@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:-TXTD8V7M-1
  skos:prefLabel "orthogonal function"@en, "fonction orthogonale"@fr ;
  a skos:Concept ;
  skos:narrower psr:-X5ZLLBMK-J .

psr: a skos:ConceptScheme .
psr:-X5ZLLBMK-J
  skos:exactMatch <https://fr.wikipedia.org/wiki/Harmonique_sph%C3%A9rique>, <https://en.wikipedia.org/wiki/Spherical_harmonics> ;
  a skos:Concept ;
  skos:definition """En mathématiques, les <b>harmoniques sphériques</b> sont des fonctions harmoniques particulières, c'est-à-dire des fonctions dont le laplacien est nul. Les harmoniques sphériques sont particulièrement utiles pour résoudre des problèmes invariants par rotation, car elles sont les vecteurs propres de certains opérateurs liés aux rotations. Les polynômes harmoniques <span class="texhtml"><i>P</i>(<i>x</i>,<i>y</i>,<i>z</i>)</span> de degré <span class="texhtml mvar" style="font-style:italic;">l</span> forment un espace vectoriel de dimension <span class="texhtml">2 <i>l</i> + 1</span>, et peuvent s'exprimer en coordonnées sphériques <span class="texhtml">(<i>r</i>, <i>θ</i>, <i>φ</i>)</span> comme des combinaisons linéaires des (<span class="texhtml">2 <i>l</i> + 1</span>) fonctions :   <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 r^{l}\\\\,Y_{l,m}(\\	heta ,\\\\varphi )}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msup>           <mi>r</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>l</mi>           </mrow>         </msup>         <mspace width="thinmathspace"></mspace>         <msub>           <mi>Y</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>l</mi>             <mo>,</mo>             <mi>m</mi>           </mrow>         </msub>         <mo stretchy="false">(</mo>         <mi>θ<!-- θ --></mi>         <mo>,</mo>         <mi>φ<!-- φ --></mi>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle r^{l}\\\\,Y_{l,m}(\\	heta ,\\\\varphi )}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e6335b348b818d0283f4bec700bc97ef503a83f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.585ex; height:3.343ex;" alt="{\\\\displaystyle r^{l}\\\\,Y_{l,m}(\\	heta ,\\\\varphi )}"></span>,  avec <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 -l\\\\leq m\\\\leq +l}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mo>−<!-- − --></mo>         <mi>l</mi>         <mo>≤<!-- ≤ --></mo>         <mi>m</mi>         <mo>≤<!-- ≤ --></mo>         <mo>+</mo>         <mi>l</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle -l\\\\leq m\\\\leq +l}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b20b822dad2fccf212f17e9d1a7a556d057e0a0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.24ex; height:2.343ex;" alt="{\\\\displaystyle -l\\\\leq m\\\\leq +l}"></span>.</dd></dl> Les coordonnées sphériques <span class="texhtml">(<i>r</i>,<i>θ</i>,<i>φ</i>)</span> sont, respectivement, la distance au centre de la sphère, la colatitude et la longitude. Tout polynôme homogène est entièrement déterminé par sa restriction à la sphère unité <span class="texhtml"><i>S</i><sup>2</sup></span>. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Harmonique_sph%C3%A9rique">https://fr.wikipedia.org/wiki/Harmonique_sph%C3%A9rique</a>)"""@fr, """In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. A list of the spherical harmonics is available in Table of spherical harmonics.
<br/>Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3). 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Spherical_harmonics">https://en.wikipedia.org/wiki/Spherical_harmonics</a>)"""@en ;
  skos:prefLabel "spherical harmonic"@en, "harmonique sphérique"@fr ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:broader psr:-TXTD8V7M-1, psr:-H7WG59FS-L ;
  skos:inScheme psr: .

psr:-H7WG59FS-L
  skos:prefLabel "fonction harmonique"@fr, "harmonic function"@en ;
  a skos:Concept ;
  skos:narrower psr:-X5ZLLBMK-J .