GermanWikipedia - /page/entry.mu

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<title>Pushforward</title>

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

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Als Pushforward wird eine Abbildung zwischen <a href="Tangentialraum" title="Tangentialraum">Tangentialräumen</a> <a href="Differenzierbare_Mannigfaltigkeit" title="Differenzierbare Mannigfaltigkeit">glatter Mannigfaltigkeiten</a> bezeichnet, die die im euklidischen Raum definierte <a href="Richtungsableitung" title="Richtungsableitung">Richtungsableitung</a> verallgemeinert.

Das duale Konzept heißt meist <a href="R%C3%BCcktransport" title="Rücktransport">Rücktransport</a> (Pullback).

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

Sind <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"> und <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/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> <a href="Differenzierbare_Mannigfaltigkeit" title="Differenzierbare Mannigfaltigkeit">glatte Mannigfaltigkeiten</a> und ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\colon M\rightarrow N}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bc71cd32e87f1986f07b8ef8db084c0b2123226.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.895ex; height:2.176ex;" alt="{\displaystyle F\colon M\rightarrow N}" loading="lazy"> eine <a href="Glatte_Abbildung" class="mw-redirect" title="Glatte Abbildung">glatte Abbildung</a>, so definiert man den Pushforward

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<annotation encoding="application/x-tex">{\displaystyle F_{\ast }\colon T_{p}M\rightarrow T_{F(p)}N}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f03fc82cfad7af2b2873ae850736de9cdc4d0729.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.046ex; height:3.009ex;" alt="{\displaystyle F_{\ast }\colon T_{p}M\rightarrow T_{F(p)}N}" loading="lazy"></dd></dl></dd></dl>

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> am Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in M}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35ad2c18a15749505c928763cd4fdb56f4982816.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.542ex; height:2.509ex;" alt="{\displaystyle p\in M}" loading="lazy"> durch

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<annotation encoding="application/x-tex">{\displaystyle (F_{*}v)(f)=v(f\circ F)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0dbb30410935bafb0003952822655aae32ac409.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.823ex; height:2.843ex;" alt="{\displaystyle (F_{*}v)(f)=v(f\circ F)}" loading="lazy"></dd></dl></dd></dl>

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8cde43d4dac32b2325e477be95a948eec8a77617.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.827ex; height:2.843ex;" alt="{\displaystyle v\in T_{p}M}" loading="lazy"> und jede glatte Funktion

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3e4a29e8fda2dce3c8399f1aa474d0e373b5df6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.111ex; height:2.843ex;" alt="{\displaystyle f\in {\mathcal {C}}^{\infty }(N)}" loading="lazy"> auf der Mannigfaltigkeit <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/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">. Hierbei werden Tangentialvektoren als Richtungsableitungen (<a href="Derivation_(Mathematik)" title="Derivation (Mathematik)">Derivationen</a>) aufgefasst, vgl. <a href="Tangentialraum" title="Tangentialraum">Tangentialraum</a>.

Auf diese Weise wird eine Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\ast }\colon TM\to TN}">

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<annotation encoding="application/x-tex">{\displaystyle F_{\ast }\colon TM\to TN}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7663e17c27b7fb589beab74ce2444b43f362a60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.293ex; margin-bottom: -0.379ex; width:14.975ex; height:2.509ex;" alt="{\displaystyle F_{\ast }\colon TM\to TN}" loading="lazy"> definiert.

<div class="mw-heading mw-heading3"><h3 id="Bezeichnungen_und_Schreibweisen">Bezeichnungen und Schreibweisen</h3></div>

Andere Bezeichnungen für den Pushforward sind Ableitung, Differential und Tangentialabbildung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy">.

