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2022-02-14 06:33:37 +01:00
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-10 Thu 08:32 -->
<!-- 2022-02-13 Sun 21:20 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1597,13 +1597,13 @@ Table of contents
<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"/>
</svg></a><span class="headline-id">ems.ca.fe.g</span></h5>
<div class="outline-text-5" id="text-ems_ca_fe_g">
<div class="info div" id="org5bc5392">
<div class="info div" id="orgc7ef55c">
<p>
<b>George Green</b>
</p>
<aside id="orgb80e217">
<aside id="org5a19053">
<p>
See a \href{short biography of George Green on Wikipedia}{https://en.wikipedia.org/wiki/George\_Green\_(mathematician)}.
See a <a href="https://en.wikipedia.org/wiki/George%5C_Green%5C_(mathematician)">short bio on wikipedia</a>
</p>
</aside>
<p>
@@ -1632,19 +1632,53 @@ and
\[
\phi {\boldsymbol \nabla} \psi \cdot {\bf n} = \phi \frac{\partial \psi}{\partial n}.
\]
Substituting this in the divergence theorem gives {\bf Green's first identity}
Substituting this in the divergence theorem gives <b>Green's first identity</b>
</p>
<div class="eqlabel" id="org713a4b7">
<p>
<a id="Green1"></a><a href="./ems_ca_fe_g.html#Green1"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<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"/>
<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"/>
</svg></a>
</p>
<div class="alteqlabels" id="orgfb4fc39">
<ul class="org-ul">
<li>J (1.34)</li>
</ul>
</div>
</div>
<p>
\[
\int_{\cal V} d\tau ~(\phi {\boldsymbol \nabla}^2 \psi + {\boldsymbol \nabla} \phi \cdot {\boldsymbol \nabla} \psi) = \oint_{\cal S} da ~\phi \frac{\partial \psi}{\partial n}.
\label{eq:GreensFirstIdentity}
\tag{Green1}\label{Green1}
\]
This first identity will prove crucial in the argument that follows.
As an aside for now, for completeness, if we do the same thing again but with \(\phi\) and \(\psi\)
interchanged, and subtract the result, we obtain another useful result known as
{\bf Green's second identity} or {\bf Green's theorem}
<b>Green's second identity</b> or <b>Green's theorem</b>
</p>
<div class="eqlabel" id="org6298bab">
<p>
<a id="Green2"></a><a href="./ems_ca_fe_g.html#Green2"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<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"/>
<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"/>
</svg></a>
</p>
<div class="alteqlabels" id="org29adc56">
<ul class="org-ul">
<li>J (1.35)</li>
</ul>
</div>
</div>
<p>
\[
\int_{\cal V} d\tau (\phi {\boldsymbol \nabla}^2 \psi - \psi {\boldsymbol \nabla}^2 \phi)
= \oint_{\cal S} da \left(\phi \frac{\partial \psi}{\partial n} - \psi \frac{\partial \phi}{\partial n} \right).
\label{eq:GreensTheorem}
\tag{Green2}\label{Green2}
\]
</p>
</div>
@@ -1653,6 +1687,8 @@ interchanged, and subtract the result, we obtain another useful result known as
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ca_fe_L.html">The Laplace Equation&emsp;<small>[ems.ca.fe.L]</small></a></li><li>Next:&nbsp;<a href="ems_ca_fe_uP.html">Uniqueness of Solution to Poisson's Equation&emsp;<small>[ems.ca.fe.uP]</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>
<br>
<hr>
<div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
@@ -1666,7 +1702,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-10 Thu 08:32</p>
<p class="date">Created: 2022-02-13 Sun 21:20</p>
<p class="validation"></p>
</div>