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frechet_derivative_chain_rule [2013/09/15 20:13] nikolaj |
frechet_derivative_chain_rule [2013/09/15 20:26] nikolaj |
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| ==== Theorem ==== | ==== Theorem ==== | ||
| | @#88DDEE: $X,Y,Z$ ... Banach spaces with topology | | | @#88DDEE: $X,Y,Z$ ... Banach spaces with topology | | ||
| - | | @#88DDEE: $F\in C(X,Y)$ | | + | | @#88DDEE: $f\in C(X,Y)$ | |
| - | | @#88DDEE: $G\in C(Y,Z)$ | | + | | @#88DDEE: $g\in C(Y,Z)$ | |
| - | | @#55EE55: $ D(G\circ F)=(DG)\circ F\ \cdot\ DF $ | | + | | @#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. | ||
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| 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 === | ||
