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

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Die Jacobi-Matrix (benannt nach <a href="Carl_Gustav_Jacob_Jacobi" title="Carl Gustav Jacob Jacobi">Carl Gustav Jacob Jacobi</a>; auch Funktionalmatrix, Ableitungsmatrix oder Jacobische genannt) einer <a href="Differentialrechnung" title="Differentialrechnung">differenzierbaren</a> <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon {\mathbb {R} ^{n}}\to {\mathbb {R} ^{m}}\,\!}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7be19350296d356bd743520627d404ae62652611.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:12.563ex; height:2.676ex;" alt="{\displaystyle f\colon {\mathbb {R} ^{n}}\to {\mathbb {R} ^{m}}\,\!}" loading="lazy"> ist die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12b23d207d23dd430b93320539abbb0bde84870d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}" loading="lazy">-<a href="Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a> sämtlicher erster <a href="Partielle_Ableitung" title="Partielle Ableitung">partieller Ableitungen</a>.

Im Falle der <a href="Totale_Differenzierbarkeit" title="Totale Differenzierbarkeit">totalen Differenzierbarkeit</a> bildet sie die <a href="Darstellungsmatrix" class="mw-redirect" title="Darstellungsmatrix">Matrix-Darstellung</a> der als <a href="Lineare_Abbildung" title="Lineare Abbildung">lineare Abbildung</a> aufgefassten ersten Ableitung der Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.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"> bezüglich der <a href="Standardbasis" title="Standardbasis">Standardbasen</a> des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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

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

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

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Genutzt wird die Jacobi-Matrix zum Beispiel zur annähernden Berechnung (Approximation) oder Minimierung mehrdimensionaler <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktionen</a> in der <a href="Mathematik" title="Mathematik">Mathematik</a>.

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

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon U\subset \mathbb {R} ^{n}\to \mathbb {R} ^{m}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2703b75767ff6fb23cc3d6ad1a25830d35eb5038.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.057ex; height:2.676ex;" alt="{\displaystyle f\colon U\subset \mathbb {R} ^{n}\to \mathbb {R} ^{m}}" loading="lazy"> eine Funktion, deren Komponentenfunktionen mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1},\ldots ,f_{m}}">

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<annotation encoding="application/x-tex">{\displaystyle f_{1},\ldots ,f_{m}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5296700cf113b5345e68e5eb462beb87350ba582.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f_{1},\ldots ,f_{m}}" loading="lazy"> bezeichnet seien und deren <a href="Partielle_Ableitung" title="Partielle Ableitung">partielle Ableitungen</a> alle existieren sollen. Für einen <a href="Raumpunkt" class="mw-redirect" title="Raumpunkt">Raumpunkt</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> im <a href="Urbild_(Mathematik)" title="Urbild (Mathematik)">Urbildraum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},\dots ,x_{n}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e5afdbc2d248d8fa9ba2c4f5188d946a0537e753.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:2.009ex;" alt="{\displaystyle x_{1},\dots ,x_{n}}" loading="lazy"> die jeweils zugehörigen Koordinaten.

Dann ist für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in U}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd1991ea9cb2ab076462a5538242321f0d0ee991.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.853ex; height:2.176ex;" alt="{\displaystyle a\in U}" loading="lazy"> die Jacobi-Matrix im Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> definiert durch

