@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:-Q10Q14NT-1
  skos:prefLabel "topologie différentielle"@fr, "differential topology"@en ;
  a skos:Concept ;
  skos:narrower psr:-FKRLMSC9-6 .

psr: a skos:ConceptScheme .
psr:-FKRLMSC9-6
  skos:exactMatch <https://en.wikipedia.org/wiki/Gaussian_curvature>, <https://fr.wikipedia.org/wiki/Courbure_de_Gauss> ;
  skos:definition """In differential geometry, the <b>Gaussian curvature</b> or <b>Gauss curvature</b> <span class="texhtml mvar" style="font-style:italic;">Κ</span> of a smooth surface in three-dimensional space at a point is the product of the principal curvatures, <span class="texhtml"><i>κ</i><sub>1</sub></span> and <span class="texhtml"><i>κ</i><sub>2</sub></span>, at the given point:
<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 K=\\\\kappa _{1}\\\\kappa _{2}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>K</mi>
<br/>        <mo>=</mo>
<br/>        <msub>
<br/>          <mi>κ<!-- κ --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <msub>
<br/>          <mi>κ<!-- κ --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle K=\\\\kappa _{1}\\\\kappa _{2}.}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76d76efdc6ce7296b3d5b3f996fbfc264a9a4e33" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -0.671ex; width:10.598ex; height:2.509ex;" alt="{\\\\displaystyle K=\\\\kappa _{1}\\\\kappa _{2}.}"></div>
<br/>The <b>Gaussian radius of curvature</b> is the reciprocal of <span class="texhtml mvar" style="font-style:italic;">Κ</span>.
<br/>For example, a sphere of radius <span class="texhtml mvar" style="font-style:italic;">r</span> has Gaussian curvature <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1050945101">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span role="math" class="sfrac tion"><span class="num">1</span>/<span class="den"><i>r</i><sup>2</sup></span></span></span> everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus.
<br/>Gaussian curvature is an <i>intrinsic</i> measure of curvature, depending only on distances that are measured “within” or along the surface, not on the way it is isometrically embedded in Euclidean space. This is the content of the <i>Theorema egregium</i>. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Gaussian_curvature">https://en.wikipedia.org/wiki/Gaussian_curvature</a>)"""@en, """La courbure de Gauss, parfois aussi appelée courbure totale, d'une surface paramétrée <i>X</i> en <i>X</i>(<i>P</i>) est le produit des courbures principales. De manière équivalente, la courbure de Gauss est le déterminant de l'endomorphisme de Weingarten. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Courbure_de_Gauss">https://fr.wikipedia.org/wiki/Courbure_de_Gauss</a>)"""@fr ;
  skos:prefLabel "courbure de Gauss"@fr, "Gaussian curvature"@en ;
  dc:modified "2023-08-16"^^xsd:date ;
  skos:inScheme psr: ;
  skos:broader psr:-Q10Q14NT-1, psr:-TX92VB3N-7 ;
  skos:altLabel "courbure totale"@fr, "Gauss curvature"@en ;
  a skos:Concept .

psr:-TX92VB3N-7
  skos:prefLabel "differential geometry of surfaces"@en, "géométrie différentielle des surfaces"@fr ;
  a skos:Concept ;
  skos:narrower psr:-FKRLMSC9-6 .