Author: Jean-Sébastien Caux</p>
1732 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1732 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1733 
 <p class="validation"></p>
 1733 
 <p class="validation"></p>
1734 
 </div>
 1734 
 </div>
1735
1735

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 <title>Pre-Quantum Electrodynamics</title>
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 <title>Pre-Quantum Electrodynamics</title>
1310 
 </summary>
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 </summary>
1311 
 <ul>
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1312 
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1313 
-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
1314 
-
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-</li>
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-<li>
1317 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
 1313 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
1318
1314
1319 
 </li>
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 </li>
1353 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
 1349 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
1354
1350
1355 
 </li>
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 </li>
1352 
+<li>
1353 
+<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
1354 
+
1355 
+</li>
1356
1356
1357 
 </ul>
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 </ul>
1358 
 </details>
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 </details>
1636 
 \]
 1636 
 \]
1637 
 so \({\bf E}\) and \({\bf B}\) are mutually perpendicular, and
 1637 
 so \({\bf E}\) and \({\bf B}\) are mutually perpendicular, and
1638 
 </p>
 1638 
 </p>
1639 
-<div class="eqlabel" id="org4747373">
1639 
+<div class="eqlabel" id="org4864134">
1640 
 <p>
 1640 
 <p>
1641 
 <a id="EBmpw"></a><a href="./emd_emw_mpw.html#EBmpw"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1641 
 <a id="EBmpw"></a><a href="./emd_emw_mpw.html#EBmpw"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1642 
   <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"/>
 1642 
   <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"/>
1643 
   <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"/>
 1643 
   <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"/>
1644 
 </svg></a>
 1644 
 </svg></a>
1645 
 </p>
 1645 
 </p>
1646 
-<div class="alteqlabels" id="org2ea2f4c">
1646 
+<div class="alteqlabels" id="org6d00653">
1647 
 <ul class="org-ul">
 1647 
 <ul class="org-ul">
1648 
 <li>Gr (9.47)</li>
 1648 
 <li>Gr (9.47)</li>
1649 
 </ul>
 1649 
 </ul>
1662 
 Generalizing to propagation in the direction of an arbitrary wavevector
 1662 
 Generalizing to propagation in the direction of an arbitrary wavevector
1663 
 \({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the
 1663 
 \({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the
1664 
 </p>
 1664 
 </p>
1665 
-<div class="core div" id="orga666428">
1665 
+<div class="core div" id="org9581019">
1666 
 <p>
 1666 
 <p>
1667 
 <b>E and B fields for a monochromatic EM plane wave</b>
 1667 
 <b>E and B fields for a monochromatic EM plane wave</b>
1668 
 </p>
 1668 
 </p>
1669 
-<div class="eqlabel" id="orge12acff">
1669 
+<div class="eqlabel" id="org98b7671">
1670 
 <p>
 1670 
 <p>
1671 
 <a id="mpw"></a><a href="./emd_emw_mpw.html#mpw"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1671 
 <a id="mpw"></a><a href="./emd_emw_mpw.html#mpw"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1672 
   <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"/>
 1672 
   <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"/>
1673 
   <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"/>
 1673 
   <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"/>
1674 
 </svg></a>
 1674 
 </svg></a>
1675 
 </p>
 1675 
 </p>
1676 
-<div class="alteqlabels" id="orgd889027">
1676 
+<div class="alteqlabels" id="orge381054">
1677 
 <ul class="org-ul">
 1677 
 <ul class="org-ul">
1678 
 <li>Gr (9.49)</li>
 1678 
 <li>Gr (9.49)</li>
1679 
 </ul>
 1679 
 </ul>
1724 
 </div>
 1724 
 </div>
1725 
 <div id="postamble" class="status">
 1725 
 <div id="postamble" class="status">
1726 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1726 
 <p class="author">Author: Jean-Sébastien Caux</p>
1727 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1727 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1728 
 <p class="validation"></p>
 1728 
 <p class="validation"></p>
1729 
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1310 
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 </summary>
1311 
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1312 
 <li>
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1313 
-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
1314 
-
1315 
-</li>
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1318
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1354
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1355 
 </li>
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 </li>
1352 
+<li>
1353 
+<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
1354 
+
1355 
+</li>
1356
1356
1357 
 </ul>
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 </ul>
1358 
 </details>
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 </details>
1641 
 Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\),
 1641 
 Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\),
1642 
 we get the
 1642 
 we get the
1643 
 </p>
 1643 
 </p>
1644 
-<div class="core div" id="orgc9c5d44">
1644 
+<div class="core div" id="org672468f">
1645 
 <p>
 1645 
 <p>
1646 
 <b>Wave equations for electric and magnetic fields in vacuum</b>
 1646 
 <b>Wave equations for electric and magnetic fields in vacuum</b>
1647 
 </p>
 1647 
 </p>
1648 
-<div class="eqlabel" id="orgb29f135">
1648 
+<div class="eqlabel" id="orgf656d76">
1649 
 <p>
 1649 
 <p>
1650 
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1651 
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 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"/>
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"/>
 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"/>
1653 
 </svg></a>
 1653 
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1654 
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 1654 
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1655 
-<div class="alteqlabels" id="orge5c16c5">
1655 
+<div class="alteqlabels" id="org1ac2ec3">
1656 
 <ul class="org-ul">
 1656 
 <ul class="org-ul">
1657 
 <li>Gr (9.41)</li>
 1657 
 <li>Gr (9.41)</li>
1658 
 </ul>
 1658 
 </ul>
1714 
 </div>
 1714 
 </div>
1715 
 <div id="postamble" class="status">
 1715 
 <div id="postamble" class="status">
1716 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1716 
 <p class="author">Author: Jean-Sébastien Caux</p>
1717 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1717 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1718 
 <p class="validation"></p>
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1640 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1640 
 <p class="author">Author: Jean-Sébastien Caux</p>
1641 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1641 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1642 
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 1640 
 <p class="author">Author: Jean-Sébastien Caux</p>
1641 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1641 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
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1616 
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 1616 
 <p>
1617 
 Maxwell's equations in vacuum:
 1617 
 Maxwell's equations in vacuum:
1618 
 </p>
 1618 
 </p>
1619 
-\begin{align}
1619 
+\begin{align*}
1620 
 (i)~~ &amp;{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0}, \hspace{1cm} &amp;\mbox{Gauss}, \nonumber \\
 1620 
 (i)~~ &amp;{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0}, \hspace{1cm} &amp;\mbox{Gauss}, \nonumber \\
1621 
 (ii)~~ &amp;{\boldsymbol \nabla} \cdot {\bf B} = 0, \hspace{1cm} &amp;\mbox{anonymous} \nonumber \\
 1621 
 (ii)~~ &amp;{\boldsymbol \nabla} \cdot {\bf B} = 0, \hspace{1cm} &amp;\mbox{anonymous} \nonumber \\
1622 
 (iii)~~ &amp;{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \hspace{1cm} &amp;\mbox{Faraday}, \nonumber \\
 1622 
 (iii)~~ &amp;{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \hspace{1cm} &amp;\mbox{Faraday}, \nonumber \\
1623 
 (iv)~~ &amp;{\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}, \hspace{1cm} &amp;\mbox{Ampère + Maxwell}.
 1623 
 (iv)~~ &amp;{\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}, \hspace{1cm} &amp;\mbox{Ampère + Maxwell}.
1624 
-\label{Gr(7.39)}
1625 
-\end{align}
1624 
+\end{align*}
1626 
 <p>
 1625 
 <p>
1627 
 are complete as they stand.  In presence of matter:  more convenient to write sources in terms of
 1626 
 are complete as they stand.  In presence of matter:  more convenient to write sources in terms of
1628 
 free charges and currents.
 1627 
 free charges and currents.
1642 
 One new feature in nonstatic case:  change in electric polarization involves flow of bound charge
 1641 
 One new feature in nonstatic case:  change in electric polarization involves flow of bound charge
1643 
 (call it \({\bf J}_p\)) which must be included in total current.
 1642 
 (call it \({\bf J}_p\)) which must be included in total current.
1644 
 Consider a small chunk of polarized material.
 1643 
 Consider a small chunk of polarized material.
1645 
-Polarization induces charge density \(\sigma_b = P\) at one end and \(-\sigma_b\) at other.  If \(P\)
1644 
+Polarization induces charge density \(\sigma_b = P\) at one end and \(-\sigma_b\) at other (refer to <a href="./emsm_esm_po.html#sigmab">sigmab</a>.  If \(P\)
1646 
 increases a bit, charge increases, giving net current
 1645 
 increases a bit, charge increases, giving net current
1647 
 \[
 1646 
 \[
1648 
 dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partial t} da_{\perp}.
 1647 
 dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partial t} da_{\perp}.
1649 
 \]
 1648 
 \]
1650 
 We therefore have the
 1649 
 We therefore have the
1651 
 </p>
 1650 
 </p>
1652 
-<div class="core div" id="orgf1835ae">
1651 
+<div class="core div" id="orge0518b1">
1653 
 <p>
 1652 
 <p>
1654 
 <b>Polarization current density</b>
 1653 
 <b>Polarization current density</b>
1655 
 </p>
 1654 
 </p>
1656 
-<div class="eqlabel" id="orgfad2da8">
1655 
+<div class="eqlabel" id="org62db796">
1657 
 <p>
 1656 
 <p>
1658 
 <a id="Jp"></a><a href="./emdm_Me_Mem.html#Jp"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1657 
 <a id="Jp"></a><a href="./emdm_Me_Mem.html#Jp"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1659 
   <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"/>
 1658 
   <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"/>
1660 
   <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"/>
 1659 
   <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"/>
1661 
 </svg></a>
 1660 
 </svg></a>
1662 
 </p>
 1661 
 </p>
1663 
-<div class="alteqlabels" id="orgf15b4cb">
1662 
+<div class="alteqlabels" id="orgebf1aab">
1664 
 <ul class="org-ul">
 1663 
 <ul class="org-ul">
1665 
 <li>Gr (7.48)</li>
 1664 
 <li>Gr (7.48)</li>
1666 
 </ul>
 1665 
 </ul>
1677
1676
1678 
 </div>
 1677 
 </div>
1679 
 <p>
 1678 
 <p>
1680 
-otherwise simply called the {\bf polarization current}.
1679 
+otherwise simply called the <b>polarization current</b>.
1681 
 This has nothing to do with the bound current \({\bf J}_b\)
 1680 
 This has nothing to do with the bound current \({\bf J}_b\)
1682 
 (the latter is associated to magnetization;
 1681 
 (the latter is associated to magnetization;
1683 
 the polarization current is the result of linear motion of charge when
 1682 
 the polarization current is the result of linear motion of charge when
1684 
 polarization changes).  We can check consistency with the continuity equation
 1683 
 polarization changes).  We can check consistency with the continuity equation
1685 
 associated to the conservation of bound charges:
 1684 
 associated to the conservation of bound charges:
1686 
 </p>
 1685 
 </p>
1687 
-<aside id="orgadb90c7">
1686 
+<aside id="orgfd82638">
1688 
 <p>
 1687 
 <p>
1689 
 Note the unfortunate labelling: it would have been nicer to have \(\rho_b\) be the charge associated to current \({\boldsymbol J}_b\) but this is not the common convention.
 1688 
 Note the unfortunate labelling: it would have been nicer to have \(\rho_b\) be the charge associated to current \({\boldsymbol J}_b\) but this is not the common convention.
