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frechet_derivative_chain_rule [2013/09/15 20:15] nikolaj |
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| - | ===== Fréchet derivative chain rule ===== | ||
| - | ==== Theorem ==== | ||
| - | | @#88DDEE: $X,Y,Z$ ... Banach spaces with topology | | ||
| - | | @#88DDEE: $F\in C(X,Y)$ | | ||
| - | | @#88DDEE: $G\in C(Y,Z)$ | | ||
| - | | @#55EE55: $ D(G\circ F)=(DG)\circ F\ \cdot\ DF $ | | ||
| - | |||
| - | where $\circ$ denotes the concatenation of functions of $X,Y$, which is taken to bind stronger than the concatenation $\cdot$ of linear operators. | ||
| - | |||
| - | ==== Discussion ==== | ||
| - | For functions in $f,g: \mathbb R\to\mathbb R$, this of course reads | ||
| - | |||
| - | ^ $\frac{\partial}{\partial x}g(f(x))=g'(f(x))\cdot f'(x)$ ^ | ||
| - | |||
| - | === Reference === | ||
| - | Wikipedia: [[http://en.wikipedia.org/wiki/Chain_rule|Chain rule]], [[http://en.wikipedia.org/wiki/Chain_rule_%28disambiguation%29|Chain rule (disambiguation)]] | ||
| - | ==== Parents ==== | ||
| - | === Requirements === | ||
| - | [[Fréchet derivative]] | ||
