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<title>Graßmann-Algebra</title>

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

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Die Graßmann-Algebra oder äußere Algebra eines Vektorraums <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/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> ist eine assoziative, <a href="Schiefsymmetrisch" class="mw-redirect" title="Schiefsymmetrisch">schiefsymmetrisch</a>-<a href="Graduierung_(Algebra)" title="Graduierung (Algebra)">graduierte</a> Algebra mit Einselement. Sie ist – je nach Definition – Unteralgebra oder eine Faktoralgebra einer antisymmetrisierten <a href="Tensoralgebra" title="Tensoralgebra">Tensoralgebra</a> von <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/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> und wird durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda V}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac8ffe9e3869983c8703db2a82e4e5645fd14812.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.4ex; height:2.176ex;" alt="{\displaystyle \Lambda V}" loading="lazy"> dargestellt. Die Multiplikation wird als äußeres Produkt, Keilprodukt, Dachprodukt oder Wedgeprodukt bezeichnet. Ein Spezialfall dieses Produkts ist mit dem <a href="Kreuzprodukt" title="Kreuzprodukt">Kreuzprodukt</a> verwandt. Anwendung findet dieser Kalkül nicht nur in der elementaren <a href="Lineare_Algebra" title="Lineare Algebra">linearen Algebra</a> (zum Beispiel in der Theorie der <a href="Determinante" title="Determinante">Determinanten</a>), sondern vor allem in der algebraischen Geometrie und der <a href="Differentialgeometrie" title="Differentialgeometrie">Differentialgeometrie</a> als Algebra der Differentialformen. In dieser Form geht die Theorie der alternierenden <a href="Differentialform" title="Differentialform">Differentialformen</a> auf <a href="%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a> zurück, der damit die bestehenden Begriffe der Flächentheorie vereinheitlichte. Antikommutative Produkte von Vektoren wie auch abstrakte Vektorräume überhaupt wurden erstmals 1844 von <a href="Hermann_Gra%C3%9Fmann" title="Hermann Graßmann">Hermann Graßmann</a><a href="#cite_note-1">[1]</a> betrachtet.

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

<div class="mw-heading mw-heading3"><h3 id="Äußere_Potenz">Äußere Potenz</h3></div>

Es sei <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/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> ein Vektorraum über einem Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">. Weiter sei

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<annotation encoding="application/x-tex">{\displaystyle T^{k}(V)=\underbrace {V\otimes \cdots \otimes V} _{k{\text{-mal}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2cfebc15cdd6d9041af33f77d175d525bdd17699.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:21.482ex; height:6.343ex;" alt="{\displaystyle T^{k}(V)=\underbrace {V\otimes \cdots \otimes V} _{k{\text{-mal}}}}" loading="lazy"></dd></dl>

(mit den Konventionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{0}(V)=K}">

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<annotation encoding="application/x-tex">{\displaystyle T^{0}(V)=K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d33fe96181b0496503100cf82425aa1e7b1d9fd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.535ex; height:3.176ex;" alt="{\displaystyle T^{0}(V)=K}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{1}(V)=V}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8500f18ff43b9010765b7463932b94ee420ec45a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.256ex; height:3.176ex;" alt="{\displaystyle T^{1}(V)=V}" loading="lazy">). Der <a href="Untervektorraum" title="Untervektorraum">Untervektorraum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{k}(V)\subseteq T^{k}(V)}">

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<annotation encoding="application/x-tex">{\displaystyle J^{k}(V)\subseteq T^{k}(V)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07c18c17058d95823f74033655d66a17881b9ddb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.715ex; height:3.176ex;" alt="{\displaystyle J^{k}(V)\subseteq T^{k}(V)}" loading="lazy"> sei erzeugt durch Elementartensoren, bei denen zwei Faktoren gleich sind:

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<annotation encoding="application/x-tex">{\displaystyle J^{k}(V):=\mathrm {span} \left\{v_{1}\otimes \cdots \otimes v_{k}{\Big |}\;\exists i,j\in \{1,\dots ,k\};\,i\neq j\ \colon v_{i}=v_{j}\right\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25d002a1b9af6abc59d4da05d7c964f86e93524c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:63.964ex; height:4.843ex;" alt="{\displaystyle J^{k}(V):=\mathrm {span} \left\{v_{1}\otimes \cdots \otimes v_{k}{\Big |}\;\exists i,j\in \{1,\dots ,k\};\,i\neq j\ \colon v_{i}=v_{j}\right\}}" loading="lazy"></dd></dl>

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-te äußere Potenz ist dann definiert als der <a href="Faktorraum" title="Faktorraum">Quotientenraum</a>

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<annotation encoding="application/x-tex">{\displaystyle \,\Lambda ^{k}(V)=T^{k}(V)/J^{k}(V)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73526ff82a8a3c9083eeaf9202ed42acad4a7edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.563ex; height:3.176ex;" alt="{\displaystyle \,\Lambda ^{k}(V)=T^{k}(V)/J^{k}(V)}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Äußere_Algebra">Äußere Algebra</h3></div>

Die direkte Summe

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<annotation encoding="application/x-tex">{\displaystyle J(V)=\bigoplus _{k=0}^{\infty }J^{k}(V)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81c3ef9abb019d212eb6d7402a867c000ce8474a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.275ex; height:7.009ex;" alt="{\displaystyle J(V)=\bigoplus _{k=0}^{\infty }J^{k}(V)}" loading="lazy"></dd></dl>

ist ein zweiseitiges, homogenes <a href="Ideal_(Ringtheorie)" title="Ideal (Ringtheorie)">Ideal</a> in der <a href="Tensoralgebra" title="Tensoralgebra">Tensoralgebra</a>

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<annotation encoding="application/x-tex">{\displaystyle T(V)=\bigoplus _{k=0}^{\infty }T^{k}(V).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d42a2fa47ae27003efb752711ffaff35cb5d247f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.281ex; height:7.009ex;" alt="{\displaystyle T(V)=\bigoplus _{k=0}^{\infty }T^{k}(V).}" loading="lazy"></dd></dl>

Die äußere Algebra ist die Faktoralgebra

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<annotation encoding="application/x-tex">{\displaystyle \,\Lambda (V):=T(V)/J(V).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83cc8450f41d5208ba5099cc11c7bbb1d0547d36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.452ex; height:2.843ex;" alt="{\displaystyle \,\Lambda (V):=T(V)/J(V).}" loading="lazy"></dd></dl>

Als Vektorraum aufgefasst ist dies isomorph zu

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<annotation encoding="application/x-tex">{\displaystyle \bigoplus _{k=0}^{\infty }\Lambda ^{k}(V)=\bigoplus _{k=0}^{\infty }T^{k}(V)/J^{k}(V).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0e63dbbdaee4a38d50ab09f6fab0e18edc982b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.618ex; height:7.009ex;" alt="{\displaystyle \bigoplus _{k=0}^{\infty }\Lambda ^{k}(V)=\bigoplus _{k=0}^{\infty }T^{k}(V)/J^{k}(V).}" loading="lazy"></dd></dl>

Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>\dim V}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cf78220a6f756f245c61e86ce47889f9848054a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.359ex; height:2.176ex;" alt="{\displaystyle k>\dim V}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda ^{k}(V)=\{0\}}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda ^{k}(V)=\{0\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d96721085e6001357c17a6fb2612f0b570cd8c7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.884ex; height:3.176ex;" alt="{\displaystyle \Lambda ^{k}(V)=\{0\}}" loading="lazy">.

Das Produkt in der äußeren Algebra wird traditionell als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\wedge b}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1dc496a5b5da3e9b94eb72f04a54167dfe022e45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.81ex; height:2.176ex;" alt="{\displaystyle a\wedge b}" loading="lazy"> geschrieben.

Analog kann man die äußere Algebra von Moduln über kommutativen Ringen definieren.

<div class="mw-heading mw-heading3"><h3 id="Alternierende_Tensoren">Alternierende Tensoren</h3></div>

Neben der oben angeführten Definition der äußeren Algebra gibt es noch weitere äquivalente Möglichkeiten die äußere Algebra zu definieren. Beispielsweise kann man die Elemente der äußeren Algebra als alternierende Tensoren auffassen. Im Folgenden sei die <a href="Charakteristik_(Algebra)" title="Charakteristik (Algebra)">Charakteristik</a> des Körpers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> gleich 0.

