GermanWikipedia - /page/entry.mu

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<title>Tangentialraum</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Tangentialraum</h1>

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In der <a href="Differentialgeometrie" title="Differentialgeometrie">Differentialgeometrie</a> ist ein Tangentialraum (auch Tangentenraum genannt) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{x}M}">

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<annotation encoding="application/x-tex">{\displaystyle T_{x}M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e9a02a3b6f9a6808be3b99d0b27d1b97b4bb025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.972ex; height:2.509ex;" alt="{\displaystyle T_{x}M}" loading="lazy"> ein <a href="Vektorraum" title="Vektorraum">Vektorraum</a>, der eine <a href="Differenzierbare_Mannigfaltigkeit" title="Differenzierbare Mannigfaltigkeit">differenzierbare Mannigfaltigkeit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> am Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" 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}" loading="lazy"> linear approximiert. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \colon (-\varepsilon ,\varepsilon )\to M}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma \colon (-\varepsilon ,\varepsilon )\to M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83d8061fd69ce1eb1d3dc4ec58c239c363395be0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.171ex; height:2.843ex;" alt="{\displaystyle \gamma \colon (-\varepsilon ,\varepsilon )\to M}" loading="lazy"> eine differenzierbare Kurve mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (0)=x}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma (0)=x}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2dab41ed8ce16471a6105c5e80458c28680f1b67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.662ex; height:2.843ex;" alt="{\displaystyle \gamma (0)=x}" loading="lazy"> und dem Kurvenparameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">, dann ist:

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<annotation encoding="application/x-tex">{\displaystyle v={\frac {d\gamma }{dt}}(0)\in T_{x}M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51819c334dd0a021c170a62db95ff367a738db30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.325ex; height:5.509ex;" alt="{\displaystyle v={\frac {d\gamma }{dt}}(0)\in T_{x}M}" loading="lazy"></dd></dl>

ein Tangentialvektor. Die Tangentialvektoren in einem Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in M}">

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<annotation encoding="application/x-tex">{\displaystyle x\in M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9df57d73e9532bb93a1439890bcddbc2806f5859.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.613ex; height:2.176ex;" alt="{\displaystyle x\in M}" loading="lazy"> spannen einen Vektorraum auf, den Tangentialraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{x}M}">

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<annotation encoding="application/x-tex">{\displaystyle T_{x}M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e9a02a3b6f9a6808be3b99d0b27d1b97b4bb025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.972ex; height:2.509ex;" alt="{\displaystyle T_{x}M}" loading="lazy">. Siehe auch <a href="Tangentialb%C3%BCndel" title="Tangentialbündel">Tangentialbündel</a>.

In der <a href="Algebraische_Geometrie" title="Algebraische Geometrie">algebraischen Geometrie</a> muss man diesen Definitionsansatz modifizieren, um singuläre Punkte und wechselnde Dimensionen zu berücksichtigen.

Dieser Artikel befasst sich nur mit dem Tangentialraum über einer differenzierbaren Mannigfaltigkeit im Sinne der Differentialgeometrie.

<div class="mw-heading mw-heading2"><h2 id="Übersicht">Übersicht</h2></div>

Am einfachsten ist eine differenzierbare Mannigfaltigkeit zu veranschaulichen, die als <a href="Untermannigfaltigkeit" title="Untermannigfaltigkeit">Untermannigfaltigkeit</a> in einen <a href="Euklidischer_Raum" title="Euklidischer Raum">Euklidischen Raum</a> (z. B. den <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f936ddf584f8f3dd2a0ed08917001b7a404c10b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}" loading="lazy">) eingebettet ist. Als Beispiel soll die <a href="Sph%C3%A4re" title="Sphäre">Sphäre</a> (= Kugeloberfläche) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle S^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b6401d5d0155afb1406770d1eb80badce4e08ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{2}}" loading="lazy"> im <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f936ddf584f8f3dd2a0ed08917001b7a404c10b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}" loading="lazy"> dienen. Der Tangentialraum in einem Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in S^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle p\in S^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34a892a26c2b469cf6bfc16a22b823b398afecdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.675ex; height:3.009ex;" alt="{\displaystyle p\in S^{2}}" loading="lazy"> ist dann die Ebene durch den Nullpunkt, die parallel zur <a href="Tangentialebene" title="Tangentialebene">Tangentialebene</a> der Kugel im Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> ist.

Ein <a href="Vektorfeld" title="Vektorfeld">Vektorfeld</a> ordnet jedem Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> einer Mannigfaltigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> einen Vektor aus dem zugehörigen Tangentialraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{p}M}">

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<annotation encoding="application/x-tex">{\displaystyle T_{p}M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71b20e0304a65ead2cafb33412a383b34fe527ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.859ex; height:2.843ex;" alt="{\displaystyle T_{p}M}" loading="lazy"> zu. Zum Beispiel könnte man mit einem Vektorfeld die Windstärke und -richtung auf der Erdoberfläche angeben.

