jscaux/pqed_html
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Update 2022-02-08 17:21

Browse Source
This commit is contained in:
Jean-Sébastien
2022-02-08 17:21:33 +01:00
parent 077433c40a
commit 3454aba504
207 changed files with 1882 additions and 1097 deletions
Download Patch File Download Diff File Expand all files Collapse all files
build/c_m_cs_cyl.html
+39 -67
View File
@@ -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>
@@ -1622,47 +1626,39 @@ Table of contents
\((r, \phi, z)\). Relation to Cartesian coordinates:
</p>
<p>
\[
\begin{equation}
x = r \cos \phi, y = r \sin \phi, z = z
\label{Gr(1.74)}
\]
</p>
\end{equation}
<p>
The unit vectors are
</p>
<p>
\[
\begin{equation}
\hat{\bf r} = \cos \phi ~\hat{\bf x} + \sin \phi~\hat{\bf y},
\hat{\boldsymbol \phi} = -\sin \theta ~\hat{\bf x} + \cos \phi~\hat{\bf y},
\hat{\bf z} = \hat{\bf z}.
\label{Gr(1.75)}
\]
</p>
\end{equation}
<p>
Infinitesimal displacement:
</p>
<p>
\[
\begin{equation}
d{\bf l} = dr ~\hat{\bf r} + r d\phi~\hat{\boldsymbol \phi} + dz ~\hat{\bf z}.
\label{Gr(1.77)}
\]
</p>
\end{equation}
<p>
Volume element:
</p>
<p>
\[
\begin{equation}
d\tau = r dr d\phi dz
\label{Gr(1.78)}
\]
</p>
\end{equation}
<p>
Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in ]-\infty, \infty[\).
@@ -1672,14 +1668,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
<div id="outline-container-c_m_cs_cyl_grad" class="outline-6">
<h6 id="c_m_cs_cyl_grad"><a href="#c_m_cs_cyl_grad">Gradient</a></h6>
<div class="outline-text-6" id="text-c_m_cs_cyl_grad">
<div class="eqlabel" id="org1f82008">
<div class="eqlabel" id="orgc4b6bac">
<p>
<a id="cylgrad"></a><a href="./c_m_cs_cyl.html#cylgrad"><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="org0d07fad">
<div class="alteqlabels" id="orgdcdbcb4">
<ul class="org-ul">
<li>Gr4(1.79)</li>
</ul>
@@ -1688,32 +1684,26 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
</div>
<p>
\[
{\boldsymbol ∇} T = \frac{\partial T}{\partial r}~\hat{\bf r}
</p>
<ul class="org-ul">
<li>\frac{1}{r} \frac{\partial T}{\partial \phi}~\hat{\boldsymbol \phi}</li>
<li>\frac{\partial T}{\partial z} ~\hat{\bf z}</li>
</ul>
<p>
\begin{equation}
{\boldsymbol \nabla} T = \frac{\partial T}{\partial r}~\hat{\bf r}
+ \frac{1}{r} \frac{\partial T}{\partial \phi}~\hat{\boldsymbol \phi}
+ \frac{\partial T}{\partial z} ~\hat{\bf z}
\tag{cylgrad}\label{cylgrad}
\]
</p>
\end{equation}
</div>
</div>
<div id="outline-container-c_m_cs_cyl_div" class="outline-6">
<h6 id="c_m_cs_cyl_div"><a href="#c_m_cs_cyl_div">Divergence</a></h6>
<div class="outline-text-6" id="text-c_m_cs_cyl_div">
<div class="eqlabel" id="org9646f8e">
<div class="eqlabel" id="orgf9d0262">
<p>
<a id="cyl_div"></a><a href="./c_m_cs_cyl.html#cyl_div"><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="orgef4d1a0">
<div class="alteqlabels" id="orgd456be1">
<ul class="org-ul">
<li>Gr4(2.21)</li>
</ul>
@@ -1722,32 +1712,26 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
</div>
<p>
\[
{\boldsymbol ∇} ⋅ {\bf v} = \frac{1}{r} \frac{\partial}{\partial r} (r v_r)
</p>
<ul class="org-ul">
<li>\frac{1}{r} \frac{∂ v_{φ}}{∂ φ} + \frac{\partial v_z}{\partial z}.</li>
</ul>
<p>
\begin{equation}
{\boldsymbol \nabla} \cdot {\bf v} = \frac{1}{r} \frac{\partial}{\partial r} (r v_r)
+ \frac{1}{r} \frac{\partial v_{\phi}}{\partial \phi} + \frac{\partial v_z}{\partial z}.
\tag{cyl_div}
\label{cyl_div}
\]
</p>
\end{equation}
</div>
</div>
<div id="outline-container-c_m_cs_cyl_curl" class="outline-6">
<h6 id="c_m_cs_cyl_curl"><a href="#c_m_cs_cyl_curl">Curl</a></h6>
<div class="outline-text-6" id="text-c_m_cs_cyl_curl">
<div class="eqlabel" id="org174b722">
<div class="eqlabel" id="orgeb922d3">
<p>
<a id="cyl_curl"></a><a href="./c_m_cs_cyl.html#cyl_curl"><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="org9f83d45">
<div class="alteqlabels" id="org66ae846">
<ul class="org-ul">
<li>Gr4(2.21)</li>
</ul>
@@ -1756,36 +1740,24 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
</div>
<p>
\[
{\boldsymbol ∇} × {\bf v} = \left( \frac{1}{r} \frac{\partial v_z}{\partial \phi} - \frac{∂ v_{φ}}{∂ z}\right) ~\hat{\bf r}
</p>
<ul class="org-ul">
<li>\left( \frac{\partial v_r}{\partial z} - \frac{\partial v_z}{\partial r} \right) ~\hat{\boldsymbol \phi}</li>
<li>\frac{1}{r} \left( \frac{\partial}{\partial r} (r v_{φ}) - \frac{\partial v_r}{\partial \phi} \right) ~\hat{\bf z}</li>
</ul>
<p>
\begin{align}
{\boldsymbol \nabla} \times {\bf v} = \left( \frac{1}{r} \frac{\partial v_z}{\partial \phi} - \frac{\partial v_{\phi}}{\partial z}\right) ~\hat{\bf r}
+ \left( \frac{\partial v_r}{\partial z} - \frac{\partial v_z}{\partial r} \right) ~\hat{\boldsymbol \phi} \nonumber \\
+ \frac{1}{r} \left( \frac{\partial}{\partial r} (r v_{\phi}) - \frac{\partial v_r}{\partial \phi} \right) ~\hat{\bf z}
\tag{cyl_curl}
\label{cyl_curl}
\]
</p>
\end{align}
</div>
</div>
<div id="outline-container-c_m_cs_cyl_lap" class="outline-6">
<h6 id="c_m_cs_cyl_lap"><a href="#c_m_cs_cyl_lap">Laplacian</a></h6>
<div class="outline-text-6" id="text-c_m_cs_cyl_lap">
<p>
\[
{\boldsymbol ∇}^2 T = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial T}{\partial r} \right)
</p>
<ul class="org-ul">
<li>\frac{1}{r^2} \frac{\partial^2 T}{\partial \phi^2} + \frac{\partial^2 T}{\partial z^2}</li>
</ul>
<p>
\begin{equation}
{\boldsymbol \nabla}^2 T = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial T}{\partial r} \right)
+ \frac{1}{r^2} \frac{\partial^2 T}{\partial \phi^2} + \frac{\partial^2 T}{\partial z^2}
\label{Gr(1.82)}
\]
</p>
\end{equation}
</div>
</div>
</div>
@@ -1795,7 +1767,7 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
<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>