@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .

psr:-JQG60V6B-G
  skos:prefLabel "Kato's inequality"@en, "inégalité de Kato"@fr ;
  a skos:Concept ;
  skos:broader psr:-GHZJHV7P-F .

psr: a skos:ConceptScheme .
psr:-P6BGQTWK-G
  skos:prefLabel "vector calculus identities"@en, "identités vectorielles"@fr ;
  a skos:Concept ;
  skos:narrower psr:-GHZJHV7P-F .

psr:-GHZJHV7P-F
  skos:inScheme psr: ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Op%C3%A9rateur_laplacien>, <https://en.wikipedia.org/wiki/Laplace_operator> ;
  a skos:Concept ;
  skos:prefLabel "laplacien"@fr, "Laplacian"@en ;
  skos:narrower psr:-JQG60V6B-G ;
  skos:broader psr:-RRBN6FVB-9, psr:-P6BGQTWK-G ;
  skos:altLabel "Laplace operator"@en, "opérateur laplacien"@fr ;
  skos:definition """L'<b>opérateur laplacien</b>, ou simplement le <b>laplacien</b>, est l'opérateur différentiel défini par l'application de l'opérateur gradient suivie de l'application de l'opérateur divergence&nbsp;:
<br/>
<br/><center><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 \\\\phi ={\\\\vec {\\
abla }}^{2}\\\\phi ={\\\\vec {\\
abla }}\\\\cdot ({\\\\vec {\\
abla }}\\\\phi )=\\\\operatorname {div} \\\\left({\\\\overrightarrow {\\\\operatorname {grad} }}~\\\\phi \\
ight).}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi mathvariant="normal">Δ<!-- Δ --></mi>
<br/>        <mi>ϕ<!-- ϕ --></mi>
<br/>        <mo>=</mo>
<br/>        <msup>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mover>
<br/>                <mi mathvariant="normal">∇<!-- ∇ --></mi>
<br/>                <mo stretchy="false">→<!-- → --></mo>
<br/>              </mover>
<br/>            </mrow>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mi>ϕ<!-- ϕ --></mi>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mover>
<br/>              <mi mathvariant="normal">∇<!-- ∇ --></mi>
<br/>              <mo stretchy="false">→<!-- → --></mo>
<br/>            </mover>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mover>
<br/>              <mi mathvariant="normal">∇<!-- ∇ --></mi>
<br/>              <mo stretchy="false">→<!-- → --></mo>
<br/>            </mover>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mi>ϕ<!-- ϕ --></mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mi>div</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mrow>
<br/>          <mo>(</mo>
<br/>          <mrow>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mover>
<br/>                <mi>grad</mi>
<br/>                <mo>→<!-- → --></mo>
<br/>              </mover>
<br/>            </mrow>
<br/>            <mtext>&nbsp;</mtext>
<br/>            <mi>ϕ<!-- ϕ --></mi>
<br/>          </mrow>
<br/>          <mo>)</mo>
<br/>        </mrow>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\Delta \\\\phi ={\\\\vec {\\
abla }}^{2}\\\\phi ={\\\\vec {\\
abla }}\\\\cdot ({\\\\vec {\\
abla }}\\\\phi )=\\\\operatorname {div} \\\\left({\\\\overrightarrow {\\\\operatorname {grad} }}~\\\\phi \\
ight).}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81c1a97c3837be78896d29ef6a771c6a8224dbee" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.598ex; height:6.176ex;" alt="{\\\\displaystyle \\\\Delta \\\\phi ={\\\\vec {\\
abla }}^{2}\\\\phi ={\\\\vec {\\
abla }}\\\\cdot ({\\\\vec {\\
abla }}\\\\phi )=\\\\operatorname {div} \\\\left({\\\\overrightarrow {\\\\operatorname {grad} }}~\\\\phi \\
ight).}"></span></center>
<br/>Intuitivement, il combine et relie la description statique d'un champ (décrit par son gradient) aux effets dynamiques (la divergence) de ce champ dans l'espace et le temps. C'est l'exemple le plus simple et le plus répandu d'opérateur elliptique. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Op%C3%A9rateur_laplacien">https://fr.wikipedia.org/wiki/Op%C3%A9rateur_laplacien</a>)"""@fr, """In mathematics, the <b>Laplace operator</b> or <b>Laplacian</b> is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols <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 \\
abla }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi mathvariant="normal">∇<!-- ∇ --></mi>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>        <mi mathvariant="normal">∇<!-- ∇ --></mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\
abla \\\\cdot \\
abla }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9464c9bf2670581ed86a15bc9c2e8ac4b6ab1484" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.551ex; height:2.176ex;" alt="\\
abla\\\\cdot\\
abla"></span>, <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}}">
<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/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\
abla ^{2}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4be87ad083e5ead48d92b0c82f2d4e719cb34a6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.99ex; height:2.676ex;" alt="\\
abla ^{2}"></span> (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 \\
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 mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="\\
abla "></span> is the nabla operator), or <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 }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi mathvariant="normal">Δ<!-- Δ --></mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\Delta }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="\\\\Delta "></span>. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian <span class="texhtml">Δ<i>f</i> (<i>p</i>)</span> of a function <span class="texhtml"><i>f</i></span> at a point <span class="texhtml"><i>p</i></span> measures by how much the average value of  <span class="texhtml"><i>f</i></span> over small spheres or balls centered at <span class="texhtml"><i>p</i></span> deviates from <span class="texhtml"><i>f</i> (<i>p</i>)</span>. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Laplace_operator">https://en.wikipedia.org/wiki/Laplace_operator</a>)"""@en .

psr:-RRBN6FVB-9
  skos:prefLabel "opérateur différentiel"@fr, "differential operator"@en ;
  a skos:Concept ;
  skos:narrower psr:-GHZJHV7P-F .