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Update 2022-02-09 07:44

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Jean-Sébastien 2 years ago
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+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
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+<img alt="Creative Commons License" style="border-width:0"
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+src="https://licensebuttons.net/l/by/4.0/80x15.png"/>
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+Except where otherwise noted, all content is licensed under a
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+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
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+target="_blank">Creative Commons Attribution 4.0 International License</a>.
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 <p class="author">Author: Jean-Sébastien Caux</p>
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-<p class="date">Created: 2022-02-08 Tue 17:21</p>
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-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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-<ul class="navigation-links"><li>Prev:&nbsp;<a href="a.html">Appendices&emsp;<small>[a]</small></a></li><li>Next:&nbsp;<a href="c.html">Compendium&emsp;<small>[c]</small></a></li><li>Up:&nbsp;<a href="a.html">Appendices&emsp;<small>[a]</small></a></li></ul><div id="outline-container-a_l" class="outline-3">
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+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="a.html">Appendices</a></li><li>Literature</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="a.html">Appendices&emsp;<small>[a]</small></a></li><li>Next:&nbsp;<a href="c.html">Compendium&emsp;<small>[c]</small></a></li><li>Up:&nbsp;<a href="a.html">Appendices&emsp;<small>[a]</small></a></li></ul><div id="outline-container-a_l" class="outline-3">
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+Except where otherwise noted, all content is licensed under a
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+target="_blank">Creative Commons Attribution 4.0 International License</a>.
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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="date">Created: 2022-02-08 Tue 17:21</p>
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-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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+target="_blank">Creative Commons Attribution 4.0 International License</a>.
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 <p class="author">Author: Jean-Sébastien Caux</p>
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-<p class="date">Created: 2022-02-08 Tue 17:21</p>
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-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li>Mathematics</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c.html">Compendium&emsp;<small>[c]</small></a></li><li>Next:&nbsp;<a href="c_m_va.html">Vector Analysis&emsp;<small>[c.m.va]</small></a></li><li>Up:&nbsp;<a href="c.html">Compendium&emsp;<small>[c]</small></a></li></ul>
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+Except where otherwise noted, all content is licensed under a
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+target="_blank">Creative Commons Attribution 4.0 International License</a>.
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 <p class="author">Author: Jean-Sébastien Caux</p>
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-<p class="date">Created: 2022-02-08 Tue 17:21</p>
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-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li>Coordinate Systems</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic_ip.html">Integration by Parts&emsp;<small>[c.m.ic.ip]</small></a></li><li>Next:&nbsp;<a href="c_m_cs_sph.html">Spherical Coordinates&emsp;<small>[c.m.cs.sph]</small></a></li><li>Up:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li></ul>
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+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1639 
+target="_blank" class="m-2">
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+<img alt="Creative Commons License" style="border-width:0"
1641 
+src="https://licensebuttons.net/l/by/4.0/80x15.png"/>
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+</a>
1643 
+Except where otherwise noted, all content is licensed under a
1644 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1645 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1646 
+</div>
1647 
+<div id="postamble" class="status">
1637 1648 
 <p class="author">Author: Jean-Sébastien Caux</p>
1638 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1639 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1649 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1650 
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+<!-- 2022-02-09 Wed 07:31 -->
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_cs_sph.html">Spherical Coordinates&emsp;<small>[c.m.cs.sph]</small></a></li><li>Next:&nbsp;<a href="c_m_cs_hyp.html">Hyperbolic Coordinates&emsp;<small>[c.m.cs.hyp]</small></a></li><li>Up:&nbsp;<a href="c_m_cs.html">Coordinate Systems&emsp;<small>[c.m.cs]</small></a></li></ul><div id="outline-container-c_m_cs_cyl" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_cs.html">Coordinate Systems</a></li><li>Cylindrical Coordinates</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_cs_sph.html">Spherical Coordinates&emsp;<small>[c.m.cs.sph]</small></a></li><li>Next:&nbsp;<a href="c_m_cs_hyp.html">Hyperbolic Coordinates&emsp;<small>[c.m.cs.hyp]</small></a></li><li>Up:&nbsp;<a href="c_m_cs.html">Coordinate Systems&emsp;<small>[c.m.cs]</small></a></li></ul><div id="outline-container-c_m_cs_cyl" class="outline-5">
1620 1620 
 <h5 id="c_m_cs_cyl">Cylindrical Coordinates<a class="headline-permalink" href="./c_m_cs_cyl.html#c_m_cs_cyl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1668,14 +1668,14 @@ Range of parameters:  \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
1668 1668 
 <div id="outline-container-c_m_cs_cyl_grad" class="outline-6">
1669 1669 
 <h6 id="c_m_cs_cyl_grad"><a href="#c_m_cs_cyl_grad">Gradient</a></h6>
1670 1670 
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1673 1673 
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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"/>
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1677 1677 
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1679 1679 
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1680 1680 
 <li>Gr4(1.79)</li>
1681 1681 
 </ul>
 
@@ -1696,14 +1696,14 @@ Range of parameters:  \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
1696 1696 
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1697 1697 
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1698 1698 
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1705 1705 
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1706 
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1707 1707 
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1708 1708 
 <li>Gr4(2.21)</li>
1709 1709 
 </ul>
 
@@ -1724,14 +1724,14 @@ Range of parameters:  \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
1724 1724 
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1725 1725 
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1726 1726 
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1729 1729 
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1730 1730 
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1733 1733 
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1735 1735 
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1736 1736 
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1737 1737 
 </ul>
 
@@ -1765,10 +1765,21 @@ Range of parameters:  \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
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1769 
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1770 
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1771 
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1772 
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1773 
+src="https://licensebuttons.net/l/by/4.0/80x15.png"/>
1774 
+</a>
1775 
+Except where otherwise noted, all content is licensed under a
1776 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1777 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1778 
+</div>
1779 
+<div id="postamble" class="status">
1769 1780 
 <p class="author">Author: Jean-Sébastien Caux</p>
1770 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1771 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1781 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1782 
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1772 1783 
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1773 1784
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_cs_cyl.html">Cylindrical Coordinates&emsp;<small>[c.m.cs.cyl]</small></a></li><li>Next:&nbsp;<a href="c_m_dd.html">Dirac delta Distribution&emsp;<small>[c.m.dd]</small></a></li><li>Up:&nbsp;<a href="c_m_cs.html">Coordinate Systems&emsp;<small>[c.m.cs]</small></a></li></ul><div id="outline-container-c_m_cs_hyp" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_cs.html">Coordinate Systems</a></li><li>Hyperbolic Coordinates</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_cs_cyl.html">Cylindrical Coordinates&emsp;<small>[c.m.cs.cyl]</small></a></li><li>Next:&nbsp;<a href="c_m_dd.html">Dirac delta Distribution&emsp;<small>[c.m.dd]</small></a></li><li>Up:&nbsp;<a href="c_m_cs.html">Coordinate Systems&emsp;<small>[c.m.cs]</small></a></li></ul><div id="outline-container-c_m_cs_hyp" class="outline-5">
1620 1620 
 <h5 id="c_m_cs_hyp">Hyperbolic Coordinates<a class="headline-permalink" href="./c_m_cs_hyp.html#c_m_cs_hyp"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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1622 1622 
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@@ -1626,10 +1626,21 @@ Table of contents
1626 1626 
 </div>
1627 1627
1628 1628
1629 
-<hr><div id="postamble" class="status">
1629 
+<hr>
1630 
+<div class="license">
1631 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1632 
+target="_blank" class="m-2">
1633 
+<img alt="Creative Commons License" style="border-width:0"
1634 
+src="https://licensebuttons.net/l/by/4.0/80x15.png"/>
1635 
+</a>
1636 
+Except where otherwise noted, all content is licensed under a
1637 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1638 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1639 
+</div>
1640 
+<div id="postamble" class="status">
1630 1641 
 <p class="author">Author: Jean-Sébastien Caux</p>
1631 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1632 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1642 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1643 
+<p class="validation"></p>
1633 1644 
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1634 1645
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_cs.html">Coordinate Systems&emsp;<small>[c.m.cs]</small></a></li><li>Next:&nbsp;<a href="c_m_cs_cyl.html">Cylindrical Coordinates&emsp;<small>[c.m.cs.cyl]</small></a></li><li>Up:&nbsp;<a href="c_m_cs.html">Coordinate Systems&emsp;<small>[c.m.cs]</small></a></li></ul><div id="outline-container-c_m_cs_sph" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_cs.html">Coordinate Systems</a></li><li>Spherical Coordinates</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_cs.html">Coordinate Systems&emsp;<small>[c.m.cs]</small></a></li><li>Next:&nbsp;<a href="c_m_cs_cyl.html">Cylindrical Coordinates&emsp;<small>[c.m.cs.cyl]</small></a></li><li>Up:&nbsp;<a href="c_m_cs.html">Coordinate Systems&emsp;<small>[c.m.cs]</small></a></li></ul><div id="outline-container-c_m_cs_sph" class="outline-5">
1620 1620 
 <h5 id="c_m_cs_sph">Spherical Coordinates<a class="headline-permalink" href="./c_m_cs_sph.html#c_m_cs_sph"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1739,10 +1739,21 @@ Infinitesimal surface element:  depends on situation.
1739 1739
1740 1740
1741 1741
1742 
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1742 
+<hr>
1743 
+<div class="license">
1744 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1745 
+target="_blank" class="m-2">
1746 
+<img alt="Creative Commons License" style="border-width:0"
1747 
+src="https://licensebuttons.net/l/by/4.0/80x15.png"/>
1748 
+</a>
1749 
+Except where otherwise noted, all content is licensed under a
1750 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1751 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1752 
+</div>
1753 
+<div id="postamble" class="status">
1743 1754 
 <p class="author">Author: Jean-Sébastien Caux</p>
1744 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1745 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1755 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1756 
+<p class="validation"></p>
1746 1757 
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1747 1758
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7 7 
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_va_pds.html">Position, Displacement and Separation Vectors&emsp;<small>[c.m.va.pds]</small></a></li><li>Next:&nbsp;<a href="c_m_dc_g.html">Gradient&emsp;<small>[c.m.dc.g]</small></a></li><li>Up:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li></ul>
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li>Differential Calculus</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_va_pds.html">Position, Displacement and Separation Vectors&emsp;<small>[c.m.va.pds]</small></a></li><li>Next:&nbsp;<a href="c_m_dc_g.html">Gradient&emsp;<small>[c.m.dc.g]</small></a></li><li>Up:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li></ul>
1620 1620 
 <h4 id="c_m_dc">Differential Calculus<a class="headline-permalink" href="./c_m_dc.html#c_m_dc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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1622 1622 
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@@ -1636,10 +1636,21 @@ Table of contents
1636 1636 
 <li><a href="c_m_dc_d2.html">Second Derivatives</a><span class="headline-id">c.m.dc.d2</span></li>
1637 1637 
 </ul>
1638 1638
1639 
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1639 
+<hr>
1640 
+<div class="license">
1641 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1642 
+target="_blank" class="m-2">
1643 
+<img alt="Creative Commons License" style="border-width:0"
1644 
+src="https://licensebuttons.net/l/by/4.0/80x15.png"/>
1645 
+</a>
1646 
+Except where otherwise noted, all content is licensed under a
1647 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1648 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1649 
+</div>
1650 
+<div id="postamble" class="status">
1640 1651 
 <p class="author">Author: Jean-Sébastien Caux</p>
1641 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1642 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1652 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1653 
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1643 1654 
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_div.html">The Divergence&emsp;<small>[c.m.dc.div]</small></a></li><li>Next:&nbsp;<a href="c_m_dc_pr.html">Product Rules&emsp;<small>[c.m.dc.pr]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul><div id="outline-container-c_m_dc_curl" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_dc.html">Differential Calculus</a></li><li>The Curl</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_div.html">The Divergence&emsp;<small>[c.m.dc.div]</small></a></li><li>Next:&nbsp;<a href="c_m_dc_pr.html">Product Rules&emsp;<small>[c.m.dc.pr]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul><div id="outline-container-c_m_dc_curl" class="outline-5">
1620 1620 
 <h5 id="c_m_dc_curl">The Curl<a class="headline-permalink" href="./c_m_dc_curl.html#c_m_dc_curl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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   <path d="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"/>
 
@@ -1638,10 +1638,21 @@ v_x &amp; v_y &amp; v_z \end{array} \right| \nonumber \\
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+<hr>
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+<div class="license">
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+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
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+target="_blank" class="m-2">
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+<img alt="Creative Commons License" style="border-width:0"
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+src="https://licensebuttons.net/l/by/4.0/80x15.png"/>
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+</a>
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+Except where otherwise noted, all content is licensed under a
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+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1650 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1651 
+</div>
1652 
+<div id="postamble" class="status">
1642 1653 
 <p class="author">Author: Jean-Sébastien Caux</p>
1643 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1644 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1654 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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+<p class="validation"></p>
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_pr.html">Product Rules&emsp;<small>[c.m.dc.pr]</small></a></li><li>Next:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul><div id="outline-container-c_m_dc_d2" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_dc.html">Differential Calculus</a></li><li>Second Derivatives</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_pr.html">Product Rules&emsp;<small>[c.m.dc.pr]</small></a></li><li>Next:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul><div id="outline-container-c_m_dc_d2" class="outline-5">
1620 1620 
 <h5 id="c_m_dc_d2">Second Derivatives<a class="headline-permalink" href="./c_m_dc_d2.html#c_m_dc_d2"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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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"/>
 
@@ -1624,9 +1624,9 @@ Table of contents
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 <div class="outline-text-5" id="text-c_m_dc_d2">
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 </div>
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1627 
-<div id="outline-container-org5fd48d1" class="outline-6">
1628 
-<h6 id="org5fd48d1"><a href="#org5fd48d1">Divergence of gradient</a></h6>
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+<div id="outline-container-orga13f3ad" class="outline-6">
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+<h6 id="orga13f3ad"><a href="#orga13f3ad">Divergence of gradient</a></h6>
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+<div class="outline-text-6" id="text-orga13f3ad">
1630 1630 
 <p>
1631 1631 
 \({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the <b>Laplacian</b> of the scalar field \(T\).
1632 1632 
 The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components
 
@@ -1635,36 +1635,36 @@ given by the Laplacian of the corresponding vector elements.
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-<div id="outline-container-org41a3448" class="outline-6">
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-<h6 id="org41a3448"><a href="#org41a3448">Curl of a gradient</a></h6>
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 <p>
1642 1642 
 This always vanishes.
1643 1643 
 </p>
1644 1644 
 </div>
1645 1645 
 </div>
1646 1646
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-<div id="outline-container-org6c4b5a7" class="outline-6">
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-<h6 id="org6c4b5a7"><a href="#org6c4b5a7">Gradient of the divergence</a></h6>
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+<h6 id="orgb23671d"><a href="#orgb23671d">Gradient of the divergence</a></h6>
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 <p>
1651 1651 
 \({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics.  No special name.
1652 1652 
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1653 1653 
 </div>
1654 1654 
 </div>
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 This always vanishes.
1661 1661 
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-<div id="outline-container-org62690be" class="outline-6">
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 <p>
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 \[
1670 1670 
 {\boldsymbol \nabla} \times ({\boldsymbol \nabla} \times {\bf v}) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v}) - {\boldsymbol \nabla}^2 {\bf v}
 
@@ -1676,10 +1676,21 @@ This always vanishes.
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+Except where otherwise noted, all content is licensed under a
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1688 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1689 
+</div>
1690 
+<div id="postamble" class="status">
1680 1691 
 <p class="author">Author: Jean-Sébastien Caux</p>
1681 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1682 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1692 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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+<p class="validation"></p>
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_g.html">Gradient&emsp;<small>[c.m.dc.g]</small></a></li><li>Next:&nbsp;<a href="c_m_dc_div.html">The Divergence&emsp;<small>[c.m.dc.div]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul><div id="outline-container-c_m_dc_del" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_dc.html">Differential Calculus</a></li><li>The \({\boldsymbol \nabla}\) Operator</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_g.html">Gradient&emsp;<small>[c.m.dc.g]</small></a></li><li>Next:&nbsp;<a href="c_m_dc_div.html">The Divergence&emsp;<small>[c.m.dc.div]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul><div id="outline-container-c_m_dc_del" class="outline-5">
1620 1620 
 <h5 id="c_m_dc_del">The \({\boldsymbol \nabla}\) Operator<a class="headline-permalink" href="./c_m_dc_del.html#c_m_dc_del"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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@@ -1637,10 +1637,21 @@ The <b>del/grad/nabla operator</b> is defined as
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+</a>
1647 
+Except where otherwise noted, all content is licensed under a
1648 
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+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1650 
+</div>
1651 
+<div id="postamble" class="status">
1641 1652 
 <p class="author">Author: Jean-Sébastien Caux</p>
1642 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1643 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1653 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_del.html">The \({\boldsymbol \nabla}\) Operator&emsp;<small>[c.m.dc.del]</small></a></li><li>Next:&nbsp;<a href="c_m_dc_curl.html">The Curl&emsp;<small>[c.m.dc.curl]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul><div id="outline-container-c_m_dc_div" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_dc.html">Differential Calculus</a></li><li>The Divergence</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_del.html">The \({\boldsymbol \nabla}\) Operator&emsp;<small>[c.m.dc.del]</small></a></li><li>Next:&nbsp;<a href="c_m_dc_curl.html">The Curl&emsp;<small>[c.m.dc.curl]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul><div id="outline-container-c_m_dc_div" class="outline-5">
1620 1620 
 <h5 id="c_m_dc_div">The Divergence<a class="headline-permalink" href="./c_m_dc_div.html#c_m_dc_div"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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1622 1622 
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@@ -1634,10 +1634,21 @@ Table of contents
1634 1634
1635 1635
1636 1636
1637 
-<hr><div id="postamble" class="status">
1637 
+<hr>
1638 
+<div class="license">
1639 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1640 
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1641 
+<img alt="Creative Commons License" style="border-width:0"
1642 
+src="https://licensebuttons.net/l/by/4.0/80x15.png"/>
1643 
+</a>
1644 
+Except where otherwise noted, all content is licensed under a
1645 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1646 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1647 
+</div>
1648 
+<div id="postamble" class="status">
1638 1649 
 <p class="author">Author: Jean-Sébastien Caux</p>
1639 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1640 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1650 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1651 
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1642 1653
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li><li>Next:&nbsp;<a href="c_m_dc_del.html">The \({\boldsymbol \nabla}\) Operator&emsp;<small>[c.m.dc.del]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul><div id="outline-container-c_m_dc_g" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_dc.html">Differential Calculus</a></li><li>Gradient</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li><li>Next:&nbsp;<a href="c_m_dc_del.html">The \({\boldsymbol \nabla}\) Operator&emsp;<small>[c.m.dc.del]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul><div id="outline-container-c_m_dc_g" class="outline-5">
1620 1620 
 <h5 id="c_m_dc_g">Gradient<a class="headline-permalink" href="./c_m_dc_g.html#c_m_dc_g"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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@@ -1658,10 +1658,21 @@ is a vector called the <b>gradient</b> of \(T\).
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1664 
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1670 
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1671 
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1672 
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1662 1673 
 <p class="author">Author: Jean-Sébastien Caux</p>
1663 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1664 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1674 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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@@ -1616,7 +1616,7 @@ Table of contents
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 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_curl.html">The Curl&emsp;<small>[c.m.dc.curl]</small></a></li><li>Next:&nbsp;<a href="c_m_dc_d2.html">Second Derivatives&emsp;<small>[c.m.dc.d2]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul><div id="outline-container-c_m_dc_pr" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_dc.html">Differential Calculus</a></li><li>Product Rules</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_curl.html">The Curl&emsp;<small>[c.m.dc.curl]</small></a></li><li>Next:&nbsp;<a href="c_m_dc_d2.html">Second Derivatives&emsp;<small>[c.m.dc.d2]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul><div id="outline-container-c_m_dc_pr" class="outline-5">
1620 1620 
 <h5 id="c_m_dc_pr">Product Rules<a class="headline-permalink" href="./c_m_dc_pr.html#c_m_dc_pr"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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   <path d="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"/>
 