Alle Tangentialräume einer Mannigfaltigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> werden als <a href="Tangentialb%C3%BCndel" title="Tangentialbündel">Tangentialbündel</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> zusammengefasst; das Tangentialbündel ist selbst eine Mannigfaltigkeit; seine Dimension ist doppelt so groß wie die von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Formale_Definitionen">Formale Definitionen</h2></div>

In der Literatur ist es üblich, gleich drei verschiedene Definitionen anzugeben, die einer geometrischen, einer algebraischen und einer theoretisch-physikalischen (auf <a href="Tensor" title="Tensor">Tensoren</a> hinarbeitenden) Sichtweise entsprechen. Der anschauliche geometrische Zugang erweist sich in der Anwendung jedoch als der am mühsamsten zu handhabende.

Die beiden auf die geometrische Definition folgenden algebraischen Definitionen des Tangentialraums funktionieren allerdings nur für Mannigfaltigkeiten der Klasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }}">

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<annotation encoding="application/x-tex">{\displaystyle C^{\infty }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/971ed05871d69309df32efdfd2020128c9cf69d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:2.343ex;" alt="{\displaystyle C^{\infty }}" loading="lazy">, aber nicht für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}}">

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<annotation encoding="application/x-tex">{\displaystyle C^{k}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/167fdb0cfb5644c4623b5842e1a9141acd83b534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle C^{k}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k<\infty }">

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<annotation encoding="application/x-tex">{\displaystyle k<\infty }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6eb6f699f0012b1d105eb5f9dce42cf6cd999f0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.633ex; height:2.176ex;" alt="{\displaystyle k<\infty }" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Geometrische_Definition:_Richtungsfelder_von_Kurven">Geometrische Definition: Richtungsfelder von Kurven</h3></div>

Gegeben seien eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" 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}" loading="lazy">-dimensionale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}}">

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<annotation encoding="application/x-tex">{\displaystyle C^{k}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/167fdb0cfb5644c4623b5842e1a9141acd83b534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle C^{k}}" loading="lazy">-Mannigfaltigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 1}">

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<annotation encoding="application/x-tex">{\displaystyle k\geq 1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d30d7dcf305b7bce39d36df72fe3985b47aa9961.svg" 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 1}" loading="lazy">, ein Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, eine <a href="Offene_Menge" title="Offene Menge">offene</a> Umgebung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> und eine <a href="Atlas_(Mathematik)" title="Atlas (Mathematik)">Karte</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \colon U\to \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi \colon U\to \mathbb {R} ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66b948859ee5f104f3154771fff9a06fcb7e9232.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.847ex; height:2.843ex;" alt="{\displaystyle \varphi \colon U\to \mathbb {R} ^{n}}" loading="lazy">.

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \colon (-1,1)\to M}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma \colon (-1,1)\to M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/328c9a77f154f978e2cff85883268e54871d29a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.329ex; height:2.843ex;" alt="{\displaystyle \gamma \colon (-1,1)\to M}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (0)=p}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma (0)=p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38920cd7608bc9fba57b4ec4582966c0ce3f936b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.502ex; height:2.843ex;" alt="{\displaystyle \gamma (0)=p}" loading="lazy"> eine differenzierbare Kurve in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \circ \gamma \colon (-1,1)\to \mathbb {R} ^{n}}">

<msup>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \varphi \circ \gamma \colon (-1,1)\to \mathbb {R} ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b61b5b212eeeddb5401d5b880f152d78d826094f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.498ex; height:2.843ex;" alt="{\displaystyle \varphi \circ \gamma \colon (-1,1)\to \mathbb {R} ^{n}}" loading="lazy"> eine differenzierbare Kurve im <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy">. Die Ableitung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\varphi \circ \gamma )'(0)}">

<msup>

</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\varphi \circ \gamma )'(0)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e93693dc87367505f00115043d2ac1a52e179f7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.443ex; height:3.009ex;" alt="{\displaystyle (\varphi \circ \gamma )'(0)}" loading="lazy"> existiert also.