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<annotation encoding="application/x-tex">{\displaystyle J_{f}(a):=\left({\frac {\partial f_{i}}{\partial x_{j}}}(a)\right)_{i=1,\ldots ,m;\ j=1,\ldots ,n}={\begin{pmatrix}{\frac {\partial f_{1}}{\partial x_{1}}}(a)&{\frac {\partial f_{1}}{\partial x_{2}}}(a)&\ldots &{\frac {\partial f_{1}}{\partial x_{n}}}(a)\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial f_{m}}{\partial x_{1}}}(a)&{\frac {\partial f_{m}}{\partial x_{2}}}(a)&\ldots &{\frac {\partial f_{m}}{\partial x_{n}}}(a)\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6913db61c168b3ab510854dd2bb6d33ec564ccc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:73.599ex; height:14.509ex;" alt="{\displaystyle J_{f}(a):=\left({\frac {\partial f_{i}}{\partial x_{j}}}(a)\right)_{i=1,\ldots ,m;\ j=1,\ldots ,n}={\begin{pmatrix}{\frac {\partial f_{1}}{\partial x_{1}}}(a)&{\frac {\partial f_{1}}{\partial x_{2}}}(a)&\ldots &{\frac {\partial f_{1}}{\partial x_{n}}}(a)\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial f_{m}}{\partial x_{1}}}(a)&{\frac {\partial f_{m}}{\partial x_{2}}}(a)&\ldots &{\frac {\partial f_{m}}{\partial x_{n}}}(a)\end{pmatrix}}}" loading="lazy">.</dd></dl>

In den Zeilen der Jacobi-Matrix stehen also gerade die (transponierten) <a href="Gradient_(Mathematik)" title="Gradient (Mathematik)">Gradienten</a> der Komponentenfunktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1},\dots ,f_{m}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6f067fd333142a9525e2bb6d75e68879724992b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f_{1},\dots ,f_{m}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.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">.

Andere übliche Schreibweisen für die Jacobi-Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{f}(a)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d602622c85d01faa032fe267d343360ccbdf98fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.466ex; height:3.009ex;" alt="{\displaystyle J_{f}(a)}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.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"> an der Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Df(a)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/17c83c98b9b68b09db509888726883bae6a68aba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.242ex; height:2.843ex;" alt="{\displaystyle Df(a)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial f}{\partial x}}(a)}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial f}{\partial x}}(a)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/744dbba1af9277b34cc5aa48c637eae4389b33a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.523ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial f}{\partial x}}(a)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {\partial (f_{1},\ldots ,f_{m})}{\partial (x_{1},\ldots ,x_{n})}}(a)}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {\partial (f_{1},\ldots ,f_{m})}{\partial (x_{1},\ldots ,x_{n})}}(a)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85d4f41511642601b0d8b389b20f46d1935712ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.705ex; height:4.843ex;" alt="{\displaystyle \textstyle {\frac {\partial (f_{1},\ldots ,f_{m})}{\partial (x_{1},\ldots ,x_{n})}}(a)}" loading="lazy">.

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

Die Funktion

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<annotation encoding="application/x-tex">{\displaystyle f\colon \mathbb {R} ^{3}\to \mathbb {R} ^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13bb78d3317a6a897796ca9023bcd7e22ab00fb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.391ex; height:3.009ex;" alt="{\displaystyle f\colon \mathbb {R} ^{3}\to \mathbb {R} ^{2}}" loading="lazy">

sei gegeben durch

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<annotation encoding="application/x-tex">{\displaystyle f(x,y,z)={\binom {x^{2}+y^{2}+z\cdot \sin x}{z^{2}+z\cdot \sin y}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/201eef5d272f72d8ae8db77d2c4c439469740412.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.868ex; height:6.343ex;" alt="{\displaystyle f(x,y,z)={\binom {x^{2}+y^{2}+z\cdot \sin x}{z^{2}+z\cdot \sin y}}}" loading="lazy"></dd></dl>

Dann ist

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\partial }{\partial x}}f(x,y,z)&={\binom {2x+z\cdot \cos x}{0}}\\{\frac {\partial }{\partial y}}f(x,y,z)&={\binom {2y}{z\cdot \cos y}}\\{\frac {\partial }{\partial z}}f(x,y,z)&={\binom {\sin x}{2z+\sin y}}\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c9be09b4c2d09d7aba35d4929c70d7175a49cd96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:32.412ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {\partial }{\partial x}}f(x,y,z)&={\binom {2x+z\cdot \cos x}{0}}\\{\frac {\partial }{\partial y}}f(x,y,z)&={\binom {2y}{z\cdot \cos y}}\\{\frac {\partial }{\partial z}}f(x,y,z)&={\binom {\sin x}{2z+\sin y}}\end{aligned}}}" loading="lazy"></dd></dl>