@@ -1631,10 +1631,21 @@ Six product rules (on inside front cover of Gr).
1631 1631
1632 1632
1633 1633
1634 
-<hr><div id="postamble" class="status">
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+<hr>
1635 
+<div class="license">
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1639 
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1641 
+Except where otherwise noted, all content is licensed under a
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+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1643 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1644 
+</div>
1645 
+<div id="postamble" class="status">
1635 1646 
 <p class="author">Author: Jean-Sébastien Caux</p>
1636 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1637 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1647 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_cs_hyp.html">Hyperbolic Coordinates&emsp;<small>[c.m.cs.hyp]</small></a></li><li>Next:&nbsp;<a href="c_m_dd_div.html">The Divergence of \(\hat{\bf r}/r^2\)&emsp;<small>[c.m.dd.div]</small></a></li><li>Up:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li></ul>
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li>Dirac delta Distribution</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_cs_hyp.html">Hyperbolic Coordinates&emsp;<small>[c.m.cs.hyp]</small></a></li><li>Next:&nbsp;<a href="c_m_dd_div.html">The Divergence of \(\hat{\bf r}/r^2\)&emsp;<small>[c.m.dd.div]</small></a></li><li>Up:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li></ul>
1620 1620 
 <h4 id="c_m_dd">Dirac delta Distribution<a class="headline-permalink" href="./c_m_dd.html#c_m_dd"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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   <path d="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"/>
 
@@ -1633,10 +1633,21 @@ Table of contents
1633 1633 
 <li><a href="c_m_dd_3d.html">The Three-Dimensional Delta Function</a><span class="headline-id">c.m.dd.3d</span></li>
1634 1634 
 </ul>
1635 1635
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-<hr><div id="postamble" class="status">
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+<hr>
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+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
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+<img alt="Creative Commons License" style="border-width:0"
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1643 
+Except where otherwise noted, all content is licensed under a
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+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1645 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1646 
+</div>
1647 
+<div id="postamble" class="status">
1637 1648 
 <p class="author">Author: Jean-Sébastien Caux</p>
1638 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1639 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1649 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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+<p class="validation"></p>
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dd_div.html">The Divergence of \(\hat{\bf r}/r^2\)&emsp;<small>[c.m.dd.div]</small></a></li><li>Next:&nbsp;<a href="c_m_dd_3d.html">The Three-Dimensional Delta Function&emsp;<small>[c.m.dd.3d]</small></a></li><li>Up:&nbsp;<a href="c_m_dd.html">Dirac delta Distribution&emsp;<small>[c.m.dd]</small></a></li></ul><div id="outline-container-c_m_dd_1d" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_dd.html">Dirac delta Distribution</a></li><li>The One-Dimensional Dirac Delta Function</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dd_div.html">The Divergence of \(\hat{\bf r}/r^2\)&emsp;<small>[c.m.dd.div]</small></a></li><li>Next:&nbsp;<a href="c_m_dd_3d.html">The Three-Dimensional Delta Function&emsp;<small>[c.m.dd.3d]</small></a></li><li>Up:&nbsp;<a href="c_m_dd.html">Dirac delta Distribution&emsp;<small>[c.m.dd]</small></a></li></ul><div id="outline-container-c_m_dd_1d" class="outline-5">
1620 1620 
 <h5 id="c_m_dd_1d">The One-Dimensional Dirac Delta Function<a class="headline-permalink" href="./c_m_dd_1d.html#c_m_dd_1d"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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1622 1622 
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@@ -1645,10 +1645,21 @@ Consequences:  for any smooth differentiable function \(f(x)\),
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+<hr>
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1655 
+Except where otherwise noted, all content is licensed under a
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1657 
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1658 
+</div>
1659 
+<div id="postamble" class="status">
1649 1660 
 <p class="author">Author: Jean-Sébastien Caux</p>
1650 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1651 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1661 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dd_1d.html">The One-Dimensional Dirac Delta Function&emsp;<small>[c.m.dd.1d]</small></a></li><li>Next:&nbsp;<a href="c_m_vf.html">Vector Fields&emsp;<small>[c.m.vf]</small></a></li><li>Up:&nbsp;<a href="c_m_dd.html">Dirac delta Distribution&emsp;<small>[c.m.dd]</small></a></li></ul><div id="outline-container-c_m_dd_3d" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_dd.html">Dirac delta Distribution</a></li><li>The Three-Dimensional Delta Function</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dd_1d.html">The One-Dimensional Dirac Delta Function&emsp;<small>[c.m.dd.1d]</small></a></li><li>Next:&nbsp;<a href="c_m_vf.html">Vector Fields&emsp;<small>[c.m.vf]</small></a></li><li>Up:&nbsp;<a href="c_m_dd.html">Dirac delta Distribution&emsp;<small>[c.m.dd]</small></a></li></ul><div id="outline-container-c_m_dd_3d" class="outline-5">
1620 1620 
 <h5 id="c_m_dd_3d">The Three-Dimensional Delta Function<a class="headline-permalink" href="./c_m_dd_3d.html#c_m_dd_3d"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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@@ -1645,14 +1645,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox:
1645 1645 
 More generally,
1646 1646 
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1654 1654 
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1655 
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1656 1656 
 <ul class="org-ul">
1657 1657 
 <li>Gr (1.100)</li>
1658 1658 
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@@ -1688,10 +1688,21 @@ we have that
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1689 1689
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1698 
+Except where otherwise noted, all content is licensed under a
1699 
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1700 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1701 
+</div>
1702 
+<div id="postamble" class="status">
1692 1703 
 <p class="author">Author: Jean-Sébastien Caux</p>
1693 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1694 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1704 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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+<p class="validation"></p>
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 </ul>
1617 1617 
 </details>
1618 1618 
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1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dd.html">Dirac delta Distribution&emsp;<small>[c.m.dd]</small></a></li><li>Next:&nbsp;<a href="c_m_dd_1d.html">The One-Dimensional Dirac Delta Function&emsp;<small>[c.m.dd.1d]</small></a></li><li>Up:&nbsp;<a href="c_m_dd.html">Dirac delta Distribution&emsp;<small>[c.m.dd]</small></a></li></ul><div id="outline-container-c_m_dd_div" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_dd.html">Dirac delta Distribution</a></li><li>The Divergence of \(\hat{\bf r}/r^2\)</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dd.html">Dirac delta Distribution&emsp;<small>[c.m.dd]</small></a></li><li>Next:&nbsp;<a href="c_m_dd_1d.html">The One-Dimensional Dirac Delta Function&emsp;<small>[c.m.dd.1d]</small></a></li><li>Up:&nbsp;<a href="c_m_dd.html">Dirac delta Distribution&emsp;<small>[c.m.dd]</small></a></li></ul><div id="outline-container-c_m_dd_div" class="outline-5">
1620 1620 
 <h5 id="c_m_dd_div">The Divergence of \(\hat{\bf r}/r^2\)<a class="headline-permalink" href="./c_m_dd_div.html#c_m_dd_div"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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@@ -1651,10 +1651,21 @@ Problem:  in Gr(1.84), we've divided by zero when \(r = 0\).
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+<hr>
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1659 
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1661 
+Except where otherwise noted, all content is licensed under a
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+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1664 
+</div>
1665 
+<div id="postamble" class="status">
1655 1666 
 <p class="author">Author: Jean-Sébastien Caux</p>
1656 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1657 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1667 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
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1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_d2.html">Second Derivatives&emsp;<small>[c.m.dc.d2]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_lsv.html">Line, Surface and Volume Integrals&emsp;<small>[c.m.ic.lsv]</small></a></li><li>Up:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li></ul>
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li>Integral Calculus</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_d2.html">Second Derivatives&emsp;<small>[c.m.dc.d2]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_lsv.html">Line, Surface and Volume Integrals&emsp;<small>[c.m.ic.lsv]</small></a></li><li>Up:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li></ul>
1620 1620 
 <h4 id="c_m_ic">Integral Calculus<a class="headline-permalink" href="./c_m_ic.html#c_m_ic"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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1622 1622 
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@@ -1636,10 +1636,21 @@ Table of contents
1636 1636 
 <li><a href="c_m_ic_ip.html">Integration by Parts</a><span class="headline-id">c.m.ic.ip</span></li>
1637 1637 
 </ul>
1638 1638
1639 
-<hr><div id="postamble" class="status">
1639 
+<hr>
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+<div class="license">
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1642 
+target="_blank" class="m-2">
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1647 
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1648 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1649 
+</div>
1650 
+<div id="postamble" class="status">
1640 1651 
 <p class="author">Author: Jean-Sébastien Caux</p>
1641 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1642 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1652 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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+<p class="validation"></p>
1643 1654 
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1644 1655
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic_lsv.html">Line, Surface and Volume Integrals&emsp;<small>[c.m.ic.lsv]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_ftg.html">The Fundamental Theorem for Gradients&emsp;<small>[c.m.ic.ftg]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul><div id="outline-container-c_m_ic_ftc" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_ic.html">Integral Calculus</a></li><li>The Fundamental Theorem of Calculus</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic_lsv.html">Line, Surface and Volume Integrals&emsp;<small>[c.m.ic.lsv]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_ftg.html">The Fundamental Theorem for Gradients&emsp;<small>[c.m.ic.ftg]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul><div id="outline-container-c_m_ic_ftc" class="outline-5">
1620 1620 
 <h5 id="c_m_ic_ftc">The Fundamental Theorem of Calculus<a class="headline-permalink" href="./c_m_ic_ftc.html#c_m_ic_ftc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1634,10 +1634,21 @@ Table of contents
1634 1634
1635 1635
1636 1636
1637 
-<hr><div id="postamble" class="status">
1637 
+<hr>
1638 
+<div class="license">
1639 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1640 
+target="_blank" class="m-2">
1641 
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1642 
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1643 
+</a>
1644 
+Except where otherwise noted, all content is licensed under a
1645 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1646 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1647 
+</div>
1648 
+<div id="postamble" class="status">
1638 1649 
 <p class="author">Author: Jean-Sébastien Caux</p>
1639 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1640 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1650 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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+<p class="validation"></p>
1641 1652 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic_ftc.html">The Fundamental Theorem of Calculus&emsp;<small>[c.m.ic.ftc]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_gauss.html">Gauss' Theorem&emsp;<small>[c.m.ic.gauss]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul><div id="outline-container-c_m_ic_ftg" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_ic.html">Integral Calculus</a></li><li>The Fundamental Theorem for Gradients</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic_ftc.html">The Fundamental Theorem of Calculus&emsp;<small>[c.m.ic.ftc]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_gauss.html">Gauss' Theorem&emsp;<small>[c.m.ic.gauss]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul><div id="outline-container-c_m_ic_ftg" class="outline-5">
1620 1620 
 <h5 id="c_m_ic_ftg">The Fundamental Theorem for Gradients<a class="headline-permalink" href="./c_m_ic_ftg.html#c_m_ic_ftg"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
   <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"/>
1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1643,10 +1643,21 @@ Table of contents
1643 1643
1644 1644
1645 1645
1646 
-<hr><div id="postamble" class="status">
1646 
+<hr>
1647 
+<div class="license">
1648 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1649 
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1650 
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1651 
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1652 
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1653 
+Except where otherwise noted, all content is licensed under a
1654 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1655 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1656 
+</div>
1657 
+<div id="postamble" class="status">
1647 1658 
 <p class="author">Author: Jean-Sébastien Caux</p>
1648 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1649 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1659 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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+<p class="validation"></p>
1650 1661 
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic_ftg.html">The Fundamental Theorem for Gradients&emsp;<small>[c.m.ic.ftg]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_stokes.html">Stokes' Theorem&emsp;<small>[c.m.ic.stokes]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul><div id="outline-container-c_m_ic_gauss" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_ic.html">Integral Calculus</a></li><li>Gauss' Theorem</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic_ftg.html">The Fundamental Theorem for Gradients&emsp;<small>[c.m.ic.ftg]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_stokes.html">Stokes' Theorem&emsp;<small>[c.m.ic.stokes]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul><div id="outline-container-c_m_ic_gauss" class="outline-5">
1620 1620 
 <h5 id="c_m_ic_gauss">Gauss' Theorem<a class="headline-permalink" href="./c_m_ic_gauss.html#c_m_ic_gauss"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
   <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"/>
1622 1622 
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@@ -1637,10 +1637,21 @@ This is know either as <b>Gauss' theorem</b>, <b>Green's theorem</b> or the <b>d
1637 1637
1638 1638
1639 1639
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1640 
+<hr>
1641 
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1642 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1643 
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1644 
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1647 
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1648 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1649 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1650 
+</div>
1651 
+<div id="postamble" class="status">
1641 1652 
 <p class="author">Author: Jean-Sébastien Caux</p>
1642 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1643 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1653 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1654 
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1644 1655 
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic_stokes.html">Stokes' Theorem&emsp;<small>[c.m.ic.stokes]</small></a></li><li>Next:&nbsp;<a href="c_m_cs.html">Coordinate Systems&emsp;<small>[c.m.cs]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul><div id="outline-container-c_m_ic_ip" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_ic.html">Integral Calculus</a></li><li>Integration by Parts</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic_stokes.html">Stokes' Theorem&emsp;<small>[c.m.ic.stokes]</small></a></li><li>Next:&nbsp;<a href="c_m_cs.html">Coordinate Systems&emsp;<small>[c.m.cs]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul><div id="outline-container-c_m_ic_ip" class="outline-5">
1620 1620 
 <h5 id="c_m_ic_ip">Integration by Parts<a class="headline-permalink" href="./c_m_ic_ip.html#c_m_ic_ip"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1644,10 +1644,21 @@ or in other words
1644 1644 
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1645 1645
1646 1646
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+<hr>
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+<div class="license">
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+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1650 
+target="_blank" class="m-2">
1651 
+<img alt="Creative Commons License" style="border-width:0"
1652 
+src="https://licensebuttons.net/l/by/4.0/80x15.png"/>
1653 
+</a>
1654 
+Except where otherwise noted, all content is licensed under a
1655 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1656 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1657 
+</div>
1658 
+<div id="postamble" class="status">
1648 1659 
 <p class="author">Author: Jean-Sébastien Caux</p>
1649 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1650 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1660 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1661 
+<p class="validation"></p>
1651 1662 
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_ftc.html">The Fundamental Theorem of Calculus&emsp;<small>[c.m.ic.ftc]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul><div id="outline-container-c_m_ic_lsv" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_ic.html">Integral Calculus</a></li><li>Line, Surface and Volume Integrals</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_ftc.html">The Fundamental Theorem of Calculus&emsp;<small>[c.m.ic.ftc]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul><div id="outline-container-c_m_ic_lsv" class="outline-5">
1620 1620 
 <h5 id="c_m_ic_lsv">Line, Surface and Volume Integrals<a class="headline-permalink" href="./c_m_ic_lsv.html#c_m_ic_lsv"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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1622 1622 
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@@ -1624,9 +1624,9 @@ Table of contents
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1626 1626
1627 
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-<h6 id="org609a3df"><a href="#org609a3df">Line Integrals</a></h6>
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1630 1630 
 <p>
1631 1631 
 \[
1632 1632 
 {\int_{\bf a}^{\bf b}}_{\cal P} {\bf v} \cdot d{\bf l}
 
@@ -1655,9 +1655,9 @@ Integral over a closed loop:
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1656 1656 
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1657 1657
1658 
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-<h6 id="org8f657b7"><a href="#org8f657b7">Surface Integrals</a></h6>
1660 
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1658 
+<div id="outline-container-org8d260ce" class="outline-6">
1659 
+<h6 id="org8d260ce"><a href="#org8d260ce">Surface Integrals</a></h6>
1660 
+<div class="outline-text-6" id="text-org8d260ce">
1661 1661 
 <p>
1662 1662 
 \[
1663 1663 
 \int_{\cal S} {\bf v} \cdot d{\bf a}
 
@@ -1677,9 +1677,9 @@ Over a closed surface:
1677 1677 
 </div>
1678 1678 
 </div>
1679 1679
1680 
-<div id="outline-container-org2ec267e" class="outline-6">
1681 
-<h6 id="org2ec267e"><a href="#org2ec267e">Volume Integrals</a></h6>
1682 
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1680 
+<div id="outline-container-org57765dc" class="outline-6">
1681 
+<h6 id="org57765dc"><a href="#org57765dc">Volume Integrals</a></h6>
1682 
+<div class="outline-text-6" id="text-org57765dc">
1683 1683 
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1684 1684 
 \[
1685 1685 
 \int_{\cal V} T d\tau
 
@@ -1705,10 +1705,21 @@ d\tau = dx dy dz
1705 1705
1706 1706
1707 1707
1708 
-<hr><div id="postamble" class="status">
1708 
+<hr>
1709 
+<div class="license">
1710 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1711 
+target="_blank" class="m-2">
1712 
+<img alt="Creative Commons License" style="border-width:0"
1713 
+src="https://licensebuttons.net/l/by/4.0/80x15.png"/>
1714 
+</a>
1715 
+Except where otherwise noted, all content is licensed under a
1716 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1717 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1718 
+</div>
1719 
+<div id="postamble" class="status">
1709 1720 
 <p class="author">Author: Jean-Sébastien Caux</p>
1710 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1711 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1721 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1722 
+<p class="validation"></p>
1712 1723 
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1713 1724
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build/c_m_ic_stokes.html View File 

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3 3 
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4 
-<!-- 2022-02-08 Tue 17:21 -->
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+<!-- 2022-02-09 Wed 07:31 -->
5 5 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1616,20 +1616,20 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic_gauss.html">Gauss' Theorem&emsp;<small>[c.m.ic.gauss]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_ip.html">Integration by Parts&emsp;<small>[c.m.ic.ip]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul><div id="outline-container-c_m_ic_stokes" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_ic.html">Integral Calculus</a></li><li>Stokes' Theorem</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic_gauss.html">Gauss' Theorem&emsp;<small>[c.m.ic.gauss]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_ip.html">Integration by Parts&emsp;<small>[c.m.ic.ip]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul><div id="outline-container-c_m_ic_stokes" class="outline-5">
1620 1620 
 <h5 id="c_m_ic_stokes">Stokes' Theorem<a class="headline-permalink" href="./c_m_ic_stokes.html#c_m_ic_stokes"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
   <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"/>
1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
1623 1623 
 </svg></a><span class="headline-id">c.m.ic.stokes</span></h5>
1624 1624 
 <div class="outline-text-5" id="text-c_m_ic_stokes">
1625 
-<div class="eqlabel" id="org4321094">
1625 
+<div class="eqlabel" id="org5b773ee">
1626 1626 
 <p>
1627 1627 
 <a id="Stokes"></a><a href="./c_m_ic_stokes.html#Stokes"><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 
   <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
1629 1629 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
1630 1630 
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1631 1631 
 </p>
1632 
-<div class="alteqlabels" id="org03c3b7b">
1632 
+<div class="alteqlabels" id="org815e278">
1633 1633 
 <ul class="org-ul">
1634 1634 
 <li>Gr (1.57)</li>
1635 1635 
 </ul>
 
@@ -1659,10 +1659,21 @@ the boundary shrinks to a point.
1659 1659
1660 1660
1661 1661
1662 
-<hr><div id="postamble" class="status">
1662 
+<hr>
1663 
+<div class="license">
1664 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1665 
+target="_blank" class="m-2">
1666 
+<img alt="Creative Commons License" style="border-width:0"
1667 
+src="https://licensebuttons.net/l/by/4.0/80x15.png"/>
1668 
+</a>
1669 
+Except where otherwise noted, all content is licensed under a
1670 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1671 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1672 
+</div>
1673 
+<div id="postamble" class="status">
1663 1674 
 <p class="author">Author: Jean-Sébastien Caux</p>
1664 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1665 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1675 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1676 
+<p class="validation"></p>
1666 1677 
 </div>
1667 1678
1668 1679

