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

psr:-WTHH9Q9G-S
  dc:modified "2023-08-10"^^xsd:date ;
  skos:definition """In mathematics, a <b>saddle point</b> or <b>minimax point</b> is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function.  An example of a saddle point is when there is a critical point with a relative minimum along one axial direction (between peaks) and at a relative maximum along the crossing axis.  However, a saddle point need not be in this form.  For example, the function <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)=x^{2}+y^{3}}">
<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 stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <msup>
<br/>          <mi>x</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>+</mo>
<br/>        <msup>
<br/>          <mi>y</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>3</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f(x,y)=x^{2}+y^{3}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10f495e0036715bf53be1d3ba382a207e4327093" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:17.144ex; height:3.176ex;" alt="{\\\\displaystyle f(x,y)=x^{2}+y^{3}}"></span> has a critical point at <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 (0,0)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo stretchy="false">(</mo>
<br/>        <mn>0</mn>
<br/>        <mo>,</mo>
<br/>        <mn>0</mn>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle (0,0)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d630d3e781a53b0a3559ae7e5b45f9479a3141a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="(0,0)"></span> that is a saddle point since it is neither a relative maximum nor relative minimum, but it does not have a relative maximum or relative minimum in the <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 y}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>y</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle y}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="y"></span>-direction. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Saddle_point">https://en.wikipedia.org/wiki/Saddle_point</a>)"""@en, """En mathématiques, un <b>point col</b> ou <b>point-selle</b> (en anglais&nbsp;: <span class="lang-en" lang="en"><i>saddle point</i></span>) d'une fonction <span class="texhtml mvar" style="font-style:italic;">f</span> définie sur un produit cartésien <span class="texhtml"><i>X</i> × <i>Y</i></span> de deux ensembles <span class="texhtml mvar" style="font-style:italic;">X</span> et <span class="texhtml mvar" style="font-style:italic;">Y</span> est un point <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 ({\\ar {x}},{\\ar {y}})\\\\in X\\	imes Y}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo stretchy="false">(</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mover>
<br/>              <mi>x</mi>
<br/>              <mo stretchy="false">¯<!-- ¯ --></mo>
<br/>            </mover>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo>,</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mover>
<br/>              <mi>y</mi>
<br/>              <mo stretchy="false">¯<!-- ¯ --></mo>
<br/>            </mover>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>∈<!-- ∈ --></mo>
<br/>        <mi>X</mi>
<br/>        <mo>×<!-- × --></mo>
<br/>        <mi>Y</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle ({\\ar {x}},{\\ar {y}})\\\\in X\\	imes Y}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2607747f4f49f2a71773ebe4bd63dd83c0155d8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:14.909ex; height:2.843ex;" alt="{\\\\displaystyle ({\\ar {x}},{\\ar {y}})\\\\in X\\	imes Y}"></span> tel que&nbsp;:
<br/>
<br/><ul><li><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 y\\\\mapsto f({\\ar {x}},y)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>y</mi>
<br/>        <mo stretchy="false">↦<!-- ↦ --></mo>
<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mover>
<br/>              <mi>x</mi>
<br/>              <mo stretchy="false">¯<!-- ¯ --></mo>
<br/>            </mover>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo>,</mo>
<br/>        <mi>y</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle y\\\\mapsto f({\\ar {x}},y)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fac123aa8c03eb17e41fd642b578ea8696f21291" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:11.377ex; height:2.843ex;" alt="{\\\\displaystyle y\\\\mapsto f({\\ar {x}},y)}"></span> atteint un maximum en <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 {\\ar {y}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mover>
<br/>              <mi>y</mi>
<br/>              <mo stretchy="false">¯<!-- ¯ --></mo>
<br/>            </mover>
<br/>          </mrow>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\ar {y}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b298744237368f34e61ff7dc90b34016a7037af" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.343ex;" alt="{\\ar  {y}}"></span> sur <span class="texhtml mvar" style="font-style:italic;">Y</span>&nbsp;;</li>
<br/><li>et <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 x\\\\mapsto f(x,{\\ar {y}})}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">↦<!-- ↦ --></mo>
<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo>,</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mover>