und damit die Jacobi-Matrix

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<annotation encoding="application/x-tex">{\displaystyle J_{f}(x,y,z)=\left({\begin{array}{ccc}2x+z\cdot \cos x&2y&\sin x\\0&z\cdot \cos y\,&2z+\sin y\end{array}}\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/806f1a19d6f7bfd667bea48a52f0938ffe654e36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:52.018ex; height:6.176ex;" alt="{\displaystyle J_{f}(x,y,z)=\left({\begin{array}{ccc}2x+z\cdot \cos x&2y&\sin x\\0&z\cdot \cos y\,&2z+\sin y\end{array}}\right)}" loading="lazy"></dd></dl>

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

<ul><li>Ist die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon U\subset \mathbb {R} ^{n}\to \mathbb {R} ^{m}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2703b75767ff6fb23cc3d6ad1a25830d35eb5038.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.057ex; height:2.676ex;" alt="{\displaystyle f\colon U\subset \mathbb {R} ^{n}\to \mathbb {R} ^{m}}" loading="lazy"> <a href="Totale_Differenzierbarkeit" title="Totale Differenzierbarkeit">total differenzierbar</a>, dann ist ihr <a href="Totales_Differential" title="Totales Differential">totales Differential</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Df_{a}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32a8678a7f42b6144a93bad7cd7890d4b732c410.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.165ex; height:2.509ex;" alt="{\displaystyle Df_{a}}" loading="lazy"> an der Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=(a_{1},\dots ,a_{n})\in U}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f061d58e1e14decdd2958a2b0b7e90b61b995fe0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.671ex; height:2.843ex;" alt="{\displaystyle a=(a_{1},\dots ,a_{n})\in U}" loading="lazy"> die lineare Abbildung</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle Df_{a}\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m},\quad Df_{a}(h)=J_{f}(a)\cdot h}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d634fb10cf86be28d8248288ed4f8166925567aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.315ex; height:3.009ex;" alt="{\displaystyle Df_{a}\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m},\quad Df_{a}(h)=J_{f}(a)\cdot h}" loading="lazy">.</dd></dl></dd>

<dd>Die Jacobi-Matrix an der Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> ist also die <a href="Abbildungsmatrix" title="Abbildungsmatrix">Abbildungsmatrix</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Df_{a}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32a8678a7f42b6144a93bad7cd7890d4b732c410.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.165ex; height:2.509ex;" alt="{\displaystyle Df_{a}}" loading="lazy">.</dd></dl>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b6100c5ebd48c6fd848709f2be624465203eb173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=1}" loading="lazy"> entspricht die Jacobi-Matrix dem <a href="Transponierte_Matrix" title="Transponierte Matrix">transponierten</a> <a href="Gradient_(Mathematik)" title="Gradient (Mathematik)">Gradienten</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.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">. Manchmal wird der Gradient auch als Zeilenvektor definiert. In diesem Fall sind Gradient und Jacobi-Matrix gleich.</li></ul>

<ul><li>Die Jacobi-Matrix kann, wenn man sie für eine Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=(a_{1},\dots ,a_{n})}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0507c57a117ff1a5c9eb8823880fefeab15270f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.048ex; height:2.843ex;" alt="{\displaystyle a=(a_{1},\dots ,a_{n})}" loading="lazy"> ausrechnet, zur Näherung der <a href="Funktionswert" class="mw-redirect" title="Funktionswert">Funktionswerte</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> verwendet werden:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle f(x)\approx f(a)+J_{f}(a)\cdot (x-a).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/947e0d845f39e4c6dd988231b7d1130cb71cdf2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -1.005ex; width:29.675ex; height:3.009ex;" aria-hidden="true" alt="{\displaystyle f(x)\approx f(a)+J_{f}(a)\cdot (x-a).}" loading="lazy"></dd></dl></dd>