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build/c_m_uf.html View File 

@@ -1,7 +1,7 @@
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2 2 
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3 3 
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4 
-<!-- 2022-02-08 Tue 17:21 -->
4 
+<!-- 2022-02-09 Wed 07:31 -->
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>
 
@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_vf_pot.html">Potentials&emsp;<small>[c.m.vf.pot]</small></a></li><li>Next:&nbsp;<a href="c_m_uf_cyl.html">Cylindrical coordinates&emsp;<small>[c.m.uf.cyl]</small></a></li><li>Up:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li></ul>
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li>Useful Formulas</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_vf_pot.html">Potentials&emsp;<small>[c.m.vf.pot]</small></a></li><li>Next:&nbsp;<a href="c_m_uf_cyl.html">Cylindrical coordinates&emsp;<small>[c.m.uf.cyl]</small></a></li><li>Up:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li></ul>
1620 1620 
 <h4 id="c_m_uf">Useful Formulas<a class="headline-permalink" href="./c_m_uf.html#c_m_uf"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1633,10 +1633,21 @@ Table of contents
1633 1633 
 <li><a href="c_m_uf_vi.html">Vector identities</a><span class="headline-id">c.m.uf.vi</span></li>
1634 1634 
 </ul>
1635 1635
1636 
-<hr><div id="postamble" class="status">
1636 
+<hr>
1637 
+<div class="license">
1638 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1639 
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1640 
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1641 
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1642 
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1643 
+Except where otherwise noted, all content is licensed under a
1644 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1645 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1646 
+</div>
1647 
+<div id="postamble" class="status">
1637 1648 
 <p class="author">Author: Jean-Sébastien Caux</p>
1638 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1639 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1649 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1650 
+<p class="validation"></p>
1640 1651 
 </div>
1641 1652
1642 1653

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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>
 
@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_uf.html">Useful Formulas&emsp;<small>[c.m.uf]</small></a></li><li>Next:&nbsp;<a href="c_m_uf_sph.html">Spherical coordinates&emsp;<small>[c.m.uf.sph]</small></a></li><li>Up:&nbsp;<a href="c_m_uf.html">Useful Formulas&emsp;<small>[c.m.uf]</small></a></li></ul><div id="outline-container-c_m_uf_cyl" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_uf.html">Useful Formulas</a></li><li>Cylindrical coordinates</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_uf.html">Useful Formulas&emsp;<small>[c.m.uf]</small></a></li><li>Next:&nbsp;<a href="c_m_uf_sph.html">Spherical coordinates&emsp;<small>[c.m.uf.sph]</small></a></li><li>Up:&nbsp;<a href="c_m_uf.html">Useful Formulas&emsp;<small>[c.m.uf]</small></a></li></ul><div id="outline-container-c_m_uf_cyl" class="outline-5">
1620 1620 
 <h5 id="c_m_uf_cyl">Cylindrical coordinates<a class="headline-permalink" href="./c_m_uf_cyl.html#c_m_uf_cyl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1646,10 +1646,21 @@ Table of contents
1646 1646
1647 1647
1648 1648
1649 
-<hr><div id="postamble" class="status">
1649 
+<hr>
1650 
+<div class="license">
1651 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1652 
+target="_blank" class="m-2">
1653 
+<img alt="Creative Commons License" style="border-width:0"
1654 
+src="https://licensebuttons.net/l/by/4.0/80x15.png"/>
1655 
+</a>
1656 
+Except where otherwise noted, all content is licensed under a
1657 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1658 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1659 
+</div>
1660 
+<div id="postamble" class="status">
1650 1661 
 <p class="author">Author: Jean-Sébastien Caux</p>
1651 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1652 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1662 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1663 
+<p class="validation"></p>
1653 1664 
 </div>
1654 1665
1655 1666

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7 7 
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_uf_cyl.html">Cylindrical coordinates&emsp;<small>[c.m.uf.cyl]</small></a></li><li>Next:&nbsp;<a href="c_m_uf_vi.html">Vector identities&emsp;<small>[c.m.uf.vi]</small></a></li><li>Up:&nbsp;<a href="c_m_uf.html">Useful Formulas&emsp;<small>[c.m.uf]</small></a></li></ul><div id="outline-container-c_m_uf_sph" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_uf.html">Useful Formulas</a></li><li>Spherical coordinates</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_uf_cyl.html">Cylindrical coordinates&emsp;<small>[c.m.uf.cyl]</small></a></li><li>Next:&nbsp;<a href="c_m_uf_vi.html">Vector identities&emsp;<small>[c.m.uf.vi]</small></a></li><li>Up:&nbsp;<a href="c_m_uf.html">Useful Formulas&emsp;<small>[c.m.uf]</small></a></li></ul><div id="outline-container-c_m_uf_sph" class="outline-5">
1620 1620 
 <h5 id="c_m_uf_sph">Spherical coordinates<a class="headline-permalink" href="./c_m_uf_sph.html#c_m_uf_sph"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
   <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"/>
1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1646,10 +1646,21 @@ Table of contents
1646 1646
1647 1647
1648 1648
1649 
-<hr><div id="postamble" class="status">
1649 
+<hr>
1650 
+<div class="license">
1651 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1652 
+target="_blank" class="m-2">
1653 
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1654 
+src="https://licensebuttons.net/l/by/4.0/80x15.png"/>
1655 
+</a>
1656 
+Except where otherwise noted, all content is licensed under a
1657 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1658 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1659 
+</div>
1660 
+<div id="postamble" class="status">
1650 1661 
 <p class="author">Author: Jean-Sébastien Caux</p>
1651 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1652 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1662 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1663 
+<p class="validation"></p>
1653 1664 
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1654 1665
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_uf_sph.html">Spherical coordinates&emsp;<small>[c.m.uf.sph]</small></a></li><li>Up:&nbsp;<a href="c_m_uf.html">Useful Formulas&emsp;<small>[c.m.uf]</small></a></li></ul><div id="outline-container-c_m_uf_vi" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_uf.html">Useful Formulas</a></li><li>Vector identities</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_uf_sph.html">Spherical coordinates&emsp;<small>[c.m.uf.sph]</small></a></li><li>Up:&nbsp;<a href="c_m_uf.html">Useful Formulas&emsp;<small>[c.m.uf]</small></a></li></ul><div id="outline-container-c_m_uf_vi" class="outline-5">
1620 1620 
 <h5 id="c_m_uf_vi">Vector identities<a class="headline-permalink" href="./c_m_uf_vi.html#c_m_uf_vi"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1649,10 +1649,21 @@ Table of contents
1649 1649 
 </div>
1650 1650
1651 1651
1652 
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+<hr>
1653 
+<div class="license">
1654 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1655 
+target="_blank" class="m-2">
1656 
+<img alt="Creative Commons License" style="border-width:0"
1657 
+src="https://licensebuttons.net/l/by/4.0/80x15.png"/>
1658 
+</a>
1659 
+Except where otherwise noted, all content is licensed under a
1660 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1661 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1662 
+</div>
1663 
+<div id="postamble" class="status">
1653 1664 
 <p class="author">Author: Jean-Sébastien Caux</p>
1654 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1655 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
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-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li><li>Next:&nbsp;<a href="c_m_va_n.html">Notation and algebraic properties&emsp;<small>[c.m.va.n]</small></a></li><li>Up:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li></ul>
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li>Vector Analysis</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li><li>Next:&nbsp;<a href="c_m_va_n.html">Notation and algebraic properties&emsp;<small>[c.m.va.n]</small></a></li><li>Up:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li></ul>
1620 1620 
 <h4 id="c_m_va">Vector Analysis<a class="headline-permalink" href="./c_m_va.html#c_m_va"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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@@ -1635,10 +1635,21 @@ Table of contents
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 <li><a href="c_m_va_pds.html">Position, Displacement and Separation Vectors</a><span class="headline-id">c.m.va.pds</span></li>
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1648 
+</div>
1649 
+<div id="postamble" class="status">
1639 1650 
 <p class="author">Author: Jean-Sébastien Caux</p>
1640 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1641 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_va_sp.html">Scalar product&emsp;<small>[c.m.va.sp]</small></a></li><li>Next:&nbsp;<a href="c_m_va_tp.html">Triple Products&emsp;<small>[c.m.va.tp]</small></a></li><li>Up:&nbsp;<a href="c_m_va.html">Vector Analysis&emsp;<small>[c.m.va]</small></a></li></ul><div id="outline-container-c_m_va_cp" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_va.html">Vector Analysis</a></li><li>Cross product</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_va_sp.html">Scalar product&emsp;<small>[c.m.va.sp]</small></a></li><li>Next:&nbsp;<a href="c_m_va_tp.html">Triple Products&emsp;<small>[c.m.va.tp]</small></a></li><li>Up:&nbsp;<a href="c_m_va.html">Vector Analysis&emsp;<small>[c.m.va]</small></a></li></ul><div id="outline-container-c_m_va_cp" class="outline-5">
1620 1620 
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@@ -1664,10 +1664,21 @@ this relation making plain that
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+</a>
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+Except where otherwise noted, all content is licensed under a
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+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
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+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1677 
+</div>
1678 
+<div id="postamble" class="status">
1668 1679 
 <p class="author">Author: Jean-Sébastien Caux</p>
1669 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1670 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_va.html">Vector Analysis&emsp;<small>[c.m.va]</small></a></li><li>Next:&nbsp;<a href="c_m_va_sp.html">Scalar product&emsp;<small>[c.m.va.sp]</small></a></li><li>Up:&nbsp;<a href="c_m_va.html">Vector Analysis&emsp;<small>[c.m.va]</small></a></li></ul><div id="outline-container-c_m_va_n" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_va.html">Vector Analysis</a></li><li>Notation and algebraic properties</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_va.html">Vector Analysis&emsp;<small>[c.m.va]</small></a></li><li>Next:&nbsp;<a href="c_m_va_sp.html">Scalar product&emsp;<small>[c.m.va.sp]</small></a></li><li>Up:&nbsp;<a href="c_m_va.html">Vector Analysis&emsp;<small>[c.m.va]</small></a></li></ul><div id="outline-container-c_m_va_n" class="outline-5">
1620 1620 
 <h5 id="c_m_va_n">Notation and algebraic properties<a class="headline-permalink" href="./c_m_va_n.html#c_m_va_n"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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+</div>
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+<div id="postamble" class="status">
1663 1674 
 <p class="author">Author: Jean-Sébastien Caux</p>
1664 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1665 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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 </ul>
1617 1617 
 </details>
1618 1618 
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1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_va_tp.html">Triple Products&emsp;<small>[c.m.va.tp]</small></a></li><li>Next:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li><li>Up:&nbsp;<a href="c_m_va.html">Vector Analysis&emsp;<small>[c.m.va]</small></a></li></ul><div id="outline-container-c_m_va_pds" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_va.html">Vector Analysis</a></li><li>Position, Displacement and Separation Vectors</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_va_tp.html">Triple Products&emsp;<small>[c.m.va.tp]</small></a></li><li>Next:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li><li>Up:&nbsp;<a href="c_m_va.html">Vector Analysis&emsp;<small>[c.m.va]</small></a></li></ul><div id="outline-container-c_m_va_pds" class="outline-5">
1620 1620 
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+</div>
1707 
+<div id="postamble" class="status">
1697 1708 
 <p class="author">Author: Jean-Sébastien Caux</p>
1698 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1699 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1709 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_va_n.html">Notation and algebraic properties&emsp;<small>[c.m.va.n]</small></a></li><li>Next:&nbsp;<a href="c_m_va_cp.html">Cross product&emsp;<small>[c.m.va.cp]</small></a></li><li>Up:&nbsp;<a href="c_m_va.html">Vector Analysis&emsp;<small>[c.m.va]</small></a></li></ul><div id="outline-container-c_m_va_sp" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_va.html">Vector Analysis</a></li><li>Scalar product</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_va_n.html">Notation and algebraic properties&emsp;<small>[c.m.va.n]</small></a></li><li>Next:&nbsp;<a href="c_m_va_cp.html">Cross product&emsp;<small>[c.m.va.cp]</small></a></li><li>Up:&nbsp;<a href="c_m_va.html">Vector Analysis&emsp;<small>[c.m.va]</small></a></li></ul><div id="outline-container-c_m_va_sp" class="outline-5">
1620 1620 
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+<div id="postamble" class="status">
1663 1674 
 <p class="author">Author: Jean-Sébastien Caux</p>
1664 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1665 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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1617 1617 
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1618 1618 
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1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_va_cp.html">Cross product&emsp;<small>[c.m.va.cp]</small></a></li><li>Next:&nbsp;<a href="c_m_va_pds.html">Position, Displacement and Separation Vectors&emsp;<small>[c.m.va.pds]</small></a></li><li>Up:&nbsp;<a href="c_m_va.html">Vector Analysis&emsp;<small>[c.m.va]</small></a></li></ul><div id="outline-container-c_m_va_tp" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_va.html">Vector Analysis</a></li><li>Triple Products</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_va_cp.html">Cross product&emsp;<small>[c.m.va.cp]</small></a></li><li>Next:&nbsp;<a href="c_m_va_pds.html">Position, Displacement and Separation Vectors&emsp;<small>[c.m.va.pds]</small></a></li><li>Up:&nbsp;<a href="c_m_va.html">Vector Analysis&emsp;<small>[c.m.va]</small></a></li></ul><div id="outline-container-c_m_va_tp" class="outline-5">
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1716 
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1717 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1718 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1719 
+</div>
1720 
+<div id="postamble" class="status">
1710 1721 
 <p class="author">Author: Jean-Sébastien Caux</p>
1711 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1712 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1722 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1723 
+<p class="validation"></p>
1713 1724 
 </div>
1714 1725
1715 1726

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4 
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+<!-- 2022-02-09 Wed 07:31 -->
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dd_3d.html">The Three-Dimensional Delta Function&emsp;<small>[c.m.dd.3d]</small></a></li><li>Next:&nbsp;<a href="c_m_vf_helm.html">The Helmholtz Theorem&emsp;<small>[c.m.vf.helm]</small></a></li><li>Up:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li></ul>
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li>Vector Fields</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dd_3d.html">The Three-Dimensional Delta Function&emsp;<small>[c.m.dd.3d]</small></a></li><li>Next:&nbsp;<a href="c_m_vf_helm.html">The Helmholtz Theorem&emsp;<small>[c.m.vf.helm]</small></a></li><li>Up:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li></ul>
1620 1620 
 <h4 id="c_m_vf">Vector Fields<a class="headline-permalink" href="./c_m_vf.html#c_m_vf"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
   <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"/>
1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1632,10 +1632,21 @@ Table of contents
1632 1632 
 <li><a href="c_m_vf_pot.html">Potentials</a><span class="headline-id">c.m.vf.pot</span></li>
1633 1633 
 </ul>
1634 1634
1635 
-<hr><div id="postamble" class="status">
1635 
+<hr>
1636 
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1637 
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1638 
+target="_blank" class="m-2">
1639 
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1643 
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1644 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1645 
+</div>
1646 
+<div id="postamble" class="status">
1636 1647 
 <p class="author">Author: Jean-Sébastien Caux</p>
1637 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1638 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1648 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1649 
+<p class="validation"></p>
1639 1650 
 </div>
1640 1651
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+<!-- 2022-02-09 Wed 07:31 -->
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_vf.html">Vector Fields&emsp;<small>[c.m.vf]</small></a></li><li>Next:&nbsp;<a href="c_m_vf_pot.html">Potentials&emsp;<small>[c.m.vf.pot]</small></a></li><li>Up:&nbsp;<a href="c_m_vf.html">Vector Fields&emsp;<small>[c.m.vf]</small></a></li></ul><div id="outline-container-c_m_vf_helm" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_vf.html">Vector Fields</a></li><li>The Helmholtz Theorem</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_vf.html">Vector Fields&emsp;<small>[c.m.vf]</small></a></li><li>Next:&nbsp;<a href="c_m_vf_pot.html">Potentials&emsp;<small>[c.m.vf.pot]</small></a></li><li>Up:&nbsp;<a href="c_m_vf.html">Vector Fields&emsp;<small>[c.m.vf]</small></a></li></ul><div id="outline-container-c_m_vf_helm" class="outline-5">
1620 1620 
 <h5 id="c_m_vf_helm">The Helmholtz Theorem<a class="headline-permalink" href="./c_m_vf_helm.html#c_m_vf_helm"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1651,10 +1651,21 @@ where
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1652 1652
1653 1653
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-<hr><div id="postamble" class="status">
1654 
+<hr>
1655 
+<div class="license">
1656 
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1661 
+Except where otherwise noted, all content is licensed under a
1662 
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1663 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1664 
+</div>
1665 
+<div id="postamble" class="status">
1655 1666 
 <p class="author">Author: Jean-Sébastien Caux</p>
1656 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1657 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1667 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1668 
+<p class="validation"></p>
1658 1669 
 </div>
1659 1670
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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>
 
@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_vf_helm.html">The Helmholtz Theorem&emsp;<small>[c.m.vf.helm]</small></a></li><li>Next:&nbsp;<a href="c_m_uf.html">Useful Formulas&emsp;<small>[c.m.uf]</small></a></li><li>Up:&nbsp;<a href="c_m_vf.html">Vector Fields&emsp;<small>[c.m.vf]</small></a></li></ul><div id="outline-container-c_m_vf_pot" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_vf.html">Vector Fields</a></li><li>Potentials</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_vf_helm.html">The Helmholtz Theorem&emsp;<small>[c.m.vf.helm]</small></a></li><li>Next:&nbsp;<a href="c_m_uf.html">Useful Formulas&emsp;<small>[c.m.uf]</small></a></li><li>Up:&nbsp;<a href="c_m_vf.html">Vector Fields&emsp;<small>[c.m.vf]</small></a></li></ul><div id="outline-container-c_m_vf_pot" class="outline-5">
1620 1620 
 <h5 id="c_m_vf_pot">Potentials<a class="headline-permalink" href="./c_m_vf_pot.html#c_m_vf_pot"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
   <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"/>
1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1678,10 +1678,21 @@ The following conditions are equivalent:
1678 1678 
 </div>
1679 1679
1680 1680
1681 
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1681 
+<hr>
1682 
+<div class="license">
1683 
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1687 
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1688 
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1689 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1690 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1691 
+</div>
1692 
+<div id="postamble" class="status">
1682 1693 
 <p class="author">Author: Jean-Sébastien Caux</p>
1683 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1684 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1694 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1695 
+<p class="validation"></p>
1685 1696 
 </div>
1686 1697
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7 7 
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@@ -1616,15 +1616,15 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Next:&nbsp;<a href="d_m.html">Diagnostics: Mathematical Preliminaries&emsp;<small>[d.m]</small></a></li></ul>
1619 
+<ul class="breadcrumbs"><li>Diagnostics</li></ul><ul class="navigation-links"><li>Next:&nbsp;<a href="d_m.html">Diagnostics: Mathematical Preliminaries&emsp;<small>[d.m]</small></a></li></ul>
1620 1620 
 <h2 id="d">Diagnostics<a class="headline-permalink" href="./d.html#d"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
1623 1623 
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1624 1624
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1627 
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1628 1628 
 Objectives
1629 1629 
 </summary>
1630 1630 
 