<br/>              <mi>y</mi>
<br/>              <mo stretchy="false">¯<!-- ¯ --></mo>
<br/>            </mover>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x\\\\mapsto f(x,{\\ar {y}})}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e172b72f82b69fc3c362c21b1dc8d573501b931" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:11.697ex; height:2.843ex;" alt="{\\\\displaystyle x\\\\mapsto f(x,{\\ar {y}})}"></span> atteint un minimum en <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 {\\ar {x}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mover>
<br/>              <mi>x</mi>
<br/>              <mo stretchy="false">¯<!-- ¯ --></mo>
<br/>            </mover>
<br/>          </mrow>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\ar {x}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/466e03e1c9533b4dab1b9949dad393883f385d80" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.009ex;" alt="{\\ar {x}}"></span> sur <span class="texhtml mvar" style="font-style:italic;">X</span>.</li></ul>
<br/>Certains auteurs inversent les maximum et minimum (<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({\\ar {x}},\\\\cdot )}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mover>
<br/>              <mi>x</mi>
<br/>              <mo stretchy="false">¯<!-- ¯ --></mo>
<br/>            </mover>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo>,</mo>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f({\\ar {x}},\\\\cdot )}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d13f71a2302a4808c00225177e87a89d2fdc801b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:6.098ex; height:2.843ex;" alt="{\\\\displaystyle f({\\ar {x}},\\\\cdot )}"></span> a un <i>minimum</i> en <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 {\\ar {y}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mover>
<br/>              <mi>y</mi>
<br/>              <mo stretchy="false">¯<!-- ¯ --></mo>
<br/>            </mover>
<br/>          </mrow>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\ar {y}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b298744237368f34e61ff7dc90b34016a7037af" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.343ex;" alt="{\\ar  {y}}"></span> et <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(\\\\cdot ,{\\ar {y}})}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>        <mo>,</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mover>
<br/>              <mi>y</mi>
<br/>              <mo stretchy="false">¯<!-- ¯ --></mo>
<br/>            </mover>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f(\\\\cdot ,{\\ar {y}})}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c8a84ef8ffb5331035ce9b5c531533bffb498d8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:6.071ex; height:2.843ex;" alt="{\\\\displaystyle f(\\\\cdot ,{\\ar {y}})}"></span> a un <i>maximum</i> en <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 {\\ar {x}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mover>
<br/>              <mi>x</mi>
<br/>              <mo stretchy="false">¯<!-- ¯ --></mo>
<br/>            </mover>
<br/>          </mrow>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\ar {x}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/466e03e1c9533b4dab1b9949dad393883f385d80" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.009ex;" alt="{\\ar {x}}"></span>), mais cela ne modifie pas qualitativement les résultats (on peut revenir au cas présent par un changement de variables).
<br/>Le terme <i>point-selle</i> fait référence à la forme de selle de cheval que prend le graphe de la fonction lorsque <span class="texhtml mvar" style="font-style:italic;">X</span> et <span class="texhtml mvar" style="font-style:italic;">Y</span> sont des intervalles de <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 \\\\mathbb {R} }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="double-struck">R</mi>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbb {R} }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="\\\\mathbb {R} "></span>. Le terme de  <i>point col</i>, renvoie, quant à lui,  à l'image du col de montagne. Dans (au moins) une direction, le point-col est un point de maximum (pour passer d'une vallée à l'autre) et dans (au moins) une autre direction, c'est un point de minimum (pour passer d'une montagne à l'autre) 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Point_col">https://fr.wikipedia.org/wiki/Point_col</a>)"""@fr ;
  skos:broader psr:-MHPG0QZH-R, psr:-TX92VB3N-7, psr:-PX7GDTC2-7 ;
  a skos:Concept ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Saddle_point>, <https://fr.wikipedia.org/wiki/Point_col> ;
  skos:inScheme psr: ;
  dc:created "2023-08-10"^^xsd:date ;
  skos:prefLabel "point col"@fr, "saddle point"@en ;
  skos:altLabel "point-selle"@fr, "minimax point"@en .

psr: a skos:ConceptScheme .
psr:-MHPG0QZH-R
  skos:prefLabel "multivariable calculus"@en, "calcul multivariable"@fr ;
  a skos:Concept ;
  skos:narrower psr:-WTHH9Q9G-S .

psr:-TX92VB3N-7
  skos:prefLabel "differential geometry of surfaces"@en, "géométrie différentielle des surfaces"@fr ;
  a skos:Concept ;
  skos:narrower psr:-WTHH9Q9G-S .

psr:-PX7GDTC2-7
  skos:prefLabel "point"@fr, "point"@en ;
  a skos:Concept ;
  skos:narrower psr:-WTHH9Q9G-S .