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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Linearisierung</span></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"><p>Bei der <b>Linearisierung</b> – ein Begriff aus der <a href="Mathematik" title="Mathematik">Mathematik</a> – werden nichtlineare <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktionen</a> oder nichtlineare <a href="Differentialgleichung" title="Differentialgleichung">Differentialgleichungen</a> durch <a href="Lineare_Funktion" title="Lineare Funktion">lineare Funktionen</a> oder durch lineare Differentialgleichungen <a href="Approximation" title="Approximation">angenähert</a>. Die Linearisierung wird angewandt, da lineare Funktionen oder <a href="Lineare_gew%C3%B6hnliche_Differentialgleichung" title="Lineare gewöhnliche Differentialgleichung">lineare Differentialgleichungen</a> einfach berechnet werden können und die Theorie umfangreicher als für nichtlineare Systeme ausgebaut ist.
</p>

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

<p>Das einfachste Verfahren zur Linearisierung ist das Einzeichnen der <a href="Tangente" title="Tangente">Tangente</a> in den Graphen. Daraufhin können die Parameter der Tangente abgelesen werden, und die resultierende lineare Funktion (<a href="Punktsteigungsform" title="Punktsteigungsform">Punktsteigungsform</a> der Geraden)
</p>
<dl><dd><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 y_{t}=f(x_{0})+{\frac {\mathrm {d} f}{\mathrm {d} x}}{\bigg |}_{x_{0}}\cdot (x-x_{0})}">
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<annotation encoding="application/x-tex">{\displaystyle y_{t}=f(x_{0})+{\frac {\mathrm {d} f}{\mathrm {d} x}}{\bigg |}_{x_{0}}\cdot (x-x_{0})}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c2eb4f36b09d9eacce2b44613427d9495da51d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.527ex; height:6.009ex;" alt="{\displaystyle y_{t}=f(x_{0})+{\frac {\mathrm {d} f}{\mathrm {d} x}}{\bigg |}_{x_{0}}\cdot (x-x_{0})}" loading="lazy"></span></dd></dl>
<p>approximiert die Originalfunktion um den Punkt <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 x_{0}}">
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<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"></span>. Dabei 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 {\tfrac {\mathrm {d} f}{\mathrm {d} x}}{\bigg |}_{x_{0}}}">
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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\mathrm {d} f}{\mathrm {d} x}}{\bigg |}_{x_{0}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/330ca5d7bd097118b992f5d8f9912ff8d47291e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:5.341ex; height:5.676ex;" alt="{\displaystyle {\tfrac {\mathrm {d} f}{\mathrm {d} x}}{\bigg |}_{x_{0}}}" loading="lazy"></span> der Anstieg im Punkt <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 x_{0}}">
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<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"></span>.
</p><p>Wenn die Funktion in analytischer Form vorliegt, kann die Gleichung der Tangente direkt angegeben werden.
</p><p>Der relative Fehler der Approximation ist
</p>
<dl><dd><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(x)={\bigg |}{\frac {f(x)-y_{t}(x)}{f(x)}}{\bigg |}}">
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<annotation encoding="application/x-tex">{\displaystyle F(x)={\bigg |}{\frac {f(x)-y_{t}(x)}{f(x)}}{\bigg |}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/580eb1e944b50db26752e24231bcc3a877444675.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.47ex; height:6.509ex;" alt="{\displaystyle F(x)={\bigg |}{\frac {f(x)-y_{t}(x)}{f(x)}}{\bigg |}}" loading="lazy"></span></dd></dl>
<p>Für die 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(x)=\sin(x)}">
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<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
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<annotation encoding="application/x-tex">{\displaystyle f(x)=\sin(x)}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fb1266b7f7718442e31e45eef3d81bef6a8b9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.511ex; height:2.843ex;" alt="{\displaystyle f(x)=\sin(x)}" loading="lazy"></span> gilt beispielsweise:
</p>
<dl><dd><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 y(x)=\sin(x_{0})+\cos(x_{0})\cdot (x-x_{0})}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
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<mo>=</mo>
<mi>sin</mi>
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<annotation encoding="application/x-tex">{\displaystyle y(x)=\sin(x_{0})+\cos(x_{0})\cdot (x-x_{0})}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/28b23779991340a0bb0dc5c0ec903f1ec92a3c9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.629ex; height:2.843ex;" alt="{\displaystyle y(x)=\sin(x_{0})+\cos(x_{0})\cdot (x-x_{0})}" loading="lazy"></span></dd></dl>
<p>Die Bestimmung der Tangente entspricht der Bestimmung des <a href="Taylor-Formel#Annäherung_durch_Tangente" title="Taylor-Formel">linearen Gliedes</a> als ersten Teil des <a href="Taylor-Formel#Annäherung_durch_Polynome_vom_Grad_n" title="Taylor-Formel">Taylorpolynoms</a> der zu approximierenden Funktion.