@@ -1673,10 +1673,21 @@ I have little doubt you'll pass the course.
1673 1673 
 <li><a href="d_red.html">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span></li>
1674 1674 
 </ul>
1675 1675
1676 
-<hr><div id="postamble" class="status">
1676 
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1677 
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1678 
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1679 
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1680 
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1683 
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1684 
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1685 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1686 
+</div>
1687 
+<div id="postamble" class="status">
1677 1688 
 <p class="author">Author: Jean-Sébastien Caux</p>
1678 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1679 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1689 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1690 
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1680 1691 
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1681 1692
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="d_emsm_msm.html">Diagnostics: Magnetostatics in Matter&emsp;<small>[d.emsm.msm]</small></a></li><li>Next:&nbsp;<a href="d_emd_ce.html">Diagnostics: Conservation Laws&emsp;<small>[d.emd.ce]</small></a></li><li>Up:&nbsp;<a href="d.html">Diagnostics&emsp;<small>[d]</small></a></li></ul><div id="outline-container-d_emd" class="outline-3">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="d.html">Diagnostics</a></li><li>Diagnostics: Electromagnetodynamics</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="d_emsm_msm.html">Diagnostics: Magnetostatics in Matter&emsp;<small>[d.emsm.msm]</small></a></li><li>Next:&nbsp;<a href="d_emd_ce.html">Diagnostics: Conservation Laws&emsp;<small>[d.emd.ce]</small></a></li><li>Up:&nbsp;<a href="d.html">Diagnostics&emsp;<small>[d]</small></a></li></ul><div id="outline-container-d_emd" class="outline-3">
1620 1620 
 <h3 id="d_emd">Diagnostics: Electromagnetodynamics<a class="headline-permalink" href="./d_emd.html#d_emd"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1653,10 +1653,21 @@ As a strict minimum, you should be able to:
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+<hr>
1657 
+<div class="license">
1658 
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1659 
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1660 
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1661 
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1662 
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1663 
+Except where otherwise noted, all content is licensed under a
1664 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1665 
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1666 
+</div>
1667 
+<div id="postamble" class="status">
1657 1668 
 <p class="author">Author: Jean-Sébastien Caux</p>
1658 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1659 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1669 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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1660 1671 
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1661 1672
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6 6 
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7 7 
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="d_emd.html">Diagnostics: Electromagnetodynamics&emsp;<small>[d.emd]</small></a></li><li>Next:&nbsp;<a href="d_emd_emw.html">Diagnostics: Electromagnetic Waves&emsp;<small>[d.emd.emw]</small></a></li><li>Up:&nbsp;<a href="d_emd.html">Diagnostics: Electromagnetodynamics&emsp;<small>[d.emd]</small></a></li></ul><div id="outline-container-d_emd_ce" class="outline-3">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="d.html">Diagnostics</a></li><li><a class="breadcrumb-link"href="d_emd.html">Diagnostics: Electromagnetodynamics</a></li><li>Diagnostics: Conservation Laws</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="d_emd.html">Diagnostics: Electromagnetodynamics&emsp;<small>[d.emd]</small></a></li><li>Next:&nbsp;<a href="d_emd_emw.html">Diagnostics: Electromagnetic Waves&emsp;<small>[d.emd.emw]</small></a></li><li>Up:&nbsp;<a href="d_emd.html">Diagnostics: Electromagnetodynamics&emsp;<small>[d.emd]</small></a></li></ul><div id="outline-container-d_emd_ce" class="outline-3">
1620 1620 
 <h3 id="d_emd_ce">Diagnostics: Conservation Laws<a class="headline-permalink" href="./d_emd_ce.html#d_emd_ce"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1648,10 +1648,21 @@ As a strict minimum, you should be able to:
1648 1648
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1651 
+<hr>
1652 
+<div class="license">
1653 
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1659 
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1660 
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+<div id="postamble" class="status">
1652 1663 
 <p class="author">Author: Jean-Sébastien Caux</p>
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-<p class="date">Created: 2022-02-08 Tue 17:21</p>
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-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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@@ -1616,7 +1616,7 @@ Table of contents
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 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="d_emd_ce.html">Diagnostics: Conservation Laws&emsp;<small>[d.emd.ce]</small></a></li><li>Next:&nbsp;<a href="d_emf.html">Diagnostics: Potentials, Gauges and Fields&emsp;<small>[d.emf]</small></a></li><li>Up:&nbsp;<a href="d_emd.html">Diagnostics: Electromagnetodynamics&emsp;<small>[d.emd]</small></a></li></ul><div id="outline-container-d_emd_emw" class="outline-3">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="d.html">Diagnostics</a></li><li><a class="breadcrumb-link"href="d_emd.html">Diagnostics: Electromagnetodynamics</a></li><li>Diagnostics: Electromagnetic Waves</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="d_emd_ce.html">Diagnostics: Conservation Laws&emsp;<small>[d.emd.ce]</small></a></li><li>Next:&nbsp;<a href="d_emf.html">Diagnostics: Potentials, Gauges and Fields&emsp;<small>[d.emf]</small></a></li><li>Up:&nbsp;<a href="d_emd.html">Diagnostics: Electromagnetodynamics&emsp;<small>[d.emd]</small></a></li></ul><div id="outline-container-d_emd_emw" class="outline-3">
1620 1620 
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@@ -1651,10 +1651,21 @@ As a strict minimum, you should be able to:
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+<hr>
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+<div class="license">
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+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
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+</a>
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+Except where otherwise noted, all content is licensed under a
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+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
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+target="_blank">Creative Commons Attribution 4.0 International License</a>.
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+</div>
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+<div id="postamble" class="status">
1655 1666 
 <p class="author">Author: Jean-Sébastien Caux</p>
1656 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
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-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="d_emd_emw.html">Diagnostics: Electromagnetic Waves&emsp;<small>[d.emd.emw]</small></a></li><li>Next:&nbsp;<a href="d_red.html">Diagnostics: Relativistic Electrodynamics&emsp;<small>[d.red]</small></a></li><li>Up:&nbsp;<a href="d.html">Diagnostics&emsp;<small>[d]</small></a></li></ul><div id="outline-container-d_emf" class="outline-3">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="d.html">Diagnostics</a></li><li>Diagnostics: Potentials, Gauges and Fields</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="d_emd_emw.html">Diagnostics: Electromagnetic Waves&emsp;<small>[d.emd.emw]</small></a></li><li>Next:&nbsp;<a href="d_red.html">Diagnostics: Relativistic Electrodynamics&emsp;<small>[d.red]</small></a></li><li>Up:&nbsp;<a href="d.html">Diagnostics&emsp;<small>[d]</small></a></li></ul><div id="outline-container-d_emf" class="outline-3">
1620 1620 
 <h3 id="d_emf">Diagnostics: Potentials, Gauges and Fields<a class="headline-permalink" href="./d_emf.html#d_emf"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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@@ -1648,10 +1648,21 @@ As a strict minimum, you should be able to:
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+<hr>
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+<div class="license">
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+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
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+<img alt="Creative Commons License" style="border-width:0"
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+Except where otherwise noted, all content is licensed under a
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+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
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+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1661 
+</div>
1662 
+<div id="postamble" class="status">
1652 1663 
 <p class="author">Author: Jean-Sébastien Caux</p>
1653 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1654 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1664 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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@@ -1616,7 +1616,7 @@ Table of contents
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 </ul>
1617 1617 
 </details>
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 </nav>
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-<ul class="navigation-links"><li>Prev:&nbsp;<a href="d_m.html">Diagnostics: Mathematical Preliminaries&emsp;<small>[d.m]</small></a></li><li>Next:&nbsp;<a href="d_ems_ca.html">Diagnostics: Calculating or Approximating the Electostatic Potential&emsp;<small>[d.ems.ca]</small></a></li><li>Up:&nbsp;<a href="d.html">Diagnostics&emsp;<small>[d]</small></a></li></ul><div id="outline-container-d_ems" class="outline-3">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="d.html">Diagnostics</a></li><li>Diagnostics: Electromagnetostatics</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="d_m.html">Diagnostics: Mathematical Preliminaries&emsp;<small>[d.m]</small></a></li><li>Next:&nbsp;<a href="d_ems_ca.html">Diagnostics: Calculating or Approximating the Electostatic Potential&emsp;<small>[d.ems.ca]</small></a></li><li>Up:&nbsp;<a href="d.html">Diagnostics&emsp;<small>[d]</small></a></li></ul><div id="outline-container-d_ems" class="outline-3">
1620 1620 
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@@ -1670,10 +1670,21 @@ As a strict minimum, you should be able to:
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+Except where otherwise noted, all content is licensed under a
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+target="_blank">Creative Commons Attribution 4.0 International License</a>.
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+</div>
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+<div id="postamble" class="status">
1674 1685 
 <p class="author">Author: Jean-Sébastien Caux</p>
1675 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1676 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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 </ul>
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 </details>
1618 1618 
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1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="d_ems.html">Diagnostics: Electromagnetostatics&emsp;<small>[d.ems]</small></a></li><li>Next:&nbsp;<a href="d_emsm.html">Diagnostics: Electromagnetostatics in Matter&emsp;<small>[d.emsm]</small></a></li><li>Up:&nbsp;<a href="d_ems.html">Diagnostics: Electromagnetostatics&emsp;<small>[d.ems]</small></a></li></ul><div id="outline-container-d_ems_ca" class="outline-3">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="d.html">Diagnostics</a></li><li><a class="breadcrumb-link"href="d_ems.html">Diagnostics: Electromagnetostatics</a></li><li>Diagnostics: Calculating or Approximating the Electostatic Potential</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="d_ems.html">Diagnostics: Electromagnetostatics&emsp;<small>[d.ems]</small></a></li><li>Next:&nbsp;<a href="d_emsm.html">Diagnostics: Electromagnetostatics in Matter&emsp;<small>[d.emsm]</small></a></li><li>Up:&nbsp;<a href="d_ems.html">Diagnostics: Electromagnetostatics&emsp;<small>[d.ems]</small></a></li></ul><div id="outline-container-d_ems_ca" class="outline-3">
1620 1620 
 <h3 id="d_ems_ca">Diagnostics: Calculating or Approximating the Electostatic Potential<a class="headline-permalink" href="./d_ems_ca.html#d_ems_ca"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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@@ -1653,10 +1653,21 @@ a point source charge \(q\) at \({\bf r}_s\) with an infinite grounded conductin
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+Except where otherwise noted, all content is licensed under a
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+target="_blank">Creative Commons Attribution 4.0 International License</a>.
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+</div>
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+<div id="postamble" class="status">
1657 1668 
 <p class="author">Author: Jean-Sébastien Caux</p>
1658 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1659 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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 </ul>
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 </details>
1618 1618 
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1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="d_emsm.html">Diagnostics: Electromagnetostatics in Matter&emsp;<small>[d.emsm]</small></a></li><li>Next:&nbsp;<a href="d_emsm_msm.html">Diagnostics: Magnetostatics in Matter&emsp;<small>[d.emsm.msm]</small></a></li><li>Up:&nbsp;<a href="d_ems.html">Diagnostics: Electromagnetostatics&emsp;<small>[d.ems]</small></a></li></ul><div id="outline-container-d_ems_ms" class="outline-3">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="d.html">Diagnostics</a></li><li><a class="breadcrumb-link"href="d_ems.html">Diagnostics: Electromagnetostatics</a></li><li>Diagnostics: Magnetostatics</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="d_emsm.html">Diagnostics: Electromagnetostatics in Matter&emsp;<small>[d.emsm]</small></a></li><li>Next:&nbsp;<a href="d_emsm_msm.html">Diagnostics: Magnetostatics in Matter&emsp;<small>[d.emsm.msm]</small></a></li><li>Up:&nbsp;<a href="d_ems.html">Diagnostics: Electromagnetostatics&emsp;<small>[d.ems]</small></a></li></ul><div id="outline-container-d_ems_ms" class="outline-3">
1620 1620 
 <h3 id="d_ems_ms">Diagnostics: Magnetostatics<a class="headline-permalink" href="./d_ems_ms.html#d_ems_ms"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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@@ -1650,10 +1650,21 @@ As a strict minimum, you should be able to:
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+<hr>
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+</a>
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+Except where otherwise noted, all content is licensed under a
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+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
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+target="_blank">Creative Commons Attribution 4.0 International License</a>.
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+<div id="postamble" class="status">
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 <p class="author">Author: Jean-Sébastien Caux</p>
1655 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1656 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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 </ul>
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 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="d_ems_ca.html">Diagnostics: Calculating or Approximating the Electostatic Potential&emsp;<small>[d.ems.ca]</small></a></li><li>Next:&nbsp;<a href="d_ems_ms.html">Diagnostics: Magnetostatics&emsp;<small>[d.ems.ms]</small></a></li><li>Up:&nbsp;<a href="d.html">Diagnostics&emsp;<small>[d]</small></a></li></ul><div id="outline-container-d_emsm" class="outline-3">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="d.html">Diagnostics</a></li><li>Diagnostics: Electromagnetostatics in Matter</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="d_ems_ca.html">Diagnostics: Calculating or Approximating the Electostatic Potential&emsp;<small>[d.ems.ca]</small></a></li><li>Next:&nbsp;<a href="d_ems_ms.html">Diagnostics: Magnetostatics&emsp;<small>[d.ems.ms]</small></a></li><li>Up:&nbsp;<a href="d.html">Diagnostics&emsp;<small>[d]</small></a></li></ul><div id="outline-container-d_emsm" class="outline-3">
1620 1620 
 <h3 id="d_emsm">Diagnostics: Electromagnetostatics in Matter<a class="headline-permalink" href="./d_emsm.html#d_emsm"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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@@ -1653,10 +1653,21 @@ As a strict minimum, you should be able to:
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+<hr>
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+<div class="license">
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+<img alt="Creative Commons License" style="border-width:0"
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+src="https://licensebuttons.net/l/by/4.0/80x15.png"/>
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+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
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+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="date">Created: 2022-02-08 Tue 17:21</p>
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-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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@@ -1616,7 +1616,7 @@ Table of contents
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1617 1617 
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1618 1618 
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-<ul class="navigation-links"><li>Prev:&nbsp;<a href="d_ems_ms.html">Diagnostics: Magnetostatics&emsp;<small>[d.ems.ms]</small></a></li><li>Next:&nbsp;<a href="d_emd.html">Diagnostics: Electromagnetodynamics&emsp;<small>[d.emd]</small></a></li><li>Up:&nbsp;<a href="d_emsm.html">Diagnostics: Electromagnetostatics in Matter&emsp;<small>[d.emsm]</small></a></li></ul><div id="outline-container-d_emsm_msm" class="outline-3">
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+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="d.html">Diagnostics</a></li><li><a class="breadcrumb-link"href="d_emsm.html">Diagnostics: Electromagnetostatics in Matter</a></li><li>Diagnostics: Magnetostatics in Matter</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="d_ems_ms.html">Diagnostics: Magnetostatics&emsp;<small>[d.ems.ms]</small></a></li><li>Next:&nbsp;<a href="d_emd.html">Diagnostics: Electromagnetodynamics&emsp;<small>[d.emd]</small></a></li><li>Up:&nbsp;<a href="d_emsm.html">Diagnostics: Electromagnetostatics in Matter&emsp;<small>[d.emsm]</small></a></li></ul><div id="outline-container-d_emsm_msm" class="outline-3">
1620 1620 
 <h3 id="d_emsm_msm">Diagnostics: Magnetostatics in Matter<a class="headline-permalink" href="./d_emsm_msm.html#d_emsm_msm"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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   <path d="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"/>
 
@@ -1651,10 +1651,21 @@ As a strict minimum, you should be able to:
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+<div id="postamble" class="status">
1655 1666 
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1656 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1657 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="d.html">Diagnostics&emsp;<small>[d]</small></a></li><li>Next:&nbsp;<a href="d_ems.html">Diagnostics: Electromagnetostatics&emsp;<small>[d.ems]</small></a></li><li>Up:&nbsp;<a href="d.html">Diagnostics&emsp;<small>[d]</small></a></li></ul><div id="outline-container-d_m" class="outline-3">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="d.html">Diagnostics</a></li><li>Diagnostics: Mathematical Preliminaries</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="d.html">Diagnostics&emsp;<small>[d]</small></a></li><li>Next:&nbsp;<a href="d_ems.html">Diagnostics: Electromagnetostatics&emsp;<small>[d.ems]</small></a></li><li>Up:&nbsp;<a href="d.html">Diagnostics&emsp;<small>[d]</small></a></li></ul><div id="outline-container-d_m" class="outline-3">
1620 1620 
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1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1661,10 +1661,21 @@ Things you should be able to do (ideally: from scratch, on a blank sheet of pape
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-<hr><div id="postamble" class="status">
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+<hr>
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1674 
+</div>
1675 
+<div id="postamble" class="status">
1665 1676 
 <p class="author">Author: Jean-Sébastien Caux</p>
1666 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1667 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1677 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="d_emf.html">Diagnostics: Potentials, Gauges and Fields&emsp;<small>[d.emf]</small></a></li><li>Next:&nbsp;<a href="a.html">Appendices&emsp;<small>[a]</small></a></li><li>Up:&nbsp;<a href="d.html">Diagnostics&emsp;<small>[d]</small></a></li></ul><div id="outline-container-d_red" class="outline-3">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="d.html">Diagnostics</a></li><li>Diagnostics: Relativistic Electrodynamics</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="d_emf.html">Diagnostics: Potentials, Gauges and Fields&emsp;<small>[d.emf]</small></a></li><li>Next:&nbsp;<a href="a.html">Appendices&emsp;<small>[a]</small></a></li><li>Up:&nbsp;<a href="d.html">Diagnostics&emsp;<small>[d]</small></a></li></ul><div id="outline-container-d_red" class="outline-3">
1620 1620 
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@@ -1655,10 +1655,21 @@ As a strict minimum, you should be able to:
1655 1655 
 </div>
1656 1656
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-<hr><div id="postamble" class="status">
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+<hr>
1659 
+<div class="license">
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1668 
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1669 
+<div id="postamble" class="status">
1659 1670 
 <p class="author">Author: Jean-Sébastien Caux</p>
1660 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1661 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1671 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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1616 1616 
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1617 1617 
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1618 1618 
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1619 
-<ul class="navigation-links"><li>Next:&nbsp;<a href="emd_Fl.html">Induction: Faraday's Law&emsp;<small>[emd.Fl]</small></a></li></ul>
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1620 1620 
 <h2 id="emd">Electromagnetodynamics<a class="headline-permalink" href="./emd.html#emd"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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1628 1628 
 Prerequisites
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@@ -1633,8 +1633,8 @@ Prerequisites
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 Objectives
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 <ul class="org-ul">
 
@@ -1660,10 +1660,21 @@ Objectives
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 <li><a href="emd_emw.html">Electromagnetic waves in vacuum</a><span class="headline-id">emd.emw</span></li>
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 </ul>
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1673 
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1674 
+<div id="postamble" class="status">
1664 1675 
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1665 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1666 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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1617 1617 
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1618 1618 
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-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emd.html">Electromagnetodynamics&emsp;<small>[emd]</small></a></li><li>Next:&nbsp;<a href="emd_Fl_Fl.html">Faraday's Law&emsp;<small>[emd.Fl.Fl]</small></a></li><li>Up:&nbsp;<a href="emd.html">Electromagnetodynamics&emsp;<small>[emd]</small></a></li></ul>
1619 
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1620 1620 
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 <li><a href="emd_Fl_e.html">Energy in Magnetic Fields</a><span class="headline-id">emd.Fl.e</span></li>
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1647 
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1648 
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1638 1649 
 <p class="author">Author: Jean-Sébastien Caux</p>
1639 
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 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Fl.html">Induction: Faraday's Law&emsp;<small>[emd.Fl]</small></a></li><li>Next:&nbsp;<a href="emd_Fl_ief.html">The Induced Electric Field&emsp;<small>[emd.Fl.ief]</small></a></li><li>Up:&nbsp;<a href="emd_Fl.html">Induction: Faraday's Law&emsp;<small>[emd.Fl]</small></a></li></ul><div id="outline-container-emd_Fl_Fl" class="outline-4">
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+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emd.html">Electromagnetodynamics</a></li><li><a class="breadcrumb-link"href="emd_Fl.html">Induction: Faraday's Law</a></li><li>Faraday's Law</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Fl.html">Induction: Faraday's Law&emsp;<small>[emd.Fl]</small></a></li><li>Next:&nbsp;<a href="emd_Fl_ief.html">The Induced Electric Field&emsp;<small>[emd.Fl.ief]</small></a></li><li>Up:&nbsp;<a href="emd_Fl.html">Induction: Faraday's Law&emsp;<small>[emd.Fl]</small></a></li></ul><div id="outline-container-emd_Fl_Fl" class="outline-4">
1620 1620 
 <h4 id="emd_Fl_Fl">Faraday's Law<a class="headline-permalink" href="./emd_Fl_Fl.html#emd_Fl_Fl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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@@ -1663,7 +1663,7 @@ Empirically:  the changing magnetic field induces an electric current around
1663 1663 
 the circuit. This current is really driven by an electric field having a component
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 along the wire.  The line integral of this field is called the
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1668 1668 
 <b>Electromotive force (or electromotance)</b>,
1669 1669 
   \[
 
