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Update 2022-03-24 08:43

master
Jean-Sébastien 2 years ago
parent
1d852e7213
commit
50704eba07
100 changed files with 595 additions and 595 deletions
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build/a.html View File 

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 <meta charset="utf-8">
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1293,7 +1293,7 @@ Table of contents
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 </summary>
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 <ul>
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 <li>
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-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
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+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
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 </li>
1299 1299 
 
@@ -1637,7 +1637,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1637 1637 
 </div>
1638 1638 
 <div id="postamble" class="status">
1639 1639 
 <p class="author">Author: Jean-Sébastien Caux</p>
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-<p class="date">Created: 2022-03-22 Tue 10:52</p>
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+<p class="date">Created: 2022-03-24 Thu 08:42</p>
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 <p class="validation"></p>
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1293,7 +1293,7 @@ Table of contents
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 </summary>
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 <ul>
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 </li>
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@@ -1720,7 +1720,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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 </div>
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 <div id="postamble" class="status">
1722 1722 
 <p class="author">Author: Jean-Sébastien Caux</p>
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 <title>Pre-Quantum Electrodynamics</title>
 
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 </summary>
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@@ -1637,7 +1637,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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 </div>
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 <div id="postamble" class="status">
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 <p class="validation"></p>
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 </div>
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 <div id="postamble" class="status">
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 <div id="postamble" class="status">
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 </div>
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 <div id="postamble" class="status">
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 <p class="author">Author: Jean-Sébastien Caux</p>
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-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
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 </li>
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 The usual Cartesian coordinates relate to spherical coordinates
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 according to
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-<div class="eqlabel" id="org309b2e9">
1626 
+<div class="eqlabel" id="org763df52">
1627 1627 
 <p>
1628 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">
1629 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 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 1631 
 </svg></a>
1632 1632 
 </p>
1633 
-<div class="alteqlabels" id="org7a6a40a">
1633 
+<div class="alteqlabels" id="orgb59869b">
1634 1634
1635 1635 
 </div>
1636 1636 
 
@@ -1655,14 +1655,14 @@ A generic vector can be expressed as
1655 1655 
 where the explicit relation between spherical and
1656 1656 
 Cartesian unit vectors is
1657 1657 
 </p>
1658 
-<div class="eqlabel" id="org43b8824">
1658 
+<div class="eqlabel" id="org923fbbc">
1659 1659 
 <p>
1660 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">
1661 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 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 1663 
 </svg></a>
1664 1664 
 </p>
1665 
-<div class="alteqlabels" id="org054cde6">
1665 
+<div class="alteqlabels" id="org2051229">
1666 1666
1667 1667 
 </div>
1668 1668 
 
@@ -1685,14 +1685,14 @@ and \(\hat{\boldsymbol \varphi} (\theta, \varphi)\).
1685 1685 
 <p>
1686 1686 
 An infinitesimal displacement \(d{\bf l}\) can be written as
1687 1687 
 </p>
1688 
-<div class="eqlabel" id="org0afd234">
1688 
+<div class="eqlabel" id="orgb032a0d">
1689 1689 
 <p>
1690 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">
1691 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 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 1693 
 </svg></a>
1694 1694 
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1695 
-<div class="alteqlabels" id="orgce4cca6">
1695 
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1696 1696
1697 1697 
 </div>
1698 1698 
 
@@ -1708,14 +1708,14 @@ d{\bf l} = dr ~\hat{\boldsymbol r} + r d\theta ~\hat{\boldsymbol \theta} + r\sin
1708 1708 
 <p>
1709 1709 
 Infinitesimal volume element:
1710 1710 
 </p>
1711 
-<div class="eqlabel" id="org260d3f6">
1711 
+<div class="eqlabel" id="orgf42297c">
1712 1712 
 <p>
1713 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">
1714 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 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 1716 
 </svg></a>
1717 1717 
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1718 
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1718 
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1719 1719
1720 1720 
 </div>
1721 1721 
 
@@ -1736,14 +1736,14 @@ Infinitesimal surface element:  depends on situation.
1736 1736 
 <div id="outline-container-c_m_cs_sph_grad" class="outline-6">
1737 1737 
 <h6 id="c_m_cs_sph_grad"><a href="#c_m_cs_sph_grad">Gradient</a></h6>
1738 1738 
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1739 
-<div class="eqlabel" id="org0a3cd9b">
1739 
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1740 1740 
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1741 1741 
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1742 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"/>
1743 1743 
   <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"/>
1744 1744 
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1745 1745 
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1746 
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1747 1747
1748 1748 
 </div>
1749 1749 
 
@@ -1760,14 +1760,14 @@ Infinitesimal surface element:  depends on situation.
1760 1760 
 <div id="outline-container-c_m_cs_sph_div" class="outline-6">
1761 1761 
 <h6 id="c_m_cs_sph_div"><a href="#c_m_cs_sph_div">Divergence</a></h6>
1762 1762 
 <div class="outline-text-6" id="text-c_m_cs_sph_div">
1763 
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1763 
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1764 1764 
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1765 1765 
 <a id="sph_div"></a><a href="./c_m_cs_sph.html#sph_div"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1766 1766 
   <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"/>
1767 1767 
   <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"/>
1768 1768 
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1769 1769 
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1770 
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1770 
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1771 1771
1772 1772 
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1773 1773 
 
@@ -1784,14 +1784,14 @@ Infinitesimal surface element:  depends on situation.
1784 1784 
 <div id="outline-container-c_m_cs_sph_curl" class="outline-6">
1785 1785 
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1786 1786 
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1787 
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1787 
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1788 1788 
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1789 1789 
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1790 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"/>
1791 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"/>
1792 1792 
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1793 1793 
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1794 
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1794 
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1795 1795
1796 1796 
 </div>
1797 1797 
 
@@ -1810,14 +1810,14 @@ Infinitesimal surface element:  depends on situation.
1810 1810 
 <div id="outline-container-c_m_cs_sph_lap" class="outline-6">
1811 1811 
 <h6 id="c_m_cs_sph_lap"><a href="#c_m_cs_sph_lap">Laplacian</a></h6>
1812 1812 
 <div class="outline-text-6" id="text-c_m_cs_sph_lap">
1813 
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1815 1815 
 <a id="sph_Lap"></a><a href="./c_m_cs_sph.html#sph_Lap"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1816 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"/>
1817 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"/>
1818 1818 
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1819 1819 
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1820 
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1820 
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1821 1821
1822 1822 
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1823 1823 
 
@@ -1851,7 +1851,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1851 1851 
 </div>
1852 1852 
 <div id="postamble" class="status">
1853 1853 
 <p class="author">Author: Jean-Sébastien Caux</p>
1854 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1854 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1855 1855 
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1856 1856 
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1857 1857

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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1642,7 +1642,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1642 1642 
 </div>
1643 1643 
 <div id="postamble" class="status">
1644 1644 
 <p class="author">Author: Jean-Sébastien Caux</p>
1645 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1645 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1646 1646 
 <p class="validation"></p>
1647 1647 
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1648 1648

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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1644,7 +1644,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1644 1644 
 </div>
1645 1645 
 <div id="postamble" class="status">
1646 1646 
 <p class="author">Author: Jean-Sébastien Caux</p>
1647 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1647 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1648 1648 
 <p class="validation"></p>
1649 1649 
 </div>
1650 1650

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3 3 
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4 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1615,9 +1615,9 @@ Table of contents
1615 1615 
 <div class="outline-text-5" id="text-c_m_dc_d2">
1616 1616 
 </div>
1617 1617
1618 
-<div id="outline-container-org78664e4" class="outline-6">
1619 
-<h6 id="org78664e4"><a href="#org78664e4">Divergence of gradient</a></h6>
1620 
-<div class="outline-text-6" id="text-org78664e4">
1618 
+<div id="outline-container-org873f1bb" class="outline-6">
1619 
+<h6 id="org873f1bb"><a href="#org873f1bb">Divergence of gradient</a></h6>
1620 
+<div class="outline-text-6" id="text-org873f1bb">
1621 1621 
 <p>
1622 1622 
 \({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the <b>Laplacian</b> of the scalar field \(T\).
1623 1623 
 The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components
 
@@ -1626,44 +1626,44 @@ given by the Laplacian of the corresponding vector elements.
1626 1626 
 </div>
1627 1627 
 </div>
1628 1628
1629 
-<div id="outline-container-orgecb53d1" class="outline-6">
1630 
-<h6 id="orgecb53d1"><a href="#orgecb53d1">Curl of a gradient</a></h6>
1631 
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1629 
+<div id="outline-container-orged6374e" class="outline-6">
1630 
+<h6 id="orged6374e"><a href="#orged6374e">Curl of a gradient</a></h6>
1631 
+<div class="outline-text-6" id="text-orged6374e">
1632 1632 
 <p>
1633 1633 
 This always vanishes.
1634 1634 
 </p>
1635 1635 
 </div>
1636 1636 
 </div>
1637 1637
1638 
-<div id="outline-container-orgdeab960" class="outline-6">
1639 
-<h6 id="orgdeab960"><a href="#orgdeab960">Gradient of the divergence</a></h6>
1640 
-<div class="outline-text-6" id="text-orgdeab960">
1638 
+<div id="outline-container-orgcf8bc6c" class="outline-6">
1639 
+<h6 id="orgcf8bc6c"><a href="#orgcf8bc6c">Gradient of the divergence</a></h6>
1640 
+<div class="outline-text-6" id="text-orgcf8bc6c">
1641 1641 
 <p>
1642 1642 
 \({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics.  No special name.
1643 1643 
 </p>
1644 1644 
 </div>
1645 1645 
 </div>
1646 1646
1647 
-<div id="outline-container-org11519c8" class="outline-6">
1648 
-<h6 id="org11519c8"><a href="#org11519c8">Divergence of a curl</a></h6>
1649 
-<div class="outline-text-6" id="text-org11519c8">
1647 
+<div id="outline-container-orgc1dc4c7" class="outline-6">
1648 
+<h6 id="orgc1dc4c7"><a href="#orgc1dc4c7">Divergence of a curl</a></h6>
1649 
+<div class="outline-text-6" id="text-orgc1dc4c7">
1650 1650 
 <p>
1651 1651 
 This always vanishes.
1652 1652 
 </p>
1653 1653 
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1654 1654 
 </div>
1655 1655
1656 
-<div id="outline-container-org8bb99ae" class="outline-6">
1657 
-<h6 id="org8bb99ae"><a href="#org8bb99ae">Curl of curl</a></h6>
1658 
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1659 
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1656 
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1657 
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1658 
+<div class="outline-text-6" id="text-org766740d">
1659 
+<div class="eqlabel" id="orga94a832">
1660 1660 
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1661 1661 
 <a id="curlcurl"></a><a href="./c_m_dc_d2.html#curlcurl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1662 1662 
   <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"/>
1663 1663 
   <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"/>
1664 1664 
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1665 1665 
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1666 
-<div class="alteqlabels" id="orgb8e88ad">
1666 
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1667 1667
1668 1668 
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1669 1669 
 
@@ -1694,7 +1694,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1694 1694 
 </div>
1695 1695 
 <div id="postamble" class="status">
1696 1696 
 <p class="author">Author: Jean-Sébastien Caux</p>
1697 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1697 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1698 1698 
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1699 1699 
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7 7 
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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
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1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1643,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1643 1643 
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1644 1644 
 <div id="postamble" class="status">
1645 1645 
 <p class="author">Author: Jean-Sébastien Caux</p>
1646 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1646 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1647 1647 
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1648 1648 
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7 7 
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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1640,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1640 1640 
 </div>
1641 1641 
 <div id="postamble" class="status">
1642 1642 
 <p class="author">Author: Jean-Sébastien Caux</p>
1643 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1643 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1644 1644 
 <p class="validation"></p>
1645 1645 
 </div>
1646 1646

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@@ -1,7 +1,7 @@
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1664,7 +1664,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1664 1664 
 </div>
1665 1665 
 <div id="postamble" class="status">
1666 1666 
 <p class="author">Author: Jean-Sébastien Caux</p>
1667 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1667 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1668 1668 
 <p class="validation"></p>
1669 1669 
 </div>
1670 1670

