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

psr: a skos:ConceptScheme .
psr:-RRBN6FVB-9
  skos:prefLabel "opérateur différentiel"@fr, "differential operator"@en ;
  a skos:Concept ;
  skos:narrower psr:-MZKTP1S1-5 .

psr:-MZKTP1S1-5
  skos:definition """En analyse vectorielle, le laplacien vectoriel est un opérateur différentiel pour les champs vectoriels. Il présente beaucoup de similitudes avec l'opérateur laplacien scalaire. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Op%C3%A9rateur_laplacien_vectoriel">https://fr.wikipedia.org/wiki/Op%C3%A9rateur_laplacien_vectoriel</a>)"""@fr, """The <b>vector Laplace operator</b>, also denoted by <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>, is a differential operator defined over a vector field. The vector Laplacian is similar to the scalar Laplacian; whereas the scalar Laplacian applies to a scalar field and returns a scalar quantity, the vector Laplacian applies to a vector field, returning a vector quantity. When computed in orthonormal Cartesian coordinates, the returned vector field is equal to the vector field of the scalar Laplacian applied to each vector component.
<br/>The <b>vector Laplacian</b> of a vector field <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 \\\\mathbf {A} }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="bold">A</mi>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbf {A} }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="\\\\mathbf {A} "></span> is defined 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}\\\\mathbf {A} =\\
abla (\\
abla \\\\cdot \\\\mathbf {A} )-\\
abla \\	imes (\\
abla \\	imes \\\\mathbf {A} ).}">
<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/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="bold">A</mi>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mi mathvariant="normal">∇<!-- ∇ --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi mathvariant="normal">∇<!-- ∇ --></mi>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="bold">A</mi>
<br/>        </mrow>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mi mathvariant="normal">∇<!-- ∇ --></mi>
<br/>        <mo>×<!-- × --></mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi mathvariant="normal">∇<!-- ∇ --></mi>
<br/>        <mo>×<!-- × --></mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="bold">A</mi>
<br/>        </mrow>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\
abla ^{2}\\\\mathbf {A} =\\
abla (\\
abla \\\\cdot \\\\mathbf {A} )-\\
abla \\	imes (\\
abla \\	imes \\\\mathbf {A} ).}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/766d34a72018a87d2ad83997ad5f98d26acb0812" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.356ex; height:3.176ex;" alt="{\\\\displaystyle \\
abla ^{2}\\\\mathbf {A} =\\
abla (\\
abla \\\\cdot \\\\mathbf {A} )-\\
abla \\	imes (\\
abla \\	imes \\\\mathbf {A} ).}"></div>
<br/>In Cartesian coordinates, this reduces to the much simpler form 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}\\\\mathbf {A} =(\\
abla ^{2}A_{x},\\
abla ^{2}A_{y},\\
abla ^{2}A_{z}),}">
<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/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="bold">A</mi>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <msup>
<br/>          <mi mathvariant="normal">∇<!-- ∇ --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <msub>
<br/>          <mi>A</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>x</mi>
<br/>          </mrow>
<br/>        </msub>
<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/>        <msub>
<br/>          <mi>A</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>y</mi>
<br/>          </mrow>
<br/>        </msub>
<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/>        <msub>
<br/>          <mi>A</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>z</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\
abla ^{2}\\\\mathbf {A} =(\\
abla ^{2}A_{x},\\
abla ^{2}A_{y},\\
abla ^{2}A_{z}),}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec913b22acd7cdb916a1ad7d3345a58a893e99c9" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.055ex; height:3.343ex;" alt="{\\\\displaystyle \\
abla ^{2}\\\\mathbf {A} =(\\
abla ^{2}A_{x},\\
abla ^{2}A_{y},\\
abla ^{2}A_{z}),}"></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 A_{x}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>A</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>x</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle A_{x}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/626cb5d94c04152accd89eee76eb7a2613376484" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.916ex; height:2.509ex;" alt="A_x"></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 A_{y}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>A</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>y</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle A_{y}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0500d93a2b28b15bb6fda15dac5a86130f41837a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.792ex; height:2.843ex;" alt="A_y"></span>, 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 A_{z}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>A</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>z</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle A_{z}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8648938c43e0b9b6445659a28118e4f0a7bdd8ff" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.745ex; height:2.509ex;" alt="A_z"></span> are the components of the vector field <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 \\\\mathbf {A} }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="bold">A</mi>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbf {A} }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="\\\\mathbf {A} "></span>, 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 \\
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> just on the left of each vector field component is the (scalar) Laplace operator. This can be seen to be a special case of Lagrange's formula; see Vector triple product.
<br/>For expressions of the vector Laplacian in other coordinate systems see Del in cylindrical and spherical coordinates. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Laplace_operator#Vector_Laplacian">https://en.wikipedia.org/wiki/Laplace_operator#Vector_Laplacian</a>)"""@en ;
  skos:inScheme psr: ;
  skos:prefLabel "vector Laplace operator"@en, "laplacien vectoriel"@fr ;
  skos:broader psr:-P6BGQTWK-G, psr:-RRBN6FVB-9 ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Laplace_operator#Vector_Laplacian>, <https://fr.wikipedia.org/wiki/Op%C3%A9rateur_laplacien_vectoriel> ;
  a skos:Concept .

psr:-P6BGQTWK-G
  skos:prefLabel "vector calculus identities"@en, "identités vectorielles"@fr ;
  a skos:Concept ;
  skos:narrower psr:-MZKTP1S1-5 .