Auf den homogenen Bestandteilen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{k}(V)}">

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<annotation encoding="application/x-tex">{\displaystyle T^{k}(V)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cfe72cf0bfc1d5e03a831bb55ffd88da2d9425d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.405ex; height:3.176ex;" alt="{\displaystyle T^{k}(V)}" loading="lazy"> operiert jeweils die <a href="Symmetrische_Gruppe" title="Symmetrische Gruppe">symmetrische Gruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{k}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd1325d812cf88e5341ac097d8bda175723da887.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.514ex; height:2.509ex;" alt="{\displaystyle S_{k}}" loading="lazy">. Ein Tensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in T^{k}(V)}">

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<annotation encoding="application/x-tex">{\displaystyle t\in T^{k}(V)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a38d4414aa87686e6b3f6e780344842277c6087d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.086ex; height:3.176ex;" alt="{\displaystyle t\in T^{k}(V)}" loading="lazy"> heißt alternierend, wenn

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<annotation encoding="application/x-tex">{\displaystyle \sigma (t)=\operatorname {sgn}(\sigma )\cdot t}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2836953cd83b4524770874882bfc0972abbce0b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.106ex; height:2.843ex;" alt="{\displaystyle \sigma (t)=\operatorname {sgn} (\sigma )\cdot t}" loading="lazy"></dd></dl>

für alle Permutationen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma \in S_{k}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd2caa5cbc962231cae2d7865bd19d5a83f82b97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.684ex; height:2.509ex;" alt="{\displaystyle \sigma \in S_{k}}" loading="lazy"> gilt (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn}(\sigma )}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn}(\sigma )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/716472e58beb0c37a9bdfbf33851309a7aae45f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.51ex; height:2.843ex;" alt="{\displaystyle \operatorname {sgn} (\sigma )}" loading="lazy"> ist das <a href="Vorzeichen_(Permutation)" title="Vorzeichen (Permutation)">Signum</a> der Permutation). Der Vektorraum der alternierenden Tensoren der Stufe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{k}(V)\subseteq T^{k}(V)}">

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<annotation encoding="application/x-tex">{\displaystyle A^{k}(V)\subseteq T^{k}(V)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e2319b570f104aa733f2c1b515380fed9e833943.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.932ex; height:3.176ex;" alt="{\displaystyle A^{k}(V)\subseteq T^{k}(V)}" loading="lazy">.

Man kann jedem Tensor mit Hilfe der Antisymmetrisierungsabbildung (auch „Alternator“) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Alt} _{k}\colon T^{k}(V)\rightarrow A^{k}(V)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Alt} _{k}\colon T^{k}(V)\rightarrow A^{k}(V)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8dff9341d17a000b053412d0b197649add4a4099.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.865ex; height:3.176ex;" alt="{\displaystyle \operatorname {Alt} _{k}\colon T^{k}(V)\rightarrow A^{k}(V)}" loading="lazy"> auf kanonische Weise einen alternierenden Tensor zuordnen. Sie ist definiert durch

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<annotation encoding="application/x-tex">{\displaystyle e_{1}\otimes \dotsb \otimes e_{k}\mapsto {\frac {1}{k!}}\sum _{\sigma \in S_{k}}\operatorname {sgn}(\sigma )(e_{\sigma (1)}\otimes \dotsb \otimes e_{\sigma (k)}).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/120c62928d147bea98e3ec382fabc12e67ab0ce6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:49.82ex; height:6.843ex;" alt="{\displaystyle e_{1}\otimes \dotsb \otimes e_{k}\mapsto {\frac {1}{k!}}\sum _{\sigma \in S_{k}}\operatorname {sgn} (\sigma )(e_{\sigma (1)}\otimes \dotsb \otimes e_{\sigma (k)}).}" loading="lazy"></dd></dl>

Sie ist eine <a href="Projektion_(Lineare_Algebra)" class="mw-redirect" title="Projektion (Lineare Algebra)">Projektion</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{k}(V)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A^{k}(V)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db3666c4b8cd9543f07782b50da449f76a767fc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.428ex; height:3.176ex;" alt="{\displaystyle A^{k}(V)}" loading="lazy">. Dabei sorgt der Faktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/k!}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/849820aa9ee7b7b81113b493b21fd19417cbe0d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.183ex; height:2.843ex;" alt="{\displaystyle 1/k!}" loading="lazy"> dafür, dass sie die Identitätsabbildung auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{k}(V)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A^{k}(V)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db3666c4b8cd9543f07782b50da449f76a767fc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.428ex; height:3.176ex;" alt="{\displaystyle A^{k}(V)}" loading="lazy"> ist, also alternierende Tensoren auf sich abbildet.

Mit dem Produkt

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\wedge b={\frac {(k+l)!}{k!\,l!}}\operatorname {Alt} _{k+l}(a\otimes b)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c7b23c08367ef39dd3eb59f848e7d8d2f983856.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.362ex; height:5.843ex;" alt="{\displaystyle a\wedge b={\frac {(k+l)!}{k!\,l!}}\operatorname {Alt} _{k+l}(a\otimes b)}" loading="lazy"></dd></dl>

für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in A^{k}(V),b\in A^{l}(V)}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\in A^{k}(V),b\in A^{l}(V)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd47182d5848a4d7733b54e54cc91fcf97b3dc26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.433ex; height:3.176ex;" alt="{\displaystyle a\in A^{k}(V),b\in A^{l}(V)}" loading="lazy"> und bilinearer Fortsetzung entsteht insgesamt im Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle A(V)=\bigoplus _{k=0}^{\infty }A^{k}(V)}">

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle A(V)=\bigoplus _{k=0}^{\infty }A^{k}(V)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d20039f379088e39cf9d9030b0e70971ccdca5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.024ex; height:3.176ex;" alt="{\displaystyle \textstyle A(V)=\bigoplus _{k=0}^{\infty }A^{k}(V)}" loading="lazy"> der alternierenden Tensoren eine assoziative, antikommutativ-graduierte Algebra. Die kanonische Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(V)\to \Lambda (V)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A(V)\to \Lambda (V)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df679e9e8ad5e3bd64eb972c23e7be0d34f6ebc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.163ex; height:2.843ex;" alt="{\displaystyle A(V)\to \Lambda (V)}" loading="lazy"> ist ein <a href="Algebrenisomorphismus" class="mw-redirect" title="Algebrenisomorphismus">Algebrenisomorphismus</a>.

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

In diesem Abschnitt wird auf die wesentlichen Eigenschaften der äußeren Algebra wie ihre <a href="Graduierung_(Algebra)" title="Graduierung (Algebra)">Graduierung</a> und die <a href="Universelle_Eigenschaft" title="Universelle Eigenschaft">universelle Eigenschaft</a> und auf ihr Produkt eingegangen. Vorausgesetzt wird dafür immer, dass <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/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</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">-dimensionaler Vektorraum ist.

<div class="mw-heading mw-heading3"><h3 id="Äußeres_Produkt">Äußeres Produkt</h3></div>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1caa4004cb216ef2930bb12fe805a76870caed94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }" loading="lazy"> der äußeren Algebra ist <a href="Assoziativgesetz" title="Assoziativgesetz">assoziativ</a>. Außerdem ist es kommutativ-graduiert, das heißt, es gilt

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\wedge b=(-1)^{kl}b\wedge a}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/102dd4899d8eba4c339e6138178a4e91c6018142.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.077ex; height:3.176ex;" alt="{\displaystyle a\wedge b=(-1)^{kl}b\wedge a}" loading="lazy"></dd></dl>

für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in \Lambda ^{k}(V)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\in \Lambda ^{k}(V)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d4038be1b78c4b2898e3bb2f3d3262e4aa061b58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.369ex; height:3.176ex;" alt="{\displaystyle a\in \Lambda ^{k}(V)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in \Lambda ^{l}(V)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b\in \Lambda ^{l}(V)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/423f2e6eeb58ccc1097ee2b90b1ea134306e39eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.77ex; height:3.176ex;" alt="{\displaystyle b\in \Lambda ^{l}(V)}" loading="lazy">. Insbesondere ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\wedge v=0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/226e81a0d394e8139e5ee97d68649ee3cdd54691.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.099ex; height:2.176ex;" alt="{\displaystyle v\wedge v=0}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in V}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99886ebbde63daa0224fb9bf56fa11b3c8a6f4fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.756ex; height:2.176ex;" alt="{\displaystyle v\in V}" loading="lazy">, aber im Allgemeinen ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\wedge a\neq 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f487266d358c9da61ac9329bff49667e9e5b6a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.303ex; height:2.676ex;" alt="{\displaystyle a\wedge a\neq 0}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in \Lambda ^{k}(V)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\in \Lambda ^{k}(V)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d4038be1b78c4b2898e3bb2f3d3262e4aa061b58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.369ex; height:3.176ex;" alt="{\displaystyle a\in \Lambda ^{k}(V)}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> gerade.