1690 
 </p>
 1689 
 </p>
1706 
 In view of this:  total charge density can be separated into 2 parts,
 1705 
 In view of this:  total charge density can be separated into 2 parts,
1707 
 <b>free</b> and <b>bound</b>:
 1706 
 <b>free</b> and <b>bound</b>:
1708 
 </p>
 1707 
 </p>
1709 
-<div class="main div" id="orgbe78924">
1710 
-<div class="eqlabel" id="org4679dd8">
1708 
+<div class="main div" id="org5bd748a">
1709 
+<div class="eqlabel" id="org94dfd02">
1711 
 <p>
 1710 
 <p>
1712 
 <a id="rhofb"></a><a href="./emdm_Me_Mem.html#rhofb"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1711 
 <a id="rhofb"></a><a href="./emdm_Me_Mem.html#rhofb"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1713 
   <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"/>
 1712 
   <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"/>
1714 
   <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"/>
 1713 
   <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"/>
1715 
 </svg></a>
 1714 
 </svg></a>
1716 
 </p>
 1715 
 </p>
1717 
-<div class="alteqlabels" id="orgb7e7ed8">
1716 
+<div class="alteqlabels" id="orgd2d4d1d">
1718 
 <ul class="org-ul">
 1717 
 <ul class="org-ul">
1719 
 <li>Gr (7.49)</li>
 1718 
 <li>Gr (7.49)</li>
1720 
 </ul>
 1719 
 </ul>
1734 
 and current can be separated into three parts, <b>free</b>, <b>bound</b> and
 1733 
 and current can be separated into three parts, <b>free</b>, <b>bound</b> and
1735 
 <b>polarization</b>:
 1734 
 <b>polarization</b>:
1736 
 </p>
 1735 
 </p>
1737 
-<div class="main div" id="org1c506a7">
1738 
-<div class="eqlabel" id="orgbf0c17d">
1736 
+<div class="main div" id="org7ad8d3b">
1737 
+<div class="eqlabel" id="org19a6c97">
1739 
 <p>
 1738 
 <p>
1740 
 <a id="Jfbp"></a><a href="./emdm_Me_Mem.html#Jfbp"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1739 
 <a id="Jfbp"></a><a href="./emdm_Me_Mem.html#Jfbp"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1741 
   <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"/>
 1740 
   <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"/>
1742 
   <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"/>
 1741 
   <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"/>
1743 
 </svg></a>
 1742 
 </svg></a>
1744 
 </p>
 1743 
 </p>
1745 
-<div class="alteqlabels" id="org33aea7e">
1744 
+<div class="alteqlabels" id="orgac73457">
1746 
 <ul class="org-ul">
 1745 
 <ul class="org-ul">
1747 
 <li>Gr (7.50)</li>
 1746 
 <li>Gr (7.50)</li>
1748 
 </ul>
 1747 
 </ul>
1768 
 \]
 1767 
 \]
1769 
 where (as in static case)
 1768 
 where (as in static case)
1770 
 </p>
 1769 
 </p>
1771 
-<div class="core div" id="org501f375">
1770 
+<div class="core div" id="org65ac5cc">
1772 
 <p>
 1771 
 <p>
1773 
 \[
 1772 
 \[
1774 
 {\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
 1773 
 {\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
1789 
 \]
 1788 
 \]
1790 
 where as before
 1789 
 where as before
1791 
 </p>
 1790 
 </p>
1792 
-<div class="core div" id="orgfaac9ca">
1791 
+<div class="core div" id="org7e8e1b5">
1793 
 <p>
 1792 
 <p>
1794 
 \[
 1793 
 \[
1795 
 {\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
 1794 
 {\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
1807 
 <p>
 1806 
 <p>
1808 
 In terms of free charges and currents, we thus get
 1807 
 In terms of free charges and currents, we thus get
1809 
 </p>
 1808 
 </p>
1810 
-<div class="core div" id="org4a1df55">
1809 
+<div class="core div" id="org089bb13">
1811 
 <p>
 1810 
 <p>
1812 
 <b>Maxwell's equations</b> <i>(in matter)</i>
 1811 
 <b>Maxwell's equations</b> <i>(in matter)</i>
1813 
 </p>
 1812 
 </p>
1814 
-<div class="eqlabel" id="orga6eb31a">
1813 
+<div class="eqlabel" id="orgfabaa36">
1815 
 <p>
 1814 
 <p>
1816 
 <a id="Max_mat"></a><a href="./emdm_Me_Mem.html#Max_mat"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1815 
 <a id="Max_mat"></a><a href="./emdm_Me_Mem.html#Max_mat"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1817 
   <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"/>
 1816 
   <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"/>
1818 
   <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"/>
 1817 
   <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"/>
1819 
 </svg></a>
 1818 
 </svg></a>
1820 
 </p>
 1819 
 </p>
1821 
-<div class="alteqlabels" id="org154a0ec">
1820 
+<div class="alteqlabels" id="orgaba63f2">
1822 
 <ul class="org-ul">
 1821 
 <ul class="org-ul">
1823 
 <li>Gr (7.55)</li>
 1822 
 <li>Gr (7.55)</li>
1824 
 </ul>
 1823 
 </ul>
1848 
 in terms of \({\bf E}\) and \({\bf B}\).
 1847 
 in terms of \({\bf E}\) and \({\bf B}\).
1849 
 For the restricted case of linear media:
 1848 
 For the restricted case of linear media:
1850 
 </p>
 1849 
 </p>
1851 
-<div class="main div" id="orge60b08f">
1852 
-<div class="eqlabel" id="org88284f8">
1850 
+<div class="main div" id="org435d93d">
1851 
+<div class="eqlabel" id="orgb5db623">
1853 
 <p>
 1852 
 <p>
1854 
 <a id="consrel"></a><a href="./emdm_Me_Mem.html#consrel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1853 
 <a id="consrel"></a><a href="./emdm_Me_Mem.html#consrel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1855 
   <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"/>
 1854 
   <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"/>
1856 
   <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"/>
 1855 
   <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"/>
1857 
 </svg></a>
 1856 
 </svg></a>
1858 
 </p>
 1857 
 </p>
1859 
-<div class="alteqlabels" id="org21479eb">
1858 
+<div class="alteqlabels" id="org6a8a0ba">
1860 
 <ul class="org-ul">
 1859 
 <ul class="org-ul">
1861 
 <li>Gr (7.56,7.57)</li>
 1860 
 <li>Gr (7.56,7.57)</li>
1862 
 </ul>
 1861 
 </ul>
1899 
 </div>
 1898 
 </div>
1900 
 <div id="postamble" class="status">
 1899 
 <div id="postamble" class="status">
1901 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1900 
 <p class="author">Author: Jean-Sébastien Caux</p>
1902 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1901 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1903 
 <p class="validation"></p>
 1902 
 <p class="validation"></p>
1904 
 </div>
 1903 
 </div>
1905
1904

+ 23
- 41
build/emdm_Me_bc.html View File

1 
 <!DOCTYPE html>
 1 
 <!DOCTYPE html>
2 
 <html lang="en">
 2 
 <html lang="en">
3 
 <head>
 3 
 <head>
4 
-<!-- 2022-03-07 Mon 20:38 -->
4 
+<!-- 2022-03-15 Tue 08:10 -->
5 
 <meta charset="utf-8">
 5 
 <meta charset="utf-8">
6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
 6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
7 
 <title>Pre-Quantum Electrodynamics</title>
 7 
 <title>Pre-Quantum Electrodynamics</title>
1310 
 </summary>
 1310 
 </summary>
1311 
 <ul>
 1311 
 <ul>
1312 
 <li>
 1312 
 <li>
1313 
-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
1314 
-
1315 
-</li>
1316 
-<li>
1317 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
 1313 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
1318
1314
1319 
 </li>
 1315 
 </li>
1353 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
 1349 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
1354
1350
1355 
 </li>
 1351 
 </li>
1352 
+<li>
1353 
+<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
1354 
+
1355 
+</li>
1356
1356
1357 
 </ul>
 1357 
 </ul>
1358 
 </details>
 1358 
 </details>
1616 
 <p>
 1616 
 <p>
1617 
 Discontinuities between different media, deduced from
 1617 
 Discontinuities between different media, deduced from
1618 
 </p>
 1618 
 </p>
1619 
-<div class="core div" id="org2152f07">
1619 
+<div class="core div" id="org31953cb">
1620 
 <p>
 1620 
 <p>
1621 
 <b>Maxwell's equations</b> <i>(in matter)</i>, <i>integral form</i>
 1621 
 <b>Maxwell's equations</b> <i>(in matter)</i>, <i>integral form</i>
1622 
 </p>
 1622 
 </p>
1623 
-<div class="eqlabel" id="org896f0bf">
1623 
+<div class="eqlabel" id="org853b142">
1624 
 <p>
 1624 
 <p>
1625 
 <a id="Max_mat_int"></a><a href="./emdm_Me_bc.html#Max_mat_int"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1625 
 <a id="Max_mat_int"></a><a href="./emdm_Me_bc.html#Max_mat_int"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1626 
   <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"/>
 1626 
   <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"/>
1627 
   <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"/>
 1627 
   <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"/>
1628 
 </svg></a>
 1628 
 </svg></a>
1629 
 </p>
 1629 
 </p>
1630 
-<div class="alteqlabels" id="org37cc7e4">
1630 
+<div class="alteqlabels" id="org734ccfd">
1631
1631
1632 
 </div>
 1632 
 </div>
1633
1633
1645 
 Applying \((i)\) to wafer-thin Gaussian pillbox straddling boundary between 2 materials:
 1645 
 Applying \((i)\) to wafer-thin Gaussian pillbox straddling boundary between 2 materials:
1646 
 \({\bf D}_1 \cdot {\bf a} - {\bf D}_2 \cdot {\bf a} = \sigma_f a\) so
 1646 
 \({\bf D}_1 \cdot {\bf a} - {\bf D}_2 \cdot {\bf a} = \sigma_f a\) so
1647 
 </p>
 1647 
 </p>
1648 
-<div class="main div" id="orgaee2676">
1649 
-<div class="eqlabel" id="org35b4f05">
1648 
+<div class="main div" id="org74cc983">
1649 
+<div class="eqlabel" id="org49821b6">
1650 
 <p>
 1650 
 <p>
1651 
 <a id="Ddisc"></a><a href="./emdm_Me_bc.html#Ddisc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1651 
 <a id="Ddisc"></a><a href="./emdm_Me_bc.html#Ddisc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1652 
   <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"/>
 1652 
   <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"/>
1653 
   <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"/>
 1653 
   <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"/>
1654 
 </svg></a>
 1654 
 </svg></a>
1655 
 </p>
 1655 
 </p>
1656 
-<div class="alteqlabels" id="orgc9c66ee">
1656 
+<div class="alteqlabels" id="orgb4f898e">
1657 
 <ul class="org-ul">
 1657 
 <ul class="org-ul">
1658 
 <li>Gr (7.60)</li>
 1658 
 <li>Gr (7.60)</li>
1659 
 </ul>
 1659 
 </ul>
1672 
 <p>
 1672 
 <p>
1673 
 Same reasoning applied to \((ii)\) gives <a href="./ems_ms_vp_mbc.html#Bdisc">Bdisc</a>
 1673 
 Same reasoning applied to \((ii)\) gives <a href="./ems_ms_vp_mbc.html#Bdisc">Bdisc</a>
1674 
 </p>
 1674 
 </p>
1675 
-<div class="main div" id="org0c24b54">
1676 
-<div class="eqlabel" id="org262505f">
1677 
-<div class="alteqlabels" id="org560c237">
1675 
+<div class="main div" id="org167c3b0">
1676 
+<div class="eqlabel" id="orgd15716f">
1677 
+<div class="alteqlabels" id="orgc56e074">
1678
1678
1679 
 </div>
 1679 
 </div>
1680
1680
1690 
 For \((iii)\):  Amperian loop straddling surface:  \({\bf E}_1 \cdot {\bf l} - {\bf E}_2 \cdot {\bf l} =
 1690 
 For \((iii)\):  Amperian loop straddling surface:  \({\bf E}_1 \cdot {\bf l} - {\bf E}_2 \cdot {\bf l} =
1691 
 -\frac{d}{dt} \int_{\cal S} {\bf B} \cdot d{\bf a}\).  Limit of small loop:  flux vanishes, therefore
 1691 
 -\frac{d}{dt} \int_{\cal S} {\bf B} \cdot d{\bf a}\).  Limit of small loop:  flux vanishes, therefore
1692 
 </p>
 1692 
 </p>
1693 
-<div class="main div" id="org2c8267d">
1693 
+<div class="main div" id="org944c540">
1694 
 <p>
 1694 
 <p>
1695 
 \[
 1695 
 \[
1696 
 {\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0
 1696 
 {\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0
1702 
 Similarly, \((iv)\) implies \({\bf H}_1 \cdot {\bf l} - {\bf H}_2 \cdot {\bf l} = I_{f_{enc}}\).