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@@ -1,7 +1,7 @@
1 1 
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2 2 
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3 3 
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4 
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4 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1622,14 +1622,14 @@ explicited as follows:
1622 1622 
 <p>
1623 1623 
 <b>Gradient of a product</b>:
1624 1624 
 </p>
1625 
-<div class="eqlabel" id="org89b1475">
1625 
+<div class="eqlabel" id="orge6e6a80">
1626 1626 
 <p>
1627 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 1628 
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1629 1629 
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1630 1630 
 </svg></a>
1631 1631 
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1632 
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1632 
+<div class="alteqlabels" id="orgf9842ae">
1633 1633 
 <ul class="org-ul">
1634 1634 
 <li>Gr (3)</li>
1635 1635 
 <li>W (1-111)</li>
 
@@ -1649,14 +1649,14 @@ explicited as follows:
1649 1649 
 <p>
1650 1650 
 <b>Gradient of a scalar product</b>:
1651 1651 
 </p>
1652 
-<div class="eqlabel" id="orgdcd103a">
1652 
+<div class="eqlabel" id="orgb8556f5">
1653 1653 
 <p>
1654 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 1655 
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1656 1656 
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1657 1657 
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1658 1658 
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1659 
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1659 
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1660 1660 
 <ul class="org-ul">
1661 1661 
 <li>Gr (4)</li>
1662 1662 
 <li>W (1-112)</li>
 
@@ -1676,14 +1676,14 @@ explicited as follows:
1676 1676 
 <p>
1677 1677 
 <b>Divergence of a product</b>:
1678 1678 
 </p>
1679 
-<div class="eqlabel" id="org16aa48b">
1679 
+<div class="eqlabel" id="org3da56b1">
1680 1680 
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1681 1681 
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1682 1682 
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1683 1683 
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1684 1684 
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1685 1685 
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1686 
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1686 
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1687 1687 
 <ul class="org-ul">
1688 1688 
 <li>Gr (5)</li>
1689 1689 
 <li>W (1-115)</li>
 
@@ -1703,14 +1703,14 @@ explicited as follows:
1703 1703 
 <p>
1704 1704 
 <b>Divergence of a cross product</b>:
1705 1705 
 </p>
1706 
-<div class="eqlabel" id="orga48567e">
1706 
+<div class="eqlabel" id="orgbfaa39d">
1707 1707 
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1708 1708 
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1709 1709 
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1710 1710 
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1711 1711 
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1712 1712 
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1713 
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1713 
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1714 1714 
 <ul class="org-ul">
1715 1715 
 <li>Gr (6)</li>
1716 1716 
 <li>W (1-116)</li>
 
@@ -1730,14 +1730,14 @@ explicited as follows:
1730 1730 
 <p>
1731 1731 
 <b>Curl of a product</b>:
1732 1732 
 </p>
1733 
-<div class="eqlabel" id="org3dec348">
1733 
+<div class="eqlabel" id="org5b32772">
1734 1734 
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1735 1735 
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1736 1736 
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1737 1737 
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1738 1738 
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1739 1739 
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1740 
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1740 
+<div class="alteqlabels" id="org22a6f09">
1741 1741 
 <ul class="org-ul">
1742 1742 
 <li>Gr (7)</li>
1743 1743 
 <li>W (1-118)</li>
 
@@ -1757,14 +1757,14 @@ explicited as follows:
1757 1757 
 <p>
1758 1758 
 <b>Curl of a cross product</b>:
1759 1759 
 </p>
1760 
-<div class="eqlabel" id="orga9f7b57">
1760 
+<div class="eqlabel" id="org81448bf">
1761 1761 
 <p>
1762 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">
1763 1763 
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1764 1764 
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1765 1765 
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1766 1766 
 </p>
1767 
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1767 
+<div class="alteqlabels" id="org5de3f60">
1768 1768 
 <ul class="org-ul">
1769 1769 
 <li>Gr (8)</li>
1770 1770 
 <li>W (1-119)</li>
 
@@ -1805,7 +1805,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1805 1805 
 </div>
1806 1806 
 <div id="postamble" class="status">
1807 1807 
 <p class="author">Author: Jean-Sébastien Caux</p>
1808 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1808 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1809 1809 
 <p class="validation"></p>
1810 1810 
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1811 1811

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@@ -1,7 +1,7 @@
1 1 
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2 2 
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3 3 
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4 
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6 6 
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7 7 
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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1639,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1639 1639 
 </div>
1640 1640 
 <div id="postamble" class="status">
1641 1641 
 <p class="author">Author: Jean-Sébastien Caux</p>
1642 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1642 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1643 1643 
 <p class="validation"></p>
1644 1644 
 </div>
1645 1645

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@@ -1,7 +1,7 @@
1 1 
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2 2 
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3 3 
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4 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1651,7 +1651,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1651 1651 
 </div>
1652 1652 
 <div id="postamble" class="status">
1653 1653 
 <p class="author">Author: Jean-Sébastien Caux</p>
1654 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1654 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1655 1655 
 <p class="validation"></p>
1656 1656 
 </div>
1657 1657

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@@ -1,7 +1,7 @@
1 1 
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2 2 
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3 3 
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4 
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4 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1636,14 +1636,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox:
1636 1636 
 More generally,
1637 1637 
 </p>
1638 1638
1639 
-<div class="eqlabel" id="orga7ea793">
1639 
+<div class="eqlabel" id="orgcf3b8a6">
1640 1640 
 <p>
1641 1641 
 <a id="divdel"></a><a href="./c_m_dd_3d.html#divdel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1642 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 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 1644 
 </svg></a>
1645 1645 
 </p>
1646 
-<div class="alteqlabels" id="org2afb48e">
1646 
+<div class="alteqlabels" id="orgd295657">
1647 1647 
 <ul class="org-ul">
1648 1648 
 <li>Gr (1.100)</li>
1649 1649 
 </ul>
 
@@ -1662,14 +1662,14 @@ More generally,
1662 1662 
 Since
1663 1663 
 </p>
1664 1664
1665 
-<div class="eqlabel" id="org0791e6b">
1665 
+<div class="eqlabel" id="org8e3f2ff">
1666 1666 
 <p>
1667 1667 
 <a id="div1or"></a><a href="./c_m_dd_3d.html#div1or"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1668 1668 
   <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"/>
1669 1669 
   <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 1670 
 </svg></a>
1671 1671 
 </p>
1672 
-<div class="alteqlabels" id="org5b1c286">
1672 
+<div class="alteqlabels" id="orgd9d8d13">
1673 1673 
 <ul class="org-ul">
1674 1674 
 <li>Gr (1.101)</li>
1675 1675 
 </ul>
 
@@ -1685,14 +1685,14 @@ Since
1685 1685 
 <p>
1686 1686 
 we have that
1687 1687 
 </p>
1688 
-<div class="eqlabel" id="org72632aa">
1688 
+<div class="eqlabel" id="org4584082">
1689 1689 
 <p>
1690 1690 
 <a id="Lap1or"></a><a href="./c_m_dd_3d.html#Lap1or"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1691 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 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 1693 
 </svg></a>
1694 1694 
 </p>
1695 
-<div class="alteqlabels" id="orgc9783d9">
1695 
+<div class="alteqlabels" id="org4002cb1">
1696 1696 
 <ul class="org-ul">
1697 1697 
 <li>Gr (1.102)</li>
1698 1698 
 </ul>
 
@@ -1724,7 +1724,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1724 1724 
 </div>
1725 1725 
 <div id="postamble" class="status">
1726 1726 
 <p class="author">Author: Jean-Sébastien Caux</p>
1727 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1727 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1728 1728 
 <p class="validation"></p>
1729 1729 
 </div>
1730 1730

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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1657,7 +1657,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1657 1657 
 </div>
1658 1658 
 <div id="postamble" class="status">
1659 1659 
 <p class="author">Author: Jean-Sébastien Caux</p>
1660 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1660 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1661 1661 
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1662 1662 
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1663 1663

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1622 1622 
 \[
1623 1623 
 {\int_{\bf a}^{\bf b}}_{\cal P} {\bf v} \cdot d{\bf l}
 
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1646 1646 
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 \int_{\cal S} {\bf v} \cdot d{\bf a}
 
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 \[
1676 1676 
 \int_{\cal V} T d\tau
 
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1712 1712 
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1713 1713 
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1714 
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1714 
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1299 1299 
 
@@ -1657,7 +1657,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1657 1657 
 </div>
1658 1658 
 <div id="postamble" class="status">
1659 1659 
 <p class="author">Author: Jean-Sébastien Caux</p>
1660 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1660 
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1661 1661 
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1662 1662 
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1663 1663

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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
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1294 1294 
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1295 1295 
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1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1662,7 +1662,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1662 1662 
 </div>
1663 1663 
 <div id="postamble" class="status">
1664 1664 
 <p class="author">Author: Jean-Sébastien Caux</p>
1665 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1665 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1666 1666 
 <p class="validation"></p>
1667 1667 
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1668 1668

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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
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1294 1294 
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1295 1295 
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1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1663,7 +1663,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1663 1663 
 </div>
1664 1664 
 <div id="postamble" class="status">
1665 1665 
 <p class="author">Author: Jean-Sébastien Caux</p>
1666 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1666 
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1667 1667 
 <p class="validation"></p>
1668 1668 
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1669 1669

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@@ -1293,7 +1293,7 @@ Table of contents
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1294 1294 
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1295 1295 
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1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
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1297 1297
1298 1298 
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1299 1299 
 
@@ -1614,8 +1614,8 @@ Table of contents
1614 1614 
 </svg></a><span class="headline-id">emd</span></h2>
1615 1615
1616 1616 
 <div class="outline-text-2" id="text-emd">
1617 
-<details class="prereq" id="orgeb3e60e">
1618 
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1617 
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1618 
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1619 1619 
 Prerequisites
1620 1620 
 </summary>
1621 1621 
 <ul class="org-ul">
 
@@ -1624,8 +1624,8 @@ Prerequisites
1624 1624 
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1625 1625 
 </details>
1626 1626
1627 
-<details class="objectives" id="org11fe244">
1628 
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1627 
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1628 
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1629 1629 
 Objectives
1630 1630 
 </summary>
1631 1631 
 <ul class="org-ul">
 
@@ -1666,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1666 1666 
 </div>
1667 1667 
 <div id="postamble" class="status">
1668 1668 
 <p class="author">Author: Jean-Sébastien Caux</p>
1669 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1669 
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1670 1670 
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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
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1295 1295 
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1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1640,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1640 1640 
 </div>
1641 1641 
 <div id="postamble" class="status">
1642 1642 
 <p class="author">Author: Jean-Sébastien Caux</p>
1643 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1643 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1644 1644 
 <p class="validation"></p>
1645 1645 
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1646 1646

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4 
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1644,18 +1644,18 @@ Empirically:  the changing magnetic field induces an electric current around
1644 1644 
 the circuit. This current is really driven by an electric field having a component
1645 1645 
 along the wire.  The line integral of this field is called the
1646 1646 
 </p>
1647 
-<div class="core div" id="orgc1ab4b8">
1647 
+<div class="core div" id="orgb0fd5b4">
1648 1648 
 <p>
1649 1649 
 <b>Electromotive force (or electromotance)</b>,
1650 1650 
 </p>
1651 
-<div class="eqlabel" id="org1ce0462">
1651 
+<div class="eqlabel" id="orgb0e7cde">
1652 1652 
 <p>
1653 1653 
 <a id="elmofo"></a><a href="./emd_Fl_Fl.html#elmofo"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1654 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 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 1656 
 </svg></a>
1657 1657 
 </p>
1658 
-<div class="alteqlabels" id="orge2b1b8b">
1658 
+<div class="alteqlabels" id="org516a196">
1659 1659 
 <ul class="org-ul">
1660 1660 
 <li>Gr (7.9)</li>
1661 1661 
 </ul>
 
@@ -1680,14 +1680,14 @@ The precise statement associated to Faraday's observations
1680 1680 
 is that the electromotive force is proportional
1681 1681 
 to the rate of change of the magnetic flux,
1682 1682 
 </p>
1683 
-<div class="eqlabel" id="orgb4b21bf">
1683 
+<div class="eqlabel" id="orgab619b1">
1684 1684 
 <p>
1685 1685 
 <a id="Fl_flux"></a><a href="./emd_Fl_Fl.html#Fl_flux"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1686 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 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 1688 
 </svg></a>
1689 1689 
 </p>
1690 
-<div class="alteqlabels" id="org4e36847">
1690 
+<div class="alteqlabels" id="org969857c">
1691 1691 
 <ul class="org-ul">
1692 1692 
 <li>Gr (7.14)</li>
1693 1693 
 </ul>
 
