Polarkoordinaten – Wikipedia

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border-bottom-width: 1px; font-size:95%; margin-bottom:1em; padding: 0.25em; overflow: hidden; word-break: break-word; word-wrap: break-word;" id="Vorlage_Dieser_Artikel"><div class="noviewer noresize" style="display: table-cell; padding-bottom: 0.2em; padding-left: 0.25em; padding-right: 1em; padding-top: 0.2em; vertical-align: middle;" id="bksicon" aria-hidden="true" role="presentation"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/25px-Disambig-dark.svg.png" decoding="async" width="25" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/38px-Disambig-dark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/50px-Disambig-dark.svg.png 2x" data-file-width="444" data-file-height="340" /></div> <div style="display: table-cell; vertical-align: middle; width: 100%;"> <div role="navigation"> Dieser Artikel behandelt Polarkoordinaten der Ebene sowie die eng damit verwandten Zylinderkoordinaten im Raum. Für räumliche Polarkoordinaten siehe den Artikel <a href="/wiki/Kugelkoordinaten" title="Kugelkoordinaten">Kugelkoordinaten</a>.</div> </div></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Datei:Polar_graph_paper.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Polar_graph_paper.svg/300px-Polar_graph_paper.svg.png" decoding="async" width="300" height="284" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Polar_graph_paper.svg/450px-Polar_graph_paper.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Polar_graph_paper.svg/600px-Polar_graph_paper.svg.png 2x" data-file-width="495" data-file-height="469" /></a><figcaption>Ein Polargitter verschiedener Winkel mit Grad-Angaben</figcaption></figure> In der <a href="/wiki/Mathematik" title="Mathematik">Mathematik</a> und <a href="/wiki/Geod%C3%A4sie" title="Geodäsie">Geodäsie</a> versteht man unter einem Polarkoordinatensystem (auch: Kreiskoordinatensystem) ein <a href="/wiki/Zweidimensional" class="mw-redirect" title="Zweidimensional">zweidimensionales</a> <a href="/wiki/Koordinatensystem" title="Koordinatensystem">Koordinatensystem</a>, in dem jeder <a href="/wiki/Punkt_(Geometrie)" title="Punkt (Geometrie)">Punkt</a> in einer <a href="/wiki/Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebene</a> durch den <a href="/wiki/Abstand" title="Abstand">Abstand</a> von einem vorgegebenen festen Punkt und durch den <a href="/wiki/Winkel" title="Winkel">Winkel</a> zu einer festen Richtung festgelegt wird. Der feste Punkt wird als Pol bezeichnet; er entspricht dem Ursprung bei einem <a href="/wiki/Kartesisches_Koordinatensystem" title="Kartesisches Koordinatensystem">kartesischen Koordinatensystem</a>. Der vom Pol in der festgelegten Richtung ausgehende <a href="/wiki/Strahl_(Geometrie)" title="Strahl (Geometrie)">Strahl</a> heißt Polarachse. Der Abstand vom Pol wird meist mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"> bezeichnet und heißt Radius oder Radialkoordinate, der Winkel wird mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> bezeichnet und heißt Winkelkoordinate, Polarwinkel, Azimut oder Argument. Polarkoordinaten bilden einen Spezialfall von <a href="/wiki/Orthogonale_Koordinaten" title="Orthogonale Koordinaten">orthogonalen Koordinaten</a>. Sie sind hilfreich, wenn sich das Verhältnis zwischen zwei Punkten leichter durch Winkel und Abstände beschreiben lässt, als dies mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">- und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}">-Koordinaten der Fall wäre. In der Geodäsie sind Polarkoordinaten die häufigste Methode zur Einmessung von Punkten (<a href="/wiki/Polaraufnahme" title="Polaraufnahme">Polarmethode</a>). In der <a href="/wiki/Funknavigation" title="Funknavigation">Funknavigation</a> wird das Prinzip oft als „<a href="/wiki/Rho-Theta" title="Rho-Theta">Rho-Theta</a>“ (für Distanz- und Richtungsmessung) bezeichnet. In der Mathematik wird die Winkelkoordinate im mathematisch positiven <a href="/wiki/Drehrichtung" title="Drehrichtung">Drehsinn</a> (Gegenuhrzeigersinn) gemessen. Wird gleichzeitig ein kartesisches Koordinatensystem benutzt, so dient in der Regel dessen Koordinatenursprung als Pol und die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">-Achse als Polarachse. Die Winkelkoordinate wird also von der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">-Achse aus in Richtung der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}">-Achse gemessen. In der Geodäsie und in der Navigation wird das <a href="/wiki/Azimut" title="Azimut">Azimut</a> von der Nordrichtung aus im <a href="/wiki/Uhrzeigersinn" class="mw-redirect" title="Uhrzeigersinn">Uhrzeigersinn</a> gemessen. <a href="/wiki/Polarkoordinatenpapier" title="Polarkoordinatenpapier">Polarkoordinatenpapier</a> ist mit einem Polarkoordinatensystem bedruckt. <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Geschichte">1 Geschichte</a></li> <li class="toclevel-1 tocsection-2"><a href="#Polarkoordinaten_in_der_Ebene:_Kreiskoordinaten">2 Polarkoordinaten in der Ebene: Kreiskoordinaten</a> <ul> <li class="toclevel-2 tocsection-3"><a href="#Definition">2.1 Definition</a></li> <li class="toclevel-2 tocsection-4"><a href="#Umrechnung_zwischen_Polarkoordinaten_und_kartesischen_Koordinaten">2.2 Umrechnung zwischen Polarkoordinaten und kartesischen Koordinaten</a> <ul> <li class="toclevel-3 tocsection-5"><a href="#Umrechnung_von_Polarkoordinaten_in_kartesische_Koordinaten">2.2.1 Umrechnung von Polarkoordinaten in kartesische Koordinaten</a></li> <li class="toclevel-3 tocsection-6"><a href="#Umrechnung_von_kartesischen_Koordinaten_in_Polarkoordinaten">2.2.2 Umrechnung von kartesischen Koordinaten in Polarkoordinaten</a> <ul> <li class="toclevel-4 tocsection-7"><a href="#Berechnung_des_Winkels_im_Intervall_(−π,_π]_bzw._(−180°,180°]">2.2.2.1 Berechnung des Winkels im Intervall (−π, π] bzw. (−180°,180°]</a></li> <li class="toclevel-4 tocsection-8"><a href="#Berechnung_des_Winkels_im_Intervall_[0,_2π)_bzw._[0,_360°)">2.2.2.2 Berechnung des Winkels im Intervall [0, 2π) bzw. [0, 360°)</a></li> <li class="toclevel-4 tocsection-9"><a href="#Verschiebung_des_Winkels">2.2.2.3 Verschiebung des Winkels</a></li> </ul> </li> </ul> </li> <li class="toclevel-2 tocsection-10"><a href="#Koordinatenlinien">2.3 Koordinatenlinien</a></li> <li class="toclevel-2 tocsection-11"><a href="#Lokale_Basisvektoren_und_Orthogonalität">2.4 Lokale Basisvektoren und Orthogonalität</a></li> <li class="toclevel-2 tocsection-12"><a href="#Metrischer_Tensor">2.5 Metrischer Tensor</a></li> <li class="toclevel-2 tocsection-13"><a href="#Funktionaldeterminante">2.6 Funktionaldeterminante</a></li> <li class="toclevel-2 tocsection-14"><a href="#Flächenelement">2.7 Flächenelement</a></li> <li class="toclevel-2 tocsection-15"><a href="#Linienelement">2.8 Linienelement</a></li> <li class="toclevel-2 tocsection-16"><a href="#Ortsvektor,_Geschwindigkeit_und_Beschleunigung_in_Polarkoordinaten">2.9 Ortsvektor, Geschwindigkeit und Beschleunigung in Polarkoordinaten</a></li> </ul> </li> <li class="toclevel-1 tocsection-17"><a href="#Räumliche_Polarkoordinaten:_Zylinder-,_Kegel-_und_Kugelkoordinaten">3 Räumliche Polarkoordinaten: Zylinder-, Kegel- und Kugelkoordinaten</a> <ul> <li class="toclevel-2 tocsection-18"><a href="#Zylinderkoordinaten">3.1 Zylinderkoordinaten</a> <ul> <li class="toclevel-3 tocsection-19"><a href="#Umrechnung_zwischen_Zylinderkoordinaten_und_kartesischen_Koordinaten">3.1.1 Umrechnung zwischen Zylinderkoordinaten und kartesischen Koordinaten</a></li> <li class="toclevel-3 tocsection-20"><a href="#Koordinatenlinien_und_Koordinatenflächen">3.1.2 Koordinatenlinien und Koordinatenflächen</a></li> <li class="toclevel-3 tocsection-21"><a href="#Lokale_Basisvektoren_und_Orthogonalität_2">3.1.3 Lokale Basisvektoren und Orthogonalität</a></li> <li class="toclevel-3 tocsection-22"><a href="#Metrischer_Tensor_2">3.1.4 Metrischer Tensor</a></li> <li class="toclevel-3 tocsection-23"><a href="#Funktionaldeterminante_2">3.1.5 Funktionaldeterminante</a></li> <li class="toclevel-3 tocsection-24"><a href="#Vektoranalysis">3.1.6 Vektoranalysis</a> <ul> <li class="toclevel-4 tocsection-25"><a href="#Gradient">3.1.6.1 Gradient</a></li> <li class="toclevel-4 tocsection-26"><a href="#Divergenz">3.1.6.2 Divergenz</a></li> <li class="toclevel-4 tocsection-27"><a href="#Rotation">3.1.6.3 Rotation</a></li> </ul> </li> <li class="toclevel-3 tocsection-28"><a href="#Ortsvektor,_Geschwindigkeit_und_Beschleunigung_in_Zylinderkoordinaten">3.1.7 Ortsvektor, Geschwindigkeit und Beschleunigung in Zylinderkoordinaten</a></li> </ul> </li> <li class="toclevel-2 tocsection-29"><a href="#Kegelkoordinaten_(Koordinaten-Transformation)">3.2 Kegelkoordinaten (Koordinaten-Transformation)</a> <ul> <li class="toclevel-3 tocsection-30"><a href="#Parameterdarstellung">3.2.1 Parameterdarstellung</a></li> <li class="toclevel-3 tocsection-31"><a href="#Flächennormalenvektor">3.2.2 Flächennormalenvektor</a></li> <li class="toclevel-3 tocsection-32"><a href="#Einheitsvektoren_der_Kegelkoordinaten_in_kartesischen_Komponenten">3.2.3 Einheitsvektoren der Kegelkoordinaten in kartesischen Komponenten</a></li> <li class="toclevel-3 tocsection-33"><a href="#Transformationsmatrizen">3.2.4 Transformationsmatrizen</a> <ul> <li class="toclevel-4 tocsection-34"><a href="#Jacobi-Matrix_(Funktionalmatrix)">3.2.4.1 Jacobi-Matrix (Funktionalmatrix)</a></li> <li class="toclevel-4 tocsection-35"><a href="#Transformationsmatrix_S">3.2.4.2 Transformationsmatrix S</a></li> </ul> </li> <li class="toclevel-3 tocsection-36"><a href="#Transformation_der_partiellen_Ableitungen">3.2.5 Transformation der partiellen Ableitungen</a></li> <li class="toclevel-3 tocsection-37"><a href="#Transformation_der_Einheitsvektoren">3.2.6 Transformation der Einheitsvektoren</a></li> <li class="toclevel-3 tocsection-38"><a href="#Transformation_von_Vektorfeldern">3.2.7 Transformation von Vektorfeldern</a></li> <li class="toclevel-3 tocsection-39"><a href="#Oberflächen-_und_Volumendifferential">3.2.8 Oberflächen- und Volumendifferential</a></li> <li class="toclevel-3 tocsection-40"><a href="#Transformierte_Vektor-Differentialoperatoren">3.2.9 Transformierte Vektor-Differentialoperatoren</a> <ul> <li class="toclevel-4 tocsection-41"><a href="#Nabla-Operator">3.2.9.1 Nabla-Operator</a></li> <li class="toclevel-4 tocsection-42"><a href="#Gradient_2">3.2.9.2 Gradient</a></li> <li class="toclevel-4 tocsection-43"><a href="#Divergenz_2">3.2.9.3 Divergenz</a></li> <li class="toclevel-4 tocsection-44"><a href="#Laplace-Operator">3.2.9.4 Laplace-Operator</a></li> <li class="toclevel-4 tocsection-45"><a href="#Rotation_2">3.2.9.5 Rotation</a></li> </ul> </li> </ul> </li> <li class="toclevel-2 tocsection-46"><a href="#Kugelkoordinaten">3.3 Kugelkoordinaten</a></li> </ul> </li> <li class="toclevel-1 tocsection-47"><a href="#n-dimensionale_Polarkoordinaten">4 n-dimensionale Polarkoordinaten</a> <ul> <li class="toclevel-2 tocsection-48"><a href="#Umrechnung_in_kartesische_Koordinaten">4.1 Umrechnung in kartesische Koordinaten</a></li> <li class="toclevel-2 tocsection-49"><a href="#Funktionaldeterminante_3">4.2 Funktionaldeterminante</a></li> </ul> </li> <li class="toclevel-1 tocsection-50"><a href="#Literatur">5 Literatur</a></li> <li class="toclevel-1 tocsection-51"><a href="#Weblinks">6 Weblinks</a></li> <li class="toclevel-1 tocsection-52"><a href="#Einzelnachweise">7 Einzelnachweise</a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Geschichte">Geschichte</h2>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=1" title="Abschnitt bearbeiten: Geschichte" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Geschichte">Quelltext bearbeiten</a>]</div> Die Begriffe Winkel und Radius wurden bereits von den Menschen des Altertums im ersten Jahrtausend vor Christus verwendet. Der griechische Astronom <a href="/wiki/Hipparchos_(Astronom)" title="Hipparchos (Astronom)">Hipparchos</a> (190–120 v. Chr.) erstellte eine Tafel von trigonometrischen <a href="/wiki/Sehne_(Mathematik)" class="mw-redirect" title="Sehne (Mathematik)">Sehnenfunktionen</a>, um die Länge der <a href="/wiki/Chord_(Mathematik)" title="Chord (Mathematik)">Sehne</a> für die einzelnen Winkel zu finden. Mit Hilfe dieser Grundlage war es ihm möglich, die Polarkoordinaten zu nutzen, um damit die Position bestimmter Sterne festlegen zu können. Sein Werk umfasste jedoch nur einen Teil des Koordinatensystems.<a href="#cite_note-milestones-1">[1]</a> In seiner Abhandlung Über <a href="/wiki/Spirale" title="Spirale">Spiralen</a> beschreibt <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> eine Spirallinie mit einer Funktion, deren Radius sich abhängig von seinem Winkel ändert. Die Arbeit des Griechen umfasste jedoch noch kein volles Koordinatensystem. Es gibt verschiedene Beschreibungen, um das Polarkoordinatensystem als Teil eines formalen Koordinatensystems zu definieren. Die gesamte Historie zu diesem Thema wird in dem Aufsatz Origin of Polar Coordinates (Ursprung der Polarkoordinaten) des <a href="/wiki/Harvard_University" title="Harvard University">Harvard</a>-Professors Julian Coolidge zusammengefasst und erläutert.<a href="#cite_note-coolidge-2">[2]</a> Demnach führten <a href="/wiki/Gr%C3%A9goire_de_Saint-Vincent" title="Grégoire de Saint-Vincent">Grégoire de Saint-Vincent</a> und <a href="/wiki/Bonaventura_Cavalieri" title="Bonaventura Cavalieri">Bonaventura Cavalieri</a> diese Konzeption unabhängig voneinander in der Mitte des 17. Jahrhunderts ein. Saint-Vincent schrieb im Jahre 1625 auf privater Basis über dieses Thema und veröffentlichte seine Arbeit 1647, während Cavalieri seine Ausarbeitung 1635 veröffentlichte, wobei eine korrigierte Fassung 1653 erschien. Cavalieri benutzte Polarkoordinaten anfangs, um ein Problem in Bezug auf die Fläche der <a href="/wiki/Archimedische_Spirale" title="Archimedische Spirale">Archimedischen Spirale</a> zu lösen. Etwas später verwendete <a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Blaise Pascal</a> Polarkoordinaten, um die Länge von <a href="/wiki/Parabel_(Mathematik)" title="Parabel (Mathematik)">parabolischen</a> Winkeln zu berechnen. In dem Werk Method of Fluxions (Fluxionsmethode) (geschrieben 1671, veröffentlicht 1736) betrachtet Sir <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> die Transformation zwischen Polarkoordinaten, auf die er sich als „Seventh Manner; For Spirals“, (Siebte Methode; Für Spiralen) bezog, und neun anderen Koordinatensystemen.<a href="#cite_note-3">[3]</a> Es folgte <a href="/wiki/Jakob_I._Bernoulli" class="mw-redirect" title="Jakob I. Bernoulli">Jacob Bernoulli</a>, der in der Fachzeitschrift <a href="/wiki/Acta_Eruditorum" title="Acta Eruditorum">Acta Eruditorum</a> (1691) ein System verwendete, das aus einer Geraden und einem Punkt auf dieser Geraden bestand, die er Polarachse bzw. Pol nannte. Die Koordinaten wurden darin durch den Abstand von dem Pol und dem Winkel zu der Polarachse festgelegt. Bernoullis Arbeit reichte bis zu der Formulierung des <a href="/wiki/Kr%C3%BCmmungskreis" title="Krümmungskreis">Krümmungskreises</a> von Kurven, die er durch die genannten Koordinaten ausdrückte. Der heute gebräuchliche Begriff Polarkoordinaten wurde von <a href="/wiki/Gregorio_Fontana_(Mathematiker)" title="Gregorio Fontana (Mathematiker)">Gregorio Fontana</a> schließlich eingeführt und in italienischen Schriften des 18. Jahrhunderts verwendet. Im Folgenden übernahm <a href="/wiki/George_Peacock" title="George Peacock">George Peacock</a> im Jahre 1816 diese Bezeichnung in die englische Sprache, als er die Arbeit von <a href="/wiki/Sylvestre_Lacroix" title="Sylvestre Lacroix">Sylvestre Lacroix</a> Differential and Integral Calculus (Differential und Integralberechnung) in seine Sprache übersetzte.<a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a> <a href="/wiki/Alexis-Claude_Clairaut" title="Alexis-Claude Clairaut">Alexis-Claude Clairaut</a> hingegen war der erste, der über Polarkoordinaten in drei Dimensionen nachdachte, deren Entwicklung jedoch erst dem Schweizer Mathematiker <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> gelang.<a href="#cite_note-coolidge-2">[2]</a> <div class="mw-heading mw-heading2"><h2 id="Polarkoordinaten_in_der_Ebene:_Kreiskoordinaten">Polarkoordinaten in der Ebene: Kreiskoordinaten</h2>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=2" title="Abschnitt bearbeiten: Polarkoordinaten in der Ebene: Kreiskoordinaten" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Polarkoordinaten in der Ebene: Kreiskoordinaten">Quelltext bearbeiten</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Definition">Definition</h3>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=3" title="Abschnitt bearbeiten: Definition" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Definition">Quelltext bearbeiten</a>]</div> Die Polarkoordinaten eines Punktes in der <a href="/wiki/Euklidische_Geometrie" title="Euklidische Geometrie">euklidischen</a> Ebene (ebene Polarkoordinaten) werden in Bezug auf einen <a href="/wiki/Koordinatenursprung" class="mw-redirect" title="Koordinatenursprung">Koordinatenursprung</a> (einen Punkt der Ebene) und eine Richtung (einen im Koordinatenursprung beginnenden <a href="/wiki/Strahl_(Geometrie)" title="Strahl (Geometrie)">Strahl</a>) angegeben. Das Polarkoordinatensystem ist dadurch eindeutig festgelegt, dass ein ausgezeichneter Punkt, auch Pol genannt, den Ursprung/Nullpunkt des Koordinatensystems bildet. Ferner wird ein von ihm ausgehender Strahl als sogenannte Polachse ausgezeichnet. Letztlich muss noch eine Richtung (von zwei möglichen), die senkrecht zu dieser Polachse ist, als positiv definiert werden, um den Drehsinn / die Drehrichtung / die Orientierung festzulegen. Nun lässt sich ein Winkel, der Polarwinkel, zwischen einem beliebigen Strahl, der durch den Pol geht, und dieser ausgezeichneten Polachse definieren. Er ist nur bis auf ganzzahlige Umdrehungen um den Pol eindeutig, unabhängig davon, was als <a href="/wiki/Winkelma%C3%9F" title="Winkelmaß">Winkelmaß</a> für ihn gewählt wird. Auf der Polachse selbst erfolgt noch eine beliebige, aber feste Skalierung, um die radiale Einheitslänge zu definieren. Nun kann jedem Paar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r,\phi )\in \mathbb {R} _{0}^{+}\times \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r,\phi )\in \mathbb {R} _{0}^{+}\times \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1a35dda6ac618687f2301b0a4a0b89faa9659df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.825ex; height:3.176ex;" alt="{\displaystyle (r,\phi )\in \mathbb {R} _{0}^{+}\times \mathbb {R} }"> ein Punkt der Ebene eindeutig zugeordnet werden, wobei man die erste Komponente als radiale Länge und die zweite als polaren Winkel ansieht. Solch ein Zahlenpaar bezeichnet man als (nicht notwendigerweise eindeutige) Polarkoordinaten eines Punktes in dieser Ebene. <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Datei:Ebene_polarkoordinaten.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Ebene_polarkoordinaten.svg/660px-Ebene_polarkoordinaten.svg.png" decoding="async" width="660" height="292" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Ebene_polarkoordinaten.svg/990px-Ebene_polarkoordinaten.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Ebene_polarkoordinaten.svg/1320px-Ebene_polarkoordinaten.svg.png 2x" data-file-width="1205" data-file-height="534" /></a><figcaption></figcaption></figure> <div style="clear:both;"></div> <dl><dd>Ebene Polarkoordinaten (mit Winkelangaben in Grad) und ihre Transformation in kartesische Koordinaten</dd></dl> Die Koordinate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}">, eine Länge, wird als Radius (in der Praxis auch als Abstand) und die Koordinate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"> als (Polar)<a href="/wiki/Winkel" title="Winkel">winkel</a> oder, in der Praxis (gelegentlich) auch als <a href="/wiki/Azimut" title="Azimut">Azimut</a> bezeichnet. In der Mathematik wird meistens der Winkel im Gegenuhrzeigersinn als positiv definiert, wenn man senkrecht von oben auf die Ebene (Uhr) schaut. Also geht die <a href="/wiki/Drehrichtung" title="Drehrichtung">Drehrichtung</a> von rechts nach oben (und weiter nach links). Als Winkelmaß wird dabei der <a href="/wiki/Radiant_(Einheit)" title="Radiant (Einheit)">Radiant</a> als Winkeleinheit bevorzugt, weil es dann analytisch am elegantesten zu handhaben ist. Die Polarachse zeigt in grafischen Darstellungen des Koordinatensystems typischerweise nach rechts. <div class="mw-heading mw-heading3"><h3 id="Umrechnung_zwischen_Polarkoordinaten_und_kartesischen_Koordinaten">Umrechnung zwischen Polarkoordinaten und kartesischen Koordinaten</h3>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=4" title="Abschnitt bearbeiten: Umrechnung zwischen Polarkoordinaten und kartesischen Koordinaten" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Umrechnung zwischen Polarkoordinaten und kartesischen Koordinaten">Quelltext bearbeiten</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Umrechnung_von_Polarkoordinaten_in_kartesische_Koordinaten">Umrechnung von Polarkoordinaten in kartesische Koordinaten</h4>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=5" title="Abschnitt bearbeiten: Umrechnung von Polarkoordinaten in kartesische Koordinaten" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Umrechnung von Polarkoordinaten in kartesische Koordinaten">Quelltext bearbeiten</a>]</div> Wenn man ein <a href="/wiki/Kartesisches_Koordinatensystem" title="Kartesisches Koordinatensystem">kartesisches Koordinatensystem</a> mit gleichem Ursprung wie das Polarkoordinatensystem, dabei die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">-Achse in der Richtung der Polarachse, und schließlich die positive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}">-Achse in Richtung des positiven Drehsinnes wählt – wie in der Abbildung oben rechts dargestellt –, so ergibt sich für die kartesischen Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> eines Punktes: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=r\cos \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=r\cos \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ec84f3b8f3196698a3693a21f258e47f270302f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.882ex; height:2.176ex;" alt="{\displaystyle x=r\cos \varphi }"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=r\sin \varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=r\sin \varphi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7ecba10b6710501c7226761c5b24b4c5c657f8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.099ex; height:2.676ex;" alt="{\displaystyle y=r\sin \varphi .}"></dd></dl> Mit komplexen Zahlen und <a href="/wiki/Komplexwertige_Funktion" title="Komplexwertige Funktion">komplexwertigen Funktionen</a> lässt sich dies schreiben als <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+iy=r\exp(i\varphi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo>=</mo> <mi>r</mi> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+iy=r\exp(i\varphi ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daa13724fe6977f080acc6a621e8102b6603f3f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.993ex; height:2.843ex;" alt="{\displaystyle x+iy=r\exp(i\varphi ).}"></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Umrechnung_von_kartesischen_Koordinaten_in_Polarkoordinaten">Umrechnung von kartesischen Koordinaten in Polarkoordinaten</h4>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=6" title="Abschnitt bearbeiten: Umrechnung von kartesischen Koordinaten in Polarkoordinaten" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Umrechnung von kartesischen Koordinaten in Polarkoordinaten">Quelltext bearbeiten</a>]</div> Die Umrechnung von kartesischen Koordinaten in Polarkoordinaten ist etwas schwieriger, weil man mathematisch gesehen dabei immer auf eine (nicht den gesamten Wertebereich des Vollwinkels umfassende) <a href="/wiki/Trigonometrische_Funktion#Umkehrung" title="Trigonometrische Funktion">trigonometrische Umkehrfunktion</a> angewiesen ist. Zunächst kann aber der Radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> mit dem <a href="/wiki/Satz_des_Pythagoras" title="Satz des Pythagoras">Satz des Pythagoras</a> einfach wie folgt berechnet werden: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r={\sqrt {x^{2}+y^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\sqrt {x^{2}+y^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66fcac81cfac010069078ce8c999bd09f285567f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:13.91ex; height:4.843ex;" alt="{\displaystyle r={\sqrt {x^{2}+y^{2}}}}"></dd></dl> Bei der Bestimmung des Winkels <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> müssen zwei Besonderheiten der Polarkoordinaten berücksichtigt werden: <ul><li>Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/894a83e863728b4ee2e12f3a999a09f5f2bf1c89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=0}"> ist der Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> nicht eindeutig bestimmt, sondern könnte jeden beliebigen reellen Wert annehmen. Für eine eindeutige Transformationsvorschrift wird er häufig zu 0 definiert. Die nachfolgenden Formeln sind zur Vereinfachung ihrer Herleitung und Darstellung unter der Voraussetzung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/034cc599221cc81da7ebd4c9090e1a988809b475" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.31ex; height:2.676ex;" alt="{\displaystyle r\neq 0}"> angegeben.</li> <li>Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/034cc599221cc81da7ebd4c9090e1a988809b475" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.31ex; height:2.676ex;" alt="{\displaystyle r\neq 0}"> ist der Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> nur bis auf ganzzahlige Vielfache von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"> bestimmt, da die Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi +2\pi k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi +2\pi k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01194f09d7fe1724d5c803053501067c8af2deb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.066ex; height:2.676ex;" alt="{\displaystyle \varphi +2\pi k}"> (für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59a12237af5f2ec5fc7c5023f439266bae1380f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {Z} }">) den gleichen Punkt beschreiben. Zum Zwecke einer einfachen und eindeutigen Transformationsvorschrift wird der Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> auf ein halboffenes <a href="/wiki/Intervall_(Mathematik)" title="Intervall (Mathematik)">Intervall</a> der Länge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"> beschränkt. Üblicherweise werden dazu je nach Anwendungsgebiet die Intervalle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\pi ,\pi ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\pi ,\pi ]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fbb1843079a9df3d3bbcce3249bb2599790de9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.058ex; height:2.843ex;" alt="{\displaystyle (-\pi ,\pi ]}"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,2\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,2\pi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ec72cfde732f42822df3cbbe175b7465887eb80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.242ex; height:2.843ex;" alt="{\displaystyle [0,2\pi )}"> gewählt.