 1702 
 Similarly, \((iv)\) implies \({\bf H}_1 \cdot {\bf l} - {\bf H}_2 \cdot {\bf l} = I_{f_{enc}}\).
1703 
 No volume current can contribute, but a surface current can.  Can write
 1703 
 No volume current can contribute, but a surface current can.  Can write
1704 
 \(I_{f_{enc}} = {\bf K}_f \cdot (\hat{\bf n} \times {\bf l}) = ({\bf K}_f \times \hat{\bf n}) \cdot {\bf l}\)
 1704 
 \(I_{f_{enc}} = {\bf K}_f \cdot (\hat{\bf n} \times {\bf l}) = ({\bf K}_f \times \hat{\bf n}) \cdot {\bf l}\)
1705 
-and thus
1705 
+and thus (as we got before in <a href="./emsm_msm_H_A.html#Hdisc">Hdisc</a>)
1706 
 </p>
 1706 
 </p>
1707 
-<div class="main div" id="orge9d0c89">
1708 
-<div class="eqlabel" id="org15b0fbc">
1709 
-<p>
1710 
-<a id="Hdisc"></a><a href="./emsm_msm_H_A.html#Hdisc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1711 
-  <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"/>
1712 
-  <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"/>
1713 
-</svg></a><a href="./emdm_Me_bc.html#Hdisc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1714 
-  <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"/>
1715 
-  <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"/>
1716 
-</svg></a>
1717 
-</p>
1718 
-<div class="alteqlabels" id="orgc1f1499">
1719 
-<ul class="org-ul">
1720 
-<li>Gr (7.63)</li>
1721 
-</ul>
1722 
-
1723 
-</div>
1724 
-
1725 
-</div>
1707 
+<div class="main div" id="org44aa4fb">
1726 
 <p>
 1708 
 <p>
1727 
 \[
 1709 
 \[
1728 
 {\bf H}_1^{\parallel} - {\bf H}_2^{\parallel} = {\bf K}_f \times \hat{\bf n}
 1710 
 {\bf H}_1^{\parallel} - {\bf H}_2^{\parallel} = {\bf K}_f \times \hat{\bf n}
1738 
 <p>
 1720 
 <p>
1739 
 In case of linear media:  can be expressed in terms of \({\bf E}\) and \({\bf B}\) alone:
 1721 
 In case of linear media:  can be expressed in terms of \({\bf E}\) and \({\bf B}\) alone:
1740 
 </p>
 1722 
 </p>
1741 
-<div class="main div" id="org8b0c892">
1742 
-<div class="eqlabel" id="org2d4783e">
1723 
+<div class="main div" id="org74a1c76">
1724 
+<div class="eqlabel" id="orgbb2af38">
1743 
 <p>
 1725 
 <p>
1744 
 <a id="disc_lm"></a><a href="./emdm_Me_bc.html#disc_lm"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1726 
 <a id="disc_lm"></a><a href="./emdm_Me_bc.html#disc_lm"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1745 
   <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"/>
 1727 
   <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"/>
1746 
   <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"/>
 1728 
   <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"/>
1747 
 </svg></a>
 1729 
 </svg></a>
1748 
 </p>
 1730 
 </p>
1749 
-<div class="alteqlabels" id="orga65ee45">
1731 
+<div class="alteqlabels" id="org98cdce0">
1750 
 <ul class="org-ul">
 1732 
 <ul class="org-ul">
1751 
 <li>Gr (7.64)</li>
 1733 
 <li>Gr (7.64)</li>
1752 
 </ul>
 1734 
 </ul>
1766 
 <p>
 1748 
 <p>
1767 
 If there is no free charge and no free current at boundary:
 1749 
 If there is no free charge and no free current at boundary:
1768 
 </p>
 1750 
 </p>
1769 
-<div class="eqlabel" id="org656d25d">
1751 
+<div class="eqlabel" id="orga2465cb">
1770 
 <p>
 1752 
 <p>
1771 
 <a id="disc_nfc"></a><a href="./emdm_Me_bc.html#disc_nfc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1753 
 <a id="disc_nfc"></a><a href="./emdm_Me_bc.html#disc_nfc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1772 
   <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"/>
 1754 
   <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"/>
1773 
   <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"/>
 1755 
   <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"/>
1774 
 </svg></a>
 1756 
 </svg></a>
1775 
 </p>
 1757 
 </p>
1776 
-<div class="alteqlabels" id="org856682c">
1758 
+<div class="alteqlabels" id="org06ac001">
1777 
 <ul class="org-ul">
 1759 
 <ul class="org-ul">
1778 
 <li>Gr (7.64)</li>
 1760 
 <li>Gr (7.64)</li>
1779 
 </ul>
 1761 
 </ul>
1810 
 </div>
 1792 
 </div>
1811 
 <div id="postamble" class="status">
 1793 
 <div id="postamble" class="status">
1812 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1794 
 <p class="author">Author: Jean-Sébastien Caux</p>
1813 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1795 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1814 
 <p class="validation"></p>
 1796 
 <p class="validation"></p>
1815 
 </div>
 1797 
 </div>
1816
1798

+ 31
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build/emdm_emwm.html View File

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+<!-- 2022-03-15 Tue 08:10 -->
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 <title>Pre-Quantum Electrodynamics</title>
 7 
 <title>Pre-Quantum Electrodynamics</title>
1310 
 </summary>
 1310 
 </summary>
1311 
 <ul>
 1311 
 <ul>
1312 
 <li>
 1312 
 <li>
1313 
-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
1314 
-
1315 
-</li>
1316 
-<li>
1317 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
 1313 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
1318
1314
1319 
 </li>
 1315 
 </li>
1353 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
 1349 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
1354
1350
1355 
 </li>
 1351 
 </li>
1352 
+<li>
1353 
+<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
1354 
+
1355 
+</li>
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1356
1357 
 </ul>
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1358 
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 </details>
1614 
 </svg></a><span class="headline-id">emdm.emwm</span></h3>
 1614 
 </svg></a><span class="headline-id">emdm.emwm</span></h3>
1615
1615
1616 
 <div class="outline-text-3" id="text-emdm_emwm">
 1616 
 <div class="outline-text-3" id="text-emdm_emwm">
1617 
+<details class="prereq" id="orgf2f0eda">
1618 
+<summary id="orged680b6">
1619 
+Prerequisites
1620 
+</summary>
1621 
+<ul class="org-ul">
1622 
+<li>Electrodynamics</li>
1623 
+<li>Convervation Laws</li>
1624 
+<li>Electromagnetic waves in vacuum</li>
1625 
+</ul>
1626 
+</details>
1627 
+
1628 
+<details class="objectives" id="orge99509d">
1629 
+<summary id="orge43e5c1">
1630 
+Objectives
1631 
+</summary>
1632 
+<ul class="org-ul">
1633 
+<li>Understand EM waves in linear media</li>
1634 
+<li>Understand reflection and transmission of EM waves at an interface</li>
1635 
+<li>Know the laws of reflection and refraction</li>
1636 
+<li>Know Fresnel's equations (parallel polarization)</li>
1637 
+<li>Know Brewster's angle</li>
1638 
+<li>Understand EM waves in conductors</li>
1639 
+<li>Understand simple waveguides and coaxial cables</li>
1640 
+</ul>
1641 
+</details>
1617 
 </div>
 1642 
 </div>
1618
1643
1619
1644
1641 
 </div>
 1666 
 </div>
1642 
 <div id="postamble" class="status">
 1667 
 <div id="postamble" class="status">
1643 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1668 
 <p class="author">Author: Jean-Sébastien Caux</p>
1644 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1669 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1645 
 <p class="validation"></p>
 1670 
 <p class="validation"></p>
1646 
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1647
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 <title>Pre-Quantum Electrodynamics</title>
1310 
 </summary>
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1311 
 <ul>
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 <ul>
1312 
 <li>
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1313 
-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
1314 
-
1315 
-</li>
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-<li>
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 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
1318
1314
1319 
 </li>
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1353 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
1354
1350
1355 
 </li>
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 </li>
1352 
+<li>
1353 
+<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
1354 
+
1355 
+</li>
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1356
1357 
 </ul>
 1357 
 </ul>
1358 
 </details>
 1358 
 </details>
1637 
 </div>
 1637 
 </div>
1638 
 <div id="postamble" class="status">
 1638 
 <div id="postamble" class="status">
1639 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1639 
 <p class="author">Author: Jean-Sébastien Caux</p>
1640 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1640 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1641 
 <p class="validation"></p>
 1641 
 <p class="validation"></p>
1642 
 </div>
 1642 
 </div>
1643
1643

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 <title>Pre-Quantum Electrodynamics</title>
1310 
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1311 
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1312 
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1313 
-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
1314 
-
1315 
-</li>
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-<li>
1317 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
1318
1314
1319 
 </li>
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1353 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
 1349 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
1354
1350
1355 
 </li>
 1351 
 </li>
1352 
+<li>
1353 
+<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
1354 
+
1355 
+</li>
1356
1356
1357 
 </ul>
 1357 
 </ul>
1358 
 </details>
 1358 
 </details>
1661 
 \[
 1661 
 \[
1662 
   {\boldsymbol E} ({\boldsymbol r}, t) = {\boldsymbol E}_0 e^{i \tilde{\boldsymbol k} \cdot {\boldsymbol r} - \omega t}
 1662 
   {\boldsymbol E} ({\boldsymbol r}, t) = {\boldsymbol E}_0 e^{i \tilde{\boldsymbol k} \cdot {\boldsymbol r} - \omega t}
1663 
 \]
 1663 
 \]
1664 
-with
1664 
+with complex-valued wavevector
1665 
 \[
 1665 
 \[
1666 
   \tilde{k}^2 \equiv \tilde{\boldsymbol k} \cdot \tilde{\boldsymbol k} = \mu \varepsilon \omega^2 + i\mu \sigma \omega.
 1666 
   \tilde{k}^2 \equiv \tilde{\boldsymbol k} \cdot \tilde{\boldsymbol k} = \mu \varepsilon \omega^2 + i\mu \sigma \omega.