@@ -1702,18 +1702,18 @@ to the rate of change of the magnetic flux,
1702 1702 
 \]
1703 1703 
 so we obtain
1704 1704 
 </p>
1705 
-<div class="core div" id="org3b354eb">
1705 
+<div class="core div" id="org5419feb">
1706 1706 
 <p>
1707 1707 
 <b>Faraday's law</b> (integral form <i>N.B.: for a stationary loop</i>)
1708 1708 
 </p>
1709 
-<div class="eqlabel" id="org3323690">
1709 
+<div class="eqlabel" id="org4696e9f">
1710 1710 
 <p>
1711 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 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 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 1714 
 </svg></a>
1715 1715 
 </p>
1716 
-<div class="alteqlabels" id="orgc991325">
1716 
+<div class="alteqlabels" id="org439dfef">
1717 1717 
 <ul class="org-ul">
1718 1718 
 <li>Gr (7.15)</li>
1719 1719 
 </ul>
 
@@ -1737,15 +1737,15 @@ for any loop (on a wire or not). Using Stokes' theorem,
1737 1737 
 \]
1738 1738 
 we obtain
1739 1739 
 </p>
1740 
-<div class="core div" id="org5eb686c">
1741 
-<div class="eqlabel" id="org7ca80cd">
1740 
+<div class="core div" id="org49ffd42">
1741 
+<div class="eqlabel" id="org5ecebad">
1742 1742 
 <p>
1743 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 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 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 1746 
 </svg></a>
1747 1747 
 </p>
1748 
-<div class="alteqlabels" id="org859ac4a">
1748 
+<div class="alteqlabels" id="org6a61522">
1749 1749 
 <ul class="org-ul">
1750 1750 
 <li>Gr (7.16)</li>
1751 1751 
 </ul>
 
@@ -1790,7 +1790,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1790 1790 
 </div>
1791 1791 
 <div id="postamble" class="status">
1792 1792 
 <p class="author">Author: Jean-Sébastien Caux</p>
1793 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1793 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1794 1794 
 <p class="validation"></p>
1795 1795 
 </div>
1796 1796

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1 1 
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2 2 
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3 3 
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4 
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4 
+<!-- 2022-03-24 Thu 08:42 -->
5 5 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1639,14 +1639,14 @@ W = \frac{1}{2} I \oint {\bf A} \cdot d{\bf l} = \frac{1}{2} \oint ({\bf A} \cdo
1639 1639 
 \]
1640 1640 
 Generalization to volume currents:
1641 1641 
 </p>
1642 
-<div class="eqlabel" id="org3409c0a">
1642 
+<div class="eqlabel" id="orge9580af">
1643 1643 
 <p>
1644 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 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 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 1647 
 </svg></a>
1648 1648 
 </p>
1649 
-<div class="alteqlabels" id="org84f1d51">
1649 
+<div class="alteqlabels" id="orgbe0c60a">
1650 1650 
 <ul class="org-ul">
1651 1651 
 <li>Gr (7.31)</li>
1652 1652 
 </ul>
 
@@ -1681,15 +1681,15 @@ W = \frac{1}{2\mu_0} \left[ \int_{\cal V} d\tau B^2 - \int_{\cal V} d\tau {\bold
1681 1681 
 \]
1682 1682 
 We can integrate over all space:  after neglecting boundary terms (assuming fields fall to zero at infinity), we are left with
1683 1683 
 </p>
1684 
-<div class="core div" id="orgced7757">
1685 
-<div class="eqlabel" id="orgd899514">
1684 
+<div class="core div" id="org9173f2d">
1685 
+<div class="eqlabel" id="org717e156">
1686 1686 
 <p>
1687 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 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 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 1690 
 </svg></a>
1691 1691 
 </p>
1692 
-<div class="alteqlabels" id="orgf7c92a2">
1692 
+<div class="alteqlabels" id="org0d5a0b3">
1693 1693 
 <ul class="org-ul">
1694 1694 
 <li>Gr (7.34)</li>
1695 1695 
 </ul>
 
@@ -1718,7 +1718,7 @@ W_{mag} &amp;= \frac{1}{2} \int d\tau ~({\bf A} \cdot {\bf J}) &amp;= \frac{1}{2
1718 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 1719 
 </p>
1720 1720
1721 
-<div class="example div" id="orge558476">
1721 
+<div class="example div" id="org34d5a7d">
1722 1722 
 <p>
1723 1723 
 <b>Example: energy in coaxial cable</b>
1724 1724 
 </p>
 
@@ -1768,7 +1768,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1768 1768 
 </div>
1769 1769 
 <div id="postamble" class="status">
1770 1770 
 <p class="author">Author: Jean-Sébastien Caux</p>
1771 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1771 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1772 1772 
 <p class="validation"></p>
1773 1773 
 </div>
1774 1774

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@@ -1,7 +1,7 @@
1 1 
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2 2 
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3 3 
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4 
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5 5 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1641,14 +1641,14 @@ so
1641 1641 
 \]
1642 1642 
 and we can write the mutual inductance as the <b>Neumann formula</b>,
1643 1643 
 </p>
1644 
-<div class="eqlabel" id="org3ec4c18">
1644 
+<div class="eqlabel" id="org1f70ff2">
1645 1645 
 <p>
1646 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 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 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 1649 
 </svg></a>
1650 1650 
 </p>
1651 
-<div class="alteqlabels" id="org7ff8d1e">
1651 
+<div class="alteqlabels" id="org57de753">
1652 1652 
 <ul class="org-ul">
1653 1653 
 <li>Gr (7.22)</li>
1654 1654 
 </ul>
 
@@ -1665,14 +1665,14 @@ M_{21} = \frac{\mu_0}{4\pi} \oint_{{\cal P}_1} \oint_{{\cal P}_2} \frac{d{\bf l}
1665 1665 
 Two things:
1666 1666 
 first, \(M_{21}\) is purely geometrical.  Second,
1667 1667 
 </p>
1668 
-<div class="eqlabel" id="orge7f5edd">
1668 
+<div class="eqlabel" id="org43668cc">
1669 1669 
 <p>
1670 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 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 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 1673 
 </svg></a>
1674 1674 
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1675 
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1675 
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1676 1676 
 <ul class="org-ul">
1677 1677 
 <li>Gr (7.23)</li>
1678 1678 
 </ul>
 
@@ -1687,7 +1687,7 @@ M_{12} = M_{21}
1687 1687 
 \]
1688 1688 
 </p>
1689 1689
1690 
-<div class="example div" id="org1bcebe3">
1690 
+<div class="example div" id="orgecf7176">
1691 1691 
 <p>
1692 1692 
 <b>Example: solenoid in solenoid</b>
1693 1693 
 </p>
 
@@ -1735,14 +1735,14 @@ What if we vary current in loop 1?  Flux in 2 will vary.  Induces EMF in loop 2:
1735 1735 
 \]
1736 1736 
 Changing current also induces EMF in the source loop itself:
1737 1737 
 </p>
1738 
-<div class="eqlabel" id="orgdaaac4e">
1738 
+<div class="eqlabel" id="org2512f69">
1739 1739 
 <p>
1740 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">
1741 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"/>
1742 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"/>
1743 1743 
 </svg></a>
1744 1744 
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1745 
-<div class="alteqlabels" id="orgb8faf94">
1745 
+<div class="alteqlabels" id="org080d683">
1746 1746 
 <ul class="org-ul">
1747 1747 
 <li>Gr (7.25)</li>
1748 1748 
 </ul>
 
@@ -1765,7 +1765,7 @@ Inductance:  measured in <b>henries</b> (\(H\)).  \(H = V s/A\).
1765 1765 
 </p>
1766 1766
1767 1767
1768 
-<div class="example div" id="org11544ba">
1768 
+<div class="example div" id="orgce99b7c">
1769 1769 
 <p>
1770 1770 
 <b>Example: self-inductance of toroidal coil</b>
1771 1771 
 </p>
 
@@ -1804,7 +1804,7 @@ Inductance (like capacitance) is intrinsically positive.  Use Lenz law.
1804 1804 
 Think of <i>back EMF</i>.
1805 1805 
 </p>
1806 1806
1807 
-<div class="example div" id="org52373ca">
1807 
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1808 1808 
 <p>
1809 1809 
 <b>Example: circuit</b>
1810 1810 
 </p>
 
@@ -1853,7 +1853,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1853 1853 
 </div>
1854 1854 
 <div id="postamble" class="status">
1855 1855 
 <p class="author">Author: Jean-Sébastien Caux</p>
1856 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1856 
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1857 1857 
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1858 1858 
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1859 1859

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6 6 
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7 7 
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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1637,7 +1637,7 @@ law in integral form:
1637 1637
1638 1638
1639 1639
1640 
-<div class="example div" id="orgfcd2e61">
1640 
+<div class="example div" id="org2badd21">
1641 1641 
 <p>
1642 1642 
 <b>Example: loop with time-dependent flux</b>
1643 1643 
 </p>
 
@@ -1664,7 +1664,7 @@ Increasing \({\bf B}\):  clockwise (viewed from above) \({\bf E}\) from Lenz.
1664 1664 
 </div>
1665 1665
1666 1666
1667 
-<div class="example div" id="orgafdd47c">
1667 
+<div class="example div" id="org6994570">
1668 1668 
 <p>
1669 1669 
 <b>Example: wheel with charged rim traversed by flux</b>
1670 1670 
 </p>
 
@@ -1705,7 +1705,7 @@ called the <b>quasistatic</b> approximation, and works provided we deal with
1705 1705 
 <i>slow enough</i> phenomena.
1706 1706 
 </p>
1707 1707
1708 
-<div class="example div" id="org6631fcc">
1708 
+<div class="example div" id="orgddce4f1">
1709 1709 
 <p>
1710 1710 
 <b>Example: field from wire with time-dependent current</b>
1711 1711 
 </p>
 
@@ -1764,7 +1764,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1764 1764 
 </div>
1765 1765 
 <div id="postamble" class="status">
1766 1766 
 <p class="author">Author: Jean-Sébastien Caux</p>
1767 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1767 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1768 1768 
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1769 1769 
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1770 1770

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7 7 
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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1640,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1640 1640 
 </div>
1641 1641 
 <div id="postamble" class="status">
1642 1642 
 <p class="author">Author: Jean-Sébastien Caux</p>
1643 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1643 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1644 1644 
 <p class="validation"></p>
1645 1645 
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1646 1646

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3 3 
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4 
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7 7 
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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1616,18 +1616,18 @@ Table of contents
1616 1616 
 <p>
1617 1617 
 Full set of equations for the electromagnetic field:
1618 1618 
 </p>
1619 
-<div class="core div" id="org71429ce">
1619 
+<div class="core div" id="org96eda60">
1620 1620 
 <p>
1621 1621 
 <b>Maxwell's equations</b> <i>(in vacuum)</i>
1622 1622 
 </p>
1623 
-<div class="eqlabel" id="org92aa36d">
1623 
+<div class="eqlabel" id="org99bc512">
1624 1624 
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1625 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 1626 
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1627 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 1628 
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1629 1629 
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1630 
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1630 
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1631 1631
1632 1632 
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1633 1633 
 
@@ -1644,7 +1644,7 @@ Full set of equations for the electromagnetic field:
1644 1644 
 <p>
1645 1645 
 Complement:
1646 1646 
 </p>
1647 
-<div class="core div" id="org93d1ad6">
1647 
+<div class="core div" id="org8bd62bd">
1648 1648 
 <p>
1649 1649 
 Force law <a href="./ems_ms_lf_pc.html#LorFo">LorFo</a>
1650 1650 
 \[
 
@@ -1667,15 +1667,15 @@ take divergence of \((iv)\) and use \((i)\).
1667 1667 
 <p>
1668 1668 
 Better way of writing:  all fields on left, all sources on right,
1669 1669 
 </p>
1670 
-<div class="core div" id="org82c9083">
1671 
-<div class="eqlabel" id="org1583626">
1670 
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1671 
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1672 1672 
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1673 1673 
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1674 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 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 1676 
 </svg></a>
1677 1677 
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1678 
-<div class="alteqlabels" id="orgfc0a16c">
1678 
+<div class="alteqlabels" id="org3e69b1d">
1679 1679 
 <ul class="org-ul">
1680 1680 
 <li>Gr (7.42)</li>
1681 1681 
 </ul>
 
@@ -1713,7 +1713,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1713 1713 
 </div>
1714 1714 
 <div id="postamble" class="status">
1715 1715 
 <p class="author">Author: Jean-Sébastien Caux</p>
1716 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1716 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1717 1717 
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1718 1718 
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1719 1719

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3 3 
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4 
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6 6 
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7 7 
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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
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1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
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1299 1299 
 
