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Update 2022-02-14 20:42

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Jean-Sébastien
2022-02-14 20:42:37 +01:00
parent 4cfe8cef59
commit 09a8ba5fb6
204 changed files with 1968 additions and 1206 deletions
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build/c_m_cs_cyl.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-13 Sun 21:20 -->
<!-- 2022-02-14 Mon 20:35 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1598,11 +1598,11 @@ Table of contents
</svg></a><span class="headline-id">c.m.cs.cyl</span></h5>
<div class="outline-text-5" id="text-c_m_cs_cyl">
<p>
\((r, \phi, z)\). Relation to Cartesian coordinates:
\((r, \varphi, z)\). Relation to Cartesian coordinates:
</p>
\begin{equation}
x = r \cos \phi, y = r \sin \phi, z = z
x = r \cos \varphi, y = r \sin \varphi, z = z
\label{Gr(1.74)}
\end{equation}
@@ -1611,8 +1611,8 @@ The unit vectors are
</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 r} = \cos \varphi ~\hat{\bf x} + \sin \varphi~\hat{\bf y},
\hat{\boldsymbol \varphi} = -\sin \theta ~\hat{\bf x} + \cos \varphi~\hat{\bf y},
\hat{\bf z} = \hat{\bf z}.
\label{Gr(1.75)}
\end{equation}
@@ -1622,7 +1622,7 @@ Infinitesimal displacement:
</p>
\begin{equation}
d{\bf l} = dr ~\hat{\bf r} + r d\phi~\hat{\boldsymbol \phi} + dz ~\hat{\bf z}.
d{\bf l} = dr ~\hat{\bf r} + r d\varphi~\hat{\boldsymbol \varphi} + dz ~\hat{\bf z}.
\label{Gr(1.77)}
\end{equation}
@@ -1631,26 +1631,26 @@ Volume element:
</p>
\begin{equation}
d\tau = r dr d\phi dz
d\tau = r dr d\varphi dz
\label{Gr(1.78)}
\end{equation}
<p>
Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in ]-\infty, \infty[\).
Range of parameters: \(r \in [0, \infty[\), \(\varphi \in [0, 2\pi[\) and \(z \in ]-\infty, \infty[\).
</p>
</div>
<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="org03db374">
<div class="eqlabel" id="org7329bb2">
<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="orgb61510b">
<div class="alteqlabels" id="org7d88a4a">
<ul class="org-ul">
<li>Gr4(1.79)</li>
</ul>
@@ -1661,7 +1661,7 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
\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{1}{r} \frac{\partial T}{\partial \varphi}~\hat{\boldsymbol \varphi}
+ \frac{\partial T}{\partial z} ~\hat{\bf z}
\tag{cylgrad}\label{cylgrad}
\end{equation}
@@ -1671,14 +1671,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
<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="orgd9435d5">
<div class="eqlabel" id="orgef5777e">
<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="org5c1c30c">
<div class="alteqlabels" id="org5bb7a68">
<ul class="org-ul">
<li>Gr4(2.21)</li>
</ul>
@@ -1689,7 +1689,7 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
\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}.
+ \frac{1}{r} \frac{\partial v_{\varphi}}{\partial \varphi} + \frac{\partial v_z}{\partial z}.
\tag{cyl_div}
\label{cyl_div}
\end{equation}
@@ -1699,14 +1699,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
<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="org695c029">
<div class="eqlabel" id="orgebbdea6">
<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="orge332907">
<div class="alteqlabels" id="org47230f3">
<ul class="org-ul">
<li>Gr4(2.21)</li>
</ul>
@@ -1716,9 +1716,9 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
</div>
\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}
{\boldsymbol \nabla} \times {\bf v} = \left( \frac{1}{r} \frac{\partial v_z}{\partial \varphi} - \frac{\partial v_{\varphi}}{\partial z}\right) ~\hat{\bf r}
+ \left( \frac{\partial v_r}{\partial z} - \frac{\partial v_z}{\partial r} \right) ~\hat{\boldsymbol \varphi} \nonumber \\
+ \frac{1}{r} \left( \frac{\partial}{\partial r} (r v_{\varphi}) - \frac{\partial v_r}{\partial \varphi} \right) ~\hat{\bf z}
\tag{cyl_curl}
\label{cyl_curl}
\end{align}
@@ -1730,7 +1730,7 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
<div class="outline-text-6" id="text-c_m_cs_cyl_lap">
\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}
+ \frac{1}{r^2} \frac{\partial^2 T}{\partial \varphi^2} + \frac{\partial^2 T}{\partial z^2}
\label{Gr(1.82)}
\end{equation}
</div>
@@ -1755,7 +1755,7 @@ 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: 2022-02-13 Sun 21:20</p>
<p class="date">Created: 2022-02-14 Mon 20:35</p>
<p class="validation"></p>
</div>