Update 2022-02-21 10:35
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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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<!-- 2022-02-17 Thu 08:42 -->
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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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@@ -602,11 +602,11 @@ Table of contents
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</summary>
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<ul>
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<li>
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<a href="./ems_ms_lf_pc.html#ems_ms_lf_pc">Point Charge</a><span class="headline-id">ems.ms.lf.pc</span>
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<a href="./ems_ms_lf_pc.html#ems_ms_lf_pc">Point Charges</a><span class="headline-id">ems.ms.lf.pc</span>
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</li>
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<li>
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<a href="./ems_ms_lf_c.html#ems_ms_lf_c">Currents</a><span class="headline-id">ems.ms.lf.c</span>
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<a href="./ems_ms_lf_sc.html#ems_ms_lf_sc">Steady Currents</a><span class="headline-id">ems.ms.lf.sc</span>
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</li>
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@@ -614,21 +614,12 @@ Table of contents
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</details>
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</li>
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<li>
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<a href="./ems_ms_ce.html#ems_ms_ce">Charge Conservation and the Continuity Equation</a><span class="headline-id">ems.ms.ce</span>
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<details>
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<summary>
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</li>
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<li>
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<a href="./ems_ms_BS.html#ems_ms_BS">Steady Currents: the Biot-Savart Law</a><span class="headline-id">ems.ms.BS</span>
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</summary>
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<ul>
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<li>
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<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>
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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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@@ -640,11 +631,15 @@ Table of contents
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</summary>
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<ul>
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<li>
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<a href="./ems_ms_dcB_sc.html#ems_ms_dcB_sc">Straight-line Currents</a><span class="headline-id">ems.ms.dcB.sc</span>
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<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>
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</li>
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<li>
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<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>
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<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>
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</li>
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<li>
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<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>
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</li>
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@@ -661,6 +656,10 @@ Table of contents
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</summary>
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<ul>
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<li>
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<a href="./ems_ms_vp_A.html#ems_ms_vp_A">Definition; Gauge Choices</a><span class="headline-id">ems.ms.vp.A</span>
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</li>
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<li>
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<a href="./ems_ms_vp_mbc.html#ems_ms_vp_mbc">Magnetic Boundary Conditions</a><span class="headline-id">ems.ms.vp.mbc</span>
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</li>
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@@ -698,10 +697,6 @@ 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_s.html#emsm_esm_s">A proper definition of "statics"</a><span class="headline-id">emsm.esm.s</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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@@ -1435,7 +1430,7 @@ Table of contents
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</li>
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<li>
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<a href="./c_m_dc_pr.html#c_m_dc_pr">Product Rules</a><span class="headline-id">c.m.dc.pr</span>
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<a href="./c_m_dc_pr.html#c_m_dc_pr">Product arguments</a><span class="headline-id">c.m.dc.pr</span>
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</li>
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<li class="toc-currentpage">
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@@ -1591,7 +1586,7 @@ Table of contents
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</ul>
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</details>
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</nav>
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<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: <a href="c_m_dc_pr.html">Product Rules <small>[c.m.dc.pr]</small></a></li><li>Next: <a href="c_m_ic.html">Integral Calculus <small>[c.m.ic]</small></a></li><li>Up: <a href="c_m_dc.html">Differential Calculus <small>[c.m.dc]</small></a></li></ul><div id="outline-container-c_m_dc_d2" class="outline-5">
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<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: <a href="c_m_dc_pr.html">Product arguments <small>[c.m.dc.pr]</small></a></li><li>Next: <a href="c_m_ic.html">Integral Calculus <small>[c.m.ic]</small></a></li><li>Up: <a href="c_m_dc.html">Differential Calculus <small>[c.m.dc]</small></a></li></ul><div id="outline-container-c_m_dc_d2" class="outline-5">
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<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">
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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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@@ -1599,9 +1594,9 @@ 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-org48d9aa2" class="outline-6">
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<h6 id="org48d9aa2"><a href="#org48d9aa2">Divergence of gradient</a></h6>
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<div class="outline-text-6" id="text-org48d9aa2">
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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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<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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@@ -1610,36 +1605,36 @@ 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-org527fbc8" class="outline-6">
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<h6 id="org527fbc8"><a href="#org527fbc8">Curl of a gradient</a></h6>
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<div class="outline-text-6" id="text-org527fbc8">
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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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<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-org9758e01" class="outline-6">
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<h6 id="org9758e01"><a href="#org9758e01">Gradient of the divergence</a></h6>
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<div class="outline-text-6" id="text-org9758e01">
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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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<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-org5e919b4" class="outline-6">
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<h6 id="org5e919b4"><a href="#org5e919b4">Divergence of a curl</a></h6>
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<div class="outline-text-6" id="text-org5e919b4">
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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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<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-org0eb22e6" class="outline-6">
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<h6 id="org0eb22e6"><a href="#org0eb22e6">Curl of curl</a></h6>
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<div class="outline-text-6" id="text-org0eb22e6">
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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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<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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@@ -1651,7 +1646,7 @@ This always vanishes.
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</div>
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<br><ul class="navigation-links"><li>Prev: <a href="c_m_dc_pr.html">Product Rules <small>[c.m.dc.pr]</small></a></li><li>Next: <a href="c_m_ic.html">Integral Calculus <small>[c.m.ic]</small></a></li><li>Up: <a href="c_m_dc.html">Differential Calculus <small>[c.m.dc]</small></a></li></ul>
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<br><ul class="navigation-links"><li>Prev: <a href="c_m_dc_pr.html">Product arguments <small>[c.m.dc.pr]</small></a></li><li>Next: <a href="c_m_ic.html">Integral Calculus <small>[c.m.ic]</small></a></li><li>Up: <a href="c_m_dc.html">Differential Calculus <small>[c.m.dc]</small></a></li></ul>
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<br>
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<hr>
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<div class="license">
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@@ -1666,7 +1661,7 @@ 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: 2022-02-17 Thu 08:42</p>
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<p class="date">Created: 2022-02-21 Mon 10:33</p>
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<p class="validation"></p>
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</div>
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