@@ -1685,7 +1685,7 @@ to the rate of change of the magnetic flux,
1685 1685 
 \]
1686 1686 
 so we obtain
1687 1687 
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1690 1690 
 <b>Faraday's law</b> (integral form <i>N.B.: for a stationary loop</i>)
1691 1691 
   \[
 
@@ -1703,7 +1703,7 @@ for any loop (on a wire or not). Using Stokes' theorem,
1703 1703 
 \]
1704 1704 
 we obtain
1705 1705 
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 <p>
1708 1708 
 <b>Faraday's law</b> (differential form)
1709 1709 
   \[
 
@@ -1725,10 +1725,21 @@ to an opposing counter-reaction.
1725 1725
1726 1726
1727 1727
1728 
-<hr><div id="postamble" class="status">
1728 
+<hr>
1729 
+<div class="license">
1730 
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1731 
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1732 
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1735 
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1737 
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1738 
+</div>
1739 
+<div id="postamble" class="status">
1729 1740 
 <p class="author">Author: Jean-Sébastien Caux</p>
1730 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1731 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1741 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1742 
+<p class="validation"></p>
1732 1743 
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1733 1744
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7 7 
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Fl_i.html">Inductance&emsp;<small>[emd.Fl.i]</small></a></li><li>Next:&nbsp;<a href="emd_Me.html">Maxwell's Equations&emsp;<small>[emd.Me]</small></a></li><li>Up:&nbsp;<a href="emd_Fl.html">Induction: Faraday's Law&emsp;<small>[emd.Fl]</small></a></li></ul><div id="outline-container-emd_Fl_e" class="outline-4">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emd.html">Electromagnetodynamics</a></li><li><a class="breadcrumb-link"href="emd_Fl.html">Induction: Faraday's Law</a></li><li>Energy in Magnetic Fields</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Fl_i.html">Inductance&emsp;<small>[emd.Fl.i]</small></a></li><li>Next:&nbsp;<a href="emd_Me.html">Maxwell's Equations&emsp;<small>[emd.Me]</small></a></li><li>Up:&nbsp;<a href="emd_Fl.html">Induction: Faraday's Law&emsp;<small>[emd.Fl]</small></a></li></ul><div id="outline-container-emd_Fl_e" class="outline-4">
1620 1620 
 <h4 id="emd_Fl_e">Energy in Magnetic Fields<a class="headline-permalink" href="./emd_Fl_e.html#emd_Fl_e"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
   <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"/>
1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1682,7 +1682,7 @@ W = \frac{1}{2\mu_0} \left[ \int_{\cal V} d\tau B^2 - \int_{\cal V} d\tau {\bold
1682 1682 
 \]
1683 1683 
 We can integrate over all space:  after neglecting boundary terms (assuming fields fall to zero at infinity), we are left with
1684 1684 
 </p>
1685 
-<div class="core div" id="org612a7d5">
1685 
+<div class="core div" id="orge86302b">
1686 1686 
 <p>
1687 1687 
 \[
1688 1688 
     W_{mag} = \frac{1}{2\mu_0} \int d\tau B^2
 
@@ -1703,7 +1703,7 @@ W_{mag} = \frac{1}{2} \int d\tau ({\bf A} \cdot {\bf J}) = \frac{1}{2\mu_0} \int
1703 1703 
 \hspace{2cm} \mbox{(7.31 and 7.34)}
1704 1704 
 \end{align}
1705 1705
1706 
-<div class="example div" id="org8806452">
1706 
+<div class="example div" id="org0c0ebda">
1707 1707 
 <p>
1708 1708 
 \paragraph{Example 7.13:}  coaxial cable (inner cylinder radius \(a\), outer \(b\)) carries current \(I\).
1709 1709 
 Find energy stored in section of length \(l\).
 
@@ -1727,10 +1727,21 @@ Note:  gives easy way to find inductance, since \(W = \frac{1}{2} L I^2\).
1727 1727 
 </div>
1728 1728
1729 1729
1730 
-<hr><div id="postamble" class="status">
1730 
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1731 
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1739 
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1740 
+</div>
1741 
+<div id="postamble" class="status">
1731 1742 
 <p class="author">Author: Jean-Sébastien Caux</p>
1732 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1733 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1743 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1744 
+<p class="validation"></p>
1734 1745 
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1735 1746
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7 7 
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Fl_ief.html">The Induced Electric Field&emsp;<small>[emd.Fl.ief]</small></a></li><li>Next:&nbsp;<a href="emd_Fl_e.html">Energy in Magnetic Fields&emsp;<small>[emd.Fl.e]</small></a></li><li>Up:&nbsp;<a href="emd_Fl.html">Induction: Faraday's Law&emsp;<small>[emd.Fl]</small></a></li></ul><div id="outline-container-emd_Fl_i" class="outline-4">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emd.html">Electromagnetodynamics</a></li><li><a class="breadcrumb-link"href="emd_Fl.html">Induction: Faraday's Law</a></li><li>Inductance</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Fl_ief.html">The Induced Electric Field&emsp;<small>[emd.Fl.ief]</small></a></li><li>Next:&nbsp;<a href="emd_Fl_e.html">Energy in Magnetic Fields&emsp;<small>[emd.Fl.e]</small></a></li><li>Up:&nbsp;<a href="emd_Fl.html">Induction: Faraday's Law&emsp;<small>[emd.Fl]</small></a></li></ul><div id="outline-container-emd_Fl_i" class="outline-4">
1620 1620 
 <h4 id="emd_Fl_i">Inductance<a class="headline-permalink" href="./emd_Fl_i.html#emd_Fl_i"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
   <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"/>
1622 1622 
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@@ -1662,7 +1662,7 @@ M_{12} = M_{21}
1662 1662 
 \]
1663 1663 
 </p>
1664 1664
1665 
-<div class="example div" id="org105c83a">
1665 
+<div class="example div" id="org6c1a2e4">
1666 1666 
 <p>
1667 1667 
 \paragraph{Example 7.10:}
1668 1668 
 short solenoid (length \(l\), radius \(a\), \(n_1\) turns per unit length) lies concentrically inside
 
@@ -1712,7 +1712,7 @@ Inductance:  measured in {\bf henries} (\(H\)).  \(H = V s/A\).
1712 1712 
 </p>
1713 1713
1714 1714
1715 
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1715 
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1716 1716 
 <p>
1717 1717 
 \paragraph{Example 7.11:}  find self-inductance of toroidal coil with
1718 1718 
 rectangular cross-section (inner radius \(a\), outer radius \(b\), height \(h\))
 
@@ -1739,7 +1739,7 @@ Total flux:  \(N\) times this, so self-inductance is
1739 1739 
 Inductance (like capacitance) is intrinsically positive.  Use Lenz law.  Think of {\bf back EMF}.
1740 1740 
 </p>
1741 1741
1742 
-<div class="example div" id="org88909b4">
1742 
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1743 1743 
 <p>
1744 1744 
 \paragraph{Example 7.12:}  circuit with inductance \(L\), resistor \(R\) and battery \({\cal E}_0\).
1745 1745 
 What is the current ?
 
@@ -1763,10 +1763,21 @@ where \(\tau \equiv L/R\) is the {\bf time constant} of the circuit.
1763 1763
1764 1764
1765 1765
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-<hr><div id="postamble" class="status">
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+<hr>
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1769 
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1774 
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1775 
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1776 
+</div>
1777 
+<div id="postamble" class="status">
1767 1778 
 <p class="author">Author: Jean-Sébastien Caux</p>
1768 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1769 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1779 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1780 
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1770 1781 
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1771 1782
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Fl_Fl.html">Faraday's Law&emsp;<small>[emd.Fl.Fl]</small></a></li><li>Next:&nbsp;<a href="emd_Fl_i.html">Inductance&emsp;<small>[emd.Fl.i]</small></a></li><li>Up:&nbsp;<a href="emd_Fl.html">Induction: Faraday's Law&emsp;<small>[emd.Fl]</small></a></li></ul><div id="outline-container-emd_Fl_ief" class="outline-4">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emd.html">Electromagnetodynamics</a></li><li><a class="breadcrumb-link"href="emd_Fl.html">Induction: Faraday's Law</a></li><li>The Induced Electric Field</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Fl_Fl.html">Faraday's Law&emsp;<small>[emd.Fl.Fl]</small></a></li><li>Next:&nbsp;<a href="emd_Fl_i.html">Inductance&emsp;<small>[emd.Fl.i]</small></a></li><li>Up:&nbsp;<a href="emd_Fl.html">Induction: Faraday's Law&emsp;<small>[emd.Fl]</small></a></li></ul><div id="outline-container-emd_Fl_ief" class="outline-4">
1620 1620 
 <h4 id="emd_Fl_ief">The Induced Electric Field<a class="headline-permalink" href="./emd_Fl_ief.html#emd_Fl_ief"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1646,7 +1646,7 @@ law in integral form:
1646 1646
1647 1647
1648 1648
1649 
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1649 
+<div class="example div" id="org547c6ca">
1650 1650 
 <p>
1651 1651 
 {\bf Example 7.7:}
1652 1652 
 \({\bf B}(t)\) points up in circular region of radius \(R\).  What is the induced \({\bf E}(t)\) ?
 
@@ -1662,7 +1662,7 @@ Increasing \({\bf B}\):  clockwise (viewed from above) \({\bf E}\) from Lenz.
1662 1662 
 </div>
1663 1663
1664 1664
1665 
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1665 
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1666 1666 
 <p>
1667 1667 
 {\bf Example 7.8:} wheel or radius \(b\) with line charge \(\lambda\) on the rim.
1668 1668 
 Uniform magnetic field \({\bf B}_0\) in central region up to \(a &lt; b\),
 
@@ -1696,7 +1696,7 @@ called the {\bf quasistatic} approximation, and works provided we deal with
1696 1696 
 'slow enough' phenomena.
1697 1697 
 </p>
1698 1698
1699 
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1699 
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1700 1700 
 <p>
1701 1701 
 {\bf Example 7.9:}  infinitely long straight wire carries \(I(t)\).  Find
1702 1702 
 induced \({\bf E}\) field as a function of distance \(s\) from wire.
 
@@ -1729,10 +1729,21 @@ Reason:  in this case, we've overstepped the quasistatic limit.  We need
1729 1729
1730 1730
1731 1731
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1741 
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1742 
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1743 
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1733 1744 
 <p class="author">Author: Jean-Sébastien Caux</p>
1734 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1735 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1745 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1746 
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1736 1747 
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1737 1748
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
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1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Fl_e.html">Energy in Magnetic Fields&emsp;<small>[emd.Fl.e]</small></a></li><li>Next:&nbsp;<a href="emd_Me_ebM.html">Electrodynamics Before Maxwell&emsp;<small>[emd.Me.ebM]</small></a></li><li>Up:&nbsp;<a href="emd.html">Electromagnetodynamics&emsp;<small>[emd]</small></a></li></ul>
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emd.html">Electromagnetodynamics</a></li><li>Maxwell's Equations</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Fl_e.html">Energy in Magnetic Fields&emsp;<small>[emd.Fl.e]</small></a></li><li>Next:&nbsp;<a href="emd_Me_ebM.html">Electrodynamics Before Maxwell&emsp;<small>[emd.Me.ebM]</small></a></li><li>Up:&nbsp;<a href="emd.html">Electromagnetodynamics&emsp;<small>[emd]</small></a></li></ul>
1620 1620 
 <h3 id="emd_Me">Maxwell's Equations<a class="headline-permalink" href="./emd_Me.html#emd_Me"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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1622 1622 
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@@ -1634,10 +1634,21 @@ Table of contents
1634 1634 
 <li><a href="emd_Me_mc.html">Magnetic Charge</a><span class="headline-id">emd.Me.mc</span></li>
1635 1635 
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1636 1636
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1637 
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1646 
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1647 
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1648 
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1638 1649 
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1639 
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1640 
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1650 
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1651 
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1641 1652 
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1642 1653
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@@ -1616,7 +1616,7 @@ Table of contents
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1617 1617 
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1618 1618 
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1619 
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1619 
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1620 1620 
 <h4 id="emd_Me_Me">Maxwell's Equations<a class="headline-permalink" href="./emd_Me_Me.html#emd_Me_Me"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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@@ -1625,7 +1625,7 @@ Table of contents
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 <p>
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 Full set of equations for the electromagnetic field:
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 </p>
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 <p>
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 {\bf Maxwell's equations} {\it (in vacuum)}
1631 1631 
 </p>
 
@@ -1641,7 +1641,7 @@ Full set of equations for the electromagnetic field:
1641 1641 
 <p>
1642 1642 
 Complement:
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 {\bf Force law}
1647 1647 
 \[
 
@@ -1665,7 +1665,7 @@ take divergence of \((iv)\).
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 <p>
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 Better way of writing:  all fields on left, all sources on right,
1667 1667 
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 \begin{align}
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   (i) &amp;{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0},
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   &amp;(iii) {\boldsymbol \nabla} \times {\bf E} + \frac{\partial {\bf B}}{\partial t} = 0, \\
 
@@ -1680,10 +1680,21 @@ Better way of writing:  all fields on left, all sources on right,
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1684 1695 
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1685 
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1686 
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1696 
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@@ -1616,7 +1616,7 @@ Table of contents
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1618 1618 
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1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Me_ebM.html">Electrodynamics Before Maxwell&emsp;<small>[emd.Me.ebM]</small></a></li><li>Next:&nbsp;<a href="emd_Me_Me.html">Maxwell's Equations&emsp;<small>[emd.Me.Me]</small></a></li><li>Up:&nbsp;<a href="emd_Me.html">Maxwell's Equations&emsp;<small>[emd.Me]</small></a></li></ul><div id="outline-container-emd_Me_dc" class="outline-4">
1619 
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1620 1620 
 <h4 id="emd_Me_dc">Maxwell's Correction to Ampère's Law; the Displacement Current<a class="headline-permalink" href="./emd_Me_dc.html#emd_Me_dc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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@@ -1632,7 +1632,7 @@ the continuity equation as
1632 1632 
 \]
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 The extra term would thus be eliminated if we were to put
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 </p>
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-<div class="core div" id="orgc09ac90">
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 <p>
1637 1637 
 \[
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     {\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}
 
@@ -1656,7 +1656,7 @@ Real confirmation of Maxwell's theory:  1888, Hertz's experiments on propagation
1656 1656 
 <p>
1657 1657 
 Maxwell baptized this term the
1658 1658 
 </p>
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-<div class="core div" id="orgd2c9b8e">
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 <p>
1661 1661 
 {\bf Displacement current}
1662 1662 
 \[
 
@@ -1689,10 +1689,21 @@ Flat surface:  OK, \(E = 0\) and \(I_{\mbox{enc}} = I\).  Balloon surface:  \(I
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1693 
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1693 1704 
 <p class="author">Author: Jean-Sébastien Caux</p>
1694 
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1695 
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1705 
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
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1618 1618 
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1619 
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1620 1620 
 <h4 id="emd_Me_ebM">Electrodynamics Before Maxwell<a class="headline-permalink" href="./emd_Me_ebM.html#emd_Me_ebM"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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@@ -1659,10 +1659,21 @@ So for non-steady currents, the 'current enclosed by a loop' is ill-defined.
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1663 1674 
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1664 
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1665 
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@@ -1616,7 +1616,7 @@ Table of contents
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1620 1620 
 <h4 id="emd_Me_mc">Magnetic Charge<a class="headline-permalink" href="./emd_Me_mc.html#emd_Me_mc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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@@ -1657,10 +1657,21 @@ Maxwell's equations {\bf beg} for magnetic charges.  But we've never found any!
1657 1657 
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1661 1672 
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1620 1620 
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1628 1628 
 Prerequisites
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 </summary>
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 <ul class="org-ul">
 
@@ -1632,8 +1632,8 @@ Prerequisites
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 Objectives
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@@ -1656,10 +1656,21 @@ Objectives
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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"/>
 
@@ -1625,7 +1625,7 @@ Table of contents
1625 1625 
 <p>
1626 1626 
 The angular momentum of EM fields is directly given by
1627 1627 
 </p>
1628 
-<div class="main div" id="org15be82e">
1628 
+<div class="main div" id="org8fc773b">
1629 1629 
 <p>
1630 1630 
 {\bf Angular momentum of EM fields}
1631 1631 
 \[
 
@@ -1640,10 +1640,21 @@ The angular momentum of EM fields is directly given by
1640 1640 
 </div>
1641 1641
1642 1642
1643 
-<hr><div id="postamble" class="status">
1643 
+<hr>
1644 
+<div class="license">
1645 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1646 
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1647 
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1648 
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1649 
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1650 
+Except where otherwise noted, all content is licensed under a
1651 
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1652 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1653 
+</div>
1654 
+<div id="postamble" class="status">
1644 1655 
 <p class="author">Author: Jean-Sébastien Caux</p>
1645 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1646 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1656 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1657 
+<p class="validation"></p>
1647 1658 
 </div>
1648 1659
1649 1660

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build/emd_ce_ce.html View File 

@@ -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 
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li><li>Next:&nbsp;<a href="emd_ce_poy.html">Poynting's Theorem; the Poynting Vector&emsp;<small>[emd.ce.poy]</small></a></li><li>Up:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li></ul><div id="outline-container-emd_ce_ce" class="outline-4">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emd.html">Electromagnetodynamics</a></li><li><a class="breadcrumb-link"href="emd_ce.html">Charge and Energy Flows</a></li><li>The Continuity Equation</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li><li>Next:&nbsp;<a href="emd_ce_poy.html">Poynting's Theorem; the Poynting Vector&emsp;<small>[emd.ce.poy]</small></a></li><li>Up:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li></ul><div id="outline-container-emd_ce_ce" class="outline-4">
1620 1620 
 <h4 id="emd_ce_ce">The Continuity Equation<a class="headline-permalink" href="./emd_ce_ce.html#emd_ce_ce"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
   <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"/>
1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1644,7 +1644,7 @@ This means that
1644 1644 
 \]
1645 1645 
 Since this is true for any volume, we have (re)derived the
1646 1646 
 </p>
1647 
-<div class="core div" id="orgf7fdcf4">
1647 
+<div class="core div" id="org068de4a">
1648 1648 
 <p>
1649 1649 
 {\bf Continuity equation}
1650 1650 
 \[
 
@@ -1670,10 +1670,21 @@ imposes a functional constraint on these sources: not {\it any} \(\rho\) and
1670 1670
1671 1671
1672 1672
1673 
-<hr><div id="postamble" class="status">
1673 
+<hr>
1674 
+<div class="license">
1675 
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1676 
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1677 
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1678 
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1679 
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1680 
+Except where otherwise noted, all content is licensed under a
1681 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1682 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1683 
+</div>
1684 
+<div id="postamble" class="status">
1674 1685 
 <p class="author">Author: Jean-Sébastien Caux</p>
1675 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1676 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1686 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1687 
+<p class="validation"></p>
1677 1688 
 </div>
1678 1689
1679 1690

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build/emd_ce_mom.html View File 