<dd>Diese <a href="Affine_Abbildung" title="Affine Abbildung">affine Abbildung</a> entspricht der <a href="Taylor-Formel" title="Taylor-Formel">Taylor-Approximation</a> erster Ordnung (<a href="Linearisierung" title="Linearisierung">Linearisierung</a>).</dd></dl>

<ul><li>Die <a href="Fehlerfortpflanzung" title="Fehlerfortpflanzung">Fortpflanzung von Messfehlern</a> in Form einer <a href="Kovarianzmatrix" title="Kovarianzmatrix">Kovarianzmatrix</a> geschieht durch die Jacobi-Matrix: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{f}=J\cdot V_{x}\cdot J^{\text{T}}}">

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<annotation encoding="application/x-tex">{\displaystyle V_{f}=J\cdot V_{x}\cdot J^{\text{T}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d74e4411db3ddacb32a4ad4a51d33a32c0c0f05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.892ex; height:3.343ex;" alt="{\displaystyle V_{f}=J\cdot V_{x}\cdot J^{\text{T}}}" loading="lazy"></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Determinante_der_Jacobi-Matrix">Determinante der Jacobi-Matrix</h2></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Jacobi-Determinante" class="mw-redirect" title="Jacobi-Determinante">Jacobi-Determinante</a></div>

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/69c9d8e54796e7de7d4738510cc10bc3fc55d48e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.676ex;" alt="{\displaystyle m=n}" loading="lazy">, es wird also eine differenzierbare Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon U\subset \mathbb {R} ^{n}\to \mathbb {R} ^{n}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab2b61ed55f4015515ede337fde29c44f68cccb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.601ex; height:2.676ex;" alt="{\displaystyle f\colon U\subset \mathbb {R} ^{n}\to \mathbb {R} ^{n}}" loading="lazy"> betrachtet. Dann ist deren Jacobi-Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{f}(a)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d602622c85d01faa032fe267d343360ccbdf98fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.466ex; height:3.009ex;" alt="{\displaystyle J_{f}(a)}" loading="lazy"> am Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in U}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd1991ea9cb2ab076462a5538242321f0d0ee991.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.853ex; height:2.176ex;" alt="{\displaystyle a\in U}" loading="lazy"> eine quadratische <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59d2b4cb72e304526cf5b5887147729ea259da78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" loading="lazy">-Matrix. In diesem Fall kann man die <a href="Determinante_(Mathematik)" class="mw-redirect" title="Determinante (Mathematik)">Determinante</a> der Jacobi-Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(J_{f}(a))}">

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<annotation encoding="application/x-tex">{\displaystyle \det(J_{f}(a))}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8116c75b42d68db7570ecfcb004c86729bb163f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.505ex; height:3.009ex;" alt="{\displaystyle \det(J_{f}(a))}" loading="lazy"> bestimmen. Die Determinante der Jacobi-Matrix wird <a href="Funktionaldeterminante" title="Funktionaldeterminante">Jacobi-Determinante</a> oder Funktionaldeterminante genannt. Ist die Jacobi-Determinante im Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> ungleich null, so ist die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> <a href="Umkehrfunktion" title="Umkehrfunktion">invertierbar</a>. Dies besagt der <a href="Satz_von_der_Umkehrabbildung" class="mw-redirect" title="Satz von der Umkehrabbildung">Satz von der Umkehrabbildung</a>. Außerdem spielt die Jacobi-Determinante eine wichtige Rolle beim <a href="Transformationssatz" title="Transformationssatz">Transformationssatz für Integrale</a>. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\neq n}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a50a2eaa8dcad2a8e19c9fd861a0fdd641bdfa46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.534ex; height:2.676ex;" alt="{\displaystyle m\neq n}" loading="lazy">, so kann man definitionsgemäß keine Determinante der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12b23d207d23dd430b93320539abbb0bde84870d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}" loading="lazy">-Jacobi-Matrix bilden. Jedoch gibt es in diesem Fall ein ähnliches Konzept. Dieses wird <a href="Gramsche_Determinante" title="Gramsche Determinante">Gramsche Determinante</a> genannt.