</p>
<div class="mw-heading mw-heading2"><h2 id="Anwendungen">Anwendungen</h2></div>
<p>Anwendung findet die Linearisierung unter anderem in der <a href="Elektrotechnik" title="Elektrotechnik">Elektrotechnik</a> und der <a href="Regelungstechnik" title="Regelungstechnik">Regelungstechnik</a> zur näherungsweisen Beschreibung nichtlinearer <a href="System" title="System">Systeme</a> durch <a href="Lineares_System_(Systemtheorie)" title="Lineares System (Systemtheorie)">lineare Systeme</a>.
</p><p>Das Ergebnis einer Netzwerkanalyse ist unter Umständen ein nichtlineares Gleichungssystem. Dies kann unter gewissen Voraussetzungen in ein <a href="Lineares_Gleichungssystem" title="Lineares Gleichungssystem">lineares Gleichungssystem</a> überführt werden. Nicht die einzige, aber die einfachste Methode der Linearisierung ist die Linearisierung in einem <a href="Arbeitspunkt" title="Arbeitspunkt">Arbeitspunkt</a> (kurz „AP“). Nur diese ist in den folgenden Abschnitten beschrieben.
</p>
<div class="mw-heading mw-heading3"><h3 id="Linearisierung_der_Multiplikation">Linearisierung der Multiplikation</h3></div><p>
In einem <a href="Signalflussplan" title="Signalflussplan">Signalflussplan</a> lassen sich <a href="Komplexes_System" title="Komplexes System">komplexe Systeme</a> durch ein Blockbild darstellen, das zur qualitativen Visualisierung von mathematischen Modellen dient. </p>
<p>Befindet sich in diesem <a href="Signalflussplan" title="Signalflussplan">Signalflussplan</a> eine Multiplikationsstelle, so lässt sich diese durch Linearisierung in eine Additionsstelle umwandeln.
</p><p>Im Folgenden bezeichnen wir mit <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 y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span> das Produkt zweier Zahlen <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 x_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8788bf85d532fa88d1fb25eff6ae382a601c308.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}" loading="lazy"></span> 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 x_{2}}">
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<annotation encoding="application/x-tex">{\displaystyle x_{2}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7af1b928f06e4c7e3e8ebfd60704656719bd766.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{2}}" loading="lazy"></span>:
</p>
<dl><dd><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 y=x_{1}\cdot x_{2}}">
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle y=x_{1}\cdot x_{2}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3fc26bcbcaebbca276f47034e16fe8cb920874c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.701ex; height:2.009ex;" alt="{\displaystyle y=x_{1}\cdot x_{2}}" loading="lazy"></span></dd></dl>
<p>Im Arbeitspunkt können wir die <a href="Multiplikation" title="Multiplikation">Multiplikation</a> linearisieren, indem wir <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 x_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8788bf85d532fa88d1fb25eff6ae382a601c308.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}" loading="lazy"></span> als Summe des Arbeitspunkts und der Differenz <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 \Delta x_{1}=x_{1}-x_{1,{\text{AP}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta x_{1}=x_{1}-x_{1,{\text{AP}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dbc9b61f62d6698b9a81994b5d30588feedac2e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.836ex; height:2.843ex;" alt="{\displaystyle \Delta x_{1}=x_{1}-x_{1,{\text{AP}}}}" loading="lazy"></span> schreiben:
</p>
<dl><dd><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 y=(x_{1,{\text{AP}}}+\Delta x_{1})\cdot (x_{2,{\text{AP}}}+\Delta x_{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
<mo>+</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
<mo>+</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y=(x_{1,{\text{AP}}}+\Delta x_{1})\cdot (x_{2,{\text{AP}}}+\Delta x_{2})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ccc9d75a82a80140a93caef9d5e33edbd42ac8f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.258ex; height:3.009ex;" alt="{\displaystyle y=(x_{1,{\text{AP}}}+\Delta x_{1})\cdot (x_{2,{\text{AP}}}+\Delta x_{2})}" loading="lazy"></span></dd></dl>
<p>Wir können dieses Produkt nach dem <a href="Distributivgesetz" title="Distributivgesetz">Distributivgesetz</a> ausmultiplizieren. Es ergibt sich die Summe:
</p>
