Wir wollen die
Kettenregel
anhand der beiden Abbildungen
-
und
-
illustrieren. Diese Abbildungen sind
stetig partiell differenzierbar
und daher nach
Fakt
auch
total differenzierbar.
Die
Jacobi-Matrizen
zu diesen Abbildungen
(in einem beliebigen Punkt
bzw.
)
sind
-
![{\displaystyle {}\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}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7def8c6b6aa659bbddc97cdb40e4554333695c48)
und
-
![{\displaystyle {}\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}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b53dc5f026aa7a2984c592cf4a0fa33376705d9)
Die zusammengesetzte Abbildung
ist
![{\displaystyle {}{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c72b19c7e9329c2b7b4baf39117387db23af056e)
die zugehörige Jacobi-Matrix in
ist
-
![{\displaystyle \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}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4d3d8b9a4423b1687e59bb9f19b5387171fc431)
Die zusammengesetzte lineare Abbildung ist
![{\displaystyle {}{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b92ae6d108771867ce287b0648c566909e0e25f)