Diese Ableitung ist ein Vektor im <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy">. Kurven <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma _{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/907e4a28946d45e5cbf3cf3c6c48de68296039c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.004ex; height:2.176ex;" alt="{\displaystyle \gamma _{i}}" loading="lazy">, für die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\varphi \circ \gamma _{i})'(0)}">

<msub>

</mrow>

</msub>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\varphi \circ \gamma _{i})'(0)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb07a42edaaa83aa9bcc2ba6fbd4b6eaa6cddb7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.185ex; height:3.009ex;" alt="{\displaystyle (\varphi \circ \gamma _{i})'(0)}" loading="lazy"> übereinstimmt, bilden eine <a href="%C3%84quivalenzklasse" class="mw-redirect" title="Äquivalenzklasse">Äquivalenzklasse</a>. Eine solche Äquivalenzklasse nennt man einen Tangentialvektor von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> und schreibt dafür <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma '(0)}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma '(0)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3db2fc37b07de2a9a0b049fd9444f571e11edc6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.936ex; height:3.009ex;" alt="{\displaystyle \gamma '(0)}" loading="lazy">. Der Tangentialraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{p}M}">

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{p}M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71b20e0304a65ead2cafb33412a383b34fe527ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.859ex; height:2.843ex;" alt="{\displaystyle T_{p}M}" loading="lazy"> ist die Menge aller dieser Tangentialvektoren; man kann zeigen, dass er nicht von der Wahl der Karte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" 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 }" loading="lazy"> abhängt.

Es bleibt zu zeigen, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{p}M}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{p}M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71b20e0304a65ead2cafb33412a383b34fe527ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.859ex; height:2.843ex;" alt="{\displaystyle T_{p}M}" loading="lazy"> durch Erklärung von Vektoraddition und <a href="Skalarmultiplikation" title="Skalarmultiplikation">Skalarmultiplikation</a> zu einem <a href="Vektorraum" title="Vektorraum">Vektorraum</a> wird. Dazu definiert man die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathrm {d} \varphi )|_{p}\colon T_{p}M\to \mathbb {R} ^{n}}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathrm {d} \varphi )|_{p}\colon T_{p}M\to \mathbb {R} ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09fde6570eb0f57e4cfe6a72c4294558152409bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.731ex; height:3.176ex;" alt="{\displaystyle (\mathrm {d} \varphi )|_{p}\colon T_{p}M\to \mathbb {R} ^{n}}" loading="lazy"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \varphi |_{p}(\gamma '(0))=(\varphi \circ \gamma )'(0)}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</msup>

<msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \varphi |_{p}(\gamma '(0))=(\varphi \circ \gamma )'(0)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be7e6266c31f08f8217145937cf6b1dc1b474e22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.806ex; height:3.343ex;" alt="{\displaystyle \mathrm {d} \varphi |_{p}(\gamma '(0))=(\varphi \circ \gamma )'(0)}" loading="lazy">, wobei die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

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<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> auf der rechten Seite ein beliebiger Repräsentant der Äquivalenzklasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma '(0)}">

<msup>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle \gamma '(0)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3db2fc37b07de2a9a0b049fd9444f571e11edc6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.936ex; height:3.009ex;" alt="{\displaystyle \gamma '(0)}" loading="lazy"> ist. Man zeigt nun, dass diese Abbildung <a href="Bijektion" class="mw-redirect" title="Bijektion">bijektiv</a> ist und überträgt mit ihrer Hilfe die Vektorraumoperationen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{p}M}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{p}M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71b20e0304a65ead2cafb33412a383b34fe527ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.859ex; height:2.843ex;" alt="{\displaystyle T_{p}M}" loading="lazy">; man zeigt außerdem, dass diese Konstruktion von der Wahl der Karte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" 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 }" loading="lazy"> unabhängig ist.

<div class="mw-heading mw-heading3"><h3 id="Erste_algebraische_Definition:_verallgemeinerte_Ableitungen">Erste algebraische Definition: verallgemeinerte Ableitungen</h3></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }}">

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<annotation encoding="application/x-tex">{\displaystyle C^{\infty }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/971ed05871d69309df32efdfd2020128c9cf69d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:2.343ex;" alt="{\displaystyle C^{\infty }}" loading="lazy">-Mannigfaltigkeit. Eine Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon M\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\colon M\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7030561275cbcb58537e6ec58e7536dfabdfda20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.884ex; height:2.509ex;" alt="{\displaystyle g\colon M\to \mathbb {R} }" loading="lazy"> gehört zur Klasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(M)}">

<msup>

</mrow>

</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{\infty }(M)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48b543485314a4426ee2ff956f206cfd802d68d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.925ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(M)}" loading="lazy">, falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ \varphi ^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle g\circ \varphi ^{-1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/28efb9908950cda9961958f439bfb6750207349f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.164ex; height:3.176ex;" alt="{\displaystyle g\circ \varphi ^{-1}}" loading="lazy"> für jede Karte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \colon U\to \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi \colon U\to \mathbb {R} ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66b948859ee5f104f3154771fff9a06fcb7e9232.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.847ex; height:2.843ex;" alt="{\displaystyle \varphi \colon U\to \mathbb {R} ^{n}}" loading="lazy"> unendlich oft differenzierbar ist. Das so definierte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(M)}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle C^{\infty }(M)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48b543485314a4426ee2ff956f206cfd802d68d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.925ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(M)}" loading="lazy"> ist eine <a href="Assoziative_Algebra" title="Assoziative Algebra">assoziative Algebra</a>.

