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Malte Winckler
Malte Winckler
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I'm looking for properties of the Newton-derivative, defined as follows: A function $F \colon X \to Y$ is Newton differentiable at $x\in X$ if there exists $\varepsilon>0$ and a function $G\colon B_\varepsilon \to L(X,Y)$$G\colon B_\varepsilon(x) \to L(X,Y)$ such that \begin{align*} \lim_{y\to0} \frac{\|F(x + y) -F(x) -G(v+y)[y]\|_Y}{\|y\|_X} = 0. \end{align*}\begin{align*} \lim_{y\to0} \frac{\|F(x + y) -F(x) -G(x+y)[y]\|_Y}{\|y\|_X} = 0. \end{align*}

$G$ is called Newton-derivative. Can Can you provide me with a proof (or any reference) for a chain rule for this differential?

Actually, I need to know about the Newton-derivative of the function \begin{align*} L^2(\Omega) \ni u \mapsto \max(-1,\min(1,u)), \end{align*} which hopefully exists... Thanks a lot, Malte.

I'm looking for properties of the Newton-derivative, defined as follows: A function $F \colon X \to Y$ is Newton differentiable at $x\in X$ if there exists $\varepsilon>0$ and a function $G\colon B_\varepsilon \to L(X,Y)$ such that \begin{align*} \lim_{y\to0} \frac{\|F(x + y) -F(x) -G(v+y)[y]\|_Y}{\|y\|_X} = 0. \end{align*}

$G$ is called Newton-derivative. Can you provide me with a proof (or any reference) for a chain rule for this differential?

Actually, I need to know about the Newton-derivative of the function \begin{align*} L^2(\Omega) \ni u \mapsto \max(-1,\min(1,u)), \end{align*} which hopefully exists... Thanks a lot, Malte.

I'm looking for properties of the Newton-derivative, defined as follows: A function $F \colon X \to Y$ is Newton differentiable at $x\in X$ if there exists $\varepsilon>0$ and a function $G\colon B_\varepsilon(x) \to L(X,Y)$ such that \begin{align*} \lim_{y\to0} \frac{\|F(x + y) -F(x) -G(x+y)[y]\|_Y}{\|y\|_X} = 0. \end{align*}

$G$ is called Newton-derivative. Can you provide me with a proof (or any reference) for a chain rule for this differential?

Actually, I need to know about the Newton-derivative of the function \begin{align*} L^2(\Omega) \ni u \mapsto \max(-1,\min(1,u)), \end{align*} which hopefully exists... Thanks a lot, Malte.

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Malte Winckler
Malte Winckler
  • 31
  • 4

Chain rule for Newton-derivative

I'm looking for properties of the Newton-derivative, defined as follows: A function $F \colon X \to Y$ is Newton differentiable at $x\in X$ if there exists $\varepsilon>0$ and a function $G\colon B_\varepsilon \to L(X,Y)$ such that \begin{align*} \lim_{y\to0} \frac{\|F(x + y) -F(x) -G(v+y)[y]\|_Y}{\|y\|_X} = 0. \end{align*}

$G$ is called Newton-derivative. Can you provide me with a proof (or any reference) for a chain rule for this differential?

Actually, I need to know about the Newton-derivative of the function \begin{align*} L^2(\Omega) \ni u \mapsto \max(-1,\min(1,u)), \end{align*} which hopefully exists... Thanks a lot, Malte.