<div class="mw-heading mw-heading2"><h2 id="Jacobi-Matrix_einer_holomorphen_Funktion">Jacobi-Matrix einer holomorphen Funktion</h2></div>

Neben Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon U\subset \mathbb {R} ^{n}\to \mathbb {R} ^{m}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2703b75767ff6fb23cc3d6ad1a25830d35eb5038.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.057ex; height:2.676ex;" alt="{\displaystyle f\colon U\subset \mathbb {R} ^{n}\to \mathbb {R} ^{m}}" loading="lazy"> kann man auch Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\colon V\subset \mathbb {C} ^{n}\to \mathbb {C} ^{m}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle h\colon V\subset \mathbb {C} ^{n}\to \mathbb {C} ^{m}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c54aea5a96cf04363934e5d543360560ccff1cee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.122ex; height:2.343ex;" alt="{\displaystyle h\colon V\subset \mathbb {C} ^{n}\to \mathbb {C} ^{m}}" loading="lazy"> auf (komplexe) Differenzierbarkeit untersuchen. Funktionen, die komplex differenzierbar sind, werden <a href="Holomorphe_Funktion" title="Holomorphe Funktion">holomorph</a> genannt, denn sie haben andere Eigenschaften als die (reell) differenzierbaren Funktionen. Auch für die holomorphe Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h:=(h_{1},\ldots ,h_{m})\colon V\subset \mathbb {C} ^{n}\to \mathbb {C} ^{m}}">

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<annotation encoding="application/x-tex">{\displaystyle h:=(h_{1},\ldots ,h_{m})\colon V\subset \mathbb {C} ^{n}\to \mathbb {C} ^{m}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9a89daeaf810495c4bd22ba07bdf3aa8d6f87b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.262ex; height:2.843ex;" alt="{\displaystyle h:=(h_{1},\ldots ,h_{m})\colon V\subset \mathbb {C} ^{n}\to \mathbb {C} ^{m}}" loading="lazy"> kann man Jacobi-Matrizen bestimmen. Hier gibt es zwei unterschiedliche Varianten. Zum einen eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12b23d207d23dd430b93320539abbb0bde84870d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}" loading="lazy"> mit komplexwertigen Einträgen und zum anderen eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2m\times 2n}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6467a4f277236af1ac041678961d766d610678a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.6ex; height:2.176ex;" alt="{\displaystyle 2m\times 2n}" loading="lazy">-Matrix mit reellwertigen Einträgen. Die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12b23d207d23dd430b93320539abbb0bde84870d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}" loading="lazy">-Jacobi-Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{h}^{\mathbb {C} }(z)}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J_{h}^{\mathbb {C} }(z)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9d93598c76e8905dab571f383e87749dc047218.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.842ex; height:3.176ex;" alt="{\displaystyle J_{h}^{\mathbb {C} }(z)}" loading="lazy"> am Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z:=(z_{1},\ldots ,z_{n})\in V\subset \mathbb {C} ^{n}}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle z:=(z_{1},\ldots ,z_{n})\in V\subset \mathbb {C} ^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0099d97031a9899606dc3a4cb0c6c4feb204d0c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.879ex; height:2.843ex;" alt="{\displaystyle z:=(z_{1},\ldots ,z_{n})\in V\subset \mathbb {C} ^{n}}" loading="lazy"> ist durch