In der Terminologie der Supergeometrie verwendet man statt kommutativ-graduiert den äquivalenten Begriff superkommutativ und mit Hilfe des Superkommutators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\cdot },{\cdot }]}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [{\cdot },{\cdot }]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e58dc6e37720306f73f8f451fdf795d655f6a1f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.621ex; height:2.843ex;" alt="{\displaystyle [{\cdot },{\cdot }]}" loading="lazy"> lässt sich die Bedingung der Superkommutativität ausdrücken als

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [a,b]=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/67fe10397a99e43e831508fe1293862ec0a9cca3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.816ex; height:2.843ex;" alt="{\displaystyle [a,b]=0}" loading="lazy"></dd></dl>

für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in \Lambda ^{k}(V)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\in \Lambda ^{k}(V)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d4038be1b78c4b2898e3bb2f3d3262e4aa061b58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.369ex; height:3.176ex;" alt="{\displaystyle a\in \Lambda ^{k}(V)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in \Lambda ^{l}(V)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b\in \Lambda ^{l}(V)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/423f2e6eeb58ccc1097ee2b90b1ea134306e39eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.77ex; height:3.176ex;" alt="{\displaystyle b\in \Lambda ^{l}(V)}" loading="lazy">.

Ist <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"> eine <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">-<a href="Multilinearform#Alternierende_Multilinearformen" title="Multilinearform">Form</a> und <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"> eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">-Form, so lautet die explizite Formel für das äußere Produkt 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"> und <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"> für beliebige endlichdimensionale Vektorräume (und für unendlichdimensionale Banachräume):

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f\wedge g)(v_{1},\ldots ,v_{p},v_{p+1},\ldots ,v_{p+q})={\frac {1}{p!q!}}\sum _{\sigma \in \operatorname {Sym} _{p+q}}\operatorname {sgn}(\sigma )f(v_{\sigma (1)},\ldots ,v_{\sigma (p)})g(v_{\sigma (p+1)},\ldots ,v_{\sigma (p+q)})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10bd482af86e057e88dfbae26ee759fe17d00e91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:93.416ex; height:7.009ex;" alt="{\displaystyle (f\wedge g)(v_{1},\ldots ,v_{p},v_{p+1},\ldots ,v_{p+q})={\frac {1}{p!q!}}\sum _{\sigma \in \operatorname {Sym} _{p+q}}\operatorname {sgn} (\sigma )f(v_{\sigma (1)},\ldots ,v_{\sigma (p)})g(v_{\sigma (p+1)},\ldots ,v_{\sigma (p+q)})}" loading="lazy">,</dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sym} _{p+q}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Sym} _{p+q}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ccaee1aa6a15c1d05316ccc657638aa24e2a13eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.55ex; height:3.009ex;" alt="{\displaystyle \operatorname {Sym} _{p+q}}" loading="lazy"> die <a href="Symmetrische_Gruppe" title="Symmetrische Gruppe">symmetrische Gruppe</a> der Ordnung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p+q}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/02fcb4cecca0b7e3116a7351e4345b48ef6de371.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.169ex; height:2.343ex;" alt="{\displaystyle p+q}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn}(\sigma )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn}(\sigma )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/716472e58beb0c37a9bdfbf33851309a7aae45f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.51ex; height:2.843ex;" alt="{\displaystyle \operatorname {sgn} (\sigma )}" loading="lazy"> das <a href="Vorzeichen_(Permutation)" title="Vorzeichen (Permutation)">Vorzeichen</a> der Permutation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.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 \sigma }" loading="lazy"> darstellen sollen.

<div class="mw-heading mw-heading3"><h3 id="Graduierung,_Basis_und_Dimension">Graduierung, Basis und Dimension</h3></div>

Die äußere Algebra

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (V)=\bigoplus _{m=0}^{n}\Lambda ^{m}(V)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03658244139e8766681b3dcd578f6b05c6c61d01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.123ex; height:6.843ex;" alt="{\displaystyle \Lambda (V)=\bigoplus _{m=0}^{n}\Lambda ^{m}(V)}" loading="lazy"></dd></dl>

ist eine <a href="Graduierte_Algebra" class="mw-redirect" title="Graduierte Algebra">graduierte Algebra</a>. Das heißt, sie kann als <a href="Direkte_Summe" title="Direkte Summe">direkte Summe</a> von <a href="Untervektorraum" title="Untervektorraum">Untervektorräumen</a>, welche durch eine <a href="Abelsche_Gruppe" title="Abelsche Gruppe">abelsche Gruppe</a> indiziert werden, dargestellt werden, sodass das Algebraprodukt mit dieser Zerlegung verträglich ist. Für die äußere Algebra folgt dies direkt aus deren Definition: für die Untervektorräume der äußeren Potenzen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda ^{m}(V)}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda ^{m}(V)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/921cb78e8228a561a67e6b9d0cc18fa8d837ef53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.885ex; height:2.843ex;" alt="{\displaystyle \Lambda ^{m}(V)}" loading="lazy"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda ^{m}(V)\wedge \Lambda ^{n}(V)=\Lambda ^{m+n}(V)}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda ^{m}(V)\wedge \Lambda ^{n}(V)=\Lambda ^{m+n}(V)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/720f411fc83a720e04d51142e4c6297954be22ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.143ex; height:3.009ex;" alt="{\displaystyle \Lambda ^{m}(V)\wedge \Lambda ^{n}(V)=\Lambda ^{m+n}(V)}" loading="lazy">.

Sei nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{1},\dotsc ,e_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{1},\dotsc ,e_{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd57f55147a3c408d8a76b61a0a4ebcfa15a8597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.618ex; height:2.009ex;" alt="{\displaystyle e_{1},\dotsc ,e_{n}}" loading="lazy"> eine Basis des <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">-dimensionalen Vektorraums <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/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">. Dann ist

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<annotation encoding="application/x-tex">{\displaystyle \{\,e_{i_{1}}\wedge \dotsb \wedge e_{i_{k}}\,|\,i_{1}<\dotsb <i_{k}\,\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/555520defce56dc8e82585d23b1a2809d94243f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.534ex; height:3.009ex;" alt="{\displaystyle \{\,e_{i_{1}}\wedge \dotsb \wedge e_{i_{k}}\,|\,i_{1}<\dotsb <i_{k}\,\}}" loading="lazy"></dd></dl>

eine Basis von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda ^{k}(V)}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda ^{k}(V)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bdafe778bbe3eabffb63bb3f8ceac909e25c0924.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.298ex; height:3.176ex;" alt="{\displaystyle \Lambda ^{k}(V)}" loading="lazy">. Die Dimension ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim(\Lambda ^{k}(V))={\tbinom {n}{k}}}">

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<annotation encoding="application/x-tex">{\displaystyle \dim(\Lambda ^{k}(V))={\tbinom {n}{k}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/248aec757a67c9a6e3cb1691dab70f7397d77bbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.197ex; height:3.343ex;" alt="{\displaystyle \dim(\Lambda ^{k}(V))={\tbinom {n}{k}}}" loading="lazy">. Insbesondere ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim(\Lambda ^{k}(V))=0}">

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<annotation encoding="application/x-tex">{\displaystyle \dim(\Lambda ^{k}(V))=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3e5fb67c6e34e745725d25f4c8363198e3f41bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.244ex; height:3.176ex;" alt="{\displaystyle \dim(\Lambda ^{k}(V))=0}" loading="lazy">, falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>n}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66e81682bf174c978e9008ffb557ba4da2cf7478.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.704ex; height:2.176ex;" alt="{\displaystyle k>n}" loading="lazy">.

