# Totale Differenzierbarkeit/R/Kettenregel/Standardbasen und Jacobimatrix/Fakt

Die Kettenregel (Jacobimatrix)

Es seien ${\displaystyle {}G\subseteq \mathbb {R} ^{n}}$ und ${\displaystyle {}D\subseteq \mathbb {R} ^{m}}$ offene Mengen, und ${\displaystyle {}f\colon G\rightarrow \mathbb {R} ^{m}}$ und ${\displaystyle {}g\colon D\rightarrow \mathbb {R} ^{k}}$ seien Abbildungen derart, dass ${\displaystyle {}f(G)\subseteq D}$ gilt. Es sei weiter angenommen, dass ${\displaystyle {}f}$ in ${\displaystyle {}P\in G}$ und ${\displaystyle {}g}$ in ${\displaystyle {}f(P)\in D}$ total differenzierbar ist.

Dann ist ${\displaystyle {}h=g\circ f\colon G\rightarrow U}$ in ${\displaystyle {}P}$ differenzierbar und zwischen den Jacobi-Matrizen gilt die Beziehung

${\displaystyle {}\operatorname {Jak} (h)_{P}=\operatorname {Jak} (g\circ f)_{P}=\operatorname {Jak} (g)_{f(P)}\circ \operatorname {Jak} (f)_{P}\,,}$

also ausgeschrieben

${\displaystyle {\begin{pmatrix}{\frac {\partial h_{1}}{\partial x_{1}}}(P)&\ldots &{\frac {\partial h_{1}}{\partial x_{n}}}(P)\\\vdots &\ddots &\vdots \\{\frac {\partial h_{k}}{\partial x_{1}}}(P)&\ldots &{\frac {\partial h_{k}}{\partial x_{n}}}(P)\end{pmatrix}}={\begin{pmatrix}{\frac {\partial g_{1}}{\partial y_{1}}}(f(P))&\ldots &{\frac {\partial g_{1}}{\partial y_{m}}}(f(P))\\\vdots &\ddots &\vdots \\{\frac {\partial g_{k}}{\partial y_{1}}}(f(P))&\ldots &{\frac {\partial g_{k}}{\partial y_{m}}}(f(P))\end{pmatrix}}\circ {\begin{pmatrix}{\frac {\partial f_{1}}{\partial x_{1}}}(P)&\ldots &{\frac {\partial f_{1}}{\partial x_{n}}}(P)\\\vdots &\ddots &\vdots \\{\frac {\partial f_{m}}{\partial x_{1}}}(P)&\ldots &{\frac {\partial f_{m}}{\partial x_{n}}}(P)\end{pmatrix}}\,.}$