Update 2022-02-21 20:42
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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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<!-- 2022-02-21 Mon 10:33 -->
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<!-- 2022-02-21 Mon 20:41 -->
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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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@@ -706,28 +706,41 @@ Table of contents
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</summary>
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<ul>
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<li>
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<a href="./emsm_esm_p.html#emsm_esm_p">Polarization</a><span class="headline-id">emsm.esm.p</span>
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</li>
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<li>
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<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
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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="./emsm_esm_fpo.html#emsm_esm_fpo">The Field of a Polarized Object</a><span class="headline-id">emsm.esm.fpo</span>
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<a href="./emsm_esm_mE.html#emsm_esm_mE">Matter Bathed in E Fields; Polarization</a><span class="headline-id">emsm.esm.mE</span>
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</summary>
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<ul>
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<li>
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<a href="./emsm_esm_fpo_pibc.html#emsm_esm_fpo_pibc">Physical Interpretation of Bound Charges</a><span class="headline-id">emsm.esm.fpo.pibc</span>
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<a href="./emsm_esm_mE_o.html#emsm_esm_mE_o">Overview</a><span class="headline-id">emsm.esm.mE.o</span>
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</li>
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<li>
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<a href="./emsm_esm_fpo_fid.html#emsm_esm_fpo_fid">The Field Inside a Dielectric</a><span class="headline-id">emsm.esm.fpo.fid</span>
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<a href="./emsm_esm_mE_P.html#emsm_esm_mE_P">Polarization</a><span class="headline-id">emsm.esm.mE.P</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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<details>
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<summary>
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<a href="./emsm_esm_po.html#emsm_esm_po">Polarized Objects; Bound Charges</a><span class="headline-id">emsm.esm.po</span>
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</summary>
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<ul>
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<li>
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<a href="./emsm_esm_po_pibc.html#emsm_esm_po_pibc">Physical Interpretation of Bound Charges</a><span class="headline-id">emsm.esm.po.pibc</span>
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</li>
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<li>
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<a href="./emsm_esm_po_fid.html#emsm_esm_po_fid">The Field Inside a Dielectric</a><span class="headline-id">emsm.esm.po.fid</span>
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</li>
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@@ -750,18 +763,34 @@ Table of contents
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</ul>
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</details>
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</li>
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<li>
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<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
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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="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
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<a href="./emsm_esm_ld.html#emsm_esm_ld">Linear Dielectrics</a><span class="headline-id">emsm.esm.ld</span>
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</summary>
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<ul>
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<li>
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<a href="./emsm_esm_di_ld.html#emsm_esm_di_ld">Linear Dielectrics</a><span class="headline-id">emsm.esm.di.ld</span>
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<a href="./emsm_esm_ld_sp.html#emsm_esm_ld_sp">Susceptibility, Permittivity, Dielectric Constant</a><span class="headline-id">emsm.esm.ld.sp</span>
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</li>
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<li>
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<a href="./emsm_esm_ld_bvp.html#emsm_esm_ld_bvp">Boundary Value Problems with Linear Dielectrics</a><span class="headline-id">emsm.esm.ld.bvp</span>
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</li>
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<li>
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<a href="./emsm_esm_ld_e.html#emsm_esm_ld_e">Energy in Dielectric Systems</a><span class="headline-id">emsm.esm.ld.e</span>
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</li>
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<li>
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<a href="./emsm_esm_ld_f.html#emsm_esm_ld_f">Forces on Dielectrics</a><span class="headline-id">emsm.esm.ld.f</span>
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</li>
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@@ -1594,9 +1623,9 @@ 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-org1a142a0" class="outline-6">
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<h6 id="org1a142a0"><a href="#org1a142a0">Divergence of gradient</a></h6>
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<div class="outline-text-6" id="text-org1a142a0">
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<div id="outline-container-org2f2fb63" class="outline-6">
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<h6 id="org2f2fb63"><a href="#org2f2fb63">Divergence of gradient</a></h6>
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<div class="outline-text-6" id="text-org2f2fb63">
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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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@@ -1605,40 +1634,52 @@ 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-org16039ac" class="outline-6">
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<h6 id="org16039ac"><a href="#org16039ac">Curl of a gradient</a></h6>
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<div class="outline-text-6" id="text-org16039ac">
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<div id="outline-container-org7940ce1" class="outline-6">
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<h6 id="org7940ce1"><a href="#org7940ce1">Curl of a gradient</a></h6>
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<div class="outline-text-6" id="text-org7940ce1">
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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-orgd64b31f" class="outline-6">
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<h6 id="orgd64b31f"><a href="#orgd64b31f">Gradient of the divergence</a></h6>
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<div class="outline-text-6" id="text-orgd64b31f">
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<div id="outline-container-org8fc30e5" class="outline-6">
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<h6 id="org8fc30e5"><a href="#org8fc30e5">Gradient of the divergence</a></h6>
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<div class="outline-text-6" id="text-org8fc30e5">
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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-orgf932975" class="outline-6">
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<h6 id="orgf932975"><a href="#orgf932975">Divergence of a curl</a></h6>
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<div class="outline-text-6" id="text-orgf932975">
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<div id="outline-container-org5323eef" class="outline-6">
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<h6 id="org5323eef"><a href="#org5323eef">Divergence of a curl</a></h6>
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<div class="outline-text-6" id="text-org5323eef">
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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-orgcb842e5" class="outline-6">
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<h6 id="orgcb842e5"><a href="#orgcb842e5">Curl of curl</a></h6>
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<div class="outline-text-6" id="text-orgcb842e5">
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<div id="outline-container-org0c116f8" class="outline-6">
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<h6 id="org0c116f8"><a href="#org0c116f8">Curl of curl</a></h6>
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<div class="outline-text-6" id="text-org0c116f8">
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<div class="eqlabel" id="org51a12dc">
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<p>
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<a id="curlcurl"></a><a href="./c_m_dc_d2.html#curlcurl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<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"/>
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<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"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="orge1da3d2">
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</div>
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</div>
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<p>
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\[
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{\boldsymbol \nabla} \times ({\boldsymbol \nabla} \times {\bf v}) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v}) - {\boldsymbol \nabla}^2 {\bf v}
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\label{Gr(1.47)}
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\tag{curlcurl}\label{curlcurl}
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\]
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</p>
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</div>
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@@ -1661,7 +1702,7 @@ 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: 2022-02-21 Mon 10:33</p>
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<p class="date">Created: 2022-02-21 Mon 20:41</p>
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<p class="validation"></p>
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</div>
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