1667 
 \]
 1667 
 \]
1733 
 </div>
 1733 
 </div>
1734 
 <div id="postamble" class="status">
 1734 
 <div id="postamble" class="status">
1735 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1735 
 <p class="author">Author: Jean-Sébastien Caux</p>
1736 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1736 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1737 
 <p class="validation"></p>
 1737 
 <p class="validation"></p>
1738 
 </div>
 1738 
 </div>
1739
1739

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 <title>Pre-Quantum Electrodynamics</title>
1310 
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 </summary>
1311 
 <ul>
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 <ul>
1312 
 <li>
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1313 
-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
1314 
-
1315 
-</li>
1316 
-<li>
1317 
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 1313 
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1318
1314
1319 
 </li>
 1315 
 </li>
1353 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
 1349 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
1354
1350
1355 
 </li>
 1351 
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1352 
+<li>
1353 
+<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
1354 
+
1355 
+</li>
1356
1356
1357 
 </ul>
 1357 
 </ul>
1358 
 </details>
 1358 
 </details>
1632 
 \]
 1632 
 \]
1633 
 If the medium is homogeneous (no spatial dependence of \(\varepsilon\) or \(\mu\)),
 1633 
 If the medium is homogeneous (no spatial dependence of \(\varepsilon\) or \(\mu\)),
1634 
 </p>
 1634 
 </p>
1635 
-<div class="main div" id="orga20c833">
1636 
-<div class="eqlabel" id="org28a8d44">
1635 
+<div class="main div" id="org7ad5a88">
1636 
+<div class="eqlabel" id="orgde6c1b4">
1637 
 <p>
 1637 
 <p>
1638 
 <a id="Max_lh"></a><a href="./emdm_emwm_plm.html#Max_lh"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1638 
 <a id="Max_lh"></a><a href="./emdm_emwm_plm.html#Max_lh"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1639 
   <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"/>
 1639 
   <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"/>
1640 
   <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"/>
 1640 
   <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"/>
1641 
 </svg></a>
 1641 
 </svg></a>
1642 
 </p>
 1642 
 </p>
1643 
-<div class="alteqlabels" id="org79eef6d">
1643 
+<div class="alteqlabels" id="org8b72259">
1644 
 <ul class="org-ul">
 1644 
 <ul class="org-ul">
1645 
 <li>Gr (9.65)</li>
 1645 
 <li>Gr (9.65)</li>
1646 
 </ul>
 1646 
 </ul>
1669 
 \]
 1669 
 \]
1670 
 where the index of refraction of the material is defined as
 1670 
 where the index of refraction of the material is defined as
1671 
 </p>
 1671 
 </p>
1672 
-<div class="main div" id="orga25a79e">
1672 
+<div class="main div" id="org56849ee">
1673 
 <p>
 1673 
 <p>
1674 
 <b>Index of refraction</b>
 1674 
 <b>Index of refraction</b>
1675 
 </p>
 1675 
 </p>
1676 
-<div class="eqlabel" id="orgc9df98d">
1676 
+<div class="eqlabel" id="orgff8c108">
1677 
 <p>
 1677 
 <p>
1678 
 <a id="n"></a><a href="./emdm_emwm_plm.html#n"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1678 
 <a id="n"></a><a href="./emdm_emwm_plm.html#n"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1679 
   <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"/>
 1679 
   <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"/>
1680 
   <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"/>
 1680 
   <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"/>
1681 
 </svg></a>
 1681 
 </svg></a>
1682 
 </p>
 1682 
 </p>
1683 
-<div class="alteqlabels" id="org574afc6">
1683 
+<div class="alteqlabels" id="orgf8eaa9d">
1684 
 <ul class="org-ul">
 1684 
 <ul class="org-ul">
1685 
 <li>Gr (9.69)</li>
 1685 
 <li>Gr (9.69)</li>
1686 
 </ul>
 1686 
 </ul>
1743 
 </div>
 1743 
 </div>
1744 
 <div id="postamble" class="status">
 1744 
 <div id="postamble" class="status">
1745 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1745 
 <p class="author">Author: Jean-Sébastien Caux</p>
1746 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1746 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1747 
 <p class="validation"></p>
 1747 
 <p class="validation"></p>
1748 
 </div>
 1748 
 </div>
1749
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1310 
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1649 
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1650 
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 <div id="postamble" class="status">
1651 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1651 
 <p class="author">Author: Jean-Sébastien Caux</p>
1652 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1652 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1653 
 <p class="validation"></p>
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1654 
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1310 
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 </summary>
1311 
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1312 
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1353 
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1354
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1352 
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1353 
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1354 
+
1355 
+</li>
1356
1356
1357 
 </ul>
 1357 
 </ul>
1358 
 </details>
 1358 
 </details>
1621 
 {\boldsymbol B}_I (z,t) = \frac{1}{v_1} E_{0_I} e^{i (k_1 z - \omega t)} \hat{\boldsymbol y}
 1621 
 {\boldsymbol B}_I (z,t) = \frac{1}{v_1} E_{0_I} e^{i (k_1 z - \omega t)} \hat{\boldsymbol y}
1622 
 \label{Gr(9.75)}
 1622 
 \label{Gr(9.75)}
1623 
 \]
 1623 
 \]
1624 
-(in which we have used \(B_0 = \frac{1}{v} E_0\), <a href="./emd_emw_mpw.html#EBmpw">EBmpw</a>.
1624 
+(in which we have used \(B_0 = \frac{1}{v} E_0\), <a href="./emd_emw_mpw.html#EBmpw">EBmpw</a>,
1625 
+adapted for the light velocity in this medium).
1625 
 </p>
 1626 
 </p>
1626
1627
1627 
 <p>
 1628 
 <p>
1670 
 Solving these coupled equations, we can write
 1671 
 Solving these coupled equations, we can write
1671 
 outgoing amplitudes in terms of incident ones:
 1672 
 outgoing amplitudes in terms of incident ones:
1672 
 </p>
 1673 
 </p>
1673 
-<div class="eqlabel" id="org653a14a">
1674 
+<div class="eqlabel" id="org1b7d37e">
1674 
 <p>
 1675 
 <p>
1675 
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 1676 
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1676 
   <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"/>
 1677 
   <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"/>
1677 
   <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"/>
 1678 
   <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"/>
1678 
 </svg></a>
 1679 
 </svg></a>
1679 
 </p>
 1680 
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1680 
-<div class="alteqlabels" id="org5690572">
1681 
+<div class="alteqlabels" id="org2907578">
1681 
 <ul class="org-ul">
 1682 
 <ul class="org-ul">
1682 
 <li>Gr (9.82)</li>
 1683 
 <li>Gr (9.82)</li>
1683 
 </ul>
 1684 
 </ul>
1743 
 </div>
 1744 
 </div>
1744 
 <div id="postamble" class="status">
 1745 
 <div id="postamble" class="status">
1745 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1746 
 <p class="author">Author: Jean-Sébastien Caux</p>
1746 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1747 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1747 
 <p class="validation"></p>
 1748 
 <p class="validation"></p>
1748 
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 <title>Pre-Quantum Electrodynamics</title>
1310 
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1311 
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1312 
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-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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-
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 </ul>
 1357 
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1358 
 </details>
 1358 
 </details>
1635 
 \]
 1635 
 \]
1636 
 All waves have the same frequency \(\omega\). Since \(\omega = k v\), the three wavevectors are related by
 1636 
 All waves have the same frequency \(\omega\). Since \(\omega = k v\), the three wavevectors are related by
1637 
 </p>
 1637 
 </p>
1638 
-<div class="eqlabel" id="org862531f">
1638 
+<div class="eqlabel" id="org792ed2b">
1639 
 <p>
 1639 
 <p>
1640 
 <a id="RTobliquek"></a><a href="./emdm_emwm_refl_oi.html#RTobliquek"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1640 
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1641 
   <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"/>
 1641 
   <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"/>
1642 
   <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"/>
 1642 
   <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"/>
1643 
 </svg></a>
 1643 
 </svg></a>
1644 
 </p>
 1644 
 </p>
1645 
-<div class="alteqlabels" id="org4435847">
1645 
+<div class="alteqlabels" id="org1997374">
1646
1646
1647 
 </div>
 1647 
 </div>
1648
1648
1665 
 <p>
 1665 
 <p>
1666 
 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
 1666 
 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
1667 
 </p>
 1667 
 </p>
1668 
-<div class="core div" id="org6f173e3">
1668 
+<div class="core div" id="orgbf226b1">
1669 
 <p>
 1669 
 <p>
1670 
 <b>First law of reflection:</b>
 1670 
 <b>First law of reflection:</b>
1671 
   the incident, reflected and transmitted wave vectors form a plane (called the plane of incidence) which also includes the normal to the surface.
 1671 
   the incident, reflected and transmitted wave vectors form a plane (called the plane of incidence) which also includes the normal to the surface.
1680 
 with the incidence (\(\theta_I\)) and reflection (\(\theta_R\)) angles
 1680 
 with the incidence (\(\theta_I\)) and reflection (\(\theta_R\)) angles
1681 
 and the angle of refraction (\(\theta_T\)) obey the following laws:
 1681 
 and the angle of refraction (\(\theta_T\)) obey the following laws:
1682 
 </p>
 1682 
 </p>
1683 
-<div class="core div" id="orgbf9459c">
1683 
+<div class="core div" id="org000f37c">
1684 
 <p>
 1684 
 <p>
1685 
 <b>Law of reflection</b>
 1685 
 <b>Law of reflection</b>
1686 
   \[
 1686 
   \[
1697 
 <p>
 1697 
 <p>
1698 
 This takes care of the spatially-dependent exponential factors in the boundary conditions. The coefficients must further obey
 1698 
 This takes care of the spatially-dependent exponential factors in the boundary conditions. The coefficients must further obey
1699 
 </p>
 1699 
 </p>
1700 
-<div class="eqlabel" id="org8d76e61">
1700 
+<div class="eqlabel" id="orgdc1db4c">
1701 
 <p>
 1701 
 <p>
1702 
 <a id="EBRT"></a><a href="./emdm_emwm_refl_oi.html#EBRT"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1702 
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1703 
   <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
 1703 
   <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
1704 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 1704 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
1705 
 </svg></a>
 1705 
 </svg></a>
1706 
 </p>
 1706 
 </p>
1707 
-<div class="alteqlabels" id="org2c4d03c">
1707 
+<div class="alteqlabels" id="orgf69a0d1">
1708
1708
1709 
 </div>
 1709 
 </div>
1710
1710
1750 
 \]
 1750 
 \]
1751 
 Writing everything in terms of the incident amplitude, we get
 1751 
 Writing everything in terms of the incident amplitude, we get
1752 
 </p>
 1752 
 </p>
1753 
-<div class="main div" id="org3866053">
1753 
+<div class="main div" id="org7e347a7">
1754 
 <p>
 1754 
 <p>
1755 
 <b>Fresnel's equations for reflection and transmission amplitudes (parallel case)</b>
 1755 
 <b>Fresnel's equations for reflection and transmission amplitudes (parallel case)</b>
1756 
 </p>
 1756 
 </p>
1757 
-<div class="eqlabel" id="org31e8ee3">
1757 
+<div class="eqlabel" id="org5c049e2">
1758 
 <p>
 1758 
 <p>
1759 
 <a id="Fresnel"></a><a href="./emdm_emwm_refl_oi.html#Fresnel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1759 
 <a id="Fresnel"></a><a href="./emdm_emwm_refl_oi.html#Fresnel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1760 
   <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"/>
 1760 
   <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"/>
1761 
   <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"/>
 1761 
   <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"/>
1762 
 </svg></a>
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 </svg></a>
1763 
 </p>
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 </p>
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-<div class="alteqlabels" id="org1f91c78">
1764 
+<div class="alteqlabels" id="org50e3086">
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1766 
 </div>
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 </div>
1767
1767
1785 
 Behaviour: for \(\theta_I = 0\) we recover <a href="./emdm_emwm_refl_ni.html#ERT">ERT</a>.
 1785 
 Behaviour: for \(\theta_I = 0\) we recover <a href="./emdm_emwm_refl_ni.html#ERT">ERT</a>.