@@ -1622,15 +1622,15 @@ The term which should be zero (but isn't) in <a href="./emd_Me_ebM.html#divcurlB
1622 1622 
 \]
1623 1623 
 The extra term would thus be eliminated if we were to put
1624 1624 
 </p>
1625 
-<div class="core div" id="orgfb244bd">
1626 
-<div class="eqlabel" id="org02e358a">
1625 
+<div class="core div" id="org65d0a3c">
1626 
+<div class="eqlabel" id="org2343c92">
1627 1627 
 <p>
1628 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 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 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 1631 
 </svg></a>
1632 1632 
 </p>
1633 
-<div class="alteqlabels" id="orgf266507">
1633 
+<div class="alteqlabels" id="org09fe122">
1634 1634 
 <ul class="org-ul">
1635 1635 
 <li>Gr (7.36)</li>
1636 1636 
 </ul>
 
@@ -1659,18 +1659,18 @@ Real confirmation of Maxwell's theory:  1888, Hertz's experiments on propagation
1659 1659 
 <p>
1660 1660 
 Maxwell baptized this term the
1661 1661 
 </p>
1662 
-<div class="core div" id="orgb3e1688">
1662 
+<div class="core div" id="org9a1c650">
1663 1663 
 <p>
1664 1664 
 <b>Displacement current</b>
1665 1665 
 </p>
1666 
-<div class="eqlabel" id="org5bc6ac9">
1666 
+<div class="eqlabel" id="orgea96ca5">
1667 1667 
 <p>
1668 1668 
 <a id="Jd"></a><a href="./emd_Me_dc.html#Jd"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1669 1669 
   <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"/>
1670 1670 
   <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"/>
1671 1671 
 </svg></a>
1672 1672 
 </p>
1673 
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1673 
+<div class="alteqlabels" id="org147727a">
1674 1674 
 <ul class="org-ul">
1675 1675 
 <li>Gr (7.37)</li>
1676 1676 
 </ul>
 
@@ -1724,7 +1724,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1724 1724 
 </div>
1725 1725 
 <div id="postamble" class="status">
1726 1726 
 <p class="author">Author: Jean-Sébastien Caux</p>
1727 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1727 
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1728 1728 
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1729 1729 
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1730 1730

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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
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1295 1295 
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1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1630,14 +1630,14 @@ Fatal inconsistency:  div of curl must always vanish.  Check on \((iii)\):
1630 1630 
 \]
1631 1631 
 But:  try same with \((iv)\):
1632 1632 
 </p>
1633 
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1633 
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1634 1634 
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1635 1635 
 <a id="divcurlB"></a><a href="./emd_Me_ebM.html#divcurlB"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1636 1636 
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1637 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"/>
1638 1638 
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1639 1639 
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1640 
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1640 
+<div class="alteqlabels" id="orga04de46">
1641 1641 
 <ul class="org-ul">
1642 1642 
 <li>Gr (7.35)</li>
1643 1643 
 </ul>
 
@@ -1682,7 +1682,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1682 1682 
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1683 1683 
 <div id="postamble" class="status">
1684 1684 
 <p class="author">Author: Jean-Sébastien Caux</p>
1685 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1685 
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1686 1686 
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1687 1687 
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1688 1688

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1293 1293 
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1294 1294 
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1295 1295 
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1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
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1297 1297
1298 1298 
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1299 1299 
 
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1663 1663 
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1664 1664 
 <div id="postamble" class="status">
1665 1665 
 <p class="author">Author: Jean-Sébastien Caux</p>
1666 
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1666 
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1667 1667 
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1668 1668 
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1293 1293 
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1294 1294 
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1295 1295 
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1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
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1299 1299 
 
@@ -1614,8 +1614,8 @@ Table of contents
1614 1614 
 </svg></a><span class="headline-id">emd.ce</span></h3>
1615 1615
1616 1616 
 <div class="outline-text-3" id="text-emd_ce">
1617 
-<details class="prereq" id="org4aea2a7">
1618 
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1617 
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1618 
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1619 1619 
 Prerequisites
1620 1620 
 </summary>
1621 1621 
 <ul class="org-ul">
 
@@ -1623,8 +1623,8 @@ Prerequisites
1623 1623 
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1624 1624 
 </details>
1625 1625
1626 
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1627 
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1626 
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1627 
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1628 1628 
 Objectives
1629 1629 
 </summary>
1630 1630 
 <ul class="org-ul">
 
@@ -1662,7 +1662,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1662 1662 
 </div>
1663 1663 
 <div id="postamble" class="status">
1664 1664 
 <p class="author">Author: Jean-Sébastien Caux</p>
1665 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1665 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1666 1666 
 <p class="validation"></p>
1667 1667 
 </div>
1668 1668

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@@ -1,7 +1,7 @@
1 1 
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2 2 
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4 
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6 6 
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7 7 
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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1616,18 +1616,18 @@ Table of contents
1616 1616 
 <p>
1617 1617 
 The angular momentum of EM fields is directly given by
1618 1618 
 </p>
1619 
-<div class="main div" id="org8982815">
1619 
+<div class="main div" id="orgc2d5343">
1620 1620 
 <p>
1621 1621 
 <b>Angular momentum of EM fields</b>
1622 1622 
 </p>
1623 
-<div class="eqlabel" id="org8297ce4">
1623 
+<div class="eqlabel" id="orgd4d0682">
1624 1624 
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1625 1625 
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1626 1626 
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1627 1627 
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1628 1628 
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1629 1629 
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1630 
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1630 
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1631 1631
1632 1632 
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1633 1633 
 
@@ -1661,7 +1661,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1661 1661 
 </div>
1662 1662 
 <div id="postamble" class="status">
1663 1663 
 <p class="author">Author: Jean-Sébastien Caux</p>
1664 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1664 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1665 1665 
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1666 1666 
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1667 1667

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4 
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6 6 
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7 7 
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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1635,7 +1635,7 @@ This means that
1635 1635 
 \]
1636 1636 
 Since this is true for any volume, we have (re)derived the
1637 1637 
 </p>
1638 
-<div class="core div" id="org1d21244">
1638 
+<div class="core div" id="orgc746662">
1639 1639 
 <p>
1640 1640 
 <b>Continuity equation</b> <a href="./ems_ms_ce.html#conteq">conteq</a>
1641 1641 
 \[
 
@@ -1675,7 +1675,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1675 1675 
 </div>
1676 1676 
 <div id="postamble" class="status">
1677 1677 
 <p class="author">Author: Jean-Sébastien Caux</p>
1678 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1678 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1679 1679 
 <p class="validation"></p>
1680 1680 
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1681 1681

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@@ -1,7 +1,7 @@
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2 2 
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3 3 
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4 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1628,18 +1628,18 @@ in which the first integral can be interpreted as the momentum stored in the EM
1628 1628 
 <p>
1629 1629 
 This is thus simply a conservation law for momentum, with
1630 1630 
 </p>
1631 
-<div class="main div" id="orgbd3d67d">
1631 
+<div class="main div" id="org9032358">
1632 1632 
 <p>
1633 1633 
 <b>Momentum density in the EM fields</b>
1634 1634 
 </p>
1635 
-<div class="eqlabel" id="org2b66251">
1635 
+<div class="eqlabel" id="org0649d73">
1636 1636 
 <p>
1637 1637 
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1638 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 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 1640 
 </svg></a>
1641 1641 
 </p>
1642 
-<div class="alteqlabels" id="org006aaa0">
1642 
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1643 1643
1644 1644 
 </div>
1645 1645 
 
@@ -1655,18 +1655,18 @@ This is thus simply a conservation law for momentum, with
1655 1655 
 <p>
1656 1656 
 In a region in which the mechanical momentum is not changing due to external influences, we then have the
1657 1657 
 </p>
1658 
-<div class="main div" id="orgd647d42">
1658 
+<div class="main div" id="org72dfdb8">
1659 1659 
 <p>
1660 1660 
 <b>Continuity equation for EM momentum</b>
1661 1661 
 </p>
1662 
-<div class="eqlabel" id="org84364f8">
1662 
+<div class="eqlabel" id="org89a6b5f">
1663 1663 
 <p>
1664 1664 
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1665 1665 
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1666 1666 
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1667 1667 
 </svg></a>
1668 1668 
 </p>
1669 
-<div class="alteqlabels" id="org03c5d23">
1669 
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1670 1670
1671 1671 
 </div>
1672 1672 
 
@@ -1699,7 +1699,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1699 1699 
 </div>
1700 1700 
 <div id="postamble" class="status">
1701 1701 
 <p class="author">Author: Jean-Sébastien Caux</p>
1702 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1702 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1703 1703 
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1704 1704 
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1705 1705

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@@ -1,7 +1,7 @@
1 1 
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2 2 
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3 3 
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4 
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4 
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5 5 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1667,18 +1667,18 @@ and similarly for \({\boldsymbol B}\). We thus get
1667 1667 
 <p>
1668 1668 
 This expression can be greatly simplified by introducing the
1669 1669 
 </p>
1670 
-<div class="main div" id="org63be843">
1670 
+<div class="main div" id="orge483101">
1671 1671 
 <p>
1672 1672 
 <b>Maxwell stress tensor</b>
1673 1673 
 </p>
1674 
-<div class="eqlabel" id="org7c75f9d">
1674 
+<div class="eqlabel" id="org42e9f16">
1675 1675 
 <p>
1676 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 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 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 1679 
 </svg></a>
1680 1680 
 </p>
1681 
-<div class="alteqlabels" id="orgd84a618">
1681 
+<div class="alteqlabels" id="orgf3e9098">
1682 1682
1683 1683 
 </div>
1684 1684 
 
@@ -1699,18 +1699,18 @@ The element \(T_{ij}\) represents the force per unit area in the \(i\) direction
1699 1699 
 <p>
1700 1700 
 We then obtain the
1701 1701 
 </p>
1702 
-<div class="main div" id="orgfeaab67">
1702 
+<div class="main div" id="org3664d29">
1703 1703 
 <p>
1704 1704 
 <b>EM force per unit volume</b>
1705 1705 
 </p>
1706 
-<div class="eqlabel" id="org9fc5057">
1706 
+<div class="eqlabel" id="org6136353">
1707 1707 
 <p>
1708 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 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 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 1711 
 </svg></a>
1712 1712 
 </p>
1713 
-<div class="alteqlabels" id="orgecbfdae">
1713 
+<div class="alteqlabels" id="org7273a9f">
1714 1714
1715 1715 
 </div>
1716 1716 
 
@@ -1726,18 +1726,18 @@ We then obtain the
1726 1726 
 <p>
1727 1727 
 where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
1728 1728 
 </p>
1729 
-<div class="main div" id="org86829bd">
1729 
+<div class="main div" id="org2a4f340">
1730 1730 
 <p>
1731 1731 
 <b>Total force on charges in volume</b>
1732 1732 
 </p>
1733 
-<div class="eqlabel" id="org2370cf3">
1733 
+<div class="eqlabel" id="org0298272">
1734 1734 
 <p>
1735 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 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 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 1738 
 </svg></a>
1739 1739 
 </p>
1740 
-<div class="alteqlabels" id="orgc55376c">
1740 
+<div class="alteqlabels" id="org1024eff">
1741 1741
1742 1742 
 </div>
1743 1743 
 
@@ -1770,7 +1770,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1770 1770 
 </div>
1771 1771 
 <div id="postamble" class="status">
1772 1772 
 <p class="author">Author: Jean-Sébastien Caux</p>
1773 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1773 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1774 1774 
 <p class="validation"></p>
1775 1775 
 </div>
1776 1776

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@@ -1,7 +1,7 @@
1 1 
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2 2 
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3 3 
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4 
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4 
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5 5 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1636,14 +1636,14 @@ done by EM forces?  From Lorentz force law:
1636 1636 
 Really, we're looking at a small volume element \(d\tau\) carrying charge \(\rho d\tau\), moving
1637 1637 
 at velocity \({\bf v}\) such that \({\bf J} = \rho {\bf v}\).  Thus,
1638 1638 
 </p>
1639 
-<div class="eqlabel" id="orge2567ac">
1639 
+<div class="eqlabel" id="org0c693e0">
1640 1640 
 <p>
1641 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">
1642 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 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 1644 
 </svg></a>
1645 1645 
 </p>
1646 
-<div class="alteqlabels" id="org4d05a80">
1646 
+<div class="alteqlabels" id="orge34a4b7">
1647 1647 
 <ul class="org-ul">
1648 1648 
 <li>Gr (8.6)</li>
1649 1649 
 </ul>
 