Die Basis der äußeren Algebra erhält man dann durch Vereinigung der Basen aller Grade. Für die Dimension von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (V)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0fe372eb86e9415930c1e3a69af01789a1b2d0d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.21ex; height:2.843ex;" alt="{\displaystyle \Lambda (V)}" loading="lazy"> gilt dann

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<annotation encoding="application/x-tex">{\displaystyle \dim(\Lambda (V))=\sum _{i=0}^{n}{\binom {n}{i}}=2^{n},}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9047c8f609058d71f0eccc608db66abe1b3a832.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.677ex; height:6.843ex;" alt="{\displaystyle \dim(\Lambda (V))=\sum _{i=0}^{n}{\binom {n}{i}}=2^{n},}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tbinom {n}{i}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e5b08ceeda71de3fe13b5e9214e3bfc7f0c26a74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.116ex; height:3.176ex;" alt="{\displaystyle {\tbinom {n}{i}}}" loading="lazy"> den <a href="Binomialkoeffizient" title="Binomialkoeffizient">Binomialkoeffizienten</a> bezeichnet. Es folgt, dass sich jedes Element der Graßmann-Algebra darstellen lässt als

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<annotation encoding="application/x-tex">{\displaystyle \sum _{I\subseteq \{1,\dotsc ,n\}}f_{I}\,e_{I},}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a2638f25fc874c37324e73c7d5e1761b70e8c2b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:14.165ex; height:6.009ex;" alt="{\displaystyle \sum _{I\subseteq \{1,\dotsc ,n\}}f_{I}\,e_{I},}" loading="lazy"></dd></dl>

wobei die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8226f30650ee4fe4e640c6d2798127e80e9c160d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:2.343ex;" alt="{\displaystyle 2^{n}}" loading="lazy"> Koeffizienten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{I}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/261d1a55930b021df6465cfd506dc13fc00b491d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.2ex; height:2.509ex;" alt="{\displaystyle f_{I}}" loading="lazy"> das Element bezüglich einer Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{1},\dotsc ,e_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{1},\dotsc ,e_{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd57f55147a3c408d8a76b61a0a4ebcfa15a8597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.618ex; height:2.009ex;" alt="{\displaystyle e_{1},\dotsc ,e_{n}}" loading="lazy"> charakterisieren und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{I}:=e_{m_{1}}\wedge \dotsb \wedge e_{m_{k}}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{I}:=e_{m_{1}}\wedge \dotsb \wedge e_{m_{k}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/893646e4379dd8b42f3049a01c80a0585ac0bc4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.986ex; height:2.676ex;" alt="{\displaystyle e_{I}:=e_{m_{1}}\wedge \dotsb \wedge e_{m_{k}}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=\{m_{1},\dotsc ,m_{k}\};\,i<j\,\Rightarrow \,m_{i}<m_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle I=\{m_{1},\dotsc ,m_{k}\};\,i<j\,\Rightarrow \,m_{i}<m_{j}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60877e0185dd83db3c29856ac2efae79038c084a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.554ex; height:3.009ex;" alt="{\displaystyle I=\{m_{1},\dotsc ,m_{k}\};\,i<j\,\Rightarrow \,m_{i}<m_{j}}" loading="lazy"> ist.

Als Beispiel kann man den Vektorraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{4}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{4}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4abb9b9dab94f7b25a4210364f0f9032704bfb9.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} ^{4}}" loading="lazy"> mit der <a href="Standardbasis" title="Standardbasis">kanonischen Basis</a> wählen. Der 3. Grad der äußeren Algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (\mathbb {R} ^{4})}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (\mathbb {R} ^{4})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d260b111b92b02d69364eaf5c2c9bdbe4dde066a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.155ex; height:3.176ex;" alt="{\displaystyle \Lambda (\mathbb {R} ^{4})}" loading="lazy"> wird aufgespannt durch:

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<annotation encoding="application/x-tex">{\displaystyle \Lambda ^{3}(\mathbb {R} ^{4})=\operatorname {span} (\{(e_{1}\wedge e_{2}\wedge e_{3}),(e_{1}\wedge e_{2}\wedge e_{4}),(e_{1}\wedge e_{3}\wedge e_{4}),(e_{2}\wedge e_{3}\wedge e_{4})\})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5a829a0772dc67d9ffca6ceab735bd362f5f706e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:75.758ex; height:3.176ex;" alt="{\displaystyle \Lambda ^{3}(\mathbb {R} ^{4})=\operatorname {span} (\{(e_{1}\wedge e_{2}\wedge e_{3}),(e_{1}\wedge e_{2}\wedge e_{4}),(e_{1}\wedge e_{3}\wedge e_{4}),(e_{2}\wedge e_{3}\wedge e_{4})\})}" loading="lazy"></dd></dl>

Durch Abzählen sieht man, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim(\Lambda ^{3}(\mathbb {R} ^{4}))=4}">

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<annotation encoding="application/x-tex">{\displaystyle \dim(\Lambda ^{3}(\mathbb {R} ^{4}))=4}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9aa8ea4b9a8dbceb9b91416a0153a35a12c94e6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.154ex; height:3.176ex;" alt="{\displaystyle \dim(\Lambda ^{3}(\mathbb {R} ^{4}))=4}" loading="lazy"> ist.

<div class="mw-heading mw-heading3"><h3 id="Universelle_Eigenschaft">Universelle Eigenschaft</h3></div>

Ist <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/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> ein Vektorraum (bzw. Modul) und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> eine <a href="Assoziative_Algebra" title="Assoziative Algebra">assoziative Algebra</a>, so gibt es eine Bijektion zwischen

<ul><li>den Homomorphismen von Vektorräumen (bzw. Moduln) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon V\to A}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8b6f9bdc18bc3cb37782a1af6f9549b4a8e94d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.457ex; height:2.509ex;" alt="{\displaystyle f\colon V\to A}" loading="lazy">, so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(v)^{2}=0}">

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<annotation encoding="application/x-tex">{\displaystyle f(v)^{2}=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61517b3cbb50a3ff6c9e831f1af1653fd85782ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.531ex; height:3.176ex;" alt="{\displaystyle f(v)^{2}=0}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in V}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99886ebbde63daa0224fb9bf56fa11b3c8a6f4fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.756ex; height:2.176ex;" alt="{\displaystyle v\in V}" loading="lazy"> gilt</li></ul>

und

<ul><li>den Algebrenhomomorphismen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (V)\to A}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda (V)\to A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7296a71f6034e7e82e4cb0477bb03220da0739ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.567ex; height:2.843ex;" alt="{\displaystyle \Lambda (V)\to A}" loading="lazy">.</li></ul>

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

Hat der Vektorraum <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/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> ein Skalarprodukt, so kann auch die äußere Algebra mit einem solchen ausgestattet werden. Dabei werden Unterräume verschiedenen Grades als <a href="Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a> definiert. Innerhalb eines Unterraums genügt es, das Skalarprodukt auf reinen Produkten zu definieren. Seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}\wedge \dots \wedge a_{m}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle a_{1}\wedge \dots \wedge a_{m}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f99c71f6e08854d6b8ed4c5c67ff34cee85cb6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.077ex; height:2.343ex;" alt="{\displaystyle a_{1}\wedge \dots \wedge a_{m}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1}\wedge \dots \wedge b_{m}}">

<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle b_{1}\wedge \dots \wedge b_{m}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a16343aa2a768c5f925adebf27e663a362f22d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.613ex; height:2.509ex;" alt="{\displaystyle b_{1}\wedge \dots \wedge b_{m}}" loading="lazy"> reine Produkte in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda ^{m}V}">

<msup>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e82da469192a80e1b823ce74414fdd94df365453.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.075ex; height:2.343ex;" alt="{\displaystyle \Lambda ^{m}V}" loading="lazy">. Ihnen kann die <a href="Gramsche_Matrix" class="mw-redirect" title="Gramsche Matrix">Gramsche Matrix</a> der Skalarprodukte zugeordnet werden. Dann kann das Skalarprodukt als Determinante der Gramschen Matrix definiert werden:

<msub>

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<msub>

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<mrow>

<mtr>

<mtd>

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<mtd>

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<mtd>

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<mtr>

<mtd>

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<mtd>

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<mtr>

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<mtd>

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<mtd>

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<annotation encoding="application/x-tex">{\displaystyle \langle a_{1}\wedge \dots \wedge a_{m},\,b_{1}\wedge \dots \wedge b_{m}\rangle :=\det {\begin{pmatrix}\langle a_{1},b_{1}\rangle &\dots &\langle a_{1},b_{m}\rangle \\\vdots &&\vdots \\\langle a_{m},b_{1}\rangle &\dots &\langle a_{m},b_{m}\rangle \end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff3d5145a4aff70a985a6e48c9701b25e6d5cc96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:64.69ex; height:10.843ex;" alt="{\displaystyle \langle a_{1}\wedge \dots \wedge a_{m},\,b_{1}\wedge \dots \wedge b_{m}\rangle :=\det {\begin{pmatrix}\langle a_{1},b_{1}\rangle &\dots &\langle a_{1},b_{m}\rangle \\\vdots &&\vdots \\\langle a_{m},b_{1}\rangle &\dots &\langle a_{m},b_{m}\rangle \end{pmatrix}}}" loading="lazy"></dd></dl>

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

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

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

</semantics>

</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 Spaltenvektorraum, so kann zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}\wedge \dots \wedge a_{m}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle a_{1}\wedge \dots \wedge a_{m}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f99c71f6e08854d6b8ed4c5c67ff34cee85cb6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.077ex; height:2.343ex;" alt="{\displaystyle a_{1}\wedge \dots \wedge a_{m}}" loading="lazy"> die Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=(a_{1},\dots ,a_{m})}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/542b9ea2c774fcdca047a351efdfdc03ca02e565.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.018ex; height:2.843ex;" alt="{\displaystyle A=(a_{1},\dots ,a_{m})}" loading="lazy"> definiert werden. Von dieser kann man die maximalen quadratischen <a href="Untermatrix" title="Untermatrix">Untermatrizen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\alpha }}">