Fixieren wir einen Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">. Eine <a href="Derivation_(Mathematik)" title="Derivation (Mathematik)">Derivation</a> an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> ist eine <a href="Lineare_Abbildung" title="Lineare Abbildung">lineare Abbildung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\colon C^{\infty }(M)\to \mathbb {R} }">

<msup>

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</msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D\colon C^{\infty }(M)\to \mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7dc17adbfacf7a5fe9599a0612d9cb9fbc33bffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.175ex; height:2.843ex;" alt="{\displaystyle D\colon C^{\infty }(M)\to \mathbb {R} }" loading="lazy">, die für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(M)}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle C^{\infty }(M)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48b543485314a4426ee2ff956f206cfd802d68d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.925ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(M)}" loading="lazy"> die (analog zur <a href="Produktregel" title="Produktregel">Produktregel</a>) folgende Eigenschaft hat:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D(gh)=D(g)h(p)+g(p)D(h)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e9a2b91552e87e023ef20fe5c2dc5345347c185.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.462ex; height:2.843ex;" alt="{\displaystyle D(gh)=D(g)h(p)+g(p)D(h)}" loading="lazy">.

Diese Derivationen bilden auf natürliche Weise einen reellen <a href="Vektorraum" title="Vektorraum">Vektorraum</a>; dies ist der Tangentialraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{p}M}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{p}M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71b20e0304a65ead2cafb33412a383b34fe527ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.859ex; height:2.843ex;" alt="{\displaystyle T_{p}M}" loading="lazy">.

Die Beziehung zwischen den zuvor definierten Tangentialvektoren und den Derivationen ist wie folgt: falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> eine Kurve mit Tangentialvektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma '(0)}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma '(0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3db2fc37b07de2a9a0b049fd9444f571e11edc6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.936ex; height:3.009ex;" alt="{\displaystyle \gamma '(0)}" loading="lazy"> ist, dann ist die entsprechende Derivation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(g)=(g\circ \gamma )'(0)}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D(g)=(g\circ \gamma )'(0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e12ed3c1694e443226dd3fdce6f49533f12cf1dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.987ex; height:3.009ex;" alt="{\displaystyle D(g)=(g\circ \gamma )'(0)}" loading="lazy"> (mit der Ableitung im üblichen Sinne, da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\circ \gamma }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45ca0cef02248fb2c5de1f6b49e34d521dc1a221.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.573ex; height:2.176ex;" alt="{\displaystyle g\circ \gamma }" loading="lazy"> eine Funktion von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1,1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (-1,1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e120a3bd60fc89b495dd7ef6039465b7e6a703b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle (-1,1)}" loading="lazy"> nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" 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} }" loading="lazy"> ist).

<div class="mw-heading mw-heading3"><h3 id="Zweite_algebraische_Definition:_Dualraum_von_I/I²">Zweite algebraische Definition: Dualraum von I/I²</h3></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> wieder eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{\infty }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/971ed05871d69309df32efdfd2020128c9cf69d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:2.343ex;" alt="{\displaystyle C^{\infty }}" loading="lazy">-Mannigfaltigkeit und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> ein Punkt in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">. Betrachten wir nun das <a href="Ideal_(Ringtheorie)" title="Ideal (Ringtheorie)">Ideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(M)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{\infty }(M)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48b543485314a4426ee2ff956f206cfd802d68d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.925ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(M)}" loading="lazy">, das aus allen glatten Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> besteht, die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> abbilden. Dann sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I^{2}=\left\{x\cdot y\mid x,y\in I\right\}}">

<msup>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I^{2}=\left\{x\cdot y\mid x,y\in I\right\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/747a9bcb865aeb7da03139b8464aa080fceb510d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.327ex; height:3.176ex;" alt="{\displaystyle I^{2}=\left\{x\cdot y\mid x,y\in I\right\}}" loading="lazy"> reelle Vektorräume, und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{p}M}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{p}M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71b20e0304a65ead2cafb33412a383b34fe527ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.859ex; height:2.843ex;" alt="{\displaystyle T_{p}M}" loading="lazy"> wird als der <a href="Dualraum" title="Dualraum">Dualraum</a> des <a href="Faktorraum" title="Faktorraum">Quotientenraums</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I/I^{2}}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I/I^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19d4b3432c9980cba099ba225470ef439bd08262.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.605ex; height:3.176ex;" alt="{\displaystyle I/I^{2}}" loading="lazy"> definiert.

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I/I^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19d4b3432c9980cba099ba225470ef439bd08262.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.605ex; height:3.176ex;" alt="{\displaystyle I/I^{2}}" loading="lazy"> wird auch als <a href="Kotangentialraum" title="Kotangentialraum">Kotangentialraum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{p}^{*}M}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{p}^{*}M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33810140102467366639d5039de84037468eb4f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.216ex; height:2.843ex;" alt="{\displaystyle T_{p}^{*}M}" loading="lazy"> bezeichnet (siehe <a href="#Kotangentialraum">unten</a>).