<dl><dd><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 y=x_{1,{\text{AP}}}\cdot x_{2,{\text{AP}}}+x_{1,{\text{AP}}}\cdot \Delta x_{2}+x_{2,{\text{AP}}}\cdot \Delta x_{1}+\Delta x_{1}\cdot \Delta x_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo>=</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y=x_{1,{\text{AP}}}\cdot x_{2,{\text{AP}}}+x_{1,{\text{AP}}}\cdot \Delta x_{2}+x_{2,{\text{AP}}}\cdot \Delta x_{1}+\Delta x_{1}\cdot \Delta x_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/284cde526333144b61a4b45d760f79f80ee0e5ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:57.543ex; height:2.843ex;" alt="{\displaystyle y=x_{1,{\text{AP}}}\cdot x_{2,{\text{AP}}}+x_{1,{\text{AP}}}\cdot \Delta x_{2}+x_{2,{\text{AP}}}\cdot \Delta x_{1}+\Delta x_{1}\cdot \Delta x_{2}}" loading="lazy"></span></dd></dl>
<p>Wir nehmen nun an, dass das Verhältnis der Abweichungen vom Arbeitspunkt <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 \Delta x_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta x_{i}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d03804c96ca38a2bff889eaf13470785aca25fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.065ex; height:2.509ex;" alt="{\displaystyle \Delta x_{i}}" loading="lazy"></span> und dem Arbeitspunkt selber klein ist:
</p><p><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 {\frac {\Delta x_{i}}{x_{i,{\text{AP}}}}}\ll x_{i,{\text{AP}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
</mfrac>
</mrow>
<mo>≪<!-- ≪ --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\Delta x_{i}}{x_{i,{\text{AP}}}}}\ll x_{i,{\text{AP}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/36f5470f57bb29f4a72c1704aad6015e071de6ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.327ex; height:6.009ex;" alt="{\displaystyle {\frac {\Delta x_{i}}{x_{i,{\text{AP}}}}}\ll x_{i,{\text{AP}}}}" loading="lazy"></span> und somit auch das Produkt <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 e_{y}=\Delta x_{1}\cdot \Delta x_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msub>
<mo>=</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e_{y}=\Delta x_{1}\cdot \Delta x_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2efa579a82dc18bd4465e025352e6c3082a39e0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.55ex; height:2.843ex;" alt="{\displaystyle e_{y}=\Delta x_{1}\cdot \Delta x_{2}}" loading="lazy"></span> klein ist. Die <b>linearisierte Multiplikation</b> lautet also:
</p>
<dl><dd><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 y\approx x_{1,{\text{AP}}}\cdot x_{2,{\text{AP}}}+x_{1,{\text{AP}}}\cdot \Delta x_{2}+x_{2,{\text{AP}}}\cdot \Delta x_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo>≈<!-- ≈ --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y\approx x_{1,{\text{AP}}}\cdot x_{2,{\text{AP}}}+x_{1,{\text{AP}}}\cdot \Delta x_{2}+x_{2,{\text{AP}}}\cdot \Delta x_{1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/22e72ce85bbd963b1ea112965c1786d6f771faeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.384ex; height:2.843ex;" alt="{\displaystyle y\approx x_{1,{\text{AP}}}\cdot x_{2,{\text{AP}}}+x_{1,{\text{AP}}}\cdot \Delta x_{2}+x_{2,{\text{AP}}}\cdot \Delta x_{1}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading4"><h4 id="Beispiel">Beispiel</h4></div>
<p>Wähle die Zahlen:
</p>
<dl><dd><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 x_{1}=2{,}4;\ x_{2}=110\Rightarrow y=x_{1}\cdot x_{2}=264.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>,</mo>
</mrow>
<mn>4</mn>
<mo>;</mo>
<mtext>&nbsp;</mtext>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>110</mn>
<mo stretchy="false">⇒<!-- ⇒ --></mo>
<mi>y</mi>
<mo>=</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>264.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{1}=2{,}4;\ x_{2}=110\Rightarrow y=x_{1}\cdot x_{2}=264.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0bf913c8df636e6d8fe12e7ccf7678736e79058.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:40.586ex; height:2.509ex;" alt="{\displaystyle x_{1}=2{,}4;\ x_{2}=110\Rightarrow y=x_{1}\cdot x_{2}=264.}" loading="lazy"></span></dd></dl>
<p>Nun stellt sich, die Frage, wie die Arbeitspunkte zu wählen sind. Um die Rechnung zu vereinfachen, runden wir <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 2{,}4}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>,</mo>
</mrow>
