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Jean-Sébastien
2022-02-14 20:42:37 +01:00
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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>
@@ -1598,64 +1598,117 @@ Table of contents
</svg></a><span class="headline-id">c.m.cs.sph</span></h5>
<div class="outline-text-5" id="text-c_m_cs_sph">
<p>
\((r, \theta, \phi)\). \(\theta\) is the <b>polar angle</b>, \(\phi\) the <b>azimuthal angle</b>.
In this system, we use coordinates \((r, \theta, \varphi)\) in
which \(r\) is the distance from the chosen origin,
\(\theta\) is the <b>polar angle</b> and \(\varphi\) is the <b>azimuthal angle</b>.
</p>
<p>
The usual Cartesian coordinates relate to spherical coordinates
according to
</p>
<div class="eqlabel" id="org9fd0b35">
<p>
<a id="sph_xyz"></a><a href="./c_m_cs_sph.html#sph_xyz"><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="org9bb0d80">
</div>
</div>
<p>
\[
x = r \sin \theta \cos \phi,
y = r \sin \theta \sin \phi,
x = r \sin \theta \cos \varphi, \hspace{5mm}
y = r \sin \theta \sin \varphi, \hspace{5mm}
z = r \cos \theta.
\label{Gr(1.62)}
\tag{sph_xyz}\label{sph_xyz}
\]
</p>
<p>
Unit vectors: \(\hat{\boldsymbol r}, \hat{\boldsymbol \theta}, \hat{\boldsymbol \phi}\).
</p>
<p>
The unit vectors are written \(\hat{\boldsymbol r}\),
\(\hat{\boldsymbol \theta}\) and \(\hat{\boldsymbol \varphi}\).
A generic vector can be expressed as
\[
{\bf A} = A_r \hat{\bf r} + A_{\theta} \hat{\bf \theta} + A_{\phi} \hat{\boldsymbol \phi}
\label{Gr(1.63)}
{\bf v} = v_r \hat{\bf r} + v_{\theta} \hat{\bf \theta} + v_{\varphi} \hat{\boldsymbol \varphi}
\]
where the explicit relation between spherical and
Cartesian unit vectors is
</p>
<div class="eqlabel" id="orgb78b624">
<p>
In terms of Cartesian unit vectors:
<a id="sph_uv"></a><a href="./c_m_cs_sph.html#sph_uv"><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="org44deec6">
</div>
</div>
\begin{align}
\hat{\boldsymbol r} &amp;= \sin \theta \cos \phi \hat{\bf x} + \sin \theta \sin \phi \hat{\bf y} + \cos \theta \hat{\bf z}, \nonumber \\
\hat{\boldsymbol \theta} &amp;= \cos \theta \cos \phi \hat{\bf x} + \cos \theta \sin \phi \hat{\bf y} - \sin \theta \hat{\bf z}, \nonumber \\
\hat{\boldsymbol \phi} &amp;= -\sin \phi \hat{\bf x} + \cos \phi \hat{\bf y}.
\label{Gr(1.64)}
\hat{\boldsymbol r} &amp;= \sin \theta \cos \varphi ~\hat{\bf x} + \sin \theta \sin \varphi ~\hat{\bf y} + \cos \theta ~\hat{\bf z}, \nonumber \\
\hat{\boldsymbol \theta} &amp;= \cos \theta \cos \varphi ~\hat{\bf x} + \cos \theta \sin \varphi ~\hat{\bf y} - \sin \theta ~\hat{\bf z}, \nonumber \\
\hat{\boldsymbol \varphi} &amp;= -\sin \varphi ~\hat{\bf x} + \cos \varphi ~\hat{\bf y}.
\tag{sph_uv}\label{sph_uv}
\end{align}
<p>
Careful: these unit vectors are direction dependent, <i>i.e.</i> we should really
write \(\hat{\boldsymbol r} (\theta, \phi), \hat{\boldsymbol \theta} (\theta, \phi), \hat{\boldsymbol \phi} (\theta, \phi)\).
Do be careful: these unit vectors are direction dependent, <i>i.e.</i> we should really
write \(\hat{\boldsymbol r} (\theta, \varphi)\),
\(\hat{\boldsymbol \theta} (\theta, \varphi)\)
and \(\hat{\boldsymbol \varphi} (\theta, \varphi)\).
