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Update 2022-02-09 07:44

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Jean-Sébastien
2022-02-09 07:44:58 +01:00
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-08 Tue 17:21 -->
<!-- 2022-02-09 Wed 07:31 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1616,7 +1616,7 @@ Table of contents
</ul>
</details>
</nav>
<ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_pr.html">Product Rules&emsp;<small>[c.m.dc.pr]</small></a></li><li>Next:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</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_d2" class="outline-5">
<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>Second Derivatives</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_pr.html">Product Rules&emsp;<small>[c.m.dc.pr]</small></a></li><li>Next:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</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_d2" class="outline-5">
<h5 id="c_m_dc_d2">Second Derivatives<a class="headline-permalink" href="./c_m_dc_d2.html#c_m_dc_d2"><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"/>
@@ -1624,9 +1624,9 @@ Table of contents
<div class="outline-text-5" id="text-c_m_dc_d2">
</div>
<div id="outline-container-org5fd48d1" class="outline-6">
<h6 id="org5fd48d1"><a href="#org5fd48d1">Divergence of gradient</a></h6>
<div class="outline-text-6" id="text-org5fd48d1">
<div id="outline-container-orga13f3ad" class="outline-6">
<h6 id="orga13f3ad"><a href="#orga13f3ad">Divergence of gradient</a></h6>
<div class="outline-text-6" id="text-orga13f3ad">
<p>
\({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the <b>Laplacian</b> of the scalar field \(T\).
The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components
@@ -1635,36 +1635,36 @@ given by the Laplacian of the corresponding vector elements.
</div>
</div>
<div id="outline-container-org41a3448" class="outline-6">
<h6 id="org41a3448"><a href="#org41a3448">Curl of a gradient</a></h6>
<div class="outline-text-6" id="text-org41a3448">
<div id="outline-container-orgf350618" class="outline-6">
<h6 id="orgf350618"><a href="#orgf350618">Curl of a gradient</a></h6>
<div class="outline-text-6" id="text-orgf350618">
<p>
This always vanishes.
</p>
</div>
</div>
<div id="outline-container-org6c4b5a7" class="outline-6">
<h6 id="org6c4b5a7"><a href="#org6c4b5a7">Gradient of the divergence</a></h6>
<div class="outline-text-6" id="text-org6c4b5a7">
<div id="outline-container-orgb23671d" class="outline-6">
<h6 id="orgb23671d"><a href="#orgb23671d">Gradient of the divergence</a></h6>
<div class="outline-text-6" id="text-orgb23671d">
<p>
\({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics. No special name.
</p>
</div>
</div>
<div id="outline-container-org68b62ab" class="outline-6">
<h6 id="org68b62ab"><a href="#org68b62ab">Divergence of a curl</a></h6>
<div class="outline-text-6" id="text-org68b62ab">
<div id="outline-container-org0373292" class="outline-6">
<h6 id="org0373292"><a href="#org0373292">Divergence of a curl</a></h6>
<div class="outline-text-6" id="text-org0373292">
<p>
This always vanishes.
</p>
</div>
</div>
<div id="outline-container-org62690be" class="outline-6">
<h6 id="org62690be"><a href="#org62690be">Curl of curl</a></h6>
<div class="outline-text-6" id="text-org62690be">
<div id="outline-container-org21d697e" class="outline-6">
<h6 id="org21d697e"><a href="#org21d697e">Curl of curl</a></h6>
<div class="outline-text-6" id="text-org21d697e">
<p>
\[
{\boldsymbol \nabla} \times ({\boldsymbol \nabla} \times {\bf v}) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v}) - {\boldsymbol \nabla}^2 {\bf v}
@@ -1676,10 +1676,21 @@ This always vanishes.
</div>
<hr><div id="postamble" class="status">
<hr>
<div class="license">
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
target="_blank" class="m-2">
<img alt="Creative Commons License" style="border-width:0"
src="https://licensebuttons.net/l/by/4.0/80x15.png"/>
</a>
Except where otherwise noted, all content is licensed under a
<a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
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: 2022-02-08 Tue 17:21</p>
<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
<p class="date">Created: 2022-02-09 Wed 07:31</p>
<p class="validation"></p>
</div>