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Update 2022-02-08 17:21

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Jean-Sébastien
2022-02-08 17:21:33 +01:00
parent 077433c40a
commit 3454aba504
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build/c_m_dd_3d.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-08 Tue 06:55 -->
<!-- 2022-02-08 Tue 17:21 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -272,6 +272,10 @@ Table of contents
</summary>
<ul>
<li>
<a href="./in_t_l.html#in_t_l">Section and equation labelling</a><span class="headline-id">in.t.l</span>
</li>
<li>
<a href="./in_t_c.html#in_t_c">Contextual colors</a><span class="headline-id">in.t.c</span>
</li>
@@ -736,7 +740,7 @@ Table of contents
</li>
<li>
<a href="./emsm_esm_d.html#emsm_esm_d">Dielectrics</a><span class="headline-id">emsm.esm.d</span>
<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
</li>
<li>
@@ -1618,43 +1622,37 @@ 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">c.m.dd.3d</span></h5>
<div class="outline-text-5" id="text-c_m_dd_3d">
<p>
\[
\begin{equation}
\delta^{(3)} ({\bf r} - {\bf r}') = \delta (x - x') \delta (y - y') \delta (z - z')
\label{Gr(1.96)}
\]
</p>
\end{equation}
<p>
\[
\begin{equation}
\int d\tau f({\bf r}) \delta^{(3)} ({\bf r} - {\bf a}) = f({\bf a})
\label{Gr(1.97)}
\]
</p>
\end{equation}
<p>
Resolution of divergence of \(\hat{\bf r}/r^2\) paradox:
</p>
<p>
\[
\begin{equation}
{\boldsymbol \nabla} \cdot \left(\frac{\hat{\bf r}}{r^2} \right)= 4\pi \delta^{(3)} ({\bf r}).
\label{Gr(1.99)}
\]
</p>
\end{equation}
<p>
More generally,
</p>
<div class="eqlabel" id="org99c1900">
<div class="eqlabel" id="org276d185">
<p>
<a id="divdel"></a><a href="./c_m_dd_3d.html#divdel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<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="orge7a14d6">
<div class="alteqlabels" id="org676b181">
<ul class="org-ul">
<li>Gr (1.100)</li>
</ul>
@@ -1662,43 +1660,37 @@ More generally,
</div>
</div>
<p>
\[
\begin{equation}
\boxed{
{\boldsymbol \nabla} \cdot \left(\frac{{\bf r} - {\bf r}'}{|{\bf r} - {\bf r}'|^3} \right)= 4\pi \delta^{(3)} ({\bf r}).
}
\tag{divdel}\label{divdel}
\]
</p>
\end{equation}
<p>
Since
</p>
<p>
\[
\begin{equation}
{\boldsymbol \nabla}_1 \left(\frac{1}{r_{12}}\right) = -\frac{\hat{\bf r}_{12}}{r_{12}^2}
\label{Gr(1.101)}
\]
</p>
\end{equation}
<p>
we have that
</p>
<p>
\[
\begin{equation}
{\boldsymbol \nabla}^2 \left( \frac{1}{|{\bf r} - {\bf r}'|} \right) = -4\pi \delta^{(3)} ({\bf r} - {\bf r}')
\label{Gr(1.102)}
\]
</p>
\end{equation}
</div>
</div>
<hr><div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-02-08 Tue 06:55</p>
<p class="date">Created: 2022-02-08 Tue 17:21</p>
<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
</div>