</mrow>

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<mtd>

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<mtd>

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<mrow>

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<mtr>

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<mrow>

<msub>

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</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J_{h}^{\mathbb {C} }(z):={\begin{pmatrix}{\frac {\partial h_{1}(z)}{\partial z_{1}}}&\cdots &{\frac {\partial h_{1}(z)}{\partial z_{n}}}\\\vdots &\ddots &\vdots \\{\frac {\partial h_{m}(z)}{\partial z_{1}}}&\cdots &{\frac {\partial h_{m}(z)}{\partial z_{n}}}\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d48c63f058f2b5b9d6d9b43f37c8a380b4017cd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:34.228ex; height:14.509ex;" alt="{\displaystyle J_{h}^{\mathbb {C} }(z):={\begin{pmatrix}{\frac {\partial h_{1}(z)}{\partial z_{1}}}&\cdots &{\frac {\partial h_{1}(z)}{\partial z_{n}}}\\\vdots &\ddots &\vdots \\{\frac {\partial h_{m}(z)}{\partial z_{1}}}&\cdots &{\frac {\partial h_{m}(z)}{\partial z_{n}}}\end{pmatrix}}}" loading="lazy"></dd></dl>

definiert.

Jede <a href="Komplexwertige_Funktion" title="Komplexwertige Funktion">komplexwertige Funktion</a> kann in zwei reellwertige Funktionen aufgespalten werden. Das heißt, es existieren Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u,v\colon \mathbb {C} ^{n}\to \mathbb {R} ^{m}}">

<msup>

</mrow>

</mrow>

</msup>

<msup>

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</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u,v\colon \mathbb {C} ^{n}\to \mathbb {R} ^{m}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3114b7b4f066af6b427c619b28afe3ab65a06858.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.389ex; height:2.676ex;" alt="{\displaystyle u,v\colon \mathbb {C} ^{n}\to \mathbb {R} ^{m}}" loading="lazy">, sodass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=u+iv}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/918d925a4e29199f97f6246cdf538a1f0e3ba2c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.538ex; height:2.343ex;" alt="{\displaystyle h=u+iv}" loading="lazy"> gilt. Die Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> kann man nun wieder gewöhnlich partiell differenzieren und in einer Matrix anordnen. Seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z:=(z_{1},\ldots ,z_{n})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z:=(z_{1},\ldots ,z_{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/95ee2bf6b9ac9831e1a66df33078d31b55f50086.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.256ex; height:2.843ex;" alt="{\displaystyle z:=(z_{1},\ldots ,z_{n})}" loading="lazy"> die Koordinaten in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a53b4e76242764d1bca004168353c380fef25258.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 {C} ^{n}}" loading="lazy"> und setze <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{j}:=x_{j}+iy_{j}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z_{j}:=x_{j}+iy_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8221b88aadd092db5d6f3d45d21b564405866bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.667ex; height:2.843ex;" alt="{\displaystyle z_{j}:=x_{j}+iy_{j}}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy">. Die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2m\times 2n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6467a4f277236af1ac041678961d766d610678a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.6ex; height:2.176ex;" alt="{\displaystyle 2m\times 2n}" loading="lazy">-Jacobi-Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{h}^{\mathbb {R} }(z)}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9727c2ea4e4f96ff11df1a669dbaed1f7c5854d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.842ex; height:3.176ex;" alt="{\displaystyle J_{h}^{\mathbb {R} }(z)}" loading="lazy"> der holomorphen Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy">

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4e530fb380ba488705c57df90a2631e9055b7e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.716ex; height:2.176ex;" alt="{\displaystyle z\in V}" loading="lazy"> ist dann definiert durch

