Totales Differential/R/Kettenregel/(uv^3w^2,u^2-v^2w) und (xy-y^2,cos x,x-y)/Beispiel

Wir wollen die Kettenregel anhand der beiden Abbildungen

$f\colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{2},\,\left(u,\,v,\,w\right)\longmapsto \left(uv^{3}w^{2},\,u^{2}-v^{2}w\right)$

und

$g\colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{3},\,\left(x,\,y\right)\longmapsto \left(xy-y^{2},\,\cos x,\,x-y\right)$

illustrieren. Diese Abbildungen sind stetig partiell differenzierbar und daher nach Fakt auch total differenzierbar. Die Jacobi-Matrizen zu diesen Abbildungen (in einem beliebigen Punkt ${}P=(u,v,w)\in \mathbb {R} ^{3}$ bzw. ${}Q=(x,y)\in \mathbb {R} ^{2}$) sind

${}\operatorname {Jac} (f)_{P}={\begin{pmatrix}{\frac {\partial f_{1}}{\partial u}}(P)&{\frac {\partial f_{1}}{\partial v}}(P)&{\frac {\partial f_{1}}{\partial w}}(P)\\{\frac {\partial f_{2}}{\partial u}}(P)&{\frac {\partial f_{2}}{\partial v}}(P)&{\frac {\partial f_{2}}{\partial w}}(P)\end{pmatrix}}={\begin{pmatrix}v^{3}w^{2}&3uv^{2}w^{2}&2uv^{3}w\\2u&-2vw&-v^{2}\end{pmatrix}}\,$

und

${}\operatorname {Jac} (g)_{Q}={\begin{pmatrix}{\frac {\partial g_{1}}{\partial x}}(Q)&{\frac {\partial g_{1}}{\partial y}}(Q)\\{\frac {\partial g_{2}}{\partial x}}(Q)&{\frac {\partial g_{2}}{\partial y}}(Q)\\{\frac {\partial g_{3}}{\partial x}}(Q)&{\frac {\partial g_{3}}{\partial y}}(Q)\end{pmatrix}}={\begin{pmatrix}y&x-2y\\-\sin x&0\\1&-1\end{pmatrix}}\,.$

Die zusammengesetzte Abbildung ${}g\circ f$ ist

${}{\begin{aligned}g(f(u,v,w))&=\left(uv^{3}w^{2}{\left(u^{2}-v^{2}w\right)}-{\left(u^{2}-v^{2}w\right)}^{2},\,\cos {\left(uv^{3}w^{2}\right)},\,uv^{3}w^{2}-u^{2}+v^{2}w\right)\\&=\left(u^{3}v^{3}w^{2}-uv^{5}w^{3}-u^{4}-v^{4}w^{2}+2u^{2}v^{2}w,\,\cos {\left(uv^{3}w^{2}\right)},\,uv^{3}w^{2}-u^{2}+v^{2}w\right),\end{aligned}}$

die zugehörige Jacobi-Matrix in ${}P=(u,v,w)$ ist

$\operatorname {Jac} (g\circ f)_{P}={\begin{pmatrix}3u^{2}v^{3}w^{2}-v^{5}w^{3}-4u^{3}+4uv^{2}w&3u^{3}v^{2}w^{2}-5uv^{4}w^{3}-4v^{3}w^{2}+4u^{2}vw&2u^{3}v^{3}w-3uv^{5}w^{2}-2v^{4}w+2u^{2}v^{2}\\-v^{3}w^{2}\sin {\left(uv^{3}w^{2}\right)}&-3uv^{2}w^{2}\sin {\left(uv^{3}w^{2}\right)}&-2uv^{3}w\sin {\left(uv^{3}w^{2}\right)}\\v^{3}w^{2}-2u&3uv^{2}w^{2}+2vw&2uv^{3}w+v^{2}\end{pmatrix}}\,.$

Die zusammengesetzte lineare Abbildung ist

${}{\begin{aligned}\operatorname {Jak} (g)_{f(P)}\circ \operatorname {Jak} (f)_{P}&=\operatorname {Jak} (g)_{(uv^{3}w^{2},u^{2}-v^{2}w)}\circ \operatorname {Jak} (f)_{P}\\&={\begin{pmatrix}u^{2}-v^{2}w&uv^{3}w^{2}-2u^{2}+2v^{2}w\\-\sin \left(uv^{3}w^{2}\right)&0\\1&-1\end{pmatrix}}\circ {\begin{pmatrix}v^{3}w^{2}&3uv^{2}w^{2}&2uv^{3}w\\2u&-2vw&-v^{2}\end{pmatrix}}\\&={\begin{pmatrix}{\left(u^{2}-v^{2}w\right)}v^{3}w^{2}+{\left(uv^{3}w^{2}-2u^{2}+2v^{2}w\right)}2u&{\left(u^{2}-v^{2}w\right)}3uv^{2}w^{2}-{\left(uv^{3}w^{2}-2u^{2}+2v^{2}w\right)}2vw&{\left(u^{2}-v^{2}w\right)}2uv^{3}w-{\left(uv^{3}w^{2}-2u^{2}+2v^{2}w\right)}v^{2}\\-v^{3}w^{2}\sin {\left(uv^{3}w^{2}\right)}&-3uv^{2}w^{2}\sin {\left(uv^{3}w^{2}\right)}&-2uv^{3}w\sin {\left(uv^{3}w^{2}\right)}\\v^{3}w^{2}-2u&3uv^{2}w^{2}+2vw&2uv^{3}w+v^{2}\end{pmatrix}}\\&={\begin{pmatrix}3u^{2}v^{3}w^{2}-^{5}w^{3}-4u^{3}+4uv^{2}w&3u^{3}v^{2}w^{2}-5uv^{4}w^{3}-4v^{3}w^{2}+4u^{2}vw&2u^{3}v^{3}w-3uv^{5}w^{2}-2v^{4}w+2u^{2}v^{2}\\-v^{3}w^{2}\sin {\left(uv^{3}w^{2}\right)}&-3uv^{2}w^{2}\sin {\left(uv^{3}w^{2}\right)}&-2uv^{3}w\sin {\left(uv^{3}w^{2}\right)}\\v^{3}w^{2}-2u&3uv^{2}w^{2}+2vw&2uv^{3}w+v^{2}\end{pmatrix}}.\,\end{aligned}}$