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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
207 changed files with 1882 additions and 1097 deletions
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build/c_m_cs_sph.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>
@@ -1688,70 +1692,46 @@ Infinitesimal surface element: depends on situation.
<div id="outline-container-c_m_cs_sph_grad" class="outline-6">
<h6 id="c_m_cs_sph_grad"><a href="#c_m_cs_sph_grad">Gradient</a></h6>
<div class="outline-text-6" id="text-c_m_cs_sph_grad">
<p>
\[
{\boldsymbol ∇} T = \frac{\partial T}{\partial r} \hat{\boldsymbol r} + \frac{1}{r} \frac{\partial T}{\partial \theta} \hat{\boldsymbol \theta}
</p>
<ul class="org-ul">
<li>\frac{1}{r\sin \theta} \frac{\partial T}{\partial \phi} \hat{\boldsymbol \phi}.</li>
</ul>
<p>
\begin{equation}
{\boldsymbol \nabla} T = \frac{\partial T}{\partial r} \hat{\boldsymbol r} + \frac{1}{r} \frac{\partial T}{\partial \theta} \hat{\boldsymbol \theta}
+ \frac{1}{r\sin \theta} \frac{\partial T}{\partial \phi} \hat{\boldsymbol \phi}.
\label{Gr(1.70)}
\]
</p>
\end{equation}
</div>
</div>
<div id="outline-container-c_m_cs_sph_div" class="outline-6">
<h6 id="c_m_cs_sph_div"><a href="#c_m_cs_sph_div">Divergence</a></h6>
<div class="outline-text-6" id="text-c_m_cs_sph_div">
<p>
\[
{\boldsymbol ∇} ⋅ {\bf v} = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 v_r) + \frac{1}{r\sin \theta} \frac{\partial}{\partial \theta} (sinθ v_{θ})
</p>
<ul class="org-ul">
<li>\frac{1}{r \sin \theta} \frac{∂ v_{φ}}{∂ φ}</li>
</ul>
<p>
\begin{equation}
{\boldsymbol \nabla} \cdot {\bf v} = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 v_r) + \frac{1}{r\sin \theta} \frac{\partial}{\partial \theta} (\sin\theta v_{\theta})
+ \frac{1}{r \sin \theta} \frac{\partial v_{\phi}}{\partial \phi}
\label{Gr(1.71)}
\]
</p>
\end{equation}
</div>
</div>
<div id="outline-container-c_m_cs_sph_curl" class="outline-6">
<h6 id="c_m_cs_sph_curl"><a href="#c_m_cs_sph_curl">Curl</a></h6>
<div class="outline-text-6" id="text-c_m_cs_sph_curl">
<p>
\[
{\boldsymbol ∇} × {\bf v} = \frac{1}{r\sin \theta} \left[ \frac{\partial}{\partial \theta} (sin θ v_{φ}) - \frac{∂ v_{θ}}{∂ φ} \right] \hat{\bf r}
</p>
<ul class="org-ul">
<li>\frac{1}{r} \left[ \frac{1}{\sin \theta} \frac{\partial v_r}{\partial \phi} - \frac{\partial}{\partial r} (r v_{φ}) \right] \hat{\boldsymbol \theta}</li>
<li>\frac{1}{r} \left[ \frac{\partial}{\partial r} (r v_{θ}) - \frac{\partial v_r}{\partial \theta} \right] \hat{\boldsymbol \phi}</li>
</ul>
<p>
\begin{equation}
{\boldsymbol \nabla} \times {\bf v} = \frac{1}{r\sin \theta} \left[ \frac{\partial}{\partial \theta} (\sin \theta v_{\phi}) - \frac{\partial v_{\theta}}{\partial \phi} \right] \hat{\bf r}
+ \frac{1}{r} \left[ \frac{1}{\sin \theta} \frac{\partial v_r}{\partial \phi} - \frac{\partial}{\partial r} (r v_{\phi}) \right] \hat{\boldsymbol \theta}
+ \frac{1}{r} \left[ \frac{\partial}{\partial r} (r v_{\theta}) - \frac{\partial v_r}{\partial \theta} \right] \hat{\boldsymbol \phi}
\label{Gr(1.72)}
\]
</p>
\end{equation}
</div>
</div>
<div id="outline-container-c_m_cs_sph_lap" class="outline-6">
<h6 id="c_m_cs_sph_lap"><a href="#c_m_cs_sph_lap">Laplacian</a></h6>
<div class="outline-text-6" id="text-c_m_cs_sph_lap">
<p>
\[
{\boldsymbol ∇}^2 T = \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial T}{\partial r}\right)
</p>
<ul class="org-ul">
<li>\frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( sin θ \frac{\partial T}{\partial \theta}\right)</li>
<li>\frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 T}{\partial \phi^2}</li>
</ul>
<p>
\begin{equation}
{\boldsymbol \nabla}^2 T = \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial T}{\partial r}\right)
+ \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial T}{\partial \theta}\right)
+ \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 T}{\partial \phi^2}
\label{Gr(1.73)}
\]
</p>
\end{equation}
</div>
</div>
</div>
@@ -1761,7 +1741,7 @@ Infinitesimal surface element: depends on situation.
<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>