<mn>4</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2{,}4}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/130012a794d8dda06fd48652903a38af3bc44ec9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.972ex; height:2.509ex;" alt="{\displaystyle 2{,}4}" loading="lazy"></span> auf <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 2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/901fc910c19990d0dbaaefe4726ceb1a4e217a0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}" loading="lazy"></span> ab 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 110}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>110</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 110}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61d2b914b923dff8ccc0236742b94ae745f1fed8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle 110}" loading="lazy"></span> auf <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 100}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>100</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 100}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0572cd017c6d7936a12737c9d614a2f801f94a36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle 100}" loading="lazy"></span> ab:
Wähle also: <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 x_{1,{\text{AP}}}=2;\ x_{2,{\text{AP}}}=100\Rightarrow \Delta x_{1}=0{,}4;\ \Delta x_{2}=10.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
<mo>=</mo>
<mn>2</mn>
<mo>;</mo>
<mtext>&nbsp;</mtext>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
<mo>=</mo>
<mn>100</mn>
<mo stretchy="false">⇒<!-- ⇒ --></mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>,</mo>
</mrow>
<mn>4</mn>
<mo>;</mo>
<mtext>&nbsp;</mtext>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>10.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{1,{\text{AP}}}=2;\ x_{2,{\text{AP}}}=100\Rightarrow \Delta x_{1}=0{,}4;\ \Delta x_{2}=10.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/867b83c94b50aefd9d0236aad6c7f1af2e0b5f7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.856ex; height:2.843ex;" alt="{\displaystyle x_{1,{\text{AP}}}=2;\ x_{2,{\text{AP}}}=100\Rightarrow \Delta x_{1}=0{,}4;\ \Delta x_{2}=10.}" loading="lazy"></span>
Das linearisierte Produkt ist also
</p>
<dl><dd><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 \Rightarrow y\approx 2\cdot 100+2\cdot 10+100\cdot 0{,}4=260}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">⇒<!-- ⇒ --></mo>
<mi>y</mi>
<mo>≈<!-- ≈ --></mo>
<mn>2</mn>
<mo>⋅<!-- ⋅ --></mo>
<mn>100</mn>
<mo>+</mo>
<mn>2</mn>
<mo>⋅<!-- ⋅ --></mo>
<mn>10</mn>
<mo>+</mo>
<mn>100</mn>
<mo>⋅<!-- ⋅ --></mo>
<mn>0</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>,</mo>
</mrow>
<mn>4</mn>
<mo>=</mo>
<mn>260</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Rightarrow y\approx 2\cdot 100+2\cdot 10+100\cdot 0{,}4=260}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38eaf9120419a2460fdf42461c8218162f314e74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:39.123ex; height:2.509ex;" alt="{\displaystyle \Rightarrow y\approx 2\cdot 100+2\cdot 10+100\cdot 0{,}4=260}" loading="lazy"></span></dd></dl>
<p>mit dem Fehler <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 e_{y}=0{,}4\cdot 10=4}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>,</mo>
</mrow>
<mn>4</mn>
<mo>⋅<!-- ⋅ --></mo>
<mn>10</mn>
<mo>=</mo>
<mn>4</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e_{y}=0{,}4\cdot 10=4}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8b60b3adcf62379168b19c7140dc53ed8dbbd81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.468ex; height:2.843ex;" alt="{\displaystyle e_{y}=0{,}4\cdot 10=4}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Linearisierung_der_Division">Linearisierung der Division</h3></div>

<p>Wir betrachten nun den Quotienten <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 y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span> zweier Zahlen <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 x_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8788bf85d532fa88d1fb25eff6ae382a601c308.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}" loading="lazy"></span> 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 x_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7af1b928f06e4c7e3e8ebfd60704656719bd766.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{2}}" loading="lazy"></span>:
</p>
<dl><dd><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 y={\frac {x_{1}}{x_{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y={\frac {x_{1}}{x_{2}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/41061ddadfb4d1fb5faa2b0350aca6f139b464d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.474ex; height:5.009ex;" alt="{\displaystyle y={\frac {x_{1}}{x_{2}}}}" loading="lazy"></span></dd></dl>
<p>Analog wie zur Multiplikation entwickeln wir <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 x_{i}=x_{i,{\text{AP}}}+\Delta x_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