@@ -1683,18 +1683,18 @@ so we get
1683 1683 
 Substituting this in <a href="./emd_ce_poy.html#dWdt_intEJ">dWdt_intEJ</a> and using the divergence theorem,
1684 1684 
 we obtain
1685 1685 
 </p>
1686 
-<div class="main div" id="org36db96e">
1686 
+<div class="main div" id="orgb7d7869">
1687 1687 
 <p>
1688 1688 
 <b>Poynting's theorem</b>
1689 1689 
 </p>
1690 
-<div class="eqlabel" id="org21ce6b8">
1690 
+<div class="eqlabel" id="org1a4b16b">
1691 1691 
 <p>
1692 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">
1693 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"/>
1694 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"/>
1695 1695 
 </svg></a>
1696 1696 
 </p>
1697 
-<div class="alteqlabels" id="org09a112b">
1697 
+<div class="alteqlabels" id="org4feadff">
1698 1698 
 <ul class="org-ul">
1699 1699 
 <li>Gr (8.9)</li>
1700 1700 
 </ul>
 
@@ -1720,18 +1720,18 @@ energy is carried by EM fields out of \({\cal V}\) across its boundary surface.
1720 1720 
 <p>
1721 1721 
 Energy per unit time, per unit area carried by EM fields: given by the
1722 1722 
 </p>
1723 
-<div class="core div" id="org0bf3f6c">
1723 
+<div class="core div" id="orgdc56ce9">
1724 1724 
 <p>
1725 1725 
 <b>Poynting vector</b>
1726 1726 
 </p>
1727 
-<div class="eqlabel" id="org327095e">
1727 
+<div class="eqlabel" id="orga61d997">
1728 1728 
 <p>
1729 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">
1730 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"/>
1731 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"/>
1732 1732 
 </svg></a>
1733 1733 
 </p>
1734 
-<div class="alteqlabels" id="org7684a8e">
1734 
+<div class="alteqlabels" id="orgb7a4cb9">
1735 1735 
 <ul class="org-ul">
1736 1736 
 <li>Gr (8.10)</li>
1737 1737 
 </ul>
 
@@ -1750,18 +1750,18 @@ Energy per unit time, per unit area carried by EM fields: given by the
1750 1750 
 <p>
1751 1751 
 We can thus express Poynting's theorem more compactly:
1752 1752 
 </p>
1753 
-<div class="core div" id="org4255191">
1753 
+<div class="core div" id="orgd48d57f">
1754 1754 
 <p>
1755 1755 
 <b>Poynting's theorem</b> (integral form)
1756 1756 
 </p>
1757 
-<div class="eqlabel" id="orgd0524d3">
1757 
+<div class="eqlabel" id="org32c9daf">
1758 1758 
 <p>
1759 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">
1760 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 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 1762 
 </svg></a>
1763 1763 
 </p>
1764 
-<div class="alteqlabels" id="org8acaa16">
1764 
+<div class="alteqlabels" id="org59d2761">
1765 1765 
 <ul class="org-ul">
1766 1766 
 <li>Gr (8.11)</li>
1767 1767 
 </ul>
 
@@ -1780,18 +1780,18 @@ We can thus express Poynting's theorem more compactly:
1780 1780 
 <p>
1781 1781 
 where we have defined the total
1782 1782 
 </p>
1783 
-<div class="core div" id="org85e77e2">
1783 
+<div class="core div" id="org5e10531">
1784 1784 
 <p>
1785 1785 
 <b>Energy in electromagnetic fields</b>
1786 1786 
 </p>
1787 
-<div class="eqlabel" id="org7d4ae3d">
1787 
+<div class="eqlabel" id="orgad6f50f">
1788 1788 
 <p>
1789 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">
1790 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"/>
1791 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"/>
1792 1792 
 </svg></a>
1793 1793 
 </p>
1794 
-<div class="alteqlabels" id="org7a55647">
1794 
+<div class="alteqlabels" id="orga3df7bc">
1795 1795 
 <ul class="org-ul">
1796 1796 
 <li>Gr (8.5)</li>
1797 1797 
 </ul>
 
@@ -1820,18 +1820,18 @@ Then,
1820 1820 
 \]
1821 1821 
 so we get the
1822 1822 
 </p>
1823 
-<div class="core div" id="orgbdb096d">
1823 
+<div class="core div" id="orgb7a11e9">
1824 1824 
 <p>
1825 1825 
 <b>Poynting theorem</b> (differential form)
1826 1826 
 </p>
1827 
-<div class="eqlabel" id="org9c56948">
1827 
+<div class="eqlabel" id="orga2e5c79">
1828 1828 
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1829 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">
1830 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"/>
1831 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"/>
1832 1832 
 </svg></a>
1833 1833 
 </p>
1834 
-<div class="alteqlabels" id="orgdfe3994">
1834 
+<div class="alteqlabels" id="org0cf780d">
1835 1835 
 <ul class="org-ul">
1836 1836 
 <li>Gr (8.14)</li>
1837 1837 
 </ul>
 
@@ -1854,7 +1854,7 @@ and has a similar for to the continuity equation
1854 1854
1855 1855
1856 1856
1857 
-<div class="example div" id="org15b725b">
1857 
+<div class="example div" id="org1083447">
1858 1858 
 <p>
1859 1859 
 <b>Example:  Joule heating</b>
1860 1860 
 </p>
 
@@ -1911,7 +1911,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1911 1911 
 </div>
1912 1912 
 <div id="postamble" class="status">
1913 1913 
 <p class="author">Author: Jean-Sébastien Caux</p>
1914 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1914 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1915 1915 
 <p class="validation"></p>
1916 1916 
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1917 1917

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@@ -1,7 +1,7 @@
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4 
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7 7 
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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1614,8 +1614,8 @@ Table of contents
1614 1614 
 </svg></a><span class="headline-id">emd.emw</span></h3>
1615 1615
1616 1616 
 <div class="outline-text-3" id="text-emd_emw">
1617 
-<details class="prereq" id="org17a8cef">
1618 
-<summary id="org45e360a">
1617 
+<details class="prereq" id="org92d8d0c">
1618 
+<summary id="orgb97228e">
1619 1619 
 Prerequisites
1620 1620 
 </summary>
1621 1621 
 <ul class="org-ul">
 
@@ -1624,8 +1624,8 @@ Prerequisites
1624 1624 
 </ul>
1625 1625 
 </details>
1626 1626
1627 
-<details class="objectives" id="org91615eb">
1628 
-<summary id="orgec366bf">
1627 
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1628 
+<summary id="orge645495">
1629 1629 
 Objectives
1630 1630 
 </summary>
1631 1631 
 <ul class="org-ul">
 
@@ -1659,7 +1659,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1659 1659 
 </div>
1660 1660 
 <div id="postamble" class="status">
1661 1661 
 <p class="author">Author: Jean-Sébastien Caux</p>
1662 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1662 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1663 1663 
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1664 1664 
 </div>
1665 1665

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1 1 
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4 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1644,18 +1644,18 @@ so for a monochromatic EM plan wave,
1644 1644 
 \]
1645 1645 
 or more succinctly:
1646 1646 
 </p>
1647 
-<div class="main div" id="orgdf4b8bc">
1647 
+<div class="main div" id="orga5c9575">
1648 1648 
 <p>
1649 1649 
 <b>Poynting vector of a monochromatic EM wave</b>
1650 1650 
 </p>
1651 
-<div class="eqlabel" id="org3913519">
1651 
+<div class="eqlabel" id="org72a5a32">
1652 1652 
 <p>
1653 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 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 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 1656 
 </svg></a>
1657 1657 
 </p>
1658 
-<div class="alteqlabels" id="org8b59092">
1658 
+<div class="alteqlabels" id="org7e5d7a7">
1659 1659 
 <ul class="org-ul">
1660 1660 
 <li>Gr (9.57)</li>
1661 1661 
 </ul>
 
@@ -1678,7 +1678,7 @@ This has a transparent physical interpretation: the energy density \(u\) flows w
1678 1678 
 <p>
1679 1679 
 Similary, we get the
1680 1680 
 </p>
1681 
-<div class="main div" id="org21912bb">
1681 
+<div class="main div" id="orgd6eada9">
1682 1682 
 <p>
1683 1683 
 <b>Momentum density of a monochromatic EM wave</b>
1684 1684 
 \[
 
@@ -1729,7 +1729,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1729 1729 
 </div>
1730 1730 
 <div id="postamble" class="status">
1731 1731 
 <p class="author">Author: Jean-Sébastien Caux</p>
1732 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1732 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1733 1733 
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1734 1734 
 </div>
1735 1735

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3 3 
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4 
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4 
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6 6 
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7 7 
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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1636,14 +1636,14 @@ From Faraday:  \({\boldsymbol \nabla} \times {\bf E} = -\partial {\bf B}/\partia
1636 1636 
 \]
1637 1637 
 so \({\bf E}\) and \({\bf B}\) are mutually perpendicular, and
1638 1638 
 </p>
1639 
-<div class="eqlabel" id="org619c73f">
1639 
+<div class="eqlabel" id="org44bb1e2">
1640 1640 
 <p>
1641 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 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 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 1644 
 </svg></a>
1645 1645 
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1646 
-<div class="alteqlabels" id="orgaad096d">
1646 
+<div class="alteqlabels" id="org0064233">
1647 1647 
 <ul class="org-ul">
1648 1648 
 <li>Gr (9.47)</li>
1649 1649 
 </ul>
 
@@ -1662,18 +1662,18 @@ B_0 = \frac{k}{\omega} E_0 = \frac{1}{c} E_0.
1662 1662 
 Generalizing to propagation in the direction of an arbitrary wavevector
1663 1663 
 \({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the
1664 1664 
 </p>
1665 
-<div class="core div" id="org9c77e17">
1665 
+<div class="core div" id="org83aeece">
1666 1666 
 <p>
1667 1667 
 <b>E and B fields for a monochromatic EM plane wave</b>
1668 1668 
 </p>
1669 
-<div class="eqlabel" id="org54506e2">
1669 
+<div class="eqlabel" id="org82e71ea">
1670 1670 
 <p>
1671 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 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 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 1674 
 </svg></a>
1675 1675 
 </p>
1676 
-<div class="alteqlabels" id="orgb384c79">
1676 
+<div class="alteqlabels" id="orgb1b901e">
1677 1677 
 <ul class="org-ul">
1678 1678 
 <li>Gr (9.49)</li>
1679 1679 
 </ul>
 
@@ -1724,7 +1724,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1724 1724 
 </div>
1725 1725 
 <div id="postamble" class="status">
1726 1726 
 <p class="author">Author: Jean-Sébastien Caux</p>
1727 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1727 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1728 1728 
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1729 1729 
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1730 1730

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2 2 
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3 3 
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4 
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7 7 
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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1641,18 +1641,18 @@ These take the form of coupled first-order partial differential equations for \(
1641 1641 
 Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\),
1642 1642 
 we get the
1643 1643 
 </p>
1644 
-<div class="core div" id="org0b3ebdc">
1644 
+<div class="core div" id="orge19d256">
1645 1645 
 <p>
1646 1646 
 <b>Wave equations for electric and magnetic fields in vacuum</b>
1647 1647 
 </p>
1648 
-<div class="eqlabel" id="orge465a34">
1648 
+<div class="eqlabel" id="orgff19c15">
1649 1649 
 <p>
1650 1650 
 <a id="WaveEq"></a><a href="./emd_emw_we.html#WaveEq"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1651 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 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 1653 
 </svg></a>
1654 1654 
 </p>
1655 
-<div class="alteqlabels" id="org3ff5ea0">
1655 
+<div class="alteqlabels" id="orga8c7165">
1656 1656 
 <ul class="org-ul">
1657 1657 
 <li>Gr (9.41)</li>
1658 1658 
 </ul>
 
@@ -1714,7 +1714,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1714 1714 
 </div>
1715 1715 
 <div id="postamble" class="status">
1716 1716 
 <p class="author">Author: Jean-Sébastien Caux</p>
1717 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1717 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1718 1718 
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1719 1719 
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1720 1720

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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
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1294 1294 
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1295 1295 
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1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
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1297 1297
1298 1298 
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1299 1299 
 
@@ -1638,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1638 1638 
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1639 1639 
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1640 1640 
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1641 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1641 
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1642 1642 
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1643 1643 
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1644 1644

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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
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1294 1294 
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1295 1295 
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1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
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1299 1299 
 
@@ -1638,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1638 1638 
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1639 1639 
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1640 1640 
 <p class="author">Author: Jean-Sébastien Caux</p>
1641 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1641 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1642 1642 
 <p class="validation"></p>
1643 1643 
 </div>
1644 1644