1786 
 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
 1786 
 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
1787 
 </p>
 1787 
 </p>
1788 
-<div class="main div" id="org328f2a5">
1789 
-<div class="eqlabel" id="org3b9cec2">
1788 
+<div class="main div" id="org73df67c">
1789 
+<div class="eqlabel" id="org8295eee">
1790 
 <p>
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 <p>
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 <a id="Brewster"></a><a href="./emdm_emwm_refl_oi.html#Brewster"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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 <a id="Brewster"></a><a href="./emdm_emwm_refl_oi.html#Brewster"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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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="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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   <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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1848 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1848 
 <p class="author">Author: Jean-Sébastien Caux</p>
1849 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
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+<p class="date">Created: 2022-03-15 Tue 08:10</p>
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 <p class="author">Author: Jean-Sébastien Caux</p>
1636 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
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+<p class="date">Created: 2022-03-15 Tue 08:10</p>
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 <p class="author">Author: Jean-Sébastien Caux</p>
1642 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
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 <p class="author">Author: Jean-Sébastien Caux</p>
 1661 
 <p class="author">Author: Jean-Sébastien Caux</p>
1662 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1662 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
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 \]
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 These fields must obey Maxwell's equations. Unlike waves in vacuum, it will turn out that EM waves in waveguides are not purely transverse. We will thus start with a general ansatz for the fields:
 1630 
 These fields must obey Maxwell's equations. Unlike waves in vacuum, it will turn out that EM waves in waveguides are not purely transverse. We will thus start with a general ansatz for the fields:
1631 
 \[
 1631 
 \[
1632 
-  {\boldsymbol E}_0 = \sum_i E_i \hat{\boldsymbol i}, \hspace{10mm}
1633 
-  {\boldsymbol B}_0 = \sum_i B_i \hat{\boldsymbol i}, \hspace{10mm}
1632 
+  {\boldsymbol E}_0 = \sum_i E_i ~\hat{\boldsymbol i}, \hspace{10mm}
1633 
+  {\boldsymbol B}_0 = \sum_i B_i ~\hat{\boldsymbol i}, \hspace{10mm}
1634 
   i = x, y, z.
 1634 
   i = x, y, z.
1635 
 \]
 1635 
 \]
1636 
 Maxwell's equations (iii) and (iv) then give
 1636 
 Maxwell's equations (iii) and (iv) then give
1683 
 </div>
 1683 
 </div>
1684 
 <div id="postamble" class="status">
 1684 
 <div id="postamble" class="status">
1685 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1685 
 <p class="author">Author: Jean-Sébastien Caux</p>
1686 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1686 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1687 
 <p class="validation"></p>
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1636 
 \[
 1636 
 \[
1637 
   X(x) = A \sin (k_x x) + B \cos (k_x x)
 1637 
   X(x) = A \sin (k_x x) + B \cos (k_x x)
1638 
 \]
 1638 
 \]
1639 
-and to fit the boundary conditions we must have that \(B_x = 0\) so
1639 
+and to fit the boundary conditions we must have that
1640 
 \(\partial B_x/\partial x\) must vanish at \(x=0, a\),
 1640 
 \(\partial B_x/\partial x\) must vanish at \(x=0, a\),
1641 
-so \(dX/dx\) must also vanish at \(x = 0, a\) and thus \(A = 0\) and
1641 
+so \(dX/dx\) must vanish at \(x = 0, a\) and thus \(A = 0\) and
1642 
 \[
 1642 
 \[
1643 
   k_x = m \pi/a, \hspace{5mm} m = 0, 1, \cdots
 1643 
   k_x = m \pi/a, \hspace{5mm} m = 0, 1, \cdots
1644 
 \]
 1644 
 \]
1693 
 </div>
 1693 
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1694 
 <div id="postamble" class="status">
 1694 
 <div id="postamble" class="status">
1695 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1695 
 <p class="author">Author: Jean-Sébastien Caux</p>
1696 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1696 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
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+<summary id="org9fe7d9c">
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 Prerequisites
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 Prerequisites
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 Objectives
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 Objectives
1629 
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 </summary>
1630 
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1660 
 </div>
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 </div>
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 <div id="postamble" class="status">
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 <div id="postamble" class="status">
1662 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1662 
 <p class="author">Author: Jean-Sébastien Caux</p>
1663 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1663 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
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1621 
 <p>
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 <p>
1622 
 There is however a certain ambiguity left when representing fields in terms of potentials. Namely, it is easy to verify that new fields
 1622 
 There is however a certain ambiguity left when representing fields in terms of potentials. Namely, it is easy to verify that new fields
1623 
 \[
 1623 
 \[
1624 
-  V^\prime = V - \frac{\partial \lambda}{\partial t}, \hspace{10mm}
1624 
+  \phi^\prime = \phi - \frac{\partial \lambda}{\partial t}, \hspace{10mm}
1625 
   {\boldsymbol A}^\prime = {\boldsymbol A} + {\boldsymbol \nabla} \lambda
 1625 
   {\boldsymbol A}^\prime = {\boldsymbol A} + {\boldsymbol \nabla} \lambda
1626 
 \]
 1626 
 \]
1627 
-give the same \({\boldsymbol E}\) and \({\boldsymbol B}\) fields as \(V, {\boldsymbol A}\) for any scalar function \(\lambda ({\boldsymbol r}, t)\).
1627 
+give the same \({\boldsymbol E}\) and \({\boldsymbol B}\) fields as \(\phi, {\boldsymbol A}\) for any scalar function \(\lambda ({\boldsymbol r}, t)\).
1628 
 </p>
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 </p>
1629
1629
1630 
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 <p>
1657 
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1658 
 <div id="postamble" class="status">
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 <div id="postamble" class="status">
1659 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1659 
 <p class="author">Author: Jean-Sébastien Caux</p>
1660 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1660 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1661 
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1616 
 <p>
 1616 
 <p>
1617 
-The {\bf Coulomb Gauge} is specified by taking
1617 
+The <b>Coulomb Gauge</b> is specified by taking
1618 
 \[
 1618 
 \[
1619 
   {\boldsymbol \nabla} \cdot {\boldsymbol A} = 0
 1619 
   {\boldsymbol \nabla} \cdot {\boldsymbol A} = 0
1620 
 \]
 1620 
 \]
1621 
-in which case (\ref{eq:LaplacianV}) becomes simply
1621 
+in which case <a href="./emf_svp.html#Lapphi">Lapphi</a> becomes simply
1622 
 \[
 1622 
 \[
1623 
-  {\boldsymbol \nabla}^2 V = -\frac{\rho}{\varepsilon_0}
1623 
+  {\boldsymbol \nabla}^2 \phi = -\frac{\rho}{\varepsilon_0}
1624 
 \]
 1624 
 \]
1625 
-{\it i.e.} Poisson's equation, whose solution we already know:
1625 
+<i>i.e.</i> Poisson's equation, whose solution we already know:
1626 
 \[
 1626 
 \[
1627 
-  V({\boldsymbol r}, t) = \frac{1}{4\pi \varepsilon_0} \int d\tau' \frac{\rho({\boldsymbol r}^\prime, t)}{|{\boldsymbol r} - {\boldsymbol r}^\prime|}
1627 
+  \phi({\boldsymbol r}, t) = \frac{1}{4\pi \varepsilon_0} \int d\tau' \frac{\rho({\boldsymbol r}^\prime, t)}{|{\boldsymbol r} - {\boldsymbol r}^\prime|}
1628 
 \]
 1628 
 \]
1629 
-Note that this is an equal-time relationship (it does not mean instantaneous action at a distance, since \(V\) by itself is not physically measurable).
1629 
+Note that this is an equal-time relationship (it does not mean instantaneous action at a distance, since \(\phi\) by itself is not physically measurable).
1630 
 </p>
 1630 
 </p>
1631
1631
1632 
 <p>
 1632 
 <p>
1633 
-Although Gauss's law looks nice in the Coulomb gauge, Amp{\`e}re-Maxwell does not:
1633 
+Although Gauss's law looks nice in the Coulomb gauge, Ampère-Maxwell does not:
1634 
 \[
 1634 
 \[
1635 
-  {\boldsymbol \nabla}^2 {\boldsymbol A} - \mu_0 \varepsilon_0 \frac{\partial^2 {\boldsymbol A}}{\partial t^2} = -\mu_0 {\boldsymbol J} + \mu_0 \varepsilon_0 {\boldsymbol \nabla} \left( \frac{\partial V}{\partial t} \right).
1635 
+  {\boldsymbol \nabla}^2 {\boldsymbol A} - \mu_0 \varepsilon_0 \frac{\partial^2 {\boldsymbol A}}{\partial t^2} = -\mu_0 {\boldsymbol J} + \mu_0 \varepsilon_0 {\boldsymbol \nabla} \left( \frac{\partial \phi}{\partial t} \right).
1636 
 \]
 1636 
 \]
1637 
 </p>
 1637 
 </p>
1638 
 </div>
 1638 
 </div>
1656 
 </div>
 1656 
 </div>
1657 
 <div id="postamble" class="status">
 1657 
 <div id="postamble" class="status">
1658 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1658 
 <p class="author">Author: Jean-Sébastien Caux</p>
1659 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1659 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1660 
 <p class="validation"></p>
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 <p class="validation"></p>
1661 
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 </div>
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build/emf_g_Lg.html View File

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 <title>Pre-Quantum Electrodynamics</title>
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1311 
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1312 
 <li>
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1313 
-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
1314 
-
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 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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1318
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1354
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+<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
1354 
+
1355 
+</li>
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 </ul>
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 </details>
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 </details>
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 </svg></a><span class="headline-id">emf.g.Lg</span></h4>
 1614 
 </svg></a><span class="headline-id">emf.g.Lg</span></h4>
1615 
 <div class="outline-text-4" id="text-emf_g_Lg">
 1615 
 <div class="outline-text-4" id="text-emf_g_Lg">
1616 
 <p>
 1616 
 <p>
1617 
-A more aesthetic choice is the {\bf Lorenz gauge}:
1617 
+A more aesthetic choice is the <b>Lorenz gauge</b>:
1618 
+</p>
1619 
+<div class="eqlabel" id="orgb11b2c1">
1620 
+<p>
1621 
+<a id="LorenzG"></a><a href="./emf_g_Lg.html#LorenzG"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1622 
+  <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"/>
1623 
+  <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"/>
1624 
+</svg></a>
1625 
+</p>
1626 
+<div class="alteqlabels" id="org9cfa6f0">
1627 
+
1628 
+</div>
1629 
+
1630 
+</div>
1631 
+<p>
1618 
 \[
 1632 
 \[
1619 
-  {\boldsymbol \nabla} \cdot {\boldsymbol A} + \mu_0 \varepsilon_0 \frac{\partial V}{\partial t} = 0
1620 
-  \label{eq:LorenzGauge}
1633 
+{\boldsymbol \nabla} \cdot {\boldsymbol A} + \mu_0 \varepsilon_0 \frac{\partial \phi}{\partial t} = 0
1634 
+\tag{LorenzG}\label{LorenzG}
1621 
 \]
 1635 
 \]
1622 
-which is chosen to put the second term in the left-hand side of (\ref{eq:LaplacianA}) to zero. What remains is then
1636 
+which is chosen to put the second term in the left-hand side of <a href="./emf_svp.html#LapA">LapA</a> to zero. What remains is then
1637 
+</p>
1638 
+<div class="eqlabel" id="org8bdd983">
1639 
+<p>
1640 
+<a id="ALorenzG"></a><a href="./emf_g_Lg.html#ALorenzG"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1641 
+  <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"/>
1642 
+  <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"/>
1643 
+</svg></a>
1644 
+</p>
1645 
+<div class="alteqlabels" id="orgda10464">
1646 
+
1647 
+</div>
1648 
+
1649 
+</div>
1650 
+<p>
1623 
 \[
 1651 
 \[
1624 
   {\boldsymbol \nabla}^2 {\boldsymbol A} - \mu_0 \varepsilon_0 \frac{\partial^2 {\boldsymbol A}}{\partial t^2} = -\mu_0 {\boldsymbol J}
 1652 
   {\boldsymbol \nabla}^2 {\boldsymbol A} - \mu_0 \varepsilon_0 \frac{\partial^2 {\boldsymbol A}}{\partial t^2} = -\mu_0 {\boldsymbol J}
1653 
+  \tag{ALorenzG}\label{ALorenzG}
1625 
 \]
 1654 
 \]
1626 
-while the equation for \(V\) becomes
1655 
+while the equation for \(\phi\) becomes
1656 
+</p>
1657 
+<div class="eqlabel" id="org92ed269">
1658 
+<p>
1659 
+<a id="phiLorenzG"></a><a href="./emf_g_Lg.html#phiLorenzG"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1660 
+  <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"/>
1661 
+  <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"/>
1662 
+</svg></a>
1663 
+</p>
1664 
+<div class="alteqlabels" id="orgf853d29">
1665 
+
1666 
+</div>
1667 
+
1668 
+</div>
1669 
+<p>
1627 
 \[
 1670 
 \[
1628 
-  {\boldsymbol \nabla}^2 V - \mu_0 \varepsilon_0 \frac{\partial^2 V}{\partial t^2} = -\frac{\rho}{\varepsilon_0}.
