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Jean-Sébastien
2022-02-21 10:35:02 +01:00
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-17 Thu 08:42 -->
<!-- 2022-02-21 Mon 10:33 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -602,11 +602,11 @@ Table of contents
</summary>
<ul>
<li>
<a href="./ems_ms_lf_pc.html#ems_ms_lf_pc">Point Charge</a><span class="headline-id">ems.ms.lf.pc</span>
<a href="./ems_ms_lf_pc.html#ems_ms_lf_pc">Point Charges</a><span class="headline-id">ems.ms.lf.pc</span>
</li>
<li>
<a href="./ems_ms_lf_c.html#ems_ms_lf_c">Currents</a><span class="headline-id">ems.ms.lf.c</span>
<a href="./ems_ms_lf_sc.html#ems_ms_lf_sc">Steady Currents</a><span class="headline-id">ems.ms.lf.sc</span>
</li>
@@ -614,21 +614,12 @@ Table of contents
</details>
</li>
<li>
<a href="./ems_ms_ce.html#ems_ms_ce">Charge Conservation and the Continuity Equation</a><span class="headline-id">ems.ms.ce</span>
<details>
<summary>
</li>
<li>
<a href="./ems_ms_BS.html#ems_ms_BS">Steady Currents: the Biot-Savart Law</a><span class="headline-id">ems.ms.BS</span>
</summary>
<ul>
<li>
<a href="./ems_ms_BS_sc.html#ems_ms_BS_sc">The Magnetic Field issuing from a Steady Current</a><span class="headline-id">ems.ms.BS.sc</span>
</li>
</ul>
</details>
</li>
<li>
@@ -640,11 +631,15 @@ Table of contents
</summary>
<ul>
<li>
<a href="./ems_ms_dcB_sc.html#ems_ms_dcB_sc">Straight-line Currents</a><span class="headline-id">ems.ms.dcB.sc</span>
<a href="./ems_ms_dcB_iw.html#ems_ms_dcB_iw">Simplistic case: infinite wire</a><span class="headline-id">ems.ms.dcB.iw</span>
</li>
<li>
<a href="./ems_ms_dcB_BS.html#ems_ms_dcB_BS">Divergence and Curl of \({\bf B}\) from Biot-Savart</a><span class="headline-id">ems.ms.dcB.BS</span>
<a href="./ems_ms_dcB_d.html#ems_ms_dcB_d">Divergence of \({\bf B}\) from Biot-Savart</a><span class="headline-id">ems.ms.dcB.d</span>
</li>
<li>
<a href="./ems_ms_dcB_c.html#ems_ms_dcB_c">Curl of \({\bf B}\) from Biot-Savart; Ampère's Law</a><span class="headline-id">ems.ms.dcB.c</span>
</li>
@@ -661,6 +656,10 @@ Table of contents
</summary>
<ul>
<li>
<a href="./ems_ms_vp_A.html#ems_ms_vp_A">Definition; Gauge Choices</a><span class="headline-id">ems.ms.vp.A</span>
</li>
<li>
<a href="./ems_ms_vp_mbc.html#ems_ms_vp_mbc">Magnetic Boundary Conditions</a><span class="headline-id">ems.ms.vp.mbc</span>
</li>
@@ -698,10 +697,6 @@ Table of contents
</summary>
<ul>
<li>
<a href="./emsm_esm_s.html#emsm_esm_s">A proper definition of "statics"</a><span class="headline-id">emsm.esm.s</span>
</li>
<li>
<details>
<summary>
@@ -1435,7 +1430,7 @@ Table of contents
</li>
<li class="toc-currentpage">
<a href="./c_m_dc_pr.html#c_m_dc_pr">Product Rules</a><span class="headline-id">c.m.dc.pr</span>
<a href="./c_m_dc_pr.html#c_m_dc_pr">Product arguments</a><span class="headline-id">c.m.dc.pr</span>
</li>
<li>
@@ -1591,14 +1586,177 @@ Table of contents
</ul>
</details>
</nav>
<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_dc.html">Differential Calculus</a></li><li>Product Rules</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_curl.html">The Curl&emsp;<small>[c.m.dc.curl]</small></a></li><li>Next:&nbsp;<a href="c_m_dc_d2.html">Second Derivatives&emsp;<small>[c.m.dc.d2]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul><div id="outline-container-c_m_dc_pr" class="outline-5">
<h5 id="c_m_dc_pr">Product Rules<a class="headline-permalink" href="./c_m_dc_pr.html#c_m_dc_pr"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_dc.html">Differential Calculus</a></li><li>Product arguments</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_curl.html">The Curl&emsp;<small>[c.m.dc.curl]</small></a></li><li>Next:&nbsp;<a href="c_m_dc_d2.html">Second Derivatives&emsp;<small>[c.m.dc.d2]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul><div id="outline-container-c_m_dc_pr" class="outline-5">
<h5 id="c_m_dc_pr">Product arguments<a class="headline-permalink" href="./c_m_dc_pr.html#c_m_dc_pr"><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><span class="headline-id">c.m.dc.pr</span></h5>
<div class="outline-text-5" id="text-c_m_dc_pr">
<p>
Six product rules (on inside front cover of Gr).