Andere Schreibweisen sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\,'(p)v}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57d617a2955ca82591ab14de5850745f6db8bfc2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.919ex; height:3.009ex;" alt="{\displaystyle F\,'(p)v}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DF_{p}(v)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89a8c5d5f1da5bbecca8aacec771a00e06bb5643.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.415ex; height:3.009ex;" alt="{\displaystyle DF_{p}(v)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{p}F(v)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b341999f1c1deae9d67960ac2cf335ba040263d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.661ex; height:3.009ex;" alt="{\displaystyle D_{p}F(v)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dF_{p}(v)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/69b47c7849fb4ae55ca61d005fc5d5d015b076ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.707ex; height:3.009ex;" alt="{\displaystyle dF_{p}(v)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{p}F(v)}">

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<annotation encoding="application/x-tex">{\displaystyle d_{p}F(v)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cf94e82842726d2e582a4070cf53c0666967864.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.946ex; height:3.009ex;" alt="{\displaystyle d_{p}F(v)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{p}F(v)}">

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<annotation encoding="application/x-tex">{\displaystyle T_{p}F(v)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8aae108d6ee3978e4638273de7bc92da1a7acccd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.094ex; height:3.009ex;" alt="{\displaystyle T_{p}F(v)}" loading="lazy">.

Oft werden die Klammern um das Argument <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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

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</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"> auch weggelassen.

<div class="mw-heading mw-heading2"><h2 id="Bedeutung_für_Tangentialvektoren_von_Kurven">Bedeutung für Tangentialvektoren von Kurven</h2></div>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v={\dot {c}}(t)\in T_{p}M}">

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<annotation encoding="application/x-tex">{\displaystyle v={\dot {c}}(t)\in T_{p}M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d54fc15912d8f126fc1ac2e19ba3ba7caa43f83e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.866ex; height:3.009ex;" alt="{\displaystyle v={\dot {c}}(t)\in T_{p}M}" loading="lazy"> der Tangentialvektor einer differenzierbaren Kurve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\colon I\to M}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0a22b3a75201b08c2f770ec4c43b36cf26f695e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.269ex; height:2.176ex;" alt="{\displaystyle c\colon I\to M}" loading="lazy"> (hierbei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

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

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</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"> ein Intervall in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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

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</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">) im Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=c(t)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/877861d69413a3601eda3e5cc6c13e7d78d25234.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.013ex; height:2.843ex;" alt="{\displaystyle p=c(t)}" loading="lazy">, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{*}v\,}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5a87383556d773e17c5d8f9dc974c805f52b108f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.293ex; margin-bottom: -0.379ex; width:4.064ex; height:2.509ex;" alt="{\displaystyle F_{*}v\,}" loading="lazy"> der Tangentialvektor der Bildkurve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {c}}=F\circ c\colon I\to N}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {c}}=F\circ c\colon I\to N}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2df27b411f7909e98324e729b00eae8ef27ec75e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.216ex; height:2.176ex;" alt="{\displaystyle {\tilde {c}}=F\circ c\colon I\to N}" loading="lazy"> im Bildpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(p)={\tilde {c}}(t)}">

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<annotation encoding="application/x-tex">{\displaystyle F(p)={\tilde {c}}(t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8c7f26f554d9e1e023ff57796b5880c30743cbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.758ex; height:2.843ex;" alt="{\displaystyle F(p)={\tilde {c}}(t)}" loading="lazy">, also

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<annotation encoding="application/x-tex">{\displaystyle F_{*}v={\dot {\tilde {c}}}(t)=(F\circ c)^{\cdot }(t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b376c516d6074104a7303a3a36271b546843929f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.904ex; height:3.176ex;" alt="{\displaystyle F_{*}v={\dot {\tilde {c}}}(t)=(F\circ c)^{\cdot }(t)}" loading="lazy">.</dd></dl></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Darstellung_in_Koordinaten">Darstellung in Koordinaten</h2></div>

Sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},\dots ,x_{m})}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c50c5c887d855729e7025b54a5e44c312dc396b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.376ex; height:2.843ex;" alt="{\displaystyle (x_{1},\dots ,x_{m})}" loading="lazy"> lokale Koordinaten auf <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"> um <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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y_{1},\dots ,y_{n})}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d943c76e1905585868262d12b81fd3277e333f63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.539ex; height:2.843ex;" alt="{\displaystyle (y_{1},\dots ,y_{n})}" loading="lazy"> lokale Koordinaten auf <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/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> um den Bildpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(p)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f96aa8fc46e2bc3f1a5a1f612c3af35609d5802d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.72ex; height:2.843ex;" alt="{\displaystyle F(p)}" loading="lazy">, so haben die Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in T_{p}M}">