</li></ul> Für die Berechnung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> kann jede der Gleichungen <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \varphi ={\frac {x}{r}};\quad \sin \varphi ={\frac {y}{r}};\quad \tan \varphi ={\frac {y}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>r</mi> </mfrac> </mrow> <mo>;</mo> <mspace width="1em" /> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>r</mi> </mfrac> </mrow> <mo>;</mo> <mspace width="1em" /> <mi>tan</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \varphi ={\frac {x}{r}};\quad \sin \varphi ={\frac {y}{r}};\quad \tan \varphi ={\frac {y}{x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d4041331c2ce772e8513461447afa11a0301072" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.38ex; height:4.843ex;" alt="{\displaystyle \cos \varphi ={\frac {x}{r}};\quad \sin \varphi ={\frac {y}{r}};\quad \tan \varphi ={\frac {y}{x}}}"></dd></dl> benutzt werden. Allerdings ist der Winkel dadurch nicht eindeutig bestimmt, auch nicht im Intervall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\pi ,\pi ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\pi ,\pi ]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fbb1843079a9df3d3bbcce3249bb2599790de9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.058ex; height:2.843ex;" alt="{\displaystyle (-\pi ,\pi ]}"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,2\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,2\pi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ec72cfde732f42822df3cbbe175b7465887eb80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.242ex; height:2.843ex;" alt="{\displaystyle [0,2\pi )}">, weil keine der drei Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee55beec18afd710e7ab767964b915b020c65093" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.856ex; height:2.176ex;" alt="{\displaystyle \sin }">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e473a3de151d75296f141f9f482fe59d582a7509" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.111ex; height:1.676ex;" alt="{\displaystyle \cos }"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b57c91704d9ab0366a6436869e2968491efc5155" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.36ex; height:2.009ex;" alt="{\displaystyle \tan }"> in diesen Intervallen <a href="/wiki/Injektive_Funktion" title="Injektive Funktion">injektiv</a> ist. Die letzte Gleichung ist außerdem für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"> nicht definiert. Deshalb ist eine Fallunterscheidung nötig, die davon abhängt, in welchem Quadranten sich der Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"> befindet, das heißt von den Vorzeichen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}">. Mit komplexen Zahlen und <a href="/wiki/Komplexwertige_Funktion" title="Komplexwertige Funktion">komplexwertigen Funktionen</a> lässt sich die Transformation schreiben als <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(r)+i\varphi =\ln(x+iy).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>φ</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(r)+i\varphi =\ln(x+iy).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0d95b0bb9946d7923e05b3d24f0d0583d54787c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.582ex; height:2.843ex;" alt="{\displaystyle \ln(r)+i\varphi =\ln(x+iy).}"></dd></dl> <div class="mw-heading mw-heading5"><h5 id="Berechnung_des_Winkels_im_Intervall_(−π,_π]_bzw._(−180°,180°]">Berechnung des Winkels im Intervall (−π, π] bzw. (−180°,180°]</h5>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=7" title="Abschnitt bearbeiten: Berechnung des Winkels im Intervall (−π, π] bzw. (−180°,180°]" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Berechnung des Winkels im Intervall (−π, π] bzw. (−180°,180°]">Quelltext bearbeiten</a>]</div> Mit Hilfe des <a href="/wiki/Arkustangens" class="mw-redirect" title="Arkustangens">Arkustangens</a> kann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> wie folgt im Intervall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\pi ,\pi ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\pi ,\pi ]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fbb1843079a9df3d3bbcce3249bb2599790de9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.058ex; height:2.843ex;" alt="{\displaystyle (-\pi ,\pi ]}"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-180^{\circ },180^{\circ }]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>,</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-180^{\circ },180^{\circ }]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52564e11cb8ecd24997d7929ef3ff9a87052b335" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.477ex; height:2.843ex;" alt="{\displaystyle (-180^{\circ },180^{\circ }]}"> bestimmt werden: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ={\begin{cases}\arctan {\frac {y}{x}}&\mathrm {f{\ddot {u}}r} \ x>0\\\left(\arctan {\frac {y}{x}}\right)+\pi &\mathrm {f{\ddot {u}}r} \ x<0,\ y\geq 0\\\left(\arctan {\frac {y}{x}}\right)-\pi &\mathrm {f{\ddot {u}}r} \ x<0,\ y<0\\+\pi /2&\mathrm {f{\ddot {u}}r} \ x=0,\ y>0\\-\pi /2&\mathrm {f{\ddot {u}}r} \ x=0,\ y<0\\\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>x</mi> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">u</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mi>x</mi> <mo>></mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>(</mo> <mrow> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>x</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>π</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">u</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mi>x</mi> <mo><</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mi>y</mi> <mo>≥</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>(</mo> <mrow> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>x</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>π</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">u</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mi>x</mi> <mo><</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mi>y</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">u</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mi>y</mi> <mo>></mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">u</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mi>y</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ={\begin{cases}\arctan {\frac {y}{x}}&\mathrm {f{\ddot {u}}r} \ x>0\\\left(\arctan {\frac {y}{x}}\right)+\pi &\mathrm {f{\ddot {u}}r} \ x<0,\ y\geq 0\\\left(\arctan {\frac {y}{x}}\right)-\pi &\mathrm {f{\ddot {u}}r} \ x<0,\ y<0\\+\pi /2&\mathrm {f{\ddot {u}}r} \ x=0,\ y>0\\-\pi /2&\mathrm {f{\ddot {u}}r} \ x=0,\ y<0\\\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2003113a6e30d58f149e4d59bb126f5ad9971c89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.17ex; margin-bottom: -0.334ex; width:40.962ex; height:16.009ex;" alt="{\displaystyle \varphi ={\begin{cases}\arctan {\frac {y}{x}}&\mathrm {f{\ddot {u}}r} \ x>0\\\left(\arctan {\frac {y}{x}}\right)+\pi &\mathrm {f{\ddot {u}}r} \ x<0,\ y\geq 0\\\left(\arctan {\frac {y}{x}}\right)-\pi &\mathrm {f{\ddot {u}}r} \ x<0,\ y<0\\+\pi /2&\mathrm {f{\ddot {u}}r} \ x=0,\ y>0\\-\pi /2&\mathrm {f{\ddot {u}}r} \ x=0,\ y<0\\\end{cases}}}"></dd></dl> Einige <a href="/wiki/Programmiersprache" title="Programmiersprache">Programmiersprachen</a> (so zuerst <a href="/wiki/Fortran" title="Fortran">Fortran 77</a>) und Anwendungsprogramme (etwa <a href="/wiki/Microsoft_Excel" title="Microsoft Excel">Microsoft Excel</a>) bieten eine Arkustangens-Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arctan2} (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan2</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arctan2} (x,y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e476a05274fd52b59be5106d852d765eb4192fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.957ex; height:2.843ex;" alt="{\displaystyle \operatorname {arctan2} (x,y)}"> mit <a href="/wiki/Arctan2" title="Arctan2">zwei Argumenten</a> an, welche die dargestellten Fallunterscheidungen intern berücksichtigt und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> für beliebige Werte von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> berechnet. Zum selben Ergebnis kommt man, wenn man den Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"> in der kartesischen Ebene als <a href="/wiki/Komplexe_Zahl#Darstellung_von_komplexen_Zahlen_in_der_komplexen_Zahlenebene" title="Komplexe Zahl">komplexe Zahl</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=x+\mathrm {i} y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=x+\mathrm {i} y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f034e7d7b0ff90492fb520f9af4687b508f8e9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.159ex; height:2.509ex;" alt="{\displaystyle z=x+\mathrm {i} y}"> auffasst und den Winkel <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =\arg(z)=\Im (\ln z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mi>arg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">ℑ</mi> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =\arg(z)=\Im (\ln z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e0b3b790cf41b820116b5d3f90a9911022b3111" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.363ex; height:2.843ex;" alt="{\displaystyle \varphi =\arg(z)=\Im (\ln z)}"></dd></dl> mittels der Argument-Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arg }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arg</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arg }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec03a9c123925f400a40064ca491d268f9312956" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.237ex; height:2.009ex;" alt="{\displaystyle \arg }"> berechnet oder den Imaginärteil des Logarithmus von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> nimmt. Mit Hilfe des <a href="/wiki/Arkuskosinus" class="mw-redirect" title="Arkuskosinus">Arkuskosinus</a> kommt man mit nur zwei Fallunterscheidungen aus: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ={\begin{cases}+\arccos {\frac {x}{r}}&\mathrm {f{\ddot {u}}r} \ y\geq 0\\-\arccos {\frac {x}{r}}&\mathrm {f{\ddot {u}}r} \ y<0\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>+</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>r</mi> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">u</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mi>y</mi> <mo>≥</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>r</mi> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">u</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mi>y</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ={\begin{cases}+\arccos {\frac {x}{r}}&\mathrm {f{\ddot {u}}r} \ y\geq 0\\-\arccos {\frac {x}{r}}&\mathrm {f{\ddot {u}}r} \ y<0\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64da1cee9ca86b7fa578eb4ba31bbd88c96b0544" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.078ex; height:6.176ex;" alt="{\displaystyle \varphi ={\begin{cases}+\arccos {\frac {x}{r}}&\mathrm {f{\ddot {u}}r} \ y\geq 0\\-\arccos {\frac {x}{r}}&\mathrm {f{\ddot {u}}r} \ y<0\end{cases}}}"></dd></dl> Durch Ausnutzen der Tatsache, dass in einem <a href="/wiki/Kreis_(Geometrie)" class="mw-redirect" title="Kreis (Geometrie)">Kreis</a> ein <a href="/wiki/Mittelpunktswinkel" class="mw-redirect" title="Mittelpunktswinkel">Mittelpunktswinkel</a> stets doppelt so groß ist wie der zugehörige <a href="/wiki/Umfangswinkel" class="mw-redirect" title="Umfangswinkel">Umfangswinkel</a>, kann das Argument <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> auch mit Hilfe der Arkustangens-Funktion mit weniger Fallunterscheidungen berechnet werden: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ={\begin{cases}2\arctan {\frac {y}{r+x}}&\mathrm {f{\ddot {u}}r} \ r+x\neq 0\\\pi &\mathrm {f{\ddot {u}}r} \ r+x=0\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>2</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mrow> <mi>r</mi> <mo>+</mo> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">u</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mi>r</mi> <mo>+</mo> <mi>x</mi> <mo>≠</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>π</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">u</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mi>r</mi> <mo>+</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ={\begin{cases}2\arctan {\frac {y}{r+x}}&\mathrm {f{\ddot {u}}r} \ r+x\neq 0\\\pi &\mathrm {f{\ddot {u}}r} \ r+x=0\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/314eb4cebeeb63a88cbf735588e9ce6635e59d0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.765ex; height:6.509ex;" alt="{\displaystyle \varphi ={\begin{cases}2\arctan {\frac {y}{r+x}}&\mathrm {f{\ddot {u}}r} \ r+x\neq 0\\\pi &\mathrm {f{\ddot {u}}r} \ r+x=0\end{cases}}}"></dd></dl> <div class="mw-heading mw-heading5"><h5 id="Berechnung_des_Winkels_im_Intervall_[0,_2π)_bzw._[0,_360°)">Berechnung des Winkels im Intervall [0, 2π) bzw. [0, 360°)</h5>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=8" title="Abschnitt bearbeiten: Berechnung des Winkels im Intervall [0, 2π) bzw. [0, 360°)" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Berechnung des Winkels im Intervall [0, 2π) bzw. [0, 360°)">Quelltext bearbeiten</a>]</div> Die Berechnung des Winkels <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>φ</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi '}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41699a838e24ad24212d37ff18eabb15a7765cd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.205ex; height:3.009ex;" alt="{\displaystyle \varphi '}"> im Intervall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,2\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,2\pi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ec72cfde732f42822df3cbbe175b7465887eb80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.242ex; height:2.843ex;" alt="{\displaystyle [0,2\pi )}"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0^{\circ },360^{\circ })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>,</mo> <msup> <mn>360</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0^{\circ },360^{\circ })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0df35806bdeebb21aaa709288550ef0334081f10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.344ex; height:2.843ex;" alt="{\displaystyle [0^{\circ },360^{\circ })}"> kann im Prinzip so durchgeführt werden, dass der Winkel zunächst wie vorstehend beschrieben im Intervall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\pi ,\pi ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\pi ,\pi ]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fbb1843079a9df3d3bbcce3249bb2599790de9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.058ex; height:2.843ex;" alt="{\displaystyle (-\pi ,\pi ]}"> berechnet wird und, nur falls er negativ ist, noch um <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"> vergrößert wird: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi '={\begin{cases}\varphi +2\pi &\mathrm {falls} \ \varphi <0\\\varphi &\mathrm {sonst} \end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>φ</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>φ</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">s</mi> </mrow> <mtext> </mtext> <mi>φ</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>φ</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi '={\begin{cases}\varphi +2\pi &\mathrm {falls} \ \varphi <0\\\varphi &\mathrm {sonst} \end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b6adfd5fa8674afc079a372f03a1e96a511b729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.575ex; height:6.176ex;" alt="{\displaystyle \varphi '={\begin{cases}\varphi +2\pi &\mathrm {falls} \ \varphi <0\\\varphi &\mathrm {sonst} \end{cases}}}"></dd></dl> Durch Abwandlung der ersten obenstehenden Formel kann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>φ</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi '}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41699a838e24ad24212d37ff18eabb15a7765cd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.205ex; height:3.009ex;" alt="{\displaystyle \varphi '}"> wie folgt direkt im Intervall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,2\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,2\pi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ec72cfde732f42822df3cbbe175b7465887eb80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.242ex; height:2.843ex;" alt="{\displaystyle [0,2\pi )}"> bestimmt werden: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi '={\begin{cases}\arctan {\frac {y}{x}}&\mathrm {f{\ddot {u}}r} \ x>0,\ y\geq 0\\\arctan {\frac {y}{x}}+2\pi &\mathrm {f{\ddot {u}}r} \ x>0,\ y<0\\\arctan {\frac {y}{x}}+\pi &\mathrm {f{\ddot {u}}r} \ x<0\\\pi /2&\mathrm {f{\ddot {u}}r} \ x=0,\ y>0\\3\pi /2&\mathrm {f{\ddot {u}}r} \ x=0,\ y<0\\\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>φ</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>x</mi> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">u</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mi>x</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mi>y</mi> <mo>≥</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>x</mi> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mi>π</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">u</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mi>x</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mi>y</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>x</mi> </mfrac> </mrow> <mo>+</mo> <mi>π</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">u</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mi>x</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">u</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mi>y</mi> <mo>></mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">u</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mi>y</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi '={\begin{cases}\arctan {\frac {y}{x}}&\mathrm {f{\ddot {u}}r} \ x>0,\ y\geq 0\\\arctan {\frac {y}{x}}+2\pi &\mathrm {f{\ddot {u}}r} \ x>0,\ y<0\\\arctan {\frac {y}{x}}+\pi &\mathrm {f{\ddot {u}}r} \ x<0\\\pi /2&\mathrm {f{\ddot {u}}r} \ x=0,\ y>0\\3\pi /2&\mathrm {f{\ddot {u}}r} \ x=0,\ y<0\\\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1103544cb922ab7399f49cd1ff2ac65a830e34df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.17ex; margin-bottom: -0.334ex; width:40.679ex; height:16.009ex;" alt="{\displaystyle \varphi '={\begin{cases}\arctan {\frac {y}{x}}&\mathrm {f{\ddot {u}}r} \ x>0,\ y\geq 0\\\arctan {\frac {y}{x}}+2\pi &\mathrm {f{\ddot {u}}r} \ x>0,\ y<0\\\arctan {\frac {y}{x}}+\pi &\mathrm {f{\ddot {u}}r} \ x<0\\\pi /2&\mathrm {f{\ddot {u}}r} \ x=0,\ y>0\\3\pi /2&\mathrm {f{\ddot {u}}r} \ x=0,\ y<0\\\end{cases}}}"></dd></dl> Die Formel mit dem Arkuskosinus kommt auch in diesem Fall mit nur zwei Fallunterscheidungen aus: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi '={\begin{cases}\arccos {\frac {x}{r}}&\mathrm {f{\ddot {u}}r} \ y\geq 0\\2\pi -\arccos {\frac {x}{r}}&\mathrm {f{\ddot {u}}r} \ y<0\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>φ</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>arccos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>r</mi> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">u</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mi>y</mi> <mo>≥</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>π</mi> <mo>−</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>r</mi> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">u</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mi>y</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi '={\begin{cases}\arccos {\frac {x}{r}}&\mathrm {f{\ddot {u}}r} \ y\geq 0\\2\pi -\arccos {\frac {x}{r}}&\mathrm {f{\ddot {u}}r} \ y<0\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b04edf391e39829b26cc0231b412d5f4235ffc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.903ex; height:6.176ex;" alt="{\displaystyle \varphi '={\begin{cases}\arccos {\frac {x}{r}}&\mathrm {f{\ddot {u}}r} \ y\geq 0\\2\pi -\arccos {\frac {x}{r}}&\mathrm {f{\ddot {u}}r} \ y<0\end{cases}}}"></dd></dl> <div class="mw-heading mw-heading5"><h5 id="Verschiebung_des_Winkels">Verschiebung des Winkels</h5>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=9" title="Abschnitt bearbeiten: Verschiebung des Winkels" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Verschiebung des Winkels">Quelltext bearbeiten</a>]</div> Bei geodätischen oder anderen Berechnungen können sich Azimute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> mit Werten außerhalb des üblichen Intervalls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\text{min}}\leq \varphi <\varphi _{\text{min}}+2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>min</mtext> </mrow> </msub> <mo>≤</mo> <mi>φ</mi> <mo><</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>min</mtext> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{\text{min}}\leq \varphi <\varphi _{\text{min}}+2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e84b35e9b0e42ba3953e071b95337ff6072460" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.037ex; height:2.676ex;" alt="{\displaystyle \varphi _{\text{min}}\leq \varphi <\varphi _{\text{min}}+2\pi }"> mit der unteren Grenze <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\text{min}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>min</mtext> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{\text{min}}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7573a50560753fae254fb4629adeb6831a6a6e0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.754ex; height:2.676ex;" alt="{\displaystyle \varphi _{\text{min}}=0}"> (oder auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\text{min}}=-\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>min</mtext> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{\text{min}}=-\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86d3664f84855a59e444f35f7541289d9bc28969" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.731ex; height:2.509ex;" alt="{\displaystyle \varphi _{\text{min}}=-\pi }">) ergeben. Die Gleichung <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi =\varphi -2\pi \cdot {\bigl \lfloor }{\frac {\varphi -\varphi _{\text{min}}}{2\pi }}{\bigr \rfloor }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mi>φ</mi> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">⌊</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo>−</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>min</mtext> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">⌋</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi =\varphi -2\pi \cdot {\bigl \lfloor }{\frac {\varphi -\varphi _{\text{min}}}{2\pi }}{\bigr \rfloor }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2652ca2b7a1e5a9537990609cca079c9ca89f055" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.902ex; height:5.176ex;" alt="{\displaystyle \phi =\varphi -2\pi \cdot {\bigl \lfloor }{\frac {\varphi -\varphi _{\text{min}}}{2\pi }}{\bigr \rfloor }}"></dd></dl> verschiebt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> in das gewünschte Intervall, sodass also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \in \left[\varphi _{\text{min}},\,\varphi _{\text{min}}+2\pi \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>min</mtext> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>min</mtext> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mi>π</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi \in \left[\varphi _{\text{min}},\,\varphi _{\text{min}}+2\pi \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9c312fe020b1eaa79e2dcbe1d626c1a2380a3ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.519ex; height:2.843ex;" alt="{\displaystyle \phi \in \left[\varphi _{\text{min}},\,\varphi _{\text{min}}+2\pi \right)}"> gilt. Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto \lfloor x\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto \lfloor x\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f3b60078378682c77f591f9e387cbea7151dbe8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.338ex; height:2.843ex;" alt="{\displaystyle x\mapsto \lfloor x\rfloor }"> die zur nächsten Ganzzahl abrundende <a href="/wiki/Floor-Funktion" class="mw-redirect" title="Floor-Funktion">Floor-Funktion</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor x\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor x\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738c94c88678dd08a289f90a47a609ce44eedf14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.394ex; height:2.843ex;" alt="{\displaystyle \lfloor x\rfloor }"> also für jedes reelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> die größte ganze Zahl, die nicht größer als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> ist. <div class="mw-heading mw-heading3"><h3 id="Koordinatenlinien">Koordinatenlinien</h3>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=10" title="Abschnitt bearbeiten: Koordinatenlinien" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Koordinatenlinien">Quelltext bearbeiten</a>]</div> Die beiden <a href="/wiki/Koordinatenlinie" title="Koordinatenlinie">Koordinatenlinien</a> durch den Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r_{0}\mid \varphi _{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∣</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r_{0}\mid \varphi _{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a5c73165f6a4577da2a4bb7ad7380d3e858621f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.424ex; height:2.843ex;" alt="{\displaystyle (r_{0}\mid \varphi _{0})}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{0}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{0}\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d7071cab3295d8920e5ef6c6139e638f51124fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.364ex; height:2.676ex;" alt="{\displaystyle r_{0}\neq 0}"> sind die <a href="/wiki/Kurve_(Mathematik)" title="Kurve (Mathematik)">Kurven</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {k}}_{1}(r)={\begin{pmatrix}r\cos \varphi _{0}\\r\sin \varphi _{0}\end{pmatrix}},r\in [0,\infty [\quad und\quad {\vec {k}}_{2}(\varphi )={\begin{pmatrix}r_{0}\cos \varphi \\r_{0}\sin \varphi \end{pmatrix}},\varphi \in [0,2\pi [}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mi>r</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">[</mo> <mspace width="1em" /> <mi>u</mi> <mi>n</mi> <mi>d</mi> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mi>φ</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">[</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {k}}_{1}(r)={\begin{pmatrix}r\cos \varphi _{0}\\r\sin \varphi _{0}\end{pmatrix}},r\in [0,\infty [\quad und\quad {\vec {k}}_{2}(\varphi )={\begin{pmatrix}r_{0}\cos \varphi \\r_{0}\sin \varphi \end{pmatrix}},\varphi \in [0,2\pi [}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06ac39216d3fbbf35dcb058ff6008606f46cd18f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:70.979ex; height:6.176ex;" alt="{\displaystyle {\vec {k}}_{1}(r)={\begin{pmatrix}r\cos \varphi _{0}\\r\sin \varphi _{0}\end{pmatrix}},r\in [0,\infty [\quad und\quad {\vec {k}}_{2}(\varphi )={\begin{pmatrix}r_{0}\cos \varphi \\r_{0}\sin \varphi \end{pmatrix}},\varphi \in [0,2\pi [}">,</dd></dl> also eine <a href="/wiki/Strahl_(Geometrie)" title="Strahl (Geometrie)">Halbgerade</a>, die im Koordinatenursprung beginnt, sowie ein Kreis mit dem Radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb12fcfddb65e3d1e6a044215f6e833f0cd4337b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{0}}"> und dem Koordinatenursprung als Mittelpunkt. <div class="mw-heading mw-heading3"><h3 id="Lokale_Basisvektoren_und_Orthogonalität">Lokale Basisvektoren und Orthogonalität</h3>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=11" title="Abschnitt bearbeiten: Lokale Basisvektoren und Orthogonalität" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: Lokale Basisvektoren und Orthogonalität">Quelltext bearbeiten</a>]</div> In geradlinigen Koordinatensystemen gibt es eine Basis für den gesamten <a href="/wiki/Vektorraum" title="Vektorraum">Vektorraum</a>, in krummlinigen muss an jedem Punkt eine lokale Basis berechnet werden. Die lokalen Basisvektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\vec {b}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\vec {b}}_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2db0611cc9cff68458e90d4e2d857bf0cfa40772" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.148ex; height:3.176ex;" alt="{\displaystyle \textstyle {\vec {b}}_{1}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\vec {b}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\vec {b}}_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/744ea3904fd207c10ad23e1c1b686a689b690874" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.148ex; height:3.176ex;" alt="{\displaystyle \textstyle {\vec {b}}_{2}}"> an einem Punkt sind Tangentenvektoren an die Koordinatenlinien und ergeben sich aus den Kurvengleichungen durch Ableitung nach dem Kurvenparameter. Zum selben Ergebnis gelangt man auch durch partielle Ableitung der Koordinatentransformation für den Ortsvektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aec3c9ce13b53e9e24c98e7cce4212627884c91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}"> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}={\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}r\cos \varphi \\r\sin \varphi \end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}={\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}r\cos \varphi \\r\sin \varphi \end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93bfb832c5bebcd7939e3ee36fd0027f084ab1d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.549ex; height:6.176ex;" alt="{\displaystyle {\vec {r}}={\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}r\cos \varphi \\r\sin \varphi \end{pmatrix}}}"></dd></dl> nach den Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}_{1}={\frac {\partial {\vec {r}}}{\partial r}}={\begin{pmatrix}\cos \varphi \\\sin \varphi \end{pmatrix}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}_{1}={\frac {\partial {\vec {r}}}{\partial r}}={\begin{pmatrix}\cos \varphi \\\sin \varphi \end{pmatrix}}\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bfc430c7abd3bee3fa25f93fd78b29871b05b39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.236ex; height:6.176ex;" alt="{\displaystyle {\vec {b}}_{1}={\frac {\partial {\vec {r}}}{\partial r}}={\begin{pmatrix}\cos \varphi \\\sin \varphi \end{pmatrix}}\quad }"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad {\vec {b}}_{2}={\frac {\partial {\vec {r}}}{\partial \varphi }}={\begin{pmatrix}-r\sin \varphi \\r\cos \varphi \end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad {\vec {b}}_{2}={\frac {\partial {\vec {r}}}{\partial \varphi }}={\begin{pmatrix}-r\sin \varphi \\r\cos \varphi \end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a600cfeafb8f24677740aec0e5139985a6082c09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.521ex; height:6.176ex;" alt="{\displaystyle \quad {\vec {b}}_{2}={\frac {\partial {\vec {r}}}{\partial \varphi }}={\begin{pmatrix}-r\sin \varphi \\r\cos \varphi \end{pmatrix}}}">.</dd></dl> Die Basisvektoren haben die Längen <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {b}}_{1}|={\sqrt {{\vec {b}}_{1}{\vec {b}}_{1}}}=1\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </msqrt> </mrow> <mo>=</mo> <mn>1</mn> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\vec {b}}_{1}|={\sqrt {{\vec {b}}_{1}{\vec {b}}_{1}}}=1\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c98216bded77911cea9187533a0a8e8be49ddf5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.744ex; height:4.676ex;" alt="{\displaystyle |{\vec {b}}_{1}|={\sqrt {{\vec {b}}_{1}{\vec {b}}_{1}}}=1\quad }"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad |{\vec {b}}_{2}|={\sqrt {{\vec {b}}_{2}{\vec {b}}_{2}}}=r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </msqrt> </mrow> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad |{\vec {b}}_{2}|={\sqrt {{\vec {b}}_{2}{\vec {b}}_{2}}}=r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e3d70f30a7162e03215faa28d9220c4304d83fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.63ex; height:4.676ex;" alt="{\displaystyle \quad |{\vec {b}}_{2}|={\sqrt {{\vec {b}}_{2}{\vec {b}}_{2}}}=r}"></dd></dl> und sind zueinander orthogonal, denn es gilt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}_{1}{\vec {b}}_{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}_{1}{\vec {b}}_{2}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f51c4a4677ce15e5163110e8cfdf5ed6ed5f2b10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.557ex; height:3.176ex;" alt="{\displaystyle {\vec {b}}_{1}{\vec {b}}_{2}=0}">.</dd></dl> Die entsprechenden Koordinatenlinien schneiden sich also rechtwinklig, die Polarkoordinaten bilden somit ein <a href="/wiki/Orthogonale_Koordinaten" title="Orthogonale Koordinaten">orthogonales Koordinatensystem</a>. In der <a href="/wiki/Tensor" title="Tensor">Tensorrechnung</a> werden die lokalen Basisvektoren, die tangential zu den Koordinatenlinien verlaufen, wegen ihres Verhaltens bei <a href="/wiki/Koordinatentransformation" title="Koordinatentransformation">Koordinatentransformationen</a> als <a href="/wiki/Tensor#Ko-_und_Kontravarianz_von_Vektoren" title="Tensor">kovariant</a> bezeichnet. <div class="mw-heading mw-heading3"><h3 id="Metrischer_Tensor">Metrischer Tensor</h3>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=12" title="Abschnitt bearbeiten: Metrischer Tensor" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=12" title="Quellcode des Abschnitts bearbeiten: Metrischer Tensor">Quelltext bearbeiten</a>]</div> Die Komponenten des kovarianten <a href="/wiki/Metrischer_Tensor" title="Metrischer Tensor">metrischen Tensors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=(g_{ij})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=(g_{ij})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e36129d1e65f26b765daad53fc02c24f1a275aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.61ex; height:3.009ex;" alt="{\displaystyle g=(g_{ij})}"> sind die Skalarprodukte der kovarianten lokalen Basisvektoren: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{ij}={\vec {b}}_{i}{\vec {b}}_{j}\quad (i,j\in \{1,2\})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{ij}={\vec {b}}_{i}{\vec {b}}_{j}\quad (i,j\in \{1,2\})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c966f2ec22ae66c770b460f17ae7d01b721580cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.033ex; height:3.509ex;" alt="{\displaystyle g_{ij}={\vec {b}}_{i}{\vec {b}}_{j}\quad (i,j\in \{1,2\})}">.</dd></dl> Nach den Rechnungen im vorigen Abschnitt ist damit <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g={\begin{pmatrix}1&0\\0&r^{2}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g={\begin{pmatrix}1&0\\0&r^{2}\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f07a35cac7ab9fd531042bcb5365032291cbe64b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.975ex; height:6.176ex;" alt="{\displaystyle g={\begin{pmatrix}1&0\\0&r^{2}\end{pmatrix}}}">.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Funktionaldeterminante">Funktionaldeterminante</h3>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=13" title="Abschnitt bearbeiten: Funktionaldeterminante" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=13" title="Quellcode des Abschnitts bearbeiten: Funktionaldeterminante">Quelltext bearbeiten</a>]</div> Aus den Umrechnungsformeln von Polarkoordinaten in kartesische Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=r\cos \varphi ,\,y=r\sin \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>y</mi> <mo>=</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=r\cos \varphi ,\,y=r\sin \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90615acc5063bc3e40fe72dd13c6711ecc14cc64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.756ex; height:2.676ex;" alt="{\displaystyle x=r\cos \varphi ,\,y=r\sin \varphi }"> erhält man für die <a href="/wiki/Funktionaldeterminante" title="Funktionaldeterminante">Funktionaldeterminante</a> als <a href="/wiki/Determinante_(Mathematik)" class="mw-redirect" title="Determinante (Mathematik)">Determinante</a> der <a href="/wiki/Jacobi-Matrix" title="Jacobi-Matrix">Jacobi-Matrix</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det J=\det {\frac {\partial (x,y)}{\partial (r,\varphi )}}={\begin{vmatrix}{\frac {\partial x}{\partial r}}&{\frac {\partial x}{\partial \varphi }}\\{\frac {\partial y}{\partial r}}&{\frac {\partial y}{\partial \varphi }}\end{vmatrix}}={\begin{vmatrix}\cos \varphi &-r\sin \varphi \\\sin \varphi &r\cos \varphi \end{vmatrix}}=r\cos ^{2}\varphi +r\sin ^{2}\varphi =r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mi>J</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> <mtd> <mo>−</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> <mtd> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mi>r</mi> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>r</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det J=\det {\frac {\partial (x,y)}{\partial (r,\varphi )}}={\begin{vmatrix}{\frac {\partial x}{\partial r}}&{\frac {\partial x}{\partial \varphi }}\\{\frac {\partial y}{\partial r}}&{\frac {\partial y}{\partial \varphi }}\end{vmatrix}}={\begin{vmatrix}\cos \varphi &-r\sin \varphi \\\sin \varphi &r\cos \varphi \end{vmatrix}}=r\cos ^{2}\varphi +r\sin ^{2}\varphi =r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0081b2012454c348d20d8f8a811a2ed0d46e8a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:77.725ex; height:9.176ex;" alt="{\displaystyle \det J=\det {\frac {\partial (x,y)}{\partial (r,\varphi )}}={\begin{vmatrix}{\frac {\partial x}{\partial r}}&{\frac {\partial x}{\partial \varphi }}\\{\frac {\partial y}{\partial r}}&{\frac {\partial y}{\partial \varphi }}\end{vmatrix}}={\begin{vmatrix}\cos \varphi &-r\sin \varphi \\\sin \varphi &r\cos \varphi \end{vmatrix}}=r\cos ^{2}\varphi +r\sin ^{2}\varphi =r}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Flächenelement">Flächenelement</h3>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=14" title="Abschnitt bearbeiten: Flächenelement" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=14" title="Quellcode des Abschnitts bearbeiten: Flächenelement">Quelltext bearbeiten</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Fl%C3%A4chenelement_Polarkoordinaten.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Fl%C3%A4chenelement_Polarkoordinaten.svg/220px-Fl%C3%A4chenelement_Polarkoordinaten.svg.png" decoding="async" width="220" height="239" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Fl%C3%A4chenelement_Polarkoordinaten.svg/330px-Fl%C3%A4chenelement_Polarkoordinaten.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Fl%C3%A4chenelement_Polarkoordinaten.svg/440px-Fl%C3%A4chenelement_Polarkoordinaten.svg.png 2x" data-file-width="106" data-file-height="115" /></a><figcaption> Flächenelement der Breite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\cdot \mathrm {d} \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\cdot \mathrm {d} \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa3e0ed47c25d67215f548e1211bd0c952ce0ea9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.54ex; height:2.676ex;" alt="{\displaystyle r\cdot \mathrm {d} \varphi }"> und der Höhe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6061c91ecd3f8919bce745870bb757249fe26bc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.341ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} r}"> in Polarkoordinaten</figcaption></figure> Mit der Funktionaldeterminante ergibt sich für das Flächenelement in Polarkoordinaten: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} A=\mathrm {d} x\,\mathrm {d} y=|J|\,\mathrm {d} r\,\mathrm {d} \varphi =r\,\mathrm {d} r\,\mathrm {d} \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>r</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>φ</mi> <mo>=</mo> <mi>r</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>r</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} A=\mathrm {d} x\,\mathrm {d} y=|J|\,\mathrm {d} r\,\mathrm {d} \varphi =r\,\mathrm {d} r\,\mathrm {d} \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/533e020801aa71fa17d6f708bc978f5abd2c00ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.458ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} A=\mathrm {d} x\,\mathrm {d} y=|J|\,\mathrm {d} r\,\mathrm {d} \varphi =r\,\mathrm {d} r\,\mathrm {d} \varphi }"></dd></dl> Wie das nebenstehende Bild zeigt, lässt sich das Flächenelement als ein differentielles Rechteck mit der Breite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\cdot \mathrm {d} \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\cdot \mathrm {d} \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa3e0ed47c25d67215f548e1211bd0c952ce0ea9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.54ex; height:2.676ex;" alt="{\displaystyle r\cdot \mathrm {d} \varphi }"> und der Höhe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6061c91ecd3f8919bce745870bb757249fe26bc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.341ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} r}"> interpretieren. <div class="mw-heading mw-heading3"><h3 id="Linienelement">Linienelement</h3>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=15" title="Abschnitt bearbeiten: Linienelement" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=15" title="Quellcode des Abschnitts bearbeiten: Linienelement">Quelltext bearbeiten</a>]</div> Aus den obigen Transformationsgleichungen <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=r\cos \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=r\cos \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ec84f3b8f3196698a3693a21f258e47f270302f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.882ex; height:2.176ex;" alt="{\displaystyle x=r\cos \varphi }"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=r\sin \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=r\sin \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/134779a8ff7881855e5fee80bf520c212414de56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.453ex; height:2.676ex;" alt="{\displaystyle y=r\sin \varphi }"></dd></dl> folgen <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} x=\cos \varphi \,\mathrm {d} r-r\,\sin \varphi \,\mathrm {d} \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>r</mi> <mo>−</mo> <mi>r</mi> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} x=\cos \varphi \,\mathrm {d} r-r\,\sin \varphi \,\mathrm {d} \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/550847c63db138c295457ddf492f46b95a140979" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.093ex; height:2.676ex;" alt="{\displaystyle \mathrm {d} x=\cos \varphi \,\mathrm {d} r-r\,\sin \varphi \,\mathrm {d} \varphi }"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} y=\sin \varphi \,\mathrm {d} r+r\,\cos \varphi \,\mathrm {d} \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>r</mi> <mo>+</mo> <mi>r</mi> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} y=\sin \varphi \,\mathrm {d} r+r\,\cos \varphi \,\mathrm {d} \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dcf9e5e5736a7a3079ff2645c57b17588de270d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.919ex; height:2.676ex;" alt="{\displaystyle \mathrm {d} y=\sin \varphi \,\mathrm {d} r+r\,\cos \varphi \,\mathrm {d} \varphi }"></dd></dl> Für das kartesische <a href="/wiki/Linienelement" class="mw-redirect" title="Linienelement">Linienelement</a> gilt <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} s^{2}=\mathrm {d} x^{2}+\mathrm {d} y^{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} s^{2}=\mathrm {d} x^{2}+\mathrm {d} y^{2}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f12bc531ebfb47e19fe1924f95ed5c45001f61fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.947ex; height:3.009ex;" alt="{\displaystyle \mathrm {d} s^{2}=\mathrm {d} x^{2}+\mathrm {d} y^{2}\,}"></dd></dl> wofür in Polarkoordinaten folgt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} s^{2}=\mathrm {d} r^{2}+r^{2}\,\mathrm {d} \varphi ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} s^{2}=\mathrm {d} r^{2}+r^{2}\,\mathrm {d} \varphi ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa530dc43613039ad600c77f4778d0a6c757d2d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.128ex; height:3.176ex;" alt="{\displaystyle \mathrm {d} s^{2}=\mathrm {d} r^{2}+r^{2}\,\mathrm {d} \varphi ^{2}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Ortsvektor,_Geschwindigkeit_und_Beschleunigung_in_Polarkoordinaten">Ortsvektor, Geschwindigkeit und Beschleunigung in Polarkoordinaten</h3>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=16" title="Abschnitt bearbeiten: Ortsvektor, Geschwindigkeit und Beschleunigung in Polarkoordinaten" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=16" title="Quellcode des Abschnitts bearbeiten: Ortsvektor, Geschwindigkeit und Beschleunigung in Polarkoordinaten">Quelltext bearbeiten</a>]</div> Mit den lokalen Basiseinheitsvektoren <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{r}={\begin{pmatrix}\cos \varphi \\\sin \varphi \end{pmatrix}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{r}={\begin{pmatrix}\cos \varphi \\\sin \varphi \end{pmatrix}}\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b7a63865de83671bfbf8e13ee689dcbf620a68e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.809ex; height:6.176ex;" alt="{\displaystyle {\vec {e}}_{r}={\begin{pmatrix}\cos \varphi \\\sin \varphi \end{pmatrix}}\quad }"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad {\vec {e}}_{\varphi }={\begin{pmatrix}-\sin \varphi \\\cos \varphi \end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad {\vec {e}}_{\varphi }={\begin{pmatrix}-\sin \varphi \\\cos \varphi \end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ec82886bdaf268eebeecf8e9843a5cba3ea865e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.082ex; height:6.176ex;" alt="{\displaystyle \quad {\vec {e}}_{\varphi }={\begin{pmatrix}-\sin \varphi \\\cos \varphi \end{pmatrix}}}"></dd></dl> ergibt sich für den Ortsvektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aec3c9ce13b53e9e24c98e7cce4212627884c91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}={\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}r\cos \varphi \\r\sin \varphi \end{pmatrix}}=r\,{\vec {e}}_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mi>r</mi> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}={\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}r\cos \varphi \\r\sin \varphi \end{pmatrix}}=r\,{\vec {e}}_{r}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8728b699d70736c92f990e5d5b7241fea5161c66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.28ex; height:6.176ex;" alt="{\displaystyle {\vec {r}}={\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}r\cos \varphi \\r\sin \varphi \end{pmatrix}}=r\,{\vec {e}}_{r}}">.</dd></dl> Ist der Ortsvektor abhängig von der Zeit, so müssen die Variablen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> und damit auch die lokalen Basisvektoren abgeleitet werden: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {{\vec {e}}_{r}}}={\begin{pmatrix}-\sin \varphi \\\cos \varphi \end{pmatrix}}\,{\dot {\varphi }}={\dot {\varphi }}\,{\vec {e}}_{\varphi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {{\vec {e}}_{r}}}={\begin{pmatrix}-\sin \varphi \\\cos \varphi \end{pmatrix}}\,{\dot {\varphi }}={\dot {\varphi }}\,{\vec {e}}_{\varphi }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ed69aeecf341cafd0aeb597eb25efd53993ea37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.898ex; height:6.176ex;" alt="{\displaystyle {\dot {{\vec {e}}_{r}}}={\begin{pmatrix}-\sin \varphi \\\cos \varphi \end{pmatrix}}\,{\dot {\varphi }}={\dot {\varphi }}\,{\vec {e}}_{\varphi }}">.</dd></dl> Mit der Produktregel ergibt sich somit für den Geschwindigkeitsvektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\vec {r}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\vec {r}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d0622c7d4647179011ad0bc43548fc6abd756db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.257ex; height:2.843ex;" alt="{\displaystyle {\dot {\vec {r}}}}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\vec {r}}}={\dot {r}}\,{\vec {e}}_{r}+r\,{\dot {{\vec {e}}_{r}}}={\dot {r}}\,{\vec {e}}_{r}+r\,{\dot {\varphi }}\,{\vec {e}}_{\varphi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>+</mo> <mi>r</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>+</mo> <mi>r</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\vec {r}}}={\dot {r}}\,{\vec {e}}_{r}+r\,{\dot {{\vec {e}}_{r}}}={\dot {r}}\,{\vec {e}}_{r}+r\,{\dot {\varphi }}\,{\vec {e}}_{\varphi }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04be98724028b6d4f82d0a90a2bd4ad9a2a9ea50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.407ex; height:3.509ex;" alt="{\displaystyle {\dot {\vec {r}}}={\dot {r}}\,{\vec {e}}_{r}+r\,{\dot {{\vec {e}}_{r}}}={\dot {r}}\,{\vec {e}}_{r}+r\,{\dot {\varphi }}\,{\vec {e}}_{\varphi }}">.</dd></dl> Eine entsprechende Rechnung führt für die Beschleunigung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {\vec {r}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ddot {\vec {r}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f02da80a6ce3388352e6efdec76e57ecf79f87ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.257ex; height:2.843ex;" alt="{\displaystyle {\ddot {\vec {r}}}}"> zu dem Ergebnis <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {\vec {r}}}=({\ddot {r}}-r{\dot {\varphi }}^{2})\,{\vec {e}}_{r}+(2{\dot {r}}{\dot {\varphi }}+r{\ddot {\varphi }})\,{\vec {e}}_{\varphi }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>r</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ddot {\vec {r}}}=({\ddot {r}}-r{\dot {\varphi }}^{2})\,{\vec {e}}_{r}+(2{\dot {r}}{\dot {\varphi }}+r{\ddot {\varphi }})\,{\vec {e}}_{\varphi }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a74fdca6eba29170aa4c2515e7969fdf348b3f8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.145ex; height:3.509ex;" alt="{\displaystyle {\ddot {\vec {r}}}=({\ddot {r}}-r{\dot {\varphi }}^{2})\,{\vec {e}}_{r}+(2{\dot {r}}{\dot {\varphi }}+r{\ddot {\varphi }})\,{\vec {e}}_{\varphi }.}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Räumliche_Polarkoordinaten:_Zylinder-,_Kegel-_und_Kugelkoordinaten">Räumliche Polarkoordinaten: Zylinder-, Kegel- und Kugelkoordinaten</h2>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=17" title="Abschnitt bearbeiten: Räumliche Polarkoordinaten: Zylinder-, Kegel- und Kugelkoordinaten" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=17" title="Quellcode des Abschnitts bearbeiten: Räumliche Polarkoordinaten: Zylinder-, Kegel- und Kugelkoordinaten">Quelltext bearbeiten</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Zylinderkoordinaten">Zylinderkoordinaten</h3>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=18" title="Abschnitt bearbeiten: Zylinderkoordinaten" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=18" title="Quellcode des Abschnitts bearbeiten: Zylinderkoordinaten">Quelltext bearbeiten</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Cylindrical_Coordinates.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Cylindrical_Coordinates.svg/220px-Cylindrical_Coordinates.svg.png" decoding="async" width="220" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Cylindrical_Coordinates.svg/330px-Cylindrical_Coordinates.