Während diese Definition die abstrakteste ist, ist sie auch diejenige, die man am leichtesten auf andere Situationen übertragen kann, beispielsweise auf <a href="Algebraische_Variet%C3%A4t" title="Algebraische Varietät">Varietäten</a>, wie sie in der <a href="Algebraische_Geometrie" title="Algebraische Geometrie">algebraischen Geometrie</a> betrachtet werden.

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> eine Derivation an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">. Dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(g)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D(g)=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a34221ff4bb114486cfbb87b9b92bbedc0668fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.11ex; height:2.843ex;" alt="{\displaystyle D(g)=0}" loading="lazy"> für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b35524ccbf943a202cfbb05ea9aaf6609a29ca0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.271ex; height:2.676ex;" alt="{\displaystyle I^{2}}" loading="lazy"> (denn es existieren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x,y\in I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3378f65d191a92b42edb1020f8fede1ce40a7d2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.532ex; height:2.509ex;" alt="{\displaystyle x,y\in I}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=xy}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g=xy}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7cc45b7497eb7cf0691f8cbeb3ab4a04385e1ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.7ex; height:2.009ex;" alt="{\displaystyle g=xy}" loading="lazy">, somit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(g)=D(xy)=D(x)y(p)+x(p)D(y)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D(g)=D(xy)=D(x)y(p)+x(p)D(y)=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2406196552a84efad15c06ac35b76c247825d78a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.761ex; height:2.843ex;" alt="{\displaystyle D(g)=D(xy)=D(x)y(p)+x(p)D(y)=0}" loading="lazy">), womit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> eine lineare Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I/I^{2}\to \mathbb {R} }">

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I/I^{2}\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b153da65a0f0686986963e09d62d948aa741a2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.897ex; height:3.176ex;" alt="{\displaystyle I/I^{2}\to \mathbb {R} }" loading="lazy"> induziert. Umgekehrt ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(g)=r((g-g(p))+I^{2})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D(g)=r((g-g(p))+I^{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c1f3304e4ba78eb8fb3f8da215d2503a985efc2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.777ex; height:3.176ex;" alt="{\displaystyle D(g)=r((g-g(p))+I^{2})}" loading="lazy"> eine Derivation, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\colon I/I^{2}\to \mathbb {R} }">

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r\colon I/I^{2}\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4578ae5fe2797a6b6f41a850a86c48acb9217f3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.98ex; height:3.176ex;" alt="{\displaystyle r\colon I/I^{2}\to \mathbb {R} }" loading="lazy"> eine lineare Abbildung ist.

Dies zeigt, dass sich der über Derivationen und der über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I/I^{2}}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I/I^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19d4b3432c9980cba099ba225470ef439bd08262.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.605ex; height:3.176ex;" alt="{\displaystyle I/I^{2}}" loading="lazy"> definierte Tangentialraum entsprechen.

<div class="mw-heading mw-heading3"><h3 id="Tangentialraum_in_der_algebraischen_Geometrie">Tangentialraum in der algebraischen Geometrie</h3></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Zariski-Tangentialraum" title="Zariski-Tangentialraum">Zariski-Tangentialraum</a></div>

Die beiden algebraischen Definitionen funktionieren genauso auch für algebraische Varietäten, wobei hier der Tangentialraum auch als <a href="Zariski-Tangentialraum" title="Zariski-Tangentialraum">Zariski-Tangentialraum</a> bezeichnet wird. Im Unterschied zu Mannigfaltigkeiten können algebraische Varietäten aber Singularitäten haben, dort hat dann der Tangentialraum eine höhere Dimension als in glatten Punkten.

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>

Wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> eine <a href="Offene_Teilmenge" class="mw-redirect" title="Offene Teilmenge">offene Teilmenge</a> des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> ist, so kann man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> in natürlicher Weise als eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{\infty }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/971ed05871d69309df32efdfd2020128c9cf69d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:2.343ex;" alt="{\displaystyle C^{\infty }}" loading="lazy">-Mannigfaltigkeit betrachten. Alle Karten sind hierbei die Identität, und die Tangentialräume werden mit dem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> identifiziert.

