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frechet_derivative_chain_rule [2013/09/15 20:13] nikolaj |
frechet_derivative_chain_rule [2014/03/21 11:11] (current) |
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| ===== Fréchet derivative chain rule ===== | ===== Fréchet derivative chain rule ===== | ||
| ==== Theorem ==== | ==== Theorem ==== | ||
| - | | @#88DDEE: $X,Y,Z$ ... Banach spaces with topology | | + | | @#55CCEE: context | @#55CCEE: $X,Y,Z$ ... Banach spaces with topology | |
| - | | @#88DDEE: $F\in C(X,Y)$ | | + | | @#55CCEE: context | @#55CCEE: $f\in C(X,Y)$ | |
| - | | @#88DDEE: $G\in C(Y,Z)$ | | + | | @#55CCEE: context | @#55CCEE: $g\in C(Y,Z)$ | |
| - | | @#55EE55: $ D(G\circ F)=(DG)\circ F\ \cdot\ DF $ | | + | | @#55EE55: postulate | @#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. | where $\circ$ denotes the concatenation of functions of $X,Y$, which is taken to bind stronger than the concatenation $\cdot$ of linear operators. | ||
| Line 12: | Line 12: | ||
| For functions in $f,g: \mathbb R\to\mathbb R$, this of course reads | For functions in $f,g: \mathbb R\to\mathbb R$, this of course reads | ||
| - | ^ $\frac{\partial}{\partial x}g(f(x))=\left(\frac{\partial}{\partial y}g(y)\right)_{y=f(x)}\cdot \frac{\partial}{\partial x}f(x)$ ^ | + | ^ $\frac{\partial}{\partial x}g(f(x))=g'(f(x))\cdot f'(x)$ ^ |
| === Reference === | === Reference === | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Chain_rule|Chain rule]], [[http://en.wikipedia.org/wiki/Chain_rule_%28disambiguation%29|Chain rule (disambiguation)]] | Wikipedia: [[http://en.wikipedia.org/wiki/Chain_rule|Chain rule]], [[http://en.wikipedia.org/wiki/Chain_rule_%28disambiguation%29|Chain rule (disambiguation)]] | ||
| ==== Parents ==== | ==== Parents ==== | ||
| - | === Requirements === | + | === Context === |
| [[Fréchet derivative]] | [[Fréchet derivative]] | ||