<mo>+</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{i}=x_{i,{\text{AP}}}+\Delta x_{i}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79b91f2d9aa7580f7958237d48cd27fbc0572edd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.072ex; height:2.843ex;" alt="{\displaystyle x_{i}=x_{i,{\text{AP}}}+\Delta x_{i}}" loading="lazy"></span> um den Arbeitspunkt <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 x_{\text{AP}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{\text{AP}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ccc236e653fc2309f8c74bf8e9f87acbfedbc53d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.914ex; height:2.009ex;" alt="{\displaystyle x_{\text{AP}}}" loading="lazy"></span>. Damit können wir den Quotienten wie folgt schreiben:
</p>
<dl><dd><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 y={\frac {x_{1,{\text{AP}}}+\Delta x_{1}}{x_{2,{\text{AP}}}+\Delta x_{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
<mo>+</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
<mrow>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
<mo>+</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y={\frac {x_{1,{\text{AP}}}+\Delta x_{1}}{x_{2,{\text{AP}}}+\Delta x_{2}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86e6f0c8af59e9f46cb223023e705fa62a42f76c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.443ex; height:6.509ex;" alt="{\displaystyle y={\frac {x_{1,{\text{AP}}}+\Delta x_{1}}{x_{2,{\text{AP}}}+\Delta x_{2}}}}" loading="lazy"></span></dd></dl>
<p>Ausklammern der Arbeitspunkte liefert für Division:
</p>
<dl><dd><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 y={\frac {x_{1,{\text{AP}}}}{x_{2,{\text{AP}}}}}\cdot {\frac {1+{\frac {\Delta x_{1}}{x_{1,{\text{AP}}}}}}{1+{\frac {\Delta x_{2}}{x_{2,{\text{AP}}}}}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
</mfrac>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
</mfrac>
</mrow>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
</mfrac>
</mrow>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y={\frac {x_{1,{\text{AP}}}}{x_{2,{\text{AP}}}}}\cdot {\frac {1+{\frac {\Delta x_{1}}{x_{1,{\text{AP}}}}}}{1+{\frac {\Delta x_{2}}{x_{2,{\text{AP}}}}}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/973c47bc137109c24d5bfefb5b1da5dd6a6a7127.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:21.69ex; height:9.843ex;" alt="{\displaystyle y={\frac {x_{1,{\text{AP}}}}{x_{2,{\text{AP}}}}}\cdot {\frac {1+{\frac {\Delta x_{1}}{x_{1,{\text{AP}}}}}}{1+{\frac {\Delta x_{2}}{x_{2,{\text{AP}}}}}}}}" loading="lazy"></span></dd></dl>
<p>Wir wollen nun den Zähler und den Nenner des Bruches linearisieren. Dazu verwenden wir die <a href="Geometrische_Reihe" title="Geometrische Reihe">geometrische Reihe</a>. Für eine Nullfolge <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 q^{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle q^{k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc9b1525f5a653e19f9fd37fd2701a768e171632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.168ex; height:3.009ex;" alt="{\displaystyle q^{k}}" loading="lazy"></span> gilt:
</p>
<dl><dd><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 \sum _{k=0}^{n}q^{k}={\frac {1-q^{n+1}}{1-q}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<msup>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msup>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>q</mi>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{n}q^{k}={\frac {1-q^{n+1}}{1-q}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd6827706031aaaaa25f2267eea17215d8a71e32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.246ex; height:7.009ex;" alt="{\displaystyle \sum _{k=0}^{n}q^{k}={\frac {1-q^{n+1}}{1-q}}}" loading="lazy"></span></dd></dl>
<p>Hierbei ist entsprechend <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 q=-{\tfrac {\Delta x_{2}}{x_{2,{\text{AP}}}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>q</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mrow>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
</mfrac>
</mstyle>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle q=-{\tfrac {\Delta x_{2}}{x_{2,{\text{AP}}}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0efa2c984fbef3f6b331da434fb0c1fac5bfaaac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:10.865ex; height:4.509ex;" alt="{\displaystyle q=-{\tfrac {\Delta x_{2}}{x_{2,{\text{AP}}}}}}" loading="lazy"></span> mit <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 \vert q\vert \ll 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">|</mo>
<mi>q</mi>
<mo fence="false" stretchy="false">|</mo>
<mo>≪<!-- ≪ --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \vert q\vert \ll 1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/37d2b3fd74c72469853394e6f0a2fccf740262bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.14ex; height:2.843ex;" alt="{\displaystyle \vert q\vert \ll 1}" loading="lazy"></span> zu wählen.
