@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: a skos:ConceptScheme .
psr:-XJ77SGF5-S
  skos:prefLabel "Laplace's equation"@en, "équation de Laplace"@fr ;
  a skos:Concept ;
  skos:related psr:-DPZFVZF1-V .

psr:-DPZFVZF1-V
  skos:prefLabel "cylindrical harmonic"@en, "harmonique cylindrique"@fr ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Cylindrical_harmonics>, <https://fr.wikipedia.org/wiki/Harmonique_cylindrique> ;
  dc:modified "2023-08-16"^^xsd:date ;
  skos:inScheme psr: ;
  skos:broader psr:-FH1H1FB9-1 ;
  skos:definition """En mathématiques, les <b>harmoniques cylindriques</b> sont un ensemble de solutions linéairement indépendantes de l'équation différentielle de Laplace
<br/>
<br/><dl><dd><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 ^{2}V={\\rac {1}{\\
ho }}{\\rac {\\\\partial }{\\\\partial \\
ho }}\\\\left(\\
ho {\\rac {\\\\partial V}{\\\\partial \\
ho }}\\
ight)+{\\rac {1}{\\
ho ^{2}}}{\\rac {\\\\partial ^{2}V}{\\\\partial \\\\varphi ^{2}}}+{\\rac {\\\\partial ^{2}V}{\\\\partial z^{2}}}=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/>        <mi>V</mi>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mn>1</mn>
<br/>            <mi>ρ<!-- ρ --></mi>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mi mathvariant="normal">∂<!-- ∂ --></mi>
<br/>            <mrow>
<br/>              <mi mathvariant="normal">∂<!-- ∂ --></mi>
<br/>              <mi>ρ<!-- ρ --></mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mrow>
<br/>          <mo>(</mo>
<br/>          <mrow>
<br/>            <mi>ρ<!-- ρ --></mi>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mfrac>
<br/>                <mrow>
<br/>                  <mi mathvariant="normal">∂<!-- ∂ --></mi>
<br/>                  <mi>V</mi>
<br/>                </mrow>
<br/>                <mrow>
<br/>                  <mi mathvariant="normal">∂<!-- ∂ --></mi>
<br/>                  <mi>ρ<!-- ρ --></mi>
<br/>                </mrow>
<br/>              </mfrac>
<br/>            </mrow>
<br/>          </mrow>
<br/>          <mo>)</mo>
<br/>        </mrow>
<br/>        <mo>+</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mn>1</mn>
<br/>            <msup>
<br/>              <mi>ρ<!-- ρ --></mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mn>2</mn>
<br/>              </mrow>
<br/>            </msup>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <msup>
<br/>                <mi mathvariant="normal">∂<!-- ∂ --></mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mi>V</mi>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi mathvariant="normal">∂<!-- ∂ --></mi>
<br/>              <msup>
<br/>                <mi>φ<!-- φ --></mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>+</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <msup>
<br/>                <mi mathvariant="normal">∂<!-- ∂ --></mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mi>V</mi>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi mathvariant="normal">∂<!-- ∂ --></mi>
<br/>              <msup>
<br/>                <mi>z</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\
abla ^{2}V={\\rac {1}{\\
ho }}{\\rac {\\\\partial }{\\\\partial \\
ho }}\\\\left(\\
ho {\\rac {\\\\partial V}{\\\\partial \\
ho }}\\
ight)+{\\rac {1}{\\
ho ^{2}}}{\\rac {\\\\partial ^{2}V}{\\\\partial \\\\varphi ^{2}}}+{\\rac {\\\\partial ^{2}V}{\\\\partial z^{2}}}=0}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7e1f23c2c2526375f3bee5945fcfedad3dce096" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.297ex; height:6.343ex;" alt="{\\\\displaystyle \\
abla ^{2}V={\\rac {1}{\\
ho }}{\\rac {\\\\partial }{\\\\partial \\
ho }}\\\\left(\\
ho {\\rac {\\\\partial V}{\\\\partial \\
ho }}\\
ight)+{\\rac {1}{\\
ho ^{2}}}{\\rac {\\\\partial ^{2}V}{\\\\partial \\\\varphi ^{2}}}+{\\rac {\\\\partial ^{2}V}{\\\\partial z^{2}}}=0}"></span></dd></dl>
<br/>exprimées en coordonnées cylindriques <span class="texhtml mvar" style="font-style:italic;">ρ</span> (rayon), <span class="texhtml mvar" style="font-style:italic;">φ</span> (azimut) et <span class="texhtml mvar" style="font-style:italic;">z</span> (cote). Chaque fonction <span class="texhtml"><i>V<sub>n</sub></i>(<i>k</i>)</span> est le produit de trois termes, chacun ne dépendant que d'une coordonnée. Le terme dépendant de <span class="texhtml mvar" style="font-style:italic;">ρ</span> s'exprime avec les fonctions de Bessel (qui sont parfois également appelées harmoniques cylindriques). 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Harmonique_cylindrique">https://fr.wikipedia.org/wiki/Harmonique_cylindrique</a>)"""@fr, """In mathematics, the <b>cylindrical harmonics</b> are a set of linearly independent functions that are solutions to Laplace's differential equation, <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 ^{2}V=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/>        <mi>V</mi>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\
abla ^{2}V=0}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ff020357f18502f7c5337e57eb075267fdf4350" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:9.038ex; height:2.676ex;" alt="\\
abla ^{2}V=0"></span>, expressed in cylindrical coordinates, <i>ρ</i> (radial coordinate), <i>φ</i> (polar angle), and <i>z</i> (height). Each function <i>V</i><sub><i>n</i></sub>(<i>k</i>) is the product of three terms, each depending on one coordinate alone. The <i>ρ</i>-dependent term is given by Bessel functions (which occasionally are also called cylindrical harmonics). 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Cylindrical_harmonics">https://en.wikipedia.org/wiki/Cylindrical_harmonics</a>)"""@en ;
  skos:related psr:-XJ77SGF5-S ;
  dc:created "2023-08-16"^^xsd:date ;
  a skos:Concept .

psr:-FH1H1FB9-1
  skos:prefLabel "special function"@en, "fonction spéciale"@fr ;
  a skos:Concept ;
  skos:narrower psr:-DPZFVZF1-V .