<msub>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7bce019389c665fc524839b04cf9674ce661d4c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="{\displaystyle A_{\alpha }}" loading="lazy"> betrachten. Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> ein <a href="Multiindex" title="Multiindex">Multiindex</a> aus

<msub>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle I_{m}:=\{\alpha \in \mathbb {N} ^{m}:\;1\leq \alpha (1)<\dots <\alpha (m)\leq n\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b6c9fe1ccdd9435f30b7d3a0d056a2e5539d6f93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.503ex; height:2.843ex;" alt="{\displaystyle I_{m}:=\{\alpha \in \mathbb {N} ^{m}:\;1\leq \alpha (1)<\dots <\alpha (m)\leq n\}}" loading="lazy"></dd></dl>

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

<msub>

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</mstyle>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7bce019389c665fc524839b04cf9674ce661d4c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="{\displaystyle A_{\alpha }}" loading="lazy"> besteht aus genau diesen Zeilen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">.

Es gilt folgende Identität nach dem <a href="Satz_von_Binet-Cauchy" title="Satz von Binet-Cauchy">Satz von Binet-Cauchy</a>, im Falle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=2}">

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b32de1b0dc05f6e525ad6a3e8ddeeb4321fd79e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=2}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=B}">

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/045cafe35b1e9c9ac889481fd7178d6f59a77fdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A=B}" loading="lazy"> auch „Flächenpythagoras“ genannt:

<msub>

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<msup>

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</msup>

<msub>

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</munder>

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<annotation encoding="application/x-tex">{\displaystyle \det(\;(\langle a_{i},b_{k}\rangle )\;)=\det(A^{t}B)=\sum _{\alpha \in I_{m}}\det A_{\alpha }\cdot \det B_{\alpha }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/503410610c3dac3960ebc59d2f45a7c4221c4815.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:50.248ex; height:5.843ex;" alt="{\displaystyle \det(\;(\langle a_{i},b_{k}\rangle )\;)=\det(A^{t}B)=\sum _{\alpha \in I_{m}}\det A_{\alpha }\cdot \det B_{\alpha }}" loading="lazy"></dd></dl>

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

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

Das Hauptanwendungsgebiet der äußeren Algebra liegt in der Differentialgeometrie. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<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 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 <a href="Differenzierbare_Mannigfaltigkeit" title="Differenzierbare Mannigfaltigkeit">differenzierbare Mannigfaltigkeit</a>. So wählt man den <a href="Kotangentialraum" title="Kotangentialraum">Kotangentialraum</a> dieser Mannigfaltigkeit als zugrundeliegenden Vektorraum und bildet die äußere Algebra. Eine Differentialform ist ein <a href="Schnitt_(Faserb%C3%BCndel)" title="Schnitt (Faserbündel)">Schnitt</a> im <a href="Vektorb%C3%BCndel" title="Vektorbündel">Bündel</a> dieser Vektorräume, also eine Abbildung, die jedem Punkt der Mannigfaltigkeit ein Element der äußeren Algebra über dem Kotangentialraum an diesem Punkt zuordnet. Diese Formen haben den großen Vorteil, dass man mit ihrer Hilfe <a href="Atlas_(Mathematik)" title="Atlas (Mathematik)">kartenunabhängig</a> auf einer Mannigfaltigkeit integrieren kann.

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

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Hodge-Stern-Operator" title="Hodge-Stern-Operator">Hodge-Stern-Operator</a></div>

Sei <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/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> (wie oben) ein Vektorraum und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda ^{n}V}">

<msup>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bea28f2771f7ac7f5a7e58b291c634258e057625.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.619ex; height:2.343ex;" alt="{\displaystyle \Lambda ^{n}V}" loading="lazy"> die äußere Algebra von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">. Weiterhin sei <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/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"><a href="Orientierung_(Mathematik)" title="Orientierung (Mathematik)">orientiert</a> und mit einem Skalarprodukt versehen. Der Hodge-Operator oder Hodge-Stern-Operator ist ein natürlicher Isomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *:\Lambda ^{k}V\rightarrow \Lambda ^{n-k}V}">

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<annotation encoding="application/x-tex">{\displaystyle *:\Lambda ^{k}V\rightarrow \Lambda ^{n-k}V}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e01d8cf784cd8db256e6a06ae727175c8acf3064.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.956ex; height:2.676ex;" alt="{\displaystyle *:\Lambda ^{k}V\rightarrow \Lambda ^{n-k}V}" loading="lazy">. Der Hodge-Operator ordnet also jedem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \in \Lambda ^{k}V}">

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<annotation encoding="application/x-tex">{\displaystyle \omega \in \Lambda ^{k}V}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9afb8342b3c81021aada2c5549d3b28af6826cc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.775ex; height:2.676ex;" alt="{\displaystyle \omega \in \Lambda ^{k}V}" loading="lazy"> auf eindeutige Weise ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *\omega \in \Lambda ^{n-k}V}">

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<annotation encoding="application/x-tex">{\displaystyle *\omega \in \Lambda ^{n-k}V}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9284c23aa253734304643193033f45b4b4476e78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.203ex; height:2.676ex;" alt="{\displaystyle *\omega \in \Lambda ^{n-k}V}" loading="lazy"> zu, das sog. „duale Element“ zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy">. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (e_{1},\dots ,e_{n})}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83ccf3398cfc17c5754c3ad49d1cb98191660797.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.427ex; height:2.843ex;" alt="{\displaystyle (e_{1},\dots ,e_{n})}" loading="lazy"> eine <a href="Orientierung_(Mathematik)" title="Orientierung (Mathematik)">orientierte Basis</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *\omega }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74950c3e2e6d527d7911338d15890b046c1e2df6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.608ex; height:1.676ex;" alt="{\displaystyle *\omega }" loading="lazy"> eindeutig durch die Formel

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<annotation encoding="application/x-tex">{\displaystyle \forall \eta \in \Lambda ^{k}V:\;\eta \wedge *\omega =\langle \eta ,\omega \rangle \cdot e_{1}\wedge \dots \wedge e_{n},}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/62f7492bcc466ec8dce39bce50ab5c2083d8489c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.945ex; height:3.176ex;" alt="{\displaystyle \forall \eta \in \Lambda ^{k}V:\;\eta \wedge *\omega =\langle \eta ,\omega \rangle \cdot e_{1}\wedge \dots \wedge e_{n},}" loading="lazy"></dd></dl>

festgelegt. Zum Beispiel gilt, falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (e_{1},\dots ,e_{n})}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83ccf3398cfc17c5754c3ad49d1cb98191660797.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.427ex; height:2.843ex;" alt="{\displaystyle (e_{1},\dots ,e_{n})}" loading="lazy"> zusätzlich eine <a href="Orthonormalbasis" title="Orthonormalbasis">Orthonormalbasis</a> ist,

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<annotation encoding="application/x-tex">{\displaystyle *(e_{1}\wedge \dots \wedge e_{k})=e_{k+1}\wedge \dots \wedge e_{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5646a926fc5a2f193455235b389ac6d5ab15d583.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.732ex; height:2.843ex;" alt="{\displaystyle *(e_{1}\wedge \dots \wedge e_{k})=e_{k+1}\wedge \dots \wedge e_{n}}" loading="lazy">

für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0,\dots ,n}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d5cf3345af67f4c5717ed8d7ca653e130d152da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.045ex; height:2.509ex;" alt="{\displaystyle k=0,\dots ,n}" loading="lazy"> (wobei das leere Produkt, für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6307c8a99dad7d0bcb712352ae0a748bd99a038b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=0}" loading="lazy"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=n}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c4b8cc6fcba0c0b24656b0fb33414d2e6cffb83c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.704ex; height:2.176ex;" alt="{\displaystyle k=n}" loading="lazy">, als 1 zu interpretieren ist). Der Hodge-Operator kann also als algebraische Verallgemeinerung des geometrischen Begriffs des orthogonalen Komplements von Unterräumen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> aufgefasst werden.