</p><p>Einsetzen liefert die Linearisierung
</p>
<dl><dd><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 {\frac {1}{1+{\frac {\Delta x_{2}}{x_{2,{\text{AP}}}}}}}\approx 1-{\frac {\Delta x_{2}}{x_{2,{\text{AP}}}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
</mfrac>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>≈<!-- ≈ --></mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{1+{\frac {\Delta x_{2}}{x_{2,{\text{AP}}}}}}}\approx 1-{\frac {\Delta x_{2}}{x_{2,{\text{AP}}}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c51b1d1ceae01105e8b649e6ccfcf57a04dc425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:22.858ex; height:7.843ex;" alt="{\displaystyle {\frac {1}{1+{\frac {\Delta x_{2}}{x_{2,{\text{AP}}}}}}}\approx 1-{\frac {\Delta x_{2}}{x_{2,{\text{AP}}}}}}" loading="lazy"></span></dd></dl>
<p>Analog lässt sich der Nenner des obigen Bruchs linearisieren. Die <b>linearisierte Division</b> lässt sich schreiben durch:
</p>
<dl><dd><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 y\approx {\frac {x_{1,{\text{AP}}}}{x_{2,{\text{AP}}}}}\cdot \left(1+{\frac {\Delta x_{1}}{x_{1,{\text{AP}}}}}-{\frac {\Delta x_{2}}{x_{2,{\text{AP}}}}}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo>≈<!-- ≈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
</mfrac>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>AP</mtext>
</mrow>
</mrow>
</msub>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y\approx {\frac {x_{1,{\text{AP}}}}{x_{2,{\text{AP}}}}}\cdot \left(1+{\frac {\Delta x_{1}}{x_{1,{\text{AP}}}}}-{\frac {\Delta x_{2}}{x_{2,{\text{AP}}}}}\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6245bbe2a8afede78f9c3d8fe96c11f95c4af7dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.285ex; height:6.176ex;" alt="{\displaystyle y\approx {\frac {x_{1,{\text{AP}}}}{x_{2,{\text{AP}}}}}\cdot \left(1+{\frac {\Delta x_{1}}{x_{1,{\text{AP}}}}}-{\frac {\Delta x_{2}}{x_{2,{\text{AP}}}}}\right)}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Linearisieren_gewöhnlicher_Differentialgleichungen"><span id="Linearisieren_gew.C3.B6hnlicher_Differentialgleichungen"></span>Linearisieren gewöhnlicher Differentialgleichungen</h2></div>
<p>Ein bekanntes Beispiel für die Linearisierung einer nichtlinearen Differentialgleichung ist das <a href="Mathematisches_Pendel" title="Mathematisches Pendel">Pendel</a>. Die Gleichung lautet:
</p>
<dl><dd><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 {\ddot {y}}(t)+D\cdot {\dot {y}}(t)+\omega ^{2}\sin(y(t))=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>y</mi>
<mo>¨<!-- ¨ --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>D</mi>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>y</mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<msup>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\ddot {y}}(t)+D\cdot {\dot {y}}(t)+\omega ^{2}\sin(y(t))=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18f9e22fbfb039be2055d13fce22446e727e7dca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.804ex; height:3.176ex;" alt="{\displaystyle {\ddot {y}}(t)+D\cdot {\dot {y}}(t)+\omega ^{2}\sin(y(t))=0}" loading="lazy"></span></dd></dl>
<p>Der nichtlineare Teil 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 \sin(y)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sin(y)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75422364ab166eb574ec319096ae43980b1297d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.82ex; height:2.843ex;" alt="{\displaystyle \sin(y)}" loading="lazy"></span>. Dieser wird für kleine Schwankungen um einen Arbeitspunkt <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 y_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y_{0}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d943dbbb0b56ca750c4d62c5b54b4ae29a773da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{0}}" loading="lazy"></span> approximiert durch:
</p>
<dl><dd><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 \sin(y)\approx \sin(y_{0})+\cos(y_{0})\cdot (y-y_{0})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>≈<!-- ≈ --></mo>
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>cos</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo>−<!-- − --></mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sin(y)\approx \sin(y_{0})+\cos(y_{0})\cdot (y-y_{0})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3b21712dba973bcbdc2248e6fbd0381597c0fa8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.409ex; height:2.843ex;" alt="{\displaystyle \sin(y)\approx \sin(y_{0})+\cos(y_{0})\cdot (y-y_{0})}" loading="lazy"></span></dd></dl>
<p>Mit dem Arbeitspunkt <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 y_{0}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y_{0}=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f952fc15fe931e15f2f4a766b3ce68dc52f64842.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.454ex; height:2.509ex;" alt="{\displaystyle y_{0}=0}" loading="lazy"></span> gilt:
</p>
<dl><dd><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 \sin(y)\approx y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>≈<!-- ≈ --></mo>