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</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8cde43d4dac32b2325e477be95a948eec8a77617.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.827ex; height:2.843ex;" alt="{\displaystyle v\in T_{p}M}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=F_{*}v\in T_{F(p)}N}">

<msub>

</mrow>

</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle w=F_{*}v\in T_{F(p)}N}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b9951b81aa418864e58b75c6ffc049418e4c492.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.27ex; height:3.009ex;" alt="{\displaystyle w=F_{*}v\in T_{F(p)}N}" loading="lazy"> die Darstellungen

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<mfrac>

<mrow>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle v=\sum _{j}v^{j}\,{\frac {\partial }{\partial x^{j}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ccb112fd333d586dfd80cb9c10bb19f9fa5ac227.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:14.786ex; height:6.843ex;" alt="{\displaystyle v=\sum _{j}v^{j}\,{\frac {\partial }{\partial x^{j}}}}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=\sum _{i}w^{i}\,{\frac {\partial }{\partial y^{i}}}}">

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</munder>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

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</msup>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle w=\sum _{i}w^{i}\,{\frac {\partial }{\partial y^{i}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6746987928fba13d5aad3a0308befcab5b0e9185.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.47ex; height:6.509ex;" alt="{\displaystyle w=\sum _{i}w^{i}\,{\frac {\partial }{\partial y^{i}}}}" loading="lazy">.</dd></dl></dd></dl>

Wird weiter die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\colon M\to N}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d5441b4074702ac361fc3026ff30deb826a3f4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.895ex; height:2.176ex;" alt="{\displaystyle F\colon M\to N}" loading="lazy"> durch die Funktionen

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<annotation encoding="application/x-tex">{\displaystyle f^{1}(x_{1},\dots ,x_{m}),\dots ,f^{n}(x_{1},\dots ,x_{m})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4bc17ef087b9ac0f0273509416a7df1f9bf71a6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.844ex; height:3.176ex;" alt="{\displaystyle f^{1}(x_{1},\dots ,x_{m}),\dots ,f^{n}(x_{1},\dots ,x_{m})}" loading="lazy"> dargestellt, so gilt

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<annotation encoding="application/x-tex">{\displaystyle w^{i}=\sum _{j}{\frac {\partial f^{i}}{\partial x^{j}}}\,v^{j}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ff3fa669e29bdffecaa7c8cf5c16f6ea445776c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:16.122ex; height:7.176ex;" alt="{\displaystyle w^{i}=\sum _{j}{\frac {\partial f^{i}}{\partial x^{j}}}\,v^{j}}" loading="lazy">.</dd></dl></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Pushforward_im_euklidischen_Raum">Pushforward im euklidischen Raum</h2></div>

Liegt der Spezialfall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\colon \mathbb {R} ^{m}\rightarrow \mathbb {R} ^{n}}">

<msup>

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<msup>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e70b16b96dae6c1ba70d5e072a077c71b6885d3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.638ex; height:2.343ex;" alt="{\displaystyle F\colon \mathbb {R} ^{m}\rightarrow \mathbb {R} ^{n}}" loading="lazy"> vor, so stellt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{*}\,}">

<msub>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e95497941d80911a08d925d0b4443b35b5098785.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.293ex; margin-bottom: -0.379ex; width:2.936ex; height:2.509ex;" alt="{\displaystyle F_{*}\,}" loading="lazy"> nichts anderes als die <a href="Totale_Ableitung" class="mw-redirect" title="Totale Ableitung">totale Ableitung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DF(p)\colon \mathbb {R} ^{m}\rightarrow \mathbb {R} ^{n}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle DF(p)\colon \mathbb {R} ^{m}\rightarrow \mathbb {R} ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/edcd028aa4c66921472069d2eca3b72b03e16fc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.541ex; height:2.843ex;" alt="{\displaystyle DF(p)\colon \mathbb {R} ^{m}\rightarrow \mathbb {R} ^{n}}" loading="lazy"> dar, wobei der <a href="Euklidischer_Raum" title="Euklidischer Raum">euklidische Raum</a> in natürlicher Weise mit seinem <a href="Tangentialraum" title="Tangentialraum">Tangentialraum</a> identifiziert wird (die Unterscheidung zwischen Richtungsableitung und totaler Ableitung spielt hier keine Rolle, da die Funktion bereits als hinreichend glatt vorausgesetzt ist).