<div class="mw-heading mw-heading3"><h3 id="Beziehung_zum_Kreuzprodukt_und_Spatprodukt_(Hodge-Dualität_von_Vektoren)_und_Begriffen_der_Physik">Beziehung zum Kreuzprodukt und Spatprodukt (Hodge-Dualität von Vektoren) und Begriffen der Physik</h3></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b830763a1f552a9c87e7a97e3fb34924c92f3ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.906ex; height:2.009ex;" alt="{\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}}" loading="lazy"> die kanonische Basis des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">

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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"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+a_{3}\mathbf {e} _{3},\beta =b_{1}\mathbf {e} _{1}+b_{2}\mathbf {e} _{2}+b_{3}\mathbf {e} _{3}\in \Lambda ^{1}(\mathbb {R} ^{3})}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha =a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+a_{3}\mathbf {e} _{3},\beta =b_{1}\mathbf {e} _{1}+b_{2}\mathbf {e} _{2}+b_{3}\mathbf {e} _{3}\in \Lambda ^{1}(\mathbb {R} ^{3})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d07a9dd577ef16df7d6b71834232677a0684875e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.146ex; height:3.176ex;" alt="{\displaystyle \alpha =a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+a_{3}\mathbf {e} _{3},\beta =b_{1}\mathbf {e} _{1}+b_{2}\mathbf {e} _{2}+b_{3}\mathbf {e} _{3}\in \Lambda ^{1}(\mathbb {R} ^{3})}" loading="lazy"> seien zwei Elemente aus der äußeren Algebra (bzw. äußeren Potenz) des reellen Vektorraumes. Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e9972f426d9e07855984f73ee195a21dbc21755.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" loading="lazy"> wird der Hodge-Operator bezüglich des Standard- (euklidischen) Skalarprodukts und der Standardorientierung bezeichnet. Für das äußere Produkt von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"> gilt mithilfe des Distributivgesetzes

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rl}*(\alpha \wedge \beta )=&*((a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+a_{3}\mathbf {e} _{3})\wedge (b_{1}\mathbf {e} _{1}+b_{2}\mathbf {e} _{2}+b_{3}\mathbf {e} _{3}))\\[0.5em]=&*((a_{2}\mathbf {e} _{2}\wedge b_{1}\mathbf {e} _{1})+(a_{3}\mathbf {e} _{3}\wedge b_{1}\mathbf {e} _{1})+(a_{1}\mathbf {e} _{1}\wedge b_{2}\mathbf {e} _{2})\\&+(a_{3}\mathbf {e} _{3}\wedge b_{2}\mathbf {e} _{2})+(a_{1}\mathbf {e} _{1}\wedge b_{3}\mathbf {e} _{3})+(a_{2}\mathbf {e} _{2}\wedge b_{3}\mathbf {e} _{3}))\\[0.5em]=&*((a_{1}b_{2}-a_{2}b_{1})(\mathbf {e} _{1}\wedge \mathbf {e} _{2})+(a_{2}b_{3}-a_{3}b_{2})(\mathbf {e} _{2}\wedge \mathbf {e} _{3})+(a_{3}b_{1}-a_{1}b_{3})(\mathbf {e} _{3}\wedge \mathbf {e} _{1}))\,.\end{array}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/95a62e54b51ea8c4c7cb506a71bd7f71e2bbee9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:90.404ex; height:15.509ex;" alt="{\displaystyle {\begin{array}{rl}*(\alpha \wedge \beta )=&*((a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+a_{3}\mathbf {e} _{3})\wedge (b_{1}\mathbf {e} _{1}+b_{2}\mathbf {e} _{2}+b_{3}\mathbf {e} _{3}))\\[0.5em]=&*((a_{2}\mathbf {e} _{2}\wedge b_{1}\mathbf {e} _{1})+(a_{3}\mathbf {e} _{3}\wedge b_{1}\mathbf {e} _{1})+(a_{1}\mathbf {e} _{1}\wedge b_{2}\mathbf {e} _{2})\\&+(a_{3}\mathbf {e} _{3}\wedge b_{2}\mathbf {e} _{2})+(a_{1}\mathbf {e} _{1}\wedge b_{3}\mathbf {e} _{3})+(a_{2}\mathbf {e} _{2}\wedge b_{3}\mathbf {e} _{3}))\\[0.5em]=&*((a_{1}b_{2}-a_{2}b_{1})(\mathbf {e} _{1}\wedge \mathbf {e} _{2})+(a_{2}b_{3}-a_{3}b_{2})(\mathbf {e} _{2}\wedge \mathbf {e} _{3})+(a_{3}b_{1}-a_{1}b_{3})(\mathbf {e} _{3}\wedge \mathbf {e} _{1}))\,.\end{array}}}" loading="lazy"></dd></dl>

Der Hodge-Operator ordnet im dreidimensionalen Raum dem Produkt der Basisvektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{1}\wedge \mathbf {e} _{2}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24cbe2f305c3398371155c651acadd5789addaf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.279ex; height:2.009ex;" alt="{\displaystyle \mathbf {e} _{3}}" loading="lazy"> zu. Durch <a href="Zyklische_Permutation" title="Zyklische Permutation">zyklisches Vertauschen</a> der Indizes ergeben sich die Zuordnungen der anderen Basisvektoren. Damit ergibt sich das <a href="Kreuzprodukt" title="Kreuzprodukt">Kreuzprodukt</a> im dreidimensionalen reellen Raum. Also kann man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *(\alpha \wedge \beta )}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8686c938653e3b8d4b991f7797412a28d332b620.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.374ex; height:2.843ex;" alt="{\displaystyle *(\alpha \wedge \beta )}" loading="lazy"> auf der äußeren Algebra als Verallgemeinerung des Kreuzproduktes verstehen. Mit Hilfe dieser Verallgemeinerung lässt sich ebenfalls der aus der Vektoranalysis bekannte <a href="Differentialoperator" title="Differentialoperator">Differentialoperator</a> <a href="Rotation_(Mathematik)" class="mw-redirect" title="Rotation (Mathematik)">Rotation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {rot} }">

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Das <a href="Spatprodukt" title="Spatprodukt">Spatprodukt</a> dreier <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f13f068df656c1b1911ae9f81628c49a6181194d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}" loading="lazy"> Vektoren im <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">

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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"> lässt sich entsprechend als Element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\wedge b\wedge c}">

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<annotation encoding="application/x-tex">{\displaystyle a\wedge b\wedge c}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d355d8ef25d2601a5ccdc05dfc92dd8e647b41be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.399ex; height:2.176ex;" alt="{\displaystyle a\wedge b\wedge c}" loading="lazy"> der dritten äußeren Potenz auffassen. Man beachte, dass der Hodge-Stern-Operator nur bezüglich eines <a href="Skalarprodukt" title="Skalarprodukt">Skalarprodukts</a> und einer Orientierung definiert ist. Das äußere Produkt dagegen lässt sich unabhängig von einer solchen Wahl definieren.

Der klassischen Physik entstammende Größen, die in der Physik <a href="Pseudovektor" title="Pseudovektor">Pseudovektoren</a> genannt werden, wie zum Beispiel eine <a href="Magnetische_Feldst%C3%A4rke" title="Magnetische Feldstärke">magnetische Feldstärke</a> oder ein <a href="Drehimpuls" title="Drehimpuls">Drehimpuls</a>, lassen sich als Elemente von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda ^{2}(\mathbb {R} ^{3})}">

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<annotation encoding="application/x-tex">{\displaystyle \Lambda ^{2}(\mathbb {R} ^{3})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e81c939d4849d76db39140333215257f47fc32d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.209ex; height:3.176ex;" alt="{\displaystyle \Lambda ^{2}(\mathbb {R} ^{3})}" loading="lazy"> auffassen. Mit einem <a href="Pseudoskalar" title="Pseudoskalar">Pseudoskalar</a> ist in vielen Fällen eine Größe gemeint, die sich als Element von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda ^{3}(\mathbb {R} ^{3})}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68e84efee4abbb9729cc4f3c9a823277fb8d4045.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.209ex; height:3.176ex;" alt="{\displaystyle \Lambda ^{3}(\mathbb {R} ^{3})}" loading="lazy"> verstehen lässt.