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sin(y)\approx y}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04715116e815a08c824eec72c24047fbd729d5bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.074ex; height:2.843ex;" alt="{\displaystyle \sin(y)\approx y}" loading="lazy"></span> und damit die linearisierte Differenzialgleichung</dd>
<dd><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 {\ddot {y}}(t)+D\cdot {\dot {y}}(t)+\omega ^{2}\cdot y(t)=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>y</mi>
<mo>¨<!-- ¨ --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>D</mi>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>y</mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<msup>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>⋅<!-- ⋅ --></mo>
<mi>y</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\ddot {y}}(t)+D\cdot {\dot {y}}(t)+\omega ^{2}\cdot y(t)=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f821a18d6757d5212594036a2bf381dff1b86037.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.431ex; height:3.176ex;" alt="{\displaystyle {\ddot {y}}(t)+D\cdot {\dot {y}}(t)+\omega ^{2}\cdot y(t)=0}" loading="lazy"></span>.</dd></dl>
<p>Diese linearisierten Differentialgleichungen sind meist deutlich einfacher zu lösen. Für ein mathematisches Pendel (wähle <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 D=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>D</mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle D=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d375dfda80ee8df1d1d7aa8b962114044e464305.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.185ex; height:2.176ex;" alt="{\displaystyle D=0}" loading="lazy"></span>) lässt sich die Gleichung durch einfache Exponentialfunktionen lösen, wobei die nicht-linearisierte nicht analytisch lösbar ist. Weitere Details über das Linearisieren von Differentialgleichungen sind in dem Artikel über die <a href="Zustandsraumdarstellung" title="Zustandsraumdarstellung">Zustandsraumdarstellung</a> beschrieben.
</p>
<div class="mw-heading mw-heading2"><h2 id="Tangentialebene">Tangentialebene</h2></div>

<p>Soll eine gegebene 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(x_{1},x_{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x_{1},x_{2})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb3c833c2c02e9f772ff207d8dcc47a5784b2084.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.89ex; height:2.843ex;" alt="{\displaystyle f(x_{1},x_{2})}" loading="lazy"></span> in einem Punkt <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 x_{10},x_{20}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>10</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>20</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{10},x_{20}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ca99a9154604052b73daa07b3f4a8684bddf84f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.446ex; height:2.009ex;" alt="{\displaystyle x_{10},x_{20}}" loading="lazy"></span> linearisiert werden, wird sich der <a href="Taylor-Formel" title="Taylor-Formel">Taylor-Formel</a> bedient. Das Ergebnis entspricht der <a href="Tangentialebene" title="Tangentialebene">Tangentialebene</a> in diesem Punkt.
</p><p>Für die 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(x_{1},x_{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x_{1},x_{2})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb3c833c2c02e9f772ff207d8dcc47a5784b2084.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.89ex; height:2.843ex;" alt="{\displaystyle f(x_{1},x_{2})}" loading="lazy"></span> gilt in der Umgebung des Punktes <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 x_{10},x_{20}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>10</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>20</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{10},x_{20}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ca99a9154604052b73daa07b3f4a8684bddf84f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.446ex; height:2.009ex;" alt="{\displaystyle x_{10},x_{20}}" loading="lazy"></span>:
</p>
<dl><dd><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 y=\underbrace {f(x_{10},x_{20})} _{={\text{const.}}}+\underbrace {{\frac {\partial f(x_{1},x_{2})}{\partial x_{1}}}{\bigg |}_{x_{10},x_{20}}\cdot (x_{1}-x_{10})+{\frac {\partial f(x_{1},x_{2})}{\partial x_{2}}}{\bigg |}_{x_{10},x_{20}}\cdot (x_{2}-x_{20})} _{=\Delta y}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo>=</mo>
<munder>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<munder>
<mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>10</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>20</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mrow>
<mo>⏟<!-- ⏟ --></mo>
</munder>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>const.</mtext>
</mrow>
</mrow>
</munder>
<mo>+</mo>