<div class="mw-heading mw-heading3"><h3 id="Tangentialvektoren_als_Richtungsableitungen">Tangentialvektoren als Richtungsableitungen</h3></div>

Eine Sichtweise von Tangentialvektoren ist, sie als Richtungsableitungen zu sehen. Für einen Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> im <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> definiert man die Richtungsableitung einer glatten Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6991fd928a18a993d719e93137af6240a9b6545a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.501ex; height:2.676ex;" alt="{\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} }" loading="lazy"> an einem Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> durch

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{v}f(p)={\frac {d}{dt}}{\bigg |}_{t=0}f(p+tv)=\sum _{i=1}^{n}v^{i}{\frac {\partial f}{\partial x^{i}}}(p)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64e609a513d1b052b28066d7b7066a3dd99ad9ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.87ex; height:6.843ex;" alt="{\displaystyle D_{v}f(p)={\frac {d}{dt}}{\bigg |}_{t=0}f(p+tv)=\sum _{i=1}^{n}v^{i}{\frac {\partial f}{\partial x^{i}}}(p)}" loading="lazy"></dd></dl>

Diese Abbildung ist offenbar eine Derivation. Tatsächlich ist sogar jede Derivation von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{\infty }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/971ed05871d69309df32efdfd2020128c9cf69d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.673ex; height:2.343ex;" alt="{\displaystyle C^{\infty }}" loading="lazy">(<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy">) von dieser Form. So existiert eine Bijektion zwischen Vektoren (als Tangentialvektor am Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> gedacht) und den Derivationen.

Da Tangentialvektoren an einer allgemeinen Mannigfaltigkeit als Derivationen definiert werden können, ist es nur natürlich, sie auch als Richtungsableitungen zu sehen. Konkret kann man für einen Tangentialvektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> an einem Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> (als Derivation gesehen) die Richtungsableitung in Richtung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon M\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon M\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b28c87bb68bdeda0931c6fa7210a19a7da47528a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.047ex; height:2.509ex;" alt="{\displaystyle f\colon M\to \mathbb {R} }" loading="lazy"> Element von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(M)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{\infty }(M)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48b543485314a4426ee2ff956f206cfd802d68d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.925ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(M)}" loading="lazy"> wie folgt definieren:

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{v}(f)=v(f)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32260760c5980d45226b2cae95015a5b2713c09e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.356ex; height:2.843ex;" alt="{\displaystyle D_{v}(f)=v(f)}" loading="lazy"></dd></dl>

Sehen wir <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> im Sinne der geometrischen Definition des Tangentialraums als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=\gamma '(0)}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v=\gamma '(0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2720acd204fd0bf59df01609d5edc934709ba1c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.162ex; height:3.009ex;" alt="{\displaystyle v=\gamma '(0)}" loading="lazy"> für eine Kurve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy">, schreiben wir

<msub>

</mrow>

</msub>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{v}(f)=(f\circ \gamma )'(0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8fe31ee185360a4e108b0e5847e52f0ed3c083ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.342ex; height:3.009ex;" alt="{\displaystyle D_{v}(f)=(f\circ \gamma )'(0)}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Die_Totalableitung_einer_Abbildung">Die Totalableitung einer Abbildung</h3></div>

<div class="sieheauch" role="navigation" style="font-style:italic;">Siehe auch: <a href="Pushforward" title="Pushforward">Pushforward</a></div>

Jede differenzierbare Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon M\to N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon M\to N}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/26c86d8613d6babf1a7cd6dd5909a3b480840c56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.432ex; height:2.509ex;" alt="{\displaystyle f\colon M\to N}" loading="lazy"> zwischen zwei differenzierbaren Mannigfaltigkeiten induziert eine <a href="Lineare_Abbildung" title="Lineare Abbildung">lineare Abbildung</a>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} f_{p}\colon T_{p}M\to T_{f(p)}N}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e1078ab0b54721f315dca77a23157c32c5bfa99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.662ex; height:3.009ex;" alt="{\displaystyle \mathrm {d} f_{p}\colon T_{p}M\to T_{f(p)}N}" loading="lazy"></dd></dl>

zwischen den entsprechenden Tangentialräumen, definiert durch

</mrow>

<msub>

</mrow>

</msub>

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} f_{p}(\gamma '(0)):=(f\circ \gamma )'(0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba1410a03c2367df5160a0086ae72ec7a1b13f13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.183ex; height:3.176ex;" alt="{\displaystyle \mathrm {d} f_{p}(\gamma '(0)):=(f\circ \gamma )'(0)}" loading="lazy"></dd></dl>

für die geometrische Definition des Tangentialraums und

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} f_{p}(D)(g):=D(g\circ f)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbe1e966a919c41d85902e78d688c5dfe6ae1942.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.218ex; height:3.009ex;" alt="{\displaystyle \mathrm {d} f_{p}(D)(g):=D(g\circ f)}" loading="lazy"></dd></dl>

für die Definition mittels Derivationen.

