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

Die Kettenregel (Jacobimatrix)

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

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

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

also ausgeschrieben

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

Zum Beweis, Alternativen Beweis erstellen