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4 
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5 5 
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6 6 
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7 7 
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_ce_mst.html">Maxwell's Stress Tensor&emsp;<small>[emd.ce.mst]</small></a></li><li>Next:&nbsp;<a href="emd_ce_amom.html">Angular Momentum&emsp;<small>[emd.ce.amom]</small></a></li><li>Up:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li></ul><div id="outline-container-emd_ce_mom" class="outline-4">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emd.html">Electromagnetodynamics</a></li><li><a class="breadcrumb-link"href="emd_ce.html">Charge and Energy Flows</a></li><li>Momentum</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_ce_mst.html">Maxwell's Stress Tensor&emsp;<small>[emd.ce.mst]</small></a></li><li>Next:&nbsp;<a href="emd_ce_amom.html">Angular Momentum&emsp;<small>[emd.ce.amom]</small></a></li><li>Up:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li></ul><div id="outline-container-emd_ce_mom" class="outline-4">
1620 1620 
 <h4 id="emd_ce_mom">Momentum<a class="headline-permalink" href="./emd_ce_mom.html#emd_ce_mom"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
   <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"/>
1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1637,7 +1637,7 @@ in which the first integral can be interpreted as the momentum stored in the EM
1637 1637 
 <p>
1638 1638 
 This is thus simply a conservation law for momentum, with
1639 1639 
 </p>
1640 
-<div class="main div" id="orgeb408ef">
1640 
+<div class="main div" id="orgb401242">
1641 1641 
 <p>
1642 1642 
 {\bf Momentum density in the EM fields}
1643 1643 
 \[
 
@@ -1649,7 +1649,7 @@ This is thus simply a conservation law for momentum, with
1649 1649 
 <p>
1650 1650 
 In a region in which the mechanical momentum is not changing due to external influences, we then have the
1651 1651 
 </p>
1652 
-<div class="main div" id="org94bfc0c">
1652 
+<div class="main div" id="orgd377c8a">
1653 1653 
 <p>
1654 1654 
 {\bf Continuity equation for EM momentum}
1655 1655 
 \[
 
@@ -1663,10 +1663,21 @@ In a region in which the mechanical momentum is not changing due to external inf
1663 1663
1664 1664
1665 1665
1666 
-<hr><div id="postamble" class="status">
1666 
+<hr>
1667 
+<div class="license">
1668 
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1669 
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1670 
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1671 
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1672 
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1673 
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1674 
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1675 
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1676 
+</div>
1677 
+<div id="postamble" class="status">
1667 1678 
 <p class="author">Author: Jean-Sébastien Caux</p>
1668 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1669 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1679 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1680 
+<p class="validation"></p>
1670 1681 
 </div>
1671 1682
1672 1683

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4 
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6 6 
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7 7 
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_ce_poy.html">Poynting's Theorem; the Poynting Vector&emsp;<small>[emd.ce.poy]</small></a></li><li>Next:&nbsp;<a href="emd_ce_mom.html">Momentum&emsp;<small>[emd.ce.mom]</small></a></li><li>Up:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li></ul><div id="outline-container-emd_ce_mst" class="outline-4">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emd.html">Electromagnetodynamics</a></li><li><a class="breadcrumb-link"href="emd_ce.html">Charge and Energy Flows</a></li><li>Maxwell's Stress Tensor</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_ce_poy.html">Poynting's Theorem; the Poynting Vector&emsp;<small>[emd.ce.poy]</small></a></li><li>Next:&nbsp;<a href="emd_ce_mom.html">Momentum&emsp;<small>[emd.ce.mom]</small></a></li><li>Up:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li></ul><div id="outline-container-emd_ce_mst" class="outline-4">
1620 1620 
 <h4 id="emd_ce_mst">Maxwell's Stress Tensor<a class="headline-permalink" href="./emd_ce_mst.html#emd_ce_mst"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
   <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"/>
1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1679,7 +1679,7 @@ and similarly for \({\boldsymbol B}\). We thus get
1679 1679 
 <p>
1680 1680 
 This expression can be greatly simplified by introducing the
1681 1681 
 </p>
1682 
-<div class="main div" id="org516af43">
1682 
+<div class="main div" id="orgf0f558d">
1683 1683 
 <p>
1684 1684 
 {\bf Maxwell stress tensor}
1685 1685 
 \[
 
@@ -1702,7 +1702,7 @@ The element \(T_{ij}\) represents the force per unit area in the $i$th direction
1702 1702 
 <p>
1703 1703 
 We then obtain
1704 1704 
 </p>
1705 
-<div class="main div" id="org38f0a89">
1705 
+<div class="main div" id="orgcd1da6b">
1706 1706 
 <p>
1707 1707 
 {\bf EM force per unit volume}
1708 1708 
 \[
 
@@ -1714,7 +1714,7 @@ We then obtain
1714 1714 
 <p>
1715 1715 
 where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
1716 1716 
 </p>
1717 
-<div class="main div" id="orgf9a42ff">
1717 
+<div class="main div" id="orgf2db445">
1718 1718 
 <p>
1719 1719 
 {\bf Total force on charges in volume}
1720 1720 
 \[
 
@@ -1728,10 +1728,21 @@ where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
1728 1728
1729 1729
1730 1730
1731 
-<hr><div id="postamble" class="status">
1731 
+<hr>
1732 
+<div class="license">
1733 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1734 
+target="_blank" class="m-2">
1735 
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1736 
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1737 
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1738 
+Except where otherwise noted, all content is licensed under a
1739 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1740 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1741 
+</div>
1742 
+<div id="postamble" class="status">
1732 1743 
 <p class="author">Author: Jean-Sébastien Caux</p>
1733 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1734 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1744 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1745 
+<p class="validation"></p>
1735 1746 
 </div>
1736 1747
1737 1748

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2 2 
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3 3 
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6 6 
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7 7 
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_ce_ce.html">The Continuity Equation&emsp;<small>[emd.ce.ce]</small></a></li><li>Next:&nbsp;<a href="emd_ce_mst.html">Maxwell's Stress Tensor&emsp;<small>[emd.ce.mst]</small></a></li><li>Up:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li></ul><div id="outline-container-emd_ce_poy" class="outline-4">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emd.html">Electromagnetodynamics</a></li><li><a class="breadcrumb-link"href="emd_ce.html">Charge and Energy Flows</a></li><li>Poynting's Theorem; the Poynting Vector</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_ce_ce.html">The Continuity Equation&emsp;<small>[emd.ce.ce]</small></a></li><li>Next:&nbsp;<a href="emd_ce_mst.html">Maxwell's Stress Tensor&emsp;<small>[emd.ce.mst]</small></a></li><li>Up:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li></ul><div id="outline-container-emd_ce_poy" class="outline-4">
1620 1620 
 <h4 id="emd_ce_poy">Poynting's Theorem; the Poynting Vector<a class="headline-permalink" href="./emd_ce_poy.html#emd_ce_poy"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
   <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"/>
1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1691,7 +1691,7 @@ so we get
1691 1691 
 Substituting this in \ref{Gr(8.6)} and using the divergence theorem,
1692 1692 
 we obtain
1693 1693 
 </p>
1694 
-<div class="main div" id="org33b0d12">
1694 
+<div class="main div" id="org75886ff">
1695 1695 
 <p>
1696 1696 
 {\bf Poynting's theorem}
1697 1697 
 \[
 
@@ -1716,7 +1716,7 @@ energy is carried by EM fields out of \({\cal V}\) across its boundary surface.
1716 1716 
 <p>
1717 1717 
 Energy per unit time, per unit area carried by EM fields:
1718 1718 
 </p>
1719 
-<div class="core div" id="orgb00d4be">
1719 
+<div class="core div" id="orgb6d912b">
1720 1720 
 <p>
1721 1721 
 {\bf Poynting vector}
1722 1722 
 \[
 
@@ -1729,7 +1729,7 @@ Energy per unit time, per unit area carried by EM fields:
1729 1729 
 <p>
1730 1730 
 We can thus express Poynting's theorem more compactly:
1731 1731 
 </p>
1732 
-<div class="core div" id="org618d06c">
1732 
+<div class="core div" id="org7131f77">
1733 1733 
 <p>
1734 1734 
 {\bf Poynting's theorem}
1735 1735 
 \[
 
@@ -1742,7 +1742,7 @@ We can thus express Poynting's theorem more compactly:
1742 1742 
 <p>
1743 1743 
 where we have defined the total
1744 1744 
 </p>
1745 
-<div class="core div" id="orgb4f2685">
1745 
+<div class="core div" id="org44b728f">
1746 1746 
 <p>
1747 1747 
 {\bf Energy in electromagnetic fields}
1748 1748 
 \[
 
@@ -1765,7 +1765,7 @@ Then,
1765 1765 
 \]
1766 1766 
 so we get the
1767 1767 
 </p>
1768 
-<div class="core div" id="org1eaf77c">
1768 
+<div class="core div" id="org11d0bdb">
1769 1769 
 <p>
1770 1770 
 {\bf Poynting theorem (differential form)}
1771 1771 
 \[
 
@@ -1782,7 +1782,7 @@ and has a similar for to the continuity equation
1782 1782
1783 1783
1784 1784
1785 
-<div class="example div" id="org545c704">
1785 
+<div class="example div" id="orgd2f58ae">
1786 1786 
 <p>
1787 1787 
 \paragraph{Example 8.1}  Current in a wire:  Joule heating.  Energy per unit time delivered to wire:  from Poynting.
1788 1788 
 Assuming that the field is uniform, the electric field parallel to the wire is
 
@@ -1812,10 +1812,21 @@ and the value is as expected.
1812 1812
1813 1813
1814 1814
1815 
-<hr><div id="postamble" class="status">
1815 
+<hr>
1816 
+<div class="license">
1817 
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1818 
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1819 
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1820 
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1821 
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1822 
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1823 
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1824 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1825 
+</div>
1826 
+<div id="postamble" class="status">
1816 1827 
 <p class="author">Author: Jean-Sébastien Caux</p>
1817 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1818 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1828 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1829 
+<p class="validation"></p>
1819 1830 
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1820 1831
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1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_ce_amom.html">Angular Momentum&emsp;<small>[emd.ce.amom]</small></a></li><li>Next:&nbsp;<a href="emd_emw_we.html">The Wave Equation&emsp;<small>[emd.emw.we]</small></a></li><li>Up:&nbsp;<a href="emd.html">Electromagnetodynamics&emsp;<small>[emd]</small></a></li></ul>
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emd.html">Electromagnetodynamics</a></li><li>Electromagnetic waves in vacuum</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_ce_amom.html">Angular Momentum&emsp;<small>[emd.ce.amom]</small></a></li><li>Next:&nbsp;<a href="emd_emw_we.html">The Wave Equation&emsp;<small>[emd.emw.we]</small></a></li><li>Up:&nbsp;<a href="emd.html">Electromagnetodynamics&emsp;<small>[emd]</small></a></li></ul>
1620 1620 
 <h3 id="emd_emw">Electromagnetic waves in vacuum<a class="headline-permalink" href="./emd_emw.html#emd_emw"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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1622 1622 
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1623 1623 
 </svg></a><span class="headline-id">emd.emw</span></h3>
1624 1624
1625 1625 
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1628 1628 
 Prerequisites
1629 1629 
 </summary>
1630 1630 
 <ul class="org-ul">
 
@@ -1633,8 +1633,8 @@ Prerequisites
1633 1633 
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 </details>
1635 1635
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1638 1638 
 Objectives
1639 1639 
 </summary>
1640 1640 
 <ul class="org-ul">
 
@@ -1660,10 +1660,21 @@ Objectives
1660 1660 
 <li><a href="emd_emw_ep.html">Energy and Momentum</a><span class="headline-id">emd.emw.ep</span></li>
1661 1661 
 </ul>
1662 1662
1663 
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1672 
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1673 
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1674 
+<div id="postamble" class="status">
1664 1675 
 <p class="author">Author: Jean-Sébastien Caux</p>
1665 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1666 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1676 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1677 
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1667 1678 
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1668 1679
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7 7 
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_emw_mpw.html">Monochromatic Plane Waves&emsp;<small>[emd.emw.mpw]</small></a></li><li>Next:&nbsp;<a href="emdm.html">Electromagnetodynamics in Matter&emsp;<small>[emdm]</small></a></li><li>Up:&nbsp;<a href="emd_emw.html">Electromagnetic waves in vacuum&emsp;<small>[emd.emw]</small></a></li></ul><div id="outline-container-emd_emw_ep" class="outline-4">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emd.html">Electromagnetodynamics</a></li><li><a class="breadcrumb-link"href="emd_emw.html">Electromagnetic waves in vacuum</a></li><li>Energy and Momentum</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_emw_mpw.html">Monochromatic Plane Waves&emsp;<small>[emd.emw.mpw]</small></a></li><li>Next:&nbsp;<a href="emdm.html">Electromagnetodynamics in Matter&emsp;<small>[emdm]</small></a></li><li>Up:&nbsp;<a href="emd_emw.html">Electromagnetic waves in vacuum&emsp;<small>[emd.emw]</small></a></li></ul><div id="outline-container-emd_emw_ep" class="outline-4">
1620 1620 
 <h4 id="emd_emw_ep">Energy and Momentum<a class="headline-permalink" href="./emd_emw_ep.html#emd_emw_ep"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1653,7 +1653,7 @@ so for a monochromatic EM plan wave,
1653 1653 
 \]
1654 1654 
 or more succinctly:
1655 1655 
 </p>
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-<div class="main div" id="orgeb0be24">
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1657 1657 
 <p>
1658 1658 
 {\bf Poynting vector of a monochromatic EM wave}
1659 1659 
 \[
 
@@ -1669,7 +1669,7 @@ This has a transparent physical interpretation: the energy density \(u\) flows w
1669 1669 
 <p>
1670 1670 
 Similary, we get the
1671 1671 
 </p>
1672 
-<div class="main div" id="org372457d">
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1673 1673 
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1674 1674 
 {\bf Momentum density of a monochromatic EM wave}
1675 1675 
 \[
 
@@ -1705,10 +1705,21 @@ The {\it radiation pressure} is the momentum transfer per unit area per unit of
1705 1705 
 </div>
1706 1706
1707 1707
1708 
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1708 
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1709 
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1718 
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1719 
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1709 1720 
 <p class="author">Author: Jean-Sébastien Caux</p>
1710 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1711 
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1721 
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1722 
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1712 1723 
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1713 1724
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_emw_we.html">The Wave Equation&emsp;<small>[emd.emw.we]</small></a></li><li>Next:&nbsp;<a href="emd_emw_ep.html">Energy and Momentum&emsp;<small>[emd.emw.ep]</small></a></li><li>Up:&nbsp;<a href="emd_emw.html">Electromagnetic waves in vacuum&emsp;<small>[emd.emw]</small></a></li></ul><div id="outline-container-emd_emw_mpw" class="outline-4">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emd.html">Electromagnetodynamics</a></li><li><a class="breadcrumb-link"href="emd_emw.html">Electromagnetic waves in vacuum</a></li><li>Monochromatic Plane Waves</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_emw_we.html">The Wave Equation&emsp;<small>[emd.emw.we]</small></a></li><li>Next:&nbsp;<a href="emd_emw_ep.html">Energy and Momentum&emsp;<small>[emd.emw.ep]</small></a></li><li>Up:&nbsp;<a href="emd_emw.html">Electromagnetic waves in vacuum&emsp;<small>[emd.emw]</small></a></li></ul><div id="outline-container-emd_emw_mpw" class="outline-4">
1620 1620 
 <h4 id="emd_emw_mpw">Monochromatic Plane Waves<a class="headline-permalink" href="./emd_emw_mpw.html#emd_emw_mpw"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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1622 1622 
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@@ -1654,7 +1654,7 @@ B_0 = \frac{k}{\omega} E_0 = \frac{1}{c} E_0.
1654 1654 
 Generalizing to propagation in the direction of an arbitrary wavevector
1655 1655 
 \({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the
1656 1656 
 </p>
1657 
-<div class="core div" id="org22a7fdd">
1657 
+<div class="core div" id="org9f5b19f">
1658 1658 
 <p>
1659 1659 
 {\bf E and B fields for a monochromatic EM plane wave}
1660 1660 
 \[
 
@@ -1683,10 +1683,21 @@ or if you prefer explicit real parts (adding a possible phase shift \(\delta\)):
1683 1683
1684 1684
1685 1685
1686 
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1686 
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1687 
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1688 
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1695 
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1696 
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1697 
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1687 1698 
 <p class="author">Author: Jean-Sébastien Caux</p>
1688 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1689 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1699 
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1700 
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1690 1701 
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1691 1702
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
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1617 1617 
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1618 1618 
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1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_emw.html">Electromagnetic waves in vacuum&emsp;<small>[emd.emw]</small></a></li><li>Next:&nbsp;<a href="emd_emw_mpw.html">Monochromatic Plane Waves&emsp;<small>[emd.emw.mpw]</small></a></li><li>Up:&nbsp;<a href="emd_emw.html">Electromagnetic waves in vacuum&emsp;<small>[emd.emw]</small></a></li></ul><div id="outline-container-emd_emw_we" class="outline-4">
1619 
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1620 1620 
 <h4 id="emd_emw_we">The Wave Equation<a class="headline-permalink" href="./emd_emw_we.html#emd_emw_we"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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@@ -1651,7 +1651,7 @@ These take the form of coupled first-order partial differential equations for \(
1651 1651 
 Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\),
1652 1652 
 we get the
1653 1653 
 </p>
1654 
-<div class="core div" id="org1160663">
1654 
+<div class="core div" id="orgfcad186">
1655 1655 
 <p>
1656 1656 
 {\bf Wave equations for electric and magnetic fields in vacuum}
1657 1657 
 \[
 
@@ -1692,10 +1692,21 @@ the actual electric and magnetic fields are given by the real part.
1692 1692
1693 1693
1694 1694
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1696 1707 
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1697 
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@@ -1632,10 +1632,21 @@ Table of contents
1632 1632 
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1633 1633 
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1634 1634
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1636 1647 
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@@ -1632,10 +1632,21 @@ Table of contents
1632 1632 
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1633 1633 
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1636 1647 
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1637 
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1638 
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1648 
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1618 1618 
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1619 
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1619 
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1620 1620 
 <h4 id="emdm_Me_Mem">Maxwell's Equations in Matter<a class="headline-permalink" href="./emdm_Me_Mem.html#emdm_Me_Mem"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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1622 1622 
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@@ -1658,7 +1658,7 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
1658 1658 
 \]
1659 1659 
 We therefore have the
1660 1660 
 </p>
1661 
-<div class="core div" id="orgff46cb4">
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+<div class="core div" id="org15d7510">
1662 1662 
 <p>
1663 1663 
 {\bf Polarization current density}
1664 1664 
 \[
 
@@ -1676,7 +1676,7 @@ the polarization current is the result of linear motion of charge when
1676 1676 
 polarization changes).  We can check consistency with the continuity equation
1677 1677 
 associated to the conservation of bound charges:
1678 1678 
 </p>
1679 
-<aside id="orgbce7d73">
1679 
+<aside id="org9c1ef52">
1680 1680 
 <p>
1681 1681 
 Note the unfortunate labelling: it would have been nicer to have \(\rho_b\) be the charge associated to current
1682 1682 
   \({\boldsymbol J}_b\) but this is not the convention used here.
 