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Cylindrical_Coordinates.svg/440px-Cylindrical_Coordinates.svg.png 2x" data-file-width="748" data-file-height="745" /></a><figcaption>Zylinderkoordinaten</figcaption></figure> Zylinderkoordinaten oder zylindrische Koordinaten sind im Wesentlichen ebene Polarkoordinaten, die um eine dritte Koordinate ergänzt sind. Diese dritte Koordinate beschreibt die Höhe eines Punktes senkrecht über (oder unter) der Ebene des Polarkoordinatensystems und wird im Allgemeinen mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> bezeichnet. Die Koordinate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\rho } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\rho } }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79eaf0a86fad190175ebd4b205e4de3c96f231d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \mathbf {\rho } }"> beschreibt jetzt nicht mehr den Abstand eines Punktes vom Koordinatenursprung, sondern von der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}">-Achse. <div class="mw-heading mw-heading4"><h4 id="Umrechnung_zwischen_Zylinderkoordinaten_und_kartesischen_Koordinaten">Umrechnung zwischen Zylinderkoordinaten und kartesischen Koordinaten</h4>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=19" title="Abschnitt bearbeiten: Umrechnung zwischen Zylinderkoordinaten und kartesischen Koordinaten" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=19" title="Quellcode des Abschnitts bearbeiten: Umrechnung zwischen Zylinderkoordinaten und kartesischen Koordinaten">Quelltext bearbeiten</a>]</div> Wenn man ein kartesisches Koordinatensystem so ausrichtet, dass die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}">-Achsen zusammenfallen, die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">-Achse in Richtung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/192287b02f5764a18fe39f37b8199d72000aa220" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.781ex; height:2.676ex;" alt="{\displaystyle \varphi =0}"> zeigt und der Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> von der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">-Achse zur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}">-Achse wächst (rechtsgerichtet ist), dann ergeben sich die folgenden Umrechnungsformeln: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\rho \,\cos \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>ρ</mi> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\rho \,\cos \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c441e9c1027e3f3c6a62ecf34b22d16e98932738" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.423ex; height:2.176ex;" alt="{\displaystyle x=\rho \,\cos \varphi }"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\rho \,\sin \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>ρ</mi> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\rho \,\sin \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0869f374369d0dc969a427ddd42d168542142782" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.993ex; height:2.676ex;" alt="{\displaystyle y=\rho \,\sin \varphi }"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68fd4b155af06002932cef8fc09cbc67743cde8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.275ex; height:1.676ex;" alt="{\displaystyle z=z}"></dd></dl> Für die Umrechnung von kartesischen Koordinaten in Zylinderkoordinaten ergeben sich für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> die gleichen Formeln wie bei den Polarkoordinaten. Für Punkte auf der z-Achse gibt es keine eindeutige Koordinatendarstellung: hier ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ba6310b27df5f9c9b0b1732e08cce27b99d68cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.463ex; height:2.676ex;" alt="{\displaystyle \rho =0}">, aber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> beliebig. <div class="mw-heading mw-heading4"><h4 id="Koordinatenlinien_und_Koordinatenflächen">Koordinatenlinien und Koordinatenflächen</h4>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=20" title="Abschnitt bearbeiten: Koordinatenlinien und Koordinatenflächen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=20" title="Quellcode des Abschnitts bearbeiten: Koordinatenlinien und Koordinatenflächen">Quelltext bearbeiten</a>]</div> Für die Koordinatentransformation als Vektorgleichung mit dem Ortsvektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aec3c9ce13b53e9e24c98e7cce4212627884c91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}"> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}={\begin{pmatrix}x\\y\\z\end{pmatrix}}={\begin{pmatrix}\rho \cos \varphi \\\rho \sin \varphi \\z\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>ρ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>ρ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}={\begin{pmatrix}x\\y\\z\end{pmatrix}}={\begin{pmatrix}\rho \cos \varphi \\\rho \sin \varphi \\z\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f577bc2b16609fd1fcbb8d59b0389c53816973f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:24.994ex; height:9.509ex;" alt="{\displaystyle {\vec {r}}={\begin{pmatrix}x\\y\\z\end{pmatrix}}={\begin{pmatrix}\rho \cos \varphi \\\rho \sin \varphi \\z\end{pmatrix}}}"></dd></dl> ergeben sich für einen Punkt <ul><li>die <a href="/wiki/Koordinatenlinie" title="Koordinatenlinie">Koordinatenlinien</a>, indem man jeweils zwei der drei Koordinaten fest lässt und die dritte den Kurvenparameter darstellt</li> <li>die <a href="/wiki/Koordinatenfl%C3%A4che" title="Koordinatenfläche">Koordinatenflächen</a>, indem man eine der drei Koordinaten fest lässt und die beiden anderen die Fläche parametrisieren.</li></ul> Jeweils zwei Koordinatenflächen schneiden sich in einer Koordinatenlinie. Koordinatenlinien und Koordinatenflächen dienen dazu, die lokalen Basisvektoren (siehe unten) zu berechnen. Durch den Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\rho _{0}\mid \varphi _{0}\mid z_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∣</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∣</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\rho _{0}\mid \varphi _{0}\mid z_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f684e6a49839115c08cc2520866e7833108d3a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.65ex; height:2.843ex;" alt="{\displaystyle (\rho _{0}\mid \varphi _{0}\mid z_{0})}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\rho _{0}\neq 0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\rho _{0}\neq 0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c1f55342f40b43a6ca6d81e8e30f05f89ba46a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.326ex; height:2.843ex;" alt="{\displaystyle (\rho _{0}\neq 0)}"> verlaufen drei Koordinatenlinien. Es handelt sich dabei <ul><li>für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"> als Kurvenparameter um eine <a href="/wiki/Strahl_(Geometrie)" title="Strahl (Geometrie)">Halbgerade</a>, die im Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0,z_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0,z_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e6f62ef19eb7b885f293be4f41da15ab2ac0068" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.337ex; height:2.843ex;" alt="{\displaystyle (0,0,z_{0})}"> beginnt und senkrecht zur z-Achse verläuft</li> <li>für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> als Kurvenparameter um einen Kreis senkrecht zur z-Achse mit dem Mittelpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0,z_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0,z_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e6f62ef19eb7b885f293be4f41da15ab2ac0068" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.337ex; height:2.843ex;" alt="{\displaystyle (0,0,z_{0})}"> und Radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9c04a9d26b86af8c6205ba2a6287fd655b6b714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.256ex; height:2.176ex;" alt="{\displaystyle \rho _{0}}"></li> <li>für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> als Kurvenparameter um eine Gerade parallel zur z-Achse.</li></ul> Als Koordinatenflächen durch den Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\rho _{0}\mid \varphi _{0}\mid z_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∣</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∣</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\rho _{0}\mid \varphi _{0}\mid z_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f684e6a49839115c08cc2520866e7833108d3a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.65ex; height:2.843ex;" alt="{\displaystyle (\rho _{0}\mid \varphi _{0}\mid z_{0})}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\rho _{0}\neq 0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\rho _{0}\neq 0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c1f55342f40b43a6ca6d81e8e30f05f89ba46a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.326ex; height:2.843ex;" alt="{\displaystyle (\rho _{0}\neq 0)}"> ergeben sich <ul><li>für konstanten Radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9c04a9d26b86af8c6205ba2a6287fd655b6b714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.256ex; height:2.176ex;" alt="{\displaystyle \rho _{0}}"> eine Zylinderfläche mit der z-Achse als Zylinderachse</li> <li>für festen Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> eine Halbebene mit der z-Achse als Rand</li> <li>für konstanten Wert von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e72d1d86e86355892b39b8eb32b964834e113bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}"> eine Ebene senkrecht zur z-Achse.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Lokale_Basisvektoren_und_Orthogonalität_2">Lokale Basisvektoren und Orthogonalität</h4>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=21" title="Abschnitt bearbeiten: Lokale Basisvektoren und Orthogonalität" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=21" title="Quellcode des Abschnitts bearbeiten: Lokale Basisvektoren und Orthogonalität">Quelltext bearbeiten</a>]</div> In geradlinigen Koordinatensystemen gibt es eine Basis für den gesamten <a href="/wiki/Vektorraum" title="Vektorraum">Vektorraum</a>, in krummlinigen muss an jedem Punkt eine lokale Basis berechnet werden. Die lokalen Basisvektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\vec {b}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\vec {b}}_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2db0611cc9cff68458e90d4e2d857bf0cfa40772" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.148ex; height:3.176ex;" alt="{\displaystyle \textstyle {\vec {b}}_{1}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\vec {b}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\vec {b}}_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/744ea3904fd207c10ad23e1c1b686a689b690874" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.148ex; height:3.176ex;" alt="{\displaystyle \textstyle {\vec {b}}_{2}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\vec {b}}_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\vec {b}}_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/793b74d9c69d7c2122c05a222e1242dc297efeb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.148ex; height:3.176ex;" alt="{\displaystyle \textstyle {\vec {b}}_{3}}"> an einem Punkt sind Tangentenvektoren an die Koordinatenlinien und ergeben sich aus deren Kurvengleichungen durch Ableitung nach dem Kurvenparameter. Zum selben Ergebnis gelangt man auch durch partielle Ableitung der Koordinatentransformation für den Ortsvektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aec3c9ce13b53e9e24c98e7cce4212627884c91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}"> nach den Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}_{1}={\frac {\partial {\vec {r}}}{\partial \rho }}={\begin{pmatrix}\cos \varphi \\\sin \varphi \\0\end{pmatrix}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}_{1}={\frac {\partial {\vec {r}}}{\partial \rho }}={\begin{pmatrix}\cos \varphi \\\sin \varphi \\0\end{pmatrix}}\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9f4659779498b2eabcd73ae5202f4b136ea390" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:23.882ex; height:9.509ex;" alt="{\displaystyle {\vec {b}}_{1}={\frac {\partial {\vec {r}}}{\partial \rho }}={\begin{pmatrix}\cos \varphi \\\sin \varphi \\0\end{pmatrix}}\quad }">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad {\vec {b}}_{2}={\frac {\partial {\vec {r}}}{\partial \varphi }}={\begin{pmatrix}-\rho \sin \varphi \\\rho \cos \varphi \\0\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mi>ρ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>ρ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad {\vec {b}}_{2}={\frac {\partial {\vec {r}}}{\partial \varphi }}={\begin{pmatrix}-\rho \sin \varphi \\\rho \cos \varphi \\0\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20f3301c25566e857de2ad0f8ec579c95a3e1b09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:27.32ex; height:9.509ex;" alt="{\displaystyle \quad {\vec {b}}_{2}={\frac {\partial {\vec {r}}}{\partial \varphi }}={\begin{pmatrix}-\rho \sin \varphi \\\rho \cos \varphi \\0\end{pmatrix}}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad {\vec {b}}_{3}={\frac {\partial {\vec {r}}}{\partial z}}={\begin{pmatrix}0\\0\\1\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad {\vec {b}}_{3}={\frac {\partial {\vec {r}}}{\partial z}}={\begin{pmatrix}0\\0\\1\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e12f93b8c7d242f58ee093603214909730524fc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:20.026ex; height:9.176ex;" alt="{\displaystyle \quad {\vec {b}}_{3}={\frac {\partial {\vec {r}}}{\partial z}}={\begin{pmatrix}0\\0\\1\end{pmatrix}}}">.</dd></dl> Die Basisvektoren haben die Längen <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {b}}_{1}|={\sqrt {{\vec {b}}_{1}{\vec {b}}_{1}}}=1\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </msqrt> </mrow> <mo>=</mo> <mn>1</mn> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\vec {b}}_{1}|={\sqrt {{\vec {b}}_{1}{\vec {b}}_{1}}}=1\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c98216bded77911cea9187533a0a8e8be49ddf5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.744ex; height:4.676ex;" alt="{\displaystyle |{\vec {b}}_{1}|={\sqrt {{\vec {b}}_{1}{\vec {b}}_{1}}}=1\quad }">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad |{\vec {b}}_{2}|={\sqrt {{\vec {b}}_{2}{\vec {b}}_{2}}}=\rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </msqrt> </mrow> <mo>=</mo> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad |{\vec {b}}_{2}|={\sqrt {{\vec {b}}_{2}{\vec {b}}_{2}}}=\rho }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5573b1f6837d20501cfb1ad853458356eb9c287" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.783ex; height:4.676ex;" alt="{\displaystyle \quad |{\vec {b}}_{2}|={\sqrt {{\vec {b}}_{2}{\vec {b}}_{2}}}=\rho }">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad |{\vec {b}}_{3}|={\sqrt {{\vec {b}}_{3}{\vec {b}}_{3}}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </msqrt> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad |{\vec {b}}_{3}|={\sqrt {{\vec {b}}_{3}{\vec {b}}_{3}}}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2054ca42642806e6c829f783f133f5445def2ec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:19.744ex; height:4.843ex;" alt="{\displaystyle \quad |{\vec {b}}_{3}|={\sqrt {{\vec {b}}_{3}{\vec {b}}_{3}}}=1}"></dd></dl> und sind zueinander orthogonal. Eine Normierung ergibt die Einheitsvektoren: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{\rho }={\frac {\frac {\partial {\vec {r}}}{\partial \rho }}{\left|{\frac {\partial {\vec {r}}}{\partial \rho }}\right|}}={\begin{pmatrix}\cos \varphi \\\sin \varphi \\0\end{pmatrix}},\quad {\vec {e}}_{\varphi }={\frac {\frac {\partial {\vec {r}}}{\partial \varphi }}{\left|{\frac {\partial {\vec {r}}}{\partial \varphi }}\right|}}={\begin{pmatrix}-\sin \varphi \\\cos \varphi \\0\end{pmatrix}},\quad {\vec {e}}_{z}={\frac {\frac {\partial {\vec {r}}}{\partial z}}{\left|{\frac {\partial {\vec {r}}}{\partial z}}\right|}}={\begin{pmatrix}0\\0\\1\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ρ</mi> </mrow> </mfrac> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{\rho }={\frac {\frac {\partial {\vec {r}}}{\partial \rho }}{\left|{\frac {\partial {\vec {r}}}{\partial \rho }}\right|}}={\begin{pmatrix}\cos \varphi \\\sin \varphi \\0\end{pmatrix}},\quad {\vec {e}}_{\varphi }={\frac {\frac {\partial {\vec {r}}}{\partial \varphi }}{\left|{\frac {\partial {\vec {r}}}{\partial \varphi }}\right|}}={\begin{pmatrix}-\sin \varphi \\\cos \varphi \\0\end{pmatrix}},\quad {\vec {e}}_{z}={\frac {\frac {\partial {\vec {r}}}{\partial z}}{\left|{\frac {\partial {\vec {r}}}{\partial z}}\right|}}={\begin{pmatrix}0\\0\\1\end{pmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca5f25231d701c18cd00be959ef306cbec919731" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:75.179ex; height:10.009ex;" alt="{\displaystyle {\vec {e}}_{\rho }={\frac {\frac {\partial {\vec {r}}}{\partial \rho }}{\left|{\frac {\partial {\vec {r}}}{\partial \rho }}\right|}}={\begin{pmatrix}\cos \varphi \\\sin \varphi \\0\end{pmatrix}},\quad {\vec {e}}_{\varphi }={\frac {\frac {\partial {\vec {r}}}{\partial \varphi }}{\left|{\frac {\partial {\vec {r}}}{\partial \varphi }}\right|}}={\begin{pmatrix}-\sin \varphi \\\cos \varphi \\0\end{pmatrix}},\quad {\vec {e}}_{z}={\frac {\frac {\partial {\vec {r}}}{\partial z}}{\left|{\frac {\partial {\vec {r}}}{\partial z}}\right|}}={\begin{pmatrix}0\\0\\1\end{pmatrix}}.}"></dd></dl> Die Basisvektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{\rho }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{\rho }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6553b63581d8e91eb8047a6365e405c4927bbd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.305ex; height:3.009ex;" alt="{\displaystyle {\vec {e}}_{\rho }}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{\varphi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{\varphi }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f4e6d6e1f4339badfe82648ea46a8b4909d9206" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.53ex; height:3.009ex;" alt="{\displaystyle {\vec {e}}_{\varphi }}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{z}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94d7dc5dd50fc55f2d715e40e466e29e28a56ffb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.225ex; height:2.676ex;" alt="{\displaystyle {\vec {e}}_{z}}"> sind zueinander <a href="/wiki/Orthonormal" class="mw-redirect" title="Orthonormal">orthonormal</a> und bilden in dieser Reihenfolge ein Rechtssystem. In der <a href="/wiki/Tensor" title="Tensor">Tensorrechnung</a> werden die lokalen Basisvektoren, die tangential zu den Koordinatenlinien verlaufen, wegen ihres Verhaltens bei <a href="/wiki/Koordinatentransformation" title="Koordinatentransformation">Koordinatentransformationen</a> als <a href="/wiki/Tensor#Ko-_und_Kontravarianz_von_Vektoren" title="Tensor">kovariant</a> bezeichnet. Die <a href="/wiki/Krummlinige_Koordinaten#Duale_Basis:_Kontravariante_Basis" title="Krummlinige Koordinaten">kontravarianten Basisvektoren</a> stehen senkrecht auf den Koordinatenflächen. <div class="mw-heading mw-heading4"><h4 id="Metrischer_Tensor_2">Metrischer Tensor</h4>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=22" title="Abschnitt bearbeiten: Metrischer Tensor" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=22" title="Quellcode des Abschnitts bearbeiten: Metrischer Tensor">Quelltext bearbeiten</a>]</div> Die Komponenten des kovarianten <a href="/wiki/Metrischer_Tensor" title="Metrischer Tensor">metrischen Tensors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=(g_{ij})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=(g_{ij})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e36129d1e65f26b765daad53fc02c24f1a275aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.61ex; height:3.009ex;" alt="{\displaystyle g=(g_{ij})}"> sind die Skalarprodukte der kovarianten lokalen Basisvektoren: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{ij}={\vec {b}}_{i}{\vec {b}}_{j}\quad (i,j\in \{1,2,3\})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{ij}={\vec {b}}_{i}{\vec {b}}_{j}\quad (i,j\in \{1,2,3\})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2164cb46fab0b063d1f3176bb9cc89ce3317f12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.229ex; height:3.509ex;" alt="{\displaystyle g_{ij}={\vec {b}}_{i}{\vec {b}}_{j}\quad (i,j\in \{1,2,3\})}">.</dd></dl> Nach den vorangegangenen Rechnungen ist damit <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g={\begin{pmatrix}1&0&0\\0&\rho ^{2}&0\\0&0&1\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g={\begin{pmatrix}1&0&0\\0&\rho ^{2}&0\\0&0&1\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d391f51932ff08a8962a4ef976da217903b9b1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:18.259ex; height:9.509ex;" alt="{\displaystyle g={\begin{pmatrix}1&0&0\\0&\rho ^{2}&0\\0&0&1\end{pmatrix}}}">.</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Funktionaldeterminante_2">Funktionaldeterminante</h4>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=23" title="Abschnitt bearbeiten: Funktionaldeterminante" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=23" title="Quellcode des Abschnitts bearbeiten: Funktionaldeterminante">Quelltext bearbeiten</a>]</div> Die Hinzunahme der geradlinigen Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> hat keinen Einfluss auf die Funktionaldeterminante: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det {\frac {\partial (x,y,z)}{\partial (\rho ,\varphi ,z)}}={\begin{vmatrix}\cos \varphi &-\rho \sin \varphi &0\\\sin \varphi &\rho \cos \varphi &0\\0&0&1\end{vmatrix}}=\rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>,</mo> <mi>φ</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> <mtd> <mo>−</mo> <mi>ρ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> <mtd> <mi>ρ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det {\frac {\partial (x,y,z)}{\partial (\rho ,\varphi ,z)}}={\begin{vmatrix}\cos \varphi &-\rho \sin \varphi &0\\\sin \varphi &\rho \cos \varphi &0\\0&0&1\end{vmatrix}}=\rho }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44280f1e65c56f52e1e05d4f102e15785c5506fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:41.888ex; height:9.509ex;" alt="{\displaystyle \det {\frac {\partial (x,y,z)}{\partial (\rho ,\varphi ,z)}}={\begin{vmatrix}\cos \varphi &-\rho \sin \varphi &0\\\sin \varphi &\rho \cos \varphi &0\\0&0&1\end{vmatrix}}=\rho }"></dd></dl> Folglich ergibt sich für das Volumenelement <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b80507190aa9d38a279909db47b63657f2b62ba7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.08ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} V}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} V=\rho \,\mathrm {d} \rho \,\mathrm {d} \varphi \,\mathrm {d} z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mo>=</mo> <mi>ρ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ρ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>φ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} V=\rho \,\mathrm {d} \rho \,\mathrm {d} \varphi \,\mathrm {d} z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dea06ffdd733ce0fee83f72c8661a362c3ed4314" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.229ex; height:2.676ex;" alt="{\displaystyle \mathrm {d} V=\rho \,\mathrm {d} \rho \,\mathrm {d} \varphi \,\mathrm {d} z}"></dd></dl> Das entspricht auch der Quadratwurzel des Betrags der Determinante des <a href="/wiki/Metrischer_Tensor" title="Metrischer Tensor">metrischen Tensors</a>, mit dessen Hilfe die Koordinatentransformation berechnet werden kann (siehe dazu <a href="/wiki/Laplace-Operator" title="Laplace-Operator">Laplace-Beltrami-Operator</a>). <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}\mathrm {d} x\\\mathrm {d} y\\\mathrm {d} z\end{pmatrix}}={\begin{pmatrix}\cos \varphi &-\rho \sin \varphi &0\\\sin \varphi &\rho \cos \varphi &0\\0&0&1\end{pmatrix}}\cdot {\begin{pmatrix}\mathrm {d} \rho \\\mathrm {d} \varphi \\\mathrm {d} z\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> <mtd> <mo>−</mo> <mi>ρ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> <mtd> <mi>ρ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ρ</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}\mathrm {d} x\\\mathrm {d} y\\\mathrm {d} z\end{pmatrix}}={\begin{pmatrix}\cos \varphi &-\rho \sin \varphi &0\\\sin \varphi &\rho \cos \varphi &0\\0&0&1\end{pmatrix}}\cdot {\begin{pmatrix}\mathrm {d} \rho \\\mathrm {d} \varphi \\\mathrm {d} z\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55c62217a6aa566566d84a4cf116181116719765" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:43.653ex; height:9.509ex;" alt="{\displaystyle {\begin{pmatrix}\mathrm {d} x\\\mathrm {d} y\\\mathrm {d} z\end{pmatrix}}={\begin{pmatrix}\cos \varphi &-\rho \sin \varphi &0\\\sin \varphi &\rho \cos \varphi &0\\0&0&1\end{pmatrix}}\cdot {\begin{pmatrix}\mathrm {d} \rho \\\mathrm {d} \varphi \\\mathrm {d} z\end{pmatrix}}}"></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}\mathrm {d} \rho \\\mathrm {d} \varphi \\\mathrm {d} z\end{pmatrix}}={\begin{pmatrix}{\frac {x}{\sqrt {x^{2}+y^{2}}}}&{\frac {y}{\sqrt {x^{2}+y^{2}}}}&0\\{\frac {-y}{x^{2}+y^{2}}}&{\frac {x}{x^{2}+y^{2}}}&0\\0&0&1\end{pmatrix}}\cdot {\begin{pmatrix}\mathrm {d} x\\\mathrm {d} y\\\mathrm {d} z\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ρ</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>y</mi> </mrow> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}\mathrm {d} \rho \\\mathrm {d} \varphi \\\mathrm {d} z\end{pmatrix}}={\begin{pmatrix}{\frac {x}{\sqrt {x^{2}+y^{2}}}}&{\frac {y}{\sqrt {x^{2}+y^{2}}}}&0\\{\frac {-y}{x^{2}+y^{2}}}&{\frac {x}{x^{2}+y^{2}}}&0\\0&0&1\end{pmatrix}}\cdot {\begin{pmatrix}\mathrm {d} x\\\mathrm {d} y\\\mathrm {d} z\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52d00ea9cc6427c248860ed93672d552ba0331a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:44.839ex; height:13.509ex;" alt="{\displaystyle {\begin{pmatrix}\mathrm {d} \rho \\\mathrm {d} \varphi \\\mathrm {d} z\end{pmatrix}}={\begin{pmatrix}{\frac {x}{\sqrt {x^{2}+y^{2}}}}&{\frac {y}{\sqrt {x^{2}+y^{2}}}}&0\\{\frac {-y}{x^{2}+y^{2}}}&{\frac {x}{x^{2}+y^{2}}}&0\\0&0&1\end{pmatrix}}\cdot {\begin{pmatrix}\mathrm {d} x\\\mathrm {d} y\\\mathrm {d} z\end{pmatrix}}}"></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Vektoranalysis">Vektoranalysis</h4>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=24" title="Abschnitt bearbeiten: Vektoranalysis" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=24" title="Quellcode des Abschnitts bearbeiten: Vektoranalysis">Quelltext bearbeiten</a>]</div> Die folgenden Darstellungen des <a href="/wiki/Nabla-Operator" title="Nabla-Operator">Nabla-Operators</a> können in der gegebenen Form direkt auf skalare oder vektorwertige Felder in Zylinderkoordinaten angewendet werden. Man verfährt hierbei analog zur <a href="/wiki/Vektoranalysis" title="Vektoranalysis">Vektoranalysis</a> in kartesischen Koordinaten. <div class="mw-heading mw-heading5"><h5 id="Gradient">Gradient</h5>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=25" title="Abschnitt bearbeiten: Gradient" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=25" title="Quellcode des Abschnitts bearbeiten: Gradient">Quelltext bearbeiten</a>]</div> Die Darstellung des <a href="/wiki/Gradient_(Mathematik)" title="Gradient (Mathematik)">Gradienten</a> überträgt sich wie folgt von kartesischen in Zylinderkoordinaten: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla f={\frac {\partial f}{\partial \rho }}{\vec {e}}_{\rho }+{\frac {1}{\rho }}{\frac {\partial f}{\partial \varphi }}{\vec {e}}_{\varphi }+{\frac {\partial f}{\partial z}}{\vec {e}}_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ρ</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla f={\frac {\partial f}{\partial \rho }}{\vec {e}}_{\rho }+{\frac {1}{\rho }}{\frac {\partial f}{\partial \varphi }}{\vec {e}}_{\varphi }+{\frac {\partial f}{\partial z}}{\vec {e}}_{z}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7f648f66e1a7dbbbe3b3ddf253e74800f9c8b9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.632ex; height:6.176ex;" alt="{\displaystyle \nabla f={\frac {\partial f}{\partial \rho }}{\vec {e}}_{\rho }+{\frac {1}{\rho }}{\frac {\partial f}{\partial \varphi }}{\vec {e}}_{\varphi }+{\frac {\partial f}{\partial z}}{\vec {e}}_{z}}"></dd></dl> <div class="mw-heading mw-heading5"><h5 id="Divergenz">Divergenz</h5>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=26" title="Abschnitt bearbeiten: Divergenz" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=26" title="Quellcode des Abschnitts bearbeiten: Divergenz">Quelltext bearbeiten</a>]</div> Bei der <a href="/wiki/Divergenz_eines_Vektorfeldes" title="Divergenz eines Vektorfeldes">Divergenz</a> kommen noch weitere Terme hinzu, die sich aus den Ableitungen der von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> abhängigen Einheitsvektoren ergeben: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot {\vec {A}}={\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}(\rho A_{\rho })+{\frac {1}{\rho }}{\frac {\partial A_{\varphi }}{\partial \varphi }}+{\frac {\partial A_{z}}{\partial z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>ρ</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot {\vec {A}}={\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}(\rho A_{\rho })+{\frac {1}{\rho }}{\frac {\partial A_{\varphi }}{\partial \varphi }}+{\frac {\partial A_{z}}{\partial z}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec4dccff8fb9000ee6ae157f979d1f8b4f7e0551" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.51ex; height:6.343ex;" alt="{\displaystyle \nabla \cdot {\vec {A}}={\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}(\rho A_{\rho })+{\frac {1}{\rho }}{\frac {\partial A_{\varphi }}{\partial \varphi }}+{\frac {\partial A_{z}}{\partial z}}}"></dd></dl> <div class="mw-heading mw-heading5"><h5 id="Rotation">Rotation</h5>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=27" title="Abschnitt bearbeiten: Rotation" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=27" title="Quellcode des Abschnitts bearbeiten: Rotation">Quelltext bearbeiten</a>]</div> Die Darstellung der <a href="/wiki/Rotation_eines_Vektorfeldes" title="Rotation eines Vektorfeldes">Rotation</a> ist wie folgt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \times {\vec {A}}=\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \varphi }}-{\frac {\partial A_{\varphi }}{\partial z}}\right){\vec {e}}_{\rho }+\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right){\vec {e}}_{\varphi }+{\frac {1}{\rho }}\left({\frac {\partial }{\partial \rho }}(\rho A_{\varphi })-{\frac {\partial A_{\rho }}{\partial \varphi }}\right){\vec {e}}_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ρ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>ρ</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \times {\vec {A}}=\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \varphi }}-{\frac {\partial A_{\varphi }}{\partial z}}\right){\vec {e}}_{\rho }+\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right){\vec {e}}_{\varphi }+{\frac {1}{\rho }}\left({\frac {\partial }{\partial \rho }}(\rho A_{\varphi })-{\frac {\partial A_{\rho }}{\partial \varphi }}\right){\vec {e}}_{z}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a12aa31731b437a3a5aa55984ee27d6dbe1340" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:81.147ex; height:6.343ex;" alt="{\displaystyle \nabla \times {\vec {A}}=\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \varphi }}-{\frac {\partial A_{\varphi }}{\partial z}}\right){\vec {e}}_{\rho }+\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right){\vec {e}}_{\varphi }+{\frac {1}{\rho }}\left({\frac {\partial }{\partial \rho }}(\rho A_{\varphi })-{\frac {\partial A_{\rho }}{\partial \varphi }}\right){\vec {e}}_{z}}"></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Ortsvektor,_Geschwindigkeit_und_Beschleunigung_in_Zylinderkoordinaten">Ortsvektor, Geschwindigkeit und Beschleunigung in Zylinderkoordinaten</h4>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=28" title="Abschnitt bearbeiten: Ortsvektor, Geschwindigkeit und Beschleunigung in Zylinderkoordinaten" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=28" title="Quellcode des Abschnitts bearbeiten: Ortsvektor, Geschwindigkeit und Beschleunigung in Zylinderkoordinaten">Quelltext bearbeiten</a>]</div> Mit den lokalen Basiseinheitsvektoren <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{\rho }={\begin{pmatrix}\cos \varphi \\\sin \varphi \\0\end{pmatrix}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{\rho }={\begin{pmatrix}\cos \varphi \\\sin \varphi \\0\end{pmatrix}}\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/925425bda72a759e495e354dde61aed9e7453243" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:17.563ex; height:9.509ex;" alt="{\displaystyle {\vec {e}}_{\rho }={\begin{pmatrix}\cos \varphi \\\sin \varphi \\0\end{pmatrix}}\quad }">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad {\vec {e}}_{\varphi }={\begin{pmatrix}-\sin \varphi \\\cos \varphi \\0\end{pmatrix}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad {\vec {e}}_{\varphi }={\begin{pmatrix}-\sin \varphi \\\cos \varphi \\0\end{pmatrix}}\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a0927165f5cf56549b70a5bfe05cdda7dc09ee5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:22.05ex; height:9.509ex;" alt="{\displaystyle \quad {\vec {e}}_{\varphi }={\begin{pmatrix}-\sin \varphi \\\cos \varphi \\0\end{pmatrix}}\quad }"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad {\vec {e}}_{z}={\begin{pmatrix}0\\0\\1\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad {\vec {e}}_{z}={\begin{pmatrix}0\\0\\1\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/971cb23000849dc7d08748346a010017c005e3a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.627ex; height:9.176ex;" alt="{\displaystyle \quad {\vec {e}}_{z}={\begin{pmatrix}0\\0\\1\end{pmatrix}}}"></dd></dl> ergibt sich für den Ortsvektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aec3c9ce13b53e9e24c98e7cce4212627884c91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}={\begin{pmatrix}x\\y\\z\end{pmatrix}}={\begin{pmatrix}\rho \cos \varphi \\\rho \sin \varphi \\z\end{pmatrix}}=\rho \,{\vec {e}}_{\rho }+z\,{\vec {e}}_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>ρ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>ρ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mi>ρ</mi> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> <mo>+</mo> <mi>z</mi> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}={\begin{pmatrix}x\\y\\z\end{pmatrix}}={\begin{pmatrix}\rho \cos \varphi \\\rho \sin \varphi \\z\end{pmatrix}}=\rho \,{\vec {e}}_{\rho }+z\,{\vec {e}}_{z}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb7ffd02008c6386bc8be3eb5df4bbf4db254fce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:38.527ex; height:9.509ex;" alt="{\displaystyle {\vec {r}}={\begin{pmatrix}x\\y\\z\end{pmatrix}}={\begin{pmatrix}\rho \cos \varphi \\\rho \sin \varphi \\z\end{pmatrix}}=\rho \,{\vec {e}}_{\rho }+z\,{\vec {e}}_{z}}">.</dd></dl> Ist der Ortsvektor abhängig von der Zeit, so müssen die Variablen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> und damit auch die davon abhängigen lokalen Basisvektoren abgeleitet werden: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {{\vec {e}}_{\rho }}}={\begin{pmatrix}-\sin \varphi \\\cos \varphi \\0\end{pmatrix}}\,{\dot {\varphi }}={\dot {\varphi }}\,{\vec {e}}_{\varphi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {{\vec {e}}_{\rho }}}={\begin{pmatrix}-\sin \varphi \\\cos \varphi \\0\end{pmatrix}}\,{\dot {\varphi }}={\dot {\varphi }}\,{\vec {e}}_{\varphi }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a2197c9a8960939f4cd905e20df2f2c9c97f41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:26.653ex; height:9.509ex;" alt="{\displaystyle {\dot {{\vec {e}}_{\rho }}}={\begin{pmatrix}-\sin \varphi \\\cos \varphi \\0\end{pmatrix}}\,{\dot {\varphi }}={\dot {\varphi }}\,{\vec {e}}_{\varphi }}">.</dd></dl> Mit der <a href="/wiki/Produktregel" title="Produktregel">Produktregel</a> ergibt sich somit für den Geschwindigkeitsvektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\vec {r}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\vec {r}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d0622c7d4647179011ad0bc43548fc6abd756db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.257ex; height:2.843ex;" alt="{\displaystyle {\dot {\vec {r}}}}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\vec {r}}}={\dot {\rho }}\,{\vec {e}}_{\rho }+\rho \,{\dot {{\vec {e}}_{\rho }}}+{\dot {z}}\,{\vec {e}}_{z}={\dot {\rho }}\,{\vec {e}}_{\rho }+\rho \,{\dot {\varphi }}\,{\vec {e}}_{\varphi }+{\dot {z}}\,{\vec {e}}_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> <mo>+</mo> <mi>ρ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> <mo>+</mo> <mi>ρ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\vec {r}}}={\dot {\rho }}\,{\vec {e}}_{\rho }+\rho \,{\dot {{\vec {e}}_{\rho }}}+{\dot {z}}\,{\vec {e}}_{z}={\dot {\rho }}\,{\vec {e}}_{\rho }+\rho \,{\dot {\varphi }}\,{\vec {e}}_{\varphi }+{\dot {z}}\,{\vec {e}}_{z}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71b4c2497abde01723d5d227e51746e13a5cd1d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.703ex; height:3.509ex;" alt="{\displaystyle {\dot {\vec {r}}}={\dot {\rho }}\,{\vec {e}}_{\rho }+\rho \,{\dot {{\vec {e}}_{\rho }}}+{\dot {z}}\,{\vec {e}}_{z}={\dot {\rho }}\,{\vec {e}}_{\rho }+\rho \,{\dot {\varphi }}\,{\vec {e}}_{\varphi }+{\dot {z}}\,{\vec {e}}_{z}}">.</dd></dl> Eine entsprechende Rechnung führt für die Beschleunigung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {\vec {r}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ddot {\vec {r}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f02da80a6ce3388352e6efdec76e57ecf79f87ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.257ex; height:2.843ex;" alt="{\displaystyle {\ddot {\vec {r}}}}"> zu dem Ergebnis <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {\vec {r}}}=({\ddot {\rho }}-\rho {\dot {\varphi }}^{2})\,{\vec {e}}_{\rho }+(2{\dot {\rho }}{\dot {\varphi }}+\rho {\ddot {\varphi }})\,{\vec {e}}_{\varphi }+{\ddot {z}}\,{\vec {e}}_{z}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>ρ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ddot {\vec {r}}}=({\ddot {\rho }}-\rho {\dot {\varphi }}^{2})\,{\vec {e}}_{\rho }+(2{\dot {\rho }}{\dot {\varphi }}+\rho {\ddot {\varphi }})\,{\vec {e}}_{\varphi }+{\ddot {z}}\,{\vec {e}}_{z}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef755bcd0b9a74c1148c7cc4c3fce79cdb3cbdb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.477ex; height:3.509ex;" alt="{\displaystyle {\ddot {\vec {r}}}=({\ddot {\rho }}-\rho {\dot {\varphi }}^{2})\,{\vec {e}}_{\rho }+(2{\dot {\rho }}{\dot {\varphi }}+\rho {\ddot {\varphi }})\,{\vec {e}}_{\varphi }+{\ddot {z}}\,{\vec {e}}_{z}.}">.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Kegelkoordinaten_(Koordinaten-Transformation)">Kegelkoordinaten (Koordinaten-Transformation)</h3>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=29" title="Abschnitt bearbeiten: Kegelkoordinaten (Koordinaten-Transformation)" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=29" title="Quellcode des Abschnitts bearbeiten: Kegelkoordinaten (Koordinaten-Transformation)">Quelltext bearbeiten</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Parameterdarstellung">Parameterdarstellung</h4>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=30" title="Abschnitt bearbeiten: Parameterdarstellung" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=30" title="Quellcode des Abschnitts bearbeiten: Parameterdarstellung">Quelltext bearbeiten</a>]</div> Die Parameterdarstellung des Kegels kann man wie folgt beschreiben. Mit der Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {P}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a1dd77291be4fd776d0aadf3d35cc3285095490" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.682ex; height:3.676ex;" alt="{\displaystyle {\overrightarrow {P}}}"> lassen sich die Kegelkoordinaten in kartesische Koordinaten umrechnen. Mit der Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {Q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {Q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a94458af2af429ff8899d573506358c9694cc63c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.517ex; height:4.009ex;" alt="{\displaystyle {\overrightarrow {Q}}}"> lassen sich die kartesischen Koordinaten in Kegelkoordinaten umrechnen. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {P}}(\gamma ,\varphi ,\chi )={\begin{pmatrix}x\\y\\z\end{pmatrix}}=\chi \cdot {\begin{pmatrix}\gamma \cos(\varphi )\\\gamma \sin(\varphi )\\1\end{pmatrix}}\quad \quad \quad {\overrightarrow {Q}}(x,y,z)={\begin{pmatrix}\gamma \\\varphi \\\chi \end{pmatrix}}={\begin{pmatrix}{\frac {1}{z}}{\sqrt {x^{2}+y^{2}}}\\\operatorname {arctan2} (x,y)\\z\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo>→</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>γ</mi> <mo>,</mo> <mi>φ</mi> <mo>,</mo> <mi>χ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mi>χ</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>γ</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>γ</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>→</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>γ</mi> </mtd> </mtr> <mtr> <mtd> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>χ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>arctan2</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {P}}(\gamma ,\varphi ,\chi )={\begin{pmatrix}x\\y\\z\end{pmatrix}}=\chi \cdot {\begin{pmatrix}\gamma \cos(\varphi )\\\gamma \sin(\varphi )\\1\end{pmatrix}}\quad \quad \quad {\overrightarrow {Q}}(x,y,z)={\begin{pmatrix}\gamma \\\varphi \\\chi \end{pmatrix}}={\begin{pmatrix}{\frac {1}{z}}{\sqrt {x^{2}+y^{2}}}\\\operatorname {arctan2} (x,y)\\z\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e057fd947e5d8e722076530b76a90bb0a26f5b7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:86.43ex; height:10.176ex;" alt="{\displaystyle {\overrightarrow {P}}(\gamma ,\varphi ,\chi )={\begin{pmatrix}x\\y\\z\end{pmatrix}}=\chi \cdot {\begin{pmatrix}\gamma \cos(\varphi )\\\gamma \sin(\varphi )\\1\end{pmatrix}}\quad \quad \quad {\overrightarrow {Q}}(x,y,z)={\begin{pmatrix}\gamma \\\varphi \\\chi \end{pmatrix}}={\begin{pmatrix}{\frac {1}{z}}{\sqrt {x^{2}+y^{2}}}\\\operatorname {arctan2} (x,y)\\z\end{pmatrix}}}"> <dl><dt>Umrechnung eines gegebenen Kegelsegments in Kegelkoordinaten</dt></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Kegelsegment.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Kegelsegment.png/220px-Kegelsegment.png" decoding="async" width="220" height="170" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Kegelsegment.png/330px-Kegelsegment.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Kegelsegment.png/440px-Kegelsegment.png 2x" data-file-width="9660" data-file-height="7480" /></a><figcaption>Kegelsegment mit Höhe h und den Radien r1 und r2</figcaption></figure> Die Parameter eines Kegelsegments seien gegeben durch (siehe nebenstehende Abbildung): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}\leq r\leq r_{2}\quad \quad \quad 0\leq \varphi \leq 2\pi \quad \quad \quad h=z_{2}-z_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <mi>r</mi> <mo>≤</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mn>0</mn> <mo>≤</mo> <mi>φ</mi> <mo>≤</mo> <mn>2</mn> <mi>π</mi> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mi>h</mi> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}\leq r\leq r_{2}\quad \quad \quad 0\leq \varphi \leq 2\pi \quad \quad \quad h=z_{2}-z_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f25dab9b53be80a93c281c6bea2aacce3f1d6fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.31ex; height:2.676ex;" alt="{\displaystyle r_{1}\leq r\leq r_{2}\quad \quad \quad 0\leq \varphi \leq 2\pi \quad \quad \quad h=z_{2}-z_{1}}">,</dd></dl> Dann lassen sich die Grenzen in Kegelparametern wie folgt ausdrücken: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{1}={\frac {r_{2}-r_{1}}{h}}\quad \quad \chi _{1}={\frac {r_{1}}{\gamma _{1}}}=h\cdot {\frac {r_{1}}{r_{2}-r_{1}}}\quad \quad \chi _{2}={\frac {r_{2}}{\gamma _{1}}}=h\cdot {\frac {r_{2}}{r_{2}-r_{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mi>h</mi> </mfrac> </mrow> <mspace width="1em" /> <mspace width="1em" /> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mi>h</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mspace width="1em" /> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mi>h</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{1}={\frac {r_{2}-r_{1}}{h}}\quad \quad \chi _{1}={\frac {r_{1}}{\gamma _{1}}}=h\cdot {\frac {r_{1}}{r_{2}-r_{1}}}\quad \quad \chi _{2}={\frac {r_{2}}{\gamma _{1}}}=h\cdot {\frac {r_{2}}{r_{2}-r_{1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e27cb9f1cacc1f5291a3d1d76a5e567aa12d739" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:67.932ex; height:5.509ex;" alt="{\displaystyle \gamma _{1}={\frac {r_{2}-r_{1}}{h}}\quad \quad \chi _{1}={\frac {r_{1}}{\gamma _{1}}}=h\cdot {\frac {r_{1}}{r_{2}-r_{1}}}\quad \quad \chi _{2}={\frac {r_{2}}{\gamma _{1}}}=h\cdot {\frac {r_{2}}{r_{2}-r_{1}}}}">.</dd></dl> Die Parameter eines soliden Kegelsegmentes bewegen sich also im Bereich: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq \gamma \leq \gamma _{1}\quad \quad \quad 0\leq \varphi \leq 2\pi \quad \quad \quad \chi _{1}\leq \chi \leq \chi _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>γ</mi> <mo>≤</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mn>0</mn> <mo>≤</mo> <mi>φ</mi> <mo>≤</mo> <mn>2</mn> <mi>π</mi> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <mi>χ</mi> <mo>≤</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq \gamma \leq \gamma _{1}\quad \quad \quad 0\leq \varphi \leq 2\pi \quad \quad \quad \chi _{1}\leq \chi \leq \chi _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f14636ab1c9049d7a296834a6c905936a593efa0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.86ex; height:2.676ex;" alt="{\displaystyle 0\leq \gamma \leq \gamma _{1}\quad \quad \quad 0\leq \varphi \leq 2\pi \quad \quad \quad \chi _{1}\leq \chi \leq \chi _{2}}">.</dd></dl> Für die entsprechende Mantelfläche dieses Kegelsegmentes gilt folgende Parameterdarstellung: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =\gamma _{1}\quad \quad \quad 0\leq \varphi \leq 2\pi \quad \quad \quad \chi _{1}\leq \chi \leq \chi _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <mn>0</mn> <mo>≤</mo> <mi>φ</mi> <mo>≤</mo> <mn>2</mn> <mi>π</mi> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <mi>χ</mi> <mo>≤</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma =\gamma _{1}\quad \quad \quad 0\leq \varphi \leq 2\pi \quad \quad \quad \chi _{1}\leq \chi \leq \chi _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3762c79ab76aea300f50a4d453875a5c7301d6ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.6ex; height:2.676ex;" alt="{\displaystyle \gamma =\gamma _{1}\quad \quad \quad 0\leq \varphi \leq 2\pi \quad \quad \quad \chi _{1}\leq \chi \leq \chi _{2}}">.</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Flächennormalenvektor">Flächennormalenvektor</h4>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=31" title="Abschnitt bearbeiten: Flächennormalenvektor" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=31" title="Quellcode des Abschnitts bearbeiten: Flächennormalenvektor">Quelltext bearbeiten</a>]</div> Der <a href="/wiki/Normalenvektor" title="Normalenvektor">Flächennormalenvektor</a> ist orthogonal zur Mantelfläche des Kegels. Er wird benötigt, um z. B. Flussberechnungen durch die Mantelfläche durchzuführen. Den Flächeninhalt der Mantelfläche lässt sich als Doppelintegral über die Norm des Flächennormalenvektors berechnen. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {n}}={\frac {\partial {\overrightarrow {P}}}{\partial \varphi }}\times {\frac {\partial {\overrightarrow {P}}}{\partial \chi }}=\chi \gamma \cdot {\begin{pmatrix}\cos(\varphi )\\\sin(\varphi )\\-\gamma \end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>→</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo>→</mo> </mover> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo>→</mo> </mover> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>χ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>χ</mi> <mi>γ</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>γ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {n}}={\frac {\partial {\overrightarrow {P}}}{\partial \varphi }}\times {\frac {\partial {\overrightarrow {P}}}{\partial \chi }}=\chi \gamma \cdot {\begin{pmatrix}\cos(\varphi )\\\sin(\varphi )\\-\gamma \end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bfdf14846318fffa55e7b28518d180144b0673d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:36.689ex; height:9.509ex;" alt="{\displaystyle {\overrightarrow {n}}={\frac {\partial {\overrightarrow {P}}}{\partial \varphi }}\times {\frac {\partial {\overrightarrow {P}}}{\partial \chi }}=\chi \gamma \cdot {\begin{pmatrix}\cos(\varphi )\\\sin(\varphi )\\-\gamma \end{pmatrix}}}"> <div class="mw-heading mw-heading4"><h4 id="Einheitsvektoren_der_Kegelkoordinaten_in_kartesischen_Komponenten">Einheitsvektoren der Kegelkoordinaten in kartesischen Komponenten</h4>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=32" title="Abschnitt bearbeiten: Einheitsvektoren der Kegelkoordinaten in kartesischen Komponenten" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=32" title="Quellcode des Abschnitts bearbeiten: Einheitsvektoren der Kegelkoordinaten in kartesischen Komponenten">Quelltext bearbeiten</a>]</div> Die Einheitsvektoren in kartesischen Komponenten erhält man durch <a href="/wiki/Norm_(Mathematik)" title="Norm (Mathematik)">Normierung</a> der <a href="/wiki/Tangentenvektor" class="mw-redirect" title="Tangentenvektor">Tangentenvektoren</a> der Parametrisierung. Der Tangentenvektor ergibt sich durch die erste <a href="/wiki/Partielle_Ableitung" title="Partielle Ableitung">partielle Ableitung</a> nach der jeweiligen Variablen. Diese drei Einheitsvektoren bilden eine Normalbasis. Es handelt sich hierbei nicht um eine <a href="/wiki/Orthonormalbasis" title="Orthonormalbasis">Orthonormalbasis</a>, da nicht alle Einheitsvektoren <a href="/wiki/Orthogonalit%C3%A4t" title="Orthogonalität">orthogonal</a> zueinander sind. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {e_{\gamma }}}={\frac {\partial _{\gamma }{\overrightarrow {P}}}{\left\|\partial _{\gamma }{\overrightarrow {P}}\right\|}}={\begin{pmatrix}\cos(\varphi )\\\sin(\varphi )\\0\end{pmatrix}}\quad \quad {\overrightarrow {e_{\varphi }}}={\frac {\partial _{\varphi }{\overrightarrow {P}}}{\left\|\partial _{\varphi }{\overrightarrow {P}}\right\|}}={\begin{pmatrix}-\sin(\varphi )\\\cos(\varphi )\\0\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo>→</mo> </mover> </mrow> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo>→</mo> </mover> </mrow> </mrow> <mo symmetric="true">‖</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="1em" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo>→</mo> </mover> </mrow> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo>→</mo> </mover> </mrow> </mrow> <mo symmetric="true">‖</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {e_{\gamma }}}={\frac {\partial _{\gamma }{\overrightarrow {P}}}{\left\|\partial _{\gamma }{\overrightarrow {P}}\right\|}}={\begin{pmatrix}\cos(\varphi )\\\sin(\varphi )\\0\end{pmatrix}}\quad \quad {\overrightarrow {e_{\varphi }}}={\frac {\partial _{\varphi }{\overrightarrow {P}}}{\left\|\partial _{\varphi }{\overrightarrow {P}}\right\|}}={\begin{pmatrix}-\sin(\varphi )\\\cos(\varphi )\\0\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c92cf993698c2967349ff93242ee0aa25dfebd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:62.928ex; height:10.509ex;" alt="{\displaystyle {\overrightarrow {e_{\gamma }}}={\frac {\partial _{\gamma }{\overrightarrow {P}}}{\left\|\partial _{\gamma }{\overrightarrow {P}}\right\|}}={\begin{pmatrix}\cos(\varphi )\\\sin(\varphi )\\0\end{pmatrix}}\quad \quad {\overrightarrow {e_{\varphi }}}={\frac {\partial _{\varphi }{\overrightarrow {P}}}{\left\|\partial _{\varphi }{\overrightarrow {P}}\right\|}}={\begin{pmatrix}-\sin(\varphi )\\\cos(\varphi )\\0\end{pmatrix}}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {e_{\chi }}}={\frac {\partial _{\chi }{\overrightarrow {P}}}{\left\|\partial _{\chi }{\overrightarrow {P}}\right\|}}={\frac {1}{\sqrt {1+\gamma ^{2}}}}{\begin{pmatrix}\gamma \cos(\varphi )\\\gamma \sin(\varphi )\\1\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>χ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>χ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo>→</mo> </mover> </mrow> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>χ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo>→</mo> </mover> </mrow> </mrow> <mo symmetric="true">‖</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>γ</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>γ</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {e_{\chi }}}={\frac {\partial _{\chi }{\overrightarrow {P}}}{\left\|\partial _{\chi }{\overrightarrow {P}}\right\|}}={\frac {1}{\sqrt {1+\gamma ^{2}}}}{\begin{pmatrix}\gamma \cos(\varphi )\\\gamma \sin(\varphi )\\1\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f55247efdbb4d8bc66543fe3ae02c2a450513de7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:39.285ex; height:10.509ex;" alt="{\displaystyle {\overrightarrow {e_{\chi }}}={\frac {\partial _{\chi }{\overrightarrow {P}}}{\left\|\partial _{\chi }{\overrightarrow {P}}\right\|}}={\frac {1}{\sqrt {1+\gamma ^{2}}}}{\begin{pmatrix}\gamma \cos(\varphi )\\\gamma \sin(\varphi )\\1\end{pmatrix}}}"> <div class="mw-heading mw-heading4"><h4 id="Transformationsmatrizen">Transformationsmatrizen</h4>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=33" title="Abschnitt bearbeiten: Transformationsmatrizen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=33" title="Quellcode des Abschnitts bearbeiten: Transformationsmatrizen">Quelltext bearbeiten</a>]</div> <div class="mw-heading mw-heading5"><h5 id="Jacobi-Matrix_(Funktionalmatrix)">Jacobi-Matrix (Funktionalmatrix)</h5>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=34" title="Abschnitt bearbeiten: Jacobi-Matrix (Funktionalmatrix)" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=34" title="Quellcode des Abschnitts bearbeiten: Jacobi-Matrix (Funktionalmatrix)">Quelltext bearbeiten</a>]</div> Die <a href="/wiki/Jacobi-Matrix" title="Jacobi-Matrix">Funktionalmatrix</a> und ihre Inverse werden benötigt, um später die partiellen Ableitungen zu transformieren. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{f}={\frac {\partial \left(x,y,z\right)}{\partial \left(\gamma ,\varphi ,\chi \right)}}={\begin{pmatrix}\partial _{\gamma }x&\partial _{\varphi }x&\partial _{\chi }x\\\partial _{\gamma }y&\partial _{\varphi }y&\partial _{\chi }y\\\partial _{\gamma }z&\partial _{\varphi }z&\partial _{\chi }z\end{pmatrix}}={\begin{pmatrix}\chi \cos(\varphi )&-\chi \gamma \sin(\varphi )&\gamma \cos(\varphi )\\\chi \sin(\varphi )&\chi \gamma \cos(\varphi )&\gamma \sin(\varphi )\\0&0&1\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow> <mo>(</mo> <mrow> <mi>γ</mi> <mo>,</mo> <mi>φ</mi> <mo>,</mo> <mi>χ</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mi>x</mi> </mtd> <mtd> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mi>x</mi> </mtd> <mtd> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>χ</mi> </mrow> </msub> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mi>y</mi> </mtd> <mtd> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mi>y</mi> </mtd> <mtd> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>χ</mi> </mrow> </msub> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mi>z</mi> </mtd> <mtd> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mi>z</mi> </mtd> <mtd> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>χ</mi> </mrow> </msub> <mi>z</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>χ</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>−</mo> <mi>χ</mi> <mi>γ</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>γ</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>χ</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>χ</mi> <mi>γ</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>γ</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{f}={\frac {\partial \left(x,y,z\right)}{\partial \left(\gamma ,\varphi ,\chi \right)}}={\begin{pmatrix}\partial _{\gamma }x&\partial _{\varphi }x&\partial _{\chi }x\\\partial _{\gamma }y&\partial _{\varphi }y&\partial _{\chi }y\\\partial _{\gamma }z&\partial _{\varphi }z&\partial _{\chi }z\end{pmatrix}}={\begin{pmatrix}\chi \cos(\varphi )&-\chi \gamma \sin(\varphi )&\gamma \cos(\varphi )\\\chi \sin(\varphi )&\chi \gamma \cos(\varphi )&\gamma \sin(\varphi )\\0&0&1\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0d2ced3847b59f1032d0d384d46c36c288dfa75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:80.161ex; height:10.176ex;" alt="{\displaystyle J_{f}={\frac {\partial \left(x,y,z\right)}{\partial \left(\gamma ,\varphi ,\chi \right)}}={\begin{pmatrix}\partial _{\gamma }x&\partial _{\varphi }x&\partial _{\chi }x\\\partial _{\gamma }y&\partial _{\varphi }y&\partial _{\chi }y\\\partial _{\gamma }z&\partial _{\varphi }z&\partial _{\chi }z\end{pmatrix}}={\begin{pmatrix}\chi \cos(\varphi )&-\chi \gamma \sin(\varphi )&\gamma \cos(\varphi )\\\chi \sin(\varphi )&\chi \gamma \cos(\varphi )&\gamma \sin(\varphi )\\0&0&1\end{pmatrix}}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{f}^{-1}={\frac {\partial \left(\gamma ,\varphi ,\chi \right)}{\partial \left(x,y,z\right)}}={\begin{pmatrix}\partial _{x}\gamma &\partial _{y}\gamma &\partial _{z}\gamma \\\partial _{x}\varphi &\partial _{y}\varphi &\partial _{z}\varphi \\\partial _{x}\chi &\partial _{y}\chi &\partial _{z}\chi \end{pmatrix}}={\begin{pmatrix}{\frac {\cos(\varphi )}{\chi }}&{\frac {\sin(\varphi )}{\chi }}&-{\frac {\gamma }{\chi }}\\-{\frac {\sin(\varphi )}{\chi \gamma }}&{\frac {\cos(\varphi )}{\chi \gamma }}&0\\0&0&1\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow> <mo>(</mo> <mrow> <mi>γ</mi> <mo>,</mo> <mi>φ</mi> <mo>,</mo> <mi>χ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>γ</mi> </mtd> <mtd> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>γ</mi> </mtd> <mtd> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>γ</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>φ</mi> </mtd> <mtd> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>φ</mi> </mtd> <mtd> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>χ</mi> </mtd> <mtd> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>χ</mi> </mtd> <mtd> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>χ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mi>χ</mi> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mi>χ</mi> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mi>χ</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>χ</mi> <mi>γ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>χ</mi> <mi>γ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{f}^{-1}={\frac {\partial \left(\gamma ,\varphi ,\chi \right)}{\partial \left(x,y,z\right)}}={\begin{pmatrix}\partial _{x}\gamma &\partial _{y}\gamma &\partial _{z}\gamma \\\partial _{x}\varphi &\partial _{y}\varphi &\partial _{z}\varphi \\\partial _{x}\chi &\partial _{y}\chi &\partial _{z}\chi \end{pmatrix}}={\begin{pmatrix}{\frac {\cos(\varphi )}{\chi }}&{\frac {\sin(\varphi )}{\chi }}&-{\frac {\gamma }{\chi }}\\-{\frac {\sin(\varphi )}{\chi \gamma }}&{\frac {\cos(\varphi )}{\chi \gamma }}&0\\0&0&1\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84b06f97a70ab47da170809e2f2f2692b4943972" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:70.306ex; height:12.343ex;" alt="{\displaystyle J_{f}^{-1}={\frac {\partial \left(\gamma ,\varphi ,\chi \right)}{\partial \left(x,y,z\right)}}={\begin{pmatrix}\partial _{x}\gamma &\partial _{y}\gamma &\partial _{z}\gamma \\\partial _{x}\varphi &\partial _{y}\varphi &\partial _{z}\varphi \\\partial _{x}\chi &\partial _{y}\chi &\partial _{z}\chi \end{pmatrix}}={\begin{pmatrix}{\frac {\cos(\varphi )}{\chi }}&{\frac {\sin(\varphi )}{\chi }}&-{\frac {\gamma }{\chi }}\\-{\frac {\sin(\varphi )}{\chi \gamma }}&{\frac {\cos(\varphi )}{\chi \gamma }}&0\\0&0&1\end{pmatrix}}}"> <div class="mw-heading mw-heading5"><h5 id="Transformationsmatrix_S">Transformationsmatrix S</h5>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=35" title="Abschnitt bearbeiten: Transformationsmatrix S" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=35" title="Quellcode des Abschnitts bearbeiten: Transformationsmatrix S">Quelltext bearbeiten</a>]</div> Die <a href="/wiki/Basiswechsel_(Vektorraum)" title="Basiswechsel (Vektorraum)">Transformationsmatrix</a> wird benötigt, um die Einheitsvektoren und Vektorfelder zu transformieren. Die Matrix setzt sich aus den Einheitsvektoren der Parametrisierung als Spaltenvektoren zusammen. Genaueres findet man unter dem Artikel <a href="/wiki/Basiswechsel_(Vektorraum)" title="Basiswechsel (Vektorraum)">Basiswechsel</a>. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S={\begin{pmatrix}{\overrightarrow {e_{\gamma }}}&{\overrightarrow {e_{\varphi }}}&{\overrightarrow {e_{\chi }}}\end{pmatrix}}={\begin{pmatrix}\cos(\varphi )&-\sin(\varphi )&{\frac {\gamma \cos(\varphi )}{\sqrt {1+\gamma ^{2}}}}\\\sin(\varphi )&\cos(\varphi )&{\frac {\gamma \sin(\varphi )}{\sqrt {1+\gamma ^{2}}}}\\0&0&{\frac {1}{\sqrt {1+\gamma ^{2}}}}\end{pmatrix}}\ ,\quad S^{-1}={\begin{pmatrix}\cos(\varphi )&\sin(\varphi )&-\gamma \\-\sin(\varphi )&\cos(\varphi )&0\\0&0&{\sqrt {1+\gamma ^{2}}}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>χ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mtext> </mtext> <mo>,</mo> <mspace width="1em" /> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>−</mo> <mi>γ</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S={\begin{pmatrix}{\overrightarrow {e_{\gamma }}}&{\overrightarrow {e_{\varphi }}}&{\overrightarrow {e_{\chi }}}\end{pmatrix}}={\begin{pmatrix}\cos(\varphi )&-\sin(\varphi )&{\frac {\gamma \cos(\varphi )}{\sqrt {1+\gamma ^{2}}}}\\\sin(\varphi )&\cos(\varphi )&{\frac {\gamma \sin(\varphi )}{\sqrt {1+\gamma ^{2}}}}\\0&0&{\frac {1}{\sqrt {1+\gamma ^{2}}}}\end{pmatrix}}\ ,\quad S^{-1}={\begin{pmatrix}\cos(\varphi )&\sin(\varphi )&-\gamma \\-\sin(\varphi )&\cos(\varphi )&0\\0&0&{\sqrt {1+\gamma ^{2}}}\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/071e60a3ada47629df2818a397e5747abe4d3421" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:97.847ex; height:18.509ex;" alt="{\displaystyle S={\begin{pmatrix}{\overrightarrow {e_{\gamma }}}&{\overrightarrow {e_{\varphi }}}&{\overrightarrow {e_{\chi }}}\end{pmatrix}}={\begin{pmatrix}\cos(\varphi )&-\sin(\varphi )&{\frac {\gamma \cos(\varphi )}{\sqrt {1+\gamma ^{2}}}}\\\sin(\varphi )&\cos(\varphi )&{\frac {\gamma \sin(\varphi )}{\sqrt {1+\gamma ^{2}}}}\\0&0&{\frac {1}{\sqrt {1+\gamma ^{2}}}}\end{pmatrix}}\ ,\quad S^{-1}={\begin{pmatrix}\cos(\varphi )&\sin(\varphi )&-\gamma \\-\sin(\varphi )&\cos(\varphi )&0\\0&0&{\sqrt {1+\gamma ^{2}}}\end{pmatrix}}}"> <div class="mw-heading mw-heading4"><h4 id="Transformation_der_partiellen_Ableitungen">Transformation der partiellen Ableitungen</h4>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=36" title="Abschnitt bearbeiten: Transformation der partiellen Ableitungen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=36" title="Quellcode des Abschnitts bearbeiten: Transformation der partiellen Ableitungen">Quelltext bearbeiten</a>]</div> Die <a href="/wiki/Partielle_Ableitung" title="Partielle Ableitung">partiellen Ableitungen</a> lassen sich mit der inversen Jacobi-Matrix transformieren. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\end{pmatrix}}^{T}=(J_{f}^{-1})^{T}\cdot {\begin{pmatrix}{\frac {\partial }{\partial \gamma }}&{\frac {\partial }{\partial \varphi }}&{\frac {\partial }{\partial \chi }}\end{pmatrix}}^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msubsup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>γ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>χ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\end{pmatrix}}^{T}=(J_{f}^{-1})^{T}\cdot {\begin{pmatrix}{\frac {\partial }{\partial \gamma }}&{\frac {\partial }{\partial \varphi }}&{\frac {\partial }{\partial \chi }}\end{pmatrix}}^{T}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f67709b7cf82dc5a620de08ee621c0537a56797" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:47.09ex; height:5.176ex;" alt="{\displaystyle {\begin{pmatrix}{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\end{pmatrix}}^{T}=(J_{f}^{-1})^{T}\cdot {\begin{pmatrix}{\frac {\partial }{\partial \gamma }}&{\frac {\partial }{\partial \varphi }}&{\frac {\partial }{\partial \chi }}\end{pmatrix}}^{T}}"> Als Ergebnis erhält man: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial }{\partial x}}={\frac {\cos(\varphi )}{\chi }}{\frac {\partial }{\partial \gamma }}-{\frac {\sin(\varphi )}{\gamma \chi }}{\frac {\partial }{\partial \varphi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mi>χ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>γ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>γ</mi> <mi>χ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial x}}={\frac {\cos(\varphi )}{\chi }}{\frac {\partial }{\partial \gamma }}-{\frac {\sin(\varphi )}{\gamma \chi }}{\frac {\partial }{\partial \varphi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24481f86587789e96a45cf5b07609c46b2a5209f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.811ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial }{\partial x}}={\frac {\cos(\varphi )}{\chi }}{\frac {\partial }{\partial \gamma }}-{\frac {\sin(\varphi )}{\gamma \chi }}{\frac {\partial }{\partial \varphi }}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial }{\partial y}}={\frac {\sin(\varphi )}{\chi }}{\frac {\partial }{\partial \gamma }}+{\frac {\cos(\varphi )}{\gamma \chi }}{\frac {\partial }{\partial \varphi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mi>χ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>γ</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>γ</mi> <mi>χ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial y}}={\frac {\sin(\varphi )}{\chi }}{\frac {\partial }{\partial \gamma }}+{\frac {\cos(\varphi )}{\gamma \chi }}{\frac {\partial }{\partial \varphi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2055182ef7dc1beb39cb72fd7bb0096009fb8218" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.637ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial }{\partial y}}={\frac {\sin(\varphi )}{\chi }}{\frac {\partial }{\partial \gamma }}+{\frac {\cos(\varphi )}{\gamma \chi }}{\frac {\partial }{\partial \varphi }}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial }{\partial z}}={\frac {\partial }{\partial \chi }}-{\frac {\gamma }{\chi }}{\frac {\partial }{\partial \gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>χ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mi>χ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>γ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial z}}={\frac {\partial }{\partial \chi }}-{\frac {\gamma }{\chi }}{\frac {\partial }{\partial \gamma }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9516260d2c932022c9321c96a11530d1378a0e52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.498ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial }{\partial z}}={\frac {\partial }{\partial \chi }}-{\frac {\gamma }{\chi }}{\frac {\partial }{\partial \gamma }}}"> <div class="mw-heading mw-heading4"><h4 id="Transformation_der_Einheitsvektoren">Transformation der Einheitsvektoren</h4>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=37" title="Abschnitt bearbeiten: Transformation der Einheitsvektoren" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=37" title="Quellcode des Abschnitts bearbeiten: Transformation der Einheitsvektoren">Quelltext bearbeiten</a>]</div> Die <a href="/wiki/Einheitsvektor" title="Einheitsvektor">Einheitsvektoren</a> lassen sich mit der inversen Transformationsmatrix transformieren. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}{\overrightarrow {e_{x}}}&{\overrightarrow {e_{y}}}&{\overrightarrow {e_{z}}}\end{pmatrix}}={\begin{pmatrix}{\overrightarrow {e_{\gamma }}}&{\overrightarrow {e_{\varphi }}}&{\overrightarrow {e_{\chi }}}\end{pmatrix}}\cdot S^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>χ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>⋅</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}{\overrightarrow {e_{x}}}&{\overrightarrow {e_{y}}}&{\overrightarrow {e_{z}}}\end{pmatrix}}={\begin{pmatrix}{\overrightarrow {e_{\gamma }}}&{\overrightarrow {e_{\varphi }}}&{\overrightarrow {e_{\chi }}}\end{pmatrix}}\cdot S^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ece4c0c7ff8af1f96d314287a4be7df38eec1cc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:39.137ex; height:4.843ex;" alt="{\displaystyle {\begin{pmatrix}{\overrightarrow {e_{x}}}&{\overrightarrow {e_{y}}}&{\overrightarrow {e_{z}}}\end{pmatrix}}={\begin{pmatrix}{\overrightarrow {e_{\gamma }}}&{\overrightarrow {e_{\varphi }}}&{\overrightarrow {e_{\chi }}}\end{pmatrix}}\cdot S^{-1}}"> Als Ergebnis erhält man: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {e_{x}}}=\cos(\varphi )\cdot {\overrightarrow {e_{\gamma }}}-\sin(\varphi )\cdot {\overrightarrow {e_{\varphi }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {e_{x}}}=\cos(\varphi )\cdot {\overrightarrow {e_{\gamma }}}-\sin(\varphi )\cdot {\overrightarrow {e_{\varphi }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a47a2668c0f86e64ccff4b9d2af03bd047020e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-top: -0.34ex; width:29.091ex; height:3.843ex;" alt="{\displaystyle {\overrightarrow {e_{x}}}=\cos(\varphi )\cdot {\overrightarrow {e_{\gamma }}}-\sin(\varphi )\cdot {\overrightarrow {e_{\varphi }}}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {e_{y}}}=\sin(\varphi )\cdot {\overrightarrow {e_{\gamma }}}+\cos(\varphi )\cdot {\overrightarrow {e_{\varphi }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {e_{y}}}=\sin(\varphi )\cdot {\overrightarrow {e_{\gamma }}}+\cos(\varphi )\cdot {\overrightarrow {e_{\varphi }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b9958572a14773b83a1bb373716409eb1e101c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-top: -0.34ex; width:29.091ex; height:3.843ex;" alt="{\displaystyle {\overrightarrow {e_{y}}}=\sin(\varphi )\cdot {\overrightarrow {e_{\gamma }}}+\cos(\varphi )\cdot {\overrightarrow {e_{\varphi }}}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {e_{z}}}={\sqrt {1+\gamma ^{2}}}\cdot {\overrightarrow {e_{\chi }}}-\gamma \cdot {\overrightarrow {e_{\gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>χ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>−</mo> <mi>γ</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {e_{z}}}={\sqrt {1+\gamma ^{2}}}\cdot {\overrightarrow {e_{\chi }}}-\gamma \cdot {\overrightarrow {e_{\gamma }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fbbd68ebf2a1923f60b0f253b55be0e34c2ad79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:26.212ex; height:4.843ex;" alt="{\displaystyle {\overrightarrow {e_{z}}}={\sqrt {1+\gamma ^{2}}}\cdot {\overrightarrow {e_{\chi }}}-\gamma \cdot {\overrightarrow {e_{\gamma }}}}"> <div class="mw-heading mw-heading4"><h4 id="Transformation_von_Vektorfeldern">Transformation von Vektorfeldern</h4>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=38" title="Abschnitt bearbeiten: Transformation von Vektorfeldern" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=38" title="Quellcode des Abschnitts bearbeiten: Transformation von Vektorfeldern">Quelltext bearbeiten</a>]</div> <a href="/wiki/Vektorfeld" title="Vektorfeld">Vektorfelder</a> lassen sich durch Matrixmultiplikation mit der Transformationsmatrix transformieren. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}F_{x}\\F_{y}\\F_{z}\end{pmatrix}}=S\cdot {\begin{pmatrix}F_{\gamma }\\F_{\varphi }\\F_{\chi }\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mi>S</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>χ</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}F_{x}\\F_{y}\\F_{z}\end{pmatrix}}=S\cdot {\begin{pmatrix}F_{\gamma }\\F_{\varphi }\\F_{\chi }\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eeb1d4b86e56ce442ae68ae2212a25f708e788d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:21.382ex; height:10.176ex;" alt="{\displaystyle {\begin{pmatrix}F_{x}\\F_{y}\\F_{z}\end{pmatrix}}=S\cdot {\begin{pmatrix}F_{\gamma }\\F_{\varphi }\\F_{\chi }\end{pmatrix}}}"> Als Ergebnis erhält man: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{x}=\cos(\varphi )\cdot F_{\gamma }-\sin(\varphi )\cdot F_{\varphi }+{\frac {\gamma \cos(\varphi )}{\sqrt {1+\gamma ^{2}}}}\cdot F_{\chi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>χ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{x}=\cos(\varphi )\cdot F_{\gamma }-\sin(\varphi )\cdot F_{\varphi }+{\frac {\gamma \cos(\varphi )}{\sqrt {1+\gamma ^{2}}}}\cdot F_{\chi }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb278f47698e502bb5abefaa6aea04f151da3828" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.783ex; height:7.009ex;" alt="{\displaystyle F_{x}=\cos(\varphi )\cdot F_{\gamma }-\sin(\varphi )\cdot F_{\varphi }+{\frac {\gamma \cos(\varphi )}{\sqrt {1+\gamma ^{2}}}}\cdot F_{\chi }}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{y}=\sin(\varphi )\cdot F_{\gamma }+\cos(\varphi )\cdot F_{\varphi }+{\frac {\gamma \sin(\varphi )}{\sqrt {1+\gamma ^{2}}}}\cdot F_{\chi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>χ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{y}=\sin(\varphi )\cdot F_{\gamma }+\cos(\varphi )\cdot F_{\varphi }+{\frac {\gamma \sin(\varphi )}{\sqrt {1+\gamma ^{2}}}}\cdot F_{\chi }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c464f3b0c77ef5e9018d7492f84ea00be557daa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.66ex; height:7.009ex;" alt="{\displaystyle F_{y}=\sin(\varphi )\cdot F_{\gamma }+\cos(\varphi )\cdot F_{\varphi }+{\frac {\gamma \sin(\varphi )}{\sqrt {1+\gamma ^{2}}}}\cdot F_{\chi }}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{z}={\frac {1}{\sqrt {1+\gamma ^{2}}}}\cdot F_{\chi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>χ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{z}={\frac {1}{\sqrt {1+\gamma ^{2}}}}\cdot F_{\chi }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3149cdd6ac9837d0e6fc5164043cb2e6c4363e7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.526ex; height:6.509ex;" alt="{\displaystyle F_{z}={\frac {1}{\sqrt {1+\gamma ^{2}}}}\cdot F_{\chi }}"> <div class="mw-heading mw-heading4"><h4 id="Oberflächen-_und_Volumendifferential">Oberflächen- und Volumendifferential</h4>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=39" title="Abschnitt bearbeiten: Oberflächen- und Volumendifferential" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=39" title="Quellcode des Abschnitts bearbeiten: Oberflächen- und Volumendifferential">Quelltext bearbeiten</a>]</div> Das Volumendifferential lässt sich über die Determinante der Jacobi-Matrix angeben. Dies bietet die Möglichkeit z. B. das Volumen eines Kegels per <a href="/wiki/Volumenintegral" title="Volumenintegral">Dreifachintegral</a> zu berechnen. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dV=\det J_{f}\cdot d\gamma d\chi d\varphi =\chi ^{2}\gamma \cdot d\gamma d\chi d\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>V</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>⋅</mo> <mi>d</mi> <mi>γ</mi> <mi>d</mi> <mi>χ</mi> <mi>d</mi> <mi>φ</mi> <mo>=</mo> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>γ</mi> <mo>⋅</mo> <mi>d</mi> <mi>γ</mi> <mi>d</mi> <mi>χ</mi> <mi>d</mi> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dV=\det J_{f}\cdot d\gamma d\chi d\varphi =\chi ^{2}\gamma \cdot d\gamma d\chi d\varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77413a726b8402b1a760c84e71eb3846bd8e191f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.143ex; height:3.343ex;" alt="{\displaystyle dV=\det J_{f}\cdot d\gamma d\chi d\varphi =\chi ^{2}\gamma \cdot d\gamma d\chi d\varphi }"> Das Oberflächendifferential lässt sich mit der Norm des Flächennormalenvektors angeben. Damit kann man z. B. per Doppelintegral den Flächeninhalt der Mantelfläche bestimmen. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dA=\|{\overrightarrow {n}}\|\cdot d\chi d\varphi =\chi \gamma {\sqrt {1+\gamma ^{2}}}\cdot d\chi d\varphi \quad {\text{wobei}}\quad \gamma ={\text{const.}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>A</mi> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>→</mo> </mover> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <mi>d</mi> <mi>χ</mi> <mi>d</mi> <mi>φ</mi> <mo>=</mo> <mi>χ</mi> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>⋅</mo> <mi>d</mi> <mi>χ</mi> <mi>d</mi> <mi>φ</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>wobei</mtext> </mrow> <mspace width="1em" /> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>const.</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dA=\|{\overrightarrow {n}}\|\cdot d\chi d\varphi =\chi \gamma {\sqrt {1+\gamma ^{2}}}\cdot d\chi d\varphi \quad {\text{wobei}}\quad \gamma ={\text{const.}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edbd7b5db839add05abe0958e5b9cd5552d3e1ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:60.128ex; height:4.843ex;" alt="{\displaystyle dA=\|{\overrightarrow {n}}\|\cdot d\chi d\varphi =\chi \gamma {\sqrt {1+\gamma ^{2}}}\cdot d\chi d\varphi \quad {\text{wobei}}\quad \gamma ={\text{const.}}}"> <div class="mw-heading mw-heading4"><h4 id="Transformierte_Vektor-Differentialoperatoren">Transformierte Vektor-Differentialoperatoren</h4>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=40" title="Abschnitt bearbeiten: Transformierte Vektor-Differentialoperatoren" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=40" title="Quellcode des Abschnitts bearbeiten: Transformierte Vektor-Differentialoperatoren">Quelltext bearbeiten</a>]</div> <div class="mw-heading mw-heading5"><h5 id="Nabla-Operator">Nabla-Operator</h5>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=41" title="Abschnitt bearbeiten: Nabla-Operator" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=41" title="Quellcode des Abschnitts bearbeiten: Nabla-Operator">Quelltext bearbeiten</a>]</div> Eine Darstellung des <a href="/wiki/Nabla-Operator" title="Nabla-Operator">Nabla-Operators</a> in Kegelkoordinaten erhält man, indem man die transformierten Einheitsvektoren und partielle Ableitungen in den kartesischen Nabla-Operator einsetzt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla =\left({\frac {1+\gamma ^{2}}{\chi }}{\frac {\partial }{\partial \gamma }}-\gamma {\frac {\partial }{\partial \chi }}\right)\cdot {\overrightarrow {e_{\gamma }}}+\left({\frac {1}{\gamma \chi }}{\frac {\partial }{\partial \varphi }}\right)\cdot {\overrightarrow {e_{\varphi }}}+{\sqrt {1+\gamma ^{2}}}\left({\frac {\partial }{\partial \chi }}-{\frac {\gamma }{\chi }}{\frac {\partial }{\partial \gamma }}\right)\cdot {\overrightarrow {e_{\chi }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>χ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>γ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>χ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>γ</mi> <mi>χ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>χ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mi>χ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>γ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>χ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla =\left({\frac {1+\gamma ^{2}}{\chi }}{\frac {\partial }{\partial \gamma }}-\gamma {\frac {\partial }{\partial \chi }}\right)\cdot {\overrightarrow {e_{\gamma }}}+\left({\frac {1}{\gamma \chi }}{\frac {\partial }{\partial \varphi }}\right)\cdot {\overrightarrow {e_{\varphi }}}+{\sqrt {1+\gamma ^{2}}}\left({\frac {\partial }{\partial \chi }}-{\frac {\gamma }{\chi }}{\frac {\partial }{\partial \gamma }}\right)\cdot {\overrightarrow {e_{\chi }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8102d8ae6ba3e274b4d0ad5d2409d7c1db60f0da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:79.94ex; height:6.343ex;" alt="{\displaystyle \nabla =\left({\frac {1+\gamma ^{2}}{\chi }}{\frac {\partial }{\partial \gamma }}-\gamma {\frac {\partial }{\partial \chi }}\right)\cdot {\overrightarrow {e_{\gamma }}}+\left({\frac {1}{\gamma \chi }}{\frac {\partial }{\partial \varphi }}\right)\cdot {\overrightarrow {e_{\varphi }}}+{\sqrt {1+\gamma ^{2}}}\left({\frac {\partial }{\partial \chi }}-{\frac {\gamma }{\chi }}{\frac {\partial }{\partial \gamma }}\right)\cdot {\overrightarrow {e_{\chi }}}}"></dd></dl> <div class="mw-heading mw-heading5"><h5 id="Gradient_2">Gradient</h5>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=42" title="Abschnitt bearbeiten: Gradient" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=42" title="Quellcode des Abschnitts bearbeiten: Gradient">Quelltext bearbeiten</a>]</div> Den <a href="/wiki/Gradient_(Mathematik)" title="Gradient (Mathematik)">Gradienten</a> in Kegelkoordinaten erhält man, indem man den transformieren Nabla-Operator auf ein Skalarfeld in Kegelkoordinaten anwendet. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {grad} \phi =\nabla \phi =\left({\frac {1+\gamma ^{2}}{\chi }}{\frac {\partial \phi }{\partial \gamma }}-\gamma {\frac {\partial \phi }{\partial \chi }}\right)\cdot {\overrightarrow {e_{\gamma }}}+\left({\frac {1}{\gamma \chi }}{\frac {\partial \phi }{\partial \varphi }}\right)\cdot {\overrightarrow {e_{\varphi }}}+{\sqrt {1+\gamma ^{2}}}\left({\frac {\partial \phi }{\partial \chi }}-{\frac {\gamma }{\chi }}{\frac {\partial \phi }{\partial \gamma }}\right)\cdot {\overrightarrow {e_{\chi }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>grad</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mi>ϕ</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>χ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>γ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>χ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>γ</mi> <mi>χ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>χ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mi>χ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>γ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>χ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {grad} \phi =\nabla \phi =\left({\frac {1+\gamma ^{2}}{\chi }}{\frac {\partial \phi }{\partial \gamma }}-\gamma {\frac {\partial \phi }{\partial \chi }}\right)\cdot {\overrightarrow {e_{\gamma }}}+\left({\frac {1}{\gamma \chi }}{\frac {\partial \phi }{\partial \varphi }}\right)\cdot {\overrightarrow {e_{\varphi }}}+{\sqrt {1+\gamma ^{2}}}\left({\frac {\partial \phi }{\partial \chi }}-{\frac {\gamma }{\chi }}{\frac {\partial \phi }{\partial \gamma }}\right)\cdot {\overrightarrow {e_{\chi }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36e103226b815b8a9ae248cfa465585ac6eb70f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:90.972ex; height:6.343ex;" alt="{\displaystyle \operatorname {grad} \phi =\nabla \phi =\left({\frac {1+\gamma ^{2}}{\chi }}{\frac {\partial \phi }{\partial \gamma }}-\gamma {\frac {\partial \phi }{\partial \chi }}\right)\cdot {\overrightarrow {e_{\gamma }}}+\left({\frac {1}{\gamma \chi }}{\frac {\partial \phi }{\partial \varphi }}\right)\cdot {\overrightarrow {e_{\varphi }}}+{\sqrt {1+\gamma ^{2}}}\left({\frac {\partial \phi }{\partial \chi }}-{\frac {\gamma }{\chi }}{\frac {\partial \phi }{\partial \gamma }}\right)\cdot {\overrightarrow {e_{\chi }}}}"></dd></dl> <div class="mw-heading mw-heading5"><h5 id="Divergenz_2">Divergenz</h5>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=43" title="Abschnitt bearbeiten: Divergenz" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=43" title="Quellcode des Abschnitts bearbeiten: Divergenz">Quelltext bearbeiten</a>]</div> Den Operator für die <a href="/wiki/Divergenz_eines_Vektorfeldes" title="Divergenz eines Vektorfeldes">Divergenz eines Vektorfeldes</a> erhält man, indem man den Nabla-Operator auf das Vektorfeld in Kegelkoordinaten anwendet: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {div} {\overrightarrow {F}}=\nabla \cdot {\overrightarrow {F}}={\frac {1}{\gamma \chi }}\cdot \left({\frac {\partial \left(F_{\gamma }\cdot \gamma \right)}{\partial \gamma }}+{\frac {\partial F_{\varphi }}{\partial \varphi }}\right)+{\frac {1}{\chi ^{2}{\sqrt {1+\gamma ^{2}}}}}{\frac {\partial \left(F_{\chi }\cdot \chi ^{2}\right)}{\partial \chi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>div</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo>→</mo> </mover> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo>→</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>γ</mi> <mi>χ</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mo>⋅</mo> <mi>γ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>γ</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>χ</mi> </mrow> </msub> <mo>⋅</mo> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>χ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {div} {\overrightarrow {F}}=\nabla \cdot {\overrightarrow {F}}={\frac {1}{\gamma \chi }}\cdot \left({\frac {\partial \left(F_{\gamma }\cdot \gamma \right)}{\partial \gamma }}+{\frac {\partial F_{\varphi }}{\partial \varphi }}\right)+{\frac {1}{\chi ^{2}{\sqrt {1+\gamma ^{2}}}}}{\frac {\partial \left(F_{\chi }\cdot \chi ^{2}\right)}{\partial \chi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6636e5f88db2b462639a8c3794360412eafda3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:71.544ex; height:7.343ex;" alt="{\displaystyle \operatorname {div} {\overrightarrow {F}}=\nabla \cdot {\overrightarrow {F}}={\frac {1}{\gamma \chi }}\cdot \left({\frac {\partial \left(F_{\gamma }\cdot \gamma \right)}{\partial \gamma }}+{\frac {\partial F_{\varphi }}{\partial \varphi }}\right)+{\frac {1}{\chi ^{2}{\sqrt {1+\gamma ^{2}}}}}{\frac {\partial \left(F_{\chi }\cdot \chi ^{2}\right)}{\partial \chi }}}"></dd></dl> <div class="mw-heading mw-heading5"><h5 id="Laplace-Operator">Laplace-Operator</h5>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=44" title="Abschnitt bearbeiten: Laplace-Operator" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=44" title="Quellcode des Abschnitts bearbeiten: Laplace-Operator">Quelltext bearbeiten</a>]</div> Der Laplace-Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }"> ist die Divergenz eines Gradienten. In Kegelkoordinaten ergibt dies den folgenden Operator: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \phi =\operatorname {div} (\operatorname {grad} \phi )=\left({\frac {1+\gamma ^{2}}{\chi ^{2}}}\right){\frac {\partial ^{2}\phi }{\partial \gamma ^{2}}}+\left({\frac {1+2\gamma ^{2}}{\gamma \chi ^{2}}}\right){\frac {\partial \phi }{\partial \gamma }}+\left({\frac {1}{\gamma \chi }}\right)^{2}{\frac {\partial ^{2}\phi }{\partial \varphi ^{2}}}+{\frac {\partial ^{2}\phi }{\partial \chi ^{2}}}-\left({\frac {2\gamma }{\chi }}\right){\frac {\partial ^{2}\phi }{\partial \gamma \partial \chi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>ϕ</mi> <mo>=</mo> <mi>div</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>grad</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ϕ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mn>2</mn> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>γ</mi> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>γ</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>γ</mi> <mi>χ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ϕ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ϕ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>γ</mi> </mrow> <mi>χ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ϕ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>γ</mi> <mi mathvariant="normal">∂</mi> <mi>χ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \phi =\operatorname {div} (\operatorname {grad} \phi )=\left({\frac {1+\gamma ^{2}}{\chi ^{2}}}\right){\frac {\partial ^{2}\phi }{\partial \gamma ^{2}}}+\left({\frac {1+2\gamma ^{2}}{\gamma \chi ^{2}}}\right){\frac {\partial \phi }{\partial \gamma }}+\left({\frac {1}{\gamma \chi }}\right)^{2}{\frac {\partial ^{2}\phi }{\partial \varphi ^{2}}}+{\frac {\partial ^{2}\phi }{\partial \chi ^{2}}}-\left({\frac {2\gamma }{\chi }}\right){\frac {\partial ^{2}\phi }{\partial \gamma \partial \chi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cc168fb64d22b6c008ff0bf17b10e3fab3839db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:94.121ex; height:6.509ex;" alt="{\displaystyle \Delta \phi =\operatorname {div} (\operatorname {grad} \phi )=\left({\frac {1+\gamma ^{2}}{\chi ^{2}}}\right){\frac {\partial ^{2}\phi }{\partial \gamma ^{2}}}+\left({\frac {1+2\gamma ^{2}}{\gamma \chi ^{2}}}\right){\frac {\partial \phi }{\partial \gamma }}+\left({\frac {1}{\gamma \chi }}\right)^{2}{\frac {\partial ^{2}\phi }{\partial \varphi ^{2}}}+{\frac {\partial ^{2}\phi }{\partial \chi ^{2}}}-\left({\frac {2\gamma }{\chi }}\right){\frac {\partial ^{2}\phi }{\partial \gamma \partial \chi }}}"></dd></dl> <div class="mw-heading mw-heading5"><h5 id="Rotation_2">Rotation</h5>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=45" title="Abschnitt bearbeiten: Rotation" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=45" title="Quellcode des Abschnitts bearbeiten: Rotation">Quelltext bearbeiten</a>]</div> Die <a href="/wiki/Rotation_eines_Vektorfeldes" title="Rotation eines Vektorfeldes">Rotation eines Vektorfeldes</a> lässt sich als <a href="/wiki/Kreuzprodukt" title="Kreuzprodukt">Kreuzprodukt</a> des Nabla-Operators mit den Elementen des Vektorfelds auffassen: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {rot} {\overrightarrow {F}}=\nabla \times {\overrightarrow {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>rot</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo>→</mo> </mover> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {rot} {\overrightarrow {F}}=\nabla \times {\overrightarrow {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df6c61175ea3ae8dd326e54f96eb1231ff5a44c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.595ex; height:3.676ex;" alt="{\displaystyle \operatorname {rot} {\overrightarrow {F}}=\nabla \times {\overrightarrow {F}}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad \quad =\left({\frac {\sqrt {1+\gamma ^{2}}}{\gamma \chi }}{\frac {\partial F_{\chi }}{\partial \varphi }}+{\frac {1}{\chi }}{\frac {\partial F_{\gamma }}{\partial \varphi }}-{\frac {1}{\chi }}{\frac {\partial \left(F_{\varphi }\cdot \chi \right)}{\partial \chi }}\right){\overrightarrow {e_{\gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <mspace width="1em" /> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <mrow> <mi>γ</mi> <mi>χ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>χ</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>χ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>χ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>⋅</mo> <mi>χ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>χ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad \quad =\left({\frac {\sqrt {1+\gamma ^{2}}}{\gamma \chi }}{\frac {\partial F_{\chi }}{\partial \varphi }}+{\frac {1}{\chi }}{\frac {\partial F_{\gamma }}{\partial \varphi }}-{\frac {1}{\chi }}{\frac {\partial \left(F_{\varphi }\cdot \chi \right)}{\partial \chi }}\right){\overrightarrow {e_{\gamma }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d24670179c21d63f7ac0637d6cb6b6edab67ce91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:55.543ex; height:7.509ex;" alt="{\displaystyle \qquad \quad =\left({\frac {\sqrt {1+\gamma ^{2}}}{\gamma \chi }}{\frac {\partial F_{\chi }}{\partial \varphi }}+{\frac {1}{\chi }}{\frac {\partial F_{\gamma }}{\partial \varphi }}-{\frac {1}{\chi }}{\frac {\partial \left(F_{\varphi }\cdot \chi \right)}{\partial \chi }}\right){\overrightarrow {e_{\gamma }}}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad \quad +\left({\frac {\partial F_{\gamma }}{\partial \chi }}+{\frac {\gamma }{\sqrt {1+\gamma ^{2}}}}{\frac {\partial F_{\chi }}{\partial \chi }}-{\frac {\gamma }{\chi }}{\frac {\partial F_{\gamma }}{\partial \gamma }}-{\frac {\sqrt {1+\gamma ^{2}}}{\chi }}{\frac {\partial F_{\chi }}{\partial \gamma }}\right){\overrightarrow {e_{\varphi }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <mspace width="1em" /> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>χ</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>χ</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>χ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mi>χ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>γ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <mi>χ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>χ</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>γ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad \quad +\left({\frac {\partial F_{\gamma }}{\partial \chi }}+{\frac {\gamma }{\sqrt {1+\gamma ^{2}}}}{\frac {\partial F_{\chi }}{\partial \chi }}-{\frac {\gamma }{\chi }}{\frac {\partial F_{\gamma }}{\partial \gamma }}-{\frac {\sqrt {1+\gamma ^{2}}}{\chi }}{\frac {\partial F_{\chi }}{\partial \gamma }}\right){\overrightarrow {e_{\varphi }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c83648548bccb50ecabf393b2bfb0eb6d7f716c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:64.925ex; height:7.509ex;" alt="{\displaystyle \qquad \quad +\left({\frac {\partial F_{\gamma }}{\partial \chi }}+{\frac {\gamma }{\sqrt {1+\gamma ^{2}}}}{\frac {\partial F_{\chi }}{\partial \chi }}-{\frac {\gamma }{\chi }}{\frac {\partial F_{\gamma }}{\partial \gamma }}-{\frac {\sqrt {1+\gamma ^{2}}}{\chi }}{\frac {\partial F_{\chi }}{\partial \gamma }}\right){\overrightarrow {e_{\varphi }}}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad \quad +\left({\frac {\sqrt {1+\gamma ^{2}}}{\chi \gamma }}{\frac {\partial (\gamma F_{\varphi })}{\partial \gamma }}-{\frac {\sqrt {\gamma ^{2}+1}}{\chi \gamma }}{\frac {\partial F_{\gamma }}{\partial \varphi }}-{\frac {1}{\chi }}{\frac {\partial F_{\chi }}{\partial \varphi }}\right){\overrightarrow {e_{\chi }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <mspace width="1em" /> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <mrow> <mi>χ</mi> <mi>γ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>γ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> <mrow> <mi>χ</mi> <mi>γ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>χ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>χ</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>χ</mi> </mrow> </msub> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad \quad +\left({\frac {\sqrt {1+\gamma ^{2}}}{\chi \gamma }}{\frac {\partial (\gamma F_{\varphi })}{\partial \gamma }}-{\frac {\sqrt {\gamma ^{2}+1}}{\chi \gamma }}{\frac {\partial F_{\gamma }}{\partial \varphi }}-{\frac {1}{\chi }}{\frac {\partial F_{\chi }}{\partial \varphi }}\right){\overrightarrow {e_{\chi }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b65ed5e3aae67dc9b94ae71b8a428281b4221842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:60.253ex; height:7.509ex;" alt="{\displaystyle \qquad \quad +\left({\frac {\sqrt {1+\gamma ^{2}}}{\chi \gamma }}{\frac {\partial (\gamma F_{\varphi })}{\partial \gamma }}-{\frac {\sqrt {\gamma ^{2}+1}}{\chi \gamma }}{\frac {\partial F_{\gamma }}{\partial \varphi }}-{\frac {1}{\chi }}{\frac {\partial F_{\chi }}{\partial \varphi }}\right){\overrightarrow {e_{\chi }}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Kugelkoordinaten">Kugelkoordinaten</h3>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=46" title="Abschnitt bearbeiten: Kugelkoordinaten" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=46" title="Quellcode des Abschnitts bearbeiten: Kugelkoordinaten">Quelltext bearbeiten</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/Datei:Kugelkoord-def.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Kugelkoord-def.svg/300px-Kugelkoord-def.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Kugelkoord-def.svg/450px-Kugelkoord-def.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/Kugelkoord-def.svg/600px-Kugelkoord-def.svg.png 2x" data-file-width="261" data-file-height="261" /></a><figcaption>Kugelkoordinaten</figcaption></figure> <div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="/wiki/Kugelkoordinaten" title="Kugelkoordinaten">Kugelkoordinaten</a></div> Kugelkoordinaten sind im Wesentlichen ebene Polarkoordinaten, die um eine dritte Koordinate ergänzt sind. Dies geschieht, indem man einen Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \in [0,\pi ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta \in [0,\pi ]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c833964ea08aa30df8b6f56664461a5499b38144" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.753ex; height:2.843ex;" alt="{\displaystyle \theta \in [0,\pi ]}"> für die dritte Achse spezifiziert. Diese dritte Koordinate beschreibt den Winkel zwischen dem Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aec3c9ce13b53e9e24c98e7cce4212627884c91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}"> zum Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> und der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}">-Achse. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> ist genau dann null, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> in der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}">-Achse liegt. <div class="mw-heading mw-heading2"><h2 id="n-dimensionale_Polarkoordinaten">n-dimensionale Polarkoordinaten</h2>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=47" title="Abschnitt bearbeiten: n-dimensionale Polarkoordinaten" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=47" title="Quellcode des Abschnitts bearbeiten: n-dimensionale Polarkoordinaten">Quelltext bearbeiten</a>]</div> Es lässt sich auch eine Verallgemeinerung der Polarkoordinaten mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73136e4a27fe39c123d16a7808e76d3162ce42bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 3}"> für einen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensionalen Raum mit kartesischen Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}\in \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c16c64c292c38a4a3ebfee3be0ade520d4463413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.648ex; height:2.509ex;" alt="{\displaystyle x_{i}\in \mathbb {R} }"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,\dotsc ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,\dotsc ,n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7f2132430a61b900cf2c4380774394ca9f09c8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.636ex; height:2.509ex;" alt="{\displaystyle i=1,\dotsc ,n}"> angeben. Dazu führt man für jede neue Dimension (induktiver Aufbau über selbige) einen weiteren Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vartheta _{n-2}\in [0,\pi ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta _{n-2}\in [0,\pi ]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd757b4fc3635a37579afba4382be3d4cda87789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.356ex; height:2.843ex;" alt="{\displaystyle \vartheta _{n-2}\in [0,\pi ]}"> ein, der den Winkel zwischen dem Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c520ee2cb6ccf8a93c89a8c58a8378796bd52e53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.067ex; height:2.343ex;" alt="{\displaystyle x\in \mathbb {R} ^{n}}"> und der neuen, positiven Koordinatenachse für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5ea190699149306d242b70439e663559e3ffbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}"> angibt. Mit demselben Vorgehen kann in konsistenter Weise die Winkelkoordinate des 2-dimensionalen Raumes mittels <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =\vartheta _{0}\in [0,\pi ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =\vartheta _{0}\in [0,\pi ]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/049d26ceee7c1903d4aefcec69374863275f25f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.709ex; height:2.843ex;" alt="{\displaystyle \varphi =\vartheta _{0}\in [0,\pi ]}"> induktiv aus dem Zahlenstrahl konstruiert werden, sofern für die radiale Koordinate auch negative Werte, also somit ganz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }">, zugelassen wären. <div class="mw-heading mw-heading3"><h3 id="Umrechnung_in_kartesische_Koordinaten">Umrechnung in kartesische Koordinaten</h3>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=48" title="Abschnitt bearbeiten: Umrechnung in kartesische Koordinaten" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=48" title="Quellcode des Abschnitts bearbeiten: Umrechnung in kartesische Koordinaten">Quelltext bearbeiten</a>]</div> Eine Umrechnungsvorschrift von diesen Koordinaten in kartesische Koordinaten wäre dann: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{lcr}x_{1}&=&r\ \cos \varphi \ \sin \vartheta _{1}\ \sin \vartheta _{2}\ \cdots \ \sin \vartheta _{n-3}\ \sin \vartheta _{n-2}\\x_{2}&=&r\ \sin \varphi \ \sin \vartheta _{1}\ \sin \vartheta _{2}\ \cdots \ \sin \vartheta _{n-3}\ \sin \vartheta _{n-2}\\x_{3}&=&r\ \cos \vartheta _{1}\ \sin \vartheta _{2}\ \cdots \ \sin \vartheta _{n-3}\ \sin \vartheta _{n-2}\\x_{4}&=&r\ \cos \vartheta _{2}\ \cdots \ \sin \vartheta _{n-3}\ \sin \vartheta _{n-2}\\\vdots &\vdots &\vdots \qquad \qquad \qquad \quad \\x_{n-1}&=&r\ \cos \vartheta _{n-3}\ \sin \vartheta _{n-2}\\x_{n}&=&r\ \cos \vartheta _{n-2}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left center right" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mi>r</mi> <mtext> </mtext> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mtext> </mtext> <mo>⋯</mo> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msub> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mi>r</mi> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mtext> </mtext> <mo>⋯</mo> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msub> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mi>r</mi> <mtext> </mtext> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mtext> </mtext> <mo>⋯</mo> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msub> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mi>r</mi> <mtext> </mtext> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mtext> </mtext> <mo>⋯</mo> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msub> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mi>r</mi> <mtext> </mtext> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msub> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mi>r</mi> <mtext> </mtext> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{lcr}x_{1}&=&r\ \cos \varphi \ \sin \vartheta _{1}\ \sin \vartheta _{2}\ \cdots \ \sin \vartheta _{n-3}\ \sin \vartheta _{n-2}\\x_{2}&=&r\ \sin \varphi \ \sin \vartheta _{1}\ \sin \vartheta _{2}\ \cdots \ \sin \vartheta _{n-3}\ \sin \vartheta _{n-2}\\x_{3}&=&r\ \cos \vartheta _{1}\ \sin \vartheta _{2}\ \cdots \ \sin \vartheta _{n-3}\ \sin \vartheta _{n-2}\\x_{4}&=&r\ \cos \vartheta _{2}\ \cdots \ \sin \vartheta _{n-3}\ \sin \vartheta _{n-2}\\\vdots &\vdots &\vdots \qquad \qquad \qquad \quad \\x_{n-1}&=&r\ \cos \vartheta _{n-3}\ \sin \vartheta _{n-2}\\x_{n}&=&r\ \cos \vartheta _{n-2}\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47cca68a0f512b76a38dee9233cd2be42f7b66cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.338ex; width:54.05ex; height:23.676ex;" alt="{\displaystyle {\begin{array}{lcr}x_{1}&=&r\ \cos \varphi \ \sin \vartheta _{1}\ \sin \vartheta _{2}\ \cdots \ \sin \vartheta _{n-3}\ \sin \vartheta _{n-2}\\x_{2}&=&r\ \sin \varphi \ \sin \vartheta _{1}\ \sin \vartheta _{2}\ \cdots \ \sin \vartheta _{n-3}\ \sin \vartheta _{n-2}\\x_{3}&=&r\ \cos \vartheta _{1}\ \sin \vartheta _{2}\ \cdots \ \sin \vartheta _{n-3}\ \sin \vartheta _{n-2}\\x_{4}&=&r\ \cos \vartheta _{2}\ \cdots \ \sin \vartheta _{n-3}\ \sin \vartheta _{n-2}\\\vdots &\vdots &\vdots \qquad \qquad \qquad \quad \\x_{n-1}&=&r\ \cos \vartheta _{n-3}\ \sin \vartheta _{n-2}\\x_{n}&=&r\ \cos \vartheta _{n-2}\end{array}}}"></dd></dl> Wie man nachweisen kann, gehen diese Polarkoordinaten für den Fall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a02c8bd752d2cc859747ca1f3a508281bdbc3b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=2}"> in die gewöhnlichen Polarkoordinaten und für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c5a5a42ced00df920fad4ab2d4acdb960a4105b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=3}"> in die Kugelkoordinaten über.<a href="#cite_note-Amann-Escher-6">[6]</a> <div class="mw-heading mw-heading3"><h3 id="Funktionaldeterminante_3">Funktionaldeterminante</h3>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=49" title="Abschnitt bearbeiten: Funktionaldeterminante" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=49" title="Quellcode des Abschnitts bearbeiten: Funktionaldeterminante">Quelltext bearbeiten</a>]</div> Die <a href="/wiki/Funktionaldeterminante" title="Funktionaldeterminante">Funktionaldeterminante</a> der Transformation von Kugelkoordinaten in kartesische Koordinaten beträgt:<a href="#cite_note-Amann-Escher-6">[6]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det {\frac {\partial (x_{1},\dotsc ,x_{n})}{\partial (r,\vartheta _{1},\dotsc ,\vartheta _{n-2},\varphi )}}=r^{n-1}\sin \vartheta _{1}\left(\sin \vartheta _{2}\right)^{2}\dotsm \left(\sin \vartheta _{n-2}\right)^{n-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋯</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det {\frac {\partial (x_{1},\dotsc ,x_{n})}{\partial (r,\vartheta _{1},\dotsc ,\vartheta _{n-2},\varphi )}}=r^{n-1}\sin \vartheta _{1}\left(\sin \vartheta _{2}\right)^{2}\dotsm \left(\sin \vartheta _{n-2}\right)^{n-2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19d557bad09c4eeebebd49ca81ee17cad430a9ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:63.135ex; height:6.509ex;" alt="{\displaystyle \det {\frac {\partial (x_{1},\dotsc ,x_{n})}{\partial (r,\vartheta _{1},\dotsc ,\vartheta _{n-2},\varphi )}}=r^{n-1}\sin \vartheta _{1}\left(\sin \vartheta _{2}\right)^{2}\dotsm \left(\sin \vartheta _{n-2}\right)^{n-2}}"></dd></dl> Damit beträgt das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensionale Volumenelement: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}\mathrm {d} V&=&r^{n-1}\sin \vartheta _{1}\left(\sin \vartheta _{2}\right)^{2}\dotsm \left(\sin \vartheta _{n-2}\right)^{n-2}\mathrm {d} r\ \mathrm {d} \varphi \ \mathrm {d} \vartheta _{1}\dotsm \mathrm {d} \vartheta _{n-2}\\&=&r^{n-1}\ \mathrm {d} r\ \mathrm {d} \varphi \ \prod \limits _{j=1}^{n-2}(\sin \vartheta _{j})^{j}\ \mathrm {d} \vartheta _{j}\end{matrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋯</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>r</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>φ</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋯</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>r</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>φ</mi> <mtext> </mtext> <munderover> <mo movablelimits="false">∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}\mathrm {d} V&=&r^{n-1}\sin \vartheta _{1}\left(\sin \vartheta _{2}\right)^{2}\dotsm \left(\sin \vartheta _{n-2}\right)^{n-2}\mathrm {d} r\ \mathrm {d} \varphi \ \mathrm {d} \vartheta _{1}\dotsm \mathrm {d} \vartheta _{n-2}\\&=&r^{n-1}\ \mathrm {d} r\ \mathrm {d} \varphi \ \prod \limits _{j=1}^{n-2}(\sin \vartheta _{j})^{j}\ \mathrm {d} \vartheta _{j}\end{matrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74e4e58d7088d81960a4aba16b4442475366208c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.364ex; margin-bottom: -0.308ex; width:65.971ex; height:10.509ex;" alt="{\displaystyle {\begin{matrix}\mathrm {d} V&=&r^{n-1}\sin \vartheta _{1}\left(\sin \vartheta _{2}\right)^{2}\dotsm \left(\sin \vartheta _{n-2}\right)^{n-2}\mathrm {d} r\ \mathrm {d} \varphi \ \mathrm {d} \vartheta _{1}\dotsm \mathrm {d} \vartheta _{n-2}\\&=&r^{n-1}\ \mathrm {d} r\ \mathrm {d} \varphi \ \prod \limits _{j=1}^{n-2}(\sin \vartheta _{j})^{j}\ \mathrm {d} \vartheta _{j}\end{matrix}}.