1671 
+  {\boldsymbol \nabla}^2 \phi - \mu_0 \varepsilon_0 \frac{\partial^2 \phi}{\partial t^2} = -\frac{\rho}{\varepsilon_0}.
1672 
+  \tag{phiLorenzG}\label{phiLorenzG}
1629 
 \]
 1673 
 \]
1630 
 These can be written compactly upon introducing a new operator: the
 1674 
 These can be written compactly upon introducing a new operator: the
1631 
 </p>
 1675 
 </p>
1632 
-<div class="core div" id="org0aaab77">
1676 
+<div class="core div" id="orgd77f9a6">
1677 
+<p>
1678 
+<b>d'Alembertian operator</b>
1679 
+</p>
1680 
+<div class="eqlabel" id="orgc902ec7">
1681 
+<p>
1682 
+<a id="dAl"></a><a href="./emf_g_Lg.html#dAl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1683 
+  <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"/>
1684 
+  <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"/>
1685 
+</svg></a>
1686 
+</p>
1687 
+<div class="alteqlabels" id="orgac35877">
1688 
+
1689 
+</div>
1690 
+
1691 
+</div>
1633 
 <p>
 1692 
 <p>
1634 
-{\bf d'Alembertian operator}
1635 
 \[
 1693 
 \[
1636 
-    \square^2 \equiv {\boldsymbol \nabla}^2 - \mu_0 \varepsilon_0 \frac{\partial^2}{\partial t^2}
1637 
-    \label{eq:dAlembertian}
1638 
-  \]
1694 
+\square^2 \equiv {\boldsymbol \nabla}^2 - \mu_0 \varepsilon_0 \frac{\partial^2}{\partial t^2}
1695 
+\tag{dAl}\label{dAl}
1696 
+\]
1639 
 </p>
 1697 
 </p>
1640
1698
1641 
 </div>
 1699 
 </div>
1642 
 <p>
 1700 
 <p>
1643 
 so we get the
 1701 
 so we get the
1644 
 </p>
 1702 
 </p>
1645 
-<div class="core div" id="orgc8e8b42">
1703 
+<div class="core div" id="org12b2ff9">
1704 
+<p>
1705 
+<b>Inhomogeneous Maxwell equations</b> <i>(Lorenz gauge)</i>
1706 
+</p>
1707 
+<div class="eqlabel" id="orge48b8c0">
1708 
+<p>
1709 
+<a id="MaxLor"></a><a href="./emf_g_Lg.html#MaxLor"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1710 
+  <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"/>
1711 
+  <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"/>
1712 
+</svg></a>
1713 
+</p>
1714 
+<div class="alteqlabels" id="org140821b">
1715 
+
1716 
+</div>
1717 
+
1718 
+</div>
1646 
 <p>
 1719 
 <p>
1647 
-{\bf Inhomogeneous Maxwell equations (Lorenz gauge)}
1648 
 \[
 1720 
 \[
1649 
-    \square^2 V = -\frac{\rho}{\varepsilon_0}, \hspace{10mm}
1650 
-    \square^2 {\boldsymbol A} = -\mu_0 {\boldsymbol J}
1651 
-    \label{eq:InhomogeneousMaxwellLorenzGauge}
1652 
-  \]
1721 
+\square^2 \phi = -\frac{\rho}{\varepsilon_0}, \hspace{10mm}
1722 
+\square^2 {\boldsymbol A} = -\mu_0 {\boldsymbol J}
1723 
+\tag{MaxLor}\label{MaxLor}
1724 
+\]
1653 
 </p>
 1725 
 </p>
1654
1726
1655 
 </div>
 1727 
 </div>
1656 
 <p>
 1728 
 <p>
1657 
 This gauge is especially nice in the context of special relativity.
 1729 
 This gauge is especially nice in the context of special relativity.
1658 
 The whole of electrodynamics has thus reduced to solving the inhomogeneous
 1730 
 The whole of electrodynamics has thus reduced to solving the inhomogeneous
1659 
-wave equations (\ref{eq:InhomogeneousMaxwellLorenzGauge})
1731 
+wave equations <a href="./emf_g_Lg.html#MaxLor">MaxLor</a>
1660 
 in terms of specified sources.
 1732 
 in terms of specified sources.
1661 
 </p>
 1733 
 </p>
1662
1734
1665 
 Without choosing the Lorenz gauge, we can still write the inhomogeneous
 1737 
 Without choosing the Lorenz gauge, we can still write the inhomogeneous
1666 
 Maxwell equations in a simpler form. Defining
 1738 
 Maxwell equations in a simpler form. Defining
1667 
 \[
 1739 
 \[
1668 
-  L \equiv {\boldsymbol \nabla} \cdot {\boldsymbol A} + \mu_0 \varepsilon_0 \frac{\partial V}{\partial t},
1740 
+  L \equiv {\boldsymbol \nabla} \cdot {\boldsymbol A} + \mu_0 \varepsilon_0 \frac{\partial \phi}{\partial t},
1669 
 \]
 1741 
 \]
1670 
 we have by direct inspection
 1742 
 we have by direct inspection
1671 
 \[
 1743 
 \[
1672 
-  \square^2 V + \frac{\partial L}{\partial t} = -\frac{\rho}{\varepsilon_0}, \hspace{10mm}
1744 
+  \square^2 \phi + \frac{\partial L}{\partial t} = -\frac{\rho}{\varepsilon_0}, \hspace{10mm}
1673 
   \square^2 {\boldsymbol A} - {\boldsymbol \nabla} L = -\mu_0 {\boldsymbol J}.
 1745 
   \square^2 {\boldsymbol A} - {\boldsymbol \nabla} L = -\mu_0 {\boldsymbol J}.
1674 
 \]
 1746 
 \]
1675 
 </p>
 1747 
 </p>
1692 
 </div>
 1764 
 </div>
1693 
 <div id="postamble" class="status">
 1765 
 <div id="postamble" class="status">
1694 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1766 
 <p class="author">Author: Jean-Sébastien Caux</p>
1695 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1767 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1696 
 <p class="validation"></p>
 1768 
 <p class="validation"></p>
1697 
 </div>
 1769 
 </div>
1698
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+ 64
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build/emf_svp.html View File

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 <meta name="viewport" content="width=device-width, initial-scale=1">
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 <meta name="viewport" content="width=device-width, initial-scale=1">
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 <title>Pre-Quantum Electrodynamics</title>
 7 
 <title>Pre-Quantum Electrodynamics</title>
1310 
 </summary>
 1310 
 </summary>
1311 
 <ul>
 1311 
 <ul>
1312 
 <li>
 1312 
 <li>
1313 
-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
1314 
-
1315 
-</li>
1316 
-<li>
1317 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
 1313 
 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
1318
1314
1319 
 </li>
 1315 
 </li>
1353 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
 1349 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
1354
1350
1355 
 </li>
 1351 
 </li>
1352 
+<li>
1353 
+<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
1354 
+
1355 
+</li>
1356
1356
1357 
 </ul>
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 </ul>
1358 
 </details>
 1358 
 </details>
1628 
 </p>
 1628 
 </p>
1629
1629
1630 
 <p>
 1630 
 <p>
1631 
-Easiest:
1631 
+Easiest: as we already saw (<a href="./ems_ms_vp_A.html#BcurlA">BcurlA</a>), we can write the magnetic
1632 
+field as a pure curl (since its divergence always vanishes):
1632 
 </p>
 1633 
 </p>
1633 
-<div class="core div" id="org5bf4193">
1634 
+<div class="core div" id="orgb63eb69">
1634 
 <p>
 1635 
 <p>
1635 
 \[
 1636 
 \[
1636 
-    {\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}
1637 
-  \]
1637 
+{\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}
1638 
+\]
1638 
 </p>
 1639 
 </p>
1639
1640
1640 
 </div>
 1641 
 </div>
1641
1642
1642 
 <p>
 1643 
 <p>
1643 
-Putting this into Faraday's law gives
1644 
+Putting this into Faraday's law <a href="./emd_Fl_Fl.html#Fl">Fl</a> gives
1644 
 \[
 1645 
 \[
1645 
   {\boldsymbol \nabla} \times \left({\boldsymbol E} + \frac{\partial {\boldsymbol A}}{\partial t} \right) = 0
 1646 
   {\boldsymbol \nabla} \times \left({\boldsymbol E} + \frac{\partial {\boldsymbol A}}{\partial t} \right) = 0
1646 
 \]
 1647 
 \]
1647 
-so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \nabla} V\)) so we get
1648 
+so this can be written as the gradient of a scalar.