@@ -1699,7 +1699,7 @@ Changing magnetization does not lead to analogous accumulation of charge and cur
1699 1699 
 In view of this:  total charge density can be separated into 2 parts,
1700 1700 
 {\it free} and {\it bound}:
1701 1701 
 </p>
1702 
-<div class="main div" id="org4df7865">
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+<div class="main div" id="org2c2e80a">
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 <p>
1704 1704 
 \[
1705 1705 
     \rho = \rho_f + \rho_b = \rho_f - {\boldsymbol \nabla} \cdot {\bf P}
 
@@ -1712,7 +1712,7 @@ In view of this:  total charge density can be separated into 2 parts,
1712 1712 
 and current can be separated into three parts, {\it free}, {\it bound} and
1713 1713 
 {\it polarization}:
1714 1714 
 </p>
1715 
-<div class="main div" id="orgcc32502">
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1716 1716 
 <p>
1717 1717 
 \[
1718 1718 
   {\bf J} = {\bf J}_f + {\bf J}_b + {\bf J}_p = {\bf J}_f + {\boldsymbol ∇} × {\bf M}
 
@@ -1736,7 +1736,7 @@ Gauss's law:  can be rewritten
1736 1736 
 \]
1737 1737 
 where (as in static case)
1738 1738 
 </p>
1739 
-<div class="core div" id="org7600f00">
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1740 1740 
 <p>
1741 1741 
 \[
1742 1742 
     {\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
 
@@ -1762,7 +1762,7 @@ or
1762 1762 
 \]
1763 1763 
 where as before
1764 1764 
 </p>
1765 
-<div class="core div" id="orgb942ad8">
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1766 1766 
 <p>
1767 1767 
 \[
1768 1768 
     {\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
 
@@ -1780,7 +1780,7 @@ bound parts, since they don't involve \(\rho\) or \({\bf J}\).
1780 1780 
 <p>
1781 1781 
 In terms of free charges and currents, we thus get
1782 1782 
 </p>
1783 
-<div class="core div" id="orgd9d30d5">
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1784 1784 
 <p>
1785 1785 
 {\bf Maxwell's equations {\it (in matter)}}
1786 1786 
 </p>
 
@@ -1806,7 +1806,7 @@ Must be complemented by the {\bf constitutive relations} giving \({\bf D}\) and
1806 1806 
 in terms of \({\bf E}\) and \({\bf B}\).
1807 1807 
 For the restricted case of linear media:
1808 1808 
 </p>
1809 
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 <p>
1811 1811 
 \[
1812 1812 
     {\bf P} = \varepsilon_0 \chi_e {\bf E}, \hspace{1cm}
 
@@ -1828,10 +1828,21 @@ where \(\varepsilon \equiv \varepsilon_0(1 + \chi_e)\) and \(\mu \equiv \mu_0 (1
1828 1828
1829 1829
1830 1830
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1832 1843 
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1833 
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@@ -1625,7 +1625,7 @@ Table of contents
1625 1625 
 <p>
1626 1626 
 Discontinuities between different media, deduced from
1627 1627 
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1628 
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1629 1629 
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1630 1630 
 {\bf Maxwell's equations {\it (in matter)}, integral form}
1631 1631 
 </p>
 
@@ -1701,10 +1701,21 @@ These are basis of theory of reflection and refraction.
1701 1701 
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1702 1702
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@@ -1635,10 +1635,21 @@ Table of contents
1635 1635 
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1636 1636 
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1639 1650 
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@@ -1631,10 +1631,21 @@ Table of contents
1631 1631 
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1632 1632 
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1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emdm.html">Electromagnetodynamics in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm.html">Electromagnetic Waves in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm_ad.html">Absorption and Dispersion</a></li><li>EM Waves in Conductors</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_ad.html">Absorption and Dispersion&emsp;<small>[emdm.emwm.ad]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_wg.html">Waveguides&emsp;<small>[emdm.emwm.wg]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_ad.html">Absorption and Dispersion&emsp;<small>[emdm.emwm.ad]</small></a></li></ul><div id="outline-container-emdm_emwm_ad_c" class="outline-5">
1620 1620 
 <h5 id="emdm_emwm_ad_c">EM Waves in Conductors<a class="headline-permalink" href="./emdm_emwm_ad_c.html#emdm_emwm_ad_c"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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@@ -1727,10 +1727,21 @@ so the final form of the fields is
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 </div>
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-<hr><div id="postamble" class="status">
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+<hr>
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+<div class="license">
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1733 
+target="_blank" class="m-2">
1734 
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1735 
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1736 
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1737 
+Except where otherwise noted, all content is licensed under a
1738 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1739 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1740 
+</div>
1741 
+<div id="postamble" class="status">
1731 1742 
 <p class="author">Author: Jean-Sébastien Caux</p>
1732 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1733 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1743 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1744 
+<p class="validation"></p>
1734 1745 
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1735 1746
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6 6 
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7 7 
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refr.html">Refraction&emsp;<small>[emdm.emwm.refr]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul><div id="outline-container-emdm_emwm_plm" class="outline-4">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emdm.html">Electromagnetodynamics in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm.html">Electromagnetic Waves in Matter</a></li><li>Propagation in Linear Media</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refr.html">Refraction&emsp;<small>[emdm.emwm.refr]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul><div id="outline-container-emdm_emwm_plm" class="outline-4">
1620 1620 
 <h4 id="emdm_emwm_plm">Propagation in Linear Media<a class="headline-permalink" href="./emdm_emwm_plm.html#emdm_emwm_plm"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
   <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"/>
1622 1622 
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@@ -1660,7 +1660,7 @@ v = \frac{1}{\sqrt{\mu \varepsilon}} = \frac{c}{n}
1660 1660 
 \]
1661 1661 
 where the index of refraction of the material is defined as
1662 1662 
 </p>
1663 
-<div class="main div" id="orgeaea7cb">
1663 
+<div class="main div" id="org72bb44e">
1664 1664 
 <p>
1665 1665 
 {\bf Index of refraction}
1666 1666 
 \[
 
@@ -1702,10 +1702,21 @@ I = \frac{1}{2} \varepsilon v E_0^2
1702 1702
1703 1703
1704 1704
1705 
-<hr><div id="postamble" class="status">
1705 
+<hr>
1706 
+<div class="license">
1707 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1708 
+target="_blank" class="m-2">
1709 
+<img alt="Creative Commons License" style="border-width:0"
1710 
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1711 
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1712 
+Except where otherwise noted, all content is licensed under a
1713 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1714 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1715 
+</div>
1716 
+<div id="postamble" class="status">
1706 1717 
 <p class="author">Author: Jean-Sébastien Caux</p>
1707 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1708 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1718 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1719 
+<p class="validation"></p>
1709 1720 
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1710 1721
1711 1722

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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refr.html">Refraction&emsp;<small>[emdm.emwm.refr]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refl_ni.html">Normal Incidence&emsp;<small>[emdm.emwm.refl.ni]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul>
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emdm.html">Electromagnetodynamics in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm.html">Electromagnetic Waves in Matter</a></li><li>Reflection and Transmission</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refr.html">Refraction&emsp;<small>[emdm.emwm.refr]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refl_ni.html">Normal Incidence&emsp;<small>[emdm.emwm.refl.ni]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul>
1620 1620 
 <h4 id="emdm_emwm_refl">Reflection and Transmission<a class="headline-permalink" href="./emdm_emwm_refl.html#emdm_emwm_refl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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1622 1622 
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@@ -1646,10 +1646,21 @@ Detailed study:  starts from boundary conditions \ref{Gr(7.64)},
1646 1646 
 <li><a href="emdm_emwm_refl_Ba.html">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span></li>
1647 1647 
 </ul>
1648 1648
1649 
-<hr><div id="postamble" class="status">
1649 
+<hr>
1650 
+<div class="license">
1651 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1652 
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1653 
+<img alt="Creative Commons License" style="border-width:0"
1654 
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1655 
+</a>
1656 
+Except where otherwise noted, all content is licensed under a
1657 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1658 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1659 
+</div>
1660 
+<div id="postamble" class="status">
1650 1661 
 <p class="author">Author: Jean-Sébastien Caux</p>
1651 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1652 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1662 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1663 
+<p class="validation"></p>
1653 1664 
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1654 1665
1655 1666

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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_Fe.html">Fresnel's Equations&emsp;<small>[emdm.emwm.refl.Fe]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_ad.html">Absorption and Dispersion&emsp;<small>[emdm.emwm.ad]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul><div id="outline-container-emdm_emwm_refl_Ba" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emdm.html">Electromagnetodynamics in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm.html">Electromagnetic Waves in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm_refl.html">Reflection and Transmission</a></li><li>Brewster's Angle</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_Fe.html">Fresnel's Equations&emsp;<small>[emdm.emwm.refl.Fe]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_ad.html">Absorption and Dispersion&emsp;<small>[emdm.emwm.ad]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul><div id="outline-container-emdm_emwm_refl_Ba" class="outline-5">
1620 1620 
 <h5 id="emdm_emwm_refl_Ba">Brewster's Angle<a class="headline-permalink" href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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1622 1622 
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@@ -1626,10 +1626,21 @@ Table of contents
1626 1626 
 </div>
1627 1627
1628 1628
1629 
-<hr><div id="postamble" class="status">
1629 
+<hr>
1630 
+<div class="license">
1631 
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1632 
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1633 
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1634 
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1635 
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1636 
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1637 
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1638 
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1639 
+</div>
1640 
+<div id="postamble" class="status">
1630 1641 
 <p class="author">Author: Jean-Sébastien Caux</p>
1631 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1632 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1642 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1643 
+<p class="validation"></p>
1633 1644 
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1634 1645
1635 1646

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7 7 
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_oi.html">Oblique Incidence&emsp;<small>[emdm.emwm.refl.oi]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refl_Ba.html">Brewster's Angle&emsp;<small>[emdm.emwm.refl.Ba]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul><div id="outline-container-emdm_emwm_refl_Fe" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emdm.html">Electromagnetodynamics in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm.html">Electromagnetic Waves in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm_refl.html">Reflection and Transmission</a></li><li>Fresnel's Equations</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_oi.html">Oblique Incidence&emsp;<small>[emdm.emwm.refl.oi]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refl_Ba.html">Brewster's Angle&emsp;<small>[emdm.emwm.refl.Ba]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul><div id="outline-container-emdm_emwm_refl_Fe" class="outline-5">
1620 1620 
 <h5 id="emdm_emwm_refl_Fe">Fresnel's Equations<a class="headline-permalink" href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
   <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"/>
1622 1622 
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@@ -1627,10 +1627,21 @@ Table of contents
1627 1627
1628 1628
1629 1629
1630 
-<hr><div id="postamble" class="status">
1630 
+<hr>
1631 
+<div class="license">
1632 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1633 
+target="_blank" class="m-2">
1634 
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1635 
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1636 
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1637 
+Except where otherwise noted, all content is licensed under a
1638 
+<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
1639 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1640 
+</div>
1641 
+<div id="postamble" class="status">
1631 1642 
 <p class="author">Author: Jean-Sébastien Caux</p>
1632 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1633 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1643 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1644 
+<p class="validation"></p>
1634 1645 
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1635 1646
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7 7 
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refl_oi.html">Oblique Incidence&emsp;<small>[emdm.emwm.refl.oi]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul><div id="outline-container-emdm_emwm_refl_ni" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emdm.html">Electromagnetodynamics in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm.html">Electromagnetic Waves in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm_refl.html">Reflection and Transmission</a></li><li>Normal Incidence</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refl_oi.html">Oblique Incidence&emsp;<small>[emdm.emwm.refl.oi]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul><div id="outline-container-emdm_emwm_refl_ni" class="outline-5">
1620 1620 
 <h5 id="emdm_emwm_refl_ni">Normal Incidence<a class="headline-permalink" href="./emdm_emwm_refl_ni.html#emdm_emwm_refl_ni"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
   <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"/>
1622 1622 
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@@ -1718,10 +1718,21 @@ R + T = 1.
1718 1718
1719 1719
1720 1720
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1721 
+<hr>
1722 
+<div class="license">
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1730 
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1731 
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1732 
+<div id="postamble" class="status">
1722 1733 
 <p class="author">Author: Jean-Sébastien Caux</p>
1723 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1724 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1734 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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+<p class="validation"></p>
1725 1736 
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7 7 
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_ni.html">Normal Incidence&emsp;<small>[emdm.emwm.refl.ni]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refl_Fe.html">Fresnel's Equations&emsp;<small>[emdm.emwm.refl.Fe]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul><div id="outline-container-emdm_emwm_refl_oi" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emdm.html">Electromagnetodynamics in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm.html">Electromagnetic Waves in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm_refl.html">Reflection and Transmission</a></li><li>Oblique Incidence</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_ni.html">Normal Incidence&emsp;<small>[emdm.emwm.refl.ni]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refl_Fe.html">Fresnel's Equations&emsp;<small>[emdm.emwm.refl.Fe]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul><div id="outline-container-emdm_emwm_refl_oi" class="outline-5">
1620 1620 
 <h5 id="emdm_emwm_refl_oi">Oblique Incidence<a class="headline-permalink" href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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1622 1622 
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@@ -1660,7 +1660,7 @@ These forms for incident, reflected and transmitted wave can be substituted in t
1660 1660 
 <p>
1661 1661 
 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
1662 1662 
 </p>
1663 
-<div class="core div" id="orga1858e0">
1663 
+<div class="core div" id="org699621d">
1664 1664 
 <p>
1665 1665 
 {\bf First law of reflection:}
1666 1666 
 the incident, reflected and transmitted wave vectors form a plane (called the plane of incidence) which also includes the normal to the surface.
 
@@ -1675,7 +1675,7 @@ Specializing (\ref{eq:RTObliquek}) to our notations, we have
1675 1675 
 with the incidence (\(\theta_I\)) and reflection (\(\theta_R\)) angles
1676 1676 
 and the angle of refraction (\(\theta_T\)) obey the following laws:
1677 1677 
 </p>
1678 
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1679 1679 
 <p>
1680 1680 
 {\bf Law of reflection}
1681 1681 
 \[
 
@@ -1733,7 +1733,7 @@ while the third equation becomes
1733 1733 
 \]
1734 1734 
 Writing everything in terms of the incident amplitude, we get
1735 1735 
 </p>
1736 
-<div class="main div" id="org4da7748">
1736 
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1737 1737 
 <p>
1738 1738 
 {\bf Fresnel's equations for reflection and transmission amplitudes (parallel case)}
1739 1739 
 \[
 
@@ -1753,7 +1753,7 @@ Amplitudes for transmitted and reflected wave: depend on angle of incidence:
1753 1753 
 Behaviour: for \(\theta_I = 0\) we recover (\ref{Gr(9.82)}).
1754 1754 
 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
1755 1755 
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1756 
-<div class="main div" id="orge2d9787">
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1757 1757 
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1758 1758 
 {\bf Brewster's angle {\it (at which the reflected wave amplitude vanishes)}}
1759 1759 
   \[
 
@@ -1788,10 +1788,21 @@ Of course, we get \(R + T = 1\) as expected.
1788 1788
1789 1789
1790 1790
1791 
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1791 
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1792 
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1793 
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1799 
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1800 
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1801 
+</div>
1802 
+<div id="postamble" class="status">
1792 1803 
 <p class="author">Author: Jean-Sébastien Caux</p>
1793 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1794 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1804 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1805 
+<p class="validation"></p>
1795 1806 
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1796 1807
1797 1808

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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_plm.html">Propagation in Linear Media&emsp;<small>[emdm.emwm.plm]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul><div id="outline-container-emdm_emwm_refr" class="outline-4">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emdm.html">Electromagnetodynamics in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm.html">Electromagnetic Waves in Matter</a></li><li>Refraction</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_plm.html">Propagation in Linear Media&emsp;<small>[emdm.emwm.plm]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul><div id="outline-container-emdm_emwm_refr" class="outline-4">
1620 1620 
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1622 1622 
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@@ -1627,10 +1627,21 @@ Table of contents
1627 1627
1628 1628
1629 1629
1630 
-<hr><div id="postamble" class="status">
1630 
+<hr>
1631 
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1632 
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1633 
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1638 
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1639 
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1640 
+</div>
1641 
+<div id="postamble" class="status">
1631 1642 
 <p class="author">Author: Jean-Sébastien Caux</p>
1632 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1633 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1643 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1644 
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1634 1645 
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1635 1646
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_ad_c.html">EM Waves in Conductors&emsp;<small>[emdm.emwm.ad.c]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_wg_gw.html">Guided waves&emsp;<small>[emdm.emwm.wg.gw]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul>
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emdm.html">Electromagnetodynamics in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm.html">Electromagnetic Waves in Matter</a></li><li>Waveguides</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_ad_c.html">EM Waves in Conductors&emsp;<small>[emdm.emwm.ad.c]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_wg_gw.html">Guided waves&emsp;<small>[emdm.emwm.wg.gw]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul>
1620 1620 
 <h4 id="emdm_emwm_wg">Waveguides<a class="headline-permalink" href="./emdm_emwm_wg.html#emdm_emwm_wg"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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1622 1622 
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@@ -1633,10 +1633,21 @@ Table of contents
1633 1633 
 <li><a href="emdm_emwm_wg_c.html">Coaxial Lines</a><span class="headline-id">emdm.emwm.wg.c</span></li>
1634 1634 
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1635 1635
1636 
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1636 
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1637 
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1645 
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1646 
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1647 
+<div id="postamble" class="status">
1637 1648 
 <p class="author">Author: Jean-Sébastien Caux</p>
1638 
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1639 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1649 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1650 
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1640 1651 
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1641 1652
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_wg_r.html">Rectangular Waveguides&emsp;<small>[emdm.emwm.wg.r]</small></a></li><li>Next:&nbsp;<a href="emf.html">Electromagnetic Fields&emsp;<small>[emf]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_wg.html">Waveguides&emsp;<small>[emdm.emwm.wg]</small></a></li></ul><div id="outline-container-emdm_emwm_wg_c" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emdm.html">Electromagnetodynamics in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm.html">Electromagnetic Waves in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm_wg.html">Waveguides</a></li><li>Coaxial Lines</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_wg_r.html">Rectangular Waveguides&emsp;<small>[emdm.emwm.wg.r]</small></a></li><li>Next:&nbsp;<a href="emf.html">Electromagnetic Fields&emsp;<small>[emf]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_wg.html">Waveguides&emsp;<small>[emdm.emwm.wg]</small></a></li></ul><div id="outline-container-emdm_emwm_wg_c" class="outline-5">
1620 1620 
 <h5 id="emdm_emwm_wg_c">Coaxial Lines<a class="headline-permalink" href="./emdm_emwm_wg_c.html#emdm_emwm_wg_c"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1653,10 +1653,21 @@ where \(A\) is a constant amplitude. Substituting and taking the real part,
1653 1653 
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1654 1654
1655 1655
1656 
-<hr><div id="postamble" class="status">
1656 
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1657 
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1658 
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1665 
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1666 
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1667 
+<div id="postamble" class="status">
1657 1668 
 <p class="author">Author: Jean-Sébastien Caux</p>
1658 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1659 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1669 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1670 
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
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1617 1617 
 </details>
1618 1618 
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1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_wg.html">Waveguides&emsp;<small>[emdm.emwm.wg]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_wg_r.html">Rectangular Waveguides&emsp;<small>[emdm.emwm.wg.r]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_wg.html">Waveguides&emsp;<small>[emdm.emwm.wg]</small></a></li></ul><div id="outline-container-emdm_emwm_wg_gw" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emdm.html">Electromagnetodynamics in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm.html">Electromagnetic Waves in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm_wg.html">Waveguides</a></li><li>Guided waves</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_wg.html">Waveguides&emsp;<small>[emdm.emwm.wg]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_wg_r.html">Rectangular Waveguides&emsp;<small>[emdm.emwm.wg.r]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_wg.html">Waveguides&emsp;<small>[emdm.emwm.wg]</small></a></li></ul><div id="outline-container-emdm_emwm_wg_gw" class="outline-5">
1620 1620 
 <h5 id="emdm_emwm_wg_gw">Guided waves<a class="headline-permalink" href="./emdm_emwm_wg_gw.html#emdm_emwm_wg_gw"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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@@ -1677,10 +1677,21 @@ with an identical equation for \(B_z\). If \(E_z = 0\) the waves are called {\bf
1677 1677
1678 1678
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1680 
+<hr>
1681 
+<div class="license">
1682 
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1683 
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1685 
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1686 
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1687 
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1688 
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1689 
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1690 
+</div>
1691 
+<div id="postamble" class="status">
1681 1692 
 <p class="author">Author: Jean-Sébastien Caux</p>
1682 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1683 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
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1617 1617 
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1618 1618 
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1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_wg_gw.html">Guided waves&emsp;<small>[emdm.emwm.wg.gw]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_wg_c.html">Coaxial Lines&emsp;<small>[emdm.emwm.wg.c]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_wg.html">Waveguides&emsp;<small>[emdm.emwm.wg]</small></a></li></ul><div id="outline-container-emdm_emwm_wg_r" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emdm.html">Electromagnetodynamics in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm.html">Electromagnetic Waves in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm_wg.html">Waveguides</a></li><li>Rectangular Waveguides</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_wg_gw.html">Guided waves&emsp;<small>[emdm.emwm.wg.gw]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_wg_c.html">Coaxial Lines&emsp;<small>[emdm.emwm.wg.c]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_wg.html">Waveguides&emsp;<small>[emdm.emwm.wg]</small></a></li></ul><div id="outline-container-emdm_emwm_wg_r" class="outline-5">
1620 1620 
 <h5 id="emdm_emwm_wg_r">Rectangular Waveguides<a class="headline-permalink" href="./emdm_emwm_wg_r.html#emdm_emwm_wg_r"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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@@ -1686,10 +1686,21 @@ which is {\it greater} than \(c\). The energy of the wave however propagates at
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1699 
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1700 
+<div id="postamble" class="status">
1690 1701 
 <p class="author">Author: Jean-Sébastien Caux</p>
1691 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1692 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1702 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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1619 
-<ul class="navigation-links"><li>Next:&nbsp;<a href="emf_svp.html">Scalar and Vector Potentials&emsp;<small>[emf.svp]</small></a></li></ul>
1619 
+<ul class="breadcrumbs"><li>Electromagnetic Fields</li></ul><ul class="navigation-links"><li>Next:&nbsp;<a href="emf_svp.html">Scalar and Vector Potentials&emsp;<small>[emf.svp]</small></a></li></ul>
1620 1620 
 <h2 id="emf">Electromagnetic Fields<a class="headline-permalink" href="./emf.html#emf"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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1628 1628 
 Prerequisites
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 Objectives
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1658 1669 
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1659 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1660 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1670 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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1618 1618 
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1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emf_svp.html">Scalar and Vector Potentials&emsp;<small>[emf.svp]</small></a></li><li>Next:&nbsp;<a href="emf_g_Cg.html">Coulomb Gauge&emsp;<small>[emf.g.Cg]</small></a></li><li>Up:&nbsp;<a href="emf.html">Electromagnetic Fields&emsp;<small>[emf]</small></a></li></ul>
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emf.html">Electromagnetic Fields</a></li><li>Gauge Freedom and Choices</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emf_svp.html">Scalar and Vector Potentials&emsp;<small>[emf.svp]</small></a></li><li>Next:&nbsp;<a href="emf_g_Cg.html">Coulomb Gauge&emsp;<small>[emf.g.Cg]</small></a></li><li>Up:&nbsp;<a href="emf.html">Electromagnetic Fields&emsp;<small>[emf]</small></a></li></ul>
1620 1620 
 <h3 id="emf_g">Gauge Freedom and Choices<a class="headline-permalink" href="./emf_g.html#emf_g"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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@@ -1651,10 +1651,21 @@ implementing a <b>gauge transformation</b>.
1651 1651 
 <li><a href="emf_g_Lg.html">Lorenz Gauge; d'Alembertian; Inhomogeneous Maxwell Equations</a><span class="headline-id">emf.g.Lg</span></li>
1652 1652 
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1664 
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1665 
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1655 1666 
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1656 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1657 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1667 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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1617 1617 
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1618 1618 
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1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emf_g.html">Gauge Freedom and Choices&emsp;<small>[emf.g]</small></a></li><li>Next:&nbsp;<a href="emf_g_Lg.html">Lorenz Gauge; d'Alembertian; Inhomogeneous Maxwell Equations&emsp;<small>[emf.g.Lg]</small></a></li><li>Up:&nbsp;<a href="emf_g.html">Gauge Freedom and Choices&emsp;<small>[emf.g]</small></a></li></ul><div id="outline-container-emf_g_Cg" class="outline-4">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emf.html">Electromagnetic Fields</a></li><li><a class="breadcrumb-link"href="emf_g.html">Gauge Freedom and Choices</a></li><li>Coulomb Gauge</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emf_g.html">Gauge Freedom and Choices&emsp;<small>[emf.g]</small></a></li><li>Next:&nbsp;<a href="emf_g_Lg.html">Lorenz Gauge; d'Alembertian; Inhomogeneous Maxwell Equations&emsp;<small>[emf.g.Lg]</small></a></li><li>Up:&nbsp;<a href="emf_g.html">Gauge Freedom and Choices&emsp;<small>[emf.g]</small></a></li></ul><div id="outline-container-emf_g_Cg" class="outline-4">
1620 1620 
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@@ -1650,10 +1650,21 @@ Although Gauss's law looks nice in the Coulomb gauge, Amp{\`e}re-Maxwell does no
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1654 1665 
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1655 
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1656 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1666 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
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1617 1617 
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1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emf_g_Cg.html">Coulomb Gauge&emsp;<small>[emf.g.Cg]</small></a></li><li>Next:&nbsp;<a href="red.html">Relativistic Electrodynamics&emsp;<small>[red]</small></a></li><li>Up:&nbsp;<a href="emf_g.html">Gauge Freedom and Choices&emsp;<small>[emf.g]</small></a></li></ul><div id="outline-container-emf_g_Lg" class="outline-4">
1619 
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1620 1620 
 <h4 id="emf_g_Lg">Lorenz Gauge; d'Alembertian; Inhomogeneous Maxwell Equations<a class="headline-permalink" href="./emf_g_Lg.html#emf_g_Lg"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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@@ -1638,7 +1638,7 @@ while the equation for \(V\) becomes
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 \]
1639 1639 
 These can be written compactly upon introducing a new operator: the
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1643 1643 
 {\bf d'Alembertian operator}
1644 1644 
 \[
 