+ 21
- 21
build/emdm_Me_Mem.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
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3 3 
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4 
-<!-- 2022-03-22 Tue 10:52 -->
4 
+<!-- 2022-03-24 Thu 08:42 -->
5 5 
 <meta charset="utf-8">
6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1648,18 +1648,18 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
1648 1648 
 \]
1649 1649 
 We therefore have the
1650 1650 
 </p>
1651 
-<div class="core div" id="org64396a0">
1651 
+<div class="core div" id="orgc00904a">
1652 1652 
 <p>
1653 1653 
 <b>Polarization current density</b>
1654 1654 
 </p>
1655 
-<div class="eqlabel" id="orgf37fc4f">
1655 
+<div class="eqlabel" id="org3742df8">
1656 1656 
 <p>
1657 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">
1658 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"/>
1659 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"/>
1660 1660 
 </svg></a>
1661 1661 
 </p>
1662 
-<div class="alteqlabels" id="org12af1ad">
1662 
+<div class="alteqlabels" id="org6073635">
1663 1663 
 <ul class="org-ul">
1664 1664 
 <li>Gr (7.48)</li>
1665 1665 
 </ul>
 
@@ -1683,7 +1683,7 @@ the polarization current is the result of linear motion of charge when
1683 1683 
 polarization changes).  We can check consistency with the continuity equation
1684 1684 
 associated to the conservation of bound charges:
1685 1685 
 </p>
1686 
-<aside id="org0322793">
1686 
+<aside id="orgff2c75f">
1687 1687 
 <p>
1688 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.
1689 1689 
 </p>
 
@@ -1705,15 +1705,15 @@ Changing magnetization does not lead to analogous accumulation of charge and cur
1705 1705 
 In view of this:  total charge density can be separated into 2 parts,
1706 1706 
 <b>free</b> and <b>bound</b>:
1707 1707 
 </p>
1708 
-<div class="main div" id="org56027a6">
1709 
-<div class="eqlabel" id="org6fd2aed">
1708 
+<div class="main div" id="org7866a36">
1709 
+<div class="eqlabel" id="orge94a27e">
1710 1710 
 <p>
1711 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">
1712 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 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 1714 
 </svg></a>
1715 1715 
 </p>
1716 
-<div class="alteqlabels" id="org9e03f13">
1716 
+<div class="alteqlabels" id="orgbb91da6">
1717 1717 
 <ul class="org-ul">
1718 1718 
 <li>Gr (7.49)</li>
1719 1719 
 </ul>
 
@@ -1733,15 +1733,15 @@ In view of this:  total charge density can be separated into 2 parts,
1733 1733 
 and current can be separated into three parts, <b>free</b>, <b>bound</b> and
1734 1734 
 <b>polarization</b>:
1735 1735 
 </p>
1736 
-<div class="main div" id="org96443a3">
1737 
-<div class="eqlabel" id="orga6b8e4a">
1736 
+<div class="main div" id="org765eeee">
1737 
+<div class="eqlabel" id="org49803b2">
1738 1738 
 <p>
1739 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">
1740 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"/>
1741 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"/>
1742 1742 
 </svg></a>
1743 1743 
 </p>
1744 
-<div class="alteqlabels" id="orgddcfaf9">
1744 
+<div class="alteqlabels" id="orgc87ffc7">
1745 1745 
 <ul class="org-ul">
1746 1746 
 <li>Gr (7.50)</li>
1747 1747 
 </ul>
 
@@ -1767,7 +1767,7 @@ Gauss's law:  can be rewritten
1767 1767 
 \]
1768 1768 
 where (as in static case)
1769 1769 
 </p>
1770 
-<div class="core div" id="orgdbd92f1">
1770 
+<div class="core div" id="org91a1e38">
1771 1771 
 <p>
1772 1772 
 \[
1773 1773 
 {\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
 
@@ -1788,7 +1788,7 @@ or
1788 1788 
 \]
1789 1789 
 where as before
1790 1790 
 </p>
1791 
-<div class="core div" id="org7839f4c">
1791 
+<div class="core div" id="org0c2e84f">
1792 1792 
 <p>
1793 1793 
 \[
1794 1794 
 {\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
 
@@ -1806,18 +1806,18 @@ bound parts, since they don't involve \(\rho\) or \({\bf J}\).
1806 1806 
 <p>
1807 1807 
 In terms of free charges and currents, we thus get
1808 1808 
 </p>
1809 
-<div class="core div" id="org6597738">
1809 
+<div class="core div" id="orgeeb0b35">
1810 1810 
 <p>
1811 1811 
 <b>Maxwell's equations</b> <i>(in matter)</i>
1812 1812 
 </p>
1813 
-<div class="eqlabel" id="orgffcbbe9">
1813 
+<div class="eqlabel" id="orgb9e6675">
1814 1814 
 <p>
1815 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">
1816 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"/>
1817 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"/>
1818 1818 
 </svg></a>
1819 1819 
 </p>
1820 
-<div class="alteqlabels" id="org5c6c62d">
1820 
+<div class="alteqlabels" id="org6367cef">
1821 1821 
 <ul class="org-ul">
1822 1822 
 <li>Gr (7.55)</li>
1823 1823 
 </ul>
 
@@ -1847,15 +1847,15 @@ This must all be complemented by the <b>constitutive relations</b> giving \({\bf
1847 1847 
 in terms of \({\bf E}\) and \({\bf B}\).
1848 1848 
 For the restricted case of linear media:
1849 1849 
 </p>
1850 
-<div class="main div" id="org2d4f8dd">
1851 
-<div class="eqlabel" id="org03c06a4">
1850 
+<div class="main div" id="orgd78e0b5">
1851 
+<div class="eqlabel" id="org86e1a29">
1852 1852 
 <p>
1853 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">
1854 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"/>
1855 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"/>
1856 1856 
 </svg></a>
1857 1857 
 </p>
1858 
-<div class="alteqlabels" id="org45b4fff">
1858 
+<div class="alteqlabels" id="org7ea2c8c">
1859 1859 
 <ul class="org-ul">
1860 1860 
 <li>Gr (7.56,7.57)</li>
1861 1861 
 </ul>
 
@@ -1898,7 +1898,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1898 1898 
 </div>
1899 1899 
 <div id="postamble" class="status">
1900 1900 
 <p class="author">Author: Jean-Sébastien Caux</p>
1901 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1901 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1902 1902 
 <p class="validation"></p>
1903 1903 
 </div>
1904 1904

+ 19
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@@ -1,7 +1,7 @@
1 1 
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2 2 
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3 3 
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4 
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4 
+<!-- 2022-03-24 Thu 08:42 -->
5 5 
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6 6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1616,18 +1616,18 @@ Table of contents
1616 1616 
 <p>
1617 1617 
 Discontinuities between different media, deduced from
1618 1618 
 </p>
1619 
-<div class="core div" id="orgb7aedd2">
1619 
+<div class="core div" id="org876a60b">
1620 1620 
 <p>
1621 1621 
 <b>Maxwell's equations</b> <i>(in matter)</i>, <i>integral form</i>
1622 1622 
 </p>
1623 
-<div class="eqlabel" id="org6c267cc">
1623 
+<div class="eqlabel" id="orga776629">
1624 1624 
 <p>
1625 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 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 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 1628 
 </svg></a>
1629 1629 
 </p>
1630 
-<div class="alteqlabels" id="org8952960">
1630 
+<div class="alteqlabels" id="org6c05186">
1631 1631
1632 1632 
 </div>
1633 1633 
 
@@ -1645,15 +1645,15 @@ Discontinuities between different media, deduced from
1645 1645 
 Applying \((i)\) to wafer-thin Gaussian pillbox straddling boundary between 2 materials:
1646 1646 
 \({\bf D}_1 \cdot {\bf a} - {\bf D}_2 \cdot {\bf a} = \sigma_f a\) so
1647 1647 
 </p>
1648 
-<div class="main div" id="orgf92353e">
1649 
-<div class="eqlabel" id="orgb9a9a3d">
1648 
+<div class="main div" id="org8d20540">
1649 
+<div class="eqlabel" id="org7926d65">
1650 1650 
 <p>
1651 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 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 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 1654 
 </svg></a>
1655 1655 
 </p>
1656 
-<div class="alteqlabels" id="orga6c409d">
1656 
+<div class="alteqlabels" id="orge1fcf9c">
1657 1657 
 <ul class="org-ul">
1658 1658 
 <li>Gr (7.60)</li>
1659 1659 
 </ul>
 
@@ -1672,9 +1672,9 @@ D^{\perp}_1 - D^{\perp}_2 = \sigma_f
1672 1672 
 <p>
1673 1673 
 Same reasoning applied to \((ii)\) gives <a href="./ems_ms_vp_mbc.html#Bdisc">Bdisc</a>
1674 1674 
 </p>
1675 
-<div class="main div" id="orga22be6d">
1676 
-<div class="eqlabel" id="orge0d8321">
1677 
-<div class="alteqlabels" id="orgb94a831">
1675 
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1676 
+<div class="eqlabel" id="orgddc00f3">
1677 
+<div class="alteqlabels" id="org80b4003">
1678 1678
1679 1679 
 </div>
1680 1680 
 
@@ -1690,7 +1690,7 @@ B^{\perp}_1 - B^{\perp}_2 = 0
1690 1690 
 For \((iii)\):  Amperian loop straddling surface:  \({\bf E}_1 \cdot {\bf l} - {\bf E}_2 \cdot {\bf l} =
1691 1691 
 -\frac{d}{dt} \int_{\cal S} {\bf B} \cdot d{\bf a}\).  Limit of small loop:  flux vanishes, therefore
1692 1692 
 </p>
1693 
-<div class="main div" id="org55a1d42">
1693 
+<div class="main div" id="org2dd2bbb">
1694 1694 
 <p>
1695 1695 
 \[
1696 1696 
 {\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0
 
@@ -1704,7 +1704,7 @@ No volume current can contribute, but a surface current can.  Can write
1704 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 1705 
 and thus (as we got before in <a href="./emsm_msm_H_A.html#Hdisc">Hdisc</a>)
1706 1706 
 </p>
1707 
-<div class="main div" id="org4a2ee70">
1707 
+<div class="main div" id="orgd223e2f">
1708 1708 
 <p>
1709 1709 
 \[
1710 1710 
 {\bf H}_1^{\parallel} - {\bf H}_2^{\parallel} = {\bf K}_f \times \hat{\bf n}
 
@@ -1720,15 +1720,15 @@ These are the general boundary conditions for electrodynamics.
1720 1720 
 <p>
1721 1721 
 In case of linear media:  can be expressed in terms of \({\bf E}\) and \({\bf B}\) alone:
1722 1722 
 </p>
1723 
-<div class="main div" id="org9e90e47">
1724 
-<div class="eqlabel" id="org8bbc727">
1723 
+<div class="main div" id="orgaafde6e">
1724 
+<div class="eqlabel" id="orge514e83">
1725 1725 
 <p>
1726 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">
1727 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"/>
1728 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"/>
1729 1729 
 </svg></a>
1730 1730 
 </p>
1731 
-<div class="alteqlabels" id="orga51b616">
1731 
+<div class="alteqlabels" id="org786e6ec">
1732 1732 
 <ul class="org-ul">
1733 1733 
 <li>Gr (7.64)</li>
1734 1734 
 </ul>
 
@@ -1748,14 +1748,14 @@ In case of linear media:  can be expressed in terms of \({\bf E}\) and \({\bf B}
1748 1748 
 <p>
1749 1749 
 If there is no free charge and no free current at boundary:
1750 1750 
 </p>
1751 
-<div class="eqlabel" id="orgf07dc3e">
1751 
+<div class="eqlabel" id="orga74521d">
1752 1752 
 <p>
1753 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">
1754 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"/>
1755 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"/>
1756 1756 
 </svg></a>
1757 1757 
 </p>
1758 
-<div class="alteqlabels" id="orgdd74046">
1758 
+<div class="alteqlabels" id="orgac68624">
1759 1759 
 <ul class="org-ul">
1760 1760 
 <li>Gr (7.64)</li>
1761 1761 
 </ul>
 
@@ -1792,7 +1792,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1792 1792 
 </div>
1793 1793 
 <div id="postamble" class="status">
1794 1794 
 <p class="author">Author: Jean-Sébastien Caux</p>
1795 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1795 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1796 1796 
 <p class="validation"></p>
1797 1797 
 </div>
1798 1798