}"></dd></dl> Anmerkung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensionale Zylinderkoordinaten sind einfach ein Produkt / eine Zusammensetzung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">-dimensionaler Kugelkoordinaten und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n-k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66e9f25b19fdb4fc155f89d4f926e9d9e4e440da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.256ex; height:2.843ex;" alt="{\displaystyle (n-k)}">-dimensionaler kartesischer Koordinaten mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\geq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c797a67c0a51167d373c013a9a020f4568a11754" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.472ex; height:2.343ex;" alt="{\displaystyle k\geq 2}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-k\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-k\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20a93b4034aae24097ffc20fbd657bed11e37200" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.707ex; height:2.343ex;" alt="{\displaystyle n-k\geq 1}">. <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=50" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=50" title="Quellcode des Abschnitts bearbeiten: Literatur">Quelltext bearbeiten</a>]</div> <ul><li>W. Werner: <cite style="font-style:italic">Vektoren und Tensoren als universelle Sprache in Physik und Technik</cite>. Band 1. Springer Vieweg, <a href="/wiki/Spezial:ISBN-Suche/9783658252717" class="internal mw-magiclink-isbn">ISBN 978-3-658-25271-7</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Polarkoordinaten&rft.au=W.+Werner&rft.btitle=Vektoren+und+Tensoren+als+universelle+Sprache+in+Physik+und+Technik&rft.genre=book&rft.isbn=9783658252717&rft.pub=Springer+Vieweg&rft.volume=1" style="display:none"> </li></ul> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=51" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=51" title="Quellcode des Abschnitts bearbeiten: Weblinks">Quelltext bearbeiten</a>]</div> <ul><li><a href="/wiki/Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/PolarCoordinates.html">Polar Coordinates.</a> In: <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a> (englisch).</li> <li><a rel="nofollow" class="external text" href="http://fooplot.com/">FooPlot: Funktionsplotter mit Polarkoordinaten</a></li> <li><a rel="nofollow" class="external text" href="https://additive.industrie.de/news/altana-verkauf-beteiligung-dp-polar-3d-systems/">3D-Drucker mit Polarkoordinatensteuerung</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2>[<a href="/w/index.php?title=Polarkoordinaten&veaction=edit&section=52" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Polarkoordinaten&action=edit&section=52" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise">Quelltext bearbeiten</a>]</div> <ol class="references"> <li id="cite_note-milestones-1"><a href="#cite_ref-milestones_1-0">↑</a> Michael Friendly: <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060925115456/http://www.math.yorku.ca/SCS/Gallery/milestone/sec4.html">Milestones in the History of Thematic Cartography, Statistical Graphics, and Data Visualization.</a> Archiviert vom <style data-mw-deduplicate="TemplateStyles:r235239667">.mw-parser-output .dewiki-iconexternal>a{background-position:center right;background-repeat:no-repeat}body.skin-minerva .mw-parser-output .dewiki-iconexternal>a{background-image:url("https://upload.wikimedia.org/wikipedia/commons/a/a4/OOjs_UI_icon_external-link-ltr-progressive.svg")!important;background-size:10px;padding-right:13px!important}body.skin-timeless .mw-parser-output .dewiki-iconexternal>a,body.skin-monobook .mw-parser-output .dewiki-iconexternal>a{background-image:url("https://upload.wikimedia.org/wikipedia/commons/3/30/MediaWiki_external_link_icon.svg")!important;padding-right:13px!important}body.skin-vector .mw-parser-output .dewiki-iconexternal>a{background-image:url("https://upload.wikimedia.org/wikipedia/commons/9/96/Link-external-small-ltr-progressive.svg")!important;background-size:0.857em;padding-right:1em!important}</style><a class="external text" href="https://redirecter.toolforge.org/?url=http%3A%2F%2Fwww.math.yorku.ca%2FSCS%2FGallery%2Fmilestone%2Fsec4.html">Original</a> am 25. September 2006; abgerufen am 10. September 2006.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3APolarkoordinaten&rft.title=Milestones+in+the+History+of+Thematic+Cartography%2C+Statistical+Graphics%2C+and+Data+Visualization&rft.description=Milestones+in+the+History+of+Thematic+Cartography%2C+Statistical+Graphics%2C+and+Data+Visualization&rft.identifier=https%3A%2F%2Fweb.archive.org%2Fweb%2F20060925115456%2Fhttp%3A%2F%2Fwww.math.yorku.ca%2FSCS%2FGallery%2Fmilestone%2Fsec4.html&rft.creator=Michael+Friendly&rft.source=http://www.math.yorku.ca/SCS/Gallery/milestone/sec4.html">  </li> <li id="cite_note-coolidge-2">↑ <a href="#cite_ref-coolidge_2-0">a</a> <a href="#cite_ref-coolidge_2-1">b</a> Julian Coolidge: <cite style="font-style:italic">The Origin of Polar Coordinates</cite>. In: <cite style="font-style:italic">American Mathematical Monthly</cite>. Nr. 59, 1952, S. 78–85 (<a rel="nofollow" class="external text" href="http://www-history.mcs.st-and.ac.uk/Extras/Coolidge_Polars.html">www-history.mcs.st-and.ac.uk</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Polarkoordinaten&rft.atitle=The+Origin+of+Polar+Coordinates&rft.au=Julian+Coolidge&rft.date=1952&rft.genre=journal&rft.issue=59&rft.jtitle=American+Mathematical+Monthly&rft.pages=78-85" style="display:none">  </li> <li id="cite_note-3"><a href="#cite_ref-3">↑</a> C. B. Boyer: <cite style="font-style:italic">Newton as an Originator of Polar Coordinates</cite>. In: <cite style="font-style:italic">American Mathematical Monthly</cite>. Nr. 56, 1949, S. 73–78.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Polarkoordinaten&rft.atitle=Newton+as+an+Originator+of+Polar+Coordinates&rft.au=C.+B.+Boyer&rft.date=1949&rft.genre=journal&rft.issue=56&rft.jtitle=American+Mathematical+Monthly&rft.pages=73-78" style="display:none">  </li> <li id="cite_note-4"><a href="#cite_ref-4">↑</a> Jeff Miller: <a rel="nofollow" class="external text" href="http://jeff560.tripod.com/p.html">Earliest Known Uses of Some of the Words of Mathematics.</a> Abgerufen am 30. August 2009.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3APolarkoordinaten&rft.title=Earliest+Known+Uses+of+Some+of+the+Words+of+Mathematics&rft.description=Earliest+Known+Uses+of+Some+of+the+Words+of+Mathematics&rft.identifier=http%3A%2F%2Fjeff560.tripod.com%2Fp.html&rft.creator=Jeff+Miller">  </li> <li id="cite_note-5"><a href="#cite_ref-5">↑</a> David Eugene Smith: <cite style="font-style:italic">History of Mathematics</cite>. Band 2. Ginn and Co., Boston 1925, S. 324.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Polarkoordinaten&rft.au=David+Eugene+Smith&rft.btitle=History+of+Mathematics&rft.date=1925&rft.genre=book&rft.pages=324&rft.place=Boston&rft.pub=Ginn+and+Co.&rft.volume=2" style="display:none">  </li> <li id="cite_note-Amann-Escher-6">↑ <a href="#cite_ref-Amann-Escher_6-0">a</a> <a href="#cite_ref-Amann-Escher_6-1">b</a> Herbert Amann, Joachim Escher: Analysis III. 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aria-labelledby="p-wikibase-otherprojects-label" > <h3 id="p-wikibase-otherprojects-label" class="vector-menu-heading " > In anderen Projekten </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Polar_coordinate_system" hreflang="en">Commons</a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q62494" title="Link zum verbundenen Objekt im Datenrepositorium [g]" accesskey="g">Wikidata-Datenobjekt</a></li> </ul> </div> </nav> <nav id="p-lang" class="mw-portlet mw-portlet-lang vector-menu-portal portal vector-menu" aria-labelledby="p-lang-label" > <h3 id="p-lang-label" class="vector-menu-heading " > In anderen Sprachen </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af badge-Q17437796 badge-featuredarticle mw-list-item" title="exzellenter Artikel"><a href="https://af.wikipedia.org/wiki/Poolko%C3%B6rdinatestelsel" title="Poolkoördinatestelsel – Afrikaans" lang="af" hreflang="af" data-title="Poolkoördinatestelsel" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%A7%D9%85_%D8%A7%D9%84%D8%A5%D8%AD%D8%AF%D8%A7%D8%AB%D9%8A%D8%A7%D8%AA_%D8%A7%D9%84%D9%82%D8%B7%D8%A8%D9%8A%D8%A9" title="نظام الإحداثيات القطبية – Arabisch" lang="ar" hreflang="ar" data-title="نظام الإحداثيات القطبية" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Coordenaes_polares" title="Coordenaes polares – Asturisch" lang="ast" hreflang="ast" data-title="Coordenaes polares" data-language-autonym="Asturianu" data-language-local-name="Asturisch" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Q%C3%BCtb_koordinat_sistemi" title="Qütb koordinat sistemi – Aserbaidschanisch" lang="az" hreflang="az" data-title="Qütb koordinat sistemi" data-language-autonym="Azərbaycanca" data-language-local-name="Aserbaidschanisch" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D1%8F%D1%80_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B0%D0%BB%D0%B0%D1%80_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0%D2%BB%D1%8B" title="Поляр координаталар системаһы – Baschkirisch" lang="ba" hreflang="ba" data-title="Поляр координаталар системаһы" data-language-autonym="Башҡортса" data-language-local-name="Baschkirisch" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%B0%D0%BB%D1%8F%D1%80%D0%BD%D0%B0%D1%8F_%D1%81%D1%96%D1%81%D1%82%D1%8D%D0%BC%D0%B0_%D0%BA%D0%B0%D0%B0%D1%80%D0%B4%D1%8B%D0%BD%D0%B0%D1%82" title="Палярная сістэма каардынат – Belarussisch" lang="be" hreflang="be" data-title="Палярная сістэма каардынат" data-language-autonym="Беларуская" data-language-local-name="Belarussisch" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D1%8F%D1%80%D0%BD%D0%B0_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%BD%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0" title="Полярна координатна система – Bulgarisch" lang="bg" hreflang="bg" data-title="Полярна координатна система" data-language-autonym="Български" data-language-local-name="Bulgarisch" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A7%8B%E0%A6%B2%E0%A6%BE%E0%A6%B0_%E0%A6%B8%E0%A7%8D%E0%A6%A5%E0%A6%BE%E0%A6%A8%E0%A6%BE%E0%A6%82%E0%A6%95_%E0%A6%AC%E0%A7%8D%E0%A6%AF%E0%A6%AC%E0%A6%B8%E0%A7%8D%E0%A6%A5%E0%A6%BE" title="পোলার স্থানাংক ব্যবস্থা – Bengalisch" lang="bn" hreflang="bn" data-title="পোলার স্থানাংক ব্যবস্থা" data-language-autonym="বাংলা" data-language-local-name="Bengalisch" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Polarni_koordinatni_sistem" title="Polarni koordinatni sistem – Bosnisch" lang="bs" hreflang="bs" data-title="Polarni koordinatni sistem" data-language-autonym="Bosanski" data-language-local-name="Bosnisch" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-ca badge-Q17437796 badge-featuredarticle mw-list-item" title="exzellenter Artikel"><a href="https://ca.wikipedia.org/wiki/Coordenades_polars" title="Coordenades polars – Katalanisch" lang="ca" hreflang="ca" data-title="Coordenades polars" data-language-autonym="Català" data-language-local-name="Katalanisch" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%B3%DB%8C%D8%B3%D8%AA%D9%85%DB%8C_%D9%BE%DB%86%D8%AA%D8%A7%D9%86%DB%8C_%D8%AC%DB%95%D9%85%D8%B3%DB%95%D8%B1%DB%8C" title="سیستمی پۆتانی جەمسەری – Zentralkurdisch" lang="ckb" hreflang="ckb" data-title="سیستمی پۆتانی جەمسەری" data-language-autonym="کوردی" data-language-local-name="Zentralkurdisch" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Pol%C3%A1rn%C3%AD_soustava_sou%C5%99adnic" title="Polární soustava souřadnic – Tschechisch" lang="cs" hreflang="cs" data-title="Polární soustava souřadnic" data-language-autonym="Čeština" data-language-local-name="Tschechisch" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D1%81%D0%B5%D0%BD_%D0%BF%D0%BE%D0%BB%D1%8F%D1%80%D0%BB%D0%B0_%D1%82%D1%8B%D1%82%C4%83%D0%BC%C4%95" title="Координатсен полярла тытăмĕ – Tschuwaschisch" lang="cv" hreflang="cv" data-title="Координатсен полярла тытăмĕ" data-language-autonym="Чӑвашла" data-language-local-name="Tschuwaschisch" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/System_cyfesurynnau_polar" title="System cyfesurynnau polar – Walisisch" lang="cy" hreflang="cy" data-title="System cyfesurynnau polar" data-language-autonym="Cymraeg" data-language-local-name="Walisisch" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Pol%C3%A6rt_koordinatsystem" title="Polært koordinatsystem – Dänisch" lang="da" hreflang="da" data-title="Polært koordinatsystem" data-language-autonym="Dansk" data-language-local-name="Dänisch" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%BF%CE%BB%CE%B9%CE%BA%CF%8C_%CF%83%CF%8D%CF%83%CF%84%CE%B7%CE%BC%CE%B1_%CF%83%CF%85%CE%BD%CF%84%CE%B5%CF%84%CE%B1%CE%B3%CE%BC%CE%AD%CE%BD%CF%89%CE%BD" title="Πολικό σύστημα συντεταγμένων – Griechisch" lang="el" hreflang="el" data-title="Πολικό σύστημα συντεταγμένων" data-language-autonym="Ελληνικά" data-language-local-name="Griechisch" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Polar_coordinate_system" title="Polar coordinate system – Englisch" lang="en" hreflang="en" data-title="Polar coordinate system" data-language-autonym="English" data-language-local-name="Englisch" class="interlanguage-link-target">English</a></li><li class="interlanguage-link interwiki-eo badge-Q17437796 badge-featuredarticle mw-list-item" title="exzellenter Artikel"><a href="https://eo.wikipedia.org/wiki/Polusa_koordinatsistemo" title="Polusa koordinatsistemo – Esperanto" lang="eo" hreflang="eo" data-title="Polusa koordinatsistemo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-es badge-Q17437798 badge-goodarticle mw-list-item" title="lesenswerter Artikel"><a href="https://es.wikipedia.org/wiki/Coordenadas_polares" title="Coordenadas polares – Spanisch" lang="es" hreflang="es" data-title="Coordenadas polares" data-language-autonym="Español" data-language-local-name="Spanisch" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Polaarkoordinaadid" title="Polaarkoordinaadid – Estnisch" lang="et" hreflang="et" data-title="Polaarkoordinaadid" data-language-autonym="Eesti" data-language-local-name="Estnisch" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Koordenatu_polar" title="Koordenatu polar – Baskisch" lang="eu" hreflang="eu" data-title="Koordenatu polar" data-language-autonym="Euskara" data-language-local-name="Baskisch" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D8%B3%D8%AA%DA%AF%D8%A7%D9%87_%D9%85%D8%AE%D8%AA%D8%B5%D8%A7%D8%AA_%D9%82%D8%B7%D8%A8%DB%8C" title="دستگاه مختصات قطبی – Persisch" lang="fa" hreflang="fa" data-title="دستگاه مختصات قطبی" data-language-autonym="فارسی" data-language-local-name="Persisch" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Napakoordinaatisto" title="Napakoordinaatisto – Finnisch" lang="fi" hreflang="fi" data-title="Napakoordinaatisto" data-language-autonym="Suomi" data-language-local-name="Finnisch" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Coordonn%C3%A9es_polaires" title="Coordonnées polaires – Französisch" lang="fr" hreflang="fr" data-title="Coordonnées polaires" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Polaarkoordinaaten" title="Polaarkoordinaaten – Nordfriesisch" lang="frr" hreflang="frr" data-title="Polaarkoordinaaten" data-language-autonym="Nordfriisk" data-language-local-name="Nordfriesisch" class="interlanguage-link-target">Nordfriisk</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Comhordan%C3%A1id%C3%AD_polacha" title="Comhordanáidí polacha – Irisch" lang="ga" hreflang="ga" data-title="Comhordanáidí polacha" data-language-autonym="Gaeilge" data-language-local-name="Irisch" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Coordenadas_polares" title="Coordenadas polares – Galicisch" lang="gl" hreflang="gl" data-title="Coordenadas polares" data-language-autonym="Galego" data-language-local-name="Galicisch" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A7%D7%95%D7%90%D7%95%D7%A8%D7%93%D7%99%D7%A0%D7%98%D7%95%D7%AA_%D7%A7%D7%95%D7%98%D7%91%D7%99%D7%95%D7%AA" title="קואורדינטות קוטביות – Hebräisch" lang="he" hreflang="he" data-title="קואורדינטות קוטביות" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A7%E0%A5%8D%E0%A4%B0%E0%A5%81%E0%A4%B5%E0%A5%80%E0%A4%AF_%E0%A4%A8%E0%A4%BF%E0%A4%B0%E0%A5%8D%E0%A4%A6%E0%A5%87%E0%A4%B6%E0%A4%BE%E0%A4%82%E0%A4%95_%E0%A4%AA%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%A4%E0%A4%BF" title="ध्रुवीय निर्देशांक पद्धति – Hindi" lang="hi" hreflang="hi" data-title="ध्रुवीय निर्देशांक पद्धति" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Polarni_koordinatni_sustav" title="Polarni koordinatni sustav – Kroatisch" lang="hr" hreflang="hr" data-title="Polarni koordinatni sustav" data-language-autonym="Hrvatski" data-language-local-name="Kroatisch" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Pol%C3%A1rkoordin%C3%A1ta-rendszer" title="Polárkoordináta-rendszer – Ungarisch" lang="hu" hreflang="hu" data-title="Polárkoordináta-rendszer" data-language-autonym="Magyar" data-language-local-name="Ungarisch" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sistem_koordinat_polar" title="Sistem koordinat polar – Indonesisch" lang="id" hreflang="id" data-title="Sistem koordinat polar" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Polala_koordinati" title="Polala koordinati – Ido" lang="io" hreflang="io" data-title="Polala koordinati" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Skauthnitakerfi" title="Skauthnitakerfi – Isländisch" lang="is" hreflang="is" data-title="Skauthnitakerfi" data-language-autonym="Íslenska" data-language-local-name="Isländisch" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Sistema_di_coordinate_polari" title="Sistema di coordinate polari – Italienisch" lang="it" hreflang="it" data-title="Sistema di coordinate polari" data-language-autonym="Italiano" data-language-local-name="Italienisch" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%A5%B5%E5%BA%A7%E6%A8%99%E7%B3%BB" title="極座標系 – Japanisch" lang="ja" hreflang="ja" data-title="極座標系" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D1%8F%D1%80%D0%BB%D1%8B%D2%9B_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B0%D0%BB%D0%B0%D1%80" title="Полярлық координаталар – Kasachisch" lang="kk" hreflang="kk" data-title="Полярлық координаталар" data-language-autonym="Қазақша" data-language-local-name="Kasachisch" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-ko badge-Q17437796 badge-featuredarticle mw-list-item" title="exzellenter Artikel"><a href="https://ko.wikipedia.org/wiki/%EA%B7%B9%EC%A2%8C%ED%91%9C%EA%B3%84" title="극좌표계 – Koreanisch" lang="ko" hreflang="ko" data-title="극좌표계" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Systema_polare_coordinatarum" title="Systema polare coordinatarum – Latein" lang="la" hreflang="la" data-title="Systema polare coordinatarum" data-language-autonym="Latina" data-language-local-name="Latein" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Polin%C4%97_koordina%C4%8Di%C5%B3_sistema" title="Polinė koordinačių sistema – Litauisch" lang="lt" hreflang="lt" data-title="Polinė koordinačių sistema" data-language-autonym="Lietuvių" data-language-local-name="Litauisch" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Pol%C4%81r%C4%81_koordin%C4%81tu_sist%C4%93ma" title="Polārā koordinātu sistēma – Lettisch" lang="lv" hreflang="lv" data-title="Polārā koordinātu sistēma" data-language-autonym="Latviešu" data-language-local-name="Lettisch" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D0%B0%D1%80%D0%B5%D0%BD_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B5%D0%BD_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Поларен координатен систем – Mazedonisch" lang="mk" hreflang="mk" data-title="Поларен координатен систем" data-language-autonym="Македонски" data-language-local-name="Mazedonisch" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Sistem_koordinat_berkutub" title="Sistem koordinat berkutub – Malaiisch" lang="ms" hreflang="ms" data-title="Sistem koordinat berkutub" data-language-autonym="Bahasa Melayu" data-language-local-name="Malaiisch" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%9D%E1%80%AD%E1%80%AF%E1%80%84%E1%80%BA%E1%80%B8%E1%80%9C%E1%80%BE%E1%80%8A%E1%80%B7%E1%80%BA_%E1%80%A1%E1%80%99%E1%80%BE%E1%80%90%E1%80%BA%E1%80%81%E1%80%BB%E1%80%A1%E1%80%AD%E1%80%99%E1%80%BA" title="ဝိုင်းလှည့် အမှတ်ချအိမ် – Birmanisch" lang="my" hreflang="my" data-title="ဝိုင်းလှည့် အမှတ်ချအိမ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Birmanisch" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Poolco%C3%B6rdinaten" title="Poolcoördinaten – Niederländisch" lang="nl" hreflang="nl" data-title="Poolcoördinaten" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Polarkoordinatsystem" title="Polarkoordinatsystem – Norwegisch (Nynorsk)" lang="nn" hreflang="nn" data-title="Polarkoordinatsystem" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegisch (Nynorsk)" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Polarkoordinatsystem" title="Polarkoordinatsystem – Norwegisch (Bokmål)" lang="nb" hreflang="nb" data-title="Polarkoordinatsystem" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegisch (Bokmål)" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AA%E0%A9%8B%E0%A8%B2%E0%A8%B0_%E0%A8%A8%E0%A8%BF%E0%A8%B0%E0%A8%A6%E0%A9%87%E0%A8%B8%E0%A8%BC%E0%A8%BE%E0%A8%82%E0%A8%95" title="ਪੋਲਰ ਨਿਰਦੇਸ਼ਾਂਕ – Punjabi" lang="pa" hreflang="pa" data-title="ਪੋਲਰ ਨਿਰਦੇਸ਼ਾਂਕ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Uk%C5%82ad_wsp%C3%B3%C5%82rz%C4%99dnych_biegunowych" title="Układ współrzędnych biegunowych – Polnisch" lang="pl" hreflang="pl" data-title="Układ współrzędnych biegunowych" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Coordenadas_polares" title="Coordenadas polares – Portugiesisch" lang="pt" hreflang="pt" data-title="Coordenadas polares" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro badge-Q17437798 badge-goodarticle mw-list-item" title="lesenswerter Artikel"><a href="https://ro.wikipedia.org/wiki/Coordonate_polare" title="Coordonate polare – Rumänisch" lang="ro" hreflang="ro" data-title="Coordonate polare" data-language-autonym="Română" data-language-local-name="Rumänisch" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D1%8F%D1%80%D0%BD%D0%B0%D1%8F_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82" title="Полярная система координат – Russisch" lang="ru" hreflang="ru" data-title="Полярная система координат" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Polar_coordinate_seestem" title="Polar coordinate seestem – Schottisch" lang="sco" hreflang="sco" data-title="Polar coordinate seestem" data-language-autonym="Scots" data-language-local-name="Schottisch" class="interlanguage-link-target">Scots</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Polarne_koordinate" title="Polarne koordinate – Serbokroatisch" lang="sh" hreflang="sh" data-title="Polarne koordinate" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbokroatisch" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Polar_coordinate_system" title="Polar coordinate system – einfaches Englisch" lang="en-simple" hreflang="en-simple" data-title="Polar coordinate system" data-language-autonym="Simple English" data-language-local-name="einfaches Englisch" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Pol%C3%A1rna_s%C3%BAstava_s%C3%BAradn%C3%ADc" title="Polárna sústava súradníc – Slowakisch" lang="sk" hreflang="sk" data-title="Polárna sústava súradníc" data-language-autonym="Slovenčina" data-language-local-name="Slowakisch" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Polarni_koordinatni_sistem" title="Polarni koordinatni sistem – Slowenisch" lang="sl" hreflang="sl" data-title="Polarni koordinatni sistem" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Sistemi_koordinativ_polar" title="Sistemi koordinativ polar – Albanisch" lang="sq" hreflang="sq" data-title="Sistemi koordinativ polar" data-language-autonym="Shqip" data-language-local-name="Albanisch" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D0%B0%D1%80%D0%BD%D0%B8_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%BD%D0%B8_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Поларни координатни систем – Serbisch" lang="sr" hreflang="sr" data-title="Поларни координатни систем" data-language-autonym="Српски / srpski" data-language-local-name="Serbisch" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Pol%C3%A4ra_koordinater" title="Polära koordinater – Schwedisch" lang="sv" hreflang="sv" data-title="Polära koordinater" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%BE%E0%AE%B3%E0%AF%8D%E0%AE%AE%E0%AF%81%E0%AE%A9%E0%AF%88_%E0%AE%86%E0%AE%B3%E0%AF%8D%E0%AE%95%E0%AF%82%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AF%81_%E0%AE%AE%E0%AF%81%E0%AE%B1%E0%AF%88%E0%AE%AE%E0%AF%88" title="வாள்முனை ஆள்கூற்று முறைமை – Tamil" lang="ta" hreflang="ta" data-title="வாள்முனை ஆள்கூற்று முறைமை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B0%E0%B8%9A%E0%B8%9A%E0%B8%9E%E0%B8%B4%E0%B8%81%E0%B8%B1%E0%B8%94%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B8%82%E0%B8%B1%E0%B9%89%E0%B8%A7" title="ระบบพิกัดเชิงขั้ว – Thailändisch" lang="th" hreflang="th" data-title="ระบบพิกัดเชิงขั้ว" data-language-autonym="ไทย" data-language-local-name="Thailändisch" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kutupsal_koordinat_sistemi" title="Kutupsal koordinat sistemi – Türkisch" lang="tr" hreflang="tr" data-title="Kutupsal koordinat sistemi" data-language-autonym="Türkçe" data-language-local-name="Türkisch" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk badge-Q17437798 badge-goodarticle mw-list-item" title="lesenswerter Artikel"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D1%8F%D1%80%D0%BD%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82" title="Полярна система координат – Ukrainisch" lang="uk" hreflang="uk" data-title="Полярна система координат" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%E1%BB%87_t%E1%BB%8Da_%C4%91%E1%BB%99_c%E1%BB%B1c" title="Hệ tọa độ cực – Vietnamesisch" lang="vi" hreflang="vi" data-title="Hệ tọa độ cực" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%9E%81%E5%9D%90%E6%A0%87%E7%B3%BB" title="极坐标系 – Wu" lang="wuu" hreflang="wuu" data-title="极坐标系" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%9E%81%E5%9D%90%E6%A0%87%E7%B3%BB" title="极坐标系 – Chinesisch" lang="zh" hreflang="zh" data-title="极坐标系" data-language-autonym="中文" data-language-local-name="Chinesisch" class="interlanguage-link-target">中文</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%A5%B5%E5%9D%90%E6%A8%99%E7%B3%BB%E7%B5%B1" title="極坐標系統 – Kantonesisch" lang="yue" hreflang="yue" data-title="極坐標系統" data-language-autonym="粵語" data-language-local-name="Kantonesisch" class="interlanguage-link-target">粵語</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q62494#sitelinks-wikipedia" title="Links auf Artikel in anderen Sprachen bearbeiten" class="wbc-editpage">Links bearbeiten</a></div> </div> </nav> </div> </div> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Diese Seite wurde zuletzt am 3. 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Polarkoordinaten – Wikipedia