When applied to arguments which are products of scalar/vector fields,
the action of \(\nabla\), \(\nabla \cdot\) and \(\nabla \times\) can be
explicited as follows:
</p>
<p>
<b>Gradient of a product</b>:
</p>
<div class="eqlabel" id="org77098c1">
<p>
<a id="grad_prod"></a><a href="./c_m_dc_pr.html#grad_prod"><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="org098750f">
<ul class="org-ul">
<li>Gr (3)</li>
<li>W (1-111)</li>
</ul>
</div>
</div>
<p>
\[
\nabla (uv) = u \nabla v + v \nabla u
\tag{grad_prod}\label{grad_prod}
\]
</p>
<br>
<p>
<b>Gradient of a scalar product</b>:
</p>
<div class="eqlabel" id="org623c0b6">
<p>
<a id="grad_sprod"></a><a href="./c_m_dc_pr.html#grad_sprod"><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="org9c465bf">
<ul class="org-ul">
<li>Gr (4)</li>
<li>W (1-112)</li>
</ul>
</div>
</div>
<p>
\[
\nabla ({\bf a} \cdot {\bf b}) = ({\bf a} \cdot \nabla) {\bf b} + ({\bf b} \cdot \nabla) {\bf a} + {\bf a} \times (\nabla \times {\bf b}) + {\bf b} \times (\nabla \times {\bf a})
\tag{grad_sprod}\label{grad_sprod}
\]
</p>
<br>
<p>
<b>Divergence of a product</b>:
</p>
<div class="eqlabel" id="org27a4a20">
<p>
<a id="div_prod"></a><a href="./c_m_dc_pr.html#div_prod"><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="org3259c68">
<ul class="org-ul">
<li>Gr (5)</li>
<li>W (1-115)</li>
</ul>
</div>
</div>
<p>
\[
\nabla (\psi {\bf a}) = {\bf a} \cdot \nabla \psi + \psi \nabla \cdot {\bf a}
\tag{div_prod}\label{div_prod}
\]
</p>
<br>
<p>
<b>Divergence of a cross product</b>:
</p>
<div class="eqlabel" id="orgaa0862c">
<p>
<a id="div_xprod"></a><a href="./c_m_dc_pr.html#div_xprod"><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="org1672361">
<ul class="org-ul">
<li>Gr (6)</li>
<li>W (1-116)</li>
</ul>
</div>
</div>
<p>
\[
\nabla \cdot ({\bf a} \times {\bf b}) = {\bf b} \cdot (\nabla \times {\bf a}) - {\bf a} \cdot (\nabla \times {\bf b})
\tag{div_xprod}\label{div_xprod}
\]
</p>
<br>
<p>
<b>Curl of a product</b>:
</p>
<div class="eqlabel" id="org8a2530a">
<p>
<a id="curl_prod"></a><a href="./c_m_dc_pr.html#curl_prod"><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="orgdb6a233">
<ul class="org-ul">
<li>Gr (7)</li>
<li>W (1-118)</li>
</ul>
</div>
</div>
<p>
\[
\nabla \times (\psi {\bf a}) = \nabla \psi \times {\bf a} + \psi \nabla \times {\bf a}
\tag{curl_prod}\label{curl_prod}
\]
</p>
<br>
<p>
<b>Curl of a cross product</b>:
</p>
<div class="eqlabel" id="org21daf74">
<p>
<a id="curl_xprod"></a><a href="./c_m_dc_pr.html#curl_xprod"><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="org87e2222">
<ul class="org-ul">
<li>Gr (8)</li>
<li>W (1-119)</li>
</ul>
</div>
</div>
<p>
\[
\nabla ({\bf a} \times {\bf b}) = {\bf a} (\nabla \cdot {\bf b}) - {\bf b} (\nabla \cdot {\bf a}) + ({\bf b} \cdot \nabla) {\bf a} - ({\bf a} \cdot \nabla) {\bf b}
\tag{curl_xprod}\label{curl_xprod}
\]
</p>
</div>
</div>
@@ -1606,6 +1764,11 @@ Six product rules (on inside front cover of Gr).
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_curl.html">The Curl&emsp;<small>[c.m.dc.curl]</small></a></li><li>Next:&nbsp;<a href="c_m_dc_d2.html">Second Derivatives&emsp;<small>[c.m.dc.d2]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul>
<br>
<hr>
@@ -1621,7 +1784,7 @@ 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: 2022-02-17 Thu 08:42</p>
<p class="date">Created: 2022-02-21 Mon 10:33</p>
<p class="validation"></p>
</div>