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@@ -1,7 +1,7 @@
1 1 
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2 2 
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3 3 
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4 
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4 
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5 5 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1614,8 +1614,8 @@ Table of contents
1614 1614 
 </svg></a><span class="headline-id">emdm.emwm</span></h3>
1615 1615
1616 1616 
 <div class="outline-text-3" id="text-emdm_emwm">
1617 
-<details class="prereq" id="org5c9f997">
1618 
-<summary id="orgcac0ec5">
1617 
+<details class="prereq" id="org499ec58">
1618 
+<summary id="org504cdbb">
1619 1619 
 Prerequisites
1620 1620 
 </summary>
1621 1621 
 <ul class="org-ul">
 
@@ -1625,8 +1625,8 @@ Prerequisites
1625 1625 
 </ul>
1626 1626 
 </details>
1627 1627
1628 
-<details class="objectives" id="orgde042c7">
1629 
-<summary id="org6ca3c40">
1628 
+<details class="objectives" id="org76e4576">
1629 
+<summary id="org77d80f7">
1630 1630 
 Objectives
1631 1631 
 </summary>
1632 1632 
 <ul class="org-ul">
 
@@ -1666,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1666 1666 
 </div>
1667 1667 
 <div id="postamble" class="status">
1668 1668 
 <p class="author">Author: Jean-Sébastien Caux</p>
1669 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1669 
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1670 1670 
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1671 1671 
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1672 1672

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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
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1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1637,7 +1637,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1637 1637 
 </div>
1638 1638 
 <div id="postamble" class="status">
1639 1639 
 <p class="author">Author: Jean-Sébastien Caux</p>
1640 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1640 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1641 1641 
 <p class="validation"></p>
1642 1642 
 </div>
1643 1643

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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1733,7 +1733,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1733 1733 
 </div>
1734 1734 
 <div id="postamble" class="status">
1735 1735 
 <p class="author">Author: Jean-Sébastien Caux</p>
1736 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1736 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1737 1737 
 <p class="validation"></p>
1738 1738 
 </div>
1739 1739

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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1632,15 +1632,15 @@ For linear medium:
1632 1632 
 \]
1633 1633 
 If the medium is homogeneous (no spatial dependence of \(\varepsilon\) or \(\mu\)),
1634 1634 
 </p>
1635 
-<div class="main div" id="org3abedbf">
1636 
-<div class="eqlabel" id="org6c3c992">
1635 
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1636 
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1637 1637 
 <p>
1638 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 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 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 1641 
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1642 1642 
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1643 
-<div class="alteqlabels" id="org4747e62">
1643 
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1644 1644 
 <ul class="org-ul">
1645 1645 
 <li>Gr (9.65)</li>
1646 1646 
 </ul>
 
@@ -1669,18 +1669,18 @@ v = \frac{1}{\sqrt{\mu \varepsilon}} = \frac{c}{n}
1669 1669 
 \]
1670 1670 
 where the index of refraction of the material is defined as
1671 1671 
 </p>
1672 
-<div class="main div" id="org92a77a7">
1672 
+<div class="main div" id="org5237a1a">
1673 1673 
 <p>
1674 1674 
 <b>Index of refraction</b>
1675 1675 
 </p>
1676 
-<div class="eqlabel" id="orgb3ac891">
1676 
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1677 1677 
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1678 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 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 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 1681 
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1682 1682 
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1683 
-<div class="alteqlabels" id="orge10b1d5">
1683 
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1684 1684 
 <ul class="org-ul">
1685 1685 
 <li>Gr (9.69)</li>
1686 1686 
 </ul>
 
@@ -1743,7 +1743,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1743 1743 
 </div>
1744 1744 
 <div id="postamble" class="status">
1745 1745 
 <p class="author">Author: Jean-Sébastien Caux</p>
1746 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1746 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1747 1747 
 <p class="validation"></p>
1748 1748 
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1749 1749

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7 7 
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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1649,7 +1649,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1649 1649 
 </div>
1650 1650 
 <div id="postamble" class="status">
1651 1651 
 <p class="author">Author: Jean-Sébastien Caux</p>
1652 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1652 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1653 1653 
 <p class="validation"></p>
1654 1654 
 </div>
1655 1655

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7 7 
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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1671,14 +1671,14 @@ E_{0_I} - E_{0_R} = \beta E_{0_T}, \hspace{1cm}
1671 1671 
 Solving these coupled equations, we can write
1672 1672 
 outgoing amplitudes in terms of incident ones:
1673 1673 
 </p>
1674 
-<div class="eqlabel" id="org0451818">
1674 
+<div class="eqlabel" id="orgf6ae25d">
1675 1675 
 <p>
1676 1676 
 <a id="ERT"></a><a href="./emdm_emwm_refl_ni.html#ERT"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1677 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 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 1679 
 </svg></a>
1680 1680 
 </p>
1681 
-<div class="alteqlabels" id="orgca80cb9">
1681 
+<div class="alteqlabels" id="org17eb76a">
1682 1682 
 <ul class="org-ul">
1683 1683 
 <li>Gr (9.82)</li>
1684 1684 
 </ul>
 
@@ -1744,7 +1744,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1744 1744 
 </div>
1745 1745 
 <div id="postamble" class="status">
1746 1746 
 <p class="author">Author: Jean-Sébastien Caux</p>
1747 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1747 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1748 1748 
 <p class="validation"></p>
1749 1749 
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1750 1750

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6 6 
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7 7 
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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1635,14 +1635,14 @@ Transmitted wave:
1635 1635 
 \]
1636 1636 
 All waves have the same frequency \(\omega\). Since \(\omega = k v\), the three wavevectors are related by
1637 1637 
 </p>
1638 
-<div class="eqlabel" id="orgd898837">
1638 
+<div class="eqlabel" id="org512368e">
1639 1639 
 <p>
1640 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">
1641 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 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 1643 
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1644 1644 
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1645 
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1645 
+<div class="alteqlabels" id="org1361b67">
1646 1646
1647 1647 
 </div>
1648 1648 
 
@@ -1665,7 +1665,7 @@ These forms for incident, reflected and transmitted wave can be substituted in t
1665 1665 
 <p>
1666 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 1667 
 </p>
1668 
-<div class="core div" id="org4e7e42a">
1668 
+<div class="core div" id="org86dbe79">
1669 1669 
 <p>
1670 1670 
 <b>First law of reflection:</b>
1671 1671 
   the incident, reflected and transmitted wave vectors form a plane (called the plane of incidence) which also includes the normal to the surface.
 
@@ -1680,7 +1680,7 @@ Specializing <a href="./emdm_emwm_refl_oi.html#RTobliquek">RTobliquek</a> to our
1680 1680 
 with the incidence (\(\theta_I\)) and reflection (\(\theta_R\)) angles
1681 1681 
 and the angle of refraction (\(\theta_T\)) obey the following laws:
1682 1682 
 </p>
1683 
-<div class="core div" id="org99d6843">
1683 
+<div class="core div" id="orge55fc76">
1684 1684 
 <p>
1685 1685 
 <b>Law of reflection</b>
1686 1686 
   \[
 
@@ -1697,14 +1697,14 @@ and the angle of refraction (\(\theta_T\)) obey the following laws:
1697 1697 
 <p>
1698 1698 
 This takes care of the spatially-dependent exponential factors in the boundary conditions. The coefficients must further obey
1699 1699 
 </p>
1700 
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1700 
+<div class="eqlabel" id="orga2e8f02">
1701 1701 
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1702 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">
1703 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 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 1705 
 </svg></a>
1706 1706 
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1707 
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1707 
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1708 1708
1709 1709 
 </div>
1710 1710 
 
@@ -1750,18 +1750,18 @@ while the third equation becomes
1750 1750 
 \]
1751 1751 
 Writing everything in terms of the incident amplitude, we get
1752 1752 
 </p>
1753 
-<div class="main div" id="org1929cd0">
1753 
+<div class="main div" id="org29006a2">
1754 1754 
 <p>
1755 1755 
 <b>Fresnel's equations for reflection and transmission amplitudes (parallel case)</b>
1756 1756 
 </p>
1757 
-<div class="eqlabel" id="org0b8be69">
1757 
+<div class="eqlabel" id="orgd431e6b">
1758 1758 
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1759 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 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 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 1762 
 </svg></a>
1763 1763 
 </p>
1764 
-<div class="alteqlabels" id="org617a8e4">
1764 
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1765 1765
1766 1766 
 </div>
1767 1767 
 
@@ -1785,15 +1785,15 @@ Amplitudes for transmitted and reflected wave: depend on angle of incidence:
1785 1785 
 Behaviour: for \(\theta_I = 0\) we recover <a href="./emdm_emwm_refl_ni.html#ERT">ERT</a>.
1786 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 1787 
 </p>
1788 
-<div class="main div" id="org087c316">
1789 
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1788 
+<div class="main div" id="orgff8c46e">
1789 
+<div class="eqlabel" id="org7aac922">
1790 1790 
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1791 1791 
 <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">
1792 1792 
   <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"/>
1793 1793 
   <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"/>
1794 1794 
 </svg></a>
1795 1795 
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1796 
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1796 
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1797 1797
1798 1798 
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1799 1799 
 
@@ -1846,7 +1846,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1846 1846 
 </div>
1847 1847 
 <div id="postamble" class="status">
1848 1848 
 <p class="author">Author: Jean-Sébastien Caux</p>
1849 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1849 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1850 1850 
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1851 1851 
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1293 1293 
 </summary>
1294 1294 
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1295 1295 
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1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1633,7 +1633,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1633 1633 
 </div>
1634 1634 
 <div id="postamble" class="status">
1635 1635 
 <p class="author">Author: Jean-Sébastien Caux</p>
1636 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1636 
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1637 1637 
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1638 1638 
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1639 1639

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1293 1293 
 </summary>
1294 1294 
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1639 1639 
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1640 1640 
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1641 1641 
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1642 
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1642 
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1297 1297
1298 1298 
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1299 1299 
 
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1659 1659 
 </div>
1660 1660 
 <div id="postamble" class="status">
1661 1661 
 <p class="author">Author: Jean-Sébastien Caux</p>
1662 
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1662 
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-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
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1297 1297
1298 1298 
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1299 1299 
 
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1683 1683 
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1684 1684 
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1685 1685 
 <p class="author">Author: Jean-Sébastien Caux</p>
1686 
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1686 
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-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
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1297 1297
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1299 1299 
 
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1693 1693 
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1694 1694 
 <div id="postamble" class="status">
1695 1695 
 <p class="author">Author: Jean-Sébastien Caux</p>
1696 
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1614 1614 
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1615 1615
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 Prerequisites
1620 1620 
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1660 1660 
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1661 1661 
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1662 1662 
 <p class="author">Author: Jean-Sébastien Caux</p>
1663 
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1663 
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-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
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1297 1297
1298 1298 
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1299 1299 
 
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1657 1657 
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1658 1658 
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1659 1659 
 <p class="author">Author: Jean-Sébastien Caux</p>
1660 
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1660 
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-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
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1297 1297
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1656 1656 
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1657 1657 
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1658 1658 
 <p class="author">Author: Jean-Sébastien Caux</p>
1659 
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1659 
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1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
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1297 1297
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1616 1616 
 <p>
1617 1617 
 A more aesthetic choice is the <b>Lorenz gauge</b>:
1618 1618 
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1619 
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1621 1621 
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1622 1622 
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1623 1623 
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1624 1624 
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1625 1625 
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1629 1629 
 
@@ -1635,14 +1635,14 @@ A more aesthetic choice is the <b>Lorenz gauge</b>:
1635 1635 
 \]
1636 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 1637 
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1640 1640 
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1641 1641 
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1642 1642 
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1643 1643 
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1644 1644 
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@@ -1654,14 +1654,14 @@ which is chosen to put the second term in the left-hand side of <a href="./emf_s
1654 1654 
 \]
1655 1655 
 while the equation for \(\phi\) becomes
1656 1656 
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1659 1659 
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1660 1660 
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1661 1661 
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1662 1662 
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1663 1663 
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@@ -1673,18 +1673,18 @@ while the equation for \(\phi\) becomes
1673 1673 
 \]
1674 1674 
 These can be written compactly upon introducing a new operator: the
1675 1675 
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1676 
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1676 
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1678 1678 
 <b>d'Alembertian operator</b>
1679 1679 
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1680 
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1683 1683 
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1684 1684 
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1685 1685 
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@@ -1700,18 +1700,18 @@ These can be written compactly upon introducing a new operator: the
1700 1700 
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1701 1701 
 so we get the
1702 1702 
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1703 
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1703 
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1705 1705 
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1706 1706 
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1707 
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1707 
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1709 1709 
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1712 1712 
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@@ -1764,7 +1764,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1764 1764 
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1765 1765 
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1766 1766 
 <p class="author">Author: Jean-Sébastien Caux</p>
1767 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1767 
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1293 1293 
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1294 1294 
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1295 1295 
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1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
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1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1631,7 +1631,7 @@ Useful strategy: represent fields in terms of potentials.
1631 1631 
 Easiest: as we already saw (<a href="./ems_ms_vp_A.html#BcurlA">BcurlA</a>), we can write the magnetic
1632 1632 
 field as a pure curl (since its divergence always vanishes):
1633 1633 
 </p>
1634 
-<div class="core div" id="org0868f0a">
1634 
+<div class="core div" id="org913ae4e">
1635 1635 
 <p>
1636 1636 
 \[
1637 1637 
 {\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}
 