Die lineare Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} f_{p}}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} f_{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df01051e1b972bf6ad289df346543a838018ec97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.491ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} f_{p}}" loading="lazy"> wird mit Differential, Ableitung, Totalableitung oder auch Tangentialabbildung bezeichnet. Auch hier variieren die Notationen stark. Benutzt werden vor allem: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} f_{p}}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} f_{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df01051e1b972bf6ad289df346543a838018ec97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.491ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} f_{p}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} f_{p}}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {D} f_{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/389f280e9948eef19bedc3f4b012802eafc6355e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.974ex; height:2.843ex;" alt="{\displaystyle \mathrm {D} f_{p}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{*}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{*}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/903e97c260c5d157bff320b5e38e0e4b42cd20ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.509ex;" alt="{\displaystyle f_{*}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(p)}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f'(p)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71159b6c94f0c2f8a4a88e0486a2bd0e95dafcbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.984ex; height:3.009ex;" alt="{\displaystyle f'(p)}" loading="lazy">.

In einem gewissen Sinne ist die Totalableitung die beste lineare Approximation von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> in einer Umgebung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">. In lokalen Koordinaten kann man die Totalableitung als Jacobische Matrix darstellen.

Ist die Tangentialabbildung <a href="Surjektive_Funktion" title="Surjektive Funktion">surjektiv</a>, hat also die <a href="Jacobi-Matrix" title="Jacobi-Matrix">Jacobi-Matrix</a> überall vollen Rang, so nennt man die zugrundeliegende Funktion <a href="Submersion" title="Submersion">Submersion</a>; ist die Tangentialabbildung injektiv, <a href="Immersion_(Mathematik)" title="Immersion (Mathematik)">Immersion</a>.

Ein wichtiges Resultat bezüglich Tangentialabbildungen ist der Satz:

<dl><dd>Genau dann, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon M\to N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon M\to N}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/26c86d8613d6babf1a7cd6dd5909a3b480840c56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.432ex; height:2.509ex;" alt="{\displaystyle f\colon M\to N}" loading="lazy"> ein lokaler <a href="Diffeomorphismus" title="Diffeomorphismus">Diffeomorphismus</a> bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> ist, ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} f_{p}\colon T_{p}M\to T_{f(p)}N}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} f_{p}\colon T_{p}M\to T_{f(p)}N}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e1078ab0b54721f315dca77a23157c32c5bfa99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.662ex; height:3.009ex;" alt="{\displaystyle \mathrm {d} f_{p}\colon T_{p}M\to T_{f(p)}N}" loading="lazy"> ein linearer <a href="Isomorphismus" title="Isomorphismus">Isomorphismus</a>.</dd></dl>

Dies ist eine Verallgemeinerung des Satzes über inverse Funktionen auf Abbildungen zwischen Mannigfaltigkeiten.

<div class="mw-heading mw-heading3"><h3 id="Richtungen_der_Tangentialvektoren">Richtungen der Tangentialvektoren</h3></div>

Falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> eine glatte <a href="Mannigfaltigkeit_mit_Rand" title="Mannigfaltigkeit mit Rand">Mannigfaltigkeit mit Rand</a> ist und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \partial M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in \partial M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b683abece9f6df4e913b310d51e3ecaa0c96fba9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.86ex; height:2.509ex;" alt="{\displaystyle p\in \partial M}" loading="lazy">, dann können die Vektoren im Tangentialraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{p}M}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{p}M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71b20e0304a65ead2cafb33412a383b34fe527ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.859ex; height:2.843ex;" alt="{\displaystyle T_{p}M}" loading="lazy"> in drei Klassen aufgeteilt werden:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v\in T_{p}M\setminus T_{p}\partial M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6aaf84745462a3c6757470bcd32f24fac748de47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.199ex; height:3.009ex;" alt="{\displaystyle v\in T_{p}M\setminus T_{p}\partial M}" loading="lazy"> heißt nach innen gerichtet für ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e04ec3670b50384a3ce48aca42e7cc5131a06b12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}" loading="lazy">, wenn eine glatte Kurve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \colon [0,\varepsilon )\to M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma \colon [0,\varepsilon )\to M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a7a5a25d5979de9976736f7615a2a8bd01111dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.184ex; height:2.843ex;" alt="{\displaystyle \gamma \colon [0,\varepsilon )\to M}" loading="lazy"> existiert mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (0)=p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma (0)=p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38920cd7608bc9fba57b4ec4582966c0ce3f936b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.502ex; height:2.843ex;" alt="{\displaystyle \gamma (0)=p}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma '(\varepsilon )=v}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma '(\varepsilon )=v}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d00b525a919d6a388f3b5e48ec57441c3d59726e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.083ex; height:3.009ex;" alt="{\displaystyle \gamma '(\varepsilon )=v}" loading="lazy">.</li>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v\in T_{p}M\setminus T_{p}\partial M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6aaf84745462a3c6757470bcd32f24fac748de47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.199ex; height:3.009ex;" alt="{\displaystyle v\in T_{p}M\setminus T_{p}\partial M}" loading="lazy"> heißt nach aussen gerichtet für ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e04ec3670b50384a3ce48aca42e7cc5131a06b12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}" loading="lazy">, wenn eine glatte Kurve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma \colon (-\varepsilon ,0]\to M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma \colon (-\varepsilon ,0]\to M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/306a9fe431290b16c6cbab5f348b6285f0b8bc5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.992ex; height:2.843ex;" alt="{\displaystyle \gamma \colon (-\varepsilon ,0]\to M}" loading="lazy"> existiert mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (0)=p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma (0)=p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38920cd7608bc9fba57b4ec4582966c0ce3f936b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.502ex; height:2.843ex;" alt="{\displaystyle \gamma (0)=p}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma '(-\varepsilon )=v}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma '(-\varepsilon )=v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68d05cb162ac2282451152858bc3a7e3cdddaea3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.892ex; height:3.009ex;" alt="{\displaystyle \gamma '(-\varepsilon )=v}" loading="lazy">.</li>

