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<title>Niveaumenge</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Niveaumenge</span></h1>

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<p>In der <a href="Mathematik" title="Mathematik">Mathematik</a> bezeichnet eine <b>Niveaumenge</b> oder <b>Levelmenge</b> die Menge aller Punkte des <a href="Definitionsbereich" class="mw-redirect" title="Definitionsbereich">Definitionsbereichs</a> einer <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktion</a>, denen ein gleicher Funktionswert zugeordnet ist. Eng verwandte Begriffe für Funktionen mit Werten in einer geordneten Menge sind die der <b>Subniveaumenge</b>, die alle Punkte enthält, deren Funktionswerte einen vorgegebenen Wert nicht überschreiten, und der <b>Superniveaumenge</b>, die alle Punkte enthält, deren Funktionswerte einen vorgegebenen Wert nicht unterschreiten.

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

<p>Es seien <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U,V}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7681409ec5fffdb272f536757c1211fe0151a9b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.604ex; height:2.509ex;" alt="{\displaystyle U,V}" loading="lazy"></span> <a href="Menge_(Mathematik)" title="Menge (Mathematik)">Mengen</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon U\to V}">

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be070144e24c3c17096f7aba9e4ffb5f3d352e80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.496ex; height:2.509ex;" alt="{\displaystyle f\colon U\to V}" loading="lazy"></span> eine Funktion und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in V}">

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f7bf917b6ed2410cd944cff7b24aeb7b8d78934.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.635ex; height:2.176ex;" alt="{\displaystyle c\in V}" loading="lazy"></span> ein Wert aus der <a href="Zielmenge" title="Zielmenge">Zielmenge</a>, dann heißt

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}_{f}(c):=f^{-1}(c)=\{x\in U\mid f(x)=c\}\subseteq U}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58774817b2a5a397eade2b35595341550a509cb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.062ex; width:42.852ex; height:3.343ex;" alt="{\displaystyle {\mathcal {N}}_{f}(c):=f^{-1}(c)=\{x\in U\mid f(x)=c\}\subseteq U}" loading="lazy"></span></dd></dl>

<p>die <i>Niveaumenge</i> der Funktion <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math></span><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"></span> zum Niveau bzw. Level <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span>.

</p><p>Trägt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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"></span> eine <a href="Ordnungsrelation" title="Ordnungsrelation">Ordnungsrelation</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq }">

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/440568a09c3bfdf0e1278bfa79eb137c04e94035.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \leq }" loading="lazy"></span> (mit <a href="Relation_(Mathematik)" title="Relation (Mathematik)">Umkehrrelation</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \geq }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bcef7c0e95bb77a35fd1a874ca91f425215f3c26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \geq }" loading="lazy"></span>), können folgende Begriffe definiert werden:

</p><p>Als <i>Subniveaumenge</i> wird die Menge

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{f}^{\leq }(c):=\{x\in U\mid f(x)\leq c\}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8fd4151691dcb151f11353d471741e8cde21966c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:28.414ex; height:3.676ex;" alt="{\displaystyle {\mathcal {L}}_{f}^{\leq }(c):=\{x\in U\mid f(x)\leq c\}}" loading="lazy"></span></dd></dl>

<p>bezeichnet, im Falle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\mathbb {R} }">

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</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63d897bc226139b01f3467c554249c117a944f4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.564ex; height:2.176ex;" alt="{\displaystyle V=\mathbb {R} }" loading="lazy"></span> ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{f}^{\leq }(c)=f^{-1}(\left(-\infty ,c\right])}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{f}^{\leq }(c)=f^{-1}(\left(-\infty ,c\right])}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/011e4d2bc2493d5684b0f5618122ed5a57c47161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:22.216ex; height:3.676ex;" alt="{\displaystyle {\mathcal {L}}_{f}^{\leq }(c)=f^{-1}(\left(-\infty ,c\right])}" loading="lazy"></span>.

</p><p>Als <i>Superniveaumenge</i> wird die Menge

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{f}^{\geq }(c):=\{x\in U\mid f(x)\geq c\}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e9dc751c830ee7f72f995dcc9e6c433af8fac18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:28.414ex; height:3.676ex;" alt="{\displaystyle {\mathcal {L}}_{f}^{\geq }(c):=\{x\in U\mid f(x)\geq c\}}" loading="lazy"></span></dd></dl>

<p>bezeichnet, im Falle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\mathbb {R} }">

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</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63d897bc226139b01f3467c554249c117a944f4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.564ex; height:2.176ex;" alt="{\displaystyle V=\mathbb {R} }" loading="lazy"></span> ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{f}^{\geq }(c)=f^{-1}(\left[c,\infty \right))}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{f}^{\geq }(c)=f^{-1}(\left[c,\infty \right))}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9253d26b7bb8e51512fe7cd7dbb820e607705db4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:20.408ex; height:3.676ex;" alt="{\displaystyle {\mathcal {L}}_{f}^{\geq }(c)=f^{-1}(\left[c,\infty \right))}" loading="lazy"></span>.