@@ -1648,15 +1648,15 @@ Putting this into Faraday's law <a href="./emd_Fl_Fl.html#Fl">Fl</a> gives
1648 1648 
 so this can be written as the gradient of a scalar.
1649 1649 
 Making the choice \(-{\boldsymbol \nabla} \phi\) for this, we get
1650 1650 
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1651 
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1653 1653 
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1654 1654 
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1655 1655 
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1656 1656 
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1657 1657 
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1658 1658 
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1660 1660
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1662 1662 
 
@@ -1673,15 +1673,15 @@ Making the choice \(-{\boldsymbol \nabla} \phi\) for this, we get
1673 1673 
 <p>
1674 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
1675 1675 
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1676 
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1679 1679 
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1680 1680 
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1681 1681 
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1682 1682 
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1687 1687 
 
@@ -1701,15 +1701,15 @@ whereas Ampère-Maxwell becomes
1701 1701 
 \]
1702 1702 
 which becomes after simple rearrangement and use of the <a href="./c_m_dc_d2.html#curlcurl">curlcurl</a> identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\),
1703 1703 
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1710 1710 
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1711 1711 
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1715 1715 
 
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1744 1744 
 </div>
1745 1745 
 <div id="postamble" class="status">
1746 1746 
 <p class="author">Author: Jean-Sébastien Caux</p>
1747 
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1747 
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1293 1293 
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1294 1294 
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1295 1295 
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1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
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1299 1299 
 
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1640 1640 
 </div>
1641 1641 
 <div id="postamble" class="status">
1642 1642 
 <p class="author">Author: Jean-Sébastien Caux</p>
1643 
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1643 
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1644 1644 
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1645 1645 
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1293 1293 
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1294 1294 
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1295 1295 
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1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
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1297 1297
1298 1298 
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1299 1299 
 
@@ -1618,8 +1618,8 @@ Table of contents
1618 1618 
 <li>Gr 3</li>
1619 1619 
 </ul>
1620 1620
1621 
-<details class="prereq" id="orgd0e4e8d">
1622 
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1621 
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1623 1623 
 Prerequisites
1624 1624 
 </summary>
1625 1625 
 <ul class="org-ul">
 
@@ -1627,8 +1627,8 @@ Prerequisites
1627 1627 
 </ul>
1628 1628 
 </details>
1629 1629
1630 
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1631 
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1630 
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1631 
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1632 1632 
 Objectives
1633 1633 
 </summary>
1634 1634 
 <ul class="org-ul">
 
@@ -1666,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1666 1666 
 </div>
1667 1667 
 <div id="postamble" class="status">
1668 1668 
 <p class="author">Author: Jean-Sébastien Caux</p>
1669 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1669 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1670 1670 
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1671 1671 
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7 7 
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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1624,7 +1624,7 @@ A generic configuration of static charges coupled via the Coulomb interaction
1624 1624 
 defines an electrostatic problem, whose solution is in principle obtained
1625 1625 
 from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_vcd">E_vcd</a>
1626 1626 
 </p>
1627 
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1628 1628 
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1629 1629
1630 1630 
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@@ -1638,7 +1638,7 @@ from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_v
1638 1638 
 or (often simpler) by calculating the electrostatic potential, using either the
1639 1639 
 explicit construction <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>
1640 1640 
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1641 
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1642 1642 
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1643 1643
1644 1644 
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@@ -1658,7 +1658,7 @@ condition <a href="./ems_es_ef_cE.html#curlE0">curlE0</a> can be expressed as th
1658 1658 
 <a href="./ems_es_ep_PL.html#Poi">🐟</a>
1659 1659 
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1660 1660
1661 
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1661 
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1662 1662 
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1663 1663
1664 1664 
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@@ -1674,7 +1674,7 @@ condition <a href="./ems_es_ef_cE.html#curlE0">curlE0</a> can be expressed as th
1674 1674 
 <p>
1675 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 1676 
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1677 
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1677 
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1678 1678 
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1679 1679
1680 1680 
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@@ -1712,7 +1712,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1712 1712 
 </div>
1713 1713 
 <div id="postamble" class="status">
1714 1714 
 <p class="author">Author: Jean-Sébastien Caux</p>
1715 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1715 
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1716 1716 
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1717 1717 
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7 7 
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@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
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1294 1294 
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1295 1295 
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1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
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1297 1297
1298 1298 
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1299 1299 
 
@@ -1630,14 +1630,14 @@ In one dimension, the potential is a single-variable
1630 1630 
 function \(\phi (x)\) and the Laplace equation reads
1631 1631 
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1632 1632
1633 
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1633 
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1634 1634 
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1635 1635 
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1636 1636 
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1637 1637 
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1638 1638 
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1639 1639 
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1641 1641
1642 1642 
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1643 1643 
 
@@ -1652,14 +1652,14 @@ function \(\phi (x)\) and the Laplace equation reads
1652 1652 
 <p>
1653 1653 
 The solution to this is
1654 1654 
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1655 
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1655 
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1656 1656 
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1657 1657 
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1658 1658 
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1659 1659 
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1660 1660 
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1661 1661 
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1663 1663 
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1664 1664 
 <li>Gr (3.6)</li>
1665 1665 
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@@ -1718,14 +1718,14 @@ In two dimensions, the potential becomes a function
1718 1718 
 of two variables (here: \(x\) and \(y\)), so Laplace's
1719 1719 
 equation now reads
1720 1720 
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1721 
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1722 1722 
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1723 1723 
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1724 1724 
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1725 1725 
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1726 1726 
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1731 1731 
 
@@ -1778,14 +1778,14 @@ a point equals its value averaged over a sphere
1778 1778 
 \(S_R({\bf r})\) of any radius \(R\) centered on this point
1779 1779 
 (and of course not containing any charges),
1780 1780 
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1781 
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1782 1782 
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1783 1783 
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1784 1784 
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1786 1786 
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1790 1790 
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1791 1791 
 
@@ -1797,8 +1797,8 @@ a point equals its value averaged over a sphere
1797 1797 
 \]
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1799 1799
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1802 1802 
 <strong>Physicist's proof</strong>
1803 1803 
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1804 1804 
 <p>
 
@@ -1860,8 +1860,8 @@ proving the theorem.
1860 1860 
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1861 1861 
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1862 1862
1863 
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1864 
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1865 1865 
 <strong>Formal proof</strong>
1866 1866 
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1867 1867 
 
@@ -1911,14 +1911,14 @@ we get the following general
1911 1911 
 <p>
1912 1912 
 <b>Theorem</b>:
1913 1913 
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1914 
-<div class="eqlabel" id="org6d0244a">
1914 
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1915 1915 
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1916 1916 
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1917 1917 
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1918 1918 
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1919 1919 
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1920 1920 
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1921 
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1922 1922
1923 1923 
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1924 1924 
 
@@ -1971,19 +1971,19 @@ are necessarily positive, we thus require \(f_x &gt; 0\), \(f_y &gt; 0\) and \(f
1971 1971 
 of the \(f_x + f_y + f_z = 0\) condition above.
1972 1972 
 </p>
1973 1973
1974 
-<div class="eqlabel" id="orga00aa93">
1974 
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1975 1975 
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1976 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">
1977 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 1978 
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1979 1979 
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1980 1980 
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1981 
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1982 1982
1983 1983 
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1984 1984
1985 1985 
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1986 
-<div class="info div" id="org81fcf04">
1986 
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1987 1987 
 <p>
1988 1988 
 <b>Earnshaw's theorem (physical version)</b> <br>
1989 1989 
 </p>
 
@@ -2102,7 +2102,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
2102 2102 
 </div>
2103 2103 
 <div id="postamble" class="status">
2104 2104 
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2105 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
2105 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
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2107 2107 
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1294 1294 
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1296 
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1613 1613 
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1616 
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1618 1618 
 <b>George Green</b>
1619 1619 
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1620 
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1620 
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1621 1621 
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1622 1622 
 See a <a href="https://en.wikipedia.org/wiki/George%5C_Green%5C_(mathematician)">short bio on wikipedia</a>
1623 1623 
 </p>
 
@@ -1650,14 +1650,14 @@ and
1650 1650 
 \]
1651 1651 
 Substituting this in the divergence theorem gives <b>Green's first identity</b>
1652 1652 
 </p>
1653 
-<div class="eqlabel" id="orgfd30f5d">
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1654 1654 
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1655 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 1656 
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1657 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 1658 
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1660 
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1661 1661 
 <ul class="org-ul">
1662 1662 
 <li>J (1.34)</li>
1663 1663 
 </ul>
 
@@ -1675,14 +1675,14 @@ As an aside for now, for completeness, if we do the same thing again but with \(
1675 1675 
 interchanged, and subtract the result, we obtain another useful result known as
1676 1676 
 <b>Green's second identity</b> or <b>Green's theorem</b>
1677 1677 
 </p>
1678 
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1679 1679 
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1680 1680 
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1681 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 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 1683 
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1684 1684 
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1685 
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1686 1686 
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1687 1687 
 <li>J (1.35)</li>
1688 1688 
 </ul>
 
@@ -1718,7 +1718,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1718 1718 
 </div>
1719 1719 
 <div id="postamble" class="status">
1720 1720 
 <p class="author">Author: Jean-Sébastien Caux</p>
1721 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1721 
+<p class="date">Created: 2022-03-24 Thu 08:42</p>
1722 1722 
 <p class="validation"></p>
1723 1723 
 </div>
1724 1724

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@@ -1,7 +1,7 @@
1 1 
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4 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1293,7 +1293,7 @@ Table of contents
1293 1293 
 </summary>
1294 1294 
 <ul>
1295 1295 
 <li>
1296 
-<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
1296 
+<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
1297 1297
1298 1298 
 </li>
1299 1299 
 
@@ -1635,14 +1635,14 @@ This yields
1635 1635 
 The first term on the left-hand side vanishes since \(\Phi\) satisfies Laplace.
1636 1636 
 The right-hand side can be made to vanish if \(\Phi\) obeys either
1637 1637 
 </p>
1638 
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1639 1639 
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1640 1640 
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1643 1643 
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1645 
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1646 1646
1647 1647 
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1648 1648 
 
@@ -1654,14 +1654,14 @@ The right-hand side can be made to vanish if \(\Phi\) obeys either
1654 1654 
 <p>
1655 1655 
 or
1656 1656 
 </p>
1657 
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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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1663 1663 
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1665 1665
1666 1666 
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1667 1667 
 
@@ -1689,19 +1689,19 @@ additive constant.
1689 1689 
 <p>
1690 1690 
 We can thus finally state the
1691 1691 
 </p>
1692 
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1693 1693 
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1694 1694 
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1696 1696 
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1699 
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1706 1706 
 <b>Uniqueness Theorem</b>
1707 1707 
 </p>
 
@@ -1740,7 +1740,7 @@ Reading other books, you might be misled into thinking that there are numerous c
1740 1740 
 and corollaries and that things are
1741 1741 
 </p>
1742 1742
1743 
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1743 
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1744 1744 
 <p>
1745 1745 
 <b>Comment/warning</b>: <b>uniqueness theorem on uniqueness theorems</b> <br>
1746 1746 
   Do not be misled: there is a <i>unique</i> uniqueness theorem for the
 
@@ -1767,7 +1767,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1767 1767 
 </div>
1768 1768 
 <div id="postamble" class="status">
1769 1769 
 <p class="author">Author: Jean-Sébastien Caux</p>
1770 
-<p class="date">Created: 2022-03-22 Tue 10:52</p>
1770 
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1771 1771 
 <p class="validation"></p>
1772 1772 
 </div>
1773 1773

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