@@ -1651,7 +1651,7 @@ These can be written compactly upon introducing a new operator: the
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1652 1652 
 so we get the
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1656 1656 
 {\bf Inhomogeneous Maxwell equations (Lorenz gauge)}
1657 1657 
 \[
 
@@ -1686,10 +1686,21 @@ we have by direct inspection
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1699 
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1700 
+<div id="postamble" class="status">
1690 1701 
 <p class="author">Author: Jean-Sébastien Caux</p>
1691 
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1692 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1702 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
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1617 1617 
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1618 1618 
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1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="emf.html">Electromagnetic Fields&emsp;<small>[emf]</small></a></li><li>Next:&nbsp;<a href="emf_g.html">Gauge Freedom and Choices&emsp;<small>[emf.g]</small></a></li><li>Up:&nbsp;<a href="emf.html">Electromagnetic Fields&emsp;<small>[emf]</small></a></li></ul><div id="outline-container-emf_svp" class="outline-3">
1619 
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 <h3 id="emf_svp">Scalar and Vector Potentials<a class="headline-permalink" href="./emf_svp.html#emf_svp"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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   <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
1622 1622 
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@@ -1639,7 +1639,7 @@ Useful strategy: represent fields in terms of potentials.
1639 1639 
 <p>
1640 1640 
 Easiest:
1641 1641 
 </p>
1642 
-<div class="core div" id="org45e0ed0">
1642 
+<div class="core div" id="org25c4e2b">
1643 1643 
 <p>
1644 1644 
 \[
1645 1645 
     {\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}
 
@@ -1655,7 +1655,7 @@ Putting this into Faraday's law gives
1655 1655 
 \]
1656 1656 
 so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \nabla} V\)) so we get
1657 1657 
 </p>
1658 
-<div class="core div" id="orgb9810af">
1658 
+<div class="core div" id="orgad63bf4">
1659 1659 
 <p>
1660 1660 
 \[
1661 1661 
     {\boldsymbol E} = -{\boldsymbol \nabla} V - \frac{\partial {\boldsymbol A}}{\partial t}
 
@@ -1668,7 +1668,7 @@ so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \
1668 1668 
 <p>
1669 1669 
 Using this potential representation for \({\boldsymbol E}\) and \({\boldsymbol B}\) automatically fulfills the two homogeneous Maxwell equations. For the inhomogeneous equations, substituting (\ref{eq:E_from_Potentials}) into Gauss's law gives
1670 1670 
 </p>
1671 
-<div class="main div" id="org1c236f0">
1671 
+<div class="main div" id="org64e1b1a">
1672 1672 
 <p>
1673 1673 
 \[
1674 1674 
     {\boldsymbol \nabla}^2 V + \frac{\partial}{\partial t} {\boldsymbol \nabla} \cdot {\boldsymbol A} = -\frac{\rho}{\varepsilon_0}
 
@@ -1684,7 +1684,7 @@ whereas Amp{\`ere}-Maxwell becomes
1684 1684 
 \]
1685 1685 
 which becomes after simple rearrangement and use of the identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\),
1686 1686 
 </p>
1687 
-<div class="main div" id="orgf3f4dcd">
1687 
+<div class="main div" id="org184a5ba">
1688 1688 
 <p>
1689 1689 
 \[
1690 1690 
   \left( {\boldsymbol ∇}^2 {\boldsymbol A} - μ_0 ε_0 \frac{∂^2 {\boldsymbol A}}{∂ t^2} \right)
 
@@ -1705,10 +1705,21 @@ which becomes after simple rearrangement and use of the identity \({\boldsymbol
1705 1705
1706 1706
1707 1707
1708 
-<hr><div id="postamble" class="status">
1708 
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1709 
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1710 
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1711 
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1713 
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1716 
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1717 
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1718 
+</div>
1719 
+<div id="postamble" class="status">
1709 1720 
 <p class="author">Author: Jean-Sébastien Caux</p>
1710 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1711 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1721 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1722 
+<p class="validation"></p>
1712 1723 
 </div>
1713 1724
1714 1725

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7 7 
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Next:&nbsp;<a href="ems_es.html">Electrostatics&emsp;<small>[ems.es]</small></a></li></ul>
1619 
+<ul class="breadcrumbs"><li>Electromagnetostatics</li></ul><ul class="navigation-links"><li>Next:&nbsp;<a href="ems_es.html">Electrostatics&emsp;<small>[ems.es]</small></a></li></ul>
1620 1620 
 <h2 id="ems">Electromagnetostatics<a class="headline-permalink" href="./ems.html#ems"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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1622 1622 
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@@ -1634,10 +1634,21 @@ Table of contents
1634 1634 
 <li><a href="ems_ms.html">Magnetostatics</a><span class="headline-id">ems.ms</span></li>
1635 1635 
 </ul>
1636 1636
1637 
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1637 
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1646 
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1647 
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1648 
+<div id="postamble" class="status">
1638 1649 
 <p class="author">Author: Jean-Sébastien Caux</p>
1639 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1640 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1650 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1651 
+<p class="validation"></p>
1641 1652 
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1642 1653
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2 2 
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3 3 
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6 6 
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7 7 
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@@ -1616,15 +1616,15 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_es_c_cap.html">Capacitors&emsp;<small>[ems.es.c.cap]</small></a></li><li>Next:&nbsp;<a href="ems_ca_fe.html">Fundamental Equations for the Electrostatic Potential&emsp;<small>[ems.ca.fe]</small></a></li><li>Up:&nbsp;<a href="ems.html">Electromagnetostatics&emsp;<small>[ems]</small></a></li></ul>
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="ems.html">Electromagnetostatics</a></li><li>Calculating or Approximating the Electrostatic Potential</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_es_c_cap.html">Capacitors&emsp;<small>[ems.es.c.cap]</small></a></li><li>Next:&nbsp;<a href="ems_ca_fe.html">Fundamental Equations for the Electrostatic Potential&emsp;<small>[ems.ca.fe]</small></a></li><li>Up:&nbsp;<a href="ems.html">Electromagnetostatics&emsp;<small>[ems]</small></a></li></ul>
1620 1620 
 <h3 id="ems_ca">Calculating or Approximating the Electrostatic Potential<a class="headline-permalink" href="./ems_ca.html#ems_ca"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
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1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
1623 1623 
 </svg></a><span class="headline-id">ems.ca</span></h3>
1624 1624
1625 1625 
 <div class="outline-text-3" id="text-ems_ca">
1626 
-<details class="prereq" id="orgbb80f83">
1627 
-<summary id="org379b6a1">
1626 
+<details class="prereq" id="org813707e">
1627 
+<summary id="org67fafaf">
1628 1628 
 Prerequisites
1629 1629 
 </summary>
1630 1630 
 <ul class="org-ul">
 
@@ -1632,8 +1632,8 @@ Prerequisites
1632 1632 
 </ul>
1633 1633 
 </details>
1634 1634
1635 
-<details class="objectives" id="orgd1100e2">
1636 
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1635 
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1636 
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1637 1637 
 Objectives
1638 1638 
 </summary>
1639 1639 
 <ul class="org-ul">
 
@@ -1656,10 +1656,21 @@ Objectives
1656 1656 
 <li><a href="ems_ca_me.html">The Multipole Expansion</a><span class="headline-id">ems.ca.me</span></li>
1657 1657 
 </ul>
1658 1658
1659 
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1659 
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1660 
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1661 
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1668 
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1669 
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1670 
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1660 1671 
 <p class="author">Author: Jean-Sébastien Caux</p>
1661 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1662 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1672 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1673 
+<p class="validation"></p>
1663 1674 
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1664 1675
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2 2 
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3 3 
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6 6 
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7 7 
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ca.html">Calculating or Approximating the Electrostatic Potential&emsp;<small>[ems.ca]</small></a></li><li>Next:&nbsp;<a href="ems_ca_fe_L.html">The Laplace Equation&emsp;<small>[ems.ca.fe.L]</small></a></li><li>Up:&nbsp;<a href="ems_ca.html">Calculating or Approximating the Electrostatic Potential&emsp;<small>[ems.ca]</small></a></li></ul>
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="ems.html">Electromagnetostatics</a></li><li><a class="breadcrumb-link"href="ems_ca.html">Calculating or Approximating the Electrostatic Potential</a></li><li>Fundamental Equations for the Electrostatic Potential</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ca.html">Calculating or Approximating the Electrostatic Potential&emsp;<small>[ems.ca]</small></a></li><li>Next:&nbsp;<a href="ems_ca_fe_L.html">The Laplace Equation&emsp;<small>[ems.ca.fe.L]</small></a></li><li>Up:&nbsp;<a href="ems_ca.html">Calculating or Approximating the Electrostatic Potential&emsp;<small>[ems.ca]</small></a></li></ul>
1620 1620 
 <h4 id="ems_ca_fe">Fundamental Equations for the Electrostatic Potential<a class="headline-permalink" href="./ems_ca_fe.html#ems_ca_fe"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1621 
   <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"/>
1622 1622 
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
 
@@ -1628,7 +1628,7 @@ A generic configuration of static charges coupled via the Coulomb interaction
1628 1628 
 defines an electrostatic problem, whose solution is in principle obtained
1629 1629 
 from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_vcd">E_vcd</a>
1630 1630 
 </p>
1631 
-<div class="main div" id="org14f8ad7">
1631 
+<div class="main div" id="orgcced719">
1632 1632 
 \begin{equation*}
1633 1633 
   {\bf E} ({\bf r}) = \frac{1}{4\pi\varepsilon_0} \int_{\mathbb{R}^3} d\tau' \rho({\bf r}') \frac{{\bf r} - {\bf r}'}{|{\bf r} - {\bf r}'|^3}
1634 1634 
 \end{equation*}
 
@@ -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 (\ref{eq:V_from_rho})
1640 1640 
 </p>
1641 
-<div class="main div" id="org83b53d5">
1641 
+<div class="main div" id="org68ed8a6">
1642 1642 
 <p>
1643 1643 
 \[
1644 1644 
     V({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' \frac{\rho({\bf r}')}{|{\bf r} - {\bf r}'|}.
 
@@ -1654,7 +1654,7 @@ condition (\ref{Gr(2.20)}) can be expressed as the single
1654 1654 
 'local' (differential) condition (Poisson's equation) (\ref{eq:Poisson})
1655 1655 
 </p>
1656 1656
1657 
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1658 1658 
 <p>
1659 1659 
 \[
1660 1660 
     {\boldsymbol \nabla}^2 V = -\frac{\rho}{\varepsilon_0}.
 
@@ -1668,7 +1668,7 @@ condition (\ref{Gr(2.20)}) can be expressed as the single
1668 1668 
 In the specific case where the charge density vanishes, we fall back onto the simpler
1669 1669 
 Laplace equation
1670 1670 
 </p>
1671 
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1672 1672 
 <p>
1673 1673 
 \[
1674 1674 
     {\boldsymbol \nabla}^2 V = 0
 
@@ -1688,10 +1688,21 @@ Laplace equation
1688 1688 
 <li><a href="ems_ca_fe_uP.html">Uniqueness of Solution to Poisson's Equation</a><span class="headline-id">ems.ca.fe.uP</span></li>
1689 1689 
 </ul>
1690 1690
1691 
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1691 
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1692 
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1693 
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1694 
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1695 
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1696 
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1697 
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1698 
+Except where otherwise noted, all content is licensed under a
1699 
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1700 
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1701 
+</div>
1702 
+<div id="postamble" class="status">
1692 1703 
 <p class="author">Author: Jean-Sébastien Caux</p>
1693 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1694 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1704 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1705 
+<p class="validation"></p>
1695 1706 
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1696 1707
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@@ -1616,7 +1616,7 @@ Table of contents
1616 1616 
 </ul>
1617 1617 
 </details>
1618 1618 
 </nav>
1619 
-<ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ca_fe.html">Fundamental Equations for the Electrostatic Potential&emsp;<small>[ems.ca.fe]</small></a></li><li>Next:&nbsp;<a href="ems_ca_fe_g.html">Green's Identities&emsp;<small>[ems.ca.fe.g]</small></a></li><li>Up:&nbsp;<a href="ems_ca_fe.html">Fundamental Equations for the Electrostatic Potential&emsp;<small>[ems.ca.fe]</small></a></li></ul><div id="outline-container-ems_ca_fe_L" class="outline-5">
1619 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="ems.html">Electromagnetostatics</a></li><li><a class="breadcrumb-link"href="ems_ca.html">Calculating or Approximating the Electrostatic Potential</a></li><li><a class="breadcrumb-link"href="ems_ca_fe.html">Fundamental Equations for the Electrostatic Potential</a></li><li>The Laplace Equation</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ca_fe.html">Fundamental Equations for the Electrostatic Potential&emsp;<small>[ems.ca.fe]</small></a></li><li>Next:&nbsp;<a href="ems_ca_fe_g.html">Green's Identities&emsp;<small>[ems.ca.fe.g]</small></a></li><li>Up:&nbsp;<a href="ems_ca_fe.html">Fundamental Equations for the Electrostatic Potential&emsp;<small>[ems.ca.fe]</small></a></li></ul><div id="outline-container-ems_ca_fe_L" class="outline-5">
1620 1620 
 <h5 id="ems_ca_fe_L">The Laplace Equation<a class="headline-permalink" href="./ems_ca_fe_L.html#ems_ca_fe_L"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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   <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
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   <path d="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"/>
 
@@ -1702,7 +1702,7 @@ are necessarily positive, we thus require \(f_x &gt; 0\), \(f_y &gt; 0\) and \(f
1702 1702 
 of the \(f_x + f_y + f_z = 0\) condition above.
1703 1703 
 </p>
1704 1704
1705 
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1706 1706 
 <p>
1707 1707 
 <b>Earnshaw's theorem</b> <br>
1708 1708 
 Since solutions to Laplace's equation have no local minimum,
 
@@ -1790,10 +1790,21 @@ This all feels a bit amateurish and not very systematic. Can we be more precise
1790 1790
1791 1791
1792 1792
1793 
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1793 
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1794 
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1795 
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1796 
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1797 
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1798 
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1799 
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1800 
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1801 
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1802 
+target="_blank">Creative Commons Attribution 4.0 International License</a>.
1803 
+</div>
1804 
+<div id="postamble" class="status">
1794 1805 
 <p class="author">Author: Jean-Sébastien Caux</p>
1795 
-<p class="date">Created: 2022-02-08 Tue 17:21</p>
1796 
-<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
1806 
+<p class="date">Created: 2022-02-09 Wed 07:31</p>
1807 
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1797 1808 
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1798 1809
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