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle J_{h}^{\mathbb {R} }(z):={\begin{pmatrix}{\frac {\partial u_{1}(z)}{\partial x_{1}}}&\cdots &{\frac {\partial u_{1}(z)}{\partial x_{n}}}&{\frac {\partial u_{1}(z)}{\partial y_{1}}}&\cdots &{\frac {\partial u_{1}(z)}{\partial y_{n}}}\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots \\{\frac {\partial u_{m}(z)}{\partial x_{1}}}&\cdots &{\frac {\partial u_{m}(z)}{\partial x_{n}}}&{\frac {\partial u_{m}(z)}{\partial y_{1}}}&\cdots &{\frac {\partial u_{m}(z)}{\partial y_{n}}}\\{\frac {\partial v_{1}(z)}{\partial x_{1}}}&\cdots &{\frac {\partial v_{1}(z)}{\partial x_{n}}}&{\frac {\partial v_{1}(z)}{\partial y_{1}}}&\cdots &{\frac {\partial v_{1}(z)}{\partial y_{n}}}\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots \\{\frac {\partial v_{m}(z)}{\partial x_{1}}}&\cdots &{\frac {\partial v_{m}(z)}{\partial x_{n}}}&{\frac {\partial v_{m}(z)}{\partial y_{1}}}&\cdots &{\frac {\partial v_{m}(z)}{\partial y_{n}}}\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1865b05f5d5c05a3f8ef9f8109c3cf3dec152c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.338ex; width:56.347ex; height:29.843ex;" alt="{\displaystyle J_{h}^{\mathbb {R} }(z):={\begin{pmatrix}{\frac {\partial u_{1}(z)}{\partial x_{1}}}&\cdots &{\frac {\partial u_{1}(z)}{\partial x_{n}}}&{\frac {\partial u_{1}(z)}{\partial y_{1}}}&\cdots &{\frac {\partial u_{1}(z)}{\partial y_{n}}}\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots \\{\frac {\partial u_{m}(z)}{\partial x_{1}}}&\cdots &{\frac {\partial u_{m}(z)}{\partial x_{n}}}&{\frac {\partial u_{m}(z)}{\partial y_{1}}}&\cdots &{\frac {\partial u_{m}(z)}{\partial y_{n}}}\\{\frac {\partial v_{1}(z)}{\partial x_{1}}}&\cdots &{\frac {\partial v_{1}(z)}{\partial x_{n}}}&{\frac {\partial v_{1}(z)}{\partial y_{1}}}&\cdots &{\frac {\partial v_{1}(z)}{\partial y_{n}}}\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots \\{\frac {\partial v_{m}(z)}{\partial x_{1}}}&\cdots &{\frac {\partial v_{m}(z)}{\partial x_{n}}}&{\frac {\partial v_{m}(z)}{\partial y_{1}}}&\cdots &{\frac {\partial v_{m}(z)}{\partial y_{n}}}\end{pmatrix}}}" loading="lazy">.</dd></dl>

Gilt bei den Jacobi-Matrizen für holomorphe Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/69c9d8e54796e7de7d4738510cc10bc3fc55d48e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.676ex;" alt="{\displaystyle m=n}" loading="lazy">, so kann man natürlich die Determinanten der beiden Matrizen betrachten. Diese beiden Determinanten stehen in Beziehung zueinander. Es gilt nämlich

<mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

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</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \det \left(J_{h}^{\mathbb {R} }(z)\right)=\left|\det(J_{h}^{\mathbb {C} }(z))\right|^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/588def9a3f4cf8aa35e9e26841d99c3558d97079.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.916ex; height:3.676ex;" alt="{\displaystyle \det \left(J_{h}^{\mathbb {R} }(z)\right)=\left|\det(J_{h}^{\mathbb {C} }(z))\right|^{2}}" loading="lazy">.</dd></dl>

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

<ul><li><a href="Mehrdimensionale_Kettenregel" title="Mehrdimensionale Kettenregel">Mehrdimensionale Kettenregel</a></li>

<li><a href="Hesse-Matrix" title="Hesse-Matrix">Hesse-Matrix</a></li>

<li><a href="Gradient_(Mathematik)" title="Gradient (Mathematik)">Gradient (Mathematik)</a></li></ul>

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

<ul><li><a href="Konrad_K%C3%B6nigsberger" title="Konrad Königsberger">Konrad Königsberger</a>: Analysis 2. Springer-Verlag, Berlin/Heidelberg, 2000, ISBN 3-540-43580-8. (für Jacobi-Matrizen reeller Funktionen).</li>

<li>Klaus Fritzsche, <a href="Hans_Grauert" title="Hans Grauert">Hans Grauert</a>: From Holomorphic Functions to Complex Manifolds. Springer-Verlag, ISBN 0-387-95395-7. (S. 30–31; für Jacobi-Matrizen holomorpher Funktionen).</li></ul></div><div><div class="zim-footer">

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