<div class="mw-heading mw-heading3"><h3 id="Beziehung_zur_Determinanten-Theorie;_Ausdehnungsmaß_von_m-Vektoren">Beziehung zur Determinanten-Theorie; Ausdehnungsmaß von m-Vektoren</h3></div>

Noch einfacher ist der mit dem Hodge-Operator einhergehende Begriff der Dualität bei Skalaren: Diese sind dual zur Determinante einer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59d2b4cb72e304526cf5b5887147729ea259da78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" loading="lazy">-Matrix.<a href="#cite_note-2">[2]</a> Im Einzelnen:

Es sollen die gleichen Voraussetzungen wie im vorigen Abschnitt gelten; nur sei jetzt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\geq 3}">

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<annotation encoding="application/x-tex">{\displaystyle m\geq 3}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4610f29d2708d1febf1c0090f58ddd3986593545.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.301ex; height:2.343ex;" alt="{\displaystyle m\geq 3}" loading="lazy"> zugelassen, und es sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq m\,.}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58a4d5054d4c2d79bb1e2844501001c23ee1e936.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.568ex; height:2.176ex;" alt="{\displaystyle n\geq m\,.}" loading="lazy"> Wenn nunmehr, für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq i_{\nu }\leq n\,,}">

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<annotation encoding="application/x-tex">{\displaystyle 1\leq i_{\nu }\leq n\,,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a462df83a30adefd269996180d5580653ff2f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.694ex; height:2.509ex;" alt="{\displaystyle 1\leq i_{\nu }\leq n\,,}" loading="lazy"> ein <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_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">-Bein der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \gamma :=\sum _{\,i_{1}<i_{2}<\ldots <i_{m}}\,(a_{i_{1}}^{(1)}a_{i_{2}}^{(2)}\ldots a_{i_{m}}^{(m)})_{\,asy}\,\mathbf {e} _{i_{1}}\wedge \mathbf {e} _{i_{2}}\wedge \ldots \wedge \mathbf {e} _{i_{m}}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \gamma :=\sum _{\,i_{1}<i_{2}<\ldots <i_{m}}\,(a_{i_{1}}^{(1)}a_{i_{2}}^{(2)}\ldots a_{i_{m}}^{(m)})_{\,asy}\,\mathbf {e} _{i_{1}}\wedge \mathbf {e} _{i_{2}}\wedge \ldots \wedge \mathbf {e} _{i_{m}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3036ec96a55a0338462cc501f50845fc52743a0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:58.556ex; height:4.009ex;" alt="{\displaystyle \textstyle \gamma :=\sum _{\,i_{1}<i_{2}<\ldots <i_{m}}\,(a_{i_{1}}^{(1)}a_{i_{2}}^{(2)}\ldots a_{i_{m}}^{(m)})_{\,asy}\,\mathbf {e} _{i_{1}}\wedge \mathbf {e} _{i_{2}}\wedge \ldots \wedge \mathbf {e} _{i_{m}}}" loading="lazy"> gegeben ist (also eine Summe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\binom {n}{m}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/288e34459a013228450ef61cbd5b1485a6f0b3d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.573ex; height:3.176ex;" alt="{\displaystyle \textstyle {\binom {n}{m}}}" loading="lazy"> elementaren <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_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">-Beinen<a href="#cite_note-3">[3]</a>), dann ergibt wie oben das antisymmetrisierte<a href="#cite_note-4">[4]</a> Produkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{i_{1}}^{(1)}a_{i_{2}}^{(2)}\ldots a_{i_{m}}^{(m)})_{\,asy}}">

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<annotation encoding="application/x-tex">{\displaystyle (a_{i_{1}}^{(1)}a_{i_{2}}^{(2)}\ldots a_{i_{m}}^{(m)})_{\,asy}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8bde869e34d094e378130cac63032f873690d029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.695ex; height:4.009ex;" alt="{\displaystyle (a_{i_{1}}^{(1)}a_{i_{2}}^{(2)}\ldots a_{i_{m}}^{(m)})_{\,asy}}" loading="lazy">, bis auf ein alternierendes Vorzeichen, das von der jeweiligen Orientierung abhängt („Rechtshändigkeit“ versus „Linkshändigkeit“), das Hyperflächenmaß des <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_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">-Beins dual zur jeweiligen „Basisrichtung“, also dessen <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_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">-dimensionales „Volumen“ im <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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<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"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}\,.}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09ebb3bdfed6e2308935549cc69d169550c9e219.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.93ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}\,.}" loading="lazy"> Zugleich stellt dieser Ausdruck eine Unterdeterminante einer Matrix mit <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_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> Spalten 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/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"> Zeilen dar. Man erhält so auf elementare Weise, nämlich wegen der Multilinearität und Multi-Assoziativität des angegebenen Ausdrucks, die bekannten Determinanten-Entwicklungsätze. Insbesondere ist das so erzeugte Volumenmaß (=Grundflächenmaß mal Höhe) des jeweiligen Parallel-Epipeds invariant gegen Verschiebungen parallel zur Grundfläche<a href="#cite_note-5">[5]</a>, weil Determinanten von linear abhängigen Vektoren verschwinden.<a href="#cite_note-6">[6]</a>

<div class="mw-heading mw-heading2"><h2 id="Beziehung_zur_Clifford-Algebra">Beziehung zur Clifford-Algebra</h2></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\colon V\times V\to K}">

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<annotation encoding="application/x-tex">{\displaystyle q\colon V\times V\to K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/489d6864617b11119f561133bc6c0442c823baa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.198ex; height:2.509ex;" alt="{\displaystyle q\colon V\times V\to K}" loading="lazy"> eine <a href="Symmetrische_Bilinearform" class="mw-redirect" title="Symmetrische Bilinearform">symmetrische Bilinearform</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">.

Nun sei die zweistellige, bilineare Verknüpfung

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<annotation encoding="application/x-tex">{\displaystyle \circ :\Lambda (V)\times \Lambda (V)\to \Lambda (V)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1b797304c239390d6526eae6f920215b50cf15f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.183ex; height:2.843ex;" alt="{\displaystyle \circ :\Lambda (V)\times \Lambda (V)\to \Lambda (V)}" loading="lazy"></dd></dl>

definiert durch

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&(v_{1}\wedge \cdots \wedge v_{i})\circ (w_{1}\wedge \cdots \wedge w_{j})\\=&v_{1}\wedge \cdots \wedge v_{i}\wedge w_{1}\wedge \cdots \wedge w_{j}\\+&\sum _{k=1}^{\min\{i,j\}}\sum _{\overset {1\leq m_{1}<\cdots <m_{k}\leq i}{1\leq n_{1}<\cdots <n_{k}\leq j}}\;\sum _{\sigma \in P_{k}}(-1)^{ik+\sum _{\nu =1}^{k}(m_{\nu }+n_{\nu })}\;\mathrm {sign} \,\sigma \left(\prod _{\nu =1}^{k}q(v_{m_{\sigma (\nu )}},w_{n_{\nu }})\right)\\\cdot &v_{1}\wedge \cdots \wedge {\hat {v}}_{m_{1}}\wedge \cdots \wedge {\hat {v}}_{m_{2}}\wedge \cdots \wedge v_{i}\wedge w_{1}\wedge \cdots \wedge {\hat {w}}_{n_{1}}\wedge \cdots \wedge w_{j}\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83156c33fc4a19062fbd05d34b8a8594407fab0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.915ex; margin-bottom: -0.257ex; width:72.314ex; height:19.509ex;" alt="{\displaystyle {\begin{aligned}&(v_{1}\wedge \cdots \wedge v_{i})\circ (w_{1}\wedge \cdots \wedge w_{j})\\=&v_{1}\wedge \cdots \wedge v_{i}\wedge w_{1}\wedge \cdots \wedge w_{j}\\+&\sum _{k=1}^{\min\{i,j\}}\sum _{\overset {1\leq m_{1}<\cdots <m_{k}\leq i}{1\leq n_{1}<\cdots <n_{k}\leq j}}\;\sum _{\sigma \in P_{k}}(-1)^{ik+\sum _{\nu =1}^{k}(m_{\nu }+n_{\nu })}\;\mathrm {sign} \,\sigma \left(\prod _{\nu =1}^{k}q(v_{m_{\sigma (\nu )}},w_{n_{\nu }})\right)\\\cdot &v_{1}\wedge \cdots \wedge {\hat {v}}_{m_{1}}\wedge \cdots \wedge {\hat {v}}_{m_{2}}\wedge \cdots \wedge v_{i}\wedge w_{1}\wedge \cdots \wedge {\hat {w}}_{n_{1}}\wedge \cdots \wedge w_{j}\end{aligned}}}" loading="lazy"></dd></dl>

für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{m},w_{n}\in V}">

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<annotation encoding="application/x-tex">{\displaystyle v_{m},w_{n}\in V}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db7f7bf5b46909c8596ff43ee5f5de6f183615c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.347ex; height:2.509ex;" alt="{\displaystyle v_{m},w_{n}\in V}" loading="lazy">. Die Hüte über den Faktoren bedeuten hier deren Auslassung im Produkt. Durch Einführen dieser neuen Verknüpfung als Multiplikation erhält man die <a href="Clifford-Algebra" title="Clifford-Algebra">Clifford-Algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Cl} (V,q)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Cl} (V,q)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47eafe6a6c807dd76c1798e302d244c2863b8cf9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.025ex; height:2.843ex;" alt="{\displaystyle \mathrm {Cl} (V,q)}" loading="lazy">. Insbesondere erhält man mit der Nullbilinearform wieder die Graßmann-Algebra: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Cl} (V,0)=\Lambda (V)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Cl} (V,0)=\Lambda (V)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8aa35df9c76d29044e5600951e1b4432f67771eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.426ex; height:2.843ex;" alt="{\displaystyle \mathrm {Cl} (V,0)=\Lambda (V)}" loading="lazy">, da der Zusatzterm in der obigen Gleichung wegfällt und somit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ =\wedge }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee5812b3ce71ac8c6e8cbee54e5a4913e1a2c628.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.811ex; height:2.009ex;" alt="{\displaystyle \circ =\wedge }" loading="lazy"> gilt.