<munder>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<munder>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mo maxsize="2.047em" minsize="2.047em">|</mo>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>10</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>20</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>10</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mo maxsize="2.047em" minsize="2.047em">|</mo>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>10</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>20</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>20</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mrow>
<mo>⏟<!-- ⏟ --></mo>
</munder>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>=</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>y</mi>
</mrow>
</munder>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y=\underbrace {f(x_{10},x_{20})} _{={\text{const.}}}+\underbrace {{\frac {\partial f(x_{1},x_{2})}{\partial x_{1}}}{\bigg |}_{x_{10},x_{20}}\cdot (x_{1}-x_{10})+{\frac {\partial f(x_{1},x_{2})}{\partial x_{2}}}{\bigg |}_{x_{10},x_{20}}\cdot (x_{2}-x_{20})} _{=\Delta y}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2fcc3f27e83ddd7aa515fb912a55b477fb6870f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; margin-right: -0.028ex; width:78.851ex; height:10.176ex;" alt="{\displaystyle y=\underbrace {f(x_{10},x_{20})} _{={\text{const.}}}+\underbrace {{\frac {\partial f(x_{1},x_{2})}{\partial x_{1}}}{\bigg |}_{x_{10},x_{20}}\cdot (x_{1}-x_{10})+{\frac {\partial f(x_{1},x_{2})}{\partial x_{2}}}{\bigg |}_{x_{10},x_{20}}\cdot (x_{2}-x_{20})} _{=\Delta y}}" loading="lazy"></span></dd></dl>
<p>Beispiel:
</p>
<dl><dd><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(x_{1},x_{2})=x_{1}\cdot x_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x_{1},x_{2})=x_{1}\cdot x_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d71d29d7ec8fd40251419169af02b6e4cdb68763.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.435ex; height:2.843ex;" alt="{\displaystyle f(x_{1},x_{2})=x_{1}\cdot x_{2}}" loading="lazy"></span></dd></dl>
<p>ergibt die Tangentialebene
</p>
<dl><dd><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 y=\underbrace {x_{10}\cdot x_{20}} _{={\text{const.}}}+\underbrace {x_{20}\cdot (x_{1}-x_{10})+x_{10}\cdot (x_{2}-x_{20})} _{=\Delta y}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
<mo>=</mo>
<munder>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<munder>
<mrow>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>10</mn>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>20</mn>
</mrow>
</msub>
</mrow>
<mo>⏟<!-- ⏟ --></mo>
</munder>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>const.</mtext>
</mrow>
</mrow>
</munder>
<mo>+</mo>
<munder>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<munder>
<mrow>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>20</mn>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>10</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>+</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>10</mn>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>20</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mrow>
<mo>⏟<!-- ⏟ --></mo>
</munder>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>=</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>y</mi>
</mrow>
</munder>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y=\underbrace {x_{10}\cdot x_{20}} _{={\text{const.}}}+\underbrace {x_{20}\cdot (x_{1}-x_{10})+x_{10}\cdot (x_{2}-x_{20})} _{=\Delta y}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f90ea20ee09c4bdd348f07abe6e6b1016bcf6f6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; margin-right: -0.028ex; width:48.303ex; height:6.676ex;" alt="{\displaystyle y=\underbrace {x_{10}\cdot x_{20}} _{={\text{const.}}}+\underbrace {x_{20}\cdot (x_{1}-x_{10})+x_{10}\cdot (x_{2}-x_{20})} _{=\Delta y}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><span class="mw-default-size" typeof="mw:File"><span title="Wikibooks"></span></span></div><b><a href="https://de.wikibooks.org/wiki/Linearisierung_von_resistiven_Sensoren" class="extiw external" title="b:Linearisierung von resistiven Sensoren">Wikibooks: Linearisierung von resistiven Sensoren</a></b>&nbsp;– Lern- und Lehrmaterialien</div>
<ul><li>Skript der <style data-mw-deduplicate="TemplateStyles:r261891140">
/* start https://de.wikipedia.org/ */


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


/* end https://de.wikipedia.org/ */
</style><a rel="nofollow" class="external text" href="https://web.archive.org/web/20060723104452/http://www.iemw.tuwien.ac.at/riedling/Linearisierung.pdf">TU Wien</a> (<span class="webarchiv-memento"><a href="Webarchivierung#Begrifflichkeiten" title="Webarchivierung">Memento</a></span> vom 23. Juli 2006 im <i><a href="Internet_Archive" title="Internet Archive">Internet Archive</a></i>)</li>
<li>Skript der <a rel="nofollow" class="external text" href="http://www.eeh.ee.ethz.ch/lehre/vorlesungen/nus/slides/nus1_vorlesung_181007.pdf">ETH Zürich</a></li></ul>
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Normdaten&nbsp;(Sachbegriff): <a href="Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a>: <span class="-print"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4199872-8">4199872-8</a></span> </div>
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