Oft wird der Tangentialraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{p}\mathbb {R} ^{m}}">

<msub>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c28e59b13a2e86bae6ad942596d8cdebf24f9866.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.77ex; height:3.009ex;" alt="{\displaystyle T_{p}\mathbb {R} ^{m}}" loading="lazy"> des euklidischen Raums <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{m}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a87a024931038d1858dc22e8a194e5978c3412e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.353ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{m}}" loading="lazy"> im Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \mathbb {R} ^{m}}">

<msup>

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</msup>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/779c69e0b377721e1d867399d74906fc20949c49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.453ex; height:2.676ex;" alt="{\displaystyle p\in \mathbb {R} ^{m}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{p\}\times \mathbb {R} ^{m}}">

<msup>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4bc66a3442e36fbe4c08d62a94b23e413fb85fdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.688ex; height:2.843ex;" alt="{\displaystyle \{p\}\times \mathbb {R} ^{m}}" loading="lazy"> identifiziert, das Tangentialbündel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\mathbb {R} ^{m}}">

<msup>

</mrow>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab15bb92bc42f2721b5220c304ce9e940b4e7764.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.989ex; height:2.343ex;" alt="{\displaystyle T\mathbb {R} ^{m}}" loading="lazy"> also mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{m}\times \mathbb {R} ^{m}}">

<msup>

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</mrow>

</msup>

<msup>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5643361bcea4d3b8cd616e7378de45530e5dae57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.547ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{m}\times \mathbb {R} ^{m}}" loading="lazy">. In diesem Fall ist der Pushforward die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\ast }\colon (p,v)\mapsto (F(p),DF(p)(v))}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle F_{\ast }\colon (p,v)\mapsto (F(p),DF(p)(v))}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7066e1d58fff0d71a7aee26e9cf2b4461bb57b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.481ex; height:2.843ex;" alt="{\displaystyle F_{\ast }\colon (p,v)\mapsto (F(p),DF(p)(v))}" loading="lazy">.

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

Für den Pushforward einer Verkettung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\circ F\colon M\to P}">

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<annotation encoding="application/x-tex">{\displaystyle G\circ F\colon M\to P}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec150ddf53f6782715dfdf9fd4edd414e826bd63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.598ex; height:2.176ex;" alt="{\displaystyle G\circ F\colon M\to P}" loading="lazy"> zweier Abbildungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\colon M\to N}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d5441b4074702ac361fc3026ff30deb826a3f4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.895ex; height:2.176ex;" alt="{\displaystyle F\colon M\to N}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\colon N\to P}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/28f927840a6deb7639ccb8a2a1263669cc4bc0ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.284ex; height:2.176ex;" alt="{\displaystyle G\colon N\to P}" loading="lazy"> gilt die <a href="Mehrdimensionale_Kettenregel" title="Mehrdimensionale Kettenregel">Kettenregel</a>:

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<annotation encoding="application/x-tex">{\displaystyle (G\circ F)_{\ast }=G_{\ast }\circ F_{\ast }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb7eaa833852c3ae44f1168f494372b17a0a20c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.349ex; height:2.843ex;" alt="{\displaystyle (G\circ F)_{\ast }=G_{\ast }\circ F_{\ast }}" loading="lazy"></dd></dl>

bzw. punktweise

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<annotation encoding="application/x-tex">{\displaystyle (G\circ F)_{\ast p}=G_{\ast F(p)}\circ F_{\ast p}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f02d52007022589315a2d1eb4fd16bf585e65e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.987ex; height:3.176ex;" alt="{\displaystyle (G\circ F)_{\ast p}=G_{\ast F(p)}\circ F_{\ast p}.}" loading="lazy"></dd></dl>

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

<ul><li>John M. Lee: Introduction to Smooth Manifolds (= Graduate Texts in Mathematics 218). Springer-Verlag, New York NY u. a. 2003, ISBN 0-387-95448-1.</li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2025-07-01" href="https://de.wikipedia.org/wiki/?title=Pushforward&oldid=257543341">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

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