@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-DPZFVZF1-V
  skos:prefLabel "harmonique cylindrique"@fr, "cylindrical harmonic"@en ;
  a skos:Concept ;
  skos:related psr:-XJ77SGF5-S .

psr:-F0LR2RP4-6
  skos:prefLabel "elliptic partial differential equation"@en, "équation aux dérivées partielles elliptique"@fr ;
  a skos:Concept ;
  skos:narrower psr:-XJ77SGF5-S .

psr:-XJ77SGF5-S
  skos:broader psr:-H7WG59FS-L, psr:-BQTC43FX-J, psr:-F0LR2RP4-6 ;
  skos:definition """In mathematics and physics, <b>Laplace's equation</b> is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
<br/><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\
abla ^{2}\\\\!f=0}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi mathvariant="normal">∇<!-- ∇ --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mspace width="negativethinmathspace"></mspace>
<br/>        <mi>f</mi>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\
abla ^{2}\\\\!f=0}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aec7da91b5ad8bf096df35e6d673975e0b64ce1e" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -0.671ex; width:8.143ex; height:3.009ex;" alt="{\\\\displaystyle \\
abla ^{2}\\\\!f=0}"></div> or <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\Delta f=0,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi mathvariant="normal">Δ<!-- Δ --></mi>
<br/>        <mi>f</mi>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\Delta f=0,}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a22747a487dee31758194b3d90f3242172acc25" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -0.671ex; width:8.122ex; height:2.509ex;" alt="{\\\\displaystyle \\\\Delta f=0,}"></div>
<br/>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\Delta =\\
abla \\\\cdot \\
abla =\\
abla ^{2}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi mathvariant="normal">Δ<!-- Δ --></mi>
<br/>        <mo>=</mo>
<br/>        <mi mathvariant="normal">∇<!-- ∇ --></mi>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>        <mi mathvariant="normal">∇<!-- ∇ --></mi>
<br/>        <mo>=</mo>
<br/>        <msup>
<br/>          <mi mathvariant="normal">∇<!-- ∇ --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\Delta =\\
abla \\\\cdot \\
abla =\\
abla ^{2}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bee487b31b9e1ef453359b842ef87322009b3402" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:16.674ex; height:2.676ex;" alt="{\\\\displaystyle \\\\Delta =\\
abla \\\\cdot \\
abla =\\
abla ^{2}}"></span> is the Laplace operator, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\
abla \\\\cdot }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi mathvariant="normal">∇<!-- ∇ --></mi>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\
abla \\\\cdot }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6df6024211b717870f07844116e116b2eb314d12" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.583ex; height:2.176ex;" alt="\\
abla \\\\cdot"></span> is the divergence operator (also symbolized "div"), <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\
abla }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi mathvariant="normal">∇<!-- ∇ --></mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\
abla }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="\\
abla "></span> is the gradient operator (also symbolized "grad"), and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f(x,y,z)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo>,</mo>
<br/>        <mi>y</mi>
<br/>        <mo>,</mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f(x,y,z)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5d48dce2c4341575269f1709237a2e18923237a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:8.729ex; height:2.843ex;" alt="{\\\\displaystyle f(x,y,z)}"></span> is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Laplace%27s_equation">https://en.wikipedia.org/wiki/Laplace%27s_equation</a>)"""@en, """En analyse vectorielle, l'équation de Laplace est une équation aux dérivées partielles elliptique du second ordre, dont le nom est un hommage au physicien mathématicien Pierre-Simon de Laplace. Introduite pour les besoins de la mécanique newtonienne, l'équation de Laplace apparaît dans de nombreuses autres branches de la physique théorique : astronomie, électrostatique, mécanique des fluides, propagation de la chaleur, diffusion, mouvement brownien, mécanique quantique. Les fonctions solutions de l'équation de Laplace sont appelées les fonctions harmoniques. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/%C3%89quation_de_Laplace">https://fr.wikipedia.org/wiki/%C3%89quation_de_Laplace</a>)"""@fr ;
  dc:modified "2023-08-16"^^xsd:date ;
  skos:related psr:-DPZFVZF1-V ;
  skos:prefLabel "Laplace's equation"@en, "équation de Laplace"@fr ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Laplace%27s_equation>, <https://fr.wikipedia.org/wiki/%C3%89quation_de_Laplace> ;
  skos:inScheme psr: ;
  a skos:Concept .

psr:-H7WG59FS-L
  skos:prefLabel "fonction harmonique"@fr, "harmonic function"@en ;
  a skos:Concept ;
  skos:narrower psr:-XJ77SGF5-S .

psr: a skos:ConceptScheme .
psr:-BQTC43FX-J
  skos:prefLabel "analyse vectorielle"@fr, "vector calculus"@en ;
  a skos:Concept ;
  skos:narrower psr:-XJ77SGF5-S .