1649 
+Making the choice \(-{\boldsymbol \nabla} \phi\) for this, we get
1648 
 </p>
 1650 
 </p>
1649 
-<div class="core div" id="orgffce8dc">
1651 
+<div class="core div" id="org23c61e0">
1652 
+<div class="eqlabel" id="org51cbd1b">
1653 
+<p>
1654 
+<a id="E_phiA"></a><a href="./emf_svp.html#E_phiA"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1655 
+  <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"/>
1656 
+  <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"/>
1657 
+</svg></a>
1658 
+</p>
1659 
+<div class="alteqlabels" id="org25412c6">
1660 
+
1661 
+</div>
1662 
+
1663 
+</div>
1650 
 <p>
 1664 
 <p>
1651 
 \[
 1665 
 \[
1652 
-    {\boldsymbol E} = -{\boldsymbol \nabla} V - \frac{\partial {\boldsymbol A}}{\partial t}
1653 
-    \label{eq:E_from_Potentials}
1654 
-  \]
1666 
+{\boldsymbol E} = -{\boldsymbol \nabla} \phi - \frac{\partial {\boldsymbol A}}{\partial t}
1667 
+\tag{E_phiA}\label{E_phiA}
1668 
+\]
1655 
 </p>
 1669 
 </p>
1656
1670
1657 
 </div>
 1671 
 </div>
1658
1672
1659 
 <p>
 1673 
 <p>
1660 
-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
1674 
+Using this potential representation for \({\boldsymbol E}\) and \({\boldsymbol B}\) automatically fulfills the two homogeneous Maxwell equations. For the inhomogeneous equations, substituting <a href="./emf_svp.html#E_phiA">E_phiA</a> into Gauss's law gives
1661 
 </p>
 1675 
 </p>
1662 
-<div class="main div" id="org26743e8">
1676 
+<div class="main div" id="orgf26abb4">
1677 
+<div class="eqlabel" id="org986b9b3">
1663 
 <p>
 1678 
 <p>
1664 
-\[
1665 
-    {\boldsymbol \nabla}^2 V + \frac{\partial}{\partial t} {\boldsymbol \nabla} \cdot {\boldsymbol A} = -\frac{\rho}{\varepsilon_0}
1666 
-    \label{eq:LaplacianV}
1667 
-  \]
1679 
+<a id="Lapphi"></a><a href="./emf_svp.html#Lapphi"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1680 
+  <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"/>
1681 
+  <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"/>
1682 
+</svg></a>
1668 
 </p>
 1683 
 </p>
1684 
+<div class="alteqlabels" id="orgcd04ad9">
1685 
+
1686 
+</div>
1669
1687
1670 
 </div>
 1688 
 </div>
1671 
 <p>
 1689 
 <p>
1672 
-whereas Amp{\`ere}-Maxwell becomes
1673 
 \[
 1690 
 \[
1674 
-  {\boldsymbol \nabla} \times \left({\boldsymbol \nabla} {\boldsymbol A}\right) = \mu_0 {\boldsymbol J} - \mu_0 \varepsilon_0 {\boldsymbol \nabla} \left(\frac{\partial V}{\partial t}\right) - \mu_0 \varepsilon_0 \frac{\partial^2 {\boldsymbol A}}{\partial t^2}
1691 
+{\boldsymbol \nabla}^2 \phi + \frac{\partial}{\partial t} {\boldsymbol \nabla} \cdot {\boldsymbol A} = -\frac{\rho}{\varepsilon_0}
1692 
+\tag{Lapphi}\label{Lapphi}
1675 
 \]
 1693 
 \]
1676 
-which becomes after simple rearrangement and use of the identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\),
1677 
 </p>
 1694 
 </p>
1678 
-<div class="main div" id="orga86c8c2">
1695 
+
1696 
+</div>
1679 
 <p>
 1697 
 <p>
1698 
+whereas Ampère-Maxwell becomes
1680 
 \[
 1699 
 \[
1681 
-  \left( {\boldsymbol ∇}^2 {\boldsymbol A} - μ_0 ε_0 \frac{∂^2 {\boldsymbol A}}{∂ t^2} \right)
1700 
+  {\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = \mu_0 {\boldsymbol J} - \mu_0 \varepsilon_0 {\boldsymbol \nabla} \left(\frac{\partial \phi}{\partial t}\right) - \mu_0 \varepsilon_0 \frac{\partial^2 {\boldsymbol A}}{\partial t^2}
1701 
+\]
1702 
+which becomes after simple rearrangement and use of the <a href="./c_m_dc_d2.html#curlcurl">curlcurl</a> identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\),
1682 
 </p>
 1703 
 </p>
1683 
-<ul class="org-ul">
1684 
-<li>{\boldsymbol ∇} \left({\boldsymbol ∇} ⋅ {\boldsymbol A} + μ_0 ε_0 \frac{\partial V}{\partial t} \right) = -μ_0 {\boldsymbol J}</li>
1685 
-</ul>
1704 
+<div class="main div" id="org135124e">
1705 
+<div class="eqlabel" id="org73667ac">
1686 
 <p>
 1706 
 <p>
1687 
-  \label{eq:LaplacianA}
1707 
+<a id="LapA"></a><a href="./emf_svp.html#LapA"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1708 
+  <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"/>
1709 
+  <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"/>
1710 
+</svg></a>
1711 
+</p>
1712 
+<div class="alteqlabels" id="org3f09a58">
1713 
+
1714 
+</div>
1715 
+
1716 
+</div>
1717 
+<p>
1718 
+\[
1719 
+\left( {\boldsymbol \nabla}^2 {\boldsymbol A} - \mu_0 \varepsilon_0 \frac{\partial^2 {\boldsymbol A}}{\partial t^2} \right) - {\boldsymbol \nabla} \left({\boldsymbol \nabla} \cdot {\boldsymbol A} + \mu_0 \varepsilon_0 \frac{\partial \phi}{\partial t} \right) = -\mu_0 {\boldsymbol J}
1720 
+\tag{LapA}\label{LapA}
1688 
 \]
 1721 
 \]
1689 
 </p>
 1722 
 </p>
1690
1723
1711 
 </div>
 1744 
 </div>
1712 
 <div id="postamble" class="status">
 1745 
 <div id="postamble" class="status">
1713 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1746 
 <p class="author">Author: Jean-Sébastien Caux</p>
1714 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1747 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1715 
 <p class="validation"></p>
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1716 
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-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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1642 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1642 
 <p class="author">Author: Jean-Sébastien Caux</p>
1643 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1643 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1644 
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 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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-<details class="prereq" id="org192cbf5">
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-<summary id="org78425f6">
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+<details class="prereq" id="org1512362">
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+<summary id="orgf95a20b">
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 Prerequisites
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 Prerequisites
1624 
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 Objectives
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1633 
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1666 
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 <div id="postamble" class="status">
1668 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1668 
 <p class="author">Author: Jean-Sébastien Caux</p>
1669 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1669 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1670 
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 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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+<li>
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+<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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1358 
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 </details>
1624 
 defines an electrostatic problem, whose solution is in principle obtained
 1624 
 defines an electrostatic problem, whose solution is in principle obtained
1625 
 from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_vcd">E_vcd</a>
 1625 
 from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_vcd">E_vcd</a>
1626 
 </p>
 1626 
 </p>
1627 
-<div class="main div" id="orgb661984">
1627 
+<div class="main div" id="org91ccc03">
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 <p>
1629
1629
1630 
 </p>
 1630 
 </p>
1638 
 or (often simpler) by calculating the electrostatic potential, using either the
 1638 
 or (often simpler) by calculating the electrostatic potential, using either the
1639 
 explicit construction <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>
 1639 
 explicit construction <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>
1640 
 </p>
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 </p>
1641 
-<div class="main div" id="org5d15b9f">
1641 
+<div class="main div" id="orge494253">
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 <p>
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 <p>
1643
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 </p>
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 </p>
1658 
 <a href="./ems_es_ep_PL.html#Poi">🐟</a>
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 <a href="./ems_es_ep_PL.html#Poi">🐟</a>
1659 
 </p>
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 </p>
1660
1660
1661 
-<div class="core div" id="org7aef84c">
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+<div class="core div" id="org4db04a9">
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 <p>
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 <p>
1663
1663
1664 
 </p>
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 </p>
1674 
 <p>
 1674 
 <p>
1675 
 In the specific case where the charge density vanishes, we fall back onto the simpler Laplace equation <a href="./ems_es_ep_PL.html#Lap">Lap</a>
 1675 
 In the specific case where the charge density vanishes, we fall back onto the simpler Laplace equation <a href="./ems_es_ep_PL.html#Lap">Lap</a>
1676 
 </p>
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 </p>
1677 
-<div class="core div" id="org3f412f2">
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+<div class="core div" id="orgd28c357">
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1679
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1680 
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1712 
 </div>
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 </div>
1713 
 <div id="postamble" class="status">
 1713 
 <div id="postamble" class="status">
1714 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1714 
 <p class="author">Author: Jean-Sébastien Caux</p>
1715 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1715 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1716 
 <p class="validation"></p>
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 <p class="validation"></p>
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 </div>
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build/ems_ca_fe_L.html View File

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+<!-- 2022-03-15 Tue 08:10 -->
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 <meta name="viewport" content="width=device-width, initial-scale=1">
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 <meta name="viewport" content="width=device-width, initial-scale=1">
7 
 <title>Pre-Quantum Electrodynamics</title>
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 <title>Pre-Quantum Electrodynamics</title>
1310 
 </summary>
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 </summary>
1311 
 <ul>
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 <ul>
1312 
 <li>
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-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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-
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-</li>
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-<li>
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 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
1318
1314
1319 
 </li>
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 </li>
1353 
 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
1354
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 </li>
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 </li>
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+<li>
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+<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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+
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+</li>
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 </ul>
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 </ul>
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 </details>
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 </details>
1630 
 function \(\phi (x)\) and the Laplace equation reads
 1630 
 function \(\phi (x)\) and the Laplace equation reads
1631 
 </p>
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 </p>
1632
1632
1633 
-<div class="eqlabel" id="org5c6cddc">
1633 
+<div class="eqlabel" id="orgc211f0a">
1634 
 <p>
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 <p>
1635 
 <a id="Lap_1d"></a><a href="./ems_ca_fe_L.html#Lap_1d"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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 <a id="Lap_1d"></a><a href="./ems_ca_fe_L.html#Lap_1d"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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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="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"/>
1637 
   <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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   <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"/>
1638 
 </svg></a>
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 </svg></a>
1639 
 </p>
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-<div class="alteqlabels" id="org5f00f86">
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+<div class="alteqlabels" id="org36bff82">
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 </div>
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 </div>
1643
1643
1652 
 <p>
 1652 
 <p>
1653 
 The solution to this is
 1653 
 The solution to this is
1654 
 </p>
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 </p>
1655 
-<div class="eqlabel" id="orgc4fd451">
1655 
+<div class="eqlabel" id="orgf5f202e">
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 <p>
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 <p>
1657 
 <a id="Lap_1d_sol"></a><a href="./ems_ca_fe_L.html#Lap_1d_sol"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1657 
 <a id="Lap_1d_sol"></a><a href="./ems_ca_fe_L.html#Lap_1d_sol"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1658 
   <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="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"/>
1659 
   <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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   <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"/>
1660 
 </svg></a>
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 </svg></a>
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 </p>
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 </p>
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-<div class="alteqlabels" id="org1f2c6cb">
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+<div class="alteqlabels" id="org25a65ef">
1663 
 <ul class="org-ul">
 1663 
 <ul class="org-ul">
1664 
 <li>Gr (3.6)</li>
 1664 
 <li>Gr (3.6)</li>
1665 
 </ul>
 1665 
 </ul>
1718 
 of two variables (here: \(x\) and \(y\)), so Laplace's
 1718 
 of two variables (here: \(x\) and \(y\)), so Laplace's
1719 
 equation now reads
 1719 
 equation now reads
1720 
 </p>
 1720 
 </p>
1721 
-<div class="eqlabel" id="orge167f78">
1721 
+<div class="eqlabel" id="org4bb6b92">
1722 
 <p>
 1722 
 <p>
1723 
 <a id="Lap_2d"></a><a href="./ems_ca_fe_L.html#Lap_2d"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1723 
 <a id="Lap_2d"></a><a href="./ems_ca_fe_L.html#Lap_2d"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1724 
   <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="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"/>
1725 
   <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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   <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"/>
1726 
 </svg></a>
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1727 
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1728 
-<div class="alteqlabels" id="org9aae751">
1728 
+<div class="alteqlabels" id="org482a454">
1729
1729
1730 
 </div>
 1730 
 </div>
1731
1731
1778 
 \(S_R({\bf r})\) of any radius \(R\) centered on this point
 1778 
 \(S_R({\bf r})\) of any radius \(R\) centered on this point
1779 
 (and of course not containing any charges),
 1779 
 (and of course not containing any charges),
1780 
 </p>
 1780 
 </p>
1781 
-<div class="eqlabel" id="orge400754">
1781 
+<div class="eqlabel" id="org96c7b40">
1782 
 <p>
 1782 
 <p>
1783 
 <a id="p_ball_avg"></a><a href="./ems_ca_fe_L.html#p_ball_avg"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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 <a id="p_ball_avg"></a><a href="./ems_ca_fe_L.html#p_ball_avg"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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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="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"/>
1785 
   <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"/>
 1785 
   <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"/>
1786 
 </svg></a>
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1787 
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-<div class="alteqlabels" id="orgf37e10c">
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1789
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1790 
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1791
1791
1797 
 \]
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 \]
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 </p>
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1799
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-<details id="org4bb0c5f">
1801 
-<summary id="orgaf332f4">
1800 
+<details id="org8644d79">
1801 
+<summary id="orgbe634df">
1802 
 <strong>Physicist's proof</strong>
 1802 
 <strong>Physicist's proof</strong>
1803 
 </summary>
 1803 
 </summary>
1804 
 <p>
 1804 
 <p>
1860 
 </p>
 1860 
 </p>
1861 
 </details>
 1861 
 </details>
1862
1862
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-<details id="orgfce7dc3">
1864 
-<summary id="org452b0d4">
1863 
+<details id="orgb4be215">
1864 
+<summary id="orgb09f919">
1865 
 <strong>Formal proof</strong>
 1865 
 <strong>Formal proof</strong>
1866 
 </summary>
 1866 
 </summary>
1867
1867
1911 
 <p>
 1911 
 <p>
1912 
 <b>Theorem</b>:
 1912 
 <b>Theorem</b>:
1913 
 </p>
 1913 
 </p>
1914 
-<div class="eqlabel" id="orgfba2553">
1914 
+<div class="eqlabel" id="org51f491a">
1915 
 <p>
 1915 
 <p>
1916 
 <a id="dfdR_intLap"></a><a href="./ems_ca_fe_L.html#dfdR_intLap"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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 <a id="dfdR_intLap"></a><a href="./ems_ca_fe_L.html#dfdR_intLap"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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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"/>
 1917 
   <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"/>
1918 
   <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"/>
 1918 
   <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"/>
1919 
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 1919 
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1920 
 </p>
 1920 
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1921 
-<div class="alteqlabels" id="orgc5c6de6">
1921 
+<div class="alteqlabels" id="org9bebf97">
1922
1922
1923 
 </div>
 1923 
 </div>
1924
1924
1971 
 of the \(f_x + f_y + f_z = 0\) condition above.