<li>die Restlichen sind tangent zum Rand.<a href="#cite_note-1">[1]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Kotangentialraum">Kotangentialraum</h2></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Kotangentialraum" title="Kotangentialraum">Kotangentialraum</a></div>

Da der Tangentialraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{p}M}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{p}M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71b20e0304a65ead2cafb33412a383b34fe527ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.859ex; height:2.843ex;" alt="{\displaystyle T_{p}M}" loading="lazy"> am Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> der Mannigfaltigkeit die Struktur eines Vektorraums trägt, kann man den <a href="Dualraum" title="Dualraum">Dualraum</a> von ihm bilden. Dieser Raum wird Kotangentialraum genannt und gewöhnlicherweise mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{p}^{*}M}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{p}^{*}M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33810140102467366639d5039de84037468eb4f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.216ex; height:2.843ex;" alt="{\displaystyle T_{p}^{*}M}" loading="lazy"> notiert. Der letzten Definition folgend ist der Raum also isomorph zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle I/I^{2}}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle I/I^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d62616160bff8edc27da70d2e29b531f165d0bd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.605ex; height:3.009ex;" alt="{\displaystyle \textstyle I/I^{2}}" loading="lazy">. Der Kotangentialraum spielt in der Differentialgeometrie ebenfalls eine sehr wichtige Rolle. So kann man zum Beispiel das <a href="Totales_Differential" title="Totales Differential">totale Differential</a>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\rm {d}}f(p)\colon T_{p}M\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d734e4df7fc1e1d379353093f32586beedb42d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.735ex; height:3.009ex;" alt="{\displaystyle {\rm {d}}f(p)\colon T_{p}M\to \mathbb {R} }" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in C^{\infty }(M)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\in C^{\infty }(M)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e422033d69a8a3ef7672c25747f14268cf7f499.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.044ex; height:2.843ex;" alt="{\displaystyle f\in C^{\infty }(M)}" loading="lazy"></dd></dl>

als eine lineare Abbildung verstehen, welche jedem Tangentialvektor die <a href="Richtungsableitung" title="Richtungsableitung">Richtungsableitung</a> in seiner Richtung zuordnet. Das totale Differential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {d}}f(p)}">

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\rm {d}}f(p)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4d11a60f9e1b9483c06cab18f87dbd06a3c8411.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.55ex; height:2.843ex;" alt="{\displaystyle {\rm {d}}f(p)}" loading="lazy"> ist somit ein Element des Kotangentialraums <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{p}^{*}M}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{p}^{*}M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33810140102467366639d5039de84037468eb4f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.216ex; height:2.843ex;" alt="{\displaystyle T_{p}^{*}M}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> am Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li><a href="Theodor_Br%C3%B6cker" title="Theodor Bröcker">Theodor Bröcker</a>: Analysis. Band 3. BI-Wissenschafts-Verlag, Mannheim u. a. 1992, ISBN 3-411-15851-4.</li>

<li><a href="Klaus_J%C3%A4nich" title="Klaus Jänich">Klaus Jänich</a>: Vektoranalysis. 5. Auflage. Springer Verlag, Berlin u. a. 2005, ISBN 3-540-23741-0 (Springer-Lehrbuch).</li>

<li>R. Abraham, <a href="Jerrold_Marsden" title="Jerrold Marsden">Jerrold E. Marsden</a>, T. Ratiu: Manifolds, Tensor Analysis and Applications. Springer-Verlag, Berlin u. a. 2003, ISBN 0-201-10168-8.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> John M. Lee: <cite style="font-style:italic">Introduction to Smooth Manifolds</cite>. Hrsg.: Springer. 2. Auflage. S. 118.<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:Tangentialraum&rft.au=John+M.+Lee&rft.btitle=Introduction+to+Smooth+Manifolds&rft.edition=2.&rft.genre=book&rft.pages=118" style="display:none">

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