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

<div class="mw-heading mw-heading3"><h3 id="Physik">Physik</h3></div>

<p>Für zweidimensionale <a href="Skalarfeld" title="Skalarfeld">Skalarfelder</a> ist eine Niveaumenge zumeist eine Linie und man spricht von einer <i><a href="Isolinie" title="Isolinie">Isolinie</a></i> oder <i>Niveaulinie.</i> Für dreidimensionale Skalarfelder (zum Beispiel für skalare <a href="Potential_(Physik)" title="Potential (Physik)">Potentialfelder</a>) ist diese Menge zumeist eine gekrümmte <a href="Fl%C3%A4che_(Mathematik)" title="Fläche (Mathematik)">Fläche</a> und man nennt sie <i><a href="Isofl%C3%A4che" title="Isofläche">Isofläche</a></i> oder <i><a href="Niveaufl%C3%A4che" title="Niveaufläche">Niveaufläche</a></i> (z. B. <i>Höhenlinien</i>).<br>

Der Begriff <i>Niveaufläche</i> wird aber auch für Kraftfelder wie das <a href="Elektrisches_Feld" title="Elektrisches Feld">elektrische Feld</a> oder <a href="Magnetismus" title="Magnetismus">Magnetfelder</a> verwendet.

</p>

<div class="mw-heading mw-heading3"><h3 id="Wirtschaftswissenschaften">Wirtschaftswissenschaften</h3></div>

<p>Für eine <a href="Produktionsfunktion" title="Produktionsfunktion">Produktionsfunktion</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon (0,\infty )^{n}\to (0,\infty )}">

<msup>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f\colon (0,\infty )^{n}\to (0,\infty )}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8baa363e01c21c6fb534d9e2facc6ad2f77b630.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.804ex; height:2.843ex;" alt="{\displaystyle f\colon (0,\infty )^{n}\to (0,\infty )}" loading="lazy"></span> sowie ein Produktionsniveau <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in (0,\infty )}">

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<annotation encoding="application/x-tex">{\displaystyle c\in (0,\infty )}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92c6376dc05e47e0dbf4cd0e89e77dae99c1f50d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.177ex; height:2.843ex;" alt="{\displaystyle c\in (0,\infty )}" loading="lazy"></span> ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}_{f}(c)=f^{-1}(c)}">

<msub>

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<msup>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}_{f}(c)=f^{-1}(c)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0b628385e3ec7149d964c739d707a462460d03e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.062ex; width:15.488ex; height:3.343ex;" alt="{\displaystyle {\mathcal {N}}_{f}(c)=f^{-1}(c)}" loading="lazy"></span> die Menge aller Bündel von Produktionsfaktoren, mit denen sich die Menge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span> generieren lässt. Die Menge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}_{f}(c)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}_{f}(c)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee6da8bf8ba5649dc80279987682e625ac5718e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.062ex; width:5.92ex; height:3.176ex;" alt="{\displaystyle {\mathcal {N}}_{f}(c)}" loading="lazy"></span> wird als <i>Isoquante</i> zum Produktionsniveau <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span> bezeichnet.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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

<p>Ist die Funktion <a href="Reell-vektorwertige_Funktion" class="mw-redirect" title="Reell-vektorwertige Funktion">reell-vektorwertig</a>, hat also als Bildraum den <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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"></span> und ist dieser mit einer <a href="Verallgemeinerte_Ungleichung" title="Verallgemeinerte Ungleichung">verallgemeinerten Ungleichung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \preccurlyeq _{K}}">

<msub>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9bef9e870b49c8958943bdcc35e59a25b31466b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.501ex; height:2.343ex;" alt="{\displaystyle \preccurlyeq _{K}}" loading="lazy"></span> versehen, so lässt sich die Subniveaumenge verallgemeinern zu

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{f}^{\leq }(c):=\{x\in U\mid f(x)\preccurlyeq _{K}c\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2659f282ac2e34b845bba9bbc7cdc27ad60a9923.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:30.107ex; height:3.676ex;" alt="{\displaystyle {\mathcal {L}}_{f}^{\leq }(c):=\{x\in U\mid f(x)\preccurlyeq _{K}c\}}" loading="lazy"></span></dd></dl>

<p>und die Superniveaumenge zu

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{f}^{\geq }(c):=\{x\in U\mid f(x)\succcurlyeq _{K}c\}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68f8d4f105e50d930d5340f8935a3304da07728f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:30.107ex; height:3.676ex;" alt="{\displaystyle {\mathcal {L}}_{f}^{\geq }(c):=\{x\in U\mid f(x)\succcurlyeq _{K}c\}}" loading="lazy"></span>.</dd></dl>

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

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Klaus D. Schmidt: <cite style="font-style:italic">Mathematik. Grundlagen für Wirtschaftswissenschaftler</cite>. 2., überarbeitete Auflage. Springer, Berlin u. a. 2000, ISBN 3-540-66521-8, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>369</span> (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=LE_RQAbHZbEC">eingeschränkte Vorschau</a> in der Google-Buchsuche).<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:Niveaumenge&rft.au=Klaus+D.+Schmidt&rft.btitle=Mathematik.+Grundlagen+f%C3%BCr+Wirtschaftswissenschaftler&rft.date=2000&rft.edition=2.%2C+%C3%BCberarbeitete&rft.genre=book&rft.isbn=3540665218&rft.pages=369&rft.place=Berlin+u.+a.&rft.pub=Springer" style="display:none"> </span></span>

</li>

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