Update 2022-02-09 22:41
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Jean-Sébastien
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<!-- 2022-02-09 Wed 07:31 -->
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<!-- 2022-02-09 Wed 22:40 -->
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<meta charset="utf-8">
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<meta name="viewport" content="width=device-width, initial-scale=1">
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<title>Pre-Quantum Electrodynamics</title>
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@@ -408,17 +408,13 @@ Table of contents
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<li>
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<a href="./ems_es_ep_fp.html#ems_es_ep_fp">Field in terms of the potential</a><span class="headline-id">ems.es.ep.fp</span>
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</li>
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<li>
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<a href="./ems_es_ep_c.html#ems_es_ep_c">Comments on the Electrostatic Potential</a><span class="headline-id">ems.es.ep.c</span>
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</li>
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<li>
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<a href="./ems_es_ep_ex.html#ems_es_ep_ex">Example calculations for the potential</a><span class="headline-id">ems.es.ep.ex</span>
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</li>
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<li>
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<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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<a href="./ems_es_ep_PL.html#ems_es_ep_PL">Poisson's and Laplace's Equations</a><span class="headline-id">ems.es.ep.PL</span>
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</li>
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<li>
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@@ -430,29 +426,8 @@ Table of contents
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</details>
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</li>
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<li>
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<details>
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<summary>
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<a href="./ems_es_e.html#ems_es_e">Electrostatic Energy from the Potential</a><span class="headline-id">ems.es.e</span>
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</summary>
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<ul>
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<li>
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<a href="./ems_es_e_pcd.html#ems_es_e_pcd">The Energy of a Point Charge Distribution</a><span class="headline-id">ems.es.e.pcd</span>
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</li>
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<li>
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<a href="./ems_es_e_ccd.html#ems_es_e_ccd">The Energy of a Continuous Charge Distribution</a><span class="headline-id">ems.es.e.ccd</span>
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</li>
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<li>
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<a href="./ems_es_e_c.html#ems_es_e_c">Comments on Electrostatic Energy</a><span class="headline-id">ems.es.e.c</span>
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</li>
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</ul>
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</details>
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</li>
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<li>
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@@ -1624,9 +1599,9 @@ Table of contents
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<div class="outline-text-5" id="text-c_m_dc_d2">
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</div>
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<div id="outline-container-orga13f3ad" class="outline-6">
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<h6 id="orga13f3ad"><a href="#orga13f3ad">Divergence of gradient</a></h6>
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<div class="outline-text-6" id="text-orga13f3ad">
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<div id="outline-container-orgddf1a80" class="outline-6">
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<h6 id="orgddf1a80"><a href="#orgddf1a80">Divergence of gradient</a></h6>
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<div class="outline-text-6" id="text-orgddf1a80">
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<p>
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\({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the <b>Laplacian</b> of the scalar field \(T\).
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The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components
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@@ -1635,36 +1610,36 @@ given by the Laplacian of the corresponding vector elements.
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</div>
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</div>
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<div id="outline-container-orgf350618" class="outline-6">
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<h6 id="orgf350618"><a href="#orgf350618">Curl of a gradient</a></h6>
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<div class="outline-text-6" id="text-orgf350618">
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<div id="outline-container-org1c6301d" class="outline-6">
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<h6 id="org1c6301d"><a href="#org1c6301d">Curl of a gradient</a></h6>
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<div class="outline-text-6" id="text-org1c6301d">
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<p>
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This always vanishes.
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</p>
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</div>
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</div>
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<div id="outline-container-orgb23671d" class="outline-6">
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<h6 id="orgb23671d"><a href="#orgb23671d">Gradient of the divergence</a></h6>
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<div class="outline-text-6" id="text-orgb23671d">
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<div id="outline-container-orgce82ec3" class="outline-6">
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<h6 id="orgce82ec3"><a href="#orgce82ec3">Gradient of the divergence</a></h6>
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<div class="outline-text-6" id="text-orgce82ec3">
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<p>
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\({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics. No special name.
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</p>
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</div>
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</div>
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<div id="outline-container-org0373292" class="outline-6">
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<h6 id="org0373292"><a href="#org0373292">Divergence of a curl</a></h6>
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<div class="outline-text-6" id="text-org0373292">
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<div id="outline-container-org820bb5e" class="outline-6">
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<h6 id="org820bb5e"><a href="#org820bb5e">Divergence of a curl</a></h6>
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<div class="outline-text-6" id="text-org820bb5e">
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<p>
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This always vanishes.
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</p>
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</div>
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</div>
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<div id="outline-container-org21d697e" class="outline-6">
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<h6 id="org21d697e"><a href="#org21d697e">Curl of curl</a></h6>
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<div class="outline-text-6" id="text-org21d697e">
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<div id="outline-container-org75d969a" class="outline-6">
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<h6 id="org75d969a"><a href="#org75d969a">Curl of curl</a></h6>
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<div class="outline-text-6" id="text-org75d969a">
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<p>
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\[
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{\boldsymbol \nabla} \times ({\boldsymbol \nabla} \times {\bf v}) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v}) - {\boldsymbol \nabla}^2 {\bf v}
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@@ -1689,7 +1664,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="date">Created: 2022-02-09 Wed 07:31</p>
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<p class="date">Created: 2022-02-09 Wed 22:40</p>
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<p class="validation"></p>
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</div>
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