Für einfache <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v,w\in V}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4799d1b026506616abad8de636f46738b8a55a96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.454ex; height:2.509ex;" alt="{\displaystyle v,w\in V}" loading="lazy"> meint obige Definition die elementare Beziehung

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v\circ w:=v\wedge w+v\cdot w}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2741669285f0de22ebdd6cd3e28b616c6b6fb84b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.417ex; height:2.176ex;" alt="{\displaystyle v\circ w:=v\wedge w+v\cdot w}" loading="lazy">,</dd></dl>

wonach das „geometrische“<a href="#cite_note-7">[7]</a> Produkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99add39d2b681e2de7ff62422c32704a05c7ec31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }" loading="lazy"> zweier Vektoren in einen antisymmetrischen Keilprodukt- und einen symmetrischen Skalarproduktanteil <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\cdot w:=-q(v,w)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v\cdot w:=-q(v,w)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77b42f1575c8c459296e3da07c39409ae8a4ba45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.729ex; height:2.843ex;" alt="{\displaystyle v\cdot w:=-q(v,w)}" loading="lazy"> zerlegt werden kann. Die Summe ist hier in der Graßmannalgebra definiert, wobei das Vorzeichen eine Frage der Konvention ist.

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Symmetrische_Algebra" title="Symmetrische Algebra">Symmetrische Algebra</a></li></ul>

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

<ul><li>Hans-Joachim Kowalsky, Gerhard O. Michler: <cite style="font-style:italic">Lineare Algebra</cite>. De Gruyter, Berlin 2003, ISBN 3-11-017963-6.<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:Gra%C3%9Fmann-Algebra&rft.au=Hans-Joachim+Kowalsky%2C+Gerhard+O.+Michler&rft.btitle=Lineare+Algebra&rft.date=2003&rft.genre=book&rft.isbn=3110179636&rft.place=Berlin&rft.pub=De+Gruyter" style="display:none"> </li>

<li>Ralph Abraham, Jerrold E. Marsden, Tudor S. Rațiu: <cite style="font-style:italic">Manifolds, tensor analysis, and applications</cite>. Addison-Wesley, Reading MA 1983, ISBN 0-201-10168-8.<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:Gra%C3%9Fmann-Algebra&rft.au=Ralph+Abraham%2C+Jerrold+E.+Marsden%2C+Tudor+S.+Ra%C8%9Biu&rft.btitle=Manifolds%2C+tensor+analysis%2C+and+applications&rft.date=1983&rft.genre=book&rft.isbn=0201101688&rft.place=Reading+MA&rft.pub=Addison-Wesley" style="display:none"> </li>

<li><a href="Herbert_Federer" title="Herbert Federer">Herbert Federer</a>: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Geometric measure theory</cite>. 1. Auflage. Springer, Berlin 1996, ISBN 3-540-60656-4 (englisch).<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:Gra%C3%9Fmann-Algebra&rft.au=Herbert+Federer&rft.btitle=Geometric+measure+theory&rft.date=1996&rft.edition=1.&rft.genre=book&rft.isbn=3540606564&rft.place=Berlin&rft.pub=Springer" style="display:none"> </li>

<li><a href="Igor_Shafarevich" class="mw-redirect" title="Igor Shafarevich">I. R. Shafarevich</a>, A. O. Remizov: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Linear Algebra and Geometry</cite>. Springer, 2012, ISBN 978-3-642-30993-9 (englisch).<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:Gra%C3%9Fmann-Algebra&rft.au=I.+R.%26%2332%3BShafarevich%2C%26%2332%3BA.+O.+Remizov&rft.btitle=Linear+Algebra+and+Geometry&rft.date=2012&rft.genre=book&rft.isbn=9783642309939&rft.pub=Springer" style="display:none"> </li></ul>

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

<ul><li><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ExteriorAlgebra.html">Exterior Algebra.</a> In: <a href="MathWorld" title="MathWorld">MathWorld</a>. Abgerufen am 12. Januar 2023 (englisch).<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AGra%C3%9Fmann-Algebra&rft.title=Exterior+Algebra&rft.description=Exterior+Algebra&rft.identifier=https%3A%2F%2Fmathworld.wolfram.com%2FExteriorAlgebra.html&rft.language=en"> </li>

/* start https://de.wikipedia.org/ */

.mw-parser-output .webarchiv-memento a{color:inherit}

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</style><a rel="nofollow" class="external text" href="https://web.archive.org/web/20081017021025/http://planetmath.org/encyclopedia/GrassmannAlgebra.html">Exterior Algebra</a> (<a href="Webarchivierung#Begrifflichkeiten" title="Webarchivierung">Memento</a> vom 17. Oktober 2008 im <a href="Internet_Archive" title="Internet Archive">Internet Archive</a>) <a href="PlanetMath" title="PlanetMath">PlanetMath</a> (englisch).</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> Hermann Grassmann: <cite style="font-style:italic">Die lineale Ausdehnungslehre</cite>. Otto Wiegand, Leipzig 1878 (<a rel="nofollow" class="external text" href="https://archive.org/details/dieausdehnungsl04grasgoog">archive.org</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Gra%C3%9Fmann-Algebra&rft.au=Hermann+Grassmann&rft.btitle=Die+lineale+Ausdehnungslehre&rft.date=1878&rft.genre=book&rft.place=Leipzig&rft.pub=Otto+Wiegand" style="display:none">

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> In der Physik wird in diesem Zusammenhang von pseudoskalaren Größen gesprochen.

</li>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1f6db407eafe843411135e28d53aa20b2e050f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.398ex; height:2.009ex;" alt="{\displaystyle p=m}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=n-m}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08b80a4973fc386cda3416977b1faadc5fd13973.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.633ex; height:2.343ex;" alt="{\displaystyle p=n-m}" loading="lazy"> ergeben also duale <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">-Beine.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> In der Antisymmetrisierung der angegebenen Produkte liegt keine Beschränkung der Allgemeinheit, weil Zusatzterme sich automatisch zu Null aufsummieren würden.

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> Das sind sog. „Scherungen“, z. B. Transformationen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}\to a_{n}+\lambda a_{i}\,,}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{n}\to a_{n}+\lambda a_{i}\,,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2077b393758857c6b0a528f80dce8a7d68d07368.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.77ex; height:2.509ex;" alt="{\displaystyle a_{n}\to a_{n}+\lambda a_{i}\,,}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\leq (n-1)\,.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4293575696acb543a3b3043706d15e576ce36925.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.142ex; height:2.843ex;" alt="{\displaystyle i\leq (n-1)\,.}" loading="lazy">

</li>

<li id="cite_note-6"><a href="#cite_ref-6">↑</a> Präzise gilt für das Ausdehnungsmaß des <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_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">-Beins <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"> :

<msqrt>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V(\gamma )={\sqrt {\sum _{i_{1}<\ldots <i_{m}}|(a_{i_{1}}^{(1)}\dots a_{i_{m}}^{(m)})_{\,asy}|^{2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/525f34c7d12fc694edebf457999579362fde4309.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:36.932ex; height:7.676ex;" alt="{\displaystyle V(\gamma )={\sqrt {\sum _{i_{1}<\ldots <i_{m}}|(a_{i_{1}}^{(1)}\dots a_{i_{m}}^{(m)})_{\,asy}|^{2}}}}" loading="lazy">. Das ist erneut ein „verallgemeinerter <a href="Satz_von_Pythagoras" class="mw-redirect" title="Satz von Pythagoras">Satz von Pythagoras</a>.“

</li>

<li id="cite_note-7"><a href="#cite_ref-7">↑</a> D. Hestenes: A Unified Language for Mathematics and Physics. In: J.S.R. Chisholm/A.K. Common (eds.): Clifford Algebras and their Applications in Mathematical Physics (Reidel: Dordrecht/Boston, 1986), S. 1–23.

</li>

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