 1971 
 of the \(f_x + f_y + f_z = 0\) condition above.
1972 
 </p>
 1972 
 </p>
1973
1973
1974 
-<div class="eqlabel" id="org8465450">
1974 
+<div class="eqlabel" id="org02999c2">
1975 
 <p>
 1975 
 <p>
1976 
 <a id="Earnshaw"></a><a href="./ems_ca_fe_L.html#Earnshaw"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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 <a id="Earnshaw"></a><a href="./ems_ca_fe_L.html#Earnshaw"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1977 
   <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"/>
 1977 
   <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"/>
1978 
   <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"/>
 1978 
   <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"/>
1979 
 </svg></a>
 1979 
 </svg></a>
1980 
 </p>
 1980 
 </p>
1981 
-<div class="alteqlabels" id="org0fbdf78">
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+<div class="alteqlabels" id="org3cd554f">
1982
1982
1983 
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1984
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1986 
-<div class="info div" id="org9dd741d">
1986 
+<div class="info div" id="org8708715">
1987 
 <p>
 1987 
 <p>
1988 
 <b>Earnshaw's theorem (physical version)</b> <br>
 1988 
 <b>Earnshaw's theorem (physical version)</b> <br>
1989 
 </p>
 1989 
 </p>
2102 
 </div>
 2102 
 </div>
2103 
 <div id="postamble" class="status">
 2103 
 <div id="postamble" class="status">
2104 
 <p class="author">Author: Jean-Sébastien Caux</p>
 2104 
 <p class="author">Author: Jean-Sébastien Caux</p>
2105 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
2105 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
2106 
 <p class="validation"></p>
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 <p class="validation"></p>
2107 
 </div>
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 </div>
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-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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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"/>
 1613 
   <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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-<div class="info div" id="orga6b382e">
1616 
+<div class="info div" id="orgce63324">
1617 
 <p>
 1617 
 <p>
1618 
 <b>George Green</b>
 1618 
 <b>George Green</b>
1619 
 </p>
 1619 
 </p>
1620 
-<aside id="org8e26888">
1620 
+<aside id="org3d8c3b4">
1621 
 <p>
 1621 
 <p>
1622 
 See a <a href="https://en.wikipedia.org/wiki/George%5C_Green%5C_(mathematician)">short bio on wikipedia</a>
 1622 
 See a <a href="https://en.wikipedia.org/wiki/George%5C_Green%5C_(mathematician)">short bio on wikipedia</a>
1623 
 </p>
 1623 
 </p>
1650 
 \]
 1650 
 \]
1651 
 Substituting this in the divergence theorem gives <b>Green's first identity</b>
 1651 
 Substituting this in the divergence theorem gives <b>Green's first identity</b>
1652 
 </p>
 1652 
 </p>
1653 
-<div class="eqlabel" id="orgeb35436">
1653 
+<div class="eqlabel" id="org86176bc">
1654 
 <p>
 1654 
 <p>
1655 
 <a id="Green1"></a><a href="./ems_ca_fe_g.html#Green1"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1655 
 <a id="Green1"></a><a href="./ems_ca_fe_g.html#Green1"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1656 
   <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"/>
 1656 
   <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"/>
1657 
   <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"/>
 1657 
   <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"/>
1658 
 </svg></a>
 1658 
 </svg></a>
1659 
 </p>
 1659 
 </p>
1660 
-<div class="alteqlabels" id="org6a05402">
1660 
+<div class="alteqlabels" id="org8e9389d">
1661 
 <ul class="org-ul">
 1661 
 <ul class="org-ul">
1662 
 <li>J (1.34)</li>
 1662 
 <li>J (1.34)</li>
1663 
 </ul>
 1663 
 </ul>
1675 
 interchanged, and subtract the result, we obtain another useful result known as
 1675 
 interchanged, and subtract the result, we obtain another useful result known as
1676 
 <b>Green's second identity</b> or <b>Green's theorem</b>
 1676 
 <b>Green's second identity</b> or <b>Green's theorem</b>
1677 
 </p>
 1677 
 </p>
1678 
-<div class="eqlabel" id="org3cd71e0">
1678 
+<div class="eqlabel" id="orgee2e293">
1679 
 <p>
 1679 
 <p>
1680 
 <a id="Green2"></a><a href="./ems_ca_fe_g.html#Green2"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
 1680 
 <a id="Green2"></a><a href="./ems_ca_fe_g.html#Green2"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1681 
   <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"/>
 1681 
   <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"/>
1682 
   <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"/>
 1682 
   <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"/>
1683 
 </svg></a>
 1683 
 </svg></a>
1684 
 </p>
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 </p>
1685 
-<div class="alteqlabels" id="org2a9c8de">
1685 
+<div class="alteqlabels" id="org20f4498">
1686 
 <ul class="org-ul">
 1686 
 <ul class="org-ul">
1687 
 <li>J (1.35)</li>
 1687 
 <li>J (1.35)</li>
1688 
 </ul>
 1688 
 </ul>
1718 
 </div>
 1718 
 </div>
1719 
 <div id="postamble" class="status">
 1719 
 <div id="postamble" class="status">
1720 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1720 
 <p class="author">Author: Jean-Sébastien Caux</p>
1721 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1721 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1722 
 <p class="validation"></p>
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 <p class="validation"></p>
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 </div>
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 </div>
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1 
 <!DOCTYPE html>
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 <!DOCTYPE html>
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 <html lang="en">
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 <head>
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-<!-- 2022-03-07 Mon 20:38 -->
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+<!-- 2022-03-15 Tue 08:10 -->
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 <meta name="viewport" content="width=device-width, initial-scale=1">
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 <meta name="viewport" content="width=device-width, initial-scale=1">
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 <title>Pre-Quantum Electrodynamics</title>
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 <title>Pre-Quantum Electrodynamics</title>
1310 
 </summary>
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 </summary>
1311 
 <ul>
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 <ul>
1312 
 <li>
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 <li>
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-<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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-
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-</li>
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-<li>
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 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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 <a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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 </li>
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 </li>
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 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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 <a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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 </li>
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 </li>
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+<li>
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+<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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+
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+</li>
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 </ul>
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 </ul>
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 </details>
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 </details>
1635 
 The first term on the left-hand side vanishes since \(\Phi\) satisfies Laplace.
 1635 
 The first term on the left-hand side vanishes since \(\Phi\) satisfies Laplace.
1636 
 The right-hand side can be made to vanish if \(\Phi\) obeys either
 1636 
 The right-hand side can be made to vanish if \(\Phi\) obeys either
1637 
 </p>
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 </p>
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-<div class="eqlabel" id="orgdd9023d">
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+<div class="eqlabel" id="org13e35fc">
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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="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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   <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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-<div class="alteqlabels" id="org85a3f6c">
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+<div class="alteqlabels" id="orgccd338e">
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1648
1648
1654 
 <p>
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 <p>
1655 
 or
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 or
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 </p>
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-<div class="eqlabel" id="orgf116b4b">
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+<div class="eqlabel" id="org68968dd">
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 <p>
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 <p>
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 <a id="Neumann"></a><a href="./ems_ca_fe_uP.html#Neumann"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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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="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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   <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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-<div class="alteqlabels" id="org018d4be">
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+<div class="alteqlabels" id="org062fea2">
1665
1665
1666 
 </div>
 1666 
 </div>
1667
1667
1689 
 <p>
 1689 
 <p>
1690 
 We can thus finally state the
 1690 
 We can thus finally state the
1691 
 </p>
 1691 
 </p>
1692 
-<div class="eqlabel" id="org6b09b4c">
1692 
+<div class="eqlabel" id="org154b9ea">
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 <p>
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 <p>
1694 
 <a id="uniq_thm"></a><a href="./ems_ca_fe_uP.html#uniq_thm"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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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="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"/>
1696 
   <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"/>
 1696 
   <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"/>
1697 
 </svg></a>
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 </p>
1699 
-<div class="alteqlabels" id="org2d6da13">
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+<div class="alteqlabels" id="org43b26fb">
1700
1700
1701 
 </div>
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 </div>
1702
1702
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-<div class="core div" id="org10be721">
1704 
+<div class="core div" id="orgc742981">
1705 
 <p>
 1705 
 <p>
1706 
 <b>Uniqueness Theorem</b>
 1706 
 <b>Uniqueness Theorem</b>
1707 
 </p>
 1707 
 </p>
1740 
 and corollaries and that things are
 1740 
 and corollaries and that things are
1741 
 </p>
 1741 
 </p>
1742
1742
1743 
-<div class="info div" id="orgf42bc42">
1743 
+<div class="info div" id="org2adbb79">
1744 
 <p>
 1744 
 <p>
1745 
 <b>Comment/warning</b>: <b>uniqueness theorem on uniqueness theorems</b> <br>
 1745 
 <b>Comment/warning</b>: <b>uniqueness theorem on uniqueness theorems</b> <br>
1746 
   Do not be misled: there is a <i>unique</i> uniqueness theorem for the
 1746 
   Do not be misled: there is a <i>unique</i> uniqueness theorem for the
1767 
 </div>
 1767 
 </div>
1768 
 <div id="postamble" class="status">
 1768 
 <div id="postamble" class="status">
1769 
 <p class="author">Author: Jean-Sébastien Caux</p>
 1769 
 <p class="author">Author: Jean-Sébastien Caux</p>
1770 
-<p class="date">Created: 2022-03-07 Mon 20:38</p>
1770 
+<p class="date">Created: 2022-03-15 Tue 08:10</p>
1771 
 <p class="validation"></p>
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 <p class="validation"></p>
1772 
 </div>
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 </div>
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