</p>
<p>
Infinitesimal displacement \(d{\bf l}\):
An infinitesimal displacement \(d{\bf l}\) can be written as
</p>
<div class="eqlabel" id="org11b03bf">
<p>
<a id="sph_dl"></a><a href="./c_m_cs_sph.html#sph_dl"><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="org814a5c0">
</div>
</div>
<p>
\[
d{\bf l} = dr \hat{\boldsymbol r} + r d\theta \hat{\boldsymbol \theta} + r\sin \theta d\phi \hat{\boldsymbol \phi}.
\label{Gr(1.68)}
d{\bf l} = dr ~\hat{\boldsymbol r} + r d\theta ~\hat{\boldsymbol \theta} + r\sin \theta d\varphi ~\hat{\boldsymbol \varphi}.
\tag{sph_dl}\label{sph_dl}
\]
</p>
<p>
Infinitesimal volume element:
</p>
<div class="eqlabel" id="orgf5fa0db">
<p>
<a id="sph_dtau"></a><a href="./c_m_cs_sph.html#sph_dtau"><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="orgf5f8360">
</div>
</div>
<p>
\[
d\tau = dl_r dl_{\theta} dl_{\phi} = r^2 \sin \theta dr d\theta d\phi
\label{Gr(1.69)}
d\tau = dl_r dl_{\theta} dl_{\varphi} = r^2 \sin \theta dr d\theta d\varphi
\tag{sph_dtau}\label{sph_dtau}
\]
</p>
@@ -1667,10 +1720,23 @@ 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">
<div class="eqlabel" id="orgb804960">
<p>
<a id="sph_grad"></a><a href="./c_m_cs_sph.html#sph_grad"><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="org3f7449e">
</div>
</div>
\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)}
+ \frac{1}{r\sin \theta} \frac{\partial T}{\partial \varphi} \hat{\boldsymbol \varphi}.
\tag{sph_grad}\label{sph_grad}
\end{equation}
</div>
</div>
@@ -1678,10 +1744,23 @@ Infinitesimal surface element: depends on situation.
<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">
<div class="eqlabel" id="org13ae106">
<p>
<a id="sph_div"></a><a href="./c_m_cs_sph.html#sph_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="org8664749">
</div>
</div>
\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)}
+ \frac{1}{r \sin \theta} \frac{\partial v_{\varphi}}{\partial \varphi}
\tag{sph_div}\label{sph_div}
\end{equation}
</div>
</div>
@@ -1689,23 +1768,50 @@ Infinitesimal surface element: depends on situation.
<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">
\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)}
\end{equation}
<div class="eqlabel" id="orge67612f">
<p>
<a id="sph_curl"></a><a href="./c_m_cs_sph.html#sph_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="org1716ad0">
</div>
</div>
\begin{align}
{\boldsymbol \nabla} \times {\bf v} = &amp;\frac{1}{r\sin \theta} \left[ \frac{\partial}{\partial \theta} (\sin \theta v_{\varphi}) - \frac{\partial v_{\theta}}{\partial \varphi} \right] \hat{\bf r}
\nonumber \\
&amp;+ \frac{1}{r} \left[ \frac{1}{\sin \theta} \frac{\partial v_r}{\partial \varphi} - \frac{\partial}{\partial r} (r v_{\varphi}) \right] \hat{\boldsymbol \theta}
+ \frac{1}{r} \left[ \frac{\partial}{\partial r} (r v_{\theta}) - \frac{\partial v_r}{\partial \theta} \right] \hat{\boldsymbol \varphi}
\tag{sph_curl}\label{sph_curl}
\end{align}
</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">
<div class="eqlabel" id="org99dc986">
<p>
<a id="sph_Lap"></a><a href="./c_m_cs_sph.html#sph_Lap"><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="org75cb0ac">
</div>
</div>
\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)}
+ \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 T}{\partial \varphi^2}
\tag{sph_Lap}\label{sph_Lap}
\end{equation}
</div>
</div>
@@ -1729,7 +1835,7 @@ 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>